GIFT OF V MECHANICS' AND ENGINEERS' POCKET-BOOK OF TABLES, RULES, AND FORMULAS PERTAINING TO MECHANICS, MATHEMATICS, AND PHYSICS: INCLUDING AREAS, SQUARES, CUBES, AND ROOTS, ETC.; LOGARITHMS, HYDRAULICS, HYDRODYNAMICS, STEAM AND THE STEAM-ENGINE, NAVAL ARCHITECTURE, MASONRY, STEAM VESSELS, MILLS, ETC. ; COMPRESSED AIR, GAS, AND OIL ENGINES; LIMES, MORTARS, CEMENTS, ETC.; ORTHOGRAPHY OF TECHNICAL WORDS AND TERMS, ETC, ETC Seventy-second. Ifidition. 146th. Thousand.. BY CHAS. H. HASWELL, CIVIL, MARINE, AND MECHANICAL LNOINEHS. HONORARY MBMBBR UP TCB AM. SOC. OF NATAL ARCHI AL ARCHITECTS OF MBMBBR OF THK AM. INSTITUTE OF ARCHITECTS, AND ASSOCIATE MEMBER OF THE NEW YORK MI- CROSCOPICAL SOCIETY, ETC., ETC. An examination of facts is the foundation of science. A resultant effect, physical or mechanical, cannot be renewed without an expenditure of power in addition to that which originated it. NEW YORK: HARPER & BROTHERS, PUBLISHERS, FRANKLIN SQUAB K. 1906. By the Author of "Mechanics' and Engineers' Pocket-Book." MENSURATION. For Tuition and Reference, containing Tables of Weights and Measures; Mensura- tion of Surfaces, Lines, and Solids, and Conic Sections, Centres of Gravity, etc. To which is added Tables ot the Areas of Circular Segments, Sines of a Circle, Circular and Semi elliptical Arcs, etc. By CHAS. H. HASWKLL, Civil and Marine Engineer. Sixth Edition. 12mo, 90 cents. American Book Concern, N. Y. REMINISCENCES OF AN OCTOGENARIAN. 1816 to 1860. Second Edition $3 00 Copyright, 1884, 1887, 1890, 1892, 1893, 1894, 1895, 1896, 1897, 1898, 1899, 1901, 1903, by ILvKi'KR & BROTHERS. INSCRIBED TO CAPTAIN JOHN ERICSSON, LL.D., AS A SLIGHT TRIBUTE TO HIS GENIUS AND ATTAINMENTS, AND IN TESTIMONY OF THE SINCERE REGARD AND ESTEEM OF HIS FRIEND, THE AUTHOR 380362 PREFACE To the ITorty-fiftlx Edition. THE First Edition of this work, consisting of 284 pages, was submitted to the Mechanics and Engineers of the United States by one of their number in 1843, who designed it for a convenient reference to Rules, Results, and Tables con- nected with the discharge of their various duties. The Twenty-first Edition was published in 1867, consisted of 664 pages, and, in addition to the original design of the work, it was essayed to embrace some general information upon Mechanical and Physical subjects. The Tables of Areas and Circumferences of Circles have been extended, and together with those of Weights of Metals, Balls, Tubes, Pipes, etc., of this and some preceding editions were computed and verified by the author. This edition is a revision and an entire reconstruction of all preceding, embracing amended and much new matter, as Masonry, Strength of Girders, Floor Beams, Logarithms, etc., etc. To the young Mechanic and Engineer it is recommended to cultivate a knowledge of Physical Laws and to note re- sults of observations and of practice, without which eminence in his profession can never be attained ; and if this work shall assist him in the attainment of these objects, one great purpose of the author will be well accomplished. NOTE i. Mechanical and Physical subjects, commencing at p. 427 and ending at p. 870, are given in alphabetical order. 2. Tons are given and computed at 2240 Ibs. 3. Degrees of temperature are given by the Scale of Fahrenheit. INDEX. A. Pag ABUTMENTS AND ARCHES (See Arches and Abutments) 604-605 ACCELERATED BODY, Distances, Ve- locities, and Acceleration of. .921-92 Acids 18! Acreage, To Compute 337 Adulteration in Metals, Proportion of Two Ingredients in a Compound. 21 A erodynamics 61 AEROMETRY 673-676 (See also Pneumatics.) " Course of Wind, etc 675 " Cyclones, Direction of. 675 " Degree of Rarefaction 67; " Discharge Pipes, Diameter of. 676 " Distance of Audible Sounds. .. 674 " ForceofWindonaSurface.bn-tijs " Height of a Column of Mer- cury to Induce an Efflux of Air, To Compute 675 " Resistance of A Plane Surface to Air, To Compute 675 " Resistance to a Steam Vessel in Air or Water 911 " Weight or Pressure of Air under a Given Height of Barometer and Temperature Discharged in One Second. . 675 " Wind, Velocity and Pressure of. 674, 911, 924 " Volume of Air discharged Through an Opening, etc., into a Vacuum 674 AEROSTATICS 427-431, 614 " Clouds, Classification of. 430 " Distances, by Velocity of Sound 428 ' ' Elevations, by a Barometer. 428-429 " Thermometer.. 429 " Lightning, Classification of. .. 430 " Velocity of Air flowing into a Vacuum, To Compute. 42 " of Sound 428 " Weather Glasses and Barom- eter Indications 430-431 Age of Horses, To Ascertain 186 Ages of Animals IQ2) Ig 6 AIR, ATMOSPHERIC 431-432, 912 " and Steam, Mixture of. 737 " Carbonic Acid Exhaled by Man. 432 " Compressors 940 " Consumption of, by Candle 432 " Decrease of Temperature by Al- titudes 522 " Discharge of, Coefficients of Ef- , flux. 674 " Expansion of. 520 " Flow of, in Pipes, Loss of. .745, 909 " Head of, to Resist Friction in Long and Rectilineal Pipes . . 925 Pressure of, in Rear ofaProjectile 6481 Page AIR, ATMOSPHERIC, Pressure,Velocity, and Resistance of a Plane Sur- face, To Compute 648 " Pressure of a Weight of, or other Gas at 62 and 14. 7 Lbs. Press- ure, with Constant Volume for a Given Temperature 522 " Proportion of Oxygen and Car- bonic Acid at Various Loca- tions 432 " Rarefaction of. 430 " Resistance of different Figures in, at different Velocities, and to Falling Bodies 646-647, 949 " Specific Heat of, and Other Gases 505-506 " Temperature, for a Given Lati- tude and Elevation 676 " Volume of a Weight of, or Per- manent Gas for any Pressure, To Compute. . 520 ' ' and Weight of Vapor in . 68-69 " " of a Weight of, or Perma- nent Gasforany Pressure and Temperature. . .521-522 " " of and Gas in a Furnace.. 760 " " of Enclosed, at o that may be Heated by One Sq. Foot of Iron Surface. . . 925 " " of, or Gas Discharged through an Opening and under a Pressure above that of External Air. . . 676 " " Pressure, and Density of at Various Tempera- tures 521-522 " " Required per Hour, for each Occupant in an Enclosed Space 525 Ajutage, Cylindrical 549 ALCOHOL 194 " Elastic Force and Tempera- ture, of Vapor of. 707 ' ' Proportion of, in Liquors. 191, 204 Ale and Beer Measures 45 Water in 201 ALGEBRAIC SYMBOLS AND FORMULAS, 22-25 Alimentary Principles 200 Alligation 106 Alloy, Expands in Cooling 952 ALLOYS AND COMPOSITIONS 634-637 " Bronze 637 " Compounds, Fusible and Solders 634 " Of Steel 643 " Soldering Fluid and Fluxes. . . 636 " Solders 636 < ' Welding Cast Steel 634 ALMANAC, Epacts and Dominical Let- ters, 1800 to 1901 73 Vlll INDEX. Page ALMANAC, Altitude of Sun at New York 932 Altitudes, Decrease of Temperature by 522 Aluminum J55 } 938, 976 Amalgam 054 Ammeters and Voltmeters 961-962 ANALYSIS OF ORGANIC SUBSTANCES.. 190 " of Foods and Fruits 201 " of 'Meat, Fish, and Vegetables, etc. 200 ANCHORS AND KEDGES and Units, To Determine Weight and Number of, U.S.N. . . 174-175 " "Number and Weight of, U.S.N. 174 " Cables, Chains, etc., for a Given Tonnage, Am. Ship M. Ass'n I73-I74 * Experiments on and Compar- ative Resistance to Dragging 175 " Proof Strain of. 175 Ancient and Scripture Lineal Meas- ures and Weights 53 ANGLES (See also Trigonometry).. 385-389 " and Distances Corresponding to Opening of a Rule of Two Feet 160 Chord of, to Plot and Compute. 359 Critical and Visual 669 Sines of, To Compute 402 To Describe, etc 222 To Plot without a Protractor, 359 ANIMAL and Human Sustenance 203 FOOD 200-207 Power and Work 432-440 Birds and Insects. . 438, 440 Camel and Crocodile. . 438 Coursing and Chasing . 440 Day's Work 434 Dog 43 8 Horse. 435-437, 439-440,918 Llama and Ox 438 Men 433-435, 438-439 Mule and Ass 437, 918 on Street Rails or Tram- ways 435 " Stage Coaching 440 ANIMALS, Proportion of Food for. . . - 205 " Ages of, etc 192,196 Annealing 7 86 ANNUITIES no-m Amount of, To Compute. . . no " at Compound Interest in Present Worth of. . . . . no-m " Yearly Amount, that will Liquidate a Debt... . . no-m Anti-attrition Metal, Babbitt's 636 ANTIDOTKS for Poisons 185, 03 s Anvils, Weight of. 0x8 Apartments, Buildings, Ventilation of 524 Apothecaries' or Fluid Measure 46 Weight 32 Appold's Pump and Wheel 579-580 APPENDIX 1. .919-955 1024-1029 Aqueducts, Roads, and Railroads,i 7 8, 939 Page Arc, To Describe 225, 227-228 Arch, Radius of, To Compute 604 " Depth of, To Compute 605 ARCHES AND ABUTMENTS 604-605 (See also Masonry, 604-605, ) " " Chords, etc. , Safe Weight of 7 76 " " Minimum Thickness of , for Bridges 605 " AND WALLS 602-603 Area and Population of Divisions and Countries 188 Arenes , 589 AREAS of CIRCLES by Sths 231-236 " " and Circumferences of Cir- cles by ioths 243-252 " " and Circumferences Greater than in Table, To Compute 252 ' byi2ths 253-257 ' by Birmingham Wire Gauge 236 ' by Logarithms 236, 252 " ' Greater than in Table. .235, 252 " ' In Feet and Inches 235-236 " ' When Diameter is composed of an Integer and a Frac- tion, To Compute 236 1 OF SEGMENTS OF A CIRCLE. . 267-269 ' OF ZONES OF A CIRCLE 269-271 " " To Compute 271 ARITHMETICAL PROGRESSION 101-103 Artesian Well 179, 198 ASBESTOS 913 " Felting, Cement, etc 1032 Ash, Proportion of, in Woods 482 ASPHALT 481, 515, 689-690 and Pavement 944-945 " Composition 593 " Mortars and Concrete 913 Astronomical Day 70 Atlantic and Pacific Oceans 937 " " Tides of. 191 Atmosphere 912 AVOIRDUPOIS WEIGHT. . 32 Axle, Compound, or Chinese Windlass, 627 (See also Wheel and Axle, 626-627.) B. Babbitt's A nti-attrition Metal 636 Bacteria in Earth soil 942 Baking of Meats, Loss by 206 Balances, Fraudulent 65 Balks and Battens, Dimensions of. . . 62 Balloons, Capacity and Diameter of.. 218 BALLS, Cast Iron and Lead 153 Balls, Lead, Weight and Dimensions of 501 BARBED WIRE Fencing 947 Barker's Mill 577 Barley, Value of, Compared to 100 Lbs. of Hay 203 BAROMETER (See also Aerostatics). 427-431 Elevations by Readings . 429 Height of, To Compute. . 429 Indications 429-430 Weather Glasses 430 Weather Indications.., , 431 Barrel, Dimensions of. , 30 INDEX. IX Page BARS, Wro't Iron, Square and. Rolled, 125-128 " Steel, Flat and Rolled 134-135 BEAMS, BARS, OR GIRDKRS, Transverse Strength of. . . ._. 802-8 1 1 , 813-820 (See also Girders, Beams, etc.) " Box, Plate and Bars. . 806, 817, 827 " Comparative Strength and De- flection of Cast Iron 809 44 Comparative Value of Bars, Girders, or Tubes 824 44 Common Centre of Gravity, etc. 819 44 Cylinder and Cylindrical, Di- ameter and Weight, To Com- pute 805 44 Cylindrical, or Tubes, Trans- verse Strength of. 810 4 ' Deflection of. 770-782, 840-8 \ i " Depths and Weight, etc., of 'Steel 134 ' ' Dimensions and Weigh t of Roll- ed Steel, and To Compute. 134, 644 44 Rolled Steel Beams and Bulb Angles 807, 808 " Elliptical Sided, To Determine Side or Curve of. 826 41 Factors of Safety 821,841 44 Flanged, Dimensions and Pro- portions of Wro't Iron 809 41 Floor Beams, Headers, Trim- mers, etc., To Compute. . .835-838 Formulas for TransverseStress, To Compute 801, 816 1 General Deductions 824-825 4 Inclined, Formula for 811 ' Lintels, etc 822-823 ' Moments and Stress of... . .621-622 ' " of Resistance 818 ' Rectangular, Girders or Tubes. 809 4 Scarfs, Resistance of 841 4 Shearing Stress 623 1 Solid Cylinder, Diameter of, To Compute 804-805 4 Symmetrical Forms and Sec- tions, Conditions of. 825-826 4 Trussed, Notes on 823-824 4 Unequally Loaded 810 ' Various Figures and Sections, Dimensions of. . 147, 805, 813, 814 1 Wrought Iron Rolled 809 BEAMS, Inertia and Resistance, Mo- ments of and Neutral Axis, To Compute 818-820 4 Circular or Elliptic 815 ' Hollow or Annular 815 ' Miscellaneous Illustrations. . . 826 4 of Unsymmetrical Section, and Ultimate Strength 0/817, 820-821 44 OR TUBES, Elliptical 810, 815 Bearings, for Propeller Shaft 473 Beef, Lean, Water in 201 Beer and Ale Measure 45 44 Water in 201 Beet Root, Ratio of Flesh -formers and Sugar, Analysis of. 207 Beeves and Beef, Weights of 35 Bell Ringing 433 Page Bells, Weight of. 180, 936 Belt, Equivalent, and Wire Rope 167 Belting, and Destructive Stress of, 907, 935 " or Hose, To Preserve 877 BELTS AND BELTING, 441-443,877.960,1018 " Adhesive Compound for Rubber. 877 Dressing for 878 Dynamo and Link Leather 960 General Notes and Memoranda, 442-443, 872, 877, 989 India Rubber 442 Width of, To Compute 44i~44 2 Bench Marks 85, 1035 Beton or Concrete 593 Bibliotheque National 936 Birds, Flying 440 Incubation of. 192 and Insects 196, 438 Bissextile or Leap Year 70 Blacking 877 Blast Furnace 529 44 Pipe of a Locomotive.. 907 BLASTING 443-445, 9'3, 9 l6 Boring Holes in Granite . . 444 44 Charge of Gunpowder for, and Effects of. 444 44 Weight of Explosive Mate- rials in Holes 444 44 Gelatine, Composition of... 916 Paper, Composition of. 912 44 Tunnels and Shafts, Cost of '. 444 44 Drilling and Mining 445 Blasts and Draughts, Comparative 746 Blower and Exhausting Fan 89 BLOWERS, Fan 447-448, 898, 1015 14 Elements of, To Compute 448 44 Memoranda 448-449 44 Power of a Centrifugal Fan. 448 44 Pressure of Blast, To Compute 447 BLOWING ENGINE, Friction of Air in Long Pipes, etc 925 ENGINES 445-449, 898, 1018 44 Dimensions of a Driving Engine, To Compute 446 44 Elements of, To Compute. . . 447 44 Memoranda 448 44 Power of, and Power Re- quired to Drive 446 44 Root's Rotary 449 44 Volume of Air transmitted. 447 Blowing Off of Steam 726-727 Board and Timber Measure 61 BOARDS, Volume that can be sawed from a Round Log 947 BOILER, Steam 526, 739-745 " and Ship Plates 828 ' 4 Abut Straps and Stays. . . 753-754 4 4 A reas and Ratio of Grate and Heating Surface, Volume of Water andWeightqf 'Fuel. 741-742 44 Coal, Utilization of, in a Ma- rine 726 4 4 Comparative Result of Experi- ments with a Steam Jet. . . . 746 INDEX. Page BOILER, Consumption of Fuel in a Furnace, To Compute. . . 725-726 " Draught 739, 744-74^ * " and Blasts, Compar- ative Effect of 746 " " Velocity of 746 u Efficiency, Nominal and IIP and Economy of. 758, 976 44 Evaporative Capacity of Tubes 742 " Evaporative Effects of, for Different Rates of Combus- tion and Surface Ratios. . . 743 " Evaporation, Power of.... 757-758 " Eyes, Stays, Rods, etc 754 44 Fuel that may be Consumed per Sq. Foot of Grate 742 44 Girders for Furnaces 754 " Grate, Heating Surface,Water, Fuel, etc., To Compute. .741, 927 44 Heating Surfaces and Rela- tive Value of. 740 " Mean Strength of Riveted Joints Compared to Plate.j 51-752 44 Minimum Volumes of Fuel Consumed per Sq. Foot of Grate 740-741 " Plates and Bolts 749-750 " u Thickness of for a Given Pressure and Pitch, etc. 753 44 Power and IPof. 526, 760 " Proportion and Capacities of, and Firing 739-74 44 Rate of Combustion 760 44 Relation of Grate, Heating Surface, and Fuel 741 '* Results of Operation of and under Varying Proportions of Grate, Surface, Draught, Combustion, etc 743, 924 " Results of Operation of Va- rious Designs of Boiler and Varying Proportions of Area of Grate Surface, etc 744 44 Return Tubular, Elements of a Test 726 " Riveting 755-757, 97 11 " General Formulas and Illustrations 757 " Safety Valves 746-747 " Saline Saturation in 726 44 Scale, Removal of Incrusta- tion of. 726 ' Stay Bolts, Diameter and Pitch of, To Compute 754 " Tensile Strength of. 753-754 44 Steam Heating 526, 957-958 41 Steam Room 748 " Tubes, Evaporating Capacity of Various Lengths. . . . 742 44 " Lap-welded Charcoal, Di- mensions of. 139 ' Volume of Water per Lb. of Coal, To Compute 725 " Weights of , and with Water. 7 59. 929 BOILBRS, Area of Grate per Lb. of Coal...,, 748 BOILERS, Blowing off, To Compute Loss of by ................. 727 " Bottoms of, To Preserve ....... 878 Corrugated Flues ............ 941 Cylindrical Shells ---- 751-752,91 Elements of. ................. Flued, Arched, or Circular 13 58 Furnaces, U. S. Law ____ 754=755 44 Shell Plates, Pressure ana Thickness of, U. S. Law. 7 50-751 44 Horse-Power of. .......... 914, 936 44 Incrustation or Scale, To Re- move ...................... 726 44 Magnesia Covering of. ---- 918, 921 44 Plates, Straps, and Stays ...... 753 44 Proportion of Grate and Heat- ing Surfaces, Result of Ex- periments .................. 926 44 Saline, Saturation in ......... 726 * 4 " Matter, Proportionate Volumes of. ......... 727 44 Smoke Pipes and Chim- neys.. ................. 748-749 14 Steam in Foreign Countries . . 935 44 Steam Water Tube, Efficiency and Results of. ............ 947 4 4 Volume of Furnace Gas per Lb. of Coal .............. 7 6o 44 4 ' of Water Slowed off to that Evaporated ...... 727 44 Water Tube and Efficiency of, 926, 947 BOILING - POINTS of Various Sub- stances ............... 517 44 " at Different Degrees of Saturation ............ 815 BOLTS, Adhesion of Drifted ......... 949 4 ' and Plates, U. S. Test ____ 749-750 4 4 Rods of Copper, Weight of.... 148 44 Round and Square, Relative Driving Resistance of Steel. 970 44 Tenacity and Resistance of, Round, Square, and Screiv. . 970 9*7 CHAIN, To Set out a Right Angle with . 69 14 Cable, Diameter and Length foraGiven Weight of Anchor, 4 4 CABLES, Breaking Strain, Proof and Strength of. . 169, 930 44 " Anchors, etc. , for a Given Tonnage, Am. S.Ass'n. 173-174 " " Length of, for Anchors ____ 175 44 u Stud - Link, Weight and Strength of. ......... 168, 930 41 " Stowage of ............... 913 Chaining over an Elevation ..... 69-1034 CHAINS AND RopES/or Cranes, Weight of and Proof. ......... 457 14 " of Equal Strength ........ 165 * ' Safe Working Load of. ....... 168 CHARACTERS AND SYMBOLS ..... 21-22, 973 CHARCOAL ........... 33, 194. 480-481, 485 44 Produce of from Woods. . . 481 Cheese, Composition of Different Countries ........................ 205 Chemical Composition of some Com- pound Combustibles ..... 461 " Formulas, To Convert ..... 190 Chimney Draught and Chimney s.goj, 918 44 Velocities of Current of Air in One of 100 Feet ........ 749 CHIMNEYS ...... 179-180, 904, 916, 918, 925 44 and Smoke Pipes ...... 748-749 44 Height of and Commercial fffor a Given Diameter of Flue. . 925 Chinese Wall 179, 93 6 44 Windlass, To Compute 627 44 or Indian Ink 907 CHRONOLOGICAL ERAS and Cycles 26 CHRONOLOGY 70-74 44 To Ascertain Years of Coinci- dence of a Given Day of the Week, etc 74 Churches, Opera-Houses, and Thea- tres, Capacity of. 180 CIRCULAR Av.cs,from i to 180 26* 44 Lengths of up to a Semi-circle 260-261 44 Length of, To Ascer- tain 261 44 Motion 618 44 Measure of an Angle 113-114 Circulating Pumps 749 CIRCLE, to Ascertain Square that has the Same Area of. 259 44 Side* of a Square Equal in Area to 258-259 CIRCLES, Areas of, by 8ths, ioths, and i2ths 231-257 44 by Logarithms 236 44 44 by Wire Gauge 236 CIRCUMFERENCE of a Circk when Greater than Contained in Tables, To Compute. 24 1-242, 252 44 of Birmingham Wire Gauge. 242 4 4 Wh en Diameter consists of an Integer and a Fraction. . . . 242 CIRCUMFERENCES OF CIRCLES by Sths, 237-242 44 and Area of by 10^5.243-252 44 44 44 " by Inches and izths 252-257 44 by Logarithms 242, 252 14 44 In Feet and Inches. .241-242 Cisterns and Wells, Excavation, Lin- ing, and Capacity of. 63 Civil Day 37,70 44 Year 70 Cloth Measure 27 Clouds, Classification of. 430 Clover, Value of, Compared to 100 Lbs. of Hay 203 COAL, Anthracite 33, 480, 483, 485-486 44 Composition of Average., 485 44 Fields of U. S. , Areas of. 191 COALS, Average Composition of and Fuels, Heat of Combustion and Evaporative Power of. . . 486 44 Bituminous 33, 479, 483, 485-486 44 44 and Natural Gas, Relative Water Evaporating Powers 913 44 " Caking, Splint or Hard, Cherry or Soft, and Cannel 479 44 44 Classification, Chemical Composition and Varieties of. 479 ; 4 Consumed per Hour, to Heat 100 Feet of Pipe 527-528 44 Effective Value of 908 COALS, Elements of Various American 480 " Fields, Areas of U '. S. 191 INDEX. Xlll COALS, Gas 484 44 Japan 909 " Lignite 479> 481 " Measures 33, 46 " Mine 93 6 " Miscellaneous Experiments 487 44 Production and Consumption of the World 955 Coast and Bay Service and Scour. . . 908 Cocks, Composition, and Copper Pipes, Dimensions of. 150 Coffee and Tea, Water in 201 COHESION 614 " Modulus of 763-764 Coins, British Standards 38 " To Convert U. S. to British Currency, and Contrariwise 31 " Tolerance of. 3! " U. S., Weight and Fineness of. 38 " Value of, To Compute 39 " Foreign Silver and Gold, and Weight, Fineness, and Mint Value of 39, 43 COKE, Evaporative Power of, etc 480 Cold, Extremes of, in Various Coun- tries and Snow Line 191-192 " Greatest, Artificial. 908 College, Oxford 179 COLLISION OR IMPACT 580-582 " Velocities of .. 58 1-582 Color Blindness 195 Colors for Drawings 913 * Proportion of, for Paints. .... 66 Columns, Towers, Domes, Spires, etc. , 1 80, 936 41 Crushing and Safe Load of. . . 766 " Long Solid, Comparative Value of. 769 " of Cast Iron 768-769 " " Weight Borne with Safety, 768-769 COMBINATION 112-113 COMBUSTION 458-466 4 4 Chemical Composition of some Compound Combustibles. . . 461 ~ x ** Composition and Equivalents of Gases Combined in Com- bustion of Fuel 460 M Consumption of Fuel to Ileat Air, To Compute 466 ** Evaporative Power of i Lb. of a Given Combustible. . . 462 44 Heat of. 463 44 Heating Powers of Combus- tibles,and To Compute. 461-462 " Of Fuel, Ratio, etc 463-465 *' Products of Decomposition in the Furnace 458-459 11 Rate of, in a Furnace 760 ** Relative Evaporation of Sev- eral Combustibles 465 ' " Volumes of Gases or Products of, per Lb. of Fuel 465 " Temperature of, To Ascer- tain. 462-463 COMBUSTION, Volume of Air Chemical- ly Consumed in Complete Combustion of i Lb.of Coal 459 1 ' Volumes of Air Required for Combustion 464-465 44 Weight and Specific Heat of Products and Temperature of Combustion, etc. 462-463 4 4 Weight and Volume, of Gaseous Products of i Lb. Fuel 460 Compass, Degrees, < Graduation, 54, 1023 COMPOSITION Cocks, Dimensions of. . . 150 COMPOSITION for Welding Steel. 634 COMPOSITIONS AND ALLOTS 634-637 (See Alloys and Compositions.) Composition Sheathing Fails 135 Compound Axle or Chinese Windlass 627 * INTEREST 108-109 44 PROPORTION 95-96 .'*.. Weights of Ingredients... 218 CONCRETE 588-597 (See Limes, Cements, Mortars, and Concretes. ) 44 CoigneVs 914 " Compositions of. 593 44 or Beton. 593 Concretes, Cements, etc 800 CONES. 353-354 CONDENSATION, Surface 967 44 4t Experiments on. 911 CONDENSER, Results of an Operation of.. 967 CONIC SECTIONS 379-380 4 4 Conoid and Ellipse, Elements of 380 44 44 To Describt.and Area, Or - dinate, Abscissa, Diam- eters, Circumference, Seg- ment, and Length of Curve, To Compute. 380-382 44 General Definitions 379 44 Hyperbola, To Describe, and Abscissa*, Diameters, Length of an Arc, and Area, To Com- pute 379-38o 44 Parabola 379-380, 382-383 44 To Describe Ordinate, Abscissa, Curve, Area, and Segment of.. . .382-383 Constructions and Natural Forma- tions, Largest 936 Contractility and Elasticity 614 Cooking of Meats 206 COPPER, Tensile Strength of. 750, 788 and Iron Riveted Pipes, Wevght of. 148 Braziers 1 and Sheathing 131 " Rods or Bolts, Weight of . 148 Given Sectional Area, Weight of 1 36 Plates, Thickness of. 121 44 Weight of. 118-119 44 per Sq. Feet. . 146 Seamless Drawn Tubes. Weight of. 140-142, 144-M5 Sheathing and nraziers*. . .131-155 Sheet, Weight of a Sq. Foot ... 135 XIV INDEX. Page COPPER, Weight of, and To Compute, i3 6 > " Wire, Cord 123 " " Weight of 120-121 Copying, Words in a Folio Cord, Copper Wire 123 CORDAGE, Friction andRigidity 0/472-47; Corn Measure 19! 11 Value of, Compared to 100 Lbs. of Hay 203 Corrosive Effects of Salt-water on Steel and Iron 916, 971 CO-SECANTS AND SECANTS 403-414 " " To Compute, etc. 414 COSINES AND SINES 390-402 it ic T O compute, etc. 401-402 CO-TANGENTS AND TANGENTS 415-426 " "To Compute, etc. 426 COTTON FACTORIES 8 Couple, Constitution of. 6 Coupling or Sleeves of Shafts 71 Coursing and Chasing 440 CRANE, Railroad 962 " Steam Dredgers, Elements of and Dredging 899-900 CRANEB 179, 433, 455-457, 9 62 ' Chains and Ropes for 457 " Dimensionsof Post, To Compute 456 " Machinery and Proportion of. 457 " Post, Stress and Conditions of. 455 " Stress on Jib, Stay, or Strut, 455-457 Crank, Turning 433 Cream, Percentage of, in Milk 205 Creosoting, Effects of. 869 Crocodile, Power of. 438 CROPS, Mineral Constituents Absorbed or Removed from an Acre of Soil. . 189 Cross-ties, Railroad, Duration of. . . 970 Croton Aqueduct 178, 939 Crusher, Ore and Stone Breaker 957 CRUSHING STRENGTH 764-769, 1021 (See Strength of Materials.) Cube Measures 30-31 CUBE ROOT, To Extract. 97 " AND SQUARE ROOT of a dum- ber consisting of Integers and Decimals, To Ascer- tain 301-302 " "of Decimals alone, To As- certain 302 " "of any Number over 1600, To Ascertain 301 *' "or Square Root of Roots, Whole Numbers and of Integers and Decimals, To Ascertain 97-98 " " of a Higher Number than is Contained in Table 301 CUBES, SQUARES, AND ROOTS 272-302 (See Squares, Cubes, and Roots.) " To Compute and to As- certain, etc 300-302 Cucumber, Water in 207 Currency, To Convert U. S- to British 39 Page Current Wheel 570 " of Rivers 193 Curvature and Refraction of Earth. . 55 Curves, Caustic, or Lines 669 Cut Nails, Tacks, Spikes, etc 154 Cutters, Yachts, Pilot Boats, Launches 895 Cycle, Dominical or Sunday Letter. . 70 " Lunar or Golden Number. ... 71 " of the Sun, To Compute 70-71 CYCLES and Chronological Eras 26 Cycloid, To Describe 228 CYCLONES, Direction of. 675 CYLINDERS, FLUES, AND TUBES, Hollow 827 u Solid and Hollow, of Various Metals 801 D. DAMS, EMBANKMENTS, AND WALLS (See Embankments, etc.) 700-703 DAY, Astronomical, Marine or Sea. .37, 70 " Sidereal, Solar, and Civil 37, 70 Day^s Work 434 Dead Sea and Valley of the Jordan. . 934 Deals and Local Standards of 62 DECIMALS 92-94 Deer Park, Copenhagen 1 79 DEFLECTION (See Strength of Mate- rials. ) 770-781 Delta Metal 384, 913 Departures, Table of. 54 Depths, Sea 184 Derrick Guys 163 Desert of Sahara 936 Desiccation 513 DETRUSIVE OR SHEARING STRENGTH (SeeStrength of Materials, 782-783. ) " and Transverse, Comparison of. 782 " Strength of Woods 782 " Wood, Surface of Resistance of. 782 Dew Point, and To Ascertain 68 Diamond Weight 32 Diamonds, Weight of. 193 Diet, Daily, of a Man 202, 207 u " of an Esquimau 914 Differentiation, Integration, and Cal- culus 24-25 DIGESTION OF FOOD, Time Required for 206-207 DISCOUNT or Rebate 109 DISPLACEMENT of a Vessel 653 DISTANCES, STEAMING 86 and Angles, Corresponding to Opening of a Rule of 2 Feet . 1 60 between Cities of U. S. 184 " " East and West. . 187 " Principal Ports of World 87 " " of U.S.. 88 " Various Ports of Eng- land, Canada, and U. S.,and N. Y. and London 86 Velocities and Acceleration of a Body, To Compute 921-922 Geographic, and Measures 54 Distemper ( Coloring) 593 DISTILLATION 514 of Fresh Water 955 INDEX. XV Page Distillers and Evaporators, Capaci- ties of. 950 Dog, Power of, Coursing and Chasing, 438, 440 DOMES and Towers, Diameter and Heights of. 179-180, 932 Domestic Remedials 938 Dominical Letters and Epacts 73 u or Sunday Letter 70 DOVETAILS, Tenacity of. 948 DRAINAGE OF LANDS by Pipes 691 Drains, Diameter and Grade of, to Discharge Rainfall 906 4 ' and Sewers, Velocity and Grade of. 692 DRAUGHT, A rtificial 745-746 Natural 739-74Q, 744 Steam Jet and Blast, Com- parative Effects of, and Result of Experiments with 746 Drawing and Tracing Paper 29, 964 u or Pushing 433 Drawings, Colors for 196, 913 ' 4 Dimensions of, for U. S. Patents 198 Dredger, Steam Hopper, and Ma- chines 899-900 DREDGING, and Cost of. 197 44 Machines and Crane. ... .. 899 DRILLING in Rock 445, 940 " in Metals 477 Drills, Mountings, etc 940 DROWNING PERSONS, Treatment of. . . 187 DRY MEASURES 30, 31 DUALIN 503 DUODECIMALS 94 DYNAMITE and Cellulose 443-444 DYNAMICS 614, 616-620 " Circular Motion 618 44 Decomposition of Force 620 44 Motion on an Inclined Plane 619 44 Uniform Motion 617-618 " Work A ccumulated in Moving Bodies, an d To Compute 619 44 " By Percussive Force. . . . 620 DYNAMO Leather Belts 960 E. EARTH, Diameters and Density. . . 188, 198 " and Rock Excavation and Embankment 192 44 Area and Population of. 188 " Boring and Heat of Mines 955 " Conductivity of Temperature in 914 44 Curvature and Refraction of. 55 44 Elements of Figure of. 6* " Influence of the Rotation of, on Moving Bodies 942 44 Motion of. 70 " Weight of, per Cube Yard 468 " Weights of. 33 EARTHWORK 467-468 44 Bulk of Rock, etc., Original Excavation Assumed at i. . 468 44 Number of Barrow and Horse- cart Loads and Shovelling, and Volume of, Transported per Day.. 908 Page Easter Day , 71 Ecclesiastical Year 70 Egg, Fowls 1 , Composition of. 207 Egyptian and Hebrew Measures 53 ELASTIC FLUIDS, Specific Gravity of. 215 ELASTICITY and Strength. 195, 614, 761-763 44 Coefficient of. 761 " Modulus of, and To Compute 762-763 4 4 Relative, of Materials 780 ELECTRIC AND GAS LIGHT 198 44 Dynamo Engine 954 44 Elevator s, Power Required. .. 959 44 Launch 900 " Light, Candle- Power of. 908 44 FANS, MOTORS, Power, Pumps. 959 " WIRES AND CABLES, Tele- graph, Telephone, and Light Wires and Cables 960 Electrical Engineering, Units in, Re- sistance and Expressions. 987-988, 1033 ELEMENTARY BODIES, with their Sym- bols and Equivalents 190 Elephant, Power and Weight of. 918 Elevations by a Barometer 428-429 44 and Heights of Various Places above the Sea 183,1035 ELLIPSE, To Describe and Construct, etc 226-227, 380 (See Conic Sections, 379-380.) ELLIPTIC ARCS, Lengths up to a Semi- 44 " ellipse of. 263-266 44 " To Ascertain Length of 266 EMBANKMENTS, WALLS, AND DAMS,7- ments of. 700-703 (See also Revetment Walls, 694- 699, and Stability, 693-703. ) u " Equilibrium, Stability and Moment, To Compute 701 44 " Form of a Pier, To Determine 700 14 " High Masonry Dams 703 44 " Materials, Weight of a Cube Foot of. 694 14 " Surcharged Revetments 699 44 " Various Elements, To Com- pute and Determine. . . . 702-703 Endless Ropes 167 ENGINES AND MACHINES, Elements and Cost of. 898- 44 and Sugar -mills, Weights of '. Engravings, To Clean Soiled 875 Ensigns, Pennants, and Flags, U.S . 199 EPACTS, AND DOJIINICAL LETTERS. ... 73 EQUATION OF PAYMENTS 109 EQUILIBRIUM, Angles of, at which Va- rious Substances will Repose 694 41 Of Forces 616 Ericsson's Caloric 903 Esquimau, Daily Food of. 914 Establishment of the Port for Several Locations in Europe 85 Ether, Elastic Force of Vapor 707 Evaporation 747-748, 1024 44 of Water per Sq. Foot 514 44 u per Month of Year 916 XVI INDEX. Evaporative Power of Tubes per De- gree of Heat, etc 51 Evaporators and Distillers, Capaci- ties of. 95 EVOLUTION " To Extract Square and Cube Roots 9- " any Root 97-9; EXCAVATION AND EMBANKMENT, Ele- ments of, etc 466-468 " " Bulk of Rock, Earthwork, etc., Original Excavation Assumed as i 468 ' " Cost of, per Cube Yard. ... 467 " Earth, Rocks, etc., Weight of. 468 '* " Earthwork by Carts and Barrow Loads Removed by a Laborer per Day. 467-468 " " Labor and Work upon and Estimate of Cost 07.466-467 " " *' in Blasting and Hauling Stone or Earthwork, etc. 468 " " Loads or Tripsin Cube Yards per Cart per Day 466-467 " " of Earth and Rock 192 Expansion 614 ** and Contraction of Building Stones, etc 184 Expenditures in England for Various Purposes and of Articles, compared with Spirituous Liquors 938 EXPLOSIVES, Relative Strength of, Fired under Water. 946 " High, Firing Point and Rela- tive Strength 953)9^6 F. FALLING BODIES, Resistance of Air to, 941 Family of Mechanics, Costof,inFrance,gol FAN BLOWERS , ...... . .447-448 " Elements of, and Power of. 448 " Exhausting and Blower 898 " Memoranda 448 Farms, Sustaining Production of... 207 Fascines 690 FELLOWSHIP 99 FELTING, Covering, Lagging, etc. . . . 1032 FENCE WIRE, Strength and Weight of Single Thread and Cable Laid. ... 164 FENCING, Barbed Steel Wire 947 ffig, Value of. 207 files, Repair of. 878 Filter Beds 851 Filtering Stone 909 FILTERS for Waterworks 184 Fire Bricks 515, 600 " Clay 597 FIRE-ENGINE, Steam.. 904, 909 F\sh,Meat,and Vegetables,Analysis of 200 Flags, Ensigns, and Pennants, U. S.. 199 FLAX MILL 476 Floating Bodies, Velocities of 909 Flexible Paint for Canvas 915 Flood Wave of Ohio River 912 FLOOR BEAMS, of Wrought Iron, and Distances from Centres, . . . , 931 Page FLOORS AND LOADS, Factor of Safety, and Weights of and on 841 (See Strength of Materials, 795-841.) FLOUR, Consumption and Tests of. 206-207 " Mills 900 FLUES, TUBES, and Cylinders 747, 827 " Arched or Circular Furnaces . 754 " and Furnace, Corrugated, and Formulas for 909, 941 Fluid and Liquid Measures 30, 46 Fluids, Candles, Lamps, and Gas 584 " Percussion of. 579 FLUTTER WHEEL 571 Fluxes for Soldering or Welding 636 FLY WHEEL, Weight and Dimensions of Rim, To Compute 451 FLYING of Birds 440 Fontaine Turbine 574 FOOD, Animal 200-207 " Comparative Values, for Sheep. 938 " Daily, of an Esquimau 206, 914 " Digestion of, and Time Re- quired for 200 ' ' Milk, Relative Richness of. 207 " Nutrient Value and Ratio of 100 Parts 202 " " Equivalents of from Amount of Nitrogen in, when Dried 205 598 GUDGEONS, Diameter of, To Com- pute 795 Gun Barrels, Length of. 198 " Browning 875 44 Cotton 443 44 Steel, Kruppi s .... 913 Gun-metal, Weight of, and of a Given Sectional Area 136, 149 INDEX. XIX Page ' , Elements of, etc 457-503 * ' Charge, Range, Elevation, and Velocity, and To Compute^gj^gg " Comparison of Force of a Charge in Various Arms. . 502 " Experiments with Ordnance and Penetration of Shot and Shells 49 8 , 5 " Initial Velocity and Ranges of Shot and Shells 498-499 " Lead Balls, Weight and Di- mensions of. 5 01 " Number of Percussion Caps corresponding to B Gauge 502 " " of Pellets in an Oz. of Lead Shot of all Sizes 501 Penetration of Lead BaUs in Small Arms 5 4 ' Ranges for Small A rms 502 * Report of Board of Engineers, U.S. A., Fortifications, etc. 499 " Summary of Practice in Eu- rope with Heavy Guns 500 Time of Flight, Rute for 497 " Velocity of a Shot or Shell ... 497 " Windage and Waddings, Loss and Effect of. 501 GUNPOWDER 443) 5 02 u Charges of, and To Compute. 444 11 Heat and Explosive Power of. 503 " Manufacture of. 503 " Proof of. 502 " Properties and Results of, De- termined by Experiments .. 503 " Proportion of, to Shot 502 " Relative Strength of Different, for Use under Water GUTTER'S CHAIN Guys, Derrick, Strength^ etc 162-163 Gwynne's Pump, Centrifugal 579, 917 GYRATION, and Centres of. 609-611 " Centre of, of a Water-wheel. . . 6n " General Formulas 61 1 ' ' Moment of Inertia of a Revolv- ing Body, To Compute 609 " Radius, To Compute 609 H. HAMMERS, Steam 179 Hancock Inspirator 901 Hand- cars and Portable Railroad. . 908 HAWSERS, WIRE, AND HEMP ROPES AND CABLES, Comparison of. 169 (See Cables, Ropes, etc. , 163-178. ) " and Warps, Length of 173 u Cables and Ropes 170 " Circumference of. To Compute 171 u Units for Computing Safe Strain Borne by and for New Ropes and Hawsers, 170-171 ' Weight of, To Compute 172 HAY, Relative Value of Foods com- pared with TOO Lbs. of. 912 " and Straw, Weights, etc 198 Page HEAT, Elements of, etc ........... 504-529 ' Absolute Temperature ......... 504 " Absorption of , in Generation of i Lb. Steam at 212 ......... 705 u Altitudes, Decrease of by ....... 522 1 ' A vailable Expended per IIP 909 " Boiling- Points of Pure Water, etc., Corresponding to Alti- tudes of Barometer .......... 518 " Capacity for ............... 505, 507 " Communication and Transmis- sion of, and Relative Power of, of Various Substances ---- 510 u CONDENSATION and of Steam in Cast-iron Pipes ...... 515-516 " " ofSteamper Sq. Foot and per Degree per Hour ......... 516 4 ' Conducting Powers, Relative, of Various Substances and De- ductions from Results ---- 514-515 " CONDUCTION or Convection of. . . 514 " CONGELATION and LIQUEFACTION 516 " Degrees of Different Scales, To Reduce, and Contrariwise ---- 523 " Densities of Some Vapors ...... 521 " Density of Water, To Compute. . 520 DESICCATION ............... 513-5*4 " DISTILLATION ................. 514 " Effect upon Various Bodies by. 518 " EVAPORATION or Vaporization, Elements of, etc. .512-513 " " A rea of Grate and Fuel for, To Compute ..... 513 ' ' of Water per Sq. Foot of Surface per Hour. . . 514 " " To Evaporate i Lb. of Water .............. 5" * ' Evaporative Power of Tubes per Degree of Heat, etc .......... 513 " Expansion of Water, Liquids, Gases, and Air ........... 519-520 " Extremes of and Cold in Va- rious Countries ........... 191-192 " Fluids, Expansion of, in Vol- ume, To Compute ........ 523-524 ' ' Frigorific Mixtures ........ 193, 516 " Heating and Evaporating Water by Steam in Pipes and Boilers 511 " Height Corresponding to Boil- ing-Point of Water .......... 519 " LATENT, and To Compute. . . 508-509 " " of Fusion of Solids, and of a Non- Metallic Substance. 509 " " of Steam, To Compute ..... 707 " Length of /(-Inch Pipe to Heat looo Cube Feet of Air ....... 526 ' ' Linear Expansion or Dilatation of a Bar, Prism, or Substance. 519 " Liquids, Volume of Several at their Boiling- Point .......... 518 " Mean Temperatures of Various Localities ................... 192 " Mechanical Equivalent (Joules). 504 " " " of , Contained in Steam. 705 " Melting and Boiling Points of Various Substances ......... 517 < of Mines XX INDEX. Page HEAT, of the Sun 193 < Proper Temperatures of Enclosed Spaces 526 " RADIATION of. 509-510, 1027 4 ' Radiating or A bsorben I and Re- flecting Powers of Substances, and in Units of. 509-510 " Reduction of, by Surfaces 525 " REFLECTION of. 510-512 " Refrigerator, Surface of. 512 " Relative Capacities of Various Bodies for 507 " Required to Evaporate i Lb. Wa- ter Below 212 from Air at 32. 512 " Saturated Vapors, Pressure of, under Various Temperatures 518 " " Steam, Total of. 705, 707 " SENSIBLE 504,507 " Sensible and Latent, Sum of, and Latent of Vaporization. . 508 " Snow Line, or Perpetual Con- gelation 192 " SPECIFIC, To Ascertain, etc. . 505-507 " " for Equal Volumes of Gas and Air, To Compute 507 " Temperature by Agitation 524 " u of a Mixture of Like and Unlike Substances 506 " "of Solidification of Several Gases 516 ' "to which a Substance of a Given Length must be Sub- mitted or Reduced to Give it a Greater or Less Length or Volume by Expansion or Contraction 522-523 *' Transmission of, through Glass of Different Colors 511 * " Quantities Transmitted from Water to Water through Metals and other Solid Bodies i Inch Thick, per Sq.Foot. 511 " Units of, To Compute. . . . 511-512 " Underground 519 " Unit 504 " Units of, in Fuels 927 *' Vegetation, Limit of 192 " Volume of Water Evaporated in a Given Time, To Compute. . .513 HEATING,^ ir, Length of Pipe Required to Heat Air in an Enclosed Space by Water 525 ' by Steam, Illustration of. 527 1 by Hot-air Furnaces, Stoves, or or Open Fires, 528-529 ' by Hot Water 524, 1028 * by Steam 527, 913, 1027 * Coal Consumed per Hour to Heat ioo Feet of Pipe 527-528 " Length of Pipe Required to Heat Air by Steam at 5 L bs. per Sq. Inch 527 " Temperature of Enclosed Spaces 526 " VENTILATION of Buildings, Apartments, etc 524-525 HBATING, Volume of Air by i Sq. Foot of Iron Surface 925 " " of Air Heated by Ra- diators, Fuel, Grate, and Heating Surfaces 528 " Warming Buildings, etc. . . 524-529 Hebrew and Egyptian Measures, etc. 53 Height Corresponding to Boiling- Points of Water 519 HEIGHTS and Elevations of Various Places Above the Sea 183 " Measurement of 60 HEMP AND WIRE ROPE, Circumference ?/> for Rigging, U.S.N.... 172 " " Circumference and Breaking Weight of, U.S.N. 168 " General Notes 167 u " Hawsers and Cables, Com- parison of. 169 *' " Weight and Strength of. 172 ' " Weight of 166 " ROPE, IRON AND STEEL, Safe Load and Strength of. 164-165 " " Iron and Steel, Relative Di- mensions of. 172 " " Safe Strain Borne by, Units for Computing. 170-171 " Shrouds and Wire 173 * * Tarred, Destructive and Break- ing Strength of. 171 ' ' ROPES (See Ropes, Hawsers, and Cables) 166-173 Hewing and Sawing Timber, Loss in. 62 High Water, Time of, To Compute. .74-75 Hills or Plants in Area of an Acre. . 193 Historical Events and Notable Facts. 939 Hitches, Knots, etc 972 Hoggin 690 Hoisting Engines, Details of, etc 901 Honey, Analysis of. 207 Hoop Iron, Weight of. 129,131 Hopper Dredgers, Steam 899-900 Horizon, Dip of. 60 Horizontal Wheels 572 HORSE 435-437 ' Cart, Volume of Earth Trans- ported 908 Transportation 918 POWER 441, 733-734, 1028 " British Admiralty Rule. 734 Cost of, by Steam 950-951 Notes on 758 of Boilers 914 On a Canal 848 Transmission of. 188 TEAM, Tractive Power of. 436 HORSES, Age of. 186 " Labor of, etc 435-437 ' ' Performance of. 439-440 u and Cattle, Transportation of 192 " Weight of. 35 Horseshoe Nails, Length of. 1 53 Horseshoes and Spikes 152 HOSE, Delivery and Friction in 922 877 INDEX. XXI HOT-AIR Furnaces or Stove* ......... 528 44 Gas, and Steam Engines, Rel- ative Cost of. .............. 909 Human and Animal Sustenance ..... 203 HYDRAULIC RADIUS or Mean Depth. . 552 " Cement .................. 958 " " or Turkish Plaster . 591 " Paint .................... 872 4 4 RAM, Elements and Efficien- cy of . ....... 561, 917,923 " " per Cent, of Volume of Water Expended, To Compute ............ 917 HYDRAULICS, Elements of, etc ---- 529-557 " Canal Locks, and Times of Filling and Discharg- ing, To Compute. .553-555 " " M iscell. Illustrations.szfi-ssj 44 Circular Bent or Angular, Circular or Cylindrical Curved Pipes, Valve Gates or Slide Valves, Throttle, Clack or Trap Valve Cock, or Imperfect Contraction, Circular Openings or Sluices, Coefficients of. ............ 536 Circular Sluices, etc ........ 537 Circular, Triangular, Trape- zoidal, Prismatic Wedges, Sluices, Slits, etc ......... 538 Curvatures, Radii of. ....... 544 Curves and Bends .......... 545 Cylindrical Ajutage ......... 549 Deductions from Experiments on Discharge, from Reser- voirs, Conduits, or Pipes. 529-531 Depth of Flow over a Sill that will Discharge a Given Vol- ume, To Compute ......... 534 Discharge from a Notch in Side of a Vessel ...... 541 "from Conduits or Pipes, and Friction of. . .530-531 " from Irregular - shaped Vessels, as a Pond, Lake, etc., Time, Flow, Fall, Velocity, and Vol- ume Discharged ...... 542 "from Vessels not Receiv- ing any Supply. . .538-539 " from Vessels of Commu- nication ............. 541 " of inPipesf or any Length and Head, etc., and Elements of, To Com- pute .............. 547-548 " of under Variable Press- ures, and Time, Rise, Fall, and Volume ____ 540 " of when Form and Di- mensions of Vessel of Efflux are not Known 539 *' or Effiux for Various Openings and Aper- tures, and Relative Ve- locity under like Heads 532 HYDRAULICS, Distance of a Jet of Water, Projected from an Opening in Side of a Vessel, 14 Fall of a Canal or Conduit to Conduct and Discharge a Given Quantity of Water per Second, To Compute. . . 920 <4 Flow and Velocity in Rivers, Canals, and Streams, and To Compute Ele- ments of. 550-553 44 " and Velocities at which Materials will Move 916 44 " in Lined Channels.. .551-552 u " of Water in .Beds, Fall and Velocity of, as in Rivers, Canals, Streams, etc., and Coefficients of Friction of. 542-543 11 Flowing Water, Head of, To Compute 552 44 Forms of Transverse Sections of Canals, etc 543 4 4 Friction in Pipes and Sewers, and Head Necessary to Over- come, To Compute 543-544 u " of Liquids through Pipes . 531 " "of Water in Beds, as Rivers, etc. , Coefficients of. 543 " Head and Discharge in Pipes of Great Length 920 44 4 ' from Surface of Supply to Centre of Discharge 544 44 Height of a Jet in a Conduit Pipe, To Compute 919 44 Inspirator, Hancock 901 " Jets d'Eau, and Formulas for 550 44 Journals or Gudgeons, Fric- tion of. 571 44 Miner's and Water Inch 557 44 Obstruction in Rivers 551 4 4 Pipe, Inclination of, and Ele- ments of Long, To Com- pute 548 44 Prismatic Vessels 539-54 44 " Fall of in a Given Time, To Compute 540 44 IP under Different Heads. . . 557 44 Rectangular Notches or Ver- tical Apertures or Slits. 534 44 " Openings or Sluices or Horizontal Slits, and Discharge, To Compute. 535 44 " Weir, Volume of Dis^Jiarge, To Compute 532-534 44 Relative Velocity of Effiux, through Different Apertures and under Like Heads 532 44 Reservoirs or Cisterns, Time of Filling and Emptying, To Compute 541 4 4 Short Tubes, Mouthpieces, and Cylindrical Prolongations or Aiutagts,and Coefficients for Discharge of. 536-537 XX11 INDEX. Page HTDRAUJJCS, Sluice Weirs or Sluices . 535 44 u Impeded or Unim- peded Discharge of. . . .535-536 44 Submerged or Drowned Ori- fices and Weirs 553 44 Variable Motion of Water in Beds of Rivers or Streams. 543 " Velocity in Profile of a Navigable River, To Com- pute 551 4 4 Velocity of Water or of Fluids, Coefficients oj 'Discharge. 531-532 44 Vena Contracta 529 44 Vertical Height of a Stream Projected from Pipe of a Fire-Engine, To Compute. . 549 44 Volume of Water Flowing in a River, To Compute. . 543 44 Weirs, Gauging of. 922 44 " or Notches 539 HYDRODYNAMICS AND HYDROSTATICS, Elements of, 610.558-580, 614 44 44 AppoWs Wheel 580 44 " Barker's Mill 577 44 u Boy den Turbine 574 44 44 Breast Wheel, Proportion, Effect, and Power of. 569-570 44 44 Centrifugal Pumps 579-580 44 " Current Wheel 570 44 " Flutter or Saw-mill Wheel 571 44 4t Fontaine Turbine 574 44 " Horizontal Wheels 572-577 44 44 Hydraulic Ram, Elements of and Operation 561-562 44 44 Hydrostatic Press 561, 901 tt it u Thickness of Metal. .. 561 44 x 4 ' Motors, Effective Pow- er of Water 563 44 " Impact and Reaction Wheel. 576 44 " Impulse and Resistance of Fluids 577-578 44 4k Inward -Flow Turbines, Description of. 575 44 " Jonval Turbine, Elements and Results 575 44 " Low-Pressure Turbines. . . 575 44 44 Memoranda on Water- Wheels 571-572 44 44 Overshot Wheel, Elements, Power of, etc. . 563-566 44 4< " Power and Effect of, To Compute. . . 565-566 44 44 Percussion of Fluids 579 44 44 Pipes. Elements and Weight of, etc., To Compute. 560-561 44 " PonceleVs Wheel, Propor- tion and Power of. 567-568 44 ' 4 " Turbine, Elements of. . 574 44 44 Power of a Fall of Water, To Compute 562 44 44 Pressure and Centre of '. 558-560 44 44 " of a Fluid upon Bottom of Vessel, Vertical, In- clined, Curved, or any Surface, and also on a Sluice 559-560 ftp HYDRODYNAMICS AND HYDROSTATICS, u u pressure of a Column of a Fluid per Sq. Inch . . . 560 44 " Rankine Wheel 580 41 " Ratio of Effect to Power of Several Turbines 577 44 u Reaction Wheel 576 44 u Swain Turbine 575-576 4 ' 44 Tangential Wheel 576 u u Tremont Turbine 576 44 44 Turbine and Water Wheels, Comparison Between. . . 579 44 " Turbines, Elements, Power, and Results.. .572-577 44 " " High Pressure and Downward Flow. 574 44 4 ' Undershot Wheel, Power of 566 44 4C Victor Turbine 576 44 44 Water Power 562 44 " " Fall of. 563 44 " 44 Motors, Ratio of Ef- fective Power 563 u pressure Engine 579 44 " 4 ' Wheels, Dimensions of Arms, etc 571 44 " 44 Wheels, Divisions of, etc 563 44 " Whitelaw's Wheel 576-577 Hydrometers 67 4 Strength of or Volume of a Spirit, To Compute, etc 67 HYDROSTATIC RAIL OR SLIP RAILWAY, Power Required to Draw a Vessel. 910 HYDROSTATIC PRESS 561, 901 Hygrometer 68 " Dew -point, and to Ascertain Volume of Vapor in Atmos- phere 68 %4 Existing Dryness, To Ascertain 68 44 Vapor, Weight of, in Air 69 44 Volume of Vapor in Air 68 Hyperbola, To Describe 230 HYPERBOLIC LOGARITHMS 331-334, 712 ICE, Strength of, etc. ... 195, 912, 939, 943 44 and Snow, Weight and Volume. . . 849 44 Boats and Speed of. 896, 909 4 4 Making and Refrigeration. 943 , 965 -968 44 Manufacture of. 943, 967 IMPACT and Reaction Wheel 576 44 OR COLLISION 580-582 44 Velocities of Inelastic Bodies after, To Compute 581-582 Impenetrability 195 INCLINED PLANE, Motion on 619 44 Elements of, To Compute. 625-630 Incubation of Birds, Periods of. 192 India- Rubber, To Cut 877 Indicator, To Compute Pressure by. . 724 INERTIA, Moment of a Revolving Body. 609 " Moment of, Approximately to Ascertain 659 " 44 of a Solid Beam 819 44 of a Revolving Body, To Com- pute 616 INDEX. XX111 Page Injector, Steam 736 " Size of, To Compute 736 " Volume of Water required per IIP per Hour, To Compute 736 " " of Feed Water required per IIP per Hour 736 Ink, Chinese or India 907 " Stains, To Remove 935 Inks, Indelible, etc 875 Insects and Birds 196 Inspirator, Hancock's 901 Integration 24-25 INTEREST, Simple and Compound. 107-109 INVENTIONS, Origin and Period of Great 937 Involute, To Describe 229 INVOLUTION 96 IRON, Elements of, etc 637-640 (See Wr -ought-iron, 130-136, 639-640, 765,768,773,780,785-786.) " and Steel, Corrosion of. 908 " Bolts in Wood, Tenacity of. 198 * ' Bridges, and Iron Pipe Bridge .. 178 " Mold, To Remove 871 OR STEEL, Corrosive Effects of Salt Water on 916 " Pig, Ton of, Requirement of Air. 445 " Preservation of. 955 ' ' Rust, To Remove 935 Iron Steamer, First Built 915 Irregular Body, Volume of, To Compute 870 IRRIGATION, Cost of, per Acre 952 J. Jarrah Wood 913 Jets d'Eau 550 Jewish Measures 53 Jonval Turbine 575 Jordan, Valley of, and Dead Sea. . . . 934 Joules' Equivalent 504 Julian Calendar 70 Jumping, Leaping, etc., by Men 439 K. KEDGES AND ANCHORS, Weight and Number of, Units to Determine. ... 174 Kerosene Lamps, etc 872 Khorassar, or Turkish Mortar 592 KNOT 27 Knots, Hitches, etc 9 go LABOR, Man and Horse. 433-434, 436, 468 Lacquers 875 Laitance 592 Lake, Highest Elevation of. 96? Lakes, Areas of, in Europe, Asia, and Africa, and Depths and Heights of Great Northern of U. S. 181-182 Lamps, Candles, Fluids, and Gas. . . 584 Laud Measure 29 Larrying 598 Laths, Dimensions, etc 60 LATITUDE, Length of, etc 19! " and Longitude of Principal Locations and Observations.j6-&; Page LATITUDE N. reached by Explorers. . . 939 Launching Vessels, Friction of 478 EAD, Sheet, Cast or Milled 640 " Balls, Weight and Dimensions of. 501 " Encased Tin Pipe, Weight of... 151 " Given Sectional Area, Weight of- 136 " Measure 32 " Pipe, Resistance, Thickness, Weight, and Bursting Press- ure of, To Compute 831 " Pipes and Tin lined, Weight of , per Foot and Thickness. .137, 150 " Plates, Weight of per Sq. Foot . 146 " " Thickness of. 121 " Sheet, Weight of. 151 " Shot, Number of Pellets in an Ounce 501 " AND CAST IRON BALLS, Weight and Volume of 153 " Weight of, To Compute 155 cap or Bissextile Year 70 raping. Jumping, etc 439-440 .eaves, Value of, etc 207, 481 ,ee-way or Drift of a Vessel 910 Legal Tenders 38 lENSESAND MlRRORS, Element* 0/670-67 1 iCvel, Apparent, of Objects at or upon Surface of Land or Sea 56 Bevelling, Geographic 55-57. 1035 " by Boiling - Point of Water, Table of, etc 55~57 ' ' Height of Above or Below Level of Sea, To Compute 55 ..EVER, Elements of, To Compute.. 624-626 lifting by Men 439 LIGHT, Elements of, etc 583-587 " Sun's Rays 195 " Candles, Lamps, Fluids, and Gas 584 " Consumption and Comparative Intensity of, of Candles 583 ' ' Decomposition of. 583 " Electric, Candle Power of. 908 " Gas and Electric 198 " " Consumption, Volume, and Flow of. 585-587 " Intensity of, with Equal Volumes of Gas from Different Burners 585 " Loss of, by Use of Glass Globes. 584 " Penetration of, in Water 915 ' ' Refraction, Mean Indices of. . . . 584 " Relative Intensity, Consumption, Illumination, and Cost of Va- rious Modes of Illumination . . 584 " Services for Lamps 587 " Standard of. 910 " Volume of Gas from a Ton of Coal, Resin, etc 586 Lighting Power in Streets, To Deter- mine Coefficients of. 969 Lightning.'CTossi/icatfon of. 430 ' ' Protection of Buildings . 907-956 Lignite 479 , 481 Lime, Hydraulic ofTeil 589 XXIV INDEX. Page LIMES, CEMENTS, MORTARS, AND CON- CRETES, Elements of 588-597 and Cements 594-595 Asphalt Composition 593 Cements and Mortars, Experi- ments of Gen. Gillmore 596 " Transverse Strength of. . . 596 Conclusions from Experiments. . 590 Concrete or Beton 593 General Deductions and Notes, by Gens.Totten and Gilmore. 596-59 7 Limestones, Indication of, etc. . 588 Mortars 590-592, 595 Mural Efflorescence 593 Pozzuolana 589 Slaking Lime, etc 594 Stucco, Exterior Plaster. . .591-592 Trass or Terras 589 Turkish Plaster or Hydraulic Cement 591 LINES, To Draw, Bisect, etc 221 Linseed Cake, Value of, Compared to 100 Lbs. of Hay 203 LIQUID MEASURE SQ-S 1 * 46 LIQUIDS, Expansion of. 520 u Volume of, at Boiling- Point. . 518 LIQUORS, Proportion of Alcohol in. . . 204 " Proof of Spirituous 218 Lithro-fracteur 443 Llama, Load of. 438 LOCOMOTIVE, Elements of 902 u Axles, Friction of. 910 ' ' Brakes, Operation of. 923 ' ' Comparative Operations of a Simple and Compound. . . 953 LOCOMOTIVES, Operation of. .681-685, 912 " Adhesion 681-685 " in Foreign Countries 935 " Tractive Power 681 " Freight Train Resistance. .. 682 Log Lines 27 Logarithm of a Number 23-24 LOGARITHMS 35-3 IO > 1030-1031 " Hyperbolic 33i~334> 712 " of Numbers 311-330 LONGITUDE, To Reduce Time into 54 (See Latitude, 76-80. ) ' ' Lengths of a Degree 60 Lucifer Match, First in Use 915 Luminous Point 195 Lunar Cycle or Golden Number. ... 71 Lunar Month 70 M. MACADAMIZED ROADS 687-690 Machinery, Friction of. 475 MACHINES AND ENGINES, Mills, etc., Elements of. 898-904 Magnesia Covering of Steam -pipes and Boilers 918, 921 " Boilers, Pipes, etc 921 Magnetic Variation 57-59, i35 " Bearings of N. Y. 184 Magnetism 614 Magnifying Power 669, 968 Malleable Castings 639 Page Malleable Cast Iron 785 Maltha, or Greek Mastic 873 Manganese Bronze 832 Manures, Fertilizing Properties of, and Relative Value of Covered and Uncovered 188 Marble, To Clean, etc 873, 878 Marine Day 37 MARINE STEAMERS AND ENGINES. . 886-893 " Composite Yachts 891 " Electric Launch 900 " Ferry, Passenger, Team, and Tow-Boats 890 " Fire-Boat 891 " Glue 874 " Launches, Wood and Steel 895 ' ' Naval Cruisers, Iron-dads, and Protected 886-887 " Oil- Engine Launch 893 " Passenger and Freight 888-890 " Petroleum, Refrigerator, Fruit, and Fishing 888 " River and Inland 888-889 " Side Wheels, Wood 892 ' ' Stern Wheels, Iron and ^00^.892-893 Torpedoes and Dynamite. . . 886-887 " SAILING VESSELS, 'Yachts, Wood. 895 " " Cutter, Wood 895 " " Pilot-Boat 895 " " Steel, Iron, and Wood. 894 MASONRY 197, 597-605, 913 " Arch, Depth of. To Compute. . . 605 " Arches and Abutments, Depth, Radius, and Thickness of, To Compute 602, 604-605 " " Walls, and Abutments. 602, 604-605 " Brick, Stone, and Granite. . 595-600 " Brickwork 597-600 u Designation of. 602-603 " Estimate of Materials and La- bor for ioo Sq. Yds. Lath and Plaster 604 " Grouting 594 " Plastering 604 " Pointing 598 " Rubble 600-601 " Stone 600-603 " Technical Terms. .597-599, 602-603 " Volume of Bricks and Number of in a Cube Foot of. 599 " Walls, Thickness of Brick for Warehouses 600 " Working Load 781 Mason and Dixon's Line 188 Mastic 593 MATERIALS, Strength of. 761-841 " Non- Conducting 911, 914 ' ' Relative Non- Conductibility 0/911 MATTER and Minuteness of. 194 Mean Proportion 94 MEASURES, Length, etc 26-54 " and WEIGHTS 2 7~35 " and Mint Values 38-43 " and Weights, Foreign ofValue.^g-^ " " " Memoranda 43 INDEX. XXV Page MKASURBS, a Barrel. 30 " Ale and Beer. 45 " Apothecaries' 32, 46-47 " Avoirdupois 32, 47 " Board and Timber 61 " British and Metric, and Com- mercial Equivalents of. 906 " Builder's 46 " by Act of Congress and Metric Computation 36 11 Cables and Ropes 26 " Circular. 113-114, 34- " Cloth I..? 27 " Coal 33, 46 " Copying. 29 " Corn 198 " Cube 30,934 " Day, Civil, Solar, and Sidereal. 37 " Diamond 32 " Dry 3 o " English and French 44-45 " Electrical, British Ass'n 34 " Equivalents of Old and New U. S. and of Metric Denom- inations, To Compute 36 " Fluid 30 " Foreign 48-53 " French Old System 47 u Geographic, and Distances. . . 54 ' ' and Nautical. ... 26 Grain 32, 45 Grecian 53 Gunter's Chain 26 Hebrew and Egyptian 53 Land 29 Lead 32 Lineal and U. S. Standard. 26, 934 Liquid 30, 46, 934 Log Lines 27 Metric 27-33, 36, 44, 47 ' ' Equivalent Value, U. S. , and Old and New U.S. 28, 33 ' Power and Work 36 ' Temperatures 37 ' Velocities 37 ' Volumes 36, 46 ' Weights and Pressures 36 Mint, and Weight of Value. .40-43 Miscellaneous 27, 29, 31, 44, 46 Nautical 26, 30, 44 of Earth, Clay, Sand, etc 33 of Length 26,44 of Surface 29, 44, 934 of Timber, Local Standards, and To Compute 61-62 of Value 38-39 of Vessels 45 of Volume 30-31, 45, 934 of Weight 32, 47, 934 Old and New U. S. ; Approx- imate Equivalents. . . 33 u " Equivalents of. 36 Paper 29 Pendulum 27 Roman. 53 Ropes and Cables 26 Scriptw e and Ancient 53 Page MEASURES, Shoemaker's 27 " Timber, English 62 "Time 37 " Troy 32,47 " Vernier Scale 27 " Wine and Spirit 45 " Wood 33,47 Meat, Analysis of, and of Fish and Vegetables 200-202 Meat, To Preserve ... 196 Meats, Loss of, by Boiling, Roasting, or Baking 206 MECHANICAL CENTRES 605-614 " and Physical Elements, Con- structions and Results. . .907-918 " Centres of Gravity 605-608 ' ' Centre of Gravity of any Plane Figure, To Ascertain. . . 605 " of Gyration 609-611 " of a Water-wheel 6n " " " General Formulas for 6n " Elements of Gyration 610 u Moment of Inertia of a Revolv- ing Body and Radius of Gy- ration of, To Compute 609 11 POWERS 624-634 " Compound Axle, or Chinese Windlass 627 " Screw 631 ( ' Inclined Plane 628-630 " Lever. 624-626 " Pulley 632-634 " Rack and Pinion 628 " Screw and Wedge 630-631 " " Differential 632 " Wheel and Axle 626-627 " " and Pinion, Combina- tion, Chinese Windlass, etc 627-628 MECHANICS 614-634 " Accumulated Work in Moving Bodies, and To Compute 619 " Couple 614 u Decomposition of Forces 620 " Dynamics 614, 616-620 " Moment 614 " Moments of Stress on Girders, 621-623 u Motion on an Inclined Plane. 619 " Solid, Fluid, and Aeriform Bodies 614 " Uniform and Variable Motion 617 " Work by Percussive Force 620 Mechanic's, Cost of Family in France 908 Melting- Points 517 MEMORANDA . Physical and Mechanical Elements and Results. .907-918 ' ' Cast and Wro ' t Iron and Steel 832 >f en and Women, Weights of 35 " Performances of 438-439 " Power of. 433-435 Meniscus and Concave- Convex 669 MENSURATION OP AREAS, LINES, SUR- FACES, AND VOLUMES 335-378 " Acreage, To Compute 337 XXVI INDEX. Page MENSURATION OP AREAB, etc., Any Figure of Revolution 358, 376 " Arc and Chord, etc. , of a Circle, 343-345 " Area Bounded by a Curve 342 " " of any Plane Figure. . . 359 " Capillary Tube 358 " Cask Gauging and Ullaging, S77-37 8 " Chord of an Angle, To Compute 359 " Circle 342 " " Sector and Segment of. 346-347 " Circular Zone 349 " Cones 353-354, 3 6 3, 365 " Cubes and Parallelopipedon. . 360 " Cycloid 352 " Cylinder 350, 363 " " Sections 357 " Ellipsoid, Paraboloid, or Hy- perboloid of Revolution, 357> 375-376 " Gnomon 335 " Helix (Screw) 354-355 " Irregular Bodies 377 u " Figures 341, 358 " Link 353,370 " Lune 352 " Parallelograms 335 " Plot Angles without a Pro- tractor 359 " Polyaons 338-341 " Polyhedrons 362 " Prismoids 351, 361 " Prisms 350, 360 14 Pyramids 354, 365-366 " Reduction of Ascending or De- scending Line to Horizontal Measurement 359 " Regular Bodies. . .340-341, 362-364 " Rings, Circular and Cylin- drical 353, 368 " Side of Greatest Square in a Circle 343 " Sphere 347-348, 367-368 " " Segment of. 347 " Spherical Sector 370 " " Triangle 387 " " Zone 368 " Spheroids or Ellipsoids, 348-349, 368-369 " Spindles 355, 370-374 u Spirals 355 " To Plot Angles without a Pro- traction 359 " Trapezium 337-338 " Trapezoid 338 " Triangles 335~337 (see Trigonometry, 385-389. ) " Ungulas 351-352, 366-367 " Useful Factors 343 " Volume of an Irregular Body . 870 u Wedge 350, 361-362 " Zone, Spherical and Circular, 348-349 Mercurial Gauge. 910 Meta-Centre of Hull of a Vessel. .659, 919 Metal Products of U.S. 910 Pugg METALS, Alloys and Compositions. 634-637 " Adulteration in, To Discover. . . 216 " and Elements of. 637-644 " Comparative Quality of Various 821 " Lustre, Degrees of. 194 " Values of some Precious 938 " Various Weight of. 155 " Weight of, To Compute 131 u u by Pattern, To Compute 217 " of a Given Sectional Area, Weight of. 149 METER,cmcZ Value of. 27, 934 Meters, Water 942 METRIC Measures 27-33, IOI 3 " Factors 923 MILK, Nutritive Values and Constit- uents of. 202 Percentage of Cream 205 and Relative Richness of, of Sev- eral Animals 207 To Detect Starch in 196 Mineral Constituents Absorbed from an Acre of Soil 189 " Waters, Analysis of, etc. .850-851 Minerals, Relative Hardness of. 193 Miner's Inch 557 Mines, Temperature of 918, 955 Mining, for Blasting 445 '' Engines and Boilers 901 " Flat Ropes . . 165 Mirage 195, 669 Mirrors and Lenses 670 Miscellaneous Elements 188-198 ' Mixtures, Cements, Glue, Inks, Lacquers, Soldering, Varnish, Staining, etc 871-879 ' Operations and Illustrations, 879-885, 935 Mississippi River, Silt in 910 Models, Strength of. 644-645 " Bridge, Resistance of, from 645 " Dimensions of a Beam, etc.,ivhich a Structure can Bear 644-645 Molasses, Analysis of. 207 u Sugar and Water in., 201 Molding and Planing 476 Molecules, Velocity, Weight, and Vol- ume of. 194 Moment, Quantity of, etc 614 Momentum 195 Monoliths 179 Month, Mean Lunar 70 Months, Numbers of. 74 MOON'S AGE, To Compute 74-75 Mortar 590-592, 951, 971 " Sugar in 951 MORTARS, Limes, Cements, and Con- cretes, 588-597 MOTION, Accelerated, Retarded, and Uniform Variable. . .494-495, 617 " of Bodies in Fluids 645-648 " Pressure, Velocity, Time, etc.. . . 648 " Resistances of Areas and Dif ferent Figures in Water or Air. . . 646 ; Motive Power 910 j " of the World 935 INDEX. XXV11 Page Motors. Experiments on, for Street Railways 915 Mountains, Volcanoes, and Passes, Heights of. 182-183 Mowing 433 u Machine 910 Mule, Load and Work of. 437, 918 Mural Efflorescence 593 N. NAILS, Length and Number of. . . 153-154 " and Spikes, Retentiveness of 159 " Composition Sheathing 135 " Tacks, Spikes, etc 154 National Road, 178 Natural Formations and Construc- tions, Largest 936 " Powers 198 NAUTICAL MEASURE 30 NAVAL ARCHITECTURE 649-667 u Angles of Course and Sails. . . 665 " Bottom, Side, and Immersed Surface of Hull 653 *' Centre of Gravity of Bottom Plating of a Vessel 658 *' " " Common of Hull, Ar- mament, Engines, etc., To Compute. . 656 u u tl Depth of or Buoyancy below Meta- Centre, and Approximately. 656-657 " " of Effort, and Lateral Re- sistance, Relative Posi- tions of. 659 " Centres of Lateral Resistance and Effort, To Compute. 6 58-6 59 "Dead Flat 22 " Displacement, and its Centre of Gravity, To Compute, 653-655 " Approximately, and Coeffi- cients of, To Compute. . . 655 " " Coefficients of. 655, 657 " " Curve of, To Delineate, and Coefficients of. 657 " Elements of a Vessel, To Com- pute 653-660 " " of Capacity and Speed of Several Types of Steam- ers of R.N. 660 " " of a Steam Frigate, Weight, Moments of, etc 656 " Experiments upon Forms of Vessels, Results of. 649 " Freeboard 666, 913 " Heel and Steady Heel, An- gles of, To Compute. . . .664-665 " Lee- way, Angle, Ardency and Slackness 666 " Length of Vessel 909 " Masts and Spars 667 " " Location of. 664 " Memoranda of Weights and Elements 667 " Meta- Centre of Hull of a Ves- sel, To Compute 659, 919 Page NAVAL ARCHITECTURE, Metalling, Loss of Weight per Sq. Foot of on a Vessel's Bottom 667 " Moment of Inertia Approx- imately, To Ascertain. . .659-660 " Pitch of Screw Propeller and Slip of Side Wheels 662 " Plating Iron Hulls 667 '- Proportion of Power Utilized in a Steam Vessel and Fric- tion of Engines 663 " Resistance of Bottoms of Hulls. 662 " " of Air to a Vessel 666 " " to Wet Surface of Hull. 653 " Rudder Head 667 " Sailing Power and Careening Power of a Vessel 665 " " Ratio of Effective Area of Sails, etc., and of Vessel 1 s Speed to Wind. 663 " Sails, Propulsion and Area of 663 " " Area and Trimming of. 664-665 " Screw Propeller, Experiments upon Resistance of at High Velocities, etc 666 " Slip of Propeller and Side- wheels 666 " Speeds, Relative of Forms of Vessels 649 " Stability, Elements of, 010.649-653 " " and Speed of Models of different Sections, etc. . 650 *' " Elements of Power Re- quired to Careen a Body or Vessel, To Compute, 652-653 " " Power Required in a Steam -vessel, Capacity of another being given. 66 1 " u Results of Experiments upon Bodies 649-650 " " Statical, Statical Surface and Dynamical Surface Stability, To Compute, 651-652 " " Measure of, of Hull of a Vessel, ToDetermine.65&-652 " Steam Vessels, Approximate Rule for Speed and IP of, To Compute 662-663 " Trim, Change of. 655-656 " Weight, Curve of. 657 " Wind, Effective Impulse of. . . 665 " " Course and Apparent Course of. 666 Needle, Magnetic, Variation of. 1035 " Decennial Variation of. 58 " Variation of it in U. S. and Canada 59 Needles, First Introduced 72 Neutral Axis of a Beam, To Compute 820 New and Old Style 37, 70 Niagara, Falls, Height of, etc. 198, 930, 952 u Volume of Water and Power .. 952 Nitro-Glycerine 443 Non-conductibility of Materials. .911, 914 " -condensing Engine, Friction of '. 918 XXV111 INDEX. Page Non-conductors of Temperature, and Comparative Efficiency of. 933 NOTATION 25 Number of Direction 71 Numbers, Properties and Powers of. 98 " ^th and $th Powers of. 303, 304 u ^th, $th, and 6th Power, and ^th and ^th Root of, To Compute. 304 Nutritive Equivalents, Computedfrom amount of Nitrogen in Human Milk at i 205 0. Oats and Oat Straw, Value of com- pared to ioo Lbs. of Hay , 203 Obelisks, Egypt and New York 179 Objective Glasses, Diameter of the Principal 942 Observatories, Latitude and Longi- tude 80 Ocean, Depth of. 912 Oceans and Seas, Depths and Areas. 182 " Atlantic and Pacific 937 Offal, Weight of, in a Beef and Sheep. 35 Oil, Yield of, from Seeds 189, 939 " Cake and Vegetables, Nutritious Properties of Compared 204 " -Engine Launch 893 " Proportions of, in Air-dry Seeds. 203 " To Remove from Leather 878 " Watchmakers' 878 Oils, Petroleum, Schist, and Pine- wood 484 Old and New Style 37, 70 Omnibus, Weight, etc 844 Onion, Proportion of Gluten, and Ra- tio of Flesh-formers 207 Opera-Glasses, Telescopes, etc 671 " -Houses 180, 936, 954 Operations, Miscellaneous 879-885 OPTICS, Elements, etc 668-671 u Critical and Visual Angles, Mi- rage and Caustic Curves or Lines 669 1 ' Elemen ts of Mirrors and Lenses, To Compute 670 " Focus and Focal Distance, etc.. 668 " Refraction, Index of and Indices of, To Compute 668-669 44 Dimensions or Volume of an Image, To Compute 668 Ordnance, Energy of.. 910 Ore and Stone Breaker .903, 951 ORGANIC SUBSTANCES, Analysis of, by Weight 190 ORTHOGRAPHY OF TECHNICAL WORDS AND TERMS 1042-1052 OSCILLATION AND PERCUSSION, Centres of. 612-614 " " Centre of, in Bodies of Va- rious Figures 613 " " Centres of, To Compute. 612-61 4 44 " " of , Experimentally, To Ascertain 613 OVBRSHOT WHEEL 563 Oxford College 179 Oxidation of Cast-iron Pipe, To Resist 927 P. Pacific and Atlantic Oceans 937 Paint, Flexible for Canvas 915 " for Window-Glass 879 " Hydraulic 872 44 To Clean and Remove 878 Painting, and Proportion of Colors for 66 " Iron Rods 956 Paper, Blasting 912 44 Drawing, Tracing, Profile, Photo-printing, Cloths, etc 29, 964 PARABOLA, To Describe 229 Park, Deer 179 Parsnips, Ratio of Flesh-formers 207 Paste, Durable 878 il Preservation of. 878 Passages of Steamboats 896 ' l Ice-boats 896 44 Steamer and Sailing Vessels . 897 Passes, Mountains, and Volcanoes. . . 182 Pastils for Fumigating 879 PAVEMENT, Asphalt 690, 944-945 " Block Stone 689-690, 944-945 " Comparative Merits of. 945 u Granite 690 44 Macadam and Brick 944 " Miscellaneous Notes 690 44 Neufchatel 945 " Rubble Stone 689 " Telford 688 44 Voids in a Cube Yara of Stone 690 " Wood 689, 690, 944 PAVEMENTS, Roads, and Streets. . . 686-690 Payments, Equation of. 109 PEAT 482 Pendulum Measure 27 PENDULUMS, Elements of, etc 452-454 44 Centre of Gravity of. 453-454 " Conical, Number of Revolu- tions of, To Compute 454 44 Lengths and Number of Vibra- tions of, To Compute 453, 454 " Vibrations, Number and Time of, To Compute 454 Pennants, Ensigns, and Flags, U. S.. 199 PERCUSSION AND OSCILLATION, Centres of (see Oscillation and Per- cussion) 612-614 1 ' Caps, Number of, Correspond- ing to Birmingham Gauge. 502 PERFORMANCES of Men, Horses, 010.438-440 Perimeter of a Figure 912 PERMUTATIONS ioo PERPETUITIES 112 Petroleum, Elastic Force of Vapor. . 707 " Evaporative Effects of. 910 PHYSICAL AND MECHANICAL ELEMENTS, Construction and Results 907 PILE DRIVING 433, 671-673, 902, 972 44 Coefficient of Resistance of Earth 972 44 Pneumatic 673, 972 " Resistance of Formations 673 INDEX. XXIX Page PILE DRIVING, Ringing Engine 972 " Safe Load, To Compute 672 " Sheet Piling 672 " Sinking. 673 " Weight of Ram 972 PILES, Foundation 198, 781, 909 " Extreme Load a Pile will Bear 912 " Retaining Walls of Iron 196 PILING OF SHOT AND SHKLLS 65 Pillar at Delphi 179 Pillars or Columns 936 Pins, First in Use 915 PIPES, Dimensions, etc 747 and Tubes, Weight of. 147-148 Copper, Dimensions of. 150 Gas,Thickness of 123 " Threads 160 Lead, and Tin-lined, Weight of. . 137 " Encased 151 " Weight of. 150 " Metal, and Weight of. 147 u of Cast-iron to Resist Oxidation. 927 u or Cylinders of Cast-iron 132-133 " Riveted Iron and Copper, Weight of One Foot in Length 148 " Steam, Gas, and Water 138 " Tin, Weight of. 151 " Thickness of, To Compute 560 u Water, Standard of Cast-iron. . . 147 Pise 593 Pivots, Friction of. 472 Planing, Cast-iron and Molding. .476-477 PLANK ROADS 688 Plants or Hills, in an Acre 193 u Weather Foretelling 185 Plaster, Turkish 591 PLASTERING, Measuring of. 197 " Volumes Required, Materials and Labor for 100 Sq. Yards of. . . 604 PLATE BENDING, Iron 476 PLATES and Bolts, U.S. Test Q/j etc. 749-753 " of Metals, by Gauge 121 " Thickness of, To Compute 751 " Wrought-iron Shell 750 PLATING IRON HULLS 667 Ploughing 433 PNEUMATICS. AEROMETRY 673-676 (See also Aerometry.) POINTING in Masonry. ., 598 POISONS, Antidotes and Treatment. 185, 935 POLES AND SPARS 62 PONCELET'S WHEEL 567-568 " Turbine 574 POPULATION, and Area of Divisions and Countries 188 " Comparative Density of, and Number of Persons in a House in Different Cities. .910 ' * of Principal Cities 187 POSITION 98-99 Pozzuolana, Elements of. 589 Potato, Anti - Scorbutic Power and Ratio of Flesh-formers 206-207 POWDER, Smokeless 952 " Forcite 966 " Gun, Proportions of to Shot.. 502 POWER AND WORK, Metric 36 Motive 910 " of the World 935 Movers and Transmitters of... . 797 of a Quantity, Value of. 359 Ordinary Distribution of, in a Propeller Steamer 911 Required to Draw a Vessel up an Inclined Plane 910 Thermometric and Mechanical Energy of 10 Grains of Va- rious Substances when Oxi- dized in Human Body 205 To Sustain a Vehicle on an In- clined Road, To Compute.. 84 5-846 Transmission, Elements of 176 POWERS, Natural 198 ' ^th, $th, and 6th of a Number. 304 ' of first 9 Numbers 98 ' of ^th and $th Numbers. . . . 303-304 " of 6th Number 304 PRECIOUS METALS, Values of some . . . 938 Pressures and Weights, Metric 36, 923 PROBABILITY and Illustrations. . .114-117 " Odds between Results and Chances, etc 117 PROGRESSION 101-105 Proof of Spirituous Liquors , . 218 Propeller Steamers, Ordinary Distri- bution of Power in 911 PROPELLERS . . . .730-731, 886-891 Properties of Numbers 98 PROPORTION 94-96 PULLEY 433 " Compound, etc 633-634 " Power, Weight it will Raise, and No.of Cords to Sustain Lower Block 632 PUMP, Working of a 433 Appold's and Gwynne's 579 Steam, Elements and Capacities of 738 Water, First in Use 932 Pumping 433 " ENGINES. . . 738, 902-903, 954, 963 PUMPS, Direct Acting 738 ' Centrifugal. . 579-580, 911, 917, 1031 ' Circulating 749 ' Water and Vacuum 932, 963 ' Worthington 738 Pushing or Drawing 433 Pyramids, Statues, etc 178, 936 Q. Quartermasters, Service Train of. . . 198 R. Race-Courses, English, Length of.... 930 Rack and Pinion, Power of. 628 Radius Vector 449 RAIL, Weight of, To Compute 679 RAILROAD Crane 962 " Horse, First in Use 915 " Portable, and Hand Cars 908 " Signals and Significations 954 " Speed 969 " " in England 930, 951 u Ties, Duration of. 968 XXX INDEX. Page RAILROADS, Street, and Cost of Main- tenance of. 915, 918 " Result of Experiments on Mo- tors for 915 9 J 4 (See also Torsion, 790-797.) " and Gudgeons 790 " Deflection of Shafts 778-779 " Diameter and Journal of, Stress Uniformly along its Length, To Compute 571 " Journals or Bearings of, etc 796 " Loaded Transversely and Jour- nals of. 914 SHEARING or Detrusive Strength. 782-783 u Experiments in Cast-iron,Steel, Treenails, and Wood 783 " Power to Punch Iron, Brass, or Copper, To Compute ... 782 " Results of Experiments on, with a Punch 782 " " " with Parallel Cutters, Wro't - iron Bolts, Riveted Joints, and Various Materials 783 Sheathing and Braziers' 1 Sheets 155 " Copper.... 131 " Nails, Weight of. 135 Sheet- Iron, Blackand Galvanized. 124, 129 " Weight of. 129 SHEET PILING 672 SHELLS and Shot, Piling of. 65 Shingles 63 Shoemaker's Measure 27 SHOT and Shells, Piling of. 65 " Chilled and Drop 906 " No., Diameter, and Numbers of. 906 " Number of Pellets in an Oz 501 Shrinkage of Castings 218 Shrouds, Hemp and Wire 173 SIDE LIGHTS, Visibility of a Vessel's, 918 *' WHEELS, A rea of Blades and Slip 662 " Friction of Engines 662 Sidereal Day and Year. 37 Page Sides of Squares, Equal in Area to a Circle 258-259 Signals, Night, U.S.N. 199 " Railroad, and Significations.. 954 Silt, in Mississippi River 910 Silver Sheet, Thickness of. 1 19 SIMPLE INTEREST 107 Simpson's Rule, Area, To Compute.. 342 '' " Volume of an Irregular Body 870 SINES AND COSINES 390-402 " Number of Degrees, Min- utes, etc. , of, To Compute 402 SIPHON, Steam 1010 Sixth Power of a Number 304 Skating Performances 439 Slackwater, Canal, etc., Traction on. 848 Slaking of Lime 594 Slate, Surface of, and Number of Squares, To Compute 64 Slating, Weight of One Sq. Foot 64 SLATES and Slating 64 " Dimensions of. 64 ' ' English 64 " Weight per 1000 and Number Required to Cover a Square 64 SLIDE VALVES, Elements, etc 731-733 Smelting of Iron Ore 445 Slotting 477 SMOKE PIPES AND CHIMNEYS 748-749 SNOW, Pressure of, on Roofs 952 " and Ice 849 " Flakes 195 " Line or of Perpetual Congela- tion 192 " Melted, Volume of. 195 Solar Day and Year 37, 70 Solders 634-636 Soldering 875 SOUND, Velocity of. 195 " Distances by Velocity of, To Com- pute 428 " Velocity of, in Several Solids. . . 428 Soundings, to Reduce to Low Water. 60 Spars and Poles 62 SPECIFIC GRAVITY AND WEIGHT. . .208-215 " Given Weight of a Body, To Compute 215 " of a Body Soluble in Water. 209 ' ' of a Body Heavier or Lighter than Water. 209 " of Elastic Fluids 215 " of a Fluid 209 " of Liquids 214-215 " of Miscellaneous Substances 214 " of Solids 210-214 " or Density of Steam 706 " Proportions of Two Ingre- dients in a Compound, or to Discover A duller ution . 216 " Weight of Ingredients, that of Compound being Given, To Compute 218 ' ' Weights and Volumes of Va- rious Substances in Ordi- nary Use .216-215 INDEX. XXX111 P*.] Speed of Vessels 971, 1010 SPIKES, SHIP, BOAT, and Railroad. 152, 154 and Horseshoes 152 and Nails, Retentiveness of.... 159 General Remarks 159-160 Ship and Railway 970 WroH-iron Nails and Tacks. .. 154 Spiral, To Describe 230 Spires, Towers, Columns, etc 180, 932 Spirits, Strength of, To Compute 67 Spirituous Liquors, Dilution per Cent. 191 " Dilution, To Reduce 191 " Proof of. 218 44 Proportion of Alcohol. .... 191 Springs, Deflection of 779 Spur Gear 911 SQUARE AND CUBE ROOT, Square or Cube, and when Number is an Odd Number 300-302 ** of Decimals alone,To A scertain,302 " of a Higher Number than con- tained in Table, To Compute 301 44 of a Number consisting of In- tegers and Decimals, To As- certain 301-302 " ROOT, To Extract. 97 " *' or CUBE ROOTS of Roots, Whole Numbers, and of Integers and Decimals, To Ascertain 97-98 " To Ascertain One that has Same Area as a given Circle 259 SQUARES, CUBES, AND SQUARE AND CUBE ROOTS 272-302 " Sides of, Equal in Area to a Circle 258-259 STABILITY, Elements, etc 693-703 44 Angles of Equilibrium of..... 694 44 Dynamical and Statical 651 44 Earthwork, Centre of Pressure of. 696 44 Equilibrium and Stability, To Compute 701 44 Memoranda 695-696 " Moment of, and To Compute. 693, 701 44 of a Body on a Horizontal Plane or on an Inclination, 694-695 44 of a Fixed or Floating Body. . 693 44 of Hull of a Vessel or Floating Body, To Determine 650 " of Varying Models 649 " Statical and Dynamical, To Compute 651 " Weight of a Body, To Sustain a Given Thrust 693-694 Staging, Coach 440 Staining, Wood and Ivory 876 Stains, To Remove 878 Starch, Proportion of, in Vegetables. 205 Stars, Velocity of. 198 STATICS 615-616 " Composition and Resolution of Forces 615 Page STATICS, Equilibrium of Force 616 " Inertia of a Revolving Body. . . 616 " Spherical Triangles 387 " Specific Heat 505-507 " of Air and Gases.. 505 Statues, Pyramids, etc 178 STAY BOLTS, Diameter, Pitch, etc. ... 754 STEAM, Elements of, etc.. 640-643, 704-727 44 and Air, Mixture of. 737 " Blowing off, Saturated Water, Loss of Heat by, To Compute. . 726-727 " " Volume Btoivn off to that Evaporated, To Compute. . . 727 " Clearance, Effect of..... 715 44 Coal, Utilization of, in a Boiler. 726 44 Combined Ratio of Expansion and Final Pressure in zd Cyl- inder, To Attain 723 " Condensation of, in Cast-iron Pipes 515-516 " Condensed per Sq. Foot and per Degree per Hour 516 " " of Expanded, per H of Ef- fect per Hour 716 " Consumption of Fuel in a Fur- nace, To Compute 725-726 " Cutting Off, Point of, for a Given Ratio of Expansion 711 " Point of, to Attain Limit of Expansion 710 44 Cylinder, Net Volume of for Given Weight of Steam, etc., To Compute 715 44 Density or Specific Gravity of. . 706 " Effect for One Stroke and a Given Combined Ratio of Expansion 723-724 " * Relative,of Equal Volumes. 714 1 Total ofiLb. of Expanded, 714-715 " Effective Work in One Stroke as Given by an Indicator Dia- gram, To Compute 714 44 Efficiency, Actual, Conclusions on 724 u Elastic Force and Temperature of Vapors of Alcohol, Ether, Sulphuret of Carbon, Petro- leum, and Turpentine 707 44 Expanded, Consumption of per IP of Effect per Hour 716 44 Expansion, Points of 712 " " Effects of. 713 " " Point of Cutting off, Actual Ratio of, Pressure at any Point of. Mean or Average, and Final Effective or Initial, To Com- pute 710-711 44 Expansive Force of. 704 44 Feed Water, Gain in at High Temperature, To Compute 719 u *' Gain in, and Initial Pressure, when Acting Expansively, compared with Non-eoepan- sion or Full Stroke 725 41 Gaseous, Total Heat, and Veloc- ity of, To Compute 710 XXXIV INDEX. Page STKAM Heating Co. of N. T. 904 44 " and Bolters. 913, 957, 1025, 1027 44 Incrustation of Scale or Sedi- ment, To Remove 726 44 Indicator, Mean Pressure by, To Compute 724 44 -Injector 736 44 Mean Pressure by Hyperbolic Logarithms, To Compute. .712-713 " Mechanical Equivalent of. 705 " Notes of. 936, 954 14 Pipes and Casing 515 44 Pipes, Gas, etc., Dimensions and Weights of. 138 44 Plant, Cost of Coal and Labor in Operation of 1000 IP 951 44 Pressure in Ins. of Mercury 706 " " Weight of a Cube Foot, Pressure and Tempera- ture, To Compute 705 44 *' in a Cylinder, at any Point of Expansion, or at End of Stroke, To Compute.. 711 " Pressures, Mean, Final. Effective, Initial, or Total Average, To Compute 711 14 Saline Matter, Proportion of in Sea-water 727 44 " Saturation in Boilers 726 44 Specific Gravity of. 704, 706 " " of, compared with Air, To Compute 706 44 Surface Condensation, Experi- ments on 911 44 Velocity of., 704, 913, 936 44 of, into a Vacuum 704 44 Volume of Cylinder for a Given Effect, etc 715 4< of Water at any Given Temperature Mixed with it, to Raise or Reduce Mixture to any Required Temperature, To Compute 707 " " of Water Evaporated per Lb. of Coal, To Com- pute 725 " of Water in a Given Volume of and of a Cube Foot of, To Compute 706 * " of to Raise a Given Volume of Water to any Given Temperature. 706-707 Weight and Effect of, for other Pressures than looLbs.j Multi- pliers for 719 " Wire drawing of. 718 " Yachts, Relative Velocities of, from Elements of their Con- struction, To Compute 928 44 COMPOUND EXPANSION, Elements of, etc 712, 720-724 " " Combined Ratio of and Final Pressure in 2d Cylinder, To Attain 723 * *' Comparative Effect in Receiv- er and Woolf Engine 724 Pag STEAM, COMPOUND EXPANSION, Effect for One Stroke and a Given Ratio of in ist Cylinder, To Compute 721-722 44 " Effect for One Stroke and a Given Combined Actual Ratio of, To Compute. .713-724 41 " Expansion in a Compound Engine, To Compute. . 721 " " " From Receiver 720-724 44 4 ' Final Pressure 720-72 1 44 " Woolf Engine, Ratio of Ex- pansion, etc 722 44 SATURATED, Total Heat and Ab- sorption of. 705 44 " Energy and Efficiency of, To Compute 716-717 44 * 4 Latent and Total Heat of, To Compute 707 44 " Pressure, Temperature,Vol- ume. and Density. . . . 708-709 u it Properties of, of Maximum Density 717 44 " Vapors, Pressure of. 518 44 SUPERHEATED, Energy and Effi- ciency of, To Compute. 7 1 7-7 1 8 44 " Expansion, Effects with Equal Volumes and One Lb. of ioo Lbs. Pressure 718-719 STEAMBOAT, Iron, First built 915 STEAMBOATS, RIVER, AND ENGINES. 892-893 44 Passages of. 896 44 Wood and River Side-wheels. 892,919 44 " Ferry, Passenger, Team, and Tow-Boats 890 44 4 ' Passenger and Deck Freight. 893 44 " Stern-wheels 892-893 STEAM-ENGINE, Elements of, etc. . 727-760 * and Sugar- Mill, Weights of. . . 908 44 -Boilers in Foreign Countries. 935 44 and Boilers, Cost of Operating per Day of 10 Hours .... 904 44 Circulating Pumps, Volume of, etc 749 44 Condenser or Reservoir, Tem- perature of Water in, To Compute 707 4 4 Dimensions of Cylinder, Grate, and Heating Surfaces, To Compute 927, 1024 44 Distance of Piston from End of Stroke, when Lead pro- duces its Effect, and when Steam is Admitted for Re- turn Stroke 732-735 44 Feed Pump, Area >/, To Com- pute 736 44 Fire, Elements, etc 904 44 General Rules for 728-730 " H* of, To Compute 733-734 44 4t Admiralty and French .. 734 44 Injection Pipe, Area of, To Compute 73S-73 6 44 Notes of 936, 954 INDEX. XXXV Page STEAM - ENGINE, Portable, Standard Operation of, and Elements of 737 " Propeller Slip and Thrust, To Compute 730-73 1 44 Proportion of Parts, Condens- ing and Non-condensing. 727-729 44 Receiver 721 " Results of Experiments on Operation of. 933 " Screw, Friction of 478 44 Steam- Injector and Volume of Water Discharged per Hour 736 44 " -Pumps, Elements and Ca- pacities of. 738 44 Vertical Beam, Jet Condens- ing, Weight of, To Compute. . 759 44 Volume of Water Required to be Evaporated in.. .734-735 " Volume of Circulating Water Required in 735 " Volume of Feed Water and In- jection Water Required per IP per Hour 736 " " of Flow through an Injec- tion Pipe 735 " Water-wheels, Radial and Feathering, and Elements of. 730 w SLIDE VALVES, To Compute and Ascertain Lap and Breadth of Ports 731 " Distance of Piston from End of Stroke given, To Compute Lead, etc 732-733 " Lap and Lead of Locomotive Valves 733 44 Part of Stroke any Given Lap will Cut off, To Compute. ... 731 44 Stroke at which Exhausting Port is Closed, etc 732 44 " of, To Compute 732 STEAM ENGINES, Results of Operation f- 737. 9 2 4 933. 954 " and Boilers, Weights of with Water 929 44 " Results of Performances of. 924, 927 " Duty of and Relative Cost of for Equal Effects 757 44 Practical Efficiency of. 737 " Side- wheels, Propeller and Ma- rine, Weights of. 758-759 " Weights of 758-759, 9" STEAM VESSEL, Power Utilized in... 662 4 ' Resistance to, in Air and Water 9 1 1 " -PROPELLER, Ordinary Distri- bution of Power in 91 1 " Velocity of. 1010 Steamer "Great Eastern " 173 STEAMERS 478 u Iron, First Built 915 " Relative Velocities of Yachts, from Elements of Construction, and Large 928 Steaming Distances 86 TEEL 640-643, 750, 783, 787-788, 827 44 and Iron, Corrosion of. 908 ' 4 and Iron, Corrosive Effects of Salt Water 916 44 Guns 913 4 4 Hemp, Iron and Steel Wire Ropes, Relative Dimensions of... 172 44 l4 and Iron Rope, Round and Flat and Safe Load. . . 164-166 1 4< Iron and Steel Wire Rope. .. 164 " Hexagonal, Octagonal, and Oval 135 ' 4 Locomotive Tubes 1 38 44 Manufacture of, Remarks on. ... 642 44 of a Given Section, Weight of. 136, 149 " Plates 750, 830 44 " Thickness of 121 44 44 Weight of ...118-119,146 " and Iron, Rolled Bars, and Weight of. . . 125, 126, 128, 134, 135 44 Wire, Weight of. 120-121 Sterling, Pound, etc 38 Stings and Burns, Application for. . . 196 Stirling's Mixed Iron 785 Stirrups or Bridles, for Beams 838 STONE and Ore Breakers and Crusher, 903, 957 4 4 Dressed, Modes of. 603 44 Hauling 468 44 Load per Sq. Foot 915 44 Masonry, Elements of. 595-600 44 " A shlar and Rubble 600-601 44 Resistance of, to Freezing 184 44 Sawing 196, 904 4 44 and Dressing, Cost of..... 949 44 Voids in a Cube Yard of. 690 STONES, CEMENTS, etc., Crushing of. . 766 (See Crushing Strength, 764-769.) ' ' Building, Expansion and Con- traction of. 184 Straw and Hay, Weight of. 198 Streams, Rivers, and Canals, Flow of Water in 550 STREET RAILS OR TRAMWAYS. 435, 915, 918 44 RAILROADS, Experiments on Mo- tors, Result of. 915 44 44 Cost of Maintenance .... 918 41 ROADS and Pavements 686-690 STRENGTH OF MATERIALS, Elements of. 761-841 " ELASTICITY AND STRENGTH, Co- hesion and Resilience. 761-763 44 44 Coefficient of. 761-762 44 44 Modulus, Height of, Weight of, Various Materials. 762-763 u <( o j- rp Compute 762 44 4t of Elasticity, Height of, To Compute 763-764 44 " Resilience, Comparative, of Woods 763 41 4 4 Weight a Material will Bear without Permanent Alter- ation of its Length 763 " COHESION, Modulus of and Weight. To Compute. . =763-764 XXXVI INDEX. Page STRENGTH OP MATERIALS, CRUSHING, ly\ \ments of, etc 764-769 " " Bricks 908 " " Ca* tnd Wro't Iron, Woods a \ Various Metals 765 " " Coll.: ins of Iron and Steel, Sn.'e Load of and Coeffi- cients of, To Compute. . . 769 " " Cohans, Arches, Chords, etc. , of Cast Iron, Safe Load of. 766-767 M u " Weight of, To Compute. 769 14 " Cylinders and Rectangular ' Tubes of Wro't Iron 767 11 " Elastic Limit compared to 764 " " Granite, Limestone, Marble, and Sandstone 767, 1029 " " Hollow Columns or Tubes, Safe Load of. 768-769 " " Ice 912 * " Long Solid Columns, Com- parative Value of. 976 " " Notes and Effects, Season- ing, etc 764 " " of Cements and Mortars. .. 596 " Relative Value of Woods, Strength and Stiffness Combined 976 ** " Sandstones, Stones, Cements, Masonry, etc 765-766 " " Various Materials 765-769 " DEFLECTION, Elements of, etc. , 770-781 " " and Weight Borne by a Bar or Beam of Wro't Iron. . 773 " " Bars, Beams, Girders, etc., 770-771 " " Beams of Rectangular Sec- tion, Formulas for. 771-773 " " "and Comparative Strength of Flanged 778 " " " and Girders 840-841 " " " Elastic Strength of, of Unsymmetrical Section 778 " " " Flanged, and Weight of that may be Borne by One of Cast Iron. . 777-778 ii ii (t q^ @ as t an ft Wro^t-iron, and Woods 77 2 -774 it u or Girders, Continuous. 772 " Bearings, Admissible Dis- tances between 778 " " Cast Iron Flanged Beams and Comparative Strength of. 809 * *' General Deductions 779-780 Girders, Tubular, ofWro't Iron 775, 809 " " Rails, Flanged, Iron and Steel 775-776 " " Rectangular Beam of Iron and Woods, Load that may be Borne by 773 M " " Bars and Beams of Cast Iron and Various Sec- on*,etc 777 STRENGTH OF MATERIALS, Results of Experiments on Subjec- tion of Cast-iron Bars to Continued Strains 780 " " Riveted Beams of Wro't Iron and Weight of. . 774-775 " " Rolled Beams of WroH Iron 774 " " Shaft from its Weight alone 778 " " Shafts and Distributed Weight for Limit 778 9 8 Sulphuret of Carbon, Elastic Force and Temperature of Vapor of. .. . 707 Sun, Heat, Diameter, etc 188 " Heat of. 193 Sunday Cycle or Cycle of the Sun. . . 70 u or Dominical Letter 70 Sun dial, To Set 69 Sunstroke Remedy 938 Surveying, Useful Number in 69 SUSPENSION BRIDGES 178, 199, 842 u " Elements, Stress, and Pressure, To Compute 842 " " Horizontal Stress and Vertical Pressure on Piers, To Compute . . 842 " " Steel Cables for 163 INDEX. XXXIX Page SUSPENSION BRIDGES, Ratio of Stress on Chains or Cables at Point of Sus- pension Bears to ivhole Weight of Structure and Load, To Compute. . 842 Sustenance, Human and Animal 203 4 ' Requirements of a Workman 207 Sweet Potato 207 Swimming 439 SYMBOLS, Algebraic, and Formulas. 22-23 41 and Characters 21-22 44 and Equivalents 190 44 for Elements and Formulas. . . 981 Symbolic Hatching and Designa- tions 93 2 T. Tacks, Nails, Spikes, etc., Wro't Iron 154 Tan, Elements of. 482 Tangential Wheel 576 TANGENTS AND CO-TANGENTS 415-426 44 44 To Compute 426 44 44 Degrees, Minutes, etc. , of Given 4 26 Tannin, Quantity of, in Substances. . 190 Tape Line or Chain, to Set out a Right Angle with 69 TECHNICAL TERMS, Orthography of. 1042-51 44 44 in Masonry 597-599 Tee and Angle Iron, Weight of. 130 Teeth of Wheels 859-862 44 Dimensions of a Tooth, etc., To Compute 860-861 " IP and Stress of, To Compute. . . 86 1 1 ' Involute 859 Telegraph Wire, Span of. 179, 936 44 Telephone, Wires and Cables. . 960 Telescopes, Opera- Glasses, etc 671, 942 Telford Roads 688, 690 TEMPERATURE and Extremes of. .914, 952 44 Absolute 504 44 Artificial and of Earth 195 44 by Agitation 524 44 Decrease of, by Altitude 522 44 Extremes of. 952 44 Non-conductors of. 933 44 of Enclosed Spaces. ........ 526 44 of Mines 918 " of Saturated Steam, Latent and Total Heat of, To Compute 707 " of Steam, To Compute 705 44 of Various Localities 192 * 4 To Reduce Degrees of Differ- ent Scales 523 44 Transmission or Conductiv- ity of 914 44 Underground 519 Temperatures, Metric 37 Tempering Boring Instruments 197 Tenacity of Iron Bolts in Woods 198 TENSILE STRENGTH (See Strengtfi of Materials) 784-790 Terne Plates 124 Terra Cotta 602 Tests, Simple, of Water 928 Theatres and Opera-Houses 180 Thermometers, Reduction of. 523 Throwing Weights by Men 439 Thrust, Weight of a Body, to Sustain a Given. To Compute 693 Tidal Phenomena and Current. .75-1010 Tide Table for Cocat of U. S 84-85 TIDES 84, 198 of Atlantic and Pacific 191, 198 of Pacific Coast 85 Rise and Fall of, Gulf of Mexico. 85 Time of High Water 74-75 Tie-rods, Experiments on 787 TIMBER AND BOAKD MEASURE 61 ;i AND WOODS, Elements, Notes, Treatment, etc 865-870 4 Comparative Weight of Green and Seasoned 217 (See Wood and Timber, 865-870.) 4 Measure, and Volume of, To Com- pute 61-62 1 Strengthof. 870 1 Volume of Squared, To Compute 62 ' Waste in Hewing or Sawing. ... 62 TIME, after Apparent Noon, before Moon next passes Meridian. .. 75 Civil and Marine 37 Difference in 81-83 44 of, between New York and Greenwich, and any Location, To Compute* 83 Measures of, and New Style 37 Sidereal and Solar 37 To Reduce to Longitude 54 TIN, Plate and Block 644 " Lined, and Lead Pipes, Weights of per Foot. i37> X 5i 44 Pipes, Lead Encased, Weight of.. 151 4 k Plates, Marks and Weights 137 TOBIN BRONZE, YachtShafling t etc.. 929 Tolerance, of Coins 38 Tonite, or Cotton Powder 443 TONNAGE, of Vessels, To Compute. 175-177 u Approximate Rule 176 41 Builder's Measurement 176 44 Corinthian, New Thames, and Royal Thames Yacht Clubs. . 177 " English Registered 1 75-176 44 Freight or Measurement 177 41 of Suez Canal 177 44 To Compute 173 44 Units for Measurement, and Dead- weight Cargoes 176-177 " Weight of Cargo, To Ascertain. 177 Tools, Friction of. 47 6 Tornado, Pressure of. 911 TORPEDOES, Submarine 946 TORSIONAL STRENGTH (See Strength of Materials) 79~797 Towers, Spires, and Domes 1 80, 932 Towing, on Erie Canal and Hudson River 193 TRACTION, Elements of, etc 843-849 44 and Statical Resistance of Ele- vations 846 4 4 Coefficients of, for Roads 845 xl INDEX. Page TRACTION, Friction of Roads and Coefficients of in Propor- tion to Load 847 44 Maximum Power of a Horse on a Canal 848 44 of Omnibus and Speed 844 " on Canal, Slack-water, River, and on Street Railroads 848 u on Various Roads and of Va- rious Vehicles 845, 847-848 u Power Necessary to Sustain a Vehicle on an Inclined Road, To Compute 845-846 44 Power Necessary to Move and Sustain a Vehicle Ascending or Descending an Elevation, To Compute 846 " Resistance of a Car 849 44 u of Gravity and Grade.. 847 44 " on an Inclined Road 846 44 u on Paved, Rough, and Common Roads 843-844 44 " to on Common Roads. 843-845 44 Results of Experiments of, on Roads and Pavements 843 Tramways or Street Railroads, 435,848,915 Transportation, Canal 193 44 of Horses and Cattle 192 TRANSVERSE STRENGTH (See Strength of Materials) 798-841 Trass or Terras. 589 Treadmill 433 Treenails, Strength of, etc 783 Trees, Large, in California 184 " u in Australia 971 Trigonometrical Equivalents 387 Trigonometry, Plane, Angles, Sides, etc., To Compute 385-389 41 Distances of Inaccessible Ob- jects, To Ascertain 388-389 44 Height of an Elevated Point, To Compute 389 Tripolith, Composition of. 198 Troops, Marine Transportation of... 914 Trotting. 439-44 Troy Measure 32 Truss, Iron and Stress on 178-1041 Tubers, Ratio of Flesh-formers 207 Tubes, and Flues 747, 827 14 and Pipes, Weight of, To Compute, 147-148 * 4 Brass and Seamless Brass, Weight of. 142 44 Copper, Seamless Drawn, Weight of. 140-142, 144 " English Wro'tlron, Weight 0/143, *45 44 Evaporative Capacity :>f Varying Length 742 44 Lap-welded Iron Boiler 139 41 or Girders, Dimensions and Pro- portions of. 809 44 Steel and Semi- Steel Locomotive 138 " Thickness of B W G 748 Tubular Bridge, Britannia 178 Tunnels, Lengths of. 179, 936 Page TURBINES (See Hydrodynamics). .572-577 " and Water -Wheels Compared. 579 Turkish Plaster and Mortar 591-592 Turning, Friction of. 477 44 and Boring Metal 197 Turnips, Ratio of Flesh-formers 207 Turpentine, Elastic Force and Tem- perature of Vapor of. 707 U. Underground Temperature 519 UNDERSHOT - WHEEL (See Hydrody- namics) 566-571 Unguents, Relative Value of. 471 Units for Computing Safe Strain of New Ropes, Hawsers, etc 170 V. Value of Coins, To Compute 39 and Weight of Foreign Coins . 40-45 of the Metre in terms of the Brit- ish Imp. Yard 934 VAPOR IN ATMOSPHERE, Volume and Weight of, To Compute 68-69 " Elastic Force of, of Alcohol, Ether, Sulphuret of Carbon, Petro- leum, and Turpentine 707 VAPORS, Relative Density of Some. . . 521 Variation of Magnetic Needle. . . 57-i39 1 ' Decennial, of Needle 58 rt of, in U. S. and Canada 59 Varnishes 876 Vegetable Marrow, Composition of. . 207 Vegetables, Analysis of Meat and Fish and Foods 200-201 4 4 and Oil-cake, Nutritious Prop- erties of Compared 204 44 Elements of Various 207 44 Proportion of Starch in 205 " Tubers, Ratio of Flesh-formers 207 Vegetation, Limits of. 192 Velocities, Metric 37,923 44 Acceleration and Distance of a Body, To Compute 921-922 41 of Different Figures in Air, ' Resistance of. 646-648 Velocity Lost by a Projectile, To Com- pute 648 " and Time, To Compute 648 44 and Volume of Molecules 194 44 of Current of a Bay or River. . 971 VENTILATION, Buildings, Apartments, etc 524 " Length of Iron Pipe required to Heat Air, To Compute. 52 5-526 4 ' of Mines 449 44 Proper Temperatures of En- closed Spaces 526 44 Volume of Air Discharged through a Ventilator. 524 44 4i of Air per Hour for each Occupant of a House. 525 Vernier Scale 27 VESSEL, Elements of, To Compute. . . 653 44 Hulls of Iron, Thickness of Plates and Rivets 830 INDEX. xli Vcuel'i Side Lights, Visibility of.... Vessels, Mean Speed of. 971 Veterinary, Treatment in 186 Victor Turbine. 576 Volcano, Power of. 910 Volcanoes, and Heights of. 182, 936 Voltmeters and Ammeters 961-962 VOLUME AND WEIGHT of Various Sub- stances in Ordinary Use 216 " of Molecules 194 VOLUMES, Mensuration of. 360-378 W. Walking 433,438 Wall, Chinese 179 WALLS and Arches, Elements of. .602-603 (See Walls, Dams and Embank- ments, 700-703. ) 4 ' and Earth, Friction of, To Ascer- tain and Compute 698 " ' Moment of, To Compute 701 44 u " of Pressure, Point of, To Ascertain 698 " of Buildings, Thickness of. i&g, 1020 44 or Dams, Centre of Gravity of. . 702 44 Retaining, of Iron Piles 196 14 "or Dam Stability of, To Determine 702 44 Revetment, Elements of. 694 Warehouses, Brick Walls for 603 WARMING BUILDINGS 527-528 44 by Hot-air Furnaces or Stoves 528 44 by Hot Water 524 4 by Steam 527 4 Coal Consumed per Hour. . . . 527 4 Furnaces and Open Fires. . . . 528 4 Illustrations of Heating 527 4 Volume of Air Heated by Ra- diators, Consumption of Coal, Areas of Grate, and Heating Surface of Boiler, etc. , per 100 Sq. Feet 528 Warps and Hawsers 173 Washington Aqueduct 178 Watches, First Constructed 915 WATER, Elements of 849-852, 916 * 4 Approximate Bottom Velocities of Flow of in Channels, at which Materials are Moved. . . 916 44 Boiling - Points of, at Different Degrees of Saturation 851 41 Column, Height of. 849 ; 4 ' Density of To Compute 520 44 Deposits of. at Different Degrees of Saturation and Tempera- ture 852 44 Evaporation of. 916 4 Expansion of. 519 4 Fresh and Sea 849-851 14 Distillation of. 955 4 Friction of in Pipes 925 4 Head and Discharge of in Pipes 920 ' Inch, Miner's 557 4 Motors, Ratio of Effective Power 563 4 Pipe, Dimensions and Weight of, from .375 to 5 Inches 137-138 Page WATER and Metal Pipes, To Compute Weightof. 147 44 Power and of a Fall of 562 44 44 Cost of on Driving Shaft. .. 950 44 Pressure Engine 579 44 Rainfall and Volume of 850 44 Resistance of, to an Area of One Sq. Foot Moving through, or Contrariwise 646 44 Saline Contents of 852 4 4 Salt, Corrosive Effects of, on Steel or Iron 916 44 Sea, Composition of. 851 44 Tests, Simple 928, 974 44 Velocity of a Falling Stream of. 49 6 Volumes of Pure, and at 32. . . 849 44 Weights of, and To Compute. 852, 923 44 -METERS, IVorthington's 942 44 -RAISING, Cost of. 949 44 -TUBE BOILER, Elements, Tests, and Average Results of. . 926 44 44 Efficiency of 947 " -WHEEL, Centre of Gyration... 611 44 4 4 Diameter and Journal of a Shaft, etc 571 44 Dimensions of Arms 571 44 -WHEELS, of Steamboats. 730 44 44 Compared with Turbines. . 579 (See Hydrodynamics and Hydrostatics, 563-579.) Waterfalls and Cascades 184 Watermelon, Water in 207 Waterproof, To Render, Wood, Iron, Walls, Paper, etc 875 Waters, Mineral, Analysis, etc. . .850-852 Waterworks, Filters for 184 Wave, Flood 912 Waves of the Sea 852-853 44 Height of, in Reservoirs, etc., To Compute 853 44 Tidal, and Length of. 853 44 Velocity of To Compute 853 Weather-Foretelling Plants 185 " Glasses 430 44 Indications 431 WEDGE, and To Compute Power 630 Weighing without Scales 66 WEIGHT, Diameter and Volume of, oj Cast-iron and Lead Balls.... 153 44 Aluminum 155 44 Anchors, Cables, Chains, etc... 173 44 and Dimensions of Lead Balls . 501 44 44 of Gas and Water Pipe.. 138 44 44 of Water Pipe 137 " and Fineness of U. S. Coins. . . 38 44 and Mint Value of Coins 40-43 44 and Mint Value of Foreign Coin, 1888 40-43 44 and Specific Gravity 208 44 and Strength of Wire, Iron. etc. 114 44 44 of Stud-link Chain Cable per Fathom 168, 930 44 4l of Hemp and Wire Ropes 172 44 Angle and T Iron and Steel, 125, 126, 130 xlii INDEX. Page WEIGHT, Apothecaries' 47 " Brass and Various Metals per Cube Inch and Foot, and Wire i " " and Gun- Metal of a Given Sectional Area 136, 149 u 1 1 1 1 yjr ro it and Cast Iron and Steel, Lead, Copper, and Zinc Plates per Sq. Foot. . 1 46 " Cast and Wro't Iron, Steel and Gun-Metal, of a Given Sectional Area. . , 149 ' " Castings 155 " " Copper, Wro't and Cast Iron, and Steel, of a Given Sectional Area 136 " " * ' Iron, etc. , Wire. . . 1 20-1 2 1 " u of Sheets 142 " *' Pipes corresponding to Iron and Iron Pipe Fittings .. 142 " " Wrought Iron, Steel, and Copper Plates 1 18-1 1 " Cables, Galvanized Steel, for Bridges 163 " Cattle, To Compute Dressed Weightof. 35 " Centrifugal Pump 917 " Crane Chains and Ropes 457 " Diamond, and of Diamonds. 32, 193 " Earth 33 " Electrical Resistance 34 " Fire-Engine 904 " Gun-Metal 155 ; Hay and Straw i Hemp and Wire Rope 162, i Lead, and of, To Compute. 32,151,155 " Pipes 137, 150, 831 " Plates, Weight of Sq. Foot. 146 " Sheet. 151 Length and Gauge of Iron Wire 172 of Anvils 918 of Articles of Food Consumed in Human System to Develop Power of Raising 140 Lbs. to a Height of 10,000 Feet 204 " of a Body or Substance when Specific Gravity is given, To Compute 215 " of Beeves and Beef, Comparative 35 " of Bolts and Nuts .-156-157, 159 " of Cast and Wrought Iron Bar or Rod, To Compute 131 " u Pipes or Cylinders. . . 132-133 41 of Cast Metal by Weight of Pat- tern 217 " of Composition Sheathing Nails 135 " of Copper, Cast and Wro'tlron, and Lead, To Compute.. 155 " of Copper, Braziers' and Sheath- ing 131 * ' Pipes and Composition Cocks 1 50 " " Rods or Bolts. 148 ** " Seamless Tubes 140-142, 144 " " Sheet per Sq. Foot 135 " of Corrugated Roof Plates. .... 131 " of a Cube Foot of Oak and Yellow Pine 870 , Page WEIGHT of a Cube Foot of Steam, To Compute 705 u of a Solid or Liquid Substance or an Elastic Fluid, To Ascer- tain 217 " of Cube Foot of Gases at 32.. . 215 " of Embankments, Walls, and Dams, per Cube Foot 694 " of Fence Wire 164 " of Flat Mining Ropes 165 " of Flat Rolled Iron and Steel, and Steel Angles 126, 127, 128 " of Food, Articles of. 204 " of Foods, to Furnish Nitroge- nous Matter 202 " of Galvanized Iron Wire. ..162, 163 Sheet Iron. . . 124, 129 " of Gaseous Products of Combus- tion, To Compute 460 " of Gun- Metal of a Given Sec- tional Area, and per Cube Inch or Foot 149, 155 " of Hemp, Iron, and Steel Rope. 164 ** of Hoop and Sheet Iron 129, 131 ** of Horses 35 * ' of Ingredients, that of Compound being given, To Compute 218 " of Iron and Steel, Round Rolled 126 " Square Rolled 125 " " Wire and Strength of '.. 124 " of Lead and Tin-lined Pipe.. . 137 " of Men and Women 35 " of Metals of a Given Sectional Area, per Lineal Foot.. . 136, 149 " of Molecules, Volume and Ve- locity 194 " of Oak and Yellow Pine 870 " f Offal in a Beef and Sheep.. 35 u of Products of Combustion. ... 462 " of Riveted Iron and Copper Pipes. 148 " of Ropes, Hawsers, and Cables, To Compute 172 " of Silver and Tin 155 " of Steam- Engine, Vertical Beam, Condensing, To Compute 759 " of Steel, Round, Hexagonal, Octagonal, and Oval 135 " of Stud-link Chain Cables 168 " of Timber, Green and Seasoned 217 " of Tin Pipes 151 " " Plates and Marks 137 " of Tubes of Copper, Brass, and Iron 140-147 " 4 * of Brass, To Compute 142 " of Various Materials 763 " of Various Substances per Cube Foot in Bulk 217 ' ' of Volume of A ir Consumed per Lb. of Combustible 461 " of Water-Pipes, To Compute. . . 561 " of Wire and Wire Rope 162 " " Length and Gauge 163 " u Iron, and Steel 164, 172 " of Wro't and Cast Iron per Sq. Foot 146 " of Zinc. Rolled 146, 155 INDEX. xliii Page WEIGHT of Wrought and Cast Iron Tubes I43-M5 " " " Steel, Copper, and Brass Plates, per Sq. Ft. per Gauge, 118-119 " on Floors and of Structures 841 " Rocks, Earth, etc 467-468 " Silver 155 44 Special, Locomotive 138 44 Steel, Copper, Iron, and Brass. 136 41 " Copper, Brass, and Wro't Iron Plates 118-119, 155 " " Copper, Brass, and Iron Wire 120-121 44 Terne Plates, and Thickness. .. 124 " Tin Cast 155 " U.S. and Standard Measures. . 934 " Water Pipe 137 " Wrought and Cast Iron, Steel, Copper, Lead, Brass, and Zinc Plates, per Sq. Foot. 146 " "Steel, Copper, and Brass Wire 120-121 44 Wire, and Strength 124 44 Zinc Sheets 123, 151-152 " " Plates and Dimensions of. 1 46, 151 44 u Rolled 146, 155 WEIGHTS, Various Materials 118-175 44 AND MEASURES, U. S. Stand- ard 26-36, 934 44 A-ncient and Scripture 53 44 and Pressures, Metric 923 44 and Volumes of Various Sub- stances in Ordinary Use 216 Apothecaries 1 32 Avoirdupois 32 Bushel, Pounds in 34 Coal, Earth, and Wood 33 Engine and Sugar Mill. 908 English and French 44, 47 Foreign Countries 48-53 Grain, Lead, and Troy 32, 47 Grecian 53 Hebrew, Jewish , and Egyptian . 5 3 Measures of. 32-47 44 and Pressures, Metric. . 36 Metric. . . .27-33, 36-37, 46-4?, 9 2 3 Miscellaneous Articles and Substances. 33, 46, 214-217 of Bells. 180 of Boilers 759 of Brain, Relative 192 of Chains and Anchors 174 of Fuels, Coals and Woods, 483-484, 486 of Guns (Ordnance) 498 of Lead Balls, and Shot. . . 500-502 " Pipe, To Compute 831 of Rope, Hemp, Iron and Steel 164-166, 172 of Slating per Sq. Foot 64 of Steam-Engines, 758-759, 9", 9 2 9i 954 of Steam- Engines and Boilers 929 of Steamers' Engines 911, 954 WEIGHTS of Steamers, Steamboats, Engines, Boilers, Launches, Yachts, Cutter, Pilot -Boat, Sailing Vessels,and Dredgers, 888-895,900 " of Substances in Bulk. 217 " of Sugar-Mills 903, 908 " of U.S. Coins. 3 8 44 of Wire, Iron 162-164 44 on Roofs. 952 " on Structures per Sq. Foot. . . 841 44 Pipes, Steam, Gas,and Water, Standard Dimensions. 138 Cast-iron 132-133 Iron and Copper, Riveted 148 Lap-welded, Steam, Gas, and Water 138 Lead and Tin.. . 137, 150-151 Metal, To Compute. . . 147-148 Seamless Brass, to Corre- spond with Iron 142 Roman 53 Tubes, Lap- welded, Steel, Semi- Steel, Special Locomotive, and Boiler 138-139 44 Charcoal Iron, Boiler 139 If. S. Old <& New, Approximate, Equivalents, 1 1 < L signify Inequality, or greater, or less than, and are put between two quantities ; as a 1 6 reads a greater than b, and a L 6 reads a less than 6. ( ) [ ] Parentheses and Brackets signify that all figures, etc., within them are to be operated upon as if they were only one ; thus, (3 -h 2) x 5 = 25 ; [8-2] X5 = 3o. ip signify that the formula is to be adapted to two distinct cases, as c =p v = a, either diminished or increased by v. Here there are expressed two values : first, the difference between c and v ; second, the sum of c and v. In this and like expressions, the upper symbol takes preference of the lower. p or TT is used to express ratio of circumference of a circle to its diameter = 3.1416; ^p=.7854,and-g-/> = .5236. '" signify Degrees, Minutes, Seconds, and Thirds. ' " set superior to a figure or figures, signify, hi denoting dimensions, Feet and Inches. a' a" a'" signify a prime, a second, a third, etc. 1, 2, added to or set inferior to a symbol, reads sub i or sub 2, and is used to designate corresponding values of the same element, as h, hi, h 2 , etc. 2 , 3 , 4 , added or set superior to a number or symbol, signify that that num- ber, etc., is to be squared, cubed, etc. ; thus, 4 2 means that 4 is to be multi- plied by 4 ; 4^, that it is to be cubed, as 4 3 = 4 X 4 X 4 = 64. The pmoer, or number of times a number is to be multiplied by itself, is shown by the number added, as 2 , 3, *, s, etc. 22 ALGEBRAIC SYMBOLS AND FORMULAS. 2, *, etc., aet superior to a number, signify square or cube root, etc., of the i J? 4 4. j>. number; as 2 s * signified square, root of 2} aiso 3 , *, 3 , 3 , etc., set superior to a number, signify two thirds power, etc., or cube root of square, or square or cube root of 4th power, or cube root of sixth power ; as 8 3 = V~& or I -7, 3-6, etc., set superior to a number, signify tenth root of iyth power, etc. 02^ .= 059 x log. of zoo = .059 X 2=r.n8; the number corresponding to log. .118, is 1.3122; hence, ioo-59rr 1.3122. That is, if 100 is raised to sgth power, and the loooth root is extracted, the result will be 1.3122. Differential and. Integral Calculus. In Equation, u=.^x e 2 a?, u is termed a function of x. If it is desired to indicate the fact that u thus depends for its value upon value of x, without expressing exact value of u in terms of #, following notation is used : M =/(#), u=.F(x'), or M =:(). Each of these notations is read, u is a function of x. If in such function of x value of x is assumed to commence with o and to increase uniformly, the notation indicating rate of increase is dx, and is read " the differential of x" Differentiation, d is its symbol, and it is the process of ascertaining the ratio existing between the rate of increase or decrease of a function of a variable and the rate of increase or decrease of the variable itself. If y = 3 as 8 , y or its equal 3 x 2 is the function of a?, and x is the independent variable, while the exponent of the variable or the primitive exponent is 2, By the operation of Calculus, such expressions are differentiated by di- minishing the exponent of the variable by unity, multiplying by the prim- itive exponent, and attaching the d x. Hence, dy=zz Xsxdx bxdx. This indicates the relation between the differential of y, the function of #, and the differential of x itself. Assume that x increasing at rate of 3 per second becomes 4 ; that is, x = 4, and d x=. 3; hence dy = 6 X 4 X 3 = 72. That is, if x is increasing at rate of 3 per second, at the time that x =. 4, the function itself is increasing at rate of 72 per second. To differentiate an expression of two or more terms, it is necessary to differentiate them separately and connect the results with the signs with which the terms are connected. Thus, differentiating u = 3 x 2 2 #, we have d u = d (3 x 2 2 x) = 6 x d x 2dx=(6x z) dx. Assuming x = 4 and d x= 3, we have c? M = (6 x 4 2) X 3 = 66. This indicates that when x = 4, and is increasing at rate of 3 per second, the func- tion M, or 3 x 2 2 a:, is at same instant increasing at rate of 66 per second. Integration. Its symbol / was originally letter S, initial of sum, the symbol of an operation the reverse of differentiation ; and when the oper- ation of integration is to be performed twice, thrice, or more times, it is written //, ///,etc. By the operation of Calculus, expressions are integrated by increasing the exponent of the variable by unity, dividing by the new exponent, and de taching the dx. Hence, integrating the differential 6 x d a*, we have /* 6 x dx = 3 or 2 . This result is the function, the differential of which is 6xd'x. To integrate an expression of two or more terms, it is necessary to inte- grate the terms separately and connect the results with the signs with which the terms are connected. Thus, integrating (6 #2) d#, we have f (6x 2) dx = f(6xdxzdx) = 3 a; 2 2x. This result is the function the differential of which is (6x 2) dx or (6x 2 a?) d x. NOTE. A quantity with the exponent , as or 3, is equal to unity. NOTATION. 25 The operation of summation may also be illustrated in use of the sym- bol / . Assuming x = 4, the former of the preceding results becomes / 6 x d x = 3 x 2 =. 48, the latter / (6 x 2) d x = 3 x 2 2 x = 40. Here x is assumed to commence at o and to continue to increase by in- finitely small increments of dx until it becomes 4. The summation is the addition of all these values of x from o to 4. Arithmetically. The first formula may be written 6 (aj' -f- x" -\- x'" -+- etc.) d x. If then x is to advance from o to 4 by in- crements of i, we have 6 (o +1 + 2 + 3-!- 4) x 1=60, which exceeds 48. If da; is assumed to be .5, the result is 54. The correct result is obtained only when d x is taken infinitely small. By Arithmetic this is approximated, but it is reached by the operations of Calculus alone. The second formula may be written (6 [x' + x" + x"' + etc.] 2 O ' + x" + x"' etc.] ) d x. Assuming x = 4, and d x = i, we have (6 [i + 2 + 3 + 4] -2 [i + i + i + i]) x i = 52, which exceeds 40. If dx= .25, the result would be 43, and if .125 it would be 41.5, ever approaching but never reaching 40, so long as a finite value is assigned to dx. A, Delta, when put before a quantity, signifies an absolute and finite in- crement of that quantity, and not simply the rate of increase. 2, Sigma, signifies the summation of finite differences or quantities. Thus, Zy 2 A#=(y' 2 + y" 2 +y" 2 + etc.) A x. Assume y = 6, y" = 8, y " = 4, and A x the common increment of x = 5, then 2y 2 A x = (36 + 64+16) x 5 = 580. NOTATION. 1 = I. 2o = XX. iooo = M,orCIO. 2 = 11. 3 o=XXX. 2ooo = MM. 3 = III. 40 = XL. 5000 = ^01100. 4 = IV. so = L. 6ooo = VI. 5 = V. 6o=LX. ioooo = X,orOCIOa 6 = VI. 70 = LXX. 50 ooo = I, or 1000. 7 = VII. 8o = LXXX. 6oooo = LX. . 8 = VIII. 9o = XC. iooooo = C,orCCCIOOO. 9 = IX. 100 = C. i ooo ooo = M,or CCCCIOOOD. 10 = X. 500 = D, or 10. 2 ooo ooo = MM. As often as a character is repeated, so many times is its value repeated, its CC = 200. A less character before a greater diminishes its value, as IV = V I. A less character after a greater increases its value, as XI = X + I. For every annexed to 10 the sum as 500 is increased 10 times. If C is placed on left side of I as many times as is on the right, tht number is doubled. A bar, thus ~~, over any number, increases it 1000 times. Illustration i.~i88o, MDCCCLXXX. 18 560, XVlUDLX. 2. - 10 = 500. CIO = 500 x 2 = looo. 100 = 500 x 10 = 5000. CCIOO = 5000 x 2 = 10 ooo. 1000 = 500 x 10 x 10 = 50 ooo. CCCIOOO = SO OOO X 2 = 100 000. 2 6 CHRONOLOGICAL ERAS. MEASURES AND WEIGHTS. CHRONOLOGICAL ERAS AND CYCLES FOR 1906. The year 1906, or the 1300/1 year of the Independence of the United States of America, corresponds to The year 7414-15 of the Byzantine Era; u 6619 of the Julian Period; " 5666-67 of the Jewish Era; u 2071 of the Olympiads, or the second year of the 67181 Olympiad, com- mencing in July (1892), the era of the Olympiads being placed at 775-5 years before Christ, or near the beginning of July of the 3938th year of the Julian Period; " 2659 since the foundation of Rome, according to Varro; 44 2218 of the Grecian Era, or the Era of the Seleucidse; 44 1622 of the Era of Diocletian. The year 1323-24 of the Mohammedan Era, or the Era of the Hegira, begins on the 26th of July, 1906. The first day of January of the year 1906 is the 2,412,115^ day since the com- mencement of the Julian Period. Dominical Letter G I Lunar Cycle or Golden Number 7 Epact 5 I Solar Cycle n Roman Indiction 3. was a period of 15 years, in use by the Romans. The precise time of its adoption is not known beyond the fact that the year 313 A.D. was a first year of a Cycle of Indiction. Julian Period is a cycle of 7980 years, product of the Lunar and Solar Cycles and the Indiction, and it commences at 4714 years B.C. 6513 + (given year 1800) = year of Julian Period, extending to 3267. MEASURES OF LENGTH. Standard of measure is a brass scale 82 inches in length, and the yard is measured between the 27th and 63d inches of it, which, at tem- perature of 62, is standard yard. Lineal. 12 inches = i foot. 3 feet = i yard. 5.5 yards = i rod. 40 rods = i furlong. 8 furlongs = i mile. Inches. Feet. Yarda. Rods. Furl. 36= 3- 198= 16.5= 5.5. 7920= 660 = 220 = 40. 63360=5280 =1760 =320 = 8. Inch is sometimes divided into 3 barleycorns, or 12 lines. A hair's breadth is .02083 (48th part) of an inch, i yard = .ooo 568, and i inch = .00001 5 8 of a mile. Grianter's Chain. 7.92 inches = i link. | 100 links = i chain, 4 rods, or 22 yards. 80 chains = i mile. Ropes and Cables, x fathom = 6 feet. j i cable's length = 120 fathoms. Greographical and ]N~au.tical. I degree, assuming the Equatorial radius at 6967 459.893 yards (3958.784 miles), as given by U. S. Coast Survey, = 69.094 Statute miles, i mile = 2026. 7566 yards or 6080.27 feet, i league = 3 Nautical miles. MEASURES AND WEIGHTS. 2J Log Lines. Estimating a mile at 6080.27 feet, and using a 30" glass, i knot = 50 feet 8.03 inches. | i fathom = 5 feet .08 inch. If a 28" glass is used, and 8 divisions, then i knot = 47 feet 5 inches. | i fathom = 5 feet 1 1.25 inches. The line should be about 150 fathoms long, having 10 fathoms between chip and first knot for stray line. NOTE. This estimate of a mile or knot is that of U. S. Coast Survey, assuming Equatorial radius of Earth to be 6967459.893 yards and a Meter to be 39.370433 inches of the Troughton scale at 62. Cloth. i nail = 2.25 inches. | i quarter = 4 nails. | 5 quarters = i ell. 3?endiilnm. 6 points = i line. | 12 lines = i inch. Shoemakers'. No. i is 4.125 inches, and every succeeding number is .333 of an inch. There are 28 numbers or divisions, in two series or numbers viz., from i to 13, and i to 15. Miscellaneous. 12 lines or 72 points = i inch. i hand = 4 inches, i palm = 3 inches. i span = 9 niches, i cubit = 1 8 inches. "Vernier Scale. Vernier Scale is ^ divided into 10 equal parts ; so that it divides a scale of loths into looths when two lines of the two scales meet. Metric, "by -A-ct of Congress of July S8, 1866. Unit of Measurement i$ the METER, which by this Act is declared to be 39. 37 ins. Denominations. Meters. Inches. Feet. Yards. Miles. Millimeter .OOI .0394 .OI 3937 Decimeter . i 7. 037 .728083 Meter I. 39.37 7.28083 1.09361 IO. 393-7 32.80833 10.936 ii Hektamet^ IOO. 328.083 33 109.361 ii I OOO. 3280.833 33 1093.611 ii .621 77 Mvriameter . . . 10 OCX). 6.2177 In METRIC system, values of the base of each measure viz. , Meter, Liter, Stere, Are, and Gramme are decreased or increased by following prefix. Thus, Milli, loooth part or .001. Centi, looth " .ox. Deci, loth part or .x. Deka, 10 times value. Myria, 10000 times value. Hekto, loo times value. Kilo, looo " NOTE. The Meter, as adopted by England, France, Belgium, Prussia, and Russia, is that determined by Capt. A. R. Clarke, R.E., F.R.S., 1866, which at 32 in terms of Imperial standard at 62 F. is 39.370432 inches or 1.09362311 yards, its legal equivalent by Metric Act of 1864 being 39.3708 inches, the same as adopted in France. Captain Rater's comparison, and the one formerly adopted by the U. S. Ordnance Corps, was = 39. 370 797 i inches, or 3. 280 899 76 feet, and the one adopted by the U. S. Coast Survey, as above noted, is = 39.370 432 35 inches. 28 MEASURES AND WEIGHTS. Denominations. Value in Meters. Denominations. Values in Meters. Inch ....... .025 4 Rod 5 029 2OQ Q Foot .304 800 6 Furlong 201. 168 396 Yard..., .014401 8 Mile... 1600. ^47 168 Approximate Eqxii.valen.ts of Old. and, Metric U. S. .Measures of .Length. i Chain = 20 meters. i Furlong . . . = 200 " 5 Furlongs . . . = i kilometer. i Kilometer . . . . = .625 mile. i Mile = 1.6 kilometers. i Pole or Perch . = 5 meters. i Foot =3 decimeters or 30 centimeters. i Metre =3.280833^/66^ = 3^66^ 3 ins. and 3 eighths. ii Meters =12 yards. \ i Decimeter ... =4 inches. i Millimeter . . = i thirty-second of an inch. To Convert Meters into Inches. Multiply by 40; and to Convert Inches into Meters. Divide by 40. Approximate rule for Converting Meters or parts, into Yards. Add one eleventh or .0909. Inches Decimally = Millimeters. Inches. Milli- meters. Inches. Milli- meters. Inches. Milli- meters. Inches. Milli- meters. Inches. Milli- meters. .01 25 .2 5-08 .48 12.2 .76 19-3 2 50.8 .02 51 .22 5-59 5 12.7 .78 19.8 3 7 6.2 03 .76 .24 6.1 52 13.2 .8 20.3 4 101.6 .04 1.02 .26 6.6 13-7 .82 20.8 5 127 1.27 .28 7.11 56 14.2 .84 21.3 6 152.4 .07 1.52 I. 7 8 3 32 7.62 8.13 :i 8 14.7 15-2 .86 .88 21.8 22.4 1 177.8 .08 2.03 8.64 .62 15-7 9 22.9 9 228.6 .09 2.2 9 36 9.14 .64 16.3 .92 23-4 10 254 .1 2-54 .38 9- 6 5 .66 16.8 94 23-9 ii 279-4 .12 3-05 4 10.2 .68 17-3 .96 24.4 12 304-8 .14 3-66 .42 10.7 7 17.8 .98 24.9 = i foot !i8 4.06 4-57 $ II. 2 II-7 .72 74 18.3 18.8 l ' 25-4 Inches in Fractions = Millimeters. 79 1-59 2.38 3-i7 3-97 4.76 5-56 6-35 If H if be 3 ( 33 g it J, 1 w 9 7.14 17 13-5 5 7-94 9 14-3 ii 8-73 '9 I 5- 1 9-52 5 iS-9 7 13 10.32 21 16.7 7 ii. ii ii 17.5 15 11.91 23 18.3 12.7 6 19 8 4 29 19.8 20. 6 21.4 22.2 23 8 23.0 2 4 .6 25-4 By means of preceding tables equivalent values of inches and millimeters, equivalent values of inches in centimeters, decimeters, and meters, may be ascertained by altering position of decimal point. ILLUSTRATION. Take i millimeter, and remove decimal point successively by one figure to the right; the values of a centimeter, decimeter, and meter become i millimeter 0394 I i decimeter 3-94| .32 inch = 8. 13 millimetera i centimeter 394 (imeter... 39.4 (3.2 MEASURES AND WEIGHTS. 29 MEASURES OF SURFACE. 144 square inches = i square foot. | 9 square feet = i square yard. Architect's Measure, 100 square feet = i square. Land. 30.25 square yards = i square rod. Yards. 40 square rods = i square rood. I2IO. 4 10 square roods square chains > = i acre. 4840: 640 acres = i square mile. 3097600: Rood*. 43 560 square feet, or 208.710326 feet square, or 220 x 198 feet = t Acre. IPaper. 84 sheets = i quire. | 20 quires = i ream, j 21.5 quires = i printer's ream. 2 reams = i bundle. | 5 bundles = i bale. Dra\ving. Columbier 23 x Atlas 26 x Theorem 28 x Doub. Elephant. 27 X 40 Antiquarian ... 31 x 53 Emperor 40 x 60 Uncle Sam 48 x 120 " Peerless 18 X 52 " Tracing. Grand Royal 18 X 24 inches. Grand Aigle 27 x 40 " Vellum Writing, 18 to 28 ins. in width. Cap 13 X 17 inches. Universal 14 X 17 u Demy 15 X 20 Medium 17 X 22, Royal 19 X 24 Super-royal .... 19 X 27 Imperial 22 X 30 Elephant 23 X 28 34 inches. 34 34 Double Crown 20 x 30 inches. Double D. Crown . . 30 X 40 Double D. D. Crown, 40 X 60 Mounted on cloth, 38 ins. in width. ^Miscellaneous. i sheet = 4 pages. I quarto = 8 i octavo = 16 " i duodecimo = 24 pages, i eighteenmo = 36 " i bundle = 2 reams. i piece wall-paper, 20 ins. by 12 yards, i " u " French, 4.5 sq. yards. Roll of Parchment = 60 sheets. Copying. too Words = i Folio. Metric, by A.ct of Congress of July 28, 1866. Unit of Surface is Are or Square Dekameter. A square meter (39.372) = 1549.9969 sq. ins., but by this Act is declared to be 1550 sq. ins. Denominations. Sq. Meters. Sq. Inches. Sq. Feet, Sq. Yards. Acres. Centimeter .0001 '55 Decimeter I CO 107 638 Centare or ) Square Meter) **"* Are I. too i55o. 10.763888 1076 388 88 1.196 - Hectare... 10 000. 11060. 2.A7I MEASURES AND WEIGHTS. Equlval Denominations. ent "Value Sq. Meters. s in Metr Denominations. LC Denom Sq. Meters. illations Sq. Hectares. of XJ. S. Sq. Ares. Sq. Inch " Foot " Yard " Rod... .000645 16 .09290323 .83612907 25.2Q2Q04 Sq. Chain . . . " Rood " Acre .... " Mile... 404.68647 1011.716 175 4046.864699 .404686 258.09934 4.046865 10.117 J 62 40.468647 25899.934074 Approximate Equivalents of Old. and. Metric U. S. Square M.easu.res. 6. 5 square centimeters = i sq. inch. I i acre 1. 16 per cent, over 4000 sq. meters i " meter = io.7ssq.feet. \ i square mile = 259 hectares. MEASURES OF VOLUME. Standard gallon measures 231 cube ins., and contains 8.3388822 avoirdupois pounds, or 58 373 Troy grains of distilled water, at temper- ature of its maximum density (39.1), barometer at 30 ins. Standard bushel is the WincJiester, which contains 2150.42 cube ins., or 77.627 413 Ibs. avoirdupois of distilled water at its maximum density. Its dimensions are 18.5 ins. diameter inside, 19.5 ins. outside, and 8 ins. deep ; and when heaped, the cone must not be less than 6 ins. high, equal 2747.715 cube ins. for a true cone. A struck bushel contains 1.24445 cube feet. Liquid. Cube Ins. 28.875 Gills. Pints. 57-75 8. 231. 32 = 8. Dry. Cube Ins. 67.2006 Pints. Quarts. Galls. 268.8025 8. 537.605 16 = 8. 2150.42 64 = 32 = 8. 4 gills = i pint. 2 pints = i quart. 4 quarts = i gallon. 2 pints = i quart. 4 quarts = i gallon. 2 gallons = i peck. 4 pecks = i bushel. Cube. 1728 cube inches = i foot. inches. 27 cube feet = i yard. I 46656 NOTE. A cube foot contains 2200 cylindrical inches, or 3300 spherical inches. IFTuid. 60 minims = i dram. 8 drams = i ounce. 16 ounces = i pint. 8 pints = i gallon. Minims. Drams. Ounces. 480. 7680=128. 61 240 = 1024 = 128. !N~au.tical. i ton displacement in salt water =35 cube feet. i " registered internal capacity =40 " " Dimensions of a Barrel. Diameter of head, 17 ins. ; bung, 19 ins. ; length, 28 ins. ; volume, 7689 cube ina = 3- 575 6 bushela MEASURES AND WEIGHTS. Miscellaneous. cube foot 7.480 5 gallons. bushel 9-309 18 gallons. chaldron = 36 bushels, or 57.244 cube feet. cord of wood 128 cube feet. perch of stone 24.75 cube feet. i quarter = 8 bushels. Galls. i Barrel 32 i Tierce 42 Butt of Sherry 35X50 108 Pipe of Port 34X58.... 115 Pipe of Teneriffe 100 Butt of Malaga 33X53 i load hay or straw = 36 trusses. Galls. Puncheon of Scotch Whisky. . no to 130 Puncheon of Brandy 34X52. .no to 120 Puncheon of Rum 100 to no Hogshead of Brandy 28X40. . 55 to 60 Pipe of Madeira 92 Hogshead of Claret 46 A Hogshead is one half, a Quarter cask is one fourth, and an Octave is one eighth of a Pipe, Butt, or Puncheon. Metric, toy Act of Congress of Jnly 28, 1866. Unit or Base of Measurement is a cube Decimeter or Liter, which is declared to be 61.022 cube ins. Cu."be M!ea Denominations. Values. snres. Cube Inches. Cube Feet. ' Cube Yards. Cube Centimete " Decimetei " Meter . . . Denominations. r .001 cube re illiliter .o6l 022 61.022 -035313657 35-313657 mres. QuarU. Pecks. Bushels. 1.308 Cube Yards. * i cube liter . Kiloliter or Dry Values. stere.. Meas Cube Ins. Milliliter i cube centimeter. 10 " .1 decimeter.. .061 .6102 6. 1022 6l.022 .908* 9.08 "35 I-I35 "35 28375 2-8 3 75t 28.375 .l7o8 1.308 Centiliter Deciliter Liter Dekaliter Hektoliter.... Kiloliter I or Stere j 10 " .1 meter i " " * Or .227 gallon. \ 3.531 365 7 cube fed. NOTE. In practice, term cube Centimeter, abbreviated to cc, is used instead of Milliliter, and cube Meter instead of Kilometer. Equivalent "Values in. Metric Denominations of U. S. Dry Measures. Denominations. Centiliters. Deciliters. Liters. Inch Pint Quart no is? Gallon ** > Peck 0881 88s 8 81 Bushel... .112A. 1. Z2A. 1C.2A 11.0125 44-05 Liquid. Measures. Denominations. Liters. Drams. Ounces. Pints. Quarts. Gallon*. MPliliter 27 Centiliter .01 2. 7 o Deciliter 27 oR Liter _ 00 8 f- _/r t Dekaliter 10 .204 I" Hektoliter........ IOO 2.641 7 Kiloliter ) or Stere } IOOO 264.17 3 2 MEASURES AND WEIGHTS. Approximate Equivalents of Old and Metric TJ. S. Pleasures of Volume. i Gallon = 4-5 liters. I i cube meter = 1.33 cube yards i Liter = .26 gallon. i " yard = .75 " meter. i cube foot = 28. 3 liters. \ i " kiloliter == 2240 Ibs. nearly of water. MEASURES OF WEIGHT. Standard avoirdupois pound is weight of 27.7015 cube inches of dis< tilled water weighed in air, at (39.83) barometer at 30 inches. A. cube inch of such water weighs 252.6937 grains. 16 drams = i ounce. 16 ounces = i pound. 112 pounds = i cwt. 20 cwt. = i ton. Ounces. Pounda. Drams. 256. 28 672 = I 792. 573 440 = 35 840 = 2240. i pound = 14 oz. ii dwts. i6grs. Troy, or 7000 grains. i ounce = 18 diets. 5.5 grains Troy, or 437.5 grains. i dram = i dwt. 3.343 75 grains Troy, or. 53.5 grains. i stone = 14 pounds. Dwt. 20 grams 3 scruples 8 drams 12 ounces 45 drops Troy. Grains. 480. 5760 = 240. = i lb. avoirdupois. = IOZ. ins = i dram " = 144 Ibs. " ounces = 192 oz. " ounce = 480 grs. " " pound = .822 857 lb. avoirdupois pound = 1.215 2 7& Ibs. Troy. -A.poth.ecaries. = i scruple. = i dram. = i ounce. 480 = 24. 5 760 =1288 = 96, 24 grains = i dwt. 20 dwt. = i ounce. 12 ounces = i pound. 7000 Troy grains 437-5 " " 27-343 75 Troy grai 175 Troy pounds 175 i i i Grains. Scruples. Dram. 60. = i pound. = i teaspoonf ul or a fluid dram. 2 tablespoonfuls = i ounce. The pound, ounce, and grain are the same as in Troy weight Diamond. i grain = 16 parts. 16 parts = .8 i roy grain. 4 grains = 3.2 Troy grains, i carat = 4 grains. 150 carats = i Troy ounce. Lead.. A Fodder of lead = 8 pigs. Sheet lead rolls = 6.5 to 7.5 feet in width and from 30 to 35 feet in length, G-rain.. Standard Weights per Bushel. Lbi. I Lbs. I Lbs. I Lbs. I Lbt. Wheat.... 60 I Corn.... 5 6 and 58 I Rye 56 I Oats 32 I Barley 48 MEASURES AND WEIGHTS. Miscellaneous. jper Cn"be Foot in Bvilk and per Ton. For additional, see page 217. 33 MATERIALS. Per Cube Foot. In Lbs. Cube Feet. In Tons. 50 to 55 44 " 50 50-3 42.2 50 52 46.6 18.5 18 80 23 to 28 97 109 80 107 21 26 41 to 45 45 " 5i 44-5 53-8 45 43 43 48 I2I.8 124.4 28 80 to 97.4 23 20.5 28 21 107 86 u Cannel " Welsh " fl ne , " Southern. . . NOTE. These weights are commercial, not computed from the specific gravity of the material. Metric, toy Act of Congress of July 38, 1866. Unit of Weight is the GRAM, which is weight of one cube centimeter of pure water weighed in vacuo at temperature of 4 C., or 39.2 F., which is about its tem- perature of maximum density = 15.432 grains. Denominations. Values. Grains. Ounces. Lba. Ton. Milligram i cube i [0 " .1 " C i " 10 " i deciliter i liter. . . 10 "... nillimeter 14 entimeter u 15432 1-5432 I5-432 -03527 3527 3-527 35-27 .22046 2.2046 22.046 220. 46 2204.6 in. tions of Grama. K .098419 .984 196 TJ. S. ilograms. Centigram Gram . . . Pekagram Hektogram Kilogram or Kilo . . Quintal z hektolit i cube me 17 Ibs. Troi r alues i Grams. er. Millier orTonneau. Kilogram = 2. 679 Equivalent ~\> Denominations. >ter r, or 2 Ibs. 8 oz. 3 dwts. . 3072 gra n Metric Denomina Dekagrams. II Denominations. Grain .0648 1.296 1-5552 J'SS 87 3. Boo S& 28.3502 31.104. 453.6028 373.2504 .02835 .0 3I i 4536 37325 IOI6.0S728 Scruple . " Troy Pound Drachm . . " Troy Ton . . " (Apoth.)... Approximate Eq.uivaleiats of Old and New TJ. S. IMeasxires of "Weight. The ton and the gram are at nearly equal distances above and below the kilogram. Thus, i ton . . . . = 1 016057.28 grama. | i kilogram = 1000 grams. i gram is nearly 15.5 grains (about .5 per cent. less). i kilogram about 2*2 pounds avoirdupois (about .25 per cent. more). looo kilograms, or a metric ton, nearly i Engl. ton (about 1.5 per cent. less). 34 MEASURES AND WEIGHTS. Absorption, toy Vegetation. Daily Consumption of Water by Vegetation. Crop. Inches o f Water. Minimum. Crop. Inches o Maximum. r Water. Minimum, .14 .11 .02 134 .122 Oats 193 055 .038 .091 .035 Risler.) Corn, Indian Fir-tree i-57 .043 .267 .287 Potato Rye . . Lucern grass Meadow grass Vineyard .031 .11 (M. E Wheat Composition of Common Food Substances. Albu- min. Fat. Carbo- hydrate. Albu- u.in. Fat. Carbo- hydrate. Asparagus. . . Beans Per cent. 2 IQ "> Per cent. 3 Per cent. 2-5 52 Oysters Peas Per cent. 4-95 Per cent. 37 Per cent. Beer . tj 52C . Potatoes Butter Oatmeal c oft fift it Buttermilk. . Cheese 3 33 i-3 91 3 ej Barleymeal.. Poultry 12.5 8.31 .81 75-19 Egg 12 ^ Milk cows' Game 23 Wheat bread 5 Kumyss 3 i-3 3 Rye do. 4-5 46 (Kcenig, Munk, & Ujflemann.) "Weights of Q-rain and. Roots. Following weights have been fixed by statute in many of the States ; and these weights govern in buying and selling, unless a specific agreement to the contrary has been made. founds in a Bxasliel. ARTICLES. California. ^ P CJ Delaware. 1 1 5 c * Kentucky. 1 I j % j S Michigan. Minnesota. | 1 N. Hampshire. 1 | Y< ^ 1 1 1 1 4 ~. ! d 1 H +* t* C q 1 1 > * Barley 50 40 45 - & 14 4 * 46 60 f 60 H 5 46 60 fi H 52 t DO & i4 52 60 32 - 46 46 48 42 48 42 f 60 14 I* - 48 So t 48 48 46 42 47 4^ - 4 6 45 46 42 Beans Blue Grass Seed. Buckwheat Castor Beans Clover Seed Dried Apples Dried Peaches . . - ~ - 60 60 oO 60 - 64 60 60 60 nO - - 60 a8 52 56 - 33 56 44 52 S 33 56 44 56 68 50 S^ 56 68 56~ 44 56 - ~ - 28 28 33 56 55 55 56 28 _ - 28 Hemp Seed Corn 56 56 ~ 56 56 56 52 - 56 58 56 56 56 - 5656 Corn in ear Corn Meal Coal So - 5 50 - ~ n - - - - - - 5 Oats 32 28 32 57 % 35 57 33* 57 32 30 30 52 32 32 35 57 30 30 32 6n 32 34 32 5 32 36 5 Onions Potatoes 54 60 56 60 54 60 56 60 56 60 56 60 60 56 56 56 60 56 60 60 56 60 56 56 60 56 56 60 6060 5656 Rye Rye Meal Salt - 56 - 45 60 20 50 45 60 So 45 60 20 50 45 60 20 _ rfi Timothy Seed. . . Wheat 60 60 60 60 60 60 & 20 60 00 60 60 60 - 6060 Wheat Bran MEASURES AND WEIGHTS. 35 "Weight of IVIen and. "Women. Average weight of 20000 men and women, weighed in Boston, 1864, was men, 141.5 Ibs. ; women, 124.5 Ibs. Average of men, women, and chil- dren, 105.5 Ibs. A mass of people, densely packed, weighs 85 Ibs. per sq. foot, each occupying .8 of one sq. foot of area = 54 450 per acre. Weight of Horses. (TJ. S.) Weight of horses ranges from 800 to 1200 Ibs. WEIGHT OF CATTLE. To Compute Dressed "Weight of Cattle. RULE. Measure as follows in feet: 1. Girth close behind shoulders, that is, over crop and under plate, immediately behind elbow. 2. Length from point between neck and body, or vertically above junction of cervical and dorsal processes of spine, along back to bone at tail, and in a vertical line with rump. Then multiply square of girth in feet by length, and multiply product by factors in following table, and quotient will give dressed weight of quarters. Condition. Heifer, Steer, or Bullock. Ball. Condition. Heifer, Steer, or Bullock. Ball. Half fat a. 1C 3.36 Very prime fat . . . 3.64 3.85 Moderate fat 3 06 3' 5 Extra fat 0.78 406 Prime fat.. . 3. 5 1.64, ILLUSTRATION. Girth of a prime fat bullock is 7 feet 2 ins., and length measured as above 4 feet 5 ins. 7' 2" = 7. 17, and 7. 17* =51. 4, which x 4' 5" and by 3.5 = 794.5 Ibs. Exact weight was 799 Ibs. NOTE. i. Quarters of a beef exceed by a little, half weight of living animal 2. Hide weighs about eighteenth part, and tallow twelfth part of animal. Comparative "Weights of* Live Beeves and of Beef*. Lbs. Per cent. Lbs. Per cent. Bullocks 2800 2600 2600 2400 2400 2IOO 2100 I800 72 to 78 } 70 to 76 | 66 to 70 J 64 to 68 63 to 66 Bullocks 1550 1550 I2OO 1200 1050 1050 9 80 Q50 | 61 to 64 | 58 to 61 } 57 to 58 | 50 to 56 Heifers Heifers Bullocks Bullocks Heifers Heifers Bullocks Bullocks . . Heifers Heifers Bullocks Bullocks Heifers Heifers . . . "Weight of* Offal in a Beef and Sheep. BEEF. SHEEP. Lba. Lbs. Hide and Hair .... 56 to 98 8 to 16* Tallow 42 " 140 5 " 14 Head and Tongue . 28 " 49 6 " nf Feet 21 35 2 " 3 * Including 3 to 6 Iba. for fleece. BEEF. Lbs. Kidneys. Heart.) Liver, etc.... 7 3' to 62 Stomach, En trails, etc., 126 " 196 Blood 6 to 10 42 " 56 t Including 2 to 5 Ibs. for horns. 36 MEASURES, WEIGHTS, PRESSURES, ETC. To Compute Equivalents of Old and. New TJ. S. and of Metric t)enominations. By Act of Congress, July 28, 1866. RULE. Divide fourth term by second, multiply quotient by first term, and divide product by third term. Or, Ascertain relative ratio of first and second terms, and multiply result by ratio of third and fourth terms. NOTE. When result is required in French or other Metric denominations thai those of U.S., use exact denominations, as, 61.025 387 for 61.022, 39.370432 for 39^ etc. EXAMPLE i. If one gallon (ist), per sq. foot, yard, acre, etc. (2d) ; how many liters, (3d), per sq. foot, yard, acre, etc. (4th) ? X 231-:- 61.022 = 3-7851 liters or 3.7848 Uteri. Or, ^3 = 1.604, and ^- = 2.3598; hence, 1.604 X 2.3598 = 3.7851 liters. 144 OI.O22 NOTE. In computing ratios, first term is to be dividedby second, and fourth by third EXAMPLE 2. If one ton per cube foot, how many kilograms per cube decimeter? Il02 -X 2240-7-2.2046 = 35.881 liters, or 35.882 litres. 1728 MEASURES. By Act of Congress of U. 8. By Metric Computation* i Liter per sq. foot, etc. = .2642 Gallon per sq.foot, or .264 2 gallon. i Liter per sq. meter . = .0245 Gallon per sq.foot, or .024 5 gallon. i Gallon per sq. foot . =40.746 Liters per sq. meter, or 40.745 4 litres. i Sq. foot per acre . . . = .2296 Sq. meters per hectare, or 2.29609 metres. WEIGHTS AND PRESSURES. By Act of Congress of U. 8. By Metric Computation. Per sq. inch. Per sq. inch. i Centimeter = .3937 In*> or -393 74 32 /*. i Atmosphere . . . . = 6.6679 Kilograms, or 6.667 8 kilogrammes. i Inch mercury . . = 2.54 Centimeters, or 2.54 centimetres. i Pound = 453.6029 Grams, or 453.592 6 grammes. i Kilogram = 317.4624 Lbs.per sq.foot, or 317.465 Ibs. NOTE. 30 ins. of mercury at 62 = 14.7 Ibs. per sq. inch ; hence, i Ib. = 2.0408 in*., and a centimeter of mercury = 30-4- .3937 for U. S. computation, and 30--- .393 704 3* for French or Metric. POWER AND WORK. i Horse - power = Cheval or Cheval - vapeur = 4500 k X m = 33 ooo 4 (4500 x 2.2046 X 39.37 -4- 12) = 1.013 88 chevaux. i Cheval or Cheval- vapeur (75 Tcxm per second) = horse-power. (4500 x 2.2046 X 39.37 -4- 12) -f- 33000 = .9863 horse-power. By A ct of Congress of U. 8. By Metric Computation. Kilograinmeter k x w = 7.233 jfroi-/6s. ; hence, i -=-(2.2046x3.280 833) = .13826 Kilogr ammeter, or .13825 kilogrammetre. i Cube foot per IP . . . . = .0279 Cube meter per cheval, or .0279 cheval. I Pound " " . . = .447 38 Kilogram per cheval, or .447 38 kilogramme i, Cube meter per cheval = 35.8038 Cube feet per IP, or 35.8058 IP. PRESSURES, ETC. MEASURES OF TIME. 37 TEMPER ATURES. i Caloric or French unit = 3.968 Heat-units, and i heat-unit = i -r- 3.968 = .252 caloric. i U. S. Mechanical equivalent ( 772 foot-lbs. ) = 772 -r- 7.233 = 106.733 Kilo gr ammeters and 106.733 kilogrammetres. i French Mechanical equivalent (423.55 k X w) = 3.280833 x 2.2046 X 423.55 = 3063.505 foot-lbs., or 3063. 566 foot-lbs. Metric. i Heat-unit per pound = .5556 Kilogram, or .5556 kilogramme. i Heat-unit per sq. foot = .2715 Caloric per sq. meter, or .zjizper sq. metre. VELOCITIES. i Foot per second = .3047 Meter per second, or .3047 metres. i Mile per hour = .447 " " " or .447 MEASURES OF TIME. 60 thirds = i second. 60 seconds = i minute. 60 minutes = i degree. 30 degrees = i sign. 360 degrees = i circle. True or apparent time is that deduced from observations of the Sun, and is same as that shown by a properly adjusted sun-dial. Mean Solar time is deduced from time in which the Earth revolves on its axis, as compared with the Sun ; assumed to move at a mean rate in its orbit, and to make 365.242 218 revolutions in a mean Solar or Gregorian year. Sidereal time is period which elapses between time of a fixed star being in meridian of a place and time of its return to that place. Standard unit of time is the sidereal day. Sidereal day = 23 h. 56 m. 4.092 sec. in solar or mean time. Sidereal year, or revolution of the earth, 365 d. 5 h. 48 m. 47.6 sec. in solar or mean time = 365.242 218 solar days. Solar day, mean = 24 h. 3 m. 56.555 sec. in sidereal time. Sol'ir year (Equinoctial, Calendar, Civil or Tropical) = 365.242 218 solar days, or 365 d. 5 h. 48 m. 47.6 sec. Civil day commences at midnight. Astronomical day commences at noon of the civil day, having same designation, that is, 12 hours later than the civil day. Marine or sea day commences 12 hours before civil time or i day before astronomical time. New Style was introduced in England ii \ 1 752. NOTE. In Russia days are reckoned by Old Style, and are consequently 12 days behiud Gregorian record. J> MEASURES OF VALUE. MEASURES OF VALUE. 10 mills = i cent. I 10 dimes = i dollar. 10 cents = i dime. 10 dollars = i eagle. Standard of gold and silver is 900 parts of pure metal and 100 of alloy in 1000 parts of coin. Fineness expresses quantity of pure metal in 1000 parts. Remedy of the Mint is allowance for deviation from exact standard fineness and weight of coins. Nickel cent (old) contained 88 parts of copper and 12 of nickel. Bronze cent contains 95 parts of copper and 5 of tin and zinc. Pure Gold 23.22 grains = $100. Hence value of an ounce is $20.67.1834-. Standard Gold, $18.60.465+ per ounce. WEIGHT, FINENESS, ETC., OF U. S. COINS. G-old. Denomination. Weigh of Coin. t of Pure Metal. Denomination. Weigh of Coin. * of Pare Metal. Dollar Oz. 053 75 134375 .161 25 080375 .16075 .2009375 Gra. 25-8 64-5 77-4 38.58 77.16 9 6 -45 Gra. 23.22 58-05 69.66 Sil 34.722 69.444 86.805 Half Eagle Eagle Oz. .26875 5375 1.075 .401 875 .875 859375 Gn. 1 29 258 5i6 192.9 420 412-5 Grs. 116.1 232.2 464.4 173.61 378 37 I - 2 5 Quarter Eagle.. Three Dollar... Dime Double Eagle. . . ver. | Half Dollar Trade Dollar. . . . 1 Silver Dollar . . . 20 Cent Quarter Dollar . Copper and. INTicltel. Weight. Copper. Tin and II Zinc. || Weight. Copper. Tin and Zinc. One Cent Two Cents . . . Grains. 48 96 Per cent. 95 95 Per cent. || 5 N Three Cents. 5 || Five Cents.. Grains. 77.16 Per cent. 75 75 Per cent. 25 25 Tolerance. Gold, Dollar to Half Eagle, .25 grains. Eagles, .5 grains. Silver, 1.5 grains for all denominations. Copper, i to 3 cents, 2 grains ; 5 cents, 3 grains. Legal Tenders. Gold, unlimited. Silver. Dollars of 412.5 grains unlimited ; for subdivisions of dollar, $10. (Trade dollars [420 grains] are not legal tender.) Copper or cents, 25 cents. NOTE. Weight of dollar up to 1837 was 416 grains, thence to 1873, 412.5. Weight of $1000, @ 412.5 gr. =859.375 oz. BRITISH standards are : Gold, || of a pound,* equal to u parts pure gold and i of alloy ; Silver, fff of a pound, or 37 parts pure silver and 3 of alloy = .925 fine. A Troy ounce of standard gold is coined into 3 175. lod. 2/, and an ounce of standard silver into 55. 6d. i Ib. silver is coined into 66 shillings. Copper is coined in proportion of 2 shillings to pound avoirdupois. Sterling (1880) $486.65; hence -^ of this = value of i penny = 2.027 708 33 cents. * A pound is assumed to be divided into 24 equal parts or carats, hence the pro portion is equal to 22 carats. FOREIGN MEASURES OF VALUE. 39 To Compute "Valne of Coins. RULE. Divide product of weight in grains and fineness, by 480 (grains in an ounce), and multiply result by value of pure metal per ounce. Or, Multiply weight in ounces by fineness and by value of pure metal per ounce. EXAMPLE i. When fine gold is $20.67.183+ per oz., what is value of a British sovereign ? By following tables, p. 40, Sovereign weighs .2567 oz., and .2567 X 480 = 123.216 grains, and has a fineness of .9165. Hence, x.o.6 7 ..8 3 + = $4.86.34. EXAMPLE 2. When fine silver is $i. 1 5. 5 per oz. , what is value of U. S. Trade dollar f By table, p. 38, Dollar weighs .875 oz. and has a fineness of .900. Hence, .87$x -QooX 1.15.5 = 90.95625 cents. EXAMPLE 3. A 4-Florin (Austrian) weighs 49.92 grains and has a fineness of .900, What is its value? 4,^900 X2o67i8j+ = To Convert TJ. S. to British Currency and Contrari- wise. RULE i. Divide Cents by 2.027 71 (2.027 78 33), or, Multiply by 493 12 (.493 118 26), and result is Pence. 2. Multiply Pence by 2.02771 , or divide by .49312, and result is Cents. EXAMPLE. What are 100 cents in pence? loo x 49312 ==49.312 pence 4*. i. 3 1 2d a. What is a Pound sterling in cents? 20 X 12 = 240 pence, which x 2.02771 =: $486.65. FOREIGN MEASURES OP VALUE. "Weight, Fineness, and. Mint "Values of Foreign Silver and. GJ-old Coins. By Laws of Congress, Regulations of the Mint, and Reports of its Directors. Current Value of silver coins is necessarily omitted, as the value of silver is a variable element. Hence, in order to compute current value of a silver coin, the price of fine or a given standard of silver being known, Proceed as per above rule to compute value of coins. The price of silver should be taken as that of the London market for British standard (925 fine), it being recognized as the standard value, and governing rates in all countries. EXAMPLE. If it is required to determine value of a Mexican dollar in cents. Weight 867.5 oz. .go^Jine. Value of Silver in London 52.75 pence per ounces 106.96164- cents. ^ = .846 867 and 106.9616 X .846 867 = 90.5822 cenfc. 9 2 5 3 FOREIGN MEASURES OF VALUE. Weight and Mint Values of Foreign Coin. (Value is based on their Value on April, 1901.) Countries given in Italics have not a National Coinage. Countr}' and Denomination. Weight. Fine- ness. Pure Silver or Gold. Current or Nominal. VALDB Gc U.S. Id. British. Arabia. Piastre or Mocha Dollar. .... Argentine Republic. Dollar = 100 Centisimos .... Peso Oz. :l s :i .104 "hous'a. 916 916.5 900 900 900 986 9 00 870 916.66 918.5 9*7-5 914 925 9 2 5 875 853-1 833 900 870 901 Grains. Cents. 83.14 50-69 $ c. .965 4-85-7 5-32-37 34-5 1.93.49 2.28.3 6.75.4 451 15-59-3 54-59 10.90.6 4.92 i 3-97-43 3-99-97 7es 3.68.8 14.96.39 ^65 9-!5-4 15-59-3 .682 X 6. d. 1911.5 i i 10.5 .2 7" 9 4-6 i 7 9.1 3' 3 4 * 26.9' 2 4 9.8, I O 2.6 o. 5 7. 4 2 16 4 16 5.2 15 i 8! 3 i 5-9' 4. 1 17 7-4. 3 4 ' .0! 3-32 Australasia. Same as British. Australia. Sovereign 1855 171.47 257-47 362.06 12.67 393-6 66.6 83.25 "34 37^8 456 Austria. Kreutzer (copper) Dollar, " 75 547 .1 1.014 1.52 9 .14 6.75 07. C2 Souverain .363 .86~ 7 .028.8 .82 575 .261 *5 1875 .027 .209 .869 .492 -867 .087 Belgium. Same aa France. Bolivia. Centena Boliviano Brazil. Rei Milreis Double Milreis Moidore 4000 Reis .... Canada. Cent " 20 Cent currency Penny u Shilling " 4 " =20 shillings, currency Pound Cape of Good Hope. Same as British. Central America. 4 Reals Colon 2 Escudos Doubloon ante 1834 . . Chili. Centaro Dollar, new 10 Pesos . ... Doubloon China. Cash, Le 10 Cents, Leang Tael Hankow Cochin China. Mas, 60 Sapeks jo Mas, i Quan... FOREIGN MEASURES OF VALUE. "Weight and Mint Values. Country and Denomination. Weight. Fine- ness. Pare Silver or Gold. Current or Nominal. V A L U . Go ' U.S. Id. British. Cuba. Ox. : .025 .927 .427 .304 .060.4 .182.5 .178 454-5 .363-6 .256.7 .256.2 .04 275 275 i- 375 .032 .161 .161 .804 .207.5 .012.8 .128 595 .1X2 .OIO.4 '719 185 012.8 2 .a 844 870 900 877 895 925 9 2 4-5 925 925 925 916.5 916.5 755 875 875 875 900 900 900 900 986 900 900 900 900 27 71 cent D* 390.23 26.82 80.99 79- 3 201.8 161.44 M-5 69-55 347-76 257-04 310.61 8. I. OX 8.94 1. 01 2.02+* .2 1. 01 2. 3 8 .405 .926 .451 7-55-5 15-59-3 26.8 7.90 451 4.86.65 4.85.1 4-9 5-0-52 4 94-3 25. 2.6 I!' 3 96-45 3.85.8 ^3-8 2.38.24 2.28.38 iy-3 344.2 5- 6. ii 23.8 5 in 0.58 3 4 i 4-39 13-22 i 12 5.6 5 ..:* 100 100 I 6.84 i o 5.3 5 2 10.2 .1 -5 15 10.26 1.17 11.74 9 9-5 9 4-63 9-J H i-75 I 9.6 11.74 .2 Colombia. Centaro Doubloon old Costa Rica. Same as Mexico. Denmark. Mark i6Skilling Crown 2 Rigsdaler j O Thaler East Indies. See Hindustan and Japan. Ecuador. Centaro Sucre England. ' ' average Half Crown Florin Sovereign or Pound, new . . . " " average. Egypt Piastre 40 Paras . Guinea, Bedidlik Pound Purse 5 Guineas France. Sou 5 Centimes . Franc 100 Centimes 5 Francs 20 Francs, Napoleon, new . . . 25 Francs 20 centimes =1 Stg. Germany. Groschen 10 Pfenning Mark 10 Groschen 10 Marks Thaler , Ducat Greece and Ionian Islands. Same as France. Drachma 100 Lepta . . . 5 Drachmas Pound . Guatemala. Same as Mexico. Guiana, British, French, and Dutch. Same as that of their Countries. Hanse Towns. Mark Holland. Cent FOREIGN MEASURES OF VALUE. \Veight and. Mint "Values. Country and Denomination. Weight. Fine- ness. Pure Silver or Gold. Current or Nominal. VALUI G U.S. >ld. British. Holland. Florin or Guilder, 100 cents. 10 Guilders Ox. .021.6 .215 374 .16 .864 375 375 .279 .866.7 053.6 .289 362 1.072 .861 .867.5 1.081 .844 .245 .803 .867 Stg., nomi Thous". 916.5 835 900 916.5 916.5 890" 900 900 572 568 900 902.5 870.5 873 8 3 o 996 896 858 nal value Grains. 164-53 65.12 373-24 16? 119.19 374-4 372.98 336-25 = 2 shill Cents. 25 3-03 I 21.63 .083 ings sterlin * c. 40.2 3-99-7 19.3 32.9 4.86.65 6.84.36 75-3 99-72 3-57-6 4.44 19.94.4 4.86.65 49.0 15. 6.1 19-51-5 5- 4-4 iS-37-8 .83 g. s. d. i 8 16 5.1* x 10.5 .13 '5 I O O i 8 1.5 .49* 4 ~j.x8 14 8.35 18 2.96 4 x ii. 6 X O 2 3 4 1-88 4 o 2.4 x o 8.75 3 3 3-39 xo.66 Hindostan. Honduras. Sam* M Mexico. Italy. Same as France. Lira 100 Centimes . . Scudo Indian Empire. Pic nominal Anna * ' Rupee,* 16 Annas 10 Rupees, and 4 Annas Mohur 15 Rupees Japan. Yen Itzebu new Yen, 100 Sen Cobang old *' new Java. Same as Holland. Liberia. U. S. Currency. Malta. 12 Scudi = i Sovereign Mexico. Peso new *' Maximilian 20 Pesos Republic . . Morocco. 10 Ounces Mitkeel Naples. Scudo 6 Ducati Netherlands. Same as Holland. New Brunswick. Same as Canada. Newfoundland. Same as Canada. New Granada. Dollar 1857 Norway. Alike to Denmark. Mark 24 Skillingen Nova Scotia. Same as Canada. Persia. Reran, 20 Shahis 10 Keran Toman Paraguay. Foreign coins. * .092 76 of a FOREIGN MEASURES OF VALUE. Weight and Mint "Values. 43 Country and Denomination. Weight. Fine- ness. Para Silver or Gold. Current or Nominal. V ALUB 04 U.S. British. Peru. Dollar 1858 q*. 766 Thoas'. Grains. Cents. * C- s. d. gol ,0- .867 868 I e. cc 7 Portugal. Corda, 1838, 10000 Reis .308 .005 912 912 - I0.8l.78 10.8 a 4 5-S Roumania. 2 Lei 322 81* 129 06 Russia. 500 77 18 ioo Copek Rouble . 66? Kin _j - e Roubles . ..... 6 6 2 OO Sandwich Islands. U. S. Currency. Sardinia. Lira .16 8i< Spain. 16 81* 6^ ii .193 Dollar 5 Peseta 3 A I9.3 ioo Reals . .. 268 rT 1048 10 Escudos 270 8 806 r re 20 Reals vellons=i U.S. Dollar. Sweden. Riksdaler, ioo Ore 271 98 28 Rixdollar I OQ2 Carolin 10 Francs 75 Switzerland. Same as Franc*. St. Domingo. Gomdes ioo Cents. 900 6 11 Tunis. Piastre 16 Karubs ii 81 c8i 5 Piastre m i 808 <; 220. 18 .511 .161 2 OO S 12 17 Turkey. Piastre 40 Paras 20 Piastre 8 TO >. 88 ioo Piastre, Medjidie 211 18 o Tuscany. . 112 2 11 1 96. i Tripoli. 20 Piastres Mahbub ?6 i o So Uruguay. Dollar ioo Centimes. 74.8 West Indies, British. Same as England. Venezuela. Centaro j .5 Bolivar, i Franc. . . 10. 1 O.4 Memoranda. FRANCE. Bronze coins 9.5 copper, 4 tin, and i zinc. HAXSE TOWNS. Monetary system same as that of German Empire. SWITZERLAND. The Centime is termed a Rappe. SPAIN. 25 Peseta piece is 195. 9.5^. Stg. ; Real vellon was 2.5^. Stg. ITALY. All coins same weight and fineness as those of France. MALTA. 7 Tari and 4 Grani = i Shilling Sterling. EGYPT. A Para =. .061 55 es or 1 u u u cheonsj 4 quails (282 cube ins.) . . = 1.017 9 gallons = i firkin = 9- I 53 2 firkins = i kilderkin . . . = 18.306 -Ale and Beer Aleasvires. Imp'l pall's. Imp'l gall' B 2 kilderkins = i barrel = 36.612 54 gallons = i hogshead = 54.918 108 " = i butt = 109.836 46 ENGLISH AND FRENCH MEASURES AND WEIGHTS. Apothecaries' or Fluid. Measures. i drop = i grain. 60 drops = i drachm. 4 drachms = i tablespoon, 2 ounces (875 grains) =: i wineglass. Coal Measures. 50 pounds . . . . = i cube foot. 88 " = i bushel. 9 bushels . . . . = i vat. 90 or 94 " =i Cornish " 93 pounds . . . . = i Welsh bushel. 3 heaped bush. = i sack. 10 sacks = i ton. 12 sacks = i chaldron = 5.25 chaldrons . . = i London chaldron: i Newcastle " = i ton = i room = 21 chaldrons = i barge or keel . . = : i chaldron. : 58.6548 cube ft. : i room. : 26.5 CWts. '53 " 144.5 cube feet. : 7 tons. : i score. :2i.2 tons. Miscellaneous. . . 80 bushels. . . 35.9 cube j eft . dicker hides . . . . = 10 skins. . . : 20 dickers. x * 26.5 gallons, 6 bushels wheat . .= i sack flour. . . =r 7 pounds. . . 28.2 " truss straw . . . ..=36 " 35.9 cube feet = i ton water. i truss old hay = 50 noun Is. i " new " = 60 u i bushel oats = 40 " i " barley . . . . = 47 " i " wheat = 60 " i cube yard new hay = 84 " i " " old " =126 " i quintal = 100 " i boll = 140 u i sack wool = 364 " LIQUID. i wine gallon = 231 cube ins. i beer " = 282 " " i litre = .220 09 gallon. i gallon = 4-544 litres. i cube foot . . = 6.2321 gallons. i auker . . . . = 8.333 " i hogshead wine . . : i " beer . . . : i puncheon wine . . : i pipe or butt wine : i " " " beer : i tun : : 52.5 gallons. 54-9 l8 " : 7 : 105 : 109.836 " :2io " i ton water 62 = 224 gallons. BUILDERS. i solid part = 12 cube ins. 12 u parts = i " inch." 12 " inches " = i cube foot. i load timber, rough = 40 " feet. i " u hewn = 50 " " i " lime = 32 bushels. z " sand = 36 " i square i bundle laths . . . i rod brickwork . i rood masonry . Batten, in section Deal, * u Plank, " " . = loo sq.fect. . = 120 laths. . = 306 cube feet, . = 648 " u . =: 7 X 2.5 ins. = 9X3 " . = 11 X 3 Volumes in Cube Iiiolies, Feet, etc. Denominations. Litres. Gills. Pints. Quarts. Gallons. Bushels. Quarters. Centilitre OI .0704 0176 Decilitre . . . 1761 Litre* i 7 0420 .lyui I 7607 .8804 22OI Dekalitre IO 8.8036 2. 2OOQ . 275 II Hectolitre IOO 22 OOQI 2.751 13 O^-JQ Kilolitre... 1000 220.0008 27.^11 **> ^. 4180 * Equal 61.025 24 cube int. ENGLISH AND FRENCH MEASURES AND WEIGHTS. 47 "Wood !Meas\ire. i Stere or cube metre = 35.3150 cube feet or 1.308 cube yards. i Voie de bois (Paris) = 70.6312 cube feet ; i voie de charbon (charcoal) = 7.063 cube feet ; i corde = 4 cube metres = 141.26 cube feet. MEASURES OF WEIGHT. BRITISH. i Troy grain = .003 961 cube inches of distilled water, i Troy pound =22.815 68 9 cu be inches of water, i Avoir, drachm = 27.343 75 Troy grains. 16 drachms, or) 437-5 grains j 16 ounces, or 1 7000 grains J 20 hundredweights . The grain, of which there are 7000 to the pound avoirdupois, is same as Troy grain, of which there are by the revised table 7000 to the Troy pound. Hence Troy pound is equal with the Avoirdupois pound. In Wales, the iron ton is 20 cwt. of 120 Ibs. each. Avoirc i ounce. lupois. 8 pounds . . = i stone (for meat). 14 " . . = i stone. 28 " . . = i quarter. 112 " . . = I CWt. . . . . i pound. . = i ton. Troy. 16 ounces = i pound. 25 pounds = i quarter. 4quarters, or loo pounds = i cwt. 24 grains = i dwt. 20 pennyweights, or j 437-5 grains f"~ By this are weighed gold, silver, jewels, and such liquors as are sold by weight. The old Troy ounce to the Avoirdupois ounce was as 480 grains, the weight of the former, to 437.5 grains, weight of the latter; or, as i to .9115. Apothecaries.* 437.5 grains = i ounce. | 16 ounces = i pound. FRENCH. M.etrio "Weights in Avoirdupois. Denominations. Grammes. Grains. Ounces. Pounds. Ton. Milligramme 001 .01 .1 ,' .-"'i i i 10 100 1000 IOOOO 100 000 I OOOOOO ogramme = 2 Ibs. 01543 IS432 i 543 23 I5-43235 154.32349 1543.23487 15432.34874 3 oz. 4 drachms, lo.t 3527 3-5274 35-2739 734 pram*. .22046 2.20462 22.04621 220.462 12 2204.621 25 .9842 Centigramme Decigramme Dekagramme Kilogramme^ . . Myriagramme Quintal Millier or Ton tKi NOTE. For the values of the prefixes, as Milli, Centi, etc. , see p. 27. Old. System. i grain . . = 0.8188 grains Troy. I i ounce = 1.0780 oz. Avoirdupois. i gross . . = 58.9548 | i livre = 1.0780 Ibs. * As by revised Pharmacopoeia. 4 8 FOREIGN MEASURES AND WEIGHTS. FOREIGN MEASURES AND WEIGHTS. It being wholly impracticable to give all the denominations of measures and weights of all countries, the following cases are selected as essential and as exponents. With parent countries, as England, France, etc., their denominations ex tend to their colonies and dependencies. Thus, the denominations of England extend to Canada, a large portion of the East a,nd West Indies, and parts of South America, and those of France to a part of the West Indies, Algiers, eta Abyssinia. Pic, Stambouili 26.8 ins. " geometrical 30.37 " Madega 3.466 bush, Ardeb 34.66 " " Musuah 83.184 " Wakea 400 grains. Mocha i Troy oz. Rottolp 10 u " Also, same as in Egypt and Cairo. Africa, Alexandria, Cairo, and. Egypt. Cubit 20.65 ins - Derah 25.49 " Pic, cloth 26.8 " " geometrical 29.53 " Kassaba, 4.73 Pics 11.65 ft. Miie 2146 yds. Feddan al-risach 552 48 acre. Roobak 1.684 galls. Ardeb 4.9 bush. Rottol 9821 Ib. Distances are measured by time. A Maragha 15 Ddreghe* or i hour. Aleppo and. Syria. Dra Mesrour 21.845 ins. Pic 26.63 " Road Measures are computed by time. Algeria. Rob, Turkish 3. n ins. Pic, " 24.92 u " Arabic 18.89 " Also Decimal System. Alicante. Palmo 8.908 ins. Vara 35.632 " Amsterdam. Voet ii. 144 ins. El 1.979 Faden 5.57 ft. Lieue 6.383 yds. Maat 1.6728 acres. Morgen 2.0095 " Vat 40 cub. ft Also Decimal System. Ant\verp. Fuss 1 1- 275 ins. Elle, cloth 26.94 u Corde 24.494 cub. ft. Bonnier 3-2507 acres. Also Decimal System. Arabia, Bassora, and IMoclia. Foot, Arabic 1.0502 ft. Covid, Mocha 19 ins. Guz, " 25 " Kassaba 12. 3 ft. Mile, 6000 feet 2146 yds. Baryd, 4 farsakh 21 120 " Feddan 57 600 sq. ft Noosfla, Arabic 138 cub. ins. Gudda 2 galls. Maund 3 Ibs. Tomand 168 " Other Measures like those of Egypt. Argentine Confederation, Paraguay, and Uruguay. Fanega 1.5 bush. Arroba, 25. 35 Ibs. Quintal 101,4 " Also Decimal System in Argentine Con< federation and Paraguay. Australasia. Land Section 80 acres. Other Measures same as English. Axistria. Zoll i-37i i ns - Fuss i 0371 ft. Meile 24000 ft Klafter, quadrat 35-854 sq. yds. Jochart 6.884 " Cube Fuss i-"55 cu ^- ft Achtel 1.692 galls. Eimer 12.774 u Viertel 3. 1143 " Metze i. 6918 bush. Unze 8642 grains. Pfund (1853, 500 grammes), 1.2347 Ibs. Centner 123. 47 u Also Decimal System. Babylon. Pachys Metrics 18. 205 ins Baden. Fuss 11.81 ins. Klafter 5.9055 ft. Ruthe 9.8427 " Stunden 4860 yds. Morgen 8896 acre. Stutze. 3-3014 galls. Malter o ....... 4. 1268 bush. Pfund 1. 1023 Ibs. Also Decimal System. FOREIGN MEASURES AND WEIGHTS. 49 Bagdad. Guz 31-665 ins. Barbary States. Pic, Tunis linen. 18.62 ins. " " cloth 26.49 " " Tripoli 21.75 " Batavia. Foot 12. 357 ins. Covid 27 El... .....27.75 Bavaria. Fuss 11.49 ins - Klafter 5-745 36 ft. Ruthe 3- *9 l8 y ds - Meile 8060 " Ruthe, quadrat 10. 1876 sq yds. Morgen or Tagwerk 8416 acre. Klafter, cube 4. 097 cub. yds. Eimer 15-05856 galls. Scheffel 6.119 Metze 1.0196 bush. Pfund 8642 grains. Also Decimal System. Belgiu.ni. Meile 2.132 yds. Also Decimal System. Benares. Yard, Tailor's 33 ins. Bengal, Bombay, and. Cal- cutta. Moot 3 ins - Span 9 " Ady, Malabar 10.46 ins. Hath 18 Guz, Bombay 27 " Bengal 36 Corah, minimum 3.417 ft. Coss, Bengal 1. 136 miles. " Calcutta 1-2273 " Kutty. 9.8175 sq. yds. Biggah, Bengal 3306 acre. " Bombay 8114 u Seer, Factory 68 cub. ins. Covit, Bombay 12.704 cub. ft. Seer, Bombay 1-234 pints. Parah 4. 4802 galls. Mooda 112.0045 kt Liquids and Grain measured by weight. Bohemia. Foot, Prague u.88 ins. " Imperial 12.45 " Also same as Austria. Bolivia, Chili, and Peru. Vara 33-333 ins - Fanegada 1.5888 acres. Gallon 74 gall. Fanega 1.572 " Libra 1.014 Ibs. Arroba 25.36 " Originally as in Spain ; now Decimal System in Chili and Peru. Brazil. Palmo, Bahia 8.5592 ins. Vara 3.566 ft Braca 7. 132 * * Geira 1.448 acres. Also same as Portugal, and sometimes as in England. Buenos Ayres. Vara 2. 84 ft Legua 3.226 miles Suertes de Estancia .... 27 ooo sq. varas. Also same as Spain. Burmah. Paulgat i inch. Dain 4. 277 yds. Viss 3.6 Ibs. Taim 5.5 " Saading 22 Also same as England. Canary Isles. Onza 927 inch. Pic, Castiliaii 11.128 ins. Almude 0416 acre. Fanegada 5 " Libra 1.0148 Ibs. Also same as Spain. Cape of Grood Hope. Foot 11.616 ins. Morgen 2. 116 54 acres. Also same as in England. Ceylon. Seer i quart Parrah 5.62 galls. Also same as in England. China. Li 486 inch. Chih, Engineer's. 12. 71 ins. " or Covid 13-125 " " legal 14.1 Chang 131-25 ' " legal 141 " Pu 4.05 ft Chang, fathom 10.9375 ft Li 486 yds. Pu or Rung 3.32 sq. yds King, loo Mau 16.485 acres. Tau 1. 13 galls. Tael 1.333 oz. Catty 1.333 Ibs. Cochin China. Thuoc or Cubit 19.2 ins. Sao 64 sq. yds. Mao 1.32 acres. Hao 6. 222 galls. Shita 12.444 " Nen 8594 Ib. Colombia and Venezuela. Libra 1. 102 lb& Oncha 25 " Also Decimal System. FOREIGN MEASURES AND WEIGHTS. Denmarlt,* Greenland, Ice- land, and. Norway. romme 1.0297 ins Fod 1.0297 ft. Favn, 3 Alen 6. 1783 " Mil 4- 68 55 miles. " nautical 4.61072 ' Anker 8.0709 galls. Skeppe 47 8 DUstL Fjerdingkar 9558 " Fund 1. 1023 IDS. Lispund i7-3 6 7 ' Centner 110.23 * Also Decimal System. Ecuador. Decimal System. Genoa, Sardinia, and Turin. Palmo 9.8076 ins. Piede, Manual, 8 oncie. . . 13.488 ' " Liprando, 12 " ...20.23 Trabuco or Tesa 10. 113 ft. Miglio 1.3835 miles. Starello 9804 acre. Giomaba 9394 * Germany. The old measures of the different States differ very materially ; generally, how- ever, Foot, Rhineland 12-357 i QS - Meile 4-603 miles. Decimal System made compulsory in 1872. Greece. Stadium 6155 mile. Also Decimal System. Guinea. Jachtan 12 ft. Hamburg. Fuss 11.2788 ins. Klafter 5.641311. Morgen 2.386 acres. Cube Fuss 831 1 cub. ft. Tehr 99.73 Viertel i. 594 7 galls. Pfund (500 grammes) ... 1. 102 32 Iba. Ton 2135.8 Ibs. Also Decimal System. Hanover. Fuss 11.5 ins. Morgen 6476 acre. Hindostan. Borrel 1.211 ins. Gerah 2.387 u Haut 19.08 " Kobe 29. 065 ' ' Coss 3.65 miles. Tuda 1. 184 cub. ft Candy 14. 209 ' Hungary. Fuss 12.445 ins. lle 3- 6 7 ' Meile 9- *39 y ds - Also as in Vienna. Indian Empire. Guz 27.125 ins. Cowrie i sq. yd. Sen 61.025 39 cub. ina. 2.204 737 Ibs. Uniform standard of multiples of the Sen adopted in 1871. Italy. -.> IVIilan and "Venice. Decimal System. The Metre is termed Metra; the Are, Ara; the Stere, Stero; the Litre, Litro; the Gramme, Gramma, and the Tonneau, Tonnelata de Mare. Naples and Two Sicilies. Palmo 10.381 ins. Canna 6.921 ft. Miglio 1. 1506 miles. Migliago 7467 acre. Moggia 86 Pezza, Roman 6529 Roman States. Old Measure. Foot ii. 592 ina " Architect's 11.73 * Braccio 30.73 * Palmo 8.347 u Miglio 1628 yds. Quarta 1.1414 acres. Lucca and Tuscany. Pie 11.94 ins. Palmo "-49 " Braccio 22.98 " Passetto 3.829 ft Passo 5.74 " Miglio 1.0277 miles. Quadrato 8413 acre. Saccato i. 324 " Japan. Sun, .303 03 Metre. . 1.193* ins. Shaku, 3. 030 3 Metres.. 11.9305* ins. Jo, 30.303 " .. 9.9421* ft. Ken, 5.5 ' .. 5-9 6 53* " Ri, 11880 " .. 2.4403 miles. Kai-ri 6o8ofeet.t Hiro 4-971* fe et - Momme 3.756 521 7 grammes Fr. Hiyaku me 828 17 Ibs. Kwam-me 8.28171 Hiyak-kin 132.507 32 Man's load S7-97 2 Koku 331.26831 Hiyak-koku 33 126. 830 8 * These are as equivalent aa they are practi- cable of reduction, f Admiralty knot. FOKEIGN MEASUEES AND WEIGHTS. Java. Duim 1.3 ins. Ell 27.08 " Djong 7-015 acres. Kan 328 galls. Tael 593.6 grains. Sach 61.034 Ibs. PecuL 122.068 " Catty 1.356 " Madras. Ady. 10.46 ins. Covid 18.6 " Guz 33 " Culy 20.92 ft. League 3472 yds. Puddy 338 galls. Marcal 2.704 " Tola 180 grains. Seer 625 Ibs. Viss 3.086 " Maund 24.686 " jNXalabar. Ady 10.46 ins. Malacca. Hasta or Covid 18. 125 ins. Depa 6ft. Orlong 80 yds. Malta. Palmo 10.3125 ins. Pie 11.167 " Canna 82. 5 Salma 4-44 acres. Also as in Sicily. Moldavia. Foot Sins. Kot, silk 24.86 ins. Fathom 8 ft Molucca Islands. Covid 18.333 ins. Morocco. Tomin 2. 810 25 ins. Cadee 20.34 ins. Cubit 21 " Muhd. 3.081 35 galls. Kula, oil 3.356 " Rotal or Artal 1.12 Ibs. Liquids other than oil are sold by weight. Mysore. Angle 2.12 ins. Haut 19. i ' Guz 38. 2 " Candy 500 Ibs. Netherlands. Elle 39.370432 ins. Decimal System since 1817. JPersIa. Gereh 2. 375 ins. Gueza, common 25 " 44 llonkelrer 37.5 " Archin, Schah 31.55 ins. " Arish 38.27 " Parasang 6076 yds. Chenica 80. 26 cub. ins Artaba 1.809 bush. Mi seal. 71 grains. Ratel 2.1136 Ibs. Batman Maund 6.49 u Liquids are measured by weight. Polando Trewiee 14.03 ins. Precikow 17 ins. Pretow 4-7245 yds. Mile, short 6075 yds. Morgen i. 3843 acres. ^Portugal and ^dozana"biq-ue. Foot 13 ins. Milha 1.2788 miles Almude 3.7 galls. Fanga 1.488 bush. Alguieri 3.6 " Libra 1.012 Ibs. Also Decimal System. Prussia. Fuss 12.358 ins. Ruthe 4. 1 192 yds. Meile 24 ooo feet Quadrat Fuss 1.0603 sq. ft. Morgen 631 03 acre. Cube Fuss 1.092 cub. ft Scheffel i. 5121 bush. Anker 7-559 galls. Pound 7217 grains. Zollpfund 1. 1023 IDS. Centner "3-43 Ibs. Russia. Vershok 1.75 ina Foot 12 ins. Arschine 28 " Rhein Fuss 1.03(1. Sajene 7 ft. Verst 3500 " Mila 5. 5574 milea Dessatina 2.4954 acres. Vedro 2. 7049 galls. Tschel- werha 1-4424 " Pajak 1.4426 bush. Tschetwert 5-7704 " Pound 6317 grains. Funt 902 85 Iba Decimal System adopted in 1872. Siam. K'up 9-75 ins. Covid 18 ins. Ken 39 " Jod 098 48 mile. Roeneng 2.462 miles. Silesia. Fuss 11.19 ins - Ruthe 4. 7238 yds. Meile 7086 yds. Morgen 1-3825 acrea FOREIGN MEASURES AND WEIGHTS. Singapore. Hasta or Cubit 18 ins. Dessa. 6 ft. Orlong 80 yds. Smyrna. Pic 26. 48 ins. Indise 24.648 " Berri 1828 yds. Spain, Cu"ba, Malaga, Ma- nilla, GJ-u.atemala, tdondu.- ras, and. Mexico. Pie 11.128 ins. Vara 33-384 " Milla 865 mile. Legua, 8000 varas 4.2151 miles. Fanegada 1.6374 acres. Vara, cubo 21. 531 cub. ft. Cuartilla 888 gall. Arroba, Castile 3. 554 galls. Fanega 1-5077 bush. Libra 1.0144 N>s. Tonelada 2028.2 Ibs. Also Decimal System. Stettin. Fuss ii. 12 ins. Foot, Rhineland 12. 357 " Elle 25.6 ins. Morgen 1-5729 acres. Sumatra. Jankal or Span o ins. Elle 18 Hailoh 36 " Fathom 6 ft Tung 4 yds. Snrat. Tussoo, cloth. 1.161 ins. Guz, " 27.864 " Hath 20. Q " Covid 18.5 " Biggah 51 acre. Tunnland 1.2198 acres Anker - .... 8.641 galls. Spaun 1.962 bush. Centner 112.05 ^ & Also Decimal System. S\vitzerland. Fuss, Berne 11.52 ins. u 11.54 " Vaud 11.81 " Klafter 5.77 ft. Meile 4. 8568 miles Juchart, Berne 85 acre. Haas 2.6412 pints. Eimer 8.918 galls. Malter 4. 1268 bush, Pfund 1. 1023 Ibs. Also Decimal System. Tripoli. Pik, 3 palmi 26.42 ins. Almud 319. 4 cub. ins. Killow 2023 " " Barile 14.267 galls. Temer 7383 bush Rottol 7680 grains. Oke 2.8286 Ibs. Tnrlzey. Pic, great 27.9 ins. " small 27.06" Berri 1.828 yds. Alma 1. 154 galls. Also Decimal System. Wiirtemtoerg. Fuss 11.29 ins. Elle 2.015(1. Meile 8146.25 yds. Morgen 7793 acre. Cube Fuss 830 45 cub. ft Eimer 64.721 galls. Scheffel 4. 878 bush. Pound 7217 grains. Zurich. Fuss ii. 812 ins. Elle 23.625 " Klafter 5.9062 ft. Meile , . . 4. 8568 miles Jachart 808 acre. Cube Klafter 144 cub. ft Sweden. Fot 11.6928 ins. Ref 32.4703 yds. Faden 5.845 ft. league 3. 3564 miles. Meile 6.6417 Holland. Denominations corresponding to the French are as follows: Length. Millimetre, Streep; centimetre, Duim; decimetre. Palm; metre, El; decametre, Roede; kilometre, Mijle. Surface. Square millimetre, Vierkante Streep; square centimetre, Vierkante Duim; and so on. Hectare, Vierkante Bunder. Cube Measure. Millistere, Kubicke Streep, and so on. Capacity. Centilitre, Vingerhoed; decilitre, Maatje; liquid litre, Kan; dry litre, Kop; decalitre, Schepel; liquid hectolitre, Vat or Ton; dry hectolitre, Mud or Zak; 30 hectolitres = i Lastr= 10.323 quarters. Weight. Decigramme, Korrel; gramme, Wigteje; decagramme, Lood; hecto- gramme, Onze; kilogramme, Pond. Belgium. Metric system. The term Livre is substituted for kilogramme, Litron for lltre> and Anne for metre. SCRIPTURE MEASURES. ANCIENT WEIGHTS. 53 SCRIPTURE AND ANCIENT LINEAR MEASURES. Scripture. Digit 912 inch. I Span, 3 palms 10.944 Palm, 40 digits 3-648 ins. j Cubit. 2 spans 21.888 Fathom, 4 cubits 7 feet 3. 552 ins. Hebrew and. Egyptian. ins Nahud cubit i-475 feet Royal " 1.7216 Egyptian finger 06145 Babylonian foot 1. 140 feet Hebrew u 1.212 " " cubit 1.817 " Hebrew sacred cubit 2.002 feet. Digit 7554 inch. Pous (foot). 1.0073 feet - Cubit i.i33 2 " Pythic or natural foot 814 foot. Attic or Olympic " 1.009 feet. Grecian. Ancient Greek foot ) Q . . (16 Egyptian fingers) f 94i wot Arabian foot 1-095 feet Stadium 604.0375 u Olympic stadium 606.29 " Mile, 8 stadium ......... 4835 feet Alexandrian or Phileterian stadium (600 Phil, feet) = 708.65 feet Volume. Keramion or Metretes .............. 8.488 gallons. Cubit. . . ............ i. 824 feet Sabbath day's journey ____ 3648 " I Mile, 4000 cubits ........ 7296 feet | Day's journey ............ 33-164 milea Digit... .......... 72575 ins - Uncia (inch) ............... 967 Pes (foot) . ............. 11.604 " Roman Long Measures. Cubit ...................... -1.4505 feet Passus. ... . . .............. .4-835 Mile, milliarmm ......... 4842 ANCIENT WEIGHTS. He"brew and Egyptian. Troy grains, Atticobolus \ X'Ct Denarius, Roman { |J;9* " Nero 54$ Shekel 151-9* " drachma \ 54-6t (69* Lesser mina 3. 892 Greater mina 5.46 Egyptian mina 8. 326* Ptolemaic " 8.985* Alexandrian " 9-992* Obolus 4.63 Ounce (431- Drachm 146. 5 Libra 4086. i Pound 12 Roman ouncee. Talub c 581-71 ounces. Talent (60 minae) 56 Ibs. avoirdupois. GJ-recian. Mina. Troy grains. Obolus, ancient 8.33 " "-57 Gramme 23. 15 Drachma 50-01 --'#< S reat 6 9-47 Roman. Ounce 416.82 grains. | Pound 10.41 ouncea Troy ounces. 10.41 " great 14-47* Talent 625. 19 u Attic , 868.32 t Arbuthnot. E* t Paucton. 54 GEOGRAPHIC MEASURES AND DISTANCES. GEOGRAPHIC MEASURES AND DISTANCES. To Reduce Ijongitncle into Time. RULE. Multiply degrees, minutes, and seconds by 4, and product is the time. EXAMPLE. Required time corresponding to 50 31'. 50 31' X 4 = 3^- 22m. 45. To Reduce Time into Longitude. RULE. Reduce hours to minutes and seconds, divide by 4, and quo- tient is the longitude. Or, Multiply them by 15. EXAMPLE. Required longitude corresponding to $h. 8m. 11.2$. 5/i. 8m. ii. 2$. = 3o8m. 11.2$., which -=- 4 = 77 2' 8". Or, multiplying by 15: 5/1. 8m. 11.25. X 15 = 77 2/ 8 ". Table of Departures for a Distance rnn of* 1 Mile. Course. Departure. II Course. Departure. || Course. Departure. 3.5 points. .773 4.5 points. .634 I 5.5 points. .471 4 -707 II 5 -556 II 6 .383 Thus, if a vessel holds a course of 4 points, that is without leeway, for distance of i mile, she will make .707 of a mile to windward. Or, a vessel sailing E.N.E. upon a course of 6 points for 100 miles will make 38.3 (100 X .383) miles of longitude. .Minutes, and. Seconds of* eacn IPoint of the Compass -with. Meridian. Degrees, NOBTH. SOOTH. Points. o i n Sin. A.* Cos. A.* Tan. A.* | 25 2 48 45 .0489 .9988 .0491 N. .5 c 07 OQ .008 oo8< "t 75 O j/ y* 8 26 15 .uyo .1467 9952 .9891 vy5 .1484 ( i 11 J 5 .195 .9808 .1989 N.by E.... N. by W S.by E 1 S.by W 1 1.25 '4 3 45 16 52 30 .2429 2903 9569 .2504 3034 I J -75 19 4i 15 3368 9415 3578 N.N.E..., N.N.W S.S.E .. 2 2.25 2-5 22 30 25 18 45 27 7 30 .3827 4275 .4714 9239 .904 .8819 .4142 .4729 5345 S.S.W. 1 I 2.75 30 56 15 8577 5994 N.E. byN. .. N.W. byN... S.E.byS. ... S.W.byS.... 1 3 3-25 3-5 33 45 36 33 45 39 22 30 5556 5957 6 344 8315 8032 773 .6682 .7416 .8207 I 3-75 42 ii 15 6715 7409 .0063 r 4 45 .7071 .7071 i N.E S.E I 4-25 47 48 45 7404 1.103 N.W S. W 1 A. 5 4-75 So 37 3 53 26 15 773 .8032 U J44 5957 1.348 N.E. by E. .. S.E. by E... 5 5-25 56 15 59 3 45 8315 8577 5556 1.497 1.668 N.W.byW... S.W. by W... 1 5-5 61 52 30 .8819 .4714 1.871 I 5-75 64 41 15 .904 4275 2.114 E.N.E... W.N.W E.S.E. .. w.s.w 1 6 6.25 6-5 67 30 70 18 45 73 7 30 9239 9569 3827 3368 2903 2.414 2-795 3-296 I 6-75 75 56 15 97 .2429 3-941 E.byN... W. by N E.byS J W.byS 1 7 7-25 7-5 78 45 81 33 45 84 22 30 .9808 .9891 9952 195 .1467 .098 5.027 6.741 I 7-75 87 ii 15 .9988 .0489 20-555 East or West. East or West. . . 8 90 i .0000 00 * A, representing course or points from the meridian. GEOGRAPHIC LEVELLING. 55 GEOGRAPHIC LEVELLING. Curvature and. Refraction.0 Correction for Curvature of Earth, to be subtracted from reading of a levelling-staff, is determined as follows : Divide square of distance in feet from level to staff, by Earth's Equa- torial diameter viz., 41 852 124 feet. Or, Two thirds of square of distance in statute miles equal the cur- vature in feet. Correction for Refraction is to be subtracted from reading, and as a mean may be taken at about one sixth of that for curvature. Correction for Curvature and Refraction combined, is to be added to reading on staff. Formulas of Capt. T. J. Lee, U. 8. Engineers. T) 2 D 2 = correction for curvature, -^ m = correction for refraction, and 2 R K D 2 (i 2 m) ^ = correction for curvature and refraction. D representing distance, R radius of earth, and m a coefficient of refraction = .075, all in feet. ILLUSTRATION. A distance is 3 statute miles, what is correction for curvature and refraction? ';;.... (.-2 x. 075) ^^=.85x5.996=5.097/0*. Approximately, D 2 = curvature in feet. Hie veiling by Boiling IPoint of TVater. To Compute Height A_"bove or Belo\v Level of Sea* 517 (212 - T) + (212 - T) 2 = Height. ILLUSTRATION. What is height of an elevation, when boiling point of water is 182? 517 X 212 182 + 212 182 = 517 X 30 + so 2 = 16 410 feet. Corrections for Temperature to be made in Connection with Formula. Temp. Correc- Temp. Correc- tion. Temp. Correc- tion. Temp. Correc- tion. Temp. Correc- tion. Temp. Correc- tion. o .936 18 .972 36 008 54 .046 72 1.083 90 12 2 94 20 .976 38 012 56 05 74 1.087 9 2 124 6 8 944 .948 952 22 It .98 40 42 44 016 02 024 62 054 .058 .062 76 1.091 1.096 i.i 9 1 96 98 128 132 136 10 956 28 .992 46 028 64 .066 82 1.104 100 .14 12 .96 30 .996 48 032 66 .071 8 4 1.108 IO2 .144 14 .964 32 I So 036 68 075 86 1. 112 104 .148 16 .968 34 1.004 52 .041 / 70 .079 88 1. 116 106 .152 ILLUSTRATION. Assume temperature in preceding illustration to have been 80*. Then i64ioX i.i = 18051 feet. GEOGRAPHIC LEVELLING A1TD DISTANCES. H Is ** w &a o (P HI Appa ifsi .IOON oo co vo ON tx o txoo * rx M M co so -*oo CM co rj- ON OO CO ON COOO CMMOOtOCMCMtxON to\O O M lOOO ON COVO ONOO VO CO CO H H M 11*^ J#?S?#RSRB88VSV3# < 8K. J?5 1? $ft o f H *S "- rxod 06 ON ON d H M CM covd oo covd ON CM to r^ o N ON co - d J5 MMWMMHi-iaCN]CMCOCOCO"<*--. d ioO od OMOO M co "t-vo t^. N vo vo CM CO CO * ^- 10 IOVO txOO W t^ 10 * N M ONOO VQ IO O * O lOOO vS * to M M N CO * IOVO txOO tx 10 Tj- N M CO 10 M CO Tj- to O >0 c e a S 5 "3 tr> Pv ON N * 10 O 1000 CO 10 tx O VO tx COVO M H O COOO CO CO ON g ONcot^Nvd d ONt^io -00 NVO VOCON O ONt^lO -^-OO CO t^ N CO t^ CO to H M cs co T}- 10 vovo r^oo vo to co N M N -4- M N CO ^- to O to hoars of to A. w g txvo vovo to^-^cow M tx Ti-vo Tt- to ON w to J? ^-ON-'j-dN^-dNdNdNdN ONOO oo t>.r>.ioto-4-cocoN \otxo w o H ON CO CO T- -^- to IOVO txOO ON-<1-ONONONONONONONON ONOO t^ t^vo -J-OO M to H H CO TJ- iovo t^OO ONONONONONO O M n. onary between "o 73 M M* CM CO -OtxiOM OOO ON ONOO' vo <* vo N 00 COOO N VO COVO JS O COOO CMVO ONCS O>M ^lOfxONM CO -^-VO (x ON O N CO IOVO N 11 *ti 79 SI'S 1 S ON CO ON M txOO t CO tx W N N M ONVO co vOwvOMioONCOr-x ^ 5^ COOO COVO ON CM to tx N T-VO OO ON M CO IOVO OO ON H CO IOVO CO .2 ^ gHMMCMCMCMCOCOCO^-^^^^^^^^^^^VO VOVO VOVO tx 15 | J ^iVO-^- lOHVO VO tOMVO'-l'OIHVO'^-CO ^VO tv CM tv M tx .od ON d M M CM co -4- iovo t>. t^.oo ON d to l| 3 c El S *** - vo* co ON COOO CO tx M TJ-00 COOO IN1OMIOCM \OONMNIO COOO tx elevation ater and i C5 S& H 5. . N li "J VO ONONONONONONONONONOON ONOO OOOOOOOOOOOOOOOOOOOOvO t^tx to ' ' M IN CO 4- OVO t^OO ON O M CM CO -* tovd t^OO ON O H N CO -4- ON il II if to ' M CNl CO -^ IO>O txOO O\ O M CM CO * tovo rxOO ON M N fO-i-u->O fcc* Q JIN CO N to Th -*VO t^ tvVO <* M 00 * * ON COOO M M- txCO CM CM t> H Tf- t> CM -**-vO OOOCMTj-lOtxONOMCO Tj-VO r^-OO ON 11 tx g H H M CNl CNl CM COCOCOCOCO'4--4 1 -4 1 4-' < i-'4 i lOXOXOtOAOlO IOVO V} GEOGRAPHIC LEVELLING. MAGNETIC VARIATION. 57 ILLUSTRATION. Curvature of Earth independent of refraction is computed at .667 foot = 8. 004 ms. for i geographical mile, and as refraction on land is taken as .104 foot or 1.248 ins., and on ocean at .099 foot or 1.188 ins., relative visible dis- tances of an object, including curvature and refraction, for an elevation of .667 foot is 1.09 miles on land, and i. 08 miles at sea. I " 1.33 " " ^32 u <( 9 feet " 4 " 3.08 i mile " 104.03 " " " " 103.54 " " " Difference between two levels in feet is as square of their distance in miles. ILLUSTRATION. At what elevation can an object be seen, at surface of ocean, when it is 2 miles distant? i 2 : 2 2 : : . 667 .099 : 2. 272 feet = 2 feet 3. 25 -f- ins. Difference between two distances in miles is as square root of their heights ILLUSTRATION i. At an elevation of 9 feet above level of sea at what distance can an object be seen upon its surface? V-667 .099 = .754 : i :: -y/g : 3.98 miles. 2 VT If a man , at the fore 1 -V ) pg allant mast-head of a vessel, 100 feet from water, sees another and a large vessel "hull to," how far are the vessels apart? A large vessel's bulwarks are at least 20 feet from water Then, by table, 100 feet ,, 27 1 5 o <* _ 5.03! 19-20 miles distance. When an observation for distance is taken from elevation, as a light-house a vessel's mast, etc., of an object that intervenes between observer and hori^ zon, or contrariwise, observer being at a horizon to elevated object, distance of observer from intervening object is determined by ascertaining or esti- mating its elevation from horizon, and subtracting its distance from whole distance between observer and point from which observation is taken and remainder is distance of object from observer. ILLUSTRATION. Top of smoke pipe of a steamer, assumed to be 50 feet above sur- face of water, is in range with horizon from an elevation of 100 feet- what is dis tance to steamer from elevation ? 100 feet ,..=1^27) 50 ti _ Q 3 sj 3 9 mites distance. MAGNETIC VARIATION OF NEEDLE. A merica. Needle reached a Westerly maximum in 1660, and then varied to Fast until 1800, when it reversed to West. London (Eng.). From 1576 to 1815 variation ranged from 11 15' East to 24 27' West, when it receded gradually to 21 in 1865. Jamaica (W. I.). No variation from year 1660. Diurnal Variation. There is a small diurnal variation, being greatest in sum- mer (15') and least in winter (7' 30"), added to which a change of temperature affects a needle. Variation in U.S. Professor Loomis concludes that the Westerly variation is increasing and Easterly diminishing in every part of United States ; that this change occurred between i 793 and 1819, and that present annual change is about 2 in Southern and Western States, from 3 ' to 4' in Middle States and c' to 7 ' in Eastern States. NOTB. Rules for computation of variation are empirical, except in each par. ticular locality, as annual and diurnal variations of needle, added to local attrar tiou, render it altogether unreliable. MAGNETIC VARIATION OF NEEDLE. Decennial Magnetic Variation in the IT. S. and. some Foreign 1 Magdalena B. , L. Cal. Magnetic St'n, Idaho IO 17.6 Sheboygan, Wis Shreveport, La 2.2 6.6 Fort Bowie, Ariz.. .. Fort Garland, Cal... Fort Gibson, Ind. T Ft Leavenw'th Kan X> OOOJ OJ Marelsl'd, N.Y., Cal. Mazatlan, Mex Memphis, Tenn Mexico City Mex Ta o.o 5-3 7 .4 Sitka, Par'e G, Alaska Springfield, 111 Tallahassee, Fla Tampa Fla 29.4 4.2 2 2.2 Fort Lyon Col 13 .2 Michigan City, Ind. . T 8 Tuscaloosa, Ala 4.6 Ft. McKiuney, Wyo.. Gainesville, Flo Galena 111 16 2. I 7 * Milledgeville, Ga.... Minneapolis, Minn.. Montgomery, Ala. . . 2.7 *:l VicksburgC.H.,Miss. Vincennes, Ind Yankton, S. Dak.... 5.6 ii Acapulco Mex WEST. Ithaca NY .... 7 " Richmond Va .... 3.7 Alleghany, Pa Atlantic City, N.J... Auburn, N. Y 3-6 7.2 8.6 Keeseville, N. Y.... Kittery, Me Knoxville, Tenn .... 12.4 13.3 .2 Rochester, N. Y. Rockland, Me Rome, N. Y 3 9-4 Bath Me . . . 14. 7 Little Falls N. Y Rutland, C. P'k, Vt.. 12 .4 Beaver Pa. 2 8 Lowell Mass . Saginaw Mich.... Belfast Me Lynn Mass . . . 12 2 Sandusky J Bellows Falls, Vt. . . . Bridgeport Conn 12.4 Mackinac, Mich Madison O .... 1.6 3 2 Saybrook, Conn Schenectady N Y 10.4 jO Buffalo, N. Y Calais, Me Cape May N J r T 5 Marietta, Newark, N. J Newbern N'l C'y N C 1:1 2 6 South Bethlehem, Pa. Springfield, Mass Stamford Conn 7.2 II. 2 Carlisle, Pa Chambersburg, Pa. . Cheboygan Mich 5-2 5.03 Newburyport, Mass.. NewLondon,G.Pt.,Ct. Newport R. I 12.8 11. 1 12 Stonington, Conn... Tappan&PTdes,N.Y. Toledo M'n Line, O. . II. 2 9.2 1.5 Columbus 'O New Rochelle N Y 8 s 7 *9 Concord, N. H Daubury, Conn Delaware City Del 12.4 12.6 Norfolk C. H., Va... Norwalk, Conn Machinac Mich 4 IO i 6 Troy, N. Y Union town, Pa Utica N Y TR Dunkirk, N. Y Geneva N Y 4.6 Oswego, N. Y Ottawa Can 8-5 Wash'tonN.Ob.,D.C. Wheeling Va 3-9 Gettysburg, Pa Greenport, N. Y. . . . Hackensack, N.J... . Hanover N H 6.1 10.8 8.7 12 8 Owego, N. Y Penobscot, Me Perth Amboy, N.J... Pittsburg Pa 7.8 19 8-5 3-6 Williamsburg, Va Wilmington, N. C... Wilmington, Del York Pa 3-9 1.6 Hudson 'N Y IO.2 Provincetown Mass. 12.9 Zanesvilie. i . i Huntington, Pa 5.6 Raleigh, nr Cap'l,N.C. 1.8 Ypsilanti, Mich 2.2 GEOGRAPHIC LEVELLING. BASE LINE. SOUNDINGS. Dip of Horizon- Approximate, 57.4 VR=-dip in seconds, varying with temperature air. H representing height of observer's eye in feet. .66 7 w 2 =H: 498s 2 =H: 1.42 -/H = : n representing distance in geographical miles and s in statute. Multi- plier. 3Vea Angle. sureir Multi- plier. lent o Angle. C Hei Multi- plier. gilts \vith H Multi- Angle. II pii er . a Sext Angle ant. Multi- plier. Angle. 1-5 2 si 63 26 2.5 3 3-5 68 II 7i 34 4 4-5 75 58 5-5 77 29 6 79 42 8d 32 81 52 8 9 10 82 52 83 40 84 17 Operation. Set sextant to any angle in table, and height will equal distance multiplied by number opposite to it ILLUSTRATION. When sextant is set at 80 32', and horizontal distance from ob- ject in a vertical line is 100 feet, what is its height? 100 X 6 = 600 feet By Trigonometry: 1:100:: 5.997 (tan. angle) ; 599.7 feet. To Reduce a Sounding: to Low Water. - f i q: cos. - -- J = h'. h representing vertical rise of tide, and h' sound* ing or depth at low water, both in feet ; t time between high and low water, and t time from time of sounding to low water, in hours. cos. when - <9Q, and + cos. when >9o. ILLUSTRATION. Low water occurring at 3.45, and high water at 10.15 P.M., a sounding taken at 5.30 P.M. was 18.25 feet; what was depth at low water, vertical rise being 10 feet? h = 10 feet ; t' = 5^. ymi. 3^. 45W,. = ih. 45*11. = i. 75 hours. t == loh. ism. 3/1. 45 m. = 6/1. 307/1. = 6. 5 hours. Then i ipcos. =5 (i-coa 48 27' 4 i")= 5 x (1663 i2 4 )=i.68 43 8/tffe Sounding 18. 25 feet Reduction 1.68407 feet = 16.565 93 feet Lengths of a Degree of Longitude on parallels of Lati- tude, for eaclx of its Degrees from Equator to IPole. Lat. Miles. Lat. Miles. Lat. Miles. Lat. Miles. Lat. Miles. Lat. Miles. i 59-99 1 6 57-67 3i 51-43 4 60 41.68 61 29.09 "76" I4-52 2 59- 9 6 17 57-38 32 50.88 47 40.92 62 28.17 77 13-5 3 59-92 18 57.06 33 50-32 48 40.15 63 27.74 78 12.48 4 59- 8 5 19 56.73 34 49-74 49 39-36 64 26.3 79 "-45 5 59-77 20 56.38 35 49- J 5 50 38.57 65 25-36 80 10.42 6 59- 6 7 21 56-01 36 48.54 5i 37-76 66 24.4 81 9-38 7 59-55 22 55-63 37 47.92 .52 36.94 67 23-44 82 8-35 8 59-42 23 55-23 38 4728 53 36.11 68 22.48 83 7-3i 9 59-26 24 S4-8i 39 46-63 54 35-27 69 21.5 84 6.27 10 59- 9 25 54-38 40 45-96 55 i 34 4i 7 20.52 85 5-23 ii 58-89 26 53-93 4i 45-28 56 33-45 7i 1953 86 4.18 12 58.69 27 53-46 42 44-59 57 32-68 72 18.54 87 3-H 13 58.46 28 52.97 43 43-88 58 ' 31 79 73 17-54 88 2 14 58.22 29 52.48 44 43.16 59 i 30-9 74 16.54 89 1.05 IS 57-95 30 51.96 45 42.43 60 30 75 15-53 90 .00 NOTE. Degrees of longitude are to each other in length as Cosines of theii latitudes. FIGURE OF EARTH. BOAED AND TIMBER MEASURE. 6 1 Klements of Figure of ttie Earth. Capt. A. R. Clarke, 1866. Feet. Mile*. Major semi-axis of Equator (longitude 15 34' E. ) 2 o 926 350 3 963. 324. Minor " ' " " ( 105 34' E.) 20919972 3962.115. Polar " " 20853429 3949.513. Equatorial semi-axis 20 926 062 3 963. 269. Circumference, mean 24 898. 562. Diameter, " 79 J 6- BOARD AND TIMBER MEASURE. BOARD MEASURE. In Board Measure, all boards are assumed to be i inch in thickness To Compute Measure or Surface. When all Dimensions are in Feet. RULE. Multiply length by breadth, and product will give surface in square feet. When either of Dimensions are in Inches. EXAMPLE. What are number of square feet in a board 15 feet in length and 16 inches in width? 15 X 16 = 240, and 240 -f- 12 = 20 sq. feet. When all Dimensions are in Inches. RULE. Multiply as before, and divide product by 144. TIMBER MEASURE. To Compute "Volume of Round Timber. When all Dimensions are in Feet. RULE. Add together squares of diameters of greater and lesser ends, and product of the two diameters ; multiply sum by .7854, and product by one third of length. Or, a + a'-\-a" x - = V, and c 2 + c' 2 -f c x c' x .07958 X - = V. a and a' representing areas of ends, a" area of mean proportional, I length, and c and c' circumference of ends. NOTE. Mean proportional is square root of product of areas of both ends. ILLUSTRATION. Diameters of a log are 2 and 1.5 feet, and length 15 feet. (a a 4-x.s a +2X 1.5) = 9-25, which X .7854 and .36.3245 cube feet. When Length in Feet, and Areas or Circumferences in Inches. RULE. Proceed as above, and divide by 144. When all Dimensions are in Inches. RULE. Proceed as before, and divide by 1728. NOTE. Ordinary rule of Hutton, Ordnance Manual of U. S., and Molesworth, of I X c-f- 4, giVv s a result of about .25 less than exact volume, or what it would be if the log was hewn or sawed to a square, c representing mean circumferences. F 62 BOARD AND TIMBER MEASURE. To Compute "Volume of Squared. Tim"ber. When all Dimensions are in Feet. RULE. Multiply product of breadth by depth, by length, and product will give volume in cube feet. When either Dimension is in Inches. RULE. Multiply as above, and divide product by 12. When any two Dimensions are in Inches. RULE. Multiply as before, and divide by 144. EXAMPLE. A piece of timber is 15 inches square, and 20 feet in length; required its volume in cube feet. 15 X X20 lao deals ......... = i hundred. Allowance is to be made for bark, by deducting from each girth from .5 inch in logs with thin bark, to 2 inches in logs with thick bark. 3VIeasu.res of Timlber. (English.) 50 cube feet of squared > i , timber J loaa * 40 feet of unhewn timber = i load. 600 superficial feet of inch planking = i load. Deals. Deals. Boards exceeding 7 ins. in width, and if less than 6 feet in length, are termed deal ends. Battens are similar to deals, but only 7 inches in width. Balk. Roughly squared log or trunk of a tree. Planks are boards 12 ins. in width. Country. Long. Broad. Loc Thick. al St Volume. andards. Country. Long. Broad. Thick. Volume. Russia and Prussia . . Sweden . . . Ft. 12 14 Ins. II 9 Ins. i-5 3 Cub. ft. 1-375 2.625 Norway . . Christiana Quebec. . . Ft. 12 II 12 Ins. 9 9 II Ins. 3 1.25 2-5 Cub. ft. 2.25 859 2.292 ioo Petersburgh standard deals equal 60 Quebec deals. SPARS AND POLES. Pine and Spruce Spars, from 10 to 4.5 inches in diameter inclusive, are to be measured by taking their diameter, clear of bark, at one third of their length from abut or large end. Spars are usually purchased by the inch diameter ; all under 4 inches are termed Poles. Spars of 7 inches and less should have 5 feet in length for every inch of diameter, and those above 7 inches should have 4 feet in length for every inch of diameter. IJOBS or Waste in Hewing or Sawing of Timber. (C. Mackrow.) Oak, English 200 per cent. " African ioo " " Dantzic 50 " " " American 10 " " Yellow Pine from planks. . 10 per cent. Teak 15 " " Elm, English 200 " " American 15 ** '* CISTERNS. SHINGLES. CISTERNS. Capacity of Cisterns in Cu.be Feet and GJ-allons. For each 10 Indies in Depth. Diam. Cub. ft. Gallons. Diara. Cub. ft. Gallons. Diam. | Cub. ft. | Gallons. Feet. Feet. Feet. 2 2.618 19.58 9-5 59.068 441.8 ! 17 189.15 1414.94 2-5 4.091 30.6 10 65449 489.6 17-5 200.432 J499-33 3 5.89 44.07 10.5 72.158 539.78 18 212.056 1586.28 3-5 8.018 59-97 ii 79.194 592-4 19 236.274 1 767.45 4 10.472 78.33 "5 86.558 647-5 20 261.797 1958.3 4-5 i3- 2 54 99.14 12 94.248 705 21 288.632 2159.11 5 16.362 122.4 12.5 102.265 764.99 22 316.776 2369.64 5-5 19.798 148.1 13 110.61 827.4 23 346.23 2589-97 6 23.562 176.24 13-5 119.282 892.29 24 376.992 2820.09 6.5 27.652 206.84 14 128.281 959-6 25 409.062 3059-8 7 32.07 239-88 14-5 137.608 1029.38 26 442.44 3309-67 7-5 36.816 275-4 15 147.262 noi.6 27 47LI3 3569-17 8 41.888 3!333 15.5 I57-243 1176.26 28 513.126 3838.44 8-5 9 47.288 530H 353-72 39^-55 16 16.5 rfj-SS 2 178.187 1253-37 1332.93 29 550.432 30 | 589.048 4"7.5i 4406.08 Excavation and Lining of "Wells or Cisterns. For each 10 Inches in Depth. 1 1 Bricks. Masonry. 1 1 Bricks. Masonry. 9 5 Num- Laid 8 inches i foot 1 09 Num- Laid 8 inches i foot 5 w ber. dry. thick. thick. g i ber. dry. thick. thick. Feet. Cub. ft. Cub. ft. Cub. ft. Cub. ft. Feet. Cub. ft. Cub. ft. Cub. ft. Cub.ft. 3 12.29 126 5-24 6.4 10.47 8.5 63.29 356 14.83 16 24.87 3-5 15.29 M7 6. ii 7.27 11.78 9 69.89 377 16.87 26.18 4 18.62 168 6.98 8.14 13.09 9-5 76.81 398 16^58 17-75 27.49 4-5 22.27 1 88 7.85 9.02 14.4 10 84.07 419 17-45 18.62 28.8 5 26.25 209 8-73 9.89 10.5 91.65 440 18.33 19.49 30.11 V 30-56 35-2 230 251 9.6 10.47 10.76 11.64 17.02 18.33 ii 12 99-56 1 16. 36 461 503 19.2 20.94 20.36 22.11 31.42 34-03 6.5 40.16 272 "34 12.51 19.63 13 134.46 545 22.69 23.85 36.65 7 45-45 293 12.22 I3-38 20.94 14 153-88 586 24-43 25-6 39-27 7-5 51.07 13.09 14.25 22.25 15 174.61 628 26.18 27-34 41.89 8 57-02 335 13.96 I5-I3 23-56 16 196.64 670 27.92 29.09 44-51 Number of bricks and width of curb are taken at dimensions of ordinary brick viz., 8 by 4 by 2.25 ins. = 72 cube ins. In computing number of bricks required, an addition of 5 per cent, should be added for waste. It is to be considered, also, that diameter of excavation necessarily exceeds that of masonry. SHINGLES. Usually of white Cedar and Cypress ; 27 inches in length and 6 to 7 inches m width, dressed to light .25 inch at point and .3125 inch at abut. Laid in three thicknesses and courses of about 8 inches, so that less than .33 of a shingle is exposed to air, or about 2.25 shingles are re- quired per square foot of roof. Shingles, alike to Slates, are laid upon boards or battens. 6 4 SLATES AND SLATING. SLATES AND SLATING. A Square of Slate or Slating is 100 superficial feet. - Gauge is distance between the courses of the slates. Lap is distance which each slate overlaps the slate lengthwise next but one below it, and it varies from 2 to 4 inches. Standard is assumed to be 3 inches. Margin is width of course exposed or distance between tails of the slates. Pitch of a slate roof should not be less than i in height to 4 of length. Po Compnte Surface of a Slate when, laid, arid. Num- ber of Sq.vi.ares of Slating. RULE. Subtract lap from length* of slate, and half remainder will give length of surface exposed, which, when multiplied by width of slate, will give surface required. Divide 14 400 (area of a square in inches) by surface thus obtained, and quotient will give number of slates required for a square. EXAMPLE. A slate is 24 X 12 inches, and lap is 3 inches; what will be number required for a square? 24 3 = 21, and 21 -f- 2 = 10.5, which X 12 = 126 inches; and 14 400 -=- 126 = 114.29 slates. Dimensions of* Slates. [AMERICAN.] Ins. Ins. Ins. Ins. Ins. Ins. Ins. 14 X 7 14 X 8 14X9 14 X 10 i6x 8 i6x 9 16 X 10 18 X 9 18 X 10 18 X ii 18 X 12 20 X 10 20 X II 20 X 12 22 X II 22 X 12 22 X 13 24 X 12 24 X 13 24 X 14 24 X 16 ENGLISH. Ins. Ins Ins. Doubles 13 X 10 C I2X 8 Marchioness . . 22X22 u 11 V 7 1 I4X 8 Duchess 24 X 12 Small doubles . iiX 6 iox 5 Ladies ( 14X12 15 X 8 Imperial Rags 30X24 36X24 f 12 X IO i6x 8 ->6X 24 Plantations . . j Viscountess . . . 13X10 18X10 Countess 16x10 20X10 Empress Princess 26X15 24X14 Thickness of slates ranges from .125 to .3125 of an inch, and their weight varies from 2 to 4.53 Ibs. per sq. foot. Weight of One Square Foot of Slating, 125 in. thick on laths 4.75 Ibs, " " " " i in. boards. . 6.75 " 1875 in. thick on laths 7 " " t: " " i in. boards. 9 " .25 in. thick on laths 9.25 Ibs- " " " " i in. boards.. 11.25 " .3125 in. thick on laths 11.15 " " " " u i in. boards, 14. 10 " Slate weighs from 167 to 181 Ibs. per cube foot, and in consequence of laps, it requires an average of nearly 2.5 square feet of slate to make one of slating. Weights per 1000 and Number Required to Cover a Square. Doubles 13 x 6 Lbs. 1680 480 Countess . . . 20 x 10 Lbs. 6720 No. 171 2800 2A.O Duchess . . . 24 x 12 44.8o i2: * Length of a slate is iaken from nail-hole to tail. SHOT AND SHELLS. FRAUDULENT BALANCES. 65 PILING OF SHOT AND SHELLS. To Compute NnmlDer of Sh.ot. Triangular Pile. RULE. Multiply continually together, number of shot in one side of bottom course, and that number increased by i, and again by 2, and one sixth of product will give number. EXAMPLE. What is number of shot in a triangular pile, each side of base contain- ing 30 shot? 30X30+1X30 + 2 = 976o shQt 6 6 Square Pile. RULE. Multiply continually together, number in one side of bottom course, and that number increased by i, double same number in- creased by i, and one sixth of product will give number. EXAMPLE. How many shells are there in a square pile of 30 courses? Oblong Pile. RULE. From 3 times number in length of base course sub- tract one less than number in breadth of it ; multiply remainder by number in breadth, and again by breadth, increased by i, and one sixth of product will give number. EXAMPLE. Required number of shells in an oblong pile, numbers in base course being 16 and 7 ? ,6X3-7=7X 7 XJTI = *J = Incomplete Pile. RULE. From number in pile, considered as complete, subtract number conceived to be in that portion of pile which is wanting, and remainder will give number. FRAUDULENT BALANCES. To Detect Them. After an equilibrium has been established between weight and article weighed, transpose them, and weight will preponder- ate if article weighed is lighter than weight, and contrariwise if it is heavier. To Ascertain True Weight. RULE. Ascertain weight which will produce equilibrium after article to be weighed and weight have been transposed; reduce these weights to same denomination, multiply them together, and square root of their product will give true weight EXAMPLE. -If first weight is 32 Ibs., and second, or weight of equilibrium after transposition, is 24 Ibs. 8 oz., what is true weight? 24 Ibs. 8 oz. =24.5 Ibs. Then 32 X 24. 5 = 784, and v/7 8 4 = a8 lbs - Or, when a represents longest arm, \ A greatest weight, and b " shortest arm, \ B least weight. Then Wa= Aft, and W6=:Ba; multiplying these two equations, W 2 a6 = ABa6, or W 2 = AB, and W = V AB - ILLUSTRATION. A = 32 ; B = 24. 5 ; W = 28. Assume length of longest arm -.= i* Then 32 : 28 :: 10 : 8.75. He^jce, a = 10, 6 = 8. 75, or 28 2 = 32 X 24- 5, and ' 66 WEIGHING WITHOUT SCALES. PAINTING. "Weighing wi.th.oirt Scales. To Ascertain AVeiglit of a Bar, Beam, etc., t>y Aid of a known "VVeignt. OPERATION. Balance bar, etc., over a fulcrum, and note distance between it and end of its longest arm. Suspend a known weight from longest arm, and move bar, etc., upon fulcrum, so that bar with attached weight will be in equilibrio ; subtract distance between the two positions of fulcrum from longest arm first obtained ; multiply this remainder by weight suspended, divide product by distance between f ulcrums, and quotient will give weight. EXAMPLE. A piece of tapered timber 24 feet in length is balanced over a fulcrum when 13 feet from less end; but when the body of a man weighing 210 Ibs. is sus- pendecTfrom extreme of longest arm, the piece and weight are balanced when ful- crum is 12 feet from this end. What is weight of the timber? 13 12 = i, and 13 i = 12 feet. Then 12 X 210-:- 1 = 2520 Ibs. PAINTING. i pound of paint will cover about 4 square yards for a first coat and about 6 yards for each additional coat. Proportions of Colors for ordinary Paints. By "Weight. COLORS. -3 J ti HJ3 li i m i* l! COLORS. II White . . . IOO Lead 08 Black (ireen 25 IOO 75 Red Chocolate. . These are the colors alone, to which boiled linseed oil, litharge, Japan varnish, and spirits turpentine are to be added according to the application of the paint. Lamp-black and litharge are ground separately with oil, then stirred into the lead and oil. Thus for black paint: Lamp-black 25 parts, litharge i, Japan varnish i, boiled lin- seed oil 72, and spirits turpentine i. Tar Paint. Coal tar 9 gallons, slaked lime 13 Ibs., turpentine or naphtha 2 or 3 quarts. A GALLON OF PAINT WILL COVER Superficial 1 feet. A GALLON OF PAINT WILL COVBR Superficial feet. On stone or brick, about On composite, etc., from On wood, from . . , 190 to 225 300 " 375 37*: " $2* On well-painted surface or iron One gallon tar, first coat " " " second coat ... 600 90 160 Boiled Oil. Raw linseed oil 91 parts, copperas 3, and litharge 6. Put litharge and copperas in a cloth bag and suspend in middle of a kettle. Boil oil four hours and a half over a slow fire, then let it stand and deposit the sediment. \Vhite Paint. Inside work. Outside work. White lead, in oil . . 80 80 Raw oil Boiled oil 14. 5 9 Spirits turpentine. New wood- work requires i Ib. to square yard for three coats. Coats for 100 Square Yards New White Pine. Inside work. Outside work. 9 4 8 INSIDE. White lead. Raw Turpen- tine. Drier. OUTSIDE. White lead. Raw oil. Boiled oil. Turpen- tine. Priming 2d coat 3d " Lba. 16 15 13 Pt3. 3-5 2-5 Pt8. 6 i-5 i-5 Lbs. 25 25 25 Priming 2d and 3d ) coats j Lbs. 18.5 15 Pts. 2 2 Pte. 2 2 Pts. 5 .1 Ib. of drier with priming and coating for outside. HYDROMETERS. HYDROMETERS. U. S. Hydrometer (Tralle's) ranges from o (water) to 100 (pure spirit) ; it has not any subdivision or standard termed "Proof," but 50, upon stem of instrument, at a temperature of 60, is basis upon which com- putations of duties are made. In connection with this instrument, a Table of Corrections, for differences in tem- perature of spirits, becomes necessary ; and one is furnished by the Treasury De- partment, from which all computations of value of a spirit are made. ILLUSTRATION. A cask contains 100 gallons of whiskey at 70, and hydrometer sinks in the spirit to 25 upon its stem. Then, by table, under 70, and opposite to 25, is 22.99, showing that there are 22.99 gallons of pure spirit in the 100. Commercial Hydrometer (Gendar's) has a " Proof " at 60, which is equal to 50 upon U. S. Instrument and its gradations, run up to 100 with it, and down to 10 below proof, at o upon U. S. Instrument ; or o of the Commercial Instrument is at 50 upon U. S. Instrument, from which it progresses numerically each way, each of its divisions being equal to two of latter. In testing spirits, Commercial standard of value is fixed at proof; hence any difference, whether higher or lower, is added or subtracted, as case may be, to or from value assigned to proof. A scale of Corrections for temperature being necessary, one is fur- nished with a Thermometer. Application of Thermometer. Elevation of the mercury indicates correction to be added or subtracted, to or from indication upon stem of hydrometer. When elevation is above 60, subtract correction ; and when below, add it. ILLUSTRATION. A hydrometer in a spirit indicates upon its stem 50 below proof, and thermometer indicates 4 above 60 in appropriate column. Then 50 4 = 46 = strength below proof, To Compute Strength, of a Spirit, or ^Volume of* its Pure Spirit, "by Commercial Hydrometer, and. Convert it to Indication of a TJ. S. Hydrometer. When Spirit is above Proof. RULE. Add 100 to indication, and divide sum by 2. When Spirit is below Proof. RULE. Subtract indication from 100, and divide remainder by 2. EXAMPLE. A spirit is n above proof by a Commercial Hydrometer; what pro- portion of pure spirit does it contain ? n-f-ioo-f-2 = 55.5 per cent. To Compute Strength, etc., by a U. S. Hydrometer. When Spirit is above Proof RULE. Multiply indication by 2, and subtract 100. When Spirit is below Proof. RULE. Multiply indication by 2, and subtract it from ioo. EXAMPLE, A spirit is 55.5; what is its per centage above proof? 55.5X2 ioo = n per cent. Commercial practice of reducing indications of a hydrometer is as follows: Multiply number of gallons of spirit by per centage or number of degrees above or below proof, divide by ioo, and quotient will give number of gallons to be added er subtracted, as case may be. ILLUSTRATION. 50 gallons of whiskey are 1 1 per cent, above proof. Then 50 X 1 1 -=- ioo = 5. 5, which added to 50 = 55. 5 gallons. 68 HYGROMETER. HYGROMETER. Dew-point. When air is gradually lowered in its temperature at a constant pressure, its density increases, and ratio of increase is sensibly same for the vapor as for the air with which it is combined, until a point is reached at which the density of the vapor becomes equal to the maximum density corresponding to the temperature. This temperature is termed dew-point of given mass, and any further re- duction of it will induce the condensation of a portion of the vapor in form of dew, rain, snow, or frost, according as temperature of surface is above or below freezing point. Mason's or like Hygrometer. To Ascertain Dew-point. RULE. Subtract absolute dryness from temperature of air, and remainder is dew-point. EXAMPLE. Temperature of air 57, and absolute dryness 7. Hence 57 7 = 50 dew-point. To Ascertain Absolute Existing Dryness. RULE. Subtract temperature of wet bulb from temperature of air, as indicated by a dry bulb, add excess of dryness from following table, multiply sum by 2, and product will give absolute dryness in degrees. EXAMPLE. Temperature of air 57, wet bulb 54 Then 57 54 = 3, and 3 + -5 (from table) X 2 = 7 absolute dryness. Observed j Excess of Dryness. | Dryness. Observed Dryness. Excess of Dryness. Observed Dryness. Excess of Dryness. Observed Dryness. Excess of Dryness. Observed Dryness. Excess of Dryness. 5 I .083 .166 5 5-5 .8 33 .9165 9-5 1.583 1.666 14 M-5 8-333 2.4165 18.5 '9 3.0*83 3.166 i-5 2495 6 o-5 1-7495 15 2-5 19-5 3-2495 2 333 6-5 .083 i 1-833 15-5 2.583 20 3-333 2-5 .4165 7 .166 i-5 1-9165 16 2.666 20.5 3-4165 3 5 7-5 2495 2 2 16.5 2-7495 21 3-5 3-5 583 8 333 2-5 2.083 17 2-833 2I -5 3-583 4 .666 8-5 4165 3 2.166 * 7 o S 2.9165 22 3.666 4-5 7495 9 5 3-5 2.2495 18 3 22.5 3-7495 To Coxnpiate "Volume of "Vapor in Atmosphere. By a Hygrometer. When temperature of atmosphere in shade, and of dew-point are given. If temper- ature of air and dew-point correspond, which is the case when both thermometers are alike, and air consequently saturated with moisture, then in table* opposite to temperature will be found corresponding weight of a cube foot of vapor in grains. ILLUSTRATION. Assume temperature of air and dew-point 70. Then opposite temperature weight of a cube foot of vapor = 8. 392 grains. But if temperature of air is different from dew-point, a correction is necessary to obtain exact weight. ILLUSTRATION. Assume dew-point 70 as before, but temperature of air in shade 80, then the vapor has suffered an expansion due to an excess of 10, which re- quires a correction. In table of corrections for 10 is 1.0208. Then divide 8.392 grains at dew-point viz., 70 by correction corresponding to degrees of absolute dryness viz., 10. ' 392 8.221 grains of existing vapor, which, subtracted from weight of vapor I.O2OO corresponding to temperature of 80, will give number of grains required for satu- ration at that temperature. "333 grains at temperature of 80 8.221 contained in the air = 3. 112 required for saturation. * For table, see Mason's as published by Pike & Sons, New York, and compared with Sir Jobs Leslie's and Professor Daniel's. HYGROMETER. SUN-DIAL. CHAINING. 69 To ascertain relations of these conditions on natural scale of humidity (complete saturation being 1000), divide weight of vapor at dew-point by weight at tempera- ture of air, and quotient will give degrees of saturation. ILLUSTRATION. Dew-point = 70, we ight = 8.392. Then 8. 392 -7-11.333 ( at 8o ) = -745 degrees of humidity; saturation = 1000. To Compute, Weight of Vapor in a Cube Foot of Air. See Pressures, Temperatures, Volumes, and Density of Steam, p. 708. Thus, Required weight of vapor in a cube foot of saturated air at 212. At a temperature of 212 density or weight of i cube foot of air = .038 /&. If density is required for any temperatures not in table, see rule, p. 706. Humidity. Condition of air in respect to its moisture involves amount of vapor present in air and ratio of it to amount which would saturate it at its temperature, and it is this element which is denoted by term humidity, and it is expressed as a per centage; thus, if weight of vapor present is .7 of that required for saturation, the humidity is 70. Dry Air is air, humidity of which is below zero, but it is customary to term it dry when its humidity is below the average proportion. NOTE. Air in a highly heated space contains as much vapor (when weight of it is equal) as a like volume of external air, but it is drrv as its capacity for vapor is greater. SUN-DIAL. To Set a Sun-dial. ,tr . Set column on which dial is to be placed perpendicular to horizon. Ascertain by spirit level that upper surface is perfectly horizontal ; screw on plate loosely by means of centre screw, and bring gnomon as nearly as practicable to its proper direction. On a bright day set dial at 9 A.M. and 3 P.M. exactly, with a correctly regulated watch; observe difference between them, and correct dial to half difference. Pro- ceed in same manner till watch and dial are found to agree perfectly. Then fix plate firmly in that situation, and dial will be correctly set. This is obvious; for, if there were any defects, the Sun's shadow would not agree with time indicated by watch, both before and after he passed meridian. Take care, however, to allow for equation of time, or you may set dial wrong. Best day in the year to set a dial is isth of June, as there is no equation to allow for, and no error can arise from change of declination. A dial may be set without a watch, by drawing a circle around centre, and marking spot where top of shadow of an upright pin or piece of wire, placed in centre, just torches circle in A.M., and again in P.M. A line should be drawn from one spot to the other, and bisected exactly; then a line drawn from centre of dial through that bisection will be a true meridian line, on which the XII hours' mark should be set. CHAINING OVER AN ELEVATION. I C == L, and C = cos. angle. I representing length of line chained, C cos. angle of elevation with horizon, and L length of line reduced to horizontal. ILLUSTRATION. Length of an elevation at an angle of 30 17' is 100 feet; what is horizontal distance ? By Table of Cosines, 30 17' = . 863 54. Hence, 100 X . 863 54 = 86. 354 feet. To set out a Riglxt Angle witla. a Chain, Tape-line, etc. Take 40 links on chain or feet of line for base, 30 links or feet for perpendicular, and 50 for hypothenuse, or in this ratio for any length or distance. USEFUL NUMBERS IN SURVEYING. For Converting Multiplier. Converse. II For Converting Multiplier. Convert*. Feet into links.. Yardb " " .. 4-545 .66 Square feet into acres.. . 0000229 43560 [I Square yards .0003066 4840 7 3' Thus, if Monday is the day determined by the year given, the following dates are the Mondays in that year. Epacts, Dominical Letters, and an .AJmanac, from 1834 to 1935. USK or TABLE. To ascertain day of the week on which any given day of the month falls in any year from 1800 to 1901. ILLUSTRATION. The great fire occurred in New York on i6th of December, 1835; what was day of the week ? Opposite 1835 is Sunday; and by preceding table, under December, it is ascertained that 1 3th was Sunday; consequently, i6th was Wednesday. Years. Days. Dom. Let- ten. *i tt. Yeara. Day*. Dom. Let- ten. I Yean. Dyi. Dom. Let- ten. 1 1834 Saturday. B 20 1868 Sunday. * ED 6 1902 Saturday. E 22 1835 Sunday. D *!' 1869 Monday. C *7 1903 Sunday. D 3 1836 Tuesday.* CB 1? 1870 Tuesday. B 28 1904 Tuesday.* CB '4 1837 Wednesd. A 23 1871 Wednesd. A 9 '90S Wednesd. A 2S 1838 Thursday. G ' 4 1872 Friday.* GF 20 1906 Thursday. G 6 1839 Friday. F is 1873 Saturday. E I 1907 Friday. F If 1840 Sunday.* ED rf 1874 Sunday. D 19 1908 Sunday.* ED ii 1841 Monday. C 7 1875 Monday. C -'3 1909 Monday. C 9 1842 Tuesday. B ii 1876 Wednesd.* BA 4 1910 Tuesday. B 20 1843 Wednesd. A 29 1877 Thursday. G 15 1911 Wednesd. A I 1844 Friday.* GF ii 1878 Friday. F 20 1912 Friday.* ~GF 12 1845 Saturday. E 22 1879 Saturday. E 7 I9 f 3 Saturday. E 2 3 1846 Sunday. D 3 1880 Monday. * DC 18 1914 Sunday. D 4 1847 Monday. C '4 1881 Tuesday. B 20 1915 Monday. C IS 1848 Wednesd.* BA 25 1882 Wednesd. A II 1916 Wednesd. * BA 26 1849 Thursday. G 6 1883 Thursday. G 22 1917 Thursday. G 7 1850 Friday. F *z 1884 Saturday.* FE 3 1918 Friday. F 18 1851 Saturday. E 28 1885 Sunday. D 14 1919 Saturday. E 29 1852 Monday. * DC 9 1886 Monday. C *3 1920 Monday. * DC ii 1853 Tuesday. B 20 1887 Tuesday. B e 1921 Tuesday. T B 22 1854 Wednesd. A I 1888 Thursday.* AG 17 1922 Wednesd. A 3 1855 Thursday. G 12 1889 Friday. F 28 1923 Thursday. G M 1856 Saturday.* FE 3 1890 Saturday. E 9 1924 Saturday.* FE 25 1857 Sunday. D 4 1891 Sunday. D 20 1925 Sunday. D 6 1858 Monday. C 15 1892 Tuesday.* CB I 1926 Monday. C '7 1859 Tuesday. B 26 1893 Wednesd A 12 1927 Tuesday. B 28 1860 Thursday. * AG 7 ,8^4 Thursday. G 23 1928 Thursday.* AG 9 1861 Friday. F iS 1895 Friday. F 4 1929 Friday. F 20 1862 Saturday. E 29 1896 Sunday.* ED IS 1930 Saturday. E I 1863 Sunday. D ii 1897 Monday. C 26 1931 Sunday. D 12 1864 Tuesday. * CB 22 1898 Tuesday. B 7 1932 Tuesday. * CB 2 3 1865 Wednesd. A 3 1899 Wednesd. A ,8 1933 Wednesd. A 4 1866 Thursday. G M 1900 Thursday. G 29 1934 Thursday. G 15 1867 Friday. F 25 1901 Friday. F ii 1935 Friday. F 26 * In leap-year, January and February must be taken In columns marked * 74 CHRONOLOGY. MOON'S AGE. TIDES. To Ascertain Year or Years of Coincidences of a given Day of the Week with a given Z>ay of a M.oiith. Look in preceding table and ascertain day of week opposite to year of occurrence, and every year in which same day is given will be year of coin- cidences required. ILLUSTRATION. If a child was born on Saturday, igth Sept. 1829, when could and can his birthdays be celebrated, that occurred or are to occur on same day of week and date of month ? Opposite to 1829 is Sunday, and in preceding table the Sundays for September of that year were 6th, i3th, 2oth; hence, if 2oth was Sunday, the igth was Saturday. Hence, every year in table opposite to which is Sunday are the years of the coin- cidence required, as 1835, 1840, 1846, 1857, 1863, 1868, 1874, 1885, etc. MOON'S AGE. To Compute Moon's ^Lge. RULE. To day of month add Epact and Number of month ; from sum subtract 29 days, 12 hours, 44 min. and 2 sec., as often as sum exceeds this period, and result will give Moon's age approximately at 6 o'clock A.M. in d. h. .7 16 .9 4 9 15 EXAMPLE. Required age of Moon on 25th February, 1877 ? Given day 25 -j-epact i5-f- num l>er of month 1.22 = 41 d. 22 h. 29 d. 12 h. 44 m. 2 sec. = 12 d. 9 h. 15 min. 58 sec. In Leap-years add i day to result after 28th February. To Compute Age of 3VEoon at Mean Moon at any other Location than, that Given. RULE. Ascertain age, and add or subtract difference of longitude or time, according as place may be West or East of it, to or from time given. 0r, when time of new Moon is ascertained for a location, and it is required to ascertain it for any other, add difference of longitude or time of the place, if East, and subtract it if it is West of it. Moon's Southing, as usually given in United States Almanacs, both Civil and Nau- tical, is computed for Washington. United States, d January February i east h. 5 22 of Mississippi Riv Nurnlaers of d. h. April i 21 May 2 8 er. the !Mon July August September . ths. d. h. October November December.. . March. . . Q June. . . . . * IQ To Compute Time of High--water "by Aid. of American Nautical Almanac. RULE. Ascertain time of transit of Moon for Greenwich, preceding time of the high-water required. For any other location (west of Greenwich), multiply the time in column " diff. for one hour " by longitude of location west of fcrreenwich, expressed in hours, and add product to time of transit. NOTE. It is frequently necessary to take the transit for preceding astronomical day, as the latter does not end until noon of day under computation. EXAMPLE. Required time of high- water at New York on 2$th of August, 1864. Longitude of New York from Greenwich = 4 h. 56 m. 1.65 sec., which, multiplied by 2. 17 min., the difference for i hour 10.71 min. for correction to be added to time of transit, to obtain time of transit at New York. TIDES. MOON'S SOUTHING. 75 Time of transit, 18 h. 38.8 m. ; then 18 h. 38.8 m. + 10.71 TO. = 18 Aowr* 49.51 mm. Time of transit at New York, 24 d. 18 h. 50 m. Establishment of the Port, 8 13 25 d. 3 A. 3 m. = time of high-water. NOTE. Time of 2$th at 3 A. 3 m. Astronomical computation = 25th at 3 h. 3 m. I. M. Civil Time. To Compute Time of Higli-water at Full and Change of Moon. Time of High-water and Age of Moon on any Day being given. RULE. Note age of Moon, and opposite to it, in last column of following table, take time, which subtract from time of high-water at this age of Moon, added to 12 h. 26 m., or 24 h. 52 m., as case may require (when sum to be subtracted is greatest), and remainder is time required. EXAMPLE. What is time of high-water at full and change of Moon at New York? Time of high- water at Governor's Island on 25th of Jan. 1864, was 9 h. 20 m. A.M. civil time. Age of Moon at 12 M. on that day was 16 d. 8 h. 59 m. Opposite to 16 days, in following table, is 13 h. 28 m. , and difference between 16 d. and 16 d. 12 h. =(16.5 16, or 13.53 13.28) is 25 m. : hence, if 12 h. =25 w., 16 d. 8 h. 59 w. 16 d. 8 h. 59 m. = 18.71 or 19 m., which, added to 13 h. 28 m. = 13 h. 47 m. Then 9 h. 20 m. -f 12 h. 26 m. (as sum to be subtracted is greater than time) 13 h. 47 m. = 21 h. 46 m. 13 h. 47 m. =. 7 A. 50 m. This is a difference of but 13 minutes from Establishment of Port. Time after apparent Noon "before Moon next passes Meridian, A-ge at Noon being given. (S. H. Wright, A.M., Ph.D.) Age of Moon. Moon at Meridian. Age of Moon. Moon at Meridian. Age of Moon. Moon at Meridian. Are of Moon. Moon at Meridian. Age of Moon. Moon at Meridian. Days. B. M. Days. H. M. Days. H. M. Days. H. M. Days. H. M. P. M. P.M. . M. A. M. A. M. .0 o 6 5 3 12 o 6 18 15 8 24 20 II 5 25 6-5 528 12.5 o 31 18.5 15 34 24-5 20 37 I 50 7 5 53 13 o 56 19 15 59 25 21 2 i-5 i 16 7-5 6 19 13-5 I 21 i9-5 16 24 25-5 21 27 2 i 41 8 644 14 i 47 20 16 49 26 21 52 . M. 2-5 2 6 8.5 7 9 H-5 2 12 20.5 17 i5 26.5 22 IJ 3 2 31 9 7 34 15 2 37 21 17 40 27 22 43 3-5 2 57 9-5 7 59 15-5 3 2 21-5 18 5 27-5 23 8 4 3 22 10 8 25 16 3 28 22 18 30 28 23 33 4-5 3 47 10.5 8 50 16.5 3 53 22-5 18 56 28.5 23 58 5 4 12 ii 9 '5 i7 4 28 23 19 21 29 24 24 5-5 438 "5 9 40 i7-5 4 43 23-5 19 46 29-5 24 48 Tidal Phenomena. The elevation of a tidal wave towards the Moon slightly exceeds that of the op- posite one, and the intensity of it diminishes from Equator to the Poles. The Sun by its action twice elevates and depresses the sea every day, following the action of the Moon, but with less effect. Spring Tides arise from the combined action of the Sun and Moon when they are on both sides of the Earth. Neap Tides are the consequence of the divided action of the Sun and Moon, when they are on opposite sides of the Earth, and the greatest elevations and depressions do not occur until the 2d or ^d day after a full or a new Moon. When Sun and Moon are in conjunction, and the time is near to the Equinoxes, the tides are fullest. The mean effect of the Moon on the tidal wave is 4.5 times that of the Sun. If, therefore, the Moon caused a tide of 6 feet, the Sun will cause one of i. 33 feet; hence a spring tide will be 7.33 feet, and a neap tide 4.67 feet. Particular locations as to contour of shores, straits, capes, and rivers, lengths and depths of channels, shoals, etc., disturb these general rules. LATITUDE AND LONGITUDE. LATITUDE AND LONGITUDE. Latitude and. Longitude of Principal Locations and Observatories. Compiled from Records of If. S. Coast and Geodetic Survey and Topograph- ical Engineer Corps, Imperial Gazetteer, and Bowditch's Navigator. Longitude computed from Meridian of Greenwich. A., represents Academy; Az., Azimuth; A. S., Astronomical Station; C., College; Cap. , Capitol ; Ch. , Church ; C. H. , City Hall ; C. S. , Coast Survey ; Ct. , Court-house ; Cy., Chimney; F.S., Flagstaff; G.S., Geodetic Station; Hos., Hospital; L. Light- house; Obs. , Observator y ; S.H., State-house; Sp. Spire ; Sq., Square; S.S., Signal Station ; T. , Telegraph ; T. H. , Town Hall ; U. , University; Un. , Union ; B. , Baptist ; Con., Congregational; E., Episcopal; P., Presby. ; and M.Ch., JtfeM. Churches. LOCATION. Latitude. Longitude. LOCATION. Latitude. Longitude. WORTH AND SOUTH AMERICA. Acapulco Mex. N. o / // ID 50 IQ W. / // 99 49 9 NORTH AND SOUTH AMERICA. Canandaigua ... .N.Y. N. 1 tl 42 54 9 W. o / // 77 17 Albany, P. Ch....N.Y. Ann Arbor Mich 42 39 3 42 1 6 48 73 45 24 83 43 3 Cape Ann, S. L. .Mass. Cape Breton .... Va. 42 38 ii AS. C? 70 34 10 38 58 42 76 29 6 Cape Canaveral. . . Fla. 28 27 30 80 33 Apalachicola,F.S.Fla. Astoria, F. S Or. 29 43 30 46 II IQ 8459 123 49 42 Cape Cod, L. P. L... Ms. Cape Fear N.C. 42 2 33 4 8 70 9 48 77 57 Atlanta C H .. Ga. 00 AA C7 84 23 22 Cape Flattery L W T 4.8 23 m Auburn N.Y 42 55 76 28 Cape Florida, L. . . Fla. 2C QQ 54 Augusta Ga TO 28 81 54 Cape Hancock Colo R f> f> ' Augusta, B.Ch.... Me. Austin Tex 44 18 52 69 46 37 Cape Hatteras, L. , N. C. Caoe Henlopen L Del 35 15 2 og A.6 6 75 3 54 Balize La 2Q 8 5 Cape Henry L. .'.Va ^6 5^ 3O 75 4 7 Baltimore, Mon't . Md. Bangor Tho's Hill Me y o 39 i7 48 76 36 59 68 46 59 Cape Horn, S. Pt. , Her- mit's Island . . 6V 33 J u o. 67 16 Barbadoes, S.Pt. . W. I. Barnegat, L N. J. Bath, W S Ch.... Me 13 3 39 46 4q CA CC 5937 74 6 Cape May, L N.J. Cape Race .... N S 55 ft 38 55 48 46 3Q 24 74 57 18 Baton Rouge La QI 18 Cape Sable N S 53 4 3 Beaufort Ct . . N C ~ Cape Sable C S Fla 43 2 4 81 is Beaufort, E.Ch...S.C. Belfast, M.Ch Me. Benicia, Ch Cal 32 26 2 44 25 29 og o C 80 40 27 69 19 Cape St. Roque, Brazil " 5 s. 53 5 N 8 35 17 Benington Vt. Bismarck, S. S . . .Neb 42 40 4.6 AS 73 18 Carthagena N.G. Castine . . . Me 10 26 7538 68 AC Boston, L Mass. Boston, S H. ... a 42 19 6 7053 6 Cedar Keys, Depot Isl. Chagres N G 29 7 3 83 245 80 I 21 Brazos Santiago. .Tex. Bridgeport Conn. Bristol R. I 26 6 41 10 30 97 12 7 l\l 4 Charleston, C.Ch.,S.C. Charlestown,Mon.,Ms. Cheboygan L Mich 32 46 44 42 22 36 79 55 39 7i 3 18 Brooklyn, C.H... N.Y. Brownsville, S.S. .Tex. Brunswick, A Ga. Brunswick, C.Sp. .Me. Buffalo L NY 40 41 31 26 3i 8 51 43 54 2 9 73 59 2 7 97 30 81 29 26 69 57 24 ?8 CO Chicago, C.Ch 111. Chickasaw Miss. Cincinnati, Obs O. Cleveland, Hos " Colorado Springs Col 4i 53 48 35 53 3 39 6 26 41 30 25 18 co 87 37 47 88 6 25 84 29 45 81 40 30 Burlington N. J. Burlington, C Vt. Burlington,Pub.Sq.,Ia. Bushnell Neb 40 4 52 44 28 52 40 48 22 74 52 37 78 10 91 6 25 Columbia, S.H....S.C. Columbus, Cap 0. Concord, S.H....N.H. Corpus Christi Tex 33 59 58 39 57 40 43 12 29 81 2 3 82 59 40 71 29 Cairo Ill Council Bluffs Neb T Qe ^8 Calais, C.S. Obs... Me. Callao, F.S Peru Cambridge, Obs. Mass 45 s! 5 12 N 4 6716 5 77 13 71 7 4.3 Crescent City, L... Cal. Cumberland Md. Darien,W.H Ga. Davenport, S. S la. Dayton 0. 4 1 44 34 39 39 H 3i 21 54 4i 32 3Q A A 124 II 22 78 45 25 81 25 39 90 38 84 ii Camden S C 80 33 Deadwood S S Dak Campeachy . .Yucatan 19 49 9033 Decatur, S. S Tex. 33 10 9730 LATITUDE AND LONGITUDE. 77 LOCATION. Latitude. Longitude. LOCATION. Latitude. Longitude. NORTH AND SOUTH AMERICA. Denver, S. H.Sp..Col. Des Moines,C.H...Ia. Detroit, St. P. Ch., Mich. Dover Del N. Q 1 II 3945 4i35 42 19 46 39 I0 43 13 42 29 55 4648 44 54 15 3 6 3 24 36 17 58 42 843 40 48 ii 44 58 40 30 40 i 8 34 47 13 35 47 35 36 30 22 42 12 10 39 2I 14 38 14 39 2 4 3818 6 4 6 3 29 18 17 32 22 2 3322 8 42 30 46 43 5 44 39 4 40 ID 4i 45 59 23 9 25 5i 5 41 27 13 4214 34 36 39 55 28 32 28 32 23 30 19 43 19 30 8 38 36 40 43 28 46 26 40 23 24 33 3i 1758 44 8 35 59 43 58 50 4 2 36 28 37 36 39 2 9 38 6 S. 12 N 3 344<> W. 4 II 104 59 33 93 37 16 83 223 75 30 ?o 54 9 39 57 92 8 76 36 31 76 13 23 80 4 12 124 9 41 93 10 30 81 27 47 87 41 40 95 15 10 88 340 104 47 43 94 44 8440 77 '8 77 27 38 66 38 15 94 47 26 64 37 6 79 l6 49 70 39 59 86 18 6335 76 50 72 40 45 82 21 23 77 10 6 70 35 59 7346 8657 86 5 9631 i 90 8 81 39 14 96 54 30 92 8 74 2 24 122 50 39 9125 81 48 31 7646 76 28 37 8354 91 14 40 76 20 33 96 37 21 94 58 84 18 77 6 92 12 1 NORTH AND SOUTH AMERICA. Lockport N.Y. N. O 1 II 43 " U! 5 42 38 46 4443 i 32 50 25 43 4 33 42 30 14 14 27 28 41 29 25 52 50 23 3 35 7 9 25 45 43 2 24 445838 30 22 54 30 41 26 36 1 5 " 34 g. 32 22 45 45 3i 37 4 47 41 23 24 4 16 57 36 933 25 5 2 3 34 4i S 5 41 38 10 41 18 28 41 21 16 29 57 46 40 42 44 35 6 21 41 30 6 42 48 30 39 39 36 41 29 12 36 50 47 41 2 50 41 33 35 628 44 45 37 2 47 3 4i 15 43 43 28 32 45 23 8 57 9 39 l6 2 30 20 42 30 24 33 37 *J 47' W. ?8 4 '6" 118 14 33 8530 71 19 2 67 2 7 21 83 37 36 8924 3 7 50 39 6055 95 57 56 97 27 50 81 40 90 7 99 5 6 8754 4 93 '4 8 .89 i 57 88 2 28 121 52 59 56 13 86 18 73 32 56 89 12 70 2 24 70 557 8649 3 77 21 2 91 24 42 101 21 24 70 55 36 72 55 45 72 5 29 90 328 74 24 77 5 74 33 70 52 28 75 33 48 71 18 49 76 7 22 73 25 35 72 7 75 58 5 75 30 7618 6 122 55 f 95 56 14 7635 5 75 42 79 2 7 7 81 34 12 88 32 45 87 is 53 77 24 16 Los Angeles Cal. Louisville Ky. Lowell, St Ann's Ch., Mass. Machias Th Me. Dover N. H. Dubuque . . la. Duluth, S.S Min. Eastpoii, Con.Ch. . Me. Edenton, C.H N.C. Elizabeth City, Ct. u Erie L Penn M aeon, Arsenal Ga. Madison, Dome... Wis. Marblehead, L. ..Mass. Martinique, S. P't . W. I. Matagorda,G.S...Tex. Matamoras " Eureka, M.Ch....Cal. Falls St. Anth'y.. Minn. Fernandina, A.S. .Fla. Florence Ala. Fort Gibson Ind. T. Fort Henry Tenn. Fort Laramie.Wyo. T. Fort Leaven worth, Ks. Frankfort . Ky Matanzas Cuba Memphis, S.S. ..Tenn. Mexico Mex. Milwaukee Mich. Minneapolis,U.C.,Min. Mississippi City, G. S., Miss. Mobile, E.Ch Ala. Monterey, Az.S... Cal. Montevideo... Rat Is' d Montgomery, S. H., Ala. Montreal C E Frederick Md. Fredericksburg, E. Ch. , Va. Fredericton N B Galveston,Cath'l. .Tex. Georgetown Ber. G eorgeto wn, E. Ch., S.C. Gloucester, U. Ch. . . Ms. Grand Haven, S. S., Mich. Halifax, Obs N.S. Harrisburg Penn. Hartford, S.H... Conn. Havana, Moro . . .Cuba Hole in the Wall, L., Bahamas Holmes's Hole, Ch., Ms. Hudson . . .NY Mound City 111. Nantucket,L....Mass. Nantucket, S. Tower, Mass. Nashville, U. . . .Tenn. Nassau L N P Natchez Miss Nebraska, Junction of Forks of Platte Riv. New Bedford, B. Ch., Mass. New Haven, Col., Conn. New London, P. Ch. " New Orleans, Mint, La. NEW YORK, C.H., N.Y. Newbern, E. Sp. . .N.C. Newburg, A. Sp.,N.Y. Newburyport, L. , Mass. New Castle, E. Ch. , Del. Newport, Sp R. I. Norfolk C H Va. Huntsville. .. . Ala. Indianapolis Ind. Indianola,G.S Tex. Jackson Miss Jacksonville, M. Ch., Fla! Jalapa Mex Jefferson City . . . .Mo. Jersey City, Gas Ch'y. Kalama,M.Ch...WT. Keokuk S S la Norwalk Conn Key West, T. Obs., Fla. Kingston Jamaica Kingston, C. H. . .C. W. Knoxville Tenn Norwich. .... " Ocracoke, L N.C. Ogdensburg, L. . .N.Y. Old Point Comfort, Va. Olympia Wash.T. Omaha, P. Ch Neb. Oswego, S. S N.Y. Ottawa Can La Crosse, Ct.S. . .Wis. Lancaster Penn. Lavaca, A. S Tex. Leavenworth, S. S. . Ks. Panama, Cath'l. ..N.G. Parkersburg. . . . W. Va. Pascagoula Miss. Pensacola, Sq're. .Fla. Petersburg, C. H. . . Va. Lima Peru Little Rock Ark. 78 LATITUDE AND LONGITUDE. Latitude and. Longitu.de Continued. LOCATION. Latitude. | Longitude. LOCATION. Latitude. Longitude, NORTH AND SOUTH AMERICA. Philadelphia, S.H., Pa. Pike's Peak, S.S.. Col. Pittsburg Penn N. O 1 II 39 56 53 3848 40 32 44 4 1 57 41 58 44 48 7 3 1833 48 6 56 43 39 28 45 30 9 34 10 28 43 4 16 43 2 40 20 40 41 49 26 42 3 '9 i5 46 49 12 43 9 35 46 50 37 32 16 22 56 N. 43 8 17 44 6 6 4355 38 34 4i 42 31 10 40 46 4 25 26 22 29 25 22 34 '5 46 32 42 42 3748 37 19 58 35 10 38 33 43 20 41 32 30 40 27 40 34 26 10 37 20 49 36 57 3i 35 4i 6 32 4 52 42 48 4i 7 So 32 30 33 54 58 39 47 57 42 6 20 4.8 ^o W. / II 75 9 3 104 59 80 2 73 26 54 70 39 12 122 44 33 72 16 3 122 44 58 70 15 i 122 27 30 7940 68 7 70 42 34 9i 835 74 39 55 71 24 19 70 ii 18 98 2 21 71 12 15 79 8 7838 5 77 26 4 43 9 69 6 52 75 57 121 27 44 70 53 58 i" 53 47 101 4 45 98 29 15 119 15 56 "7 9 43 122 23 19 121 53 39 120 43 31 118 16 3 82 42 15 74 9 119 42 42 121 26 56 122 I 29 IO6 I 22 81 5 26 73 55 105 23 33 93 45 78 i 8 89 39 20 72 36 . 81 K NORTH AND SOUTH AMERICA. St. Augustine, P. Ch., Fla. St. Bartholomew, S. Point W. I. N. 1 II 29 53 20 17 53 3 17 24 i7 44 30 18 29 17 29 19 58 45 M 6 23 3 i3 38 8 3 30 9 i 18 5 30 43 12 44 52 46 18 21 13 9 38 8 51 30 50 41 19 36 42 27 18 46 12 43 3 30 28 27 36 22 15 30 41 54 ii II 20 43 39 35 40 13 10 1039 42 43 3312 43 s 649 33 N 2 3850 19 ii 52 32 23 28 46 57 3843 45 20 38 53 20 42 21 41 41 23 26 40 7 34 14 2 39 44 27 42 16 17 42 45 33 5 39 58 37 17 W. o / u 81 18 41 62 56 54 62 50 64 40 42 69 52 63 75 52 66 330 109 40 44 90 12 17 8 4 12 3 63 3 81 32 53 95 4 54 64 55 18 61 14 79 4 15 102 50 7i 54 107 45 27 60 12 76 9 16 8436 82 45 15 97 5i Si 7i 5 55 60 27 79 23 21 74 45 50 61 32 73 2 16 8742 75 13 71 41 89 2 96 8 36 9 54 97 * 87 25 112 3 77 36 7i 9 45 73 57 80 42 77 56 38 75 33 * 7i 4 8 13 97 3 90 20 76 40 76 -3A Plattsburg, Sp....N.Y. Plymouth, Pier ...Ms. Point Hudson W.T. Port au Prince. ..W. I. Port Townshend, A.S., Wash. T. Portland, C.H Me. Portland S S 0. St. Christopher, N. Pt., W.I. St. Croix, Obs " St. Domingo u St.Eustatia,Town. " St. Jago de Cuba, En- Porto Bello N G St John N B Porto Cabello, Mara- caibo Portsmouth, L. . .N. H. Prairie du Chien..Wis. Princeton, S. Cap., N.J. Providence, U.Ch.,R. I. Provincetown, Sp., Mass. Puebla de los Angelos, Mex. Quebec, Citadel. .Can'a Queenstown .... " Raleigh, Square.. N.C. Richmond, Cap. . . . Va. Rio de Janeiro, S. Loaf. Rochester, R.H..N.Y. Rockland,E.Ch...Me. Sackett's Harbor, N.Y. Sacramento Cal. Salem, So Mass. Salt Lake City, Obs., Utah Saltillo Mex St. Joseph L. CaL St. Louis, W.U Mo. St. Mark's, Fort.. Fla. St. Martin's, Fort, W. I. St. Mary's, M. H. ..Ga. St Paul Minn St. Thomas, Fort Ch'n, W. I. St. Vincent's, S. Point, W.I. Staunton . . . .Va Stockton, S.S Tex. Stonington, L. . .Conn. Sweetwater River, Mouth of. ..Wyo.T. Sydney S.S N.S. Syracuse N Y Tallahassee . . . Fla Tampa Bay, E. Key " Tampico, Bar. . . .Mex. Taunton,T. C.Ch., Mass. Tobago, N.E.P'r. W.I. Toronto Can Trenton, P. Ch ...N.J. Trinidad, Fort... W. I. Troy, D.Ch N.Y. Tuscaloosa Ala. San Antonio Tex. San Buenaventura, G S Cal Utica,Dut.Ch....N.Y. Valparaiso, Fort. .Chili Vandalia 111. San Diego, B.C ... " San Francisco, C. S. Station Cal. San Jose, Sp " San Luis Obispo.. " Vera Cruz Mex. Vicksburg, S.S... Miss. Victoria Tex. Vincennes ....hid Sandusky, L 0. Sandy Hook, L... N.J. Santa Barbara, M.Ch., Cal. Santa Clara, C.Ch.. " Santa Cruz, F. S.. " Santa F4 N. Mex. Savannah, Sp Ga. Schenectady N.Y. Sherman, R. R. D. , Wy. Shreveport, S. S La. Smithville, G.S. ..N.C. Springfield Mass. Springfield, S.H.... 111. Springfield, S.S " St. Augustine Fla. Virginia City,S.S.,M.T. WASHINGTON. . . Capitol Watertown, Ars'l. .Ms. West Point N.Y Wheeling Va Wilmington, E. Ch., N.C: Wilmington, T.H. .Del. Worcester, Ant. H. . Ms. Yankton, S.S Dak. Yazoo Miss York Penn Yorktown... ...Va. LATITUDE AND LONGITUDE. 79 Latitude and Longitude Continued. LOCATION. Latitude. Longitude. LOCATION. Latitude. Longitudf KUROPE, ASIA, AFRICA, AND THE OCEANS. N. o ^ // E. / // EUROPE, ASIA, AFRICA, AND THE OCEANS. N. O y // 44 24 E. 8 53 " Alexandria L .... 31 12 29 53 W? Algiers L 06 4.7 3 4 Gibraltar. 36 7 ^ 22 55 S^ 4 IO Antwerp 5 1 *3 4 24 CT oR -R Archangel 64 32 4 33 "^ Athens 37 58 23 44 E. 53 33 ">8 41 s 3 40 20 9 6 Batavia Obs 6 8 106 50 49 I W. Bencoolen, Fort, Su'a. Berlin Obs 3 i 8 102 19 Hawaii or Owyhee Hongkong 20 23 tf 22 I 6 30 155 1 4 114 14 45 Bombay F S ?! S l6 21 l8 12 IC7 ao 36 Botany Bay, C. Roads. Bremen s. 34 N 2 53 5 151 13 8 4Q Hood Isrd,Gallapagos. Hood's Island, Mar- 'I 3 Q 26 "w. 3 ^ 138 57 Bristol 51 27 w 9 2 3S Jeddo or Tokio 9 N. 3C 40 139 40 E 31 48 37 20 Brussels Obs 50 5' ii 4 22 43 3 2 10 18 Bussorah -}Q 3O 4 8 Leipsic 51 20 20 12 22 4 V C.2 Q 28 42Q IS Cadiz 36 32 6 18 W W. E Lisbon 38 42 9Q Cairo 30 3 IT 18 Liverpool Obs 53 24 48 3 Calais 5 58 i 5 1 E. Calcutta 88 20 Madras 80 I ^ 4>t 35 3 1 25 8 w 5 4S Canton . . 23 7 113 14. Madrid 40 25 3 42 Cape Clear 51 26 V O 2O Majorca Castle 5Q -34 3 E. 2 23 Cape of G. Hope Obs & 33 S^ 3 9 E 9 18 28 45 36 43 W. 4 26 Cape St. Mary, Mad'r. . *53p 45 7 Malta Valetta 35 54 E. 14 30 Ceylon Port Pedro . . 940 80 23 Manila. 14 36 121 2 Christiana CQ CC Marseilles . .... 43 *8 5 32 5955 Messina L 38 12 15 35 Congo River 6 8 12 Q 13 20 43 12 N Moscow je o-a Constantinople St S 41 i 28 co Muscat 23 37 58 35 Copenhagen Corinth 55 4i 37 54 12 34 22 52 Naples,!, 40 50 14 16 W. Cronstadt CQ CQ 2Q 47 New astle c.4 ;8 I 37 Dover 5I g I IQ New Hebrides Table S K w 9 Island ic. 28 167 7 Dublin co 23 12 6 20 30 Niphon Cape Idron N. Edinburgh cc e,7 5557 3 12 Japan 34 3^ 3 138 50 35 Falkland Islands St Odessa 46 28 30 44 Helena Obs 1C ee e 4c 38 8 13 22 N Paris Obs 48 50 13 2 2O Fayal S E Point 38 3O 28 42 Pekin OQ C.4. 116 28 Feejee Group, Ovolau, Obs 3 s. E. I?8 C.3 Plymouth 50 21 W. 4 Q Florence 17 K 43 46 ii 16 Port Jackson . . N. S. W. S. 3C Cl -52 4 E 9 151 18 Funchal Madeira 02 38 W. 16 c,^ Porto Praya, Cape Verd 35 N. 14 "54 W. 23 3 Geneva . . . J.6 II C.Q E 5 6 o !=; Prince of Wales Island. 1 s! 10 46 3 E 3 142 13 8o LATITUDE AND LONGITUDE. Latit LOCATION. rule Latitude. and. TJ Longitude. ongitnde Contini LOCATION. ted. Latitude. Longitude, EUROPE, ASIA, AFRICA, AND THE OCEANS. Quct'iistowu N. o / // 5' 47 4i 54 5 54 28 28 49 54 16 i 4437 36 59 14 I s 8 f i 17 38 26 51 s. 15 55 'vatoi Lo titude. w. V 12 27 4 ? 16 16 621 16^2 33 30 558 100 W. 13 18 E. 103 50 "V i 30 5 45 ries. JV ngitude g Longitude. EUROPE, ASIA, AFRICA, AND THE OCEANS. St. Petersburg N. , 59 56 29 59 21 II S. 33 33 4' 17 ff 35 47 43 7 3454 36 47 40 50 48 13 52 s 3 41 N 4 35 26 6 28 Table. itude. E. O / ft 30 19 32 34 72 47 " 5 w 3 M930 5 54 5 22 13 " 10 6 14 26 16 23 5 21 2 9 '7444 13939 3933 Longitude. Rome, St. Peter's Suez Surat Castle Sydney N.J 3.W. Santa Cruz Ten'fe Scilly, St. Agnes, L.... Senegal Fort Tahiti or Otaheite Tangier Sevastopol Seville Tripoli Siam Tunis City Sierra Leone Vienna Singapore Warsaw Obs Wellington... New Z'd St. Helena . . Zanzibar Island, Sp. . . ot included in previous wen in Time. LOCATION. La' Obser LOCATION. La Albany, Dudley . . Alleghany, Fenn. . Birr Castle, Earl of Rosse N. O 1 II in 42 39 49.55 40 27 36 53 5 47 42 22 52 52 12 51.6 S. 33 V 55 4<> 53 4i 44 43 53 23 13 55 57 23-2 43 46 41.4 46 ii 59.4 38 53 39 32 47 7 5i 28 38 53 33 5 51 20 20.1 52 9 28.2 53 24 47.8 40 43 49 W. h. m. ,. 4 54 59- 52 5 20 2.9 31 40.9 4 44 30-9 22-75 i 13 55 50 19.8 W. 3 1 6 49.1 25 22 "E 43 ' 6 45 3-6 *V 37 - 7 5 8 12.5 5 19 44-7 E. 39 54-i 49 28 -5 ' 7 w 57 ' 5 12 O. II 4 55 57 Madras N. t ii m 13 4 8.1 43 '7 So 39 6 26 55 45 19-8 48 845 38 644 5048 3 46 48 30 4i 53 52.2 4046 4 374JM 33 26 24.8 i7 44 30 59 56 29.7 59 20 31 33 5 N 4 '" 24 33 3i 49 35 4 38 53 39 41 23 26 E. A. m. i. 5 20 57.3 21 29 W. 537 E 59 2 30 16.96 46 26.5 5^.7 4 23.9 4 44 49-2 4^54-7 7 27 35-i 8 9 38. i 4 42 18.9 4 1 8 42.8 E. 2 I 13-5 I 12 24.8 10 4 59.86 5 27 14-1 E. 1 V ' 1 5 8 12.03 45548 Mitchell' s,Cin.,0. Moscow Cambridge, U. S. . . Cambridge, Eng. . . Cape oft?. Hope.. Copenhagen, Un'y. Crescent City, A. S.,Cal Dublin Munich, Bogenh'n Palermo Portsmouth Quebec Rome, College Salt Lake City, Utah San Francisco, Sq., Cal Edinburgh .... Florence Santiago de Chili. St. Croix, W. I.... St. Petersburg, A. . Stockholm .... Geneva Georgetown, U.S. . Gibbes's, Charles- ton U S GREENWICH Sydney Hamburg Tifft's, Key West. Fla Leipsic Ley den Unkrechtsberg,01- Liverpool Washington West Point, N. Y. . L. M. Rutherfurd, New York DIFFERENCE IN TIME. 8l DIFFERENCE IN TIME. Difference in. Time at following Locations. Longitude computed both from New York and Greenwich. Exact Difference of Time between New York and Greenwich is 4 h. 56 m. 1.6 sec., but in following table 2 seconds are given when the decimal in any reduction exceeds .5 seconds. F representing Fast, and S Slow. LOCATION. New York. Greenwich. LOCATION. New York. Greenwich. Acapulco A. m. *. .43.58- lilt ' 5 16 5 5U38 4354S. 3 19 '7 4 1 32 950 3i 34 i6 5 5F. i 34558. 10 26 2054F 5734 228. 16 46 F. i 9 108. 1038 2639 20 iF. 312368. 5 49 37 F. i 46 30 S. 9 47 38 F. ii 47.6 53n8 3 17 16 12 29 56 S. 5 13 3 F. i 234 19 54 S. i 824 3 2 9 1638 i 5930 4 305oF. 7 i 14 ' 43. 26 5 7F. 10 49 22 12 508. ii 30 F. 12 28 58 I 2 10 S. 6 957F. 26 58 2568. i 23 46 F. 6 308. 21 2F. A. m. . 6 39 178. 4 55 2 i 59 32 F. 12 IO I 9 3 2 17 36 5 39 568. 815 19 5 37 33 5 552 527 36 4 39 6 6 3027 5 6 28 435 8 35828 4 56 24 4 39 l6 6 5 12 5 6 39 5 22 40 4 36 i 8 838 5335F. 6 42 32 S. 45i 36 F. 444 148. 35 16 F. 4 52 448. 455 58 4 39 50 52558 17 28 F 353288. 5 i5 56 6 4 26 4 59 3 5 12 40 6 55 32 25 12 2 5 12 F. 556458. 4 29 4 5 5320F 5 8528. 4 44 3i 7 3256F. 5 58 128 i '355F. 4 29 48. 45858 3 32 16 5 2 32 4 35 Cedar Keys A. m. . 36 98. 24 3 234i n 4 8F. 41 378. 54 30 56 24 4i 57 3040 2 3 '5 28 7 35 57 10 6F. 651 58 5 46 18 i 33 47 S. i 27 10 3 2044 2941 I 12 30 4042 I 58 3 2 3^ 3558 12 26 F. 4 3 ? 4 2o i 6 388. I 12 10 28 6F. 10 248. 443 M F. 8 528. 24 i5 32037 i ID 40 29 50 3 10 F. 54458. i 2459 56 13 239 i 22 54 13 10 3 49 29 29 F. 3 48 22 i 23 88. 5 20 39 F. 12 148. 53i 34 F. 37 33 21 68. 4 34 34 F. *. m. ,. 532IIS. 5 20 5 5 1943 4 44 '3 53738 s 5031 5 52 26 5 37 59 5 26 42 6 59 '7 5 24 8 5 3 1 59 4 45 56 i 55 56 P. 50 16 6 29 48 S. 6 23 12 8 1645 5 2543 6 2 32 5 3644 654 32 65958 5 32 10 4 43 36 2522 6 2 40 6 8 32 4 27 S^ 5 6 26 12 48 5 454 52017 8 1639 6 12 42 5 25 5' 4 52 5' 5 5047 6 21 I 5 52 15 6 59 6 18 56 5 9 I2 5 9 5i 4 26 33 i 7 40 6 19 10 slliS 35 32 F. 4 18 288. 517 7 21 38 Albany Alexandria. .Egypt Algiers Charleston Charlestown Amsterdam Cheboygan Antwerp Chicago. Chickasaw. Astoria Cincinnati Atlanta Colorado Springs. . Columbia .... Augusta Ga. Augusta Me Columbus Austin Concord, Baltimore . . Constantinople.... Copenhagen Bangor Barbadoes, 8. Ft... Corpus Christ! Council Bluffs Crescent City Darien Bath Baton Rouge Beaufort N.C. Beaufort S C Davenport Dayton Belfast Deadwood Denver Berlin Detroit Bismarck Dover Del. Bombay F S Dover . . N H Boston S H Dublin Bremen. Bridgeport Duluth Brooklyn, N. Yard. Brunswick Me. Brunswick Ga. Brussels Eastport Eden ton Edinburgh Elizabeth City, N.C. Erie ' Buenos Ayres Buffalo L Eureka Burlington la. Burlington N. J. Burlington Vt. Bushnell Neb. Cadiz Falls St. Anthony.. Fire Island, L Florence Ala Fort Gibson Cairo Fort Henry..Tenn. Fort Laramie Fort Leaven worth . Cairo 111. Calais Me. Calcutta. Callao Fredericksb'g..Va. Fredericton. . . N. B. Funchal Cambridge. . . Mass. Canton Cape Gircrdeau. . . Cape of Good Hope. Cape Horn Galveston-. Geneva Geneva N.Y. Cape May Cape Race Georgetown... Ber. Georgetown... S.C. Gibraltar Carthagena Castine . . . 82 DIFFERENCE IN TIME. Difference in Time Continued. LOCATION. New York. Greenwich. LOCATION. New York. Greenwich. Glasgow A 43858F. 13 22 2 4 5S. 49 I0 4 56 i .6 4I 42 F. 5 35 54 ii i8S. 5 19 F. 33248. 4 56 26 F. 5 27 34 S. 12 27 i F. 15 27 30 I 12 5I46S. 48 18 I 30 2 I 430 30 35 i 31 S 6 14 16 2F. I 12 308. 8 7 25 22 F. 3 15 21 S. i 938 3i i3 9 53 II 2 39 34 i 8 58 27 54 F. 19 21 b. I 30 27 I 2 3 4 5 37 i4 F. 41 10 S. 12 22 4 19 26 F. I 12 46 S. 444 2F. 19 2 o. 2 56 16 4558 10 45 F. 26 12 38 28 S. 1 1 35 ^ 441 14 F. 4 38 18 5 54 2 . 13 10 9 2 12 41 5 17 20 52 22 27 50 S. 33 So 3038 4 26 40 19 A. m. s. 17 48. 4 42 40 5 20 7 5 45 12 4 14 40 3952*. 5 7 208. 4 50 43 5 29 26 24 F. 10 23 36 S. 7 36 59 F. 10 31 28 4 54 40 S. 547 48 5 44 20 6 26 4 6 o 32 5 26 37 62738 9 20 F. 6 8 32 S. 4 5 6 10 2 29 20 F. 8 ii 238. 6 5 40 5 27 14 5 5 54 5 7 4 5 35 36 6 4 59 4 28 8 .5 *5 22 6 26 29 6 19 52 41 12 F. 5 37 "S. 5 824 36 36 F. 6 8 48 S. 12 5 i5 4 7 52 18 542 4 45 16 4 29 49 5 34 30 5 57 36 1448 i7 44 58 F. 848 447 S. 4 43 21 21 28 F. 4 3 40 8. 6 23 52 6 29 51 5 26 40 6 o 28 6 36 20 Milwaukee h. m. i. 55 35 S. i 16 55 i 6 $1*. 3 ii 308. i ii 10 F. 49 10 S. i 50 F. 47 M 7 18 14 i 268. i 53 8F. 5 53 6 5i 158. 13 23 1 9 37 i 49 24 12 19 F. 4 19 7 40 i 4 128. 12 18 i 12 32 F. 449 34 6 138. 10 46 F. 9 88. 2 19 F. 7 34 ^ 6 58 58 F. 5588. 9 ii 3 1538 i 27 43 10 19 646 58 22 5 49 30 F. I 21 48 S. 5 522F. 30 i5 S. 12 41 54 F. 52 508. i3 35. 4 346 2 3 50 24 6 2 14 F. 4 39 26 13 25 16 34 3 I458S. 15 2F. 3 I348S. 3 23 50 F. 33 26 13 " A. m. . 55i 36S, 6 12 57 556 8 5 52 9 4 37 40 8 7 32 344 52 5 45 12 4 54 '2 4 848 2 22 12 F. 5 56 28 S. 4 40 24 57 4F. 5 47 16 S. 5 9 24 6 45 26 4 43 42 4 5i 43 44822 6 14 4 56 1.6 5 8 20 4 56 2 44330 6 28 5 2 15 4 45 15 559 4 53 42 44828 5 3 55^ 2 2 56 F. 5 2 S. 5 5 12 8 ii 40 6 23 45 5 6 20 5 2 48 5 54 24 5328F. 5 17 49 s - 9 20 F. 5 26 37 S. 7 45 52 F. 5 48 52 S. 5 9 37 5 36'* 6 59 52 5 20 8 45348 16 36 4 42 37 4 39 28 8 ii 4 4i 8 9 50 I 32 12 4 22 36 4 42 Si Gloucester Grafton . . Minneapolis Mississippi City.. Mobile Grand Haven GREENWICH Halifax Montauk Point... Monterey Montevideo Montgomery Montreal Hamburg Harrisburg Hartford Montserrat Havana, Morro . . . Havre Moscow Mound City Hawaii or Owyhee Hongkong .... Nantucket Naples Honolulu Nashville Hudson Nassau Huntsville Natchez Indianapolis Indianola Nebraska New Bedford New Haven New London New Orleans NEW YORK . . Jacksonville Jalapa .'.... Jeddo or Tokio. .. Jefferson City.... Jersey City Newbern Jerusalem Kalama Keokuk Newburg Newburyport New Castle New Castle. . . Del. Newport Key West Kingston Can. Kingston Jam. Knoxville .... Norfolk Norwalk La Crosse Norwich La Guayra Ocracoke Lancaster Odessa Lavaca Leaven worth Leghorn Ogdensburg Old Point Comfort Olympia Lexington Omaha Lima . . Oswego Lisbon Ottawa Little Rock . Paducah Liverpool .... Palermo Lockport Panama Los Angeles Louisville Paris Parkersburg Lowell Machias Bay Macon Pekin Pensacola Petersburg Madison Philadelphia Pike's Peak Pittsburg Madrid Malaga Malta Plattsburg Manila Plymouth Maracaibo Plvmouth...Mass. Port Au Prince, St. Domingo Port Townshend. . Portland Portland Or. Porto Praya Porto Rico . . . Marblehead, L Marseilles Martinique Matagorda Matamoras Matanzas Memphis . Mexico Portsmouth DIFFERENCE IN TIME. Differe LOCATION. j New York. Mice in. Greenwich. Time Contin LOCATION. ued. New York. Greenwich. Prairie du Chien . Princeton .... h. m. 9. i 8 33 S. 238 10 24 F. 15 16 ii 13 442 46 18 31 S. 13 43 2 3 26 F. 15 228. 19 34 F. 5 45 50 5 1358 7468. 3 9 49 12 26 F. 2 31 34 S. 148 17 i 37 55 3 i 2 2 52 37 3 13 32 3 13 So 3 ii 33 344 ?F. 3 2 49 S. 3 9 4 6 3 12 4 3 So 58 F. 2 8 48. 28 20 22 F. 432 10 2 533S. i 18 58 II 36 2F. 4 2 50 S. II 51 22 F. 16 38. 6 44 30 F. 4 50 2 i 2368. 539 F - 29 13 S. 37I9F. 4 33 2 7 268. 3 i 4 8F. 2 22 AT S. A. m. . 6 4 34 S. 4 58 40 4 45 37 4 4 45 4 44 49 33 16 5 H 32 5 9 44 2 52 36 5 ii 24 4 36 27 49 48 F. 17 56 5 348S. 8 55i 4 43 36 7 27 35 6 44 i9 633 57 7 57 4 7 47 39 8 933 8 95i 8 7 35 5 30 49 4 56 i 7 58 50 8 548 886 i 5 4 745 5 24 22 4 55 40 23 52 7 134 6 15 6 40 F. 53 12 S. 6 55 20 F. 5 12 58- i 48 28 F. 6 S. 5 5837 4 5 24 5 25 15 4 18 43 5 328 4 24 14 7 l8 A1 St Louis A. m. t. t 4478. 4048 30 10 ii 24 18 6 57 18 F. 36 20 20 I 5 S. 6 8 26 F. 8 26 7 6 18 2 15 S. 8 2 F. 5 i 34 8358. 14 44 2 F. 4 2 22 S. 34 59 1 35 25 ii 38 F. 21 32 S. 5 17 30 F. 328. 5 48 36 F. S^ II 22 54 4 6 S. 4 50 9 18 5 53 46 F. i 28 33 S. i 7 34 I 32 2 6 i 34 F. 53 38 S. 23210 6 2O II -r. 12 loS. 14 F. 26 46 S. 6 ii 15 45 8 4 9F. i 33588. i 5 18 14 14 42 F. H 14 43 10 38 S. IO 14. A. m. t. 6 488. 5 36 50 5 26 12 6 20 20 2 i 16 F. 4 19 4i S. 5 16 17 i 12 25 F. 4 37 36 S. 2 10 16 F. 7 ii 28. 448 10 5 32 F. 5 4 37 S. 9 48 F. 538243. 5 3i ' 6 31 27 4 44 24 5 17 33 21 28 F. 4 59 38. 52 44 F. 4 52 98. 40 24 F. 4 44 40 S. 5 5048 5 52 4 46 44 556 8 57 44 F. 624348. 6 336 6 28 4 i 5 32 F. 5 49 40 S. 7 28 12 i 24 9 F. 5 8 128. 45548 5 22 48 5 2 12 5 " 47 4 47 13 6 30 6 I 20 9 18 40 F. 9 18 41 5 6408. s 6 16 St. Mark's Providence Province town .... Quebec . . St. Mary's St. Paul St. Petersburg St. Thomas, Fort.. St aunt >n Queenstown,L.... Raleigh Richmond Stockholm Rio de Janeiro... Rochester Stonington Suez Rockland Sweetwater River, Mouth of. Rome Rotterdam Sydney N.S Sackett's Harbor. Sacramento Sydney... N.S.W. Syracuse Salem Tahiti or Otaheite. Tallahassee Tampa Bay Tampico Bar Taunton . Salt Lake City... Saltillo San Antonio San Buenaventura San Diego Toronto San Francisco, c. s. s. San Francisco, P. San Jose Toulon Trenton Tripoli Troy Sandusky Tunis Sandy Hook Turk's Island Tuscaloosa Santa Barbara. . . . Santa Clara Utica Santa Cruz Valparaiso Santa Cruz. Ten' fe Santa Fe* Vandalia Venice Savannah Schenectady Seville Vicksburg Victoria Tex. Vienna Sherman Shreveport Vincennes Siam Virginia City Warsaw WASHINGTON, Obs.. West Point Sierra Leone Singapore Smithville Smyrna Southampton Springfield 111. Springfield. .Mass. St. Augustine St. Croix,0bs St Helena Wilmington.. Del. Wilmington. .N.C. Worcester Yank ton Yazoo Yeddo St. Jago de Cuba. . St. John St. Joseoh..L.Cal Yokohama York Yorktown. . . To Compute Difference of Time "between !N~ew Y'orlc and Greenwich, and. any Location not given in Table. RULE. Reduce longitude of location to time, and if it is W. of as- sumed meridian it is Sloio ; if E., it is Fast. If difference for New York is required, and it exceeds 4 h. 56 m. 2 sec., subtract this time, and remainder will give difference of time, S. ; and if it (4 h. 56 m. 2 sec.) does not exceed it, iubtract difference from it, and remainder will give difference of time, F. TIDES. TIDES. Tide-Table for Coast of United States, Showing Time of High-water at Full and New Moon, termed Establish- ment of the Port, being Mean Interval between Time of Moon' 1 s Transit and Time of High-water. (U. S. Coast and Geodetic Survey.) LOCATIONS AND TIME. | I LOCATIONS AND TIME. si a 1 1 COAST FROM BASTPORT TO NEW YORK. Eastport Me. A. TO. 3 25 30 23 22 13 3 12 27 2 24 2 l6 s;i 7 So 7 57 8 13 3 3 7 45 7 32 8 20 8 25 9 7 9 38 9 28 16 ii 7 13 22 20 9 35 7 32 S3 8 19 9 34 8 833 9 4 " 53 i3 44 o r se am Feet. 15 25 9.9 9-9 10.6 6 10.9 10.3 3-6 2-5 1.8 2.8 4-7 4.6 5-4 i 4.6 3-7 2.4 3-i 5 3-2 2.9 8 9.2 8.6 r r 5 tf L 6.8 fall of Feet. 7.6 U 7.6 8.1 8-5 2.6 1.6 13 r.8 I 2.8 3-4 3-i i.' 8 2.4 2.2 2-3 2.1 5-2 4-7 6.6 6.1 3.6 4 4-3 3 3-9 11 tide a CHESAPEAKE BAY AND RIVERS. OldPt. Comfort..Va. Cape Henry* u Point Lookout. . . Md. Annapolis " h. m. 8 17 12 58 'A 4 s 18 59 H 37 16 58 7 4 9 \ 7 26 7 19 7 26 IT 3 8 21 8 84 9 22 II 21 13 *5 9 38 9 39 9 25 o 8 22 o 37 2 6 3 40 4 10 2 36 I 17 2 2 2 4 2 2 33 3 49 2 230 p. 85. Feet. 3 6 1.9 i i 3 i-5 3 3-4 2.2 5 3-3 5-5 6 8 7.6 4-9 1.8 1.6 1.8 3-2 5 4-7 4-3 4-4 4-3 5-2 5-i 7-3 4-7 5-5 7-4 7-4 5-5 60 12 30 1 2 7-5 1.5 1.6 Feet 2 :1 .8 9 2-5 2-3 1.8 2.2 3-S 4.1 5-S 1.2 I I 1.6 2-3 2,2 2.8 2.J 2. == 9.1 2.8 4- ] 3-7 2.7 . t Campo Bello*.... " Portland u Cape Ann* " Portsmouth N.H. Newburyport. . .Mass. Salem . " Bodkin Light. ... " Baltimore . . " James R. (City Pt. ), Va Richmond u Cape Cod* ' ' Boston Light ... " Bostonf " COASTS OF N. AND S. CAROLINA, GEORGIA, AND FLORIDA. Hatteras Inlet .. N.C Cape Hat teras ... " Beaufort " Nantucket " Edgartown " Holmes' s Hole .. u Tarpaulin Cove . " Wood's Hole, n. side. N. Bedford (Dump-) ing Rock) J New York* N.Y. Albany* " Smithv'le(C.Fear) " Charlestonll (C. H. ) Wharf S C. j FortPulaski Ga, Savannah " LONG ISLAND SOUND. Newport R I St. Augustine Fla. Cape Florida " Key West " Point Judith " Montauk Point. . N.Y. Watch Hill . . R I Tampa Bay " Cedar Keys " Providence* " Stonington Ct WESTERN COAST. San Diego Cal. San Pedro " Little Gull Isl'd. N.Y. New London Ct New Haven ... " Cuyler's Harbor . " San Luis Obispo.. " Monterey " South Farallone . " San Francisco. . . u Mare Island " Benicia . . . " Bridgeport " Oyster Bay N.Y. Sand's Point.... " New Rochelle. . . " Throg's Neck... " Hell Gate* ... " COAST OF NEW JERSEY. Cold Spring Inlet, N.J. Sandy Hook N.J. Amboy " Ravenswood " Bodega " Humboldt Bay. . . " Astoria Or Nee- ah Harbor, Wash. Port Townshend " MISCELLANEOUS. Bay of Fundy*..N.S. Blue Hill Bay*.. " St. John's* " Kingston* Jam. Halifax* N. S. Cape May Landing " Egg Harbor* u DELAWARE BAY AND RIVER. Delaware Breakwater Higbee's (Cape May). . Egg Isl'd Light.. N.J. New Castle Del. Philadelphia. . . .Penn. * Refers t Pensacola* Fla Galveston* Tex. one. t t II see in. (half a mean lunar day) for som ports in Del- aware River and Chesapeake Bay. to give succession of times from the mouth ; hence 12 A. 26 min. is. to be ubtrcUd from the Establishments which are greater than that, to give the interval required, TIDES. Bench Maries referred, to in preceding Table. t BOSTON. Top of wall or quay, at entrance to dry- dock in Charlestown navy- j-ard, 14.76 feet above mean low- water. i NEW YORK. Lower edge of a straight line, cut in a stone wall, at head of wooden wharf on Governor's Island, 14.56 feet above mean low- water. OLD POINT COMFORT, Va. A line cut in wall of light-house, one foot from ground, on southwest side, n feet above mean low- water. II CHARLESTON, S. C. Outer and lower edge of embrasure of gun No. 3, at Castle Pinckney, 10.13 feet above mean low- water. Establishment of the Fort for several Locations in Europe, etc. PORT. TIME. PORT. TIME. PORT. TIME. Am sterdaw A. m. Chatham h. TO. ii 16 Antwerp Beachy Head Eng 4 25 Cherbourg Clear Cape 7 49 London Bridge Newcastle 2 7 I 22 Belfast 10 Cowes 10 46 Portsmouth D.-yard, Bordeaux .... 6 1O Dover Pier. II 12 Eng. II 41 Bremen 6 Dublin Bar Quebec 8 Brest Harbor . Funchal ... II 3O Ramsgate Pier 10 27 Bristol Bristol Quay 7 21 6 27 Gravesend Eng. Greenock I 14 g Rye Bay Eng. Sheerness II 20 57 Cadiz Holyhead 8 15 Calais ii 40 Hull Eng. 6 20 Southampton. .Eng. ii 40 Calf of Man Land's End Thames R mo'th " 12 Caoe St. Vincent... 2 ^O Lisbon.. . 2 10 Woolwich ... . . " 2 IS Rise and Fall of Tides in Q-ulf of Mexico. 1 1 1 LOCATIONS. 1 I m i" Feet. Feet, i 8 Fee Isle Derniere La. Feet. i 4 Feet. 1.2 Feet. .7 I i-5 4 Entrance to Lake Cal-) casieu La \ i-5 !* .6 , Aransas Pass " 1. 1 T 8 6 i.i 1.4 .5 Brazos Santiago " 9 1.2 5 St. George's Island.. . .Fla. Fort Morgan ( Mobile ) Bay) Ala. j Cat Island Miss. Southwest Pass La. Tides of Q-nlf of Mexico. On Coast of Florida, from Cape Florida to St. George's Island, near Cape San Bias, the tides are of the ordinary kind, but with a large daily inequality. From St. George's Island, Apalachicola entrance, to Derniere Isle, the tides are usually of the single-day class, ebbing and flowing but once in 24 (lunar) hours. At Calcasieu en- trance, double tides reappear, and except for some days about the period of Moon's greatest declination, tides are double at Galveston, Texas. At Aransas and Brazos Santiago the single-day tides are as perfectly well marked as at St. George's, Pensa- cola, Fort Morgan, Cat Island, and the mouths of the Mississippi. For some 3, to 5 days, however, about the time when the Moon's declination is nothing, there are generally two tides at all these places in 24 hours, the rise and fall being quite small. Highest high and lowest low waters occur when greatest declination of Moon happens at full or change. Least tides when Moon's declination is nothing at first or last quarter. Tides of Pacific Coast. On Pacific coast there is, as a general rule, one large and one small tide during each day, heights of two successive high- waters occurring, one A.M., and other P.M. of same 24 hours, and intervals from next preceding transit of Moon are very different. These inequalities depend upon Moon's declination. When Moon's de- clination is nothing, they disappear, and when it is greatest, either North or South, they are greatest. The inequalities for low water are not same as for high, though they disappear, and have greatest value at nearly same time. When Moon's declination is North, highest of two high tides of the 24 hours oc- curs at San Francisco, about 11.5 hours after Moon's southing (transit); and when declination is South, lowest of the two high tides occurs about this interval. Lowest of two low- waters of the day is the one which follows next highest high- water. H 86 STEAMING DISTANCES. STEAMING DISTANCES. Distances "between various Ports of "United. States and. Canada. By Lake, River, arid Canal. LOCATIONS. Lake and River. Canal. Total. LOCATIONS. Lake and River. Canal. Total. Duluth to Buffalo... Chicago to Buffalo . . Miles. 1024 Q2S Miles. I Miles. 1025 925 Chicago to New York, via Oswego Miles. IJ 95 Miles. 232 Miles. 1427 Duluth to Oswego. . . Chicago to Oswego.. Duluth to New York, via, Buffalo via Oswego Duluth to Montreal 133 034 166 294 280 2 7 26 353 233 72 1160 1060 1519 1527 1361 Chicago to Montreal. Buffalo to Colborne, via Welland Canal. Buffalo to New York. Welland Canal to Montreal 1190 142 304. 5 7i 26.77 352 70.5 1261 26.77 494 aye Chicago to New York, via Buffalo . 1067 352 1419 Montreal to Kingston Ottawa to Kingston . 126.25 1 20 126.25 246.25 126.25 Distances between varioias IPorts and ISTew York and London. Not included in preceding Table. PORTS. Miles. Miles. Alexandria. . . Amsterdam . . Barbadoes . . . Batavia Bermudas . . . Bombay Boston Bremen Bristol Buenos Ayres 6oio Cadiz Calcutta N.Y. 4893 3291 1855 8972 682 8522 356 3428 2979 3125 9350 Lond. 3 102 262 3812 11492 3M2 10703 3030 408 50i 6280 i "5 Cape Race Cowes Funchal Galway Gibraltar Glasgow Halifax Havana Hobart Town . . Kingston, Jam. Lima Madras Miles. Miles. N.Y. 1 004 3092 2 760 2720 3260 2913 590 1 161 9187 M56 10050 8707 Lond. 2249 200 1303 721 1325 765 2 706 41 113 4305 PORTS. New Orleans Norfolk Pensacola . . . Philadelphia. Quebec Queenstown . Rio Janeiro. . St. Johns Southampton Swan River. . Tortugas Washington . N.Y. 1790 308 1623 262 1360 2780 4970 1064 3103 8480 U5I 461 Lond. 4730 3447 4654 3404 3080 55i 5200 2214 211 I066I 4182 Distances between various Ports of England, Canada, TJnited States, etc. Not included in preceding Table. PORTS. Miles. PORTS. Miles. PORTS. Miles. Halifax to Liverpool 2 563 Liverpool to Havana . Panama to San Diego St Thomas I 563 Portland * Monterey St Johns N. F . . 520 Baltimore .... San Francisco Quebec to Glasgow . Liverpool to Boston 2563 2 955 N Orleans to Havana Cape Race to Fastnet 570 1711 San Francisco to San Juan del Sud . Acapulco 2685 184.1 Quebec 2 855 Halifax Manzanilla Philadelphia 3 *47 Boston goc San Diego J 543 Callao II 37Q St Johns N F to Monterey Fastnet 283 Quebec 80 1 Humboldt Cape Race . . . I QQ2 Boston 890 Columbia R Bar Aspinwall 4 650 Greenock 1848 Vancouver . 638 Port Said 3 290 Bermudas to Nassau. 804 Portland Melbourne 13 290 Panama to Port Townshend r Rio Janeiro 5 I2 5 San Juan del Sud . 57 Victoria San Francisco. . . 13800 Gulf of Fonseca. . . 739 Yokohama . via Panama. . . . tnaTehuantepec 7378 6400 Acapulco Manzanilla 1416 1724 Honolulu Honolulu to Callao.. 2080 5HS STEAMING DISTANCES. P* 3 ~ CJ rt ^ ^ E3 fe. c5 "SS^pce^s 3 | U 02^oSSfW 88 STEAMING DISTANCES. e Q. M m o vo vo ro ON vo V mmvo (^ i^ ON o> t^ oo t^< 00 VO m n en co M tx tx ^ o O M tx o\ * * t> m vo vo in vo m oo H ON ro ON M 5- "8. * I - : < ^ O t it 45 e w< 3 ^2^ K i-lsl ii fjlfiliii o n M * os 8 vg g 2 2- O VO VO t^ ^ H N d | g 1 s s ' VO N N s i S- 5- 5 SCJ VO CO m m S " , a|||aUSI|ll|p i FT o 3 I : : : : : : " : : : : i g ' ft | *0 ' -1 .! ' fl s i c VO w ON in Q i? 2- vf 3; S J-. bN TT l>x Tj- * OO ON * tx O"*- O\vo w m moovo N vb O COM o wvo o\ H H ft *J|l| | lll.r' 5 -- J - 111 0) o fl B J R J 1 .2 t^ fO N - ! 6 1 ^ *ag|'SSS2 i^o,'^7^J^= 1111 Q ------^s ^o >- : : : S-Jfl W I > ' ^*S^ 5 ^ ^ oo 2 ??l Illflf I O Ps | ; O- I *<2 *& fl .- S* S * =5 -5 to

|| Jlajl i x t x i 4 by cancellin 9 2 ' s and 3 ' s - To Reduce Fractions of different Denominations to Equivalents having a Common Denominator. RULE. Multiply each numerator by all denominators except its own for new nu- merators; and multiply all denominators together for a common denominator. NOTE. In this, as in all other operations, whole numbers, mixed or compound fractions, must first be reduced to form of simple fractious. 2. When many of denominators are same, or are multiples of each other, ascertain their least common multiple, and then multiply the terms of each fraction by quo- tient of least common multiple divided by its denominator. EXAMPLE. Reduce ^, J, and J to a 1X3X4 = 12) common denominator. 2X2X4 = l6 ( 27 = ^.r = ^T> 3X2X3^,8) 6 8 d ^ 2X3X4 = 24 Addition.. RULE. If fractions have a common denominator, add all numerators together, and place sum over denominator. NOTE. If fractions have not a common denominator, they must be reduced to one. Also, compound and complex must be reduced to simple fractions. EXAMPLE i. Add ^ and J together. ^ + f = f = ! 2. Add i of | of T % to 2 J of f . ixfxA^if. 2tor| = vx* = Tben, J J + f | = f f | o + ^^ = j^ reduced to equivalent fractions having a common denominator and thence to its lowest terms. FRACTIONS. 91 Subtraction. RULB. Prepare fractions same as for other operations, when necessary; then subtract one numerator from the other, and set remainder over common denom- inator. EXAMPLE. -What is difference 6 *i = 54 ) 64 84 so 15 B between* and f? 3X8 = 24 =ff-?t = ff = if = ff- 8X9 = 72) Multiplication. RULE. Prepare fractions as previously required; multiply all numerators to- gether for a new numerator, and all denominators together for a new denominator. EXAMPLE i. What is product of f and f? f x f = ^ = J. 2. What is product of 6 and f of 5? * X f of 5 = * x ^ = = 2a Division. RULE. Prepare fractions as before; then divide numerator by the numerator, and denominator by the denominator, if they will exactly divide; but if not, invert the terms of divisor, and multiply dividend by it, as in multiplication. EXAMPLE i. Divide *-f by f . 2 5 -r- f = f = if. 2. -Dividef by ^. f-& = fx^ .Application of Reduction of Fractions. To Compute Value of a Fraction in IParts of a "Whole Nvimloer. RULE. Multiply whole number by numerator, and divide by denominator; then, if anything remains, multiply it by the parts in next inferior denomination, and divide by denominator, as before, and so on as far as necessary; so shall the quo- tients placed in order be value of fraction required. EXAMPLE i. What is value of J of f of 9? a. Reduce 4 of a pound to an avoirdupois ounce. 4) 3 ( &* j.6 ounces in a Ib. 4) 48 (12 ounces. To Reduce a Fraction from one Denomination to another. RULE. Multiply number of required denomination contained in given denomina- tion by numerator if reduction is to be to a less name, but by denominator if to a greater. EXAMPLE i. Reduce of a dollar to fraction of a cent JXIOO = i^ = ^. 2. Reduce ^ of an avoirdupois pound to fraction of an ounce. iXi6 = = ! = 2|- 3. Reduce f of % of a mile to the fraction of a foot 2of*=&Xsrto=*lp- For Rule of Three in Vulgar Fractions, see Decimals, page 94. 92 DECIMALS. DECIMALS. A DECIMAL is a fraction, having for its denominator a UNIT with as many ciphers annexed as the numerator has places ; it is usually ex- pressed by writing the numerator only, with a point at the left of it. Thus, T% is -4; TO is - 8 5; AVoV is -ooys ; and T^Vrnnr is -00125. When there is a deficiency of figures in the numerator, prefix ciphers to make up as many places as there are ciphers in denominator. Mixed numbers consist of a whole number and a fraction; as, 3.25, which is the same as 3 T 8 ^, or M. Ciphers on right hand make no alteration in their value; for .4, .40, .400 are deci- mals of same value, each being T ^, or J. .A-clclition. RULE. Set numbers under each other according to value of their places, as in whole numbers, in which position the decimal points will stand directly under each other; then begin at right hand, add up all the columns of numbers as in integers, and place the point directly below all the other points. EXAMPLE. Add together 25. 125 and 293.7325. 25. 125 293-7325 318.8575 sum. Subtraction. RULE. Set numbers under each other as in addition; then subtract as in whole numbers, and point off decimals as iu last rule. EXAMPLE. Subtract 15. 15 from 89. 1759. 89. 1759 74.0259 remainder. Multiplication. RULE. Set the factors, and multiply them together same as if they were whole numbers; then point off in product just as many places of decimals as there are decimals in both factors. But if there are not so many figures in product, supply deficiency by prefixing ciphers. EXAMPLE. Multiply 1.56 by .75. 1.56 75 780 1092 1. 1700 product. By Contraction. To Contract the Operation so as to retain only as many Decimal places in Irrod.vict as may "be required. RULE. Set unit's place of multiplier under figure of multiplicand, the place of which is same as is to be retained for the last in product, and dispose of the rest of figures in contrary order to which they are usually placed. In multiplying, reject all figures that are more to right hand than each multiply- ing figure, and set down the products, so that their right-hand figures may fall in a column directly below each other, and increase first figure in every line with what would have arisen from figures omitted; thus, add i for every result from 5 to 14, 2 from 15 to 24, 3 from 25 to 34, 4 from 35 to 44, etc., and the sum - 6 of all the lines will be the product as required. 8 J 44 go-}- 2 for 18 EXAMPLE. Multiply 13.57493 by 46. 2051, and retain only four places of decimals in the product. 627.23 ii NOTE. When exact result is required, increase last figure with what would have arisen from all the figures omitted. DECIMALS. 93 Division. RULB. Divide as in whole numbers, and point off in quotient as many places for decimals as decimal places in dividend exceed those in divisor; but if there are not so many places, supply deficiency by prefixing ciphers. EXAMPLE. Divide 53 by 6.75. 6.75) 53.00000 (=7.851+. Here 5 ciphers are annexed to dividend to extend division. By Contraction. RULE. Take only as many figures of divisor as will be equal to number of figures, both integers and decimals, to be in quotient, and ascertain how many times they may be contained in first figures of dividend, as usual. Let each remainder be a new dividend; and for every such dividend leave out one figure more on right-hand side of divisor, carrying for figures cut off as in Con- traction of Multiplication. NOTE. When there are not so many figures in divisor as there are required to be in quotient, con- tinue first operation until number of figures in divisor are equal to those remaining to be found in quo- tient, after which begin the contraction. EXAMPLE. Divide 2508.92806 92.410315)2508.928106(27.1498 13.849 912 by 92.41035, so as to have only 1848207-}-! 9*41 832 + 4 four places of decimals in quo- 660721 ~T6o8 ~8o~ 646872 + 2 3696 J4 + 2 13 849 912 6 Reduction, of Decimals. To Reduce a Vulgar Fraction to its Equivalent Decimal. RULE. Divide numerator by denominator, annexing ciphers to numerator to ex- tent that may be necessary. EXAMPLE. Reduce to a decimal. 5) 4.0 ~8 To Compute Value of a Decimal in Terms of an Inferior Denomination. RULE. Multiply decimal by number of parts in next lower denomination, and cut off as many places for a remainder, to right hand, as there are places in given decimal. Multiply that remainder by the parts in next lower denomination, again cutting off for a remainder, and so on through all the parts of integer. EXAMPLE i. What is value of .875 dollars? .875 100 Cents, 87.500 Mills, 5.000 = 87 cent* 5 milk. 2 . What is volume of .140 cube feet in inches? .140 1728 cube inches in a cube foot. 2 4 1. 920 cube ins. 3. What is value of .00129 of a foot? .01548 ins. To Reduce a Decimal to an Equivalent Decimal of a Higher Denomination. RULE. Divide by number of parts in next higher denomination, continuing op* eration as far as required. EXAMPLE i. Reduce i inch to decimal of a foot. i2|i.ooooo I o8 33 3+/oot .Reduce 14" 12'" to decimal of a minute. 14" 12'" .236 66'+ minute. 94 DECIMALS. DUODECIMALS. MEAN PROPORTION. When there are several numbers, to be reduced all to decimal of highest. RULE. Reduce them all to lowest denomination, and proceed as for one denomi- nation. Feet. Ins. Be. EXAMPLE. Reduce 5 feet 10 inches and 3 5 10 3 barleycorns to decimal of a yard. I2 70 3 5-9166 1.9722-}- ycuds. Rule of Three. RULE. Prepare the terms by reducing vulgar fractions to decimals, compound numbers to decimals of the highest denomination, first and third terms to same denomination, then proceed as in whole numbers. EXAMPLE. If .5 of a ton of iron cost .75 of a dollar, .5 : .75 :: .625 what will .625 of a ton cost? .625 .5) -468 75 .9375, dollar. DUODECIMALS. In Duodecimals, or Cross Multiplication, the dimensions are taken in feet, inches, and twelfths of an inch. RULE. Set dimensions to be multiplied together one under the other, feet under feet, inches under inches, etc. Multiply each term of multiplicand, beginning at lowest, by feet in multiplier, and set result of each immediately under its corresponding term, carrying i for every 12 from one term to the other. In like manner, multiply all multiplicand by inches of multiplier, and then by twelfth parts, setting result of each term one place farther to right hand for every multiplier. And sum of products will give result. EXAMPLE. How many square inches are Feet. Ins. Twelfths. there in a board 35 feet 4.5 inches long and 12 35 4 6 feet 3^ inches wide? I2 3 4 424 6 o 8 10 i 6 ii 960 434 3 ii o o "Value of Duodecimals in Sq.uare Feet and Inches. Sq. Ft. Sq. Ins. y* of i twelfth = yy 1 ^ or .083 333, etc. " .006 944, etc. Sq. Ft. Sq. 1 i Foot = i or 144. i Inch i Twelfth ILLUSTRATION. What number of square inches are there in a floor 100 feet 6 inches long and 25 feet 6 inches and 6 twelfths broad? 2566 feet ii ins. 3 twelfths = 2566 feet 135 ins. MEAN PROPORTION. MEAN PROPORTION is proportion to two given numbers or terms. RULE. Multiply two numbers or terms together, and extract square root of their product. EXAMPLE. What is mean proportionate velocity to 16 and 81 ? 1 6 X 8 1 = 1296, and -^1296 36 mean velocity. RULE OF THREE. COMPOUND PROPORTION. 95 RULE OF THREE. RULE OF THREE. It is so termed because three terms or numbers are given to ascertain a fourth. It is either DIRECT or INVERSE. It is Direct when more requires more, or less requires less ; thus, if 3 bar- rels of flour cost $18, what will 10 barrels cost? In this case Proportion is Direct, and stating must be, As 3 : 10 : : 18 60. It is Inverse when more requires less, or less requires more; thus, if 6 men build a certain quantity of wall in 10 days, in how many days will 8 men build like quan- tity? Or, if 3 men dig 100 feet of trench in 7 days, in how many days will 2 men perform same work ? Here the Proportion is Inverse, and stating must be, As 8 : 6 : : 10 : 7. 5, and 2 : 3:17: 10. 5. The fourth term is always ascertained by multiplying 2d and 3d terms together, and dividing their product by ist term. Of the three given numbers necessary for the stating, two of them contain the supposition, and the third a demand. RULE. State question by setting down in a straight line the three necessary numbers in following manner : Let third term be that of supposition, of same denomination as the result, or 4th term is to be, making demanding number 2d term, and the other number ist term when question is in Direct Proportion, but contrariwise if in Inverse Proportion; that is, let demanding number be ist term. Multiply 2d and 3d terms together, and divide by ist, and product will give re- sult, or 4th term sought, of same denomination as 2d term. NOTE. If first and third terms are of different denominations, reduce them to same. If, after divis- ion, there is any remainder, reduce it to next lower denomination, divide by divisor as before, and quotient will be of this last denomination. Sometimes two or more statings are necessary, which may always be known by nature of question. EXAMPLE i. If 20 tons of iron cost $225, what will Tons. Tons. Dolls. 500 tons cost? 20 : 500 :: 225 500 2|0) II 250|0 '' v 5625 dollars. 2. A wall that is to be built to height of 36 feet, was raised 9 feet by 16 men in 6 days; how many men could finish it in 4 days at same rate of working? Days. Days. Men. Men. 4 : 6 '.'. 16 : 24 Then, if 9 feet requires 24 men, what will 27 feet require? 9 : 27 : : 24 : 72 men. COMPOUND PROPORTION. COMPOUND PROPORTION is rule by means of which such questions as would require two or more statings in simple proportion (Rule of Three) can be resolved in one. As rule, however, is but little used, and not easily acquired, it is deemed prefer- able to omit it here, and to show the operation by two or more statings in Simple Proportion. ILLUSTRATION i. How many men can dig a trench 135 feet long in 8 days, when 16 men can dig 54 feet in 6 days? Feet, jrtei. Men. Men. First As 54 : 135 : : 16 : 40 Days. Days. Men. Men. Second As 8 : 6 :: 40 : 30 gO COMPOUND PROPORTION. INVOLUTION. EVOLUTION. 2. If a man travel 130 miles in 3 days of 12 hours each, how many days of 10 hours each would he require to travel 360 miles? Miles. Miles. Days. Days. First As 130 : 360 :: 3 : 8.307+ Hours. Hours. Days. Days. Second As 10 : 12 :: 8.307 : 9.9684 3. If 12 men in 15 days of 12 hours build a wall 30 feet long, 6 wide, and 3 deep, in how many days of 8 hours will 60 men build a wall 300 feet long, 8 wide, and 6 deep? 120 days. By Cancellation. RULE. On right of a vertical line put the number of same denomination as that of required answer. Examine each simple proportion separately, and if its terms demand a greater result than $d term, put larger number on right and lesser on left of line; but if its terras demand a less result than $d term, put smaller number on right and larger on left of line. Then Cancel the numbers divisible by a common divisor, and evolve the 4th term or result required. Take Illustration i, page 95 : 3d term, or term of supposition of same denomination as required result, 16 men. Statement. 135 feet require more men than 54 feet. Result by Cancellation. 54 8 16 and 8 days less men than 6 days. 135 6 2 X 5 X 3 = 30 men. 2 m 5 3 ILLUSTRATION 3. 3d term, 15 days. Statement. 60 men require less days than 12 men, Result by Cancellation. 15 8 hours more days than 12 hours, 300 feet 60 8 30 6 3 1 2 more days than 30 feet, 8 feet more days than 6 feet, and 6 feet more days than 300 8 3 feet. 6 3X4X10 = 120 days. 3 4 10 INVOLUTION. INVOLUTION is multiplying any number into itself a certain number of times. Products obtained are termed Powers. The number is termed the Root, or first power. When a number is multiplied bv itself once, product is square of that number ; twice, cube ; three times, biquadrate ; etc. Thus, of the number 5. 5 is the Root, or ist power. 5 X 5 = 25 " Square, or 2d power, and is expressed 5*. 5 x 5 X 5 = 125 " Cube, or 3d power, and is expressed 53. 5X5X5X5 = 625 " Biquadrate, or 4th power, and is expressed 54. The lesser figure set superior to number denotes the power, and is termed the Index or Exponent. ILLUSTRATION i. What is cube of 9 ? 729. 2. What is cube of j ? ff . 3. What is 4th power of i. 5 ? 5.0625. EVOLUTION. EVOLUTION is ascertaining Root of any number. Sign ^J placed before any number indicates that square root of that number it re- quired or shown. Same character expresses any other root by placing the index above it. 7 = 3, and ^64 = 4. Roots which only approximate are termed Surd Roots. EVOLUTION. 97 To Extract Square Root. BULB. Point off given number from units' place, into periods of two figures each. Ascertain greatest square in left-hand period, and place its root in quotient; sub- tract square number from this period, and to remainder bring down next period for a dividend. Double this root for a divisor; ascertain how many times it is contained in divi- dend, exclusive of right-hand figure, which, when multiplied by number to be put to right hand of this divisor, product will be equal to, or next less than dividend; place result in quotient, and also at right hand of divisor. Multiply divisor by last quotient figure, and subtract product from dividend; bring down next period, and proceed as before. NOTE. Mixed decimals must be pointed off both ways from units. EXAMPLE i. What is square root of 2? 2.000000 (MI+. 2 - What is S( l uare root of '44' 144 (12 24 ioo 4l 9 6 22(044 981! 400 I 44 i 281 Square Roots of Fractions. RULE. Reduce fractions to their lowest terms, and that fraction to a decimal, and proceed as in whole numbers and decimals. NOTE. When terms of fractions are squares, take root of each and set one above the other ; aa . is square root of |~|. EXAMPLE. What is square root of ^ ? . 866 025 4. To Compute 4th. or 8th. Root of a Number, etc. RULE. For the 4th root extract square root twice, and for 8th root thrice, etc. To Extract Cube Root. RULE. From table of roots (page 272) take nearest cube to given number, and term it the assumed cube. Then, as given number added to twice assumed cube, is to assumed cube added to twice given number, so is root of assumed cube to required root, nearly ; and by using in like manner the root thus found as an assumed cube, and proceeding in like manner, another root will be found still nearer; and in like manner as far as may be deemed necessary. EXAMPLE. What is cube root of 10517.9? Nearest cube, page 272; 10648, root 22. 10648. 10517.9 21296 2I035-8 1 10517-9 10648. 31813.9 : 3 l68 3- 8 ' 22 : 21.9+- To Ascertain or to Compute the Square or Cube Roots of Roots, \Vhole Numbers, and of Integers and Decimals, see Table of Squares and Cubes, and Rules, pp. 272, 300. To Extract any Root whatever. Let P represent number. I Let A represent assumed power, r ita root n " index of the power. | R " required root of P. Then, as sum of w+i x A and ni x P is to sum of n-f-i X P and n i X A so is assumed root r to required root R. ILLUSTRATION. What is cube root of 1500? Nearest cube, page 272, is 1331, root u. then, w-f-i x A = 5324, n-f-i X P = 6ooo n i X P = 3000, n i X A 2662 8324 8662 :: u : 11.446-!-. 98 EVOLUTION. PROPERTIES OP NUMBERS. POSITION. To Compute the Root of an Even IPower greater than any given in Table of Square and. Cube Roots. RULE. Extract square or cube root of it, which will reduce it to half the given power; then square or cube root of that power; and so on until required root is ob- tained. EXAMPLE i. Suppose a i2th power is given; the square root of that reduces it to a 6th power, and the square root of 6th power to a cube. 2. What is biquadrate, or 4th root, of 2560000? -^/2 560 ooo = 1600, and -1/1600=40. NOTB. For other rules for extraction of roots sae pp. 301-4. PROPERTIES OF NUMBERS. 1. A Prime Number is that which can only be measured (divided without a re- mainder) by i or unity. 2. A Composite Number is that which can be measured by some number greater than unity. 3. A Perfect Number is that which is equal to the sum of all its divisors or ali- quot parts ; as 6 = |-, ^ , |r. 4. If sum of the digits constituting any number be divisible by 3 or 9, the whole Is divisible by them. 5. A square number cannot terminate with an odd number of ciphers. 6. No square number can terminate with two equal digits, except two ciphers or two fours. 7. No number, the last digit of which is 2, 3, 7, or 8, is a square number. IPowers of tlie first Nine Numbers. ISt. 2d. 3d. 4 th. 5 th. 6th. 7 th. 8th. 9th. I I i i i i i i i 2 3 4 8 16 32 64 128 256 512 9 27 81 243 729 2187 6561 19083 4 16 64 256 1024 4096 16384 65536 262144 5 6 25 125 625 3125 15625 78125 390625 1953125 36 216 1296 7776 46656 279936 i 679616 10077 696 7 49 343 2401 16807 117649 823 543 5 764 801 40 353 607 8 9 64 512 4096 32768 262 144 2097152 16777216 -134217728 81 729 6561 59049 531 441 4 782 969 43046721 387420489 POSITION. POSITION is of two kinds, SINGLE and DOUBLE, and it is determined by number of SUPPOSITIONS. Single Position. RULE. Take any number, and proceed with it as if it were the correct one; then, as result is to given sum, so is supposed number to number required. EXAMPLE i. A commander of a vessel, after sending away in boats A, J, and J of his crew, had left 300; what number had he in command? Suppose he had 600. i of 600 is 200 ^ of 600 is 100 of 600 is 150 450 150 : 300 : : 600 : 1200 men. POSITION. FELLOWSHIP. 99 2. A person asked his age, replied, if ^ of my age be multiplied by 2, and that product added to half the years I have lived, the sum will be 75. How old was he ? 37. 5 year*. Double ^Position. RULE. Assume any two numbers, and proceed with each according to conditions of question ; multiply results or errors by contrary supposition ; that is, first posi- tion by last error, and last position by first error. If errors are too great, mark them -f ; and if too little, . Then, if errors are alike, divide difference of products by difference of errors; but if they are unlike, divide sum of the products by sum of errors. EXAMPLE i. A asked B how much his boat cost; he replied, that if it cost him 6 times as much as it did, and $30 more, it would have cost him $300. What was price of the boat? Suppose it cost him. . 60 30 6 times. 6 timet. 360 180 and 30 more and 30 more 390 210 300 300 90-}- 90 30 2d position. 60 ist position. 90 2700 5400 9 54oo 180) 8100 (45 dollars. . 2. What is length of a fish when the head is 9 inches long, tail as long as its head and half its body, and body as long as both head and tail ? 6 feet. FELLOWSHIP. FELLOWSHIP is a method of ascertaining gains or losses of individuals engaged in joint operations. Single Fellowship. RULE. As the whole stock is to the whole gain or loss, so is each share to the gain or loss on that share. EXAMPLE. Two men drew a prize in a lottery of $9500. A paid $3, and B $2 for the ticket; how much is each share? 5 : 9500 1:3: 5700, A's share. 5 : 9500 :: 2 : 3800, B's share. Double Fellowship, Or Fellowship with Time. RULE. Multiply each share by time of its interest ; then, as sum of products is to product of each interest, so is whole gain or loss to each share of gain or loss. EXAMPLE. A cutter's company take a prize of $10000, which is to be divided ac- cording to their rate of pay and time of service on board. The oflScers have been on board 6 months, and the crew 3 months; pay of lieutenants is $100, ensigns $50, and crew $10 per month; and there are 2 lieutenants, 4 ensigns, and 50 men; what is each one's share ? 2 lieutenants $100 = 200 x 6 = 1200 4 ensigns 50 = 200 X 6 = 1200 50 men 10 = 500 x 3 = 1500 3900 Lieutenants 3900 : 1200 : : 10000 : 3076.92 -f- 2 = 1538.46 dolls. Ensigns 3900 : 1200 :: 10000 : 3076.92-:- 4= 769.23 ' Men 3900 : 1500 :: 10 ooo : 3846.16-7-50= 76.92 " 100 PERMUTATION. PERMUTATION. PERMUTATION is a rule for ascertaining how many difl arent ways auy given number of numbers of things may be varied in their position. Permutation of the three letters abc, taken all together, are 6 ; taken twe and two, are 6 ; and taken singly, are 3. RULE. Multiply all the terms continually together, and last product will give result. EXAMPLE i. How many variations will the nine digits admit of? 1X2X3X4X5X6X7X8X9 = 362 880. 2. How many years would there be required to elapse before 10 persons could be seated in a varied position collectively, each day at dinner, including one day in every 4 years for a leap year? 9935 years, 42 days. When only part of the Numbers or Elements are taken at once. RULE. Take a series of numoers, 'beginning with number of things given, decreasing by i, until number of terms equals number of things or quantities to be taken at a time, and product of all the terms will give sum required. EXAMPLE i. How many changes can be made with 2 events in 5? 5 1 = 4, and 4X5 = 2 terms. Hence, 5 x 4 = 20 changes. 2. How many changes of 2 will 3 playing cards admit of? 31 = 2, and 2X3 = 2 terms. Hence, 2X3 = 6 changes. 3. How many changes can be rung with 4 bells (taken 4 and 4 together) out of 6 ? 4 1 = 3, and 3X4X5X6 = 4 terms or changes. [ence, 3X4X5X6 = 360 changes. When several of the Elements are alike. RULE. Ascertain the permutations of all the numbers or things, and of all that can be made of each separate kind or division; divide number of permutations of whole by product of the several partial permutations, and quotient will give number of permutations. EXAMPLE. How many permutations can be made out of the letters of the word persevere (9 letters, having 4 e's and 2 r's)? i X2X 3X4X 5X6X7X8X9 = 362880; i X 2 X 3 X 4 = 24 for the e's ; 1X2 = 2 for the r's, and 24 X 2 = 48. Hence, 362 880 -4- 48 = 7560. Or, Add logarithms of all the terms together, and number for the sum will give result. EXAMPLE i. How many permutations can be made with three letters or figures? Log. i = .oo 2 = .3oio3 3 = . 4771213 .7781513 = log. of number 6. a. How many variations will 15 numbers in 16 places admit of? Add logarithms of numbers i to 16 and take logarithm of their sum viz. , 13. 320 661 97 = 20 922 789 888 ooo. Number of positions of the blocks in the " 15 puzzle " is as above for their 16 permutations. IPermutatioxis, Whereby any questions of Permutation may be solved by Inspection, number of terms not exceeding 20. I 5 120 2 6 720 6 7 5040 24 8 40320 362880 3628800 39916800 479001600 6227020800 87178291200 1307674368000 20922789888000 355687428096000 6402373705728000 121645100408832000 2432902008176640000 ARITHMETICAL PROGRESSION. IOI ARITHMETICAL PROGRESSION. ARITHMETICAL PROGRESSION io a series of numbers increasing or de- creasing by a constant number ov difference ; 'Is, i,<$, 3, 7/1.2, 9, 0, 3. The numbers which form the series are designated Terms ; the first and last are termed Extremes, and the others Means. When any three of following elements are given, the remaining two can be ascer- tainedviz., First term, Last term, Number of terms, Common Difference, and Sum of all the terms. To Compute First Term. When Last term, Number of terms, and Sum of series are given. RULE. From quotient of twice sum of series, divided by number of terms, subtract last term. I d S dn i , , Or, ; ; and y (I + . 5 d) 2 zdS-5d = a. a represent- n i n 2 ing ist, I last, n number of, and S sum of all terms, and d common difference. ILLUSTRATION. A man travelled 390 miles in 12 days, travelling 60 miles last day. How far did he travel first day ? - = 65, and 65 60 = 5 first term. To Compute Last Term. When First term, Common Difference, and Number of terms are given. RULE. Multiply the number of terms less i, by common difference, and to product add first term. EXAMPLE. A man travelled for 12 days, at the rate of 5 miles first day, 10 second, and so on ; how far did he travel the last day ? 12 i X 5 = 55> an d 55 + 5 = 60 miles. When First term, Number of terms, and Sum of series are given. RULE. Divide twice sum of series by number of terms, and from quotient subtract first term. Or, -a; V 2 dS + (a-. 5 d) 2 . 5 d ; and - + li!Lnl> = i n n 2 ILLUSTRATION. A man travelled 360 miles in 12 days, commencing with 5 miles first day; how far did he travel last day? - = 65, and 65 - 5 = 60 miles. To Compute Number of* Terras. When Common Difference and Extremes, or First and Last term, are given. RULE. Divide difference of extremes by common difference, and add i to quotient. EXAMPLE. A man travelled 3 miles first day, 5 second, 7 third, and so on, till he went 57 miles in one day ; how many days had he travelled at close of last day ? 57 3-7-2 = 27, and 27+ i =28 days. When Sum of series and Extremes are given. RULE. Divide twice sum of series by sum of first and last terms. r l ~ a i /2 r > +'> v~ S /2 a d\* d za ILLUSTRATION. A man travelled 840 miles, walking 3 miles first day and 57 last day; how many days was he travelling? IO2 ARITHMETICAL PROGRESSION. To Compiate Common Difference. When Number of terms and Extremes are given. RULE. Divide difference of extremes by i less than number of terms. 28 2 an ' l-^a/Xl a. 2 nl 2 S r '"w(n~ir ; TtT-T^-a ' ~n~(n ^TjT ~~ ILLUSTRATION. Extremes are 3 and 15, and number of terms 7 ; what is common difference ? *5 3-M7 i) = j, and ^ = 2 com - di f- To Compute Sum of* the Series or of* all Terms. When Extremes and Number of terms are given. RULE. Multiply number of terms by half sum of extremes. l+ax(l-a) . l + a Or, 2 a + d (n-i) X .5 n; M ^ - + - and 2 J (d X n i ) X . 5 n = S. . ILLUSTRATION. How many times does hammer of a clock strike in 12 hours? 12 X 12+ J JS^, and 156-7-2 = 78 fo'wes. To Comp-ute any N"um"ber of* Arithmetical Means or Terms between t\vo Extremes. RULE. Subtract less extreme from greater, and divide difference by i more than number of means or terms required to be ascertained, and then proceed as in rule. To Compute T\vo Arithmetical IMeans or Terms "bet-ween two given Extremes. RULE. Subtract less extreme from greater, and divide difference by 3, quotient will be common difference, which being added to less extreme, or taken from great- er, will give means. EXAMPLE i. Compute two arithmetical means between 4 and 16. 16 4-7-31= 4 com. dif. 4 -}- 4 = 8 one mean. 1 6 4 = 12 second mean. 2. Compute four arithmetical means between 5 and 30. 30 5 = 25, and 25 ^-44-1=5=: com. dif. 5 4- 5 = 10= i st mean. 15-}- 5 = 20 = 3*2 mean. io-j-5 = i5 2d " 20 -j- 5 = 25 = ^th " !M!iscellaneoi*s Illustrations,, 1. A steamer having been purchased upon following terms viz.: $5000 upon transfer of bill of sale and balance in monthly instalments, commencing at $4500 for first month, and decreasing $500 in each month, until whole sum is paid. ist. How many months must elapse before final payment? 2d. What was amount of purchase money, or sum of series? Here are first and last terms viz., 500 and 5000, and common difference, 500. Hence, To compute number of terms and amount of purchase, 5000 500 -r- 500 = 9, and 9 -f- 1 = 10 = number of terms or months, and 10 X ' ' 5 = 10 X 2750 = $ 27 500, amount of purchase. 2. If TOO stones are placed in a right line, one yard apart; how many yards must a person walk, to take them up one at a time and put them into a basket, one yard from first stone? JFirst term 2, last term 200, and number of terms 100. Hence, 100 x = 10 100 yards. GEOMETRICAL PROGRESSION. IO3 3. If in the sinking of curb of a well, $3 is to be given for first foot in depth, $5 for second, $7 for third, and increasing in like manner to a depth of 20 feet, what would it cost? First term 3, common difference 2, and number of terms 20. Hence, 20 1X2-1-3 = 41, last term. Then, 3 -|- 41 X = 440, sum of all terms, or cost of curb. 4. If a contractor engaged to sink a curb to depth of 20 feet for $400, and the contract was annulled when he had reached a depth of 8 feet; how much had he earned? 400 -r- 20 =: number of terms. But inasmuch as 400 may be divided into 20 terms in arithmetical proportion in many different ways, according to value of ist term, it becomes necessary to assume the value of the first foot as value of ist term. Assuming it at $5, the required proportion will be, ist term 5, number of terms 20, rum of series 400. Hence, 400-^X^X2 = & 1 11 common difference. _20X(20 l) 380 ld ' Then, 5 -f- i4-J x 7 = 16^ = ist term + product of common difference and Sth term less i, which added to 5 21-^, and X 4 = half number of terms for which cost is sought = 84^ dollars, sum earned. GEOMETRICAL PROGRESSION. GEOMETRICAL PROGRESSION is any series of numbers continually in- creasing by a constant multiplier, or decreasing by a constant divisor, as i, 2, 4, 8, 16, etc., and 15, 7.5, 3.75, etc. The constant multiplier or divisor is the Ratio. When any three of following elements are given, remaining two can be computed, viz. : first term, Last term, Number of Terms, Ratio, and Sum of all Terms. To Compute First Terxn. When Ratio, Last Term, and Number of Terms are given. RULE. Divide last term by ratio raised to a power denoted by number of terms less i. Or, K~ and rl S (r i) = a. a representing ist term, I last, n number of, S sum of all terms, and r ratio. ILLUSTRATION. Last term is 4374, number of terms 8, and ratio 3; what is first term? To Compute Last Term. When First Term and Ratio are Equal. RULE. Write a few of leading terms of series and place their indices over them, beginning with a unit. Add together the most convenient and least number of indices to make the index to term required. Multiply terms of the series of these indices together, and product will give term required. Or, Multiply first term by ratio raised to a power, denoted by number of terms less i. EXAMPLE i. First term is 2, ratio 2, and number of terms 13; what is last term? Indices, 12345 Terms, 2, 4, 8, 16, 32. Then, 5-j-s-j-3=:i3z= sum of indices, and 32 X 32 X 8 = 8192 = last term. Or, 2 X 2 J 3 x = 8192. Also by inspection of table, page 105, isth term = 8192. IO4 GEOMETRICAL PROGRESSION. 2. The price of 12 horses being 4 cents for first, 16 for second, and 64 for third, and so on; what is price of last horse? Indices, 1234 Terms, 4, 16, 64, 256. Then, 4 + 4 + 4 = i2=swm of indices, and 256 X 256X256 = 2563 = 1167772.16. When First Term and Ratio are Different. RULE. Write a. few of leading terms of series, and place their indices over them, beginning with a cipher. Add together the most convenient indices to make an index less by i than term sought. Multiply terms of these series belonging to these indices together, and take product for a dividend. Or, Raise first term to a power, index of which is i less than number of terms multiplied; take result for a divisor; proceed with their division, and quotient will give term required. EXAMPLE i. First term is i, ratio 2, and number of terms 23; what is the last term? Indices, 01234 5 Terms, i, 2, 4, 8, 16, 32. Then, 5 + 5 + 5 + 5-1-2 = 22 = sum of indices, and 32 X 32 X 3 2 X 3 2 X 4 = 4 194 304, and 4 194 304 -r- the sth power (6 i) of i = i = 4 194 304. Or, i X 2 2 3 I = 4 194 304. By inspection of table, page 105, 23d term = 4 194 304. 2. If i cent had been put out at interest in 1630, what would it have amounted to in 1834, if it had doubled its value every 12 years? 1834 1630 = 204, which -r- 12 ,= 17, and 17 + 1 = 18 = number of terms. Indices, 01234 7 Terms, i, 2, 4, 8, 16, 128. 16 X 8 X 4 X 2 X i = 131 7 2 > and J 3i 072 Then, 7 + 4 + 3 + 2 + 1 = 17, and 128 X 16 -r- 1, the 4th power (5 i) of i $ 1310.72. When First Term, Ratio, and Sum of the series are given. RULE. From sum of series subtract quotient of first term subtracted from sum of series, divided by ratio. Oraxr n ~ I =Z EXAMPLE. First term is 2, ratio 3, and sum of series 2186; what is last term? ai 86 = 2186 728 = 1458, last term. To Compute !N~umlber of Terms. When Ratio, First, and Last Terms are given. RULE. Divide logarithm of quo- tient of product of ratio and last term, divided by first term, by logarithm of ratio. ft log- (q + Sr i) log, q log. I log, q log. r log. (S - q) - log. (S - 1) "*" and ' ^ 1- i = n. log. r EXAMPLE. Ratio is 2, and first and last terms are i and 131072; what is num- ber of terms? log. 2X I 3 I 7 2 _ i g 262 I44 = 5 . 4 i8 54, and 5. 418 54 -r- log. of 2 = 5 ' 41 54 = 18. i .30103 To Compute Sunn of Series. When First Term, Ratio, and Number of Terms are given. RULE. Raise ratio to a power index of which is equal to number of terms, from which subtract i ; thei divide remainder by ratio less i, and multiply quotient by first term. GEOMETRICAL PROGRESSION. 105 ILLUSTRATION i. First term is 2, ratio 2, and number of terms 13; what is sum of series ? 2*3 1 = 8192 1 = 8191, and 8191-^(2 1) = 8191, and 8191 x 2 = 16382. 2. If a man were to buy 12 horses, giving 2 cents for first horse, 6 cents for second, and so on, what would they cost him ? $5314.40. To Compute Ratio. When First Term, Last Term, and Numbers of Terms are given. RULE. Divide last term by first, and quotient will be equal to ratio raised to power denoted by i less than number of terms; then extract root of this quotient ^ S o ILLUSTRATION. First term is 2, last term 4374, and number of terms 8; what is ratio ? 4074 8i, = 2i87,and -^2187 = 3, ratio. Miscellaneous Illustrations. 1. What is gth term in geometrical progression 3, 9, 27, 81, etc.? and what is sum of terms? ist term = 3, number of terms 9, and ratio 3. Hence, by rule to compute last term, ist term and ratio being equal- Indices, 1234 Terms, 3, 9, 27, 81. Then, 2-1-3-1-4 = 9 = sum of indices, and 9 X 27 X 81 = 19 683 = last term. By rule to compute sum of terms 3 ~* X 3 = I9 a 2 = 9841 X 3 = 29 523, sum of terms. 2. First term is i, ratio 2, and last term 131072; what is sum of series? 131 072 X 2 i = 262 143, and 262 143 -r- 2 i = 262 143. 3. What are the proportional terms between 2 and 2048 ? 4 -f- 2 = 6, and 6 1 = 5, and A/ = 4. Hence, 2 : 8 : 32 : 128 : 512 : 2048. 4. Sum of series is 6560, ratio 3, and number of terms 8; what is first term? 6 ^x~f^: Greoxnetrical ^Progressions, Whereby any questions of Geometrical Progression and of Double Ratio may solved by Inspection, number of terms not exceeding 56. I 1 'I 16384 2 9 268 435 456 43 4398046511104 2 2 16 32768 30 536870912 44 8 796 093 022 208 3 4 17 65536 3i 1 073 741 824 45 17 592 186044416 4 8 18 131 072 32 2147483648 46 35184372088838 5 16 J 9 262 144 33 4 294 967 296 47 70368744177664 6 32 20 524288 34 8589934592 48 140737488355328 7 64 21 I 048 576 35 17179869184 49 281474976710656 8 128 22 2097152 36 34359738368 5 562949953421 312 9 256 23 4 194 304 37 68719476736 51 1 125 899 906 842 624 10 512 2 4 8 388 608 38 i3743 8 953472 52 2251799813685248 ii 1024 25 16777216 39 274 877 906 944 53 4503599627370496 12 2048 26 33554432 40 549755813888 54 9007199254740992 13 4096 27 67 108 864 4 1 1099511627776 55 18 014 398 509 481 9^4 14 8192 28 134217728 42 2199023255552 56 36 028 797 oi 8 963 968 ILLUSTRATIONS. i2th power of 2 = 4096, and 7th root of 128 = 2. IO6 ALLIGATION. ALLIGATION. ALLIGATION is a method of finding mean rate or quality of different ma- terials when mixed together. To Compute UVIean JPrice of a, Mixture. When Prices and Quantities are known. RULE. Multiply each quantity by its mte, divide sum of products by sum of quantities, and quotient will give rate of the composition. EXAMPLE. If 10 Ibs. of copper at 20 cents per lb., i Ib. of tin at 5 cents, and i Ib. of lead at 4 cents, be mixed together, what is value of composition? 10 X 20 = 200 iX 5= 5 _iX 4= 4 12 ) 209 (17. 416 cents. To Compute Quantity of each, Article. When Prices and Mean Price are given. RULE. Write prices of ingredients, one under the other in order of their values, beginning with least, and set mean price at left. Connect with a line each price that is less than mean rate with one or more that is greater. Write difference between mixture rate and that of each of simples opposite price with which it is connected; then sum of differences against any price will express quantity to be taken of that price. EXAMPLE. How much gunpowder, at 72, 54, and 48 cents per pound, will compose a mixture worth 60 cents a pound? (48 \ 12, at 48 cents. 60 ] 54\/ 12, at 54 cents. (727 12 -f 6 = 18, at 72 cents. Here, 72 60 = 12 at 48, 72 60 = 12 at 54, 60 48 = 12, and 60 54 = 6 = 12 + 6 = 18 at 72. Then 12 X 48-4-12 X 54 -f 18 X 72 = 2520, and 2520^-12 + 12+ 12 -f 6 60 cents. NOTE. Should it be required to mix a definite quantity of any one article, the quantities of each, determined by above rule, must be increased or decreased in proportion they bear to defined quantity. Thus, had it been required to mix 18 pounds r\t 48 cents, result would be 18 at 48, 18 at 54, and 27 at 72 cents per pound. When the whole Composition is limited. RULE As sum of relative quantities, as ascertained by above rule, is to whole quantity required, so is each quantity so ascertained to required quantity of each. EXAMPLE. Required 100 pounds of abore mixture Then, 12 -f 12+ 18 = 42. Then, 42 : 100 :: 12 : 28.571 pounds. 42 : loo :: 12 : 28.571 pounds. 42 : loo :: 18 : 42.857 pounds. When Price of Several Articles and Quantity of one of them is given. RULE. As- certain proportionate quantities of ingredients by previous rule. Then, as number opposite ingredients, quantity of which is given, is to given quantity; so is number opposite to each ingredient to quantity required of that in- gredient. EXAMPLE. Having 35 Ibs. of tobacco, worth 60 cents per pound, how much of other qualities, worth 65, 70, and 75 cents per pound, must be mixed with it, so as to sell mixture at 68 cents per pound? By previous rule, it is ascertained there must be 7 Ibs. at 60, 2 at 65, 3 at 70, and 8 at 75 cents; but as there are 35 Ibs. at 60 cents to be taken, other quantities and kinds must be increased in like manner. Hence, 7 : 35 : : 2 : 10 = 10 at 65 cents. 7 : 35 : : 3 : i5 15 " 7 cents. 7 : 35 : : 8 : 40 = 40 " 75 cents. SIMPLE INTEREST. IO/ SIMPLE INTEREST. To Compvite Interest on. any Griven Sum fbr a Period of One or more Years. RULE. Multiply given sum or principal by rate per cent, and number of years; point off two figures to right of product, and result will give interest in dollars and cents for the period. EXAMPLE. What is interest upon $ 1050 for 5 years at 7 per cent. ? 1050 X 7 X 5 = 3 6 750, and 3 6 7- 5o $ 3 6 7-5<>-* ^ ^ When Time is less than One Year. RULE. Proceed as before, multiplying by number of months or days, and dividing by following units viz., 12 for months, and 365 or 366, as the case may be, for days. EXAMPLE. What is interest upon $1050 for 5 months and 30 days at 7 per cent.? 5 months and 30 days = 183 days. * 5 X J X 183 = ^ ftn The operation of computing interest may be performed thus : Assuming interest upon any sum at 6 per cent. i per cent for 2 months. Interest at 5 per cent, is ^th less than at 6 per cent. Interest at 7 per cent, is ^th greater than at 6 per cent. Taking preceding example 2 months = i per cent= 10.50 2 " =i " 10.50 " =i " 5-25 30 days = i month = 5.25 31-50 Add J for 7 per cent.= 5.25 $36- 75 NOTB. Difference between this amonnt and preceding arises from 183 days being taken in one case, and half a year, or 182.5 days, in the other. In every computation of interest there are four elements viz. , Principal, Time, Rate, and Interest or Amount, any three of which being given, remaining one can be ascertained. To Compute Principal. When Time, Rate per Cent., and Interest are given. RULE. Divide given interest by interest of $i, etc., for given rate and time. EXAMPLE. What sum of money at 6 per cent, will in 14 months produce $ 14? 14 -T- .07 = 200 dollars. To Compute Rate per Cent. When Principal, Interest, and Time are given. RULE. Divide given interest by interest of given sum, for time, at i per cent. EXAMPLE. If $ 32.66 was discounted from a note of $400 for 14 months, whaf was that per cent. ? Interest on 400 for 14 months at i per cent. = 4. 66. Then 32.66 -=- 4.66 = 7 per cent. To Compute Time. When Principal, Rate per Cent., and Interest are given. RULE. Divide given in I erest by interest of sum, at rate per cent, for one year. EXAMPLE. In what time will $ 108 produce $ 11.34, at 7 P er cent. ? Interest on 108 for one year is 7.56. 1 1. 34 ^-7. 56 =1.5 years. ILLUSTRATION i. If an amount of $ 2175 is returned for a period of 15 months rate of interest having been 7 per cent., what was principal invested? $2000. . If $ looo in 1 8 months will produce $ 1090, what is rate ? 6 ver cent. io8 COMPOUND INTEREST. COMPOUND INTEREST. If any Rrincipal be multiplied by number (in following table) opposite years, and under rate per cent., sum will be amount of that principal at com- pound interest for time and rate taken. EXAMPLE. What is amount of $500 for 10 years at 6 per cent. ? Tabular number. . . . i. 790 84, and i. 790 84 x 500 = 895.42 dollars. i 3 Per Cent. 4 Per Cent. 5 Per Cent. 6 Per Cent. 1 3 Per Cent. 4 Per Cent. 5 Per Cent. 6 Per Cent, i 3 .04 05 i. 06 13 1.46853 1.66507 1.88564 2.1329* 2 .0609 .0816 .1025 1.1236 14 1-51529 1.73167 1-97993 2.2609 3 .09273 .12486 .15762 1.191 01 15 1-55797 1.80095 2.07892 2.39 6 55 4 12551 .16986 2155 1.26247 16 1.60471 1.87298 2.18287 2-54 35 IS9 2 7 .21668 276 28 1.338 22 17 1.65285 1.94799 2.29201 2.69277 6 194 05 26532 34 1.41851 18 1.70244 2.025 8 1 2.40661 2-85433 7 .22987 31593 .4071 L50363 J 9 1-7535 2.10684 2.52695 3-02559 8 .26677 .36857 47745 I-59384 20 i. 806 1 1 2.191 13 2.65329 3.207 13 9 30477 42331 55132 1.68947 21 1.86029 2.27876 2. 785 96 3-39956 10 34392 .48024 .62889 1.79084 22 1.916 i 2.36992 2.925 26 3-603 53 ii .38424 53945 7 I0 33 1.89829 23 I-9736 2.464 21 3.07152 3.81974 12 4257 6 . . 601 03 79585 2.OI2 19 24 2.03279 2-5633 3.22509 4.04873 For any other Rate or Period. Multiply logarithm of rate + 1 by period, and number for logarithm will give tabular amount as above. ILLUSTRATION. What is tabular number for 4 per cent, for 10 years? Log. of i. 04 = .017 033 3, which x io = . 170 333, and number for log. = 1.48024. Time in "years in "which a Sum of ]Vtoiiey "will "foe doubled at Several Rates of Interest. Rate. Time. Rate. Time. Rate. Time. | Rate. Time. Per cent. i 2 3 69.68 35 23-44 Per cent. 4 I 17.67 14.21 11.88 Per cent. 9 10.34 9.01 8.04 Per cent. JO 20 30 7-27 3-8 2.64 "Value of $1, etc., Computed Semi-annually for a Period of 13 Years. Years. Per Cent. 4 Per Cent. 5 Per Cent. 6 Per Cent. Years. 3 Per Cent. Per Cent. 5 Per Cent. 6 Per Cent, 5 .015 i. 02 .025 1.03 6-5 .2134 2936 3785 1.4684 i .0302 1.0404 .0506 1.0609 7 2317 3195 413 i 5102 i-5 0457 i. 0612 .0769 1.0927 7-5 3459 -4483 1.558 2 .0614 1.0824 .1038 1-1255 8 .3728 4845 1.6047 2-5 0773 1. 1041 1314 I-I593 8-5 .4002 .5216 1.6528 3 0934 1.1262 1597 1.1941 9 3073 .4282 5597 1.7024 3-5 .1098 1.1487 .1887 1.2299 9-5 .3269 .4568 5987 1-7535 4 .1265 1.1717 .2184 1.2668 o 3469 .486 .6386 i. 8061 4-5 1434 1.1951 .2489 1.3048 0-5 3671 5157 6796 1.8603 5 .1604 1.219 .2801 1-3439 i .3876 546 .7216 1.9161 5-5 .178 1.2434 .3121 1.3842 I -5 .4084 5769 .7606 I-9736 6 .1956 1.2689 3449 1-4258 2 4295 .6084 .8087 2.0356 ILLUSTRATION. What is amount of $500 at semi-annual interest of 5 per cent compounded for 10 years ? Tabular number i. 6386. Then, 500 X i. 628 89 = $ 814. 44. 5. To Compute Interest on any GHven Sum. 'A /A For a Period of Years. P (i + r) n = A ; log. A log. P (i-f-r) = n. P representing principal, r rate per cent.-^-ioo per annum, log. (1-r-r) n number of years, and A amount of principal and interest. DISCOUNT OK REBATE. EQUATION OF PAYMENTS. 1 09 ILLUSTRATION. Assume as preceding, $500 at 5 per cent, for 10 years. 500 x i. 05 10 = 500 X i. 628 89 = $814. 44. 5, amount. ( ^' 4 ^ JO = 590, principal 10 7814.44.5 log. 814. 44. 5 log. 500 ~ ' = -5, '<* = I0 ' num For any Period. Assume elements of preceding case, interest payable semi- annually. 10 X 2 = 20, number of payments ; = .025, rate. Then,-5oox i. 0252= 500 X 1.63862 = $819.31. When term of payments and rate are not given in table. [log. (^ + i) x n P] = log. A. ILLUSTRATION. Assume $1000 for 30 years, at 7 per cent, half-yearly. log. -f i = .014 940 3, and log. .0149403 X 30 X 1000 = $ 28o6.7& DISCOUNT OR REBATE. DISCOUNT or REBATE is a deduction upon money paid before it is due. To Compute R,e"bate upon any Sum. RULE. Multiply amount by rate per cent, and by time, and divide product by sum of product of rate per cent, and time, added to 100. EXAMPLK i. What is discount upon $ 12075 for 3 years, 5 months, and 15 days at 6 per cent. ? 3 years 5 months and 15 days = 3. 4574 years. I2075X6X 3-4574 250488.63 ,oo+,6x 3-4574) =-T^^- = 20 "t- 2. What is present value of a note for $963.75, payable in 7 months, at 6 per cent. ? 6rate. 7 months= T 7 ^of i year = 6 X 7 4-12 = 3.5, and 3 5 + 100 = 103. 5 -r- 100 = i-035 963 75 4" i 035 = $931.16 To Compute tlie Sum for a given Time and Rate, to yield a Certain. Sum. RULE. Divide given sum by proceeds of $ i for given time and rate. EXAMPLE. For what sum should a note be drawn at 90 days, that when dis- counted at 6 per cent, it will net $ 200 ? Discount on $ i for 90+3 days at 6 per cent. = $ .0155. Hence, $i . 0155 = .9845, proceeds, and $200--- .9845 = $203. 14.9. EQUATION OF PAYMENTS. RULE. Multiply each sum by its time of payment in days, and divide sum of products by sum of payments. EXAMPLE. A owes B $300 in 15 days, $60 in 12 days, and $350 in 20 days; when is the whole due ? 300 X 15 = 4500 60X1*= 720 350 X 20 = 7000 710 ) izaao (ij + dayi. Jf IIO ANNUITIES. ANNUITIES. To Compute Amount of Annuity. When Time and Ratio of Interest are Given. RULE. Raise the ratio to a power denoted by time, from which subtract i ; divide remainder by ratio less i, and quo- tient, multiplied by annuity, will give amount. NOTE. $ i added to given rate per cent, is ratio, and preceding table in Compound Interest is a table of ratios. EXAMPLE. What is amount of an annual pension of $100, interest 5 percent., which has remained unpaid for four years? 1.05 ratio; then 1.054 1 = 1.21550625 i =.21550625, and .215 506 25-7- (1.05 i). 05 = 4. 310 125, which x zoo = $431. 01. 25. To Compute ^Present ^Worth. of an. Aiinuity., When Time and Rate of Interest are Given. RULE. Ascertain amount of it for whole time; divide by ratio, involved to time, and result will give worth. EXAMPLE. What is present worth of a pension or salary of $500, to continue 10 years at 6 per cent, compound interest ? $ 500, by last rule, is worth $6590.3975, which, divided by i.o6 10 (by table, page 108, is 1.79084) = $ 3680.05. Or, Multiply tabular amount in following table by given annuity, and product will give present worth. ILLUSTRATION i. As above; 10 years at 6 per cent. =j. 360 08, and 7.36008 x 50 = 3680.04 dollars. 2. What is present worth of $150 due in one year at 6 per cent, interest per annum ? . 943 39 X 150 $141.50.85. Present "Worth, of* an Annuity of $1, at 4, >, and. > 3?er Cent. Compound. Interest for ^Periods under 25 Years. Years. 4 Per Cent. 5 Per Cent. 6 Per Cent. Years. 4 Per Cent. 5 Per Cent. 6 Per Cent. i .961 54 .95238 94339 13 9.98562 9-39357 8.85268 2 1.88609 1.85941 I-83339 H 0.56307 9. 898 64 9.29498 3 2-7751 2.72325 2.67301 15 1.11843 10.37966 9.71225 4 3.6299 3-54595 3-465I 16 1.651 28 10.83778 o. 105 89 M52Q3 4.32948 4.21236 17 2.16626 11.27407 0.47726 6 5.24215 5-07569 4.91732 18 2.659 26 11.68958 0.8276 - 6.00203 5-78637 5-58238 J 9 3.13388 12.08532 1.158 ii 8 6.73176 6.46321 6.20979 20 3-59029 12.46221 1.46992 9 7-4364 7. 107 82 6.80169 21 4.029 12 12.821 15 1.76407 10 8.11085 7.72173 7.36008 22 4.451 12 13.163 2.041 58 ii 8.76044 8.30641 7.88687 23 4.85682 13.48807 2.30338 12 9-38S05 8.86325 8.38384 24 5-24695 13.79864 2.55035 For a Rate of Interest and Term of Years not given in either Table. i ' = A. Notation as preceding. ILLUSTRATION. Take $ i at 4 per cent, for 24 years. Log. 1.04 = .017033, which x 24 = .408 799. log. .408 799 = 2.5633 = ratio raised to power of 24. Then, X (i -- ^ ) = 25 X i 390122 = $ 15. 24.695. .04 \ 2.56337 To Compute Yearly Amount tliat -will Liquidate a Delot in a Q-iven NunVber of Years at Compound Interest. n A. ILLUSTRATION. What is amount of an annual payment that x _1_ Y \rill liquidate a debt of $100 in 6 years at 5 per cent, compound interest? ANNUITIES. Ill (i -f .os) 6 per table, page 108, = 1.34. 4_-7 = | 6 -34 (i+.o 5 ) 6 -i i-34- 1 When Annuities do not commence till a certain period of time, they are said to be in Reversion. To Compute Present \Vorth of an Annuity in Reversion. RULE. Take two amounts under rate in above table viz., that opposite sum of two given times and that of time of reversion; multiply their difference by an- nuity, and product will give present worth. EXAMPLE. What is present worth of the reversion of a lease of $40 per annum, to continue for 6 years, but not to commence until end of 2 years, at rate of 6 per cent. ? 6 + 2 = 8 years 6.209 79 2 * 1-83339 4.37640X40 = 1175-05-6. Amount of Annuity of $1, etc., Compound Interest, from. 1 to SO Years. E * 4 5 6 7 i 4 5 6 7 > Per Cent. Per Cent. Per Cent. Per Cent. Per Cent. Per Cent. Per Cent. Per Cent. : I I. i. I. i. ii I3-48635 14.20679 14.97164 15.7836 ,, 3 2.04 2.05 2.06 2.07 12 15.0258 15.91713 16.86994 17.88845 3 3.1216 3-I525 3-1836 3.2149 13 16.62684 17.71298 18.88214 20.14064 4 4.24646 4.310 12 4.37462 4-439 94 14 18.291 91 19.59863 21.01507 22.55049 5.41632 5-52563 5-63709 5-75074 15 20.023 59 21.57856 23-27597 25.12902 7 6.63297 7.89829 6.801 91 8.14201 6-97532 8.39384 7-15329 8.65402 16 17 21.824 53 23-69751 23-65749 25.84037 25-67253 28.21288 27.88805 30. 840 22 8 9.21423 9.54911 9.89747 10.259 8 18 25.64541 28.13238 30.90565 33.99903 9 10 10. 582 79 12.006 ii 11.02656 12.57789 11.49132 13-18079 11.97799 13.81645 *9 20 27.671 23 29.77808 30-539 33-o6595 33-75999 36.78559 37-37896 40.99549 ILLUSTRATION. What is amount of $ 1000 for 20 years at 5 per cent? 5 per cent, for 20 years = 33.065 95 ; hence, 1000 x 33.065 95 = $ 33.06.595. To Compute Amount of an Annuity for any Period and Rate. RULE. From table for Compound Interest, page 108, take value for rate per cent, for i year, and raise it to a power determined by time in years, from which subtract i, divide remainder by rate, and quotient multiplied by annuity will give amount required. EXAMPLE. What will an annuity of $ 50, payable yearly, amount to in 4 years, at 5 per cent. ? By table, page 108, 1.054 = 1.2155. i. 2155 i -f- (1-05 i) = 4-31. and 4. 31 X 50 = $ 215.50. For Half-yearly and Quarterly Payments. Multiply annuity for given time by amount in following table: ite per cent. Half-yearly. Quarterly. Rate per cent. Half-yearly. Quarterly. 3 3-5 4 45 5 .007445 .008675 .009902 .on 126 .012 348 .on 181 .013031 .014877 .016729 .018559 5-5 6 6-5 7 7-5 .013567 .014781 015993 .017 204 .018414 .020395 .022227 .024055 .02588 .027704 ILLUSTRATION i. Annuity as determined in previous case = $21 5. 50. Hence, 215. 50 X 1.012348 from above table = $21 8.16 for half yearly payments. 2. A person 30 years of age has an annuity for 10 years, present worth of it being $1000, provided he may live for 10 years. What is annuity worth, assuming that 60 persons out of every 3550, between the ages of 30 and 40, die annually? 3550 600 (60 X 10) = 2950 would therefore be living. And, 3550 : 2950 :: 1000 = $830. 98. 1 1 2 PERPETUITIES. COMBINATION. PERPETUITIES. PERPETUITIES are such Annuities as continue forever. To Compute Value of a I?erpetu.al Annuity. RULE. Divide annuity by rate per cent., and multiply quotient by unit in pre- ceding table. EXAMPLE. What is present worth of an annuity for $ 100, payable semi-annually, at 5 per cent. ? 100^- .05 = 2, and 2 X i. 01 * 348, from preceding table = 2.024.70. To Compute "Value of* a Perpetuity in. Reversion. RULE. Subtract present worth of annuity for time of reversion from worth of annuity, to commence immediately. EXAMPLE. What is present worth of an estate of $50 per annum, at 5 per cent., to commence in 4 years ? 50 -T- .05 = 1000 $50, for 4 years, at 5 per cent. = 3. 545 95 (from table, page no) X 50= 177.2975 822.7025 which in 4 years, at 5 per cent, compound interest, would produce $1000. COMBINATION. COMBINATION is a rule for ascertaining how often a less number of num- bers or things can be chosen varied from a greater, or how many different collections may be formed without regard to order of each collection. Combinations of any number of things signify the different collections which may be formed of their quantities, without regard to the order of their arrangement. Thus, 3 letters, a, 6, c, taken all together, form but one combination, abc. Taken two and tivo, they form 3 combinations, as ab, ac, be. NOTE. Class of the combination is determined by number of elements or things to be taken ; if two are taken, the combination is of ad class, and so on. RULE. Multiply together natural series i, 2, 3, etc., up to the number to be taken at a time. Take a series of as many terms, decreasing by i, from number out of which combination is to be made, ascertain their continued product, and divide this last product by former. EXAMPLE i. How many single combinations, as a&, ac, may be made of 2 letters out of 3? i X 2 _ 2 _ 6 _ 3~x~^ ~~ ~6~ ~~ V ~ 3 ' 2. How many combinations may be made of 7 letters out of 12? iX 2X 3X4X5X6X7 = 5040 and 399i68o _ 12X11X10X9X8X7X6 3991680' 5040 3. How many different hands of cards may be held, as at whist, combinations 13 out of 52? 635013559600. "When t\vo Nu.xn'bers or Tilings are Combined. RULE. Multiply together natural series i, 2, 3, etc., to one less term than numbei of combinations ; ascertain their continued product, and proceed as before. EXAMPLE. There are 3 cards in a box, out of which two are to be drawn in a re quired order. How many combinations are there? Here there are 2 terms ; hence, 2 1 = 1, and - = =6-7-1 = 6. 3X2 6 To Compxite Number of "Ways in -which, any Number of Distinct Objects can "be Divided, among any Number. RULE. Multiply together numbers equal to number given, as often as objects are to be divided among them. EXAMPLE. In how many different ways can 10 different cards be divided amon/ 3 persons? 3X3X3X^*3X3X3X3X3X30 r 3 IO = 5949- COMBINATION. CIRCULAR MEASURE. 113 Combinations -with Repetitions. In this case the repetition of a term is considered a new combination. Thus, i, 2, admits of but one combination, if not repeated; if repeated, however, it admits of three combinations, as i, i ; i, 2; 2, 2. RULE. To number of terms of series add number of class of combination, less i ; multiply sum by successive decreasing terms of series, down to last term of series; then divide this product by number of permutations of the terms, denoted by class of combination. EXAMPLE. How many different combinations of numbers of 6 figures can be made out of n? -i. ( 6 _ j) - ,6 = sum of number of terms, and number of class, Uss i. 16 X 15 X 14 X 13 X 12 X ii = 5 765 7f>=product of sum, and successive terms to last term. 1X2X3X4X5X6 = 720 permutations of class of combination. Then, 5 -7 = 8008. 720 Variations with. Repetitions. Every different arrangement of individual number or things, including repeti- tions, is termed a Variation. Class of Variation is denoted by number of individual things taken at a time. RULE. Raise number denoting the individual things to a power, the exponent of which is number expressing class of variation. EXAMPLE i. How many variations with 4 repetitions can be made out of 5 fig- ures ? 54 = 625. 2. How many different combinations of 4 places of figures can be made out of the 9 digits ? 12 X ii X 10X9 Il88 = = 495- Combination witho-ut Repetitions. RULE. From number of terms of series subtract number of class of combination, less i ; multiply this remainder by successive increasing terms of series, up to last term of series; then divide this product by number of permutations of the terms, denoted by class of combination. EXAMPLE i. How many combinations can be made of 4 letters out of 10, exclud- ing any repetition of them in any second combination? 10 (4 i) = 7 = number of terms number of class, less i. 7X8X9X10 = 5040 = prod, of remainder 7, and successive terms up to last term. 1X2X3X4 = 24 = permutations of class of combination. Then, 5i = 2I0 . 24 2 . How many combinations of the sth class, without repetitions, can be made of 12 different articles? / x a .8X9X10X11X12 85040 12 (5 i) = 8, and - - --- = = 702. 1X2X3X4X5 120 /y CIRCULAR MEASURE. Unit of Circular Measure is an angle which is subtended at centre of a circle by an arc equal to radius of that circle, being equal to 1 80 Circular measure of an angle is equal to a fraction which has for its numerator the arc subtended by that angle at centre of any circle, and for its denominator the radius of that circle. K* 114 CIRCULAR MEASURE. - PROBABILITY. To Compnte Circular ^Measure of an Angle. RULE. Multiply measure of angle in degrees by 3.1416, and divide by 180. EXAMPLE. What is circular measure of 24 10' 8"? 24 10' 8" X 3-1416 _ 87008 x 3-1416 180 ~ - To Compute ]Measu.re of an Angle, its Circular Measure being Q-iven. RULE. Multiply circular measure of angle by 180, and divide by 3.1416. PROBABILITY. Probability of any event is the ratio of the favorable cases, to all the cases which are similarly circumstanced with regard to the occurrence. If an event have 3 chances for occurring and 2 for failing, sum of chances being 5, the fraction f will represent probability of its occurring and is taken as measure of it. Thus, from a receptacle containing i white and 2 black balls, the probability of drawing a white ball, by abstraction of i, is ; prob- ability of throwing ace with a die is ^ : in other words, the odds are 2 to i against first, and 5 to i against second. If m -f- n = whole number of chances, m representing number which are favorable, and n unfavorable. Therefore = probability of event. m-f-w ' Probabilities of two or more single events being known, probability of their oc- curring in succession may be determined by multiplying together the probabilities of their events, considered singly. Thus, probability of one event in two is expressed by ^; of its occurring twice in succession, ^ X J, or ; of thrice in succession, J x ^ X J? or J, etc. ILLUSTRATION i. If a cent is thrown twice into the air, the probability of its fall- ing with its head up, twice in succession, is as i to 4. Thus, it may fall: 1. Head up twice in succession. \ 2. Head up ist time and wreath 2d time, f i i 3 . Wreath up ist time and bead 2 d time, f Hence, jq _ .25 = = 4 times. 4. Wreath up twice in succession. These are the only results possible, and being all similarly circumstanced as to probability, the probability of each case is as i to 4, or odds are as 3 to i. Probability of either head or wreath being up twice in succession is as i to i, or chances are even, because ist and 4th cases favor such a result; probability of head once and wreath once in any order is as i to 2, because 2d and 3d cases favor such a result; and probability of head or wreath once is as 3 to 4, or odds are as 3 to i, be- cause ist, ad, and 3d, or 2d, 3d, and 4th cases favor such a result. NOTE. i to 2 is an equal chance, for i out of 2 chances = i to i, being an equal chance ; again, i to 5 is 4 to i, for i out of 5 chances is i to 4. 2. If there are 4 white balls and 6 black in a bag, what is the chance of a person drawing out 2 black at two successive trials? This is a combination without repetition. Hence, 6 (2 i) = 5, and , which x 2 for successive trials = or . 1X221' , 2 15 3. Suppose with two bags, one containing 5 white balls and 2 black, and the other 7 white and 3 black. Number of cases possible in one drawing from each bag is (5 -4- 2) x (7 -f- 3) = 7 X 10 = 70, because every ball in one bag may be drawn alike to one in the other. PROBABILITY. 115 Number of cases which favor drawing of a white ball from both bags is 5 X 7 = 35, for every one of the 5 white balls in one bag may be drawn in combination with every one of the 7 in the other. For a like cause, number of cases which favor drawing of a white ball from ist bag and a black one from 2d is 5 x 3 15 ; a black ball from ist bag and a white ball from 2d is 7 X 2 = 14 ; and a black ball from both is 3 X 2 =6. Probability, therefore, of drawing is as 2l!-3i = J. = x to i, a white ball from both bags. 1*J = 1* = A = 3 to n, 70 70 2 70 70 14 a white ball from ist, and a black from 2d. 1^2. 11 = = x to 4, a black 7 7 5 ball from ist. and a white from zd. - - = = = 3 to 32, a black ball from 70 70 35 j a white i a nf rom one, and a black from other, for both 2 d and 3 d cases favor this result : hence, -f - = . 5 14 7 7 = = = 32 to 3, at least one white ball, for the ist, 2d, and 3d cases favor this 7> 35 result ; hence, + A + _L = 3. 2 14 ^ 5 35 Again, if number of white and black balls in each bag are same, say 5 white and 2 black, 5 + 2 X 5 + 2 = 49, then probability of drawing is as i-*- 5 . = ?S _ 25 to 24 a white b a u from both. 5211 = I0 to 39, a wAife ball 49 49 jo 49 49 from ist, and a black from zd. - = = 10 to 39, a black ball from ist, and a white from id. ^2H = A 4 to 45, a fcZacfc ball from both. 49 49 4. When two dice are thrown, probability that sum of numbers on upper sides is any given number, say 7, is as follows: As every one of the six numbers on one die may come up alike to, or in combi- nation with the other, number of throws is 6 X 6 = 36. ! i and 61 2 " 5} ; and as these numbers may be 3 " 4) upon either die, there are 3 x 2 = 6 throws in favor of the combination of 7 ; hence 6 i probability of throwing 7 is ? = , or as i to 5. 5 . Probability of a player's partner at Whist holding a given card is as follows: Number of cards held by the other 3 players is 3 x 13 = 39; probability, there- fore, that it is held by partner is , but it may be one of the 13 cards which he holds; hence probability is X 13 = = , or as i to 2. 39 39 3 6. Probability of a player's partner at Whist holding two given cards is as follows: Number of combinations of 39 things, taken 2 and 2 together, is = 741 ; therefore, probability that these 2 cards are in partner's hand is 39 x 38 = ~7^7~ 39Xl9 -^- = i to 740; but they may be any 2 cards in partner's hand; therefore, since number of combinations of 13 cards, taken 2 and 2 together, is = = 7 8, 78 2 probability required is -f = , or as 2 to 17. Similarly, probability that he holds any 3 given cards is as , or as 22 to 681. Il6 PROBABILITY. Probabilities at a game of Whist upon following points are : 9 to 7, that one hand has two honors, and two hands one; g to 55, that two hands have each two honors ; 3 to 29, that each hand holds an honor; 3 to 13, that one hand has three honors, and one hand one; i to 63, that four honors are held by one hand. 7 . If 3 half-dollars are thrown into the air, probability of any of the possible com binations of their falling is determined as follows : Hence, (^-) 3 = .125 = i to 7 in favor of 3 heads. JL /Z. j . 375 = 3 to 5 " " 2 heads and i tail. 3X2 ( ) 3 = -375 = 3 to 5 " " i head and 2 tails. 3x2x1/^3 u u IX2X 3\2/ And in like manner, if 5 were thrown up, probability of any of their possible combinations would be determined as follows : \*\ 5X4/'\5 5X 4 X3/i\5, 5 X 4 X 3 X 2 / i \5 /+o^W~ i ~ix 2x3 W + 1X2X3 X 4 W , 5X4X3X2X i /_i_\S """1X2X3X4X5 \2/ 5 -' I5 - 5 1X2X3X4X5 Hence, ( ) = .031 25 i to 31 in favor of 5 heads ; ( ) = .15625 = 5 to 27 " " 4 heads and i tail ; - - ( ) = .3125 = 10 to 22 " " 3 heads and 2 tails; I X 2 \ 2 / - 5 = .312 5 = 10 to 22 u " 2 heads and 3 tails; All Wagers are founded upon the principle of product of the event, and contingent gain, being equal to amount at stake. ILLUSTRATION i. Suppose 3 horses, A, B, and C, are entered for a race, and X wagers 12 to 5 against A, n to 6 against B, and 10 to 7 against C. If A wins, X wins 6 4- 7 12 = i. " X X , B X s--n^L 5-1-6-10=1. Hence, X wins i, whichever horse wins, from having taken field against each horse at odds named. . ( A are 5 to 12 ) ( -^ in favor of A, Odds given m fa- J ^ ( ; corresponding probabil- ) V vorof 1 B 6 f ity is 1 ff B ' ( C 7 " 10 ) ( A " * nd x"^i"^ == i" = I ' o6==Xi 6 <0 J in f avor of taker of odds. PROBABILITY. 2 . Odds given upon first seven favorite horses for Oaks Stakes of 1828 were so great, that probability in favor of taker of the odds when reduced was as follows : ist, 5 to 2 ; 2d, 5 to 2 ; 3d, 4 to i ; 4th, 7 to i ; sth, 14 to i ; 6th, 14 to i ; 7th, 15 to i ( 4X3 X 161=192 _2 2 , _L _i_ _L i _L _ 1 _i_ _l-j_ - 3 _ == !_i_-i-JL : _ J 1X7X16 = 112 7 7 ( 7 X 3 X 16 "3^6 _ 3 6 7 _:_ 33 6 = LQQ2 ,[.092 to i, in favor of taker of odds, yet neither of the horses upon which these odds were given won. 3. if odds are 3 to i against a horse in a race, and 6 to i against another horse in a second race, probability of ist horse winning is , and of other ^. Therefore probability of both races being won is ^, and odds against it 27 to i , or 1000 to 37.037. Odds upon such an event were given in 1828 at 1000 to 60, or 16.67 to i. 4 ._Two persons play for a certain stake, to be won by winner of three games or results. One having won one and the other two, they decide to divide the sum, proportioaate to their interest. How much of it should each one receive? OPERATION. If winner of two games should win game to be played, he would be entitled to the whole sum; if he lost, he would be entitled to half of it. Now as one event is as probable as the other, + = , half of which = , or sliare 122 4 of winner of two games. When events are wholly independent, so that occurrence of one does not affect that of the other, probability that both will occur is product of proba- bilities that each will occur. NOTE. It is indifferent whether events are to occur together or consecutively. ILLUSTRATION i. Assume three boxes, each containing white and black balls as follows : 6 white, 5 black ; 7 white, 2 black ; 8 white, 10 black. What is chance of drawing from them a white, black, and a white ball ? Probabilities are , , and , product of which = ii 9 10 + 2-1-8 297 = 17.625 to i. 2. A gives an answer correctly 3 times out of 4, B 4 times out of 5, and C 6 out of 7. What is probability of an event which A and B declare correct and C denies? OPERATION. Compound probability that A and B answer correctly and C denies (all i of which are in favor of event) is X X = = . 4 5 7 140 35 Compound probability that A and B deny and C is correct (all 3 of which are against event) is X X = = . 4 5 7 140 70 Then correct, divided by sum of correct and incorrect, .8714 35 \35 "85714 . =.68 or-. 428 57 3 Odds "between "Results or Chances, and bet-ween any Number and "Whole Num'ber, at various Odds against each., also "Value of each Chance in parts of 1OO. Odds against each. Value of Chance. Odds against each. Value of Chance. Odds against ; Value of each. i Chance. Odds against each. Value of Chance. Even 50 2 tO 33-33 6.5 tc 13-33 15 tc 6.25 IX tO 10 47.62 2-5' 28.57 7 12-5 18 5-26 6 5 45-45 3 ' 25 7-5 11.76 20 4.76 5 4 44.44 3-5 ' 22.22 8 ; ii. ii 25 3-84 5-5 4 42.1 4 ' 2O 8.5 10.52 3 3-22 6 4 40 4-5' 18.18 9 i I0 40 2.44 6-5 4 38.1 5 ' 16.66 9-5 9-52 5 1.96 7 4 36.36 5-5' 15-38 10 9.09 60 1.64 7-5 4 34-78 6 ' 14.28 12 i 7-7 IOO 99 OPERATION. Divide 100, or unit, as case may be, by sum of odds, and multiply quotient by lesser chance or odds. ILLUSTRATION. 6 to 4. 6 + 4 = 10, and 100 -r- 10 x 4 = 40, value of chance. Il8 WEIGHTS OF IRON/ STEEL, COPPER, AND BRASS. WEIGHTS OF IRON, STEEL, COPPER, AND BRASS. "Wrought Iron, Steel, Copper, and Brass T*lates t u Stand No. of Gauge. S. Law, ard Gau THI Approxi- mate Fractions. March $d, 3 ge. Iron ai CKNBSB. Approxi- mate Decimals. 893- id Steel WEIGHT. Wro't Iron Per Sq. Ft. No. of Gauge. American THICKNESS. Approximate Decimals. Gauge. WEI PER Squ Copper. GHT. ARE FOO* Brass Inch. Inch. Lbs. Inch. Lbs. Lbs. DOOOOOO 1-2 5 20. oooo .46 or % f. 20.838 19.688 000000 15-32 .46875 iS-75 ooo .40964 18.5567 I7-5326 00000 7-16 4375 17-5 00 . 3 6 4 8or% 1. 16.5254 I5-6I34 oooo 13-32 .40625 16.25 .324 86 or %1. 14.7162 13.904 000 3-8 375 15- i .2893 13-1053 12.382 oo 11-32 34375 13-75 2 257 63 or X f- 11.6706 11.0266 o 5-16 3125 12.5 3 .229-42 10.3927 9.819 2 X 9-32 .28125 11.25 4 .20431 or'/ 5 f. 9-2552 8-7445 2 17-64 .265625 10.625 5 .18194 orjl^ 1. 8.2419 7.787 3 1-4 25 10. 6 . 162 02 7-3395 6.934 5 4 15-64 234375 9-375 7 .14428 6-5359 6.1751 5 7-32 .21875 8-75 8 .1284901 > f. 5.8206 5-4994 6 13-64 .203125 8.125 9 "443 5-1837 4.8976 1 11-64 -1875 .171875 6-875 10 ii . 101 89 or Vio f. .090742 4.6156 4.1106 4. 360 9 9 15625 6.25 12 .080808 3.6606 3-4586 xo 9-64 .140625 5-625 13 .071 961 3-2598 3.0799 II 1-8 .125 5- 14 .064084 2-903 2.7428 12 7-64 109375 4-375 15 .057068 2.5852 2.4425 3 3-32 9375 3-75 16 .050 82 or Vaof. 2. 302 I 2.1751 14 5-64 .078 125 3-125 17 045257 2.050 I 1-937 X5 9-128 .0703125 2.8125 18 .040303 1.8257 1-725 16 1-16 .062 5 2-5 '9 03589 1.6258 1.5361 17 9-160 -05625 2.25 20 .031 961 1.4478 1-3679 18 1-20 5 2. 21 .028462 1.2893 1.2182 19 7-160 04375 i-75 22 025347 1.1482 1.0849 20 3-8o 375 23 .022571 1.0225 .96604 21 x 1-320 034375 i-375 24 .0201 .910 53 .86028 22 1-32 03125 1.25 25 .0179 .81087 .766 12 23 9-320 .028 125 1.125 26 -01594 .72208 .68223 24 1-40 .025 i. 2 7 .014195 64303 60755 25 7-320 .021875 875 28 .012641 57264 .541 03 26 3-160 .01875 75 2 9 .011257 .50994 .4818 27 i 1-640 .0171875 6875 3 .010025 454 J 3 42907 28 1-64 .015625 .625 3 1 .008928 404 44 .382 12 29 9-640 .014062 5 5625 32 .00795 .36014 .34026 30 1-80 .012 5 5 33 .00708 .32072 .30302 31 7-640 .0109375 4375 34 .006304 28557 .26981 32 13-1280 .010 156 25 .40625 35 .005614 25431 .24028 33 3-320 009375 375 36 .005 .226 5 .214 34 11-1280 00859375 34375 37 004453 .201 72 .19059 35 5-640 .0078125 3125 38 .003965 .17961 .1697 36 9-1280 .007031 25 .28125 39 .003531 15995 I5H3 37 17-2560 .006640625 .265 625 40 .003144 .14242 13456 38 1-160 .00625 25 In the practical use and application of the U. S. Gauge, a variation of two and one-half per cent, either way may be allowed. Wr't Iron. Specific Gravities 7.704 Weights of a Cube Foot 481.75 " Inch 2787 Steel. 7.806 Copper. .2823 Brass. 8.218 543- 6 .3146 i WEIGHTS OF IRON, STEEL, COPPEE, ETC. Iron, Steel, Copper, and. Brass (Birmingham Gauge.) IPlates. No. of Gauge. Thickness. Iron. PER SQCAI Steel. IE FOOT. Copper. Brass. Inch. Lbs. Lb8. Lbs. Lbs. 0000 .454 or ^ f uU 18.2167 18.4596 20.5662 | 19.4312 000 425 17.0531 17.2805 19.2525 i 18.19 00 .38 or f full 15.2475 15.4508 17.214 | 16.264 34 or*. " 13-6425 13.8244 15402 14.552 i 3 12.0375 12.198 13-59 12.84 2 .284 11-3955 n-5474 12.8652 12.1552 3 .259 or i full 10.3924 10.5309 H.7327 11.0852 4 .238 9-5497 9.6771 10.7814 10.1864 5 .22 8.8275 8.9452 9.966 9.416 6 .203 or i full 8.1454 8.254 9.1959 8.6884 7 .18 or T \ light 7.2225 7.3188 8.154 7.704 8 .165 or J " 6.6206 6.7089 7-4745 7.062 9 .148 or i full 5.9385 6.0177 6.7044 6.3344 JO .134 5.3707 5.4484 6.0702 5-7352 ii .12 or ^ light 4-815 4.8792 5.436 5> I 3 6 12 .109 4.3736 4.4319 4.9377 4.6652 13 .095 or ^ light 3.809 3-8627 4.3035 4.066 14 .083 3.3304 3.3748 3-7599 3.5524 15 .072 2.889 2.9275 3.2616 3.0816 16 .065 2.6081 2.6429 2-9445 2.782 i? .058 2.3272 2.3583 2.6274 2.4824 18 049 or ihj light 1.9661 1.9923 2.2197 2.0972 19 .042 1.6852 1.7077 1.9026 1.7976 20 : .035 1.4044 1.4231 1.5855 1.498 21 .032 1.284 1.3011 1.4496 1.3696 22 .028 I -i235 1.1385 1.2684 1.1984 2 3 025 or ^ 1.0031 1.0165 1.1325 1.07 24 .022 .8827 .8945 .9966 .9416 25 .02 or^ .8025 .8132 .906 .856 36 i .018 .7222 .7319 .8i54 .7704 27 : .Ol6 .642 .6506 .7248 .6848 28 .014 5617 .5692 -6342 .5992 29 .013 5216 .5286 5889 5564 30 .012 .4815 .4879 5436 5136 31 .01 or^ .4012 .4066 453 .428 32 .009 .3611 3659 .4077 .3852 33 .008 .321 3253 3624 -3424 34 .007 .2809 .2846 .3171 .2996 35 .005 or 2W .2006 .2033 .2265 .214 36 ; .004 or ^ .1605 .1626 .1812 .1712 Thickness of Sheet Silver, Grold, etc. By Birmingham Gauge for these Metals. No. Inch. No. Inch. No. Inch. No. Inch. No. Inch. No. Inch. i 2 3 4 5 6 .004 .005 .008 .01 .013 .013 7 8 9 10 ii 12 .015 .016 .019 .024 .029 -034 13 T 4 15 16 J 7 18 .036 .041 .047 .051 -057 .061 19 20 21 22 23 24 .064 .067 .072 .074 -077 .082 j 25 26 27 28 29 30 095 .103 "3 .12 .124 .126 31 32 33 34 35 36 133 143 .145 . I4 8 .158 .167 I2O WEIGHTS OF IKON, STEEL, COPPER, ETC. Wrought Iron, Steel, Copper, and Brass Wire. American Gauge, f. full, 1. light. No. of Gauge. Diameter. Iron. PER LINE.A Steel. L FOOT. Copper. Brass. Inch. Lbs. Lbs. Lbs. Lba. oooo .46 or T 7 ^ f. .56074 .56603 .640513 .605 176 ooo .400. 64*" .444683 .448 879 .507 946 .479 908 00 .364 8 or 1 1. .352 659 .355986 .402 83 .380666 o .324 86 orfg f. .279665 .282 303 .319451 .301 816 i .2893 .221 789 .223 891 253 342 239 353 2 .257 63 or i f. .175888 .177 548 .200911 .189818 3 .229 42 .13948 .140 796 .159323 .150522 4 .204 31 or \ f. .H06l6 .11166 .126353 .119376 5 . 18194 or A I- .087 72 .088 548 .1002 .094666 6 .16202 .069565 .070 221 .079 462 075 075 7 .14428 055 165 .055 685 -063 013 .059 545 8 .128 49 or \ f. 043 751 .044 l6 4 .049 97 .047219 9 .11443 .034699 .035 026 .039 636 037 437 10 .101 89 or ^ f. .027 512 .027 772 .031 426 .029 687 ii .090 742 .021 82 .022 026 .024 924 .023 549 12 .080808 .017304 .017468 .019766 .018 676 J 3 .071 961 .013 7 22 .013 851 .015 674 .014809 14 .064084 .010 886 .010989 .OI2435 .on 746 15 .057068 .008631 .008712 .009859 .009315 16 i? .050 82 or ^j. f . .045 257 .006845 .005 427 .006 009 .O07 819 .005 478 .006 199 .007 587 .005 857 18 .040 303 .004 304 .004 344 .004 916 .004 645 19 035 89 .003 413 .003 445 .003 899 .003684 20 .031 961 .002 708 .002 734 .003094 .00292 21 .028 462 .002 147 .002 167 j .002 452 .002317 22 025 347 .001 703 .001 719 ! .001 945 .001 838 23 .022 571 00135 .001 363 , .001 542 .001 457 24 .020 i or ^ f . .OOI 071 .001 081 ! .001 223 .001 155 25 .0179 .000 849 i .000 857 i .000^69 9 .000 916 3 26 .015 94 .0006734 .000-679 7 .000 769 2 .000 726 7 2 7 .014 195 .000534 .000 539 i .000 609 9 .000 576 3 28 .012 641 .000 423 5 .000 427 5 .000 483 7 .000 457 29 .on 257 .000 335 8 .000 338 9 .0003835 .000 362 4 30 .010 025 or i f. .000 266 3 .000 268 8 .000 304 2 .000 287 4 31 .008 928 .000211 3 .0002132 .000 241 3 .000228 32 .00795 .000 167 5 .000 169 i .000 191 3 .0001808 33 00708 .000 132 8 .000 134 i .000 151 7 .000 143 4 34 .006304 .0001053 .0001063 .0001204 .0001137 35 .005 614 .000 083 66 .000 084 45 .000 095 6 ; .000 090 i 36 005 or ^ .00006625 .00006687 .0000757 .0000715 37 .004 453 .000 052 55 .000 053 04 .000 060 03 .000 056 7 38 .003965 .000 041 66 .000 042 05 .000 047 58 .000 044 9 39 03 53i .000 033 05 .000 033 36 .000 037 75 .000 035 6 40 .003144 .000 026 2 .000 026 44 .000 029 92 .000 028 2 8.88 8.^86 Weights of a Cube Foot . . 485.87 490.45 554.988 524.16 " " Inch.. .2812 .2838 .3212 3033 Specific Gravities to determine the computations of these weights were made by author for Messrs. J. R. Browne & Sharpe, Providence, R. I. WEIGHTS OF IBOX, STEEL, COPPER, ETC. 121 Iron, Steel, Copper, and. Brass AV^ire. Birmingham Wire Gauge, f. full, 1. light. T PER LINEAL FOOT auge. Diameter. Iron. | Steel. Copper. Brass. Inch. Lbs. Lbs. Lbs. Lbs. xxx) .454 or ^ f. .546 207 551 36 .623 913 .589 286 ooo .425 .478 656 .483 172 .546 752 .516407 oo .38 or | f. .38266 .38627 437099 .41284 o .34 or J f. 30634 .30923 .349921 .3305 i -3 2385 .240 75 .272 43 257 3 1 2 ' .284 .213738 215 755 .244 146 .230 596 3 .259 or * f. .177765 .179442 .203 054 .191 785 4 .238 .150 107 .151523 .171461 .161 945 5 .22 .12826 .12947 .146507 138376 6 .203 or i f . .109204 .110234 -12474 .117817 7 .18 or&L .08586 .086 667 .098 075 .092 632 8 .165 or J 1. .072 146 .072 827 .08241 .077 836 9 .148 or iL f. .058 046 058 593 .066303 .062 624 io .134 .047 583 .048 032 .054 353 .051 336 n .12 or ^ 1. .038 1 6 .038 52 .043 589 .041 17 12 .109 .031 485 .031 782 .035 964 .033968 13 .095 or T i ff 1. .023 916 .024 142 .027 319 .025 802 14 .083 .018 256 .018428 .020 853 .019696 15 .072 .CU3 728 .013 867 .015692 j .014821 16 .065 .on 196 .on 302 .012 789 .012 079 J 7 .058 .008915 .008999 .010 183 .009618 18 .049 or -^ 1. .006363 .006423 .007268 .006864 19 .042 .004675 .004 719 .005 34 .005 043 20 035 .003 246 .003 277 .003708 .003 502 21 .032 .002 714 .002 739 .003 I .002928 22 .028 .OO2 078 .002 097 .002 373 .002 241 23 025 or ^ .001 656 .001 672 .001 892 .001 787 24 .022 .001 283 .001 295 .001 465 .001 384 25 .02 or -I* .001 06 .OOI 070 .OOI 211 .001 144 26 .018 .000 858 6 .000 866 7 .000 980 7 .000 926 3 27 1 .Ol6 .0006784 .000 684 8 .000 774 9 .000 731 9 28 .014 .000 5194 .000 524 3 .000 593 3 . 0005604 29 .013 .0004479 .0004521 1.0005116 0004832 30 .012 .000 381 6 .000 385 2 ! .000 435 9 .000 4117 3 1 .01 or ^j .000265 .000 267 5 .000 302 7 1 . 0002859 32 .009 .0002147 .000 216 7 .000 245 2 j .000 231 6 33 .008 .000 169 6 .000 1712 .000 193 7 ooo 183 34 .007 .000 129 9 .000 131 i -ooo 148 3 ooo 140 i 35 .005 or YOU .00006625 1 .00006688 .00007568 1 .00007148 36 .004 or ^ .000 042 4 j .000 042 8 ! .000 048 43 .000 045 74 Thickness of IPlates. No. Inch. No. Inch. No. Inch. No. Inch. I 3 12 5 9 .156 25 17 .056 25 25 .02344 2 .281 25 io .140625 18 05 26 .021 875 3 .25 n .125 19 043 75 27 .020312 4 234 375 I2 .1125 20 0375 28 .018 75 5 .218 75 13 .1 21 034 375 29 .017 19 6 .203 125 14 .0875 22 031 25 30 .015 625 7 -1875 15 075 23 .028 125 3 1 .014 06 8 .171 875 16 .0625 2 4 .025 32 .0125 122 WIKE GAUGES. WIKE GAUGES. (English.) Warrington (Rylands Brothers). No. Inch. No. Inch. No. Inch. No. Inch. No. Inch. 053 .047 .041 .036 0315 .028 7/o 6/0 5/o 4/o 3/0 2/0 i i 2 3 4 5 .326 3 .274 25 .229 .209 6 8 9 10 10.5 .191 .174 159 .146 133 .125 ii 12 13 H 15 16 .117 .1 .09 .079 * .0625 i? 18 J 9 20 21 22 Sir Joseph Whitworth & Co.'s. No. Inch. No. Inch. No. Inch. No. Inch. No. Inch. i .001 14 .014 34 034 85 085 240 .24 2 .002 15 .015 36 .036 90 .09 200 .26 3 .003 16 .016 38 .038 95 .09 280 .28 4 .004 17 .017 40 .04 100 .1 300 3 5 .005 18 .018 45 045 no .11 325 325 6 .006 19 .019 50 05 120 .12 350 35 7 .007 20 .02 55 055 135 135 375 375 8 .008 22 .022 60 .06 150 15 400 4 9 .009 2 4 .024 65 -065 165 .165 425 425 10 .01 26 .026 70 .07 180 .18 450 45 ii .on 28 .028 75 075 200 .2 475 475 12 .012 30 03 80 .08 22O .22 500 5 13 .013 32 .032 Sir Joseph Whitworth, in 1857, introduced a Standard Wire-Gauge, rang- ing from half an inch to a thousandth, and comprising 62 measurements. It commences with least thickness, and increases by thousandths of an inch up to half an inch. Smallest thickness, YirVff ^ an mcn i ^ s No. i ; No. 2 is YJJ%^, and so on, increasing up to No. 20 by intervals of y<^ ' ^ rom No. 20 to No. 40 by YI&TT ; and from No. 40 to No. 100 by y^^. The thicknesses are designated or marked by their respective numbers in thou- sandths of an inch. This gauge is entering into general use in England. !N"ew Standard Wire Grange of Grreat Britain, 1884. No. Inch. No. Inch. No. Inch. No. Inch. 7/o .5 8 .100 22 .028 36 .0076 6/0 .464 9 .144 23 .024 37 .0068 5/0 432 10 .128 24 .022 38 .006 4/ 4 ii .Il6 25 .02 39 .0052 3/0 372 12 .104 26 .018 40 .0048 2/0 348 13 .092 27 .0164 4i .0044 O 324 14 .08 28 .0148 42 .004 I .3 15 .072 29 .0136 43 .0036 2 .276 16 .064 30 .OI24 44 .0032 3 .252 J 7 .056 31 .OIl6 45 .0028 4 .232 18 .048 32 .0108 46 .0024 5 .212 !9 04 33 .OI 47 .002 6 .192 20 .036 34 .0092 48 .OOl6 7 .I 7 6 21 .032 35 .0084 49 .0012 No. 50, .001 inch. WIKE GAUGES. GAS PIPES AND WIRE COKD. 123 French (Jauges de Fits de Fer). French wire-gauges, alike to the English, have been subjected to variation, Following table contains diameters of the numbers of the Limoges gauge. 'Wire-Q-aiage (Jauge de Limoges). Number. ; Millimetre. Inch. Number. Millimetre. Inch. Number. Millimetre. Inch. 39 .0154 9 1.35 0532 18 3-4 134 I 45 .0177 10 1.46 0575 19 3-95 .156 2 56 .0221 ii 1.68 .0661 20 4-5 .177 3 .67 .0264 12 1.8 .0706 21 .201 4 79 .0311 13 1.91 .0752 22 5^5 .222 5 9 0354 14 2.02 0795 23 6.2 .244 6 I.OI .0398 15 2.14 .0843 24 6.8 .268 7 1. 12 .0441 16 2.25 .0886 8 1.24 .0488 17 2.84 .112 Number. Millimetre. For Inch. (3-alva Number. mized Millimetre. Iron "V Inch. Vire. Number. Millimetre. Inch. I .6 .0236 9 1.4 0551 17 3- ! .us 2 7 .0276 10 i-5 .0591 18 3-4 134 3 .8 0315 ii 1.6 .06 3 19 39 154 4 9 0354 12 1.8 .0709 20 44 173 5 i. 394 13 2. .0787 21 4.9 193 6 i.i 0433 !4 2.2 .0866 22 5-4 .213 7 1.2 0473 15 2. 4 0945 23 5-9 .232 8 i-3 .0512 16 2-7 .106 For TVire and Bars. Mark. Millimetre. Mark. {Millimetre. Mark. Millimetre. Mark. Millimetre. Mark. Millimetre. ~~P~ 5 7 12 13 20 19 39 25 70 I 6 8 13 14 22 20 44 26 7 6 2 7 9 14 15 24 21 49 27 82 3 i 8 10 15 16 27 22 54 28 88 4 9 ii 16 17 30 23 59 29 94 5 10 12 18 18 34 24 64 30 100 6 ii Diameter. Thickness of Thickness. II Diameter. G-as !> Thickness. Lpes. Diameter. Thickneu. i-5 to 3 4 "6 25 375 (I 8 to 10 12 " 13 ^625 14 to 15 16 " 48 75 .875 Copper "Wire Cord.. Circumference and Safe Load. Inch. Inch. Inch. Inch. Inch. Inch. Ins. Ina Circumference 25 .375 .5 .625 .75 i 1.125 1-25 Safe load in Lbs 34 50 75 112 168 224 336 448 Zinc sheets. Thickness and 'Weight per Square Foot. Inch. i Inch. i inch. .031 1 = 10 oz. .0534 = 14 oz. .0686 = 18 oz. .0457 = 12 OZ. .O6l I = l6 OZ. .0761 = 20 OZ. 124 WEIGHT AND STRENGTH OF WIRE, IRON, ETC. WEIGHT AND STRENGTH OF WIRE, IRON, ETC. "Weight arid. Strength, of "Warring-ton Iron "Wire, Manufactured by Rylands Brothers. (England.) Weight per 100 Lineal Feet. No. Diame- ter. Weight Breaking An- nealed. Weight. Bright. No. Diameter. Weight. Breaking An- nealed. Weight Bright. Gauge. Inch. Lbs. Lbs. Lbs. Gauge. Inch. Lbs. Lbs. Lbs. 7/0 X 64.46 3490 5233 9 .146 5-5 208 447 6/0 %Z 56.66 3 066 4603 10 133 4-43 247 | 370 5/o %6 49-36 2673 4OOO 10.5 .125 4-03 218 327 4/0 /Is 42-53 2303 3457 ii .117 3-53 191 288 3/o % 36.26 1963 2945 12 .1 2.66 145 2I 7 2/O /&, 30.46 1653 2473 13 .09 2.1 H3 169 O .326 27.36 I486 2226 14 .079 1.6 8 7 130 I 3 23-3 1257 1885 15 .069 1.23 66 99 2 .274 19.36 1046 1572 16 .0625 .96 53 77 3 25 16.13 873 1309 17 053 73 39 59 4 .229 13-53 732 1098 18 .047 .56 3i 46 5 .209 11.26 610 913 19 .041 43 23 35 6 .191 94 509 763 20 .036 33 18 27 7 .174 7.8 422 633 21 .031 25 .26 14 21 8 159 6-53 353 519 22 .028 .2 ii 16 To Compute Length of 1OO JPoxiiids of Wire of a Griveii Diameter. RULE. Divide following numbers by square of diameter, in parts of an inch, and quotient is length in feet. 37.68 for wrought iron. I 33.42 for copper. I 28 for silver. 37.45 for steel. 34-4* for brass. | 15.3 for gold. 13.64 for platinum. WINDOW GLASS. Thiolrness and "Weight per Sqnare Foot. No. Thickness. Weight. No. Thickness. Weight. No. Thickness. Weight. 12 13 15 16 Inch. 59 .063 .071 .077 Oz. 12 13 15 17 9 21 24 Inch. .083 .091 .1 .III Oz. I? 19 21 24 26 32 36 42 Inch. .125 154 .167 .2 Oz. 26 $ 42 Terne Plates. Teme Plates Are of iron covered with an amalgam of lead. Thickness and 'Weight of Gralvanized. Sheet Iron. Sheet 2 Feet in Width by from 6 tog Feet in Length (M. Le/erts). .?! &5 Weight per Sq. Foot. t* t> * o Weight || & per J: Sq. Foot. a Weight per Sq Foot. & Si Weight per Sq/Foot. hi t No. 17 16 1 J 5 Weight per Sq. Foot. 1 & II Weight per Sq. Foot. No. 2 9 28 2 7 Oz, 12 13 J 4 No. 26 25 24 Oz. H No. 15 23 l6 22 18 H 21 Oz. 20 22 2 4 No. 20 19 18 Oz. 27 30 35 Oz. 36 42 4 6 No. 14 13 12 j Oz. 53 61 70 WEIGHTS OF METALS. WROUGHT IKON AND STEEL. 125 Weights of Square Rolled. Iron and. Steel, From .125 to 10 Inches. ONE FOOT IN LENGTH. Iron, 485 Ibs. Steel, 489.6 Ibs. PER CUBE FOOT. SIDE. IRON. STEEL. SIDE. IRON. STEEL. SIDE. IROX. STEEL. Ins. Lbs. Lbs. Ins. Lbs. Lbs. Ins. Lbs. Lbs. .125 053 053 2-75 25-47 25.71 6.25 I3I.6 132.8 1875 .118 .119 875 27.84 28.1 375 137 138.2 25 .21 .212 3 30.3 1 30.6 -5 142.3 143.6 3125 329 333 .125 32.89 33-2 .625 147-9 149.2 -375 474 .478 25 35-57 35-92 75 153-5 154.9 4375 .645 .651 375 38.57 38.73 875 159.2 160.8 5 .812 85 5 41.26 41.65 7 165 166.6 5625 1.066 1.076 .625 44.26 44-68 .125 171 172.6 .625 1.316 1.328 75 47-37 47.82 25 177 178.7 .6875 1.592 1.608 875 50.37 51-05 175 183.2 184.9 75 1.895 I -9 I 3 4 53.89 54-4 5 189.5 191.3 .8125 2.223 2.245 .125 57.85 .625 195.8 197.7 875 2-579 2.608 25 60.84 61.41 75 202.3 204.2 9375 2.96 2.989 375 64.17 65.08 .875 208.9 210.8 i 3-368 3-4 ' 5 68.2 68.85 8 215.6 217.6 .125 4.263 4-303 -625 72.05 72.73 .125 222.4 224.5 25 5.263 5-312 -75 75-99 76-71 .25 229.3 231.4 375 6.368 6.428 875 80.05 80.8 1 375 236 238.5 5 7.578 7-65 5 84.20 85 5 2434 245.6 5625 8.893 8.978 .125 88.47 89-3 .625 250.6 252.9 *75 10.31 10.41 25 92.83 93.72 75 257.9 260.3 875 11.84 "95 375 98.23 875 265.3 267.9 2 13.37 13-6 5 101.9 102.8 9 272.8 275.4 .125 15.21 15.35 .625 106.6 107.6 25 288.2 290.9 .25 17.08 17.22 75 111.4 112.4 5 304 306.8 375 19 19.18 875 116.3 117.4 75 320.2 323.2 5 21.05 21.25 6 121.3 122.4 875 328.6 331.6 .625 23.21 23-43 .125 127.6 10 336.8 340 Weight of .A^ngle Iron, From 1.25 to 4.5 Inches. ONE FOOT IN LENGTH. Thickness measured in Middle of each Side. L EQUAL SIDES ' 1 -L UNEQUAL SIDES. L UNEQUAL SIDES. Thick- Thick- Thick- Sides. ness. Weight Side.. * Weight. Sides. ness. Weight Ins. Inch. Lbs. Ins. Inch. Lbs. Ins. Inch. Lbs. I.25XI.25 .1875 3 X2.5 .375 6.2 5 6 X3-5 .625 18 1.5 Xi.5 1875 2 3-5X3 -4375 7-75 6 X4-5 -625 20 I.75XI.75 .25 3 3.5X3 -4375 9.6 2 X2 .25 3-5 4 X3 -5 ii TT 2.25X2.25 .3125 4-5 4 X3-5 -5 "5 2 X 2.375* 375 5-5 2.5 X2.5 3125 5 4 X3-5 -5 "75 2.5X2.875 375 6-5 3 X3 375 7 4.5X3 -5 "75 3-5X3.5 4375 10.5 3-5 X3-5 4375 9 5 X3 -5 12.65 V, f 4375 11 4 X4 5 12.5 5 X3 -5625 13.7 4 3o "^ 75 A O 4-5 X4-5 5 14 5-5X3.5 -5 14-5 4 X3-5 .75 13.5 4-5 X4-5 16 5-5X3.5 .5625 15-6 * This column gives depth of web added to the thickness of base or flange. L* 126 WEIGHTS OF METALS. WROUGHT IRON AND STEEL. "Weights of Round Rolled Iron, and Steel, From .125 to 10 Inches. ONE FOOT IN LENGTH. Iron, 485 Ibs. Steel, 489.6 Ibs. PER CUBE FOOT. Diameter. IRON. STEEL. Diameter IRON. STEEL. Diameter IRON. STEEL. Ins. Lbs. Lbs. Ins. Lbs. Lbs. Ins. Lbs. Lbs. .125 .041 .042 2-75 20. 01 20.2 6.25 103-3 104.3 -1875 093 .094 .875 21.87 22.07 375 107.7 108.5 2 5 165 .167 3 23.81 24.03 5 in. 8 II2.8 3125 258 .261 .125 25-83 26.08 .625 116.4 117.2 375 372 -375 25 27.94 28.2 75 120.5 121.7 4375 506 .511 375 30-13 30.42 875 124.9 126.2 5 .661 .667 5 32.41 32.71 7 129.6 130.9 837 .845 .625 34-76 35-09 .125 134.2 135-6 .625 1-033 1.043 75 37-2 37-56 25 139 140.4 .6875 1-25 1.262 .875 39-72 40.1 -375 143-8 145-3 75 1.488 1.502 4 42.33 42.73 .5 148.8 150.2 .8125 1.746 '.763 .125 45.01 45-44 .625 153-8 155-2 .875 2.025 2.044 25 47.78 48.24 75 158.9 160.3 9375 2-325 2-347 375 50.63 51.11 875 164.1 165.6 i 2.645 2.67 5 53-57 54-07 8 169.3 171 .125 3-348 3-379 .625 56.59 57-12 .125 174.6 '76.3 25 4'*33 4- I 73 75 59-69 60.25 -25 1 80. i 181.8 -375 5 5-049 875 62.87 63-46 375 185-5 187-3 5.952 6.008 5 66.13 66.76 5 191.1 .625 6.985 7.051 .125 69.48 70.14 .625 196.6 198.7 75 8.104 8.178 25 72.91 73-6 75 202.5 204.4 .875 9-3 9-388 -375 76.43 77.16 .875 208.1 210.3 2 .125 10.58 "95 10.68 12.06 :L S 80.02 83-7 80.77 84.49 9 25 214.3 226.3 216.3 228.5 25 13-39 I3-52 75 87.46 88.29 5 238.7 241 375 14.92 15.07 875 91.31 92.17 -75 251-5 253-9 16.53 16.69 6 95-23 96.14 .875 259.5 260.4 625 18.23 18.4 .125 103.3 100.2 10 264.5 267 "Weight of Steel Angles. From .75 to 7 X 3.5 Inches. ONE FOOT IN LENGTH. Thickness measured in middle of each side. SIDE. EQUAL Thick- ness. SIDES Area. Weight. SIDES. Thick- ness. U Area. NEQUAL Weight. SIDES. SIDES. Thick- Area. Weight. Ins. Ins. Sq.Ins Lbs. Ins. Ins. Sq. Ins Lbs. Ins. Ins. Sq.Ins Lbs. 75 .125 17 .6 i.375Xi 25 53 1.8 5X3-5 5 4 13.6 .875 .125 .31 7 2 Xi-375 25 .78 2-7 5X3-5 .625 4.92 16.8 i .125 .24 .8 2.25X1.5 -25 .88 3 5X3-5 .75 19.8 1.25 I2 5 30 i 2.25X1.5 5 1.63 5-5 5X3-5 .875 6.' 67 22.7 1.5 25 .69 2.4 2.5 X2 25 i. 06 3-7 5X4 5 4-25 14-5 1.75 .25 .8l 2.8 2-5 X2 5 2 6.8 5X4 625 5-23 17.8 2 25 94 3-2 3 X2 25 I.I9 4 5X4 75 6.19 21. I 2.25 25 1.06 3-7 3 X2 5 2.25 7-7 5X4 875 7.11 24.2 2-5 25 1.19 4.1 3.25X2 25 1.25 4-3 6X3-5 5 4-5 15-3 2-75 25 4-5 3.25X2 5 2. 3 8 8.1 6X3.5 .625 5-55 18.9 3 5 2-75 9-4 3-5 X2.5 25 1.44 4-9 6X3-5 75 6.56 22.3 3-5 5 3-25 ii. i 3-5 X3 75 4-31 14.7 6X3-5 i 8-5 28.9 4 5 3-75 12.8 4 X 3 5 3-25 ii 6X4 5 4-75 16.2 4 75 5-44 18.5 4 X 3 75 4.69 16 6X4 625 5-86 20 5 5 4-75 16.2 4 X3-5 5 3-5 11.9 6X4 75 6-94 23.6 5 75 6-94 23.6 4 X 3 -5 75 5-o6 17.2 6X4 i 9 30.6 5 i 9 30.6 4-5 X3 5 3-5 11.9 7X3-5 5 5 17 6 5 5-75 19.6 4-5 X 3 75 5-o6 17.2 7X3-5 .625 6.17 21 6 75 8-44 28.7 5 X3 5 3-75 12.8 7X3-5 75 24.9 $ ii 37-4 5 X 3 75 5-44 18.5 7X3-5 i 9-5 32-3 WEIGHTS OF METALS. 127 :> 00 OOVI ON ONOn On*.OJOJMMMQ vOOO OOvl O\ ONOn On*.OJOJMMM < O vj M On 00 M ON OJ vl w on 00 M O\ OJ vi w on OO M ON OJ vl M on CO M J On M vi on M vionM vionM vionM vlonM VJCT. M vl on ON ON ON ONOn OnOnOnOn^^^ vj On OJ M VO ON*. M O OO ON M 1 On * OJ M Q "O OOVI ONOi * to MM \O OOVI ONOn OOOJ MM ^ ^ __ O vl on OJ M vO vl On OJ M VO O^*- 10 O OO ON*. O VO OOVI ONOn 4k. OJ M M vo vl ONOn *. Co M ? \O OOVI VO On ONOJ *. M M vl v) on On OJ M ON ON ON ONOn OnOnOnOn*.*.*.*.*.OJOJOJOJM; *t ) J 3WMMMM . . . | OOOn OJ MOvionOJ M OO ON* M O OO ON*. MVOvlOnOJ M vO v| *. (0 O GO ON*. ' VO 00 ONOn *. M K) VO vl ONOn *. OJ M VO OO ONOn *. M 10 NO CO ONOn OJ 10 i OOOn 00 OO^J -Ul t/lOlCn^- tM vO ONOJ M OO O\OJ O OOt/l 10 O ^J - COtO(JivONO\ OJVlO4>-OOMLn CC4- ON N> 004^ O\ tO 004>> O\ M ^ ^ . \O O\OJ M OOLn u> O *^> VTI ONVO W OO4>->J M4>. OOMCnvO tO. O\ 10 OO4^ ON 10 OO-U ON M OO 00 00v ^1 jl vl p\ ON Ox O\Ut y tn W vb vj 4w M vO OstO M ooi/i >v O vp vp vp 00 00 povj xj vi ON ON ON OM/t -n^4>.f'4>.OJOJOJIOK)(OMMM ^ ^ M OOC.JJ M 0001 N) vb ON K) vb ONOJ O vj OJ O vj 4>- M OO-J M OOOi N vb (In io vb O\ g* -f. to O OOvl on OJ N O CQV) On OJ M O 00 ONOn OJ - O OO OxOn OJ N O OO ONOn OJ ." M4-V|M4.v) OJvj OJ ONVO M ^vO OJOnOOK)OnOONOnOOM4i.vJ 4>. O vp VO -4>.OnOnO\ON ON VI vi 00 OO OOVO vO M . M M P P OJOJMMMMpppvpVppOOO OOVl vj ON ONOn OnOn4>.*.OJOJMtOMMjH f. On M ON M OO*. VO On M vj M OO*. O ONMVJOJVO*. O ONM OOOJ vb On M ON 5o OO ^ M VI 01 OJ 00 ONOJ M VO vj *. (0 O 0001 OJ M \O ON*- M O VI on OJ M OO ON*. * OOOn 00 ON 00 ON*. M 00 ON*. M\OVJ4^ NvOvj*. (0 VO vj on OJOJMMMMMppvpvpOOpO OOVJ VI ON ONOn OnOn4k.*.OJOJMMMMH < ^ ONMVJOJVO*. O ONMvlOJVOOn O ONM OOOJ VO On M ON M 00*. vb On M vj M OO O" OOOn OJ vl on OJ OOOn OJ OOOn M OOOn OJ v) On OJ OOOn M OOOn ? ON ONOn V^^ OJt i JNMMM PP v P? P ^ 1 "^ P" 1 P^V On-^-fk.OJOJMMMM vb OJ OOOJ vl (0^1 K) ONM ONOOn bonvb*.vboj OOOJ OOMvj !o ONM ONMOn b VI *. M OOOn OJ v) *. M vO ONOJ O vj *. to vO ONOJ O OOOn M VO ON*. M OOOn * ONOOM M*. ONOO ON*. ON OO M OJ j ONOn on*.*-OJOJ iO M M M p OvOvp OOvl -sj QN ONOn On*.*.OJOJMIOMMl *. VO * OOOJ OOMvl M ONM ONMOn boivO*-VOOJ OOOJ vlIovlMONMOnb' vi 4*. OC*. M OO'/i M vO ONOJ vl OJ vl *. OO*. M OOOn M VO On M vO ON MOn OOJvi M4*. QOMOnvO . 4*. QOMOnvOOJ ONO*- OOMOn OOM ONVO OJVJ O*. OOMOivb M cr OOOJ MV I *.MVJ4^MVJ*. vl*.Ov|OJ ONOJ O vj OJ Ox " H *. v) VOONMOJOn V) O vp VD OOvl vi ONOn On*.*.OJMMMppvppO OOv) vi ONOn On *. OJ OJ M Mf 4^.^l M* OEMOnvOOJ ONOOJvi M4>- OOMOnVO M ONOOJVI M4^ oOMOnvb M vi OJ VO On M 00*- ONOJ vO On M OO*- VI OJ O Oi M OO*- ONOJ vO On M OO ' OJ M M M p vO vO OOVI vl ONOn *.*.OJMM MQvp OO OOV1 ONOn Ol *. OJ M M M M ONVO M*vlvO M*-VI O MOnv] OOJOn OOOOJ ONOOMOJ ONvO M *. ONVO (0 *. ONMOO*. vjtovO*-M ONOJ vOOiMVIojvOon MvjoJvOOnwvJOJ VO On M ^o ." OJ M*.OJOn*. ONON OOVI OOVO OJ OJ M M O O vO povl VI ONOn -^*.OJMMHpvpOO povl ONOn On 4. OJ M (0 M E ON"M vj OJ 00*. On M ONMV!OJVOOn O^M vToj' QO*^ ^ ^ ^ ^ "(0 ^OJ >O ' " jj S-* s a u. H 3 128 WEIGHTS OF METALS. i 00 GO a ap H 2 rf I o^-fS (^ " & rH M * 5j S ,2 ^ vo o 4 $ I A s M moo PJ moo * inoo PI moo PJ moo T- moo <* moo PJ moo j PJ ro, mvo txoo H N ro mvo r^oo <-> PJ ro mvo t^oo M PJ ro invo t-^oo 00 M H M ^ M pi PJ p? N" T ro ro ro'ro <* 5- ^^^5- m m in in mvo TJ- invo OO N ro ^-vo vo oo ON N ro TJ-VO vo oo N ro -**-vo vo oo Os f^vo in -* TJ- ro N M o ^oo two vO oo tx\o 10 * ro N N iot^ON'H roint^-ONw N 'i-vd 06 d N 4-vO 06 d N Tt-inti.oXw roint^ONi-S H H M M IH mio in\O oo t-svq m * ro PJ M ONOO txvq m -^- ro PJ H ONOO t>.vq m <* ro N M ONOO oo txvo m Tt- ro PI M ONOO r^vo m <* ro PJ M ONOO i>.vo m * ro PJ M TJ-OO IN * ON * ON rooo rooo rooo rooo Pi fx PJ tx Pi' t~. PJ tx M'VO"^ vo' M vb' w vo S lAod d H ro 4-vd t*> ON d pi ro mvo* oo O^ M PJ 4- m t-od d -* ro 4-vd i>. m ro 2"oNtxrn ro oo vo Tt- PJ ON t^ m ro H oo vo -4- PJ ON t^ m ro M oo vo < PJ i ON << ON rooo rooo rotxpj I>-PJVO MVO nvq O inq m o\ * ON * ON rooo rooo < ! PJ PJ P) PJ I ro oomro oo> ci ro invd t~od d H N -4- in\d t^. ON d H ci -4- i I pTr^ro! roovo roHoo mnoo innoo in H t>.inpjoo mpjoompjoo mpi ONVO PJ ONVO J pi ro mvo t>od d M PI* ro invd i>> o\ d * PJ 4- mvo r>.od d H pi -4- mvd i> ON d Mwi-,MMMMMpjpJPJPJPJPJpjpirorororororororoi*- B -MMpjroro-*m mvo vo t^oo COON owpjroro^m mvo vo t^oo oo ON PJ OO * vo PJ ON -5 pj ro 4- mvd i>.00 CJN O M pj ro 4- invd l>> ON O* ~ Pi ro 4- mvd t^OO d^ O H PJ ro MMMHMMWMwpipjpjpjpjpjpjpjpJpjrorororo jg ONOO oo tr- lr^ t-^vo voinin-^--Tj-rororoNPj M M q O ONON ONOO oo t^ txvq vq vO J M pj ro 4- mvd i>sod ON d ** PJ ro 4- nvd i>.od ON d M M pi ro 4- mvo i>.oo' ON d MMHMMMMMlHMPlPlPJPJPJPJPJPJPJPJPlrO . ONOO oo t^ t^-vo vomin-^--^-roropjpiMMq ON ONOO oo t^ t^vq vo m m ON * M pi ro 4- mvd i>.od ON d M pi ro 4- mvd t^-.od ON ON d H pj ro 4- mvd i>-oo ON d . t^ m TJ- N M ONOO vq m ro PJ q ON txvq < f f ^ w oo>m-.od ONO^^^M^M''^'^^' ^'S 2 s N ci PJ PJ et pt c?^ pT . ON rooo TJ- t~< M in <*- ON ^-oo ro t^ PJ vo w m <* rooo PJ i>> M vo m ON * J vo m ro PJ q ON t^vq TJ- pj H ONOO vq m ro PJ q q\ t^vq * PJ M q>oo vq m ro H q pjooini-ifxro vo9 2O 23-7 24.74 25.78 26.83 27.87 29-95 32.03 34.12 36.2 38.28 40.36 28.13 29.37 30.63 31.88 33-13 35.63 38.13 40.63 41.13 43.63 46.13 32.45 33-91 35.36 36.82 38.28 41.19 44.12 47.02 49.95 52.87 55.78 36.6 7 38.33:40 141.67 43.33 46.67 50 53-33 56.67160 [63.33 NOTE. American rolled is slightly heavier. WEIGHT OP HOOP IRON. CAST IRON. METALS. Weight of HOOT> Iron. (D. K. Clark.) ONE FOOT IN LENGTH. As by Wire-gauge used in outh Staffordshire (England). THICKNESS. .625 75 875 i ., ID! 1. 25 H IN INCHES. '25 1-375 '5 i.625| 1.75 2 No. Inch. Lb. Lb. Lb. Lb. Lb Lb. Lbs. Lbs. Lbs. Lbs. Lbs. 21 .0344 .0716 .0861 .1 "5 .129 .144 .158 .173- .197 .201 .229 2O 0375 .0781 .0938 .109 .125 .141 .156 .172 .188 .203 .219 25 19 .0438 .OQII .109 .128 .146 .164 .l82J .2 .219 .238 257 .292 18 OS .104 .125 .146 .167 .188 .208 .229 25 .271 .292 333 *7 0563 .117 .141 .164 .188 .211 234 .258 .28l| .305 .328 375 16 .0625 .13 .156 .182 .208 234 .26 .286 313 -339 -365 .417 15 075 .156 .188 .219 .25 .281 313 344 375 -307 -438 .5 14 .0875 I8 3 .219 .256 .293 329 .366 .402 438 -475 .512 585 13 .1 .208 25 .292 ! .333 375 .416 .458 -5 543 .584 .667 12 .1125 234 .281 328 .375 422 .469 .516 .563 .609 .656 75 II .125 .26 313 365 -417 .469 521 573 .625 .677 729 -833 10 .1406 293 352 .41 .469 527 .586 .645 .703; .762 .82 .938 9 .1563 .326 391 456 .522 587 .652 .717 .783 .848 913 1.04 8 I 7 I 9 -358 43 .501 573 .644 . 7 l6 .78.8 859 -931 1-15 7 .1875 ! .391 .469 547 - 62 5 703 .781 859 ^38 1-02 .09 1.25 6 .2031 423 508 593 -677 .762 .836 .931 1.02 i.i .19 i-35 5 .2188 .456 547 ,638 .729 .82 .912 I 1.09 1.19 .28 1.46 4 -2344 .488 .586 .683 .781 .879 977 1.07 1.17 1.27 37 1.56 CAST IRON. To Compute "Weight of a Cast Iron Bar or Hod, Ascertain weight of a wrought iron bar or rod of same dimensions hi preceding tables, or by computation, and from weight deduct ^th part. Or, As .1000 : .9257 :: weight of a wrought bar or rod : to weight re- quired. Thus, what is weight of a piece of cast iron 4 x 3.75 X 12 inches? In table, page 128, weight of a piece of wrought iron of these dimensions is 50.692 Ibs. Tnen? I000 . 9 Braziers' and Sheathing Copper. BRAZIERS' SHEETS, 2X4 feet from 5 to 25 Ibs., 2.5 x 5 feet from 9 to 150 Ibs., and 3X5 feet and 4X6 feet, from 16 to 300 Ibs. per sheet. SHEATHING COPPER, 14 x 48 inches, and from 14 to 34 oz. per square foot YELLOW METAL, 14 x 48 inches, and from 1 6 to 34 oz. per square foot. Weight o Corrugated Steel Root Plates. Carnegie Steel Co. Dimensions, Thickness. Per Sq. Ft. | Dimensions. Thickness. Per Sq. Ft. Ins. Ins. Lbs. Ins. Ins. Lbs. 8.75X1.5 8.75 X 1.5625 25 3125 ,ol 12.1875X2.75 12.1875 X 2.8125 375 4375 17-75 20.71 8.75 X 1.625 375 !2 1 12.1875X2.875. 5 23.67 METALS. To Com.pu.te "Weight of Mietals of any Dimen- sions or Form- By rules in Mensuration of Solids (page 360 ), ascertain volume of the piece, multiply it by weight of a cube inch, and product will give weight in pounds. 132 WEIGHT OF CAST IRON PIPES. "Weight of Cast Iron 3?ipes or Cylinders. From i to 70 Inches in Internal Diameter. ONE FOOT IN LENGTH. Diameter. Thickn. Weight. Diameter. Thickn. Weight. Diameter. Thickn. Weight. IllS. Inch. Lba. Ins. Inch. Lba. lus. Inch. Lks. Z 25 3-06 4-75 375 18.84 II .875 IOI.85 -375 5-05 5 25.72 "5 5 58.81 1.25 25 3-68 625 32.93 .625 74.28 3125 4-79 75 40-43 75 OX).o6 375 5-97 5 375 19.76 875 106.13 i-5 375 6.89 5 26.95 12 5 61.26 4375 8.31 625 34-46 .625 77-34 5 9.8 75 42.27 75 93-73 x -75 375 7.81 5-5 375 21.59 875 110.42 4375 9-38 5 29-4 12.5 5 63.7I 5 11.03 625 37-52 625 80.4 2 375 8-73 75 45-95 75 97-4 4375 10.45 6 375 23-43 875 114.71 5 12.25 5 31.86 13 5 66.16 2.25 375 965 625 40-59 .625 8347 4375 11.52 75 49.62 75 101.08 5 13.48 6-5 375 25.27 875 II 9 2-5 375 10.57 5 34-31 13-5 5 68.61 4375 12.6 .625 4365 .625 86.53 5 14.7 75 53-3 75 104.76 2 -75 375 11.49 7 5 36-76 875 123.29 4375 14.67 5625 41.7 14 5 71.06 5 '5-93 .625 46.71 .625 89.6 3 375 12.4 75 56-97 75 108.43 5 17-15 7-5 -5 39.21 875 127.58 .625 22.2 5625 44-45 14.5 5 73-51 75 27-57 625 49-77 625 92.66 3-25 375 I3-32 75 60.65 75 112. II 5 18.38 8 5 41.66 875 131.87 .625 2 3-74 5625 47-21 15 5 75-06 75 29-4 625 52.84 625 95-72 3-5 375 14.24 75 64.32 75 115.78 5 19.6 9 5 46.56 875 136.16 .625 25.27 -5625 52.72 15.5 5 78.47 75 31.24 625 58.96 625 98.78 3-75 375 I5.l6 75 71.67 75 119.46 5 20.83 9-5 5 49.01 875 140.44 .625 26.8 5625 55.48 16 625 101.85 75 33-08 625 62.06 75 123.14 4 375 16.08 75 75-35 875 144.73 5 22.05 10 5 51.45 i 166.63 .625 28.33 .625 65.09 16.5 625 104.9 75 34-92 75 79-03 75 126.75 *25 375 17 875 93-2; 875 149.02 5 23.28 10.5 5 53.9i i I7L53 .625 29.86 625 68.15 !7 625 107.97 75 36-76 75 82.7 75 130.48 4-5 375 17.92 875 97.56 875 153-3 5 23.88 n 5 56-36 i 176.43 .625 31.4 .625 71.21 17.5 .625 IJI.03 75 1 38-59 75 86.38 75 I34-I6 WEIGHT OP CAST IKON PIPES. 133 tmeter. Thickn. Weight. Diameter. Thickn. Weight. I Diameter. f Thickn. Weight. Ins. Inch. Lbs. Ins. Ina. Lbs. Ins. Ins. Lbs. J 7-5 .875 157.59 29 218.7 40 875 350.56 I I8L33 .875 256.23 ! I 401.86 *8 .625 II4.I I 294.05 I.I25 453.46 75 137-84 30 75 226.05 : 1.25 505.41 .875 I6I.88 .875 264.8 ; 42 .875 367-69 I 186.23 i 303.86 r 421.45 19 .625 120.23 1.125 343-22 1.125 472.52 75 145.19 3 1 -75 233-41 1.25 529.87 .875 1 70.46 .875 273-38 44 875 384.88 i 106.03 i 313.66 i 441.1 20 .625 126.35 1.125 354.24 1.125 497-58 75 152.54 32 75 240.75 1.25 554-42 .875 179.03 875 281.95 46 875 402.OI i 205.84 i 323-46 1 i 460.07 21 625 132.48 1.125 365.27 1.125 519.64 75 159.89 33 75 248.11 1-25 578.88 .875 187.61 875 290.53 48 875 419.17 i 215.64 i 333-26 i 480.29 22 625 138.61 1-125 376.29 1.125 541.69 75 167.24 34 75 255-46 1.25 603.44 .875 196.19 875 299.11 50 875 436.43 I 225.44 i 343.06 i 499.89 23 .625 144-73 1.125 387.33 1.125 563.75 75 174-59 35 75 262.81 1.25 627.93 .875 204.76 875 307.68 52 875 453-49 i 235.24 i 352.87 i 5^-5 24 .625 150.86 1.125 398.35 ! 1.125 585-81 75 181.95 36 75 270.16 : 1.25 654.42 875 213.34 875 316.26 j 55 875 479-23 i 245-04 i 362.67 | i 548.9 25 625 156.98 1.125 409.28; 1.125 618.91 75 189.3 fcfer-rttt- 1.25 456-37 { 1.25 689.21 .875 221.92 37 75 277-5I 1 58 i 578.29 i 254.85 875 324.84 1.125 651.96 26 .625 163.11 i 372.47 | 1.25 725.93 75 i 196-65 1.125 420.4 1-375 800.22 875 230.5 1.25 468.65 60 i 597-92 i 264.65 38 75 284.86 1.125 674.01 27 .625 169.23 875 333-41 1-25 i 750.45 75 204 i 382.27 1-375 827.17 875 239.07 I.I25 43I-4I 65 i 646.93 i 274-45 1.25 480.89 1.125 729.18 28 .625 175.36 39 75 292.21 1.25 811.73 75 211-35 875 341-97 i 1-375 894.6 875 247.65 i 392.08 ! 70 i 695.92 i 284.25 1.125 442.44 1.25 872.98 29 625 181.49 1.25 493-14 i 1*5 1051.25 Equivalent Length of Pipe for a Socket. 7 -J = 1. d representing diameter of pipe and I length in inchet. Additional weight of two flanges for any diameter is computed equal to a line&J foot of the pipe. NOTB. These weights do not include any allowance for spigot and socket ends. 2. For rule to compute thicknesses of pipes, flanges, etc., see page 560. 134 WEIGHT OF STANDARD ROLLED STEEL BEAMS. 05 g o g S * 03 D O 2 1^9 ,1" ^ O ^ : M GC i *d rf ^ 0) 0^ ^0 ^ r^ ! I' -P ^ s ^m ^ - : ^ fc Piiii''i "Sod d\ T}- pi ci rfr^t-Jvd pied 10 pi 6\ t^ ir> r*) & d +* N OOO t>vO iO*O^^-rOrOfOP4C4NNPJPI k ii: a i . vO *O 00 IOH H r*5 fO w 10 fO^O PJ H 10 vO ^ ft *** H A . ^ ^ ^j lOOO M ONOO vQ H M vQ ^ IOOO Tj- M Q\ O^ N lO ;^ | 10 ood M I> O ^ON'^-Mod iOf*5w ONOO I I 2 ^ lOiori-rororoPlNNiNwM H -e | I ' ^ I s * .^? i r ^. t " .^? ex - 1 i i i i i i i i i i "3 ^ - J g fe 9 IN 8 S s | o5 J ^i o Q*H 10 2J l i illi^s^i i H M M 1 1 i ^ loco M vo oo . S M M 10 ON ^ | ld?d5 l l I I I I I I I I I I I I C< M M 1 1 - iovO t^OO ^ O V fQ S V S 8 4i ^ ? i ft ^ C v. 1 1 H WEIGHT OF ROLLED STEEL, SHEET COPPER, ETC. 135 Weight of Round Rolled. Steel. From .125 Inch to 12 Inches Diameter. ONE FOOT IN LENGTH. Diam. Weight. Diameter. Weight. Diameter. Weight. Diam. Weight. Diam. Weight. Inch. Lbs. Ins. Lbs. Ins. Lbs. Ins. Lbs. Ins. Lbs. .125 .0417 .875 2.04 1.625 7-05 2.875 22 5-75 88. 3 1875 0939 9375 2-35 1.6875 7 .6l 3 24 .1 6 96.1 25 .167 i 2.6 7 1-75 8.18 3.25 28.3 6.5 II3.2 .26 1.0625 3 I.8I25 8.77 3-5 32 7 7 i I3C-8 375 375 1.125 3.38 1.875 9.38 3-75 34-2 7-5 1 136.8 4375 5" 1.1875 3.76 2 10.7 4 42 7 8 170.8 5 .667 1-25 4.17 2.125 12 4-25 48.3 8-5 193.2 5625 .845 1.3125 4.6 2.25 13.6 4-5 54.6 9 218.4 .625 1.04 1.375 5-05 2-375 I5.I 4-75 60.3 9-5 241.2 6875 1.27 1-4375 2-5 I6. 7 5 66.8 10 267.2 75 1 6.01 2.625 18.4 5.25 73-6 ii 323 .8125 1^6 1.5625 6.52 2-75 20. 2 5-5 80.8 12 384-3 Weiglit of* Hexagonal, Octagonal, and Oval Steel. ONE FOOT IN LENGTH. HEXAGONAL. OCTAGONAL. OVAL. Diam. Diam. Diam. Diam. over Sides. Weight. over Sides. Weight over Sides. Weight over Sides. We ght. Diam. over Sides. Area. Weight. Inch. Lb. l ns< Lbs. Inch. Lbs. Ins. Lbs. Ins. Sq.In. Lbs. % .414 I 2-94 % .396 I 2 82 % X % .251 853 .736 *% 3-73 % .704 1% 356 % x 5| 344 I.I7 3 I-I5 1% 4-6 % I.I 1% 4.4 !.; .446 1.52 % 1.66 I /8 5-57 % 1.58 1% 5-32 J / x % .697 2-37 % 2.25 1% 6.63 % 2.l6 & 6-34 I^X % .884 3 Weight of a Square Foot of* Sheet Copper. Wire Gauge of Win. Foster $ Co. (England.) Thickness. Weight. Thickness. Weight. Thickness. Weight. w.o. Inch. Lbs. W.G. Inch. Lbs. W.G. Inch. Lbs. I .306 I 4 II .123 5.65 21 .034 i-55 a .284 13 12 .109 5 22 .029 1-35 3 .262 12 13 .098 4-5 23 025 4 .24 II 14 .088 4 24 .022 i 5 .222 10.15 15 .076 3-5 25 .019 .89 6 .203 93 16 .065 3 26 .017 79 7 ,186 8-5 17 057 2.6 27 .015 7 8 .168 7-7 18 .049 2.25 28 .013 .62 9 .153 7 19 .044 2 29 .012 .56 10 .138 6-3 20 038 1-75 30 .Oil 5 "Weight of Composition. Sheathing INTails. ; Number Number Number Number No. Length. in a No. Length. in a No. Length. in a No. Length. in a Pound. Pound. Pound. Pound. Inch. Ins. Ins. Ins. I 75 290 4 I.I25 201 7 I.I25 I8 4 10 1.625 101 2 .875 260 5 1.25 199 8 1.25 168 II 1-75 74 3 i 212 6 100 9 no 12 2 64 136 WEIGHT OP IRON, STEEL, COPPER, ETC. Weight of Cast and. 'Wrcmglit Iron, Steel, Copper, and Brass, of a given Sectional Area. PER LINEAL FOOT. Sectional Area. Wrought Iron. Cast Iron. Steel. Copper. Lead. Brass. Gun-metal Sq. Ins. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. .1 .336 .313 339 .385 .492 357 .38 .2 .6 7 I .626 .677 .771 .984 713 759 3 I,OO7 "939 1.016 1.156 1.476 1.07 I.I39 4 1-343 I.25I J-355 1.542 1.967 1.427 L5I9 5 1.678 1.564 1.694 1.927 2.461 1.783 1.894 .6 2.014 1.877 2.032 2.312 2-953 2.14 2.279 .7 2-35 2.19 2.371 2.698 3-445 2.497 2-658 .8 2.685 2.503 2.71 3-083 3-937 2.853 3.038 9 3.021 2.816 3-049 3-469 4.429 3.21 3.418 i 3-357 3.129 3-387 3-854 4.922 3-567 3-798 i.i 3.692 3.442 3.726 4.24 5-4I4 3-923 4.177 1.2 4.028 3-754 4-065 4.625 5.906 4.28 4-557 i-3 4-364 4.067 4-404 5-01 6.398 4-636 4-937 1.4 4.699 4-38 4-742 5.396 6.89 4-993 5-3 J 7 J-5 5-035 4-693 5.081 5-781 7.383 5-35 5-6o6 1.6 5-37i 5.006 5.42 6.167 7.875 5-707 6.076 i-7 5.706 5-3I9 5-759 6.552 8.367 6.063 6.456 1.8 6.042 5-632 6.097 6-937 8.859 6.42 6.836 1.9 6.378 5-945 6.436 7-323 9-351 6-777 7-215 2 6.714 6.258 6-775 7.708 9.843 7.133 7-595 2.1 7.049 6-57 7.114 8.094 10.33 7-49 7-97 2.2 7-385 6.883 7-452 8.474 10.83 7.847 8-35 2 -3 7.721 7.196 7.791 8.864 11.32 8.203 8-73 2.4 8.056 7-509 8.13 9-25 n.8i 8.56 9.11 25 8.392 7.822 8.469 9-635 12.3 8.917 9-49 26 8.728 8-135 8.807 10.02 12.8 9-273 9.87 2.7 9.063 8.448 9.146 10.41 13.29 963 10.25 2.8 9-399 8.76 9-485 10.79 13.78 9-98 10.63 2. 9 9-734 9.073 9.824 ii. 18 14.27 10.34 1 1. 01 3 10.07 9-386 10. 16 11.56 14.76 10.7 "39 3-i 10.41 9.699 10.5 ".95 15.26 11.06 11.77 3-2 10.74 IO.OI 10.84 12.33 !5.75 11.41 12.15 3-3 11.08 10.32 11.18 12.72 16.24 11.77 J2-53 3-4 11.41 10.64 11.52 I3-I 16.73 12.13 12.91 3-5 "75 10.95 11.86 13.49 17.22 12.48 13.29 3-6 12.08 11.26 12.19 13-87 17.72 12.84 13-67 3-7 12.42 11.58 12.53 14.26 18.21 13.2 14.05 3-8 12.76 11.89 12.87 14.64 18.7 13-55 14-43 3-9 13.09 12.2 13.21 15-03 19.19 J3-9 1 14.81 4 13.43 12.51 13-55 15.42 19.69 14.27 15.19 4.1 I3-76 12.83 13.89 15.8 20. 1 8 14.62 15-57 4.2 14.1 I3.J4 14.23 16.19 20.67 14-98 15.95 4-3 1443 13-45 14.57 16.57 21. l6 15-34 16.33 4.4 14.77 13.77 14.91 16.96 21.65 15.69 16.71 4-5 15-11 14.08 15.24 !734 22.15 16.05 17.09 4.6 15-44 J 4-39 15-58 !7-73 22.64 16.41 17-47 4-7 15-78 14.7 15.92 i8.ii 23-13 16.76 17-85 4.8 16.11 15.02 16.26 18.5 23.62 17.12 18.23 4.9 16.45 !533 16.6 18.88 24.12 17.48 18.61 5 16.78 1564 16.94 19.27 24.61 17-83 18.99 WEIGHT OF LEAD AND TIN PIPE AND TIN PLATES. 137 "Weigh.! of Lead. and. Tin Lined IPipe per Foot. From .375 Inch to 5 Inches in Diameter. (Tatham fy Bros.) Diam. WASTE Weight. -PIPE. Diam. | Weight. Diam. Weight. ) BLOCK-l Diam. ^IN PIPE. Weight. | Diam. Weight. Ins. Lbs. Ins. Lbs. Inch. Lb. Inch. Lbs. Ins. Lbs. 4 8 375 3594 -625 5 1.25 1.25 2 " *3* ; 4-5 6 375 375 625 625 1.25 1-5 3 3-5 4-5 8 375 5 75 .625 1-5 2 3 5 5 8 5 375 75 75 1 '5 2-5 4 5 5 IO 5 .5 i 9375 2 2-5 4 6 5 12 5 .625 i 1.125 2 3 WATER-PIPE. From .375 Inch to 5 Inches in Diameter. Diam. Thick- ness. Weight. Diam. Thick- ness. Weight. Diam. Thick- ness. Weight. Diam. Thick- ness. Weight. Inch. Inch. Lbs. Ins. Inch. Lbs. Ins. Inch. Lbs. Ins. Inch. Lbs. 375 .08 .625 625 25 3-5 1.25 .19 4-75 2-5 3125 14 375 .12 I 75 .1 1.25 1.25 25 6 2-5 375 17 375 .16 1-25 75 .12 i-75 i-5 .12 3 3 1875 9 375 .19 i-5 75 .16 2.25 i-5 .14 3-5 3 25 12 375 34 2-5 75 .2 3 i-5 17 4-25 3 3125 16 5 .07 0545 75 23 3-5 i-5 .19 5 3 375 20 5 .09 75 75 3 4-75 i-5 23 6-5 3-5 1875 9-5 5 .11 i i; t j;i ^ i-5 i-5 .27 8 3-5 25 15 5 13 1.25 2*1 2 i-75 13 4 3-5 3125 18.5 5 .16 *-75 .14 2-5 i-75 17 5 3-5 375 22 5 .19 2 I? 3-25 i-75 .21 6-5 1875 12.5 5 25 3 .21 4 i-75 27 8-5 4 25 16 .625 .08 .0727 .24 4-75 2 15 4-75 4 3125 21 .625 .09 i 3 6 2 .18 6 4 375 25 .625 13 i-5 25 .1 2 2 .22 7 4-5 1875 14 625 .16 2 25 .12 2-5 2 .27 9 4-5 25 18 625 .2 2-5 25 .14 3 2-5 1875 8 5 2 5 20 .625 .22 2-75 25 .16 3-75 2.5 25 ii 5 375 31 JNlarks and Weight of* Tin-plates. (English.) MARK OB BRAND. Plates per Box. Dimensions. Weight! per Box. MARK OR BRAND. Plates per Box. Dimensions. Weight per Box. No. 1 88 209 230 251 272 293 112 140 112 140 168 105 126 112 126 i C or i Com. 2 C No. 22.5 225 225 225 225 225 225 225 225 225 225 225 225 100 IOO IOO IOO Ins. 13.75X10 I3.25X 9-75 i2. 75 X 9.5 i3-75Xio i3-75Xio 13.75X10 I3-25X 9-75 I2-75X 95 i3-75Xio i3-75Xio i3-75Xio 13.75X10 13.75X10 16.75X12.5 16.75X12.5 16.75X12.5 16.7^X12.5 No. 112 I0 2 9 8 II 9 157 140 133 126 161 182 203 224 245 9 l .126 M7 168 DXXXX SDC No. 100 2OO 200 200 2OO 200 200 2OO 112 112 225 225 200 IOO IOO 450 4^0 Ins. 16.75X12.5 15 Xi 15 Xi 15 Xi 15 Xi 15 Xi 15 Xi 15 Xi 20 Xl 20 Xl i3-75Xi i3-75Xi 15 Xi 16.75X1 .5 16.75X1 .5 13.75X10 13.7^X10 , C SDX .. HC'"::::' SDXX H X SDXXX SDXXXX. . . . SDXXXXX. SDXXXXXX. Leaded 1C... " IX... ICW i X 2 X q X .. XX . . XXX xxxx. . . . xxxxx . . xxxxxx. DC IXW CSDW CIIW DX XIIW DXX TT. . . . DXXX. . XTT. . . , When the plates are 14 by 20 inches, there are 112 in a box. 138 STEAM, GAS, AND WATER PIPES. Iron. and. Steel Welded Steam., Gras, and 'Water Pipes. STANDARD DIMENSIONS. National Tube Co. D Nomi- nal In ternal. iamete Act'a Ex- t'nal r. Act'a In- t'nal | Thickness. Circ ei Ex- t'nal umfer- ce. Inter nal. Tran Ex- ternal averse A In- ternal reas. Metal Len{ Sq. Su Ex- tvi Cth per ?oot o rface. t'nal. Pi .= ..* 11 Nominal Weight per Foot. a a % ftl *" 2 Ins. I2 1 Ins. Ins. Ins 7 Ins. 1.27 Ins. .85 Sq. Ins *3 Sq. Ins 06 S.Ins Ft. 9 43 Ft. Ft. 2513.1 Lbs. . 24 Int. 25 54 36 .09 '7 i.i- .2' .1 .12 7.07 10.49 1383-1 .42 25 -375 67 -49 .09 2.12 i-55 36 .19 .17 5-68 7-73 751-2 -56 375 -5 .84 .62 . ii 2.64 1.96 55 -3 .25 4-55 6.13 472-4 .84 5 75 1.05 .82 ii 3-3 2-59 .87 53 33 3-64 4.63 270 1. 12 75 i I *3 I 1.05 13 4-13 3-29 1-36 .86 49 2-9 3-64 166.9 1.6 7 i 1-25 '5 1.66 1.9 1.38 1.61 M M 5-21 5-97 4-33 5-o6 2.16 2-83 2.04 t 2-3 2.OI 2-77 2-37 70.66 2.24 2.68 1.25 2 2-37 2.07 7.46 6.49 4-43 3-36 1.07 x.6i 1.85 42.91 3-6i 2 2-5 2.87 2-47 2 9-3 7-75 6-49 4-78 1.71 i-33 i-55 30.1 5-74 2-5 3 3-5 3-5 4 3-07 3-55 22 2 3 ii 12.57 11.15 9.62 '2-57 7-39 9.89 2.23 2.68 1-09 95 13 19.49 14.56 7-54 9 3 3-5 4 4-5 4-03 2 4 14.14 12.65 iS-9 12.73 3- 1 7 95 ii. 3 1 10.66 4 4-5 5 25 14-16 19.63 15.96 3-67 .76 85 9.02 12.49 4-5 5 5-56 5-04 26 17^8 15-85 24.31 4-32 .69 -76 7-2 H-5 5 6 6.62 6.06 28 20.81 19.05 34-47 28.85 5-58 .58 -63 4.98 18.76 6 7 7.62 7.02 3 23-95 22.06 45-66 38.74 6-93 5 54 3-72 23-27 7 8 8.62 7-98 32 27.1 25.08 58-43 50.04 8-39 44 .48 2.88 28 18 8 9 9.62 8-94 34 30.24 28.08 72.76 62.73 10.3 4 -43 2-3 33-7 9 10 IO -75 O.O2 37 33-77 31.48 90.76 78.84 11.92 35 .38 1.83 40 ii "75 I 37 36.91 14.56 108.43 95-03 13.4 32 -35 1-52 45 ii 12 I2 -75 2 37 0.05 37-7 127.68 113.1 14.58 3 32 1.27 49 2 3-25 37 43.98 41-63 153-94 137.89 16.05 27 .29 1.05 53-89 '5 14.25 37 47.12 44-77 176.71 159.48 17-23 25 27 9 57-8i 16 15-25 37 50-27 47.91 201.06 182.65 18.41 .24 25 79 6i-77 18 '7-25 37 56.55 54-19 254-47 233-78 20.76 .21 .22 .62 69.66 20 '9-25 37 62.83 60.48 114. 16 291.04 23.12 .19 .2 -49 77-57 22 21.25 . 37 69.11 66.76 380. 13 354-66 25-4 17 .18 4 35-47 2 4 23.25 37 75-4 3-4 452.39 424. 56 27.83 .16 .16 34 93-37 26 25-25 37 81.68 9-32 30-93 00.74 30. 19 15 .15 .29 02 28 27.12 44 7.96 5-22 77-87 37-88 .14 .14 25 27-34 30 29 5 34-25 I. II 06^86 60.52 46.34 13 13 .22 56 .Lap-welded Steel, Semi-Steel, Special Locomotive and Franklinite Boiler Tubes. STANDARD DIMENSIONS. National Tube Co. Dia Ex- t'ual. meter. Inter- nal. |i & *3 Circum Exter- nal. ference. Inter- nal. Trai Exter- nal. isverse Ar Inter- nal. eas. Metal. Length Foot of Exter- nal. perSq. Surface Inter- nal. 111 fffi *n !l Q l^sT Ins. Ins. Ins. Ins. Ins. Sq. Ins. Sq. Ins. Sq.Ins Feet. Feet. Lbs. Ins. i -834 .083 4 3I42 2.62 .785 .546 239 3-82 4-58 .81 i 1.25 1.084 -083 4 3-927 3-405 1.227 923 304 3-056 3-524 1.02 1-25 '5 I-3I 095 3 4.712 4-iiS 1.767 1.348 .419 2-547 2.916 1.4 i-5 '75 1-532 .109 2 5.498 4.8i3 2.405 1.843 .|62 2.183 2-493 1.8 7 i-75 2 1.782 .109 2 6.283 5-598 3.142 2-494 .648 1.91 2.144 2.1 7 2 2.25 2.032 .109 2 7.069 6.384 3-976 3-243 733 1.698 1.88 2-45 2.25 2-5 2.26 .12 I 7.854 7.1 4.909 4.011 897 1.528 1.69 3 2-5 2-75 2.51 .12 I 8.639 7.885 594 49.48 .992 1-389 1.522 3-3i 2-75 3 2.76 .12 I 9-425 8.671 7.069 5-9 8 3 1.086 1-273 1.384 3-63 3 3 25 2.982 -134 10.21 9.368 8.296 6.984 1-312 I-I75 1.281 4-39 3-25 3-5 3-232 134 O 10.996 10.154 9.621 8.204 1.417 1.091 1.182 4-74 3-5 3-75 3482 134 o 11.781 10.939 11.045 9.522 1.523 1.019 1.097 5-09 3-75 4 3-74 .I 4 8 9 12.566 11.636 12.566 io- 775 K.OXD 955 1.031 6 4 NOTE i For diameters from 13 up to and including 30 ins. O. D., details are n conformance with the circumstances, as there is not a standard, the thickness varying. NOTB 2. In estimating effective heating or evaporating surface of tubes, as heating liquids by steam, superheating steam, or trans- ferring heat from one liquid or one gas to another, mean surface of tubes is to be computed. IKON BOILER TUBES. Lap - welded. Charcoal Iron. Tubes. and x Steel STANDARD DIMENSIONS. National Tube Co. Diarr Exter- nal. eter. Inter, nal. Thickness. y Circ en Exter nal. umfer- ce. Inter- nal. Tran Exter- nal. averse A Inter. nal. reas. Metal. Leng Squar of Su Exter- nal. thper 9 Foot rface. Inter- i! External Diameter. Ins. Ins. Ins. No. Ins. Ins. Sq. Ins. Sq. Ins. Sq.Ins Ft. Ft. Lba. Ins. .86 .072 '5 3-14 2.69 .78 57 .21 3-82 4.463 71 .125 .98 .072 15 3-53 3.08 99 75 .24 3.396 .8 125 25 i. ii .072 15 3-93 3-47 1.23 .96 -27 3-056 3-453 .89 25 1.15 .083 14 4.12 3.6 1.35 1.03 -32 .911 3-333 .08 -312 375 I. 21 .083 14 4-32 3-8 1.48 34 -778 3.16 .13 -375 5 I *3' .083 M 4.71 4.19 1.77 1.4 37 547 2.863 24 .5 .625 1.4; .095 13 2.07 1.62 .46 352 2.662 53 .625 75 1.56 095 13 5-5 4-9 2-4 1.91 49 183 2-448 .66 75 875 1.68 095 13 5-89 5-29 2.76 2.23 53 037 2.267 .78 -875 1.81 095 '3 6.28 5-69 3-M 2.57 57 .91 2. II .91 .125 1.9" .095 6.68 6.08 355 2-94 .61 797 1.974 .04 125 25 2.06 95 13 7.07 6-47 3.98 3-33 .64 .698 1.854 .16 .25 375 2.16 .109 12 7-46 6.78 4-43 3-65 .78 .608 I.77I .61 375 5 2.28 .109 12 7.85 7.17 4.91 4.09 .82 .528 1.674 75 5 2-53 .109 12 8.64 7-95 5-94 5-03 9 389 1.508 3.04 75 875 2.66 . 109 12 9-3 8-35 6-49 5-54 95 329 1.438 3.18 875 3 2.78 . 109 12 9-42 8.74 7.07 6.08 99 273 1-373 3-33 3 3-25 3.01 .12 II 10.21 9.46 8-3 7.12 1.18 '75 1.269 3.96 3-25 3-5 3-26 .12 I IX 10.24 9.62 8.35 1.27 .091 I.I72 4.28 3-5 3-75 3-51 .12 I 11.78 11.03 11.04 9.68 '37 .019 1. 088 4.6 3-75 4 3-73 -134 12.57 11.72 12.57 10.94 1.63 955 1.024 5-47 4 4-25 3.98 .'34 O 13-35 12.51 14.19 12.45 i-73 899 '959 5.82 4-25 4-5 4.23 34 14.14 13-29 '5-9 14.07 1.84 .849 -93 6.17 4-5 4-75 4.48 '34 10 14.92 14.08 17.72 15.78 1.94 .804 .852 6-53 4-75 5 4-7 .148 9 15.71 14.78 19.63 17.38 2.26 764 .812 7.58 5 5-125 4-95 .148 9 16.49 15-56 21.65 19.27 2-37 .728 .771 7-97 5-25 5.25 5-2 .148 9 17.28 16.35 23.76 21.27 2-49 694 734 8.36 5-5 6 5.67 .165 8 18.85 17.81 28.27 25-25 3.02 637 674 10.16 6 8 6.67 7-67 .165 .165 8 8 21.99 25.13 20.95 24.1 38.48 50.27 34-94 46 2 3-J4 4.06 546 477 573 .498 xx.o 13.65 7 8 9 8.64 .18 7 28.27 27.14 63.62 58.63 4-99 .424 .442 16.76 9 o 9-59 203 6 31.42 30.14 78.54 72.29 6.25 382 20.99 10 i 10.56 22 5 34-56 33-17 95.03 87.58 7-45 347 .362 25.03 iz 2 "54 229 4-5 37-7 36.26 113.1 104.63 8-47 .318 33 28.46 12 3 12.52 2 3 8 4 40.84 39-34 132.73 123.19 9-54 294 35 32.06 13 4 13.5 2 4 8 3-5 43-98 42-42 153-94 143.22 10.71 273 283 36 14 5 14.48 259 3 47.12 45-5 176.71 164.72 11.99 255 .264 40.3 6 15.46 2 7 I 2-5 50.27 48.56 201.06 187.67 X 3'39 239 247 45-2 1 6 8 '7-43 284 2 56.55 54.76 254.47 238.66 15.81 .212 .219 52.87 18 20 19.38 3 I2 .31 62.83 60.87 314.16 294.86 i9.3 .191 .197 64.84 20 22 21.31 343 03 69.11 66.96 380.13 356.8 23-34 .174 .179 78.5 22 24 23.25 375 37 75-4 73-04 452.39 424.56 27.83 '59 .164 93-37 24 26 25.25 375 37 81.68 r 9 . 3 2 530.93 500.74 30.19 .147 'S 1 102 26 28 30 27.25 29.25 375 375 37 37 87.96 94-25 91.89 706.' 86 583.21 671.96 32.54 34.9 .136 .127 .14 no 118 28 |O NOTE i. For diameters from 13 up to and including 30 ins. O. D., details are in conformance with the circumstances, as there is not a standard, the thickness varying. NOTE 2. In estimating effective heating or evaporating surface of tubes* as heating liquids by steam, superheating steam, or transferring heat from one liquid or one gas to another, mean surface of tubes is to be computed. M* 140 WEIGHT OF COPPER TUBES. "Weiglit of Seamless Drawn Copper Tiabes. American Tu.be "Works. (Boston.) BY EXTERNAL DIAMETER. ONE FOOT IN LENGTH. Stubs 1 W. G. From .25 Inch to 12 Ins. f full, I light. No. 20 19 18 17 16 15 14 13 12 11 Ins. '/W 3/64* 3/6 4 / Vi6 / x /x6/ 5/6 4 l 5/6 4 / 3/32 / 7/6 4 i/8* Diamet'r. Lbs. Lbs. Lbs. Lbs. Lbs. | Lbs. Lbs. Lbs. Lbs. Lbr. 25 .09 .1 .12 13 .14 15 17 .18 .19 .19 375 .14 .16 .19 23 .24 .26 .29 32 35 37 5 .2 23 .27 31 34 37 .42 47 52 56 .625 25 .29 34 4 44 .48 55 .61 .69 74 75 3 36 .42 49 54 59 .67 .76 -85 .92 875 36 .42 49 58 .64 7 .8 9 i. 02 i. ii i .41 .48 57 .67 74 .81 93 1.05 1.18 1.29 1.125 .46 55 .64 .76 83 .92 1.05 1.19 i-35 1.47 1.25 52 .61 7i .84 93 1.03 1.18 i-34 1.52 1-65 1-375 57 .68 79 93 1.03 1.14 i-3i 1.48 1.68 1.84 i-5 .62 74 .86 1.02 J -i3 1.25 i-43 1.63 1.85 2.02 1.625 .68 .8 94 I. II -23 1.36 1.56 1.77 2.02 2.2 !-75 73 .87 1. 01 1.2 33 1.47 1.69 1.92 2.18 2-39 1-875 .78 93 1.09 1.29 43 1.58 1.81 2.06 2-35 2-57 2 .84 .16 i-37 53 1.69 1.94 2.21 2.51 2-75 2.125 .89 .06 .24 .46 63 1.8 2.07 2-35 2.68 2-93 2.25 94 13 3i 55 73 1.91 2.19 2-5 2.85 3.12 2 -375 .19 39 .64 .82 2.02 2.32 2.64 3.01 3-3 2-5 05 25 .46 73 1.92 2.13 2-45 2.79 3 .l8 3.48 2.625 .1 32 54 .82 2.02 2.23 2-57 2-93 3-35 3>6 7 2-75 .16 38 .61 9 2.12 2-34 2.7 3.08 3-5i 3-85 2.875 .21 45 1.68 99 2.22 2-45 2.83 3.22 3.68 4-03 3 .26 5i 1.76 2.08 2.32 2.56 2-95 3-37 3-84 4.22 3-25 37 .64 1.91 2.26 2.52 2. 7 8 3.21 3.66 4.18 4.58 3-5 .48 77 2.06 2-43 2. 7 2 3 3'46 3-95 4-5i 4-95 3-75 58 9 2.21 2.61 2.92 3-22 3-7 1 4.24 4.84 5-31 4 .69 2.02 2.36 2.79 3-n 3-44 3-97 4-53 5-i7 5-68 4-25 .8 2.15 2.51 3-14 3-3i 3-66 4.22 4.82 5-5i 6.05 4-5 1.9 2.28 2.6 5 6-32 3-5i 3-88 4-47 5-" 5-84 6.41 4-75 2.01 2. 4 I 2.8 3-49 3-7i 4.1 4-73 5-4 6.17 6.78 5 2.12 2-54 2-95 3-67 3-9i 4-32 4.98 5-69 6-5 7.14 5-25 2.23 2.66 3-i 3.85 4.11 4-54 5-23 5-98 6.84 7-51 5-5 2-34 2.79 3-25 3.85 4-3 4.76 5-49 6.27 7.17 7.87 5-75 2-44 2.92 3-4 4.02 4-5 4.98 5-74 6.56 7-5 8.24 6 2-55 3-05 3-55 4.2 4-7 5-2 5-99 6.85 7-83 8.61 6.25 2.66 3-i8 3-7 4.38 4.9 5-4i 6.25 7.14 8.17 8-97 6.5 2.76 3-3i 3-85 4.55 5-i 5-63 6.5 7-43 8-5 9-34 6-75 2.87 3-44 4 4-73 5-3 5-85 6-75 7.72 8.83 9-7 7 2.98 3.56 4-15 4.91 5-49 6.07 7.01 8.01 9.16 10.07 7-25 3-09 3-69 4-3 5-09 5-69 6.29 7.26 8.30 9-5 i 10.44 7-5 3-19 3-82 4-45 5.26 5-89 6.51 7-51 8-59 9.83 10.8 8 34 1 4.08 4-74 5-62 6.29 6-95 8.02 9.17 10.49 "-53 8-5 3-62 4-33 5-04 1 5-97 6.68 7-39 8.52 9-75 ii. 16 , 12.26 9 3-83 4-59 5-34 6.33 7.08 7-83 9-03 10-33 11.82 i3 9-5 4-05 4-85 5.64 6.68 7.48 8.26 9-54 10.91 12.49 *3-73 10 4.26 5- 11 5-94 7-03 7.87 8-7 10.05 11.49 13.15 14.46 10.5 4-47 5-37 6.24 7.39 8.27 9.14 10.55 12.07 13.82 15.19 ii 4.69 5-62 6-54 ! 7-74 8.67 9.58 11.06 12.65 14.48 ! 15.92 "5 4.9 5-88 6.84 -8.1 9.06 10.02 11.56 13-23 15.15 16.66 12 5-" 6.13 7-13 ! 8-45 9.46 IO.45 ' 12.07 13.81 15.81 17.29 WEIGHT OF COPPER TUBES. 141 No. 10 9 8 7 6 5 4 3 2 1 Ins. 9/641 9/64 / /6 4 / 3/i6 / 13/64 7/32 / x 5/6 4 / '/4/ 9/32 / '9/64 / Dumet'r. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. 375 4 .41 .42 .44 5 .61 .64 .6 7 -71 73 75 . 7 6 .625 .81 .86 .92 99 1.04 1.09 1. 12 I - I 3 1.18 75 I.OI 1.09 I.I7 1.26 i-35 1.42 1.49 1-53 1.61 1.63 .875 1.22 i-3i 1.42 i-53 1.66 1.76 1.8 5 1.92 2.04 2.09 i 1.42 1-54 1.6 7 1.81 i-97 2.09 2.21 2.32 2.48 2-55 1.125 1.6 3 1.78 i-93 2.08 2.28 2-43 2.58 2.71 2.91 3 1.25 1.8 3 2 2.18 2.36 2-59 2.76 2.94 3-" 3-34 3-46 J -375 2.03 2.22 2-43 2.63 2.9 3-i 3-3 3-5 3-77 3-92 i-5 2.24 2.44 2.68 2.91 3.21 3-43 3-67 3-9 4.21 4-38 1.625 2.44 2.67 2-93 3-i8 3.52 3-77 4-03 4.29 4.64 4.83 i-75 2.65 2.89 3.18 3-45 3.83 4.11 4-39 4.69 5-07 5-29 1.875 2.8 5 3.12 3-44 3-73 4.14 4-44 4.76 5-o8 5-51 5-75 2 3-06 3-34 3-69 4 4-45 4-78 5-12 5-48 5-94 6.21 2.125 3.26 3-57 3-94 4.28 4-75 5-" 5-48 5-87 6 -37 6.66 2.25 3.46 3-8 4.19 4-55 5.06 5-45 5-84 6.27 6.81 7.12 2 -375 367 4.02 4.44 4.82 5-37 5-78 6.21 6.66 7.24 7-57 2-5 3.87 4-25! 4-69 5-i 5-68 6.12 6 -57 7.06 7.67 8.04 2.625 4.08 4-47! 4-95 5-37 6 6-45 6 -93 7-45 8.1 8.49 2-75 4.28 4-7 5-2 5-65 6.3 6.79 7.29 7-85 8-54 8.95 2.875 4.48 4.92 5-45 5-92 6.61 7.12 7.66 8.24 8-97 9.41 3 4.69 5-i5 5-7 6.2 6.92 7.46 8.02 8.64 9.4 9.87 3-25 5-i 5-6 6.2 6.74 7-54 8.13 8.75 9-43 10.27 10.78 3-5 5-51 6.05 6.71 7.29 8.16 8.8 9-47 IO.22 11.14 11.7 3-75 5-9i 6-5 7.21 7-84 8.78 9-47 10.2 II.OI 12 12.61 4 6.32 6-95 7.71 8-39 9-4 10.14 10.92 II.8 12.87 I 353 4-25 6-73 7-4 8.22 8-94 IO.O2 10.81 11.65 12.59 13.73 14.44 4-5 7.14 7-85 8. 7 2 9-49 10.64 11.48 12.37 13.38 14.6 '5.36 4-75 7-55 8-3 9.22 10.04 11.26 12. 16 J 3-i 14.17 15.46 16.27 5 7-96 8-75 9-73 10.58 11.88 12.83 13-83 14.96 16.33 17.19 5-25 8.36 9-2! 10.23 11.13 12.49 13-5 J 4-55 !5-75 17.2 18.1 5-5 8.77 9.66 10.73 11.68 13.11 14.17 15.28 16.54 18.06 19.02 5-75 9.18 IO.II 11.24 12.23 13.73 14.84 16 17-33 18.93 T 993 6 9-59 10.56 11.74 12.78 14.35 i5-5i 16.73 18.12 19.79 20.85 6.25 10 II.OI 12.24 !3-33 14-97 16.18 i 17.46 18.91 20.66 21.76 6-5 10.41 11.46 12.75 ,3.88 15.59 l6 -85 18.18 19.7 21-53 22.68 6-75 10.82 11.91 13-25 14.42 16.21 I7-52 18.91 20.49 22.39 23-59 7 11.22 12.36 13-75 14.97 16.83 18.19 19.63 21.28 23.26 24.51 7-25 7-5 11.63 12.04 12.81 13.26 14.26 14.76 15-52 16.07 17.45 J 8.86 18.07 19-54 20.36 22.07 24.13 21.08 22.86 25 25.42 26.34 7-75 12.45 13.71 15.26 16.62 1 8.68 20.21 21. 81 23.65 25.86 27-25 8 12.86 14.17 i 15.77 17.17 19.3 20.88 22.54 24.44 26.72 28.17 8.25 13.27 14.62 16-27 17.71 19.92 21.55 23.26 25.23 27-59 29.08 8-5 13.67 15.07 16.77 18.26 20-54 22.22 23.99 26.02 28.45 30 8-75 14.08 : 15.52 17.28 18.81 21. 16 22.89 24.71 26.81 29.32 30.91 9 14.49 15-97 ! 17-78 19.36 21.78 23.56 25.44 27.6 30.18 31-83 9-25 9-5 9-75 10 10.5 14.9 i5-3i J 5-72 16.12 16.94 16.42 18.28 16.87 l8 -79 17.32 19.29 17.77 19-79 18.68 20.8 19.91 20.46 21.01 2L55 22.65 22.4 24.23 23.02 24.9 23.64 25.57 24.26 26.24 25-5 27.59 26.17 26.89 27.62 28.34 29.79 28.39 31.05 29.18 31.92 29.97 ! 32.78 30.76 1 33.65 32.34 35.38 32 'Z2 33-66 34-57 35-49 37-32 ii 17.76 19.58 21.81 23.75)26.73 28.93 31-25 33-92 37-11 39.15 "S 18.57 1 20.48 22. 8l 24.84 27.97 30.27 32.7 35.5 38.84 40.98 12 19.39 21.38 23.82 25-94 29.21 3I.6l 34.15 37.08 40.58 ! 42.81 142 WEIGHT OF COPPER AND BRASS TUBES, ETC. By Internal Diameter. Add following Units to Weights for External Diameter in preceding tables. No. 1 2 3 4 5 6 7 8 9 10 2.21 1.97 1.66 1.38 1.18 1. 01 .78 .67 53 43 11 12 13 14 15 16 17 18 19 20 35 .29 .22 I? 13 .11 .08 .06 05 03 ILLUSTRATION. What is weight of a copper tube 6 ins. in internal diameter, No. 3 gauge, and one foot in length ? By preceding table 6 ins. external, No. 3 gauge = 18.12, and 18.12 + 1.66 = 19.78 Ibs. WEIGHT OF BRASS TUBES. To Compute "Weight of* Brass Tiobes. American Tnt>e \Vorlis. (Boston.) RULE. Deduct 5 per cent, from weight of Copper tubes. EXAMPLE. What is weight of a brass tube 6 ins. in external diameter, No. 3 gauge, and one foot in length ? By preceding table 6 ins. = 18.12, from which deduct 5 per cent. = 17.21 Ibs. By Internal Diameter. RULE. Proceed as above for internal diameter, and deduct 5 per cent. EXAMPLE. Weight of a copper tube 6 ins. internal diameter, No. 3 gauge, and i foot in length 19.78 Ibs. Hence, 19.78 5 per cent. = 18.79 ^ s - NOTE. Diameter of Tubes, as for Boilers, is given externally, and that for Pipes internally. Weights of English are essentially alike to the preceding. (D. K. Clark.") Seamless Brass Pipe. American. Txi"be "Worlrs. (Boston.) Made to correspond with Iron Pipe and to Jit Iron Pipe fittings. Diameters. 3s Diameters. |j| Diameters. f 1.8 Same as Iron Pipe. Ex Inter- nal. act Exter- nal. i Same as Iron Pipe. Ex Inter- nal. act Exter- nal. P Same Ex Inter- nal. act Exter- nal. if! Ins. Ins. Ins. Lbs. Ins. Ins. Ins Lbs. Ins Ins. Ins. Lbs. X .281 .405 25 1.368 1.66 2-5 4 4 4-5 12.7 3^ 375 54 43 1% 1.6 1.9 3 4% 45 5 % .484 .625 675 .84 .62 9 2 2.062 2-5 2-375 2.875 4 5-75 5 6 5.062 6.125 6^625 15-75 18.31 X .808 1.05 1.25 3 3.062 3-5 8-3 i 1.062 I-3I5 1 7 3> 3-5 4 10.9 amless Copper Pipe of like diameter is 5 per cent, heavier "Weight of Sheet Brass. ONE SQUARE FOOT. (HoltzapJfeVs Gauge.) Thickness. Weight. Thickness. Weight. Thickness. Weight. Thickness. Weight. No. 3 4 6 1 Inch. 259 .238 .22 .203 .18 .165 Lbs. 10.9 10 9.26 8-55 7.58 6-95 No. 9 10 ii 12 13 H Inch. .148 134 .12 .109 095 .083 Lbs. 6.23 5-64 5-05 4-59 4 3-49 No. 15 16 i? 18 19 20 Inch. .072 .065 .058 .049 .042 035 Lbs. 3-03 2.74 2.44 2.06 1.77 1.47 No. 21 22 23 2 4 25 Inch. .032 .028 .025 .022 .02 Lbs. i-35 1.18 1.05 .926 .842 WEIGHT OF WEOUGHT IKON TUBES. 143 "Weight of "Wrought Iron Tiobes. (English.) EXTERNAL DIAMETER. ONE FOOT IN LENGTH. HoltzapJeVs Wire-Gauge, ffull, I light. No. ~ 4 5 6 7 8 9 Ins. .3125 .281 .238 .22 .203 .18 .165 .148 5/i6 9/32 15/64 / 7/32 I 3/6 4 3/i6 / "/6 4 / 9/64 / Diam. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. 7 21-9 19.8 16.9 15-6 14.5 12 9 II.8 10.6 7-5 235 21.3 18.1 16.8 15-5 13.8 I2. 7 11.4 8 25.2 22.7 19.3 17.9 16.6 14.7 13-5 12.2 85 26.8 24.2 20.6 19.1 17.6 15-7 14.4 12.9 9 284 25.7 21.8 20.2 18.7 166 15-3 13-7 9-5 30.1 27.1 23.1 21.4 19.8 17.6 16.1 14-5 10 31.7 28.6 24.3 22.5 20.8 18.5 17 15-3 No. ! 7 8 9 IO 1 1 12 3 4 5 18 .165 .148 134 . 12 .109 095 .083 .072 3/161 /6 4 I 9/64 / 9/64 / '/8J 7/64 3/32 / 5/64 / 5/64 / Diam Lbs. Lbs. Lbs. Lbs. Lba. Lbs Lbs. Lbs. Lbs. i-55 1.44 1.32 1.22 I. II .02 9 797 7 I.I25 1.78 1.66 1-39 1.26 .16 .906 .794 .25 2 O2 1.88 I.7I 1.57 I. 4 2 3 . .15 i.oi .888 375 2.25 2.09 1.9 1.74 1.58 45 .27 1. 12 983 5 2.49 2.31 2.1 1.92 I. 73 59 4 1.23 i. 08 .625 272 2.52 2.29 2.09 1.89 73 52 1-34 i 1. 17 75 2.96 2.74 2. 4 8 2.27 2.05 .87 .65 1.45 1.27 1-875 319 2.96 2.68 2.45 2.21 2.02 .77 i 1.56 1.36 2 343 3-17 2.87 2.62 2. 3 6 2.16 1.9 1.67 x -45 2125 367 3-39 3.06 2.8 2.52 2-3 2.O2 1.78 J 55 2 25 39 3-6 3-26 2.97 2.68 2.44 2.14 1.88 i!6 4 2375 4.14 3.82 3-45 3-15 2.83 2-59 2.27 * 1.99 1.74 2-5 4-37 4.04 3-65 3-32 2.99 2-73 2.39 i 2.1 1.83 2625 4.61 4-25 3-84 3-5 3- 15 2.87 2.52 2.21 1-93 275 4.84 4-47 4-03 3-67 3i 3-02 2.64 2.32 2.02 2875 5-08 4.68 4-23 3.85 +6 3.16 2.77 2-43 2.II 3 5-32 4.9 4.42 4.02 3-62 3-3 2.89 2-54 2.21 3-25 5-79 5-33 4.81 4-37 3-94 3-59 3-14 2-75 2.4 3-5 6.26 5-76 5-2 4.72 425 387 3-39 2.97 2-59 3-75 6-73 6.19 5-58 5-07 4-57 4.16 3.64 3-19 2.77 4 7-2 6.63 5-97 5-43 4-88 4.44 i 3.89 3.4 2.96 4-25 7.67 7.06 6.36 5.78 5-2 4-73 4-i3 3-62 3-15 4-5 8.14 7-49 6-45 6.13 5-5i 5-oi 408 3.84 3-34 4-75 8.61 7.91 7-13 6.48 5-82 5-3 4.63 4.06 3-53 5 9.08 8-35 7.52 6.83 6.13 5.58 | 4.88 4.27 3-72 5-25 9'56 8.79 7.91 7.18 6.44 5-87 i 5 13 4-49 3-9 5-5 10 9.22 8-3 7-53 6.76 6.15 j 5-38 4-71 4.09 5-75 10.5 965 8.68 7-88 7.07 6-44 5-63 4-93 4.28 6 6.25 ii 11.4 IO.I 10.5 9.07 9.46 8.23 8.58 7-39 ! 6.73 i 5-87 5.14 7.7 i 7.01 6.12 5.316 4-47 4.66 6-5 11.9 109 9-85 8-93 8.02 7-3 6-37 5.58 4-85 6-75 12.4 11.4 10.2 9.28 8.33 7.58 ] 6.62 5.79 5-03 7 12.9 n.8 10.6 9-63 8.64 7.87 6.87 6.01 5-22 725 13-3 12.2 ii 999 8.96 8.15 7.12 6.23 75 138 12.7 11.4 10.3 9.27 8.44 7.37 | 6.45 5-6 1 7-75 14-3 I 3- 1 n.8 10.7 9-59 8.72 7.62 ! 6.66 5-79 14-7 13-5 12.2 ii 99 9.01 7.86 6.88 5.98 144 WEIGHT OF COPPEK TUBES. Weigh-t of Seamless Drawn. Copper Tubes. (English. For Diameters and Thicknesses not given in preceding Tables. (D. K. Clark.) INTERNAL DIAMETER. ONE FOOT IN LENGTH. HoltzapffeVs Wire-Gauge, ffull, I light. Specific Weight = 1.16. Wrought Iron = i. No. 0000 ooo oo No. oooo ooo oo o Ins. 454 29/64 425 27/64 / .38 3/8 / 34 11/32 Ins. 454 29/64 425 27/64 / 38 3/8 / 34 , "/32 Diam. Lbs. Lbs. Lbs. Lbs. Diam. Lbs. Lbs. Lbs. Lbs. 75 4-5 5-75 34-2 31-9 28.3 25.2 875 5-79 5.02 6 35.6 33-2 29-5 26.2 8.02 7.36 6-37 5-53 6.5 38.4 35-8 31.8 28.3 .125 8.71 8 6-95 6.05 7 41.1 38.3 34-i 30.3 25 9-4 8.65 7-52 6-57 7-5 43-9 40.9 36.4 32.4 375 IO.I 9-3 8.1 7.08 8 46.6 43-5 38-7 34-5 5 10.8 994 8.68 7.6 9 52.1 48.7 43-3 38.6 .625 "5 10.6 9.26 8.12 10 57-7 53-8 47-9 42.7 75 12. 1 II. 2 9-83 8.63 ii 63.2 59 52-5 46.8 875 12.8 11.9 10.4 9-15 12 68.7 64.2 57-2 5i 2 13-5 12.5 ii 9.66 13 74-2 69-3 61.8 55-i 2.125 14.2 13-3 n.6 10.2 14 79-7 74-5 66.4 59-2 2.2 5 14.9 13.8 - 12. 1 10-7 15 85-2 79.6 7i 63-4 2 -375 15-6 14.5 I2. 7 II. 2 16 90.7 84.8 75-6 67.7 2.5 I6. 3 15-1 13-3 II.7 17 96.3 90 80.2 71.8 2.625 J 7 15.8 13-9 12.2 18 101.8 95- 1 84.9 76 2-75 17.7 16.4 14-5 12.8 19 107.3 100.3 89-5 80,1 3 19.1 17.7 15-6 I3'8 20 II2.8 105-5 94.1 84.2 3-25 20.4 19 16.8 14.8 21 118.3 110.7 98-7 88.3 3-5 21.8 20.3 17.9 15-9 22 123.8 115.8 103-3 92-5 3-75 23.2 21.6 19.1 16.9 23 129.3 120.9 107.9 96.6 4 24.6 22.9 2O.2 17.9 2 4 134.8 126.1 II2.6 100.6 4-25 25.9 24.2 21.4 19 26 146 136.4 I2I.8 108.8 4-5 27-3 25-4 22.5 20 28 157-2 146.7 131 117.1 4-75 28.7 26.7 23-7 21 30 168.4 i57-i 140.2 125.4 5 30.1 28 24.8 22.1 32 179.6 167.4 149-5 133-6 5-25 31.5 29-3 26 23.1 34 190.7 177.7 158.7 141.9 5-5 32.8 30.6 27.1 24.1 136 201.9 188 167.9 150.1 13 14 15 16 *7 18 19 20 21 22 23 2 4 For Diameters from 13 to 24 Inches. 5 i 2 3 4 3 J 9/6 4 / .284 9/32 / 259 x /4/ 238 I 5/6 4 / Lbs. Lbs. Lbs. Lbs. 48.5 45-8 41.7 38.3 52.1 49-3 44-9 41.2 55-8 52.7 48 44.1 59-4 56.2 51-2 46.9 63 59-6 54-3 49-8 66.7 63.1 57-4 52.7 70.3 66.5 60.6 55-6 74 70 63-7 58.5 77-6 73-4 66.9 61.4 81.3 76.9 70 64-3 84.9 80.3 73-2 67.2 88.6 83.8 76.3 70.1 7/32 / 35-3 38 40.7 43-4 46 48.7 5i-4 54 56.7 59-4 62.1 64.7 6 7 8 9 10 203 13/64 .18 3/i6 / .165 /64 I .148 9/6 4 / 134 9/6 4 / Lbs. Lbs. Lbs. Lbs. Lbs. 32-6 28.8 26.4 23.6 21.4 35-i 31 28.4 25-4 23 37-6 33-2 30-4 27.2 24.6 40 35-4 32-4 29 26.3 42.5 37-5 34-4 30.8 27.9 45 39-7 3 6 -4 32.6 29-5 47-4 41.9 38-4 34-4 31.2 49.9 44.1 40.4 36.2 3 2.8 52.4 46-3 42.4 38 34-4 54-9 48.5 44-4 39-8 36 57-3 50-7 46.4 41.6 37-7 59-8 52.9 48.5 43-4 39-3 WEIGHT OF COPPER AND WROUGHT IKON TUBES. 145 For Diameters from 13 to 24 Inches. No. 1 1 12 "3 '4 "5 16 17 8 | 19 20 Ins. .12 .109 095 .083 .072 .065 058 .049 j .042 035 1/8 I 7/64 3/32 / 5/64 / 5/64 / Vi6/ Vi6 / 3/64/1 3/64 / x /32/ Diam. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. 13 19.1 17.4 I5.I 13.2 II.4 10.3 9.3 7-77 6.65! 5.55 14 20. 6 I8. 7 16.3 14.2 12.3 II. I 9.9 8.37 7.16 5 98 15 22.1 20 17.4 15-2 13.2 II.9 10.6 8.96 7.67 6.4 16 23-5 21-3 18.6 16.2 I4.I 12.7 "3 9.56, 8.18 6.82 17 25 22.7 19.7 17.2 14.9 13-5 12. 1 10.2 8.69 7.27 18 26.4 24 20.9 18.2 15.8 -14-3 I2. 7 10.7 9.2 7.69 19 27.9 25-3 22 19.2 I6. 7 I5-I 13.4 "3 9.71 8.12 20 29-3 26.6 23.2 20.2 17.6 15.9 11.9 10.2 8.54 21 30.8 27.9 24-3 21.3 I8. 4 16.6 14.8 12.5 10.7 8.96 22 32.3 290 25-5 22.3 19.3 17.4 15-5 13-1 II. 7. 9-39 23 33-7 30.6 26.7 23-3 20.2 18.2 16.2 13-7 u.8 9.81 34 35-2 31.9 i 27.8 24-3 21. 1 19 16.9 14-3 12.3 IO.2 "Weight of Wrought Iron Tn"bes. (English.) For Diameters and Thicknesses not given in preceding Tables. (D. K. Clark.) INTERNAL DIAMETER. ONE FOOT IN LENGTH. HoltzapffeVs Wire-Gauge, ffull, I light. No. 4 5 6 7 THICKNESS IN INCHES. .238 .22 .203 .18 Ins. 5/8 9/i6 /2 7/i6 3/8 5/i6 j /4 '5/64 / 7/32 / 3/6 4 3/i6 / Diam. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. 1 Lbs. Lbs. Lbs. Lb. Lbs. 19 | 128.5 II5.2| 102. 1 89.1 7 6.1 63.2 50.4 4 8 44-2 140.8 36.2 20 135 121. 1 107.3 93.6 80 66.5 53 i 50-4 46.5 42.9 38 21 '1141.5 127 II2.6 98.2 83.9 69.7 55.6 52.9 48.8 45-1 39-9 22 i 148.1 132.9 117.8 102.8 87.9 73 58.3 55-4 51-1 47.2 41.8 23 i 154.6 138.8 I23.I 107.4 91.8 76.3 60.9 57.9 53-4 49-3 43-7 24 ; l6l.2 144.7 128.3 112 95-7 79.6 63.5 60.4 55-7 5i.5 456 26 : 174.3 156.5 138.8 121. 1 IO3.6 86.1 68.7 05-4 60.3 55-7 49-3 28 ! 187.4 168.3 149.2 130.3 III.4 92.7 74 70.4 64.9 60 53-i 30 200.4 180 , 159.7 139-5 1 19-3 99.2 79.2 75-4 09-5 64.2 56-8 32 213.5 191.8 170.2 148.6 I27.I 105.7 84.4 80.4 74.1 68.5 60.6 34 226.6 203.6 180.6 157-8 135 112.3 89.7 85-4 78.7 72.8 64.4 36 239.7 215.4! 191.1 I6 7 142.9 118.8 94.9 90.4 83-4 77 68.1 No. 8 9 IO 1 1 12 3 '4 5 16 i? 18 Ins .165 .148 134 .12 .109 .095. .083 .072 .065 .058 .049 /64 / 9/64 / 9/64 / 1/8 I 7/64 3/32 / 5/6 4 / 5/641 x /i6/ Vi6 / 3/64 / Diam. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbt. 19 33-i 29.7 26.9 24 21.8 19 16.6 14.4 13 u.6 9.78 20 34.8 31.2 28.3 25-3 22.9 20 17-5 I5-I 13.7 12.2 10.3 21 36.6 32-8 29.7 26.5 24.1 21 18.3 15-9 14.3 12.8 10.8 22 38.3 34-3 3 1 - 1 2 7 .8 25.2 22 19.2 16.6 15 13-4 "3 23 40 1 35-9 32.5 29.1 26.4 23 20.1 17.4 15-7 14 u.8 2 4 41.8 37-4 33.9 30.3 27.5 24 2O.9 iS.i 16.4 14.6 12.6 26 45-2 40-5 36.7 32.8 29.8 26 22.6 19.7 17.7 15.8 13-4 28 48.7 43-6 39-5 35-3 32.1 28 24.4 21.2 I 9 .I 17 14.4 30 52.1 46.7 42-3 37-8 34-4 3 26.1 22-7 20.5 I8. 3 i5-4 32 55-5 49.8 45-i 40.4 36.7 32 27.9 24.2 21.8 19-5 16.5 34 59 52.9 48 42.9 39 34 29.7 25.8 23.2 20-7 i7-5 36 62.4 56 50.8 45-4 41-3 36 3 T -4 27-3 2 4 .6 21. 9 18.6 146 WEIGHT OF IRON, STEEL, COPPER, ETC. "Weight of a Square IToot of "Wrought and Cast Iron, Steel, Copper, Lead, Brass, and Zinc 3?lates. From .0625 to i Inch in Thickness. Thickness. Wrought Iron. Cast Iron. Steel. Copper. Lead. Brass. Gun- metal. Zinc. Inch. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. .0625 2.5I7 2.346 2.541 2.89 3.691 2.675 2.848 2-34 .125 5-035 4-693 5-o8l 5.781 7.382 5-35 5-696 4.68 1875 7.552 7-039 7.6s2 8.672 11.074 8.025 8-545 7-02 25 IO.07 9-386 10.163 11.562 14.765 10.7 ".393 9-36 3125 12.588 "733 12.703 14-453 18.456 J 3-375 14.241 II. 7 375 I5.I06 14.079 15.244 17.344 22.148 16.05 17.089 14.04 4375 17.623 16.426 17-785 20.234 25^39 18.725 19.938 16.34 5 20.141 18.773 20.326 23.125 29-53 21.4 22.786 18.72 5625 22.659 21.119 22.866 26.0l6 33-222 24-075 25-634 21.06 .625 25.176 23.466 25.407 28.906 36.913 26.75 28.483 23.4 .6875 27.694 25.812 27.948 3L797 40.604 29-425 3i.33i 25.74 75 30.211 28.159 30.488 34.688 44.296 32.1 34-179 28.68 .8125 32.729 30-505 33.029 37-578 47.987 34-775 37.027 30.42 875 35-247 32.852 35-57 40.469 51.678 36.656 39-875 32.76 9375 37.764 35- J 99 38.11 43-359 55-37 39-331 42.723 35-i i 40.282 37-545 40.651 46.25 59.061 42.8 45-572 37-44 From One Twentieth Inch to Two Inches in Thickness. Thickness. Wrought Iron. Cast Iron. Steel. Copper. Lead. Brass. metal. Zine. Inch. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. .05 2.014 1.877 2-033 2.312 2-593 2.1 4 2.279 1.872 .1 4.028 3-754 4.065 4.625 5.906 4.28 4-557 3-744 15 6.042 5.632 6.098 6.938 8.859 6.42 6.836 5-6i6 .2 8.056 7-509 8.13 9-25 II.8I2 8. 5 6 9.114 7.488 25 10.071 9.386 10.163 11.562 14.765 10.7 "-393 9-36 3 12.085 11.264 12.195 I3-875 17.718 12.84 13.672 11.232 35 14.099 13.141 14.228 16.187 20.671 14-98 15-95 13.104 .4 16.113 I5.0I8 16.26 I8. 5 23.624 17.12 18.229 14.976 45 18.127 16.895 18.293 20.812 26.577 19.26 20.507 16.848 5 20.141 18.773 20.325 23.125 29-53 21. 4 22.786 18.72 55 22.155 20.65 22.358 25-437 32.484 23.54 25.065 20.592 .6 24.169 22.527 24.391 27.75 35-437 25.68 27-343 22.464 65 26.183 24.409 26.423 30.063 38.39 27.82 29.622 24-336 7 28.197 26.281 28.456 32.375 41-343 29.96 31-9 26.208 75 30.211 28.154 30.488 34.687 44.296 32.1 34.179 28.08 .8 32.226 30.035 32.521 37 47-249 34.24 36.458 29-95 85 34-24 31.912 34-553 39-312 50.202 36,38 38.736 31.824 9 36.254 33-79 36.586 41.625 53-154 3852 41.015 33.696 95 38.268 35-668 38.628 43-937 56.108 40.66 43-293 35.568 i 40.282 37-545 40.651 46.25 59.o6l 42.8 45.572 37-44 1.125 45.317 42.238 45.732 52.031 66.443 48.15 51.268 42.12 1.25 50.352 46.931 50.814 57.8i3 73.826 53-5 56.065 46.8 1-3125 52.87 49.278 53-354 60.703 77.517 56-17 59-8i3 49.14 1-375 55.387 51.624 55-895 63.594 81.209 58.85 62.661 51.48 1-4375 57-905 53.971 58.436 66.484 84.9 6i.53 65-51 53-82 i-5 60.422 56-3 1 7 60.976 69.375 88.591 64.2 68.358 56-16 1.5625 62.94 58-663 63-517 72.266 92.283 66.88 71.206 58.5 1.625 65.458 61.011 66.058 75-156 95.974! 69.55 74-054 60.84 i-75 70-493 65.704 7i-i39 80.938 !03-356 74>9 79-75 1 65-52 1-875 75.528 70.397 76.22 86.719 110.739 80.25 85447 70.2 a 80.564 75-09 81.3 92.5 118.122 85.6 91.144 74.88 WEIGHT OF ROLLED STEEL T. PIPES AND TUBES. "Weights, etc., of Rolled Steel T. Safe Load for One Foot Uniformly Distributed. Dimen- sions. Area. Weight per foot. Load. Tensile Strength per Sq. Inch. Dimen- sions. Area. Weight per foot. Load. Tensile Strength per Sq. Inch. 12500 16000 12500 16000 Ins. Sq. Ins. Lbs. Lbs. Lbs. Ins. Sq.Ins. Lbs. Lbs. Lbs. 4-5X2.5 4-5X2.5 2-79 2.4 r 3 5220 4520 6950 6030 4X4 4X4 3.21 4.02 10.9 J 3-7 13 loo 16 170 17470 21550 4-5X3 3 10 754 10050 4X4.5 3.36 11.4 15840 21 120 4-5X3 2-55 8.5 6490 8650 4X4.5 4-29 14.6 20400 27200 4-5X3.5 4-65 15.8 17020 22690 4X5 3-54 12 19410 25880 5 X2.5 3-24 II 6900 9 200 4X5 4.56 15.6 24800 337 5 X 3 3-99 13.6 9410 12550 To Compute Weight of Metal Pipes. D 2 d 2 C. D and d representing external and internal diameters in inches, and C coefficient. Cast Iron 2.45. Wrought Iron 2.64. Brass 2.82. Copper 3.03. Lead 3.86. To Compute "Weight of Metal Tu.bes and Pipes per .Lineal iToot. From .5 Inch to 6 Inches Internal Diameter. Diam. Area of Plato. Diam. Area of Plate. Diam. Area of Plate. Diam. Area of Plat* Ins. Sq. Foot. Ins. Sq. Foot. Ins. Sq. Feet. In*. Sq. Feot. 5 .1309 L3I25 .3436 2-75 .7199 4-5 1.1781 .5625 1473 1-375 .36 2-875 .7526 4.625 1. 2108 .625 1636 i 4375 3764 3 .7854 4-75 1-2435 .6875 .18 i-5 3927 3-125 .8l8l 4875 1.2763 75 .1964 1.625 .4254 3-25 .8508 5 1.309 .8125 .2127 i-75 .4581 3-375 .8836 5-125 I'34I7 .875 .2291 1-875 .4909 3-5 .9163 5-25 1-3744 9375 2454 2 5236 3625 949 5-375 1.4072 .2618 2.125 5543 3-75 .9818 5-5 J -4399 .0625 .2782 2.25 587 4 1.0472 5-625 1.4726 .125 2945 2-375 .6198 4.125 1.0799 5-75 I-5053 i875 3*05 2-5 6545 4-25 1.1126 5.875 i-538i '25 .3272 2.625 .6872 4-375 I-I454 6 1.5708 Application, of Table. When Thickness of Metal is given in Divisions of an Inch. To internal diameter of tube or pipe add thickness of metal ; take area of the plate in square feet, from table for a diameter equal to sum of diameter and thickness of tube or pipe, and multiply it by weight of a square foot of metal for given thickness (see table, page 146), and again by its length in feet. ILLUSTRATION. Required weight of 10 feet of copper tube i inch in diameter and 125 of an inch in thickness. i + .125 = 1.125 X 3. 1416 -f- 12 = .2945 square feet for ifoot of length. Weight of i square foot of copper .i2sth of an inch in thickness, per table, page 135, =5.781 Ibs.; then, .2945 (from table above) x 5.781 x 10 = 17.025 Ibs. When T/iickness of Metal is given in Numbers of a Wire -Gauge. To internal diameter of tube or pipe add thickness of number from table, pp. 120 or 121 ; multiply sum by 3.1416, divide product by 12, and quotient will give area of plate in square feet. Then proceed as before, 148 WEIGHT OF IRON AND COPPER PIPES, BOLTS, ETC. ILLUSTRATION. Required weight of 10 feet of copper pipe 2 inches in diameter and No. 2 American wire-gauge in thickness. 2 + -257 63X3. 1416 -f- 12 = 2. 257 63 X 3. 1416 -r- 12 = .591 square feet; then, .591 X 11.6706 (weight from table, page 118) =6.897 Ibs. of Riveted. Iron and. Copper IPipes, From 5 to 30 Inches in Diameter. ONE FOOT IN LENGTH. Diameter. Thicknesi. Iron. Copper. Diameter. Thickness. Iron. Copper. Ins. Inch. Lbs. Lbs. Ins. Inch. Lbs, Lbs. 5 .125 7.12 8.1 4 9 25 25.01 28.58 1875 10.68 12.21 -25 26.33 30.09 25 14.25 16.28 10 25 27-75 3I-7I 5-5 .125 7.78 8.89 10.5 2 5 29.19 33-22 1875 11.66 13-33 ii 25 30-49 34.85 25 15-56 I 7 .78 12 -25 33-!3 37-86 6 .125 8.44 9.64 J 3 25 35-88 4i 1875 12.65 14.46 14 25 38-52 44.02 25 16.88 19.29 15 -25 41.26 47- J 5 6-5 125 9,1 10.4 3125 51-57 58.94 .1875 13-65 15-6 16 25 43-9 50-17 25 18.2 20.8 3125 54-87 62.71 7 125 9.78 ii. 18 17 25 46.53 53-iS 1875 14.68 16.78 3125 58.17 66.48 25 19-57 22.37 18 25 49.17 56.2 7-5 125 10.49 11.99 3125 61.47 70.25 1875 15-73 17.98 20 3 I2 5 68.07 77-79 25 20.89 23.87 24 3125 Si-33 92.95 8 1875 16.7 19.08 25 3125 84-57 96.65 25 22.26 25-44 28 3125 94-56 107.95 8-5 25 23-59 26.96 30 3 I2 5 101.14 "5-59 Above weights include laps of sheets for riveting and calking. Weights of the rivets are not added, as number per lineal foot of pipe depends upon the distance they are placed apart, and their diameter and length depend upon thickness of metal of the pipe. "Weight of Copper Rods or Bolts, From .125 Inch to 4 Inches in Diameter. ONE FOOT IN LENGTH. Diameter. Weight. Diameter. Weight. Diameter. Weight. Diameter. Weight. Inch. Lbs. Ins. Lb. Ins. Lbs. Ins. Lbs. .125 047 .8125 1.998 i-5 6.8II 2-75 22.891 1875 .106 875 2.318 5625 7-39 .875 25.019 25 .189 9375 2.66 .625 7-993 3 27.243 3125 .206 i 3-03 75 9.27 .125 29-559 375 .426 1.0625 342 .875 10.642 25 31.972 4375 579 .125 3-831 2 12.108 375 34.481 5 757 1875 4.269 .125 13.668 5 37.081 5625 .958 25 4-723 25 15-325 .625 39-777 .625 1.182 3 I2 5 5-2i 375 17.075 75 42.568 .6875 L43 1 375 5.723 5 18.916 .875 45-455 75 1.703 4375 ! 6.255 .625 20.856 4 48.433 WEIGHT OF METALS. Weiglit of M!etals of a Griven. Sectional From .1 Square Inch to 10 Square Inches. PER LINEAL FOOT. (Z). K. Clark) 149 &flCI. AREA. Wrought Iron. Cast Iron. 9375' Steel. I.O2. Brass. 1.052. Gun- metal. 1.092. SECT. ARKA. Wrought Iron. ! 'l.' ' ' Cast 9375- Steel. B^l^a I.O2. I.O52. 1.092. Sq.Ins. Lbs. Lbs. Lbs. Lbs. Lbs. Sq.Ins. Lbs. Lbs. Lbs. Lbs. Lbs. ,1 33 3 1 34 35 .36 5-i 17 15-9 17-3 17.9 18.6 ,2 .67 .62 .68 7 73 5-2 17-3 I6. 3 17.7 18.2 j 18.9 .3 i .94 i. 02 1.05 1.09 5-3 17.7 16.6 18 18.6 19.3 4 5 i 1.25 1.56 1.36 i-7 1-43 1-75 1.46 1.82 5-4 5-5 18 18.3 16.9 17.2 18.4 18.7 18.9 ! 19.7 I 9 .3 20 .6 2 1.88 2.04 2.II 2.18 5-6 18.7 17-5 19 19.6 20.4 7 2.33 2.19 2.38 2.46 2-55 5-7 19 17.8 19.4 20 26.8 .8 2.67 2.5 2.72 2.81 2.91 5-8 19-3 18.1 19.7 20-3 21. 1 9 3 | 2.81 3.06 3.16 3-28 5-9 19.7 18.4 20.1 2O.7 21.5 i 3-33 3-i5; 3-4 3-51 3.6 4 6 20 ! 18.8 2O.4 21 21.8 i.i 3-67 3-44 3-74 3-86 4 6.1 20.3 19.1 20.7 21.4 22.2 1.2 4 3-75 4-o8 4.21 4-37 i 6.2 20.7 19.4 21. 1 21.7 22.6 J -3 4.33 4.06 4.42 4-56 4-73 6-3 21 19.7 21.4 22.1 22-9 1.4 4.67 4.38 4-76 4.91 5-i 1 6.4 21-3 ^20 21.8 22-4 23'3 i-5 5 4-69 5-1 5-26 5-46; 6.5 21.7 20.3 22.1 22.8 23-7 1.6 5-33 5 5-44 5-6i 5-82; 6.6 22 2O.6 22-4 23.1 24 i-7 5-67 5-31 5.78 5-96 6.19 6.7 22.3 20-9 22.8 23-5 244 1.8 6 5.63 6.12 6.31 6-55 6.8 22.7 i 21.3 23.1 23-9 2 4 .8 1.9 6 -33 5-94 6 -4o 6.66 6.92 6.9 23 21.6 23-5 24.2 25.1 2 6.67 6.25 6.8 7.01 7.28, 7 23.3 21.9 23.8 24.6 25-5 2.1 7 6.56 7-14 7-36 7-64; 7*i 23.7 22.2 24-1 24.9 25.8 2.2 7.33 6.88 7-48 7-73 8.01, 7.2 24 22-5 24-5 25-3 26.2 2.3 7.67 7.19 7-82 8.67 8.37J 7-3 24.3 22.8 24.8 25.6 26.6 2.4 i 8 7-5 8.16 8.42 8-74, 7-4 24.7 23.1 25.2 26 26.9 2-5 8. 33 7.81 8.5 8.77 9.1 | 7-5 25 23-4 25-5 26.3 27-3 2.6 8.67 8.13 8.84 9.12 9.46 7.6 25-3 23.8 25-9 26.7 27.7 2.7 9 8.44 9.18 9-47 9-83 7-7 25.7 24.1 26.2 27 28 2-8 i 9-33 8-75 9-52 9.82 10.2 7.8 26 24.4 26.5 27.4 28.4 2.9 ! 9.67 9.06 9.86 10.2 10.6 7-9 26.3 24.7 26.9 27.7 28.8 3 10 9.38 10.2 10.5 10.9 8 26.7 25 2 7 .2 28-1 29.1 3-i 10.3 9.69 10.5 10.9 "3 8.1 2 7 25-3 27-5 28.4 29-5 3.2 10.7 IO 10.9 II.2 11.7 8.2 27-3 25-6 27.9 28-8 29.9 3-3 10.3 II. 2 n.6 12 8.3 27.7 25-9 28.2 29.1 30.2 3-4 "-3 10.6 n.6 11.9 I2. 4 8.4 28 26.3 28.6 29-5 30.6 3-5 "-7 10.9 11.9 12.3 I2. 7 8-5 28.3 26.6 28.9 29-8 30-9 3-6 12 "3 12.2 12.6 I3-I 8.6 28. 7 26.9 29.2 30.2 31-3 3-7 12.3 n.6 12.6 13 13-5 8.7 2 9 27.2 29.6 30-5 31-7 3-8 12.7 11.9 12.9 13-3 13-8 8.8 29-3 27-5 29.9 30-9 32 3-9 J 3 12.2 13.3 13.7 14.2 8.9 29.7 27.8 30-3 31-2 32.4 4 13-3 12-5 13.6 14 14.6 9 30 28.1 30.6 31.6 32.8 4.1 13.7 12.8 13.9 14.4 14.9 9.1 30-3 28.4 30-9 3 J -9 33-i 4.2 14 13.1 14.3 14.7 15-3 9.2 30.7 28.8 31-3 32-3 33-5 4-3 J 4-3 J 3-4 !4-6 15-1 !5-7 9-3 31 29.1 31.6 32.6 33.9 4.4 14.7 13.8 15 15-4 16 9-4 31-3 29.4 32 33 34-2 4-5 15 I4-I 15-3 15.8 16.4 9-5 31-7 29.7 32.3 33-3 34-6 4.6 15.3 14.4 15.6 16.1 16.7 9.6 32 30 32.6 33-7 34-9 4.7 15.7 14.7 16 16.5 17.1 9-7 32-3 30-3 33 34 35-3 4-8 16 15 16.3 16.8 17-5 9.8 32-7 30.6 33-3 j 34-4 35-7 4.9 16.3 15-3 16.7 17.2 17.8 9.9 33 30-9 33-7 34-7 36 5 16.7 15.6 17 17-5 18.2 IO 33-3 3i-3 34 35- 1 3^4 150 LEAD PIPES. COPPEK PIPES AND COCKS. Diam. Thick- ness. "TO Weight. r eigl Diam. it o ONE Thick- ness. f Lej FOOT Weight. ad i IN LE Diam. D ipe NGTH Thick- ness . (Eng Weight. lish.) Diam. Thick- ness. Weigkt Inch. Inch. Lbs. Ins. Inch. Lbs. Ins. Inch. Lbs. Ins. Inch. Lbs. -5 .097 93 I .136 2. 4 i-75 .166 5 3 275 H .112 .07 .156 2.8 .199 6 3-5 .225 13 .124 .2 .2 3-73 .228 7 273 16 .146 47 .225 4.27 .256 8 4 257 i7 625 .089 1.25 139 3 2 .I 7 8 6 3125 20.5 .101 13 .16 3-5 .204 7 327 22 .121 4 .18 4 .231 8 4-25 3125 22.04 .14 2 193 4-33 .266 9-33 4-5 .232 17 75 .112 .6 i-5 .156 4 2-5 .2 8.4 295 22 .147 1.87 .179 4.67 .227 9.6 3125 23.25 .l8l 2.13 .224 6 .261 II.2 4-75 3 I2 5 24-45 .215 2.4 257 7 3 .218 II.2 5 3125 25.66 Dimensions of Copper Pipes and. Composition Cocks. From i Inch to 23 Inches in Diameter. s * Flange I Pipe. >iameter. Cock. Thick- ness. B No. olts. Diam. "8 -8 a.S.3 **1 Flange Diam. Pipe. Thick- ness. B No. >lta. Diam. Ins. Ins. Ins. Inch. Inch. Ins. Ins. Inch. Inch. I 3-375 3-5 -375 3 5 9 12-75 .625 9 .625 1.25 3-625 3-75 375 3 -5 9-25 I3.I25 .625 10 .625 i-5 3.875 4-25 375 3 5 9-5 13-375 .6875 IO .625 i-75 4.125 4-375 4375 4 5 9-75 I3-625 .6875 10 .625 2 4-375 4-75 -4375 4 5 10 13.875 .6875 10 .625 2.25 4.625 5.25 4375 5 5 10.5 14-5 .6875 10 .62 5 2-5 4.875 5-5 4375 5 -5 ii 15 .6875 IO .625 2-75 5.25 5-75 4375 5 5 "5 15.625 75 IO 75 3 6 6.25 5 5 625 12 16.125 75 IO 75 3-25 6.125 6.625 -5 6 625 12.5 16.625 75 10 75 35 6 -375 6.875 -5 6 625 13 I7-25 75 10 75 3-75 6.625 7.25 5 6 625 13-5 17.875 75 10 75 4 6.875 7-375 5 6 625 14 18.375 75 IO 75 425 7-125 7.625 5 6 625 14-5 18.875 75 10 75 45 7-375 8.25 5 6 625 15 19-5 -75 10 75 4-75 7.625 8-5 5 6 625 15-5 20 75 10 75 5 8 9 5 6 625 16 20.5 75 10 75 5-25 8.25 9-25 5 6 625 16.5 21.125 75 10 75 5-5 8-5 9-5 5 6 625 17 21.625 75 II 75 5-75 9 9-875 5 6 625 17-5 22.125 75 II 75 6 9-25 .625 8 625 18 22-75 75 II 75 6.25 9-75 -625 8 625 18.5 23-25 75 II 75 6-5 10 .625 8 .625 19 23-75 75 12 75 6-75 10 .625 8 625 19-5 24-375 75 12 -75 7 10.5 .625 8 625 20 24.875 75 12 .75 7-25 10.75 -625 8 625 205 25-375 75 13 -75 7-5 11.125 -625 8 625 21 26 75 13 1 -75 7-75 11 -375 .625 8 625 21-5 26.5 75 13 ! -75 8 11.625 -625 9 625 22 27 75 J 3 -75 8.25 12 -625 9 625 22.5 27-625 75 14 -75 8-5 12.25 .625 9 625 23 28.125 75 14 -75 8-75 12.5 .625 9 .625 WEIGHT OF SHEET LEAD, LEAD AND TIN PIPES, ETC. I 5 I Weight of Sheet Lead. PER SQUARE FOOT. Thickness. Weight. || Thickness. Weight. Thickness. Weight. || Thickness. Weight. Inch. Lbs. Inch. Lbs. Inch. Lbs. Inch. Lbs. .017 I .068 4 .118 7 .169 10 .034 2 .085 5 135 .186 II 051 3 II .101 6 .152 9 II -203 12 Weight of Tin !Pipe. ONE FOOT IN LENGTH. Diam. THICKNESS. Diam THICKNESS. Diam. THICKN. Diam. THICKN. External. & inch. % inch. External. X inch % inch. External. % inch. External . % inch. Inch. Lb. Lbs. Ins. Lbs. Lbs. Ins. Lbs. Ins. Lbs. 25 .148 1.25 1.095 1.417 2.25 5-04 3.25 7.56 5 384 .472 1.328 1-732 2-5 5.67 35 8.19 75 .62 .787 1.75 1.564 2.047 2-75 6-3 3-75 8.82 i .856 I.I03 2 1. 802 2.362 3 6-93 4 9-45 "Weight of Lead Encased Tin 3?ipes. Diameter. Light Weights. ForS 50 feet and under. upply of Water Head 51 to 250 feet. .* 251 to 500 feet. Ins. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. 375 I 1-5 2 2.5 to 4 3 to 4.5 3-5 to 5 5 2 2-5 3 3-5 " 5 4 " 6 4-5 " 7 .625 3 3-5 4 4-5 " 7 5-25 " 8 6 9 75 3-5 4 4-5 5.5 " 8 6 " 9 7 " 10 i 4-5 5 5-5 7.25 " 10 8 " ii 9 " 13 1.25 6-5 7 8 9 " 12.5 10 " 14 ia " 16 8 9 10 ii " 16 12.5 " 18 14 " 21 2 ii 13 16 " 23 18.5 " 26 21 "30 * The extreme weights are for extra heavy pipe with less proportion of tin. Dimensions and "Weight of Sheet Zinc. (Vielle-Montagne.) PER SQUARE FOOT. No. Thickness. sX. 5 metres; area, i square metre. 6.56X1.64 feet; area, 10.76 square feet. 2X .65 metres ; area, 1.3 sq. metres. 6.56X2.13 feet; area, 13.99 square feet. 2X.8 metres; area, 1.6 sq. metre*. 6 56X2.62 ft. ; area, 17.22 square feet. Weight. Millim. Inch. Kilom. Lbs. Kilom. Lbs. Kilom. Lbs. Lbs. 9 .41 .Ol6l 2. 9 6-39 3-7 8.16 4.6 10.14 .589 10 51 .O2OI 3-45 7 .6l 4-45 9.8l o-5 12.12 .704 ii .6 .0236 4-05 8-93 5-3 11.68 6-5 14-33 .832 12 .69 .0272 4-65 10.25 6.1 13-45 7-5 l6 -53 .96 13 .78 .0307 5-3 11.68 6.9 15.21 8.5 18.74 1.088 14 .87 0343 5-95 13.12 7-7 16.94 9-5 20.94 1.216 15 .96 .0378 6-55 14.44 8-55 18.85 10.5 23.I5 1-344 16 i.i 0433 7-5 16.53 9-75 21.5 12 26.46 I-536 17 1.23 .0485 8-45 18.63 10.95 24.14 13-5 29.97 1.74 18 1.36 0536 9-35 20.61 12.2 26.9 15 33-07 1.92 J 9 1.48 .0583 10.3 22.71 !3-4 29-54 I6. 5 36.38 2. 112 20 1.66 .0654 11.25 24.8 14.6 32.19 18 39.68 2.304 21 1.85 .0729 12.5 27.56 16.25 35.82 20 44-09 2. 5 6 22 2. 02 0795 13-75 30-31 17.9 39.46 22 48.5 2.816 23 2.19 .0862 15 33-07 19-5 42.99 24 52.91 3-073 24 2-37 0933 16.25 35-82 21. 1 46.52 26 57-32 3-329 25 2.52 .0992 17-5 38.58 22.75 50.15 28 61.73 3-585 26 2.66 .1047 18.8 41.44 24.4 53-79 31 68.34 3-060 152 SHIP AND RAILROAD SPIKES, HORSESHOES. Railroad. Spikes. (Dilworth, Porter & Co., Pittsburg, Pa.} Dimensions. In keg Of 200 Weight of Rail per Dimensions. In keg Of 200 Weight of Rail per Dimensions In keg Of 200 Weight of Rail per Lbs. Yard. Lbs. Yard. Lbs. Yard. Ins. No. Lbs. tns. No. Lbs. Ins. No. Lbs. 2-5X-3I25 3 X-3I25 2-5X-375 2230 1880) 1650) 8 to 12 12 to 16 4-5X.375 3-5X-4375 4 X.4375 780) 8 9 o| 780) 16 to 25 4-5X-5 5 X-S 4-5X-5625 5i8 475} 460) 28 to 35 35 to 40 3 X-375 3-5X.375 1380) 1250) 16 to 20 4-5X-4375 3-5X-5 690 670) 20 tO 30 5 X-5625 5-5X-5625 405 360 40 to 56 45 to zoo 4 X-375 1025 16 to 25 4 X-5 605 24 to 35 Street Railway Spikes. From .25 to .625 Inch. Have Countersunk Heads. Square Bolt Spikes. .25 In. .25 In. .3i25ln. 375 In. 4375 In. Sin. .625 In. .752 In. Length. Length. Length. Length. Length. Length. Length. Length. Ins. Ins. Ins. Ins. Ins. Ins. Ins. Ins. 3 to 3-5 4 to 8 4 to 8 4 to 12 6 to 12 6 to 16 8 to 16 12 to 24 Ship and. Railroad Spikes. DIMENSIONS AND NUMBER PER POUND. (P. C. Page, MOSS.) Sliip Spikes. X In. Sq. %, In. Sq. % In. Sq. tf In. Sq. % In. Sq. ^ In. Sq. Xln.Sq. 1 fif S 1 fl T3 43 gi s-g i e ^ i ID S"g | C "C J II 1 -0 fcfc ,2 o fc< ! <3 | ^&H a ^p^ J_ il. J M Ins. Ins. Ins. Ins. Ins. Ins. Ins. 3 19 3 10 4 5-4 5 3-4 6 2.2 8 1.4 10 .8 3-5 15-8 3-5 9.6 4-5 5 5-5 3-i 6-5 2 9 1.2 15 .6 4 13.2 4 8 5 4.6 6 3 7 1.9 10 I.I 4-5 12.2 4-5 6 5-5 4.2 6-5 2.8 7-5 1.8 ii I 5 10.2 5 5-8 6 4 7 2.6 8 i7 6 5-2 6.5 3-2 7-5 2.4 8-5 1.6 8 2.2 9 i-5 > 10 1.4 Railroad Spikes 5 inch square X 5 r ine o r>pr IK " " . .$621 " " * c.e " r.fi " Spikes and Horseshoes. LENGTH AND NUMBER PER POUND. (//. Burden, Troy, N. F.) d be 1 Boat g pikes. 1 3 d . 0-3 i I Ship Spikes. I .S Hook Hea Length. d. c . Horse 1 shoes. . 3 3-5 4 4-5 5 5-5 6 17-5 14.68 12-57 9 .2 7-2 6. 3 4-97 Ins. 6.5 7 7-5 8 8-5 9 10 4.78 3-62 3-37 2-95 2.9 2.1 1.08 Ins. 4 4.5 5 5-5 6 6-5 7 8 6.5 4-37 4-3 4.2 3-77 2-75 Ins. 7-5 8 8-5 9 10 2-5 1.74 1.63 Ins. 4 X.375 4-5 X. 4375 5-5X-5 5.5 X. 5625 6 X .5625 6 X .625 5-55 4.14 2.52 2.41 1.87 1.72 1.38 Ins. I 2 3 4 5 .84 75 65 56 39 CAST IEON AND LEAD BALLS. NAILS. 153 \Veight and "Volume of Cast Iron and. Lead Sails. From i Inch to 20 Inches in Diameter. Diameter. Volume. Cast Iron. Lead. Diameter. Volume. Cast Iron. Lead. lus. Cube Ins. Lbs. Lbs. Ina. Cube Ins. Lbs. Lbs. I .5 2 3 .136 .215 9 381.703 99-51 156.553 i-5 1.767 .461 .725 9-5 448.92 117.034 184.121 a 4.189 1.092 1.718 JO 523-599 136.502 214.749 2-5 8.181 2.133 3-355 10.5 606.132 158.043 j 248.587 3 14.137 3-685 5.798 II 696.91 181.765 285.832 3-5 22.449 5-852 9.207 11.5 796.33 207.635 326.591 4 33-5 i 8.736 13-744 12 904.778 235.876 371.096 4-5 47-7 J 3 12.439 19.569 12.5 1022.656 266.647 419.512 5 6545 17.063 26.843 13 1150.346 299.623 471.806 5-5 87.114 22.721 35-729 14 1436.754 374-563 589-273 6 113.097 29.484 46-385 15 1767.145 460.696 724.781 6-5 143-793 i 37-453 58.976 i 16 2144.66 559- "4 879.616 7 179-594 46.82 73-659!; 17 2572.44 670.717 1055.066 7-5 220.893 57.587 90.598 18 3053.627 796.082 1252.422 8 268.082 69.889 109.952 19 359I-363 936.271 1472.97 8-5 321.555 83-84 131.883 20 4188.79 1092.02 1717-995 NOTE. To compute weight of balls of other metals, multiply weight given in table by following multipliers: Steel i 088 Gun-metal... .. i.i6c. Weight and Diameter of Cast Iron Balls. Weight. Diameter. Weight. Diameter. Weight. Diameter. Weight. Diameter. Weight. Diameter. Lbs. Ins. Lbs. Ins. Lbs. Ins. Lbs. Ins. Lbs. Ins. I 1.94 12 4-45 50 7 .l6 224 n.8 1344 21.44 2 2-45 14 4.68 56 7-43 336 13.51 1568 22.57 3 2.8 16 4.89 DO 7-6 448 14-87 1792 23-6 4 3-08 18 5-09 70 8.01 560 16.02 20l6 24-54 5 3-3 2 20 5-27 80 8-37 6 7 2 17.02 2240 25.42 6 3-53 25 5-68 90 8.71 784 17.91 2800 27.38 7 3-72 28 5-9 IOO 9.02 896 18.73 3360 29.1 8 3-89 30 6.04 112 9-37 I008 19.48 3920 30.64 9 4.04 40 6.64 168 10.72 1 120 20.17 4480 32.03 No. 5 , " 6 Length, of Horseshoe Nails. By Numbers. . 1.5 Ins. I No. 7 1.875 Ins. I No. 9 2.25 Ins. , 1.75 " 1 " 8 2 " I "10 2.5 " Lengths of Iron Nails, and Number in a 3d. L'gth. 1.25 420 270 L'gth. i-75 L'gth, 2-5 Size. L'gth. ,20 Ins. 3-25 3-5 40 L'gth. 4 4.25 1 54 NAILS, SPIKES, TACKS, ETC. Wrough-t Iron. Cut Nails, Tacks, Spikes, etc. (Cumberland Nail and Iron Co.) Lengths and Number per Lb. c Size. >rdinar Length. y- No. per Lb. I Size. ~*inish Length. ing. No. per Lb. Size. Shingl Length. 3. No. per Lb Ina. Ina. Ina. 2 d 7 l6 4 d 1-375 384 5 d i-75 I 7 8 3 fine 1.0625 588 5 256 8 2-5 74 3 1.0625 448 6 2 204 9 2-75 00 4 1-375 336 8 2-5 102 10 3 52 5 i-75 216 10 3 80 Tacks. 6 7 2 2.25 166 118 12 2O 3.625 3.875 65 4 6 I OZ. .125 1875 16000 10666 8 2-5 94 Core. 2 25 8000 10 12 2-75 3-5 72 50 6d 8 2 143 68 2.5 3 3125 375 - 6400 5333 20 30 4 50 00 4 d 3-75 4-25 4.75 5 5-5 Light 1-375 32 20 14 10 373 10 12 20 30 40 WH WHL 2333 3-125 3-75 4.25 4-75 2-5 2 25 60 42 25 18 14 69 72 6 8 10 12 16 4375 -5625 .625 .6875 75 .8125 .875 4000 2666 2000 I 000 1333 H43 I 000 5 *-75 272 18 9375 888 6 2 196 Clinch. 20 i 800 Brads. 6 d 2 152 Boat. 6d 2 163 2.25 133 Size. No. per Lb. 8 2-5 96 2-5 9 2 Ina. 10 2-75 74 10 2-75 7 2 206 12 3-125 Fence 50 '. 3 3-25 60 43 Spi 3-5 fees. 6d 2 96 Slate. 4 15 7 2.25 66 3 d 1.625 288 4-5 13 8 2-5 56 4 1-4375 244 5 10 10 2-75 50 5 i-75 187 5-5 9 3 40 6 2 146 6 7 Railroad. Spikes. Number in a Keg of 150 Ibs. Length. No. Length. No. Length. No. Length. No. No. 3 X .375 3-5 X .375 4 X -375 930 890 7 60 5-5 X 3-5 X .4375 4 X .4375 4-5 X .4375 .5625 standai 675 540 510 d for Ins. 4X.5 4-5 X .5 5X .5 a, gauge of * 450 400 340 1-feet Ins. 5 X .5625 5-5 x .5625 3.5 ins. 300 280 Sliip and. Boat Spikes. Number in a Keg of 150 Ibs. Length. No. Length. No. Length. No. 455 424 390 384 300 Length. No. Ina. 4 X.25 4-5 X.25 5 X.25 6 X.25 7 X.25 1650 1464 I 3 80 1292 1161 Ina. 5 X. 3125 6X-3I25 7 X. 3125 6X.375 7X-375 930 868 662 570 482 Ins. 8X.375 9X-375 10 X. 375 8 X. 4375 9X-4375 Ina. 10 X, 4375 8X.5 9X.5 iox.5 iiX-5 270 256 240 222 203 VAEIOUS METALS. 155 "Weight of "Various Mletals. Per Cube Inch and Foot. AfCTALS. Spec. Gravi- ty. W'ght in an Inch. Ins. Lb! Weight in a Foot. METALS. Specific Gravi- ty. W'ght in an Inch. ini. in* Lb. Weight in a Foot. Wrought-iron plates " wire. Cast iron Steel plates.. " wire... Copper, ( . . . rolled {... Gun -metal,) cast j Wrought iron Cast iron .... Steel 7734 7774 7209 gj 8750 7.698 7.217 7.852 8.805 8.404 Lb. 2797 .2812 .2607 .2823 .2838 .3146 .3212 3^5 .278 .26 .283 .318 304 3-57 3-55 3-84 3-54 3-52 3-i9 3-" 3.16 En* 3 1 ' 3-84 3-53 3-i5 3-29 Lba. 483-38 485.87 450- 54 487.8 490.45 543-6 555 546-875 ylish. 4 8o 450 489.6 549 524 Brass, rolled. " cast. . . Lead, rolled . Tin, cast Zinc, rolled.. Alumini- ) um, cast j Silver 8217 8080 11340 7292 7188 2560 10480 8379 7.409 7.008 11.418 8.099 8.548 Lb. .2972 .2922 .4101 2673 .26 .0926 379 1 .3031 .268 253 .412 .292 .308 3-37 3-42 2-44 3-74 3-85 10.8 2.64 3.299 3-74 3-95 2-43 3-42 3-24 Lb. 5i3- 6 505 708.73 462 449.28 1 60 655 523.69 462 437 712 5<>5 533 Tobin Bronze. (D.K. Clark.) Tin Zinc Lead. . . . Copper plates Gun-metal. . . Brass, cast. . . u wire.. WROUGHT AND CAST IRON. To Compute "Weight of "Wrought or Cast Iron. RULE. Ascertain number of cube inches in piece; multiply sum by .2816* for wrought iron and .2607* for cast, and product will give weight in pounds. Or, for cast iron multiply weight of pattern, if of pine, by from 18 to 20, accord- ing to its degree of dryness. EXAMPLE. What is weight of a cube of wrought iron 10 inches square by 15 inches in length ? 10 X 10 X 15 X- 2816 = 422.4 Its. COPPER. To Compute "Weight of Copper. RULE. Ascertain number of cube inches in piece ; multiply sum by .321 18,* and product will give weight in pounds. Sheathing and Braziers' Sheets. For dimensions and weights see Measures and Weights, pages 118-121, 131, 142. LEAD. To Compute Weight of Lead. RULE. Ascertain number of cube inches in piece; multiply sum by .41015,* and product will give weight in pounds. EXAMPLE. What is weight of a leaden pipe 12 feet long, 3.75 inches in diameter, and i inch thick? By Rule in Mensuration of Surfaces, to ascertain Area of Cylindrical Rings. Area of (3.75 + 1 + 1) = 25. 967 " 3-75 ".044 Difference, 14.923 (area of ring) X 144 (12 feet) = 2148.912 X. 410 15 = 881.376 Ibs. BRASS. To Compute "Weight of Ordinary Brass Castings. RULE. Ascertain number of cube inches in piece; multiply sum by .2922,* and product will give weight in pounds. * Weights of a cube inch as here given are for the ordinary metals ; when, however, the specific gravity of the metal under consideration is accurately known, the weight of a cube inch of it should be substituted for the units here given. 156 DIMENSIONS AND WEIGHTS OF BOLTS AND NUTS. Dimensions and "Weights of "Wrought Iron Bolts and. IN"irts. SQUARE AND HEXAGONAL HEADS AND NUTS. ftcmgli, and from .25 Inch to 4 Inches in Diameter. Square Head and JSTut. Diameter of Bolt. Wid Head. th. Nut. Diagc Head. nal. Nut. De Head. P th. Nut. Weight. Head Bolt and Nut. jper Inch. Threads per Inch. Ins. Ins. Ins. Ins. Ins. Ins. Ins. Lbs. Lbs. No. 25 .36 49 51 .69 25 25 .024 .014 20 3125 45 .58 .64 .82 3 3125 043 .022 18 375 -54 .67 .76 95 -34 375 .068 .031 16 4375 63 .76 .8 9 1.07 4 4375 .104 .042 14 5 .72 .84 1.02 1.19 44 5 145 055 13 5 62 5 .82 94 1.16 i-33 .48 5625 .204 .07 12 .625 .91 1.03 1.29 1.46 53 .625 273 .086 II .6875 i 1. 12 1.41 1.58 58 .6875 356 .IO4 II 75 1.09 1. 21 1.54 1.71 63 75 454 .124 10 .8125 1.18 3 1.67 1.84 .67 .8125 565 145 10 .875 1.27 39 1.8 1.96 .72 .875 .696 .168 9 i J -45 57 2.05 2.22 .81 i 1.013 .22 8 1.125 1.63 75 2-3 2.47 9 1.125 1.416 .278 7 1.25 1.81 94 2.56 2.74 i 1.25 1.923 344 7 1-375 1.99 2.12 2.81 3 i.i 1-375 2-543 .416 6 i-5 2.17 2-3 3-7 3-25 1.18 i-5 3-234 495 6 1.625 2.36 2.48 3-34 3.5I 1.28 1.625 4.105 .581 5-5 1-75 2-54 2.66 3-59 3.76 i-37 1-75 5.087 .674 5 1.875 2.72 2.84 3-85 4.02 1.46 1.875 6.182 773 5 2 2.9 3.02 4.1 4.27 1.56 2 7.491 .88 4-5 2.125 3.08 3.21 4-35 4-54 1.65 2.125 8.936 993 4-5 2.25 3-26 3-39 4.61 4-79 1.75 2.25 iQ-543 1.113 4-5 2-375 3.44 3-57 4.86 5-05 1.84 2.375 12-335 1.24 4-375 2-5 3-62 3-75 5.12 5-3 1.94 2-5 14-359 1-375 4-25 2.625 3-81 3-93 5-49 5-56 2.03 2.625 16.549 i.5i5 4 2-75 3-99 4.11 5-64 5-8i 2.12 2-75 18.897 1.663 4 2.875 4.17 4.29 5-9 6.07 2.22 2.875 21-545 1.818 3-75 3 4-35 4-47 6.15 6-32 2-3 1 3 24.464 1.979 3-5 3-25 4.71 4.84 6.66 6.84 2.5 325 30.922 2.323 3-5 3-5 5-07 5-2 7.17 7-35 2.68 3-5 38-391 2.694 3-25 3-75 5-44 5.56 7.69 7.86 2.87 3-75 47.168 3-093 3 4 5-8 5-92 8.2 8-37 3.06 I 4 56.882 3-518 3 FINISHED. Deduct .0625 from diameters of bolts and depths of all heads and nuts. For Steel Bolts, add i. 3 per cent. Screws with square threads have but one half number of threads of those with triangular threads. NOTE. The loss of tensile strength of a bolt by cutting of thread is, for one of i. 25 ins. diameter, 8 per cent. The safe stress or capacity of a wrought iron bolt and nut may be taken at 5000 Ibs. per square inch. Preceding width, depth, etc., are for work to exact dimensions, whether forged or finished. To Compute AVeiiglit of a, Bolt and Nxit. Operation. Ascertain from table weight of head and nut for given di ameter of bolt, and add thereto weight of bolt per inch of its length, multi- plied by full length of its body from inside of its head to end. NOTE. Length of a bolt and nut for measurement, as such, is taken from inside of head to inside of nut, or its greatest capacity when in position. DIMENSIONS AND WEIGHTS OF BOLTS AND NUTS. 157 ILLUSTRATION. A wrought-iron bolt and nut with a square head and nut is i inch in diameter and 10 inches in length; what is it's weight? Weight of head and nut 1.013 \ 7/10 " bolt per inch of length .22 X 10 = 2.2 ) 3 ' 213 For Steel Bolts, add 1.3 per cent. Hexagonal Head. and. Diameter of Bolt. Wic Head. Ith. Nut. Diagonal. Head. I Nut. D Head. ;pth. Nut. Weig Head and Nut. ht. Bolt jer Inch. Threads per Inch. Ins. Ins. Ins. Ins. Ins. Ins. Ins. Ibs. Lbs. No. 25 375 5 43 58 25 25 .022 .OI4 20 3 I2 5 4375 .5625 5 65 3 .3125 037 .022 18 375 .5625 6875 .65 79 34 375 .002 .031 16 4375 .625 75 72 87 4 4375 .094 .042 14 -5 75 875 .87 i 44 5 134 055 13 5625 .8125 9375 94 i. 08 .48 5625 .18 .07 12 625 9375 1.0625 i. 08 1.23 53 .625 .249 .086 II 6875 i 1.125 1.16 58 .6875 318 .104 II 75 1.125 1.25 1.3 1.44 63 75 413 .124 IO 8125 1.25 1-375 1.44 1-59 .67 .8125 522 145 IO .875 1-3125 1-4375 1.52 1.66 .72 875 639 .168 9 1-5 1.625 i-73 1.88 .81 931 .22 8 .125 1.6875 1.8125 i-95 2.09 9 .125 1.299 .278 7 25 1.875 2 2.17 2.31 i 25 1-759 344 7 375 2 2.1875 2.31 2-53 i.i 375 2.263 .416 6 5 2.25 2-375 2.6 2.74 1.18 .5 2.958 495 6 1.625 2-4375 2.5625 2.81 2.96 1.28- .625 3-741 581 5-5 1.75 2.625 2-75 3-03 i 3.i8 i-37 ! -75 4-654 .674 5 1.875 2.8125 2-9375 3.25 3-39 ! 1-46 1.875 5.675 773 5 2 3 3-125 3-46 3.61 1.56 2 6.854 .88 4-5 2.125 3-1875 3-3I25 3.68 3.83 1.65 2.125 8.163 993 4-5 2.25 3-375 3-5 3-9 4.04 i-75 2.25 9-658 4-5 2-375 3-5625 3.6875 4.11 4.26 1.84 2-375 11.263 .24 4-375 2-5 3-75 3*875 4-33 4-47 1-94 2.5 13-149 375 4-25 2.625 3-9375 4.0625 4-55 4.69 ; 2.03 ! 2.625 4 2-75 4.125 4-25 4.77 4.91 2.12 2.75 17-285 ^663 4 2.875 4-3125 4-4375 4-99 5-12 , 2.22 2.875 1 975 I .818 3-75 3 4-5 4.625 5-2 5-34 2-31 3 22.378 979 3-5 3-25 4.875 5 5-63 5-77 2-5 3-25 28.258 2.323 3-5 3-5 5-25 5-375 6.06 6.21 2.68 3.5 35-o8i 2.694 3.25 3-75 5-625 5-75 6.5 6.64 2.87 ; 3.75 43.178 3.093 3 4 6 6.125 6-93 7-07 3o6 4 51.942 3.518 3 FINISHED. Deduct .0625 from diameters of bolts and depths of all heads and nuts. For Wood or Carpentry. Head and Nut (Square), 1.75 diameter of bolt. Depth of Head, .75, and of Xut, .9. Washer. Thickness, .35 to .4 of diameter of bolt, on Pine 3.5 diameter, and Oak 2.5. English. Moleswcn*th gives following elements of Thread of Bolts : Angle of thread, 55. Depth of thread = Pitch of screw. Number of threads per Inch. Square, half number of those in angular threads. Depth of thread. .64 pitch for angular and .475 for square threads. IJ8 DIMENSIONS AND WEIGHTS OF BOLTS AND NUTS. ITrencli Standard. Bolts and RTirts. (Armengau&t.) HEXAGONAL, HEADS AND NUTS. Equ Diainet of Bolt. ilatei sr Si c( alT si ft 'iangt Thicl Head. ilar 1 ness. Nut. r hrea ll *! i. Safe Tensile Stress. & Diameter of Bolt. ^quare ""S ! Thn s-g I* ei ad. 2- c a P Safe Tensile Stress. Mm. Ins. Ins. No. Ins. Ins. Ins. Lbs. Mm. Ins. Ins. No. Ins. Lbs. 5 .2 J 3 18.1 .24 .2 55 44 20 79 .072 6-57 1.82 717 7-5 3 .22 16 3 3 .68 99 25 .98 .O8l 5-97 2.01 I 142 IO 39 31 14.1 38 39 .88 178 30 1.18 093 5-4 2.22 1635 12.5 49 39 12.7 44 49 .04 277 35 1.38 .1 4-93 2.41 22l8 15 59 .4811.5 52 59 .2 400 40 J -57 .106 4-53 2.63 2912 17.5 .69 .58 10.6 .58 .69 4 545 45 1.77 .II 4 4.2 2.8 5 3074 20 79 .66 9.8 .66 79 5 7i3 50 1.97 .128 3-9i 3-07 4547 22.5 .89 .76 9.1 .72 .89 .68 902 55 2.17 13 3-65 3-3 5288 25 .98 .84 8.5 .8 .08 .84 I 120 00 2.36 .14 343 3-5 6540 30 1.18 i. 02 7-5 94 1.18 2.16 1035 65 2.56 15 3-23 3-7 7660 35 1.38 12 6.7 i. 08 1.38 2.48 22l8 70 2.76 .158 3-o6 3-9 2 8893 40 1.58 1.4 6 .22 158 2.8 2912 75 2-95 .166 2.92 4-13 10214 45 1.77 1-56, 5-5 .36 1.77 3-2 3674 80 3-15 .174 2.76 4-36 11603 50 1.97 i-74 5-i 5 1.97 3-44 4547 85 3-35 .183 2.63 4.58 13100 55 2.17 1.92 4.7 .64 2.17 3-76 5288 90 3-54 .192 2.51 4.78 14794 60 2.36 2.08, 4.4 74 2.36 4.08 6540 95 3-74 .2 2.41 5 16352 65 2.56 2.26 4.1 .92 2.56 4.4 7660 100 3-94 .209 2.31 5.22 18144 70 2.76 2.44' 3-8 2.06 2.76 4-7 8893 105 4-i3 .22 2.22 5-43 20000 75 295 2-6 | 35 2.2 2-95 5 10214 no 4-33 .226 2.13 5-66 21950 80 3-15 2.78, 3-4 2-34 3-15 5-35 11468 "5 4-53 23 2.O6 5-87 23990 English. Bolts and iNTuts. (WUtworWs.) Hexagonal Heads arid. N"nts, and Triangular Threads. Diame Bolt. ter. s-i II l"i ei D Head. sp tb. Nut. Width of Head and Nut. Diana Bolt. eter. Base of Thread. Threads per Inch. De F Head. th. Nut. Width of Head and Nut. Ins. Inch. No. Inch. Ins. Ins. Ins. Ins. No. Ins. Ins. Ins. - I 25 093 4 .109 .125 338 1.25 1.067 7 1.094 1.25 2.048 .1875 134 24 .164 1875 .448 1-375 1.161 6 1.203 1-375 2.215 .2187 24 i-5 1.286 6 1.312 1.5 2.413 25 .186 20 .219 25 525 1.625 1.369 5 1.422 1.625 2.576 3 I2 5 .241 18 273 3125 .001 i-75 1.494 5 L53I i-75 2.758 375 .295 16 .328 375 .709 1-875 !-59 4-5 1.641 1-875 3.018 4375 346 14 383 4375 .82 2 I-7I5 4-5 i-75 2 3- T 49 5 393 12 437 5 .919 2.125 1.84 4-5 1.859 2.125 3-337 5625 456 12 .492 5625 .Oil 2.25 i-93 4 1.969 2.25 3-546 .625 .508 II 547 .625 .101 2-375 2.055 4 2.078 2-375 3-75 .6875 57 1 II .60! .6875 .201 2-5 2.18 4 2.187 2-5 3.894 75 .622 10 .656 75 .301 2.625 2.305 4 2.297 2.625 4.049 .8125 .684 10 .711 .8125 39 2-75 2.384 3-5 2.406 2-75 4.181 8/5 733 9 .766 875 479 2.875 2.509 3-5 2.516 2.8 7 5 4.346 9375 795 9 .82 9375 574 3 2.634 3-5 2.625 3 4-531 i .84 8 875 i .67 3-25 2.84 3-25 1,125 942 7 .984 1.125 .86 3-5 3-o6 3-25 RETENTION OF SPIKES AND NAILS. Square Heads and l^uts. ( Whitwortk* s. ) 159 Dia Bolt. meter. Base of Thread. Threads per Inch. Dia Bolt. meter. Base of Thread. Threads per Inch. Dia Bolt. meter. Base of Thread. Threads, per Inch. Ins. 3-75 4 4-25 Ins. 3-25 3-5 3-75 No. 3 3 2-875 Ins. 4-5 4-75 5 Ins. 3.875 4.0625 4-25 No. 2.875 2-75 2-75 Ins. 5-25 5-5 6 Ins. 4-4375 4.625 4.875 No. 2.625 2.625 2-5 "Weight of* Heads and Nuts in. Lt>s. (Molesworth.) Hexagonal, 1.07 D 3 . Square, 1.35 D 3 . D representing diameter of bolt in inches. Retentiveness of 'Wrought Iron Spikes and. Nails. Deduced from Experiments of Johnson and Sevan. SPIKES. SFIKI. WOOD. 1 | Depth of Insertion. Force re- auired to raw it. Ratio of force to weight. REMARKS. Ins. Ins. Ins. Lbs. Square Hemlockf Chestnut 39 37 38 3-5 3-5 1297 1873 2.16 Seasoned in part. Unseasoned. " * .... " * .... Yellow pine 375 375 3-375 2052 2-37 Seasoned. i r*f? ' * .... White oak 375 375 3-375 3910 4-52 u Locust 4 4 3.5 5967 6.33 ii ef^' Flat narrow. . Chestnut 39 25 3-5 2223 3-93 Unseasoned. it White oak 39 25 3-5 3990 7-05 Seasoned. (t U Locust 39 25 3-5 5673 9-32 " " broad . . Chestnut 539 .288 3-5 2394 2.66 Unseasoned. u u White oak 539 .288 3-5 5330 5.71 Seasoned. U U Locust 539 .288 3-5 7040 7.84 u Square} ^ Hemlockf 4 39 3-5 1638 Seasoned in part. " > 2,2 Chestnutf -4 39 3-5 1790 i.Si Unseasoned. " J Q Locust f 4 39 3-5 3990 4.17 Seasoned in part. Round and) grooved..) Ash Diam. .5 3-5 2052 2.21 Seasoned. M M " -5 3-5 245 * 2.41 u M White oak .48 3-5 3876 3-2 u * Burden's patent. t Soaked in water after the spikes were driven. NAIL. Length. Depth of Insertion. Pine. NAILS Force r& Hemlock. ;. quired to Elm. draw it. Oak. Beech. Pressure required to force them into Pine. Sixpenny u Ins. 2 2 2 Ins. I 1-5 2 Lba. I8 7 327 530 Lbs. 3 I2 539 857 Lbs. 327 571 899 Lbs. 507 675 1394 Lbs. 66 7 889 1834 Lbs. 235 4OO 610 General Remarks. With a given breadth of face, a decrease of depth will increase retention. In soft woods, a blunt-pointed spike forces the fibres downwards and backwards so as to leave the fibres longitudinally in contact with the faces of the spike. l6f Manila Rope of Equal Strength. Ins. Ins. Lbs. No. Ins. Ins. Ins. Lbs. No. Ins. 75 5-5 4-85 44 ii 25 4 2.55 23 8 .6875 5-25 4-4 40 10.5 1875 3-75 2.25 20 7-5 .625 5 4 36 10 .125 3-5 i-95 18 6.5 5 4-75 3-6 32 9-5 .0625 3-^5 i-7 15 6 4375 4-5 3-25 2 9 9 3 i-44 13 5-75 375 4-25 2.9 26 8-5 '-875 2.75 i. 21 II 5-25 WIRE ROPES, HAWSERS, AND CABLES. i6 3 Gralvanized Charcoal Iron "Wire Rope. Vessels' Rigging and Derriclz Griiys. John A. Roebling' l s Sons C"o., Trenton, N. J. 7 Wires in a Strand. Approx- imate Diam- eter. Circum- ference. Weight Fwt. Breaking Strain in Tons of POOO Lbs. Circum. of Manila Rope of Equal Strength. Approx- imate Diam- eter. Circum- ference. Weight Breaking Strain in Tons Of 2080 Lbs. Circum. ofManila Rope of Equal Strength. Ins. Ins. Lbs. No. Ins. Ins. Ins. Lbs. No. Ins. .8125 2.5 i 9 5 375 1.125 .2 1.8 2.25 75 2.25 .81 73 4-75 3125 i .16 1.4 2 .625 2 .64 5.8 4-5 .2812 .875 .123 i.i I -75 5625 1.75 4.4 3-75 .25 .81 1-5 .5 1.5 36 3-2 3 .2188 .625 .063 56 1.25 4375 1.25 .25 2-3 2.5 1875 5 .04 36 1.125 Gralvanized. Steel Hawsers. For Sea and. Lalze Ter Foot. Breaking Strain in Tons Of 2000 Circum. ofManila Rope of Equal eter. Lbs. Strength. eter. Lbs. Strength. Ins. Ins. Lbs. No. Ins. Ins. Ins. Lbs No. Ins. 1-75 5-5 3.25 61 13-5 I -4375 4-5 2.1 3 42 "5 1.6875 5.25 2-95 57 13 1.375 4.25 1.94 39 ii 1.625 5 2.7 53 12.5 i 25 4 1-7 2 32 10 1-5 4-75 2.42 45 12 1.1875 3-75 1.5 29 9.25 GJ-alvanized Steel Cables for Suspension Bridges. Diam- Weight Breaking Strain in Diam- Weight Breaking Stra n in Diam- Weight Breaking Strain in eter. Foot. Tons of 2000 eter Lbs. BE Tons of 2000 eter Lbs. Ct. Tons of 2000 Lbs. Ins. Lbs. No. Ins. Lbs. No. Ins. Lbs. No 2-75 12.7 3o 2.375 9-5 232 2 6-73 164 2.625 n. 6 283 2.25 8.52 208 I .875 5-9 144 2-5 10.5 256 2.125 7-6 185 I 75 5-' 124 Grange, Weight, and Length of Iron "Wire. 1 Diam. Weight per loo Feet. Weight of one Mile. 63 Ibs. Bundle. Area. 1 I ' 1 > Diam. Weight per 100 Feet Weight of one Mile. 63 Ibs. Bundle. Area. N Inch. Lbs. Lbs. Feet. Sq. Inch. NoT Inch. Lbs Lbs. Feet. Sq. Inch. 6/0 .46 56.1 2962 112 .16619 16 .063 1.05 55 6000 . 003 117 5/o 43 49.01 2588 129 .14522 I .054 77 41 8182 .00229 4/0 393 40.94 2162 154 .1 21304 I 8 .04 7 -58 31 10 36 2 .001734 3/o .362 34-73 1834 .102921 I .041 45 24 14000 .00132 2/0 331 29.04 1533 217 . 086 049 20 035 32 17 19687 .000962 I/O 37 27.66 1460 228 .0 74023 2 i -03 2 .27 14 23 333 .000804 i .283 21.23 II2I 296 .062901 22 .028 .21 ii 30000 .000615 2 .263 18.34 9 68 343 .0 54325 2 3 .02 5 17 5 9.: '4 36 DOO .000491 3 .244 I5-78 833 399 .046759 24 .023 .14 7-39 45000 .000415 4 .225 13-39 7 7 470 .03976 25 .02 5 6.124 54310 .000314 5 .207 "35 599 555 .0 33653 2 5 .OI 8 .09 3 )i 67 742 .000254 6 .192 9-73 647 .0 28952 2 7 .01 7 .08 3 4-: 582 75 ?3 .000227 7 .177 8.03 439 759 .024605 2 S .Ol6 .074 3-97 85135 .000201 8 .162 6.96 367 95 .0- 20612 2 9 .01 5 .06 r 3-' 22 103 278 .000 176 9 .148 5.08 306 1086 .017 203 30 .014 054 2.851 116666 .000154 o 135 4-83 255 1304 014313 31 0135 2.64 126000 .000133 i .12 3.82 202 1649 .on 309 32 .013 .046 2.428 136956 .000132 2 .105 2.92 154 2158 .0< 38659 3 3 .01 i 03 7 I. ?53 170 270 .000095 3 .092 2.24 118 2813 .006647 34 .01 03 1.584 2IOOOO .000078 4 .08 1.69 89 3728 .01 35026 3 5 .oc Q5 .02 5 I. 32 252 000 .000071 5 .072 7 2 459 8 .004071 36 .009 .02 1.161 286363 .000064 164 IKON, STEEL, AND HEMP HOPE. "Weight and. Strength, of Single Strand and Cable laid Fence TV^ire. (F. Morton & Co.} Strands. N , Single Wire of equal Diameter. Len per 10 Of a Strand. gth oo Ibs. Of Rope. Strands. No. Single Wire of equal Diameter. Len per io< Of a Strand. gth DO Ibs. Of Rope. No. No. Inch. Feet. Feet. No. No. Inch. Feet. Feet. 3 2A 8 159 20090 15270 7 00 4 .229 8300 7366 4 2 7 .174 14730 12790 7 3/o 3 -25 80 3 6 6228 7 I 6 .191 I3I25 10580 7 4/o 2 .274 7500 5156 7 5 .209 10446 8928 7 5/o I 3 5090 4286 No. and diameter of wire is that of Ryland's Bros., pp. 122-4. Hemp, Iron, and Steel. (R. S. Newall & Co.) HEMP. Circumference. Weight per Foot. IRON. Circumference. Weight per Foot. STEEL. Circumference. Weight Foot. Tensile 5 Safe Load. >trength. Ultimate Strength. Ins. Lbs. Ins. Lbs. Ins. Lbs. Lbs. Lbs. 2.75 33 I .16 672 4480 i-5 25 I .16 1008 6720 3-75 .66 1.625 33 J 344 8960 1-75 .42 i-5 25 1680 II 20O 4-5 .83 1-875 5 2016 13440 2 58 1.625 33 2352 15680 5-5 1.16 2.125 .66 i-75 .42 2688 17920 2.25 75 3024 20 160 6 i-5 2-375 83 1-875 5 336o 22400 2.5 .92 3696 24640 6-5 1.66 2.625 2 58 4032 26680 2-75 .08 2.125 .66 4368 29 1 20 7 2 2.875 .16 2.25 75 4704 31360 3 25 5040 33600 7-5 2-33 3-125 33 2-375 -83 5376 36840 3-25 .41 5672 38080 8 2.66 3-375 -5 2-5 .92 6048 40320 3-5 1.66 2.625 i 6720 44800 8.5 3 3-625 1-83 2-75 i. 08 7392 49280 3-75 2 8064 5376o 9-5 3-66 3-875 2.16 3-25 i-33 8736 58 240 10 4-33 4 2-33 9408 62 720 4-25 2-5 3-375 i-5 10080 67200 ii 5 4-375 2.66 10752 71680 4-5 3 3-5 1.66 12096 80640 12 5-66 4-625 3-33 3-75 2 13440 89600 FLAT. Dimensions. Dimensions. Dimensions. 4 X .5 3-33 2.25 X.5 1-85 4928 44800 5 Xi.25 4 2.5 X-5 2.l6 5824 51520 5-5 Xi. 375 4-33 2.75 X.625 2-5 6720 60480 5-75XI.5 4.66 3 X .625 2.66 2 X.5 1.66 7168 62720 6 Xi.5 5 3.25 X.625 3 2.25 x-5 1.83 8064 71680 7 Xi. 875 6 3-5 X.625 3-33 2.25 x. 5 2 8960 80640 8.25X2.125 6.66 3-75 X.68 75 3-66 2.5 X-5 2.16 9850 89600 8.5 X2.25 7-5 4 X.68 75 4.16 2-75 X. 375 2-5 II 200 100800 9 X2.5 8-33 4-25 X-75 4.66 3 X.375 2.66 12544 II2000 9-5 X 2.375 9.16 4-5 X-75 5-33 3-25 X. 375 3 14336 125 440 10 X2.5 10 1 4.625 X. 75 5-66 3-5 X.375 3-23 15232 134400 HOPES AND CHAINS. 165 Ultimate Strength, arid. Safe Loads of Hemp, Iron, and. Steel. Ultimate SAFB LOAD < | Ultimate SAFE LOAD Strength per Lb. Weight per Foot. perLb. Weight per Foot. per Square in InXT/l Strength perLb.Weigbt per Foot. per Lb. Weight per Foot. per Square ofCircum. in Inches. Lba. Lbs. Lbs. 1 Lbs. Lbs. Lbs. Hemp . Iron . . . 15000 22OOO 4550 5000 100 ! 600 Steel. f 30000 (45500 f6ooo (5000 f 1000 (1300 PLOUGH STEEL FLAT MINING ROPES. John A . Roebling's Sons Co. , New York Width. Thickness. Weight per Foot. Ultimate Strength. Width. Thickness. Weight per Foot. Ultimate Strength. Ins. Ins. Lbs. Lbs. Ins. Ins. Lbs. Lbs. 2 375 I.I9 63000 5-5 375 3-9 156000 2-5 375 1.86 i 74000 5-5 5 4.8 193000 3 375 2.32 93000 6 375 4-34 I73OOO 3-5 5 2.97 118000 6 4375 4-5 lOOOOO 4 375 2.86 114000 6 5 5- 1 210000 4 -5 3.3 130000 6-5 5 5-5 224000 4-5 375 3.12 125000 7 5 5-9 238000 4-5 -1 V5 1;I 4 160000 7-5 5 6.25 250000 5 375 3-4 125000 8 5 6-75 270000 sJ$uifri 5 4.27 I7QOOO For Cast- Steel Flat Ropes see page 1029. - Ropes and Chains of Equal Strength. CIRCUMFERENCE. WEIGHT PER FOOT. Diameter of Iron Chain. Hemp Rope. Crucible Steel Rope. Charcoal Iron Rope. Steel Rope. Iron Rope. Hemp Rope. Iron Chain. Safe Load. Ins. Ins. Ins. Ins. Lbs. Lbs. Lbs. Lbs. Tous. 218 75 2-75 I . 14 34 5 -3 25 3 1.18 , 21 .46 65 4 .28125 3-5 I 1.39 17 28 .67 .81 5 3125 4-25 1.26 1.57 25 33 75 .96 .6 375 4-5 1.45 1.77 3 45 83 1.38 .8 4375 5 1.57 1.97 35 57 1.16 I. 7 6 i .46875 5-5 1.77 2.19 45 7 1.2 2.2 1.3 5 5-75 1.06 2.36 59 -83 1.6 2.63 1.5 -625 6-75 2. 3 6 2-75 85 i. 08 2 4.21 2-3 .6875 7-75 2-75 3-14 I .1 1-43 2.6 5 4.83 75 8-75 2-95 3-53 I .28 1.8 3-35 5-75 3'8 875 9-75 3.14 3-93 I 45 2-3 4.6 7-5 4.8 9375 10.5 3-53 4-32 1.8 3 2-94 4-92 9-33 5-9 1.0625 n-75 3-93 4.71 2-33 3-56 5-83 10.6 7 1.125 12.75 4-32 5.1 2.98 4 6.2 11.9 8.2 1.25 J4-75 4.71 5-5 3-58 4.8 8-7 14-5 9-5 1-375 15-25 4.81 5-89 3-65 5-6 9 17.6 ii 15-75 5.1 6.28 4.04 6.3 10. 1 20 12.5 1.625 17-75 5-8 7.07 5-65 7-95 13-7 22.3 J -75 19-5 6-35 7.85 6-5 9 81 16.4 24-3 19.6 By experiments of U. S. Navy, hemp rope of this circumference has weight of 71 309 Z&s., and a wire rope of 5.34 ins. has equivalent strength. a breaking 1 66 WEIGHT, STRESS, AND TENSION OF EOPES. \Veiglit of* Hemp and Wire Rope. (Molesworth.) In Lbs. per Fathom. Circum- ference. H Common. HP. Good. Wi Iron. SE. Steel. Circum- ference. Hi Common. HP. Good. Ins. Lbs. Lbs. Lbs. Lbs. Ins. Lbs. Lbs. I .18 .24 .87 .89 5 4-5 6 i-5 .41 54 1.96 2 5-5 5-45 7.26 55 74 2.66 2-73 6 6.48 8.64 2 .72 .96 3.48 3.56 6-5 7.61 10.14 2.25 .91 1.22 4.4 4-51 7 8.82 11.76 2-5 1*13 **5 5-44 5.56 7-5 10.13 13*5 2-75 1.36 1.82 6.58 6-73 8 11.52 15-36 3 1.62 2.16 7.83 8.01 8-5 I3-05 17-34 3-25 1.9 2-54 9.19 9-4 9 14.58 19.44 3-5 2.21 2.94 10.66 10.9 10 18 24 3-75 2-53 3.38 12.23 12.52 12 26 34-56 4 2.88 3-84 13.92 14.24 15 40.52 54 To Compvite Stress upon a Rope set at an Inclination. RULE. Multiply sine of angle of elevation by strain in Ibs., add an allow- ance for rolling friction and weight of rope, and multiply by factor of safety. Factor of safety. For standing rope 4, for running 5, and for inclined planes from 5 to 7. ILLUSTRATION. Inclination of rope 92.5 feet in 100, velocity 1500 feet per minute, and strain 2000 Ibs. ; what should be diam. of iron rope, 7 wires to a strand ? Angle of 92.5 feet in 100 = 43, and sine of 43 .682. .682 X 2000= 1364, to which is to be added rolling friction and weight of rope, assumed to be u ; hence, 1364 + 11 = 1375. Factor of safety assumed at 6, consequently 1375 X 6 = 8250 Ibs. , capacity or break- ing weight or stress of rope. By table, page 162, 8200 Ibs. is breaking weight of a wire rope of 7 strands, .625 inch in diam. To Compvite Tension of a Rope. TD = t. v representing velocity of rope in feet per minute, EP horses 1 power, and t tension in Ibs. ILLUSTRATION. Assume wheel 7 feet in diameter, revolution 140 per minute, and IP as per preceding table, 29.6. _, 29. 6 X 33 ooo 976 800 7X3- Hi6 X 140 , 3079 To Compvite Operative Deflection of a Rope. = d. D representing distance between centres of wheels or drums in feet, w weight of rope in feet per lb., t tension, or power required to produce required power or tension of rope when at rest, and d deflection in feet. ILLUSTRATION. Take elements of preceding case: diam. of wire rope of 7 strands = .5625 inch, and by table, page 162, w = .41 lb., and D = 300 feet. Then 10.7 X 317-2 = 10.87 feet. Capacity. At the Falls of the river Rhine there is a wire rope in operation that transmits the power of 600 horses for a distance exceeding one mile. TRANSMISSION OF POWEK AND EQUIVALENT BELT. Endless Ropes. Wire Ropes, when practicable and proper for application, can be used for transmission of power at a less cost than belting or shafting. Transmission, of IPower. Diameter of Wheel. Jifc Is! tf^S So ll Diameter of Wheel. *N III Diamater of Rope. || 1 Diameter ( of Wheel. Revolu- tions per Minute. Diameter of Rope. II Feet. Ins. Feet. Ins. Feet. Ins. 4 80 375 3-3 7 IOO .5625 21. 1 II I4O 68 75 132.1 4 IOO 375 4.1 7 140 .5625 29.6 12 80 75 99-3 4 1 20 375 5 8 80 .625 22 12 IOO 75 124.1 4 140 375 5-8 8 IOO .625 27-5 12 140 75 173-7 5 80 4375 6.9 8 140 .625 38.5 13 80 75 122.6 5 IOO 4375 8.6 9 80 .625 41-5 13 IOO 75 153-2 5 120 4375 10.3 9 IOO .62 5 51-9 13 120 75 183.9 5 I4O 4375 I2.I 9 140 .625 72.6 14 80 875 I 4 8 6 80 5 IO.7 10 80 6875 58.4 14 IOO 875 I 7 6 6 IOO 5 13-4 10 IOO .6875 73 14 120 875 222 6 120 5 16.1 10 140 .6875 102.2 15 80 875 2I 7 6 I 4 5 18.7 ii 80 .6875 75-5 15 IOO .875 259 7 80 5625 16.9 ii IOO 6875 94-4 15 120 875 300 Wire Rope and. Equivalent Belt. In substituting wire rope for an ordinary flat belt, the diameter is deter- mined by rule in practice for estimating power transmitted by a belt viz., One horse power for every 70 square feet of running belt surface per minute. Thus, a belt 15 inches wide running at rate of 1400 feet per min- ute, its power would be equal to (1400 x 15) (70 x 12) = 25 horses' power. The same result is obtained by the use of a wire rope .5625 inch in diam- eter, running over a wheel 6 feet in diameter, making 130 revolutions per minute. Average life of iron wire rope with good care is from 3 to 5 years t and that of steel rope is greater. Wear increases rapidly with velocity. GJ-eiieral Notes. Hemp and. "Wire Ropes. White Rope, 2 inches in circumference, of different manufactures, parted at a stress of from 4413 to 6160 Ibs. Specimens of Italian, Russian, and French manufacture parted with an average stress of 5128 Ibs. = 1633 Ibs. per square inch of rope. Bearing capacity of a hemp rope is proportional to its thickness, number of its strands, slackness with which they are twisted, and quality of the hemp. Hemp and Wire Ropes. Ultimate Strength is 2240 Ibs. per Ib. per fathom for round hemp, 3300 Ibs. for iron, 7000 Ibs. for cast-steel, and 10000 Ibs. for plough-steel. Working Load is 336 Ibs. per Ib. weight per fathom for round hemp, 660 Ibs. for iron, 1400 Ibs. for cast-steel, and 2000 Ibs. for plough-steel. Or, .83 times square of circumference in inches for round hemp, 5 times square of circumference for iron, and 9 times square of circumference for steel. (D. K. Clark.} Steel Ropes may be one half less in weight than iron or hemp for like working loads. 1 68 ROPES AND CHAINS. IRON WIRE AND UNITED STATES NAVY HEMP ROPE. Wire 6 Strands-, Hemp Core. Rope 4 Strands. Ci Actual. rcumferenc Nominal. WIRE, e. Core. Wires. Breaking Weight. Circun Actual. HEl ference. Nominal. [p. Yarns. Breaking Weight. Ins. Ins Ins. No. Lbs. Ins. Ins. No. Lbs. 7 7 2-35 108 187400 12 I3-25 1168 75966 6 6 2.25 108 104050 II 12.25 1036 77633 4-937 4-9 i-57 114 65409 10.5 11.875 928 76933 4-375 4-5 i-57 114 55316 10 n-375 876 7533 3-5 3-36 1.27 114 34480 95 10.5 800 58766 3-187 2.98 1.17 114 28606 9 10.312 712 56466 2-75 2.68 .78 114 21846 8-5 9-437 640 42866 2-5 2-45 .78 114 15692 8 8.812 560 4OOOO 2-375 2.4 .78 42 I57I8 7-5 8-437 484 35500 2 2.06 39 114 10925 ! 7 7.812 436 32 166 Weight and. Strength, of Stud-link Chain Cable. (English.) D Diam. of each Side. MKNSION Length of Link. 3. Width of Link. Weight Fathom. Admiralty Proof-stress (adopted by Lloyds'). D Diam. of each Side. IMKNSION Length of Link. 8. Width of Link. Weight per Fathom. Admiralty Proof-stress (adopted by Lloyds'). Ina. Ins. Ins. Lbs. Tons. Ins. Ins. Ins. Lbs. Tons. 4375 2.625 1-575 "3 3-5 IJ 9 5-4 121 405 5 3 1.8 134 4-5 1.625 9-75 585 I 4 2 47-5 5625 3-375 2.025 17.2 5-5 i-75 o-5 6-3 164.6 55-125 .625 3-75 2.25 21 7 1-875 1.25 6-75 189 6325 .6875 4.125 2.475 25.4 8-5 2 2 7-2 215 72 75 4-5 ; 2.7 30.2 10.125 2.125 2?5 7-65 242.8 81.25 .875 5-25 3-!5 41.2 13-75 2-25 ! 35 8.1 276.2 91-125 i 6 3-6 53-8 18 2-375 425 8-55 303.2 IOI.5 1.125 6-75 4-05 69 22.75 2-5 5 336 112 5 1.25 75 4-5 84 28.125 2-75 16.5 9-9 406.6 ; 136.125 1-375 [8.25 4.95 101.6 34 i NOTE i. Safe Working-stress is taken at half Proof-stress, 3.82 tons per sq. inch of section. 2. Proof -stress and Safe Working - stress for close-link chains are respectively two-thirds of those of stud-link chains. 3. Proof-stress averages 72 per cent, ultimate strength, and Ultimate Strength averages 8 tons per square inch of section of rod or one side of a link. Weight of close-link chain is about three times weight of bar from which it is made, for equal lengths. Karl von Ott, comparing weight, cost, and strength of the three materials, hemp, iron wire, and chain iron, concludes that the proportion between cost of hemp rope, wire rope, and chain is as 2 : i : 3 , and that, therefore, for equal resistances, wire rope is only half the cost of hemp rope, and a third of cost of chains. Safe "Working Load of Chains. (Molesworth). Diameter of Iron. Load. Diameter of Iron. Load. Diameter of Iron. Load. Diameter Load. Ins. 375 5625 .625 Lbs. 2240 3800 4900 6270 Ins. .6875 75 .8125 875 Lbs. 7390 8960 10280 12320 Ins. 9375 I 1.0625 1.125 Lbs. 13700 15680 17920 20 160 Ins. 1.1875 1.25 I-3I25 1-375 Lbs. 22 4OO 24640 26680 30240 ROPES AND CHAINS. 169 Breaking Strain and IProof of Chain Cables. Diam. of Chain. Breaking Strain. Diam. .of Chain. Breakinf Strain. Diam. of Chain. Breaking Strain. Diam. of Chain. Breaking Strain. Ins. I 1.0625 I.I25 Lbs. 67700 75640 84 100 I.l8 7 5 1.25 1-375 Lbs. 92940 102 160 121 840 Ins. 1-5 1.625 i-75 Lbs. I43IOO 165 920 2l6 120 Ins. 2 2.125 2.25 Lbs. 243 1 80 272580 303280 Proof-stress is 50 per cent, of estimated strength of weakest link and 46 per cent, of strongest. Comparison, of" "Wire Ropes and Tarred. Hemp Rope, Hawsers, and Cables. COARSE LAID. FINE LAID. Ropes. Haws'rs. Cables. Ropes. Haws'rs. Cables. Diam- | Safe 11 4 a Ii if Diam- Safe rt H II eter. g Load. g tajl g eter. Load. (2S F 5 H 35 02 H 55 H 02 02 H H o5 Ins. Ins. Lbs. Ins. Ins. Ins. Ins. Ins. Lbs. Ins. Ins. Ins. 25 .78 425 1.25 5 1875 3-12 2.87 3125 i 690 2-43 2.25 3-32 5625 2420 3-56 3-25 4.87 375 1.25 825 2.68 2-375 3-5 625 2900 3-93 3-62 5-25 5 1600 2.87 2.62 3-87 75 4320 4.81 4-37 6-37 -5625 !-75 2800 3-8i 3-5 875 5700 5-5 5 7-25 .6875 2.125 3800 4-75 4-25 6^2 i 8200 7-25 6.25 8-75 75 2-375 4400 5-25 4.87 7 1.125 IO IOO 8.18 7 9-5 875 2.625 6 150 6.12 5-75 8 8 1.25 13600 8.81 8.06 ii 3 8400 6.62 6.12 8.62 8.62 ! 17500 10 9-75 12.5 25 3-75 13400 8.81 8-5 10.93 10.93 1.625 21800 n.x8 10.93 375 4-25 16800 9.87 9-56 12.25 12.12 i-75 27000 12.5 12.12 .5 4.625 20 160 10.75 10.5 13 13.12 1875 32500 .625 5 24600 - 11.87 ii 56 "75 2 37000 In above table, determination of circumference of rope, etc., is based upon Breaking Weight or Tensile resistance of wire being reduced by one fourth, and ultimate resistances of rope, etc., are reduced one third. Result of Experiments upon "Wire Rope at TJ. S. Navy- Yard, Washington. -(J A Roebling's Sons.) Circumfe Actual. rence. Nom- inal. ill 1 Weight Foot. Breaking Weight. Circumft Actual. rence. Nom- inal. tii rf * *~Z jj!^ No. 13 14 14 17 20 18 19 III Breaking Weight. Ins. 4-9375 4-375 3-9375 3-5 3-1875 2-75 2.6875 2-5 Ins. 4-9 4-5 3*36 2.98 2.68 2.56 2-45 No. 19 19 19 19 19 19 7 19 No. II 13 14 14 15 17 13 18 Lbs. 3-14 2.15 2.0875 I-I525 1.09 1.0275 1.0225 .14 Lbs. 65409 55 3 l6 44420 34840 28606 21846 18810 15692 Ins. 2-375 2.1875 2 1-9375 1-4375 1-3125 1.125 Ins. 2. 4 2.12 2.06 1.9 1.8 5 I. II 7 7 7 7 19 7 7 Lbs. .14 .11 .1 .1 .07 .06 05 -035 Lbs. I57I8 14478 10925 10 118 7880 5687 4428 3729 To Compute Circumference of Wire Rope with Hemp Core, of Corresponding Strength to Hemp Rope, and of Hemp Rope to Circumference of "Wire Rope. RULE i. Multiply square of circumference of hemp rope by .223 for iron wire and .12 for steel, and extract square root of product. 2. Multiply square of circumference of hemp-core wire rope by 4.5 for iron wire and 8.4 for steel wire. EXAMPLE. What are the circumferences of an iron and steel wire rope corre- sponding to one of hemp-core, having a circumference of 8 ins. ? /8 2 X .223 = 3. 78 in*, trow, and i '8 2 x.i2 = 2. 77 ins. steel. ROPES, HAWSEKS, AND CABLES. ROPES, HAWSERS, AND CABLES. Ropes of hemp fibres are laid with three or four strands of twisted fibres, and are made up to a circumference of 12 ins., and those of four strands up to 8 ins. are fully i6per cent, stronger than those of three strands. Hawsers are laid with three or four strands of rope. Cables are laid with but three strands of rope. Hawsers and Cables, from having a less propor- tionate number of fibres, and from the irregularity of the resistance of their fibres in consequence of the twisting of them, have less strength than ropes, difference varying from 35 to 45 per cent., being greatest with least circum- ference, and those of three strands up to 12 ins. are fully loper cent, strong- er than those having four strands. Tarred ropes, hawsers, etc., have 25 per cent, less strength than white ropes ; this is in consequence of the injury fibres receive from the high tem- perature of the tar, viz. 290. Tarred hemp and Manila ropes are of about equal strength, and have from 25 to 30 per cent, less strength than white ropes. White ropes are more durable than tarred. The greater degree of twisting given to fibres of a rope, etc., less its strength, as exterior, alone resists greater portion of strain. Ultimate strength of ropes varies from 7000 to 12000 Ibs. per square inch of section, according as they are wetted, tarred, or dry. One sixth of ulti- mate strength is a safe working load= 1166 to 2000 Ibs. per square inch. Units for computing Safe Strain th.at may "be "borne "by N~e\v Ropes, Hawsers, and. Catoles. (t^. S. Navy.) DESCRIP- TION. Circumference. Wl 3 strands. Ron ite. 4 strands. ts. Tar 3 str'ds. red. 4 str'ds. HAW White. 3 str'ds. SEES. Tarred. 3 str'ds. CAI White. 3 str'ds. LES. Tarred. 3 str'ds. Ins. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. White 2.5 to 6 II4O 1330 600 u 6 " 8 1090 I20O 570 510 u 8 " 12 1045 880 530 530 u 12 " 18 550 550 it 18 " 26 500 Tarred 2-5 " 5 855 1005 460 44 5 " 8 82 5 940 4 80 u 8 " 12 7 80 820 505 505 u 12 " 18 5 2 5 It 18 " 26 __ 55 Manila 2.5 " 6 810 950 440 tt 6 " 12 760 835 465 510 : 41 12 U l8 535 u 18 " 26 560 ILLUSTRATION. What weight can be borne with safety by a Manila rope of 3 strands, having a circumference of 6 inches ? (See Rule, page 167. ) 6 2 = 27360^5. When it is required to ascertain weight or strain that can be borne by ropes, etc., in general use, preceding Units should be reduced from one third to two thirds, in order to meet their condition or reduction of their strength by chafing and exposure to weather. Molesworth's table is based upon a reduction of three fourths. ILLUSTRATION. What weight can be borne by a tarred hawser of 3 strands. 10 inches in circumference, in general use? io 2 X (505 505 -r- 3) = zoo X 336.67 = 33 667 #* HOPES, HAWSERS, AND CABLES. Destructive Strength, of* Tarred. Hemp Ropes. (D. K. Clark.) Circum. Diam. Res Common Cold. ister. Russian Warm. Circum. Diam. Reg Common Cold. wter. Russian Warm. Ins. 3 3-5 4 4-5 5 Ins. 95 i. ii 1.27 1-43 i-59 Lbs. 7390 II 200 I3IOO 16330 19580 Lbs. 8620 II 760 15340 19440 23990 Ins. 5-5 6 6-5 7 8 1-75 1.91 2.07 2.24 2-54 Lbs. 24800 28985 34030 40320 52480 Lbs. 29120 33150 40550 47041 61420 Specimens furnished by National Association of Rope and Twine, Spinner*, As tested by Mr. Kirkaldy. Ron. Circum- ference. Weight perLb. Extreme Strength. Breaking Weight per Ib.per Fathom. Extend atStre 1000 Ibs. >n In 50 in ss per Ib. r Fathom 2000 IDS. s. Length W*h. 3000 Ibs. Rus Ma< Hai Br a 9 5 sian 3hine id-sp eaki 5 rope ... 48 thr yarn. . . 50 ' an yarn, 51 * ing Streng Old Method. Common Best Hemp. Russian. ds. tn I Cc Ins. 5-26 5-37 5-39 of a y Regist Id. W Lbs .92 .85 1. 00 ^arr sr. arm. 6 i 6 ec i ! I I I 1 > Lbs. 1088 1514 8278 Hei 5 Lbs. 1933 2152 3024 np R OldM Common Hemp. Ins. 5-29 4-53 4.46 opes. ethod. Best Russian. Ins. 6.56 5-91 (Mr. By Re Cold. Ins. 6.63 Plynn.) gister. Warm. Ins. 3 3-5 4 4-5 5 Ins. 95 i. ii 1.27 J -43 i-59 Lbs. 5056 74 66 8780 10300 13328 Lbs. 6248 8668 10460 12432 15859 Lbs. 7392 II 2OO I3I04 16330 20496 Lbs. 8624 II 760 17810 19443 23990 Ins. 5 6 5 6.5 8 Ins. 1-75 I. 9 I 2.O7 2.24 2-54 Lbs. 15456 18144 20518 22938 26680 Lbs. 18414 2l6lO 23 610 27462 32032 Lbs. 24797 28986 34630 40320 52483 Lbs. 29120 33150 40544 47040 61420 To Compute Strain that may "be borne witn safety t>y new Ropes, Hawsers, and. Cables. Deduced from experiments of Russian Government upon relative strength of different Circumferences of Ropes, Hawsers, etc. U. 8. Navy test is 4200 Ibs. for a White rope of three strands of best Riga hemp, of 1.75 inches in circumference (= 17000 Ibs. per square inch ofjibre), but in preceding table (page 166) 14000 Ibs. is taken as unit of strain that may be borne with safety. RULE. Square circumference of rope, hawser, etc., and multiply it by Units in table. To Compute Circumference of a Rope, Hawser, or Cat>le for a GKven Strain. RULE. Divide strain in pounds by appropriate units in preceding table, and square root of product will give circumference of rope, etc., in ins. EXAMPLE i. Stress to be borne in safety is 165 550 Ibs. ; what should be circum- ference of a tarred cable to withstand it? 165 552 -r- 550 = 301, and -v/30 1 = J 7- 35 ins - 2. What should be circumference of a Manila cable to withstand a strain, in general ttse, of 149 336 Ibs. ? Assuming circumference to exceed 18 ins., unit = 560. 149 336 -r- (560 560-7-3) = 400, and V4 00 = 20 *'* 172 KOPES, HAWSERS, AND CABLES. To Compute "Weignt of Ropes, Hawsers, and Catoles. RULE. Square circumference, and multiply it by appropriate unit in following table, and product will give weight per foot in Ibs. : HAWSERS. HOPES. CABLES. 3 strand Hemp 032 .031 .031 3-strand tarred Hemp, .042 .041 .041 3-strand Manila 032 .031 .031 4-strand Hemp 033 4-strand tarred Hemp, .048 4-strand Manila 035 .034 .034 Units for Thread Ropes is same as that for Ropes of like material. EXAMPLE. What is weight of a coil of lo-inch Manila hawser of 4 strands of 120 fathoms? io 2 X .034 = 3. 4, and 120 X 6 X 3- 4 = 2448 Ibs. Weignt and. Strength, of Hemp and Wire Ropes. (Molesworth.) C representing circumference in ins., W weight of rope in Ibs. per fathom, L working load in tons, and S destructive stress in tons. VALUES OF y, X, AND k. ROPES. y X k ROPES. y X k Hawser, hemp ' '3 1 Warm register, hemp .7 .116 Cable " Manila hawser *77 .27 045 Tarred hawser hemp ' ' cable IQ 03 3 ' ' cable ' ' Iron rope .87 1.8 ,2Q Cold register. " . .6 .1 Steel " . .80 2.8 .4^ To Compute Circumference of Hemp or "Wire Rope for Fore or ]Vtaiii Standing Rigging. (V. S. Navy.) RULE. To length of mast between partners and deck, add half extreme breadth of beam of vessel and divide sum by half extreme breadth. Mul- tiply quotient by half square root of tonnage (OM) and extract square root of product. For Mizzen, take .74 of Fore and Main. EXAMPLE. Required circumference of hemp rope, for main - mast of a vessel having a breadth of beam of 45 feet and a burden of 3213 tons? Extreme length of mast ____ .................. 94.4 feet. Depth of hold, or total bury of mast, 21.4 feet. Head .............................. 15 " 36.4 " Breadth of beam, 45 feet. 5 8 + 1 5 --:- 1 s - = 3. 58, and 58 " 6 = io.n ins. Then if circumference for a wire rope is required, see table, page 164. Thus, a hemp rope 10 ins. in circumference has equivalent strength of an iron wire rope of 4 ins. and a steel rope of 3.25+ ins. Galvanized Iron Wire. Experiments at Navy Yard, Washington, gave for flex- ibility a mean loss of 30 per cent, and for tensile strength a like loss of 13.5 per cent. Relative Dimensions of Kemp Rope and Iron and Steel \Vire Rope. (U. S. Navy.) Circumference in Inches. Hemp. 2.5 3.125 4 4.5 5.25 6.5 7.75 8.5 9.5 Iron.. 1.25 1.625 2 2.125 2.5 3 3-5 4 4-5 Steel.. .875 1.125 i-5 1-625 I - 8 75 2 - I2 5 2.5 2.75 3.25 11.75 13- 5 '6.5 5 5-5 6 7 3.5 4 4.375 5.25 ANCHORS, CABLES, ETC. 173 ANCHORS, CABLES, ETC. Anchors, Chains, etc., for a Griven Tonnage. (American Shipmasters' Association.) SAILS. til |fi Bow With- out Stock. j en. Admi- ralty Test. ANCHORS lucl Stream. ading Stc Ked K e. ck. 3d Kedge. Diameter. CHJ M ! JN CABLI Admi- ral ty Test. t. STUD Weigh Stud. t per Fa Short Link. bom. Eng- lish.t Lbs. Tons. Lbs. Lbs. Lbs. IllS. Fnths. Tons. Lbs. Lbs. 75 616 7 168 8 4 .8125 90 II 4 4 2 35 IOO 728 8 106 112 .875 105 13 44 4 8 125 840 9 224 112 9375 105 15 51 55 48 150 952 IO 280 I4O i 1 20 17-5 59 63 54 i?5 1036 II 336 168 1.0625 1 20 20 66 70 200 1 120 12 392 196 1.125 1 20 22-5 75 79 68 250 1288 13 448 224 112 1.1875 135 25 82 88 300 1456 14 504 252 126 1.25 135 28 9 1 98 84 350 1624 T 5-5 560 280 140 I -3 I2 5 150 31 IOO 106 4OO 1848 17 616 308 154 i-3 I2 5 150 31 IOO 106 450 1904 18.5 672 336 168 i-375 165 37 H5 118 102 500 20l6 20 784 392 196 1-4375 165 40 120 600 2352 22 896 448 224 i-5 180 44 132 122 700 2688 24 1008 504 252 1-5625 180 47 145 800 3024 26 II2O 560 280 1.625 180 5i 156 143 900 32 4 8 28 1232 616 308 1.6875 180 55 162 1000 3584 29-5 X 344 672 336 i-75 180 59 175 166 1200 3 808 3 1 I45 6 738 364 1.875 180 63 !80 191 1400 4032 32.5 1568 784 392 *-9375 180 67 205 1000 4256 34 1680 840 420 2 180 72 219 I800 4480 35-5 1792 896 448 2 180 72 240 217 2OOO 4704 37 1904 952 504 2.0025 180 81 2500 5040 39 2128 II2O 560 2.125 180 86 244 3000 | 5376 4i 2353 1232 616 2.1875 180 96 t Brown, Lennox, & Co. Xo CompxTte Tonnage. Take dimensions as follows : Length. From after-side of stem to for- ward-side of stern-post, measured on spar or upper deck in vessels having two decks and under, and on main deck in vessels having three or more decks. Breadth. Extreme at widest point. Depth. At forward coaming of main hatch, from top of ceiling at side of keelson to under side of deck. Then multiply these dimensions together, divide product by 100, and take .75 of quotient. All vessels to have 2 bowers and i each stream and kedge anchor, and for a tonnage exceeding 1400 a third bower is recommended. Hawsers and Warps to be 90 fathoms in length. Snroxids. SQUARE-RIGGED. Hemp. 5.75 ins. in diameter for a tonnage of 75, ini creasing progressively up to 12.75 ins. for 3000 tons. FORE-AND-AFT RIGGED. From .25 to i inch hi diameter progressively greater than for square-rigged. Wire. One half diameter of hemp, increasing very slightly as tonnage increases. Thus, for 3000 tons, 12.75 ins. for hemp and 6.875 ins. for wire. '74 ANCHOKS, CABLES, ETC. (American Shipmasters' Association.) STEAM. Tonnage computed as per Rule preceding. Bow ill 4 en. m NCHORS luclu ding Stc ! ck. '! Diam- eter. C i Admiral- > tyTest. * O ABLE. ST Diam. Stream. UD. Weig 1 02 it per . 5*3 3 Path. Ii Lbs. Tons. Lbs. Lbs. Lbs. Ins. Paths. Tons. Ins. Lbs. Lbs. 100 336 4.9 112 .6875 165 8.1 5 25 ISO 448 6.4 106 .8125 120 11.9 .5625 4 42 35 200 616 7.6 22 4 .875 120 13-8 5625 44 48 250 672 8.2 280 9375 120 15.8 .625 5i 55 48 300 812 9-5 308 i 1 2O 18 .625 59 63 54 350 924 10.4 336 1.0625 120 20.3 .68 7 5 66 70 400 1 120 12 532 252 1.125 135 22.8 .6875 75 79 68 450 1344 13-9 500 280 1.1875 135 25-4 75 82 88 500 1512 15-2 672 336 1.25 150 28.1 75 9 1 98 84 600 i',o8 I6. 7 738 364 1-3125 150 31 .8125 100 106 700 1876 18 784 392 1-375 I6 5 34 .8125 "5 118 104 800 2026 19 896 448 22 4 1-4375 I6 5 37-2 .875 120 QOO 2352 21.6 I008 504 252 i-5 180 40.5 .875 I 3 2 122 1000 2632 23-5 1 120 560 280 1-5625 1 80 44 -9375 J 45 1200 2856 25.2 1176 588 3 08 1.625 180 47-5 9375 J 56 !43 1400 3108 26.9 1232 616 308 1.6875 180 51.2 162 I6OO 3360 28.6 1344 672 336 1-75 180 55-1 175 166 I800 2000 3584 3808 30.1 31.6 1456 1512 738364 766 364 1.8125 1-875 180 180 59-i 633 .0625 .0625 189 205 191 2300 4088 33-4 1568 784 392 1-9375 180 67.6 .125 215 2000 4256 34.5 1624 812 392 2 270 72 .125 240 217 3000 4480 35-7 1680 840 420 2.0625 270 76.6 1875 3500 4592 37 1792 896 476 2.125 270 81.3 i .1875 244 4000 4816 38 1000 952 i 504 2.l8 7 5 270 86.1 .25 . 4500 5040 39-2 2128 1064 532 2.25 270 91.1 .25 -^ SQOO 5264 4i 2352 1 1 20 560 2.3125 270 96 .3125 * Brown, Lennox, & Co. ANCHORS AND KEDGES. (U. 8. Navy.) To Compute "Weight of a Bovver A.nchor for a Vessel of* a given Character and Rate. RULE. Multiply approximate displacement in tons, by unit in following table, and product will give weight in Ibs., inclusive of stock. "Units to determine "Weights and. !N"u.m"ber of Anchors or Kl edges. Displacement of Vessel in Tons. "3 p 1 1 35 ! Displacement of Vessel in Tons. "a D ! 2 2 2 *s s S. 3 Over 3700 " 2400 " 1900 i-75 2 2.25 2 2 2 2 2 2 i i i 4 3 3 Over 1500 . . " 900 .. 900 and under 2-5 2-75 3 2 I I 3 3 2 EXAMPLE. Tonnage of a bark- rigged steamer is 1500. 1500 X 2. 5 = 3750 Ibs. , weight of anchor. Bower and Sheet Anchors should be alike in weight. Stream A nchors and Kedges are proportional to weight of bowers. Thus, Stream Anchor .25 weight. Kedges. If i, .125 weight; if 2, .16 and .1 weight; if 3, .16, .125, and .1 weight. ANCHORS, CABLEis, ETC. TONNAGE. 175 To Com.pu.te Diameter of a Chain Ca"ble corresponding to a Q-iveii ^Weight of Anchor. (U. S. Navy.) RULE. Cut off the two right-hand figures of the anchor's weight in Ibs., multiply square root of remainder by 4, and result will give diameter of chain in sixteenths of an inch. EXAMPLE. The weight of an anchor is 2500 Ibs. -^25.00 x 4 = 20 sixteenths = 1.25 ins. NOTE. Diam. of a messenger should be .66 that of the cable to which it is applied. Lengths of Chain Cables for each Anchor. (U. S. Navy.) Weight of Anchor. Bower. Sheet. Stream. Weight of Anchor. Bower. Sheet. Stream. Lbs. Fathoms. Fathoms. Fathoms. Lba. Fathoms. Fathoms. Fathom* Under 800 60 60 60 Over 2000 120 120 90 Over 800 90 90 60 " 3000 120 120 00 " 1200 90 90 75 " 5000 120 120 105 " I600 105 105 75 " 75oo 135 135 105 ANCHORS. From Experiments of a Joint Committee of Representatives of Ship- owners and Admiralty of Great Britain. An anchor of ordinary or Admiralty pattern, Trotman or Porter's im- proved (pivot fluke), Honiball, Porter's, Aylin's, Rodgers's, Mitcheson's, and Lennox's, each weighing, inclusive of stock, 27 ooo Ibs., withstood without injury a proof strain of 45000 Ibs. Breaking weights between a Porter and Admiralty anchor, as tested at Woolwich Dock-yard, were as 43 to 14. Comparative Resistance to Dragging. Trotman 's dragged Aylin's, Honiball's Mitcheson's and Lennox's ; Aylin's and Mitcheson's dragged Rodgers's ; and Rodgers's and Lennox's dragged Admiralty's. (TONNAGE OF VESSELS. To Compute Tonnage of Vessels. For Laws of United States of America, with amendments of 1882 relative to Steam- vessels, see Mechanics' Tables, with rule and illustrated diagrams, by Chas. H. Haswell, 3d edition, Harper & Bros., New York, 1878. English Registered Tonnage. (New Measurement.) Divide length of upper deck between after-part of stem and fore-part of stern- post into 6 equal parts, and note foremost, middle, and aftermost points of division. Measure depths at these three points in feet and tenths of a foot; also depths from under-side of upper deck to ceiling of limber-strake; or in case of a break in the upper deck, from a line stretched in continuation of the deck. For breadths, divide each depth into 5 equal parts, and measure the inside breadths at following points, viz. : At .2 and .8 from upper deck of foremost and aftermost depths; and from .4 and .8 from upper deck of amidship depth. Take length at half amidship depth from after-part of stem to fore-part of stern-post. , Then, to twice amidship depth add foremost and aftermost depths for sum of depths, and add together foremost upper and lower breadths, 3 times upper breadth with lower breadth at amidship, and upper and twice lower breadth at after division for sum of breadths. Multiply together sum of depths, sum of breadths, and length, and divide product I by 35o, which will give number of tons. If the vessel has a poop or half deck, or a break in upper deck, measure inside mean length, breadth, and height of such part thereof as may be included within the bulkhead ; multiply these three measurements together, divide product by 92.4, and quotient will give number of tons to be added to result as above ascertained. 176 TONNAGE OP VESSELS. For Open Vessels. Depths are to be taken from upper edge of upper strake. For Steam Vessels. Tonnage due to engine-room is deducted from total tonnage computed by above rule. To determine this, measure inside of the engine-room from foremost to aftermost bulkhead; then multiply this length by amidship depth of vessel, and product by inside amidship breadth at .4 of depth from deck, and divide final product by 92.4. The volume of the poop, deck-houses, and other permanently enclosed spaces, available for cargo or passengers, is to be measured and included in the tonnage, but following deductions are allowed, the remainder being the Register tonnage. Deductions. Houses for the shelter of passengers only; space allotted to crew (12 square feet in surface and 72 cube feet in volume for each person); and space occupied by propelling power. Approximate Rule. Gross Register. Tonnage of a vessel expresses her entire cubical volume in tons of loo cube feet each, and is ascertained by following formula : = Gross tonnage, and c = Register tonnage. L representing length of keel between perpendiculars, B breadth of vessel, and D depth of hold, all in feet. Builders' ^Measurement. (L-.6B)XBX.5B = e 94 Fore-perpendicular is taken at fore-part of stem at height of upper deck. Aft-perpendicular is taken at back of stern-post at height of upper deck. In three-deckers, middle deck is taken instead of upper deck. Breadth is taken as extreme breadth at height of the wales, subtracting differ- ence between thickness of wales and bottom plank. Deductions to be made for rake of stem and stern. 1 8 / Girth + Breadth\ 2 Iron Vessels. I ) X length = Gross tonnage. I0000\ 2 ) Length measured on upper deck, between outside of outer plank at stem and the after-side of stern post and rabbet of stern-post, at point where counter plank crosses it. Girth measured by a chain passed under bottom from upper deck at extreme breadth, on one side, to corresponding point on the other. Register tonnage = X C. C representing a coefficient for vessels as IOO follows : of usual form ................. 7 Clippers and Steamers { Yachts above 60 tons 5 5 Smanve S se,s{^ a ;- ;::::::::; -45 "Units for Measurement and Dead --weight Cargoes. (C. Mackrow, M. S. N. A.) To Compute Approximately for an Average Length of Voyage the Measure- ment Cargo, at 40 feet per Ton, which a Vessel can carry. RULE. Multiply number of register tons by unit 1.875, ai *d product will give approximate measurement cargo. To Compute Approximately Dead-weight Cargo in Tons ivhich a Vessel can carry on an Average Length of Voyage. RULE. -Multiply number of register tons by 1.5, and product will give approximate dead-weight cargo required. With regard to cargoes of coasters and colliers, as ascertained above, about 10 per cent, may be added to said results, while about 10 per cent, may be deducted in cases of larger vessels on longer voyages. TONNAGE OF VESSELS. 177 In case of measurement cargoes of steam-vessels, spaces occupied by ma- chinery, fuel, and passenger cabins under the deck must be deducted from space or tonnage under deck before application of measurement unit thereto. In case of dead-weight cargoes, weight of machinery, water in boilers, and fuel must be deducted from whole dead weight, as ascertained above by application of dead-weight unit. The deductions necessary for provisions, stores, etc., are allowed for in selection of the two units. To A scertain Weight of Cargo for an A verage Length of Voyage. (Moorsom. ) Deduct tonnage of spaces of passenger accommodations from net register tonnage, and multiply remainder by 1.5. Average space for each ton weight of cargo on such a voyage 67 cube feet. Freight Tonnage or Measurement Cargo. Freight Tonnage or Measurement Cargo is 40 cube feet of space for cargo, and it is about 1.875 times net register tonnage less that for passenger space. Royal Thames Yacht Cluto. Measure length of yacht in a straight line at deck from fore-part of stem to after- part of stern-post, from which deduct extreme breadth (measured from outside of outside planking), both in feet; remainder is length for tonnage. Multiply length for tonnage by extreme breadth, that product by half extreme breadth, divide re- sult by 94, and quotient will give tonnage. If any part of stem or stern-post projects beyond length as taken above, such projection or projections shall, for purpose of computing tonnage, be added to length taken as before mentioned. All fractional parts of a ton are to be considered as a ton. Measurements to be taken either above or below main wales. ' = Tons. L representing length and B breadth, in feet. Corinthian and. New Thames Yacht Civil). Measure length and breadth as in foregoing rule, and depth to top of covering board; multiply length, breadth, and depth together, divide result by 200, and quo- tient will give tonnage. Suez Canal Tonnage. Gross Tonnage. Spaces under tonnage deck, below tonnage and uppermost deck, all covered or closed - in spaces, such as poop, forecastle, officers' cabins, galley, cook, deck, and wheel houses, and all inclosed or covered-in spaces for working the vessel. From which are to be deducted berthing accommodations for crew, not including spaces for stewards and passengers' servants ; berthing accommodations for officers, except captain; galleys, cook-houses, etc., used exclusively for crew, and inclosed spaces above uppermost deck, designed for working the vessel. In none of these spaces can passengers be berthed or cargo carried, and total deduction under all of these spaces must not exceed 5 per cent, of gross tonnage. In steamers witb standing coal-bunkers, English rule may be followed, or owner may elect to have tonnage of his vessel computed by "Danube rule," which is an allowance of 50 per cent, above space allowed to machinery in side- wheel steamers and 75 in screw steamers. In no case, however, except with tow-boats, must deduction for propelling power exceed 50 per cent, of gross tonnage. 1^8 WORKS OF MAGNITUDE. WORKS OF MAGNITUDE. American. Aqueducts, Roads, and Railroads. Croton Aqueduct, N. Y. Has a section of 53.34 square feet and capacity of loooooooo to 118000000 gallons per day. and from Dam to Receiving Reservoir is 38. 134 miles in length. Aqueduct, Washington. Cylinder of masonry 9 feet in diameter. Stone arch over Cabin John's Creek, 220 feet span, 57.25 feet rise. National Road. Over the Alleghany Mountains, Cumberland to Illinois Town. 650.625 miles in length, and 80 feet in width. Macadamized for a width of 30 feet. Illinois Central Railroad. Chicago to Cairo, length 365 miles, Centralia to Dun- leith 344 miles, total 709 miles. Bridges. Suspension Bridge, Niagara River. Wire, Span 1042 feet 10 ins. Suspension Bridge, New York and Brooklyn. Length of river span 1595 feet 6 ins. ; of each land span 930 feet; length of Brooklyn approach 971 feet; of N. Y. approach 1562 feet 6 ins. ; total length of bridge 5989 feet; width 85 feet; number of cables 4; diameter of each cable 15.5 ins. ; each consisting of 6300 parallel steel wires No. 7 gauge, closely laid and wrapped to a solid cylinder; ultimate strength of each cable 11200 tons; depth of tower foundation below high water, Brooklyn, 45 feet New York 78 feet; towers at high-water line 140X59 feet; towers at roof course 136x53 feet; total height of towers above high water 277 feet; clear height of bridge in centre of river span above high water, at 50, 135 feet; height of floor at towers above high water 119 feet 3 ins. ; grade of roadway 3 feet in 100; anchor- ages, at base i2gX 119 feet, at top ii7X 104 feet; weight of each anchor-plate 23 tons. Iron Pipe Bridge over Rock Creek. 200 feet span, 20 feet rise. Arch of 2 lateral courses of cast-iron pipe, 4 feet internal diameter, and i inch thick. These pipes conveying the water not only sustain themselves over the great span, but support a street road and railway. Iron Bridge over Kentucky River near Shakers' Ferry, Md. 3 spans, each 375 feet, and 275.5 feet above low water. Bridge on line of New York, Erie, and Western Railroad across the Kinxua,.- Of iron; length 2060 feet; central span 301 feet in height. Iron Truss. Cincinnati and Southern Railway, over Ohio River, 519 feet, Foreign. Pyramids, Statues, etc. Pyramid of Cheops, Egypt. Length of side at base 762 feet; height to present summit 453.3 feet; to original summit 485.2 feet; inclined length 568.25 feet; angle of side 51 51' 14"; area of each face = square of height; weight 5272600 tons; built 2 1 70 years B.C. Peter the Great, St. Petersburg. Russia. Bronze; height of horse 17 feet; of man ii feet; base of rock 42 feet at bottom, 36 at top, 21 wide, and 17 high, weighing ti oo tons. Liberty, New York Harbor. Bronze; no feet in height from head to foot and 151.1 feet to flambeau ; including base, 305.6 feet. Weight of statue 225 tons. Daibutsu, of stone, Japan. Sitting posture; height 44 feet^ circumference 87 feet; face 8.5 feet; circumference of thumb 3 5 feet. Colossus of Rhodes. Height, 105 feet. Bridge. Britannia Tubular Bridge. Of iron, with a double line of Railway, 964 feet in length, with two approaches of 230 feet each. Weight 3658 tons. WORKS OF MAGNITUDE. UVIonolitlis. 179 Obelisk at Karnak, Egypt Of granite, 108 feet 10 ins. ; pedestal 13 feet 2 ins.: weight 400 tons. Obelisk in Central Park, N. Y. Of granite, 68 feet n ins.; weight 168 tons. U. S. Treasury, Washington. Some stones of, are heavier than any in the Pyra* mids of Egypt. Steam. Hammers. At workshops of Herr Krupp, at Essen, there is a steam hammer weighing 50 tone having a fall of 3 metres; and at Creusot there is a hammer weighing between 75 and 80 tons having a fall of 5 metres. Crane. At Creusot there is a steam crane having a capacity to lift 150 tons. Ch.imn.eys. J. Townsend's chemical works, Glasgow, diameter at foundation 50 feet; at top 12 feet 8 ins. ; height from foundation 488 feet; from ground 474 feet. Metropolitan Traction Company, N. Y., diameter at base 85 feet; at top 25 feet, and height 353 feet. Pillar. At a gate near Delhi is a wrought-iron pillar having diameters of 16.4 ins. at 22 feet in its height above ground and 12 ins. at its top. It is estimated from the re- sult of excavations at its base to be 60 feet in length or height and to weigh 17 tons. Its period of structure is assigned to the 3d or 4th century A.D. Roofs. Midland Railway Station, London. 240(1. I Union Railway Station, Glasgow. 195 ft Imperial Riding-School, Moscow. 235 " | Grand Central Station, N. Y 200 " JDiameters of Domes. DOMES. Feet. DOMES. Feet. DOHXS. Capitol, Washington Glasgow W. Railw'y 124-75 198 St. Paul's, London. St. Peter's, Rome.. 112 139 MidrndRail'y,Lon. Great North'n, Eng. Lengths of Tnnnels. TtTNNILS. Feet. TUNNKLS. Feet. TUNNILS. Feet. Blaizy 13 455 Gunpowder, Md.. . 06 EJOO Nerthe Blue Ridge 4280 Sutro 20028 Nochistongo I5IS3 Hoosac 25 031 Semmering . . , 1610 Riauivel. . . 18627 Thames and Medway , 1 1 880 feet. Weehawken, 4000 feet Mont Cenis 7.5 miles 242 yards, rises i in 45, and descends i in 2000. St. Gothard Tunnels and Roads o miles 477 yards in length ; tunnels 116 156.5 feet, and rises i in 233 in whole length; 26.5 feet in width; 19 feet 10 ins. in height Maximum grade 2.7 feet per 100. Schemnitz, 10.27 miles in length, 9 feet 10 ins. in height by 5.25 feet in width. Miscellaneous. Fortress Monroe, Old Point Comfort, Va. Largest fortress. Telegraph Wire. Span over river Kistnah between Bezorah and Sectanagran, 6000 feet in length. Deer Park, Copenhagen. 4200 acres. Oxford College, England. Largest University; said to have been founded by Alfred. Cathedral St. Peter's, Rome. Width of front 216 feet; of the cross 251 feet; total height 469. 5 feet. Steamer Great Eastern. Of iron, 680 feet in length; 83 feet width of beam- 60 feet depth of hold; 22927 tons; built at Millwall, England, 1857. Chinese Wall. 25 feet at base; 15 at top; height, with a parapet of 5 feet, 20 feett length 1250 miles. ArUsian Wett, Perth. 3050 feet in depth; temperature of watr 99: volume of discharge 18000 gallons per day. ISO BELLS, CHURCHES, COLUMNS, TO WEBS, ETC. Weights of Bells. BULLS. Lbs. BELLS. Lbs. BELLS. Lbs. Pekin I2OOOO Oxford, "Great St. Peter's, Rome. 1 8 ooo Lewiston Me Tom " Eng. .... 17 O24. Vienna 40 200 Montreal, Can 10233 28560 Olmutz. Bohemia. / "-'* / t 40320 Westm'ster, "Big Moscow, Russia. . . 44S77 2 Sac'd Heart, Paris 55 o Ben," England. 35620 Erfurt, Saxony. . . . Notre Dame, Paris 30800 28670 St. Paul's, Eng. . . St. Ivan's, Moscow 42000 127 830 York State House, Phila. 24080 13000 Rangoon, Burmah, 201 600 Ibs. Capacity- of Principal Ch/urohes and. Opera Houses. Estimating a person to occupy an Area 0/19.7 Ins. Square. Clinrones. St. Peter's 54 ooo I St. John, Lateran 22 900 Milan Cathedral 37 ooo Notre Dame, Paris 21 ooo St. Paul's, Rome 32000 I Pisa Cathedral 13000 St. Paul's, London 25 600 I St. Stephen's, Vienna 12 400 St. Petronio, Bologna 24 400 | St. Dominic's, Bologna 12 ooo Florence Cathedral 24 300 Antwerp Cathedral 24 ooo St. Sophia's, Constantinople 23 ooo Tabernacle, London 7 ooo " Brooklyn 5500 St. Mark's, Venice 7 ooo Opera Houses and. Theatres. Carlo Felice, Genoa 2560 Opera House, Munich 2370 Alexander, St. Petersburg 2332 San Carlos, Naples 2240 Imperial, St. Petersburg 2160 La Scala, Milan 2113 Academy of Paris 2092 Teatro del Liceo, Barcelona 4000 Covent Garden, London 2684 Opera House, Berlin 1636 New York Academy 2526 Metropolitan Opera, N. Y 5000 Philadelphia Academy 3124 Chicago " 3000 Heights of Columns, Towers, I3oraes, Spires, etc. LOCATIONS. Feet. CHIMNEYS. Townsend's Glasgow. . . St Rollox u 474 455- 5 Musprat's Liverpool 406 Gas Works Edinburgh New E ngland Glass Co . Boston Steam Heating Co New York. Metropolitan Tract. Co. ' * COLUMNS. Alexander St. Peters'g 230 220 353 *75 Bunker Hill Mass City London... July Paris 221 202 157 132 Nelson's London . . . Place Vendome Paris 171 Pompey's Pillar Egypt Trajan Rome 114 145 Washington . . Wash'gton 555 York ' London . . . I3 8 TOWERS AND DOMES. Babel 680 Balbec Capitol Wash'gton 287 5 St. Peters Rome Cathedral Cologne . . . " Cremona.. " Escurial. .. 469-5 524-9 392 200 LOCATIONS. Feet. TOWERS AND DOMES. Cathedral Florence . . " .Maedeb'rer 390-5 4$ 9 363 188 200 328 355-1 325 465-9 4048 216 410 450 210 2OO 4 4 286 216 232.9 344 473 443-8 486 464 314-9 .Milan .Petersburg Leaning Pisa Porcelain St. Mark's 3t. Paul's SPIRES. Catbedral .China .Venice. . .. . London . . . .New York. Strasburg Grace Church .Antwerp . . New York Freiburg Salisbury St John's New York. fSt Paul's St. Mary's Trinity Church .Lubeck... .New York. Balustrade of Notre Dame Paris Towers of ditto u HAtel des Invalides.. St. Nicholas St. Stephen Strasburg. Utrecht .Hamburg.. .Vienna Votive Church . Vienna BRIDGES, CANALS, BREAKWATERS, ETC. Areas of Lakes in. Europe, Asia, and Africa. LAKES. Mi?es. LAKES. Mrfes. LAKES. Sq. Miles. 400 11600 L Feet. Dembia, Abyssinia. Loch Lomond engths of B BRIDGES. 13000 27 ridge Feet. Lough Neagh,lrel'd Tonting, China S. Petcha 15 ooo Orizaba . . 18870 Sinai .... Potosi Ben Nevis 4 38 Sierra Nevada . . . Elbrus Caucasus AFRICA. Tahiti T^ R Guadarama Spain 8 520 Atlas 10 400 White Mountains Hacla 5147 Compass, Cape of * Ida f> Good Hope IOOOO VOLCANOES. Jungfrau Switz'd. . 13 725 Dianai Peak, St He- Cotopaxi 18887 Mont Blanc je 707 lena 2 700 Etna IO8?A " Cenis 6780 20 ooo Hecla 5 ooo Mont d' Or, France. Mulahassen,Gren'a. Neph in, Ireland Olympus 6510 II 663 2634 6 cio Ruivo, Madeira. ... Teneriffe Peak AMERICA. 5160 12300 Popocatapetl Sahama St. Helen's, Oregon. Vesuvius 17784 22350 13320 a Q'IO Parnassus 6 ooo Aconcagua (highest Plynlimmon, Wales. The Cylinder, Pyr. . Wetterhorn 2463 10930 12 154 in America) Blue Mount, Jam 'a. Catskill 23910 8000 3 804 PASSES. Cordilleras { 13525 Chimborazo Mont Cenis . 6778 ASIA. Ararat Correde, Potosi .... Crows' Nest Hi^h- 21 441 16036 " Cervis Pont d' Or 11 100 0843 16 433 lands N. Y I 37O St Bernard, Great. . 8 172 Phawalagheri 28077 8 500 Great Peak, New Mexico 19 788 " Little. . St Gothard 7192 6808 Mount Lebanon... 12000 Mauna Lou. Hawaii liSot: Simnlon. . . 63 CANAL LOCKS, ELEVATIONS, AND RIVERS. 183 Dimensions of* Canal Locks. (U. S.) CANAL. ja ! Breadth- ja & Length of Canal. CANAL. 5 ! 1 ! UJ^ CanaL Albemarle and ) Feet. Ft. Ft. Miles. Cham plain Feet, no Feet. 18 Feet. Miles. 66 7< Chesapeake. . J Black River ] 220 40 6 '4 Cayuga and | Seneca ) no 18 7 *4-75 Crook'd L'ke, Chenango ll Delaware and I Raritan . . . . } 220 24 7 43 Chemung, and Gencsce 90 15 4 P Dismal Swamp. . . Erie . . 90 '7-5 18 5-5 44 Valley I "3- 75 Falls of Ohio Ky 35 80 7-6o Chesapeake and ) Oswego 18 18 Delaware ) 220 24 9 M Welland, Canada. . 270 45 M 28 Length of vessel that can be transported is somewhat less than lengths of locks. Suez Canal. Width 196 to 328 feet at surface, 72 at bottom, and 26 deep. Length 99 miles. Heights of* obtained. Elevations, and. various I?laces and ^Points above the Sea. LOCATIONS. Feet. LOCATIONS. Feet. LOCATIONS. Feet. Aconcagua Chili 2-3 QIO Geneva city I 22O Mont Rosa Alps Antisana highest Geneva Lake I Oo6 Mount Adams established eleva- Gibraltar Mount Katahdin C -jfvj tion (Farmhouse) 1-1 A "3 A Humboldt's highest Mount Pitt Balloon (Gay Lussac) 22 900 19 4OO Mount Washington 9549 6420 u (Green, 1837) 27 ooo Isthmus of Darien. 645 us " (Glaisher and Coxwell) Jungfrau, Switz'd. . La Paz Bolivia 13725 Pont d' Oro, Pyr's. . Posthouse Ap Peru 9843 Brazil, Quito, and ( Mexico plains. . \ 6000 8000 Laguna,Teneriffe. .. London, city 2OOO 64 Potosi, Bolivia Quito 13223 Condor's flight Madrid . . St Bernard's Mon'y 8 Eagle's " 16 500 Mexico city of 7 WJ Vegetation ... . ' Everest, Himalaya. 29003 Mont Blanc, Alps. . . 15797 White Mountain . . . 17000 6230 Lengths of Rivers. RlVKRS. Miles. RIVERS. EUROPE. Danube 1800 400 '780" 442 545 420 760 510 45o 050 5io 220 100 630 2400 2500 I 7 86 Ganges Hoang Ho Indus Jordan Lena . . .... Dnieper. Douro . Dwina Tigris Elbe Yenesei and lenga .... Se- Garonne Loire Yang-Tse Po AFRICA. Gambia Rhone Seine Shannon Nile NORTH AMERICA. Thames . Tiber Vistula Colorado Columbia. . . . Volga, Russia ASIA. Connecticut. . Delaware . . . Hudson and hawk Mo- Euphrates Miles. RIVERS. Mile*. *5*4 Kansas I4OO 3040 La Platte g 1800 176 Mackenzie Mississippi 2440 3160 Missouri 1 1 60 Ohio and Allegheny Potomac 1480 42O Red 37 A 2300 Rio Grande .... l8oo 700 St. Lawrence Susquehanna Tennessee 2173 62O 4000 SOUTH AMERICA. Amazon 2O7O Essequibo 5 2O 1050 I2OO Magdalena Orinoco T6 410 Platte 2300 4 20 32 5 Rio Madeira Rio Negro 2300 1650 IIOO 1 84 SEA DEPTHS, BUILDING STONES, ETC. Large Trees in California. "Keystone State. " Calavera Grove, is 325 feet in height. "Father of the Forest." Felled, is 385 feet in length, and a man on horseback can ride erect 90 feet inside of its trunk. " Mother of the Forest. "Is 315 feet in height, 84 feet in circumference (26.75 feet in diameter) inside of its bark, and is computed to contain 537 ooo feet of sound i inch lumber. Sea Depths. Feet. Feet. Feet. Baltic Sea 120 I 3 JOO 3000 Of Atl Pac B Cod, Uasc Feet. Coast of Spain West of St. Helena. Tortugas to Cuba . . Gulf of Florida.... Off Cape Florida... 6000 27000 4200 3720 1950 Off Cape Canaveral. u Charleston " Cape Hatteras. . " Cape Henry " Sandy Hook 2400 4200 3120 4200 2400 t-U Feet. Adriatic English Channel. . . Straits of Gibraltar. Eastward of " Estimated depth 250 miles off Cap < LOCATION. iflc no bottom at 7800 feet. >ades and. "Waterfalls. LOCATION. Feet. LOCATION. Arve Savoy 1600 2400 (30 34 (40 362 197 1000 1260 Genesee, N. Y Lidford, England . . . Lulea Sweden . . IOO . IOO 600 Niagara Great Fall 164 152 74 74 800 798 125 Cascade Alps Cataracts of the Nile. Chachia Asia Passaic Mohawk . 68 Missouri (So . '80 Ribbon, Yosemite) Valley J Foyers, Scotland Garisha, India . ^00 (94 . 2^0 Ruican, Norway Staubbach, Switz'd. . Tendon, France Gavarny, Pyrenees . . Nant d'Apresias. . . . . 800 Sandstone 000009 532 Whitepine 000002 55 Yosemite Valley 2600 feet. Expansion, and. Contraction of Building Stones for each Degree of Temperature. (Lieut. W. H. C Barllett, U. S. E.) For One Inch. Granite. 000004 825 Marble 000005 6 8 Resistance of Stones, etc., to the Effects of Freezing. Various experiments show that the power of stones, etc., to resist effects of freez- ing is a fair exponent of that to resist compression. Magnetic Bearings of New York. The Avenues of the City of New York bear 28 50' 30" East of North. Filters for Waterworks. i square yard of filter for each 840 U. S. and 700 Imp'l gallons in 24 hours ; formed of 2.5 feet of fine sand or gravel and 6 inches of common sand or shells. Led off by perforated pipes laid in lowest stratum. Distances between New York, Boston, Philadelphia, Baltimore, and Western Cities of U. S. Assuming Boston as standard, New York averages 12 per cent, nearer to these cities, Philadelphia 18 per cent., and Baltimore 22 per cent. Between New York and Chicago the line of the Pennsylvania Railroad is 47 miles shorter than that by the Erie and its connections, 50 miles shorter than that by the N. Y. Central and Hudson River and its connections, and 114 miles shorter than that by the Baltimore and Ohio and its connections. For Distances between these and other cities of the U. S., see page 88. WEATHER-PLANTS, ANTIDOTES, .ETC. 1 8$ "Weather-foretelling [Plants. (Hanneman.) If Rain is imminent. Chickweed,* Stellaria media ; its flowers droop and do not open. Crowfoot anemone, Anemone ranunculoides ; its blossoms close. Bladder Ketmia, Hibiscus trionum ; its blossoms do not open. Thistle, Ccirduus acaulis ; its flowers close. Clover, Trifolium pratense, and its allied kinds, and Whitlow grass, Draba verna; all droop their leaves. Nipple- wort, Lampsana communis ; its blossoms will not close for the night. Yel- low Bedstraw, Galium verum ; it swells, and exhales strongly ; and Birch, Betula alba, exhales and scents the air. Indications of Rain. Marigold, Calendula pluvialis ; when its flowers do not open by 7 A. M. Hog Thistle, Sonchus arvensis and oleraceus ; when its blossoms open. Rain of shoi*t duration. Chickweed, Stellaria media ; if its leaves open but partially. If cloudy. Wind-flower, or Wood Anemone, Anemone memorasa; its flowers droop. Termination of Rain. Clover, Trifolium pratense ; if it contracts its leaves. Birdweed and Pimpernel, Convolvulus and Anagallis arvensis; if they spread their leaves. Uniform Weather. Marigold, Calendula pluvialis ; if its flowers open early in the A. M. and remain open until 4 P. M. Clear Weather. Wind-flower, or Wood Anemone, Anemone memorasa; if it bears its flowers erect. Hog Thistle, Sonchus arvensis and oleraceus ; if the heads of its blossoms close at and remain closed during the night. Treatment and. Antidotes to Severe Ordinary I?oisons. Antidotes in very small doses. Chloroform and Ether. Cold affusions on head and neck, and ammonia to nostrils. Antidote. Camphor, .petroleum, sulphur. Toadstools. (Inedible mushroom). Antidote. Same as for chloroform. Arsenic or Fly Powder. Emetic ; after free vomiting give calcined mag- nesia freely. If poison has passed out of stomach, give castor oil. Antidote. Camphor, nux vomica, ipecacuanha. Acetate of Lead (Sugar of lead). Mustard emetic, followed by salts, Large draughts of milk with white of eggs. Antidote. Alum, sulphuric acid alike to lemonade, belladonna, strychnine. Corrosive Sublimate (Bug poison). White of eggs in i quart of cold water, give cupful every two minutes. Induce vomiting without aid of emetics. Soapsuds and wheat flour is a substitute for white of eggs. Antidote. Nitric acid, camphor, opium, sulphate of zinc. Phosphorus Matches. Rat Paste. Two teaspoonfuls of calcined magne- sia, followed by mucilaginous drinks. Antidote. Camphor, coffee, mix vomica, Carbonic Acid (Charcoal fumes), Chlorine, Nitrous Oxide, or Ordinary Gas. Fresh air, artificial respiration, ammonia, ether, or vapor of hot water. Antidote. Camphor, coffee, nux vomica. Belladonna (Nightshade). Emetic and stomach pump, morphine and strong coffee. Antidote. Camphor. Opium. Stomach pump or emetic of sulphate of zinc, 20 or 30 grains, or mustard or salt. Keep patient in motion. Cold water to head and chest. Antidote. Strong coffee freely and by injection, camphor, ether, and nux vomica. Strychnine (Nux vomica). Stomach pump or emetic, chloroform, cam- phor, animal charcoal, lard, or fat. Antidote. Wine, coffee, camphor, opium freely, and alcohol in small doses. Vegetable Poisons. As a rule, an emetic of mustard and drink freely of warm water. * Spreads ita leaves about 9 A. M., and they remain open until noon. o* 186 VETEBINABY. "Veterinary. . Horses. Cathartic Ball Cape Aloes, 6 to 10 drs. ; Castile Soap, x dr. ; Spirit of Wine, i dr. ; Sirup to form a ball. If Calomel is required, add from 20 to 50 grains. During its operation, feed upon mashes and give plenty of water. Cattle. Cathartic. Cape Aloes, 4 drs. to i oz. ; Epsom Salts, 4 to 6 oz. ; Gin- ger, 3 drs. Mix, and give in a quart of gruel. For Calves, one third will be sufficient. Horses and. Cattle. Tonic. Sulphate of Copper, i oz. to 12 drs. ; Sugar, .5 oz. Mix, and divide into 8 powders, and give one or two daily in food. Cordial. Opium, i dr.; Ginger, 2 drs.; Allspice, 3 drs., and Caraway Seeds, 4 drs., all powdered. Make into a ball with sirup, or give as a drench in gruel. Cordial Astringent Drench, for Diarrhoea, Purging, or Scouring. Tincture of Opium, .5 oz. ; Allspice, 2.5 drs. ; powdered Caraway, .5 oz. ; Catechu Powder, 2 drs. ; strong Ale or Gruel, i pint Give every morning till purging ceases. For Sheep .25 this quantity. Alterative. Ethiop's Mineral, .5 oz. ; Cream of Tartar, i oz. ; Nitre, 2 drs. Divide into from 16 to 24 doses, one morning and evening in all cutaneous diseases. Diuretic Ball. Hard Soap and Turpentine, each 4 drs. ; Oil of Juniper, 20 drops; and powdered Resin to form a ball. For Dropsy, Water Farcy, Broken Wind, or Febrile Diseases, add to above, All- spice and Ginger, each 2 drs. Divide into 4 balls, and give one morning and evening. Alterative or Condition Powder. Resin and Nitre, each 2 oz. ; levigated Anti- mony, i oz. Mix for 8 or 10 doses, and give one morning and evening. When given to Cattle, add Glauber Salts, i Ib. Fever Ball. Cape Aloes, 2 oz. ; Nitre, 4 oz. ; Sirup to form a mass. Divide into 12 balls, and give one morning and evening until bowels are relaxed; then give an Alterative Powder or Worm Ball. Hoof Ointment Tar and Tallow, each i Ib. ; Turpentine .5 Ib. Melt and mix. Dogs. Cathartic. Cape .Moos. .5 dr. to i oz. ; Calomel, 2 to 3 grs. ; Oil of Caraway, 6 drops; Sirup to form a ball. Repeat every 5 hours till it operates. Emetic. 2 to 4 grs. of Tartar Emetic in a meat ball, or a teaspoonful or two of common salt. Give twice a week if required. Distemper Powder. Antimonial Powder, 2, 3, or 4 grs. ; Nitre, 5, 10, or 15 grs. ; powdered Ipecacuanha, 2, 3, or 4 grs. Make into a ball, and give two or three times a day. If there is much cough, add from .5 gr. to i gr. of Digitalis, and every 3 or 4 days give an Emetic. Mange Ointment. Powdered Aloes, 2 drs.; White Hellebore, 4 drs.; Sulphur, 4 oz. ; Lard, 6 oz. Red Mange, add i oz. of Mercurial Ointment, and apply a muzzle. NOTE: Physic, except in urgent cases, should be given in morning, and upon an empty stomach; and, if required to be repeated, there should be an interval of sev- eral days between each dose. of Horses. To Ascertain, a Horse's Age. A foal of six months has six grinders in each jaw, three in each side, and also six nippers or front teeth, with a cavity in each. At age of one year, cavities in front teeth begin to decrease, and he has four grinders upon each side, one of permanent and remainder of milk set. At age of two years he loses the first milk grinders above and below, and front teeth have their cavities filled up alike to teeth of horses of eight years of age. At age of three years, or two and a half, he casts his two front uppers, and in a short time after the two next. At four, grinders are six upon each side; and, about four and a half, his nippers are permanent by replacing of remaining two corner teeth; tushes then appear, and he is no longer a colt. At five, a horse has his tushes, and there is a black-colored cavity in centre of all his lower nippers. At six, this black cavity is obliterated in the two front lower nippers. At seven, cavities of next two are filled up, and tushes blunted; and at eight, that of the two corner teeth. Horse may now be said to be aged. Cavities in nippers of upper jaw are not obliterated till horse is about ten years old, after which time tushes become round, and nippers project and change their surface. DISTANCES, POPULATION, DROWNING, ETC. 187 Distances "between Principal Cities of* East and. "West. In Miles. CITIES. Boston. New York. Phila- delphia. Balti- more. CITIES. Boston. New York. Phila- delphia. Balti- more. Burlington, la. Chicago .... I2l6 1009 1106 900 1030 823 995 802 Louisville Memphis 1161 1438 870 1247 794 1171 706 1083 Cincinnati.. . . Cleveland 743 ego 667 CQ4. 576 4 8q Milwaukee Omaha >3 I5O3 947 1393 908 1^17 887 1294 Columbus, O.. Detroit 807 724 500 623 673 547 682 512 661 St. Joseph St. Louis I 47 8 1212 1356 1050 1280 973 1223 917 Indianapolis . . Kansas Citv. . . 951 1487 810 1324. 735 1248 700 1 1 02 St. Paul Toledo... 1418 784 1308 603 Z232 6l7 I2II SQ6 Population of some Principal Cities of tne World. London New York. 1901 4 536 063 Warsaw St. Louis. 1897 638 208 Copenhagen. Dublin 1901 378 235 Paris..,..!!! 1896 3 437 2O 3 2 536 834 Naples 1899 575 23 544057 Cologne 1900 373 Z 79 372 229 Berlin 1900 i 888 326 Boston 1900 560 892 Belfast 1901 348965 Chicago Vienna 1900 1900 i 698 575 i 662 269 Manchester.. Amsterdam.. 1901 1899 5439 6 9 523 557 SanFrancis'o Cincinnati.. . 1900 1900 342 782 325902 Peking Est. i 650000 Madrid 1897 512 150 Pittsburgh . . 1900 21* 321 020 Tokio Philadelphia 1898 I 440 121 I 293 697 Baltimore. . . Munich 1900 508 957 Edinburgh.. . Lisbon* 1901 1890 316479 St. Petersb'g 1807 Milan 1900 499 959 491 460 Stockholm.. . * x- Constant'ple toy 7 Est. 7 3 Lyons 1806 466 028 Frankfort. .. IQOO 288 480 iSm ^ > Rome * y New Orleans. iy_iv " ^D >Ju y 442239 402000 395 349 381 768 Washington. Montreal Manilla Havana I9OO I9OI 1901 l8 99 278718 265 826 244 732 235981 Treatment of 3Dr owning Persons. Practice adopted toy Board of Health, New York. Place patient face downward, with one of his wrists under his forehead. Cleanse his mouth. If he does not breathe, turn him on his back with shoulders raised on a support. Grasp tongue gently but firmly with fingers covered with end of a hand- kerchief or cloth, draw it out beyond lips, and retain it in this position. To Produce and Imitate Movements of Breathing. Raise patient's extended arms upward to sides of his head, pull them steadily, firmly, slowly, outwards. Turn down elbows by patient's sides, and bring arms closely and firmly across pit of stomach, and press them and sides and front of chest gently but strongly for a mo- ment, then quickly begin to repeat first movement. Let these two movements be made very deliberately and without ceasing until patient breathes, and let the two movements be repeated about twelve or fifteen times in a minute, but not more rapidly, bearing in mind that to thoroughly fill the lungs with air is the object of first or upward and outward movement, and to expel as much air as practicable is object of second or downward motion and pressure. This artificial respiration should be maintained for forty minutes or more, when the patient appears not to breathe; and after natural breathing begins, let same motion be very gently continued, and give proper stimulants in intervals. What Else is to be Done, and What is Not to be Done, while the Movements are Being Made. If help and blankets are at hand, have body stripped, wrapped in blankets, but not allow movements to be stopped. Briskly rub feet and legs, press- ing them firmly and rubbing upward, while the movements of the arms and chest are in progress. Apply hartshorn, or like stimulus, or a feather within the nostrils occasionally, and sprinkle or lightly dash cold water upon face and neck. The legs and feet should be rubbed and wrapped in hot blankets, if blue or cold, or if weather is cold. What to Do when Patient Begins to Breathe. Give stimulants by teaspoonful two or three times a minute, until beating of pulse can be felt at wrist, but be careful and not give more of stimulant than is necessary. Warmth should be kept up in feet and legs, and as soon as patient breathes naturally, let him be carefully removed to an enclosure, and placed in bed, under medical care. i88 MISCELLANEOUS ELEMENTS. MISCELLANEOUS ELEMENTS. Earth.. Polar diameter 7899. 3 miles. Mean density or specific gravity of mass 5.672. Mass 5 272 600 ooo ooo ooo ooo ooo tons. Apparent diameter as seen from Sun 17 seconds. Sun. Heat of Sun equal to 322 794 thermal units per minute for each sq. foot of pho- tosphere or solar surface. Diameter of Sun 882000 miles, tangential velocity 1.25 miles per second or 4.41 times greater than that of the Earth. Distance from Earth 91.5 to 92 millions of miles. ]VEason and. Dixon's Line. 39 43' 26.3" N. mean latitude. 68.895 miles. Area and Population. (Behm and Wagner. ) Divisions. Area. Population. Divisions. Area. Population. America Sq. Miles. 14 491 ooo 95 495 5^o Oceanica Sq. Miles. 4 500000 Europe 3 760000 315 929 ooo Greenland ) Asia 16313000 834 707 ooo Iceland j * * ' 82000 10936000 Total 50000000 1455923500 38-oco China 434 626 ooo France 37000000 (United States. Countries. Germany 43 900000 Great Britain.. 34 ooo ooo f Russia 66 ooo ooo } Territories ... 22 ooo ooo .50000000 I India, British . .240298000 Canada 3 839 ooo Mexico 9485000 Brazil n 106000 (Turkey 8866000 \Indians 300000 | ( " in Asia. .16320000 About one thirtieth of whole population are born every year, and nearly an equal number die in same time ; making about one birth and one death per second. Earlier authority estimated population at i 288000000, divided as follows Caucasians 360 ooo ooo Mongolians .... 552 ooo ooo Ethiopians 190000000 Asiatics 60000000 Malays and Indo-Amer's Protestants.. Israelites. . . . \ 177000000 . . 80 ooo ooo .. 5000000 Mohammedans. 190000000 Pagans 300 ooo ooo Catholics ) Rom. & Greek I 250000000 Descent of \Vestern Rivers. Slope of rivers flowing into Mississippi from East is about 3 inches per mile; and from West 6 inches. Mean descent of Ohio River from Pittsburgh to Mississippi, 975 miles, is about 5.2 inches per mile; and that of Mississippi to Gulf of Mexico, 1180 miles, about 2.8 inches. Transmission, of Horse 3?ower. Largest, and perhaps most successful, wire rope transmission is one at Schaff- hauseu, at Falls of the Rhine. Here, power of a number of turbines, amounting to over 600 IP, is conveyed across the stream, and thence a mile to a town, where it is distributed and utilized. At mines of Falun, Sweden, a power of over 100 horses is transmitted in like manner for a distance of three miles. A.cids. Acetic Acid (Vinegar), acid of Malt beer, etc. Tartaric Acid, acid of Grape wine. Lactic Acid, acid of Milk, Millet beer, and Cider. IVtan-ures. Relative Fertilizing Properties of Various Manures. Peruvian Guano x I Horse 048 I Farm-yard 0298 Human, mixed 069 | Swine 044 | Cow 0259 Or, i Ib. guano = 14.5 human, 21 horse, 22.5 swine, 33.5 farm-yard, and 38.5 cow. Relative Value, Covered and Uncovered, on an A ere of Ground. Covered n tons 1665 Ibs. potatoes, 61 Ibs. wheat, 215 Ibs. straw< Uncovered 7 " 1307 " " 61.5" " 156 " " MISCELLANEOUS ELEMENTS. Yield of Oil of Several Seeds. PerCent. I Per Cent. I Per Cent. I Per Cent. I Per Cent. Poppy. . 56 to 63 I Castor . . 25 I Sunflower. 15 I Hemp. 14 to 25 I Linseed, n to 22 Thickness of \Valls of Buildings. (English.) (Molesworth.) OUTER WALLS. Maximum Height of Wall. Width of Footings. Ground Floor. Mi ist Floor. nimum Width c 2d I 3 d Floor. | Floor. f Walls 4th Floor. 5th Floor. 6th Floor. ist class dwelling. Feet. 85 38.5 21.5 21.5 17-5 T 7-5 17.5 13 J 3 2(1 " " 70 30-5 17-5 17-5 17-5 13 13 13 3 d " 52 30-5 17-5 13 13 13 13 4 th " v^; ! 38 21.5 13 13 8-5 8-5 PARTY WALLS. ist class dwelling. 85 38.5 21.5 21-5 17-5 17-5 17-5 13 13 2d " " 70 30-5 17-5 17-5 17-5 13 13 13 3d " 52 30.5 17-5 13 I 3 13 8.5 4 th " " 38 iv~~ __ r 21.5 13 n/vtV. 41 8-5 8.5 8.5 .!.,_ J If walls are more than 70 feet in length, those of lower stories must be widened by half a brick. Minimum Width of Wall. Minimum Width "Warehouses 1st Class. of Wall. For a height of 36 feet from ins. topmost ceiling 17.5 For a height of 40 feet lower . . 21.5 " " 24 feet lower . . 26 For footings 43.5 3d Class. For a height of 28 feet below topmost ceiling 13 For a height of 16 feet lower . . 17.5 For footings 30.5 "Wooden "Warehouses Sd Class. For a height of 22 feet below topmost ceiling 13 For a height of 36 feet lower . . 17.5 8 feet lower.. 21.5 For footings 34.5 4th Class. For a height of 9 feet below topmost ceiling 8.5 For a height of 13 feet below . . 13 For footings 21.5 Roofs . (English. ) Span in Feet. Principal Beam. Tie Beam. King Posts. Queen Posts. Small Queens. Straining Beam. Struts. 20 4X4 9X4 4x4 3X3 25 5X4 10 x 5 5x5 5 X3 30 6x4 ii X 6 6X6 6 X 3 35 5X4 ii X 4 4X4 7X4 4X2 45 6X5 13 X 6 6x6 7X6 5X3 50 8x6 13x8 8x8 8X4 9X6 5 X3 55 8X7 H X 9 9X8 9X4 10 X 6 5-5X3 60 | 8x 8 15 X 10 10 X 8 10 X 4 ii X 6 6 X 3 Mineral Constituents absorbed or removed from an Acre of Soil t>y several Crops. (Johnson.) CROPS. 1J n H .ii gj g *i K 10 CROPS. H 14 5 &2 E5 a o H -1 PC 10 Potassa Soda Los. 29.6 3 Lbs. 17-5 5-2 Lbs. IS Lbs. 38.2 12 Sulphuric ) Acid...) " Lbs. 10.6 Lbs. 2.7 Lbs. 13-3 Lbs. 9 .2 Lime 17 20. Q A A. g Chlorine 2 16 3.6 4. 1 Magnesia Oxide of Iron. 12.9 10.6 2.6 9.2 2.1 19.7 7-i 7 :l Silica Alumina 118.1 129.5 2.4 247.8 7 8.2 Acid j 20. 6 2 5 .8 46.3 15-1 Total.... 210 213 423 209 190 MISCELLANEOUS ELEMENTS. Average Quantity of* Tannin, in Several Su.batan.ce0. (Morfit.) Catechu. Per Cent. Oak. Per Cent. Young, inner b'k 15.2 u entire b'k. 6 " spring- ) cut bark} 22 " root bark. 8.9 Chestnut. Amer. rose, bark 8 Horse, " 2 Sassafras, root bark <;8 Sumac. p ( Sicily and Malaga Virginia rCemt, 16 10 5 16 16 16 13 2A Bengal 44 Carolina Nutgalls. Willow. Inner bark Weeping Sycamore bark .... Tan shrub " .... Cherry-tree . . . Oak. Old, inner bark { I4 ' 2 Alder bark 36 per cent. To Convert Chemical Formulae into a ^Mathematical Expression. RULE. Multiply together equivalent and exponent of each substance, and product will give proportion in compound by weight. Divide 1000 by sum of their products, and multiply this quotient by each of these products, and products will give re- spective proportion of each part by weight in 1000. EXAMPLE. Chemical formula for alcohol is tional parts by weight in 1000? C 4 Carbon =6.1X4 = 24.4) (525-82) #6 Hydrogen = i X 6= 6 '/ X 21.55 {129.3 02 Oxygen = 8X2 = 16 ) ( 344.8 looo -7-46.4= =21.55 999-92 Required their propor- by weight. Elementary Bodies, Avith their Symbols and. Equivalents. BODY. Symb. Equiv. BODY. Symb. Equiv. BODY. Symb. Equir. Aluminium Al jo. 7 Gold Au 196 6 Platinum .... Pt 08 8 Antimony Arsenic Sb As 64-6 37 7 Hydrogen Iodine H I I 126 5 Potassium . . . Rhodium . . . K R 39-2 Barium Bismuth .... Ba Bi 68.6 71 e Iridium Iron Ir Fe 98.5 28 Selenium .... Silicon Se Si 4 Boron B II Lead Pb 103* 7 Silver Ag 108 3 Bromine Cadmium. ... Calcium Carbon ...... Chlorine ... Br Cd Ca C Cl 78.4 55-8 20.5 6.1 oe e Lithium Magnesium . . Manganese. . . Mercury Molybdenum L Mg Mn Hg Mo 7 II 7 200 Sodium Strontium Sulphur Tellurium Tin Na Sr S Te Sn 23-5 43-8 16.1 64.2 58 o Chromium. . . Cr 35-5 26.2 Nickel Ni Titanium .... Ti Cobalt Co Nitrogen N Tungsten W Columbium. . Copper Fluorine Ta Cu F 184.8 31-7 18.7 Osmium Oxygen Palladium Os Pd T Uranium Yttrium Zinc U Y Zn 92 60 32 Glucinum G '1 6.9 Phosphorus. . P i5-9 Zirconium. . . Zr 34 Analysis of* certain Organic Substances by \Veight BODY. Car- bon. Hydro- gen. Oxy- gen. Nitro- gen. BODY. Car- bon. Hydro- gen. Oxy- gen. Nitro gen. Albumen 5 2- 9 7' 5 23.9 ic 7 Morphi ne 6 A. 16 1 Alcohol 52.7 12 Q 34 4 Narcotine 7^-3 6c. Atmospheric air Camphor Caoutchouc Casein 73-4 87.2 CQ.8 10.7 12.8 7 - 4 lit 11.4 23 3 21.4 Oil, Castor. Linseed Spermaceti. Quinine 2 10.3 "3 xi.8 iS-7 12.7 IO. 2 8 6 8 i Fibrin eo A IQ 7 Starch f>7 Gelatine Gum 47-9 7-9 6 4. 27.2 5O Q i7 Strychnine Sugar 76.4 7 6.7 6 6 ii. i 5-8 Hordein 44. 2 0.4 6 4 ? 47.6 1.8 Tannin 42.2 52 6 *.8 13* Lifirnin. . . 52. 5 ;.7 41.8 Urea. . . 18.0 O. 7 &, r In 100 Specific Gravity. oportic Parts oj Alcohol. >n of A.] 'Spirit, by Specific Gravity. Looliol Weight c Alcohol. !er Cei r Volume, Specific Gravity. It. at 60. Alcohol. Specific Gravity. 5 10 I .991 .984 20 30 40 972 .958 94 50 60 70 .918 .896 .872 80 90 100 .848 .823 794 In TOO Parts of Alcohol and Water, by Weight, at 60. Alcohol. Specific Gravity. Alcohol. Specific Gravity. Alcohol. Specific Gravity. Alcohol. Specific Gravity. 53 1.02 f -JjOi 1.99 3-02 4.02 .996 994 993 S-oi 6.02 7.02 .991 .988 7-99 90S 10.07 .987 .985 .984 Tides of Atlantic and Pacific Oceans at Isthmus of Panama. (Totten.) Atlantic, Navy Bay. Highest tide i.$feet; lowest .63 feet. Pacific, Panama Bay. Highest tide 17.72 to 21. 3 feet; lowest 9.7 feet. STATE. A.res Sq. Miles, is of TJ. S. C STATE. oal IT Sq. Miles. ields. STATE. Sq. Miles. Illinois 44000 I 21000 15437 13500 1 tuminous a Ohio ii 900 7700 6000 5000 Tennessee 4300 3400 55<> 150 Virginia Pennsylvania* . . . Kentucky Indiana. Missourif Alabama Maryland Michiganf Georgia *B nd Anthracite. f Anthracite. Extremes of Heat in Various Countries. England 96 France 106.5 Holland 1 _ Belgium 99-5 Egypt Africa Asia Denmark Sweden Norway Russia ...'... 102 Germany 103 | Manilla 113-5 | N. America 102 Extremes of temperature upon the Earth 240. Extremes of Cold in "Various Countries. Greece 105 Italy 104 Spain 102 Tunis 112.5 116.1 133-4 120 Suez ....... 126.5 England... 5 Holland ) Belgium } Denmark Sweden Norway France 24 Russia 46 Germany. .32 Italy 10 Fort Reliance, N. A. .70 Semipalatinsk, " ..76 I 9 2 MISCELLANEOUS ELEMENTS. Mean. Temperatures of "Various Localities. London 51 I Rome 60 I Poles 13 I Polar Regions. . 36 Edinburgh 41 | Equator 82 | Torrid Zone. 75 | Globe 50 Line of* Perpetual Congelation, or Sno\v Line. Latitude. Height. Latitude. Height. Latitude. Height. Latitude. Height. 10 15 20 25 14764 14760 13478 12557 30 35 40 45 11484 10287 9000 7670 50 g 65 6334 5020 3818 2230 70 75 80 85 1278 1016 45i 327 At the Equator it is 15260 feet; at the Alps 8120 feet; and in Iceland 3084 feet. At Polar Regions ice is constant at surface of the Earth. Limits of "Vegetation in Temperate Zone. The Vine ceases to grow at about 2300 feet above level of the sea, Indian Corn at 2800, Oak at 3350, Walnut at 3600, Ash at 4800, Yellow Pine at 6200, and Fir at 6700. Periods of Q-estation and Number of Young.- Weeks. No. Weeks. No. Elephant. 100 i Horse . . Camel. . Weeks. No. We eks. No. Sheep ... 21 2 Dog 9 6 Goat .... 22 2 Fox 5 Beaver.. 17 3 Cat 6 Pig 17 12 Rat 5 8 Wolf 10 5 Squirrel . . 4 6 Guinea Pig. 3 3 Cow 41 {43 Buffalo . . 40 50 Stag 36 45 i Bear 30 43 i Deer 24 Rabbit. . . 4 Periods of Inculoation of Birds. Swan, 42 days; Parrot, 40 days; Goose and Pheasant, 35 days; Duck, Turkey, and Peafowl, 28 days; Hens of all gallinaceous birds, 21 days; Pigeon and Canary, 14 days. Temperature of incubation is 104. -A.ges of Animals, etc. Whale, estimated 1000 years; Elephant, 400; Swan, 300; Camel, 100; Eagle, 100; Raven, 100; Tortoise, 100 to ; Lion, 70; Dolphin, 30; Horse, 30; Porpoise, 30; Bear, 20; Cow, 20; Deer, 20; Rhinoceros, 20; Swine, 20; Wolf, 20; Cat, 15; Fox, 15; Dog, 15; Sheep, 10; Hare, Rabbit, and Squirrel, 7. Relative "Weights of Brain. Man, 154.33; Mammifers, 29.88; Birds, 26.22; Reptiles, 4.2; Fish, i. Buoyancy of Casks. Buoyancy of a submerged cask in fresh water in lbs.=62.425 times the volumn of it in cube feet, 7.48 times the volume in U. S. gallons, and 6.2355 times in Imperial gallons, less the weight of the cask. Transportation of Horses and Cattle. Space required on board of a Marine Transport is: for Horses, 30 ins. by 9 feet; Beeves, 32 ins. by 9 feet. Provender required per diem is: for Horses, Hay, 15 Ibs. ; Oats, 6 quarts; Water, 4 gallons. Beeves, Hay, 18 Ibs. ; Water, 6 gallons. Rods and Eartn. Excavation and Embankment. Number of Cube Feet of various Earths in a Ton. Loose Earth 24 I Clay 18.6 I Clay with Gravel 14.4 Coarse Sand 18.6 | Earth with Gravel. . . 17.8 | Common Soil 15.6 The volume of Earth and Sand in embankment exceeds that in a primary ex- cavation in following proportions: Rock, large 1.5 I Rock, ballast 1.2 I Clay m " medium 1.25 | Sand 143 | Gravel 09 Clay and Earth will subside about .12. MISCELLANEOUS ELEMENTS. 193 Hills or Plants in an Area of* One Acre. From i to 40 feet apart from centres. Feet apart. No. Feet apart. No. Feet apart. No. Feet apart. No. X 4356o 5 1742 9 538 16 171 i-5 19360 5-5 1440 9-5 482 17 IS' 2 2-5 10890 6 6.5 1210 IO3I IO 10.5 III 18 20 135 108 3 4840 7 889 12 302 25 69 3-5 4 3556 2722 ? 8 5 ffi 13 H 258 223 30 35 48 35 4-5 2 151 85 692 i5 193 40 27 Number of several Seeds in a Bushel, and Number per Square Foot per Acre. Timothy. Clover... 41 823 360 16400960 Sq. Foot. No. Sq. Foot. Rye 888390 20 4 Wheat... q =16200 12.8 "Volumes. Permanent gases, as air, etc., are diminished in their volume in a ratio direct with that of pressure applied to them. With vapor, as steam, etc., this rule is varied in consequence of presence of the temperature of vaporization. Minerals. Relative Hardness of some Minerals. Talc i Gypsum 2 Mica 2.5 Carbonate of lime. 3 Bary tes 3. 5 Fluor-spar.... 4 Feldspar 6 Lapis Lazuli . 6 Opal , 6 Quartz 7 Tourmalin .... 7 Garnet 7. 5 Emerald 8 Topaz 8 Ruby 9 Diamond. 10 Weight of Diamonds. Regent or Pitt Carats. Carats. Dresden 765 Star of the SouthJ 125 Sancy 53 5 Koh i-Noort Piggott Napac . 106.06 . 82.25 . 78 62*; Eugenie, brilliant . 51 Hope (blue) 48.5 Polar Star 40 25 t Rough 254.5. t Originally 793. Carats. Mattam .367 Grand Mogul* 279.9 Orloff. 194. 25 Florentine, brilliant . 139.5 Crown of Portugal. . . 138.5 * Rough 900. Heat of the Sun. Sir Isaac Newton 3138740 I Waterston 16000000 Capt. John Ericsson 4909860 | Soret 10443323 Sundry others ranging from 2520 to 183600. Moon. Distance of Moon from Earth 237000 miles. Krigorinc Mixture. Lowest temperature yet procured. Faraday obtained 166 by evaporation of a mixture of solid carbonic acid and sulphuric ether. Current of Rivers. A fall of . i of an inch in a mile will produce a current in rivers. Sandstones. Structures of sandstone erected in England in i2th century are yet in good condition. Canal Transportation. Erie Canal and Hudson River. From Buffalo to New York, 495 miles, cost of transportation 2.46 mills per ton (inclusive of tolls) per mile. Transportation of wheat costs when it reaches New York 4.72 cents per bushel, and .61 cents per bushel for elevating and trimming. Towing. Erie Canal. Four mules will tow 230 tons of freight down and 100 tons back, involving a period of 30 days, at a cost of 8 cents per mile for a course of 690 miles. 1 94 MISCELLANEOUS ELEMENTS. Matter. Unit of the Physicist is a molecule, and a mass of matter is composed of them, having same physical properties as parent mass. It exists in three forms, known as solid, liquid, and gaseous. Solids have indi- viduality of form, and they press downward alone. Liquids have not individuality of form, except in spherical form of a drop, and they press downward and sideward. Gases are wholly deficient in form, expanding in all directions, and consequently they press upward, downward, and sideward. Liquids are compressible to a very moderate degree. Water has been forced through pores of silver, and it may be compressed by a pressure of one pound per square inch to the 3 soooooth part of its volume. Gases may be liquefied by pressure or by reduction of their temperature. Combustible matter (as coal) may be burned, a structure (as a house) may be destroyed as such, and the fluid (of an ink) may be evaporated, yet the matter of which coal and house were composed, although dissipated, exists, and the water and coloring matter of the ink are yet in existence. Spaces between the particles of a body are termed pores. All matter is porous. Polished marble will absorb moisture, as evidenced in its discoloration by presence of a colored fluid, as ink, etc. Silica is the base of the mineral world, and Carbon of the organized. Miuiiteriess of Matter. A piece of metal, stone, or earth, divided to a powder, a particle of it, however minute, is yet a piece of the original material from which it was separated, retain- ing its identity, and is termed a molecule. It is estimated there are 120000000 corpuscles in a drop of blood of the musk-deer. Thread of a spider's web is of a cable form, is but one sixth diameter of a fibre of silk, and 4 miles of it is estimated to have a weight of but i grain. One imperial gallon (277.24 cube ins.) of water will be colored by mixture therein of a grain of carmine or indigo. A grain of platinum can be drawn out the length of a mile. Film of a soap-and- water bubble is estimated to be but the aoooooth part of an inch in thickness. It is computed that it would require 12000 of the insect known as the twilight monad to fill up a line one inch in length. A drop of water, or a minute volume of gas, however much expanded even to the volume of the Earth would present distinct molecules. Gold leaf is the zSooooth part of an inch in thickness. A thread of silk is 25ooth of an inch in diameter. A cube inch of chalk in some places in vicinity of Paris contains 100000 of shells of the foraminifera. There are animalcules so small that it requires 75 ooo ooo of them to weigh a grain. Velocity, ^Weight, and "Volume of Molecules. Velocity. Collisions among the particles of Hydrogen are estimated to occur at the rate of 17 million-million-million per second, and in Oxygen less than half this number. Weight. A million-million-million-million molecules of Hydrogen are estimated to weigh but 60 grains. Volume. 19 million-million-million molecules of Hydrogen have a volume of .061 cube ins. Diameter. Five millions in a line would measure but . i inch. Charcoal, Alcohol. Charcoal as yet has not been liquefied, nor has Alcohol been solidified. Metals. Metals have five degrees of lustre splendent, shining, glistening, glimmering, and All metals can be vaporized, or exist as a gas, by application to them of their ap- propriate temperature of conversion. Repeated hammering of a metal renders it brittle ; reheating it restores its tenacity. Repeated melting of iron renders it harder, and up to twelfth time it becomes stronger. Platinum is the most ductile of all metals. MISCELLANEOUS ELEMENTS. 19$ Impenetrability. Impenetrability expresses the inability of two or more bodies to occupy same space at same time. A mixture of two or more fluids may compose a less volume than that due to sum of their original volume, in consequence of a denser or closer occupation of their molecules. This is evident in the mixture of alcohol and water in the proportion of 16. 5 volumes of former to 25 of latter, when there is a loss of one volume. Elasticity. Elasticity is the term for the capacity of a body to recover its former volume, after being subjected to compression by percussion or deflection. Glass, ivory, and steel are the most elastic of all bodies, and clay and putty are illustrations of bodies almost devoid of elasticity. Caoutchouc (India rubber) is but moderately elastic; it possesses contractility, however, in a great degree. Momentum. Momentum is quantity of motion, and is product of mass and its velocity. Thus, the momentum of a cannon-ball is product of its velocity in feet per second and its weight, and is denominated foot-pounds. A foot-pound is the power that will raise one pound one foot. Sound. Velocity of sound is proportionate to its volume; thus, report of a blast with 2000 Ibs. of powder passed 967 feet in one second, and one of 1200 Ibs. 1210 feet. It passes in water with a velocity of 4708 feet per second. Conversation in a low tone has been maintained through cast-iron water pipes for a distance of 3120 feet, and its velocity is from 4 to 16 times greater in metals and wood than air. T.jignt. Sun's rays have a velocity of 185000 miles per second, equal to 7.5 times around the Earth. Color Blindness Is absence of elementary sensation corresponding to red. JLmminous JPoint. To produce a visual circle, a luminous point must have a velocity of 10 feet in a second, the diameter not exceeding 15 ins. All solid bodies become luminous at 800 degrees of heat. IVIirage. When air near to surface of Earth becomes so highly heated, as upon a sandy plain, that its density within a defined distance from it increases upwards, a line of vision directed obliquely downwards will be rendered by refraction, gradually increasing, more and more nearly horizontal as it advances, until its direction is so great as to produce a total reflection, and the reflected ray then, by successive re- fractious, is gradually elevated until it meets the eye of the observer. Looming is inverted mirage, frequently seen over calm water, and is effect of lower or surface stratum of air being colder than that above it. Snow Flakes. 96 forms of snow flakes have been observed. ^^JMQ Melted Snow Produces from .25 to .125 of its bulk in water. Strength, of Ice. Two inches thick will support men in single file on planks 6 feet apart; 4 inches will support cavalry, light guns, and carts; and 6 inches wagons drawn by horses. Temperature. Sulphuric acid and water produce a much greater proportionate contraction than alcohol and water. Both of these mixtures, however low their temperature, pro- duce heat which is in a direct proportion to their diminution in volume. At the depth of 45 feet, the temperature of the Earth is uniform throughout the year. Temperature of Earth increases about i for every 50 to 60 feet of depth, and its crust is estimated at 30 miles. A body at Equator weighs two hundred and eighty-nine parts less than at the Poles, 196 MISCELLANEOUS ELEMENTS. Ages of* Animals, Fishes, etc. (Additional to page 192.) Tiger, Leopard, Jaguar, and Hyena (in confinement), 25 years; Beaver, 50; Stag, under 50; Ox and Ass, 30; Chamois, 25; Llama, Monkey, and Baboon, 15 to 18; Par- rot, 200; Tortoise, 100 to 200; Crocodile, 100; Carp, 70 to 150; Goose, 80; Pelican, 45; Hawk, 30 to 40; Crane, 24; Peacock, Goldfinch, Chaffinch, from 10 to 25; Do- mestic Fowls, Pigeons, Blackbird, Nightingale, and Linnet, 10 to 16; Thrush, Robin, and Starling, 8 to 12; Wren, 2 to 3] Salmon, 16; Eel, 10; Codfish, 4 to 17; Pike, 30 to 40; Queen Bee, 4; Bee, 6 months, and Drones, 4 months. (Houghtaling.) Birds and. Insects. (M. De Lacy.) Elements of Flight. Resistance of air to a body in motion is in ratio of surface of body and as square of its velocity. Wing Surface. Extent or area of winged surface is in an inverse ratio to weight of bird or insect. A Stag-beetle weighs 460 times more than a Gnat, and has but one fourteenth of its wing surface; 150 times more than a Lady Bird (bug), and has but one fifth. An Australian Crane weighs 339 times more than a sparrow, and has but one sev- enth; 3000000 times more than a Gnat, and has but one hundred and fortieth. A Stork weighs eight times more than a Pigeon, and has but one half. A Pigeon weighs ten times more than a Sparrow, and has but one half; 97 ooo times more than a Gnat, and has but one fortieth. A resisting surface of 30 sq. yards will enable a man of ordinary weight to descend safely from a great elevation. Strength of Insects. Insects are relatively strongest of alf animals. A Cricket can leap 80 times its length, and a F'ea 200 times. Application for Stings and Bnrns. Sting of Insects. Ammonia, or Soda moistened with water, and applied as a paste. Burns. Hot alcohol or turpentine, and afterwards bathed with lime water and sweet oil. Cold water not to be applied. To .Preserve Meat. Meat of any kind may be preserved in a temperature of from 80 to 100, for a period of ten days, after it has been soaked in a solution of i pint of salt dissolved in 4 gallons of cold water and .5 gallon of a solution of bisulphate of calcium. By repeating this process, preservation may be extended by addition of a solution f gelatin or white of an egg to the salt and water. To Detect Starch, in Milk. Add a few drops of acetic acid to a small quantity of milk ; boil it, and after it has cooled filter the whey. If starch is present, a drop of iodine solution will produce a blue tint. This process is so delicate that it will show the presence of a milligram of starch in a cube centimeter of whey (i grain of starch in 2.16 fluid-ounces). Retaining "Walls of Iron Piles. Sheet Piles. 7 feet from centres, 18 ins. in width and 2 ins. in thickness, strength- ened with 2 ribs 8 ins. in depth. Plates. 7 feet in length by 5 feet in width and i inch in thickness, with on diagonal feather i by 6 ins. Tie-rods 2 ins. in diameter Stone Sawing. Diamond Stone Sawing. (Emerson.) Alabama marble 6 feet X 2.5 feet in 22 min- utes = 41 sq. feet per hour. "Wood Sa-wing. 7722 feet of poplar, board measure, from g round logs in i hour. Engine 12 ins. diameter by 24 ins. stroke. MISCELLANEOUS ELEMENTS. 197 Cost oi* Dredging. Actual cost, if on an extended work, inclusive of Delivery, if dredging into or on a vessel alongside of dredger. (Trautwine.) Labor at $ i per day and Repairs of Plant included. Depth. Cents. Depth. Cents. Depth. Cents. Depth. Cents. Feet. 10 Cube Yards. 6 Feet. 20 Cube Yards. 8 Feet. 25 Cube Yards. 10 Feet. 35 Cube Yard*. 18 15 7 22 9 30 13 40 25 Discharge of Scows or Camels. Towing .25 mile 4 cents per cube yard, .5 mile 6 cents, .75 mile 8 cents, and i mile 10 cents. NOTE. A Scow is a flat-bottomed vessel or boat A Camel is a shallow, flat- bottomed and decked vessel, designed for the transportation of heavy freight or the sustaining of attached bodies, as a vessel, by its buoyancy. Dredging. A steam dredge will raise 6 cube yards, or 8.5 tons, per hour per H*. ^Eetal Boring and. Turning. BORING. Cost iron. Divide 25 by the diameter of the cylinder in inches for the revolutions per minute. Wrought iron. The speed is one fourth to one fifth greater than for cast iron. Brass. The speed is about twice that for cast iron. TURNING Cast iron. The speed is twice that of boring. Wrought iron. Theispeed is one fourth to one fifth greater than that for cast iron. Brass. The speed is twice that of boring. Vertical boring. The speed may be twice that of horizontal boring. The feed depends upon the stability of the machine and depth of the cut. "Well Boring. At Coventry, Eng., 750000 galls, of water per day are obtained by two borings of 6 and 8 ins., at depths of 200 and 300 feet. At Liverpool, Eng., 3000000 galls, of water per day are obtained by a bore 6 ins. in diameter and 161 feet in depth. This large yield is ascribed to the existence of & fault near to it, and extending to a depth of 484 feet. At Kentish Town, Eng., a well is bored to the depth of 1302 feet. At Passy, France, a well with a bore of i meter in diameter is sunk to a depth of 1804 feet, and for a diameter of 2 feet 4 ins. it is further sunk to a depth of 109 feet 10 ins. , or 1903 feet 10 ins. . froii which a y 'eld of 5 582 ooo galls, of water are obtained per day. Tempering Boring Instruments. Heat the tool to a bloca-red heat; hammer it until it is nearly cold; reheat it to a blood-red heat, and plunge it into a mixture of 2 oz. each of vitriol, soda, sal-am- moniac, and spirits of nitre, i oz. of oil of vitriol, .5 oz. of saltpetre, and 3 galls, of water, retaining it there until it is cool. Circular Sa-ws. Revolutions per Minute. 8 ins. 4500, 10 ins. 3600, and 36 ins. xcxxx ^Easonry. Concrete or Beton should be thrown, or let fall from a height of at least 10 fet, r well beaten down. The average weight of brickwork in mortar is about 102 Ibs. per cube foot. Plastering. In measuring Plasterers' work all openings, as doors, windows, etc., are com- puted at one half of their areas, and cornices are measured upon their extreme edges, including that cut off by mitring. GJ-lazing. In Glaziers' work, oval and round windows are measured as squares. R* 198 MISCELLANEOUS ELEMENTS. Coi'n Measure. Two cube feet of corn in ear will make a bushel of corn when shelled. Tenacity of* Iron. Bolts in "Woods. Diameter 1.125 ins - and 12 ins. in length required for Hemlock 8 tons, and for Pine 6 tons to withdraw them. Length of* G-vin Barrels. (C. T. Coathupe.) The length of the barrel of a gun, to shoot well, measured from vent-hole, should not be less than 44 times diameter of its bore, nor more than 47. Hay and. Stra\v. Hay, loose, 5 Ibs. per cube foot. Ordinarily pressed, as in a stack or mow, 8 Ibs. Close pressed, as in a bale, 12 to 14 Ibs. Ordinarily pressed, as in a wagon load, 450 to 500 cube feet will weigh a ton. Straw in a bale 10 to 12 Ibs. per cube foot. Natural Powers. Sun. The power or work performed by the Sun's evaporation is estimated at 90000000000 IP. Niagara. Volume of water discharged over the falls is estimated at 33000000 tons per hour, and the entire fall from Lake Erie at Buffalo to Lake Ontario is 323.35 feet. Velocity of Stars. According to computation of Mr. Trautwine a Star passes a range in 3' 55.91" less time each day. Service Train of a Quartermaster. Quartermaster's train of an army averages i wagon to every 24 men; and a well- equipped army in the field, with artillery, cavalry, and trains, requires i horse or mule, upon the average, to every 2 men. Tides. The difference in time between high water averages about 49 minutes each day. Atlantic and Pacific Oceans. Rise and fall of tide in Atlantic at Aspinwall 2 feet, in Pacific at Panama 17.72 to 21.3 feet. Dimensions of Drawings and Paper for TJ. S. Patents. Drawings, 8 x 12 inches, one inch margin. Paper, 8 x 12.5 inches. Latitude. One minute of latitude, mean level of Sea, nearly 6076 feet = 1. 1508 Statute miles. Artesian Well. White Plains, Nev., Depth 2500 feet. IToxindation Piles. A pile, if driven to a fair refusal by a ram of i ton, falling 30 feet, will bear i ton weight for each sq. foot of its external or frictional surface, or a safe load of 750 Ibs. per sq. foot of surface. Eartlu- Density of its mass 5.67. Tripolith. A new building material, compounded of Coke, Sulphate of Lime, and Oxide of Iron. It has increased tensile strength after exposure to the air, being much in excess of that of lime and cement. Gras and Electric Light. Gas light of 16 candle power costs 5 cent per hour; Electric, 4.15 cents. Niagara. Discovered, 1678. Falls have receded 76 feet in 175 years. Height, American Falls, 164 feet; Horseshoe, 158 feet. BRIDGES. U. S. ENSIGNS, PENNANTS, AND FLAGS. 1 99 U. S. ENSIGN, PENNANTS, AND FLAGS, (From April 20, 1896.) Ensign. Head (Depth, or Hoist}. Ten nineteenths of its length. Field. Thirteen horizontal stripes of equal breadth, alternately red and white, beginning with red. Union. A blue field in upper quarter, next the head, .4 of length of field, and seven stripes in depth, with white stars ranged in equidistant, horizontal lines and set staggered, equal in number to number of States of the Union. Pennants (Narrow). Head. 6.24 ins. to a length of 70 feet; 5.04 ins. to a length of 40 feet; 4.2 ins. to a length of 35 feet. Night, 3.6 ins. to a length of 20 feet, and 3 ins. to a length of 9 feet. Boat, 2.52 ins. to a length of 6 feet. Union. A blue field at head, one fourth the length, with 13 white stars in a hori- zontal line. Field. A red and white stripe uniformly tapered to a point, red up- permost. Night and Boat Pennants. Union to have but 7 stars. Union Jacli. Alike to the Union of an Ensign in dimensions and stars. Flags. President. Rectangle, with arms of the U. S. in centre of a blue field, over which are 13 stars in an arc. Secretary- of Navy. Rectangle, with a vertical white foul anchor in centre of a blue field, with four white stars in a rectangle, set quadrilateral around a foul anchor. Admiral. Rectangle, with 4 white stars in centre of a blue field, set as a lozenge. Vice- Admiral. Same as Admiral's, with 3 white stars set as an equilateral triangle? Rear- Admiral. Same as Admiral's, with 2 white stars set vertically. If two or more Rear-Admirals in command afloat should meet, their seniority is to be indicated respectively by a Blue flag, a Red with White stars, and a White with Blue stars, and another or all others, a White flag with Blue stars. Commodore. (Broad Pennant.) Blue, Red, or White, according to rank, with one star in centre of field, being white in blue and red pennants, and blue in white. Swallow-tailed, angle at tail, bisected by a line drawn at a right angle from centre of depth or hoist, and at a distance from head of three fifths of length of pennant; the lower side rectangular with head or hoist; upper side tapered, running the width of pennant at the tails . i the hoist. Head. .6 length. Fly 1.66 hoist. Divisional Marks. Triangle, ist Blue, 2d Red, 3d White, Blue vertical. Reserve Division. Yellow, Red vertical. Division mark is worn by Commander of a division of a squadron at mizzen, when not authorized to wear Broad Pennant of a Commodore or Flag of an Admiral. Fly .8 hoist. Signal Numbers. Fly 1.25 hoist. Signal Pennants, Fly 4.6 hoist. Repeaters 1.89 hoist. Distinctive Pennants. Of a Senior Officer Present, is the Dis- tinctive Mark of the First Division of a fleet. Nignt Signals. Very's System. International, Signal Number, Square, and Signal Pennants. Fly, 3 hoist. Suspension Bridges. Length of Spans in Feet. You-Mau, China 330 Schuylkill (Phila.) 342 Hammersmith, Eng. 422 Pesth (Danube) 660 Niagara 822 Lewistown and Queenstown 1040 Cincinnati 1057 Niagara Falls 1280 New York and Brooklyn, 930, 1595.5, and 930; clear height of Bridge above high water, at 90, 135 feet. 2OO ANIMAL FOOD. Alimentary ^Principles. Primary division of Food is into Organic and Inorganic. Organic is subdivided into Nitrogenous and Non-Nitrogenous ; Inorganic is composed of water and various saline principles. The former elements are destined for growth and maintenance of the body, and are termed " plas- tic elements of nutrition." The latter are designed for undergoing oxidation, and thus become source of heat, and are termed " elements of respiration," or " Calorificient." Although Fat is non -nitrogenous, it is so mixed with nitrogenous matter that it becomes a nutrient as well as a calorificient. Alimentary Principles. i. Water; 2. Sugar; 3. Gum; 4. Starch; 5. Pectine; 6. Acetic Acid; 7. Alcohol; 8. Oil or Fat. Vegetable and Animal. 9. Albumen; 10. Fibrine; n. Caseine; 12. Gluten; 13. Gelatine; 14. Chloride of Sodium. These alimentary principles, by their mixture or union, form our ordinary foods, which, by way of distinction, may be denominated compound aliments ; thus, meat is composed of fibrine, albumen, gelatine, fat, etc. ; wheat consists of starch, gluten, sugar, gum, etc. Analy FOOD. sis o Water. f IVte Nitro- genous Matter. ats, I Fat. "isli, Saline Matter. Vegeta Non-Nitro- kfenoua Matter. bles, < Sugar. 3 to. Cellu- lose. Ash, etc. 18 15 9-9 54 Si 72 49.1 9 1 13 15 9 1 83 36.8 6~ 3 25-5 *7.t 14.6 19-3 29.6 .1 13-1 2 1-3 33-5 2<4 > 2.8 15-45 29.8 3-6 .2 sf 5 .2 2 *' 3 2 82 69.4 55-7 8~7 64-5 5~8 7-4 57~6 1.26 1.38 58~4 48.2 9.6 50.2 16.8 78" 69-5 95 4-3 61.1 4 weight of fl 4.9 6.1 4 2.8 5-2 5-4 5-8 2 3-2 4 3-7 2.1 4-2 3-6 our, the b 2-9 3-5 5-9 7.6 3-i i 3-5 est qualit 3-2 25 i i. a 3-3 9.1 '*/"" I .8 '7 2 3 y absorb Barley Meal .... Beans White Beef roast. 2.Q5 44 5-1 21. 1 .2 4 2 7 i 5-4 fat lean . . . salt Beer and Porter. . . . Buckwheat Butter and Fats Cabbage .... Carrots Cheese Corn Meal 66 74 52 78 76.6 80.39 It'* 53 15 21 82 15 39 is 75 74 i3 i5 68 91 63 i5 37 '3 our var field 130 2.7 11 18.1 9.9 19.17 14.01 20.55 4.1 12.4 12.6 14.4 i.i 1.8 8.8 2.1 f 13.2 1.2 l6. 5 10.8 8.1 18 es from 4c Ibs. brear 26.7 10.5 30-7 2. 9 13-8 I.I7 1.52 5.58 3-9 3 X 5-5 5 2.1 48.9 73-3 .2 3-8 7 2 I&i 15^8 2 1.6 6 > to 60 pe 1.8 i-5 i-3 i 1.3 2-7 "f 3-5 3 i 2-5 2-3 2.9 7 1.2 1.8 2.4 .6 4-7 i-7 2-3 r cent, ol Egg yolk Fish, white flesh... Eels Lobster, flesh. Oysters Liver, Calf's Milk Cow's . Mutton, fat Oatmeal .... Oats Parsnips ... Peas Pork, fat Bacon, dry. . . Potatoes Poultry Rice Rye Meal Tripe Turnips... . Veal Wheat Flour Bread* Bran * Water absorbed by 1 ing most. 100 Ibs. flour ANIMAL FOOD. 201 Analysis of Different Foods In their Natural Condition. Nf- trates. Carbon- ates. Phos- phates. Water. Ni- trates. Carbon- ates. Phos- phates. Water, Apples 84 Milk of cow . . 5 8 i 86 Barley 60 < a e Mutton 12 S 4.0 BeanS JA 8 Oats 66 , tie Beef Ttf 3Q e KO Parsnips 9-2 1 i 82 8 86 is Pork _o c Cabbage Chicken Corn North 'n 4 '9 5 3-5 i 4-5 90 73 Potatoes " sweet Rice 2.4 g 22.5 28.4 J c 3-5 5 South'n 12 Turnips 35 48 3 Veal j 16 s 6? b Lamb. . . ii 35. ; *.; 97 so Wheat... is 60.2 in 14.2 Nitrates Are that class which supplies waste of muscle. Carbonates Are that class which supplies lungs with fuel, and thus furnishes heat to the system, and supplies fat or adipose substances. Phosphates Are that class which supplies bones, brains, and nerves, and gives vital power, both muscular and mental. From above it appears, that Southern corn produces most muscle and least fat, and contains enough of phosphates to give vital power to brain, and make bones strong. Mutton is the meat which should be eaten with Southern corn. The nitrates in all the fine bread which a man can eat will not sustain life beyond fifty days; but others, fed on unbolted flour bread, would continue to thrive for an indefinite period. It is immaterial whether the general quantity of food be reduced too low, or whether either of the muscle-making or heat-producing principles be withdrawn while the other is fully supplied. In either case the effect will be the same. A man will become weak, dwindle away and die, sooner or later, according to the deficiency ; and if food is eaten which is deficient in either principle, the appe- tite will demand it in quantity till the deficient element is supplied. All food, be- yond the amount necessary to supply the principle that is not deficient, is not only wasted, but burdens the system with efforts to dispose of it. Analysis of Fruits. FRUIT. Water. Sugar. Acid. Albumi- nous sub- stances. Insoluble matter. Pectous sub- stances. Ash. 85 III IJ 85*6 85-4 80 85 83-5 80.8 79-7 88.7 85-3 81.3 83-9 87 73-9 er in tn T M 7 Le 6.4 Bi 7.6 1.8 4.44 13 i 8.77 10.7 7 If 3-4 2 2.25 6.73 3-6 Sugar, Varic e Tat> classes . i i.i 1.19 *| .56 i-7 i-35 1.2 I .07 . 9 6 .87 1.27 ^84 2 *-5 Pectin, >ns f? le. (1 .22 51 % I .36 44 .46 3- i 4 4 43 83 :I 5 Salt, Acic rod vie *er Cent.) Water. . . 01 1.83 5'.8 3 5-91 6.04 3-74 2.92 l:ll 5-49 3-54 3-98 4-23 4.01 8-37 5-5 I, etc., 26 ts not Cabbage Ale and Coffee a 3.88 7-55 1.72 3-73 2.07 i-33 2.4 1.26 2.4 1.44 6.4 4.8 10.48 "3 4 Inoluc 11 .48 S i 7 43 37 47 .46 34 34 .42 !66 4 i Led in Water. Apricot, average Blackberry ......... sour black Currant red ... Gooseberry red yellow .... Grape white Peach, Dutch Pear red Plum, yellow gage.... large " black blue " red Italian, sweet . . Raspberry wild Strawberry " Banana Sugar and "Wat Su Sugar crude c an beef, ittermilk T<> Beer . . 91 Buttermilk. . . ::::::::::: 11 ndTea 100 2O2 ANIMAL FOOD Relative "Val vies of "Vegetable K"oods to procure an Equal "Volume of Flesh, in. Beeves or Sneep. (Ewart.) ARTICLE. Beeves. Sheep. ARTICLE. Beeves. Sheep. ARTICLE. Beeves. Sheep. Peameal Bean meal. . . Oatmeal Cornmeal. . . Barley Wheat braD 3 .06 .09 1.87 32C Meadow hay. Oat straw. . . . Turnips Oats Bean straw. . Potatoes 3.12 3-98 6.24 8 7 3-12 12.48 2.18 6.24 6 24 Parsnips Beans or Peas Buckwheat.. Pea straw. . . Cabbage Beets .... 18.72 l8 72 6.24 '7 2.03 6.24 7-8 Linseed cake 50 Me'dow grass 10.7 12.5 Carrots 10.72 19.67 NOTE. When these values express weight in Ibs., then such food will produce about 4 to 5 Ibs. beef or mutton. Relative Nutritive "Value of 1OO parts of Human Food. Nutrient Ratio Is determined by the ratio of albuminoids to the digestible carbo-hydrates and oil, considered as starch. Nutrient Value Is the percentage of starch, albuminoids, oil, and sugar converted into their equivalents of starch. (A. H. Church.) Ratio. Value Ratio Value. Ratio. Value. Almond, Sweet \ pple 5-3 158 II "i Fig, Dried Grape 10 20 6 5 16 Milk, Human. . " Skim.... 9 I 7S Gooseberry Rice 8? Barley 13 8s Ground-nut. . . 5' ^ Rye Flour 7 85 Beans . . 2 ? fio Macaroni I fan 88 } Tomato e 8 5 Buckwheat 86 Maise Corn 8 < RT Turn ip 6 Beet- root . Oatmeal r 8 IO2 Marrow Veg'e Carrot Celery Cabbage 14 4-5 7-5 5 Onion Parsnip Pea 3-5 12 2 S 65 16 7Q Moss, Iceland. " Irish... Walnut 8 " 70 64 Cocoanut Cheese Glos'r 16 90 Pistachio-nut. Potato . . 5-7 M3 Wheat, Indian. " Flour 15 7 T. 84.6 86 5 Date Eee. . . 10 Q 40 " Sweet.. Milk, Cow. . . '3 4 22 1 ' Bran . . Bread... 4.8 67 S3 The Nutrient ratio generally adopted for Standard diet is i to 4.75, and the proportion of fat or oil to starch is i to 3. 5. The Full Daily Diet of a man is held to be 12 oz. bread, 8 oz. potatoes, 6 oz. meat, 4 oz. boiled rice with milk, .375 pint of broth or pea soup, i pint milk, and i pint of beer. Nutritive Values and Constituents of Miillr. (Payen.) Nitrogenous Lactic Nitrogenous Lactic ANIMAL. Matter and insoluble Butter. and soluble Water. ANIMAL. Matter and insoluble Butter. and soluble Water. Salts. Salts. Salts. Salts. Goat 4-5 4.1 ~*8~~ 85-6 Ass 1 -7 1.4 6.4 9-5 Cow Woman. 4-55 3-35 3-7 3-34 5-35 3-77 86.4 89-54 Mare . . . Ewe 1.62 4.68 .2 4.2 8. 75 5-5 89.43 85.62 \Veight of some Different Foods required to furnish 122O GJ-rains of Nitrogenous ^Matter. Pease 7 Meat, lean o Fish, White... i Lbs. Meat, fat 1.3 Oatmeal 1.5 Corn Meal 1.6 Wheat Flour.. 1.7 Bread 2.1 Rye Meal 2. 3 Rice 2.8 Turnips, 15.9 Ibs. ; Beer or Porter, 158.6 Ibs. Barley Meal. Milk Potatoes Parsnips Lb. 2.9 8.3 15-9 ANIMAL FOOD. 203 Proportion FRUIT. of Sug Sugar. ar and (Fresi Acid. Acid in "Vari< 'm'Ms. ) FKUIT. MIS Fr Sugar. tiits. Acid. Apple Per Cent. ' 8.4 1.8 7.2 14.9 9.2 1.6 f Oil i Mustard . Flax. ... Per Cent. .8 i.i 1.2 2 7 1-9 7 n Vari 30 34 Plum Per Cent. 2.1 6.3 4 5-7 10.8 5-8 eeds. < Orange. . . Per Cent. 9 .1 13 Berjot.) Apricot . . Prune Blackberry . . . Currants Raspberry Red Pear Gooseberry . . . Sour Cherry Grape Strawberry. . . . Mulberry Sweet Cherry Peach Whortleberry Proportion c Beechnut .... ">* ou.s Air-dry S Almond 40 Colza ( 4 Hemp ?8 ( 40 Watermelon . . 36 Peanut 38 " US Is Analysis of different Articles of Food, with. Reference only to tlieir Properties for giving Heat and Strength. (Payen.) In 100 Parts. SUBSTANCES. Cr- b n. Nitro- gen. SUBSTANCES. Car- bon. Nitro- gen. SUBSTANCES. Car- bon. Nitro- gen. Alcohol Barley 52 40 I Q Coflee Corn 9 44 I.I 1.7 Oil, Olive Oysters 98 o 7.18 2 11 Beans 4e Eels . ... JQ O^ 2 Pease 166 Beef meat a Eggs 13. e I.Q Potatoes II Beer strong oR Pitrg dried Rice i 3 Bread, stale. . . Buckwheat 28 42. 5 1.07 2.2 Herring, salt- ed 2 3 3- n Rye Flour Salmon ?6 i-7S 2 OQ Butter 83 .64 Liver, Calf's.. 15.68 3.93 Sardines 2O 6^ 5-5 .31 Lobster IO.OO 2-93 Tea 2. 1 .2 Caviare .... 27.41 4.. 4.Q Mackerel IQ. 26 3. 74 Truffes Q. Af I. ^"J Cheese Chest Y Milk Cow's I 66 Wheat Chocolate 5 8 ! 52 Nuts 10 65 " Flour 38 c; ?6, Codfigtusalt'd 16 t;.02 Oatmeal. . . 44 l.QS Wine . . , A. .01* NOTE. Multiply figures representing nitrogen by 6.5, and equivalent amount of nitrogenous matter is obtained. Hximaii and Animal Sustenance. Least Quantity of Food required to Sustain Life. (E. Smith, M.D.) Carbon. Hydrogen. Grs. Grs. Adult Man, 4300) 200) M Adult Woman, 3900} Moan ' 4IOC 180} Moan ' I9 * An adult man, for his daily sustenance, requires about 1220 grs. nitrog- enous matter or 200 of nitrogen, and bread contains 8.1 per cent, of it. Hence, ^^ = 15 062 grains which -f- 7000 in alb. =2 Ibs. 2.43 oz. of bread. .081 These quantities and proportions are also contained in about 16 Ibs. of turnips. Thus, by table of nutritive values, page 202, turnips have 263 grains of carbon and 13 of nitrogen. Hence, ^ and = 16.35 Ibs. for the necessary carbon and 15.4 Ibs. for the nitrogen. Relative "Value of Foods compared -with. 1OO l"bs. of Q-ood May. Lbe. Clover, green . . 400 Corn, green ... 275 Wheat straw . . 374 Lb8. Rye straw 442 Oat straw 195 Cornstalks .... 400 Lb. Carrots ....... 276 Barley ........ 54 Oats 57 Corn 59 Linseed cake . . 69 Wheat bran. ... 105 204 ANIMAL FOOD. "Weight of* Articles of* Food, required, to "be consumed in the human system to develop a power eq.ua! to rais- ing 14O Ifos. to a height of* 1O OOO feet. (Frankland.) SUBSTANCES. Weight. SUBSTANCES. Weight. SUBSTANCES. Weight. Cod-liver oil Lbs. 553 Rice Lbs. I-34 1 Salt Beef Lbs. 3.65 Beef fat eee Isinglass . ... Veal lean . . . Bacon 67" Sugar lump i- 55 Porter 4 615 Butter 693 Cream Potatoes 5 068 Cocoa .7Q7 Egg boiled 2 2OQ Fish 6 316 Fat of Pork O7 Bread , 2. 34^ 7.8iq Cheese ... ft Salt Pork 2 826 Milk 8 O2 1 Oatmeal 1 'Si Ham, lean, boiled. . 3.OOI Egg white of. 8.745 Arrowroot Wheat flour. . . 1.287 I ^11 Mackerel Ale. bottled... 3.124 3.461 Carrots Cabbage . . . 9-685 12.02 Relative Value of Varioiis Foods as Productive of Force when Oxidized in the Body. Cabbage .... i Porter ^ 6 Egg hard boil'd 5 4 Oatmeal .... 93 Carrots x 2 Veal lean .... > 8 Cream 5 o Cheese 10 4 Skimmed Milk i 2 Salt Beef a. q Egg yolk 79 Fat of Pork 12 4 White of Egg.. 1.4 3.3 Cocoa 16.3 Milk i^ Lean Beef .... O A Isinglass . . .87 Pemmican 16 o Apples 1.5 Mackerel 3 Rice 89 Butter 173 Ale 1.8 Ham, lean 4 Pea Meal 9. Bacon J 7-94 Fish i. Q Salt Pork 4.0 Wheat Flour . 9 i Fat of Beef 21 6 Potatoes 2.4 Bread, crumb. . 5-i Arrowroot 9. 3 Cod-liver Oil. 21.7 Nutritious Properties of different "Vegetables and Oil- cake, compared \vith each other in Quantities. Oil-cake i Pease and Beans 1.5 Wheat, flour. .. 2 " grain ..2.5 Oats 2.5 Rye 2.5 Bran, wheat j*' 75 Corn 3 Barley 3 Clover hay 4 Hay 5 Potatoes 14 old. ... 20 Wheat straw.. 26 Barley " 26 Oat " 27.5 Carrots 17. 5 | Turnips 30 ILLUSTRATION. i Ib. of oil-cake is equal to 18 Ibs. of cabbage. Volume of Oxygen required to Oxidize 100 parts of following Foods as con- sumed in the Body. Grape Sugar . . 106 | Starch 120 | Albumen 150 | Fat 293 Hence, assuming capacity for oxidation as a measure, albumen has half value of fat as a food-producing element, and a greater value than either starch or sugar. Proportion of A.lcohol in 1OO Parts of follo\ving Liquors. Small Beer. .. i and i. 08 Porter 3.5 and 5.26 Cider 5.2 and 9.8 Brown Stout. 5.5 and 6.8 Ale 6. 87 and 10 Rhenish 7.58 Moselle 8.7 Johannisberger 8.71 Elder Wine 8.79 Claret ordinaire 8.99 Tokay 9. 33 Rudesheimer 10.72 Marcobrunner n.6 Gooseberry Wine ... 11.84 Hockheimer. 12.03 Via de Grave 12.08 (Brande. ) Herm i tage red Lisbon 1 8 QA. Champagne 12. 6l Lachryma . . IQ. 7 Amontillado Frontignac 12.63 12 89 Teneriffe Currant Wine *9-79 Barsac TO 86 Madeira OT Sauterne 14 22 Port . . 23 Champagne Burg'dy Sherry old 23 86 White Port Marsala Bordeaux *5' * Raisin Wine Malmsey. 16 4 Madeira Sercial . . Sherry 17. 17 Cape Madeira . . 20 m Malaga Gin ^y-5 A 51 6 Alba Flora. Hermitage white 17.2 17.26 Brandy Rum 53-39 co 68 Cape Muscat Constantia, red 18.25 18.92 Irish Whiskey Scotch Whiskey . . 53-9 54-3 ANIMAL FOOD. 205 Proportion of Food Appropriated and Expended "by following Animals. . Proportion appropriated .................. 6.2 " in manure ..................... 36.5 " respired ....................... 57.3 Sheep. 8 31.9 60. i Swine. 17.6 16.9 65.5 Specific Gravity of IVlillz and [Percentage of Cream, etc. MILK. Specific Gravity. Volume of Cream. Volume Curd. Specific Gravity when skimmed. Milk, pure* 1030 1027 1024 102 1 12 10.5 I 5 6-3 5-6 4-9 4.2 1032 1029 1026 1023 10 per ( 20 " 3Q " (C (f * For a method of testing the purity of milk, see Pavy on Food (Philadelphia, 1874), page 196. NOTE. The average proportion of cream is 10, or ioper cent. Proportion Per cent, of Starch, in sundry Vegetables. Arrowroot 82 | Wheat flour. . . 66.3 I Oatmeal 58.4 I Potatoes 18.8 Rice. . 79. i I Corn meal .... 64.7 | Pease. 55.4 j Turnips 5.1 Compos ition Fat. of 01 Nitrogen . iees< Salt. 4-25 7.09 5-63 6.21 J Of Water. Different Con Fat. n tries Nitrogen. (Pc Salt. TTS" 4-29 5-93 4-45 tyen.) Water. 3-39 32-05 40.07 26.53 Neufchatel. . Parmesan .. Brie Holland .... 18.74 21.68 24.83 25.06 2.28 5-48 2-39 4.1 61.87 30-3I 53-99 41.41 Chester Gruyeres . . . Marolles .... Roquefort. . . 25.41 28.4 28.73 32.31 5-56 5-4 3-73 5-07 Nutritive Equivalents. Compnted from Amount of Ni- trogen in Sn"bstances -when Dried. Unman jVlillc at 1. Rice . . 81 Bread White i 42 Cheese a QI Lamb 8 M Potatoes Corn .84 Milk, Cows' . . Pease 2-37 Eel Mussel .... 4-34 e 28 Egg, White:.. Lobster 8-45 8 so Rye 1.06 Lentils .... 076 Liver Ox . 5 7 Veal 873 Wheat Egg Yolk Pigeon 7 f.6 Beef !U Barley Oysters o QC Mutton 7 73 Pork 8 01 Oats . . . I.,* Beans. . . 1.2 Salmon . . . 7-73 7.76 Ham. . . O.I Herring, 9.14. Thermometric 3?ower and Mechanical Energy of 1O GJ-rains of "Various Sntostances in their Natural Con- dition, when Oxidized in the A.nimal Body into Car- bonic A.oid, Water, and "Urea. (Frankland.) SUBSTANCE. Water raised Lifted i foot high. SUBSTANCE. Water raised i. Lifted 1 i foot high. SUBSTANCE. Water raised I*. Lifted i foot high. Ale, Bass's . . Lbs. 1.99 i j.8 Lbs. 1-54 Cheese Cocoa-nibs Lbs. II. 2 Lbs. 8.65 7 7 Mackerel. . . . Milk .... Lbs. 4.14 i 64 Lbi. 3-2 I.2S Arrowroot. . . Beef, lean . . . Bread Butter Cabbage Carrots 10.06 3.66 1 8. 68 1.08 '33 7-77 2.83 4.26 14.42 8 3 1.03 Cod- liver oil. Egg, h'd boil. " yolk.... " white... Flour, wheat. Ham, boiled . II 5.86 8.5 !. 4 8 9.87 4-3 18.12 *fl g 3-32 Oatmeal Pea meal Potatoes Porter Rice, ground. Sugar, grape. IO. I 9-57 2.56 2-77 9.52 8.42 7 .8 7-49 1.99 2.19 7-45 6.51 206 ANIMAL FOOD. Digestion- Time required, for Digestion of several Articles of Food. (Beaumont, M.D.) FOOD. Time. FOOD. Time. Apple, sweet and mellow .... sour End mellow A. m. * 50 2 2 50 2 2 3 3 45 3 3 30 3 2 45 3 30 5 30 4 4 15 3 45 3 15 3 30 3 30 2 30 2 4 4 30 3 15 4 15 3 30 2 45 2 45 4 4 30 3 2 I 30 3 30 3 3 30 3 30 2 4 i 30 4 3 2 70 A. m. 4 2 30 2 2 30 2 2 15 3 15 3 2 55 3 15 3 30 2 30 2 30 I 5 15 4 30 4 15 3 15 3 3 30 3 20 2 30 I i 45 3 20 i 30 4 3 3 30 2 30 5 30 4 30 2 I 2 18 2 3 2 25 3 30 4 4 50 i 45 i .is Lamb boiled sour and hard Liver, Beefs, boiled Meat and Vegetables, hashed . TV/Till?- KrvJlrt 1 /-* Broccoli > and Leaves are g enerall y ri ch in gluten, while the Ratio of Flesli-forxxiers of Tubers. Per Cent. TUBERS. Flesh- formers. Starch, etc. Ratio to Heat-giv'rs. TUBERS. Flesh- formers. Starch, etc. Ratio to Heat-gw'rt. Beet root . Turnip 4 . 5 13-4 1:30 i'8 Parsnip 1.2 ii i: 10 Carrot . c * Sweet Potato 4.8 i: 3-5 Potato I. 2 g i* 16 Yam z -5 -f. _ M3 2.2 10.3 i: 7.$ 2O8 GRAVITY OF BODIES. GRAVITY AND WEIGHT. GRAVITY OF BODIES. GRAVITY acts equally on all bodies at equal distances from Earth's centre ; its force diminishes as distance increases, and increases as dis- tance diminishes. Gravitating forces of bodies are to each other, 1. Directly as their masses. 2. Inversely as squares of their distances. Gravity of a body, or its weight above Earth's surface, decreases as square of its distance from Earth's centre in semi-diameters of Earth. ILLUSTRATION i. If a body weighs goo Ibs. at surface of the Earth, what will ii weigh 2000 miles above surface ? Earth's semi-diameter is 3963 miles (say 4000). Then 2000 + 4000 = 6000 = i. 5 semi-diam's, and 900 -r- x. s 2 = ^ = 4 los - Inversely, If a body weighs 400 Ibs. at 2000 miles above Earth's surface, what will it weigh at surface ? 400 X i.5 2 = 9ooZ6s. 2. A body at Earth's surface weighs 360 Ibs. ; how high must it be elevated to weigh 40 Ibs.? ^ = 9 semi-diameters, if gravity acted directly; but as it is inversely as square 4 of the distance, then -^/g = 3 semi-diameters = 3 X 4000= 12000 miles. 3. To what height must a body be raised to lose half its weight? As ^/i : T/2 : : 4000 : 5656 = as square root of one semi-diameter is to square root of two semi-diameters, so is one semi-diameter to distance required. Hence 5656 4000 = 1656 = distance from Earth's surface, Diameters of two Globes being equal, and their densities different, weight of a body on their surfaces will be as their densities. Their densities being equal and their diameters different, weight of them will be as their diameters. Diameters and densities being different, iveight will be as their product. ILLUSTRATION. If a body weighs 10 Ibs. at surface of Earth, what will it weigh at surface of Sun, densities being 392 and 100, and diameters 8000 and 883000 miles? 883 ooo x ioo -r- 8000 x 392 = 28. 157 = quotient of product of diameter of Sun and its density, and product of diameter of Earth and its density. Then 28.157 X 10 = 281.57 Ibs. NOTE. Gravity of a body is .00346 less at Equator than at Poles. SPECIFIC GRAVITY AND WEIGHT. Specific Gravity or Weight of a body is the proportion it bears to the weight of another body of known density or of equal volume, and which is adopted as a standard. If a body float on a fluid, the part immersed is to whole body as specific gravity of body is to specific gravity of fluid. When a body is immersed in a fluid, it loses such a portion of its own weight as is equal to that of the fluid it displaces. An immersed body, ascending or descending in a fluid, has a force equal to difference between its own weight and weight of its bulk of the fluid, less resistance of the fluid to its passage. Water is well adapted for standard of gravity ; and as a cube foot of it at 62 F. weighs 997.68 ounces avoirdupois, its weight is taken as the unit, or approximately 1000. SPECIFIC GRAVITY AND WEIGHT. 2OQ French standard temperature for comparison of density of solid bodies and determination of their specific gravities, is that of maximum density of water, at 4 C. or 39.1 F., and for gases and vapors under one atmosphere or .76 centimeters of mercury is 32 F. or o C., and specific gravity of a body is expressed by weight in kilogrammes of a cube decimeter of that body. Densities of metals vary greatly. Potassium, Sodium, Barium, and Lithium are lighter than water. Mercury is heaviest liquid, and Iridium heaviest metal. Volcanic scoriae are lighter than water. Pomegranate and Lignum-vitas are heaviest of woods. Pearl is heaviest of animal substances, and Flax and Cotton are heaviest of vegetable sub- stances, former weighing nearly twice as much as water. Zircon is heaviest of precious stones, being 4.5 times heavier than water. Garnet is 4 times heavier, Diamond 3.5 times, and Jet, lightest of all, is but .3 heavier than water. To Ascertain. Specific Grravity of a Solid. Body heavier than. "Water. RULE. Weigh it both in and out of water, and note difference ; then, as weight lost in water is to whole weight, so is 1000 to specific gravity of body. Or, _ = G, W and w representing weights out and in water, and G specific gravity. EXAMPLE. What is specific gravity of a stone which weighs in air 15 Ibs., in water 10 !bs.? 15 10 = 5; then 5 : 15 :: 1000 : 3000 Spec. Grav. To Ascertain Specific Grravity of a Body lighter than \Vater. RULE. Annex to lighter body one that is heavier than water, or fluid used ; weigh piece added and compound mass separately, both in and out of water, or fluid ; ascertain how much each loses, by subtracting its weight from its weight in air, and subtract less of these differences from greater. Then, as last remainder is to weight of light body in air, so is 1000 to specific gravity of body. EXAMPLE. What is specific gravity of a piece of wood that weighs 20 Ibs. in air; annexed to it is a piece of metal that weighs 24 Ibs. in air and 21 Ibs. in water, and the two pieces in water weigh 8 Ibs.? 20 -f- 24 8 = 44 8 = 36 = loss of compound mass in water; 24 21 = 3 = loss of heavy body in water. 33 : 20 :: 1000 : 606.06 Spec. Grav. To Ascertain Specific Grravity of a Fluid. RULE. Take a body of known specific gravity, weigh it in and out of the fluid ; then, as weight of body is to loss of weight, so is specific gravity of body to that of fluid. EXAMPLE. What is specific gravity of a fluid in which a piece of copper (spec, grav. = 9000) weighs 70 Ibs. in, and 80 Ibs. out of it ? 80 : 80 70 = 10 : : 9000 : 1125 Spec. Grav. To Ascertain Specific Grravity of a Solid Body -which is solnble in "Water. RULE. Weigh it in a liquid in which it is not soluble, divide its weight out of the liquid by loss of its weight in the liquid, and multiply quotient by specific gravity of liquid ; the product is specific gravity. EXAMPLE. What is specific gravity of a piece of clay, which weighs 15 Ibs. in air and 5 Ibs. in a liquid of a specific gravity of 1500, in which it is insoluble ? 15 -f- 10 X 1500= 2250 Spec. Grav s* 2IO SPECIFIC GRAVITY AND WEIGHT. SOLIDS. SCBSTANCIB. Specific Gravity. Weight of a Cube Inch. SUBSTANCES. Specific Gravity. Weight of a Cub* Inch. IMetals. 2560 2670 7700 6712 5763 470 9823 2000 8450 8300 8200 8380 8lOO 8214 Lb. .0926 .0906 2785 .2428 .2084 .017 3553 .0723 3056 2997 .2966 .3026 .2930 . 2Q 7 2 Metals. Mercury 60 13569 13370 8600 8800 8279 10 000 11350 20337 16000 22069 865 8940 10650 I 520 8600 4500 10474 10511 970 7700 7900 7806 7833 7818 7847 7823 7842 7848 7852 7834 2540 6 no 11850 739 7291 53oo 17000 18330 7119 6861 7191 800 793 845 690 400 822 852 690 567 720 898 1031 1328 912 928 176 Lb. .4908 .4836 3183 .2994 -3613 .4105 7356 5787 .7982 0313 324 3852 055 .3111 .1627 .3788 .3802 0351 .2785 .2857 .2823 .2833 .2828 2838 283 .2836 .2839 .284 .2916 .0918 .221 .4286 .2673 .2637 .1917 *I 49 .6629 2575 .2482 .26 Cube Foot. 5 49.562 52.812 43-125 25 51-375 53-25 43- I 25 35-437 56.125 6 4-437 83 H 2*.* " wrought.... " Bronze Antimony Arsenic " 212 ... Nickel " cast Barium . . ........ Bismuth Palladium Platinum, hammered . . " native Brass. Sheet, cop. 75, zinc 25. Yellow " 66, " 34. Muntz " 60, " 40. plate " rolled. Potassium 59 Red lead Cast Rubidium Wire . . Ruthenium Bromine 3000 8750 8217 88 3 2 8700 8379 8060 739 8650 1580 5900 8098 8600 6000 8608 8698 8880 8880 19258 19361 17486 15709 18680 23000 7308 6900 75oo 7207 7217 7065 7218 7788 7774 7704 7698 7540 7808 8140 7744 H352 11388 590 1750 8000 15632 n<;a8 .1085 3l65 .2972 3194 .2929 .3021 .291 .2668 3129 057 .2134 .2929 SI" .217 33 .3146 .3212 .3212 .6965 .7003 6325 .5682 .6756 .8319 .264 .2491 .2707 .2607 .2609 2555 .2611 .2817 2811 .2787 .2779 .2722 2819 .2938 .2801 .4106 .4119 .0213 0633 .2894 .5661 .4018 Selenium Bronze gun metal Silver pure cast . . . ' ordinary mean . cop. 84, tin 16 . . " 81, " 19 .. Tobin " " hammered. Sodium Steel minimum maximum je tin 6? plates, mean soft " 21 tin 74. Cadmium temper'd and hard- ened Calcium Chromium . wire . . Cinnabar blistered Cobalt crucible . Columbium cast Copper cast Bessemer u plates ordinary mean " wire and bolts.. " ordinary mean. Gold pure cast Tellurium Thalium u hammered Tin, Cornish, hammered. " " pure " 22 carats fine U 20 " 4i Iridium Titanium Tungsten " hammered Iron, Cast, gun metal. . . " minimum Wolfram " rolled u ordinary mean .... " mean Eng . Woods (Dry). Alder " cast hot blast " " cold " ** Wrought, bars " wire Ash 1 " rolled plates " average Eng. rails . . Lowmoor. . . Bamboo Bay tree Beech j " ordinary mean I Birch | ' ' rolled Blackwood, India Boxwood Brazil Lithium Magnesium " France Manganese " Holland Bullet-wood Butternut, . . Mercury 40 U +12... SPECIFIC GRAVITY AND WEIGHT. 211 SUBSTANCES. Specific Gravity. Weight of a Cube Foot. SUBSTANCES. Specific Gravity Weight of a Cube Foot. Woods (Dry). Campeachy 9J3 56l 1315 441 3 8o 1573 280 1380 7'5 OIO 726 1040 240 644 756 i33i 1209 695 570 671 800 1014 600 512 582 970 1055 843 IOOO 592 910 860 368 792 990 770 566 1171 720 544 560 703 650 i|33 4 604 728 913 11 1063 560 852 750 576 849 56i 897 823 872 759 *U. Lbs. 57-062 35-062 82.157 27.562 23-75 98-312 S2 5 44-687 38.125 45-375 65 '5 40.25 47-25 83-187 75562 43-437 35625 4i 937 63-375 37-5 32 36-375 60.625 65-95 52.687 62.5 37 56.875 53-75 23 49-5 43- 125 47-5 61.875 48.125 35-375 73-187 45 34 35 43-937 40.625 83-312 50-25 37-75 45-5^ 57062 &7 35 53-25 46.875 S 6 53.062 56.062 Si-437 54-5 47-437 5. Ordnanc ^Voods (Dry). Oak, English J 858 932 1146 1170 1260 1068 860 680 75 66x 710 785 660 59<> 554 461 740 1354 58o 383 529 705 728 482 885 g 8 500 486 585 788 807 722 624 606 441 838 720 473 54 >' 587 6B 7 759 2730 2699 1714 1078 866 3073 2250 4000 4865 2305 Lbs. 53-625 58-25 71-625 73-125 78-75 66.75 53-75 4 2 '5 44.062 41.312 44-375 49.062 41-25 36-875 34-625 28.812 46-25 84.625 3625 23-937 33.062 44.062 45-5 30-125 55-3" 3i 25 38.937 23-937 41 062 61.25 41937 31-25 30-375 36.562 4925 50-437 45-125 39 37-875 27.562 52-375 29-562 33-812 36-687 42.937 42.437 170.625 168.687 107.125 67-375 192.062 140.625 250 ' 304.062 144.06 " Indian ........ " heart, 60 years.... u live green " fresh burned.. " " seasoned " white ** soft wood .... " triturated Olive Chestnut, sweet Citron Pear Cocoa Plum ... . . Cork Pine pitch Cypress, Spanish " red Dog- wood " white Ebony, American " yellow. " Norway Elder Pomegranate Elm { Poon ' ' rock Poplar " white Quince Filbert Rosewood Fir, Norway Spruce. . . . Satinwood Fustic Spruce Greenheart or Sipiri. . . . Gum, blue . . , ' ' water Teak (African oak).. . . { Walnut Hackmatack Hazel " black Willow Hickory, pig-nut. shell-bark Holly Yew Dutch Iron-wood. (Well Seasoned.*) Ash Juniper .. Khair, India Beech Larch I Cypress I Lemon Hickory, red Mahogany, St. Domingo. Pine, white Lignum-vitse .. .. | Lime " yellow Poplar White Oak, upland " u James River Stones, Earths, etc. Locust " Honduras. . . . " Spanish Maple *' yellow . " bird's-eye Mastic Mulberry { Asbestos, starry Oak African Bary tes, sulphate .... J Beton, N. Y. StCon'g Co. e Manual, 1841. ' ' Canadian ......... u Dantzic 212 SPECIFIC GKAVITY AND WEIGHT. SUBSTANCES. Specific Gravity. Weight of a Cube Foot. SUBSTANCES. Specific Gravity. Weight of a Cub Foot. Stones, Earths, etc. Basalt | 2740 Lbs. 171.25 Stones, Earths, etc. Glass, green 2642 Lbs. 165.125 Bitumen red 2864 179 72. 3 " optical 3450 2802 215.625 180.75 " brown 830 u window 2642 165. 125 Borax 107 125 u soluble 1250 78.125 Brick 1 1714 1367 85-437 Gniess, common 2700 167.4 ' ' pressed 1900 2400 118.75 ISO Granite, Egyptian red. . " Patapsco 2654 2640 165.875 165 " fire " Quincy . . .. 2652 165.7^ *' work in cement 1800 II2.5 " Scotch 2625 164.062 it u 41 mortar. 4 Carbon . . . 1600 2000 IOO 125 218 ?; " Susquehanna . . gray Graphite 2704 2800 2 2OO 169 JOT C Cement Portland I3OO 81.25 Gravel, common J 749 109. 312 " Roman Grindstone J-3-J Q07 Chalk 1520 95 Gypsum, opaque 2168 135-5 Clay 2784 Hone, white, razor Hornblende 2876 3^4O 179-75 221.25 u with gravel 2480 155 Iodine 4Q4O 1350 84.375 Lava, Vesuvius | 1710 o Tr . 106.875 1436 89-75 ... I Lias 175.625 146 875 " Borneo * 80 625 Lime quick 804 50. 25 " Cannel { 1238 77-375 " hydraulic 2745 171.562 u Caking I3l8 82.375 7Q 8l2 Limestone, white ' ' green 3156 3180 197-25 198.75 " Cherry 1277 Magnesia carbonate 150 Chili ' 80 625 Magnetic ore ^OQ4. 317.6 " Derbyshire I2Q 80 7S Marble Adelaide 169 687 " Lancaster * ' African 2708 169 25 '355 84.687 ' Biscayan, black. 260^ 168.437 " Newcastle ' Carrara. . . 2716 169 75 " Riv&deGier I7OO 8l.25 ' common 2686 167.871; " Scotch | 78.687 ' Egyptian 2668 f>f\tc\ ,i '-> 166.75 Splint 1302 81 O 7 e ' Italian, white. . . 2049 2708 169. 25 >-> iraint, and compared with water at 62 specific gravity = .000 612 3. Weight of a Cube Foot of Gases at 32 F., and under Pressure of one Atmos- phere, or 2116.4 Ibs. per Square Foot. Lbs. Air, at 32 080728 " " 62 076097 Alcohol 1302 Carbonic acid 12344 Carburet. Hydrog. . 044 62 Lbs. Chlorine "... .197 Chloroform 428 Coal gas 035 36 Ether, Sulphuric. . .2093 Gaseous steam 050 22 Lbs. Hydrogen 005 594 Nitrogen 078 596 Olefiant gas 079 5 Oxygen 089 256 Steam 05022 Sulphurous acid 1814 Ibs. To Compute "Weight of a Body or Sul3staii.ee when Specific Gravity is given. RULE. Multiply specific gravity by unit or standard of body or sub- stance, and product is the weight. Or, Divide specific gravity of body or substance by 16, and quotient will give weight of a cube foot of it in Ibs. EXAMPLE. Specific gravity is 2250; what is weight of a cube foot of it? 2250 x 62.5 = 140.625 Ibs. 216 WEIGHTS OF VARIOUS SUBSTANCES. "Weights and. A^olumes of various Substances in Ordinary Use. SUBSTANCES. Cube Foot. Cube Inch. SUBSTANCES. Cube Foot. Cube Feet in a Ton. Mietals. B-s. .{SSTg} u gun metal. u sheets " wire Lbs. 488.75 543-75 5I3-6 524-16 547-25 543-62S 450-437 466.5 479-5 481.5 486.75 709-5 7"-75 848.7487 487-75 489. 562 455-687 428.812 449-437 52.812 Si-375 64-3 35.062 38-125 49-5 43-125 83-312 57-062 35 66-437 54-5 & 25 66-75 53-75 4^-937 41.25 36-875 34-625 29. 562 33-8i2 -Tobin Br< Lbs. .2829 3147 .297 3033 3179 3167 .2607 .27 2775 .2787 .2816 ,4106 .4119 .491174 .2823 2833 2637 .2482 .2601 Cube Feet in a Ton. 42.414 43-6oi 34.837 gin 58.754 45-252 51.942 26.886 39.255 64 33.714 41.101 38.455 33-558 41.674 52.169 54-303 60.745 64.693 75-773 66. 348 mze. . . Woods. Spruce Lbs. 31-25 31-25 36-562 30-375 .075291 137.562 1 02 89-75 102.5 80 94.875 84.687 81.25 62.5 14-5 20 *s 120.625 137- J 25 109.312 1 2O 93-75 128.125 128.125 101.875 126.25 I65-75 l6 9 135-5 12 25 57-5 56.437 197-25 167.875 97.98 73-5 62.5 64.312 62.355 ibe incn. )2i IbS. 71.68 71 68 61.265 73-744 12.8 16.284 21.961 24.958 21.854 28 23.609 26.45! 27-569 35-84 154.48 114 89.6 18.569 16.335 20.49 18.667 23-893 17.482 17.482 21.987 17.742 13-514 13-254 16.531 186.66 89.6 38.95 39-69 "355 13-343 22.862 30.476 35-84 34-83 35-955 Walnut, black, dry... Willow u drv Miscellaneous. Air. Copper cast " plates Iron cast BaS'ilt mean u gun metal Brick fire " heavy forging.. *' plates u mean Coal, anthracite j u bitumin., mean. u Cannel " wrought bars. . . Lead cast " rolled Mercury 60 . . . " Cumberland u Welsh, mean. . . Coke Steel plates " soft . . . Tin Cotton, bale, mean . . . " " pressed J Earth, clay " rolled . .. Woods. Ash *' common soil. . " gravel dry, sand ' loose Bay Blue Gum Cork ' moist, sand. .. ' mold Cedar . .. Chestnut 1 mud Hickory, pig-nut " shell-bark.. " with gravel. .. Granite, Quiucy " Susquehanna Gypsum Logwood Mahoga'y,Hondur's { Oak, Canadian u hard pressed Ice at 32 " English India rubber " live, seasoned... " white, dry " " vulcanized Limestone " " upland... Pine, pitch Marble mean Mortar, dry, mean . . . Plaster of Paris Water, rain " red " white . " well seasoned.. Pine, yellow " salt " at 62 Metals. - Cube foot. Ci . . S22.02 IbS. .^c To Compnte Proportions of Two Ingredients in a Com- pound, or to Discover Adxdteration in IMetals. RULE. Take differences of each specific gravity of ingredients and spe- cific gravity of compound, then multiply gravity of one by difference of other ; and, as sum of products is to respective products, so is specific gravity of body to proportions of the ingredients. EXAMPLE. A compound of gold (spec. grav. = 18.888) and silver (spec. grav. = 10.535) has a specific gravity of 14; what is proportion of each metal? 18.88814=4.888X10.535=51-495. 1410.535=3-465X18.888=65.447. 65-447+51-495: 65-447 -4: 7-835 0old, 65.447+51.495: 5i.495"H : 6-165 suvtfft WEIGHTS OP VARIOUS SUBSTANCES IN BULK. 2I 7 "Weights of "Various Substances per Cn/be Foot in Sulk. Lba. Lead, in pigs 567 Iron, " 360 Marble, in blocks) Limestone, " ) Trap 172 170 Granite, in blocks .... 164 Sandstone 141 Ash, dry, ioo feet BM " white, " *' Cement, struck bushel and packed* ioo Cement, Portland, bushel, no Cherry, dry, ioo BM Chestnut, dry, I0 o BM . . . . Coal, anthracite, i cub. yd. broken and loose ... i :c " " .1 ton.. 41. Coke, ton = 80 to 97 Earth, common soil 137 Potters' clay 130 Loam 126 Gravel 109 Sand 95 Bricks, common 93 Ice, at 32 57.5 Oak, seasoned 52 Lba. Coal, caking 50 Wheat 48 Barley 3 8 Fruit and vegetables . . 22 Cotton seeds 12 Cotton 10 Hay, old 8 175 ton. 141 Ibs. Ibs. 156 ton. i53 " ,75 yds 5 cub. feet cub. feet. . 125 Ibs. * One packed bushel = 1.43 loose. Earth, loose 93.75105. Elm, dry, ioo feet BM 13 ton. Gypsum, ground, str. bush. 70 Ibs. " well shaken 80 " Hemlock, dry, ioo feet BM. .093 ton. Hickory, " " " . .197 " Masonry, Granite, dressed.. 165 Ibs. " rough... 126 Limestone, dres'd 165 Sandstone 135 Brick, pressed . . . 140 " com'n, rough, ioo Comparative ^ TIMBER. Weight Weight of a Green. of G-i Cube Foot. Seasoned. een and. Seaso TIMBER. ned. T Weight of a Green. Lm"ber. Cube Foot. Seasoned. American Pine Lbs. 44-75 58.18 60 Lbs. 30-7 50 53.17 Cedar Lbs. & 48. 7 ; Lbs. 28.25 43-5 is. 5 Ash English Oak Beech . . . Riga Fir. . . -Application, of th.e Ta~bles. When Weight of a Solid or Liquid Substance is required. RULE. Ascer- tain volume of substance in cube feet ; multiply it by unit in second column of tables (its specific gravity), and divide product by 16; quotient will give weight in Ibs. When Volume is given or ascertained in Inches. RULE. Multiply it by unit in third column of tables (weight of a cube inch), and product will give weight in Ibs. EXAMPLE. What is weight ol a cube of Italian marble, sides being 3 feet? 33 x 2708 = 73 1 16 oz. , which -7-16 = 4569. 75 Ibs. Or of a sphere of cast iron 2 inches in diameter ? 2 3 X .5236 X -2607 weight of a cube inch=i 1.092 Ibs. When Weight of an Elastic Fluid is required. RULE. Multiply specific gravity of fluid by 532.679 (weight of a cube foot of air at 62 hi grains), divide product by 7000 (grams hi a Ib. Avoirdupois), and quotient will give weight of a cube foot hi Ibs. EXAMPLE. What is weight of a cube foot of hydrogen? Specific gravity of hydrogen .0692. 532.679 x -0692 -7- 7000 = .005 265 9 Ibs. To Compute "Weight of Cast iMetal "by "Weight of Pattern. When Pattern is of White Pine. RULE. Multiply weight of pattern in Ibs. by following multipliers, and product will give weight of casting : Iron, 14 ; Brass, 15 ; Lead, 22 ; Tin, 14 ; Zinc, 13.5. When the Cores or Prints are of White Pine. Multiply the product of their area and length in inches by .0175 or .02, according to the dryness of the wood, and proportionately for other woods, and result is weight of core or print to be deducted from weight of pattern. V 2l8 BALLOONS, SHRINKAGE OP CASTINGS, ETC. To Compxite Weights of Ingredients, that of Compound being given. RULE. As specific gravity of compound is to weight of compound, so are each of the proportions to weight of its material. EXAMPLE. Weight, as p. 216, being 28 Ibs., what are weights of the ingredients? 14:28:: {7-835: 15-67 gold, (0.165 ' I2 -33 silver. N TI ?- Specific gravity of alloys does not usually follow ratio of their compo- nents, it being sometimes greater and sometimes less than their mean. To Compnte Capacity of a Balloon. RULE. From specific gravity of air in grains per cube foot, subtract that of the gas with which it is inflated ; multiply remainder by volume of bal- loon in cube feet; divide product by 7000, and from quotient subtract weight ot balloon and its attachments. EXAMPLE. Diameter of a balloon is 26.6 feet, its weight is 100 Ibs and soecifir gravity of the gas with which it is inflated is .07 (air being assumed at if what is its capacity, specific gravity of air assumed at 527.04 grain! 527-04 (527-04 X .07) X 26.63 x .5236 I00 . 500.04 Ibt. 7000 To Compute Diameter of a Balloon. Weight to be raised being given. Ey inversion of preceding rule. ~/W -T- 7000-f- * -T- *' ,, ^7 = a . s and s representing weight of air and gas in grains per cube foot, W weight to be raised in Ibs., and d diameter of bal- loon in feet. ILLUSTRATION. Given elements in preceding case. Then 3/590-04 + 100X7000^527.04-36.89^ 3 /9854.69 = 266feet v .5236 V -5236 IProof of* Spirituous Liquors. A cube inch of Proof Spirits weighs 234 grains ; then, if an immersed cube inch of any heavy body weighs 234 grains less in spirits than air, it shows that the spirit in which it was weighed is Proof. If it lose less of its weight, the spirit is above proof; and if it lose more, it is below proof. ILLUSTRATION. A cube inch of glass weighing 700 grains weighs 500 grains when weighed in a certain spirit; what is the proof of it? 700 500 = 200 = grains = weight lost in spirit. Then 200 : 234 :: i : i.ij= ratio of proof of spirits compared to proof spirits, or i = . 17 above proof. NOTE. For Hydrometers and Rules for ascertaining Proof of Spirits, see page 67 ; and for a very full treatise on Specific Gravities and on Floatation, see Jamie- son's Mechanics of Fluids. Lond., 1837. Shrinkage of Castings. It is customary, in making of patterns for castings, to allow for shrinkage per lineal foot of pattern as follows : Iron, small cylinders ... = ^ in. per ft. Ditto in length. . . . = % in 16 ins. " Pipes =K " ' Girders, beams, etc. = % in 15 ins. " Large cylinders,") the contraction > = %& P 61 " of diam.at top.J Ditto at bottom . . = T V " Brass, thin ........ = % in 9 ins. " thick ....... = % in 10 ins. Zinc ............. = j in a foot. Lead Copper Bismuth GEOMETRY. GEOMETRY. Definitions. Point has position, but not magnitude. Line is length without breadth, and is either Right, Curved, or Mixed. Right Line is shortest distance between two points. Curved Line is one that continually changes its direction. Mixed Line is composed of a right and a curved line. Superficies has length and breadth only, and is plane or curved. Solid has length, breadth, and thickness, or depth. Angle is opening of two lines having different directions, and is either Right, A cute, or Obtuse. Right A ngle is made by a line perpendicular to another falling upon it. Acute Angle is less than a right angle. Obtuse Angle is greater than a right angle. Triangle is a figure of three sides. Equilateral Triangle has all its sides equal. Isosceles Triangle has two of its sides equal. Scalene Triangle has all its sides unequal. Right-angled Triangle has one right angle. Obtuse-angled Triangle has one obtuse angle. Acute-angled Triangle has all its angles acute. Oblique-angled Triangle has no right angle. Quadrangle or Quadrilateral is a figure of four sides, and has following particular designations viz., Parallelogram, having its opposite sides parallel. Square, having length and breadth equal. Rectangle, a parallelogram having a right angle. Rhombus or Lozenge, having equal sides, but its angles not right angles. Rhomboid, a parallelogram, its angles not being rig-nt angles. Trapezium, having unequal sides. Trapezoid, having only one pair of opposite sides parallel. NOTE. Triangle is sometimes termed a Trigon, and a Square a Tetragon. Gnomon is space included between the lines forming two similar parallelo- grams, of which smaller is inscribed within larger, so as to have one angle in each common to both. Polygons are plane figures having more than four sides, and are either Regular or Irregular, according as their sides and angles are equal or un- equal, and they are named from number of their sides or angles. Thus : Pentagon has five sides. Hexagon " six " Heptagon " seven" Octagon " eight " Nonagon has nine sides. Decagon " ten " Undecagon " eleven " Dodecagon " twelve" Circle is a plane figure bounded by a curved line, termed Circumference or Periphery. Diameter is a right line passing through centre of a circle or sphere, and terminated at each end by periphery or surface. Arc is any part of circumference of a circle. Chord is a right line joining extremities of an arc. Segment of a circle is any part bounded by an arc and its chord. Radius of a circle is a line drawn from centre to circumference. Sector- is any part of a circle bounded by an arc and its two radii. Semicircle is half a circle. Quadrant is a quarter of a circle. Zone is a part of a circle included between two parallel cords. Lune is space between the intersecting arcs of two eccentric circles. 22O GEOMETRY. Secant is line running from centre of circle to extremity of tangent of arc. Cosecant is secant of complement of an arc, or line running from centre of circle to extremity of cotangent of arc. Sine of an arc is a line running from one extremity of an arc perpendicu- lar to a diameter passing through other extremity, and sine of an angle is sine of arc that measures that angle. Versed Sine of an arc or angle is part of diameter intercepted between sine and arc. Cosine of an arc or angle is part of diameter intercepted between sine and centre. Coversed Sine of an arc or angle is part of secondary radius intercepted between cosine and circumference. Tangent is a right line that touches a circle without cutting it. Cotangent is tangent of complement of arc. Circumference of every circle is supposed to be divided into 360 equal parts, termed Degrees ; each degree into 60 Minutes, and each minute into 60 Seconds, and so on. Complement of an angle is what remains after subtracting angle from 90 degrees. Supplement of an angle is what remains after subtracting angle from 180 degrees. To exemplify these definitions, let Acb, in following Figure, be an assumed arc of a circle described with radius B A : A g A c 6, an Arc of circle AGED. A 6, Chord of that arc. B A, an Initial radius. B C, a Secondary radius. e D d, a Segment of the circle. A B &, a Sector. A D E, a Semicircle. C B E, a Quadrant. AedK.ii Zone. n A, a Lime. B g, Secant of arc A c &; written Stec. b k, Sine of arc A c 6 ; written Siu. A k, Versed Sine of arc A c 6; written Versin. B k or m 6, Cosine of arc Acb. A ff, Tangent of arc A c b. C B 6, Complement, and b B E, Supplement of D arc A c 6. C*, Cotangent of arc, written Cot. B s, Cosecant of arc; written Cosec. m C, Coversed sine of arc, or, by convention, of angle A B 6 ; written Coversin. Vertex of a figure is its top or upper point. In Conic Sections it is point through which generating line of the conical surface always passes. Altitude, or height of a figure, is a perpendicular let fall from its vertex to opposite side, termed base. Measure of an angle is an arc of a circle contained between the two lines that form the angle, and is estimated by number of degrees in arc. Segment is a part cut off by a plane, parallel to base. Frustum is the part remaining after segment is cut off. Perimeter of a figure is the sum of all its sides. Problem is something proposed to be done. Postulate is something required. Theorem is something proposed to be demonstrated. Lemma is something premised, to render what follows more easy. Corollary is a truth consequent upon a preceding demonstration. Scholium is a remark upon something going before it. For other definitions see Mensuration of Surfaces and Solids, and Conic Sectiona GEOMETRY. 221 Leng ths of* To Angle 45. llo-wing Angle 60. Elements, 3 Radius = Angle 45. 1. Angle 60. Sine . . .707 107 .707 107 .292893 .292893 1.414214 a Line, of ] -?- .866025 5 5 133975 2 Sea as A B, ^ Equal 3? rr t=j. Cosecant 1.414214 I '.765366 .785398 quired. ] draw two p definite lengt equired nurn 5, 4, and B i, i A o, join o arallel there I.I547 1.73205 577349 i 1.0472 S'um'ber irallel lines, h, and upon ber of equal 2,3,4; Joifc B, and draw to. Cosine Versed Covers( Secant To 3D 1. Sine. . id " . . Cotangent . . . Chord ivide jL les. ;vitli any re arts. Fig. 1 > From A and E Ao, Br, to an in them point off r parts, as A i, 2, - o B, 4 i, etc. Or, point off 01 the other lines p Apcc 1 i j j /- x .-t.~ -i "3 2 n A To Construct a 33iagonal Scale, as A B. Fig. 2. Divide a line into as many di- visions as there are hundreds of feet, spaces of ten feet, feet, or inches required. Draw perpendiculars from each division to a parallel line, C D. Divide one of divisions, A E, C F, into spaces of ten if for feet and hundredths, and twelve if for feet and inches; draw the lines Ai, A to E, E to G, etc. , will measure one foot;' A to"a"c to",7ttT^etc7^n measure i-ioth of a foot The several lines A i, a 2, etc, will measure upon lines fc, Z, etc, i-iooth of a foot; and op will measure upon fc, I, etc, divisions of i-ioth of a foot. Lilies. To 33ra\v a Perpendicular to a Right Line, o as or, Fig. 3, c A, Fig. 4, or *%Scr' from a 3?oint external to it, as A, Fig. S, and from any two 3?oints, as c d, Fig. 6. With any radius as r A, r B, cut line at A and B ; then with a longer radius, as A o, B o, describe arcs cutting each _j other at o, and connect o r. (Fig. 3. ) B Or, from A, set off A B equal to 3 B parts by scale ; from A B, with radii g $ of 4 and 5 parts, describe arcs cut- ting at c, and connect c A. (Fig. 4. ) NOTE. This method is useful where straight edges are inappli- cable. Any multiples of numbers 3, 4, 5 may be taken with same ef- fect, as 6, 8, 10, or 9, 12, 15. From A, with a sufficient radius, " cut line at o c, and from them de- scribe arcs cutting at ?', and connect Ar. (Fig. 5.) From any two points, as c d, at a proper distance apart, describe arcs cutting at A B, and connect them. (Fig. 6.) T * 222 GEOMETRY. To Bisect a Flight Line or an Arc of* a Circle, and. to Draw a, l^erpeivdicvi- lar to a, Circular or Right Line, or a, Radial Are. Fig. 7. From A B as centres describe arcs cutting each other at c and d, connect c d, and line and arc are bisected at e and o. Line cd is also perpendicular to a right line as A B, and radial to a circular arc as A o B. To Draw a Line Parallel to a Given Right Line, as c d, Fig. 8. From A B describe arcs Ac, Bd,and draw a line par- allel thereto, touching arcs c and d. To Describe Angles of 3O and, 6O, Fig. D, and. 45 Fig. 10. From A, with ar r radius, A o, de- scribe or, and from o with a like ra- dius cut it at r, let fall perpendicular rs; then o Ar = 6o, and Ars = 3o. (Fig. 9-) Set off any distance, as A B, erect perpendicular Ao = A B, and connect o B. (Fig. 10.) To Bisect Inclination of Two Lines. when IPoint of Intersection is Inac- cessible. Fig. 11. m Upon given lines, A B, C D, at any points draw perpen- diculars e o, s r, of equal lengths, and from o and s draw parallels to their respective lines, cutting at n; bisect angle ons, connect n m, and line will bisect lines as re- quired. To 12. Rectilineal Describe an Octagon upon a Line, as A B. Fig. IS. From points A B erect indefinite perpendiculars A/, Be; produce A B to m and w, and bisect angles m A o and n Ep With A u and B r. Make A u and B r equal to A B, and draw u z, r v parallel to A/, and equal to A B. From z and v, as centres, with a radius equal to A B, de- scribe arcs cutting A/, B e, in / and e. Connect /,/, and e v. To Inscribe any Reg-alar Polygon in a Circle, or to Divide Circumference into a given !N"umber of EcLnal .Parts. Fig. 13. If Circle is to contain a Heptagon. Draw angle A o B at centre o for 360 -=-7 = 51 42' 51"+, or 51^, then set off upon circumference distance A B or remaining angles A o B. GEOMETRY. 223 Xo Inscribe a Hexagon in. a Circle. Fig. 14. H. c^ Draw a diam- eter, AoB. From A and B as cen- tres, with A o and B o, cut circle at cm and ew, and connect. To Describe a Hexagon about a Circle. Fig. l5. Draw a diam- eter as a o b ; and with a o cut circle at c ; join a c, and bisect it with ra- dius o r, through r draw e r paral- lel to c a, cutting diameter at m ; then with radius o m describe circle, within which describe a hexagon as above. To Inscribe a Pentagon in a Circle. Fig. 16. 16. A Draw diameters A c and m n, at right angles to each other; bisect o n in ?-, and with r A describe A s ; from A with A * describe s B. Connect A B, and distance is equal to one side of a pentagon. To Describe a Pentagon upon a Line, as A B. Fig. ir. 17 n Draw B m per- pendicular to A B, and equal to one half of it ; extend A m until m n is equal to B m. From A and B, with radius Bn, de- scribe arcs cutting each other in o; then from o, with radius o B, describe circle A C B, and line A B is equal to one side of a pentagon upon circle described. To Describe a Regular Polygon of* any required Number of Sides. Fig. 18. From point o, with distance o B, describe semicircle B 6 A, which divide into as many equal parts, A a, a 6, b c, etc. , as the polygon is to have sides. Thus, let a Hexagon be required : From o to second point 6 of six divisions draw o &, and a/ \\ /^\ // through other points, c, d, and e, draw o C, o D, etc. / N \!/,-^ v/ Apply distance o B, from B to E, from E to D, from D to Jf"~ o*- ''JJ C, etc. Join these points, as b C, C D, etc. To Construct a Square or a Rectangle on a given 3L.irie.-Fig. 19. 19. mx- X,n On A B as cen tres, with ABas radius, describe __________ arcs cutting at 9y Ordinates. Fig. 45. Divide semi transverse axis, as A 6, into 8 or 10 divisions, as may be convenient, and erect ordi- nates, the lengths of which are equal to semi-con- jugate, multiplied by the units for each division as follows: Eighths. 1 .48412 5 .92703 2 . 661 44 6 . 968 24 3 .78063 7 .99216 4 .86603 8 i Divisions. i -435385 2 .6 3 .71414 4 -.8 Tenths. 5 .86602 6 .91651 7 -95394 8 .97979 9 .99499 10 i C To Construct an Ellipse -when Diameters do not Inter- sect at Right Angles. Fig. 46. Let A B and C D be given diameters. Draw boundary lines parallel to diameters, divide longest diameter into any number of equal parts, and divide shortest boundary lines into same number of equal parts. From one end of shortest diameter, D, draw radial lines through divisions of longest diame- ter, and from opposite end, C, draw radial lines to divisions on shortest boundary lines ; the intersection of these lines will give points in the curve. To Describe a Grothic Arc. Fig. 47'. Take line A B. At points A and B draw arcs B a and A c, and it will describe arc required. [ B To Describe an Elliptic Arc, Chord and Height being given. Fig. 48. Bisect A B at c ; erect perpendicular A g, and draw line q D equal and parallel to A c. Bisect A c and A q in r and n; make c I equal to c D, and draw line I r q ; draw also line n s D ; bisect s D with a line at right angles, and cutting line c D at o; draw line o q; make cp equal to c A:, and draw line o p i. Then, from o as a centre, with radius o D, describe arc s D i; and from k and p as centres, with radius A k, describe arcs A s and B i. 228 GEOMETRY. To Describe a Grothie Arc. Figs. 4:9 and SO. Divide line A B into three equal parts, e c ; from points A and B let fall perpendiculars A o and B r, equal in length to two of divisions of line A B; draw lines o h and r g from points e, c; with length of cB, describe arcs Ag and B/i, and from points o and r describe arcs g i and i h. (Fig. 49. ) ^1 Or, divide line A B into three equal parts at a and b, and on points A, a, b, and B, with distance of two divisions, make four arcs intersect- ing at c and o. Through points c, o, and divisions a, &, draw lines c/and o e, on points a and fc describe arcs A e and B/, and on points c o arcs/ 5 and e s. (Fig. 50.) Cycloid, and. Epicycloid. To Describe a Cycloid. Fig. si. When a circle, as a wheel, rolls over a straight right line, beginning as at A and ending at B, it completes one revolution, and measures a straight line, A B, exactly equal to circumference of circle cer, which is termed the generating circle, and a point or pencil fixed at point r in circumference traces out a curvilinear path, A r B, termed a cycloid. A B is its base ar?d cr its axis. Place generating circle in middle of Cy- cloid, as in figure; draw a line, m n, paral- lel to base, cutting circle at e; and tangent n i to curve at point n. The following are some of properties of Cycloid : Horizontal line e n = arc of circle e r. Half-base A c half-circumference cer. Arc of Cycloid r n twice chord r e. Half-arc of cycloid Ar=twice diameter of circle r c. Or, whole arc of Cycloid A r B = four times axiscr. Area of Cycloid A r B A = three times area of generating circle re. Tangent n i is parallel to chord e r. To Describe Cnrve of a Cycloid. Fig. 53. On an indefinite line, AB, set off co- circumference of generating circle, di- vide this line into any number of equal parts (8 in figure), and at points of divis- ion erect perpendiculars thereto. Upon eacl1 of these lines describe a circle = generating circle. On c i take i x .25 C i, and with as a centre, with radius a? c = .75 c x, describe an arc cutting circle at x'; from 2 on next circle, with two distances of i i', measured as chords, cut circle at 2'; from 3 on next circle, with three distances of i i', cut circle at 3', and proceed in like manner from each side until figure is complete. To Describe an Interior Epicycloid or Hypocycloid. Fig. S3. If generating circle is rolled on inside of fundamental circle,* as in Fig. 53, it forms an interior epicycloid, or hypocycloid, A c B, which becomes in this case nearly a straight line. Other points of reference in figure cor- respond to those in Fig. 51. When diameter of generat- ing circle is equal to half that of fundamental circle, epicycloid becomes a straight line, being diameter or the larger circle. Se explanation, Fig. 54. GEOMETRY. 229 To Describe an Exterior Epicycloid. Fig. 54. An Epicycloid differs from a Cycloid in this, that it is generated by a point, o'", in one circle, o r, rolling upon circumference of another, A r , instead of upon a right line or horizontal surface, former being generating circle and latter fundamental circle. Generating circle is shown in four positions, in which generating point is indicated by oo'o^'o'". A.O'" * is an Epicycloid. Involute. To Describe an Involute. Fig. 55. Assume A as centre of a circle, b c o ; a cord laid partly upon its circumference, as be; then the curve eimn, described by a tracer at end of cord, when unwound from a circle, is an involute. This curve can also be defined by a batten, x, rolling on a circle, as s u. 3?ara"bola. To Construct a 3Para"bola "by Ordinates or -Abscissa. Figs. 56 and 57". By Ordinates. Divide ordinate a b into 10 equal parts, and erect perpendicu- lars, length of which will be determined by multiplying abscissa c by respective units for each perpendicular, as follows: Divisions. 3 -Si 5 -75 4-64 6-. 84 7 .91 8 .96 9 -99 10 I By Abscissa. Divide abcissa a c into 8 or 10 equal parts, as may be convenient, and draw ordinates thereto, the lengths of which will be deter- mined by multiplying half ordinate a & by respective units for each ordinate, as follows: Eighths. Divisions. 5 -79 57 7 8 i -93541 6 .7746 7 .83666 8 .89443 9 .94868 Tenths. 1 31623 2 .44721 3 -54772 4 .63245 5 .70711 10 i "With a Sq.-u.are and Cord. Fig. 58. Place a straight edge to directrix A B, and apply to it a square, c o. Attach to end o end of a cord equal to o A, and attach other end to focus e; slide square along straight edge, maintaining cord taut against edge of square, by a point or pencil, and curve will be traced. (Fig. 58.) When Height and mr ._ r _^ ___ r .^ Base are given. j ' - L -~y Fig. 59. "]-"/ Assume A B axis and c d a double ordinate or base. ' ' Through A draw m n parallel to c d, and through c and d draw cm,dn, parallel to axis A B. Divide c m, d n into any number of equal parts, as at a c e o, also cB,Bd, into a like number of parts. Through points c X i, 2, 3, and 4 draw lines parallel to axis, and through a c e o draw lines to vertex A, cutting these perpendiculars, and through these pointe curve may be traced. (Fig. 59.) c ' GEOMETRY. To Describe Curve of a Parabola, Base and Height being given. Fig. 6O. Draw an isosceles triangle, as a 6 d, base of which shall be equal to, and its height, c 6, twice that of proposed parabola. Divide each side, a &, d 6, into any number of equal parts ; then draw lines, i i, 2 2, 3 3, etc., and their intersection will define curve. (Fig. 60.) To Describe a Parabola, any Ordinate to Axis and its Abscissa being given. Fig. 61. Bisect ordinate, as A o in r; join m c ji B r, and draw r s perpendicular to it, meeting axis continued to s. Set off a B c, B e, each equal to o s; draw m c u perpendicular to B s, then m w is directrix and B e focus; through e and any number of points, i, i, i, etc., in axis, draw double ordinates v i v, and on centre e, with radii e c, i c, etc., cut respective or- dinates at v v, etc., and trace curve through these points. NOTE. Line vev passing through focus is parameter. Spiral. To Draw a Spiral about a given Point. Fig. 62. Assume c the centre. Draw A h, divide it into twice number of parts that there are to be revolutions of line. Upon c de- scribe re, os, Ah, and upon e describe rs,os, etc. Hyperbola. To Describe a Hyperbola, Transverse and Conjugate Diameters being given. Fig. 63. Let A B represent transverse diameter, and C D conjugate. Draw C e parallel to A B, and e r parallel to C D ; draw o e, and with radius o e, with o as a centre, describe circle F e r, cutting transverse axis pro- duced in F and /; then will F and /be foci of fig- ure. In o B produced take any number of points, n, w, etc., and from F and /as centres, with A n and B n as radii, describe arcs cutting each other in s, s, etc. Through s, s, etc. , draw curve ssssBssss. NOTE. If straight lines, as o ey and o r y, are drawn from centre o through ex- tremities e r, they will be asymptotes of hyperbola, property of which is to ap- proach continually to curve, and yet never to touch it. When Foci and Conjugate Axis are given. Let F and /be foci, and C D conjugate axis, as in preceding figure. Through C draw g C e parallel to F and /; then, with o as a centre and o F as a radius, describe an arc cutting g C e at g and e; from these points let fall perpen- diculars upon line connecting F and/ and part intercepted between them, as A B, will be transverse axis. Catenary. To Delineate a Catenary, Span and Versed Sine being given. Fig. 64. ( W. Hildenbrand. ) Divide half span, as A B, into any required number of equal parts, as i, 2, 3, and let fall B C and A o, each equal to versed sine of curve ; divide Ao into like number of parts, i', 2', 3', as A B. Connect C i', C 2', and C 3', and points of intersec- tion of perpendiculars let fall from A B will give points through which curve is to be drawn. Or, suspend a finely linked chain against a ver- tical plane, trace curve from it on the plane in accordance with conditions of given length and height, or of given width or length of arc. NOTE. For other methods see D, B, Clark's Manual, pp. 18, 19. AREAS OF CIRCLES. 231 DlAM. . .OOOI92 3 7.0686 7 38-4846 ; 14 & .000767 $ 7.3662 ; 7.6699 % 39-8713 41.2826 , ^ X* .003068 7-9798 i % 42.7184 ; % % .012272 .027612 % 8.2958 ; 8.618 8.9462 \ 44.1787 45-6636 47.1731 \ % .049087 % 9.2807 % 48.7071 % % .076699 % 9.6211 9.968 8 50.2656 51.8487 15 M .110447 % 10.3206 % 53-4563 P Y .15033 % 10.679 % 55.0884 /to % 11.0447 K 56.7451 /I} % .19635 % 11.416 58.4264 % %t .248505 % n-7933 % 60.1322 i % % .306796 4 12.177 12.5664 K 9 61.8625 ' 63.6174 16 8 % .371 22 4 12.962 65.3968 % % .441 787 ^ 13.772 % 67.2008 69.0293 % % .518487 % 14.1863 % 70.8823 i % .601 322 /5 14.606 % 72.7599 % .690292 S I5-033 15-465 % 74.6621 76.5888 % ! .7854 % I5-9043 10 78.54 17 Ae .8866 %& 16.349 % 80.5158 % .99402 % 16.8002 % 82.5161 % /J6 1.1075 % 17-257 % 84.5409 % 3^ 1.2272 % 17.7206 % 86.5903 1 A %& 1-353 % 18.19 % 88.6643 % % 1.4849 % 18.6655 % 90.7628 % %& 1.6229 % 19.147 % 92.8858 % K 1.7671 5 19-635 ii 95-0334 18 /M I-9I75 x& 20.129 <^J 97-2055 ^3 % 2.0739 % 20.629 % 99.4022 /^ % 2.2365 %> 21.135 % 101.6234 % M 2.4053 % 21.6476 % 103.8691 % % 2.58 %& 22.166 % 106.1394 % % 2.761 2 % 22.6907 % 108.4343 % M^6 2.9483 %B 23.221 110.7537 % 2 3.1416 % 23-7583 12 113.098 19 xie 3.338 %> 24.301 % 115.466 /^ % 24.8505 % 117.859 % %& 3.7584 % ! 25.406 % 120.277 % % 3.9761 % 1 25-9673 % 122.719 K 4 .2 % 26.535 % 125.185 % B* 4.4301 % 27.1086 % 127.677 % %s 4 .666 4 % 27.688 % 130.192 % K 4.9087 6 28.2744 13 132-733 20 %L 5- I 573 % 29.4648 y* 135-297 K % 5-4 JI 9 30-6797 137.887 y* A& 5-6723 % 31.9191 % 140.501 % 5-9396 % 33-1831 H i43'i39 K 6.2126 % 34-4717 145.802 % 6.491 8 % 35-7848 % 148.49 % % 6.7772 % 1 37-1224 % 151.202 % 232 AREAS OP CIRCLES. DlAM. AREA. DlAM. AREA. DlAM. AKIA. DlAM. AREA. 21 346.361 28 6I5-754 35 962.115 42 1385.45 % 350.497 % 621.264 H 069 % 1393-7 & 354.657 y 626.798 % 975-909 % 1401.99 i 358.842 % 632.357 % 982.842 % 1410.3 3 6 3-05I X 637.941 % 989.8 y* 1418.63 % 367.285 % 643-549 % 996.783 % 1426.99 % 371-543 % 649.182 % 1003.79 % 1435-37 % 375.826 % 654.84 % 1010.822 % 1443-77 22 380.134 29 660.521 36 1017.878 43 1452.2 /^ 384.466 % 666.228 % 1024.96 % 1460.66 % 388.822 x 671.959 H 1032 065 g 1469.14 % 393-203 % 677.714 % 1039.195 % 1477.64 % 397.609 y* 683.494 % 1046.349 % 1486.17 % 402.038 689.299 % 1053.528 % 1494-73 % 406.494 % 695.128 % 1060.732 % I503-3 % 410.973 % 700.982 % 1067.96 % I5II-9I 23 415477 30 706.86 37 1075.213 44 1520.53 420.004 % 712.763 % 1082.49 % 1529.19 / 424.558 y 718.69 % 1089.792 y 1537-86 % 429'135 % 724 642 % 1097.118 % 1546.56 % 433-737 % 730.618 i 1104.469 % 1555.29 % 438-364 $ 736.619 1111.844 % 1564.04 % 443.015 742.645 1119.244 H 1572.81 % 447.69 % 748.695 I 1126.669 % 1581.61 24 452-39 31 754.769 38 1134.118 45 1 I590-43 457-115 760.869 Ys 1141.591 1599.28 % 461.864 % 766.992 % 1149.089 /^ 1608. 16 % 466.638 % 773-14 % 1156.612 i 1617.05 X 471.436 % 779-3!3 % 1164.159 & 1625.97 % 476.259 % 785-51 % 1171.731 % 1634.92 H 481.107 % 791.732 % 1179.327 % 1643.89 % 485-979 % 797.979 % 1186.948 % 1652.89 25 490.875 32 i 804.25 39 1194.593 46 1661.91 495.796 810.545 1202.263 % 1670.95 % 500.742 3^ 816.865 % 1209.958 1680.02 % 505-712 ft 823.21 % 1217.677 1689. 1 1 x 510.706 i 829.579 % 1225.42 % 1698.23 % 515.726 835-972 % 1233.188 % 1707.37 % 520.769 % 842.391 % 1240.981 K 1716.54 % 525.838 % 848.833 % 1248.798 % I725-73 26 530-93 33 855.301 40 1256.64 47 1734-95 X^j 536.048 % 861.792 H 1264.506 % 1744.19 % 54 I - I 9 y 868.309 H 1272.397 y 1753-45 % 546.356 % 874.85 % 1280.312 % 1762.74 55L547 % 881.415 % 1288.252 y* 1772.06 556-763 888.005 1296.217 % 1781.4 % 562.003 % 894.62 1304.206 % 1790.76 % 567-267 % 901.259 1312.219 % 1800.15 27 572.557 34 907.922 41 1320.257 48 1809.56 K 577.87 914.611 H 1328.32 % 1819 M 583.209 % 921.323 % 1336.407 y 1828.46 X 588.571 % 928.061 % 1344.519 % I837-95 K 593-959 % 934-822 H 1352.655 % 1847.46 % 599-371 941.609 % 1360.816 1856.99 % 604.807 % 948.42 % 1369.001 % 1866.55 K 610.268 % 955-255 % 1377.211 % 1876.14 AREAS OF CIRCLES. 233 DlAM. ARIA. DIAM. AREA. DIAM. ARIA. DIAM. ARIA. 49 1885.75 56 2463.01 63 3" 7-25 70 3848.46 % 1895.38 % 2474.02 % 3129.64 % 3862.22 U 1905.04 2485.05 & 3142.04 % 3876 1914.72 % 2496. 1 1 % 3I54-47 M 3889.8 % 1924.43 X 2507.19 x 3166.93 K 3903-63 % 1934.16 % 2518.3 % 3I79-4I % 39I7.49 % 1943.91 % 2529-43 % 3191.91 % 393 I -37 % 1953-69 % 2540.58 % 3204.44 % 3945-27 So 1963-5 57 2551-76 6 4 3217 7* 3959-2 X 1973-33 % 2562.97 X 3229.58 % 3973-15 % 1983.18 & 2574.2 M 3242 18 % 3987.13 % 1993.06 % 2585-45 H 3254.81 % 4001.13 X 2002.97 X 2596-73 X 3267.46 B 4015.16 2012.89 % 2608.03 /N* 3280.14 78 4029.21 a| 2022.85 % 2619.36 % 3292.84 74 4043.29 H 2032.82 % 2630.71 % 3305-56 % 4057-39 51 2042.83 58 2642.09 65 33I8.3I 72 4071.51 % 2052.85 2653.49 333L09 X 4085.66 2062.9 % 2664.91 % 3343-89 % 4099.84 2072.98 % 2676.36 % 3356.71 % 4114.04 % 2083.08 X 2687.84 % 3369-56 X 4128.26 % 2093.2 2699.33 % 3382.44 % 4142.51 % 2103.35 M 2710.86 % 3395-33 % 4156.78 % 2113.52 % 2722.41 % 3408.26 % 4171.08 S 2 2123.72 59 2733-98 66 3421.2 73 4185.4 % 2133-94 H 2745-57 H 3434-17 % 4199.74 X 2144.19 2757.2 H 3447-!7 % 4214.11 2154.46 % 2768.84 % 3460.19 % 4228.51 % 2164.76 & 2780.51 % 3473-24 % 4242.93 % 2175.08 % 2792.21 % 3486.3 % 4257-37 % 2185.42 % 2803.93 % 3499-4 % 4271.84 % 2195-79 % 2815.67 % 3512.52 % 4286.33 S3 2206.19 60 2827.44 67 3525.66 74 4300.85 % 2216.61 /^ 2839.23 3538.83 % 43 1 5 -39 & 2227.05 % 2851.05 /^ 3552.02 y 432996 2237.52 % 2862.89 H 3565-24 % 4344-55 % 2248.01 H 2874.76 3578.48 H 4359- * 7 % 2258.53 % 2886.65 % 3591-74 % 4373 81 % 2269.07 H 2898.57 % 3605.04 % 4388.47 % 2279.64 % 2910.51 % 3618.35 % 4403.16 54 2290.23 61 2922.47 68 3631.69 75 4417.87 X 2300.84 % 2934.46 % 3645-05 y* 4432.61 2311.48 % 2946.48 y 3658.44 % 4447-38 2322.15 % 2958.52 % 3671.86 % 4462.16 % 2332-83 % 2970.58 H 3685.29 y* 4476.98 % 2343-55 2982.67 % 3698-76 M 4491.81 % 2354-29 % 2994.78 % 3712.24 M 4506.67 % 2365-05 % 3006.92 % 3725'75 % 4521.56 55 2375-83 62 3019.08 69 3739-29 7 6 4536.47 2386.65 H 3031.26 % 3752.85 H 455L4I % 2397.48 H 3043-47 x 3766.43 4566.36 % 2408.34 % 3055-71 % 3780.04 458i.35 X 2419.23 % 3067.97 % 3793-68 % 4596 36 2430.14 % 3080.25 % 3807-34 % 4611.39 8 2441.07 % 3092.56 % 3821.02 % 4626 45 % 2452.03 \ % 3104.89 % 3834-73 % 464L53 234 AREAS OF CIRCLES. 77 4656.64 84 i 554 J -78 9i 6503.9 98 7542-c H 4671.77 5558.29 H 6521.78 % 7562.: X 4686.92 % 5574-82 y 653968 % 758i.. % 4702.1 % 559 I -37 M 6557.61 % 7600.? X 47I7-3 1 X 5607.95 M 6575.56 X 7620. i 473 2 -54 % 5624.56 % 6593-54 % 7639-. % 4747-79 M 5641.18 % 6611.55 % 7658.* % 4763.07 H 5657-84 % 6629.57 % 7678.2 78 4778.37 85 5674-51 92 6647.63 99 7697.- 4793-7 5691.22 y* 6665.7 K 7717.1 3 4809.05 3^ 5707.94 % 66838 H 7736.< H 4824.43 % 5724.69 % 6701.93 % 7756.] % 4839.83 % 574 I -47 X 6720 08 % 7775 ( 4855.26 % 5758.27 6738.25 % 7795- % 4870.71 % 5775-1 % 675645 M 7814.' % 4886.18 % 579L94 % 6774.68 % 7834< 79 4901.68 86 5808.82 93 6792.92 100 7854 X 4917.21 X 5825.72 % 68u.2 H 7893-: 4932.75 X 5842.64 % 6829.49 X 7932.' 4948.33 % 5859.59 % 6847.82 % 7972- K 403-92 X 5876.56 X 6866.16 IOI Son.J % 4979-55 5893'55 % 6884.53 3 805 1 v % 4995-19 % 5910.58 % 6902.93 x 8091.: % 5010.86 % 5927.62 % 6921.35 % 8131-: so 5026.56 87 5944.69 94 6939.79 102 8i 7 i. : % 5042.28 H 5961.79 % 6958.26 y 8211.. % 5058.03 % 5978.91 y 6976.76 X 825i.( % 5073.79 % 5996.05 % 6995.28 % 8291. c X 5089.59 H 6013.22 X 7013.82 103 8332-, JNs 5 I 05-4 I % 6030.41 H 7032-39 H 8 37 2.{ M 5121.25 % 6047.63 *A 7050.98 X 8413. % 5I37-I2 % 6064.87 % 7069.59 H 8454.0 81 5i53-oi 88 6082.14 95 7088.23 104 8494.5 ^ 5168.93 H 6099.43 H 7106.9 y 8535.' 3^ 5184.87 X 6116.74 X 7 12 5-59 % 8576.' K 5200.83 % 6134.08 % 7I44.3I % 8617.5 X 5216.82 x 6151-45 X 7163.04 105 86 59 .( % 5232.84 6168.84 % 7181.81 H 8700.: % 5248.88 % 6186.25 % 7200.6 X 8741.- % 5264.94 % 6203.69 % 7219.41 % 8783.1 82 5281.03 89 i 6221.15 & 7238.25 106 8824.- K 5297.14 6238.64 7257.11 y 8866.. 3^ 53I3-28 % 6256.15 % 7275-99 X 8908.: X 5329-44 % 6273.69 % 7294.91 % 8950.C K 5345.63 6291.25 % 73 I 3-84 107 8992. c % 5361.84 6308.84 % 7332-8 x 9034.] % 5378.08 % 6326.45 % 735L79 X 9076.2 % 5394.34 % 6344.08 % 7370.79 % 9118.' 83 5410.62 90 6361.74 97 7389-83 108 9 i6o.c K 5426.93 H 6379.42 X 7408.89 % 9203-: % 5443-26 % 6397.13 7427-97 y> 9245.0 K 5459-62 % 6414 86 7447.08 M 9288.= K 5476.oi H 6432.62 % 7466.21 109 9331.: % 5492.41 % 6450.4 % 7485-37 y 9374- ] % 5508.84 % 6468.21 % 7504.55 X 9417.] % 5525.3 % 6486.04 % 7523-75 % 9460.] AREAS OF CIRCLES. 235 no 9 503-34 1 20 11309.76 130 13273.26 140 1539384 y 9546.59 y "356.93 y 13324.36 # 15448.87 x 9589-93 % n 404.2 H 13375-56 H 15503-99 % 9 6 33-37 % II45I-57 % 13426.85 % 15559-22 in 9676.91 121 11499.04 131 13478.25 141 15614-54 y 9 7 2 o-55 X 1 1 546.61 y 13529-74 15669.96 K 9 764-29 K 11594.27 y* 13581.33 15 725-48 H 9808.12 H 11642.03 % 13633.02 % 15781.09 112 9852.06 122 11689.89 132 13684.81 142 15836.81 M 9896.09 u 1I737-85 X 13736.69 u 15892.62 2 9940.22 X 11785.91 m 13788.08 X 15948.53 % 9984.45 X 11834.06 % 13840.76 % 16004.54 1J 3 10028.77 123 11882.32 133 13892.94 143 16060.64 y 10073.2 y 11930.67 X 13945.22 X 16116.85 y* 10117.72 X 11979.12 y* 13997.6 K 16173.15 % 10162.34 X 12027.66 % 14050.07 % 16229.55 114 10207.06 124 12076.31 134 14 102.64 144 16 286.05 y 10251.88 % 12 125.05 X 14 155-31 Sf 16342.65 8 10296.79 X I2I73.9 K 14208.08 H i639935 % 10341.8 12222.84 g 14253.09 % 16456.14 "5 10386.91 125 12271.87 135 14313.91 145 16513-03 10432.12 k I232I.OI K 14366.98 M 16570.02 > 10477.43 x 12370.25 fc 14420.14 X 16627.11 8 10522.84 X 12419.58 B A 14473-4 % 16684.3 116 10568.34 126 12469.01 136 14526.76 146 16741.59 K 10613.94 * 12518.54 % 14580.21 X 16798.97 H 10659.65 12568.17 y* 14633-77 % 16856.45 H 10705.44 % I2Ol8.09 % 14687.42 % 16914.03 117 10751.34 127 12667.72 137 14741.17 147 16971.71 3*: 10 797-34 K 12717.64 K 14795.02 H 17029.48 X 10843.43 H 12 767.66 % 14848.97 H 17087.36 % 10889.62 % 12817.78 H 14903.01 % i7 I 45-33 118 10935.91 128 12867.99 138 14957.16 148 17203.4 X 10982.3 12918.31 H 15011.4 3^ 17261.57 K 11028.78 5 12968.72 K 1506574 k 17319.84 M "075.37 M 13019.23 X 15120.18 % 17378.2 119 ii 122.05 129 13069.84 139 i5 I 74-7i 149 17436.67 K n 168.83 X 13120.55 15 229.35 3^ I7495-23 11215.71 5 i3 I 7i-35 S 15284.08 H I7553.89 % 11262.69 % 13222.26 M I5338.9 1 J 50 17671.5 To Compvite Area of a Circle greater than any in Tat>le. RULE. Divide dimension by two, three, four, etc., if practicable to do so, until it is reduced to a diameter to be found in table. Take tabular area for this diameter, multiply it by square of divisor, and product will give area required. EXAMPLE. What is area for a diameter of 1050? 1050-4-7 = 150; tab. area, 150 = 17671.5, which x 7 2 = 865 903.5, area. To Compute Area of* a Circle in Feet and Inches, etc., "by preceding Table. RULE. Reduce dimension to inches or eighths, as the case may be, and take area in that term from table for that number. AREAS OF CIRCLES. Divide this number by 64 (square of 8) if it is in eighths, and quotient will give area in inches, and divide again by 144 (square of 12) if it is in inches, and quotient will give area in feet. EXAMPLE. What is area of i foot 6.375 ms< ? i foot 6. 375 ins. = 18. 375 ins. = 147 eighths. Area of 147 = 16 971.71, which -f- 64 = 265. 181 25 ins.; and by 144 = 1.84 125 feet. To Compute Area of a Circle Composed, of an Integer and. a Fraction. RULE. Double, treble, or quadruple dimension given, until fraction is in- creased to a whole number, or to one of those in the table, as >g, %, etc., provided it is practicable to do so. Take area for this diameter ; and if it is double of that f 01 which area is required, take one fourth of it ; if treble, take one sixteenth of it, etc. EXAMPLE. Required area for a circle of 2.1875 ins. 2.1875 x 2 = 4.375, area for which = 15.0331, which -1-4 = 3. 758 **. When Diameter is composed of Integers and Fractions contained in Table. RULE. Point off a decimal to a diameter from table, and add twice as many figures or ciphers to the right of the area as there are figures cut off from the diameter. EXAMPLE i. What is area of 9675 feet diameter? Area of 96. 75 = 7351.79 ; hence, area = 73 517 goo feet. 2. What is area of 24 375 feet diameter? Area of 2. 4375 4. 6664 ; hence, area = 466 640 ooo feet. To Ascertain. Area of a Circle as 3OO, 3OOO, etc., not contained in Table. RULE. Take area of 3 or 30, and add twice the excess of ciphers to the result. EXAMPLE. What is area of a circle 3000 feet in diameter? Area of 30 = 706. 86, hence area of 3000 = 7 068 600 feet. To Compute Area of a Circle toy Logarithms. RULE. To twice log. of diameter add 7.895091 (log. of .7854), and sum is log. of area, for which take number. EXAMPLE. What is area of a circle 1200 feet in diameter? Log. 1200 x 2 -j-J 895 091 = 6.158 362 + ^895091 = 6.053453, and number for which = 1 130 976 feet. Diam. Aret Area. is of Diam. Birming Area. lam ^ Diam. Wire Q-a Area. uge. Diam. Area. No. Sq. Inch. No. Sq. Inch. No. Sq. Inch. No. Sq. Inch. I .070686 IO .014 103 19 .001 385 28 .000154 2 063347 II .011309 20 .000962 29 .000 133 3 .052 685 12 .009331 21 .000804 3 .OOO 113 4 .044 488 13 .007 088 22 .000616 3i .000 078 5 .038013 14 .005411 23 .000491 32 .000064 6 032365 15 .004071 2 4 .00038 33 .00005 7 .025 447 16 .003318 25 .000314 34 .000038 8 .021 382 17 .002 642 26 .OOO 254 35 .00002 9 .017203 18 .001 886 2 7 O0020I 36 .000013 CIRCUMFERENCES OF CIRCLES. 237 Ci DlAM. rcnmfe ClRCUM. rence DlAM. s of Circles, ClRCUM. I i DlAM. from ^ ClRCUM. ^ to 1 DlAM. 50. ClRCUM. ^ ^.04909 3 9.4248 8 25.1328 15 47.124 jy 9.62II X 25.5255 47.5167 /te .0961 X 9-8I75 i % 25.9182 % 47.9094 & .19635 /M 10.014 % 26.3IO9 i % 48.3021 K 7Q2 7 % 10.2102 i % 26.7036 % 48.6948 /o %& 10.406 I % 27.0963 49.0875 _ -589 % 10.6029 ' % 27.489 49.4802 % .7854 %& 10.799 ! % 27.8817 49.8729 & -981 75 M 10.9956 | II.I9I 9 28.2744 % 28.6671 16 50.2656 50.6583 % 1.1781 11.3883 j % 29.0598 / 5L05I y 1.37445 11.584 % 29.4525 M 5L4437 18 29.8452 X 51.8364 A 1.5708 % 11.977 ff 30.2379 % 52.2291 %L 1.767 15 % 12.1737 M 30.6306 % 52.6218 6/ % 12.369 % 3I-0233 % 53-0145 /B 935 4 12.5664 10 3I.4I6 17 53.4072 % 2.15985 12.762 X 31.8087 X 53-7999 R/ 2.3562 X 12.9591 % 32.2014 / 54.1926 /tt % 32.5941 /H) 54.5853 % 2-552 55 % I3-35I8 /% 32.0868 /5 54.978 % 2.7489 /5s 13-547 % 33-3795 /N* 55.3707 % 2-945 25 r 13-7445 13-94 % 33-7722 34.1649 % 55.7634 56.1561 I 3.1416 % 14.1372 ii 34-5576 18 8 56.5488 3-3379 %& H-333 X 349503 X 56.0415 3-5343 % 14.5299 % 35-343 3^ 57.3342 3.7306 % I4'725 % 35-7357 ^ 57.7269 /? % 14.9226 i 36.1284 % 58.1196 i 4-1233 % 15.119 36-5211 % 58.5123 4-3I97 JB I5-3I53 M 36.9138 y 58-905 /is 4.516 % 15-5" % 37-3065 % 59.2977 H 4.7124 5 15-708 12 37.6992 19 59.6904 /16 49087 16.1007 X 38.0919 60.0831 % 5-1051 * 16.4934 3 38.4846 x 60.4758 % 5.301 4 16.8861 % 38.8773 ^^ 60.8685 % 54978 X 17.2788 M 39-27 /2 61.2612 % 5.6941 % 17.6715 % 39.6627 % 61.6539 % 5.8905 % 18.0642 % 40-0554 M 62.0466 % 6.0868 % 18.4569 K 40.4481 % 62.4393 2 6.283 2 6 18.8496 40.8408 2O 62.832 /ie 6-4795 X 19.2423 I3 x 4 1 -2335 X 63.2247 X 6.675 9 3 19.635 41.6262 63.6174 % 6.872 2 ^ 20.0277 % 42.0189 M 64.0101 /^ 7.0686 /? 20.4204 X 42.4116 3>| 64.4028 %& 7.2649 % 20.8131 M 42.8043 M 64-7955 % 7.461 3 M 21.2058 % 43- J 97 M 65.1882 %& 7.6576 % 21.5985 K % 65.5809 % 7.854 7 21.9912 14 43.9824 21 65.9736 %> 8.050 3 X 22.3839 X 44-3751 X 66.3663 % 8.2467 22.7766 % 44.7678 % 66.759 % 8-443 % 23.1693 % 45.1605 % 67.1517 % 8.6394 23.562 % 45.5532 H 67.5444 % 8-835 7 % 23.9547 % 45-9459 % 67.9371 K 9.032 i M 24-3474 M 46.3386 M 68.3298 % 9.228 4 K 24.7401 % 46.7313 % 68.7225 238 CIRCUMFERENCES OF CIRCLES. DlAM. ClECUM. DlAM. ClHCUM. DlAM. ClRCUM. DlAM. CrRcuu. 22 69.1152 29 9I.IO64 3 6 113.098 43 135.089 % 69.5079 % 91.4991 % H3.49 ^ I35.48I X 69.9006 % 9I.89I8 % 113.883 y 135.874 % 70.2933 % 92.2845 % 114.276 % 136.267 y, 70.686 92.6772 X 114.668 % 136.66 71.0787 93.0699 % II5.o6l % 137.052 % 71.4714 % 93.4626 % 115-454 % 137-445 % 71.8641 % 93.8553 % 115.846 % 137.838 23 72.2568 30 94.248 37 Il6.239 44 138.23 72.6495 94.6407 % 1 16.632 % 138.623 /^ 73.0422 % 95-0334 % II7.O25 % 139.016 % 73.4349 % 954261 % II7.4I7 % 139.408 % 73.8276 % 95.8188 y* II7.8I ! 139.801 % 74.2203 % 96.2115 % II8.203 % 140.194 H 74.6I3 % 96.6042 % H8.595 % 140.587 % 75-0057 % 96.9969 % 118.988 % 140-979 24 75-3984 31 97.3896 38 II9.38I 45 141.372 % 75- 79 11 % 97.7823 /& 119.773 & 141.765 U 76.1838 X 98.175 M I20.I66 % 142.157 % 76.5765 % 98.5677 H 120.559 % 142.55 76.9692 % 98.9004 % 120.952 % 142.943 % 77.3619 993531 % 121.344 % 143-335 M 77-7546 % 99'7458 % 121.737 % 143.728 % \ 78.1473 % 100.1385 % 122.13 % I44.I2I 25 i 78.54 32 100.5312 3 V 122.522 46 144.514 % 78.9327 100.9239 122.915 % 144.906 M 79-3254 /^ 101.3166 % 123.308 % 145.299 % 79.7181 H 101.7093 % 123.7 % 145.692 % 80.1108 y* IO2.IO2 y* 124.093 X 146.084 % 80.5035 % 102.4947 % 124486 % 146.477 % i 80.8962 % 102.8874 M 124.879 % i 146.87 % 81.2889 % 103.2801 K 125.271 % I 147.262 26 81.6816 33 103.673 4 1 125.664 47 147.655 > 82.0743 104.065 126.057 147.048 3 82.467 % 104.458 % 126.449 % 148.441 82.8597 % 104.851 % 126.842 % I48.8 33 M 83.2524 * 105.244 y> 127.235 k 149.226 % 83.6451 105.636 % 127.627 149.619 M 84.0378 % 106.029 M 128.02 M I5O.OII ji 84-4305 % 106.422 % 128.413 % 150.404 27 84.8232 34 106.814 41 128.806 4 8 150.797 85.2159 107.207 % 129.198 X 151.189 /^ 85.6086 / 107.6 % 129.591 % 151.582 M 86.0013 % 107.992 % 129.984 % I5I-975 /^ 86.394 % 108.385 y* 130.376 y* 152.368 % 86.7867 108.778 % 130.769 % 152.76 K 87.1794 % 109.171 % I3I.I62 % I53-I53 X 87.5721 % 109.563 % I3L554 % 153.546 28 87.9648 35 109.956 42 I3L947 49 \, 153.938 /^ 88.3575 H 110.349 % 132.34 H 154.331 M 88.7502 IIO.74I % 132.733 % 154.724 % 89.1429 III.I34 % I33-I25 % I55-II6 K 89.5356 K III.527 % I33-5I8 K 155.509 % 89.9283 % III.9I9 % 133-9" If 155.902 % 90.321 % II2.3I2 /^ 134303 % 156.295 X 90.7137 % 112.705 % 134.696 % 156.687 CIRCUMFERENCES OF CIRCLES. 239 ClRCUM. DIAM. ( Cracim. [1 DIAM. ClRCtTM. DIAM. 1 CiKcrv. 157.08 57 1 179.071 64 2OI .062 71 223.054 J57-473 179.464 y& 201.455 /^J . 223.446 157-865 % 179.857 % 201.848 /^ 223.839 158.258 % 180.249 %> 202.24 % 224.232 158.651 % 180.642 y* 202.633 224.624 159.043 % 181.035 % 203.026 % 225.017 159.436 % 181.427 % 203.419 % 225.41 159.829 H 181.82 % 203.8II % 225.802 160.222 58 182.213 65 i 204.204 72 226.195 160.614 182.605 204.597 X 226.588 161.007 % 182.998 x 204989 % 226.981 161.4 % 183.39! % 205.382 % 227.373 161.792 X 183.784 X 205.775 % 227.766 162.185 184.176 206.167 % 228.159 162.578 % 184.569 % 206.56 % 228.551 162.97 % 184.962 % 206.953 % 228.944 163.363 59 185.354 66 207.346 73 229.337 163.756 185.747 X 207.738 % 229.729 164.149 3^ 186.14 X 208.131 % 230.122 164.541 % 186.532 208.524 % 230.515 164.934 /& 186.925 % 208.916 X 230.908 165.327 187.318 % 209.309 % 231-3 165.719 % 187.711 % 209.702 % 231.693 166.112 % 188.103 % 210.094 % 232.086 166.505 60 188.496 67 210.487 74 232.478 166.897 % 188.889 X 210.88 232.871 167.29 % 189.281 211.273 % 233-264 167.683 % 189.674 % 211.665 % 233-656 168.076 ii 3^ 190.067 X 212.058 % 234-049 168.468 % 190.459 % 212.451 % 234.442 168.861 j % 190.852 % 212.843 % 234.835 169.254 % 191.245 % 213-236 % 235.227 169.646 61 191.638 68 213.629 75 235.62 170.039 X 192.03 X 214.021 236.013 170.432 192.423 214.414 % 236.405 170.824 % 192 816 % 214.807 % 236.798 171.217 X 193.208 % 215.2 X 237.191 171.61 193.601 % 215-592 237.583 172.003 % 193-994 % 215-985 % 237.976 172.395 % 194.386 % 216.378 % 238.369 172.788 62 194.779 69 216.77 76 238.762 173.181 /> 195.172 217.163 % 239.154 1 73-573 % 195.565 % 217-556 X 239.547 173.966 % J95-957 % 217.948 % 239.94 174-359 X X 218.341 X 240.332 174.751 196.743 218.734 240.725 I75.I44 % i97- I 35 % 219.127 % 241.118 175.537 % 197.528 % 219.519 % 241.51 175-93 63 197.921 70 219.912 77 241.903 176.322 198.313 % 220.305 X 242.296 176.715 /^ 198.706 % 220.697 242.689 177.108 % 199.099 %$ 221.09 % 243.081 177-5 X 199.492 sv 221.483 % 243.474 177.893 % 199.884 % 221.875 % 243.867 178.286 % 200.277 % 222.268 % 244.259 178.678 % 200.67 y* 222.661 % 244.652 240 CIRCUMFERENCES OF CIRCLES. DlAM. ClRCUM. DlAM. ClRCUM. DlAM. ClRCUM. DlAM. ClRCUM. 78 245.045 85 267.036 92 289.027 99 3II.OI8 H 245-437 267.429 H 289.42 % 3".4II X 245.83 , X 267.821 % 289.813 % 311.804 H 246.223 268.214 % 290.205 % 3I2.I06 X 246.6l6 % 268.607 % 200.598 % 312.589 % 247.008 % 268.999 % 290.991 78 312.082 H 247.401 % 209.392 % 291.383 % 313.375 % 247.794 % 269.785 % 291.776 % 313.767 79 248.186 86 270.178 93 292.169 IOO 314.16 y* 248.579 % 270.57 292.562 & 3T4.945 X 248.972 % 270.963 /^ 292.954 H 315.731 % 249.364 % 271.356 % 293-347 H 3l6.5l6 % 249.757 x 271.748 x 293.74 IOI 317.302 % 250.15 % 272.141 % 294.132 H 318.087 % 250.543 % 272.534 % 294'525 X 318.872 % 250.935 % 272.926 % 294.918 % 3I9.6 5 8 so 251.328 87 273.3I9 94 295-3 1 102 320.443 y* 251.721 H 273.712 H 295.703 % 321.229 % 252.113 X 274.105 % 296.096 % 322.014 % 252.506 % 274.497 % 296.488 H 322.799 X 252.899 % 274.89 y& 296.881 IQ 3 323.585 % 253.291 % 275.283 % 297.274 % 324.37 % 253.684 K 275-675 % 297.667 % 325.156 % 254.077 % 276.068 % 298.059 H 325.941 81 254-47 88 276.461 95 298.452 104 326.726 /^ 254.862 H 276.853 X 298.845 H 327.512 / 255.255 H 277.246 X 299.237 H 328.297 % 255.648 % 277.629 % 299.63 H 329.083 % 256.04 278.032 300.023 105 329.868 % 256.433 278.424 300.415 H 330.653 % 256.826 % 278.817 % 300.808 X 33 1 -439 K 257.218 % 279.21 % 301.201 % 332.224 82 257.6II 8 9 279.0O2 96 3 OI 594 106 333-01 H 258.004 279-995 H 301.986 % 333-795 H 258.397 % 280.388 X 302.379 x 334-58 % 258.789 % 280.78 % 302.772 % 335-306 % 259.182 % 281.173 303.164 107 336.151 % 259.575 281.566 6/ 303.557 fc 336.937 % 259.967 % 281.959 % 303.95 X 337.722 K 260.36 % 282.351 % 304.342 H 338.507 83 i 200.753 90 282.744 97 304.735 108 339.293 26l.I45 283.137 H 305.128 % 340.078 % 261.538 /^ 283.529 % 305.521 X 340.864 % 26I.93I % 283.922 % \ 305-913 % 341.649 % 262.324 % 284.315 Jf 306.306 109 342.434 % 262.716 % 284.707 306.699 y 343-22 % 263.109 % 285.1 !% 307.091 X 344.005 % 263.502 % 285493 K 307.484 % 344- 79 1 84 263.894 91 i 285.886 9 8 i 307.877 no 345.576 H 264.287 y% 286.278 308.27 % 346.361 264.68 % 286.671 % 308.662 X 347-147 265.072 % 287.064 % 309055 % 347.932 % 265.465 K 287.456 X 309.448 in 348.718 % 265.858 287.849 309.84 H 349-503 % 266.251 % 288.242 % 3J0.233 X 350.288 n/. 266.643 % \ 288.634 % 310.626 % 35L074 CIBCUMFEKENCES OF CIRCLES. 241 DlAM. ClECUM. DlAM. ClRCUM. DlAM. ClECUM. DlAM. ClBCTM. 112 35L859 121 380.134 130 408.408 T 39 436.682 / 352.645 / 380.919 409.192 /^ 437.467 X 353-43 x5 381.704 ff 409.979 /^ 438.253 354-215 K 382.49 M 410.763 M 439.037 "3 355-001 122 383.275 131 4"-55 140 439.824 355.786 % \ 384.061 u 412.334 / 440.608 356.572 K ! 384-846 % 413.12 /^ 441-395 % 357-357 % I 385.63I % 413.905 2i 442.179 114 358.142 123 386.417 i32 i 414.691 141 442.966 358.928 387.202 415.476 /^ 443-75 % 359- 7 J 3 i 387.988 M 416.262 i^ 444-536 % 360.499 388.773 M 417.046 K 445-321 "5 361.284 124 389.558 133 417-833 142 446.107 362.069 390-344 M 446.891 % 362.855 K 391.129 % 419.404 447.678 H 363-64 39 I -9 I 5 H 420.188 M 448.462 116 364.426 125 392.7 134 420.974 143 449-249 % 365.211 * 393.484 X 421.759 450.033 % 365.996 394.271 B 422.545 K 450.82 % 366.782 % 395-055 M 423-33 % 451-604 117 367-567 126 395.842 135 424.II6 144 452.39 368.353 3^ 396.626 M 424.9 /^ 453-175 M 369.138 K 397.412 x^ 425.687 K 453.961 369.923 398.197 M 426.471 454-745 118 370.709 127 398-983 136 427^58 J 45 455.532 X 371-494 * 399.768 i 428.042 456.316 % 372.28 400.554 428.828 % 457.103 % 373.065 9! 401.338 % 429.613 146 458.674 119 373.85 128 402.125 137 430.399 /^ 460.244 374.636 3^ 402.909 431.183 I47 i 461.815 x^ 375-421 i 403.696 /? 43L97 463.386 % 376.207 M 404.48 ^ 432-754 148 464.957 1 20 376.992 129 405.266 138 433-541 % 466.528 xi 377-777 406.051 434-325 149 468.098 x 378.563 i 406.837 )2 435- "2 K 469.669 X 379-348 407.622 K 435.8o6 150 471,24 To Compute Circnmference of* a Diameter greater tlian any in preceding Ta"ble. RULE. Divide dimension by two, three, four, etc., if practicable to do so, until it is reduced to a diameter in table. Take tabular circumference for this dimension, multiply it by divisor, according as it was divided, and product will give circumference required. EXAMPLE. What is circumference for a diameter of 1050? I050 -j_ 7 = 150; tab. circum., 150 = 471.24, which X 7 = 3298.68, circumference. To Compute Circumference of* a Diameter in Feet and. Indies, etc., toy preceding Tatole. RULE. Reduce dimension to inches or eighths, as the case may be, and take circumference in that term from table for that number. Divide this number by 8 if it is in eighths, and by 12 if hi inches, and quotient will give circumference hi feet. 242 CIRCUMFERENCES OF CIRCLES. EXAMPLE. Required circumference of a circle of i foot 6.375 ins. i foot 6.375 ins. = 18.375 ins. = 147 eighths. Circum. of 147 = 461.815, which -f- 8 = 57.727 ins.; and by 12 = 4.8106 feet. To Compute Circumference for a Diameter composed of an Integer and. a Fraction. RULE. Double, treble, or quadruple dimension given, until fraction is h> creased to a whole number or to one of those in the table, as %, %, etc., pro- vided it is practicable to do so. Take circumference for this diameter ; and if it is double of that for which circumference is required, take one half of it ; if treble, take one third of it ; and if quadruple, one fourth of it. EXAMPLE. Required circumference of 2.21875 ins. 2.21875 X 2 = 4.4375, which X 2 = 8. 875; circum. for which = 27. 8817, which -7-4 = 6.9704 ins. When Diameter consists of Meyers and Fractions contained in Table. RULE. Point a decimal to a diameter in table, take circumference from table, and add as many figures to the right as there are figures cut off. EXAMPLB. What is circumfe?snce of a circle 9675 feet in diameter? Circumference of 96. 75 = 303.95 ; hence, circumference of 9675 = 30 395 feet. To Ascertain Circumference for a Diameter, as 5OO, COOO, etc., not contained in Table. Rule. Take circumference of 5 or 50 from table, and add the excess of ciphers to the result. EXAMPLE. What is circumference of a circle 8000 feet in diameter? Circumference of 80 = 251. 38 ; hence, circumference of 8000 = 25 138 feet. To Compute Circumference of* a Circle "by Logarithms. RULE. To log. of diameter add .497 15 (log. of 3.1416), and sum is log. of circumference, from which take number. EXAMPLE. What is circumference of a circle 1200 feet in diameter? Log. 1200 = 3.079 18 + .497 15 = 3. 57633, and number for which = 3769.92 /. Circumferences of Birmingham \Vire Grauge. Diam. Circum. Diam. Circum. Diam. Circum. Diam. Circum. No. Ins. No. Ins. No. IDS. No. Ins. I 2 .94248 .89221 10 II .42097 .37699 19 20 .13195 .10995 28 29 .04398 .040 84 3 .81367 12 .34243 21 .10053 30 0377 4 7477 I 3 .29845 22 .08796 31 .031 41 5 .691 15 14 .260 75 23 .078 54 32 .028 27 6 637 74 15 j .226 19 24 .06911 33 025 13 7 .565 49 16 .2042 25 .06283 34 .021 99 8 .51836 I 7 .18221 26 056 55 35 .015 71 9 .46495 18 15394 27 .050 26 36 .01257 AREAS AND CIRCUMFERENCES OF CIRCLES. 24J .A^reas and Circumferences. (Advancing by Tenths.) DlAM. AREA. ClRCUM. DlAM. AREA. ClRCUM. .1 .007854 .31416 .6 24.6301 17-593 .2 .031 416 .62832 7 25.5176 17.0071 .3 .070686 .94248 .8 26.4209 I8.22I3 4 .125664 1.2566 9 27.3398 18.5354 5 .19635 1.5708 6 28.2744 18.8496 .6 .282744 1.885 .1 29.2247 I9.I638 7 .384846 2.I99I .2 30.1908 19.4779 .8 .502656 2.5133 3 3 I I 7 2 5 19.7921 9 .636174 2.8274 4 32.17 20.1062 i .7854 3.I4I6 20.4204 .1 9503 3-455 8 .6 34.212 20.7346 .2 I.I3I 3.7699 7 35.2566 2I.O487 3 L3273 4.0841 .8 36.3169 21.3629 4 1-5394 4.3982 9 37.3929 21.677 I 1.7671 2.0IO6 2.2698 5.0266 5-3407 7 .1 .2 38.4846 39.592 40.7151 21.9912 22.3054 22.6195 .8 9 2 .1 .2 3 2-5447 2.8353 3.1416 3-4636 3-80I3 4.1548 5-969 6.2832 6-5974 6.9115 7-225 7 3 4 i .8 41.854 43.0085 44.1787 45-3647 46.5664 47-7837 22.9337 23.2478 23-502 23.8762 24.1903 24-5045 4 4-5239 4.9087 5.3093 7*854 8.1682 9 8 49.0168 50.2656 24.8l86 25 1328 9 3 .1 .2 3 4 i 6.1575 6.6052 7.0686 75477 8.0425 8-553 9.0792 9.621 1 10.1788 8.4823 8.7965 9.1106 9.4248 9-739 10.053 * 10.3673 10.681 4 10.9956 11.3098 .1 .2 3 -4 .6 '.8 9 9 51.5301 52.8103 54.1062 55-4*7* 56.7451 58.0882 59-4469 60.8214 62.2115 63.6174 25-447 25.76II 260753 26.3894 26 7036 27.0178 27.3319 276461 27 9602 28.2744 .7 10.752 i 11.6239 .1 65-039 28.5886 .8 11.3412 11.9381 .2 664763 28.OXD27 9 11.9459 12.2522 3 67.9292 29.2169 4 12.5664 12.5664 4 69-3979 29531 .1 13.2026 12.8806 5 70.8823 298452 .2 J 3-8545 .6 72-3825 3- 1 594 3 14.522 I 3-589 7 73-8983 30-4735 4 15-2053 13.823 .8 75.4298 30-7877 5 I5-9043 14.1372 9 76.9771 31.1018 .6 16.619 1 14.4514 10 78.54 31.416 7 1 7-349 5 14-765 5 .1 80.1187 31.7302 .8 18.0956 15.0797 .2 81.713 32-0443 9 18.8575 I5-3938 3 83.3231 32.3585 5 19-635 15-708 -4 84.9489 32.6726 .1 20.4283 l6.022 2 5 86.5903 32.9868 .2 21.2372 16.3363 .6 88.2475 33-301 3 22.0619 16.650 5 7 89.9204 336151 4 22.9023 16.9646 .8 91.6091 33-9293 5 23-7583 17.2788 9 93-3134 34-2434 244 AREAS AND CIRCUMFERENCES OF CIRCLES. DlAM. AREA. CIRCUM. DlAM. AREA. CIRCUM. II 95-0334 34.5576 5 213.8251 51.8364 .1 96.7691 34.8718 .6 216.4248 52.1505 .2 98.5206 35-I859 7 219.0402 52.4647 .3 100.2877 35.5001 .8 221.6713 52.7789 4 102.0700 35.8142 9 224.3181 53-093 5 .6 103.8691 105.6834 36.1284 36.4426 17 .1 226.9806 229.6588 53-4072 53-72I4 7 107.5134 36.7567 .2 232.3527 54-0355 .8 109.3591 37-0709 3 235.0624 54-3497 9 III .22O5 37-385 -4 237.7877 54.6638 12 113.0976 37.6992 5 240.5287 54.978 .1 114.9904 38.0134 .6 243.2855 55.2922 .2 116.8989 38.3275 7 246.058 55.6063 3 118.8232 38.6417 .8 248.8461 55-9205 4 120.7631 38.9558 9 251-65 56.2346 5 122.7187 39-27 18 254.4696 56.5488 .6 .'s 9 124.6901 126.6772 128.6799 130.6984 39.8983 40.2125 40.5266 .1 .2 3 4 257.3049 260.1559 263.0226 265.905 56.863 57-I77I 57"49 r 3 57.8054 13 132.7326 40.8408 5 268.8031 58.1196 .1 134.7825 41.155 .6 271.717 58.4338 .2 136.8481 41.4691 7 274.6465 58.7479 -3 138.9294 4L7833 .8 277.5918 59.0621 4 141.0264 42.0974 -9 280.5527 59.3762 i I43-I39I 145.2676 42.7258 19 .1 283.5294 286.5218 59.6904 60.0046 7 I47.4II7 43-0399 .2 289.5299 60.3187 .8 149.5716 43-3541 3 292.5536 606329 9 I5I.747I 43.6682 4 295-593I 60.947 14 153-9384 43.9824 5 298.6483 61.2612 .1 156.1454 44.2966 .6 301.7193 6i.5754 .2 158.3681 446107 -7 304.806 61.8895 3 160.6064 44.9249 .8 307.9082 62.2037 4 162.8605 45239 9 311.0263 62.5178 5 165.1303 45-5532 20 314.16 62.832 .6 167.4159 45.8674 .1 317.3094 63.1462 7 169.7171 46.1815 .2 320 4746 63.4603 .8 172034 46.4957 .3 63.7745 9 174.3667 46.8098 4 326.8521 64.0886 15 176.715 47.124 5 330.0643 64.4028 .1 179.0791 47.4382 .6 333.2923 64.717 .2 181.4588 47.7523 7 336.536 65.0311 3 183.8543 48.0665 .8 339-7955 65.3453 4 186.2655 48.3806 9 343.0706 65-6594 5 188.6924 48.6948 21 346.3614 65.9736 .6 I9I.I349 49.009 .1 349.6679 66.2878 is 193.5932 196.0673 49.3231 49.6373 .2 3 352.9902 356.3281 66.6019 66.9161 9 108.557 49-95 14 -4 359.6818 67.2302 16 201.0624 50.2656 5 3630511 67.5444 .1 203.5835 50.5797 .6 366.4362 67.8586 .2 2O6.I2O4 50.8939 7 369.837 68.1727 3 208.6729 51.2081 .8 373-2535 68.4869 -4 211.2412 5L5222 9 376.6857 68.801 AREAS AND CIRCUMFERENCES OF CIRCLES. 245 DlAM. ARIA. ClRCTJM. DlAM. ASIA. ClRCUM. 22 380.1336 69.1152 5 593-9587 86.394 .1 69.4294 .6 598.2863 86.7082 .2 387.0765 69.7435 7 602.6296 87.0223 3 390.5716 70.0577 .8 606.9885 87.3365 4 394.0823 70.3718 9 611.3632 87.6506 5 397.6087 70.686 28 6I5-7536 87.9648 .6 401.1509 71.0002 .1 620.1597 88.279 7 404.7088 7L3I43 .2 624.5815 88.59 3 I .8 408.2823 71.6285 3 629.019 9 411.8716 71.9426 4 633.4722 89.2214 2 3 415.4766 72.2568 5 637.9411 89.5356 .1 419.0973 72.571 .6 642.4258 89.8498 .2 422.7337 72.8851 7 646.9261 90.1639 .3 426.3858 73-1993 .8 651.4422 90.4781 4 430.0536 73-5I34 9 655-9739 9O.7922 5 433-7371 73.8276 29 660.5214 9I.IO64 .6 437.4364 74.I4I8 .1 665.0846 91.4206 7 441.1513 74-4559 .2 669.6635 9 J -7347 .8 9 444.882 448.6283 74.7701 75.0842 3 4 674.258 678.8683 92.0489 92.363 24 452.3904 75-3984 5 683.4943 92.6772 .1 456.1682 75.7126 .6 688.1361 92.9914 .2 459.9617 76.0267 7 692.7935 93-3055 3 463.7708 76.3409 .8 697.4666 93.6197 4 467-5957 76.655 9 702.1555 93-9338 47I-4363 76.9692 30 706.86 94.248 .6 475.2927 77-2834 .1 711.5803 94.5622 7 479.1647 77-5975 .2 716.3162 94.8763 .8 483.0524 77.9117 3 721.0679 95-I905 -9 486.9559 78.2258 4 725-8353 95-5046 25 490.875 ^8.54 5 730.6183 95.8188 .1 494.8099 78-8542 .6 735.4171 96.133 .2 498.7604 79.1683 7 740.2316 96.4471 3 502.7267 79-4825 .8 745.0619 96.7613 4 506.7087 79.7966 9 749.9078 97-0754 510.7063 80.1108 754.7694 97.3896 .6 514.7196 80.425 .1 759.6467 97.7038 7 518.7488 80.7391 .2 764.5398 98.0179 .8 522.7937 81.0533 3 769.4485 98.3321 -9 526.8542 81.3674 4 774-373 98.6462 26 530.9304 81.6816 5 779-3I3I 98.9604 .1 535.0223 81.9958 .6 784.269 99.2746 .2 539- J 3 82.3099 7 789.2406 99-5887 3 543-2533 82.6241 .8 794.2279 99.9029 4 547.3924 82.9382 9 799.2309 100.217 :1 7 555-7176 559.9038 83.2524 83.5666 83.8807 32 .1 .2 804.2496 809.284 814.3341 100.5312 100.8454 101.1595 .8 564.1057 84.1949 .3 819.4 101.4737 '9 568.3233 84.509 4 824.4815 101.7878 27 572.5566 84.8232 5 829.5787 102.102 .1 576.8056 85.1374 .6 834.6917 102.4162 .2 581.0703 85-4515 7 839.8204 102.7303 -3 585.3508 85-7657 .8 844.9647 : 103.0445 4 589.6469 86.0798 9 850.1248 ! 103.3586 246 AREAS AND CIRCUMFERENCES OF CIRCLES. DlAM. AREA. CincuM. DlAM. AREA. Cinctrat. 33 855.3006 103.6728 5 1164.1591 120.9516 .1 860.4921 103.987 .6 1170.2146 121.2658 .2 865.6993 104.3011 7 1176.2857 121.5799 .3 870.9222 104.6153 .8 1182.3726 121.8941 4 876.1608 104.9294 9 1188.4751 122.2082 5 881.4151 105.2436 39 1194.5934 122.5224 .6 886.6852 105.5578 .1 1200.7274 122.8366 7 891.9709 105.8719 .2 1206.8771 123.1507 .8 897.2724 106.1861 3 1213.0424 123.4649 -9 902.5895 106.5002 4 1219.2235 123.779 34 907.9224 106.8144 5 1225.4203 124.0932 .1 913.271 107.1286 .6 1231.6329 124.4074 .2 918.6353 107.4427 7 1237.8611 124.7215 3 924.0152 107.7569 .8 1244.105 125-0357 4 929.4109 108.071 9 1250.3647 125.3498 5 934.8223 108.3852 40 1256.64 125.664 .6 940.2495 108.6994 .1 1262.9311 125.9782 7 945.6923 109.0135 .2 1269.2378 126.2923 .8 951.1508 109.3277 3 1275.5603 126.6065 9 956.6251 109.6418 4 1281.8985 126.9206 35 962.115 109.956 5 1288.2523 127.2348 .1 967.6207 IIO.27O2 .6 1294.6219 127.549 .2 973.142 110.5843 7 1301.0072 127.8631 .3 978.6791 110.8985 .8 1307.4083 128.1773 4 984.2319 III.2I26 9 I3I3825 128.4914 5 989.8003 III.5268 4i 1320.2574 128.8056 .6 995.3845 111.841 .1 1326.7055 129.1198 !s 1000.9844 1006.6001 112.1551 112.4693 .2 3 I333- 16 94 1339.6489 129.4339 129.7481 9 1012.2314 112.7834 4 1346.1442 130.0622 36 1017.8784 113.0976 5 1352-6551 130.3764 .1 1023.5411 113.4118 .6 1359.1818 130.6906 .2 1029.2196 113.7259 7 1365.7242 131.0047 3 1034.9137 1 14.0401 .8 1372.2823 *3*'3*&9 4 1040.6236 "4.3542 9 1378.8561 131-633 1046.3491 114.6684 42 1385-4456 131.9472 '.8 1052.0904 1057.8474 1063.6201 114.9826 115.2967 115.6109 .1 .2 3 1392.0508 1398.6717 1405.3084 132.2614 132.5755 132.8897 9 1069.4085 II5.925 4 1411.9607 133.2038 37 1075.2126 116.2392 5 1418.6287 i33-5i8 .1 1081.0324 "6.5534 .6 1425.3125 133.8322 .2 1086.8679 116.8675 7 1432.012 134.1463 3 1092.7192 117.1817 .8 1438.7271 v i 34.4605 4 1098.5861 117.4958 9 1445.458 134.7746 5 1104.4687 117.81 43 1452.2046 135.0888 .6 7 1110.3671 IIl6.28l2 118.1242 118.4383 .1 .2 1458.9669 1465.7449 135.403 I 35-7 I 7 I .8 9 II22.2IO9 1128.1564 118.7525 119.0666 3 4 1472.5386 1479.348 136.0313 136.3454 38 1134.1176 119.3808 5 1486.1731 136.6596 .1 1140.0945 119.695 .6 1493.014 136.9738 .2 1146.0871 120.0091 7 1499.8705 137.2879 3 1152.0954 120.3233 .8 1506.7428 137.6021 4 1158.1194 120.6374 9 1513-6307 137.9162 AREAS AND CIRCUMFERENCES OF CIRCLES. 247 DlAM. AREA. ClRCCM. DlAM. ARIA. ClRCTM. 44 1520.5344 138.2304 5 1924.4263 155.5092 .1 I527-4538 138.5446 .6 1932.2097 155.8234 .2 1534.3889 138.8587 -7 1940.0087 156.1375 3 1 54 1 -3396 139.1729 .8 1947.8234 156.4517 4 1548.3061 139.487 9 1955.6539 156.7658 5 1555-2883 139.8012 50 1963.5 I57-08 .6 1562.2863 I40.II54 .1 1971.3619 157.3942 7 1569.2999 140.4295 .2 1979.2394 157-7083 .8 1576.3292 140.7437 3 1987.1327 I58.O225 9 I583.3743 141.0578 4 1995.0417 158.3366 45 I590-435 141.372 -5 2002.9663 158.6509 .1 I597.5H5 141.6862 .6 2010.9067 158.965 .2 1604.6036 142.0003 7 2018.8628 159.2791 3 1611.7115 142.3145 .8 2026.8347 1 59-5933 4 1618.8351 142.6286 9 2034.8222 159.9074 5 1625.9743 142.9428 51 2042.8254 160.2216 .6 1633.1293 I43.257 .1 2050.8443 160.5358 7 1640.3 I43.57H .2 2058.879 160.8499 .8 1647.4865 I43.8853 3 2066.9293 161.1641 9 1654.6886 144.1994 4 2074.9954 161.4782 46 1661.9064 144.5136 5 2083.0771 161.7924 .1 1669.1399 144.8278 .6 2091.1746 162.1066 .2 1676.3892 I45.I4I9 7 2099.2878 162.4207 3 1683.6541 145.4501 .8 2107.4167 162.7349 4 1690.9348 145.7702 9 2H5-56I3 163.049 .5 1698.2311 146.0844 52 2123.7216 163.3632 .6 I705-543 2 146.3986 .1 2131.8976 163.6774 7 1712.871 146.7127 .2 2140.0893 1639915 .8 1720.2145 147.0269 -3 2148.2968 164.3057 9 I727-5737 I47'34I 4 2156.5199 164.6198 47 1734.9486 I4 7 .6552 5 2164.7587 164.934 .1 1742.3392 147.9694 .6 2173.0133 165.2482 .2 1749-7455 148.2835 7 2181.2836 165.5623 3 1757.1676 148.5977 .8 2189.5695 165.8765 4 1764.6053 I48.9II8 9 2197.8712 166.1906 5 1772.0587 149.226 53 2206.1886 166.5048 .6 I779.5279 149.5402 .1 2214.5217 166.819 7 1787.0128 149.8543 .2 2222.8705 167.1331 .8 1794-5133 I50.I685 .3 2231.235 167.4473 9 1802.0296 150.4826 4 2239.6152 167.7614 48 1809.5616 150.7968 5 2248.0111 168.0756 .1 1817.1093 I5I.III .6 2256.4228 168.3898 .2 1824.6727 I5L425I 7 2264 8501 168.7039 3 1832.2518 I5L7393 .8 2273.2932 169.0181 4 1839.8466 152.0534 9 2281.7519 169.3322 5 1847.4571 152.3676 54 2290.2264 169.6464 .6 1855.0834 I52.68l8 .1 2298.7166 169.9606 7 1862.7253 152.9959 .2 2307.2225 170.2747 .8 1870.383 I53.3IOI 3 23*5.744 170-5889 9 1878.0563 153.6242 4 2324.2813 170.903 49 1885.7454 I53-9384 5 2332-8343 171.2172 .1 1893.4502 154.2526 .6 2341.4031 171-5314 .2 1901.1707 154.5667 7 2349.9875 171.8455 3 1908.9068 154.8809 .8 2358-5876 172.1597 4 1916.6587 155.195 9 2367-2035 372.4738 248 AREAS AND CIRCUMFERENCES OF CIRCLES. DlAM. ABBA. ClRCUM. DlAM. ABBA. ClRCUM. 55 2375-835 172.788 5 2874.7603 190.066 .1 2384.4823 173.1022 .6 2884.2715 190.381 .2 2393-I452 I73-4I63 7 2893.7984 190.695 3 2401.8239 I73-7305 .8 2903.3411 191.005 4 2410.5183 174.0446 9 2912.8994 I9I.32J 5 2419.2283 174.3588 61 2922.4734 191.637 .6 7 2427.9541 2436.6957 174.673 174.9871 .1 .2 2932.0631 2941.6686 I9I.95I I92.26C .8 2445.4529 I75.3OI3 .3 2951.2897 I92.58C 9 2454.2258 I75-6I54 4 2960.9266 192.894 56 2463.0144 I75.92Q6 5 2970.5791 I93.20S .1 2471.8187 176.2438 .6 2980.2474 I93-522 .2 2480.6388 176.5579 7 2989.9314 193.83^ 3 2489.4745 176.8721 .8 2999.6311 I94.I5C 4 2498.326 I77.I862 9 3009.3465 194.46 = .5 2507.1931 177.5004 62 3019.0776 194.779 .6 2516.076 177.8146 .1 3028.8244 195.09: .7 2524.9736 178.1287 .2 3038.5869 195.40^ .8 9 2533.8889 2542.8189 178.4429 178.757 -3 4 3048.3652 3058.1591 195.72] 196.03 = 57 2551.7646 179.0712 5 3067.9687 106.35 .1 2560.726 I79-3854 .6 3077.7941 196.664 .2 2569-703I 179.6995 7 3087.6341 196.97? 3 2578.696 180.0137 .8 3097.4919 197.292 4 2587.7045 180.3278 9 3107.3644 5 2596.7287 180.642 63 3117.2526 I97-92C .6 2605.7687 180.9562 .1 3127.1565 i98.23 v 7 2614.8244 l8l.27O3 .2 3137.0761 i98.54< .8 2623.8957 181.5845 .3 3147.0114 198.86; -9 2632.9828 I8l.8o86 4 3156.9624 199.17- 58 2642.0856 182.2128 -5 3166.9291 199.49 .1 2651.2041 182.527 .6 3176.9116 199.80^ .2 2660.3383 I82.84II 7 3186.9097 200. IK 3 2669.4882 183.1553 .8 3196.9236 200.43/ 4 2678.6538 183.4694 9 3206.9531 200. 74* '.6 2687.8351 2697.0322 183.7836 184.0978 64 .1 3216.9984 3227.0594 201 .06; 201. 37< 7 2706.2449 184-4119 .2 3237.1361 20I.6gj( .8 27 J 5-4734 184.7261 3 3247.2284 202.00, 9 2724.7175 I85.O4O2 4 202.3 1 < 59 2733-9774 185.3544 5 3267.4603 202 .63, .1 2743-253 185.6686 .6 3277.5999 202.94 .2 2752.5443 185.9827 7 3287.7551 203.26 3 2761.8512 186.2969 .8 3297.9261 203.57. 4 2771.1739 I86.6II -9 3308.1127 20 3 .88< 5 2780.5123 186.9252 65 3318.315 .6 2789.8665 187.2394 .1 3328.5331 204.5 1 < 7 2799.2363 187.5535 .2 3338.7668 204.83 .8 2808.6218 187.8677 .3 3349.0163 205. 1 4< 9 2818.0231 I88.l8l8 4 3359.2815 205. 4cx 60 2827.44 188.496 -5 3369.5623 205.77, .1 2836.8727 I88.8I02 .6 3379'8589 2o6.o8< .2 2846.321 189.1243 7 3390.1712 206.40, 3 2855.7851 189 4385 .8 3400.4993 206.71 4 2865.2649 189.7526 9 3410.843 207.03 AREAS AND CIRCUMFERENCES OF CIRCLES. 249 DlAM. ARIA. ClRCUM. DlAH. AREA. CntctiM. 66 3421.2024 207.3456 5 40I5.l6lI 224.6244 .1 343 I -5775 207.6598 .6 4026.4002 224.9386 .2 3441.9684 207.9739 .7 4037.655 225.2527 3 3452.3749 208.2881 .8 4048.9255 225.5669 -4 3462.7972 208.6022 9 4060.2117 225.881 3473'235i 208.9164 72 4071.5136 226.1952 .6 3483.6888 209.2306 .1 4082.8312 226.5094 7 3494.1582 209.5447 .2 4094.1645 226.8235 .8 3504.6433 209.8589 -3 4105.5136 227.1377 9 3515.1441 210.173 4 4116.8783 227.4518 67 3525.6606 2IO.4872 5 4128.2587 227.766 .1 3536.1928 2IO.8OI4 .6 4 J 39-655 228.0802 .2 3546.7407 2II.II55 7 4151.0668 228.3943 3 3557-3044 211.4297 .8 4162.4943 228.7085 4 3567.8837 211.7438 9 4I73-9376 229.0226 -5 3578.4787 2I2.O58 73 4185.3966 229.3368 .6 -7 3589.0895 3599.716 212.3722 212.6863 .1 .2 4196.8713 4208.3617 229.651 2299651 .8 9 3610.3581 3621.016 213.0005 213.3146 3 4 4219.8678 4231.3896 230.2793 230.5934 68 3631.6896 213.0288 -5 4242.9271 2309076 .1 3642.3789 213.943 .6 4254.4804 231.2218 .2 3653-0839 214.2571 7 4266.0493 231 5359 3 3663.805 214.5713 .8 4277.634 231.8501 4 3674-54I 214-8854 9 4289.2343 232.1642 '.6 7 3685.2931 3696.061 3706.8445 215.1996 215.5138 215.8279 74 .1 .2 4300.8504 4312.4822 4324.1297 232.4784 232.7926 233.1067 .8 9 3717.6438 3728.4587 2I6.I42I 216.4562 3 4 4335.7928 4347-47I7 233.4209 233-735 69 3739.2894 216.7704 -5 4359.1663 234.0492 .1 3750.1358 217.0846 .6 4370.8767 234.3634 .2 3760.9979 2X7.3987 7 4382.6027 234-6775 3 3771.8756 217.7129 .8 4394-3444 234.9917 -4 3782.7691 218.027 9 4406.1019 235-3058 3793.6783 218.3412 7er 4417 875 235.62 .6 3804.6033 218.6554 lO 4429.6639 -7 3815.5439 218.9695 .2 4441.4684 236.2483 .8 3826.5002 219.2837 3 4453.2887 236.5625 9 3837.4722 219.5978 4 4465.1247 236.8766 70 3848.46 219.912 5 4476.9763 237.1908 ,i 3859.4635 22O.2262 .6 4488.8437 237-505 .2 3870.4826 220.5403 7 4500.7268 237.8191 3 3881.5175 220.8545 .8 4512.6257 238.1333 4 3892.5681 221.1686 9 4524.5402 238.4474 5 3903.6343 221.4828 76 4536.4704 238.7616 .6 3914.7163 221.797 .1 4548.4163 239.0758 7 3925.814 222.IIII .2 4560.378 239-3899 .8 3936.9275 222.4253 3 4572.3553 239.7041 9 3948.9566 222.7394 4 4584.3484 240.0182 7 1 3959.2014 223.0536 5 4596.3571 240.3324 .1 3970.3619 223.3678 .6 4608.3816 240.6466 .2 3981.5382 223.6819 7 4620.4218 240.9607 3 3992.7301 223.9961 .8 4632.4777 241.2749 4 4003.9378 224.3IO2 9 4644.5493 241.589 AREAS AND CIRCUMFERENCES OF CIRCLES. DlAM. ARIA. ClRCUM. DtAM. AREA. ClRCUM. 77 4656.6366 241.9032 5 5345.6287 259.182 .1 4668.7396 242.2174 .6 5358.5957 259.4962 .2 4680.8583 242.5315 7 5371.5784 259.8103 .3 4692.9928 242.8457 .8 5384.5767 200.1245 4 4705.1429 I 243.1598 9 5397.5908 260.4386 -5 4717.3087 243.474 83 5410.6206 200.7528 .6 4729.4903 243.7882 .1 5423.6661 261.067 7 4741.6876 244.1023 .2 5436.7273 26l.38ll .8 4753.9005 244.4165 3 54498042 261.6953 9 4766.1292 244.7306 4 5462.8968 262.0094 78 4778.3736 245.0448 5 5476.0051 262.3236 .1 4790.6337 245-359 .6 5489.1292 262.6378 .2 4802.9095 245.6731 7 5502.2689 262.9519 3 4815.201 245.9873 .8 5515.4244 263.2661 4 4827.5082 246.3014 9 5528.5955 263.5802 5 .6 4839.8311 4852.1698 246.6156 246.9298 84 .1 5541.7824 5554-985 263.8944 264.2086 7 4864.5241 247.2439 .2 5568.2033 264.5227 .8 4876.8942 247.5581 3 5581.4372 264.8369 9 4889.2799 247.8722 4 5594 6869 265.151 79 4901.6814 248.1864 5 5607.9523 265.4652 .1 4914.0986 248.5006 .6 5621.2335 265.7794 .2 4926.5315 248.8147 7 5634-5303 266.0935 3 4938.98 249.1289 .8 5647-8428 266.4077 4 4951.4443 249.443 -9 5661.1711 266.7218 5 4963.9243 249.7572 85 5674.515 267.036 .6 7 4976.4201 4988.9315 250.0714 250.3855 .1 .2 5687.8747 57 OI -25 267.35O2 267.6643 .8 5001.4586 250.6997 3 5714.6411 267.9785 9 5014.0015 251.0138 4 5728.0479 268.2926 80 5026.56 251.328 5 5741.4703 268.6o68 .1 5039.1343 251.6422 .6 5754.9085 268.921 .2 5051.7242 2 5 J " 9563 7 5768.3624 209.2351 3 5064.3299 252.2705 .8 5781.8321 269.5493 4 5076.9513 252.5846 9 5795-3 I 74 269.8634 5 5089.5883 252.8988 86 5808.8184 270.1776 .6 5102.2411 253-2I3 .1 5822.3351 270.4918 7 5114.9096 253-527I .2 5835.8676 270.8059 .8 5127.5939 253.8413 3 5849.4157 27I.I2OI 9 5140.2938 254.1554 4 5862.9796 271.4342 81 5153.0094 254.4696 5 5876-5591 271.7484 .1 5165.7407 254.7838 .6 5890.1544 272.0626 .2 5178.4878 255.0979 7 5903.7654 272.3767 -3 5191.2505 255.4121 .8 5917.3921 272.6909 4 5204.0289 255.7262 -9 5931.0345 273.005 .6 5216.8231 5229.633 256.0404 256.3546 8 7 .1 5944.6926 5958.3644 273.3192 7 5242.4586 256.6687 .2 5972.0559 2739475 .8 5255.2999 256.9829 3 5985.7612 274.2617 9 5268.1569 257.297 4 5999.4821 274.5758 82 5281.0296 257.6lI2 5 6013.2187 274.89 .1 5293.918 257.9254 .6 6026.9711 275.2042 .2 5306.8221 258.2395 7 6040.7392 275.5183 3 53I9-742 258.5537 .8 6054.5229 275 8325 4 5332-6775 258.8678 9 6068.3224 276.1466 AEEAS AND CIRCUMFERENCES OF CIRCLES. 251 DlAM. AREA. ClKCUM. DlAM. AREA. ClECUM. 88 6082.1376 276.4608 5 6866.1631 293.7396 .1 6095.9685 276.775 .6 6880.858 294.0538 .2 6109.8151 277.0891 7 6895.5685 294.3679 .3 6123.6774 277-4033 .8 6910.2948 294.6821 4 6I37-5554 277.7174 9 6925.0367 294.9962 .6 6151.4491 6165.3586 278.0316 278.3458 $4 .1 6939-7944 6954.5678 295.3104 295.6246 i 6179.2837 6193.2246 278.6599 278.9741 .2 3 6969-3569 6984.1616 295.9387 296.2529 -9 6207.1811 279.2882 4 6998.9821 296.567 89 6221.1534 279.6024 5 7013.8183 296.8812 .1 6235 1414 279.9166 .6 7028.6703 297.1954 .2 6249 1451 280.2307 -7 7043-5379 297-5095 3 6263.1644 280.5449 .8 7058.4212 2978237 4 6277.1995 280.859 9 7073.3203 208.1378 .6 6291.2503 6305 -3 l6 9 28I.I732 281.4874 95 .1 7088.235 7103.1655 298.452 298.7662 7 63I9-399I 28I.80I5 .2 299.0803 .8 9 6333497 6347.6107 282.1157 282.4298 3 4 7I33-0735 7148.0511 299-3945 299.7086 9 6361.74 282.744 5 7163.0443 300.0228 .1 6375.8851 283.0582 .6 7178.0533 300.337 .2 6390.0458 283.3723 7 7193.078 300.6511 3 6404.2223 283.6865 .8 7208.1185 300.9653 4 64184144 284.0006 9 7223.1746 301.2794 5 .6 6432.6223 6446.8459 284.3148 284.629 96 .1 7238.2464 72533339 301.5936 301.9078 !s 9 6461.0852 6475.3403 6489.61 1 284.9431 285.2573 285.5714 .2 3 4 7268.4372 7283.5561 7298.6908 302.2219 302.5361 302.8502 9 1 6503.8974 285.8856 5 7313.8411 303.1644 .1 6518.1995 286.1998 .6 7329.0072 303.4786 .2 6532.5174 286.5139 7 7344.189 303.7927 3 6546.8509 286.8281 .8 7359-3865 3041009 4 6561.2002 287.1422 9 7374-5997 304.421 7 65755651 6589.9458 6604.3422 287.4564 287.7706 288.0847 97 .1 .2 7389.8286 7405.0732 7420.3335 304-7352 305.0494 305.3635 .8 6618.7543 288.3989 3 7435.6096 305-6777 9 6633.1821 288.713 4 7450.9013 305.9918 92 6647.6256 289.O272 5 7466.2087 306.306 .1 6662.0848 289.3414 .6 7481.5319 306.6202 .2 6676.5598 289.6555 7 7496.8708 306.9343 3 6691.0504 289.9697 .8 7512.2253 307.2485 4 6705-5567 290.2838 9 7527-5956 307.5626 5 6720.0787 290.598 98 7542.9816 307.8768 .6 6734.6165 29O.9I2I .1 7558.3833 308.J9I 7 674917 291.2263 .2 7573.8007 308.5051 .8 6763.739 1 291.5405 3 7589.2338 3O8.8l93 9 6778 324 291.8546 4 7604.6826 309.1334 93 6792.9246 292.1088 5 7620.1471 309.4476 .1 6807.5409 292.483 .6 7635.6274 .2 6822.1729 292.7971 7 7651.1233 310.0759 3 6836.8206 , 293.1113 .8 7666.635 310.3901 4 6851.484 293.4254 9 7682.1623 310.7042 252 AREAS AND CIRCUMFERENCES OF CIRCLES. DlAM. ABBA. ClRCUM. DlAM. AREA. CTRCUM. 99 ,i .2 3 4 7697.7054 7713.2642 7728.8337 7744.4288 7760.0347 31I.OI84 311.3326 311.6467 311.9009 312.275 .6 '.8 9 7775.6563 7791.2937 7806.9467 7822.6154 7838.2999 312.5892 312.9034 3I3-2I75 3 I 353 I 7 313.8458 To Compute Area or Circumference of a Diameter greater than, any in preceding Table. See Rules, pages 235-6 and 241-2. Or, If Diameter exceeds 100 and is less than 1001. Put a decimal point, and take out area or circumference as for a Whole Number by removing decimal point, if for an area, two places to right , and if for a circumference, one place. EXAMPLE. What is area and what circumference of a circle 967 feet in diame- ter? Area of 96.7 is 7344.189; hence, for 967 it is 734 418.9; and circumference of 96.7 is 303.7927, and for 967 it is 3037.927 To Compute Area and Circumference of a Circle by JJog- aritlims. See Rules, pages 236, 242. .A^reas and. Circumferences of Circles. FROM i TO 50 FEET (advancing by an Inch). OR, FROM i TO 50 INCHES (advancing by a Twelfth). DlAM. AREA. ClRCUM. DlAM. AREA. ClKCUM. Feet. Feet. Feet. Feet. i/fc .7854 3.1416 3ft- 7.0686 9.4248 I .9217 3-4034 I 7.4668 9,6866 2 1.069 3.6652 2 7.8758 9.9484 3 1.2272 3.927 3 8.2958 IO.2IO2 4 1.303 4.1888 4 8.7267 10.472 5 L5763 4.4506 5 9.1685 10.7338 6 1.7671 4.7124 6 9.62II 10.9956 7 1.969 4.9742 7 10.0848 11.2574 8 2.1817 5.236 8 IO -5593 11.5192 9 2.4053 5.4978 9 11.0447 II.78I 10 2.6398 5.7596 10 11.541 12.0428 ii 2.8853 6.O2I4 ii 12.0483 12.3046 2 ft. 3.1416 6.2832 4.A 12.5664 12.5664 I 3.4088 6545 i T 30955 12.8282 2 3 .68 7 6.8068 2 T 3"6354 13.09 3 3.9761 7.0686 3 14.1863 I3.35I8 4 4.2761 7.3304 4 14.7481 13.6136 5 4.5869 7.5922 5 15.3208 I3-8754 6 4.9087 7.854 6 I5-9043 14.1372 7 5.24I5 8.1158 7 16.4989 14.499 8 5.5852 8.3776 8 17.1043 14.6608 9 5.9396 8.6394 9 17.7206 14.9226 10 6.305 8.9012 10 18.3478 15.1844 ii 6.6814 9.163 ii 18.9859 15.4462 AKEAS AND CIKCUMFEEEXCES OF CIRCLES. 253 DiAM ARIA. ClRCVH. DlAM. AREA. ClRCUM. Feet. Feet. Fr 3 t. Feet. 5A I9-635 I5-708 6 70.8823 29.8452 I 20.2949 15-9698 7 72.1314 30.107 2 20.9658 16.2316 8 7339 I 3 30.3688 3 21.6476 16.4934 9 74.6621 30.6306 4 22.3403 16.7552 10 75-9439 30.8924 5 23-0439 17.017 ii 77-2365 31.1543 6 8 9 10 ii 23-7583 24.4837 25.22 25.9673 26.7254 27.4944 17.2788 17.5406 17.8024 18.0642 18.326 18.5878 10 ft. i 2 3 4 5 78.54 79- 8 545 81.1798 82.5161 83-8633 85.2214 3I-4I6 31.6778 3I-9396 32.2014 32.4632 32.725 3ft. 28.2744 18.8496 6 86.5903 32.9868 1 29.0653 I9.III4 7 87.9703 33.2486 2 29.867 I9-3732 8 89.3611 ' 33-5I04 3 30.6797 19.635 9 90.7628 33-7722 4 3I-5033 19.8968 10 92.1754 34-034 5 32.3378 20.1586 ii 93-599 34-2958 6 I 9 10 ii 33-I83I 34-0394 34.9067 35.7848 36.6738 37-5738 20.4204 20.6822 20.944 21.2058 21.4676 21.7294 ii ft. i 2 3 4 5 95-0334 96.4787 97-935 99.4022 100.8803 102.3693 34.5576 34.8194 35.o8l2 35-343 35-6048 35-8666 7A 38.4846 21.9912 6 103.8691 36.1284 I 39.4064 22.253 7 105.38 36.3902 2 40.339 22.5148 8 106.9017 36.652 3 41.2826 22.7766 9 108.4343 36.9138 4 42.2371 23.0384 10 109.9778 37-1756 5 43.2025 23.3002 ii 111.5323 374374 6 7 8 44.1787 45.1659 46.164^ 23.562 23.8238 24.0856 12 ft. I 2 113.0976 114.6739 116.261 37-6992 37.o6i 38.2228 9 10 ii 47-I73I 48.193 49.2238 24-3474 24.6092 24.871 3 4 5 117.8591 1 19.468 121.088 38.4846 38.7464 39.0082 9ft. 50.2656 25.1328 6 122.7187 39-27 I 51-3183 25.3946 7 124.3605 39-53 I 8 2 52.3818 25.6564 8 126.0131 39-7936 3 534563 25.9182 9 127.6766 40.0554 4 54.5417 26.18 10 129.351 40.3172 5 55-638 26.4418 ii 131.0366 40.579 6 8 9 10 ii 56.7451 57-8632 58-9923 60.1322 61.283 62.4448 26.7036 26.054 27.2272 27.489 27.7508 28.0126 13 ft- i 2 3 4 5 132.7326 134.4398 136.1578 137.8868 139.6267 141.3774 40.8408 41.1026 41.3644 41.6262 41.888 42.1498 w* 63.6174 28.2744 6 I43-I39I 42.4116 i 64.801 28.5362 7 144.9117 42.6734 2 65.9954 28.798 8 146.6953 42.9352 3 67.2008 29.0598 9 148.4897 43-197 4 68.417 29.3216 10 150.295 43.4588 5 69.6442 29.5834 ii 152-1113 43-7206 254 AREAS AND CIRCUMFERENCES OP CIRCLES. DlAM. AKKA. ClRCUM. DlAM. AREA. ClKCUM. Feet. Feet. Feet. Feet. 14 ft. I53-9384 43.9824 6 268.8031 58.1196 I 155.7764 44.2442 7 271.2302 58.3814 2 157.6254 44.506 8 273.6683 58.6432 3 I59-4853 44.7678 9 276.1172 58.905 4 161.3561 45.0296 10 278.577 59.1668 5 163.2378 45.2914 ii 281.0477 59.4286 6 8 9 165.1303 167.0338 I70-8736 45-5532 45-8I5 46.0768 46.3386 19 ft. i 2 283.5294 286.0219 288.5255 59.6904 59-9522 60.214 10 ii 172.8098 I74-7569 46.6004 46.8622 4 5 293.5651 296.IOI2 60.7376 60.9994 15 ^ 176.715 47.124 6 298.6483 6l.20I2 i 178.684 7 301.2064 61.523 2 180.6638 47.6476 8 303-7753 61.7848 3 182.6546 47-9094 9 306.3551 62.0466 4 184.6563 48.1712 10 308.9458 62.3084 5 186.6689 48.433 ii 3U-5475 62.5702 6 7 188.6924 190.7267 48.6948 48.9566 20 ft. I 314.16 316.7834 62.832 63.0938 8 9 10 ii 192.7721 194.8283 196.8954 198.9734 49.2184 49.4802 49-742 50.0038 2 3 4 5 319.4178 322.0631 324.7193 327.3864 63.3556 63.6174 63.8792 64.141 i6ft. 20I.0624 50.2656 6 330.0643 64.4028 i 203.1622 50.5274 7 332.7532 64.6646 2 205.273 50.7892 8 335-4531 64.9264 3 207.3947 9 338.1638 65.1882 4 209.5273 51.3128 10 340.8854 6545 5 211.6707 5L5746 ii 343.6l8 65.7118 6 8 9 10 ii 213.8252 215.9904 218.1667 220.3538 222.5518 224.7607 51.8364 52.0982 52.36 52.6218 52-8836 53- I 454 21 ft. I 2 3 4 5 346.3614 349- "57 351.881 357.4442 360.2422 65-9736 66.2354 66.4972 66.759 67.0208 67.2826 17 A 226.9806 53-4072 6 363-0511 67.5444 i 229.2113 53-669 7 365.8709 67.8062 2 2 3I-453 53-9308 8 368.7017 68.068 3 233-7056 54.1926 9 371-5433 68.3298 4 235.9691 54-4544 10 374-3958 68.5916 5 238.2434 54.7162 ii 377.2592 68.8534 6 8 9 240.5287 242.8249 245.1321 247.4501 55-5oi6 55-7634 22ft. I 2 3 380.1336 383.0188 385915 388.8221 69.1152 69.377 69.6388 69.9006 19 249.779 56.0252 7QI 74. 7O.l62J II 252.II88 56.287 5 394.6689 70.4242 '8 ft. 254.4696 56.5488 6 397.6087 70.686 i 256.8312 56.8106 7 400.5594 70.9478 2 259.2038 57.0724 8 403.5211 71.2096 3 261-5873 57-3342 9 406.4936 7 T 47 J 4 4 263.9817 57-596 10 409-477 5 266.3869 57-8578 ii 412.4713 71-995 AREAS AND CIRCUMFERENCES OF CIRCLES. 255 DIAM. AREA. ClROCM. DIAM. AREA. ClRCUM. Feet. Feet. Feet. Feet. 23 A 4I5-4766 72.2568 6 593-9587 86.394 I 418.4927 72.5186 7 597.5639 86.6558 2 421.5198 72.7804 8 601.18 86.9176 3 424.5578 73.0422 9 604.8071 87.1794 4 427.6067 73'304 10 608.445 87.4412 5 430.6664 73.5658 II 612.0938 87.703 6 433-7371 436.8187 73.8276 740894 28ft. I 6i5-7536 619.4242 87.9648 88.2266 8 439-9J 74-3512 2 623.1058 88.4884 9 10 ii 443.0147 446.129 449.2542 74-6I3 74.8748 75-I366 3 4 5 626.7983 630.5016 634.2159 88.7502 89.012 89.2738 24ft. 452.3904 75.3984 6 637.9411 89.5356 I 455-5374 75.6602 7 641.6772 89-7974 2 458.6954 75.922 8 645-4243 90.0592 3 461.8643 76.1838 9 649.1822 90.321 4 465.044 76.4456 10 652.951 90.5828 5 468.2347 76.7074 ii 656-7307 90.8446 6 8 9 10 ii 47I-4363 474.6488 477 8723 481.1066 484.3518 487.6076 76.9692 77.231 77.4928 77-7546 78.0164 78.2782 29 A i 2 3 4 5 660.5214 664.3229 668.1354 671.9588 675.7931 679.6382 91.1064 91.3682 91.63 91.8918 92.1536 92.4154 25 A 490.875 78.54 6 683.4943 92.6772 i 494.1529 78.8018 7 687.3613 92.939 2 497.4418 79.0636 8 691.2393 93.2008 3 500.7416 79-3254 9 695.1281 93.4626 4 504.0523 79.5872 10 699.0278 937244 r 507-3738 79.849 ii 702.9384 93-9862 6 9 10 ii .510.7063 514.0485 517.404 520.7693 524.1454 527.5324 80.1108 80.3726 80.6344 80.8962 81.158 81.4198 30 A i 2 3 4 5 706.86 710.7924 7I4.7358 718.6901 722.6553 726.6313 94.248 94.5098 94.7716 95.0334 95.2952 95.557 26ft. 530-9304 816816 6 730.6183 958188 I 534-3397 81.9434 7 734.6162 96.0806 2 537-759 82.2052 8 738-625! 96.3424 3 4 5 541.1897 544-6313 548.0837 82.467 82.7288 82.9906 9 10 ii 742.6448 746.6754 750 7164 96.6042 96.866 97.1278 6 1 9 10 ii 27 A 55I.547I 555.0214 558.5066 562.0028 565.5098 569.0277 572.5566 83-2524 83.5142 83-776 84.0378 84.2996 84.5614 84.8232 3'/<. I 2 3 4 754.7694 758-8327 762907 766.9922 771.0883 775.1952 779-3I3I 97.3896 97.6514 97.9132 98.175 98.4368 98.6986 98.9604 i 576.0963 85.085 7 783.4419 99-2222 2 579.6467 85.3468 8 787.5817 99.484 3 4 583.2086 586.781 85.6086 85.8704 9 10 79^7323 795.8938 99-7458 100.0076 5 590.364^ 86.1322 II 800.0662 100.2694 256 AREAS AND CIRCUMFERENCES OF CIRCLES. DlAM. AREA. ClRCtTM. DlAM. AREA. ClRCtTM. Feet. Feet. Feet. Feet. J2.A 804.2496 100.5312 6 1046.3491 114.6684 I 808.4439 100.793 7 I05I.I324 114.9302 2 812.649 101.0548 8 1055.9266 II5.I92 3 816.8651 101.3166 9 1060.7318 II5-4538 4 821.092 101.5784 10 1065.5478 II5-7I56 5 825.3299 101.8402 ii 1070.3747 115.9774 6 8 9 10 ii 829.5787 833-8384 838.1091 842.3906 846.683 850.9863 102.102 102.3638 102.6256 102.8874 103.1492 103-411 37 A i 2 3 4 5 1075.2126 1080.0613 1084.921 1089.7916 1094.6731 10995654 116.2392 Il6.5OI 116.7628 117.0246 117.2864 117.5482 33 A 85S-3006 103.6728 6 1 104.4687 II7.8I i 859.6257 103.9346 7 1109.3829 Il8.07l8 2 863.9618 104.1964 8 III4.308 118.3336 3 868.3088 104.4582 9 1119.2441 1185954 4 872.6667 104.72 10 1124.191 1188572 5 877-0354 104.9818 ii 1129.1489 II9.II9 6 8 9 10 ii 881.4151 885.8057 890.2073 894.6197 899.043 903.4772 105.2436 105.5054 105.7672 IO6.O29 106.2908 106.5526 3A i 2 3 4 5 II34.II76 II39.0972 1144.0878 11490893 II54.IOI7 II59.I249 II93808 119.6426 II99044 1 2O l662 120.428 120 6898 34 .A 907.9224 106.8144 6 1164.1591 I20.95I6 i 912.3784 107.0762 7 1169.2042 I2I.2I34 2 916.8454 107.338 8 1174.2603 121.4758 3 921.3233 107.5998 9 1179.3272 121.737 4 925.812 I07.86l6 10 1184.405 121.9988 5 930.3117 108.1234 ii 1189.4937 122.2606 6 8 9 10 ii 934.8223 939-3439 943.8763 948.4196 952.9738 957-5392 108.3852 108.647 108.9088 109.1706 109.4324 109.6942 39A i 2 3 4 5 1194.5934 1199.7039 1204.8254 1209.9578 I2I5.IOI 1220.2552 122.5224 122.7848 123.046 123.3078 123.5696 123.8314 35 A 962.115 109.956 6 1225.4203 124.0932 i 966.7019 110.2178 7 1230.5063 124-355 2 971.2998 110.4796 8 I235-7833 I24.6l68 3 975.9086 110.7414 9 1240.9811 124.8786 4 980.5287 III.0032 10 1246.1898 125.1404 5 985.1588 III.265 ii 1251.4094 125.4022 6 989.8005 994-45 2 7 III.5268 III.7886 40 ft. i 1256.64 I26l.88l4 125.664 125.9258 8 999.116 112.0504 2 1267.1338 126.1876 9 10 ii 1003.7903 10084754 1013.1714 II2.3I22 112.574 112.8358 3 4 5 1272.3971 1277.6712 1282.9563 126.4494 I26.7II2 126.973 16 A 1017.8784 113.0976 6 1288.2523 127.2348 i 1022.5962 "3-3594 7 1293.5592 127.4966 a 1027.325 113.6212 8 1298.877 127.7584 3 1032.0647 113.883 9 1304.2058 128 0202 4 1036.8153 114.1448 10 I309-5454 128.282 5 1041.5767 114.4066 ii 1314.8959 I28.5 43 8 AREAS AND CIRCUMFERENCES OF CIRCLES. 257 DlAli. AREA. ClECCM. DlAM. ABBA. ClRCUM. Feet. Feet. Feet. Feet. 41 ft. 1320.2574 128.8056 6 1625.9743 142.9428 I 1325.6297 129.0674 7 1631.9357 143.2046 2 i33 I - OI 3 129.3292 8 1637.9081 143.4664 3 1336.4072 129.591 9 1643.8913 143.7282 4 1341.8123 129.8528 10 1649.8854 143-99 5 1347.2282 130.1146 ii 1655.8904 144.2518 6 1352.6551 130.3764 46A 1661.9064 144.5136 7 1358.0929 130.6382 i 1667.9332 144-7754 8 1363-5416 I30. 9 2 1673.971 145.0372 9 1369.0013 I3I.l6l8 3 1680.0197 145.299 10 1374.4718 131.4236 4 1686.0792 145.5608 ii I379-9532 131.6854 5 1692.1497 145.8226 42 y*. i 2 3 4 I385-4456 1390.9488 1396.463 1401.9881 1407.5241 131.9472 132.209 132.4708 132.7326 132.9944 6 8 9 10 1698.2311 1704.3195 1710.4267 1716.5408 1722.6658 146.0844 146.3462 146.608 146.8698 147.1316 5 6 14130709 1418.6287 133.2562 I33-5I8 ii 47 A 1728.8017 1734.9486 147-3934 147.6552 7 1424.1974 I33-7798 I 1741.1063 147917 8 9 10 1429.777 1435-3676 1440.969 134.0416 I34-3034 I34-5652 2 3 4 I747-275 I753-4546 I759-645I 148.1788 148.4406 148.7024 ii 1446.5813 134.827 5 1765.8464 148.9642 43 A i 2 3 4 6 8 9 10 ii 1452.2046 1457.8387 1463.4838 1469.1398 1474.8066 1480.4844 1486.1731 1491.8717 I497-5833 I503-3047 1509-037 1514.7802 1350888 I35-3506 135.6124 135.8742 136.136 136.3978 1366596 136.9214 137.1832 137-445 137.7068 137.9686 6 8 9 10 ii 48A i 2 3 4 1772.0587 1778.2819 1784.516 1790.7611 1797.017 1803.2838 1809.5616 1815.8502 1822.1498 1828.4603 1834.7817 1841.1139 1847.4571 149.226 149.4878 149.7496 150.0114 150.2732 150.535 150.7968 151.0586 151.3204 151.5822 151.844 152.1058 152.3676 44 ft. 1520.5344 138.2304 7 I853.8II2 152.6294 i 1526.2994 138.4922 8 1860.1763 152.8912 2 1532.0754 138.754 9 1866.5522 153-153 3 1537.8623 139.0158 10 1872.939 153.4148 4 1543.66 139.2776 ii I879-3367 153.6766 5 1549.4687 139-5394 49 A 1885.7454 1539384 6 1555-2883 139.8012 i 1892.1649 154.2002 7 1561.1188 140.063 2 1898.5954 154462 8 1566.9603 140.3248 3 19050368 154.7238 9 1572.8126 140.5866 4 1911.4897 154.9856 10 1578.6756 140.8484 5 1917.9522 155.2474 ii 1584.5499 I4I.II02 6 1924.4263 155.5092 45 ft- I590-435 141.372 7 1930.9113 I55-77I i 1596.3309 141.6338 8 I937-4073 156.0328 2 1602.2378 141.8956 9 1943.9142 156.2946 3 1608.1556 142.1574 10 1950.4318 156.5564 4 1614.0843 142.4192 ii 1956.9604 156.8182 5 1620.0238 I42.68I So ft. 1963-5 157-08 2 5 8 SIDES OF SQUARES EQUAL TO AREAS. Sides of Squares equal in. .A^rea to a Circle. Diameter from i to 100. Diaro. Side of Sq. Diam. Side of Sq. Diam. Side of Sq. Diam. I .8862 14 12.4072 27 23.9281 40 X .1078 H 12.6287 K 24.1497 y .3293 H 12.8503 24.3712 y* % .5509 H 13.0718 % 24.5928 M 2 .7724 15 !3.2934 28 24.8144 41 ^ 994 # 13-515 M 25.0359 X 2.2156 n 13.7365 Y2 25-2575 . X % 2.4371 i3.958i % 25-479 % 3 2.6587 16 14.1796 29 25.7006 42 K 2.8802 3^ 14.4012 % 25.9221 y 3.1018 S 14.6227 K. 26.1437 % 2i 3-3233 14.8443 n 26.3653 % 4 3-5449 17 15.0659 30 26.5868 43 X 3-7665 15.2874 X 26.8084 K X 3-988 M 15-509 H 27.0299 X H 4.2096 % 15.7305 H 2 7- 2 5 I 5 H 5 4-43" 18 15-9521 31 27-473 44 % 4.6527 i 16.1736 H 27.6946 % K 4.8742 16.3952 y* 27.9161 % % 5-0958 % 16.6168 % 28.1377 % 6 5-3*74 19 16.8383 32 28.3593 45 3 5.5389 3^ 17.0599 & 28.5808 k M 5.7605 3^ 17.2814 y* 28.8024 x % 5-982 % 17-503 % 29.0239 % 1 6.2036 20 17.7245 33 29-2455 4 6 % 6.4251 y 17.9461 % 29.467 % X 6.6467 y* 18.1677 y* 29.6886 % % 6.8683 % 18.3892 % 29.9102 M 8 7.0898 21 18.6108 34 30.1317 47 % 7.3H4 % 18.8323 & 30.3533 % 7.5329 % 19.0539 X 30.5748 % 7-7545 % 19.2754 % 30.7964 % 9 7.976 22 19.497 35 31.0179 4 8 X 8.1976 ^ 19.7185 y 3L2395 k K 8.4192 K 19.9401 y* 31.4611 ^ % 8.6407 % 20.1617 % 31.6826 % 10 8.8623 23 20.3832 36 31.9042 49 k 9.0838 | 20.6048 y 32.1257 k 9.3054 20.8263 y* 32.3473 H % 9-5269 M 21.0479 % 32.5688 % II 9.7485 24 21.2694 37 32-7904 50 k 9-97 3 21.491 y 33.0112 3^ % 10.1916 i 21.7126 % 33.2335 B % 10.4132 % 21.9341 % 33-4551 x 12 10.6347 25 22.1557 38 33.6766 51 * 10.8563 y 22.3772 * 33-8982 11.0778 y* 22.5988 34.H97 % 11.2994 % 22.8203 M 34.3413 ^i 13 11.5209 26 23.0419 39 34.5628 52 ^ 11.7425 x 23.2634 3 34.7884 K M 11.9641 K 23-485 ^ 35.006 ^ % 12.1856 % 23.7066 M 35-2275 % SIDES OF SQUARES EQUAL TO AEEAS. 259 Side of Sq. | Diam. Side of Sq. Diam. Side of Sq. Diam. Side of Sq. 46.97 65 57.6047 77 68.2395 89 78.8 74 2 47.1916 / 57.8263 X 68461 M 79.0957 47-4I3 1 %" 58.0479 % 68.6826 79-3 J 73 M 58.2694 % 68.9041 M 79.5389 47.8562 48.0778 48.2994 48.5209 66 58.491 58.7125 58.9341 59.1556 78 % 69.1257 69-3473 69.5688 69.7904 if 79.7604 79-982 80.2035 80.4251 48.7425 48.964 6 \ 59-3772 79 7O.OI 19 70.2335 80.6467 49.1856 X 59.8203 % 70455 M 80.8682 49.4071 60.0419 % 70.6766 81.0898 49.6287 68 * 60.2634 80 70.8981 A 81.3113 49.8503 /^ 60.485 /^ 71.1197 92 81.5329 50.0718 K 60.7065 M 7L34I3 8i.7544 50.2934 % 60.9281 M 71.5628 i 81.976 50.5149 69 61.1497 81 71.7844 82.1975 50.7365 50.958 X 61.3712 61.5928 i 72-0059 72.2275 93 i 82.4191 82.6407 51.1796 X 61.8143 /i 72.4491 i^ 82.8622 51.4012 70 62.0359 82 72.6706 M 83.0838 51.6227 5I.8443 52.0658 j 52.2874 I 7i % 62.2574 62.479 62.7006 62.9221 i H 83 72.8921 73."37 73-3353 73-5568 94 83-3053 83.5269 83.7484 Si O7 52.5089 k 63.1437 /4 o j.y/ 52.7305 B 63.3652 M 73-9999 95 84.1916 52.9521 63.5868 74.2215 % 84.4131 53.1736 72 63-8083 84 74-4431 % ^6347 53-3952 64.0299 74.6647 A 84.8562 53.6167 % 64.2514 M 74.8862 96 85.0778 53-8383 X 64.4730 g 75.1077 85-2993 54.0598 73 64.6946 85 75.3293 g 85.5209 54.2814 64.9161 75.5508 M 85-7425 54-503 - 54.7245 1 65.I377 65.3592 1 75.7724 75-9934 97 85.9646 86.185 54.946i 74 65.5808 86 76.2155 M 86.4071 55-1676 k 65.8023 % 76.4371 86.6289 55-3892 55-6107 H 66.0239 66.2455 X K 76.6586 76.8802 9 4 86.8502 O_ f^iQ 55.8323 56.0538 56.2754 1 66.467 66.6886 66.9104 \ X 77.1017 77.3233 77-5449 1 57.O7IO 87.2933 87.5449 56497 67.1317 77.7664 99 87.7364 56.7185 56.9401 ^ 67.3532 67.5748 88 77.988 78.2095 K 87.958 88.1796 57.1616 xl 67.7964 x^ 78.4316 M 88.40X1 57-3832 % 68.0179 % 78.6526 100 88.6227 -A.pplicati.on of Ta~ble. To A.scrtain. a Sq.xi.are tliat lias same ^Vrea as a Q-iven. Circle. ILLUS. If side of a square that has same area as a circle of 73.25 ins. is required. By Table of Areas, page 233, opposite to 73.25 is 4214.11; and in this table is 64.9161, which is side of a square having same area as a circle of that diameter. 26o LENGTHS OF CIRCULAR ARCS. Lengths of Circular .A^rcs, Tip to a Semicircle. Diameter of a Circle = i, and divided into 1000 equal Parts. H'ght. Length. H'ght. Length. H'ght. Length. H'ght. Length. H'ght. Length. .1 1.02645 -15 1.05896 .2 .10348 25 .15912 3 1.22495 .IOI 1.02698 .151 1.05973 .201 .10447 .251 .16033 .301 1.22635 .102 1.02752 .152 1.06051 .202 .10548 .252 .16157 .302 1.227 76 .103 1.02806 -153 1.0613 .203 .1065 .253 .162 79 .^03 1.22918 .104 1.0286 .154 1.06209 .204 .10752 254 .16402 304 1.23061 .105 ^.02914 .155 1.06288 205 10855 255 .16526 305 1.23205 .106 i .029 7 .156 1.06368 .206 10958 .256 .16649 .306 1-23349 .107 1.03026 -57 1.06449 .207 .11062 "257 .16774 -307 1.23494 .108 1.03082 .158 1-0653 .208 .11165 .258 .16899 .308 1.23636 .109 1-03^39 159 1.06611 .209 .11269 .259 .17024 309 1.2378 .11 i. 0310 .16 1.06693 .21 .H374 .26 .1715 31 1.23925 .III 1.03254 .161 1.06775 .211 .11479 .261 I 7 2 75 3" 1.2407 .112 1.03312 .162 1.06858 .212 .11584 .262 .17401 .312 1.242 16 "3 1.03371 .163 1.06941 213 .11692 .26 3 17527 .313 1.2436 .114 1-0343 .164 1.07025 .214 .11796 .264 17655 .314 1.24506 US 1.0349 .165 1.07109 215 .11904 .265 .17784 .315 1.24654 .116 1.03551 .166 1.07194 .2l6 .12011 .266 .17912 .316 1.24801 .IT? 1.03611 .167 1.07279 .217 .121 l8 .267 .1804 317 1.24946 .118 1.03672 .168 1.07365 .218 .12225 .268 .18162 .318 1.25095 .119 1-03734 .169 1.07451 .219 .12334 .269 .18294 .319 1.25243 .12 1.03797 .17 1.07537 .22 .12445 .27 .18428 32 I-2539 1 .121 1.0386 .171 1.07624 .221 .12556 .271 18557 .321 1-25539 .122 1.03923 .172 1.07711 .222 .12663 .272 .18688 .322 1.25686 .123 1.03987 J 73 1.07799 .223 .12774 273 .18819 323 1.25836 .124 1.04051 .174 1.07888 .224 .12885 .274 .18969 324 1.25987 125 1.041 16 .175 1.07977 .225 .12997 275 .19082 .325 1.26137 .126 1.04181 .176 1.08066 .220 .13108 .276 .19214 .326 1.26286 .127 1.04247 .177 1.08156 .227 .13219 .277 !9345 327 1.26437 .128 1-04313 .1.78 1.08246 .228 I333I .278 .19477 328 1.26588 .129 1.0438 .179 1-08337 .229 .13444 .279 .1961 329 1.2674 J 3 1.04447 .18 1.08428 23 I 3557 .28 J 9743 33 1.26892 131 1.04515 .181 1.08519 .231 .13671 .281 .19887 .331 1.27044 .132 1.04584 .182 i. 086 1 1 .232 .13786 .282 .20011 .332 1.27196 133 1.04652 .183 1.08704 233 13903 283 .201 46 333 1.27349 .134 1.04722 .184 1.08797 234 .1402 .284 .202 82 334 1.27502 .135 1.04792 .185 1.0889 235 .14136 285 .204 19 335 1.27656 .136 1.04862 .186 1.08984 .236 .14247 .286 .20558 .336 1.2781 137 1.04932 .187 1.09079 237 14363 .28 7 .20696 337 1.27964 .138 1.05003 .188 1.09174 .238 .1448 .288 .20828 338 1.281 18 .139 1-05075 .189 1.09269 239 14597 .289 .20967 339 1.28273 .14 1.05147 .19 1.09365 .24 .14714 .29 .21202 34 1.28428 .141 1.0522 .191 1.09461 .241 .14831 .291 .21239 341 1.28583 .142 1.05293 .192 1-09557 .242 .14949 .292 .21381 .342 1.28739 .143 1.05367 .193 1.09654 243 .15067 293 .2152 343 1.28895 .144 1.05441 .194 1.09752 .244 .15186 .294 .21658 344 1.29052 145 1.05516 195 1.0985 .245 .15308 .295 .21794 345 1.29209 .146 I -0559 l .196 1.09949 .246 .15429 .296 .21926 346 1.29366 .147 1.05667 .197 1.10048 .247 15549 .297 .22061 347 1.29523 .148 1-05743 .198 1.10147 .2 4 8 1567 .2 9 8 .222O3 348 1.29681 .149 1.058 19 .199 1.10247 249 .15791 .299 .22347 349 1.29839 LENGTHS OF CIRCULAR ARCS. 261 H'ght. Length. H'ght. Length. H'ght. Length. H'ght. Length. H'ght. Length. 35 1.29997 38 1.34899 .41 1.40077 44 L455I2 47 I.5"85 .351 1.30156 .381 1.35068 .411 1.40254 .441 1.45697 .471 L5I378 352 L303I5 .382 L35237 .412 1.40432 .442 1.45883 .472 353 1.30474 .383 1.35406 .413 i 1.4061 443 1.46069 -473 1.51764 354 1.30634 384 1-35575 .414 1 1.40788 444 1.46255 474 1.51958 355 1.30794 385 1-35744 4*5 1.40966 445 1.46441 475 I.52I 52 .356 1.30954 386 I-359I4 .416 1.41145 .446 1.46628 .476 I - 523 46 357 387 1.36084 .417 1.41324 447 1.46815 477 ^525 41 358 1.31276 .388 1.36254 .418 1.41503 .448 1.47002 .478 L527 36 359 L3I437 389 1.36425 .419 1.41682 449 1.47189 479 I-5293I n I.3I599 1.31761 39 391 1.36596 1.36767 .42 .421 1.41861 1.42041 45 451 1-47377 L47565 .48 .481 1.53126 1.53322 .362 1.31923 392 1.36939 .422 1 .422 22 452 1 -477 53 2 I 535 J 8 .363 1.32086 393 1.37111 423 1.42402 453 1.47942 483 J -537 J 4 364 .365 .366 1.32249 L324I3 L32577 394 -3 ?l 396 1.37283 1-37455 1.37628 .424 425 .426 1.42583 1.42764 1.42945 454 455 .456 1.48131 1.4832 1.48509 485 .486 I -539 I 1.54106 L54302 .367 1.32741 1.32905 1.33069 397 398 399 1.37801 1 -379 74 1.38148 .427 .428 .429 I.43I27 L43309 I.4349I 457 458 459 1.48699 1.48889 1.49079 .488 .489 49 1.54499 1.54696 1.54893 37 I-33234 4 1.38322 43 1.43673 .46 1.49269 .491 1.55288 371 1-33399 .401 1.38496 431 1.43856 .461 1.4946 .492 1.55486 372 1-33564 .402 1.38671 432 1.44039 .462 1.49651 493 1-55685 373 1-3373 403 1.38846! 433 1.44222 463 1.49842 494 I.55854 374 1.33896 .404 1.39021 434 1.44405 .464 1.50033 495 1.56083 375 1.34063 405 1.39196 435 1.44589 465 1.50224 496 1.56282 376 1.34229 .406 1-39372; .436 1.44773 466 1.50416 497 1.56481 377 I.34396 .407 i .395 48 j .437 1-44957 467 1.50608 .498 1.5668 378 1.34563 .408 I-39724 438 L45I42 .468 1.508 499 1.56879 379 I-3473I .409 1-399 I 439 I-45327 469 1.50992 5 1.57079 To Ascertain Length, of an -A.ro of a Circle "by pre- ceding Table. RULE. Divide height by base, find quotient in column of heights, take length for that height opposite to it in next column on the right hand. Multiply length thus obtained by base of arc, and product will give length. EXAMPLE. What is length of an arc of a circle, base or span of it being 100 feet, and height 25? 25-7- too = .25; and .25, per table, = 1.15912, length of base, which, multiplied by loo u$.gi2feet. When, in division of a height by base, the quotient has a remainder after third place of decimals, and great accuracy is required. RULE. Take length for first three figures, subtract it from next following length ; multiply remainder by this fractional remainder, add product to first length, and sum will give length for whole quotient. EXAMPLE. What is length of an arc of a circle, base of which is 35 feet, and height or versed sine 8 feet? 8-^35 = .228 571 4; tabular length for .228 = 1.13331, and for .229 = 1. 13444 the difference between which is .001 13. Then .5714 X .001 13 = .000645 682. Hence .228 =1.13331, and .0005714= .000645682 i-i33955682~, the sum by which base of arc is to be multiplied ; and 1.133 955 682 X 35 = 39.688 45 feet. 262 LENGTHS OF CIRCULAR ARCS. Lengths of Circular .A.rcs from 1 to 18O. (Radius = i.) Degrees. Length. Degrees. Length. Degrees. Length. Degrees. | Length. I .0174 46 .8028 91 1.5882 I 3 6 2.3736 2 0349 47 .8203 92 1.6057 137 2.3911 3 .0524 48 8377 93 1.6231 138 2.4085 4 .0698 49 .8552 94 1.6406 T AeRr 139 2.426 5 6 7 .0873 1047 .1222 50 52 .8727 .8901 .9076 96 97 98 I.O5OI 1.6755 1.693 I.7IO4 140 142 2-4435 2.4609 2.4784 8 T 396 53 .925 99 1.7279 J 43 2.4958 .TCT7I 54 9424 144 2-5133 ' ' Cl / 55 9599 100 1-7453 145 2-5307 10 1745 56 9774 101 1.7628 146 2.5482 ii .192 57 9948 102 1.7802 147 2.5656 12 .2094 58 1.0123 103 1.7977 148 2.5831 13 .2269 59 1.0297 104 1.8151 149 2.6005 15 16 17 18 2443 .26l8 .2792 .2967 33l6 60 61 62 1.0472 1.0646 1.0821 1.0995 1.117 105 106 107 108 109 1.8326 1.85 1.8675 1.8849 1.9024 150 152 153 154 2.618 2-6354 2.6529 2.6703 2.6878 65 I-I345 no 1.9199 155 2-7053 21 22 349 1 3665 384 66 67 68 1.1519 1.1694 1.1868 III 112 "3 1-9373 1.9548 1.9722 156 157 158 2.7227 2.7402 2.7576 2 3 24 .4014 .4180 69 1.2043 114 1.9897 159 2.7751 25 26 27 28 2 9 ^ "^ 4363 4538 .4712 .4887 .5061 70 72 73 74 1.2217 1.2392 1.2566 1.2741 1.2915 "5 116 117 118 119 2.0071 2.0246 2.042 2.0595 2.0769 160 161 162 163 164 2.7925 2.81 2.8274 2.8449 2.8623 3 .5236 75 1.309 120 2.0944 165 2.8798 54 1 76 1.3264 121 2.1118 166 2.8972 32 .5585 77 1-3439 122 2.1293 167 2.9147 33 5759 78 1-3613 123 2.1467 168 2.9321 34 5934 79 1.3788 124 2.1642 169 2.9496 35 36 37 38 39 .6109 .6283 .6458 .6632 .6807 80 81 82 83 84 1.3963 I-4I37 1.4312 1.4486 1.4661 125 126 I2 7 128 I2 9 2.1817 2.1991 2.2166 2.2304 2.2515 170 171 172 173 174 2.967 2.9845 3.002 3.0194 3.0369 40 .6981 85 1-4835 I 3 2.2689 3-0543 41 7156 86 1.501 2.2864 ^76 3-0718 42 733 87 1.5184 I 3 2 2.3038 177 3.0892 43 7505 88 1.5359 133 2.3213 178 3.1067 44 .7679 89 1-5533 134 2.3387 179 3.1241 45 7854 90 1.5708 135 2.3562 180 3.1416 To Ascertain Length, of a Circular Arc "by Table. RULE. From column opposite to degrees of arc, take length, and multii ply it by radius of circle. EXAMPLE. Number of degrees in an arc are 45, and diameter of circle 5 feet. Then .7854 tab. length X 5-7- 2 = 1.363$ feet. ' LENGTHS OF ELLIPTIC ABCS. Lengths of Elliptic -A^rcs. Up to a Semi-ellipse. Transverse Diameter =. i, and divided into 1000 equal Parts. 263 H'ght. Length. H'ghL LMgth. H'ght. Length. H'ght Length. H'ght. Length. tl 1.04202 15 .0933 .2 I.I50I4 25 1.21136 3 1.27669 .101 1.04262 .151 .09448 .2OI I5I3I .251 1.21263 .301 1.27803 .102 1.04362 .152 09558 .202 .15248 .252 1.2139 .302 1.27937 .103 1.04462 153 .09669 .203 .15366 253 I.2I5I7 303 1.28071 .104 1.04562 154 .0978 .204 .15484 254 1.21644 304 1.28205 .105 1.04662 155 .09891 .205 .15602 255 1.21772 305 1.28339 .106 1.04762 .156 .10002 .206 1572 .256 I.2I9 .306 1.28474 .107 1.04862 !57 .101 13 .207 15838 257 I.22O28 307 1.28609 .108 1.04962 .158 .10224 .208 15957 .258 1.22156 .308 1.28744 .109 1.05063 159 10335 .209 .16076 259 1.22284 309 1.28879 .11 1.05164 .16 .10447 .21 .16196 .26 1.22412 .31 I.290I4 .III 1.05265 .161 .1056 .211 .16316 .261 1.22541 3" 1.29149 .112 1.05366 .162 .10672 .212 .16436 .262 1.2267 .312 1.29285 "3 1.05467 .163 .10784 .213 16557 263 1.22799 .313 1.29421 .114 1.05568 .164 .10896 .214 .16678 .264 1.22928 3*4 1-29557 115 1.05669 .165 .11008 .215 .16799 .265 I-23057 315 1.29603 .116 1-0577 .166 .1112 .216 .1692 .266 1.23186 .316 1.29829 .117 1.05872 .167 .11232 .217 .17041 .26 7 I-233I5 317 1.29965 .118 1.05974 .168 "344 .218 .17163 .268 1-23445 .318 I.30I 02 .119 1.06076 .169 .11456 .219 17285 .269 1-23575 .319 1.30239 .12 1.06178 i? .11569 .22 .17407 .27 1-23705 32 1.30376 .121 1.0628 .171 .11682 .221 17529 .271 I-23835 .321 I-305I3 .122 1.06382 .172 "795 .222 .17651 .272 1.23966 .322 1.3065 .123 1.06484 173 .11908 .223 17774 273 1.24097 323 1.30787 .124 1.06586 .174 .12021 .224 .17897 .274 1.24228 324 1.30924 .125 1.06689 175 .12134 .225 .1802 275 1-24359 325 I.3I06I .126 1.06792 .176 .12247 .226 .18143 .276 1.2448 .326 1.31198 .127 1.06895 .177 .1236 .227 .18266 277 1.24612 327 L3I335 ,128 1.06998 .178 12473 .228 .1839 .278 1.24744 .328 I.3I472 .129 I.0700I .179 .12586 .229 .18514 279 1.24876 329 I.3l6l 13 1 .072 04 .18 .12699 23 .18638 .28 I.250I 33 I.3I748 131 1.07308 .181 .12813 .231 .18762 .281 1.25142 331 I.3I886 .132 1.07412 .182 .12927 .232 .188-86 .282 1.25274 332 1.32024 133 1.07516 .183 .13041 233 .1901 283 1.25406 333 I.32I62 134 1.07621 .184 13*55 234 .19134 .284 I-25538 334 L323 135 1.07726 .185 .13269 235 .19258 .285 1-2567 335 1.32438 .I 3 6 1.07831 .186 13383 .236 .19382 .286 1.25803 336 1.32576 137 1.07937 .187 13497 *J T 7 1 237 .19506 .28 7 1.25936 337 L327I5 .I 3 8 1.08043 .188 .I36ll .238 .1963 .288 1.26069 -338 1.32854 139 1.08149 .189 .13726 239 19755 .289 1.26202 339 L32993 .14 1.08255 .19 .13841 .24 .1988 .29 1-26335 34 L33I32 .141 1.08362 .191 I 3956 .241 .20005 .291 1.26468 341 1.33272 .142 1.08469 .192 .14071 .242 2013 .292 1.26601 342 I-334I2 .143 1.08576 193 .14186 243 .20255 293 1.26734 343 1-33552 .144 1.08684 .194 .14301 244 .2038 .294 1.26867 344 1.33692 MS 1.08792 195 .14416 245 .20506 295 1.27 345 L33833 .146 1.08901 .196 i453i .246 .20632 .206 1.27133 346 1 -339 74 .147 I.090I .197 .14646 .247 20758 297 1.27267 -347 i-34i 15 .148 1.091 19 .198 .14762 .248 .20884 .208 1.27401 -348 1.34256 J 49 T.09228 .199 .14888 .249 .2101 99 L27535 349 1-34397 264 LENGTHS OF ELLIPTIC ARCS. H'ght. Length. H'ght Length. H'ght Length. H'ght. Length. H'ght. Length. 35 1-34539 405 I.42533 .46 1.50842 .515 1.59408 57 1.68036 351 1.34681 .406 1.42681 .461 1.50996 5l6 I.59564 571 1.68195 352 1-34823 407 1.42829 .462 I.5II5 .517 1.5972 .572 1.68354 353 1-34965 .408 1.42977 463 L5I304 .518 1.59876 573 1.685 13 354 1.35108 .409 L43I25 464 I-5I458 .519 1.60032 574 1.68672 355 1-35251 .41 1.43273 465 1.51612 .52 1. 60188 575 1.68831 356 1-35394 .411 1.43421 .466 I.5I766 .521 1.60344 576 1.6899 357 1-35537 .412 1.43569 467 1.5192 .522 1.605 577 1.69149 358 1.3568 413 1.43718 .468 1.52074 .523 1.60656 578 1.69308 359 1-35823 .414 1.43867 .469 1.52229 524 1. 608 12 579 1.69467 36 i-359 6 7 415 1.44016 47 1.52384 .525 1.60968 58 1.69626 361 1.361 ii .416 1.44165 .471 1.52539 .526 I.6II24 581 1.69785 .362 1-36255 .417 1.443 J 4 .472 1.52691 527 I.6I28 582 1.69945 363 1 -363 99 .418 1.44463 473 1.52849 .528 1.61436 583 1.70105 364 I-36543 .419 1.44613 474 1.53004 529 1.61592 .584 1.70264 365 1.36688 .42 1.44763 475 I-53I 59 53 1.61748 585 1.70424 366 1.36833 .421 1.44913 476 i-533 I 4 531 1.61904 .586 1.70584 .367 1.36978 .422 1.45064 477 1.53469 .S3 2 1.0206 .587 L70745 .368 1-37123 .423 1.4. 52 14. 478 1.53625 533 I.622I6 .588 1.70905 369 1.37268 .424 1.45364 479 i.5378i 534 1.62372 589 1.71065 37 I.374I4 .425 1.45515 .48 1 -539 37 535 1.62528 59 1.71225 371 1.37662 .426 ' 1.45665 .481 I-54093 .536 1.62684 591 1.71286 372 1.37708 427 I.458I5 .482 1.54249 537 1.6284 592 1.71546 373 I.37854 .428 1.45966 483 1-54405: .538 1.62996 j-593 1.71707 374 1.38 429 1.46167 484 i.5456i 539 1.63152 594 1.71868 375 1.38146 43 1.46268 485 1.54718 54 1.63300 595 1.72029 376 1.38292 431 1.46419 .486 1.54875 541 x- f 1.63465 596 1.7219 377 1.38439 .432 1.4657 .487 1-55032 .542 1,636 23 597 I-7235 378 1-38585 433 1.46721 .488 1-55189 543 1.6378 598 1.72511 379 1-38732 434 1.46872 .489 I.55346 544 1.63937 599 1.72672 38 1-38879 435 1.47023 49 I.55503 -545 1.64094 .6 1.72833 381 1.39024 436 1.47174 .491 1.5566 546 1.64251 .601 1.72994 .382 1.39169 437 1.47326 .492 1.55817 547 1.64408 .602 L73I55 383 I-393I4 438 1.47478 493 1-55974 .548 1.64565 .603 I-733I6 384 1-39459 439 M763 494 1.56131 549 1.64722 .604 1-73477 385 1.39605 44 1.47782 -495 1.56289 55 1.64879 .605 1.73638 .386 387 I-3975I 1.39897 .441 .442 1-47934 1.48086 496 497 1.56447 1.56605 551 552 1.65036 1.65193 .606 .607 1-73799 I.739 6 388 1.40043 443 1.48238 .498 1.56763 553 1.6535 .608 1.74121 389 1.40189 444 1.48391 499 1.56921 554 1.65507 .609 1.74283 39 L40335 445 1.48544 5 1.57089 555 1.65665 .61 1.74444 391 1.40481 446 1.48697 .501 I.57234 556 1.65823 .611 1.74605 392 1.40627 447 1-4885 502 I.57389 557 1.65981 .612 1.74767 393 1.40773 .448 1.49003 503 1-57544 558 1.66139 -613 1.74929 394 1.40919 449 I-49I57 504 1.57699 559 1.66297 .614 1.75091 395 1.41065 45 1.49311 505 I-57854 56 1.66455 .615 1-75252 396 1.41211 45 ! 1.49465 506 1.58009 -561 1.66613 .616 i754 I 4 397 I-4I357 452 1.49618 507 1.58164 562 1.66771 .617 I-7557 6 398 1.41504 453 1.49771 -508 1-583 J 9 .563 1.66929 .618 I-75738 399 1.41651 454 1.49924 509 1.58474 .564 1.67087 .619 1-759 4 1.41798 455 1.50077 5i 1.58629 565 1.67245 .620 1.76062 .401 1.41945 456 1.5023 5" 1-58784 -566 1.67403 .621 1.76224 .402 i .420 92 457 1-50383 .512 1.5894 -567 1.67561 .622 1.76386 403 1.42239 458 1-50536 513 1.59096 .568 1.67719 .623 1.76548 .404 1.42386 459 1.50689 .514 1.59252 569 1.67877; .624 1.7671 LENGTHS OP ELLIPTIC ARCS. 265 H'ght. Length. H'ght. Length. H'ght. Length. H'ght. Length. H'ght. Length. .625 1.76872 .68 1.85874 -735 L95059 79 2.04462 .845 2-i4 I 55 .626 1.77034 .681 1.86039 -736 1.95228 .791 2.04635 .846 2.14334 .627 1.77197 .682 1.86205 737 1-95397 .792 2.04809 .847 2.14513 .628 !77359 683 1.8637 .738 1.95566 793 2-04983 .848 2.14692 .629 1.77521 .684 1-86535 739 1-95735 794 2.05157 849 2.14871 63 4631 1.77684 1.77847 .685 .686 1.867 1.86866 74 741 1.95994 1.96074 795 2.05331 796 2.05505 85 .851 2.1505 2.15229 .632 633 .634 1.78009 1.781 72 1.78335 .687 .688 .689 1.87031 1.87196 1.87362 .742 743 744 1.96244 1.96414 1.96583 797 .798 799 2.05079 2-05853 206027 .852 853 854 2.15409 2.15589 2.1577 635 1.78498 .69 1.87527 745 i.o6753 .8 2.O02O2 .855 2.1595 .636 1.7866 .691 1.87693 .746 1.96923 .801 2.06377 .856 2.1613 .637 1.78823 .692 1.87859 747 1.97093 .802 2.065 52 857 2.16309 .638 1.78986 .693 1.88024 748 1.97262 .803 2.06727 858 2.16489 639 1.79149 .694 1.8819 749 1.97432 .804 2.06901 859 2.16668 .64 1.79312 695 1.88356 75 1.97602 805 2.070 76 .86 2.16848 .641 1-79475 .696 1.88522 751 1.97772 .806 2.07251 .861 2.17028 'f*3 1.70038 1.79801 .697 .698 1.88688 1.88854 752 753 1-97943 1.981 13 .807 .808 2.07427 2.07602 .862 .863 2.17209 2.17389 644 1.79964 .699 1.8902 754 1.98283 .809 2.07777 .864 2.1757 645 1.80127 7 1.89186 755 1.08453 .81 2.07953 .865 2.17751 .646 1.8029 .701 1.89352 756 1.98623 .811 2.081 28 .866 2.17932 .647 1.80454 .702 1.89519 757 1.98794 .812 2.08304 .867 2.18113 .648 1.80617 703 1.89685 758 1.98964 813 2.0848 .868 2.18294 649 1.8078 704 1.89851 759 1.99134 .814 12.08656 .869 2.18475 .65 1.80943 705 1.90017 .76 1.99305 -815 2.08832 87 2.18656 .651 1.81107 .706 1.901 84 .761 1.994 76 |-8i6 2.09008 .871 2.18837 .652 1.81271 .707 1-9035 .762 i.996 47 ||.8i7 2.09198 .872 2.19018 653 i.8i435 .708 1.905 17 763 1.99818 .818 2.0936 .873 2,102 654 1.81599 .709 1.90684 .764 1.99989 .819 2.09536 .874 2.19382 655 1.81763 .71 1.90852 765 2.0016 .82 2.097I2H.875 2.19564 .656 1.81928 .711 1.91019 .766 2.00331 .821 2.09888 .876 2.19746 657 1.82091 .712 1.91187 767 2.005 02 1 .822 2.10065 877 2.19928 658 1.82255 .713 L9I355 .768 2.00673 823 2.10242 .878 2.201 I 659 1.82419 .714 1-91523 .769 2.00844 .824 2.IO4 19 .879 2.20292 .66 1.82583 715 1.91691 77 2.OIOI6 -825 2.10596 .88 2.204 74 .661 1.82747 .716 1.91859 .771 2.0II87 .826 2.10773 .881 2.20656 ,662 1.82911 .717 1.92027 772 2.01359 .827 2.1095 .882 2.20839 663 1-83075 .718 1.92195 773 2.0I53I .828 2.III27 .883 2.21022 .664 1.8324 .719 1.92363 774 2.01702 -829 2.11304 .884 2.2I2O5 665 1.83404 .72 1-92531 775 2.018 74 1 1 .83 2.11481 .885 2.21388 .666 1.83568 .721 1.927 .776 202045 -83 1 2.11659 .886 2.2I57I .667 I.83733 .722 1.92868 -777 2.02217 .832 2.11837 .887 2.21754 .668 1.83897 723 1.93036 .778 2.02389 -833 2.I2OI5 .888 2.21937 ,669 1.84061 .724 1.93204 779 2.02561 834 2.12193 .889 2.2212 .67 1.84226 725 193373 .78 2.02733 835 2.12371 .89 2.22303 .671 1.84391 .726 I9354I .781 2.02907 .836 2.12549 .891 2.22486 .672 1.84556 727 1 937 I .782 2.0308 .837 2.12727 .892 2.22O7 .673 1.8472 .728 1.93878 -783 2.03252 .838 2.12905 .893 2.22854 .674 1.84885 .729 1.94046 .784 2.03425 -839 2.13083 .894 2.23038 675 1.8505 73 1.94215 .785 2.03598! .84 2.13261 .895 2.23222 .676 1.85215 731 I-94383 .786 | 2.037 71 .841 2.13439 .896 2.23406 677 I.85379 732 1.94552 787 2.03944 .842 2.13618 .897 2.2359 .678 I-85544 733 1.94721 .788 2.04117 .843 2.13797 .898 2-237 74 67911.85709 734 1.9489 789 2.0429 1 .844 2.13976 899 2.23958 Z LENGTHS OF ELLIPTIC ARCS. (Tght. Length, H'ght. Length. H'ght. Length. H'ght. Length. H'ght. Length. 9 2.241 42 .92 2.27803 94 2.31479 .96 2.35241 .98 2.39055 .901 2.24325 .921 2.27987 .941 2.31666 .061 2.35431 .081 2.39247 .902 2.24508 .922 2.281 7 .942 2.31852' .962 2.35621 .082 2-39439 93 2.24691 923 2.28354 943 2.32038 963 2.3581 .983 2.39631 .904 2.248 74 .924 2.28537 944 2.322 24 .964 2.36 .984 2.39823 95 2.25057 925 2.2872 945 2.32411 .965 2.36191 .985 2.400 16 .906 2.2524 .926 2.28903 .946 2.32598 .966 2.36381 .086 2.40208 .907 2.25423 .927 2.29086 947 2.32785 .967 2.305 71 .087 2.404 .908 2.25606 .928 2.292 7 .948 2.32972 .968 2.367 62 .088 2.40592 .909 2.25789 .929 2-29453 949 2.3316 .969 2.36952 .989 2.407 84 .99 2.409 76 .91 2.25972 93 2.29636 95 2.33348 97 2.371 43 .991 2.41169 .911 2.261 55 931 2.298 2 951 2-33537 .971 2-37334 .992 2.41362 .912 2.26338 932 2.30004 952 2.337 26 972 2.37525 993 2.41556 9i3 2.265 21 933 2.301 88 953 2.33915 973 2.37716 994 2.41749 .914 2.26704 934 2.30373 954 2.34104 974 2.37908 995 2.41943 -9*5 2.26888 935 2.30557 955 2.34293 975 2.381 .996 2.421 36 .916 2.27071 936 2.30741 956 2.34483 .976 2.382 91 997 2.42329 .917 2.27254 937 2.30926 957 2.34073 977 2.38482 998 2.425 22 .918 2.27437 938 2.31111 958 2.34862 .978 2.38673 999 2.42715 .919 2.2762 939 2.31295 959 2.35051 979 2.38864 i. 2.42908 To Ascertain. Length of an Elliptic Arc (right Semi- Ellipse) "by preceding Table. RULE. Divide height by base, find quotient in column of heights, and take length for that height from next right-hand column. Multiply length thus obtained by base of arc, and product will give length. EXAMPLE. What is length of arc of a semi-ellipse, base being 70 feet, and height 30. 10 feet? 30. 10 -7-70 = .43; and 43, per table, = 1.462 68. Then 1.462 68 X 70 = 102. 3876 feet When Curve is not that of a right Semi-Ellipse, Height being half of Trans- verse Diameter. RULE. Divide half base by twice height, then proceed as in preceding example ; multiply tabular length by twice height, and product will give length. EXAMPLE. What is length of arc of a semi-ellipse, height being 35 feet, and base 60 feet? 60 -4- 2 = 30, and 30 -- 35 x 2 = . 428, tabular length of which = 1.459 66. Then 1.45966 X 35X2 = 102.1762 feet. When, in Division of a Height by Base, Quotient has a Remainder after third Place of Decimals, and great A ccuracy is required, RULE. Take length for first three figures, subtract it from next following length; multiply remainder by this fractional remainder, add product to first length, and sum will give length for whole quotient. EXAMPLE. What is length of an arc of a semi-ellipse, base being 171.1 feet and height 125 feet? 171.3-7-2 = 85.65, and 125 X 2 = 250. 85. 65 -7-250:=. 3426 ; tabular length for ,342 = 1.334 12 > and f r -343 = '-335 52, the difference between which is .0014. Then . 6 x. 0014 = .0084. Hence, .342 =1.33412 .0006= .0084 1.342 52, the sum by which base of arc vstobe multiplied; and 1.34252 X 171- 3 = 229. 973676/0*. AKEAS OF SEGMENTS OF A CIKCLE. 267 A.reas of* Segments of a Circle. The Diameter of a Circle = i, and divided into 1000 equal Parts. Versed Sine. Seg. Area. Versed Sine. Seg. Area. Versed Sine. 1! Versed Seg. Area, j Sine. Seg. Area. Versed Sine. Seg. Area. .001 .00004 .052 .01556 .103 .04269 154 .07675 205 ."584 .OO2 .00012 053 .Ol6oi .104 0433 155 .07747 .206 .11665 .003 .O0022 054 .01646 .105 .04391 156 .0782 .207 .11746 .004 .00034 .055 .01691 .106 .04452 157 .07892 .208 .11827 -005 .00047 .056 01737 .107 045 14 .158 .07965 .209 .11908 .006 .00002 057 .01783 .108 045 75 159 .08038 .21 .1199 .007 .00078 .058 .0183 .109 .04638 .16 .O8l II .211 .12071 .008 .00095 059 .01877 .11 047 .161 .08185 .212 I2i 53 .009 .00113 .06 .019 24 .III .04763 .162 .08258 .213 12235 .01 00133 .061 .01972 .112 .04826 .163 .08332 .214 .12317 .Oil 00153 .062 .02O2 "3 .04889 .164 .08406 .215 .12309 .012 .00175 .063 .02068 .114 04953 .165 .0848 .216 .12481 .013 .00197 .064 .O2I 17 US .050 16 .166 .08554 .217 12563 .014 .0022 065 .021 65 .116 .0508 .167 .08629 .218 .12646 .015 .00244 .066 .02215 .117 05145 .168 .08704 .219 .12728 .Ol6 .00268 .067 .02265 .118 .05209 .169 .08779 .22 .12811 .017 .O0294 .068 .02315 .II 9 .05274 17 .08853 .221 .12894 .018 .0032 .069 .02366 .12 .05338 .171 .08929 .222 .12977 .OI9 .00347 .07 .02417 .121 .05404 .172 .09004 223 .1306 .02 00375 .071 .02468 .122 .05469 173 .0908 .224 I3I44 .021 .00403 .072 .025 19 .123 .05534 .174 09155 .225 .13227 .022 .00432 073 .02571 .124 .056 175 .09231 .226 133" .023 .00462 .074 .02624 .125 .05666 .176 .09307 .227 13394 .024 .00492 075 .02676 .126 05733 .177 .09384 .228 13478 .025 .00523 .076 .02729 .127 05799 .178 .0946 .229 13562 .O26 '00555 .077 .02782 .128 .05866 .179 09537 .23 .13646 .027 | .00587 .078 .02835 .129 05933 .18 .09613 .231 I 373 I .028 .006 19 .079 .02889 13 .06 .181 .0969 .232 138 15 .029 .00653 .c3 .02943 131 .06067 .182 09767 .233 139 03 .00686 .081 .02997 .132 06135 183 09845 234 .13984 031 .00721 .082 .03052 133 .06203 .184 .09922 .14069 .032 .00756 .083 .03107 134 .06271 185 .1 236 .14*54 033 .00791 .084 .031 62 135 .06339 .186 .10077 237 .14239 034 .00827 .085 .032 1 8 .136 .06407 .187 IOI55 .238 .14324 035 .00864 .086 .032 74 137 .06476 .188 .10233 .239 .14409 .036 .009OI .087 0333 .138 06545 .189 .103 12 .24 .14494 037 .00938 .088 03387 139 .06614 .19 .1039 .241 .1458 .038 .00976 .089 .03444 .14 .06683 .191 .10468 .242 .14665 039 .01015 .09 03501 .141 06753 .192 10547 .243 .14751 .04 .01054 .091 03558 .142 .06822 193 .10626 .244 .14837 .041 .01093 .092 .036 16 143 .06892 .194 .10705 .245 .14923 .042 .OH33 093 .03674 .144 .06962 195 .10784 .246 .15009 043 .01173 .094 .03732 145 07033 . I9 6 .10864 .247 .15095 .044 .01214 095 0379 .146 .071 03 .197 .10943 .248 .15182 045 .01255 .096 .03849 .147 .071 74 .198 .11023 .249 .15268 .046 .01297 .097 .03908 .148 .07245 .199 .11102 2 5 .15355 .047 01339 .098 .03968 .149 .073 16 .2 .11182 .251 .15441 .048 .01382 .099 .04027 15 .07387 .201 .11262 .252 15528 .049 .01425 .1 .040 87 151 .07459 .202 "343 .253 15615 05 .01468 .101 .041 48 .152 .07531 .203 .11423 .254 .15702 .051 .015 12 .102 .04208 153 .07603 .204 ."503 .255 15789 268 AREAS OF SEGMENTS OF A CIECLE. Versed Sine. Seg. Area. Versed Sine. Seg. Area. Versed Sine. Seg. Area. Versed Sine. Seg. Area. Versed Sine. Seg. Area. ^56" .15876 305 .202 76 354 .2488 403 29631 452 344 77 257 .15964 .306 .20368 355 .24976 .404 .29729 453 345 77 .258 .16051 307 .2046 356 .25071 405 .298 27 454 346 76 259 .16139 .308 20553 357 .25167 .406 .29925 455 347 76 .26 .16226 309 .20645 358 25263 .407 .30024 456 34875 .261 .16314 3 1 .20738 359 25359 .408 .301 22 457 34975 .262 .16402 3 11 .2083 36 25455 .409 .3022 458 35075 .263 .1649 .312 .20923 .361 25551 .41 30319 459 .351 74 .264 16578 3i3 .21015 362 .25647 .411 30417 .46 352 74 .265 .16666 314 .2IIO8 363 25743. .412 30515 .461 353 74 .266 16755 3i5 .2I2OI 364 25839 413 .30614 .462 354 74 .267 .16844 .316 .21294 365 25936 .414 .307 12 463 355 73 .268 .16931 317 21387 .366 .26032 415 .308ll 464 356 73 .269 .1702 318 .2148 367 .261 28 .416 .30909 465 357 73 .27 .17109 3i9 21573 368 .262 25 .417 .31008 .466 35872 .271 .17197 32 .21667 369 .26321 .418 .31107 .467 35972 .272 .17287 .321 .2176 37 .264 1 8 .419 31205 .468 36072 273 .17376 .322 .21853 371 .265 14 .42 31304 469 .361 72 .274 17465 323 .21947 372 .26611 .421 3!403 47 .362 72 275 17554 324 .22O4 373 .26708 .422 31502 47 1 363 7i .276 17643 325 .221 34 374 .26804 423 3i6 472 36471 .277 17733 326 ,22228 375 .26901 .424 .31699 473 365 7i .278 .17822 327 .22321 .376 .26998 425 31798 474 .36671 .279 .17912 328 .22415 377 27095 .426 31897 475 .36771 .28 .I8OO2 329 .22509 378 .27192 427 .31996 476 36871 .281 .18092 33 .22603 379 .27289 .428 32095 477 36971 .282 .18182 331 .22697 38 .27386 .429 .32194 478 .37071 283 .18272 332 .22791 381 .27483 43 .32293 479 3717 .284 .18361 333 .22886 382 .27580 431 32391 .48 3727 .285 .18452 334 .2298 383 .27677 432 3249 .481 3737 .286 .18542 335 .23074 384 277 75 433 3259 .482 3747 .287 18633 336 .23169 .385 .27872 434 .32689 483 3757 .288 18723 337 .23263 386 .27969 435 .32788 484 3767 .289 .18814 338 .23358 .387 .28067 436 .32887 485 3777 .29 .18905 339 23453 .388 .281 64 437 .32987 .486 3787 .291 .18995 34 23547 389 .28262 438 .33086 487 3797 .292 .19086 341 .23642 39 28359 439 33185 .488 3807 293 .191 77 342 23737 391 28457 44 33284 .489 3817 .294 .19268 343 .23832 392 28554 .441 .33384 49 .3827 295 .1936 344 .23927 393 .28652 .442 33483 .491 3837 .296 I945I 345 .240 22 394 2875 443 .33582 .492 3847 .297 .19542 346 .241 I 7 395 .28848 444 .33682 493 3857 .298 .19634 347 .242 12 396 .28945 445 .33781 494 .3867 .299 19725 348 .24307 397 .29043 .446 .3388 495 .3877 3 .19817 349 .24403 398 .291 41 447 3398 .496 .3887 .301 .19908 35 .24498 399 .29239 448 34079 497 3897 .302 .2 351 24593 4 .29337 449 341 79 .498 3907 .303 .20O92 352 .24689 .401 29435 45 .34278 499 3917 .304 .201 84 353 .24784 .402 29533 451 34378 5 3927 To Compnte A.rea of a Segment of a Circle "by preceding Tatole. RULE.- Divide height or versed sine by diameter of circle ; find quotient in column of versed sines. Take area for versed sine opposite to it in next col- umn on right hand, multiply it by square of diameter, and it will give area. AREAS OF ZONES OF A CIRCLE. 269 EXAMPLE. Required area of a segment of a circle, its height being 10 feet and diameter of circle 50. 10 -r- 50 = . 2, and . 2, per table, = . 1 1 1 82 ; then . 1 1 1 82 x so 2 = 279. 55 feet When, in Division of a Height by Base, Quotient has a Remainder after Third Place of Decimals, and great Accuracy is required. RULE. Take area for first three figures, subtract it from next following area, multiply remainder by said fraction, add product to first area, and sum will give area for whole quotient. EXAMPLE. What is area of a segment of a circle, diameter of which is 10 feet, and height of it 1.575? i. 575 -4- 10 = .1575 ; tabular area for . 157 = .078 92, and for . 158 = .079 65, the dif- ference between which is .00073. Then . 5 X -ooo 73 = .000 365. Hence, .157 =.07892 .0005 = .ooo 365 .079 285, sum by which square of diameter 0f circle is to be multiplied ; and .079 285 X io 2 = j.g of Zones of a Circle. The Diameter of a Circle = i, 302500 166375000 23.452 078 8 8.193212' 55 1 303601 167 284 151 23-4733892 8.198175: 552 304704 168196608 23.494 680 2 8.203 131 < 553 305809 169112377 23-515952 8.208 082 1 554 306916 170031464 23-5372046 8.213027 555 308025 170953875 23-558438 8.217965- 556 309136 171 879616 23-5796522 8.222 898 . 557 310249 172808693 23.600 847 4 8.227 825 <. 558 311364 1 73 74 1 1 1 2 23.622 023 6 8.232746^ A A* 282 SQUARES, CUBES, AND ROOTS. NUMBER SQUARE. CUBE. SQUARE ROOT. CUBE ROOT- 559 31 24 81 174676879 23.6431808 8.237 661 4 560 313600 175616000 23.6643191 8.2425706 56i 31 47 21 176558481 23-6854386 8.247 474 562 315844 1 77 504 328 23.7065392 8.2523715 563 316969 178453547 23.727621 8.257 263 3 564 318096 179406144 23.7486842 8.262 149 2 565 31 92 25 180362125 23.7697286 8.2670294 566 320356 181 321 496 23.7907545 8.2719039 567 32 14 89 182 284 263 23.811 7618 8.2767726 5 68 32 26 24 183250432 23.832 750 6 8.281 625 5 569 323761 184220009 23-8537209 8.2864928 570 324900 185 193 ooo 23.874 672 8 8.2913444 57i 326041 186169411 23.8956063 8.201903 572 32 71 84 187 149 248 23.9165215 8.301 030 4 573 328329 188132517 23.9374184 8.305 865 i 574 329476 189119224 23.9582971 8.3106941 575 330625 190109375 23.9791576 8.3I55I75 576 33 17 76 191 102 976 24 8.3203353 577 332929 I92I00033 24.020 824 3 8.325 147 5 578 334084 193 ioo 552 24.041 6306 8.3299542 579 33 52 41 194 104 539 24.0624188 8-3347553 580 336400 195 1 12 000 24.083 189 i 8.3395509 58i 337561 196122941 24.1039416 8.344 34i 582 338724 197 137 368 24.1246762 8.3491256 583 584 339889 34 10 56 198 155 287 199176704 24.1453929 24.1660919 8-353 94 7 8.3586784 585 342225 200 201 625 24.1867732 8.3634466 586 34330 2O I 230 056 24.2074369 8.3682095 587 344569 202 262 003 24.228 082 9 8.3729668 588 345744 203297472 24.2487113 8.3777188 589 346921 204336469 24.2693222 8.3824653 590 348100 205379000 24.2899156 8.3872065 59i 349281 206425071 24.3104916 8.391 942 3 592 350464 207 474 688 24-331 050 1 8.3966729 593 351649 208 527 857 24-35I 59 1 3 8.401 398 i 594 352836 209 584 584 24.3721152 8.406118 595 35 40 25 210644875 24.392 621 8 8.410 832 6 596 35 52 16 211708736 24.4131112 8.4155419 597 356409 212 776 173 24-4335834 8.420 246 598 357604 213 847 192 24-4540385 8.424 944 8 599 3588oi 214921799 24.474 476 5 8.429 638 3 600 360000 216000000 24.494 897 4 8.434 326 7 601 361201 217081801 2 4-5 1 5 3 01 3 8.439 9 8 602 36 24 04 218 167 208 24-5356883 8.4436877 603 363609 219256227 24-5560583 8.4483605 604 36 48 16 220 348 864 24.5764115 8.4530281 605 366025 221 445 125 24.5967478 8.4576906 606 36 72 36 222 545 016 24.6170673 8.462 347 9 607 368449 223648543 24-637 37 8.467 608 369664 224 755 712 24.657656 8.471 647 i 609 370881 225 866 529 24.6779254 8.476 289 2 610 372100 226981000 24.6981781 8.480 926 I 611 373321 228099131 24.718414 2 8.4855579 612 374544 229 220 928 24.7386338 8.4901848 613 375769 230346397 24-7588368 8.4948065 614 376996 231475544 34.7790234 8.4994233 SQUARES, CUBES, AND ROOTS. 28 3 NUMBER. SQUARE. CUBE. SQUARE ROOT. CUBE ROOT. 615 37 82 25 232608375 24.7991935 8.504035 616 37 94 56 233744896 24.8193473 8.508641 7 617 380689 234885113 24.8394847 8.5132435 618 38 19 24 236029032 24.859 605 8 8.5178403 619 383161 237176659 24.8797106 8.522 432 I 620 384400 238 328 ooo 24.8997992 8.5270189 621 385641 239 48306 1 24.9198716 8.5316009 622 386884 240641 848 24.9399278 8.536178 623 388129 241804367 24.9599679 8.5407501 624 38937 6 242 970 624 24.979992 8.5453173 625 390625 244140625 25 8.5498797 626 39 18 76 245134376 25.019992 8.5544372 627 393129 246491883 25.0399681 8.5589899 628 394384 247 673 152 25-0599282 8.5635377 629 39564 1 248858189 25.0798724 8.5680807 630 396900 250 047 ooo 25.0998008 8.5726189 631 398161 251239591 25.1197134 8.5771523 632 399424 252435968 25.1396102 8.5816809 633 400689 253636137 25.1594913 8.5862047 634 40 19 56 254 840 104 25-I793566 8.5907238 635 403225 256047875 25.1992063 8.595238 636 404496 257259456 25.2190404 8.5997476 637 405769 258474853 25.2388589 8.604 252 5 638 407044 259694072 25.2586619 8.6087526 639 408321 260917 119 25.2784493 8.613 248 640 409600 262 144000 25.298 221 3 8.6177388 641 410881 263374721 25-3I79778 8.622 224 8 642 41 21 64 264 609 288 25.3377189 8.6267063 643 413449 265 847 707 25-3574447 8.631 183 644 4i 47 36 267 089 984 25-377 155 I 8.6356551 645 41 6025 268336125 25-3968502 8.6401226 646 4i 73 16 269585136 25.4165301 8.644 585 5 647 41 8609 270840023 25.436 194 7 8.6490437 648 419904 272097792 25.4558441 8.6534974 949 42 12 01 273359549 25-4754784 8.6579465 650 422500 274625000 25.4950976 8.662 391 i 651 42 38 oi 275894451 25.5147016 8.666831 652 425104 277167808 25-534 290 7 8.6712665 653 426409 278445077 255538647 8.6756974 654 42 77 16 279726264 25-5734237 8.6801237 655 429025 281011375 25.5929678 8.6845456 656 430336 282300416 25.6124969 8.688963 657 431649 283593393 25.6320112 8.6933759 658 432964 284890312 25.6515107 8.6977843 659 434281 28^191 179 25.6709953 8.7021882 660 43 56 oo 287 496 ooo 25.6904652 8.7065877 661 436921 288804781 25.7099203 8.7109827 662 43 82 44 290 117528 25.7293607 8-7J53734 663 439569 291 434 247 25.7487864 8.7197596 664 440896 292754944 25.768 197 5 8.724 141 4 665 442225 294079625 25-7875939 8.7285187 666 443556 295 408 296 258069758 8.7328918 667 444889 296740963 25.8263431 8.7372604 668 446224 298077632 25.845696 8.7416246 669 44756i 299418309 25.865 034 3 8.7450846 670 448900 300 763 ooo 25-8843582 8.7503401 284 SQUARES, CUBES, AND BOOTS. NUMBER. SQUARE. CUBE. SQUARE ROOT. CUBE ROOT. 6 7 I 45 02 41 302 III 711 25.9036677 8.7546913 6 7 2 45 15 84 33 404 448 25.9229628 8.7590383 673 452929 304821 217 25.942 243 5 8.7633809 674 454276 306182024 25.96151 8.7677192 675 45 56 25 307546875 25.980 762 i 8.77 2 0532 676 456976 308915776 26 8.776383 6 77 458329 310288733 26.019 223 7 8.7807084 678 459684 311665752 26.038 433 i 8.7850296 679 461041 313046839 26.057 628 4 8.7893466 680 462400 314432000 26.0768096 8.7936593 681 463761 315821241 26.0959767 8.7979679 682 465124 317214568 26.115 1297 8.8022721 683 466489 318611987 26.1342687 8.8065722 684 46 78 56 320 013 504 26.1533937 8.8108681 685 469225 321419125 26.1725047 8.8151598 686 470596 322 828 856 26.191601 7 8.8194474 687 471969 324242703 26.2106848 8.823 730 7 688 473344 325660672 26.229 754 i 8.8280099 689 474721 327082769 26.2488095 8.832 285 690 476100 328509000 26.267851 i 8.8365559 691 477481 329939371 26.2868789 8.8408227 692 478864 331373888 26,305 892 9 8.8450854 693 48 02 49 332812557 26.324 893 2 8.849,344 694 481636 334 255 384 26.343 879 7 8.8535985 695 483025 335702375 26.362 852 7 8.8578489 696 484416 337153536 26.381 8119 8.862 095 2 697 485809 338608873 26.4007576 8.8663375 698 487204 340 068 392 26.4196896 8-8705757 699 488601 341532099 26.4386081 8.8748099 700 490000 343000000 26.4575131 8.87904 701 49 14 01 344472101 26.476 404 6 8.883 266 I 702 49 28 04 345948408 26.495 282 6 8.887 488 2 703 494209 347 428 927 26.5141472 8.8917063 704 49 56 16 348913664 26.5329983 8.895 920 4 705 497025 350402625 26.551 836 i 8.9001304 706 498436 351895816 26.5706605 8.9043366 707 499849 353393243 26.5894716 8.9085387 708 501264 354894912 26.608 269 4 8.9127369 709 502681 356400829 26.6270539 8.9169311 710 504100 357911000 26.645 825 2 8.921 1214 711 505521 359425431 26.6645833 8.925 307 8 712 506944 360944128 26.6833281 8.929 490 2 713 508369 362 467 097 26.7020598 8.9336687 7 J 4 509796 303994344 26.7207784 8.9378433 7J5 51 1225 365 525 875 26.7394839 8.942 014 716 512656 367 061 696 26.7581763 8.946 1809 717 514089 368601813 26.7768557 8.9503438 718 5 J 5524 370146232 26.795 522 8.9545029 719 516961 371694959 26.814 J 754 8.9586581 720 518400 373248000 26.8328157 8.9628095 721 519841 374 805 361 26.8514432 8.966957 722 521284 376367048 26.8700577 8.971 1007 723 52 27 29 377933067 26.8886593 8.975 240 6 724 52 41 76 379503424 26.907 248 i 8.9793766 725 52 56 25 381 078 125 26.925 824 8.9835089 726 52 70 76 382657 176 26.9443872 8-9876373 SQUARES, CUBES, AND ROOTS. If UMBER. SQUABB. CCBK. SQUARE ROOT. CCBB ROOT. 727 52 85 29 384 240 583 26.9629375 8.991 762 7 28 529984 385828352 26.981 475 i 8.9958829 729 53*441 387420489 27 9 73 532900 389017000 27.0185122 9.0041134 73 1 53436i 3906I789I 27.0370117 9.OO8 222 9 73 2 53 58 24 392 223 168 27.055 498 5 9.0123288 7.33 537289 393832837 27-073 972 7 9.0164309 734 538756 395446904 27.092 434 4 9.0205293 735 540225 397065375 27.1108834 9.024 623 9 736 541696 398688256 27.1293199 9.0287149 737 543 l6 9 400315553 27.1477439 9.032 802 I 738 544644 401 947 272 27.1661554 9.036 885 7 739 5461 21 403583419 27.1845544 9.0409655 740 547600 405 224 ooo 27.202941 9.045 041 7 741 549081 406869021 27.2213152 9.0491142 742 550564 408518488 27.2396769 9.053 183 i 743 552049 410172407 27.2580263 9.057 248 2 744 55 35 36 411830784 27.2763634 9.061 309 8 745 55 50 25 413493625 27.294688 i 9-o65 367 7 746 55 65 16 415160936 27.3130006 9.069 422 747 558009 416832723. 27-3313007 9.0734726 748 559504 418 508 992 27-349 588 7 9.0775197 749 561001 420 189 749 27.3678644 9.081 563 1 750 562500 421 875000 27.3861279 9.085 603 75 * 564001 423564751 27.4043792 9.089 639 2 75 2 565504 425259008 27.4226184 9.0936719 753 567009 426957777 27.4408455 9 097 701 754 568516 428 661 064 27.4590604 9.101 7265 755 570025 430368875 27.4772633 9.1057485 756 571536 432081 216 27-495 454 2 9.1097669 757 * 57 30 49 433798093 27-513633 9.1137818 758 57 45 64 4355J95I2 27-53I 7998 9.1177931 759 576081 437 245 479 27-5499546 9.121 801 760 577600 438976000 27.5680975 9.1258053 761 579121 440711081 27.5862284 9.1298061 762 580644 442450728 27.6043475 9.1338034 763 582169 444194947 27.6224546 9.1377971 764 583696 445 943 744 27.6405499 9.1417874 765 585225 447697125 27.6586334 9-H57742 766 586756 449 455 096 27.676705 9^497576 767 58 82 89 451217663 27.6947648 9-1537375 768 589824 452 984 832 27.7128129 9-I577I39 769 59i36i 454756609 27.7308492 9.161 6869 770 592900 456533000 27.7488739 9.1656565 771 594441 458314011 27.7668868 9.1696225 772 595984 460 009 648 27.784888 9- J 73 585 2 773 59 75 29 461 889917 27.8028775 9-1775445 774 599076 463 684 824 27.8208555 9.1815003 775 600625 465 484 375 27.838 821 8 9.1854527 776 6021 76 467288576 27.8567766 9.1894018 777 603729 469097433 27.8747197 9-1933474 778 605284 470910952 27.8926514 9.1972897 779 606841 472 729 139 27.9105715 9.201 2286 780 608400 474552000 27.928 480 i 9.205 164 i 781 609961 476379541 27.9463772 9.2090962 782 61 15 24 478211768 27.964 262 9 9.213025 286 SQUARES, CUBES, AND ROOTS. NUMBER. SQUABS. CUBE. SQUARE ROOT. 783 61 30 89 480048687 27.982 137 2 784 6 1 46 56 481890304 28 785 61 62 25 483736625 28.0178515 7 86 617796 485587656 28.0356915 7 8 7 619369 487 443 403 28.0535203 7 88 620944 489303872 28.0713377 789 62 25 21 491 169069 28.089 143 8 790 624100 493039000 28.1069386 791 625681 494913671 28.124 722 2 792 62 72 64 496793088 28.1424946 793 62 88 49 498677257 28.1602557 794 630436 500566184 28.1780056 795 63 20 25 502459875 28.1957444 796 63 36-16 504358336 28.213472 797 635209 506 261 573 28.231 1884 798 63 68 04 508169592 28.2488938 799 63 84 01 510082399 28.2665881 800 640000 512000000 28.284271 2 801 64 16 01 513922401 28.3019434 802 64 32 04 515849608 28.3196045 803 644809 517781627 28.3372546 804 64 64 16 519718464 28.3548938 805 64 80 25 521660125 28.3725219 806 649636 523606616 28.390 139 I 807 651249 525 557 943 28.407 745 4 808 65 28 64 527514112 28.4253408 809 654481 529475129 28.4429253 810 656100 531 441 ooo 28.460 498 9 811 65 77 21 533 4 11 73i 28.478061 7 812 659344 535387328 28.4956137 813 660969 537367797 28.5131549 814 662596 539353144 28.530 685 2 8i5 664225 541343375 28.548 204 8 816 665856 543338496 28.565 713 7 817 667489 545338513 28.5832119 818 6691 24 547343432 28.6006993 819 67 07 61 549353259 28.618176 820 672400 551368000 28.635 642 i 821 67 40 41 55338766i 28.653 097 6 822 675684 555412248 28.670 542 4 823 67 73 2 9 557 441 767 28.6879766 824 678976 559476224 28.7054002 825 680625 561515625 28.7228132 826 68 22 76 563559976 28.7402157 827 683929 565609283 28.7576077 828 685584 567 663 552 28.7749891 829 68 72 41 569 722 789 28.7923601 830 688900 571787000 28.8097206 831 690561 573856191 28.8270706 832 692224 57593 368 28.8444102 833 693889 578009537 28.861 739 4 834 695556 580093704 28 879058 2 835 697225 582182875 28.8963666 836 698896 584277056 289136646 837 700569 586376253 28.9309523 838 702244 588 480 472 28.948 229 7 CUBE ROOT. SQUARES, CUBES, AND ROOTS. 287 SQUARE. CUBE. SQUARE ROOT. CUBE ROOT. 703921 590589719 28.965 496 7 9.431 642 3 705600 592704000 28.982 753 5 9-43538 70 72 81 59482332! 2 9 9.4391307 708964 596947688 29.0172363 9.4428704 710649 599077107 29.034 462 3 9.446 607 2 712336 601211584 29 051 678 I 9.450341 714025 603351125 29.0688837 9.4540719 71 57 16 605495736 29.086 079 1 9-4577999 717409 607645423 29.1032644 9.4615249 71 91 04 609800192 29.1204396 9-465 247 720801 611960049 29.1376046 9.468 966 i 722500 614 125000 29- 1 54 759 5 9.4726824 72 42 01 616295051 29.1719043 9-4763957 725904 618 470 208 29.189039 9.480 1 06 i 727609 620650477 29.206 163 7 9.4838136 729316 622835864 29.2232784 9.4875182 73*025 625026375 29.240383 9.491 22 73 27 3 627 222 Ol6 29.2574777 9.4949188 734449 629422793 29.2745623 9.4986147 736164 631 628 712 29.291 637 9.502 307 8 737881 633839779 29.308 701 8 9.505 998 739600 636056000 29-3257566 9.5096854 74I32I 638277381 29.342 801 5 9-5I33699 74344 640 503 928 29-3598365 9-5I705I5 744769 642735647 29.3768616 9.5207303 746496 644972544 29-3938769 9.5244063 748225 647214625 29.410 882 3 9.5280794 749956 649 461 896 29.427 877 9 9-53 1 749 7 751689 651714363 29.4448637 9-5354I72 75 34 24 653972032 29.461 839 7 9.5390818 75 5i 61 656234909 29.4788059 9.542 743 7 756900 658 503 ooo 29.495 762 4 9.5464027 758641 660776311 29.512 709 i 9-5500589 760384 663054848 29.5296461 9-5537 J 23 7621 29 665338617 29-5465734 9-557363 763876 667 627 624 29563491 9.5610108 765625 669921875 29.5803989 95646559 76 73 76 672 221 376 29.597 297 2 9.5682982 769129 674 526 133 29614 1858 9-57 1 937 7 770884 676836152 29.631 064 8 9-5755745 772641 679 !5 J 439 29.647 934 2 95792085 774400 681 472000 29.6647939 9.582 839 7 77 61 61 683 797 841 29,681 644 2 9.586 468 2 777924 686128968 29.698 484 8 9.5900937 779689 688465387 29-7153159 95937^9 781456 690 807 104 29.7321375 9-597 337 3 783225 693 154 125 29.7489496 9.6009548 784996 695 506 456 29-765 752 1 9.604 569 6 786769 697 864 103 29.7825452 9.608 181 7 788544 700 227 072 29.7993289 9.611 791 i 790321 702595369 29.816 103 9-6i53977 7921 oo 704969000 29.832 867 8 9.619001 7 793881 707 347 97i 29.8496231 9.622 603 795664 709 732 288 29.866369 9.626 201 6 797449 712121957 29.883 105 6 9.629 797 5 799236 714516984 29.8998328 0-6333907 288 SQUARES, CUBES, AND BOOTS. NUMBER. SQUARE. CUBK. | SQUARE ROOT. CUBB ROOT. 895 801025 716917375 29.9165506 9.6369812 896 802816 719323136 29-933 259 i 9.640 569 8 97 804609 721734273 29.9499583 9.644154: 8 9 8 806404 724 150 792 29.966 648 i 9.647 736 ' 899 808201 726572699 29.9833287 9.651 316 (. 900 SlOOOO 729000000 . 3 9.654893* 901 81 1801 73 I 432 7 OI 30.016662 9.658 468 i 902 81 3604 733870808 30.033 314 8 9.662 040 v 93 815409 7363*4327 30.0499584 9. 665 609 i 904 81 72 16 738 763 264 30.066 592 8 9.669176: 95 819025 741217625 30.0832179 9.672740. 906 82 08 36 743677416 30.0998339 9.676 301 907 82 26 49 746 142 643 30.1164407 9.679 860 < 908 824464 748613312 30.1330383 9.683 416 ( 909 82 62 81 751089429 30.1496269 9.686 970 910 828100 753571000 30.1662063 9.690521 911 829921 756 058 031 30.1827765 9.694069 912 831744 758550528 30.1993377 9.697615 9*3 833569 761 048 497 30.2158899 9.701 158 914 835396 763551944 30.2324329 9.704698 9i5 83 72 25 766060875 30.2489669 9.708 236 916 839056 768 575 296 30.2654919 9.711772 917 84 08 89 771095213 30.282 007 9 9-7J5305 918 84 27 24 773620632 30.2985148 9-718835 919 84 45 61 776i5i559 30.3150128 9.722363 920 846400 778 688 ooo 30.331 501 8 9.725888 921 848241 781 229961 30.3479818 9.729410 922 850084 783777448 30.3644529 9-732 930 923 851929 786330467 30.380915 i 9.736448 924 853776 788889024 30-3973683 9-739963 925 855625 79M53I25 30.4138127 9-743475 926 85 74 76 794022776 30.430 248 i 9.746985 927 859329 796 597 9 8 3 30.4466747 9.750493 928 861184 799178752 30.4630924 9-753997 929 86 30 41 801 765 089 30.4795013 9-757 5oo 930 864900 804357000 30.495 901 4 9.761 ooo 93 i 866761 806954491 30.5122926 9.764497 932 868624 809557568 30.528675 9.767992 933 87 04 89 812 166237 30.545 048 7 9.771 484 934 872356 814 780 504 30.5614136 9-774974 935 874225 817400375 30.577 769 7 9.778461 936 876096 820 025 856 30.5941171 9.781 946 937 877969 822 656 953 30.6104557 9.785428 938 879844 825 293 672 30.626 785 7 9.788908 939 881721 827936019 30.643 1069 9.792386 940 883600 830 584 ooo 30.6594194 9.795 861 941 885481 833237621 30.6757233 9-799333 942 887364 835896888 30.6920185 9.802 803 943 88 92 49 838 561 807 30.708 305 i 9.806271 944 891136 841 232 384 30.724583 9.809 736 945 89 ..025 843 908 625 30.7408523 9.813 198 946 89 49 16 846 590 536 30.757113 9.816659 947 896809 849278123 30.7733651 9.820 u6< 948 89 87 04 851971392 30.7896086 9-823572, 949 900601 854670349 30.805 843 6 9.827025 950 902500 857375ooo 30.822 07 9'830475 SQUARES, CUBES, AND BOOTS. 289 NUMBER. SQUARE. CUBE. SQUARE ROOT. CUBE ROOT. 951 904401 860085351 30.8382879 9.8339238 95 2 906304 862 801 408 30.8544972 9.8373095 953 908209 865523177 30.870 698 I 9.840812 7 954 91 01 16 868250664 30.8868904 9.8442536 955 91 20 25 870983875 30.903 074 3 9.847 692 956 9*3936 873 722 816 30.919 247 7 9.851 128 957 915849 876467493 30.9354i66 9.8545617 958 91 77 64 879217912 30.951 575 i 9.8579929 959 919681 881974079 30.07 725 i 9.861 421 8 960 92 1600 884 736 ooo 30.9838668 9.8648483 01 92 35 21 887503681 31 9.868 272 4 02 9 2 5444 890277 128 31.0161248 9.8716941 963 927369 893056347 31.0322413 9.875"35 964 9 2 9 20 .-895841344 -31.0483494 9.8785305 05 93 J 2 25 898632125 31.0644491 9.881 945 i 06 933156 901 428 60 31.0805405 9-885 357 4 07 935089 904231063 31.006236 9.8887673 968 93 70 24 907039232 31.1126984 9.8921749 969 938961 909853209 31.1287648 9.895 580 i 970 940900 912673000 31.144823 9-898983 971 942841 915498611 31.1608729 9.9023835 972 944784 918330048 31.1769145 9.905 ?8i 7 973 946729 921167317 31.1929479 9.9091776 974 94-8676 924 oio 424 31.2089731 9.9125712 975 950625 926859375 31.22499 9.9159624 976 95 25 76 929714176 31.2409987 9-9I935I3 977 954529 932574833 31.2569992 9.9227379 978 956484 935441352 31.2729915 9.926 122 2 979 958441 938313739 31.2889757 9.9295042 980 00400 941 192 ooo 31.304951? 9.9328839 981 962361 944076141 31.3209195 9.9362613 982 04324 946966168 31.3368792 9.9396363 983 06289 949 862 087 31-3528308 9.9430092 984 08256 952763904 31.3687743 9.9463797 985 970225 955671625 31.3847097 9.9497479 986 97210 958 585 256 31.4006369 9.953II38 987 974169 01504803 31.4165561 9.9564775 988 976i44 04430272 31.4324673 9.9598389 989 9781 21 07361669 31.4483704 9.03 198 i 990 98OIOO 970299000 31.4642654 9.9665549 991 982081 973242271 31.4801525 9.9699095 992 984064 976 191 488 31400315 9-973 261 9 993 986049 979146657 31.5119025 9.976612 994 988036 982 107 784 31.5277655 9-9799599 995 990025 985074875 31.5436206 9.9833055 90 992016 988047936 31.5594677 9.9866488 997 994009 991026973 31.5753068 9.98999 998 996004 994011992 31.591 138 9.9933289 999 998001 997002999 31. 606 0i 3 9-9966656 1000 I OOQOOO 1000 000 OOO 31.6227766 10 1001 I 00 20 01 1003003001 31.638584 IO.O03 322 2 1002 i oo 40 04 1006012008 31.6543836 10.006 662 a 1003 1006009 1009027027 31.6701752 10.009 989 g 1004 I0o8oi6 1012048064 31.685959 10.0133155 1005 IOI0025 1015075125 31.7017349 10.016 638 5 1006 I OI 20 36 1018108216 i 31-717503 10.0199601 BB 2QO SQUARES, CUBES, AND BOOTS. NUMBER. SQUARE. CUBE. SQUARE ROOT. CUBE ROOT. 1007 i 01 4049 i 021 147 343 31.7332633 10.023 2 79 I 1008 i 01 6064 i 024 192 512 31.7490157 10.026 595 8 lOOQ I0l8o8l i 027 243 729 31.7647603 10.0299104 1010 I 02 01 00 1 030301 ooo 31.7804972 10.033 22 2 8 IOII I O2 21 21 1033364331 31.796226 2 10.036533 IOI2 I024I44 1036433728 31.8119474 10.039 841 1013 I 02 6l 69 1039509197 31.8276609 10.043 ^69 1014 I O2 8l 96 i 042 590 744 31.8433666 10.046 450 6 1015 I 03 02 25 1045678375 31.8590646 10.049 752 I 1016 I 03 22 56 i 048 772 096 31.8747549 10.053051 4 1017 I 03 42 89 i 051 871 913 31.890437.4 10.056 348 5 1018 J 03 63 24 1054977832 31.9061123 0.0596435 1019 1038361 i 058 089 859 31.9217794 0.062 936 4 IO2I n^ojoo 1042441 i 064 332 261 3' -937*138 8 31.9530906 0.066 227 i 0.0695156 IO22 i 04 44 84 i 067 462 648 31.9687347 0.072 802 1023 i 04 65 29 1070599167 31.9843712 0.076 086 3 1024 1 04 85 76 1073741824 32 10.079 368 4 1025 i 05 06 25 i 076 890 625 32.0156212 10.082 648 4 1026 i 05 26 76 1080045576 32.031 234 8 10.085 926 2 1027 1 05 47 29 1083206683 32.046 840 7 10.089 2O1 9 1028 i 05 67 84 1086373952 32.062 439 i 10.092 475 5 1029 1058841 1089547389 32.078 029 8 10.0957469 1030 1060900 1 092 727 ooo 32.0936131 10.0990163 1031 i 062961 1095912791 32.1091887 10.1022835 1032 1 06 50 24 1 099 104 768 32.1247568 10.1055487 1033 i 06 70 89 1102302937 32.1403173 10. 10881 1 7 J 034 10691 56 1 105 507 304 32.1558704 10.1120726 1035 I 07 12 25 1 108717875 32.1714159 10.1155314 1036 1073206 i 1 1 1 934 656 32.1869539 10.1185882 103? 1075369 1115157653 32.2024844 10. 121 8428 1038 1077444 i 118386872 32.2180074 10.126095.3 1039 1 07 95 21 i 121622319 32.2335229 10.1283457 1040 1081600 1 124864000 32.249031 10.1315941 1041 1083681 1 128 in 921 32.2645316 10.1348403 1042 1 08 57 64 1131366088 32.2800248 10.1380845 1043 1087849 1 134 626 507 32.2955105 IO.T4I 3266 1044 i 08 99 36 1137893184 323109888 10.1445667 1045 1 09 20 25 I 141 166 125 32.3264598 10.1478047 1046 10941 16 1144445336 32.3419233 IO.I5I 0406 1047 i 096209 1147730823 3 2 -3S73794 10.1542744 1048 1 09 83 04 1151022592 32.372 828 i 10.1575062 1049 i 100401 I 154320649 32.3882695 IO.I007359 1050 1 10 25 oo 1 157625000 32.403 703 5 10.1639636 1051 1 104601 1 160935651 32.419 130 1 10.1671893 1052 1 10 67 04 i 164 252 608 32.4345495 10.1704129 1053 1 108809 1167575 877 32.4499615 10.1736344 1054 i ii 09 16 1170905464 32.4653662 10.1768539 1055 1 113025 1174241375 32.480 763 5 10.1800714 1056 1 115136 1177583616 32.4961536 10.1832868 1057 1 117249 i 180932193 32.5115364 IO.I865002 1058 1119364 I 184287 112 32.5269119 10.1897116 1059 I 12 1481 i 187 648 379 32.542 280 2 10.1929209 1060 I 123600 i 191016000 32.5576412 10.1961283 1061 I I2572I 1194389981 32.5729949 10.1993336 1062 1127844 1197770328 32.5883415 10.2025369 SQUARES, CUBES, AND BOOTS. 2 9 I NUMBKE. SqUABK. CUBE. SQUARE ROOT. CUBE ROOT. 1063 I 129969 I 201 157047 32.6036807 10.205 738 2 1064 I 132096 I 204 550 144 32.6190129 10.208 937 5 1065 I 134225 i 207 949 625 32.6343377 10.2121347 1066 I I3 6 356 1211355496 32.6496554 10.21533 1067 1138489 I 214 767 763 32.6649659 10.2185233 1068 I 140624 1218186432 32.6802693 10.221 7146 1069 I 142761 I 221 6ll 509 32-695 565 4 10.2249039 IO7O i 1 4 49 oo i 225 043 ooo 32.7108544 I0.22809I 2 1071 1147041 I 228480911 32.7261363 IO.23I 2766 1072 i 14 91 84 I 231 925 248 32.7414111 10.2344599 1073 1151329 1235376017 32.7566787 10.2376413 1074 i 15 34 76 I 238 833 224 32.7719392 10.240 820 7 1075 1 15 56 25 1242296875 32.7871926 10.2439981 1076 1157776 1245766976 32.8024389 10.2471735 1077 i I599 2 9 1249243533 32.8176782 10.250347 1078 1 16 20 84 I 252 726 552 32.8329103 10.2535186 1079 1 164241 1256216039 32.8481354 10.256 688 i I080 i 166400 I 259 7I2OOO 32.8633535 10.2598557 1081 1 168561 I 263214441 32.8785644 10.2630213 1082 1 170724 I 266 723 368 32.8937684 10.266 185 1083 1172889 I 270 238 787 32.9089653 10.2693467 1084 H75056 I 273 760 704 32.9241553 10.2725065 1085 1177225 I 277 289 125 32.9393382 10.2756644 1086 i 179396 I 280 824 056 32.9545141 10.2788203 1087 i 181569 1284365503 32.969683 10.2819743 1088 1183744 1287913472 32.984845 10.285 1264 1089 i 185921 I 2 9 I 467 969 33 10.288 276 5 1090 1 188100 i 295 029 ooo 33.015 148 10.291 424 7 1091 1 190281 1298596571 33.030 289 1 10.2945709 1092 1 192464 1302 170688 33.0454233 10.2977153 1093 1 194649 1 305 75i 357 ' 33-0605505 10.3008577 1094 1 196836 1309338584 33-0756708 10.3039982 1095 1 199025 1312932375 33.0907842 10.307 136 8 1096 I 20 12 l6 1316532736 33.1058907 10.310 273 5 1097 1203409 1320139673 33.1209903 10.3134083 1098 I 205604 1 323 753 192 33.136083 10.316541 1 1099 I 20 78 OI i 327 373 299 33.1511689 10.3196721 IIOO I2IOOOO 1331000000 33.1662479 10.322 801 2 1 1 01 I 21 22 OI I33463330I 33-18132 10.3259284 IIO2 I 214404 1338273208 33.1963853 10.3290537 1103 I 216609 1341919727 33.2114438 10.332 177 1104 i 21 88 16 1 345 572 864 33.2266955 10.3352985 1105 I 22 IO 25 1 349 232 625 33.2415403 10.3384181 1106 I 22 32 36 1352899016 33-2565783 10.341 535 8 1107 1225449 1356572043 33.2716095 10.344651? 1108 I 22 76 64 1360251712 33.2866339 10.347 765 7 1109 I 22 98 8l i 363 938 029 33-301 651 6 10.3508778 I IIO I 23 21 OO 1 367 631 ooo 33.3166625 10.353988 mi I23432I i 37 J 330 631 33.3316666 10.3570964 III2 1236544 1375036928 33.346664 10.360 202 Q III3 I 23 87 69 1378749897 33.3616546 10.363 307 6 III4 I 240996 i 382 469 544 33.3766385 10.3664103 mS I 24 32 25 i 386 195 875 33.391 615 7 10.3695113 1116 i 24 54 56 1389928896 33.406 586 2 10.3726103 1117 i 24 76 89 1393668613 33.4215499 !0.375 707 6 m8 1249924 1 397 4 J 5 3 a 33436b07 10.378 803 292 SQUARES, CUBES, AND ROOTS. NUMBER. } SQUARE. CtTBH. SVTARB ROOT. CUBK ROOT. IIIQ i 25 21 61 i 401 168 159 33-45I4573 10.3818965 1 120 1254400 i 404 928 ooo 33.466401 I 10.384 988 2 II2I 1256641 i 408 694 561 3348l 338 I 10.388 078 I 1122 i 25 88 84 1412467848 33.4962684 10.391 166 i 1123 i 26 1 1 29 i 416 247 867 33.511 1921 10.3942523 1124 1263376 1 420 034 624 33.5261092 10.3973366 1125 i 26 56 25 1423828125 33.5410196 10.4004192 1126 i 26 78 76 1 427 628 376 33-5559234 10.4034999 1127 1270129 i 431 435 383 33.5708206 10.4065787 1128 i 27 23 84 i 435 249 152 33-585 7" 2 10.4096557 1129 i 274641 1439069689 33.6005952 10.412 73,1 II3O i 276900 1 442 897 ooo 33.6154726 10.415 8044 II3I i 279161 i 446 731 091 33.6303434 10.418 876 1132 i 28 14 24 .145057108 33.645 207 7 10.421 945 8 "33 i 28 36 89 1454419637 33.6600653 10.4250138 "34 i 28 59 56 i 458 274 104 33.674 916 5 10.428 08 "35 i 28 82 25 1462135375 33.689 761 10.431 1443 1136 i 29 04 96 i 466 003 456 33.7045991 10.4342069 "37 i 292769 1469878353 33.7194306 10.437 267 7 1138 1295044 1473760072 33.7342556 10.440 326 7 "39 i 29 73 21 1477648619 33.7490741 10.4433839 1140 i 299600 1481 544000 33.7638860 10.4464393 1141 1301881 I 485 446 221 33.7786915 10.4494929 1142 1304164 M89355288 33-7934905 10.4525448 "43 1306449 i 493 271 207 33.808283 10.4555948 1144 1308736 1497193984 33.8230691 10 458 643 i "45 131 1025 1501 123625 33.8378486 10.461 6896 1146 i 3 1 33 16 i 505 060 136 33.8526218 10.4647343 "47 1315609 1509603523 33.8673884 10.4677773 1148 1317904 1 512 953 792 33.882 148 7 10.4708185 "49 i 32 02 01 i 516910949 33.8969025 104738579 1150 i 32 25 oo 1520875000 33.9116499 10.4768955 "5i i 32 48 oi 1524845951 33.9263909 10.4799314 "52 1 32 71 04 i 528 823 808 33.9411255 10.482 965 6 "53 1329409 1532808577 33-955 853 7 10.485 998 "54 1331716 i 536 800 264 33-9705755 10.489 028 6 "55 i 33 40 25 1540798875 33.985 291 10.4920575 1156 1336336 i 544 804 416 34 10.4950847 "57 i 33 86 49 1548816893 34.014 702 7 10.4981101 1158 1340964 1552836312 34.029399 10.501 133 7 "59 1343281 i 556 862 679 34.044089 10.504 155 6 1160 1345600 i 560 896 ooo 34.058 772 7 10.5071757 1161 1347921 i 564 936 281 34.0734501 10.5101942 1162 i 35 2 44 1568983528 34.088 121 I 10.5132109 1163 i 35 25 69 1573037747 34.1027858 10.5162259 1164 i 35 48 9 6 1577098944 34.1174442 10.5192391 1165 i 35 72 25 i 581 167 125 34.1320963 10.5222506 1166 i 35 95 56 i 585 242 296 34,1467422 10.5252604 1167 1361889 1589324463 34.1613817 10.528 268 5 168 i 36 42 24 15934*3632 34.176015 10.5312749 169 i 36 65 61 1597509809 34.190642 10.534 279 5 170 i 36 89 oo 1601613000 34.205 262 7 10.537 282 5 171 1371241 1605723211 34.2198773 10.540 283 7 172 1 37 35 84 i 609 840 448 34.2344855 10.543 283 2 "73 1 37 59 29 1613964717 34.2490875 10.546 281 "74 1378276 1618096024 34.263 683 4 10.5492771 SQUARES, CUBES, AND BOOTS. 293 NCMBBB. SQUARE. CUBB. SQUARE ROOT. CCBB ROOT. "75 1380625 1622234375 34.278273 10.552 271 5 I 7 6 I 38 29 76 1626379776 34.2928564 10.555 264 2 177 1385329 1630532233 34-3074336 10.5582552 I 7 8 1387684 163469X752 34.3220046 10.5612445 179 1390041 1638858339 34-3365694 10.564 232 2 180 i 39 24 oo I 643 032000 34.351 128 I 10.567218 I 181 1394761 I 647 212 741 34.3656805 10.5702024 182 1397124 1651400568 34.380 226 8 10.5731849 183 1399489 1655595487 34-394 767 10.576 165 8 1184 1 40 18 56 i 659 797 504 34.409301 i 10.5791449 1185 i 40 42 25 i 664 006*625 34.4238289 10.5821225 1186 i 40 65 96 i 668 222 856 34.4383507 10.5850983 1187 1408969 1 672 446 203 34.4528663 10.588 072 5 1188 1411344 1 676 676 672 34-467 375 9 10.591 045 1189 1413721 1680914269 34.4818793 10.5940158 1190 i 41 61 oo 1685159000 344963766 10.596985 1191 1418481 1689410871 34.5108678 10.5999525 1192 1420864 1693669888 34.525353 10.6029184 "93 i 42 32 49 1697936057 34-5398321 10.605 882 6 1194 i 42 56 36 I 702 209 384 34-554305 1 10.608 845 i ii95 i 42 80 25 1706489875 34.568 772 10.611 806 1196 1430416 1710777536 34-5832329 IO.6l4 765 2 1197 1432809 1715072373 34.5976879 10.617 722 8 1198 i 43 52 04 1719374392 34.612 1366 10.620 678 8 1199 1437601 1723683599 34.6265794 10.6236331 1200 1440000 I 728000000 34.641 016 2 10.626 585 7 1 201 1442401 1732323601 34.6554469 10.629 536 7 1202 1444804 1 736654408 34.6698716 10.632 486 1203 1447209 1740992427 34.6842904 10.6354338 1204 1449616 i 745 337 664 34.698 703 1 10.6383799 1205 1452025 1749690125 34.7131099 10.6413244 I2O6 1454436 i 754 049 816 34.7275107 10.644 267 2 1207 1456849 1758416743 34.741 905 5 10.647 208 5 1208 i 45 92 64 i 762790912 34.7562944 10.650 148 I2O9 1461681 1767172329 34.7706773 10.653 086 I2IO i 46 41 oo i 771561000 34-7850543 10.656 022 3 I2II 1466521 17759569^ 34-7994253 10.658957 1212 1468944 1780360128 34.8137904 10.661 890 2 1213 1471369 i 784 770 597 34.8281495 10.664821 7 1214 147370 1789188344 34.842 502 8 10.667 751 6 1215 1476225 I7936i3375 34.8568501 10.6706799 1216 i 47 86 56 1798045696 34.871 191 5 10.6736066 1217 1 48 10 89 1 802 485 313 34.885 527 i 10.6765317 1218 1483524 i 806 932 232 34.8998567 10.6794552 1219 1485961 1811386459 34.9141805 10.6823771 1220 1 48 84 oo 1815848000 34.9284984 106852973 1221 1490841 i 820 316 861 34.9428104 10.688 216 1222 1493284 i 824 793 048 34.9571166 10.691 133 i 1223 M95729 i 829 276 567 34.9714169 10.694 048 6 1224 i 49 81 76 1833767424 34.9857114 10.6969625 1225 1500625 i 838 265 625 35 10.699 874 8 1226 1503076 i 842 771 176 35.0142828 10.7027855 I22 7 1505529 i 847 284 083 35.0285598 10.7056947 1228 i 50 79 84 1851 804352 35.0428309 10.7086023 1229 1510441 1856331989 35.0570963 10.7115083 1230 1512900 i 860 867 ooo 35.07i.3558 10.7144127 BB* 294 SQUARES^ CUBES, AND BOOTS. NUMBER. SQUARE. CUBE. SQUARE ROOT. CUBK ROOT. 1231 I5I536I 1865409391 35.0856096 10.7173155 1232 I5I7824 1869959168 35.0998575 10.7202168 1233 I 52 02 89 1874516337 35.1140997 10.7231165 1234 I 52 27 56 I 879 080 904 35.1283361 10.7260146 1235 I 52 52 25 1883652875 35.1425568 10.728911 2 1236 1527696 i 888 232 256 35- I 5679 I 7 10.731 8062 1237 1530169 1892819053 35.1710108 10.7346997 1238 1532644 i 897 413 272 35.1852242 10.737 59 1 6 1239 i 53 5 1 21 1902014919 35.1994318 10.740481 9 1240 1537600 1 906 624 ooo 35-2I36337 10.7433707 1241 i 540081 1 911 240521 35.2278299 10.7462579 1242 i 54 25 64 1915864488 35.2420204 10.7491436 1243 1545049 1920495907 35.2562051 10.7520277 1244 1547536 1 925 134 784 35.2703842 10.7549103 1245 i 55 oo 25 1 929 781 125 35.284 557 5 10.7577913 1246 i 55 25 16 1934434936 35.298 725 2 10.7606708 1247 1555009 1939096223 35.3128872 I0 - 763 548 8 1248 i 55 75 04 i 943 764 992 35.3270435 10.766425 2 1249 i 56 oo 01 1948441249 35.341 194 i 10.7093001 1250 i 56 25 oo 1953125000 35-355 339 i 10.7721735 1251 1565001 i 957816251 35.3694784 10.7750453 1252 i 56 75 04 1962515008 35.383612 10.7779156 1253 i 570009 1967221 277 35-397 74 10.7807843 1254 i 57 25 16 1971935064 35.4118624 10.783651 6 1255 1575025 1976656375 35.4259792 10.7865173 1256 1577536 i 981 385 216 35.4400903 10.7893815 1257 1580049 i 986 121 593 35-454 195 8 10.792 244 i 1258 i 58 25 64 1990865512 35.4682957 10.7951053 1259 1585081 1995616979 35.48239 10.7979649 1260 i 58 76 oo 2000376000 35.4964787 10.800 823 1261 1590121 2005 142581 35.5105618 10.803 679 7 1262 1592644 2009916728 35-5246393 10.806 534 8 1263 1 59 5i 69 2014698447 35.5387113 10.8093884 1264 1597696 2019487744 35.5527777 10.8122404 1265 1 60 02 25 2 O24 284 625 35.5668385 10.8150909 1266 1602756 2 O29 089 096 35.5808937 10.81794 1267 1 60 52 89 2033901 163 35-5949434 10.820 787 6 1268 1 60 78 24 2 038 720 832 35.6089876 10.8236336 1269 1 61 03 61 2043548109 35.6230262 10.826 478 2 1270 161 2900 2 048 383 OOO 35.6370593 10.8293213 1271 1 61 5441 20532255II 35.6510869 10.832 162 9 1272 1 61 79 84 2 058 075 648 35.665 109 10.835 003 1273 i 62 05 29 2062933417 35.679 125 5 10.837841 6 1274 i 62 30 76 2 067 798 824 35-6931366 10.8406788 1275 i 62 56 25 2072671875 35.707 142 i 10.8435144 1276 i 62 81 76 2077552576 35.7211422 10.846 348 5 1277 1630729 2 082 440 933 35-735 136 7 10.849 J 8i 2 1278 i 63 32 84 2087336952 35.7491258 10.8520125 1279 1635841 2 092 240 639 35.7631095 10.854 842 2 1280 i 63 84 oo 2097 152000 35.7770876 10-8576704 I28l 1640961 2 102 071 041 35.7910603 10.860 497 2 1282 1643524 2 106 997 768 35.805 027 6 10.863 322 5 1283 1 64 60 89 2 III932 187 35.8189894 10-866 146 4 1284 i 64 86 56 2116874304 35.8329457 10.8689687 1285 i 65 12 25 2 121 824 125 35.8468966 10-871 789 7 ,'286 1653796 2 126781656 35.8608421 10.874609 i SQUARES, CUBES, AND BOOTS. 2 9 5 SQUARB. CUBB. SCIUARH ROOT. CUBE ROOT. 1656369 2I3I746003 35.874 782 2 10.877427 I 1658944 2 136 719 872 35.8887169 10.880 243 6 1661521 2 141 700 569 35.002 646 I 10.8830587 16641 oo 2 146 689 000 35.9165699 10.8858723 l66668l 2151685 171 35.9304884 10.8886845 1 66 92 64 2156689088 35.9444015 10.891 495 2 167 1849 2 161 700 757 35.9583092 10.894 304 4 1674436 2 166 720 184 35.9722115 10.8971123 1 67 70 25 2171747375 35.9861084 10.8999186 i 6796 16 2176782336 36 10.902 723 5 1682209 2 181 825 073 36.0138862 10.905 526 9 i 68 48 04 2186875592 36.027 767 I 10.908329 1687401 2191933899 36.041 642 6 10.911 1296 1690000 2 197000000 36.0555128 10.913 928 7 i 69 26 01 2 202 073 901 36.0693776 10.916 726 5 i 69 52 04 2 207 155 608 36.0832371 10.9195228 1697809 2212 245 127 36.0970913 10.9223177 i 700416 2217342464 36. 1 10 940 2 10.925 in i 1703025 2 222 447 625 36.1247837 10.9279031 1 70 5 6 36 2227560616 36.138622 10.9306937 i 70 82 49 2 232 681 443 36.152455 10.933 482 9 1710864 2237810112 36.1662826 10.9362706 1713481 2 242 946 629 36.180105 10.939 OS 6 9 i 716100 2 248 091 OOO 36.1939221 10.941 841 8 1718721 2253243231 36.207 734 10.9446253 1721344 2 258 403 328 36.221 5406 10.9475074 1723969 2263571297 36.2353419 10.950 1 88 i 72 65 96 2268747144 36.249 137 9 10.9529673 i 72 92 25 2273930875 36.262 928 7 10.955 745 * 1731856 2 279 122 496 36.2767143 10.9585215 1734489 2 284 322 013 36.2904946 10.961 296 5 i 73 71 24 2289529432 36.3042697 10.964070 i 1739761 2 294 744 759 36.3180396 10.966 842 3 1742400 2299968000 36.3318042 10.9696131 1745041 2 305 199 161 36.345 563 7 10.972 382 5 1747684 2 310 438 248 36.3593179 10.9751505 1750329 2315685267 36.373067 10.9779171 1752976 2320940224 36.3868108 10.980 682 3 1755625 2326203125 36.4005494 10.9834462 1758276 2 33 1 47397 6 36.414 282 9 10.986 208 6 i 76 09 29 2336752783 36.428 01 1 2 10.9889696 1763584 2342039552 36.4417343 10.9917293 i 766241 2347334289 36.4554523 10.9944876 i 768900 2 352 637 ooo 36.469 165 10.997 244 5 1771561 2357947691 36.482 872 7 ii 1774224 2363266368 36.4965752 11.0027541 1776889 2368593037 36.5102725 11.0055069 1 77 95 56 2373927704 36.5239647 11.0082583 i 78 22 25 2379270375 36.5376518 n.oii 0082 i 78 48 96 2384621056 36.5513338 11.0137569 1787569 2389979753 36.5650106 11.016504 i 1790244 2395346472 36.5786823 11.01925 i 792921 2400721 219 36.5923489 11.0219945 1795600 2 406 104 000 36.6060104 11.0247377 i 798281 2411494821 36.6196668 11.0274795 18009647 2416893688 36.6333181 11.0302199 296 SQUARES, CUBES, AND BOOTS. NUMBKR. SQUARE. CUBE. SQUARE ROOT. CUB ROOT. 1343 1803649 2 422 3OO 607 36.646 964 4 11.032959 1344 1806336 2427715584 36.660 605 6 11.0356967 1345 I 809025 2433138625 36.674 241 6 11.038433 1346 i 81 17 16 2 438 569 736 36.687 872 6 11.041 168 1347 i 81 44 09 2444008923 36.7014986 11.043901 7 1348 i 81 71 04 2449456192 36.7151195 11.0466339 1349 1819801 2454911549 36.7287353 11.0493649 1350 i 82 25 oo 2 460 375 ooo 36.7423461 11.0520945 1351 i 82 52 01 2465846551 36.7559519 11.054822 7 1352 1827904 2471326208 36.7695526 11.0575497 1353 1830609 2476813977 36.7831483 11.060275 2 1354 1833316 2482309864 36.796739 11.0629994 1355 1836025 2487813875 36.8103246 II.o657222 1356 1838736 2493326016 36.823 905 3 11.0684437 1357 i 84 14 49 2 498 846 293 36.8374809 11.071 1639 1358 i 84 41 64 2504374712 36.8510515 11.0738828 1359 1846881 2509911279 36.8646172 11.0766003 1360 1849600 2 5i5 456 ooo 36.8781778 11.0793165 I36l I 85 23 21 2521008881 36.891 733 5 11.082031 4 1362 1855044 2 526 569 928 36.905 284 2 11.0847449 1363 1857769 2 532I39I47 36.9188299 11.0874571 *3 6 4 1860496 2537710544 36.9323706 11.090 1679 1365 i 86 32 25 2543302125 36.9459064 11.0928775 1366 1865956 2548895896 36.9594372 11.0955857 1367 1 86 86 89 2554497863 36.9729631 11.0982926 1368 1 87 14 24 2 560 108 032 36.986484 11.1009982 1369 1874161 2565726409 37 11.1037025 1370 1876900 2571353000 37.013511 11.106405 4 J37i 1879641 2576987811 37.0270172 11.109107 1372 1882384 2582630848 37.0405184 ii. in 8073 1373 1885129 2588282117 37.0540146 11.1145064 1374 i 88 78 76 2593941624 37.067 506 11.117204 i 1375 1890625 2599609375 37.0899924 11.1199004 1376 1893376 2605285376 37.094474 11.1225955 1377 i 89 61 29 2610969-633 37.1079506 11.1252893 1378 1898884 2 6l6662 152 37.1214224 11.127981 7 1379 1,901641 2622362939 37.1348893 11.1306729 1380 1 90 44oo 2 628 072 OOO 37.1483512 11.1333628 1381 1 9071*61 263378934! 37.1618084 11.1360514 1382 1909924 2-639514968 37.1752606 11.1387386 1383 191.2689 2-645248887 37.1887079 11.1414246 1384 1 9 I% 54 5 6 2650-991 104 37.2021505 11.1441093 1385 i 91 82 25 2 656 741 625 37.2155881 1 1 . 146 792 6 1386 1920996 2 662 50O 456 37.2290209 11.1494747 1387 1923769 2 668 267 603 37.2424489 11.1521555 1388 1926544 2 674 043 072 37.255 872 11.154835 1389 1929321 2 679 826 869 37.2692903 "IS7SI33 1390 1932100 2685619000 37.282 703 7 11.1601903 1391 1 93 48 81 2691419471 37.2961124 11.162865 9 1392 1937664 2 697 228 288 37.3095162 11.1655403 1393 1940449 2703045457 37.3229152 11.1682134 1394 1943236 2 708 870 984 37-3363094 11.1708852 J 395 1 94 60 25 2714704875 37.3496988 ".1735558 1396 1948816 2 7 20 547 136 37.3630834 11.176225 1397 1951609 2726397773 37.3764632 11.178893 J 398 1954404 2 732 256 792 37.3898382 11.1815598 SQUARES, CUBES, AND ROOTS. NfMBKR, SVARK. CCBB. SQUARE ROOT. CUBB ROOT. 1399 I9S720I 2 738 I2 4 199 37.403 208 4 11.1842252 I4OO 1960000 2744000OOO 37-4I65738 11.1868894 1401 i 96 28 01 2 749 884 201 37.4299345 11.1895523 1402 I 96 56 04 27557768o8 37.4432904 11.1922139 1403 1968409 2 761 677 827 37.4566416 11.1948743 1404 I 97 12 16 2 767 587 264 37.469988 II I 975334 1405 1974025 2773505125 37.4833296 11.200 191 3 1406 i 97 68 36 277943I4IO 37.4966665 11.2028479 1407 1979649 2 785 366 143 37.5099987 11.2055032 1408 1982464 279I3093I2 37.5233261 11.208 1573 1409 1985281 2 797 260 929 37.5366487 11.2108101 I4IO 1988100 2803221000 37.5499667 11.213461 7 I 4 II 1990921 2809189531 37-5632799 II.2IOII2 1412 1993744 2815166528 37.5765885 11.218761 I 1413 1996569 2821 151997 37.5898922 11.2214089 1414 1999396 2827145944 37.603 191 3 11.2240054 MIS 2 OO 22 25 2 833 148 375 37.6164857 11.2267007 I4l6 2 OO 50 56 2 839 159 296 37.629 775 4 11.2293448 1417 2007889 2845178713 37.6430604 11.2319876 I4l8 2 OI 07 24 2851 206632 37.6563407 11.2346292 1419 20I356I 2857243059 37.6696164 11.2372696 1420 2OI 6400 2863288000 37.6828874 11.2399087 1421 2OI 9241 2869341461 376961536 11.2425465 1422 2 02 20 84 2 875 403 448 37.7094153 II.245I83I 1423 2 02 49 29 2881473967 37.722672 2 11.2478185 1424 2 02 77 76 2887553024 37.7359245 11.2504527 1425 2 03 06 25 2893640625 37.7491722 11.2530856 1426 2033476 2899736.776 37.7624152 11.2557173 1427 2 03 63 29 2 005 841 483 37-7756535 11.2583478 1428 2 03 91 84 29II954752 37.7888873 11.200977 1429 2 O4 2O 41 2918076589 37.802 1163 11.263605 143 2044900 2 924 2O7 OOO 37.8153408 11.266231 8 1431 2047761 2930345991 37.8285606 11.2688573 1432 2 O5 00 24 2936493568 37.8417759 11.271481 6 1433 2053489 2942649737 37.8549864 11.2741047 1434 2 05 63 56 2948814504 37.8681924 11.2767266 1435 2 05 92 25 2954987875 37.8813938 11.2793472 1436 2062096 2 O6l 169 856 37.8945906 11.2819666 1437 2064969 2967360453 37.9077828 11.2845849 1438 2067844 2973559672 37.9209704 11.287201 9 1439 2 O7 07 2 1 2979767519 37-934I535 11.289817 7 1440 2 07 36 00 2 985 984 000 37-9473319 11.2924323 1441 2076481 2992 2O9 121 37.960 505 8 11.2950457 1442 2079364 2 998 442 888 37-973 675 * 11.2976579 1443 2 O8 22 49 3004685307 37.9868398 11.3002688 1444 2085136 3 oio 936 384 38 11.3028786 1445 2 08 80 25 3017196125 38.0131556 11.3054871 1446 20909 16 3023464536 38.026 306 7 11.3080945 1447 2 09 38 09 3029741623 38.0394532 11.3107006 1448 2 09 67 04 3036027392 38.052 595 2 "3133056 1449 209060I 3042321849 38.065 732 6 11.3159094 1450 2 10 25 00 3 048 625 ooo 38.0788655 11.3185119 1451 2 10 54 01 3054936851 38.0919939 11.321 1132 1452 2 10 83 04 3061257408 38.1051178 n-3237 I 34 M53 2 II 1209 3067586677 38.1182371 11.3263124 1454 2 1141 1 6 3073924664 38.1313519 11.3289102 298 SQUARES, CUBES, AND ROOTS. NUMBBB. SQUARE. CUBB. SQUARE ROOT. CUBE ROOT. 1455 211 7025 3080271375 38.1444622 11.3315067 1456 2119936 3086-626816 38.1575681 11.3341022 1457 2 122849 3092990993 38.1706693 11.3366964 1458 2125764 3099363912 38.1837662 11.3392894 1459 2I2868I 3 ^S 745 579 38.1968585 11.3418813 1460 2 13 I60O 3112 136000 38.2099463 11.3444719 1461 2I3452I 3118535181 38.2230297 11.3470614 1462 2*37444 3124943128 38.2361085 11.3496497 1463 2 14 03 69 3I3I359847 38.249 182 9 11.3522368 1464 2 14 32 96 3137785344 38.262 252 9 11.3548227 1465 2 14 62 25 3144219625 38.2753184 11-3574075 1466 2 14 91 56 3150662696 38.2883794 "359991 I 1467 2152089 3157114563 38.301 436 11.3625735 1468 2155024 3163575232 38.3144881 II-365I547 1469 2157961 3170044709 38.3275358 II-3677347 1470 2 160900 3*76 523 ooo 38.340579 "3703136 1471 2163841 3183010111 38-3536178 11.372891 4 1472 2166784 3189506048 38.3666522 II-3754679 1473 2 16 97 29 3196010817 38.3796821 11.3780433 1474 2 172676 3202524424 38.3927076 11.3806175 1475 2175625 3209046875 38.405 728 7 11.3831906 1476 2178576 3215578176 38.4187454 11.3857625 M77 2181529 3222118333 38.431 757 7 "3883332 1478 2 18 44 84 3228667352 38.444 765 6 11.3909028 1479 2187441 3235225239 38.457 769 i 11.3934712 1480 2 190400 3241792000 38.470 768 i 11.3960384 1481 2193361 3248367641 38.483 762 7 11.3986045 1482 2196324 3254952168 38.496 753 11.401 1695 1483 2 19 92 89 3261545587 38-509 739 11.4037332 1484 2 20 22 56 3268147904 38.522 720 6 11.4062959 1485 2205225 3274759125 38.535 697 7 11.4088574 1486 2208196 3 281 379 256 38.5486705 11.4114177 1487 221 IIO9 3288008303 38.5616389 11.4139769 1488 2214144 3 294 646 272 38.574603 11.4165349 1489 221 7121 3301293169 38.587 562 7 11.4190918 1490 2220IOO 3307949000 38.6005181 1 1 .420 647 6 1491 2223081 3314613771 38.6134691 11.4242022 1492 2226064 3321287488 38.6264158 11.4267556 1493 2 22 90 49 3327970157 38-6393582 11.4293079 1494 2 23 20 36 3334661 784 38.652 296 2 11.4318591 1495 2235025 3341362375 38.6652299 11.4344092 1496 2 23 80 16 3348071936 38.6781593 11.4369581 1497 2 24 10 09 3354790473 38.6910843 11.4395059 1498 2 24 40 04 3361517992 38.704005 11.4420525 1499 2 24 70 OI 3368254499 38.7169214 11.444598 1500 2 25OOOO 3375000000 38.7298335 11.4471424 1501 2 25 30 01 338i7545oi 38.742 741 2 11.4496857 1502 2 25 OO 04 3388518008 38.7556447 11.4522278 1503 2259009 3395290527 38.7685439 11.4547688 1504 2 26 20 16 3 402 072 064 38.7814389 11.4573087 1505 2 26 50 25 3 408 862 625 38.7943294 11.4598474 1506 2268036 3415662216 38.807 2158 11.462385 1507 2 27 10 49 3422470843 38.820 097 8 11.4649215 1508 2 27 40 64 3429-288512 38.832 975 7 1 1 .467 456 8 1509 2 27 70 81 3436115229 38.845 849 i 11.4699911 1510 2280100 3442951000 38.8587184 11.4725243 SQUARES, CUBES, AND ROOTS. 299 DUMBER. SQUARE. CUBE. SQUARE ROOT. CUBK ROOT. 15" 22831 21 3 449 795 831 38.8715834 11.4750562 1512 2286144 3456649728 1 38.8844442 11.4775871 I 5 I 3 2289169 3463512697 38.8973006 11.4801169 1514 2299196 3470384744 38.9101529 11.4826455 2 29 52 25 3477265875 38.923 ooo 9 11.485 1731 1516 2 29 82 56 3484156096 38.935 844 7 11.4876995 J 5 J 7 2301289 3491055413 38.9486841 11.4902249 1518 2304324 3497963832 38.9615194 11.4927491 1519 2307361 3 504 88 1 359 38.9743505 11.4952722 1520 2 31 04 00 3511808000 38.9871774 11.4977942 1521 2313441 3518743761 39 11.5003151 1522 2316484 3525688648 39.0128184 11.5028348 1523 2 3 1 95 29 3532642667 39.0256326 ".5053535 1524 2 32 25 76 3539605824 39.0384426 11.507871 I 1525 2325625 3 546 578 125 39.0512483 11.5103876 1526 2328676 3 553 559 576 39.0640499 11.512903 J 527 2331729 3560558183 39.0768473 11.5154173 1528 2 33 47 84 3567549952 39.0896406 11.5179305 1529 2337841 3574558889 39.1024296 11.5204425 *530 2340900 3581577000 39.1152144 11.5229535 1531 2343961 3 588 604 291 39.1279951 ".5254634 1532 2 34 70 24 3 595 640 768 39.1407716 11.5279722 J 533 2350089 3 602 686 437 39- J 535439 ".5304799 1534 2 35 3i 56 3609741304 39.166312 11.5329865 1535 2356225 3616805375 39.179076 "535492 1536 2359296 3623878656 39.1918359 "5379965 J 537 2362369 3630961 153 39.2045915 11.5404998 1538 2365444 3 638 052 872 39.2173431 11.5430021 J 539 2368521 3645153819 39.2300905 ".5455033 1540 237 1600 3652264000 39.242 833 7 11.5480034 2374681 3659383421 11.5505025 1542 2377764 3666512088 39-268307 8 11.5530004 1543 2380849 3673650007 39.281 038 7 "5554973 1544 2383936 3680797184 39.2937654 "-557993I t545 2387025 3687953625 39.306488 11.5604878 1546 23901 16 3695119336 39.3192065 11.5629815 1547 2393209 3702294323 39.3319208 11.565474 1548 2396304 3709478592 39.344631 i 11.5679655 1549 2399401 3716672149 39-3573373 ".5704559 1550 2 40 25 OO 3 723 875 ooo 39.3700394 11.5729453 2 40 56 OI 3731087151 39-3827373 "5754336 1552 2 40 87 04 3738308608 39-3954312 11.5779208 1553 241 1809 3 745 539 377 39.408 121 11.5804069 1554 2 41 49 16 3752779464 39.4208067 11.5828919 1555 2418025 3760028875 39.4334883 "5853759 1556 2421136 3 767 287 616 39.446 165 8 11.5878588 1557 2424249 3 774 555 693 39.4588393 11.5903407 1558 2427364 3781833112 39.471 508 7 11.5928215 1559 2 43 04 81 3789119879 39.484174 "5953013 1560 2433600 3 796 416 ooo 39.496 835 3 "5977799 1561 2436721 3 803 721 481 39.5094925 11.6002576 1562 2439844 3811036328 39.522 145 7 11.6027342 1563 2442969 3818360547 39-5347948 11.6052097 1564 2446096 3 825 641 144 39-5474399 11.6076841 1565 2449225 3833037125 39.5600809 11.6101575 1566 2452356 3840389496 39.5727179 11.6126292 300 SQUAKES, CUBES, AND ROOTS. NUMBER. SQUABB. CUBB. SQUARE ROOT. CUBE ROOT. 1567 2455489 3847751263 39-585 350 8 11.615 IQI 2 1568 2 45 86 24 3 855 122 432 39-597 979 7 11.6175715 1569 246 1761 3 862 503 009 39.6106046 11.620040 7 1570 2 46 49 oo 3 869 893 ooo 39.623 225 5 11.6225088 1571 2 46 80 41 3877292411 39.635 842 4 11.6249759 I57 2 2 47 1 1 84 3 884 701 248 39.648 455 2 11.627442 T 573 2 47 43 29 3892119517 39.661 064 11.629907 1574 2 47 74 76 3899547224 39.673 668 8 11.632371 1575 2 48 06 25 3 906 984 375 39.686 269 6 11.6348339 1576 2 48 37 76 3914430976 39.6988665 11.6372957 1577 2 48 69 29 3921887033 39.7114593 11.6397566 1578 2 49 oo 84 3929352552 39.7240481 11.642 2164 1579 2493241 3936827539 39.736 632 9 11.6446751 1580 2 49 64 oo 3944312000 39.7492138 11.6471329 1581 2 49 95 61 3951805941 39.7617907 11.6495895 1582 2 50 27 24 3959309368 39-7743636 11.6520452 1583 2 50 58 89 3 966 822 287 39.7869325 11.6544998 1584 2 50 90 56 3 974 344 704 39-7994975 11.6569534 1585 2 SI 22 25 3981876625 39.8120585 11.6594059 1586 2515390 3 989 418 056 39.8246155 11.661 8574 1587 2 51 85 69 3996969003 39.8371686 11.6643079 1588 2521744 4004529472 39.8497177 11.6667574 1589 2 52 49 21 4012099469 39.862 262 8 1 1 .669 205 8 1590 2 52 81 oo 4 019 679 ooo 39.874 804 11.6716532 I59i 253 1281 4027268071 39.887 341 3 11.6740996 159? 2534464 4 034 866 688 39.899 874 7 11.6765449 1593 2537649 4 042 474 857 39.912 404 i 11.6789892 1594 2 54 08 36 4 050 092 584 39.924 929 5 11.6814325 1595 2544025 4057719875 39-937451 i 11.6838748 150 2 54 72 16 4065356736 39.9499687 11.686316 i 1597 2 55 04 09 4073003173 399^24824 11.6887563 1598 2 55 3 6 04 4 080 659 192 39-974 992 2 11.691 1955 1599 2 55 68 01 4088324799 39.987 498 11.6936337 1600 2560000 4096000000 40 11.6960709 Uses of preceding table may be extended by aid of following Rules, tc Compute Square or Cube of a higher Number than is contained in it. To Compute Sq.ta.are. When Number is an Odd Number. RULE. Take the two numbers nearest to each other, which, added together, make that sum ; then from sum of squares of these two numbers, multiplied by 2> subtract i, and remainder will give result. To Compute Square or Cxi"be. When Number is divisible by a Number without leaving a Remainder. RULE. If number exceed by 2, 3, or any other number of times, any numb^i contained in table, multiply square or cube of that number in table by square of a j. etc., and product will give result. EXAMPLE. Required square of 1700. 1700 is 10 times 170, and square of 170 is 2 8900. Then, 2 89 oo X io 2 = 2 89 oo oo. a.- What is cube of 2400? 2400 is twice 1200, and cube of 1200 is 1 728000000. Then 1728 ooo ooo x 23 = 13824000000. SQUABES, CUBES, AND BOOTS. JO I EXAMPLE. What is square of 1745? Two nearest numbers are { g 7 ^ 1 = 1745. Then, per table, 8 8 : i 52 25 13 X 2 =r 3 045 026 i = 3 04 50 25. To Compute Sq.uare or Cube Root of a high.er Number th.au is contained, in Table. When Number is divisible by 4 or 8 without leaving a Remainder. RULE. Divide number by 4 or 8 respectively, as square or cube root is required; take root of quotient in table, multiply it by 2, and product will give root required. EXAMPLE. What are square and cube roots of 3200? 3200 -4- 4 = 800, and 3200 H- 8 = 400. Then, square root for 800, per table, is 28. 28 42 71 2, which, being X 2 = 56. 56 85 42 4 root. Cube root for 400, per table, is 7. 368 063, which, being x 2 = 14. 736 126 root. When the Root (which is taken as Number) does not exceed 1600. The Numbers in table are roots of squares or cubes, which are to be taken as numbers. ILLUSTRATION. Square root of 6400 is 80, and cube root of 51200x3 is 80. When a Number has Three or more Ciphers at its right hand. RULE. Point off'number into periods of two or three figures each, according as square or cube root is required, until remaining figures come within limits of table; then take root for these figures, and remove decimal point one figure for every pe- riod pointed off. EXAMPLE. What are square or cube roots of 1 500 ooo? 1 500000 = 150, remaining figure, square root of which= 12. 247 45; hence 1224.745, square root. 1500000=1500, remaining figures, cube root of which = 11.447 14 ; hence 14.4714, c^&e root. To Ascertain Cube Root of* any Number over 16OO. RULE. Find by table nearest cube to number given, and term it assumed cube; multiply it and given number respectively by 2 ; to product of assumed cube add given number, and to product of given number add assumed cube. Then, as sum of assumed cube is to sum of given number, so is root of assumed cube to root of given number. EXAMPLE. What is cube root of 224809? By table, nearest cube is 216000, and its root is 60. 216 ooo X 2 -f- 224 809 = 656 809, And 224 809 X 2 -}- 216 ooo = 665 618. Then 656809 : 665618 :: 60 : 60.804+, r t- To Ascertain Square or Cube Root of a Number con sisting of Integers and. Decimals. RULE. Multiply difference between root of integer part and root of next higher integer by decimal, and add product to root of integer given; the sum will give root of number required. This is correct for Square root to three places of decimals, and for Cube root to seven. Co 3 nr o figures as snnare or ouhft root, is rennired- and ormosit.fi t,o it, in column of roots, take root and point off i or 2 additional places of decimals to those in root, as square or cube root is required, and result is root required. EXAMPLE i. What are square roots of .15, 1.50, and 15.00? In table, 15 has for its root 3.87 298; hence .38 7298 = square root for .15. 150 has for its root 12. 24 74 5 ; hence i. 22 47 45 = square root for 1.50. 1500 has for its root 38.72 98 ; hence 3.87 29 8 = square root for 15. 2 . What are cube roots of .15, 1.50, and 15.00? Add a cipher to each, to give the numbers three places of figures, as .150, 1.500, and 15.000. In table 150 has for its root 5.3133; hence .531 33 = cube root o/.is. 1500 has for its root 11.447 ; hence 1.1447 = cu^ root 0/1.50. 15 has for its root 2.4662; and 15.000, by addition of 3 places of figures, has 24.662 ; hence 2.4662 = cube root of 15.00. To Ascertain. Square or Cu."be Roots of* Decimals alone. RULE. Point off number from decimal point into periods of two or three figures each, as square or cube root is required. Ascertain from table or by calculation root of number corresponding to decimal given, the same being read off by remov- ing the decimal point one place to left for every period of 2 figures if square root is required, and one place for every period of 3 figures if cube root is required. EXAMPLE. What are square and cube roots of .810, .081, and .0081 ? .810, when pointed off = .8i, and ^/.Bi =.9. .081, " " " = .081, " V- 08 ! =.2846. .0081, " " " = .oo8i, " V-8i=.o9. .810, when pointed off = .810, and -^.810 =93217. .081, " " " =.o8i, " ^.081 =.43267. .0081, " " " =.0081, " -^.0081 = .200 83. To Compute 4th Root of* a CT umber* RULE. Take square root of its square root. EXAMPLE. What is the $/ of 1600? ^1600 = 40, and V4= 6 -3 2 45553* To Compute 6th. Root of a IST RULE. Take cube root of its square root. EXAMPLE. What is the $ of 441 ? V44i = 21, and ^21 = 2.7 589 243. FOURTH AND FIFTH POWEBS OF KUMBEES. 303 4th. and. 5th. IPowers of Nnnibers. From i to 150. Number. 4th Power. 5th Power. Number. 4th Power. 5th Power. :'. 4 -^,t I i 64 16777 216 i 073 741 824 2 16 32 65 17850625 i 160290625 3 81 243 66 18974736 i 252 332 576 4 256 1024 67 20I5I 121 i 350 125 107 5 625 3 I2 5 68 21 381 376 1 453 933 568 6 i 296 7 776 69 22667 I2 I 1564031349 7 2401 16807 70 24 oio ooo i 680 700 ooo 8 4096 32768 7i 25 411 681 1804229351 9 6561 5949 72 26 873 856 1934917632 10 IOOOO IOOOOO 73 28398241 2073071593 ii 14641 161051 74 29 986 576 2219006624 12 20736 248 832 75 31640625 2373046875 '3 28561 371 293 76 33 362 176 2 535 525 376 4 38416 537 824 77 35153041 2706784157 15 50625 759375 78 37015056 2887174368 16 65 536 1 048 576 79 38950081 3077056399 X 7 83521 1419857 80 40960000 3276800000 18 104976 i 889 568 81 43046721 3486784401 J 9 130321 2476099 82 45212176 3 707 39 8 432 20 160000 3200000 83 47 458 321 3 939 040 643 21 194481 4084 101 84 49787136 4182119424 22 234 256 5153632 8s 52200625 4437053125 23 279841 6436343 86 54 708 016 4 704 270 176 24 331 776 7962624 87 57 289 761 4984209207 25 390625 9765625 88 59969536 5277319168 26 456 976 11881376 89 62 742 241 5584059449 27 531 441 14348907 9 65610000 5904900000 28 614656 17210368 9 1 68574961 6 240 321 451 29 707 281 20511 149 9 2 71639296 6590815232 30 810000 24300000 93 74 805 201 6956883693 31 923521 28629 1 5 I 94 78074896 7339040224 32 1 048 576 33554432 95 81450625 7 737 809 375 33 1185921 39 '35 393 96 84 034 656 8153726976 34 1336336 45 435 424 88 529 281 8587340257 35 1 500625 52521875 98 92 236 816 9039207968 36 1679616 60466176 99 96 059 601 9509900499 37 38 I 874 161 2085136 69343957 79235168 oo 01 100 OOO OOO 104060401 IOOOOOOOOOO 10510100501 39 231344! 90224199 02 108243216 11*040808032 4 2560000 102400000 03 112550881 11592740743 4i 2 825 761 115856201 04 116985856 12 166 529024 42 3111696 130691232 05 121 550625 12762815625 43 34I880I 147 008 443 06 126 247 696 13382255776 44 3748096 164916224 07 131079601 14025517307 45 4 100625 184528125 08 136048896 14693280768 46 4 477 456 205 962 976 09 141 158 i6i 15386239549 47 4879681 229345007 10 146410000 1 6 105 looooo 48 ' 5308416 254803968 II 151 807041 16850581551 49 5 764 801 282 475 249 12 1 57 35i 936 17623416832 So 5i 6250000 6765201 312500000 345025251 13 14 163047361 168896016 18424351793 19 254 145 824 52 7311616 380204032 15 174900625 20113581875 53 7890481 418195493 16 181 063936 21 003416576 54 8 503 056 459165024 i7 187388721 21924480357 55 9150625 503 284 375 18 193 877 776 22 877 577 568 56 57 9 8 34 49 6 10556001 550731776 601 692 057 '9 20 200 533 921 207360000 23 863 536 599 24883200000 58 11316496 656 356 768 21 214358881 25937424601 59 12117361 714924299 22 221533456 27027081632 60 12960000 777600000 23 228886641 28153056843 61 62 13845841 14776336 844 596 301 916132832 24 25 236421376 244140625 29316250624 30517578125 63 15752961 992 43 6 543 26 252 047 376 31757969376 304 POWERS OF NUMBERS. RECIPROCALS. Number. 4th Power. 5th Power. Number. 4th Power. 5th Power. 127 128 129 260 144641 268 435 456 276922881 33038369407 34 359 738 3 6 8 35723 5i649 139 140 141 37330I64I 384 160000 395 254 161 51 888 844 699 53782400000 55730836701 130 285610000 37129300000 142 406 586 896 57735339232 IS' 294499921 38579489651 143 418 161 601 59 797 108 943 132 303 595 776 40074642432 144 429981696 61 917 364224 133 312900721 41615795893 145 442050625 64097340625 '34 322417936 43 204 003 424 146 454 37 1 8 56 66 338 290976 135 332150625 44 840 334 375 147 466 948 881 68641485507 136 342102016 46 525 874 176 148 479785216 71008211 968 37 352 275 361 48261724457 149 492 884 401 73439775749 1 3 8 362 673 936 50 049 003 1 68 150 506250000 759375ooooo To Compxite 4th Power of a Nximber greater than is contained, in Table. RULE. Ascertain square of number by preceding table or by calculation, and square it; product is power required. EXAMPLE What is 4tb power of 1500? i5oo 2 2250000, and 2 2 50 ooo 2 = 5 062 500000000. To Compnte Sth Power of a Number greater than is contained, in Table. RULE. Ascertain cube of number by preceding table or by calculation, and mul- tiply it by its square; product is power required. To Compxxte 4th and 5th Powers by another Method. RULE. Reduce number by 2 until it is one contained within table. Take power which is required of that number, and multiply it by 16, i6 2 , or i63 respectively for each division, by 2 for 4th power, and by 32, 32% or 323 respectively for each division by 2 for sth power. EXAMPLE. What are the 4th and sth powers of 600? 600 -4- 2 300, and 300 -r- 2 = 150. The 4tti power of 150, per table, 506 250000, w r hich x i6 2 , multiplier for a second division 256 = 129600000000, ^th power. Again, the sth power of 150 = 75937 500000, which X 32 2 , multiplier for a second division 1024 = 77 760000000000 = power. To Compute Gth Power of a Number. RULE. Square its cube. EXAMPLE. What is the 6th power of 2? 2l 2 = 64. To Compute 4th or Sth Root of a N"xxmber per Table. RULE. Find in column of 4th and sth powers number given, and number from which that power is derived will give root required. EXAMPLE. What is the sth root of 3 200000? 3200000 in table is sth power of 20; hence 20 is root required. RECIPROCALS. Reciprocal of a number is quotient arising from dividing i by number; thus, re- ciprocal of 2 is i -f- 2 = . 5 Product of a number and its reciprocal is always equal to i ; thus, 2 x .5 r. Reciprocal of a vulgar fraction is denominator divided by numerator . thus, - = . 5, LOGARITHMS. LOGARITHMS. Hiogarith-ms of Numbers. Logarithms are a series of numbers adapted to facilitate the operation of numerical computation, Addition being substituted for Multiplication, Subtraction for Division, Multiplication for Involution, and Division for Evolution. The Logarithm of a number is the exponent of a power to which 10 must be raised to give that number. It is not necessary, however, that the base should be 10, it may be any other num- ber; but Tables of Logarithms, in common use, are computed with 10 as the base. Thus, Number 100 Log. = 2, as io 2 base and exponent = 100. " 10000 " = 4, " io 4 " " " =10000. The Unit or Integral part of a Logarithm is termed the Index, and the Decimal part the Mantissa; the sum of the index and mantissa is the Logarithm. The Index of the Logarithm of any number, Integral or Mixed, when the base is io, is equal to the number of digits to the left of the decimal point less i. From o to 9, it is o; from io to 99, it is i, and from 100 to 999, it is 2, etc. Thus, logarithm of 3304 = 3.51904, 3 being the index and .51904 the mantissa. The Index of the Logarithm of a Decimal Fraction is a negative number, and is equal to the number of places which the first significant figure of the decimal is re- moved from the place of units. Thus, index of logarithm .005 is 3 or 3, the first significant figure, 5, being re- moved three places from that of units. The bar or minus sign is placed over an index to indicate that this alone is negative, while the decimal part is positive. The Difference is the tabular difference between the two nearest logarithms. The Proportional Part is the difference between the given and the nearest less tabular logarithm. The Arithmetical Complement of a number is the remainder after subtracting it from a number consisting of i, with as many ciphers annexed as the number has integers. When the index of a logarithm is less than io, its complement is ascer- tained by subtracting it from io. II Ki s t r at i on s . Number. Logarithm. 4743 3-676053 474-3 2.676053 47-43 1-676053 4-743 676 053 Number. Logarithm. 4743 -676053 04743 2.676053 .004743 3-676053 Computation of* Negative Indices. To add two Negative Indices. Add them and put the sum negative. As 5 -f- 3 = 8. To add a Positive and Negative Index. Subtract the less from the greater, and to remainder give the positive or negative sign, according as the positive or nega- tive index is the greater. As 6 -\- 2 =; 4, and 6 + 2 = 4. ILLUSTRATION. Add 6. 387 57 and 2. 924 59. 6. 387 57 2.92459 5.31216 Here the excess of i from 13 in the first decimal place, being positive, is carried to the positive 6, which makes 7, and 7 2=5. To Subtract a Negative Index. Change its sign to plus or positive, and then add it as in addition. _ As 3 from 2, = 3 -f 2 = 5. And 5 from 2, = 5 -}- 2 = 3 ; also 3 from 5, =3 + 5 = 2". ILLUSTRATION. Subtract 5. 765 52 from 2. 346 74. 2. 346 74 5-76552 2.581 22 Here, excess of i in the first decimal place used with the .3 in subtracting the .8 from the 1.3 is to be subtracted from the upper number 2, which makes it 3; then 3 + 5=2- CG * 306 LOGARITHMS. To Subtract a Positive Index. Change its sign to negative, and then add as in addition. As 2 2 = 2-1-2 = 4. To Multiply a Negative Index. Multiply the fractional parts by the ordinary rule, then multiply the negative index, which will give a negative product, and when an excess over 10 is to be carried, subtract the less index from the greater, and the re- mainder gives the positive or negative index, according as the positive or negative index is the greater. As 2 X 5 = io, and i to be carried = 9. ILLUSTRATION. Multiply 2.3681 by 2, and 3.7856 by 6. 2.3681 3.7856 2 6 4.7362 14-7136 Here 2X2 = 4, also 3" X 6 = 18, with a positive excess of 4 = "14. To Divide a Negative Index. If index is divisible by divisor, without a remain- der, put quotient with a negative sign. If negative exponent is not divisible by divisor, add such a negative number to it as will make it divisible, and prefix an equal positive integer to fractional part of logarithm; then divide increased nega- tive exponent and the other part of logarithm separately by ordinary rules, and for- mer quotient, taken negatively, will be index to fractional part of quotient. As 6 -=- 3 = 2. io-r-3 requires 2 to be added or 2 to be subtracted, to make it divisible without a remainder, then io-f s^ = 12,72-^-3 = 4, and 2 (the sum subtracted) -r- 3 = .66, the quotient therefore is 4.66. ILLUSTRATION i. Divide 6.324282 by 3. 6. 324 282 -7-3 = 2. 108 094. a. Divide 14.326745 by 9. I4-326745 -f- 9 = 18 -J- 4.326 745 -r- 9 = 2.480749+. Here 4 is added toTJ, that the sum ~i8 may be divided by 9, and as 4 is added, 4 must be prefixed to the fractional part of the logarithm, and thus the value of the logarithm is unchanged, for there is added 4, and 4 = o, or 4 is subtracted and 4 added. To Ascertain Logarithm of a !N~umber by Table. When the Number is less than 101. Look into first page of table, and opposite to number is its logarithm with its index prefixed. ILLUSTRATION. Opposite 7 is .845098, its logarithm; hence 70=1.845098, .7 = 7. 845 098, and . 07 = 2. 845 098. When the Number is between 100 and 1000. RULE. Find the given number in left-hand column of table headed No., and un- der o in next column is decimal part of its logarithm, to which, is to be prefixed a whole number for an index, of i or 2, according as the number consists of 2 or 3 figures. EXAMPLE. What is logarithm of 450, and what of .45 ? Log. 450 = 2.653213, and of .45 = 1.653213. When the Number is between 1000 and io ooo. RULE. Find the three left-hand figures of the number in the left-hand column of the table headed No., and under the 4th figure at top of table is the four last figures of the decimal part of logarithm, to which is to be prefixed the proper index. EXAMPLE. What is logarithm of 4505, and what of .04505? Log. 4505 = 3.653 695, and of .045 05 = 2.653 6 95- LOGARITHMS. 307 When the Number consists of Five Figures. RULE. Find the logarithm of the number composed of the first four figures as preceding, then take the tabular difference from the right-hand column under D and multiply it by the fifth figure; reject the right-hand figure of the product and add the other figures, which are, and are termed, a proportional part to the logarithm found as above, observing that the right-hand figure of the proportional part is to be added to that of the logarithm, and the rest in order. EXAMPLE. Required logarithm of 83 407 ? NOTE. When the number consists of less than 4 figures conceive a cipher an- nexed to make it four. Log. of 8340 (83 407) = 4.921 166 Tabular difference 52, which x 7 (sth figure) = 364 = 364 4.921 202 4 logarithm. The difference of the numbers is nearly proportionate to the difference of their logarithms. Thus, difference between the numbers 8340 and 8341, the next in order, is i, and the difference between their logarithms or tabular difference is 52. The log. of this i in the 4th place is therefore 52. The correction then, for the 7 of the sth place, which is .7 of i in the 4th place, is ascertained by the proportion i : 52 :: .7 : 36.4. The correction is obtained by multiplying the tabular difference by 7, rejecting the right hand figure of the product, if the log. is to be confined to six decimal places. When the Number consists of any Number over Four Figures. RULE. Proceed as for four figures for the first four, multiplying the tabular dif- ference by the excess of figures over 4 and rejecting one right-hand figure of the product for a number of five figures, and two for one of six, and so on. EXAMPLE L Required logarithm of 834079? Log. of 8340 (834079)=: 5.921 166 Tabular difference 52, which X 79= 4108 5.92120708 logarithm. 2. Required logarithm of 8340794? Log. of 8340 (8 340 794) = 6.921 166 Tab. diff. 52, which x 794 (5th, 6th, and jth figures) = 41288 6. 921 207 288 logarithm. Or, Mantissa of 8340 = .921166 u " 7 (sth figure) X 52 tab. dif. = 364 " " 9 (6th )X52 u " = 468 " 4( 7 th " )X52 " " = 208 Log. with index for 7 figures 6.921 207 288 To Ascertain. Logarithm of a Mixecl :N"xnn"ber. RULB. Take out logarithm of the number as if it were an integer or whole num- ber, to which prefix the index of the integral part of the number. EXAMPLE. What is logarithm of 834.0794? Mantissa of log. of 834.0794 = 9 212 073 ; hence log. of 834.0794 = 2.921 207 3. To Ascertain Logarithm of* a Decimal Fraction.. RULE. Take logarithm from table as if the figures were all integers, and prefix index as by previous rules. EXAMPLE. Logarithm of . 1234 = 1.091 305. To Ascertain Logarithm of* a "Vxilgar Fraction. RULE. Reduce the fraction to a decimal, and proceed as by preceding rule. Or, subtract logarithm of denominator from that of numerator, and the difference will give logarithm required. EXAMPLE. Logarithm of jj^? ^ = .1875. Log. .1875 ="1.273001 logarithm. Or, Log. 3 = .477121 1 16 = 1.20412 7.273001 logarithm. 308 LOGARITHMS. To .A-scertain tne Nu.m"ber Corresponding to a Q-iven Logarithm. When the given or exact Logarithm is in the Table. OPERATION. Opposite to first two figures of logarithm, neglecting the index, in column o, look for the remaining figures of the log. in that column or in any of the nine at the right thereof; the first three figures of the number will be found at the left in column under No., and the fourth at top directly over the log. The number is to be made to correspond to index of logarithm, by pointing off decimals or prefixing ciphers. ILLUSTRATION. What is number corresponding to log. 3-963977 ? Opposite to 963977, in page 329, is 920, and at top of column is 4; hence, num- ber = 9204. When the given or exact Logarithm is not in the Table. OPERATION. Take the number for the next less logarithm from table, which will give first four figures of required number. To ascertain the other figures, subtract the logarithm in table from the given logarithm, add ciphers, and divide by the difference in column D opposite the logarithm. Annex quotient to the four figures already ascertained, and place deci- mal point. ILLUSTBATION i. What is number corresponding to log. 5.921 207? Given log. = 5.921 207 Next less in table 5.921166 8340 0=52)4100(78-!- 78 3 6 4 834078 460 _4 l6 Hence, number = 834 078. 44 a. What is number corresponding to log. 3.922853? Given log. = 3.922853 Next less in table 3.922829 8372 D = 52) 2400 (46 + 46 *_ 837^6 320 3'2 iO Hence, number = 8372.46. 8 3VInltiplioation. RULE. Add together the logarithms of the numbers and the sum will give the logarithm of the product. EXAMPLE i. Multiply 345.7 by 2.581. Log. 345.7 =2.538699 2.581= .411788 2. 950 487 log. of product. Number = 892. 251. *. Multiply .03902, 59.71, and .003147. Log. .03902 =2.591287 59.71 =.776047 " .003147 = 3.497897 3. 865 231 log. of product. Number = .007 332 15. Division. RULE. From logarithm of dividend subtract that of divisor, and remainder will give logarithm of the quotient. EXAMPLE. Divide 371.4 by 52.37. Log. 371.4 =2.569842 52.37 = 1.719083 850 7 59 log. of guotient. Number = 7. 091 85. LOGARITHMS. 309 Rule of Three, or Proportion. RULE. Add together the logarithms of the second and third terms, from their sum subtract logarithm of the first, and the remainder will give logarithm of the fourth term. Or, instead of subtracting logarithm of first term, add its Arithmetical Comple- ment, and subtract 10 from its index. EXAMPLE i. What is fourth proportional to 723.4, .025 19, and 3574? As 723.4 log. = _ 2.859379 Is to .02519 *' =2.401228 So is 3574 " =3-553iS5 i-9543 8 3 First term 2-859379 7. 095 004 log. of 4th term. Number = . 124 453. By Arithmetical Complement. ILLUSTRATION. As 723.4 log. = 2.859 379, Ar. com. = 7. 140621 Is to .025 19 u = 2.401 228 So is 3574 " = 3-553I55 i 095 004 log. of tfh term. Number = .124 453. 2. If an engine of 67 IP can raise 57 600 cube feet of water in a given time, what HP is required to raise 8 575 ooo cube feet in like time ? Log. 8 575 ooo = 6.933 234 67 = 1.826075 8.759309 57600 = 4.760422 3. 998 877 log. of 4th term. Number = 9974. 4 IP. 3. If 14 men in 47 days excavate 5631 cube yards, what time will it require to excavate 47 280 at same rate of excavation ? 394. 626 days. Involution. RULE. Multiply logarithm of given number by exponent of the power to which it is to be raised, and the product will give the logarithm of the required power. EXAMPLE. What is cube of 30.71 ? Log. 30.71 = 1.48728 3 4. 461 84 log. of power. Number = 28 962. 73. Evolution. RULE. Divide logarithm of given number by exponent of the root which is to be estracted, and quotient will give logarithm of required root. EXAMPLE i. What is cube root of 1234? Log. 1234 = 3.091315 Divide by 3 = 1.030 438 log. of root. Number = 10. 72601. 2. What is 4th root of .007 654? Log. .007654 = 3.883888 Divide by 4 (here 3 -}- 1 + i) = 1.470 972 log. of root. Number = .295 78. To Ascertain. Reciprocal of a Number. RULE. Subtract decimal of logarithm of the number from .000000; add i to in- dex of logarithm and change its sign. The result is logarithm of the reciprocal EXAMPLE. Required reciprocal of 230? .000000 Log. 230 = 2.361728 3.638 272 = log. of .004 348 reciprocal. 3 io LOGARITHMS. Simple Interest. RULE. Add together logarithm of principal, rate per cent., and time in years, from the sum subtract 2, and the remainder will give logarithm of the interest. EXAMPLE. What is interest on $500, @ 6 per cent., for 3 years? Log. 500 = 2. 698 97 6= .778151 3= .477121 3-954 242 i. 954 242 log. of interest Number = 90 dollar 9. Compound. Interest. RULE. Compute amount of $ i or i, etc., at the given rate of interest for one year for the first term, which is termed the ratio. Multiply logarithm of ratio by the time, add to product logarithm of the principal, and sum is logarithm of the amount. .Logarithms of* Ratios at given Rates Per Cent. Rate. Log. of Ratio. Rate, i Log. of Ratio. Rate. Log. of Ratio. Rate. Log. of Ratio, I .0043214 3-25 .013 890 i 5-5 .0232525 7-75 .0324373 1.25 005 395 3-5 0149403 5-75 .024 2804 8 0334238 i-5 .006466 3-75 .0159881 6 0253059 8.25 0344279 i-75 .0075344 4 0170333 6.25 .0263289 8-5 0354297 2 .0086002 4-25 .018076 i 6-5 .0273496 8.75 .0364293 2.25 .0096633 4-5 .0191163 6.75 .0287639 9 .0374265 2-5 .0107239 4-75 .020154 7 .0293838 9.25 .0384214 2.75 .0117818 5 .021 1893 7-25 0303973 9-5 .0394141 3 .0128372 5-25 .022 222 1 7-5 .031 408 5 9-75 .040 404 3 EXAMPLE. What will $364, at 6 per cent, per annum, compounded yearly, amount to in 23 years? Log. of ratio from above table .025 305 9 *3 364 .5820357 2.561101 3.1431367 log. of amount. Number = 1390. 39 doll Miscellaneous Illustrations. i. What is area and circumference of a circle of 21.72 feet in diameter? Log. of 21.72 1.336860 Log. of 21. 72 2 =2.673720 " .7854=1.895091 " " 2. 568 81 1 = 370. 54 feet area. Log. of 21. 72 =1.33686 ' 3.1416= -497^5 " " j.834 ox = 68. 236 feet circum. a. Sides of a triangle are 564, 373, and 747 feet; what is its area? Log. of sides 564 + 373 + 747 = 2 925 3I2 " " .5 side 61 = 842 564 = 2.444045 " .5 side 6 = 842 373 = 2.671 173 " .5 side c =842 747 = 1.977724 2^10.018 254 Area = Number of 5.009 127 = 102120. 4 feet 3. What is logarithm of 8 3 ' 6 ? Log. X 3 = X log. 8 = 3.6 X. 903 09 = 3. 251 124. Number =1782.89= LOGARITHMS OF NUMBBBS. Logarithms of iN'uxxi'bers. From i to 10000. No. Logarithm. No. 1 Logarithm. No! Logarithm. No. Logarithm. 1 .O 26 1.414973 51 1.70757 76 1.880814 2 3 01 3 27 1.431 364 52 1.716003 77 1.886491 3 477 "i 28 1.447 158 53 1.724276 78 1.892095 4 .60206 29 1.462398 54 I-73394 79 1.897627 5 .69897 30 1.477 J2i 55 1.740363 80 1.90309 6 .778 151 31 1.491 362 56 1.748 188 81 1.908485 7 .845098 32 L505 15 57 1-755 875 82 1.913814 8 .90309 33 1.518514 58 1.763428 83 1.919078 9 954 2 43 34 I-53I 479 59 1.770852 84 1.924279 10 i 35 1.544068 60 1.778 151 85 1.929419 11 1.041 393 36 1.556303 61 1.78533 86 1.934498 12 1.079 J 8i 37 1.568202 62 1.792392 87 I.9395I9 13 I.II3943 38 I-579784 63 I-79934I 88 1.944483 14 1.146 128 39 1.591065 64 1.80618 89 1-94939 15 1.176091 40 1.60206 65 1.812913 90 1.954243 16 1.204 12 41 i. 612 784 66 1.819544 91 1.959041 i? 1.230449 42 1.623249 67 1.826075 92 1.963788 18 1.255273 43 1.633468 68 1.832 509 93 1.968483 19 1.278 754 44 1.643453 69 1.838849 94 1.973 128 20 1.301 03 45 1.653213 70 1.845098 95 1.977 724 21 1.322219 46 1.662 758 71 1.851 258 96 1.982 271 22 1.342423 47 1.672098 72 1.857332 1.986772 23 1.361 728 48 1.681 241 73 1.863323 98 1.991 226 2 4 1.380211 49 1.690 196 74 1.869232 99 1.995635 25 1-39794 50 1.69897 75 1.875061 100 2 No. \o .1 2 3 4 5 6 7 8 9 D 100 00- OOOO 0434 0868 1301 1734 2166 2598 3029 346i 3891 43 161 oo- 43*21 4751 5l8l 5609 6038 6466 6894 7321 7748 8174 428 102 oo- 86 9026 9451 9876 425 102 01- i- 03 0724 1147 157 1993 2415 424 103 01- 2837 3259 368 41 4521 494 536 5779 6i97 6616 420 IO4 oi- 7033 7451 7868 8284 8 7 9116 9532 9947 417 IO4 02- 0361 0775 416 105 02- 1189 1603 2Ol6 2428 2841 3252 3664 4075 4486 4896 412 106 02- 5306 5715 6l25 6533 6942 735 7757 8164 8571 8978 408 107 02- 9384 9789 405 107 03- 0195 06 1004 1408 1812 2216 2619 3021 404 108 03- 3424 3826 4227 4628 5029 543 583 623 6629 7028 400 109 03- 7426 7825 8223 862 9017 9414 9811 398 109 04- 0207 0602 0998 397 110 04- 1393 1787 2l82 2576 2969 3362 3755 4148 454 4932 393 in 04- 5323 5714 6l05 6495 6885 7275 7664 8053 8442 883 389 112 04- 9218 9606 9993 388 112 05- 038 0766 "53 1538 1924 2309 2694 386 "3 05- 3078 3463 3846 423 4613 4996 5378 576 6142 6524 383 114 05- 6905 7286 7666 8046 8426 8805 9185 9563 9942 383 114 06- 032 379 No. O 1 t a 3 4 5 6 7 8 9 D 312 LOGARITHMS OF NUMBERS. No. 0123456789 o6- 0698 1075 1452 1829 2206 ; 2582 2958 3333 3709 4083 06- 4458 4832 5206 558 5953 6326 6699 7071 7443 7815 06- 8186 8557 8928 9298 9668 | 07- 0038 0407 0776 1145 1514 07- 1882 225 2617 2985 3352 3718 4085 4451 4816 5182 07- 5547 59 12 6276 664 7004 7368 7731 8094 8457 8819 07- 9181 9543 9904 08- 0266 0626 0987 1347 1707 2067 2426 08- 2785 3144 3503 3861 4219 4576 4934 5291 5647 6004 08- 636 6716 7071 7426 7781 8136 849 8845 9198 9552 08- 9905 09- 0258 0611 0963 1315 1667 2018 237 2721 3071 09- 3422 3772 4122 4471 482 5169 5518 5866 6215 6562 09- 691 7257 7604 7951 8298 8644 899 9335 9681 10- 0026 10- 0371 0715 1059 1403 1747 2091 2434 2777 3119 3462 10- 3804 4146 4487 4828 5169 551 5851 6191 6531 6871 io- 721 7549 7888 8227 8565 8903 9241 9579 9916 jj_ 0253 ii- 059 0926 1263 1599 1934 227 2605 294 3275 3609 "- 3943 4277 4611 4944 5278 5611 5943 6276 6608 694 ii- 7271 7603 7934 8265 8595 8926 9256 9586 9915 12- . 0245 12- 0574 0903 1231 156 1888 2216 2544 2871 3198 3525 12- 3852 4178 4504 483 5156 5481 5806 6131 6456 6781 12- 7105 7429 7753 8076 8399 8722 9045 9368 969 13- 0012 J 3- 0334 o^SS 0977 I2 98 1619 1939 226 258 29 3219 *3- 3539 3858 4177 449 4814 5i33 545i 5769 86 6403 13- 6721 7037 7354 7671 7987 8303 8618 8934 9249 9564 13- 9879 14- 0194 0508 0822 1136 145 1763 2076 2389 2702 J 4- 3015 3327 3639 395 1 4263 4574 4885 5196 5507 5818 14- 6128 6438 6748 7058 7367 7676 7985 8294 8603 8911 14- 9219 9527 9835 15- 0142 0449 0756 1063 137 1676 1982 15- 2288 2594 29 3205 351 3815 412 4424 4728 5032 15- 5336 564 5943 6246 6549 6852 7154 7457 7759 8061 15- 8362 8664 8965 9266 9567 9868 16- j 0168 0469 0769 1068 16- 1368 1667 1967 2266 2564 2863 3161 346 3758 4055 l6 ~ 4353 465 4947 5244 554 1 5838 6134 643 6726 7022 16- 7317 7613 7908 8203 8497 8792 9086 938 9674 9968 17- 0262 0555 0848 1141 1434 1726 2019 2311 2603 2895 17- 3186 3478 3769 406 4351 4641 4932 5222 5512 5802 17- 6091 6381 667 6959 7248 7536 7825 8113 8401 8689 17- 8977 9264 9552 9839 18- 0126 0413 0699 0986 1272 1558 18- 1844 2129 2415 27 2985 327 3555 3839 4123 4407 18- 4691 4975 5259 5542 5825 6108 6391 6674 6956 7239 18- 7521 7803 8084 8366 8647 8928 9209 949 9771 TO- ______ 0051 o r a 3 4 56789 tOGAEITHHS OF NUMBEBS. No. o i 2 3 4 5 6 7 8 9 D 155 19- 0332 0612 0892 1171 1451 173 201 2289 2567 2846 279 156 19- 3125 3403 3681 3959 4237 45H 4792 5069 5346 5623 278 19- 59 6176 6453 6729 7005 7281 7556 7832 8107 8382 276 158 19- 8657 8932 9206 9481 9755 275 158 20- 0029 0303 577 085 1124 274 159 20- 1397 167 1943 2216 2488 2761 3033 3305 3577 3848 272 160 20 412 439i 4663 4934 5204 5475 5746 6016 6286 6556 271 161 20 6826 7096 7365 7634 7904 8i73 8441 871 8979 9247 269 l62 20 9515 9783 268 l62 21 - 0051 0319 0586 0853 1121 1388 1654 1921 267 163 21- 2l88 2454 272 2986 3252 3783 4049 43H 4579 266 164 21 4844 5109 5373 5638 5902 6166 643 6694 6957 7221 264 165 21- 7484 7747 801 8273 8536 8798 906 9323 9585 9846 262 166 22- 0108 037 0631 0892 "53 1414 1^75 1936 2196 2456 261 167 22- 2716 2976 3236 340 3755 4015 4274 4533 4792 259 168 22 5309 5568 5826 6084 6342 66 68 5 8 7"5 7372 763' 258 169 22- 7887 8144 84 8657 8913 917 9426 9682 9938 257 169 23- 0193 256 170 23 0449 0704 096 1215 147 1724 1979 2234 2488 2742 255 171 23- 2996 325 3504 3757 4011 4264 4517 477 5023 5276 253 172 23- 5528 578i 6033 6285 6537 6789 7041 7292 7544 7795 252 173 1 23- 8046 8297 8548 8799 9049 9299 955 98 251 173 24- 005 03 250 174 24- 0549 0799 1048 1297 1546 1795 2044 2293 2541 279 249 175 24- 3038 3286 3534 3782 403 4277 4525 4772 5019 5266 248 176 24- 5513 5759 6006 6252 6499 6745 6991 7237 7482 7728 246 177 24- 7973 8219 8464 8709 8954 9198 9443 9687 9932 246 177 25- ~ 0176 245 178 25- 042 0664 0908 1151 1395 1638 1881 2125 2368 261 243 179 25- 2853 3096 3338 358 3822 4064 4306 4548 479 5031 242 180 25- 5273 55H 5755 590 6237 6477 6718 6958 7198 7439 241 181 25- 7679 7918 8158 8398 8637 8877 9116 9355 9594 9833 239 182 26- 0071 031 0548 0787 1025 1263 1501 1739 1976 2214 238 183 26- 2451 2688 2925 3162 3399 3636 3873 4109 4346 4582 237 184 26- 4818 5054 529 5525 5996 6232 6467 6702 6937 235 185 26- 7172 7406 7641 7875 811 8344 8578 8812 9046 9279 234 186 26- 9513 9746 998 234 186 27- 0213 0446 0679 0912 "44 1377 1609 233 187 27- 1842 2074 2306 2538 277 3001 3233 3464 3696 3927 232 188 27- 4158 4389 462 485 5081 53" 5542 5772 6002 6232 230 189 27- 6462 6692 6921 7151 738 7609 7838 8067 8296 8525 229 190 27- 8754 8982 9211 9439 9667 9895 228 190 28- 0123 0351 0578 0806 228 191 28- 1033 1261 1488 1715 1942 2169 2396 2622 2849 3075 227 192 i 28- 3301 3527 3753 3979 4205 443 1 4656 4882 5107 5332 226 193 ! 28- 5557 5782 6007 6232 6456 6681 6905 7*3 7354 7578 225 194 28- 7802 8026 8249 8473 8696 892 9M3 9366 9589 9812 223 /95 29- 0035 0257 048 0702 0925 "47 1369 1591 1813 2034 222 196 29- 2256 2478 2699 292 3363 3584 3804 4025 4246 221 197 29- 4466 4687 4907 5127 5347 5567 5787 6007 6226 6446 220 198 29- 6665 6884 7104 7323 7542 7761 7979 8198 8416 8635 2I 9 199 29- 8853 9071 9289 9507 9725 9943 218 I99|30- - 0161 0378 0595 0813 218 No. 1 o 9. 3 4 f 6 7 8 9 ~D" 314 LOGARITHMS OF NUMBERS. 01234 56789 30- 103 I2 47 J 4^4 1681 1898 2114 2331 2547 2764 298 30- 3196 3412 3628 3844 4059 4275 449i 4706 4921 5136 30- 535i 5566 5781 590 6211 6425 6639 6854 7068 7282 30- 7496 771 7924 8137 8351 8564 8778 8991 9204 9417 30- 963 9843 31- 0056 0268 0481 0693 0906 1118 133 1542 31- 1754 1966 2177 2389 26 2812 3023 3234 3445 3656 31- 3867 4078 4289 4499 471 492 513 534 555i 576 31- 597 618 639 6599 6809 7018 7227 7436 7646 7854 31- 8063 8272 8481 8689 8898 9106 9314 9522 973 9938 32- 0146 0354 0562 0769 0977 1184 1391 1598 1805 2012 32- 2219 2426 2633 2839 34 6 3252 3458 3665 3871 4077 32- 4282 4488 4694 4899 5105 531 5516 5721 5926 6131 32- 6336 6541 6745 695 7155 7359 7563 7767 7972 8176 32- 838 8583 8787 8991 9194 9398 9601 9805 33- 0008 O2II 33- 0414 0617 0819 1022 1225 1427 163 1832 2034 2236 33- 2438 264 2842 3044 3246 3447 3649 385 4051 4253 33- 4454 4655 4856 5057 5257 5458 5658 5859 6o59 626 33- 646 666 686 706 726 7459 7659 7858 8058 8257 33- 8456 8656 8855 9054 9253 9451 965 9849 34- 0047 0246 34- 0444 0642 0841 1039 1237 *435 l6 3 2 183 2028 2225 34- 2423 262 2817 3014 3212 3409 3606 3802 3999 4196 34- 4392 4589 4785 4981 5178 5374 557 576 59 2 6157 34- 6 353 6 549 6 744 6939 7^5 733 7525 772 7915 811 34- 8305 85 8694 8889 9083 9278 9472 9666 986 35- 0054 35- 0248 0442 0636 0829 1023 1216 141 1603 1796 1989 35- 2183 2375 2568 2761 2954 3*47 3339 353 2 3724 39 l6 35- 4108 4301 4493 4685 4876 5068 526 5452 5643 5834 35- 6026 6217 6408 6599 679 6981 7172 7363 7554 7744 35- 7935 8125 8316 8506 8696 8886 9076 9266 9456 9646 35- 9835 36- 0025 0215 0404 0593 0783 0972 1161 135 1539 36- 1728 1917 2105 2294 2482 2671 2859 3048 3236 3424 36- 3612 38 3988 4176 4363 455i 4739 4926 5113 530i 36- 5488 5675 5862 6049 6236 6423 66 1 6796 6983 7169 3- 7356 7542 7729 79*5 8101 8287 8473 8659 8845 903 36- 9216 9401 9587 9772 9958 37- 0143 0328 0513 0698 0883 37- 1068 1253 1437 1622 1806 1991 2175 236 2544 2728 37- 2912 3096 328 3464 3647 3831 4015 4198 4382 4565 37- 4748 4932 5115 5298 5481 5664 5846 6029 6212 6394 37- 6 577 6 759 6942 7124 7306 7488 767 7852 8034 8216 ; 37- 8398 858 8761 8943 9124 9306 9487 9668 9849 38 _____ _____ oo 3 38- 021 1 0392 0573 0754 0934 1115 1296 1476 1656 1837 3 8- 2017 2197 2377 2557 2737 2917 3097 3277 3456 3636 38- 3815 3995 4*74 4353 4533 47 12 489! 57 5 2 49 5428 38- 5606 5785 5964 6142 6321 6499 6677 6856 7034 7212 38- 739 7568 7746 7923 8101 ; 8279 8456 8634 8811 8989 01234 56789 LOGARITHMS OF NUMBERS. 315 No. o i 2 3 4 5 6 7 8 9 D 245 38- 9166 9343 952 9698 9875 177 2 45 39- 0051 0228 0405 0582 0759 177 246 39- 0935 III2 1288 1464 1641 1817 1993 2169 2345 2521 176 247 39- 2697 2873 3048 3224 34 3575 3751 3926 4101 4277 176 248 39- 4452 4627 4802 4977 5152 5326 5501 5676 585 6025 175 249 39- 6199 6374 6548 6722 6896 7071 7245 7419 7592 7766 174 250 39- 794 8lI4 8287 8461 8634 8808 8981 9*54 9328 9501 173 2 5I 39- 9674 9847 *73 251 40- 002 0192 0365 0538 0711 0883 1056 1228 173 252 40- 1401 1573 1745 1917 2089 2261 2433 2605 2777 2949 172 253 40- 3121 3292 3464 3635 3807 3978 4149 432 4492 4663 171 254 40- 4834 5005 5176 5346 5517 5688 5858 6029 6199 637 171 255 40- 654 6 7 I 688l 7051 7221 7391 7501 773i 7901 807 170 256 40- 824 841 8579 8749 8918 9087 9257 9426 9595 9764 169 257 40- 9933 169 257 41- 0102 O27I 044 0609 0777 0946 1114 1283 1451 169 258 41- 162 1788 1956 2124 2293 2461 2629 2796 2964 3*32 168 259 4i- 33 3467 3635 3803 397 4137 4305 4472 4639 4806 167 260 4i- 4973 514 5307 5474 5641 5808 5974 6141 6308 6474 167 261 41- 6641 6807 6973 7139 7306 7472 7638 7804 797 8i3S 166 262 41- 8301 8467 8633 8798 8964 9129 9295 946 9625 9791 165 263 41- 9956 165 263 42- OI2I 0286 0451 0616 0781 0945 in 1275 1439 165 264 42- 1604 1768 1933 2097 2261 2426 259 2754 2918 3082 164 265 42- 3246 341 3574 3737 3901 4065 4228 4392 4555 4718 164 266 42- 4882 5045 5208 537i 5534 5697 586 6023 6186 6 349 163 267 42- 6511 66 74 6836 6999 7161 7324 7486 7648 7811 7973 162 268 42- 8135 8297 8459 8621 8783 8944 9106 9268 9429 9591 162 269 42- 9752 9914 162 269 43- 0075 0236 0398 0559 072 0881 1042 1203 161 270 43- 1364 1525 1685 1846 2007 2167 2328 2488 2649 2809 161 271 43- 2969 313 329 345 3 DI ! 377 393 409 4249 4409 160 272 43- 4569 4729 4888 5048 5207 5367 5526 5685 5844 6004 159 273 43- 6163 6322 6481 664 6799 6957 7116 7275 7433 759 159 274 43- 775i 7909 8067 8226 8384 i 8542 8701 8859 9017 9175 158 275 43- 9333 9491 9648 9806 9964 iss 275 44- OI22 0279 0437 0594 0752 158 276 44- 0909 1066 1224 1381 1538 1695 1852 209 2166 2323 157 277 44- 248 2637 2793 295 3106 3263 3419 3576 3732 3889 157 278 44- 4045 42OI 4357 4513 4669 4825 4981 5137 5293 5449 156 279 44- 5604 576 5915 6071 6226 6382 6537 6692 6848 7003 J55 280 44- 7158 73 r 3 7468 7623 7778 7933 8088 8242 8397 8552 155 281 44- 8706 8861 9 I 5 917 9324 9478 9633 9787 9941 154 281 45- 0095 154 282 45- 0249 0403 0557 0711 0865 IOT8 1172 1326 J479 1633 154 283 i 45- 1786 194 2093 2247 24 2553 2706 2859 3012 3165 153 284 45- 33i8 347i 3624 3777 393 4082 4235 4387 454 4692 153 285 45- 4845 4997 5i5 5302 5454 Sooo 5758 59 1 6062 6214 152 286 45- 6366 6518 667 6821 6973 7125 7276 7428 7579 7731 152 287 45- 7882 8033 8184 8336 8487 8638 8789 894 9091 9242 151 288; 45-9392 9543 9694 9845 9995 151 288 46- 0146 0296 0447 597 0748 I5 1 289 ; 46- 0898 1048 1198 1348 J 499 1649 1799 1948 2098 2248 150 No. i o 2 3 4 5 6 7 8 9 D 316 LOGARITHMS OF NUMBERS. No. 01234 56789 D 46- 2398 2548 2697 2847 2997 46- 3893 4042 4 I 9 I 434 449 46- 5383 5532 568 5829 5977 46- 6868 7016 7164 7312 746 46- 8347 8495 8643 879 8938 3146 3296 3445 3594 3744 4639 4788 4936 5085 5234 6126 6274 6423 6571 6719 7608 7756 7904 8052 82 9085 9233 938 9527 9675 150 149 149 148 148 46- 9822 9969 47- 0116 0263 041 47- 1292 1438 1585 1732 1878 47- 2756 2903 3049 3195 3341 47- 4216 4362 4508 4653 4799 47- 5671 5816 5962 6107 6252 0557 074 0851 0998 1145 2025 2171 2318 2464 261 3487 3633 3779 3925 4071 4944 509 5235 5381 5526 6397 6542 6687 6832 6976 147 147 146 146 1,46 4s 47- 7121 7266 7411 7555 77 47- 8566 8711 8855 8999 9143 48- 0007 0151 0294 0438 0582 48- 1443 1586 1729 1872 2016 48- 2874 3016 3159 3302 3445 7844 7989 8133 8278 8422 9287 9431 9575 9719 9863 0725 0869 IOI2 1156 1299 2159 2302 2445 2588 2731 3587 373 3872 4015 4i57 145 144 144 143 143 48- 43 4442 4585 4727 4869 48- 5721 5863 6005 6147 6289 48- 7138 728 7421 7563 7704 48- 8551 8692 8833 8974 9114 48- 9958 - 49- 0099 0239 038 052 Son 5153 5295 5437 5579 643 6572 6714 6855 6997 7845 7986 8127 8269 841 9255 9396 9537 9677 9818 O66l 080I 0941 I08l 1222 142 142 141 141 140 140 49- 1362 1502 1642 1782 1922 49- 276 29 304 3179 3319 49- 4155 4294 4433 4572 47ii 49- 5544 5683 5822 596 6099 49- 693 7068 7206 7344 7483 2O62 22OI 2341 2481 2621 3458 3597 3737 3 8 76 4015 485 4989 5128 5267 5406 6238 6376 6515 6653 6791 7621 7759 7897 8035 8173 140 139 139 139 138 49- 8311 8448 8586 8724 8862 49- 9687 9824 902 50- 1059 1196 1333 147 1607 50- 2427 2564 27 2837 2973 50- 3791 3927 4063 4199 4335 8999 9137 9275 9412 955 0374 5 IT 0648 0785 0922 1744 188 2017 2154 2291 3109 3246 3382 35x8 3655 4471 4607 4743 4878 5014 138 137 137 137 136 50- 515 5286 5421 5557 5693 50- 6505 664 6776 6911 7046 50- 7856 7991 8126 826 8395 50- 9203 9337 9471 9606 974 51- 0545 0679 0813 0947 1081 5828 5964 6099 6234 637 7181 7316 7451 7586 7721 853 8664 8799 8934 9068 9874 ~ ~ 0009 0143 0277 0411 1215 1349 1482 1616 175 136 135 135 134 134 134 51- 1883 2017 2151 2284 2418 51- 3218 3351 3484 3617 375 51- 4548 4681 4813 4946 5079 51- 5874 6006 6139 6271 6403 51- 7196 7328 746 7592 7724 2551 2684 2818 2951 3084 3883 4016 4149 4282 4415 i 5211 5344 5476 5609 5741 6535 6668 68 6932 7064 7855 7987 8119 8251 8382 133 133 133 132 132 51- 8514 8646 8777 8909 904 51- 9828 9959 S 2- 009 0221 0353 52- 1138 1269 14 153 1661 52- 2444 2575 2705 2835 2966 52- 3746 3876 4006 4136 4266 9171 9303 9434 9566 9697 0484 0615 0745 0876 1007 1792 1922 2053 2183 2314 3096 3226 3356 3486 3616 4396 4526 4656 4785 4915 130 130 01234 56789 LOGABITHMS OP NUMBERS. 317 No. 1 I 2 3 4 5 6 7 8 9 D 335 52- 5045 5174 5304 5434 5563 5693 5822 5951 6081 621 129 336 52- 6339 6469 6598 6727 6856 6985 7114 7243 7372 7501 129 337 " y v ^-' 52- 763 7759 7888 8016 8i45 8274 8402 8531 866 8788 129 338 52- 8917 945 9 J 74 9302 943 9559 9687 98i5 9943 128 338 53- 0072 128 339 53- 02 0328 0456 0584 0712 084 0968 1096 1223 1351 128 340 53- 1479 1607 1734 1862 199 2117 2245 2372 25 2627 128 34i 53- 2754 2882 3009 3136 3264 3391 3518 3645 3772 3899 127 342 53- 4026 4153 428 4407 4534 4661 4787 4914 5041 5167 127 343 53- 5294 542i 5547 5674 58 5927 6053 618 6306 6432 126 344 53- 6558 6685 6811 6937 7063 7189 7315 744i 7567 7693 126 545 53- 7819 7945 8071 8197 8322 8448 8574 8699 8825 8951 126 346 53- 9076 9202 9327 9452 9578 9703 9829 9954 126 346 54- 0079 0204 125 347 54- 0329 0455 058 0705 083 0955 108 1205 133 1454 125 348 54- 1579 1704 1829 J 953 2078 2203 2327 2452 2576 2701 125 349 54- 2825 295 3074 3199 3323 3447 357i 3696 382 3944 124 350 54- 4068 4192 43i6 444 4564 4688 4812 4936 506 5183 124 3Si 54- 5307 543i 5555 5678 5802 5925 6049 6172 6296 6419 124 352 54- 6 543 6666 6789 6913 7036 7159 7282 7405 7529 7652 123 353 54- 7775 7898 8021 8i44 8267 8389 8512 8635 8758 8881 123 354 54- 9003 9126 9249 937i 9494 9616 9739 9861 9984 123 354 55- 0106 123 355 55- 0228 O35i 0473 0595 0717 084 0962 1084 1206 1328 122 356 55- 145 J 57 2 1694 1816 1938 206 2181 2303 2425 2547 122 357 55- 2668 279 2911 3033 3155 3276 3398 3519 364 3762 121 358 55- 3883 4004 4126 4247 4368 4489 461 473i 4852 4973 121 359 55- 5094 5215 5336 5457 5578 5699 582 594 6061 6182 121 360 55- 6303 6423 6544 6664 6785 6905 7026 7146 7267 7387 120 36i 55- 7507 7627 7748 7868 7988 8108 8228 8349 8469 8589 120 362 55- 8709 8829 8948 9068 9188 9308 9428 9548 9667 9787 120 363 55- 997 120 363 56- - 0326 0146 0265 0385 0504 0624 0743 0863 0982 II 9 364 56- uoi 1221 134 1459 1578 1698 1817 1936 2055 2174 "9 365 56- 2293 2412 2531 265 2769 2887 3006 3125 3244 3362 II 9 366 56- 3481 36 37i8 3837 3955 4074 4192 43" 4429 4548 II 9 367 56- 4666 4784 4903 5021 5139 5257 5376 5494 5612 573 118 368 56- 5848 59 66 6084 6202 632 6437 6555 6673 6791 6909 118 369 56- 7026 7 J 44 7262 7379 7497 7614 7732 7849 7967 8084 118 370 56- 8202 8319 8436 8554 8671 8788 8905 9023 914 9257 117 37i 56- 9374 9491 9608 9725 9842 9959 117 37 1 57- 0076 0193 0309 0426 117 372 57- 0543 066 0776 0893 101 1126 1243 1359 1476 1592 117 373 57- 1709 1825 1942 2058 2174 2291 2407 2523 2639 2755 116 374 57- 2872 2988 3*04 322 3336 3452 3568 3684 38 3915 116 375 57- 4031 4 J 47 4263 4379 4494 461 4726 4841 4957 5072 116 376 57- 5i88 5303 5419 5534 565 5765 588 590 6m 6226 I1 5 377 57- 6341 6457 6572 6687 6802 6917 7032 7147 7262 7377 "5 378 57- 7492 7607 7722 7836 7951 8066 8181 8295 841 8525 "5 379 57- 8639 8754 8868 8983 9097 9212 9326 9441 9555 9669 114 No. o i 2 3 4 5 6 7 8 9 , D D 0* LOGARITHMS OF NUMBEBS. No. 01234 56789 380 57- 9784 9898 _ _ 380 58- 0012 0126 0241 0355 0469 0583 0697 0811 38i 58- 0925 1039 1153 1267 1381 1495 1608 1722 1836 195 382 58- 2063 2177 2291 2404 2518 j 2631 2745 2858 2972 3085 383 58- 3199 3312 3426 3539 3652 3765 3879 399 2 4*05 4218 384 58- 4331 4444 4557 467 4783 4896 5009 5122 5235 5348 385 58- 5461 5574 5686 5799 5912 6024 6137 625 6362 6475 386 58- 6587 67 6812 6925 7037 7149 7262 7374 7486 7599 387 58- 7711 7823 7935 8047 816 8272 8384 8496 8608 872 388 58- 8832 8944 9056 9167 9279 939 1 9503 9 I 5 9726 9838 389 58-995 389 59- 0061 0173 0284 0396 0507 0619 073 0842 0953 390 59- 1065 1176 1287 1399 151 1621 1732 1843 T 955 2066 39i 59- 2177 2288 2399 251 2621 2732 2843 2954 3064 3175 39 2 59- 3286 3397 3508 3618 3729 384 395 4061 4171 4282 393 59- 4393 4503 4614 4724 4834 4945 5055 5165 5276 5386 394 59- 5496 5606 5717 5827 5937 6047 6l 57 6267 6377 6487 395 59- 6597 6707 6817 6927 7037 7146 7256 7366 7476 7586 396 59- 7695 7805 7914 8024 8134 8243 8353 8462 8572 8681 397 59- 8791 89 9009 9119 9228 9337 9446 9556 9605 9774 398 59- 9883 9992 398 60- OIOI O2I 0319 0428 0537 0646 0755 0864 399 00- 0973 1082 II9I 1299 1408 1517 1625 1734 1843 1951 400 60- 2O6 2169 2277 2386 2494 2603 2711 2819 2928 3036 401 60- 3144 3253 3361 3469 3577 3686 3794 3902 401 4118 402 60- 4226 4334 4442 455 4658 4766 4874 4982 5089 5197 403 60- 5305 5413 5521 5628 5736 5844 5951 6059 Ol66 6274 404 60- 6381 6489 6596 6704 68 n 6919 7026 7133 7241 7348 405 60- 7455 7562 7669 7777 7884 7991 8098 8205 8312 8419 406 60- 8526 8633 874 8847 8954 9061 9167 9274 9381 9488 407 60- 9594 9701 9808 9914 407 61- 0021 0128 0234 0341 0447 0554 408 61- 066 0767 0873 0979 1086 1192 1298 1405 1511 1617 409 61- 1723 1829 1936 2042 2148 2254 236 2466 2572 2678 410 61- 2784 289 2996 3102 3207 33 J 3 3419 3525 3 6 3 3736 411 61- 3842 3947 4053 4159 4264 437 4475 458 1 4686 4792 412 61- 4897 5003 5108 5213 5319 5424 5529 5634 574 5845 413 61- 595 6055 616 6265 637 6476 6581 6686 679 6895 414 61- 7 7105 721 7315 742 7525 7629 7734 7839 7943 415 61- 8048 8153 8257 8362 8466 8571 8676 878 8884 8989 416 61- 9093 9198 9302 9406 9511 9615 9719 9824 9928 416 62- 0032 4i7 62- 0136 024 0344 0448 0552 0656 076 0864 0968 1072 418 62- 1176 128 1384 1488 1592 1695 1799 1903 2007 211 419 62- 2214 2318 2421 2525 2628 2732 2835 2939 3042 3146 420 62- 3249 3353 3456 3559 3663 3766 3869 3973 4076 4179 421 62- 4282 4385 4488 4591 4695 4798 4901 5004 5107 521 422 62- 5312 5415 5518 5621 5724 5827 5929 6032 6135 6238 4 2 3 62- 634 6443 6546 6648 6751 6853 6956 7058 7161 7263 424 62- 7366 7468 7571 7673 7775 7878 798 8082 8185 8287 No. 01234 56789 LOGARITHMS OF NUMBERS. 319 No. o i 2 3 4 56789 62- 8389 8491 8593 8695 8797 62- 941 9512 9613 9715 9817 63- 63- 0428 053 0631 0733 0835 63- 1444 1545 1647 1748 1849 63- 2457 2559 266 2761 2862 89 9002 9104 9206 9308 99i9 0021 0123 0224 0326 0936 1038 1139 1241 1342 1951 2052 2153 2255 2356 2963 3064 3165 3266 3367 63- 3468 3569 367 3771 3872 63- 4477 4578 4679 4779 488 63- 5484 5584 5685 5785 5886 63- 6488 6588 6688 6789 6889 63- 749 759 769 779 79 3973 4074 4*75 4276 43?6 4981 5081 5182 5283 5383 5986 6087 6187 6287 6388 6989 7089 7189 729 739 799 809 819 829 8389 63- 8489 8589 8689 8789 8888 63- 9486 9586 9686 9785 9885 64- - 64- 0481 0581 068 0779 0879 64- 1474 1573 1672 1771 1871^ 64- 2465 2563 2662 2761 280 8988 9088 9188 9287 9387 9984 0084 0183 0283 0382 0978 1077 1177 1276 1375 197 2069 2168 2267 2366 2959 3058 3156 3255 3354 64- 3453 355i 365 3749 3847 64- 4439 4537 4636 4734 4832 64- 5422 5521 5619 5717 5815 64- 6404 6502 66 6698 6796 64- 7383 748i 7579 7676 7774 3946 4044 4143 4242 434 4931 5029 5127 5226 5324 5913 6011 611 6208 6306 6894 6992 7089 7187 7285 7872 7969 8067 8165 8262 64- 836 8458 8555 8653 875 64- 9335 9432 953 9627 9724 65- ----- 65- 0308 0405 0502 0599 0696 65- 1278 1375 1472 1569 1666 65- 2246 2343 244 2536 2633 8848 8945 9043 914 9237 9821 9919 00l6 0113 021 0793 089 0987 1084 1181 1762 1859 X 956 2053 215 273 2826 2923 3019 3116 65- 3213 3309 3405 3502 3598 65- 4i77 4273 4369 4465 4562 65- 5138 5235 533i 5427 5523 65- 6098 6194 629 6386 6482 65- 7056 7*52 7247 7343 7438 3695 379i 3888 3984 408 4658 4754 485 4946 5042 5619 5715 581 5906 6002 6577 6673 6769 6864 696 7534 7629 7725 782 7916 65- Son 8107 8202 8298 8393 65- 8965 906 9155 925 9346 65- 9916 66- ooi i 0106 0201 0296 66- 0865 096 1055 115 1245 66- 1813 1907 2002 2096 2191 8488 8584 8679 8774 887 9441 9536 9631 9726 9821 0391 0486 0581 0676 0771 1339 1434 1529 1623 1718 2286 238 2475 2569 2663 66- 2758 2852 2947 3041 3135 66- 370i 3795 3889 3983 4078 66- 4642 4736 483 4924 5018 66- 5581 5675 5769 5862 5956 66- 6518 6612 6705 6799 6892 323 3324 3418 3512 3607 4172 4266 436 4454 4548 5112 5206 5299 5393 5487 605 6143 6237 6331 6424 6986 7079 7173 7266 736 66- 7453 7546 764 7733 7826 66- 8386 8479 8572 8665 8759 66- 9317 941 9503 9596 9689 67- ----- 67- 0246 0339 0431 0524 0617 67- 1173 1265 1358 1451 1543 792 8013 8106 8199 8293 8852 8945 9038 9131 9224 9782 9875 9967 006 0153 071 0802 0895 0988 108 1636 1728 1821 1913 2005 01 a 3 4 56789 320 LOGARITHMS OF NUMBERS. No. 01234 56789 470 67- 2098 219 2283 2375 2467 256 2652 2744 2836 2929 i 47 1 67- 3021 3113 3205 3297 339 3482 3574 3666 3758 385 472 67- 3942 4034 4126 4218 431 4402 4494 4586 4677 4769 473 67- 4861 4953 5045 5137 5228 532 5412 5503 5595 5687 474 67- 5778 587 5962 6053 6145 6236 6328 6419 6511 6602 475 67- 6694 6785 6876 6968 7059 7151 7242 7333 7424 7516 476 67- 7607 7698 7789 7881 7972 8063 8154 8245 8336 8427 477 67- 8518 8609 87 8791 8882 8973 9064 9155 9246 9337 478 67- 9428 9519 961 97 9791 9882 9973 -- 478 og_ 0063 0154 0245 479 68- 0336 0426 0517 0607 0698 0789 0879 097 106 1151 480 68- 1241 1332 1422 1513 1603 1693 1784 1874 1964 2 o55 481 68- 2145 2235 2326 2416 2506 2596 2686 2777 2867 2957 482 68- 3047 3137 3227 3317 3407 3497 3587 3 6 77 37 6 7 3857 483 68- 3947 4037 4127 4217 4307 439 4486 4576 4666 4756 484 68- 4845 4935 5025 5114 5204 5294 5383 5473 5563 5652 485 68- 5742 5831 5921 601 61 6189 6279 6368 6458 6547 486 68- 6636 6726 6815 6904 6994 7083 7172 7261 7351 744 487 68- 7529 7618 7707 7796 7886 7975 8064 8153 8242 8331 488 68- 842 8509 8598 8687 8776 8865 8953 9042 9131 922 489 68- 9309 9398 9486 9575 9664 9753 9841 993 489 69 - - - 0019 0107 490 69- 0196 0285 0373 0462 055 0639 0728 0816 0905 0993 491 69- 1081 117 1258 1347 1435 1524 1612 17 1789 1877 492 69- 1965 2053 2142 223 2318 2406 2494 2583 2671 2759 493 69- 2847 2935 3023 3111 3199 3287 3375 3463 355i 3639 494 69- 3727 3815 3903 399i 4078 4166 4254 4342 443 4517 495 69- 4605 4693 4781 4868 4956 5044 5131 5219 5307 5394 496 69- 5482 5569 5657 5744 5832 5919 6007 6094 6182 6269 497 69- 6356 6444 6531 6618 6706 6793 688 6968 7055 7142 498 69- 7229 7317 7404 7491 7578 7665 7752 7839 7926 8014 499 69- 8101 8188 8275 8362 8449 8535 8622 8709 8796 8883 500 69- 897 9057 9144 9231 9317 9404 9491 9578 9664 9751 50 1 69- 9838 9924 501 70- ooi i 0098 0184 0271 0358 0444 0531 0617 502 70- 0704 079 0877 0963 105 1136 1222 1309 1395 1482 503 70- 1568 1654 1741 1827 1913 1999 2086 2172 2258 2344 504 70- 2431 2517 2603 2689 2775 286l 2 947 3033 3119 3205 505 70- 3291 3377 3463 3549 3635 3721 3807 3893 3979 4065 506 70- 4151 4236 4322 4408 4494 4579 4665 4751 4837 4922 507 70- 5008 5094 5179 5265 535 543 6 5522 5607 5693 5778 508 70- 5864 5949 6035 612 6206 6291 6376 6462 6547 6632 509 70- 6718 6803 6888 6974 7059 7144 7229 7315 74 7485 510 70- 757 7655 774 7826 7911 7996 8081 8166 8251 8336 5ii 70- 8421 8506 8591 8676 8761 8846 8931 9015 91 9185 512 70- 927 9355 944 9524 9609 9694 9779 9863 9948 512 yi- 0033 513 71- 0117 0202 0287 0371 0456 054 0625 071 0794 0879 5H 71- 0963 1048 1132 1217 1301 J 385 147 J554 l6 39 J 723 No. 1 234 56789 LOGARITHMS OF NUMBERS. 321 No. 01234 56789 515 71- 1807 1892 1976 206 2144 2229 2313 2397 2481 2566 5i6 71- 265 2734 2818 2902 2986 307 3154 3238 3323 3407 517 71- 3491 3575 3659 3742 3826 391 3994 4078 4162 4246 5i8 71- 433 4414 4497 4581 4665 4749 4833 49i6 5 5084 519 71- 5167 5251 5335 5418 5502 5586 5669 5753 5836 593 520 71- 6003 6087 617 6254 6337 6421 6504 6588 6671 6754 521 71- 6838 6921 7004 7088 7171 7254 7338 742i 7504 7587 522 71- 7671 7754 7837 792 8003 8086 8169 8253 8336 8419 523 71- 8502 8585 8668 8751 8834 8917 9 9083 9165 9248 524 7i- 933* 94H 9497 958 9663 9745 9828 9911 9994 524 72- __ __ 0077 525 72- 0159 0242 0325 0407 049 0573 o655 0738 0821 0903 526 72- 0986 1068 1151 1233 1316 1398 1481 1563 1646 1728 527 72- 1811 1893 1975 2058 214 2222 2305 2387 2469 2552 528 72- 2634 2716 2798 2881 2963 3045 3127 3209 3291 3374 529 72- 3456 3538 362 3702 3784 3866 3948 403 4112 4194 530 72- 4276 4358 444 4522 4604 4685 4767 4849 4931 5013 53* 72- 5095 5176 5258 534 5422 5503 5585 5667 5748 583 532 72- 5912 5993 6075 6156 6238 632 6401 6483 6564 6646 533 72- 6727 6809 689 6972 7053 7134 7216 7297 7379 746 534 72- 7541 7623 7704 7785 7866 7948 8029 811 8191 8273 535 72- 8354 8435 8516 8597 8678 8759 8841 8922 9003 9084 536 72- 9165 9246 9327 9408 9489 957 9651 9732 9813 9893 537 72- 9974 537 73- 0055 0136 0217 0298 0378 0459 054 0621 0702 538 73- 0782 0863 0944 1024 1105 1186 1266 1347 1428 1508 539 73- 1589 1669 175 183 1911 1991 2072 2152 2233 2313 540 73- 2394 2474 2555 2635 2715 2796 2876 2956 3037 3117 54i 73- 3197 3278 3358 3438 3518 3598 3679 3759 3839 3919 542 73- 3999 4079 4i6 424 432 44 448 456 464 472 543 73- 48 488 496 504 512 52 5279 5359 5439 55*9 544 73- 5599 5679 5759 5838 59*8 5998 6078 6157 6237 6317 545 73- 6397 6476 6556 6635 6715 6795 6874 6954 7034 7113 546 73- 7193 7272 7352 743i 75ii 759 767 7749 7829 7908 547 73- 7987 8067 8146 8225 8305 8384 8463 8543 8622 8701 548 73- 8781 886 8939 9018 9097 9177 9256 9335 9414 9493 549 73- 9572 05i 973i 981 9889 9968 - 549 74- 0047 0126 0205 0284 550 74- 0363 0442 0521 06 0678 0757 0836 0915 0994 1073 551 74- 1152 123 1309 1388 1467 1546 1624 1703 1782 186 552 74- 1939 2018 2096 2175 2254 2332 2411 2489 2568 2647 553 74- 2725 2804 2882 2961 3039 3118 3196 3275 3353 3431 554 74- 35i 3588 3667 3745 3823. 3902 398 4058 4136 4215 555 74- 4293 4371 4449 4528 4606 4684 4762 484 4919 4997 556 74- 5075 5153 5231 5309 5387 5465 5543 5621 5699 5777 557 74- 5855 5933 6on 6089 6167 6245 6323 6401 6479 6556 558 74- 6634 6712 679 6868 6945 7023 7101 7179 7256 7334 559 74- 7412 7489 7567 7645 7722 78 7878 7955 8033 811 No. 01234 56789 322 LOGARITHMS OF NUMBERS. No. 560 56i 562 562 563 564 01234 56789 D 77 77 77 77 77 77 74- 8188 8266 8343 8421 8498 74- 8963 904 9118 9195 9272 74- 9736 9814 9891 9968 75- 0045 75- 0508 0586 0663 074 0817 75- 1279 1356 1433 151 1587 8576 8653 8731 8808 8885 935 9427 9504 9582 9659 0123 02 0277 0354 0431 0894 0971 1048 1125 1202 1664 1741 1818 1895 1972 565 5 66 567 568 569 75- 2048 2125 2202 2279 2356 75- 2816 2893 297 3047 3123 75- 3583 366 3736 3813 3889 75- 4348 4425 45oi 4578 4^54 75- 5112 5189 5265 5341 5417 2433 2509 2586 2663 274 32 3277 3353 343 356 3966 4042 4119 4195 4272 473 4807 4883 496 5036 5494 557 5646 5722 5799 77 77 77 76 76 570 57 1 572 573 574 75- 5875 5951 6o 2 7 6l 3 6l 8 75- 6636 6712 6788 6864 694 75- 7396 7472 7548 7624 77 75- 8155 823 8306 8382 8458 75- 8912 8988 9063 9139 9214 6256 6332 6408 6484 656 7016 7092 7168 7244 732 7775 7851 7927 8003 8079 8533 8609 8685 8761 8836 929 9366 9441 9517 9592 76 76 76 76 76 575 575 576 577 578 579 75- 9668 9743 9819 9894 997 76- 76- 0422 0498 0573 0649 0724 76- 1176 1251 1326 1402 1477 76- 1928 2003 2078 2153 2228 76- 2679 2754 2829 2904 2978 0045 0121 0196 0272 0347 0799 0875 095 1025 noi 1552 1627 1702 1778 1853 2303 2378 2453 2529 2604 3053 3128 3203 3278 3353 76 75 75 75 75 75 580 58i 582 583 584 76- 3428 3503 3578 3653 3727 76- 4176 4251 4326 44 4475 76- 4923 4998 5072 5147 5221 76- 5669 5743 5818 5892 5966 76- 6413 6487 6562 6636 671 3802 3877 3952 4027 4101 455 4624 4699 4774 4848 5296 537 5445 552 5594 6041 6115 619 6264 6338 6785 6859 6933 77 782 75 75 75 74 74 585 586 587 588 588 589 76- 7156 723 7304 7379 7453 76- 7898 7972 8046 812 8194 76- 8638 8712 8786 886 8934 76- 9377 945i 9525 9599 9^73 77 _ _____ _ 77- 0115 0189 0263 0336 041 7527 7601 7675 7749 7823 8268 8342 8416 849 8564 9008 9082 9156 923 9303 9746 982 9894 9968 0042 0484 0557 0631 0705 0778 74 74 74 74 74 74 590 59i 592 593 594 77- 0852 0926 0999 1073 1146 77- 1587 1661 1734 1808 1881 77- 2322 2395 2468 2542 2615 77- 3055 3 J 28 3201 3274 3348 77- 3786 386 3933 4006 4079 122 1293 1367 144 1514 1955 2028 2102 2175 2248 2688 2762 2835 2908 2981 3421 3494 3567 364 3713 4152 4225 4298 4371 4444 74 73 73 73 73 505 596 597 598 599 77- 4517 459 4663 4736 4809 77- 5246 5319 5392 5465 5538 77- 5974 6047 6l2 6193 6265 77- 6701 6774 6846 6919 6992 77- 7427 7499 7572 7644 7717 4882 4955 5028 51 5173 73 561 5683 5756 5829 5902 ; 73 6338 6411 6483 6556 6629 1 73 7064 7137 7209 7282 7354 73 7789 7862 7934 8006 8079 1 72 600 601 602 602 603 604 77- 8151 8224 8296 8368 8441 77- 8874 8947 9019 9091 9163 77- 959 9^ 974 1 9 8 *3 9885 78- 78- 0317 0389 0461 0533 0605 78- 1037 1109 1181 1253 1324 8513 8585 8658 873 8802 9236 9308 938 9452 9524 9957 0029 oioi 0173 0245 0677 0749 0821 0893 0965 1396 1468 154 1612 1684 72 I 72 72 7 2 72 72 No. 01234 56789 D LOGABITHMS OF NUMBERS. 323 01234 56789 78- 1755 1827 1899 1971 2042 2114 2186 2258 2329 2401 78- 2473 2544 2616 2688 2759 2831 2902 2974 3046 3117 78- 3189 326 3332 3403 3475 3546 3618 3689 3761 3832 78- 3904 3975 4046 4118 4189 4261 4332 4403 4475 4546 78- 4617 4689 476 4831 4902 4974 545 5"6 5187 5259 ?8- 533 540i 5472 5543 5615 5686 5757 5828 5899 597 78- 6041 6112 6183 6254 6325 6396 6467 6538 6609 668 78- 6751 6822 6893 6964 7035 7106 7177 7243 7319 739 78- 746 7531 7602 7673 7744 7815 7885 7956 8027 8098 78- 8168 8239 831 8381 8451 8522 8593 8663 8734 8804 78- 8875 8946 9016 9087 9157 9228 9299 9369 944 951 78- 9581 9651 9722 9792 9863 9933 79- 0004 0074 0144 0215 79- 0285 0356 0426 0496 0567 0637 0707 0778 0848 0918 79- 0988 1059 1129 1199 1269 134 141 148 155 162 79- 1691 1761 1831 1901 1971 2O4I 21 1 1 2l8l 2252 2322 79- 2392 2462 2532 2682 2672 2742 28l2 2882 2952 3022 79- 3092 3162 3231 3301 3371 3441 35H 3581 3651 3721 79- 379 386 393 4 407 4139 4209 4279 4349 4418 79- 4488 4558 4627 4697 4767 4836 4906 4976 5045 5115 79- 5185 5254 5324 5393 5463 5532 5602 5672 5741 5811 79- 588 5949 6019 6088 6158 6227 6297 6366 6436 6505 79- 6574 6644 6713 6782 6852 6921 699 706 7129 7198 79- 7268 7337 7406 7475 7545 7614 7683 7752 7821 789 79- 796 8029 8098 8167 8236 8305 8374 8443 8513 8582 79- 8651 872 8789 8858 8927 8996 9065 9134 9203 9272 79- 9341 9409 9478 9547 9616 9685 9754 9823 9892 90i 80- 0029 0098 0167 0236 0305 0373 0442 0511 058 0648 80- 0717 0786 0854 0923 0992 1061 1129 1198 1266 1335 80- 1404 1472 1541 1609 1678 1747 l8l5 1884 1952 2021 80- 2089 2158 2226 2295 2363 2432 25 2568 2637 2705 80- 2774 2842 291 2979 3047 3116 3184 3252 3321 3389 80- 3457 3525 3594 3662 373 3798 3867 3935 4003 4071 80- 4139 4208 4276 4344 4412 448 4548 4616 4685 4753 80- 4821 4889 4957 5025 5093 5161 5229 5297 5365 5433 80- 5501 5569 5637 5705 5773 5841 5908 5976 6044 6112 80- 618 6248 6316 6384 6451 6519 6587 6655 6723 679 80- 6858 6926 6994 7061 7129 7197 7264 7332 74 7467 80- 7535 7603 767 7738 7806 7873 794 1 8008 8076 8143 80- 8211 8279 8346 8414 8481 8549 8616 8684 8751 8818 80- 8886 8953 9021 9088 9156 9223 929 9358 9425 9492 80- 956 9627 9694 9762 9829 9896 9964 81- _ _ 0031 0098 0165 81- 0233 03 0367 0434 0501 0569 0636 0703 077 0837 81- 0904 0971 1039 1106 1173 124 1307 1374 1441 1508 81- 1575 1642 1709 1776 1843 191 1977 2O44 2III 2178 81- 2245 2312 2379 2445 2512 2579 2646 2713 278 2847 81- 2913 298 3047 3114 3181 3247 3314 3381 3448 3514 81- 358i 3648 3714 3781 3848 3914 3981 4048 4114 4l8l 81- 4248 4314 4381 4447 4514 458l 4647 4714 478 4847 81- 4913 498 5046 5113 5179 5246 5312 5378 5445 5511 81- 5578 5644 57" 5777 5843 591 5976 6042 6109 6175 0-234 56789 324 LOGARITHMS OF NUMBERS. No. o i 2 3 4 5 6 7 8 9 D 655 81- 6241 6308 6374 644 6506 6573 6639 6705 6771 6838 66 656 81- 6904 697 7036 7102 7169 7235 7301 7367 7433 7499 66 657 81- 7565 7631 7698 7764 783 7896 7962 8028 8094 816 66 658 81- 8226 8292 8358 8424 849 8556 8622 8688 8754 882 66 6 59 81- 8885 8951 9017 9083 9149 9215 9281 9346 9412 9478 66 660 81- 9544 961 9676 9741 9807 9873 9939 66 660 82- 0004 007 0136 66 661 82- O20I 0267 0333 0399 0464 053 0595 0661 0727 0792 66 662 82- 0858 0924 0989 1055 112 1186 1251 *3!7 1382 1448 66 663 82- 1514 J 579 1645 171 1775 1841 1906 1972 2037 2103 65 664 82- 2l68 2233 2299 2364 243 2495 256 2626 2691 2756 65 665 82- 2822 2887 2952 3018 3083 3H8 3213 3279 3344 3409 6 5 666 82- 3474 3539 3605 367 3735 38 3865 393 3996 4061 65 667 82- 4126 4191 4256 4321 4386 445i 45*6 458i 4646 4711 6 5 668 82- 4776 4841 4906 4971 5036 5101 5166 5231 5296 536i 65 669 82- 5426 549 r 555 6 5621 5686 575i 5815 588 5945 601 6 5 670 82- 6075 614 6204 6269 6 334 6399 6464 6528 6593 6658 65 671 82- 6723 6787 6852 6917 6981 7046 7111 7 J 75 724 7305 6 5 672 82- 7369 7434 7499 7563 7628 7692 7757 7821 7886 795i 65 673 82- 8015 808 8144 8209 8273 8338 8402 8467 8531 8595 64 674 82- 866 8724 8789 88 53 8918 8982 9046 9111 9i75 9239 64 675 82- 9304 9368 9432 9497 956i 9625 969 9754 9818 9882 64 676 82- 9947 64 676 83- oon 0075 0139 0204 0268 0332 0396 046 0525 64 677 83- 0589 0653 0717 0781 0845 0909 0973 1037 IIO2 1166 64 678 83- 123 1294 1358 1422 1486 155 1614 1678 1742 1806 64 679 83- 187 1934 1998 2062 2126 2189 2253 2317 2381 2445 64 680 83- 2509 2573 2637 27 2764 2828 2892 2956 302 3083 64 68 1 83- 3^47 3211 3275 3338 3402 3466 353 3593 3657 3721 64 682 83- 3784 3848 3912 3975 4039 4103 4166 423 4294 4357 64 683 83- 442i 4484 4548 4611 4675 4739 4802 4866 4929 4993 64 684 83- 5056 5^2 5^83 5247 53i 5373 5437 55 5564 5627 63 685 83- 5691 5754 5817 5881 5944 6007 6071 6i34 6197 6261 63 686 83- 6324 6387 6451 6514 6577 6641 6704 6767 68 3 6894 63 687 83- 6957 702 7083 7146 721 7273 7336 7399 7462 7525 63 688 83- 7588 7652 7715 7778 7841 7904 7967 803 8093 8156 63 689 83- 8219 8282 8345 8408 8471 8534 8597 866 8723 8786 63 690 83- 8849 8912 8975 9038 9101 9164 9227 9289 9352 9415 63 691 83- 9478 954i 9604 9667 9729 9792 9855 9918 9981 63 691 84- 0043 63 692 84 0106 0169 0232 0294 0357 042 0482 0545 0608 0671 63 693 ; 84- 0733 0796 0859 0921 0984 1046 1109 1172 1234 1297 63 694 ; 84- 1359 1422 1485 1547 161 1672 1735 1797 186 1922 63 695 84- 1985 2047 211 2172 2235 2297 236 2422 2484 2547 62 696 84- 2609 2672 ^734 2796 2859 2921 2983 3046 3108 3i7 62 697 ! 84- 3233 3295 3^57 342 3482 3544 3606 3669 373i 3793 62 698 84- 3855 39i8 39? 4042 4104 4166 4229 4291 4353 4415 ! 62 699 84- 4477 4539 4601 4664 4726 4788 485 4912 4974 5036 , 62 700 84- 5098 5i6 5222 5284 5346 i 54o8 547 5532 5594 5656 ! 62 701 84- 57i8 578 5842 5904 5966 ! 6028 609 6151 6213 6275 62 702 84- 6337 6399 6461 6523 6585 6646 6708 677 6832 6894 62 703 84- 6955 7017 7079 7141 7202 7264 7326 7388 7449 75" I 62 704 84- 7573 7634 7696 7758 7819 | 7881 7943 8004 8066 8127 62 No. 1 2 3 4 5 6 7 8 9 1 D LOGARITHMS OP NTTMBEKS. 325 No. l 2 3 4 5 6 7 8 9 D 705 84- 8189 8251 8312 8374 8435 8497 8559 862 682 8743 ~62 706 84- 8805 8866 8928 8989 9051 9112 9 J 74 9235 9297 9358 61 707 84- 9419 9481 9542 9604 9665 9726 9788 9849 9911 9972 61 708 8fi- 0033 0095 0156 0217 0279 034 0401 0462 0524 0585 61 709 85- 0646 0707 0769 083 0891 0952 1014 1075 1136 1197 61 710 85- 1258 132 T38l 1442 1503 1564 1625 1686 1747 1809 61 711 85- 187 I93 1 1992 2053 2114 2175 2236 2297 2358 2419 61 712 85- 248 2541 2602 2663 2724 2785 2846 2907 2968 3029 61 7 T 3 85- 309 315 3 2II 3272 3333 3394 3455 3516 3577 3637 61 85-3698 3759 382 3881 3941 , 4002 4063 4124 4185 4245 61 715 85- 4306 4367 4428 4488 4549 461 467 4731 4792 4852 61 716 85- 4913 4974 5034 5095 5216 5277 5337 5398 5459 61 717 85- 5519 558 564 570 1 5761 ' 5822 5882 5943 6003 6064 61 718 85- 6124 6185 6245 6306 6366 6427 6487 6548 6608 6668 60 719 85- 6729 6789 68 5 691 697 7031 7091 7152 7212 7272 60 720 85- 7332 7393 7453 7513 7574 7634 7694 7755 7815 7875 60 721 85- 7935 7995 8056 8116 8176 8236 8297 8357 8417 8477 60 722 85- 8537 8597 8657 8718 8778 8838 8898 8958 9018 9078 60 723 i 85- 9138 9198 9258 9318 9379 9439 9499 9559 9619 9679 60 724 85- 9739 9799 9859 9918 9978 60 724 86- 0038 0098 0158 0218 0278 60 725 86- 0338 0398 0458 0518 0578 0637 0697 0757 0817 0877 60 726 86- 0937 0996 1056 1116 1176 1236 1295 1355 1415 1475 60 727 86- 1534 1594 1654 1714 1773 1833 1893 1952 2012 2072 60 728 86- 2131 2191 2251 231 237 243 2489 2549 2608 2668 60 729 86- 2728 2787 2847 2906 2966 3025 3085 3 J 44 3204 3263 60 730 86- 3323 3382 3442 3501 3561 362 368 3739 3799 3858 59 73 l 86- 3917 3977 4036 4096 4155 4214 4274 4333 4392 4452 59 732 86- 4511 457 463 4689 4748 4808 4867 4926 4985 5045 59 733 86- 5104 5163 5222 5282 534i 54 5459 5519 5578 5637 59 734 86- 5696 5755 5814 5874 5933 5992 6051 611 6169 6228 59 785 86- 6287 6346 6405 6465 6524 6583 6642 6701 676 6819 59 73 6 86- 6878 6937 6996 7055 7114 7173 7232 7291 735 7409 59 737 86- 7467 7526 7585 7644 7703 7762 ',32 1 788 7939 7998 59 738 86- 8056 8115 8i74 8233 8292 835 8409 8468 8527 8586 59 739 86- 8644 8703 8762 8821 8879 8938 8997 9056 9114 9173 59 740 86- 9232 929 9349 9408 9466 9525 9584 9642 9701 976 59 741 86- 9818 9877 9935 9994 59 741 87- - 0053 OIII 017 O2Cc 0287 0345 59 742 87- 0404 0462 0521 0579 0638 0696 0755 0813 0872 093 58 743 87- 0989 1047 1106 1164 1223 1281 1339 1398 1456 1515 58 744 87- 1573 1631 169 1748 1806 1865 1923 1981 204 2098 58 745 87- 2156 2215 2273 2331 2389 2448 2506 2564 2622 2681 58 746 87- 2739 2797 2855 2913 2972 303 3088 3146 3204 3262 58 747 87- 332i 3379 3437 3495 3553 3611 3669 3727 3785 3844 58 748 87- 3902 396 4018 4076 4192 425 4308 4366 4424 58 749 87- 4482 454 4598 4656 4714 4772 483 4888 4945 5003 58 750 87- 5061 5119 5177 5235 5293 5351 5409 5466 5524 5582 58 751 87- 564 5698 5756 5813 5871 5929 5987 6045 6102 616 58 752 87- 6218 6276 6333 6391 6449 6507 6564 6622 668 6737 58 753 87- 6795 6853 691 6968 7026 7083 7141 7199 7256 73M 58 754 87- 7371 7429 7487 7544 7602 7659 7717 7774 7832 7889 58 No. i 2 3 4 5 6 7 8 9 D EE 326 LOGARITHMS OF NUMBERS. 01234 56789 87- 7947 8004 8062 8119 8177 8234 8292 8^49 8407 8464 87- 8522 8579 86 37 8694 8752 8809 8866 8924 8981 9039 87- 9096 9153 9211 9268 9325 9383 944 9497 9555 9 12 87- 9669 9726 9784 9841 9898 9956 88- 0013 007 0127 0185 88- 0242 0299 0356 0413 0471 0528 0585 0642 0699 0756 88- 0814 0871 0928 0985 1042 1099 IJ 56 1213 1271 1328 88- 1385 1442 1499 1556 1613 167 1727 1784 1841 1898 88- 1955 2012 2069 2126 2183 224 2297 2354 2411 2468 88- 2525 2581 2638 2695 2752 2809 2866 2923 298 3037 88- 3093 315 3207 3264 3321 3377 3434 349 1 3548 3605 88- 3661 3718 3775 3832 3888 3945 4002 4059 4115 4172 88- 4229 4285 4342 4399 4455 4512 4569 4625 4682 4739 88- 4795 4852 4909 4965 5022 5078 5i35 5192 5248 530S 88- 5361 54i8 5474 553i 5587 5644 57 5757 5813 587 88- 5926 5983 6039 6096 6152 6209 6265 6321 6378 6434 88- 6491 6547 6604 666 6716 6773 6829 6885 6942 6998 88- 7054 7111 7167 7223 728 733 6 7392 7449 75O5 7 561 88- 7617 7674 773 7786 7842 7898 7955 8011 8067 8123 88- 8179 8236 8292 8348 8404 846 8516 8573 8629 8685 88- 8741 8797 8853 8909 8965 9021 9077 9134 919 9246 88- 9302 9358 9414 947 9526 9582 9638 9694 975 9806 88- 9862 9918 9974 gg_ 003 0086 0141 0197 0253 0309 0365 89- 0421 0477 0533 0589 0645 07 0756 0812 0868 0924 89- 098 1035 1091 1147 1203 1259 1314 137 1426 1482 89- 1537 1593 1649 1705 176 1816 1872 1928 1983 2039 89- 2095 215 2206 2262 2317 2373 2429 2484 254 2595 89- 2651 2707 2762 2818 2873 2929 2985 304 3096 3151 89- 3207 3262 3318 3373 3429 3484 354 3595 3 6 5* 376 89- 3762 3817 3873 3928 3984 4039 4094 415 4205 4261 89- 4316 4371 4427 4482 4538 4593 4648 4704 4759 4814 89- 487 4925 498 5036 5091 5146 5201 5257 5312 5367 89- 5423 5478 5533 5588 5644 5699 5754 5809 5864 592 89- 5975 603 6085 614 6195 6251 6306 6361 6416 6471 89- 6526 6581 6636 6692 6747 6802 6857 6912 6967 7022 89- 7077 7132 7187 7242 7297 7352 7407 7462 7517 7572 89- 7627 7682 7737 7792 7847 7902 7957 8012 8067 8122 89- 8176 8231 8286 8341 8396 8451 8506 8561 8615 867 89- 8725 878 8835 889 8944 8999 954 9 I0 9 9^4 9218 89- 9273 9328 9383 9437 9492 9547 9602 9656 9711 9766 89- 9821 9875 993 9985 90- 0039 0094 0149 0203 0258 0312 90- 0367 0422 0476 0531 0586 064 0695 0749 0804 0859 90- 0913 0968 1022 1077 II3I 1 1 86 124 1295 1349 1404 90- 1458 1513 1567 1622 1676 1731 1785 184 1894 1948 90- 2OO3 2O57 21 12 2l66 2221 2275 2329 2384 2438 2492 90- 2547 2001 2655 271 2764 2818 2873 2927 2981 3036 90- 309 3144 3199 3253 3307 3361 3416 347 3524 3578 90- 3633 3687 374i 3795 3849 3904 3958 4012 4066 412 90- 4174 4229 4283 4337 4391 4445 4499 4553 4607 4661 90- 4716 477 4824 4878 4932 4986 504 5094 5148 5202 90- 5256 53 1 53 6 4 54i8 5472 5526 558 5634 5688 5742 012^4 56789 LOGARITHMS OF NUMBERS. 327 No. 01234 56789 805 90- S79 6 585 594 5958 6012 6066 6119 6173 6227 6281 806 9- 6 335 6389 6443 6497 6551 6604 6658 6712 6766 682 807 90- 6874 6927 6981 7035 7089 7143 7196 725 7304 7358 808 90- 7411 7465 7519 7573 7626 768 7734 7787 7841 7895 809 90- 7949 8002 8056 811 8163 8217 827 8324 8378 8431 810 90- 8485 8539 8592 8646 8699 8753 8807 886 8914 8967 811 90- 9021 9074 9128 9181 9235 9289 9342 9396 9449 9503 812 90- 9556 9609 9663 9716 977 9823 9877 993 9984 812 91- 0037 813 91- 0091 0144 0197 0251 0304 0358 0411 0464 0518 0571 814 91- 0624 0678 0731 0784 0838 0891 0944 0998 1051 1104 815 91- 1158 I2II 1264 1317 1371 1424 1477 153 1584 1637 816 91- 169 1743 1797 185 1903 1956 2009 2063 2116 2169 817 91- 2222 2275 2328 2381 2435 2488 2541 2594 2647 27 818 91- 2753 2806 2859 2913 2966 3019 3072 3125 3178 3231 819 91- 3284 3337 339 3443 3496 3549 3602 3655 3708 3761 820 91- 3814 3867 392 3973 4026 4079 4132 4184 4237 429 821 91- 4343 4396 4449 4502 4555 4608 466 4713 4766 4819 822 91- 4872 4925 4977 503 5083 5136 5189 5241 5294 5347 823 9i- 54 5453 5505 5558 5611 5664 5716 5769 5822 5875 824 91- 5927 598 6033 ooSS 6138 6191 6243 6296 6349 6401 825 91- 6454 6507 6559 6612 6664 6717 677 6822 6875 6927 826 91- 698 7033 7085 7138 719 7243 7295 7348 74 7453 827 91- 7506 7558 7611 7663 7716 7768 782 7873 7925 7978 828 91- 803 8083 8135 8188 824 8293 8345 8397 845 8502 829 91- 8555 8607 8659 8712 8764 8816 8869 8921 8973 9026 830 91- 9078 913 9183 9235 9287 934 9392 9444 940 9549 831 91- 9601 9653 9706 9758 981 9862 9914 9967 831 92- 0019 0071 832 92- 0123 0176 0228 028 0332 0384 0436 0489 0541 0593 833 92- 0645 0697 0749 0801 0853 0906 0958 101 1062 1114 834 92- 1166 1218 127 1322 1374 1426 1478 153 1582 1634 835 92- i686 v 1738 179 1842 1894 1946 1998 205 2102 2154 836 92- 2206 2258 231 2362 2414 2466 2518 257 2622 2674 837 92- 2725 2777 2829 2881 2933 2985 337 3089 314 3192 838 92- 3244 3296 3348 3399 3451 3503 3555 3607 3658 371 839 92- 3762 3814 3865 3917 3969 4021 4072 4124 4176 4228 840 92- 4279 4331 4383 4434 4486 4538 4589 4641 4693 4744 841 92- 4796 4848 4899 4951 5003 5054 5106 5157 5209 5261 842 92- 53i2 5364 5415 5407 55i8 557 5621 5673 5725 5776 843 92- 5828 5879 5931 5982 6034 6085 6137 6188 624 6291 844 92- 6342 6394 6445 6497 6548 66 6651 6702 6754 6805 845 i 92- 6857 6908 6959 7011 7062 7114 7165 7216 7268 7319 846 92- 737 7422 7473 7524 7576 7627 7678 773 7781 7832 847 92- 7883 7935 7986 8037 8088 814 8191 8242 8293 8345 848 92- 8396 8447 8498 8549 8601 8652 8703 8754 8805 8857 849 92- 8908 8959 901 9061 9112 9163 9215 9266 9317 9368 850 92- 9419 947 9521 9572 9623 9674 9725 9776 9827 9879 851 92- 993 998i 851 93- 0032 0083 0134 0185 0236 0287 0338 0389 852 93- 044 049 1 0542 0592 0643 0694 0745 0796 0847 0898 853 93- 0949 i 1051 1 102 1153 1203 1254 1305 1356 1407 854 93- 1458 1509 156 161 1661 1712 1763 1814 1865 1915 No. 01234' 56789! 328 LOGARITHMS OF NUMBERS. 01234 56789 93- 1966 2017 2068 21 18 2169 222 2271 2322 2372 2423 93- 2474 2524 2575 2626 2677 2727 2778 2829 2879 2 93 93- 2981 3031 3082 3133 3183 3234 3285 3335 3386 3437 93- 34S7 3538 3589 3639 369 374 379 1 3841 3892 3943 93- 3993 4044 4094 4145 4195 4246 4296 4347 4397 4448 93- 4498 4549 4599 465 47 4751 4801 4852 4902 4953 93- 5003 5054 5104 5154 5205 5255 53o6 5356 5406 5457 93- 5507 5558 5608 5658 5709 5759 5809 586 591 50 93- 6011 6061 6m 6162 6212 6262 6313 6363 6413 6463 93- 6514 6564 6614 6665 6715 6765 6815 6865 6916 6966 93- 7016 7066 7117 7167 7217 7267 7317 7367 74i8 7468 93- 75i8 7568 7618 7668 7718 7769 7819 7869 7919 7969 93- 8019 8069 8119 8169 8219 8269 8319 837 842 847 93- 852 857 862 867 872 877 882 887 892 897 93- 902 907 912 917 922 927 932 9369 9419 9469 93- 9519 9569 9619 9669 9719 9769 9819 9869 9918 9968 94- 0018 0068 01 18 0168 0218 0267 0317 0367 0417 0467 94- 0516 0566 0616 0666 0716 0765 0815 0865 0915 0964 94- 1014 1064 1114 1163 1213 1263 1313 1362 1412 1462 94- 1511 1561 1611 166 171 176 1809 1859 1909 1958 94- 2008 2058 2107 2157 2207 2256 2306 2355 2405 2455 94- 2504 2554 2603 2653 2702 2752 2801 2851 2901 295 94- 3 3049 3099 3 T 48 3J98 3247 3297 3346 3396 3445 94- 3495 3544 3593 3^43 3^2 3742 379 1 3841 389 3939 94- 3989 4038 4088 4137 4186 4236 4285 4335 4384 4433 94- 4483 4532 4581 4631 468 4729 4779 4828 4877 4927 94- 4976 5025 5074 5124 5173 5222 5272 5321 537 5419 94- 5469 5518 5567 5616 5665 5715 57 6 4 5813 5862 5912 94- 5961 601 6059 6108 6157 6207 6256 6305 6354 6403 94- 6452 6501 6551 66 6649 6698 6747 6796 6845 6894 94- 6943 6992 7041 709 714 7189 7238 7287 7336 7385 94- 7434 7483 7532 758i 763 7679 7728 7777 7826 7875 94- 7924 7973 8022 807 8119 8168 8217 8266 8315 8364 94- 8413 8462 8511 856 8609 8657 8706 8755 8804 8853 94- 8902 8951 8999 9048 9097 9146 9195 9244 9292 9341 94- 939 9439 9488 9536 9585 9634 9683 9731 978 9829 94- 9878 9926 9975 95- 0024 0073 OI2T 017 0219 0267 0316 95- 03 6 5 0414 0462 0511 056 0608 0657 0706 0754 0803 95- 0851 09 0949 0997 1046 1095 1143 1192 124 1289 95- 1338 1386 1435 1483 1532 158 1629 1677 1726 1775 95- 1823 1872 192 1969 2017 2066 2114 2163 2211 226 95- 2308 2356 2405 2453 2502 255 2599 2647 2696 2744 95- 2792 2841 2889 2938 2986 3034 3083 3131 318 3228 95- 3276 3325 3373 342i 347 3518 3566 3615 3663 3711 95- 376 3808 3856 3905 3953 40OI 4049 4098 4146 4194 95- 4243 4291 4339 4387 4435 4484 4532 458 4628 4677 95- 4725 4773 4821 4869 4918 4966 5014 5062 511 5158 95- 5207 5255 5303 5351 5399 5447 5495 5543 559 2 564 95- 5688 5736 5784 5832 588 5928 5976 6024 6072 612 95- 6168 6216 6265 6313 6361 6409 6457 6505 6553 6601 o i 2 3 . 4 56789 OP NUMBERS. 329 No. 01234 5 6-7 8 9 905 95- 6649 6697 6745 6793 684 6888 6936 6984 7032 708 906 95- 7128 7176 7224 7272 732 7368 7416 7464 7512 7559 907 95- 7607 7655 7703 775* 7799 7847 7894 7942 799 8038 908 95- 8086 8134 8181 8229 8277 8325 8373 8421 8468 8516 909 95- 8564 8612 8659 8707 8755 8803 885 8898 8946 8994 910 95- 9041 9089 9137 9185 9232 928 9328 9375 9423 9471 911 95- 95i8 9566 9614 9661 9709 9757 9804 9852 99 9947 912 95- 9995 912 96- 0042 009 0138 0185 0233 028 0328 0376 0423 9*3 96- 0471 0518 0566 0613 0661 0709 0756 0804 0851 0899 914 96- 0946 0994 1041 1089 1136 1184 1231 1279 1326 1374 915 96- 1421 1469 1516 1563 1611 1658 1706 1753 1801 1848 916 96- 1895 1943 I 99 2038 2085 2132 218 2227 2275 2 3 22 917 96- 2369 2417 2464 2511 2559 2606 2653 2701 2748 2795 918 96- 2843 289 2937 2985 3032 3079 3126 3174 3221 3268 919 96- 33i6 3363 34i 3457 3504 3552 3599 3646 3693 374i 920 96- 3788 3835 3882 3929 3977 4024 4071 4118 4165 4212 921 96- 426 4307 4354 4401 4448 4495 4542 459 4^37 4684 922 96- 4731 4778 4825 4872 49*9 4966 5013 5061 5108 5155 9 2 3 96- 5202 5249 5296 5343 539 5437 5484 553i 5578 5625 924 96- 5672 5719 5766 5813 586 597 5954 6001 6048 6095 925 96- 6142 6189 6236 6283 6329 6376 6423 647 6517 6564 926 96- 6611 6658 6705 6752 6799 6845 6892 6939 6986 7033 927 96-708 7 1 27 7 1 73 722 7267 7314 7361 7408 7454 7501 928 0- 7548 7595 7642 7688 7735 7782 7829 7875 7922 7969 929 96- 8016 8662 8109 8156 8203 8249 8296 8343 839 8436 930 96- 8483 853 8576 8623 867 8716 8763 881 8856 8903 93i 96- 895 8996 9043 909 9136 9183 9229 9276 9323 9369 932 96- 9416 9463 9509 9556 9602 9649 9695 9742 9789 9835 933 96- 9882 9928 9975 933 97- OO2I OOO8 0114 0161 0207 0254 03 934 97- 0347 0393 44 0486 0533 0579 0626 0672 0719 0765 935 97- 0812 0858 0904 0951 0997 1044 109 1137 1183 1229 93 6 97- 1276 1322 1369 1415 1461 J 5o8 1554 1601 1647 l6 93 937 97- 174 1786 1832 1879 1925 1971 2018 2064 211 2157 938 97- 2203 2249 2295 2342 2388 2434 2481 2527 2573 2619 939 97- 2666 2712 2758 2804 2851 2897 2943 2989 3035 3082 940 97- 3128 3174 322 3266 3313 3359 3405 345i 3497 3543 94i 97- 359 3636 3682 3728 3774 382 3866 3913 3959 4005 942 07- 4051 4097 4i43 4189 4235 4281 4327 4374 442 4466 943 1 97- 4512 4558 4604 465 4696 4742 4788 4834 488 4926 9*4 97- 4972 5018 5064 511 5156 5202 5248 5294 534 5386 945 97- 5432 5478 5524 557 5616 5662 5707 5753 5799 5845 946 97- 5891 5937 5983 6029 6075 6121 6167 6212 6258 6304 947 97- 6 35 6 396 6 442 6488 6533 6579 6625 6671 6717 6763 948 97- 6808 6854 69 6946 6992 7037 7083 7129 7175 722 949 97- 7266 7312 7358 7403 7449 7495 7541 7586 7632 7678 950 Q7- 7724 7769 7815 786i 7906 7952 7998 8043 8089 8135 951 97- 8181 8226 8272 8317 8363 8409 8454 85 8546 8591 952 97- 8637 8683 8728 8774 8819 8865 8911 8956 9002 9047 953 97- 9093 9 J 3 8 9 l8 4 9 2 3 9 2 75 9321 9366 9412 9457 9503 954 1 97- 9548 9594 9639 9685 973 9776 9821 9867 9912 9958 No. 01234 56789 EE* 330 LOGARITHMS OF NUMBERS. No. i 2 3 4 5 6 7 8 9 D 955 98- 0003 0049 0094 014 0185 0231 0276 0322 0367 0412 45 956 98- 0458 0503 0549 0594 064 0685 073 0776 0821 0867 i 45 957 98- 0912 0957 1003 1048 1093 "39 1184 1229 1275 132 45 958 98- 1366 1411 1456 1501 1547 1592 1637 1683 1728 1773 45 959 98- 1819 1864 1909 1954 2 2045 209 2135 2181 2226 45 960 98- 2271 2316 2362 2407 2452 2497 2543 2588 2633 2678 45 961 98- 2723 2769 2814 2859 2904 2949 2994 304 3085 313 45 962 98- 3175 322 3265 331 3356 340i 3446 3491 3536 358i 45 963 98- 3626 3671 37l6 3762 3807 3852 3897 394 2 3987 4032 45 964 98- 4077 4122 4167 4212 4257 4302 4347 4392 4437 4482 45 965 98- 4527 4572 4617 4662 4707 475 2 4797 4842 4887 4932 45 966 98- 4977 5022 5067 5112 5*57 5202 5247 5292 5337 5382 45 967 98- 5426 547 1 55l6 SS^I 5606 5651 5696 5741 5786 583 45 968 98- 5875 592 5965 601 6055 61 6144 6189 6234 6279 45 969 98- 6324 6369 6413 6458 6503 6548 6593 6637 6682 6727 45 970 98- 6772 6817 686l 6906 6951 6996 704 7085 713 7175 45 97 1 98- 7219 7264 7309 7353 7398 7443 7488 7532 7577 7622 45 972 98- 7666 7711 7756 78 7845 789 7934 7979 8024 8068 45 973 98- 8113 8i57 8202 8247 8291 8336 8381 8425 847 8514 45 974 98- 8559 8604 8648 8693 8737 8782 8826 8871 8916 896 45 975 98- 9005 9049 9094 9138 9183 9227 9272 9316 9361 9405 45 976 98- 945 9494 9539 9583 9628 9672 9717 9761 9806 985 44 977 98- 9895 9939 9983 i 44 977 99- 0028 0072 0117 0161 0206 025 0294 44 978 99- 0339 0383 0428 0472 0516 0561 0605 065 0694 0738 44 979 99- 0783 0827 0871 0916 096 1004 1049 1093 "37 1182 i 44 980 99- 1226 127 1315 1359 1403 1448 1492 i53 6 158 1625 44 981 99- 1669 1713 1758 1802 1846 189 1935 1979 2023 2067 44 982 99- 2IH 2156 22 2244 2288 2333 2377 2421 2465 2509 44 933 99- 2 554 2598 2642 2686 273 2774 2819 2863 2907 295 1 44 984 99- 2995 3039 3083 3127 3172 3216 326 3304 3348 3392 44 985 99- 3436 348 3524 3568 3613 3657 3701 3745 3789 3833 44 986 99- 3877 3921 3965 4009 4053 4097 4141 4185 4229 4273 44 987 99- 43!7 436i 4405 4449 4493 4537 458i 4625 4669 4713 44 988 99- 4757 4801 4845 4889 4933 4977 5021 5065 5108 5152 44 989 99- 5196 524 5284 5328 5372 54i6 546 5504 5547 559 1 44 090 99- 5635 5679 5723 5767 5811 5854 5898 5942 5986 603 44 991 99- 6074 6117 6161 6205 6249 6293 6337 638 6424 6468 44 992 99- 6512 6555 6599 6643 6687 6731 6774 6818 6862 6906 44 993 99- ^49 6993 7037 708 7124 7168 7212 7255 7299 7343 44 994 99- 7386 743 7474 7517 756i 76o5 7648 7692 7736 7779 44 995 99- 7823 7867 791 7954 7998 8041 8085 8129 8172 8216 44 996 99- 8259 8303 8347 839 8434 8477 8521 8564 8608 8652 44 997 99- 8695 8739 8782 8826 8869 8913 8956 9 9043 9087 44 998 99- 9131 9174 9218 9261 9305 9348 9392 9435 9479 9522 44 999 99- 9565 9609 9652 9696 9739 9783 9826 987 9913 9957 43 No. I 2 3 4 5 6 7 8 9 D HYPERBOLIC LOGARITHMS OF NUMBERS. S3' Hyperbolic Logarithms of Numbers. From i. 01 to 30. In following table, the numbers range from i.oi to 30, advancing by .01, up to the whole number 10 ; and thence by larger intervals up to 30. The hyperbolic logarithms of numbers, or Neperian logarithms, as they are some- times termed, are computed by multiplying the common logarithms of num- bers by the constant multiplier, 2.302 585. The hyperbolic logarithms of numbers intermediate between those which are given in the table may be readily obtained by interpolating proportional differences. No. Log. No. Log. No. Log. I No. Log. No. Log. I.OI .0099 I. 4 I .3436 1.81 5933 2.21 793 2.61 9594 1.02 .0198 1.42 .3507 1.82 .5988 2.22 7975 2.62 .9632 1.03 .0296 i-43 3577 1.83 .6043 2.23 .802 2.63 .967 1.04 .0392 1.44 .3646 1.84 .6098 2.24 .8065 2.64 .9708 1.05 .0488 1-45 .3716 1.85 6152 2.25 .8109 2.65 .9746 1.06 .0583 1.46 .3784 1.86 .6206 2.26 8i54 2.66 9783 1.07 .0677 1.47 .3853 1.87 .6259 2.27 .8198 2.67 .9821 .08 .077 1.48 392 1.88 6 3 J 3 2.28 .8242 2.68 .9858 .09 .0862 1.49 .3088 1.89 .6366 2.29 .8286 2.69 9895 .1 0953 i-5 .4055 1.9 .6419 2-3 .8329 2.7 9933 .11 .1044 i-5i .4121 1.91 .6471 2.31 .8372 2.71 .9969 .12 "33 1.52 .4187 1.92 6523 2.32 .8416 2.72 i. 0006 13 .1222 1-53 4253 i-93 6575 2-33 8458 2-73 1.0043 .14 .131 i-54 .4318 1.94 .6627 2-34 .8502 2.74 1.008 15 .1398 i-55 .4383 i-95 .6678 2-35 8544 2-75 i .0116 .16 .1484 1.56 4447 1.96 .6729 2. 3 6 8587 2.76 1.0152 - 1 ? 157 i-57 45U 1.97 .678 2-37 .8629 2.77 i. 01 88 .18 1655 1.58 4574 1.98 .6831 2. 3 8 .8671 2.78 1.0225 .19 .174 i-59 -4637 1.99 .6881 2-39 8713 2.79 1.026 .2 .1823 1.6 47 2 .6931 2.4 8755 2.8 1.0296 .21 .1906 1.61 .4762 2.OI .6981 2.41 .8796 2.81 1.0332 .22 .1988 1.62 .4824 2. 02 7031 2.42 .8838 2.82 1.0367 23 .207 1.63 .4886 2.03 .708 2-43 .8879 2.83 1.0403 .24 .2151 1.64 4947 2.04 .7129 2.44 .892 2.84 1.0438 25 .2231 1.65 .5008 2.05 .7178 2-45 .8961 2.85 1-0473 .26 .2311 1.66 .5068. 2.06 .7227 2.46 .9002 2.86 1.0508 .27 239 1.67 .5128 2.07 7275 2.47 .9042 2.87 1-0543 .28 .2469 1.68 .5188 2.08 7324 2.48 9083 2.88 1.0578 .29 .2546 1.69 5247 2.O9 7372 2.49 .9123 2.89 1.0613 3 .2624 i-7 5306 2.1 .7419 2-5 .9163 2.9 1.0647 1-31 .27 1.71 5365 2. II .7467 2.51 .9203 2.91 1.0682 1.32 .2776 1.72 5423 2.12 75*4 2.52 9243 2.92 1.0716 1-33 .2852 i-73 .5481 2.13 7501 2-53 .9282 2-93 1-075 1-34 .2927 1.74 5539 2.14 .7608 2-54 .9322 2.94 1.0784 1-35 .3001 i-7S 559 2.15 7655 2-55 .9361 2-95 i. 0818 1.36 .3075 1.76 5653 2.16 .7701 2.56 94 2.96 1.0852 i-37 .3148 1.77 571 2.17 7747 2-57 9439 2.97 1.0886 1.38 .3221 1.78 .5766 2.18 7793 2.58 9478 2.98 1.0919 i-39 3293 1.79 .5822 2 19 7839 2-59 9517 2-99 1-0953 1.4 .3365 1.8 .5878 2.2 .7885 2.6 9555 3 1.0986 HYPEEBOLIC LOGARITHMS OF NUMBERS. Log. No. Log. [ No. Log. No. Log. No. Log. I.IOI9 3-51 1.2556 4-01 1.3888 4-5 1 1.5063 5-oi 1.6114 I -1053 3-52 1.2585 4.02 I.39I3 4-52 1.5085 5-02 1.6134 1. 1086 3-53 1.2613 4-03 I-3938 4-53 1.5107 5.03 1.6154 I.IIIQ 3-54 1.2641 4.04 1.3962 4-54 1.5129 5-04 1.6174 I.IISI 3-55 1.2669 4-05 L39S7 4-55 1.5151 5-05 1.6194 1.1184 3.56 1.2698 4.06 1.4012 4-56 I.5I73 5.o6 1.6214 I.I2I7 3-57 1.2726 4.07 1.4036 4-57 LS^S 5.07 1 1.6233 1.1249 3.58 1.2754 4.08 1.4061 4-58 1.5217 5.08 1.6253 I.I282 3-59 1.2782 4.09 1.4085 4-59 I-5239 5-09 1.6273 1.1314 3-6 1.2809 4.1 1.411 4.6 1.5261 5-i 1.6292 1.1346 3-6i 1.2837 4.II I.4I34 4.61 1.5282 5-n 1.6312 I.I378 3-62 1.2865 4.12 I.4I59 4.62 I-5304 5.12 1.6332 I.I4I 3-63 1.2892 4.13 1.4183 4-63 1-5326 5-13 I-635I I.I442 3-64 1.292 4.14 1.4207 4.64 1-5347 5.i4 1.6371 1.1474 3-65 1.2947 4-15 1.4231 4-65 1-5369 5-15 1.639 I.I506 3-66 1-2975 4 .l6 1.4255 4.66 1-539 5-i6 1.6409 I-I537 3-67 1.3002 4.17 1.4279 4-67 1.5412 5-17 1 .6429 1.1569 3.68 1.3029 4.18 1.4303 4.68 1-5433 5-i8 1.6448 1.16 3-69 1.3056 4.19 1.4327 4.69 1-5454 5-19 1 .6467 1.1632 3-7 1.3083 4.2 L435I 4-7 I-5476 5-2 1.6487 1.1663 3-7i I.3II 4.21 1-4375 4.71 1-5497 5-21 1.6506 1.1694 3-72 I.3I37 4.22 1.4398 4.72 i-55i8 5-22 1.6525 1.1725 3-73 1.3164 4-23 1.4422 4-73 1-5539 5-23 1.6514 1.1756 3-74 1.3191 4-24 1.4446 4-74 I-556 5-24 1.6563 1.1787 3-75 1.3218 4-2S 1.4469 4-75 i.558i 5-25 1.6582 1.1817 3.76 1.3244 4.26 1-4493 4.76 1.5602 5-26 I.660I 1.1848 3-77 1.3271 4.27 1.4516 4-77 1-5623 5.27 1.662 1.1878 3-78 1.3297 4.28 1-454 4.78 1.5644 5-28 1.6639 1.1909 3-79 L3324 4.29 1-4563 4-79 1.5665 5.29 1.6658 I-I939 3-8 1-335 4-3 1.4586 48 1.5686 5-3 1.6677 1.1969 3-8i 1.3376 4-31 1.4609 4.81 I-5707 5-31 1.6696 1.1999 382 1.3403 4.32 1-4633 4.82 1.5728 5-32 I.6 7 I5 1.203 3-83 1.3429 4-33 1.4656 4-83 1.5748 5-33 1-6734 1.206 3-84 1-3455 4-34 1.4679 4.84 1-5769 5-34 1.6752 1.209 3-85 1.3481 4-35 1.4702 4-85 1-579 5-35 1.6771 1.2119 3-86 L3507 4-36 1.4725 4.86 1.581 5.36 1.679 1.2149 3-87 1-3533 4-37 1.4748 4-87 1.5831 5-37 1. 6808 1.2179 3-88 1-3558 4-38 1-477 4.88 1-5851 5.38 1.6827 1.2208 3.89 I-3584 4-39 1-4793 4.89 1.5872 5-39 1.68 4 5 1.2238 3-9 1.361 4-4 1.4816 4.9 1.5892 5-4 1.6864 1.2267 3-9i 1-3635 4.41 1.4839 4.91 I -59 I 3 5-41 1.6882 1.2296 3-92 1.3661 4-42 1.4861 4.92 1-5933 5-42 1.6901 1.2326 3-93 1.3686 4-43 1.4884 4-93 1-5953 5-43 1.6919 1.2355 3-94 1.3712 4.44 1.4907 4.94 1-5974 5-44 1.6938 1.2384 3-95 1-3737 4-45 1.4929 4-95 1-5994 5-45 1.6956 1.2413 3-90 1.3762 4.46 i-495i 4.96 1.6014 5-46 1.6974 1.2442 3-97 1.3788 4-47 1.4974 4-97 1.6034 5-47 1.6993 1.247 3-98 1-3813 4.48 1.4996 4-98 1.6054 5-48 I.7OII 1.2499 3-99 1.3838 4-49 1.5019 4.99 1.6074 5-49 1.7029 1.2528 4 1.3863 4-5 1.5041 5 1.6094 5-5 1.7047 HYPERBOLIC LOGARITHMS OF NUMBERS. 333 No. Log. No. Log. No. Log- f No. | Log. 1 No. Log. 5.5i 1.7066 6.01 1-7934 6. 5 I 1-8733 7.01 1-9473 7-51 2.0162 5-52 1.7084 6.02 I-795I 6.52 1.8749 7.02 1.9488 7-52 2.0176 5-53 1 1.7102 6.03 1.7967 :: 6-53 1.8764 7.03 1.9502 7-53 2.0189 5-54 1.712 6.04 1.7984 j 6-54 1.8779 7.04 I.95I6 7-54 2.0202 5-55 1.7138 6.05 1.8001 6.55 1.8795 7-05 1-953 7-55 2.0215 5.56 1.7156 6.06 1.8017 6.56 I.88I 7.06 1-9544 7.56 2.0229 5-57 ' 1.7174 6.07 1.8034 6.57 1.8825 7.07 1-9559 7-57 2.0242 5-58 1.7192 6.08 1.805 6.58 1.884 7.08 1-9573 7-58 2.0255 5-59 1.721 6.09 i. 8066 6-59 1.8856 7.09 L9587 7-59 2.0268 5-6 1.7228 6.1 1.8083 6.6 1.8871 7-1 1.9601 7.6 2.O28I 5-6i 1.7246 6.ii 1.8099 6.61 1.8886 7.II I.96I5 7.61 2.0295 5-62 1.7263 6.12 1.8116 6.62 1.8901 7.12 1.9629 7.62 2.0308 5-63 1.7281 6.13 1.8132 6.63 1.8916 7-13 1.9643 7-63 2.032J 5-64 1.7299 6.14 1.8148 6.64 1.8931 7.14 1.9657 7.64 2.0334 5-65 I.73I7 6.15 1.8165 i 6.65 1.8946 7-15 1.9671 7.65 2.0347 566 1-7334 6.16 1.8181 6.66 1.8961 7 .l6 1.0685 7.66 2.036 5-6? 1-7352 6.17 1.8197 6.67 1.8976 7.17 1.9699 7.67 2-0373 5-68 1-737 6.18 1.8213 ; 6.68 1.8991 7 .l8 I-97I3 7.68 2.0386 5-69 1.7387 6.19 1.8229 6.69 1.9006 7.19 1.9727 7.69 2.0399 5-7 I.7405 6.2 1.8245 | 6-7 1.9021 7.2 1.9741 7-7 2.0412 5-7i 1.7422 6.21 1.8262 6.71 1.9036 7.21 1-9755 7.71 2.0425 5-72 1.744 6.22 1.8278 6.72 1.9051 7.22 1.9769 7.72 2.0438 5-73 1-7457 6.23 1.8294 6.73 1.9066 7-23 1.9782 7-73 2.0451 5-74 1-7475 6.24 1.831 6.74 1.9081 7.24 1.9796 7-74 2.0464 5-75 1.7492 6.25 1.8326 6.75 1.9095 7.25 1.981 7-75 2.0477 5.76 1.7509 6.26 1.8342 6.76 1.911 7 .36 1.9824 7.76 2.049 5-77 1.7527 6.2 7 1-8358 6.77 1.9125 7-27 1.9838 7-77 2.0503 5.78 1-7544 6.28 1-8374 6.78 1.914 7 .28 1.9851 7.78 2.0516 5-79 1.7561 i 6.29 1.839 6.79 i.9i55 7.29 1.9865 7-79 2.O528 5-8 1-7579 6-3 1.8405 6.8 1.9169 7-3 1.9879 7.8 2.0541 5-8i 1.7596 6. 3 I 1.8421 6.81 1.9184 7.3i 1.9892 7.81 2.0554 5-82 1.7613 6. 3 2 1-8437 6.82 1.9199 7-32 1.9906 7.82 2.0567 5-83 1-763 6-33 1.8453 6.83 1.9213 7-33 1.992 7.83 2.058 5-84 1.7647 6-34 1.8469 6.84 1.9228 7-34 1-9933 7.84 2.0592 5.85 1.7664 ;6. 35 1.8485 6.85 1.9242 7-35 1.9947 7.85 2.0605 5-86 1.7681 6.36 1.85 6.86 1.9257 7.36 1.9961 7.86 2.o6l8 5-87 1.7699 6.37 1.8516 6.87 1.9272 7-37 1.9974 7.87 2.0631 5-88 1.7716 6. 3 8 1-8532 6.88 1.9286 7.38 1.9988 7.88 2.0643 5-89 1-7733 6.39 1-8547 6.89 1.9301 7-39 2.0001 7.89 2.0656 5-9 1-775 6.4 1.8563 6.9 i.93i5 7-4 2.0015 7-9 2.0669 5.9i 1.7766 i 6.41 1-8579 6.91 1-933 7.41 2.0O28 7.91 2.o68l 5-92 1.7783 6.42 1.8594 6.92 1-9344 7.42 2.0042 7.92 2.0694 5-93 1.78 6-43 1.861 6-93 t-9359 7-43 2.0055 7-93 2.0707 5-94 1.7817 6.44 1.8625 6.94 1-9373 7-44 2.O009 7-94 2.O7I9 5-95 1.7834 6.45 1.8641 6-95 I-9387 7-45 2.O082 7-95 2.0732 S.96 1.7851 i 6.46 1.8656 6.96 i .9402 7.46 2.0096 7.96 2.0744 5-97 1.7867 ! 6.47 1.8672 6.97 1.9416 7-47 2.OIO9 7-97 2.0757 598 1.7884 6.48 1.8687 6.98 1-943 7.48 2.0122 7.98 2.0769 5-99 1.7901 6.49 1.8703 6-99 1-9445 7-49 2.0136 7-99 2.0782 6 1.7918 6.5 1.8718 7 1.9459 7-5 2.OI49 1 8 2.0794 334 HYPERBOLIC LOGARITHMS OF NUMBERS. Log. No. Log. No. Log. No. Log. No. Log. 2.0807 8.41 2.1294 8.81 2.1759 9.21 2.22O3 9.61 2.2628 2.0819 8. 4 2 2.1306 8.82 2.177 9.22 2.2214 9.62 2.2638 2.0832 8-43 2.1318 8.83 2.1782 9-23 2.2225 9-03 2.2649 2.0844 8.44 2.133 8.84 2.1793 9.24 2.2235 9.64 2.2659 2.0857 8-45 2.1342 8.85 2.1804 9-25 2.2246 9-05 2.267 2.0869 8.46 2.1353 8.86 2.1815 9.26 2.2257 9.66 2.268 2.0882 8.47 2.1365 8.87 2.1827 9.27 2.2268 9.67 2.269 2.0894 8.48 2.1377 8.88 2.1838 9.28 2.2279 9.68 2.2701 2.0906 8-49 2.1389 8.89 2.1849 9.29 2.2289 9.69 2.2711 2.0919 8.5 2.1401 8.9 2.1861 9-3 2.23 9-7 2.2721 2.0931 8. 5 I 2.1412 8.91 2.1872 9-31 2.2311 9.71 2.2732 2.0943 8.52 2.1424 8.92 2.1883 9-32 2.2322 972 2.2742 2.0956 8-53 2.1436 8-93 2.1894 9-33 2.2332 973 2.2752 2.0968 8-54 2.1448 8.94 2.1905 9-34 2.2343 974 2.2762 2.098 8-55 2.1459 8-95 2.1917 9-35 2.2354 975 2.2773 2.0992 8.56 2.1471 8.96 2.1928 9.36 2.2364 9.76 2.2783 2.1005 8-57 2.1483 8.97 2.1939 9-37 2-2375 9-77 22793 2.IOI7 8.58 2.1494 8.98 2.195 9-38 2.2386 9.78 2 2803 2.IO29 8-59 2.1506 8.99 2.1961 9-39 2.2396 9-79 2.2814 2.IO4I 8.6 2.1518 9 2.1972 9.4 2.2407 98 2.2824 2.1054 8.61 2.1529 9.01 2.1983 9.41 2.2418 9.81 2.2834 2.I066 8.62 2.1541 9.02 2.1994 9.42 2.2428 ! 9.82 2.2844 2.1078 8.63 2.1552 9-03 2.2006 9-43 22439 983 22854 2.IO9 8.64 2.1564 9.04 2.2017 9-44 2245 984 2.2865 2.II02 8.65 2.1576 9-05 2.2028 9-45 2.246 9-85 2.2875 2.III4 8.66 2.1587 9.06 2.2039 9.46 2.2471 9.86 2.2885 2.II20 8.67 2.1599 9.07 2.205 9-47 2.2481 9.87 2.2893 2.II38 8.68 2.161 9.08 2.2061 948 2 2492 ! 9.88 2.2003 2.II5 8.69 2.1622 9.09 2.2072 9-49 2.2502 9.89 2.2913 2.1163 8.7 2.1633 9.1 2.2083 9-5 2.2513 9-9 2.2923 2.II75 8.71 2.1645 9.11 2.2094 9.5i 2.2523 9.91 2.2933 2.1187 8.72 2.1656 9.12 2.2105 9-52 2-2534 9.92 2.294(j 2.II99 8-73 2.1668 9-i3 2. 2Il6 9-53 2.2544 9-93 2.295^ 2.I2II 8.74 2.1679 9.14 2.2127 9-54 2.2555 9-94 2.206 2.1223 8-75 2.1691 9-i5 2.2138 9-55 2.2565 9-95 2-297 remainders. 60 50 = 10 ) Whence, 30 X 20 X 10 X 60 = 360000, and ^360000 = 600 square feet When all Sides are Equal. RULE. Square length of a side, and multi- ply product by .433. Or, S 2 x .433 = area, S representing length of a side. To Compute Length of One Side of a Right-Angled Triangle. When Length of the other Tiro Sides are given. To Ascertain Hypotenuse. Fig. C. RULE. Add together squares of the two legs, and take square root of sum. _ _ Or, Va 6 2 -f 6 c 2 = hypotenuse. Or, Vb 2 + h*. EXAMPLE. Base, a &, Fig. 5. is 30 ins., and height, & c, 40; what is length of hy- potenuse ? 3o 2 + 4o 2 = 2500, and V 2 5oo = 50 ins. To Ascertain, other Leg. When Hypotenuse and One of the Legs are given.Fig. 5. RULE. Sub- tract square of given leg from square of hypotenuse, and take square root of remainder. EXAMPLE. Base of a triangle, a &, Fig. 5, is 30 feet, and hypotenuse, a c, 50; what is height of it? 5o 2 so 2 = 1600, and 1/1600 = 40/66*. To Compute Length of a Side. When Hypotenuse of a Right-angled Triangle of Equal Sides alone is given. Fig. 8. RULE. Divide hypotenuse by 1.414213. Or, hyp ' = length of a side. Fig. 8. - '- 4 ' 4213 / EXAMPLE. Hypotenuse a c of a right-angled triangle, Fig. 8, is 300 feet; what is length of its sides? 300-:- 1. 414 213 = 212. 1321 feet. a~~ & To Compute Perpendicular or Height of a Triangle. When Base and Area alone are given. Fig. 9. RULE. Divide twice area by its base. Or, za -4- 6 = h. EXAMPLE. Area of a triangle, Fig. 9, is 10 feet, and length of its base, a &, 5; what is its perpendicular, c d? TO X 2 = 20, and 20-7- 5 = 4 feet. MENSURATION OF AREAS, LINES, AND SURFACES. 337 To Compnte Perpendicular or Height of a Triangle. When Base and Two Sides are given. RULE. As base is to sum of the sides, so is difference of sides to difference of divisions of base. Half this difference being added to or subtracted from half base will give the two di- visions thereof. Hence, as the sides and their opposite division of base con- stitute a right-angled triangle, the perpendicular thereof is readily ascertained by preceding rules. tc + ox ba = ad; whence -\ zab EXAMPLE. Three sides of a triangle, a b c, Fig. 9, are 9.928, 8, and 5 feet; what is length of perpendicular on longest side ? As 9.928 : 8 -\- 5 : : 8 t ce *n = area, n representing number of sides. *& *4- EXAMPLE. What is area of a pentagon, side a 6, Fig. 14, being 5 feet, and distance c e 4.25 feet? 5 x 4- 25 X 5 (n) = 106. 25 = product of length of a side, dis- tance to centre, and number of sides. Then, 106.25 -f- 2 = 53.125 square feet. To Compute Radius of a Circle that contains a Q-iven Polygon. When Length of a Perpendicular from Centre alone is given. RULE. Multiply distance from centre to a side of the polygon, by unit in column A of following Table. EXAMPLE. What is radius of a circle that contains a hexagon, distance to centre being 4.33 inches? 4-33X 1.156 = 5 ins. To Compute Length of a Side of a Polygon that is con- tained in a GHven Circle. When Radius of Circle is given. RULE. Multiply radius of circle, by unit in column B of following Table. EXAMPLE. What is length of side of a pentagon contained in a circle 8.5 feet in diameter? 8.5-7-2 = 4.25 radius, and 4.25 x 1.1756 = 5 fiet. MENSURATION OF AREAS, LINES, AND SURFACES. 339 To Compute Radius of* a Circumscribing Circle. When Length of a Side is given. RULE. Multiply length of a side of the polygon, by unit in column C of following Table. EXAMPLE. What is radius of a circle that will contain a hexagon, a side being 5 inches? 5X1 = 5 ins. To Compute Radius of a Circle that can "be Inscribed in a Griven IPolygon. When Length of a Side is given. RULE. Multiply length of a side of polygon, by unit in column D of following Table. EXAMPLE. What is radius of the circle that is bounded by a hexagon, its sides being 5 inches? 5 X- 866 = 4.33 ins. To Compute A.rea of* a Regular IPolygon. Wlten Length of a Side only is given. RULE. Multiply square of side, by multiplier opposite to term of polygon in following Table : No. of Sides. POLYGON. AKEA. A. Radius of Circumscribed B. Length of a C. Radios of Circumscrib- D. Radius of Inscribed Circle. Side. ing Circle. Circle. 3 Trigon 43301 i 73 2 5773 .2887 4 Tetragon I .414 1.4142 .7071 5 5 Pentagon 1.72048 .238 1.1756 .8506 .6882 6 Hexagon 2.59808 .156 I .866 7 Heptagon 3-633 9 1 .11 .8677 .1524 0383 8 Octagon 4.82843 .083 7653 .3066 .2071 9 Nonagon 6.18182 .064 .684 .4619 3737 10 Decagon 7.69421 .051 .618 .618 .5388 ii Undecagon 9-36564 .042 5634 7747 7028 12 Dodecagon 11.196 15 037 5176 .9319 .866 EXAMPLE. What is area of a square (tetragon) when length of its sides is 7 0710678 inches? 7.071 067 8 2 = 50, and 50 X i = 50 square ins. To Compute Length of a Side and Radii of a Regular Polygon. When Area alone is given. RULE. Multiply square root of area of poly- gon by multiplier in column E of the following table for length of side ; by multiplier in column G for radius of circumscribing circle, and by multiplier in column H for radius of inscribed circle or perpendicular. No. of Sides. POLYGON. E. Length of Side. G. Radius of Circumscrib- ing Circle. H. Radius of Inscribed Circle. Angle. Angle of Polygon. Tangent. 3 Trigon I.5I97 .8774 4387 120 60 5774 4 Tetragon I .7071 5 90 9 I 5 Pentagon .7624 .6485 5247 72 1 08 I-3764 6 Hexagon .6204 -6204 5373 60 120 1.7321 7 Heptagon .5246 .6045 5446 51 25.71' 128 34.29' 2.0765 8 Octagon 4551 .5946 5493 45 135 2.4142 9 Nonagon .4022 .588 5525 40 140 2-7475 10 Decagon .3605 5833 5548 36 144 3-0777 ii Undecagon .3268 5799 5564 32 43-64' 147 16.36' 3-4057 12 Dodecagon .2989 5774 5577 30 'SO 3-7321 EXAMPLE i. Area of a square (tetragon) is 16 inches; what is length of its side? V/i6 = 4, and 4X1=4 ins. 2. Area of an octagon is 70.698 yards; what is diameter of its circumscribing circle? ^70.698 X . 5946 = 5, and 5 x 2 = 10 yards. 34O MENSURATION OF AREAS, LINES, AND SURFACES. Additional Uses of foregoing Table. 6th and ;th columns of table facilitate con- struction of these figures with aid of a sector. Thus, if it is required to describe an octagon, opposite to it in column 6th, is 45; then, with chord of 60 on sector as radius, describe a circle, taking length 45 on same line of sector; mark this dis- tance off on the circumference, which, being repeated around the circle, will give points of the sides. yth column gives angle which any two adjoining sides of the respective figures make with each other; and 8th gives tangent of .5 angle in column yth. To Coxxipu.te Radius of Inscribed, or Circumscribed. Circles. When Radius of Circumscribing Circle is given. RULE. Multiply radius given by unit in column E, in following Table, opposite to term of polygon for which radius is required. When Radius of Inscribed Circle is given. RULE. Multiply radius given by unit in column F, in following Table, opposite to term of polygon for which radius is required. To Compute Area. When Radii of Inscribed or Circumscribing Circles are given. RULE. Square radius given, and multiply it by unit in columns G or H, in following Table, and opposite to term of polygon for which area is required. When Length of a Side is given. RULE. Square length of side and multiply it by unit in column I, in following Table, opposite to term of polygon for which area is required. To Compute T-jength. of* a Side. When Radius of Inscribed Circle is given. RULE. Multiply radius given by unit in column K, in following Table, and opposite to term of polygon for which length is required. E. F. G. H. I. K. Radius of Radius of Area. Length of No. of Sides. POLYGON. Inscribed by Circum- scribing Circumscrib- ing by Inscribing Area. By Radius of Inscribed Circle. By Radius of Circum- scribing Area. By Length of Side. Side. By Radius of Inscribed Circle. Circle. Circle. Circle. 3 Trigon 5 5- 196 15 1.29904 43301 3.4641 4 Tetragon .707 II .41421 4 2 i 2 I I 9 Pentagon Hexagon Heptagon Octagon Nonagon .80902 .86602 .90097 .92388 .93969 .23607 1547 .10992 .08239 .064 18 3.63272 3.4641 3.37102 3-3I37I 3- 2 7573 2.37764 2.59808 2.73641 2.82842 2.89254 1.72048 2.59808 3-6339 1 4.81843 6.18282 1.45308 I-I547 9 6 3I5 .82843 .727 94 10 ii 13 Decagon Undecagon Dodecagon .95106 95949 9 6 593 .05146 .04222 .035 28 3.2492 3.22989 3-21539 2 -93893 2-97353 3 7.69421 9-36564 11.19615 .649 84 58725 5359 Regular Bodies. To Compnte Surface or Linear Edge of Regular Body. RULE. Multiply square of linear edge, or radius of circumscribed or in- scribed sphere, by units in following table, under head of dimension used : No. of Sides. BODY. Surface by Linear Edge. Radius of Circumscribed Sphere. Radius of Inscribed Sphere. Linear Edge by Surface. I Tetrahedron Hexahedron 1.73205 6 1.63299 J - 154 7 4. 898 98 2 759 8 4 8 12 2O EVrii Octahedron Dodecahedron Icosahedron 3.4641 20.645 78 8.66025 1.41421 71364 1.051 46 2-44949 .89806 1-32317 53729 .22008 3398i EXAMPLE. What is surface of a hexahedron or cube, having sides of 5 inches? S 2 X 6 = 25 X 6 = 150 square ins. MENSURATION OF AREAS, LINES, AND SURFACES. 34! To Compute Linear Edge. When Surface alone ig given. RULE. Multiply square root of surface, by multiplier opposite to term of body under head of Linear Edge by Sur- face in preceding Table. EXAMPLE. What is linear edge of a hexahedron, surface being 6 inches? V6 X -408 25 = i inch. When Radius of an Inscribed or Circumscribed Sphere is given. RULE. Multiply radius given, by multiplier opposite to term of body in preceding Table, under head of the Radius given. EXAMPLE. Radius of circumscribing sphere of a hexahedron is 10 inches; what is its linear edge? 10 X i.iS47 = II -547 ins* To Compute Surface. When Linear Edge is given. RULE. Multiply square of edge, by multi- plier opposite to term of body hi preceding Table, under head of Surface. EXAMPLE. Linear edge of a hexahedron is i inch; what is its surface? i 2 x 6 = 6 square ins. Irregidar Polygons. DEFINITION. Figures with unequal sides. To Compute .A^rea of an Irregular Polygon. Figs. 15 and. 16. RULE. Draw diagonals and per- pendiculars, as. d /*, dg, a, and c, Fig. 15, and/d, g d, g\ g M, and i, o, r, and *, Fig. 16, to divide the figures into triangles and quadrilaterals: ascer- tain areas of these separately, and take their sum. NOTE. To ascertain area of mixed or compound figures, or such as are composed of rectilineal and curvilineal figures to- gether, compute areas of the several figures of which the whole is composed, then add them together, and the sum will give area of compound figure. In this manner any irregular surface or field of land may be measured by dividing it into trapeziums and triangles, and computing area of each separately. When any Part of a Figure is bounded by a Curve the Area may be ascer- tained as follows: Erect any number of perpendiculars upon base, at equal distances, and ascertain their lengths. Add lengths of the perpendiculars thus ascertained together, and their sum, divided by their number, will give mean breadth ; then multiply mean breadth by length of base. To Compute Area of a Long, Irregular Figure. Fig. 17". Fig. 17. RULE. Take mean breadths at several places, at equal distances apart, as i, 2, 3, b d r etc. ; add them together, divide their sum by number of breadths for total mean breadth, and multiply quotient by length of figure. 6", etc. x I = area. 342 MENSURATION OF AREAS, LINES, AND SURFACES. To Coiio.pu.te an Area "bounded, toy a Curve. 3T"ig. 18 (Simpsons Rale.) OPERATION. Divide line a b into any number of equal parts, D 7 perpendiculars from base, as i, 2, 3, etc., which will give an odd number of points of division. Measure lengths of a i 2 3 4 5 these perpendiculars or ordinates, and proceed as follows: To sum of lengths of first and last ordinates, add four times sum of lengths of all even numbered ordinates and twice sum of odd; multiply their sum by one third of distance between ordinates, and product will give area required. ILLUSTRATION. Water-line of a vessel has a length of 80 feet, and ordinates o, i, 1.2, 1.5, 2, i. 9, 1.5, i.i, and o, each 10 feet apart; what is its area? Ordinatt*. Even. Odd. Sums, i 1.2 first o 1.5 2 last o 1.9 1.5 even 22 i.i odd 9.4 575X4 = 22. 4.7X2 = 9.4 31- 4 X 10 = 314, Vfh\ch--r 3 = 104. 66 square feet Circle. Diameter is a right line drawn through its centre, bounded by its periphery. Radius is a right line drawn from its centre to its circumference. Circumference is assumed to be divided into 360 equal parts, termed degrees; each degree is divided into 60 parts, termed minutes; each minute into 60 parts, termed seconds ; and each second into 60 parts, termed thirds, and so on. To Compute Circumference of a Circle. RULE. Multiply diameter by 3.1416. Or, as 7 is to 22, so is diameter to circumference. Or, as 113 is to 355, &Q is diameter to circumference. EXAMPLE. Diameter of a circle is 1.25 inches; what is its circumference? 1.25 X 3.1416 = 3.927 ins. To Compute Diameter of a Circle. RULE. Divide circumference by 3.1416. Or, as 22 is to 7, so is circumference to diameter. NOTE. Divide area by .7854, and square root of quotient will give diameter of circle. To Compute A_rea of a Circle. RULE. Multiply square of diameter by .7854. Or, multiply square of circumference by .07958. Or, multiply half circumference by half diameter. Or, multiply square of radius by 3. 1416. Or, p r 2 = area, r representing radius. EXAMPLE. The diameter of a circle is 8 inches; what is the area of it? 8 2 = 64, and 64 X .7854 = 50.2656 ins. Proportions of a Circle, its Kc^ual, Inscribed, and Cir cumscrilbed Sq.uares. CIRCLE. 1. Diameter X .8862) id f F , q niiarft 2. Circumference X .2821 } - Ol an Lquai b( l uare - 3. Diameter X .7071) 4. Circumference X .2251 [ =Side of Inscribed Square. 5. Area x .9003 -i-diam. ) 6. Diameter X 1-3468 = Side of an Equilateral Triangle. SQUARE. 7. A Side X 1.4142 = Diameter of its Circumscribing Circle. 8. " X 4.443 = Circumference of its Circumscribing Circle. 9. u X 1-128 = Diameter ) 10. " X3-545 = Circumference^ of an Equal Circle. u. Square inches X 1.273 = Circle inches ) NOTE. Square described within a circle is one half area of one described without it. MENSURATION OF AREAS, LINES, AND SURFACES. 343 To Compute Side of* C3-reatest Square that can. "be In- scribed, in a Circle. RULE. Multiply diameter by .7071, or take twice square of radius. TJsefUl Factors. In -wliicn. p or it represents Circumference of a Circle. Diameter = i. p= 3. 141 592653 589+ 2J> = 6.283185307179-!- 4^=12.566370614359+ %p i. 570 79 6 326 794+ XP= -785 39 8 i 6 3 397+ % P= -39 26 99+ TSP= -261799+ ?&)P= -008726+ Diameter = 10. VP= 1-772453 yj= -797884 Log. p = .49714987 %y/p= .886226-)- 36 p= 113.097 335+ x. Chord of arc of semicircle =10 2. Chord of half arc of semicircle = 7.071067 3. Versed sine of arc of semicircle. = 5 4. Versed sine of half arc of semicircle = 1.464 466 5. Chord of half arc, of half of arc of semicircle = 3 . 826 83 6. Half chord, of chord of half arc = 3-535533 7. Length of arc of semicircle =15-707963 8. Length of half arc of semicircle = 7.853981 9. Square of chord, of half arc of semicircle (2) = 50 10. Square root of versed sine of half arc (4) = 1.210151 11. Square of versed sine of half arc (4) = 2.144664 12. Square of chord of half arc, of half arc of semicircle (5) = 14.64467 13. Square of half chord, of chord of half arc (6) = 12. 5 NOTE. In all computations # is taken at 3.1416, % p at .7854, %p at .5236; and whenever the decimal figure next to the one last taken exceeds 5, one is added. Thus, 3.141 59 for four places of decimals is taken as 3.1416. To Compute Length, of an Arc of a Circle. Fig. 19. When Number of Degrees and Radius are given. RULE i. Multiply number of degrees in the arc by 3.1416 times the radius, and divide by 180. 2. Multiply radius of circle by .01745329, and product by degrees in the arc. If length is required for minutes, multiply radius by .000 290 889 ; if for seconds, by .000004848. * 9 EXAMPLE i. Number of degrees in an arc, o a &, Fig. 19, are 90, and radius, o 6, 5 inches; what is length of arc? 90 X (3.1416 X 5) = 1413-72, which-:- 180 = 7.854 ins, 2. Radius of an arc is 10, and measure of its angle 44 30* 30"; what is length of arc? 10 X .017 453 29 = . 174 532 g, which X 44 = 7-679 447 6, length for 44. xo X -ooo 290 889 = .002 908 89, which X 30 = .087 266 7, length for 30'. xo X .000004 848 = .000048 48, which X 30 = .001 454 4, length for 30''. Then 7. 679 447 6) .087 266 7 > = 7.768 168 7 ins. .0014544) Or, reduce minutes and seconds to decimal of a degree, and multiply by it. See Rule, page 93. 30' 30" = .5083, and .1745329 from above X 44.5083 = 7.768163 ins. 344 MENSURATION OF AREAS, LINES, AND SURFACES. When Chord of Half Arc and Chord of Arc are given. RULE. From eight times chord of half arc subtract chord of arc, and one third of remainder will give length nearly. Or, , c' representing chord of half arc, and c chord of arc. EXAMPLE. Chord of half arc, a c, Fig. 19, is 30 inches, and chord of arc, a 6, 48; what is length of arc ? 30 X 8 = 240 8 times chord of half arc ; 240 48 = 192, and 192 -4- 3 = 64 ins. When Chord of Arc and Versed Sine of Arc are given. RULE. Mul- tiply square root of sum of square of chord, and four times square of the versed sine (equal to twice chord of half arc), by ten times square of versed sine ; divide this product by sum of fifteen times square of chord and thirty- three times square of versed sine ; then add this quotient to twice chord of half arc,* and sum will give length of arc very nearly. . Vc 2 4- 4 v - sin- 2 X 10 v. sin. 2 . Or, -^ c z~ v sin 2 1~ 2 c ' v ' ww ' ^Presenting versed sine. EXAMPLE. Chord of an arc is 80, and its versed sine, c r, 30 ; what is length of arc ? 8o 2 = 6400 = square of chord ; 3o 2 = 900 = square of versed sine. v / (64oo -{- 900 x 4) = ioo = square root of square of chord and four times square of versed sine = twice chord of half arc. Then ioo X so 2 X 10 = goo ooo = product o/io times square of versed sine and root above obtained. And 8o 2 X 15 = 96 ooo = 15 times square of chord. 3o 2 x 33 = 29 700 = 33 times square of versed sine. 12570 Hence 2^22 = 7.1599, and 7.1599 + 100, or twice chord of half arc = 107.1599 125700 length. When Diameter and Versed Sine are given. RULE. Multiply twice chord of half the arc by 10 times versed sine ; divide product by 27 times versed sine subtracted from 60 times diameter, add quotient to twice chord of half arc, and the sum will give length of arc very nearly. |00._ 6od 27 v. sin. EXAMPLE. Diameter of a circle is ioo feet, and versed sine, cr, of arc 25 ; what is length of arc? V25 x ioo = 50 = chord of half arc. See Rule, page 345. 50 X 2 X 25 x io = 25 ooo = twice chord of half arc by 10 times versed sine. ioo X 60 25 X 27 = 5325 = 27 times versed sine from 60 times diameter. Then 2500 = 4.6948, and 4.6948 -f 50X2 = 104. 6948 feet. To Compute Chord, of* an Arc. When Chord of Half the Arc and Versed Sine are given. RULE. From square of chord of half arc subtract square of versed sine, and take twice square root of remainder. Or, V (c' 2 v. sin. *)X2 = c. EXAMPLE. Chord of half arc, a c, is 60, and versed sine, c r, 36; what is length of chord of arc? 6o 2 s6 2 = 2304, and v / 234 X 2 = 96. * Square root of sum of square of chord and four times square of the versed sine is equal to twicr chord of half arc. MENSURATION OF AREAS, LINES, AND SURFACES. 345 When Diameter and Versed Sine are given. Multiply versed sine by 2, and subtract product from diameter; subtract square of remainder from square of diameter, and take square root of that remainder. Or, V d 2 (d v. sin. x 2) 2 = c. EXAMPLE. Diameter of a circle is 100, and versed sine of the arc is 36; what la length of chord of arc ? (100 36 X 2) 2 ioo 2 = 92i6, and -^9216 96. To Compute Chord, of Half an Arc. When Chord of the Arc and Versed Sine are given. RULE i. Divide square root of sum of square of chord of the arc and four times square of versed sine by two. 2. Take square root of sum of squares of half chord of arc and versed sine. When Diameter and Versed Sine are given. RULE. Multiply diameter by versed sine, and take square root of their product. Or, Vdx v. sin. = c'. To Compute Diameter. RULE i. Divide square of chord of half arc by versed sine. Or, c' 2 -i- v. sin. =z diameter. 2. Add square of half chord of arc to the square of versed sine, and divide this sum by versed sine. v. sin. To Compute Versed Sine. RULE. Divide square of chord of half arc by diameter. c' 2 Or, = v. sin. d When Chord of the Arc and Diameter are given. RULE. From square of diameter subtract square of chord, and extract square root of remainder ; subtract this root from diameter, and divide remainder by 2. When it is greater than a Semidiameter. RULE. Proceed as before, but add square root of remainder (of squares of diameter and chord) to diam- eter, and halve the sum. EXAMPLE. Diameter of a circle is 100, and chord of arc 97.9796; what is its versed sine? ioo -j- Vioo 2 97. 9796 2 ioo-j-2o 2 2 To Compute Ordinate of a Circular Curve. Fig. 2O. Vr 2 x 2 (r v) = ordinate. ILLUSTRATION. Radius of circle 5 ins., versed sine \ 2, and distance x 2 ; what is length of ordinate o ? V 5 2 _2 2 (5-2) = 4 . 58-3 = 1.58 ins. X 346 MENSURATION OF AREAS, LINES, AND SURFACES. Sector of a Circle. DEFINITION. A part of a circle bounded by an arc and two radii. To Compute Area of* a Sector of a Circle. When Degrees in the Arc are given. Fig. 21. RULE. As 360 is to num- ber of degrees in a sector, so is area of circle of which sector is a part to area of sector. da Or, = area, d representing degrees in arc, and a area 36o x / of circle. \^ / EXAMPLE. Radius of a circle, o a, Fig. 21, is 5 ins., and \^/ number of degrees of sector, a b o, is 22 30' ; what is area ? o Area of a circle of 5 ins. radius = 78. 54 ins. Then, as 360 : 22 30' :: 78.54 : 4-9 8 75 *'. When Length of the Arc, etc., are given. RULE. Multiply length of arc by half length of radius, and product is area. Or, b X r -r- 2 = area, 6 representing arc, and r radius. Segment of a Circle. DEFINITION. A part of a circle bounded by an arc and a chord. To Conapnte Area of a Segment of a Circle. When Chord and Versed Sine of Arc, and Radius or Diameter of Circle are given. When Segment is less than a Semicircle, as a b c, Fig. 21. RULE. Ascer- tain area of sector having same arc as segment ; then ascertain area of tri- angle formed by chord of segment and radii of sector, and take difference of these areas. NOTE Subtract versed sine from radius; multiply remainder by one half of chord of arc, and product will give area of triangle. Or, a a' = area, a and a' representing areas of sector and triangle. When Segment is greater than a Semicircle. RULE. Ascertain, by pre- ceding rule, area of lesser portion of circle ; subtract it from area of whole circle, and remainder will give area. Or, a a' = area, a and a' representing areas of circle and lesser portion. See Table of Areas of Segments, page 267. Fig. 22. , EXAMPLE. Chord, a c, Fig. 22, is 14.142; diameter, 6 e, is 20 ins. ; and versed sine, br, is 2.929; what is area of segment? 14. 142 -r- 2 = 7.071 = half chord of arc. V7-o7i 2 - T -2.929 2 = 7.654 = square root of sum of squares of half chord of arc and versed sine, which is chord a b of half arc ab c. By Rule, page 344, 7.654 X 2 X 2.929 X 10 = 448.371 = fr/nce chord of half arc by 10 times versed sine. 20X60 2.929X27 = 1120.917=60 times diameter subtracted from 27 times versed sine. Then 448. 371 -4-1120.917 = .4, and .4 added to 7.654 x 2 (twice chord of half arc) = 15.708 inches, length of arc. By Rule above, 15.708 X = 78.54 = toe arc multiplied by half length of radius, = area of sector. I0 2 . 929 = 7. 071 = versed sine subtracted from a radius, which is height oftri angle a o c, and 7.071 x * 4 ' 142 = 5 = area of triangle. Consequently, 78. 54 50 = 28. 54. MENSURATION OF AREAS, LINES, AND SURFACES. 347 When the Chords of Arc ^ and of half of Arc, and Versed Sine are given. RULE. To chord of whole arc add chord of half arc and one third of it more ; multiply this sum by versed sine, and this product, multiplied by .404 26, will give area nearly. Or, c + c' H v. sin. x 404 26 = area nearly. EXAMPLE. Chord of a segment, a c, Fig. 22, is 28 feet; chord of half arc, a 6, is 1 5 ; and versed sine, b r, 6 ; what is area of segment ? 28 -f- 15 -f- 15 = chord of arc added to chord of half arc and one third of it more. 48 x 6 = 288 = product of above sum and versed sine. Hence 288 x .404 26 = 116.427 square feet. When the Chord of Arc and Versed Sine only are given. RULE. Ascer- tain chord of half arc, and proceed as before. To Compute Chord, and Height of* a Segment of a Circle. When Area is given. RULE. Divide area by square of diameter of circle, take tab. height for area from table of Areas of Segments of a Circle, p. 267, multiply it by diameter, and product will give required height. From diameter subtract height, multiply remainder by height, take square root of product and multiply it by 2 for required chord. Or, J^ = (tab. area for height) Xd = h, and Vd hxh. X 2 = c. Circular Measure. (See Rule, page 1 13.) Sphere. DEFINITION. A figure, surface of which is at a uniform distance from centre. To Compnte Convex Surface of a Sphere. Fig. S3. Fig. 23. RULE. Multiply diameter by circumference, and prod* uct will give surface. Or, 4 p r 2 = surface. * Or, p d 2 = surface. EXAMPLE. What is convex surface of a sphere, Fig. 23, hav- ing a diameter, a 6, of 10 ins? 10 x 31-416 = 314.16 square ins. Segment of a Sphere. DEFINITION. A section of a sphere. To Compute Surface of a Segment of a Sphere. Fig. 34r. RULE. Multiply height by the circumference of sphere, and add product to the area of base. Or, 2 p r h = cor: : zx surface alone. Fig. 24. EXAMPLE. Height, b o, of a segment, a b c, Fig. 24, is 36 ins., and diameter, 6 e, of sphere 100 ; what is convex surface, and what whole surface ? 36 X ioo X 3- 1416 = n 309. 76 = height of segment multiplied by circumference of sphere. To ascertain area of base ; diameter and versed siie being given, diameter of base of segment, being equal to chord of arc, is, by Rule, page 344 , ioo 36 X 2 = 28 ; Vioo 2 28 2 = 96. 96* X. 7854 =7238. 2464 = convex surface, and 7238.24644-11309.76=18548.0064 = convex surface added to area of base = square ins. NOTE. When convex surface of a figure alone is required, area or areas of base or ends must be omitted. * p or IT represent* in this, and in all cases where it is used, ratio of circumference of a circle to itf diameter, or 3.1416. 348 MENSURATION OF AREAS, LINES, AND SURFACES. When the Diameter of Base of Segment and Height of it are alone given. RULE. Add square of half diameter of base to the square of height ; divide this sum by height, and result will give diameter of sphere. Or, d-r-2 + h 2 -r-h = diameter. Spherical Zone (or IBVustrum of a Sphere). DEFINITION. The part of a sphere included between two parallel chords. To Compute Surface of a Spherical Zone. Wig. 25* Fig. 25. ^ RULE. Multiply height by the circumference of sphere, and add product to area of the two ends. Or, h c -f- a -f- a' = surface. Or, 2prJi = convex surface alone. EXAMPLE. Diameter of a sphere, a &, Fig. 25, from which a zone, c g, is cut, is 25 inches, and height, c g, is 8 ; what is convex surface ? 25 X 3.1416 X 8 = 628. 32 = height X circumference of sphere = square ins. When the Diameter of Sphere is not given. RULE. Multiply mean length of the two chords by half their difference ; divide this product by breadth of zone, and to quotient add breadth. To square of this sum add square of lesser chord, and square root of their sum will give diameter of sphere. Spheroids or Ellipsoids. DEFINITION. Figures generated by the revolution of a semi- ellipse about one of its diameters. When revolution is about Transverse diameter they are Prolate, and when it is about Conjugate they are Oblate. To Compute Surface of a Spheroid. Fig. Q6. When Spheroid is Prolate. RULE. Square diameters, and multiply square root of half their sum by 3.1416, and this product by conjugate diameter. /d 2 I d' 2 Fig. 26 c Or, / X3-i4i6xd surface, d and d' represent- ing conjugate and transverse diameters. r EXAMPLE. A prolate spheroid, Fig. 26, has diameters, c d and a&, of io and 14 inches; what is its surface? io 2 -f- 142 = 296 sum of squares of diameters. 296-7-2 = 148, and -v/H^ = 12. 1655 = square root of half sum of squares of diameters. 12.1655 X 3-1416 X 10= 382.191 in*. = product of root above obtained X 3-1416, and by conjugate diameter. When Spheroid is Oblate. RULE. Square diameters, and multiply square root of half their sum by 3.1416. and this product by transverse diameter. /d 2 -\-d' 2 Or, / - x 3. 1416 X d' = surface. EXAMPLE. An oblate spheroid has diameters of 14 and io inches; what is its surface ? i2 2 -f- io 2 = 296 = sum of squares of diameters. 296 -=-2 = 148, and -,/ 14% = 12. 1655 = square root of half sum of squares of di- ameter. 12.1655 X 3.1416 x 14 = 535.0679 in.=product of root above obtained X 3.1416, and by transverse diameter. MENSURATION OF AREAS, LINES, AND SURFACES. 349 To Compute Convex Surface of a Segment of a Sphe- roid. Figs. 27 and 3S. RULE. Square diameters, and take square root of half their sum ; then, as diameter from which the segment is cut is to this root, so is the height of segment to proportionate height required. Multiply product of other di- ameter and 3.1416 by proportionate height of segment, and this last product will give surface. Yd 2 -I- d' 2 2 - x h x d' or d X 3- 1416 = surface. Fig. 27. Fig. *& dor a, EXAMPLE. Height, a o, of a seg- ment, ef, of a prolate spheroid, Fig. 27, is 4 inches, diameters being 10 and 14; what is convex surface of it? Square root of half sum of squares a[ of diameters, 12.1655. Then 14 : 12. 1655 : : 4 : 3. 4758 = height of segment, proportionate to mean of diameters, and 10 X 3- 1416 X 3-4758 = 109. 1957 ins. 2. Height, co, of a segment of an oblate spheroid, Fig. 28, is 4 inches, the diam eters being 14 and 10; what is convex surface of it? 214.0272 square ins. To Compute Convex Sur/ace of a Frustum or Zone of a Spheroid. Figs. 39 and 3O. RULE. Proceed as by previous rule for surface of a segment, and obtain proportionate height of frustum ; then multiply product of diameter par- allel to base of frustum and 3.1416 by proportionate height of frustum, and it will give surface. Fig. 29. EXAMPLE. Middle frustum, o e, of a prolate spheroid, Fig. 29, is 6 inch- es, diameters of spheroid being 10 and 14; what is its convex surface? Mean diameter, as per preceding example, is 12.1655. Diameter parallel to base of frus- tum is 10. Fig. 30. Then 14 : 12.1655 : ; 6 : 5.2138, and 10 X 3.1416 X 5.2138=1 163.7967 square ins. 2. Middle frustum of an oblate spheroid, as o e, Fig. 30, is 2 inches in height, diameters of spheroid, as in preceding examples, being 10 and 14 ; what is its con- vex surface? 107.0136 square ins. Circular Zone. DEFINITION. A part of a circle included between two parallel chords. To Compute Area of a Circular Zone. RULE. From area of circle subtract areas of segments. Or, see Table of Areas of Zones, page 269. When Diameter of Circle is not given. Multiply mean length of the two chords by half their difference ; divide this product by breadth of zone, and to quotient add the breadth. To square of this sum add square of lesser chord, and square root of their sum will give diameter of circle. EXAMPLE. Greater chord, Jig, is 90 inches; lesser, a c, is 80; and breadth of zone, ao, is 72.526; what is its diameter? e = 8s x 5 = 4*5, and - + ,,^8.385. Then A/78. 385 2 -f8o 2 = V 12 544- 2 = 112 = diameter. GQ 35O MENSURATION OF AREAS, LINES, AND SURFACES. Cylinder. DEFINITION. A figure formed by revolution of a right-angled parallelogram around one of its sides. To Compute Sxirface of a Cylinder. -Fig. 31. RULE. Multiply length by circumference, and add product to area of the two ends. Or, I c -f- 2 a =. . s, a representing area of end. NOTE. When internal or convex surface alone is wanted, areas of ends are omitted. EXAMPLE. Diameter of a cylinder, b c, Fig. 31, is 30 inches, and its length, a b, 50; what is its surface? 30 X 3.1416 = 94.248, and 94.248 X 50 = 4712.4. Then 3o 2 X .7854 706.86 =area of one end; 706.86 X 2 = 1413.72 = area of both ends, and 4712.4 + 1413.72 = 6125. 12 square ins. Frisrns. DEFINITION. Figures, sides of which are parallelograms, and ends equal and parallel. NOTE. When ends are triangles, they are termed triangular prisms ; when they are square, square or right prisms ; and when they are a pentagon, pentagonal prisms, etc. To Compute Surface of a Right 3?rism. Figs. 33 and. 33. Fig. 32. Fig. 33- RULE. Ascertain areas of ends and sides, and add them together. Or, 2 a-}- n a' .= s, a representing area of ends, a' area of sides, and n their number. EXAMPLE. Side, a b, Fig. 32, of a square prism is 12 inches, and length, & c, 30; what is surface? 12X12 = 144 = area of one end ; 144 x 2 = 288 = area of both ends ; 12 X 30 = 360 = area of one side ; 360 X 4 = 1440 = area of four sides, and 288 + 1440= 1728 sq. ins. To Compute Surface of an Otolique or Irregular iPrism. Fig. 34. RULE. Multiply perimeter of one end, by perpendic- ular height, a o. Or, multiply perimeter as at c, at a right angle to sides by actual length of figure, and add area of ends. EXAMPLE. Sides, a c, of an oblique hexagonal prism, Fig. 34, are 10 inches, and perpendicular height, a o, is 5 feet; what is its sur- face? 10 X 6 = 60 ins. =. length of sides. 60 X 5 X 12 = 3600 square ins.=:area of sides, and by table, page 339, zoo X 2. 598 08 X 2 = 519.616 square ins., which added to 3600 = 4119.616 square ins. DEFINITION. A wedge is a prolate triangular prism, and its surface is computed by rule for that of a right prism. To Compute Surface of a "Wedge. Fig. 35. Fig. 35. EXAMPLE. Back of a wedge, abed, Fig. 35, is 20 by 2 inches, and its end, ef, 20 by 2; what is its surface? i 2o 2 -}- 2 j = 401 = sum of squares of half base, af, and height, ef, of triangle, efa. 1/401 = 20.025 = square root of above sum =. length ofea. Then 20.025 X 20 X 2 = 801 = area of sides. And 20X2 = 40 = area of back; and 20 X 2-^-2 X2 = 4o = area of ends. Hence 801 -f- 40 -f- 40 = 88 1 square ins. MENSURATION OF AREAS, LINES, AND SURFACES. 351 3?rism.oicls. . DEFINITION. Figures alike to a prism, having only one pair of sides parallel To Compute Surface of a IPrismoid. Fig. 36. RULE. Ascertain area of sides and ends as by rules for squares, triangles, etc., and add them together. EXAMPLE. Ends of a prismoit', efg h and a b c d, Fig. 36, are 10 and 8 inches square, and its slant height, d h, 25; what is its surface? 10 X 10= ioo area of base ; 8 X 8=64 = area of top. :> - X 25 =: 225, and 225 X 4 = 900= area of sides. Then ioo -f- 64 + 9 = 1064 = square ins. To Compute Surface of an. Ot>liq.ne or Irregular IPrismoid. Proceed as directed for an Oblique or Irregular Prism, page 350. TJngulas. DEFINITION. Cylindrical ungulas are the parts (including all or part of the base) left by a plane cutting a cylinder through any portion and at any angle. TJngxila. Figs. 37, RULE i. Mul- Fig-37- Fig. 38. To Compute Curved Surface of an 38, 39, and 4O. When Section is parallel to Axis of the Cylinder, Fig. 37, tiply length of arc of one end by height. EXAMPLE. Diameter of a cylinder, a c, from which an ungula, Fig. 37, is cut, is 10 inches, its length, 6 d, 50, and versed sine or depth of ungula is 5 inches; what is curved surface ? 10 -f- 2 = 5 = radius of cylinder. Hence radius and versed sine are equal; the arc, there- fore, of ungula is one half circumference of the cylinder, which is 31.416-7- 2 = 15-708, und 15.708 X 50 = 785.4 square ins. When Section passes obliquely through opposite Sides of Cyl- inder, Fig. ^8. RULE 2. Multiply circumference of base of cylinder by half sum of greatest and least heights of ungula. EXAMPLE. Diameter, cd, of a cylindrical ungula, Fig. 38, is 10 inches, and great- er and less heights, b d and a c, are 25 and 15 inches; what is its curved surface? 10 diameter = 31.416 circumference; 25+15 =40, and 40-^ 2 = 20. Hence 31.416 X 20 = 628.32 square ins. When Section passes through Base of Cylinder and one of its Sides, and Versed Sine does not exceed Sine, or Base is equal to or less than a Semi- circle, Fig. 39. RULE 3. Multiply sine, a d, of half arc, d g, of base, d g f, by diameter, e g, of cylinder, and from this product subtract product * of arc and cosine, a o. Multiply difference thus found by quotient of height, g c, divided by versed sine, dg. NOTE. The sine of base is half of the longest chord that can be drawn in base. EXAMPLE. Sine, a d, of half arc of base of an ungula, Fig. 39, is 5, diameter of cylinder, eg, is 10, and height, eg, of ungula 10 inches; what is curved surface ? 5 x 10 = 50 = sine of half arc by diameter. Length of arc, versed sine and radius being equal, under Rule, page 346 = 15.708, and as versed sine and radius are equal, cosine is o. Hence, when cosine is o, product is o. Therefore 50 = 50 = dif ference of product before obtained and product of arc and cosine, and 50 X 10 -r- 5 = 50 X 2 ioo square ins. * When the cosine is o. this product is o. 352 MENSURATION OF AREAS, LINES, AND SURFACES. When Section passes through Base of Cylinder, and Versed Sine, a g, ex- ceeds Sine, or when Base exceeds a Semicircle, Fig. 40. RULE 4. Multiply Fig. 40. sme f half the arc of base by diameter of cylinder, and to thfs product add product of arc and the excess of versed sine over the sine of base. Multiply sum thus found by quotient of height divided by versed sine. EXAMPLE. Sine, a d, of half arc of an ungula, Fig. 40, is 12 inches; versed sine, ag, is 16; height, c g, 16; and diameter of cylinder, h g, 25 inches; what is curved surface? g 12X25 = 300 =. sine of half arc by diameter of cylinder, and length of arc of base, Rule, page 344 = arcofd hf circumference of base = Then 46.392X16 12.5 = 162.372, and 300-1-162.272 = 462.372; 16-:- 16 = 1, and 462.372 X i = 462.372 square ins. Fig 41 NOTE. When sine of an arc is o, the versed sine is equal to diameter. When Section passes obliquely through both Ends of Cylinder, Fig. 41. RULE 5. Conceive section to be continued to m, till it meets side of cylinder produced ; then, as difference of versed sines, a e and d o, of arcs of two ends of ungula is to versed sine, a e, of arc of the less end, so is height of cylinder, a d, to the part of side produced. Ascertain surface of each of ungulas thus found by Rules 3 and 4, and their difference will give curved surface. DEFINITION. Space between intersecting arcs of two eccentric circles. To Compute Area of a Lune. T^ig. 43. RULE. Ascertain areas of the two segments from which lune is formed, and their difference will give area. EXAMPLE. Length of chord a c, Fig. 42, is 20 inches, height c d is 3, and e b 2 ; what is area of lune ? By Rule 2, page 345, diameters of circles of which lune is formed are thus ascertained: Then, by Rule for Areas of Segments of a Circle, page 267, Area of a d c is 70.5577 sq. ins. " aec " 27.1638 " Their difference 43.3939 * in * Cycloid. DEFINITION. A curve generated by revolution of a circle on a plane. To Compute Area of* a Cycloid. Fig. -4:3. i - 43- __ * _^ /X ,--< A RULE. Multiply area of generating circle by 3. EXAMPLE. Generating circle of a cycloid, a 6 c, Fig. 43, oid ? "A / has an area of 115.45 sq. Inches; what is area of cycloid "'c 1 1 5. 45 X 3 = 346. 35 sgware in*. To Compute Length of a Cyoloidal Curve. RULE. Multiply diameter of generating circle by 4. EXAMPLE. Diameter of generating circle of a cycloid, Fig. 43, is 8 inches; what is length of curve d s c ? 8 X 4 = 32 = product of diameter and 4 = ins. NOTE. The curve of a cycloid is line of swiftest descent; that is, a body will fall through arc of this curve, from one point to another, in less time than through any other path. MENSURATION OF AREAS, LINES, AND SURFACES. 353 Circular Rings. DEFINITION. Space between two concentric circles. To Compute Sectional Area of a Circular Ring. Fig. 44. RULE. From area of greater circle subtract that of less. Cylindrical Rings. DEFINITION. A ring formed by curvature of a cylinder. To Compute Surface of a Cylindrical Ring. Fig. 4.4.. RULE. To diameter of body of the ring add inner diameter of the ring ; multiply this sum by diameter of the body, and product by 9.8696. Fig. 44. Or, c X I = surface. EXAMPLE. Diameter of body of a cylindrical ring, a 6, Fig. 44, is 2 inches, and inner diameter, b c, is 18; what is surface of it? 2 -j- 1 8 = 20 = thickness of ring added to inner diameter. 20 x 2 X 9 8696 = sum above obtained X thickness of ring, and that product by 9. 8696 = 394. 784 ins. Link. DEFINITION. An elongated ring. To Compute Surface of a H.inlz. Figs. 45 and 46. RULE. Multiply length of axis of link Tby circumference of a section of body, a b. Or, I X c = surface. To Compute Length of Axis and Circumference. When Ring is Elongated. RULE. To less diameter add the diameter of the body of the link, and multiply sum by 3.1416; subtract less diameter from greater, multiply remainder by 2, and sum of these products is length Fig. 45. of axis. Fig. 4 6. a EXAMPLE. Link of a chain, Fig. 45, is i inch in diameter a of body, a 6, and its inner diameters, b c and ef y are 12.5 and 2.5 inches; what is its circumference? 2. 5 -}- i X 3. 1416 = 10.9956 = length of axis of ends. 1 2. 5 2. 5 X 2 = 20 = length of sides of body. Then 10.9956 -(- 20 = 30.9956 = length of axis of link, and 30.9956 X 3- 1416 (cir. of i inch) = 97. 3758 square ins. When Ring is Elliptical, Fig. 46. RULE. Square diameters of axes of ring, multiply square root of half their sum by 3.1416, and product is length of axis. Cones. DEFINITION. A figure described by revolution of a right-angled triangle about one of its legs. For Sections of a Cone, see Conic Sections, page 379. To Compute Surface of a Cone. Fig. 47'. RULE. Multiply perimeter or circumference of base by slant height, or side of cone ; divide product by 2, and add the quotient to area of the base. Fig. 47. c Or, c X & -r- 2 -}- a' = surface, c representing perimeter. EXAMPLE. Diameter, a &, Fig. 47, of base of a cone is 3 feet, and slant height, a c, 15; what is surface of cone? Circum. of 3 feet = 9. 4248, and ^^ =7o.686 = swr- face of side; area of base 3=7.068, and 70.686+7.068 = 77.754 square feet. GG* 354 MENSURATION OF AREAS, LINES, AND SURFACES. To Compute Surface of the Frustum of a Cone. ITig. 48. RULE. Multiply sum of perimeters of two ends by slant height of frus- tum ; divide product by 2, and add it to areas of two ends. Or, ^t-^ \-a + a' = surface, EXAMPLE. Frustum, abed, Fig. 48, has a slant height, c d, of 26 inches, and . s circumferences of its ends are 15.71 and 22 inches respectively; ^ A what is its surface? JLrf + ( j 2 l6 ) 2 X .7854 = 58.119 = areas of ends. Then 490.23 -+ 58. 119 = 548.349 square ins. Pyramids. DEFINITION. A figure, base of which has three or more sides, and sides of whicli are plane triangles. To Compute Surface of a Pyramid.. Figs. 49 and. aO. RULE. Multiply perimeter of base by slant height ; divide product by 2, and add it to area of base. Fig. 49. c EXAMPLE. Side of a quadrangular pyramid, a b, Fig. 49, is 12 inches, and its slant height, o c, 40; what is its surface? 12X4 48= perimeter of base. = 960 = ] b area of sides, and 1 2 X 'i 2 -j- 960 = 1 104 square ins. To Compute Surface of Frustum of a Pyramid. Fig. SI. RULE. Multiply sum of perimeters of two ends by slant height ; divide product by 2, and add it to areas of ends. F i g . 5I . Or, C -^-*lL + a + a - = sur face. %- f EXAMPLE. Sides ab,cd, Fig. 51, of frustum of a quadrangular pyramid are 10 and 9 inches, and its slant neight is 20 ; what is its surface? 10 x 4 = 40, and 9 X 4 = 36 ; 40 + 36 = 76 = sum of perimeters. , 76 x 20:= 1520, and = 760 = area of sides; 10 X 10= 100, and 9X9 = 81. Then 100 -J- 81 -f 760 = 941 = square ins. When Pyramid is Irregular sided or Oblique. RULE. The surfaces of each of the sides and ends must be computed and added together. Helix (Screw). DEFINITION. A line generated by progressive rotation of a point around an axis and equidistant from its centre. To Compute Length, of a Helix. Fig. 52. RULE. To square of circumference described by generating point, add square of distance advanced in one revolution, extract square root of their sum, and multiply it by number of revolutions of generating point. MENSURATION OF AEEAS, LINES, AND SURFACES. 355 Fig. 52. Or, -\/(P 2 -\-l 2 )n = length, n representing number of revolutions. EXAMPLE. What is length of a helical line, Fig. 52, running 3.5 times around a cylinder of 22 inches in circumference, and advancing 16 inches in each revolution ? 22 2 -J- 16 2 = 740 = sum of squares of circumference and of distance advanced. * Then V74 X 3- 5 = 95- 21 ins. To Compete Length, of a Revolution of Thread of a Screw. RULE. Proceed as above for length and omit number of revolutions. Spirals. DEFINITION. Lines generated by the progressive rotation of a point around a fixed axis. A Plane Spiral is when the point rotates around a central point. A Conical Spiral is when the point rotates around an axis at a progressing dis- tance from its centre, as around a cone. To Compnte Length of a Plane Spiral Line. Fig. 543. RULE. Add together greater and less diameters ; divide their sum by 2 ; multiply quotient by 3.1416, and again by number of revolutions. Or, when circumferences are given, take their mean length, and multiply it by number of revolutions. Or, d -\- d' -f- 2 K 3- 1416 n = length of line; Pxn = radius, and * } & 53- p r 2 _i- 1 =pitch. P representing the pitch. EXAMPLE. Less and greater diameters of a plane spiral spring, as a 6, c d, Fig. 53, are 2 and 20 inches, and number of revolutions \d 10} what is length of it? sT^flzo -T- 2 = ii = sum of diameters -r- 2 ; u X 3.1416 = Then 34.5576 X 10= 345.576 inches. NOTE. Above rule is applicable to winding engines, see page 862, where it is re- quired to ascertain length of a rope, its thickness, number of revolutions, diameter of drum, etc. To Comp-ute Length of a Conical Spiral Line. Fig. S4r. RULE. Add together greater and less diameters; divide their sum by 2, and multiply quotient by 3.1416. To square of product of this circumference and number of revolutions of spiral, add square of height of its axis, and take square root of the sum. Or, V(d -f d' -4- 2 X 3. 1416 n -f h 2 ) = length of line. EXAMPLE. Greater and less diameters of a conical spiral, Fig. 54, are 20 and 2 inches; its height, cd, 10; and number of revolutions 10; what is length of it? 20 -f 2 -f- 2 = ii x 3.1416 = 34.5576 = sum of diameters -r- 2, and X 1.1416; 34.5576 X 10 = 345.576. I l,i Then V345-576 2 -f- io 2 = 345. 7 2 inches. Spindles. DEFINITION. Figures generated by revolution of a plane area, when the curve is revolved about a chord perpendicular to its axis, or about its double ordinate, and they are designated by the name of the arc or curve from which they are generated, as Circular, Elliptic, Parabolic, etc. * When the spiral is other than a line, measure diameters of it from middle of body composing it. 356 MENSURATION OP AREAS, LINES, AND SURFACES. Circular Spindle. To Compute Con-vex Srarface of a Circular Spindle, Zone } or Segment of it. figs. (5S, 56, and. ST. RULE. Multiply length by radius of revolving arc ; multiply this arc by central distance, or distance between centre of spindle and centre of revolv- ing arc ; subtract this product from former, double remain- der, and multiply it by 3.1416. I / c \ 2 -(a.^J r 2 ( J ) 2 p = surf ace, a representing length I c the spindle chord. Or, Ir of arc, andcthespi, EXAMPLE. What is surface of a circular spindle, Fig. 55, length of it,/c, being 14.142 inches, radius of its arc, o c, 10, and central distance, oe, 7.071? 14. 142 X 10 = 141.42 = length x radius. Length of arc, fa c, by Rules, page 344 = 15.708. 15-708 X 7-071 = 111.0713 = length of arc X central distance ; 141.42 111.0713 of products. Then 30. 3487 X 2 X 3- 1416 = 190.687 square ins. Zone. EXAMPLE. What is convex surface of zone of a circular spindle, Fig. 56, length of it, i c, being 7.653 inches, radius of its arc, o g, 10, central distance, o e, 7.071, and length of its side or arc, d b, 7.854 inches? 7. 653 X 10= 76. $3=length X radius ; 7. 854 X 7-071 = 55. 5356 = length of 'arc x central distance ; 76. 53 55. 5356 = 20.9944 = difference of products. Then 20.9944 X 2 X 3. 1416 = 131.912 square ins. d lg ' S7 ' Segment. EXAMPLE. What is convex surface of a segment of a cir- cular spindle, Fig. 57, length of it, ic, being 3.2495 inches, radius of its arc, og, 10, central distance, o e, 7.071, and length of its side, id, 3.927 inches? 3.2495 X 10 = 32.495 == length X radius; 3.927 X 7- ? 1 = 27.7678 = length of arc X central distance ; 32. 495 27. 7678 = 4. 7272 = difference of products. Then 4.7272 X 2 X 3.1416 = 29.702 square ins. GENERAL FORMULA. 8 = 2 (lr ac)p = surface, I representing length of spindle, segment, or zone, a length of its revolving arc, r radius of generating circle, and c central distance. ILLUSTRATION. Length of a circular spindle is 14.142 inches, length of its revolv- ing arc is 15.708, radius of its generating circle is 10, and distance of its centre from centre of the circle from which it is generated is 7.071 ; what is its surface? 2 X (14.142 X 10 15.708X7-071) X 3.1416= 190.687 square inches. NOTE. Surface of a frustum of a spindle may be obtained by division of the surface of a zone. Cycloidal Spindle. To Compnte Convex Snrflaee of a Cycloidal Spindle. Fie. 58. RULE. Multiply area of generating circle by 64, and divide it by 3. Fig. 58.^ ' 3 ' c EXAMPLE. Area of generating circle, a 6 c, of a cycloidal spindle, d e, is 32 inches; what is surface of spindle? 32 X 64 = 2048 = area of circle x 64 , and 2048 -4- 3 = 682. 667 square ins. NOTE. Area of greatest or centre section of a cycloidal spindle is twice area of the cycloid. MENSURATION OF AREAS, LINES, AND SURFACES. 357 Ellipsoid., Paraboloid., or Hyper"boloid of Rev- olution. DEFINITION. Figures alike to a cone, generated by revolution of a conic section around its axis. NOTE. These figures are usually known as Conoids. When they are generated by revolution of an ellipse, they are termed Ellipsoids, and when by a parabola, Paraboloids, etc. Revolution of an arc of a conic section around the axis of the curve will give a segment of a conoid. Ellipsoid. To Compute Convex Surface of an. Ellipsoid. Fig. 59. RULE. Add together square of base and four times square of height ; multiply square root of half their sum by 3.1416, and this product by radius of the base. Or ' 3-4i6r ^surface. EXAMPLE. Base, a 6, of an ellipsoid, Fig. 59, is 10 inches, and vertical height, c d, 7; what is its surface? io 2 -f- 7 2 X 4 = 296 = sum of square of base and 4 times square of height; 296 -4- 2 = 148, and \/ I 4^ I2 - J ^55 = square root of half above sum. Then 12.1655 X 3.1416 X = 191.0957 square ins. To Compute Convex Surfa.ce of a Segment, Frustum, or Zone of an Ellipsoid.. Fig. 59. See Rules for Convex Surface of a Segment, Frustum, or Zone of a Spheroid or Ellipsoid, pages 348-9. d or d' X 3- 1416 X h = surface, and = h ; then d X 3. 4 6 X = surf**. 3?ara"boloid. To Compute Convex Surface of a IParalDoloid.. Fig. 6O. RULE. From cube of square root of sum of four times square of height, and square of radius of base, subtract cube of radius of base ; multiply re- mainder by quotient of 3.1416 times radius of base divided by six times square of height. Flg - fe 1 _ E. = surface, Or, 3_ r 3 x EXAMPLE. Axis, 6 d, of a paraboloid, Fig. 60, is 40 inches; ra- dius, a d, of its base is 18 inches; what is its convex surface? 4o 2 X 4 = 6400 = 4 times square of height ; 6400 + i8 2 = 6724 sum of above product and square of radius of base; (-^6724)3 18 3 545 536 = remainder of cube of radius of base subtracted from cube of square root of preceding sum ; 3. 1416 X i8-f- (6 X 4o 2 ) = .005 8905 =. quotient 0/3.1416 times radius of base -4-6 times square of height. Then 545 536 X .005 890 5 = 3213.48 square ins. Fig. 61. t a Cylinder Sections. To Compute Surface of a Cylinder Section. -Fig. 61. RULE. From entire surface of cylinder a o subtract surface of the two ungulas, ? o, o c, as per rule, page 35 1> and multiply result by 4. 358 MENSURATION OF AREAS, LINES, AND SURFACES. Figure of Revolution.. To Ascertain Convex Surface of any Figure of Revolu- tion. Figs. 6J, G3 9 and 64. RULB. Multiply length of generating line by circumference described by its centre of gravity. Or, I 2 r p = surface, r representing radius of centre of gravity. EXAMPLE i. If generating line, a c, of cylinder, a cdf, 10 inches * '- . in diameter, Fig. 62, is 10, then centre of gravity of it will be in b, radius of which is 6 r = 5. \ / i J-V- n V Hence 10 X 5 X 2 X 3- 1416 = 314. 16 ins. V'\ 1 Again, if generating line is e a c g, and it is (e a = 5, a c 10, ' ^ L and c g = s) = 2o, then centre of gravity, o, will be in middle of A. -j line j oming centres of gravity of triangles e a c and a c y = 3. 75 from r. Hence 20 X 3.75 X 2 X 3.1416 = 471.24 square ins.=entire surface. w J Convex surface as above 314. 16 N. { Area of each end, io*x .7854 X 2 = .157.08 471.24 inches. Fig. 63. 2. If generating elements of a cone, Fig. 63, are a d 10, d c = 10, and ac, generating line,= 14.142, centre of gravity of which is in o, and o r = 5, Then 14.142 XsX2X 3. 1416 =444. 285, con- vex surface, and 10 x 2 X .7854 = 314.16, area of base. , Hence 444.285 -\- 314. 16 758.445, entire surface. 3. If generating elements of a sphere, Fig. 64, are a c = so, a & c will be 15.708, centre of gravity of which is in o, and by Rule, page 606, o r = 3. 183. Hence 15.708 X 3-183 X 2 X 3.1416 = 314.16 square ins. Capillary TulDe. To Compute Diameter of a Capillary Tube. RULE. Weigh tube when empty, and again when filled with mercury ; subtract one weight from the other ; reduce difference to grains, and divide it by length of tube in inches. Extract square root of this quotient, multi- ply it by .019 224 5, and product will give diameter of tube in inches. Or, I x .0192245 = diameter, w representing difference in weights in grqins and I length of tube. EXAMPLE. Difference in weights of a capillary tube when empty and when filled with mercury is 90 grains, and length of tube is 10 inches; what is diameter of it? 90 -r- 10 9 = weight of mercury -=- length of tube ; -\/Q = 3, and 3 x .019 224 5 = .057 673 5 square root of above quotient x .019 224 5 inches = diameter of tube. PROOF. Weight of a cube inch of mercury is 3442.75 grains, and diameter of a circular inch of equal area to a square inch is 1.128 (page 342). If, then, 3442.75 grains occupy i cube inch, 90 grains will require .026 141 9 cube inch, which, -4- 10 for height of tube .002 614 19 inch for area of section of tube. Then ^-002 61419 = . 05 11 29 = side of square of a column of mercury of this area. Hence .051 129 x 1.128 (which is ratio between side of a square and diameter of a circle of equal area) = .057 673 5 ins. To Ascertain A.rea of an Irregular Figure. RULE. Take a uniform piece of board or pasteboard, weigh it, cut out figure of which area is required, and weigh it ; then, as weight of board or pasteboard is to entire surface, so is weight of figure as cut out to its surface, Or, see rule page 341, or Simpson's rule, page 342. MENSURATION OF AREAS, LINES, SURFACES, ETC. 359 To Ascertain Area of any Plane Figure, RULE. Divide surfaces into squares, triangles, prisms, etc. ; ascertain their areas and add them together. Reduction of an -Ascending or Descending Line to Hor- izontal ^Measurement. In Link and Foot. Degrees. Link. Foot. Degrees. Link. Foot. Degrees. I Link. Foot. .000099 .000403 .000904 .001 61 .002515 .003617 .00015 .00061 00137 .00244 .00381 .00548 9 10 ii 12 .004917 .006421 .008 125 .OIOO25 .012 124 .014421 .00745 .00973 .01231 .015 19 .01837 .02285 13 3 016915 019602 022 486 025569 O28 925 0323 02563 0297 03407 03874 0437 04894 O .003017 .00540 12 .014421 .O2205 10 0323 040 ILLUSTRATION i. In an ascending grade of 14, what is reduction in 500 feet? 14 = 500 X .0297 = 14.85 feet = 14 feet 10.2 ins. 2. What is reduction in 500 links? 14 = 500 X -019 602 = 9. 801 feet = 9 feet 9.6 ins. Reduction of Oracle of an Ascending or Descending Line to Degrees. Per 100 Links, Feet, etc. Grade. Degrees. Grade. Degrees. Grade. Degrees. Grade. Degrees. 25 / // 8 35-2 1-75 I o 10.3 4-5 2 34 45-5 10 5 44 20.7 5 17 10.3 2 i 8 45.5 5 2 5i 57-6 ii 6 18 55.8 75 25 47-6 2.5 i 25 57.6 6 3 26 22.7 12 6 53 3i x-25 34 22.7 42 57-9 3 3-5 i 43 8.3 2 O 2O.7 7 8 4 o 49.6 4 35 18-6 13 14 7 28 10.3 8 2 51-7 i-5 5i 35-2 4 2 17 33-i 9 5 9 49- 6 15 8 37 37- To Plot Angles -without a Protractor. On a given line prick off 100 with any convenient scale, and from the point so pricked off lay off at right angle with the same scale the natural tangent due to the angle (see table of Natural Tangents and Sines) ; or strike out a portion of a circle with radius 100 and lay off a chord = 2 sin. of half the angle required. To Compute Chord, of an Angle. Double sine of half angle. ILLUSTRATION. What is the chord of 21 30'? Sine of 3 = 10 45', and sine of 10 45' = . 186 52, which, X 2 = .373 04 chord. To Ascertain Value of a IPower of a Qu.an.tity. RULE. Multiply logarithm of quantity by fractional exponent, and prod- uct is logarithm of required number. EXAMPLE. What is the value of 16% ? X X log. 16 = & x x. 204 la = . 903 09. Number for which = & 360 MENSURATION OF VOLUMES. MENSURATION OF VOLUMES. Cubes and IParallelopipedons. Cube. DEFINITION. A volume contained by six equal square sides. Fig. i. h To Compute Volume of a Cube. T^ig. 1. RULE. Multiply a side of cube by itself, and that product again by a side. Or, 3 V, s representing length of a side, and V volume. EXAMPLE. Side, a b, Fig. i, is 12 inches; what is volume of it? 12 X 12 X 12 1728 cube ins. IParallelopipedon. DEFINITION. A volume contained by six quadrilateral sides, every opposite two of which are equal and parallel. To Compute "Volume of a IParallelopipedon. Fig. 2. RULE. Multiply length by breadth, and that product again by depth. Or, lbd = V. IPrisms, IPrismoids, and "Wedges. Prisms. DEFINITION. Volumes, ends of which are equal, similar, and parallel planes, and sides of which are parallelograms. NOTE. When ends of a prism or prismoid are triangles, it is termed a triangular prism or prismoid; when rhomboids, a rhomboidal prism, and when squares, a square prism, etc. Fig 3 Fig 5- To Compute Volume of a Prism. Figs. 3 and. 4. RULE. Multiply area oPbase by height. Or, a h = V. EXAMPLE. A triangular prism, a b c, Fig. 4, has sides of 2. 5 feet, and a length, c b, of 10; what is its volume ? By Rule, page 339 , 2.52 x .433 = 2.70625 = area of end a 6 v and 2.706 25 X 10 = 27.0625 cube feet. When a Prism is Oblique or Irregular. RULE. Multiply area of an end by height, as a o ; or, multiply area taken at a right angle to sides, as at c, by actual length. To Compute Volume of any ITrustxxm of a IPrism, whether Right or Oblique. Figs. Fig. 6. Fig. 7 . RULE. Multiply area of base by perpendicular distances between it and centre of gravity of upper or other end. EXAMPLE. Area of base, a o, of frustum of a rectan- gular or cylindrical prism, Fig. 6, is 15 inches, and height to centre of gravity, c, is 12 ; what is its vol- ume? 10 X 12 = 120 cube ins. MENSURATION OF VOLUMES. IPrismoids. * To Compute "Volume of* a Irismoid. Fig. 8. RULE. To sum of areas of the two ends add four times area of middle section, parallel to them, and multiply this sum by one sixth of perpendicu- lar height. NOTE. This is the general rule, and known as the Prismoidal Formula, and it applies equally to all figures of proportionate or dissimilar ends. Fig. 8. Or, a + a' -f 4 ra x h -r- 6 = V, a and a' representing areas of ends, and m area of middle section. EXAMPLE. What is volume of a rectangular prismoid, Fig. 8, lengths and breadths, e g and g h, a b and b d, of two ends being 7X6 and 3X2 inches, and height 15 feet ? 7X6+3X2 = 42 -f6 = 48 = mm of areas of two ends ; 7 -f 3 -r- 2 = 5 = length of middle section ; 6 + 2-^2 = 4 = breadth of middle section ; 5X4X4 = 80 =four times area of middle section. Then 48 + 80 X 15 X 12 = 128 X 30 = 3840 cube ins. NOTE i. Length and breadth of middle section are respectively equal to half sum of lengths and breadths of the two ends. 2. Prismoids, alike to prisms, derive their designation from figure of their ends, as triangular, square, rectangular, pentagonal, etc. When it is Irregular or Oblique and their ends are united by plane or curved surfaces, through which and every point of them, a right line may be drawn from one of the ends or parallel faces to the other. Figs. 9, io,and n. Fig. 10. Fig. ii. Fig. 9. EXAMPLE. Areas of ends, a c and o r s, Fig. 10, a b c d, and i m n w, Fig. n, and abce and v x w z, Fig. 9, are each 10 and 30 inches, that of their middle section 20, and their perpendicular heights 18; what is their volume? 10 + 30+ 20 x 4 = 120 = sum of areas of ends -\- 4 times middle section. And 120 X -T- = 360 cube ins. 6 TVedge. To Compnte "Volvitne of a "Wedge. Fig. 12. RULE. To length of edge add twice length of back ; multiply this sum by perpendicular height, and then by breadth of back, and take one sixth of product. Fig. 12. Or, (I -f- V X 2 X h b) -i- 6 = V. EXAMPLE. Length of edge of a wedge, eg, is 20 inches, back, abed, is 20 by 2, and its height, e/, 20; what is its volume? 20 + 20 X 2 = 60 length of edge added to twice length of back ; 60 X 20 X 2 = 2400 above sum multiplied by height, and that product by breadth of back. Then 2400 -r- 6 = 400 cube ins. NOTE. When a wedge is a true prism, as represented by 2, volume of it is equal to area of an end multiplied by its length. * An excavation or embankment of a road, when terminated by parallel cross sections, is a rectan- gular prismoid. H H 362 MENSURATION OF VOLUMES. To Compute Frustum of a. "Wedge. Fig. 13. RULE. To sum of areas of both ends, add 4 times area of section parallel to and equally distant from both ends, and multiply sum by one sixth of length. Or,A-J-a-f- 4 a / X T - = V. o EXAMPLE. Lengths of edge and back of a frustum of a wedge a b and c d are 20 X i and 20 X 2 ins. , and height o r is 20 ins. ; what is its volume ? X 2 + 4 X (20 X fe^ X ^ = 60 + 120 X = 600 cube ins. 2 \ 2 / 6 6 NOTE. When frustum is a true prism, as represented Fig. 13, volume of it is equal to mean area of ends multiplied by its length. Regular Bodies (3Polyh.ed.rons). DEFINITION. A regular body is a solid contained under a certain number of simi- lar and equal plane faces,* all of which are equal regular polygons. NOTE i. Whole number of regular bodies which can possibly be formed is five. 2. A sphere may always be inscribed within, and may always be circumscribed about a regular body or polyhedron, which will have a common centre. Fig. 14. Fig. 16. Fig. 17- 1. Tetrahedron, or Pyramid, Fig. 14, which has four triangular faces. 2. Hexahedron, or Cube, Fig. i, which has six square faces. 3. Octahedron, Fig. 15, which has eight triangular faces. 4. Dodecahedron, Fig. 16, which has twelve pentagonal faces. 5. Icosahedron, Fig. 17, which has twenty triangular faces. Xo Compute Klements of any Regular Body. Figs. 14:, 15, 16, and IT. To Compute Radius of a Sphere that will Circumscribe a given Regular Body, or that may be Inscribed within it. When Linear Edge is given. RULE. Multiply it by multiplier opposite to body in columns A and B in following Table, under head of element re- quired. EXAMPLE. Linear edge of a hexahedron or cube, Fig. i, is 2 inches; what are radii of circumscribing'and inscribed spheres? 2 x . 866 02 = i . 732 04 inches = radius of circumscribing sphere ; 2 X . 5 = i inch = radius of inscribed sphere. When Surface is given. RULE. Multiply square root of it by multiplier opposite to body in columns C and D in following Table, under head of element required. When Volume is given. RULE. Multiply cube root of it by multiplier opposite to body in columns E and F in following Table, under head of ele- ment required. * Angle of adjacent facei of a polygon is termed diedral angle. MENSURATION OF VOLUMES. 363 When one of the Radii of Circumscribing or Inscribed Sphere alone is re-< quired, the other being given. RULE. Multiply given radius by multiplier opposite to body in columns G and H in Table, page 364, under head ef other radius. To Compute Linear Edge. When Radius of Circumscribing or Inscribed Sphere is given. RULE. Multiply radius given by multiplier opposite to body in columns I and K in Table, page 364. When Surface is given. RULE. Multiply square root of it by multiplier opposite to body in column L in Table, page 364. When Volume is given. RULE. Multiply cube root of it by multiplier opposite to body hi column M in Table, page 364. To Compute Surface. When Radius of Circumscribing Sphere is given. RULE. Multiply square of radius by multiplier opposite to body hi column N hi Table, page 364. When Radius of Inscribed Sphere is given. RULE. Multiply square of radius by multiplier opposite to body in column O in Table, page 364. When Linear Edge is given. RULE. Multiply square of edge by multi- plier opposite to body hi column P in Table, page 364. When Volume is given. RULE. Extract cube root of volume, and multi- ply square of root by multiplier opposite to body in column Q hi Table, page 364. To Compute "Volume. When Linear Edge is given. RULE. Cube linear edge, and multiply it by multiplier opposite to body in column R in Table, page 364. When Radius of Circumscribing Sphere is given. RULE. Multiply cube of radius given by multiplier opposite to body hi column S hi Table, page 364. When Radius of Inscribed Sphere is given. RULE. Multiply cube of radius given by multiplier opposite to body in column T in Table, page 364. When Surface is given. RULE. Cube surface given, extract square root, and multiply the root by multiplier opposite to body hi column U in Table, page 364. Fig. 18. Cylinder. To Compute Volume of a Solid Cylinder. Fig. 18. RULE. Multiply area of base by height. EXAMPLE. Diameter of a cylinder, b c, is 3 feet, and its length, a 6, 7 feet; what is its volume? Area of 3 feet = 7.068. Then 7.068 X 7 = 49.476 cube feet. To Compute "Volume of* a Hollow Cylinder. RULE. Subtract volume of internal cylinder from that of cylinder. Fig. ig. a Cone. To Compute "Volume of a Cone. Fig. 19. RULE. Multiply area of base by perpendicular height, and take one third of product. EXAMPLE. Diameter, a &, of base of a cone is 15 inches, and height, c e, 32. 5 inches ; what is its volume f Are* of 15 inches = 176.7146. Then 176 ' 7I5X32 ' 5 = 1914.4125 cube int. 3 6 4 MENSURATION OF VOLUMES. gjgqdg pgquos [ JTJ9UI'! ^ Saiquosumo ON t^ W vO -t N ^f ^t 1 fO H CO O M M IO NO M rJ-C^. O M vg vO CO 1-1 tT) n pgquosai Xg gjgqdg 9 aiquos -uinoaiojos'nipT?H ro fO >O O Sntquosiuno gjgqdg SatquosranoaiQ ^fg gagqdg pgqiaosaj jo snip^a f*3 fO f^vO vO f) tOCO P. P3 *9J9qdg Sui vO N vO o ON ON O O C^ W 00 vQ CO O O Tj"O M CO W CO ON C> t> "93pg M IO t^ M H H CO O 00 O Tj- ON OvO paquosui jo s Q -nj jo sntp^i ^fg goring Xg H Q '9J9qdg guiquos Sniquosuino jTJoniT Xg pgquosaijosmpBy -gjgqdg Saiquos -uinoaio jo snip^a t*" N H VO VO COO H N O N \O t^ M M M VQ O O IO \O 00 t^ rh ON hedron. . hedron . edron . . ahedron Te H Oc D Ic MENSURATION OF VOLUMES. Fig. 20. To Compute "Volume of Frustum of a Cone. Fig. 2O. RULE. Add together squares of the diameters or circumferences of greater and lesser ends and product of the two diameters or circumferences ; mul- tiply their sum respectively by .7854 or .07958, and this product by height; then divide this last product by 3. Or, d 2 + a"' 2 -f dxtf X . 7854 h -r- 3 = V. Or, c 2 + c /2 + c X c' X -079 58 h-r- 3 = V. EXAMPLE. What is volume of frustum of a cone, diameters of greater and lesser ends, bd,ac, being 5 and 3 feet, and height, eo, 9? 5 2 -f- 3 2 H-5X3 = 49 ; and 49 X .7854 = 38-4846 = above sum by .7854; and 38 ' 4 ^ 6 X 9 = 115.4538 cube feet. 3Pyrain.icU NOTE. Volume of a pyramid is equal to one third of that of a prism having equal bases and altitude. Fig. si. Fig. 22. To Compute "Volume of a Pyramid.- ITig. SI. RULE. Multiply area of base by perpendicular height, and take one third of product. EXAMPLE. What is the volume of a hexagonal pyramid, Fig. 21, a side, a b, being 40 feet, and its height, e c, 60? 40* X 2.5981 (tabular multiplier, page 339) =4156.96 = area of base. 4156 9 6 X *> = 83 x 39 . 2 ate fed. To Compute Volume of Frustum of a Pyramid. Fig. 22. RULE. Add together squares of sides of greater and lesser ends, and product of these two sides ; multiply sum by tabular multiplier for areas in Table, page 339 and this product by height ; then divide last product by 3. Or, * 2 + s' 2 -fsxs' X tab. mult, x h -r- 3 = V. When A reas of Ends are known, or can be obtained without reference to a tabular multiplier, use following. Or, a + a'-f- Vax a'X A-r-s = V. EXAMPLE. What is the volume of the frustum of a hexagonal pyramid, Fig. 22, the lengths of the sides of the greater and lesser ends, ab,cd, being respectively 3.75 and 2.5 feet, and its perpen- dicular height, e o, 7.5? 3.75 2 -r-2.5 2 = 2o.3i25=$Mw of squares of sides of greater and lesser ends; 20.3125+ 3.75 x 2. 5 = 29. 6875 = above sum added to product of the two sides ; 29.6875 X 2.5981 X 7-5 578.48 x tab. mult., and again by the height, which, -7-3192.83 cube feet. When Ends of a Pyramid are not those of a Regular Polygon, or when Areas of Ends are given RULE. Add together areas of the two ends and square root of their prod- uct ; multiply sum by height, and take one third of product. Or, a -\- a' + Vaa' X h -r- 3 = V. EXAMPLE. What is the volume of an irregular sided frustum of a pyramid, the areas of the two ends being 22 and 88 inches, and the length 20? 22-j-88=no = 5wm of areas of ends; 22 X 88 = 1936, and ^1936 = 44 = square root of product of areas. Then 3 HH* = 1026.66 cube int. 366 MENSURATION OF VOLUMES. Spherical 3?yramid. A Spherical Pyramid is that part of a sphere included within three or more ad- joining plane surfaces meeting at centre of sphere. The spherical polygon defined by these plane surfaces of pyramid is termed the base, and the lateral faces are sectors of circles. NOTE. To compute the Elements of Spherical Pyramids, see Docharty and Hack- ley's Geometry. Cylindrical TJngnlas. DEFINITION. Cylindrical Ungulas are frusta of cylinders. Conical Ungulas arc frusta of cones. To Coiripxite "Volume of a Cylindrical TJngula. Fig. 23 C i. When Section is parallel to Axis of Cylinder. RULE. Multiply area Fig. 23. of base by height of the cylinder. Or, a h = V. EXAMPLE. Area of base, d ef, Fig. 23, of a cylindrical ungula is 15.5 inches, and its height, a e, 20; what is its volume? 15. 5 X 20 = 310 cube ins. 2. When Section passes Obliquely through opposite sides of Cylinder, Fig. 24. RULE. Multiply area of base of cylinder by half sum of greatest and least lengths of ungula. Fig. 24. Or, ax J-f r-5-2 = V. EXAMPLE. Area of base, c d, of a cylindrical ungula, Fig. 24, is 25 iV^^'ft inches, and the greater and less heights of it, ac,bd, are 15 and 17: what is its volume? 25 X Ig = 400 cube ins. 3. When Section passes through Base of Cylinder and one of its Sides, and Versed Sine does not exceed Sine, or the Base is equal to or less than a Semicircle, Fig. 25. RULE. From two thirds of cube of sine of half arc of base subtract product of area of base and cosine * of half arc. Multiply difference thus found by quotient arising from height divided by versed sine. Fiff. 25. ^2 sin.3 h Or, a c X : = v , v. sin. representing versed sine. 3 v. sin. EXAMPLE. Sine, a d, of half arc, d ef, of base of an ungula, Fig. 25, is 5 inches, diameter of cylinder is 10, and height, e g, of ungula 10; what is its volume ? Two thirds of 53 83.333 = ^0 thirds of cube of sine. As versed sine and radius of base are equal, cosiae is o. Henoe, area or base x cosine = o, and 83. 333 o x 10 -r- 5 = 166.666 cube ins. 4. When Section passes through Base of Cylinder, and Versed Sine exceeds Radius, r when the Base exceeds a Semicircle, Fig. 26. RULE. To two thirds of cube of sine of half arc of base add product of area of base and cosine. Multiply sum thus found by quotient arising from height, divided by versed sine. 2_wn.3 _A__ V 3 v. sin. EXAMPLE. Sine, a d, of half arc of an ungula, Fig. 26, is 12 inches, versed sine, a g, is 16, height, g c, 10, and diameter of cylinder 25 ; c what is its volume? Two thirds of i23 = n 52 two thirds of cube of sine of half arc of base. Area of base =331. 78; 1152 + 331.78 x 16 12.5 = 2313.23 = sum of two thirds of cube of sine of half the arc of base, and product of area of base and cosine. Then 2313. 23 x 20-^-16 = 2891. 5375 cube ins. * Whn ih ouu is o t the product is o. MENSURATION OF VOLUMES. 367 5. When Section passes Obliquely through both Ends of Cylinder, Fig. 27. RULE. Conceive section to be continued till it meets side of cylinder produced ; then, as the difference of versed sines of the arcs of the two ends of ungula is to the versed sine of arc of less end, so is the height of cylinder to the part of side produced. Ascertain volume of each of the ungulas by Rules 3 and 4, and take their difference. Or, '- : ; = h', v. sin. and v. sin. ' representing versed sines v. sin. v. sin. of arcs of the two ends, h height of cylinder, and h' height of part pro- duced. EXAMPLE. Versed sines, ae,do, and sines, e and o, of arcs of two ends of an ungula, Fig. 27, are assumed to be respectively 8.5 and 25, and 11.5 and o inches, length of ungula, bo, within cylinder, cut from one having 25 inches diameter, d o, is 20 inches; what is height of un- gula produced beyond cylinder, and what is volume of it? 25 <-o 8.5 : 8.5 :: 20 : 10.303 = height of ungula produced beyond cyl- inder. Greater ungula, sine o being o, versed sine = the diameter. Base of ungula being a circle of 25 inches diameter, area = 490.875. Versed sine and diameter of base being equal (25), sine = o. 490.875 X 25 = 6135.9375 =product of area of base and cosine, or excess of versed sine over sine of base. 30. 303 -4-25 = 1.21212 = quo- tient of height -r- versed sine. Then 6135.9375 x 1.212 12 = 7437.4926 cube inches; and by Rules 3 and 4, volumes of less and greater ungulas = 515.444, and 6922.0486 = 7437.4926 cube inches. Sphere. DEFINITION. A solid, surface of which is at a uniform distance from the centre. Fig ' 2 i--5=^ To Compute Volume of a Sphere. Fig. 28. RULE. Multiply cube of diameter by .5236. a Or,

,0 arc, o a, is 10; and length of segment, i c, is 3.535 53; what is .-;/ its volume ? _J.. *'*/ j / ^ 3. 535 53 X 2 = 7.071 07 = double remainder of **% length of segment subtracted from half length of spindle length of middle frustum. NOTK. Area of revolving or generating segment of whole spindle is 28.54 inches, and that of middle frustum is 19.25. The volume of whole spindle is 212.9628 cube ins. " " middle frustum is 162.8982 " " Hence 50.0646 H- 2 = 25.0323 cube ins. Cycloidal Spindle.* To Compute Volume of a Cycloidal Spindle. Fig. 4:3. RULE. Multiply product of square of twice diameter of generating circle and 3.927 by its circumference, and divide this product by 8. _. or, .9..4. = v\ 'c d (- ^==^-'"' f c*VcZe, or half width of spindle. EXAMPLE. Diameter of generating circle, a 6 c, of a cy- cloid, Fig. 43, is 10 inches; what is volume of spindle, d e? 2 10 X 2 X 3-927 = 1570.8 =1 product of twice diameter squared and 3.927. Then 1570.8 x iox 3.1416-7-8 = 6168.5316 cube ins. Elliptic Spindle. To Compute Volume of an Elliptic Spindle. Fig. 44. RULE. To square of its diameter add square of twice diameter at one fourth of its length ; multiply sum by length, and product by .1309-! . 2 Or, d 2 -f- 2 d' 1. 1309 = V, d and d' representing diameters as above. Fig. 44. EXAMPLE. Length of an elliptic spindle, a 6, Fig. 44, \& 75 inches, its diameter, cd, 35, and diameter, e/, at .25 of its length, 25 ; what is its volume ? 35 2 -f- 25 X 2 = 3725 = sum of squares of diameter of spindle and of twice its diameter at one fourth of its length; 3725 X 75 279 375 = above sum X length of spindle. Then 279 375 x . 1309 = 36 570. 1875 cube ins. NOTK. For all such solid bodies this rule is exact when body is formed by a conic section, or a part of it, revolving about axis of section, and will always be rery near when figure revolves about another line. To Compute "Volume of* Middle FmstxTm or Zone of an Elliptic Spindle. Fig. 45. RULE. Add together squares of greatest and least diameters, and square of double diameter in middle between the two ; multiply the sum by length, and product by .13094 2 Or, d 2 -f- d' 2 + 2 d" 1. 1309 = V, d, d', and d" representing different diameters. * Volume of a Cycloidal Spindle is equal to .625 of ite circumscribing cylinder, t See preceding Note. J See Note abore. MENSURATION OF VOLUMES. 373 Fig. 45- qa EXAMPLE. Greatest and least diameters, a b and cd, of the frustum of an elliptic spindle, Fig. 45, are 68 and 50 inches, its middle diameter, g h, 60, and its length, g/, 75; what is its volume? 68 2 -\- so 2 -|- 60 X 2 = 21 524 = sum of squares of greatest and least diameters and of double middle diameter. Then 21 524 X 75 X .1309 = 211 311.87 cube ins. To Compute "Volume of* a Segment of an. Elliptic Spin- dle. Fig. 46. RULE. Add together square of diameter of base of segment and square of double diameter in middle between base and vertex ; multiply sum by length of segment, and product by .1309.* 2 Or, d 2 + 2 d" I X . 1309 = V, d and d" representing diameters. EXAMPLE. Diameters, c d and g h, of the segment of an elliptic spindle, Fig. 46, are 20 and 12 inches, and length, *"\ % o e y is 16 ; what is its volume ? ""/ 2o 2 -f 12 x 2 = 976 = sum of squares of diameter at base ^**' and in middle. >J "* Then 976 x 16 X . 1309 = 2044. 134 cube ins. 3?ara"bolic Spindle. To Compute "Volume of a ^Parabolic Spindle. Fig. 47'. RULE i. Multiply square of diameter by length, and the product by .41888.1 RULE 2. To square of its diameter add square of twice diameter at one fourth of its length ; multiply sum by length, and product by .13094 Or, d 2 + ~zd f I X -1309 V. EXAMPLE. Diameter of a parabolic spindle, a b t Fig. 47, is 40 ins., and its length, cd, 10; what is its volume? 40 2 X 10 = 16 ooo = square of diameter X length. Then 16000 X .418 88 = 6702.08 cube ins. Again, If middle diam. at .25 of its length is 30, Then, by Rule 2, 4o 2 + 30 X 2 X 40 X . 1309 = 6806.8 cube ins. To Compute "Volume of ^Middle Frustum of a 3?ara"bolio Spindle. Fig. 48. RULE i. Add together 8 times square of greatest diameter, 3 times square of least diameter, and 4 times product of these two diameters ; mul- tiply sum by length, and product by .052 36. RULE 2. Add together squares of greatest and least diameters and square of double diameter in middle between the two ; multiply the sum by length, and product by .1309. Or, d 2 -f- d' 2 -f- 2 d" 2 I X 1309 = V, d" representing diameter between the two. pj ff o EXAMPLE. Middle frustum of a parabolic spindle, Fig. 48, has diameters, a b and ef, of 40 and 30 inches, and its Ill>y length, cd, is 10; what is its volume? 4o 2 X 8 + 30-' X 3 + 40 X 30 X 4 = 20 300 = sum of 8 times square of greatest diameter, 3 times squeire of least diameter, and 4 times product of these. Then 20 300 x 10 X -052 36 = 10629.08 cube ins. * See Note, page 372. t 8-15 of .7854. !-i5 of .; Il J See Note, page 372. 374 MENSURATION OF VOLUMES. To Compute "Volume of a Segment of a IParatoolic Spindle. Fig. 49. RULE. Add together square of diameter of base of segment and square of double diameter in middle between base and vertex ; multiply sum by height of segment, and product by .1309. EXAMPLE. Segment of a parabolic spindle, Fig. 49, has diameters, e/and g h, of 15 and 8.75 inches, and height, c d, is 2.5 ; what is its volume ? 152 -f- 8. 75 x 2 = 531.25 = sum of square of base and of double diameter in middle of segment. Then 531.25 X 2.5 X .1309 = 173.852 cube ins. Hyperbolic Spindle. To Compute Volume of* a Hypertoolie Spindle. Fig. SO. RULE. To square of diameter add square of double diameter at one fourth of its length ; multiply sum by length, and product by .1309.* Fig. 50. a Or, d 2 +l^o?l x . 1309 = V. EXAMPLE Length, aft, Fig. 50, of a hyperbolic spindle is ioo inches, and its diameters, cd and ef, are 150 and no; what is its volume? I 5 2 H~ II0 X 2 X 109 = 7090000 = product of sum of squares of greatest diameter and of twice diameter at one fourth of length of spindle and length. Then 709ooooX . 1309 = 928 08 1 cube inches. To Compute Volume of Middle Frustum of a Hyper- toolie Spindle. Fig. SI. RULE. Add together squares of greatest and least diameters and square of double diameter in middle between the two ; multiply this sum by length, and product by .1309^ Fi Or, d 2 + d' 2 -f (2 d") 2 I X . 1309 = V. EXAMPLE. Diameters, a b and c d", of middle frustum of a hyperbolic spindle, Fig. 51, are 150 and no inches; diam- eter, g h, 140; and length, ef t 50; what is its volume? i5o 2 + iio 2 -|- 140 X 2 = 113 000 = sum of squares ofgreat- eg t an d feast diameters and of double middle diameter. Then 113000 X 5 X 1309 = 739 585 cube ins. To Compute Volume of a Segment of a Hyper"bolic Spin- dle. Fig. S3. RULE. Add together square of diameter of base of segment and square of double diameter in middle between base and vertex ; multiply sum by length of segment, and product by .1309. Or, d 2 + d" 2 lX.i3og = V. EXAMPLE. Segment of a hyperbolic spindle, Fig. 52, has diameters, e/and g h, of no and 65 inches, and its length, a 6, 25; what is its volume? i io 2 -f- 65 x 2 = 29 ooo = sum of squares of diameter of base and of double middle diameter. Then 29 ooo X 25 x 1309 = 94 902.5 cube ins. * See Note, page 372. MENSURATION OF VOLUMES. 375 Ellipsoid, I?ara"boloid, and. Hypertooloid of Revo- lution* (Conoids). DEFINITION. Figures like to a cone, described by revolution of a conic section around and at a right angle to plane of their fixed axes. Ellipsoid of* Revolution (Spheroid). DEFINITION. An ellipsoid of revolution is a semi-spheroid. (See page 368.) IParaboloid of* Revolntion.t To Compute Volume of a Paraboloid. of Revolution. Fig. S3. RULE. Multiply area of base by half height Fig. 53. Or,afc-i-2 = V. NOTE. This rule will hold for any segment of paraboloid, whether base be perpendicular or oblique to axis of solid. EXAMPLE. Diameter, a 6, of base of a paraboloid of revolution, Fig. 53, is 20 inches, and its height, d c, 20; what is its volume? Area of 20 inches diameter of base := 314.16. Then 314. 16 X 20 -i- 2 3141.6 cube ins. of a I?ara"boloid of Revolution. To Compute "Volume of a Frustum of a IParatooloid. of Revolvition. Fig. 54. Fig. 54. RULE. Multiply sum of squares of diameters by height of frustum, and this product by .3927. Or,(d 2 + and '75 104 X .06566 = ii 497.329, which-:- 231 = 49.77 gallons. Generally. Dd-f-M 2 .061 692 L = U. S. gallons, and .001 416 2 = Imperial gallons. D, d, and M representing interior, head and bung diameters, and L length of cask in inches. To Ascertain. Mean Diameter of a Cask. RULE. Subtract head diameter from bung diameter in inches, and mul- tiply difference by following units for the four varieties ; add product to heacl diameter, and sum will give mean diameter of varieties required. ist Variety 7 I sd Variety 56 2d Variety 68 | 4 th Variety 52 EXAMPLE. Bung and 1 head diameters of a cask of ist variety are 24 and 20 inch- es; what is its mean diameter? 24 20 = 4, and 4 x .7 = 2.8, which, added to 20, = 22.8 ins. ULLAGE CASKS. To Compute Volume of* Ullage Casks. When a cask is only partly filled, it is termed an ullage cask, and is com sidered in two positions, viz., as lying on its side, when it is termed a /Seg- ment Lying, or as standing on its end, when it is termed a Segment Standing. To Ullage a Lying Cask. RULE. Divide wet inches (depth of liquid) by bung diameter ; find quo- tient in column of versed sines in table of circular segments, page 267, and take its corresponding segment ; multiply this segment by capacity of cask in gallons, and product by 1.25 for ullage required. EXAMPLE. Capacity of a cask is 90 gallons, bung diameter being 32 inches; what is its volume at 8 inches depth? 8-:- 32 = .25, tab.seg. of which is. 153 55, which x 90 = 13.8195, and again x 1.25 = i7.2744.ya/Zons. To Ullage a Standing Cask. RULE. Add together square of diameter at surface of liquor, square of head diameter, and square of double diameter taken in middle between the two; multiply sum by wet inches, and product by .1309, and divide by 231 for result in gallons. To Compute "Volume of a Cask "by Four Dimensions. RULE. Add together squares of bung and head diameters, and square of double diameter taken in middle between bung and head ; multiply the sum by length of cask, and product by .1309, and divide this product by 231 for result in gallons. To Compute Volume of any Cask from Tliree Dimen- sions only. RULE. Add into one sum 39 times square of bung diameter, 25 times square of head diameter, and 26 times product of the two diameters ; mul- tiply sum by length, and product by .008 726 ; and divide quotient by 231 for result in gallons. For Rules in Gauging in all its conditions and for description and use of instruments, see HaswelVs Mensuration, pages 307-23. CONIC SECTIONS. 379 Fig. i. CONIC SECTIONS. A Cone is a figure described by revolution of a right-angled triangle about one of its legs, or it is a solid having a circle for its base, and terminated in a vertex. Conic Sections are figures made by a plane cutting a cone. If a cone is cut by a plane through vertex and base, section will be a triangle, and if cut by a plane parallel to its base, section will be a circle. Axis is line about which triangle revolves. Base is circle which is described by revolving base of triangle. An Ellipse is a figure generated by an oblique plane cut- ting a cone above its base. Transverse axis or diameter is longest right line that can be drawn in it, as a fc, Fig. i. Conjugate axis or diameter is a line drawn through centre of ellipse perpendicular to trans- verse axis, as c d. A Parabola is a figure generated by a plane cutting a cone parallel to its side, as a o c, Fig. 2. Axis is a right line drawn from vertex to middle of base, as bo. NOTE. A parabola has not a conjugate diameter. A Hyperbola is a figure generated by a plane cutting a cone at any angle with base greater than that of side of cone, as a b c, Fig. 3. Transverse axis or diameter, o 6, is that part of axis, e b, which, if continued, as at o, would join an opposite cone, ofr. Conjugate axis or diameter is a right line drawn through centre, g, of transverse axis, and perpendicular to it. Straight line through foci is indefinite transverse axis; that part of it between vertices of curves, as o b, is definite transverse axis. Its middle point, g, is centre of curve. Eccentricity of a hyperbola is ratio obtained by dividing distance from centre to either focus by semi-transverse axis. Parameter is cord of curve drawn through focus at right angles to axis. Asymptotes of a hyperbola are two right lines to which the curve continually ap- proaches, touches at an infinite distance but does not pass; they are prolongations of diagonals of rectangle constructed on extremes of the axes. Two hyperbolas are conjugate when transverse axis of one is conjugate of the other, and contrariwise. <3-eneral Definitions. An Ordinate is a right line from any point of a curve to either of diameters, as a e and do, Fig. 4, and a b and d/, are double ordinates; cb } Fig. 5, is an ordinate, and a 6 an abscissa. Fig. 4 c d An Abscissa is that part of diameter which is contained between vertex and an ordinate, as ce. go, Fig. 4, and a b. Wia Fig. 5- Parameter of any diameter is equal to four times j distance from focus to vertex of curve; parameter * of axis is least possible, and is termed parameter of curve. Parameter of curve of a conic section is equal _^ to chord of curve drawn through focus perpendic- ^ / ular to axis. & Parameter of transverse axis is least, and is termed parameter of curve. Parameter of a conic section and foci are sufficient elements for construction of curve. CONIC SECTIONS. d _\ s A Focus is a point on principal axis where double ordinate to axis, through point, is equal to parameter, as ef, Fig. 5. It may be determined arithmetically thus: Divide square of ordinate by four times abscissa, and quotient will give focal distances, as and s, in preceding figures. Fig. 6. Directrix of a conic section is a right line at right angles to -f* major axis, and it is in such a position that f:g:\u: o. Here a d, Fig. 6, is directrix, and o is offset to directrix. Latus Rectum, or principal parameter, passes through a focus; it is a double ordinate, which is a third proportion to the axis. Or, A : a : : a : L. A and a representing major and minor axes. (See HasweWs Mensuration, page 232. ) A Conoid is a warped surface generated by a right line being moved in such a manner that it will touch a straight line and curve, and continue parallel to a given plane. Straight line and curve are called di- rectrices, plane a plane directrix, and moving line the generatrix. Thus, let a b a', Fig. 7, be a circle in a horizontal plane, . and d d' projection of right lines perpendicular to a ver- tical plane, r' b e ; if right lines, d a, r s, r' b, r" s, and d' a, be moved so as to touch circle and right line d d' and be constantly parallel to plane r' b e, it will generate conoid dab a' d'. Radii vectores are lines drawn from the foci to any point in the curve; hence a radius vector is one of these lines. Traced angle is angle formed by the radii vectores and the transverse diameter. Ellipsoid, Paraboloid, and Hyperboloid of Revolution Figures generated by the revolution of an ellipse, parabola, etc., around their axes. (See Men- suration of Surf aces and Solids, pages 357-75.) NOTE i. All figures which can possibly be formed by cutting of a cone are men- tioned in these definitions, and are five following viz., a Triangle, a Circle, an El- lipse, a Parabola, and a Hyperbola ; but last three only are termed Conic Sections. 2. In Parabola parameter of any diameter is a third proportional to abscissa and ordinate of any point of curve, abscissa and ordinate being referred to that diameter and tangent at its vertex. 3. In Ellipse and Hyperbola parameter of any diameter is a third proportional to diameter and its conjugate. To Determine Parameter of an Ellipse or Hyperbola. Fig. 8. RULE. Divide product of conjugate a Fig. 9. c diameter, multiplied by itself, by trans- verse, and quotient is equal to para- meter. rts ' \ In annexed Figs. 8 and 9, of an Ellipse and Hyperbola, transverse and conjugate c ~ diameters, ab, cd, are each 30 and 20. Then 30 : 20 :: 20 : 13. 333= parameter. Parameter of curve = ef, a double ordinate passing through focus, s. o Ellipse. To Describe Ellipses. (See Geometry, page 226.) To Compute Terms of an Ellipse. When any three of four Terms of an Ellipse are given, viz., Transverse and Conjugate Diameters, an Ordinate, and its Abscissa, to ascertain remain- ing Terms. CONIC SECTIONS. 381 To Compute Ordinate. Transverse and Conjugate Diameters and Abscissa being given. RULE. As trans- verse diameter is to conjugate, so is square root of product of abscissae to ordinate which divides them. Fig. 10. EXAMPLE. Transverse diameter, a b, of an ellipse, Fig. 10, is 25; conjugate, c d, 16; and abscissa, ai, 7; what is length of ordinate, t e? 25 7 = 18 less abscissa ; Vy X 18 = 1 1. 225. Hence 25 : 16 :: 11.225 : 7.184 ordinate. Or, A/c 2 I ^ J = any ordinate, c and t representing semi-conjugate and transverse diameters, and x distance of ordinate from centre of figure. To Compute .AJbscissae. Transverse and Conjugate, Diameters and Ordinate being given. RULE. As conju- gate diameter is to transverse, so is square root of difference of squares of ordinate and semi -conjugate to distance between ordinate and centre; and this distance be- ing added to, or subtracted from, semi-transverse, will give abscissae required. EXAMPLE. Transverse diameter, a b, of an ellipse, Fig. 10, is 25; conjugate, c d, 16; and ordinate, ie, 7.184; what is abscissa, t6? V8 2 7. i84 2 = 3-519 943- Hence, as 16 : 25 :: 3.52 : 5.5. Then 25 -4- 2 = 12. 5, and 12. 5 + 5. 5 = 18 = 6 t, ) abscissce 25-7-2 = 12.5, and 12.5 5.5= 7 = at,)" To Compute Transverse Diameter. Conjugate, Ordinate, and Abscissa being given. RULE. To or from semi-conju. gate, according as great or less abscissa is used, add or subtract square root of dif- ference of squares of ordinate and semi-conjugate. Then, as this sum or difference is to abscissa, so is conjugate to transverse. EXAMPLE. Conjugate diameter, c d, of an ellipse, Fig. 10, is 16; ordinate, ie, 7.184; and abscissae, 6 i, i a, 18 and 7 ; what is length of transverse diameter? V(i6-=-2) 2 - 7 .i8 4 2 = 3- 52. 16 -i- 2 -f- 3. 52 : 18 :: 16 : 25; 16-7-2 3.52 : 7 :: 16 : 25 transverse diameter. To Compute Conjugate Diameter. Transverse, Ordinate, and Abscissa being given. RULE. As square root of prod- uct of abscissae is to ordinate, so is transverse diameter to conjugate. EXAMPLE. Transverse diameter, a 6, of an ellipse, Fig. 10, is 25; ordinate, i t, 7. 184 ; and abscissae, b i and i a, 18 and 7 ; what is length of conjugate diameter ? Vi8 X 7 = ii 225. Hence 11.225 ' 7- 184 :: 25 : 16 conjugate diameter. To Compute Circumference of an Ellipse. RULE. Multiply square root of half sum of the squares of two diameters by 3- 1416- EXAMPLE. Transverse and conjugate diameters, a b and cd, of an ellipse, Fig. 10, are 24 and 20; what is its circumference? ' = 488, and x/488 = 22.09. Hence 22.09 X 3. 1416 = 69. 398 circumference. To Compute Area of an Ellipse. RULE. Multiply the diameters together, and the product by .7854. Or, multiply one diameter by .7854, and the product by the other. EXAMPLE. The transverse diameter of an ellipse, a b, Fig. 10, is 12, and its con jugate, c d, 9 ; what is its area ? 12 x 9 X .7854 = 84.8232 area. NOTE. Area of an ellipse is a mean proportional between areas of two circles, diameter of one being major axis and of the other minor axis. ILLUSTRATION. Area of circle of 40 = 1256.64; area of ellipse 40 X 20 = 628.32; area of circle of 20 = 314.16, mean of the two circles 1256.644-314.16 = 785.4. Therefore the conjugate diameter of an ellipse of an area of 785.4 sq. ins., its trans verse being 40, is 25 feet, as 40 X 25 x .7854 = 785.4 sq. ins. 382 CONIC SECTIONS. Segment of an Ellipse. To Compute Area of a Segment of an Ellipse. When its Base is parallel to either Axis, as e if. RULE. Divide height of seg- ment, b t, by diameter or axis, a &, of which it is a part, and find in Table of Areas of Segments of a Circle, page 267, a segment having same versed sine as this quo- tient; then multiply area of segment thus found and the axes of ellipse together. EXAMPLE. Height, b t, Fig. n, is 5, and axes of ellipse are 30 and 20; what is area of segment? 5 -r- 30 = . 1666 tabular versed sine, the area of which (page 267) is .085 54. Hence .085 54 X 30 X 20 = 51.324 area. To Ascertain Length of an Elliptic Curve which is less than, half of entire Figure. Fig. ja. ^jjfc Let curve of which length is required be A 6 C, Fig. 12. Extend versed sine 6 d to meet centre of curve in e. A/, __ ola. (See Geometry, page 229.) To Compute either Ordinate or Abscissa of a Parabola. When the other Ordinate and Abscissa, or other Abscissa and Ordinates are given. RULE. As either abscissa is to square of its ordinate, so is other abscissa to square of its ordinate. Or, as square of any ordinate is to its abscissa, so is square of other ordinate to its abscissa. EXAMPLE i. Abscissa, a 6, of parabola, Fig. 13, is 9; its ordi- nate, b c, 6; what is ordinate, d e, abscissa of which, a d, is 16 ? Hence 9 : 6 2 :: 16 : 64, and 1/64 = 8 length. 2. Abscissae of a parabola are 9 and 16, and their correspond- ing ordinates 6 and 8; any three of these being taken, it is re- quired to compute the fourth. x . _ = g ordinate. 9 I6X6 2 3. = 9 less abscissa. - ID = 6 ordinate. = rf abscissa. !Para"bolic Curve. To Compute Length of Curve of a IParatoola out off t>y a Double Ordinate. Fig. 13. RULE. To square of ordinate add of square of abscissa, and square root of this sum, multiplied by two, will give length of curve nearly. EXAMPLE. Ordinate, d e, Fig. 13, is 8, and its abscissa, a d, 16; what is length of curve, fa e ? 8 2 + 4Xl62 = 405- 333, and ^405. 333X2 = 40. 267 length. CONIC SECTIONS. 383 jrjg x . To Compute Area of a JParabola. RULE. Multiply base by height, and take two thirds of product Corollary. A parabola is two thirds of its circumscribing par- allelogram. EXAMPLE. What is area of parabola, a b c, Fig. 14, height, &, being 16, and base, or double ordinate, a c, 16 ? 16 x 16 = 256, and 0/256 = 170.667 area. To Compute Area of a Segment of a Farabola. RULE. Multiply difference of cubes of two ends of segment, a c, df, by twice Hi height, e o, and divide product by three times difference of squares of ends. EXAMPLE. Ends of a segment of a parabola, a c and df, Fig. 14, are 10 and 6, and height, e o, is 10; what is its area? io 3 a.6 3 X 10 X 2 = 15680, and + io 2 ^ 6 2 X 3 = 81.667 area. NOTE. Any parabolic segment is equal to a parabola of the same height, the base of which is equal to base of segment, increased by a third proportional to sum of the two ends and lesser end. Hyperbola. To Describe a Hyperbola. (See Geometry, page 230.) To Compute Ordinate of a Hyperbola, Transverse and Conjugate Diameters and Abscissa being given. RULE. As trans- verse diameter is to conjugate, so is square root of product of abscissae to ordinate required. Fig. 15. 5 EXAMPLE. Hyperbola, a be, Fig. 15, has a transverse diameter, a t, of 120; a conjugate, df, of 72 ; and abscissa, ae, 40; what is the length of ordinate, ec? 40-}- 120 = 160 greater abscissa, and 120 172:: V(4 X 160) : 48 ordinate. NOTE i. In hyperbolas lesser abscissa, added to axis (the transverse diameter), gives greater. 2. Difference of two lines drawn from foci of any hyperbola to any point in curve is equal to its transverse diameter. To Compute Abscissae, Transverse and Conjugate Diameters and Ordinate being given. RULE. As con- Jugate diameter is to transverse, so is square root of sum of squares of ordinate and semi conjugate to distance between ordinate and centre, or half sum of abscissae. Then the sum of this distance and semi-transverse will give greater abscissa, and their difference the lesser abscissa. EXAMPLE. Transverse diameter, a t, of a hyperbola, Fig. 15, is 120; conjugate, d/, 72 ; and ordinate, e c, 48 ; what are lengths of abscissae, t e and ae? 72 : 120 :: -\/48 2 + (72 ~- 2) 2 = 60 : 100 half sum of abscissa, and ioo-f-(i20-r-2) = 160 greater abscissa, and 100 (120-?- 2) = 40 lesser abscissa. To Compute Conjugate Diameter, Transverse Diameter, Abscissa, and Ordinate being given. RULE. As square root of product of abscissae is to ordinate, so is transverse diameter to conjugate. EXAMPLE. Transverse diameter, at, of a. hyperbola, Fig. 15, is 120; ordinate, ec, 48; and abscissae, te and ae, 160 and 40; what is length of conjugate, df? 1/40 X 160 = 80 : 48 :: 120 : 72 conjugate. 384 CONIC SECTIONS. To Compute Transverse Diameter, Conjugate, Ordinate, and an Abscissa being given. RULE. Add square of ordinate to square of semi conjugate, and extract square root of their sum. Take sum or difference of semi-conjugate and this root, according as greater or lesser abscissa is used. Then, as square of ordinate is to product of abscissa and conjugate, so is sum or difference above ascertained to transverse diameter required. NOTE. When the greater abscissa is used, the difference is taken, and con- trariwise. EXAMPLE. Conjugate diameter, df, of a hyperbola, Fig. 15, is 72; ordinate, e c, 48 ; and lesser abscissa, a e, 40 ; what is length of transverse diameter, at? v/48 2 -p- (72 -r- 2) 2 = 60, and 60 -f- 72 -r- 2 = 96 lesser abscissa, and 40 X 72 = 2880. Hence, 48 2 : 2880 :: 96 : 120 transverse diameter. To Compute Length of any Arc of a HyperTaola, com- mencing at Vertex. RULE. To 19 times transverse diameter add 21 times parameter of axis. To 9 times transverse diameter add 21 times parameter, and multiply each of these sums respectively by quotient of lesser abscissa divided by transverse di- ameter. To each of products thus ascertained add 15 times parameter, and divide former by latter; then this quotient, multiplied by ordinate, will give length of arc, nearly. NOTE. To Compute Parameter, divide square of conjugate by transverse diam- eter. Fig. 16. 5 EXAMPLE. In hyperbola, a be, Fig. 16, transverse diameter is 120, conjugate, 72, ordinate, e c, 48, and lesser abscissa, a e, 40; what is length of arc, a b ? = 43. 2 parameter. 120 X 19 + 43.2X21 X = 1062. 4. 40 120 X 9 + 43. 2 X 21 X =662.4. Th en 1062.4 + 43.2X15-^662.4 + 43. 2 X 15 = 1.305, which x 48 = 62.64 length. NOTE. As transverse diameter is to conjugate, so is conjugate to parameter. (See Rule, page 380.) To Compete A.rea of a Hypert>ola, Transverse, Conjugate, and Lesser Abscissa being given. RULE. To product of transverse diameter and lesser abscissa add five sevenths of square of this abscissa, and multiply square root of sum by 21. Add 4 times square root of product of transverse diameter and lesser abscissa to product last ascertained, and divide sum by 75. Divide 4 times product of conjugate diameter and lesser abscissa by transverse diameter, and this last quotient, multiplied by former, will give area, nearly. EXAMPLE. Transverse diameter of a hyperbola, Fig. 16, is 60, conjugate 36, and lesser abscissa or height, a e, 20; what is area of figure ? 60 X 20 H of 2o 2 = 1485. 7143, and 1/1485. 7143 X 21 = 809. 43, and V6o X 20 X 4 + 809.43 =901.02, which -r- 75 = 12.0136 and 3 X ^ X 4 X 12.0136 = 576.653 area. NbM. For ordinates of a parabola in divisions of eighths and tenths, see page 229. Delta Metal. Delta Metal is an improved composition of Aluminium and its alloys ; it is non-corrosive, capable of being cast, forged, and hot rolled. Tensile Strength per Sq. Inch. Cast in green sand 48 380 Ibs. I Rolled, annealed 60 920 lb& Rolled, hard 75260 u | Wire, No. 22 WG 140000 " PLANE TBIGONOMETRY. 385 PLANE TRIGONOMETRY. By Plane Trigonometry is ascertained how to compute or determine four of the seven elements of a plane or rectilinear triangle from the other three, for when any three of them are given, one of which being a side or the area, the remaining elements may be determined ; and this operation is termed Solving the Triangle. The determination of the mutual relation of the Sines, Tangents, Secants, etc., of the sums, differences, multiples, etc., of arcs or angles is also classed under this head. For Diagram and Explanation of Terms, see Geometry, pp. 219-21. Right-angled. Triangles. For Solution by Lines and Areas, see Mensuration of Areas, Lines, and Surf aces, pp. 335-39. To Compute a Side. When a Side and its Opposite Angle is given. RULE. As sine of angle opposite given side is to sine of angle opposite required side, so is given side to required side. To Compute an Angle. RULE. As side opposite to given angle is to side opposite to required angle, so is sine of given angle to sine of required angle. To Compute Base or Perpendicular in a Right-angled Triangle. When A ngles and One Side next Right A ngle are given. RULE. As ra- dius is to tangent of angle adjacent to given side, so is this side to other side. To Compute the other Side. When Two Sides and Included Angle are given. RULE. As sum of two given sides is to their difference, so is tangent of half sum of their opposite angles to tangent of half their difference ; add this half difference to half sum, to ascertain greater angle ; and subtract half difference from half sum, to ascertain less angle. The other side may then be ascertained by Rule above. To Compute Angles. When Sides are given. RULE. As one side is to other side, so is radius to tangent of angle adjacent to first side. To Compute an Angle. When Three Sides are given. RULE i. Subtract sum of logarithms of sides which contain required angle, from 20 ; to remainder add logarithm of half sum of three sides, and that of difference between this half sum and side opposite to required angle. Half the sum of these three logarithms is logarithmic cosine of half required angle. The other angles may be ascer- tained by Rule above. 2. Subtract sum of logarithms of two sides which contain required angle, from 20, and to remainder add logarithms of differences between these two sides and half sum of the three sides. Half result is logarithmic sine of half required angle. NOTE. In all ordinary cases either of these rules will give sufficiently accurate results. Rule i should be used when required angle exceeds 90 ; and Rule 2 when it ts less than 00. Ks 3 86 PLANE TRIGONOMETRY. EXAMPLE. The sides of a triangle are 3, 4, and 5 ; what are the angles of the hypothenuse ? 20 (Log. 4 = .602 06 + Log. 5 .698 97) = 18.698 97 ; Log. 3 + 4 + 5-^2 4 = .301 03; and Log. 3 -}- 4 + 5 -r- 2 - 5:= o. Then 18.698 97 + . 301 03 = 19, which -f- 2 = 9. 5 = log. sin. of half angle = 18 26', which X 2 = 36 52' angle. Hence 90 36 52' = 53 8' remaining angle. In following figures, i and 2 : A = 9o, B = 45, = 45, Radius = i, Secant = 1.4142, Cosine = .7071, Sin. 45 .7071, Tangent = i, Area = .25. By Sin., Tan., Sec., etc., A B, etc., is expressed Sine, Tangent, Secant, etc., of angles, A, B, etc. To Compute Sides A C arid. B C. Figs. 1 and. 3. When Hyp., Side B A, and Angles B and C are given. Fig. i. Sin. B X B A _ Sin. C B A X Cot. C = A C. Hyp. X Cos. C = A C. Hyp. X Sin. B = A C. BA Fig. 2. I O Cosine. A Vers? Sin. C AC Sin. B = BC. To Compute Side A C and Angles. When Hyp. and Side B A are given. Fig. i and 2. Hyp. = Sin. B. BA BAXSJD.B HTpT sln7c B C X Sin. B = A C. To Compute Side B C and Hyp. or Angles. When both Sides are given. Fig. 2. Sin. C BA BC = BC. = Sin. C. AC = Tan. C. Fig. 3- To Compute Sides. Figs. 3 and When a Side and an Angle are given. B C x Cos. B = B A. B C X Sin. B = A C. A B X Sec. B = B C.' = BA. ACxSin.C Sin. B = BA. Rad. = BC. - = Sin. B Tangent. In BAG, Fig. 5, a right-angled triangle, C A, is assumed to be radius ; B A tangent of C, and B C secant to that radius ; Or, dividing each of these by base, there is obtained the tangent and secant of C respectively to radius i. PLANE TRIGONOMETRY. 387 Fig. 5- Radius C A = Secant C B = Tangent A B = Co-secant CB = Co -tangent e B = VAC 2 H-BA2 = hyp. B C. A C -r- Cos. C = hyp. B C. 4142 4142 O Radius, ff B C X Cos. C = Rad. B A X Tan. B = Rad. BC-^-BA = Sec. B. B C X Cos. B = B A. Cos. G Sin. C Sine dg .7071 Cosine Cgorod= .7071 Versed sine g A = . 2929 Co- versed sine o e = .2929 Angle C A B = 90 BA-:-Sin. C = hyp. BC. i -f- Tan. C = Cot. C. B C 2 X Sin. 2 C = CotC. - =Area, B A x Sec. B = B C. B AX Cot. C = Rad. B C X Sin. B = Rad. B C X Sin. C = B A. A C X Tan C = B A. i -i- Sin. C = Cosec. C. i Sin. C = Co-ver. sin. Cos. C-=-Sin. C^Cot. C. C B x Sin. B = AC. Trigonometrical Equivalents. Perp. -r- hyp. = Sin. C. Hyp. -r- base = Sec. C. Perp. -r- base = Tan. C. Base -r- perp. = Tan. B. Hyp. -=- perp. = Sec. B. Perp. -4- hyp. = Cos. B. Hyp. -r- perp. Cosec. C. Hyp. Base = Versin. Hyp. Perp. = Co-ver. sin. Ci Tan. -4- sin. = Sec. Tan. -4- sec. = Sin. Tan. X cot. = Rad. V(i cos. 2 ) = Sin. i -f- cot. = Tan. i -4- sin. = Cosec. ILLUSTRATIONS. Assume side A B of a right-angled triangle is 100, and angle C 53 8'; what are its elements? Fig. 6. B Oblique-angled. Triangles. To Compute Sides B A and B C. When Side A C and Angles are given. Fig. 6. Base -4- hyp. = Cos. C. Base -4- hyp. = Sin. B. Base -4- perp. = Cotan. C. V (i sin. 2 ) = Cos. Sin. -4- tan. = Cos. Sin. x cot. = Cos. Sin. -4- cos. =Tan. Cos. -T- cot = Sin. Cos. -4- sin. = Cot. -f- cos. = Sec. -f- cosec. = Sin. -r- sec. = Cos. cos. = Versiu. sin. = Co-ver. sin. -r- tan. = Cotan. Sin. C X A C Sin.B = B A. Sin. Ax AC Sin B : Sin. C X B C Sin. A = To Compute Angles and Side A C. When Sides A B, B C, and one of the Angles are given. Fig. 6. B C X Sin. B AJC Fig. 7. = Sin. A. Sin. C X A C B A Sin. BxBC = Sin. B. A B x Sin. B ~ACT = Sin. C. = AC. Sin. A To Compute Sides B A and B C. When Side A C and A ngles are given. Fig. 7. Sin. CXBC Sin. AxAC_ Sin. A Sin. B When Side B C and Angles are given. Fig. 7. B C x Sin. C Sin. C X A C Sin. A Sin. B NOTE. Sine and Cosine of an arc are each equal to sine and cosine of their sup- plements. Spherical Triangles, Right-angled and Oblique. For full formulas See Molesworth, Lond., 1878, pp. 435-6. 388 PLANE TRIGONOMETBY. To Compute Angles and Side AC. When Sides A B, B C, and A ngle B are given. Fig. 7. Sin. BxBC BCxSin. B Sin. A AC X Sin. A BC = Sin. B. AC B A X Sin. A BC = Sin. A. = Sin. C. To Compute all the Angles. When all the Sides are given, Figs. 6 and 7. RULE. Let fall a perpen- dicular, B d, opposite to required angle. Then, as A C : sum of A B, B C : : their difference : twice d g, the distance of perpendicular, B d, from middle of the base. Hence A d, C g are known, and triangle, A B C, is divided into two right- angled triangles, B C d, B A d ; then, by rules for right-angled triangles, ascertain angle A or C. OPERATION. AC, Fig. 6, .5014 : AB-j-BC, 1.1174+ 1.4142 = 2.5316:^ B GO BC, 1.4142 1. 1174 = . 2968 : a x d g = i 4986. Consequently, triangle B d C, Fig. 6, is divided into two triangles, BAG and B d A. To Compute Side A B and Angles. When Two Sides and One Angle, or One Side and Two Angles, are given. Fig. 6. A C x Sin. C = Sin. B. = AB. B C X Sin. B AC ABxSin. B = Sin. A. = Sin. C. A C X Sin. A AB (ACxCos. A)" AC X Sin. C = Tan. B 2 Area AC BC (ACXCOS.C)" To Compute Area of a Triangle. Fig. 8. BAxBCxSin. B ACxBCxSin. C BAx AC X Sin. A 222 Sin. 2 C, B C 2 A C 2 , Tan. C . B A 2 , Cot. C , , and = Area. 42 2 NOTE. For other rules, see Mensuration of Areas, Lines, and _ Surfaces, page 335. To Compute Sides. When Areas and Angles are given. Figs. 6 and 7. 2 Area : = AC. - = BA. B C, Sin. C ~" AC, Sin. A To Ascertain Distance of Inacces- si"ble Otjects on a Level IPlane. Figs. 9 and 1O. / 2 Area, Sin. A _ V Sin. C,Sin. (A + C)~ Fig. ia Fig. 9. OPERATION. Lay off perpendic- ulars to line A B, Fig. 9, as B c, d e, on line A d, terminating on line e A. Then ed cB:cB::Bd: BA. When there are Tioo Inacces- sible Objects, as Fig. 10. OPERATION. Measure a base line, A B, Fig. 10, and angles c A B, dBA, dAB, cBA,eto. Then pro- ceed by formulas, page 387, to deduce cd. NOTE. If course of cd is required, take difference of angles d c A and c d JB from course A B. PLANE TKIGONOMETBT. 389 When the Objects can be aligned. Fig. ii. OPERATION. Align c B, Fig. n, at A, measure a base line at any angle there- to, as A o, and angles o A c, c o A, and B o A. Then proceed as per formula, page 386, to deduce c B. To Compute Distance from, a Griveii feint to an In- accessible Object. Fig. 12. Fig. 13. OPERATION. Measure a level line, A c, Fig. 12, and ascertain angles, B A c, B c A, Hence, having side, A c, and two angles, proceed as per formula, page 386, to de- termine A B. To Compute Height of an. Elevated Point. Fig. 13. Fig. 13. OPERATION. Measure B\ Fig. 14. distance on a horizontal line, A c, Fig. 13 ; ascertain Angle B A c. Then pro- ceed as per formulas, pp. 386-8, to ascertain B c. When a Horizontal Base is not Attainable. Fig. 14. OPERATION. Measure or compute distance A c, Fig. 14; ascertain angle of depression cAo and of elevation B A c. Then proceed as per formula, page 386, to ascertain B c. Fig. 15- When a Full Base Line is not Attain* able. Fig. 15. OPERATION. Measure a base line, A c, Fig. 15, and ascertain angles A c B, c A B. Then proceed as per for- mula, page 386, to ascer- tain d B. Fig. 16. Without Use of an Instrument. Fig. 16. OPERATION. Lay off any suitable and level distance, d d, set up a staff at each ex- tremity at like elevation from base line d d, and note distances y and z, at which the lines of sight of object range with tops of the staffs; deduct height of eye from length of staffs, and ascertain heights h. D h Then 1- h -\- s = height. $ representing height of line of sight from base d d, and D length of lined d. 390 NATURAL SINES AND COSINES. Natural Sines and. Cosines. QO 1 2 30 N. sine. N. cos. N. sine. N. cos. N. sine. N. cos. N. sine. N. cos. o .00000 I 01745 99985 0349 99939 05234 .99863 I .00029 I .01774 99984 03519 .99938 .05263 .99861 2 .00058 X 01803 .99984 03548 99Q37 .05292 .9986 3 .00087 I .01832 .99983 03577 .99936 05321 .99858 4 .00116 I .01862 .99983 03606 99935 0535 99857 5 .00145 I .01891 .99982 03635 99934 05379 99855 6 .00175 X .0192 .99982 .03664 99933 .05408 .99854 7 .00204 I .01949 99981 03693 99932 05437 .99852 8 9 .00262 I I .01978 .02007 .9998 .9998 03723 03752 99931 9993 .05466 05495 .99851 .99849 10 .00291 I .02036 99979 03781 .99929 05524 99847 ii 12 .0032 .00349 .99999 .99999 02065 .02094 99979 .99978 .0381 .03839 .99927 .99926 05553 .05582 .99846 .99844 '3 .00378 .99999 .02123 99977 03868 .99925 .05611 .99842 X 4 .00407 99999 .02152 99977 .03897 .99924 .0564 .99841 5 .00436 99999 .02181 .99976 .03926 .99923 .05669 .99839 16 .00465 99999 .O22II .99976 03955 .99922 .05698 .99838 '7 .00495 99999 .0224 99975 .03984 .99921 .05727 .99836 18 .00524 99999 . 02269 99974 .04013 .99919 05756 .99834 *9 00553 .99998 .02298 99974 .04042 .99918 05785 99833 20 .00582 9999 s .02327 99973 .04071 .99917 .05814 .99831 21 .00611 .99998 .02356 .99972 .041 .99916 .05844 .99829 22 .0064 .99998 .02385 99972 .04129 99915 05873 .99827 23 .00669 .99998 .02414 .99971 .04159 999 X 3 .05902 .99826 2 4 .00698 .99998 .02443 9997 .04188 .99912 05931 .99824 25 .00727 99997 .02472 .99969 .04217 .99911 .0596 .99822 26 .00756 99997 .02501 .99969 .04246 .9991 .05989 .99821 2 7 .00785 99997 0253 .99968 04275 .99909 .06018 .99819 28 .00814 99997 .0256 .99967 .04304 .99907 .06047 .99817 29 .00844 .99996 .02589 .99966 04333 .99906 .06076 .99815 3o .00873 .99996 .026l8 .99966 .04362 .99905 .06105 .99813 3i .00902 .99996 .02647 .99965 .04391 .99904 .06134 .99812 32 .00931 .99996 .02676 .99964 .0442 .99902 .06163 .9981 33 .0096 99995 .02705 .99963 .04449 .99901 .06192 .99808 34 .00989 99995 .02734 .99963 .04478 999 .06221 .99806 35 01018 99995 02763 .99962 .04507 .99898 .0625 .99804 36 .01047 99995 .02792 .99961 04536 .99897 .06279 .99803 37 .01076 99994 .02821 .9996 04565 .99896 .06308 .99801 38 .01105 99994 0285 99959 .04594 99894 06337 99799 39 .01134 .99994 .02879 99959 .04623 .99893 .06366 99797 40 .01164 99993 .02908 .99958 .04653 .99892 06395 99795 4i .01193 99993 .02938 99957 .04682 %o642 4 99793 42 .01222 99993 .02967 .99956 .O 47 n .99889 06453 .99792 43 .01251 .99992 02996 99955 .0474 .99888 .06482 9979 44 .0128 .99992 .03025 99954 .04769 .99886 .06511 .99788 45 01309 .99991 .03054 99953 .04798 .99885 .0654 .99786 46 .01338 .99991 .03083 .99952 .04827 .99883 .06569 .99784 47 .01367 .99991 .63112 .99952 .04856 .99882 06598 .99782 48 .01396 9999 .03141 99951 04885 .99881 06627 9978 49 .01425 .9999 0317 9995 04914 99879 .06656 .99778 So .01454 .99989 .03199 99949 04943 .99878 06685 .99776 5i .01483 99989 .03228 .99948 .04972 .99876 .06714 99774 52 OI5I3 .99989 .03257 99947 .05001 99875 .06743 99772 53 .01542 .99988 .03286 .99946 0503 .99 8 73 .06773 9977 54 .01571 .99988 .03316 99945 05059 .99872 .06802 .99768 5 I .Ol6 .99987 03345 99944 .05088 9987 .06831 99766 56 .01629 .99987 03374 99943 .05117 .99869 .0686 .99764 11 .01658 .01687 .99986 .99986 03403 03432 99942 .99941 ,05146 05175 .99867 .99866 .06889 .06918 .99762 9976 f 9 .01716 .99985 .03461 9994 .05205 .99864 .06947 99758 60 01745 .99985 0349 99939 05234 .99863 .06976 99756 N. cos. N. sine. N. cos. N. sine. N. cos. N. sine. N. cos. N. sine. 89 88 ' 87 [ 86 NATUBAL SINES AND COSINES. 4 5 o C o 1 ro ' N. sine. N. cos. N. sine. N. cos. N. sine. N. cos. N. sine. N. cos. .06976 99756 ^>87i6 .99619 0453 99452 .12187 99255 I .07005 99754 .08745 .99617 . 0482 99449 .12216 .99251 2 .07034 99752 .08774 .99614 . 0511 .99446 .12245 .99248 3 .07063 9975 .08803 .99612 054 -99443 . 12274 99244 4 .07092 .99748 .08831 .99609 . 0569 9944 . 12302 9924 5 .07121 .99746 .0886 .99607 0597 99437 12331 99237 6 0715 99744 .08889 .99604 . 0626 -99434 .1236 .99233 7 .07179 .99742 .08918 .99602 0655 9943 1 12389 9923 8 9 .07208 .07237 9974 9973 8 ^08976 99599 .99596 . 0684 . 0713 .99428 99424 .12418 .12447 .99226 .99222 10 .07266 99736 .09005 -99594 . 0742 .99421 .12476 .99219 ii .07295 99734 09034 9959 1 . 0771 .99418 .12504 .99215 12 .07324 99731 .09063 .99588 . 08 99415 12533 .99211 13 07353 .99729 .09092 .99586 . 0829 .99412 .12562 .99208 *4 .07382 .99727 .09121 99583 . 0858 .99409 .12591 .99204 J 5 .07411 99725 .0915 9958 . 0887 .99406 .1262 99 2 16 .0744 99723 .09179 99578 . 0916 .99402 .12649 .99197 '7 .07469 .99721 .09208 99575 0945 99399 .12678 99193 18 .07498 .99719 .09237 99572 0973 99396 .12706 .99189 X 9 .07527 .95716 .09266 9957 . IOO2 99393 12735 .99186 20 07556 .99714 .09295 99567 1031 9939 .12764 .99182 21 07585 .99712 .09324 99564 . 106 .99386 12793 .99178 22 .07614 .9971 .09353 .99562 . 1089 99383 .12822 99*75 2 3 .07643 .99708 .09382 99559 . 1118 .9938 .12851 .99171 2 4 .07672 99705 .09411 99556 "47 99377 .1288 .99167 2 5 .07701 99703 .0944 99553 1176 99374 . 12908 .99163 26 0773 .99701 .09469 99551 . 1205 9937 12937 .9916 2 Z 07759 .99699 .09498 .99548 1234 .99367 .12966 .99156 28 .07788 .99696 .09527 99545 1263 99364 .12995 .99152 29 .07817 .99694 09556 99542 . 1291 9936 .13024 .99148 3o .07846 .99692 09585 9954 . 132 99357 13053 .99144 3i .07875 .99689 .09614 99537 1349 99354 13081 .99141 32 .07904 .99687 .09642 99534 . 1378 99351 .13" 99'37 33 07933 .99685 .09671 99531 . 1407 99347 .13139 99133 34 35 .07962 .07991 .99683 .9968 .097 .09729 .99528 .99526 . 1436 . 1465 99344 99341 .13168 13197 .99129 .99125 36 .0802 .99678 .09758 .99523 . 1494 -99337 .13226 .99122 37 .08049 .99676 .09787 9952 1523 99334 13254 .99118 38 .08078 99673 .09816 99517 1552 99331 13283 .99114 39 .08107 .99671 .09845 .99514 99327 .13312 .9911 40 .08136 .99668 .09874 995" . 1609 .99324 .13341 .99106 4i .08165 .99666 .09903 .99508 1638 9932 1337 .99102 42 .08194 .99664 09932 .99506 . 1667 99317 13399 .99098 43 .08223 .99661 .09961 .99503 . 1696 99314 13427 .99094 44 .08252 99659 999 995 1725 9931 13456 45 .08281 99657 .10019 99497 1754 99307 13485 .99087 46 .0831 .99654 .10048 99494 1783 99303 I35I4 .99083 47 08339 .99652 10077 .99491 . 1812 993 13543 .99079 48 .08368 99649 .10106 .99488 . 184 .99297 .13572 .99075 49 .08397 .99647 10135 .99485 . 1869 .99293 136 .99071 50 .08426 .99644 .10164 .99482 . 1898 .9929 .13629 .99067 5i 08455 .99642 . 10192 .99479 i9 2 7 99286 .13658 .99063 52 .08484 99639 .IO22I 99476 1956 .99283 .13687 .99059 53 .08513 99637 .1025 99473 . 1985 .99279 .13716 9955 54 .08542 99635 . 10279 9947 . 2014 .99276 13744 .99051 55 .08571 99632 .10308 99467 . 2043 .99272 13773 99047 56 .086 '99 6 3 10337 .99464 . 2071 .99269 .13802 .99043 57 .08629 .99627 .10366 .99461 . 21 .99265 13831 .99039 58 .08658 .99625 10395 .99458 , 2129 .99262 ^386 9935 59 .08687 .99622 . 10424 99455 . 2158 .99258 .13889 99031 60 .08716 .99619 10453 .99452 . 2187 99255 i39 I 7 .99027 N. cos. N. sine. N. cos. N. sine. N.C08. N. sine. N. cos. N. sine. a - i # 1 s: JO 82 ! 392 NATURAL SINBS AND COSINES. 8 90 100 11 ' N.sine. N. cos. N. sine. N. cos. N. sine. N. cos. N. sine. N. cos. 139*7 .99027 15643 .98769 .17365 .9848* . 19081 .98163 I .13946 .99023 .15672 .98764 17393 .98476 .19109 2 13975 .99019 .15701 .9876 .17422 .98471 .19138 98152 3 .14004 .99015 1573 98755 I745I .98466 .19167 .98146 4 H033 .99011 15758 98751 17479 .98461 .9814 5 .14061 .99006 15787 .98746 .17508 9 8 455 .19224 6 .1409 .15816 .98741 17537 9 8 45 .19252 .98129 7 .14119 . 98998 .15845 98737 17565 .98445 .19281 .98124 8 .14148 98994 15873 .98732 17594 9844 .19309 .98118 9 .14177 .9899 .15902 .98728 .17623 98435 .19338 .98112 10 .14205 .98986 *593 .98723 .17651 9843 .19366 .98107 ii .14234 .98982 *S959 .98718 .1768 98425 .98101 12 .14263 .98978 .15988 .98714 .17708 .9842 .19423 .98096 13 . 14292 98973 .16017 .98709 17737 .98414 .19452 .9809 14 .1432 .98969 . 16046 .98704 .17766 .98409 .19481 .98084 15 i 4349 98965 .16074 987 17794 .98404 19509 .98079 16 14378 .98961 .16103 .98695 .17823 .98399 19538 .98073 17 I 447 98957 .16132 .9869 17852 .98394 . 19566 .98067 18 .14436 98953 .1616 .98686 .1788 98389 .98061 19 .14464 .98948 .16189 .98681 .17909 98383 19623 .98056 20 H493 .98944 .16218 .98676 17937 98378 .19652 9805 21 .14522 .9894 . 16246 .98671 . i 7966 9 8 373 .1968 .98044 22 I455I .98936 .16275 .98667 17995 .98368 .19709 .98039 23 .1458 .98931 . 16304 .98662 .18023 .98362 19737 98033 24 . 14608 .98927 16333 .98657 .18052 .98357 .19766 .98027 25 14637 .98923 .16361 .98652 .18081 98352 .19794 . 9802 i 26 . 14666 .98919 .1639 .98648 .18109 9 8 347 .19823 .98016 27 . 14695 .98914 .16419 98643 .18138 .98341 .19851 .9801 28 14723 .9891 .16447 .98638 .18166 98336 .1988 .98004 2 9 14752 .98906 .16476 98633 .18195 98331 .19908 .97988 3 .14781 . 98902 16505 .98629 .18224 98325 .97992 31 .1481 .98897 16533 .98624 .18252 .9832 .19965 .97987 S 2 .14838 .98893 .16562 .98619 .18281 98315 19994 .97981 33 .14867 .98889 .16591 .98614 .18309 .9831 . 20022 97975 34 .14896 .98884 .1662 .98609 18338 98304 .2OO5I .97969 35 .14925 .9888 .16648 .98604 .18367 .98299 .20079 .97963 36 14954 .98876 .16677 .986 i8395 .98294 -2OIO8 97958 37 .14982 .98871 .16706 98595 .18424 .98288 .20136 97952 38 .15011 .98867 16734 9859 18452 .98283 .20l65 .97946 39 .1504 .98863 16763 98585 .18481 .98277 .20193 9794 40 .15069 .98858 .16792 9858 .18509 .98272 .20222 97934 41 15097 .98854 .1682 98575 18538 .98267 .2O25 .97928 42 .15126 .98849 .16849 9857 .18567 .98261 .20279 .97922 43 I5I55 98845 .16878 98565 18595 .98256 .20307 .97916 44 .15184 .98841 .16906 .98561 .18624 .9825 .20336 .9791 45 .15212 .98836 .16935 98556 .18652 .98245 . 20364 . 97905 46 .15241 98832 .16964 98551 .18681 .9824 20393 .97899 47 1527 .98827 .16992 .98546 .1871 .98234 .2O42I 97 8 93 48 .15299 .98823 .17021 .98541 .18738 .98229 .2045 .97887 49 .15327 .98818 1705 9 8 536 .18767 .98223 .20478 .97881 50 15356 .98814 17078 98531 .98218 .20507 97875 51 15385 .98809 .17107 .98526 .18824 .98212 20535 .97869 52 I54H .98805 .17136 .98521 .18852 .98207 20563 .97863 53 .15442 .988 .17164 98516 .18881 .98201 . 20592 97857 54 .98796 17193 98511 .1891 .98196 .2062 97851 55 .155 .98791 .17222 .98506 .18938 .9819 .20649 97845 56 15529 .98787 1725 .98501 .18967 .98185 .20677 97839 57 15557 .98782 .17279 .98496 .18995 .98179 . 20706 97833 58 .15586 .98778 17308 .98491 .19024 .98174 20734 .97827 59 15615 98773 17336 .98486 19052 .98168 .20763 .97821 60 15643 .98769 17365 .98481 .19081 .98163 .20791 97815 N. cos. N. sine. N. cos. I N.sine. N. cos. N. sine. N. cos. N. sine. 81 80 790 78 NATURAL SINES AND COSINES. 393 II 12 130 14 15 27 1 N. sine. N. cos N. sine. N. cos. N. sine N. cos N. sine N. cos. o o \ .20791 .97815 .22495 97437 .24192 9703 .25882 96593 I . 2082 .97809 .22523 9743 .2422 .97023 .2591 96585 X 2 . 20848 .97803 22552 97424 .24249 97015 .25938 .96578 I 3 .20877 97797 .2258 97417 .24277 .97008 .25966 9657 2 4 .20905 .97791 .22608 .97411 24305 .97001 .25994 .96562 2 5 20933 .97784 .22637 .97404 24333 .26022 96555 3 6 .20962 97778 .22665 9739 s .24362 . 9698- .2605 96547 3 7 .2099 .97772 .22693 9*739! 2439 .9698 .26079 9654 4 8 .21019 .97766 .22722 97384 .24418 96973 . 26107 96532 4 9 .21047 9776 .2275 97378 .24446 .96966 26135 .96524 5 o .21076 97754 .22778 97371 24474 .96959 .2616; .96517 5 i .21104 .97748 .22807 97365 24503 .96952 .26191 .96509 2 .21132 .97742 22835 97358 24531 96945 .26219 .96502 6 3 .21161 97735 .22863 97351 24559 96937 . 26247 96494 61 4 .21189 .97729 .22892 97345 .24587 9693 26275 .96486 7 5 .21218 97723 .2292 97338 .24615 96923 .26303 .96479 7 6 .21246 .97717 .22948 97331 .24644 .96916 26331 .96471 8 7 .21275 .97711 .22977 9732; .24672 .96909 26359 .96463 8 8 .21303 97705 .23005 .247 .26387 96456 9 9 21331 .97698 23033 973" .24728 . 96894 26415 .96448 9 j o 2136 .97692 .23062 97304 24756 .9688; .26443 .9644 9 ' .21388 .97686 .2309 .97298 .24784 .9688 .26471 96433 10 2 .21417 .9768 .23118 .97291 .24813 .96873 265 .96425 10 j 3 | .21445 97673 23146 .97284 .24841 .96866 .26528 .96417 ii | 4 .21474 .97667 23175 .97278 .24869 .96858 .26556 .9641 11 I 5 .21502 .97661 23203 .97271 .24897 96851 .26584 .96402 12 26 2153 97655 23231 .97264 24925 .96844 .26612 .96394 12 27 13 i 28 21559 21587 .97648 97642 .2326 .23288 97257 97251 24954 .24982 .96837 .96829 .2664 .26668 .96386 96379 13 ; 29 .2l6l6 .97636 .23316 .97244 2501 -96822 .26696 96371 14 30 .21644 9763 2 3345 97237 .25038 96815 .26724 96363 X 4 3 1 .21672 97623 23373 9723 .25066 .96807 26752 96355 14 32 .21701 .97617 23401 .97223 .25094 .968 .2678 96347 S 33 .21729 .97611 .23429 .97217 .25122 96793 .26808 9634 15 34 .21758 .97604 23458 .9721 25151 .96786 .26836 .96332 16 35 .21786 97598 .23486 .97203 25179 .96778 .26864 .96324 16 36 .21814 97592 23514 .97196 .25207 .96771 .26892 .96316 17 37 | .21843 97585 23542 .97189 25235 .96764 .2692 .96308 17 38 .21871 97579 23571 .97182 .25263 .96756 .26948 .96301 18 39 ; .21899 97573 23599 .97176 25291 96749 .26976 96293 18 40 ! .21928 .97566 .23627 .97169 2532 .96742 .27004 96285 18 41 .21956 9756 .23656 97162 25348 96734 .27032 96277 19 42 21985 97553 .23684 97155 25376 .96727 .2706 96269 19 43 -22013 97547 .23712 97148 .25404 96719 .27088 96261 20 44 .22041 97541 2374 97141 254J2 96712 .27116 96253 20 45 .2207 97534 23769 97134 2546 96705 27144 96246 21 46 .22098 97528 23797 97127 25488 96697 27172 96238 21 47 j .22126 97521 23825 9712 272 9623 22 48 22155 97515 23853 97"3 25545 96682 27228 96222 22 49 .22183 975o8 23882 97106 25573 96675 27256 96214 23 50 .22212 .97502 2391 971 25601 96667 27284 96206 23 51 .2224 97496 23938 97093 25629 9666 27312 96198 23 52 . 22268 97489 23966 97086 25657 96653 2734 9619 24 53 22297 97483 23995 97079 25685 96645 27368 96182 24 54 .22325 97476 24023 97072 25713 96638 27396 96174 25 55 22353 9747 24051 97065 25741 9663 27424 96166 25 56 .22382 97463 24079 .97058 25769 96623 27452 96158 26 57 .2241 97457 24108 .97051 25798 2748 96i5 26 58 .22438 9745 24136 .97044 25826 96608 27508 96142 27 59 .22467 97444 24164 97037 25854 966 27536 96i34 27 60 .22495 97437 24192 9703 25882 96593 27564 96126 N. cos. N. sine. N. cos. | N. sine. N. cos. J N. sine. N. cos. N. sine. 770 76 75 74 1 394 NATURAL SINES AND COSINES. 16 170 18 igo ' N. sine. N. cos. N. sine. N. COB. N. sine. N. cos. N. sine. N. cos. o .27564 .96126 29237 9563 .30902 .95106 32557 94552 I .27592 .96118 . 29265 .95622 .30929 95097 32584 94542 2 .2762 .9611 29293 95613 30957 .95088 .32612 94533 3 .27648 .96102 29321 95605 30985 95079 .32639 94523 4 .27676 .96094 29348 95596 .31012 9507 . 32667 945H 5 .27704 .96086 .29376 .95588 .3104 .95061 .32694 .94504 6 27731 .96078 .29404 95579 .31068 .95052 .32722 94495 7 27759 .9607 .29432 95571 31095 95043 .32749 .94485 8 .27787 .96062 .2946 .95562 .3"23 95033 32777 .94476 9 .27815 .96054 .29487 95554 3 IJ 5i .95024 .32804 .94466 10 .27843 . 96046 29515 95545 .31178 95015 .32832 94457 ii .27871 .96037 29543 .95536 .31206 .95006 32859 94447 12 .27899 .96029 29571 95528 31233 94997 32887 94438 13 .27927 .96021 .29599 95519 .31261 .94988 .32914 .94428 14 27955 .96013 . 29626 955" .31289 94979 .32942 .94418 '5 27983 .96005 29654 95502 31316 9497 .32969 .94409 16 .28011 95997 29682 95493 31344 .94961 32997 94399 17 . 28039 .95989 .2971 95485 31372 94952 33024 9439 18 28067 9598i 29737 95476 31399 94943 .33051 9438 J 9 .28095 95972 29765 95467 3H27 94933 33079 9437 20 .28123 .95964 29793 95459 3H54 .94924 .33106 .94361 21 .2815 95956 29821 9545 .31482 949 J 5 33134 94351 22 .28178 .95948 29849 95441 3i5i .94906 33161 .94342 23 . 28206 9594 .29876 95433 31537 .94897 33i89 94332 2 4 .28234 95931 29904 95424 31565 .94888 .33216 94322 25 .28262 95923 29932 95415 31593 .94878 .33244 94313 26 .2829 959'S 2996 95407 .3162 .94869 33271 94303 2 7 .28318 9597 .29987 95398 .31648 .9486 33298 94293 28 .28346 .30015 .95389 31675 .94851 .33326 .94284 29 .28374 9589 .30043 | .9538 3i7 3 .94842 33353 .94274 30 . 28402 .95882 .30071 .95372 3i73 94832 3338i .94264 31 . 28429 95874 .30098 .95363 31758 .94823 33408 .94254 32 .28457 95865 .30126 | .95354 .31786 .94814 33436 94245 33 .28485 95857 .30154 -95345 31813 94805 33463 94235 34 28513 .95849 .30182 95337 .31841 94795 3349 94225 35 .28541 .95841 30209 .95328 .31868 .94786 33518 .94215 36 .28569 95832 30237 95319 .31896 94777 33545 . 94206 37 .28597 .95824 .30265 9531 3i9 2 3 .94768 33573 .94196 38 .28625 95816 .30292 95301 3i95i 94758 336 .94186 39 40 .28652 .2868 .95807 95799 3032 30348 95293 .95284 31979 .32006 94749 9474 33627 33655 .94176 .94167 4i .28708 9579 1 30376 95275 32034 9473 .33682 94157 42 .28736 95782 30403 .95266 .32061 94721 3371 .94147 43 .28764 95774 30431 95257 .32089 .94712 33737 94137 44 .28792 .95766 30459 .95248 .32116 .94702 33764 .94127 45 .2882 95757 . 30486 .9524 32144 .94693 33792 .94118 46 .28847 95749 .30514 .95231 .32171 94684 33819 .94108 47 28875 9574 30542 .95222 .32199 94674 33846 .94098 48 .28903 95732 3057 95213 .32227 .94665 33874 .94088 49 .28931 95724 30597 .95204 32254 .94656 339 01 .94078 50 28959 95715 .30625 95195 .32282 .94646 339 2 9 .94068 5i .28987 95707 30653 .95186 32309 94637 33956 .94058 52 29015 .95698 .3068 .95177 32337 94627 33983 .94049 53 .29042 9569 .30708 .95168 32364 .94618 .34011 .94039 54 .2907 .95681 30736 95159 32392 .94609 34038 .94029 55 .29098 .95673 30763 95i5 .32419 94599 34065 .94019 56 .29126 .95664 .30791 .95142 32447 9459 34093 .94009 57 .29154 95656 .30819 .95133 32474 9458 .3412 58 .29182 95647 .30846 .95124 32502 94571 34H7 . 93989 59 .29209 95639 30874 95"5 32529 .94561 34175 93979 60 .29237 9563 .30902 .95106 32557 94552 .34202 93969 N. cos. N. sine. N. cos. N. sine. N. cos. N. sine. N. cos. N. sine. 73 72 710 70 NATURAL SINES AND COSINES. 395 if 2( ) 21 21 jo 23 27 ' N. sine. N. cos. N. sine. N. cos. N. sine. N. eofl. N. sine. N. cos. O .34202 .93969 35837 93358 >374 oo .92718 3973 .9205 o I .34229 93959 35864 9334 s 37488 .92707 39* .92039 I 2 34257 93949 3589* 93337 375*5 .92697 39*27 .92028 I 3 34284 93939 359 l8 93327 37542 .92686 39*53 .92016 2 4 343" 93929 35945 37569 92675 39*8 .92005 2 5 34339 939*9 35973 .93306 37595 .92664 .39207 9*994 3 6 34366 36 93295 37622 92653 39234 .91982 3 7 34393 .93899 36027 93285 37649 . 92642 .3926 .91971 4 8 34421 .93889 36054 93274 37676 .92631 39287 9*959 4 9 .34448 93879 .36081 .93264 37703 .9262 393*4 .91948 5 10 34475 93869 .36108 93253 3773 .92609 3934* .91936 5 ii 34503 93859 36135 93243 37757 92598 39367 9*925 5 12 3453 .93849 .36162 93232 37784 .92587 39394 9*9*4 6 13 34557 93839 .3619 .93222 .37811 92576 .39421 .91902 6 14 34584 .93829 36217 .93211 37838 92565 .39448 .91891 7 15 34612 93819 36244 .93201 37865 92554 39474 .91879 7 1 6 34639 93809 36271 93*9 .37892 92543 39501 .91868 8 17 : .34666 93799 .36298 93*8 379*9 92532 39528 .91856 8 i 8 .34694 93789 36325 93*69 37946 92521 39555 .91845 9 '9 34721 93779 36352 93*59 37973 .9251 3958i 9 l8 33 9 20 34748 93769 36379 .93148 92499 .39608 .91822 9 21 -34775 93759 .36406 93*37 . 38026 .92488 .39635 .9181 10 22 34803 93748 36434 93*27 38053 .92477 .39661 9*799 10 23 3483 93738 .36461 93"6 .3808 .92466 .39688 9*787 ii 24 34857 93728 .36488 .93106 .38107 92455 397*5 9*775 ii 12 3 .34884 349*2 .93708 365*5 . 36542 93095 .93084 .38161 92444 .92432 3974* .39768 9*764 9*752 12 27 34939 36569 93074 .38188 .92421 39795 9*74* 13 28 34966 .93608 36596 .93063 38215 .9241 .39822 .91729 *3 29 34993 93677 36623 93052 .38241 .92399 .39848 .91718 30 35021 93667 3665 93042 .38268 .92388 39875 .91706 *4 3* 35048 93657 36677 9303* 38295 92377 .39902 .91694 *4 32 35075 93647 36704 9302 .38322 .92366 .39928 9*683 15 33 35*02 93637 3673* ,9301 38349 92355 39955 .91671 15 34 35*3 .93626 36758 .92999 38376 92343 .39982 .9166 16 35 35*57 .93616 36785 . 92988 38403 .92332 .40008 .91648 16 36 35*84 .93606 .36812 .92978 3843 92321 .40035 \ .91636 *7 37 352" 93596 36839 .92967 38456 .9231 .40062 \ .91625 38 35239 93585 . 36867 .92956 38483 .92299 .40088 .91613 18 39 35266 93575 36894 92945 385* .92287 .40115 .91601 18 18 40 4* 35293 3532 93565 93555 .36921 .36948 92935 .92924 38564 .92276 .92265 .40141 .40168 9*59 9*578 *9 42 35347 93544 36975 929*3 3859* .92254 40195 .91566 *9 43 35375 93534 37002 38617 .92243 .40221 9*555 20 44 .35402 93524 .92892 .38644 .92231 .40248 9*543 20 4 I 35429 935*4 . 37056 .92881 .38671 .9222 .40275 9*53* 41 46 35456 93503 . 37083 .9287 .38698 .92209 .40301 9*5*9 1 47 35484 93493 37" .92859 38725 .92198 . 40328 .91508 22 48 355" 93483 37*37 .92849 38752 .92186 40355 9*496 22 49 35538 93472 37*64 .92838 .38778 92*75 .40381 .91484 23 50 35565 .93462 37*9* .92827 38805 .92164 .40408 9*472 23 5* 35592 93452 37218 .92816 38832 .92152 .40434 .91461 23 52 35619 9344* 37245 92805 .38859 .92141 .40461 9*449 24 24 53 54 35647 35674 9343* 9342 37272 37299 .92794 92784 .38886 .38912 92*3 .92119 .40488 .40514 9*437 9*425 25 55 357* 934* 37326 92773 38939 .92107 .40541 9*4*4 25 56 35728 934 37353 .92762 .38966 .92096 40567 .91402 26 57 35755 93389 3738 9275* 38993 .92085 .40594 9*39 26 58 35782 93379 37407 9274 .3902 92073 .40621 9*378 27 59 93368 37434 .92729 39046 .92062 .40647 9*366 27 60 35837 93358 3746i .92718 3973 .9205 .40674 9*355 N. cos. N. sine. N. cos. N. sine. N. cos. N. sine. N. COB. N. Bine. 690 680 67 66 39 6 NATUKAL SINKS AND COSINES. ti 24 25 26 27 X| 0. 26 ' N. sine. N. cos. N. sine. N. cos. N. sine. N. cos. N. sine. N. cos. o o .40674 9*355 .42262 .90631 43837 .89879 45399 .89101 i .407 9 I 343 .42288 .90618 .43863 .89867 45425 .89087 I 2 .40727 9*33* 42315 .90606 .43889 .89854 45451 .89074 I 3 40753 9*3*9 .42341 9594 .43916 .89841 45477 .89061 a 4 .4078 -9 1 3<>7 .42367 .90582 .43942 .89828 45503 .89048 2 5 .40806 .91295 .42394 .90569 .43968 .89816 45529 89035 3 6 40833 .91283 .4242 90557 43994 .89803 45554 .89021 3 3 i .4086 .40886 .91272 .9126 .42446 42473 9545 90532 .4402 .44046 ,8979 .89777 45f .45606 ;fg*| 4 9 .40913 .91248 42499 .9052 .44072 .89764 45632 .88 9 8i 4 10 .40939 .91236 42525 .90507 .44098 .89752 45658 .88968 5 ii .40966 .91224 42552 .90495 .44124 89739 .45684 88955 12 .40992 .91212 .42578 .90483 44i5i .89726 457 1 .88942 6 13 .41019 .912 .42604 947 44177 89713 45736 .88928 6 H .41045 .91188 .42631 .90458 .44203 897 45762 .88915 7 IS .41072 .91176 42657 .90446 .44229 .89687 45787 .88902 7 16 .41098 .91164 .42683 9433 44255 .89674 458i3 .88888 7 17 .41125 .91152 .42709 .90421 .4428! .89662 45839 .88875 8 18 .41151 .9114 42736 .90408 44307 .89649 45865 .88862 8 i9 .41178 .91128 .42762 .90396 44333 .89636 .45891 .88848 9 20 .41204 .91116 .42788 90383 44359 .89623 459*7 88835 9 21 .41231 .91104 .42815 9 37i .44385 .8961 .45942 .88822 10 22 41257 .91092 .42841 90358 .44411 89597 45968 .88808 10 23 .41284 .9108 .42867 .90346 44437 i 89584 45994 .88795 10 24 4i3i .91068 .42894 90334 44464 89571 .4602 .88782 II II 11 41337 41363 .91056 .91044 .4292 .42946 .90321 90309 4449 .44516 .89558 89545 .46046 .46072 .88768 88755 12 12 11 4139 .41416 .91032 .9102 .42972 .42999 .90296 .90284 44542 .44568 89532 .89519 .46097 .46123 .88741 .88728 13 29 41443 .91008 .43025 90271 44594 .89506 .46149 .88715 13 30 .41469 .90996 43051 .90259 .4462 89493 46175 .88701 13 31 .41496 .90984 43077 .90246 .44646 .8948 46201 .88688 H 32 .41522 .90972 .43104 .90233 .44672 .89467 .46226 .88674 14 15 33 34 41549 41575 9096 .90948 4313 43156 .90221 .90208 .44698 .44724 .89454 .89441 46252 .46278 .88661 .88647 15 35 .41602 .90936 43182 .90196 4475 .89428 .46304 .88634 16 36 .41628 .90924 .43209 .90183 .44776 .89415 4633 .8862 16 37 41655 43235 .90171 .44802 .89402 46355 .88607 16 38 .41681 . 90899 43261 90158 .44828 .89389 .46381 88593 17 39 .41707 . 90887 .43287 .90146 .44854 .89376 .46407 .8858 17 40 41734 9 8 75 43313 9 OI 33 .4488 89363 46433 .88566 18 4i .4176 .90863 4334 .9012 .44906 8935 .46458 88553 18 42 .41787 .90851 43366 .90108 44932 89337 .46484 88539 J 9 43 .41813 .90839 43392 90095 .44958 .89324 .4651 .88526 ^9 20 44 45 .4184 .41866 . 90826 .90814 .43418 43445 90082 .9007 .44984 .4501 89311 .89298 .46536 .46561 .88512 .88499 20 46 .41892 .90802 43471 .90057 45036 .89285 .46587 .88485 2O 47 .41919 .9079 43497 .90045 .45062 .89272 .46613 .88472 21 48 41945 .90778 43523 .90032 .45088 .89259 .88458 21 49 .41972 .90766 43549 .90019 45"4 .89245 . 46664 .88445 22 50 .41998 9753 43575 4514 .89232 .4669 .88431 22 5i .42024 .90741 .43602 89994 .45166 .89219 .46716 .88417 23 52 42051 .90729 .43628 .89981 .45192 .89206 .46742 .88404 23 53 .42077 .90717 43654 .89968 452i8 .89193 46767 .8839 23 54 .42104 .90704 4368 .89956 45243 .8918 46793 88377 24 55 4213 .90692 .43706 .89943 .45269 .89167 .46819 .88363 24 56 .42156 .9068 43733 8993 45295 89153 .46844 .88349 25 57 .42183 .90668 43759 .89918 45321 .8914 .4687 .88336 25 58 .42209 90655 .43785 .89905 45347 .89127 .46896 .88322 26 26 59 60 42235 .42262 .90643 .90631 .43811 43837 i .89892 .89879 45373 45399 .89114 .89101 46921 46947 88308 88295 N. cot. N. sine. N. cot. N. sine. N. cos. N. tine. N. cos. N. tine. 65 64 630 62 NATURAL SINES AND COSINES. 397 2 BO 2 50 3 30 3] LO ' N. sine. N. cos. N. sine. N. cos. N. sine. N. cos. N. sine. N. COB. o .46947 .88295 .48481 .87462 5 .86603 51504 .85717 I 46973 .88281 .48506 .87448 .50025 .86588 51529 .85702 2 .46999 .88267 48532 87434 5005 86573 51554 .85687 3 .47024 .88254 .48557 .8742 .50076 86559 51579 .85672 4 4705 .8824 .48583 .87406 .50101 .86544 .51604 85657 5 .47076 .88226 .48608 87391 .50126 8653 .51628 .85642 6 .47101 .88213 .48634 .87377 50151 .86515 51653 .85627 7 .47127 .88199 48659 87363 50176 86501 .51678 .85612 8 47 I 53 .88185 .48684 87349 . 50201 .86486 51703 .85597 9 .47178 .88172 .4871 87335 .50227 .86471 .51728 .85582 10 .47204 .88158 .48735 87321 . 50252 .86457 51753 .85567 ii .47229 .88144 .48761 .87306 .50277 .86442 .51778 .85551 12 47255 .8813 .48786 .87292 .50302 .86427 .51803 .85536 13 .47281 .88117 .48811 .87278 50327 .86413 .51828 .85521 J 4 .88103 .48837 .87264 50352 .86398 51852 .85506 15 47332 .88089 .48862 .8725 50377 .86384 51877 .85491 16 47358 .88075 .48888 87235 .50403 .86369 51902 .85476 17 47383 .88062 .48913 .87221 . 50428 86354 51927 .85461 18 .47409 .88048 .48938 .87207 50453 8634 51952 .85446 '9 47434 .88034 .48964 87193 50478 86325 51977 .85431 20 4746 .8802 .48989 .87178 50503 86 3I .52002 .85416 21 .47486 .88006 .49014 .87164 .50528 86295 . 52026 .85401 22 475H 87993 .4904 8715 .50553 .8628! 52051 .85385 23 47537 .87979 .49065 .87136 50578 .86266 52076 8537 2 4 47562 .87965 499 .87121 .86251 .52101 .85355 25 .47588 87951 .49116 .87107 .50628 86237 .52126 8534 26 .47614 87937 .49141 87093 50654 .86222 52I5I 85325 11 47639 .47665 .87923 .87909 .49166 .49192 .87079 .87064 50679 .50704 .86207 .86192 52175 522 8531 .85294 2 9 .4769 .87896 .49217 .8705 50729 .86178 52225 85279 30 .47716 47741 .87882 .87868 . 49268 .87036 .87021 50754 50779 .86163 .86148 5225 52275 .85264 .85249 32 47767 87854 .49293 .87007 .50804 .86133 52299 85234 33 47793 .8784 .49318 .86993 .50829 .86119 52324 .85218 34 .47818 .87826 49344 .86978 50854 .86104 .52349 .85203 35 .47844 .87812 49369 .86964 .50879 .86089 .52374 .85188 36 .47869 .87798 49394 .86949 .50904 .86074 .52399 85173 37 47895 .87784 .49419 86935 .50929 .86059 52423 85157 38 .4792 .8777 49445 .86921 .50954 .86045 .52448 .85142 39 47946 87756 4947 .86906 50979 .8603 52473 .85127 40 4797 1 87743 49495 .86892 .51004 .86015 .52498 .85112 41 47997 .87729 .49521 .86878 .51029 .86 .52522 .85096 42 .48022 87715 49546 .86863 51054 85985 52547 .85081 43 .48048 .87701 49571 .86849 51079 8597 .85066 44 .48073 .87687 .49596 .86834 .51104 .85956 .52597 .85051 45 .48099 .87673 .49622 .8682 .51129 .85941 .52621 85035 46 .48124 .87659 49647 .86805 .85926 .52646 .8502 47 .4815 .87645 .49672 .86791 51179 .85911 52671 .85005 48 48i75 87631 .49697 .86777 .51204 .85896 .52696 .84989 49 48201 .8 7 6 I7 49723 .86762 .51229 .85881 5272 .84974 So .48226 .87603 .49748 .86748 51254 .85866 52745 84959 51 .48252 .87589 49773 86733 .51279 .85851 5277 84943 52 .48277 87575 .49798 .86719 51304 85836 52794 .84928 53 48303 .87561 .49824 .86704 51329 .85821 .52819 .84913 54 .48328 87546 .49849 .8669 .51354 .85806 52844 .84897 55 48354 87532 .49874 86675 51379 .85792 .52869 .84882 56 .48379 87518 .49899 .86661 .51404 85777 52893 .84866 u .48405 .4843^ .87504 .8749 .49924 4995 .86646 ,86632 .51429 51454 85762 85747 52918 52943 .84851 .84836 59 .48456 .87476 49975 .86617 5H79 85732 52967 .8482 60 .48481 .87462 5 .86603 51504 85717 52992 .84805 N. cos. N. sine. N. cos. N. sine. N. cos. N. sine. N. cos. N. Bin*. 6 1 6 5 JO 51 P L L 398 NATURAL SINES AND COSINES. 35 ,o 33 o 34 t 35 o ' N. sine. N. cos. N.sine. N. cos. N. sine. N. cos. N. sine. N. cos. o 52992 .84805 .54464 .83867 559 J 9 .82904 .57358 .81915 I 53017 .84789 .54488 .83851 . 82887 .57381 .81899 2 84774 54513 83835 . 5598 .82871 57405 .81882 3 ^ 53066 84759 54537 .83819 55992 82855 57429 .81865 4 53091 84743 5456i .83804 .56016 .82839 57453 .81848 5 53"5 .84728 54586 83788 .5604 .82822 57477 .81832 6 5314 .8 47I2 .5461 .83772 .56064 .82806 57501 .81815 7 53164 .84697 54635 .83756 .56088 .8279 57524 .81798 8 53189 .84681 54659 8374 .56112 .82773 57548 .81782 9 10 53214 53238 .84666 8465 .54683 .54708 .83724 .83708 .56136 .5616 .82757 .82741 57572 57596 81765 .81748 ii 12 53263 .53288 84635 .84619 54732 54756 .83692 .83676 56184 .56208 .82724 .82708 -57619 .57643 .81731 .81714 13 53312 .84604 5478i .8366 56232 .82692 57667 .81698 14 53337 .84588 .54805 83645 .56256 82675 57691 .81681 15 5336i .84573 .54829 .83629 .5628 .82659 57715 .81664 16 53386 84557 .54854 .83613 56305 .82643 .57738 .81647 !7 534" 84542 .54878 .83597 56329 .82626 .57762 .81631 18 53435 .84526 .54902 .83581 56353 .8261 .57786 .81614 19 5346 84511 .549 2 7 .83565 56377 82593 .5781 8l597 20 53484 84495 .54951 .83549 .56401 .82577 57833 .8158 21 53509 .8448 54975 .83533 56425 .82561 .57857 81563 22 53534 .84464 54999 .83517 56449 .82544 57881 .81546 23 53558 .84448 .55024 .83501 56473 .82528 57904 .8i53 24 53583 .84433 .55048 83485 56497 .82511 .57928 81513 25 53607 .84417 55072 .83469 56521 .82495 57952 .81496 26 53632 .84402 .55097 .83453 56545 .82478 57976 .81479 2 7 53656 .84386 55121 83437 56569 .82462 57999 .81462 28 53681 .8437 55145 .83421 56593 .82446 .58023 8i445 2 9 53705 84355 55169 83405 .56617 .82429 .58047 .81428 30 5373 84339 55194 83389 .56641 82413 5807 .81412 31 53754 .84324 55218 .83373 . 56665 .82396 .58094 .81395 32 53779 .84308 55242 .83356 .56689 .8238 .58118 81378 33 53804 .84292 55266 .8334 56713 .82363 .58141 .81361 34 53828 .84277 55291 .83324 56736 .82347 .58165 .81344 35 53853 .84261 .83308 .0676 8233 .58189 .81327 36 .53877 .84245 55339 .83292 .56784 .82314 .58212 .8131 37 53902 .8423 55363 83276 .56808 .82297 58236 .81293 38 53926 .8 42I4 55388 .8326 .56832 .82281 .5826 .81276 39 53951 .84198 55412 .83244 .56856 .82264 .58283 .81259 40 53975 .84182 .55436 .83228 .5688 .82248 .58307 .81242 54 .84167 .5546 .83212 .56904 .82231 5833 .81225 42 54024 .84151 55484 .83195 .56928 .82214 .58354 .81208 43 .54049 84135 55509 .83179 56952 .82198 58378 .81191 44 54073 .8412 55533 .83163 56976 .82181 .58401 .81174 45 54097 .84104 55557 .83147 57 .82165 58425 .81157 46 .54122 .84088 5558i 83131 57024 .82148 . 58449 .8114 47 .54146 .84072 55605 83115 57047 .82132 .58472 .81123 48 54 I 7 I .84057 .5563 .83098 -57071 .82115 .58496 .81106 49 54 I 95 .84041 55654 .83082 57095 .82098 58519 .81089 50 .5422 .84025 55678 .83066 57"9 .82082 .58543 .81072 Si .54244 .84009 55702 8305 .57143 .82065 58567 81055 52 . 54269 .83994 55726 83034 57167 .82048 5859 .81038 53 .54293 .83978 5575 83017 57191 .82032 586,4 .81021 54 54317 83962 55775 .83001 57215 .82015 58637 .81004 55 54342 .83946 55799 .82985 57238 .81999 .58661 .80987 56 54366 8393 55823 . 82969 57262 .81982 .58684 .8097 57 5439 1 83915 55847 82953 .57286 81965 .58708 80953 58 544 I 5 .83899 55871 .82936 5731 .81949 58731 .80936 59 5444 83883 55895 .8292 57334 81932 58755 .80919 60 54464 .83867 559*9 .82904 57358 .81915 58779 .80902 N. cos. N. sine. N. cos. N. sine. N cos N. sine. N. cos. N sine 5 70 Si 5 5 5 5< 1 NATURAL SINES AND COSINES. 399 360 370 380 390 ' N. sine. N. cos. N. sine. N. cos. N. sine. N. cos. N. sine. N. cos. 58779 .80902 .60182 .79864 .61566 .78801 -.62932 77715 j I .58802 .80885 .60205 .79846 .61589 .78783 62955 77696 2 .58826 .80867 .60228 .79829 .61612 .78765 .62977 .77678 3 .58849 .8085 .60251 .79811 61635 .78747 .63 .7766 4 58873 .80833 .60274 79793 .61658 .78729 .63022 .77641 5 .58896 .80816 .60298 .79776 .61681 .78711 63045 77623 6 .5892 .80799 .60321 79758 .61704 .78694 .63068 .77605 7 58943 .80782 60344 79741 .61726 .78676 .6309 77586 8 58967 .80765 .60367 79723 .61749 .78658 .63113 77568 X 9 5899 .80748 .6039 797o6 .61772 .7864 63135 7755 10 .59014 8073 .60414 .79688 6i795 .78622 .63158 77531 ii 5937 80713 60437 .79671 .61818 .78604 .6318 77513 12 .59061 .80696 .6046 79653 .61841 78586 .63203 77494 13 .59084 .80679 .60483 79635 .61864 .78568 63225 .77476 .59108 .80662 .60506 .79618 .61887 7855 .63248 77458 15 .80644 60529 .796 .61909 78532 .63271 77439 16 59 I 54 .80627 60553 79583 61932 78514 .63293 .77421 '7 .59178 .8061 .60576 79565 61955 .78496 .63316 .77402 18 .59 201 80593 .60599 79547 .61978 .78478 .63338 77384 19 .59225 80576 .60622 7953 62001 .7846 .63361 77366 20 . 59248 80558 .60645 79512 .62024 78442 .63383 77347 21 .59272 80541 .60668 79494 .62046 .78424 .63406 .77329 22 .59295 .80524 .60691 79477 .62069 .78405 63428 -773I 23 .80507 .60714 79459 .62092 .78387 63451 .77292 24 59342 . 80489 60738 .79441 .62115 .78369 63473 .77273 25 59365 .80472 .60761 .79424 .62138 78351 63496 .77255 26 59389 80455 .60784 .79406 .6216 78333 .63518 77236 3 .59412 59436 .80438 .8042 .60807 .6083 .79388 79371 .62183 .62206 78315 .78297 6354 63563 .77218 .77199 2 9 59459 .80403 60853 79353 .62229 .78279 63585 .77181 30 .59482 .80386 .60876 79335 .62251 .78261 .63608 .77162 595o6 .80368 .60899 .62274 .78243 6363 77*44 32 59529 80351 .60922 793 .62297 .78225 63653 77125 33 59552 80334 60945 .79282 .6232 .78206 63675 .77107 34 59576 .80316 60968 .79264 62342 .78188 .63698 .77088 35 59599 .80299 .60991 .79247 .62365 .7817 6372 .7707 36 .59622 .80282 .61015 .79229 .62388 78152 63742 77051 37 .59646 .80264 .61038 .79211 .62411 78134 63765 .77033 38 .59669 .80247 .61061 79 J 93 62433 78116 63787 .77014 39 59693 .8023 .61084 .79176 .62456 .78098 6381 .76996 40 597i6 .80212 .61107 79158 .62479 .78079 63832 76977 41 59739 .80195 .6113 .7914 .62502 .78061 63854 76959 42 .59763 .80178 6u53 .79122 .62524 .78043 .63877 7694 43 .59786 .8016 .61176 79 I0 5 62547 .78025 .63899 .76921 44 .59809 .80143 .61199 .79087 .6257 .78007 .63922 76903 45 59832 .80125 .61222 .79069 62592 .77988 63944 .76884 46 59856 .80108 .61245 79051 .62615 7797 .63966 .76866 47 59879 .80091 .61268 79033 .62638 77952 .63989 .76847 48 .59902 .80073 .61291 .79016 .6266 77934 .64011 .76828 49 50 .59926 .80056 .59949 ! .80038 61314 61337 .78998 .7898 .62683 .62706 .77916 .77897 64033 -64056 . 7 68x 76791 Si .59972 : .80021 6136 .78962 .62728 .77879 .64078 .76772 52 59995 .80003 61383 78944 .62751 .77861 .641 76754 53 .60019 -79986 61406 .78926 .62774 77843 .64123 76735 54 .60042 .79968 .61429 .78908 .62796 .77824 64145 1 .76717 55 60065 .79951 .61451 .78891 .62819 .77806 .64167 .76698 56 60089 .79934 .61474 .78873 62842 .77788 .6419 .76679 57 .60112 .79916 .61497 78855 .62864 .77769 64212 .76661 58 .60135 .79899 .6152 .78837 .62887 .77751 64234 .76642 59 .60158 .79881 6i543 78819 .62909 .77733 64256 .76623 60 .60182 .79864 .61566 .78801 .62932 777*5 .64279 .76604 N. cos. N. sine. N. cos. N.sine. N. cos. N. sine. N. cos. N. sine. 53 I 52 I 61 60 i 400 NATURAL SINES AND COSINES. si 4 3 4 L 4 go 4 3 22 N. sine. N. cos. N. sine. N. cos. N. sine. N. cos. N. sine. N. cos. O .64279 .76604 .65606 75471 669x3 743*4 .682 73135 O I .64301 76586 .65628 75452 .66935 74295 .68221 .73116 2 64323 .76567 6565 75433 .66956 74276 .68242 .73096 3 76548 .65672 75414 .66978 74256 .68264 .73076 4 . 64368 7653 65694 75395 .66999 74237 .68285 73056 5 6439 .76511 65716 75375 .67021 .74217 .68306 73036 6 .64412 '.76492 65738 75356 .67043 74198 68327 .73016 7 64435 76473 65759 75337 .67064 .74178 68349 .72996 3 8 64457 76455 65781 753i8 .67086 74 J 59 6837 .72976 3 9 64479 76436 65803 75299 .67107 74139 .68391 72957 4 10 .64501 .76417 .65825 .7528 .67129 .7412 .68412 72937 4 ii .64524 .76398 .65847 .75261 .67151 .741 .68434 .72917 4 5 12 13 .64546 .64568 .7638 .76361 .65869 .65891 75241 75222 .67172 .67194 .7408 .74061 68455 .68476 .72897 .72877 5 14 6459 76342 .75203 .67215 .74041 .68497 .72857 6 15 .64612 76323 65935 75184 .67237 . 74022 .68518 .72837 6 16 .64635 .76304 65956 75165 .67258 .74002 68539 .72817 6 17 .64657 .76286 .65978 75146 .6728 73983 .68561 72797 7 18 .64679 .76267 .66 .75126 67301 73963 .68582 .72777 7 19 .64701 .76248 .66022 75107 67323 73944 .68603 72757 7 20 .64723 .76229 .66044 .75088 67344 73924 .68624 72737 8 21 .64746 .7621 .66066 .75069 .67366 7394 .68645 .72717 8 22 .64768 .76192 .66088 7505 .67387 73885 .68666 . 72697 8 23 6479 76173 .66109 7503 .67409 73865 .68688 72677 9 2 4 .64812 .66131 .75011 6743 73846 .68709 .72657 9 25 .64834 .76135 66153 .74992 .67452 .73826 6873 .72637 10 26 .64856 .76116 .66175 74973 67473 .73806 .68751 .72617 10 2 7 .64878 .76097 .66197 74953 67495 73787 .68772 72597 xo 28 .64901 .76078 .66218 74934 67516 73767 68793 72577 II 29 64923 .76059 .6624 67538 73747 .68814 72557 II 30 .64945 . 76041 .66262 .74896 67559 73728 68835 72537 II 31 .64967 . 76022 .66284 .74876 .6758 .73708 68857 72517 12 32 .64989 .76003 .66306 74857 .67602 .73688 .68878 72497 12 33 .65011 75984 66327 .74838 67623 73669 .68899 72477 12 34 65033 75965 .66349 .74818 .67645 73649 .6892 72457 13 35 65055 75946 .66371 74799 .67666 .73629 .68941 72437 13 36 .65077 75927 66393 .7478 .67688 .68962 .72417 14 37 .651 .664x4 7476 .67709 7359 .68983 72397 14 38 .65122 .75889 .66436 7474i 6773 7357 .69004 72377 14 39 .65144 7587 .66458 74722 .67752 73551 .69025 72357 IS 40 .65166 75851 .6648 7473 67773 73531 .69046 72337 15 4 1 .65188 75832 .66501 74683 67795 735" . 69067 72317 15 42 6521 .66523 .74664 .67816 7349 1 . 69088 .72297 16 43 65232 75794 66545 74644 .67837 73472 .69109 .72277 16 44 65254 75775 .66566 .74625 67859 73452 .6913 72257 17 45 .65276 75756 .66588 .74606 .6788 73432 69151 .72236 17 46 65298 75738 .6661 .74586 .67901 73413 .69172 .72216 17 47 6532 75719 .66632 74567 .67923 73393 .69193 .72196 18 48 .65342 757 .66653 74548 67944 73373 .69214 .72176 18 49 65364 .7568 66675 .74528 67965 73353 69235 72156 18 50 .65386 75661 .66697 74509 .67987 73333 .69256 .72136 19 51 .65408 75642 .66718 .74489 .68008 73314 .69277 .72116 19 52 6543 75623 .6674 7447 .68029 73294 .69298 72095 19 53 65452 .75604 .66762 74451 .68051 73274 .69319 72075 20 54 65474 75585 66783 7443 1 .68072 73254 6934 72055 2O 55 65496 75566 .66805 .74412 .68093 7323'. .69361 72035 21 56 65518 75547 .66827 7439 2 .68115 73215 .69382 .72015 21 57 6554 75528 .66848 74373 .68136 73195 69403 71995 21 58 .65562 75509 .6687 74353 68157 73175 .69424 71974 22 59 65584 7549 .66891 74334 68179 73155 . 69445 71954 22 60 .65606 7547 1 66913 74314 .682 73135 .69466 71934 N. cos. N.sine. N. cos. N. sine. N. cos. N. sine. N. cos. N. sin*. 4< > 4i ti 4' 1 4C > NATUKAL SINES AND COSINES. 401 n ii " , 44 N. sine. 1 N. cog. 3 &l 19 it 22 , 44 N. sine. 1 N. cos. if 9 o o .69466 .71934 60 19 II 31 .70112 71305 2 9 9 o I I 2 .69487 .69508 .71914 .71894 59 58 19 18 12 12 32 33 .70132 70153 .71284 .71264 28 27 9 9 I 3 .69529 71873 57 18 12 34 .70174 71243 26 8 I 4 .69549 71853 56 18 13 35 .70195 .71223 25 8 2 5 6957 71833 55 17 13 36 .70215 .71203 24 8 2 6 .69591 .71813 54 17 14 37 .70236 .71182 23 7 3 7 .69612 .71792 53 i7 14 38 .70257 .71162 22 7 3 8 .69633 .71772 52 16 14 39 .70277 .71141 21 7 3 9 -69654 71752 51 16 15 40 .70298 .71121 20 6 4 10 69675 71732 So 16 15 4i 70319 .711 19 6 4 ii .69696 .71711 49 16 15 42 70339 .7108 18 6 4 12 .69717 .71691 48 15 16 43 7036 -71059 17 5' 5 13 69737 .71671 47 15 16 44 .70381 7 I0 39 16 5 5 14 .69758 7165 46 15 17 45 .70401 .71019 IS 5 6 15 .69779 7163 45 14 17 46 .70422 .70998 14 4 6 16 .698 .7161 4* *4 '7 47 70443 .70978 13 4 6 7 ;? .69821 .69842 7159 71569 43 42 14 13 18 18 48 49 70463 .70484 70957 70937 12 II 4 3 7 19 .69862 7 I 549 4i 13 18 So 70505 .70916 10 3 7 20 .69883 71529 4 13 J 9 Si .70525 .70896 9 3 8 21 .69904 .71508 39 12 J 9 52 .70546 .70875 8 3 8 22 .69925 .71488 38 12 X 9 53 .70567 70855 7 2 8 23 .69946 .71468 37 12 20 54 .70587 .70834 6 2 9 24 .69966 7H47 36 II 20 55 .70608 .70813 5 2 9 25 .69987 .71427 35 II 21 56 .70628 70793 4 I o 26 .70008 .71407 34 II 21 57 .70649 .70772 3 I o 2 7 .70029 71386 33 10 21 58 7067 .70752 2 I o 28 .70049 .71366 32 10 22 59 .7069 70731 I I 29 .7007 7 I 345 3i IO 22 60 70711 .70711 O O I 30 .70091 71325 30 10 N. cos. N. sine. ' N.C08. N. sine. ~~' 4i > 4 5 Preceding Table contains Natural Sine and Cosine for every minute of the Quadrant to Radius i. If Degrees are taken at head of columns, Minutes, Sine, and Cosine must be taken from head also ; and if they are taken at foot of column, Minutes, etc., must be taken from foot also. ILLUSTRATION .3173 is sine of 18 30', and cosine of 71 30'. To Compute Sine or Cosine for Seconds. When Angle is less than 45. RULE. Ascertain sine or cosine of angle for degrees and minutes from Table; take difference between it and sine or cosine cf angle next below it. Look for this difference or remainder,* if Sine is required, at head of column of Proportional Parts, on left side ; and if Cosine is required, at head of column on right side; and in these respective columns, opposite to number of seconds of angle in column, is number or correction in seconds to be added to Sine, or subtracted from Cosine of angle. ILLUSTRATION i. What is sine of 8 9' 10"? Sine of 8 9', per Table = .141 77;) ,.~ ^ Sine of 80 ,o', =.14205!} - 00028 *ff*<*> In left side column of proportional parts, under 28, and opposite to 10', is 5, cor- rection for 10', which, being added to .141 77 = .141 82 Sine. * Tke table in some instances will give a unit too much, but this, in general, is of little importance, L L* 4O2 NATURAL SINES AND COSINES. 2. What is cosine of 8 9' 10"? In right-side column of proportional parts, under 4, and opposite to 10', is i, tho correction for 10', which, being subtracted from .9899098989 cosine. When Angle exceeds 45. RULE. Ascertain sine or cosine for angle in degrees and minutes from Table, taking degrees at the foot of it ; then take difference between it and sine or cosine of angle next above it. Look for re- mainder, if Sine is required, at head of column of Proportional Parts, on right side ; and if Cosine is required, at head of column on left side ; and in these respective columns, opposite to seconds of angle, is number or correction in seconds to be added to Sine, or subtracted from Cosine of angle. ILLUSTRATION. What is the Sine and Cosine of 81 50' 50"? 1SS SIS ; ""P" 8 ::^'} ** *&*. In right-side column of proportional parts, and opposite to 50', is 3, which, added to . 989 86 = . 989 89 Sine. In left-side column of proportional parts, and opposite to 50', is 24, which, sub- tracted from . 142 05 = . 141 81 Cosine. To Ascertain, or Compute 3S"um"ber of Degrees, Minutes, and Seconds of* a given Sine or Cosine. When Sine is given. RULE. If given sine is in Table, the degrees of it will be at top or bottom of page, and minutes in marginal column, at left or right side, according as sine corresponds to an angle less or greater than 45. If given sine is not in Table, take sine in Table which is next less than the one for which degrees, etc., are required, and note degrees, etc., for it. Sub- tract this sine from next greater tabular sine, and also from given sine. Then, as tabular difference is to difference between given sine and tabu- lar sine, so is 60 seconds to seconds for sine given. EXAMPLE. What are the degrees, minutes, and seconds for sine of .75? Next less sine is .74992, arc for which is 48 35'. Next greater sine is .75011, difference between which and next less is .75011 .749 92 = .00019. Difference be- tween less tabular sine and one given is . 75 . 749 92 = 8. Then 19 : 8 :: 60 : 25+, which, added to 48 35' = 48 35' 25". When Cosine is given. RULE. If given cosine is found in Table, degrees of it will be found as in manner specified when sine is given. If given cosine is not in Table, take cosine in Table which is next greater than one for which degrees, etc., are required, and note degrees, etc., for it. Subtract this cosine from next less tabular cosine, and also from given cosine. Then, as tabular difference is to difference between given cosine and tabu- lar cosine, so is 60 seconds to seconds for cosine given. EXAMPLE. What are the degrees, minutes, and seconds for cosine of .75? Next greater cosine is .750 u, arc for which is 41 24'. Next less cosine is .749 92, difference between which and next greater is .750 u -74992 = .000 19. Difference between greater tabular cosine and one given is .750 n .75000= u. Then 19 : n : : 60 : 35 , which, added to 41 24' = 41 24' 35". To Compnte Versed Sine of an Angle. Subtract cosine of angle from i. ILLUSTRATION. What is the versed sine of 21 30'? Cosine of 21 30' is .930 42, which, i = .069 58 versed sine. To Compnte Co-versed Sine of an Angle. Subtract sine of angle from i. ILLUSTRATION. What is the co-versed sine of 21 30'? The sine of 21 30' is .3665, which, i = .6335 co-versed sine. NATUKAL SECANTS AND CO-SECANTS. 403 N"atnral Secants and Co -secants. 0< > .. l c 2 C 11 V > CO-SKC'T. Infinite. 1. 0001 57-299 i. 0006 28.654 i 1.0014 19.107 I i 3437-7 .0001 6-359 .0006 8.417 i .0014 9.002 2 j 1718.9 .0002 5-45 .0606 8.184 .0014 8.897 3 i 145-9 .0002 4-57 .0006 7-955 .0014 8.794 4 859.44 .0002 3.718 .0006 7-73 .0014 8.692 5 687-55 1.0002 52.891 1.0007 27-508 1.0014 18.591 6 572-96 .OOO2 2.09 .0007 7.29 .0015 8.491 7 491.11 .0002 1.313 .0007 7-075 .0015 8-393 s 29.72 .0002 0.558 .0007 6.864 .0015 8.295 9. .OOO2 49.826 .0007 6.655 .0015 8.198 10 343-77 1.0002 49.114 1.0007 26.45 1.0015 18.103 rl 12.52 .OOO2 8.422 .0007 6.249 .0015 8.008 12 13 286.48 64.44 45-55 .0002 .0002 0002 7-75 7.096 6.46 .0007 .0007 .0008 6.05 5-854 5.661 .0016 .0016 .0016 7.914 7.821 7-73 j - 229.18 1.0002 45.84 1.0008 25.471 1.0016 17-639 ii 14.86 .OOO2 5-237 .0008 5-284 .0016 7-549 17 02.22 .0002 4-65 .0008 .0016 7-46 18 190.99 .OOO2 4.077 .0008 4 .' 9 i8 .0017 7-372 J 9 80.73 .0003 3-52 .0008 4-739 .0017 7.285 20 171.89 1.0003 42.976 1.0008 24.562 1.0017 17.198 21 63-7 .0003 2-445 .0008 4.358 .0017 7-"3 22 56.26 .0003 1.928 .0008 4.216 .0017 7.028 23 49-47 .OOO3 1.423 .0009 4.047 .0017 6-944 2 4 43-24 .0003 40-93 .0009 3.88 .0018 6.861 25 I37-5I 1.0003 40.448 1.0009 23.716 1.0018 16.779 26 32.22 .0003 39.978 .0009 3-553 .0018 6.698 2 7 28 27.32 22.78 .0003 .OOO3 9.518 9.069 .0009 .0009 3-393 3-235 .0018 .0018 6.617 6.538 29 18.54 .0003 8.631 .0009 3-079 .0018 6-459 30 "4-59 1.0003 38.201 1.0009 22.925 1.0019 16.38 31 10.9 .0003 7.782 .001 2.774 .0019 6.303 32 07-43 .0003 7-371 .001 2.624 .0019 6.226 33 04.17 .0004 6.969 .001 2.476 .0019 6.15 34 OI.II .0004 6.576 .001 233 .0019 6-075 35 98.223 1.0004 36- 19 1 I.OOI 22.186 1.0019 16 36 5-495 .0004 5.814 .001 2.044 .002 5.926 37 2.914 .0004 5-445 .001 1.904 .002 5-853 38 .0001 2.469 .OOO4 5-084 .001 1-765 .OO2 578 39 .0001 88.149 .0004 4-729 .OOII 1.629 .002 5-708 40 1. 0001 85.946 1.0004 I.OOII 21.494 I.OO2 15.637 41 .0001 3.849 .0004 4.042 .OOII 1.36 .0021 5.566 42 .0001 1-853 .0004 3.708 .OOII 1.228 .0021 5-496 43 .0001 7995 .0004 3-38i .OOII 1.098 .OO2I 5-427 44 .0001 8.133 .0004 3-o6 .OOII 20.97 .0021 5.358 45 1. 0001 76-396 I.OOO5 32-745 I.OOII 20.843 I.OO2I 15.29 46 .0001 4-736 .0005 2-437 .0012 0.717 .0022 5-222 .0001 3.146 .0005 2.134 [ .0012 0.593 .OO22 5-155 48 .0001 1.622 .0005 1.836 .0012 0.471 .0022 5-089 49 .0001 1.16 .0005 1-544 .0012 o.35 .0022 5-023 1. 0001 68.757 I.OOO5 3I-257 I.OOI2 20.23 1.0022 14.958 51 .0001 7-409 .0005 30. 976 .0012 0. 112 .0023 4-893 52 0001 6.113 .OOO5 0.699 .OOI2 19.995 .0023 4.829 53 .0001 4.866 .0005 0.428 .0013 9-88 .0023 4.765 54 .0001 3.664 OOO5 o. 161 .0013 9.766 .0023 4.702 55 1. 0001 62.507 1.0005 29.899 I.OOI3 1.0023 14.64 56 .0001 I-39 1 .0006 9.641 .0013 9-54 1 .0024 4578 57 .0001 .OOO6 9.388 .OOI3 943' .OO24 4-5I7 58 .0001 59- 274 .0006 9- r 39 .0013 9.322 .0024 59 .0001 8.27 .0006 8.894 .0013 9.214 .OO24 4-395 60 1. 0001 57-299 I. OOO6 28.654 I.OOI4 19.107 1.0024 14-335 / CO-SKC'T SECANT. CO-SEC 'T SECANT CO-SKC'T SECANT. CO-SEC'T SECANT. 890 88 87 86 404 NATURAL SECANTS AND CO-SECANTS. 4 1 50 6 70 SECANT. CO-SIC'T. SKCANT. CO-SKC'T. SKCANT. CO-SKC'T. SXCANT. | CO-SKC'T. 1.0024 .0025 14-335 4.276 1.0038 .0038 i .474 436 1.0055 0055 9.5668 5404 1.0075 .0075 8.2055 .1861 .0025 4.217 .0039 .398 .0056 SHI .0076 .1668 .0025 4.159 .0039 36 .0056 .488 .0076 .1476 .0025 4.101 .0039 323 .0056 .462 .0076 .1285 1.0025 14.043 1.0039 i .286 1.0057 9.4362 1.0077 8.1094 .0026 3-986 .004 .249 .0057 .4105 .0077 .0905 .0026 3-93 .004 .213 0057 .385 .0078 .0717 .0026 3-874 .004 .176 .0057 3596 .0078 .0529 .0026 3.818 .004 .14 .0058 3343 .0078 .0342 1.0026 13-763 1.0041 i .104 1.0058 9.3092 1.0079 8.0156 .0027 3.708 .0041 .069 .0058 .2842 .0079 7.9971 .0027 3' 6 54 .0041 033 .0059 2593 .0079 .9787 .0027 3.6 .0041 0.988 0059 2346 .008 .9604 .0027 3-547 .0042 0.963 .0059 .21 .008 .9421 1.0027 13-494 1.0042 10.929 1.006 9-I855 i. 008 7.924 .0028 3-441 .0042 0.894 .006 .l6l2 -0081 9059 .0028 3-389 .0043 0.86 .006 137 .0081 .8879 .0028 3-337 .0043 0.826 .0061 .1129 .0082 .87 .0028 3.286 .0043 0.792 .0061 .089 .0082 .8522 1.0029 13-235 1.0043 10.758 1.0061 9.0651 1.0082 7-8344 .0029 3.184 .0044 0.725 .0062 .0414 .0083 .8168 .0029 3-*34 .0044 0.692 .0062 .0179 .0083 .7992 .0029 0029 3-084 3-034 .0044 .0044 -659 0.626 .0062 .0063 8.9944 -97II .0084 .0084 .7817 .7642 1.003 12.985 1.0045 10.593 1.0063 8-9479 1.0084 7.7469 .003 2-937 .0045 0.561 .0063 .9248 .0085 .7296 .003 2.888 .0045 0.529 .0064 .9018 .0085 .7124 .003 2.84 .0046 0.497 .0064 .879 .0085 6953 .0031 z-793 .0046 0.465 .0064 .8563 .0086 .6783 1.0031 12.745 1.0046 10-433 1.0065 8-8337 1.0086 7.6613 .0031 2.698 .0046 0.402 .0065 .8112 .0087 .6444 .0031 2.652 .0047 0.371 .0065 .7888 .0087 .6276 .0032 2.606 .0047 o-34 .0066 .7665 .0087 .6108 .0032 2.56 .0047 0.309 .0066 7444 .0088 594 2 1.0032 12.514 1.0048 10.278 1.0066 8.7223 1.0088 7-5776 .0032 2.469 .0048 0.248 .0067 .7004 .0089 .5611 .0032 2.424 .0048 0.217 .0067 .6786 .0089 5446 0033 2-379 .0048 0.187 .0067 .6569 .0089 .5282 .0033 2-335 .0049 o.i57 .0068 6353 .009 5"9 1.0033 12.291 1.0049 10.127 i. 0068 8.6138 1.009 7-4957 0033 2.248 .0049 0.098 .0068 5924 .009 -4795 .0034 2.204 .005 0.068 .0069 57" .0091 4634 .0034 2.161 .005 0.039 .0069 5499 .0091 4474 .0034 2.118 .005 0.01 .0069 .5289 .0092 43*5 1.0034 12.076 1.005 9.9812 1.007 8.5079 1.0092 7-4156 0035 2.034 0051 9525 .007 .4871 .0092 .3998 0035 1.992 .0051 .9239 .007 4663 .0093 .384 .0035 i-95 .0051 8955 .0071 4457 0093 3683 .0035 1.909 .0052 .8672 .0071 .4251 .0094 3527 1.0036 n.868 1.0052 9.8391 1.0071 8.4046 1.0094 7-3372 .0036 1.828 .0052 .8112 .0072 3843 .0094 3217 .0036 1.787 0053 -7834 .0072 .3640 .0095 .3063 .0036 1-747 0053 .7558 .0073 3439 .0095 .2909 .0037 1.707 0053 .7283 .0073 3238 .0096 2757 1-0037 n.668 1-0053 9.701 1.0073 8-3039 1.0096 7.2604 .0037 1.628 0054 6739 .0074 .2840 .0097 2453 .0037 1.589 .0054 .6469 .0074 .2642 .0097 .2302 .0038 '55 0054 .62 .0074 .2446 .0097 .2152 .0038 1.512 0055 -5933 .0075 .225 .0098 .2002 1.0038 11.474 1.0055 9.5668 1.0075 8.2055 .0098 7-1853 CO-8KC'T. SKCANT. CO-BKC'T. SKCANT. CO-SKC'T. SKCANT. CO-BEC'T. SECANT. 85 84 83 82 NATURAL SECANTS AND CO-SECANTS. 405 8 9 c II )0 13 L SKCANT. CO-SKC'T. SKCANT. CO-SKC'T. SKCANT. CO-SKC'T. SKCANT. Co-SBC'T. ' 1.0098 7-^853 1.0125 6.3924 1.0154 5.7588 1.0187 5.2408 60 .0099 .0099 .1704 .1557 .0125 .0125 .3807 369 0155 0155 7493 7398 .0188 .0188 233 .2252 ti .0099 .01 .1409 1263 .0126 .0126 3574 3458 .0156 .0156 7304 .721 .0189 .0189 .2174 .2097 57 56 1. 01 7.1117 1.0127 6-3343 1.0157 5>7"7 1.019 5.2019 55 .0101 .0972 .0127 .3228 0157 7023 .0191 .1942 54 .0101 .0827 .0128 .3113 .0158 693 .0191 .1865 53 .0102 .0683 .0128 .2999 .0158 .6838 .0192 .1788 52 .0102 0539 .0129 .2885 .0159 6745 .0192 .1712 51 I.OIO2 7.0396 1.0129 6. 2772 1-0159 5-6653 1.0193 5.1636 50 .OIO3 .0103 .0254 .0112 013 .013 .2659 2546 .016 .016 .6561 .647 .0193 .0194 .156 .1484 49 43 .OIO4 6.9971 .0131 2434 .0161 6379 .0195 .1409 47 .0104 983 .0131 .2322 .0162 .6288 .0195 '333 46 I.OI04 6.969 1.0132 6.22II 1.0162 5.6197 1.0196 5-1258 45 0105 955 .0132 .21 .0163 .6107 .0196 .1183 44 .0105 94" 0133 .199 .0163 .6017 .0197 .1109 43 .OIO6 9273 0133 .188 .0164 .5928 .0198 .1034 42 .OI06 9135 .0134 .177 .0164 5838 .0198 .096 I.OIO7 6.8998 1.0134 6.1661 1.0165 5-5749 1.0199 5.0886 40 .OIO7 .8861 0135 1552 .0165 .566 .0199 .0812 39 .0107 .8725 0135 1443 .0166 5572 .02 -0739 38 .OI08 8589 .0136 1335 .0166 5484 .0201 .0666 37 .OI08 8454 .0136 .1227 .0167 539 6 .O2OI 0593 36 I.OI09 6.832 1.0136 6. 1 12 1.0167 5-53o8 1.0202 5-052 35 .0109 .8185 0137 .1013 .0168 .5221 .O2O2 .0447 34 .Oil .8052 0137 .0906 .0169 5134 .0203 0375 33 .Oil .7919 .0138 .08 .0169 5047 .0204 .0302 32 .0111 .7787 .0138 .0694 .017 .496 .O2O4 .023 I.OIII 6-7655 1.0139 6.0588 1.017 5-4874 1.0205 5-0158 30 .0111 7523 .0139 0483 .0171 .4788 .0205 .0087 29 .0112 7392 .014 0379 .0171 .4702 .0206 .0015 28 .OII3 7*32 .0141 .017 .0172 4532 .0207 9873 26 I.OII3 6.7003 1.0141 6.0066 1.0173 5-4447 I. O2O8 4.9802 25 .OII4 .6874 .0142 5*9963 .0174 .4362 .0208 9732 2 4 .0114 6745 .0142 .986 .0174 .4278 .0209 .9661 23 .OII5 .6617 .0143 9758 0175 .4194 .021 959 1 22 .0115 649 0143 9655 0175 .411 .021 .9521 21 I.OII5 6.6363 1.0144 5-9554 1.0176 5.4026 Z.02II 4.9452 20 .OIl6 6237 .0144 9452 .0176 3943 .O2II 9382 '9 .OIl6 .6m .0145 9351 .0177 .386 .0212 9313 18 .0117 5985 .0145 925 .0177 3777 .O2I3 9243 17 .0117 586 .0146 9*5 .0178 3695 .0213 9*75 16 1.0118 6.5736 1.0146 5.9049 1.0179 5-3612 I.O2I4 4.9106 15 .0118 .5612 .0147 895 .0179 353 .0215 937 14 .0119 5488 .0147 .885 .018 3449 .O2I5 .8969 13 .0119 5365 .0148 8751 .018 3367 .02l6 .8901 12 .0119 5243 .0148 .8652 .0181 .3286 .O2l6 8833 II 1. 012 6.5121 1.0149 5-8554 1.0181 5-3205 I.02I7 4-8765 IO .012 4999 .015 .8456 .0182 .3124 .02l8 .8697 9 .0121 .4878 .015 8358 .0182 .02l8 .863 8 .0121 4757 .0151 .8261 .0183 2963 .0219 8563 7 .OI22 4637 .0151 .8163 .0184 .2883 .022 .8496 6 1. 0122 6.4517 1.0152 5-8067 1.0184 5-2803 1.022 4.8429 5 .0123 .4398 .0152 797 .0185 .2724 .0221 .8362 4 0123 4279 0153 .7874 .0185 .2645 .0221 .8296 3 .0124 .416 0153 7778 .0186 .2566 .0222 .8229 2 .0124 .4042 .0154 .7683 .0186 .2487 .0223 ,8163 I I.OI25 6.3924 1.0154 5-7588 1.0187 5.2408 1.0223 4.8097 O CCHEC'T. SECANT. CO-SEC'T. SECANT, j CO-SEC'T. SECANT. CO-SEC'T. SECANT. ' ' 8] 80 7S >o 7 o 406 NATURAL SECANTS AND CO-SECANTS. u SECANT. H CO-BBC 'T. 1 SECANT. 3 CO-SKC'T. 1' SECANT. t CO-SEC'T. 150 SECANT. | CO-SEC'T. 1.0223 4.8097 1.0263 4-4454 1.0306 4.I336 1-0353 3-8637 .0224 .8032 .0264 .4398 .0307 .1287 0353 8595 .0225 .7966 .0264 4342 .0308 .1239 0354 .8553 .0225 .7901 .0265 .4287 .0308 .1191 0355 .8512 ,0226 7835 .0266 .4231 .0309 .1144 OSS 6 .847 1.0226 4-777 1.0266 4.4176 1.031 4. 1096 1.0357 3.8428 .0227 .7706 .0267 .4121 .0311 .1048 0358 8387 .0228 .7641 .0268 .4065 .0311 .1001 -0358 .8346 .0228 7576 .0268 .4011 .0312 0953 0359 .8304 .0229 -7512 .0269 3956 0313 .0906 .036 .8263 1.023 4.7448 1.027 4.3901 1.0314 4.0859 1.0361 3.8222 .023 '7384 .0271 3847 .0314 .0812 .0362 .8181 .0231 732 .0271 3792 0315 .0765 .0362 .814 .0232 7257 .0272 .3738 .0316 .0718 .0363 \ .81 .0232 7*93 .0273 .3684 0317 .0672 .0364 .8059 1.0233 4-7!3 1.0273 4-363 1.0317 4.0625 1.0365 3.8018 0234 .7067 .0274 3576 .0318 0579 .0366 | .7978 0234 .7004 .0275 3522 .0319 0532 0367 j .7937 0235 .6942 .0276 3469 .032 .0486 .0367 j .7897 0235 .6879 .0276 .3415 .032 .044 .0368 7857 1.0236 4.6817 1.0277 4-3362 1.0321 4.0394 1.0369 3-7816 .0237 6754 .0278 3309 .0322 .0348 037 .7776 .0237 .6692 .0278 3256 ^0323 .0302 0371 7736 .0238 .6631 .0279 3203 .0323 .0256 Q37 1 .7697 .0239 .6569 .028 315 .0324 .021 1 .0372 7657 1.0239 4.6507 1.028 4.3098 1.0325 4.0165 1.0373 3-76i7 .024 .6446 .0281 3045 .0326 .012 0374 7577 .0241 .6385 .0282 2993 .0327 .0074 0375 7538 .0241 .6324 .0283 .2941 .0327 .0029 .0376 .7498 .0242 .6263 .0283 .2888 .0328 3.9984 .0376 7459 1.0243 4. 6202 1.0284 4-2836 1-0329 3-9939 1-0377- 3-742 .0243 6142 0285 .2785 033 .9894 .0378 .738 .0244 6081 .0285 2733 033 985 0379 -734I .0245 .6021 .0286 .2681 0331 .9805 .038 .7302 .0245 .5961 .0287 .263 0332 .976 .0381 .7263 1.0246 4- 59 01 1.0288 4-2579 1-0333 3-97i6 1.0382 3.7224 .0247 .5841 .0288 .2527 0334 .9672 .0382 .7186 .0247 .5782 .0289 .2476 0334 .9627 0383 7H7 .0348 .5722 .029 .2425 0335 9583 .0384 .7108 .0249 5663 .0291 2375 0336 9539 .0385 .707 1.0249 4- 5604 1.0291 4.2324 1-0337 3-9495 1.0386 3-7 3i .025 5545 .0292 .2273 0338 9451 .0387 6993 ,0251 .5486 .0293 .2223 0338 .9408 .0387 6955 .0251 5428 .0293 2173 0339 9364 .0388 .6917 .0252 5369 .0294 .2122 034 932 .0389 .6878 1-0253 4-53" 1.0295 4.2072 1.0341 3-9 2 77 1.039 3-684 0253 5253 .0296 .2022 .0341 9234 .0391 .6802 .0254 5*95 .0296 .1972 .0342 .919 .0392 .6765 0255 5137 .0297 .1923 0343 .9147 0393 .6727 0255 5079 .0298 l8 73 0344 .9104 0393 .6689 1.0256 4.5021 1.0299 4.1824 1-0345 3.9061 1.0394 3-6651 .0257 4964 .0299 1774 0345 .9018 0395 .6614 .0257 .4907 03 1725 .0346 .8976 .0396 .6576 .0258 485 .0301 .1676 0347 8933 0397 6539 .0259 4793 0302 . 1627 .0348 .899 0398 .6502 1.026 4-4736 1.0302 4-I578 1.0349 3.8848 1.0399 3-6464 .026 .4679 0303 .1529 0349 .8805 0399 .6427 .0261 .4623 .0304 .1481 035 .8763 .04 639 .0262 .4566 0305 .1432 0351 .8721 .0401 .6353 .0262 451 0305 .1384 0352 .8679 .0402 .6316 1.0263 4-4454 1.0306 4-I336 1-0353 3-8637 1.0403 3.6279 CO-SEC 'T. SECANT. CO-SEC'T. SECANT. CO-SEC'T. SECANT. CO-SEC'T. SECANT. 770 76 75 74 NATURAL SECANTS AND CO-SECANTS. 407 | 160 17 180 19 ' |. SECANT. CO-SEC'T. SKCANT. CO-SEC'T. SECANT, j COHBEC'T. SECANT. CO-SBC'T. o 1.0403 3.6279 1-0457 3-4203 1-0515 3-2361 1.0576 3-0715 (> l .0404 .0458 .417 .0516 2332 0577 .069 2 .0405 .6206 0459 .4138 0517 .2303 .0578 .0664 3 .0406 .6169 .046 .4106 .0518 .2274 579 .0638 4 .0406 6i33 .0461 4073 .0519 .2245 .058 .0612 5 1.0407 3.6096 1.0461 3.4041 1.052 3.2216 1.0581 3-0586 6 .0408 .606 .0462 .4009 .0521 .2188 .0582 .0561 7 .0409 .6024 .0463 3977 .0522 .2159 .0584 0535 8 .041 .5987 .0464 3945 .0523 .2131 0585 .0509 9 .0411 5951 .0465 39*3 .0524 .2102 .0586 .0484 10 1.0412 3-59*5 1.0466 3-3881 1.0525 3.2074 1.0587 3-0458 ii .0413 5879 .0467 3849 .0526 .2045 .0588 0433 12 .0413 5843 .0468 3817 0527 .2017 0589 .0407 13 .0414 5807 .0469 3785 .0528 .1989 059 .0382 14 .0415 5772 .047 3754 .0529 .196 .0591 0357 15 1.0416 3.5736 1.0471 3-3722 1-053 3.1932 1.0592 3-033I 16 .0417 57 .0472 369 0531 .1904 0593 .0306 ;i .0418 .0419 -5665 .5629 0473 0474 3659 3627 0532 533 .1876 .1848 0594 0595 .0281 .6256 IQ .042 5594 475 3596 534 .182 .0596 .0231 20 1.042 3-5559 1.0476 1-0535 3-I792 1.0598 3.0206 21 .0421 5523 .0477 3534 0536 .1764 0599 .0181 22 .0422 .5488 .0478 3502 0537 1736 .06 .0156 23 .0423 5453 0478 347* 0538 .1708 .0601 .0131 24 .0424 .5418 0479 344 0539 .1681 .0602 .0106 25 1.0425 3.5383 1.048 3-3409 1.054 3-1653 1.0603 3.0081 26 .0426 5348 .0481 3378 .0541 1625 .0604 .0056 27 .0427 5313 .0482 3347 .0542 .1598 .0605 0031 28 .0428 5279 0483 0543 *57 .0606 .0007 29 .0428 5244 .0484 !3286 0544 1543 .0607 2.9982 30 1.0429 3.5209 1.0485 3-3255 1-0545 1.0608 i 29957 31 043 5175 .0486 3224 .0546 .1488 0609 -9933 32 .0431 .0487 3*94 547 .1461 .0611 -9908 33 34 .0432 0433 .5106 5072 .0488 .0489 3163 0548 0549 .1406 .0612 .9884 9859 35 1.0434 3-5037 1.049 3-3102 1-055 3-1379 1.0614 2.9835 36 0435 .5003 .0491 .3072 0551 1352 .0615 .981 37 .0436 .4969 .0492 3042 0552 1325 .0616 .9786 38 0437 4935 0493 .3011 0553 .1298 .0617 .9762 39 .0438 .4901 .0494 .2981 0554 .1271 .0618 9738 40 1.0438 3-4867 1.0495 3-2951 1-0555 3.1244 1.0619 2.9713 4i 0439 4833 .0496 .2921 05fi6 .1217 .062 .9689 42 044 4799 .0497 .2891 0557 .119 .0622 .9665 43 .0441 .4766 .0498 .2861 0558 .1163 .0623 .9641 44 .0442 4732 0499 .2831 0559 H37 .0624 .9617 45 1.0443 3. 4698 1.05 3.2801 1.056 1.0625 2-9593 46 0444 .4665 .0501 .2772 .0561 .1083 .0626 9569 47 -0445 .4632 .0502 .2742 .0562 1057 .0627 9545 48 .0446 4598 .0503 .2712 .0563 .103 .0628 .9521 49 0447 -4565 0504 .2683 0565 .1004 .0629 9497 50 1.0448 3-4532 1.0505 3-2653 1.0566 3-0977 1.063 2-9474 51 .0448 .4498 .0506 .2624 .0567 .0951 .0632 945^ 52 .0449 4465 .0507 2594 .0568 .0925 .0633 .9426 53 045 4432 .0508 2565 .0569 .0898 .0634 .9402 54 .0451 .0509 2535 057 .0872 0635 9379 55 1.0452 3-4366 1.051 3.2506 1.0571 3.0846 1.0636 2-9355 56 0453 4334 .0511 .2477 0572 .082 .0637 9332 57 0454 .4301 .0512 .2448 0573 0793 .0638 .9308 58 0455 .4268 0513 .2419 0574 .0767 0639 .9285 59 .0456 4236 .0514 2 39 0575 .0741 .0641 .9261 60 1 -457 3.4203 1-0515 3.2361 1.0576 3-0715 1.0642 2.9238 ' CO-SEC'T. SECANT. CO-SBC'T. SECANT. CO-SBC'T. SECANT. | CO-SBC'T. SECANT. 73 720 710 |i 70 o 408 NATURAL SECANTS AND CO-SECANTS. 20 21 22 23 SECANT. CO-SEC'T SECANT. CO-SEC'T SECANT. CO-SEC'T SECANT. CO-SEC'T. 1.0642 2.9238 1.0711 2.7904 1.0785 2.6695 1.0864 2-5593 .0643 .0644 .9215 .9191 0713 .0714 .7883 .7862 .0787 .0788 .6675 .6656 .0865 .0866 5575 5558 .0645 .9168 0715 7841 .0789 .6637 .0868 554 .0646 9 J 45 .0716 .782 .079 .6618 .0869 5523 1.0647 2.9122 1.0717 2-7799 1.0792 2.6599 1.087 2. 5506 .0648 .9098 .0719 .7778 0793 .658 .0872 -5488 .065 975 .072 7757 .0794 .6561 .0873 547i .0651 .9052 .0721 7736 0795 .6542 .0874 5453 .0652 .9029 .0722 7715 .0797 6523 .0876 5436 1-0653 2.9006 1.0723 2.7694 1.0798 2.6504 1.0877 2.5419 .0654 .8983 .0725 .7674 .0799 6485 .0878 .5402 0655 .896 .0726 7653 .0801 .6466 .088 5384 .0656 8937 .0727 .7632 .0802 .6447 .0881 53 6 7 .0658 .8915 .0728 .7611 .0803 .6428 .0882 535 1.0659 2.8892 1.0729 2-7591 1.0804 2.641 1.0884 2-5333 .066 .8869 0731 757 .0806 6391 .0885 53i6 .0661 .8846 .0732 755 .0807 6372 .0886 5299 .0662 .8824 0733 7529 .0808 6353 .0888 .5281 .0663 .8801 .0734 7509 .081 -6335 .0889 .5264 1.0664 2.8778 1.0736 2.7488 1.0811 2.6316 1.0891 2-5247 .0666 .8756 0737 .7468 .0812 .6297 .0892 523 .0667 8733 .0738 7447 .0813 .6279 .0893 5213 .0668 .8 7 n 0739 .7427 .0815 .626 .0895 .5196 .0669 .8688 .074 .7406 .0816 .6242 .0896 5179 1.067 2.8666 1.0742 2.7386 1.0817 2.6223 1.0897 2.5163 .0671 .8644 0743 .7366 .0819 .6205 .0899 .5146 .0673 .8621 .0744 .7346 .082 .6186 .09 .5129 .0674 8599 0745 7325 .0821 .6168 .0902 .5112 ^ 7 i 1.0676 .8577 2-8554 .0747 1.0748 7305 2-7285 .0823 1.0824 .615 2.6131 .0903 1.0904 5095 2.5078 .0677 8532 .0749 .7265 .0825 .61x3 .0906 .5062 .0678 .851 075 7245 .0826 .6095 .0907 5045 .0679 .8488 0751 .7225 .0828 .6076 .0908 .5028 .0681 .8466 753 7205 .0829 .6058 .091 .5011 1.0682 2.8444 i-754 2-7185 1.083 2.604 1.0911 2.4995 .0683 .8422 0755 7165 .0832 .6022 .0913 .4978 .0684 84 .0756 7145 0833 .6003 .0914 .4961 \ .0685 .8378 .0758 7125 0834 5985 .0915 4945 ! .0686 8356 0759 7 I0 5 .0836 59 6 7 .0917 .4928 1.0688 2-8334 1.076 2.7085 1.0837 2-5949 1.0918 2.4912 .0689 .8312 .0761 .7065 .0838 5931 .092 4895 .069 .829 .0763 7045 .084 59*3 .0921 .4879 .0691 .8269 .0764 .7026 .0841 5895 .0922 .4862 .0692 .8247 .0765 .7006 .0842 5877 .0924 .4846 1.0694 2.8225 1.0766 2.6986 1.0844 2-5859 1.0925 2.4829 .0695 .8204 .0768 .6967 .0845 .5841 .0927 .4813 .0696 .8182 .0769 .6947 .0846 .5823 .0928 4797 .0697 .816 .077 .6927 .0847 5805 .0929 478 .0698 8139 .0771 .6908 .0849 5787 .0931 .4764 1.0699 2.8117 I -Q773 2.6888 1.085 2-577 1.0932 2.4748 .0701 .8096 .0774 .6869 .0851 5752 0934 4731 .0702 .8074 0775 .6849 0853 5734 0935 47i5 .0703 8053 .0776 .683 .0854 57i6 .0936 .4699 .0704 -8032 .0778 .681 0855 5699 .0938 -4683 1.0705 2.801 1.0779 2.6791 1.0857 2.5681 1.0939 2.4666 .0707 .7989 .078 .6772 .0858 5663 .0941 465 .0708 .7968 .0781 .6752 .0859 .5646 .0942 4634 .0709 7947 .0783 6733 .0861 .5628 0943 .4618 .071 79 2 5 .0784 .6714 .0862 .561 0945 .4602 1.0711 2.7904 1.0785 2.6695 1.0864 2- 5593 .0946 2.4586 CO-SEC'T. SECANT. CO-SEC'T. SECANT. CO-SEC'T. SECANT. CO-SEC'T. SECANT, j 69 68 6?o 660 j NATUBAL SECANTS AND CO-SECANTS. 409 : 24 25 260 27 ' SECANT. CO-SKC'T SECANT. CO-SKC'T. SECANT. CO-SKC'T. SECANT. | CC-BKC'T. o 1.0946 2.4586 1-1034 2.3662 1.1126 2.2812 1.1223 2. 2027 i 60 I .0948 457 1035 3647 .1127 .2798 .1225 .2014 59 2 .0949 4554 .1037 3632 .1129 .2784 .1226 . 2OO2 i 58 3 .0951 .4538 .1038 .3618 .1131 .2771 .1228 .1989 57 4 .0952 4522 .104 3603 "32 2757 .123 .1977 56 5 1-0953 2.4506 1.1041 2.3588 1.1134 2.2744 1.1231 2.1964 55 6 0955 449 .1043 3574 "35 -273 1233 .1952 54 7 .0956 4474 .1044 3559 "37 .2717 -1235 1 939 53 8 0958 4458 .1046 3544 "39 .2703 1237 .1927 52 9 .0959 4442 .1047 353 .114 .269 .1238 .1914 5I 10 1.0961 2.4426 1. 1049 2.3515 1.1142 2.2676 1.124 2. 1902 50 ii .0962 44" .105 3501 "43 .2663 .1242 .1889 49 12 .0963 4395 .1052 .3486 "45 .265 .1243 .1877 48 '3 .0965 4379 1053 3472 ."47 .2636 .1245 .1865 47 14 .0066 4363 1055 3457 .1148 .2623 .1247 -1852 46 15 ! 1.0968 a- 4347 1. 1056 2-3443 1.115 2.261 1.1248 | 2.184 i 45 16 .0969 4332 .1058 3428 1151 .2596 .125 .1828 44 17 .0971 .4316 .1059 34H "53 .2583 .1252 .1815 \ 43 18 .0972 0973 .4285 .1061 .1062 3399 3385 ."55 ."56 257 2556 1253 1255 .1803 42 .1791 41 20 | 1.0975 2.4269 1.1064 2-337 1 1.1158 2-2543 1.1257 2.1778 40 21 .0976 4254 .1065 3356 "59 253 .1258 .1766 j 39 22 .0978 .4238 .1067 3342 .1161 .2517 .126 1754 38 23 .0979 .4222 .1068 3328 .1163 -2503 .1262 1742 37 24 .0981 .4207 .107 .1164 .249 .1264 173 36 25 1.0982 2.4191 1.1072 2.3299 1.1166 2.2477 1.1265 2.1717 35 26 .0984 .4176 1073 3285 .1167 .2464 .1267 .1705 34 2 7 .0985 .416 1075 3271 .1169 .2451 .1269 .1693 33 28 .0986 4145 .1076 3256 .1171 2438 .127 .1681 32 29 .0988 413 .1078 3242 .1172 .2425 .1272 .1669 30 1.0989 2.4114 1.1079 2.3228 1.1174 2.2411 1.1274 2.1657 30 31 .0991 .4099 .1081 .3214 .1176 .2398 1275 .1645 2 9 32 .0992 4083 .1082 .32 "77 -2385 .1277 1633 28 33 .0994 .4068 .1084 3186 "79 -2372 .1279 .162 27 34 0995 -4053 .1085 3172 .118 2359 .1281 .1608 26 35 I 0997 2-4037 1.1087 2.3158 1.1182 2.2346 1.1282 2.1596 25 36 .0998 .4022 .1088 3143 .1184 2333 .1284 .1584 24 37 *x .4007 .109 .3129 1185 232 .1286 1572 23 38 .1001 3992 .1092 .1187 .2307 .1287 156 22 39 .1003 3976 .1093 .3101 .1189 .2294 .1289 .1548 21 40 1*1004 2.3961 1.1095 2.3087 1.119 2.2282 1.1291 2.1536 20 4* .1005 3946 .1096 3073 .1192 .2269 .1293 1525 19 42 .1007 3931 .1098 -359 "93 .2256 ,1294 1513 18 43 .1008 .3916 .1099 3046 "95 2243 .1296 .1501 '7 44 .101 .3901 .1101 3032 "97 223 .1298 .1489 16 45 I.IOII 2.3886 1. 1 102 2.3018 1.1198 2.2217 1.1299 2.1477 15 46 .1013 3871 .IIO4 .3004 .12 .2204 .1301 .1465 14 47 .1014 .3856 .II06 299 .I2O2 .2192 1303 1453 13 48 .I0l6 .3841 : .1107 .2976 .1203 2179 1305 .1441 12 49 .1017 .3826 .1109 .2962 I2O5 .2166 .1306 .143 II 50 1.1019 2.3811 I. Ill 2.2949 I.I207 2.2153 1.1308 2.1418 10 51 52 .102 .1022 379 6 378i ! .1112 .1113 .2935 .2921 .1208 .121 .2141 .2128 ll3I2 1406 1394 I 53 .IO23 .3766 .1115 .2907 .1212 2115 I3I3 .1382 7 54 I02 5 I .3751 .1116 .2894 .1213 .2103 I3I5 I37I 6 55 I.I026 | 2.3736 1.1118 2.288 I.I2I5 2.209 1-1317 2-1359 5 56 .1028 j 3721 112 .2866 .1217 .2077 1319 1347 4 57 .1029 .1121 .2853 .I2l8 .2065 .132 1335 3 58 .1031 .3691 "23 2839 .122 .2052 .1322 .1324 2 59 .1032 3 6 77 "24 .2825 .1222 2039 1324 1312 I 00 I.I034 2.3662 I.II26 2.2812 I.I223 2.2027 1.1326 2.13 T~ CO-SBC'T. SECANT. CO-SKC'T. SECANT. CO-SKC'T. SECANT. CO-SEC'T. SECANT. 9 65 64 63 620 MM 4io NATURAL SECANTS AND CO-SECANTS. m 1 2< ) 3( ) 3] L SECANT. CO-BEC'T. SECANT. CO-SBC'T. SECANT. CO-SEC'T. SECANT. CO-SEC'T. 1.1326 2- 3 I-I433 2.0627 I-I547 2 1. 1666 1.9416 1327 . 289 1435 .0616 1549 1.999 .1668 9407 .1329 277 1437 .0605 .998 .167 9397 . 266 1439 0594 .1553 997 .1672 .9388 i333 254 .1441 0583 '555 996 .1674 9378 I-I334 2. 242 I-I443 2-0573 I-I557 1-995 i 1676 i 9369 .1336 . 231 M45 .0562 1559 994 1678 936 1338 . 219 .1446 0551 .1561 993 .1681 935 134 . 208 .1448 054 .1562 992 .1683 9341 I 34 I . I 9 6 145 053 .1564 .991 1685 9332 I-I343 2. 185 1.1452 2.0519 1.1566 1.99 1.1687 1.9322 1345 173 1454 .0508 .1568 989 .1689 .9313 1347 . 162 .1456 .0498 .157 .988 1691 934 '349 15 .1458 .0487 .1572 987 .1693 9295 135 139 .1459 .0476 1574 .986 .1695 9285 I-I352 2. 127 1.1461 2.0466 1.1576 1.985 1.1697 1.9276 1354 . 116 .1463 0455 1578 -984 .1699 .9267 .1356 . 104 .1465 .0444 .158 .983 .1701 .9258 1357 . 093 .1467 0434 .1582 .982 1703 .9248 1359 . 082 .1469 .0423 .1584 .9811 1705 9239 1.1361 2. 07 1.1471 2.0413 1.1586 1.9801 1.1707 1.923 1363 059 1473 .0402 1588 .9791 .1709 .9221 1365 . 048 .1474 0392 '59 .9781 .1712 .9212 .1366 036 .1476 .0381 .1592 .9771 .1714 .9203 1368 02 5 .1478 037 1594 .9761 .1716 9 J 93 I-I37 2. 014 1.148 2.036 1.1596 1-9752 1.1718 1.9184 1372 . OO2 .1482 0349 .1598 .9742 .172 9*75 '373 .0991 .1484 0339 .16 9732 .1722 .9166 '375 .098 .1486 .0329 .1602 .9722 .1724 9 I 57 '377 .0969 .1488 .0318 .1604 .1726 .9148 2.0957 1.1489 2.0308 1. 1606 I-9703 1.1728 .1381 .0946 .1491 0297 .1608 9693 173 9'3 .1382 -0935 M93 .0287 .161 .9683 1732 .9121 .1384 .0924 .1495 .0276 .1612 .9674 1734 .9112 .1386 .0912 .1497 .0266 .1614 .9664 1737 .9102 1.1388 2.0901 1.1499 2.0256 1. 1616 1.9654 I-I739 1.9093 139 .089 .1501 .0245 .1618 9645 .1741 .9084 .0879 1503 0235 .162 9635 1743 975 1393 .0868 1505 .0224 .1622 .9625 1745 .9066 1395 0857 1507 .0214 .1624 .9616 1747 957 I-I397 2.0846 1.1508 2.0204 1.1626 1.9606 1.1749 1.9048 '399 0835 IS' 0194 .1628 .9596 .1751 939 .1401 .0824 .1512 .0183 .163 9587 -1753 93 .1402 .O8l2 1514 .0173 .1632 9577 1756 .9021 .1404 .O8oi .1516 .0163 .1634 .9568 .1758 .9013 1.1406 2.079 1.1518 2.0152 1.1636 I.9558 1.176 .1408 .0779 .152 .0142 .1638 9549 .1762 .141 .0768 .1522 .0132 .164 9539 .1764 . .1411 .0757 .1524 .0122 .1642 953 .1766 .8977 .0746 .1526 .OIII .1644 952 .1768 .8968 1.1415 2.0735 1.1528 2.0IOI 1.1646 1.177 1.8959 .1417 .0725 153 .0091 .1648 .9501 .1772 .895 .1419 .0714 .1531 .0081 .165 .9491 1775 .8941 .1421 .0703 1533 .0071 .1652 .9482 .1777 .8932 .1422 .0692 1535 .Oo6l .1654 9473 .1779 .8924 1.1424 2.0681 I-I537 2.005 1.1656 1-9463 1.1781 1.8915 .1426 .06 7 1539 .004 .1658 9454 -1783 .8906 .! 4 28 0659 .1541 .003 .166 9444 1785 .8897 143 .0648 1543 .002 .1662 9435 .1787 .8888 .1432 .0637 1545 .OOI .1664 -9425 179 .8879 2.0627 i- 1547 2 1. 1666 1.9416 1.1792 1.8871 CO-BEC'T. SECANT. CO-BEC'T. SECANT. CO-SEC'T. SECANT. CO-SEC'T. SECANT. 6 L 6( ) 5< 61 1? NATURAL SECANTS AND CO-SECANTS. 411 32 33 34 35 SECANT. CO-SEC'T. SECANT. CO-SEC'T. SECANT. CO-SBC'T. SECANT. CO-SEC'T. 1.1792 1.8871 1.1924 1.8361 1.2062 1.7883 1.2208 1-7434 .1794 .1796 .8862 8853 .1926 .1928 8352 .8344 .2064 .2067 7875 .7867 .221 .2213 .7427 .742 .1798 .8844 193 8336 .2069 .786 .2215 7413 .18 .8836 -1933 .8328 .2072 .7852 .22l8 745 1. 1802 1.8827 I-I935 1.832 1.2074 1.7844 1.222 I-739 8 .1805 .8818 1937 .8311 2076 -7837 .2223 -7391 .1807 .1809 .8809 .8801 .1939 .1942 -8303 .8295 .2079 .2081 .7829 .7821 .2225 .2228 -7384 7377 .1811 .8792 .1944 .8287 .2083 .7814 223 7369 1.1813 1.8783 1.1946 1.8279 1.2086 1.7806 1.2233 1-7362 .1815 .8785 .1948 .8271 .2088 .7798 -2235 7355 .1818 .8766 .8263 .209! .7791 .2238 7348 .182 -8757 1953 -8255 .2093 .7783 .224 7341 .1822 .8749 1955 .8246 .2095 .7776 .2243 -7334 1.1824 .1826 1.874 8731 1.1958 .196 1.8238 823 21 1.7768 .776 1-2245 .2248 1-7327 .1828 .8723 .1962 .8222 .2103 7753 .225 -7312 .1831 .8714 .1964 .8214 2105 -7745 2253 7305 1833 .8706 .1967 8206 .2IO7 7738 2255 .7298 1-1835 1.8697 1.1969 1.8198 1. 211 1-773 1.2258 1.7291 -1837 .8688 .1971 .819 .2112 7723 .226 .7284 .1839 .868 .1974 .8182 .2115 7715 .2263 .7277 .1841 .1844 .8671 .8663 .1976 .1978 -8174 .8166 .2II 7 .2119 .7708 77 .2265 .2268 .727 .7263 1.1846 1.8654 i. I9 8 1.8158 I. 2122 1-7693 1.227 1.7256 .1848 .8646 .1983 .815 .2124 .7685 .2273 7249 .185 .8637 .1985 .8142 .2127 .7678 .2276 .7242 .1852 .8629 .1987 .8134 .2129 .767 .2278 7234 1855 .862 .199 ,8126 .2132 i' 7 f 3 .228l .7227 1.1857 1.8611 1.1992 1.8118 I.2I34 1.2283 1.722 .1859 .8603 .1994 .811 .2136 .7648 .2286 7213 .1861 .8595 .1997 .8102 .2139 -764 .2288 .7206 1863 .8586 .1999 .8094 .2141 7633 .2291 .7199 .1866 .8578 .2001 .8086 .2144 .7625 .2293 .7192 1.1868 1.8569 I.2OO4 1.8078 1.2146 1.7618 1.2296 1.7185 .187 .8561 .2006 .807 .2149 .761 .2298 .7178 .1872 .8552 .2008 .8062 .2151 7603 .2301 .7171 .1874 .8544 .201 .8054 2153 7596 .2304 .7164 .1877 .8535 .2OI3 .8047 2156 .7588 .2306 .7157 1.1879 1.8527 I.20I5 1.8039 I.2I58 1.7581 1.2309 .1881 .8519 .2017 .8031 .2l6l -7573 .2311 .7144 .1883 .851 .202 .8023 .2163 .7566 .2314 .1886 .8502 .2022 .8015 .2166 7559 .2316 7'3 .1888 8493 .2024 .8007 .2168 7551 .2319 7123 1.189 1.8485 I.2O27 1.7999 I.2I7I 1-7544 J.2322 1.7116 .1892 .8477 .2029 .7992 2173 7537 2324 .7109 .1894 .8468 .2031 .7984 2175 7529 2327 .7102 .1897 .846 .2034 7976 .2178 .7522 .2329 7095 .1899 .8452 .2036 .7968 .218 7514 .2332 .7088 1.1901 1.8443 1.2039 1.796 I.2I83 I-7507 1-2335 1.7081 .1903 ' -8435 .2041 7953 .2185 75 2337 775 .1906 .8427 .2043 7945 .2188 7493 234 .7068 .1908 .8418 .2046 7937 .219 7485 2342 .7061 .191 .841 .2048 .7929 .2193 .7478 2345 7054 1.1912 1.0402 1.205 1.7921 I.2I95 1.7471 1.2348 1.7047 .1915 8394 2053 .7914 .2198 7463 235 .704 .1917 8385 2055 .22 7456 2353 7033 .1919 8377 .2057 .7898 .2203 7449 2355 .7027 1921 .8369 .206 .7891 2205 7442 2358 .702 1.1922 1.8361 1.2062 1-7883 1. 2208 1-7434 1.2361 1.7013 CO-SBC'T. SECANT. CO-SEC'T. SECANT. CO-SEC'T. SECANT. CO-SEC'T. SECANT. 570 56 55 54 412 NATUKAL SECANTS AND CO-SECANTS. 3* SECANT. > CO-SEC'T. & SECANT. ro CO-SEC'T. 3E SECANT. 1 CO-SEC'T. 35 SECANT. >o CO-SEC'T. 1.2361 1.7013 1.2521 i. 6616 1.269 1.6243 1.2867 1.589 2363 .7006 2524 .661 .2693 6237 .2871 .5884 .2366 .6999 2527 .6603 .2696 .6231 .2874 5879 .2368 6993 253 6597 .2699 .6224 .2877 -5873 2371 .6986 .2532 .6591 .2702 .6218 .288 .5867 1-2374 1.6979 1-2535 1.6584 1.2705 1.6212 1.2883 1.5862 .2376 .6972 .2538 .6578 .2707 .6206 .2886 .5856 2379 .6965 .2541 .6572 .271 .62 .2889 -585 .2382 .2384 6959 .6952 2543 .2546 6565 6559 2713 .2716 .6194 .6188 .2892 .2895 -5845 5839 1.2387 1.6945 1.2549 1.6552 1.2719 1.6182 1.2898 I-5833 .2389 .6938 2552 .6546 .2722 .6176 .2901 .5828 2392 .6932 2554 654 .2725 .617 .2904 .5822 2395 .6925 2557 6533 .2728 .6164 .2907 j .5816 : 2397 .6918 .256 6527 2731 .6159 .291 .5811 ; 1.24 1.6912 1.2563 1.6521 1-2734 1.6153 1.2913 1.5805 .2403 .6905 .2565 .6514 2737 .6147 .2916 -5799 2405 .6898 .2568 .6508 2739 .6141 .2919 .5794 .2408 .6891 2571 .6502 .2742 6i35 .2922 | .5788 .2411 .6885 2574 .6496 2745 .6129 .2926 .5783 1.2413 1.6878 1-2577 1.6489 1.2748 1.6123 1.2929 1-5777 .2416 .6871 2579 .6483 2751 .6117 .2932 5771 .2419 .6865 .2582 6477 2754 .6111 2935 .5766 .2421 .6858 2585 .647 2757 .6105 .2938 576 .2424 6851 .2588 .6464 .276 .6099 .2941 5755 1.2427 1.6845 1.2591 1.6458 1.2763 1.6093 1.2944 1-5749 .2429 .2432 .6838 .6831 2593 .2596 .6452 6445 .2766 .2769 .6087 .6081 2947 295 5743 .5738 2435 6825 2599 6439 .2772 .6077 2953 5732 2437 .6818 .2602 6433 2775 .607 .2956 5727 1.244 i. 6812 1.2605 1.6427 1.2778 1.6064 1.296 1.5721 2443 .6805 .2607 .642 .2781 .6058 .2963 .5716 2445 .6798 .261 .6414 .2784 .6052 .2966 571 .2448 .6792 2613 .6408 .2787 .6046 .2969 5705 .2451 .6785 .2616 .6402 .279 .604 .2972 -5699 1-2453 1.6779 1.2619 1.6396 1-2793 1.6034 1-2975 1.5694 .2456 .6772 .2622 .6389 2795 .6029 .2978 .5688 2459 .6766 .2624 6383 .2798 .6023 .2981 -5683 .2461 6759 .2627 6377 .2801 .6017 .2985 5677 .2464 .6752 .263 .6371 .2804 .6011 .2988 .5672 1.2467 1.6746 1.2633 1.6365 i. 2807 1.6005 1.2991 1.5666 .247 6739 .2636 6359 .281 .6 2994 .5661 .2472 2639 6352 .2813 5994 2997 5655 2475 .6726 .2641 .6346 .2816 .5988 3 -565 .2478 .672 .2644 -634 .2819 .5982 3003 5644 1.248 1.6713 1.2647 I-6334 1.2822 I-5976 1.3006 I-5639 .2483 .6707 .265 .6328 .2825 5971 .301 5633 .2486 .67 2653 .6322 .2828 5965 3013 .5628 .2488 .6694 .2656 .6316 .2831 5959 .3016 .5622 .249 .6687 2659 6309 2834 5953 .3019 5617 1.2494 I- 6681 1.2661 1.6303 1-2837 1-5947 1.3022 1.5611 .2497 .6674 .2499 .6668 .2664 .2667 .6297 .6291 .284 .2843 5942 5936 3025 .3029 .5606 56 .2502 .6661 .267 .6285 .2846 593 .3032 5595 2505 6655 2673 .6279 .2849 5924 3035 559 1.2508 1.6648 1.2676 1.6273 1.2852 1-5584 .251 .6642 .2679 .6267 -2855 59*3 .3041 5579 2513 .6636 .2681 .6261 .2858 597 3044 5573 2516 .6629 .2684 6255 .2861 .3048 .5568 .2519 .6623 .2687 .6249 .2864 .5896 5563 1.2521 i. 6616 1.269 1.6243 1.2867 1.589 1-3054 1-5557 CO-SEC'T. SECANT. CO-SEC'T. SECANT. CO-SEC'T. SECANT. CO-SEC'T. SECANT. 53 520 510 50 NATURAL SECANTS AND CO-SECANTS. 413 4C 1 ! 41 42 jo 42 1 SECANT. CO-SEC'T. SECANT. CO-SEC'T. SECANT. CO-SKC'T. SECANT. CO-SBC'T. I-3054 1-5557 I-325 1-5242 I-3456 1-4945 I-3673 1-4663 3057 5552 3253 5237 .346 494 3677 .4658 -306 5546 3257 5232 3463 4935 -3681 4654 .3064 5541 .326 5227 3467 493 .3684 4649 .3067 5536 3263 .5222 347 4925 .3688 .4644 1-307 1-553 1.3267 1.5217 1-3474 1.4921 1.3692 1.464 3073 5525 327 .5212 3477 .4916 3695 4635 3076 552 3274 .5207 .3481 49" 3699 .4631 .308 5514 .3277 .5202 .3485 .4906 3703 .4626 -3083 5509 328 5197 .3488 .4901 3707 .4622 1.3086 I-5503 1.3284 1.5192 1.3492 1.4897 1.4617 -3089 5498 .3287 5187 3495 .4892 3714 .4613 .3092 5493 329 .5182 -3499 .4887 .4608 .3096 5487 3294 5177 3502 .4882 3722 .4604 3099 -5482 3297 .5171 35o6 .4877 3725 4599 1.3102 1-5477 I-330I 1.5166 I-3509 1.4873 I-3729 1-4595 3105 5473 3304 .5161 3513 .4868 3733 459 .3109 .5466 3307 5156 3517 .4863 3737 .4586 .3112 .5461 33" 5 I 5 1 352 .4858 374 .4581 3"5 5456 3314 .5146 .3524 4854 3744 4577 1.3118 1-545 1.5141 1-3527 1.4849 I-3748 1-4572 .3121 5445 3321 5136 3531 4844 3752 .4568 3125 544 3324 5131 3534 -4839 3756 4563 .3128 5434 3328 .5126 3538 4835 3759 4559 3 I 3 I 5429 3331 .5121 3542 483 3763 4554 1.3134 1.5424 1-3335 1-5116 1-3545 1-4825 1-3767 1-455 3138 3338 .5111 3549 .4821 4545 5413 3342 .5106 3552 .4816 3774 -3144 .5408 3345 .5101 3556 .4811 3778 4536 .3148 5403 1.5398 3348 1-3352 .5096 1-5092 356 1-3563 .4806 1.4802 .3782 1-3786 4532 M527 3154 539 2 3355 5087 3567 4797 379 4523 5387 3359 .5082 3571 4792 3794 .4518 .3161 5382 3362 5077 3574 .4788 3797 4514 .3164 5377 -3366 .5072 3578 4783 .3801 451 1.3167 .5366 3372 .5062 3585 4774 1-3805 3809 ^SOS .4501 -3174 3376 5057 3589 .4769 3813 .4496 5177 .318 5356 5351 3379 -3383 5052 5047 3592 359 4764 476 .3816 -382 4492 -4487 1-3184 1-5345 1-3386 1-5042 1-36 1-4755 1-3824 1.4483 .3187 534 339 5037 -3603 475 3828 4479 5335 3393 5032 .3607 4746 3832 4474 3193 533 3397 .5027 .3611 .4741 3836 447 3197 5325 34 .5022 .3614 4736 3839 4465 1.32 I-53I9 1.3404 1.5018 1.3618 1-4732 1-3843 1.4461 -3203 53H 3407 5013 .3622 4727 3847 4457 .3207 5309 34" .5008 3625 4723 3851 4452 .321 534 34H .5003 -3629 .4718 3855 .4448 3213 5299 .3418 4998 3633 3859 4443 1.3217 1.5294 1.3421 1-4993 1.3636 1.4709 1.3863 1-4439 322 -5289 3425 .4988 364 .4704 3867 4435 3223 -5283 3428 4983 3644 4699 387 443 .3227 5278 3432 4979 3647 4695 3874 .4426 323 5273 3435 4974 3651 .469 .3878 .4422 1-3233 1.5268 1-3439 1.4969 1.4686 1.3882 1.4417 3237 5263 3442 .4964 ''3658 .4681 .3886 44*3 324 -5258 3446 4959 .3662 .4676 389 .4408 3243 5253 3449 4954 .3666 .4672 3894 .4404 3247 -5248 3453 4949 -3669 .3667 .3898 44 1-325 1.5242 I-3456 1-4945 1-3673 1-4663 1.3902 1-4395 CO-BHC'T. SKCANT. CO-SKC'T. SECANT. CO-SEC'T. SECANT. CO-SEC'T. SKCANT. 4< ) 41 30 4' 7 > M M* NATURAL SECANTS AND CO-SECANTS. 44 t<> 44 t 44 ' SECANT. CO-SEC'T. ' ' SECANT. CO-SEC'T. r 1 SECANT. CO-SKC'T. '.- 1.3902 1-4395 60 21 1.3984 i-435 39 41 1.4065 1.4221 ^0 I 395 439* 59 22 3988 .4301 38 42 .4069 .4217 18 2 399 4387 58 23 3992 .4297 37 43 473 .4212 l l 3 39*3 .4382 56 24 399 6 .4292 1.4288 36 44 .4077 .4208 16 4 5 39 1 7 1.3921 4378 1-4374 5 U 55 11 .4004 .4284 35 34 45 46 .4085 .42 H 6 39 2 5 437 54 27 .4008 .428 33 47 .4089 .4196 13 7 39 2 9 4365 53 28 .4012 .4276 32 48 493 4I9 ,o 12 8 3933 .4361 5 2 29 .4016 .4271 3i 49 .4097 .4188 11 9 3937 4357 Si 3 1.402 1.4267 30 50 1.4101 1.4183 10 i-394i 1-4352 5 3i .4024 .4263 2 9 5i .4105 .4179 9 i 3945 4348 49 > 32 .4028 '4259 28 52 .4109 4175 8 2 3949 4344 48 33 .4032 4254 27 53 4"3 .4171 7 3 3953 4339 47 34 .4036 425 26 54 .4117 .4167 6 4 3957 4335 46 35 1.404 1.4246 25 55 1.4122 1.4163 5 1.396 39 6 4 I-433I 43 2 7 45 44 36 37 .4044 .4048 .4242 4238 24 23 5* 57 .4126 .413 4159 4!54 ' 4 3 7 .3968 .4322 43 38 .4052 4333 22 58 4i34 415 2 8 3972 .4318 42 39 .4056 .4229 21 39 .4138 .4146 ^ * 9 3976 4314 4i 40 1.406 1.4225 20 60 1.4142 1.4142 20 1.398 J-43I 40 / CO-SEC'T. SECANT. i / CO-SEC'T. SECANT. 1 i CO-SEC'T. SBCANT. I 4i 5 4 5 4, 50 Preceding Table contains Natural Secants and Co-secants for every minute of the Quadrant to Radius i. If Degrees are taken at head of column, Minutes, Secant, and Co-secant must be taken from head also; and if they are taken at foot of column, Minutes, etc., must be taken from foot also. ILLUSTRATION. 1.05 is secant of 17 45' and co-secant of 72 15'. To Compute Secant or Co-secant of any Angle. RULE. Divide i by Cosine of angle for Secant, and by Sine for Co-secant. EXAMPLE i. What is secant of 25 25'? Cosine of angle = .903 21. Then i -f- .903 21 = 1.1072, Secant. 2. What is co secant of 64 35'? Sine of angle = .903 21. Then i -r- .903 21 = 1. 1072, Co-secant. To Compute Degrees, IVTiiiTites, and Seconds of* a Secant or Co-secant. When Secant is given, Proceed as by Rule, page 402, for Sines, substituting Secants for Sines. EXAMPLE. What is secant for 1.1607? The next less secant is 1.1606, arc for which = 30 30'. Next greater secant is 1.1608, difference between which and next less is 1.1608 1. 1606 = .0002. Difference between less tab. secant and one given is 1. 1607 1. 1606 = .0001. Then .0002 : .0001 :: 60 : 30, which, added to 30 30' = 30 30' 30". When Co-secant is given, Proceed as by Rule, page 402, Substituting Co-secants for Cosines. NATURAL TANGENTS AND CO-TANGENTS. Natural Tangents and. Co-tangents. || 1 2 3 TANO. CO-TANG. 1 1 TANG. CO-TANG. TANG. CO-TANG. TANG. | CO-TANG. .00000 Infinite. .01746 57-29 .03492 28.6363 05241 19.0811 .00029 3437-75 01775 6.3506 .03521 8.3994 .0527 8-9755 .00058 1718.87 .01804 5-44I5 0355 8. 1664 05299 8.8711 .00087 145.92 01833 4-56i3 035 79 7-9372 05328 8.7678 .001 16 859-436 01862 3-7086 036 09 7.7117 05357 8.6656 .00145 687.549 01891 52.8821 .03638 27. 4899 05387 18.5645 .00175 .00204 572-957 491.106 0192 01949 2.0807 1.3032. .03667 .03696 7-27 *l 7.0566 .05416 05445 8.4645 8-3655 .00233 29.718 01978 0-5485 037 25 6-845 05474 8.2677 .00262 381.971 02007 49.8157 037 54 6.6367 05503 8.1708 .00291 343-774 02036 49- 10 39 037 83 26.4316 05533 18.075 .OO3 2 12.521 02066 8.4121 .038 12 6.2296 05562 7.9802 .00349 286.478 .02095 7-7395 .03842 6.0307 05591 7.8863 00378 64.441 .021 24 7-0853 .03871 5-8348 .0562 7-7934 .00407 45-552 .021 53 6.4489 039 5-6418 05649 7-70I5 .00436 229. 182 .021 82 45.8294 .039 29 25-45I7 .05678 17.6106 .00465 14-858 .02211 5.2261 03958 5-2644 05708 7-5205 .00495 02.219 .O224 4.6386 .03987 5.0798 05737 7-43I4 .00524 190.984 .02269 4.0661 .040 16 4.8978 05766 7-3432 00553 .OO5 82 80.932 171-885 .02298 .02328 3-5081 42.9641 .04046 .040 75 4-7i85 24.5418 05795 .05824 7-2558 17.1693 .006 II 63-7 023 57 2-4335 .041 04 4-3675 05854 7.0837 .0064 56- 259 .023 86 1.9158 041 33 4.1957 .05883 6.999 ,00669 49.465 02415 1.4106 .041 62 4.0263 .05912 6.915 .00698 43-237 .024 44 0.9174 04191 3-8593 .05941 6.8319 .00727 I37-507 .02473 40.4358 .0422 23-6945 0597 16.7496 .00756 32.219 .025 02 39-9655 0425 3-5321 05999 6. 668 1 .00785 27.321 .02531 9- 5059 .04279 3-37I8 .06029 6.5874 .00814 22-774 .0256 9.0568 .04308 3-2137 .06058 6-5075 .00844 18.54 .025 89 8.6177 04337 3-0577 .06087 6.4283 .00873 114-589 .026 19 38.1885 .043 66 22.9038 .06116 16.3499 .00902 10. 892 .02648 7.7686 043 95 2.7519 06145 6.2722 .00931 07.426 .026 77 7-3579 .044 24 2.602 06175 6.1952 .0096 04.171 .02706 6.956 .04454 2-4541 .06204 6.119 .00989 01.107 02735 6.5627 .04483 2. 3081 06233 6-0435 .OIOlS 98.2179 .02764 36. 1776 .04512 22. 164 .06262 15-9687 .01047 5-4895 02793 5.8006 .04541 2.0217 .06291 5-8945 .01076 2.9085 .02822 5-43I3 .0457 I.88I3 .06321 5.8211 .01105 0-4633 .02851 5-0695 .04599 1.7426 0635 5-7483 01135 88.1436 .02881 4-7 x 5i .04628 1.6056 .06379 5.6762 .on 64 85.9398 .029 i 34.3678 .046 58 21.4704 .06408 15.6048 .01193 3-8435 .02939 4-0273 .04687 L3369 .06437 5-534 .OI2 22 1.847 .02968 3-6935 .047 16 1.2049 .06467 5.4638 .01251 79-9434 .02997 3.3662 04745 1.0747 .06496 5-3943 .0128 8.1263 .03026 3-0452 04774 0.946 .06525 5-3254 .01309 7 6 -39 03055 32-7303 .04803 20.8l88 065 54 15-2571 01338 4.7292 .03084 2.4213 .04832 0.6932 .06584 5-1893 .01367 3- J 39 .031 14 2.1181 .048 62 0.5691 .066 I3 5.1222 .01396 1.6151 031 43 1.8205 .04891 0.4465 .06642 5-0557 .01425 0.1533 031 72 1.5284 .0492 0.3253 .06671 4.9898 01455 68.7501 .03201 31.2416 04949 20. 2056 .067 14.9244 .01484 7.4019 0323 0-9599 .049 78 0.0872 .0673 4-8596 OI5I3 6. 1055 032 59 0.6833 .05007 19.9702 .067 59 4-7954 .01542 4-858 .03288 0.4116 05037 9.8546 .06788 4-73I7 .01571 3-6567 03317 o. 1446 .05066 9-7403 .06817 4.6685 .Ol6 62.4992 03346 29.8823 .05095 19.6273 .06847 14.6059 .01629 1.3829 033 76 9.6245 .05124 9-5I56 .06876 4-5438 .01658 o. 3058 03405 9-37" 05153 9.4051 .06905 4-4823 .01687 59- 26 59 034 34 9.122 .051 82 9.2959 .06934 4.4212 ;.OI7l6 8.2612 03463 8.8771 .052 12 9.1879 .06963 4.3607 .017 46 57-29 .03492 28.6363 05241 19.0811 06993 14.3007 CO-TANG TANG. CO-TANG TANG. CO-TANG TANG. CO-TANG TANS. 89 88 87 86 416 NATURAL TANGENTS AND CO-TANGENTS. 4 i 5 C 3 3 TANS. CO-TANG. TANG. CO-TANG. TANG. CO-TANG. TANG. CO-TANS. .06993 14.3007 .08749 i -430I .105 i 9.5H36 .12278 8.M435 .07022 4.2411 .08778 39 T 9 105 4 .48781 .12308 .12481 .07051 4.1821 .08807 354 105 69 .46141 12338 105 36 .0708 4-1235 08837 3163 105 99 43515 .12367 .086 .071 I 4- 6 55 .08866 .2789 . 106 28 .40904 12397 .06674 071 39 14.0079 .08895 i .2417 .10657 9-38307 . 124 26 8.047 S 6 .071 68 3-9507 .089 25 .2048 . 106 87 35724 .12456 .02848 .07197 3-894 08954 .1681 .107 16 33 I 54 .12485 8.00948 .07227 3-8378 .08983 .1316 .10746 30599 12515 7.99058 .07256 3.7821 .09013 0954 10775 .28058 .12544 .97176 .072 85 13.7267 .09042 i .0594 . 108 05 9-2553 12574 7-95302 073 14 3- 6 7i9 .09071 1.0237 .108 34 .23016 .12603 93438 .07344 3-6i74 .09101 0.9882 .10863 .205 16 12633 .91582 .073 73 3.5634 .0913 0.9529 .10893 .18028 .12662 89734 .074 02 3. 5098 09159 0.9178 .10922 15554 .12692 .87895 j 074 3i 13.4566 .091 89 10. 882 9 .10952 9- I 393 .12722 7.86064 ; .07461 3-4039 .092 18 0.8483 .10981 . 106 46 .12751 .84242 .0749 3.35I5 .09247 0.8139 .110 1 1 .08211 .127 81 .82428 07519 3.2996 .09277 0.7797 .1104 057 89 .1281 . 806 22 .07548 3-248 .09306 0-7457 .1107 03379 .1284 78825 075 78 13-1969 093 35 10.711 9 .11099 9.00983 .12869 7-77035 .07607 3.1461 09365 0.6783 .11128 8.98598 .12899 75254 .076 36 3-0958 .09394 0.645 .in 58 .96227 .12929 7348 .07665 3-0458 .09423 0.6118 .11187 .93867 .12958 7I7I5 .07695 2.996? 09453 0.5789 .11217 .9152 .12988 69957 .07724 12.9469 .09482 10. 546 2 .11246 8.89185 .13017 7.68208 07753 2.8981 09511 0.5136 .11276 .86862 13047 . 664 66 .077 82 2.8496 .09541 0.4813 11305 84551 .13076 64732 .078 12 2.8014 957 0.4491 "335 .822 52 .13106 .63005 .07841 2.7536 .096 0.4172 .11364 .79964 13136 .61287 .0787 12.7062 .09629 10. 385 4 "394 8.77689 13165 7-59575 .07899 2.6591 .09658 0-3538 .11423 75425 i3i95 .57872 .07929 2.6124 .09688 0.3224 .11452 73 r 7 2 .13224 56176 079 58 2.566 .09717 0.2913 .11482 70931 13254 54487 .07987 2.5199 .09746 O. 260 2 .11511 .68701 13284 .52806 .08017 12.4742 .09776 10.2294 .11541 8.66482 -133*3 7.51132 .08046 2.4288 .09805 0.1988 "57 64275 !3343 .49465 .08075 2.3838 .09834 o. 168 3 .116 .62078 13372 .47806 .081 04 2-339 .09864 0.1381 .11629 59893 .13402 .46154 .081 34 2. 2946 .09893 0.108 .11659 577i8 13432 .44509 .081 63 12.2505 .099 23 10.078 .11688 8-55555 .13461 7.42871 .081 92 2. 2067 .09952 0.048 3 .11718 53402 I349 1 .4124 .08221 2.1632 .09981 0.018 7 .11747 51259 13521 .39616 .08251 2. 1 201 .100 ii 9.9893 .11777 .491 28 1355 37999 .0828 2.0772 .1004 .96007 .11806 .47007 1358 36389 .08309 12.0346 .10069 9.93101 .11836 8.44896 .13609 7.34786 08339 1.9923 .10099 .902 ii .11865 42795 13639 33i9 08368 1.9504 .101 28 87338 .11895 40705 .316 .08397 1.9087 .10158 .84482 .11924 38625 .13698 .30018 .084 27 .084 56 18673 11.8262 .10187 .102 l6 .81641 9-788x7 ."954 .11983 36555 8.34496 13728 13758 .28442 7.26873 .08485 I-7853 .10246 .76009 2013 32446 13787 2531 .085 14 1.7448 102 75 73217 . 2042 . 304 06 .13817 23754 .08544 I-7045 10305 .70441 . 2072 .28376 .13846 .22204 08573 1.6645 103 34 .6768 . 21 01 26355 .13876 .20661 .08602 11.6248 .10363 9-64935 2I 3 I & 243 45 .13906 7.19125 .08632 I-5853 103 93 .62205 . 21 6 .22344 13935 17594 .08661 I.546I . 104 22 5949 . 219 .20352 13965 .16071 .0869 1.5072 . 104 52 56791 .122 19 1837 I 3995 14553 .0872 1.4685 .10481 .541 06 .12249 .16398 . 140 24 .13042 .08749 11.4301 .1051 9-5I436 . 122 78 8.14435 .14054 7-"537 CO-TANS. TANG. CO-TANG. TANG. CO-TANG. TANG. CO-TANG. TANG. 8, 5 8 40 8 JO 8 2 NATUBAL TANGENTS AND CO-TANGENTS. 417 h 8 I ! s 1( 30 1 LO * TANG. CO-TANG. ( TANG. CO-TANG. TANG. CO-TANG. TANG. CO-TANG. o - 140 54 7-"537 .15838 6.31375 17633 5.67128 .19438 5- I 4455 I .14084 .10038 .15868 .30189 .17663 .66165 .19468 .13658 'a .14113 .08546 .15898 .29007 17693 .65205 .19498 .12862 3 I4M3 .07059 .15928 .27829 17723 .64248 19529 .12069 i 4 I4I73 055 79 15958 .26655 '7753 63295 19559 .11279 5 .14202 7.041 05 .15988 6.25486 17783 5-62344 .19589 5- 104 9 6 14232 .02637 .16017 .24321 .17813 6i397 .19619 .09704 7 .14262 .01174 .16047 .231 6 17843 .60452 . 196 49 .0892! 8 .14291 6.997 1 8 .16077 .22003 17873 595" .1968 .081 39 9 .14321 .98268 .161 07 .20851 i79 3 .1971 .0736 10 I435I 6.96823 .16137 6. 197 03 '7933 5-57638 .1974 5.06584 ii 14381 .95385 .16167 18559 17963 .56706 .1977 .05809 13 .1441 939 52 .161 96 .17419 17993 55777 .19801 05037 *3 .1444 92525 .16226 . 162 83 .18023 54851 .19831 .04267 H '447 .911 04 .162 56 15151 .18053 53927 .19861 .03499 15 .14499 6. 896 88 .16286 6.14023 .18083 5-53007 .19891 5.02734 16 14529 .88278 .16316 .12899 .18113 .5209 .19921 .01971 17 14559 .86874 163 46 .11779 18143 .51176 . 199 52 .0121 18 .14588 .85475 163 76 .10664 .18173 .50264 .19982 .00451 i9 .14618 .84082 .16405 09552 .18203 49356 .20012 4.99695 20 .14648 6.82694 16435 6.08444 18233 5-48451 .20042 4.9894 21 .14678 .81312 .16465 0734 .18263 47548 1 -20073 .98188 : 22 147 07 79936 .16495 .0624 18293 .46648 1 .20103 97438 23 *47 37 .78564 165 25 051 43 18323 45751 20133 .9669 24 .14767 .77199 i6555 040 51 .18353 44857 ! . 201 64 95945 25 . 147 96 6.75838 16585 6.02962 .18383 5-43966 .20194 4-95201 26 . 148 26 74483 .166,5 .01878 .18414 43077 . 2O2 24 9446 2 7 14856 73133 .16645 .00797 .18444 .42192 202 54 93721 28 .14886 .71789 .16674 5-9972 .18474 .41309 .20285 .92984 2 9 .14915 7045 .16704 .98646 .18504 .40429 20315 92249 30 I 4945 6.691 16 16734 5-97576 18534 5-39552 20345 4.91516 31 Z 4975 .67787 .16764 .9651 .18564 .38677 .20376 .90785 32 .15005 .66463 .16794 .95448 .18594 37805 .20406 33 15034 .651 44 .16824 9439 .18624 .36936 .20436 8 933 34 .15064 63831 16854 93335 .18654 3607 . 204 66 . 886 05 35 .15094 6.62523 .16884 5.92283 .18684 5.35206 .20497 4.87882 36 .15124 .612 19 .16914 9 I2 35 .18714 34345 .20527 .87162 37 I5I53 .59921 .16944 .901 91 18745 33487 20557 .86444 38 .15183 .58627 .16974 .891 51 .18775 .32631 .20588 85727 39 .15213 57339 .17004 .881 14 .18805 31778 .20618 .85013 40 15243 6.56055 17033 5.8708 18835 5.30928 .20648 4-843 4i .15272 54777 .17063 .86051 .18865 .3008 .20679 .8359 42 .15302 53503 .17093 .85024 .18895 29235 .20709 .82882 43 15332 52234 .17123 .84001 18925 28393 .20739 .82175 44 15362 5097 I 7i53 .82982 18955 27553 .2077 .81471 45 1539 6.4971 .17183 5.81966 .18986 5-26715 .208 4.80769 46 -1542 48456 .17213 80953 .19016 .2588 .2083 .80068 47 1545 .47206 17243 79944 .19046 .25048 .20861 7937 48 .1548 .45961 17273 .78938 19076 .24218 .20891 .78673 49 1551 4472 17303 77936 .19106 2339 1 .20921 7797 8 50 1554 6.43484 17333 5-76937 .19136 5-22566 .20952 4.77286 5i 52 1557 156 422 53 .41026 17363 17393 75941 74949 .19166 .19197 .21744 .20925 .20982 2 013 76595 .75906 53 1563 .39804 17423 7396 .19227 .201 07 .2 043 75219 54 1566 38587 17453 72974 .19257 .19293 2 73 74534 55 .15689 6-37374 17483 5.71992 .19287 5-1848 .2 I 04 4-7385I 56 .15719 36165 I75I3 .71013 I93I7 .17671 .2 I 34 7317 57 15749 .34961 17543 70037 19347 .16863 .2 I 64 7249 58 15779 3376i '7573 .69064 19378 .16058 .2 195 .71813 59 .15809 .32566 .17603 .68094 .19408 15256 2 225 7"37 60 .15838 6.31375 17633 5.671 28 194 38 5-14455 .2 256 4- 74 63 ' CO-TANG. TAMO. CO-TANG. TANG. CO-TANG. TANG. CO-TANG. TANG. * 8 1 8 7 90 7 8 418 NATURAL TANGENTS AND CO-TANGENTS. 1 TANG. 20 3 L3 I J i~ CO-TANG. ] TANG. L50 CO-TANG. .21256 4.70463 .23087 4-33148 24933 4.01078 26795 3-73205 .21286 .69791 .23117 32573 .24964 .005 82 .26826 .72771 .21316 .691 21 .23148 .32001 24995 .00086 26857 .72338 21347 68452 23179 3i43 .25026 3-99S9 2 .26888 .71907 21377 .67786 .23209 .3086 .25056 9999 .2692 .71476 .21408 4.67121 .2324 4.30291 25087 3.98607 26951 3.71046 .21438 .66458 .23271 .29724 .25118 .981 17 .26982 .70016 .21469 .65797 .23301 .29159 25M9 .97627 27013 .70188 .21499 65138 23332 28595 .2518 97 1 39 .27044 .69761 .21529 .6448 23363 .28032 .25211 .96651 .27076 69335 .215 6 4.63825 23393 4.27471 .25242 3.961 65 .27107 3.68909 2159 .63171 234 24 .26911 25273 .9568 27138 .68485 | .21621 .62518 23455 26352 .25304 .95196 .27169 .68061 1 .21651 ! .21682 .61868 6l2 19 23485 23516 25795 .25239 25335 .25366 94713 94232 .27201 .27232 .67638 .67217 j .217 12 4.60572 23547 4.24685 25397 3-93751 .27263 3.66796 21743 .59927 .23578 24132 .25428 93271 27294 .66376 21773 .59283 | . 236 08 2358 25459 9 2 793 27326 65957 .21804 .58641 23639 2303 2549 92316 27357 65538 -21834 .58001 .2367 .22481 25521 .91839 .27388 651 21 .21864 4-57363 237 4-21933 25552 3-9 I 3 6 4 .27419 3.64705 21895 56726 23731 .21387 .25583 .9089 .27451 .64289 .21925 .56091 .23762 . 208 42 .25614 .90417 .27482 .63874 21956 .55458 23793 .20298 25645 .89945 27513 .63461 ! .21986 . 548 26 .23823 19756 .25676 .89474 27545 .63048 .22017 4.54196 -23854 4.19215 .25707 3.89004 27576 3.62636 22047 53568 .23885 .18675 25738 88536 .27607 .622 24 .22078 52941 .23916 .18137 .25769 .88068 .27638 .61814 .22108 .52316 .23946 . I7 6 258 ^ .87601 .2767 .61405 .22139 ! .221 69 51693 4.51071 23977 .24008 .17064 4- 165 3 .25862 .87136 3.86671 .27701 27732 . 609 96 3-60588 .222 50451 .24039 .15997 .25893 .86208 .27764 .60181 .22231 .49832 24069 15465 .25924 .85745 27795 59775 .22261 49215 .241 .14934 25955 .85284 .27826 5937 .22292 .486 24131 .14405 .25986 .84824 .27858 .58966 .22322 4.47986 .241 62 4- * 3 8 77 .26017 3-84364 .27889 3-58562 22353 47374 .24193 1335 .26048 .83906 2792 .5816 .22383 .46764 .24223 .12825 . 260 79 .83449 .27952 57758 .22414 46155 24254 .12301 .261 1 .82992 .27983 57357 .22444 .45548 24285 .11778 .261 41 82537 .28015 569 57 22475 4.44942 24316 4.11256 .261 72 3.82083 .28046 3-56557 22505 .44338 24347 .10736 .26203 .8163 .28077 56159 .22536 -43735 24377 . 102 l6 .26235 .81177 .28109 5576i .22567 43134 .24408 .09699 .26266 .80726 .2814 55364 .22597 42534 .24439 .09182 .26297 . 802 76 .28172 .54968 .22628 4.41936 .2447 4.08666 . 263 28 3.79827 .28203 3-54573 ; .22658 4134 24501 .081 52 26359 79378 .28234 54i 79 , .22689 40745 24532 07639 2639 .78931 .28266 53785 22719 .401 52 24562 .071 27 .26421 .78485 .28297 53393 .2275 3956 24593 .06616 .264 52 .7804 .283 29 .53001 .22781 4.38969 24624 4.061 07 .26483 3-77595 .2836 3-52609 .22811 .38381 .24655 05599 26515 .77*52 28391 .52219 .22842 .22872 37793 .37207 . 246 86 .24717 . 050 92 .045 86 .26546 26577 ! 762 68 .28423 .28454 .51829 5H4 1 .22903 36623 .24747 .04081 .26608 .75828 .28486 51053 22934 4.3604 .24778 4-035 78 .26639 3.75388 .28517 3-50666 .22964 35459 .24809 03075 .2667 7495 .28549 502 79 22995 34879 .2484 .02574 .26701 74512 .2858 .49894 . 230 26 343 .24871 .02074 26733 .74075 .28612 .49509 .23056 33723 .24902 01576 .26764 7364 .28643 49 I2 5 23087 4-33M8 24933 4.01078 26795 3-73205 .28675 3.48741 CO-TANG . TANG. CO-TANG. TANG. CO^TANG. TANG. CO-TANG. TANG. 7 7 7 50 1 7i )0 7< 1 NATURAL TANGENTS AND CO-TANGENTS. 419 f ; 16 170 18 190 | TANG. CO-TANG. TANG. CO-TANG. TANG. CO-TANG. TANG. Co-T.lNG. .28675 3.48741 30573 3.27085 .32492 3.07768 34433 2.90421 . j. .28706 48359 30605 26745 3 2 524 .07464 34465 .90147 2 28738 47977 30637 .26406 32556 .071 6 .34498 89873 3 .28769 475 96 .30669 26067 .32588 .06857 3453 .896 4 .288 .47216 307 25729 .32621 . -065 54 34563 89327 5 .28832 3-46837 30732 3-25392 32653 3.062 52 34596 2.89055 6 .28864 .46458 .30764 25055 .32685 0595 .34628 .887 83 7 .28895 .4608 .30796 .24719 327 17 .05649 .34661 .88511 8 .28927 45703 .30828 24383 32749 053 49 34693 .8824 9 .28958 45327 .3086 .24049 .32782 .05049 34726 .8797 le .2899 3-44951 .30891 3-237I4 .32814 3-04749 34758 2.877 ii .29021 44576 .30923 .23381 .32846 0445 3479 1 8743 13 29053 .44202 30955 .23048 .32878 .041 52 .34824 .87161 X 5 .29084 .43829 .30987 .22715 32911 .038 54 34856 .86892 M .291 16 43456 .31019 .22384 32943 035 56 .34889 .86624 15 .29147 3-43084 31051 3-22053 32975 3.0326 .34922 2.86356 16 .291 79 42713 31083 .217 22 33007 .02963 34954 .86089 *7 .292 i 42343 3" 15 .21392 3304 .02667 34987 .85822 iS . 292 42 41973 3" 47 .21063 33072 .02372 35019 .85555 1 9 .29274 .41604 .31178 20734 33x04 .02077 35052 .85289 20 29305 3.41236 .3121 3. 204 06 33136 3-01783 35085 2.85023 i 31 29337 .40869 .31242 .20079 33169 .014 89 35117 84758 1 22 .29368 .40502 31274 19752 .33201 .on 96 3515 84494 23 .294 .40136 .31306 .19426 33233 .00903 35183 .84229 24 294 32 39771 31338 .191 .33266 .00611 .35216 .83965 i 25 .29463 3-39406 3137 3-I8775 33298 3.00319 35248 2.83702 ' 26 29495 39042 .31402 .18451 3333 .00028 35281 8 3439 , 2 7 .29526 .38679 31434 .18127 33363 2.99738 35314 .83176 28 29558 38317 .31466 .17804 33395 99447 .35346 .82914 2 9 2959 37955 .31498 .I7 4 8l 33427 .991 58 35379 .82653 3" .29621 3-37594 3153 3-I7I59 3346 2.98868 35412 2.82391 3i 29653 37234 31562 .16838 33492 .9858 35445 8213 32 29685 36875 31594 .16517 33524 .98292 35477 .8187 33 .29716 .36516 .31626 .16197 33557 .98004 3551 .8161 34 .29748 .36158 31658 15877 33589 .97717 35543 8135 35 .2978 3-358 .3169 3-I5558 33621 2-9743 35576 2.81091 36 . 298 1 1 35443 31722 .1524 33654 .97144 .35608 .80833 37 .29843 35087 31754 . 149 22 .33686 .96858 35641 .80574 38 .29875 34732 31786 .14605 337i8 96573 .35674 .80316 39 .29906 34377 .31818 .14288 33751 .96288 35707 .80059 40 .29938 3-34023 3185 3-I3972 33783 2.96004 3574 2.79802 4i .2997 3367 .31882 . 136 56 33816 95721 35772 79545 42 .30001 33317 3I9I4 I334I 33848 95437 35805 .79289 43 44 30033 .30065 32965 -32614 .31946 31978 .13027 .12713 .33881 33913 95155 .94872 35838 35871 79033 .78778 45 .30097 3-32264 3201 3.124 33945 2-9459 3594 2.78523 46 .301 28 319x4 .32042 .12087 33978 94309 35937 .78269 47 .3016 31565 32074 "775 3401 .94028 359 6 9 .78014 48 .30192 .31216 .32106 .11464 34043 .93748 .36002 7776i 49 .30224 .30868 32139 "i 53 34075 .93468 36035 77507 50 30255 3-30521 .32171 3. 108 42 .34108 2.93189 .36068 2.77254 5i 30287 30174 .32203 .10532 3414 .929 i .361 01 .77002 52 30319 .29829 32235 . 102 23 34173 .92632 36i34 7675 53 30351 .29483 .32267 .099 14 34205 .92354 .36167 .76498 54 . 303 82 .29139 .32299 .09606 34238 .92076 36i99 .76247 55 . 304 14 3-28795 32331 3.092 98 3427 2.91799 .36232 2.75996 56 .30446 .28452 32363 .089 91 34303 9x523 .36265 75746 57 .30478 .28109 .32396 . 086 85 34335 .91246 .36298 75496 58 30509 .27767 .32428 .08379 34368 .90971 36331 .75246 59 305 41 .27426 .3246 .08073 344 .90696 36364 74997 &> -30573 3-27085 .32492 3.07768 34433 2.90421 36397 2.74748 ' CO-TANG. TANG. CO-TANG. TANG. CO-TANG. TANG. CO-TANG. TANG. 730 720 71 70 420 NATURAL TANGENTS AND CO-TANGENTS. 2 TANG. DO CO-TANG. 2 TANG. LO CO-TANG. 2 TANG. 20 CO-TANG. 2 TANG. 30 CO-TANG. o 36397 2.74748 38386 2. 605 09 .40403 2.47509 .42447 2.35585 1 3643 74499 .3842 .602 83 .40436 .47302 .42482 35395 2 36463 74251 38453 60057 4047 47095 .425 16 35205 3 364 9 6 .74004 .38487 59831 .40504 .46888 42551 35015 4 36529 73756 3852 .59606 40538 .46682 42585 34825 5 .36562 2.73509 38553 2.59381 .40572 2.46476 .426 19 2.34^36 6 36595 .73263 38587 591 56 .40606 .4627 .42654 34447 7 .36628 73 J 7 .3862 58932 .4064 .46065 .42688 .34258 8 .36661 .72771 38654 .58708 .40674 .4586 .42722 .34069 9 36694 .725 26 .38687 .58484 .40707 45655 42757 .33881 '/O .36727 2.72281 .38721 2.58261 .40741 2-45451 .42791 2-33693 ii .3676 .72036 .38754 58038 .40775 45246 .42826 .33505 12 367 93 .71792 .38787 .40809 45043 .4286 .33317 *3 .36826 .71548 .38821 57593 .40843 .44839 .42894 3313 U 36859 .71305 .38854 57371 .40877 44636 .42929 32943 i5 .36892 2.71062 .38888 2-57I5 .40911 2-44433 .42963 2.32756 16 36925 .70819 .389 21 .56928 .40945 4423 .42998 .3257 17 36958 .70577 .38955 .56707 ,40979 .44027 43032 32383 18 .36991 70335 .38988 .56487 .41013 43825 43067 32197 19 .37024 .70094 .39022 .56266 .41047 43623 .43101 .32012 20 37057 2.69853 .39055 2. 560 46 .41081 2.43422 431 36 2.31826 21 3709 .696 12 .39089 55827 .41115 .4322 437 31641 22 37124 .69371 .39122 .55608 .41149 .43019 .43205 3M56 23 37157 .69131 39 1 56 .5538 9 4" 83 .428 19 43239 .31271 24 .68892 39*9 .5517 .41217 .42618 43274 .31086 25 .37223 2.68653 .39223 2.54952 .41251 2.42418 43308 2.30902 26 37256 .68414 .39 2 57 54734 .41285 .42218 43343 .30718 27 .37289 .68175 .39 2 9 545i6 41319 .42019 43378 .30534 28 .37322 67937 .39324 .54299 41353 .41819 43412 30351 2 9 37355 .677 39357 .54082 41387 .416 2 43447 .30167 3" 37388 2.67462 .39391 2.53865 .41421 2.41421 .43481 2.29984 31 37422 .67225 .39425 .53648 41455 .41223 435 16 ,29801 32 37455 .66989 39458 53432 .4149 .41025 4355 .29619 33 37488 .66752 39492 53217 .41524 .40827 43585 29437 34 37521 .66516 39526 53001 41558 .40629 4362 .29254 35 37554 2.66281 39559 2.52786 .41592 2.40432 43054 2.29073 36 37588 .66046 39593 52571 .41626 40235 .43689 .28891 37 37621 .65811 . 396 26 .52357 .4166 .40038 43724 .2871 38 37654 65576 3966 .52142 .41694 .39841 43758 .28528 39 37687 65342 .39694 .51929 41728 .39645 .28348 40 3772 2.65109 39727 2-51715 41763 2.39449 .43828 2.28167 41 37754 .64875 3976i 51502 41797 39253 .43862 .27987 42 43 37787 3782 .64642 .6441 39829 .51289 .51076 .41831 .41865 .39058 .38862 43897 43932 .27626 44 37853 .64177 .39862 .50864 .41899 .38668 .43966 27447 45 37887 2-63945 .39896 2.50652 .41933 2.38473 .44001 2.27267 46 3792 637H 3993 .5044 .41968 .38279 44036 .27088 47 37953 .63483 39963 .50229 . 420 02 .38084 .44071 . 269 09 48 .37986 .63252 39997 .50018 .42036 .3789* .44105 .2673 49 .3802 .63021 .40031 .49807 .4207 .37697 .4414 26552 50 38053 2.627 9 1 .40065 2-49597 .42105 2.37504 .44i 75 2.26374 51 .38086 .625 61 .40098 .49386 .42139 373" .4421 .261 96 52 .3812 62332 .40132 49 1 77 42173 .37118 44244 .26018 53 54 ! 381 86 .621 03 .61874 .401 66 .402 .48967 .48758 .42207 .42242 36925 36733 44279 44314 2 584 25663 55 .3822 2.61646 . 402 34 2. 485 49 .42276 2.36541 44349 2.25486 56 38253 .61418 .40267 4834 .4231 36349 .44384 25309 57 .38286 .611 9 .40301 .48132 42345 36158 .44418 .25132 58 3832 .60963 .40335 .47924 42379 35967 44453 .24956 59 38353 .60736 .40369 .47716 .42413 35776 .44488 2478 60 .38386 2.60509 .40403 2.47509 42447 2-35585 44523 2.24604 *^~ CO-TANG. TANG. CO-TANG. TANG. CO-TANG. TANG. CO-TANG. TANG. 690 68 67 66<> NATURAL TANGENTS AND CO-TANGENTS. 421 24 25 260 27 TANG'. ' CO-TANK. TANG. CO-TANG. TANG. CO-TANG. TANG. CO-TANG. .44523 2.24604 .46631 2.14451 48773 2.0503 50953 1.96261 .44558 .24428 .46666 . 142 88 .48809 .04879 .50989 .961 2 44593 .24252 .46702 .14125 .48845 .04728 .51026 95979 .44627 24077 -46737 13963 .48881 04577 .51063 95838 .44662 .23902 .46772 .13801 -489 17 .04426 .51099 .95698 .44697 2.23727 : .46808 2.13639 48953 2.04276 .51136 1-95557 447 32 23553 46843 13477 .48989 .041 25 5" 73 .95417 .44767 .23378 .46879 -13316 .49026 3975 51209 .95277 .44802 .23204 .46914 i3i54 .49062 03825 .51246 95137 44837 2303 ; 4695 .12993 .49098 03675 .51283 94997 .44872 2.22857 .46985 2.12832 49' 34 2.03526 51319 1.94858 .44907 .22683 .47021 .12671 .4917 03376 51356 .94718 .44942 .2251 47056 .125 ii .49206 .03227 .51393 94579 44977 22337 .47092 1235 .49242 .03078 SMS 9444 .45012 .221 64 .47128 .121 9 49278 .02929 51467 .94301 45047 2.21992 47163 2.1203 49315 2.0278 51503 1.941 62 .45082 ,2l8 19 .47199 .11871 49351 .02631 5154 .94023 4i>u7 .21647 47234 .117 ii 49387 .02483 51577 .93885 45152 -21475 .4727 "552 49423 02335 .51614 93746 45i87 .21304 47305 .11392 -49459 .021 87 .51651 .93608 .45222 2.2II32 47341 2.11233 -49495 2.02039 .51688 1-9347 45257 .20961 47377 .11075 | .49532 .01891 51724 93332 45292 .2079 .47412 .109 16 .49568 01743 .51761 93195 45327 .206 19 .47448 .10758 .49604 01596 51798 93057 45362 .20449 47483 .106 .4964 .01449 51835 .9292 45397 2. 2O2 78 47519 2. 104 42 49677 2.01302 .51872 1.92782 45432 . 201 08 47555 . 102 84 49713 OII 55 .51909 .92645 -45467 19938, 4759 . 101 26 49749 .01008 .51946 .92508 45502 .19769 .47626 .09969 .49786 .00862 51983 9 2 3 7 1 45537 '95V9 .47662 .09811 .49822 00715 .5202 92235 45573 2 - 194 3 .47698 2.09654 49858 2.00569 52057 1.92098 .45608 .19261 47733 .09498 .49894 .00423 .52094 .91962 45643 .19092 .47769 .09341 49931 .00277 .52131 .91826 .45678 .18923 .47805 .091 84 .49967 0)1 31 .52168 .9169 45713 18755 4784 .09028 .50004 1-99986 .52205 9 I 554 45748 2.18587 .47876 2.08872 .5004 1-99841 .52242 1.91418 45784 .18419 .47912 .087 16 .50076 99695 .52279 .91282 ; .45819 .18251 .47948 .0856 50113 9955 52316 .91147 45854 .18084 .47984 .08405 .50149 994 06 52353 .910 12 45889 .17916 .48019 .0825 50185 .99261 5239 .90876 45924 2.17749 48055 2.08094 .50222 1.991 X 6 52427 1.90741 4596 17582 .48091 07939 .50258 .98972 52464 .90607 45995 .17416 .481 27 .07785 50295 52501 .90472 4603 .17249 48163 .0763 50331 '98684 52538 9337 ! .46065 .17083 .48198 .07476 50368 985 4 .52575 .90203 j .461 01 2.169 17 .48234 2.07321 .50404 1-98396 .52613 1.90069 .461 36 .167 51 .4827 .07167 .50441 98253 5265 .89935 .46171 16585 .48306 .07014 50477 9811 .52687 .89801 .46206 .1642 48342 .0686 50514 97966 .52724 .89667 .46242 -16255 .48378 .06706 5055 .97823 52761 89533 .46277 2.1609 .48414 2.06553 50587 1.9768 .52798 1.894 ; -46312 15925 4845 .064 50623 9753 s .52836 .89266 .46348 .1576 .48486 .06247 .5066 97395 52873 89133 1 -46383 I559 6 .48521 .06094 .50696 97253 .5291 .89 .46418 15432 48557 .05942 50733 .97111 52947 .88867 ! -46454 2.15268 48593 2-0579 . 507 69 1.96969 .52984 1.88734 ! .46489 .15104 .48629 05637 . 508 06 .96827 .53022 .88602 46525 .1494 .48665 05485 50843 .96685 53059 .88469 .4656 14777 .48701 05333 .50879 96544 53096 88337 46595 .14614 48737 .051 82 .50916 .96402 53134 . 882 05 .46631 2.14451 48773 2.0503 50953 1.96261 53I7I 1.88073 CO-TANG, i TANG. CO-TANG. TANG. CO-TANG TANG. CO-TANG TANG. 65 64 63 62 422 NATURAL TANGENTS AND CO-TANGENTS. 28 29 30 31 \ TAWO. CO-TANO. TANG. CO-TANG. TANG. CO-TANG. TANG. CO-TANG. 53 I 7 I 1.88073 55431 1.80405 57735 1.73205 .60086 1.66428 53208 .87941 55469 .80281 57774 .73089 .601 26 .66318 .87809 55507 . 801 58 57813 72973 .60165 .66209 '53283 .87677 55545 .80034 57851 .72857 .60205 .66099 .5332 .87546 55583 .79911 5789 .72741 .60245 6599 53358 1.87415 55621 1.79788 579 2 9 1.72625 .60284 1.65881 53395 .87283 55659 .79665 .57968 72509 60324 .65772 53432 55697 79542 .58007 72393 .60364 65663 5347 .87021 55736 .79419 .58046 .72278 .60403 65554 53507 .868 9 ! 55774 .79296 .58085 .72163 .60443 65445 53545 1.8676 558i2 1.79174 58124 1.72047 .60483 I-65337 53582 .8663 5585 79051 .58162 .71932 .60522 .65228 5362 53657 .86499 .86369 55888 559 26 .78929 .78807 .58201 5824 .71817 .71702 .60562 .60602 .651 2 .65011 53694 .86239 55964 78685 .58279 .71588 .60642 .64903 53732 1.86109 .56003 1.78563 58318 I-7I473 .60681 1.64795 53769 .85979 .56041 .78441 58357 .71358 .60721 .64687 53807 8585 56079 .78319 58396 .71244 .607 61 64579 53844 .8572 561 17 .78198 58435 .71129 .60801 64471 53882 85591 .56156 .78077 58474 .71015 .60841 .64363 .5392 1.85462 56194 '77955 58513 1.70901 .60881 1.64256 53957 85333 .56232 77834 58552 .70787 .60921 .64148 53995 .85204 .5627 77713 .5859 1 .70673 .6096 .64041 54032 85075 56309 .77592 58631 .7056 .61 63934 547 .84946 56347 77471 .5867 .70446 .6104 .63826 54107 !. 84818 56385 I-7735I .58709 1.70332 .6108 1.63719 54M5 .84689 .56424 7723 .58748 .70219 .6112 .63612 54183 84561 .56462 7711 .58787 .70106 .6116 63505 .5422 84433 565 7699 .58826 .69992 .612 .63398 .54258 .84305 56539 .76869 -58865 .69879 .6124 .63292 54296 I.8 4 I7 7 56577 1.76749 .58904 1.69766 .6128 1.63185 54333 .84049 .56616 .7663 .58944 .69653 6132 .63079 54371 .83922 56654 7651 .58983 .69541 6136 .62972 .54409 54446 .83794 .83667 .56693 56731 7639 .76271 .59022 596i .69428 693 16 .614 .6 I44 .62866 6276 .54484 I-8354 .56769 1.7615! 59 101 1.69203 6148 1.62654 54522 83413 . 568 08 .76032 59*4 .69091 6152 .62548 5456 .83286 .56846 .75913 59179 .68979 .61561 .62442 54597 83159 .56885 75794 .59218 .68866 .61601 62336 54635 54673 .83033 1.82906 .56923 .56962 75675 I-75556 59258 59297 .68754 1.68643 .61641 .61681 6223 I.62I 25 547" .8278 -57 75437 593 36 .68531 .61721 .620 19 54748 .82654 57039 75319 59376 .68419 .61761 .61914 54786 .82528 .57078 752 59415 .68308 61801 .61808 .54824 .82402 .57116 .75082 594 54 .68196 61842 .61703 54862 1.82276 57155 1.74964 59494 1.68085 61882 1.61598 549 8215 57193 .74846 59533 .67974 61922 .61493 54938 .82025 .57232 .74728 59573 .67863 61962 .61388 54975 .81899 57271 .7461 59612 67752 , 620 03 .61283 55013 .81774 .57309 74492 59651 .6 7 6 4I 62043 .61179 55051 1.81649 57348 '74375 .59691 1-6753 .62083 1.61074 .55089 .81524 57386 74257 5973 .67419 .621 24 .6097 55127 .81399 57425 .7414 5977 .67309 621 64 .60865 55165 .8 I274 57464 .74022 .59809 .67198 .62204 .60761 55203 8115 57503 73905 598 49 .67088 .62245 60657 .55241 1.81025 57541 1.73788 .59888 1.66978 62285 1-60553 .55279 .80901 5758 73671 . 599 28 .66867 .62325 .60449 .55317 .80777 57619 -73555 599 67 66757 .62366 .60345 55355 80653 57657 .73438 .60007 .66647 . 624 06 .60241 5539? .80529 .57696 .73321 .60046 .66538 .62446 .60137 55431 1.80405 57735 1.73205 .60086 1.66428 .62487 1.60033 ' CO-TANG. TANG. CO-TANG. TANG. CO-TANG. TANG. CO-TANG . TANG. 610 60 _ 59 58 NATURAL TANGENTS AND CO-TANGENTS. 423 32 33 340 | 35 ] TAKO. CO-TANO. TANG. CO-TANO. TANG. CO-TANO. TANO. CO-TANO. .62487 .62527 1.60033 .649 82 1.53986 53888 6745* 67493 1.48256 .48163 .70021 .70064 1.42815 .42726 .62568 . 598 26 .65023 5379* 67536 .4807 .70107 .42638 .62608 59723 65065 53693 .67578 47977 .70151 .4255 .62649 .5962 .65106 53595 .6762 .47885 .70194 .42462 .62689 1-595*7 .65148 1-53497 67663 1.47792 .70238 1.42374 .6273 -594*4 .65189 534 .67705 .47699 .70281 .42286 .6277 .62811 593" .59208 .65231 .65272 53302 53205 67748 .6779 47607 475*4 70325 70368 .42198 .4211 .62852 59*05 653*4 53*07 .67832 47422 .70412 .42022 .62892 1.59002 65355 1.5301 67875 1-4733 70455 1.41934 62933 589 .65397 529*3 679*7 47238 .70499 4*847 .62973 58797 .65438 .52816 .6796 47*46 7 542 4*759 .63014 .58695 .6548 527*9 .68002 4753 .70586 .41672 63055 58593 65521 .526 22 .68045 . 469 62 .70629 4*584 63095 1.5849 65563 I.52525 .68088 1.4687 .70673 1-4*497 i 63136 58388 .65604 .52429 6813 .46778 .70717 4*49 : 63177 .58286 .65646 52332 .68! 73 .46686 .7076 .41322 .632 17 .58184 || .65688 52235 .68215 46595 .70804 4*235 .63258 .58083 65729 .52139 .68258 .46503 .70848 .41148 .63299 1.57981 6577* 1.52043 .68301 1.46411 .70891 1.41061 6334 57879 658 I3 .51946 .68343 4632 70935 .40974 .6338 57778 65854 5*85 .68386 .46229 .70979 .408 87 ., 63421 57676 .65896 5*754 .68429 .408 63462 57575 65938 .5*658 .6847* .46046 .71066 407*4 63503 1-57474 .6598 1.51562 .68514 *-45955 .7111 1.40627 63544 57372 .66021 .5*466 68557 .45864 7** 54 4054 63584 5727* .66063 5*37 .686 45773 .71198 .40454 '. 636 66 57*7 .57069 .66105 .66147 51*79 .68642 .68685 45682 .45592 .71242 .7*285 .40367 .40281 63707 1.56969 .66189 1.51084 .68728 I-4550I 7*329 1.40195 63748 .56868 .6623 .50988 .68771 4541 7*373 .40109 63789 567 67 .66272 508 93 .688 14 4532 7*4*7 .40022 6383 .56667 .66314 50797 .68857 45229 .71461 39936 .63871 56566 66356 .50702 .689 45*39 3985 63912 1.56466 .66398 1.50607 .68942 1.45049 7*549 I-39764 63953 .56366 .6644 505*2 .68985 44958 7*593 39679 63994 .56265 .66482 504*7 .69028 .44868 7*637 39593 64035 .56165 .66524 50322 .69071 44778 71681 39507 .64076 .56065 .66566 . 502 28 .69114 .44688 7*725 .39421 .641 17 1.55966 .66608 i-5oi33 69*57 1.44598 7*769 I-39336 641 58 .55866 .6665 .50038 .692 .445o8 7*8*3 .3925 .64199 .55766 .66692 499 44 69243 .44418 7*857 .39*65 .6424 .55666 .66734 49849 .69286 44329 .71901 .39079 .64281 .55567 .66776 49755 .69329 44239 .71946 .38994 64322 1-55467 .66818 1.49661 69372 1.44149 7*99 1.38009 .64363 .55368 1 .6686 .49566 .69416 .4406 72034 .38824 .55269 1 .66902 .49472 69459 4397 .72078 .38738 .64446 55*7 .66944 4937 s .69502 .43881 .72122 .38653 . 644 87 5507* .66986 .492 84 69545 4379 2 .72166 .38568 .64528 1.54972 .67028 1.4919 .69588 1-43703 .722 ii 1.38484 64569 54873 1 .67071 4997 .69631 .43614 72255 38399 .6461 54774 -671*3 .49003 69675 43525 .72299 383*4 .646 52 54675 67*55 .48909 .69718 43436 72344 . 382 29 .646 93 54576 67*97 .488 16 .69761 43347 .72388 38*45 64734 1.54478 67239 1.48722 .69804 1.43258 72432 1.3806 64775 54379 .67282 .48629 .69847 43*69 .72477 37976 .64817 .64858 .54281 54*83 .67324 -67366 48536 .48442 .69891 .69934 .4308 .42992 .72521 72565 .3789* .37807 .64899 54085 .67409 .48349 .69977 .42903 .7261 .37722 .64941 1.53986 6745* 1.48256 .70021 1.42815 .72654 *-37638 CO-TANG. 'IANG. CO-TANG. TANG. CO-TANG. TANO. CO-TANG. TANG. 57 56 550 54 424 NATURAL TANGENTS AND CO-TANGENTS. S6 37 |1 38 39 J TANG. CO-TANG. TANG. CO-TANG. 1 TANG. CO-TANG. TANG. CO-TANG. .72654 1-37638 75355 1.32704 .78129 1.27994 .80978 1.2349 .72699 37554 75401 .326 24 .27917 .81027 .23416 72743 3747 75447 32544 .782 22 .27841 .81075 23343 .72788 37386 75492 .32464 .78269 .27764 .81123 .2327 .72832 37302 75538 32384 .78316 .27688 .81171 .23196 .72877 1.37218 75584 1.32304 78363 1.276 ii .8122 1.231 23 .72921 37134 .75629 .32224 .7841 27535 .81268 2305 .72966 3705 75675 .32144 .78457 .27458 .81316 .22977 7301 75721 .32064 .78504 27382 .81364 .22904 73055 . 368 83 75767 .31984 78551 27306 81413 .22831 731 1.368 .75812 1.31904 .78598 1.2723 .81461 1.22758 731 44 .367 16 75858 .31825 .78645 27153 .8151 .22685 .36633 75904 31745 .78692 .27077 .81558 .226 12 73234 36549 7595 .31666 78739 .27001 .81606 22539 73278 1.36466 75996 31586 .78786 .269 25 .816 55 | .22467 73323 .36383 .76042 1-31507 .78834 1.26849 .81703 1.22394 73368 .363 .76088 3 I 4 2 7 .78881 .26774 81752 .223 21 73413 36217 76134 31348 .78928 .26698 .818 .22249 73457 36133 .7618 .31269 .78975 .2662-2 .81849 .221 76 73502 36051 .76226 .79022 .26546 .81898 .22104 73547 1.35968 .76272 1.311 i 797 1.26471 .81946 I.2203I 73592 .35885 .76318 31031 .79117 26395 .81995 21959 73637 35802 76364 30952 .79164 .26319 .82044 .21886 .73681 .35719 .7641 30873 .79212 .26244 .82092 .2l8l4 73726 35637 .76456 30795 79259 .261 69 .82141 .21742 73771 1-35554 76502 1.30716 .79306 1.26093 .8219 1.2167 .73816 35472 1-76548 .30637 79354 .26018 .82238 .21598 .73861 35389 76594 30558 .79401 25943 .82287 .21526 .73906 35307 .7664 .3048 -79449 .25867 .82336 21454 73951 -35224 .76686 .30401 79496 .25792 .82385 .213 82 73996 I-35I42 76733 1.30323 79544 1.25717 82434 I.2I3I 74041 .3506 .76779 30244 79591 .25642 .82483 .212 38 .74086 34978 .76825 . 301 66 79639 25567 82531 .2Il66 .741 31 .34896 .76871 .30087 .79686 .25492 .8258 .21094 .74176 34814 . 7 6 9I 8 .30009 -79734 -25417 .82629 .2IO23 i .74221 L34732 .76964 1.29931 .79781 1-25343 .82678 1.20951 -742 67 3465 .7701 .298 53 .79829 .25268 .82727 .20879 74312 34568 77057 29775 79877 25193 .82776 .20808 74357 34487 77103 . 296 96 799 24 .25118 .82825 .207 36 .74402 34405 77M9 .29618 .79972 .25044 .82874 .20665 74447 I-34323 .77196 1.29541 .8002 1.24969 .82923 1.20593 .74492 34242 .77242 29463 .80067 .24895 .82972 .20522 74538 .3416 .77289 29385 .80115 .2482 .83022 20451 74583 34079 '77335 2 93 07 .80163 .24746 .83071 .20379 .74628 3399 8 77382 .292 29 .80211 .24672 .8312 . 2O3 08 .74674 1.33916 .77428 1.29152 .80258 1-24597 .83169 1.20237 74719 33835 77475 .29074 . 803 06 24523 .83218 . 201 66 .74764 33754 77521 .28997 80354 24449 .83268 .20095 748i .33673 77568 .28919 . 804 02 24375 83317 . 200 24 74855 .33592 77615 .28842 .8045 24301 83366 .19953 749 1-335" .77661 1.28764 .80498 1.24227 83415 1.19882 74946 3343 .77708 .28687 .80546 -24153 83465 .19811 .74991 33349 77754 .2861 .80594 .24079 83514 .1974 75037 .33268 .77801 -28533 .80642 .24005 83564 .I 9 66 9 .75082 33187 .77848 28456 .8069 2393 1 .83613 19599 .75128 1-33107 77895 1.28379 .807 38 1.23858 .83662 I.I95 28 75173 .33026 77941 .28302 . 807 86 .23784 .83712 19457 75219 .32946 .77988 .28225 .80834 2371 83761 .19387 .75264 .32865 78035 .28148 .80882 23637 .83811 .19316 7531 32785 .78082 .28071 .8093 23563 .8386 .19246 75355 1.32704 .78129 1.27994 .80978 1.2349 .8391 i- 19* 75 CO-TAKG. TANG. CO-TANG. TANG. CO-TANG. TANG, CO-TANG. TANG. 53 52 510 50 NATURAL TANGENTS AND CO-TANGENTS, 425 40 410 42 43 TANG. CO-TANG. TANG. CO-TANG. TANG. CO-TANG. TANG. CO-TANG. 8391 i- 19 1 75 .86929 I-I5037 .9004 1. 11061 93252 1.072 37 8396 .19105 .8698 .14969 90093 .10996 93306 .071 74 .84009 87031 .14902 .90146 .10931 9336 .071 12 .84059 .18964 .87082 .14834 .90199 . 108 67 93415 .07049 .84108 .18894 87133 .14767 .90251 . 108 02 .93469 .06987 .84158 1.18824 .87184 1.14699 .90304 1.10737 93524 1.06925 .84208 .18754 .87236 .14632 9357 .10672 93578 .06862 .84258 .18684 .87287 14565 .9041 .10607 93633 .068 .84307 .18614 .87338 .14498 90463 10543 .93688 .06738 84357 .18544 87389 1443 .90516 . 104 78 93742 .06676 .84407 1.18474 .87441 1.14363 .90569 1.10414 93797 1. 066 13 84457 .18404 .87492 . 142 96 .90621 .10349 93852 06551 84507 18334 87543 . 142 29 .90674 .10285 .93906 .06489 84556 .18264 87595 .141 62 .90727 .IO2 2 .93961 .064 27 .84606 .18194 .87646 .14095 .90781 .10156 .94016 .06365 .84656 1.18125 .87698 1.14028 .90834 I.IOOgi 94071 1.06303 .84706 .18055 .87749 .13961 .90887 .10027 94 1 25 .06241 .84756 .17986 .87801 .13894 994 .09963 .9418 .06179 .84806 .17916 .87852 .13828 .90993 .09899 94235 .061 17 .84856 .17846 .87904 .13761 .91046 .09834 .9429 .06056 .84906 1.17777 87955 1.13694 .91099 1.0977 94345 1.05994 .84956. .17708 .88007 .13627 9" 53 .09706 944 05932 .85006 .17638 .88059 .13561 .91206 .09642 94455 0587 85057 17569 .8811 13494 .91259 .09578 9451 .05809 .85107 175 .88162 .13428 913*3 .09514 94565 05747 85157 I-I743 .88214 1.13361 .91366 1.0945 .9462 1.056 85 .85207 .17361 .88265 .13295 .91419 .09386 .94676 .056 24 85257 .17292 .88317 .13228 9M73 .09322 94731 .05562 85307 .17223 .88369 .131 62 .91526 .09258 .94786 .0550! 85358 .17154 .88421 .13096 .9158 .09195 .94841 05439 .85408 1.17085 88473 1.13029 9 l6 33 1.09131 .94896 1-05378 .85458 .17016 .88524 12963 .91687 .09067 94952 05317 85509 .16947 .88576 .12897 .9174 .09003 .95007 05255 85559 .16878 .88628 .12831 .91794 .0894 .95062 .051 94 .85609 .16809 .8868 .12765 .91847 .088 76 .95118 051 33 .8566 1.16741 88732 1.12699 .91901 1.08813 95173 1.05072 8571 .16672 12633 91955 .08749 95229 0501 .85761 .16603 .88836 12567 .92008 .08686 .95284 .04949 .85811 I 6s35 .88888 .12501 .92062 .08622 9534 .04888 .85862 .16466 .8894 12435 .921 16 08559 95395 .048 27 .85912 1.16398 .88992 1.12369 .9217 1.08496 95451 1.04766 85963 .86014 .16329 .16201 .89045 .89097 .12303 .12238 .92223 .92277 .08432 .08369 955o6 95562 .04705 .04644 .86064 .16192 .891 49 .121 72 92331 .08306 .95618 04583 .861 15 .16124 .89201 .121 06 92385 .08243 95673 .045 22 .86166 1.16056 .89253 I.I204I .92439 1.081 79 957 29 1.04461 .86216 .15987 .89306 "975 .92493 .081 16 95785 .04401 .86267 .15919 .89358 .11909 92547 08053 .95841 0434 .86318 .158 51 .8941 .11844 .92601 0799 95897 .042 79 .86368 15783 .89463 .11778 92655 .07927 95952 .042 18 .86419 89515 1.11713 .92709 1.07864 .96008 1.041 58 .8647 15647 .89567 .11648 .92763 .07801 .96064 .04097 .8652! 15579 .8962 .11582 .92817 077 38 .961 2 . 040 36 .86572 155" .89672 "5 X 7 .92872 .07676 .961 76 03976 .86623 15443 .89725 .11452 .92926 .076 13 .96232 03915 .86674 I-I5375 .89777 1.11387 .9298 1-0755 .96288 1-03855 .86725 .15308 .8983 .113 21 93034 07487 96344 03794 .86776 .1524 .89883 .11256 .93088 .07425 .964 3734 .86827 15172 .89935 .III 91 93M3 .07362 96457 .03674 .86878 .15104 .89988 .III 26 93197 .07299 96513 036 13 .86929 .9004 I. Ilo6l .93252 1.07237 .96569 1-03553 CO-TANG. TANG. CO-TANG. TANG. CO-TING. TANG. CO-TANG. TANG. 490 480 470 460 NX* 426 NATURAL TANGENTS AND CO-TANGENTS. 4 4 ' TANG. CO-TANG. o .96569 035 53 X .96625 03493 2 .96681 03433 3 .96738 03372 4 96794 .033 12 5 .9685 .032 52 6 .96907 03192 7 96963 .03132 8 .9702 .03072 9 .97076 .03012 10 971 33 .02952 ii .97189 .02892 12 .972 46 .02832 13 .97302 .02772 14 97359 .02713 15 .97416 .02653 16 .97472 02593 i7 97529 025 33 18 .97586 .024 74 J 9 97643 .024 14 20 ~ 977 023 55 CO-TAUG. TANG. 4 5 4 4 4* t o ' ' TANG. CO-TANG. ' TANG. CO-TANG. i ~6o~ 21 977 56 .02295 39 41 .98901 .Oil 12 9 59 22 .022 36 38 42 .98958 .01053 18 58 23 [9787 .021 76 37 43 .99016 .00994 i7 57 24 979 2 7 .021 17 36 44 99073 09935 16 56 25 .97984 .02057 35 45 99 1 3i .00876 15 55 26 .98041 .01998 34 46 .99189 .00818 J 4 54 27 .98098 .01939 33 47 .99247 .00759 13 53 28 98155 .01879 32 48 99304 .00701 12 52 2 9 .98213 .Ol82 49 .99362 .00642 II 30 .9827 .01761 30 50 9942 .00583 10 50 49 31 3 2 98327 .98384 .OI7O2 .01642 2 9 28 5i 52 .99478 99536 00525 .00467 8 48 33 .98441 .01583 27 53 99594 .00408 7 47 34 98499 01524 26 54 .99652 0035 6 46 35 985 56 .01465 25 .00291 5 45 36 .98613 .01406 24 56 .99768 .00233 4 44 37 .98671 01347 23 57 .99826 00175 3 43 38 .98728 .01288 22 58 .99884 .001 16 2 42 39 .98786 .01229 21 59 .99942 .00058 ; .i 41 4 .98843 .on 7 20 60 i o 40 / f CO-TANG. TANG. 1 ~7~ CO-TANG. TANG. ~T 4 50 4 50 Preceding Table contains Natural Tangents and Co-tangents for every minute of the quadrant, to the radius of i. If Degrees are taken at head of columns, Minutes, Tangents, and Co-tan- gents must be taken from head also ; and if they are taken at foot of col- umns, Minutes, etc., must be taken from foot also. ILLUSTRATION. .1974 is tangent for n 10', and co-tangent for 78 50'. To Compxite Tangents and. Co-tangents for Seconds. Ascertain tangent or co-tangent of angle for degrees and minutes from Table ; take difference between it and tangent or co-tangent next below it. Then as 60 seconds is to difference, so are seconds given to result required, frhich is to be added to tangent and subtracted from co-tangent. ILLUSTRATION. What is the tangent and co- tangent of 54 40' 40"? SSSSX S$# P "? bto = ' i ;j$} .-8, Serene, Then 60 : .00087 ''' 4 : -00 5 8 , which, added to 1.41061 = 1.41119 tangent. Co-tangent of 54 40', per Table .70801 ) ,.. Co-tangent of 5X1, = . 708 48 } difference. Then 60 : .00043 : : 4 : .000 29, which, subt'd from .70891 = .70862 co-tangent. To Compute Tangent or Co-tangent of* any A.ngle in Degrees, IMinxites, and Seconds. Divide Sine by Cosine for Tangent, and Cosine by Sine for Co-tangent. EXAMPLE. What is tangent of 25 18'? Sine = .427 36 ; cosine = .904 08. Then ' 4273 = .4727 tangent. .90408 To Compute Number of* Degrees, Minutes, and Seconds of a given Tangent or Co-tangent. When Tangent is given. Proceed as by Rule, page 402, for Sines, substi- tuting Tangents for Sines. EXAMPLE. What is tangent for 1.411 19? Next less tangent is 1.41061, arc for which is 54 40'. Next greatest tangent is 1.411 48, difference between which and next less is .00087. Difference between less tabular tangent and one given is 1.41061 1.411 19 = .00058. Then .00087 : -00058 :: 60 : 40, which, added to 54040 =54 40' 40". When Co-tangent is given. Proceed as by Rule, page 402, for Cosines, substituting Co-tangents for Cosines. AEROSTATICS. 427 AEROSTATICS. Atmospheric Air consists, by volume, of Oxygen 21, and Nitrogen 79 parts; and in 10000 parts there are 4.9 parts of Carbonic acid gas. By weight, it consists of 23 parts of Oxygen, and 77 of Nitrogen. One cube foot of Atmospheric Air at surface of Earth, when barome- ter is at 30 ins., and at a temperature of 32, weighs 565.0964 grains = ,080 728 Ibs. avoirdupois, being 773.19 times lighter than water. Specific gravity compared with water, at 62.418 = .obi 293 345. Mean weight ot a column of air a foot square, and of an altitude equal to height of atmosphere (barometer 30 ins.), is 2124.6875 Ibs. = 14.7548 Ibs. per sq. inch = support of 34.0393 feet of water. Standard pound is computed with a mercurial barometer at 30 ins. ; hence, as a cube inch of mercury at 60 weighs .4907769 Ibs., pressure of atmos- phere at 60 = 14.723307 Ibs. per square men. 12.3873 cube feet of air weigh a pound, and its weight varies about I gr. for each degree of heat. Extreme height of barometer in latitude 30 to 35 N. = 3o.2i his. Rate of expansion of Air, and all other Elastic Fluids for all temperatures, is essentially uniform. From 32 to 212 they expand from i to 1.3665 volumes = .002036 or ^[rY^th part of their bulk for every degree of heat From 212 to 680 they expand from 1.3665 to 2.3192 = .002036 for each degree of heat. Thus, if volume of air at 132 is required, 132 32 = 100, and i + 100 X .002036 = 1.2036 volumes. Height, at Equator is estimated at 300 feet greater than at Poles, its mean height at 45 latitude. In like latitudes, air loses i for every 340 feet in height above level of sea. Below surface of Earth, temperature increases. Elasticity of air is inversely as space it occupies, and directly as its density. When altitude of air is taken in arithmetical proportion, its Rarity will be in geometric proportion. Thus, at 7 miles above surface of Earth, ah* is 4 times rarer or lighter than at Earth's surface; at 14 miles, 16 times; at 21 miles, 64 times, and so on. Density of an aeriform fluid mass at 32 and at t will be to each other as i + .002 088 (t 32) is to i. For Volume, Pressure, and Density of Air, see Heat, page 521. Altitude of Atmosphere at ordinary density is = a column of mercury 30 ins. in height, divided by specific gravity of air compared with mercury. Hence 30 ins. = 2.5 feet, which, divided by .000094987, specific gravity of air compared with mercury, = 263 igfeet=: 4.985 miles. Gay Lussac, Humboldt, and Boussingault estimated it at a minimum of 30 miles, Sir John Herschell 83, Bravais 66 to 100, Dalton 102, and Liais at 1 80 or 204 miles. The aqueous vapor always existing in air, in a greater or less quantity, being lighter than ah*, diminishes its weight in mixing with it ; and as, other things equal, its quantity is greater the higher the temperature of the air, its effect is to be considered by increasing the multiplier of t by raising it to .002 22. Glaisher and Coxwell, in i862 4 ascended hi a balloon to a height of 37 ooo feet 428 AEROSTATICS. At temperature of 32, mean velocity of sound is 1089 feet per second. It is increased or diminished about one foot for each degree of temperature below or above 32. Velocity of sound in water is estimated at 4750 feet per second. Velocity of /Sound at Various Temperatures. Per Second. Per Second. Per Second. Per Sec.oi 5 i4 23 Feet. 1056 1070 1079 32 50 59 Feet. 1089 IIO2 III2 68 g Feet. 1122 1132 1142 95 104 3 Feet. 1152 1161 1171 Motions of air and all gases, by force of gravity, are precisely alike to those of fluids. Sensation of hearing, or sound, cannot exist in an absolute vacuum. The human voice can be heard a distance of 3300 feet. Echo. At a less distance than 100 feet there is not a sufficient interval between the delivery of a sound and its reflection to render one perceptible. To Compute Distances "by Velocity of Sound in Air. 1089 X T V 1 + [.002 088 (t 32)] = distance in feet per second, T representing time before report was heard, and t temperature of air. ILLUSTRATION. Flash of a caunou from a vessel was observed 13 seconds before report was heard; temperature of air 60; what was distance to vessel? 1089 X 13 V i + [-002088(60 32)] 1089 X 13 X 1.029 J 4 5 6 7- 5$feet=2.j6 miles. Theoretical velocity with which air will flow into a vacuum, if wholly un- obstructed, is Vvgh = 1347.4 /eetf per second. In operation, however, it is 1347.4 X .707 = 952.6i feet. To Compute "Velocity of A.ir FloAving into a "Vacuum. \/2 g h X c = v in feet per second, c representing coefficient of efflux. Coefficients for openings are as follows : Circular aperture in a thin plate 65 (0.7 Cylindrical adjutage 92 | Conical adjutage 93 Velocity of Sound, in Several Solids. Velocity in Air=i. Lead 3.9 I Zinc 9.8 I Pine 12.5 I Glass 11.9 I Steel 14.3 Gold 5.6 I Oak 9.9 | Copper ... 11.2 | Pine 12.5 | Iron 15.1 To Compute Elevations "by a Barometer. Approximately * 60 ooo (log. B log. 6) C height in feet ; B and b representing heights of barometer at lower and upper stations, and C correction due to T -}- 1 or temperatures of lower and upper stations. Values of C or T-f t. C C C o C C C O C 4 973 60 .996 80 .018 oo .04 20 1.062 140 1.084 1 60 . 06 42 .976 62 .998 82 .02 02 .042 22 1.064 142 1.087 162 . 08 44 .978 64 84 .022 04 .044 24 1.067 144 1.089 164 . ii 4 6 .98 66 .002 86 .024 06 .047 26 1.069 146 1.09! 166 13 48 .982 68 .004 88 .027 08 .049 28 1.071 148 1-093 168 So .984 70 .007 90 .029 IO .051 30 1-073 ISO 1.096 170 *7 52 987 72 .009 92 .031 12 053 32 1.076 1*2 1.098 172 . 2 54 .989 74 .Oil Q4 033 14 .0*6 134 1.078 r ^4 i.i I. 22 56 .991 76 .013 96 .036 16 1.058 136 1. 08 is6 I.IO2 176 I. 24 58 993 7 .016 98 .038 118 i. 06 1.082 i5 y I.IO4 178 I. 26 For more exact formulas, see Tables and Formulas, by Capt. T. S. Lee, U. S. Top. Eng., 1853. AEROSTATICS. 429 Their values vary approximately .001 1 per degree. Lower Station. 77.6 30.05 = 77.6 + 70.4 1.093, log. 6 = 1.4778, log. 6 = 1.374. Then 6ooooX (1.4778 1.374) X 1.093 = 6807.2/6^. Upper Station. ILLUSTRATION. Thermometer 70.4 Barometer 23.66 To Compute Elevations Toy a Thermometer. 520 B + B 2 X C = height in feet. B representing temperature of water boiling at elevated station deducted from 212. Correction for temperatures of air at lower and upper stations, or T + t, to be taken from table, page 428, as before. ILLUSTRATION. Temperature of water boiling at upper station 192; temperature of air 50 and 32. C = 1.02. Then 520X212 192 + 212 192 X 1.02=: io8o8/eefc To Compute Capacity of a Balloon, etc., see page 218. B ar oxnet er . Elevations by Barometer Readings, (Astronomer Royal.) Mean Temperature of Air 50. For correction for temperature, see note at foot. Height. Barom. Height. Barom. Height. Barom. Height. Barom. Height. Barom. Feet. Ins. Feet. Ins. Feet. Ins. Feet. Ins. Feet. Ins. o 31 600 30.325 1500 29-34 4000 26.769 7000 23-979 50 30-943 650 30.269 1600 29. 233 4250 26. 524 7 5 23-543 too 30.886 700 30.214 1750 29.072 4500 26. 282 8000 23.115 150 30- 8 3 750 30.159 1800 29.019 4750 26 042 8500 22.695 2OO 30-773 800 30. 103 2000 28.807 5000 25.804 9000 22.282 250 30.717 850 30.048 2250 28.544 5250 25-569 9500 21.877 300 30.661 900 29.993 25OO 28.283 5500 25-335 0000 21.479 35 30.604 IOOO 29.883 2750 28.025 5750 25. 104 0500 21.089 400 30.548 1 100 29.774 3000 27.769 6000 24.875 IOOO 20.706 450 30.492 I2OO 29.665 3250 27-5I5 6250 24.648 1500 20.329 500 30-436 1300 29-556 3500 27.264 6500 24.423 2000 19.959 550 30-381 1400 29.448 375 27.015 6750 24.2 2 500 I9-952 Barometer. Correction for Capillary Attraction to be added in Inches. Diameter of tube Correction, unboiled Correction, boiled 5 .007 004 4 55 3 25 .2 .1 .014 .02 .025 04 .059 .08 7 .007 OI .014 .02 029 .044 To Compute Heignt. RULE. Subtract reading at lower station from reading at upper station, difference is height in feet. Table assumes mean temperature of atmosphere to be 50 F. or 10 C. For other temperatures following correction must be applied. Add together temperatures at upper and lower station If this sum, in degrees in P., is greater than 100, increase height by 1 fa Q part for every degree of excess above 100; if sum is less than 100, diminish height by 10 1 00 part for every degree of defect from 100. Or if sum in C is greater than 20, increase height by ^1^ part for every degree of excess above 20; if sum is less than 20, diminish height by g i ff part for every degree of defect from 20. Barometer Indications. Increasing storm. If mercury falls during a high wind from S. W., S. S. W., W., or S. Violent but short If fall be rapid. Less violent but of longer continuance. If fall be slow. Snow. If mercury falls when thermometer is low. Improved weather. When a gradual continuous rise of mercury occurs with a falling thermometer. 43O AEROSTATICS.^ Heavy gales from N. Soon After first rise of mercury from a very low point Unsettled weather. With a rapid rise of mercury. Settled weather. With a slow rise of mercury. Very tine weather. With a continued steadiness of mercury with dry air. Stormy weather with rain (or snow). With a rapid and considerable fall of men cury. Threatening, unsettled weather. With an alternate rising and falling of mercury Lightning only. When mercury is low, storm being beyond horizon. Fine weather. With a rosy sky at sunset. Wind and rain. When sky has a sickly greenish hue. Rain. When clouds are of a dark Indian red. Foul weather or much wind. When sky is red in morning. "Weather GJ-lasses. Explanatory- Card. Vice- Admiral Fitzroy, F. R. S. Barometer Rises for Northerly wind (including from N. W. by N. to E.), for dry, or less wet weather, for less wind, or for more than one of these changes Except on a few occasions when rain, hail, or snow comes from N. with strong wind. Barometer Falls for Southerly wind (including from S. E. by S. to W.), for wet weather, for stronger wind, or for more than one of these changes Except on a few occasions when moderate wind with rain (or snow) comes from N. For change of wind toward Northerly directions, a Thermometer falls. For change of wind toward Southerly directions, a Thermometer rises. Moisture or dampness in air (shown by a Hygrometer) increases before rain, fog, or dew. Add one tenth of an inch to observed height for each hundred feet Barometer is above half- tide level. Average height of Barometer, in England, at sea-level, is about 29.94 inches; and average temperature of air is nearly 50 degrees (latitude of London). Thermometer falls about one degree for each 300 feet of elevation from ground, but varies with wind. " When the wind shifts against the sun, Trust it not, for back it will run." First rise after very low Long foretold long last, Indicates a stronger blow. Short notice soon past. Rarefaction of Air. In consequence of rarefaction of air, gas loses of its illuminating power i cube Inch for each 2.69 feet of elevation above the sea. (M. Bremond.) Clouds. Classification. i. Cirrus Like to a feather, commonly termed Mare's tails. 2. Cirro-cumulus Small round clouds, termed mackerel sky. 3. Cirro-stratus Concave or undulated stratus. 4. Cumulus Conical, round clusters, termed wool-packs and cotton balls. 5. Cumulo-stratus Two latter mixed. 6. Nimbus A cumulus spreading out m arms, and precipitating rain beneath it. 7. Stratus A level sheet. NOTE. Cirrus is most elevated. Height. Clouds have been seen at a greater height than 37000 feet. Velocity. At an apparent moderate speed, they attain a velocity of 80 miles per hour. Lightning. Classification. i. Striped or Zigzag Developed with great rapidity. 2. Sheet Covering a large surface. 3. Globular When the electric fluid appears condensed, and it is developed at a comparatively lower Telocity. 4. Phosphoric When the flash appears to rest upon the edges of the clouds. AEROSTATICS. ATMOSPHERIC AIR. 431 WEATHER INDICATIONS. Sky. Gray in morning and light, delicate tints and low dawn. High dawn, and sunset of a bright yellow. Sunset of a pale yellow. Orange or copper color. Gaudy unusual hues. Weather. Clouds. Fine and Soft or delicate looking and in- Fair, definite outlines. Wind. Hard edged, oily - looking, and tawny or copper-colored, and the more hard, "greasy," and ragged, the more wind. Wind only. Light scud alone. Rain. Small and inky. Wind and Light scud driving across heavy Rain. masses. Rain and Hard defined outlines. Wind. Change of High upper, cross lower in a di- Wind. rection different to their course or that of wind. G-eneral* Fair. When sea-birds fly early and far out, when dew is deposited, and when a leech, confined in a bottle of water, will curl up at the bottom. Rain. Clear atmosphere near to horizon and light atmospheric pressure, or a good "hearing day," as it is termed. Storm. When sea-birds remain near to shore or fly inland. Rain, Snow, or Wind. When a leech, confined in a bottle of water, will rise ex- citedly to the surface. Thunder. When a leech, confined as above, will be much excited and leave the water. Value of Indications of Fair Weather, in Days, Coin- pared, to one of Rain. From an extended series of observations. (Lowe.\ Profuse Dew. 4. 5 I Mock Sun or Moon. . White Stratus in a valley 7. 2 | Stars falling abundant 3.2 Colored Clouds at sunset 2.9 Stars bright 3.4 Solar Halo. . 1.9 Sun red and rayless 10. 3 Sun pale and sparkling i White Frost 4.2 Lunar Halo i Lunar burr, or rough-edged 2.8 Moon dim 2 Moon rising red 7 Stars dim.. Stars scintillated. ......... ....... 6 " " Aurora borealis., Toads in evening. 2. 4 Landrails noisy. 13 Ducks and Geese noisy 2.3 Fish rising i. 5 Smoke rising vertically. 5 For weather-foretelling plants, see page 185. ATMOSPHERIC AIR. Very pure air contains Oxygen 20.96, Nitrogen 79, and Carbonic Acid .04. Air respired by a human being in one hour is about 15 cube feet, produc- ing 500 grains of carbonic acid, corresponding to 137 grains carbon, and during this time about 200 grains of water will be exhaled by the lungs. During this period there would be consumed about 415 grains of oxygen. In one hour, then, there would be vitiated 73 cube feet pure air. A man, weighing 150 Ibs., requires 930 cube feet of air per hour, in order that the air he breathes may not contain more than i per 1000 of carbonic acid (at which proportion its impurity becomes sensible to the nose): he ought, therefore, to have 800 cube feet of well ventilated space. 432 ATMOSPHERIC AIR. ANIMAL POWER. An adult human being consumes in food from 145 to 165 grains of carbon per hour, and gives off from 12 to 16 cube feet of carbonic acid gas. An assemblage of 1000 persons will give off in two hours, in vapor, 8.5 gallons water, and nearly as much carbon as there is in 56 Ibs. of bitumi- nous coal. Proportion of Oxygen and. Carbonic A.cid. at following .Locations. Pure Air represented by Oxygen 20.96. Street in Glasgow 20. 895 Regent Street, London 20. 865 Centre Hyde Park 21.005 Metropolitan Railway (underground) . . 20.6 Pit of a Theatre ...................... 20.74 Gallery of a Theatre .................. 20.63 Carbonic Acid .04 Per cent. Open field, Manchester 0383 Churchyard 0323 Market, Smithfield 0446 Factory mills 283 School-rooms 097 Pitt of theatre, n P. M 32 Boxes " 12 " 218 Gallery " 10 " 101 * Roscoe. Top of Monument, London 0398 Hyde Park 0334 Metropolitan Railway (underground).. .338 Lake of Geneva 046 Boys 1 school 31* Girls' " 723! Horse stable 7 Convict prison 045 t Peltenhoffer. Consumption of Atmospheric A.ir. (Coathupe.) One wax candle (three in a Ib.) destroys, during its combustion, as much oxygen per hour as respiration of one adult. A lighted taper, when confined within a given volume of atmospheric air, will become extinguished as soon as it has converted 3 per cent, of given volume of air into carbonic acid. Carbonic Acid Exhaled per Minute by a Man. (Dr. Smith.) During sleep 4.99 per cent., lying down 5.91, walking at rate of 2 miles per hour 18.1, at 3 miles 25.83, hard labor 44.97. ANIMAL POWER. "Work. Work is measured by product of the resistance and distance through which its point of application is moved. In performance of work by means of mechanism, work done upon weight is equal to work done by power. Unit of Work is the moment or effect of i pound through a distance of i foot, and it is termed a foot-pound. In France a kilogrammetre is the expression, or the pressure of a kilogramme through a distance of i meter = 7.233 foot-pounds. Result of observation upon animal power furnishes the following as maximum daily effect: 1. When effect produced varied from .2 to .33 of that which could be produced without velocity during a brief interval. 2. When the velocity varied from .16 to .25 for a man, and from .08 to .066 for a horse, of the velocity which they were capable for a brief interval, and not involv- ing any effort. 3. When duration of the daily work varied from .33 to .5 for a brief interval, during which the work could be constantly sustained without prejudice to health of man or animal; the time not extending beyond 18 hours per day, however lim- ited may be the daily task, so long as it involved a constant attendance. ANIMAL POWER. 433 Men. Mean effect of power of men working to best practicable advantage, is raising of 70 Ibs. i foot high in a second, for 10 hours per day = 4200 foot- pounds per minute. Windlass. Two men, working at a windlass at right angles to each other, can raise 70 Ibs. more easily than one man can 30 Ibs. Labor. PL man of ordinary strength can exert a force of 30 Ibs. for m hours in a day, with a velocity of 2.5 feet in a second = 4500 Ibs. raised one foot in a minute = .2 of work of a horse. A man can travel, without a load, on level groundj during 8.5 hours a day, at rate of 3.7 miles an hour, or 31.45 miles a day. He can carry in Ibs. 1 1 miles in a day. Daily allowance of water, i gallon for ah 1 purposes ; and he requires from 220 to 240 cube feet of fresh air per hour. A porter going short distances, and returning unloaded, can carry 135 Ibs. 7 miles a day, or he can transport, hi a wheelbarrow, 150 Ibs. 10 miles in a day. Crane. The maximum power of a man at a crane, as determined by Mr. Field, for constant operation, is 15 Ibs., exclusive of frictional resistance, which, at a velocity of 220 feet per minute = 3300 foot-pounds, and when exerted for a period of 2.5 minutes was 17.329 foot-pounds per minute. Pile-driving. G. B. Bruce states that, in average work at a pile-driver, a laborer, for 10 hours, exerts a force of 16 Ibs., plus resistance of gearing, and at a velocity of 270 feet per minute, making one blow every four minutes. Rowing. A man rowing a boat i mile in 7 minutes, performs the labor of 6 fully-worked laborers at ordinary occupations of 10 hours per day. Drawing or Pushing. A man drawing a boat in a canal can transport 1 10 ooo Ibs. for a distance of 7 miles, and produce 156 times the effect of a man weighing 154 Ibs., and walking 31.25 miles in a day ; and he can push on a horizontal plane 20 Ibs. with a velocity of 2 feet per second for 10 hours per day. Tread-mill. A man either inside or outside of a tread-mill can raise 30 Ibs. at a velocity of 1.3 feet per second for 10 hours, = 1 404 ooo foot-pounds. Pulley. A man can raise by a single pulley 36 Ibs., with a velocity of .8 of a foot per second, for 10 hours. Walking. A man can pass over 12.5 times the space horizontally that he can vertically, and, according to J. Robison, by walking in alternate directions upon a platform supported on a fulcrum in its centre, he can, weighing 165 Ibs., produce an effect of 3 984 ooo foot-pounds, for 10 hours per day. Pump, Crank, Bell, and Rowing. Mr. Buchanan ascertained that, in work- ing a pump, turning a crank, ringing a bell, and rowing a boat, the effective power of a man is as the numbers 100, 167, 227, and 248. Pumping. A practised laborer can raise, during 10 hours, i ooo ooo Ibs. water i foot in height, with a properly designed and constructed pump. Crank. A man can exert on the handle of a screw-jack of n inches ra- dius for a short period a force of 25 Ibs., and continuously 15 Ibs., a net power of 20 Ibs. Mr. J. Field's tests gave 11.5 Ibs. as easily attained, 17.3 as difficult, and 27.6 with great difficulty. Mowing. A man can mow an acre of grass in i day. Reaping. A man can reap an acre of wheat in 2 days. Ploughing. A man and horse .8 of an acre per day. Oo 434 ANIMAL POWER. Day's Work. (D. K. Clark.) Laborer. Carrying bricks or tiles, net load 106 lbs.= 6oo Ibs. i mile. Carrying coal in a mine, net load 95 to 115 Ibs. = 342 Ibs. i mile. Loading coke into a wagon, net load 100 Ibs. = 270 Ibs. i mile. Loading a boat with coal, net load 190 lbs.= 1230 Ibs. i mile, or 20 cube yards of earth in a wagon. Digging stubble land .055 of an acre per day, or 2000 cube feet of superficial earth. Breaking 1.5 cube yards hard stone into 2 inch cubes. Quarrying. A man can quarry from 5 to 8 tons of rock per day. A foot-soldier travels in i minute, in common time, 90 steps = 70 yards. He occupies in ranks a front of 20 inches, and a depth of 13, without a knapsack: interval between the ranks is 13 inches. Average weight of men, 150 Ibs. each, and five men can stand in a space of i square yard. Effective iPovsrer of !M!en for a Short JPeriod.. Manner of Application. Force. Manner of Application. I orce. Lbs. Screw-driver one-hand Lbs. 8. Drawing-knife or Auger IOO 5 Small screw-driver H Hand saw. .. afi Windlass or Pincers . . . & The muscles of the human jaw exert a force of 534 Ibs. Mr. Smeaton estimated power of an ordinary laborer at ordinary work was equiv- alent to 3762 foot-pounds per minute. But, according to a particular case made by him in the pumping of water 4 feet high, by good English laborers, their power was equivalent to 3904 foot-pounds per minute; and this he assigned as twice that of ordinary persons promiscuously operated with. Mr. J. Walker deduced from experiments that the power of an ordinary laborer, in turning a crank, was 13 Ibs., at a velocity of 320 feet per minute for 8 hours per day. A.ixioiir.t of Labor prod. viced, toy a JVtaii. (Morin.) For 10 hours per day. MANNER OF APPLICATION. Power. Velocity per Second. Weight raised. Feet per Minute. H> for Period given. Throwing earth with a shovel, a height of 5 feet. . Wheeling a loaded barrow up an inclined plane, ! tO 12 Lbs. 6 132 6 13 132 H3 26 18 140 26 88 140 140 44 n an indiv foot per s Feet. i-33 .625 2.25 2-5 I 5 2 2-5 5 5 2-5 i-75 .2 5 idual case econd; h Lbs. 480 4950 810 1950 7920 4290 3 120 2790 4 200 7800 13200 14700 1680 1320 , at 140 Ib nee 70-7- No. 8-7 9 14.7 35-5 144 62 45-2 39 61.1 "3 160.5 160.5 *9 14.4 s., at a ve- 1.3 feet ae Raising and pitching earth in a shovel 13 feet horizontally Pushing and drawing alternately in a vertical direction Transporting weight upon a barrow, and return- ing unloaded FOR 8 HOURS PER DAY. Ascending a slight elevation unloaded Walking, and pushing or drawing in a horizontal FOR 7 HOURS PER DAY. Walking with a load upon his back FOR 6 HOURS PER DAY. Transporting a weight upon his back, and return- ing unloaded Transporting a weight upon his back up a slight elevation and returning unloaded . Raising a weight by his hands * Morin gives amount of labor of a man upon tread-mill, i locity of .5 feet per second for 8 hours per day = 70 Ibs. at i ANIMAL POWEE. 435 To Compute Number of M.en to Perform Work: upon a Tread-mill or file-driver. RULE. To product of weight to be raised and radius of crank, add fric- tion of wheel, and divide sum by product of power and radius of wheel. EXAMPLE. How many men are required upon a tread-mill, 20 feet in diameter, to raise a weight of 9233.33 IDS., crank 9 inches in length, weight of wheel and its load estimated at 5000 Ibs., and friction at .015. Weight of a man assumed at 25 Ibs. Radius of crank .75 feet. Effect of a man on a tread-mill, page 433, 30 Ibs. at a velocity of 1.3 feet per second, = i. 3 X 60 = jBfeet per minute. 9 2 33-33 X .75 + 5000 x .015 = 7000 Ibs. resistance of load and wheel, and 7000-4- X 10 X 30 == 7000 = load and weight -l- product of power increased by its velocity over load, radius of wheel and power = 7000 -f- 1.241 x ioX 30 = 18. 8 men. Horse. -A.mou.nt of Labor produced by a Morse under different Circumstances. (Morin.) For 10 hours per day. MANNER OF APPLICATION. Power. Velocity per Second. Weight drawn. Feet per Minute. IP for Period given. Drawing a 4,-wheeled carriage at a walk Lbs. Feet. Lbs. No. With load upon his back at a walk 26,, 54 1080 Transporting a loaded wagon, and returning un- 184800 fi FOR 8 HOURS PER DAY. 260 8 FOR 4.5 HOURS PER DAY. Upon a revolving platform at a trot 66 3 6 7C 2l8 7 Drawing an unloaded 4-wheeled carriage at a trot. Drawing a loaded 4 wheeled carriage at a trot 97 77 7-25 7-25 43195 334 950 353-5 2741 If traction power of a horse, when continuously at a walk, is equal to 120 Ibs., and grade of road i in 30, resistance on a level being one thirtieth of load, he can draw a load of 120 X 30 -4- 2 = 1500 Ibs. Street nails or Tramways. (Henry Hughes.) Cars, 26 Ibs. per ton, or i to 86 as a mean. Performance of Horses in France. (M. CharU-Marsaines.) SEASON. Road. Weight Horse. Speed H P our. Work per Hour, drawn One Mile. Ratio of Pavement to Macadam. Winter ( Pavement Tone. 1.306 Miles. 2.05 Ton-miles. 2.677} 1.644 tO i Summer \ Macadam | Pavement .851 1-395 1.91 2.17 1.625) 3.027) 1.229 ^ x I Macadam 1.141 2.l6 2.464) Average daily work of a Flemish horse in North of France, where country is flat and loads heavy, is, on same authority, as follows: Winter, 21.82 ton-miles per day. } M for th Summer, 27.82 ) given in example = 53.8 Ibs., from which a deduction is to be made for excess of amount of labor that can be performed in 8 hours over 10. Or, as 10 : 8 ; ; 53.8 : 43.04 lt., which doe* not essentially differ from effect of 30 Ibs. for that of an average performance. 436 ANIMAL POWER. Greatest mechanical effect of an ordinary horse is produced in operating a gin or drawing a load on a railroad, when travelling at rate of 2.5 miles per hour, where he can exert a tractive force of 150 Ibs. for 8 hours per day. Horse upon Turnpike Road. At a speed of 10 miles per hour, a horse will perform 13 miles per day for 3 years. In ordinary staging, a horse will perform 15 miles per day. To Compute Tractive Power of a Horse Team, see Traction, page 848. Assuming maximum load that a horse can draw on a gravel road as a standard, he can draw, On best-broken stone road 2 to 3 times. On a well-made stone pavement 3 to 5 " On a stone trackway 7 to 8 *' On plank road 41012 " On a railway 18 to 20 " NOTE. Track of an iron railway compared with a plank-road is as 27 to 10. To Comptite Iower of Draught of a Horse at Different Elevations. Let ABC represent an inclined plane, o weight of a horse which, being resolved into two com- ponent forces, one of which, n, is perpendicular to plane of inclination, and other, r, is parallel to it. Hence, r represents force which horse must over- come to move his own weight. Then, by similar triangles, A B or I : B C or h : : o : r. Or, -T- = r. If t represents tractive power of horse, upon a level, of 100 Ibs., t' tractive power upon a plane of inclination, and r that part of force exerted by horse which is expended upon his own body, then t = t r, or t = t' in Ibs. ILLUSTRATION. If inclination is i in 50. Assume t = 100, weight of horse 900 Ibs., and I = 50.01. Then, 100 100 17.99 = 82.01 Ibs. Assuming load that a horse can draw on a level at 100, he can draw upon inclinations as follows : i m ioo ..... 91 ; in 75. 70 87 i " 60 85 i in 50 82 i " 45 80 1 " 40 77 in 35 74 " 30 70 " 25 64 i in 20 ..... 55 i " 15 ..... 40 IO On his back a horse can carry from 220 to 390 Ibs., or about 27.5 per cent. Labor. The work of a horse as assigned by Boulton & Watt, Tredgold, Rennie, Beardmore, and others, ranges from 20600 to 39320 foot-pounds per minute for 8 hours, a mean of 27 750 Ibs. A horse can travel, at a walk, 400 yards in 4.5 minutes ; at a trot, in 2 minutes ; and at a gallop, in i minute. He occupies in ranks, a front of 40 ins., and a depth of 10 feet; in a stall, from 3.5 to 4.5 feet front; and at a picket, 3 feet by 9 ; and his average weight = 1000 Ibs. . Carrying a soldier and his equipments (225 Ibs.) he can travel 25 miles m a day of 8 hours. A draught-horse can draw 1600 Ibs. 23 miles a day, weight of carriage in- cluded. ANIMAL POWER. 437 Ordinary work of a horse may be stated at 22 500 Ibs., raised i foot in & minute, for 8 hours per day. In a mill, he moves at rate of 3 feet in a second. Diameter of track should not be less than 25 feet. Rennie ascertained that a horse weighing 1232 Ibs. could draw a canal-boat at a speed of 2.5 miles per hour, with a power of 108 Ibs., 20 miles per day. This is equivalent to a work of 23 760 foot-lbs. per minute, He estimated that the average work of horses, strong and weak, is at the rate of 22 ooo foot-lbs. per minute. From results of trials upon strength and endurance of horses at Bedford, Eng., it was determined that average work of a horse -=^ 20000 foot-lbs. per minute. A good horse can draw i ton at rate of 2.5 miles per hour, from 10 to 12 hours per day. Expense of conveying goods at 3 miles per hour, per horse teams being i, expense at 4.33 miles will be 1.33, and so on, expense being doubled when speed is 5. 125 miles per hour. Strength of a horse is equivalent to that of 5 men, and his daily allowance of water should be 4 gallons. -A.inoti.iit of La"bor a Morse of average Strength is capa- ble of performing, at different "Velocities, on Canal, Railroad., and Toarnpike. Traction estimated at 83.3 Ibs. Veloci- Dura- Useful Effect, drawn i Mile. Veloci- Dura- Useful Effect, drawn i Mile. ty per Hour. tion of Work. On a Canal. On a Rail- road. On a Turn- pike. ty per Hour. tion of Work. On a Canal. On a Rail- road. On a Turn- pike. Miles. Hours. Tons. Tons. Tons. Miles. Hours. Tons. Tons. Tons. 2-5 "5 520 "5 14 6 2 30 48 6 3 8 243 92 12 7 1-5 19 41 5-1 4 4-5 102 72 9 8 I.I25 12.8 36 4-5 5 2.9 52 57 7-2 10 75 6.6 28.8 3-6 Actual labor performed by horses is greater, but they are injured by it. Tractive Power of a horse decreases as his speed is increased, and within limits of low speed, or up to 4 miles per hour, it decreases nearly in an inverse ratio. For 10 Hours per Day. Miles. Traction. Miles. Traction. Miles. Traction. Miles. Traction. Per Hour. 75 i 1-25 Lbs. 330 250 200 Per Hour. 1-5 r -75 2 Lbs. i6 5 140 5 Per Hour. 2.25 2-5 2-75 Lbs. no 100 QO Per Hour. 3 3-5 4 Lbs. 82 70 62 Fc Miles per hour Power in Ibs. . . r Ordinal y or Short Periods. (Molesworth. ) 2 33-54 4-5 5 166 121; 104 83 62 41 M:\ale. (D.K.Clark.) Load on back, 170 to 220 Ibs. day's work = 6400 Ibs. i mile ; 400 Ibs. at 2.9 miles per hour =, 5300 Ibs. i mile, and 330 Ibs. at 2 miles per hour = 5000 Ibs. i mile. Upon a revolving platform, at a velocity of 3 feet per second, = n 880 Ibs. raised one foot per minute, or 172.2 IP for 8 hours per day Load on back, 176 Ibs. carried 19 miles day's work = 3300 Ibs. i mile. In Syria an ass carries 450 to 550 Ibs. grain. Upon a revolving platform, at a velocity of 2.75 feet per second, = 5280 Ibs. raised one foot per minute, or 76.5 IP for 8 hours per day. Go* 438 ANIMAL POWER. Ox. An Ox, walking at a velocity of 2 feet in a second (1.36 miles per hour), exerts a power of 154 lbs., = 18480 Ibs. raised one foot per minute, or 268.8 IP for 8 hours per day. A pair of well-conditioned bullocks in India have performed work = 8000 foot-lbs. per minute. Camel. Load on back, 550 Ibs. carried 30 miles per day for 4 days, 4 days' work 16 500 Ibs. i mile, for 5 days 13 ooo Ibs. i mile = 44 IP for 10 hours per day, Load of a Dromedary, 770 Ibs. Llama. Load on back, no Ibs., day's work 2000 to 3000 Ibs. i mile = .5 to .75 H> for 10 hours per day. Birds and Insects. Area of their wing surface is in an inverse ratio to their weight. Assuming weight of each of the following Birds to be one pound, and each Insect one ounce, the relative area of their wing surface proportionate to that of their act- ual weight would be as follows (M. De Lucy) : Sq. ft. Swallow 4.85 Sparrow .... 2. 7 Turtle-dove.. 2.13 Sq. ft. Pigeon 1.27 Vulture 82 Crane, Australia, .41 Sq. ft. Gnat 3.05 Dragon-fly, sm'll, 1.83 Lady-bird 1.66 Sq. ft. Cockchafer ... 32 Bee 33 Meat-fly 35 Crocodile and IDog. The direct power of their jaws is estimated at 120 Ibs. for the former and 44 for the latter, which, with the leverage, will give respectively 6000 and 1500 Ibs. PERFORMANCES OF MEN, HORSES, ETC. Following are designed to furnish an authentic summary of the fastest or most successful recorded performances in each of the feats, etc., given. MAN. Walking. 1874, Wm. Perkins, London, Eng., .5 mile, in 2 min. 56 sec.; i, in 6 min. 23 sec.; 1877, 20, in 2 hours 39 min. 57 sec. 1881, C. A. Harriman, Chicago, 111., 530 miles, in 5 days 20 hours 47 mm. 1878, W. Howes, London, Eng., 50 miles, in 7 hours 57 min. 44 sec.; 1880, 57 miles, in 13 hours 7 min. 27 sec., and 100, in 18 hours 8 min. 15 sec. 1801, Capt. R. Barclay, Eng., country road, 90 miles, in 20 hours 22 min. 4 sec., in- cluding rests; 1803, .25 mile, in 56 sec., and Charing Cross to Newmarket, 64, in 10 hours, including rests; 1806, 100, in ighours, including i hour y>min. in rests, 1809, 1000, in looo consecutive hours, walking a mile only at commencement of each hour. 1877, D. O'Leary, London, Eng., 200 miles, in 45 hours 21 min. 33 sec. 1818, Jos. Eaton, Stowmarket, Eng., 4032 quarter miles, in 4032 consecutive quar- ter hours. 1877, Wm. Gale, London, Eng., 1500 miles, in 1000 consecutive hours, 1.5 miles each hour ; and 4000 quarter miles, in 4000 consecutive periods of 10 minutes. 1882, Chas. Rowell, New York, N. Y., and running, 89 miles 1640 yards, in 12 hours. 1882, Geo. Hazael, New York, N. Y., and running, 600 miles 220 yards, in 6 days. 1883, J. W. Raby, London, Eng., 2 miles in 13 min. 14 sec.; 3, in 20 min. 21.5 sec.; 4, in 27 min. 38 sec. ; 5, in 35 min. 10 sec.; and 10, in i hour 14 min. 45 sec. 1882, John Meagher, New York, N. Y., 8 miles in 58 min. 37 sec. W. Franks, London, Eng., 25 miles in 3 hours 35 min. 14 sec. 1885, W. Cummings, London, Eng., 10 miles in 51 min. 6.6 sec. 1884, J. E. Dixon, Birmingham, Eng., 40 miles in 4 hours 46 min. 54 sec. 1883, Peter Golden, Brooklyn, N. Y., 50 miles in 7 hours 29 min. 47 sec. R, tinning. 1844, Geo. Seward, of U. S., Manchester, Eng., flying start, 100 yards, in 5.25 sec. 1864, Jos. Nutall, Manchester, Eng., 600 yards, in i min. 13 sec. 1881, L. E. Myers, New York, N. Y., 1000 yards, in 2 min. 13 sec. 1863, Wm. Lang, Newmarket, Eng., i mile, in 4 min. 2 sec., descending ground; Manchester, 2, in 9 min. 11.5 tec.; 1865, n miles 1660 yards, in i hour 2 min. 2.5 see. ANIMAL POWER. 439 1852, Wm. Howilt, " American Deer," London, Eng., 15 miles in i hour 22 min. 1863, L. Bennett, '' Deerfoot," Hackney Wick, Eng., 12 m., in i hour 2 min. 2.5 sec. 1879, Patrick Byrnes, Halifax, N. S., 20 miles, in i hour 54 min. 1880, D. Donovan, Providence, R. L, 40 miles, in 4 hours 48 min. 22 sec. 17, ^4 Courier, East Indies, 102 miles, in 24 Jiowrs. 1889, H. M. Johnson, Denver, Col., 50 yards, in 5 sec. 1884, M. K. Kittleman, Oakland, Cal., 150 yards (twice), in 14 min. 6 sec. 1890, James Grant, Cambridge, Mass., 5 miles in 25 min. 22.25 sec. Jumping, Leaping, etc. 1854, J. Howard, Chester, Eng., i jump, board raised 4 ins. in front, running start, with dumb-bells, 5 Ibs., 29 feet 7 ins. 1868, Geo. M. Kelley, Corinth, Mass., running, and from a spring board, leaped oyer 17 horses standing side by side. 1879, G. W. Hamilton, Romeo, Mich., dumb-bells, 22 Ibs., standing jump, 14 feet 5.5 ins. 1886, J. Purcell, Dublin, running long jump, -23 feet 11.5 ins. 1889, J. Darby, Ashton- under- Lyne, Eng., two standing jumps, with weights, 26 feet 8. 5 ins. H. M. Johnson, St. Louis, Mo. , without weights, 22 feet 6.75 ins. 10 standing long jumps, without weights, 114 feet 8.5 ins. J. F. Kearny, Walpole, Mass., 3 standing long jumps, with weights, 4-2 feet 3 ins.; without weights, at Boston, Mass. , 35 feet 6 ins. Boston, Mass. , running high jump, with weights, 6 feet 5.25 ins.; backward jump, with weights, heel to toe, izfeet 1.25 ins. Oak Island, Mass., standing high leap, with weights, 5 feet 9.5 ins. Lifting. 1825, Thomas Gardner, of New Brunswick, N. S., a barrel of pork, 320 tes., under each arm ; also transported across a pier an anchor, 1200 Ibs. 1868, Wm. B. Curtis, New York, N. Y., 3239 Ibs., in harness. 1883, D. L. Dowd, Springfield, Mass., by hands, 1442.25 Ibs. Throwing ^Weights. 1870, D. Dinnie, New York, N. Y., light stone, 18 Ibs., 43 feet; heavy stone, 24 Ibs., 34 feet 6 ins.; heavy hammer, 24 Ibs., 83 feet 8 ins.; 1872, Aberdeen, Scotland, light hammer, i^Bfeet; run, 16 Ibs., 162 feet. 1887, Peter Foley, Milwaukee, Wis., 56 Ibs., without follow, 31 feet 5 in*. S%viinrning. 1835, S. Bruck, 15 miles, in rough sea, in 7 hours 30 min. 1846, A Native, off Sandwich Islands, 7 miles at sea, with a live pig under one arm. 1870, Pauline Rohn, Milwaukee, Wis., 650 feet, still water, in 2 min. 43 sec. 1872, J. B. Johnson, London, Eng., remained under water 3 min. 35 sec. 1875, Capt. M. Webb, Dover, Eng., to Calais, France, 23 miles, crossing two full and two half tides == 35 miles, in 21 hours 45 min. 1880, Afloat 6b hours. 1886, J. Haggerty, Blackburn Baths, Eng., 100 yards, 4 turns, in i min. 5.5 sec. 1890, J. Nuttall, London, Eng., 1000 yards, 23 turns, in 13 min. 54.5 sec. 1885, J. J. Collier, London, Eng. , i mile in 26 min. 52 sec. Skating. 1877, John Ennis, Chicago, 111., 9 laps to a mile, 100 miles, in u hours 37 min. 45 sec.; and 145 inside of 19 hours. 1887, T. Donoghue, Jun., Newburgh, N. Y., i mile, with wind, in 2 min. 12.375 *** 1882, S. J. Montgomery, New York, N. Y., 50 miles, in 4 hours 14 min. 36 sec. NOTE. The Sporting Magazine, London, vol. ix., page 135, reports a man in 1767 to have skated mile upon the Serpentine, Hyde Park, London, in 57 seconds. HORSE. Trotting. 1814, "Boston Blue," Lynn turnpike, one mile, sulky, in 2 min. 54 sec. 1875, "Steel Grey," Yorkshire, Eng., 10 nrles, saddle, in 27 min. 56.5 sec. 1867, "John Stewart," Boston, Mass., half-mile track, 20 miles, harness, in 58 min. 5.75 sec., and 20.5 miles in 59 min. 31 sec. 1830, "Top Gallant," Philadelphia, Penn., 12 miles, harness, in 38 min. 1829, "Torn Thumb," Sunbury Common, Eng., 16.5 miles, harness, 248 Ibs., in 56 min. 45 sec.; and 100 miles, in 10 hours 7 min., including 37 min. in rests. 1869, "Morning Star," Doncaster, Eng., 18 miles, harness (sulky 100 Ibs.), in 57 min. 27 sec. 1835, " Black Joke," Providence, R. L, 50 miles, saddle, 175 Ibs., in 3 hourt 57 min. 44O ANIMAL POWER. 1855, "Spangle," Long Island, N. Y., 50 miles, wagon and driver 400 Ibs., in 3 hours 59 min. 4 sec. 1837, u Mischief, " Jersey City, N. J., to Philadelphia, Penn., 84.25 miles, harness, very hot day and sandy road, in 8 hours 30 min. 1853, "Conqueror," Long Island, N. Y., 100 miles, harness, in 8 hours 55 min. 53 sec., including 15 short rests. 1873, M. Delaney's mare, St. Paul's, Minn., 200 miles, race track, harness, in 44 hours 20 min., including 15 hours 49 min. in rests. 1834, "Master Burke " and " Robin," Long Island, N. Y., 100 miles, wagon, in 10 hours, 17 min. 22 sec., including 28 min. 34 sec. in rests. Stage coaching. 1750, By the Duke of Queensberry, Newmarket, Eng., 19 miles, in 53 min. 24 sec. 1830, London to Birmingham, Eng., "Tally-ho," 109 miles, in 7 hours 50 min. including stop for breakfast of passengers. Leaping.* 1821, A horse of Mr. Mane, at Loughborough, Leicestershire, Eug., 173 Ibs., over a hedge 6 feet in height, 35 feet. 1821, A horse of Lieut. Green, Third Dragoon Guards, at Inchinnan, Eng., ridden by a heavy dragoon, over a wall 6 feet in height and ifoot in width at top. 1847, "Chandler,"" Warwick, Eng., over water, 37" feet. 1901, " Heather bloom," Chicago, 111., over a bar, 7 feet 4-5 ins. NOTE. The maximum stride of a horse is estimated to be 28 feet g ins. ; " Eclipse" has covered 25 fett. The maximum stride of an elk is 34 feet, and of an elephant 14 feet. R-u lining. 1701, Mr. Sinclair, on the Swift at Carlisle, a gelding, 1000 miles, in 1000 consecu- tive hours. 1731, Geo. Osbaldeston, Newmarket, 156 Ibs., 100 miles, by 16 horses, in 4 hours 19 mm. 40 sec., and 200, by 28 horses, in 8 hours 39 min., including i hour 2 min. 56 sec. in rests; i horse, "Tranby," 16 miles, in 33 min. 15 sec. 1752, Spedding's mare, 100 miles, in 12 hours 30 min., for 2 consecutive days. 1754, A Galloway mare of Daniel Corker's, Newmarket, 300 miles, by one rider, 67 Ibs., in 64 hours 20 min. 1761, John Woodcock, Newmarket, 100 miles per day, by 14 horses, one each day, for 29 consecutive days. 1814, An Officer of i^lh Dragoons, Blackwater, 12 miles, i horse, in 25 min. n sec. 1868, N. H. Mowry, San Francisco, Cal., race track, 160 Ibs., 300 miles, by 30 horses (Mexican), in 14 hours g min., including 40 minutes for rests; the first 200, in 8 hours 2 min. 48 sec., and the fastest mile in 2 min. 8 sec. 1869, Nell Coher, San Pedro, Texas, 61 miles, in 2 hours 55 min. 15 sec., including rests. 1870, John Faylor, Carson City, Nevada, 50 miles, by 18 horses, in i hour 58 min. 33 sec.; and Omaha, Neb., 56 miles, in 2 hours 26 min., including rests. 1876, John Murphy, New York, N. Y., 155 miles, by 20 horses, in 6 hours 45 min. 7 sec. 1878, Capt. Salvi, Bergamo to Naples, Italy, 580 miles, in 10 days. 1880, " Mr. Brown," Rancocas, N. J., aged, 160 Ibs., 10 miles, in 26 min. 18 sec. 1828, "Chapeau de Paille" (Arabian), India, 1.5 miles, 115 Ibs., in 2 min. 53 sec. 183- Capt. Home (Arabians), Madras to Bungalore, India, 200 miles, in less thaa 10 hours. DOGS. Ccmrsing and Chasing. A Greyhound and Hare ran 12 miles in 30 min. 1794, A Fox, at Brende. Eng., ran 50 miles in 6.5 hours. A Greyhound, at Bushy Park, Eng., leaped over a brook 30 feet 6 ins. BIRDS. Flying. In Miles per hour : Swallow, 65 ; Marten, 60 , Carrier Pigeon and Seal Duck, 50; Wild Goose, 45; Quail, 38; Crow, 25. 1870, Carrier Pigeons, Pesth to Cologne, Germany, 600 in 8 hours. 1875, D" n( i ee Lake to Paterson, N. J., 3 in 3 min. 24 sec. NOTE. At 50 miles the pressure on a plane surface is 12.5 Ibs. per so., foot ; and at *oo, 50 Ibs. * A Salmon can leap ix dam nfeet in height. Sporting Magazine, London, vol. xii., page 79. HOKSE-POWER. BELTS AND BELTING. 441 HORSE -POWER. Horse-power. IP is the principal measure of rate at which work is per- formed. One horse-power is computed to be equivalent to raising of 33 ooo Ibs. one foot high per minute, or 550 Ibs. per second. Or, 33000 foot-lbs. of work, and it is designated as being Nominal, Indicated, or Actual. A IP in work is estimated at 33000 Ibs., raised i foot in a minute; but as a horse can exert that force (or only 6 hours per day, one work IP is equivalent to that of 4.5 horses, at a rate of 3 miles per hour. Cheval-vapeur of France is computed to be equivalent to 75 kilogram- meters of work per second, or 7.233 foot-lbs., or 75 x 7.233 = 542.5 foot-lbs n which is 1.37 per cent, less than American or English value. BELTS AND BELTING. Capacity of belts to transmit power is determined by extent of their adhesion to surface of pulley, and it is very limited in comparison with tensile strength of belt. Resistance of a belt to slipping depends essentially upon character of surface of pulley, its degree of tension, and width, and as adhesion is in proportion to pressure on surface of pulley, long belts, by having greater weight, give greater adhesion. Ultimate Tensile Strength, of Belting per Sq.. Inch of Section. Merchantable Oak- tanned, of first quality. Belts, 6 ins. in width. Single, .2 inch in thickness, 918 Ibs. per lineal inch, and 4536 Ibs. per sq. inch of section. Double, .35 inch, 1396 Ibs. per lineal inch, and 4101 per sq. inch. Ratio of single to double, 918 -4- 1396 = .658. Elongation in two inches of length and four in width for a load of 2000 Ibs. Sin- gle, 9.09; double, 5.79. The resistance and elongation of double belting is more uniform than that of sin- gle, from the irregularities in each layer counteracting each other. Length of Lap. Rivets. Belt J Destructive Stress. "ointing. Sti per sq. inch of section. ess per lineal inch of width. Elongation. Ins. lit 7.2 5-i Cemented.. . No. 6 6 I Lbs. 4170 3610 4520 2420 4^8o Lbs. 394<> 3000 3545 1792 ss6o Lbs. 709 611 762 407 III2 Per cent. 7-5 7-5 7-5 7-5 7 Riveted joints failed at rivet holes. Riveting of double belts was shown to be objectionable. Riveted joints of single belts have one-third less strength than the average of different manners of lacing. A double staggered laced joint, i strand only in each hole (5 of . 1875 inch punched), broke in belt at a stress equal to that of resistance of it per area of section. Transmission of 3?o\ver. D to 4 ooo Ibs. per sq. inch) of 50 Ibs. per sq. inch, case, including friction, was 30.6 Ibs. per sq. inch. From Elements of Prof . Chas. H: Benjamin. 442 BELTS AND BELTING. Computation of IP. t>(S-s) 868(88X4-22X4) , .> - = EP = - = 6. 94 HP. 33000 33000 v representing velocity of belt in feet per minute, S an s stress on belt per lineal inch of width on upper or driving side and underneath or returning side. To Compute Width of a Leather Belt. Assuming a well-defined case (where limit of adhesion was ascertained), a belt of ordinary construction (laced), and 9 inches in width, transmitted the power of 15 horses over a pulley 4 feet in diameter, at a velocity of 1800 feet per minute, with an arc of adhesion of 210, or of .6 or 7.54 feet of cir- cumference, and with an area of 95 square feet of belt per H?. Hence, - ^ - = w; w representing width of belt in inches, d di- et v ameter of pulley in feet, and v velocity of belt in feet per minute. NOTE. Thickness of belt should be added to diameter of pulley. Applying these elements to the formulas of 13 different authors, the result varies from 7.85 to 13.5 ins., mean of which is 10 675. For double belting width = .66 w. ILLUSTRATIONS. If IP 25, and velocity of belt 2250 feet per minute, what should be width of belt, diameter of pulley 4 feet? - '== 12.5 ins. for ordinary thickness of 1875 in. To Compute Elements of Belting. _ = . IP 33 OOP... -p. 33 OOP IP = w . Wyto = ir . =S' =t ' ' ' _ 1000* vw v ' 33000 ' t ' S * looo laced, 550 riveted (for a thickness of .1875 inch), with variations according to the character and condition of the belt, diameter of pulley, and arc of adhesion of belt. P representing power transferred, W weight or stress in Ibs. , t thickness of belt in ins., v velocity of it in feet per minute, and S stress on belt per lineal inch of width w, in Ibs. Single belts at their relative thickness with double, of .2 to .35 inch, will sustain one-tenth more stress per sq. inch of belt. To Compute tne Angle of the Arc of Contact of a Belt. Sin- (R r-r-d) x 2-f-i8o for large pulley or driver and 180 for small. R and r representing radii of pulleys, d distance between their centres all in feet or inches. ILLUSTRATION. Assume pulleys 11.2 feet and 4 feet in diameter and distance apart 15 feet. Sin/.r I ' 2 ~ 4 -r- 15 Jx 2+ i8o = 207 46' for large pulley and 152 14' for small. India Ptn"b"ber Belting. (Vulcanized.) Results of Experiments upon Adhesion of India Rubber and Leather Belting. (J. H. Cheever). Rubber belt slipped on iron pulley at 90 " " leather " 128 Leather belt slipped on iron pulley at 48 leather " 64 Hence it appears that a Rubber Belt for equal resistances with a Leather' Belt may be reduced respectively 46, 50, and 30 per cent. tron Wire. A wire rope .375 inch in diameter, over a pulley 4 feet in di- ameter, running at a velocity of 1250 feet per minute, will transmit 4.5 IP. In order to avoid undue bending of wires, diameter of pulley should not be less than 140 times diameter of rope. By Experiments of H. R. Towne and Mr. Kirkaldy. (England.) Tensile strength of Single leather belting per square inch of section. Laced, 966 Ibs. = i. Riveted, 1740 Ibs. = 1.8. Solid, 3080 Ibs. BELTS AND BELTING. BLASTING. 443 By the experiments of F W Taylor, M. R, the tensile strength of belts Per Square Inch of Section. Oak-tanned 192 to 229 Ibs. Raw hide 253 to 284 Ibs. Greneral Notes. Leather Belts Are best when oak-tanned, should be frequently oiled * and when run with hair side over pulley will give greatest adhesion. Ordinary thickness .1875 inch, and weight 60 Ibs. per cube foot. Relative effect of different pulleys and belts : Leather surface., i. Rough iron. ... 41 Turned iron... .64 Turned wood. .. .7 Morin assigned 50 Ibs. as a proper stress per inch of width of good belting. Presence of small holes in a belt will prevent its slipping or squealing. To increase adhesion, coat driving surface with boiled oil or cold tallow, and then apply powdered chalk. When new, cut them .1875 inch short for each foot in length required, to admit of the stretch that occurs in their early operation. Belts should be set as nearly horizontal as practicable, in order that the sag may increase adhesion on pulley, and hence power should be communicated through under side. The "creeping" or lost speed by belts is about .006 per cent., hence, to maintain a uniform or required speed, driver must be increased in diameterpro rata with slip. A double belt, 75 ins. in width and 153.5 f eet in length, transmitted 650 IIP. (See page 989). BLASTING. In Blasting, rock requires from .25 to 1.5 Ibs. gunpowder per cube yard, according to its degree of hardness and position. In small blasts 2 cube yards have been rent and loosened, and in very large blasts 2 to 4 cube yards have been rent and loosened, by i Ib. of powder. Tunnels and shafts require 1.5 to 2 Ibs. per cube yard of rock. G-tin pcrwcler has an explosive force varying from 40000 to 90000 /bs. per sq. inch. That used for blasting is much inferior to that used for projectiles, the proportion being fully one third less. Nitro-gly cerine is an unctuous liquid, which explodes by concussion, an extreme pressure (2000 Ibs. per sq. inch), or a temperature exceeding 600 if quickly applied to it ; it will inflame, however, and burn gradually. At a temperature below 40 it solidifies in crystals. Its explosion is so instantaneous that in rock-blasting tamping is not nec- essary ; its explosive power by weight is from 4 to 5 times that of gun- powder. Dynamite is nitre-glycerine 75 parts, absorbed in 25 parts of a sili- ceous earth termed kieselguhr; it also explodes so instantaneously as to render tamping in blasting quite unnecessary. It is insoluble in water, and may be used in wet holes ; it congeals at 40, is rendered ineffective at 212, and has an explosive force by weight of 3 times that of gunpowder, and by bulk 4.25 times. GKm. - cotton is insoluble in water, and has an explosive force by weight of from 2.75 to 3 times that of gunpowder, and by bulk 2.5 times. It may be detonated in a wet state with a small quantity of dry material. Tonite is nitrated gun-cotton, and is known also as cotton powder. It is produced in a granulated form. Litlio-fractexir is a nitro-glycerine compound in which a portion of the base or absorbent material is made explosive by the admixture therein of nitrate of baryta and charcoal. * See Cements, etc., page 871, for compositions, etc. 444 BLASTING. Cellulose Dynamite is when gun-cotton is used as the absorbent for nitre-glycerine ; it will explode frozen dynamite, and is more sensitive tc percussion than it. To Compxite Charge of Q-tnipo^vcler for Roclr Blasting. RULE. Divide cube of line of least resistance by 25, as for limestone, tc 32 for granite, and quotient will give charge of powder in Ibs. Or, L3 -i. 32 = Ibs. EXAMPLE. When line of least resistance is 6 feet, what is charge required? 6 3 -r-32 = 6.75 Ibs. Line of least resistance should not exceed .5 depth of hole. Tamping. Dried clay is the most effective of all materials for tamping; Broken Brick the next, and Loose Sand the least. Relative Costs of a Tunnel and Shaft in England. (Sir John Burgoyne.) Smitl Fuses Diam. is and coa 1 6 18 rials Inch < Pow or G cottc I 'n L >J'L tor m- n. ..abor 48.8 lolesofL zngth. Dynamite. >iffercnl Diam. Diamete Powder or Gun- cotton. IOO *S. Dynamite Weight oj Powder or Gun- cotton. f Esplosii Dynamite. e Mate Per Diam. Ins. I 1.25 Oz. .419 .654 .942 Oz. 1.046 I-507 Ins. 2 2.25 Oz. 1.283 1-675 2.12 Oz. 2-053 2.68 3-392 Ins. 2-5 2-75 3 Oz. 2.618 3.166 Oz. 4.189 5.066 6.03 Diam. of Jumper. Depth of Hole. Men. Depth bored per Day. Ins. I i-75 2 Ins. I to 2 2. 5 to 6 4 to 7 No. I 3 3 Feet. 8 12 8 Boring Holes in Granite. 14 Drill. Width of bit compared to stock .625. Diam. of Jumper. Dep f th Hole. Men. Ins. 2.25 2-5 3 Ins. 5 to 10 9 to 12 9 to 15 No. 3 3 3 Depth bored per Day. !. Feet. 6 5 4 Lbs. 16 16 18 Charges of Powder. Usual practice of charging to one third depth of hole is erroneous, inasmuch as volume of charge increases as square of diameter of hole. Hence holes of 1.5 and 2 inches, although of equal depths, would require charges in proportion of 2.25 and 4. Line of least re- sistance. Powder. Line of least re- sistance. Powder. Line of least re- sistance. Powder. Line of least re- sistance. Powder. Feet. I 2 Oz. 75 4 Feet. 3 4 Lbs. Oz. 13-5 2 Feet. 1 Lbs. Oz. 3 14-5 6 12 Feet. I Lbs. Oz. 10 11.5 16 Effects. Gunpowder. From its gradual combustion, rends and projects rather than shatters. A hole 5.5 ins. in diameter and IQ feet 7 ins. in depth, filled to 8 feet 10 ins. with 75 Ibs. powder, has removed and rent 1200 cube yards, equal to 2400 tons. The labor expended was that of 3 men for 14 days. Temperature of gases of explosion 4000. Gun-cotton. From the rapidity of its combustion, shatters. Dynamite. From the greater rapidity of its combustion over gun cotton, ie more shattering in ite explosion. BLASTING. BLOWING ENGINES. 445 Drilling. Churn-drilling. A churn-driller will drill, in ordinary hard rock, from 8 to 12 f eet, 2 inch holes of 2.5 feet depth, per day, and at a cost of from 12 to 18 cents per foot, on a basis of ordinary labor at $i per day. Drillers receiving $2.50. One man can bore, with a bit i inch in diameter, from 50 to 100 inches per day of 10 hours in granite, or 300 to 400 inches per day in limestone. Tamping. Two strikers and a holder can bore, with a bit 2 inches in diameter, 10 feet in a day in rock of medium hardness. Composition for waterproof charger or fuse consists by weight of Pitch, 8 parts; Beeswax and Tallow each i part. Mining. (Lefroy's Handbook.) In demolition of walls line of least resistance L = half thickness, and C is a co- efficient depending on structure. Charge in Ibs. = C X L3. In a wall without counterforts, where interval between the charge is 2 L, C = .i5. In a wall with counterforts the charge to be placed in centre of each counterfort at junction with wall, C . 2. Where the charge is placed under a foundation, having equal support on both sides, C = .4. A leather bag, containing 50 to 60 Ibs. powder, hung or supported against a gate or like barrier, will demolish it. For ordinary mines in average rock charge in ounces = L3 -f- 160. BLOWING ENGINES. For Smelting. Volume of oxygen in air is different at different temperatures. Thus, dry air at 85 contains 10 per cent, less oxygen than when it is at tern- perature of 32; and when it is saturated with vapor, it contains 12 per cent. less. If an average supply of 1500 cube feet per minute is required in winter, 1650 feet will be required in summer. Smelting of Iron Ore. Coke or Anthracite Coal. 18 to 20 tons of air are required for each ton of Pig Iron, and with Charcoal 17 to 18 tons are re'quired. (i ton of air at 34 = 29 751, and at 60 = 31 366 cube feet.) Pressure. Pressure ordinarily required for smelting purposes is equal to a column of mercury from 3 to 10 inches, or a pressure of 1.5 to 5 Ibs. per square inch. Reservoir. Capacity of it, if dry, should be 15 to 20 times that of cylin- der if single acting, and 10 times rf double acting. Pipes. Their area, leading to reservoir, should be .2 that of blast cylinder, and velocity of the air should not exceed 35 feet per second. A smith's forge requires 150 cube feet of air per minute. Pressure of blast .25 to 2 Ibs. per square inch. A ton of iron melted per hour in a cu- pola requires 3500 cube feet of air per minute. A finery forge requires 100 ooo cube feet of air for each ton of iron refined. A blast furnace re- quires 20 cube feet per minute for each cube yard capacity of furnace. A Ton of Pig Iron requires for its reduction from the ore 310000 cube feet of air, or 5.3 cube feet of air for each pound of carbon consumed Pressure, .7 Ib. per square inch. P P 446 BLOWING ENGINES. N rive a Blowing Compute IPower Required. to Drive a Blowing Engine. = x / V . t? representing velocity of air in feet per sec- _ _ .93 x .7854 x _ . ond, d and d! diameters of pipe and of nozzle in feet, =V .gjx .7854 X 500 = .309. ILLUSTRATION. What should be power of a steam engine to drive 35 cube feet of air at a velocity of 500 feet per second, through a pipe i foot in diameter and 300 feet in length? c ratio between power employed and effect produced by it = ina well-constructed engine . 5, and C = .93. d = . 2974, assumed at . 3. l^5oj> x ^3 f3 + 4\ 60-4-33000 = 22631.625 X 60-5-33000 = 41-15 H. To Compute Required Power of a, Blowing Engine. P-r/X av_ jp p ygpregenting pressure of blast in Ibs. per sq. inch-f a area of cylinder in sq. ins. ; v velocity of piston injeet per minute; f fric- tion of piston and from curvatures, etc., estimated at 1.25 per sq. inch, of piston. NOTE. If cylinder is single acting, divide result by 2. ILLUSTRATION. Assume area of blast cylinder 5600 sq. ins., pressure of blast 2.25 Ibs. per sq. inch, and velocity of piston 96 feet per second. 2.25 + 1.25X5600X96^1 88! 600 h(y) . ses the exact power developed in ----- 33000 To Compute Dimensions of* a Driving Engine. RULE i. Divide power in Ibs. by product of mean effective pressure upon piston of steam cylinder in Ibs. per sq. inch, and velocity of piston in feet per minute, and quotient will give area of cylinder in sq. ins. 2. Divide velocity of piston by twice number of revolutions, and quotient will give stroke of piston in feet. Volume of air at atmospheric density delivered into reservoir, in consequence of escape through valves, and partial vacuum necessary to produce a current, will be about .2 less than capacity of cylinder. EXAMPLE. Assume elements of preceding case, with a pressure of 50 Ibs. steam, cut off at .375, and with 12 revolutions of engine per minute, what should be area of cylinder of a non-condensing engine? Mean effective pressure of steam with 5 per cent, clearance 50 Ibs., and 50 /* + 14.7 = 50 2.5 -J- 3.33-!- 14.7 = 29.47 Ibs.y and velocity of piston = 192 feet. 5600 Xa . 3S + i.5X96 = 1881600 _i9_ 29.47 X 192 5658 12 X 2 Area of cylinder in this case was 324 sq. ins. For Volume, Pressure, and Density of Air, see Heat, page 521. * See formula and note for power of non-condensing engine, page 733. BLOWING ENGINES. 447 To Compute Elements of a Blo^wing Kngine. Single Stroke. V representing volume of air in cube feet per minute, P pressure of air and f fractional resistance in Ibs. per sq. inch, A area of cylinder and a area of its valves in sq. ins., s stroke of piston in feet, n number of single strokes of piston per minute, L length of air-pipe from reservoir to discharge in feet, d diameter of air or blast pipe and 1) diameter of cylinder in ins., v velocity of blast in feet per second, and t temperature of blast consequent upon com- pression in degrees. ILLUSTRATIONS. Assume blowing cylinder 50 ins. in diam., stroke of piston 10 feet, number of single strokes 10 per minute, pressure by mercurial manometer 6.12 ins., frictional resistance .4 lb., length of pipe 25.25 feet, and area of valves 95 sq. ins. V = 1 363. 54 cube feet, P = 3 Ibs. , - A = 1963. 5 sq. ins. Then ^3J|L3S = 20 . I 6H>, and -963.5 X ' ^ X Ii = ^ g. To Compute "Volume of Air transmitted, by an Engine. When Pressure, Temperature, etc., are given. - J \/ * (* A4H* ) ^ = v ' Then r * 60 = V m cube feet per minute. tl and h representing height of barometer and pressure of blast in ins. of mercury ; t temperature of blast ; and v velocity in feet per second. ILLUSTRATION. A furnace having 2 tuyeres of 5 ins. diameter, pressure and tem- perature of blast 3 ins. and 350, and barometer 30 ins. ; what is volume of air trans- mitted per minute? C for a conical opening .94. 34-5X/3 (^^ X .94 = 34.5/3~(f) = 34-5 X -467 X .94 = *4 fat velocity per second. Then, area 5 ins.= 19.635, which X 2 = 39.27 ins., and 39.27 X 15. 14 X 6o-=- 144.-= r 47.73 cube feet. To Compute Pressure of Blast from "Water or Mercurial Grauge. RULE. Divide Water and Mercurial Gauge in ins. by 27.67 and 2.04 re- spectively, and quotient will give pressure in Ibs. per sq. inch. Proportions of Parts. Blades. Their width and length should be at least equal to .4 or .5 radius of fan. Openings. Inlet should be equal to radius of fan ; and outlet, or dis- charge, should be in depth not less than .125 diameter, its width being equaJ to width of fan. Eccentricity. .1 of diameter of fan. Journals, 4 diameters of shaft. 448 BLOWING ENGINES. By the experiments of Mr. Buckle, he deduced 1. That velocity of periphery of blades should be .9 that of their theoretical velocity ; that is, velocity a body would acquire in falling height of a homo- geneous column of air equivalent to required density. 2. That a diminution of inlet from proportions here given involved a greater expenditure of power to produce same density. 3. That greater the depth of blade, greater the density of air produced with same number of revolutions. To Compute Elements of a Fan-blower. ,/H av6o , r dav . v representing velocity of periphery of fan in feet per second, d inches of mercury, V volume of air in cube feet, and a area of discharge in sq. ins. ILLUSTRATION. Assume velocity of periphery of fan 123 feet per second, density of blast .25 inch, volume of air 1845 cube feet, and area of discharge 40 sq. ins. 244-^.25 = 122/6^. *~ - - = 1845 cub. ft. = 2.97 IB?, independent of friction of blast in pipes and tuyeres. To Compute Power of* a Centrifxigal ITan. V 2 -i- 97 300 = P. V representing velocity of tips of fan in feet per second. (See also p. 1018.) Memoranda. Operation of a blower requires about 2.5 per cent, of power of attached boiler. An increase in number of blades renders operation of fan smoother, but does not increase its capacity. Pressure or density of a blast is usually measured in ins. of mercury, a pressure of i Ib. per sq. inch at 60 = 2.0376 ins. When water is used, a pressure of i Ib. = 27.671 ins. Cupola blast .8 Ibs., and Smith's forge .25 to .3 Ibs. per sq. inch. An ordinary Eccentric Fan, 4 feet in diameter, with 5 blades 10 ins. wide and 14 in length, set 1.5 ins. eccentric, with an inlet opening of 17.5 ins. in diameter, and an outlet of 12 ins. square, making 870 revolutions per min- ute, will supply air to 40 tuyeres, each of 1.625 ms - H1 diameter, and at a pressure per sq. inch of .5 inch of mercury. An ordinary eccentric fan blower, 50 ins. in diameter, running at 1000 revolutions per minute, will give a pressure of 15 ins. of water, and require for its operation a power of 12 horses. Area of tuyere discharge 500 sq. ins. A non-condensing engine, diameter of cylinder 8 ins., stroke of piston i foot, press- ure of steam 18 Ibs. (mercurial gauge), and making 100 revolutions per minute, will drive a fan, 4 feet by 2, opening 2 feet by 2, 500 revolutions per minute. Such a blower was applied as an exhausting draught to smoke pipe of steamer Keystone State, cylinder 80 ins. by 8 feet, and evaporation was doubled over that of wiien wind was calm. In French blowing engines, volume of air discharged 75 per cent, that of volume of piston space in cylinder, stroke equal diameter of cylinder, and velocity of piston from 100 to 200 feet per minute. Area of admission valves from .066 to .083 of that of cylinder for speeds of 100 to 150 feet per minute, and from .1 to .in for higher speeds. Area of exit valves from .066 to .05 of cylinder. (M. Claudel.) BLOWING ENGINES. CENTRAL FORCES. 44^ By some experiments lately concluded in England with boilers of two iteamers, to determine relative effects of natural and forced draught furnaces, the results were as follows (/?. J. Butler) : Per Sq. Foot of Grate Surface. Natural Draught, 10 to 10.87 IH? Steam Blast, 12.5 to 13 ; Forced or Blast Draught, 15 to 16. Heating Surface per IIP. Natural Draught, 2.44 to 2.61 ; Steam Blast, 1.71 to 2.86; Forced or Blast Di*aught, 1.56 to 2.5. Tube Surf ace per IIP in Sq. Feet. Natural Draught, 2.03 to 2.18; Steam Blast, 2.02 to 2.08; Forced 01* Blast Draught, 1.3 to 2.8. IIP per Sq. Foot of Grate in these Trials. Natural Draught, 10.15 to 10.87; Steam Blast, 12.76 to 13.1 ; Forced or Blast Draught, 10.6 to 16.9. Root's Rotary Blower Is constructed from .125 to 14 nominal IP, supplying from 1 50 to 10 800 cube feet of air per minute. Delivery pipe 2.5 to 19 ins. in diameter. Efficiency 65 to 80 per cent, of power. For Ventilation of Mines From 40 to 280 revolutions per minute, equal to discharge of 12 500 to 200000 cube feet of air per minute. 15.5 to 189 H>. Steam cylinder from 14 x 18 ins. to 28 x 48 ins. For other details of Blowing Engines see page 898. CENTRAL FORCES. All bodies moving around a centre or fixed point have a tendency to fly off in a straight line: this is termed Centrifugal Force; it is op- posed to a Centripetal Force, or that power which maintains a body in /ts curvilineal path. Centrifugal Force of a body, moving with different velocities in same eircle, is proportional to square of velocity. Thus, centrifugal force of a body making 10 revolutions in a minute is 4 times as great as centrif- ugal force of same body making 5 revolutions in a minute. Hence, in equal circles, the forces are inversely as squares of times of revolution. If times are equal, velocities and forces, are as radii of circle of revolution. The squares of times are as cubes of distances of centrifugal force from axis of revolution. Centrifugal forces of two unequal bodies, having same velocity, and at same dis- '.Ance from central body, are to one another as the respective quantities of matter "n the two bodies. Centrifugal forces of two bodies, which perform their revolutions in same time, the quantities of matter of which are inversely as their distances from centre, are equal to one another. Centrifugal forces of two equal bodies, moving with equal velocities at different distances from centre, are inversely as their distances from centre. Centrifugal forces of two unequal bodies, moving with equal velocities at different distances from centre, are to one another as their quantities of matter, multiplied by their respective distances from centre. Centrifugal forces of two unequal bodies, having unequal velocities, and at differ- ent distances from their axes are in compound ratio of their quantities of matter, squares of their velocities, and their distances from centre. Centrifugal force is to weight of body, as double height due to velocity is to radius of rotation. A Radius Vector is a line drawn from centre of force to moving body. PP* 450 CENTRAL FORCES. To Compute Centrifugal Force of any Body. RULE i. Divide its velocity in feet per second by 4.01, also square of quotient by diameter of circle ; this quotient is centrifugal force, assuming the weight of body as i. Then this quotient, multiplied by weight of body, will give centrifugal force required. EXAMPLE. What is the centrifugal force of the rim of a fly-wheel having a diam- tter of 10 feet, and running with a velocity of 30 feet per second? 3o-r- 4.01 = 7. 48, and 7. 48 2 -t- 10 = 5. 59, or times weight of rim, W n 2 A/R 2 -h* 2 Or, -- ! - = C. r representing radius of inner diameter of ring. NOTE. Diameter of a fly-wheel should be measured from centres of gravity of rim, When great accuracy is required, ascertain centre of gyration of body, and take twice distance of it from axis for diameter. RULE 2. Multiply square of number of revolutions in a minute by diam- eter of circle of centre of gyration in feet, and divide product by constant number 5217 ; quotient is centrifugal force when weight of body is i. Then, as in previous Rule, this quotient, multiplied by weight of body, is centrif- ugal force required. ri 2 d Or, - x W. n representing number of revolutions per minute, d diameter of circle of gyration in feet, and W weight of revolving body in Ibs. EXAMPLE. What is centrifugal force of a grindstone weighing 1200 Ibs., 42 inches in diameter, and turning with a velocity of 400 revolutions in a minute? Centre of gyration = rad. (42-^-2) X .7071 = 14.85 ins., which -7-12 and X2 = 2.475 feet = diameter of circle of gyration. Then - - '-^^ X 1200 = 91 080 Ibs. Formulas to Determine "Various Elements. 2925 29300 ^ Wv 2 . 72930 C . /CR 32.166 -W^ ; =3^66"C ; W= V~vnT' V - W - ; : C representing centrifugal force, W mass or weight of revolving body, both in Ibs., R radius of circle of revolving body in feet, n number of revolutions per minute, and v and v' linear or circumferential and angular velocities of body in feet per second. ILLUSTRATION. What is centrifugal force of a sphere weighing 30 Ibs., revolving around a centre at a distance of 5 feet, at 30 revolutions per second? = = Centrifugal forces of two bodies are as radii of circles of revolution directly, and as squares of times inversely. ILLUSTRATION. If a fly-wheel, 12 feet in diameter and 3 tons in weight, revolves in 8 seconds, and another of like weight revolves in 6, what should be the diameter of the second when their centrifugal forces are equal ? 12 a; 12 X 6 2 . . Then 3 : 3 :: : ; or *= =6.75/6^, a? = unknown element. Centrifugal forces of two bodies, when weights are unequal, are directly as squares of times. ILLUSTRATION. What should be the ratio of the weights of the wheels in the pre- ceding case, their forces being equal ? Then 3 : x :: 6 2 : 8 2 , or a? = = = 5-333 tons. Molesworth gives .000 34 W R n 2 = C. * Thi i termed the Vi Viva, or living force. CENTRAL FORCES. FLY-WHEEL. 451 FLY-WHEEL. A FLY-WHEEL by its inertia becomes a reservoir as well as a regulator of force, and, to be fully effective, it should have high velocity, and its diam- eter be from 3 to 4 times that of stroke of driving engine. Coefficient of fluctuation of its energy ranges from .015 to .035. Weight of a fly-wheel in engines that are subjected to irregular mo- tion, as in a cotton-press, rolling-mill, etc., must be greater than in others where so sudden a check is not experienced, and its diameter should range from 3.5 to 5 times length of the stroke of the piston. A single-acting engine requires a weight of wheel about 2.5 times greater than that for a double-acting, and 5 times for double engines of double action. To Compute \Veiglit of Rim of* a Fly-wheel. RULE. Multiply mean effective pressure upon piston in Ibs. by its stroke in feet, and divide product by product of square of number of revolutions, diameter of wheel, and .000 23. NOTE. If a light wheel, multiply by .0003 ; and if a heavy one, by .000 16. EXAMPLE i. A non condensing engine (double-acting), having a diameter of cyl. inder of 14 ins., and a stroke of piston of 4 feet, working full stroke, at a pressure of 65 Ibs. mercurial gauge, and making 40 revolutions per minute, develops about 65 IP ; what should be the weight of its fly-wheel when adapted to ordinary work f Area of cylinder 154 sq. ins. Mean pressure assumed 50 Ibs. per sq. inch. Diam- eter of wheel = 4 feet stroke of piston X 3.5, assumed as above, = 14 feet. 50 X 154 X 4 = 30 800, which -j- 4o 2 X 14 X .000 23 5978 Ibs. 2. _ If a fly-wheel, 16 feet in diameter and 4 tons in weight, is sufficient to regulate an engine (double-acting), it revolving in 4 seconds, what should be the weight of a wheel of 12 feet, revolving in 2 seconds, so that it may have like centrifugal force? NOTE. The centrifugal forces of two bodies are as the radii of the circles of revo- lution directly, and as squares of times inversely. , 2 4 Z 2 2 12 X4 2 12X16 To Compute Dimensions of* Rim. RULE. Multiply weight of wheel in Ibs. by .1, and divide product by mean diameter of rim in feet ; quotient will give sectional area of rim in square inches of cast iron. Assume elements of example i. 5978 X .1 -5- 13-25 = 45. 12 sq. ins. Or, = W, and - = A. P representing pressure on piston and W weight of 4 if 10 D wheel in Ibs., S stroke of piston, and D mean diameter of wheel, both in feet, and A area of section of rim in sq. ins. Or, Ijl - = W. C coefficient, varying from 3 to 4 ordinarily, increasing to 6 with great regularity of speed and n number of revolutions per minute. NOTE. Maximum safe velocity for cast iron is assumed at 80 feet per second. For engines at high expansion of steam, or with irregular loads, as with a rolling- mill, multiply W by i. 5, or put W 100 Ibs. for each IP. (MoUsworth. ) In corn or like mills, the velocity of periphery of fly-wheel should exceed that of the stones, to arrest backlash, 452 CENTRAL FORCES. GOVEKNOKS. PENDULUMS* GOVERNORS. A GOVERNOR or CONICAL PENDULUM in its operation depends upon tha principles of Central Forces. When in a Ball Governor the balls diverge, the ring on vertical shaft raises, and in proportion to the increase of velocity of the balls squared, or the square roots of distances of ring from fixed point of arms, cor- responding to two velocities, will be as these velocities. Thus, if a governor makes 6 revolutions in a second when ring is 16 ins. from fixed point or top, the distance of ring will be 5.76 ins. when speed is increased to 10 revolutions in same time. For 10 : 6 : : V 16 : 2.4, which, squared = 5.76 ins., distance of ring from top. Or, 6 2 ; 10'* : : 5.76 : 16 ins. A governor performs in one minute half as many revolutions as a pendulum vibrates, the length of which is perpendicular distance be- tween plane in which the balls move and the fixed point or centre of suspension. To Compnte Nximtoer of Revolutions of a Ball Q-overnor per IM.in.ixte to maintain JBalls at any given Height. 188 -T- v/H = revolutions. H representing vertical height between plane of balls and points of their suspension in ins. ILLUSTRATION. If the rise of the balls of a centrifugal governor is 22 ins., what are the number of revolutions per minute ? 188 -r- ^22 = 40.09 revolutions. To Compute Vertical Height between Plane of Balls and. their Points of Svispension. (i 88 -j- r) 2 = vertical height in ins. r representing number of revolutions per minute. ILLUSTRATION. If number of revolutions of a centrifugal governor is 100, what will be rise of balls? i88-7-ioo=i.88 2 = 3.53 ins. To Compute Angle of A.rms or Plane of Balls -with Centre Shaft. r -1-1 = sin. /__. r representing distance of balls fi'om plane of centre shaft, and. I distance between balls and point of suspension measured in plane of shaft. ILLUSTRATION. Distance of balls from plane of centre shaft is 10 inches, and their distance from point of suspension is 25 ; what is the angle ? 10 -T- 25 = .4, and gin. .4 = 23 35'. When Number of Revolutions are given. [ ~~T ~ cos - - ILLUSTRATION. Revolutions of a governor per minute are 50, and length of its arms 2 feet; what is their angle with plane of shaft? (54 ' I6 ^ 50)2 = 1IZ3 = . 5865 = cos. 54 6'. 2 2 PENDULUMS. Pendulums are Simple or Compound, the former being a material point, or single weight suspended from a fixed point, about which it oscillates, or vibrates, by a connection void of weight ; and the latter, a like body or number of bodies suspended by a rod or connection. Any such body will have as many centres of oscillation as there are given points of suspension to it, and when any one of these centres are determined the others are readily ascertained. CENTRAL FORCES. PENDULUMS. 453 Thus, s o X s g = a constant product, and s r = Vs o x s g, s g o and r representing points of suspension, gravity, oscillation, and gyration. Or, any body, as a cone, a cylinder, or of any form, regular or irregular, so suspended as to be capable of vibrating, is a compound pendulum, and distance of its centre of oscillation from any assumed point of suspension is considered as the length of an equivalent simple pendulum. The Amplitude of a simple pendulum is the distance through which it passes from its lowest position to its farthest on either side. Complete Period of a pendulum in motion is the time it occupies hi making two vibrations. All vibrations of same pendulum, whether great or small, are performed very nearly in same time. Number of Oscillations of two different pendulums in same time and at same place are in inverse ratio of square roots of their lengths. T^ength of a Pendulum vibrating seconds is hi a constant ratio to force of gravity. Time of Vibration is half of a complete period, and it is proportional to square root of length of pendulum. Consequently, lengths of pendulums for different vibrations are Latitude of Washington. 39.0958 ins. for one second. I 4.344 for third of a second. 9. 774 ins. for half a second. 2.4435 for quarter of a second. ^engths of Pendulums vibrating Seconds at "Level of* the Sea in several Places. Ins. Equator 39.0152 Washington 39-0958 New York 39. 1017 Lat. 45 39. 127 In*. Paris 39-1284 London 39-1393 To Compute Length, of* a Simple Pendulum for a given Latitude. 39. 127 .099 82 cos. 2 L = I. L representing latitude. ILLUSTRATION. Required the length of a simple pendulum vibrating seconds iu the latitude of 50 31'. L = 50 31" cos. 2L = 2X5o 31' = cos. 180 50 31' X 2 = cos. 78 58' = . 191 38 39. 127 -f- . 191 38 x .099 82 (two or negative = an affirmative or -f-) = 39. 1461 ins. To Compute Length of* a Simple Pendulum for a given Number of "Vibrations. L t z = I. L representing length for latitude, t time in seconds, and I length of pen- dulum in ins. ILLUSTRATION. Required vibrations of a pendulum in a minute at New York, are 60; what should be its length? 39. 1017 x i 2 =39- 1017. Or, = I. n representing number of vibrations per second. To Compute Number of Vibrations of a Simple Pendu- lum in a given Time. = n, representing time of one vibration in seconds. v * ** To Compute Centre of Gravity of a Compound Pend.u lum of T\vo Weights connected in a. Right Line. When Weights are both on one Side of Point of Suspension. lW-\-l'w ~\ o = distance oj centre of gravity from point of suspension. W -|- w 454 CENTBAL FORCES. PENDULUMS. When Weights are on Opposite Sides of Point of Suspension. -_"~ = c = distance of centre of gravity of greater weight from point ofsus- W -j- w pension. NOTE. To obtain strictly isochronous vibrations, the circular arc must be sub- stituted for the cycloid curve, which possesses the property of having an inclina- tion, the sine of which is simply proportional to distance measured on the curve from its lowest point. For construction of a Cycloidal pendulum, see Deschaniel's Physics, Part I., pp. 71-2. To Compute Length of a, Simple !Pendul vim, Vib ration s of xvhich -will be same in. Number as Inches in its Length.. 2 = I in inches. ILLUSTRATION. What will be length of a pendulum in New York, vibrations of which will be same number as the ins. in its length ? V (>/39. 1017 X 6o) 2 = 7.21 1 2 = 52 ins. To Compute Time of Vibration of a Simple I>endvalum, Length "being given. Vl -^L = tin seconds. dulum 156.4 ILLUSTRATION. Length of a pendulum is 156.4 ins. : what is the time of its vibra- tion in New York ? J: - 2 seconds. 39- IQ i7 Or, */ X 3. 1416 = t. I representing length of a pendulum vibrating seconds in ins., g measure of force of gravity, and t time of one oscillation. ILLUSTRATION. Length of a simple pendulum vibrating seconds, and measure of force of gravity at Washington, are 39.0958 ins., and 32.155 feet. 3.1416 /-jT, ^ 3- M l6 X V X - OX 3 = 3-1416 X .3183 = i second. To Compute Number of Vibrations of a Simple Pen- dulum in a given Time. 'L x t = n. n representing number of vibrations. ILLUSTRATION. The length of a pendulum in New York is 156.4 ins., and time of its vibration is 2 seconds; what are number of its vibrations? X 2 = X 2 = .5 X 2 = z vibration. Hence, , X = 30 r- brationsper minute. To Compute Measure of Gravity, Length of IPendulum and. Number of its "Vibrations 'being given. .82246? w 2 = g. g representing measure of gravity in feet. To Compute Number of Revolutions of a Conical Pen- dulum per iMinute. %/ ^ = n. h representing distance between point of suspension and plane oj revolutions in ins. NOTE. Number of revolutions per minute are constant for any given height, and the time of a revolution is directly as square root of height. CRANES. 455 When IPost is CRANES. Usual form of a Crane is that of a right-angled triangle, the sides being post or jib, and stay or strut, which is hypothenuse of triangle. When jib and post are equal in length, and stay is diagonal of a square, this form is theoretically strongest, as the whole stress or weight is borne by stay, tending to compress it in direction of its length ; stress upon it, com- pared to weight supported, being as diagonal to side of square, or as 1.4142 to i. Consequently, if weight borne by crane is 1000 Ibs., thrust or com- pression upon stay will be 1414.2 Ibs., or as a e to e W, Fig. i. Supported at tooth Head and Foot, as Fig. 1. a Weight W is sustained by a rope or chain, and tension is equal upon both parts of it ; that is, on two sides of square, i a and e W. Conse- quently jib, i a, has no stress upon it, and serves merely to retain stay, a e. If foot of stay is set at w, thrust upon it, as t v compared with weight, will be as an to aw ; and if chain or rope from i to a is removed, and weight is suspended from a, tension on jib will be as i a to a W. If foot of stay is raised to o, thrust, as compared with weight, will be as line a o is to a W, and tension on jib will be as line ar. By dividing line representing weight, as a W or a w, into equal parts, to represent tons or pounds, and using it as a scale, stress upon any other part may be measured upon described parallelogram. Thus, a? length of a W, compared to a e, is as i to 1.4142 : if a W is di- vided into 10 parts representing tons, a e would measure 14.142 parts or tons. 'When Post is Supported at Foot only. If post is wholly unsupported at head, and its foot is secured up to line o W, then W, acting with leverage, e W, will tend to rupture post at e, with. Bailie effect as if twice that weight was laid upon middle ef a beam equal to twice length of e W, e being at middle of beam, which is assumed to be sup- ported at both ends, and of like dimensions to those of post. Or, force exerted to rupture post will be represented by stress, W, multi- plied by 4 times length of lever, e W, divided by depth of post in line of stress, squared, and multiplied by breadth of it and Value * of its material. Post of such a crane is in condition of half a beam supported at one end, weight suspended from other ; consequently, it must be estimated as a beam of twice the length supported at both ends* stress applied in middle. To Compute Stress on Jib, and on Stay or Struts-Fig. Q. On diagram of crane, Fig. 2, mark off on line of chain, a W, a distance, a , representing weight on chain; from point b draw a line, b c, parallel to jib, a e, and where this intersects stay or strut, draw a vertical line, c o, extending to jib, and distances from a to points b c and o c, measured upon a scale of equal parts, will represent proportional strain. ILLUSTRATION. In figure, weight being 10 tons, stress on stay or strut compressing, a c, will be 31 tons, and on jib or tension-rods, a o, 26 tons. * For Value of Materials, see page 77^ Fig. 2. 456 CRANES. To Compnte Dimensions of Post of a Crane. When Post is Supported at Feet only. RULE. Multiply weight or stress to be borne in Ibs. by length of jib in feet measured upon a horizontal plane ; divide product by Value of material tc >e used, and product, divided by breadth in ins., will give square of depth, aiso in ins. EXAMPLE. Stress upon a crane is to be 22400 Ibs., and distance of it from centre of post 20 feet; what should be dimension of post if of American white oak? Value of American white oak 50. Assumed breadth *" ins. When Post is /Supported at both Ends. RULE. Multiply weight or stress to be borne in Ibs. by twice length of jib in feet measured upon a horizontal plane ; divide product by Value of material to be used, and product, divided by four times breadth in ins., will give square of depth, also in ins. EXAMPLE. Take same elements as in preceding case. Assumed breadth 10 ins. 22400X20X2 17020 Then = 17 920, = 448, and ^448 = 21. 166 ins. 50 4 X 10 In Fig. 3, angle abe and e b c being equal, chain or rope is represented by a b c, and weight by W ; stress upon stay b a, as compared with weight, is as b d to a b or b c. In practice, however, it is not prudent to consider chain as supporting stay ; but it is proper to disrep d chain or rope as forming part of system, and crane should be designed to support load independent of it. It is also proper that angles on each side of diagonal stay, in this case, should not be equal. If side a b is formed of tension-rods of wrought iron, point a should be depressed, so as to lengthen that side, and decrease angle a b e ; but if it is of timber, point a should be raised, and angle abe increased. Fig. 4. g Fig. 4 shows a form of crane very generally used; ,< angles are same as in Fig. 3, and weight suspended from f S'; ! c it, being attached to point d, is represented by line b d. '',''/ The tension, which is equal to weight, is shown by leugt! / of line b c, and thrust by length of line b a, measured by / a scale of equal parts, into which line bd, repiesentiag weight, is supposed to be divided. But if & e be direction of jib, then b g will show ten- sion, and bf the thrust (df being taken parallel to b e), both of them being now greater than before ; line b d representing weight, and being same in both cases. To Ascertain Stress on Ji"b, on Strut of a Crane. Fig. 5. Through a draw a s, parallel to jib or tension -rod o r, and also s u parallel to strut a r ; then r s is a diagonal of parallelogram, sides of which are equal to ra and r u. If then r s represents a stress of 20 Ibs., the two forces into which it is decom- posed are shown by r u and r a ; or is equal to r u, as each of them is equal to a s, and r s is equal to o a. Hence, 20 represented by a o, stress on jib will be represented by o r, and that on strut by ra. Assuming then or 3 feet, a r 3.5, and o a i, stress on jib will be 60 Ibs., and on strut 70. CEANES. 457 Thus, in all cases, stress on jib or tension-rod and on strut can be deter- mined by relative proportions of sides of triangle formed. To Compttte Stress -upon Strnt of* a Crane. RULE. Multiply length of strut in feet by weight to be borne in Ibs. ; di- vide product by height of jib from point of bearing of strut in fee% and quotient will give stress or thrust in Ibs. EXAMPLE. Length of strut of a crane is 28.284 feet > height of post is 26.457 feet, and weight to be borne is 22 400 Ibs. ; what is stress ? 28.284X22400 = 633561.6 = lbg 26.457 26.457 Chains and. Ropes. Chains for Cranes should be made of short oval links, and should not ex- ceed i inch in diameter. Short -linked Crane Chains and Ropes showing T^i- mensions and "Weight of each, and 3?roof of Chain in Tons. Diam. of Chains. Weight per Fathom. Proof Strain. Circumf. of Rope. Weight of Rope per Path. Diam. of Chains. Weight Fathom. Proof Strain. Circnmf. 1 Weight of of Rope Rope, j per Path. Ins. I/'S. Tons. Ins. Lba. Ing. Lbs. Tons. Ins. Lbs. 3125 6 75 2-5 i-5 .6875 28 6.5 7 10.5 375 8-5 i-5 3-25 2-5 75 32 7-75 7-5 12 4375 ii 2-5 4 3-75 .8125 36 925 8.25 15 5 H 3-5 4-75 5 875 44 10-75 9 17-5 5625 18 4-5 5-5 7 9375 50 12.5 9-5 19-5 .625 24 5-25 6.25 8.7 i 56 H 10 22 Ropes of circumferences given are considered to be of equal strength with the chains, which, being short-linked, are made without studs. A crane chain will stretch, under a proof of 15 tons, half an inch per fathom. Machinery of Cranes. To attain greater effect of application of power to a crane, the wheel-work must be properly designed and executed. If manual labor is employed, it should be exerted at a speed of 220 feet per minute. Proportions. Capacity of Crane, 5 tons. Radius of winch or handle 15 to 1 8 ins. Height of axle from flou 36 to 39. 2d pinion, 12 teeth, 1.5 ins. pitch. 2d wheel, 96 " " " u Barrel 8 ins. x 1 1 teeth x 12 teeth X 1 1 200 Ibs. = 30 800 ., - = 20. 7 5 I os. = statical re- Winch 17 ins. x 89 teeth x 96 teeth x 4 men = 1513 sistance to each of the 4 men at winches. An experiment upon capacity of a crane, geared i to 105, develor ad that a strong man for a period of 2.5 minutes exerted a power of 27 562 foot- pounds per minute, which, when friction of crane is considered, is fully equal to the power of a horse for one minute. In practice an ordinary man can develop a power of 15 Ibs. upon a crane, handle moved at a velocity of 220 feet per minute, which is equivalent to 3300 foot-pounds. For Treatise on Cranes, see Weales' Series, No. 33. ist pinion, n teeth, 1.25 ins. pitch, ist wheel, 89 " 1.25 " " 458 COMBUSTION. COMBUSTION. Combustion is one of the many sources of heat, and denotes combi- nation of a body with any of the substances termed Supporters of Com- bustion ; with reference to generation of steam, we are restricted to but one of these combinations, and that is Oxygen. All bodies, when intensely heated, become luminous. When this heat is produced by combination with oxygen, they are said to be ignited ; and when the body heated is in a gaseous state, it forms what is termed Flame. Carbon exists in nearly a pure state in charcoal and in soot. It com- bines with no more than 2.66 of its weight of oxygen. In its combus- tion, i Ib. of it produces sufficient heat to increase temperature of 14 500 Ibs. of water i. Hydrogen exists in a gaseous state, and combines with 8 times its weight of oxygen, and i Ib. of it, in burning, raises heat of 50 ooo Ibs. of water i .* An increase in the rapidity of combustion is accompanied by a dimi- nution in the evaporative efficiency of the combustible. Mr. D. K. Clark furnishes the following: When coal is exposed to heat in a fur- nace, the carbon and hydrogen, associated in various chemical unions, as hydrocar- bons, are volatilized and pass off. At lowest temperature, naphthaline, resins, aud fluids with high boiling-points are disengaged; at a higher temperature, volatile fluids are disengaged; and still higher, defiant gas, followed by light carburetted hydrogen, which continues to be given off after the coal has reached a low red heat. As temperature rises, pure hydrogen is also given off, until finally, in the fifth or highest stage of temperature for distillation, hydrogen alone is discharged. What remains after distillatory process is over, is coke, which is the fixed or solid carbon of coal, with earthy matter or ash of the coal. The hydrocarbons, especially those which are given off at lowest temperatures, being richest in carbon, constitute the flame-making and smoke-making part of the coal. When subjected to heat much above the temperatures required to vaporize them, they become decomposed, and pass successively into more and more perma- nent forms by precipitating portions of their carbon. At temperature of low red- ness none of them are to be found, and the oleflant gas is the densest type that remains, mixed with carburetted and free hydrogen. It is during these trans- formations that the great volume of smoke is made, consisting of precipitated car- bon passing off uncombined. Even olefiant gas, at a bright red heat, deposits half its carbon, changing into carburetted hydrogen; and this gas, in its turn, may deposit the last remaining equivalent of carbon at highest furnace heats, and be converted into pure hydrogen. Throughout all this distillation and transformation, the element of hydrogen maintains a prior claim to the oxygen present above the fuel; and until it is satis- fied, the precipitated carbon remains unburned. Sximmary of J?rodTu.cts of Decomposition, in the ITnrnace. Reverting to statement of average composition of coal, page 485, it ap- pears that the fixed carbon or coke remaining in a furnace after volatile portions of coal are driven off, averages 61 per cent, of gross weight of the coal. Taking it at 60 per cent., proportion of carbon volatilized in com- bination with hydrogen will be 20 per cent., making total of 80 per cent, of constituent carbon in average coal. Of the 5 per cent, of constituent hydrogen, i part is united to the 8 per cent, of oxygen, in the combining proportions to form water, and remaining 4 parts of hydrogen are found partly united to the volatilized carbon, and partly free. COMBUSTION. 459 These particulars are embodied in following summary of condition of elements of 100 Ibs. of average coal, after having been decomposed, and prior to entering into combustion 100 Lbs. of Average Coal in a Furnace. Composition Lbs. Lbs. Decomposition. 60 fixed carbon. 24 hydrocarbons and free hydrogen 1.25 sulphur. Hydrogen ............. 5 Sulphur ............... 1.25 Oxygen ............... 8 Nitrogen .............. 1.2 Ash, etc ............... 4-55 forming 85-25 9 water or steam. 1.2 nitrogen. 4.55 ash, etc. IOO showing a total useful combustible of 85.25 per cent., of which 25.25 per cent, is volatilized. While the decomposition proceeds, combustion proceeds, and the 25.25 per cent, of volatilized portions, and the 60 per cent, of fixed carbon, successively, are burned. It may be added that the sulphur and a portion of the nitrogen are dis- engaged in combination with hydrogen, as sulphuretted hydrogen and am~ monia. But these compounds are small in quantity, and, for the sake of simplicity, they have not been indicated in the synopsis. Volume of Air chemically consumed in complete Combustion of Coal. Assume 100 Ibs. of average coal Then, by following (o\ _ 5 -J + -4 X 1.25 X 152 = 14060 cube feet of air at 62 for 100 Ibs. coal For volatilized portion, Hydrogen (H), 4 Ibs. X 457= 1828 cube feet Carbon (C), 20 " X 152= 3040 " " Sulphur (S), 1.25 " X 57= _ ij_ " " 4939 " " For fixed portion, Carbon, 60 Ibs.x 152= 9120 " " Total useful combustible, 85.25 u 14059 " " for com- plete combustion of 100 Ibs. coal of average composition at 62. To Comprite "Volume of Air at 62, nnder One At- mosphere, chemically- consumed, in. Complete Com* iDxistion of* 1 Lt>. of* a given Fuel. RULE. Express constituent carbon, hydrogen, oxygen, and sulphur, as percentages of whole weight of fuel ; divide oxygen by 8, deduct quotient from hydrogen, and multiply remainder by 3 ; multiply sulphur by .4 ; add products to the carbon, and multiply sum by 1.52. Final product is volume of air in cube feet. To compute weight of air chemically consumed. Divide volume thus found by 13.14; quotient is weight of air in Ibs. Or, 1.52 (C + 3 (H--|) +.4 S) = Air. Oxygen. NOTE. In ordinary or approximate computations, sulphur may be neglected. EXAMPLE. Assume i Ib. Newcastle coal = 82.24, 1^ = 5.42, = 6.44, an d 8 = 1.35- -^ = .805, 5.42 .805 = 4.615 X 3 = 13-845, 1.35 X .4 = - 54, 13-845 + -54+8-2.24 = 96.625, and 96.625 X 1-52 = 146.87 cube feet. Then 146. 87 -r- 13. 14 = 1 1. 18 Ibs. 460 COMBUSTION. To Compxite Total Weight of Gaseous Products of Com- plete Combustion, of 1 I-*b. of a given Fuel. RULE. Express the elements as per-centages of fuel ; multiply carbon by .126, hydrogen by .358, sulphur by .053, and nitrogen by .01, and add prod- ucts together. Sum is total weight of gases in Ibs. Or, .126 C + .358 H-f. 053 S + .oi N = Weigkt. EXAMPLE. Assume as preceding case. N = 1,61. 82.24 X .1264- 5-42 X. 358 + 1.35 X 053 + 1.61 X .01 = 12.39 Ibs. To Compute Total "Volume, at 62, of Q-aseous Products of Complete Combustion of 1 .Lb. of given Fuel. RULE. Express elements as per-centages ; multiply carbon by 1.52, hy- drogen by 5.52, sulphur by .567, and nitrogen by .135, and add products together. Sum is total volume, at 62 F.. of gases, in cube feet. Or, 1.52 C-f-5.52 H + .s67 S -f . 135 N = Volume. To Compute "Volume of the several O-ases separately from their Respective Quantities. RULE. Multiply weight of each gaseous product by volume of i Ib. in cube feet at 62, as below. Volume of i Lb. of Gases at 62 under a Pressure of 14.7 Lbs. Cube feet. Cube feet. Cube feet. Aqueous Vapor or) I Oxygen 11.887 I Nitrogen 13.501 Gaseous Steam . j , 5 | Hydrogen 190 | Carbonic Acid 8. 594 Air 13. 141 cube feet. For a Ib. of oxygen in combustion, 4.35 Ibs. air are consumed; or, by volume, for a cube foot of oxygen 4.76 cube feet of air are consumed. i Ib. Hydrogen consumes 34. 8 Ibs. , or 457 cube feet, at 62. i " Carbon, completely burned, consumes n.6 " u 152 " " " " i " " partially " u .... 5.8 " " 76 " " " " Composition and Equivalents of Gases, combined in Combustion of Fuel. OASES. Elements. By Weight. GASES. Elements. By Weight. ELEMENTS. Equ v- alents. COMPOUNDS. Equiv- alents. 0. H. 8 Light Carburetted Hydrogen C. 2 H A. ;} Hydrogen . Carbon C. S. 6 16 Carbonic Oxide ii. 4 0. i C. i 4 ) 8 1 6 j Nitrogen N. 14 Carbonic Acid O. 2 C. i 16) 6J COMPOUNDS. OlefiantGas(Bi-car-) C. 4 24) "Atmospheric Air 0.23 8 ) buretted Hyd. . . . j H. 4 4} (mech. mixture) . . Aqueous Vapor or Water. . . S: 7 , 7 H. i 26.8} Sulphurous Acid. . . 0. 2 8. z 16) 16 Weights of products in combustion of i Ib. of given fuel, are 0^.0366. H = .o9. S = .o2. N = .08930 + . 268 H + . 0335 S+. 01 N. Cube Feet. .0366 X 8.59= .315 volume carbonic acid. .09 X 190 =17.1 Cube Feet. .02 X 5.85 = .117 volume sulph. acid. .0893 4- -268 -f- .0335 -f- .01 X 13-501 5.409 volume nitrogen. Volume of Air or Gases at higher temperatures than here given (62) is ascer- tained by, V = V. V representing volume of air or gas at temperature t, t +461 and V at temperature t'. * By Volume i Oxygen, 3.762 Nitrogen. COMBUSTION. Chemical Composition of some Compound. Com- t>u.stil>les. COMBUSTIBLK. Combi Car. Ding equiv Hyd. alents. Oxy. In 100 Car. parts by weight. Hyd. | 0*y. Carbonic oxide I I Per Cent. 42.9 Per Cent. Per Cent. 57-i 21.7 34-8 4-5 9-4 9-3 Oleflant gas, Bicarburetted hyd. 4 4 4 20 4 5 6 16 I 2 III 1 81.6 77.2 79 14-3 13-5 i3-9 i3-4 ii. 7 Wax Olive oil Tallow . . . Heating powers of compound bodies are approximately equal to sum of heating powers of their elements. Thus, carburetted hydrogen, which consists of two equivalents of carbon and four of hydrogen, weighing respectively 2X6=12 and i x 4 = 4, in proportion of 3 to i, or .75 Ib. of carbon and .25 Ib. of hydrogen in one Ib. of gas. Elements of heat of combustion of one Ib. are, then Units of heat. For carbon 14 544 X -75 = 10 908 For hydrogen 62 032 X .25 = 15508 Total heat of combustion, as computed. 26 416 Total heat, by direct trial 23513 Heating IPowers of Com.~biistit>les. (MM. Favre and Silbermann, D. K. Clark and others.) i LB. OF COMBUSTIBLE. Oxygen consumed per Ib. of Com- bustible. Weight and Volume of Air consumed per Ib. of Combustible. Total Heat of Combus- tion of i Ib. of Combus- tible. Equivalent evaporative Power of i Ib. of Com- bustible, under one At' mosphere. Lbs. Lbs. Cube Feet at 62. Units. Lbs. of wa- ter at 62. Lbs. of wa- ter at 212". 8 34-8 457 62032 55-6 64.2 Carbon, making ) carbonic oxide, j i-33 5-8 76 4452 4 4.61 Carbon, making ) carbonic acid.. } 2.66 n.6 152 14500 13 15 Carbonic oxide 57 2.48 33 43 2 5 3.88 4.48 Light carburetted ) hydrogen J 4 17.4 229 23513 21.07 24.34 Oleflant gas a A"i je 196 21 343 IQ. 12 22.09 Sulphuric ether. . . . 3- 43 2.6 "3 149 16249 y * 14.56 16.82 Alcohol . . 2 78 12. 1 12 Q2Q 11.76 13.38 Turpentine Ui /W 3 2Q 14- 3 188 ** v^y 19 534 17.5 20.22 Sulphur . . . . O'^y A. 35 57 4 3 2 2 7.61 4. 17 Tallow 2 0^ 12.83 169 18028 J.V 16.15 18.66 Petroleum ^yo 17 Q^ 235 27 531 28.5 Coal (average) 4-12 2.46 *-/'yj 10.7 141 14 '33 12.67 14.62 Coke, desiccated. . . 2-5 IO.Q 143 13550 12.14 14.02 Wood, desiccated . . 1.4 6.1 80 7792 6.98 8.07 Wood - charcoal, ) desiccated J 2.25 9.8 129 13309 11.92 13-13 Peat, desiccated i-75 7.6 IOO 9951 8.91 10.3 peat- charcoal, de- ) 2.28 9.9 129 12325 11.04 12.76 2.03 8.85 116 11678 12. X Asphalt... 2.73 11.87 156 16655 I7-*4 When carbon is not completely burned, and becomes carbonic oxide, it producei less than a third of heat yielded when it is completely burned. For heating powei of carbon an average of 14 500 units is adopted. Uu* COMBUSTION. To Compute Heating Power of* 1 I_/b. of a given Corn- ID ustifole. When proportions of Carbon, Hydrogen, Oxygen, and Sulphur are given. RULE. Ascertain difference between hydrogen and .125 of oxygen; multi- ply remainder by 4.28 ; multiply sulphur by .28, add products to the carbon, multiply sum by 14 500, divide by 100, and product is total heating power hi units of heat. Or, 145 ( + 4.28 H Ox. 125 -f . 288) = heat ILLUSTRATION. Assume as preceding case. 5.42 'v 82.28 X. 125 X 4-28 -f- 1. 35 X . 28 + 82.28 X 14500-1-100^:15005. To Compute Evaporative IPo\ver of 1 Lt>. of a Griven Combustible. When Proportions of Carbon, Hydrogen, Oxygen, and Sulphur are given. RULE. Ascertain difference between hydrogen and .125 of oxygen, multiply remainder by 4.28 ; multiply sulphur by .28, add products to the carbon, and multiply sum by .13, when water is supplied at 62, and .15 when at 212 ; product is evaporative power in Ibs. of water at 212. Or, When total heating power is known, divide it by 1116 when water is at 62, or 996 when at 212. ILLUSTRATION. By table, heating power of Tallow is 18028 units. Hence, 18028 -5- 1116 = 16.15 Ms. water evaporated at 62. Tem.peratu.re of ComlmstiorL. Temperature of combustion is determined by product of volumes and specific heats of products of combustion. ILLUSTRATION. i Ib. carbon, when completely burned, yields 3.66 Ibs. carbonic acid and 8.94 of nitrogen. Specific heats .2164 and .244. 3.66 X .2164 = .792 units of heat for i. 8. 94 X. 244 = 2.181 " " " i. 12.6 2.973 " " " i. Consequently, products of combustion of i Ib. carbon absorbs 2.973 units of heat in producing i temperature. Weignt and Specific Heat of Products of Comtmstioii, and Temperature of Combustion . (D. K. Clark.) Gaseous Products for i Lb. of Combustible. i LB. OF COMBUSTIBLE. Weight. Mean specific Heat. Heat to raise the Tempera- ture i. Temperature ol Combustion. Lbs. 35-8 11.97 15-9 13.84 11.94 12.6 15-21 10.09 18.4 5-35 12. 18 22.64 Water = i. .302 .256 257 .256 .240 .236 257 .27 .268 .211 257 .242 Units. 10.814 3.063 4.089 3-54 2-935 2-973 3-9M 2.68 4-933 1.128 3-127 5-478 5744 5305 5219 5093 4879 4877 4826 4825 4766 3575 3470 2614 Ratio. 100 92 Is, 85 1 S 8 4 84 83 62 60 45 Sulphuric ether Olefiant gas (Bi-carburetted hyd.) Tallow Carbon or pure coke. Wax Light carburetted hydrogen Turpentine Coal, with double supply of air. . Whence it appears, that mean specific heat of products of combustion, omitting hydrogen .302 and sulphur .211, is about .25. Hence. To Ascertain Temperature of Combustion. Divide total heat of combustion in units by units of heat for i, and quotient will give tem- perature. COMBUSTION. 463 ILLUSTRATION'. What is temperature of combustion of coal of average composi- tion? Gaseous products as per preceding table 11.94, which x .246 specific heat = 2.935 units of heat at 1. Hence, 14 133 units of combustion (from table, page 461) -7-2.935 = 4812 temper- ature of combustion of average coal. If surplus air is mixed with products of combustion equal to volume of air chem- ically combined, total weight of gases for ore Ib. of this coal is increased to 22.64. See following table, having a mean specific heat ol .242. Then 22.64 X .242 = 5.478 units for 1. Hence, 14133 total heat of combustion -4-5. 478 = 2580 temperature of combus- tion, or a little more than half that of undiluted products. Taking averages, it is seen that the evaporative efficiency of coal varies directly with volume of constituent carbon, and inversely with volume of constituent oxygen ; and that it varies, not so much because there is more or less carbon, as, chiefly, because there is less or more oxygen. The per-cent- ages of constituent hydrogen, nitrogen, sulphur, and ash, taking averages, are nearly constant, though there are individual exceptions, and their united effect, as a whole, appears to be nearly constant also. Heat of Combustion. Or, number of times in combustion of a substance, its equivalent weight of water would be raised 1, by heat evolved in combustion of substance. Alcohol 12 930 I Ether 16 246 I Olefiant gas 21 340 Charcoal 14 545 I Olive oil 17 750 | Hydrogen 62 030 Com'bnstion of IPuel. Constituents of coal are Carbon, Hydrogen, Azote, and Oxygen. Volatile products of combustion of coal are hydrogen and carbon, the unions of which (relating to combustion in a furnace) are Carburetted hydrogen and Bi-carburetted hydrogen or Olefiant gas, which, upon com- bining with atmospheric air, becomes Carbonic acid or Carbonic oxide, Steam, and uncombined Nitrogen. Carbonic oxide is result of imperfect combustion, and Carbonic acid that of perfect combustion. Perfect combustion of carbon evolves heat as 15 to 4.55 compared with imperfect combustion of it, as when carbonic oxide is produced. i Ib. carbon combines with 2.66 Ibs. of oxygen, and produces 3.66 Ibs. of carbonic acid. Smoke is the combustible and incombustible products evolved in combustion of fuel, which pass off by flues of a furnace, and it is composed of such portions of hydrogen and carbon of the fuel gas as have not been supplied or combined with oxygen, and consequently have not been converted either into steam or carbonic acid; the hydrogen so passing away is invisible, but the carbon, upon being sepa- rated from the hydrogen, loses its gaseous character, and returns to its elementary state of a black pulverulent body, and as such it becomes visible. Bituminous portion of coal is converted into gaseous state alone, carbonaceous portion only into solid state. It is partly combustible and partly incombustible. To effect combustion of i cube foot of coal gas, 2 cube feet of oxygen are required; and, as 10 cube feet of atmospheric air are necessary to supply this volume of oxy- gen, i cube foot of gas requires oxygen of 10 cube feet of air. In furnaces with a natural draught, volume of air required exceeds that when the draught is produced artificially. An insufficient supply of air causes imperfect combustion ; an excessive supply, a waste of heat. 464 COMBUSTION. Volume of atmospheric air that is chemically required for combustion of b. of bituminous coal is 150.35 cube feet. Of this, 44.64* cube feet corn- Volume < i lb. bine with the gases evolved "from the coal, and remaining 105.71 cube feet combine with the carbon of the coal. Combination of gases evolved by combustion gives a resulting volume proportionate to volume of atmospheric air required to furnish the oxygen, as ii to 10. Hence the 44.64 cube feet must be increased in this proportion, and it becomes 44.64 -f- 4.46 = 49.1. Gases resulting from combustion of the carbon of coal and oxygen of the atmosphere, are of same bulk as that of atmospheric air required to furnish the oxygen, viz., 105.71 cube feet. Total volume, then, of the atmospheric air and gases at bridge wall, flues, or tubes, becomes 105.71 + 49.1 = 154.81 cube feet, assuming temperature to be that of the external air. Conse- quently, augmentation of volume due to increase of temperature of a fur- nace is to be considered and added to this volume, in the consideration of the capacity of flue or calorimeter of a furnace. There is required, then, to be admitted through the grates of a furnace for combustion of i lb. of bituminous coal as follows : Coal containing 80 per cent, of carbon, or .7047 per cent, of coke. i lb. coal x 44-64 cube feet of gas = 44.64 7047 lb. carbon x 150 cube feet of air ... = 105.71 150.35 cube feet. For anthracite, by observations of W. R. Johnston, an increase of 30 per cent, over that for bituminous coal is required = 195.45 cube feet. Coke does not require as much air as coal, usually not to exceed 108 cube feet, depending upon its purity. Heat of an ordinary furnace may be safely considered at 1000 ; hence air entering ash-pit and gases evolved in furnace under general law of expan- sion of permanently elastic fluids of ^-^ths of its volume (or .002087) for each degree of heat imparted to it, the 154.81 is increased in volume from 100 (assumed ordinary temperature of air at ash-pit) to 1000 = 900 ; then 9oox .002087 = 1.8783 times, or 154.81 + 154.81 x 1.8783 = 445.59 cube feet. If the combustion of the gases evolved from coal and air was complete, there would be required to give passage to volume of but 445.59 cube feet over bridge wall or through flues of a furnace ; but by experiments it ap- pears that about one half of the oxygen admitted beneath grates of a furnace passes off uncombined ; the area of the bridge wall, or flues or tubes, must con- sequently be increased in this proportion, hence the 445.59 becomes 891.18. Velocity of the gases passing from furnace of a proper-proportioned boiler may be estimated at from 30 to 36 feet per second. Then 891.18 60' x 60" x 36" .00687 sq. feet, or .99 sq. ins., of area at bridge wall for each lb. of coal con- sumed per hour. A limit, then, is here obtained for area at the bridge wall, or of flues or tubes immediately behind it, below which it must not be decreased, or com- bustion will be imperfect. In ordinary practice it will be found advan- tageous to make this area .014 sq. feet, or 2 sq. ins. for every lb. of bitu- minous coal consumed per sq. foot of grate per hour, and so on In proportion for any other quantity. Volumes of heat evolved are very nearly same for same substance, what- ever temperature of combustible. * By experiment, 4.464 cube feet of gas are evolved from i lb. of bituminous coal, requiring 44.64 cube feet of air. COMBUSTION. 465 Relative Volumes of Air required for Combustion of Fuels. Lbs. 1 Lbs. Warlich's patent. . . 13. i I Anthracite Coal .... 12. *-. Charcoal 11.16 Bituminous " .... io.gl Coke 11.28 1 Bitum. Coal, average 10.7 Lba. Bitum. Coal, lowest. . 5.92 Peat, dry 7.08 Wood, dry..., 6 Perfect combustion of i Ib. of carbon requires 11.18 Ibs. air at 62, and total weight = 1 2.39./fo. Total heat of combustion of i Ib. carbon or char- coal is 14 500 thermal units ; mean specific heat of products of combustion is .25, which, multiplied by 12.39 as above = 3.0975, and 14 500* -r- 3.0975 = 4681 temperature of a furnace, assuming every atom of oxygen that was ignited in it entered into combination. If, however, as in ordinary furnaces, twice volume of air enters, then products of combustion of i Ib. of coal will be 12.39 H~ 11.18 = 23.57, which, multiplied by its specific heat of .25 as before, and if divided into 14 500, quotient will be 2461 , which is temperature of an ordinary furnace. Ratio of Combustion. Quantity of fuel burned per hour per sq. foot of grate varies very much in different classes of boilers. In Cornish boilers it is 3.5 Ibs. per sq. foot ; in ordinary Land boilers, 10 to 20 Ibs. ; (English) 13 to 14 Ibs. ; in Marine boilers (natural draught), 10 to 24 Ibs. ; (blast) 30 to 60 Ibs. ; and in Locomotive boilers, 80 to 120 Ibs. Volumes of air and smoke for each cube foot of water converted into steam, is for coal and coke 2000 cube feet, for wood 4000 cube feet ; and for each Ib. of fuel as follows : Coal .......... 207 1 Cannel coal ... 315 | Coke ......... 216 | Wood ......... 173 Calorific power of i Ib. good coal = 14 ooo X 772 = 10 808 ooo Ibs. Relative Evaporation of Several Combnstil>les in L"bs. of Water, Heated 1 1 Ll>. of Material. Combiwtible. Composition. Water. Combustible. Composition. Water. Alcohol .... 812 (Hyd. .12) Lbs. 8 I2O Olive oil (Hyd. .13) Lbs. Bituminous coal. . . iCarb. .45} ] Hyd. .04) o 830 Peat, moist \Carb. .77} (Hyd. .04) 3481 \Carb. . 75 J 14220 " dry (Carb. .43) (Hyd. .06) 3 900 Coke Hydrogen (mean). . Oak wood, dry .... " " green... Carb. .84 /Hyd. .06) iCarb. .53} (Hyd. .08) )Carb. .r7f 9028 50854 6018 5662 Pine wood, dry.... Sulphuric ether. .7 Tallow. . . (Carb. .58) (Hyd. .06) {Carb. .7 } (Hyd. .13) \Carb. .6 } 3618 8680 14 ^60 i Ib. Hydrogen will evaporate 62.6 Ibs. water from 212 = 60.509 Ibs. heated i. i Ib. Carbon " 14.6 Ibs. " 212, or raise 12 Ibs. water at 60 to steam at 120 Ibs. pressure. i Ib. of Oxygen will generate same quantity of heat whether in combustion with hydrogen, carbon, alcohol, or other combustible. Relative Volumes of Gases or Products of Combustion per Lb. of Fuel. Temp. Airf Supply ( 12 Ibs. Volume per Ib. >f Air per Ib. x8 Ibs. Volume per Ib. of Fuel. 24 Ibs. Volume per Ib. Temp. Air. Supply < 12 Ibs. Volume per Ib. f Air per Ib. 1 8 Ibs. Volume per Ib. of Fuel. 24 Ibs. Volume per Ib. o II 104 212 392 Cube Feet. ISO 161 172 205 259 Cube Feet. 225 241 2 5 8 37 389 Cube Feet. 300 322 344 409 519 572 752 III2 1472 2500 Cube Feet. 314 369 9*1 Cube Feet. 471 553 718 882 1359 Cube Feet. 628 738 957 1176 18x2 * Mean of all experiments 13964. 466 COMBUSTION. EXCAVATION AND EMBANKMENT. To Compute Consumption of Fuel to Heat Air. RULE. Divide volume of air to be heated by volume of i Ib. of it, at its temperature of supply ; multiply result by number of heat-units necessary to raise i Ib. air through the range of temperature to which it is to be heated, and product, divided by number of heat-units of fuel used, will give result in Ibs. per hour. EXAMPLE. What is required consumption per hour of coal of an average compo- sition to heat 776400 cube feet of air at 54 to 114? Coal of an average composition (Table, page 461) = 14 133 heat-units. Volume of i Ib. air at 54 (see formula, page 522) = 4 ' g 4 = 12.94 cube feet, i X 114 54 X .2377 (specific heat of air) = 14. 262 heat-units. 776400 X 14. 262 -r- 14 133 = 60. 55 Us. Loss of heat by conduction of it to walls of apartment is to be added to this. EXCAVATION AND EMBANKMENT. Labor and "Work npon Excavation and Embankment. Elements of Estimate of Work and Cost. Per Day of 10 Hours. Cart. One horse. Distance or lead assumed at 100 feet, or 200 feet fat a trip, at a speed of 200 feet per minute. Earths. Of gravelly, loam, and sandy, a laborer will load per day into a cart respectively 10, 12, and 14 cube yards as measured in embankment, and if measured in excavation, .11 more is to be added, in consequence of the greater density of earth when placed in embankment than in excavation. NOTE. Earth, when first loosened, increases in volume about .2, but when settled in embankment it has less volume than when in bank or excavation. Carting. Descending, load .33 cube yard, Level, .28, and Ascending .25, measured in embankment ; and number of cart-loads in a cube yard of em- bankment are, Gravelly earth 3, Loam 3.5, and Sandy earth 4. Loosening. Loam, a three-horsed plough will loosen from 250 to 800 cube yards per day. Trimming. Cost of trimming and superintendence i to 2 cents per cube yard. Scooping. A scoop load measures about .1 cube yard in excavation; time lost in loading, unloading, and turning, 1.125 minutes per load ; in double scooping it is i minute. Time occupied for every 100 feet of dis- tance from excavation to embankment, 1.43 minutes. Time. Time occupied in loading, unloading, awaiting, etc., 4 minutes per load. To Compute Number of Loads or Trips in Cube Yards per Cart per Day. (= ^ : } h -r- y = n. E representing average distance of carting from em- E -r-IOO-j-4/ bankment in stations of loofeet each, y number of cart-loads to cube yard ofexcava tion, and n number of cube, yards in embankment, hauled by a cart per day to dis fancc E. EXCAVATION AND EMBANKMENT. 467 ILLUSTRATION. What is number of cube yards of loam that can be removed by one cart from an embankment on level ground for an average distance of 250 feet ? E = 250 -i- 100 = 2.5, and y = 3. 5. T X 10 -i- 3. 5 = X 10 -7-3.5 = 26. 37 cube yards. Substituting for 3, 3.5, and 4 number of cart-loads in a cube yard of embank' ment, 20, 17.14, and 15, = 60 minutes, divided respectively by these numbers. r=j =. n, in descending carting ; . ~ - - = n, in level, and -^ - =. n. in ascending, h representing number of hours actually at work. To Compute Cost of Excavating and. Embaiilsiiig per Cube Yard. -- 1 --- |- J + s = V. L representing pay of laborers, v value or result of loading in different earths, as 10, 12, and 14, c of one cart and driver per day, I cost of loosen- ing material per cube yard, and s cost of trimming and superintendence, both per cube yard, and all in cents. ILLUSTRATION. Volume of excavation in loam 30000 cube yards. Level carting 650 feet = 6. 5 trips or courses. Loosening by plough 1.7 cents per cube yard, laborers 106 cents per day, carts 160, and trimming and superintendence 1.5 cents per cube yard. v = 12, and l -^~- = 16. 33, number of loads per day by preceding formula. Then ^-+ ^+ 1-7 + 1-5 = 8.833 + 9.797 + 1.7 + 1.5 = 21.83 cents per cube yard. Earthworlz . By Carts. A laborer can load a cart with one third of a cube yard of sandy earth in 5 minutes, of loam in 6, and of heavy soil in 7. This will give a result, for a day of 10 hours, of 24, 20, and 17.2 cube yards of the respective earths, after de- ducting the necessary and indispensable losses of time, which is estimated at .4. It is not customary to alter the volume of a cart-load in consequence of any dif- ference in density of the earths, or to modify it in consequence of a slight inclina- tion in the grade of the lead. In a lead of ordinary length one driver can operate 4 carts. With labor at $i per day, the expense of a horse and cart, including harness, repairs, etc., is $1.25 per day. A laborer will spread from 50 to 100 cube yards of earth per day. The removal of stones requires more time than earth. The cost of maintaining the lead in good order, the wear of tools, superintend- ence, trimming, etc., is fully 2.5 cents per cube yard. By Wheel-barrows. A laborer in wheeling travels at the rate of 200 feet per min- ute, and the time occupied in loading, emptying, etc., is about 1.25 minutes, with- out including lead. The actual time of a man in wheeling in a day of 10 hours is .9 or 2.25 minutes per lead of 100 feet. Hence, To Compute Number of Barrow-Loads removed by a Laborer per Day. i2_* - *l2 n . n' representing number of leads ofioo feet. i.25 + n' A barrow-load is about .04 of a cube yard Rock. By Carte. Quarried rock will weigh upon an average 4250 Ibs. per cube yard, and a load may be estimated at .2 cube yard, and weighing a very little more than a load of average earth. Hence, the comparative cost of carting earth and rock is to be computed on the basis of a cube yard of earth averaging 3.5 loads and one of rock 5 loads, with the addition of an increase in time of loading, and wear of cart. 468 EXCAVATION AND EMBANKMENT. For labor of a man, see Animal Power, pp. 433-34. By Wheel-barrow. A barrow-load may be assumed at 175 Ibs. 2 cube feet of space. Blasting When labor is $ i per day, hard rock in ordinary position may be blasted and loaded for 45 cents per cube yard. The cost, however, in consequence of condition, position, etc., may vary from 20 cents to $ i See Blasting page 443. 17 cube yards of hard rock may be carted per day over a lead of 100 feet, at a cost of 7.29 cents per yard. The preceding elements are essentially deduced from notes furnished by EHwood MorriSy C.E.,and the valuable treatise of John C. Trautwine, C.E., Phila., 1872. Stone. Hauling Stone. A cart drawn by horses over an ordinary road will travel 1.15 miles per hour of trip = 2.3 miles per hour. A four-horse team will haul from 25 to 36 cube feet of stone at each load. Time expended in loading, unloading, etc., including delays, averages 35 minutes per trip. Cost of loading and unloading a cart, using a horse-crane at the quarry, and unloading by hand, when labor is $ i 25 per day, and a horse 75 cents, is 2f cents per perch 24. 75 cube feet i cent per cube foot. Work done by an animal is greatest when velocity with which he moves is .125 of greatest with which he can move when not impeded, and force then exerted .4^ of utmost force the animal can exert at a dead pull. Eartli worlr. (Molesworth.) Proportion of Getters, Fillers, and Wheelers in different soils, Wheelers being cal- culated at 50 yards run. In loose earth, sand, etc. " Compact " Marl Gett's. I Fill's. Wheel's. Gett's. Fill's. Wheel's. i I i i 2 2 I 2 2 In Hard clay " Compact gravel " Rock, from. .. . 1 1.25 i i 1.25 i Average "Weight of Eartlis, Per cube yard. Lbs. Gravel. . . . Mud. . . ... 3360 ... 2800 Marl. , Lbs. ,. 2912 Clay 3472 Chalk 4032 Sandstone . Lbs. 4368 Shale ........ 4480 Quartz. . . . 4492 Lbs. Granite 4700 Trap. .. 4700 Slate 4710 of Rock:, Eartliworlz, etc., Original Excavation ass vim ed. at 1. When in Embankment. Rock, large 1.5 1.6 1.7 Medium 1.25 to 17 Metal 1.2 to 1.8 Sand and gravel 1.07 Clay and earth after subsidence . . . 1.08 u " before u ... 1.2 In small stones, per cent, of interstices to total volume is 44 to 48, which is an increase in volume of solid rock to fragments of 79 and 92 per cent. The relative proportions of Earth in Bank and Embankments, as given by differ- nt authorities, are so varied and so opposite that it is evident the difference' is acci- dental, depending, primarily, upon the season, location, and character or condition of the earth, and then upon the height of the embankment, the manner and dura- tion of time of raising it. Thus, Ellwood Morris, ante p. 466, makes the embankment less, and Molesworth, 8 above, gives it greater. FRICTION. 469 FRICTION. Friction is the force that resists the bearing or movement of one sur- face over another, and it is termed Sliding when one surface moves over another, as on a slide or over a pin ; and Rotting when a body ro- tates upon the surface of some other, as a wheel upon a plane, so that new parts of both surfaces are continually being brought in contact with each other. The force necessary to abrade the fibres or particles of a body is termed Measure of friction ; this is determined by ascertaining what portion of the weight of a moving body must be exerted to overcome the resistance arising from this cause. Coefficient of Friction expresses ratio between pressure and resistance of one surface over or upon another, or of surfaces upon each other. Angle of Repose is the greatest angle of obliquity of pressure between two planes, consistent with stability, the tangent of which is the coefficient of friction. Experiments and Investigations have adduced the following observations and results : 1. Amount of friction in surfaces of like material is very nearly propor- tioned to pressure perpendicularly exerted on such surfaces. 2. With equal pressure and similar surfaces, friction increases as dimen- sions of surfaces are increased. 3. A regular velocity has no considerable influence on friction ; if velocity is increased friction may be greater, but this depends on secondary or inci- dental causes, as generation of heat and resistance of the air. M. Morin's experiments afford the principal available data for use. Though con- stancy of friction holds good for velocities not exceeding 15 or 16 feet per second, yet, for greater velocities, resistance of friction appears, from experiments of M. Poiree, in 1851, to be diminished in same proportion as velocity is increased. 4. Similar substances excite a greater degree of friction than dissimilar. If pressures are light, the hardest bodies excite least friction. 5. In the choice of ungnepts, those of a viscous nature are best adapted for rough or porous surfaces, as tar and tallow are suitable for surfaces of woods, and oils best adapted for surfaces of metals. 6. A rolling motion produces much less friction than a sliding one. 7. Hard metals and woods have less friction than soft. 8. Without unguents or lubrication, and within the limits of 33 Ibs. press- ure per sq. inch, the friction of hard metals upon each other may be esti- mated generally at about one sixth the pressure, 9. Within limits of abrasion friction of metals is nearly alike. 10. With greatly increased pressures friction increases in a very sensible ratio, being greatest with steel or cast iron, and least with brass or wrought iron. 11. With woods and metals, without lubrication, velocity has very little influence in augmenting friction, except under peculiar circumstances. 12. When no unguent is interposed, the amount of the friction is, in every case, independent of extent of surfaces of contact ; so that, the force with which two surfaces are pressed together being the same, their friction is the same, whatever may be the extent of their surfaces of contact. 13. Friction of a body sliding upon another will be the same, whether the body moves upon its face or upon its edge. RR 470 FRICTION. 14. When fibres of materials cross each other, friction is less than when they run in the same direction. 15. Friction is greater between surfaces of the same character than be- tween those of different characters. 1 6. With hard substances, and within limits of abrasion, friction is as pressure, without regard to surfaces, time, or velocity. 17. The influence of duration of contact (friction of rest) varies with the nature of substances ; thus, with hard bodies resting upon each other, the effect reaches a maximum very quickly ; with soft bodies, very slowly ; with wood upon wood, the limit is attained in a few minutes ; and with metal on wood, the greatest effect is not attained for some days. Coefficients of Friction and. Angles of Repose. The Coefficient of Friction is the tangent of the angle of repose from a horizontal plane. MATERIAL. Coefficient. Angle. Cotangent of Angle. Exponent of Stability. Belt on wood dry . ....... 47 Clay damp I 45 I " wet .25 to .31 14 tO 17 3.23 to 4 Earth .i to .25 14 to 43 i to 4 ' ' dry .81 1.23 3.23 17 .31 Gravel .81 to i. ii 39 to 48 .9 to 1.23 53 1.89 .-3-1 18 30' 3 Sand fine 6 3* i.6 7 Timber on stone 4 22 2-5 .25 to .5 14 to 26 30' 2 to 4 ** " 5 5.Q3 14320 15 ooo * J- S m 14.82 15 66 Van Diemen's Land Chili I. 17 65.8 go e6 T y/ 3-5 .82 .1 * /y 5.58 j-yj 22.71 II 320 11-83 11.68 Lignite, Trinidad . . U 3o u 65.2 5-43 4-25 2-5 .69 21.69 6.84 10438 10.87 " French Alps 1.28 70.02 5-2 3.01 11790 12. I " Bitum.,Cuba 1.2 75.85 7-25 3-94 14562 14.96 " Wash.Ter.*. 67 4-55 I. 12538 12.91 Asphalt 1.06 70.18 9*3 ___ __ __ 2.8 16655 17.24 Petroleum .87 n 84.7 13. i 2. 2 20 240 2O 73 " oils. "/ 75 _ _ _ 27530 * JJ 28.5 Oak bark Tan, dry. 15 6100 6. 3 I " " moist Charcoal at 302. . . 47-51 6.12 (Oa ddN A< 5.29) I5 .8 4284 8130 4 8 ;r " "572... " " 810. . . 1.4 1.71 73.24 81.64 & (OandN 21.96) (OandN 15.24) liii 11861 14916 12.27 15-43 Peat, dry, average. " moist, t " 53 42 58.18 33.38 5-9 6 1-23 1 ( O and N 2 .18 1 31.21 .08 3-43 3-3 9951 8917 52061 10.3 9.22 47.51 * Water 7. Oxygen and Nitrogen 17.36. t Moisture 27.8. Sulphur .2. Elements of I^viels not included in ^Preceding Tatoles. FUEL. Heat of Combustion of i Ib. Evaporative Power of i Ib. at 212. Coke pro- duced. Weight of i Cub. Foot. Volume of i Ton. Bituminous Coal. Welsh Units. 14858 Lbs. 0.05 Per cent. 73 Lbs. 82 Cube Feet. 42 7 14 820 8.01 61 78.3 45* 3 13018 7.04 58 70-4 AC 2 Scotch ijyio 14 164 7.7 54 78.6 42 Boghead 14478 787 30.04 British average . . 8 13 OI 7O 8 Q 85 oo 79.0 OQ 6 oe 7 Cumberland Md . . .... 837 84. O7 825 87 54 14 723 64.2 40 68 27 Anthracite. 14 500 94.82 OX 78 42 15 14038 88.83 Miscellaneous. Warlich's fuel 16 495 70. e 34 5 Coke Mickley 15 600 80 Virginia average *3 55 14 O2 45 60 8 Charcoal 12 ^25 vy.o 12.70 Lignite perfect. .. ^ ...... ii 678 12 I 47 0814 I0.l8 37' 5 " Russian 15837 Asphalt 16 555 17.24 Q _ Woods, dry, average. . . 7 702 8.07 114 FUEL. GRAVITATION. 487 ^Miscellaneous. Experiments undertaken by Baltimore and Ohio R. R. Co. determined evaporating effect of i ton of Cumberland coal equal to 1.25 tons of anthra- cite, and i ton of anthracite to be equal to 1.75 cords of pine wood; also that 2000 Ibs. of Lackawanna coal were equal to 4500 Ibs. best pine wood. One Ib. of anthracite coal in a cupola furnace will melt from 5 to 10 Ibs. of cast iron; 8 bushels bituminous coal in an air furnace will melt i ton of cast iron. Small coal produces about .75 effect of large coal of same description. Experiments by Messrs. Stevens, at Bordentown, N. J., gave following results: Under a pressure 0/30 Ibs., i Ib. pine wood evaporated 3.5 to 4.75 Ibs. of water, i Ib. Lehigh coal, 7.25 to 8.75 Ibs. Bituminous coal is 13 per cent, more effective than coke for equal weights; and in England effects are alike for equal costs. Radiation from Fuel. Proportion which heat radiated from incandescent fuel bears to total heat of combustion is, From Wood 29 | From Charcoal and Peat 5 Least consumption of coal yet attained is 1.5 Ibs. per IIP. It usually varies in different engines from 2 to 8 Ibs. Volume of pine wood is about 5.5 times as great as its equivalent of bituminous coal. GRAVITATION. GRAVITY is an attraction common to all material substances, and they are affected by it directly, in exact proportion to their mass, and inversely, as square of their distance apart. This attraction is termed terrestrial gravity -, and force with which a body is drawn toward centre of Earth is termed the weight of that body. Force of gravity differs a little at different latitudes : the law of variation, however, is not accurately ascertained ; but following theorems represent it very nearly : \l Z '^ 8 8 ^ C at' theses I -a * representing force of gravity at lati* I' [*:SaSf 7 | al uSSStorp* ** 45, and g force at other places. Or, 32.171 (lat 45) (i + . 005 133 sin. L) i ^-J = g. L representing latitude, H height of elevation above level of sea, and R radius of Earth, both in feet. NOTE. If 2 L exceeds 90, put cos. 180 2 L, and R at Equator = 20 926 062, at Poles 20853 4 2 9> and mean 20889 746. ILLUSTRATION. What is force of gravity at latitude 45, at an elevation of 209 feet, and radius = 20 900000 feet? 32.171 (i + .005 133 sin. 45) (i l__J:=32.i7i X 1.00363 X -999 9 s = 32- 287. Gravity at Various Locations at Level of Sea. Equator ........... 32.088 I New York ......... 32. 161 I London ........... 32. 189 Washington ....... 32-155 I Lat. 45 ........... 32-171 I p l es ...... ........ 32.253 In bodies descending freely by their own weight, their velocities are as times of their descent, and spaces passed through as square of the times. Times, then, being i, 2, 3, 4, etc., Velocities will be i, 2, 3, 4, etc. Spaces passed through will be as square of the velocities acquired at end of those times, as i, 4, 9, 16, etc. ; and spaces for each tune as 1,3, 5, 7, 9, etc. 483 GRAVITATION. A body falling freely will descend through 16.0833 feet m fir st second of time, and will then have acquired a velocity which will carry it through 32. 166 feet in next second. If a body descends in a curved line, it suffers no loss of velocity, and the curve of a cycloid is that of quickest descent. Motion of a falling body being uniformly accelerated by gravity, motion of a body projected vertically upwards is uniformly retarded in same manner. A body projected perpendicularly upwards with a velocity equal to that which it would have acquired by falling from any height, will ascend to the same height before it loses its velocity. Hence, a body projected up- wards is ascending for one half of time it is in motion, and descending the other half. Various Formulas here given are for Bodies Projected Upwards or Falling Freely, in Vacuo. When, however, weight of a body is great compared with its volume, and velocity of it is tow, deductions given are sufficiently accurate for ordinary purposes. In considering action of gravitation on bodies not far distant from surface of the Earth, it is assumed, without sensible error, that the directions in which it acts are parallel, or perpendicular to the horizontal plane. A distance of one mile only produces a deviation from parallelism less than one minute, or the 6oth part of a degree. Relation of Time, Space, and. "Velocities. Time from Beginning of Descent. Velocity acquired at End of that Time. Squares of Time. Space fallen through in that Time. Spaces for this Time. Space fallen through in last Second of Full. Seconds. Feet. Seconds. Feet. No. Feet. I 32.166 i 16.083 I 1 6. 08 2 64-333 4 64.333 3 48-25 3 96.5 9 144-75 5 80.41 4 5 128.665 160.832 16 25 257-33 402.08 7 9 112.58 '44-75 6 193 f 36 579 ii 176.91 7 225. 166 49 788.08 13 209. 08 8 257-333 64 1029.33 15 241.25 9 10 32?! 666 100 1302.75 1608.33 17 273-42 305-58 and in same manner this Table may be continued to any extent. Velocity acquired due to given Height of Fall and Height due to given Velocity. 8.04 ^/h = v - =fc; and 64.4 6.083 t 2 = h h representing height of fall in feet, v velocity acquired in feet per second, and t time of fall in seconds. To Compute Action of GJ-ravity. Time. When Space is given. RULE. Divide space by 16.083, and square root of quotient will give time. EXAMPLE. How long will a body be in falling through 402.08 feet? \/402.o8-r- 16.083 5 seconds. When Velocity is given. RULE. Divide given velocity by 32.166, and quotient will give time. EXAMPLE. How long must a body be iu falling to acquire a velocity of 800 feet per second ? 800 -r- 32. 166 = 24. 87 seconds. GRAVITATION. 489 Velocity. When Space w given. RULE. Multiply space in feet by 64.333, and square root of product will give velocity. EXAMPLE. Required velocity a body acquires in descending through 579 feet. Vs79 X 64.333 = i93 fat- Velocity acquired at any period is equal to twice the mean velocity during that period. ILLUSTRATION. If a ball fall through 2316 feet in 12 seconds, with what velocity will it strike ? 2316 -r- 12 = 193, mean velocity, which x 2 = 386 feet = velocity. When Time is given. RULE. Multiply time in seconds by 32.166, and product will give velocity. EXAMPLE. What is velocity acquired by a falling body in 6 seconds? 32.166 X 6 = 192.996/66*. Space. When Velocity is given. RULE. Divide velocity by 8.04, and square of quotient will give distance fallen through to acquire that velocity. Or, Divide square of velocity by 64.33. EXAMPLE. If the velocity of a cannon-ball is 579 feet per second, from what height must a body fall to acquire the same velocity? 579 -r- 8.04 = 72.014, and 72.oi4 2 = 5186. 02 feet. When Time is given. RULE. Multiply square of time in seconds by 16.083, and it will give space in feet. EXAMPLE. Required space fallen through in 5 seconds. 5 = 25, and 25 X 16.083 = 402.08 feet. Distance fallen through in feet is very nearly equal to square of time in fourths of a second. ILLUSTRATION L A bullet dropped from the spire of a church was 4 seconds in reaching the ground ; what was height of the spire ? 4 X 4 = 16, and i6 2 = 256 feet. By Rule, 4 X 4 X 16.0833 = 257.33/66?. 2. A bullet dropped into a well was 2 seconds in reaching bottom; what is the depth of the well ? Then 2X4 = 8, and 82 = 64 feet. By Rule, 2 X 2 X 16.0833 = 64.33/66*. By Inversion. In what time will a bullet fall through 256 feet? ^256 = 16, and 16 -r- 4 = 4 seconds. Space fallen through in last Second, of Fall. When Time is given. RULE. Subtract half of a second from time, and multiply remainder by 32.166. EXAMPLE. What is space fallen through in last second of time, of a body falling for 10 seconds? 10 .5 X 32. 166 = 305. 58 feet. JPromiscuovis Examples. i. If a ball is i minute in falling, how far will it fall in last second? Space fallen through = square of time, and i minute = 60 seconds. 6o 2 X 16.083 = 57 898 feet for 60 seconds. 59* X 16.083 = 55 984 " " 59 1914 i a. Compute time of generating a velocity of 193 feet per second, and whole space descended. 193 -r- 32. 166 = 6 seconds ; 6 2 X 16.083 = 579 /66*. 4QO GRAVITATION. 3. If a body was to fall 579 feet, what time would it be in falling, and how far would it fall in the last second ? /579_X2 __ ^^ _ 6 Kconds ^ ftnd 6 . 5 x 32. 166 = 5. 5 X 32. 166 = 1 76. 91 /t V 32. 160 Formulas to determine the various Elements. / S _ V _ a_S /a~S '' V-so' ~ g' - V '' -V g> /S X 2 ; =T0; =2 T representing time of fatting in seconds, V velocity acquired in feet per second, 9 space or vertical height in feet, h space fallen through in last second, g 32.166 and 5 g and .25 ^ representing 16.083 and 8.04. Retarded Motion. A body projected vertically upward is affected inversely to its motion when falling freely and directly downward, inasmuch as a like cause retards it in one case and accelerates it in the other. In air a ball will not return with same velocity with which it started. In vacuo it would. Effect of the air is to lessen its velocity both ascending and descending. Difference of velocities will depend upon relative specific grav- ity of ball and density of medium through which it passes. Thus, greater weight of ball, greater its velocity. To Compute .Auction of Grravity "by a Body projected Upward or I3ownward with, a given. Velocity. Space. When projected Upward. RULE. From the product of the given velocity and the time in seconds subtract the product of 32.166, and half the square of the time, and the remainder will give the space in feet. Or, Square velocity, divide result by 64.33, an( ^ quotient will give space in feet. EXAMPLE If a body is projected upward with a velocity of 96.5 feet per second, through what space will it ascend before it stops ? 96.5-7- 32- 166 = 3 seconds = time to acquire this velocity. Then, 96. 5 X 3 (32- 166 X ^ = 289.5 144.75 = 144.75 feet. Time. RULE. Divide velocity in feet by 32.166, and quotient will give time in seconds. EXAMPLE. Velocity as in preceding example. 96. 5 -T- 32. 1 66 = 3 secondt. "Velocity. RULE. Multiply time in seconds by 32.166, and product will give velocity in feet per second. EXAMPLE. Time as in preceding example. 3 x 32. 166 = 96. 5 feet velocity. Space fallen through in last Second.. RULE. Subtract .5 from time, multiply remainder by 32.166, and product will give space in feet per second. EXAM FIJI. Time as in preceding example. 3 -5 X 32.166 = 2.5 X 32.166 = 8o-4i6/eet GRAVITATION. 49! When projected Downward. Space. RULE. Proceed as for projection upwards and take sum of products. EXAMPLE i. If a body is projected downward with a velocity of 96.5 feet per sec- ond, through what space will it fall in 3 seconds? 96-5 X 3-1- (32-166 X ^J = 289.5-1-144.75 = 434-25/*. Or, t 2 X 16.083 + X t = * 2. If a body is projected downward with a velocity of 96.5 feet per second, through what space must it descend to acquire a velocity of 193 feet per second? 96.5-7- 32. 166 =. 3 seconds, time to acquire this velocity. 193 -r- 32. 1 66 = 6 seconds, time to acquire this velocity. Hence 6 3 = 3 seconds, time of body falling. Then 96.5 X 3 = 289.5 = product of velocity of projection and time. 16.083 X 3 2 = 144-75 =product 0/32. 166, and half square of time. Therefore 289.5 + 144.75 = 434.25/6^. Titne. RULE. Subtract space for velocity of projection from space given, and remainder, divided by velocity of projection, will give time. EXAMPLE. In what time will a body fall through 434.25 feet of space, when pro- jected with a velocity of 96.5 feet? Space for velocity of 96.5 = 144.75/^1 Then, 434. 25 144. 75 + 96. 5 = 289. 5 -=- 96. 5 = 3 seconds. Velocity. RULE. Divide twice space fallen through in feet by time in seconds. EXAMPLE. Elements as in preceding example. Space fallen through when projected at velocity of 96. 5 feet = 144. 75 feet, and 434. 25 feet = space fallen through in 3 seconds. Then, 144.75-1-434.25 = 579 feet space fallen through, and Vs79 -*- 16.083 = 6 seconds. Hence, 579 x 2 -4- 6 = 1158 -=- 6 = 193 feet. Space Fallen through, in last Second.. RULE. Subtract .5 from time, multiply remainder by 32.166, and product will give space in feet per second. EXAMPLE. Elements as in preceding example. 6 .5 X 32. 166 = 5.5 X 32. 166 = 176.91 feet Ascending bodies, as before stated, are retarded in same ratio that descending bodies are accelerated. Hence, a body projected upward is ascending for one half of the time it is in motion, and descending the other half. ILLUSTRATION i. If a body projected vertically upwards return to earth in 12 seconds, how high did it ascend ? The body is half time in ascending. 12 -f- 2 = 6. Hence, by Rule, p. 489, 6 2 X 16.083 = 579 feet = product of square of time and 16.083. 2. If a body is projected upward with a velocity of 96.5 feet per second, it is required to ascertain point of body at end of 10 seconds. 96.5-7-32.166 = 3 seconds, time to acquire this velocity, and 3 2 X 16.083 = 144.75 feet, height body reached with its initial velocity. Then 10 3 = 7 seconds left for body to fall in. Hence, by Rule, as in preceding example, 7 2 X 16.083 = 788.07, and 788.07 144. 75 = 643. 32 feet = distance below point of projection. Or, io 2 X 16.083 = 1608.3 feet, space fallen through under the effect of gravity, and 96. 5X10 = 965 feet, space if gravity did not act. Hence 1608. 3 965 = 643. 3 feet. 49 2 GRAVITATION. 3. A body is projected vertically with a velocity of 135 feet; what velocity will it have at 60 feet ? *35 2 -^ 64. 33 = 283.3/6^ space projected at that velocity, 135-7-32.16 = 4.197 sec- onds = time of projection, and 283. 3 60 = 223. 3 = space to be passed through after attainment of 60 feet. Hence, V 223. 3 X 64.33 = "9-85 feet velocity, and 223.3 + 60 = 283.3/6^. By Inversion. Velocity 119.85. Hence, ** 9 ' 5 = 223.3 f ee t space, and 283.3 223.3 = 6 Formvilas to Determine Klements of Retarded Motion. 7. = + . b. = v representing velocity at expiration of time, t any less time than T, t' less lime than t, s space through which a body ascends in time t, V, T, S, and h as in previous formu- las, page 490. ILLUSTRATION. A body projected upwards with a velocity of 193 feet per second, was arrested in 5 seconds. i _ o, t i. 1. What was its velocity when arrested? (i.) 2. What was the time of its passing through 562.92 feet of space ? (8.) 3. What space had it passed through? (5.) 4. What was the time of its projection, when it had a velocity of 96.5 feet? (4.) 5. What was the height it was projected in the last second of time? (6.) i. 193-^32.166X5 = 32.17/6^. 3. 32.17 + 32.166X5 = 1 562.92 3*. 166X5 velocit I9L jl = 3 seconds. 52 32.166 32.166 6. 6-r=l -.5X32.^66 = 48.25 7. S = t v + g t 2 -T- 2 = 562.92 feet. 8. * 93 ~ " I7 = 5 seconds. GJ-ravity and !M!otion at an Inclination. If a body freely descend at an inclination, as upon an inclined plane, by force of gravity alone, the velocity acquired by it when it arrives at ter- mination of inclination is that which it would acquire by falling freely through vertical height thereof. Or, velocity is that due to height of in- clination of the plane. Time occupied in making descent is greater than that due to height, in ratio of length of its inclination, or distance passed, to its height. Consequently, times of descending different inclinations or planes of like heights are to one another as lengths of the inclinations or planes. Space which a body descends upon an inclination, when descending by gravity, is to space it would freely fall in same time as height of inclination is to its length ; and spaces being same, times will be inversely in this pro- portion. If a body descend in a curve, it suffers no loss of velocity. If two bodies begin to descend from rest, from same point, one upon an in- clined plane, and the other falling freely, their velocities at all equal heights below point of starting will be equal. GRAVITATION. 493 ILLUSTRATION. What distance will a body roll down an inclined plane 300 feet long and 25 feet high in one second, by force of gravity alone? As 300 : 25 :: 16.083 1.34025/68*. Hence, if proportion of height to length of above plane is reduced from 25 to 300 to 25 to 600, the time required for body to fall 1.34025 feet would be determined as follows: As 25 : 600 :: 1.34025 : 32.166, and 32.166 = 16.083 X 2 = twice time or space in which it would fall freely required for one half proportion of height to length. 300 600 Or, as - : :: 1.34025 : 32.166, as above. Impelling or accelerating force by gravitation acting in a direction paral- lel to an inclination, is less than weight of body, in ratio of height of in- clination to its length. It is, therefore, inversely in proportion to length of inclination, when height is the same. Time of descent, under this condition, is inversely in proportion to accel- erating force. If, for instance, length of inclination is five times height, time of making freely descent at inclination by gravitation is five times that in which a body would freely fall vertically through height ; and impelling force down inclination is .2 of weight of body. When bodies move down inclined planes, the accelerating force is ex- pressed by h -r- /, quotient of height -r- length of plane; or, what is equivalent thereto, sine of inclination of plane, f . e., sin. a. ILLUSTRATION. An inclined plane having a height of one half its length, the space fallen through in any time would be one half of that which it would fall freely. Velocity which a body rolling down such a plane would acquire in 5 seconds is 80.416 feet. Thus, 32.166 X 5 = 160.833 f ee ^ an ^ an inclined plane, having a height one half of its length, has an angle or sine of 30. Hence, sin. 30 = .5, and 160.833 x .5 = Bo. 416 feet. Formulas to Determine various Elements of G-ravita* tion on. an Inclined. !Plane. = qr gTsin. a. . a; = ^(z 9 S sin. a) ; =~. 6. H= ' 3- A = V 2 5. S = VTqr.s gT 2 sin. a. Or,- 7 2 g sin. a v representing velocity of projection in feet per second, S space or vertical height of velocity, and projection, a angle of inclination of plane, I length, and H height of plane. ILLUSTRATION. Assume elements of preceding illustration. = 80.416, 1=5, and H = 2oi.o4. i. sX32.i66X5 2 X.5 zoi.o+feet. 2. 32.166 X 5 X -5 = 80.416/6^. 5X If projected downward with an initial velocity of 16.083 feet P er second. 4 . 16.083 + 32.166 X 5X-5 = 96. 5 feet. 5. 80.4164-16.083 X 5 -5 X 32-166 X 5 2 X .5 = 281.46 feet. T T 494 GBAVITATIOK. ILLUSTRATION. What time will it take for a ball to roll 38 feet down an inclined plane, the angle a= 12 20', and what velocity will it attain at 38 feet from its start- ing-point? X . 2136 = 22. 88 feet per second. When a body is projected upward it is retarded in the same ratio that a descending body is accelerated. ILLUSTRATION. If a body is projected up an inclined plane having a length of twice its height, at a velocity of 96.5 feet per second, Then, T = 96. 5 -r- 32. 166 = 3 seconds. S = . 5 32. 166 x 3 2 X 5 72. 375 feet, v = 32. 166 X 3 X . 5 = 48- 25 feet. Inclined Plane. Problems on descent of bodies on inclined planes are soluble by formulas i to 9, page 495, for relations of accelerating forces. As a preliminary step, however, accelerating force is to be determined by multiplying weight of descending body by height of plane, and dividing product by length of plane. ILLUSTRATION. If a body of 15 Ibs. weight gravitate freely down an inclined plane, length of which is five times height, accelerating force is 15 -r- 5 = 3 Ibs. If length of plane is ioo feet and height 20, velocity acquired in falling freely from top to bottom of plane would be v = 8 V ^p = 8 V20 = 35.776/0*. Time occupied in making descent, /i5 X ioo t = . 25 ^ = . 25 V 5 = 5- 59 seconds. Whereas, for a free vertical fall through height of 20 feet, time would be, -= 1.118 seconds, 32.166 which is .2 of time of making descent on inclined plane. Velocities acquired by bodies in falling down planes of like height will all be equal when arriving at base of plane. When Length of an Inclined Plane and Time of Free Descent are given. RULE. Divide square of length by square of time in seconds and by 16 ; the quotient is height of inclined plane. EXAMPLE. Length of plane is ioo feet, and time of descent is 5.59 seconds; then Tertical height of descent is ioo 2 . , = 20 feet. 5-59 X 16.08 Accelerated and Retarded Motion. If an Accelerating or Retarding force is greater than gravity, that is, weight of the body, the constant, 55-9 X 10 = 559 /* If the question is put otherwise What space will a weight move over before it comes to a state of rest, with an initial velocity of 60 feet per second, allowing fric- tion to be one tenth weight? The answer is that friction, which is retarding force, being one tenth of weight, or of gravity, space described will be 10 times as great as is necessary for gravity, supposing the weight to be projected vertically upwards to bring it to a state of rest. The height due to velocity being 55.9 feet; then 55. 9 X 10 = 559 feet. Average velocity of a moving body, uniformly accelerated or retarded during a given period or space, is equal to half sum of initial and final velocities. To Compute "Velocity of a Falling Stream of Water per Second at End of any given Time. When Pei*pendicular Distance is given. EXAMPLE. What is the distance a stream of water will descend on an inclined plane 10 feet high, and 100 feet long at base, in 5 seconds? 5 ' 2 X 16. 083 = 402.08 feet = space a body will freely fall in this time. Then, as 100 : 10 :: 402.08 : 40.21 feet = proportionate velocity on a plane of these dimensions to velocity when falling freely. Miscellaneous Illustrations. x. What is the space descended vertically by a falling body in 7 seconda S = .5gxt 2 . Then 16.083 X 7 2 788.067 feet. 2. What is the time of a falling body descending 400 feet, and velocity acquired at end of that time? t = - . Then ' 4 = 4.08 sec. v = Vz g X S. Then i 9 3*166 3. If a drop of rain fall through 176 feet in last second of its fall, how high was the cloud from which it fell? h* _ i 7 6 2 4. If two weights, one of 5 Ibs. and one of 3, hanging freely over a sheave, are set free, how far will heavier one descend or lighter one rise in 4 seconds. ^-j^X 16.083 X4 2 = ^X 257. 328 = 64.33/66*. 5. If length of an inclined plane is 100 feet, and time of descent of a body is 6 seconds, what is vertical height of plane or space fallen through ? IOO 2 IOOOO = = i 7 . 27 feet. 6 2 X-5P 579 6. If a bullet is projected vertically with a velocity of 135 feet per second, what velocity will it have at 60 feet? Formula 9, page 492. TS'-X/S^ ~~ GUNNERY. 497 GUNNERY. A heavy body impelled by a force of projection describes in its flight or track a parabola, parameter of which is four times height due to velocity of projection. Velocity of a shot projected from a gun varies as square root of charge directly, and as square root of weight of shot reciprocally. To Compute Velocity of a Shot or Shell. RULE. Multiply square root of triple weight of powder in Ibs. by 1600; divide product by square root of weight of shot ; and quotient will give ve- locity in feet per second. EXAMPLE. What is velocity of a shot of 196 Ibs., projected with a charge of 9 Ibs. of powder? A/9 X 3 X 1600 -4- >/ I 9 6 = 8 3 20 -* 14 = 594- 3 feet. To Compute Range for a Charge, or Charge for a Range. When Range for a Charge is given. Ranges have same proportion as charges of powder ; that is, as one range is to its charge, so is any other range to its charge, elevation of gun being same in both cases. Consequently, To Compute Range. RULE. Multiply range determined by charge in Ibs. for range required, divide product by given charge, and quotient will give range required. EXAMPLE. If, with a charge of 9 Ibs. of powder, a shot ranges 4000 feet, how far will a charge of 6.75 Ibs. project same shot at same elevation? 4000 x 6.75 -7-9 = 3000 feet. To Compute Charge. RULE. Multiply given range by charge in Ibs. for range determined, divide product by range determined, and quotient will give charge required. EXAMPLE. If required range of a shot is 3000 feet, and charge for a range of 4000 feet has been determined to be 9 Ibs. of powder, what is charge required to project same shot at same elevation ? 3000 X 9 -T- 4000 = 6. 75 Ibs. To Compute Range at one Elevation, when Range for another is given. RULE. As sine of double first elevation in degrees is to its range, so is sine of double another elevation to its range. EXAMPLE. If a shot range 1000 yards when projected at an elevation of 45, how tar will it range when elevation is 30 16', charge of powder being same ? Sine of 45 X 2 100 ooo ; sine of 30 16' X 2 = 87 064. Then, as 100 ooo : 1000 : : 87 064 : 870. 64 feet To Compute Elevation at one Range, when Elevation for another is given. RULE. As range for first elevation is to sine of double its elevation, so is range for elevation required to sine for double its elevation. EXAMPLE. If range of a shell at 45 elevation is 3750 feet, at what elevation must a gun be set for a shell to range 2810 feet with a like charge of powder? Sine of 45 X 2 = 100000. Then, as 3750 : 100000 :: 2810 : 74933 = sine for double elevation 24 16'. Approximate Rule for Time of Flight. Under 4000 yards, velocity of projectile 900 feet in one second ; under 6000 yards, velocity 800 feet ; and over 6000 yards, velocity 700 feet. Guns and Howitzers take their denomination from weights of their solid shot in round numbers, up to the 42-pounder ; larger pieces, rifled guns, and mortars, from diameter of their bore. TT* 498 GUNNERY. Initial Velocity and. Ranges of Shot and. Shells. The Range of a shot or shell is the distance of its first graze upon a horizontal plane, the piece mounted upon its proper carriage. Project Description. le. Weight. Powder. Initial Velocity. Time of Flight. Eleva- tion. Range. Grains. Grains. Feet. Seconds. / Yards. Elongated, j err* 60 06? Round. 412 no y u j 1500 Lbs. LbB. ( 6.15 1.25 e 1523 I2 " i 12.3 2. 5 1 8~2~6 1.75 I 575 i 24.25 6 5 1870 2 1147 t 32.3 8 ,' 1040 I 7*3 < 42.5 65 10.5 IO 14.19 5 15 1955 3224 8 inch Columbiad... 10 " " M 127.5 15 14.32 15 3281 10 " Mortar. Shell. 98 IO 36 45 4250 200 3 02 20 40 45 7 1948 15 " Columbiad... 15 " " 315 50 23-29 25 4680 RIFLED. io-pounder Parrott. . " 9-75 X 21 20 5000 20 u i9 2 I7-25 15 4400 30 it 29 3-25 27 25 6700 100 Elongated. 100 10 29 25 6910 100 Shell. 101 10 1250 28 25 6820 2OO " 150 16 4 2200 1 2- inch Rodman " 50 "54 5-5 40 Hall's Rockets. . . vinch. 16 47 1720 Penetration of Shot and Shell. Experiments at Fort Monroe, 1839, and at West Point, 1853. Mean Penetration. Mean Penetration. ORDNANCB. 1 6 5 i^l 1| 5 ORDNANCE. 1 1 $ || Granite. Lbs. Yds; Ins. Ins. Ins. Lbs. Yds. Ins. Ins. Ins. 32 Lbs. Shot. 8 880 I5-25 3-5 8-inch Howitz.* 6 880 8-5 I 32 " " II IOO 60 8 " Columb.if 12 200 42 42 " Shell. 10.5 7 100 IOO 54-75 40-75 18 4 10 " " t 10 " " * 18 18 114 IOO 63-5 56.75 44 7-75 1 24 ins. of Concrete. * Shell. t Shot. Solid shot broke against granite, but not against freestone or brick, and general effect is less upon brick than upon granite. Shells broke into small fragments against each of the three materials. Penetration in earth of shell from a io-inch Columbiad was 33 ins. Experiments England. (Holley. ) ORDNANCE. Charge. Projectile. Weight. Velocity. Range. Target and Effects. Lbs. Lbs. Feet. Yards. n-inch U. S. Navy. 3<> Shot. 169 1400 50 Iron plates, 14 ins. loosened. i5-inch Rodman... 60 " 400 1480 50 Iron plates, 6 ins. destroyed. RIFLED. 7-inch Whitworth. . 25 Shot. ISO 1241 200 Inglis'st destr'd. io-5-inch Armstrong 45 " 37 1228 200 u a i3-inch " 90 * u 344-5 1760 200 Sol id plates, n ins. thick destr'd. * Steel. f 8-inch vertical and 5-inch horizontal slabs, and 7-inch vertical and $-in. horizontal slabs, 9X5 ins. ribs and 3-inch ribs. GCNNBEY. 499 Elements of Report of Board of Engineers for Fortifications, U. S. A. Professional Papers No. 25. (Brev. Maj.-Gen, Z. B. Tower.) Experimental firings for penetration during the past twenty years have determined that wrought iron and cast iron, unless chilled, are unsuitable for projectiles to be used against iron armor ; that the best material for that purpose is hammered steel or Whitworth's compressed steel. 2. That cast-iron and cast-steel armor-plates will break up under the im- pact of the heaviest projectiles now in service, unless made so thick as to exclude their use in ship-protection. 3. That wrought-iron plates have been so perfected that they do not break up, but are penetrated by displacement or crowding aside of the material hi the path of the shot, the rate of penetration bearing an approximately deter- mined ratio to the striking energy of the projectile, measured per inch of shot's circumference, as expressed by the following formula : 2.035 / vil V 2 P = penetration in ins. V representing velocity in n X 2240 X- 86 feet per second, P weight of shot in Ibs., and r radius of shot in ins. That such plates can therefore be safely used in ship construction, their thickness being determined by the limit of flotation and the protection needed. 4. That, though experiments with wrought-iron plates, faced with steel, have not been sufficiently extended to determine the best combination of these two materials, we may nevertheless assume that they give a resistance of about one fourth greater* than those of homogenous iron. 5. That hammered steel in the late Spezzia trials proved superior to any other material hitherto tested for armor-plates. The icj-mch plate resisted penetration, and was only partially broken up by 4 shots, three of which had a striking energy of between 33 ooo and 34 ooo foot-tons each. Not one shot penetrated the plate. Those of chilled iron were broken up, and the steel projectile, though of excellent quality, was set up to about two thirds of its length. Velocity and [Ranges of Shot. (Krupp's Ballistic Tables.) Penetration, in "Wrought Iron. / tration in i elocity Range. 3000 6000 ns. G at Muzzle ration Range 3000 6000 22.04 23-47 21-35 21.89 12.14 5-17 V 2 GUN. 0X2 Cali- ber. rnX2'< Powder. 540 X( Shot. j^/CTO V at Muzzle per Sec. 2.53 Penet 600 Tons. Armstrong, 100 Woolwich, 8 1 Krupp, 71 " 18 U.S.* 8-inch.. Ins. 17-75 17-75 16 15-75 V s Lbs. 55 2 776 445 485 165 35 Lbs. 2O2 2 2000 1760 1715 474 180 Feet. 1715 1832 1657 1703 1450 Yds. 1424 1518 1393 H34 1351 1036 Yds. 1191 1259 1181 I2II 1113 840 Ins. 34-76 37-52 32.6 33-52 20.42 10.23 Ins. 33-2 35-8i 31-23 32.12 I9-3I 9.22 Ins. 27-55 29.66 26.24 27.04 15.46 6.72 * Unchambered. Target For loo-ton gun, steel plate 22 ins. thick, backed with 28.8 ins. of wood, 2 wrought-iron plates 1.5 ins. thick, and the frame of a vessel. Effect. Total destruction of steel plate, and backing entered to a depth of 22 ina, but not perforated. 500 GUNNERY. Summary of Record, of" Practice in Enrope with Heavy A.rm strong, \Vool\vich, and. Krvipp Grtins. Board of Engineers for Fortifications, U. S. A.,Professional Papers No. 25. ^ Energy g . o ii . si GUN. Powder. Projectile. fl II ]f 1^ * u!^ ARMSTRONG, "] loo Tons, caliber 1 i. 5- inch cubes. . Waltham Abbey Shot Lbs. 330 375 Lbs. 2OOO 2000 Feet. 1446 1543 Ft.-tons. 28990 33000 Foot-ton 544-Of 623 17 ins., bore 30.5 f feet. 400 776 2000 2OOO 1502 1832 31282 46580 835.3: WOOLWICH, 81 "1 75-inch cubes. //'-' 170 1258 1393 16922 371-5 Tons, caliber 14.5 V 1.5 " " 220 1450 1440 20842 457-57 ins. , bore 24 feet. J 2 " " ' .... 250 1260 I5 2 3 20259 444-7* caliber 16 ins 1-5 " " ' 310 I 4 66 1553 24508 520.4 38 Tons, ] 1.5 " " Pall, shell 130 800 1451 11668 297.6,! caliber 12.5 ins., \- 15 " " 11 2OO 800 1421 II 210 285.4 bore 16.5 feet. J 1.5 " " " 1 80 800 1504 12545 3!9-4 KRUPP, 71 Tons, ] caliber 15.75 ins., \- bore 28. 58 feet. J Prism A u H Plain . . . Shrapnel Shell. . . . 2 9 8 485 441 1707 1419 1184 1703 1761 16602 34503 30484 335-4- ^97-9 616.1^ " 2 inch... 1 8 Tons, 1 " i hole... Plain . . . 132 300 1873 7298 246.0 caliber 9.45 ins., V bore 17.5 feet. J " 2 inch.. . Shrapnel Shell. . . . 145 474 300 1688 1991 9307 8244 3i 5 -6( 2 77 .6( Penetration in Ball Cartridge Paper, No. i. Musket, with 134 grains, at 13.3 yards 653 sheets. Common rifle, 92 grains, at 13.3 yards 500 sheets. Penetration, of Lead. Balls in Small Arms. Experiments at Washington Arsenal in 1839, and at West Point in 1837. ARM. Diameter of Ball. Charge Powder. Distance. Weight of Ball. Peneti White Oak. ation. White Pine, Musket Inch. H4 (-64 j-5775 5775 5775 5775 55 55 Grains. 134 144 IOO 92 IOO 70 70 80 90* IOO* 51 60 70 40 60 55 Yards. 9 5 5 9 5 9 5 5 5 5 5 200 200 200 2OO 30 30 397-5 397-5 219 219 219 219 219 500 730 500 450 463 350 Ins. 1.6 3 2.05 1.8 2 .6 Ji i.i 1.2 725 Ina. II 10.5 9-33 5-75 7.17 6.15 Common Rifle Hall's rifle Hall's carbine, musket caliber Pistol Rifle musket Altered musket Rifle, Harper's Ferry. . Pistol carbine . Sharpens carbine Burnside's " * Charges too great for service. Musket discharged at 9 yards distance, with a charge of 134 grains, i ball and 3 buckshot, gave for ball a penetration of 1.15 ins., buckshot, .41 inch. GUNNERY. 501 Loss of Force toy "Windage. A comparison of results shows that 4 Ibs. of powder give to a ball without wind age nearly as great a velocity as is given by 6 Ibs. having .14 inch windage, which is true windage of a 24-lb. ball; or, in other words, this windage causes a loss of nearly one third of force of charge. Vents. Experiments show that loss of force by escape of gas from vent of a gun is altogether inconsiderable when compared with whole force of charge. Diameter of Vent in U. S. Ordnance is in all cases .2 inch. Effect of different Waddings with a Charge of 77 Grains of Powder. WAD. Velocity of Ball per Second. Feet. i felt wad upon powder and i upon ball I 377 1482 i elastic wad upon powder and i upon ball IIOO Felt wads cut from body of a hat, weight 3 grains. Pasteboard wads .1 of an inch thick, weight 8 grains. Cartridge paper 3X4-5 ins., weight 12.82 grains. Elastic wads, "Baldwin's indented," a little more than .1 of an inch thick, weight 5.127 grains. Most advantageous wads are those made of thick pasteboard, or of or- dinary cartridge paper. In service of cannon, heavy wads over ball are in all respects injurious. For purpose of retaining the ball in its place, light grommets should be used. On the other hand, it is of great importance, and especially so in use of small arms, that there should be a good wad over powder for developing full force of charge, unless, as in the rifle, the ball has but very little windage. (Capt. Mordecai.) Weight and Dimensions of I.^ead Balls. Number of Balls in a Lb.,from 1.67 to .237 of an Inch Diameter. Diam. No. Diam. No. Diam. No. Diam. No. Diam. No. Diam. No. Ins. Inch. Inch. Inch. Inch. Inch. 1.6 7 i 75 ii 57 25 388 80 301 170 259 270 1.326 2 73 12 537 30 375 88 295 180 256 280 I-I57 3 7 1 13 5i 35 372 90 29 190 .252 290 1.051 4 693 14 SOS 36 359 100 285 200 249 300 977 .919 .677 .662 \l .488 .469 40 45 .348 338 no 120 281 276 2IO 220 247 244 310 320 8?3 7 .65 17 453 50 329 130 272 230 .242 330 835 8 637 18 .426 60 .321 I 4 268 240 239 340 .802 9 .625 19 405 7 3H 150 265 250 237 350 775 10 .615 20 395 75 307 160 262 260 Heated shot do not return to their original dimensions upon cooling, but retain a permanent enlargement of about .02 per cent, in volume. Number of Pellets in an Ounce of Lead Shot of the different Sizes. B A A 40 A 50 BB 58 No. ^0-3 i35 4 i77 5 218 No. 14 No. 6 7 Xo. 9 . . 10. . , 1726 , 2140 502 GUNNERY. Proportion of" Po'wd.er to Shot for following of Snot. No. Shot. Powder. No. Shot. Powdr. No. Shot. Powder. 3 Oi. 2 i-75 Drains. 1.625 4 5 Oi. i-5 1-375 Drams. I-875 2.125 6 7 Oi. 1.25 1-125 Drams. 2-375 2.625 NOTE. 2 oz. of No. 2 shot, with 1.5 drams of powder, produced greatest effect. Increase of powder for greater number of pellets is in consequence of increased friction of their projection. Numbers of Percussion Caps corresponding with Birmingham, Numbers. Eley's. , Birmingham.. 43l44l46 48 49 18 51 and 52 | 53 and 54 | 55 and 56 Where there are two numbers of Birmingham sizes corresponding with only one of Eley's, it is in consequence of two numbers being of same size, varying only in length of caps. Comparison of Force of a Charge in various Arms. ABM. Lock. Powder, AS- Windage. Weight of Ball. Velocity. Percussion Grains. Inch. Grains. Feet. 219 Hall's rifle Flint Percussion 219 1687 Cadet's musket Flint. 7O OAK 2IQ Pistol... Percussion. JS OI* 2l8.< OA1 Ranges for Small Arms. Musket. With a ball of 17 to pound, and a charge of no grains of powder, etc., an elevation of 36' is required for a range of 200 yards; and for a range of 500 yards, an elevation of 3 30' is necessary, and at this distance a ball will pass through a pine board i inch in thickness. jRt/Ze. With a charge of 70 grains, an effective range of from 300 to 350 yards is obtained; but as 75 grains can be used without stripping the ball, it is deemed better to use it, to allow for accidental loss, deterioration of powder, etc. Pistol. Wiib a charge of 30 grains, the ball is projected through a pine board i inch in thickness at a distance of 80 yards. GKmp owder . Gunpowder is distinguished as Musket, Mortar, Cannon, Mammoth, and Sporting powder ; it is all made in same manner, of same proportions of materials, and differs only in size of its grain. Bursting or Explosive Energy. By the experiments of Captain Rodman, U. S. Ordnance Corps, a pressure of 45 ooo Ibs. per square inch was obtained with 10 Ibs. of powder, and a ball of 43 Ibs. Also, a pressure of 185000 Ibs. per sq. inch was obtained when the powder was burned in its own volume, in a cast-iron shell having diameters of 3.85 and 12 ins. Proof of IPowder. (V. S. Ordnance Manual) Powder in magazines that does not range over 180 yards is held to be unservice- able. Good powder averages from 280 to 300 yards; smaU grain, from 300 to 320 yards. Restoring Unserviceable Powder. When powder has been damaged by being stored in damp places, it loses its strength, and requires to be worked over. If quantity of moisture absorbed does not exceed 7 per cent., it is sufficient to dry it to restore it for service. This is done by exposing it to the sun. When powder has absorbed more than 7 per cent, of water it should be sent to a powder mill to be worked over. GUNNEEY. 503 ^Properties i 1C 24-Po Weight of ball an 44 * 4 powder Windage of ball and Res xperime [JNDER GUN. d wad. ... 2 i alts tits. 3 -i35 i 311. Sul- phur. of < (Cat nch. 1 \ 3-nnpowc tain A. Mord M Weight of ba " pc Windage of 1 lanufacture. Vhere from. ler, detern ecai, U. S. A.) USKET PENDULI til ainec UM. 397-5 1 iDy grains. iach. -8 )a il . Water ab- sorbed by ex- posure to Air. ^ GRAIN. c Salt- petre. jmpositi Char- coal. i--i Relative Quickness of Burning. Cannon, large. . . " small. . . Musket Rifle 76 i 77 70 76 75 Slazed. '4 "S 13 15 15 15 12 12.5 10 \ 15 ) 9 10 * Dupont's Mills, Del. t Dupont's Mills, Del. * Dupont's Mills, Del. Loomis, Hazard, & Co., Conn.* Waltham Abbey, England.* "34 6174 5344 1642 13 '52 166 103 72808 295 2378 11600 t E 275 3H 214 282 182 100 212 204 ough. Per c't. 2.77 3-35 3-55 2.09 1.91 4.42 .677 :'S .834 943 . 7 8 .756 i .82 .888 .865 Rifle Musket Rifle Cannon, uneven. " large . . . Blasting, uneven Rifle Rifle * < Manufacture of Powder. Powder of greatest force, whether for cannon or small arms, is produced by incorporation in the "cylinder mills." Effect of Size of Grain. Within limits of difference in size of grain, which occurs in ordinary cannon powder, the granulation appears to exercise but little influence upon force of it, unless grain be exceedingly dense and hard. Effect of Glazing. Glazing is favorable to production of greatest force, and to quick combustion of grains, by affording a rapid transmission of flame through mass of the powder. Effect of using Percussion Primers. Increase of force by use of primers, which nearly closes vent, is constant and appreciable in amount, yet not of sufficient value to authorize a reduction of charge. Ratio of Relative Strength of different Powders for use under water differ but little from the reciprocal of the ratio between the sizes of the grain*, showing that the strength is nearly inversely proportional thereto.* Mammoth, .08; Oliver, .09; Cannon, .18; Mortar, i; Musket, 1.57; Sporting 2.61, and Safety Compound 30.62. Du.alin is nitro-glycerine absorbed by Schultze's powder. For other powders and explosive materials see Blasting, p. 443. Heat and Explosive JPcrwer. (Capt Noble and F. A. Abel.) One gram of fired powder evolves a mean temperature of 730. Temper- ature of explosion 3970. Volume of permanent gas (which is in an in- verse ratio to units of heat evolved) at 32 =250. The explosive power of powder, as tested in Ordnance, ranges, for volumes of expansion of 1.5 to 50 times, from 36 to 170 foot-tons per Ib. burned. A charge of 70 Ibs. gave to an 180 Ibs. shot a velocity of 1694 feet per second, equal to a total energy of 3637 foot-tons, and a charge of 100 Ibs. gave a velocity of 2182 feet, and an energy of 5940 foot-tons. * Report of Experiments and Investigation* to develop a system of submarine mine*. Professional Papers, U. S. E., No. 23. 504 HEAT. HEAT. Heat, alike to gravity, is a universal force, and is referred to both as Cause and effect. Caloric is usually treated of as a material substance, though its claims to this distinction are not decided ; the strongest argument in favor of this position is that of its power of radiation. Upon touching a body having a higher temperature than our own, caloric passes from it, and excites the feeling of warmth ; and when we touch a body having a lower temperature than our own, caloric passes from our body to it, and thus arises the sensation of cold. To avoid any ambiguity that may arise from use of the same expres- sion, it is usual and proper to employ the word Caloric to signify the principle or cause of sensation of heat. Heat Unit. For purpose of expressing and comparing quantities of heat, it is convenient and customary to adopt a Unit of heat or Thermal unit, being that quantity of heat which is raised or lost in a defined period of temperature in a defined weight of a particular substance. Thus, a Thermal unit, Is quantity of heat which corresponds to an interval 0/1 in temperature of i Ib. of pure liquid water, at and near its temperature of greatest density, 39.1. Thermal unit in France, termed Caloric, Is quantity of heat which corresponds to an interval ofi C. in temperature ofi kilogramme of pure liquid water, at and near its temperature of greatest density. Thermal unit to Caloric, 3.96832; Caloric to Thermal unit, .251 996. One Thermal unit or i in i Ib. of water, 772 foot-lbs. One Caloric or i C. in i kilogramme of water, 423.55 kilogrammetres. i C. in i Ib. water, 1389.6 foot-lbs. Ratio of Fahrenheit to Centigrade, 1.8; of Centigrade to Fahrenheit, .555. Absolute Temperature, Is a temperature assigned by deduction, as an opportunity of observing it cannot occur, it being the temperature corre- sponding to entire absence of gaseous elasticity, or when pressure and vol- ume ==o. By Fahrenheit it is 461.2, by Reaumur 229.2, and by Cen- tigrade 274. Heat is termed Sensible when it diffuses itself to all surrounding bodies ; hence it is free and uncombined, passing from one substance to another, affecting the senses in its passage, determining the height of the thermometer, etc. Temperature of a body, is the quantity of sensible heat in it, present at any moment. Heat is developed by water when it is violently agitated. Heat is developed by percussion of a metal, and it is greatest at the first blow. Quantities of heat evolved are nearly the same for same substance, with- out reference to temperature of its combustion. Mechanical power may be expended in production of heat either by fric- tion or compression, and quantity of heat produced bears the same propor- tion to quantity of mechanical power expended, being i unit for power necessary to raise i Ib. 772 feet in height. This number of 772 is termed the mechanical equivalent of heat (Joules). HEAT. Specific Heat. Specific Heat of a body signifies its capacity for heat, or quantity re- quired to raise temperature of a body i, or it is that which is ab- sorbed by different bodies of equal weights or volumes when their temperature is equal, based upon the law, That similar quantities of different bodies require unequal quantities of heat at any given tempera- ture. It is also the quantity of heat requisite to change the tempera- ture of a body any stated number of degrees compared with that which would produce same effect upon water at 32. Quantity of heat, therefore, is the quantity necessary to change the tem- perature of a body by any given amount (as i), divided by Quantity of heat necessary to ciiauge an equal weight or volume of water at 32" by same amount. NOTE. Water has greater specific heat than any known body. Every substance has a specific heat peculiar to itself, whence a cliange of composition will- be attended by a change of its capacity for heat. Specific heat of a body varies with its form. A solid has a less capacity for heat than same substance when in state of a liquid; specific heat of water, for instance, being .5 in solid state (ice), .622 in gaseous (stea:n), and i in liquid. Specific heat of equal weights of same gas increases as density decreases ; exact rate of increase js not known, but ratio is less rapid than diminution in density. Change of capacity for heat always occasions a change of temperature. Increase in former is attended by diminution of latter, and contrariwise. Specific heat multiplied by atomic weight of a substance will give the constant 37.5 as an average, which shows that the atoms of all substances have equal capacity for heat. This is a result for which as yet no reason has been assigned. Thus: atomic weights of lead and copper are respectively 1294.5 and 395.7, and their specific heats are .031 and .095. Hence 1294.5 x .031 = 40.129, and 395.7 x .095 = 37.591. It is important to know the relative Specific Heat of bodies. The most conve- nient method of discovering it is by mixing different substances together at dif- ferent temperatures, and noting temperature of mixture ; and by experiments it appears that the same quantity of heat imparts twice as high a temperature to mercury as to an equal quantity of water; thus, when water at 100 and mercury at 40 are mixed together, the mixture will be at 80, the 20 lost by the water causing a rise of 40 in the mercury; and when weights are substituted for meas- ures, the fact is strikingly illustrated; for instance, on mixing a pound of mercury at 40 with a pound of water at 160, a thermometer placed in it will fall to 155. Thus it appears that same quantity of heat imparts twice as high a temperature to mercury as to an equal volume of water, and that the heat which gives 5 to water will raise an equal weight of mercury 115, being the ratio of i to 23. Hence, if equal quantities of heat be added to equal weights of water and mercury, their temperatures will be expressed in relation to each other by numbers i and 23; or, in order to increase the temperature of equal weights of those substances to the same extent, the water will require 23 times as much heat as the mercury. Capacity for Heat is relative power of a body in receiving and re- taining heat in being raised to any given temperature ; while Specific applies to actual quantity of heat so received and retained. Specific Heat of Air and. other Q-ases. Specific heat, or capacity for heat, of permanent gases is sensibly constant for all temperatures, and for all densities. Capacity for heat of each gas is U u HEAT. same for each degree of temperature. M. Regnault proved that capacity for heat for air was uniform for temperatures varying from 22 to +437; consequently, specific heat for equal weights of air, at constant pressure, averaged .2377. Metals from, 32 to Specific Heat Silver 056 . Water at 32 = i. Woods. Sulphur 2026 212. Steel 1165 Oak <7 Antimony 0^08 Tin 0562 Pear 5 Liiquias. Bismuth 0308 Wrought iron .1138 Pine 65 Brass 0939 Zinc 0955 Copper 092 MinH Substances. ijinseeu on .. . O i Cast iron 1298 Stones. Charcoal 2415 Steam 365 Gold 0324 Chalk 2I 49 Lead 03*4 Limestone... .2174 Coke 203 Mercury 03 7 ^ Masonry 2 Glass iQ77 Nickel 1086 Marble, gray. .2694 Gypsum 1966 Solid. Platinum 0124 " white. 21^8 Phosphorus.. 250^ Ice 504 Air Oxygen. Air.. Gases. ... .2377 I Hydrogen 2356 ... .2412 I Carbonic Acid 3308 For Equal Weights. .1688 | Hydrogen 2.4096 Oxygen 1559 | Carbonic Acid 1714 Metals have least, ranging from Bismuth .0308 to Cast Iron .1298. Stones and Mineral Substances have .2 that of water, and Woods about .5. Liquids, with ex- ception of Bromine, are less than water, Olive oil being lowest and Vinegar highest. ILLUSTRATION. If i Ib. of coal will heat i Ib. of water to 100, ^- of a Ib. will 33 heat i Ib. of mercury to 100. To Compute Temperature of a Mixture of lilie Sub- stances. w (if t) . \- 1' = T. W representing weight or volume of a substance of temperature T, w weight or volume of a like substance of temperature t, and t' temperature of mixture W + w. ILLUSTRATION i. When 5 cube feet of water (W) at a temperature of 150 (T) is mixed with 7.5 cube feet (w) at 50 (0, what is the resultant temperature of the mixture? 5X150 + 7-5X50 5 + 7-5 w(t'-t) T t' ' 1125 = = 90. 12.5 y 2. How much water at (T) 100 should be mixed with 30 gallons (w) at 60, for a required temperature of 8o? 30(80 60) 600 xooo-soo- = - = 3 9allon *' To Compute Temperatxire of a Mixture of Unlike Substances. WST + - = <'; ws(t t') S(T t) - = T. W and w YVS-j-ws " S(T t) ~ WS representing weights, and S and s specific heat of substances. ILLUSTRATION. To what temperature should 20 Ibs. cast iron (W) be heated to raise 150 Ibs. (w) of water to a temperature (t) of 50 to 60 ? 20 X- 1298 2.596 _ ~~ * HEAT. 5O7 To Compute Specific Heat at Constant "Volume. When Specific Heat at Constant Pressure is knoum. -^- = s. S represent- ing specific heat at constant pressure, p proportion of heat absorbed at constant vol- ume, H total heat absorbed at constant pressure, and s specific heat at constant volume. Or, ' * = s. t and t' representing initial and final tempera- ture of the gas and that to which it is raised, and V and v initial and final volumes of the gas under 14.7 Ibs. per sq. inch, and of it heated under constant pressure in cube feet. ILLUSTRATION. Assume i Ib. air at atmospheric pressure and at 32, doubled in volume by heat. S = . 2377 *, t t' = 32 a, 525 = 493 and V v = 12. 387 * cube feet. .2377X493-(2. 742X12.387) = > i688 ^ heat 493 For comparative volumes of other gases, see Table, page 506. To Compute Specific Heat for Eq.ua! Volume of Gras and Air. RULE. Multiply specific heat of the gas for equal weights of gas and air by specific gravity of gas, and product is specific heat for equal volume. EXAMPLE. What is specific heat of air at equal volume with hydrogen? Specific heat of hydrogen for equal weights at constant volume, 2.4096, and speci- fic gravity of the gas, .0692. (See Table, page 506.) Then, 2.4096 X .0692 = . 1667 specific heat for equal volumes at constant volume. Specific heat of steam, air at unity = 1.281. Capacity for Heat. When a body has its density increased, its capacity for heat is di- minished. The rapid reduction of air to .2 of its volume evolves heat sufficient to inflame tinder, which requires 550. Relative Capacity for Heat of Various Bodies. ( Water at 32 = i. ) BODIES. Equal Weights. Equal Volumes. BODIES. Equal Weights. Equal Volumes. BODIES. Equal Weights. Equal Volumes. Water.. Brass. . . Copper.. Glass... I .116 .114 .187 i .971 1.027 .448 Gold. . . . Ice Iron Lead . . . 05 :? 26 043 .966 993 .487 Mercury Silver . . Tin Zinc .036 '.06 .IO2 .83*3 To Ascertain. Relative Capacities of Different Bodies, combined, -with, experiment. RULE. Multiply weight of each body by number of degrees of heat lost or gained by mixture, and capacities of bodies will be inversely as products. Or, if bodies be mingled in unequal quantities, capacities of the bodies will be reciprocally as quantities of matter, multiplied into their respective changes of temperature. ILLUSTRATION. If i Ib. of water at 156 is mixed with i Ib. of mercury at 40, resultant temperature is 152. Thus, i x 156 152 = 4, and i x 40 ou 152 = 112. Hence capacity of water for heat is to capacity of mercury as 112 to 4, or as 28 to i. Sensible Heat. Sensible heat or temperature to raise water from 32 to 212 = 180.9, or heat units. * See Tables, pages 506 and 520-21. 508 HEAT. Latent Heat. Latent Heat is that which is insensible to the touch of our bodies, and is incapable of being detected by a thermometer. When a solid body is exposed to heat, and ultimately passes into the liquid state under its influence, its temperature rises until it attains the point of fusion, or melting point. The temperature of the body at this point remains stationary until the whole of it is melted ; and the heat mean- time absorbed, without affecting the temperature or being sensible to the touch or to the indications of a thermometer, is said to become latent. It is, in fact, the latent heat of fusion, or the latent heat of liquidity, and its func- tion is to separate the particles of the body, hitherto solid, and change their condition into that of a liquid. When, on me contrary, a liquid is solidified, the latent heat is disengaged. If to a pound of newly-fallen snow were added a pound of water at 172, the anow would be melted, and 32 would be resulting temperature. When a body is fusing, no rise in its temperature occurs, however great the additional quantity of heat may be imparted to it, as the increased heat is absorbed in the operation of fusion. The quantity of heat thus made latent varies in different bodies. A pound of water, in passing from a liquid at 212 to steam at 212, re- ceives as much heat as would be sufficient to raise it through 966.6 ther- mometric degrees, if that heat, instead of becoming latent, had been sensible. If 5.5 Ibs. of water, at temperature of 32, be placed in a vessel, communicating with another one (in which water is kept constantly boiling at temperature of 212), until former reaches temperature of latter quantity, then let it be weighed, and it will be found to weigh 6.5 Ibs., showing that one Ib. of water has been received In form of steam through communication, and reconverted into water by loweF temperature in vessel. Now this pound of water, received in the form of steam, had, when in that form, a temperature of 212. It is now converted into liquid form, and still retains same temperature of 212; but it has caused 5.5 Ibs. of water to rise from the temperature of 32 to 212, and this without losing any tempera- ture of itself. Now this heat was combined with the steam, but as it is not sensible to a thermometer, it is termed Latent. Quantity of heat necessary to enable ice to resume the fluid state is equal to that which would raise temperature of same weight of water 140 ; and an equal quantity of heat is set free from water when it assumes the solid form. Su.m of Sensible and. .Latent Heats. From Water at 32. Press- ure. Latent. Sum. Press- ure. Latent. Sum. Press- ure. Latent. Sum. Press- ure. Latent. Sum. Lbs. o Lbs. Lbs. o Lbs. o 14.7 6 4-3 146.1 26 943-7 155-3 55 912 1169 120 873-7 185.4 16 17 962.1 959-8 147.4 148.3 27 28 942.2 940.8 155-8 156.4 60 65 908 904.2 1170.7 1172.3 130 140 869.4 865-4 187-3 189 18 957-7 149.2 2 9 939-4 157- 1 70 900.8 1173.8 ISO 861.5 190.7 *9 955-7 150.1 30 937-9 157-8 75 897-5 1175.2 160 857-9 192.2 20 952.8 150.9 32 935-3 158.9 80 894-3 1176.5 170 854-5 !93-7 21 95i-3 I5I-7 35 931.6 160.5 85 891.4 1177.9 1 80 851-3 I95-J 22 23 949-9 948.5 152-5 153-2 37 40 9 2 9-3 926 161.5 162.9 90 95 888.5 885.8 1179 s1 1180.3 190 200 848 845 196.5 197.8 24 946.9 153-9 45 920.9 164.6 IOO 883.1 1181.4 220 829.2 200.3 25 945-3 154.6 So 916.3 167.1 no 878.3 "83.5 250 831.2 203.7 Latent Heat of Vaporization, or Number of Degrees of Heat required to con- vert following Substances from their 'Liquidities to Vapor at Pressure of Atmosphere. Alcohol 364 Ice 142.6 Water 966.6 Ammonia 860 Mercury 157 Zinc 493 Ether (Sulph.) 163 Carbonic Acid 298 Oil of Turpentine. . 124 HEAT. 509 Latent Heat of Fusion of Solids. (Person.) Substances. Melt- A Specifi Liquid. : Heat. Solid. In Heat- units of lib. Substances. Melt- ing Point. Specifi Liquid. Heat. Solid. In Heat- unite of ilb. o c o Tin Bismuth. Lead .... 442 507 617 .0637 0363 .0402 .0562 .0308 .0314 25-6 22.7 9.86 Ice. Phosphorus .... Spermaceti 32 112 120 I .2045 504 .1788 142-85 , 4 ? Zinc 773 .0056 50.6 Wax Silver . . . 1873 57 37.0 142 27Q 234. , 1 75 Mercury. Cast iron. 39 3400 0333 .0319 .129 o/'y 5 2 33 Nitrate of soda.. Nit. of potassia . *3y 59i 642 '*M 413 3319 .2782 .2388 1 7 '11 To Compnte Latent Heat of Fusion of a Non-metallic Substance. C 'v c (t + 256) = L. C and c representing specific heats of substance in solid and liquid state, t temperature of fusion, and L latent heat. ILLUSTRATION. What is latent heat of fusion of ice? = .504; c = i .504 ^ i X 32 -f- 256 = 142.85 units. NOTE. For Latent Heat of Fusion of some substances, see Deschanel's, New York, 1872, Heat, part 2. Radiation of Heat. Radiation of Heat is diffusion of heat by projection of it in diverging right lines into space, from a body having a higher temperature than space sur- rounding it, or body or bodies enveloping it Radiation is affected by nature of surface of body ; thus, black and rough surfaces radiate and absorb more heat than light and polished surfaces. Bodies which radiate heat best absorb it best. Radiant heat passes through moderate thicknesses of air and gas without suffering any appreciable loss or heating them. When a polished surface receives a ray of heat, it absorbs a portion of it and reflects the rest. The quantity of heat absorbed by the body from its surface is the measure of its absorbing power, and the heat reflected, that of its reflecting power. When temperature of a body remains constant it is in consequence of quantity of heat emitted being equal to quantity of heat absorbed by body. Reflecting power of a body is complement of its absorbing power ; or, sum of absorbing and reflecting powers of all bodies is the same. Thus, if quantity of heat which strikes a body = 100, and radiating and reflecting powers each 90, the absorbent would be 10. Radiating or Absorbent and. Reflecting Powers of Substances. SUBSTANCES. Radiating or Ab- sorbing. Reflect- ing. SUBSTANCES. Radiating or Ab- sorbing. Reflect- ing. IOO Wrought Iron polished Water IOO Lead polished 11 Carbonate of Lead IOO Zinc polished IQ 81 Lead, white IOO Steel polished e. Writing Paper 98 Platinum in sheet 83 Ivory, Jet, Marble 03 to 08 7 to 2 Tin .' !!..' I e Re Resin QO Copper varnished 86 Glass Brass dead polished o rt India Ink " bright polished 11 9 Ice 85 I e Copper ham'ered or cast 93 Shellac 28 93 Lead 45 ee Gold plated 93 Cast Iron, bright polished 25 75 95 Platinum, a little polish'd 24 76 Silver polished . . Mercury 23 77 ** cast polished "u u* 97 5io HEAT. Radiating and A'bisOr'bing Power of varioxxs Bodies, in Units of Heat per Sq.. Foot per Hou.r for a Difference of 1. (Peclet.) Unit. Unit. Iron, ordinary 5662 Woollen stuff 7522 Glass 5948 Oil paint , 7583 ron, cast 648 Paper 77 o6 Unit. Silver, polished 0266 Copper 0327 Tin 0439 Brass, polished 0491 Iron, sheet 092 Wood sawdust 7225 Stone, Brick, etc 7358 Lamp-black 8196 Water 1.0853 To Compute Loss of Heat toy Radiation per So;. Foot. - = R. T representing temperature of pipe, which is assumed to be .05 dv less than that of steam; t temperature of air; I length of pipe, and v velocity of heat in feet per second; d diameter in ins., and R radiation in degrees per second. ILLUSTRATION. Assume temperatures of a steam pipe, steam, 212, 200, and air 60, length of pipe 20 feet, velocity of heat (steam) 15 feet per second, and diameter of pipe 16 ins. ; what will be loss of heat by radiation? 1.7X20(200-60) i6X 15 Reflection. Reflection of Heat is passage of heat from surface of one substance to another or into space, and it is the converse of radiation. Heat is reflected from surface upon which its rays fall in same manner as light, angle of reflection being opposite and equal to that of incidence. Met- als are the strongest reflectors. Reflecting Power of various Substances. Silver. 97 Gold 95 Brass 93 Specular metal 86 I Zinc 81 Tin 85 Iron 77 Steel 83 I Lead 6 Communication and Transmission of Heat. Communication of Heat is passage of heat through different bodies with different degrees of velocity. This has led to division of bodies into Conductors and Non-conductors ; former includes such as metals, which allow caloric to pass freely through their substance, and latter comprise those that do not give an easy passage to it, such as stones f glass, wood, charcoal, etc. Velocity of cooling, other things being equal, increases with extent of sur- face compared with volume of substance ; and of two bodies of same mate- rial, temperature, and form, but differing in volume. Transmission of Heat is passage of heat through different bodies with dif- ferent degrees of intensity. Gaseous bodies and a vacuum are highest in order of transmittents. Relative Power of various Substances to Transmit Heat. All bodies capable of transmitting heat are more or less translucent, though their powers of transmitting heat and light are not in same rela- tive proportions. Nitric acid Rock-crystal . . Rape seed oil. . Heat which passes through one plate of glass is less subject to absorption in passing through a second and a third plate. Of 1000 rays, 451 were iiy tercepted by 4 plates as follows : ist. 381. 2d. 43. 3d. 1 8. 4th. 9. Air i Alcohol 15 Crown-glass.. .49 Flint-glass 67 Gypsum 2 Ice 06 Sulphuric acid. Turpentine Water HEAT. Average Results of Heati Steam in Copper Fip ng and. Evaporating "Water "by es and Boilers. (D.K.Clark.) Steam condensed Heat transmitted Per sq. foot for i* difference per hour. Heating. Evaporating. Heating. Evaporating Lbs. .077 .248 .201 Lba. 105 483 1.07 Unit*. 82 2 7 6 U2 Unite. 100 534 ion Copper-plate surface Copper- pipe surface. . . Yellow 40 Orange 44 Red 53 Whence. Efficiency of copper-plate surface for evaporation of water is double its efficiency for heating ; for copper-pipe surface efficiency is more than three times as much ; and for cast-iron-plate surface, a fourth more. Efficiency of pipe surface is a fifth more than that of plate surface for heating, and more than twice as much for evaporation. Generally, copper -plate surface condenses .5 Ib. of steam, copper-pipe i Ib., and cast-iron-plate surface .1 Ib. per sq. foot per i of temperature per. hour, for evaporation. Quantity of heat transmitted is at rate of about 1000 units per Ib. of steam condensed. Transmission of Heat through Glass of different Colors. Direct = 100. Plate 65. 5 I Blue, deep 19 Window 52 " light 42 Violet, deep 53 | Green 26 M. Peclet defines law of transmission of heat as : The flow of heat which traverses an element of a body in a unit of time is proportional to its sur- face, and to difference of temperature of the two faces perpendicular to direc- tion of flow, and is in inverse of thickness of element. ri Or, (t t') - = H. t and If representing temperatures of surfaces, C constant for material i inch thick, or quantity of heat transmitted per hour for i difference of temperature through i unit of thickness, T thickness, and H quantity of heat in unitt passed through plate per sq. foot per hour. Quantities of Heat transmitted from "Water to "Water throngh Plates or Beds of IMetals and other Solid Bodies, 1 Inch thick, per Sq[. Foot. For i Difference of Temperature between the two Faces per Hour. Selected from M. Peclet's tables. (D. K. Clark.) C or G or C or SUMTANCB. Quantity SUBSTANCE. Quantity SUBSTANCK. of Heat. of Heat. Quantity of Heat. UniU. 225 225 177 112 Marble. , Plaster. , Glass . . , Sand UniU. 6.56 2.16 Gold 620 Iron Platinum 604 Zinc Silver 596 Tin Copper 555 Lead The conditions are, that the surfaces of conducting material must be per- fectly clean, that they be in contact with water at both faces of different temperatures, and that the water in contact with surfaces be thoroughly and constantly changed. M. Peclet found that when metallic surfaces became dull, rate of transmission of heat through all metals became very nearly the same. To Compnte Units of Heat Transmitted. ILLUSTRATION i. If 2000 Ibs. beet root juice at 40 are contained in a copper boiler with a double bottom, and heated to 212, with a heating surface of 25 sq. feet, and subjected to steam at a temperature of 275, for a period of 15 minutes, what will be the total heat, and heat per degree of difference transmitted per sq. foot per hour? 512 HEAT. 212 40 x 60 4- 15 = 688 per hour, and 2000 X 688 -7-25 = 55 040 units per sq. foot per hour. (212 -f 40) -4- 2 = 126 mean temperature of juice, and 275 126 = 149 mean difference of temperature. Hence, 55040-=- 149 = 369.4 units per sq. foot per degree of difference per hour. 2. If 48.2 S}. feet of iron pipe 1.36 ins. in diameter, is supplied with steam at 275, and it raises temperature of 882 Ibs. water from 46 to 212 in 4 minutes, what will be total heat per sq. foot per hour, total heat per sq. foot per degree, and quantity condensed per sq. foot per degree per hour ? 212 46 X 60 -4- 4 = 2490 in an hour ; 46 + 212 -4- 2 = 129 mean temper- ature, and 275 129 = 146 difference of temperature. 2490 X - = 45 563 units per sq.foot per hour, 45 563 -r- 146 = 312. i units per sq. 48.2 foot per degree, and total heat of steam above 129 = 1068. Hence -I = .292 Ibs. steam condensed per sq.foot per degree per hour. Evaporation. Evaporation or Vaporization is conversion of a fluid into vapor, and it produces cold in consequence of heat being absorbed to form vapor. It proceeds only from surface of fluids, and therefore, other things equal, must depend upon extent of surface exposed. When a liquid is covered by a stratum of dry air, evaporation is rapid, even when temperature is low. As a large quantity of heat passes from a sensible to a latent state during formation of vapor, it follows that cold is generated by evaporation. Fluids evaporate in vacuo at from 120 to 125 below their boiling-point. Heat required to Kvaporate 1 Ito. \Vater at Temperatures toelcrw 313 from, a Vessel in. open air at 3S. (Thomas Box.) 13 ^ HEAT ?v.b' HEAT 1 1-1 S . \ S.-I i 3 . 11 rn feS"S Ml '3 .2 K; 11 **"! Srg-S IT? Hi 3 3 Ki Is' "- 1 o "a- H 14 fh 1 *fi 3 H SS, H S u, & S^: ES^g |l 1 * 3 H SI Lbs. Units. Units. Units. Units. Lbs. Units. Units. Units. Units. 32 .027 1091 29 132 .706 182 202 1506 1068 42 .04 270 424 1788 71 142 .916 158 162 1445 1326 52 .058 375 58i 2052 119 152 1.178 137 127 1392 1637 62 .083 45 605 2110 174 162 I-505 118 97 J 346 2039 72 .117 386 566 2055 239 172 I 895 106 72 1312 2475 82 .162 358 504 1968 319 182 2-373 92 So 1279 345 92 .223 3i9 434 1862 415 192 2.947 81 32 1253 3685 102 303 280 366 1758 533 202 3-633 7i 14 1228 4465 112 .406 245 34 l66 4 671 212 4.471 . 6 3 1209 5397 122 .528 211 250 1580 849 To Compute Surface of a Refrigerator. Illustration of Table. If it is required to cool 20 barrels, of 42 gallons each, of beer, from 202 to 82 in an hour. Result to be attained is to dissipate 42 X 8.33 (Ibs. TJ. S. gallons) X 20 X 202 82 = 840000 units of heat per hour. At 202, 4465 units are lost, and at 82, 319, hence, average loss for each temper- ature between extremes = 1850 units per sq.foot per hour. 840000 Then ~ = 454 sq.feet in a still air. 1050 The volume of air required per hour in this case would be about 100000 cube feet HEAT. 5 I 3 To Compute Area of* Q-rate and Consumption of Fuel for Evaporation. Illustration of Table. It it is required to evaporate 6 Beer gallons (282 cube ins.) of liquid per hour, at a temperature not exceeding 152. 6 gallons = 50 Ibs. At 152, water evaporated as per table = 1. 178 Ibs. per hour. -^- = 42 sq. feet. Heat required to effect this = 1392 X 50 = 6 9 6o units - 1-178 ^^ Assuming 6000 units as average economic value of coals, then ^^ = u.6 Ibs. coal, on a grate of i sq.foot. When it is practicable to evaporate at a high temperature, as at or above 212, it is most economical. Thus, water requires only 1209 units per Ib. if surface is exposed, but if enclosed, heat is reduced (1209 63) to 1146 units. Evaporative Powers of Different Tubes per Degree of Heat, per Sq. Foot of Surface. In Units. Vertical tube, 230; Double-bottomed vessel, 330; Horizontal tube or Worm, 430. To Compute "Volume of* Water Evaporated in. a given Time. ILLUSTRATION. What is volume evaporated at 212, in 15 minutes per sq. foot of surface, in a double- bottomed vessel having an area of heating surface of 17 feet, and subjected to steam at a pressure of 25 Ibs. ? Temperature of steam at 25 + 14.7 Ibs. = 269. 269 212 = 57, and latent heat 927. Then 33QX57XI7X.5 = 86.2 to*, water. 927X60 When Water is at a Lower Temperature than 212. If 120 gallons or 1000 Ibs. of water were to be evaporated from 42 in an hour, from same vessel and under like pressure as preceding : There would be requ ired 1000 X (2 1 2 42) 1 70 ooo units of heat. Mean tempera- tare of water while being heated = 4 2 +212 _ I27< j Difference between temperature of steam and water = 267 127 = 140. Then, - 10 - _ 2l6 ft(mr _ time ^ rai$e water fo 2I2 o . hence i .216 =. 330 X 140 X 17 .784 hour left for evaporation, and quantity evaporated = 33 x 57 X 17 X .784., 927 270.4 Z&*., or 32.44 gallons. Dessieeation.. Dessiccation, or the drying of a substance, is best effected in a drying chamber, and it is imperative that to attain greatest effect the hot air should be admitted at highest point of exposed substance and dis- charged at its lowest. Wood, submitted to an average temperature of 300 in an enclosed space for a period of 2.5 clays, will lose its moisture at a consumption of i Ib. of wood for 10.5 Ibs. of wood dried, and evaporating 4 Ibs. of water, equal to 2.66 Ibs. of water per Ib. of undried wood. Limit of temperature for drying of wood is 340. 514 HEAT. Kvaporation of* "Water per S ing lengths of increased and primitive substance in like denominations, T and t tern peratures o/L and I, and C expansion of substance for each degree of heat, HEAT. 523 ILLUSTRATION. A copper rod at 32 is 100 feet in length; to what temperature must it be subjected to increase its length 1.1633 ins. ? Expansion for a length of 100 feet of copper for i = .0115. . .0115 .0115 When Length is to be reduced. -- T = L ILLUSTRATION. Take elements of preceding case. ,*>. I6 33 - 1200 _ I33 . l6 =IOI . l6 _ I33 . l6 = 32 Q .0115 To Reduce Degrees of* Fahrenheit to Reaumur and. Cen~ tigrade, and Contrariwise; Fahrenheit to Reaumur. If above zero. Multiply difference between number of degrees and 32 by 4, and divide product by 9. Thus, 212 32 = 180, and 180 X 4 + 9 = 80. If below zero. Add 32 to number of degrees ; multiply sum by 4, and divide product by 9. Thus, 40 + 32 = 72, and 72 X 4 -*- 9 32- Reaumur to Fahrenheit. If above freezing-point. Multiply number of degrees by 9, divide by 4, and add 32 to quotient. Thus, 80 X 9 -P- 4 180, and 180 -f 32 = 212. If below freezing-point. Multiply number of degrees by 9, divide by 4, and subtract 32 from product. Thus, 32 X 9 -s- 4 = 72, and 72 32 = 40. Fahrenheit to Centigrade. If afiove zero. Multiply difference between number of degrees and 32 by 5, and divide product by 9. Thus, 212 32 X 5^-9 = i8oX5-:-9=ioo . If below zero. AM 32 to number of degrees, multiply sum by 5, and divide product by 9. Thus, 40 -f 32 X 5 -? 9 = 72 X 5 -=- 9 = 40- Centigrade to Fahrenheit. If above freezing -point,. Multiply number of degrees by 9, divide product by 5, and add 32 to quotient. Thus, 100 x 9-1-5 = 180, and 1 80 + 32 = 212. If below freezing-point. Multiply number of degrees by 9, divide product by 5, and take difference between 32 and quotient. Thus, 10 X 9 -r- 5 = 1 8, and 18 o.5 ; of Wate Increase of Temperature. tion. el and violentl ",59-5. Duration of Agitation. y Agitated. Increase of Temperature. Hour. 5 I o JO 14-5 Hours. 3 O 19-5 29-5 Hours. I 39-5 42-5 VENTILATION. Buildings, -A.partirLen.ts, etc. In Ventilation of Apartments. From 3.5 to 5 cube feet of air are required per minute in winter, and 5 to ip feet in summer for each occupant. In Hospitals, this rate must be materially increased. Ventilation is attained by both natural draught and artificial means. In first case the ascensional force is measured by difference in weight of two columns of air of same height, the height being determined by total difference of level between entrance for warm air and its escape into the atmosphere. The difference of weight is ascertained from difference of temperatures of ascending warm air and the external atmosphere, as by Table, page 521, or by formula, page 522. Volumes of Air Discharged through a Ventilator One Foot Sq.uare of Opening, at Various Heights and. Temperatures. Height of Ventilator Excess of Temperature of Apartment above that of External Air. Height of Ventilator from Excess of Temperature of Apartment above that of External Air. Ba 8 e-iine. 5 10 5 20 25 3 o Base-line. 5 10 15 20 25 30 Feet. C.ft. C.ft. C.ft. C.ft. C.ft. C.ft. Feet. C.ft. C.ft. C.ft. C.ft. C.ft. C.ft. 10 116 164 200 235 260 284 35 218 306 376 436 486 53i 15 142 202 245 284 3i8 348 40 235 329 403 465 5i8 57 20 164 232 285 330 368 404 45 248 348 427 493 55i 605 25 30 184 2OI 260 28 4 3l8 347 368 403 410 450 450 493 50 55 260 270 367 385 450 472 5i8 579 54i 1 605 635 663 Velocity of draft having been ascertained for any particular case, together with volume of air to be supplied per minute, sectional area of both air passages may be computed from these data. Heating "by Hot "Water. One sq. foot of plate or pipe surface at 200 will heat from 40 to 100 cube feet of enclosed space to 70 where extreme depression of temperature is 10. The range from 40 to 100 is to meet conditions of exposed or corner buildings, of buildings less exposed, as intermediate ones of a cluster or block, and of rooms intermediate between the front and rear. When the air is in constant course of change, as required for ventilation or occupation of space, these proportions are to be very materially increased as per following rules. HEAT, VENTILATION, AND HEATING. 52$ In determining length of pipe for any given space it is proper to include in the computation the character and occupancy of the space. Thus, a church, during hours of service, or a dwelling-room, will require less service of plate or length of pipe than a hallway or a public building. Reduction of Heat by Surfaces of Glass or Metal. In addition to the volume of air to be heated per minute for each occupant, 1.25 cube feet for each sq. foot of glass or metal the space is enclosed with must be added. The communicating power of the glass and metal being directly proportion- ate to difference of external and internal temperature of the air. Thus, 80 feet of glass will reduce 100 feet of air per minute. When Pipes are laid in Trenches in the Earth. The loss of heat is es- timated by Mr. Hood at from 5 to 7 per cent. Circulation of Water in Pipes. In consequence of the complex forms of heating-pipes and the roughness of their internal surface, it is impracticable to apply a rule to determine the velocity of circulation, as consequent upon difference of weights of ascending and descending columns of the water. For a difference of temperature in the two columns of 30 (190 160) and a height of 20 feet, the velocity due to the height would be 3.74 feet. In practice not .3, and hi some cases but .1, would be attained. In Churches and Large Public Rooms, with ordinary area of doors and windows and moderate ventilation, a large amount of heat is generated by the respiration of the persons assembled therein. In these cases it is not necessary to heat the air above 55, and a rule that will meet the ordinary ranges of temperature from 10 is to divide volume in cube feet by 200, and quotient will give area of plate in sq. feet or length of 4-inch pipe in lineal feet. Volume of Air required per Hour for each Occupant in an Enclosed Space, (General Morin.) Cube Feet. Hospitals .... 2100 to 3700 Workshops . . 2100 " 3500 Cube Feet. Lecture-rooms 1000 to 2100 Theatres: 1400 " 1800 Cube Feet. Prisons 1800 Schools 424 to 1060 To Compute Length of Iron IPipe reqxiired. to Heat Air in an Enclosed. Space. By Hot Water. RULE. Multiply volume of air to be heated per minute in cube feet by difference of temperatures in space and external air, divide product by differ- ence of temperatures of surface of pipe and space, multiply result by follow- ing coefficients, and product will give length of pipe in feet. For diameter of 4 ins. multiply by .5 to .55, for 3 ins. by .7 to .75, and for 2 ins. by i to i.i. A pipe 4 ins. in diameter, .375 inch thick, and i foot in length has an area of internal surface of 1.05 sq.feet. EXAMPLE. Volume of a room of a protected dwelling is 4000 cube feet; what length of 4 ins. pipe, at 200, is necessary to maintain a temperature of 70, when external air is at o ? 4000 X ^ X. 4 = 862 feet. 200 70 In computing length of pipe or surface of plate it is to be borne in mind that the coefficients here given and computation in following table are based upon a ventilation or change of air ordinarily of 3.5 to 5 cube feet per person, and from 5 to 10 cube feet in summer per minute. Hence, when the ventilation is restricted the coefficient may be correspondingly .in- creased. 5 26 HEAT AND HEATING. Ijength.8 of Four-Inch. ripe to Heat 1OOO Cube Feet of Air per IMiivute. (Chas. Hood.) Temperature of Pipe 200. Temperature Temperature of Building. External Air. 45 50 55 | 60 6 5 70 75 80 85 90 Feet. Feet. Feet, j Feet. Feet. Feet. Feet. Feet. Feet. Feet. 10 126 174 | 200 229 259 292 328 367 409 16 20 105 9 1 127 112 135 176 1 60 204 187 223 2IO 265 247 300 281 337 378 358 26 69 9 112 136 162 I 9 220 253 288 327 30 36 54 75 52 97 73 120 96 H5 1 20 173 147 202 S 269 239 307 276 40 18 37 58 80 104 129 157 187 220 255 SO 19 40 62 86 112 140 171 204 Proper Temperatures of Enclosed. Spaces. SPACES. Temper- ature required. SPACES. Temper- ature required. Work-rooms manufactories etc O Dwelling-rooms Churches and like spaces ... Hot-houses 80 Schools lecture-rooms 5* Drying-rooms when filled . . 80 Halls, shops, waiting-rooms, etc. Dwelling-rooms . . . < 6* " " for curing paper.. 7 1 20 Boiler. Boiler for steam-heating should be capable of evaporating as much water as the pipes or surfaces will condense in equal times. Mr. Hood recom- mends that 6 sq. feet of direct heating-surface of boiler should be provided to evaporate a cube foot per hour. Adopt mean weight of steam of 5 Ibs. above pressure of atmosphere, or 20 Ibs. absolute pressure, condensed per sq. foot of pipe per degree of difference of temperature per hour, viz., .002 35 Ib. (as given by D. K. Clark), the quantity of pipe or plate surface that would form a cube foot of condensed water per hour, weight of like volume of water 62.4 Ibs., would be, per i difference of temperature, 62. 4 -T-. 002 35 = 26 5 50 sq.feet, and for differences of 168, 158, 148, and 108, required surface would be respectively (26550-7-168 = 158) 158, 168, 179, and 246 sq. feet. Henoe, assuming, as previously stated, that 4 sq. feet of direct and effec- tive heating boiler-surface, or its equivalent flue or tube surface, will evap- orate i cube foot of water per hour, 158 sq. feet of steam-pipe or plate will require 4 sq. feet of direct surface, etc., for a temperature of 60, and cor- respondingly for other temperatures. Boiler-power. One sq. foot of boiler-surface exposed to direct action of fire, or 3 sq. feet of flue-surface, will suffice, with good coal, for heating 50 sq. feet of 4-inch, 66 of 3-inch, and 100 of 2-inch pipe. Mr. Hood assigns the proportion at 40 feet of 4-inch pipe for all purposes. Usual rate of com- bustion of coal is 10 or n Ibs. per sq. foot of grate-surface, and at this rate, 20 sq. ins. of grate suffice for heating 40 feet of 4-inch pipe. Four sq. feet of direct heating boiler-surface, or equivalent flue or tube surface, exposed to direct action of a good fire, are capable of evaporating i cube foot of water per hour. According to M. Grouvelle, i sq. meter of pipe-surface (10.76 sq. feet), heated to 60 an ordinary room alike to a library or office, of from 90 to 100 cube meters (3178 to 3531 cube feet). HEAT, WARMING BUILDINGS, ETC. 527 If a workshop to be heated to a high temperature, i sq. meter (10.76 sq. feet) of surface is assigned to 70 cube meters (2472 cube feet) = 4. 35 sq. feet or 5.11 lineal feet of 4 inch pipe per 1000 cube feet. For heating workshops, having a transverse section of 260 sq. feet, with a window- surface of one sixth total surface, it is customary in France to assign 1.33 sq. feet of iron pipe surface per lineal foot of shop =. 5.2 sq. feet per 1000 cube feet. Illustrations of extensive Heating by Steam. (R. Briggs, M. I. C. E.) i. Total number of rooms, including halls and vaults ............ 286 " Area of floor surface ................................ 137 370 sq. feet. " Volume of rooms .................................. i 923 500 cube feet. " Number of occupants ................................. 650 Maximum average of occupants at any time .................. 1300 Volume per occupant, excluding vaults ...................... 1443 cube feet. Boilers. 8 with 173 sq. feet of grate surface and 8000 sq. feet of heating surface. Furnishing steam in addition to the above, to operate the elevators and electric dynamos, elevating water, and supplying steam to heat a distant building, requiring one third of their capacity. By Steam. To Compute .Length of Iron Pipe req.ui.red. to Heat Air in an Enclosed. Space, with JSteain at 5 libs, per Sq.. Inch a"bove Pressure of Atmosphere. RULE. Multiply volume of air in cube feet to be heated per minute, by difference of temperature in space and external air, divide product by coeffi- cients in preceding table, and quotient will give length of 4-inch pipe in lineal feet, or area of plate-surface in sq. feet. Temperature of steam at 5 Ibs. -f- pressure = 228. Hence, if temperature of space required is 60, 70, 80, or 120, the differences will be 168, 158, 148, and 108, which for a coefficient of .5, as given in rule for hot water, would be 336, 316, 296, and 216, for a pipe 4 ins. in diameter, and for 60 70 80 120 3-inch pipe ........ 252 237 222 162 2 ' " ........ 168 158 148 108 i * " ........ 84 79 74 54 ILLUSTRATION. Volume of combined spaces of a factory is 50000 cube feet; what surface of wrought iron plate at 200 is necessary to maintain a temperature of 50 when external air is at o ? 50000 Xo 200 50 = Coal Consumed per Hour to Heat 1OO Feet of Pipe. (Chas. Hood.) Difference of Temperature of Pipe and Air in Space, in Degrees. ijiam. 01 Pipe. ISO MS 140 135 130 125 1 20 "5 no 105 IOO 95 9 85 80 IDB. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. I I.I i.i 1. 1 i I 9 9 9 .8 .8 7 7 7 .6 .6 2 2-3 2.2 2.2 2.1 2 1.9 1.8 1.8 i-7 1.6 i-5 1.4 1.4 i-3 1.2 3 3-5 3-4 3-3 3-i 3 2.9 2.8 2.7 2-5 2.4 2-3 2.2 2.1 2 1.8 4 4-7 4-5 4.4 4.2 4.1 3-9 3-7 3-6 3-4 3-2 3-i 2. 9 2.8 2.6 2-5 To warm a factory, according to M. Claude!, 43 feet in width by 10.5 high, a single line of hot- water pipe 6.25 ins. in diameter per foot of length of room, appears to be sufficient, temperature in pipe being from 170 to 180. Also, water being at 180, and air at 60, making a difference of 120, it is convenient to estimate from 1.5 to 1.75 sq. feet of water-heated surface as equivalent to one sq. foot of steam-heated surface, and to allow from 8 to 9 sq. feet of hot-water pipe-surface per 1000 cube feet of room. M. Grouvelle states that 4 sq. feet of cast-iron pipe-surface, whether heated by steam or by water at 176 to 194, will warm 1000 cube feet of workshop, main- taining a temperature of 60. Steam is- condensed at rate of .328 Ib. per sq. foot per hour. 528 HEAT, WASHING BUILDINGS, ETC. 2. (R. L. Greene.) Length of fronts of buildings. . . . , 2 ooo lineal feet Total volume of rooms '. 2 574 084 cube feet. Radiating surfaces, direct, 10804 j , 4 100 sq fee t. indirect, 23 296 > " Boilers Grate-surface 180 Heating surface 5 863 " Volnxne of Air Heated toy Radiators ; Consumption of Coal ; A.reas of GJ-rate arid Heating-svirfaee of J3 oiler. (RoPt Briggs.) Per 100 Sq. Feet of Warming-surface of Radiator. Pressure of steam per sq inch -f- atmosphere in Ibs _ 3 10 30 60 Heat from radiators per minute in units . 456 486 537 642 74i Volume of air heated i per min- ute in cube feet 25110 26772 29570 35352 40803 Efficiency of radiators in ratio. . . . Coal consumed per hour in Ibs.... Area of grate consuming 8 Ibs. i 304 38 i. 066 3-24 .405 1.178 3.58 .448 1.408 4.28 1.625 4-94 do 12 Ibs .... .208 357 .412 Heating surface of boiler ; coal consumed per hour Xa.8 in sq.feet 8 Ibs. X28 8.512 22 4 9.072 22 A, 10.02 22.4 11.98 13-83 12 Ibs. X 2. 8... ^.6 n.6 ^.6 By Hot-j^ir Furnaces or Stoves. A square foot of heating surface in a hot-air furnace or stove is held to be equivalent to 7 sq. feet of hot water pipe. M. Peclet deduced that when the flue-pipe of a stove radiated its heat directly to air of a space, the heat radiated per sq. foot per hour, for i difference of temperature, were, for: Cast iron, 3.65 units; Wrought iron, 1.45 units, and Terra cotta .4 inch thick, 1.42 units. In ordinary practice, i sq. foot of cast iron is assigned to 328 cube feet of space. Open Fires. According to M. Claudel, the quantity of heat radiated into an apart- ment from an ordinary fireplace is .25 of total heat radiated by combustible. For wood the heat utilized is but from 6 to 7 per cent., and for coal 13 per cent. In combustion of wood, chimney of an ordinary open fireplace draws from looo to 1600 cube feet of air per pound of fuel, and a sectional area of from 50 to 60 sq. ins. is sufficient for an ordinary apartment. Proportions of fuel required to heat an apartment are : For ordinary fire- places, 100 ; metal stoves, 63 ; and open fires, 13 to 16. Furnaces. By D. K. Clark, from investigations of Mr. J. Lothian Bett. Cupola. M. Peclet estimates that in melting pig-iron by combustion of 30 per cent, of its weight of coke, 14 per cent, only of the heat of combus- tion is utilized. Metallurgical. According to Dr. Siemens, i ton of coal is consumed in heating 1.66 tons of wrought iron to welding-point of 2700, and a ton of coal is capable of heating up 39 tons of iron ; from which it appears that only 4.5 per cent, of whole heat is appropriated by the iron. Similarly, he estimates 1.5 per cent, of whole heat generated is utilized in melting pot HEAT AND HEATING. HYDRAULICS. 529 steel in ordinary furnaces, whilst, in his regenerative furnace, i ton of steel is melted by combustion of 1344 Ibs. of small coal, showing that 6 per cent, of the heat is utilized. Blast-furnace. Mr. Bell has formed detailed estimates of appro- priation of the heat of Durham coke in a blast-furnace ; from which is de- duced following abstract : Durham coke consists of 92.5 per cent, of carbon, 2.5 of water, and 5 of ash and sulphur. To produce i ton of pig-iron, there are required 1232 Ibs. of limestone, and 5388 Ibs. of calcined iron-stone ; the iron-stone consists of 2083 Ibs. of iron, 1008 Ibs. of oxygen, and 2509 Ibs. of earths. There is formed 813 Ibs. of slag, of which 123 Ibs. is formed with ash of the coke, and 690 Ibs. with the limestone. There are 2397 Ibs. of earths from the iron- stone, less 93 Ibs. of bases taken up by the pig-iron and dissipated in fume, say 2314 Ibs. Total of slag and earths, 3127 Ibs. Mr. Bell assumes that 30.4 per cent, of the carbon of the fuel, which es- capes in a gaseous form, is carbonic acid; and that, therefore, only 51.27 per cent, of heating power of fuel is developed, and remaining 48.73 per cent, leaves tunnel-head undeveloped. He adopts, as a unit of heat, the heat required to raise the temperature of 112 Ibs. of water 33.8. HYDRAULICS. Descending Fluids are actuated by same laws as Falling Bodies. A Fluid will fall through i foot in .25 of a second, 4 feet hi .5 of a second, and through 9 feet in .75 of a second, and so on. Velocity of a fluid, flowing through an aperture in side of a vessel, reservoir, or bulkhead, is same that a heavy body would acquire by fall- ing freely from a height equal to that between surface of fluid and middle of aperture. Velocity of a fluid flowing out of an aperture is as square root of height of head of fluid. Theoretical velocity, therefore, in feet per sec- ond, is as square root of product of space fallen through in feet and 64.333 = ^ 19 h', consequently, for one foot it is 1/64.333 = S.ozfeet. Mean velocity, however, of a number of experiments gives 5.4 feet, or .673 of theoretical velocity. In short ajutages accurately rounded, and of form of contracted vein, (vena coniracta), coefficient of discharge = .974 of theoretical. Fluids subside to a natural level, or curve similar to Earth's convexity; apparent level, or level taken by any instrument for that purpose, is only a tangent to Earth's circumference; hence, in leveling for canals, etc., difference caused by Earth's cur vature must be deducted from apparent level, to obtain true level. Deductions from. Experiments on Discharge of IFlnids from. Reservoirs, 1. That volumes of a fluid discharged in equal times by same apertures from same head are nearly as areas of apertures. 2. That volumes of a fluid discharged in equal times by similar apertures, under different heads, are nearly as square roots of corresponding heights of fluid above surface of apertures. 3. That, on account of friction, small-lipped or thin orifices discharge pro- portionally more flaid than those which are larger and of similar figure, under same height of fluid. YY 530 HYDRAULICS. 4. That in consequence of a slight augmentation which contraction of the fluid vein undergoes, in proportion as the height of a fluid increases, the flow is a little diminished. 5. That if a cylindrical horizontal tube is of greater length than its di- ameter, discharge of a fluid is much increased, and may be increased with advantage, up to a length of tube of four times diameter of aperture. 6. That discharge of a fluid by a vertical pipe is augmented, on the prin- ciple of gravitation of falling bodies ; consequently, greater the length of a pipe, greater the discharge of the fluid. 7. That discharge of a fluid is inversely as square root of its density. 8. That velocity of a fluid line passing from a reservoir at any point is equal to ordinate of a parabola, of which twice the action of gravity (2 g) is parameter, the distance of this point below surface of reservoir being the abscissa.* Or, velocity of a jet being ascertained, its curve is a parabola, parameter of which = 4 A, due to velocity of projection.! 9. Volume of water discharged through an aperture from a prismatic vessel which empties itself, is only half of what it would have been during the time of emptying, if flow had taken place constantly under same head and corresponding velocity as at commencement of discharge ; consequently, the time in which such a vessel empties itself is double the time in which all its fluid would have run out if the head had remained uniform. 10. Mean velocity of a fluid flowing from a rectangular slit in side of a reservoir is two thirds of that due to velocity at sill or lowest point, or it is that due to a point four ninths of whole height from surface of reservoir. 11. When a fluid issues through a short tube, the vein is less contracted than in preceding case, in proportion of 16 to 13 ; and if it issues through an aperture which is alike to frustum of a cone, base of which is the aper- ture, the height of frustum half diameter of aperture, and area of small end to area of large end as 10 to 16, there will be no contraction of the vein. Hence this form of aperture will give greatest attainable discharge of a fluid. 12. Velocity of efflux increases as square root of pressure on surface of a fluid. 13. In efflux under water, difference of levels between the surfaces must be taken as head of the flowing water. 14. To attain greatest mechanical effect, or vis viva, of water flowing through an opening, it should flow through a circular aperture in a thin plate, as it has less frictional surface. From Conduits or Pipes. (Bossut.) 1. Less diameter of pipe, the less is proportional discharge of fluid. 2. Discharges made in equal times by horizontal pipes of different lengths, but of same diameter, and under same altitude of fluid, are to one another in inverse ratio of sq. roots of their lengths. 3. In order to have a perceptible and continuous discharge of fluid, the altitude of it in a reservoir, above plane of conduit pipe, must not be lees than .082 ins. for every 100 feet of length of pipe. 4. In construction of hydraulic machines, it is not enough that elbows and contractions be avoided, but also any intermediate enlargements, the in- jurious effects of which are proportionate, as in following Table, for like volumes of fluid, under like heads in pipes, having a different number of enlarged parts. No. of Parts. II No. Velocity. O f p art8 . Velocity. 741 No. of Parts. Velocity. .569 No. of Parts. 5 Velocity. '454 * See D'Aubulsson, page 66. t Humber, page 57. HYDKAUL1CS. 531 Friction. Flowing of liquids through pipes or in natural channels is materially af- fected by friction. If equal volumes of water were to be discharged through pipes of equal diameters and lengths, but of following figures : Fig. i. Fig. 2. Fig. 3 . Figs. i. 2. 3. The times would be as i, i.n, and 1.55. And velocities AS i, .72, and .64. Discharges from Compound, or Divided. Reservoirs. Velocity in each may be considered as generated by difference of heights in contiguous reservoirs ; consequently, square root of difference will rep- resent velocities, which, if there are several apertures, must be inversely as their respective areas. NOTE. When water flows into a vacuum, 32.166 feet must be added to height of it; and when into a rarefied space only, height due to difference of external and internal pressure must be added. VELOCITY OF WATER OK OF FLUIDS. Coefficients of Discharge. Coefficient of Discharge or Efflux is product of coefficients of Contraction and Velocify. It is ascertained in practice that water issuing from a Circular Aperture in a thin plate contracts its section at a distance of .5 its diameter from aperture to veiy nearly .8 diameter of aperture, so as to reduce its area from i to about .61.* Velocity at this point is also ascertained to be about .974 times theoretical velocity due to a body falling from a height equal to head of water. Mean velocity in aperture is therefore .974, which, X .61 = .594, theoretical discharge ; and in this case .594 becomes coefficient of discharge, which, if expressed generally by C, will give for discharge itself . .3 V. a representing area of aperture, and V volume discharged per second. Or, 4. 97 a -\Jh =. V. Or, 3.91 d 2 ^/h =. V. d representing diameter in feet. Hence, for cube feet per second, 4.97 a-^/h, or 3.91 d 2 ^/h. ILLUSTRATION. Assume head of water 10 feet, diameter of opening 1.127 feet, area i sq. foot, and C = .62. Then i V* g 10 X .62 = 15.72 cube feet. 4.97 X i X V 10 = I 5-7 2 CM & e f ee *"> and 3.91 X i.i27 2 X V IO = I 5-7 cube feet. For square aperture it is .615, and for rectangular .621. Volume of water or a fluid discharged in a given time from an aperture of a given area depends on head, form of aperture, and nature of approaches. _.2 h representing height to centre of opening in feet. 64.333 h = v 2 , and - - = h. NOTE. Head , or height, h, may be measured from surface of water to centre of aperture without practical error, for it has been proved by Mr. Neville that for cir- cular apertures, having their centre at the depth of their radius below the surface, and therefore circumference touching the surface, the error cannot exceed 4 pel cent, in excess of the true theoretical discharge, and that for depths exceeding threa * Bayer, .61. Observed discharge! of water coincide nearer to unit of Bayer than that of all others. 532 HYDRAULICS. times the diameter, the error is practically immaterial. For rectangular apertures it is also shown that, when their upper side is at surface of the water, as in notches, the extreme error cannot exceed 4.17 per cent, in excess; and when the upper is three times depth of aperture below the surface, the excess is inappreciable. For notches, weirs, slits, etc., however, it is usual to take full depth for head, when .666 only of above equation must be taken to ascertain the discharge. Experiments show that coefficient for similar apertures in thin plates, for small apertures and low velocities, is greater than for large apertures and high velocities, and that for elongated and small apertures it is greater than for apertures which have a regular form, and which approximate to the circle. When Discharge of a Fluid is under the Surface of another body of a like Fluid. The difference of levels between the two surfaces must be taken as the head of the fluid. Or, ^2g(h h') = v. When Outer Side of opening of a discharging Vessel is pressed by a Force. The difference of height of head of fluid and quotient of pressures on two sides of vessel, divided by density of fluid, must be taken as heads of fluid. Or, ^/2 g(h te""g) Xl *A = v . s representing density of fluid. ILLUSTRATION. Assume head of water in open reservoir is 12 feet above water- line in boiler, and pressures of atmosphere and steam are 14.7 and 19.7 Ibs. Then = 4 . 333XI2 _.M = 5 . 56/ee , When Water flows into a rarefied Space, as into Condenser of a Steam- engine, and is either pressed upon or open to Atmosphere. The height due to mean pressure of atmosphere within condenser, added to height of water above internal surface of it, must be taken as head of the water. Or, V 2 g (h -f- h r ) = v. ILLUSTRATION. Assume head of water external to condenser of a steam-engine to feet, vacuum gauge to indicate a colum a column of water of 13 Ibs. = 29. g feet. be 3 feet, vacuum gauge to indicate a column of mercury of 26.467 ins. (= 13 Ibs.), Then -v/2 g (3 + 29.9) = 1/64.333 X 3 2 -9 = V 21 16 - 57 = Relative "Velocity of Discharge of "Water through differ- ent Apertures and xinder like Heads. Velocity that would result from direct, unretarded action of the column of water which produces it, being a constant, or i Through a cylindrical aperture in a thin plate 625 A tube from 2 to 3 diameters in length, projecting outward 8125 A tube of the same length, projecting inward 6812 A conical tube of form of contracted vein 974 Wide opening, bottom of which is on a level with that of reservoir; sluice with walls in a line with orifice; or bridge with pointed piers 96 Narrow opening, bottom of which is on a level with that of reservoir; abrupt projections and square piers of bridges 86 Sluice without side walls 63 Discharge or Efflux of "Water for various Openings and Apertvires. Rectangular "Weir. Weirs are designated Perfect when their sill is above surface of natural stream, and Imperfect, Submerged, or Drowned when it is below that surfaco. HYDRAULICS. 533 Height measured from Surface of Water to Sill. (Jas. B. Francis.) Mean Head. Length of Opening. Mean Discharge per Second. Mean Coefficient. .62 to 1.55 feet. 10 feet. 32.9 cube feet. .623 Principal causes for variation in coefficients derived from most experi- ments giving discharge of water over weirs arises from, 1. Depth being taken from only one part of surface, for it has been proved that heads ora, ctf, and above a weir should be taken in order to determine true discharge. 2. Nature of the approaches, including ratio of the water-way hi channel above, to water-way on weir. When a weir extends from side to side of a channel, the contraction is less than when it forms a notch, or Poncelet weir, and coefficient sometimes rises as high as .667. When weir or notch extends only one fourth, or a less portion of width, coefficient has been found to vary from .584 to .6. When wing-boards are added at an angle of about 64, coefficient is greater than even when head is less. Computation, of "Volume of Discharge. Mean velocity of a fluid issuing through a rectangular opening in side of a vessel is two thirds of that due to velocity at sill or lower edge of opening, or it is that due to a point four ninths of whole height from surface of fluid. Height measured from Surface of Head of Water to Sill of Opening. RULE. Multiply square root of product of 64.333 and height or whole depth of the fluid in feet, by area in feet, and by coefficient for opening, and two thirds of product will give volume in cube feet per second. t representing time in seconds and V volume in cube feet. EXAMPLE. Sill of a weir is i foot below surface of water, and its breadth is 10 feet; what volume of water will it discharge in one second? C = . 623, V64-33X i X 10 X i = 80. 2, and 80. 2 X . 623 = 33. 32 cube feet. NOTE. Mean coefficient of discharge of weirs, breadth of which is no more than Ihird part of breadth of stream, is two thirds of .6 = .4 ; and for weirs which extend Vhole width of stream it is two thirds of .666 = .444. Or, 214 VP= V in cube feet per minute. When h is in ins., put 5. 15 for 214. Or, C 6 h VTgh V. C for a depth . i of length = . 417, and for . 33 of length = . 4. 3 Or, by formula of Jas. B. Francis: 3.33 (L .1 n H) H* = V. L representing length of weir and H depth of water in canal,, sufficiently far from weir to be unaffected by depression caused by the current, both in feet, and n number of end contractions. NOTE. When contraction exists at each end of weir, n = 2; and when weir is of width of canal or conduit, end contraction does not exist, and n = o. This formula is applicable only to rectangular and horizontal weirs in side of a dam, vertical on water-side, with sharp edges to current; for if bevelled or rounded off in any perceptible degree, a material effect will be produced in the discharge; it is essential also that the stream, after passing the edges, should in nowise be restricted in its flow and descent. Y Y* 534 HYDRAULICS. In cases in which depth exceeds one third of length of weir, this formula is not applicable. In the observations from which it was deduced, the depth varied from 7 to nearly 19 ins. With end contraction, a distance from side of canal to weir equal to depth on weir is least admissible, in order that formula may apply correctly. Depth of water in canal should not be less than three times that on weir for ac- curate computation of flow. ILLUSTRATION. If an overfull weir has a length of 7.94 feet and a depth of .986 (as determined by a hook gauge), what volume will it discharge in 24 hours? 3-33 (7-94 -2 X- 986) .986^ = 3. 33 X 7-94 -I972 X- 9797 = 3-33X7-7428 X .97907 = 25.243 875, which X 60 X 60 X 24 = 2 181 061 cube feet. By Logarithms. Log. 3. 33 = .522444 7.7428 = .888898 .986^ = 7.993877 3 2) 1.981631 1.990 815 = 1.990815 1.403 157 Log. 24 hours = 86 400 seconds. 4. 936 514 6.338671 Log. 6. 338 67 = 2 181 073 cube feet. C in this case = .615. Or. 2i4V / lP and s.i5v / P = V, if stream above the sill is not in motion. H representing height of surface of water above sill in feet, h in inches; and 214 VH3-J--035 v 2 H3 V, if in motion, v representing velocity of approach of water in feet per secondhand V volume in cube feet discharged over each lineal foot of sill per minute. In gauging, waste-board must have a thin edge. Height measured to level of sur- face not affected by the current of overfall. (Molesworth.) To Compute Depth, of Flow over a Sill that will Dis- charge a given. "Volume of* "Water. -j- K$ \ k = d. k representing height due to velocity (v) as it 2C&V20 / 2 9 Hows to the weir. NOTE. When back-water is raised considerably, say 2 feet, velocity of water ap- proaching weir (k) may be neglected. Rectangular Notches, or "Vertical Apertures or Slits. A Notch is an opening, either vertical or oblique, in side of a vessel, reser- voir, etc., alike to a narrow and deep weir. Vertical Apertures or Slits are narrow notches or weirs, running to or near to bottom of vessel or reservoir. Coefficient for opening, 8 ins. by 5, mean .606 (Poncelet and Lesbros). Coefficient increases as depth decreases, or as ratio of length of notch to its depth increases. When sides and under edge of a notch increase in thickness, so as to be converted into a short open channel, coefficients reduce considerably, and to an extent beyond what increased resistance from friction, particularly for small depths, indicates. Poncelet and Lesbros found, for apertures 8x8 ins., that addition of a horizontal hoot 21 ins. long reduced coefficient from .604 to .601, with a head of about 4. feet; but for a head of 4.5 ins. coefficient fell from .572 to .483. For Rule and Formulas, see preceding page. HYDRAULICS. 535 Rectangular Openings or Sluices, or Horizontal Slits. Height, measured from Surface of Head of Water to Upper Side and to Sill of Opening. (Opening, i inch by i inch. Head, 7 to 23 feet. = .621. " 3 " " 3 " 7 " 23 " =.614. " 2 feet " i foot. i " a " =.641. Poncelet and Lesbros deduced that coefficient of discharge increases with small and very oblong apertures as they approach the surface, and decreases with large and square apertures under like circumstances. Coefficients ranged, in square apertures of 8 by 8 ins., under a head of 6 ins. to rectangular apertures, 8 by 4 ins. ; under a head of 10 feet, from .572 to .745. In a Thin Plate, C = .6x6 (Bossut) ; C = .61 (Michelotti). To Compute Discharge. RULE. Multiply square root of 64.333 afl d breadth of opening in feet, by coefficient for opening, and by difference of products of heights of water and their square roots, and two thirds of whole product will give discliarge in cube feet per second. Or, b^Tg(h^hh'^/h') C = V; - - - - - = <; and 3 f bVTj (h^/h-h'^/h') C y rr rr = v. h and h' representing depth to sill and opening infect, and v velocity b (fi li) in feet per second. EXAMPLE. Sill of a rectangular sluice, 6 feet in width by 5 feet in depth, is 9 fet below surface of water; what is discharge in cube feet per second? C = .625, 9 5 = 4, and A/2~0X6x.625X(9-\/9 4X V4) = 3 8 -95 cube feet. Or, Vzgd a C = V. d representing depth to centre of opening in feet. d = 9 2.5 = 6.5, a = 6 X 5 = 30, and ^64.33 X 6.5 X 30 X .625 = 383.44 cube ft. Sluice "Weirs or Sluices. Discharge of water by Sluices occurs under three forms viz., Unimpeded, Impeded, or Partly Unimpeded. To Compute Discharge when. Unimpeded. C d b V^ gh-=V. d representing depth of opening and h taken from centre of pening to surface of water. If velocity, &, with which water flows to sluice is considered, ., d\ 7) V V v -d- and - =d. n . ., o v* h' representing height to which water is raised by dam above sill ILLUSTRATION. How high must the gate of a sluice weir be raised, to discharge 150 cube feet of water per second, its breadth being 24 feet and height, h' t 5 feet? C by experiment = .6. d approximately = i. = 1.0204.0* 5-i) I4 ' 4 X I To Compute Discharge when Impeded. CdbVTgh = V, and - - - = d. CbVzgh h representing difference of level between supply and back-water. HYDRAULICS. 536 To Compute Discharge -when partly Impeded. C b A/2T0 (d */h 1- d'VM = V. d' representing depth of back-water above upper edge of sill. ILLUSTRA RATION. Dimensions of a sluice are 18 feet in breadth by .5 in depth; height of opening above surface of water .7 feet, and difference between levels of supply and surface water is 2 feet; what is discharge per second? .6 X 18 X 8.02 } =86.62 X. 896 + .707 = 138.85 cube feet. %%Z%%&%%&%22 Coefficients of Circular Openings or Sluices. Height measured from Surface of Head of Water to Centre of Opening. Contraction of section from i to .633, and reduction of velocity to .974; hence 633 X -974 = -617 (Neville). In a Thin Plate, C = .666 (Bossut); .631 (Fenfrm); .64 (Eytelwein). Cylindrical Ajutages, or Additional Tubes, give a greater discharge than apertures in a thin side, head and area of opening being the same ; but it is necessary that the flowing water should entirely fill mouth of ajutage. Mean coefficient, as deduced by Castel, Bossut, and Eytelwein, is .82. Short Tubes, Mouth-pieces, and Cylindrical Prolonga- tions or A j \itages. F j g- 4- If an aperture be placed in side of a ft Fig. 5. vessel of from 1.5 to 2.5 diameters in thickness, it is converted thereby into a short tube, and coefficient, instead of being reduced by increased friction, is increased from mean value up to about .815, when opening is cylindrical, as in Fig. 4 ; and when junction is rounded, as in Fig. 5, to form of contracted vein, coefficient increases to .958, .959, and .975 for heads of i, 10, and 15 feet. Conically Convergent and Divergent TxVbes. Fig. 6. In conically divergent tube, Fig. 6, coeffi- cient of discharge is greater than for same tube placed convergent, fluid filling in both miirir o cases, and the smaller diameters, or those at same distance from centres, O O, being used in the computations. A tube, angle of convergence, O, of which is 5 nearly, with a head of from i to 10 a feet, axial length of which is 3.5 ins., small diameter i inch, and large diameter 1.3 ins., b gives, when placed as at Fig. 6, .921 for co- efficient ; but when placed as at Fig. 7, co- efficient increases up to .948. Coefficient of velocity is, however, larger for Fig. 6 than for Fig. 7, and discharging jet has greater amplitude in falling. If a prismatic tube project beyond sides into a vessel, coefficient will be re- duced to .715 nearly. Form of tube which gives greatest discharge is that of a truncated cone, lesser base being fitted to reservoir, Fig. 7. Venturi concluded from his ex- Fig. 7- HYDRAULICS. 537 periments that tube of greatest discharge has a length 9 times diameter of lesser opening base, and a diverging angle of 5 6' discharge being 2.5 greater than that through a thin plate, 1.9 times greater than through a short cylindrical tube, and 1.46 greater than theoretic discharge. Compound. 3Vou.tli-pieees and. Ajutages. Fig. 8. a Fig. 9. ^ Fig. 10. Coefficients for Month -pieces, Short Tntoes, and. Cyl- indrical ^Prolongations. Computed and reduced by Mr. Neville, from Venturis Experiments. Description of Aperture, Mouth-piece, or Tube. C. for C. for Diam. a b. Diain. o r, .622 .823 .611 .607 .561 .928 .823 .823 .911 1.02 I.2I5 .895 974 .823 .956 934 .948 1.4*8 .266 .266 :k 855 1-377 1. An aperture i. 5 ins. diameter, in a thin plate 2. Tube i. 5 ins. diameter, and 4. 5 ins. long, Fig. 4 3. Tube, Fig. 5, having junction rounded to form of contracted vein 4. Short conical convergent mouth piece, Fig. 6 5. Like tube divergent, with smaller diameter at junction with reservoir; length 3.5 ins.,or = i in., and ab = i.$ ins. ... 6. Double conical tube, a o, 8 T, r b. Fig. 9, when a6 = ST = i.5 ins., or = 1.21 ins., ao = -92 in., and oS = 4. i ins 7. Like tube when, as in Fig. 8, aor6 = oSTr, and a o S = 1.84 ins 8. Like tube when S T = 1.46 ins. , and o S = 2. 17 ins 9. Like tube when ST=3 ins., and 08 = 9.5 ins 10. Like tube when o S = 6.5 ins., and S T = 1.92 ins ti. Like tube when ST = 2.25 ins., and 08 = 12.125 ins 12. A tube, Fig. 10, when o s = r t = 3 ins., o r = s t = i. 21 ins., and tube o S T r, as in No. 6, S T = i. 5 ins. , and s 8=4. i ins. Mean of various experiments with tubes of .5 to 3 ins. in diameter, and with a head of fluid of from 3 to 20 feet, gave a coefficient of .813 ; and as mean for circular apertures in a thin plate is .63, it follows that under similar circumstances, .813 -f- .63= 1.29 times as much fluid flows through a tube as through a like aperture hi a thin plate. Preceding Table gives coefficients of discharge for figures given, and it will be found of great value, as coefficients are calculated for large as well as small diameters, and the necessity for taking into consideration form of junction of a pipe with a reservoir will be understood from the results. Circular Sluices, etc. To Compute Discharge. Height measured from Surface of Head of Water to Centre of Opening. RULE. Multiply square root of product of 64.333 and depth of centre of opening from surface of water, by area of opening in square feet, and this product by coefficient for the opening, and whole product will give discharge in cube feet per second. Or, -\/20d, a C = V. a representing area in sq. feet, and d depth of surface qf fluid from centre of opening in feet. 538 HYDEAULICS. EXAMPLE. Diameter of a circular sluice is i foot, and its centre is 1.5 feet below surface of the water; what is discharge in cube feet per second? Area of i foot = .7854; C = .64, and ^64.333 X 1.5 X -7854 X .64 = 4.938 cube feet. When Circumference reaches Surface of Water. Vz gr, .9604 a C = V. r representing radius of circle in feet. ILLUSTRATION. In what time will 800 cube feet of water be discharged through a circular opening of .025 sq. foot, centre of which is 8 feet below surface of water? 800 Boo C = . 63. __ - = ^ - = 2239. 58 = 37 min. 19. 6 sec. 5X.6 3 22.68 X. 025 X. 63 NOTE. For circular orifices, the formula \/2 g d a C = V is sufficiently exact for all depths exceeding 3 times diameter; the finish of openings being of more effect than extreme accuracy in coefficient. Semicircular Sluices. When Diameter is either Upward or Downward, -\fzgd a C = V. d repre- senting depth of centre of gravity of figure from surface. When Diameter as above is at Depth d, beloiv Surface. V? gd i . 188 a C = V. Circular, Semicircular, Triangular, Trapezoidal, Pris- matic Wedges, Sluices, Slits, etc. See Neville, London, 1860, pp. 51-63, and Weisbach, vol. \.p. 456. For greater number of apertures at any depth below surface of water, product of area, and velocity of depth of centre, or centre of gravity, if practicable to obtain it, will give discharge with sufficient accuracy. Discharge from "Vessels not Receiving any Supply. For prismatic vessels the general law applies, that twice as much would be discharged from like apertures if the vessels were kept full during the time which is required for emptying them. 2 A ^/h 2 A h To Compute Time. - - = = t. CaVsg v ILLUSTRATION. A rectangular cistern has a transverse horizontal section of 14 feet, a depth of 4 feet, and a circular opening in its bottom of 2 ins. in diameter; in what time will it discharge its volume of water, when supply to it is cut off aud cistern allowed to be emptied of its contents? h 4 feet, a 2 2 X -7854-:- 144 = .0218, = .613, and Vz gh x a X C = .2143 cube foot per second. Then - = 522. 6 seconds. To Compute Time and Fall. Depression or subsidence of surface of water in a vessel, corresponding to a given time of efflux, is h h 1 . h 1 representing lesser depth. Inversely, ( V h- CaV ^ t\* = h'. \ 2A/ ILLUSTRATION. In what time will the water in cistern, as given in preceding case, subside 1.6 feet, and how much will it subside in that time ? A = 14, C = .6, a = .0218, -\/2 = 8.02, ft = 4, h' =r 4 1.6 = 2.4. 2X14 X (-y/4 V 2 - 4) = -^- X (2 i. 55) = 120. i seconds. .6 X. 0218 X 8.02 ~ -.1049' / .6 X .0218 X 8.02 \ 2 IV4 Xi2o.il = 2 .4s = 2.4feet; hence, 4 2.4 = 1.6 feet \ 2 X 14 / When Supply is maintained. Divide result obtained as preceding by 2. HYDRAULICS. 539 Discharge, -when. Form. and. Dimensions of Vessel of Efflvix are not kno^wn. Volume discharged may be estimated by observing heads of the water at equal intervals of time ; and at end of half time of discharge, head of water will be .25 of whole height from surface to delivery. When t = such interval. For openings in bottom or side, C a t V^g ( M = V,/or i depth; C at^Tg ^V* + 4 V*i + V*a\ = V f&r 2 4^. and NOTK. At end of half time of discharge, head of water will be . 25 of whole height from surface to delivery. "Weirs or !N"otch.es. - C 6 t VTg (V& 3 + 4 V A3 i + V^ 3 z) = V. 6 representing breadth in feet. 9 ILLUSTRATION. A prismatic reservoir 9 feet in depth is discharged through a notch 2.222 feet wide, surface subsiding 6.75 feet in 935 seconds; what is volume discharged ? C = .6, h^ g 6.75 = 2.25/66*, and - 6X2.222X935X8.02 (V^ + t V2.253-J- Vo^) = 2221. 6 x 40. 5 = 89 974. 8 cube feet. When there is an Influx and Efflux. If a reservoir during an efflux from it has an influx into it, determination of time in which surface of water rises or falls a certain height becomes so complicated that an approximate determination is here alone essayed. A state of permanency or constant height occurs whenever head of water is in- creased or decreased by ( J = k. I representing influx in cube feet per second. Time (t) in which variable head (x) increases by volume (v) = j _ ^ and time in which it sinks height, k, by -- - . Time of efflux, in which GaV? gx I subsiding surface falls from A to Ax, etc., and head of water from h to AI, when k is represented by - = -^/Ar, is Ca Vs g h h 4 /A 4Ai 2 A 2 4A 3 A 4 \ ~ ILLUSTRATION. In what time will surface of water in a pond, as in a previous example, fall 6 feet, if there is an influx into it of 3.0444 cube feet per second? _ 20 14 _ / 600000 4X495000 . 2 X 4 10 OOP i 4X3 2 5ooo 12 X -537X .8836X 8.02 ^4.472 s" 1 " 4.301 .8 "^ 4. 123 -.8 ~*~ 3-937 -8 _|_ _? -g ) = gg- X 1 480 201 = 194 486 seconds = 54 h. , i mm , 26 sec. If vessel has a uniform transverse section, A. Then 2 V. f V* - V*x + V* X hyp. log. (^~ V ^) ] = t = time in which CaVz^L \vx v fc / J eati of water flows from h to h t . 54O HYDRAULICS. ILLUSTRATION. A reservoir has a surface of 500000 sq. feet, a depth of 20 feet; it is fed by a stream affording a supply of 3.0444 cube feet per second, and outlet has an area of .8836 sq. foot; in what time will it subside 6 feet? v/fc, as before, = .8, C = .537, and 2 X SOOOOQ x [^ 20 __^^+ 8 x nyp log 0/i 87 X 2<3 3 j = 2 3 8 4M seconds = 66 h. 13 min. 34 sec. To Compute Fall in a given Time. This is determining head hi at end of that time, and it should be sub- tracted from head h at commencement of discharge. Put into preceding equation several values of h lt until one is found to meet the condition. ILLUSTRATION. Take a prismatic pond having a surface of 38 750 sq. feet, a depth to centre of opening of sluice of 10. 5 feet, a supply of 33.6 cube feet, and a discharge of 40 cube feet per second. Vfc = .8 4 . Putting these numerical values into the equation, and assuming different values for hi, a value which nearly satisfies the equation is 4. Consequently, 10.5 4 = 6. 5 feet, fall. . ( - - )* = &; arc (tang. = y, arc tangent of which = y, and I as preceding. \t C 6 v 2 gf According as k is ^ h, and influx of water, I ^ C I V^gh^ there is a rise or fall of fluid surface, the condition of permanency occurring when h f k, and time cor- responding becomes co. ILLUSTRATION. In what time will water in a rectangular tank, 12 feet in length by 6 feet in breadth, rise from sill of a weir or notch, 6 inches broad, to 2 feet above it, when 5 cube feet of water flow into the tank per second ? ht = 2, h = o, A = 12X6 = 72, 1 = 5, b = .5, C = .6. ) J = 10.2423 X 0-9 61 (3-46i X arc, tangent of which = .56497, or 29 28' = 29.466, length of which = .5143) = 1.781] =x 10.2423 7.961 1.781 == 10.2423 X 6.18 = 63.297 seconds. Discharge of* Water -under "Variable Pressures. To Compute Time, Rise and Fall, and "Volume. V* gv = v. representing variable head, A and a areas of transverse horizon- tal section of vessel and discharge, and v theoretical velocity of efflux. To Compute "Volume. A y = V. y representing extent of fall, and V volume of water discharged, as ILLUSTRATION. Assume elements of preceding case. A = 14. y = 4 feet. Then 56 x 4 = 224 cube feet HTDEAULICS. 541 Discharge from Vessels of Communication. When Reservoir of Supply is maintained at a uniform Height. Fig. n. To Compute Time. ^- = t. ILLUSTRATION i. In what time will level of water in a receiving vessel having a section of 14 sq. feet attain height of that in supply, through a pipe 2 ins. in diam- eter/placed 4 feet below level of supply? C = .6x3. 2XHXV4 = _56_ = 522 . 3 seconds , .613 X .0218 X 8.02 . 1072 2 Assume C, vessel, Fig. n, to be a cylinder 18 ins. in diameter, head of water in A = 4 feet, at A' i foot, and 2 feet below outlet o; in what time vrill water in vessel run out and over at o through a pipe, a, 1.5 ins. diameter? = ^j='- 111611 .8X8.03 When Vessel of Supply has no Influx, and is not indefinitely great compared with Receiving Vessel. ^ = t. A' representing section of receiving vessel, t time in which C a (A -f- A') ^/Tg the two surfaces of water attain same level; and * = t, time within which level falls from h to h'. ILLUSTRATION. Section of a cistern from which water is to be drawn is 10 sq. feet, and section of receiving cistern is 4 sq. feet; initial difference of level is 3 feet, and diameter of communicating pipe is i inch; in what time will surfaces of water in both vessels attain like levels? .7854. 2 X io X 4 V3 . 82 X. 7854X^X8.02 .502 Discharge from a Notch* in Side of a Vessel. When it has no Influx. ^^7= (^r, Jr) = ' *> breadth of notch in feet. CbxVzg W * v */ ILLUSTRATION. If a reservoir of water, no feet in length by 40 in breadth, has a notch in end of 9 ins. in width; in what time will head of water of 15 ins. fall to 6? 3X110X40 /i i \ 13200 .6 X. 75 X8 os >< (71-7^) =767 X -4.4-94= V "*. NOTE. For discharge of vessels in motion, see Weisbach, vol. i, pp. 394-396. Reservoirs or Cisterns. To Compxate Time of Trilling and of Emptying a Reser- voir under Operation of "both Supply and Discharge. V V T, and ^ =. t. V representing volume of vessel, S supply of water, and D discharge of water, both per minute, and in cube feet. T time of filling wawJ, and t time of discharging it, both in minutes. * When the notch extends to the bottom of the reservoir, etc., the timt for the water to mm out i* ' 542 HYDRAULICS. Irregular-Shaped. Vessels, as a Pond, liaise, etc. To Compute Time and. "Volume Discharged. Operation. Divide whole mass of water into four or six strata of equal depths. h h4 /a , 4 01 , 2 ct2 4 a3 , a 4\ Then, for 4 Strata, x (_ + 1_ 4. -_ + 1 + -nrj = ? ; *, h', i2GaV2ff W* v" v /i2 v 713 v^ 4 / etc., representing depths of strata at a, ai, etc., commencing at surface; a*, as, etc., 6ewp area* of first, second, etc., transverse sections of pond, etc. ; and -^- ILLUSTRATION. In what time will depth of water in a lake, A 6 C, Fig. 12, subside 6 feet, sur- faces of its strata having follow- ing areas, outline of sluice being a semicircle, 18 ins. wide, 9 deep, and 60 feet in length? a at 20 feet (h ) depth of water = area of 600000 sq. feet ai "18.5 " (fti) " " = " 495000 " a 2 "17 v (h 2 ) = 410000 03 " 15.5 " (&3) " = 325000 a4 " 14 " (&4) " = " 265000 " a = area of 18 -=- 2 = .8836 sq. feet ; C = .537. 20 14 /6ooooo 4X495000 2X410000 . 4X325000 * 12 X. 537 X. 8836X8.02 \4-472 4-30i 4-123 3.937 , 26sooo\ 6 ~~ = X X I94 431 = I5 93 Se ' = 43 ' 35 mm> 38 5eC< And discharge = x (600000 + 4 X 495000-1-2X410000+4X3250004-265000) = . 5 X 4 965 ooo = 2 482 500 cube feet. For 6 Strata, put 2 a4, instead of a4, and 4 as and a6 additional, and divide by 1 8 instead of 12. Flow of TVater in. Beds. Flow of water in beds is either Uniform or Variable. It is uniform when mean velocity at all transverse sections is the same, and consequently when areas of sections are equal ; it is variable when mean velocities, and there- fore areas of sections, vary. To Compute Fall of Flow. C -? x = h. C representing coefficient of friction, I length of flow, p perimeter a 2 g of sides and bottom of bed, and hfatt in feet. ILLUSTRATION. A canal 2600 feet in length has breadths of 3 and 7 feet, a depth of 3 feet, with a flow of 40 cube feet per second; what is its fall ? C = as per table below .007565; p = -\/3 2 + 2 2 x 2 + 3 = 10.2; 0=15; and = 40-:- 15 = 2.66. Hence .007 565 X To Compute Velocity of Flow. ILLUSTRATION. A canal 5800 feet in length has breadths of 4 and 12 feet, a depth of 5, and a fall of 3 ; what is velocity and volume of flow ? Then . ooyS 6 5 xT8ooX^8 X64 - 33X3 = V ' 542 X ' 93 = 3 " 3 ^ **" volume = 40 x 3. 23 = 129. 2 cube feet. HYDRAULICS. 543 Coefficients of Friction of ITlo-w of "Water in Beds, as in Ravers, Canals, Streams, etc. In Feet per Second. Velocity. c. Velocity. C. Velocity. C. Velocity. C. 3 4 :I .00815 .00797 .00785 .00778 .1 9 i 00773 .00769 .00766 .00763 i-5 2 2-5 3 .00759 .00752 00751 .00749 8 10 12 00745 .00744 .00743 .00742 Forms of Transverse Sections of Canals, etc. Resistance or friction which bed of a stream, etc., opposes to flow of water, in consequence of its adhesion or viscosity, increases with surface of contact between bed and water, and therefore with the perimeter of water profile, or of that portion of transverse section which comprises the bed. Friction of flow of water in a bed is inversely as area of it. Of all regular figures, that which has greatest number of sides has for same area least perimeter ; hence, for enclosed conduits, nearer its trans- verse profile approaches to a regular figure, less the coefficient of its friction ; consequently, a circle has the profile which presents minimum of friction. When a canal is cut in earth or sand and not walled up, the slope of its sides should not exceed 45. "Variable Motion. Variable motion of water in beds of rivers or streams may be reduced to rules of uniform motion when resistance of friction for an observed length of river can be taken as constant. To Compute "Volume of "Water flowing in a River. - = V. A and A r representing areas of upper and lower transverse sections of flow. ILLUSTRATION A stream having a mean perimeter of water profile of 40 feet for a length of 300 feet has a fall of 9.6 ins. ; area of its upper section is 70 sq. feet, and of its lower 60; what is volume of its discharge? To obtain C for velocity due to this case, 92.35 coefficient for which, see Table above, =.00744. V - 7 o + 6oX 9.6 40X300 _ 7-174 _ f~* i~ V ^~6o^ 300X40 ~ V.ooo33o89 = 394.6 cube feet; and mean velocity = 394 ' 2 = 6.07 feet, C for which is .007 45. FRICTION IN PIPES AND SEWERS. Fi'iction in flow of water through pipes, etc., of a uniform diameter is in- dependent of pressure, and increases directly as length, very nearly as square of velocity of flow, and inversely as diameter of pipe. With wooden pipes friction is 1.75 times greater than in metallic. Time occupied in flowing of an equal quantity of water through Pipes or Sewers of equal lengths, and with equal heads, is proportionally as follows : In a Right Line as 90, in a True Curve as 100, and in a Right Angle as 140. 544 HYDRAULICS. To Compute Head necessary to overcome Friction of Pipe. (Weisbach.) .0144 4- > ) X -r X = h' representing head to overcome friction of -r a 5-4 flow in pipe, I length of pipe, and v velocity of water per second, all in feet, and d internal diameter of pipe in ins. ILLUSTRATION. Length of a conduit-pipe is 1000 feet, its diameter 3 ins., and the required velocity of its discharge 4 feet per second; what is required head of water to overcome friction of flow in pipe? .0144 + ' X ~ = .023 13 X 333-333 X 2.963 = 22.845^. Head here deduced is height necessary to overcome friction of water in pipe alone. Whole or entire head or fall includes, in addition to above, height between surface of supply and centre of opening of pipe at its upper end. Conse- quently, it is whole height or vertical distance between supply and centre of outlet. Xo Compute whole Head, or Height from Surface of Supply to Centre of Discharge. 1.5 is taken as a mean, and is coefficient of friction for interior orifice, or that of upper portion of pipe. To obtain C or coefficient. (.0144 -f ' OI j = C. For facilitating computation, following Table of coefficients of resistance is introduced, being a reduction of preceding formula : Coefficients of Friction of Water. In Iipes at Different "Velocities. V. C. V. C. V. C. V. C. V. C Ft. Ins. Ft. Ins. Ft. Ins. Ft. Ins. Ft. Ins. 4 0443 2 8 025 5 .0221 7 4 .0208 i 6 .0195 8 .0356 3 0244 5 4 .O2I9 7 8 .0206 2 .0194 X OS'? 3 4 0239 5 8 .02I 7 8 .0205 2 6 .0193 * 4 .0294 3 8 0234 6 .O2I5 8 6 0204 3 .0191 i 8 .0278 4 0231 6 4 .0213 9 0202 4 .0189 2 .0266 4 4 0227 6 8 .O2I I 10 0199 .0188 2 4 .0257 4 8 0224 7 .0209 ii 0196 6 .0187 ILLUSTRATION i. Coefficient due to a velocity of 4 feet per second is .0231. 2. Take elements of preceding case. (.0231 X IO 3 I2 + i-5) X g^ = 93.9 X g^ = 23.35 /ee<. NOTE. In preceding formula I was taken in feet, as the multiplier of 12 for ins. was cancelled by taking 5.4 for 2 g, but jn above formula it is necessary to restore this multiplier. R-adii of Curvatures. When Pipes branch off from Mains, or when they are deflected at right angles, radius of curvature should be proportionate to their diameter. Thus, Ins. Ins. Ins. Ins. Ins. Diameter . . 2 tO 7 1 tO A ,. g Radius... 8 3 2O 10 A3 60 urim'AULICS. 545 Curves and Sends. Resistance ox toss of head due to curves and bends, alike to that of friction, increases as square of velocity ; when, however, curves have a long radius and bends are obtuse, the loss is small " x ^- h. a representing angle of curve, d diameter of pipe, r radius of curve, and h height due tofi iction or resistance of curve, all in feet. Curved Circular Pipe. (Wdsbach). -^ x [.131 + 1.847 (^) vmeter of pipe all in feet. For ftility of computations, following values of .131 -f- 1.847 ( \* are intro- Coefficients of Resistance. In Curved JPipes -with Section of a Circle. .138 .158 .178 .206 44 54 .661 .806 977 1.177 9 95 1.978 .6 .244 II .65 .294 II .7 ILLUSTRATION. If in a pipe 18 ins. in diameter and i mile in length there is a right angled curve of 5 feet radius, what additional head of flow should be given to attain velocity due to a head of 20 feet ? a = 90, v for such a pipe and head = 4 feet per second; 18 = 1.5 and = . 15, and , 15 by table = . 133. NOTE. If angle is greater than 90, head should be proportionately increased. Bent or Angular Circular Pipes. Coefficient tor angle of bend = .9457 sin. 2 x + 2.047 sin - 4 * Hence, 10 20 30 4 45 50 55 60 65 70 .046 139 .364 74 .984 1.26 1.556 1.861 2.158 2-43J and x C = h. x representing half angle of bend. 00 ILLUSTRATION. Assume v = 4 feet, and angle == 90 ; x = ?- =. 45. >2 Then ^ X .984 = .2447/00^ additional head required. In "Valve Q-ates or Slide "Valves. In Rectangular Pipes. r x 9 .8 7 .6 5 4 3 .2 .x G .0 .09 39 95 2.08 4.02 8.12 x 7 .8 44-5 193 r = ratio of cross section. In Cylindrical Pipes. h .125 25 375 5 .625 75 .875 r C X .0 .948 .07 .856 .26 ;8 609 2.06 .466 5-52 3i5 17 159 97-8 h = relative height of opening. In a Throttle "Valve. In Cylindrical Pipes. 913 24 .826 52 .741 .658 i-54 25 30 35 40 45 50 60 577 2.51 5 3-9 1 .426 6.22 357 10.8 293 18.7 .234 32.6 .134 118 70 .06 75* A = angle of position. Z * 546 HYDRAULICS. In a Claelz or Trap "Valve. Angle of opening 20 60 In. a Cock. In Cylindrical Pipes. .772 .692 1.56 .613 535 35 9.68 40 '7-3 45 50 55 60 315 31.2 25 52.6 '9 1 06 i37 206 70 65 In a Conical "Valve. (1.645 , i J =C. a and a' ' = areas of pipe and opening. In Imperfect Contractions. (~~; *) = c - c = a factor, rang- ing from .624 for = .1 to i for = i, being greater the greater the ratio. ILLUSTRATION. If a slide valve is set in a cylindrical pipe 3 ins. in diameter and 500 feet in length, is opened to .375 of diameter of pipe (hence, .625 diameter closed), what volume of water will it discharge under a head of 100 feet, coefficient of en- trance of pipe assumed at .5? C, by table, p. 545, pipe being .625 closed = $.52. C from, table, p. 544, for an assumed velocity of u feet 6 ins. = .0195. V/64-33 X -y/ioo 8.03 X 10 80. 3 Then 500 X I2\ and V I 5 X 2 = 24-495/eet Fig. 13. 550 HYDRAULICS. Velocity of a jet of water flowing from a cylindrical tube is determined to be .974 to .98 of actual to theoretic velocity, or = .82 of that due to height of reservoir. Hence volume of discharge through a cylindrical opening =s= .82 a Vzgh. Fig. 14. Jets d.'Eau. (Fig. 14.) That a jet may ascend to greatest practicable height, communication with supply should be perfectly free. Short tubes shaped alike to contracted fluid vein, and conically convergent pipes, are those which give greatest velocities of efflux. Hence, to attain greatest effect, as in fire-engines, long and slightly conically convergent tubes or pipes should be applied. In order to diminish resistance of descending water, a jet must be directed with a slight inclination from vertical. Effect of combined causes which diminish height of a jet from that due to elevation of its supply can only be determined by experiments. Great jets rise higher than small ones. With cylindrical tubes, velocity being reduced in ratio of i to .82, and as heights of jets are as squares of these coefficients or ratios, or as i to .67, height of a jet through a cylindrical tube is two thirds that of head of water from which it flows. H C = h. H representing head of water, C coefficient, and h height of jet. (Moles- worth.) When d H -r- 300, C = .96. -*- 450, " = . 93 . "-:- 600, '- = .0. "-5- 800, "=.87. " -r- 1000, " = .85. When d = H -r- 1500, C = .8. , 2800, " = .6. "=''-3500, " = .5. "= " 4500, " = .25. FLOW OF WATER IN RIVERS, CANALS, AND STREAMS. Running Water. Water flows either in a natural or artificial bed or course. In first case it forms Streams, Brooks, and Rivers ; in second, Drains, Cuts, and Canals. Bed of a water-course is formed of a Bottom and two Banks or Shores. Transverse Section is a vertical plane at right angles to course of the flowing water ; Perimeter is length of this section in its bed. Longitudinal Section or Profile is a vertical plane in the course or thread of current of flowing water. Slope or Declivity is the mean angle of inclination of surface of the water to the horizon. Fall is vertical distance of the two extreme points of a denned length of the flowing course, measured upon a horizontal plane, and this fall assigns angle for defined length of the course. Line or Thread of Current is the point where flowing water attains its maximum velocity. Mid-channel is deepest point of the bed in thread of current. Velocity is greatest at surface and in middle of current ; and surface of flowing water is highest in current, and lowest at banks or shore. A River, Canal, etc., is in a state of permanency when an equal quantity of water flows through each of its transverse sections in an equal time, or when V, product of area of section, and mean velocity through whole extent of the stream, is a constant number. HYDRAULICS. 5 5 I To Compute Mean Depth of Flowing "Water. RULE. Set off breadth of the stream, etc., into any convenient number of divisions ; ascertain mean depths of these divisions ; then divide their sum by number of divisions, and quotient is the mean depth. To Compute Mean Area of Flowing "Water. RULE i. Multiply breadth or breadths of the stream, etc., by the mean depth or depths, and product is the area. 2._Divide the volume flowing hi cube feet per second by mean velocity in feet per second, and quotient is area in sq. feet. To Compute "Volume of Flo-wing "Water. RULE. Multiply area of the stream, etc., in sq. feet, by the mean velocity of its flow in feet, and product is volume in cube feet. To Compute Mean "Velocity of Flo^wing "Water. RULE. Divide surface velocity of flow in feet per second by area of the stream, etc., and quotient, multiplied by coefficient of velocity, will give mean velocity in feet. Mean velocity at half depth of a stream has been ascertained to be as .915 to i, and at bottom of it as .83 to i, compared with velocity at surface. Again, the ve- locity diminishes from line of current toward banks, and, to obtain mean superficial velocity, t>i4- i> 2 -l-v* n = '9 ist * ; hence ' To Compute Mean. Velocity in whole Profile of a Navi- gable River, etc., V+i 2 V v = velocity at bottom, and V-f .5 V v = mean velocity. In rivers of low velocities multiply mean velocity by .8. Obstruction in Rivers. (Molesworth.) ^1 1_ .05 X f V x = R. v representing velocity in ins. per second previous 58.6 \aj to obstruction, A and a areas of river unobstructed and at obstruction in sq.feet, and R rise in feet ILLUSTRATION. Velocity of obstructed flow of a river is 6 feet per second, and areas of section before and after obstruction are 100 and 90 sq. feet ; what would be rise in feet ? -|^ + -05 X f^V i = -664 X .234 = .i55/eet ITlcrw of Water in Lined Channels. (Batin.) /5J? _ v ' = C. D representing mean hydraulic depth in feet, F V F ' ( i _L\ fall, or length of cftannel to fall of i, x and \ y * D/ yas per table, and C as per table p. 543. x I y * I v Plastered 0000045 10.16 Rubble Masonry 00006 1.219 Cut Stone 000013 | 4-354 Earth .00035 | .214 For Sections of Uniform Area, as Canals, Sewers, etc. y/^ 2 D = v. A = area of flow in sq, feet, P wet perimeter of section, and D fall of stream per mile "^ILLUSTRATION. Area of transverse section of a sewer is 50 sq. feet, its wet perim- eter 20 feet, and its fall 5 feet per mile. / (. x 2 x 5) = V 2 5 = 5 f eet For Sections of Rivers. 12 ^/D p = v. ILLUSTRATION. Assume area 500 sq. feet, wet perimeter 200, and fall 5 feet per milt, 552 HYDRAULICS. Hydraulic Radius or Mean Depth is obtained by dividing area of trans- verse section by wet perimeter, both in feet. To Compute Fall per Mile for a required. Mean Velocity. r 3 - iij -=- 2 r D. r representing hydraulic radius in feet. Upper surface of flowing water is not exactly horizontal, as water at its surface flows with different velocities with respect to each other, and consequently exert on each other different pressures. If v and vi are velocities at line of current and bank of a stream, the difference of the two levels is V ~~ Vl = h. ILLUSTRATION. If v = 5 feet and v x .o v ; then ^ - = ^^- = . 0738 foot. 20 64.33 A velocity of 7 to 8 ins. per second is necessary to prevent deposit of slime and growth of grass, and 15 ins. is necessary to prevent deposit of sand. Maximum velocity of water in a canal should depend on character of bed of the channel. Thus, Mean Velocity should not exceed per second over Fine clay 6 ins. A slimy bed 8 " Common clay. 6 " River sand i ft. I Broken stones 4 ft. Small gravel i u Stones 6 U Large shingle 3 " | Loose rocks 10 " 3. W vide vo To Compute "Velocity- of Flow or Discharge of \Vater in Streaixis, 3?ipes, Canals, etc. i. When Volume discharged per Minute is given in Cube Feet, and Area of Canal, etc., in Sq. Feet. RULE. Divide volume by area, and quotient, di- vided by 60, will give velocity in feet per second. a. When Volume is given in Cube Feet, and A rea in Sq. Tns. RULE. Di- vide volume by area ; multiply quotient by 144, and divide product by 60. When Volume is given in Cube Ins., and Area in Sq. Ins. RULE. Di- volume by area, and again by 12 and by 60. To Compute Flow or "Volume of Discharge. 1. When Area is given in Sq. Feet. RULE. Multiply area of flow by its velocity in feet per second, and product, multiplied by 60, will give volume in cube feet per minute. 2. When Area is given in Sq. Tns. RULE. Multiply area by its velocity, and again by 60, and'divide product by 144. NOTE i. Velocities and discharges here deduced are theoretical, actual results de- pending upon coefficient of efflux used. Mean velocity, however, as before given, page 529, may be taken at VTg .673 = 5.4 feet, instead of 8.02 feet. 2. As a rule, with large bodies, as vessels, etc., their floating velocity is some- what greater than that of flow of water, not only because in floating they descend an inclined plane, formed by surface of the water, but because they are but slightly affected by the irregular intimate motion of water: the variation for small bodies is BO slight that it may be neglected. To Compute Height of Head of Flowing "Water. When Volume and Area of Flow are given in Feet. RULE. Divide vol- ume in feet per second by product of area, and coefficient for opening, and square of quotient, divided by 64.33, will give height in feet. EXAMPLK. Assume volume 266.48 cube feet, area 40 sq. feet, and C = $23. HYDEAULICS. 553 Submerged, or Dro^wned Orifices and. "Weirs. When wholly submerged (Fig. 15) Available pressure at any point in depth of orifice is equal to difference of pressure on F 'g- 5- & each side. Whence, C Vzgh = v, and C a \fzgh = V. a representing area of sluice in sq.feet. ILLUSTRATION. Assume opening 3 feet by 5, h = 4 feet, and C = . 5. ^ Then, 5 X 3^5 V6 4 . 33 X 4 = 7-5 X 16.04 = 120. 3 cube feet per second. When partly submerged (Fig. 16). h' h = d = submerged depth, and h :- T< e ft" = d' remaining portion of depth; whence 74 4- 9 = 3"7- o seconds. '.545X5X5-67 15-45 2 Ifd = o- ^- = t\ = ^^ , = = 366.34 seconds. ' Ca^g .545X5X5-67 15-45 NOTE /is never greater than I (lift in feet); it is equal to I when d = o; / 2 is equal to I when/i = o, never greater. In each case it is the unbalanced head abovt eluice, however far below the lowest water level the sluice is. To Fill Upper Locli or Empty Lower. To fill upper lock or empty lower, when the sluice is below the lowest water-line, in either case, takes the same time; for the head diminishes at the same rate, one from the upper surface, the other from the bottom. z _ t Here ^ j. bcing below lowest wair faa O f fafr _ g yj^ as d = o, C a yg and/= whole lift = 2 - - = - = 517. 8 seconds. 545X5X5.67 15-45 To Discharge a like Volume under a Constant Head. AX// A // 2000 / 8 4. - ^= = 7r-v/ =* = - v/z - = 2 5 8 -9 seconds, CaVTg CaVaflr .545X5V64-33 Or, one half the time given by preceding case. The times deduced by preceding formulas are in the following proportions in order, as i : V 2 : ^^ , or i : V 2 : ~r~ If sluice of upper lock, through which it is filled, is above lowest water level, then, by combining formulas 3 and 4. the time is thus deduced. To Jill from Lowest Water Level of said Lock to Level of Centre of Sluice. A \! f 5. v = t'. f representing height of centre of sluice above said lowest water C a vTg level To Jill remaining Portion of Lock above Sluice. 6. - Zz = t". f" representing depth below upper water level of centre of CaVzg sluice or remaining portion of lift. Hence, ' -f- " = -^ - -. (V/' -f 2 \//") = * C a x/2 g To Jill Lower Lock under Constant Head from Upper Canal Level, 8. If both lifts are the same, h f= I, and - A ^- L -}. * _ 2 /1\ = t CaVzg^ h V i 2 / If lower lock is filled from upper one under a constant head, when latter is drawn down to lowest level, formula 7 will apply by making h =/, and - (2 2r V which is identical with 7 , for/=/ 2 and d=f, the cases being the same. 556 HYDRAULICS. MISCELLANEOUS ILLUSTRATIONS. 1. If external height of fresh water, at 60 above injection opening in condensei of a steam-engine, is 3 feet, and the indicated vacuum at 23 ins , velocity of water flowing into condenser is thus determined. (Formula page 532.) v = -\/2 g (h + h'). h' representing height of a column of water equivalent to press- ure of atmosphere within condenser. Assuming mean pressure of atmosphere 14.7 Ibs.per sq. inch, height of a column of fresh water equivalent thereto = 33.95 feet. Then, if i inch = .4912 Ibs., 23 ins. = 11.3 Ibs.; and if 14.7 Ibs. = 33.95 feet, 11.3 Ibs. = 26. i feet. Hence v V* g (3 -f- 26. i) = 43.27 feet, less retardation due to coefficient of both influx and efflux. 2. What breadth must be given to a rectangular weir, to admit of a flow of 6 cube feet of water, under a head of 8 ins. ? (Formula page 533.) - : = - 2 ' = a. 21 feel. $X.625V20 66 -417X6.55 3. It being required to ascertain volume of water flowing in a stream, a tem- porary dam is raised across it, with a notch in it 2 feet in breadth by i in depth, which so arrests flow that it raises to a head of 1.75 feet above sill of notch; what is volume of flow per second.? (Formula page 533. ) = .635. - X- 635X2X1-75 Vzgx 1.75 = 1-481 X 10.6 15.7 cube feet. 4. A rectangular sluice 6 feet in breadth by 5 in depth, has a depth of 9 feet of water over its sill, and discharges, as per example page 535, 380.95 cube feet per second ; what is velocity of flow ? (Formula page 535. ) If volume was not given: 3 Then - X .625 X 8.02 X 7 * 9 ~~ 4 = 3- 34' X 3-8 = 12.7 feet. 3 9 4 5. If a river has an inclination of i. 5 feet per mile, is 40 feet in breadth with nearly vertical banks, and 3 feet depth ; what is volume of its discharge ? (Formula p. 542. ) Perimeter 40 -\-z X 3 = 46 feet; hydraulic mean depth 2.61 feet; 46 a = 120 feet; Cper table, page 543,/or assumed velocity of 2.5 feet =.007 5. Then ___-__ x 64. 33 x i. 5 = ^.0659X96.5 = 2. 52 feet velocity. Hence 120 X 2.52 = 302.4 vubefeet. 6. What is head of water necessary to give a discharge of 25 cube feet of water per minute, through a pipe 5 ins. in diam. and 150 feet in length? (Formula p. 548.) Tabular number for diameter 5 ins., page 547, = 263.87. Then 263. 87 -=-25 = 111.3, and 150 -5-111.3 = 1.35 /get If this pipe had 2 rectangular knees or bends, what then would be head of water required? (Formula page 545.) C, page 545,/or ^- = .984, area of 5 ins. = . 136 feet, and -25- -4- 60 = 3.06 feet 2 .130 velocity. Then j^ X .984 X 2 = .2863, which, added to 1.35 = 1.64 feet. By formulas foot of page 548, 0^.024, and c .505 velocity = 3.06 feet ; head = i.+gfeet, and volume 26.38 cube feet. 7. If a stream of water has a mean velocity of 2.25 feet per second at a breadth of 560 feet, and a mean depth of 9 feet, what will be its mean velocity when it has a breadth of 320 feet, and a mean depth of 7. 5 feet ? (Rule page 548. ) 560X9X2.25^1134? 320X7-5 2400 HYDEAULICS. 557 8. What volume will a pipe 48 feet in length and 2 ins. in diameter, under a head of 5 feet, deliver per second ? ( Formula page 547. ) Tabular number for diameter 2 ins., page 547, = 26.69. I = 3. i. Then 9 = 8.61, which -r- 60 = . 143 cube feet. If this pipe had 5 curves of 90, with radii = - =.5; what would be its dis- charge per second ? V = .i 43 ; a = 2-7- 144 =.0139; Gper tables = .294; v = ^^- = 10.29 feet. O I0 2 2 2T ' 139 Then .294 X ^5 X * 6 ' , 147 X 1.64 = 241, which x 5 for 5 curves = 1.2 = height due to resistance of curves, h = 5 1.2= 3. 8. Hence, if -v/20 5 = .143; Vzg 3.8 = .125 cube feet. g. If a slide stop valve, set in a cylindrical conduit 500 feet in length and 3 ins. in diameter, is raised so as to close .625 of conduit; what volume will it discharge under a head of 4 feet ? (Formula page 546. ) Cfor conduit = .5, for friction .025, and for slide valve .375 open, table, page 545, 5.52, d = .25, and a = 7.07 sq. ins. 2 9 h 16.06 Ihen = -j- -r- - = 2.13 feet velocity, and */ ( l + 5 + 5- 52 + 025 -J 2. 13 X 12 X 7- 07 = 180. 71 cw&e ITIS. 10. If a single lock chamber is 200 feet in length by 24 in breadth, with a depth of 10 feet, centre of upper gate, which is 4 feet in depth by 2.5 in breadth, is at middle of depth of chamber, lower gate, 5 feet in depth by 2.5 in breadth and wholly immersed; what is time required for filling and discharging it? (Formula p. 553.) C = .6is, h = $, h' = $, A = 200 X 24 = 4800, = 4X2.5 = 10, and a' = 5 X 2.5 = 12.5 = 652. 8 seconds time of filling. t 5 - = 6 = 49 x -4 seconds time of emptying. 11. In a moderately direct and uniform course of a river, the depths and velocities are as follows ; what is the volume of its flow and what its mean velocity ? (p. 551. ) Feet. Feet. Feet. Feet. Feet. Distances 5 12 20 15 7 I Area of profiles = 5 x 3 + Depths 3 6 ir 8 4 12X6 + 20X11 + 15X8 + Mean velocity 1.9 2.3 2.8 2.4 2.1 | 7 X 4 = 455 sq.feet. 15 X 1.9 + 72X2.3 + 220X2.8 + 120X2.4 + 28X2.1 = 1156.9 cube feet volume, and * 156 ' 9 = 2 . 54 feet velocity. Miner's Inch. A ' Miner's inch " is a measure for flow of water, and is an opening one i'lch square through a plank two inches in thickness, under a head of six inches of water to upper edge of opening. It will discharge 11.625 U. S. gallons water in one minute. Theoretical H? under different Reads. Heads in feet.|ioo Ins. pe in feet. 1 100 190 |8o 70 jrB?... I 3.251 3.61 1 4.06 4.64 1 60 I 5-4i 3 Water Inch (Pouce d"eau). Circular opening of i inch in a thin plate is equal to a discharge of 19.1953 cube meters per 24 hours. A* 558 HYDRODYNAMICS. HYDRODYNAMICS. Hydrodynamics treats of the force of action of Liquids or Inelastic Fluids, and it embraces Hydraulics and Hydrostatics: the former of which treats of liquids in motion, as flow of water in pipes, etc., and latter of pressure, weight, and equilibrium of liquids in a state of rest. Fluids are of two kinds, aeriform and liquid, or elastic and inelastic, and they press equally in all directions, and any pressure communicated to a fluid at rest is equally transmitted throughout the whole fluid. Pressure of a fluid at any depth is as depth or vertical height, and pressure upon bottom of a containing vessel is as base and perpendicu- lar height, whatever may be the figure of vessel. Pressure, therefore, of a fluid, upon any surface, whether Vertical, Oblique, or Horizontal, is equal to weight of a column of the fluid, base of which is equal to sur- face pressed, and height equal to distance of centre of gravity of sur- face pressed, below surface of the fluid. Side of any vessel sustains a pressure equal to its area, multiplied by half depth of fluid, and whole pressure upon bottom and against sides of a cubical vessel is equal to three times weight of fluid. Pressure upon a number of surfaces is ascertained by multiplying sum of surfaces into depth of their common centre of gravity, below surface of fluid. When a body is partly or wholly immersed in a fluid, vertical press- ure of the fluid tends to raise the body with a force equal to weight of fluid displaced ; hence weight of any quantity of a fluid displaced by a buoyant body equals weight of that body. Centre of Pressure is that point of a surface against which any fluid presses, to which, if a force equal to whole pressure were applied, it would keep surface at rest. Hence distance of centre of pressure of any given surface from surface of fluid is same as Centre of Percussion. Centres of fressnre. Parallelogram, Side, Base, Tangent, or Vertex of Figure at Surface of Fluid, is at .66 of line (measuring downward) that joins centres of two horizontal sides. Triangle, Base uppermost, is at centre of a line raised from lower apex, and join ing it with centre of base; and Vertex uppermost, it is at .75 of a line let fall from vertex, and joining it with centre of base. Right-angled Triangle, Base uppermost, is at intersection of a line extended from centre of base to extremity of triangle by a line running horizontally from centre of side of triangle. Vertex or Extremity uppermost, is at intersection of a line ex- tended from the centre of the base to the vertex, by a line running horizontally from .375 of side of triangle, measured from base. Trapezoid, either of parallel Sides at Surface, ^"v 3 6 / X a = d. b and &' repre- senting breadths of figure, d distance from surface of fluid, and a length of line join- ing opposite sides. Circle, at 1.25 of its radius, measured from upper edge. Semicircle, Diameter at Surface of Fluid, *^L d.r representing radius of circle ID and p = 3. 1416. Diam. downward, ^fl " = <* HYDRODYNAMICS. 559 Side, Base, or Tangent of Figure "belcrw Surface of* TTluid. Rectangle or Parallelog'm. - x i^^ 2 = d ; or, 3 mo + m * = d ; and - = d". 3 h' 2 h 2 30 30 h and h' representing depths of upper and lower surfaces of figure and d depth, both from surface of fluid, m half depth of figure, o depth of centre of gravity of figure from surface of fluid, d' distance from upper side of figure, and d" distance from centre of gravity. Triangle. Vertex Uppermost. -^- = d; = d'. Base Uppermost. 1 8 o 18 o "*"' = d. I representing depth of figure, d distance from surface of fluid upon Io O a line from vertex to centre of base, and d" distance from centre of gravity of figure. A o 2 -4- r 2 r 2 Circle. = d, or = distance from centre of circle. Semicircle. Diam. Horizontal and Upward or Downward. 1- o = d\ 40 9J>o^ ~ 4 = d' : = d", and = c. d representing distance from 3P 3P 4 9P surface of fluid, d' distance of centre of gravity from centre of arc, d" distance of centre of gravity from diameter when it is uppermost, and c centre of pressure. ^Pressure. To Compute Pressure of a Fluid, upon Bottom of its Containing "Vessel. RULE. Multiply area of base by height of fluid in feet, and product by weight of a cube foot of fluid. To Compute Pressure of a Fluid, upon a "Vertical, In- clined, Curved, or any Surface. RULE. Multiply area of surface by height of centre of gravity of fluid in feet, and product by weight of a cube foot of fluid. EXAMPLE i. What is pressure upon a sloping side of a pond of fresh water 10 feet square and 8 feet in depth * Centre of gravity, 8 -=- 2 = 4/0 -tfrom surface. Then io 2 X 4 X 62. 5 = 25 ooo Its. 2. What is pressure upon staves of a cylindrical reservoir when filled with fresh water, depth being 6 feet, and diameter of base 5 feet? 5X3. 1416 =15. 708 feet curved surface of reservoir, which is considered as a plane. 1 5. 708 X 6 X 6 -i- 2 = 282. 744, which X 62. 5 = 17 671. 5 Ibs. 3. A rectangular flood-gate in fresh water is 25 feet in length by 12 feet deep; what is pressure upon it? 25 X 12 X i2-r- 2 = 1800, which X 62.5 = 112500 Ibs. When water presses against both sides of a plane surface, there arises from resultant forces, corresponding to the two sides, a new resultant, which is obtained by subtraction of former, as they are opposed to each other. ILLUSTRATE- -Depth of water in a canal is 7 feet; in its adjoining lock it is 4 feet, and breadtn of gates is 15 feet; what mean pressure have they to sustain, and what is depth of point of its application below surface? 7 x 15 = 105, and 4 X 15=60 sq.feet. (105 X - 60 X 2) X 62.5 = 15468.75 Ibs., mean pressure. Then 15468.754-62.5 = 247.5 = cube feet pressing upon gates upon high side, and 247. 5-1-15 X 7 = 2.35 feet = depth of centre of gravity of mean pressure. To Compute Pressure on a Sluice. A iv d = P, and C P = I". A representing area of sluice in sq. feet, w weight of water per cube foot, d mean depth of sluice below surface, in feet, P pressure on sluice, and P' power required to operate it, both in Ibs. C = .68 when sluice is of wood, and .31 when of iron. 560 HYDRODYNAMICS. EXAMPLE What is pressure on a sluice-gate 3 feet square, its centre of gravity being 30 feet below surface of a pond of fresh water? 3 X 3 X 30=270, which x 62.5 = 16875 Ibs. To Compute IPressxire of a Colxixxixi of" a Fluid, per Sq.. Inch. RULE. Multiply height of column in feet by weight of a cube foot of fluid, and divide product by 144 ; quotient will give weight or pressure per sq. inch in Ibs. NOTB When height is given in ins., omit division by 144. PIPES. To Compute required. Thickness of* a Pipe. RULE. Multiply pressure in Ibs. per sq. inch by diameter of pipe in ins., and divide product by twice assumed tensile resistance or value of a sq. inch of material of which pipe is constructed. By experiment, it has been found that a cast-iron pipe 15 ins. in diameter, and .75 of an inch thick, will support a head of water of 600 feet; and that one of oak, of same diameter, and 2 ins. thick, will support a head of 180 feet? EXAMPLE i. Pressure upon a cast-iron pipe 15 ins. in diameter is 300 Ibs. per sq. inch; what is required thickness of metal? 300 X 15 = 45oo> which -~ 3000 X 2 = . 75 inch. NOTE. Here 3000 is taken as value of tensile strength of cast iron in ordinary small water-pipes. This is in consequence of liability of such castings to be im- perfect from honey-combs, springing of core, etc. 2 . Pressure upon a lead pipe i inch in diameter is 150 Ibs. per sq. inch; what is required thickness of metal ? Here 500 is taken as value of tensile strength. 150 X i = 150, which -r- 500 x 2 = .15 inch. Cast-iron Pipes. To Compute Thickness, etc., of* Flanged Pipes. For 75 Ibs. Pressure. For 100 Ibs. Pressure. .0250+ .25 =T 03 D + .3 =< .05 D + I.IS =1 03 P-f -35 =/ 1.05 D -f- 4.25 d 4- 1-25 = o 1.05 D + z X d 4-i = o' .03 D4- .3 =T .o 35 D4- .45 =t .05 D-j- 1. 15 =1 04 D+ -6 =/ i.i D + 5 X d+ 1.5 =o i.i D4- 2.5X^4-1.4 =o' . 7 D 4-2.2 = *; Ax *- n =q , and / 4-C = d. 4000 y -7854 D representing diam. of pipe, T thickness of metal, t thickness and I length of boss, f thickness of flange, o diam. of flange, o' diam. of centres at bolt holes, and d diam. of bolts, all in ins.; A area of pipe and a area of bolt at base of its thread, in sq. ins., p pressure in Ibs. per sq. inch, and C a coefficient due to diam. of bolt. Thus, diam. .125 + .032, .25 4-. 064, .5 4-- 107, i 4~-i6, 1.54- -214, and 24-. 285. ILLUSTRATION. What should be dimensions of a flanged pipe, 10 ins. in diameter, for a pressure of 100 Ibs. per sq. inch ? .7 X 10 -f- 2.2 = 9.2 = 10 number of bolts, and diam. 10 ins. = 78.54 ins. area = A. Z!^i^^ = . I9 635,a n d N /S + C = V. 25 = .s; hence,. 5+ ,o 7 = . 607 = . 625 Ibs. diameter of bolls ; .03 X 10 -f- . 3 = . 6 = thickness of metal ; . 035 X 10 4-. 45 = .8 thickness of flange; .05 X 10 + 1. 15 = 1.65 = length of boss; .04 X 10 4- .6 i = thickness of flange ; i.i X 10 4~ 5 X .625 4- * 5 = 15-625 = diameter of flange ; and 1. 1 X 10 4-2.5 X -625 4- 1. 4 = 13. 9625 = diameter of bolt holes. For Tables of Cast-iron Pipes, see page 132. HYDRODYNAMICS. To Compmte Elements of* 'Water-pipes. .0001245 Pd + C = t; or, .000054 H d + C t; .4336 H^P; and 02 _ d 2 x 2. 45 = W. P representing pressure of water in Ibs. per sq. inch, D and d external and internal diameters of pipe, and t thickness of metal, all in ins., C coeffi- cient for diameter of pipe, and H head of water in feet. C =s . 37 for pipes less than 12 ins. in diameter, . 5 from 12 to 30, and 6 from 30 to 50 To Compnte "Weight of Pipes. To Diameter add thickness of metal, multiply sum by 10 times thickness, and product will give weight in Ibs. per foot of length. Weight of Faucet end is equal to 8 ins. of length of pipe. Hydrostatic 3?ress. To Compute Elements of a Hydrostatic Press. ure applied, W weight or resistance in Ibs., I and If lengths of lever and fulcrum in ins. or feet, and A and a areas of ram and piston in sq. ins. ILLUSTRATION. Areas of a ram and piston are 86.6 and i sq. ins., lengths oflever and fulcrum 4 feet and 9 ins., end power applied 20 Ibs. ; what is weight that may be sustained ? _ 20 X 4X 12 X 86.6 83 136 -!>n =-V= 9*37.3 . To Compute Thickness of Metal to Resist a given Pressure. RULE. Multiply pressure per sq. inch in Ibs. by diameter of cylinder in ins., and divide product by twice estimated tensile resistance or value of metal in Ibs. per sq. inch, and quotient will give thickness of metal required. EXAMPLE. Pressure required is 9000 Ibs. per sq. inch, and diameter of cylinder is 5.3 ins. ; what is required thickness of metal of cast iron? Value of metal is taken at 6000. 9 X5 ' 3 = ^-^ = 3. 975 in*. 6000X2 12000 Values of Different Metals in Tons. (Molesworth.) Cast iron ....... 41 1 Gun metal ...... 22 | Wrought iron. . .14 | Steel .......... 06 Hydraulic Rarru Useful effect of an Hydraulic Ram, as determined by Eytelwein, varied from .9 to .18 of power expended. When height to which" water is raised compared to fall is low, effect is greater than with any other machine ; but it diminishes as height increases. Length of supply pipe should not be less than .75 of height to which water is to be raised, or 5 times height of supply ; it may be much longer. To Compute Elements. efficiency. V and v representing volumes expended and raised, in cube feet per minute, h and h' heights from which water is drawn and elevated in feet, D and d diameters of supply and discharging pipes in ins., and IP effective horse-power. ILLUSTRATION. Heights of a fall and of elevation ar3 10 and 26.3 feet, and vol- umes expended and raised per minute are 1.71 and .543 cube feet .OOH3X 1.71 X 10 = .01933*; - r iQ I93 = i.7i cube feet; 1.45-^1.71 = 1.89 = .6g6 efficiency. 562 HYDRODYNAMICS. Results of Operations of Hydraulic Rams* Strokes perM. Fall. Eleva- tion. Wat Eipen'd. er Raised. Useful Effect. Strokes per M. Fall. Eleva- tion. Wai Expen'd. er Raised. Useful Effect. No. 66 5 2 36 3' Feet. 10. 06 9-93 6.05 5-o6 Feet. 26.3 38.6 38.6 38.6 C. Ft. 1.71 1-93 J-43 1.29 C. Ft. 543 .421 .169 "3 : 5 $ No. 15 10 Feet. 3.22 1.97 22.8 8-5 Feet. 38.6 38.6 196.8 52-7 C. Ft. 1.98 1.58 38 2 C. Ft. .058 .014 .029 .186 1 .67 57 NOTE. Volume of air vessel = volume of delivery pipe. One seventh of water may be raised to about 4 times head of fall, or one fourteenth 8 times, or one twenty- eighth ib times. WATER POWER. Water acts as a moving power, either by its weight or by its vis viva, and in latter case it acts either by Pressure or by Impact. Natural Effect or Power of a fall of water is equal to weight of its volume and vertical height of its fall. IfSvater is made to impinge upon a machine, the velocity with which it impinges may be estimated in the effect of the machine. Result or effect, however, is in nowise altered ; for in first case P = V w h, and in latter = V w. V representing volume in cube feet, w weight in Ibs., and v velocity of flow in feet per second. 62. 5 V h = P, and 3. 2 * a ^h = V. P representing pressure in Ibs. , a area of open- ing in sq.feet, and h height of flow in feet per second. To Cotnpnte Power of a Fall of "Water. RULE.- Multiply volume of flowing water in cube feet per minute by 62.5, and this product by vertical height of fall in feet. NOTE. When Flow is over a Weir or Notch, height is measured from surface of tail race to a point four ninths of height of weir, or to centre of velocity or pressure *f opening of flow. When Flow is through a Sluice or Horizontal Slit, height is measured from sur- face of tail-race to centre of pressure of opening. EXAMPLE. What is power of a stream of water when flowing over a weir 5 feet in breadth by i in depth, and having a fall of 20 feet from centre of pressure of flow? By Rule, page 533, -5X1 Vzgi X .625 = 16.72 cube feet per second. 16.68 X 60 X 62. 5 X 20 = i 251 ooo Ibs. , which -r- 33 ooo = 37.91 horses' power. Or, . 1135 V h = theoretical IP. h representing height from race in feet. ILLUSTRATION. If flow of a stream is 17.9 cube feet per second, to what height and area of flow of i foot in depth should it be dammed to attain a power of 10 horses. 33 ooo X 10 _ ^ p er secona ^ an $ 5500 _ 88 cube j ee i ^ secona _. 60 62 . 5 17.9 2 feet height. Hence, - .6 /2gx 1 = 3.2, and 17.9-^-3.2 = 5.59 sg. Water sometimes acts by its weight and vis viva simultaneously, by com- bining effect of an acquired velocity with fall through which it flows upon wheel or instrument. In this case / [h -\ \ 2 ) V X 62.5 = mechanical effect. * As determined by ~ C. HYDRODYNAMICS. 563 WATER-WHEELS. WATER- WHEELS are divided into two classes, Vertical and Horizontal. Vertical comprises Overshot, Breast, and Undershot ; and Horizontal, Turbine, Impact, or Reaction wheels. Vertical wheels are limited by construction to falls of less than 60 feet Turbines are applicable to falls of any height from i foot upward. Vertical wheels applied to a fall of from 20 to 40 feet give a greater effect than a Turbine, and for very low falls Turbines give a greater effect. Sluices. Methods of admitting water to an Overshot or Breast Wheel are various, consisting of Overfall, Guide-bucket, and Penstock. An Overfall Sluice is a saddle-beam with a curved surface, so as to direct the current of water tangentially to buckets; a Guide-bucket is an apron by which water is guided in a course tangential to buckets; and a Penstock is sluice-board or gate, placed as close to wheel as practicable, and of such thickness at its lower edge as to avoid a contraction of current. Bottom surface of penstock is formed with a parabolic lip. Sh.rondlng of a wheel consists of plates at its periphery, which form the sides of the bucket. Height of fall of a water-wheel is measured between surfaces of water in penstock and in tail-race, and, ordinarily, two thirds of height between level of reservoir and point at which water strikes a wheel is lost for all effective operation. Velocity of a wheel at centre of percussion of fluid should be from .5 to .6 that of flow of the water. Total effect in a fall of water is expressed by product of its weight and height of its fall. Ratio of Effective IPower of Water IVlotors. I fw *Q * < *, Undershot, Poncelet's, from. 6 to. 4 toi jfrom.68to.6 to x Undershot ' ] Lte.Jitoi Turbine " .6 to .8 to i * Breast " . 45 to. 65 to i Hydraulic Ram " .6 to i Water-pressure engine " .8 to i Oversliot-\vlieel. OVERSHOT-WHEEL. The flow of water acts in some degree by impact, but chiefly by its weight. Lower the speed of wheel at its circumference, the greater will be mechan- ical effect of the water, in some cases rising to 80 per cent. ; with velocities of from 3 to 6.5 feet, efficiency ranges from 70 to 75 per cent. Proper ve- locity is about 5 feet per second. Number of buckets should be as great, and should retain water as long, as practicable. Maximum effect is attained when the buckets are so numerous and close that water surface in the bucket commencing to be emptied should come in contact with the under side of the bucket next above it. Moles- worth gives 12 ins. apart. Curved buckets give greatest effect, and Radial give but .78 of effect of Elbow buckets. Wheel 40 feet in diameter should have 152 buckets. Small wheels give a less effect than large, in consequence of their greater centrifugal action, and discharging water from the buckets at an earlier period than with larger wheels, or when their velocity is lower. When head of water bears to fall or height of wheel a proportion as great as i to 4 or 5, ratio of effect to power is reduced. The general law there- fore is, that ratio of effect to power decreases as proportion of head to total head and fall increases. 564 HYDRODYNAMICS. Wheel with shallow Shrouding acts more efficiently than one where it is deep, and depth is usually made 10 or 12 ins., but in some cases it has been increased to 15. Breadth of a wheel depends upon capacity necessary to give the buckets to receive required volume of water. Form of Buckets. Radial buckets that is, when the bottom is a right line in- volve so great a loss of mechanical effect as to render their use incompatible with economy; and when a bucket is formed of two pieces, lower or inner piece is termed bottom or floor, and outer piece arm or wrist. Former is usually placed in a line with radius of wheel. Line of a circle passing through elbow, made by junction of floor and arm, is termed division circle, or bucket pitch, and it is usual to put this at one half depth of shrouding. 560 When arm of a bucket is included in division angle of buckets, that is, - , n representing number of buckets, the cells are not sufficiently covered, except for very shallow shrouding; hence it is best to extend arm of a bucket over 1.2 of division angle, so as to cover or overlap elbow of bucket next in advance of it. Construction of Buckets (Fig. i). Capacity of bucket should be 3 times volume of water. Fairbairn gives area of opening of a bucket in a wheel of great diameter, compared to the volume of it, as 5 to 24. Buckets having a bottom of two planes, that is, with two bottoms, and two division circles or bucket pitches and an arm, give a greater effect than with one bottom. When an opening is made in base of buckets, so as to afford an escape of air contained within, without a loss of water admitted, the buckets are termed ven- tilated, and effective power of wheel is much greater than with closed buckets. D = distance apart at periphery = d, d depth of shrouding, s length of radial start .33 d, I length of bucket curve = 1.25 d in large wheels, and i in wheels under 25 feet, a angle of radius of curve of bucket, with radial line of wheel at points of bucket = 15. (Molesworth.) To Compute Radius and. Revolutions of an Overshot- wheel, and Height of Fall of \Vater. When whole Fall and Velocity of Flow, etc., are given, he v 2 ,. . 3.1416 nr h and i -f cos. a = c. h representing height of whole 3.1416 r fall, h' height between the centre of gravity of discharge and half depth of bucket upon which water flows, v velocity of flow in feet per second, a angle which point of entrance of water into a bucket makes with summit of wheel, n number of revolutions per minute, c velocity of wheel at its circumference per second, and r its radius. NOTE. Height of whole fall is distance between surface of water in flume and point at which lower buckets are emptied of water, and as a proportion of velocity of flow is lost, it is proper to assume height h' as above given. ILLUSTRATION. A fall of water is 30 feet, velocity of its flow is 16 feet per second, angle of its impact upon buckets is 12, and required velocity of wheel is 8 feet per -- number of revolutions, and height of fall upon second; what is required radius, wheel? 30X8 3.1416 X 12.95" ^ HYDRODYNAMICS. 565 When Number of Revolutions and Ratio between Velocities ef Flow and at Circumference of Wheel are given. y/.ooo 772 (x n) 2 h + (i -f cos, a) 2 i -f- cos, a) _ _v ^ 3.1416 nr ~~ .000386 (3571)2 ~ - C ' io" ILLUSTRATION. If number of revolutions are 5, x = 2, and fall, etc. , as in previous case; what is radius of wheel, velocity of flow, and height of fall? -518 3 ' 4 J = .000386 (2 Xs) 2 .0386 3.1416x5X13-41 _ 7 . 03/ee< Hence 7> o 3 x 2 = 14.06 velocity of flow, and I4 ' 62 30 64.33 X i i=3. 3 8/eet To Compute Width. of an Overshot-wlieel. C V ? = w. C representing a coefficient = 3, when buckets are jilted to an excess, and 5 when they are deficiently filled, V volume of water in cube feet per second, s depth of shrouding, w width of buckets, both in feet, and c' velocity of wheel at centre of shrouding, in feet per second. ILLUSTRATION A wheel is to be 31 feet in diameter, with a depth of shrouding of i foot, and is required to make 5 revolutions per minute under a discharge of 10 cube feet of water per second; what should be width of buckets? Assume C = 4 , and c' = 3 - I ~ I X j* I4 * 6 X 5 = 7.854- Then - 4 - x - IO _ 5.09/5* 60 i X 7-054 To Compute Number of Buckets. 7 ( i + -|- J -=- 12 = d, and ** S = n. D representing diameter of wheel, d dis- tance between centres of buckets, in feet, and n number of buckets. ILLUSTRATION Take elements of preceding case. Then 7 (, + -|-) = 7 X 2.2 + 12 ^ 1.283, and 3I ~* x ^ l6 X ' = 73 . 4> say 7 . 060 buckets, hence - = 5, angle of subdivision of buckets. To Compnte Kffect of an Oversliot-\vlieel. -- = P. w representing weight of cube foot of water in Ibs., v" velocity of it discharged at tail of wheel, in feet per second, V volume of flow in cube feet, and f friction of wheel in Ibs. ILLUSTRATION A volume of 12 cube feet per second has * fall of 10 feet, wheel using but 8.5 feet of it, and velocity of water discharged is o feet per second; what is effect of fall ? Friction of wheel is assumed to be 750 Ibs. = 6375 _ (r . 26 x 75o+75 o) ^ .,680 = I2XIOX62.5 7500 7500 . 624 = ratio of effect to power ; and 4680 X 60 seconds H- 33 ooo 8. 5 1 IP. To Compute 3?ower of an. Overshot--wh.eel. RULE. Multiply weight of water in Ibs. discharged upon wheel in one minute by height or distance in feet from centre of opening in gate to sur- face of tail-race ; divide product by 33 ooo, and multiply quotient by as- sumed or determined ratio of effect to power. Or, for general purposes, divide product by 50 ooo, and quotient is H*. Or, .0852 V h = IP, and ^-^ = V per second; or, ~- = V per minute. 3B 566 HYDRODYNAMICS. Mechanical Effect of water is product of its weight into height from which it falls. EXAMPLE. Volume of water discharged upon an overshot- wheel is 640 cube feet per minute, and effective height of fall is 22 feet; what is IP? = 26.67, which, x .75 = assumed ratio of effect to power 20 IP. Useful Effect of an Oversliot-\vlieel. With a large wheel running in most advantageous manner, .84 of power may be taken for effect. Velocity of a wheel bears a constant ratio, for maximum effects, to that of the flowing water, and this ratio is at a mean .55. Ratio of effect to power with radial-buckets is .78 that of elbow-buckets. Ratio of effect decreases as proportion of head to total head and fall increases. Thus, a wheel 10 feet in diameter gave, with heads of water above gate, ranging from .25 to 3.75 feet, a ratio of effect decreasing from .82 to .67 of power. Higher an overshot-wheel is, in proportion to whole descent of water, greater will be its effect. Effect is as product of volume of water and its perpendicular height. Weight of arch of loaded buckets in Ibs. is ascertained by multiplying .444 of their number by number of cube feet in each, and that product by 40. TJnd.ersh.ot-wh.eel. UNDERSHOT-WHEEL is usually set in a curb, with as little clearance for escape of water as practicable ; hence a curb concentric to this wheel is more effective than one set straight or tangential to it. Computations for an undershot-wheel and rules for construction are near- ly identical with those for a breast-wheel. Buckets are usually set radially, but they may be inclined upward, so as to be more effectively relieved of water upon their return side, and they are usually filled from .5 to .6 of their volume. Depth of shrouding should be from 15 to 18 ins., in order to prevent overflow of water within the wheel, which would retard it. Velocity of periphery should equal theoretical velocity due to head of water x .57. NOTE. When constructed without shrouding, as in a current- wheel, etc., buckets become blades. Sluice-gate should be set at an inclination to plane of curb, or tangential to wheel, in order that its aperture may be as close to wheel as practicable ; and in order to prevent partial contraction of flow of water, lower edge of sluice should be rounded. Effect of an undershot-wheel is less than that of a breast-wheel, as the fall available as weight is less than with latter. To Compute I > o\ver of* an Under shot- wheel. Proceed as per rule for an overshot-wheel, using 93 750 for 50000, and .4 for .75. Or, V h . ooo 66 = IP ; or, ^| = V. V representing volume of water in cube n feet per minute, and h head of water in feet. HYDRODYNAMICS. 567 IPoncelet's "Wheel. PONCELET'S WHEEL. Buckets are curved, so that flow of water is in course of their concave side, pressing upon them without impact; and effect is greater than when water impinges at nearly right angles to a plane sur- face or blade. This wheel is advantageous for application to falls under 6 feet, as its effect is greater than that of other undershot wheels with a curb, and for falls from 3 to 6 feet its effect is equal to that of a Turbine. For falls of 4 feet and less, efficiency is 65 per cent., for 4.25 to 5 feet, 60 per cent., and from 6 to 6.5 feet, 55 to 50 per cent. In its arrangement, aperture of sluice should be brought close to face of wheel. First part of course should be inclined from 4 to 6 ; remainder of course, which should cover or embrace at least three buckets, should be car- ried concentric to wheel, and at end of it a quick fall of 6 ins. made, t guard against effect of back-water. Sluice should not be opened over i foot in any case, and 6 ins. is a suitable height for falls of 5 and 6 feet. Distance between two buckets should not exceed 8 or 10 ins., and radius of wheel should not be less than 40 ins., or more than 8 feet. Plane of stream or head of water should meet periphery of wheel at an angle of from 24 to 30, Space between wheel and its curb should not ex- ceed .4 of an inch. Depth of shrouding should be at least .25 depth of head of water, or such as to prevent water from flowing through it and over the buckets, and width of wheel should be equal to that of stream of impinging water. Effect of this wheel increases with depth of water flow, and, therefore, other elements being equal, as filling of buckets, to obtain maximum effect, water should flow to buckets without impact, and velocity of wheel should be only a little less than half that of velocity of water flowing upon wheel. To Compute Proportions of a, I*oiioelet Wheel. NOTE. As it is impracticable to arrive at the results by a direct formula, they must be obtained by gradual approximation. EXAMPLE. Height of fall is 4.5 feet; volume of water 40 cube feet per second; radius of wheel = 2 h, or 9 feet; depth of the stream =.j 5 feet; and C assumed at. 9. V representing volume of water in cube feet per second, h height of fall, d depth of shrouding == . 1- d' ; d' opening of and e width of sluice, r radius of curva- ture of buckets , and a of wheel, all in feet ; n number of revolutions = ^-? cos. z pa per minute; c velocity of circumference of wheel and v velocity of water, both in feet per second ; C coefficient of resistance of flow of water ; x angle between plant of flowing water and that of circumference of wheel at point of contact, sin. of - = Vcos. z; z angle made by circumference of wheel with end of buckets = 2 tang, y; and y angle of direction of water from circumference of wheel =. I . V* + I . J*. Then v = -9^/2 g ( h J = .9 x 16.29 = J 4-66 feet :. velocity of wheel, being 568 HYDRODYNAMICS. less than half velocity of water ; c = '4-66 .66 _ ^ ^ ^ d = - x > x .25, angle corresponding to which = 14 30'; n = 3 7 =7.43 revolutions; z = 2 tang, i/ = 2 X- 258 62 = .517 24 .-. * = 2 7 2o'; e = ^ 66 = 3.63 feet; r - LJL __ '-5 _ x 8 ^ . x _ gin * _ ^/QQS'Z _ Vcos. 27 20' = .943 COS. 27 20 .88835 2 := sin. of 70 34' /. x = 141 8'. Effect is a maximum when c = .5 v cos. y. Fig. 2. Construction of Buckets (Fig. 2). (Molesworth.) From point of bucket, a, draw a line, a 6, at an angle of 26 with radial line, point 6, where this line cuts an imaginary cir- cle, drawn at a distance of s x 1.17 from periphery of wheel, is centre from which bucket is struck with radius, 6 a. Radius of wheel should not be less than 7, or more than 16 feet. Curb should fit wheel accurately for 18 or 20 ins., measured back from perpendicular line which passes through axis of wheel, the breast should then incline i in 10, or i in 15 towards sluice. After passing axis of wheel in tail-race, curb should make a sudden dip of 6 ins. To Compute I?o~wer of a IPoncelet "Wheel. 880 TP V h .001 13 = EP, and - = V. V = velocity of theoretical periphery = . 55. * Number of buckets 1.6 D-f- 1.6, D = diameter of wheel in feet. Shrouding .33 to .5 depth of head of water, and D = 2 h, and not less than 7 or more than 16 feet. Breast-wheel. BREAST-WHEEL is designed for fails of water varying from 5 to 15 feet, and for flows of from 5 to 80 cube feet per second. " It is constructed with either ordinary buckets or with blades confined by a Curb. Enclosure within which water flows to a breast-wheel as it leaves the sluice is termed a Curb or Mantle. When blades are enclosed in a cur J, they are not required to hold water ; hence they may be set radial, and they should be numerous, as the loss of water escaping between the wheel and the curb is less the greater their num- ber ; and that they may not lift or carry up water with them from tail-race, it is proper to give them such a plane that it may leave the water as nearly vertical as may be practicable. Distance between two buckets or blades should be from 1.3 to 1.5 times head over gate for low velocity of wheel and more for a high velocity, or equal to depth of shrouding, or at from 10 to 15 ins. It is essential that there should be air-holes in floor of buckets, to prevent air from impeding flow of water into them, as the water admitted is nearly as deep as the interval between them ; and velocity of wheel should be such that buckets should be filled to .5 or .625 of their volume. .When wheels are constructed of iron, and are accurately set in masonry, a clearance of .5 of an inch is sufficient. * -\/2gh in ft* ftf tecond. HYDRODYNAMICS. 569 High Breast-wheel is used when level of water in tall-race and penstock or forebay are subject to variation of heights, as wheel revolves in direction in which water flows from blades, and back-water is therefore less disad- vantageous, added to which, penstocks can be so constructed as to admit of an adjustable point of opening for the water to flow upon the wheel. Effect of this wheel is equal to that of the overshot, and in some instances, from the advantageous manner in which water is admitted to it, it is greater when both wheels have same general proportions. Under circumstances of a variable supply of water, Breast-wheel is better designed for effective duty than Overshot, as it can be made of a greater diameter; whereby it affords an increased facility for reception of water into its buckets, also for its discharge at bottom ; and further, its buckets more easily overcome retardation of back-water, enabling it to be worked for a longer period in back-water consequent upon a flood. In a well-constructed wheel an efficiency of 93 per cent, was observed by M. Morin, and Sir Wm. Fairbairn gives, at a velocity of circumference of wheel of 5 feet, an efficiency of 75 per cent. Velocity usually adopted by him was from 4 to 6 feet per second, both for high and low falls; a minimum of 3.5 feet for a fall of 40 and a maximum of 7 feet for a fall of 5 to 6 feet. When water flows at from 10 to 12 above horizontal centre of wheel, Fairbairn gives area of opening of buckets, compared with their volume, as 8 to 24. The capacity between two buckets or blades should be very nearly double that of volume of water expended. To Compute Proportions and Effect of a Breast-awheel. ILLUSTRATION. Flow of water is 15 cube feet per second ; height of fall, measured from centre of pressure of opening to tail race, is 8.5 feet; velocity of circumference of wheel 5 feet per second ; and depth of buckets or blades i foot, filled to .5 of their volume. Width of wheel = , d representing depth, and v velocity of buckets ; = 3 ; and as buckets are but . 5 filled, 3 -r- . 5 = 6 feet. Assume water is to flow wi th double velocity of circumference of wheel ; v = 5 X 2 = 10 feet ; and fall required to gen- erate this velocity = X i.i ^' 7 X i-i = 1.71 feet. Deducting this height from total fall, there remains for height of curb or shroud- ing, or fall during which weight of water alone acts, h h' = 8. 5 1.71 = 6.79 feet. Making radius of wheel 12 feet, and radius of bucket circle u feet, whole mechan- ical effect of flow of water = 15 x 62.5 X 8.5 = 7968.75 Ibs., from which is to be de- ducted from 10 to 15 per cent, for loss of water by escape. Theoretical effect, as determined by M. Morin, velocity of circumference about .5 of that of water, and within velocities of 1.66 to 6 feet. (( i_^_j_ ft"\ v 62.5. a representing angle of direction of velocity with which water flows to wheel at centre of thread of flow and direction of velocity of wheel at this line, and h" h h' in feet. a is here assumed at 20. See Weisbach, London, 1848, vol. ii. page 197, and for the necessarily small value of a, its cosine may be taken at i. Cos. 20 = .94. Then ^ IOX '94 5) 5_|_ 6 \ x Ig x 6a>5 _ ? 474 x I$ x 62 5 _ 700 6. 9 ibs., which \ 32.16 / is to be reduced by a coefficient of .77 for a penstock cluice, and .8 for an overfall sluice. Theoretical effect, as determined by Weisbach, 7273 Ibs., from which are to be deducted losses, which he computes as follows : Loss by escape of water between wheel and curb = 916 Loss by escape at sides of wheel and curb = 180 Friction and resistance of water = 2. 5 per cent = 160 1256 t. 3 B* HYDRODYNAMICS. Friction of wheel as per formula, page 571, = W r n G .0086; a= .048 I = .048 / - = 4. 36 ins. : and n = - - -- - = 4 revolutions. C = .08. 4 V 2 12X2X3-1416 r= 4. 36-7- 2 = 2.18. Then i6sooX 2.18 X 4 X .08 X .0086 = 98.99 Ibs. Whence, - - " = .72 efficiency, upon assumption of losses as com- puted by Weisbach. To Compute IPower of a Breast-wliee!. RULE. Proceed as per rule for an overshot-wheel, using 55 ooo and .65 with a high breast, and 62 500 and .6 for a low breast. Or, High breast, .0612 V h = IP, and ^ = V ; and Low breast .0546 V h = a-, = . ILLUSTRATION. Assume elements of preceding case. Then - 5 >< 62 -5X 8.5X60 33000 = 14.49, which X -7 = 10. 14 horses. 7006.0 1256-1-102.6x60 Or, '- - - - - = 10.27 horses. 33000 Openings of Buckets or Blades. High Breast, .33 sq. foot, and Low Breast, .2 sq. foot for each cube foot of their volume, or generally 6 to 8 in opening in a high breast and 9 to 12 in a low breast. Forms of Buckets. Two Part. d D, s = .$ d, I 1.25 d in large wheels, and =d in wheels less than 25 feet in diameter. Three Part Buckets. d divided into 3 equal parts; I = .25 d, d = D, s = .33 d, I = d in large wheels, and .75 d in wheels less than 25 feet in diameter. Ventilating Buckets (Fairbairn's). Spaces are about i inch in width. NOTES. A Committee of the Franklin Institute ascertained that, with a high breast- wheel 20 feet in diameter, water admitted under a head of 9 ins., and at 17 feet above bottom of wheel, elbow-buckets gave a ratio of effect to power of .731 at a maximum, and radial-blades .653. With water admitted at a height of 33 feet 8 ins., elbow-buckets gave .658, and radial blades .628. At 10.96 feet above bottom of wheel, with a head of 4.29 feet, elbow-buckets gave . 544, and blades . 329. At 7 feet above bottom of wheel, and a head of 2 feet, a low breast gave for elbow-buckets .62, and for blades .531. At 3 feet 8 ins. above bottom of wheel, and a head of i foot, elbow-buckets gave .555, and blades .533. Current- wheel. CURRENT-WHEEL. D. K. Clark assigns the most suitable ratio of veloc- ity of blades to that of current as 40 per cent. Depth of blades should be from .25 to .2 of radius ; it should not be less than 12 or 14 ins. Diameter is usually from 13 to 16.5 feet, with 12 blades ; but it is thought that there might be an advantage in applying 18 or even 24. The blades should be completely submerged at lower side, but not more than 2 ins. under water, and not less than 2 at one time. (v s) 2 = H*- a representing area of vertical section of immersed blades in 150 tq. feet, s velocity of wheel at circumference, and v of stream, both in feet per second. Or, .38 V 62. 5 = useful effect. Hence, efficiency = . sa HYDRODYNAMICS. 5/1 ITTu.tter-\vlieel. Flutter or Saw-mitt Wheel Is a small, low breast-wheel operating under a high head of water ; the design of its construction, water being plenty, is the attainment of a simple application to high-speed connections, as a gang or circular saw. In effect it is from .6 to .7 that of an overshot-wheel of like head of fall. V s (v s) = H?. v and s as preceding. 150 Friction of Journals or Grudgeons. A very considerable portion of mechanical effect of a wheel is lost in ef- fect absorbed by friction of its gudgeons. To Compute Friction, of* Journals or Gudgeons of a \Vater-\vheel. W rn C .0086 =/ W representing weight of wheel in Ibs., r radius of gudgeon in ins., and n number of revolutions of wheel per minute. For well-turned surfaces and good bearings, C = .o75 with oil or tallow; when best of oil is well supplied = .054; and, as in ordinary circumstances, when a black- lead unguent is alone applied = . u. ILLUSTRATION. A wheel weighing 25000 Ibs. has gudgeons 6 ins. in diameter, and makes 6 revolutions per minute; what is loss of effect? Assume C = .08. Then 25000 X - X 6 X .08 X .0086 = 309.6 Ibs. "Weights. Iron wheels of 18 to 20 feet in diameter will weigh from 800 to jooo Ibs. per H?. Wood wheels of 30 feet in diameter, 2000 to 2500 Ibs. per H*. To Compute Diameter and Journals of a Shaft, Stress laid uniformly along its Length. tXwl /IP Cast Iron, = d. Wood, 6. 12 3/ = d. W representing weight or load in Ibs., I length of shaft between journals in feet, and d diameter of shaft in its body in ins. Journals or Gudgeons. Cast Iron, .048 / =d. When Shaft has to resist both Lateral and Torsional Stress. Ascertain the diameter for each stress, and cube root of sum of their cubes will give diameter. To Compute Dimensions of Arms. Cast Iron, ^ = w. d representing diameter of shaft, and w width of arm, both yn in ins., n number of arms, = t, and t thickness of arm. When Arm is )f Oak, w should be 1.4 times that of iron, and thickness .7 that of width. Memoranda. A volume of water of 17. 5 cube feet per second, with a fall of 25 feet, applied to an undershot-wheel, will drive a hammer of 1500 Ibs. in weight from 100 to 120 blows per minute, with a lift of from i to 1.5 feet.* A volume of water of 21.5 cube feet per second, with a fall of 12.5 feet, applied to a wheel having a great height of water above its summit, being 7.75 feet in diame- ter, will drive a bammer of 500 Ibs. in weight 100 blows per minute, with a lift of a feet 10 ins. Estimate of power 31. 5 horses. * Volume of water required for a hammer increases in a much greater ratio than velocity to be given to it. it being nearly as cube of velocity. 5 72 HYDRODYNAMICS. A Stream and Overshot Wheel of following dimensions viz., height of head to centre of opening, 24.875 ins. ; opening, 1.75 by 80 ins. ; wheel, 22 feet in diameter by 8 feet face; 52 buckets, each i foot in depth, making 3.5 revolutions per minute , drove 3 run of 4.5 feet stones 130 revolutions per minute, with all attendant ma- chinery, and ground and dressed 25 bushels of wheat per hour. 4.5 bushels Southern and 5 bushels Northern wheat are required to make i bar- rel of flour. A Breast-wheel and Stream of following dimensions viz., head, 20 feet; height of water upon wheel, 16 feet; opening, 18 feet by 2 ins. ; diameter of wheel, 26 feet 4 ins. ; face of wheel, 20 feet 9 ins. ; depth of buckets, 15.75 ins. ; number of buck- ets, 70; revolutions, 4.5 per minute drove 6144 self-acting mule spindles; 160 looms, weaving printing-cloths 27 ins. wide of No. 33 yarn (33 hanks to a Ib. ), and producing 24000 hanks in a day of n hours. Horizontal "Wheels. In horizontal water-wheels, water produces its effect either by Impact, Pressure, or Reaction, but never directly by its weight. These wheels are therefore classed as Impact, Pressure, and Reaction, but are now designated by the generic term of Turbine. Tu.r~bin.es. TURBINES, being operated at a higher number of revolutions than Ver- tical Wheels, are more generally applicable to mechanical purposes ; but in operations requiring low velocities, Vertical Wheel is preferred. For variable resistances, as rolling-mills, etc., Vertical Wheel is far preferable, as its mass serves to regulate motion better than a small wheel. In economy of construction there is no essential difference between a Vertical Wheel and a Turbine. When, however, fall of water and volume of it are great, the Turbine is teast expensive. Variations in supply of water affect vertical wheels less than Turbines. Durability of a Turbine is less than that of a Vertical Wheel ; and it is indispensable to its operation that the water should be free from sand, silt, branches, leaves, etc. With Overshot and Breast Wheels, when only a small quantity of water is available, or when it is required or becomes necessary to produce only a por- tion of the power of ihsfall, their efficiency is relatively increased, from the blades being but proportionately filled ; but with Turbines the effect is con- trary, as when the sluice is lowered or supply decreased water enters the wheel under circumstances involving greater loss of effect. To produce maximum effect of a stream of water upon a wheel, it must flow without im- pact upon it, and leave it without velocity ; and distance between point at which the water flows upon a wheel and level of water in reservoir should be as short as practicable. Small wheels give less effect than large, in consequence of their making a greater number of revolutions and having a smaller water arc. In High-pressure Turbines reservoir (of wheel) is enclosed at top, and water is admitted through a pipe at its side. In Low-pressure, water flows into res- ervoir, which is open. In Turbines working under water, height is measured from surface of water in supply to surface of discharged water or race ; and when they work in air, height is measured from surface in supply to centre of wheel. In order to obtain maximum effect from water, velocity of it, when leav- ing a Turbine, should be the least practicable. HYDRODYNAMICS. 573 Efficiency is greater when sluice or supply is wide open, and it is less af- fected by head than by variations in supply of water. It varies but little with velocity, as it was ascertained by experiment that when 35 revolutions gave an effect of .64, 55 gave but .66." When Turbines operate under water, the flow is always full through them ; hence they become Reaction-wheels, which are the most efficient. Experiments of Morin gave efficiency of Turbines as high as .75 of power. Angle of plane of water entering a Turbine, with inner periphery of it, should be greater than 90, and angle which plane of water leaving reservoir makes with inner circumference of Turbine should be less than 90. When Turbines are constructed without a guide curve*, angle of plane of flowing water and inner circumference of wheel = 90. Great curvature involves greater resistance to efflux of water ; and hence it is advisable to make angle of plane of entering water rather obtuse than acute, say 100 ; angle of plane of water leaving, then, should be 50, if in- ternal pressure is to balance the external ; and if wheel operates free of water, it may be reduced to 25 and 30. If blades are given increased length, and formed to such a hollow curve that the water leaves wheel in nearly a horizontal direction, water then both impinges on blades and exerts a pressure upon them; therefore effect is greater than with an impact-wheel alone. Turbines are of three descriptions : Outward, Downward, and Inward flow. Ou.tward.-fi.ow Turbines. FOURNEYRON TURBINE, as recently constructed, may be considered as one of the most perfect of horizontal wheels; it operates both in and out of back-water, is applicable to high or low falls, and is either a high or low pressure turbine. In high-pressure, the reservoir is closed at top and the water is led to it through a pipe. In low-pressure, the water flows directly into ,an open res- ervoir. Pressure upon the step is confined to weight of wheel alone. Foumeyron makes angle of plane of water entering = 90, and angle of plane of water leaving = 30. Efficiency is reduced in proportion as sluice is lowered, for action of water on wheel is less favorably exerted. M. Morin tested a Fourneyron turbine 6.56 feet in diameter, and he found that efficiency varied from a minimum of 24, to 79 per cent., when supply of water was reduced to .25 of full supply. In practice, radial length of blades of wheel is .25 of radius, for falls not ex- ceeding 6.5 feet, .3 for falls of from 6.5 to 19 feet, and .66 for higher falls. To Compnte Elements and. Results. High Pressure, 6.6 ^/h = v: ^ = A; ^ I>77 V =Dt; 12.6 = V: and v -\/h h ,079 V h = IP. h representing head of water, v velocity of turbine at periphery per minute, and D internal diameter of turbine, all in feet, V volume of water in cube feet per second, A sum of area of orifices in sq.feet, aria IP effective horse-power. 1.2 D = external diameter of turbine in feet, when it is more than 6 feet, and 1.4 when it is less than 6 feet. Number of guides = number of blades J when less than 24, and number -H 3 when greater than 24. Area of section of supply-pipe = .4 V. For construction of blades and guides, see Molesworth, London, 1882, page 540. * Guide curves are plates upon centre body of a Turbine, which give direction to flowing water, or to blades of wheel wnich surround them. f In extreme cases of very high falls diameter given by this formula may be increased. t Fourneyron ' rule for the number of blades it constant number 36, irrespective of size of turbine. 574 HYDRODYNAMICS. Operation of High-Pressure Turbines. 30 4.2 36 60 70 80 1.6 59 1.27 66 1.05 73 140 78' 9 1 80 200 .63 94 h = head of water in feet, V volume of water in cube feet required for each 10 IP, and v velocity of periphery of turbine in feet per second. BOYDEN TURBINE. Mr. - Boy den, of Massachusetts, designed an outward-flow turbine of 75 IP, which realized an efficiency of 88 per cent. Peculiar features, as compared with a Fourneyron turbine, are, ist, and most important, the conduction of the water to turbine through a vertical trun- cated cone, concentric with the shaft. The water, as it descends, acquires a gradually increasing velocity, together with a spiral movement in direction of motion of wheel. The spiral movement is, in fact, a continuation of the motion of the water as it enters cone. 2d. Guide-plates at base are inclined, so as to meet tangentially the approaching water. 3d. A " diffuser," or annu- lar chamber surrounding wheel, into which water from wheel is discharged. This chamber expands outwardly, and, thus escaping velocity of water, is eased off and reduced to a fourth when outside of diffuser is reached. Effect of diffuser is to accelerate velocity of water through machine ; and gain of efficiency is 3 per cent. Diffuser must be entirely submerged. (D. K. Clark.) PONCELET TURBINE. This wheel is alike to one of his undershot-wheels set horizontally, and it is the most simple of all horizontal wheels. To Compu.te Elements of Q-eneral Proportion and. Results. (Lt.F.A.Mahan,U.S.A.) J) > -5l> 2 V^ = V; .1 D = H; '4.49^=1); 3 (D+io)=N; = u;; = W; D- = d V and C coefficient for V in terms of V= -. D and d representing exterior and in- terior diameters of wheel, H and h heights of orifices of discharge at outer circum- ference and of fall acting on wheel, w and w shortest distances between two adjacent blades and two adjacent guides, all in feet, V, V, and v velocities due to fall of water passing through narrowest section of wheel, and of interior circumference of wheel, all in feet per second, N and n numbers of blades and guides, and IP actual horse- power. For falls of from 5 feet to 40, and diameters not less than 2 feet, n w should be equal to diameter of wheel. H equal to . i D, n w' = d, and 4 w = width of crown. For falls exceeding this, H should be smaller, in proportion to diameter of wheel. Downward-flow Turbines. In turbines with downward flow, wheel is placed below an annular series of guide-blades, by which water is conducted to wheel. The water strikes curved blades, and falls vertically, or nearly so, into tail-race ; consequently, centrifugal action is avoided, and downward flow is more compact. FONTAINE TURBINE yields an efficiency of 70 per cent., when fully charged. When supply of water is shut off to .75, by sluice, efficiency is 57 per cent. Best velocity at mean circumference of wheel is equal to 55 per cent, of that due to height of fall. It may vary .25 of this either way, without materially affecting efficiency. In operation the water in race is in immediate contact with wheel, and its efficiency is greatest when sluice is fully opened. Its efficiency, also, is less affected by variations of head of flow than in volume of water supplied; hence they are adapted for Tide-mills. HYDRODYNAMICS. 575 JONVAL TURBINE. This wheel is essentially alike in its principal propor- tions to Fontaine's, and in principle of operation it is the same. Water in race must be at a certain depth below wheel. For convenience, it is placed at some height above level of tail-race, within an air-tight cylinder, or " draft-tube," so that a partial vacuum or reduction of pressure is induced under wheel, and effect of wheel is by so much in- creased. Resulting efficiency is same as if wheel was placed at level of tail- race ; and thus, while it may be placed at any level, advantage is taken of whole height of fall, and its efficiency decreases as volume of water is di- minished or as sluice is contracted. To Compute Elements and. Results. Low Pressure. For falls of 30 feet and less. V Vi.77 V IP 6 h representing head of water, v velocity of turbine at periphery per minute, and D internal diameter of turbine, all in feet, V volume of water in cube feet per second, A sum of area of orifices in sq.feet, and IP effective horse-power. 1.2 Dm external diameter of turbine in feet, when it is more than 6 feet, and 1.4 when it is less than 6 feet. Number of guides = number of blades t when less than 24, and number -+- 3 when greater than 24. Area of section of supply-pipe = .4 V. For construction of blades and guides, see Molesworth, London, 1882. page 540. ire Turbines. (Molesworth.) 5l P 10 HP IS rP 20 H? 30 IP 40 H> 50 ff 1 v V R V R V R V R V R V R V R n *R IOO 2-5 5 n.38 12.5 if 25 57 38 47 50 41 75 33 IOO 28 126 26 7-5 16.38 8. S 136 17 35 79 33 68 5i 56 68 48 8s 43 10 18.96 6.3 1 80 12.6 128 19 105 25 go 38 75 50 6 4 63 58 IS 20 23.22 26.82 4.2 319 8.4 6.3 226 329 12.6 9-3 i85 273 Hi 232 25 18.9 IQ 4 33 2.S "3 164 42 li 100 148 2 5 30 7-5 35 10 310 15 253 20 220 25 196 3 32.88 8.4 380 12.6 310 '7 268 21 240 v representing velocity of centre of blades in feet and V volume of water, in cube feet, both per second, R revolutions per minute, and IP effective horse-power. Vertical Shaft. 3 /-^n = diameter of sJiqft in ins. Inward-flow Turbine. INWARD-FLOW TURBINE. Inward-flow or vortex wheel is made with radiating blades, and is surrounded by an annular case, closed externally, and open internally to wheel, having its inner circumference fitted with four curved guide-passages. The water is admitted by one or more pipes to the case, and it issues centripetally through the guide-passages upon circum- ference of wheel. The water acting against the curved blades, wheel is driven at a velocity dependent on height of fall, and water having expended its force, passes out at centre. This wheel has realized an efficiency as high as 77.5 per cent. It was originally designed by Prof. James Thomson. SWAIN TURBINE. Combines an inward and a downward discharge. Re- ceiving edges of buckets of wheel are vertical opposite guide-blades, and lower portions of the edges are bent into form of a quadrant. Each bucket thus forms, with the surface of adjoining bucket, an outlet which combines an inward and a downward discharge. One, 72 ins. in diameter, was tested * In extreme cases of very high falls diameter given by this formula may be increased. t Pourneyron's rule for the number of blades is constant nmmber 36, irrespective of site of turbine. 576 HYDEODYNAMICS. by Mr. J. B. Francis, for several heights of gate or sluice, from 2 to 13.08 ins., and circumferential velocities of wheel ranging from 60 to 80 per cent, of respective velocities due to heads acting on wheel. For a velocity of 60 per cent., and for heights of gate varying within limits al- ready stated, efficiency ranged from 47.5 to 76.5 per cent., and for a velocity of 80 per cent, it ranged from 37.5 to 83 per cent. Maximum efficiency attained was 84 per cent, with a i2-inch gate and a velocity-ratio of 76 per cent. ; but from g-inch to i3-inch gate, or from .66 gate to full gate, maximum efficiency varied within very narrow limits from 83 to 84 per cent , velocity-ratios being 72 per cent, for 9-inch gate, and 76.5 per cent, for full gate. At half-gate, maximum efficiency was 78 per cent., when velocity-ratio was 68 per cent. At quarter-gate, maximum effi- ciency was 61 per cent., and velocity-ratio 66 per cent TREMONT TURBINE, as observed by Mr. Francis, in his experiments at Lowell, Mass., gave a ratio of effect to power as .793 to i. VICTOR TURBINE is alleged to have given an effect of .88 per cent, under a head of 18.34 feet, with a discharge of 977 cube feet of water per minute, and with 343.5 revolutions. Tangential Wheel. Wheels to which water is applied at a portion only of the circumference are termed tangential. They are suited for very high falls, where diameter and high tangential velocity may be combined with moderate revolutions. The Girard turbine belongs to this class. It is employed at Goeschenen station for St. Gothard tunnel , it operates under a head of 279 feet. The wheels are 7 feet 10.5 ins. in diam., having 80 blades, and their speed is 160 revolutions per minute, with a maximum charge of water of 67 gallons per second. An efficiency of 87 per cent, is claimed for them at the Paris water-works ; ordinarily it is from 75 to 80 per cent. (D. K. Clark.) Impact and. Reaction Wheel. IMPACT-WHEEL. Impact Turbine is most simple but least efficient form of impact-wheel. It consists of a series of rectangular buckets or blades, set upon a wheel at an angle of 50 to 70 to horizon ; the water flows to blades through a pyramidal trough set at an angle of 20 to 40, so that the water impinges nearly at right angles to blades. Effect is .5 entire me- chanical effect, which is increased by enclosing blades in a border or frame. If buckets are given increased length, and formed to such a hollow curve that the water leaves wheel in nearly a horizontal direction, the water then impinges on buckets and exerts a pressure upon them ; effect therefore is greater than with the force of impact alone. By deductions of Weisbach it appears that effect of impact is only half available effect under most favorable circumstances. REACTION-WHEEL. Reaction of water issuing from an orifice of less capacity than section of vessel of supply, is equal to weight of a column of water, basis of which is area of orifice or of stream, and height of which is twice height due to velocity of water discharged. Hence, the expression is 2. a w = R. to representing weight of a cube foot of water in Z6*., and a area of opening in sq.feet WHITELAW'S is a modification of Barker's ; the arms taper from centre towards circumference and are curved in such a manner as to enable the water to pass from central openings to orifices in a line nearly right and radial, when instrument is operating at a proper velocity ; in order that very little centrifugal force may be imparted to the water by the revolution of the arms, and consequently a minimum of frictional resistance is opposed to course of the water. HYDRODYNAMICS. 577 A Turbine 9.55 feet in diameter, with orifices 4.944 ins. in diameter, oper- ated by a fall of 25 feet, gave an efficiency of 75 per cent , including friction of gearing of an inclined plane. When a reaction wheel is loaded, so that height due to velocity, corresponding to velocity of rotation v, is equal to fall, or = ft, or v = Vzgh, there is a loss of 17 v 2 per cent, of available effect; and when = 2 h, there is a loss of but 10 per cent. ; v 2 and when t- = 4 h, there is a loss of but 6 per cent. Consequently, for moderate falls, and when a velocity of rotation exceeding velocity due to height of fall may be adopted, this wheel works very effectively. Efficiency of wheel is but one half that of an undershot- wheel. When sluice is lowered, so that only a portion of wheel is opened, efficiency of a Reaction-wheel is less than that of a Pressure Turbine. Ratio of Effect to Power of several Turbines is as follows: Poncelet 65 to 75 to i I Jonval , 6 to 7 to i Fourneyron 6 to 75 to i J Fontaine 6 to 7 to i BARKER'S MILL. Effect of this mill is considerably greater than that which same quantity of water would produce if applied to an undershot- wheel, but less than* that which it would produce if properly applied to an overshot-wheel. For a description of it, see Grier's Mechanics' Calculator, page 234; and for ita formulas, see London Artisan, 1845, page 229. IMPULSE AND RESISTANCE OP FLUIDS. Impulse and Resistance oi"Water. Water or any other fluid, when flowing against a body, imparts a force to it by which its condition of motion is altered. Resistance which a fluid opposes to motion of a body does not essentially differ from Impulse. Impulse of one and same mass of fluid under otherwise similar circum- stances is proportional to relative velocities c =p v of fluid. For an equal transverse section of a stream, the impulse against a surface at rest increases as square of velocity of water. Impulse against Plane Surfaces. The impulse of a stream of water de- pends principally upon angle under which, after impulse, it leaves the water ; it is nothing if the angle is o, and a maximum if it is deflected back in a line parallel to that of its flow, or 180, 2 - - V w = P*. When Surface of Resistance is a Plane, and = 90, then ^ V w = P, and for a surface at res/, 2 a h w = P. a representing area of opening in sq.feet. P = 2 A h w ; c and v representing velocities of water and of surface upon which it impinges in feet per second, w weight of fluid per cube foot in Ibs., A transverse section of stream in sq. ins. , and c=p v relative motions of water and surface. Normal impulse of water against a plane surface is equivalent to weight of a column which has for its base transverse section of stream, and for c 2 altitude twice height due to its velocity, 2^ = 2 . Resistance of a fluid to a body in motion is same as impulse of a fluid moving with same velocity against a body at rest. * Weiabach, New York, 1870, vol. i. page 1008. 57* HYDRODYNAMICS. Maximum Effect of Impulge. Effect of impulse depends principally on velocity v of impinged surface. It is, for example, p, both when v =. c and v = o ; hence there is a velocity for which effect of impulse is a maximum e= (c 1>) v; that is, v= , and maximum effect of impulse of water is ob- tained when surface impinged moves from it with half velocity of water. ILLUSTRATION. A stream of water having a transverse section of 40 sq. ins., dis charges 5 cube feet per second against a plane surface, and flows off with a velocity of 12 feet per second ; effect of its impulse, then, is ^^ V w = P; c= - = i8j 9 4 = 32.16; 10 = 62.5; - ^-^ X 5X62. 5= 58.28 Ibs. Hence mechanical effect upon surface = P v = 58.28 x 12 = 699.36 Ibs. Maximum effect would be v = ^ = i x 5 X Q * 44 = 9 feet, and - x X 5 X 62. 5 = - X 5-036 x 312.5 = 786.87 Ibs.; and hydraulic pressures- = 87.44 Ibs. When Surface is a Plane and at an Angle, then (i cos. a) V >= P. ILLUSTRATION. A stream of water, having a transverse section of 64 sq. ins., dis- charges 17.778 cube feet per second against a fixed cone, having an angle of con- vergence from flow of stream of 50, hydraulic pressure in direction of stream ; thenc = ~^ = 40; cos. 50 = . 64279. (i .64279) ^ x 17.778 X 62.5 = .357 21 X 1382.2 = 494.26 Ibs. When Surface of Resistance is a Plane at 90, and has Borders added to its Perimeter, effect will be greater, depending upon height of border and ratio of transverse section between stream and part confined. Oblique Impulse. In oblique impulse against a plane, the stream may flow in one, two, or in all directions over plane. When Stream is confined at Three Sides, (i cos. a) -^- V w = P. When Stream is confined at Two Sides, sin. a 2 V w = P. Normal impulse of a stream increases as sine of angle of incidence ; par- allel impulse as square of sine of angle ; and lateral impulse as double the angle. When an Inclined Surface is not Bm^dered, then stream can spread over it in all directions, and impulse is greater, because of all the angles by which the water is deflected, a is least ; hence each particle that does not move in normal plane exerts a greater pressure than particle in that plane, , 2 sin. a 2 c v ~, ^ and . . ~X- Vwj = P. i -f- sin. a 2 g Impulse and Resistance against Surfaces. Coefficient of resistance, C, or number with which height due to velocity is to be multiplied, to obtain height of a column of water measuring this hydraulic press- ure, varies for bodies of different figures, and only for surfaces which are at right angles to direction of motion is it nearly a definite quantity. According to experiments of Du Buat and Thibault, C = 1.85 for impulse of air or water against a plane surface at rest, and for resistance of air or water against a surface in motion, C = 1.4. In each case about .66 of effect is expended upon front surface, and .34 upon near. HYDRODYNAMICS. 579 Comparison between Turbines and. other "Water-wheels. Turbines are applicable to falls of water at any height, from i to 500 feet. Their efficiency for very high falls is less than for smaller, in consequence of the hydraulic resistances involved, and which increase as the square of the velocity of the water. They can only be operated in clear water. With Fourneyron's, the stress and pressure on the step is that of the wheel in motion ; with Fontaine's, the whole weight of the water is added to that of the wheel ; they are well adapted, however, for tide-mills. Experiments on Jouval's gave equal results with Fontaine's. Vertical Water-wheels are limited in their application to falls under 60 feet in height. For falls of from 40 to 20 feet they give a greater effect than any turbine ; for falls of from 20 to 10 feet, they are equal to them ; and for very low falls, they have much less efficiency. Variations in the supply of water effect them less than turbines. "Water-pressure Engine. By experiments of M. Jordan, he ascertained that a mean useful effect of .84 was attainable. Weisbach, London, 1848, vol. ii. page 349. PERCUSSION OF FLUIDS. When a stream strikes a plane perpendicular to its action, force with which it strikes is estimated by product of area of plane, density of fluid, and square of its velocity. Or, A d v 2 = P. A representing area in sq. feet, d weight of fluid in Ibs., and v velocity in feet per second. If plane is itself in motion, then force becomes A d (v v') 2 = P. t>' representing velocity of plane. If C represent a coefficient to be determined by experiment, and h height due to velocity v. then v 2 = 2 q h. and expression for force becomes CENTRIFUGAL PUMPS. (D. K. Clark.) Appold. 3?uinp, made with curved receding blades, is the form of centrifugal pump most widely known and accepted. M. Morin tested three kinds of centrifugal or revolving pumps : ist, on model of Appold pump; 2d, one having straight receding blades inclined at an angle of 45 with the radius, and 30!, one having radial blades. They were 12 ins. in diameter and 3.125 ins. in length, and had central open- ings of 6 ins. Their efficiencies were as follows : i. Curved blades. . 48 to 68 per cent. | 2. Inclined blades. . 40 to 43 per cent 3. Radial blades ........ 24 per cent. Height to which water ascends in a pipe, by action of a centrifugal pump, would, if there were no other resistances, be that due to velocity of circum- ference of revolving wheel, or to ~ . Results of experiments made by the author on two pumps, in 1862, yielded following data, showing height to which water was raised, without any discharge : Diameter of pump- wheel ..... .......... 4 feet. 4 feet 7 ins. Revolutions per minute ................ 177 95 . 4 Velocity of circumference per second. . . 37.05 feet. 22.9 feet Head due to the velocity ............... 21.45 " 8.194 " Actual head ........................... 18.21 " 5.833 " Do. do. in parts of head due to velocity, 85 per cent. 71.2 per cent 580 HYDKODYNAMICS. IMPACT OE COLLISION. Mr. David Thomson made similar experiments with Appold pumps of from 1.25 to 1.71 feet in diameter, the results of which showed that the actual head was about 90 per cent, of the head due to the velocity. M. Tresca, in 1861, tested two centrifugal pumps, 18 ins. in diameter, with a cen- tral opening of 9 ins. at each side. The blades were six in number, of which three sprung from centre, where they were .5 inch thick; the alternate three only sprung at a distance equal to radius of opening from centre. They were radial, except at ends, where they were curved backward, to a radius of about 2.25 ins. ; and they joined the circumference nearly at a tangent. Width of blades was taper, and they were 5.75 ins. wide at nave, and only 2.625 ins. at ends: so designed that section of outflowing water should be nearly constant. M. Tresca deduced from his experiments that, in making from 630 to 700 revolu- tions per minute, efficiency of the pump, or actual duty in raising water, through a height of 31.16 feet, amounted to from 34 to 54 per cent, of work applied to shaft; or that, in the conditions of the experiment, the pump could raise upward of 16 200 cube feet of water per hour, through a height of 33 feet, with about 30 HP applied to shaft, and an efficiency of 45 per cent. According to Mr. Thomson, maximum duty of a centrifugal pump worked by a steam-engine varies from 55 per cent, for smaller pumps to 70 per cent, for larger pumps. They may be most effectively used for low or for moderately high lifts, of from 15 to 20 feet; and, in such conditions, they are as efficient as any pumps that can be made. For lifts of 4 or 5 feet they are even more efficient than others. At same time, larger the pump higher lift it may work against. Thus, an 1 8- inch pump works well at 20- feet lift, and a 3-feet pump at 3o-feet lift. A 21 inch wheel at 4o-feet lift has not given good results: high lifts demand very high velocities. Efficiency is influenced by form of casing of pump. Hon. R. C. Parsons made exper- iments with two i4-inch wheels on Appold's and on Rankine's forms. In Rankiue's wheel blades are curved backwards, like those of Appold's, for half their length ; and curved forwards, reversely, for outer half of their length. Deducing results of performance arrived at, following are the several amounts of work done per Ib. of water evaporated from boiler : Work done per Ib. of water evaporated. Foot-lbs. Ratio. Appold wheel, in concentric circular casing u 385 1.06 " " in spiral casing 15996 1.5 Rankine wheel, in concentric circular casing 10 748 i " in spiral casing 12954 '-2 These data prove: ist, that spiral casing was better than concentric casing; 2d that Appold's wheel was more efficient than Rankine's wheel. IMPACT OR COLLISION. IMPACT is Direct or Oblique. Bodies are Elastic or Inelastic. The division of them into hard and elastic is wholly at variance with these properties ; as, for instance, glass and steel, which are among hardest of bodies, are most elastic of all. Product of mass and velocity of a body is the Momentum of the body. Principle upon which motions of bodies from percussion or collision are determined belongs both to elastic and inelastic bodies ; thus there exists in bodies the same momentum or quantity of motion, estimated in any one and same direction, both before collision and after it. A ctlon and reaction are always equal and contrary. If a body impinge obliquely upon a plane, force of blow is as the sine of angle of incidence. When a body impinges upon a plane surface, it rebounds at an angle equal to that at which it impinged the plane, that is, angle of reflection is equal to that of incidence. Effect of a blow of an elastic body upon a plane is double that of an in- elastic one, velocity and mass being equal in each^ for the force of blow IMPACT OR COLLISION. jgl from inelastic body is as its mass and velocity, which is only destroyed by resistance of the plane ; but in an elastic body that force is not only destroyed, being sustained by plane, but another, also equal to it, is sustained by plane, in consequence of the restoring force, and by which the body is repelled with an equal velocity ; hence intensity of the blow is doubled. If two perfectly elastic bodies impinge on one another, their relative ve- locities will be same, both before and after impact ; that is, they will recede from each other with same velocity with which they approached and met. If two bodies are imperfectly elastic, sum of their moments will be same, both before arid after collision, but velocities after will be less than in case of perfect elasticity, in ratio of imperfection. Effect of collision of two bodies, as B and b, velocities of which are differ- ent, as v and v', is given in f ollowing formulas, in which B is assumed to have greatest momentum before impact. If bodies move in same direction before and after impact, sum of their moments before impact will be equal to their sum after. If bodies move in same direction before, and in opposite direction after impact, sum of their moments before impact will be equal to difference of their sums after. If bodies move in opposite directions before, and in same direction after impact, difference of their moments before impact will be equal to their sum after. If bodies move in opposite directions before, and in opposite directions after impact, difference of their moments before impact will be equal to their difference after. To Compute "Velocities of Inelastic Bodies after Impact. When Impelled in Same Direction. "^ v = r. B and b representing weights of the two bodies, V and v their velocities before impact, and r velocity of bodies after impact, all in feet. Consequently, ^| x b = velocity lost by B, and ^ X B = velocity gained by b. -- -- NOTE. In these formulas it is assumed that V>r. If V the result will be negative, but may be read as positive if lost and gained are reversed in places. ILLUSTRATION. An inelastic body, b, weighing 30 Ibs., having a velocity of 3 feet, is struck by another body, B, of 50 Ibs., having a velocity of 7 feet; the velocity of b after impact will be , - 50 X 7 + 30 X 3 440 __ F - = When Impelled ir, Opposite Directions. - ~=r. B-}- b ILLUSTRATION. Assume elements of preceding case. *b 50 + 30 80 B V When One Body is at Rest. ^^ = r. ILLUSTRATION. Assume elements as preceding. 50X7 35o When Bodies are inelastic, their velocities after impact will be alike. 30* 5 82 IMPACT OK COLLISION. To Compxite Velocities of Elastic JBodios after Impact , . -. _^. A . B b V + a&v ,,26 V B bv When Impelled in One Direction. = R, and = r. ILLUSTRATION. Assume elements as preceding. =g=4^^ ax5ox r-r 55x ^fe=8^ 50+30 80 50+30 80 Or > v S^-TT V v = velocity of ri, and v + -^r V^v = velocity of r. o + i> + / TF%en Impelled in Opposite Directions. B 6 V2 fe v 2 BV+ B 6 v - B + 6 ILLUSTRATION. Assume elements as preceding. 50 30 X 7^2X 30 X 3_i4Q / vi8o_ ,2X50X7 + 50 3oX3_^ Rest. 7 = R, and =r. ILLUSTRATION. Assume elements as preceding. = ^o = pd Z == 50+30 80 50+30 To Compute "Velocities of* Imperfect Elastic Bodies after Impact. Effect of Collision is increased over that of perfectly inelastic bodies, but not doubled, as in case of perfectly elastic bodies ; it must be multiplied by i + ^ or m j^ n , when represents degree of elasticity relative to both per- fect inelasticity and elasticity. Moving in same Direction. V -- 3l_ x ( V v) = R : and v -j -_ in D-\- o m X p , . (V v) = r. m and n representing ratio of perfect to imperfect elasticity. D + O ILLUSTRATION. Assume elements as preceding. m and n = 2 and i. , and 3 + When Moving in Opposite Directions. T-=5 x = B and x When One Body is at Rest. * ' = R, and 3+6 ILLUSTRATION. Assume elements of preceding case. 50+30 3SQ Xr 5 - - = 6.56*5 /*. LIGHT. LIGHT. LIGHT is similar to Heat in many of its qualities, being emitted in form of rays, and subject to same laws of reflection. It is of two kinds, Natural and Artificial ; one proceeding from Sun and Stars, the other from heated bodies. Solids shine in dark only at a temperature from 600 to 700, and in daylight at 1000. Intensity of Light is inversely as square of distance from luminous body. Velocity of Light of Sun is 185 ooo miles per second. Standard of Intensity or of comparison of light between different methods of Illumination is a Sperm Candle "short 6," burning 120 grains per hour. Candles. A Spermaceti candle .85 of a inch in diameter consumes an inch in length in i hour. Decomposition of Light. Maximum Contrasts. Combinations. COLORS. Ray. Primary. Second'y. Tertiary. Primary. Secondary. Tertiary. Violet.. Indigo. . Chemical. - - Brown. Blue...) Yellow. } Green. . i Dark. Blue . . . Electrical. Blue. Blue...) Purple. ) Green. Green . . Green. Green. Red. . . . J Orange. ) _ Yellow. Light. Yellow. Green. . J i*ray. Orange . Red.... Heat. Red. Orange. Purple. Broken. Green. Yellow. ) Red. . . . } Purple. ) Orange. ) Brown. All colors of spectrum, when combined, are white. Consumption and. Comparative Intensity of Light of Candles. CANDLK. No. in a Lb. Diameter. Length. Consumption per Hour. Light comp'd with Carcel. Wax Inch. Ins. 12 Grains. 3 .875 } 135 .09 Q I e ) : 13.5 } 156 u 6 84 ) wy Tallow t 3 15 204 4 .07 4 .8 13.7=5 Compared with 1000 Cube Feet of Gas. CANDLB. Gas=i. Con- sump- tion. Light. Con- 1 sumption for equal Light. | CANDLE. Gaa=i. Con- sump- tion. Light. Con- sumption for equal Light. Paraffine. Sperm . . . .098 .005 Lbs. 3-5 3.Q Lbs. 35-5 41. 1 .03 120 Adamantine. Tallow .108 .074 Lbs. 5-i 5- 1 Lbs. 47.2 53-8 137 155 In combustion of oil in an ordinary lamp, a straight or horizontally cut wick gives great economy over one irregularly cut. 584 LIGHT. Relative Intensity-, Consumption, Illumination, and. Cost of various Ivlotles of Illumination. Oil at ii cents, Tallow at 14 cents, Wax at 52 cents, and Stearine at 32 cents pei Ib. ioo cube feet coal gas at 14 cents, and 100 cube feet of oil gas at 52 cents. ILLUMINATOR. Illumi- nation. Carcel Lamp = 100. Actual Cost Hour. Cost for equal Inten- sity. ILLUMINATOR. Illumi- Carcel' Lamp = IOO. Actual Cost deSir. Costfoi ssi ity. Carcel Lamp Lamp with in-) verted reserv'r. j Astral Lamp IOO 57-8 48 7 Cents. .87 .89 .56 PerH'r. .87 99 1.78 Stearine Candle 5 to Ib. Tallow u 6 " Sperm 6 " Coal Gas 66.6 3.5 Cents. 59 % 1.22 PerH'r. 4-13 2-34 5-7 .06 Wax Candle 6 to Ib. 61.6 .02 1.70 6.^1 Oil Gas... 1.25 5 1000 cube feet of 13- candle coal gas is equal to 7.5 gallons sperm oil, 52.9 Ibs. mold, and 44.6 Ibs. sperm candles. Candles, .Lamps, Fluids, and GJ-as. Comparison of several Varieties of Candles, Lamps, and Fluids, with Coal* Gas, de- duced from Reports of Com. of Franklin Institute, and of A. Frye, M.D.,etc. CANDLB. B :i .58 t Coi consum en- Li rof F ' ht. g If! Cost com- pared with Gas for CANDLB. 3 s |I| "3 1! Diaphane 5 5 .8 upar It* f ual St. 4 Bdl cub Tin Bu i of 15 16 7 vitha e foot lie of nnf Oil. I 2 5 fisl per Tallow, short 6's, ) double wick . . ] Wax, short 6's.... Palm oil i .8 is, conta Inten- sity of Light. i .61 77 ning 12 Light at Equal Cost. 7- 1 14.4 10.5 per cent. Time of Burning i Pint of Oil. Spermaceti, short Tallow, short 6's, 1 single wick . . . j *CityofPhiladelph of condensable matter LAMP AND FLUID. 5a id. and Int Bit) Li? -tail jet of Edinburgh g hour. LAMP AND FLUID. Carcel. Sperm oil, max'm ' ' mean. ' ' wuw'wi Lard oil. . . 2.15 1.22 .69 .77 1.8 i-35 1.2 .07 Hours 9.87 14.6 Ga Se So Ca S I l'* 7 6 1.75 93 i-55 1.08 Hours. 8^2 mi-solar, Sperm oil lar, Sperm oil mphene . . . Loss of Light by Use of Glass Globes. Clear Glass, 12 per cent. | Half ground, 35 per cent. | Full ground, 40 per cent. Refraction. Relative Index of Refraction Is, Ratio of sine of angle of incidence to sine of angle of refraction, when a ray of light passes from one medium into another. Absolute Index or Index of Refraction Is, When a ray passes from a vacuum into any medium, the ratio is greater than unity. Relative index of refraction from any medium, as A, into another, as B, is always equal to absolute index of B, divided by absolute index of A. Absolute index of air is so small, that it may be neglected when compared with liquids or solids; strictly, however, relative index for a ray passing from air into a given substance must be multiplied by absolute index for air, in order to obtain like index of refraction for the substance. Mean Indices of Refraction. Air at 32 i Alcohol i. 37 Canada balsam ...... i. 54 Crystalline lens x.^4 Glass, fluid J J'j? 8 " crown }i:ll Humors of eye. . Salt, rock Water, fresh 1.34 1.55 1.34 1-34 LIGHT. 585 GJ-as. Retort. A retort produces about 600 cube feet of gas in 5 hours with a charge of about 1.5 cwt. of coal, or 2800 cube feet in 24 hours. In estimating number of retorts required, one fourth should be added for being under repairs, etc. Pressure with which gas is forced through pipes should seldom exceed 2.5 ins. of water at the Works, or leakage will exceed advantages to be obtained from increased pressure. The a\ erage mean pressure in street mains is equal to that of i. inch of water. When pipes are laid at an inclination either above or below horizon, a cor- rection will have to be made in estimating supply, by adding or deducting .01 inch from initial pressure for every foot of rise or fall in the length of pipe. It is customary to locate a governor at each change of level of 30 feet. Illuminating power of coal-gas varies from 1.6 to 4.4 times that of a tallow candle 6 to a Ib. ; consumption being from 1.5 to 2.3 cube feet per hour, and specific gravity from .42 to .58. Higher the flame from a burner greater the intensity of the light, the most effective height being 5 ins. Standard of gas burning is a i5-hole Argand lamp, internal diameter .44 inch, chimney 7 ins. in height, and consumption 5 cube feet per hour, giving a light from ordinary coal-gas of from 10 to 12 candles, with Cannel coal from 20 to 24 candles, and with rich coals of Virginia and Pennsylvania of from 14 to 1 6 candles. In Philadelphia, with a fish-tail burner, consuming 4.26 cube feet per hour, illuminating power was equal to 17.9 candles, and with an Argand burner, consuming 5.28 cube feet per hour, illuminating power was 20.4 candles. Gas, which at level of sea would have a Value, of 100, would have but 60 in city of Mexico. Internal lights require 4 cube feet, and external lights about 5 per hour. When large or Argand burners are used, from 6 to 10 are required. An ordinary single-jet house burner consumes 5 to 6 cube feet per hour. Street-lamps in city of New York consume 3 cube feet per hour. In some cities 4 and 5 cube feet are consumed. Fish-tail burners for ordinary coal gas consume from 4 to 5 cube feet of gas per hour. A cube foot of good gas, from a jet .033 inch in diameter and height of flame of 4 ins., will burn for 65 minutes. Resin Gas. Jet .033, flame 5 ins., 1.25 cube feet per hour. Purifiers. Wet purifiers require i bushel of lime mixed with 48 bushels of water for 10000 cube feet of gas. Dry purifiers require i bushel of lime to 10000 cube feet of gas, and i superficial foot for every 400 cube feet of gas. Intensity of* Light -with Kqnal "Volumes of Gras from cUfferent Burners. Equal to Spermaceti Candle burning 120 Grains per Hour. BUBNIRS. **! enditi eet pe re in r Hou 3 2ube r. 4 BURNERS. Exp z enditii >etp 2 re in < r Hou 3 ?ubt r. 4 Single-jet, i foot Fish-tail No. 3 Bat's wine. . ." 2.6 3-5 q 4 4-1 4.2 4-3 4.5 Argand, 16 holes. . . . Argand, 24 holes. . . . Argand, 28 holes. . . 32 33 .34 1.9 2.2 2.3 3-3 3-4 3.* 3-8 5:8 586 LIGHT. Volume of G^as obtained from a Ton of Coal, Resin* etc. Material. Cube Feet. Material. Cube Feet. Material. Cube Feet. Boghead Cannel I -I -3-2 A Cumberland o 8bo Pittsburgh O S2O Wigan Cannel 15 42O Resin 15600 Cfinncl < 8960 Newcastle < 9500 Scotch | 10300 Cape Breton, ) 15000 Oil and Grease IOOOO 23 ooo I5OOO 8960 "Cow Bay,"} .. etc ) 9500 Pictou and Sidney. . Pine wood . . , 8000 11800 44 West'n.. Walls-end . . . 9500 12 OOO i Chaldron Newcastle coal, 3136 Ibs., will furnish 8600 cube feet of gas at a specific gravity of .4, 1454 Ibs. coke, 14.1 gallons tar, and 15 gallons am- moniacal liquor. Australian coal is superior to Welsh in producing of gas. Wigan Cannel, i ton, has produced coke, 1326 Ibs. ; gas, 338 Ibs, ; tar, 250 Ibs. ; loss, 326 Ibs. Peat, i Ib. will produce gas for a light of one hour. Fuel, required for a retort 18 Ibs. per 100 Ibs. of coal. In distilling 56 Ibs. of coal, volume of gas produced in cube feet when distillation was effected in 3 hours was 41.3, in 7, 37.5, in 20, 33.5, and in 25, 31.7- Flow of Q-as in IPipes. Flow of Gas is determined by same rules as govern that of flow of water. Pressure applied is indicated and estimated in inches of water, usually from .5 to i inch. Volumes of gases of like specific gravities discharged in equal times by a horizontal pipe, under same pressure and for different lengths, are inversely as square roots of lengths. Velocity of gases of different specific gravities, under like pressure, are in- versely as square roots of their gravities. By experiment, 30 ooo cube feet of gas, specific gravity of .42, were dis- charged in an hour through a main 6 ins. in diameter and 22.5 feet in length. Loss of volume of discharge by friction, in a pipe 6 ins. in diameter and i mile in length, is estimated at 95 per cent. Diameter and. Length of Q-as-pipes to transmit given Volumes of <3-as to Branch-pipes. (Dr. Ure.) Volume per Hour. Diameter. Length. Volume per Hour. Diameter. Length. Volume per Hour. Diameter. Length. Cube Feet. 50 250 500 700 Ins. 4 1.97 2.65 Feet. 100 200 600 IOOO Cube Feet. IOOO 1500 2000 2000 Ins. 3.16 3-87 5.- 32 6-33 Feet. IOOO IOOO 2OOO 4000 Cube Feet. 2OOO 6000 6oOO 8000 Ins. 7 7-75 9.21 8-95 Feet. 6000 IOOO 2000 IOOO Regulation of Diameter and Extreme .Length of Tub- ing, and Number of Burners permitted. Diameter Capacity Diameter Capacity of Tubing. Length. of Meters. Burners. Tubing. Length. of Meters. Burners. Ins. Feet. Light. No. Ins. Feet. Light. No. 25 6 3 9 75 50 30 90 375 5 20 30 5 10 15 30 I 1.25 70 IOO 'M .625 40 20 60 i-5 50 IOO 300 LIGHT. 587 Temperature of Gases. Combustion of a cube foot of common gas will heat 650 Ibs. of water i. Services for Lamps Lamps. Length from Main. Diameter of Pipe. Lamps. Length from Main. Diameter of Pipe. Lamps. Length from Main. Diameter of Pipe. No. 2 i Feet. 40 40 50 Ins. 375 ^625 No. 10 15 20 Feet. 100 130 150 Ins. 75 I 1.25 No. 25 30 Feet. 1 80 200 Ins. 1-75 Volumes of Q-as Discharged per Hour under a Pressure of Half an Inch of Water. Diam. of Opening. Volume. Diam. of Opening. Specific Gr Volume. avity .42. Diam. of Opening. Volume. Diam. of Opening. Volume. Ins. 25 5 Cube Feet. 80 321 Ins. 75 i Cube Feet. 723 1287 Ins. 1.125 1.25 Cub Feet. 1625 2OIO Ins. i-5 5 Cube Feet. 2885 46150 To Compute Volume of G-as Discharged through a IPipe. yd* h /V a G I = V, and 063 5 / - d. d representing diameter of pipe, and (r I \ n, h, height of water in ins. , denoting pressure upon gas, I length of pipe in yards, G specific gravity of gas, arid V volume in cube feet per hour. G may be assumed for ordinary computation at .42, and h .5 to i inch. ILLUSTRATION. Assume diameter of pipe i inch, pressure 1.68 ins., and length of pipe i yard. tooo x /r :; I00 X . /- = 2000 cube feet, and .063 X :; 40oooooX -42X i -- -- = -OS * NOTE. For tables deduced by above formulas see Molesworth, 1878, page 226. Dimensions of Mains, \vith "Weight of One Length. Diameter in ins ..... Length in feet ...... Thickness in ins. . . . Weight in Ibs ....... 375 6 9 375 400 9 9 5 454 489 '4 9 868 .625 75 1316 t 4 8 4 75 GAS ENGINES. In the Lenoir engine, the best proportions of air and gas are, for common gas, 8 volumes of air to i of gas, and for cannel gas, n of air to i of gas. The time of explosion is about the 27th part of a second. An engine, having a cylinder 4.625 ins. in diameter and 8.75 ins. stroke of piston, making 185 revolutions per minute, develops a half horse-power. Distribution of Heat Generated in the Cylinder. (M. Tresca.) Per cent. Per cent. Dissipated by the water and prod- I Losses 27 ucts of combustion 69 ~^ Converted into work 4 | Hence efficiency as determined by the brake 4 per cent. Atmospheric Gas Engine. A single-acting cylinder 6 ins. in diameter, making 81 strokes per minute, devel- oped .456 IP, and the gas consumed per minute for cylinder 20 cube feet and for in- flaming 2 cube feet. (M. Tresca.) 588 LIMES, CEMENTS, MORTARS, AND CONCRETES. LIMES/CEMENTS, MORTARS, AND CONCRETES. Essentially from a Treatise by Brig.-Geril Q. A. Gillmore, U.S.A.* Lime. Calcination of marble or any pure limestone produces lime (quick" lime). Pure limestones burn white, and give richest limes. Finest calcareous minerals are rhombohedral prisms of calcareous spar, the transparent double-reflecting Iceland spar, and white or statu- ary marble. Property of hardening under water, or when excluded from air, con- ferred upon a paste of lime, is effected by presence of foreign sub- stances as silicum, alumina, iron, etc. when their aggregate presence amounts to . I of whole. Limes are classed : i. Common or Fat limes, which do not set in water. 2. Poor or Meagre, mixed with sand, which does not alter its condition. 3. Hydraulic Lime, containing 8 to 12 per cent, of silica, alumina, iron, etc., set slowly in water. 4. Hydraulic, containing 12 to 20 per cent, of similar ingredients, sets in water in 6 or 8 days. 5. Eminently Hydraulic, containing 20 to 30 per cent, of similar ingredients, sets in water in 2 to 4 days. 6. Hydraulic Cement, containing 30 to 50 per cent, of argil, sets in a few minutes, and attains the hardness of stone in a few months. 7. Natural Pozzuolanas, including pozzuolana properly so called, Trass or Terras, Arenes, Ochreous earths, Basaltic sands, and a variety of similar substances. Indications of Limestones. They dissolve wholly or partly in weak acids with brisk effervescence, and are nearly insoluble in water. Rich Limes are fully dissolved in water frequently renewed, and they remain a long time without hardening ; they also increase greatly in vol- ume, from 2 to 3.5 times their original bulks, and will not harden without the action of air. They are rendered Hydraulic by admixture of pozzuolana or trass. Rich, fat, or common Limes usually contain less than 10 per cent, of im- purities. Hydraulic Limestones are those which contain iron and clay, so as to en- able them to produce cements which become solid when under water. Poor Limes have all the defects of rich limes, and increase but slightly in bulk, the poorer limes are invariably basis of the most rapidly - setting and most durable cements and mortars, and they are also the only limes which have the property, when in combination with silica, etc., of indurating under water, and are therefore applicable for admixture of hydraulic cements or mortars. Alike to rich limes, they will not harden if in a state of paste under water or in wet soil, or if excluded from contact with the atmosphere or carbonic acid gas. They should be employed for mortar only when it is impracticable to procure common or hydraulic lime or cement, In which case it is recommended to reduce them to powder by grinding. Hydraulic Limes are those which readily harden under water. The most valuable or eminently hydraulic set from the 2d to the 4th day after immer- sion ; at end of a month they become hard and insoluble, and at end of six months they are capable of being worked like the hard, natural limestones. They absorb less water than pure limes, and only increase in bulk from 1.75 to 2.5 times their original volume. * See also his Treatises on Limes, Hydraulic Cements, and Mortars, in Papers on Practical Engineer- ing, Engineer Department, U. S. A. LIMES, CEMENTS, MORTARS, AND CONCRETES. 589 Inferior grades, or moderately hydraulic, require a period of from 15 to 20 days' immersion, and continue to harden for a period of 6 months. Resistance of hydraulic limes increase if sand is mixed in proportion of 50 to 1 80 per cent, of the part in volume ; from thence it decreases. M. Vicat declares that lime is rendered hydraulic by admixture with it of from 33 to 40 per cent, of clay and silica, and that a lime is obtained which does not slake, and which quickly sets under water. Artificial Hydraulic Limes do not attain, even under favorable circum- stances, the same degree of hardness and power of resistance to compression as natural limes of same class. Close-grained and densest limestones furnish best limes. Hydraulic limes lose or depreciate in value by exposure to the air. Pastes of fat limes shrink, in hardening, to such a degree that they can- not be used as mortar without a large proportion of sand. Arenes is a species of ochreous sand. It is found in France. On account of the large proportion of clay it contains, sometimes as great as .7, it can be made into a paste with water without any addition of lime ; hence it is some- times used in that state for walls constructed en pise, as well as for mortar. Mixed with rich lime it gives excellent mortar, which attains great hardness under water, and possesses great hydraulic energy. Pozzuolana is of volcanic origin. It comprises Trass or Terras, the Arenes, some of the ochreous earths, and the sand of certain graywackes, granites, schists, and basalts; their principal elements are silica and alumina, the former preponderating. None contain more than 10 per cent, of lime. When finely pulverized, without previous calcination, and combined with paste of fat lime in proportions suitable to supply its deficiency in that element, it pos- sesses hydraulic energy to a valuable degree. It is used in combination with rich lime, and may be made by slightly calcining clay and driving off the water of com- bination at a temperature of 1200. Brick or Tile Dust combined with rich lime possesses hydraulic energy. Trass or Terras is a blue-black trap, and is also of volcanic origin. It requires to be pulverized and combined with rich lime to render it fit for use, and to develop any of its hydraulic properties. General Gillmore designates the varieties of hydraulic limes as follows: If, after being slaked, they harden under water in periods varying from 15 to 20 days after immersion, slightly hydraulic ; if from 6 to 8 days, hydraulic; and if from i to 4 days, eminently hydraulic. Pulverized silica burned with rich lime produces hydraulic lime of ex- cellent quality. Hydraulic limes are injured by air-slaking in a ratio vary- ing directly with their hydraulicity, and they deteriorate by age. For foundations in a damp soil or exposure, hydraulic limes must be ex- clusively employed. Hydraulic Lime of Teil is a silicious hydraulic lime ; it is slow in setting, requiring a period of from 18 to 24 hours. Cements. Hydraulic Cements contain a larger proportion of silica, alumina, magnesia, etc., than any of preceding varieties of lime ; they do not slake after calcina- tion, and are superior to the very best of hydraulic limes, as some of them set under water at a moderate temperature (65) in from 3 to 4 minutes ; others require as many hours. They do not shrink in hardening, and make an excellent mortar without any admixture of sand. 7 D 59O LIMES, CEMENTS, MORTARS, AND CONCRETES. When exposed to air, they absorb moisture and carbonic acid gas, and are rapidly deteriorated thereby. Roman Cement is made from a lime of a peculiar character, found in Eng- land and France, derived from argillo-calcareous kidney-shaped stones termed Septaria. It is about .33 strength of Portland, and is not adapted for use with sand. Rosendale Cement is from Rosendale, New York. Portland Cement is made in England, Germany, France, and the United States. It requires less water (cement i, water .29) than Roman cement, sets slowly, and can be remixed with additional water after an interval of 12 or even 24 hours from its first mixture. Property of setting slow may be an obstacle to use of some designations of this cement, as the Boulogne, when required for localities having to contend against immediate causes of destruction, as in sea constructions, having to be executed un- der water and between tides. On the other hand, a quick-setting cement is always difficult of use ; it requires special workmen and an active supervision. A slow- setting cement, however, like natural Portland, possesses the advantage of being managed by ordinary workmen, and it can also be remixed with additional water after an interval of 12 or even 24 hours from its first mixing. Conclusions derived from Mr. Grant's Experiments. 1. Portland cement improves by age, if kept from moisture. 2. Longer it is in setting, stronger it will be. 3. At end of a year, i of cement to i sand is about .75 strength of neat cement; i to 2, .5 strength; i to 3, .33; i to 4, .25; i to 5, .16. 4. Cleaner and sharper the sand, greater the strength. 5. Strong cement is heavy; blue gray, slow-setting. Quick-setting has generally too much clay in its composition is brownish and weak. 6. Less water used in mixing cement the better. 7. Bricks, stones, etc., used with cement should be well wetted before use. 8. Cement setting under stitt water will be stronger than if kept dry. 9. Bricks of neat Portland cement in a few months are equal to Blue bricks, Bramley-Fall stone, or Yorkshire landings. 10. Bricks of i cement to 4 or 5 of sand are equal to picked stock bricks. 11. When concrete is being used, a current of water will wash away the cement. A rtificial Cement is made by a combination of slaked lime with unburned clay in suitable proportions. Artificial Pozzuolana is made by subjecting clay to a slight calcination. Salt water has a tendency to decompose cements of all kinds, and their strength is considerably impaired by their mixture with it. Mortar. Lime or Cement paste is the cementing substance in mortar, and its pro- portion should be determined by the rule that Volume of cementing substance should be somewhat in excess of volume of voids or spaces in sand or coarse material to be united, the excess being added to meet imperfect manipulation of the mass. Hydraulic Mortar, if re-pulverized and formed into a paste after having once set, immediately loses a great portion of its hydraulicity, and descends to the level of moderate hydraulic limes. The retarding influence of sea-water upon initial hydraulic induration is not very great, if the cement is mixed with fresh water. The strength of mortars, however, is considerably impaired by being mixed with sea-water. Pointing Mortar is composed of a paste of finely-ground cement and clean sharp siliceous sand, in such proportions that the volume of cement paste is slightly in excess of the volume of voids or spaces in the sand. The volume LIMES, CEMENTS, MORTARS, AND CONCRETES. 59! of sand varies from 2.5 to 2.75 that of the cement paste, or by weight, i ol cement powder to 3 to 3.33 of sand. The mixture should be made under shelter, and in small quantities. All mortars are much improved by being worked or manipulated; and as rich limes gain somewhat by exposure to the air, it is advisable to work mortar in large quantities, and then render it fit for use by a second manipulation. White lime will take a larger proportion of sand than brown lime. Use of salt-water in the composition of mortar injures adhesion of it When a small quantity of water is mixed with slaked lime, a stiff paste is made, which, upon becoming dry or hard, has but very little tenacity, but, by being mixed with sand or like substance, it acquires the properties of a cement or mortar. Proportion of sand that can be incorporated with mortar depends partly upon the degree of fineness of the sand itself, and partly upon character of the lime. For rich limes, the resistance is increased if the sand is in pro- portions varying from 50 to 240 per cent, of the paste hi volume ; beyond this proportion the resistance decreases. Lime, i, clean sharp sand, 2.5. An excess of water in slaking the lime swells the mortar, which remains light and porous, or shrinks in. drying ; an excess of sand destroys the cohesive properties of the mass. It is indispensable that the sand should be sharp and clean. Stone Mortar. % parts cement, 3 parts lime, and 31 parts of sand ; or i cask cement, 325 Ibs., .5 cask of lime, 120 Ibs., and 14.7 cube feet of sand= 18.5 cube feet of mortar. Brick Mortar. % parts cement, 3 parts lime, and 27 parts of sand ; or i cask cement, 325 Ibs., .5 cask of lime, 120 Ibs., and 12 cube feet of sand= 16 cube feet of mortar. Brown Mortar. Lime i part, sand 2 parts, and a small quantity of hair. Lime and sand, and cement and sand, lessen about .33 in volume when mixed together. Calcareous Mortar, being composed of one or more of the varieties of lime or cement, natural or artificial, mixed with sand, will vary in its properties with quality of the lime or cement used, the nature and quality of sand, and method of manipulation. Tuirlzisli 3?laster, or Hydraulic Cement. 100 Ibs. fresh lime reduced to powder, 10 quarts linseed-oil, and i to 2 ounces cotton. Manipulate the lime, gradually mixing the oil and cotton, in a wooden vessel, until mixture becomes of the consistency of bread-dough. Dry, and when required for use, mix with linseed-oil to the consistency of paste, and then lay on in coats. Water-pipes of clay or metal, joined or coated with it, resist the effect of humidity for very long periods. Stucco. Stucco or Exterior Plaster is term given to a certain mortar designed for exterior plastering; it is sometimes manipulated to resemble variegated marble, and consists of i volume of cement powder to 2 volumes of dry sand. In India, to water for mixing the plaster is added i Ib. of sugar or molas- ses to 8 Imperial gallons of water, for the first coat ; and for second or finish- ing, i Ib. sugar to 2 gallons of water. Powdered slaked lime and Smith's forge scales, mixed with blood in suit- able proportions, make a moderate hydraulic mortar, which adheres well to masonry previously coated with boiled oil. 5Q2 LIMES, CEMENTS, MORTARS, AND CONCRETES. Plaster should be applied in two coats laid on in one operation, first coat being thinner than second. Second coat is applied upon first while latter is yet soft. The two coats should form one of about 1.5 inches in thickness, and when fin- ished it should be kept moist for several days. When the cement is of too dark a color for desired shade, it may be mixed w*tti white sand in whole or in part, or lime paste may be added until its volume equals that of the cement paste. Klliorassar, or Tvirlzish. Miortar, Used for the construction of buildings requiring great solidity, .33 pow- dered brick and tiles, .66 fine sifted lime. Mix with water to required con- sistency, and lay between the courses of brick or stones. Mortars. Mortars used for inside plastering are termed Coarse, Fine, Gauge or hard finish, and Stucco. Plastering. i bushel, or 1.25 cube feet of cement, mortar, etc., will cover 1.5 square yards .75 inch thick. 75 volumes are required upon brick work for 70 upon laths. When full time for hardening cannot be allowed, substitute from 15 to 20 per cent, of the lime by an equal proportion of hydraulic cement. For the second or brown coat the proportion of hair may be slightly diminished. Coarse Stu.fr. Common lime mortar, as made for brick masonry, with a small quantity of hair ; or by volumes, lime paste (30 Ibs. lime) i partj sand 2 to 2.25 parts, hair .16 part. Fine Stuff (lime putty). Lump lime slaked to a paste with a mod- erate volume of water, and afterwards diluted to consistency of cream, and then to harden by evaporation to required consistency for working. In this state it is used for a slipped coat, and when mixed with sand or plaster of Paris, it is used for finishing coat. Q-auge, or Hard Finish, is composed of from 3 to 4 volumes fine stuff and i volume plaster of Paris, in proportions regulated, by rapidity re- quired in hardening; for cornices, etc., proportions are equal volumes of each, fine stuff and plaster. Scratch Coat. First of three coats when laid upon laths, and is from .25 to 375 f an i ncn m thickness. One-coat Work. Plastering in one coat without finish, either on masonry or laths that is, rendered or laid. Two-coat Work. Plastering in two coats is done either in a laid coat and set, or in a screed coat and set. Screed coat is also termed a Floated coat. Laid first coat in two-coat work is resorted to in common work instead of screeding, when finished sur- face is not required to be exact to a straight-edge. It is laid in a coat of about .5 inch in thickness. Laid coat, except for very common work, should be hand-Jloated. Firmness and tenacity of plastering is very much increased by hand-floating. Screeds are strips of mortar 6 to 8 inches in width, and of required thick ness of first coat, applied to the angles of a room, or edge of a wall and paral- lelly, at intervals of 3 to 5 feet over surface to be covered. When these have become sufficiently hard to withstand pressure of a straight-edge, the inter- spaces between the screeds are filled out flush with them. Slipped Coat is the smoothing off of a brown coat with a small quantity of lime putty, mixed with 3 per cent, of white sand, so as to make a compar- atively even surface. this finish answers when the surface is to be finished in distemper, or paper. LIMES, CEMENTS, MORTARS, AND CONCRETES. 593 Concrete or Beton fs a mixture of mortar (generally hydraulic) with coarse materials, as gravel, pebbles, stones, shells, broken bricks, etc. Two or more of these materials, or all of them, may be used together. As lime or cement paste is the cementing substance in mortar, so is mortar the cementing substance in concrete or beton. The original distinction between cement and beton was, that latter possessed hydraulic energy, while former did not. Hydraulic. 1.5 parts unslaked hydraulic lime, 1.5 parts sand, i part gravel, and 2 parts of a hard broken limestone. This mass contracts one fifth in volume. Fat lime may be mixed with concrete, without serious prejudice to its hydraulic energy. "Various Compositions of Concrete. Hydrauli c. 308 Ibs. cement = 3.65 to 3.7 cube feet of stiff paste. 12 cube feet of loose sand = 9.75 cube feet of dense. For Superstructure. 11.75 cube feet of mortar as above, and 16 cube feet of stone fragments. Sea Wall. Boston Harbor. Hydraulic. 308 Ibs. cement, 8 cube feet of sand, and 30 cube feet of gravel. Whole producing 32.3 cube feet. Superstructure. 308 Ibs. cement, 80 Ibs. lime, and 14.6 cube feet dense sands. Whole producing 12.825 cube feet. I?ise fs made of clay or earth rammed in layers of from 3 to 4 ins. in depth. In moist climates, it is necessary to protect the external surface of a wall constructed in this manner with a coat of mortar. A.splialt Composition. Asphaltum 3 parts, residuum oil or soft bitumen i part, powdered stone or fine sand 12 parts. Ashes 2 parts, powdered clay 3 parts, sand i part. Mixed with soft bitumen makes a very fine and durable cement, suitable for external use. Flooring. 8 Ibs. of composition will cover i sup. foot, .75 inch thick. Asphaltic limestone 55 Ibs. and gravel 28.7 Ibs. will cover 10.75 sq. feet, .75 inch thick. Asphaltic Mastic. Mix hot asphaltic limestone 8 parts, asphaltum i part; add sufficient sand for density needed for floor, roof, or walk. Waterproofing. Asphaltum 4 parts, linseed oil 2 parts, sand 14 parts, pulverized limestone 14 parts, by weight. Materials to be well dried, hot, and apply to dry surface. For Roads. Asphaltum 12.5 parts, soft bitumen or maltha 2.5 parts, powdered limestone 5 parts, sand 80 parts, mixed at temperature of 300. Thickness, 2 ina Artificial Mastic. Composition of i square yard .9 inch thick: Mineral tar. 205 cube ins. I Gravel 275 cube ins. Pitch 165 " Slakedlime 55 " Sand 549 " 1249 cube ins. M: viral "Efflorescence. White alkaline efflorescence upon the surface of brick walls laid in mortar, of which natural hydraulic lime or cement is the basis. Mortar mixed with animal fat in the proportion of .025 of its weight will prevent its formation. Crystallization of these salts within the pores of bricks, into which they have been absorbed from the mortar, causes disintegration. Distemper is term for all coloring mixed with water and size. Gr0H#w<7. Mortar composed of lime and fine sand, in a semi-fluid state, poured into the upper beds and internal joints of masonry. Laitance is the pulpy and gelatinous fluid, of a milky hue, that is washed from cement upon its being deposited in water. It is produced more abun- dantly in sea water than in fresh ; it sets very imperfectly, and has a ten- dency to lessen the strength of the concrete. 594 LIMES > CEMENTS, MORTARS, AND CONCRETES. Slaking. Slaked Lime is a hydrate of lime, and it absorbs a mean of 2.5 times its volume, and 2.25 times its weight of water. Lime (quicklime) must be slaked before it can be used as a matrix for mortar. Ordinary method of slaking is by submitting the lime to its full propor- tion of water (previously known or attained by trial) in order to reduce it to the consistency of a thick pulp. The volume of water required for this pur- pose will vary with different limes, and will range from 2.5 to 3 volumes that of the lime, and it is imperative that it should all be poured upon it so nearly at one time as to be in advance of the elevation of the temperature consequent upon its reduction. This process, when the water used is in an excessive quantity, is termed " drowning," and when the volume of lime has increased by the absorption of water it is termed its " growth." If too much water is used, the binding qualities of the lime is injured by its semi-fluidity ; and if too little, it is injurious to add after the reduction of the lime has commenced, as it reduces its temperature and renders it granu- lar and lumpy. While lime is in progress of slaking it should be covered with a tarpaulin or canvas (a layer of sand will suffice), in order to concentrate its evolved heat. The essential point in slaking is to attain the complete reduction of the lime, and the greater the hydraulic energy of a lime, the more difficult it be- comes to effect it. Whitewash or Grouting. When lime is required for a whitewash or for grouting, it should be thoroughly " drowned," and then run off into tight ves- sels and closed. Slaking by Immersion is the method of suspending lime in a suitable ves' sel in water for a very brief period, and withdrawing it before reduction commences. The lime is then transferred to casks or like suitable receptacles, and tightly enclosed, until it is reduced to a fine powder, in which condition, if secured from absorption of air, it may be preserved for several months without essential deterioration. Spontaneous or Air Slaking. When lime is not wholly secured from ex- posure to the air, it absorbs moisture therefrom, slakes, and falls into a powder. Limes and Cements. A Cask of Lime = 240 Ibs., will make from 7.8 to 8.15 cube feet of stiff paste. A Cask of Cement = 300* Ibs., will make from 3.7 to 3.75 cube feet of stiff paste. A Cask of Portland Cement = 4 bushels or 5 cube feet = 420 Ibs. A Cask of Roman Cement = 3 bushels or 3.75 cube feet = 364 Ibs. .5 inch. .75 inch. i inch. A Bushel of cement will cover 2.25 yards 1.5 yards 1.14 yards. From experiments of General Totten, it appeared that i volume of lime slaked with .33 its volume of water gave 2.27 volumes of powder. i " " " .66 " " 1.74 " " i " i 2 .o6 " One cube foot of dry cement, mixed with .33 cube foot of water, will make 63 to .635 cube foot of stiff paste. Lime should be slaked at least one day before it is incorporated with the sand, and when they are thoroughly mixed, the mortar should be heaped into one volume or mass, for use as required. * 300 Ibs. net is standard ; it usually overruns 8 Ibs. LIMES, CEMENTS, MORTARS, AND CONCRETES. 595 Mortar, Cement, &c. (Molesworth.) Mortar. i of lime to 2 to 3 of sharp river sand. Or, i of lime to 2 sand and i blacksmith's ashes, or coarsely ground coke. Coarse Mortar. i of lime to 4 of coarse gravelly sand. Concrete.^i of lime to 4 of gravel and 2 of sand. Hydraulic Mortar. i of blue lias lime to 2.5 of burnt clay, ground to- gether. Or, i of blue lias lime to 6 of sharp sand, i of pozzuolana and i of calcined ironstone. Beton. i of hydraulic mortar to 1.5 of angular stones. Cement. i of sand to i of cement. If great tenacity is required, the ce- ment should be used without sand. Portland Cement Is composed of clayey mud and chalk ground together, and afterwards cal- cined at a high temperature after calcining it is ground to a fine powder. Strength, of Mortars, Cements, and Concretes. Deduced from Experiments of Vicat, Paisley, Treussart, and Voisin. Tensile Weight or Power required to Tear asunder One Sq. Inch. Cement Mortar. (42 days old.) Proportion of Sand to i of Cement. i t 2 roport 3 Roman Portland 284 142 284 142 199 "3 166 92 5 6 7 8 9 xo 128 67 116 57 106 42 99 35 92 25 95 lbs. 79 Brick, Stone, and. Q-ranite Masonry. (320 days old.) Experiments of General Gillmore, U. S. A. Cement on Bricks. Cement on Granite. Lba. Pure, average 30.8 Pure Lb6. Lbs. Sand i ) Sand i ) Cement i J * 5 ' 7 Sand i J Sand i Cement i Sand i .... 20.8 . . 12 6 Cement 4 j 7 '9 ^ ateri i . ..205 Cement 2) Water .42) .. 3725 Cement 2 J I2 ' 3 and ' 1 68 Cement 2 ' ' Sand i Cement i J " Water .33) Cements) Delafield and Baxter. Lbs. Pure cement 68 Cement 3 * ' James River. Pure cement. . Cement 4 ) Lbs. 87 .. rfv> Cement i J ' ' Lbs. Neivark and Rosendale. Cement i ) Cement 4 ) 6g Sand i J ' ' Cement 8 ) ~ Sand i f ' ' Newark Lime and Cement Co. Pure cement 93 Sand 3 ) * ' Pure, without) mortar, mean j ** Mortar. Lime paste i, sand 2.5, 6 " " I, " 2 4 I! " i, " 3 6 " " , " 3, cement paste 5 it Siftings i j ' ' Cement i ) Siftings i J * * Cement i ) Cement i ) Sand 2 } 4 Newark and Rosendale. Pure cement *e Siftings 2 J ' ' Lawrence Cement Co. Pure cement 87 Cement i ) j6 " 54 Sand i j ' ' * 596 LIMES, CEMENTS, MOKTAKS, AND CONCRETES. Pure Cement. Boulogne 100, water 50 112 Portland, natural, i year 675 " artificial, Eiig., i year... 462 " English, 320 days 1152 " " i month 393 Newark and Rosendale 339 Portland, in sea- water, 45 days 266 ' ' English, 6 months 424 Roman "Septaria," i year 191 " masonry, 5 months 77 Rosendale, o months 700 Lawrence Cement Co 1210 Transverse. Reduced to a uniform Measure of One Inch Square and One Foot in Length. Supported at Both Ends. Experiments of Greneral Grillmore. Formed in molds under a pressure of 32 Ibs. per sq. inch, applied until mortar had set. Exposed to moisture for 24 hours, and then immersed in sea- water. Prisms 2 by 2 by 8 ins. between supports. Reduced by Formula - W a : C. C coefficient of rupture, and a weight of bd 2 2 ~ portion of prism I. Cement. Mortar. MATERIAL. t Z MATERIAL. 1 "S M 1" "a ** S* 3l SI Days. Lbs. Days. Lbs. T,h. James River. Portland, Eng., stiff paste 320 13 10 Thick cream . . > 3 Q Roman " " " 2 C Thin paste 320 5.8 6 Stiff paste 6 Q Cumberland Md 12 8 8 Rosendale "Hoffman " Akron, N Y 320 8 8 8 4 Thin paste James River Va 8 6 8 8 Stiff paste 320 ii Pulverized and re- ) * " Delafield and Baxter." mixed after set i 3 3- 6 Thin paste . 320 8.5 Fresh Stiff paste 020 12 Kingston and Rosendale. 76 6.6 English. tf\ High Falls, UM 95 3-2 Stiff paste 320 320 13 Fresh water to a stiff ) Cumberland, Md., pure High Falls, UM 320 95 13-2 8. 4 paste ) 95 95 4.4 2.6 Sea-water to a stiff paste Lawrence Cement Co. sterCo.,N.Y. j Comolete calcination. . . Q< 4.2 Fresh. . . 32O IO.2 Crnsliing. Cements, Stones, etc. (Crystal Palace, London.) Reduced to a uniform Measure of One Sq. Inch. MATERIAL. Destructive Pressure. MATERIAL. Destructi Pressure Portl'd cem't, area i, height i. Lbs. 1680 Portland coment i ) Lbs. 1244 " cement ) 1244 " sand. . . ) " " cement i ) 692 Roman cement, pure. . . , 14.2 G-eneral Deductions. i. Particles of unground cement exceeding .0125 of an inch in diameter may be allowed in cement paste without sand, to extent of 50 per cent, of whole, without detriment to its properties, while a corresponding proportion of sand injures the strength of mortar about 40 per cent. LIMES, CEMENTS, MORTARS, ETC. MASONRY. 597 2 When these unground particles exist in cement paste to extent of 66 per cent, of whole, adhesive strength is diminished about 28 per cent. For a corresponding proportion of sand the diminution is 68 per cent. 3. Addition of siftings exercises a less injurious effect upon the cohesive than upon the adhesive property of cement. The converse is true when sand, instead of sift- ings, is used. 4. In all mixtures with siftings, even when the latter amounted to 66 per cent, of whole, cohesive strength of mortars exceeded their adhesion to bricks. Same re- sults appear to exist when siftings are replaced by sand, until volume of the latter exceeds 20 per cent, of who^ after which adhesion exceeds cohesion. 5. At age of 320 days (and perhaps considerably within that period) cohesive strength of pure cement mortar exceeds that of Croton front bricks. The converse is true when the mortar contains 50 per cent, or more of sand. 6. When cement is to be used without sand, as may be the case when grouting is resorted to, or when old walls are to be repaired by injections of thin paste, there is no advantage in having it ground to an impalpable powder. 7. For economy it is customary to add lime to cement mortars, and this may be done to a considerable extent when in positions where hydraulic activity and strength are not required in an eminent degree. 8. Ramming of concrete under water is held to be injurious. 9. Mortars of common lime, when suitably made, set in a very few days, and with such rapidity that there is no need of awaiting its hardening in the prosecution of work. Fire Clay. The fusibility of clay arises from the presence of impurities, such as lime, iron, and manganese. These may be removed by steeping the clay in hot muriatic acid, then washing it with water. Crucibles from common clay may be made in this manner. Notes by General Gillmore, U. S. A. Recent experiments have developed that most American cements will sustain, without any great loss of strength, a dose of lime paste equal to that of the cement paste, while a dose equal to 5 to 75 the vol- ume of cement paste may be safely added to any Rosendale cement without pro- ducing any essential deterioration of the quality of the mortar. Neither is the hydraulic activity of the mortars so far impaired by this limited addition of lime paste as to render them unsuited for concrete under water, or other submarine masonry. By the use of lime is secured the double advantages of slow setting and economy Notes by General Totten, V. S. A 240 Ibs. lime = i cask, will make from 7.8 to 8. 15 cube feet of stiff paste. i cube foot of dry cement powder, measured when loose, will measure . 78 to 8 cube foot when packed, as at a manufactory. For composition of Concretes, at Toulon, Marseilles, Cherbourg, Dover, Alderney, etc., see Treatise of General Gillmore, pp. 253-256. MASONRY. Brickwork. Bond is an arrangement of bricks or stones, laid aside of and above each other, so that the vertical joint between any two bricks or stones does not coincide with that between any other two. This is termed "breaking joints." Header is a brick or stone laid with an end to face of wall. Stretcher is a brick or stone laid parallel to face of wall. Header Course or Bond is a course or courses of headers alone. Stretcher Course or Bond is a course or courses of stretchers alone. Closers are pieces of bricks inserted in alternate courses, in order to obtain a bond by preventing two headers from being exactly over a stretcher. English Bond is laying of headers and stretchers in alternates courses. 598 MASONRY. Flemish Bond is laying of headers and stretchers alternately in each course, Gauged Work. Bricks cut and rubbed to exact shape required. String Course is a horizontal and projecting course around a building. Corbelling is projection of some courses of a wall beyond its face, in order to support wall-plates or floor-beams, etc. Wood Bricks, Pallets, Plugs, or Slips are pieces of wood laid in a wall in order the better to secure any woodwork that it may be necessary to fasten to it. Reveals are portions of sides of an opening in a wall in front of the recesses for a door or window frame. Brick Ashlar. Walls with ashlar-facing backed with brick. Grouting is pouring liquid mortar over last course for the purpose of filling all vacuities. Larrying is filling in of interior of thick walls or piers, after exterior faces are laid, with a bed of soft mortar and floating bricks or spawls in it. Rendering (Eng.) is application of first coat on masonry, Laying if one or two coats on laths, and " Pricking up " if three-coat work on laths. Bricks should be well wetted before use. Sea sand should not be used in the composition of mortar, as it contains salt and its grains are round, being worn by attrition, and consequently having less tenacity than sharp-edged grains. A common burned brick will absorb i pint or about one sixth of its weight of water to saturate it. The volume of water a brick will absorb is inversely a test of its quality. A good brick should not absorb to exceed .067 of its weight of water. The courses of brick walls should be of same height in front and rear, whether front is laid with stretchers and thin joints or not In ashlar- facing the stones should have a width or depth of bed at least equal to height of stone. Hard bricks set in cement and 3 months set will sustain a pressure of 40 tons per sq. foot. The compression to which a stone should be subjected should not exceed . i of its crushing resistance. The extreme stress upon any part of the masonry of St. Peter's at Rome is com- puted at 15.5 tons per sq. foot ; of St. Paul's, London, 14 tons ; and of piers of New York and Brooklyn Bridge, 5. 5 tons. The absorption of water in 24 hours by granites, sandstones, and limestones of a durable description is i, 8, and 12 per cent, of volume of the stone. Color of Bricks depends upon composition of the clay, the molding sand, tem- perature, of burning, and volume of air admitted to kiln. Pure clay free of iron will burn white, and mixing of chalk with the clay will produce a like effect. Presence of iron produces a tint ranging from red and orange to light yellow, according to proportion of iron. A large proportion of oxide of iron, mixed with a pure clay, will produce a bright red, and when there is from 8 to 10 per cent., and the brick is exposed to an intense heat, the oxide fuses and produces a dark blue or purple, and with a small volume of manganese and an increased proportion of the oxide the color is darkened, even to a black. Small volume of lime and iron produces a cream color, an increase of iron pro- duces red, and an increase oflime brown. Magnesia in presence of iron produces yellow. Clay containing alkalies and burned at a high temperature produces a bluish green. For other notes on materials of masonry, their manipulation, etc., see "Limes, Cements, Mortars, and Concretes," pp. 588-597. Pointing. Before pointing, the joints should be reamed, and in close ma- sonry they must be open to 2 of an inch, then thoroughly saturated with water, and maintained in a condition that they will neither absorb water from the mortar or impart any to it. Masonry should not be allowed to dry rapidly after pointing, but it should be well driven in by the aid of a calking iron and hammer. In pointing of rubble masonry the same general directions are to be observed. MASONEY. 599 Sand is Argillaceous, Siliceous, or Calcareous, according to its composition. Its use is to prevent excessive shrinking, and to save cost of lime or cement Or- dinarily it is not acted upon by lime, its presence in mortar being mechanical, and with hydraulic limes and cements it weakens the mortar. Rich lime adheres better to the surface of sand than to its own particles; hence the sand strengthens the mortar. It is imperative that sand should be perfectly clean, freed from all impurities, and of a sharp or angular structure. Within moderate limits size of grain does not affect the strength of mortar; preference, however, should be given to coarse. Calcareous sand is preferable to siliceous. Sea and River sand are suitable for plastering, but are deficient in the sharpness required for mortar, from the attrition they are exposed to. Clean sand will not soil the hands when rubbed upon them, and the presence of gait can be detected by its taste. Scoriae, Slag, Clinker, and Cinder, when properly crushed and used, make good substitutes for sand. Concrete. In the mixing of concrete, slake lime first, mix with cement, and then with the chips, etc., deposit in layers of 6 ins., and hammer down. Bricks. Variations in dimensions by various manufacturers, and different degrees of intensity of their burning, render a table of exact dimensions of different manufactures and classes of bricks altogether impracticable. As an exponent, however, of the ranges of their dimensions, following averages are given : DESCRIPTION. Ins. DESCRIPTION. In. Baltimore front Maine . 7 e y o 571: V. 2 77C Philadelphia " 8.25 X 4-125 X 2.375 Milwaukee 8. c X 4 125 X 2 375 Wilmington " Croton 8.5 X4 X2.25 North River Ordinary 8 X3-5 X2.2 5 (7-75 X 3-625X2.25 Eng. ordinary... " Lond. stock Dutch Clinker. . . 8.25 X 3-625 X 2.375 9 X4-5 X2. 5 8.75X4-25 X2. 5 6.25X3 Xi.5 Stourbridge ) fire-brick. . . . j Amer. do.,N. Y.. (8 X 4- 125X2.5 9.125X4-625X2.375 8-875X4-5 Xa.625 In consequence of the variations hi dimensions of bricks, and thickness of the layer of mortar or cement in which they may be laid, it is also impracti- cable to give any rule of general application for volume of laid brick-work. It becomes necessary, therefore, when it is required to ascertain the volume of bricks in masonry, to proceed as follows : To Compute Volume of Bricks, and. Number in a Cube Foot of 3VIason.ry. RULE. To face dimensions of particular bricks used, add one half thick- ness of the mortar or cement in which they are laid, and compute the area ; divide width of wall by number of bricks of which it is composed ; multiply this area by quotient thus obtained, and product will give volume of the mass of a brick and its mortar in ins. Divide 1728 by this volume, and quotient will give number of bricks in a cube foot. EXAMPLE. Width of a wall is to be 12.75 ins., and front of it laid with Philadel- phia bricks in courses .25 of an inch in depth; how many bricks will there be in face and backing in a cube foot? Philadelphia front brick, 8.25 x 2.375 ins. face. 8. 25 + . 25 X 2-7-2 = 8. 25 + . 25 = 8. 5 = length of brick and joint ; 2-375 -f- -25 X 2 -r- 2 = 2.375 -f- .25 = 2.625 = width of brick and joint. Then 8.5 x 2.625 = 22.3125 ins. area of face; 12.75 -5-3 (number of bricks in width of wall) = 4. 25 ins. Hence 22.3125 x 4.25 = 94.83 cube ins. ; and 1728 -7-94. 83 = 18.22 brickt. 600 MASONRY. One rod of brick masonry (Eng.)r^ 11.33 cube yards and weighs 15 tons, or 272 superficial feet by 13.5 thick, averaging 4300 bricks, requiring 3 cube yards mortar and 120 gallons water. Bricklayers' hod will contain 16 bricks or .7 cube feet mortar. Fire-clay contains Silica, Alumina, Oxide of Iron, and a small proportion of Lime, Magnesia, Potash, and Soda. Its fire-resisting properties depend- ing upon the relative proportions of these constituents and character of its grain. A good clay should be of a uniform structure, a coarse open grain, greasy to the hand, and free from any alkaline earths. The Stourbridge clay is black and is composed as follows : Silica ..... 63.3 | Alumina ..... 23.3 | Lime ...... 73 | Protoxide of iron.... 1.8 Water and organic matter ...... . . 10.3 Newcastle clay is very similar. Thickness of Brick \Valls for \Varehcmses in Feet. (Molesworth.) Height in Feet IOO 9 80 7 60 5 40 30 Length Unlimited Thickness in Ins 34 34 30 26 26 26 21.5 '7-5 Length in Feet ..... 70 7 60 45 50 70 60 5 Thickness in Ins 30 30 26 21.5 21-5 21.5 17.5 Length in Feet 55 60 45 3 35 4 3 45 Thickness in Ins 26 26 21.5 i7-5 i7-5 '7-5 13 13 Stone Masonry. Masonry is classed as Ashlar or Rubble. Ashlar consists of blocks dressed square and laid with close joints. Coursed Ashlar consists of blocks of same height throughout each course. Rubble is composed of small stones irregular in form, and rough. Rubble Ashlar is ashlar faced stone with rubble backing. Ashlar. Fig. 2. Fig. i. Fig. i. Coursed, with chamfered and rusticated quoins and plinth. Fig. 3- Fig. 2. Coursed, with rock face and draft edges. Fig. 4. ?:..*.*>.'[%: '*-;\w.,w'jt:f\ 1 1 1] 1 \ 1 1 l\ 1 1 1 ! I Fig. 3. Coursed, with rock face. Fig. 4. Regular Courted. MASONRY. 6O I 2 1 1 1 landomed Ashlar. -n Fig-* m h-M 1 ' i ' r ' i i i H 3 -^vw i * 1 1 i i \ i ' i ' i ' 3 MH HH Fig- 5- Fig. 5. Irregular Coursed. Fig. 6. Random Coursed. Fig. 7. | 1 1 T~l V Fi S 8. ffl Fig. 7. Ranged Random, level, and broken coursea Fig. 9. Fig. 8. Random, level, and broken. le. Fig. io. Fig. 9. Block Coursed. f^arge blocks in courses (regular or irregular), Beds and Joints roughly dressed. Fig. xx. Fig. io. Coursed and Ranged Random. Fig. 12. Fig. ii. Ranged Random. Squared rubble laid in level and broken courses. Dry Tl\ is a wall laid without cement or mortar. Fig- 13- (TT Fig. 12. Coursed Random. Stones laid in courses at intervals of from 12 to 18 ins. in height. Fig. 13. Dry Rubble. Without mor- tar or cement. Fig. 15. Tig. 14. Rustic or .Rap. Stones of irregular form, and dressed to make close joints. Fig. 16. Fig. 15. ITncoursed or Random. Beds and Joints undressed, projections knocked off, and laid at random. In- terstices filled with spalls and mortar. NOTE Rustic or Rag work is frequently laid in mortar 3 E Fig. 16. Laced Coursed. Horizontal bands of stone or bricks, interposed to give stability. 6O2 MASONEY. Terra Cotta. Terra Cotta in blocks should not exceed 4 cube feet in volume. When properly burned, it is unaffected by the atmosphere or by fumes of any acid. and. AValls. Singing. Point a, Fig. 15, on each side, Fig. 15. from which arch springs. Crown. Highest point of arch. Haunches. Sides of arch, from springing half-way up to crown. Spandrel. Space between extrados,a hor- izontal line drawn through crown and a ver- tical line through upper end of skewback. Skewback is upper surface of an abut- ment or pier from which an arch springs, and its face is on a line radiating from centre of arch. Abutment is outer body that supports arch and from which it springs. Pier is the intermediate support for two or more arches. Jambs are sides of abutments or piers. Voussoirs are the blocks forming an arch. Key-stone is centre voussoir at crown. Span is horizontal distance from springing to springing of arch. Rise. Height from springing line to under side of arch at key-stone. length is that of springing line or span. Ring-course of a wall or arch is parallel to face of it, and in direction of its span. String and Collar courses are projecting ashlar dressed broad stones at right angles to face of a wall or arch, and in direction of its length. Camber is a slight rise of an arch as .125 to .25 of an inch per foot of span. Quoin is the external angle or course of a wall. Plinth is a projecting base to a wall. Footing is projecting course at bottom of a wall, in order to distribute its weight over an increased area. Its width should be double that of base of wall, diminishing in regular offsets .5 width of their height. Blocking Course. A course placed on top of a cornice. Parapet is a low wall, over edge of a roof or terrace. Extrados. Back or upper and outer surface of an arch. Intrados or Soffit is underside of lower surface of arch or an opening. Groined is when arches intersect one another. Invert. An inverted arch, an arch with its intrados below axis or spring- ing line. Ashlar masonry requires .125 of its volume of mortar. Rubble, 1.2 cube yards stone and .25 cube yard mortar for each cube yard. Rubble masonry in cement, 160 feet in height, will stand and bear 20 ooo \bs. per sq. inch. Stones should be laid with their strata horizontal. When " through " or " thorough bonds " are not introduced, headers should overlap one another from opposite sides, known as dogs' tooth bond. Aggregate surface of ends of bond stones should be from .125 to .25 of area of each face of wall. Weak stones, as sandstone and granular limestone, should not have a length over 3 times their depth. Strong or hard stones may have a length from 4 to 5 times their depth. MASONRY. 603 Gdllets are small and sharp pieces of stone stuck into mortar joints, in which case the work is termed galleted. Snapped work is when stones are split and roughly squared. Quarry or Rock-faced. Quarried stones with their faces undressed. Pitch-faced. Stones on which the arris or angles of their face, with their sides and ends, is defined by a chisel, in order to show a right-lined edge. Drafted or Drafted Margin is a narrow border chiselled around edges of faces of a block of rough stone. Diamond-faced is when planes are either sunk or raised from each edge and meet in the centre. Squared Stones. Stones roughly squared and dressed. Rubble. Unsquared stones, as taken from a quarry or elsewhere, in their natural form, or their extreme projections removed. Cut Stones. Stones squared and with dressed sides and ends. Dressed. Stones. The following are the modes of dressing the faces of ashlar in engineering: Rough Pointed. Rough dressing with a pick or heavy point. Fine Pointed. Rough dressing, followed by dressing with a fine point. Crandalled. Fine pointing in right lines with a hammer, the face of which is close serried with sharp edges. Cross Crandalled. When the operation of crandalling is right angled. Hammered. The surface of stone may be finished or smooth dressed by being Axed or Bushed; the former is a finish by a heavy hammer alike to a crandall, the latter is a final finish by a heavy hammer with a face serried with sharp points at right angles. Thickness of* Brick: \Valls for "Warehouses. (Molesworth.) Length. Height. Thickness. Length. Height. Thickness. Length. Height. Thickne Feet. Feet. Ins. Feet. Feet. Ins. Feet. Feet. Ins. Unlimited. 25 '3 Unlimit'd. 100 34 45 3 13 do. 30 17-5 60 40 '7-5 30 40 13 do. 40 21.5 70 21.5 40 So 17-5 do. 50 26 50 60 21.5 35 60 17-5 do. do. 60 70 26 26 i 70 80 21.5 26 45 70 80 '7-5 21.5 do. 80 30 70 90 30 60 9 26 do. 90 34 70 100 30 55 100 26 For drawings and a description of stone-dressing tools, see a paper by J. R. Cross, W. E. Merrill, and E. B. Van Winkle, U A. S. Civil Engineer Transactions," Nov. 1877. Walls not exceeding 30 feet in height, upper story walls may be 8.5 ins. thick. From 16 feet below top of wall to base of it, it should not be less than the space defined by two right lines drawn from each side of wall at its base to 16 feet from top. Thickness not to be less in any case than one fourteenth of height of story. Laths. Laths are 1.25 to 1.5 ins. by 4 feet in length, are usually set .25 of an inch apart, and a bundle contains 100. 604 MASONRY. Plastering. Volumes required for Various Thickness. MATKEIAL. Sq 5 uare Yar 75 ds. i MATERIAL. Sq uare Yar 75 di. Cube Feet. 2.25 4-5 6-75 Ins. i-5 3 4-5 Ins. I-I5 2.25 3-33 Cube Feet. Lime i, sand 2, ) hair 3 75 i "" Ins. 75 ya dered brick rds, sup and s or 7001 Ins. '1 ren- et on i lath. Cement i,sand i... Cement i, sand 2... Kstiinate of Materials and Labor for 1OO Sq.. Yards of Lath, and blaster. Materials and Labor. Three Coats Hard Finish. Two Coats Slipped. Materials and Labor. Three Coats Hard Finish. Two Coats Slipped. Lime 4 casks. 3. 5 casks White sand 2 5 bushels Lump lime .66 " Nails 13 Ibs 13 Ibs Plaster of Paris.. 5 " Masons ....... 4 days. 3. 5 days. Hair 4 bushels -3 bushels Laborer 3 u 2 " Sand . . , 7 loads. 6 loads. Cartage... i " .?; " Rough Cast is washed gravel mixed with hot hydraulic lime and water and applied in a semi-fluid condition. -A^rclies and .AJbutments. To Compute IDepth of Keystone of Circular or Elliptic Arch. f- .25 d. R representing radius, s span, and d depth, all in feet. This is for a rise of about .25 of span ; when it is reduced, as to . 125, add .5 instead of .25. ILLUSTRATION. Arch of Washington aqueduct at "Cabin John" has a span of 220 feet, a rise of 57.25, and a radius of 134.25; what should be depth of its keystone? 4 4 Vtaducts of several arches increase results as determined above by add- ing .125 to .15 to depth. For arches of 2d class materials and work, and for spans exceeding 10 feet, add .125 to depth of keystone, and for good rubble or brick- work add .25. NOTE. It is customary to make the keystones of elliptic arches of greater depth than that obtained by above formula. Trautwine, however, who is high authority in this case, declares it is unnecessary. To Compute Radius of an Arch, Circular or Ellipse, f j -|-r 2 -:-2r = R. r representing rise. Rail-way Arches. For Spans between 25 and 70 feet. Rise .2 of span. Depth of arch .055 of span. Thickness of abutments . 2 to . 25 of span, and of pier . 14 to . 16 of span. Altmtments. When height does not exceed i. 5 times base. R-t-$-}-.ir-}-2 = thickness at spring qf arch in feet. (Trautwine.) Batter. From .5 to 1.5 ins. per foot of height of wall. MASONRY. MECHANICAL CENTRES. GRAVITY. 605 To Compute Depth of* Arch. (Hurst.) c ^R = D. c = Stone (block) .3. Brick = .4. Rubble = .45. When there are a series of arches, put .3 = .35, .4 = .45, and .45 .5. Mininaviixi Thickness of* AJtmtments for Bridge and similar Arches of ISO . (Hurst.) When depth of crown does not exceed 3 feet. Computed from formula v /6 R 4- ( ^-= ) 2 = T. H representing height of abutment to springing in feet V \2 H/ z U Radius of Arch. Heif 5 rht of At 7-5 utment 10 to Spring 20 ing. 30 Radius of Arch. Heig 5 ht of Ab 7-5 utment t 10 o Spring 20 in*. 30 Feet. 4 Feet. 3-7 Feet. 4.2 Feet. 4-3 Feet. 4.6 Feet. 4-7 12 5-6 6.4 6.9 7.6 Feet. 7-9 4-5 3-9 4.4 4.6 4-9 5 IS 6 7 7-5 8.4 8.8 5 4.2 4.6 4.8 5-1 5-2 20 6-5 7-7 8.4 9.6 10 6 4-5 4-7 5-2 5-6 5-7 25 6.9 8.2 9.1 0-5 ii. i 7 4-7 5-2 5-5 6 6.1 30 7-2 8.7 9-7 1.4 12 8 4.9 5-5 5-8 6.4 6-5 35 7-4 9.1 10.2 1.8 12.9 9 5-i 5.8 6.1 6.7 6.9 40 7.6 9.4 10.6 2.8 13-6 10 5-3 6 6.4 7-i 7-3 45 7.8 9-7 ii 3-4 14-3 ii 5-5 6.2 6.6 7-3 7.6 So 7-9 10 11.4 4 15 NOTE. Abutments in Table are assumed to be without counterforts or wing- walls. A sufficient margin of safety must be allowed beyond dimensions here given. Culverts for a road having double tracks are not necessarily twice the length tor a single track. For other and full notes, tables, etc., see Trautwine's Pocket Book, pp. 693-710. MECHANICAL CENTRES. There are four Mechanical centres of force in bodies, namely, Centre of Gravity, Centre of Gyration, Centre of Oscillation, and Centre of Percussion. Centre of GS-ravity. CENTRE OP GRAVITY of a body, or any system of bodies rigidly con- nected together, is point about which, if suspended, all parts will be in equilibrium. A body or system of bodies, suspended at a point out of centre of gravity, will rest with its centre of gravity vertical under point of suspension. A body or system of bodies, suspended at a point out of centre of gravity, and successively suspended at two or more such points, the vertical lines through these points of suspension will intersect each other at centre of gravity of body or bodies. Centre of gravity of a body is not always within the body itself. If centres of gravity of two bodies, as B C, be connected by a line, dis- tances of B and C from their common centre of gravity, c, is inversely as the weights of the bodies. Thus, B : C : : C c : c B. To Ascertain Centre of Gravity of any Plane Figure Mechanically. Suspend the figure by any point near its edge, and mark on it direction of a plumb-line hung from that point ; then suspend it from some other point, and again mark direction of plumb-line. Then centre of gravity of surface will be at point of intersection of the two marks of plumb-line. 3E* 606 MECHANICAL CENTRES. GRAVITY. Centre of gravity of parallel-sided objects may readily be found in this way. For instance, to ascertain centre of gravity of an arch of a bridge, draw elevation upon paper to a scale, cut out figure, and proceed with it as above directed, in order to find position of centre of gravity in elevation of the model. In actual arch, centre of gravity will have same relative position as in paper model. In regular figures or solids, centre of gravity is same as their geometrical centres. Line. Circular Arc. =- =. distance from centre, r representing radius, c chord, and I length of arc. Surfaces. Square, Rectangle, Rhombus, Rhoinboid, Gnomon, Cube, Regular Polygon, Circle, Sphere, Spheroid or Ellipsoid, Spheroidal Zone, Cylinder, Circular Ring, Cylindrical Ring, Link, Helix, Plain Spiral, Spindle, all Regular Fig- ures, and Middle Frusta of all Spheroids, Spindles, etc. The centre of gravity of the surfaces of these figures is in their geometri- cal centre. Triangle. On a line drawn from any angle to the middle of opposite side, at two thirds of the distance from angle. Trapezium. Draw two diagonals, and ascertain centres of gravity of each of four triangles thus formed > join each opposite pair of these centres, and it is at intersection of the lines. Trapezoid. I rT_L^ ) X = distance from B on a line joining middle of two parallel sides Eb,m representing middle line. Circular Arc. -y- = distance from centre of circle. Sector of a Circle. ,4244 r = distance from centre of circle, c representing chord. Semicircle. .4244 r = distance from centre. Semi-semicircle. .4244 r = distance from both base and height and at their inter- section. Segment of a Circle. = distance from centre, a representing area of segment. Sector of a Circular Ring. X " ^ X 2 _ t , 2 = distance from centre of arcs, r and r' representing the radii. ILLUSTRATION. Radii of surfaces of a dome are 5 and 3.5 feet, and angle ) at centre = 130. 4 sin. 65 125-42.875 _ 4 .9063 82.125 X ~ X = X 2o~ X ~ = 3-437 eec - 3 arc 130 25 12.25 3 2.2609 I2 -75 Hemisphere, Spherical Segment, and Spherical Zone, At centre of their heights. Circular Zone. Ascertain centres of gravity of trapezoid and segments comprising zone ; draw a line (equally dividing zone) perpendicular to chords; connect centres of segments by a line cutting perpendicular to chords. Then centre of gravity of figure will be on perpendicular, toward lesser chord, at such proportionate distance of difference between centres of gravity of trapezoid and line connecting centres of segments, as area of segments bears to area of trapezoid. MECHANICAL CENTRES. - GRAVITY. 6O/ Prism and Wedge. When end is a Parallelogram, in their geometrical centres ; when the end is a Triangle, Trapezium, etc., it is in middle of its length, at same distance from base, as that of triangle or trapezoid of which it is a section. Parabola in its axis = .6 distance from vertex. Prismoid.At same distance from its base as that of the trapezoid or trapezium, which is a section of it. Lum.On a line connecting centres of gravity of arcs at a proportionate point to respective areas of arcs. Co-ordinates. \r-\-r' -- ~r~>) ~ z > * Solids. Cube, Parallelopipedon, Hexahedron, Octahedron, Dodecahedron, Icosahe-* dron, Cylinder, Sphere, Right Spherical Zone, Spheroid or Ellipsoid, Cylin- drical Ring, Link, Spindle, all Regular Bodies, and Middle Frusta of all Spheroids and Spindles, etc. Centre of gravity of these figures is in their geometrical centre. Tetrahedron. In common centre of centres of gravity of the triangles made by a section through centre of each side of the figures. Cone and Pyramid. . 25 of line joining vertex and centre of gravity of base = dis- tance from base. / r _j_ r /\2_j_ 2 r 2 t Frustum of a Cone or Pyramid. ' - , X - h = distance from centre of lesser end, r and r', in a cone representing radii, and in a pyramid sides, and A height. Cone, Frustum of a Cone, Pyramid, Frustum of a Pyramid, and Ungula. At same distance from base as in that of triangle, parallelogram, or semicir- cle, which is a right section of them. Hemisphere. . 375 r = distance from centre. Spherical Segment. 3. 1416 vs 2 (**) -*- v = distance from centre, vs repre- senting versed sine, and v volume of segment. I r ~ 3 .] X h = distance from \i2r- 4 hf vertex. Spherical Sector. .75 (r . 5 h) = d istance from centre. jjt3_ distance 8 from vertex. Spirals. Plane, in its geometrical centre. Conical, at a distance from the base, .25 of line joining vertex and centre of gravity of base. r 2 _ r ' 2 Frustum of a Circular Spindle. j- =p- = distance from centre of spindle, h representing distance between two bases, D distance of centre of spindle from centre of circle, and z generating arc, expressed in units of radius. r 2 Segment of a Circular Spindle. - - = distance from centre of spindle. 2 (h U. z) Semi-spheroids. Prolate. .375 a. Oblate. .375 a = distance from centre. Semi-spheroid or Ellipsoid and its Segment. See HaswelVs Mensuration, pages 281 and 282. Frusta of Spheroids or Ellipsoids. Prolate. .75 -^^ = distance from centre of spheroid, a representing semi-transverse diameter in a prolate frustum, and semi-conjugate in an oblate frustum. 608 MECHANICAL CENTRES. GRAVITY. Segments of Spheroids. Prolate. . 75 Oblate. , 75 dietanct from centre of spheroid, d and d' representing distances of base of segments froih centre of spheroid. Any Frustum. . 75 "*" ^^-d 2 = distance ^ rom centre f *P he ~ roid, d and d' representing distances of base and end of segments from centre of the spheroid. Segment of an Elliptic Spindle at two thirds of height from vertex. Paraboloid of Revolution, at two thirds of height from vertex. Segment of a Hyperbolic Spindle, at 75 of height from vertex. 2 r 2 -\- r h Frustum of Paraboloid of Revolution. 2 _T , X - = distance from base^ r and r' representing radii of base and vertex. Segment of Paraboloid of Revolution, at two thirds of height from vertex. Segments of a Circular and a Parabolic Spindle. See HaswelVs Mensuration, pages 192 and 199. Parabola. .4 of height = distance from base. Hyperboloid of Revolution. \ , X h = distance from vertex, b representing o b -j- 4 h> diameter of base. Frustum of Hyperboloid of Revolution. . 75 _ = distance from centre of base, a representing semi-transverse axis, or distance from centre of curve to vertex of figure ; d and d' distances from centre of curve to centre of lesser and greater diameter of frustum. Segment of Hyperboloid of Revolution. ^ J] 3 X h =. distance from vertex. Of Two Bodies. = distance from V or volume or area of larger body, d rep- resenting distance between centres of gravity of bodies, and v volume or area of less body. Cycloid. .833 of radius of generating circle = distance from centre of chord of curve. Any Plane Figure. Divide it into triangles, and ascertain .centre of grav- ity of each ; connect two centres together, and ascertain their common cen- tre ; then connect this common centre and centre of a third, and ascertain the common centre, and so on, connecting the last-ascertained common centre to another centre till whole are included, and last common centre will give centre required. Of an Irregular Body of Rotation. Divide figure into four or six equidistant divisions ; ascertain volume of each, their moments with reference to first horizontal plane or base, and then connect them thus : (A + 4 Ai-f-2 A 2 -f-4 A 3 + A 4) = V, A AI, etc., representing volume of divis- ions, and h height of body from base; and ( A+ . X 4 A. + . X . A, + 3 X 4 A 3 + 4 A 4 ) ft =- ^ ^ A + 4 Ai-f-2 A 2 + 4 A 3 + A 4 4 gravity from base. MECHANICAL CENTRES. GYRATION. OOQ Centre of Gfyratioru CENTRE OP GYRATION is that point in any revolving body or system of bodies in which, if the whole quantity of matter were collected, the Angular velocity would be the same ; that is, the Momentum of the body or system of bodies is centred at this point, and the position of it is a mean proportional between the centres of Oscillation and Gravity, If a straight bar of uniform dimensions was struck at this point, the stroke would communicate the same angular velocity to the bar as if the whole bar was collected at that point. The A ngular velocity of a body or system of bodies is the motion of a line connecting any point and the centre or axis of motion : it is the same in all parts of the same revolving body. In different unconnected bodies, each oscillating about a common centre, their angular velocity is as tho velocity directly, and as the distance from the centre inversely. Hence, if their velocities are as their radii, or distances from the axis of motion, their angular velocities will be equal. When a bodv revolves on an axis, and a force is impressed upon it suffi- cient to cause "it to revolve on another, it will revolve on neither, but on a line in the plane of the axes, dividing the angle which they contain ; so that the sine of each part will be in the inverse ratio of the angular velocities with which the bodies would have revolved about these axes separately. Weight of revolving body, multiplied into height due to the velocity with which centre of gyration moves in its circle, is energy of body, or mechani- cal power, which must be communicated to it to give it that motion. Distance of centre of gyration from axis ot motion is termed the Radius of gyration ; and the moment of inertia is equal to product of square of radius of gyration and mass or weight of body. The moment of inertia of a revolving body is ascertained exactly by as- certaining the moments of inertia of every particle separately, and adding them together ; or, approximately, by adding together the moments of the small parts arrived at by a subdivision of the body. To Compute Moment of Inertia of a Revolving Body. RULE. Divide body into small parts of regular figure. Multiply mass or weight of each part by square of distance of its centre of gravity from axis of revolution. The sum of products is moment of inertia of body. NOTE. The value of moment of inertia obtained by this process will be more exact, the smaller and more numerous the parts into which body is divided. To Compute Radius of* GJ-yration of a Revolving Body at>out its Axis of Revolxition. RULE. Divide moment ol inertia of body by its mass, or its weight, and square root of quotient is length of radius of gyration. NOTE. When the parts into which body is divided are equal, radius of gyration may be determined by taking mean of all squares of distances of parts from axis of revolution, and taking square root of their sum. Or, VR 2 -f- r* -4- 2 = G. R and r representing radii. EXAMPLE. A straight rod of uniform diameter and 4 feet in length, weighs 4 Ibs. what is its inertia, and where is its radius or centre of gyration? Each foot of length weighs i lb., and if divided into 4 parts, centre of gyration of each is respectively .5, 1.5, 2.5, and 3.5 feet. Hence, . = inertia, which -4- 4 = 5. 25, and ^/$. 25 = 2. 201 feet radius. 6lO MECHANICAL CENTRES. GYRATION. Following are distances of centres of gyration from centre of motion in various revolving bodies : Straight, uniform Rod or Cylinder or thin Rectangular Plate revolving about one end; length x -5773> and revolving about their centre; length X .2886. The general expression is, when revolving at any point of its length, /" ) . I and I' representing length of the two arms. Circular Plane, revolving on its centre; radius of circle X -7071 ; Circle Plane, as a Wheel or Disc of uniform Thickness, revolving about one of its diameters as an axis ; radius X 5- Solid Cylinder, revolving about its axis; radius x -7071. Solid Sphere, revolving about its diameter as an axis; radius X .6325. Thin, hollow Sphere, revolving about one of its diameters as an axis; radius X 8 164. Surface of sphere . 86 1 5 r. Sphere and Solid Cylinder (vertical;, at a distance from axis of revolution = Vl> 2 -J- 4 r 2 for sphere, and V^ 2 -{- -5 r 2 for cylinder, I representing length of connec- tion to centre of sphere and cylinder. Cone, revolving about its axis; radius of base X .5447; revolving about its ver- tex = Vi2 /i 2 4-3 r 2 -r-2o, h representing height, and r radius of base, revolving about its base= -\/2 h 2 -t-$ r 2 -r-2o. Circular Ring, as Rim of a Fly-wheel or Hollow Cylinder, revolving about its diameter = VR 2 + r 2 -:-2, R representing radius of periphery, and r of inner circle of ring. Fly-wheel = / - - wTi^ 4 7 - - , W and w representing weights of rim and of arms and hub, and I length of arms from axis of wheel. Section of Rim. * -- \-r 2 -\-r d. d representing depth and c periphery of rim. Parallelepiped, revolving about one end, distance from end=r */- , b rep- resenting breadth. ILLUSTRATION. In a solid sphere revolving about its diameter, diameter being 2 feet, distance of centre of gyration is 12 X -6325 = 7.59 ins. To Compxvte Elements of GS-yration. GWjy_ Prtg_ n GWv_ ?rtg_ GWv_ rig ~ Wv ~ Ptg~ G t> ~ Prg~ ? = v. G representing distance of centre of gyration from axis of rotation, W weight of body, t time power acts in seconds, v velocity in feet per second acquired by revolving body in that time, and r distance of point of application of power from axis of body, as length of crank, etc. ILLUSTRATION r. What is distance of centre of gyration in a fly-wheel, power 224 Ibs., length of crank 7 feet, time of rotation 10 seconds, weight of wheel 5600 IDS., and velocity of it 8 feet per second? 224 X 7 X io X 32. 166 _ 504 373 2. What should be weight of a fly wheel making 12 revolutions per minute, its diameter 8 feet, power applied at 2 feet from its axis 84 Ibs., time of rotation 6 sec onds, and distance of centre of gyration of wheel 3.5 feet? 8X3. 1416 X 12 84 X 2 X 6 X 32. 166 - 5.0265 feet velocity. Then -- --- - - = 1843.2 Ib3 60 3-5X5-0265 MECHANICAL CENTRES. GYRATION. 6 1 I When the Body is a Compound one. RULE. Multiply weight of several particles or bodies by squares of their distances in feet from centre of mo- tion or rotation, and divide sum of their products by weight of entire mass ; the square root of quotient will give distance of centre of gyration from centre of motion or rotation. EXAMPLE. If two weights, of 3 and 4 IDS. respectively, be laid upon a lever (which is here assumed to be without weight) at the respective distances of i and 2 feet, what is distance of centre of gyration from centre of motion (the fulcrum) ? = = 2.71, and That is, a single weight of 7 IDS., placed at 1.64 feet from centre of motion, and re- volving in same time, would have same momentum as the two weights in their respective places. When Centre of Gravity is given. RULE. Multiply distance of centre of oscillation from centre or point of suspension, by distance of centre of grav- ity from same point, and square root of product will give distance of centre of gyration. EXAMPLE. Centre of oscillation of a body is 9 feet, and that of its gravity 4 feet from centre ef rotation or point of suspension; at what distance from this point is centre of gyration ? 9 x 4 = 36, and ^36 = 6 feet. To Compute Centre of Gryration. of a "Water- -wheel. RULE. Multiply severally twice weight of rim, as composed of buckets, shrouding, etc., and twice that of arms and that of water in the buckets (when wheel is in operation) by square of radius of wheel in feet ; divide sum by twice sum of these several weights, and square root of quotient will give distance in feet. EXAMPLE. In a wheel 20 feet in diameter, weight of rim is 3 tons, weight of arms 2 tons, and weight of water in buckets i ton; what is distance of centre of gyration from centre of wheel ? Rim =3 tonsx io 2 X 2= 600 3-{-2-f-i X 2 = 12 sum of weights. Buckets = 2 tons X 102 x 2 = 400 Water = iton X io 2 = 100 TT Hence GENERAL FORMULAS.? representing power, H Worses' power, F force applied to rotate body in Ibs., M mass of revolving body in Ibs., r radius upon which F acts in feet, d distance from axis of motion to centre of gyration in feet, t time force is ap- plied in seconds, n number of revolutions in time t, x angular velocity, or number of revolutions per minute at end of time t, and G = 32 '' . /4prn_ f 2 pr*x Mod 2 _ M^n d* _ 2. 5 6< 2 Fr_ V G =f| 6oG ~ ' 153.5 tr~ ' 2. 5 6< 2 F~ Md 2 ~~Wd 2 ~~ =X] ~x^~d 2 ~ == ^' t 244* " :?; i 3 4ioot = H< ILLUSTRATION. Rim of a fly-wheel weighing 7000 Ibs. has radii of 6.5 and 5.75 feet; what is its centre of gyration, and what force must be applied to it 2 feet from axis of motion to give it an angular velocity of 130 revolutions per minute in 40 seconds? how many revolutions will it make in 40 seconds? and what is its power ? i3Q 2 X 7000 X 6. i4 2 _ 4 459 862 680 134 zoo X 40 5 364 ooo 612 MECHANICAL CENTRES. OSCILLATION, ETC. Centres of Oscillation, and 3?ercnssion. CENTRE OF OSCILLATION of a body, or a system of bodies, is that point in axis of vibration of a vibrating body in which, if, as an equivalent condition, the whole matter of vibrating body was concentrated, it would continue to vibrate in same time. It is resultant point of whole vibrat- ing energy, or of action of gravity in producing oscillation. As particles of a body further from centre of its suspension have greater velocity of vibration than those nearer to it, it is apparent that centre of oscillation is further from its centre than centre of gravity is from axis of suspension, but it is situated in centre of a line drawn from axis of a body through its centre of gravity. It further differs from centre of gyration in this, that while motion of oscillation is produced by gravity of a body, that of gyration is caused by some other force acting at one place only. Radius of oscillation, or distance of centre of oscillation from axis of sus- pension, is a third proportional, to distance of centre of gravity from axis of suspension and radius of gyration. CENTRE OF PERCUSSION of a body, or a system of bodies, revolving about a point or axis, is that point at which, if resisted by an immov- able obstacle, all the motion of the body, or system of bodies, would be destroyed, and without impulse on the point of suspension. It is also that point which would strike any obstacle with greatest effect, and from this property it has been termed percussion. Centres of Oscillation and Percussion are in same point. If a blow is struck by a body oscillating or revolving about a fixed centre, percussive action is same as if its entire mass was concentrated at centre of oscillation. That is, centre of percussion is identical with centre of oscillation, and its position is ascertained by same rules as for centre of oscillation. If an ex- ternal body is struck so that the mean line of its resistance passes through centre of percussion, then entire force of percussion is transmitted directly to the external body ; on the contrary, if a revolving body is struck at its centre of percussion, its motion will be absolutely destroyed, so that the body will not incline either way. As in bodies at rest, the entire weight may be considered as collected in centre of gravity ; so in bodies in vibration, the entire force may be consid- ered as concentrated in centre of oscillation ; and in bodies in motion, the whole force may be considered as concentrated in centre of percussion. If centre of oscillation is made point of suspension, point of suspension will become centre of oscillation. Angle of Oscillation or Percussion is determined by angle delineated by vertical plane of body in vibration, in plane of motion of body. Velocity of a Body in Oscillation or Percussion through its vertical plane. is equal to that acquired by a body freely falling through a vertical line equal in height to versed sine of the arc. To Compute Centre of* Oscillation or Percussion of* a Body of* Uniform. Density and Figure. RULE. Multiply weight of body by distance of its centre of gravity from point of suspension ; multiply also weight of body by square of its length, and divide product by 3. Divide this last quotient by product of weight of body and distance of its centre of gravity, and quotient is distance of centre from point of sus- pension. MECHANICAL CENTRES. OSCILLATION, ETC. 613 Or, -- - W x 9 = distance from axis. Or, square radius of gyration of body and divide by distance of centre of gravity from axis of suspension. EXAMPLE. Where is centre of oscillation in a rod 9 feet in length from its point Of suspension, and weighing 9 Ibs. ? 9 X - = 40. 5 = product of weight and its centre of gravity ; 2 ^- = 243 = quo- tient of product of weight of body and square of its length -r- 3 ; ^^- = 6 feet. When Point of Suspension is not at End of Rod. RULE. To cube of distance of point of suspension from top of rod or bar, add cube of its dis- tance from lower end, and multiply sum by 2. Divide product by three times difference of squares of these distances, and quotient is distance of point of oscillation from point of suspension. EXAMPLE. A homogeneous rod of uniform dimensions, 6 feet In length, is sus- pended 1.5 feet from its upper end; what is distance of point of oscillation from that of suspension ? Centres of Oscillation, and. Fercnssion in Bodies of* "Various Figures. When Axis of Motion is in Vertex of Figure, and when Oscillation or Motion is Facewise. Right Line, or any figure of uniform shape and density = .661 Isosceles Triangle = .75 h. Circle = 1.25 r. Parabola = . 714 h. Cone = .8h. When Axis of Motion is in Centre of Body. Wheel = .75 radius. When Oscillation or Motion is Sidemse. Right Line, or any figure of uni- form shape and density = 66 I Rectangle, suspended at one angle = .66 of di- agonal Parabola, if suspended by its vertex = .7 14 of axis -{-.33 parameter; if suspended by middle of its base = . 57 of axis -|- . 5 parameter. Sector of a Circle = - - - , c representing chord of arc, and r radius of base. 2 r 2 Sphere = -{- r -{- c, c representing length of cord by which it is suspended. To Ascertain Centres of Oscillation and IPercussion experimentally. Suspend body very freely from a fixed point, and make it vibrate in small arcs, .noting number of vibrations it makes in a minute, and let number made in a min- ute be represented by n; then will distance of centre of oscillation from point of 140850 suspension be = 2 = ins. For length of a pendulum vibrating seconds, or 60 times in a minute, being 39.125 ins., and lengths of pendulums being reciprocally as the squares of number of vibrations made in same time, therefore n 2 : 6o 2 : : 39. 125 : -- 39^5 __ ** ^ f being length of pendulum which vibrates n times in a minute, or distance of centre of oscillation below axis of motion. 3F 6 14 MECHANICAL CENTRES. MECHANICS. To Compxite Centres of* Oscillation or 3?ercu.ssion of a System of* Particles or Bodies. RULE. Multiply weight of each particle or body by square of its distance from point of suspension, and divide sum of their products by sum of weights, multiplied by distance of centre of gravity from point of suspension, and quotient will give centre required, measured from point of suspension. W d 2 -f W d' z Or > w _LW ~ = distance of centre. w 9 -f- W g EXAMPLE i. Length of a suspended rod being 20 feet, and weight of a foot in length of it equal 100 pz., has a ball attached at under end weighing 100 oz. ; at what point of rod from point of suspension is centre of percussion ? 100 x 20 = 2000 = weight of rod ; 2000 X = 20000 momentum of rod, or prod- uct of its weight, and distance of its centre of gravity ; 25^2 _ =266666.66 = force of rod ; 1000 X 2o 2 = 400 ooo = force of ball. 266 666. 66 -4- 400 ooo Then - j-^4 = *6.66feet. 20 000 -f- 20 000 2. Assume a rod 12 feet in length, and weighing 2 Ibs. for each foot of its length, with 2 balls of 3 Ibs. each one fixed 6 feet from the point of suspension, and the other at the end of the rod; what is the distance between the points of suspension and percussion ? ^X2X^- = ^ = m omentumofrod 24 X ' g45g 3X6 = 18= " ofistball 3X12 = 36= " of zd ball. 3X 6 2 = 3X36 = io8 "ofittbalL ~i98~*tm of moments. 3X i2 2 = 3 X 144 =432 = " ofzd ball. Then 1692 -f- 198 = 8. 545 feet. 1692 sum of forces. MECHANICS. MECHANICS is the science which treats of and investigates effects of forces, motion and resistance of material bodies, and of equilibrium : it is divided into two parts STATICS and DYNAMICS. STATICS treats of equilibrium of forces or bodies at rest. DYNAMICS of forces that produce motion, or bodies in motion. These bodies are further divided into Mechanics of Solid, Fluid, and Aeri- form bodies ; hence the following combinations : 1. Statics of Solid Bodies, or Geostatics. 2. Dynamics of Solid Bodies, or Geodynamics. 3. Statics of Fluids, or Hydrostatics. 4. Dynamics of Fluids, or Hydrodynamics. 5. Statics of A eriform Bodies, or A erostatics. 6. Dynamics of Aeriform Bodies, Pneumatics or Aerodynamics. Forces are various, and are divided into moving forces or resistances ; as Gravity, Heat or Caloric, Inertia, Muscular, Magnetism, Cohesion, Elasticity and Contractility, Percussion, A dhesion, Central, Expansion, and Explosion. Couple. Two forces of equal magnitude applied to or operating upon same body in parallel and opposite directions, but not in same line of action, constitute a couple, and its force is sum or magnitude of the two equal forces. Moment. Quantity of motion in a moving body, which is always equal to product of quantity of matter and its velocity. When velocities of two moving bodies are inversely as their quantities of matter, their momenta are equal. MECHANICS. STATICS. 6i 5 Fig. i. o STATICS. Composition and. Resolution of Forces. When two forces act upon a body in same or in an opposite direc- tion, effect is same as if only one force acted upon it, being sum or difference of the forces. Hence, when a body is drawn or projected in directions immediately opposite, by two or more unequal forces, it is affected as if it were drawn or projected by a single force equal to difference between the two or more forces, and acting in direction of greater force. This single force, derived from the combined action of two or more forces, is their Resultant. The process by which the resultant of two or more forces, or a single force equivalent in its effect to two or more forces, is determined, is termed the Composition of Forces, and the inverse operation ; or, when combined effects of two or more forces are equivalent to that of a single given force, the process by which they are determined is termed the Decomposition or Resolution of Forces. Two or more forces which are equivalent to a single force are termed Components. When two forces act on same point their intensities are represented by sides of a parallelogram, and their combined effect will be equivalent to that of a single force acting on point in direction of diagonal of parallelogram, the intensity of which is proportional to diagonal. ILLUSTRATION. Attach three cords to a fixed point, c, Fig. i ; let c a and c 6 pass over fixed rollers, and suspend weights A and B therefrom. Point c will be drawn by the forces A and B in directions a c and 6 c. Now, in order to ascertain which single force, P, would produce the same effect upon it, set off the distances c w and c n on the cords in the same proportion of length as weights of A and B ; that is, so that cm: en:: A : B ; then draw par- allelogram cm on and diagonal o c, and it will represent a sin- gle force, P, acting in its direction, and having same ratio to weights A or B as it has to sides c m or c n of parallelogram. Consequently, it will produce same effect on point c as com- bined actions of A and B. A parallelogram, constructed from lateral forces, and diagonal of which is j g 2 a mean force, is termed a Parallelogram of Forces. ILLUSTRATION. Assume a weight, W, Fig. 2, to be suspended from a; then, if any distance, a o, is set / ^^^v. off m numerical value upon the vertical line, aW, and the parallelogram, o r a s, is completed, a s and o IT r i measured upon the scale, a o, will represent strain upon a c and a e in same proportion that a o W bears to weight W. If several forces act upon same point, and their intensities taken in order are represented by sides of a polygon, except one, a single force proportioned to and acting in direction of that one side will be their resultant. To Resolve a Single Force into a Pair of Forces. Figs. 3 and 4. The ends of a cord, Fig. 3, are led over two points, a and 6, and in centre of cord at c a weight of 4 Ibs. is suspended. If distances a c, b c, are each i foot, dis- tance a b should be 18 ins. Fig. 4. When cord is in this posi- tion, weight at c draws upon c a and c 6 with a force of 3 Ibs. ; hence c of 4 Ibs. is equal to two forces of 3 Ibs. each in direction of a c and b c. Apply ends of cord to /, Fig. 4, distance being 22 ins., then the strain on ce, c are each 5 Ibs. ; hence one force of 4 Ibs. is equal to two of 5 Ibs. each- B 6 1 6 MECHANICS. STATICS. DYNAMICS. Equilibrium of ITorees. Two bodies which act directly against each other in same line are in equi- librium when their quantities of motion are equal; that is, when product of mass of one, into velocity with which it moves or tends to move, is equal to product of mass of other, into its actual or virtual * velocity. When the velocities with which bodies are moved are same, their forces are proportional to their masses or quantities of matter. Hence, when equal masses are in motion, their forces are proportional to their velocities. Relative magnitudes and directions of any two forces may be represented by two right lines, which shall bear to each other the relations of the forces, PI and which shall be inclined to each other in an angle - ', V, v, v', etc., are employed. ILLUSTRATION i. Two bodies, one of 20, the other of 10 Ibs., are impelled by same momentum, say 60. They move uniformly, first for 8 seconds, second for 6; what are the spaces described by both? 60 -i- 20 = 3 = V, and 60 -f- 10 = 6 = v. Then TV = 3X8 = 24 = S, and v = 6x6 = 36 = *, spaces respectively. 2. If a power of 12 800 effects has a velocity of* 10 feet per second, what is its force ? 12 800 -f- 10 = 1280 Ibs. "Uniform. Variable Motion. Space described by a body having uniform variable motion is represented by sum or difference of velocity, and product of acceleration and time, ac- cording as the motion is accelerated or retarded. ILLUSTRATION i. A sphere rolling down an inclined plane with an initial velocity of 25 feet, acquires in its course an additional velocity at each second of time oi 5 feet; what will be its velocity after 3 seconds? 2 A. locomotive having an initial velocity of 30 feet per second is so retarded that in each second it loses 4 feet; what is its velocity after 6 seconds? 30 4X6 = 6 feet. 3** 6 1 8 MECHANICS. DYNAMICS. "Uniform Motion. Accelerated.. In this motion, velocity acquired at end of any time whatever is equal to _ uct of accelerating force into time, and space described is equal to product of half accelerating force into square of time, or half product of velocity and time of ac- quiring the velocity. Spaces described in successive seconds of time are as the odd numbers, i, 3, 5, 7, o, etc. Gravity is a constant force, and its effect upon a body falling freely in a vertical line is represented by g, and the motion of such body is uniformly accelerated. The following theorems are applicable to aii cases of motion uniformly acceler- ated by any constant force, F : When gravity acts alone, as when a body falls in a vertical line, F it omit- ted. Thus, V 2 V /2 3 V 2 S V* . = _=* ,! = ,/.,.= j=VT T = 7? = J7 = ! '- t representing time in seconds, and s velocity in feet per second. If, instead of a heavy body falling freely, it be projected vertically upward or downward with a given velocity, p, then s = tv qp .5 g t 2 ; an expression in which must be taken when the projection is upward, and + when it is downward. ILLUSTRATION i. If a body in 10 seconds has acquired a velocity by uniformly accelerated motion of 26 feet, what is accelerating force, and what space described, in that time? 26 -=-10 = 2.6 = accelerating force ; - X io 2 = 1 30 feet = space described. 2. A body moving with an acceleration of 15.625 feet describes in 1.5 seconds a 8pace = '5.625 X(i. S ) a = I7 . 578/e ^ 3 . A body propelled with an initial velocity of 3 feet, and with an acceleration of 5 feet, describes in 7 seconds a space = 3X7 + 5X = 143. 5 feet. 4. A body which in 180 seconds changes its velocity from 2.5 to 7.5 feet, trav- erses in that time a distance of 2 ' 5 *~ 7 ' 5 x 180 = 900 feet. 5. A body which rolls up an inclined plane with an initial velocity of 40 feet per second, by which it suffers a retardation of 8 feet, ascends only = 5 seconds, and 4o 2 -f-2 X 8= ioo feet in height, then rolls back, and returns, after io seconds, with a velocity of 40 feet, to its initial point; and after 12 seconds arrives at a distance of 40 X 12 4 X i2 2 = g6feet below point, assuming plane to be extended backward. Circular Motion. _ ____ = _2 = 60 t rn sprn' J ' 5500 ~~ 550 X 60 ~~ ' fzprn' 2 fo Tn = W. r representing radiiis in feet, n number of revolutions of circle per minute, n' total revolutions, f force in Ibs., t time in seconds, and BP horse-power. MECHANICS. DYNAMICS. lQ Motion on an Inclined. Plane. To Ascertain Conditions of Motion by Gravity. Fig. 6. b Assume A B, Fig. 6, an inclined plane, B C its base, ~ A C its height, and b a body descending the plane ; from dot, centre of gravity of body, draw b a perpendicular to B C, representing pressure of b by gravity ; draw 6 o \\ \r parallel and 6 r perpendicular to A B, and complete _ <**'' I parallelogram ; then force & a is equal to both 6 o, 6 r, c of which 6 r is sustained by reaction of plane, and force 6 o is wholly effective in accelerating motion of body. Let this force be represented byf and ba,byg or force of gravity, then by similar triangle,/: g::bo : ba: AC : A B. Hence, ^^L-f. Put A B = ?, A C = fc and ^_ A B C = a, then force which produces motion on the plane on/ becomes g y , and g sin. a. Therefore, accelerating force on an inclined plane is constant, and equations of motion will be obtained by substituting its value of /for g in equations i 3 2, and 3, page 618. If, 111, J 2 JL1, 9 'sin. a, , g sin. a ' /_i^- = . a representing L A B C. V sm - a VFAero a Body is projected down or up an Inclined Plane, with a given Ve- locity. The distance which it will be from point of projection hi a given time will be ^ght* t t r- , and (2 I v g h t) = s. 26 21 ILLUSTRATION i. Length of an inclined plane is 100 feet, and its angle of inclina- tion 60; what is time of a body rolling down it, and velocity acquired ? sin. 60 = .866. = ^' l8 = 2 ' 68 * eMndS > and 32 ' l6 X 2 ' 68 X 866 = 74.64 fed" 2. If a body is projected up an inclined plane, which rises i in 6, with a velocity of 50 feet per second, what will be its place and velocity at end of 6 seconds? 6 x 50 -^ 6 _^-Xj>! = 2 ^^ bo and SQ _ / x 6 x i \ _ 2X0 \ o/ 50 32. 16 = 1 7. 84 feet. To effect an ascent up an inclined plane in least time, its length, to its height, must be as twice weight to power. "Work Accoimnlateci in Gloving Codies. Quantity of work stored in a body in motion is same as that which would be accumulated in it by gravity if it fell from the height due to the velocity. Accumulated work expressed in foot-lbs. is equal to product of height so found in feet, and weight of body in Ibs. Height due to velocity is equal to square of velocity divided by 64.4, and work and velocity may be de- duced directly from each other by following rules : To Compute Accumulated. "Work. RULE. Multiply weight in Ibs. by square of velocity in feet per second, and divide by 64.4, and quotient is accumulated work in foot-lbs. Or, W = ^ - , or, =wxh. W representing work, w weight in Ibs., and 64.4 h height due to velocity in feet per second. 62O MECHANICS. DYNAMICS. "by ^Percussive Force. If a wedge is driven by strokes of a hammer or other heavy mass, effect of percussive force is measured by quantity of work accumulated in stricken body. This work is computed by preceding rules, from weight of body and velocity with which a stroke is delivered, or directly from height of fall, if gravity be percussive power. Useful work done through a wedge is equal to work expended upon it, assuming that there is no elastic or vibrating reaction from the stroke, as if the work had been exerted by a constant pressure equal to weight of strik- ing body, exerted through a space equal to height of fall, or height due to its final velocity. If elastic action intervenes, a portion of work exerted is absorbed in an elastic stress to resisting body ; and the elastic action may be, in some cases, so great as to absorb the work expended. The principle of action of a blow on a wedge is alike applicable to action of the stroke of a monkey of a pile-driver upon a pile. If there be no elastic action, the work expended being product of weight of monkey by height of its fall, is equal to work performed in driving the pile: that is, to product of resistance to its descent by depth through which it is driven by each blow of monkey. ILLUSTRATION If a horse draws 200 Ibs. out of a mine, at a speed of 2 miles per hour, how many units of work does he perform in a minute, coefficient of friction .05 ? Fig- 7- 03 '"ft- ~~ = 176 feet per minute. Hence, 176 X 200 + .05 x 200 = 35 210 units. Decomposition of Force. * By parallelogram of force it is il- lustrated how a vessel is enabled to T be sailed with a free wind and against one. Assume wind to be free or in direction of arrows, Fig. 7, and perpendicular to line A B, the course of vessel. Let line m o represent direction and B force of wind, and r s plane of sail ; from o draw o u perpendicular to r s, and from m perpendicular, m v on r s, and /m u on o u. By principle of parallelogram offerees, force m o may be decomposed into o v and OM, since they are the sides of parallelogram of which m o, representing force of wind, is diagonal. Force of wind, therefore, is measured by ow, both in magni- tude and direction, and represents actual pressure on sail. Draw un and u x parallel to oA and om, thus forming parallelogram unox. Hence force o M is equal to the two, o n and o x. Force o n acts in a direction perpendicular to vessel's course and that of o a; is to drive vessel onward. It can ^us be shown that when di- rection of sail bisects angle m o B, the effect of o a; is greater than when sail is in any other position. Assume wind to be ahead as in direc- tion of arrows, Fig. 8. Let o m repre- sent direction and force of wind, and r s direction of sail; from o draw ow, and proceed as before, and o u represents the effective force that acts upon the sail, on that which drives her to leeward, and o x that which drives her on her course. For full treatises on this subject, see John C. Trautwine's Engineer's Pocket-book, 1872 ; Bull's Ex- perimental Mechanics, London, 1871 ; and Dynamic*, Construction of Machinery, etc., by G. Finden Warr, London, 1851. MECHANICS. MOMENTS OF STRESS ON GIRDERS, ETC. 621 MOMENTS OP STRESS. To Describe and Compute Moments of Stress on <3-ird era or Beams. Supported at Both Ends. Fig. i. Loaded in Centre, Fig. i. Assume A B, a beam. At centre erect We = . Connect A c and c B, and any vertical distance between them and A B will give moment required at that point. Wa/ = M at any point. W represent- ing weight or load, I length of span, x horizontal distance from nearest support at which M, the moment of stress, is required. ILLUSTRATION*. Assume I = 10 feet, W = 10 Ibs. , and x = 3 feet. Then, W c = I0 X I0 = 25 Ibs. at centre of span ; and ^- X ~ 3 - = 15 Ibs. at x. Fig. 2. c Loaded at Any Point, Fig. 2. Proceed as for previous figure. = M between W and B. a representing least distance of W to swpportf, and 6 greatest distance. ILLUSTRATION. Take elements as before with a = 3 feet, a? 1.5, and *' 3.5 feet. Wafr 2 Wxb __ Wxa T or W c = maximum load. = M between A and W. I0 . 5 U>s. at x Then,Wc = 1 -^ -*-? = 21 Ibs. at point of stress; IoX *-5X7 = between A and W, and IoX 3-5X3 = ^ 10 NOTE. and x' must be taken from the pier, which is on the same side of W as that of the stress desired. Loaded with Two Equal Weights at Equal Distances from Supports, alike to a Transverse Girder in a Single Line of Railway. Fig. 3. Fig. 3. q d At point of stress of weights erect W c and W d, each = W a. Connect A cd and B, and vertical distances between them and AB will give moments required. Fig. 4. any point between weights. Loaded with Four Equal Weights, symmetrically bearing from Centre, alike to a Transverse Girder in a Double Line of Railway. Fig. 4. At W and w" erect We, and w" i = 2 W a, and at w and w / erect w d, w' e, each = W (2 a-f- a'). Connect Acdei and B, and or- dinates from them to A B will give moments required. W (2 a -\- a') = M at w and w'\ 2 W a = M at W and w". ILLUSTRATION. Assume W each 10 Ibs. 2 feet apart, and 1 10 feet. Then, 10 (2 x 2 -f- 2) = 60 at w or w', and 2 x 10 x 2 ^ 40 at W or to". 622 MECHANICS. MOMENTS OF STRESS ON GIRDERS, ETC. Loaded at Different Points. Fig. 5. Locate three weights, W, w, and tt>', as at a 6, a t 6^ a 2 b m . Draw A c B, A d B, and A B, as three separate cases, by formula, Wa& I -, Fig. -2. f Produce We until Wo = Wr,Ws, * and We; Wd until tow tow, wv and w; d, and w' e to w/ m in like manner. Connect A owm and B, and an or- dinate therefrom, to A B will give moment of stress at the point taken. ILLUSTRATION. Take a = 2 feet, a =4, a 2 6, 6 = 8, 6 X =:6, 6 2 = 4, W,w, and w' each 10 Ibs., and I 10 feet, carefully observing Note to Fig. 2. Then j (W 6 x + w b x x -f w" b 2 x) M at x. Take x = 2. Then (10x8x2 + 10x6x2-1-10X4X2)=:^ = 36 Z&* 360 *' = 4- (io X 2 X 6 + I0 X 6 X 4 + io X 4 X 4) = ^- = 52 *. *" = 6. j- (10 X 2 X 4 + 10 X 4 X 4 + 10 X 4 X 6) = ^ = 48 Ibs. Loaded with a Rolling Weight. Fig. 6. Define parabola A c B as deter- mined by = the ordinate at c, B and vertical distances between A B ) I mined by = c. 4 At A and B erect A e, B i = w d, connect A i and B e, and vertical i-o distances between A o B and A c B |f will give moments. Or if W and w are at any point. Position ofW at greatest moment, when x=- 2 2 equal, when x = it . ILLUSTRATION. Assume x= 3, d = 4, and W w each 10 Ibs. , and 1 10 feet Then (io-f-io X 10 3 10 X 4) = M at any point, as at W r, w r. MECHANICS. MOMENTS OF STRESS ON GIKDERS^ETC. 623 Shearing Stress. To Determine Shearing Stress at any 3?art of* a Girder or 13 earn and. -under any Distribution of Load. Fig. 8. Required to determine stress of a - ~"VB beam at an y P int as c, Fig. 8. I j^ Assume W = load between A and -^ p> Cj and w t^t, between B and c. Then S x at c = P W, or P' w. The greater of the two values to be taken. S x representing shearing stress at any point x, P and P' the reaction on supports due to total load on beam between supports, W and w loads or stress concentrated at iny point. Desoritoe and Ascertain Shearing Stress in a GHrder or Beam. To Fig. 9. Supported or Fixed at Both Ends. Loaded Uniformly. Fig. 9. At A and B, erect A c, B e. each W I equal to . Connect c and e at middle of span as at w, and vertical distances between A B and cue will i-,, give shearing stresses as determined rwl. by the ordi nates tocne. ,j >% L n _ \ _ s Si of result to \2 / It representing distributed load per unit of length. l^k X, ' ^JL L >^Ww>^ 43 ooo -f 7XI3X400QX.5 _ $2 IQQ m ^ concentrate( j i oad at w> ^^ proportion 20 of uniformly distributed load of 4000 Ibs. 624 MECHANICAL POWERS. LEVEE. MECHANICAL POWERS. MECHANICAL POWER is a compound of Weight, or Force and Velocity: it cannot be increased by mechanical means. The Powers are three in numberviz., LEVER, INCLINED PLANE, and PULLEY. NOTE. A Wheel and Axle is a continuous or revolving lever, a Wedge a double in clined plane, and a Screw a revolving inclined plane. LEVER. Levers are straight, bent, curved, single, or compound. To Compute Length, of a Lever. When Weight and Power are given. RULE.- Divide weight by power, and quotient is leverage, or distance from fulcrum at which power supports weight, w Or, ~ = p. W representing weight, P power, and p distance of power from fulcrum. EXAMPLE. A weight of 1600 Ibs. is to be raised by a power or force of 80; re- quired length of longest arm of lever, shortest being i foot. 1600 -i- 80 = 20 feet. To Compute \Veight that can "be raised, "by a Lever. When its Length, Power, and Position of its Fulcrum are given. RULE. Multiply power by its distance from fulcrum, and divide product by dis- tance of weight from fulcrum. Or, -^ = W. w representing distance of weight from fulcrum, w EXAMPLE. What weight can be raised by 375 Ibs. suspended from end of a lever 8 feet from fulcrum, distance of weight from fulcrum being 2 feet? 375X8-7-2 = 1500^5. To Compute Position of* Fulcrum. When Weight and Power and Length of Lever are given, and when Ful- crum is between Weight and Power. RULE. Divide weight by power, add i to quotient, and divide length by sum thus obtained. Or, L-f-(-p-f-i):=w>. L representing entire length of lever. EXAMPLE. A weight of 2460 Ibs. is to be raised with a lever 7 feet long and a power of 300; at what part of lever must fulcrum be placed ? 2460-7-300 = 8.2, and 8. 2-}- 1 =9-2. Then 7 X 12 -1-9.2 = 9.13 irw. When Weight is between Fulcrum and Power. RULE. Divide length by quotient of weight, divided by power. Or, LH-^ = M>. To Compiate Length of* Arm of* Lever to \vhich "Weight is attached. When Weight, Power, and Length of Arm of Lever to which Power is ap- plied are given. RULE. Multiply power by length of arm to which it is applied, and divide product by weight. MECHANICAL POWERS. LEVEK. 625 EXAMPLE. A weight of 1600 Ibs. , suspended from a lever, is supported by a power of 80, applied at other end of arm, 20 feet in length ; what is length of arm ? 80 X 20 -r- l6", etc., = Ww + W w>', etc. In a system of levers, either of similar, compound, or mixed kinds, condition is p p p' p" Vf w w' w" = W " ILLUSTRATION. Let P = i lb., p and p' each 10 feet, p" i foot; and if w and w' be each i foot, and w" i inch, then = 1200; that is, i lb. will support 1200, with levers I X 120 X 120 X 12 172 800 12 X 12 X I " 144 of the lengths above given. NOTE. Weights of levers in above formulas are not considered, centre of gravity being assumed to be over fulcrum s. GENERAL RULE, therefore, for ascertaining relation of POWER to WEIGHT j n a i ever? whether straight or curved, is, Power multiplied by its distance from fulcrum is equal to weight multiplied by its distance from fulcrum. Or,P:W::w:p,orPp = ; and W w :P. e. -^ = T W w Pp_ WHEEL AND AXLE. A. "Wh.ee! and Axle is a revolving lever. Power, multiplied by radius of wheel, is equal to weight, multiplied by radius of axle. As radius of wheel is to radius of axle, so is effect to power. R R P Or,PR = Wr. Or, PV = Wv. Or, R:r::W:P. Or, P-^W; ^f = r' t - := R. R and r representing radw, and V and v velocities of wheel and axle. MECHANICAL POWERS. WHEEL AND AXLE. 627 When a series of wheels and axles act upon each other, either by belts or teeth, weight or velocity will be to power or unity as product of radii, or circumferences of wheels, to product of radii, or circumferences of axles. ILLUSTRATION. If radii of a series of wheels are 9, 6, 9, 10, and 12, and their pin- ions have each a radius of 6 ins., and power applied is 10 Ibs., what weight will they raise? io X 9 X 6 X 9 X io X 12 _ 583200 _ lb 6X6X6X6X6 ~ 7776 ~ 75 Or, if ist wheel make io revolutions, last will make 75 in same time. Xo Coxnprite 3?o\ver of a Com"bination of "Wheels and an Axle or .A.xles, as in Cranes, etc. RULE. Divide product of driven teeth by product of drivers, and quo- tient is their relative velocity ; which, multiplied by length of lever or arm and power applied to it in pounds, and divided by radius of barrel, will give weight that can be raised. Or. ^ = W ; Or, W r = v I P; Or, -^ == P. I representing length of lever or arm, r radius of barrel, P power, v velocity, and W weight. EXAMPLE i. A power of 18 Ibs. is applied to lever or winch of a crane, length of it being 8 ins., pinion having 6 teeth, driving-wheel 72, and barrel 6 ins. diameter. ^ = 12, and 12 X 8 X 18 = 1728, which, -f- 3, radius of barrel, = 576 Ibs. o 2. A weight of 94 tons is to be raised 360 feet in 15 minutes, by a power, velocity of which is 220 feet per minute; what is power required? 360 -r- 1 5 = 24 feet per minute. Hence =. io. 2545 tons. Compound. .A.xle, or Chinese "Windlass. Axle or drum of windlass consists of two parts, diameter of one being less than that of the other. The operation is thus : At a revolution of axle or drum, a portion of sus- taining rope or chain equal to circumference of larger axle is wound up, and at same time a portion equal to circumference of lesser axle is unwound. Effect, therefore, is to wind up or shorten rope or chain, by which a weight or stress is borne, by a length equal to difference between circumferences of the two axles. Consequently, half that portion of the rope or chain will be shortened by half difference between circumferences. To Compnte Elements of a "Wheel and Conapou.net Axle, or Chinese "Windlass. Fig. G. RULE. Multiply power by radius of wheel, arm, or Fig. 6. r bar to which it is applied, and divide product by half difference of radii of axle, and -quotient is weight that a ' can be sustained. ** P R Or, 7- = W. R representing radius of wteel, etc. , and r and r' radii of axle at its greatest and least diameters. EXAMPLE. What weight can be raised by a capstan, radius of its bar, a, 5 feet, power applied 50 Ibs., and radii, r r', of axle or drum 6 and 5 ins. ? 50 X 5 X 12 _: 5(6-5) 628 MECHANICAL POWERS. INCLINED PLANE. "Wh.ee! and. IPinion Combinations, or Complex "Wheel- work. Power, multiplied by product of radii or circumferences, or number of teeth of wheels, is equal to weight, multiplied by product of radii or circum- ferences, or number of teeth or leaves of pinions. Or, P R R' R", etc., = W r r' r", etc. NOTE. Cogs on face of wheel are termed teeth, and those on surface of axle are termed leaves ; the axle itself in this case is termed a pinion. Tfcaok and. Opinion. To Compute 3?o\ver of a Rack and. Pinion. RULE. Multiply weight to be sustained by quotient of radius of pinion, divided by radius of crank, and product is power required. Or, W - = P- Jtv When Pinion on Crank Axle communicates with a Wheel and Pinion. RULE. Multiply weight to be sustained by quotient of product of radii of pinions, divided by radii of crank and wheel, and product is power required. EXAMPLE. If radii of pinions of a jack-screw are each one inch; of crank and wheel 10 and 5 ins. ; what power will sustain a weight of 750 Ibs. ? INCLINED PLANE. To Compute Length of Base, Height, or Length. When any Two of them are given, and when Line of Direction of Power or Traction is Parallel to Face of Plane. Proceed as in Mensuration or Trigonometry to determine side of a right-angled triangle, any two of thre being given. To Compute Power necessary to Support a "Weight on an. Inclined Plane. When Height and Length are given. RULE. Multiply weight by height of plane, and divide product by length. r, = P- h an d I representing height and length of plane. EXAMPLE. What is power necessary to support 1000 Ibs. on an inclined plane 4 feet in height and 6 feet in length ? looo X 4 -r- 6 = 666.67 lb s - To Compute "Weight that may "be Sustained by a given Po*wer on. an Inclined Plane. When Height and Length of Plane are given. RULE. Multiply power by length of plane, and divide product by height. EXAMPLE. What is weight that can be sustained on an inclined plane 5 feet in height and 7 feet in length by a power of 700 Ibs. ? 700 x 7 -T- 5 = 980 Ibs. NOTE. In estimating power required to overcome resistance of a body being drawn up or supported upon an inclined plane, and contrariwise, if body is de- scending; weight of body, in proportion of power of plane (i. e., as its length to its height), must be added to resistance, if being drawn up or supported, or to the wo- ment if descending. MECHANICAL POWERS. INCLINED PLANE. 62Q To Compute Heiglxt or Length of arx Inclined Plane. When Weight and Power and one of required Elements are given, and when Height is required. RULE. Multiply power by length, and divide product by weight. When Length is required. RULE. Multiply weight by height, and divide product by power. To Compute Pressure on an Inclined Plane. RULE. Multiply weight by length of base of plane, and divide product by length of face. w b Or, j- = pressure, b representing length of base of plane. EXAMPLE. Weight on an inclined plane is 100 IDS., base of plane is 4 feet, and length of it 5 ; required pressure on plane. When Two Bodies on Two Inclined Planes sustain each other, as by Connection of a Cord over a Pulley, their Weights are directly as Lengths of Planes. ILLUSTRATION. If a weight of 50 Ibs. upon an inclined plane, of 10 feet rise in 100 of an inclination, is sustained by a weight on another plane of 10 feet rise in 90, what is the weight of*the latter ? ioo : 90 :: 50 : 45 = weight that on shortest plane would sustain that on largest. When a Body is Supported by Two Planes, as Fig. 7, pressure upon them p. will be reciprocally as sines of inclinations of planes. -i Thus, weight is as sin. A B D. Pressure on A B as sin. D B i. Pressure on B D as sin. A B h. Assume angle A B D to be 90, and D B t, 60 ; then angle A B h will be 30; and as sines of 90, 60, and 30 are respec- tively .1, .866, and .5, if weight = ioo Ibs., then pressures on A B and B D will be 86.6 and 50 Ibs., centre of gravity of weight assumed to be in its centre. When Line of Direction of Power is parallel to Base of Plane, power is to weight as height of plane to length of its base. Or, P: W.:h:b. When ZAne of Direction of Poiner is neither parallel to Face of Plane nor to its Base, but in some other Direction, as P', Fig. 8, power is to weight as sine of angle of plane's elevation to cosine of angle which line of power or traction describes with face of plane. Fig. 8. \ Thus, P' : W :: sin. A : cos. P* e c. Sin. A: cos. P'ec::P': W. Cos. P' ec: sin. A : : W : P'. ILLUSTRATION. A weight of 500 Ibs. is required to be sustained on a plane, angle of elevation of which, c A B, is 10; line of direction of power or traction, A n B P'e c, is 5; what is sustaining power required? Cos. P' e c (5) = .996 19 : sin. A (10) = .173 65 : : 500 : 87. 16 Ibs. Or, draw a line, B s, perpendicular to direction of power's action from end of base line (at back of plane), and intersection. of this line on length, Ac, will determine length and height (n r) of the plane. 3 G* 630 MECHANICAL POWERS. - WEDGE. - SCREW. ILLUSTRATION. By Trigonometry (page 385), A B, assumed to be i, A r and n r are = .985 and. 171. Hence 5 X ' 171 = 86.8 Ibs. ^product of weight X height of plane -r- length of it. 9 8 5 NOTE. When line of direction of power is parallel to plane, power is least. Wedge. A WEDGE is a double inclined plane. To Compnte Power. 1. When One Body is to be Forced or Sustained. RULE. Multiply weight or resistance to be sustained by depth of back of wedge, and divide product by length of its base. EXAMPLE. What power, applied to the back of a wedge 6 ins. deep, will raise a weight of 15000 Ibs., the wedge being 100 ins. long on its base? 15000X6^90 ooo = bs IOO IOO 2. When Two Bodies or Two Parts of a Body are Forced or Sustained in a Direction Parallel to Back of Wedge. RULE. Multiply weight or resist- ance to be sustained by half depth of back of wedge, and divide product by length of wedge. Or, Wd ~ 2 = P. d representing depth of back, and I length. I NOTE. The length of a single wedge is measured on its base, and of a double wedge, from centre of its head to its point. EXAMPLE. The depth of the back of a double-faced wedge is 6 ins., and the length of it through the middle 10; what power applied to it is necessary to sus- tain or overcome a resistance of 150 Ibs. ? = 45 To Compute Elements of a "Wedge. ... = NOTE. As power of wedge in practice depends upon split or rift in wood to be cleft, or in rise of body to be raised, the above rules as regards length of wedge are only theoretical when a rift or rise exists. SCREW. A SCREW is a revolving inclined plane. To Compvite Length and Height of Plane of a Screw. As a screw is an inclined plane wound around a cylinder, length of plane is ascertained by adding square of circumference of screw to square of dis- tance between threads, and taking square root of sum. The Pitch or height of a screw is distance between its consecutive threads. To Compute IPower. RULE. Multiply weight or resistance, to be sustained by pitch of threads, and divide product by circumference described by power. W n Or, - = P. p representing pitch, and c circumference. EXAMPLE. What is power requisite to raise a weight of 8000 Ibs. by a screw of 12 ins. circumference and i inch pitch ? 8000 X i -5- ia = 666.66 Ibt. MECHANICAL POWERS. SCREW. 63 I To Compute "Weight. RULE. Multiply power by circumference described by it, and divide product by pitch of threads. Or,^=W. P To Compute Pitch.. RULE. Multiply power by circumference described by it, and divide product by weight. To "Compute Circumference. RULB. Multiply weight by pitch, and divide product by power. Or, - - = c. Or, p = r. r representing radius. r 0.20 r When Power is applied by a Lever or Wheel r substitute radius of power for circumference. ILLUSTRATION. If a lever 30 ins. in length was added to circumference of screw in preceding example, Then, 12-7-3.416 = 3.819, and ^-^-|-3o=3i.9095 = radtM of power. Compound. Screw. Fig. 9. & When a Lever and Endless Screw or a Series of Wheels are applied to a Screw, as Fig. 9. RULE. Ascertain result of each application, and take their continued product. NOTE. If there is more than one thread to a screw, pitch must be increased as many times as there are threads. ILLUSTRATION. What weight can be raised with a power of 10 Ibe., applied to a crank, c, Fig, 9, 32 ins. long, turning an end- less screw, 6, of 3.5 ins. diameter and i inch pitch, applied to a wheel, d, of 20 ins. diameter, upon an axle, a, of 5 ins. ? ioX32X .2 _ 20Q9 6 _ quotient of product of power and iV Ifr'W circumference described by it, and pitch, and 2 .9^ X20 = 8038.4 Ibs. = quotient of power applied to wheel, divided by its axle. When a Series of Wheels and Axles are in Connection with each other, Weight is to power, as continued product of radii of wheels is to continued product of radii of axles. W : P : : R n : r n. Or, rn : Rn :: P : W. n representing continued product of number of wheels or axles. ILLUSTRATION. If a power of 150 Ibs. is applied to a crank of 20 ins. radius, turn- ing an endless screw with a pitch of half an inch, geared to a wheel, pinion of which is geared to another wheel, and pinion of second wheel is geared to a third wheel, to axle or barrel of which is suspended a weight; it is required to know what weight can be sustained in that position, diameter of wheels being 18, and pinions and axle 2 ins. 150X20X2X3-1416 = 3?699 2 ifa = power applied to face of first wheel. Diameters of wheels and pinions being 18 and 2, their radii are 9 and i. Hence, i XxXx:9X9X9 :: 37 699-2 : 27482716.8 Ibs. 632 MECHANICAL POWERS. SCREWS. PULLEY. Differential Screw. When a hollow screw revolves upon one of less diameter and pitch (as designed by Mr. Hunter), effect is same as that of a single screw, in which the distance between threads is equal to difference of distances between threads of the two screws. Therefore power, to effect or weight sustained, is as difference between distances of threads of the two screws to circumference described by power. ILLUSTRATION. If external screw has 20 threads, and internal one 21 threads in pitch of j inch, and power applied describes a circumference of 35 ins., the result or 35 =.4706. power is as co = , or . 002 38. Hence 21 20 420 .00238" 600 PULLEY. PULLEYS are designated as Fixed and Movable, according as cord is passed ovfer a fixed or a movable pulley. A movable pulley is when cord passes through a second pulley or block in suspension ; a single movable pulley is termed a runner; and a combination of pulleys is termed a system of pulleys. A Whip is a single cord over a fixed pulley. To Compxite JPower Required to Raise a given "Weight. When Number of Parts of Cord supporting Lower Block are given, and when only one Cord or Hope is used. RULE. Divide weight to be raised by number of parts of cord supporting lower or movable block. Or, W -r- n = P. Or, n P = W. n representing number of parts of cord sustain- *ng lower block. EXAMPLE. What power is required to raise 600 Ibs. when lower block contains jix sheaves ? When Cord is attached to Upper or Fixed Block. - = 50 Ibs. = weight-?- number of parts of rope sustaining lower block. When Cord is attached to Lower or Movable Block, = 46.15 Ibs. = weight -r- number of parts of rope sustaining lower block. To Compute "\Veiglit a, given Power "will Raise. When Number of Parts of Cord supporting Lower Block are given. RULE. Multiply power by number of parts of cord supporting lower block. Or, P n = W. To Compute Numtoer of Cords necessary to Sustain Lower Block. When Weight, and Power are given. RULE. Divide weight by power. When more than one Cord is used. In a Spanish Burton, Fig. 10, where ends of one cord, a P, are fastened to support and power, and ends of the other, c o, to lower and upper blocks, weight is to power as 4 to i. In another, Fig. n, where there are two cords, a and o, two movable pulleys, and one fixed pulley, with ends of one rope fastened to sup- port and upper movable pulley, and ends of other fastened to lower block and power, weight is to power as 5 to i. Fig. to. MECHANICAL POWERS. PULLEY. 633 Compound or Fast and Loose IPvilleys. When Cord is attached to Fixed Block, Fig. 12. RULE. Multiply power by the power of 2, of which the index is number of movable pulleys. Or, P 2 W = W. Or, Multiply power successively by 2 for each pulley. EXAMPLE i. What weight will one pound support in a system of three movable pulleys, the cords being connected to a fixed block on Fig. 12. i x 23 = 8 Ibs EXAMPLE 2. What would a like power support, fixed block be- ing made movable and cord attached thereto? If Qxed pulleys were substituted for hooks abc, Fig. 12, power would be increased threefold; hence i x 33 = 27. In a System of Pulleys, Figs. 13 and 14, with any Number of Cords, oo,ee, Ends being fastened to Support. W F 'g- '4- W-r-2=P; 2XP = W; = 2" nrep- resenting number of distinct cords. ILLUSTRATION. What weight will a power of i Ib. sustain in a system of two movable pul- j i leys and two cords ? i x 2 X 2 4 Ibs. When fixed Pulleys, e e, are used in Place p of Hooks, to Attach Ends of Rope to Sup- port. Fig. 14. ILLUSTRATION. What weight will a power of 5 Ibs. sustain with two movable and three fixed pulleys, and two cords? 5 x 3 x 3 _ 45 fo s When Ends of Cord or Fixed Pulleys are fastened to Weic/ht, as by an Inver- sion of the last Figures, putting Supports for Weights, and contrariwise. Figs. 13 and 14. Fig. 13. Fig. 13- Fig. 14. ILLUSTRATION What weight will a power of i Ib. sustain in a system of two mov- able pulleys and two cords, and one of two movable and two fixed pulleys and two 3ords9 1X2X2 1 = 3/65. 1X3X3 i = 8/6*. When Cords sustaining Pulleys are not in a Vertical Direction. Fig. 15. Fig. 15. eo, Fig. 15, is vertical line through which weight bears, and ^- ^ from o draw or,os parallel to De and A e. Forces acting at e are represented by lines es, er, and eo; and as tension of every part of cord is same, and equal to power P, sides o s and or of parallelogram must be equal, and therefore diagonal e o divides the angle r o s into two equal portions. Hence the weight will always fall into the position in which the two parts of cord A e and e D will be equally inclined to vertical line, and it will bear to power same ratio as eo to es. Therefore W : P : : 2 cos. . 5 e : i. e representing angle A e D. Or, 2 P X cos. . 5 e W. That is, twice pewer, multiplied by cosine of half angle of cord, at point of suspension of weight, is equal to weight. 634 METALS. ALLOYS AND COMPOSITIONS. ILLUSTRATION. What weight will be sustained by a power of 5 Ibs., with an ob- lique movable pulley, Fig. 15, having an angle, A c D, of 30 ? 5 X 2 X .965 93 = 9.6593 Ibs. = twice power X cos. 15. When Direction of Cord is Irregular, Weight not resting in Centre of it. P sin. a P sin. (a-f-6) W sin. a W sin. (a -f- b) ' sin. a greater and lesser angles of cord at e. sin. (a + 6) = P. a and b representing METALS. ALLOYS AND COMPOSITIONS. Alloy is the proportion of a baser metal mixed with a finer or purer, as copper is mixed with gold, etc. Amalgam is a compound of Mercury and a metal a soft alloy. Compositions of copper contract in admixture, and all Amalgams ex- pand. In manufacture of Alloys and Compositions, the less fusible metals should be melted first. In Compositions of Brass, as proportion of Zinc is increased, so is malleability decreased. Tenacity of Brass is impaired by addition of Lead or Tin. Steel alloyed with one five-hundredth part of Platinum, or Silver, is rendered harder, more malleable, and better adapted for cutting instru- ments. Specific gravity of alloys* does not follow the ratios of those of their components ; it is sometimes greater and sometimes less than the mean. Composition for 'Welding Cast Steel. Borax, 91 parts; Sal-ammoniac, 9 parts. Grind or pound them roughly together; fuse them in a metal-pot over a clear fire, continuing heat until all spume has dis- appeared from surface. When liquid is clear, pour composition out to cool and con- crete, and grind to a fine powder; then it is ready for use. To use this composition, the steel to be welded should be raised to a bright yellow heat; then dip it in the welding powder, and again raise it to a like heat as before; it is then ready to be submitted to the hammer. F'usi'ble Compounds. COMPOUNDS. Zinc. Tin. Lead. Bismuth. Cadmium. 2 5 2C 5 33.3 33. 3 33- 4 Newton's, fusing at less than 212. Fusing at iso to 160 . . , 19 3' 33 50 50 13 Solders. Solder is an alloy used to make joints between metals, and it must be more fusible than the metals it is designed to unite, and it is distinguished as hard and soft, according to the temperature of its fusing. The addition of a small portion of Bismuth increases its fusibility. * For a table of Alloys, having densities different from Manual, London, 1877, page 201, D of their components, see D. K. Clark'a METALS. ALLOYS AND COMPOSITIONS. 635 Alloys and. Compositions. Copper. Zinc. Tin. Nickel. Lead. Anti- mony. Bis- muth. Ala- minqm. 55 95 3-7 84-3 75 79-3 92.2 90 80 88.8 74-3 5 88.9 90 10 3 4 66 s7 86 67.2 90 93 95 80 93 91.4 58.1 40.4 80 69 87.5 72 33-3 40.4 49-5 81.6 g 87-5 77-4 60 50 66.6 33-4 & 73 24 5-2 25 6.4 }-*"- 20 II. 2 22.3 3 ;.8 80 90 33 34 13 ii. i 31-2 5-5 17.2 25-4 5-6 33-4 25.4 24 7 40 45 21 5 12.3 89~ 10.5 14-3 7-8 9 3-4 8^3 10 10 25 2.9 1.6 10 7 5 20 7 1.4 2.6 10. 1 31 12.5 26.5 iZ 4 23 20 12.5 15-6 86~ 80 22 2 9 m 2 8~ 4 4.4 (Magn iSaHu 21 X 9 n.6 31-6 46 7-3 7 47 25 25 25 | 55 2 .| ? (Cobalt of Iron. , , , , , | , , j, , , , , , , , , , , , , Babbitt's metal * " " hard " instruments " locomot. bearings. " Pinchbeck " rolled " Tutenag 44 very tenacious... wheels, valves.... ** white u It " yellow, fine When fused add ...... i-7 4-3 " yellow ** Gun metal, large " " small " soft. " Medals " Statuary ....... Chinese silver " white copper... Church bells Clocks, Musical bells. . . . Clock bells , ... 33-3 31.6 24 75 16.7 i-5 2.6 2-5 12 ' ar.6.5 .. .1.3 (t 41 1 s 8.3 of tart irae. . . " " fine Gongs House bells Machinery bearings hard. Metal that expands in) cooling f Muntz metal... . ....... Pewter, best esia... nmonia 20 ..4.4 c.2.5 14 25 12.5 56.8 Cream Quickl Sheathing metal Speculum " . F u u Telescopic mirrors Temper t Type metal and stereo- ) White metal " " bard Orelde * See page 636 for direction*. i For adding small quantities of copper. 6 3 6 METALS. ALLOYS AND COMPOSITIONS. Solders. Copper. Tin. Lead. Zinc. Silver. Bis- muth. Gold. Cad- mium. Anti- mony. Tin je g 16 16 " coarse, melts ) at 500 . . . J " ordi'y, melts) at 360...} Spelter, soft 33 67 67 33 _ _ " hard Lead ' 67 Steel ' 82 Brass or Copper... Fine brass 50 - - 50 - - - - Pewterers' or Soft. Plumbers' pot- ) metal..} " coarse . . . " fine 33 50 33 25 67 45 25 67 75 - 22 25 - - - " fusible... " very " ... Gold So 25 50 25 50 " hard 66 9 " soft 66 Silver, hard 20 80 soft Pewter ' 21 66 Copper . . , c;^ J.7 T A Plastic Metallic Alloy. See Journal of Franklin Institute, vol. xxxix., for its composition and manufacture. Page 55, Soldering Fluid for use with Soft Solder. To 2 fluid oz. of Muriatic acid add small pieces of Zinc until bubbles cease to rise. Add . 5 a teaspoonful of Sal-ammoniac and two fluid oz. of Water. By the application of this to Iron or Steel, they may be soldered without their sur- faces being previously tinned. Fluxes for Soldering or Welding. Iron Borax. Tinned iron Resin. Copper and Brass Sal-ammoniac. Zinc Chloride of zinc. Lead Tallow or resin. Lead and tin Resin and sweet oil. Babbitt's Anti-attrition 3VEetal. Melt 4 Ibs. copper ; add 12 Ibs. Banca tin, 8 Ibs. Regulus of antimony, and 12 Ibs. more of Tin. After 4 or 5 Ibs. Tin have been added, reduce heat to a dull red, then add remainder of metal as above. This composition is termed hardening ; for lining, melt i Ib. of this hardening with 2 Ibs. tin, which produces the lining metal for use. As this metal was introduced in 1839, it is now maintained by engineers that the increased weight of machines and the velocity of engines and dynamos require an appropriate alloy ; and it is claimed by engineers that Phoenix Metal meets existing requirements. Brass. Brass is an alloy of copper and zinc, in proportions varying with purpose of metal required, its color depending upon the proportions. It is rendered brittle by continued impacts ; more malleable than copper when cold, but is impracticable of being forged, as its zinc melts at a low temperature. Its fusibility is governed by the proportion of zinc in it; a small quantity of phosphorus gives it fluidity. METALS. ALLOYS AND COMPOSITIONS. IRON. 637 Bronze. Bronze is an alloy of copper and tin ; it is harder, more fusible, and stronger than copper. It is usually known as Gun-metal. Aluminum Bronze contains 90 to 95 per cent, of copper, and 5 to 10 per cent, aluminum. Phosphor Bronze contains copper and tin and a small proportion of phos- phorus. It wears better than bronze. IRON. Foreign substances which iron contains modify its essential proper, ties. Carbon adds to its hardness, but destroys some of its qualities, and produces Cast Iron or Steel, according to proportion it contains. Thus, .25 per cent, renders it malleable, .5 steel, 1.75 is limit of weld- ing steel, and 2 is lowest limit of cast iron. Sulphur renders it fusible, difficult to weld, and brittle when heated, or "hot short." Phosphorus renders it " cold short" but may be present in proportion of .002 to .003, without affecting injuriously its tenacity. Antimony, Arsenic, and Copper have same effect as sulphur, the last in a greater degree. Sili- con renders it hard and brittle. Manganese, in proportion of .02, ren- ders it " cold short" and Vanadium adds to its ductility. Cast Iron. Process of making Cast Iron depends much upon description of fuel used ; whether charcoal, coke, bituminous, or anthracite coals. A larger yield from same furnace, and a great economy in fuel, are effected by use of a hot blast. The greater heat thus produced causes the iron to combine with a larger percentage of foreign substances. Cast Iron for purposes requiring great strength should be smelted with a cold blast. Pig-iron, according to proportion of carbon which it contains, is divided into Foundry Iron and Forge Iron, latter adapted only to conver- sion into malleable iron ; while former, containing largest proportion of car- bon, can be used either for castings or bars. High temperature in melting injures gun-metal. There are many varieties of Cast Iron, differing by almost insensible shades ; the two principal divisions are gray and white, so termed from color of their fracture. Their properties are very different. Gray Iron is softer and less brittle than white ; it is in a slight degree malleable and flexible, and is insonorous ; it can easily be drilled or turned, and does not resist the file. It has a brilliant fracture, of a gray, or some- times a bluish-gray, color ; color is lighter as grain becomes closer, and its hardness increases. It melts at a lower heat than white, and preserves its fluidity longer. Color of the fluid metal is red, and deeper in proportion as the heat is lower ; it does not adhere to the ladle ; it fills molds well, con- tracts less, and contains fewer cavities than white; edges of its castings are sharp, and surfaces smooth and convex. It is used for machinery and ordnance where the pieces are to be bored or fitted. Its tenacity and specific gravity are diminished by annealing. White Iron is very brittle and sonorous ; it resists file and chisel, and is susceptible of high polish ; surface of its castings is concave ; fracture pre- sents a silvery appearance, generally fine grained and compact, sometimes radiating or lamellar. When melted it is white, throws off a great number of sparks, and its qualities are the reverse of those of gray iron ; it is there- fore unsuitable for machinery purposes. Its tenacity is increased, and its specific gravity diminished, by annealing. 3 H 638 METALS. IKON. Mottled Iron is a mixture of white and gray ; it has a spotted appear- ance ; flows well, and with few sparks ; its castings have a plane surface, with edges slightly rounded. It is suitable for shot, shells, etc. A fine mot- tled is only kind suitable for castings which require great strength. The kind of mottle will depend much upon volume of the casting. A medium- sized grain, bright gray color, fracture sharp to touch, and a close, compact texture, indicate a good quality of iron. A grain either very large or very small, a dull, earthy aspect, loose texture, dissimilar crystals mixed together, indicate an inferior quality. Besides these general divisions, the different varieties of pig-iron are more particularly distinguished by numbers, according to their relative hardness. No. i. Fracture dark gray, crystals large and highly lustrous, alike to new surface of lead. It is the softest iron, possessing in highest degree the qualities belonging to gray iron ; it has not much strength, but on account of its fluidity when melted, and of its mixing advantageously with scrap iron and with the harder kinds of cast iron, it is of great use to a foundry. No. 2 is harder, closer grained, and stronger than No. i ; it has a gray color and considerable lustre. It is most suitable for shot and shells. No. 3 is harder than No. 2. Fracture white, crystals larger and brighter at centre than at the sides ; color gray, but inclining to white ; has consid- erable strength, but is principally used for mixing with other kinds of iron and for large castings. No. 4 or Bright. Fracture light gray, with small crystals and little lustre, and not being sufficiently fusible for castings it is used for conversion to wrought iron. No. 5. Mottled. Fracture dull white, with gray specks, and a line of white around edge or sides of fracture. No. 6. White. Fracture white, with little lustre, granulated with radiat- ing crystalline surface. It is hardest and most brittle of all descriptions, and is unfit for use unless mixed with other grades, or for being converted to an inferior wrought iron. Qualities of these descriptions depend upon proportion of carbon, and upon state in which it exists in the metal ; in darker kinds of iron, where propor- tion is sometimes 7 per cent., it exists partly in state of graphite or plumbago, which makes the iron soft. In white iron the carbon is thoroughly com- bined with the metal, as in steel. Cast iron frequently retains a portion of foreign ingredients from the ore, such as earths or oxides of other metals, and sometimes sulphur and phos- phorus, which are all injurious to its quality. Foreign substances, and also a portion of the carbon, are separated by melting iron in contact with air, and soft iron is thus rendered harder and stronger. Effect of remelting varies with nature of the iron and character of ore from which it has been extracted ; that from hard ores, such as mag- netic oxides, undergoes less alteration than that from hematites, the latter being sometimes changed from No. i to white by a single remelting in an air furnace. Color and texture of cast iron depend greatly upon volume of casting and rapidity of its cooling ; a small casting, which cools quickly, is almost always white, and surface of large castings partakes more of the qualities of white metal than the interior. All cast iron expands at moment of becoming liquid, and contracts in cool- ing ; gray iron expands more and contracts less than other iron. Remelting iron improves its tenacity ; thus, a mean of 14 cases for two fusions gave, for ist fusion, a tenacity of 29 284 Ibs. ; for 2d fusion, 33 790 Ibs. For two cases for first fusion, 15 129 Ibs. ; for 2d fusion, 35 786 Ibs. METALS. IRON. 639 Malleable Castings. Malleable cast iron is made by subjecting a casting to a process of anneal- ing, by enclosing it in a box with hematite iron ore or black oxide of iron, and maintaining it in an equable heat for a period depending upon form and volume of casting. "Wrought Iron- Wrought iron is made from pig-iron in a Bloomeiy Fire or in a Puddling Furnace generally in latter. Process consists in melting and keeping it exposed to a great heat, constantly stirring the mass, bringing every part of it under action of the flame until it loses its remaining carbon, when it be- comes malleable iron. When, however, it is desired to obtain iron of best quality, pig-iron should be refined. Refining. This operation deprives iron of a considerable portion of its carbon ; it is effected in a Blast Furnace, where iron is melted by means of charcoal or coke, and exposed for some time to action of a great heat ; the metal is then run into a cast-iron mold, by which it is formed into a large broad plate. As soon as surface of plate is chilled, cold water is poured on to render it brittle. A Bloomery resembles a large forge fire, where charcoal and a strong blast are used ; and the refined metal or pig-iron, after being broken into pieces of proper size, is placed before the blast, directly in contact with charcoal ; as the metal fuses, it falls into a cavity left for that purpose below the blast, where the " bloomer " works it into the shape of a ball, which he places again before the blast, with fresh charcoal ; this operation is generally again re- peated, when ball is ready for the " shingler." Shingling is performed in a strong squeezer or under a trip-hammer. Its object is to press out as perfectly as practicable the liquid cinder which a ball contains ; it also forms a ball into shape for the puddle rolls. A heavy hammer, weighing from 6 to 7 tons, effects this object most thoroughly, but not so cheaply as the squeezer. A ball receives from 15 to 20 blows of a hammer, being turned from time to time as required : it is now termed a Bloom, and is ready to be rolled or hammered ; or a ball is passed once through the squeezer, and is still hot enough to be passed through the puddle rolls. A Puddling Furnace is a reverberatory furnace, where flame of bituminous coal is brought to act directly upon the melted metal. The " puddler " then stirs it, exposing each portion in turn to action of flame, and continues this as long as he is able to work it. When it has lost its fluidity, he forms it into balls, weighing from 80 to 100 Ibs., which are then passed to the "shingler." Puddle Rolls. By passing through different grooves in these rolls, a bloom is reduced to a rough bar from 3 to 4 feet in length, its term convey- ing an idea of its condition, which is rough and imperfect. Piling. To prepare rough bars for this operation, they are cut, by a pair of shears, into such lengths as are best adapted to the volume of finished bar required ; the sheared bars are then piled one over the other, according to volume required, when pile is ready for balling. Balling. This operation is performed in balling furnace, which is similar to puddling furnace, except that its bottom or hearth is made up, from time to time, with sand ; it is used to give a welding heat to piles to prepare them for rolling. Finishing Rolls. The balls are passed successively between rollers of va- rious forms and dimensions, according to shape of finished bar required. Quality of iron depends upon description of pig-iron used, skill of the kt puddler," and absence of deleterious substances in the furnace. 640 METALS. IRON. LEAD. STEEL. Strongest cast irons do not produce strongest malleable iron. For many purposes, such as sheets for tinning, best boiler-plates, and bars for converting into steel, charcoal iron is used exclusively ; and, generally, this kind of iron is to be relied upon, for strength and toughness, with greater confidence than any other, though iron of a superior quality is made from pigs made with other fuel, and with a hot blast. Iron for gun-barrels has been lately made from anthracite hot-blast pigs. Iron is improved in quality by judicious working, reheating, hammering, or rolling : other things being equal, best iron is that which has been wrought the most. Best quality of iron has greatest elasticity. Tests. It will not blacken if exposed to nitric acid. Long silky fibres in a fracture denote a soft and strong metal ; short black fibres denote a badly refined metal, and a fine grain denotes hardness and condition known as u cold short." Coarse grain with bright and crystallized fracture, with dis- colored spots, also denotes " cold short " and brittle metal, working easily and welding well. Cracks upon edges of a bar, etc., indicate " hot short." Good iron heats readily, is worked easily, and throws off but few sparks. A high breaking strain may not be conclusive as to quality, as it may be due to a hard, elastic metal, or a low one may be due to great softness. When iron is fractured suddenly, a crystalline surface is produced, and when gradually, a fibrous one. Breaking strain of iron is increased by heat- ing it and suddenly cooling it in water. Iron exposed to a welding or white heat and not reduced by hammering or rolling is weakened. Specific gravity of iron is a good indication of its quality, as it indicates very correctly its relative degree of strength. LEAD. Sheet Lead is either Cast or Milled, the former in sheets 16 to 18 feet in length and 6 feet in width ; the latter is rolled, is thinner than the former, is more uniform in its thickness, and is made into sheets 25 to 35 feet in length, and from 6 to 7.5 feet in width. Soft or Rain Water, when aerated, Silt of rivers, Vegetable matter, Acids, Mortar, and Vitiated Air will oxidize lead. The waters which act with greatest effect on it are the purest and most highly oxygenated, also nitrites, nitrates, and chlorides, and those which act with least effect are such as con- tain carbonate and phosphate of lime. Coating of Pipes, except with substances insoluble in water, as Bitumen and Sulphide of lead, is objectionable. Lead-encased Pipes. An inner pipe of tin is encased in one of lead. STEEL. Steel is a compound of Iron and Carbon, in which proportion of latter is from I to 5 per cent., and even less in some descriptions. It is dis- tinguished from iron by its fine grain, and by action of diluted nitric acid, which leaves a black spot upon it. There are many varieties of steel, principal of which are : Natural Steel, obtained by reducing rich and pure descriptions of iron ore with charcoal, and refining cast iron, so as to deprive it of a sufficient portion of carbon to bring it to a malleable state. It is used for files and other tools. Indian Steel, termed Wootz, is said to be a natural steel, containing a small portion of other metals. METALS. STEEL. 64 1 Blistered Steel, or Steel of Cementation, is prepared by direct combination of iron and carbon. For this purpose, iron in bars is put in layers, alternating with powdered charcoal, in a close furnace, and exposed for 7 or 8 days to a high temperature, and then put to cool for a like period. The bars, on being taken out, are covered with blisters, have acquired a brittle quality, and exhibit in fracture a uniform crystalline appearance. The degree of carbonization is varied according to purposes for which the steel is intended, and the very best qualities of iron are used for the finest kinds of steel. Tilted Steel is made from blistered steel moderately heated, and subjected to action of a tilt hammer, by which means its tenacity and density are in- creased. Shear Steel is made from blistered or natural steel, refined by piling thin bars into fagots, which are brought to a welding heat in a reverberatory furnace, and hammered or rolled again into bars ; this operation is repeated several times to produce finest kinds of shear steel, which are distinguished by the terms of Half shear, Single shear, and Double shear, or steel of i, 2, or 3 mark*, etc., according to number of times it has been piled. Spring Steel is blister steel heated to an orange red color and rolled or hammered. Cast or Crucible Steel is made by breaking blistered steel into small pieces and melting it in close crucibles, from which it is poured into iron molds ; ingot is then reduced to a bar by hammering or rolling. Cast steel is best kind of steel, and best adapted for most purposes ; it is known by a very fine, even, and close grain, and a silvery, homogeneous fracture ; it is very brittle, and acquires extreme hardness, out is difficult to weld without use of a flux. Other kinds of steel have a similar appearance to cast steel, but grain is coarser and less homogeneous ; they are softer and less brittle, and weld more readily. A fibrous or lamellar appearance in fracture indicates an imperfect steel. A material of great toughness and elasticity, as well as hardness, is made by forging together steel and iron, forming the celebrated Damasked Steel, which is used for sword-blades, springs, etc. ; damask ap- pearance of which is produced by a diluted acid, which gives a black tint to the steel, while the iron remains white. With cast steel, breaking strength is greater across fibres of rolling than with them. Heath's Process is an improvement on this method, and consists in adding to molten metal a small quantity of carburet of manganese. Heaton's Process consists in adding nitrate of soda to molten pig-iron, in order to remove carbon and silica. Musket's Process. Malleable iron is melted in crucibles with oxide of manganese and charcoal. Puddled Steel is produced by arresting the puddling in the manufacture of the wrought iron before all the carbon has been removed, the small amount of carbon remaining, .3 to i per cent., being sufficient to make an inferior steel. Mild Steel contains from .2 to .5 per cent, of carbon ; when more is pres- ent it is termed Hard Steel. Bessemer Steel is made direct from pig-iron. The carbon is first removed, in order to obtain pure wrought iron, and to this is added the exact quantity of carbon required for the steel. The pig should be free from sulphur and phosphorus. It is melted in a blast or cupola, and run into a converter (a pear-shaped iron vessel suspended on hollow trunnions and lined with fire- brick or clay), where it is subjected to an air blast for a period of 20 min- utes, in order to dispel the carbon, after which from 5 to 10 per cent, of apie- geleisen is added. 3 IP 642 METALS. STEEL. The blast is then resumed for a short period, to incorporate the two metals, when the steel is run off into molds. The moment at which all the carbon has been removed is indicated by color of the flame at mouth of converter. The ingots, when thus produced, contain air holes, and it becomes necessary to heat them and render them solid under a hammer. Siemens Process. Pig-iron is fused upon open hearth of a regenerative furnace, and when raised to a steel-melting temperature, rich and pure ore and limestone are added gradually, whereby a reaction is established between the oxygen of the ferrous oxide and the carbon and silicon in the metal. The silicon is thus converted into silicic acid, which with the lime forms a fusible slag, and the carbon, combining with oxygen, escapes as carbonic acid, and induces a powerful ebullition. Modification of this process. The ore is treated in a separate rotatory furnace with carbonaceous material, and converted into balls of malleable iron, which are transferred from the rotatory to the bath of the steel-melting furnace. This process is adapted to the production of steel of a very high quality, because the sulphur and phosphorus of the ore are separated from the metal in the rotatory furnace. Siemens -Martin Process. Scrap-iron or steel is gradually added in a highly heated condition to a bath of about .25 its weight, of highly heated pig, and melted. Samples are occasionally taken from the bath, in order to ascertain the percentage of carbon remaining in the metal, and ore is added in small quantities, in order to reduce the carbon to about .1 per cent. At this stage of the process, siliceous iron, spiegeleisen, or ferro-manganese is added in such proportions as are necessary to produce steel of the required degree of hardness. The metal is then tapped into a ladle. Landore-Siemen's Steel is a variety of steel made by the Modification of Siemens Process. Its great value is due to its extreme ductility, and its having nearly like strength in both directions of its plates. Whitworth's Compressed Steel is molten steel subjected to a pressure of about 6 tons per square inch, by which all its cavities are dispelled, and it is compressed to about .875 of its original volume, its density and strength be- ing proportionately increased. Chrome and Tungsten Steel are made by adding a small percentage of Chromium or Tungsten to crucible steel, the result producing a steel of great hardness and tenacity, suitable for tools, such as drills, etc. Homogeneous Steel is a variety of cast steel containing .25 per cent, of carbon. Remarks on Manufacture of Steel, and Mode of Working it. (D. Chernoff, 1868). Steel, when cast and allowed to cool quietly, assumes a crystalline structure. Higher temperature to which it is heated, softer it becomes, and greater is liberty its particles possess to group themselves into crystals. Steel, however hard it may be, will not harden if heated to a temperature lower than what may be distinguished as dark cherry-red, a, however quickly it is cooled , on contrary, it will become sensibly softer, and more easily worked with a file. Steel, heated to a temperature lower than red, but not sparkling, &, does not change its structure whether cooled quickly or slowly. When temperature has reached 6, substance of steel quickly passes from granular or crystalline condition to amorphous, or wax-like structure, which it retains up to its melting-point, c. Points a, 6, and c have no permanent place in scale of temperature, but their posi- tions vary with quality of steel; in pure steel, they depend directly on quantity of constituent carbon. Harder the steel, lower the temperatures. Tints above speci- fied have reference only to hard and medium qualities of steel; in very soft kinds of steel, nearly approaching to wrought iron, points a and b range very high, and in wrought iron point 6 rises to a white heat. METALS. STEEL. 643 Assumption of the crystalline structure takes place entirely in cooling, between temperatures c and 6; when temperature sinks below 6 there is no change of struc- ture. For successful forging, therefore, heated ingot, after it is taken out of furnace, must be forged as quickly as practicable, so as not to leave any spot untouched by hammer, where the steel might crystallize quietly, as formation of crystals should be hindered, and the steel should be kept in an amorphous condition until tem- perature sinks below point 6. Below this temperature, if piece is cooled in quiet, mass will no longer be disposed to crystallize, but will possess great tenacity and homogeneousness of structure. When steel is forged at temperatures lower than 6, its crystals or grains, being driven against each other, change their shapes, becoming elongated in one direction, and contracted in another; while density and tensile strength are considerably in- creased. But available hammer-power is only sufficient for treatment of small steel forgings; and object of preventing coarse crystalline structure in large forgings is more easily and more certainly effected, if, after having given forging desired shape, its structure be altered to an homogeneous amorphous condition by heating it to a temperature somewhat higher than 6, and the condition be fixed by rapid cooling to a temperature lower than &, the piece should then be allowed to finish cooling gradually, so as to prevent, as far as practicable, internal strains due to sudden and unequal contraction. Alloys of steel with Silver, Platinum, Rhodium, and Aluminum have been made with a view to imitating Damascus steel, Wootz, etc., and improving fabrication of some finer kinds of surgical and other instruments. Properties of Steel. After being tempered it is not easily broken ; it welds readily ; does not crack or split ; bears a very high heat, and preserves the capability of hardening after repeated working. Hardening and Tempering. Upon these operations the quality of manu- factured steel in a great measure depends. Hardening is effected by heating steel to a cherry-red, or until scales of oxide are loosened on surface, and plunging it into a cooling liquid ; degree of hardness depends upon heat and rapidity of cooling. Steel is thus ren- dered so hard as to resist files, and it becomes at same time extremely brittle. Degree of heat, and temperature and nature of cooling medium, must be chosen with reference to quality of steel and purpose for which it is intended. Cold water gives a greater hardness than oils or like sub- stances, sand, wet-iron scales, or cinders, but an inferior degree of hardness to that given by acids. Oil, tallow, etc., prevent cracks caused by too rapid cooling. Lower the heat at which steel becomes hard, the better. Tempering. Steel in its hardest state being too brittle for most purposes, the requisite strength and elasticity are obtained by tempering or " letting down the temper " which is performed by heating hardened steel to a certain degree and cooling it quickly. Requisite heat is usually ascertained by color which surface of the steel assumes from film of oxide thus formed. Degrees of heat to which these several colors correspond are as follows : At 430, very faint yellow.. (Suitable for hard instruments; as hammer - faces, At 450, pale straw color. ... \ drills, lancets, razors, etc. At 470, full yellow ( For instruments requiring hard edges without elastici- At 490, brown color ( ty ; as shears, scissors, turning tools, penknives, etc. knots' br WD ' WUh PUrPl6 ( For tools for cuttin & wood and soft metals ; such as At 538, purple .'!!.'!!.'!!!!!( P lane - irons > saws knives, etc. At 550' dark blue (For tools requiring strong edges without extreme At 560, full blue I hardness; as cold-chisels, axes, cutlery, etc. At 600, grayish blue, verg- j For spring- temper, which will bend before breaking; ing on black \ as saws, sword-blades, etc. If steel is heated to a higher temperature than this, effect of the hardening process is destroyed. A high breaking strain may not be conclusive as to quality, as it may be due to a hard, elastic metal, or a low one may be due to great softness. 644 METALS. TIN. ZINC. MODELS. Case-h.arden.ing. This operation consists in converting surface of wrought iron into steel T by cementation, for purpose of adapting it to receive a polish or to bear fric- tion, etc. ; it is effected by heating iron to a cherry-red, in a close vessel, in contact with carbonaceous materials, and then plunging it into cold water. Bones, leather, hoofs, and horns of animals are generally used for this pur- pose, after having been burned or roasted so that they can be pulverized. Soot is also frequently used. The operation reduces strength of the iron. TIN. TIN is more readily fused than any other metal, and oxidizes very slowly. Its purity is tested by its extreme brittleness at high temperature. Tin plate is iron plate coated with tin. Block Tin is tin plate with an additional coating of tin. ZINC. ZINC, if pure, is malleable at 220 ; at higher temperatures, such as 400, it becomes brittle. It is readily acted upon by moist air, and when a film of oxide is formed, it protects the surface from further action. When, how- ever, the air is acid, as from the sea or large towns, it is readily oxidized to destruction. Iron, Copper, Lead, and Soot are very destructive of it, in consequence of the voltaic action generated, and it should not be in contact with calcareous water or acid woods. The best quality, as that known as " Vielle Montagne," is composed of zinc .995, iron .004, and lead .001. Its expansion and contraction by differences of temperature is in excess of that of any other metal. STRENGTH OF MODELS. The forces to which Models are subjected are, i. To-draw them asunder by tensile stress. 2. To break them by trans- verse stress. 3. To crush them by compression. The stress upon side of a model is to corresponding side of a structure as cube of its corresponding magnitude. Thus, if a structure is six times greater than its model, the stress upon it is as 6 3 to i = 216 to i : but resistance of rupture increases only as squares of the corresponding magnitudes, or as 6 2 to i =36 to i. A structure, therefore, will bear as much less resistance than its model as its side is greater. To Com.pu.te Dimensions of a Beam, etc., -which a Strvi.ctu.re can "bear. RULE. Divide greatest weight which the beam, etc. (including its weight), in the model can bear, by the greatest weight which the structure is required to bear (including its weight), and quotient, multiplied by length of beam, etc., in model, will give length of beam, etc., in structure. EXAMPLE. A beam in a model 7 inches in length is capable of bearing a weight of 26 Ibs., but it is required to sustain only a weight or stress of 4 Ibs. ; what is the greatest length that a corresponding beam can be made in the structure? 26 -r- 4 = 6.5, and 6. 5 X 7 = 45-5 * In structure. MODELS. MOTION OF BODIES IN FLUIDS, 645 Resistance in a model to crushing increases directly as its dimensions; but as stress increases as cubes of dimensions, a model is stronger than the structure, inversely as the squares of their comparative magnitudes. Hemce, greatest magnitude of a structure is ascertained by taking square root of quotient, as obtained by preceding rule, instead of quotient itself. EXAMPLE. If greatest weight which a column in a model can sustain is 26 Ibs., and it is required to bear only 4 Ibs. ; height of column being 18 ins., what should be height of it in structure ? /( \ = ^6.5 = 2.55, and 2.55 X 18 = 45.9 ins., height of column w If, when length or height and breadth are retained, and it is required to give to the beam, etc., such a thickness or depth that it will not break in con- sequence of its increased dimensions, Then /( ) = V 6 -5 = 2 -55> which, x square of relative size of model = thick- ness required. To Compute Resistance of a Bridge from, a Alodel. n 2 W (n i) w\ = load bridge will bear in its centre. EXAMPLE. If length of the platform of a model between centres of its repose upon the piers is 12 feet, its weight 30 Ibs., and the weight it will just sustain at its centre 350 Ibs. , the comparative magnitudes of model and bridge as 20, and actual length of bridge 240 feet ; what weight will bridge sustain ? 2o 2 X 350 [^ X (201) X 30! = 140000 3800 X 30 = 26000 Ibs. MOTION OF BODIES IN FLUIDS. If a body move "through a fluid at rest, or fluid move against body at rest, resistance of fluid against body is as square of velocity and density of fluid ; that is, R = d v*. For resistance is as quantity of matter or particles struck, and velocity with wL\ch they are struck. But quan- tity or number of particles struck in any time are as velocity and density of fluid ; therefore, resistance of a fluid is as density and square of velocity. = h, and = R. h representing height due to velocity, d density of fluid, 2 g zg and R resistance or motive force. Resistance to a plane is as plane is greater or less, and therefore resistance to a plane is as its area, density of medium, and square of velocity; that is, R = adv*. Motion is not perpendicular, but oblique, to plane or to face of body in any angle, sine of which is * to radius i ; then resistance to plane, or force of fluid against plane, in direction of motion, will be diminished in triplicate ratio of radius to sine of angle of inclination, or in ratio of i to s 3 . Hence. - R, and - ^ ^- = F. w representing weight of body, and F 2 g 2 g w retarding force. Progression of a solid floating body, as a boat in a channel of still water, gives rise to a displacement of water surface, which advances with an un- dulation in direction of body, and this undulation is termed Wave of Dis- placement. 646 MOTION OF BODIES IN FLUIDS. Resistance of a fluid to progression of a floating body increases as velocity of body attains velocity of wave of displacement, and it is greatest when the two velocities are equal. In the motion of elastic fluids, it appears from experiments that oblique action produces nearly same effect as in motion of water, in the passage of curvatures, apertures, etc. Resistance to an Area of One Sq.. Foot moving through. Water, or Contrariwise. Angle of Angle of Surface with Plane of Pressure per Sq. Foot for following Ve- locities per Foot per Minute. Surface with Plane of Pressure per Sq. Foot for following Ve- locities per Foot per Minute. Current. 120 240 480 900 Current. 1 20 240 480 900 Lbs Lbs. Lb. Lbs. Lbs. Lbs. Lbs. Lbs. 6 .09 359 '435 5.046 45 2.66 10.639 42.557 149.614 8 133 53 2.122 7-459 50 2-995 11.981 47-923 168.48 9 .156 .624 2.496 8-775 55 3-249 12.995 182.739 10 .179 .718 2.8 7 10.091 60 3-455 13.822 55-286 194.366 15 355 142 5.678 19.963 6 5 3.607 14-43 57-72 202.922 20 .608 2-434 9-734 34-222 70 3.728 14.914 59-654 209. 722 25 94 376 15038 52.869 75 3-8i 15.241 60.965 214.329 30 1-353 54 J 3 21.653 76.123 80 3-857 15-428 61.714 216.926 35 1.798 7.192 28.766 IOI.I32 85 3.892 15-569 62.275 218.936 40 2.258 9032 36-13 I27.0l8 90 3-9 15-6 62.4 219.375 Resistance to a plane, from a fluid acting in a direction perpendicular to its face, is equal to weight of a column of fluid, base of which is plane and altitude equal to that which is due to velocity of the motion, or through which a heavy body must fall to acquire that velocity. Resistance to a plane running through a fluid is same as force of fluid in motion with same velocity on plane at rest. But force of fluid in motion is equal to weight or pressure which generates that motion, and this is equal to weight or pressure of a column of fluid, base of which is area of the plane, and its altitude that which is due to velocity. ILLUSTRATION. If a plane \ foot square be moved through water at rate of 32.166 o 2 ^52 feet per second, then 16.083, space a body would require to fall to acquire a velocity of 32.166 feet per second; therefore i x 62.5 (weight of a cube foot of water) x ^^ = 1005 J&s. = resistance of plane. Resistance of* different Figures at different "Velocities in Air. Veloci- ty per Second. Co Vertex. ne. Base. Sphere. Cylin- der. Hemi- sphere. Round. Veloci- ty per Second. Co Vertex. ne. Base. Sphere. Cylin- der. Hemi- sphere. Round. Feet. Oz. Oz. Oz. Oz. Oz. Feet. Oz. Oz. Oz. Oz. Oz. 3 .028 .064 .027 05 .02 12 .376 -85 37 .826 347 4 .048 .109 .047 .09 039 14 .512 1.166 505 I-I45 .478 5 .071 .162 .068 143 .063 15 -589 1.346 .581 1-327 552 8 .168 -382 .162 -36 .16 16 673 1.546 .663 1.526 -634 9 10 .211 26 .478 -587 205 255 .456 565 .199 .242 18 20 .858 1.069 2.002 2-54 .848 1.057 1.986 2.528 .818 1.033 Diameter of all the figures was 6.375 ins., and altitude of the cone 6.625 ins. Angle of side of cone and its axis is, consequently, 25 42' nearly. From the above, several practical inferences may be drawn. T. That resistance is nearly as surface, increasing but a very little above that proportion in greater surfaces. MOTION OF BODIES IN FLUIDS. 647 9. Resistance to same surface is nearly as square of velocity, but gradu- ally increasing more and more above that proportion as velocity increases. 3. When after parts of bodies are of different forms, resistances are differ- ent, though fore parts be alike. 4. The resistance on base * of a cone is to that on vertex nearly as 2.3 to i. And in same ratio is radius to sine of angle of inclination of side of cone to its path or axis. So that, in this instance, resistance is directly as sine of angle of incidence, transverse section being same, instead of square of sine. Resistance on base of a hemisphere is to that on convex side nearly as 2.4 to i, instead of 2 to i, as theory assigns the proportion. Sphere. Resistance to a sphere moving through a fluid is but half re- sistance to its great circle, or to end of a cylinder of same diameter, moving with an equal velocity, being half of that of a cylinder of same diameter. * / 2 gx- dx = V. d representing diameter of sphere, and N and n spe~ V 3 n ciflc gravities of sphere and relisting fluid. x - d S. S representing space through which a sphere passes while acquir- n 3 ing its maximum velocity, in falling through a resisting fluid. ILLUSTRATION. If a ball of lead i inch in diameter, specific gravity 11.33, be set free in water, specific gravity i, what is greatest velocity it will attain in descend- ing, and what space will it describe in attaining this velocity ? g = 32.166, d = foot, N = n.33, and n = i. Then^2 X 32.166 x ^-of ^ x *-^ - = v 7 .i 4 8 X 10.33 = Hence, ^ x. - of = i.2$gfeet. 2^L = f= retardive force = . ' i 3 12 8#Nd ' ags Cylinder. = R, and =/ a representing area or p r 2 , and w weight of body. ILLUSTRATION. Assume a = 32 sq. feet, v = 10 feet per second, and n = .0012. Then = .o6ofa cube foot of water = . 06 of 62. 5 = 3. 75 Ibs. nav 2 s 3 npd 2 v 2 s* . npd*v 2 s 2 Conical Surface. = R, also - = R, and ?- 2 g 8 g 8 g to =/. * representing sine of inclination, and a convex surface of cone. Curved End as a Spliere or Hemispherical End. ^ io g = R, and Circle .5 of spherical end. In general, when n is to water as a standard, result is in cube feet of water, if a is in sq. feet; and in cube ins. of water, if a is in sq. ins., v in ins., and g in ins. If n is given in Ibs. in a cube foot, a is in sq. feet, v and g are in feet, result is in Ibs. To Compute Altitude of* a Column of Air, ^Pressure of -which, shall toe ec^ual to Resistance of a Body moving- through it, -with any Velocity. ^ X = = altitude in feet. ax = volume of column in feet, and ax weight in ounces, a representing area of section of body, similar to any in table, perpen- dicular to direction of motion, r resistance to velocity in table, and x altitude sought of a column of air, base of which is a, and pressure r. * This is a refutation of the popular assertion that a taper spar can be towed in water easiest when the base is foremost. 648 MOTION OF BODIES IK FLUIDS. When a of a foot, as in all figures in table, x becomes r wfcen r = re- 9 4 Distance in table to similar body. ILLUSTRATION. Assume convex face of hemisphere resistance = .634 oz. at a Te- locity of 16 feet per second. Then r = .634, and x = r = 2.3775 feet = altitude of column of air, pressure of which = resistance to a spherical surface at a velocity of 16 feet. To Compute 'when. Pressure of Air in rear of* a ^Projectile is Inferior to Pressure d.ue to its "Velocity. Assume height of barometer = 2. 5 feet, and weight of atmosphere =14. 7 Ibs. Weight of cube inch of mercury = -^ = .49 Ibs., and weight of cube inch of air = .00004357 Ibs.; hence, .49-7-. 000043 57 = 11 246, which X2-5 feet = 28 115 feet. Then Vi6.o8 : V^^S '' 32-16 : a?, and x = 32 ' l6 X ,^ 2B " 5 = 1341.6 feet. VID To Compute "Velocity -with, which a Plane Surface must "be projected, to generate a Resistance just eq.ua! to ^Pressure of Atmosphere upon it. By table, resistance on a circle with an area of .222 sq. foot (2 -1-9) = .051 oz., at a velocity of 3 feet per second. Hence 3 2 : i 2 : : .051 : .0056 oz. at a velocity of i foot, and i X 144 X 14- 7 X 16 X 2 ^-9= 7526. 4 oz. Hence, V- 0056 : V/526.4 :: i : n6ofeet. To Compute "Velocity lost toy a ^Projectile. If a body is projected with any velocity in a medium of same density with itself, and it describes a space = 3 of its diameters, Then * = 3 d, and b = j^ = . Hence, b x = -|- , and ^-- = ^- = velocity lost nearly .66 of projectile velocity. c = base of Nap. system of log. ; hence c* = number corresponding to Nap. log. b x. Hence, if & a; x -4343, result =. com. log. of c& *. b x =| = 1. 125, which x .4343 = .488 587 5, and number to this com. log. = 3.0803. Hence, velocity lost = 3 ' o8 3 ~ I = ^. 3.0803 3.08 ILLUSTRATION. If an iron ball 2 ins. diam. were projected with a velocity of 1200 feet per second, what would be velocity lost after moving through 500 feet of space ? d = = -, 05 = 500, N = 7i., and n = . 0012. 3n 3X12X500X3X6 81 1200 Hence, 6 = ^= 8X 21 x . 000 ^~ = . ** = ^ = 998 JU per second, having lost 202 feet, or nearly -J- of its initial velocity. o 12 = .0012, and = ~ and ~ inverted, because N and n are in denominator. 10 000 22 3 6 To Compute Time and "Velocity. ILLUSTRATION. If an iron ball 2 ins. in diameter were projected in air with a ve- locity of 1200 feet per second, in what time would it pass over 1500 feet, and what its velocity at end of that time ? 6 3XX 3 X6 = , 6 V>9o i2oo/ NAVAL ARCHITECTURE, 649 NAVAL ARCHITECTURE. Results of Experiments upon. Form of "Vessels. (Wm. Bland.) Cnbical Models. Head Resistance. Increases directly with area of its surface. Weight Resistance. Increases directly as weight. Vessels' Models. Lateral Resistance. About one twelfth of length of body immersed, varying with speed. Order of Superiority of Amidship Section. Rectangle, Semicircular, Ellipse, and Triangle. Centre of lateral resistance moves forward as model progresses. Centre of gravity has no influence upon centre of lateral resistance. Relative Speeds. Z^n^A. Increased length gives increased speed or less resistance. Depth of Flotation. Less depth of immersion of a vessel, less the resistance. A midship Section. Curved sections give higher speed than angled. Sides. Slight horizontal curves present less resistance than right lines. Curved sides with one fourth more beam give equal speeds with straight sides of less beam. Keel. Length of keel has greater effect than depth. Stern. Parallel-sided after bodies give greater speed than taper-sided. FORM OF Bow. Order of Speed. Isosceles triangle, sides slightly convex " right lines " slightly concave at entrance and running) out convex J Spherical equilateral triangle compared to Equilateral triangle, speed is as ii to 12. Equilateral triangle , with its isosceles sides bevelled off at an angle of 45, compared to bow with vertical sides, is as 5 to 4. When bow has an angle of 14 with plane of keel, compared with one of 7, its speed is greater. Bodies Inclined Upwards from Amidship Section. i. Model with bow inclined from &, has less resistance than model with- out any inclination. a. Model with stern inclined from 58, has less resistance than model with- out any inclination. Model i had less resistance than model 2. Model with both bow and stern inclined from &, has less resistance than either i or 2. Stability. Results of Experiments npon Stability of* Rectangular Blocks of Wood of Uniform Length, and Depth, toxit of Different Breadths. (Wm. Bland.) Length 15, Depth 2, and Depression i inch. Width. Weight. AB Observed. Ratio o With like Weights. f Stability. By Squares of Breadth. By Cubes of Breadth. Int. 3 4-5 6 7 Oz. 24 35 45 55 i 2-5 7 ii I 2.4 3-7 4.8 i 2.25 6.25 i 3-375 8 15-625 650 NAVAL ARCHITECTURE. Hence it appears that rectangular and homogeneous bodies of a uniform length, depth, weight, and immersion in a fluid, but of different breadths, have stability for uniform depressions at their sides (heeling) nearly as squares of their breadth ; and that, when weights are directly as their breadths, their stability under like circumstances is nearly as cubes of their breadth. With equal lengths, ratio of stability is at its limit of rapid increase when width is one third of length, being nearly in cube ratio ; afterwards it ap- proaches to arithmetic ratio. Results of* Experiments vipon Stability and. Speed, of Models having Amidship Sections of different Forms, t>\at Uniform Length, Breadth, and "Weights. (W. Bland.) Immersion different, depending upon Form of Section. FORM OF IMMERSED SECTION. Half-depth triangle, other half rectangle. Rectangle Right-angled triangle * Semicircle Stability. 14 7 9 Speed. 4 3 3 * Draught of water or immersion double that of rectangle. Fig. i. Statical Stability is moment of force which a body in flotation exerts to attain its normal position or that of equilibrium, it having been deflected from it, and it is equal to product of weight of fluid displaced and horizontal distances between the two centres of gravity of body and of displacement, or it is product of weight of displacement, height of Meta-centre, and Sine of angle of inclination. Dynamical Stability is amount of mechanical work necessary to deflect a body in flotation from its normal position or that of equilibrium, and it is equal to product of sum of vertical distances through which centre of grav- ity of body ascends and centre of buoyancy descends, in moving from ver- tical to inclined position by weight of body or displacement. To Determine Measure of Stability of Hull of a Vessel or Floating Body. Fig. 1. Measure of stability of a floating body depends essentially upon horizontal dis- tance, G s, of meta-centre of body from centre of gravity of body; and it is product offeree of the water, or resistance to displacement of it, acting upward, and distance of G 5, or P x G s. If distance, G M, represented by r, and angle of rolling, c M r, by M, measure of sta- bility, or S is determined by P r, sin. M = 8 ; and this is therefore greater, the greater the weight of body, the greater distance of meta- centre from centre of gravity of body, and the greater the angle of inclination of this or of cMr. Assume figure to represent transverse section of hull of a vessel, G centre of gravity of hull, w I water-line, and c centre of buoyancy or of displacement of im- mersed hull in position of equilibrium. Conceive vessel to be heeled or inclined over, so that ef becomes water-line, and s centre of buoyancy; produce s M, and point M is meta-centre of hull of vessel. Transverse meta-centre depends upon position of centre of buoyancy, for it is that point where a yertical line drawn from centre intersects a line passing through centre of gravity of hull of vessel perpendicular to plane of keel. Point of mtta-centre may be the same, or it may differ slightly for different angles of heeling. Angla of direction adopted to ascertain position of meta- centre should be greatest which, under ordinary cir- cumstances, is of probable occurrence ; in different vessels this angle ranges from 20 to 6o e . If meta-centre is above centre of gravity, equilibrium is Stable; if it coincides with it, equilibrium is Indifferent ; and if it is below it, equilibrium is Unstable. W NAVAL ARCHITECTURE. 65 I Comparative Stability of different hulls of vessels is proportionate to the distance of G M for same angles of heeling, or of distance G s. Oscillations of hull of a ves- sel may be resolved into a rolling about its longitudinal axis, pitching about its transverse axis, and vertical pitching, consisting in rising and sinking below and above position of equilibrium. Tf transverse section of hull of a vessel is such that, when vessel heels, level of centre of eravity is not altered, then its rolling will be about a permanent longi- tudinal axis traversing its centre of gravity, and it will not be accompanied by any vertical oscillations or pitchings, and moment of its inertia will be constant while it rolls But if when hull heels, level of its centre of gravity is altered, then axis about which it rolls becomes an instantaneous one, and moment of its inertia will vary as it rolls; and rolling must then necessarily be accompanied by vertical os- cillations. Such oscillations tend to strain a vessel and her spars, and it is desirable, therefore, that transverse section of hull should be such that centre of its gravity should not alter as it rolls, a condition which is always secured if all water-lines, as w I and ef, are tangents to a common sphere described about G; or, in other words, if point of their intersections, o, with vertical plane of keel, is always equidistant from centre of gravity of hull. To Compute Statical Stability. D c M sin. M = S. D representing displacement, M angle of inclination, and S stability. ILLUSTRATION i. Assume a ship weighing 6000 tons is heeled to an angle of 9, distance c M = 3 feet, Sin. 9 = .1564. Then 6000 X 3 X .1564 = 2815.2 foot-tons. 2._Weight of a floating body is 5515 Ibs., distance between its centre of gravity and meta-centre is 11.32 feet, and angle M = 20 Sin. M = . 34202. Hence 5515 X 11-32 X 34202 =21 352. 24 /oo<-Z5*. Statical Surface Stability. Moment of Statical surface stability at any angle is c z D. Assuming centre of gravity of vessel coincided with c ; coefficient of a vessel's stability at any angle of heel is expressed when the displacement is multiplied by vertical height of the meta-centre for given angle of heel above centre of gravity, or D c M. Approximately. RULE. Divide moment of inertia of plane of flotation for upright position, relatively to middle line by volume of displacement ; and quotient multiplied by sine of angle of heel will give result. Per Foot of Length of Vessel, - (B 3 sin. M). B representing half breadth. Dynamical Surface Stability. Moment of Dynamical surface stability is expressed by product of weight of vessel or displacement and depression of centre of buoyancy during the inclination, that is, for angle M. To Compute Dynamical Stability of a Vessel. Approximately. RULE. Multiply displacement by height of meta-centre above centre of gravity, and product by versed sine of angle of heel. !Or multiply statical stability for given angle by tangent of .5 angle of heel. To Compute Elements of* Stability of* a Floating Body. T a ~ ** sin M = Tj ~ M = ^ ftnd sin ' M r c - A representing area of immersed section; A' section immersed by careening of body, as fo I; s horizontal distance, c r, between centres of buoyancy ; a horizontal distance between centres of gravity, i i, of areas immersed and emerged by careening; g distance, c M, between centre of buoyancy or of water displaced and meta-centre ; r distance, G M, between centre of gravity and meta-centre; c horizontal distance, G s, between centre of grav- ity and of line of displacement of it when careened ; e vertical distance between centres of gravity and buoyancy, all in feet ; and M angle of careening. 652 NAVAL ARCHITECTURE. NOTE. When centre of gravity, G, is below that of displacement, c, then e is +; when it is above c it is ; and when it coincides with c it is o; or e is when p s. Assumed elements of figure illustrated are A = 86, A' = 21. 5, 6 = 21. 5, and e = . 5. The deduced arc 5 = 3. 7, = 3.87, # = 10.82, a =14.9, and r = 11.32. b repre- senting breadth at water-line or beam in feet, and P weight or displacement in ibs. or tons. = io.82/eeJ, c = .34202X 11.32 = 3.87 feet. Of Hull of a Vessel. ( ^ ^ e\ P, sin. M = S ; d cos. .5 M = d', b -JL-(f_^ = c; p( 6 t a + iiiOl)=S; and ^ _ - - -_ , --_ t 10.7 to 13 (11.93) A ' sin. M\P y VA ^ P (s db e sin- M) =S. d representing depth of centre of gravity of displacement un- der water in equilibrium, and d' depth when out of equilibrium, both in feet. ILLUSTRATION i Displacement of a vessel is 10000000 Ibs. ; breadth of beam, 50 feet; area of immersed section, 800 sq. feet; vertical distance from centre of grav- ity of hull up to centre of buoyancy or displacement, 1.9 feet, and horizontal dis- tance a between centres of gravity of areas immersed and emerged, when careened to an angle of 9 10' 33.4 feet, immersed area being 50 sq. feet. Sin. 9 io' = .i593. Then s = -~ X 33.4 = 2.0875 feet, 800X2.0875 = 50X33.4, t= = . B .. = g . . 1593 ii -93 X 800 \ii .93 X 8oo/ T 10 ooo ooo X.i 593 = 23 905 396 Ibs., and e = ( 23 9 5 39 -- 2.0875) = 1.9 feet. .1593 \iooooooo / 2. Assume a ship having a displacement of 5000 tons, and a height of meta-centre of 3.25 feet, to be careened to 6 12'. What is her statical stability ? Sin. 6 12'=. 1079. Then 5000 X 3.25 X -1079 = 1753.37 foot-tons. 3. Assume a weight, W, of 50 tons to be placed upon her spar deck, having a common centre of gravity of 15 feet above her load-line, Then 5000 X 3- 25 50 -j- 1 5 X . 1079 J 747- 3^ foot- tons. 4. Assume 100 tons of water ballast to be admitted to her tanks at a common centre of gravity of 15 feet below her load-line. Then 5000 X 3-25 -J- 100 X 15 X .1079 = I 9 1 5- 22 foot-tons. 5. Assume her masts, weighing 6 tons, to be cut down 20 feet, Then = foot =fall of centre of gravity, and 5000 X (3. 25 -f j X . 1079 = 1774.95 tons. To Compixte Elewnents of Power, etc., required to Careen a Body or "Vessel. Sin. M (h - n sin. M) -f n sec. M - s = I. ** . A 3/- r=m. io-7toi3*A V 64.125 L A Wlr = Pc, and W 1 = S. W representing weight or power exerted and I distance at which weight or power acts to careen body, taken from centre of gravity of displace- ment perpendicular to careening force, h vertical height from centre of gravity of dis- placement to centre of weight or power to careen body tvhen it is in equilibrium, n horizontal distance from centre of vessel to centre of weight or power, L length of vessel, m meta-centre, and S as in preceding case, all in feet. * Unit for section of a parallelogram ia 10.7 ; of a semicircle 12, and of a triangle 12.8. NAVAL ARCHITECTURE. 653 ILLUSTRATION. A weight is placed upon deck of a vessel at a mean height of 3.87 feet from centre line of hull; height at which it is placed is 11.32. and other ele- ments as in tirst case given. Sec. 20 = .342. Thenft=ii. 3 2, w 3.87, and ^ = .342 (11.3 3.87 x.342)-f- 3.87 X 1.0642 3.7 = .342X16 + 4.12 3.7 = 3.84/<*rt. Assume W = 5515. Then 5515 x 3.84 = 21 i8-,.6foot-lbs. Or P (w cos. M -j- h sin. M) = 8. w representing distance of weight from centre of vessel, and h height ofw above water-line, both, in feet. ILLUSTRATION. If a weight of 30 tons placed at 20 feet from centre of hull or deck, 10 feet above water-line, careens it to an angle of 2 9', what is its stability? cos. 2 9' = .9993 ; sin. 2 9' = -0375. 30 (20 X -9993 + 10 X .0375) = 30 X 20.361 = 610.83 foot-tons. Bottom, and. Immersed Surface of Hull of Vessels. To Compute Bottom and Side Surface of Hull. Bottom and Side. RULE. Multiply length of curve of amidship section, taken from top of tonnage or main deck beams upon one side to same point upon other (omitting width of keel), by mean of lengths of keel and be- tween perpendiculars in feet, multiply product by .85 or .9 (according to the capacity of vessel), and product will give surface required in sq. feet. EXAMPLE. Lengths of a steamer are as follows: keel 201 feet, and between per- pendiculars 210 feet, curved surface of amidship section 76 feet; what is surface? Coefficient .87. 2104-201 -=- 2 = 205.5, and 76 X 205.5 X .87 = 13 587 sq. feet. NOTE. Exact surface as measured was 13650 sq. feet. Bottom Surface. RULE. Multiply length of hull at load-line by its breadth, and this product by depth of immersion (omitting the depth of keel) in feet ; and this product multiplied by from .07 to .08 (according to capacity of vessel) will give surface required in sq. feet. EXAMPLE. Length upon load-line of a vessel is 310 feet, beam 40 feet, depth of keel i foot, and draught of water 20 feet; what is bottom or wet surface? Coefficient assumed .073. 310 X 4 X 20 1 x .073 = 17 199 sq. feet To Compute Resistance to Wet Surface of Hull. C a v 2 = R. C representing a coefficient of resistance, a area of wet surface in sq. feet, and v velocity of hull in feel per second. Values of C f <00 7> clean c PP er - I .014, iron plate. '(.or, smooth paint. | .019, iron plate, moderately foul. Power required to propel one sq. foot of immersed amidship section at gj is .073 that of smooth wet surface. To Compute Elements of a Vessel. Displacement and its Centre of Grravity. Displacement of a vessel is volume of her body below water-line. Centre of Gravity, or Centre of Buoyancy of Displacement, is centre of gravity of water displaced by hull of vessel. For Displacement. RULE. Divide vessel, on half breadth plan, into a number of equidistant sections, as one, two, or more frames, commencing at i& and running each side of it. Add together lengths of these lines in both fore and aft bodies, except first and last, by Simpson's rule for areas (see page 344) ; multiply sum of products by one third distance between sections, and product will give area of water-line between fore and aft-sections. Then compute areas contained in sections forward and aft of sections taken, in- cluding stern and rudder-post, rudder and stem, and add sum to area of body-sec* tions already ascertained. * * To Compute Area of a Water-line, see Mauau.atlon of Surfaces, 654 NAVAL ABCHITECTUEB. Compute area of remaining water-lines in like manner. Tabulate results, and multiply them by Simpson's rule in like manner as for a water-line, and again by consecutive number of water-lines, and sum of products between water-line and product will give volume between load and lower water-line. Add area of lower water-line to area of upper surface of keel ; multiply half sum by distance between them, and product will give volume; then compute areas con- tained in sections forward and aft of sections taken as before directed. If keel is not parallel to lower water-line, take average of distance between them. Compute volume of keel, rudder-post and rudder below water-line ; add to volume already ascertained; multiply product by two, for full breadth, and product will give volume required in cube feet, all dimensions being taken in feet. . EXAMPLE. -Assume a vessel 100 feet in length by 20 feet in extreme breadth, on load-line of 8 feet 9 inches immersion. Figs. 2 and 3. Distance between sections, for purpose of simplifying this example, is taken at 10 feet; usually frames are 18 to 30 ins. apart, and two or more included in a section. Water-lines 2 feet apart. Fig. 3. ist Water-line. 45 =5 3 7-7X4 = 30-8 2 9.5 X 2 = 19 i 9.9 X 4 = 39- 6 O 10 X 2 = 20 A 9.6 X 4 = 38-4 B 7.8 X 2 = 15.6 C 6.8 X 4 = 27.2 D 4 = 4 199.6 10-^-3 = 3^ zd Water-line. 4 2-7 = 2 -7 3 6.9 X 4 = 27.6 2 8.7 X 2 = 17.4 i 9.5 X 4 = 38 9.6 X 2 = 19.2 A 9 X 4 = 36 B 7 X 2 = 14 C 5 x 4 = 20 D 2 =2 176.9 10 -i- 3 = 3^ 3d Water-line. 4 i-5 = -5 3 5 X 4 = 20 2 6.6 X 2 = 13.2 i 8.7 X 4 = 34-8 o 8.9 X 2 = 17.8 A 7.6 X 4 = 3<>-4 B 7 X 2 = 14 C 3 X 4 = 12 D J. 2 = 1.2 144.9 xo-r-3 = 3 665-3 Abaft section 4, rud- der and post 25 Forward section D 589.7 Abaft section 4, rud- der and post 13.2 Forward section D and stem 9. i 483 Abaft section 4, rud- der and post 7 Forward section D 7" 4th Water-line. 4-7 = -7 3 2X4=8 2 4.3 X 2 = 8.6 i 6.5 X 4 = 26 o 6.8 X 2 = 13.6 A 5 X 4 = 90 B 3.6 X 2 = 7.2 C -9 X 4 -= 3-6 D -3 = -3 88 I0 ~*" 3 = _M- 293.3 Abaft section 4, rud- der and post 3.2 Forward section D and stem 8 "6^2" Ke Half breadth = .25 X le Rudder-post and rudde Res' ist water-line 711 2d 612 x 4 = 3d 495.4 X 2 4th 297.3 X 4 = Keel 24.8 495-4 el. ngth of 98 feet = 24. 5 r 3 24.8 ults. 711 2448 X i = 2448 990.8 X 2 = 1981.6 1189.2 X 3 = 3567-6 24.8 X 4= 99-2 5363" 8 8096.4 2 3)10727.6 Displacement, 3575.9 X 2 = jisi.Sculeft 297-3 NAVAL ARCHITECTURE. 655 To Compute Centre of* Q-ravity of Displacement. RULE. Divide sum of products obtained as above, by consecutive water- lines, by sum of products obtained in column of products by Simpson's mul- tipliers, and quotient, multiplied by distance between water-lines, will give, depth of centre below load water-line. ILLUSTRATION i. 8096.4, from above, -4- 5363.8 = 1.5, which X 2 = 3 feet Or, n = d. n representing draught of water exclusive of any drag of 2 ('-3Ti) keel, a area of immersed surface of hull in sq.feet, and D displacement in cube feet. 2 . Assume draught of water 8 feet, displacement 7152 cube feet, and area of Im- mersed surface of hull noo sq. feet. (. 25! \ IIOO) X8/ To Compute Displacement Approximately. Coefficient of Displacement of a vessel is ratio that volume of displacement bears to parallelopipedon circumscribing immersed body. y = C. V representing volume of displacement in cube feet, L length at im- L B D mersed water-line, B extreme breadth, and D draught in depth of immersion, both in feet. Coefficient of Area of A midship Section in Plane of a Water-line is ratio which their areas bear to that of circumscribing rectangle. L representing length of water-line, and D distance between water-lines, both in feet. Coefficients. (By S. M. Pook, Constructor U. S. Navy. ) RULE. Multiply length of vessel at load-line by breadth, and product by depth (from load-line to under side of garboard-strake) in feet, and this product by coefficient for vessel as follows : divide by 35 for salt water, 36 for fresh water, and quotient will give displacement in tons. Amidship sections range from .7 to .9 of their circumscribing square, and mean of horizontal lines from . 55 to . 75 of their respective parallelograms. Hence, ranges for vessels of least capacity to greatest are -7 X .55 = .385, and .9 X .75 = .675. Merchant ship, very full 6 to . 7 " " medium 5810.62 River steamer, stern- wheel. . . ,6 to .65 Ship of the line 5- to 6 Naval steamer, first class 5 to .6 " 52 to. 58 Merchant steamer, sharp 54 to .58 Half clipper 52 to .56 Brigs, barks, etc 52 to .56 River steamer, tug-boat, med'm . 52 to . 56 Merchant steamer, medium. . . . 52 to .54 Clipper 5 10.54 Schooner, medium 48 to .52 River steamer, tug-boat, sharp .45 to .5 " medium 45 to. 5 " " sharp 42 to .45 Schooner, sharp 46 to .5 Yachts, sharp 4 to .45 " very sharp 3 to .4 River steamers, very sharp. . . .36 to .42 In steam launch Miranda, when making 16.2 knots per hour, with a displace- ment of 58 tons, her coefficient was 3. To Compute Change of* Trim. W (I T -=r- X = d'. D representing displacement at line of draught in tons, L length at same line in feet, and m longitudinal meta-centre. ILLUSTRATION. " Warrior," at draught of 25. 5 feet, has L = 380 feet, m = 475 feet, and D = 8625 tons. If, then, a weight of 20 tons was shifted fore and aft 100 feet, ^?X * = . , 8625 475 NAVAL ARCHITECTURE. To Compnte Common Centre of Oravity of* Hull, Ar- mament, Engine, Boiler, etc., of a Vessel. RULE. Compute moments of the several weights, relatively to assigned horizontal and vertical planes, by multiplying weight of each part by its horizontal and vertical distance from these planes. Add together these moments, according to their position forward or aft, or above or below these planes, and difference between these sums will give po- sition forward or aft, above or below, according to which are greatest. Divide results thus ascertained by total weight of vessel, and product will give horizontal and vertical distances of centre of gravity from these planes. NOTE. To simplify computation in table, common centre of gravity of hull, ma- chinery, etc., is taken, instead of centres of individual parts, as engine, boiler, pro- peller, etc. Illustration Vertical Plane at & and Horizontal at Load-line. ELEMENTS OF A STKAM FKIOATK. Weight. HORIJ Distances. Forward. Abaft. ONTAL. Mom< Forward. nts. Abaft. Dista Above VEK nces. Below FICAL. Morr Above ents. Below Hull, bunkers, and ce- ment in bottom Tons. 1075 470 252 I3I-5 24 25 3-25 22 30 3 7-25 Feet. 1.6 16 62 27 40 40 1.2 17 Feet. 2 9 48 40 1720 4032 8l53 648 I OOO 880 360 510 13630 156 290 Feet. 2 31 16 5 7 Feet. I 6-3 4 6 3 8 263 744 52 15 210 1075 3011 1008 ISO 66 "58 Engines, boilers, water, and stores Coal Battery and ammuni tion Masts, spars, sails, and rigging . ... Anchors and cables Boats Water and ship's stores Provisions and galley. . Crew and effects Officers' and mess stores Total..., 2070 1730-3 14076 1410 1 "5^68 Moments forward g$, 17303 moments abaft, 14076 3227-7-2070 tons (weight) = 1.56 feet = distance of centre forward o/jgj. Moments above load-line, 5368 moments below, 1419 3949 -4- 2070 tons (weight) = i. 91 feet = distance of centre below load-line. NOTE. Rule, in Strength of Materials, to compute common centre of gravity, page 819, would apply in this case. To Compute Depth of Centre of Grravity or Buoy- ancy Belo-vv ^teta-Centre. g - = d. S representing statical stability, D displacement in tons, and sin. M sine of angle of heel. ILLUSTRATION. Elements of Fig. 2, page 654, are, statical stability at angle of 5.44, 90 tons, and displacement 204.33 tons. Sin. 5. 44 = .0999. Then 9 =4.41 feet. 204. 33 X. 0999 NAVAL ARCHITECTURE. 65; To Compute Centre of Q-ravity or Buoyancy Approxi- mately. - to i- of mean draught of hull, using larger coefficient for full bodied vessels. 5 20 To Delineate Curve of Displacement. This curve is for purpose of ascertaining volume of water or tons weight, displaced by immersed hull of a vessel at any given or required draught ; or weight required to depress a hull to any given or required draught. From results of computation for displacement of vessel, proceed as follows, Fig. 4 : Fig. 4. On a vertical scale of feet and ins., " B as A B, set off depths of keel and water- lines, draw ordinates thereto represent- ing displacement of keel, and at each water-line, in tons. Through points i, 2, 3, 4, and 5 df lineate curve A 5, which will represent displacement at any given or required draught. Draw a horizontal scale correspond- 1 3*5 m g ^ w e ight d ue to displacement at load-line, as A C. and subdivide it into tons and decimals thereof, and a ver- tical line let fall from any point, as r, at a given draught, will indicate weight of displacement at depth, on scale A C, and, contrariwise, a line raised from any point, as z, on A C will give draught at that weight. ILLUSTRATION. Displacement of hull (page 654) at load 1106 = 7151.8 cube feet, which -r- 35 for salt water = 204. 3 tons, hence A C represents tons, and is to be sub- divided accordingly. Assume launching draught to have been 4 feet, then a vertical let fall from 4 will indicate weight of hull in tons on A C. Coefficients. (By C. Mackrow, M. I. N. A. ) DISCBIPTION or VESSEL. Length. Breadth. Mean Draught. Displace- Coefficient. Amidahip Section. Water- lines. 225 45 15 .715 .932 755 Mail Steamers. < 325 350 38< 59 35 42 24-75 21 22 .64 .687 6^0 .81 -85 .88 $ .8 368.27 220 42.5 27 18.71 8 .516 .702 .812 .912 635 742 Gunboats | 90 125 15 23 8 .637 536 .914 .87 .704 .616 Troop Ships < 160 350 3i-3 49.12 12 23-5 .466 47 -745 -674 .603 7 Q Swift Naval Steamers. . . . j Fast Steamers. R. N.... 340-5 337-3 270 300 46.13 50.28 42 40.27 15-75 22-75 '9 14 :; 83 497 .414 .68 .787 .792 .711 .582 .6 I4 .628 711 Cnrve of "Weight. To Compute Number of* Tons required to Depress a Vessel One Inch, at any Draught of Water Parallel to a "Water-line. RULE. Divide area of plane by 12, and again by 35 or 36, as may be required for salt or fresh water. EXAMPLE. Area of load water-line of a vessel is 1422 sq. feet; what is its ca pacity per inch in salt water? 1422-7-12 = 118.5, which -=-35 = 3.38 tom. 658 NAVAL ARCHITECTURE. To Compute Centre of Gravity of Bottom Plating of a Vessel. Longitudinal. RULE. Measure half girths of plating at equidistant sections, as at two or more frames. Multiply these in accordance with Simpson's rule for areas and add products together. Multiply each of these products in their order, by number representing number of intervals of section forward and abaft of &. Divide difference of these moments by sum of products of half girths, previously obtained. Multiply product by common distance between sections, and result will give distance of centre of gravity from JS in a horizontal plane. ILLUSTRATION. Assume half-girths as in following table, and distance between sections 10 feet. Sec- tion. No. f. B. C. D. E. Half- Girths. FOR 1 ? Multi- pliers. fARD. Prod- uct. Multi- pliers. Mo- ments. Sec- tion. Half- Girths. ABJ Multi- pliers. LFT. Prod- uct. Multi- pliers. Mo- ments. 92 80 216 128 70 Feet. 25 23 21 19 17 15 A 4 2 4 2 I 25 9 2 42 76 34 15 I 2 3 4 5 i* 84 228 75 No I . 2 . 3 4 5 ~ Feet. 23 20 18 16 14 4 2 4 2 X 92 40 72 32 14 S 2 3 4 5 534 586 615 Moments forward, 615 moments abaft, 586 = 29 -4- sum of product 534 = .054, which x 10 feet = .54 feet forward of gj. Centre of Lateral Resistance. Centre of Lateral Resistance is centre of resistance of water, and as its po- sition is changed with velocity of vessel, it is variable. It is generally taken at centre of immersed vertical and longitudinal plane of vessel when upon an even keel. If vessel is constructed with a drag to her keel, the centre will be moved proportionately abaft of longitudinal centre. Yacht A merica had a drag to her keel of 2 feet, and centre of lateral re- sistance of her hull was 8.08 feet abaft of centre of her length on load-line. Centre of Effort. Centre of Effort is centre of pressure of wind upon sails of a vessel in a vertical and longitudinal plane. Its position varies with area and location of sails that may be spread, and it is usually taken and determined by the ordinary standing sails, such as can be carried with propriety iii a moderately fresh breeze. In computing this position, the yards are assumed to be braced directly fore and aft and the sails flat. NOTE. Centre of effort of sails, to produce greatest propelling effect, must accord with capacity of vessel at her load line, compared with fullness of her immersed body at its extremities. Thus, a vessel with a full load-line and sharp extremities below, will sustain a higher centre of effort than one of dissimilar capacity and con- struction. NAVAL ARCHITECTURE. 659 To, Compute Location of* Centre of Effort. RULE. Multiply area of each sail in square feet by height of its centre of gravity above centre of lateral resistance in feet, divide sum of these prod- ucts (moments) by total area of sails in square feet, and quotient will give height of centre in feet. 2. Multiply area of each sail in square feet, centre of which is forward of a vertical plane passing through centre of lateral resistance, by direct dis- tance of its centre from that plane in feet, and add products together. 3. Proceed in like manner for sails that are abaft of this plane, add their products together, and centre of effort will be on that side which has greatest moment of sail. EXAMPLE. Assume elements of yacht America as rigged when in U. S. Service. SAIL. Area. Height of Cent, of Grav- ity of Sails. Vertical Moments. Distance o of Gravity Foreward. f Centre of Sails. Abaft. Mome Foreward. nta. Abaft. Flying Jib Sq. Feet. 656 1087 1455 2185 Feet. 28 26 34 35 18368 28262 49470 76475 52 32 3 40 34112 34784 4365 87400 Jib Foresail Mainsail 5383 172575 68896 9i7 6 5 Vertical moments 17 5 I72 575 == 32.06 = height of centre above centre of lateral re- Area of sails 5 383 sistanee, Moments {^^ sistanee. 533 - ^- = 4.25 = distance of centre abaft centre of lateral re- Relative Positions of Centre of* Effort and. of Lateral Resistance. Square Riff. 4A. Fare and Aft Rig. and E'. L representing length of load-line, d distance of centre of buoyancy of vessel below it, d' distance of centre of lateral resistance abaft centre of it, d" dis- tance of centre of buoyancy before centre of it, E distance of centre of effort before centre of lateral resistance, and E' distance of centre of effort above centre of lateral resistance. Meta-Centre. Meta-centre of a vessel's hull is determined by location of centre of grav- ity or buoyancy of immersed bottom of hull, for it is that point in transverse section of hull, where a vertical line raised from its centre of gravity or buoyancy intersects a line passing through centre of gravity of hull, as Fig. i, page 650. To Compute Height of Meta-Centre. By Moment of Inertia. =r = M. I representing moment of inertia of area of water-line or plane of flotation, and D volume of displacement in cube feet. NOTE. Moment of Inertia of an area is sum of products of each element of that area, by square of its distance from axis, about which moment of area is to be computed, To Ascertain Moment of Inertia approximately. Rectangle = CLB3; C = when L = 4B; C = when L = sB; and C 12 50 when L = 6 B. With very fine lines and great proportionate length C = . 200 25 L and B measured at load-line. 66o NAVAL ARCHITECTURE. ILLUSTRATION. Assume length of vessel 233 feet, breadth 43, draught 16, and displacement 2700 tons. Length = 5.65 beams; hence C is taken at . Volume of displacement = 2700 X 35 = 92 500 cube feet. Then 21 X 233 X 43 3 400 X 92 500 = 10.51. Exact height of moment was 10.44 feet. By Ordinates. RULE. Divide a half longitudinal section of load water- line by ordinates perpendicular to its length, of such a number that area between any two may be taken as a parallelogram. Multiply sum of cubes of ordinates by respective distances between them, and divide two thirds of product by volume of immersion, in cube feet. ILLUSTRATION. Take dimensions from Figs. 2 and 3, page 654. Cube. 51460 2 3)102920 7I5I.8) 34306.6 = 4.77 ft. If there are more ordinates, their coefficients must be taken in like manner, as i 4 2 4 2 4 i. For operation of this method, see Simpson's rule for areas, page 342. Or, i y X = M. y representing ordinates of half -breadth sections at load- line, d x increment of length of load-line section or differential ofx, and D displace- ment of immersed section in cube feet. I Length. Cube. A ... Length. Cube. . 885 3 2 ... 7.7... .... 9.5-. 456 857 B ... C ... D... ...7-8 ...6.8 475 3 ;t ffi... .... 9.9.. ....10 .. 97 5146 X 10 By Areas. - = M. a, 6, c, d, and e representing ordinates of ist or load water-line, F area of irregular section between ist frame and stem, and A area of like section between last frame and stern-post, both in sq.feet, D displacement, in cube feet, and I distance between frames or sections of water-line, as may be taken, in feet. To Ascertain Areas of F and A. abXbc3--A=:F, and -de 3 3 Elements of Capacity and Speed of Several Types of Steamers of R. N. (W. H. White.) CLASSES. Length. Length to Breadth. Displacement. Speed. HP Displace- ment. to Displace- ment . IRON-CLADS. Recent types, do. twin sc. UNARMORED. Swift cruisers Corvettes Ships Feet. 300 to 330 280 to 320 270 to 340 2OO tO 22O 1 60 I25tOl70 80 to 90 400 to 500 300 to 400 250 to 350 200 tO 300 5.25105.75 4-5 tO 5 6. 5 to 6. 75 6 5.5106.25 3 to 3. 25 9 to ii 8 to 10 7. 5 to 10 7 to 9 Tons. 7500 to 9000 6000 to 9000 3000 to 5 500 1800 to 2 ooo 850 to 950 420 to 800 2OO tO 250 7OOO tO IOOOO 5000 to 7 ooo 3000 to 6000 1500 to 4000 Knots. 14 to 15 14 to 15 15 toi6 12.751013.25 ii 9. 5 to ii 8 to 9 14 to 15 13 to 14 II tO 15 9 to ii .9 to i .7 to .9 1.3*01.5 I tO I. 2 I tO I. 2 .8toi-4 .8 to 1. 1 .5 to .6 .4 to .5 3 to .5 ,3 tQ .4 1 6 to 20 15 tO 19 20 tO 24 13 to 14 10 tO 1 1 7 to ii 5 to 7 10 to ii 7 to 10 5 to o 3to 6 Gun-vessels.. Gun-boats . . . MERCHANT. Mail, Inrge... 44 smaller. Cargo, large.. " smaller. NAVAL ARCHITECTURE. 66 1 To Compute 3?o\ver Required, in a Steam Vessel, capac- ity of anotlier "Vessel "being given. , > . >, j , v A __ S 3 V rV . C In vessels of similar models. = V; = V ; = C; and > = R; u and V representing product of volumes of given and required cylinders and revo- lutions in cube feet, a and A areas of immersed section of given and required vessel in sq. feet at like revolutions and speed of given vessel, s and S speeds of given and required vessel at revolutions of given vessel, both in feet per minute, r and r' revolutions of given and required vessel per minute, and C product of volume of com,' bined cylinder and revolutions for required vessel. ILLUSTRATION. A steam vessel having an area of amidshlp section of 675 sq. feet, has two cylinders of a combined capacity of 533.33 cube feet, and a speed of 10.5 knots per hour, with 15 revolutions of her engines. Required volume of steam cylinders, with a stroke of 10 feet, for a section of 700 feet and a speed of 13 knots with 14.5 revolutions. v = 533. 33 X 15 = 8000 cube feet, 15 X 15 745.2 _ 8000X700 15745.2 cube feet, 675 = 16 288. i cube feet, = 8296. 3 cube feet, and 16288.; 133 x 8296. 3 _ 10.53 = 561.66 cube J4-5 2X14-5 feet, which -r- 10 stroke of piston, 12 for ins., and X 1728 ins. in a cube foot=. ^-1 - ^Zl_ = 8087.9 5 2- tm - area of each cylinder diameter 0/101.5 ins. Approximate Rules to Compute Speed and IBP of Steam C representing coefficient of vessel, A area of immersed amidship section in sq.feet, V velocity of vessel in knots per hour, and D displacement of vessel in tons. NOTE. When there exists rig, an unusual surface in free board,deck-houses, etc., or any element that effects coefficient for class of vessel given, a corresponding ad- dition to, or decrease of, following units is to be made: Range of Coefficients as deduced from observation is as follows : SIDE-WHEEL. PPOPELLER. C C VESSEL. A D V V s A V'D! VESSEL. A D V V 3 A V 3 D IH IIP "IH?" "Tip" Steamboat. Sq.F. T's. K'ts. Steamboat. Medium lines .... 43 73 10 470 212 Medium lines.. 45 12 500 Fine lines 150 136 465 300 13 19 570 540 219 200 T^ine lines 150 15 53 Steamer. Steamer. Medium full lines* 675 3600 IO 650 214 Medium full... 55 2532 9 194 570 390 1475 10 180 470 Fine linesf 880 5233 15 650 211 Torpedo boat.. 3600 13 20 210 170 * Full rigged. t Bark rigged. Coefficients as Determined by Several Steamers of H. B. M. (C. Mackrow, M. I. N. A.) Service. Length. Length Beam. Area of Section at gj. Displace- ment. IIP Speed. T15 = c - Feet Sq. Feet. Tons. Knots. 185 6-53 775 782 10.34 333 212 589 377 1554 1070 10.89 456 270 7-33 6.43 814 632 5898 3057 2084 2046 "5 12.3 598 574 380 6. 52 1308 9487 3205 12.05 7M 400 6-73 1198 9152 5971 13.88 536 ^62 400 7-33 6-73 778 1185 5600 9071 3945 6867 14.06 15-43 548 634 3K 662 NAVAL ARCHITECTUKE. Approximate R-vile for Speed, of Screw Propellers. (Molesworth.) IO ' V _ N . PN ioiV 88 PN and 88 "-P ~P~- N > To7- V ' ~N~-- P> -p-- N ' "88"- Wj l ~N~- V and v representing velocities in knots and miles per hour, P pitch of propeller in feet, and N number of revolutions per minute. This does not include slip, which ranges from 10 to 30 per cent. Pitch of Screw Propeller. Pitch ranges with area of circle described by diameter of screw to that of amidship section. Area of screw circle to amidship H * \ I section = i to 1,1 5 4<5 3-53 2-5 Two Blades. Pitch to diameter of screw = i to | ,8 | 1.02 | i.n | 1.2 | 1.27 1 1.31 | 1.4 | 1.47 Four Blades. | 1.08 | 1.38 | 1.5 | 1.62 | 1.71 | 1.77 1 1.89 | 1.98 Length = . 166 diameter. Slip of Side-wlieels. Radial Blades. 2 ( A ^- C )- S> Feathering. *' 5 ( ^~ c) = S. A representing length of arc of immersed circumference of blades, c length of chord of immersed arc. and S slip, all in feet. Area of Blades. JTT> TTP River Service, '-r = A. Sea Service, ~y = A, D representing diameter of wheel in feet, and A area of each blade in square feet. Length of Blades. .7 in River service and 6 in Sea service. Distances between Radial Blades. 2.25 in River service and 3 feet in Sea service; between Feathering blades, 4 to 6 feet. Proportion of Power Utilized in a Steam Vessel. p g Side Wheel. ^r - = C. P representing gross HP, z loss of .00000259

J Oblique action of wheels 201.6 18 Slip of wheels 172.14 15.37 Absorbed by propulsion of vessel 591. 36 52. 8 II2O IOO NAVAL ARCHITECTURE. 66 3 rp Per cent " of Power Screw Propeller. Friction of engines 06.06) QQ Friction of load 81.48 } " of screw surface and resistance of edges of blades 53.44 6.83 Slip of propeller. 205.55 26.27 Absorbed by propulsion of vessel 375-92 48.04 782.45 100 NOTE. From experiments of Mr Froude, he deduced that, as a rule, only 37 to 40 per cent, of whole power exerted was usefully employed. With an auxiliary propeller, essential differences are in friction of surfaces and edges of blades of propeller and slip of propeller, being as 12 to 6-83 in excess in first case, and as 13.7 to 26.27 m second case, or 50 per cent less. Resistance of Bottoms of Hulls at a Speed of one Knot per Hour. Smooth wood or painted. 01 Ib. Smooth plank 016 " Iron bottom, painted 014 " Copper. 007 Ib. Moderately foul 019 " Grass and small barnacles 06 " Sailing. Ratio of Effective Area of Sails and of Vessel's Speed under Sail to Velocity of Wind. Ratio of Ratio of Ratio of Ratio of Effective Speed of Effective Speed of COURSE. . Area Vessel COURSE. Area Vessel of Sails. to Wind. of Sails. to Wind. 5 points of wind 2 " abaft beam 59 -33 Wind abeam .82 .6 e ' ' astern ... . 6 " of wind,,. :5 5 .; ** on quarter... X .o6 :& Propulsion and Area of Sails. Plain sails of a vessel are standing sails, excluding royals and gaff topsails. Resistance of vessels of similar models but of different dimensions for equal speeds D Hence -7 = (=y) a and a' representing areas of sails of known and given ves- tels, and D and D' their displacements in tons. ILLUSTRATION. Assume D and D' = 24oo and 1600. Then i6oo/ ^/ I5 2_ I<3Ij hence area of sails a' = = .763 per cent I - I Tn Vessels of Dissimilar Models. Plain sail area should be a multiple of Df. Multiples for Different Classes of Vessels, R. N. Sailing. Ships of Line 100 to 120 Frigates ) Sloops > 120 to 160 Brigs ) Steamers. Ships, iron-clad 60 to 80 Frigates , ) Sloops J 80 to 120 Brigs ) English Yachts, designed for high speed, have multiples from 180 to 200, and when designed for ordinary speed from 130 to 180. When Area of Sail to Wet Surface of Hull is taken. American yacht Sappho had a ratio of 2.7 to i, and several English yachts nearly the same, while in some others it was but 2 to i. 664 NAVAL AKCHITECTURE. Location of IVlusts, etc. Load-line = 100. VESSEL, D Fore. stance from Ste Main. n. Miuen. Foot of Sail.* Height of Centre of Effect above Water-line = Breadth.* Ship 10 to 20 12 tO 20 17 to 20 l6 tO 22 53 to 58 54 to 60 64 to 65 55 to 61 ^6 tO A.2 80 to 90 81 to 91 125 to 160 130 to 160 160 to 165 160 to 170 170 tO IQO .5 102 5 to 1.95 5 to 1.75 5 to i. 75 .2q to i. 75 Bark Brig Schooner .... Sloop... * Measured from Tack of Jib to Clew of Spanker or Mainsail. Rake of Masts. Ships. Foremast o to .28 of length from heel, Main and Mizzen o to .25. Schooners. Foremast .1 to .25, Mainmast .63 to .77. Sloops. .08 to .u. SAILS. 3 Yards upon each Mast. Area, c 4 Yards upon each Mast. >f Sails. SAILS. 3 Yards upon ' each Mast. 4 Yards upon each Mast. jib .08 .08 Mizzenmast. . . . . 127 Foremast Mainmast 295 .417 295 .417 Spanker or } Driver. . . ) " .081 ,068 Proportional A SAIL. rea of Sails u^ Fore. oon each Mast Main. under above D Mizzen. ivisions. Proportion to x. Course "5 .105 075 .08 .097 .00 .063 045 .08 .162 ;$ .138 .127 .089 .063 075 .052 .081 .063 045 .032 .068 .389 .358 253 33 303 .215 .152 Topsail Topgallant sail . . Roval Spanker or Driver Jib 375 375 .417 .417 .208 .208 I i Balance of Sails. Effect of jib is equal to that of all sails upon main- mast, and sails upon mizzenmast balance those of foremast. Areas of sails upon masts of a ship should be in following proportion : Fore 1.414 | Main 2 | Mizzen i When, therefore, main yard has a breadth of sail of 100 feet, fore yard should have 70.71 feet, and mizzen 50 feet j topgallant and royal yards and sails being in same proportion. Angles of Heel for Different Vessels. Approximately. = S. D representing displacement of vessel in Ibs., M height of meta- centre above centre of gravity in feet, a angle of heel of vessel in cir- cular measure,* and H height of centre of effect above centre of lateral resistance, infeet. Moment of sail should be equal to moment of stability at a defined angle of heel. Angle, Frigates, etc 4 Corvettes 5 ILLUSTRATION. Assume displacement 170 tons, height of meta-centre 6.75 feet> H = 36 feet, and angle of heel 9 ; what should be area of sails ? 170 X 2240 380 800 Ibs. 9 = . 107. ._. Circular An & le ' Measure. .07 .087 Circular Measur*. Schooners, etc 6 . 105 Yachts 6 to 9 . 105 to . 107 * Se rule, page 113. NAVAL AKCHITECTURE. 665 Trimming of Sails. That a vessel's sail may have greatest effect to propel her forward, it should be so set between plane of wind and that of her course, that tangent of angle it makes with wind may be twice tangent of angle it makes with her course. Or, tan. a = 2 tan. b. a representing angle of sail with wind, and b angle of sail and course of vessel. Angles of Course and Sails \vith. Wind. Wind Ahead. Angle Course. Tan- gent. Half gei". .281 .365 .461 .707 Angles with Wind. of Sail with Course. Wind Abaft. Angle of Course. Tan- gent. Half Tan- gent. 1.082 1.368 1.781 3-754 Angle, with Wind. of Sail with Course. Points. 4 5 6 Abeam 45 56 15' 67 30' 900 .562 732 9 2 3 I-4I5 29 1 8' 36 12' 42 43' 54 45' 15 42' 20 3' 24 45' 35 16' Points. 2 3 I 112 30' 123 45' 135 157 30' 2.166 2-737 3o62 7-5" 6 5 13' 69056; 74 17 82 25' 47 17' 53 49' 60043' 75 5' Fig. 6. Effective Impulse of "Wind. Let P o, Fig. 6, represent direction by com- pass and force of wind on sail, AB; from P draw P C parallel to A B, from o draw o C per- pendicular to AB; o C is eft'ective pressure of wind on sail A B, and r C, perpendicular to plane of vessel, is component of o C, which pro- duces lateral motion, as heel and leeway, and r o is component of o C, which propels vessel. I sin. a =. P ; P cos. x L ; and P sin. x E. I representing direct impact and P effective pressure of wind on sail, L effective impact producing leeway, and E effective impact which propels vessel. NOTE. The law as usually given is sin. 2 . This is manifestly incorrect, as it gives results less than normal pressure for angles of small incidence. At an angle of in- cidence of wind of 25, the law of sin. is exact. Hence, although it may not be exact at all angles, it is sufficiently so for practical purposes. ILLUSTRATION i. Assume wind 5 points ahead, and I = 100 IDS. By preceding table angle of course with wind 56 15'; hence angle of sail a, with wind 36 12', as tan. 36 12.' = 2 tan. 20 3', and angle x 56 15' 36 12' = 20 3'. Then, 100 X sin. 36 12' = 100 X .5906 = 59.06; 59.06 X cos. 20 3' = 59.06 X .9394 = 55.48, and 59.06 x sin. 20 3' = 59.06 x .3426 = 20.23 Ibs. 2. Assume wind 4 points abaft, and I = 100 Ibs. Then, iooxsin. 2 74 17'= 100 X g626 2 =r 92. 66 ; 92.66 X cos. 180 74 17'+ 45 = 60 43' = 92. 66 X- 49 = 45.41, and 92. 66 X sin. 60 43' = 92 66 x . 8722 = 80. 82 Ibs. To Compute Sailing Power of a Vessel. F/sin. w, sin. s = P. To Compute Careening IPower of a Sailing Vessel. F/sin. w, cos. s = P. F representing area of sails in sq. feet, f force of wind in Ibs. per sq.foot, w angle of wind to sails, and s angle of sails to course of vessel. To Compute Angle of Steady Heel. Within a Range 0/8. a P E ., JJ .M = sin. H. a representing area of plain sail in sq.feet, P pressure of wind in Ibs. per sq.foot, E height of centre of effect above mid-draught, in feet, D displace- ment of hull, in Ibs., and M height of meta-centre in feet. P assumed at i Ib. per sq. foot, or that due to a brisk wind. ILLUSTRATION. Assume a =15 600, draught = 20, and = 62; hence 62-}- = 7*, D = 6 800000, and M = 3. Then 15 6oo X i X 72 1 123 200 6 800 ooo X 3 ~~ 20 400 ooo ~ ; = 3 10'. 3K* 666 NAVAL ARCHITECTURE. Course and Apparent Course of \Vind. Apparent course of a wind against sails of a vessel is resultant of normal course of wind and a course equal and directly opposite to that of vessel. Fig. 7- ILLUSTRATION. If P, Fig. 7, repre- sent direction by compass and force of wind, and a b direction and velocity of vessel, from P draw P c parallel and equal to a 6, join c a and it will repre- sent direction and force of apparent wind. Or, = ratio of velocity of apparent aP wind to that of vessel, = ==. ratio of velocity oj wind to that of vessel. Resistance of Air. (Mr. Froude.) Resistance of wind to a vessel is estimated as equivalent to square of its velocity. In a calm, resistance of air to a steamer = one thirty-fourth part of resist- ance of water, and when a steamer's course is head-to, and combined veloc- ity of vessel and wind= 15 knots, resistance is one ninth of that of the water. Resistance of air to a sq foot of surface at right angles to course of a ves- sel is about .33 lb., and when surface is inclined to direction of wind, press- ure varies as sine of angle of incidence, Mean of angles of surface of a steamer exposed to wind may be taken at 45 ; hence their resistance is about .25 lb. per sq. foot when wind has a ve- locity of 10 knots per hour. If sectional area of a steamer's hull above water is 750 sq. feet, resistance to air at a speed of 10 knots in a calm would be 750 X .25 = 187.5 Iks., and resistance to smoke-pipe, spars, and rigging (brig rigged) would be 201 Ibs. Leeway. Angle of Leeway in good sailing vessels, close hauled, varies from 8 to 12, and in inferior vessels it is much greater. Ardency is tendency of vessel to fly to the wind, a consequence of the centre of effort being abaft centre of lateral resistance. Slackness is tendency of vessel to fall off from the wind, a consequence of the centre of effort being forward centre of lateral resistance. Results of Experiments upon Resistance of Screw-propellers, at High Velocities and Immersed at Varying Depths of Water. Immersion of Screw. Resistance. Immersion of Screw. Resistance. Immersion of Screw. Resistance. Surface, i foot. I 5 2 feet. 3 u 7 7-5 4 feet, 5 " I s Slip of Propeller, 15 per cent. ; of Side-wheel (feathering blades), and tak- ing axes of blades as the centre of pressure, 23 per cent. ITree"board. Measured from Spar-deck stringer to surface of water. Depth of Hold from under- side of spar deck to top of ceiling. Hold. Per Ft. Hold. Feet. 12 M Per Ft. Hold. Per Ft. Hold. Per Ft. Hold. Per Ft. Hold. Per Ft. Feet. 8 10 Ins. * 5 2 Ins. 2.25 2-5 Feet. 16 18 Ins. 2-75 3 Feet. 20 22 Ins. 3-125 3-25 Feet. 24 26 Ins. 3-375 3-5 Feet. 28 30 Ins. 3-625 3-75 NAVAL ARCHITECTURE. 667 PL Soobd Plating Iron Hulls. = T. D representing displacement in torn, L length of hull, b breadth, and d d depth. Or, .o$f^d = T. /representing distance between centres of frames, and d depth of plate below load-line, all in feet, and T thickness of plate in ins. Masts and Spars Lower masts at spar deck. Bowsprit " stem. Topmasts " lower cap. Topgallant masts " topmast cap. Diameter for Dimensions. Jib-boom at bowi Yards in middle. Gaffs at inner end. Main and Spanker booms at taffrail. Fore and main masts, when of pieces, i inch for each 3 to 3.25 feet of whole length. Mizzenmast .66 diameter of mainmast. Masts of one piece i inch for each 3-5 to 3-75 f eet of whole length. Bowsprit, depth, equal diameter of mainmast; width, diameter equal to foremast. Main and fore topmasts Mizzen topmast Topgallant masts Royal masts Topgallant poles Jib boom Fore and main yards Topsail yards Cross -jack, Topgallant, and) Royal yards ) Main and Spanker booms Gaffs Studding-sail yards and booms. inch for each 3 to 3.25 ) " " " 3-25 " 875 feet of whole length. 2 ft. of length beyond bowsprit cap. 4 4 5 3-5 3-5 to 4 4-5 to 4.75 feet of whole length. Pd_ 2400 Rudder Head. (Mackrow.) l = T: . 196 C D3 M ; 3 / - D ; and = P. P representing press- V 196 c 2400 ure on rudder when hard over, in tons, d distance of geometrical centre of rudder from axis of motion, in ins., T stress on head, and M moment of resistance of head, both in inch-ions, A immersed area of rudder in sq. feet, v velocity of water passing rudder in knots per hour, and G coefficient 3. 5 per sq. inch for Iron, and . 125 for Oak. ILLUSTRATION. Assume area of wooden rudder 24 sq. feet, distance of its geomet- rical centre from centre of pintles 2 feet, and velocity of water 10 knots. = i ton. i X 2 X 12 = 24 inch-tons. 3 I 1 = 9-93 tw *- Memoranda. Weights. A man requires in a vessel a displacement or 488 Ibs. per month, for baggage, stores, water, fuel, etc., in addition to his own weight, which is estimated at 175 Ibs. A man and his baggage alone averages 225 Ibs. A ship, 150 feet in length, 32 beam, and 22.83 in depth, or 664 tons, C. H. (O. M.), has stowed 2540 square and 484 round bales of cotton. Total weight of cargo i 254448 Ibs., equal to 4.57 bales, weighing 1889 Ibs., per ton of vessel. A full built ship of 1625 tons, N. M., can carry 1800 tons' weight of cargo, or stow 4500 bales of pressed cotton. Hull of iron steamboat John Stevens length 245 feet, beam 31 feet, and hold ii feet; weight of iron 239440 Ibs. And of one other length 175 feet, beam 24 feet, and 8 feet deep; weight of iron 159 190 Ibs. Weight of hull of a vessel with an iron frame and oak planking (composite), com- pared with a hull entirely of wood, is as 8 to 15. An iron hull weighs about 45 per cent, less than a wooden hull. Iron ship, 254 feet in length, 42 beam, and 23.5 hold, 1800 tons register, has a stow- age of 3200 tons cargo at a draught of 22 feet. Weight of hull in service 1450 tons. Loss by Weight per Sq. Foot per Month of Metalling of a VesseVs Bottom in Service. Copper .0061 Ib. ; Muntz metal .0045 Ib. ; Zinc .007 Ib. ; and Iron .0204 Ib. Comparison between Iron and Steel plated Steamers. In a vessel of 5000 tons displacement, hull of steel-plated will weigh 320 tons less = 6.66 per centum less. 668 OPTICS. Fig. 2. OPTICS. IVTirrors, in Optics, are either Plane or Spherical. A plane mirror is a plane reflecting surface, and a spherical mirror is one the reflecting surface of which is a portion of surface of a sphere. It is concave or convex, ac- cording as inside or outside of surface is reflected from. Centre of the sphere is termed Centre of curvature. Focus Point in which % number of rays meet, or would meet if produced. Fig. i. Principal Focal Distance is half radius of curvature, and is generally termed the focal distance. Line a c is termed the principal axis, and any other right line through c which meets the mirror is termed a Secondary axis. When the incident rays are parallel to the principal axis, the reflected rays converge to a point, F. Conjugate Foci are the foci of the rays proceeding from any given point in a spherical concave mirror, and which are reflected so as to meet in an- other point, on a line passing through centre of sphere. Hence, their relation being mu- tual, they are termed conjugate. Let P be a luminous point on principal axis, Fig. 2, and P i a ray ; draw the normal line c i, whichfis a radius of the sphere; then c i P is an gle of incidence, and c i O the angle of reflection, equal to it ; hence c i bisects an angle of triangle t P c P P i 0, and therefore, = ' i cO When conjugate focus is behind a mirror, and reflected rays diverge, as if emanating from that point, such focus is termed Virtual, and a focus in which they actually meet is termed Real. Fig. 3. ^ As a luminous point, as P, Fig. 3, is ^"^ moved to the mirror, the conjugate focus moves up from an indefinite distance at back, and meets it at surface of mirror. If an incident ray converges to a point s, at back of mirror, it will be reflected to a point P in front. The conjugate foci P s having changed places. Pencil Kays which meet in a focus and are taken collectively. Objects. As regards comparative dimensions or volumes, it follows, from similar triangles, that their linear dimensions are directly as their distances from centre of curvature. To Compute Dimension, or "Vol-urne of an Image. When Dimensions and Position of Object are Given, and for either Convex or Concave Mirrors. =z - , or = ^r . L and I representing lengths of image and object, F fooal IV Li r length, and D and d respectively, distances of image and object from principal focux. Refraction . Deviation. Angle at which a ray is diverted from its original or normaJ course when subjected to refraction is thus termed. Indices^of Refraction. Katio of sine of angle of incidence to sine of angle of refraction, when a ray is diverted from one medium into another, is termed relative index of refraction from former to latter. OPTICS. 669 When a ray is diverted from vacuum into any medium, the ratio is greater than unity, and is termed absolute index or index of refraction. Mean Indices of Refraction. Eye, vitreous humor i. 339 u crystalline lens, under 1.379 " " " central...... 1.4 Diamond 2.6 Glass, flint. . '. i-57 Glass, lead, 3 flint 2.03 " lead 2, sand i 1.99 " " i, flint i 1.78 Ice 1.31 Quartz 1.54 For indices of other substances, see page 584. Heat increases refractive power of fluids and glass. Critical A ngle. Its sine is reciprocal of index of refraction, the incident ray being in the less refractive medium. Visual Angle is measure of length of image of a straight line on the retina. Total Reflection is when rays are incident in the more refractive medium, at an angle greater than the critical angle. Mirage. An appearance as of water, over a sandy soil when highly heated by the sun. Caustic Curves or Lines are the luminous intersections from curve lines, as shown on any reflective surface in a circular vessel. To Compute Index of Refraction. -|5: Index. I representing angle of incidence, and R that of refraction. Sin. R Xo Compute Refraction. Concave-Convex and Meniscus. Effect of a concave-convex in refracting light is same as that of a convex lens of same focal distance, and that of a meniscus is same as a concave lens of same focal distance. Meniscus, with parallel rays ^ - = P. Magnifying Power. In Telescopes the comparison is the ratio in which it apparently increases length. In Microscopes the comparison is between the object as seen in the instrument and by the eye, at the least distance of vision, which is assumed at 10 ins., and the magnifying power of a micro- scope is equal to the distance at which an object can be most distinctly ex- amined, divided by the focal length of the lens or sphere. Linear power is number of times it is magnified in length, and Super- ficial, number of times it is magnified in surface. Magnifying power of microscopes varies, according to object and eye- glass, from 40 to 350 times the linear dimensions of object, or from 1600 to 122500 times its superficial dimensions. Apparent Area. As areas of like figures are as the squares of their linear dimensions, the apparent area of an object varies as square of visual angle subtended by its diameter. The number expressing Magnification of Apparent Area is therefore square of magnifying power as above described. ILLUSTRATION. If diameter of a sphere subtends i as seen by the eye, and 100 as seen through a telescope, the telescope is said to have a power of 10 diameters. 6/O OPTICS. To Compute Elements of Mirrors and. Lenses. Mirrors. Spherical Concave.* = D; = L. r 2 I r 2 I Or Lr d 2 Spherical Convex.^ = =D; = ; = I. Parabolic Concave. -r F 2L-J-r 2L-j-r i6h Unequally Convex, $ /* . F. Piano- Convex. 2 B . 66 t = F. Hyperbolic Concave.]] Elliptic Concave. 1[ Sphere. = = F. O representing object i, r radius of convexity, I and L length or distance of object from vertex of curve, and from external vertex, D dimension of object, d diameter of base, F focal distance, and h depth of mirror in like dimensions, 1 index of refraction, and t thickness of lens. ILLUSTRATION i. Before a concave mirror of 5 feet radius is set an object at 1.5 feet from vertex of curve ; what is ratio of apparent dimension of image, and what is length of and distance of object from external vertex ? Object = i. 1X5 = 2. 5 feet, and -ii- 5 -*- 5 - = 3. 75 feet. 52X15 52X15 a. If object is set at 4.5 feet from vertex of a like mirror, what is length of and distance of inverted object from internal vertex ? = 1.25 feet, and 5.625 feet. 8X4-5 5 2X4-5 5 3. Before a convex mirror of 3.5 feet radius is set an object at 3 feet from ver- tex of curve; what is length of and distance of object from external curve? IX 3' 5 = .368 foot, and 3 X f 5 = 1. 105 feet. 2X3 + 3-5 2X3 + 3-5 4. A parabolic reflector has a depth of 1.25 feet and a diameter of 2 feet; what is its focal distance from vertex of internal curve ? _2 = .2 feet or 2.4 ins. 16X125 TJ y Lenses. Double Convex. F. When R r = F; m-ixR + r 2m i ^_L = D- JJL-L- ?-dbZ_p. OF _ and sj? _ Double Concave. r F ; Optical centres are in centres of lens. Piano - Convex and Piano - Concave. m __ F - Optical centres are respectively centres of convex and concave sur- faces. Convex Concave (Meniscus) and Concavo- Convex. Rr =F. m i XR r Optical Centres. Convex Concave. Delineate lens in half section, draw R from its centre to circumference of lens (intersection of radii), draw r parallel thereto and extending to its circumference, connect R and r at these external points of contact with circumference and external curve, extend line to axis of lens, and point of contact is centre required. Concavo- Convex. Proceed in like manner, but in this case r extends to, or delineates, the inner surface of the lens, and point of con- tact with axis is centre required. * D or image disappears when I .5 r. Z J T * nd 2 p i ~ = I- t When equally convei F = R. When convex side is exposed to parallel rays and when parallel rays fall upon plane side, F = 2 R. || Rays of light, heat, or sound, reflected from focus of a hyperbola, will diverge from its concave surface, f and when from the focus of an ellipw, will be refracted by surface of the other. OPTICS. PILE-DRIVING. 67 1 Wken object ii beyond focal distance (F), its image (D) will be inverted, as F = D, and LF = /. P representing magnifying power of lens , S limit of normal sight, to to 12 ins. for far-sighted eyes and 6 to S for near-sighted, ordinarily 10 ins., V limit of distinct vision, extreme distance of object from optical centre at distinct vision, and m index of refraction. ILLUSTRATION i. If a double convex lens of flint glass has radii of 6 and 6.25 ins. what is its focal distance? Index of refraction = 1.57, see page 584. ' 6x6.25 - =. = 5-37 ins - 1.57 1X6-1-6.25 2. If a double concave lens has a focal distance of 2 ins., and object is 6 ins. from vertex of curve, what is its dimension and what is its distance from vertex of inner curve ? 6X2 . 4X 2 r = 2 ins., and 1_ I>33 i n8 , 2 + 4 4 + 2 3. If focal distance of a single microscope is 4 ins., what is its limit of distinct vision, and what its magnifying power? O = 2.857 tn *- Telescopes, Opera-glasses, etc. D:o = F:/; o/-r-F = D, and ^- = Z; rA-^ = F-f/ f represent- j ij / s-\-x ing length of focal distance from object lens. ILLUSTRATION. Principal focal distance of ocular lens of a telescope is .9 in., of objective lens 90 ins. ; what is its magnifying power? 90 -r- .9 = ioo times the object. PILE-DRIVING. Effect of the impact of the ram of a pile-driver is as the square root of its velocity or height of its fall. Thus the theoretical velocity of fall is aa Vz ff h or 8 Vh. The impact or dynamic effect of the blow of a ram on a pile cannot be determined with exactness, so long as it yields under the blow, as the yield- ing cushions it and reduces its effect. By my experiments in 1852 to determine the dynamic effect of a falling body, I found it to be far greater than that given by the formula ^/ 2 g h, and upon a late repetition of them, under improved conditions of the instrument of registry, I find it to be for one pound falling two feet, 52 pounds. One pound falling 2 feet has a velocity of 11.31 feet per second, but its dynamical effect or vis viva was 52 pounds, or 4.6 times the velocity. Observation and tests of the sustaining power of piles, at different locations and under different conditions, gave it as 2, 3, and 3.7 to i times that deduced by the formula 8 ^/h, which was but the net effect, or capacity, of ram, less the friction of its operation. Wm. J. McAlpine in his operation on the foundations of the dry-dock in the Navy Yard, Brooklyn, estimated the effect of a ram weighing 2240 Ibs., falling 30 feet to a refusal, at 224000 Ibs., or 2.28 times that given by the formula w 8 ^/h. Essayists present a variety of formula, which differ in form. Some are com- paratively simple, while others embrace diameter, length, weight, and sectional area, depth driven by last blow in feet or in inches, and Modulus of Elasticity of the material of the pile, together with various factors for results. When the losses of effect in the operation of a pile-driver are duly considered viz., friction of ram in the guides of the leader, and of the hoisting line of ram in the sheave and over drum {ascertained by experiment with a very heavy ram to * + for telescopes and for opra-glasses, etc. 6 7 2 PILE-DRIVING. be equal to .2 foot of penetration: with a light ram it would be materially more), the cushioning of it on head of a pile, however square it may be dressed off, the want of verticality both of ram in falling and of plane of the pile to the blow and consequent lateral vibration of it, the buckling of it in driving, the frequent split- ting of it on a boulder, and the condition of soil, whether dry, moist, or wet; if it is imbedded or partially exposed to the air, or wholly immersed in wet soil and water, and the integrity of the driving they furnish the elements in determina, tion of a coefficient of safety. Opposed to these effects is that of the subsidence of the soil around a pile thai has been disturbed in driving, the effect of which, under favorable conditions of soil, has approached to that of the resistance of the pile at its final blow. The following formula is constructed on the basis of a pile being driven to a depression of one inch or less, as all estimates based upon a greater de- pression are not only comparatively valueless, in consequence of the cush- ioning of the ram, but if piles are not driven to such depression their utility is decreased, and a greater number are rendered necessary to support the weight to be imposed upon them, and in it I have omitted an element which is universally given in others, that of the last depression of a pile as a divi- sor, as I not only fail to recognize its connection, but hold its introduction erroneous. r J?o Compute Safe Load, of a Pile Driven to a Depres- sion, of 1 Inch, or Less. 4 W 8 v^ = L - W representing weight of ram, and L load, both in Ibs., and h height of fall in feet. From which result is to be deducted a factor of safety representing the friction and losses of effect. Hence, the formula: - == L, or safe load in pounds. \j For C, or coefficient of safety, in consideration of the several losses of effect re- cited, and especially that of brooming of the heads of a pile, it is assumed at from 3 to 6, according to the soil and the integrity of the driving. Eliminating the numerator 4 and correspondingly reducing the 3 and 6, the for- W 8 -Jh mula is, * = L. ' -75 to 1.5 ILLUSTRATION. Assume an ordinary pile driven in firm soil by a ram of 2000 Ibs. weight, falling 25 feet, with a final depression of .5 inch, and coefficient of 1.25; what would be its safe load? 2000 X 8 ^25 2000 X 8X5 , 7 r - = = 64 ooo Ibs. 1.25 1.25 In practice, in the determining the capacity of a range of piles, it is proper to reduce the result obtained by the formula, to meet incidental effects, as negligence in driving, in the superintendence of it, and the frequent and un- observed splitting or crushing of a pile on a stone or boulder. A heavy ram and a low fall is most effective condition of operation of a pile-driver, provided height is such that force of blow will not be expended in merely overcoming friction of leader and inertia of pile, and at same time not from such a height as to generate a velocity which will be essentially expended in crushing fibres of head of pile. When the soil is very soft or wet, concrete should be laid between the heads of the piles to a depth of from 1.5 to 3 feet. When the soil is of fine sand or light gravel, piles may be set two feet from their centres, but if it is saturated with moisture, a greater distance is necessary, otherwise small piles are liable to be disturbed by large, (Continued on page 972.) PILE-DRIVING. PNEUMATICS. AEROMETRT. 6/3 Pile-sinking. MitchelV* Screw Piles are constructed of a wrought-iron shaft of suitable diameter, usually from 3 to 8 ins., with 1.5 turns of a cast-iron thread of from 1.5 to 3 feet diameter. Hydraulic Process is effected by the direction of a stream of water under pressure, within a tube or around the base of a pile, by which the sand or earth is removed. Pneumatic and Plenum Process. For illustration and details, see Traut- wine's Engineer's Pocket-book, 647-8. New Edition. Dr. Whewell deduced the following results : 1. A slight increase in hardness of a pile or in weight of a ram will con- siderably increase distance a pile may be driven. 2. Resistance being great, the lighter a pile the faster it may be driven. 3. Distance driven varies as cube of the weight of ram. Relative Resistance of Formations to Driving a Pile. Coral 100 I Hard clay 60 I Light clay and sand. . . 35 Clay and gravel 83 | Clay and sand. 45 | River silt 25 PNEUMATICS. AEROMETRY. Motion of gases by operation of gravity is same as that for liquids. Force or effect of wind increases as square of its velocity. If a volume of ah* represented by i, and of 32, is heated t degrees without assuming a different tension, the volume becomes (i -f- .002088 t) =V; and if it requires a temperature in excess of t 1 32, it will then assume volume (i + .002088 t' 32). All aeriform fluids follow this law of dilatation as well as that of compression proportional to weight. When air passes into a medium of less density, its velocity is determined by difference of its densities. Under like conditions, a conduit will discharge 30.55 times more air than water. To Compnte the Degree of Rarefaction that may "be ef- fected, in. a "Vessel. Let quantity of air in vessel, tube, and pump be represented by i, and proportion of capacity of pump to vessel and tube by .33 ; consequently, it contains .25 of the air in united apparatus. Upon the first stroke of piston this .25 will be expelled, and .75 of original quantity will remain ; .25 of this will be expelled upon second stroke, which is equal to .1875 of original quantity; and consequently there remains in apparatus .5625 of original quantity. Proceeding in this manner, following Table is deduced : No. of Strokes. Air Expelled at each Stroke. Air Remaining in Vessel. . i 2 3 25 = -25 3 _ 3 16 4X4 9 3X3 75 = -75 9 _3X3 16 4X4 27 3X3X3 64 4X4X4 64 4X4X4 And so on, multiplying air expelled at preceding stroke by 3, and dividing it by 4 ; and air remaining after each stroke is ascertained by multiplying air remaining after preceding stroke by 3, and dividing it by 4. 3 L 6/4 PNEUMATICS. AEROMETEY. Distances at which Different Sounds are Audible. Feet. A full human voice speaking in open air, calm 460 In an observable breeze, a powerful human voice with the) a wind can be heard J I5 * 40 Report of a musket 16 ooo Drum 10 560 Music, strong brass band 15 840 Cannonading, very heavy 575 ooo Miles. .087 3 3.02 2 3 90 In Arctic Ocean, conversation has been maintained over water a distance of 6696 feet. In a conduit in Paris, the human voice has been heard 3300 feet. For an echo to be distinctly produced, there must be a distance of 55 feet. Coefficients of Efflux of Discharge of Air. (D' Aubuisson.) Orifice in a thin plate 65 .751 Cylindrical ajutage 93 .958 Slight conical ajutage 94 1.09 To Compute "Volume of Air Discharged, th.rou.gh an Open- ing into a "Vacuum, per Second. a C -\/2 g h = V in cube feet, a representing area of opening in square feet, C co- efficient of efflux, and Vz g h= 1347.4, as shown at page 428. ILLUSTRATION. Area of opening i foot square, and C = .707. Then i X .707 X 1347.4 = 952.61 cube feet. Inversely, V -f- a =. velocity in feet per second. Velocity and Pressure of \Vind. Pressure varies as square of velocity, or P oc V 2 . V2x.oo5 = P; V2ooP = V; w 2 X-oo23 = P; and .0023 v 2 sin. a P. V representing velocity in miles per hour, v in feet per second, P pressure in Ibs. per sq.foot, and x angle of incidence of wind with plane of surface. Table deduced from a"bove Formulas. Vel Hou'r. >city per Minute. Pressure on a Sq. Foot. Character of the Wind. Vel Hour. ocity Minute. Pressure on a Sq. Foot. Character of th Wind. Miles. Feet. Lbs. Miles. Feet. Lbs. 88 .005 Barely observable. 25 2 2OO 3-125 Very brisk. 2 3 *i 264 .02 } 045) Just perceptible. 30 35 2640 3080 4-5 \ 6.125} High wind. 4 352 .08 Light breeze. 4 3520 8 Very high wind. 5 6 440 528 125) .18 Gentle, pleasant wind 45 5 3960 4400 10.125 12.5 Gale. Storm. 8 704 32 ) 60 5280 18 Great storm. 10 880 5 Fresh breeze. 80 7040 32 Hurricane. 15 20 1320 1760 1.125 2 Brisk blow. Stiff breeze. 9 100 792O 8800 40-5} So J " Tornado. ILLUSTRATION. What is pressure per sq. foot, when wind has a velocity of 18 miles per hour? l8 2 x ^ = 1<6a lbs To Compute Force of "Wind upon a Surface, ( ^ - J = P. v representing velocity of wind in feet per second, a area, oj surface in sq.feet, and x angle of incidence of wind. At Mount Washington wind has been observed to have had a velocity of 150 milei per hour 112.5 lbs. per sq. foot. Extreme pressure of wind at Greenwich Observatory for a period of 20 years was 41 lbs. per eq. foot. PNEUMATICS. AEROMETK Y. 6/ 5 Force of wind upon a surface, perpendicular to its direction, has been ob- served as high as 57.75 Ibs. per sq. foot ; velocity = 159 feet per second. Dr. Hutton deduced that resistance of air varied as square of velocity nearly, and to an inclined surface as 1.84 power of sine x cosine. Figure of a plane makes no appreciable difference in resistance, but con- vex surface of a hemisphere, with a surface double the base, has only half the resistance. At high velocities, experiments upon railways show that the resistance becomes nearly a constant quantity. Ccmrse of "Wind.. Direction in Cvolon^s Direction in Northern Hemisphere. ^y^ Southern Hemisphere Wind has its direction nearly at right angles to line between points of highest and lowest pressure of air, or barometer readings, and its course is with the point of lowest pressure at its left, and its velocity is directly as difference of the pressures. In Northern Temperate zone, winds course around an area of low pressure in reverse direction to course of hands of a watch, and they flow away frorii a location of high pressure, and cause an apparent course of the winds in di* rection of course of the hands. To Compute Resistance of a IPlane Surface to Air. .0023 a v 2 = P in Ibs. a representing area of plane in sq.feet, v velocity in direc tion of wind in feet per second, -{-when it moves opposite, and when with the wind When Barometer Pressure = 30 Lbs. (C. F. Martin, U.S. S. S.) .004 a V = P. V representing velocity of wind in miles per hour, and a area of pressure in sq. feet. To Compute Height of a Column of Mercury to induce an ICfnux of Air through a given Nozzle. Barometer assumed at 2. 46 feet = 29.52 ins., and Temperature 52. pa -g-^ f-^ H, and 48.073 d 2 ^/R = P. d representing diameter of nozzle and H height of column of mercury, loth in feet, and P volume of air in Ibs. per one second. ILLUSTRATION. Assume d = .19, and P == .7 Ibs. <-< f ^ 48.073 X.i9 2 V.x6a6 = -7- To Compute Pressure or \Veight of Air under a given Height of Barometer and Temperature, Discharged in One Second 30. 787 d* ^E -^pressure in Ibs. Or, 48.073 d 2 ^B = Ibs. I representing height of barometer in external air, B manometer or pressure of air in reservoir in mercury, both in feet, and t temperature of air or gas in degrees. ILLUSTRATION. Assume b = 2. 5 feet ; d = . 25 foot ; B = . i foot ; and t = 1.0550. Then 30.787 X .0625 ./.i X 2 ' 5 " t "' 1 = 1.924 X V- 2 4 6 5 = -9543 * 1 6/6 PNEUMATICS. AEKOMETBY. To Compute Teinperature for a, given Latitude and. Ele- vation. 82. 8 cos. I . ooi 98 1 E .4 t. E representing elevation in feet. ILLUSTRATION. Assume 1 = 45; cos. =.707; and E = 656 feet. Then 82.8 X .707 .001 981 X 656 .4 = 58.54 1.299 .4 = 58.54 .899 = 57.641. To Compute Volume of Air or Q-as Discharged through an Opening and vinder a Pressure above that of Ex- ternal Air. A ir. 1347. 4 C VB (6' -f B) T = V in cube feet per second. T = i -j- .002 22 (t 32), and 6' = 2.5 .00009 elevation. Or, 621.28 c?2 V B = V - ILLUSTRATION. What would be volume of air that would flow through a nozzle .246 foot in diam. from a reservoir under a pressure of .098 foot of mercury, into air under a barometric pressure of 2.477 ^ ee ^ temperature of air 55.4, location 45 of latitude, and at an elevation of 650 feet above level of sea? C = -75; 6' = 2.5 .00009 X 650 = 2.4415 (2.44); and T = 1.0502. Then 1347.4 X .75 : ^ ^.098 (2.44 -f- .098) X 1.0502 = 24.689 X V- 26l 7 = 12.63 cube feet. When Densities of External Air and that in Reservoir are Equal. 1 347. 4 C , \/ B (6 -f- B ) T ' = V. &' representing height of mercury in reservoir. Gas. ^j^ . /=-] -, V.p representing specific gravity of gas compared vP V lj H- 4 2 X d with atr, and L length of pipe or conduit in feet. ILLUSTRATION. If a pipe .05 feet in diameter and 420 feet in length, communi- cates with a gasometer charged with carburetted hydrogen (illuminating gas), under a water pressure as indicated by a manometer of . 1088 foot, what would be the dis- charge per second ? d = .05 foot ; L = 420 feet ; and B = ' I0 = .008 foot. Specific gravity of gas 13.6* .5625. 4231 / .008 X -05* 4231 /. 0000000025000 ; ; / 3 =3-^- / - ^ = .01371 cwfte /oof. V-5625 V 420+ 42 X- 05 -75 V 420-1-2.1 Resistance of Curves and Angles. Curves and angles increase resistance to discharge of air or gas very materially. By experiment of D'Aubuisson 7 angles of 45 reduced discharge of gas one fourth. To Compute Diameter of Discharge-pipe or Nozzle. When Length and Diameter of Pipe, Volume, and Pressure are given. 4/ f-^ d ',_. = yfcA* V 4230 2 B d 5 L V 2 ILLUSTRATION. If a pipe 1000 feet in length, and .4 foot in diameter, leads to a reservoir of air, under a mercurial manometric pressure of .18 foot, what diameter must be given to a nozzle to discharge 4 cube feet per second ? ./ 4 2X4 2 X.4S ./ 6.88128 Then */ = 4 / _ _ = .#.0004052 = V 4 23o*Xi8X. 45-ioooX4 2 V 32 980. 19 -16000 .i4i8jfoo<= 1.703 ins. Volumes of two gases flowing through equal orifices, and under equal pressures, are in inverse ratio of square roots of their respective densities. * Specific gravity of mercury compared with waUr. BAILWAYS. 6/7 RAILWAYS. To Define a Curve. Fig. 1. (Moksworth.) I7 ' 9 or * tan. * = R ; R (cotan. ) = * '7i9 c R = a; R (cosec. x i) = d R (cosin. x) = s\ R (coversin. x) = V; ^i^ = n, and (5400 x) .000582 R = i c representing any chord, t length of tangent, d distance of centre of curve from in- tersection of tangents, s half chord of curve, and I length of curve, all in like dimensions, a tangential angle ofc in minutes, n number of chords in curve, and x half angle oj intersection, but in formulas for number of chords and length of curve to be expressed in minutes. ILLUSTRATION. Assume radius 900 and chord 400 feet; angle of intersection = 12 44' = 764 minutes, and x = 56 15' 5". Tangent of 56 15' 5" = 1.496 73. Cotangent = .668 14. - = R = goo feet ; ^^ = 764 minutes ; 900 X . 668 14 = t = 900 X 1.20269 1 = d = 182.42 feet ; 900 X . 555 55 = s 500 feet; 5400 3379 _ 2 645 times ^ and .000 582 X 900 X 7 6 4 601.33 feet; 5400 3379 = 1058.6/66*. Tangential Angles for Chords of One Chain. Radius of Curve. Tangential Angle. Radius of Curve. Tangential Angle. Radius of Curve. Tangential Angle. Radius of Curve. Tangential Angle. Chains. Chains. Chains. 5 43-8', 3 34-87 IS 20 1 54-6' i 25-95' 40 45 42. 97,' 38.2' i mile i. 25 mil's 21.48' 17.19' 9 3 "' 25 1 8.76' 50 3 t'l 8 1.5 miles 14-33' 10 2 51.9', 30 57-3' 60 28.65' i-75 " 12.28 12 2 23.25 35 49.11 70 24-55' 2 10.74' NOTE. Angle for 2 chain chords is double angle for i chain chords. Angle for .5 chain chords is .5 the angle for i chain chords. Curves of less than 20 chains radius should be set out in . 5 chain chords. Curves of more than i mile radius may be set out in 2 chain chords. Angles in above Table are in degrees, minutes, and decimals of minutes. Fig. 2. I Sidings. 2 VdR. (.5 d) 2 = 1. R representing radius of curve, I length of curve over points, and d distance between tracks, aU in feet. Turn-out of TJneq.vi.al Radii R and r representing radii of the curves re- spectively as to length, x distance between outer rails of tracks and other symbols as shown, att to feet. 678 RAILWAYS. IPoints and Crossings. fin . ' senting radius of curves, G gauge of road, a angle of crossing, ~ and x =. R G, all in feet. In horizontal curves, width required for clearance of flange of wheel, and for width of rail at heel of switch, render it necessary to make an allowance in length of /, as ascertained by formula. For other diagrams and formulas, see Molesworth's Pocket- book, pp. 208-18, 2ist edition. 1719 c To Compute Tangential A~ngle .^br Curves. - = a. c representing chord in feet, and a angle in minutes. ILLUSTRATION. What is angle for a curve with a radius of 900 feet, and a chord of 400 feet? ,7,9 X 400 = inutes 900 Curving of Rails. 5- = v. I representing length of rail in feet, v versed sine at centre, when R urved, in ins. ILLUSTRATION. What is curve for a rail 20 leet in length, with a radius of 900 feet? I. 5 X20*_ 900 Curves t>y Offsets in Equal Chorda, Fig. 5. .# Chord 2 Chord 2 ^* 2 R = o offset. - = 2, o offset ILLUSTRATION. Assume chords 150, andra dius 900 feet. 22500 To Compute 'Versed Sines and Ordinates of Curves. .; LL^ + ,, = D; and \ R 2 2 (R v) = o. D representing diameter of I 'x s \ circle, and v versed sine of curve. ^ "X v \ ILLUSTRATION. Assume radius 900, and chord 400 feet. D 900 VSioooo 40000 = 900 877.5 =22.5 ./e. Relation of Base of Driving or Rigid Wheels to Curve. ^ =: B. R representing minimum radius of curve, G gauge of road, and B base, all in feet. To Compute Elevation of Outer Rail. For any Radius or Combination of Curve with Straight Line. = c. V representing velocity of train in feet per second, G gauge of road, and c length of a chord, both in feet, the versed sine of which = elevation in ins. On Curves. V 2 - G = E. E representing elevation of outer rail in ins, 1.25 R RAILWAYS. 679 Radii of Curves set out in Tangential Angles. Angle for Chord of 100 Feet. Radius Curve. Angle for Chord of loo Feet. Radius of Curve. Angle for Chord of 100 Feet. Radius Curve. Angle for Chord of ipo Feet. Radius of Curve. o ' 30 I i 30 2 Feet. 5729.6 2864.8 1909.9 1432.4 o ' 2 30 3 3 30 4 Feet. "45-9 954-9 818.5 716.2 o ' 4 30 5 5 3 6 Feet. 636.6 573 520.9 447-5 ' 6 30 7 7 3<> 8 Feet. 440.7 409-3 382 358.1 NOTE. If chords of less length are used, radius will be proportional thereto. To Ascertain Radius of Curve in Inches for Scale, in Feet per Inch. Divide radius of curve in feet by scale of feet per inch. To Compute Required \Veight of Rail. RULE. Multiply extreme load upon one driving-wheel in Ibs. by .005, and product will give weight of rail in Ibs. per yard. To Compute Radius of Curve and "Wheel Base. P 9 B G = R. = B. B representing maximum rigid wheel base of cars, and G gauge of way, both in feet, To Determine Elevation of Outer Rail. For any Radius or Construction of Curve with Straight. Fig. 7. Fig. 7. V .5 v'G = c. V representing speed of train in feet per sec- ond, G gauge of rails in feet, and c length of chord, versed sine v of which will give at its centre the elevation required. Thus, determine chord c, align it on inner side of rail, and distance of rail from it at centre of its length will give elevation re- quired, whatever tke radius of rail. ForCums . I.7.V(NDW)]-4PB or,W-I!- = E . representing r 1) K 1.25 K diameter of wheels, W width of gauge, P lateral play between flange and rail, and R radius of curve, all in feet, i ^- N ratio of inclination of tire, V velocity of train in miles per hour, and E elevation of outer rail in ins. (Molesworth.) ' = resistance due to curve, and W representing weight of body, both in 2 R Ibs., C coefficient of friction of wheels upon rails = .\ to .27, according to condition of weather, d distance of rails apart, I length of rigid wheel base, and R radius of curve, all in feet. (Morrison.) ILLUSTRATION. Assume weight of locomotive 30 tons, radius of curve 1000 feet, distance of rails apart 4 feet 8.75 ins., length of base 10 feet, and rails, dry, C = i. ) = ^ To Compute Resistance due to GJ-ravity upon an In- clination. 2240 Ibs. per ton of train. gradient Rise per Mile, and Resistance to GJ-ravity, in Lbs. per Ton. Gradient of i inch.. Rise in feet Resistance 264 176 64 45 60 80 | 90 i 66 59 28 24.8 68o RAILWAYS. To Compute Load, -which, a Locomotive -will Draw up an Inclination. T -i- r -j- r' W = L. T representing tractive power of locomotive in Ibs. , r re- sistance due to gravity, and r' resistance due to assumed velocity of train in Ibs. per ton, W weight of locomotive and tender, and L load locomotive can draw, in tons, ex- clusive of its own weight and tender. Coefficients of Traction of Locomotives. Railroads in good order, etc., 4 to 6 Ibs. ; in ordinary condition, 8 Ibs. In coupled engines adhesion is due to load upon wheels coupled to drivers. To Compute Traction, Retraction, and. Adhesive IPower* of a Locomotive or Train. When upon a Level. asP-r-D = T. a representing area of one cylinder in sq. ins., s stroke of piston and D diameter of driving-wheels, both in feet, P mean pressure of steam in Ibs. per sq. inch, and T traction, in Ibs. When upon an Inclination. asP-r-p rwh = T. r representing resistance per ton, w weight of locomotive upon driving-wheels, in tons, h height of rise in feet per zoo of road, and R = r w h = retraction, in Ibs. C w b -f- 100 = A. 6 representing base of inclination in feet per 100 of road. C w = A. C = coefficient in Ibs. per ton, and A adhesion, in Ibs. When Velocity of a Train is considered. When upon a Level, W (C + VV) = R. When upon an Inclination, W (r h + C + y/V) = R. V representing velocity of train in miles per hour. ILLUSTRATION. A train weighing 200 tons is to be driven up a grade of 52.8 feet per mile, with a velocity of 16 miles per hour; required the retractive power? 52.8 per mile = i in 100 feet = r = 22.4 Ibs. C = 5. = 6280/65. 200 (22-4 X I-f 5 + V l6 ) = 20 X Velocity of Trains. Miles per hour 10 15 20 30 40 50 60 70 Lbs. Lba. Lba. Lbs. Lbs. Lbs. Lbs. Lbs. Resistance upon straight line per ton . 8-5 9- 2 5 10.25 I3-25 I7-25 22.5 2 9 36.5 Do., with sharp curves and strong wind*. . . . 13 14 15-5 20 26 34 43-5 55 * Equal to 50 per cent, added to resistance upon a straight line. Friction of locomotive engines is about 9 per cent., or 2 Ibs. per ton of weight. Case-hardening of wheel-tires reduces their friction from 14 to .08 part of load. To Compute Maximum Load that can Ibe drawn "by- an Engine, "up the ^Eaximnm Q-rade that it can .Attain, \Veight and G-rade being given. (Maj. McClellan, U.S.A.) .2 A - = L, and * = G. A representing adhesive weight of engine, .424204-8 .4242 L in Ibs. , G grade in feet per mile, and L load, in tons. NOTE i. When rails are out of order, and slippery, etc., for .2 A, put .143 A. 2. With an engine of 4 drivers, put .6 as weight resting upon drivers; with 6 drivers the entire weight rests upon them. ILLUSTRATION. An engine weighing 30 tons has 6 drivers; what are the maximum loads it can draw upon a level, and upon a grade of 250 feet, and what is its maxi- mum grade for that load ? .. X MO X 3 .34JQ fon , g fewt .*X2*4X3C 34g = .4242-f-8 8.4242 .4242X250-}-8 114.05 , . .2X2240X30 8XlI7.8 12497 ,. . 117. 8 tons up a grade of 250 feet. ' =. -^- = 250. i feet. Adhesion of a 4- wheeled locomotive, compared with one of 6 wheels, i? as 5 to & RAILWAYS. 68 1 OPERATION OF LOCOMOTIVES. (O. Chanute, Am. Soc. C. E.) .Axlliesion.. Adhesion of a locomotive is friction of its driving-wheels upon the rails, rarying with condition of the surface, and must exceed traction of the engine upon them, otherwise the wheels will slip. Improvements heretofore made in the construction of locomotives and tracks have gradually increased the proportion which the adhesion bears to the insistent weight upon the driving-wheels. The first accurate experiments were those of Mr. Wood upon the early English coal railways. He deduced the adhesion to be as follows: Upon perfectly dry rails 14 of weight on drivers. " damp or muddy rails 08 " " very greasy rails 04" " " In 1838, B. H. Latrobe indicated .13 as a safe working adhesion, while modern European practice assumes about .2 of weight as maximum, and . n as a minimum, except perhaps in some mountainous regions, subject to mists. Thus, on the Scem- mering line, adhesion is generally .16, and between Pontedecimo and Busalla, in Italy, it never exceeds .12 in open cuttings, or .1 in tunnels. Extensive experiments made upon French railways, 1862-67, bv Messrs. Vuille- min, Guebhard, and Dieudonne gave following coefficients in actual working; dry weather, extreme, .105 to .2; damp, .132 to .139; wet, .078 to .164; light rain, .09; extreme rain, .109 to .2, mean, . 13; rain and fog, .115 to .14; heavy rain, 16. Materially better results are obtained in United States, partly, perhaps, in con- sequence of greater dryness of the weather, and certainly because of the American method of construction and equalizing the weight between the drivers, and of mak- ing the locomotive so flexible as to adapt itself to inequalities in the track. Modern engines in America can safely be relied upon to operate up to an adhesion equal to .222 in summer and .2 in winter, of weight upon the driving wheels. From these data the following tables have been computed: Coefficients of Adhesion xipoii Driving "Wheels per Ton. Condition of Rails. European Practice. American Practice. Condition of Rails. European Practice. American Practice. C. Lbs. C. Lbs. C. Lbs. C. Lbs. Rails very dry 3 670 33 667 In misty weather . .015 350 .2 400 Rails very wet .27 600 25 500 In frost and snow. .09 200 .16 333 Ordinary working. . .2 450 .222 444 Adhesion of Locomotives, in Lbs. (.222 in Summer and .2 in Winter). Type of Locomotive. No. of Drivers. Wei Locomotive. At. On Drivers Adh esion. Winter. Lbs. Lbs. Lbs. Lbs. American 4 wheels coupled. . . . 64000 42000 9350 8400 Ten wheeled 6 i jonne cted.. ii 6 Mogul 6 88000 72 ooo 16 r>nr 14 ooo Consolidation 8 ii \' m 100 000 88000 I 955 17600 Tank switching.... 6 u 68000 68000 15100 13600 " .... 4 w 48000 48000 10650 9600 Tractive Power. Traction of a locomotive is the horizontal resultant on the track of the pressure of the steam, as applied in the cylinders. D 2 P L-:- W =:T. D representing diameter of cylinder, L length of stroke, and W diameter of driving wheels, all in ins. , P mean pressure in cylinder, in Ibs. per q. inch, and T tractive force on rails, in Ibs. ILLUSTRATION. Assume a locomotive, cylinders 18 ins. in diam., 22 ins. stroke, wheels 68 ins. in diam., and average steam pressure in cylinders 50 Ibs. per sq. inck Then 18 X 18 X 50 X 22 -=- 68 = 5241 /6s. 682 RAILWAYS. Train Resistances. Usual formula for train resistances, on a level and straight line, is yz V 2 1- 8 = R per ton of train, and \- 6 = R per ton of train alone. V renre- 171 240 senting velocity in miles per hour, and 8 constant axle friction. (D. K. Clark.) NOTE. To meet the unfavorable conditions of quick curves, strong winds, and imperfection of road, Mr. Clark estimates results as obtained by above formula should be increased 50 per cent. ILLUSTRATION. At 20 miles per hour, the resistance would be: 20 2 ~r 171 -p- 8 10.3 Ibs. per ton of train. This formula, however, is empirical. It gives results which are too large for freight trains at moderate speeds, and too small for passenger trains at high speeds. Engineers are not agreed as to exact measure and value of each of the elements of train resistances, but following approximations are sufficient for practical use: Analysis of Train Resistances. Resistance of trains to traction may be divided into four principal ele- ments : i st. Grades ; 2d. Curves ; 3d. Wheel friction ; 4th. Atmosphere. ist. Grades. Gradients generally oppose largest element of resistance to trains. Their influence is entirely independent of speed. The meas- ure of this resistance is equal to weight of train multiplied by rate of in- clination or per cent, of grade. Thus, a gradient of .5 per 100 feet (26.4 feet per mile) offers a resistance of 5 X 2240 ._. I12 lbs> t or JQ j^ s 10 X ioo per 2000 Ibs., which is to be multiplied by weight in tons of entire train. Following table shows resistance, due to gravity alone, for the most usual grades, in Ibs. per ton of train : i st. Resistance due to Grades. Lbs. per ton of 2240 Ibs. . . Rate per mile si 4-48 " <% 16 & o II. 2 26 13-44 ,5^8 17.92 Lbs. per ton of 2000 Ibs. . . Rate per ioo feet ; .0 6 8 10 3 2 12 H ?6 1.6 Lbs. per ton of 2240 Ibs. . . Rate per mile *i 20.10 22. 4 24.64 s8 26.88 61. i- O 29.12 68 3-36 * D 33-6 35-84 ge Lbs. per ton of 2000 Ibs. . . JJ 2O 22 U J 24 26 74 28 yy 3 j 32 2d. Curves. Recent European formula is that given by Baron von Weber. . 6504 -T- R 55 = W. R representing radius of curve in metres. This formula assumes that resistance due to curve increases faster than radius diminishes. It gives results varying from a resistance of .8 Ib. per 2000 Ibs. per degree for a curve of 1000 metres radius (3310 feet, or i 44') to a resistance of 1.67 Ibs. per 2000 Ibs. per degree for curves of ioo metres radius (331 feet, or 17 20'). Messrs. Vuillemin, Guebhard, and DieudonnS found curve-resistance to European rolling-stock to be from 8 to i Ib. per 2000 Ibs. per degree, on a gauge of 4 feet 8. 5 ins., while Mr. B. H. Latrobe, in 1844, found that with American cars resistance on a curve of 400 feet radius did not exceed .56 Ib. per 2000 Ibs. per degree. Resistance of same curve varies with coning given tires of wheels, elevation of outer rail, and speed of train running over it, but both reasoning and experiment indicate that the general resistance of curves increases very nearly in direct pro- portion to degree of curvature, or inversely to the radius. Recent American experiments show that a safe allowance for curve resistance may be estimated at .125 of a Ib. per 2000 Ibs. for each foot in width of gauge. Thus, for 3 feet gauge resistance would be .375 Ib. per degree of curve; for standard gauge of 4 feet 8.5 ins. .589, say .60, and for 6 feet gauge .75 Ib. per degree. For standard gauge, when radius is given in feet, resistance due to this element is: .60 X 5730 -i- R = C in Ibs. per ton of train. RAILWAYS. 683 This is somewhat reduced when curve coincides with that for which wheels are coned (generally about 3), and when train runs over it, at precise speed for which outer rail is elevated, an allowance of .5 Ib. per ton per degree is found to give good results in practice. 2d. Resistance on Curves. It follows from above estimate of curve resistance that, in order to have the same resistance on a curve as on a straight line, the gradient should be diminished by .03 per ioo feet of each degree of curve. Thus a 3 curve requires an easing of the grade by .09 per ioo feet, a 10 curve an easing of .3 per ioo, etc. This, however, need only be done upon the limiting gradients, and when sum of grade and curve resistances exceeds resistance which has been assumed as limiting the trains. 3d. Resistance due to Wheel Friction. Experimenters are not agreed whether friction of wheels increases simply with weight which they carry, but also in some ratio with the speed. Originally as- sumed as a constant at 8 Ibs. per ton, improvements in condition of track (steel rails, etc.) and in construction and lubrication of rolling stock have reduced it to 3.5 and 4 Ibs. per ton for well-oiled trains. Under ordinary circumstances, in sum- mer, it will be safe to estimate it at 5 Ibs. per ton on first-class tracks, and 6 Ibs. per ton on fair tracks. It may run up to 7 or 8 Ibs. per ton on bad tracks (iron rails) in summer, and all these amounts should be increased from 25 to 50 percent, in cold climates in winter, to allow for inferior lubrication. 4th. Resistance due to A tmosphere. Atmospheric resistance to trains, complicated as it is by the wind which may be prevailing, has not been accurately ascertained by experiment. It consists of ist. Head resistance of first car of train, which is presumably equal to its exposed area, in sq. feet, multiplied by air pressure due to speed. 2d. Head resistance of each subsequent car. This varies with distance they are coupled apart, and so shield each other from end air pressure due to speed. 3d. Friction of air against sides of each car depending upon the speed. This is generally so small that it may be neglected altogether. 4th. Effect due to prevailing wind, which modifies above three items of resistance. A head wind retards the train, a rear wind aids it, while a side wind increases re- sistance by pressing flanges of wheels against one rail, and, in consequence of curves, a train may assume all of these positions to same wind. Recent experiments on Erie Railway seem to indicate that in a dead calm re- sistance of first car of a freight train may be assumed at an exposed surface of 63 sq. feet,* multiplied by a'ir pressure due to speed, and that each subsequent car may be assumed to offer a resistance of 20 per cent, of that of first car, while in a pas- senger train first car may be assumed at an area of 90 sq. feet,t multiplied by air pressure due to speed, and that each subsequent car adds an increment equal to 40 per cent, that of first car, in consequence of greater distance they are coupled apart. This resistance is, of course, entirely independent of cars being loaded or empty. In practice it has been found that an allowance of 1.5 to 2 Ibs. per ton of weight of & freight train covers atmospheric resistance, except in very high winds. In consequence of complexity of elements above enumerated, exact formulas can- not probably be now given for train resistances, but following, if applied with judg- ment (and modified to fit circumstances), will be found to give fairly accurate results in practice. They are for standard gauge, and in making them, curve resistance has been assumed at .5 Ib. per degree, wheel friction at 5 Ibs.. exposed end area of first car at 90 sq. feet for passenger cars and 63 feet for freight cars, and increment for succeeding cars at .4 for passenger trains and .2 for freight trains. Passenger Train. W (& + + s) + ( r + ~~) 9 P=::B - Freight Train. W (G -f^ + 5) + (' +^ Zl1 ) 63 P = R * This is less than area of car, which generally measures about 71 sq. feet ; but part is shielded by tender, and parts being convex, as wheels, bolts, etc., offer lees resistance than a flat plane. t Not only is end area of passenger cars greater than that of freight cars, but in consequence of the projecting roof the end forms a hood in nature of a concave surface, and so opposes greater resistance than a flat plan*. 684 EAILWAYS. W representing weight of train, without engine, in tons (2000 Ibs. ), G resistance of gradient per ton (2000 Ibs.; see table, page 683), C curve in degrees, n number of cars in train, P pressure per sq.foot due to speed, to which an allowance must be made for wind, if existing, R resistance of train, and 5, wheel friction, both in Ibs. ILLUSTRATION i. Assume a passenger train of 5 cars, weighing 136 tons (2000 Ibs.), ascending a grade .5 per 100 (26.4 feet per mile), with curves of 4, at a speed of 60 miles per hour (for which the pressure is 18 Ibs. per sq. foot), resistance will be: 136 (10+ 2 -f- 5) + f i + J (90 X 18) = 6524 Ibs., of which 2312 Ibs. are due t& grade, curve, and wheels, and 4212 Ibs. to atmospheric resistance. 2. Assume a freight train of 31 cars, weighing 620 tons (2000 Ibs.), turning a curvt of 3, up a grade of 52.8 feet per mile (i foot per 100), at a speed of 21 miles per hour (pressure 2 Ibs. per sq. foot), resistance will be: 620(20+1. ~ ( 6 3 X 2) = 17 312 Ibs., requiring a "Consolidation engine to haul it, allowance being made for possible winds, etc. Assume conversely, it is desired to know how many tons an American engine, with an adhesion of 10650 Ibs., will draw up a grade of .9 per 100 (47 feet per mile), with curves of 4. assuming atmospheric resistance between 1.5 to 2 Ibs. per ton of train. Resistance from grade .9 x 2000-4- 100 ............. = 18 Ibs. ) u curve 4-^-2 ....................... = 2" } 27 Ibs. " wheel friction 5, atmosphere 2 ..... 7 " ) Hence, 10 650 -r- 27 := 395 tons, or about 20 cars, and in winter same engine will haul 9600-7-27 355 tons (2000 Ibs.), or about 18 cars. Following table approximates to best modern practice. For freight trains it gives aggregate resistance, in Ibs. per ton (2000 Ibs ), for various grades and curves. In usiiig it, it is sufficient to divide the adhesion in Ibs. of locomotive used by number found in table, in order to obtain number of tons of train that it will haul at or- dinary speeds on gradient and curve selected. Of course, if grade has been equated for curves, only number found in first column (for straight lines) is to be used in computing tons of train on limiting gradient. Approximate !F"reiglit-train. Resistances. Gauge 4 feet 8. 5 ins. In Lbs. per 2000 Ibs. at Ordinary Speeds. Curve Resistance assumed at .5 Ibs. per , Wheel Friction at 5 Ibs., Atmospheric Re- sistance at 2 Ibs. per Ton. GRADB. i CURVE. Per Per * Cent. Mile. qo i 2 3 4 5 6 7 8 9 10 11 12 13 14 i5 Ibs. Ibs. Ibs. Ibs. Ibs". Ibs. IbsT Ibs. IbsT Ibs. IbsT Ibs. Ibs. Ibs. Ibs. Ibs. Level. Feet. 7 7-5 8 8-5 9 9-5 10 10.5 ii "5 12 12.5 13 13-5 14 14-5 .1 5 9 9-5 10 10.5 ii "5 12 12.5 13 13-5 14 14-5 15 15-5 16 16.5 .2 ii ii "5 12 12.5 13 13-5 H 14-5 15 15-5 r6 I6. 5 i? 17-5 18 18.5 3 16 13 i3-5 14 14-5 15 15-5 16 16.5 i? 17-5 18 l8. 5 1 9 19.5 20 20.5 4 21 15 i5-5 16 16.5 i? i7-5 18 18.5 '9 19-5 20 20.5 21 21.5 22 22.5 5 26 17 i7-5 18 18.5 i9 19-5120 20.5 21 21.5 22 22-5 23 23-5 24 24-5 .6 32 9 i9-5 20 20.5 21 2I.5| 22 22.5 23 23-5 2 4 24-5 25 25-5 26 26.5 7 37 21 21.5 22 22.5 23 23.5,24 24-5 25 25-5 26 26.5 2? 27-5 28 28.5 .8 42 2 3 23-5 2 4 24-5 25 25.5(26 26.5 27 27-5 28 28.5 2 9 29-5 30 30-5 9 47 25 25-5 26 26.5 27 27-5 28 28.5 2 9 29-5 1 30 30-5 31 31-5 32 32-5 i 53 2 7 27-5 28 28.5129 29-5 30 30-5 31 3i-5 32 32-5 33 33-5 34 34-5 i.i 58 2 9 29-5 30 30- 5 i 3i 31-5 32 32.5 33 33-5134 34-5 35 35-5 36 36.5 1.2 63 31 3i-5 32 32-5 33 33-5 34 34-5 35I35.5 36 36.5 37 37-5 38 38.5 i-3 68 33 33-5 34 34-3 35 35-5 36 36.5 37 '37- 5 38 38.5 39 39-5 40 40-5 1.4 74 35 35-5 36 36-5 37 37-5 38 38.5 39 39-5 4 40-5 4i 4i-5 42 42-5 i-J 79 37 37-5 38 38- 5 39 39-5 40 40-5 41 41.5 42 42-5 43 43-5 44 44-5 1.6 85 39 1 39- 5 40 40.5:41 4i-5 4 2 42-5 ! 43 43-5 44 44-5 45 45-5 46 46.5 ILLUSTRATION. Assume a "Mogul" engine to have an adhesion of 16000 Ibs. ; what weight will it haul up a grade of 74 feet per mile, combined with a curve of 9 ? 16000-4-39.5 =405 tons (2000 Ibs.). EA1LWATS. 685 Hence, To Compute Adhesion on a Given Grade and Curve, having Weight of Train. RULE. Multiply tabular number by weight of train in tons (2000 Ibs.), and product will give adhesion, in Ibs. EXAMPLE. Assume preceding elements. Then 39. 5 X 405 = 16 ooo Ibs. NOTE. A "Consolidation" engine, by its superior adhesion (19550 Ibs.) would haul up a like grade and curve 495 tons. Memoranda .011 English. Railways. Regulations (Board of Trade). Cast-iron girders to have a breaking weight = 3 times permanent load, added to 6 times moving load. Wrought-iron bridges not to be strained to more than 5 tons per sq. inch. Minimum distance of standing work from outer edge of rail at level of carriage steps, 3.5 feet in England and 4 feet in Ireland. Minimum distance between lines of railway, 6 feet. Stations. Minimum width of platform, 6 feet, and 12 at important stations. Minimum distance of columns from edge of platform, 6 feet. Steepest gradient for stations, i in 260. Ends of platforms to be ramped (not stepped). Signals and dis- tant signals in both directions. Carriages. Minimum space per passenger 20 cube feet. Minimum area of glass per passenger, 60 sq. ins. Minimum width of seats, 15 ins. Minimum breadth of seat per passenger, 18 ins. Minimum number of lamps per carriage, 2. Requirements. Joints of rails to be fished. Chairs to be secured by iron spikes. Fang bolts to be used at the joints of flat-bottomed rails. Construction. v^T Width, single line ................................... 18 '* double line ................................... 30 ii My. top of ballast, single line ...................... 13 Broad. Feet. Ins. P 6 15 29 double line ..................... 24 Slope of cuttings from centre, i in 30. Width of land beyond bottom of slope, o to 12 feet. Ditch with slopes, i foot at bottom, i to i. Quick mound, 18 ins. in height. Post and rail-fence posts, 7 feet 6 ins. x 6 ins. x 3.5 ins., 9 feet apart, 3 feet in ground. Intermediate posts, 5 feet 6 ins. X 4 ins. X 1.5 ins., 3 feet apart. Rails 4 of 4X 1.5 ins. Parliamentary Regulations for Crossing Roads. Turnpike Road. Public Road. Occupation Road. Clear width of under bridge, or approach .... Clear height of under bridge for a width of 12 ft. (( U U U U IQ U a tc u u g a " " " at springing Feet. Ins. 3 I ~~ 16 12 4 i in 30 3 ach side Feet. Ins. 2 5 15 12 4 i in 20 3 of centre Feet. Ina. 12 14 i in 16 3 line. In Limits of Deviation. In towns, 10 yards country, 100 yards, or 5 chains nearly. Level. In towns, 2 feet. In country, 5 feet. Gradient. Gradients flatter than i in 100, deviation 10 feet per mile steeper. Do., steeper, 3 feet per mile. Curve. Curves upwards of .5 a mile radius, may be sharpened to .5 mile radius. Curves of less than .5 mile radius may not be sharpened. 3M 686 EOADS, STEEETS, AND PAVEMENTS. ROADS, STREETS, AND PAVEMENTS. Classification. o' Roads. i. Earth. 2. Corduroy. 3. Plank. 4. Gravel. 5. Broken stone (Marv ftdam). 6. Stone sub-pavement with surface of broken stone (Telford). 7. Stone sub-pavement with surface of broken stone and gravel, or gravel alone. 8. Rubble stone bottom with surface of broken stone or gravel, or both. 9. Concrete bottom with surface of broken stone or gravel, or both. Oracle of Roads. Limit of practicable grade varies with character of road and friction of ye- hide. For best carriages on best roads, limit is i in 35, or 150 feet in a mile. Maximum grade of a turnpike road is i in 30 feet. An ascent is easier for draught if taken in alternate ascents and levels, than in one continuous rise, although the ascents may be steeper than in a uniform grade. Ordinary angle of repose is i in 40 if roads are bad, and i in 30, to i in 20. When roads have a greater grade than i in 35, time is lost in descending, in order to avoid unsafe speed. Grade of a road should be less than its angle of repose. Minimum grade of a road to secure effective drainage should be i in 80. In France it is i in 125. In construction of roads the advantage of a level road over that of an in- clined one, in reduction of labor, is superior to cost of an increased length of road in the avoiding of a hill. Alpine roads over the Simplon Pass average i in 17 on Swiss side, i in 22 on Italian side, and in one instance i in 13. In deciding upon a grade, the motive power available of ascent and avoid- able of waste of power in descending are to be first considered. When traffic is heavier in one direction than the other, the grade in as- cent of lighter traffic may be greatest. When axis of a road is upon side of a hill, and road is made in parts by excavation and by embankment, the side surface should be cut into steps, in order to afford a secure footing to embankment, and in extreme cases, sustaining walls should be erected. Construction, Estimate of Labor in Construction of Roads. (M. Ancelin.) A day's work of 10 hours of an average laborer is estimated as follows: In Cube Yards. Loose Earth. WOBK. Ordinary Earth. Picking and digging Excavation and pitching ) 6 to 12 feet ) 18 to 23 8 to 12 Loading in barrows Wheeling in barrows per) 100 feet J 22 20 to 33 Loading in carts 16 to 48 Spreading and levelling. . . 44 to 88 Mud. Clay and Earth. Gravel. Blasting Rock. 9 7 to ii 2.4 7 to 16 4 2.2 8 J 9 24 to 28 _ 17 to 27 25 30 to 80 Time of pitching from a shovel is one third of that of digging. Ditches. All ditches should lead to a natural water-course, and their min- imum inclination should be i in 125. Depressions and elevations in surface of a roadway involve a material loss of power. Thus, if elevation is i inch, under a wheel 4 feet in diameter, an inclined plane of i in 7 has to be surmounted, and, as a consequence, one seventh of weight has to be raised x inch. ROADS, STREETS, A^D PAVEMENTS. 68/ An unyielding foundation and surface are indispensable for a perfect roadway. Earth in embankment occupies an average of one tenth less space than in natural bank, and rock about one third more. Ruts. Surface of a roadway should be maintained as intact as prac- ticable, as the rutting of it not only tends to a rapid destruction of it, but involves increased traction. The general practice of rutting a road displays a degree of ignorance of physical laws and mechanical effects that is as inexplicable as it is injurious and expensive. On compressible roadways, as earth, sand, etc., resistance of a wheel decreases as breadth of tire increases. Depressing of axles at their ends increases friction. Long and pliant springs de- crease effect of shock in passing over obstacles in a very great degree. Transverse Section. Best profile of section of roadway is held to be one formed by two inclined planes meeting in centre of road and slightly rounded off at point of junction. Roads having a rough surface or of broken stone should have a rise of i in 24, equal to a rise on crown of 6 ins., and on a smooth surface, as a block-stone or wood pavement, the rise may be reduced to i in 48. On roads, when longitudinal inclination is great, the rise of transverse section should be increased, in order that surface water may more readily run off to sides of roadway, instead of down its length, and consequently gullying it. Stone Breaking. A steam stone-breaking machine will break a cube yard of stone into cubes of 1.5 ins. side, at rate of i to 1.5 IP per hour. Macadainized. Roads. In construction of a Macadamized road, the stones ( road metal ) used should be hard and rough, and cubical in form, the longest diameter of which exceed 2.5 ins., but when they are very hard this may be reduced to 1.25 and 1.5 ins. The best stones are such as are difficult of fracture, as basaltic and trap, and especially when they are combined with hornblende. Flint and sili- ceous stone are rendered unfit for use by being too brittle. Light granites are objectionable, in consequence of their being brittle and liable to disinte- gration ; dark granites, possessing hornblende, are less objectionable. Lime- stones, sandstones, and slate are too weak and friable. Dimensions of a hammer for breaking the stone should be, head 6 ins. in length, weighing i lb., handle 18 ins. in length; and an average laborer can break from 1.5 to 2 cube yards per day. Stones broken up in this manner have a volume twice as great as in their original form. 100 cube feet of rock will make 190 of 1.5 ins. dimension, 182 of 2 ins., and 170 of 2.5 ins. A ton of hard metal has a volume of 1.185 cu t> e yards. Construction of a Roadway. Excavate and level to a depth of i foot, then lay a " bottom " 12 ins. deep of brick or stone spalls or chips, clinker or old concrete, etc., roll down to 9 ins, then add a layer of coarse gravel or small ballast 5 ins. deep, roll down to 3 ins., and then metal in 2 equal lay- ers of 3 ins., laid at an interval, enabling first layer to be fully consolidated before second is laid on and rolled to a depth of 4 ins. ; a surface or " blind'' of .75 inch of sharp sand should be laid over last layer of metal and rolled in with a free supply of water. 688 ROADS, STREETS,- AND PAVEMENTS. Proportion of Getters, Fillers, and Wheelers in different Soils, at a Run of 50 Yards. (Molesworth.) Wheelers computed Getters. Fillers. Wheelers. Getters. Fillers. Wheelers. Loose earth ) 1.25 1.25 Sand, etc. } Compact earth . . . Marl... I X I I 2 2 X 2 2 Compact ) gravel J "' Rock . . X 7 2 I X I Telford Roads. In construction of a Telford road, metalling is set upon a bottom course of stones, set by hand, in the manner of an ordinary block stone pavement, which course is composed of stones running progressively from 3 inches in depth at sides of road to 4, 5, and 7 inches to centre, and set upon their broadest edge, free from irregularities in their upper surface, and their in- terstices filled with stone spalls or chips, firmly wedged in. Centre portion of road to be metalled first to a depth of 4 ins., to which, after being used for a brief period, 2 ins. more are to be added, and entire surface to be covered, " blinded," with clean gravel 1.5 ins. in depth. Telford assigned a load not to exceed i ton upon each wheel of a vehicle, with a tire 4 ins. in breadth. Gravel or Earth. Roads. In construction of a gravel or earth road, selection should be made between clean round gravel that will not pack, and sharp gravel intermixed with earth or clay, that will bind or compact when submitted to the pressure of traffic or a roll. Surface of an ordinary gravel roadway should be excavated to a depth of from 8 to 12 ins. for full width of road, the surface of excavation conforming to that of road to be constructed. The gravel should then be spread in layers, and each layer compacted by the gradual pressure due to travel over it, or by a roller, the weight of it in- creasing with each layer. One of 6 tons will suffice for limit of weight. If gravel is dry and will not readily pack, it should be wet, and mixed with a binding material, or covered with a thin layer of it, as clay or loam. In rolling, the sides of road should be first rolled, in order to arrest the gravel, when the centre is being rolled, from spreading at the side. To re-form a mile of gravel or earth road, 30 feet in width between gutters, material cast up from sides, there wili be required 1640 hours' labor of men, and 20 of a double team. Cordiiroy Roads. A Corduroy road is one in which timber logs are laid transversely to its plane. TMarilz Roads. A single plank road should not exceed 8 feet in width, as any greater width involves an expenditure of material, without any equivalent advantage. If a double track is required it should consist of two single and independ- ent tracks, as with one wide track the wear would be mostly in the centre, and consequently, wear would be restricted to one portion of its surface. Materials. Sleepers should be as long as practicable of attainment, in depth 3 or 4 ins., according to requirements of the soil, and they should have a width of 3 ins. for each foot of width of road. Pine, oak, maple, or beech are best adapted for economy and wear. Planks should be from 3 to 3.5 ins. thick, and not less than 9 ins. in width, or more than 12 if of hard wood, or 15 if of soft. A plank road will wear from 7 to 12 years, according to service, material, and location, and its traction, compared with an ordinary Macadamized road, is 2.5 to 3 times less, and with a common country road in bad order 7 times. For other elements, see Earth- work, page 466. ROADS, STREETS, AND PAVEMENTS. 689 -A.sph.alt. Asphalt pavements are made in two ways, either from a mixture of asphaltum with sand and a little powdered limestone, or from natural asphaltic limestone, called sometimes "rock asphalt," which contains from 6 to 12 per cent, of asphal- tum. The asphalt pavements of America are principally made by the former, and those of Europe by the latter method. The composition of one is of about 12.5 per cent, refined asphaltum, 2. 5 residuum oil of petroleum or soft bitumen termed u maltha," 5 powdered limestone, and 80 sharp sand, by weight, mixed at about 300. The rock asphalt pavement is made by powdering the natural asphaltic lime- stone, heating the powder, and compressing it in place. Asphaltic mastic, for floors, roofs, and sidewalks, is made from rock asphalt, by adding asphaltum to it as a flux and incorporating 60 per cent, more or less of sand and gravel, according to the density needed and the temperature of the place, cel- lar or walk, and whether exposed to the sun or not. The roadway needs a convex- ity of at least . 15 of its breadth. Artificial Asphalt. Heated sand, gravel, and powdered limestone, with gas tar or coal tar, when mixed, possess some of the properties of asphalt mastic, but are much inferior. Bituminous Road may be made by breaking up asphaltic limestone, laying it 2 ins. thick, covering with coal tar and ramming. Useful in country districts near such deposits. 'Wood. 3?avexnent. Close-grained and hard woods only are suitable, such as oak, elm, ash, beech, and yellow pine, and they should be laid on a foundation of concrete. Block Stone I?avexnent. Paving-blocks, as the Belgian, etc., where crest of street or area of pave- ment does not exceed i inch in 7.5 feet, should taper slightly toward the top, and the joints be well filled, " blinded," with gravel. The common practice of tapering them downward is erroneous. The foundation or bottoming of a stone pavement for street travel should consist either of hydraulic concrete or rubble masonry in hydraulic mortar. The practice in this country of setting the stones in sand alone is at variance with endurance and ultimate economy, but when resorted to, there should be a bed of 12 ins. of gravel, rammed in three layers, covered with an inch of sand. Granite or Trap blocks should be 4 x 9 X 12 ins. Ru."b~ble Stone I?avement. Bowlders or Beach stone of irregular volumes and forms, set in a bed of sand, involves great resistance to vehicles and frequent repairs ; it is wholly at variance with requirements of heavy traffic or city use. Concrete Ttoads. Concrete roads are constructed of broken stones (road metal) 4 volumes, clean sharp sand 1.25 to .33 volumes, and hydraulic cement i volume. The mass is laid down in a layer of 3 or 4 ins. in depth, and left to harden during a period of 3 days, when a second and like layer is laid on and well rolled, and then left to harden for a period of from 10 to 20 days, according to temperature and moisture of the weather. Roads. (Molesworth.) Ordinary turnpike roads. 30 feet wide, centre 6 ins. higher than sides ; 4 feet from centre, .5 inch below centre ; 9 feet from centre, 2 ins. below centre; 15 feet from centre, 6 ins. below centre. Foot-paths 6 feet wide, inclined i inch towards road, of fine gravel, or sifted quarry chippings, 3 ins. thick. Cross-roads 20 feet wide. Foot-paths 5 feet. Side drains 3 feet below surface of road. Road material bottom layer gravel, burned clay or chalk, 8 ins. deep. Top layer, broken granite not larger than 1.5 cube ins., 6 ins. deep. 3 M* 690 ROADS, STREETS, AND PAVEMENTS. Miscellaneous !N"otes. Metalling should be from 6 ins. to i foot in depth, and in cubes of 1.5 to 1.75 ins. One layer of material of a road should be spread and submitted to traffic or roll- ing before next is laid down, and this process should be repeated in 2 or 3 layers of 3 ins. each. When new metal is laid on old, the surface of the old should be loosened with a pick. Patching is termed darning. Sand and Gravel, Blinding, should not be spread over a new surface, as they tend to arrest binding of metal. Mud should be scraped oft" of surface. Hoggin is application of a binding of surface of a metal road, composed of loam, fine gravel, and coarse sand. Metalled Roads should be swept wet. Rolling. Steam rolls are most effective and economical. 1000 sq. yards of metal- ling will require 24 hours' rolling at i. 5 miles per hour. A roller of 15 tons' weight will roll looo sq. yards of Telford or Macadam pavement in from 30 to 40 hours, at a speed of 1.5 miles per hour, equal .675 and .9 ton mile per sq. yard. Sprinkling. 60 cube feet of water with one cart will cover 850 sq. yards. 100 cube feet per day will cover 1000 sq. yards; ordinarily two sprinklings are necessary. Granite Pavement. The wear of granite pavement of London Bridge was .22 inch per year, and from an average of several streets in London, the wear per 100 vehicles per foot of width per day is equal to one sixteenth of an inch per year. Sweeping and Watering of granite pavement and Macadam road, for equal areas and under alike conditions in every respect, costs as i for former toj of latter. By men, with cart, horse, and driver, costs 3.25 times more than by a machine, one of which will sweep 16000 sq. yards of street per period of 6 hours. Asphalt Pavement. Average cost per sq. yard in London: foundation, 50 cents; surface, $3.25; cost of maintenance per sq. yard per year, 40 cents. Wear varies from .2 to 42 near curb, and .17 to .34 inch on general surface per year. Washing. Surface cleaning of stone or asphalt pavement by a jet can be effected at from i to 2 gallons per sq. yard. Wood Pavement. Wear of wood pavement in London, per 100 vehicles per day per foot of width, .083 inch per year. Macadamized Roads. Annual cost of maintenance of several such roads in London was 62 cents per sq. yard. Block Stone Pavement. Stones should be set with their tapered or least ends up- wards, with surface joints of i inch. Fascines, when used, should be in two layers, laid crosswise to each other and picketed down. Bituminous road may be made by breaking up asphalt, laying it 2 ins. thick, covering with coal tar, and ramming it with a heavy beetle. To repair a bitumi- nous surface, dissolve one part of bitumen (mineral tar) in three of pitch oil or resin oil, spread .625 of a Ib. of solution over each sq. yard of road, sprinkle 2 Ibs. pow- dered asphalt (bituminous limestone) and then sand, and sweep off the surplus. Slipping. Granite safest when wet, and asphalt and wood when dry. Gravel, alike to that of Roa Hook, from its uniformity, will bear an admixture of from .2 to .25 of ordinary gravel or coarse sand. Annual cost of a Telford pavement 4.2 cents per sq. yard, including sprinkling, repairs, and supervision. Voids in a Cube Yard of Stone. Broken to a gauge of 2. 5 ins 10 cube feet. I Shingle 9 cube feet. 2 u 10.66 " " I Thames ballast. ... 4.5 " " i-S " "-33 ' ' For further and full information, see Law and Clarke on Roads and Streets, New York, 1867; Weale's Series, London, 1861 and 1877; Roads, Streets, and Pavements, by Brev. Maj.-Gen. Q. A. Gilmore, U. S. A., New York, 1876; Engineering Notes, by F. Robertson, London and New York, 1873; and Construction and Maintenance of Roads, by Ed. P. North, C. E., see Transactions Am. Soc. of C. E., vol. viii., May, 1879. SEWERS. 691 ', and D = r. x representing SEWERS. Sewers are the courses from a series of locations, and are classed as Drains, Sewers, and Culverts. Brains are small courses, from one or more points leading to a sewer. Ctdverts are courses that receive the discharge of sewers. Greatest fall of rain is 2 ins. per hour = 54 308.6 galls, per acre. Inclination of sewers should not be less than i foot in 240, and for house or short lateral service it should be i inch in 5 feet. Fig. i. Circular. 55 Vx zf= v, and u a = V. _ r Egg. - = w, = area of sewer -4- wetted perimeter, f inclination of sewer per mile, and v velocity of flow of contents in feet per minute ; a area of flow, in sq.feet, V volume of discharge, in cube feet per minute ; D height of sewer, w and w' width at bottom and lop, and r radius of sides, in feet. For diameter of sewer exceeding 6 feet. (T. Hawksley.) D -- w'. D diameter of a circular sewer of area required. 9 Elliptic. Top and bottom internal should be of equal diam- eters. Diameter .66 depth of culvert ; intersections of top and bottom circles form centres for striking courses connect- ing top and bottom circles. Pipes or Small Sewers. Height of section = i ; diameter of arch =r .66 ; of invert = .33, and radius of sides = i. In culverts less than 6 feet internal depth, brickwork should be 9 ins. thick ; when they are above 6 feet and less than 9 feet, it should be 14 ins. thick. If diameter of top arch = i, diameter of inverted arch = .5, and total depth = sum of the two diameters, or 1.5 ; then radius of the arcs which are tangential to the top, and inverted, will be 1.5. From this any two of the elements can be deduced, one being known. Drainage of I^ands toy 3?ipes. SOILS. Depth of Pipes. Distance apart. SOILS. Depth of Pipes. Distance apart. Coarse gravel sand Light sand with gravel Ft. Ins. 4 6 Feet. 60 Loam with gravel . . . Sandy loam Ft. Ins. 3 3 3Q Feet. 27 Light loam 3 6 J-3 Soft clav 2 O 21 Loam with clav . . . 3 2 21 Stiff clav... 2 6 1C "Velocity and <3-rade of Servers and. Drains in Cities. iWickstted.) Diam. Vel. per Minute. Grade, tin Grade MTle. Diam. Vel. per Minute. Grade, i in Grade jfXe. Diam. Vel. per Minute. Grade, i in Grade Ins. Feet. Feet. Ins. Feet. Feet. Ins. Feet. Feet. 240 36 146.7 15 180 244 21.6 42 180 686 77 6 220 65 81.2 18 180 294 18 48 1 80 784 6.8 8 220 87 60.7 24 1 80 392 '3-5 54 1 80 882 6 10 210 119 44-4 30 1 80 490 10.8 60 180 980 5-4 12 190 175 30.2 36 1 80 588 9 Area of Sewers or Pipes. An area of 20 acres, miles, etc., will not re- quire 20 times capacity of pipes for one acre, mile, etc., as the discharge from the 19 acres, etc., will not flow into the main simultaneously with that from one acre, etc. Ordinarily in this country an area of sewer or pipe that will discharge a rainfall of i inch per hour (3630 cube feet per acre) is sufficient. 092 SEWERS. Sewage. The excreta per annum of 100 individuals of both sexes and all ages is estimated at 7250 Ibs. solid matter and 94 700 fluid, equal to 1020 Ibs. per capita, and in volume 16 cube feet, to which is to be added the volume of water used for domestic purposes. A velocity of flow of from 2.5 to 3 feet per second will discharge a sewer of its sewage matter and prevent deposits. The minimum velocity should not be less than 1.3 feet per second. Surface from -which Circular Sewers -with proper Curves will discharge that Proportion of* \Vater from a ITall of One Inch in Depth per Hour which would, reach them., including City Drainage. (John Roe.) INCLINATION IN FEET. 2 D 2-5 IAMETEK 3 OF SEW 4 ERS IN ] 5 ^EKT. 6 7 8 None Acres. 08 7 K Acres. Acres. Acres. Acres. Acres. Acres. Acres. in 480 . . 4.8 108 57 5 i n 240 87 ' 1318 5 63 460 2871; 78 g rgoe in 80 jfie 1188 2486 6625 in 60. . . I2S l82 318 73O moo 267"; 4^0 7125 Surface Of a Town from which small Circxilar Drains will discharge \Vater equal in "Volume to Two Inches in Depth per Hour. (John Roe.) INCLINATION. Fall of one in. Di 3 iMETE 4 B OF 5 DRAIN 6 IN INS. 7 i 8 INCLINATION. Fall of one in. DlAMI 9 TER OF 12 DRAIN i s INS. 18 Acres. Feet. F^el Feet. Feet. Feet. Feet. Acres. Feet. Feet. Feet. Fet. 125 120 2.1 120 2 5 20 120 2-5 80 4375 40 2-75 60 5 30 80 4-5 1 20 .6 20 60 5-3 80 i 2O 60 5-8 60 240 1.3 40 20 7.8 120 1.5 20 60 120 9 80 1.8 T 80 60 10 17 60 240 1 20 Dimensions, Areas, and "Volume of Material per Lineal IToot of Egg-shaped Sewers of different Dimensions. Depth. Feet. 2.25 3 3-75 45 I' 5 6.75 L 5 5 9 INTERNAL D Diam. of Top Arch. IMKNSIONS. Diam. of Invert. Area. VOLU 4.5 Ins. thick. ME OF BRICK-TI 9 Ins. thick. FORK. 13.5 Ins. thick. Feet. Feet. Sq. Feet. Cube Feet. Cube Feet. Cube Feet. 1.5 75 2-53 2.81 2 i 4-5 3-56 2-5 1.25 7-03 4-3 1 9-56 3 IO. 12 5-o6 10.87 3-5 4 2 I3-78 18 5-8i 6.56 12.75 14.25 4-5 2.25 22.78 7-3 1 15-75 24-75 5 2-5 28.12 17.06 2 7 5-5 2-75 34-03 18 28.41 6 3 40-5 19.69 30-94 Area = product of mean diameter x height. Sewer Pipes should have a uniform thickness and be uniformly glazed, both internally and externally. Fire-clay pipes should be thicker than those of stone-clay. STABILITY. 693 STABILITY. STABILITY, Strength, and Stiffness are necessary to permanence of a structure, under all variations or distributions of load or stress to which it may be subjected. Stability of a Fixed Body Is power of remaining in equilibrio without sensible deviation of position, notwithstanding load or stress to which it may be submitted may have certain directions. Stability of a Floating Body. A body in a fluid floats, or is balanced, when it displaces a volume of the fluid, weight of which is equal to weight of body, and when centre of gravity of body and that of volume of fluid dis- placed are in same vertical plane. When a body in equilibrio is free to move, and is caused to deviate in a small degree from its position of equilibrium, if it tends to return to its original position, its equilibrium is termed Stable ; if it does not tend to de- viate further, or to recover its original position, its equilibrium is termed Indifferent and when it tends to deviate further from its original position, its equilibrium is Unstable. A body in equilibrio may be stable for one direction of stress, and unstable for another. Moment of Stability of a body or structure resting upon a plane is mo- ment or couple of forces, which must be applied in a plane vertically inclined to the body in addition to its weight, in order to remove centre of resistance of body upon plane, or of the joint, to its extreme position consistent with stability. The couple generally consists of the thrust of an adjoining struct- ure, or an arch and pressure of water, or of a mass of earth against the structure, together with the equal and parallel, but not directly opposed, re- sistance of plane of foundation or joint of structure to that lateral thrust. It may differ according to position of axis of applied couple. Couple. Two forces of equal magnitude applied to same body or struct- ure in parallel and opposite directions, but not in same line of action, consti- tute a couple. NOTE. For Statical and Dynamical Stability, see Naval Architecture, page 649. To Ascertain Stability of" a Body on a Horizontal Plane. Fig. 1. Fig. i. ILLUSTRATION. Stability of a body, A, Fig. i, when a 7) L _ ^ thrust is applied as at o, to turn it on a, is ascertained by multiplying its weight by distance a s, from fulcrum a to line of centre of gravity, c s. Hence, if cubical block weighed 10 tons and its base is 6 feet, its moment would be 10 x = 30 tons. If upper part, a b d c, was removed, remainder, a e d, would weigh but 5 tons, but its centre of gravity would be a e =. 4 feet Hence its moment would be 5 x 4 = 20 tons, although it is but half the weight To Compute Weight of a GHven Body to Sustain a Q-iven Tlirnst. F h = W. F representing thrust in Ibs. , h height of centre of gravity of body c *, and I distance of fulcrum from centre of gravity = as. ILLUSTRATION. Assume figure to be extended to a height of 20 feet, and required to be capable of resisting the extreme pressure of wind. 694 STABILITY. KEVETMENT WALLS. Pressure estimated at 50 Ibs. F = 6 X 20 X 50 = 6000 Ibs. at centre of gravity of *~turface of body. 6000 X 10 Then = 20000 Ibs. 3 NOTK i. This result is to be increased proportionately with the factor of safety due to character of its material and structure. 2. If form of body has a cylindrical section, as a round tower, the thrust of wind would be but one half of that of a plane surface. When the Body is Tapered, as Frustum of Pyramid or Cone. Ascertain centres of gravity of surface for pressure or thrust, and of body for its sta- bility, and proceed as before. Fig. 2. To Ascertain Stability of a Body on. an. Inclination. Fig. 2. ILLUSTRATION. Stability of body, Fig. 2, when thrust is applied at c, is ascertained by multiplying its weight by distance a b from fulcrum, 6, to line of centre of gravity, a g. If thrust was applied at o, stability would be ascer- tained by distance s r from fulcrum r. Angles of EcLnili"bri\xm at -which, varions Sn~bstances Avill Repose, as determined, "by a Clinometer. Angle measured from a Horizontal Plane, and fatting from a spout. Degrees Common mold. . . 37 Common gravel. . 35 to 36 Stones or Coal... 43 Degrees. Lime-dust 45 Dry sand 40 Moist sand 41 Degrees. Sand, less dry 39.6 Wheat 37 Corn 37 "Weight of a Cnbe Foot of Materials of Embankments, "Walls, and Dams. Concrete in cement. . . Stone masonry ^ Gravel. . . Loam . . 125 I2 6 Clay Marl 120 Brick 1x2 Sand 120 Revetment "Walls. When a wall sustains a pressure of earth, sand, or any loose material, it is termed a Revetment wall, and when erected to arrest the fall or subsidence of a natural bank of earth, it is termed a Face wall. When earth or banking is level with top of wall, it is termed a Scarp re- vetment, and when it is above it, or surcharged, a Counterscarp revetment. When face of wall is battered, it is termed Sloping, and when back is bat- tered, Countersloping. Thrust of earth, etc., upon a wall is caused by a certain portion, in shape of a wedge, tending to break away from the general mass. The pressure thus caused is similar to that of water, but weight of the material must be rduced by a particular ratio dependent upon angle of natural slope, which varies from 45 to 60 (measured from vertical) in earth of mean density. Or, natural slope of earth or like material lessens the thrust, as the cosine of the slope. Angle which line of rupture makes with vertical is .5 of angle which line of natural slope, or angle of repose, makes with same vertical line. When earth is level at top, its pressure may be ascertained by considering it as a fluid, weight of a cube foot of which is equal to weight of a cube foot of the earth, multiplied by square of tangent of .5 angle included between natural slope and vertical. STABILITY. REVETMENT WALLS. 695 Therefore squares of the tangents of .5 of 45 and .5 of 60 = . 1716 and 3333? which are the multipliers to be used in ordinary cases to reduce a cube foot of material to a cube foot of equivalent fluid, which will have same effect as earth by its pressure upon a wall. 3?ressu.re of Earth against Revetment "Walls. Fig. 3. Let A B C D, Fig. 3, be vertical section of a revetment HA wall, behind which is a bank of earth, A D/e ; let D o -? 6 represent angle of repose, line of rupture, or natural slope c /' which earth would assume but for resistance of wall. j\ / In sandy or loose earth angle o D A is generally 30 ; in firmer earth it is 36; and in some instances it is 45. If upper surface of earth and wall which supports it are both in one horizontal plane, then the resultant, I n, of , f pressure of the bank, behind a vertical wall, is at a dis- u -i) tance, D n, of one third A D. Line of Rupture behind a wall supporting a bank of vegetable earth is at a distance A o from interior face, A D = .618 height of it. When bank is of sand, A o = .677 h ; when of earth and small gravel == .646 h ; and when of earth and large gravel = .618 h. The prism, vertical section of which is A D o, has a tendency to descend along inclined plane, o D, by its gravity; but it is retained in its place by resistance of wall, and by its cohesion to and friction upon face o D. Each of these forces may be resolved into one which will be perpendicular to o D, and into another which will be parallel to o D. The lines c i, i I represent components of the force of gravity, which is represented by vertical line c J, drawn from centre of gravity, c, "of prism. Lines nr,lr represent compo- nents of forces of cohesion and friction, which is represented by horizontal line n /. Force that gives the prism a tendency to descend is i /, and that opposed to this is r /, together with effects of cohesion and friction. Thus, i I = r I -f cohesion -f- friction. Consequently, exact solution of prob- lems of this nature must be in a great measure experimental. It has been found, however, and confirmed experimentally, that angle formed with vertical, by prism of earth that exerts greatest horizontal stress against a wall, is half the angle which angle of repose or natural slope of earth makes with vertical. Memoranda. Natural slope of dry sand = 39, moist soil = 43, very fine sand = 21, wet clay = 14, and gravel = 35. In setting or founding of retaining walls, if earth upon which wall is to rest is clayey or wet, coefficient of friction between wall and earth falls to .3; hence it is necessary, in order to meet this, that the wall should be set to such a depth in the earth that the passive resistance of it on outer face of wall, combined with its fric- tion on its bottom, may withstand the pressure or thrust on its inner face. Moment of a Retaining Wall is its weight multiplied by distance of its centre of gravity to vertical plane passing through outer edge of its base. Moment of Pressure of Earth against a retaining wall is pressure multiplied by distance of its centre of pressure to horizontal plane passing through base of wall. Equilibrium of Retaining Wall is when respective moments of wall and earth are equal. Stability of a Retaining Wall should be in excess of its equilibrium, according to character of thrust upon it, and the line of its resistance should be within wall and at a distance from vertical passing through centre of gravity of wall, at most .44 of distance of exterior axis of wall from this line. Coefficient of Stability varies with character of earth, location, exposure to vibra- tions, floods, etc. ; hence thickness of base of wall will vary from 1.4 to 2 b. Backs of retaining walls should be laid rough, in order to arrest lateral subsidence of the filling. 696 STABILITY. KEVETMENT WALLS. When filling is composed of bowlders and gravel, the thickness of wall must be increased, and contrariwise; when of earth in layers and well rammed, it may b& Courses of dry wall should be inclined inwards, in order to arrest the flow of water of subsidence in filling from running out upon face of wall. Less the natural slope, greater the pressure on wall. Sea walls should have an increased proportion of breadth, as the earth backing is not only subjected to being flooded, but the walls have at times to sustain the weight of heavy merchandise. Buttress. An increased and projecting width of wall on its front, at intervals in its length. Counterfort. An increased and projecting width of wall at its back and at in- tervals. Coefficient of Friction of masonry on masonry .67, of masonry on dry clay .51, and on wet clay . 3. Face of wall should not be battered to exceed i to 1.25 ins. in a foot of height, in consequence of the facility afforded by a greater inclination to the permeation of rain between the joints of the courses. Footing of a wall, projecting beyond its faces, is not included in its width. Pressure. Limit of pressure on masonry 12 500 to 16 500 Ibs. per sq. foot wall. t i Thickness of Walls, in Mortar, Faces vertical. For Railways or Like Stress. Cut stone or Ranged rubble ,35 j Brick or Dressed rubble 4 When laid dry, add one fourth. Friction in vegetable earths is .5; pressure in sand .4. When vegetable earths are well laid in courses, the thrust is reduced .5. When bank is liable to be saturated with water, thickness of wall should be doubled. Centre of Pressure of earthwork, etc., coincides with centre of pressure of water, and hence, when surface is a rectangle, it is at .33 of height from base. The theory of required thickness of a retaining wall, as before stated, is, that the lateral thrust of a bank of earth with a horizontal surface is that due to the prism or wedge-shaped volume, included between the vertical inner face of the wall and a line bisecting the angle between the wall and the angle of repose of the material. To Compute Elements of Revetment Walls. Fig. 4:. n* * A Let A Do represent angle of repose of material, against -^ -j ? a wall, ABC D. ADn = .sADo. Tan. A D w = -492. / / h h 2 ! / Tan. A D n. h , or tan. ADn = V; f / 2*2 / / , = m: tan. a ADn = E; - tan. 3 ADn = S; h* h tan. A D n/^ x, and h tan. A D n */-^ = *' * representing height of watt in feet, V volume of section of prism of material A D n one foot in length in cube fttt, W and w weights of a cube foot of wall and of material, P, p, and p' lateral and moments of pressure of prisms of earth A D o and A D n upon wall, M and m moments of pressure and weight on and of wall, E and S equilibrium and stability of wall, all in Ibs., and x and x', C D/or weights of wall for equilibrium and stability. ILLUSTRATION. A revetment wall, Fig. 4, of 125 Ibs. per cube foot and 40 feet in height, sustains a bank of earth having a natural slope of 52 24', and a weight of 89. 25 Ibs. per cube foot ; what is pressure or thrust against it, etc. ? STABILITY. REVETMENT WALLS. 697 Tan. 2 Then .492 X 40 X = 393.6 cube feet. 89.25X40* X .492 2 = 17278.8 Ibs. 125 X 40 X = 230400 Ibs. ''\ X J*? = **s*f eet - , and 4 o X .49V-T^ V j XN **3 For Rubble Walls in Mortar or Dry Hubble, add respectively to base as above obtained, .14 and .42 part. NOTE i. When coefficient of friction is known, use it for tan. 2 A D n. S X C D fig. 5 = moment of stability. (Molesworth.) 2 . When either relative weights of equal volumes of wall and bank of earth or their specific gravities are given, S and s may be taken for W and w. These equations involve simply the operation of a lever, the fulcrum being at the outer edge of wall C. The moment of pressure of bank is product of lateral pressure and perpendicular distance from fulcrum to line of direction of pressure. The moment of weight of wall is product of weight of wall and perpendicular distance from fulcrum to vertical line drawn through centre of gravity of wall. When Weights of Embankment and Wall are equal per Cube Foot. C for clay = .336, and for sand .267. Wlien Weights are as 4 to 5. C for clay = . 3, and for sand .239. When Watt has an Exterior Slope or Batter. Fig. 5. . ( c D -f- E C ^j = M. M representing r moment of weight of wall in Ibs. ILLUSTRATION. Assume weight of wall 120 Ibs. per cube foot, and C D and E C respectively 10 and 2. 5 feet, and all other elements as in preceding case. Hence, ? 20X4 - J X (10+2.5 M = 3700 / 1 tan. 2 ADn nh = x. x representing A B or C D. n ratio of 3 3 * difference of widths of base and top to height. In absence of tan. 2 A D n put C, co- efficient of material. C = .0424 for vegetable or clayey earth, mixed with large gravel; .0464 if mixed with small gravel; .1528 for sand, and .166 for semi-fluid earths. ILLUSTRATION. Assume elements of preceding case, n = one fortieth, and tan. 4o\/; V3^ + T^7 x - 49 **-' = ' 2 - 6/ ^ Hence, thickness of wall at base = 12.6-)- i (one fortieth of height) = 13.6 feet. NOTE. If n = one twentieth, 40 V 3 X 20 2 3X 125 Hence, wall at base = 11.63-1-2 (one twentieth of height) = 13.63 feet. IfC was used, ii. 32 feet. 3N STABILITY. REVETMENT WALLS. When Wall has an Interior Slope or Batter, B E. Fig. 6. w h^ o E r J^_Xtan.2 lJL = Mo/ / / earth for equilibrium; /,'' 2 , j , M of wall ; and X tan. 2 o E n = M of earth for sta- bility. Coefficients for Batter of following Proportions. Base = Height X Tab. number. Weight of Earth to Wall. Weight of Earth to. Wall. BATTKR OF As 4 to 5. As i to i. BATTER OF As 4 to 5. Clay. Sand. Clay. Sand. WALL. Clay. As i to i. Sand. Clay. Sand. .083 .122 .149 .029 .065 .092 "5 155 .183 054 .092 .118 i in 8 ...... I " 12 ..... Vertical ____ .184 .221 3 .125 .16 2 39 .218 .256 .336 '53 .189 .267 To Compute Pressure Perpendicular to Baclt of "Wall. Fig. 7. \ n o P # = or , and/* at right angle to back of wall, / whether vertical or inclined. LxAn 2- n ,or L x tan. A D w, or 2 x tan. 2 ADn , or =/ *. L representing weight of triangle of em- ' bankment, as A D n. This is pressure independent of friction between surfaces of wall and earth. To Ascertain, and Compute Amount and Effect of Fric- tion of \Vall and Earth. Fig. 8. Fig. 8. Draw/ * by scale to computed pressure at right angle i. ---- i]L ----- ,0 to back of wall, draw angle/* r = mDo of natural slope - -' - -' of earth with horizon, draw/r at right angle to/ #, make r c =/ *, then c r will represent by scale effect of friction against back of wall. Assume friction to act at point #, then r * will give by scale resultant of the two forces of pressure and friction, equal to pressure in force and direction, which bears = m against wall. This resultant is also equal to/ # x sec. m D o. L x A n X sec. m D o - = r #, or - X sec. m D o, or L x tan. A D n X sec. w D o. To Ascertain Point of Moment of Pressure of a Fif. 9 ._ (P By its resisting lever la,added to its weight Weight of wall as computed assumed as concentrated at its centre of gravity Draw a vertical line . o through its centre of gravity, and con- tinue line of pressure P * to I, take any distance r o by scale rep- resenting weight of wall, and r w, by same scale, for amount of pressure or thrust against wall, complete parallelogram r o w M, then diagonal ru will give resultant of pressure in amount and direction to overturn wall. For stability this diagonal should fall inside of base at a point not less than one third of its breadth. STABILITY. EEVETMENT WALLS. 699 Surcharged. Revetments. Fig. lo. / r o When the earth stands above a wall, as A B e, ~~~ 7 Fig. 10, with its natural slope, A/, A B C is termed a Surcharged Revetment. If C r is line of rupture, A/r C is the part of earth that presses upon wall, which part must be taken into the computation, with exception of portion A Be, which rests upon wall; that is, the computation must be for part C efr, which must be reduced by multiply- ing weight of a cube foot of it by square of tangent of angle e C r = angle of line of rupture, or half angle eC o, which natural slope makes with vertical, and then proceed as in previous cases for revetments. = breadth or C D. W and w representing weights of watt and 3 ft W embankment in Ibs. per cube foot, and h' height of embankment, as C e. ILLUSTRATION. Height of a surcharged revetment, BC, Fig. 10, is 12 feet, weight 130 Ibs. per cube foot; what is its width or base to resist pressure of earth of a weight of ioo Ibs. per cube foot, and a height, C e, of 15 feet, angle of repose 45 ? Tan. 2 (45 -^ = . Then 15 = I5 ^.055 = 3 To Ascertain IPoint of MLoment of DPressnre of a Sur- charged. 'Wall. ITig. 11. F*g. ii. / Draw a line, P *, parallel to slope, C r, through centre "' of gravity of sustained backing, B C r. When, as in this case, this section is that of a triangle, point * will be at .33 height of wall. ~~ When natural slope is 1.5 in length to i in height, as with gravel or sand, w x . 64 = pressure P #. In a surcharged revetment, as/B o, at its natural slope, the maximum pressure is attained when the backing reaches to r. When slope of maximum pressure, Cnr, intersects face of natural slope, B/, so that if backing is raised to /, or above it, there is theoretically no addi- tional stress exerted at back of or against wall, but prac- tically there is, from effect of impact of vibration of a passing train, proximity to percussive action, alike to that of a trip hammer, etc. When backing rests on top of wall, as A B e. Fig. 10, small triangle of it is omitted in computations. Direction of pressure against wall is same as when wall is not surcharged. When Wall is set below Surface of Earth. Tig. 12. h Fig. 12. i. 4 tan. 45 W -=d B a representing angle of repose of earth, w and W weightt of earth and wall per cube foot, f friction of wall on base A B, and V weight of wall. ILLUSTRATION. If a wall of masonry, Fig. 12, 8 feet in thickness and 13 in height, is to sustain earth level with its upper surface, earth weighing ioo Ibs. per cube foot, weight of wall 150 Ibs. per cube foot =15600 Ibs., and angle of repose of earth 30; what should be the depth of wall below surface of earth ? Tan. 45 30 -=- 2 = . 5774, and /= . 3. ST^.x.ax.sto,.,^ /936o^ V i5 Then 1.4 X- 5774X/ = 4.o2jfeet. NOTE. Coefficient of stability is assumed by French engineers for walls of forti- fications 1.4 h, and if ground is clayey or wet/=.3- 7OO STABILITY. EMBANKMENT WALLS AND DAMS. Fig. 13. In Computing Stability of a Surcharged Wall, Fig. 13, nub- stitute dfor ft, as in following illustration. (Molesworth.) d, representing depth at distance I, = h. In slopes of i to i, d = 1.71 h; of 1.5 to j,= 1.55; of 2 to i,= 1.45; of 3 to i,= 1.31, and 4 to 1,=: 1.24. To Determine Form of* a 3?ier to Snstain equal Pressure per Unit of Surface at all its Horizontal Sections, or any Height. = a, or AN = a. A and a representing areas of sections at summit of pier and at any depth, d, measured from summit, n a number the hyp. log. of which = i -4- height, H, of a column of the material of which pier is constructed, due to required pressure, and N the number, com. log. of which = 43 ^ 3 ILLUSTRATION. Height of a pier is 20 feet, and area of section of its summit = i foot; what should be its areas at 10 feet and base? i -7- 20 = .05, and number =1.0513; i X i. 0513 10 = i. 649 feet ; and i x i.o5i3 20 = a. jig feet. Counterforts are increased thicknesses of a wall at its back, at intervals of its length. Em.T3an.ls:rn.en.t ^Walls and. Dams. Thrust of water upon inner face of an Embankment wall or Dam is horizontal. When Both Faces are Vertical, Fig. 14. Assume perpendicular embankment or wall, A B C D, Fig. 14, to sustain pressure of water, B C ef. Fig. 14. Let lei be a vertical line passing through o, centre of gravity of wall, c centre of pressure of water, dis- tance C c being = .33 B C. Draw c I perpendicular to B C ; then, since section A C of wall is rectangular, centre of gravity, o, is in its geometrical centre, and therefore D i = .5 DC. Now I D i is to be consid- ered as a bent lever, fulcrum of which is D, weight of wall acting in direction of centre of gravity, o, on arm D *, and pressure of water on arm D /, or a force equal to that pressure thrusting in direction c I. Then P x D I = P X = W x , or P = - ' . P representing pressure 3 2 2 r> O of water. NOTE. When this equation holds, a wall or embankment will just be on the point of overturning; but in order that they may have complete stability, this equation should give a much larger value to P than its actual amount. The following formulas are for walls or embankments one foot in length; for if they have stability for that length they will be stable for any other length. h 2 P iv, also W = h b W, each value being for i foot in length, which, being sub- stituted in the equations, there will result h 3 ibxhb W' AW / w w , or h 2 w = i o 2 W; b A /- = h, and A / ^r = 6. h rep- 2 2 A V V 3 w resenting depth of water and wall or embankment, which are here assumed to be equal, b breadth of wall or embankment, and W and w weights of wall and wattr per cube foot in Ibs. Which gives breadth of a wall or embankment that will just sustain pressure of the water. STABILITY. EMBANKMENT WALLS AND DAMS. 70 1 To Compute Eq.uililt>riu.m. h /==&. ILLUSTRATION i. Height of a wall, B C, equal to depth of water, is 12 feet, and re- spective weights of water and wall are 62.5 Ibs. and 120 Ibs. per cube foot; required breadth of wall, so that it may have complete stability to sustain the pressure of water. I2 / = 12 X .4166 = sfeet, breadth that will just sustain pressure of the water. Therefore an addition should be made to this to give the wall complete stability, say 2 feet; hence 5 -|- 2 := 7, required width of wall. 2. Width of a wall is 3 feet, and weight of a cube foot of it is 150 Ibs. ; required height of wall to resist pressure of fresh water to the top. H X i ^o To Compute Stability. ^ A = V 3 w ILLUSTRATION. Take elements of preceding case. Or, Divide i, 2, or 3, etc., according as the nature of the ground, the mate- rial, and the character of the thrust of the water requires, by .05 weight of material of wall, per cube foot, extract the square root of quotient, and mul- tiply result by extreme height of water. EXAMPLE. What should be the thickness of a vertical faced wall of masonry, having a weight of 125 Ibs. per cube foot, to sustain a head of water of 40 feet, and to have stability ? V(2 -r- .05 X 125) 40 = -V/-32 X 40 = 22.63 feet. Or, *i/^y = 4 V-347 2 = 23.56 feet. When Dam has an Exterior Slope or Batter, as A D. Fig. 15. Fig. 15* A B^j Assume prismoidal wall, A B C D, to sustain press- ure of water, B C ef. Draw A E perpendicular to D C ; h = B C, the top breadth A B = E C = &, and bottom breadth, D E, of sloping part, A E D =. S. Then weights of portions A C and A E D respec- tively for one foot in length are hbW and .5 W S h, these weights acting at points n and i respectively. To Compute IVIoinent. h b W X (s -^ ) = moment for A C, and x = moment for A E D. yy fa / 2 S^ Hence, (S-f-6 ) moment of dam. S representing batter or base E D. ILLUSTRATION. Height of a dam, B C, Fig. 15, is 9 feet, base C E 3, and E D 4 feet; what is its moment? A C = 9 X 3 X 120 X U + |j = 3240 X 5-5 = 17820 Ibs. 2 ' 3 Hence, 17 820 + 5760 = 23 580 Ibs. moment. Or, I2 X 9 / 4 -f 3 i-\ = 54 o x 43$ = 23 580 Ibs. moment. JQ2 STABILITY. EMBANKMENT WALLS AND DAMS. To Cornpxite Elements of "Walls or Dams with. an Exterior Batter. Fig. 15. To Compute Width of Top. When Width of Batter is Given. + _ s = 6. ILLUSTRATION. Assume height of wall 9 and batter 3 feet, and W and w 120 and 62.5 Ibs. per cube foot. To Compute Widtli of Base. When Width of Batter is Given. + = B. To Compute Width of Batter. When Width of Top is Given. + - = S. /gx^qT^_3)^ = v V 120 4 2 TFAcw JFicM s - NOTB. See table, page 708. i atmosphere or 14.723307 Ibs. per sq. inch = 30 ins. of mercury. To Compute Temperature of Steam. RULB. Multiply 6th root of its force in ins. of mercury by 177.2, sub- tract ioo from product, and remainder will give temperature in degrees. EXAMPLE. When elastic force of steam is equal to a pressure of 64 ins. of mer- cury, what is its temperature? NOTE. To extract 6th root of a number, ascertain cube root of Its square root -^64 = 8, and -g/8 = 2. Hence, 2 x 177- 2 ioo = 254. 4 t. Or, 2Q38 '^ 371. 85 = t. p rejn-esenting pressure in Ibt. per tq. inch. 706 STEAM. To Compute "Volume of "Water contained, in a given "Vol- ume of Steam. When its Density is given. RULE. Multiply volume of steam in cube feet by its density, and product will give volume of water in cube feet. EXAMPLE. Density of a volume of 16420 cube feet of steam is .000609; what is the weight of it in Ibs. ? 16 420 X .000609 = 10 = volume of water, which X 62.425 = 624.25 Ibs. To Compute Pressure of Steam in Ins. of Mercury, or L"bs. per Sq.. Inch. When Temperature is given. RULE i. Add 100 to temperature, divide sum proportionally by 177.2 for temperature of 212, and by 160 for tem- peratures up to 445 ; or, 177.6 for sea-water, and 185.6 for sea-water sat- urated with salt, and 6th power of quotient will give pressure. EXAMPLE. Temperature of steam is 254; what is its pressure? loo -{-254-:- 177.2 = 1.998, and i. 9986 = 63. 62 ins. When Ins. of Mercury are given. 2. Divide ins. of mercury by 2.037 5^6, and quotient will give pressure. When Pressure in Lbs. is given. 3. Multiply pressure by 2.037 586. To Compute Specific Q-ravity of Steam compared, -with Air. RULE. Divide constant number 829.05 (1642 x .5049) by volume of steam at temperature of pressure at which gravity is required. EXAMPLE. Pressure of steam is 60 Ibs., and volume 437 ; what its specific gravity ? 829.05-7-437 = 1.898. To Compute Volume of a Cutoe Foot of "Water in Steam. When Elastic Force and Temperature of Steam are given. RULE. To 430.25 for temperature of 212, and 332 for temperatures up to 445, add temperature in degrees ; multiply sum by 76.5, and divide product by elastic force of steam in ins. of mercury. NOTE. When force in ins. of mercury is not given, multiply pressure in Ibs. per sq. inch by 2.037 5 86 - EXAMPLE. Temperature of a cube foot of water evaporated into steam is 386, and elastic force is 427.5 ins. ; what is its volume? Assume 369 for proportionate factor. 369 + 386 x 76.5-7-427.5 = 135.1 cube feet. Or, for i Ib. of steam, 2. 519 .941 log. p log. V in cube feet. Assumep = i4.7 Ibs. 2.519 .941 log. 14.7 = 2.519 1.098 = 1.421 = log. 26.34 cube feet, which X 62.425 = 164 feet. Or, When Density is given. Divide i by density, and quotient will give volume in cube feet. To Compute Density or Specific Q-ravity of Steam. When Volume is given. RULE. Divide i by volume in cube feet. EXAMPLE. Volume is 210; what is density? i -r- 210=:. 004 761. Or, for i Ib. of steam, .941 log. p 2. 519 = log. D. When Pressure is given. Take temperature due to pressure, and proceed as by rule to compute volume, which, when obtained, proceeds as above. To Compute Volume of Steam required to raise a Qiven Volume of \Vater to any Qiven Temperature. RULE. Multiply water to be heated by difference of temperatures between it and that to which it is to be raised, for a dividend ; then to temperature of steam add 965.2, from that sum take required temperature of water for a divisor, and quotient will give volume of water. STEAM. 707 EXAMPLE. What volume of steam at 212 will raise 100 cube feet of water at 80 tO 212 ? IPO X 212 _ 6g cube feet water; or, (13.68 X 1642 212) =22 250 of steam. 212-4-965.2 212 To Compute "Volume of "Water, at any Given Temper- atui*e, that must toe Mixed -with Steam to liaise or Re- duce the Mixture to any Required Temperatiire. KULE. From required temperature subtract temperature of water ; then ascertain how often remainder is contained in required temperature sub- tracted from sum of sensible and latent heat of the steam, and quotient will give volume required. Sum of Sensible and Latent Heats for a range of temperatures will be found under Heat, pages 508 and 509. EXAMPLE. Temperature of condensing water of an engine is 80, and required temperature 100 ; what is proportion of condensing water to that evaporated at a pressure of 34 Ibs. per sq. inch ? Sum of sensible and latent heats 930.12-!- 257.6 = 1187.72. 100 80 20. Then, 1187.72 100-7-20= 54.386 to i. 2-i-T t When Temperature of Steam is given. t _ w v - * representing latent heat, T and t temperatures of steam and required temperature, w temperature of condensing water, and V volume of condensing water in cube feet. ILLUSTRATION. Temperature of steam in a cylinder is 257.6, and other elements same as in preceding example ; required volume of injection water? Latent heat of steam at 230 = 930. 12. 930. 1 2 + 257. 6 ioo 100 80 20 To Compute Temperature of "Water in. Condenser or Reservoir of a Steam-engine. H-TJ-_VXW> v-f-i = t. ILLUSTRATION. Assume elements as preceding. 930. 1 2 -f- 257. 6 -f 54. 39 X 80 _ 5539 _ IOQO 54-39 + 1 55-39 To Compute Latent Heat of Saturated Steam. 1112.5 '78 t = l. ILLUSTRATION. Assume temperature 257.6 as preceding. 1112.5 .708 X 257.6 = 930.12. To Compute Total Heat of Saturated Steam. 35 ^ H- 1081.4 = H. ILLUSTRATION. Assume temperature as preceding. .305 X 257.6 -f- 1081.4 = 1160. Elastic Force and Temperature of "Vapors of -A-lcoliol, Either, Sulphuret of Carbon, Petroleum, and Tur- pentine. Force in Ins. of Mercury. o Ins. Ins. | Ins. o Ins. o In*. ALCOHOL. ALCOHOL. ETHER. SULPHURET OF PETROLEUM. 3* 4 140 13-9 34 6.2 CARBON. 316 30 50 .86 160 22.6 54 15-3 53-5 7-4 345 44.1 60 1.23 '73 3 74 16.2 72-5 12-55 375 64 70 80 1.76 2.45 1 80 200 34-73 53 96) 24.7 no 212 3 126 OIL OF 9 3-4 212 67.5 104) 30 279.5 300 TURPENTINE. IOO 4-5 22O 78.5 120 39-47 347 606 315 3 1 20 8.1 240 111.24 I5O 67.6 357 4778 130 10.6 264 166.1 212 178 37 62.4 ;o8 STEAM. Saturated. Steam. Pressure, Temperature, Volume, and Density. ISSURE | Ii. "o * PRESSURE 2 "8J5 s -*! in g I 1 s K fl| per in g *n 1* ill Mer- g ^ ^ Q. Mer- 6 c2 O a> cury. fet Hi: Inch. cury. h o Ina. Cub. ft. i Lb. Lbs. Ins. o _ o Cub. ft. Lb. 2.04 IO2. I 112.5 330-36 .003 58 118.08 290.4 1170 7.24 138 4.07 126.3 119.7 172.08 .0058 59 120.12 291.6 1170.4 7.12 .1403 6. ii 141.6 124.6 117.52 .008 5 60 122. l6 292.7 1170.7 7.01 H25 8.14 I53-I 128.1 89.62 .on 2 61 124.19 ; 293.8 1171.1 6.9 .1447 10.18 162.3 130.9 72.66 .0138 62 126.23 294.8 1171.4 6.81 .1469 12.22 I7O.2 133-3 6l.2I .0163 63 128.26 295-9 1171.7 6.7 M93 I4-25 176.9 135-3 52.94 .0189 64 130.3 296.9 1172 6.6 .1516 16.29 182.9 137-2 46.69 .021 4 65 132.34 298 1172.3 6-49 .1538 18.32 188.3 138-8 41.79 .0239 66 *34-37 2 99 1172.6 6.41 .156 20.36 193-3 140.3 37.84 .0264 67 136.4 300 1172.9 6.32 1583 22.39 197.8 141.7 34.63 .0289 68 138.44 300.9 1173.2 6.23 .1605 24-43 202 143 31.88 .0314 69 140.48 301.9 "73-5 6.15 .1627 26.46 205.9 144.2 29-57 0338 70 142-52 302.9 1173.8 6.07 .1648 28.51 209.6 145-3 27.61 .0362 144-55 303-9 1174.1 5-99 .167 29.92 212 146.1 26.36 .03802 72 146.59 304.8 "74-3 5-9 1 . 1692 30.54 2I3.I 146.4 25-85 .0387 73 148.62 305-7 1174.6 5-83 .1714 32-57 216.3 147.4 24-32 .041 1 74 150.66 306.6 1174.9 5-76 .1736 34.61 219.6 148.3 22.96 0435 75 152.69 307-5 "75-2 5-68 1759 36.65 222.4 149.2 21.78 459 76 154-73 308.4 "75-4 5-6i .1782 38.68 225.3 150.1 20.7 .0483 77 156-77 309-3 "75-7 5-54 .1804 40.72 228 150.9 19.72 .0507 78 153.8 310.2 1176 5-48 .1826 42-75 230.6 151-7 18.84 053 1 79 160.84 3"-i 1176.3 5-41 .1848 44-79 233-1 152-5 18.03 555 80 162.87 312 1176.5 5-35 .1869 46-83 235-5 153-2 17.26 .058 81 164.91 312.8 1176.8 5-29 .1891 48.86 237.8 J53-9 16.64 .0601 82 166.95 313.6 1177.1 5-23 .1913 50.9 240.1 154.6 15.99 .0625 83 168.98 3I4-5 1177.4 1935 52.93 242.3 155-3 15.38 .065 84 171.02 3I5-3 1177.6 5-" 1957 54-97 244.4 155-8 14.86 .0673 85 I73-05 316.1 1177.9 5-05 .198 57- 01 246.4 156.4 14.37 -69 6 86 175-09 316.9 1178.1 5 .2002 59-04 248.4 i57-i .0719 87 I77-I3 3I7-8 1178.4 4-94 .2O24 61.08 250.4 157.8 13.46 743 88 179.16 318.6 178.6 4.89 .2044 63-11 252.2 158.4 13-05 .0766 89 181.2 3I9-4 178-9 4.84 .2067 65-15 254.1 158.9 12.67 .0789 90 183.23 320.2 179.1 4-79 .2089 67.19 255-9 J 59-5 12.31 .0812 9 1 185.27 321 179-3 4-74 .2111 69.22 257.6 160 11.97 0835 92 187.31 321-7 179-5 4-69 2133 71.26 259-3 160.5 11.65 .0858 93 189.34 322.5 179.8 4.64 2155 73-29 260.9 161 "34 .0881 94 191-38 323-3 180 4.6 .2176 75-33 77-37 262.6 264.2 161.5 162 11.04 10.76 .0905 .0929 95 96 I93-4I 324.1 324.8 180.3 180.5 4-55 4-5i .2198 .2219 79-4 265.8 162.5 10.51 .0952 97 197.49 325-6 180.8 4.46 .2241 81.43 267-3 162.9 10.27 .0974 98 199. 52 326.3 18 4.42 .2263 83.47 268.7 163.4 10.03 .0996 99 201.56 327-1 18 .2 4-37 .2285 85-5 270.2 163.8 9.81 .102 100 203- 59 327-9 18 .4 4-33 .2307 87-54 271.6 164.2 9-59 .1042 101 205.63 328.5 18 .6 4.29 2329 89.58 273 164.6 9-39 .1065 102 207. 66 329.1 18 .8 4-25 .2351 91.61 274.4 165-1 9.18 .1089 103 209.7 329-9 182 4.21 2373 93-65 275.8 165-5 9 .in i I0 4 211.74 33-6 182.2 4.18 2393 95-69 277.1 165.9 8.82 "33 105 213-77 331-3 182.4 4.14 2414 97.72 278.4 166.3 8.65 .1156 106 215.81. 33J-9 182.6 4.11 2435 99.76 279-7 166.7 8.48 j. 1179 107 217.84 332.6 182.8 4.07 2456 01.8 2 8l 167.1 8.31 ! .1202 108 219.88 333-3 183 4.04 2477 03-83 282.3 167.5 8.17 .1224 109 221.92 334 183-3 4 2499 05.87 283.5 167.9 8.04 i .124 6 no 223-95 334-6 183-5 3-97 2521 07.9 284.7 168.3 7.88 .1269 in 225-99 335-3 183-7 3-93 2543 09-94 11.98 285.9 287.1 1 68. 6 169 7-74 -1291 7.61 1.1314 112 "3 228.02 230.06 336 336.7 1x83.9 1184.1 2564 2586 14.01 288.2 169.3 7.48 .1336 232.1 | 337.411184.3 3-83 2607 16.05 289.3 169.7 7- 3 6 1-136 4 "5 3-8 2628 STEAM. 709 per & ESSURK in Mer- i Ik if Volume of i Lb. Density, or Weight of one Cube Foot. PR . BSSURB in Mer- cury. Temperate r. Total H*at from Water atjja 6 . fe r Density, or Weight of one Cube Foot. Lbs. Ins. Cub. ft Lb. Lbs. Ins. o Cub. ft Lbs. 116 j 236.17 338.6 1184.7 3-77 .2649 149 303-35 357-8 1190.5 2.98 3357 117 238.2 339-3 1184.9 3-74 .2652 305-39 358.3 1190.7 2.96 3377 118 240.24 339-9 1185.1 3-71 .2674 155 3I5-57 361 1191.5 2.87 .3484 119 242.28 340.5 "85.3 3-68 .2696 160 325-75 363.4 1192.2 2-79 359 120 j 244.31 341.1 1185-4 3-65 2738 165 335-93 366 1192.9 2.71 .369^ 121 246.35 341-8 1185.6 3-62 2759 170 346-11 368.2 II93-7 2.63 3798 122 I 248.38 342-4 1185.8 3-59 .278 175 356.29 370.8 1194.4 2.56 3899 I2 3 250.42 343 1186 3-56 .2801 180 366,47 372.9 1195.1 2-49 .4009 124 252.45 343-6 1186.2 3-54 .2822 185 376-65 375-3 1195-8 2-43 .4117 125 254- 49 '344- 2 u86. 4 3-5i .2845 190 386.83 377-5 1196.5 2-37 .4222 126 256.53 344-8 1186.6 3-49 .2867 397-01 379-7 1197.2 2.31 4327 I2 7 258.56 345-4 1186.8 3-46 .2889 200 407. 19 381-7 1197.8 2.26 4431 128 | 2 6o.6 346 1186.9 3-44 .2911 2IO 427-54 386 1199.1 2.16 4634 129 262.64 346.6 1187.1 2933 220 447-9 389-9 1200.3 2.06 .4842 130 264.67 347.2 266.71 347-8 187-3 187-5 3-38 3-35 2955 .2977 230 240 468.26 488.62 393-8 397-5 1201.5 1202.6 1.98 1.9 5052 .5248 132 133 268.74 348.3 270.78 348.9 187.6 187.8 3-33 .2999 .302 250 260 508.98 529-34 401.1 404-5 1203.7 1204.8 1.83 1.76 .'5660 134 135 272.81 274.85 349-5 35o.i 188 1188.2 3-29 3-27 g 270 280 549-7 570-06 407.9 411.2 1205.8 ' 1.7 1206. 8 j 1.64 .*6o8i 136 276.89 350.6 1188.3 3-25 290 590-42 414.4 1207.8 i-59 6273 137 278.92 351-2 1188.5 3-22 3101 300 610.78 4I7.5 1208.7 i-54 .6486 138 280.96 351-8 1188.7 3-2 3121 350 712.57 430.1 I2I2.6 i-33 .7498 J39 282.99 352-4 1188.9 3H2 400 8i4-37 444-9 I2I7.I 1.18 .8502 140 285.03 352-9 1189 3.16 3162 450 916.17 456.7 1220.7 1.05 9499 141 287.07 353-5 1189.2 3.14 3184 500 1018 467-5 1224 95 .049 142 289.1 354 1189.4 3-12 3206 550 1119.8 477-5 1227 87 .148 143 291.14 354-5 1189.6 3228 600 I22I.6 487 1229.9 .8 245 144 293-I7 355 1189.7 loB 325 650 I323-4 495-6 1232.5 74 342 145 295-21 355-6 1189.9 3-o6 3273 700 H25.8 504.1 I235-I .69 4395 146 297-25 356.I 1190 3-04 3294 800 1628.7 5I9-5 1239.8 .61 .6322 147 299. 28 356.7 1190.2 3-02 3315 900 1832.3 533-6 1244.2 55 8235 4 8 301.32 357-2 190.3 3- 3336 1000 2035.9 546.5 1248.1 5 2.014 Saturated Steam from 32 to S1S. (Claudel) Tem- pera- ture. PRK Mercu- ry. 5SURB. Per Sq. Inch. Weight of ICQ Cub. Feet. Volume of iLb. Tem- pera- ture. PR* Mercu- ry. 5SUHK. Per Sq. Inch. Weight of zoo Cub. Feet. Volum* of iLb. o Ins. Lbs. Lb. Cub. Feet. Ins. LI-. Lbs. Cub. Feet. 3 2 35 .181 .204 .089 .1 .031 .034 3226 2941 125 130 3-933 4-509 1.932 2.215 554 63 180.5 158-7 40 .248 .122 .041 2439 135 5-174 2.542 .714 140.1 45 .299 .147 .049 2041 140 5-86 2.879 .806 124.1 50 .362 .I 7 8 059 1695 145 6.662 3-273 .909 no 55 .426 .214 .07 1429 ISO 7-548 3.708 1.022 97.8 60 Si? 254 .082 I22O 8-535 4-193 I-I45 87-3 65 .619 304 .097 1031 1 60 9-63 4-731 1-333 75 70 733 36 .114 877.2 165 10.843 1-432 69.8 75 427 I 34 746.3 170 12.183 5^985 1.602 62.4 80 1.024 503 .156 6 4 I 175 13-654 6.708 1-774 56.4 85 1.205 592 .182 549-5 1 80 15.291 7-5" 1.97 50.8 90 95 1.41 1.647 .693 .809 .212 245 408.2 190 17.041 19.001 8-375 9-335 2.181 2.411 45-9 100 1.917 .942 .283 353-4 195 21.139 0-385 2.662 37-6 105 2.229 1.095 325 37-7 200 23.461 1.526 2-933 no "5 2-579 2.976 1.267 1.462 373 .426 268.1 234-7 205 2IO 25.994 28.753 2-77 4.127 3.225 3-543 | 2 120 3-43 1.685 .488 204.9 212 29.922 4-7 3-683 27.2 7IO STEAM. GASEOUS STEAM. When saturated steam is surcharged with heat, or superheated, it is termed gaseous or steam-gas. The distinguishing feature of this condition of steam is its uniformity of rate of expansion above 230, with the rise of its tem- perature, alike to the expansion of permanent gases. To Compute Total Heat of G-aseous Steam. 1074.6 -j- .475 t = H. t representing temperature, and H total heat in degrees. Hence, total heat at 212, and at atmospheric pressure = 1175.3. Specific gravity = .622. To Compute Velocity of Steam. Into a Vacuum. RULE. To temperature of steam add constant 459, and multiply square root of sum by 60.2 ; product will give velocity in feet per second. Into Atmosphere. 3.6 ^h = V. V representing velocity as above, and h height in feet of a column of steam of given pressure and uniform density, weight of which is equal to pressure in unit of base. ILLUSTRATION. Pressure of steam 100 Ibs. per sq. inch, what is velocity *of its flow into the air? Cube foot of water = 62. 5 Ibs. , density of steam at 100 Ibs. = 270 cube feet. Henc, 62.5 : ioo :: 270 : 432 = volume at 100 Ibs. pressure, and 432 X 144 = 62208 feet=sz height of a column of steam at a pressure q/ioo Ibs. per sq. inch. Then 3.6 -^62 208 898 feet. EXPANSION. To Compute Point of Cutting off to Attain Limit of Expansion. b-\-f!j--P = point of cutting off. b representing mean back pressure for entire stroke, in lbs.&er sq. inch, f friction of engine, P initial pressure of steam, all in Ibs per sq. inch, and L length of stroke, in feet. ILLUSTRATION. Assume stroke of piston 9 feet, pressure 30 Ibs. , mean back press- ure 3 Ibs. , and friction 2 Ibs. 3 + 2X9-:- 30 = i- 5 feet. To Compute Actual Ratio of Expansion. , *" = R. c representing clearance or volume of space between valve seat and l-\-c mean surface of piston, at one or each end in feet of stroke, I length of stroke at point of cutting off, excluding clearance in feet, and R actual ratio of expansion. ILLUSTRATION. Assume length of stroke 2 feet, clearance at each end 1.2 ins., and point of cutting off i foot. 1.2 ins. = .x. Then = i.g ratio. To Compute Pressure at any Point of Period of Ex- pansion. When Initial Pressure is given. Pl^-sp. p representing pressure at period of given portion of stroke, both in Ibs. persq. inch, and s any greater portion of stroke than I. When Final Pressure is given. P' x I/ -r- s = p. P' representing final pressure, in Ibs. per sq. inch, and L' length of stroke, including clearance, in feet. ILLUSTRATION i. Assume length of stroke 6 feet, clearance at each end 1.2 ins., pressure of steam 60 Ibs., point of cutting off one third; what is pressure at 4 feet? 1.2 ins. .ifoot. 60 x 2 + .i+- 4 + -1 30.73 Ibs. t. What is pressure in above cylinder at 2.8 feet, when final pressure is 21 Iba ? 21 X 6-4- . i -i- 2. 8 -j- . i = 44. 17 Ibt, STEAM. 711 To Compute Mean or Total Average Pressure. P (I' i + hyp. log. R c) , - - - = p or mean or average pressure. I length of stroke at Li point of cutting off, including clearance. ILLUSTRATION. Assume elements of preceding cases: i -f- hyp. log. R= 2.065. 60 (2. x X 2.065 - 1) = 254^ 6 ^ 6 6 To Compute Final Pressure. ILLUSTRATION. Assume elements of preceding cases, steam cut off at 2 feet. 60 x 2 + . i -=-6 + . i = 20.65 M S - To Compute Mean. Effective Pressure. j- V, VI V/ 7 L.;. ILLUSTRATION. Assume elements of preceding cases, b 2 Ibs. per sq. inch. _ 2 = To Compute Initial Pressure to Produce a Q-iven erage Effective or Net Pressure. _ _ _ r(i+hyp.log.R) c ILLUSTRATION. Assume elements of case i. 6 =P . __ 2-|--I (2.1X2.065).! 4.2365 To Compute Point of Cutting off for a GHven Ratio of Expansion. L'-=-R c. Or, L-f-c-=-R c I ILLUSTRATION. Assume elements of preceding cases: R = -^t-- ~ 2 .o, and '"' I 2 + .1 2.9 .1 = 2 /ee. To Compute Pressure in a Cylinder, at any Point of Ex- pansion, or at End of Stroke. ILLUSTRATION. Assume elements of preceding cases: 60 X 2. i .60 ; - = 60 Ibs.. and = 20.60 ibs. 2 + .I 2. 9 To Compute Initial Pressure for a Required Net Effec- tive Pressure for a GHven. Ratio of Expansion. 6 L - Or, P - = P. W representing net- a (I' i + hyp. log. K c) I' i + hyp. log. R c work infoot-lbs. = a L p' 6, and a area of piston, in sq. ins. ILLUSTRATION. Assume elements of preceding cases: area of piston = 100 sq. ins., back pressure 2 Ibs., and net effective pressure = 42.365 Ibs. loo X 6 X 42. 365 2 = 24 219 foot-lbs. 242194-100X2X6 _ 25419 =6Q ^ 42.365X6 __254.i 9 __ 6o ^ 100X2.1 X 2.065 . i "00X4-2365 ' 2.1X2.065.! 4.2365" f\2 STEAM. IPoints of Expansion. Relative points of expansion, including clearance 5 per cent., assuming stroke of piston to be divided as follows, and initial pressure = i. Point i .75 .6875 .625 .5625 .5 .4375 .375 .333 .25 .2 .125 .1 Ratio i 1.31 1.43 1.55 1.71 1.912.15 2.43 2.74 3.5 4.46. 7. Hyp. Log. of above Ratios. .0 1.27 1.36 1.44 1.54 1.65 1.77 1.9 2 2.25 2-43 2.79 2-95 Receiver of Compound. Volume Into which the HP cylinder exhausts, should be from i to 1.5 times the volume of it, plus that of the clearance in it, when the cranks are set at angles of 120 and 90. When the cranks are opposite (180) or very nearly so, the volume may be proportionately decreased. Pressure In a Receiver should not exceed one half that of the boiler pres- sure, and usually it is operated lower. Receiver Of a Triple compound engine need not have as great a vol- ume, as the cranks are set at angles of 120 to each other. If a receiver is insufficient in volume, the result is back pressure in the B? cylinder. If otherwise, it has too great a volume, the result is that of a material reduction of the pressure, when the exhaust port of the cylinder is opened, a consequent loss of external work and of efficiency. r J?o Compute Volume of a Receiver. Single Compound. A 2 81.5 Cranks at go . ^- = volume in cube feet. A representing area of BP cylinder, and S stroke of piston, both in sq. ins. ILLUSTRATION. Assume a compound engine having a B? cylinder of 28 ins., cranks at 90, and stroke of piston 36 ins. ; what should be volume of receiver? . 1728 1728 Triple Compound. A 2 S i Cranks at 120 --- ? volume in cube feet. 1728 ILLUSTRATION. Assume a triple compound engine having a B? cylinder of 28 ins., cranks at 120, and stroke of piston 36 ins. ; what should be volume of receiver? 1 - '1 - -^ - - =i 7 .i cube feet. - - - - - 1728 1728 (The Practical Engineer.) The practice with some is to give the receiver an equal volume with that of the cylinder from which it receives the steam. To Compute Mean. ^Pressure of Steam upon a IPiston, "by Hyperbolic J-jogari.th.ms. RULE. Divide length of stroke of a piston, added to clearance in cylinder at one end, by length of stroke at which steam is cut off, added to clearance at that end, and quotient will express ratio or relative expansion of steam or number. Find in table, logarithm of number nearest to that of quotient, to which add i. The sum is ratio of the gain. Multiply ratio thus obtained by pressure of steam (including the atmos- STEAM. 7 1 3 phere) as it enters the cylinder, divide product by relative expansion, and quotient will give mean pressure. NOTE. Hyp. log. of any number not in table may be found by multiplying a common log. by 2.302 585, usually by 2.3. Proceed by referring to table pp. 331-334. EXAMPLE. Assume steam to enter a cylinder at a pressure of 50 Ibs. per sq. inch, and to be cut off at .25 length of stroke, stroke of piston being 10 feet; what wUl be mean pressure? Clearance assumed at 2 per cent. = .2 feet. 10 + 2 = 10. 2 feet, stroke 10 -f- 4-^.2 = 2. 38 feet. Then 10. 2 -r- 2. 38 = 4. 29 rela- tive expansion. Hyp. log. 4.29 (p. 332) = 1.4563, which -f- 1 = 2.4563, and 2 ' 4s6 3 x 5<> _ 2 g 6z lbg 4.29 Relative Effect of steam during expansion is obtained from preceding rule. Mechanical Effect of steam in a cylinder is product of mean pressure in Ibs., and distance through which it has passed in feet. Effects of Expansion. (Essentially from D. K. Clark.) Back Pressure is force of the uncondensed steam in a cylinder, consequent upon impracticability of obtaining a perfect vacuum, and is opposed to the course of a piston. It varies from 2 to 5 Ibs. per sq. inch. It must be deducted from average pressure. Thus: assume pressure 60 Ibs., stroke of piston as in preceding case, and back pressure 2 Ibs. At termination of. ist, 2d, 3d, 4th, sth, and 6th foot of stroke. Pressure 60 30 20 15 12 10 Ibs. per inch. Backpressure 22222 2 " " " Effective pressure 58 28 18 13 10 8 " " " Total work done by expansion at termination of each foot or assumed division of stroke of piston is represented by hyp. log. of ratio of expansion, initial work=i. Thus, for a stroke of 10 feet and a pressure of 10 Ibs. : At end of ist, 2d, 3d, 4th, sth, 6th, yth, Sth, gth, and loihfoot. Steam is expanded ) into vols., hyp. >= .69 I.I 1.39 I.6l 1.79 1.95 2.08 2.2 2.3 log. of which... ) Initial duty i Total duty ... i 1.69 2.1 2-39 2.61 2,79 2-95 3.08 3-2 3-3 Initial duty is rep- \ resented by 10.. ] [ 10 16.9 21 23-9 26.1 27.9 29-5 30.8 32 33 Resistance for each i foot of stroke... j [= . 4 6 8 10 12 i4 16 18 20 Total effective ] duty ] |= 8 12.9 '5 15-9 16.1 15-9 15-5 14.8 14 13 Gain by expansion 061.2587.598.75101.2598.7593.7585 75 62.5 The same results would be produced if expansion was applied to a non-condens- ing engine, exhausting into the atmosphere. Again, assume total initial pressure in a non-condensing cylinder 75 Ibs. per sq. inch, expanded 5 times, or down to 15 Ibs., and then exhausted against a back press- ure of atmosphere and friction of 15 Ibs. At termination of. ist, 2d, 3d, 4th, and $ih foot of stroke. Total duty. i 1.69 2.1 2.39 2.61 " " performed... 75 126.75 157.5 179.25 195.75/00^0*. u backpressure.... 15 30 45 60 75 ' " 11 effective duty 60 96.75 112.5 "9-25 120.75 " " Gain by expansion o 61.25 87.5 98.75 101^25 per cent. From which it appears that the total duty performed by expanding steam 5 times its initial volume is full 2.5 times, or as 75 to 195.75. 30* STEAM. Relative Effect of Equal "Volumes of Steam. Relative total effect or work of steam is directly as its mean or average pressure (A), and inversely as its final pressure (B), or volume of steam condensed. If former is divided by latter, quotient will give relative total effect or work (C) of a given volume of steam as admitted and cut off at different points of stroke of piston, with a clearance of 3.125 per cent. In following computations resistance of back pressure is omitted. If this press- ure is uniform with all the ratios of expansion, it is a uniform pressure, to be de- ducted from the total mean pressure in column (A). Cut off at Pres J A) Mean. ure. & Relative Effect. Cut off at Pres (A) Mean. ure. (B) Final. (C) Relative Effect. I ^6875 .625 5625 5 I .946 .924 .889 .857 V 21 .636 .576 .501 .28 35 45 54 7i 375 33 25 .2 .125 .1 .761 .702 .628 559 435 .418 394 335 273 .224 15 !3 1-93 2.09 2-3 2.05 2.9 3.21 To Compvite Total Effective Worlz in One Strolre of I>is- ton, or as Griven Tt>y an Indicator Diagram. a P (I' i -f- hyp. log. R c) = w, and a 6 L = w'. w representing total work, and w' back pressure. NOTE. Pressure of atmosphere is to be included in computations of expansion; it is therefore to be deducted from result obtained in non-condensing engines. In condensing engines, the deduction due to imperfect vacuum must also be made, usually 2. 5 Ibs. per sq. inch. ILLUSTRATION. Assume cylinder of a condensing engine 26.1 ins. in diameter, a stroke of 2 feet, pressure of steam 95 Ibs. (8o.3-}-i4-7)persq. inch, cut off at .5 stroke, with an average back pressure of 2 Ibs. per sq. inch, and a clearance of 5 per cent. Area of piston, deducting half area of rod == 530 sq. ins. 2X5-;- too = . i clear- ance, and 2 -{-. i-7- i + .1 = 1.9 = ratio of expansion, and i -j-hyp. log. 1.9 = 1.642. Then 530 X 95 X 1. 1 X i. 642 . i 530 X 2 X 2 = 50 350 X i. 706 2120 = 83 777 Ibs. ILLUSTRATION. Assume cylinder of a non-condensing engine having an area of 2000 sq. ins., a stroke of 8 feet, steam at a pressure of 50 Ibs. (35.3 -f- 1 4-7)> cut ff a ^ .25 of stroke, and clearance .25 foot. Ratio of expansion 3.66, back pressure 17 Ibs., and i -f hyp. log. 3.66 = 2. 297. 2000 X 50(2.25 X i-}- hyp. log. 3.66 .25) = loooooX 2.25 X 1 + 1.297 .25 = 460573 foot-to*- 2000 X 17 X S = 272 ooo foot-lbs. or negative effect, and 460575 272000=188575 foot-lbs. Total Effect of One Lt>. of Expanded Steam. If i Ib. of water is converted into steam of atmospheric pressure = 14.7 Ibs. per sq. inch, or 2116.8 Ibs. per sq. foot, it occupies a volume equal to 26.36 cube feet ; and the effect of this volume under one atmosphere = 2116. 8 Ibs. x 26.36 feet = 55799 foot-lbs. Equivalent quantity of heat expended is i unit per 772 foot-lbs., 55 799 -J- 772 = 72. 3 unit*. This is effect of i Ib. of steam of a pressure of one at- mosphere on a piston without expansion. Gross effect thus attained on a piston by i Ib. of steam, generated at pressures varying from 15 to 100 Ibs. per sq. inch, varies from 56000 to 62 ooo foot-lbs. , equiv- alent to from 72 to 80 units of heat. Effect of i Ib. of steam, without expansion, as thus exemplified, is reduced by clearance according to proportion it bears to volume of cylinder. If clearance is 5 per cent, of stroke, then 105 parts of steam are consumed in the work of a stroke, which is represented by too parts, and effect of a given weight of steam without ex- pansion, admitted for full stroke, is reduced in ratio of 105 to 100. Having deter- mined, by this ratio, effect of work by i Ib. of steam without expansion, as reduced by clearance, effect for various ratios of expansion may be deduced from that, in terms of relative operation of equal weights of steam. STEAM. 7 1 J Volume of i Ib. of saturated steam of 100 Ibs. per sq. inch is 4.33 cube feet, and pressure per sq. foot is 144X100= 14 400 Ibs.; then total initial work = 14400X4-33 62 is-zfoot-lbs. This amount is to be reduced for clearance assumed at 7 per cent. Then 62 352 X ioo-r- 107 = 58273/00^65., which, divided by 772 (Joule's equiva- lent), = 75.5 units of heat. Total or constituent heat of steam of 100 Ibs. pressure per sq. inch, computed from a temperature of 212, is 1001.4 units; and from 102 (temperature of condenser under a pressure of i Ib.) the constituent heat is 1111.4 units. Equivalent, then, of net simple effect 75.5 units is 7.5 per cent, of total heat from 212, or 6.7 per cent, from 102. When steam is cut off at i .75 .5 .33 .25 .2 .125 and . i of stroke, comparative effects are as i 1.26 1.616 1.92 2.14 2.27 2.51 and 2.6. Total effects as given in table, page 718. Effect of i Ib. of steam, without deduction for back pressure or other effects, vanes from about 6oooofoot-lbs., without expansion, to about double that, or 1 20000 foot- Ibs., when expanded 3 times, cutting off at about 27 per cent, of stroke; and to about 1 50 ooo foot- Ibs. when expanded about 6 times, and cut off at about 10 per cent, of stroke. Effect of Clearance. Clearance varies with length of stroke compared with diameter of cylinder, with form of valve, as poppet, slide, etc. With a diameter of cylinder of 48 ins., and a stroke of 10 feet, and poppet valves, clearance is but 3 per cent., and with a diameter of 34 ins. and a stroke of 4.5 feet and slide valves, it is 7 per cent. ILLUSTRATION OF EFFECT. Assume steam admitted to a cylinder for .25 of its stroke, with a clearance of 7 per cent Mean pressure for i Ib. = .637, and loss by clearance = 7 -=- 100 = .07, which, added to .637, =.707, which is effect of a given volume of steam, if there was not any loss by clearance, or a gain of n per cent. When steam is cut off at i .75 .5 .33 .25 .125 and .1 stroke. Loss at 7 per cent, clearance.. =7 7.2 8.1 9.6 u 15.3 17 percent. To Compute !N"et Volume of Cylinder for Q-iven "Weight of Steam, Ratio of Expansion and One Strolte. RULE. Multiply volume of i Ib. of steam, by given weight in Ibs., by ratio of expansion and by 100, and divide product by 100, added to per cent, of clearance. EXAMPLE. Pressure of steam 95 Ibs., cut off at .5, weight .54 Ibs., volume of i Ib. steam 4.55, and weight^ .2198 Ibs., stroke of piston 2 feet, and clearance 7 per cent Ratio of expansion 2 -f- . 14 -f. i-j-. I4 ^gg. 4.55 X. 54 X.. 88 X. 00 = 46iS! = 100+7 107 To Compute "Volxime of Cylinder for Given Effect witli a Q-iven Initial Pressure and Ratio of Expansion. RULE. Divide given effect or work by total effect of i Ib. of steam of like pressure and ratio of expansion, and quotient will give weight of steam, from which compute volume of cylinder by preceding rule. EXAMPLE. Assume given work at 50766 foot-lbs., and pressure and expansion as preceding. Total work by i Ib., too Ibs. steam, cut off at .5, =by table 94200/00^6*., and by table of multipliers for 95 Ibs. = .998, which x 94200 = 94012/00^6*. Then 5iZ_ = . 54 ib s . weight of steam. 94012 STEAM. Consumption of Expanded Steam per UP of Effect per Hour. IP = 33 ooo, which X 60= i 980 ooo foot - Ibs. per hour, which -4- i Ib. steam, the quotient = weight of steam or water required per IP per hour. ILLUSTRATION. Effect of i Ib., 100 Ibs. steam, without expansion, with 7 per cent, of clearance = 58273 foot-lbs., and * 9 c gg __ ^ ^ 5 steam = weight of steam con- sumed for the effect per IP per hour. When steam is expanded, the weight of it per IP is less, as effect of i Ib. of steam is greater, and it may be ascertained by dividing i 980000 by the respective effect, or by dividing 34 Ibs. by quotient of total mean pressure by final pressure, as given in table, page 718. When steam is cut off at i .75 .5 .375 .33 .25 and .2 of stroke. our | 34 26-9 21 18.5 17.6 16 14.9/65. Hence, assuming 10 Ibs. steam are generated by combustion of i Ib. coal per IP of total effect per hour, The coal consumed per \ , , EPperhour ..... ...j-3'4 2 ' 6 9 2>I x - 8 5 *-76 x.6 1.49 to*. SATURATED STEAM. To Compute Energy and Efficiency of Saturated Steam. l-p'XaRS = X; _XD = H"; = j, Y *' h P"- ' *' - P" T ' E i9oooo- = ; = e; nlap p' = x, and - = IBP; Rp jp'a=:a5; FCx6o=/; ^ = cube feet. '9 gg _ cu i e f ee i wa t er evaporated per hour per IP. pa p a 62.5 X V and v representing volumes of mass of steam entering cylinder and of it at termination of stroke of piston; S and s volumes of i Ib. steam when admitted and when at termination of expansion ; C volume of cylinder per minute for each IIP ; R and r ratios of expansion and effective cut-off; F feed water per cube foot of vol- ume of cylinder per stroke of piston, and f per IIP per hour, all in cube feet. D den- sity or weight of i cube foot of steam at temperature of operation, in Ibs. ; p mean pressure ; p' mean back pressure ; I initial pressure ; P mean effective pressure, or energy per cube foot of volume of cylinder ; P' pressure per sq. inch or that equivalent of volume of cylinder, or pressure equivalent to heat expended per sq.foot; H""heat rejected per cube foot of steam admitted; H'" heat rejected per cube foot of volume of cylinder ; A available heat per IIP per hour ; e energy per cube foot of volume of cylinder to point of cutting off, or of steam admitted; h and h' heat expended and rejected, and X energy exerted, all per Ib. of steam and infoot-lbs. E efficiency ; x en- ergy exerted per minute and per cube foot of steam admitted ; a area of piston in sq. ins. ; I length of stroke of piston in feet, and f feed water per IIP per hour* in cube feet. ILLUSTRATIONS. Assume volume of cylinder and clearance (5 percent. = .6 inch) i cube foot, steam (86.34- 14.7) 100 Ibs. per sq. inch, cut off at .5, mean pressure by rule (page 711) 86 Ibs., and back pressure 3 Ibs. V=i. v = 2. 8 = 4.33. 5 = 8.31. p = B6. p' 3. a = 144 ins. -f 4 6i.2and ioo + 46i.2. l = 2 feet. n=i. 1^157748. STEAM. -T-J = a ratio. 33000 86-3X144 4- 33 *& 3* = -52i effective cut-off. = 2.76 cube feet. 4-33 772 X- 231 (789.1 561. 2) = 99 195 foot-lbs. 2 " 2X4-33 157 748 = 198 389/00^6*. 86 3X144 = 11952 Ibs. .1154 cube feet. 86 3 X 144 X 2 X 4- 33 = 103 504 foot-lbs. 198 389 -r- .231 103 504 X .231 = 174 479 foot-lbs. 174 479 -f- 2 = 87 239 foot-lbt. 1 5. 5 X ioo X 144 X 4- 33 = 966 456 foot-lbs. 966 456 103 504 = 862 952 foot-lbs. 966456 144X86 144X3 1980000 ^ = in 6oolbs. - -=.io7E. 2X4-33 111600 Or i 980000 X 966456 = 18 504 673 foot-lbs. 103504 33000 i X 2 X 144 X 86 3 = 23 904 foot-lbs. ^ \ 9 ^ ' = . 306 cube feet. = 18 504 (>T$foot-lb\ 23 904 _ 2 X 86 3 X 144 = 23 904 foot-lbs. 62. 5 X 103 504 .1154X2.76x60 = 19.11 cube feet. 103 504 = ii 952 foot-lbs. 33000 - = 2.761 cube feet. 2X4-33" 86X144-3X144" In illustration of connection of expenditure of available heat (A) and consumption of fuel, assume coal to have a total heat of combustion of iooooooo*/oo-/&s., cor- responding to an equivalent evaporative power under i atmosphere at 212 of 13.4 los water and efficiency of furnace . 5 ; then available heat of combustion of i Ib. coal 5 ooo ooo foot-lbs. Hence, consumption of coal per IEP in an engine of like dimensions and opera- tion with that here given would be 19223000--- 5000000 = 3.8444 Ibs. Properties of Steam of Maximum Density. (Rankine.) Per Cube Foot. L o O O O o 32 248 95 1999 158 9 68 7 221 33180 284 88740 347 197700 41 348 104 2571 167 II 760 230 38700 293 100500 356 219000 50 481 "3 3277 176 14200 239 44930 302 113400 365 242000 59 655 122 4136 185 17 oio 2 4 8 51920 3" 127500 374 266600 68 88 1 131 5178 I 94 20280 257 59720 320 143000 383 293 ioo 77 1171 140 6430 203 24020 266 68420 329 159800 39 2 321400 86 1538 I 49 7921 212 28 3IO 275 78050 338 178000 401 351600 L representing latent heat of evaporation per cube foot of vapor in foot-lbs. of en- ergy. To reduce this to units of heat divide by 772, or Joule j s equivalent. SUPERHEATED STEAM. The results attained by imparting to steam a temperature moderately in excess of that due to the volume or density of saturated steam are : 1. An increase of elasticity without a corresponding increase of water evaporated. 2. Arresting or reducing passage of water, in suspension, to cylinder (foaming), as the heat contained in that water is wholly lost without affording any elastic effect. Both of these results, by increasing effect of the steam, economize fuel. Superheated steam should be treated as a gas. The product of its pressure, p in Ibs. per sq. foot, and volume v of i Ib. of it in cube feet, in the perfectly gaseous condition, is obtained by following formula: 42 140 T-r-=pt> = 8s.44T. T temperature of steam 4-461.2, and t 32 -j- 461. 2. ILLUSTRATION. Assume temperature of steam, 327.9, superheated to 341.1. Then 42 140 X 461. 2 + 341.1 -f- 324-461.2 = 68 549 foot-lbs. Hence, as pressure of steam at 327.9= ioo Ibs. per sq. inch, and at 341.1 120. 120 -T- ioo = 1.2 to i = a gain of one fifth. * Coal of average composition, 14 133 x 772 = 10910676. STEAM. To Compute Energy and. Efficiency of Superheated Steam. In following illustrations elements are same as those in preceding cases for satu- rated steam, with addition of the steam being superheated, so that 1 = 115 lbs. t < = 338-f 461. 2= 799. 2, t' 290 + 461. 2 = 751. 2, 8 = 3.8, 5 = 7.4. Efficiency of saturated steam (p. 716) . 107, and, as above, . 109 ; hence = 1.02 to i. If, then, available heat of combustion of efficiency of furnace is assumed at 5 ooo ooo , as above, consumption of coal per IIP 18 183 486 -r- 5 oooooo ^ 3.637 Ibs. NOTE. For further illustrations Rankine's " Steam-engine, 1 ' London, 1861, p. 436. "Wire-drawing. Wire drawing of steam is difference between pressure in boiler and pressure in cylinder, and is occasioned as follows: Resistance or friction in steam-pipe to passage of steam to steam-chest and piston. Resistance of throttle- valve to passage of steam, when it is partly closed or of in- sufficient area in proportion to steam -pipe. Resistance from insufficient area of valves or ports. Mr. Clark, from his experimental investigation, declared, that resistance in a steam-pipe is inappreciable, when its sectional area is not less than . i area of piston, and its velocity not exceeding 600 feet per minute. When velocity of a piston is from 200 to 240 feet per minute, area of steam may be .04th of piston. Effect of* Expansion, xvitn. Equal "Volumes, and Effect of One Llo. of 1OO Llt)s. Pressvire per Sci. Inch. Clearance at each End of Cylinder, including Volume of Steam Openings, 7 per cent, of Stroke, and iooper cent, of Admission = i. Ratio of Ex- pansion . Initial Volume = i. Point of Cut-off. Stroke = i. TOTA Final. Initial Pressure = i. L PRESSU Mean. Initial Pressure = i. RES. Initial. Mean Pressure = i. Weight of Steam of 100 Lbs. for one Stroke per Cube Foot. ACTUAL By i Lb. of 100 Lbs. Steam. EFFECT. Per Sq. Inch per Foot of Stroke by too Lbs. Steam. Volume of Steam expended per IP of Work per Hour. Heat con- verted. Lbs. Foot-lbs. Foot-lbs. Lba. Units. X j i I .247 58273 100 34 75-5 i.i 9 .909 .996 004 .225 63850 99.6 3 1 82.7 1.18 .83 .847 .986 014 .209 67836 98.6 29.2 87.9 1.23 .8 .813 .98 O2 .201 70246 98 28.2 9 1 i-3 75 .769 .969 032 19 73513 96.9 26.9 95-2 i-39 7 .719 953 049 .178 77242 95-3 25-6 100. 1 1.45 .66 .69 .942 062 17 79555 94-2 24.9 102.9 i-54 .625 .649 925 081 .l6l 83055 92-5 23.8 107.6 1.6 .6 .625 9 I 3 095 155 85125 9 r -3 23-3 110.3 1.88 .5 532 .86 163 131 94200 86 21 122 2.28 4 439 .787 271 .108 104 466 78.7 19 132.5 2-4 375 .417 .766 35 .103 107 050 76.6 l8. 5 138.6 2.65 33 377 .726 377 093 1 12 22O 72.6 17.7 145-4 2.9 .3 345 .692 445 .085 116855 69.2 16.9 I5I-4 3-35 25 .298 .637 57 .074 124066 63.7 16 160.7 4 .2 25 .567 .764 .062 I327 7 56-7 14.9 I7I.9 4-5 .16 .322 .526 .901 055 138130 52.6 14-34 178.8 5 .14 .2 .488 .049 .049 142 1 80 48.8 13.92 184.2 5-5 .125 .182 457 .188 045 146325 45-7 13-53 l8 9 .5 5-9 .11 .169 432 315 .042 148 940 43-2 13.29 192.9 6-3 .1 159 4*3 .421 039 I5I37 4i-3 13.08 196. I 6.6 .09 .152 .398 5i3 037 152955 39-8 12.98 197.7 7 .083 143 381 .625 035 155200 38-1 12.75 2OI.I 7-8 .066 .128 .348 .874 .032 158414 34-8 12.5 205.2 8 .0625 125 342 .924 .031 159433 34' 2 11.83 206.5 STEAM. Multipliers for Actual Weight and Effect for other Pressures than IOO Lbs. Pressure Multipliers. Pressure Multipliers. Pressure Multipliers. per Sq. Inch. Weight. Actual Effect. per Sq. Inch. Weight. Actual Effect. per Sq. Inch. Weight. Actual Effect. Lbs. Lba. Lbs. 6 S .666 975 9 .901 995 130 1.28 .015 70 .714 .981 95 .952 .998 140 i-37 .022 s .986 .988 100 no I 1.09 I 1.009 150 160 1.46 1-55 .025 .031 85 .855 .991 120 HZ i. on 170 1.64 033 In this illustration, in connection with preceding table, no deductions are made for a reduction of temperature of steam while expanding, or for loss by back pressure. When steam is cut off at .0625, or one sixteenth, its expansion is 16 times, but as 7 per cent, of stroke is to be added to it (.0625 -K7) -1325 132. 5 per cent , or nearly double of 16, or only a little over 7 times, as in 3d column of table on pre- ceding page. Column 7 is product of 58 273 and ratio of total effect of equal weights of steam when expanded, or average total pressure divided by average final pressure. Thus, if steam is cut off at .5, with a clearance of 7 per cent, - X I0 ~ = '532X 100 = 53.2 1.6165, and 58273 X 1.6165 = 94200/00^65. Column 9 gives volume of steam consumed per HP per hour. Thus, assume cyl- inder to have an area of 292 sq. ins., a stroke of 2 feet, and pressure of steam 100 Ibs. without expansion. 292 X ioo x 2 = 584oo/oo<-Z&*., and 292 + 7 per cent, of stroke for clearance = . 14 ; hence, 292 X 2. 14 -- 144 = 4.34 cube feet, and weight of a cube foot of such steam = .23 Ibs., and 58400 : 4.34 X .23 :: 33000 : .564, which, x 60 minutes = 33. 84, or 34 as per table. The pressures are computed on premise that steam is maintained at a uniform pressure during its admission to cylinder, and that expansion is operated correctly to termination of stroke. Column 10 is quotient of work in foot-lbs., divided by Joule's equivalent 772. Thus, 94 200-7-772 = 122. For percentage of constituent heat, converted from 102 and 212, assume 122 as in last case : Then 122 x 9 -r- ioo = 10.98 per cent for 102, and 122 X io-f- ioo = 12.2 per cent, for 212. "Wire-drawing" will cause a reduction of pressure during admission, and clear- ance will vary from 3 to 8 per cent, according to design of valve, as poppet, long or short slide. In practice, wire-drawing of steam, and opening of exhaust before termination of stroke, involve deviations from a normal condition, for which deductions must be made, added to which there is the back pressure, from insufficient condensation in condensing engines, and from pressure of air in non-condensing engines, and com- pression of exhaust steam at termination of stroke. To Compute Grain, in. Feed "Water at High Temperature. T -|-W w = H. T and t representing total heat in steam and temperature of feed water, W and w temperature, of water blown off and fed = heat lost by blowing off, and H total heat required from fuel, all in degrees. ILLUSTRATION. Assume steam at 248, feed water 100 in one case and 150 in another, and density , and total heat at 248 = 1157; what is gain ? 1157 1004-248 ioo = 1205 = total heat required from fuel. 7 1504-248 150 = 1105 = Then H H'_ 1205 1105 H 1205 = .083 = 8.3 per cent, 720 STEAM. COMPOUND EXPANSION. Compound Expansion is effected in two or more cylinders, and is tised in three forms. ist. When steam in one cylinder is exhausted into a second, pistons of the two moving in unison from opposite ends that is, steam from top or for- ward-end of first cylinder being exhausted into bottom or after-end of the other, and contrariwise this is known as the Woolf * engine. 2d. Steam from the ist cylinder is exhausted into an intermediate vessel, or " receiver," the pistons being connected at right angles to each other. 3d. Steam from receiver is exhausted into a 3d cylinder of like volume with 2d, pistons of all being connected at angles usually of 120. The two latter types are those of the compound engine of the present time. Expansion from Receiver. The receiver is filled with steam exhausted from ist cylinder, which is then admitted to 2d, or 2d and 3d, in which it is cut off and expanded to termination of stroke. Initial pressure in ad, or 2d and 3d cylinders, is assumed to be equal to final press- ure in ist, and consequently the volume cut off in the one or the other cylinders must be equal in volume to that of ist cylinder, for its full volume must be dis- charged therefrom. Inasmuch as 3d cylinder is but a division of the ad, with addition of receiver, this engine, in following illustrations, will, for simplification, be treated as having but two cylinders. In illustration, assume ist and 2d cylinders to have volumes as i to 2, with like lengths of stroke, and that steam is cut off at .5 stroke, and equally expanded in both cylinders, the ratio of expansion in each cylinder being thus equal to their volumes. Volume received into 2d cylinder would be equal to that exhausted from ist, as- suming there would not be any diminution of pressure from loss of heat by inter- mediate radiation, etc. This is based upon assumption that expansion occurs only upon a moving piston; but in operation, expansion occurs both in receiver and in intermediate passages, as nozzles and clearances; the 2d cylinder, therefore, receives steam at a reduced pressure, increased volume, and reduction of ratio of expansion. To meet this, and attain like effects, volume of 2d cylinder must be increased in proportion to increased volume of steam and its ratio of expansion. Consequently, there is no loss of effect aside from increased volume and weight of parts by inter- mediate expansion, provided primitive ratio of expansion is maintained by giving relative increased volume to 2d cylinder. ILLUSTRATION. Assume cylinders having volumes as i and 3, initial steam of ist cylinder to be 60 Ibs. per sq. inch, stroke of piston 6 feet, cut off at one third, and clearance 7 per cent. Final pressure, as per rule, page 711, =22.62 Ibs., and pressure as exhausted into receiver, reduced one fourth, = 16.97 Ibs., assuming there is no intermediate fall of pressure. The steam, therefore, is expanded to 1.33 times volume of cylinder; a like volume, therefore, must be given to 2d cylinder, to admit of this at a like press- ure. If, therefore, the increased terminal volume of the steam in the ist cylinder was augmented, including a clearance of 7 per cent., the effect would be as follows: Volume admitted to 2d cylinder is equal to volume of ist added to its clearance, or to .33 volume of 2d cylinder added to its clearance; that is, to .33 of 107 percent., or 35.66 per cent., consisting of clearance, and 35.66 7 = 28.66 per cent, stroke of 2d cylinder. The steam exhausted into 2d cylinder thus fills less than .33 of its stroke by 4.67 (33.33 28.66). As steam is expanded from volume of ist cylinder, plus its clearance, to 2d cylinder, plus its clearance, ratio of expansion in 2d cylinder is equal to ratio of volume of both cylinders, which is 3, and joo (representing full stroke) -4- j 22.62 -E- = 3 , and final pressure = 7. 54 Ibs. per sq. inch. 28.66 + 7 3 * In 1825-28 James P. Allaire, of New York, adopted this design of engine in the steamboats Henrf Eckford, Sun, Commerce, Swiflsure, Post Boy, and Pilot Boy. STEAM, 721 AaPimfng volume of receiver, or augmented terminal volume, for expansion in 2d cylinder, to have proportions of i, 1.25, 1.33, and 1.5 times volume of ist cylinder plus J ts clearance, the relations would be as follows: Augmented terminal volumes) E *1_^ 1.25 i-33 (times volume of > I ist cylinder. I.O7 1.3-37 T 4.27 !do. do. including clear Fiil volumes in 2d cylinder) added to clearance ) Ratio of expansion in 2d cyl'r. . Intermediate reductions of ) *"/ 321 3 *'33i 3-21 2-4 .2 i- T-^/ 3-21 2.25 .25 ance. ( times volume of 3- 21 \ ist cylinder. 2 (of terminal press- 33 | ure in ist cyl'r. o 4- 52 5.65 11.31 Ibs. per sq. inch. Pressures in receiver and ini- 1 tial pressure in 2d cylinder. . j Final pressure in 2d cylinder ... 22.62 7-54 18.1 7-54 16.96 7-54 11.31 do. do. 7.54 do. do. To Compute Expansion in a Compound Engine. RECEIVER ENGINE. Ratio of Expansion. In ist cylinder, as per formula, page 710. In 2d cylinder. n ~ I r = ratio. Of Intermediate Expansion. = ratio, n representing ratio of intermediate reduction of pressure between ist and zd cylinder, to final pressure in ist cylinder, and r ratio of area of ist cylinder to that of^d. ILLUSTRATION. Assume n == 4, and r = 3. Then 4 ~ I X 3 = 2.25 ratio, and 4 =: 1.33 ratto. Total or Combined Ratio of Expansion, r R' = product of ratio ofist and zd cyl- inders by ratio of expansion in ist cylinder. As when r = 3, and R' = 2.653, then 2.653 X 3 = 7-959 total ra t io - Bence, Combined Ratio of Expansion in both cylinders. *- r R'=R". R' rep- resenting ratio of expansion in ist cylinder, and R" combined ratio. ILLUSTRATION. Assume as preceding, and R' = 2.653. Then X 3 X 2.653 = 5-969 combined ratio. To Compute KfFect for One Stroke and a Given Ratio of Expansion in ITirst Cylinder. Without Intermediate Expansion. RULE. Multiply actual ratio of ex- pansion in ist cylinder by ratio of both cylinders, and to hyp. log. of com- bined ratio add i; multiply sum by period of admission to ist cylinder plus clearance, and term product A. Divide ratio of both cylinders, less i, by ratio of expansion in ist cyl- inder ; to quotient add i ; multiply sum by clearance, and term product B. Subtract B from A, and term remainder C. Multiply area of ist cylinder in sq. ins. by total initial pressure in Ibs. per sq. inch, and by remainder C. Product is net effect in foot-lbs. for one stroke. With Intermediate Expansion. Add effect thereof to result obtained above, and by following formula : Or, I' i + hyp. log. R" c ( i -f -|p- ) a P = E. a representing area in sq. ins. , P initial pressure in Ibs. per sq. inch of ist cylinder, I' length of admission or point of cutting off plus clearance, c clearance in feet, and E effect in foot-lbs. 3? 722 STEAM. ILLUSTRATION. Assume areas of cylinders i and 3 sq. ins., length of stroke 6 feet, pressure of steam 60 Ibs. per sq. inch, cut off at 2 feet, clearance 7 per cent, and area of intermediate space, as receiver, one third volume of ist cylinder. R" = ratio of expansion in 2d cylinder X 3 X 2.653 = 5-969 hyp. log. 2.653X2.25 + 1X2.42 3 i -^-2.653 4-1 X- 42 X i X6o= 1.7865 + i X 2.42 2^-2.6534-1 X-42 X 60 = 6. 743 .737 X 60 = 360. 36 foot-lbt. ist Cylinder. Effect on piston 60 Ibs. x i inch x 2 feet ....................... =120 foot-lbt. " of clearance 60 Ibs. x 42 foot ............................ = 25.2 *j _ Total initial effect = 6oX 2 X .42 .......................... =145.2 fooTlbs. Then 145.2 x i + hyp. log. 2.653 or 1.976 ....................... =286.91 foot-lbt. Less effect of clearance ....................................... = 25.2 " Net effect on piston above vacuum line .................... = 261.71 foot-lbt. Less effect of back pressure 60-1-2.653 = 22.61, which, x 3 sq. ) , , ins. and 2 feet stroke ...................................... J >' 00 _ Net effect on piston ....................................... = 126.05/00^0*. zd Cylinder. 145.2 X i + hyp. log. 2.25 or 1.81 .............. = 262.81 foot-lbs. Effect of clearance 22.61 X 3 X. 42 ..... . ...... = 28.49 " = 234.32/00^6$. 3 6 - 37 foot-lbs. Intermediate reduction of pressure, as given at page 721, = .25 X 22.61 =5.65 Ibs. per sq. inch, which, x 3 sq. ins. and by 2 per foot of stroke, = 33.9 foot-lbs. Hence 360. 36 + 33.9 = 394. 26 foot-lbs. Or, by sum of the three results, viz. : ist cylinder. . . .................................................. 126.05 foot-lbt. Intermediate expansion ....................... . ................. 33.9 " 2d cylinder ..................................................... 234. 32 * ' 394.27/00^6*.' WOOLF ENGINE. D. K Clark. Ratio of Expansion. In ist cylinder a* per formula, page 710. In id cylinder, r ^-aj^-i-f-aj = ratio, r representing ratio of area ofist cylinder to that 0/2^, I and I' lengths of stroke and of stroke added to clearance, in ins. or feet, and x ratit value of intermediate volume. ILLUSTRATION Assume I = 6 feet, V 7 per cent. = . 42, r = 3, and x = . 333. 3X 6l2 + ' 333 Then - -^ - = 2.353, ratio of expansion in zd cylinder. Total Actual Ratio of Expansion. R' (r -p -f *) = ratio. ILLUSTRATION. Assume preceding elements, R = 2.653. Then 2.653 (3 X g-^ + .333) = 2.653 X 3.137 = 8.322, total actual ratio. Combined Actual Ratio of Expansion. R' ( r -? -f- *J -r- 1 -fa; = ratio. ILLUSTRATION. Assume preceding elements. 2 - 6 53 (3 X ^ -- 1- .333 -T- i -f- 333J * 322 = 6.242, combined actual ratio. STEAM. 723 To Attain Combined Ratio of Expansion and Final IPressvire in 2d Cylinder. Assuming four cases as taken for Receiver Engine with a clearance of 7 per cent. The relations would be as follows: 5 535 1.605 Intermediate spaces are o '333 Volume of ist cylinder o .357 Add to these 1.07, the volume of iet ) cylinder plus its clearance, and.... ] I>O 7 M2? To same values of intermediate space*! add 3, the volume of 2d cylinder, I and the sums are the final volumes [ 3 3-357 3-535 by expansion in 2d cylinder J Ratios ofexpansion in 2dcyl'rarequo- ) o tients of nnal by initial volumes. . } 2 " 8 4 2. 352 2. 2O2 Intermediate falls of pressure are, in ) parts of final pressure in ist cylinder ) ( part of volume of ist cylin- ( der plus its clearance, or, total initial volumes for ex- pansion in 2d cylinder or , times volume of ist cyl'r. ( times volume of ist cyl- 4-7 j inder. 1.07 2.14 o 5.94 The initial pressures for expansion in ) ad cylinder are .........VT. J 1 '75 23.75 17-81 Etntt, final pressures in zd eyVr are. . 8. 47 7. 57 333 7.92 .66 15-83 7.19 1.902 ratios of expansion. f of final pressure ; or, as- J snming initial pressure at I 63 Ibs., and final pressure I, at 23.75 I DS -> & ae y are 11.87 Ibt. per sq. inch. (of final pressure in ist cyl- > \ inder, or 11.87 76*. per q. inch. 6. 24 Ibs. per aj. inch. Combined Ratios in these Four Cases. ist. ist ratio of expansion 2d do. do. * 33.rP1 to 2. 653 Combined Ratio, to 2.804=2.653 X 2.804 = 7.44. ad. ist do. 2d do. 3d. ist do. 2d do. 4th. ist do. 2d do. do do. OtfAJ to 2. 653 tO 2. 352 = 2. 653 X 2. 352 = 6. 24. do do to 2.653 tO 2. 202 = 2.653X2. 202 = 5. 84. do do to 2.653 to 1.905 = 2.653 X 1-905 = 5.05. Initial effect of st< pressure, admitted to ist cylinder, for 2 feet, or one third of stroke of piston, and with a clearance of 7 per cent, or .42 feet, is as follows: Effect on piston ..... 63 X 2 feet = 126 foot-lbs. f Total initial do. in clearance . . 63 X .42 foot = 26. 46 = 63 X 2. 42 = 1 52. 46 foot-lbs. ( effect. This sum is initial effect, on which effect by expansion is computed, while it is 26.46 foot-lbs. in excess of the initial effect on the piston. The total effect, then, is computed as follows: ist case. 152.46 x (i + hyp. log. 7.44) or 3.0069 = 458.27 Net Effect. Less effect of clearance .............. 26. 46 431. 81 foot-lbs. 2dcase. 3d case. 4th case. 152.46 x (i-fhyp- log. 6.24) or 2.831 =431.47 Less effect of clearance .............. 26.46 405.01 152.46 X (i 4- h 7P- log- 5-84) or 2.7647 = 421.35 Less effect of clearance .............. 26. 46 394. 89 152.46 X (i -hhyp. log. 5.05) or 2.6294 = 399.29 Less effect of clearance .............. 26. 46 372. 83 The reductions of net effect in 2d, sd, and 4th cases are 6.2, 8.6, and 13.7 per cent of effect in ist case. To Compute Effect for One Stroke and a Q-iven Com- bined Actnal Ptatio of Expansion. RULE. To hyp. log. of combined actual ratio of expansion (behind both pistons) add i ; multiply sum by period of admission of steam to ist cylin- der, added to clearance, and from product subtract clearance. Multiply area of ist cylinder in sq. ins. by initial pressure of steam in Ibs. per sq. inch and by above remainder. Product is net effect in foot-lbs. for jone stroke. ' 724 STEAM. EXAMPLE. Assume elements of ist illustration page 723. Hyp. log. 6.24 + 1 = 2.831, which, x 2.42 = 6.85, and 6.85 .42 and remainder X 60 = 385.8/00^0$. Or, V (i -f- hyp. log. R') C x a P = E. Comparative Effect of* Steam, in Receiver and. "Woolf Engines. The effect of steam in a compound engine, without clearance and without any intermediate reduction of pressure, is the same whether operated in a receiver or Woolf engine. When, however, there is an intermediate space between the two cylinders, as a receiver, there is an intermediate reduction of the pressure of the steam, conse- quent upon the increase of its volume in the receiver; the reduction of pressure, therefore, being less rapid than with the Woolf engine, the effect is greater. In illustration, the following comparative elements of the effect of both engines Is furnished. RECEIVER. (7 per cent, clearance. ) WOOLF. Ratio of Expansion. Net Effect. ist case.... ..7.96 422.3 foot-lbs. d "' 5.97 421-55 " 3 d " 5-3i 4I7-9 6 Ratio of Expansion. Net Effect. ist case 7.64 431.71 foot-lbs. 2d " 6.24 405.11 " 3 d , " 5-84....... 394-99 " 4th " 3-98 402.78 Ma " 5-05 372-93 " From wliich it appears, that although the effect of a receiver engine is the great- est, its ratio of expansion is less than with the Woolf engine. Also, that by the addition of clearance to the pistons of each engine, the actual ratios of expansion are sensibly reduced, as compared with the ratios without clearance. INDICATOR. To Compute Mean. IPressxire "by an Indicator. ;* & ff ^ oooxo RULE. Divide atmosphere line, o o in fig- ^ ' ' ' ' ' ' ure, into any convenient number of parts, as feet of stroke of piston, and erect perpendic- ulars at each point. Measure by scale of parts (alike to that of diagram) the actual mean pressure, as defined between the two lines at top and bottom of diagram, add the results, divide sum by number of points, and 3 4 5 6 7 s 9 10 quotient will give mean pressure in Ibs. per sq. inch upon piston. EXAMPLE. Pressures, as above given, are: 35 + 35 + 35 + 34 + 3^ + 25 -f 16 -f 10 -f 8 -f 6 = 236, which, -=- 10, = 23.6 Ibs. NOTE. If it were practicable to run an engine without any load, and it some- times is, the mean pressure, as exhibited by an indicator, would be an exact meas- ure of the friction of the engine. Conclusions on Actual Efficiency of Steam. For development of highest efficiencies of steam, as used in an engine, means for protecting it from cooling and condensing in the cylinder must be employed. Super- heating of it prior to its introduction into a cylinder is probably most efficient means that may be employed for this purpose. Application to cylinder of gases hotter than it is next best means; and next is the steam-jacket. In cases of locomotive and portable engines, consumption of steam per IIP per hour is less than for that of single-cylinder condensing engines for like ratios of ex- pansion, which is due to effect of temperature of non-condensing cylinders, always exceeding 212. It is dedncible from these results that the compound engine is more efficient than the single-cylinder, and that, of the two kinds of compound engines, the receiver- engine is more efficient than the Woolf. Average consumption of bituminous coal per IIP per hour, for compound engines in long voyages, as shown by Mr. Bramwell, ranged from i. 7 to 2. 8 Ibs. (D. K. Clark,) STEAM. 725 To Compute Volume of "Water Evaporated, per X^b. of Coal. ^ = volume of water, in tts. V and v representing volume of steam and relative volume of water, in cube feet, W weight of cube foot of water, and F weight of fuel consumed, both in Ibs., and d density of water, in degrees of saturation. ILLUSTRATION. Take case at foot of page. = 449887 cube feet, u = 8q8 cube feet, W = 64.3, E = i, and F = 4061 Ibs. Gain in Fuel, and Initial Pressure of Steam required, ichen Acting Expan- sively, compared with Non-Expansion or Full Stroke. Point of Cutting off. Gain in Fuel. Cutting off. 1 Point of [Cutting off. Gain in Fuel. Cutting off. | Point of Cutting off. Gain in Fuel. Cutting off. Stroke. ft Per Cent. 22.4 32 Lbs. 1.03 1.09 Stroke. 1 -375 Per Cent. 49.6 Lbs. 1.18 1,32 Stroke. 25 1 -125 PerCent. 58.2 67.6 Lbs. 1.67 2.6 ILLUSTRATION. What must be initial pressure of steam cut off at .5, to be equiv- alent to ioo Ibs. per sq. inch at full stroke ? 100 at full stroke = ioo, and ioo X 1. 18 = 118 Ibs. To Compute Gain, in Fuel. RULB. Divide relative effect of steam by number of times the steam is expanded, and divide i by quotient ; result is the initial pressure of steam required to be expanded to produce a like effect to steam at full stroke. Divide this pressure by number of times the steam is expanded, and sub- tract quotient from i, remainder will give gain per cent, in fuel. EXAMPLE. When steam is cutoff at .5, what is gain in fuel, and what mechanical effect ? Relative effect, including clearance of 5 per cent.,= 1.64; number of times of ex- pansion = 2. 1.64-5- 2 = .82, and i -5- .82 = 1.22 initial pressure. x.22-5-2 = .6i, and i .61 =.39 per cent Mechanical effects of steam at full and half strokes are 2 1.64 = .36 difference. Hence, 1.64 : .36 :: 50 (half volume of steam used) : 10.97 per cent, more fuel to produce same effect at half stroke, compared with steam at full stroke. To Compnte Consumption of* 3Txael in a IP'irnaoe. When Dimensions of Cylinder, Pressure of Steam, Point of Cut-off, Revo- lutions, and Evaporation per Lb. of Fuel pei* Minute are given. RULE. Compute volume of cylinder to point of cutting off steam, in- cluding clearance. Multiply result by number of cylinders, by twice number of strokes of piston, and by 60 (minutes) ; divide product by density of steam at its pressure in cylinder, and quotient will give number of cube feet of water expended in steam. Multiply number of cube feet by 64.3 for salt water (62.425 for fresh), divide product by evaporation per Ib. of fuel consumed, and quotient will give consumption in Ibs. per hour. EXAMPLE. Cylinder of a marine engine is 95 ins. in diameter by 10 feet stroke of piston; pressure of steam in steam-chest is 15.3 Ibs. per sq. inch, cut off at .5 stroke; number of revolutions 14.5, and evaporation estimated at ': 5 Ibs. of salt water per Ib. of coal; what is consumption of coal per hour, when density of water is maintained at 2-32? (See Saturation, page 726.) Volume of steam at above pressure, compared with water (15.3-1-14.7), = 838. Area of 95 ins. -{-2.5 per cent, for clearance -r- 144 = 50.45 cube feet. Point of cut- ting off 5 feet-}- 2.5 per cent. = 5 feet 1.5 ins., and 50.45 X 5 feet 1.5 ins, x 14.5 X a X 60 = 449 887 cube feet steam per hour. STEAM. Hence, 449 887 -=- 838 = 536. 86 cube feet water, which, X 64.3 = 34 520 Ibs. , which -f- 8. 5 = 4061 Ibs. coal per hour. NOTE. Elements given are those of one engine of steamer Arctic, and consump- tion of clean fuel (selected) for a run of 12 days (one engine) was 3820 Ibs. per hour. Utilization of Coal in a Marina Boiler. Experiment gives from .55 to .8 per cent, of the heat developed in the combustion of coal, as utilized in the generation of steam. Ordinarily it may be safely taken at ,65. SALINE SATURATION IN BOILERS. Average sea-water contains per 100 parts : Chloride of sodium (com. salt) . .2.5; Chloride of magnesium .33.. Sulphuret of magnesium 53; Sulphuret of lime ox.. Carbonate of lime and of magnesia = 2.83 = -54 .02 Saline matter. Water , Hence, sea-water contains .0339111 part of its weight of solid matter in solution, and is saturated when it contains 36.37 parts. Mean quantity of salts, or solid matter, in solution, is 3.39 per cent, three fourths of which is common salt. Removal of Incrustation of Scale or Sediment. Potatoes, in proportion of .033 weight of water. Molasses, in proportion of 1.6 Ibs. per IP of boiler. Oak, suspended in the water, and Mahogany or Oak sawdust, and Tanner's and Slippery Elm bark, renewed frequently, according to volume of it, and the evaporation of the water. Muriate of Ammonia and Carbonate of Soda, in quantity to be determined by observation. BLOWING OFF. To Compute Loss of Heat l>y Blowing Off of Saturated Water from a Steam-boiler. E "*"* = proportion of heat lost, S T x E = heat required from fuel for water evaporated in degrees, and loss of heat per cent. S representing - ,, , . b - 1 Jii -j- t sum of sensible and latent heats of water evaporated, T temperature of feed water, t difference in temperature of water blown off and that supplied to boiler, E volume of water evaporated, proportionate to that blown off, the latter being a constant quan- tity, and represented by i. Values of E at following degrees of saturation, and volumes to be blown off: ... {j I s ! 32. | 1 f 1*1 > s 32- Jrf 1 ^ l-l ^ S 32. l ?ta | a ; > S 1-25 '35 25 35 5 4 3 2 1.6 5 I! 85 65 75 85 i-33 1.18 2 2.25 2-5 I 1.25 i .8 66 2-75 3 2.25 2 2 25 57 5 45 Thus, when water in a boiler is maintained at a density of , i volume of it U 32 evaporated, and an equal volume, or i, is to be blown off. Hence i -j- i i i =. ratio of volume evaporated to volume blown off. ILLUSTRATION i. Point of blowing off is 2 (32), pressure of steam is 15 3 Ibs., me* aurial gauge, and density of feed water i (32); what is proportion of heat lost? 8 = 11578 T=ioo. t=. 15.3 + 14.7 =250.4 100 = 150.4. E = i. STEAM. STEAM-ENGINE. 727 2. Assume point of blowing off 1.75 (32); what would be loss of heat per cent, in preceding case? E = . 75. -^^ ; = 15-9 per cent, lost by blowing off. 1157-8 ioo X. 75 + i5o-4 3. Assume elements of preceding case. What is total heat required from fuel for water evaporated ? 1157.8 ioo X -75 = 793- 35- To Compute Volume of* "Water Blown. Off to that Evaporated. When Degree of Saturation is Given. RULE. Divide i by proportionate volume of water evaporated to that blown off, or value of E as above, for degree of saturation given, and quotient will give number of volumes blown off to that evaporated. ILLUSTRATION. Degree of saturation in a marine boiler is - ; what is volume of water blown off? E = i.25. Then 1-^-1.25 = .8 blown off. Proportional Volumes of Saline Matter in Sea-water. Atlantic, South. . i in 24 Dead Sea. North. . i 2-59 Baltic i in 152 I British Channel 1 in 28 Black Sea i " 46 Mediterranean i " 25 Red Sea i " 131 | Atlantic, Equator. . i " 25 When saline matter at temperature of its boiling-point is in proportion of 10 per cent. , lime will be deposited, and at 29. 5 per cent salt. Temperature of water adds much to extent of saline deposits STEAM-ENGINE. The range of proportions here given is to meet the requirements of variations in speed, pressure, length of stroke, draught of water, etc., in the varied purposes of Marine, River, and Land practice. CONDENSING. For a Range of Pressures of* from 3O to SO Itos. (Mercu- rial Grange) per Sq.. Inch., Ciit Off at Half Stroke. Piston-rod. Cylinder or Air-pump (Wrought /row), .1 to .14 of its diam. ; (Steel\ .08 diam. ; and {Copper or Brass), .11 and .125. Condenser (Jet). .Volume, .35 to .6 of cylinder. (Surface.) Brass tubes, 16 to 19 B W G, .625 to .75 in diameter by from 5 to 10 feet in length, and .75 to 1.25 in pitch, condensing surfaces, .55 to .65 area of evaporating sur- face of boiler with a natural draught; .7 to .8 with a blower, jet, or like draught. Or, for a temperature of water of 60, 1.5 to 3 sq. feet per IIP. With a very effective and sufficient circulating pump, areas may be reduced to .5 and .6. Effect of vertical tube surface, compared to horizontal, is as 10 to 7. Air-pump (Single acting and direct connection). Volume from .15 to .2 steam cylinder. Or, 2 -75> For Double acting put 4 for 2.75. V and v representing volumes of condensing and condensed water per cube foot, and n strokes of piston per minute. Foot and Delivery Valves. Area, .25 to .5 area of air-pump. Delivery Valve (Out-board). With a Reservoir. Area from .5 to .8 Foot valve. NOTE. Velocity of water through these valves should not exceed 12 feet per second. 728 STEAM-ENGINE. Steam and Exhaust Valves. (Popnet).^-^=ai'ea for steam, asn for " 24000 ' 20000 J exhaust; (Slide). - for steam, and - for exhaust, a representing " 30000 J 22 750 J urea of steam cylinder in sq. ins., s stroke of piston in ins., and n number of revolu- tions per minute. Injection Pipes. One each Bottom and Side to each condenser; area of each equal to supply 70 times volume of water evaporated when engine is ivorking at a maximum ; and in Marine and River engines the addition of a Bilge, which is properly a branch of bottom pipe. NOTE i. Proportions given will admit of a sufficient volume of water when en- gine is in operation in the Gulf Stream, where the water at times is at temperature of 84, and volume required to give water of condensation a temperature of 100 is 70 times that of volume evaporated. 2. Velocity of flow of water through cock or valve 20 feet per second in river or at shallow draught, and 30 feet in sea or deep draught service. Feed Pump.* (Single acting, Marine), Volume, .006 to .01 steam cylinder. (River and Land), or when fresh water alone or a surface condenser is used, .004 to .007. NON-CONDENSING. If or a Range of ^Pressures of from. 5O to ISO Has. GVIerou.- rial Grange) per Sq.. Inch., Cnt Off' at Half Strolie. Piston-rod. (Wrought Iron), Diam., .125 to .2 steam cylinder. (Steel), i. to 1.6 steam cylinder. Steam and Exhaust Valves. Area is determined by rules given for them in a condensing engine, using for divisors 30 ooo and 22 750. A decrease in volume of cylinder is not attended with a proportionate decrease of their area, it being greater with less volume. Feed-pump.* (Single acting, Marine), Volume, .008 to .016 steam cylin- der. (River and Land), or where fresh water alone is used, .005 to .on". Greneral Rules. Engines. Cylinder. Thickness. ( Vertical), ^ = t ; (Horizontal), - = t ; (In- clined), divide by 2000 in a ratio inversely as sine of angle of inclination. D representing diameter of cylinder, p extreme pressure injbs per sq. inch that it may be subjected to, and t thickness in ins. Shafts, Gudgeons, Journals, etc. To resist Torsion. See rules, pp. 790, 796. Coupling Bolts. /- = d. n representing number of bolts, D diameter of shaft, d' distance of centre of bolts from centre of shaft, and d diameter of bolts, all in ins. Cross-head, Wrought-iron. (Cylinder), ~ = S, and /-r- = d, or =, b. a representing area of cylinder in sq. ins., I length of cross-head between centres of its journals in feet, and S product of square of depth d, and breadth, b, of section, both in ins. (Air-pump), = S, and as above for d and b. If section of either of them is cylindrical, for S put -\Xs X 1.7. Diam. of boss twice, and of end journals same as that of piston-rod. Sec- tion at ends .5 that of centre. * See Formulas, page 736. STEAM-ENGINE. 729 Steam-pipe. Its area should exceed that of steam-valve, proportionate to its length and exposure to the air. Connecting - rod. Length, 2.25 times stroke of piston when it is at all practicable to afford the space ; when, however, it is imperative to reduce this proportion, it may be twice the stroke. Comparative friction of long and short connecting-rods is, for length of stroke ot piston, 12 per cent, additional; twice, 3 per cent. ; and for thrice, 1.33. Neck. Diam.. i to i.i that of piston-rod. Centre of body (Horizontal), 1.25 ins. ; ( Vertical), .06 inch per foot of length of rod. With two connecting-rods or links, area of necks. 65 to. 75 area of attached piston-rod. Straps of Connecting-rods, Links, etc. (Strap), area at its least section .65 neck of attached rod ; (Gib and Key), .3 diam. of neck, width, 1.25 times, (Slot) 1.35 times (Draff) of keys .6 to .8 inch per foot. Distance of Slot from end of rod .5 diam. of pin. /P I Pins (Cranks, Beams, etc.). 3/ . 355 = d. P representing pressure or thrust of rod or beam, I length of journal in ins., and C,/or Wrought iron = 640, Cast, 560. Puddled steel, 600, and Cast, 1200. Length, 1.3 to 1.5 times their diam., and pressure should not exceed 750 Ibs. per sq. inch for propeller engine, and 1000 for side-wheel. Cranks (Wrought-iron). Hub, compared with neck of shaft, 1.75 diam., and i depth. Eye, compared with pin, 2 diam., and 1.5 depth. Web, at pe- riphery of hub, width, .7 width, and in depth .5 depth of hub ; and at periph- ery of eye, width, .8 width, and in depth, .6 depth of eye. (Cast-iron.) Diameters of Hub and Eye respectively, twice diam. of neck of shaft, and 2.25 times that of crank pin. Radii for fillets of sides of web . 5 width of web at end for which fillet is designed ; for fillets at back of web, .5 that at sides of their respective ends. Beams, Open or Trussed. Length from centres 1.8 to 2 stroke of piston, and depth .5 length. If strapped, Strap at its least dimensions .9 area of piston-rod, its depth equal to .5 its breadth. End centre journals each i, and main centre journals 2.5 times area of piston or driving-rod. This proportion for strap is when depth of beam is .5 length, as above; conse- quently, when its depth is less, area of strap must be increased; and when depth of strap is greater or less than .5 width, its area is determined by product of its 6 f Joss by oblique action of wheel blades upon the water, their slip, and thrust and drag of arms and blades as they enter and leave the water. Loss by oblique action is computed by taking mean of square of sines of angles of blades when fully immersed in the water. Loss by oblique action of blades of wheel of steamer Arctic, when her wheels were immersed 7 feet 9 ins. and 5 feet 9 ins., was 25.5 and 18.5 per cent., which was the loss of useful eftect of the portion of total power developed by engines, which was applied to wheels. Feathering. Loss of effect is confined to thrust and drag of arms and blades as they enter and leave the water. Comparative Effects. In two wheels of a like diameter (26 feet, and 6 feet immer- sion), like number and depth of blades, etc., the losses are as follows : Radial 26.6 per cent. | Feathering 15.4 per cent. Loss of effect by thrust and drag in a feathering wheel, having these elements and included in the above given loss, is computed at 2 per cent. Relative loss of effect of the two wheels is, approximately, for ordinary immer- sions, 20 and 15 per cent, from circumference of wheel. 2 d* d' 3 Centre of Pressure, _ d = c. d and d' representing depths of blades below surface of water, and c centre of pressure, all in like dimensions, from bottom edge. In the cases here given, centres of pressure are as follows: Radial 6.4 ins. | Feathering 8.5 ias. Propellers. Propellers (Screw). Pitch should vary with area of circle described by screw to area of midship section of vessel. AREA, TWO-BLADED. Area of disk of propeller to mid- ) , ship section being i to } Ratio of pitch to the diameter of ) ~~T~ propeller is i to } ' 8 '- 02 *" - 2 *-*7 '-3' M '-47 For Four-bladed screws, multiply ratio of pitch to diam. as given above, by 1.35. Length, .166 diam. STEAM-ENGINE. 731 Slip. Slip of a screw propeller is directly as its pitch, and economical effect of a screw is inversely as its pitch j greater the pitch less the effect. An expanding pitch has less slip than a uniform pitch, and, consequently, is more effective. To Compute Thrust of a Propeller. IIP - = T. S representing speed of vessel in knots per hour. SLIDE VALVES. All Dimensions in Inches. To Compute Lap required on Steam End, to Cut-off at any given IPart of* Stroke of Piston. RULE. From length of stroke subtract length of stroke that is to be made before steam is cut off; divide remainder by stroke, and extract square root of quotient. Multiply this root by half throw of valve, from product subtract half lead, and remainder will give lap required. EXAMPLE. Having stroke of piston 60 ins., stroke of valve 16 Ins., lap upon ex- haust side .5 in. = one thirty-second of valve stroke, lap upon steam side 3.25 ins., lead 2 ins., steam to be cut off at five sixths stroke; what is the lap? 60 |< of 60 = 10. / = .408. .408 x * 3.264, and 3.264 -j = 2.264 * To Ascertain Lap required on Steam End, to Cut-off at various Portions of Stroke. Distance of piston from end of its stroke when steam is cut off, in parts of length of its stroke. Valve without Lead. Lap in parts of) stroke j 354 323 .286 .27 25 A t 1 A A .228 .204 "77 '44 102 ILLUSTRATION. Take elements of preceding case. Under is 204, and .204 X 16 = 3. 264 ins. lap. When Valve is to have Lead. Subtract half proposed lead from lap as- certained by table, and remainder will give proper lap to give to valve. If, then, as last case, valve was to have 2 ins. lead, then 3.264 2-7- 2 = 2.264 ins - To Compute at what Part of* Stroke any given Lap on Steam Side will Cvat off. RULE. To lap on steam side, as determined above, add lead ; divide sum by half length of throw of valve. From a table of natural sines (pages 390- 402) find the arc, sine of which is equal to quotient ; to this arc add 90, and from their sum subtract arc, cosine of which is equal to lap on steam side, divided by half throw of valve. Find cosine of remaining arc, add i to it, and multiply sum by half stroke, and product will give length of that part of stroke that will be made by piston before steam is cut off. EXAMPLE. Take elements of preceding case. Cos. (sin. *^i- 2 +9 o-cos. i) + , X |= , which, X 2 for i revolution, = .0264 IP per revolution. To Compvite Volume of "Water required to "be Evapo- rated in an Engine. RULE. Multiply volume of steam expended in cylinder and steam-chests by twice number of revolutions, and multiply product by density of steam at given pressure. * f For reference see ist and zd foot-note on previous page. STEAM-ENGINE. 735 EXAMPLE. What volume of water will an engine require to be evaporated per revolution, diam. of cylinder being 70 ins., stroke of piston 10 feet, and pressure of steam 34 Ibs. per sq. inch, including atmosphere, cut off at .5 of stroke? Area of cylinder = 3848. 5 ins. 10X12^-2 = 60 ins. , 60 X 3848. 5 = 230910 cube ins. Add, for clearance at one end, volume of nozzle, steam-chest, etc., 17 317 cube ins. Then 230910+ 17 317-=- 1728 X 2 = 287.3 cube feet, which, X .001 336, density of steam at 34 Ibs. pressure (see Note 2), = .3838 cube feet. NOTE i. This refers to expenditure of steam alone; in practice, however, a large quantity of water "foaming," differing in different cases, is carried into cylinder in combination with the steam ; to which is to be added loss by leaks, gauges, etc. 2 . Volume of steam is readily computed by aid of table, pp. 708-9. Thus, den- sity or weight of one cube foot of steam at above pressure = .0835 Ibs. Hence, as 62.5 Ibs. : i cube foot :: .0835 Ibs. : .001 336 cube foot. To Compute "Volume of Circulating "Water required toy an jiigine. " I4 /"-3 _ v T representing temperature of steam upon entering the con- denser, 1. 1', and t" temperatures of feed water, of water of condensation discharged, and of circulating water, all in degrees. ILLUSTRATION. Assume exhaust steam at 8 Ibs. per sq. inch, temperatures of dis- charge 100, feed water 120, and sea- water 75. Temperature at 8 Ibs. pressure = 183. II14 3 ~ I2 = 41.95 times. 100 75 To Compute Volume of Flow through an Injection Pipe. RULE. Multiply square root of product, of 64.33 an d depth of centre of opening into condenser, from surface of external water, added to height of a column of water due to vacuum in condenser, all in feet, by area of opening in sq. ins. ; and .6 product, divided by 2.4 (144 -j- 60) will give volume in cube feet per minute. EXAMPLE. Diameter of an injection pipe is 5.375 ins., height of external water above condenser 6.13 feet, and vacuum 24.45 ' ns - > what is volume of flow per min.? Area of 5.375 ins. = 22.69 ins -> c = .6. Vacuum 24 ' 45 1DS ' = 12 Ibs. : 12 X 2.24 2.04 feet (sea- water) = 26. 88 feet, and 26. 88 -}- 6, 13 = 33. i feet. Then V6 * 33 X 3 3^ x 2 igX^ = g^I5 = 26l . 73 ^ tf ^ To Compute Area of* an Injection Pipe. RULE. Ascertain volume of water required by rule, page 706, in cube ins. per second, multiply it by number of volumes of water required for con- densation, by rule, page 707, divide it by velocity due to flow in feet per second, and again by 12, and quotient will give area in sq. ins. EXAMPLE. An engine having a cylinder 70 ins. diam., stroke of piston 10 feet, revolutions per minute 15, and steam 19.3 Ibs., mercurial gauge cut off at .5; what should be area of its injection pipe at its maximum operation ? Volume of cylinder 267.25 cube feet, cut off at .5 = 133.625 ins. Density of steam at 34 Ibs. (19.3 -f 14-7) = -ooi 336. Velocity of flow of injected water (computed from vacuum and elevation of condensing water) 33 feet per second. Thea 133.625 X 15 X 2 X 1728 -4- 60 == 115452 cube ins. steam per second, and 115 452 X .001 336 = 154.24 cube ins. water per second. Maximum volume of water required to condense steam is about 70 times volume of that evaporated, which only occurs in the Gulf of Mexico; ordinary requirement is about 40 times. 154.24 -f- 11.59 (= 7-5 Per cent, for leakage of valves, etc.) = 165.83, which, X 70 as above, = n 608. i cube ins., and n 608. 1-^-33 X 12 = 29.31 sq. ins. 73 6 STEAM-ENGINE. Coefficient of velocity for flow under like conditions =. .6; hence, 29.31-=-. 6 = 48.8555. ins. NOTE. This is required capacity for one pipe. It is proper and customary that there should be two pipes, to meet contingency of operation of one being arrested. To Compute Area of* a Feed. 3?ump. (Sea-water. ) RULE. Divide volume of water required in cube ins. by number of single strokes of piston, both per minute, and divide quotient by stroke of pump, in ins. ; multiply this quotient by 6 (for waste, leaks, "running up," etc.), and product will give area of pump in sq. ins. EXAMPLE. Assume volume to be 5 cube feet and revolutions of engine 15 per minute, with a stroke of pump of 3.5 feet. 5 X 1728 15 = 576, which -7-3. 5 x 12 = 13.72, and 13.72 X 6 = 82.32 sq. ins. NOTB. In fresh water, this proportion may be reduced one half. STEAM-INJECTOR. "Wm. Sellers teaminLb8. II 120 | 180 || I 60 'ressure of 90 team in Lb I2O a. 180 4-3 5-4 6-5 7-5 Cub. feet. % 129 172 Cub. feet. 67 107 '54 206 Cub. feet. 75 121 I 7 6 234' Cub. feet. 69 137 199 265 8-5 9-5 10.5 n-5 Cub. feet. 221 2 7 6 338 405 Cub. feet. 265 332 407 446 Cub. feet. 301 376 460 55i Cub. feet. 340 425 520 623 Highest admissible temperature of water supply at 120 Ibs. steam, 138. Minimum capacity, 36$ to 40$ of maximum. To Compute Size of* Injector required.. One H 3 per hour will require from 15 to 40 Ibs. of water per hour, accord- ing to character of engine. When the Ibs. of coal burned per hour can be ascertained, divide them by 7.5, and quotient will give the volume of water in cube feet per hour. When the area of grate-surface is known, multiply it by 1.6 for IP. In case of plain cylindrical boilers, divide the number of sq. feet of heat- ing-surface by 10 for the IP. In case of flue boilers, divide by 12, and with multi-tubular boilers, by 15, for the nominal IP. To Compute "Volume of* Injection "Water required, per IH* per Hour. OPERATION. Assame temperature of water 80, and of condensation 100. Vol- ume of cylinder per IIP as per formula, page 716, and illustration, page 717, = 2.76 feet per minute. Then, as per rule page 707, " 4 * "V = 52-3 cube ins. per cube foot of steam. nnnu . 2.76X 52.3 X 62.5 1728 = 5.22 Ibs., which, X 60, = 313.2 Ibs. Xo Compute Net "Volume of* Feed Water required per IH per Hour. OPERATION. Assume elements of formula, page 716, and illustration, page 717. Then. 1154X2. 76X60 = 19. ii Ibs. Feed Pipes. Vv = diameter for small, and \/v, for large pumps. d representing diameter of plunger in ins., and v its velocity in feet per minute. STEAM-ENGINE. 737 Results of Operations of S team-engines. (D. K. Clark.) CONDENSING ENGINK. Actual Ratio of Expan- sion. Steam iF. cut-oflF. Coal S. Initial Pressure at cat-off. Steam perlJP per hour, SINGLE. Corliss Saltaire c 2 Lbs. 14 51 Lba. 2 5 Lbs. 04. e Lbs. 17 A * O.O7 14.27 2 2 1* 18.7 " East London 3 62 12 Q2 IO 3. o 20.72 19.6 A T-32 fe 18 62 COMPOUND. J Elder & Co J Receiver 1.8.5 14-45 i 61 c6 - J. fciaer & u> j Marine, jacketed J &E.Wood. ., . ( R e ceiver 1.852 4.01 14.85 10.94 85.5 * {stationary. n ,, ( Woolf, stationary 1.857 2.486 13-34 13.18 50-5 S =-S America, Woolf {S h cylinder 3.221 2.31 13-87 actual 9 22.51 15-37 " Jacketed { t ;; 5-03 3-77 9IO 20.71 90 14.1 NON-CONDENSING. 4. 8 16 87 76 25 Q e 14. Q3 70 2Q 6 A Si 2.Q4 21. 24. _ 87 21.24 CYLINDERS. | = e. * -O CYLINDERS. Most Efficient Ratio of Ex- pansion. M ji CONDENSING. Single cylinder, jacketed. . . 6 4 4 6 Lbs. 19-5 SM Compound, jacketed, Woolf Compound, Woolf. 10 7 4 3 Lbg. 20.5 23 24 21 " " superheated Compound, jacketed, Re-) NON-CONDENSING. Single cylinder, t jacketed. . Single cylinder. t... * From boiler. f 70 Iba. pressure. % 90 Ibs. pressure. Standard Operation of a Portable Engine. Grate 5.5 sq. feet Heating surface 220 " ** Coal per IP per hour. ... 6.25 Ibs. " u sq. foot of grate. 9 " " " hour 50 " Water evaporated from ) and at 212 per hour. } * 44 u per IP per hour 62.5 '*-* u | fi.E STEAM-ENGINE. 2* rf K m^ ^ g | gj 9 H | i .s * S ^ o-S 4 || M M M M M H ^ s. discharge o ch stroke-p |l i . |]| HS 3 3 in in 'S hr> ooi/iir)t>Nin mm m m mmgj~ ; "a S ^ gf| I fell coroint>. inmin inmminm mmm mm y**" 3 6 - | * & P Q ||{ H ^ * * .^ a* i || m m mm m mm 92 S vo oo* vo co co m indcToo w N N t^ o O " ^ ^ c_, S be fi *S|! o || OF DIREC r age in the su a uniform hei S "= -o vi & I*:* * Will S ' s^S^I 6C lUl iw vOOOOOinOOOQOOOOOOOOOOOOOO eS ' HdMtHNrO^J- -^-^O vOvOvOOOOOOOCO N CM MvO >>-rt M H H M ^ ^ S_OOOOOOOQOOOOOOOO_O OO P O 2 "2 M * * H ^ ^_J II ||| a FUS S> OH v C O . as Sfe OOOOOOOminininLnminmininininininininm j^ Pn S ,S Sfl ^ 1 !lSl 3323322222233222222,23222 |g Q ^ -S N 1 JNi S || n 1^ 1 jiiii i^KS^i^^^ll SU l W s ^> g 5~- " So d IH *ii .? \3 a > m3 M ro -4- m\o t^.r>.r>.O>O N ^cs -^-vo oo M -^vo oo ^vo oo oo g ^j 1 STEAM-ENGINE. BOILER. 739 BOILER. Its efficiency is determined by proportional quantity of heat of com- bustion of fuel used, which it applies to the conversion of water into steam, or it may be determined by weight of water evaporated per Ib. of fuel. In following results and computations, water is held to be evaporated from stand- ard temperature of 212. Proportion of surplus air, in operation of a furnace, in excess of that which is chemically required for combustion of the fuel, is diminished as rate of combustion is increased; and this diminution is one of the causes why the temperature in a furnace is increased with rapidity of combustion. When combustion is rapid, some air should be introduced in a furnace above the grates, in order the better to consume the gases evolved. Naturcd Draught. Grate (Coat) should have a surface area of i sq. foot for a combustion of 15 Ibs. of coal per hour, length not to exceed 1.5 times width of furnace, and set at an inclination toward bridge-wall of i to 1.5 ins. in every foot of length. When, however, rate of combustion is not high, in consequence of low ve- locity of draught of furnace, or fuel being insufficient, this proportion of area must be increased to one sq. foot for every 12 Ibs. of fuel. Width of bars the least practicable, spaces between them being from .5 to .75 of an inch, according to fuel used. Anthracite requiring less space than bituminous. Short grates are most economical in combustion, but generate steam less rapidly than long. Level of grate under a plain cylindrical boiler gives best effect with a fire 5 ins. deep, when grate is but' 7.5 ins. from lowest point. Depth, Cast-iron, .6 square root of length in ins. (Wood), their area should be 1.25 to 1.4 that for coal. Automatic (Vicar's). Its operation effects increased rapidity in firing and more effective evaporation. Ash-pit. Transverse area of it, for a combustion of 15 Ibs. of coal per hour, 2 to .25 area of grate surface for bituminous coal, and .25 to .3 for anthracite. Or 15 to 20 ins. in depth for a width of furnace of 42 ins. Furnace or Combustion Chamber. (Coal) Volume of it from 2.75 to 3 cube feet per sq. foot of grate surface. ( Wood) 4.6 to 5 cube feet. The higher the rate of combustion the greater the volume, bituminous coal requiring more than anthracite. Velocity of current of air entering an ash-pit may be estimated at 12 feet per second. Volume of air and smoke for each cube foot of water converted into steam is, from coal, 1780 to 1950 cube feet, and for wood, 3900. Rate of Combustion. In Ibs. of coal per sq. foot of grate per hour. Cornish Boilers, slowest, 4 ; ordinary, 10. Stationary, 12 to 16. Marine, 16 to 24. Quickest: complete combustion of dry coal, 20 to 23; of caking coal. 24 to 27 ; Blast or Fan and Locomotive, 40 to 120. Bridge-wall (Calorimeter). Cross-section of an area of 1.2 to 1.6 sq. ins. for each Ib. of bituminous coal consumed per hour, or from 18 to 24 sq. ins. for each sq. foot of grate, for a combustion of 15 Ibs. of coal per hour. Temperature of a furnace is assumed to range from 1500 to 2000, and volume of air required for combustion of i Ib. of bituminous coal, together with products of combustion, is 154.81 cube feet, which, when exposed to above temperatures, makes volume of heated air at bridge- wall from 600 to 750 cube feet for each Ib. of coal consumed upon grate. 740 STEAM-EM GINE. BOILER. Hence, at a velocity of draught of about 12 feet per second, area at bridge- wall, required to admit of this volume being passed off in an hour, is 2 to 2.5 sq. ins., and proportionately for increased velocity, but in practice it may be 1.2 to 1.6 ins. When 20 Ibs. of coal per hour are consumed upon a sq. foot of grate, 20 X 1.2 or 1.6 = 24 or 3 2 S( l- ms - are required, and in a like proportion for other quantities. Or, When area of flues is determined upon, and area over bridge-wall is required, it should be taken at from .7 to .8 area of lower flues for a natural draught, and from .5 to .6 for a blast. When one half of tubes were closed in a fire-tubular marine boiler, the evapora- tion per Ib. of coal was reduced but 1.5 per cent. Firing. Coal of a depth up to 12 ins. is more effective than at a less depth. Admission of air above the grate increases evaporative effect, but diminishes the rapidity of it. Air admitted at bridge-wall effects a better result than when admitted at door, and when in small volumes, and in streams or currents, it arrests or pre- vents smoke. It may be admitted by an area of 4 sq. ins. per sq. foot of grate. Combustion is the most complete with firings or charges at intervals of from 15 to 20 minutes. With a fuel economizer (Green's) an increased evaporative effect of 9 per cent, has been obtained. When external flues of a Lancashire boiler were closed, evaporative power was slightly increased, but evaporative efficiency was decreased; and when 25 per cent, of like surface in setting of a plain cylindrical boiler was cut off. evaporation was reduced but 1.5 per cent. When temperature at base of chimney was 630, with a fire 12 ins. in depth, it was decreased to 556 with one 9 ins. in depth, and to 539 with one 6 ins. High wind increases evaporative effect of a furnace. Stationary or Land. Set at an inclination downward of .5 inch in 10 feet. Smoke Preventing. A. test of C. Wye Williams's design of preventing smoke, at Newcastle, 1857, as reported by Messrs. Longridge, Armstrong, and Richardson, gave an increased evaporative effect with the "practical prevention of smoke. " Hence it was concluded, " That by an easy method of firing, combined with a due admission of air in front of furnace, and a proper arrangement of grate, emission of smoke may be effectually prevented in ordinary marine multi-tubular boilers, with suitable coals. 2d. That prevention of smoke increases economic value of fuel and evaporative power of boiler. 3d. That coals from the Hartley district have an evaporative power fully equal to that of the best Welsh steam-coals. " Heating Surfaces. Marine (Sea-water). Grate and heating surfaces should be increased about .07 over that for fresh water. Relative Value of Heating Surfaces. Horizontal surface above the flame = i I Horizontal beneath the flame = . i Vertical = .5 | Tubes and flues =.56 Minimum "Volumes of* Fuel Consumed per Sq. Foot of GS-rate per Hour, for given Stirface-ratios. (D. K. Clark.) DESCRIPTION OF BOILER. 10 15 Si 20 irface-r 30 atios of 40 Seating J So Surface t 60 o Grate. 75 90 100 Stationary Lbs. 7 7 .2 3 4 Lbs. \:l 4 7 i Lbs. is .8 1.3 1.8 Lbs. 6.8 6-3 1.8 2.9 4 Lbs. 12. 1 II. 2 3-2 5-2 7 Lbs. 18.9 17-5 II Lbs. 26 24 11.7 16 Lbs. isT 3 25 Lbs. I" 3 36 Lba. 32-5 44 Marine Portable Locomotive (coal) . " (coke). At extreme consumption of fuel (120 Ibs.) coke will withstand disturbing effect of a blast better than coal. STEAM-ENGINE. BOILER. 74! A scale of sediment one sixteenth of an inch thick will effect a loss of 14.7 per sent, of fuel. One sq. foot o{Jlre surface is held to be as effective as three of heating. Relation of Grrate, Heating Surface, and. Fuel. When Grate and Heating Surface are constant, greater the weight of fuel consumed per hour, greater the volume of water evaporated ; but the volume is in a decreased proportion to fuel consumed. In treating of relations of grate, surface, and fuel, D. K. Clark, in his valuable treatise, submits, that in 1852 he investigated the question of evaporative perform- ance of locomotive-boilers, using coke; and he deduced from them, that, assuming a constant efficiency of fuel, or proportion of water evaporated to fuel, evaporative effect, or volume of water which a boiler evaporates per hour, decreases directly as grate-area is increased; that is to say, larger the grate, less the evaporation of water, at same rate of efficiency of fuel, even with same heating surface. 2d. That evaporative effect increases directly as square of heating surface, with same area of grate and efficiency of fuel. 3d. Necessary heating surface increases directly as square root of effect viz., for four times effect, with same efficiency, twice heating surface only is required. 4th. Necessary heating surface increases directly as square root of grate, with same efficiency; that is, for instance, if grate is enlarged to four times its first area, twice heating surface would be required, and would be sufficient, to evaporate same vol- ume of water per hour with same efficiency of fuel. Result of 40 experiments with a stationary boiler (fresh water), with an evaporation of 9 Ibs. water per Ib. of fuel consumed, the coefficient .002 22 was deduced. Hence, ( \ .00222 = W. W representing volume of water in cube feet, and g and h areas of grate and heating surfaces in sq. feet. ILLUSTRATION. Assume a heating surface of 90 feet, and a grate of 3; what will be the evaporation ? Then 90-4- 3 x .002 22 = 1.998 cube feet. NOTE. A Galloway stationary boiler, with a ratio of grate area of 34.3 and a con- sumption of 21.8 Ibs. coal per hour, evaporated 2.9 cube feet of water per sq. foot of grate. Hence the coefficient in this case would be .002 466. To Compute .A^reas of Grrate and. Heating Surfaces, Volume of ^Water, and. AVeight of 3Tuel. For a Temperature 0/281, or Pressure 0/50 Ibs. per Sq. Inch. To Compute \Veignt of Fuel. When Water per Sq. Foot of Grate per Hour and Surface Ratio are Given. -^ = F, and x R 2 = (E C) F. ILLUSTRATION. Assume elements as preceding. 200 .02 X 50 2 /200 \ 15} and <02 x 50 2 _ i 10 \ x 15 = 50. To Compute Ratio of Heating Surface to Area of Grate, and. to Effect a <3-iven Evaporation. When Water and Fuel per Sq. Foot of Grate are Given. ^/ W ~ CF = R. W representing water evaporated per sq. foot of grate, and F fuel consumed, both in Ibs. per hour. C and x specific constants for each type of boiler, and R (h ~- g) ratio of heating surface to grate. ILLUSTRATION. Assume W = 200, C = 10, F = 15, and x = .02. /200-ioXis 200-.02X5Q 2 d /('3. 33-xo) Xij_ V -03 10 V .02 742 STEAM-ENGINE. BOILER. When Efficiency of Fuel and Fuel consumed per Sq. Foot of Grate per Hour are given. = = E or efficiency of fuel or weight of water evaporated per Ib. w /^=*. To Comptite Fnel that Tnay "be consumed, per Sq.. Foot of Grate per Hour, corresponding to a Gfiven Effi- ciency, When Efficiency of Fuel, that is, Weight of Water evaporated per Lb. of Fuel, and the Surface Ratio, are given. - ILLUSTRATION. Assume elements as preceding. Combustion of Coal per sq. foot of grate. Natural Draught, from 20 to 25 Ibs. can be consumed per hour. Steam-jet, 30 Ibs., and Exhaust-blast 65 to 80 Ibs. From Results of Experiments upon Marine Boilers, see Manual of D. K. Clark, page 808; he deduced the following formula, as applicable to all surface ratios in such boilers. Newcastle 021 56 R 2 + 9.71 F, and for Wigan .01 R 2 + 10.75 F = W in Ibs. And the general formulas he deduced from all the various experiments are as follows. From and at 212. Portable 008 R 2 + 8.6 F = W Marine 016 R 2 + 10.25 F = W. Stationary. , . .0222 R 2 + 9-s6F = W, | Locomotive, coal, .009 R 2 + 9.7 F = W. Locomotive, coke 0178 R 2 -f- 7.94^ = W. As the maximum evaporative power of fuel is a fixed quantity, the preceding formulas are not fully applicable in minimum rates of its consumption and evapo- rative quality. With coal and coke the limits of evaporative efficiency may be taken respectively at 12.5 and 12 Ibs. water from and at 212. ILLUSTRATION i. Assume a marine fire-tubular boiler with a surface ratio of heat- ing surface to grate of 30 and a consumption of coal of 15 Ibs. per sq. foot of grate per hour, what will be its evaporation per sq. foot of grate? .016 X 30* -f- 10. 25 X 15 = 168. 15 Ibs, 2 Assume a like boiler, using fresh water, to have a ratio of heating surface to grate of 30 and an evaporation of 165 Ibs. water per sq. foot of grate per hour, what would be consumption of coal per sq. foot of grate per hour? 165 .016 x so 2 = 14.69 Ibs. 10.25 Tube Surface (Iron) per Ib. of coal 1.58, per sq. foot of grate 32, and per IBP 4.27 sq. feet. Locomotive Boiler has from 60 to 90 sq. feet per foot of grate, and consumes 65 Ibs. coal per sq. foot per hour. Evaporative Capacity of Tu/bes of* "Varying Length. By Temperatures. (A J. Dutton, Eng. -in- Chief, R. N.) Diameter, external, 2.75 in*. Length, 6 feet 8 ins. Combustion Chamber, 1644. IN TUBES. 2 ins. . . 1426 | 5 ins. . . 1398 3 " ...1405 6 " ...1406 4 "... 1412 ' 7 "... 1400 8 ins... 1410 14 " ...1368 20 "... 1295 32 ins... 1 198 44 "...1106 56 " ...1015 68 ins 926 80 " 887 Connection. 782 STEAM-ENGINE. BOILEE. 743 Results of Operation of* Boilers under "Varying Propor- tions of 03- rate, Area, and. Length of Heating Surface, Dratignt of Furnace, and. Rate of Combustion. DESCRIPTION'. Area of Grate. Heating Surface. Grate to Heating Surface. Coal per Sq. Foot of Grate per Hour. Evapor Water fi per sq. ft. of grate. ation of om 212 per Ib. of Coal. FUEL. Fire-tubular. Agricultural and Hoisting Sq.Feet. 4-7 (2|25 |i6 10.5 10.6 22 18 10.3 10.3 10.8 Sq. Feet. 158 220 9 6 3-5 818 788 1056 748 749 9*5 508 151-2 945 767 Ratio. Si 75 100 & 50 49-3 14 30 24.4 Lbs. 3.1 30.86 38 45 *57 24-3 23-6 41.25 27.63 27.76 28.87 14 Lbs. 119 151 327 375 419 1401 265 264 468 309.8 205 293-7 141.4 Lbs. 9-33 1.83 0.6 0.47 1.04 0.41 0.7 1.2 I. 3 6 i-54 7-39 0.17 O.I Welsh. f < t n Lanc'r Anth'e Welsh. English } i 2 2* 3 " 2 3i-5 i New Castle. 3 Experimented at New York. 2 and 4 Wigan. * Effect of reducing the tube-surfaces was tried by stopping one half the number of tubes in alter- nate diagonal rows, so that the tube surface was reduced 206.5 sq. feet. The results with fires 12 ins. deep were as follows t Tubes open. Tubes half closed. Coal per sq. foot of grate per hour .................. 25 Ibs. 24 Ibs. Water from 212 per Ib. of coal ..................... 12.41" 12.23" Smoke per hour, very light. .... .................... 2.8 minutes. 8 minutes. Evaporative Effects of Boilers for Different Rates of Comoustion, and Surface Ratios. (D. K. Clark.) Water from and at 212 per Hour. Surface Ratio SO. STATIONARY. MARINE. PORTABLE. LOCOMOTIVB. Fuel per Sq. Foot Water Water Water Coal. Water Coke. Water of Grate per Hour. per Sq. foot. per Ib. of Coal. per Sq. foot. per Ib. of Coal. Sq. P foot. perlb. of Coal. per Sq. foot. per Ib. of Coal. per Sq. foot. per Ib. of Coal. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. 10 116 1.6 117 11.7 93 9-3 105 10.5 95 9-5 15 163 0.9 168 XX. 2 I 3 6 9 154 10.3 135 9 20 211 0.6 219 IO.9 179 9 202 10. I 175 8.7 30 307 O.2 322 10.7 265 8.8 209 10 254 8.5 Surface Ratio SO. i5 l8 7 2.5 187.5 12.5 149 9-9 168 II. 2 !6 4 10.9 20 247 2.3 248 J2.5 192 9.6 217 10.9 203 IO. 2 3 342 1.4 348 ii. 6 278 9-3 3i4 10-4 283 9-4 40 438 0.9 450 "3 364 9.1 411 10.3 362 9 50 534 0.7 552 ii 450 9 508 XO.I 442 8.8 Surface Ratio ?5. Water. Fuel per Sq. Foot of Grate per Hour in Lbs. 30 40 So 60 75 90 IOO Lba. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs, LOCOMOTIVE, coal.. Per sq. foot. 342 439 536 633 778 927 IO1O ti U " coke . u Ib. coal. " sq. foot. 11.4 338 ii 418 10.7 497 10.7 576 10.4 695 10.3 8i5 10.2 894 .. " Ib. coal. "3 10.4 9.9 9-6 9-3 9 8. 9 When a heater is used, and temperature of feed-water is raised above that ob- tained in a condensing engine, the proportions of surfaces may be correspondingly reduced. 744 STEAM-ENGINE. BOILER, Results of Operation of varioxis Designs of Boiler, tin- der varying IPz-oportioiis of Grrate, Calorimeter, Area and. Length of Heating Surface, Draught, Firing, and. Rate of Comtoustion.. STATIONARY. Area of Grate. Heat- ing Surface. Grate to Heating Surface. Circuit of Heating Surface. * li F "3 Coal per Sq. Foot of Grate per Hour. Water E from 212 per Ib. of Coal. vaporated per Sq. Foot of Grate per Hour. Sq. Ft. Sq. Ft. Ratio. Feet. Lbs. Lbs. Lbs. Lancashire double) internal and ex- > 20.5 612 29.8 79 5" 15-35 8.33 125 4 ternalflued 1 ...) " 2 21 767 36.5 80 55 21.5 10.88 204 4 Galloway vertical water-tubular 2 . 21 719 22.8 79 55 22.7 10.77 212 4 " " 2 . . 31-5 719 34-3 80 630 18.3 10.17 162 4 Fairbairn 1 1!'. 20. 5 -^87 T c 27 8 67 133 20.1 I 4 .2 1017 607 377-5 49' 5 30-3 26.8 56 o"/ 510 292 */ 16.42 7-43 J.U/ 8.12 9,08 133 59 5 Cylindrical flued3. . . MARINE. At Pressure of Atmosphere. Horizontal fire-tub. 2 10.3 508 49-3 27-5 11.92 328? U 11 2 10.3 508 49-3 41.25 11.36 4698 U ( 2 10.3 302 30 24 12.23 2689 (i I 19-3 749 _ 21 10 182 > 11 ( 28.5 749 26.3 21.15 8.94 164 > (t ( 28.5 749 26.3 J 9 11.13 3-35" U I (C. Wye William ) iS-5 749 48.3 - 600 37-4 10.63 39 8 u ! 22 749 34 600 17.27 11.7 202" tt . ( t 4 2 749 17.6 16 9- 6 5 154 J 3 " 4 3 10.8 150 13-9 8.5 10.99 8-95 88 I4 4 3 4-32 147 34 8.5 27.58 7.24 40 i4 1 Trial in France. 2 At Wigan, 1866-68, height of chimneys 100 feet. 3 Navy. yard, Washington, U. S., chimney 61 feet. 4 At pressure of atmosphere, fires 12 ins. deep, at 40 Ibs. pressure, evaporation was reduced 12 per cent. 5 Bituminous coals. 6 Anthracite, at pressure of 6.5 Ibs. above atmosphere. 7 Fires 14 ins. deep, air ad- mitted through furnace - doors. 8 Ditto do., jet blast. 9 Half tubes closed up. i Air through grate only. " Air through grate and door, no smoke. I2 One open- ing in door, temp. 625, with two 633, with four 638, and with six 600. J 3 Long grates, air spaces fully open, no smoke. J 4 One furnace, anthracite coal, 5 ins. deep. Draught. Draught of Furnace. Volume of gas varies directly as its absolute tem- perature, and draught is best when absolute temperature of gas in chimney is to that of external air as 25 to 12. T -|- 4.61 2 V ^ = 7 = V". V, V, and V" representing absolute temperatures at T 32 + 461. 2 V or temperature given, and at 32, in degrees and volume of furnace gas at tempera- ture T in cube feet. ILLUSTRATION. Assume temperature of furnace or T 1500, and 12 Ibs. air per Ib. of fuel. volume of gas per Ib. of fuel at i I5 o-4-~f *'o ~ 3 '9 8 ' Ibs. supply of air, 150 X 3-98 = 597 cube feet. W v t ;- = C. W representing weight of fuel consumed in furnace per second in Ibs., v volume of air at 32 supplied per Ib. of fuel in cube feet, t absolute temperature oj gas discharged by chimney in degrees, a area of chimney in sq.feet, and C velocity oj current in chimney in feet per second. STEAM-ENGINE. BOILEB. 745 ILLUSTRATION. Assume W = .i6, t> = i5o, t=. 1000, and a = 5. .16 X 150 X iooo_ 24000 5X493-2 2466 V .084 to .087 = D. D representing weight of a cube foot of gas discharged by 4Q3 2 thimney, in Ibs. ILLUSTRATION. 2*- X .086 = .0424 Ib. * r* 2 / /*z\ ( ! _|_ G + ) = H. G representing a coefficient of resistance and friction of 2 g \ m/ tr through grate and fuel,* f coefficient of friction of gas through flues and over sooty surfaces,^ I length of flues and chimney, m hydraulic mean depth,l and H height of chimney, all in feet. ILLUSTRATION i. Assume C = 9.73, I 60, and m = .72, all in feet. = : 64.33 * y3 g * x*^' = -' 6 04-33 V -7 2 2. Assume preceding elements. When H is given. ^/Tn 2flf-j-i + G + ^j=C ILLUSTRATION. Assume preceding elements. V2o. 6 x 64. 33 -r- 14 = 9. 73 feet. . 192 x pressure in Ibs. per sq. foot = head in ins. of water. Temperature at base of smoke-pipe or chimney, or termination of flues or tubes, is estimated at 500 ; and base of chimney, or its calorimeter, should have an area of 1.3 to 1.6 sq. ins. for every Ib. of coal consumed per hour. With tubes of small diameter, compared to their length, this proportion may be reduced to i and 1.2 ins. Admission of air behind a bridge-wall increases temperature of the gases, but it must be at a point where their temperature is not below 800. Loss of* I*ressTire "by Flow of Air in. Pipes. Length 3280 Feet, or 1000 Meters. Velocity at Pi] Feet per Second. Entrance of >e. Meter per Second. 4 1 6 Loss c Diameter of 8 f Pressure i Pipe in Int. 10 | 12 | 14 n Lbs. per Sq. Inch. 3.28 6.56 9.84 ,3-12 16.4 19.68 i 2 3 4 5 6 .114 1.183 2.06 4.446 .076 343 1-374 2.16 2.964 057 25 SQ 2 1.03 1.61 2.223 057 .21 477 .84 1.29 1.778 038 172 394 687 i i 1.482 .038 153 :l 43 ,ll 3 At Mount Cenis Tunnel, the loss of pressure from 84 Ibs. per sq. inch, in a pipe 7.625 ins. in diameter and i mile 15 yards in length, was but 3.5 per cent. Artificial I3ranglit. In production of draught in an ordinary marine boiler, from 20 to 33 per cent, of total heat of combustion of fuel is expended. Blast. By experiments of D. K. Clark and others it was deduced that the vacuum in back connection is about .7 of blast pressure, and in the furnace from .33 to .5 of that in back connection; that rate of evaporation varies nearly as square root of vacuum in back connection; that best proportions of chimney and passages thereto are those which enable a given draught to be produced with greatest diameter of blast pipe; for the manifest reason, that the greater that diameter, the less the back- pressure due to resistance of orifice, and that these proportions are best at all rates of expansion and speeds. * Which, in furnaces consuming from 20 to 24 Ibs. coal per sq. foot of grate per hour, is assigned by Peclet at 12. t Estimated by tame authority at .012. t For square or circular flue i .25 its diameter. 746 STEAM-ENGINE. DRAUGHT. SAFETY VALVES. Velocity of Draught. Locomotive. 36.5 VH (T t) = V. H representing height of chimney or pipe in feet, T awe? < temperatures of air at base and top of chim- ney, and V velocity in feet per second. Sectional area of tubes within ferrules ................. 2 grate. " " of smoke-pipe .......................... 066 " Area of blast-pipe (below base of smoke-pipe) ........... 015 " Volume of back connection ................ 3 feet X area of grate. Height of smoke- pipe 4 times its diameter. Steam-jet. Rings set above base of smoke-pipe, and should equally divide' the area ; jets .06 to .1 inch in diameter, 3 ins. apart at centres. A Steam-jet, involving 50 per cent, increased combustion of coal, produced 45 per cent, more evaporation at nearly same evaporation per Ib. of coal. Fan Blowers. See page 447. Comparative Result of Experiments with a Steam -jet in a Marine Boiler, with Bituminous Coal. (Nicoll and Lynn, Eng.) Without Jet. With Jet. Area of grate .................... sq. feet ......... . 10.3 10.3 Coal per sq. foot of grate per hour. . . . Ibs ........... 27. 5 41. 25 Water " " u .......... 293.1 419-37 from 212 per Ib. of coal " .......... 11.9 11.36 Comparative EJffect of Draught and. Blasts. By late experiments in England, with boilers of two steamers, to deter- mine relative effects of the different methods of combustion, the results were: Natural draught i, Jet 1.25, and Blast 1.6. In CylMM Pipes. Flow of Air. (Hawksley.) In Conduits of Various Sections. 796 /^ = IP - I* which xmch water is taken as equivalent to a pressure of 5.2 Ibs. per sq. inch for any passage. V representing velocity in feet per second, h head of water in ins. , d diameter of pipe, I length, and C perimeter, all in feet, a area of section in sq.feet, Q (V a) volume of air discharged per second in cube feet, and IP horse-power. Safety "Valves. Up to a pressure of 100 Ibs. per sq. inch, area in sq. ins. equal product of weight of water evaporated in Ibs. per hour by .006. Act of Congress (U. S.). For boilers having flat or stayed surfaces, 30 ins. for every 500 sq. feet of effective heating surface; for cylindrical boilers, or cylindrical flued, 24 sq. ins. Board of Trade, Eng. Two of .5 inch area per sq. foot of grate. Or, / - = diameter. G representing area of grate in sq. ins. Locked Safety-valves. Effective heating surface, less than 700 sq. feet, valve 2 ins. in diameter; less than 1500, 3 ins. in diameter; less than 2000, 4 ins. in diameter; less than 2500, 5 ins. in diameter; and above 2500, 6 ins. in diameter. Or, (.05 G -}- .005 S) /T5r area f eac h of two valves. G representing sq. t'ncft, per sq. foot of grate, and S sq. inch, per sq.foot of heating surface. STEAM-ENGINE. FLUES AND TUBES. 747 ILLUSTRATION. Assume G = 50 sq. feet, S = 1600 sq. feet, and P = 80 Ibs. (m. g.) Then, (.05X50 + .005X1600) X V 100-^-80 =^-2. 5 + 8 X 1.118=11.73 sq. ins. Pipes. Area. .25 G -f- .01 S / . G representing area of grate and S area of heat- ing surface, both in sq.feet, and P pressure per mercurial gauge in Ibs. (Copper), Thickness. Steam, . 125 -f ^^ ; Feed, . 125 -f- g^ ; Blow (Bottom and Surface), . 125 -^- ; Supply, .i-\ ; Discharge, . i -f - ; Feed, Suction, 9000 300 200 and Bilge discharge, .09 +^, and Steam Blow-off, -05 + . d representing internal diam. of pipe, and p internal pressure per sq. inch in Ibs. Flanges. Of brass, thickness 4 times that of pipe ; breadth, 2.25 times diam. of bolt ; bolts, diam. equal to and pitch 5 times thickness of flange. For lower pressure or stress, pitch of bolts 6 times. ; ITln.es and. Tubes. Flues and Tubes. Cross section, for 15 Ibs. of coal consumed per hour, an area of from .18 to .2 area of grate, area being measurably inverse to diameter, and directly increased with length. Thus, in Horizontal Tubular Boilers, area .18 to .2 area per sq. foot of grate, and in Vertical Tubular .22 to .25, area, decreasing with their length, but not in proportion to reduction of temperature of the heated air, area at their termination being from .7 to .8 that of calorimeter or area immediately at bridge-wall. Large flues absorb more heat than small, as both volume and intensity of heat is greater with equal surfaces. Tubes. Surface i sq. foot, if brass, and 1.33, if iron, for each Ib. of coal consumed per hour ; or 20 of brass and 27 of iron for each sq. foot of grate, and 2.6 sq. feet of brass and 3.7 of iron per IIP. Set in vertical rows, and spaces between them increased in width with number of the rows. Temperature of base of Chimney or Smoke-pipe, or termination of the flues or tubes, is estimated at 500 ; and base of chimney, or its calorimeter, with natural draught, should have an area of 1.33 sq. ins. for every Ib. of coal consumed per hour. With tubes of small diameter, compared to their length, this proportion may be reduced to i and 1.2 ins. When combustion in a furnace is very complete, the flues and tubes may be shorter than when it is incomplete. Evaporation. i sq. foot of grate surface, at a combustion of 15 Ibs. coal per hour, will evaporate 2.3 cube feet of salt water per hour. A sq. foot of heating surface, at a like combustion of fuel, will evaporate from 5 to 6.2 Ibs. of salt water per hour ; and at a combustion of 40 Ibs. coal per hour (as upon Western rivers of U. S.), from 10 to n Ibs. fresh water, exclusive of that lost by being blown out from boilers. 13.8 to 17.2 sq. feet of surface will evaporate i cube foot of salt water per hour, at a combustion of 15 Ibs. coal per hour per sq. foot of grate. Relative evaporating powers of Iron, Brass, and Copper are as i, i 32, and 1.56. NOTE. Boilers of Steamer Arctic, of N. Y., vertical tubular, having a surface of 33.5 to i of grate, consuming 13 Ibs. of coal per sq. foot of grate per hour, evapo- rated 8. 56 Ibs. of salt water per Ib. of coal, including that lost by blowing out of saturated water. 743 STEAM-ENGINE. SMOKE-PIPES AND CHIMNEYS. Water Surface. At low evaporations, 3 sq. feet are required for each sq. foot of grate siu> face, and at high evaporation 4 to 5 sq. feet. Steam Room. From 15 to 18 times volume that there are cube feet of steam expended for each single stroke of piston for 25 revolutions per minute, increasing directly with their number. Or, .8 cube feet per IIP for a side-wheel engine, and .65 for an ordinary and .55 for a fast-running screw-propeller. Space is required proportionate to volume of steam per stroke of piston Tli us, with like boilers, the space may be inversely as the pressures. Steam-drums and steam-chimneys, by their height, add to the effect of their volume, by furnishing space for water that is drawn up mechanically by the current of steam, to gravitate before reaching the steam-pipe. Grate. Area in sq. feet per Ib. of coal per hour for following boilers. Width, 1.5 diameter of furnace: Cornish and Lancashire, slow I Portable, moderate forced . . 03 sq. foot combustion .2 sq. foot. Locomotive and like, strong Marine, tubular 0510.066" " | blast... 01 " *' Thickness of Tubes per B W G. External diameter in ins 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 Thickness for pressure of 50 Ibs., number.. 12 12 n n n 10 10 10 9 " " " ioo u " ..it 10 99 98 88 7 Smoke-pipes and. diimneys. Area at their base should exceed that of extremity of flue or flues, to which they are connected. In Marine service smoke-pipe should be from .16 to .2 area of grate. In Locomotive, it should be .1 to .083. Intensity of their draught is as square root of their height. Hence, rela- tive volumes of their draught is determined by formula: yh . i a = volume in sq. feet, h representing height of pipe or chimney in feet, and a its area in sq. feet. When wood is consumed their area should be 1.6 times that of coal. Chimneys (Masonry). Diameter at their base should not be less than from .1 to .n of their height. Batter or inclination of their external surface .35 inch to a foot, which is about equal to i brick (.5 brick each side) in 25 feet. Diameter of base should be determined by internal diameter at top, and necessary batter due to height. Thickness of walls should be determined by internal diameter at top ; thus, for a diameter of 4 feet and les, thickness may be i brick, but for a diameter in excess of that 1.5 bricks. Area. ~r = <* C representing weight of coal consumed per hour in Ibs.. and T/h a area of ditto at top, in sq. ins. (Brick masonry.) 25 tons weight per sq. foot of brickwork in height is safe if laid in hydraulic mortar. Less the height of a smoke-pipe or chimney, the higher the temperature ol its gases is required. STEAM-ENGINE. PUMPS. PLATES AND BOLTS. 749 Velocities of Current of Heated Air in a Chimney 100 Feet in Height. In Feet per Second. External Air. Air 150 at Base 250 of Chirnr 350 ey. 45o e External Air. Air 150 at Base 250' of Chimr JSP" ey. 450* Feet Feet. Feet. Feet. Feet. Feet. Feet. Feet. 10 24 30 33 35 60 *9 26 29 33 32 22 28 3i 34 70 18 25 29 32 50 20 27 30 33 80 17 24 28 32 When Height of Chimney is less than 100 feet. Multiply velocity as ob- tained for temperature by .1 square root of height of chimney in feet. Draught consequent upon a steam -jet in a smoke-pipe or chimney is nearly equal to that of a moderate blast. The most effective draught is when absolute temperature of heated air or gas is to that of external air as 25 to 12, or nearly equal to temperature of melting lead. In chimneys of gas retorts, ovens, and like furnaces, the draught is more intense for a like height of chimney than in ordinary furnaces, in con- sequence of the great mass of brick masonry, which, becoming heated, adds to intensity of draught. Chimneys. Lawrence Manufacturing Co., Mass. Octagonal. Height above ground 211 feet. Diameters 15, and 10 feet 1.5 ins. Wall at base 93.5, and at top 11.5 ins. Shell at base 15 ins., at top 3.75 ins. Foundation 22 feet deep. England. Square. Height 190 feet Diameter at base. .., ...20 feet " 300 " 29 " Round. " 312 " " " 30 " 3 oo " " " 20 u Diameter at base usually i of height above ground. Vacuum at base of chimney ranges from .375 to 43 his. of water. Circulating 3?u.mps. Single-acting. .6 volume of single-acting air-pump and .32 of double* acting. Double-acting. .53 volume of double-acting air-pumpb Volume of Pump compared to Steam Cylinder or Cylinders. Engine. Pump Volume. Expansive, 1.5 to 5 times Single-acting 08 to .045. Compound do. 04510.035. Expansive, 1.5 to 5 times Double-acting .045 to .025. Compound do 02510.02. Valves. Area such as to restrict the mean velocity of the flow to 450 feet per minute. PLATES AND BOLTS. Wrouglit-irori. Tensile strength ranges from 45500 to 70000 Ibs. per sq. inch for plates, and 60000 to 65000 Ibs. for bolts, being increased when subjected to a moderate temperature. English plates range from 45000 to 56000 Ibs., and bolts from 55000 to 59 ooo Ibs. D K. Clark gives best quality of Yorkshire 56 ooo Ibs., of Staffordshire 44 800 Ibs. Test of IPlates. (U. S.) All plates to be stamped at diagonal corners at about four ins. from edge, and also in or near to their centre, with name of manu- facturer, his location, and tensile stress they will bear. Plates subjected to a tensile stress under 45 ooo Ibs. per sq. inch, should contract in area of section 12 per cent., 45000 and under 50000, 15, and 50000 and over, 25, at point of rupture. 750 STEAM-ENGINE. PLATES. Brands. (C No. i) Charcoal No. i. Plates, will sustain a stress of 40000 Ibs. per sq. inch; hard and unsuited for flanging or bending. (C No. i R H) Reheated, hard and durable, suited for furnaces, unsuited for con- tinued bending. (C H No. i S) Shell, will sustain a stress of 50000 to 54000 Ibs. in direction of fibre, and 34000 to 44000 across it: hard and unsuited for flanging or even bending with a short radius. (C H No. i F) Flange, will sustain a stress of 50000 to 54000 Ibs., soft and suited for flanging. (C H No. i F B) Furnace and (C H No. i F F B) Flange Furnace. The first is hard, but capable of being flanged, the other is hard, and suited for flanging. The especial brands are Sligo, Eureka, Pine, etc. The best English plates known are the Yorkshire, as Low Moor, Bowling, Farnley, Monk Bridge, Cooper & Co., etc. (See Steam-boilers, W. H. Shock, U. S. N., 1880.) Steel. Tensile strength ranges from 75000 to 96000 Ibs. Mr. Kirkaldy gives 85 966 Ibs. as a mean. When used in construction of boiler-plates should be mild in quality, containing but about .25 to .33 per cent, of carbon; for when it contains a greater proportion, although of greater tensile strength, it is unsuited for boilers, from its hardness and consequent shortness in its resistance to bending. Crucible steel may be used, but that obtained by the Bessemer or Siemens-Martin process is best adapted for boiler-plates. Its strength becomes impaired by the processes of punching and shearing, rendering it proper thereafter to submit it to annealing. Steel rivets, when of a very mild character and uniformly heated to a bright red, are superior to iron in their resistance to concussion and stress. Copper. Tensile strength is 33 ooo Ibs., being reduced when subjected to a temperature exceeding 120. At 212 being 32 ooo, and at 550 25 ooo Ibs. "Wrouglit-iron Shell IPlates. Pressure and. Thickness. ((7. S. Law.) Based upon a Standard of One Sixth of Tensile Strength of Plates. Iron or Steel Results with a Tensile Strength 0/50000 Lbs. Thick- Diameters in Ins. ness. 36 38 40 42 44 46 48 54 60 66 7 2 78 Inch. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Tbs7 Lbs. Lbs. Lba. .25 116 no 104 99 95 9 1 87 77 69 63 58 53 3125 145 137 130 124 118 IJ 3 109 96 87 79 72 67 375 '74 165 156 149 142 136 130 116 104 87 80 5 232 220 208 198 190 182 174 *54 138 126 116 106 84 90 96 IO2 108 "4 120 126 132 i35 140 144 Inch. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. 375 4375 69 80 3 61 58 68 52 61 49 57 47 55 46 53 44 43 50 5625 99 in 92 103 98 81 87 I 69 78 65 73 63 61 69 i 57 64 148 138 130 121 115 109 103 97 94 88 85 875 172 160 152 142 136 128 122 114 no 106 102 TOO i 198 184 l62 154 146 138 130 126 122 118 114 To which 20 per cent, is to be added for double riveting and drilled holes. Iron plates .375 inch in thickness will bear, with stay bolts at 4, 5, and 6 ins. apart from centres, respectively 170, 150, and 120 Ibs. per sq. inch. Iron plates, as tested by Mr. Phillips at Plymouth Dockyard, .4375 inch in thick- ness, with screw stay bolts 1.375 ins. in diameter riveted over heads, 15. 75 and 15.25 ins. from centre = 240 sq. ins. of surface for each bolt; bulged between bolts and drew from bolts at a pressure of 105 Ibs. per sq. inch of plate. Iron plates .5 inch in thickness, under like conditions with preceding case, bulged and drew from bolts at a pressure of 140 Ibs. per sq. inch of plate. Hence, it ap- pears, resistances of plates are as squares of their thickness. When nuts were applied to ends of bolt through .4375 inch plate, its resistance in- creased to 165 Ibs. per sq. inch of ulate. STEAM-ENGINE. SHELLS. PLATES. 75 1 Cylindrical Shells. (& S. Law.) Xo Compute Pressure for a Griven Thickness and Diameter, or Thickness for a Griven. Pressure and Diameter. For Pressure. RULE. Multiply thickness of plate in ins. by one sixth of tensile strength of metal, and divide product by radius or half diameter of shell in ins. When rivet-holes are drilled, and longitudinal courses are double riveted, add one fifth to result as above attained. EXAMPLE. Assume boiler 8 feet in diam., and plates .5 inch thick; what work- ing pressure will it sustain, tensile strength of plates equal to a stress of 60000 Ibs.? 8 X 12 5 X 60000-:- one sixth-: = ^-^- = 104.16 Ibs. 2 48 For Thickness. RULE. Multiply pressure by radius of shell, and divide product by one sixth of tensile strength of metal. EXAMPLE. Assume pressure, radius, and tensile strength as preceding. 104.16x96-^2 = 5000 _ 60 ooo-r- one sixth 10000 For Evaporation of Salt Water. Add one sixth to thickness of plates and sec- tional area of stay bolts. For Freight and. River Steaniboats. Standard. 150 Ibs. pressure for a boiler 42 ins. in diameter and plates .25 inch thick. For Pressure. RULE. Multiply thickness of plate by 12 600, and divide result by radius of boiler in ins. EXAMPLE. Assume a boiler 42 ins. in diameter, and plates .25 inch in thickness; what working pressure will it sustain? .25 X 12 600 -7-42 -r- 2 = 150 Ibs. Proof. All boilers by U. S. Law to be tested to a hydrostatic pressure of 50 per cent, above that of their working pressure. Relative Mean. Strength of Riveted Joints compared to that of IPlates. Allowances being made for Imperfections of Rivets, etc. Plates, 100 ; Triple, .72 to .75; Double or Square, .68 to .72; Double with double abut straps, .7 to .75 ; Staggered, .65 ; Single, .56 to .6. Board of Trade, England. Coefficient or Factor of Safety. When shells are of best material and workmanship, rivet-holes drilled when plates are in place, abut strapped, plates at least .625 inch in thickness and double riveted, with rivets com- puted at a resistance not to exceed 75 per cent, over the single shear,* the coefficient is taken at 5. Boilers must be tested by hydrostatic pressure to twice that of working pressure. Tensile strengths of plates are taken, with fibre 47 ooo Jbs. per sq. inch, across it 40000 Ibs., and when in superheaters from 30000 to 22 400 Ibs. 47 - = P, and = t. P representing pressure that shell will sus- tain per sq. inch in Ibs., B least per cent, of strength of rivet or plate (whichever it least) at lap, D diam. of shell and t thickness of plate, both in ins., and C coefficient of safety. * Shearing or detruaive resistance of wrought iron U from 70 to 80 per cent, of its tensile strength* 752 STEAM-ENGINE. SHELLS. PLATES. ox 6x5X5 = . 5 inch . 50 ooo X- 75X2 Pd _ 2u = ,. 6200 Ordinary plates. ) (" = 3000 3700 Working stress not to exceed .2 tensile strength of joint or riveted plate. Then for a pressure of no Ibs., and a diameter of 42 ins., as given for a standard U. S. boiler. Taking C as above for best single-riveted plate at 6200, " x 42 = .372 -f ins. 2 X O2OO in thickness, or . 122 inch in excess of U. S. Law for a plain cylindrical boiler, single riveted. Lloyd's. Thickness of shells to be computed from strength of longitudinal joints. /JC , ; PD t JC p d . na = P, = t, = D, = x, and = *. t representing thick- ness of plate, D diameter of shell, p pitch and d diameter of rivets, all in ins. ; J per cent, of strength of joint or rivets, the least to be taken; C a constant as per table ; P working pressure in Ibs. per sq. inch ; n number and a area of rivet ; x per cent, of strength of plate at joint compared with solid plate, and zper cent, of strength of tivets compared with solid plate. When plates are drilled, take .9 of z, and when rivets are in double shear, put 1.75 a for a. Constants. IRON PLATES. STEEL PLATES. JOINT. .5 inch and under. .75 inch and under. Above . 7 Sinch. 3 sr under. inland under. .75 inch and under. Above 75inch. T ( punched holes .. 155 170 165 1 80 170 IQO 200 215 230 240 Lap i drilled do ....... Double abut } punched holes strap ( drihed do. 170 1 80 1 80 190 Jrf IQO 200 215 230 250 260 When plates, as in steam-chimneys, superheaters, etc., are exposed to direct ac- tion of the flame, these constants are to be reduced .33. ILLUSTRATIONS. Assume pitch 4 ins., diam. of rivet 1.375 ins., and thickness of plate i inch, both single and double riveted. Area 1.375 = 1.48 sq. ins. 4 ~'' 375 = . 656 per cent, strength of joint compared to solid plate. ?-^4^- = . 37 4 4X1 d r ' 75 = .647 4X per cent, strength of rivet to solid plate when single riveted, an per cent, when do by go with drilled. per cent, when double riveted. Rivets at Joint. X 100 with punched holes and STEAM-ENGINE. PLATES. ABUT STRAPS, ETC. 753 Flates. To Co**ipnte Thickness of I>lates for a Griven Pressure and /i>itcn, and. I*ressu.re and IPitch for GJ-iven Thicli- ness - rrcp. /-p, and /- = t- t representing thickness of metal in p 2 V " V vy sixteeH'is of an inch, p pitch of stays or distance apart at centres in ins., P working pressure in Ibs. per sq. inch, and C a constant, as follows : For a Tensile Strength of Metal of 50 ooo Lbs. per Sq. Inch. Screw Stay-bolts with Riveted Heads. Plates up to .4375 inch in thickness C = 90, and above that 100. Screiv Stay-bolts with Nuts. Plates up to .4375 inch in thickness C = uo, and above that 120. Screw Stay-bolts with Double Nuts and Washers. Up to 4.375 ins. in thickness C as 140, and above that 160. When stay-bolts are not exposed to corrosion, these constants may be reduced .2. Resistance of a flat surface decreases in a higher ratio than space between stays. Hence, C must be decreased in proportion to increase of pitch above that of ordinary boiler-plates. ILLUSTRATION i. Assume pressure no Ibs. per sq. inch, and pitch of stays 5 ins. ; what should be thickness of plate for screw-bolts and riveted heads? C = 95. Then J 1 -^^- = y/ = 5- 38- sixteenth, 2. Assume thickness of metal 5 sixteenths inch thick, stay-bolts screwed and riveted over its threads, and working pressure of steam 80 Ibs. per sq. inch, C = 95 . Then S ** 9S = 5-45 ins. pitch. A-font Straps. Double Abuts should be at least .625 thickness of plate covered. Single, .125 thicker than plate covered, and Double, .625. Stays. Direct. Tensile stress should not exceed 5000 Ibs. per sq. inch for Iron, and 7000 for Steel. Diagonal or Oblique. Ascertain area of direct stay required to sustain the surface ; multiply it by length of diagonal stay, and divide product by length of a line drawn at a right angle to surface stayed, to end of diagonal 8ta}% and quotient will give area of stay increased to that which is required. Stress upon an oblique stay is also equal to stress which a perpendicular stay supporting a like surface would sustain, divided by cosine of angle which it forms with perpendicular to surface to be supported. ILLUSTRATION. Assume pressure no Ibs. per sq. inch, area of supported surface 36 sq. ins., and angle of stay 45; what would be pressure or stress upon stay? Cosine 45 = . 707 1 1. Then 1 10 x 36 -5- . 707 1 1 = 5600 tot. 754 STEAM-EN GLNE. GIKDEKS. FLUES, ETC. Iroporti< DIMENSIONS. 5ns of* Eyes of* Stays, Rod No. i. No. 2. FORGED AND WELDED. s, etc. No. 3- DRILLED FROM BAI SL c C No. i. a and a = x inch. rrt b= .9 *>( ) ~" t>( @ ) = -75 " \ *_./ 1 V ^-S J No. 2. a and a = i " T^ / \ Nk 6= .6 " = -75 " No. 3. a and a = i " a % a ^ n a i & = -75 " """ = .875 " m >-~ bss . - 6 L- When drilled from upset bar, dimensions same as for No. i. Pins when of steel .66 neck of rod. Stay-bolts. /row, are not to be subjected to a greater stress than 6000 Ibs. per sq. inch of section ; Steel, 8000 Ibs., both areas computed from weakest part of rod, and when of steel they are not to be welded. To Compute Diameter and. IPitch of* Stay - "bolts, and Resistance they will Snstain. Screwed. = d, ^=p, and (^-) 2 = P- Socket. =|7, and (^ ) =P. d representing diameter in ins. \ P / 95 ILLUSTRATION. Assume pitch of stay bolts 6 ins., and working pressure 100 Ibs. per sq. inch; what should be diameters of bolts, both screw and socket? 6 X -v/ioo , 6 X -v/ IO( ^-^ = .857 inch Screwed, and 4 70 95 GJ-irders. (Lloyd j s.) P(L p)DL_ /P(L = .6 3 -j-tncA Socket. Cd*t (L-p)DL"' OS*-"? V 0* = * L ^^ ew ^ 2en^A of girder, d its depth, t its thickness at centre or sum of its thicknesses, D its distance apart from centre to centre, and p pitch of stays, all in ins., and C a constant as per following : One stay to each girder, C = 6000. If two or three = 9000. If four 10 200. ILLUSTRATION. Assume triple stayed girder, 24 ins. in length, 3 ins. in depth, j inch thick, and stayed at intervals of 6 ins. ; what working pressure will it sustain? _ 9000 X 6 2 X i 324 ooo C = 9000. Then -^ = 222L = 125 Ibs. (24 6)X6X24 2592 Flues, Arched or Circular innrnaces. U. S. Law. .3125 inch for each 16 ins. of diameter. English iron, being harder than American, is better constructed to resist compression, and consequently a less thickness of metal is required for like stress. Lloyd's. /PLD - = L - ^representing external diameter of flue or furnace, and t thickness of plate, both in ins., L length of flue or furnace between its ends or between its rings, in feet, and P working press- ure in Ibs. per sq. inch. ILLUSTRATION. Assume diameter of flue 16 ins., length 6 feet, and working press use of steam 80 Ibs. per sq. inch. _, /8o x 6 x 16 Then / - = J = .29 inch. Furnace. P not to exceed STEAM-ENGINE. RIVETING. 75 5 ILLUSTRATION. Assume diameter of a circular furnace or width of a semicircular one 48 ins., working pressure of steam 80 IDS., and length 6 feet. Then - = ^-257 = .507 inch thickness. RIVETING. Plates. The strength of a joint is determined by ascertaining which of the two, the plate or the rivets, has the least resistance ; the stress on the first being tensile and the latter detrusive. The tensile strength is to be taken from that of the article under consider- ation, making due allowances for construction and location of the joint, and the consequent variation of stress, as with or across the fibre of the metal, or exposed to high heat as in a superheater. With or Across the Fibre. From experiments of Mr. D. Kirkaldy and others, the difference in strength of Iron plates is ascertained to be from 6.5 to 18 per cent., the average 10 per cent. Steel Plates. The relative strength of plates with or across the fibre, as determined by Mr. Kirkaldy, for "Fagersta" is 9 per cent., and for "Siemens" it is without material difference. Holes. The relative strength of plates when subjected to drilled or punched holes, as determined by the experiments of Mr. Kirkaldy, is shown to be 15 per cent. In Riveted Joint exposed to a tensile stress, area of rivets should be equal to area of section of plates through line of rivets, running a little in excess up to .5625 inch diameter of rivet, and somewhat less beyond that, area be- ing determined by relative shearing and tensile resistances of rivet and plate. NOTE. For Riveting of Hulls of Vessels, see pp. 828-30. Essentially by Nelson Foley. Single Lap Riveting. =b for plate, ^ = V for rivets, -A- = p, ptV = a, and ' - 1 = d . p representing pitch, t thickness of plate, and d diameter oj rivets, i o all in ins., a sectional area of rivets in sq. ins. , n number oj rivets, and b and b per cent, of plate between holes and of section of rivets to solid plate, i. e. plate before being punched. ILLUSTRATION, Assume p = 3 ins. , d = i inch, a = . 7854 inch, and t = 5 inch. 3 ""' = . 66 per cent, strength of lap, ' 7 54 = . 523 per cent, of rivet to solid plate, 3 3X5 3 X .5 X . 5 2 3 -f- = .7 When Shearing Strength of Rivet is not Equal to Tensile Strength of Plate. Then diameter of rivet must be increased in ratio of excess of strength of plate over rivet. Or, ^^ T- t = d. T and S representing tensile and shearing strengths, which may i o S be takzn at 5 and 4 for Iron and ^ and 6 for Steel. When full value of rivet sectiom is not allowed as by Lloyd's rules for drilled holes, b'=:&'x.9- 756 STEAM-ENGINE. RIVETING. Pitches as Determined by Diameter of Rivets. Plate between Edges of Holes. Pitch = Diam. of Rivet X Plate between Edges of Holes. Pitch = Diam. of Rivet X Plate between Edges of Holes. Pitch = Diam. of Rivet X Plate between Edges of Holes. Pitch = Diam. of Rivet X Per Cent. 50 52 55 2 2.08 2.22 Per Cent. 62 2. 3 8 2-5 2.63 Per Cent. 65 68 70 2.86 3-13 3-33 Per Cent. 72 75 78 3-57 4 4-55 OPERATION. If distance between edges of holes, or p d, =65 per cent, of solic plate, and diam. of rivet i inch, then 2.86 X i = 2.86 ins. pitch. When Plate and Rivets are of equal strength in ultimate tension, &' = 6, = B. Hence, *'^ t = a. In illustration of B, assume p = 3, d = 1. 1, and t = .5. Then 3 i.i = 1.9, and ^= .633 = Z>, or per cent, of strength of punched to tolid plate. Area 1. 1 = .95, and ' 9 = .633 = &', or per cent, of section of rivet to tolid plate. Hence, B = . 633. ILLUSTRATION. -Assume as shown, 6=1.633. Diameter of Rivets as Determined toy- Plate. B Or Strength at Joint. Diam. = Thickness of Plate X B Or Strength at Joint. Diam. = Thickness of Plate X B Or Strength at Joint. Diam. = Thick ness of Plate X Per Cent. T = S. .9 per cent, of Section Per Cent. T = S. .9 per cent. ofSection Per Cent. T = S. .9 per cent, ofSection of Rivet. of Rivet. of Rivet. 52 53 1-38 1.44 i-53 i-59 55 S^ It I:P f 60 1.76 1.91 i-95 2.12 54 i-5 1.66 57 1.69 i.8 7 62 2.08 2.31 OPERATION. If thickness of plate = .5 inch and plate and rivet have equal resist- ance, or B = 62 per cent., then .5 X 2.08 = 1.04 ins. diameter. 9 = . 44l8amt()/d , Donble T-jap Riveting. Preceding formulas for single lap riveting apply to this, with substitution of 2d for a and .64 for 1.27. ILLUSTRATION. Assume p = 3 ins. , t = .5 inch, and b' = .589. 3X.5X. 5 8 9 . l . ., i -75 3 Diameter of Rivets as Determined by Plate. B r Strength at Joint. Diam. = Thickness of Plate X B Or Strength at Joint. Diam. = Thickness of Plate X B Or Strength at Joint. Diam. = Thickness of Plate X Per Cent. T = S. .0 per cent. ofSection Per Cent. T = S. .9 per cent, of Section Per Cent. T = S. .9 per cent, of Section of Rivet. of Rivet. of Rivet. 68 1-35 1-5 7 1 1.56 1-73 74 1.81 2 69 1.42 i-57 72 1.64 1.82 75 1.91 2.12 70 1.48 1.65 73 1.72 1.91 76 2 2.25 OPERATION. Assume t . 5 inch and B 70 per cent. , tensile strength compared to shearing being as 7 to 6. What should be diameter of the rivets? .5 X 1.48 x = .863 inch. When rivets are in double shear, put 1.9 a for a. STEAM-ENGINE. DUTY. EVAPOEATION. 757 Triple Lap Riveting. Preceding formulas for single lap riveting apply to this, with substitution >f 3 a for a and .42 for 1.27. ILLUSTRATION. Assume p = 3 in*., t .5 inch, and b' = .883. - = 75 1, and i=.88 3 6'. 3X-5 Diameter of* Rivets as Determined by Plate. B Or Strength at Joint. Diam. = Thickness of Plate X B Or Strength at Joint. Diam. = Thickness of Plate X B Or Strength at Joint. Diam. =Thlcknei of Plate X Per Cent. t = S. .9 per cent. ofSection Per Cent. T = S. .9 per cent. ofSection Per Cent. T = S, .9 per cent ofSection of Riret. of Rivet. of Rivet. 70 99 I.I 73 I.I5 1.27 76 i-34 1.49 7i 1.04 'IS 74 I. 21 '34 77 1.42 1.58 72 1.09 1. 21 75 1.2 7 1.41 78 i-5 1.67 OPERATION. As shown by preceding tables. Q-eneral Formulas and. 1.27 BT Rivets in Single Shear. Rivets in Double Shear. Rivets in Triple Shear. Illxist rations. '=*- FTf= 6 ' i (i B)S i^S.-*- i.2 7 BT , fcS (.-B)S < = Lbs. 1 100 I04of 1225 1480$ 1089! 850 55o 1 100 River. Coast. Coast. Coast. Sea. Sea. Sea. Sea. t( u H Oscillating Inclined... * Without frame. t With frame 1109. t Including boilers. Single frame. Screw Propellers. American Marine (Condensing). ENGINE. Cyli No. nders. Volume. Engine. WEIGHTS. Boilers. Per C. Ft. Cylinder. SEH- VICX. Vertical direct, Jet Condens'g . . " " Surface Cond'g . " " Jet " . i< <( n it U (( U Horizontal back-action Cube Feet. 4 12.5 12.5 $ 68 67 4.8 24-3 42 3 5 .6 3 5.86 2.77 Lbs. 22040 59000 48130 120450 i 523 060 289 680 201000 24705 94I 9 6 I O22 400 30534 172028 I44IO 14759 Lbs. 12 100 32OOO 35000 98000 985600 200800 200593 26372 88050 840000 27301 100065 22481 22417 Lbs. 8535 7280 6650 6620 4958 7212 6009 10641 75oo 4380 16066 7774 i9 8 34 13421 Sea. Sea. Sea. Coast Sea. Sea. Sea. Coast. Sea. Sea. Coast. Sea. River. Coast. Vertical compound ( *> ;; ;; JJ| " direct. . 1 3 a ::::::::: i^a " Non-Condensing. it U English Marine (Condensing). DESCRIPTION. Cy No. linders. Volume. Engines. WKJ Propeller and Shafting. GHTS. Boilers and Water. Total. Per IIP Per Cube Ft Cylinder. Trunk 2 2 2 2 2 6 2 2 Cube Ft. 230 382 393 440 24 707 52 '43 Tons. 121 223 105 117 4-2S 497 55 130 Tons. 47 85 48 43 W 75 15 27 Tons. 257 303 144 135 7-25 656 no 162 Tons. 425 6n 357 295 12.25 1320 1 80 319 Lbs. 465 338 781 560 60 368 35* 309 Tons. 1.85 z. 6 9 7 52 1.87 3-44 2.23 Horizontal direct . . Vertical direct Oscillating Vertical compound Horizontal compound. . . STEAM-ENGINE. WEIGHT OF BOILERS. Land-engines. (Non-condensing. ) 759 ENGINE. Volume of Cyl'r. Engine. Spur-wheel and Connections. Sugar-Mill Complete. Boilers, Grates, etc. Engine per Cube Foot of Cylinder. Vertical) 18 ins. X4 feet beam f 30 ins. X 5 feet Horizon'l, 14 ins. X 2 feet " 22 ins. X 4 feet 7 24-5 2.2 10.6 LbB. 67200 105000 10914 56000 Lbs. 37800 137 '79 Lbs. 89600 265 879 Lbs. 26880 75000 8200 30140 Lba. 9600 4290 5100 5600 To Compxite Weight of* a Vertical Beam and Side-wheel Jet Condensing Engine. (T. F. Rowland, A.S.C.E.) Including all Metals, Boiler and A ttackments. Smoke-pipe, Grates, Iron Floors, and Iron in Wooden Water-wheels, omitting Coal-bunkers. For a Pressure per Mercurial Gauge of 40 Ibs.per Sq. Inch. For surface condenser add 10 to 15 per cent RULE. Multiply volume of cylinder in cube feet by Coefficient in follow- ing table corresponding to length of stroke, and product will give rough weight in Ibs. For finished weight deduct 6 per cent. Stroke. Coefficient. I Stroke. Coefficient. Stroke. Coefficient. Stroke. Coefficient. Feet. ,4 2467 2340 2213 2OOO Feet. 9 1865 1730 1619 1546 EXAMPLE i. What are the rough and finished weights of a vertical beam engine, cylinder 80 ins. in diameter and 12 feet stroke of piston ? Area of 80 ins. = 5026. 56, which x 12 feet = 419 cube feet, and x 1546 for 12 feet, stroke = 647 774 Ibs. rough weight. Then 647 774 X .06 = 38 866, and 647 774 38 866 = 608 908 Ibs. finished weight. WEIGHTS OF BOILERS. Weights of Iron Boilers (including Doors and Plates, and exclusive of Smoke- pipes and Grates) per Sq. Foot of Heating Surface. Surface Measured from Grates to Base of Smoke-pipe or Top of Steam Chimney. BOILER. For a Working Pressure of 40 Lb. Weight. Single return, Flue * water bottom. . " " " Multi-flue'*!*.!! '.".".water bottom!! Horizontal return, Tubularf water bottom . . " t Vertical Horizontal direct, Tubular*. t water bottom. . . Lbs. 25.6 to 32.9 24 to 30 27 to 45 25 to 43 22.5 to 35 21 to 33 17.7 to 26.7 18.5 to 26. 5 19.8 to 23.8 17 tO 21 23. 5 to 24 18.1 to 18.6 16.3 to 17.3 24 to 26 Cylindrical, external furnace, t 36 ins. in diam., .25 inch thick. . " Flue " $361042 " .25 " " .. Horizontal direct, Tubular Locomotive Vertical Cylinder direct, Tubular Weight of Cylindrical Furnace and Shell Boilers, all complete for Sea Service and for a pressure of 60 Ibs. steam, 200 Ibs. per IIP. * Section of furnace square. Shell cylindrical. t Section of furnace and shell square. t Wrought-iron heads, .375 inch thick, flues, .25 inch, and surface computed to half diameter of shell. NOTES. i, The range in the units of weight arises from peculiarities of construc- tion, consequent upon proportionate number of furnaces, thicknesses of metal, vol ume of shell compared with heating surface, character of staying, etc. 2. If pressure is increased the above units must be proportionately increased. 760 STEAM-ENGINE. BOILEB-POWER, COMBUSTION. Boiler-power. The power of a boiler is the volume or weight of steam alone (indepen- dent of any water that it may hold in suspension) that it will generate at its operating pressure in a unit of time,, Marine boilers of the ordinary type and proportions, with natural draught, burn- ing anthracite coal, produce 3.5 to 5.5 IIP per sq. foot of grate per hour; with a free burning or a semi-bituminous coal, 5 to 7.5 IIP; and with a foroad draught, with 25 to 30 Ibs. best coal per sq. foot of grate per hour, 8 to 10 IIP. Marine engines, operating with a steam-pressure of 35 Ibs. (m. g.), and with mod- erate expansion, consume 30 Ibs. steam per IIP per hour, and with a high rate of expansion, under a pressure of 70 Ibs. , 20 Ibs. steam. With a blast draught and consuming 30 to 40 Ibs. of a fair quality of coal per sq. foot of grate per hour, 7 to 10 IP per hour can be attained. In locomotive boilers, having from 50 to 90 sq. feet of heating surface per sq. foot of grate, and at a rate of combustion of from 45 to 125 Ibs. of coke, an average evap- oration of 9 Ibs. of water per Ib. of coke has been attained at ordinary temperatures and pressure. To Compute Volume of* Air and Q-as in a Furnace. When Volume at a Given Temperature is known. RULE. Multiply given volume by its absolute temperature, and divide product by the given abso- lute temperature. NOTE. Absolute temperature is obtained by adding 461 to given or acquired temperature. EXAMPLE. Assume volume of air entering a furnace at i cube foot, its tempera- ture^ , and temperature of furnace 1623; what would be the increase of volume? * X 16230 + 461 _ 2084 600 + 461 - 521 ~ "Volume of Furnace Q-as per Lib. of Coal. (Rankine.) Tempera- ture. 12 Lbs. A.ir Supplied 18 Lbs. 24 Lbs. Tempera- ture. 12 Lb8. Air Supplied 18 Lbs. 24 Lbs. If ISO 161 225 241 300 322 752 III2 369 479 553 718 738 957 104 172 2 5 8 344 1472 588 882 1176 212 205 3<>7 409 1832 697 1046 1395 572 3H 471 628 2500 906 1357 1812 Temperature of ordinary boiler furnaces ranges from 1500 to 2500. The opening of a furnace door to clean the fire involves a loss of from 4 to 7 per cent, of fuel. For other illustrations, see ante, page 744-6. Rate of ConVbustion. The rate of combustion in a furnace is computed by the Ibs. of fuel consumed per sq. foot of grate per hour. In general practice the rate for a natural draught is, for anthracite coal from 7 to 16 Ibs., for bituminous, from 10 to 25 Ibs., and with artificial or forced draught, as by a blower, exhaust-blast, or steam -jet, the rate may be increased from 30 to 120 Ibs. The dimensions or size of coal must be reduced and the depth of the fire increased directly, as the intensity of the draught is increased. Temperature of gases at base of chimney or pipe should be 600, and frictional resistance of surface of chimney is as square of velocity of current of gases. Ordinarily from 20 to 32 per cent, of total heat of combustion is expended in tlie production of the chimney draught in a marine boiler, to which is to be added the Josses by incomplete combustion of the gaseous portion of the fuel and the dilution of the gases by an excess of air, making a total of fully 60 per cent. (Steam-boilers. Wm, H. Mock, U. S. N. y 1881.) STRENGTH OF MATERIALS. ELASTICITY. 761 STRENGTH OF MATERIALS. Strength of a material is measured by its resistance to alteration of form, when subjected to stress and to rupture, which is designated as Crushing, Detrusive, Tensile, Torsion, and Transverse, although trans- verse is a combination of tensile and crushing, and detrusive is a form of torsion at short lengths of application. ELASTICITY AND STRENGTH. Strength of a material is resistance which a body opposes to a per- manent separation of its parts, and is measured by its resistance to alteration of form, or to stress. Cohesion is force with which component parts of a rigid body adhere to each other. Elasticity is resistance which a body opposes to a change of form. Elasticity and Strength, according to manner in which a force is exerted upon a body, are distinguished as Crushing Strength, or Resistance to Com- pression ; Detrusive Strength, or Resistance to Shearing ; Tensile Strength, or Absolute Resistance ; Torsional Strength, or Resistance to Torsion ; and Transverse Strength, or Resistance to Flexure. Limit of Stiffness is flexure, and limit of Resistance is fracture. Neutral Axis, or Line of Equilibrium, is the line at which extension ter- minates and compression begins. Resilience, or toughness of bodies, is strength and flexibility combined ; hence, any material or body which bears greatest load, and bends most at time of fracture, is toughest. Stiffest bar or beam that can be cut out of a cylinder is that of which depth is to breadth as square root of 3 to i ; strongest, as square root of 2 to i ; and most resilient, that which has breadth and depth equal. Stress expresses condition of a material when it is loaded, or extended in excess of its elastic limit. General law regarding deflection is, that it increases, cceteris paribus, di- rectly as cube of length of beam, bar, etc., and inversely as breadth and cube of depth. Resistance of Flexure of a body at its cross-section is very nearly .9 of its tensile resistance. Coefficient of* Elasticity. Elasticity of any material subjected to a tensile or compressive force, within its limits, is measured by a fraction of the length, per unit of force per unit of sectional area, termed a constant, and coefficient of elasticity is usually defined as the weight which would stretch a perfectly elastic bar of uniform section to double its length. Unit of force and area is usually taken at one Ib. per sq. inch. E represent- ing denominator of fraction. EXAMPLE. If a bar of iron is extended one i2ooooooth part of its length per Ib. of stress per sq. inch of section, t r 12000000 E ' The bar would, therefore, be stretched to double its normal length by a force ot 12000000 Ibs. per sq. inch, if the material were perfectly elastic. 762 STRENGTH OF MATERIALS. ELASTICITY. The same method of expressing coefficient of elasticity is applied to re- sistance to compression. That is, coefficient, in weight, is expressed by de- nominator of fraction of its length by which a bar is compressed per unit of weight per sq. inch of section. Ultimate extension of cast iron is sooth part of its length. Extension of Cast-iron Bars, when suspended Vertically. i Inch Square and 10 Feet in Length. Weight applied at one End. Weight. Extension. Set. Weight. Extension. Set. Weight. Extension. Set. Lba. 529 Ins. .0044 .0092 Lb3. 2117 4234 Ins. .0190 .0397 Ins. .000059 .00265 Lbs. 8468 14820 Ins. .0871 Ins. .00855 Q25S5 "Woods. MM. Chevaudier and Wertheim deduced that there was no limit of elasticity in woods, there being a permanent set for every extension. They, however, adopted a set of .00005 of length as limit of elasticity. This is empirical. MODULUS OF ELASTICITY. Modulus or Coefficient of Elasticity of any material is measure of its elastic reaction or force, and is height of a column of the material, pressing on its base, which is to the weight causing a certain degree of compression as length of material is to the diminution of its length. It is computed by this analogy : As extension or diminution of length of any given material is to its length in inches, so is the force that pro- duced that extension or diminution to the modulus of its elasticity. P I Or, x : P : : I : w = . x representing length a substance i inch square and i foot in length would be extended or diminished by force P, and w weight of modulus in Ibs. To Compute "Weight of" M.odulns of Elasticity. RULE. As extension or compression of length of any material i inch square, is to its length, so is the weight that produced that extension or com- pression, to modulus of elasticity in Ibs. EXAMPLE. If a bar of cast iron, i inch square and 10 feet in length, is extended .008 inch, with a weight of 1000 Ibs., what is the weight of its modulus of elasticity? .008 : 120 (10 X 12) :: 1000 : 15000000 Ibs. To Compete Modnlns of Elasticity. When a Bar or Beam is Supported at Both Ends and Loaded in Centre. RULE. Multiply weight or stress per sq. inch in Ibs. by length of material in ins., and divide product by modulus of weight. Or, = E; = M; W. I representing length in ins., M modulus, W weight in Ibs. per sq. inch, and E compression or extension. EXAMPLE i. If a wrought-iron rod, 60 feet in length and .2 inch in diameter, is subjected to a stress of 150 Ibs., what will it be extended? Modulus of elasticity of iron wire is 28 230 500 Ibs. (see following table), and area ofit.2 2 X .7854 = . 31416. = 477.46 Ibs. per sq. inch, and 60 X 12 = 720 ins. .31416 Then 477.46 X 720 = = OI2 18 inch " 28 230 500 28 230 500 a. Take elements of preceding case under rule for weight of modulua 120X1000 .., . 008 x 15 ooo ooo _ 15000000 - = .008 inch. - = 1000 Ibs. STRENGTH OP MATERIALS. COHESION, 763 Modulus of 3 SUBSTANCES. Slastici Height. ty and "> Weight. TVeight of Var: SUBSTANCES. LOU* 1M Height. iterials. Weight. Ash Feet. 4970000 4600000 2 460000 4 II2000 4800000 5680000 8330000 4440000 2 79OOOO 5000000 6OOOOOO 5750000 7550000 8 377 OOO Lbs. i 656 670 1345000 8464 ooo 14 632 720 18240000 1499500 2Ol6 DOO 5550000 8 844 300 170000 2370000 17 968 500 25 820 ooo 28 230 qoo Larch Feet. 4415000 146000 i 850000 2400000 6570000 2 150000 4750000 8700000 8970000 8530000 9OOOOOO 1672000 1053000 4480000 Lbs. 1074000 720000 1080400 3 300000 2071 ooo 2508000 i 710000 2430000 i 830000 26650000 28689000 1718800 3510000 1 3 44O OOO Beech Brass, yellow " wire Lignum-vitae Limestone Copper cast Mahogany . . . Elm Marble, white Oak Fir red Glass Pine, pitch Gun-metal " white Hempen fibres. . . . Ice Steel, cast " wire Iron cast Stone, Portland . . . Tin cast " wrought " wire... Zinc "Weight a Material will "bear per Sq.. Inch, -without Permanent Alteration of its Length. MATERIAL. Lbs. MATERIAL. Lbs. MATERIAL. Lbs. Metals. Brass Stones, etc. Marble Woods. Beech Gun metal IOOOO Limestone* Elm . . 3 15 coo Portland 1500 Fir red 42QO ' * wrought 17 800 Larch . . 060 Lead I 500 Woods. Mahogany JOOO Steel . . 4^000 Ash ... TUO Oak . . JUOU 3060 * Tensile strength 2800. Comparative Resilience of "Woods. Ash i Beech 86 Cedar 66 Chestnut Elm Fir Larch 84 Oak 63 Pitch Pine 57 Spruce 64 Teak. 59 Yellow Pine. .. .64 MODULUS OF COHESION. To Compute Length of a Prism of a Material -which would. t>e Severed toy its own Weight when Sxaspended. RULE. Divide tensile resistance of material per sq. inch by weight of a foot of it in length, and quotient will give length in feet. ILLUSTRATION. Assume tensile resistance of a wrought- iron rod to be 60000 Ibs. per sq. inch. Weight of i foot = 3.4 Ibs. Then 60000-4-3.4 = 17647.06/66*. length in Feet required to Tear Asunder the following Substances: Rawhide 15 375 feet. | Hemp twine. . . 75 ooo feet. | Catgut 25 ooo feet. Elasticity of Ivory as compared with Glass is as .95 to i. When Height is given. RULE. Multiply weight of i foot in length and i inch square of material by height of its modulus in feet, and product will give weight. To Compute Height of Modulus of Elasticity. RULE. Divide weight of modulus of elasticity of material by weight of i foot of it, and quotient will give height in feet. EXAMPLE. Take elements of preceding case (page 762), weight of i foot being 3 Ibs. ; what is height of its modulus of elasticity ? 15000000-:- 3 = 5000 ooo feet 764 STRENGTH OF MATERIALS. CRUSHING. From a series of elaborate experiments by Mr. E. Hodgkinson, for the Railway Structure Commission of England, he deduced following formulas for extension and compression of Cast Iron : Extension : 13 934 040 290 743 200 -^ = W. Compression : 12 931 560 - 522 979 200 -^ = W. e and c representing extension and compression, and I length in ins. ILLUSTRATION. What weight will extend a bar of cast iron, 4 inc. square and 10 feet in length, to extent of .2 inch? .2 2*~ 13 934 040 X 290 743 200 = 23 223.4 807.62 = 22 415. 78, which X 4 ins. = 89666.12 Ibs. CRUSHING STRENGTH. Crushing Strength of any body is in proportion to area of its section, and inversely as its height. In tapered columns, it is determined by the least diameter. When height of a column is not 5 times its side or diameter, crushing strength is at its maximum. Cast Iron. Experiments upon bars give a mean crushing strength of 100 ooo Ibs. per sq. inch of section, and 5000 Ibs. per sq. inch as just sufficient to overcome elasticity of metal ; and when height exceeds 3 times diameter, the iron yields by flexure. When it is 10 times, it is reduced as i to 1.75 ; when it is 15 times, as i to 2 ; when it is 20 times, as i to 3 ; when it is 30 times, as i to 4 ; and when it is 40 times, as i to 6. Experiments of Mr. Hodgkinson have determined that an increase of strength of about one eighth of destructive weight is obtained by enlarging diameter of a column in its middle. In columns of same thickness, strength is inversely proportional to the I>6 3 power of length nearly. A hollow column, having a greater diameter at one end than the other, has not any additional strength over that of an uniform cylinder. Wrought Iron. Experiments give a mean crushing stress of 47 ooo Ibs. per sq. inch, and it will yield to any extent with 27 ooo Ibs. per sq. inch, while cast iron will bear 80 ooo Ibs. to produce same effect. Effects. A wrought bar will bear a compression of -g^-g of its length, with- out its utility being destroyed. With cast iron, a pressure beyond 27 ooo Ibs. per sq. inch is of little, if any, use in practice. Glass and hard Stones have a crushing strength from 7 to 9 times greater than tensile ; hence an approximate value of their crushing strength may be obtained from their tensile, and contrariwise. Various experiments show that the capacity of stones, etc., to resist effects of freezing is a fair exponent of that to resist compression. Seasoning. Seasoned woods have nearly twice crushing strength of un- seasoned. Elastic Limit compared to Crxxsliing Resistance. Wrought- iron Commerce 545 Bessemer steel 615 Cast steel 473 Cast steel 692 Fagersta steel { ' 25 STRENGTH OF MATERIALS. CRUSHING. 765 Crushing Strength, of various Materials, deduced from Experiments of Maj. \Vade, Hodglzinson, Capt. IVIeigs, TJ S. .A.., Stevens Institute, and by Gr. JL*. Vose. Meduced to a Uniform Measure of One Sq. Inch. CAST IRON. FlGURBS AND M ATSRIAL. Crushing Weight. FIGURES AND MATERIAL. Crushing Weight. Gun- metal American Lbs. Lbs. " J 85000 Stirling, mean of all, English .. 12* 395 " mean 125000 IOOOOO '* extreme, English 134400 53760 Low Moor, No. i, English " No. 2. *' ...... 62450 O2 33O Average (Hodgkinson), English 153200 84 240 Clyde. No, * " zoo oqo IOQ7OO WEOUGHT IEOH. American, extreme.. . 47040 VARIOUS METALS. averaga. 65200 40000 37850 Aluminium bronze, 95 cop. .... I2QQ2O p eoooo 164800 " " soft 66 I " tempered Steel, cast ( I0500O " Siemens 335000 I " Fagersta 250000 IS4SOO Lead IS500 77^0 Elastic Crushing Strength of Wrought Iron and Crucible Steel is equal to its tea eile, of Bessemer Steel, 50 per cent of its transverse strength, WOODS. Ash 6663 6061 Birch { 3300 Box 7900 IO SI3 Cedar red .... 6 " seasoned 6 noo 5 35 Elm 6831 4 ' seasoned 802=; Larch . { 3200 Locust. . . 5500 O III Mahogany, Spanish 8 1 08 Maple { 8100 Oak, American white. .... IOOOO ' ' Canadian white ' Q " " live 6 850 " English { 95oo Pine, pitch 6484 " white " yellow 8000 Spruce, white Teak Walnut. .. 66^s Chestnut.. .... 900 Hemlock 600 Pine, white 800 Crosswise of Fibre. Pine, Yellow- South... 1400 " Oregon 1200 ' ' Northern 1000 Redwood 800 Spruce 700 White Oak 2000 Increase in Strength of Oabes of Sandstone, per Sq. Inch (under Blocks of Wood\ as Area of Surface is increased. (Gen'l Gillmore, U. S. A.) STONE. 5 I i-5 INC 2 HKS. 2.25 2-75 3 Yellow Berea sandstone . . Blue " " Lbs. 6080 Lbs. 6990 9500 Lbs. 8226 10730 Lbs. 8955 12000 Lbs. 9130 12500 Lbs. 9838 13200 Lbs. 10125 STRENGTH OF MATERIALS. CRUSHING. Stones, Ce FIGURES AND MATERIAL. jment Crushing Weight. s, etc. (Per Sq. Inch.) FIGURES AND MATERIAL. Crushing Weight. Basalt, Scotch Lbs. 8300 16800 800 I 400 6 222 I02I9 14 216* 3630 800 4000 1440 1650 7200 2250 5600" 808 2228 1543 17000 32000 1280 600 3800 2464 5980 2330 2650 1800 342 750 3270 1280 460 775 3522 33J9 3069 2991 31 ooo 19600 10760 6339 10450 12850 B, N. J. 3 atent Offic Eleilly, Ord Lbs. 5340 457 15583 15000 18800 4000 9000 7800 3 'So 3600 14000 8057 18061 '39 J 7 18941 17440 12624 9630 22702 8950 3 360 10382 u 156 18248 10124 500 800 760 240 460 595 120 3850 12000 5340 7850 11789 5825 3136 2554 10762 5710 13890 23744 5000 8300 v York. stitute. " Welsh Beton, N. Y. S. ConcretiDg Co. j 11 Quincy Mass . . Greenstone, Irish Limestone J " hard burned " compact, Eng " coniiDon < ' { yellow-faced burned, Eng. " Stourbridge fire-clay, ' " Staffordshire blue, u " stock English " Anglesea " " Irish " Marble Baltimore Md { " East Chester, N.Y.f... " Hastings N. Y *' Fareham, English " red, English " Irish " Sydney N. S " Italian " " white Cement, Hydraulic, pure, Eng. | " Portland, sand i ' Lee Mass 1 Montgomery Co. , Pa. ... " " sand 3 ' Stockbridge, Mass.*.... ' Symington, large " fine crystal " strata horizontal Masonry, brick, common { " " in cement Mortar good " " 3 mos " u i sand, 3 mos " ** 9 mos *' " i sand, 9 mos. .. . " " 12 inch cubes, ) 12 mos. i sand and gravel ) " 3 " Roman " lime and sand " u u beaten. . . " common " pure, Eng u Rosendale Oolite Portland ' ' Sheppey Eng Pottery- pipe Chelsea . Concrete, lime i, gravel 3. ... { Freestone Belleville N J Sandstone Aquia Creek " Connecticut II " Connecticut ' Craigleth, Eng " Dorchester, Mass " Little Falls, N. Y.... Glass crown ' Derby grit " .. . 1 Holyh'd quartz, Eng. ' Seneca IT . . . Gneiss 4 Yorkshire, Eng. Slate Irish { Granite, Aberdeen, Eng u Cornish, " ....;... " Dublin " Terra Cotta . . " Newry, " Whinstone Scotch * Tested by author at Stevens Institut Capitol, Treasury Department, and '. 11 Cn.mwell, Conn. Tested by J. W, ] t Post-office, Wash. \ City Hall, Ne\ e, Washington, D. C. nance Dept., U.S.A. f Smithsonian Ir Safe .Load of* Hollow, Cylindrical, and Solid Columns, A-rches, Chords, etc., of Cast Iron. Hollow Columns. Per Sq. Inch. (F. W. Shields, M. I. C. E.) Length. Thick- ness. Load. Length. Thick- ness. Load. Length. Thick- ness. Load. Length. Thick- ness. Load. Inch. Lbs. Inch. Lbs. Inch. Lbs. Inch. Lbs. 20 to 24 diam's. 375 5 2800 336o 20 tO 24 diam's. .625 75 3920 4480 25 to 30 diam's. 375 5 2240 2800 25 to 3 diam's. 625 75 336o 3920 Solid Colnmns, etc. 3360 IDS. per sq. inch. (Brunei.) Arclies. 5600 Ibs. per sq. inch. STRENGTH OF MATERIALS. CRUSHING. 7 6 7 Chords and Posts. i inch diameter and not more than 15 diameters in length . 2 of breaking weight of metal. .625 inch diameter and not more than 25 diameters in length .5 of breaking weight of metal, and when more than 25 diameters in length from .1 to .025 of breaking weight of metal (Baltimore Bridge Co.) Wrought-iron Cylinders and Rectangular TiVbes. LENGTH. External Diameter. Internal Diameter. Thickness, Area. Crushing Weight per Sq. Inch. CYLINDERS. Ins. Ins. Ins. Sq. Ins. Lbs. 10 feet 1-495 1.292 .1 444 14661 10 " 2.49 2.275 .107 .804 29779 10 " 6.366 6.106 2-547 35886 RECTANGULAR TUBES. 10 feet 4.1 X 4.1 03 504 10980 5 " 4-1 X 4-1 03 504 li 514 TO d 4.1 X 4.1 .06 1.02 19261 to " "5 4.25 X 4-25 134 2-395 21585 7-5 " 4.25 X 4-25 134 2-395 23203 10 " i 8.4 X 4-25 (.26 (.126 6.89 29981 10 " 8. i X 8. i .06 2.07 13276 7.66" 8.1 X 8.1 .06 2.07 13300 10 " ) Internal 8.1 X 8.1 .0637 3-551 5 " . j diaphrag's 8.1 X 8.x .0637 3-55' 23208 Strength, per Sq.. Inch, of 3 -Inch Cntoes under Bloclts of Wood. (Gertl Gillmore, U. S. A.) Surfaces Worked to a Clear Bed. GRANITE. Lbs. 22250 15000 17750 14750 18250 16187 12500 21250 14100 12423 19500 14937 *5937 13370 17750 12875 17500 20750 24040 "475 25000 20700 13900 18500 12600 16900 18000 25000 21 500 LIMESTONE. Bardstown, Ky. , dark Cooper Co., Mo., dark drab. .... Erie Co. , N. Y. , blue Lbi. 16250 6650 12250 3650 13504 13062 7612 9687 20025 9850 u 700 6950 4350 17725 7250 10250 8850 6250 7450 9687 6800 13500 10700 6250 12000 ?i5o 750 5000 " light Westchester Co N Y MARBLE. East Chester, N. Y. Millstone Point, Conn New London, Conn " gray Dorset, Vt Mill Creek, 111., drab North Bay Wis drab . . . Westerly, R. I., gray Fall River Mass gray SANDSTONE. Little Falls, N. Y. , brown Belleville N J , gray Garrisons, Hudson River, gray. . Duluth Minn dark Middletown, Conn., brown Keene, N. H., bluish gray Used in Central Park, N. Y., red Jersey City, N. J., soap Medina N Y pink LIMESTONE. Glen's Falls, N. Y Vermillion, O. , drab Fond du Lac, Wis., purple Marquette, Mich., u Seneca red brown Lake Champlain, N. Y. Cana.job.arie N Y Cleveland, 0. , olive green Albion, N. Y, brown Kasota, Minn. , pink Fontenac, Minn., light buff. .... Craigleth Edinburgh Garrisons '* Marblehead, 0. , white Joliet 111 white Lime Island, Mich. , drab . . . . | Sturgeon Bay, Wis. , bluish drab Dorchester, N. B., freestone. . . . Massillon, O., yellow drab Warrensburg, Mo., bluish drab. 768 STRENGTH OF MATERIALS. CRUSHING. To Compute Crushing "Weight of Columns, Deduced by Mr. L. D. B. Gordon from Results of Experiments of various Authors METALS. Cast Iron. (Hodgkinson.) Solid or Hollow. Round, r = W. Rectangxalar, = W. ' 400 500 For L, T, U, T, etc., put ' 9 a 2 (Unwin). "Wr ought Iron. (Stoney.) ~1 Solid or Hollow. Round, ^- = W. Rectangular, = W. 2400 7000 Steel. (Baker.) Solid. Strong steel. 51 a 51 a Round, = W. Rectangular, i=W. i-\ i-f 900 1600 Solid. MM steel. 30 a 30 a Round, ^~ = w - Rectangular, = W. 1400 "~ 2480 a representing area of section of metal in sq. ins., r ratio of length to least external diameter or side in like terms, and W crushing weight in tons. ILLUSTRATION. What is the crushing weight of a hollow cylindrical column of cast iron, 10 ins. in diameter, 20 feet in length, and i inch in thickness? a =. area of 10 ins. area of 10 i X 2 = 28.28 ins. r = = 24, and 24* 36X28.28 1018.08 = 576. Then, - = 417. 25 tons = 934 640 Ibs. Safe Loads. Cast Iron, one fifth. Wrought Iron or Steel, one fourth. WOODS. (C. Shaler Smith. ) C a \V C representing coefficient of material, a area of section (~yr^ ' in sq. ins. , I length, and d diameter or least side, both in ) X -004 like terms, and W crushing weight in Ibs. Coefficients.* For Crushing Stress per Sq. Inch of Section. Hemlock 3100 I White Pine 3500 I Georgia Pine 5000 Spruce 3500 I Yellow Pine 5000 | Oak, White 6000 (Hodgkinson.) Ash 9000 I Beech 7050 I Elm 7000 " Canadian 7000 | Cedar. 5100 | " rock 10000 ILLUSTRATION. Assume a Yellow-pine column 10 ins. square and ia ft. in length. 5000 X io 2 500000 2 73 373 * I+ ("/<^) 2 X.oo4 Safe Load.* One fifth. (Department of Buildings, City of New York.) STRENGTH OF MATERIALS. CRUSHING. 769 To Compute Safe Load, of Columns, it Iron. ) 80000 a Cai Round, or Rectangular, Solid or Hollow. "Wrought Iron. 40000 a = W. For Mild Steel put 48 ooo, and for Strong or Hard put 60000. a representing area of section in sq. ins. , I length of column in ins. , r radius oj Gyration =/ , I moment of Inertia (see p. 819), C coefficient, and W safe load in Ibs. Coefficients. Rou Solid. ND. Hollow. RECTANC Solid. ULAR. Hollow. Cast Iron .000164 .000047 .000022 .oooo .000272 .000059 .000035 .000087 .000189 .000049 .000033 . ooo 006 .000267 .000047 .000081 .OOO TCC Wrought Iron Steel, Mild Do. Strong..., ILLUSTRATION. What is the safe load for a Cylindrical and Hollow Cast-iron col- umn, 10 ins. in external diameter, 8 ins. internal, and 20 feet in length ? Area = 28. 28 sq. ins. I = 5 4 4 4 X . 7854 = 289. 8. r = /^f = 3. 22. V 20.20 80 000 X 28. 28 2 262 400 - -- . 00027 + SSSoX.ooo,7] -=181 zoo 2. Assume a solid column of Strong Steel of like diameter and 15 feet in length. Area = 78. 54 sq. ins. , and r = 2. 5. 78.54 X 60000 4712400 4712400 fi , 4 X[i<" 5184 X -000053] 4 X> 7 2 5 Ibs. 4 X i-f I -)X -ooo For Relative Value of various Woods and Comparison of Long and Short Col- umns, see page 976. Weight TDorne with Safety- t>y Solid Cast-iron Columns. In looo Lbs. (New Jersey Steel and Iron Co.) Length. Feet. 2 Ins. & Ins. In 5 , 6 Ins. m 7 , iJlJLl 8 Ins. 1KTKK In 9 s. 10 Ins. ii Ins. 12 Ins. 13 Ins. 14 Ins. 15 Ins. 5 12.4 44 I O2 184 288 414 560 728 916 1126 1354 _ 6 9.4 36 88 264 S3 2 884 1082 1320 157 - 7 7.2 30 76 146 242 360 502 660 850 1056 1282 '530 1798 2086 8 24 66 130 218 332 470 630 812 1016 1240 1486 1754 2040 9 20 56 114 198 306 44 59 6 774 974 1196 1440 1706 1992 18 48 102 1 80 282 410 5 6o 739 932 1152 1392 1656 1940 2 38 80 136 238 354 494 658 846 1056 1292 1550 1828 4 6 - 28 6 4 52 122 100 2OO 170 304 262 432 378 586 520 966 878 1192 1094 1440 1332 1712 8 44 8 4 144 226 332 462 616 796 1000 1:228 1482 20 72 124 196 292 410 552 720 912 1130 1372 For Tn"bes or Hollow Columns. Subtract weight that may be borne by a column, of diameter of internal diameter of tube from external diameter, and remainder will give weight that may be borne. Thickness of metal should not be less than one twelfth diameter of column. ILLUSTRATION. Required the safe load of a solid cast-iron column 6 ins. in diam- eter and 20 feet in length. Under 6 and in a line with 20 is 72, which x 1000 = 72 ooo Ibs. NOTE. This is about one sixth of destructive weight. 3 A 77O STRENGTH OF MATERIALS. DEFLECTION. DEFLECTION. Deflection of Bars, Beams, Grirders, etc. Experiments of Barlow upon deflection of wood battens determined, that deflection of a beam from a transverse strain, varied directly as cube of length and inversely as breadth and cube of depth, and that with like beams and within limits of elasticity it was directly as the weight. In bars, beams, etc., of an elastic material, and having great length com- pared to their depth, deductions of Barlow will apply with sufficient accu- racy for all practical purposes ; but in consequence of varied proportions of depth to length, of varied character of materials, of irregular resistance of beams constructed with scarphs, trusses, or riveted plates, and of unequal deflection at initial and ultimate strains, it is impracticable to deduce any exact laws regarding degrees of deflection of different and dissimilar figures and proportions. From an experiment of Mr. Tredgeld it was shown that deflection of cast iron is exactly proportionate to load until stress reaches a certain magnitude, when it becomes irregular. In experiments of Hodgkinson, it was further shown that sets from de- flections were very nearly as squares of deflections. In a rectangular bar, beam, etc., position of neutral axis is in its centre, and it is not sensibly altered by variations in amount of strain applied. In bars, beams, etc., of cast and wrought iron, position of neutral axis varies in same beam, and is only fixed while elasticity of beam is perfect. When a bar, beam, etc., is bent so as to injure its elasticity, neutral line changes, and continues to change during loading of beam, until its elasticity is destroyed. When bars, beams, etc., are of same length, deflection of one, weight being suspended from one end, compared with that of a beam Uniformly Loaded, is as 8 to 3 ; and when bars, etc., are supported at both ends, deflection in like case is as 5 to 8. Whence, if a bar, etc., is in first case supported in middle, and ends permitted to deflect, and in second, ends supported, and middle permitted to descend, deflection in the two cases is as 3 to 5. Of three equal and similar bars or beams, one inclined upward, one down- ward, at same angle, and the other horizontal, that which has its angle up- ward is weakest, the one which declines is strongest, and the one horizontal is a mean between thetwo. When a bar, beam, etc., is Uniformly Loaded, deflection is as weight, and approximately as cube of length*or as square of length ; and element of de- flection and strain upon beam, weight being the same, will be but half of that when weight is suspended from one end. Deflection of a bar, beam, etc., Fixed at one End, and Loaded at other, compared to that of a beam of twice length, Supported at both Ends, and Loaded in Middle, strain being same, is as 2 to i ; and when length and loads are same, deflection will be as 16 to i, for strain will be four times greater on beam fixed at one end than on one supported at both ends ; there- fore, ah 1 other things being same, element of deflection will be four times greater ; also, as deflection is as element of deflection into square of length, then, as lengths at which weights are borne in their cases are as i to 2, de- flection is as i : 2 2 x 4= i to 16. Deflection of a bar, beam, etc., having section of a triangle, and supported at its ends, is .33 greater when edge of angle is up than when it is down. In order to counteract deflection of a beam, etc., under stress of its load, where a horizontal surface is required, it should be cambered on its upper surface, equal to computed deflection. STRENGTH OF MATERIALS. DEFLECTION. 77! Safe Deflection. One fortieth of an inch for each foot of span, with a factor of safety for load of .33 of destructive weight = y^u, but for ordinary loads and purposes, Cast Iron, y^^ to -^^ ; and Wrought Iron, y^j- to ^rinF or TTDTF after beam, etc., has become set. When Length is uniform, with same weight, deflection is inversely as breadth and square of depth into element of deflection, which is inversely as depth. Hence, other things being equal, deflection will vary inversely as breadth and cube of depth. ILLUSTRATION. Deflections of two pine battens, of uniform breadth and depth, and equally loaded, but of lengths of 3 and 6 feet, were as i to 7.8. Deflection of different bars, beams, etc., arising from their own weight, having their several dimensions proportional, will be as square of either of their like dimensions. NOTE. In construction of models on a scale intended to be executed in full di- mensions, this result should be kept in view. When a continuous girder, uniformly loaded, is supported at three points by two equal spans, middle portion is deflected downwards over middle bear- ing, and it sustains by suspension the extreme portions, which also have a bearing on outer bearings. Middle portion is, by deflection, convex up- wards, and outer portions are concave upwards ; and there is a point of "contrary flexure," where curvature is reversed, being at junction of con- vex and concave curves, at each side of middle bearing. This point is dis- tant from middle bearing, on each side, one fourth of span. Of remaining three fourths of each span, a half is borne by suspension by middle portion, and a half is supported by abutment. Hence, distribution of load on bear- ings is easily computed, as given above. Deflection of each span is to that of an independent beam of same length of span as 2 to 5. In a beam of three equal spans, deflection at middle of either of side spans is to that of an independent beam as 13 to 25. In a long continuous beam, supported at regular intervals, deflection of each span is to that of an independent beam of one span as i to 5. Cylinder. If a bar or beam is cylindrical, Barlow gives the deflection 1.7 times that of a square beam, other things being equal ; D. K. Clark puts it at 1.47. Formulas fbr Deflection of Beams of Rectangular Sec- tion, etc. Loaded at One End. ^^ n = D. Loaded Uniformly. * b ^ = D. Both i s.) Loaded in Middle. = D Looted Uniformly. Ends. Supported at Both Ends. 5 = Loaded in Middle. - = D. Loaded Uniformly. = D. 16 6 di C 8 X 16 6 d3 C m 2 n 2 W Loaded at any one Point. ^ = D. Supported in Middle. Ends Loaded Uniformly. - 3 f , W l7 - = D. 5 X ID o u j O I representing length in feet, b breadth, and d depth, both in ins., W weight or stress in Ibs., m and n distances of weight between supports, C a constant, and D deflection in ins. 7/2 STRENGTH OF MATERIALS. DEFLECTION. Deflection of Beams or Bars of Rectangular iSection. To Compute Deflection of a Rectangular Beam or Bar. Supported at Both Ends. Loaded in Middle. CAST IRON. 73 vy Rectangular Beams. . 7-77 = D - Cylindrical. For 36 ooo put 24 ooo. 36 ooo b d* I representing length in feet, b and d in inches. ILLUSTRATION Assume a rectangular bar of cast iron, i inch square and loaded with 224 Ibs., 4.5 feet between its supports. 224 X 4- 5 _ 20412 _ inc ^ 36 ooo X i X i 3 36 ooo By actual experiment of Mr. Hodgkinson the deflection was .561 inch. WROUGHT IRON. 3 W Rectangular Beams. =- T . - D. Cylindrical. For 60000 put 42000. 60 ooo & d3 WOODS. 73 TIT ^ = D. I representing length in inches, and W weight in tons. Mean of LasleWs, Barlow, etc. Ash Canadian.... c . . . 1476 c Iron- wood 4228 Oak, French .... C ... 2656 " Eng Larch ... . 2100 Beech 2418 Mahogany Honduras 2118 Pitch pine ... 2968 Blue Gum " Mexican 3608 Elm 1227 u Spanish.. 3360 Fir Dantzic Oak Baltimore 2761 Spruce . . . 3300 " Riga " Canadian .... 3445 Greenheart. . . ... 1888 " Eng. .. .. 1848 Yellow pine ... 2084 ILLUSTRATION. What is the deflection of a floor beam of Yellow pine, 3 by 12 ins., 12 feet between its supports, under a uniformly distributed load of 3000 Ibs.? 8 X 3 X i23 x 2084 X .216 18668415 zr: ;= 1.25 tons. 5X2985984 14929920 * % for being uniformly distributed. By a test of a like beam, the deflection was .2125. Z3 W For Cylindrical Beams deduct one-third from these constants, or -~. _ = D. 3.14^4 C For Torsional Deflection of Iron Shaft. (D. K. Clark.) Cast Iron, - = D. Wrought Iron, ^= D. 044 a4 I 37 4 W in tons and r radius or distance of applied power. Deflection of Continuous Girders or Beams. Beams of Uniform Dimensions, Supported at Three or More Bearinga. (D. K. Clark.) 2. Three Equal Spans or 4 Bearings. Weight on ist and 4th bearing .4 W I " " 2 d " sd " = i.i Wl 3. Four Equal Spans or 5 Bearings. Weight on ist and sth bearing = .39 W I \ Weight on 2d and 4th bearing = 1.14 W I Weight on sd bearing = .93 W I. i. Two Equal Spans or 3 Bearings. Weight on ist and 3d bearing . 375 W I " 2d bearing STRENGTH OF MATERIALS. DEFLECTION. 773 To Compxite IVTaximnm Load, that may "be "borne "by a Rectangular Beam. Deflection not to exceed Assigned Limit of one hundred and twentieth of an Inch for each Foot of Span. Supported at Both Ends. Loaded in Middle. j = W. b and d representing breadth and depth in ins., I length in feet, C con- stant, and W weight or load in Ibs. Constants. Cast Iron . 0003 Oak white 027 Oak red oto Wrought Iron 0021 Hickory 018 Pine pitch ... . 033 Pine white. " O3Q Teak 024 u vellow . .006 Chestnut, horse . . . borne by a beam o ed in its middle? = ^m.^ Ibs. .. .051 r white ILLUSTRATION. What is maximum load that may be pine, 3 by 12 ins., 20 feet between its supports, and load 3 X i2 3 5184 C .0^0. Then r-=-P ^4 = WROUGHT IRON. Deflection of "Wrovight-iron Bars. Supported at Both Ends. Weight applied in Middle. i. A Weight and Deflection all ' No. FORM. s 1 by Actual Observation. at one sixth of Destruc- tive Weight. at 1 thof an Inch for each Foot of Span. Constant Reduced W and Deflec W/3 Feet. Ins. Ins. Lbs. Ins. Lbs. Ins. Lba. Ins. C i. American. K 1.83 I i 600 .06 266 .027 148 .015 X 2. English... " 2-75 2 2 4480 .08 1310 .022 1310 .022 1.29 3. " .... 2.75 1-5 2.5 8960 .104 2128 .025 1873 .022 1.25 4 . .... " 2.75 i-5 3 8960 .088 3800 .037 2259 .022 .88 To Compute Deflection of*, and 'Weight that may "be toorne Toy, a .Rectangular Bar or Beam of Wrought Iron. - C. 6oooo6d 3 D~~ 60000 b'd*'C ILLUSTRATION. What weight will a beam 2 ins. in breadth, 5 ins. in depth, and 15 feet between its supports, bear with safe deflection of -j-W of an inch for each foot of space, or -j-^ of its length ? C from table = .88. D = y^ of 15 = . 12 inch. 6ooooX2X5 3 X.88x.i2 -- D. K. Clark gives for Elastic deflection, 47 ooo for Rectangular bars, and 32 ooo for Cylindrical. NOTE. Deflection of ^-^ to ^-J-^ of the length may be allowed under special cir- cumstances ; but under ordinary loads the deflection should not exceed one fourth of these, as ^-^ to -^far. Practice in U. S. is to allow i2 1 OQ after girder has taken its permanent set. In small bridges there is a slight increase in deflection from high speeds, about .166 or . 144 of the normal deflection, with the same load moving at slow speed. In large girders there is no perceptible difference between the deflection at high and low speeds. 774 STRENGTH OF MATERIALS. DEFLECTION. Deflection of Wronght-iron Rolled. Beai Supported at Both Ends. Weight applied in Middle. 70000 d 2 (4 a + 1.155 a') D . Flanges. Weight and Deflection No. FORM. I Width. Mean Thick- ness. Web. Depth. by Actual Observation. at one sixth of Destructive Weight. C I Feet. 10 IDS. 3 Inch. .485 Inch. 5 Ins. 7 Lbs. 12000 4 Lbs. 3800 Inch. .127 1.05 2. u 3- " 20 2O 4.6 5-7 .8 643 5 .6 9- 8 5 11.75 l6oOO 20000 1.15 .85 6300 8000 453 34 .92 .98 To Compute Deflection of, and. "Weight that may "be "borne "by,a"Wrovight-iron. Rolled Beam of TJnifbrxn and Sym- metrical Section. Supported at Both Ends. Weight applied in Middle. (D. K. Clark.) 70000 d 2 (4 a + i . 155 a') D _ -=W. 70000 d 2 (4 a-f- 1.155 ') I representing span in feet, d reputed depth, or depth less thickness of lower flange in ins., a area of section of lower flange, a' area of section of web for reputed depth of beam, both in sq. ins., and W weight or stress in Ibs. ILLUSTRATION. What is deflection of a wrought-iron rolled beam of New Jersey Steel and Iron Co., 10.5 ins. in depth, flanges 5 by .5 ins., and width of web .47 inch, when loaded in its middle with 8000 Ibs., and supported over a span of 20 feet? d = 10.5 .5 = 10 ins., a 5 X -5 = 2. 5 sq. ins., and a' = 10 X -47 4-7 sq. ins. 8oOO X 2O 3 64OOOOOO Then - = ' = .59 inch. 70000 X io 2 X (4 X 2.5 + 1.155 X 4-7) I07999 5 If weight is uniformly distributed, divide by 112 500 instead of 70000. A like beam 6 ins. in depth, loaded with 2608 Ibs., and supported over a span of 12 feet, gave by actual test a deflection of .3 inch, and by above formula it is also .3 inh. NOTE. Deflection for such a beam, for a statical weight or stress of 17 100 Ibs., uniformly distributed, by rules of N. J. Steel and Iron Co. , would be . 54 inch, which, with difference in weights, will make deflections alike. Deflection of \Vronght-iron Riveted 13 earns. Supported at Both Ends. Weight applied in Middle. -J4- T, = C at Reduced Weight and Deflection. No. FORM. Length. Flanges. Angles. Web. Feet. Ins. Ins. 2.125X2 Inch. I 'I 7 X-28 2.125X2 X.2 9 }, 4-5X 2X2 'I n.66 5 4-5X 375 X-3I25 2X2 X-3'25 25 4-5X 2X2 3- " 28-5 5 7 X X-375 3X3 375 5 X-4375 16.5 Weight and bv Actual Observation. Deflection at one sixth of Destructive Weight. C Lbs. Inch. Lbs. Inch. 4216 .1 4062 .096 63 77280 .46 12880 075 1.96 "5584 .875 19265 .148 3-86 STBENGTH OF MATERIALS. DEFLECTION. 775 To Compnte Deflection of, and "Weight that may "be "borne by, a Riveted Beam of "Wrought Iron. r--, -7K = D. ,68000 (^-^- + WcD W. a, a', and a" representing areas of upper and lower flanges with their angle pieces, and of web for its entire depth, all in sq. ins. NOTE. If there are not any flanges, as in No. i, angle pieces alone are to be computed for flange Area. ILLUSTRATION. What weight will a riveted and flanged beam of following dimen- sions sustain, at a distance between its supports of 25 feet, and at a safe deflection of . 2 inch or ^^ of its length ? Top flange 6 X -5 ins. \ Web 5 in*. Bottom flange 6X-5 " | Depth 17 " Angles * 2.25 X 2.25 X .5 ins. a and a' each = 6 X 5 = 3 + 2.25 + 2.25 .5 x .5 X 2 = 7 sq. ins. a" . 5 x 17 = 8. 5 sq. ins. C, as per No. 2,= .43, but inasmuch as flanges in thii case are much heavier, assume .5. 168000 ( Then- 2X.5 25 s Strength of a Riveted beam compared to a Solid beam is as i to 1.5, while for qual weights its deflection is 1.5 to i. Tu.~bular GJ-irders. "Wrought Iron. Supported at Both Ends. Weight applied in Middle. c. g g 1 til j Depth. ^ 1 111 8 No. SECTION. tL-.'C o *& ? S 1 * Inter- Ex- ' q~ C 00 "SCO ^ J M nal. ternal. 1 Q ojj Feet. Ins. Ins. Ins. Lbs. Ins. Inch. ( t. Thickness .03 inch 3-75 1.9 2-94 3 448 .z .03 28! a. " " .525 " 30 15-5 22.95 24 33685 -56 .24 47: top .372 " ) 3. ' bottom .244 " 30 16 23.28 24 32538 i. ii .24 22. sides .125 " ) 4. " Thickness .75 " 45 24 34-25 35-75 128850 1.85 .36 362 5. fj Thickness .0375" '7 12 11.925 12 2755 .65 .136 6s 6. f\ " .0416" 7- U " -'43 " 17 17 9.25 , 9- 2 5 13-535 14.714 13.62 15 2262 16800 .62* 1.39* .136 .136 4' "S * Destructive weight. To Compnte Deflection of, and "Weight that may "be loorne by, a \Vroxight-iron Txabxilar G-irder. 16 & d 3 C D W I 3 ILLUSTRATION. What weight may be safely borne by a wrought-iron tube, alike > No. 3 in preceding table, for a length of 30 feet, and a deflection of .32 inch ? to N < 24 3 x 224 Xv2 4 _ 190253629 _ ^ 3 27000 ' Flanged Rails. Deflection of Iron and Steel Flanged Rails within their elastic limit, comparad with their transverse strength, is as 17 to 20, and with double-headed it is as n to 23. 7/6 STRENGTH OF MATERIALS, DEFLECTION. RAILS. Supported at Both Ends. Weight applied in Middle. Iron. No. FORM. rl Head. Bottom. Weight Y?rd. I Ares. Observed Weight and Deflection. Destructive Weight and Deflection. Feet. Ins. Ins. Lbs. Ins. S 1 " Bessemer. .875 75 ( 103 600 51800 5 i 17 (184800 II 950 82870 92400 43900 Pi 2 1.06 297400 44300 ??^ To Compute IPower to Punch, Iron, Brass, or Copper. RULE. Multiply product of diameter of punch and thickness of metal by 150000 if for wrought iron, by 128000 if for brass, and by 96000 if for cast iron or copper, and product will give power required, in Ibs. EXAMPLE. What power is required to punch a hole .5 inch in diameter in a plato of brass 25 inch thick? >g x -25 x I28ooo = l6ooo lbs , Comparison, "between IDetrusive and. Transverse Strengths. Assuming compression and abrasion of metal in application of a punch of one inch in diameter to extend to .125 of an inch beyond diameter of punch, comparative resistance of wrought iron to detruswe and transverse strain, latter estimated at 600 Ibs. per sq. inch, for a bar i foot in length, is as 3 to i. WOODS. Detrusive Strength of* Woods. Lbs. Spruce 470 Pine, white 490 Lba. Pine, pitch... 510 Hemlock 540 Ash Chestnut 690 Per Sq. Inch. Lbs. Lbs. 650 Oak 780 Locust 1180 To Compute Length of Surface of Resistance of "Wood to Horizontal Thrust. RULE. Divide 4 times horizontal thrust in Ibs. by product of breadth of wood in ins., and detrusive resistance per sq. inch in Ibs. in direction of fibre, and quotient will give length required. EXAMPLE. Thrust of a rafter is 5600 Ibs., breadth of tie beam, of pitch or Georgia pine, is 6 ins. ; what should be length of beyond score for rafter ? Assume strength 510 as above. Then ^ ^-^ = 224 = 7.32 ins. ^ 6 X 510 3060 STRENGTH OF MATERIALS. DETRUSIVE. 783 Shearing. "Wrought Iron. Resistance to shearing of American is about 75 per cent., and of English 80 per cent., of its tensile strength. Resistance to shearing of plates and bolts is not in a direct ratio. It ap- proximates to that of square of depth of former, and to square of diameter of latter. Results of* Experiments upon. Shearing Strength, of Various Metals t>y Parallel Cutters. Wrought Iron. Thickness from .5 to i inch, 50000 Ibs. per sq. inch. Made try Inclined Cutters, angle = 7. PLATES. ; ; Thickness. Power. BOLTS. Diam. Power. Brass Ins. 05 Lbs. 54O Brass Ins. I. II Lbs. 29700 6 775 ii 310 Steel 24. 14 030 Steel 775 28 720 Wrought iron ] 51 39 x 5o Wrought iron i 32 3093 I 44800 \ 1.142 354"> Result of Experiments in Shearing, made at tJ. S. Navy Yard, Washington, on "Wrought-iron Bolts. Diam. Minimum. Stress. Maximum. PerSq.Inch. Diam. Minimum. Stress. Maximum. PerSq.Inch. Inch. 5 75 Lbs. 8900 18400 Lbs. 9400 19650 Lbs. 44149 39553 Mean 4 Inch. .875 i L 033 Ibs. Lbs. 25500 32900 Lbs. 27600 35800 Lbs. 41503 40708 Result of Experiments on .87"5 Inch "Wrought-iron. Bolts. (E. Clark.) Lbs. Single shear 54096 Double " 46904 Tons. 24.15 Double shear of two .625-inch plates riveted together (one section) .... 45 696 Tensile strength 50 176 Ibs. Riveted Joints. Experiments on strength of riveted joints showed that while the plates were destroyed with a stress of 43 546 Ibs., the rivets were strained by a stress of 39088 Ibs. Cast Iron. Resistance to shearing is very nearly equal to its tensile strength. An average of English being 24000 Ibs. per sq. inch. Steel. Shearing strength of steel of all kinds (including Fagersta) is about 72 per cent, of its tensile strength. Treenails. Oak treenails, i to 1.75 ins. in diameter, have an average shearing strength of 1.8 tons per sq. inch, and in order to fully develop their strength, the planks into which they are driven should be 3 times their diameter. Woods. When a beam or any piece of wood is let in (pot mortised) at an inclina- tion to another piece, so that thrust will bear in direction of fibres of beam that is cut, depth of cut at right angles to fbres should not be more than .a of length of piece, fibres of which, by their cohesion, resist thrust. 784 STRENGTH OF MATERIALS. PENSILE. TENSILE STRENGTH. Tensile Strength is resistance of the fibres or particles of a body to separation. It is therefore proportional to their number, or to area of its transverse section, and in metals it varies with their temperature, generally decreasing as temperature is increased. In silver, tenacity decreases more rapidly than temperature ; and in copper, gold, and plat- inum less rapidly. Cast Iron- Experiments on Cast-iron bars give a tensile strength of from 4000 to 5000 Ibs. per sq. inch of its section, as just sufficient to balance elasticity of the metal; and as a bar of it is extended the 12300^1 part of its length for every 1000 Ibs. of direct strain, or one sixteenth of an inch in 64.06 feet per sq. inch of its section, it is deduced that its elasticity is fully excited when it is extended less than the 24 tons (14 per cent.) per sq. inch of section. Hot and Cold Blast. Mr. Hodgkinson deduced from experiments that relative strength o* 1.2 and 3 ins. square was as 100, 80, and 77, and that hot blast had less tensile strength than cold blast, but greater resistance to a crushing stress. Captain James ascertained that tensile strength of .75 inch bars, cut out of 2 and 3 inch bars, had only half strength of a bar cast i inch square. Mr. Robert Stephenson concluded, from experiments of recent date, that average strength of hot blast was not much less than that of cold blast ; but that cold blast, or mixtures of cold blast, were more regular, and that mixt- ures of cold blast and hot blast were better than either separate. Stirling's M.ixed or Toxighened Iron. By mixture of a portion of malleable iron with cast iron, carefully fused in a crucible, a tensile strain of 25 764 Ibs. has been attained. This mixt- ure, when judiciously managed and duly proportioned, increases resistance of cast iron about one third ; greatest effect being obtained witfc a propor- tion of about 30 per cent, of malleable Iron. Malleable Cast Iron. Tensile strength of annealed malleable is guaranteed by some Manufact- urers of it at 56000 Ibs. ; it is capable of sustaining 22400 Ibs. without per- manent set. "Wrought Iron. Experiments on English bars gave a tensile strength of from 22000 Ibs. to 26 400 Ibs. per sq. inch of its section, as just sufficient to balance elasticity of the metal; and as a bar of it is extended the 2 8oooth part of its length for everv 1000 Ibs. of direct strain, or one sixteenth of an inch in 116.66 teet per sq. inch of its section, it is deduced that its elasticity is fully excited when it is extended the loooth part of its length, and extension of it at its limit of elasticity, which is from .45 to .5 of its destructive weight, is esti- mated at i52oth"part of its length. A bar will expand or contract .000006614 inch, or 151 200 part of its length for each degree of heat; and assuming, as before stated for cast iron, that extreme range of temperature in air in this country is 140, it will contract or expand with this change .000926, or io8oth of its length, which is equiva- lent to a force of 20 740 Ibs. (9.25 tons) per sq. inch of section. Mean tensile strength of American bars and plates (45000 to 76000), 60 500 Ibs. (27 tons) per sq. inch of section ; as determined by Prof. Johnson in 1836, is 55 900 Ibs. ; and mean of English, as determined by Capt. Brown, Barlow, Brunei, and Fairbairn, is 53 900 Ibs. ; and by Mr. Kirkaldy, bars and plates (47040 to 55910) 51 475 Ibs. (22.97 tons). a U* ;86 STRENGTH OF MATERIALS. TENSILE. Greatest strength observed 73449 Ibs. (3 2 -79 tons). Ultimate strength, as given by Mr. D. K. Clark, 59 732 Ibs. (26.66 tons). Average ultimate extension is 6ooth part of its length. Strength of plates, as determined by Sir William Fairbairn, is fully 9 per cent, greater with fibre than across it. Resistance of wrought iron to crushing and tensile strains is, as a mean, as 1.5 to i for American; and for English 1.2 to i. Reheating. Experiments to determine results from repeated heating and laminating, furnished following : From i to 6 reheatings and rollings, tensile stress increased from 43904 Ibs. to 61 824 Ibs., and from 6 to 12 it was reduced to 43904 again. Effect of Temperature. Tensile strength at different temperatures is as follows: 60, i; 114, 1.14; 212, 1.2; 250, 1.32; 270, 1.35; 325, 1.41; 435, 1-4- Experiments of Franklin Institute gave at 80 56000 Ibs. I 720 55 ooo Ibs. I 1240 22000 Ibs. 570 66500 " | 1050 32000 " | 1317 9000 " Annealing. Tensile strength is reduced fully i ton per sq. inch by an- nealing. Cold Rolling. Bars are materially stronger than when hot rolled, strength being increased from one fifth to one half, and elongation reduced from 21 to 8 per cent. Hammering increases strength in some cases to one fifth. Welding. Strength is reduced from a range of 3 to 44 per cent. 20 per cent., or one fifth, is held to be a fair mean. Temperature. From o to 400 strength is not essentially affected, but at high temperature it is reduced. When heated to redness its strength is re- duced fully 25 per cent. Tensile strength at 23 was found to be .024 per cent, less than at 64, Cutting Screw Threads reduces strength from n to 33 per cent. Hardening in water, oil, etc., reduces elongation, but does not essentially increase the strength. Case Hardening reduces strength fully 10 per cent. Galvanizing does not affect strength of plates. Angled Bars, etc. Their strength is fully 10 per cent, less than for bolts and plates. Elements connected with Tensile Resistance of various Substances. af| M If gf 1? If rf SUBSTANCES. ^1 *$$ SUBSTANCES. S-g^ 25| *~ NS w.S ||l as li ii 1 Lbs. Lbs. Beech 3 355 .5 Wrought iron ordinary. Cast iron English J " " Swedish .... ' , 3 ** " American Oak 5000 2856 .2 " " English 24 400 (18850 34 35 Steel plates, . 5 inch 52000 '.11 " " American. . . 15000 .26 75 700 ' "wire, No. 9, unannealed Yellow pine. . . " " " annealed . . 47 53 Z ID 1OO !'*< Turning. Removing outer surface does not reduce the strength of bolts. STRENGTH OF MATERIALS. TENSILE. 787 TIE-BODS. Results of Experiments on. Tensile Strength. of Wrough.t~ iron Tie-rods. Common English Iron, 1. 1875 Ins. in Diameter, DESCRIPTION OF CONNECTION. I Breaking Weight. Semicircular hook fitted to a circular and welded eye Two semicircular hooks hooked together Right-angled hook or goose-neck fitted into a cylindrical eye Two links or welded eyes connected together Straight rod without any connective articulation Lbs. 14000 16 220 29120 48160 56000 Ratio of Ductility and. IMallea'bility- of Metals. In order of Wire-drawing Ductility. In order of Laminable Ductility. In order of Wire-drawing Ductility. In order of Laminable Ductility. In order of Wire-drawing Ductility. In order of Laminable Ductility. Gold. Silver. Platinum. Iron. Copper. Zinc. Tin. Lead. Nickel. Gold. Silver. Copper. Tin. Platinum. Lead. Zinc. Iron. Nickel. Relative resistance of Wrought Iron and Copper to tension and compres- sion is as 100 to 54.5. Steel. Experiments of Mr. Kirkaldy, 1858-61, give an average tensile strength for bars of 134 400 Ibs. (60 tons) per sq. inch for tool-steel, and 62 720 Ibs. (28 tons) for puddled. Greatest observed strength being 148 288 Ibs. (66.2 tons). Plates, mean, 86800 Ibs. (32 to 45.5 tons) with fibre, and 81 760 Ibs. (36.5 tons) across it. Its resistance to crushing compared to tension is as 2.1 to i. Hardening. Its strength is very materially increased by being cooled in oil, ranging from 12 to 55 per cent. Crucible. Experiments by the Steel Committee of Society of Civil Engineers, England, 1868-70, give a tensile strength of 91 571 Ibs. per sq. inch (40.88 tons), with an elongation of .163 per cent., or i part in 613, and an elastic extension of .000034 7th part for every 1000 Ibs. per sq. inch, or i part in 28818. Bessemer. Experiments by same Committee give a tensile strength of 76653 Ibs. per sq. inch (34.22 tons) with an elongation of .144 per cent., or i part in 695, and an elastic extension of .oooo3482d part for every IOOG Ibs. per sq. inch, or i part in 28 719. Result of Experiments by Committe of Society of Civil Engineers of England, 1868-T'O, and Mr. Daniel Kir- Isaldy, IS-rS. Per Sq. Inch. STIEL. Elastic Strength. Elastic in Parts of Length. xtension per 1000 Lbs. Ratio of Elastic to Ultimate Strength. Destrnctire Weight. Crucible Lbs. 49840 44800 48608 39200 32080 28784 Tons. 22.25 20 21.7 '7-5 14.56 12.85 Per Cent. .225 .204 In Length. .0005 .00045 Per Cent. 58.2 59 59- 2 5i-5 46.4 44-4 Lbs. 86464 75757 78176 72576 69888 64512 Tom. 38.6 33-82 34-9 32.4 31.2 28.8 Bessemer Fagersta, unannealed . " annealed.... Siemens, unannealed. . " annealed.... 788 STRENGTH OF MATERIALS. TENSILE. Average Tensile Elasticity of Steel Bars and. Plates. (jCom. of Civil Engineers, 1870. ) DESCRIPTION. Elasticity per Sq. Inch. Elastic Exten- sion in Parts of Length. Ratio of Elas- tic to Destruc- tive Strength. Bars. Lbs. 5 557 Parts. i in 485 Per Cent. 58.2 43814 i in 675 55 56 560 64.8 34048 55-6 " hciiiitrH'rcd and roll6d 55 574 64.7 *' " * 4 annealed AO 8t:8 54 " plates unannealed- ao 710 i in 980 SO- 2 26 O4O i in 1020 56.5 32 500 46.4 28780 44.4 40 174 58.8 Kruno's shaft. .. 4.2 112 i in 181; Tensile strength of steel increases by reheating and rolling up to second operation, but decreases after that. Tensile Strength, of Various Materials, deduced from Experiments of TJ. S. Ordnance department, ITair- 'bairn, liodgkinson, KLirkaldy, and Iby the Author. Power or Weight required to tear asunder One Sq. Inch, in Lbs. METALS. Lbs. Antimony, cast 1 053 Bismuth, cast 3 248 Cast Iron, Greenwood 45 970 " mean, Major Wade. . . 31 829 " gun- metal, mean 37232 " malleable, annealed.. 56000 " Eng. , strong 29 ooo " " weak 13400 \ 21 280 " gun-metal 23 257 " *' mean* 19484 " * Low Moor, No. 2 14076 " " Clyde, No. i.... 16125 * " " No. 3 23468 " " Stirling, mean.. 25764 Copper, wrought 34 ooo " rolled. 36 ooo " cast 24250 " bolt 36800 " wire 61200 Gold 20 384 Lead, cast 1 800 " pipe 2240 " " encased 3759 " rolled sheet 3320 Platinum wire 53000 Silver, cast 40000 Steel, cast, maximum 142 ooo " ** mean 88560 41 puddled, maximum 173 817 " Amer. Tool Co 179 980 (I wi - e f 210000 ^ \ 300000 1 plates, lengthwise 96 300 u crosswise .... 93 700 " Chrome bar 180 ooo METALS. Lbs. Steel, Pittsburgh, mean 94 450 Bessemer, rolled [-'#* " " hammered.. Eng., cast " plates, mean. plates puddled plates 152900 134000 93500 86800 62720 crucible 91570 homogeneous 96 280 blistered, bars 104 ooo Fagersta bars " plates... Whitworth's " Siemens's plates. . " " Krupp's shaft Tin, cast " Banca 2100 Wire rope, per Ib. w't per fathom 4 480 " " galvanized steel, " 6720 Wrought Iron, boiler plates. . . { ^ 500 rivets 63 ooo bolts, mean 60 500 " inferior 30000 hammered 54 ooo shaft 44 750 wire 73 600 ' No. 9 100000 u No. 20 120000 " diam. .0069 inch 301 168 " galv'ized.osS " 64960 Eng., heavy forging. 33600 a plates, lengthw'e 53800 " 4< crosswise 48800 By Comru's on application of Iron to Railway Structure. STRENGTH OF METALS. Wrought Iron, Eng., mean MAT Lba. 51000 57600 48800 65920 40000 3130 56000 63000 51760 595oo 49000 72000 48900 7000 16000 85120 IIOOO 71600 96320 3670 18000 49000 509 1 5 344 6 4 23500 33000 38080 36000 29000 18000 33600 42040 7 ooo 48700 14000 9500 16000 6300 14000 ii 500 15000 7000 23000 19000 ii 400 7500 ii 600 12500 10000 6000 12400 27000 6000 13000 18000 15860 12 000 II OOO 16000 17350 23000 ERIALS. TENSILE. WOODS. Larch | 789 LbB. 4200 9500 11800 16000 20500 21 OOO 8000 I2OOO 20333 2522Z 16500 16380 IO25O 9500 4500 4200 9860 19200 I4OOO 13000 IlSoO 13000 13300 7OOO 10833 IO29O 12400 9600 13000 15000 21000 7800 13000 8000 2300 550 1469 300 500 77 750 100 290 860* 393 7i3 948 1152 201 319 310 214 284 104 102 7 00 Lignum vitas ti Thames Locust " " armor-plates .... Mahogany, Honduras " bar { " " charcoal " " rivet, scrap 41 Russian, bar, best " Spanish { Oak, Pa. seasoned " Va., " " white " Swedish, " best.... u u u u live, Ala. " red " African " sheet { " English { ALLOYS OR COMPOSITIONS. Alloy, Cop. 60, Iron 2, Zinc 35, Tin 2 . " Dantzic Pear Pine, Va " Riga Aluminium Cop 90 ' ' yellow. " maximum " white Bell- metal " red Brass cast Poon " wire Poplar Bronze Phosphor extreme Redwood, Cal " mean..; Spruce, white... .. j " ordinary Cop. 10, Tin i I Sycamore \ f '' ;'-:::::: Teak, India *' 2, Zinc i Gun-metal ordinary Walnut, Eng " mean ' ' bars " Mich Speculum metal . .... Willow Yellow metal Yew WOODS. Ash, white Across Fibre. Oak Pine MISCELLANEOUS. Basalt, Scotch " English Bamboo Bay Beton, N. Y. Stone Con'g Co { Beech English Birch " Amer black Brick extreme " inferior < Bullet Cedar, Lebanon Cement Portland 7 days | " West Indian " American u pure, i mo Chestnut sand 2, 320 days. . pure, " " sand i, in water ) i mo. j " " 3, i year.... ;< 5,i " " " 7,1 " ..-. Cypress Ebony Elm { " Alabama Hickory " Rosedale, Ulst. Co. , 7 days " " sand i, 30 " (i it ( Holly 1 I 790 STRENGTH OF MATERIALS. TORSION. MISCELLANEOUS. Cement, Roman, in water 7 days Lbs. 90 MISCELLANEOUS. Lbs. " " i mo.. " " i year "5 286 " hydraulic .., .. f 8s " sand i, 42 days. 284 "^ UI *' t 130 " u 3 " 199 IDO Flax 25 ooo Glue 4OOO Granite 578 " fine green 1260 3 5 (( Ai-KrrtotVi 5^3 12000 Arbroatn 1261 16000 " Caithness x ^3 Leather belting 33 ** PnrflanH "57 i Limestone \ 070 rtland looo ( Marble statuary Silk fibre 52 ooo " 'Italian 5200 Slate.., 9 Marble, white " Irish... 9000 17600 Whalebone 7 ooo TOKSIONAL STRENGTH. SHAFTS AND GUDGEONS. Shafts are divided into Shafts and Spindles, according to their mag- nitude, and are subjected to Torsion and Lateral Stress combined, or to Lateral Stress alone. A Gudgeon is the metal journal or Arbor upon which a wooden shaft revolves. Lateral Stiffness and Strength. Shafts of equal length have lateral stiff- ness as their breadth and cube of their depth, and have lateral strength as their breadth and square of their depths. Shafts of different lengths have lateral stiffness directly as their breadth and cube of their depth, and inversely as cube of their length ; and have lateral strength directly as their breadth and as square of their depth, and inversely as their length. Hollow Shafts having equal lengths and equal quantities of material have lateral stiffness as square of their diameter, and have lateral strength as their diameters. Hence, in hollow shafts, one having twice the diameter of an- other will have four times the stiffness, and but double the strength ; and when having equal lengths, by an increase in diameter they increase in stiff- ness in a greater proportion than in strength. When a solid shaft is subjected to torsional stress, its centre is a neutral axis, about which both intensity and leverage of resistance increase as radius or side ; and the two in combination, or moment of resistance per sq. inch, increase as square of radius or side. Round Shaft. Radius of ring of resistance is radius of gyration of sec- tion, being alike to that of a circular plate revolving on its axis, viz., .7071 radius. The ultimate moment of resistance then is expressed by product of sectional area of shaft, by ultimate shearing resistance per sq. inch of material by radius, and by .7071. Or, .7854 d 2 r S X .7071 = .278 d* S = R W. (D. K. Clark.) d representing diameter of shaft and r radius, S ultimate shearing stress of mate- rial in Ibs. per sq. inch, R radius through which stress is applied, in in*., and W moment of load or destructive stress, in Ibs. STKENGTH OF MATERIALS. TORSION. 791 Round Shaft. Strength, compared to a square of equal sectional area, is about as i to .85. Diameter of a round section, compared to side of square section of equal resistance, is as i to .96. Square Shaft. Moment of torsional resistance of a square shaft exceeds that of a round of same sectional area, in consequence of projection of cor- ners of square ; but inasmuch as material is less disposed to resist torsional stress, the resistance of a square shaft, compared to a round one of like area of section, is as i to 1.18, and of like side and diameter, as 1.08 to i. Hence, !l * 8x * ol "' 8 = W. Hott Round ShajU. -^ (d-d') S = w When Section is comparatively Thin. 7 = W. * representing side, d and d' external and internal diameters, and t thickness of metal in ins. Torsional Angle of a bar, etc., under equal stress, will vary as its length. Hence, torsional strength of bars of like diameters is inversely as their lengths. Stress upon a shaft from a weight upon it is proportional to product of the parts of shaft multiplied into each other. Thus, if a shaft is 10 feet in length, and a weight upon centre of gravity of the stress is at a point 2 feet from one end, the parts 2 and 8, multiplied together, are equal to 16 ; but if weight or stress were applied in middle of the shaft, parts 5 and 5, multiplied together, would produce 25. When load upon a shaft is uniformly distributed over any part of it, it is consid- ered as united in middle of that part; and if load is not uniformly distributed, it is considered as united at its centre of gravity. Deflection of a shaft produced by a load which is uniformly distributed over its length is same as when .625 of load is applied at middle of its length. Resistance of body of a shaft to lateral stress is as its breadth and square of its depth ; hence diameter will be as product of length, of it, and length of it on one side of a given point, less square of that length. ILLUSTRATION. Length of a shaft between centres of its journals is 10 feet; what should be relative cubes of its diameters when load is applied at i, 2, and 5 feet from one end? and what when load is uniformly distributed over length of it? I x I 1 I*=d3; and when uniformly distributed, d3^- 2 = d 1 . ioX 1 = 10 i* = g=zcube of diameter at i foot; ioX 2 = 20 2 2 = i6 = cufc of diameter at 2 feet ; 10 X 5 = 50 s 2 = 25 = cube of diameter at 5 feet. When a load is uniformly distributed, stress is greatest at middle of length, and is equal to half of it ; 25 -r- 2 = 12. 5 = cube of diameter at 5 feet. Torsional Strength of any square bar or beam is as cube of its side, and of a cylinder as cube of its diameter. Hollow cylinders or shafts have great- er torsional strength than solid ones containing same volume of material. To Compute IDiameter of* a Solid Shaft of Cast or "Wrought Iron to Resist Lateral Stress alone. When Stress is in or near Middle. RULE. Multiply weight by length of shaft in feet ; divide product by 500 for cast iron and 560 for wrought iron, and cube root of quotient will give diameter in ins. EXAMPLE. Weight of a water-wheel upon a cast-iron shaft is 50000 Ibs., its length 30 feet, and centre of stress of wheel 7 feet from one end; what should be diameter of its body? 3/ /50QOO X 3; i = = 2816.6 Ibs. To Compute Torsional Strength of Ronnd and Sqnai-e Shafts. RULE. Multiply Coefficient in preceding Table by cube of side or of diameter of shaft, etc., and divide product by distance from axis at which stress is applied in feet ; quotient will give resistance in Ibs. ILLUSTRATION. What torsional stress may be borne by a cast-iron shaft of best material, 2 ins. in diameter, power applied at 2 feet from its axis. C from table = 130. I3 X23 = ^i? = S20 Ibs 2 2 For steamers, when from heeling of vessel or roughness of sea the stress may be confined to one wheel alone, diameter of journal of its shaft should be equal to hat of centre shaft. STRENGTH OF MATERIALS. TORSION. /Q5 GUDGEONS. To Compute Diameter of a Single Q-udgeon of Cast Iron, to Support a given. \Veight or Stress. RULE. Divide square root of weight in Ibs. by 25 for Cast iron, and 2tS for Wrought iron, and quotient will give diameter in ins. EXAMPLE. Weight upon a gudgeon of a cast-iron water-wheel shaft is 62 500 Ibs. ; what should be its diameter ? 25 25 To Compute Diameter of Two Q-udgeons of Cast Iron, to Support a given. Stress or "Weight. RULE. Proceed as for two shafts, page 792. To Compute "Ultimate Torsional Strength, of Round, and Square Shafts. (D. K. Clark.) Cast Iron. Round. ^M1? = W; x. 5 34^r = *; and ^ = S. Sguare . *** = W , and ..363/^5 = * Hollow. ^ ^^ S = W. S representing ultimate shearing strength, and W moment of load, both in Ibs., s side of square shaft, and R radius of stress, both in ins. ILLUSTRATION. What is ultimate torsional strength of a round cast-iron shaft 4 ins. in diameter, stress applied at 5 feet from its axis ? Assume S = 20 ooo Ibs. Then ' 2yB X 4 * X 2 = 5930 Ibs. 5 X 12 By experiments of Major Wade, ordinary foundry iron has a torsional strength of 7725 Ibs., or 644 Ibs. per sq. inch at radius of one foot. Thus, take preceding illustration. Then ^^ = 8240 Ibs. "Wrought Iron. Round. -^^ - = W. Square. -^-^ = W. When Torsional Strength per sq. inch for radius of i inch is ascertained, substitute C for .278, .4, .2224, or .32. Stress which will give a bar a permanent set of .5 is about .7 of that which will break it, and this proportion is quite uniform, even when strength of material may vary essentially. Wrought Iron, compared with Cast Iron, has equal strength under a stress which does not produce a permanent set, but this set commences under a less force in wrought iron than cast, and progresses more rapidly thereafter. Strongest bar of wrought iron acquired a permanent set under a less strain than a cast-iron bar of lowest grade. Strongest bars give longest fractures. cj, , , .g d^ S When S is not known, substitute for ' na ' ~1T~ ' 872 5 =72 per cent, of tensile strength. Torsional Strength of Cast Steel is from 2 to 3 times that of Cast Iron. Following rules are purposed to apply in all instances to diameters of journals of shafts, or to diameter or side of bearings of beams, etc., where length of journal or distance upon which strain bears does not greatly ex- ceed diameter of journal or side of beam, etc. ; hence, when length or distance is greatly increased, diameter or side must be correspondingly increased. Coefficients for torsional breaking stress of Iron, Bronze, and Steel, as de- termined by Major Wade, are: Wrought Iron, 640; Cast Iron, 560; Bronze, 460 ; Cast Steel, 1120 to 1680. Puddled Steel does not differ essentially from that of cast iron 796 STRENGTH OF MATERIALS. TORSION. Formulas for Minimum and Maximum Diam. of Wrought-iron Shafts. (A. E. Seaton, London, 1883, and Board of Trade, Eng.) Compound Engines. ^J S = diameter. D and d representing diam- eter of low and high pressure cylinders, and S half stroke, all in ins., p pressure of steam in boiler, in Ibs. per sq. inch, and C a coefficient, as follows : A of le Crank. Sha Crank. ts. Pro- peller. Angle of Crank. Sba Crank. ts. Pro- peller. Angle of Crank. Shaf Crank. ts. Pro- peller. An ? ,e Crank. Sha Crank. 'ts. Pro- peller. 900 (2468 (4000 2880 5400 100 (2279 (4000 2659 5400 110 (2131 Uooo 2487 5400 120 (2016 (4000 2352 5400 I/ - C = diameter. A. E. Seaton, London, 1883. Side-wheel Engines, Sea Service. One cylinder crank journal, C = 8o; outboard loo ; Two cylinder crank journal 50; outboard 65; and centre shaft 58. Propeller Engines. One cylinder crank journal 150; Tunnel 130; Two cylinder compound crank 130; Tunnel no; Two cranks, crank 100; Tunnel 85; Three cranks, crank 90; and Tunnel 78. River Service. C may be reduced one fifth. ILLUSTRATION. With a compound propeller engine, steam cylinders 20 and 40 ins. in diameter, by 40 ins. stroke, operating under a pressure of 80 Ibs. steam (mercurial gauge), what should be the diameter of the shafts of wrought iron? =8. 24 ins. crank and A60 V 54< - x 40 = 7.46 ins. propeller shaft. Journals of Shafts, etc. Journals or bearings of shafts should be proportioned with reference to pressure or load to be sustained by the journal. Simplest measure of bear- ing capacity of a journal is product of its length by its diameter, in sq. ins. ; and axial area or section thus obtained, multiplied by a coefficient of pressure . per sq. inch, will give bearing capacity. Sir William Fairbairn and Mr. Box give instances of weights on bearings of shafts, etc., from which following deductions are made, showing pressure per sq. inch of axial section of journal : Crank pins, 687 to 1150 Ibs. per sq. inch. Link bearings, 456 to 690 Ibs. per sq. inch. Pressure on bearings, as a general rule, should not exceed 750 Ibs. per sq. inch of axial area. Length of Journals should be 1.12 to 1.5 times diameter. Journals of Locomotives or Like Axles are usually made twice diameter, and to sustain a pressure of 300 Ibs. per sq. inch of axial area, or 10 sq. ins. per ton of load. Solid. Cylindrical Couplings or Sleeves. ^-f-\/5-5 d=D; $d = L', .8d = J; .25 d-f~-i2 k. d representing diameter, and L length of sleeve, I length of lap or scarf of shaft, k breadth of key, its depth be- ing half its breadth, and D diameter of coupling or sleeve, all in ins. Flanged Couplings. d + VS-Sd^D; 3 d4-i F; . 3 d-j-. 4 = ? ; d+i = L; Z-^- 4 s. senting diameter of body of coupling, F diameter of flanges, I thickness of b L length of each coupling, s projection of end of one shaft and retrocessi from centre of coupling, and d diameter of shaft, all in ins. Supports for Shafts. (Molesworth.) 5 tyd* = L. L representing distance of supports apart, in feet. . D repre- oth flanges, retrocession of other STRENGTH OF MATERIALS. TOBSION. 797 To Resi*t Lateral Stress. fg- = D. W representing weight or pressure sit centre of length in Ibs., and D diameter or side, if square, in ins. Value o/C. Wrought Iron, 560; Cast Iron, 500; Cast Steel, 1000 to 1500; Bronze, 420 ; and Wood, 40. When Weight is distributed put 2 C. Values of C for Shafting of Various Metals, as observed by different Authorities, and deduced from Formulas of Navier. -^r^r = c - Ultimate Resistance. METAL. c METAL. C METAL. C WROUGHT IRON. American, Pemb,Me. " Ulster.... " mean English, refined Swedish 61673 61815 66436 49148 54585 61909 and , CAST IRON. American, mean j 44 1 8 trials English, mean.. | Factory Sliai 36846 38300 42821 44957 22132 38217 ts. (. STEEL. American, Conn . . " Spindle " Nash. I. Co. English, Shear. . , . Bessemer j 82926 102131 95213 III 191 73060 79662 Cylindrical r I T. B. Francis.) Square. mean 35000 " Eng. 30000 i6WR Mean value of T. Wrought Iron.... { W 000 I Steel f 76000 ( 94000 i } iiiooo ;< mean 50000! u mean 86000 " " Eng. 45000 1 *' " Bessemer 78000 ILLUSTRATION. What is the ultimate or destructive weights that may be borne by a Round Cast-iron shaft 2 ins in diameter, and by a Square shaft 1.75 ius. side stress applied at 25 ins from axis ? Assume T = 36 ooo. Round Square. 6000 , _ /,. 75 x 36000^., l ^, /7=i8i9 , \ 3 J 3 -f- * Ibs. Their lengths should be reduced, and diameter increased, in following cases : ist. At high velocities, to admit of increased diameter of journals, thereby rendering them less liable to heating, ad. As they approach extremity of a line of shafting. 3d. Attachment of intermediate pulleys or gearing. Prime Movers of Power. 3/P = *, and .ox n d* = IH>. Transmitters of Power. ^ and . 02 n d , = IH >. -- 5 n IH * = d, and .016 n d* = Iff . 3/ 3I ' 2 ^ IIP = d, and .032 n d = IH. Iron. Steel Cast Iron. IIP representing horse-power transmitted, n number of revolutions, and d diameter of shaft in ins. ILLUSTRATION i. What should be diameter of a wrought-iron shaft, to simply transmit 128 H? at 100 revolutions per minute? 3 / 5 X I2 = 3 /_i2 = 4 ins. V 10 V I0 2. What H? will a steel shaft of 4 ins. diameter transmit at 100 revolutions per minute? .032 X ioo X 4 3 = 204.8 horses. 3X* STRENGTH OF MATERIALS. TRANSVERSE. TRANSVERSE STRENGTH. Transverse or Lateral Strength of any Bar, Beam, Rod, etc., is in proper- tion to product of its breadth and square of its depth; in like-sided bars s beams, etc., it is as cube of side, and in cylinders as cube of diameter ol section. When One End is Fixed and the Other Projecting, strength is inversely a* distance of weight from section acted upon ; and stress upon any section ii directly as distance of weight from that section. When Both Ends are Supported only, strength is 4 times greater for an equal length, when weight is applied in middle between supports, than if one end only is fixed. When Both Ends are Fixed, strength is 6 times greater for an equal length, when weight is applied in middle, than if one end only is fixed. When Ends Rest merely upon Two Supports, compared to one When Ends are Fixedj strength of any bar, beam, etc., to support a weight in centre of it, is as 2 to 3. When Weight or Stress is Uniformly Distributed, weight or stress that can be supported, compared with that when weight or stress is applied at one end or in middle between supports, is as 2 to i. Metals. In Metals, less dimension of side of a beam, etc., or diameter of a cylinder, greater its proportionate transverse strength, in consequence of their having a greater proportion of chilled or hammered surface, compared to their ele- ments of strength, resulting from dimensions alone. Strength of a Cylinder, compared to a Square of like diameter or sides, is as 5.5 to 8. Strength of a Hollow Cylinder to that of a Solid Cylinder, of same area of section, is about as 1.65 to i, depending essentially upOE the proportionate thickness of metal compared to diameter. Strength of an Equilateral Triangular Beam, Fixed at One End and Loaded at the Other, having an edge up, com oared to a Square of the same area, is as 22 to 27 ; and strength of one, having an edge down, compared to one with an edge up, is as 10 to 7. NOTE. In Barlow and other authors the comparison in this case is made when the beam, etc., rested upoa supports. Hence the stress is contrariwise. Strongest rectangular bar or beam that can be cut out of a cylinder is one of which the squares of breadth and depth of it, and diameter of the cylinder, are as i, 2, and 3 respectively. Cast Iron. Mean transverse strength of American, as determined by Major Wade, U 681 Ibs. per sq. inch, suspended from a bar fixed at one end and loaded at the other ; and mean of English, as determined by Fairbairn, Barlow, and others, is 500 Ibs. Experiments upon bars of cast iron, i, 2, and 3 ins. square, give a result of transverse strength of 447, 348, and 338 Ibs. respectively ; being in the ratio of i, .78, and .756. ^Woods. Beams of wood, when laid with their annular layers vertical, are stronger than when they are laid horizontal, in the proportion of 8 to 7. Relative Stiffness of Materials to Resist a Transverse Stress. Ash ........... 080 I Cast Iron ---- i I Oak ........... 095 J Wrought iron i. 3 Beecb ......... 073 I Elm .......... 073 | White pine... .1 | Yellow pine.. .087 STRENGTH OF MATERIALS. TRANSVERSE. 799 Strength of a Rectangular Beam in an Inclined position, to resist a vertical stress, is to its strength in a horizontal position, as square of radius to square of cosine of elevation ; that is, as square of length of beam to square of dis- tance between its points of support, measured upon a horizontal plane. WOODS. California Red Pine. California Spruce. . . Canadian Red Pine.. Cedar 5000 Ultimate Resistan Chestnut ce. Lbs. 5000 7000 35oo 6000 Oregon Pine Lbe. 6500 4000 6000 AOOO Georgia Pine Spruce Hemlock White Oak Northern Pine. . . White Pine. . . Transverse Strength of* "Various Materials. (U. S. Ordnance Department, Hodgkinson, Fairbaim, Kirkaldy, by the Author, and Digest of Physical Tests.) Power reduced to uniform Measure of One Inch Square, and One Foot in Length; Weight suspended from one End. Safe Stress. METALS. Brass 260 Cast Iron, mean (Maj. Wade) 681 ordinary 575 extreme, West P't F'dry 980 gun -metal,* " " 740 Eng. , Low Moor, cold blast. 472 Ronkey, 581 Ystalyfera, " 770 mean, 65 kinds.. .... 500 " 1 5 kinds, cold blast 641 planed bar. 518 Copper 244 Steel, hammered, mean 1500 cast, soft 1540 " hard 4200 hematite, hammered 1620 Krupp's shaft 2096 Fagersta, hammered 1200 Wrought Iron, mean 600 " " English 475 " Swedisht 665 WOODS. Ash 220 " English 160 " Canada. 120 Balsam, Canada 87 Beech 130 " white 112 Birch 137 Cedar, white 160 " Cuba, mean 84 Chestnut 160 Elm. 125 " Canada, red 170 Fir, Baltic, mean 153 " Canada, yellow J ^ " " red 120 Ureenheart, Guiana 160 WOODS (Continued). Gum, blue 136 Hackmatack 102 Hemlock 100 Hickory 210 Larch, Russian ng Lignumvitse 162 Locust 295 Mahogany u 2 Maple 202 Oak, white 150 4 live 160 ' African 207 ' English 130 ' French 160 ' Canada. 146 ' Spanish 105 Pine, white 125 * Pitch 137 ' yellow 130 4 Georgia 200 Poon 184 Spruce, Canada 125 " black 87 Sycamore 125 Tamarack 100 Teak 165 Walnut 112 STONES, BRICKS, ETC. Brick, common, mean Cement, mean Portland " hydraulic, Portland. . " Roman " Puzzuolana " Portland, i year Concrete, Eng., fire-brick beam,) cement f 20 40 15 10.2 37-5 5 2 4-5 * This was with a tensile strength of 27 ooo Ibs. t With 840 Ibs. the deflection was i inch, and the elasticity of the metal destroyed. 8oo STRENGTH OF MATERIALS. TRANSVERSE. STONES, BRICKS, ETC. Concrete, Eng. , fire-brick, sand 3, ) lime i J ' 7 " Eng., clay and chalk. ... 5.4 Flagging, blue, New York 3*-2S freestone, Conn 13 Dorchester 10. 8 New Jersey, mean 19 New York 24 Eng. , Craigleth 10. 7 " Darby, Victoria. .. 1.3 " Park Spring 4. 3 Glass, flooring 42. 5 Granite, blue, coarse 18 " Quincy 26 " mean 25 " Eng., Cornish 22 Limestone n to 15 " English it Marble, Vermont, mean 92 STONES, BRICKS, ETC. Marble, Adelaide 4. 5 " Italian, white IX< 6 Mortar, lime, 60 days 2.5 " i lime, i sand , . . 2 i " 2 " i-75 i 4 1-25 Oolite, English, Portland 21.2 Paving, Scotch, Caithness 68 ' ' Ireland, Valentia 68. 5 " Welsh 157 " English, Yorkshire, blue.. 10.4 " " Arbroath ? 7 Slate 81 " Bangor 90 " English, Llangollen 43 Stones, English, Bath 5.2 " Kentish, Rag 35.8 " Yorkshire, landing 22.5 " Caen 12.5 Elastic Transverse Strength of Woods, compared with their Breaking Weight, is as follows : PerCent. I Ash 29 Norwa; Beech 25 Elm 32 Larch 38 Per Cent. iy Spruce ____ 30 Oak, Dantzic ...... 36 English ...... 33 Pitch Pine ......... 24 Per Cent. Red Pine 29 Riga Fir 30 Teak 32 Yellow Pine 30 Increase in. Strength of several "Woods "by Seasoning. Per Cent Ash 44.7 | Beech 61.9 | Elm 12.3 | Oak 26.1 | White pine.... 9 Concretes, Cements, etc. MATERIALS. Breaking Weight. MATERIALS. Breaking Weight. CONCRETES (English). Fire-brick beam Portl'd cement Lbs. BRICKS (English). Best stock Lbs. u sand 3 parts, lime i part .7 Fire-brick CEMENTS (English). New brick 5-4 Old brick Q I Portland t 37-5 Stock-brick, well burned *i Sheppev. . . IO.2 inferior, burned. . . 2-5 Transverse Strength, of "Variovis Figures of Cast Iron. Reduced to Uniform, Measure of Sectional Area, of One, Inch Square and One Foot in Length. Fixed at one End ; Weight suspended from the other. Form of Bar or Beam. Breaking Weight. Form of Bar or Beam. Breaking Weight. Wi Square Lbs. 673 1 Rectangular prism. Lbs. ^^ Square, diagonal vertical. . . 568 i 2 X . 5 ins. in depth . . . ** 3 X .33 " in depth. . . " 4 X .25 " in depth. . . A Equilateral triangle, an) edge up j 1456 2392 2652 560 Cylinder j 573 WW Equilateral triangle, an ) 958 O Hollow cylinder; greater) diameter twice that of \ lesser. . . . . ) 794 T2 ins. in depth X 2 X ) .268 inch in width. .. } \ 2 ins. in depth x 2 x ) JL .268 inch in width . . j 2068 555 STRENGTH OF MATERIALS. TRANSVERSE. 80 1 Solid and. Hollow Cylinders of various IVIaterialSo One Foot in Length. Fixed at one End ; Weight suspended from the other. MATERIALS. External Diam. Internal Diam. Breaking Weight. MATERIALS. External Diam. Internal Diam. Breaking Weight. WOODS. Ash Ins. 2 Inch. Lbs. 685 METAL. Cast iron cold) las. Ina. Lbs. 60 A. blast ) 3 12000 Fir* 2 772 STONE -WARE White pine. . I 2 610 Rolled pipe of ) fine clay j 2.87 1.928 IQO * An inch-square batten, from same plank as this specimen, broke at 139 Ibs. Formulas for Transverse Stress of Rectangular Bars, Beams, Cylinders, etc. Fixed at One End. Loaded at the Other. Bars, Beams, etc. = /z w and Cylinder 3/=b and d. Fixed at Both Ends. Loaded in Middle. IW 6S6d 2 6 S 6 d 2 Bars, Beams, etc. ^^ = 8; ^ = W; -^-= / - = d ; and Cylinder 3 / _ = b and d. V 6 S b V 6 b Fixed at Both Ends. Loaded at any Other Point than in Middle. Bars, Beams, etc. 2MnW - 3 ' 678> which ._ = 89 107 Ibs. MOLESWORTH. 4 * **' = W. C = 7616 Hw. , and /or b d 2 put bd' ab' d' 3 . 6 = 5.7 : fe /=:: 5-7 -6-T-2 = 2.55 : ^ = 11.75 : d' = 11.75 -6 X 2=10.55 : and 5.7 x u-75 2 2 X 2.55 x 10. 55^ = 786.94 567.63 = 219.31. Then, 4X7616x219.3x^6781060^ ^ IO X 12 I2O 6 and d representing exterior and V and d' interior dimensions, and I length all in ins. Fairbairn's formula would give a result less than half of the first, and Hodgkin- son's alike to that of Molesworth. 8o6 PLATE AND BOX GIRDERS. Steel Plate Grirders. Steel Box Grirders. Safe Loath in Tons of 2000 Ibs. Uniformly Distributed. Tensile Strength, 15000 Ibs. Plate. Carnegie Steel Co. 33 ox. SOX. 5 Ins. 33X.G Ins. SOX. 3 Ins. 33X.C5 Ins. \Vet> Plate. Welo Plates. Flanges, 12 X .375 ins. Anglet, 5 X .5 Flanges, 16 X .375 Flanges, 20 X .4375 ins. ins. X 3.5 ins. Angles, 3.5 X 3-5 X -5 ins. c" LOAD. c* LOAD. LOAD. (= S. LOAD. ill 5 {!| ill ^ ||| 2 i 'o . ||1 *" fii ijj %*i S|| 111 s'J! ||| ^fl flf lj| WJ ill H-, o '-' 2 Feet. Tons. Tons. Feet. Tons. Tons. Feet. Tons. Tons. Feet. Tons. Tons. 20 93.67 4.62 20 105.82 5 08 20 II2.6 6.61 2O 150.2 9.17 21 89.22 4.38 21 100.78 4.85 21 107.24 6-3 21 1 43- 1 8-75 22 85-15 4.19 22 96.2 4.62 22 102.37 6 22 136-5 8-33 23 81.46 4 23 92.01 4.42 23 97.92 5-75 23 130.6 7-96 24 78.07 3-83 2 4 88. 1 8 4-23 24 93.83 5-52 24 125.2 7-64 25 74-94 3-68 25 84-65 4.06 25 90.08 5-3 2 5 1 20. i 7-33 26 72.06 3-54 26 81.39 3-9 1 26 86.62 5-09 26 "5-5 7.06 3 69.4 66.91 3-42 3-29 2 7 28 78.38 75.58 3-76 3-63 27 28 83.41 80.42 4.9 4-73 27 28 III. 2 107.3 6.8 6-54 29 64.6 29 72.98 3-5 29 77-65 4-57 29 103.6 6.32 30 62.45 3-7 30 3-39 30 75.07 4.41 30 100.2 6.1 31 60.44 2-97 31 68.' 26 3-29 31 72.65 4.27 31 96.9 592 32 58.55 2.88 32 66.14 32 70.38 4-13 3 2 93-9 5-73 33 56.77 2-79 33 64.13 3*o8 33 68.24 4.02 33 9 1 5.56 55-" 34 62.24 2-99 34 66.23 3-9 34 88.4 5-39 36 52.04 2! 56 36 58.79 2-83 36 62.56 3-67 36 83-4 5-og 38 49-3 2.42 38 55-7 2.67 59.26 3-48 38 79 4.82 40 46.83 2.31 40 52-9 2-55 40 56-31 3-3 4 4-58 36X.G Ins. 4rSX.6QO Ins. 36X.S Ins. 4SX.S Ins. "W"el>. \Vebs. Flanges, 12 X. 37 5 Flanges, 14 X -625 Flanges, 24 X .5625 'Flanges, 30 X .6875 ins. Angles, 5 X ins. Angles, 6 X ins. Angles, 4 X ins. Angles, 5 X 3.5 X. 5 * 6 X -625 ins. 3.5 X.5*f*. 3.5X.5*?. 20 "8.35 5-54 20 176.01 7-74 20 213-3 12.22 20 329-2 18.29 21 112.7 5.28 21 167.63 7-37 21 203.3 11.65 21 3I3-5 17.41 22 107.57 5-04 22 1 60. 02 7-03 22 194.1 II. 12 22 299.2 16.63 23 102.9 4.82 23 153.06 6-73 23 185-5 10.64 23 286.2 15-9 24 98.61 4-63 24 146.68 6-44 24 177.9 10.2 2 4 274-3 15.24 25 94.66 4-44 25 140.82 6.18 25 170.8 9.78 25 263.3 14.63 26 91.02 4.27 26 135.39 5-95 26 164.3 9-44 26 253-2 14.07 2 7 87.65 4 ii 27 130.38 5-73 27 158-1 9.06 27 243.8 '3-55 28 84-53 3.96 28 125-73 5-52 28 152-4 8-73 28 235.2 13.06 2 9 81.61 3.82 2 9 121.38 5-34 2 9 147.2 8-43 29 227 12.61 30 78.89 3-7 30 "7.35 5-17 30 142.3 8-15 3 219.5 12.18 31 76-34 3-58 31 113-56 4.98 31 137-7 7.88 31 212.3 11.79 32 73-96 3.46 32 III.OI 4-85 32 133-4 7-65 32 205.7 11.42 33 71.72 3.36 33 106.67 4-7 33 129.3 7.42 33 199.5 10.08 34 69.61 3-27 34 103-55 4-55 34 125-5 7-2 34 193.6 10.75 36 65-75 3-07 36 97.78 4-3 36 118.6 6.81 36 182.9 10. 15 38 62.28 2.91 38 92.64 4-07 38 112.4 6-44 38 173.2 9.62 40 59-14 2-77 40 88 3-87 40 106.7 6.12 40 164.5 9.14 BUCKLING. To arrest the buckling of these girders, strips of plate, termed Fillers, are set vertical on the outer sides only of a web plate, together with other vertical angles, termed Sti/eners, both of which are riveted to the web plate, and both of these additions are set at intervals, dependent upon the length of the girder and the character of its stress. STRENGTH OF MATERIALS. TRANSVERSE. 8O/ Rolled Steel Beams. Safe Load for One Foot, Uniformly Distributed and Supported Sidewise. Carnegie Steel Co., Pittst>u.rg, !>a. Index. Depth. Designation. Wi Flange. ith. Web. Area. Section. Weight Foot. Loa Tensile perSq 12500 ds. Strength . Inch. 16000 Ins. Ins. Ins. Sq. Ins. Lbs. Lbe. Lbs. B 77 3 Light 2.423 .263 1.91 6-5 15000 19 loo Heavy 2.521 .361 2.21 7-5 16200 20700 Standard 2-33 '7 1.6 3 5-5 13800 17600 623 4 Light 2-733 .263 2-5 8.5 26 500 33900 Heavy 2.88 .41 3.09 10.5 29800 38100 Standard 2.66 .19 2.21 7-5 24900 31800 B 21 5 Light 3-147 357 3 .6 12.25 45400 58100 Heavy 3-294 504 4-34 '4-75 50500 64600 Standard 3 .21 2.87 9-75 40300 51 600 B 19 6 Light 3-452 352 4-34 14-75 66600 86300 Heavy 3-575 475 5-07 17.25 72800 93100 Standard 3-33 .23 3-6i 12.25 60500 77500 Bi 7 7 Light Heavy 3-763 3.868 353 458 5-15 5-88 17-5 20 93300 100400 119400 128600 Standard 3-66 .25 4.42 15 86300 110400 Bis 8 Light 4.087 357 6.03 20.5 126200 161600 Heavy 4.271 541 7-5 25-5 142600 182 500 Standard 4 .27 5-33 18 118500 15170 Bi 3 9 Light 4.446 .406 7-35 25 170300 217000 Heavy 4.772 732 10.29 35 207 ooo 265000 Standard 4-33 .29 6-31 21 157300 20 1 300 B ii 10 Light 4-805 455 8.82 30 223600 286300 Heavy 5.099 749 11.76 40 264 500 338500 Standard 4.66 7-37 25 203500 260500 B 9 12 Light 5.086 ^436 10.29 35 317000 405800 Standard 5 35 9.26 299700 383700 B 8 12 Light 5.366 .576 13.24 45 396800 5o 7 9oo Heavy 5.612 .822 16.18 55 445800 5 7 o6oo Standard 5-25 .46 11.84 4 373500 4 7 8 loo B 7 15 Light 5.55 .46 13-24 45 506400 648200 Heavy 5-746 656 16.18 55 567 800 726800 Standard 5-5 .41 12.48 42 490800 628300 B 5 15 Light 6.096 .686 19.12 6 5 706700 904600 Heavy 6.292 .882 22.06 75 768000 983000 Standard 6 59 17.67 60 676600 866100 B 4 15 Light 6-479 .889 25 85 908600 i 163000 Heavv 6-774 1.184 29.41 IOO I 000600 i 280 700 Standard 6.4 .81 23.81 80 883900 i 131 300 B8o 18 Light 6.095 555 17-65 60 779600 997700 Heavy 6.259 .719 20.59 70 853000 1091 900 Standard 6 .46 !5-93 55 736700 943000 B 3 20 Light 6-325 -575 20.59 70 I 016600 I 301 200 Heavy 6-399 649 22.06 75 1 057 400 i 353 5oo Standard 19.08 65 974700 i 247600 B 2 20 Light 7-063 .663 25 85 I 257 200 1609300 Heavy 7.284 .884 29.41 IOO 1379800 i 766 loo Standard 7 .6 23-73 80 I 222 IOO i 564 300 B i 24 Light 7.07 57 25 85 I 505 900 i 927 600 Heavy 7.254 754 29.41 IOO I 653 3 00 2115900 Standard 7 5 23.32 80 1449900 1855900 Index refers to Illustration in Catalogue. For safe load of IRON deduct 25 per cent. For permanent stress, absolutely free from vibration, a greater strain would be allowable, and, contrariwise, if the stress is vibrative or mainly that of a live load, the loads here given should be relatively reduced. 8o8 STRENGTH OF MATERIALS. TRANSVERSE. A difference of 25 per cent. In either direction should be made, according to char- acter of load to be supported or stress to be borne. Elastic Transverse Strength of Wrought-iron Bars is about 45 per cent, of their transverse strength, of Solid rolled beams, 50 per cent. ; and of double - headed rails, 46 per cent, of their transverse strength ; of Fagersta Steel, 56 per cent, of its transverse strength ; of double-headed Steel rails, 47 per cent. ; of Bessemer Steel, 37.5 to 48 per cent. ; and of Steel flanged, 68 per cent. Transverse strength of Solid Cast-iron Beams or Girders is about 50 per cent, of ultimate strength. NOTE. Actual breaking weight of a 10.5 ins. beam of New Jersey Steel and Iron Co., weight 35 Ibs. per foot, for a length of span of 20 feet, is 60000 Ibs. Rolled. Steel Deck Beams. Safe Load for One Foot, Uniformly Distributed, Supported Sidewise. Depth. Web. Flange. Area. Add to Web for each Ib. increase. Weight. Loa Tensile perSq I2OOO ds. Strength 16000 Ins. Ins. Ins. Sq. Ins. Ins. Lbs. Lbs. Lbs. 6 2.8 4.38 4.1 .049 14.1 48800 65 100 6 4-3 4-53 5 17.16 57600 76800 7 3- 1 4.87 5-3 .042 i8.ii 77300 103000 7 5-4 6.9 23.46 93400 124600 8 3.1 5 5-9 037 20.15 97400 1 29 800 8 9 4-7 4-4 4-94 033 3*. 112600 141 800 150 100 189 100 9 5-7 5-07 8.8 30 156400 208 500 10 3-8 5-25 8 .029 27.23 169600 226 100 10 6-3 5-5 10.5 35-7 205 600 274100 "5 4.2 9-5 .026 32-2 221 000 294700 "5 5-5 5-3 10.9 37 244800 326500 Steel Bulb Angles. 5 .31 2-5 2.94 10 32500 43300 6 .31 3 3-62 12.3 45300 60400 6 38 3 4.04 13-75 52 800 70400 6 7 .50 34 3 3 4.71 17.2 16 60400 69600 80500 92 800 7 44 3 5-37 18.25 76700 IO2 3OO 8 .41 3-5 5.66 19.23 93600 124800 9 44 3-5 6.41 21.8 115700 154200 10 .48 3-5 7.8 26.5 158800 211 700 10 .63 3-5 9.41 32 172500 23OOOO Operation of Tables. To Compute Depth of a Beam to Support a Uniformly Distributed. Load. RULE. Multiply load in Ibs. by length of span in feet, and take from table the beam, load of which is nearest and in excess of product obtained. ILLUSTRATION. What should be depth of a steel beam to sustain with safety a uniformly distributed load of 30000 Ibs., over a span of 15 feet? 30000 X 15 = 450000, which is load for a heavy beam 12 ins. in depth. Weight of beam should be added to load. Inversely. If the load is required, divide load in table by span of beam in feet, and subtract weight of beam. To Compute Deflection of Like Beams. RULE. Divide square of span in feet by 70 times depth of beam in ins. ILLUSTRATION. Assume beam as preceding. 70X12.25 857.5 22 ., = -?-=. 262 ins. STRENGTH OP MATERIALS. TRANSVERSE. Comparative Strength, and. Deflection of Cast-iron. Flanged. Beams. DESCRIPTION OF BEAM. Comp. Strength. DESCRIPTION OF BEAM. Comp. Strength. Beam of equal flanges Beam with flanges as i to 4 5 78 " with only bottom flange . . " with flanges as i to 2 u with flanges as i to 4. ... "f* 63 73 u with flanges as i to 5. 5 ... u with flanges as i to 6 " with flanges as i to 6. 73 . . : i .92 Rolled "Wrought-iron Bea'fiis Itinglish. Safe Stress fa a Span of 10 Feet. (D. K. Clark.) Depth. Breadth of Flanges. Thic Web. mess. Flanges. Weight per Lineal Foot. Ultimate Strength. Loaded in Middle. Safe Stress Uniformly Distributed. Ins. Ins. Inch. Inch. Lbs. Lbs. Lbs. 3 2 -1875 .2187 5-5 2800 910 3 3 2 5 3 I2 5 10 5600 i860 3-i 2 5 1-625 1875 .2187 5-5 2490 830 4 2 25 3*25 8 549 1830 4 3 25 375 12 8 510 2830 4-75 2 25 3125 8 6940 2310 5 3 3 I2 5 4375 13 13440 4480 5 4-5 375 5 23 19270 6420 5-5 2 375 4375 10 11880 3960 6 5 4375 5625 30 23830 7940 6.25 2 3125 4375 u 13440 4440 6.25 2.25 3 I2 5 375 18 13000 4330 6.25 3-25 3125 .4062 12.5 17470 5820 7 2.25 .281 375 14 14790 4930 7 2.25 3125 4375 14 17020 5670 7 3-625 3125 4375 '9 23300 7760 7 3.625 3125 5 J 9 25980 8660 8 2-375 -3125 4375 15 20830 6940 8 2-5 375 375 15 21 280 7090 8 4 375 5 21 34500 11500 8 5 375 5625 29 44800 H930 8 5-125 4375 5625 2 9 47040 15680 9-25 95 3-75 4-5 4375 375 .'6875 24 30 41560 59360 13850 19750 4-5 4375 5625 32 56000 18660 o 4-75 4375 5625 32 58240 19410 4-75 75 625 36 76160 2539 2 5 5625 8175 42 IOO8OO 33600 2 6 5625 9375 56 136640 45530 4 5-5 5625 875 60 150020 50000 4 6 5625 8175 60 152 260 50750 6 5625 75 8175 62 188 160 62720 TVr ought-iron Rectangular Grirders or Tu.~bes. (Riv'd.) Supported at Both Ends. Loaded in Middle. - = W. A representing area of section in sq. ins., d depth in ins., I length be- tween supports in feet, and W destructive weight in Ibs. ILLUSTRATION. What is the destructive weight of a rectangular girder, 35.75 ins. in depth by 24 in breadth, metal .75 inch thick, and length between supports 45 feet ? Assume C or coefficient = 37 oo, as per case (18) in preceding table, page 806, andaiea = 87.375 ins. Then 8 ?-375 X 35-75 X 37QQ = ' 557 528 = ^ ^ ^ By experiment it was 257 080 Us. By Inversion 77-7 = A, and = d. \j a A O HODGKINSON'S formula would give a result of 259373 Ibs., and MOLESWOBTH'S 303907 Ibs. 8lO STRENGTH OF MATERIALS. TRANSVERSE. TJneqvially Loaded. Beains, etc. I 3 W - = w. I representing length between supports, and m and n distances from points of support, all in like denomination, and W and w destructive and safe weights, also in tike denomination. To Compute .Destructive Weight and Area of Bottom IPlate. - = W; -^rr = A; 5Z?d !- = A. A representing area of plate in sq. ins., d and I depth and length, in and n distances of ioad at other points than in middle, all in feet, and W weight in Ibs. NOTE. Sufficient metal should be provided in sides to resist transverse and shearing stress, and in upper flange to resist crushing. ILLUSTRATION. What area of wrought iron is necessary in bottom plate of a rec- tangular tubular girder, 3 feet in depth, supported at both ends, and loaded in middle with 130000 Ibs. ? C, ascertained by experiment for destructive stress, 180000 Ibs., and area 7. i sq.ins. 130000X30 ~ - 7. 22 sq. ins. 180000X3 Wrougnt-iron. Cylindrical Beams or Tubes. - = W. ILLUSTRATION. What is destructive weight of a cylindrical tube, 12.4 ins. in diameter, .131 inch in thickness, and 10 feet between its supports? Area of metal = 5.05 sq. ins., and C = 2856, as in the igth case of table, page 806. Then 5.05X12.4X2856 = 884 2 ^ 10 D. K. CLARK. ^-^ - = W. d representing diameter, t thickness of metal, and I length, all in ins., S tensile strength of metal per sq. inch, and W weight, both in Ibs, MOLESWORTH'S formula gives a result of 23 286.1 Ibs. "Wrouglit-iron Elliptical Beams or Tubes. - -. W. ILLUSTRATION. -Assume diameter of tube 9.75 and 15 ins., metal 143 inch in thickness, and distance between supports 10 feet. A = 5.56 sq. ins. C = 3147, as per case (20) in preceding table, page 806. Then S.56XI5X3I47 = ^^? = 26 245.9 * 10 10 D. K. CLARK. *' 57 * - - = W. 6 and d representing conjugate and trans- verse diameter, I length between supports, t thickness of metal, all in ins.,S tensile strength of metal per sq. inch, and W destructive weight, buth in Ibs. NOTE. B. Baker, in his work on Strength of Beams, etc., London, 1870, page 26, shows that ordinary method of computing transverse strength of a hollow shaft by difference of diameter alone is erroneous, in consequence of Joss of resistance to flexure in a hollow beam. Grirders and Beams of TJnsymmetrical Section. 4 =W. S representing tensile resistance of metal, and W destructive weight, both in Ibs., d distance between centres of compression and extension, or crushing and tensile resistances, in ins., and I length between supports, in feet. NOTE. To ascertain d, see Rule, page 819, STRENGTH OF MATERIALS. - TRANSVERSE. 8ll ILLUSTRATION. Dimensions of a rolled wrought-iron girder, n feet in length be- tween its supports, is as follows : Top flange ................ 2.5 X i inch, j Bottom flange ............ 4 X .38 inch. Web ...................... 325 ** j Depth ................... 7 ins. What is its destructive weight ? d = 5 .22 ins S assumed at 45000 IDS. Then 4 X 4 f^* 5 '** = 7118.18 Ibs. Strength of Riveted Beams or Girders, compared with Solid, is less, and deflec- tion is greater "Wrought-iron. Inclined .Beams, etc. = w L and I representing lengths or inclination, and horizontal line, in like denominations, and W and w destructive and safe weights on horizontal line and in- clination, also in like denominations. Plate Grirders. - = W. A representing section in sq. ins. , d depth in ins. t and I length be- tween supports in feet. ILLUSTRATION What load will destroy a wrought-iron plate girder or beam of following dimensions, 10 feet in length between its supports? Top flange .............. 4.5 X .375 inch. Bottom flange ......... 4-5 X. 375 " Angle pieces ........... 2 X -3125 u Area of Section = 13 sq. ins. Assume coefficient of 5180 as per case (14) in preceding Table, page 806. Then I3 X *+** X 5l MOLESWORTH. = S. L representing load equally distributed, and S stress on 8 d centre, both in tons, and d effective depth of girder in feet. By actual experiment L = 48 tons for 16.5 feet between supports; hence, 10:16 5:148 79.2 tons 39. 6 when supported in middle, and 14.25 ins. = i.iSjsfset. Then Q 39' 6 x 10 _ 39$ _. ^^ which x 224Q _ ^ ^ 2 ^ 8 X 1-1075 9.5 D. K CLARK. d '4 q +^55) _ w d representing &> p fa O f gi rar or beam less depth of lower flange in ins., a and a' areas of sections oj bottom flange and of web, at its reputed depth, both in sq. ins., and I length between supports in feet. d = 14. 25 .375 = 13.875 ins. a 3, and a = 5 sq. ins. I Widthof web 375 inch. Depthofweb 13.5 ins. Depthofbeam 14.25 Mr Clark assumes, however, that for girders of like construction the destructive stress should be taken at two thirds of that deduced by the formula. I Girders or Beams mthout Upper and Lower Flanges. ILLUSTRATION. Assume angles 2.125 X .28 above, 2.125 X .3 below, web .25, depth 7 ins., and length between supports 7 feet Area of section = 6.35 sq. ins., and C = 3840, as per case (15) in preceding Table, page Sod. Approximate. - ^ -- W. a representing area of sections of upper and lower angles, a' area of section of web for total depth, both in sq. ins., d depth of girder in ins. , and W load or stress in Ibs. 8 12 STRENGTH OF MATERIALS. TRANSVERSE. a = 4.6 sq. ins. , and a'= 7 X -25 = 1.75 sg. ins. Then - - - - = 21 = I3 .68 7 , which x 2240 = 30658.8 Ibs. IRON AND STEEL RAILS. Symmetrical Section. To Compute Transverse Strength. (D. K. dark.) - - !l_- - - = W, and - -^ - - = S. S representing ten ( 4 a + i.i55<< 2 ) ftte strength in Ibs. or tons per sq. inch, a area of one head or flange exclusive of cen- tral portion composing web, in sq. ins.,d' depth or distance between centres of heads, d depth of rail, t thickness of web, I distance between supports, all in ins., and W weight in Ibs. or tons, alike to S. ILLUSTRATION i. What is destructive weight of a wrought-iron double-headed rail, 5.4 ins. deep, having a web of .8 ins., an area of head of i.o sq. ins., distance between centres of its heads 4.2 ins., and between its supports 5 feet? S assumed at 50 ooo Ibs. soooo = 5 X 12 60 43 125 Ibt. 2 . What is destructive weight of a Bessemer steel double-headed rail, 5.4 ins. deep, having a web of .75 inch, an area of head of 2 sq. ins., and distance between heads 4. 2 ins.? S assumed at 80000 Ibs. W ^A * -r ,.+ 80000X51.39 Then -= 3 ** = 68 520 Z6s. 5X12 60 NOTE. Transverse strength of Bessemer Rails increases very generally, in direct proportion with the proportion of Carbon in it. TJn symmetrical Section. ' 92 = W. d" representing vertical distance between centres of tension I h and compression, h height of neutral axis above base of section, and I length between supports, all in ins., and A sum of products, obtained by multiplying areas of strips of reduced section under tensile stress, by their mean distances, respectively, that is, the distances of their centres of gravity, from the neutral axis, in ins. Bowstring Grirder. To Compute Diameter of a "Wrought-iron Tie-rod of an Arched or 33o\vstring G-irder of Cast Iron. j- = d. W representing weight distributed over beam in Ibs., I length 4500 X n> between piers or supports in feet, and h height between centre of area of section of girder and centre of rod in ins. ILLUSTRATION. Required diameter of tie-rod for an arched girder, 25 feet be- tween its piers, and 30 ins. between centres of its area and of rod, to safely support a uniformly distributed load of 25 ooo Ibs. ? 725000X25 7625000 . , = A / = -v/4-62 = 2. 15 ins. 4500X30 V 135000 If two rods are used Then / - = 1.52 ins. = diameter of each rod. STRENGTH OF MATERIALS. TRANSVERSE. CAST IRON. Transverse Strength, of GJ-irders and. Beams. (Deduced from Experiments of Barlow, Hodgkinson, Hughes, Bramah, Cubitt, Tredgold, and others.) Reduced to a Uniform Measure of One Foot in Length. Supported at Both Ends. Stress or Weight applied in Middle. SKCTIOX. Flanges. Web. Depth. Distance. Area. Destrncti For Dis- tance. re Weight, of One Foot. JW_ A-5 a'b'=. 7.4 X 11.5= 85.1 18.4 145.6 = 7.91 ins. Showing that centre of gravity of reduced section, being neutral axis of whole section, is 7.91 ins. below upper edge, in line ii. Centre of gravity of entire section at , it may be added, is 8.65 ins. below upper edge, or .74 inch lower than that of reduced section. Neutral axes of other sections, Figs. 3 to 7, found by same process, are marked on the figures. Section of a flange rail, No. 7, which is very various in breadth, may be treated in two ways: either by preparatorily averaging projections of head and flange into rectangular forms, or, by taking it as it is, and dividing it into a con- siderable number of strips parallel to base, for each of which the moment, with re- spect to assumed datum-line, is to be ascertained. First mode of treatment is ap- proximate; second is more nearly exact. To Compute Ultimate Strength of Homogeneous Beams of TJnsymmetrical Section. OPERATION. Resuming section, Fig. 9, for which neutral axis has been ascertained, To Compute Tensile Resistance, Divide portion below neutral axis t z, Fig. 9, with reduced width of flange, a! 6", into parallel strips, say .5 inch deep, as shown, and multiply area of each strip by its mean distance from neutral axis for proportional quantity of resistance at strip. Divide sum of products, amounting in this case to 31.3, by extreme depth below neutral axis = 4.09 ins., and multiply quotient by 1.73 S (ultimate tensile resist- ance at lower surface). The final product is total tensile resistance of section ; or, 31.3X1.738 S representing ultimate tensile strength of material per sq. inch. Again, multiply area of each strip by square of its mean distance from neu- tral axis, and divide sum of these new products, amounting to 104.64, by sum of first products. The quotient is distance of resultant centre of tensile stress, d, from neutral axis. Or, resultant centre is, *f' * = 3. 34 ins. below neutral axis. This process is that of ascertaining centre of gravity of all the tensile resistances 8TBENGTH OF MATEBIALS. TKANSVEESE. 821 By a similar process for upper portion in compression, sum of first products is ascertained to be same as for lower portion =: 31.3. But maximum compress! ve stress at upper portion is greater than maximum tensile stress at lower portion, in ratio of their distances from neutral axis, or as -73SX =3.34 S, and 3I ' 3 X 3 ' 34 S = 13.24 S total compressive resistance, 4.09 7.91 which is same as total tensile resistance, in conformity with general law of equal ity of tensile and corapressive stress in a section. Sum of products of areas of stress, divided by squares of their distances respec- tively from neutral axis, is 164.9, *&& resultant centre c, Fig. 9, is ins. above neutral axis. Sum of distances of centres of stress or of resistance from neutral axis, 3.34 -4- 5.27 = 8.61 in*. = distance apart of these centres as represented by central line, c' d'. Abbreviated Computation. As upper part of section is a rectangle, its resultant centre = f of height, or 7.91 x j= 5.27 ins. above neutral axis. Average resist- ance is half maximum stress, viz., that at upper portion, which is 3.34 S per sq. inch. Area of rectangle therefore = 7. 91 x i = 7.91 * ( Least. . . . 7-978 17698 1852 4-57 lze (Greatest. 8-953 56786 2656 94 Factors of Safety. Girders, Beams, etc., of cast iron should not be subjected to a greater stress than one sixth of their destructive weight, and they should not be subjected to an impulsive stress greater than one eighth. The following are submitted by English Board of Trade, Commission- ers, etc. STRUCTTEB. Stress. Factor. STBUCTURB. Stress. FactoT. CAST IRON. Dead q tO 6 WROUGHT IRON. Girders Dead Columns. t 3 6 Live it Bridges Mixed 4.. ,....,,,', Machinery Live 8 STEEL. Shock 10 Bridges. Mixed 3 Z* 822 STRENGTH OF MATERIALS. TRANSVERSE. GJ-irders, Beams, Lintels, etc. Transverse or Lateral Strength of any Girder, Beam, Breast-summer, Lintel, etc., is in proportion to product of its breadth and square of its depth, and area of its cross-section. Best form of section for Cast-iron girders or beams, etc., is deduced from experiments of Mr. E. Hodgkinson, and such as have this form of section T are known as Hodgkiuson's. Rule deduced from his experiments directs, that area of bottom flange should be 6 times that of top flange flanges connected by a thin ver- tical web, sufficiently rigid, however, to give the requisite lateral stiff- ness, tapering both upward and downward from the neutral axis ; and in order to set aside risk of an imperfect casting, by any great dispro- portion between web and flanges, it should be tapered so as to connect with them, with a thickness corresponding to that of flange. As both Cast and Wrought iron resist compression or crushing with a greater force than extension, it follows that the flange of a girder or beam of either of these metals, which is subjected to a crushing strain, according as the girder or beam is supported at both ends, or fixed at one end, should be of less area than the other flange, which is subjected to extension or a ten- sile stress. When girders are subjected to impulses, and sustain vibrating loads, as in bridges, etc., best proportion between top and bottom flange is as i to 4 ; as a general rule, they should be as narrow and deep as practicable, and should never be deflected to more than .002 of their length. In Public Halls, Churches, and Buildings where weight of people alone are to be provided for, an estimate of 175 Ibs. per sq. foot of floor surface is sufficient to provide for weight of flooring and load upon it. In comput- ing other weight to be provided for it should be that which may at any time bear upon any portion of their floors ; usual allowance, however, is for a weight of 280 Ibs. per sq. foot of floor surface for stores and factories. In all uses, such as in buildings and bridges, where the structure is ex- posed to sudden impulses, the load or stress to be sustained should not ex- ceed from .2 to .16 of breaking weight of material employed ; but when load is uniform or stress quiescent, it may be increased to .3 and .25 of breaking weight. An open-web girder or beam, etc., is to be estimated in its resistance on the same principle as if it had a solid web. In cast metals, allowance is to be made for loss of strength due to unequal contraction in cooling of web and flanges. In Cast Iron, the mean resistances to Crushing and Extension are, for American as 4.55 to i, and for English as 5.6 to 7 to i ; and in Wrought Iron are, for American as 1.5 to i, and for English as 1.2 to i ; hence the mass of metal below neutral axis will be greatest in these proportions when stress is intermediate between ends or supports of girders, etc. Wooden Girders or Beams, when sawed in two or more pieces, and slips are set between them, and whole bolted together, are made stiffer by the operation, and are rendered less liable to decay. Girders cast with a face up are stronger than when cast on a side, in the proportion of i to .96, and they are strongest also when cast with bottom flange up. Most economical construction of a Girder or Beam, with reference to at- taining greatest strength with least material, is as follows : The outline of STRENGTH OF MATERIALS. TRANSVERSE. 823 top, bottom, and sides should be a curve of various forms, according as breadth or depth throughout is equal, and as girder or beam is loaded only at one end, or in middle, or uniformly throughout. Breaking Weights of Similar Beams are to each other as Squares of their like Linear Dimensions. By Board of Trade regulations hi England, iron may be strained to 5 tons per sq. inch in tension and compression, and by regulation of the Fonts et Chausse'es, France, 3.81 tons. Rivets .75 and i inch in diameter, and set 3 ins. from centre in top .o t^Q t>-O t-^0 t^O t^O t^O t^-O joo mcxi O oo m m o oo moo O oo rooo o M IN f) -^- m rxOO O N Tl- tv ON N ERSE. <2 o i il , m V 4X 112.5 V 450 )o, Spruce 550, White Pine 500, Chestnut 480, and Hemlock 450. ' , tit 1 ~ N fl ^s JOOOOOQOOQQQQOOQ OOOQOOOOOOOOOOO * "*-^O O \O ^ -^-VO O \0 * -^-vO O VO i-3 M in osin M o\oo ooONvoMtv.xnro M w N ro ^-\O txoo O w ro in * ** Is 8 rf - 32 | 'S i wo wo mo mo mo ^o m g in o s o/ TaWe. 2 ._What should be depth of a like beam, 4 ins. between its supports, to bear a statical weight of 4 / IoX4 5oo_ /45000 jginroit cj roi^Htoo mm moo t^ f> M ?^^^?*^fifS h ^ . I ** ! ~ * 2 * 1 ^ M g "5 s^ ONOO n N t^ CM moo o o o\oo m CN t^ CM 2 M ^o f f^^ -> _, n ro 03 vf w ^j flJdJX^jS crtcsO "5151 s^ inmininmininm .O t^O *^Q t~O t^>Q S 1 ^ M in c? CM i- ON Koo ^? tv.\o^oo cT c* o^ CM ^ w m TJ-VO oo H ro^o o, mvo o m V^ Illustration i. What is safe statical load for a White-pine beam, 4 ins. by 12, and 15 feet between its supports, loaded in middle? A like beam, i inch in width, 12 ins. in depth, and x foot between its Knrmnrts will hpar JIR nor tahlp -,f\ o Ihc Hence 16 200 x 4 -r- 15 = 4320 Ibs. Hatfield gives Georgia Pine 850, White Oak 650, Canadian Oak 5c ?j H ^ 3^11 13 I s ^i Bw .OOOOOOOOOQOOOOOO 'S 8c?8c?8c?8cM ) 8cM 1 8 ^ M CM co TJ-VO oo o CM inoo w ^oo CM MMMIHCMCMCMCO .NOCMQCMOCMOCMOCMQCMOCMO MiniHOMini-(Oi^ini-iOi-(inMO 2 w - TT^T = b- w representing weight in Ibs. per sq. foot. 2 (j d EXAMPLE. What should be breadth of a Georgia pine header, 13 ins. in depth, 10 feet in length, supporting tail beams 12 feet in length, bearing 200 Ibs. per sq. foot of area ? C, a? per preceding table, 112.5, and depth = 13 i = 12 ins. 12 X 10 X 200-7-2 X 10 120000 2 X II2-5 X I2 2 32400 = 3. ^ ins. To Compnte the Capacity of a ITloor Uniformly loaded when one of its Sides rests npon a Header Beam. 1. Determine the capacity of a trimmer and header beam at the point of their connection. Assume the less, as the limit of their capacity to sus- tain a load, and twice this capacity will represent that of one half of the floor at the points of connection of the header and trimmers, the other half resting on the wall. 2. Compute area of floor in square feet, first by its length from wall of building to face of header beam, and its width from the centre of the spaces between the trimmers and the beams beyond ; then add that determined by the width of the trimmer and the centre of the space between it and the beams, and the length of it by the width of the opening between the face of the header and the wall, as hatch or stairway, and this combined area will be that which rests upon the header and trimmer. 3. Divide the capacity of the header and trimmers as obtained, by the half area of the floor resting thereon, less the area required or allotted for passage way (but not considered by the Department of Buildings), and the quotient will give the capacity of the floor in Ibs. per square foot, from which is to be deducted the weight per square foot of the beams, flooring, ceiling, etc. STRENGTH OF MATERIALS. TRANSVERSE. 837 To Compute Depth of a Header Beam. RULE. See rule for depth of a floor beam, page 835, with the exception that a header, alike to a trimmer, is assumed to be always uniformly loaded. V 4&C ~ To Compute Breadth of a Trimmer Beam. With One Header and One Set of Tail Beams. RULE. Proceed as for computation of dimension of a beam loaded at any other point than middle. Uniformly Loaded. 1 e' X -\-n c X W = L, product of area of floor and load per sq. foot, and L -; -- - = breadth. H representing length of header, c distance between centres of beams, m and n, lengths of tail beams and width of hatch or stairway, c' sum of half distance ofc, added to half of an assumed width of trimmer, and I length of trimmer, all infect, W load per sq. foot on floor, and C coefficient in Ibs., b breadth of one sq. inch of the material, and d depth of beam in ins. EXAMPLE. What should be breadth of a trimmer or carriage beam of Georgia pine. 23 feet in length, 15 ins. in depth, sustaining a header 10 feet in length, with tail beams 19 feet, distance between centres one foot, and designed fora load of 270 Ibs. per sq foot of floor? Assume C = 275, as assigned by the Department of Buildings, N. Y.,m and n = ig and 4 feet, (715 1 14, and c' = .75. io-f .75~xf^ X4-75 + 4X 273=: 15828.75 and * 5 = 6. 75 in*. I X 196 X 275 -r- 23 NOTE i. Depth of trimmer beams is usually determined by depth of floor beams; where not, proceed to determine it as for a header. 2. When a trimmer beam is mortised to receive headers, it is proper to deduct i inch from its depth, as in preceding illustrations. When bridle or stirrup irons are used to suspend headers, a deduction of the thickness of the iron only is neces- sary, usually .5 inch. With Two Headers and One Set of Tail Beams. Fig. i. OPERATION. Proceed for each weight or load as for a beam, when weights are sustained or stress borne at other point than the middle. = W and w. a representing area of floor in sq. feet, L load per sq.foot, and W and w weights or loads at points of rest on trimmers. NOTE. Hatfleld and some other authors give complex and extended formulas, to deduce the dimensions of a Girder or Beam, under a like stress. Upon consideration, however, it will be readily recognized that a beam loaded at more than one point is simply two or more beams of proportionate width, as the case may be, loaded at different points, and connected together. Fig. i . -....-m--.fr- -/- ILLUSTRATION. What should be breadth of a trimmer beam of Yellow pine 25 feet in length, u ins. in depth, sustaining two headers 12 feet in length, set at 15 feet from one wall and 5 feet from the other, to support with safety 300 Ibs. per sq. foot of floor ? w W 12 X 10 5 X 3o _ l8 and d = n i = iofor loss by mortising. 12 X 5 X 300 18000 4 4 = 4500 Ibs. at w. , at W, and 838 STRENGTH OF MATERIALS. TRANSVERSE. Then aud > 25Xio 2 Xi25 312500 tns. t and 2.i6-|- 1.44 = 3.6 ins. combined breadth. _ = 25X10^X125 312500 Fig. 2. With Two Headers and Two Sets of Tall Beams. Fig. 2. OPERATION. Proceed as directed for Fig. i. ILLUSTRATION. What should be breadth of a trimmer beam of Yellow or Georgia pine, 25 feet in length, 12 ins. in depth, sustaining two headers 12 feet in length, one set at 15 feet from one wall and the other at 5 feet from the other, to support with safety 300 Ibs. per sq. foot of floor? W 112.5, and d = i2 1 = 11 for loss by mortising. 54000 . at Then '5 X io X 13 SOP 2 025 OOP = ^ breadth for load on header at 15 feet. or load on header at 5 feet, and 25 X n 2 X 112.5 340312 _ 20X5X4500 ._ 450000 _ ins 25 X ii 2 X 112.5 340312 5.94 -f- 1.32 = 7.26 ins. combined breadth. With Three Headers and Two Sets of Tail JSeaws. Fig. 3. w' w W OPERATION. Proceed as directed for Fig. i. ILLUSTRATION. What should be breadth of a trim- mer beam of Yellow pine, 20 feet in length, 13 ins. in depth, sustaining 3 headers 15 feet iu length, set at 3, 7, and 13 feet from one wall, to sustain a load of 200 lbs. per sq. foot of floor ? = i2 ins., and C= 125. 15 X 7 X 200 2 4 = 5250 a* at w; and 20Xl2 bined breadth. = . 360000 oo i ins.; 7X13X3000^000 20Xl2 2 Xl25 3 6 000 H 3 = *5a ins. com. 360000 Stirrnps or Bridles. Stirrups are resorted to in flooring designed for heavy loads, in order to avoid the weakening of the trimmers by mortising. Average wrought iron will sustain from 40000 to 50000 lbs. per sq. inch. Hence 45 ooo lbs. as a mean, which -=- 5 for a factor of safety, = 9000 lbs. A stirrup supports one half weight of header, and being doubled (looped), the stress on it is but .5 -=- 2 = .25 of load on header. To Compete Dimensions of Stirrnps or JBridles. W-r-2 area - area. Hence - = width. 2 x C thickness ILLUSTRATION. What should be area and width of .75 inch wrought-iron stirrup irons for a weight on a header beam of 240000 lbs. ? oooo 24 00 - f " 2 = '-^^ 2X9000 18000 = 6. 66 sq. ins. , and ^ 8. 8 ins. = width. -75 STRENGTH OF MATERIALS. TRANSVERSE. 839 Grirder. Condition of stress borne by a Girder is that of a beam fixed or supported at both ends, as the case may be, supporting weight borne by all beams resting thereon, at the points at which they rest To Coxnpvite Dimensions of a Girder. RULE. Multiply length in feet by weight to be borne in Ibs., divide product by twice * the Coefficient, and quotient will give product of breadth and square of depth in ins. j vrr I I TTT Or, = b and d 2 , and A /-^- = d. 2 C V 2 6 C EXAMPLE. It is required to determine dimensions of a Yellow-pine girder, 15 feet between its supports, to sustain ends of two lengths of beams, at distances of 5 feet, each resting upon it and adjoining wall, 15 feet in length, having a superincumbent weight, including that of beams, of 200 Ibs. per sq. foot. Condition of stress upon such a girder is that of a number of beams, 15 feet in length, supported at their ends, and sustaining a uniform stress along their length, of 200 Ibs. upon each superficial foot of their supporting area. Coefficient = 137. 5. 15X15X200^-2 (for half support on the uall) 22 500 Ibs. Then ^= 1227.2 = & and d 2 . Assume b = 12 ins., then / I22 7- 2 I0 x 2X137-5 V 12 ins. the depth. To Compute Greatest .Load, upon a Girder, and Dimen- sions thereof'. Fig. 1. When a Beam is Loaded at Two Points. Fi&. i. < 4- _ ee = effect of weight at 2, (W n -f w s) the two effects W w l at 1, and j (w r -f W m) = tivo effects at 2. Then, for weight and dimensions, same formulas will apply. ILLUSTRATION. Assume weight of 8000 Ibs. at 3 feet from one end of a white-pine beam io inches in depth and 12 feet in length between its bearings, and another weight of 3000 Ibs. at 5 feet from other end. C = 112.5. 8000 X 3 X 12 3 = 216 ooo effect of weight at location i, and 3000 X 5 X 12 5 fe 105 ooo effect of weight at location 2. Hence i, being greatest, = AV, and 2 = w. Then, - - x 8000 = 18 ooo at W, and - - x 7000 = 8750 at w ; and 'TO TO * ' J (8000 x 9 + 3000 x 5) = 21 750 = total effect at W, and 4- (3000 X 7 + 8000 x 3 ) 1 8 750 = total effect at w. Hence, to ascertain total effect and dimensions. 2x750x3x9 ^.fr^afc 12 X io 2 X 112.5 Verification. Breadth at W. l8oo X3X9 =3.6 ins. Then 21 750 18 ooo 12 X io 2 x 112.5 * For being uniformly loaded. 840 STRENGTH OF MATERIALS. TRANSVERSE. Beam Loaded Uniformly and at Several .Points. To Determine Equal Weiyht at Centre Fig. 2. Fig. 2.4 7. ^ ILLUSTRATION. What should be breadth of a beam of Georgia pine, 20 feet in length, 15 ins. in depth, ' uniformly loaded with 4000 Ibs., and sustaining 3 headers or con- centrated loads of 6000 Ibs., at re- 20' ^L spective distances of 4 and 9 feet from one end and 7000 Ibs. at 6 feet from the other end ? m = g, 7i= 1 1, r = i6, = 4, s 20 u 14, d 15 1 = 14, L = 4ooo, and _ _ su m n o r . L I C = 800 -r- 4 = 200. ^=w; =W; = w; and -r-2 = load in centre uniformly distributed. 4 X 16 X 6000 9 X 1 1 X 6000 6 X 14 X 7000 =19 200 \ = 29700; 20400 76s. on r>r\ * r>n Then -5- (6000 X i 1 + 7000 X 6) + 6000 = -^-x 108000+ 13200 = 61 800 7bs. omitting uniformly distributed load = = 2000 Ibs. concentrated at centre, of A B. Then to obtain total effect at W, 10 : 9 :: 2000 : 1800 effect of load. Hence, 9 X. X 1800 which 61800 = 70710 Ibs., and 7 7I X9Xl1 20 20 X I4 2 X 200 = 8.93 ins. breadth of beam. Operation deduced by Graphic Delineation of Greatest Stress without uni- form Load. Fig. 3. < -j . . }. Moments of weights = At i 2 3 T B 40' or W mn ^ w s u _; -- .; and = 19 200, 29 700, and 29 400, and let fall perpendiculars i, 2, and 3 proportionate thereto. Connect w', W, and w with A B, and sum of distances of in- tersections of these lines upon perpendiculars, from x, 2, and 3, respectively, will give stress upon A B at these points. Whence, greatest stress at greatest load will be ascertained to be 61 800 Ibs. as in Fig. 2. I ILLUSTRATION. Take elements of above case, omitting uniformly distributed load. (6000 X ii X 7000 X 6) -f- 6000 " X4 = X 108 000+ 13 200 = 61 800 Ibs. 20 20 20 Deflection of Girders and Beams. ing length in feet, b and d breadth and depth, and D deflection in ins. Values of C for Various Woods. (Hatfield.) Ash.. , 4000 Chestnut 2550 Hemlock 2800 Hickory 3850 Larch 2093 Oak, white 3100 " English, mean. . 2686 Spruce 3500 Pine, Georgia 5900 " pitch 2836 u white 2900 " red 4259 ILLUSTRATION. What would be deflection of a floor beam of white pine, 10 feet in length, 4 ins. in breadth, and 8 in depth, with 4000 Ibs. loaded in its middle? 4000X10' 4000000 . C = 2900. ^ ^ ^ g3 = = -674 tncA. 2900 X 4 X 8 3 5 939 20 * Load uniformly distributed. STRENGTH OF MATERIALS. TRANSVERSE. 84! When Weight is Uniformly Distributed. .625 W? " D 3 / 29 * 3X r* X * = 3/^ V -625 x 6000 v Hence, Deflection in preceding illustration would be ,674 x .625 = .421 ins. ILLUSTRATION. What should be length of a white-pine beam 3 by 10 ins., to sup- port 6000 Ibs. uniformly distributed, with a deflection of 2 ins. ? C = 2900. ^400000 3750 A fair allowance for deflection of floor beams, etc., is .03 inch per foot of length; .04 inch may be safely resorted to. "Weights of Floors and. of Loads. Dwellings. Weight of ordinary floor plank of white pine or spruce, 3 Iba. per sq. foot, and of Georgia pine, 4.5 Ibs. Plastering, Lathing, and Furring will average 9 Ibs. per sq. foot. Clay Blocks (Flat Arch) 5.25 x 7.25 ins. in depth and i foot in length, 21 Ibs. = 80 Ibs. per cube foot of volume. Floors of dwellings will average 5 Ibs. per sq. foot for white pine or spruce, and on iron girders will average from 17 to 20 Ibs. per sq. foot. Weight of men, women, and children over 5 years of age, 105.5 Ibs., and one third of each will occupy an average area of 12 x 16 ins. = 192 sq. ins. = 78.5 Ibs. per sq. foot. Of men alone 15 x 20 ins. = 300 sq. ins. =48 in 100 sq. feet. Bridges, etc. Weight of a body of men, as of infantry closely packed, = 138 Ibs. each, and they will occupy an area of 20 X 15 ins. = 300 sq. ins. = 66.24 Ibs. per sq. foot of floor of bridge, and as a live or walking load, 80 Ibs. per sq. foot. Weight of a dense and stationary crowd of men, 120 Ibs. per sq. foot. Bridging of Floor Beams increases their resistance to deflection in a very essential degree, depending upon the rigidity and frequency of the bridges. Weight on Floors, etc., in addition to Weight o-f Struct- ure, per Sq.. Foot. Ballrooms 85 Ibs. Brick or stone walls 115 to 150 Churches and Theatres. . . 80 Dwellings 40 Factories 200 to 400 Grain 100 Roofs, wind and snow 30 to 35 Ibs. Slate roofs 45 " Snow, per inch . 5 Ib. Street bridges 80 Ibs. Warehouses 250 to 50x3 " Wind 50 " Scarfs. Relative resistance of scarfs in Oak and Pine, 2 ins. square, and 4 feet in length, by experiments of Col. Beaufoy. Scarf 12 ins. in Length and 13 ins. from End, or i inch from Fulcrum. Vertical. no Ibs. gave away in scarf. Horizontal, large end uppermost and towards fulcrum. 101 Ibs. fastenings drew through small end of scarf ; small end uppermost, etc., 87 Ibs. gave away in thick part of scarf. Factors of Safety. Statical or Dead Load at .2 of destructive stress, but for ordinary pur- poses it may be increased to .25, and in some cases with good materials to .3. Live Load at .1 to .125 of destructive stress. See also page 802. 842 SUSPENSION BRIDGE. SUSPENSION BKIDGE. To Compute Elements. sin. z = stress at . C representing chord or span, a half chord, and v versed sine of chord or curve of deflection, in feet, L distributed load inclusive of suspended struct- ure, Q load per lineal foot, and S stress at centre, all in tons, x distance of any point from centre of curve, and h height of chain at x above centre of it, both in feet, s stress on chain at any point, as x,from centre of span, s stress on any tension-rod, and t stress at abutments, all in tons, n number of tension-rods, o angle of tangent of chain with horizon at any point, as x, r angle of chain with vertical at abutments, I length of chain, in feet, and z angle of direction of chain. Assume C =. 300 feet, L = 1000 tons, v = 25 feet, x = ioofeet, n = 30, r 71 34', and o = 12 32'. 1500 tan. 4X ii.n 2X 100 = . 2222 = I2 U 32 = 4 >_ = cot angle r . /7T V \ s = , ; and- 2X25 - = .3162=18 26'. For a deflection of .125 of span, horizontal stress is equal to total load. To Construct curve, see Geometry, page 230. To Compute Ratio which Stress on Chains or Cables at either Point of Suspension Bears to -whole Suspended "Weight of Structxire and Load. sjp 9 = R. R representing ratio. ILLUSTRATION. Assume elements of preceding case. = 1.58 ratio. By a preceding formula it would be 1.536. Stress on Back Stays. The cables being led over rollers, having free mo- tion, tension upon them is same, whether angle i is same as that of r or not. Stress on Piers. When angles r and i are alike, stress on piers will be vertical, but when angle of i is greater or less than r, stress will be oblique. To Compute Horizontal Stress and 'Vertical Pressure on Piers. S cos. z S i, S cos. n = S o, S sin. z P i, and S sin. n P o. Si and S o representing stress, and P i and P o pressure, inward and outward. NOTE. Span of New York and Brooklyn Bridge 1595.5 feet, deflection 128 feet, angle of deflection at piers from horizontal 15 10'. TRACTION. 843 TRACTION. Results of Experiments on Traction of Roads and. IPavements. (At. Morin.) i st. Traction is directly proportional to load, and inversely proportional to diameter of wheel. 2d. Upon a paved or Macadamized road resistance is independent of width of tire, when it exceeds from 3 to 4 ins. 3d. At a walking pace traction is same, under same circumstances, for carriages with or without springs. 4th. Upon hard Macadamized, and upon paved roads, traction increases with velocity : increments of traction being directly proportional to incre- ments of velocity above velocity of 3.28 feet per second, or about 2.25 miles per hour. The equal increment of traction thus due to each equal increment of velocity is less as road is more smooth, and carriage less rigid or better hung. 5th. Upon soft roads of earth, sand, or turf, or roads thickly gravelled, traction is independent of velocity. 6th. Upon a well-made and compact pavement of dressed stones, traction at a walking pace is not more than .75 of that upon best Macadamized roads under similar circumstances ; at a trotting pace it is equal to it. yth. Destruction of a road is in all cases greater as diameters of wheels are less, and it is greater in carriages without springs than with them. Experiments made with the carriage of a siege train on a solid gravel road and on a good sand road gave following deductions : 1. That at a walk traction on a good sand road is less than that on a good firm gravel road. 2. That at high speeds traction on a good sand road increases very rapidly with velocity. Thus, a vehicle without springs, on a good sand road, gave a traction 2.64! tunes greater than with a similar vehicle on same road with springs. Results -with, a Dynamometer. Wagon and Load 2240 Ibs. * ROADWAY. Relat'e num- ber of horses for like effect. On railway, 8 Ibs On best stone tracks, 12.5 Ibs. Good plank road, 32 to 50 Ibs. . 4 to 6.25 Telford road, 46 Ibs Broken stone or con'te, 46 Ibs. Gravel or earth, 140-147 Ibs. I Common earth road, 200 Ibs. . Relat'e nnm- ber of horses for like effect. 5-75 5-75 17-5 18.37 25 Stone block pavement, 32.5 " 4.06 Macadamized road, 65 Ibs. ... 8.12 NOTE. By recent experiments of M. Dupuit, he deduced that traction is inversely proportional to square root of diameter of wheel. Relation of force or draught to weight of vehicle and load over 6 different con^ Et ructions of road, gave for different speeds as follows: Walk. Trot. Walk. Trot. Stage coach, 5 tons. .1.3 x | Carriage, seats only, on springs. .1.29 i Resistance to Traction on Common Roads. On Macadamized or Uniform Surfaces. (M. Dupuit.) 1. Resistance is directly proportional to pressure. 2. It is independent of width of tire. 3. It is inversely as square root of diameter of wheel. 4. It is independent of speed. * See Treatise on Roads, Streets, and Pavements, by Brev. Maj.-Gen'l Q. A. Gillmore, U. 8. A. t Telford estimated it at 3.5. 844 TRACTION. On Paved and Rough Roads. Resistance increases with speed, and is diminished by an enlargement of tire up to a moderate limit. Traction on Various Roads. Traction of a wheeled vehicle is to its weight upon various roads as follows : Per Ton. 46 to 78 46 to 90 Per Ton. Stone track, best 12.5 to 15 " " .... 28 to 39 " pavement. 14 to 36 Asphalted 22 to 28 Per 100 Ibs. 55 to .58 1.25 to 1.3 .5 to 1.5 i to 1.25 Plank 22 to 45 .98 tO 2 Block stone } t pavement.... J 32 1.4 to 1.6 Tel ford road Macadamized. . . " loose 67 to 1 Gravel 134 to 180 Sandy 140 to 313 Earth 200 to 290 Per ito Ibi. 2.1 tO 3.5 2 to 4 3 t( > 5 6 to 8 6.3 to 14 9 to 13 Hence, a horse that can draw 140 Ibs. at a walk, can draw upon a gravel road X JOG = 2000 Ibs. Resistance on Common. Roads or Fields (Bedford Experiments, 1874.*) GRAVELLED ROAD. (Hard and dry, rising i in 430.) Maxi- mum Draft. Average Draft. Average Speed dour. IP de- veloped per Minute. Draft per Ton on Level. Work per HP Horse. Lbs. Lbs. Miles. IP. Lbs. IP. 2 horse wagon without springs. 320 159 2-5 i. 06 43. 5 or. 0192 3 4 " 400 251 2.6 1.74 44-5 "-2 .87 2 " " with 300 133 2.47 .88 34.7 " .015 44 i " cart without " 180 49-4 2.65 35 28 " .0125 35 ARABLE FIELD. (Hard and dry, rising i in 1000.) 2 horse wagon without springs. IOOO 700 2-35 4-36 210 or .099 2.18 4 " " I2OO 997 2.52 6.7 194 " .083 3-35 I " with " IOOO 710 2-35 4-45 210 " .099 1.22 i " cart without " 400 212 2.61 1.48 140 " .0625 I. 4 8 Fore wheels of wagons were 39 ins., and hind 57 ins. in diam. ; tires varying from 2.25 to 4 ins. ; and wheels of cart were 54 ins. in diam., and tires 3.5 and 4 ins. Springs reduced resistance on road 20 per cent., but did not lessen it in the field. From these data it appears, that on a hard road, resistance is only from 25 to . 16 of resistance in field. Lowest resistance is that of cart on road 28 Ibs. per ton ; due, no doubt, to absence of small wheels alike to those of the wagons. Assuming average power without springs to be .6 IP on road, as average for a day's work, it represents .6 X 33000 = 19800 foot-lbs. per minute for power of a horse on such a road. Resistance of a smooth and well-laid granite track (tramway), alike to those in London and on Commercial Road, is from 12.5 to 13 Ibs. per ton. Omnibus. t ( Weight 5758 Ibs.) Average Speed per Hour. Per Ton. Total. Granite pavement (courses 3 to 4 ins.) 2.87 miles. 17.41 Ibs. 44.75 Ibs Asphalt roadway 3.56 " 27.14 " 69.75 " Wood pavement 3.34 " 41.6 ' 106.88 " Macadam road, gravelly 3.45 " 44-48 " 114-32 " " " granite, new 3.51 " 101.09 " 2598 " NOTE. The resistance noted for an asphalt roadway is apparently inconsistent with that for a granite pavement, for when it is properly constructed it is least resistant of all pavements. * See report in Engineering, July 10, 1874, pagt 23. t Report Soc. Arts, London, 1875. TRACTION. 845 Wagon . (Sir John Macneil. ) Weight 2342 Ibs. Speed 2.5 Miles per Hour. Resistance. Per Ton. Total. Well-made stone pavement 31.2 Ibs 33 Ibs. Road made with 6 ins. of broken hard stone, on a foundation) 4< , u of stoues in pavement, or upon a bottom of concrete > 44 Old flint road, or a road made with a thick coating of broken ) , u 6 tt stone, on earth f Road made with a thick coating of gravel, on earth 140 " 147 " Stage Coach. (Sir John Macneil.) Weight 3192 Ibs. Gradients i to 20 to 600. Speed. Metalled Road. At 6 miles per hour 62 Ibs. per ton. " 8 73 " " 10 79 ' NOTE. It was found that, from some unexplained cause, the net frictional resistance at equal speeds varied considerably, according to gradient, resistances being a maximum for steepest gradient, and a minimum for gradients of i in 30 to i in 40 ; for these they are less than i in 600. Mode of action of the horses on the carriage may nave been an influential element. (I). K. Clark.) To Compxite Resistance to Traction on "Various Roads. (Sir John Macneil.) ON A LEVEL. RULE. Divide weight of vehicle and load in Ibs. by its unit in following table, and to quotient add .025 of load ; add sum to product of velocity of vehicle in feet per second, and Coefficient in following table for the particular road, and result will give power required in Ibs. Or, . W -{- w .025 -|- C v = T. W and w representing weights of vehicle and load, unit Coefficients for Traction of Various Vehicles. Stage coach too I 2 horse wagon without springs 54 Heavy wagon 93 2 " " with " 42 4 horse wagon without springs 55 I i " cart without " 36 Coefficients for Roads of Various Construction. Pavement 2 Broken stone, dry and clean " covered with dust. . . . " " muddy 10 Macadamized road 4.3 Gravel, clean 13 " muddy 32 Stone tramway 1.2 Sand and Gravel 12.1 ILLUSTRATION. What is the traction or resistance of a stage coach weighing 2200 Ibs., with a load of 1600 Ibs., when driven at a velocity of 9 feet per second over a dry and clean broken stone road ? 22OO-4-l6oO . , 1- 1600 X .025 -f 5 X 9 = 123 Ibs. To Compute JPower necessary to Sustain a "Vehicle upon an Inclined Road, and also its Pressure thereon ^omit>- ting Effect of Friction. AT AN INCLINATION. W : A C :: o : B C, and W : A C :: p : A B. Or, r e : e o : : A B : B C; W : e o : : I : h: whence, W j = eo. Assume A B of such a length that vertical rise BC = i foot; then, x/.. W representing 1 l vV 2 +i Vr 2 +i weights of vehicle and load o, and Y power or force necessary to sustain load on road, p pressure of load on surface, all in Ibs., h height of plane, I inclined length of road or plane, and V horizontal length, all in feet. ILLUSTRATION. What is power required to sustain a carriage and its load, weigh- ing 380x3 Ibs., upon a road, inclination of which is i in 35, and what is its pressure upon road? Sin. A = .028 56. Cos. A = .999 59. I = 35.014. Then 3800 X-Q28 56= 108.53 Jt)S - power, and 3800 X .99959=3798.44 Ibs. pressure. To Compute Resistance of a Load, on an Inclined. Road. RULE. Ascertain the tractive power required, and add to it the power necessary to sustain load upon inclination, if load is to ascend, and subtract it if to descend. EXAMPLE i. In preceding example tractive power required is 123 Ibs., and sus- taining power for that inclination 108.53; hence 123 + 108.53^231.53 Ibs. 2. If this load was to be drawn down a like elevation. Then 123 108.53 = 14.47 Ms. To Compute Power necessary to Move and Sustain a "Vehicle either Ascending or Descending an Elevation, and at a given Velocity, omitting Effect of ifriction. f -f- J cos. L =F (W -f w) sin. L. -j- Fc = R. L representing angle of elevation for a stage wagon and a stage coach, and t units as preceding ; upper sign taken when vehicle descends the plane, and lower when it ascends. ILLUSTRATION. Assume a stage coach to weigh 2060 Ibs., added to which is a load of i zoo Ibs., running at a speed of 9 feet per second over a broken stone road covered with dust, and having an inclination of i in 30; what is power necessary to move and sustain it up the inclination, and what down it ? v = <), c = 8, sin. of L. =sin. of i 54'-J-:r=.o333, and cos. L. =.9995. Then ( 2 ^oo" 00 -T- ^) X .9995 + (2060+ i loo )X. 0333 + 8X9 = 59-7 + 105.23 -j- 72 3= 236.3 Ibs. up inclination. " " ADd " 100 + r X * 9995 + 8X9 (2060 + i ioo) X. 0333 = 59.07 -f 72 105.23 = 25.84 Ibs. down inclination. Tractive and Statical Resistance of Elevations. (Gillmore.) T : = g'. T representing traction in Ibs. per ton, W weight of load in Ibs., VW 2 T 2 and g' grade of road. ILLUSTRATION. Assume traction as per preceding table, page 844, 200, and weight of vehicle 2 tons; what should be least grade of road? 200 X 2 I . , =^ = .0897 = -+. V 448o 2 200 X 2 Showing that, for a road upon which traction is 200 Ibs. per ton, the grade should not exceed one in height to one eleventh fall of base; hence, generally, the proper grade of any description of road will be equal to force necessary to draw load upon like road when level. Practically, greatest grade of a Telford or Macadamized road in good condition = .05, and a horse can attain at a walk a required height upon this grade, without more fatigue and in nearly same time that he would require to attain a like height over a longer road with a grade of .033, that he could ascend at a trot. For passenger traffic, grades should not exceed .033. TRACTION. 847 Resistance of* Gravity at Different Inclinations of Grade. For a Load of ioo Lbs. Grade. R Grade. R Grade. R Grade. R i in 5 i in 10 i in 15 i in 20 Lbs. 19.61 9-95 6.65 4.99 i in 25 i in 30 i in 35 i in 40 Lbs. 4 3-33 2.85 2-5 i in 45 i in 50 i in 55 i in 60 Lbs. 2.22 2 1.82 1.6 7 i in 70 i in 80 i in 90 . i in ioo Lbs. 1-43 1-25 i. ii I Inclination of Roads. Power of draught at different inclinations and velocities Is as follows (Sir John Macneil) : Inclination. Angle. Feet per Mile. Tractioi 6 Miles. i at Speed Hour of 8 Miles. a of per 10 Miles. Frictior Ton at S 6 Miles. al Resists jeedsofpt 8 Miles. nee per r Hour of 10 Miles. in 20 in 26 in 30 in 40 in 60 2 o 5 < 2 12 i55' I26' 57-5' 264 203.4 176 132 88 268 2I 3 *s 1 60 III 296 219 196 166 120 318 225 200 I 7 2 128 % 4 56 7 2 P 61 78 112 & ii Grade. Grade of a road should be reduced to least of practicable attainment, and ae a general rule should be as low as i in 33, and steepest grade that is ad- missible on a broken stone road is i in 20. The condition of traction is /-f sin. a L, which should not exceed P, and sin. a P should not exceed -= /, or/ f representing coefficient of friction, a angle ofin- Lt clination, L load, and P power in Ibs. ILLUSTRATION. In case, page 846, weight or load = 2060 4-1100=3160 Ibs., Co- efficient of friction for such a road = .042 per ioo Ibs. , and sin. i 54' = .033 16. Then .042 + .033 16 X 3160 = 237. 5 Ibs. Traction of a Vehicle compared to its Weight on Different Roads. (F. Robertson, F. R. A. S.) Stone pavement i in 68 I Flint foundation i in 34 Macadamized road i " 49 | Gravel road i " 15 Sandy road i in 7. Assuming a horse to have a tractive force of 140 Ibs. continuously and steadily at a walk, he can draw at a walk on a gravel road 15 x 140 = 2100 Ibs. Friction of Roads. Friction of Roads. According to Babbage and others, a wagon and load weighing 1000 Ibs. requires a traction as follows : Of Load. Fresh earth 125 ( -035 1 .067 jw, iron Friction Per Ton. 1 86 157 141 117 63 34 101 74 Sled, hard sn Coefficients of , Per ioo Ibs. Gravel road new 083 Common road, bad order. . .07 Sand road 063 Broken stone, rutted 052 " " fair order. . . .028 " " perfect order .015 Macadamized new 045 " gravelly 02 Warth, good ovder. 0.15 Macadamized Dry high road. .. Well paved road. Railroad shod 033 of load. in Proportion to Load. Per ioo Ibs. Wood pavement 019 Asphalt roadway 012 Stone pavement 015 Granite " 008 Stone " very smooth .006 Plank road 01 Of Load. 033 025 014 f -0035 I -0059 Stone track 05 Per Ton 42 27 34 13 us 848 TRACTION. To Compute Friction al Resistance to Traction of a Stage Coach, on a Metalled Road, in Grood Condition. 30 -|- 4 v + \/ 10 v = & v representing speed in miles per hour, and R frictional resistance to traction per ton. NOTE. Formula is applicable to wagons at low speeds. Canal, Slackwater, and. River. On a canal and water, resistance to traction varies as square of velocity, from that of 2 feet per second to that of 11.5 feet. When velocity is less than .33 miles per hour, resistance varies in a less degree. In towing, velocity is ordinarily i to 2.5 miles per hour. Resistance of a boat in a canal depends very much upon the comparative areas of transverse sections of it and boat, it being reduced as difference increases. In a mixed navigation of canal and slack-water, 3 horses or strong mules will tow a full-built, rough-bottomed canal boat, with an immersed sectional area of 94.5 sq. feet, and a displacement of 240 tons, 1.75 to 2 miles per hour for periods of 12 hours. With a section of but 24.5 sq. feet, or a displacement of 65 tons, an aver- age speed of 2.5 miles is attained for a like period. By the observations of Mr. J. F. Smith, Engineer of the Schuylkill Navigation Co., a canal boat, with an immersed section alike to that above given, can be towed for 10 hours per day as follows: Per Hour. By i horse or mule. By 2 horses or mules. By 3 horses or mules. By 4 horses or mules. By 8 horses or mules. i mile. i. 5. miles. 1.75 miles. 1.875 miles. 2.5 miles. Assuming then, the tractive power of a horse as given in table, page 437, the above elements determine results as follows: Horses. Miles. Tractive Power divided by Load. in Lbs. per Ton. fraction in Lbs. per Sq. Foot of immersed Section. 250 240 1.04 2.65 165 X 2 240 1.38 3-49 3 I 7^ 140 X 3 240 4. 44 1.871; 132 X 3 240 1.65 4.19 125 X 3 240 i 56 ^.08 ? (fight) 2. S loo X ^ 6q 4.61 12.24 Upon a canal of less section and depth, a displacement of 105 tons, with a:i im- mersed section of 43 sq. feet, a speed of 2 miles with 2 borses was readily obtained, which would give a traction of 2.38 Ibs. per ton, and of 5.71 Ibs. per sq. foot of im- mersed section. Maximum Power of a Horse on a Canal. (Molesworth.) Miles per hour 2.5 3 3.5 4 5 678 9 10 D "..: n } -5 * M 4-5 *.9 - .S 5 .9 -73 Load drawn in tons .. 520 243 153 102 52 30 19 13 9 6.5 Street Railroads or Tram-ways. (Gen'l Gillmore.*) Upon a level road, and at a speed of 5 miles per hour, the power required to draw a car and its load is from -g^-Q to ^^ of total weight, varying with condition of rails and dryness or moisture of their surface. * Treatise on Roads, Streets, and Pavements. D. Van Notrand, 1876, N. Y. TJRACTION. WATEK. 849 To Compute Resistance of* a Car. TX6=/; - = c; - = r; and /+ c + r = R. T representing weight in tons, f friction in Ibs., v speed in miles per hour, a area of front or section of car in sq.feet, c concussion, r resistance of atmosphere, and R total resistance, all in Ib*. ILLUSTRATION. Assume a car and load of 8960 Ibs., with an area of section of 56 sq. feet, and a speed of 5 miles per hour. Then = 4 tons ; 4 x 6 = 24 Ibs. friction ; 1- = 6.66 Ibs. concwrion ; 5 - 5_ = 3. 5 Ibs. resistance of air ; and 24 + 6. 66 + 3. 5 = 34. 16 Ibs. 400 In average condition of a road, the resistance of a car may be taken at T ^, which, in preceding case, would be 74.66 Ibs. On a descending grade, therefore, of i in 74 66, the application of a brake would mot be required. WATER. FRESH WATER. Constitution of it by weight and measure is By Weight. By Measure. I By Weight. By Mewur*. Oxygen... 88.9 i | Hydrogen., n.i 2 Cube inch of distilled water at its maximum density of 39. i, barom- eter at 30 ins., weighs 252.879 grains, and it is 772.708 times heavier than atmospheric air. Cube foot (at 39.1) weighs 998.8 ounces, or 62.425 Ibs. NOTE. For facility of computation, weight of a cube foot of water is usually taken at 1000 ounces and 62.5 Ibs. At a temperature of 32 it weighs 62.418 Ibs., at 62 (standard tem- perature) 62.355 Ibs., and at 212 59.64 Ibs. Below 39.1 its density decreases, at first very slow, but progressing rapidly to point of conge- lation, weight of a cube foot of ice being but 57.5 Ibs. Its weight as compared with sea-water is nearly as 39 to 40. It expands .085 53 its volume in freezing. From 40 to 12 it ex- pands .00236 its volume, and from 40 to 212 it expands .0467 times = .ooo 271 5 for each degree, giving an increase in volume of i cube foot in 21.41 feet. Volumes, Height, and. Pressure of Pure Water. Cube Ins. Feet. At 32 27.684 2.307 ) i Lb. At 62 i Ton 35-9 2 3 cu be feet. " 39-i 2 7 .68 2.3067 _ Pressure " " i Lb. = 27.71 ' ins. " 62 27.712 ss 2-393 per \\ 39 u l0 i Tonneai I == 35.3156 " feet. " 212 28.978 = 2.4148 sq. inch. i Kilogr. = 0353 (( U Height of a Column of Water at 62 or 62.355 Ibs. i Ib. per sq. inch = 2. 3093 feet, and at pressure of atmosphere = 33. 947 feet = 10.347 meters. Ice and Snow. Cube foot of Ice at 32 weighs 57.5 Ibs., and i Ib. has a volume of 30.067 cube ins. Volume of water at 32, compared with ice at 32, is as i to 1.085 53? ex- pansion being 8.553 per cent. Cube foot of new fallen snow weighs 5.2 Ibs., and it has 12 times bulk of water. 850 WATER. Rainfall. Annual Fall at different Places. LOCATIOW. Ins. LOCATION. Ins. LOCATION. Ins. Alabama Ft Crawford Wis.. 29 54 Michigan 3-1 e Albany 41. 3^ Ft Gibson, Ark 30. 64 45 7. ye Ft. Snelling, Iowa.. 30. 32 Mobile, 1842 S4. Q4 Allegheny ....... ,666 Fortr Monroe, Va.. 52. 53 Naples 41 8 Antigua Florence O.C Q Newburg Archangel .... 14 ^2 Frankfort Oder... 21.3 New York 4-5 36 " Main 6 A Ohio 3.6 Bahamas Geneva iu.4 02 6 Palermo 22 8 Baltimore OQ Q Gibraltar 47- 20 Paris 23. 1 55.87 21.3 Philadelphia 49 Bath Me 1,4. ;8 31 Plymouth (Eugl ) . Belfast * Gordon Castle Sc'd Port Philip 20 16 Biskra .2 Granada | 105 Poughkeepsie 32.06 Bombay Bordeaux 1 10 Great Britain ..... 126 Providence Rochester 36.74 Boston OQ 27 Greenock 61.8 Rome on Brussels Halifax Santa Fe Le Buffalo Hanover 22.4. Savannah 55 52 Schenectadv 47 - 77 Calcutta RT Hong-kong 8l 3S Siberia .... Cape St. Francois. . Cape Town 150 23-31 Hudson India { 39-32 60 Sierra Leoiie Sitka 84 85.79 Charleston Cherbourg 54 OQ. 7 Jamaica I 3 3 A -31 St. Bernard St. Domingo 48 1 20 Cologne Copenhagen 24 23 Jerusalem Key West J4-3 1 65 31 3Q St. Petersburg State of N Y 17.6 OO 7Q Cracow 10. oo Khassaya, India. . . 610 Sydney 4Q Demerara ... Lewiston Tasmania " 1849 10,2. 21 Liverpool 34 *2 Trieste 46.4 Dover (Engl. ) Dublin 37-52 30.87 London 25-2 51 85 Ultra Mullay, India Utica 263.21 on -3 Dumfries 36.92 22 Venice 34- I East Hampton 38 S2 4Q Vera Cruz 62 / 49 ofi IA. Vienna 10 6 Edinburgh J Manchester j Washington . . 19.0 Fairfield... 32.03 West Point . . . 48.7 Average rainfall in England for a number of years was, South and East, 34 ins.; West and hilly, 43.02 to 50 ins., and percolation of it was estimated at 30 per cent. Mean volume of water in a cube foot of air in England is 3.789 grains. Globe, mean depth 36 ins. Cape of Good Hope in 1846 in 3 hours, 6.2 " At Khassaya, in 6 rainy months 550 ins. ; in i day, 25. 5 " Evaporation. Mean daily evaporation, in India .22 inch; greatest .56; in Eng- land .08. At Dijon, when mean depth of rainfall was 26.9 ins. in 7 years, evapora- tion was for a like period 26.1 ins., and in Lancashire, Eng., when fall was 45.96 ins., evaporation was 25.65. "Volume of Rainfall. Rainfall, depth in ins. , X 2 323 200 = cube feet per sq. mile. X 17.378 74 = millions of gallons per sq. raila X 3630 = cube feet per acre. ' " " X 27 154.3 = gallons per acre. Mineral Waters are divided into 5 groups, viz. : 1. Carbonated, containing pure Carbonic acid as, Seltzer, Germany; Spa, Bel- gium; Pyrmont, Westphalia; Seidlitz, Bohemia; and Sweet Springs, Virginia. 2. Sulphurous, containing Sulphuretted hydrogen as, Harrowgate and Chelten- ham, England; Aix-la-Chapelle, Prussia; Blue Lick, Ky. ; Sulphur Springs, Va., etc 3. Alkaline, containing Carbonate of soda these are rare, as, Vichy, Ems. WATER. 4. Chalybeate, containing Carbonate of iron as, Hampstead, Tunbridge, Chelten- ham, and Brighton, England; Spa, Belgium; Ballston and Saratoga, N. Y. ; and Bedford, Penn. 5. Saline, containing salts as, Epsom, Cheltenham, and Bath, England; Baden- Baden and Seltzer, Germany; Kissingen, Bavaria; Plombieres, France; Seidlitz, Bohemia ; Lucca, Italy ; Yellow Springs, Ohio ; Warm Springs, N. C. ; Congress Springs, N. Y. ; and Grenville, Ky. Brief Rules for Qualitative Analysis of Mineral Waters. First point to be determined, in examination of a mineral water, is to which of above classes does water in question belong. 1. If water reddens blue litmus paper before boiling, but not afterwards, and blue color of reddened paper is restored upon warming, it is Carbonated. 2. If it possesses a nauseous odor, and gives a black precipitate, with acetate of lead, it is Sulphurous. 3. If, after addition of a few drops of hydrochloric acid, it gives a blue precipitate, with yellow or red prussiate of potash, water is a Chalybeate. 4. If it restores blue color to litmus paper after boiling, it is Alkaline. 5. If it possesses neither of above properties in a marked degree, and leaves a large residue upon evaporation, it is a Saline water. Re-agents. When water is pure it will not become turbid, or produce a precipitate with any of following Re-agents : Baryta Water, if a precipitate or opaqueness appear, Carbonic Acid is present. Chloride of Barium, indicates Sulphates, Nitrate of Silver, Chlorides, and Oxalate of Ammonia, Lime salts. Sulphide of Hydrogen, slightly acid, Antimony, Arsenic, Tin, Copper, Gold, Platinum, Mercury, Silver, Lead, Bismuth, and Cadmium; Sul- phide of Ammonium, solution alkaloid by ammonia, Nickel, Cobalt, Manganese, Iron, Zinc, Alumina, and Chromium. Chloride of Mercury or Gold and Sulphate of Zinc, organic matter. Filter Beds. Fine sand, 2 feet 6 ins. ; Coarse sand, 6 ins. ; Clean shells, 6 ins. , and Clean gravel 2 feet, will filter 700 gallons water in 24 hours per square foot, by gravitation. SEA WATER. Composition of it per volume : Chloride of Sodium (common salt). . 2.51 Sulphuret of Magnesium 53 Chloride of " 33 Carbonate of Lime ) " of Magnesia J 2 Sulphate of Lime 01 Water 96.6 By analysis of Dr. Murray, at specific gravity of 1.029, it contains Muriate of Soda 220.01 I Muriate of Magnesia 42.08 Sulphate of Soda 33. 16 | Muriate of Lime 7.84 303.09 Or, i part contains .030309 parts of salt = ^ part of its weight. Mean volume of solid matter in solution is 3.4 per cent., .75 of which is common salt. Boiling Points at Different Degrees of Saturation. Salt, by Weight, in 100 Parts. Boiling Point. Salt, by Weight, in 100 Parts. Boiling Point. Salt, by Weight, in loo Parts. Boiling Point. 3-03 = A 213.2 5-5 = & 217.9 27 . 2 8 = inf 222.5 6.06 = ^ 214.4 18.18 = -^ 2I 9 30-31 = if 223.7 9-9 = inr 215.5 "=* 22O.2 33-34 = -H- 224.9 ""-A 2l6. 7 24.25 = ^ 221.4 *3-37 = if 236 852 WATER. WAVES OF THE SEA. J3epcsits at Different Degrees of Saturation and. Tern peratnre. When 1000 Parts are reduced by Evaporation. Volume of Water. Boiling Point. Salt in 100 Parts. Nature of Deposit. 1000 299 102 214 217 228 10 29-5 None. Sulphate of Lime. Common Salt It contains from 4 to 5.3 ounces of salt in a gallon of water. Saline Contents of Water from several Localities. Baltic 6.6 Black Sea 21.6 Arctic 28.3 British Channel 35.5 Mediterranean 39-4 Equator 39.42 South Atlantic 41,2 North Atlantic .... 42.6 Dead Sea 385 There are 62 volumes of carbonic acid in 1000 of sea- water. Cube foot at 62 weighs 64 Ibs. Its weight compared with fresh water being very nearly as 40 to 39. Height of a Column of Water at 60 or 64.3125 Ibs. At 62, i Ton = 35 cube feet, i Lb. per sq. inch = 2.239 ^ ee ^ au( ^ at pressure of atmosphere = 32.966 feet = 10.048 meters. Weights. A ton of fresh water is taken at 36, and one of salt at 35 cube feet WAVES OF THE SEA. Arnott estimated extreme height of the waves of an ocean, at a distance from land sufficiently great to be freed from any influence of it upon their culmination, to be 20 feet. French Exploring Expedition computed waves of the Pacific to be 22jeet in height. By observations of Mr. Douglass in 1853, ne deduced that when waves had heights of 8 feet, there were 35 in number in one mile, and 8 per minute. 15 " " 5 and 6 " 5 " 20 " 3 4 J. Scott Russell divides waves into 2 classes viz. : Waves of Translation, or of ist order; of Oscillation, or of 2d order. "Waves of tlxe First Order. 1. Velocity not affected by intensity of the generating impulse. 2. Motion of the particles always forward in same direction as wave, and same at bottom as at surface. 3. Character of wave, a prolate cycloid, in long waves, approaching a true cycloid. When height is more than one third of length, the wave breaks. "Waves of the Second Order. 1. Ordinary sea waves are waves of second order, but become waves of the first order as they enter shallow water. 2. Power of destruction directly proportional to height of wave, and great- est when crest breaks. 3. A wave of 10 feet in height and 32 feet in length would only agitate the water 6 ins. at 10 feet below surface ; a wave of like height and 100 feet in length would only disturb the water 18 ins. at same depth. Average force of waves of Atlantic Ocean during summer months, as de- termined by Thomas Stevenson, was 611 Ibs. per sq. foot; and for winter months 2086 Ibs. During a heavy gale a force of 6983 Ibs. was observed, WAVES OF THE SEA. 853 J. Scott Riissell deduced that a wave 30 feet in height exerts a force of i ton per sq. foot, and that, in an exposed position in deep water, 1.75 tons may be exerted upon a vertical surface. At Cassis, France, when the water is deep outside, blocks of 15 cube me- ters were found insufficient to resist the action of waves. Breakwaters with vertical walls, or faces of an angle less than i to i, will reflect waves without breaking them. Waves of oscillation have no effect on small stones at 22 feet below the surface, or on stones from 1.5 to 2 feet, 12 feet below surface. A roller 20 feet high will exert a force of about i ton per sq. foot. Greatest force observed at Skerryvore, 3 tons per sq. foot ; at Bell Rock, 1.5 tons per sq. foot. Waves of the second order, when reflected, will produce no effect at a depth of 12 feet below surface. Action of waves is most destructive at low-water line. Waves of first order are nearly as powerful at a great depth as at surface. To Compnte "Velocity. When I is less than d. . 55 ^/l or i. 818 ^Jl = V. When I exceeds 1000 d. -^32.17 d = V, and When Height of Wave becomes a sen- sible Proportion to Depth, */32. 17(1 + 3 \=V. To Compute Height of* "Waves in Reservoirs, etc. (2. 5 Vf 1 ) height in feet, L representing length of Reservoir, Pond, etc., exposed to direction of wind, in miles. Tidal Waves. Wave produced by action of sun and moon is termed Free Tide Wave. Semi-diurnal tide wave is this, and has a period of 12 hours 24+ minutes. ' Professor A iry declared that when length of a wave was not greater than depth of the water, its velocity depended only upon its length, and was pro- portionate to square root of its length. When length of a wave is not less than 1000 times depth of water, velocity of it depends only upon depth, and is proportionate to square root of it; velocity being same that a body falling free would acquire by falling through a height equal to half depth of water. For intermediate proportions, velocity can be obtained by a general equation. Under no circumstances does an unbroken wave exceed 30 or 40 feet in height. A wave breaks when its height above general level of water is equal to general depth of it. Diurnal and other tidal waves, so far as they are free, may be all considered as running with the same velocity, but the column of the length of wave must be doubled for diurnal wave. Length of Wave. Depth of Water. Feet. i Feet. 10 Feet. ZOO Feet. 1000 Feet. IOOOO Feet. IOO 000 Velocity per Second. Feet. Feet. Feet. Feet. Feet. Feet. Feet. i 2.26 5-34 5-67 10 2.26 7-iS 16.88 17.92 '7-93 IOO 7-5 22.62 53-19 56.67 56.71 1000 22.62 71-54 168.83 179.21 KOOOO 71-54 326.714 533-9 854 WHEEL GEARING. WHEEL GEARING. Pitch Line of a wheel is circle upon which pitch is measured, and it is circumference by which diameter, or velocity of wheel, is measured. Pitch is arc of circle of pitch line, is determined by number of teeth in wheel, and necessarily an aliquot part of pitch line. True or Chordial Pitch, or that by which dimensions of tooth of a wheel are alone determined, is a straight line drawn from centres of two contiguous teeth upon pitch line. Line of Centres is line between centres of two wheels. Radius of a wheel is semi - diameter bounded by periphery of the teeth. Pitch Radius is semi-diameter bounded by pitch line. Length of a Tooth is distance from its base to its extremity. Breadth of a Tooth is length of face of wheel. Depth of a Tooth is thickness from face to faoe at pitch line. Face / a Tooth, or Addendum, is that part of its side which extends from its pitch line to its top or Addendum line. Flank of a Tooth is that part of its side which extends from pitch line to line of space at base of and between adjacent teeth ; its length, as well as that of face of tooth, is measured in direction of radius of wheel, and is a little greater than face, of tooth, to admit of clearance between end of tooth and periphery of rim of wheel or rack. Cog Wheel is general term for a wheel having a number of cogs or teeth set in or upon, or radiating from, its circumference. Mortice Wheel is a wheel constructed for reception of teeth or cogs, which are fitted into recesses or sockets upon face of the wheel. Plate Wheels are wheels without arms. Rack is a series of teeth set in a plane. Sector is a wheel which reciprocates without forming a full revolution. Spur Wheel is a wheel having its teeth perpendicular to its axis. Bevel Wheel is a wheel having its teeth at an angle with its axis. Crown Wheel is a wheel having its teeth at a right angle with its axis. Mitre Wheel is a wheel having its teeth at an angle of 45 with its axis. Face Wheel is a wheel having its teeth set upon one of Hs sides. Annular or Internal Wheel is a wheel having its teeth convergent to its centre. Spur Gear. Wheels which act upon each other in same plane. Bevel Gear. Wheels which act upon each other at an angle. Inside Gear or Pin Gearing. Form of acting surfaces of teeth for a pitch-circle in inside gearing is exactly same with those suited for same pitch-circle in outside gearing, but relative position of teeth, spaces, and flanks are reversed, and- adden- dum-circle is of less radius than pitch-circle. A Train is a series of wheels in connection with each other, and consists of a series of axles, each having on it two wheels, one is driven by a wheel on a preced- ing axis and other drives a wheel on following axis. Idle Wheel. A wheel revolving upon an axis, which receives motion from a pre- ceding wheel and gives motion to a following wheel, used only to affect direction of motion. Trundle, Lantern, or Wallower is when teeth of a pinion are constructed of round bars or solid cylinders set into two disks. Trundle with less than eight staves can- not be operated uniformly by a wheel with any number of teeth. Spur, Driver, or Leader is term for a wheel that impels another; one impelled is Pinion, Driven, or Follower. WHEEL GEARING. 855 Teeth of wheels should be as small and numerous as is consistent with Strength. When a Pinion is driven by a wheel, number of teeth in pinion should not be less than 8. When a Wheel is driven by a pinion, number of teeth in pinion should not be less than 10. When 2 wheels act upon one another, greater is termed Wheel and lesser Pinion. When the tooth of a wheel is made of a material different from that of wheel it is termed a Cog ; in a pinion it is termed a Leaf, in a trundle a Stave, and on a disk a Pin. Material of which cogs are made is about one fourth strength of cast iron. Hence, product of their b d 2 should be 4 times that of iron teeth. Number of teeth in a wheel should always be prime to number of pinion ; that is, number of teeth in wheel should not be divisible by number of teeth in pinion without a remainder. This is in order to prevent the same teeth coming together so often and uniformly as to cause an irregular wear of their faces. An odd tooth introduced into a wheel is termed a Hunting tooth or Cog. The least number of teeth thai; it is practicable to give to a wheel is regu- lated by necessity of having at least one pair always in action, in order to provide for the contingency of a tooth breaking ; and least number that can be employed in pinions having teeth of following classes is : Involute, 25 ; Epicycloidal, 12; Staves or Pins, 6. Velocity Ratio in a Train of Wheels. To attain it with least number of teeth, it should, in each elementary combination, approximate as near as practicable to 3.59. A convenient practical rule is a range from 3 to 6. ILLUSTRATION. i 6 36 216 1296 velocity ratio. 123 4 elementary combination. To increase or diminish velocity in a given proportion, and with least quantity of wheel-work, number of teeth in each pinion should be to number of teeth in its wheel as i : 3.59. Even to save space and expense, ratio should never exceed i : 6. (Buchanan.) To Compute Fitch. RULE. Divide circumference at pitch-line by number of teeth. EXAMPLE. A wheel 40 ins. in diameter requires 75 teeth; what is its pitch? 3.1416X40-^-75 = 1-6755 * To Compute True or Ch.ordial Pitcli. RULE. Divide 180 by number of teeth, ascertain sine of quotient, and multiply it by diameter of wheel. EXAMPLE. Number of teeth is 75, and diameter 40 ins. ; what is true pitch? i8o-f- 75 = 2 24', and sin. of 2 24' = .041 88, which x 40 = 1.6752 ins. To Compute Diameter. RULE. Multiply number of teeth by pitch, and divide product by 3.1416. EXAMPLE. Number of teeth in a wheel is 75, and pitch 1.6755 ins. ; what is di- ameter of it? 75 x ,.6755-:- 3. 1416 = 4 o ins. When the True Pitch is given. RULE. Multiply number of teeth hi wheel by true pitch, and again by .3184. EXAMPLE. Take elements of preceding case. 75 X 1.6752 X .3184 = 40 ins. Or, Divide 180 by number of teeth, and multiply cosecant of quotient by pitch. 180-:- 75 = 2 24', and cos. 2 24' = 23.88, which x 1.6752 = 40 int. WHEEL GEARING. To Compute Number of Teeth. RULE. Divide circumference by pitch. To Compute Number of Teeth in. a Pinion or Follo-wer to have a given "Velocity. RULE. Multiply velocity of driver by its number of teeth, and divide product by velocity of driven. EXAMPLE i. Velocity of a driver is 16 revolutions, number of its teeth 54, arid velocity of pinion is 48; what is number of its teeth? 16X54-:- 48=18 teeth. 2. A wheel having 75 teeth is making 16 revolutions per minute; what is num- ber of teeth required in pinion to make 24 revolutions in same time? 16X75-^24 = 50 teeth. To Compute Proportional Radius of a Wheel or Pinion. RULE. Multiply length of line of centres by number of teeth in wheel, for wheel, and in pinion, for pinion, and divide by number of teeth in both wheel and pinion. EXAMPLE. Line of centres of a wheel and pinion is 36 ins., and number of teeth in wheel is 60, and in pinion 18 ; what are their radii ? _ vx _O = 27.691715. To Compxite Diameter of* a Pinion. When Diameter of Wheel and Number of Teeth in Wheel and Pinion are given. RULE. Multiply diameter of wheel by number of teeth in pinion, and divide product by number of teeth in wheel. EXAMPLE. Diameter of a wheel is 25 ins., number of its teeth 210, and number of teeth in pinion 30; what is diameter of pinion? 25X30-^-210 = 3.57 ins. To Compute Number of Teeth required in a Train of "Wheels to produce a given Velocity. RULE. Multiply number of teeth in driver by its number of revolutions, and divide product by number of revolutions of each pinion, for each driver and pinion. EXAMPLE. If a driver in a train of three wheels has 90 teeth, and makes 2 revo- lutions, and velocities required are 2, 10, and 18, what are number of teeth in each of other two? 10 : 90 :: 2 : 1 8 = teeth in 2d wheel 18 : 90 :: 2 : 10 = teeth in $d wheel To Compute "Velocity of a Pinion. RULE. Divide diameter, circumference, or number of teeth in driver, as case may be, by diameter, etc., of pinion. When there are a Series 01* Train of Wheels and Pinions. RULE. Divide continued product of diameter, circumference, or number of teeth in wheel* by continued product of diameter, etc., of pinions. EXAMPLE i Tf a wheel of 32 teeth drives a pinion of 10, upon axis of which there is one of 30 teeth, driving a pinion of 8, what are revolutions of last? ^X^ = 9 ^= 12 revolutions. IO O OO 2. Diameters of a train of wheels are 6, 9, 9, 10, and 12 ins. ; of pinions, 6, 6, 6, 6, and 6 ins. ; and number of revolutions of driving shaft or prime mover is 10; what are revolutions of last pinion? 6 X 9 X 9 X 10 X 12 X io _ 583 200 _ "~" ""~ ~ - = 7S revolutlons ' WHEEL GEARING. 857 To Compute Proportion, that Velocities of Wheels in a Train, should, bear to one another. RULE. Subtract less velocity from greater, and divide remainder by one less than number of wheels in train ; quotient is number, rising in arithmet- ical progression from less to greater velocity. EXAMPLE. What should be velocities of 3 wheels to produce 18 revolutions, the driver making 3? = 7. 5 = number to be added to velocity of driver = 7. 5 -f- 3 = 10. 5, and jo. 5 -j- 7.5 = 18 revolutions. Hence 3, 10. 5, and 18 are velocities of three wheels. Pitch, of "Wheels. To Compute Diameter of a "Wheel for a given Pitch,, or Pitch for a given Diameter. From 8 to 192 Teeth. No. of Teoth. Diame- ter. No. of i Teeth. Diame- ter. No. of Teeth. Diame- ter. No. of Teeth. Diame- ter. No. of Teeth. Diame- ter. 8 2.61 45 14-33 82 26.11 119 37-88 156 49-66 9 2.93 4 6 14.65 83 26.43 120 38.2 157 49.98 10 3-24 47 14.97 84 26.74 121 38-52 158 50.3 ii 3-55 48 15.29 85 27.06 122 38.84 159 50.61 12 3.86 49 I5.6l 86 27.38 123 39.16 160 50.93 13 4.18 50 15-93 8 7 27.7 124 39-47 161 i 51.25 14 4-49 5i 16.24 88 28.02 125 39-79 162 5L57 15 4.81 52 16.56 89 28.33 126 40.11 163 51.89 16 5-12 53 16.88 90 28.65 I2 7 40-43 164 52.21 17 5-44 54 17.2 9i 28. 97 128 40-75 165 52.52 18 5.76 55 17.52 92 29.29 I2 9 41.07 166 52.84 19 6.07 56 I 7 .8 93 29.61 130 41.38 167 53-i6 20 6-39 57 I8.I5 94 29-93 131 41.7 168 53.48 21 6.71 58 18.47 95 30.24 132 42.02 169 | 53-8 22 7-03 59 18.79 96 30-56 133 42.34 170 54-12 23 7-34 60 19.11 97 30.88 134 42.66 171 54-43 2 4 7.66 61 19.42 98 31.2 135 42.98 172 54-75 25 7.98 62 19.74 99 31-52 136 43-29 J 73 55-07 26 8-3 63 20.06 100 31.84 137 43.61 \ 174 55-39 27 8.61 64 20.38 IOI 32.15 138 43-93 | 175 55-71 28 8-93 65 20.7 102 3247 139 44-25 176 56.02 29 9- 2 5 66 2I.O2 103 32.79 I4O 44-57 177 56.34 30 9-57 67 21-33 104 33-ii 141 44.88 178 56.66 31 9.88 68 21.65 105 33-43 142 45-2 179 56.98 32 10.2 69 21.97 106 33-74 *43 45-52 180 57-23 33 10.52 70 22.29 107 34.06 144 45.84 181 5762 34 10.84 7 1 22.6l 108 34.38 145 46.16 182 5793 35 ii. 16 72 22.92 109 34-7 146 46.48 183 58.25 36 11.47 73 23.24 no 35-02 147 46.79 184 58.57 37 11.79 74 23.56 III 35-34 148 47.11 185 58.89 38 12. II 75 23.88 112 35.65 149 47-43 186 59.21 39 12.43 76 24.2 "3 35-97 150 47-75 187 59-53 40 12.74 77 24.52 114 36.29 151 48.07 188 59.84 4i 13.06 78 24.83 "5 36.61 152 48.39 189 60. 16 42 13.38 79 25.15 116 36.93 153 48.7 190 60.48 43 13-7 80 25-47 117 37.25 154 49.02 191 60.81 44 14.02 81 25.79 "8 37.56 155 49-34 192 61.13 Pitch in this table is true pitch, as before described. To Compute Circumference of a "Wheel. RULE. Multiply number of teeth by their pitch. 4. C* 858 WHEEL GEARING. Xo Compute Revolxitions of a Wheel or Pinion. RULE. Multiply diameter or circumference of wheel or number of its teeth in ins., as case may be, by number of its revolutions, and divide prod- uct by diameter, circumference, or number of teeth in pinion. EXAMPLE. A pinion 10 ins. in diameter is driven by a wheel 2 feet in diameter, making 46 revolutions per minute; what is number of revolutions of pinion? 2Xi2X46-=-io=:iio.4 revolutions. To Compute Numtoer of Teeth of a Wheel for a given Diameter and. Pitch. RULE. Divide diameter by pitch, and opposite to quotient in preceding table is given number of teeth. EXAMPLE. Diam. of wheel is 40 ins., and pitch 1.675; what is number of its teeth? 40-7-1.675 = 23.88, and opposite thereto in table is 75 = number of teeth. To Compute Diameter of a Wheel for a given Pitch and Number of Teeth. RULE. Multiply diameter in preceding table for number of teeth by pitch, and product will give diameter at pitch circle. EXAMPLE. What is diameter of a wheel to contain 48 teeth of 2.5 ins. pitch? 15. 29 X 2. 5 = 38. 225 ins. To Compute Pitch of a Wheel for a given Diameter and. Number of Teeth. RULE. Divide diameter of wheel by diameter in table for number of teeth, and quotient will give pitch. EXAMPLE. What is pitch of a wheel when diameter of it is 50.94 ins., and num- ber of its teeth 80? 5a94 _._ 25>47 _. 2 intm G-eiieral Illustrations. i. A wheel 96 ins. in diameter, making 42 revolutions per minute, is to drive a shaft 75 revolutions per minute ; what should be diameter of pinion ? 96 X 42 -J-75 = 53- 7 6 in*. 2. If a pinion is to make 20 revolutions per minute, required diameter of an- other to make 58 revolutions in same time. 58 -r- 20 = 2.9 = ratio of their diameters. Hence, if one to make 20 revolutions is given a diameter of 30 ins., other will be 30-^2.9 = 10.345 ins. 3. Required diameter of a pinion to make 12.5 revolutions in same time as one of 32 ins. diameter making 26. 32 X 26 -t- 12. 5 = 66. 56 ins. 4. A shaft, having 22 revolutions per minute, is to drive another shaft at rate of 15, distance between two shafts upon line of centres is 45 ins. ; what should be diameter of wheels? Then, ist. 22 + 15 : 22 :: 45 : 26.75 = ins. in radius of pinion. 2d. 22 -}- 15 : 15 :: 45 : 18.24 = *w*. in radius of spur. 5. A driving shaft, having 16 revolutions per minute, is to drive a shaft 81 revo- lutions per minute, motion to be communicated by two geared wheels and two pul- leys, with an intermediate shaft; driving wheel is to contain 54 teeth, and driving pulley upon driven shaft is to be 25 ins. in diameter; required number of teeth in driven wheel, and diameter of driven pulley. Let driven wheel have a velocity of Vi6x 81 = 36, a mean proportional between extreme velocities 16 and 81. Then, ist. 36 : 16 :: 54 : 24 = teeth in driven wheel. 2d. 81 : 36 :: 25 : u.u =ins. diameter of driven pulley. 6. If, as in preceding case, whole number of revolutions of driving shaft, num- ber of teeth in its wheel, and diameters of pulleys are given, what are revolutions of shafts ? Then, ist. 18 : 16 :: 54 : 48 = revolutions of intermediate shaft. 2d. 15 : 48 :: 25 : 80 = revolutions of driven shaft. WHEEL GEARING. TEETH OF WHEELS. 859 Teeth, of TVTieels. Epicycloidal. In order that teeth of wheels and pinions should work evenly and without unnecessary rubbing friction, the face (from pitch line to top) of the outline should be determined by an epicycloidal curve (see page 228), and that of the flank (from pitch line to base) by an hypocycloidal (see also page 228). When generating circle is equal to half diameter of pitch circle, hypocy- cloidal described by it is a straight diametrical line, and consequently out- line of a flank is a right line, and radial to centre of wheel. If a like generating circle is used to describe face of a tooth of other wheel or pinion respectively, the wheel and pinion will operate evenly. ILLUSTRATION. Determine all elements of wheel viz. , Pitch circle, Number of teeth, Pitch, Length, Face, and Flank. Cut a template A to pitch circle c c of wheel, and secure it temporarily to a board. Having determined depth of tooth, set it off on pitch line, as a o, Fig. i, and above it apply a sec- ond template, a; radius of wheel is equal to half radius of pinion; insert into, or attach exactly at its edge, a tracer ., roll template a along A, and tracer will describe an epicycloidal curve, a r, and by inverting a describe o ?, and faces of a tooth are delineated. To describe flanks, define pitch line c c, Fig. 2, and arc n n, drawn at base of teeth or board A (as in Fig. i), secure a strip of wood, w, equal in length to radius of wheel, and locate centre of it, a;, draw radii x a and * o, and they will define flanks, which should be filleted, as shown at ss. Define arc zz, and length of tooth is determined. Proceed in like manner conversely for teeth of pinion, and wheel and pinion thus constructed will operate truly. In construction of the teeth of a wheel or pinion in the pattern-shop, it is customary to construct the wheel or pinion complete, out to face of wheel at base of teeth, and then to insert the teeth in rough, approximately shaped blocks, by a dovetail at their base, fitting into face of wheel, and then the outline of a tooth is described thereon ; the block is then removed, fin- ished as a tooth, replaced, fastened, and filleted. Involute. Teeth of two wheels will work truly together when their face is that of an involute (see page 229), and that two such wheels should work truly, the circles from which the involute lines for each wheel are generated must be concentric with the wheels, with diameters in same ratio as those of the wheels. Assume Ac, Be, Fig. 3. pitch radii of two wheels designed t& work together, through c, draw a right line, e i, and with perpendiculars e c, i c, describe arcs n o, r s, and involutes n c o and res define a face of each of the teeth. To describe teeth of a pair of wheels of which Ac, Be, Fig. 4, are pitch radii, draw c t, c e, per- pendicular to radials B i and A e, and they are to be taken as the radials of the involute arcs from which the faces of the teeth are to be defined ; then fillet flanks at base, as before described, Fig. 2. Involute teeth will work with truth, even at varying distances apart of the centres of the wheels, and any wheels of a like pitch will work in union, however varied their diameters. 86o WHEEL GEARING. TEETH OF WHEELS. Circular teeth are defined as follows : Assume A A, Fig. 5, pitch-line, 6 b line of base of teeth, and t t face line. Set off on pitch-line divisions both of pitch and depth of teeth, then with a radius of 1.25 pitch describe arcs as o s upon pitch line for faces of teeth, then draw ra- dial lines o v, r , to centre of wheel for flanks, strike arc 1 1 to define length of tooth, and fillet flanks at base as before described. Proportions of Teeth. In computing dimensions of a tooth, it is to be considered as a beam fixed at one end, weight suspended from other, or face of beam \ and it is essential to consider the element of velocity, as its stress in opera- tion, at high velocity with irregular action, is increased thereby. Dimensions of a tooth should be much greater than is necessary to resist direct stress upon it, as but one tooth is proportioned to bear whole stress upon wheel, although two or more are actually in contact at all times ; but this requirement is in consequence of the great wear to which a tooth is sub- jected, shocks it is liable to from lost motion, when so worn as to reduce its depth and uniformity of bearing, and risk of the loss of a tooth from a defect. A tooth running at a low velocity may be materially reduced in its dimen- sions, compared with one running at a high velocity and with a like stress. Result of operations with toothed wheels, for a long period of time, has determined that a cast-iron (Eng.) tooth with a pitch of 3 ins. and a breadth of 7.5 ins. will transmit, at a velocity of 6.66 feet per second, power of 59.16 horses. To Compute Dimensions of a Tooth to Resist a given Stress. RULE. Multiply extreme pressure at pitch-line of wheel by length of tooth in decimal of a foot, divide product by Coefficient of material of tooth, and quotient will give product of breadth and square of depth. S I Or, = 6 d 2 . S representing stress in Ibs., and I length in feet. The Coefficient of cast iron for this or like purposes may be taken at from 50 to 70. Pitch A B = i. Length c o = . 75. Working length c c = .7. Clearance e to o .05. Depth r s = .+^ Space s v = .55. Play s v r s = .i Face B c = .3s. NOTE. It is necessary first to determine i order to obtain either length or depth of a tootb itch, io EXAMPLE. Pressure at pitch line of a cast- iron wheel (at a velocity of 6.66 feet per sec- ond) is 4886 Ibs. ; what should be dimensions of teeth, pitch being 3 ins. ? 3 X 75 2. 25 length of tooth, which -f- 12 = . 1875 = length in decimal of a foot. Coefficient of material is taken at 60. '' 7 =15. 27. If length = 2. 25, pitch = 3, and depth = i. 35 ins. Pitches of Equivalent Strength for Cast Iron and Wood. Iron i. Hard wood 1.26. Then = 8.39 ins. breadth. When Product ofbd 2 is obtained, and it is required to ascertain eithet dimension. ~ = depth, and b -- = breadth. WHEEL GEAKING. TEETH OF WHEELS. 86 1 To Compute Depth of a Tooth. 1. When Stress is given. RULE. Extract square root of stress, and mul* tiply it by .02 for cast iron, and .027 for hard wood. 2. When H? is given. RULE. Extract square root of quotient of EP di- vided by velocity in feet per second, and multiply it by .466 for cast iron, and .637 for hard wood. EXAMPLE. H* to be transmitted by a tooth of cast iron is 60, and velocity of it at its pitch-line is 6.66 feet per second; what should be depth of tooth? / 60 \/6~66 * -4 66 = Zl 39 8 *** To Compute KP of a Tooth. RULE. Multiply pressure at pitch-line by its velocity in feet per minute, and divide product by 33 ooo. EXAMPLE. What is H? of a tooth of dimensions and at velocity given in preced- ing example. 4886 X 6. 66 X 60" -f- 33 ooo 59. 16 horses. To Compute Stress that may be borne "by a Tooth. RULE. Multiply Coefficient of material of tooth to resist a transverse strain, as estimated for this character of stress, by breadth and square of its depth, and divide product by extreme length of it in decimal of a foot. EXAMPLE. Dimensions of a cast-iron tooth in a wheel are 1.38 ins. in depth, 2.1 ins. in length, and 7.5 ins. in breadth; what is the stress it will bear? Coefficient assumed at 60. ' - 4^97 deductions by the rules of different authors for like cast-iron tooth: Pitch ....... 3 ins. \ Depth. ... 1.38 ins. \ Breadth. . . 7.5 ins. \ Length. ... 2.1 ins. Following deductions by the rules of different authors for like elements are sub- mitted for a cast-iron tooth: ACTUAL POWER IN STRESS EXERTED at a velocity of 400 feet per wan., 4886 Ibs. Depth of Tooth. ACTUAL POWER IN STRESS EXERTED at a velocity of 400 feet per min. , 4886 Ibs. Depth of Tooth. /H By Above Rule 1 X 446 Ins. 1.398* i-75 1.76 By Rankine / . . . In*. 1.8 2.25 2.24 V 1500 " Tredgold /... " Imperial Journal / -; .. 4 V "Buchanan / 556H .. V v H representing horse-power (60), W stress in Ibs., and v velocity in feet per second. Depth, IPitoh, and Breadth. (M. Aforin.) Cast iron 028 <^W = d. .057 V w = p - Hard wood 038 v\Y = d. .079 v w = p - W representing weight or stress upon tooth in Ibs., d depth of tooth, and P pitch i ins in ins When velocity of pitch-circle does not exceed 5 feet per second b = 4 d, when it exceeds 5 feet b = 5 d, and if wheels are exposed to wet 6 = 6 d. b representing breadth. ILLUSTRATION. Assume pressure at pitch-line of a cast-iron wheel upon a tooth equal 6000 Ibs., and velocity 5 feet per second. Then .028 ^6000= 2. 17 ins. Depth, and .057 v / 6ooo = 4.41 ins. Pitch. NOTE. For farther Illustrations of Formation of Teeth, Bevel Gearing, Willis's OdontoprapW, Staves, Trundles, etc., see Mosely's Engineering, Shelton's Mechanic's Guide, Fairbairn's Mechanism and Machinery of Construction, etc. * This depth, with a breadth of 7.5 int., is .1 of ultimate strength of average strength of America* Cast Iron. 862 TEETH OF WHEELS. WINDING ENGINES. PROPORTIONS OP WHEELS. With six flat A rms and Ribs upon one side of them, as cmmfy ; or a Web in centre, as e&^aa. Rim. Depth, measured from base of teeth, .45 to .5 of pitch of teeth, hav- ing a web upon its inner surface .4 of pitch in depth and .25 to .3 of it in width. NOTE. When face of wheel is mortised, depth of rim should be 1.5 times pitch, and breadth of it 1.5 times breadth of tooth or cog. Hub. When eye is proportionate to stress upon wheel, hub should be twice diameter of eye. In other cases depth around eye should be .75 to .8 of pitch. Arm. Depth .4 to .45 of pitch. Breadth at rim 1.5 times pitch, increas- ing .5 inch per foot of length toward hub. Rib upon one edge of arm, or Web in its centre, should be from .25 to .3 pitch in width, and .4 to .45 of it in depth. When section of an arm differs from those above given, as with one with a plane section, as . Feet. 8-5 10 14 18 20 25 No. 70 to 75 60 1065 So to 55 40 to 45 35 to 4> 30 to 35 Gallons. 6.16 19.18 45-14 97.68 124.95 212.38 Gallons. 3.02 9-56 22.57 52.16 63 t* 7 f 106.96 Gallons. 4-75 11.25 24.42 3i- 2 5 49-73 Gallons. 5 12.21 15-94 26.74 IP .04 .12 .28 .6l .78 1.34 Cents. .60 70 1-63 2.83 3.56 4.26 Cents. 15 5-8 5-8 4.6 4-5 3-3 * Including interest at 5 per cent, per annum. WOOD AND TIMBER. Selection of Standing Trees. Wood grown in a moist soil is lighter, and decays sooner, than that grown in dry, sandy soil. Best Timber is that grown in a dark soil, intermixed with gravel. Poplar, Cypress, Willow, and all others which grow best in a wet soil, are exceptions. Hardest and densest woods, and least subject to decay, grow in warm climates ; but they are more liable to split and warp in seasoning. Trees grown upon plains or in centre of forests are less dense than those from edge of a forest, from side of a hill, or from open ground. Trees (in U. S.) should be selected in latter part of July or first part of August; for at this season leaves of sound, healthy trees are fresh and green, while those of unsound are beginning to turn yellow. A sound, healthy tree is recognized by its top branches being well leaved, bark even and of a uniform color. A rounded top, few leaves, some of them turned yellow, a rougher bark than common, covered with parasitic plants, and with streaks or spots upon it, indicate a tree upon the de- cline. Decay of branches, and separation of bark from the wood, are infallible indications that the wood is impaired. Green timber contains 37 to 48 per cent, of liquids. By exposure to air in seasoning one year, it loses from 17 to 25 per cent., and when seasoned it retains from 10 to 15 per cent. According to M. Leplay, green wood contains about 45 per cent, of its weight of moisture. In Central Europe, wood cut in winter holds, at end of following summer, fully 40 per cent, of water, and when kept dry for sev' eral years retains from 15 to 20 per cent, of water. Felling Timber. Most suitable time for felling timber is in midwinter and in midsummer. Recent experiments indicate latter season and month of July. 4D 866 WOOD AND TIMBER. A tree should be allowed to attain full maturity before being felled. Oak matures at 75 to 100 years and upwards, according to circumstances ; Ash, Larch, and Elm at 75 ; and Spruce and Fir at 80. Age and rate of growth of a tree are indicated by number and width of the rings of annual increase which are exhibited in a cross-section of its body. A tree should be cut as near to the ground as practicable, as the lower part furnishes best timber. Dressing Timber. As soon as a tree is felled, it should be stripped of its bark, raised from the ground, reduced to its required dimensions, and its sap-wood removed. Inspection of Timber. Quality of wood is in some degree indicated by its color, which should be nearly uniform, and a little deeper towards its cen- tre, and free from sudden transitions of color. White spots indicate decay. Sap-wood is known by its white color ; it is next to the bark, and soon rots. Defects of Timber. Wind-shakes are serious defects, being circular cracks separating the con- centric layers of wood from each other. Splits, Checks, and Cracks, extending toward centre, if deep and strongly marked, render timber unfit for use, unless purpose for which it is intended will admit of its being split through them. Brash is when wood is porous, of a reddish color, and breaks short, with- out splinters. It is generally consequent upon decline of tree from age. Belted is that which has been killed before being felled, or which has died from other causes. It is objectionable. Knotty is that containing many knots, though sound ; usually of stinted growth. Twisted is when grain of it winds spirally ; it is unfit for long pieces. Dry-rot is indicated by yellow stains. Elm and Beech are soon affected, if left with the bark on. Large or decayed knots injuriously affect strength of timber. Heart-shake. Split or cleft in centre of tree, dividing it into segments. Star-shake. Several splits radiating from centre of timber. Cup-shake. Curved splits separating the rings wholly or in part. Rind-gall. Curved swelling, usually caused by growth of layers over spot where a branch has been removed. Upset. Fibres injured by crushing. Foxiness. Yellow or red tinge, indicating incipient decay. Doatiness. A speckled stain. Seasoning and. Preserving Timber. Seasoning is extraction or dissipation of the vegetable juices and moisture or solidification of the albumen. When wood is exposed to currents of air at a high temperature, the moisture evaporates too rapidly, and it cracks ; and when temperature is high and sap remains, it ferments, and dry-rot ensues. Wood requires time in which to season, very much in proportion to density of its fibres. Water Seasoning is total immersion of timber in water, for purpose of dissolving the sap, and when thus seasoned it is less liable to warp and crack, but is rendered more brittle. WOOD AND TIMBER. 867 For purpose of seasoning, it should be piled under shelter and kept dry; should have a free circulation of air, without being exposed to strong cur- rents. Bottom pieces should be placed upon skids, which should be free from decay, raised not less than 2 feet from ground ; a space of an inch should intervene between pieces of same horizontal layers, and slats or piling- strips placed between each layer, one near each end of pile, and others at short distances, in order to keep the timber from winding. These strips should be one over the other, and in large piles should not be less than i inch thick. Light timber may be piled in upper portion of shelter, heavy timber upon ground floor. Each pile should contain but one description of timber, and they should be at least 2.5 feet apart. It should be replied at intervals, and all pieces indicating decay should be removed, to prevent their affecting those which are still sound. It requires from 2 to 8 years to be seasoned thoroughly, according to its dimensions, and it should be worked as soon as it is thoroughly dry, for it deteriorates after that time. Gradual seasoning is most favorable to strength and durability of timber. Various methods have been proposed for hastening the process, as Steaininy, which has been applied with success; and results of experiments of va ions processes of saturating it with a solution of Corrosive sublimate and Anti- septic fluids are very satisfactory. Such process hardens and seasons wood, at the same time that it secures it from dry-rot and from attacks of worms. Woods are densest and strongest at the roots and at their centres. Their strength decreasing with the decrease of their density. Oak timber loses one fifth of its weight in seasoning, and about one third in becoming perfectly dry. Pitch pine, from the presence of pitch, requires time in excess of that due to the density of its fibre. Mahogany should be seasoned slowly, Pine quickly. Whitewood should not be dried artificially, as the effect of heat is to twist it. Salt water renders wood harder, heavier, and more durable than fresh. Condition of timber, as to its soundness or decay, is readily recognized when struck with a quick blow. Timber that has been for a long time immersed in water, when brought into the air and dried, becomes brashy and useless. When trees are barked in the spring, they should not be felled until the foliage is dead. Timber cannot be seasoned by either smoking or charring ; but when it is exposed to worms or to the production of fungi, it is proper to smoke or char it, and it may be partially seasoned by being boiled or steamed. Timber houses are best provided with blinds which keep out rain and snow, but which can be turned to admit air in fine weather, and the houses should be kept entirely free from any pieces of decayed wood. Kiln-drying is suited only for boards and pieces of small dimensions, as it is apt to cause cracks and to impair the strength, unless performed very slowly. Charring, Painting, or covering the surface is highly injurious to any but seasoned wood, as it effectually prevents drying of the inner part of the wood, in consequence of which fermentation and decay soon take place. Timber is subject to Common or Dry-rot, former occasioned by alternate exposure to moisture and dry ness, and as progress of it is from the exterior, covering of the surface, if seasoned, with paint, tar, etc., is a preservative. 868 WOOD AND TIMBER. Common-rot is the consequence of its being piled in badly-ventilated sheds. Outward indications are yellow spots upon ends of pieces, and a yellowish dust in the checks and cracks, particularly where the pieces rest upon pil- ing-strips. Dry or Sap-rot is inherent in timber, and it is the putrefaction of the veg- etable albumen. Sap wood contains a large proportion of fermentable ele- ments. Insects attack wood for the sugar or gum contained in it, and fungi subsist upon the albumen of wood ; hence, to arrest dry-rot, the albumen must be either extracted or solidified. Most effective method of preserving timber is that of expelling or ex- hausting its fluids, solidifying its albumen, and introducing an antiseptic liquid. Strength of impregnated timber is not reduced, and its resilience is improved. In desiccating timber by expelling its fluids by heat and air, its strength is increased fully 15 per cent. The saturation of wood with creosote, tar, antiseptics, etc., preserves it from the attack of worms. Jarrow wood, from Australia, is not subjected to their attack. In a perfectly dry atmosphere durability of woods is almost unlimited. Rafters of roofs are known to have existed 1000 years, and piles submerged in fresh water have been found perfectly sound 800 years from period of their being driven. Resistance of woods to extension is greater than that of compression. Impregnation of "Wood.. Several of the successful processes are as follows : Kyan, 1832. Saturated with corrosive sublimate. Solution i Ib. of chlo- ride of mercury to 4 gallons of water. Burnett (Sir Wm.), 1838. Impregnation with chloride of zinc by sub- mitting the wood endwise to a pressure of 150 Ibs. per sq. inch. Solution, i Ib. of the chloride to 4 gallons of water. Boucheri. Impregnation by submitting the wood endwise to a pressure of about 15 Ibs. per sq. inch. Solution, i Ib. of sulphate of copper to 12.5 gallons of water. Bethel. Impregnation by submitting the wood endwise to a pressure of 150 to 200 Ibs. per sq. inch, with oil of creosote mixed with bituminous matter. Robbins, 1865. Aqueous vapor dissipated by the wood being heated in a chamber, the albumen solidified, then submitted to vapor of coal tar, resin, or bituminous oils, which, being at a temperature not less than 325, readily takes the place of the vapor expelled by a temperature of 212. Hayford, 187-. Aqueous vapor dissipated by the wood being heated in a chamber to a temperature of from 250 to 270, the albumen solidified, then air introduced to assist the splitting of the outer surfaces. When vapor is dissipated, dead oils are introduced under a pressure of 75 Ibs. per sq. inch. Planks, Deals, and Battens. When cut from Northern pine (Plnus Sylve- stris) are termed yellow or red deal, and when cut from spruce (Abies, alba, etc.) they are termed white deal. Desiccated wood, when exposed to air under ordinary circumstances, ab- sorbs 5 per cent, of water in the first three days ; and will continue to absorb it until it reaches from 14 to 16 per cent., the amount varying according to condition of the atmosphere. WOOD AND TIMBER. 869 Durability of Various Woods, Pieces 2 feet in Length, 1.5 ins. Square, driven 28.5 ins. into the Earth. WOOD. C After 2.5 Years. ondition After 5 Years. Good (Externally decayed, rest per- ( fectly sound. Decayed. Sound as when driven. Tolerable. Entirely decayed. Decayed. Much decayed. ( Attacked in part only, rest fair \ condition. Very rotten. (Some moderately, most very \ much, decayed. ( Attacked in part only, rest fair \ condition. Much decayed. Very rotten. Somewhat soft, but good. Ash Amer Much decayed Cedar, Va ' * Lebanon Very good Good Elm Eng ' : Ca.ii Fir " attacked Surface only attacked Very much decayed Oak Can " ' Memel " Dautzic " Chestnut Pine pitch Surface only attacked Attacked 4 ' yellow " white . . . Very much decayed Teak . . Very good Effect of Creosoting. Results of Experiments with Various Woods (E. R. Andrews). WOOD. Water absorbed. WOOD. Water absorbed. Spruce Onlr f dried Per cent. 2543 .0261 .2 .0 .714 347 Hard pine. . . . Gum, black . . Birch, white . dried creosoted. dried .... creosoted. dried creosoted. Per cent. 16 o i 5 43 124 \ creosoted . . . (dried Cotton-wood \ creosoted . . . | dried { creosoted . . . Sesquoia Gigantea of California, dried, .4722; creosoted, .o. Fluids will pass with the grain of wood with great facility, but will not enter it except to a very limited extent when applied externally. .Absorption, of Preserving Solntioii "by different "Woods for a Period of 7" Days. Average Lbs. per Cube Foot. Black Oak 3.6 I Hemlock 2.6 I Rock Oak 3.9 Chestnut 3 | Red Oak 3.9 | White Oak 3.1 Proportion of "Water in various "Woods. Alder (Betula alnus) 41.6 Ash (Fraxinus excelsior) 28.7 Beech (Fagus sylvatica) 33 Birch (Betula alba) 30.8 Elm ( Ulmus campestris) 44. 5 Horse-chestnut (^Esculus hippocast. ) 38. 2 Larch (Pinus larix). 48.6 Mountain Ash (Sorbus aucuparia). . 28.3 Oak (Quercus robur) 34.7 Pine (Pinus Sylvestris L.) 39.7 Red Beech (Fagus sylvatica) 39.7 Red Pine (Pinus picea dur) 45.2 Spruce (Abies, alba, nigra, rubra, \ excelsa) ) 35 Sycamore (Acer pseudo-platanus) . . 27 White Oak (Quercus alba) 36.2 White Pine (Pinus abies dur) 37. i White Poplar (Populus alba) 50.6 Willow (Salis caprea) 26 Decrease in Dimensions of Timber t>y Seasoning. Ins. WOODS. Ins. Int. to 13.25 Pitch Pine, South 18.375 to 18.25 to 10.75 Spruce 8.5 to 8.375 to 11.625 White Pine, American.. 12 1011.875 WOODS. Cedar, Canada. Elm Oak, English 12 Pitch Pine, North... ioXioto 9.75X9-75 Yellow Pine, North 18 to 17.875 Weight of a beam of English oak, when wet, was reduced by seasoning from 972.25 to 630.5 Ibs. 4 D* 8/o WOOD AND TIMBER. Weight of a Cube Foot of Oak and Yellow !Pine. AOB. White < Round. 3ak, Va. Square. Yellow 1 Round. Pine, Va. Square. Live Oak. 78.7 66. T Green 64.7 53-6 4.6 67.7 53-5 AQ.Q 47-8 39-8 n. ^ 39-2 34-2 7Q. C i Year 2 Years. . . In England, Timber sawed into boards is classed as follows : 6.5 to 7 ins. in width, Battens; 8.5 to 10 ins., Deals; and n to 12 ins., Planks. (See also page 62.) Distillation. 'From a single cord of pitch pine distilled by chemical ap- paratus, following substances and in quantities stated have been obtained : Charcoal 50 bushels. Illuminating Gas about 1000 cu. feet. Illuminating Oil and Tar. . . 50 gallons. Pitch or Resin 1.5 barrels. Pyrol igneous Acid 100 gallons. Spirits of Turpentine 20 " Tar i barrel. Wood Spirit 5 gallons. Strength, of Timber. Results of experiments have satisfactorily proved: That deflection was sensibly proportional to load ; That extension and compression were nearly the same, though former being the greater ; That, to produce equal deflection, load, when placed in the centre, was to a load uniformly distributed, as .638 to i ; That deflection under equal loads is inversely as breadths and cubes of the depths, and directly as cubes of the spans. ( M. Morin.) It has also been shown, that density of wood varies very little with its age. That coefficient of elasticity diminishes after a certain age, and that it de- pends also on the dryness and the exposure of the ground where the wood is grown. Woods from a northerly exposure, on dry ground, have a high coefficient, while those from swamps or low moist ground have a low one. That tensile strength is influenced by age and exposure. The coefficient of elasticity of a tree cut down in full vigor, or before it arrives at this condition, does not present any sensible difference. That there is no limit of elasticity in wood, there being a permanent set for every extension. Average Result of Experiments on Tensile Strength of Wood in Various Positions per Sq. Inch. (MM. Chevandier and Wertheim.) With the fibre, 6900 Ibs. Radially, 683 Ibs., and Tangentially, 723 Ibs. To Compute Volume of an Irregular Body. By " Simpson's Rule." OPERATION. Take a right line in the figure for a base line, as A B, divide the fig- ure into any number of equal parts, and compute the areas of their plane sections as i, 2, 3, etc. , at the points of division, by rules applicable to area of a plane. Then, operate these areas as if they were the ordinates of a plane curve or figure of same length as the figure, and result will give volume required. ILLUSTRATION. Assume a figure having areas as follows, and A B 24 feet. Sections, i Areas, 3 feet Multiplier, i Products, 3 14 36 ii 84" and 84 X>4-J- 4-7-3 = 168 cube feet. MISCELLANEOUS MIXTURES. 8/1 MISCELLANEOUS MIXTURES. Cements. Much depends upon manner in which a cement is applied as upon the cement itself, as best cement will prove worthless if improperly applied. Following rules must be rigorously adhered to to attain success : 1. Bring cement into intimate contact with surfaces to be united. This is best done by heating pieces to be joined in cases where cement is melted by heat, as with resin, shellac, marine glue, etc. Where solutions are used, cement must be well rubbed into surfaces, either with a brush (as in case of porcelain or glass), or by rubbing the two surfaces together (as in making a glue joint between pieces of wood). 2. As little cement as practicable should be allowed to remain between the united surfaces To secure this, cement should be as liquid as practicable (thoroughly melted if used with heat), and surfaces should be pressed closely into contact until cement has hardened. - 3. Time should be allowed for cement to dry or harden, and this is particularly the case in oil cements, such as copal varnish, boiled oil, white lead, etc. When two surfaces, each .5 inch across, are joined by means of a layer of white lead placed between them, 6 months may elapse before cement in middle of joint be- comes hard. At the end of a month the joint will be weak and easily separated; at end of 2 or 3 years it may be so firm that the material will part anywhere else than at joint. Hence, when article is to be used immediately, the only safe cements are those which are liquefied by heat and which become hard when cold. A joint made with marine glue is firm an hour after it has been made. Next to cements that are liquefied by heat are those which consist of substances dissolved in water or alcohol. A glue joint sets firmly in 24 hours; a joint made with shellac varnish becomes dry in 2 or 3 days. Oil cements, which do not dry by evaporation, but harden by oxidation (boiled oil, white lead, red lead, etc. ) are slowest of all. Stone. Resin, Yellow Wax, and Venetian Red, each i oz. ; melt and mix. Aquarium. Litharge, fine white dry Sand, and Plaster of Paris, each i gill ; finely pulverized Resin, .33 gill. Mix thoroughly and make into a paste with boiled linseed oil to which drier has been added. Beat well, and let stand 4 or 5 hours before using it. After it has stood for 15 hours, however, it loses its strength. Glass cemented into a frame with this cement will resist percolation for either salt or fresh water. Adhesive for Fractures of all Kinds. White Lead ground with Linseed-oil Varnish, and kept from contact with the air. Requires a few weeks to harden. t Stone or Iron. Compound equal parts of Sulphur and Pitch. Brass to G-lass. Electrical. Resin, 5 ozs. ; Beeswax, i oz. ; Red Ochre or Venetian Red, in pow- der, i oz. Dry earth thoroughly on a stove at above 212. Melt Wax and Resin together and stir in powder by degrees. Stir until cold, lest earthy matter settle to bottom. Used for fastening brass-work to glass tubes, flasks, etc. Chinese "Waterproof. Schio-liao.To 3 parts of Fresh Beaten BJood add 4 parts of Slaked Lime and a little Alum; a thin, pasty mass is produced, which can be used immediately. Materials which are to be made specially waterproof are painted twice, or at most three times. Wooden public buildings of China are painted with gchio-liao, which gives them an unpleasant red- dish appearance, but adds to their durability. Pasteboard treated with it receives appearance and strength of wood. China. Curd of milk, dried and powdered, 10 ozs. ; Quicklime, i oz. ; Camphor, 2 drachma Mix, and keep air-tight. When used, a portion is to be mixed with a little water into a paste. Cisterns and "Water-casks. Melted Glue, 8 parts; Linseed oil, boiled into a varnish with Litharge, 4 parts. This cement hardens in about 48 hours, and renders the joints of wooden cisterns and casks air and water tight. 8/2 MISCELLANEOUS MIXTURES. Cloth or Leather. Shellac, i part; Pitch, 2 parts; India Rubber, 4 parts; and Gutta Percha, 10 parts; cut small; Linseed oil, 2 parts; melted together and mixed Earthen and Glass Ware. Heat article to be mended a little above 212, then apply a thin coating of gum Shellac upon both surfaces of broken vessel. Or, dissolve gum Shellac in alcohol, apply solution, and bind the parts firmly to- gether until cement is dry. Or, dilute white of egg with its bulk of water and beat up thoroughly. Mix to consistence of thin paste with powdered Quicklime. Use immediately. Entomologists'. Thick Mastic Varnish and Isinglass size, equal parts. Gutta Percha. Melt together, in an iron pan, 2 parts Common Pitch and i part Gutta Percha. Stir well together until thoroughly incorporated, and then pour liquid into cold water. When cold it is black, solid, and elastic ; but it softens with heat, and at 100 is a thin fluid. It may be used as a soft paste, or in liquid state, and answers an excellent purpose in cementing metal, glass, porcelain, ivory, etc. It may be used instead of putty for glazing. GUass. SoreVs. Mix commercial Zinc White with half its bulk of fine Sand, add a solu- tion of Chloride of Zinc of 1.26 spec, grav., and mix thoroughly in a mortar. Apply immediately, as it hardens very quickly. Holes in Castings. Sulphur in powder, i part; Sal-ammoniac, 2 parts; powdered Iron turnings, 80 parts. Make into a thick paste. Make only as required for immediate use. Hydraulic Paint. Hydraulic cement mixed with oil forms an incombustible and waterproof paint for roofs of buildings, outhouses, walls, etc. Iron "Ware. Sulphur, 2 parts; fine Black-lead, i part. Heat sulphur in an iron pan until it melts, then add the lead ; stir well, and remove. When cool, break into pieces as required. Place upon opening of the ware to be mended, and solder with an iron. [Kerosene Lamps, etc. Resin, 3 parts; Caustic Soda, i; Water, 5, mixed with half its weight of Plaster of Paris. It seta firmly in about three quarters of an honr. Is of great adhesive power, not permeable to kero ene, a low conductor of heat, and but superficially attacked by hot water. Leather to Iron, Steel, or G-lass. i. Glue, i quart, dissolved in Cider Vinegar; Venice Turpentine, i oz. ; boil very gently or simmer for 12 hours. Or, Glue and Isinglass equal parts, soak in water 10 hours, boil and add tannin until mixture becomes "ropy;" apply warm. Remove surface of leather where it is to be applied. 2. Steep leather in an infusion of Nutgall, spread a layer of hot Glue on sur- face of metal, and apply flesh side of leather under pressure. Leather Belting. Common Glue and Isinglass, equal parts, soaked for 10 hours in enough water to cover them. Bring gradually to a boiling heat and add pure Tannin until whole be- comes ropy or appears alike to white of eggs. Clean and rub surfaces to be joined, apply warm, and clamp firmly. ^Molding and Temporary- Adhesion. Soft. Melt Yellow Beeswax with its weight of Turpentine, and color with finely powdered Venetian red. When cold it has the hardness of soap, but is easily softened and molded with the fingers. MISCELLANEOUS MIXTURES. 8/3 Maltha, or Grreek Mastic. Lime and Sand mixed in manner of mortar, and made into a proper consistency with milk or size without water. Marble. Plaster of Paris, in a saturated solution of Alum, baked in an oven, and reduced to powder. Mixed with water, and color if required. JMetal to Glass. Copal Varnish, 15 parts; Drying Oil, 5; Turpentine, 3. Melt in a water bath and add 10 of Slaked Lime. Mending Shells, etc. Gum Arabic, 5 parts; Rock Candy, 2; and White Lead, enough to color. Large Objects. Wollastori's White. Beeswax, i oz. ; Resin, 4 ozs. ; powdered Plaster of Paris, 5 oz. Melt together. Warm the edges of the object and apply warm. By means of this cement a piece of wood may be fastened to a chuck, which will hold when cool ; and when work is finished it may be removed by a smart stroke with tool. Any traces of cement may be removed by Benzine. Marble Workers and Coppersmiths. White of egg, mixed with finely-sifted Quicklime, will unite objects which are not submitted to moisture. Porcelain. Add Plaster of Paris to a strong solution of Alum till mixture is of consistency of cream. It sets readily, and is suited for cases in which large rather than small surfaces are to be united. Rust Joint. (Quick Setting.) Sal-ammoniac in powder, i Ib. ; Flour of Sulphur, 2 Ibs. ; Iron borings, 80 Ibs. Made to a paste with water. (Slow Setting.) Sal-ammoniac, 2 Ibs. ; Sulphur, i Ib. ; Iron borings, 200 Ibs. The latter cement is best if joint is not required for immediate use. Steam Boilers, Steam-pipes, etc. Finely powdered Litharge, 2 parts; very fine Sand, i; and Quicklime slaked by exposure to air, i. This mixture may be kept for any length of time without -'njuring. In using it, a portion is mixed into paste with linseed oil, boiled or crude. Apply quickly, :-.s it soon becomes hard. Soft. Red or White Lead in oil, 4 parts; Iron borings, 2 to 3 parts. Hard. Iron borings and salt water, and a small quantity of Sal-ammoniac with fresh water. Transparent GHass. India-rubber, i part in 64 of chloroform ; gum Mastic in powder, 16 to 24 parts. Digest for two days, with frequent shaking. Or, pulverized Glass, 10 parts; powdered Fluor-spar, 20; soluble Silicate of Soda, 60. Both glass and fluor-spar must be in finest practicable condition, which is best done by shaking each in fine powder, with water, allowing coarser particles to de- posit, and then by pouring off remainder, which holds finest particles in suspension. The mixture must be made very rapidly, by quick stirring, and applied immediately. TJniting Leather and. Mietal. Wash metal with hot Gelatine; steep leather in an infusion of Nutgalls, hot, and bring the two together. "Waterproof Mastic. Red Lead, i part; ground Lime, 4 parts; sharp Sand and boiled Oil, 5 parts. Or, Red Lead, i part; Whiting, 5; and sharp Sand and boiled Oil, 10. "Wood to Iron. Litharge and Glycerine. Finely powdered Oxide of Lead (litharge) and Concen- trated Glycerine. The composition is insoluble in most acids, is unaffected by action of moderate heat, sets rapidly, and acquires an extraordinary hardness. Turner's. Melt i Ib. of Resin, and add .25 Ib. of Pitch. While boiling add Brick dust to give required consistency. In winter it may be necessary to add a little Tallow. 874 MISCELLANEOUS MIXTURES. GLUES. [Marine. Dissolve India Rubber, 4 parts, in 34 parts of Coal-tar Naphtha; add powdered Shellac, 64 parts. While mixture is hot pour it upon metal plates in sheets. When required for use, heat it, and apply with a brush. Or, India Rubber, i part; Coal Tar, 12 parts; heat gently, mix, and add powdered Shellac, 20 parts. Cool. When used, heat to about 250 Or, Glue, 12 parts; Water, sufficient to dissolve; add Yellow Resin, 3 parts; and, when melted, add Turpentine, 4 parts. Strong Glue. Add Powdered Chalk to common Glue. Mix thoroughly. 3VEvicilage. Curd of Skim Milk (carefully freed from Cream or Oil), washed thoroughly, and dissolved to saturation in a cold concentrated solution of Borax. This mucilage keeps well, and, as regards adhesive power, far surpasses gum Arabic. Or, Oxide of Lead, 4 Ibs. ; Lamp-black, 2 Ibs. ; Sulphur, 5 ozs. ; and India Rubber dissolved in Turpentine, 10 Ibs. Boil together until they are thoroughly combined. Preservation of Mudlage. A small quantity of Oil of Cloves poured into a bottle containing Gum Mucilage prevents it from becoming sour. To Resist Moistixre. Glue, 5 parts; Resin, 4 parts; Red Ochre, 2 parts; mixed with least practicable quantity of water. Or, Glue, 4 parts; Boiled Oil, i part, by weight, Oxide of Iron, i part. Or, Glue, i lb., melted in 2 quarts of skimmed Milk. JParchxn exit . Parchment Shavings, i lb. ; Water, 6 quarts. Boil until dissolved, then strain and evaporate slowly to proper consistence. Rice, or Japanese. Rice Flour; Water, sufficient quantity. Mix together cold, then boil, stirring it during the time. Glue, Water, and Vinegar, each 2 parts. Dissolve in a water-bath, then add Al- cohol, i part. Or, Cologne or strong Glue, 2.2 Ibs. ; Water, i quart; dissolve over a gentle heat; add Nitric Acid 36, 7 ozs. , in small quantities. Remove from over fire, and cool. Or, White Glue, 16 ozs. ; White Lead, dry, 4 ozs. ; Rain Water, 2 pints. Add Al- cohol, 4 ozs., and continue heat for a few minutes. Elastic and Sweet. Stamps or Rolls. Elastic. Dissolve good Glue in water by a water-bath. Evaporate to a thick con- sistence, and add equal weight of Glycerine to Glue; submit to heat until all water is evaporated, and pour into molds or on plates. Sweet. Substitute Sugar for the Glycerine. To Adhere Engravings or Lithographs upon Wood. Sandarach, 250 parts; Mastic in tears, 64 parts; Resin, 125 parts; Venice Tur- pentine, 250 parts; and Alcohol, 1000 parts by measure. BROWNING, OR BRONZING, LIQUID. Sulphate of Copper, i oz. ; Sweet Spirit of Nitre, i oz. ; Water, i pint Mix. Let stand a few days before use. MISCELLANEOUS MIXTURES. 8/5 Q-tm. Barrels. Tincture of Muriate of Iron, i oz. ; Nitric Ether, i oz. ; Sulphate of Copper, 4 scruples ; rain water, i pint. If the process is to be hurried, add 2 or 3 grains of Oxy muriate of Mercury. When barrel is finished, let it remain a short time in lime-water, to neutralize any acid which may have penetrated ; then rub it well with an iron wire scratch-brush. After Browning. Shellac, i oz. ; Dragon's-blood, .25 oz. , rectified Spirit, i qt. Dissolve and filter. Or, Nitric Acid, spec. grav. 1.2; Nitric Ether, Alcohol, and Muriate of Iron, each i part. Mix, then add Sulphate of Copper 2 parts, dissolved in Water 10 parts. [" LACQUERS. Small Arms, or Waterproof* IPaper. Beeswax, 13 Ibs. ; Spirits Turpentine, 13 gallons; Boiled Linseed Oil, i gallon. All ingredients should be pure and of best quality. Heat them together in a copper or earthen ves- sel over a gentle fire, in a water-bath, until they are well mixed. Bright Iron Work. Linseed Oil, boiled, 80.5 parts; Litharge, 5.5 parts; White Lead, in oil, 11.25 parts; Resin, pulverized, 2.75 parts. Add litharge to oil ; simmer over a slow fire 3 hoars ; strain, and add resin and white lead , keep it gently warmed, and stir until resin is dissolved. Or, Amber, 6 parts; Turpentine, 6 parts; Resin, i part; Asphaltum, i -part; and Drying Oil, 3 parts; heat and mix well. Or, Shellac, i Ib. ; Asphaltum, 6 Ibs. ; and Turpentine, i gallon. Iron and. Steel. Clear Mastic, 10 parts; Camphor, 5 parts; Sandarac, 15 parts; and Elimi Gum, 5 parts. Dissolve in Alcohol, filter, and apply cold. Brass. Shellac, 8 ozs. ; Sandarac, 2 ozs. ; Annatto, 2 ozs. ; and Dragon's-blood Resin, .25 oz. ; and Alcohol, i gallon. Or, Shellac, 8 ozs. ; and Alcohol, i gallon. Heat article slightly, and apply lacquer with a soft brush. Wood, Iron, or "Walls, and. rendering Cloth, Paper, etc., 'Waterproof. Heat 120 Ibs. Oil Varnish in one vessel, 33 Ibs. Quicklime in 22 Ibs. water in an- other. Soon as lime effervesces, add 55 Ibs. melted India Rubber. Stir mixture, and pour into vessel of hot Varnish. Stir, strain, and cool When used, thin with Varnish and apply, preferably hot. To Clean Soiled Engravings. Ozone Bleach, i part; Water, 10; well mixed. INKS. Indelible, for Marking Linen, etc. i. Juice of Sloes, i pint; Gum, .5 oz. This requires no " preparation " or mordant, and is very dnrable. 2 Nitrate of Silver, i part; Water, 6 parts; Gum, i part; Dissolve. 3. Lunar Caustic, 2 parts; Sap Green and Gum Arabic, each i part; dissolve with distilled water. "Preparation." Soda, i oz. ; Water, i pint; Sap Green, .5 drachm. Dissolve, and wet article to be marked, then dry and apply the ink. Perpetual, for Tomb-stones, Marble, etc. Pitch, n parts; Lamp-black, i part; Turpentine sufficient. Warm and mix. Copying Ink. Add i oz. Sugar to a pint of ordinary Ink. SOLDERING. Base for Soldering. Strips of Zinc in diluted Muriatic, Nitric, or Sulphuric Acid, until as much is de- composed as acid will effect. Add Mercury, let it stand for a day; pour off the Water, and bottle the Mercury. When required, rub surface to be soldered with a cloth dipped in the Mercury. 8/6 MISCELLANEOUS MIXTURES. VARNISHES. 'Waterproof. Flour of Sulphur, i Ib. ; Linseed Oil, i gall. ; boil them until they are thoroughly combined. Good for waterproof textile fabrics. Harness. India Rubber, .5 Ib. ; Spirits of Turpentine, i gall. ; dissolve into a jelly; then mix hot Linseed Oil, equal parts with the mass, and incorporate them well over a slow fire. Fastening I-jeather on Top Rollers. Gum Arabic, 2.75 ozs., and a like volume of Isinglass, dissolved in Water. To IPreserve GHass from the Su.n. Reduce a quantity of Gum Tragacanth to fine powder, and dissolve it for 24 hours in white of egg well beat up. "Water-color Drawings. Canada Balsam, i part; Oil of Turpentine, 2 parts. Mix and size drawing before applying. Objects of Natural History, Shells, Fish, etc. Mucilage of Gum Tragacanth and of Gum Arabic, each i oz. Mix, and add spirit with Corrosive Sublimate, to precipitate the more stringy por- tion of the Gum. Iron and. Steel. Mercury, 120 parts; Tin, 10 parts; Green Vitriol, 20 parts; Hydrochloric Acid of 1.2 sp. gr., 15 parts, and pure Water, 120 parts. Blackboards. Shellac Varnish, 5 gallons; Lamp-black, 5 ozs.; fine Emery, 3 ozs.; thin with Alcohol, and lay in 3 coats. Black. Heat, to boiling, Linseed Oil Varnish, 10 parts, with Burnt Umber, 2 parts, and powdered Asphaltum, i part. When cooled, dilute with Spirits of Turpentine as may be required. Balloon. Melt India Rubber in small pieces with its weight of boiled Linseed Oil. Thin with Oil of Turpentine. Transfer. Alcohol, 5 ozs. ; pure Venice Turpentine, 4 ozs. ; Mastic, i oz. To render Canvas "Waterproof and Pliable, Yellow Soap, i Ib , boiled in 6 pints of Water, add, while hot, to 112 Ibs. of oil Paint 'Waterproof Bags. Pitch, 8 parts, Wax and Tallow, each i part. To Clean Varnish. Mix a lye of Potash or Soda, with a little powdered Chalk. STAINING. Wood and Ivory. Yellow. Dilute Nitric Acid will produce it on wood. Red. An infusion of Brazil Wood in Stale Urine, in the proportion of i Ib. to a gallon, for wood, to be laid on when boiling hot, also Alum water before it dries. Or, a solution of Dragon's-blood in Spirits of Wine. Black. Strong solution of Nitric Acid. Blue. For Ivory: soak it in a solution of Verdigris in Nitric Acid, which will turn it green; then dip it into a solution of Pearlash boiling hot. Purple. Soak Ivory in a solution of Sal-ammoniac into four times its weight of Nitrous Acid. Jf dhogany. Brazil, Madder, and Logwood, dissolved in water and put on hot. MISCELLANEOUS MIXTURES. 8/7 MISCELLANEOUS. Blacking for Harness. Beeswax, .5 Ib. ; Ivory Black, 2 ozs. ; Spirits of Turpentine, i oz. ; Prussian Blue ground in oil, i oz. ; Copal Varnish, .25 oz. Melt wax and stir it into other ingredients before mixture is quite cold; make it into balls. Rub a little upon a brush, and apply it upon harness, then polish lightly with silk. To Clean Brass Ornaments. Brass ornaments that have not been gilt or lackered may be cleaned, and a very brilliant color given to them, by washing them in Alum boiled in strong Lye, in the proportion of an ounce to a pint, and afterwards rubbing them with strong Tripoli. To Harden Drills, Chisels, etc. Temper them in Mercury. To Clean Coral. Brush with equal parts Spirits of Salts and cold water. Or, dip in a hot solution of Potash or Chloride of Lime. If much discolored, let it remain in solution for a few hours. Blacking, ^vithout Polishing. Molasses, 4 ozs. ; Lamp-black, .5 oz. ; Yeast, a table-spoonful; Eggs, 2; Olive Oil, a teaspoonful; Turpentine, a teaspoonful. Mix well. To be applied with a sponge, without brushing. Dubbing. Resin, 2 Ibs. ; Tallow, i Ib. ; Train-oil, i gallon. Anti-friction Q-rease. Tallow, ioo Ibs. ; Palm-oil, 70 Ibs. Boiled together, and when cooled to 80, strain through a sieve, and mix with 28 Ibs. of Soda, and 1.5 gallons of Water. For Winter, take 25 Ibs. more oil in place of the Tallow. Or, Black Lead, i part; Lard, 4 parts. To Attach, Hair Felt to Boilers. Red Lead, i Ib. ; White Lead, 3 Ibs. ; and Whiting, 8 Ibs. Mixed with boiled Lin^ seed Oil to consistency of paint. IPastils for Fumigating. Gum Arabic, 2 ozs. ; Charcoal Powder, 5 ozs. ; Cascarilla Bark, powdered, .75 oz. ; Saltpetre, .25 drachm. Mix together with water, and make into shape. For "Writing -upon Zinc Labels. Horticultural. Dissolve ioo grains of Chloride of Platinum in a pint of water; add a little Mu- cilage and Lamp-black. Or, Sal-ammoniac, i dr. ; Verdigris, i dr. ; Lamp-black, .5 dr. ; Water, 10 drs. Mix. To Remove old. Ironmold. Rempisten part stained with ink, remove this by use of Muriatic Acid diluted by 5 or 6 times its weight of water, when old and new stain will be removed. To Cut India Rubber. Keep blade of knife wet with water or a strong solution of Potash. Adhesive for Rubber Belts. Coat driving surface with Boiled Oil or Cold Tallow, and then apply powdered Chalk. Liard. 50 parts of finest Rape-oil, and i part of Caoutchouc, cut small. Apply heat until it is nearly all dissolved. To Preserve Leather Belting or Hose. Apply warm Castor Oil. For hose, force it through it. To Oil Leather Belting. Apply a solution of India Rubber and Linseed Oil. * 8;8 MISCELLANEOUS MIXTURES. Dressing for Leather Belts. i. Beef Tallow, i part, and Castor Oil, 2 parts. Apply warm. 2. Beef Tallow, 3 Ibs. , Beeswax, i Ib. Healed and applied warm to botL sides Files. Lay dull files in diluted Sulphuric Acid until they are bitten deep enough. To Remove Oil from Leather. Apply Aqua-ammonia. To Clean Paint. Wash with a solution of Pearlash in water. If greasy, use Quicklime. Or, Extract of Litherium diluted with from 200 to 300 parts of water. To Remove Paint. Mix Soft Soap, 2 ozs., and Potash, 4 ozs., in boiling Water, with Quicklime, .5 Ib. Apply hot, and let remain for i day. Or, Extract of Litherium, thinly brushed over the surface 2 or 3 times. To Clean Marble. Chalk, powdered, and Pumice-stone, each i part; Soda, 2 parts. Mix with water. Wash the spots, then clean and wash off with Soap and Water. Paste for Cleaning Metals. Oxalic Acid, i part; Rottenstone, 6 parts. Mix with equal parts of Train Oil and Spirits of Turpentine. "Watchmaker's Oil, which never Corrodes or Thickens. Place coils of thin Sheet Lead in a bottle with Olive Oil. Expose it to the sun for a few weeks, and pour off the clear oil. Durable Paste. Make common Flour paste rather thick (by mixing some Flour with a little cold water until it is of uniform consistency, and then stir it well while boiling water is being added to it); add a little Brown Sugar and Corrosive Sublimate, which will prevent fermentation, and a few drops of Oil of Lavender, which will prevent it be- coming moldy. When dried, dissolve in water. It will keep for two or three years in a covered vessel. To Extract Orease from Stone or Marble. Soft Soap, i part ; Fuller's Earth, 2 parts ; Potash, i part. Mix with boiling water. Lay it upon the spots, and let it stand for a few hours. Stains. To Remove. Stains of Iodine are removed by rectified Spirit; Ink stains by Ox- alic or Superoxalate of Potash; Ironmolds by same; but if obstinate, moisten them with Ink, then remove them in the usual way. Red spots upon black cloth, from acids, are removed by Spirits of Hartshorn, or other solutions of Ammonia. Stains of Marking -ink, or Nitrate of Silver. Wet stain with fresh solution ol Chloride of Lime, and, after 10 or 15 minutes, if marks have become white, dip the part in solution of Ammonia or of Hyposulphite of Soda. In a few minutes wash with clean water. Or, stretch the stained linen over a basin of hot water, and wet mark with Tinc- ture of Iodine. Preservative Paste for Objects of Natural History. White Arsenic, i Ib. ; Powdered Hellebore, 2 Ibs. To Preserve Bottoms of Iron Steam-boilers. Red Lead, 75 parts; Venetian Red, 17 parts; Whiting, 6.5 parts; and Litharge. 1.5 parts by weight. To Preserve Sails. Slacked Lime, 2 bushels. Draw off the lime-water, and mix it with 120 gallons water, and with Blue Vitriol, .25 Ib. MISCELLANEOUS OPERATIONS AND ILLUSTRATIONS. 8/9 "Whitewash. For outside exposure, slack Lime, .5 bushel, iu a barrel; add common Salt, i Ib. , Sulphate of Zinc, .5 Ib. ; and Sweet Milk, i gallon. To ^Preserve "Wood. -work. Boiled Oil and finely powdered Charcoal, each i part; mix to the consistence of paint. Apply 2 or 3 coats. This composition is well adapted for casks, water-spouts, etc. To JPolish. "Wood. Rub surface with Pumice Stone and water until the rising of the grain is removed Then, with powdered Tripoli and boiled Linseed Oil, polish to a bright surface. P*aint for "Window Grlass. Chrome Green, .25 oz. ; Sugar of Lead, i Ib. ; ground fine, in sufficient Linseed Oil to moisten it. Mix to the consistency of cream, and apply with a soft brush. The glass should be well cleansed before the paint is applied. The above quantity is sufficient for about 200 feet of glass. To Alake 13 rain Tiles IPorous. Mix sawdust with the clay before burning. MISCELLANEOUS OPERATIONS AND ILLUSTRATIONS. i. It is required to lay out a tract of land in form of a square, to be en- closed with a post and rail fence, 5 rails high, and each rod of fence to con- tain 10 rails. What must be side of this square to contain just as many acres as there are rails in fence? OPERATION, i mile =: 320 rods. Then 320 X 320 -r- 160, sq. rods in an acre = 640 acres; and 320 X 4 sides and X 10 rails = 12 800 rails per mile. Then, as 640 acres : 12800 rails :: 12800 acres : 256000 rails, which will enclose 256000 acres, and V 2 5 6 ooX 69.5701 = number of yards in side of a sq. acre, and -r- 1760, yards in a mile = 20 milts. 2. How many fifteens can be counted with four fives? OPERATION. 4 X 3 X 2 X f = *-* = <. 1X2X3 6 3. What are the chances in favor of throwing one point with three dice? OPERATION. Assume a bet to be upon the ace. Then there will be 6 X 6 x 6 = 216 different ways which the dice may present themselves, that is, with and without an ace. Then, if the ace side of the die is excluded, there will be 5 sides left, and 5X5X5 = 125 ways without the ace. Therefore, there will remain only 216 125 = 91 ways in which there could be an ace. The chance, then, in favor of the ace is as 91 to 125 ; that is, out of 216 throws, the probability is that it will come up 91 times, and lose 125 times. 4. The hour and minute hand of a clock are exactly together at 12; when are they next together ? OPERATION. As the minute hand runs n times faster than the hour hand, then, as ii : 60 :: i : 5 win. 27^- sec. = time past i o'clock. 5. Assume a cube inch of glass to weigh 1.49 ounces troy, the same of sea-water .59, and of brandy .53. A gallon of this liquor in a glass bottle, which weighs 3.84 Ibs., is thrown into sea-water. It is proposed to deter- mine if it will sink, and, if so, how much force will just buoy it up? OPERATION. 3.84 X 12 -r- 1.49 = 30.92 cube ins. of glass in bottle. 231 cube ins. in a gallon X .53 = 122.43 ounces of brandy. Then, bottle and brandy weigh 3.84 X 12 + 122.43 = 168.51 ounces, and contair 961.92 cube ins., which X .59 = 154- 53 ounces, weight of an equal bulk of sea-water And, 168.51 154.53 = 13.98 ounces, weight necessary to support it in the water 88O MISCELLANEOUS OPERATIONS AND ILLUSTRATIONS. 6. A fountain has 4 supply cocks, A, B, C, and D, and under it is a cis- tern, which can be tilled by the cock A in 6 hours, by B in 8 hours, by C in 10 hours, and by D in 12 hours ; now, the cistern has 4 holes, designated E, F, G, and H, ami it can be emptied through E in 6 hours, F in 5 hours, G in 4 hours, and H in 3 hours. Suppose the cistern to be full of water, and that all the cocks and holes were opened together, in what time would the cistern be emptied? OPERATION. Assume the cistern to hold 120 gallons. bra. gall. hrs. gall. If 6 : 120 8 : 120 xo : 120 12 : 120 20 at A. 15 at B. 12 at G. 10 at D. Run in in i hour, 57 gallons. hrs. gall. hrs. gall. If 6 5 4 3 20 at E. 24 at F. 30 at G. 40 at H. Run out in i hour, 114 gallons. 57 Run out in i hour more than run in, 57 gallons. Then, as 57 gallons : i hour :: 120 gallons : 2.105-!- hours. 7. A cistern, containing 60 gallons of water, has 3 cocks for discharging it ; one will empty it in i hour, a second in 2 hours, and a third in 3 hours ; in what time wilHt be emptied if they are all opened together? OPERATION. ist, .5 would run out in i hour by the 2d cock, and .333 by the 3d-, consequently, by the 3 would the reservoir be emptied in i hour. .5 -|- .333 + 1 = ^ -f- 1^ -f- ^, being reduced to a common denominator, the sum of these 3 = ^- ; whence the proportion, 1 1 : 60 : : 6 : 32 T 8 T minutes. 8. A reservoir has 2 cocks, through which it is supplied ; by one of them it will fill in 40 minutes, and by the other in 50 minutes ; it has also a dis- charging cock, by which, when full, it may be emptied in 25 minutes. If the 3 cocks are left open, in what time would the cistern be filled, assuming the velocity of the water to be uniform ? OPERATION. The least common multiple of 40, 50, and 25, is 200. Then, the ist cock will fill it 5 times in 200 minutes, and the 2d, 4 times in 200 minutes, or both, 9 times in 200 minutes ; and, as the discharge cock will empty it 8 times in 200 minutes, hence 9 8 = i, or once in 200 minutes = 3. 2 hours. 9. The time of the day is between 4 and 5, and the hour and minute hands are exactly together ; what is the time? OPERATION. Difference of speed of the hands is as i to 12 = 11. 4 hours X 60 = 240, which -4-11 = 21 min. 49.09 sec., which is to be added to 4 hours. 10. Out of a pipe of wine containing 84 gallons, 10 were drawn off, and the vessel refilled with water, after which 10 gallons of the mixture were drawn off, and then 10 more of water were poured in, and so on for a third and fourth time. It is required to compute how much pure wine remained in the vessel, supposing the two fluids to have been thoroughly mixed. OPERATION. 84 10 = 74, quantity after the ist draught. Then, 84 : 10 :: 74 : 8.8095, and 74 8.8095 = 65.1905, quantity after zd draught. 84 : 10 : : 65. 1905 : 7. 7608, and 65. 1905 7. 7608 = 57. 4297, quantity after yd draught. 84: 10 :: 57. 4297 -.6.8367, and 57.42976.8367 = 50.593, quantity after $th draught, =. result required. ii. A reservoir having a capacity of 10000 cube feet, has an influx of 750 and a discharge of 1000 cube feet per day. In what time will it be emptied? 10000 OPERATION. - = 4 o days. 1000 750 Contrariwise : The discharge being 1000 and the influx 1250 cube feet per houc In what time will it be filled? OPERATION. = 40 hours = z day 16 hours. 1250 1000 MISCELLANEOUS OPEBATIONS AND ILLUSTRATIONS. 88 1 12. A son asked his father how old he was. His father answered him thus : If you take away 5 from my years, and divide the remainder by 8, the quotient will be one third of your age ; but if you add 2 to your age, and multiply the whole by 3, and then subtract 7 from the product, you will have )the number of years of my age. What were the ages of father and son? OPERATION. Assume father's age 37. Then 37 5 = 32, and 32 -f- 8 = 4, and 4 X 3 = 12, son's age. Again: 12 -f 2 = 14, and 14 X 3 = 42, and 42 7 = 35. Therefore 37 35 = 2, error too little. Again : Assume father's age 45 ; then 45 5 = 40, and 40 -5- 8 = 5. Therefore 5 x 3 = 15, son's age. Again: 15 + 2 = 17, and 17 X 3 = 51, and 51 7 = 44. There- fore 45 44 = i, error too little. Hence (45 sup. X 2 error) (37 sup. x i error) = 90 37 = 53, and 2 i = i. Consequently, 53 is father's age. Then 53 5 = 48, and 48 -j- 8 = 6 = . 333 of son's age, and 6 x 3 = 18 years, son's age. 13. Two companions have a parcel of guineas. Said A to B, if you will give me one of your guineas I shall have as many as you have left. B re- plied, if you will give me one of your guineas I shall liave twice as many as you will 'have left. How many guineas had each of them ? OPERATION. Assume B had 6. Then A would have had 4, for 6 i = 4 -f- 1 = 5- Again : 4 ( A's parcel) 1=3, and 6 -f- 1 = 7, and 9X2 = 6. Therefore 7 6 = 1, error too little. Again: Assume B had 8. Then A would have 6, for 8 i = 6 + i = 7. Again : 6 (A's parcel) 1 = 5, and 8 -f- 1 = 9> aQ d 5 X 2 = 10. Therefore 10 9=1, error too great. Hence 8 X i = 8, and 6 X i = 6. Then 8 -f 6 = 14, and i + i = 2 . Whence, di- viding products "by sum of errors, 14 -f- 2 = 7 = B's parcel, and 7 i = 5 -J- 1 = 6 for A when he had received i ofB ; also 5 1 X2 = 7 + i=8 = B's parcel when he had received i of A. 14. If a traveller leaves New York at 8 o'clock in the morning, and walks towards New London at the rate of 3 miles per hour, without intermission; and another traveller starts from New London at 4 o'clock in the 'vening, and walks towards New York at the rate of 4 miles per hour continuously; assuming distance between the two cities to be 130 miles, whereabouts upon the road will they meet ? OPERATION. From 8 to 4 o'clock is 8 hours; therefore, 8 X 3 = 24 miles, per- formed by A before B set out from New London; and, consequently, 130 24 = 106 are the miles to be travelled between them after that. Hence, as (3 -f 4) 7 : 3 : : 106 . ^fi = 45^ more miles travelled by A. at the meeting; consequently, Z4 -f- 45y = 69^ miles from New York is place of their meeting. 15. If from a cask of wine a tenth part is drawn out and then it is filled with water ; after which a tenth part of the mixture is drawn out ; again is filled, and again a tenth part of the mixture is drawn out : now, assume the fluids to mix uniformly at each time the cask is replenished, what frac- tional part of wine will remain after the process of drawing out and replen- ishing has been repeated four times ? OPERATION. Since .1 of the wine is drawn out at first drawing, there must remain g. After cask is filled with water, .1 of whole being drawn out, there will remain 9 of mixture; but .9 ofthi* mixture is wine; therefore, after second drawing, there o 2 will remain .gof.g of wine, or -fra ; and after third drawing, there will remain .9 Q* f -9 f -9 of wine, or -2-j. Hence, the part of wine remaining is expressed by the ratio .9, raised to a powef txponent of which is number of times cask has been drawn from. Therefore, fractional part of wine is ^ = .6561. 1E* 882 MISCELLANEOUS OPERATIONS AND ILLUSTRATIONS. 16. Tnere is a fish, the head of which is 9 ins. long, the tail as long as the head and half the body, and the body as long as both the head and tail. Required the length of the fish. OPERATION. Assume body to be 24 ins. in length. Then 24-4-24-9 = 21, Length of tail. Hence 21 + 9 = 30, length of body, which is 6 ins. too great. Again : assume the body to be 26 ins. in length. Then 26 -5- 2 -f- 9 = 22, length of tail. Hence 22 + 9 = 31, length of body, which is 5 ins. too great. Therefore, by Double Position, divide difference of products (see rule, page 99) by difference of errors (the errors being alike), 26 X 6 24 X 5 = 36 = difference of products, and 6 5 = 1= difference of errors. Consequently, 36 -r- 1 = 36, length of body, and 36 -r- 2 + 9 = 27, length of tail, and 36 + 27-1-9 = 72 ins. , length required. 17. A hare, 50 leaps before a greyhound, takes 4 leaps to the greyhound's 3, but 2 leaps of the hound are equal to 3 of the hare's. How many leaps must the greyhound take before he can catch the hare ? OPERATION. As 2 leaps of the greyhound equal 3 of the hare, it follows that 6 of the greyhound equal 9 of the hare. While the greyhound takes 6 leaps, the hare takes 8; therefore, while the hare takes 8, the greyhound gains upon her i. Hence, to gain 50 leaps, she must take 50 X 8 = 400 leaps ; but, while hare takes 400 leaps, greyhound takes 300, since number of leaps taken by them are as 4 to 3. 1 8. If a basket and 1000 eggs were laid in a right line 6 feet apart, and 10 men (designated from A to J) were to start from basket and to run alter- nately, collect the eggs singly, and place them in basket as collected, and each man to collect but 10 eggs in his turn, how many yards would each man run over, and what would be entire distance run over ? OPERATION. A's course would be 6 X 2 feet (first term) -f- 10 X 6 X 2 feet (last term) = 132 = sum, of first and last terms of progression. Then 132-7-2 X 10 = 660 feet = number of times x half sum of extremes = sum of all the terms, or the distance run by A in his first turn. B's course would be u X6x 2 = 132/6^ (first term) + 20 X 6 X 2 = 240 feet (last term) = 372 = sum of first and last terms. Then 372 -=- 2 X 10 = 1860 = sum of all the times, or B's first turn. A's last course would be 901 X 6 X 2 = 10812 feet for the first term, and 910X6X2 = 10920 feet for the last term of his last turn. Then 10 812 -f- 10 920 -f- 2 X 10 = 108 660 sum of the terms, or distance run. B's last course would be 911 x 6 X 2 = 10932 feet for the first term, and 920X6X2 = 11 040 feet for the last term of his last turn. Then 10 032 -f- n 040-7-2 X 10 = 109 860 = sum of the terms or distance run. Therefore, if A'S first and last runs = 660 and 108 660 feet, and the number of terms 10, then, by Progression, the sum of all the terms = 546600/6^. And if B's first and last runs=: 1860 and 109860 feet, and the number of terms 10, then the sum of all the terms = 5586oo/ee<. Consequently, 558 600 546 600 = 12 ooo = common difference of runs, which, be- ing added to each man's run = sum of all runs, or entire distance run over. A's run, 546 600 = 182 200 yds. B's " 558600=186200 " C's " 570600 = 190200 " D's " 582600 = 194200 u E's " 594600=198200 " F's run, 606 600 = 202 200 yds. G's " 618600 = 206200 " H's " 630600 = 210200 " I's " 642600 = 214200 " J's " 654600 = 218200 " 6oo6ooo./ee, which -7-5280 = 1137.5 miles. 19. If, in a pair of scales, a body weighs 90 Ibs. in one scale, and but 40 Ibs. in the other, what is the true weight ? -/(4Q X 90) = 60 IbS. MISCELLANEOUS OPERATIONS AND ILLUSTRATIONS. 883 20. If a steamboat, running uniformly at the rate of 15 miles per hour through the water, were to run for i hour with a current of 5 miles per hour, then to return against that current, what length of time would she require to reach the place from whence she started ? OPERATION. 15 -f- 5 = 20 miles, the distance run during the hour. Then 15 5 = 10 miles is her effective velocity per hour when returning, and 20-r- 10 = 2 hours, the time of returning, and 2 + i = 3 hours, or the whole time oc- cupied. Or, Let d represent distance in one direction, t and t f greater and less times of run- ning in hours, and c current or tide. Then, -^velocity of boat through the water, and V t X t 2i. Flood-tide wave in a given river runs 20 miles per hour, current of it is 3 miles per hour. Assume the air to be quiescent, and a floating body set free at commencement of flow of the tide ; how long will it drift in one direction, the tide flowing for 6 hours from each point of river ? OPERATION. Let x be the time required; 202; = distance the tide has run up, to- gether with the distance which the floating body has moved; 3* = whole distance which the body has floated, Then 20 x 3 x = 6 X 20, or the length in miles of a tide. - X 6 = 7 hours, 3 minutes, 31.765 seconds. 20 3 22. A steamboat, running at the rate of 10 miles per hour through the water, descends a river, the velocity of which is 4 miles per hour, and re- turns in 10 hours ; how far did she proceed ? OPERATION. Let x = distance required, - = time of going, = time of 104-4 10 4 returning. Then, -f- = 10 ; 6x -f 1 4* = 840 ; 20* = 840 ; 840 -r- 20 = 42 miles. 14 6 23. From CaldwelTs to Newburgh (Hudson River) is 18 miles ; the cur- rent of the river is such as to accelerate a boat descending, or retard one ascending, 1.5 miles per hour. Suppose two boats, running uniformly at the rate of 15 miles per hour through the water, were to start one from each place at the same time, where will they meet ? OPERATION. Let 2; = the distance from N. to the place of meeting; its distance from C., then, will be 18 x. Speed of descending boat, 15 -f- 1. 5 = 16. 5 miles per hour ; of ascending boat, 15 1.5 = 13.5 miles per hour. = time, of boat descending to point of meeting. 10.5 13.5 = time of boat ascending to point of meeting. These times are of course equal; therefore, = . Then, 13. 53 = 297 io-5 *3-5 16. 53, and 13. 53 -f- 16. 53 = 297, or 30* = 297. Hence * = = 9.9 miles, the distance from Newburgh. 3 24. There is an island 73 miles in circumference ; 3 men start together to walk around it and in the same direction : A walks 5 miles per hour, B 8, and C 10 ; when will they all come aside of each other again ? OPERATION. It is evident that A and C will be together every round gone by A ; hence it remains to ascertain when A and B will be in conjunction at an even round, as 3 miles are gained every day by B. Therefore, as 3 : i :: 73 : 24.33-!-; ^ Ut 5 as the conjunction is a fractional number, it is necessary to ascertain what number of * multiplier will make the division a whole number. 73 -f- 24. 33-J- = 3, the number of days required in which A will go round 5 time^ B 8, and C 10 times. 884 MISCELLANEOUS OPEKATIONS AND ILLUSTRATIONS. 25. Assume a cow, at age of 2 years, to bring forth a cow-calf, and then to continue yearly to do the samej and every one of her produce to bring forth a cow-calf at age of 2 years, and yearly afterward in like manner ; how many would spring from the cow and her produce in 40 years ? OPERATION. The increase in ist year would be o, in 2d year i, in 3d i, in 4th 2, in $th 3, in 6th 5, and so on to 40 years or terms, each term being = sum of the two preceding ones. The last term, then, will be 165580141, from which is to be sub- tracted i for the parent cow, and the remainder, 165 580140, will represent increase required. 26. The interior dimensions of a box are required to be in the propor- tions of 2, 3, and 5, and to contain a volume of 1000 cube ins. ; what should be the dimensions V And what for a box of one half the volume, or 500 cube ins., and retaining same proportionate dimensions ? OPERATION. 2 X 3 X 5 = 30, and = 15. Then, 7. V 5X6.43 = 6 and 3 A5*2fi! = V 3 3 o V 30 27. The chances of events or games being equal, what are the odds for or against the following results ? Ive Events. rcmr Events. Odds. Against. In favor. Odds. Against. In favor. 31 to i 4-33 to i 5 to 3 in fa ing 3 and 2. Tl Odds. All the 5 4 out of 5 ivor of the 5 e tree Ever Against. i out of 5 2 out of 5 vents result- its. In favor. 15 to i 2.2 tO I 5 to 3 ag{ the 4 events T Odds. All the 4 3 out of 4 linst 2 events do not result wo Even Against. i out of 4 2 out of 4 only, or that 2 and 2. ts. In favor. 7 to i Even 3 to i in ft ing 2 and i. All the 3 (2 or all out \ of 3 ivor of the 3 e i out of 3 ( 2 or all out { of 3 vents result- 3 to i Even Even that Both events {i only out Of 2 the events re I OUt Of 2 {i only out of 2 suit i and i. 28. Required the chances or probabilities in events or games, when the chances or probabilities of the results, or the players, are equal. Events or Games. That a named event occurs a majority or more of times. Against a named event occurring an exact majority of times. Against each event occur- ring an equal number of times. Events or Games. That a named event occurs a majority or more of times. Against a named event occurring an exact majority of times. Against each event occur- ring an equal number of times. 21 Even 5 to i II Even 3-4 to i 90 19 1.33 to i Even 4-5 to i 4.66 tO i 10 9 1.7 to i Even 3 to i 3.06 to i ii 1.55 to i 4.4 tO I 8 1-75 tO I 2.66 to i 17 Even 4.4 to i 7 Even 2.7 to i 16 1.5 to i 4. 1 tO I 6 2 tO I 2.2 tO I 15 Even 4 to i 5 Even 2.2 tO I __ 14 1.5 to i 3.8 to i 4 2.2 tO I 1.66 to i 3 Even 3-7 to i 3 Even 1.66 to i __ 12 1.6 to i 3-44 to i 2 3 to i Even. 29. The chances of consecutive events or results are as follows : &x. 2047 to i. I 10. 1023 to i. I 9. 511 to i. I 8. 255 to i. | 7. 127 to i. | 6. 63 to i. Hence it will be observed that the chances increase with the number of events very nearly in a duplicate ratio. ILLUSTRATION. The chances of n consecutive events compared with 10, are as 2047 to 1023, or 2 to i. MISCELLANEOUS OPERATIONS AND ILLUSTRATIONS. 885 30. Required the chances or probabilities of events or results in a given number of times. The numerator of a fraction expresses the chance or probability either for the re- sult or event to occur or fail, and the denominator all the chances or probabilities both for it to occur or fail. Thus, in a given number of events or games, if the chances are even, the proba- bility of any particular result is as ^ = ; ; , etc., being i out of i + i 2 2-1-2' 3 + 3' 2, 2 out of 4, etc. , or even. If the number of events or games are 3, then the probability of any par- ticular result, as 2 and i, or i and 2, is determined as follows : Number of permutations of 3 events are i x 2 X 3 = 6, which represents number of times that number of events can occur, 2 and i, or i and 2, to which is to be added the 2 times or chances they can occur all in one way or the reverse thereto. Hence, = = = , or 3 to i in favor of result; and probability of '2+6 443 i one party naming or winning two precise events or results, as winning 2 out of 3, is determined as follows: Number of permutations and chances, as before shown, are 8. Hence, number of his chances being 3, f = -| = 2 = , or 3 to 5 in 3 + 5 8 8 3 5 favor of result; and probability of one party naming or winning all, or 3 events or results, is determined as follows: Number of permutations and chances being also, as before shown, 8. Hence, as there is but one chance of such a result, L_ = __ = , or i to 7 in favor of result i + 7 8 8-1 7' If number of events, etc., are 4, then probability of any particular result, as 2 and 2, or of winning 2 or more of them, is determined as follows : Number of permutations and chances of 4 events are 16. Hence, as number of chances of such a result are n, " = ^ = -^ = , or as n to 5 in favor ' 5 + n 16 16 ii 5 of the result, and that the results do not occur precisely 2 and 2. The number of chances of such a result being 10, . ' = -|- = -^ = , or 5 to 3 against it. o-j-io o o 5 3 If number of events, etc., are 5, then probability of any particular result, as 3 and 2, is determined as follows : Number of permutations and chances being 32, and number of chances of such a result being 20, ^ = ^ = -^ = ~ = , or as 5 to 3 in favor of the '12 + 20 16 16 10 6 3 result; and that it may occur precisely 3 out of 5, the number of chances are - 5; = , or ii to 5 against it ic + 22 32 16 16 5" 31. What is the dilatation of the iron in a railway track per mile, be- tween the temperatures of 20 and +130? OPERATION. 20 + 130 = 150. The dilatation of wrought iron (as per table, page 519) is, from 32 to 212 = 180 = .001 257 5 times its length. Hence, as 180 : 150 : : .001 257 5 : .001 047 9 = 479 of 5280 (feet in a mile) = 5-53 f eet P er mile - 32. A steamer having an immersed amidship section of 125 sq. feet, has a speed of 15 miles per hour with 300 H?. What power would be required for one of like model, having a section of 150 sq. feet for a speed of 20 miles? As power required for like models is as cube of speeds. Then = 1.2 relative sections, and ooo _ relative powers. 125 i5 3 = 3375 Hence, i : 1.2 :: 2.37 : 2.844 times IP. 886 MARINE STEAMEKS AND ENGINES. * ife . 1 'Ml- :$ iJ^S AB!8*ftRa8 S ; N S 2 f> ^HMinoo^o a -s ^H .- &P MARINE STEAMERS AND ENGINES. 887 fe 1 2 ( 2 m l*!J i = s;^8"sii ro M M m BO o. 31 *> I * I ^ .1 ! : : : B -g : 3 C3 72 .2 * * |g2- M 'p.- gg'g a &! a I S r= 03 -2 g o i -" . 888 MARINE, RIVER, ^Passenger Compound ami Lengths and Hull, in feet and tenths; Draught, Propeller, and Side Wheels Surfaces, in sq.feet; Weights and Displacements, in Tons 0/2240 Ibs.; Speed, in. Knots per Hour. DIMENSIONS AND CAPACITIES. City of Paris and New York. Columbia. Nairn- Bhire. Brerner- haven. Tyne- sicler. Simon Durnois and Ma- nagua. Eleciric end Frolic. El Sol. Service 1. Steel. P ami F 2. Steel. P and F 3. Steel. Refrig'tor 4. Steel. 5. Iron. Fand P 6. Steel. Fruit 7. Iron. 8. Iron. F and P Length on deck . . . " bet. perp'rs " tonnage . . . Beam, do Hold, do 527.6 525 527.6 63-2 22 474 463 463-5 55-6 35.8 350.5 350 350.6 47-7 24. 2 350 340 339- 6 42.6 27.3 260.5 260 260 33-7 15. 2 184 174.8 175-2 27.8 19 111.9 106.9 107.4 20.5 !! 5 388.75 377-2 375 48 24 Decks 2 2 2 2 Tons . . ..I''*' 558l 3737 2428 2179 6 9 2 5M 7 o 9 3021 \.... Draught, load Displacement do. . . Imm'd Sec'n at do. 10499 25 7363 24 10000 14.. 7 3720 24-3 7880 1058 3393 21.3 6600 870 e 1290 I 74 2560 530 7 I 7 16.2 1462 404 A e 183 "5 230 135 21 6760 934 TO Cylinders, IP " Int " L.P..... Stroke of piston... Steam pressure. . . . Revolutions .... 2 Of 45 2" 7 I i"xx3 60 150 86 < 20f4I 2" 66 2 " IOI 66 150 27 44 I 1 60 25 40 66 160 62 28.5 4 6 75 42 160 U J 26 40 12.75 20 32 22 ISO 32 52 8 4 & nf\ Q 3 Grate surface 1293 5O 625 I22O 209 606 o i54 4800 2?6 6618 63 29 800 I 44 r 7 6 Condensing do Propeller, diam. . . . Pitch 33000 2 Of l8 32 3032 16.5 t 8 5 2383 17.6 17 6 345o 16 10 39 7-9 8 6 6400 18 Side wheels, diam.. Breadth - - Coal, weight Consumption Combustion Cargo . . . . 3300 Blast Natural 1266 3000 Natural 821 Natural 130 2400 Natural 1080 H3 IOO5 Natural Natural IOOO 5000 NandB Passengers I 37 2 1006 * 1180 4000 Crew 48 IIP IQ 17^ 13 ooo - 388 20 20 ^ Bte Bark 'ne 7-m Sch'r Brier j-m Rrh'r d-tn Sr.h firh'r rnTifii7 A -m SMi Remarks. No. 1. J. & G. Thomson, Glasgow, Scotland; Area of Immersed Horizontal Section at Load-line, 16500 nfeet = coefficient .5. No. 2. Laird Bros., Birkenhead, Eng. No. 3. R. & W. Hawthorn, Leslie & Co., New- castle, Eng. ; Hull 2375 tons, Engines 180, and Boilers 156.- -No. 4. Russell & Co. and G. Stewart & Co., Greenock, Eng. ; Hull 1950 tons. Engines and Boilers 33 o. No. 5. Tyne Steam Shipping Co., Newcastle, England ; Engines 135 tons, Boilers 220, and Water 60, "Well deck." No. 6. Grangemouth Dock- yard Co. & Hudson & Corbett, Glasgow, Scotland; Hull 391 tons, Engines 52, Boilers 40, and Water 27; Area of Load-line 3850 Gfeet, and of Sails 2310. No. 7. Earle's Co., Hull,Eng. No. 8. The Wm. Cramp & Sons S. and E. B. Co., Phil- adelphia, Penn. No. 9 and 10. Delaware River I. S. B. and E. Co., Chester, AND INLAND STBAMEES. 889 and. Triple Expansion. in feet and ins.; Engines, in ins.; Pressure, in Ibs. ; Revolutions, per minute; Fuel, in Ibs. per Hour ; P Passengers and F Freight. Speed, in Allies per Hour. Santa Rosa. Puritan. Tuscaro- ra. City Racine. John F. Smith. New York. Ata- lanta. Susque- hanna. Robert E.Lee. Mary- lani L 9. 10. 11. 12. 13. 14. 16. 16. 17. lisT Iron. Steel. Steel. Iron. Iron. Iron. Iron. Iron. Wood. Steel. PandF PandF F FandP PandF P Yacht Yacht PandF FandP 342-5 306.7 220 130 3*5 240 166.5 315 326 402 296.7 122 228.5 150 306 332.5 40.6 403-5 289.3 40 203.5 35 122 42 301 40.2 222.9 26.33 164 22 315.8 48.5 316.4 42 23 1 8. i 23 9 ii 15-2 13 9.2 20.4 2 i 2 i I i 2 I i 2 '335 3075 1937 802 I42-39 1092 284 117 l8 9 2 2416 4593 2669 1041 I35-60 '553 568 233 J 479 2419 13-7 '3 16 5-3 6-33 12 9-3 16 3"5 5" 4775 643 357<> 624 218 'o 55 83 1000 235 1042 2 4 6 310 128 4690 650 8-5 7 9-7 3 6 4-75 4-5 8 45 75 2 f 28 14 30 2 Of 40.5 22 86 no 38 61 50 26 75 60 28 4 2 35 56 54 9 and 14 42 36 18 12 30 22 IO 44 90 no 160 no 125 50 160 I 4 8 160 80 2 o 4 90 100 30 128 164 21 85 4 8 3 2 i 3 2 i 9 2 480 850 162 90 230 I 4 6 65 118 152 12 OOO 26000 5574 4000 1258 5360 4534 2180 3360 4656 15000 Jet 624 5700 2226 1470 '5 14 10.5 6.4 8 13.2 24 17-5 16.5 9 16 35 30.16 39 14 12.5 17 200 200 230 IIO 15 50 170 5 200 X75p.rP 1.9 p. H> 2340 1400 4 2-5P- H Natural Natural Natural Natural Natural Blast Blast Blast Natural Natural 600 900 2140 310 H25 3100 165 1200 150 2100 18 300 8 2OO 29 8 50 45 150 *9 3000 7500 1800 750 300 3700 1950 925 1 200 19 21 16 14.5 13 17-45 18 21 12.5 Sch'r Sch'r Sch'r 3-m Sch'r Sch'r Sch'r Perm., and W. and A. Fletcher Co., Hoboken, N. J. - -No. 11. Globe Iron Works, Cleveland, Ohio; Hull 1240 tons, Engines 200, and Boilers 70 tons. No. 12. Chas. F. Elmes, Chicago, 111., and Burger & Burger, Wis. ; Freight and Cabin on deck ; Hull 350 tons, Engines 40, Boilers 36, and Water 23. No. 1?. Pusey & Jones Co., Wilmington, Del. No. 14. W. & A. Fletcher Co., Hoboken, N. J. ; Water-wheel blades, 13 of 45 ins. No. 15. Same builders as No. 8. No. 16. The Harlan & Hollingworth Co., Wilmington, Del. ; Hull 136 tons, Engines 20, and Boilers 25. - -No. 17. Jas. Howard & Co., Jef- fersonville and American Foundry, New Albany, Ind. ; Water - wheel blades, 22 of 35 ins. No. 18. Detroit Dry Dock Co., Detroit, Mich. ; Hull 1250 tons, Engines 140, and Boilers 70. 4 F 890 STEAM-VESSELS. FERRY AND TOWING. IPassenger and. Team, and. Tow-tooats. Single, Compound, and. Triple Expansion. Length and Hull, in feet and tenths ; Draught, Propeller, and Side Wheels, in feet and ins.; Engines, in ins.; Pressure, in Ibs.; Revolutions, per minute ; Surfaces, in sq. feet ; Weights and Displacements, in Tons of 2240 Ibs. ; Fuel, in Ibs. per Hour ; P Passengers and T Teams. Speed in Miles per Hour. DIMENSIONS AND CAPACITIES. Montauk and Whitehall. John G. McCul- lough. Bergen. In- trepid. Maine. Ferry. Inter- nation- al. Meteor. "~7~ Iron. Towiiitf. 8 7 95 66 ! 8.6 Pat- erson* and Mate. Sleel. PandT 222 217 4 62 16.6 Service 1. Iron. PandT 209 I9 I 196 37-4 65 14.1 2. Steel. TandP 215 198.5 198.5 14-5 3. Steel. PandT 203 200 220.4 11 16.6 4. Iron. Towing. 118 no 114 23-5 ii. 6 6. Iron. TandP 189.2 175 174 S^.S 62.5 13-3 6. Iron. Towing. 140 129.6 130 26 16.2 Length on deck " bet. perp'rs. . " tonnage Beam do .... u over guards. . . Hold tonnage Decks Tons I 839 1088 880 215 7-5 50 10 50 32 168 1380 Jet 20.5 8.66 5 Natural IOIO 35oo 4130 47 104 5i 29-5 12 1008 1310 ii 1340 450 7-75 22 50 36 100 1 2O 2 140 2 Of 734 1117 9-5 560 225 6.9 18.5 2 7 42 2 * 4 1 60 162 2 81 3462 2 Of 8 8.91 108.9 217.8 9-5 33 164 4 22 4 2 36 100 9 i 71-5 2503 1105 9-5 i4&i6 545-7 850.3 7.2 678.5 206 5&6. 5 46 120 22 24 I 7 6 2259 Jet 400 2OO 12 llo V6 5 24 41 3 160 2 80 2400 IIOO 9-5 55-6 95-6 8 150 16 32 28 IOO 100 I 45-5 1318 553 14 10.6 75 6 T 9 2 Of 20 2 Of 36 28 125 120 2 9 1 3332 2224 2 Of 8.6 ii Draught load Displacement do ImmersedSec'n at do. Freeboard .... Cylinders. IP Int " L.P Stroke of Piston Steam Pressure Revolutions Boilers Grate surface Heating do Condensing do Propeller, ilium. . . J Pitch Blades Side-wheel diam " width.... Coal weight Natural 453 5200 12 1580 Natural 1007 3448 4330 321 & ,'!? 60 Natural 450 *3 20.5 17.6 n.s 20.5 8.6 40 Natural 650 3420 2896 12 39-75 29.6 12 270 Natural 800 IS 16 420 Natural 280 50 26 21 12 12 Natural 1250 3760 4750 3 8o no 80 40 14. 5 Combustion IIP Team space Passenger do Weight Hull Engine Boilers Water Speed . . Remarks. No. 1. Side-wheel, T. S. Marvel & Co., Newburgh, and Quintard Iron. Works, N. Y. ; Double ends. No. 2. Neafle and Levy, Penn Works, Phila., Pa. ; Propeller at each end. No. 8. Hull same as 1, and Delamater Iron Works, N.Y.; Propeller at each end ; Weights: of Hull as launched; Engines, not including steering and ventilating; donkey pumps, piping and chimney ; augmented surface, 7524 ^feet. Nos. 4. and 5. The Harlan and Hollingsworth Co., Wil- mington, Del., Propeller and Side-wheel. No. 6. Neafie and Levy, Phila.; one Wrecking pump, 16 and 20X18 ins., three 8-inch suctions on each side, capacity 700 tons water per hour; one fire-pump, eight 2.5-inch streams ; Electric search- lights, 6000 candle power, several of 2000 candle arc -lights. No. 7. The Pusey & Jones Co., Wilmington, Del. No. 8. Hull same as Nos. 1 and 3, and engines W. & A. Fletcher Co., Hoboken, N. J. ; Propeller at each end; No. 8 and these * designed bv Col v. A Stevens, Hoboken, N. J. MARINE STEAM VESSELS AND ENGINES. 89! "Wood IPropellers. HERRHSHOFF, R. N., VERTICAL DIRECT ENGINE (Compound). Length on deck, 46 feet; over all, 48 feet ; beam, 9 feet; hold, 5 feet. Displacement at load-line, 7.44 tons. Area of section at load-line, 217.8 sq.feet. Area of wetted surface, 365.5 sq.feet. Coefficient of fineness, .396. Cylinder. 8 and 14 ins. in diam. by 9 ins. stroke of piston. Condenser, External. Surface. Propeller. 4 blades, 3 feet in diam. by 4 feet i inch pitch. Blower, 42 ins. in diam. Boiler (vertical coil). Heating surface, 174 sq. feet. Grates, 12.5 sq: feet. Pressure of Steam, 53 Ibs. per sq. inch. Revolutions, 333 per minute. IIP, 68.4. Speed. 10. 18 knots per hour. With 129 Ibs. and 466 revolutions, 14.26 knots. IIP, 169.5. Weight of Engines, Boiler, and Water, 5300 Ibs. HERRKSHOPP, VERTICAL DIRECT ENGINE (Compound). Length over all, 86 feet; beam, nfeet. Displacement, 27 tons. Cylinder. 13 and 22 ins. in diam. by 12 ins. stroke of piston. Surface Condensing. Pressure, 130 Ibs. per sq. inch. Revolutions, 460 per minute. Speed, 20 knots per hour. IH*, 425. Propeller, 3 blades. Pitch, 5 feet. HERRESHOFF, R. I. N. VERTICAL DIRECT ENGINE (Compound). Length over aW, 60 feet; beam, 7 feet; Jwld, s.^feet. Displacement at load-draught of 32 ins., 7 toni (2240 Ibs.). Cylinders. 8 and 14 ins. in diam. by 9 ins. stroke of piston. Surface condenser. Pressure of Steam. 140 Ibs. per sq. inch, cut off at .5. Revolutions, 6ob per minute. Speed, 19.875 knots per hour. Cable or Rope Towing. "NYITRA." HORIZONTAL DIRECT ENGINES (Condensing). Length of boat, 138 feet; beam, 24. 5 feet ; hold, 7. 5 feet. Immersed section, 74.4 sq.feet. Displacement, 200 tons at load-line of 3. 7$ feet. Immersed section, 263.7 S 2- f eet - Displacement, 949 tons. Tow. 3 barges. Cylinders. 2 of 14.18 ins. in diam. by 23.625 ins. stroke of piston. IIP, net effective, 100. Speed, 7.73 miles per hour. Propellers. Twin, 4 feet 2 ins. in diam. Stress. Cable, 7485 Ibs. Per ton of displacement, 6.5 Ibs. ; per sq. foot of im- mersed section, 22 Ibs. Fuel. Per mile and ton of displacement (1149), -7^ I DS - Towing. "Wood. Side "Wheels. "WM. H. WEBB." HARBOR AND COAST. VERTICAL BEAM ENGINES (Condensing). ^Length upon deck, 185. $ feet ; beam, 30.25 feet ; hold, 10.8 feet. Immersed Section at load-line, 194 sq.feet. Displacement 498.25 tons, at load- draught of 7. 25 feet. Cylinders. 2, of 44 ins. in diam. by 10 feet stroke of piston ; volume, 211 cube feet. Condensers. Jet, 2, volume 105 cube feet. Air-pumps. 2, volume 45 cube feet. Water-wheels. Diam., 30 feet. Blades (divided), 21; breadth of do., 4.6 feet; depth of do., 2.33 feet. Dip at load line, 3.75 feet. Boilers. 2 (return flue). Heating surface, 3280 sq. feet. Grates, 147.5 sq. feet Smoke-pipe. Area, n.6 sq. feet, and 35 feet in height above the grate level Pressure of Steam. 35 Ibs. per sq. inch, cut off at .5 stroke. Revolutions^ 22 pel minute. IIP, 1500. Fuel. Anthracite or Bituminous. Consumption, 1680 Ibs. per hour. Speed. 20 miles per hour. Weights. Engines, Wheels, Frame, and Boilers, 310 579 Ibs. RIVER STEAMBOATS, SIDE AND STERN WHEEL. Wood Side "Wheels. Passenger. " MARY POWELL," HUDSON RIVER. VERTICAL BEAM ENGINE (Condensing). Length on water-line, 286 feet; over all, 294 feet; beam, 34 feet 3 ins. ; over all, 64 feet; hold, gfeet. Deck to promenade deck, lofeet. Immersed section at load line of 6 feet, 200 sq. feet Displacement, 800 tons at mean load-araught of 6 feet. Area of transverse head surface of hull above water, 2000 sq. feet. Cylinder. 72 ins. in diam. by 12 feet stroke of piston; volume, 338 cube feet. Clearance at each end, 12.5 cube feet. Steam and Exhaust Valves, 14.75 ins. in diam. Air-pump, 40 ins. in diam. by 5 feet 2 ins. stroke of piston. Condenser. Jet, 128 cube feet. Crank-pin, 8.75 ins. in diam. x 10.75 ins. Beam, 22.5 feet in length; centre, 9.75 in diam. Water-wheels Diam. 31 feet; blades (divided), 26; breadth of do., 10 feet 6 ins. ; width, i foot 6 ins. ; immersion, 3 feet 6 ins. Shafts. Journal, 15.625 ins. by 17 ins. Boilers. 2 (flue and return tubular), of steel, u feet front by 26 feet in length; shell, 10 feet in diam. and 16 feet i inch in length. Furnaces, 2 in each, of 4 feet 10 ins. by 8 feet in length. Heating Surface, 2660 sq. feet; and Superheating, 340 sq. feet in each. Grates, 152 sq. feet. Flues, 10 in each, transverse area, n feet 7 ins. Tubes, 80 in each, 4.5 ins. in diam., 6 feet 6 ins. in length, and 8 feet 7 ins. in transverse area. Steam Chimneys, 8 feet in diam. x 12 feet in height. Smoke-pipe, 4 feet 6 ins. in diam. and 68 feet in height from grates. Combustion, Blast. Blowers, 4 feet in diam. and 3 feet in width. Revolutions, 78 per minute. Fuel (anthracite), 6280 Ibs. per hour, or 40 Ibs. per sq. foot of grate per hour. Per sq. foot of heating surface, 2. 25 Ibs. Speed, 23.65 miles per hour. Pressure of Steam, 28 Ibs. per sq. inch, cut off at .47 stroke; terminal pressure, 16.4 Ibs. ; throttle, .625 open. Vacuum, 25 ins. Revolutions, 22.75 per minute. Temperatures. Reservoir, 120. Feed water, 120. Chimney, 740. IP. Total, 1900. IIP, 1560. Net, 1450. Evaporation. Water per Ib. of coal, from 120, 7 Ibs.; per Ib. of combustible, from 120, 8.2 Ibs. Steam per total IP per hour, 21.1 Ibs. Coal per do. do., 3. 14 Ibs. Weights. Engine. Frame, keelson, out -board wheel -frames donkey engine, and boiler, blower engines and blowers, all complete, 360000 Ibs. Boilers. Iron return flue, 120000 Ibs. Steel return tubular, 116000 Ibs. Water, 128000 Ibs. Capacity. 2000 passengers and their baggage. Memoranda. This vessel was originally but 266 feet in length, and when length- ened the cylinder of 62 ins. in diam. wag removed and replaced with one of 72 ins. Engine designed throughout for original cylinder and a pressure of from 50 to 55 Ibs., cutting off at .625 of stroke, with throttle wide open. Engines and Boilers built by Fletcher, Harrison, & Co., New York, 1861 and 1875. Iron Stern "Wheels. Passenger and. Freight. HORIZONTAL ENGINES (Non-condensing). Length upon deck, no feet; beam, 14 feet (deck projecting over, 4 feet) ; hold, 3. 5 feet. Immersed section at load-line, 10. 25 sq. feet. Displacement at load-draught of 1. 1 feet, 33 tons. Cylinders. Two, of 10 ins. in diam. by 3 feet stroke of piston; volume of piston space, 1.6 cube feet. Wheel Diam. 13 feet. Blades, 13; breadth of do., 8.5 feet; depth of do., 8 ins. Revolutions, 33 per minute. Boiler. One (horizontal tubular). Tubes, 100 of 2 ins. in diam. Fuel. Bituminous coal. Consumption, 4480 Ibs. in 24 hours. Hull Plates, keel, No. 3; bilges, No. 4 ; bottom, No. 5; sides, Nos. 6 and 7. Frames, 2.5 x .5 ins,, and 20 ins. apart from centres. RIVER STEAMBOATS, STERN WHEELS. OIL LAUNCH. 893 "Wood Stern "Wheels. Passenger and Deck Freight. " MONTANA. "HORIZONTAL ENGINES (Non-condensing). Length upon deck (over att), 248 feet; at water-line, 245 feet; beam, 48 feet 8 ins. (over all, 50 feet 4 ing.); hold, 6 feet; draught of water at load-line, 5. 5 feet. Immersed section at load-line, 244 sq. feet. Displacement at mean light draught of 22 ins., 594 tons (2000 Ibs.) Cylinders. Two, 18 ins. in diam. by 7 feet stroke of piston. Valves, 4.5 and 5 ins. in diam. Piston-rod, 4 ins. Steam-pipe, 4.5 ins. Connect- ing-rod, 30 feet in length. Water-wheel, 19 feet in diam. by 35 feet face; blades, 3 feet in depth. Shaft, 10.25 ins. in diam. Boilers. Four (horizontal tubular), 42 ins. in diam. by 26 feet in length. Two flues in each, 15 ins. in diam. Heating surface, effective, 1023, total 1431 sq. feet Furnace, 6.5 X 17 feet. Grates, 4.16 X 17 feet; surface, 70.8 sq. feet. Smoke-pipes. Two, 3 feet in diam. by 55 feet 3 ins. in height. Exhaust or Blower draught. Calorimeter. Of Bridge, 15.27; of Flues, 9.82; and of Chimneys, 14.14 sq. feet. Areas of grate, compared to calorimeter of flues, 7.2; to ditto, of chimneys, 5; and of bridge, 4.6 sq. feet. Steam-room, 562 ; and water space, 294 cube feet. Hull Frames, 4X6 ins. and 15 ins. apart at centres. Intermediate do., 4 x 6 ins., and running for 7. 5 feet each side of keelson. Planking. Bottom, oak, 4 ins. ; side do., 2.5 to 4 ins. Deck beams, pine, 3 X 6 ins. Deck plank, 2. 5 ins. Keelson, oak; side do., eight each side, one each 7, 8.75, and 9 ins., and five 6.75 ins. Wales, one each side, 9 and 7 ins. by 3, and one 10 X 2.5 ins. Deck posts, 3.5 X 3 ins. and 4 feet apart. Deck beams, 5.5 X 3 ins. Knuckles, oak, 6 X 12 ins. Bulkheads, one longitudinal and one athwartship at shear of stern. Sheathing of wrought iron, .0625 to .125 inch from just below light to load-line. Hog Posts. White pine, 8.5 and n ins. square. Chains, 1.5 ins. in diam. Weights. Boilers, 29 264; water, 18 351 ; and boilers, chimneys, grates, and water, 55672 Ibs. Hull, oak, 520560; Pine, 91 437; Bolts, spikes, etc., 8000, and Deck and guards, 76000 Ibs. ; Hull alone, 310 tons. Weight of hull compared to one of iron as 8 to 5, effecting a difference of about loo tons. "PITTSBURGH." HORIZONTAL ENGINES (Non-condensing). Length on deck, 252 feet; beam, 39 feet; hold, 6 feet; draught of water at load-line, 2 feet. Immersed section at load-line, 75 sq.feet. Displacement at load draught of 2 feet, 380 tons (2000 Ibs.). Cylinders. Two, 21 ins. in diam. by 7 feet stroke of piston. Water-wheel 21 feet in diam. by 28 feet face. Boilers. a (horizontal tubular), 47 ins. in diam. by 28 feet in length. Two fires in each. Oil Engine Hianncn. Bilemeiits of Engine and Dimensions of Hiannoh. Consumption .9 pint ordinary Mineral Oil per IB? per Hour. Typ. H>* Lau Length. nch. Breadth. Weight^ Type. H?* Lau Length. Breadth. Weight^ No. 6 5 4 No. I 2 3 Feet. 16 SI 27 Feet. 4 1 Lb8. 896 1332 1568 No. 3 2 I No. 5 10 IS Feet. 30 40 45 Feet. 7 7 7-5 Lbs. 1848 2688 3136 Developed by Brake. 4F* t Of engine without oil. 894 RIVER STEAMBOATS. SAILING VESSELS. Passenger and. Deck Freight. "PITTSBURGH." HORIZONTAL ENGINES (Non-condensing). Length on deck, 252 feet; beam, 39 feet; hold, 6 feet; draught of water at load-line, 2 feet. Immersed section at load-line, 75 sq.feet. Displacement at load-draught of 2 feet, 380 tons (2000 Ibs.). Cylinders. Two, 21 ins. in diam. by 7 feet stroke of piston. Watcr-vjliccl. 21 feet in diam. by 28 feet face. Boilers. 2 (horizontal tubular), 47 ins. in diam. by 28 feet in length. Two fires in each. Iron. Stern. Wheels. HORIZONTAL ENGINES (Non-condensing). Length upon deck, no feet; beam, 14 feet (deck projecting over, 4 feet) ; hold, 3. 5 feet. Immersed section at load-line, 10.25 sq.feet. Displacement at load-draught o/i.i feet, 33 tons. Cylinders. Two, of 10 ins. in diam. by 3 feet stroke of piston; volume of piston space, 1.6 cube feet Wheel Diam. 13 feet. Blades, 13; breadth of do., 8.5 feet; depth of do., 8 ins. Revolutions, 33 per minute. Boiler. One (horizontal tubular). Tubes, 100 of 2 ins. in diam. Fuel Bituminous coal. Consumption, 4480 Ibs. in 24 hours. Hull Plates, keel, No. 3; bilges, No. 4; bottom, No. 5; sides, Nos. 6 and 7. Frames, 2.5 X .5 ins., and 20 ins. apart from centres. Steel. " CHATTAHOOCHEE. "INCLINED ENGINES (Non- condensing). Length on deck, 157 feet; beam, 31. 5 feet; hold, 5 feet. Immersed section at load-line, 153 sq.feet. Freight capacity, 400 tons (2000 Ibs.). Cylinders. Two, 15 ins. in diam. by 5 feet stroke; volume of piston space, 12.26 cube feet. Wheel One, 18 feet in diam. ; blades, 2 feet in depth. Boilers. Three (cylindrical flued). Diam. 42 ins. ; length, 22 feet; 2 flues of 10 ins. in each. Heating surface, 690 sq. feet. Grates, 48 sq. feet. Pressure of Steam, 160 Ibs. per sq. inch, cut off at . 375. Revolutions, 22 per min. Consumption of Fuel, 12 tons (2000 Ibs.) in 24 hours. Plating of Hull, .1875 to .25 inch. Light draught, 21 ins. Iron. Propellers. VERTICAL DIRECT ENGINES (Non-condensing). Length on deck, jofeet; beam, 10.5 feet; draught, 12 ins. Propellers, 2. 2 blades, 16. ins. in diam., set u ins. below water-Hne. Boiler (tubular coil). Revolutions, 480 per minute. Speed, 10. 49 miles per hour. Water led to propellers through tunnels in bottom at sides. " LOUISE. "VERTICAL TANDEM ENGINES (Compound). Length, 60 feet; beam, 12 feet; hold, 4.25 feet. Displacement at load-draught of 2.5 feet, 8 tons. Cylinders, 5 and 10 ins. in diam. by 8 ins. stroke of piston. Surface Condenser. Boiler (vertical tubular), 4 feet in diam. by 8.5 in length. Iron Sailing Vessels. Passenger and Freight. ENGLISH. SHIP. Length upon deck, ijSfeet ; do. at mean load-line ofig. i6feet, 177 feet; keel, 171 feet; beam, ^2.8Bfeet; depth of hold, 21.75/66*; keel (mean), 2-75/eefc Immersed section at load-line, 387 sq.feet. Displacement at load-draught of 19.16 feet, 1385 tons; at deep load-draught of 20 feet, 1495 tons; and, in proportion to its circumscribing parallelopipedon, . 524. Load-line. Area at load-draught, 4557 sq. feet. Angle of entrance, 57; of clear- ance, 64. Area in proportion to its circumscribing parallelogram, .784. YACHTS. CUTTERS. PILOT BOAT. 895 Centre of Gravity, 6.416 feet below mean load-line. Centre of Displacement (grav- ity of), 6.25 feet below load-line; and 4.33 feet before middle of length of load line. Immersed Surface. Bottom, 7370 sq. feet. Keel, 1130 sq. feet. Sails, 13 282 sq. feet. Mcta-centre, 6.66 feet above centre of gravity of displacement. Centre of Effort before centre of displacement, 3.5 feet; height of do. above mean load-line, 55.5 feet. Laimcli. "Wood.. STEAM LAUNCH " HERRESHOFF. " VERTICAL ENGINE (Compound). Length, 33 feet i inch ; beam, %.j$feet. Displacement at mean load-draught o/(to rabbet of keel) 19 ins., 8929 Ibs. Weights. Hull and Machinery, 6555 Ibs. Coal, 1120 Ibs. Yachts. Wood. "AMERICA," SCHOONER. Length over all, oB feet; upon deck, 94 feet; at load-line, 90. 5 feet ; beam, 22. 5 feet ; at load-line, 22 feet ; depth of hold, 9. 25 feet. Height ait tide from under side ofgarboard stroke, 1 1 feet. Sheer, forward, 3 feet ; aft, i. 5 feet, Immersed section at load-line, 121.8 sq.feet. Displacement at load-draught 0/8.5 feet, from under side ofgarboard stroke and of u feet aft, 191 tons; and, in pro- portion to Volume of circumscribing parallelopipedon, .375. Displacement at 4 feet (from garboard stroke), 43 tons ; at 5 feet, 66 tons ; at 6 ftet, 93 tons ; at 7 feet, 127 tons ; and at 8 feet, 167 tons. Centre of Gravity. Longitudinally, 1.75 feet aft of centre of length upon load- line. Sectional, 2.58 feet below load line. Of Fore body, 14.25 feet forward; and of After body, 19 feet aft. Meta-centre, 6. 72 feet above centre of gravity. Centre of Effort, 31 17 feet from load-line. Centre of Lateral Resistance, 6.33 feet abaft of centre of gravity. Area of Load-line, 1280 sq. feet. Mean girths of im- mersed section to load-line, 25 feet. Load-draught. Forward, 4.91 feet; aft, u 5 feet. Rake of Stem, 17 feet Spars. Mainmast, 81 feet in length by 22 ins. in diam. Foremast, 70.5 feet in length by 24 ins. in diam. Main boom, 58 feet in length. Main gaff, 28 feet. Fore gaff, 24 feet. Rake, 2.7 ins. per foot. Drag of Keel, 3 feet Tons, 170.56. " JULIA," SLOOP. Length for tonnage, 72.25 feet; on water-line, 70 feet ^ inf.; beam, 19 feet 8 ins.; hold, 6 feet 8 ins. Tons, O. M. 83.4; N. M. 43.98. Load-draught, 6.25 feet. Sails. Mainsail, hoist, 49.75 feet, foot 54.25, and gaff 27.66; Jib, hoist, 49.75 feet, foot 39. 5, and stay 63. 5. Gaff topsail, hoist. 24. 5 feet. Areas. Mainsail, 2322 sq. feet. Jib, 986, and Topsail, 454. - Cntters. "TAR A" (English) SLOOP. Length on load-line, 66 feet; beam, 11.5 feet. Immersed section at load-line, 11.5 sq.feet. Displacement, 75 tons. Spars. Mast, deck to hounds, 42 feet. Boom, 58 feet. Gaff, 39 feet. Bowsprit outside of stem, 30 feet. Mast to stem, 26 feet. Topmast, foot to hounds, 25 feet Balloon topsail yard, 46 feet. Canvas, area, 3450 sq. feet. Tons, C. H., 9. Ballast At Keel, 38.5 tons. Hull, 1.5 tons. "MISCHIEF" (English), SLOOP. Length on load-line, 6ifeet; beam, ig.gfeet Immersed section at load-line, 60 sq.feet. Displacement, 55 tons. Pilot Boat. "WM. H. ASPINWALL," SCHOONER. Length of keel, 74 feet; upon beam, 19 feet; hold, 7. 6 feet. Draught of water, 6 feet forward ; aft, 9.5 feet. Keel, 22 ins. in depth. False keel, 12 ins. in depth at centre. Spars. Mainmast, 77 feet in length. Foremast, 76 feet Main doom, 46 feet Main gaff, 21 feet. Fore gaff, 20 feet Tons. N. M., 46.32. 896 PASSAGES OF STEAMBOATS. ICE-BOATS. PASSAGES OF STEAMBOATS. Distances in Statute Miles. 1807, Clermont, of N. Y., New York to Albany, 145 miles, in 32 hours = 4. 53 miles per hour, neglecting effect of the tide. 1811, New Orleans, of Pittsburgh, Penn. (non-condensing and stern- wheel), Pitts- burgh to Louisville, Ky., 650 miles, in 2 days 22 hours. 1849, Alida, of N. Y., Caldwell's, N. Y., to Pier i, North River, 43.25 miles, in i hour 42 min. , ebb tide = 2. 75 miles per hour. Speed = 22. 19 miles per hour. 1860, 3oth Street, N. Y., to Cozzens's Pier, West Point, 50.5 miles, in 2 hours 4 min., and to Poughkeepsie, 74.25 miles, in 3 hours 27 mm., 5 landings, flood tide. And 1853, Robinson Street to Kingston Light, 90.375 miles, in 4 hours, making 6 landings, flood tide. 1850, Buckeye State, of Pittsburgh, Penn. (non - condensing), Cincinnati to Pitts- burgh, 500 miles (200 passengers), 53 landings, in i day 19 hours; 4 miles per hour adverse" current. Speed = 15.63 miles and 1.23 landings per hour. Average depth of water in channel 7 feet. 1852, Reindeer, of N. Y., New York to Hudson, 116.5 miles, in 4 hours 57 min., making 5 landings. Flood tide. 1853, Shotwell, of Louisville, Ky. (non-condensing), New Orleans to Louisville, 1450 miles, 8 landings, in 4 days 9 hours; 4.5 to 5.5 miles per hour adverse cur- rent. Speed = 18.81 miles per hour. NOTE. In 1817-18 the average duration of a passage from New Orleans to Louisvill- was 27 days, 12 hours; the shortest, 25 days. 1855, New Princess, of New Orleans (non-condensing), New Orleans, La., to Natchez, Miss., 310 miles, in 17 hours 30 min.; 3.5 to 4 miles per hour adverse current. Speed = 20.98 miles per hour. 1864, Daniel Drew, of N. Y, Jay Street, N. Y., to Albany, 148 miles, in 6 hours 51 min. , 9 landings. Flood tide. Speed of boat = 22.6 miles per hour. 1867, Mary Powell, of N. Y., Desbrosses Street, N. Y., to Newburgh, 60.5 miles, in 2 hours 50 min., 3 landings; from Poughkeepsie to Rondout Light, 15.375 miles, in 39 mm., flood tide. 1873, Milton to Poughkeepsie, light draught and flood tide, 4 miles, in 9 min.; and 1874, Desbrosses Street to Piermont, 24 miles, in i hour ; to Caldwell's, 43.25 miles, in i hour 50 min. Speeds 22.77 to 23 miles per hour. Runs from New York to Albany, 146 miles, by different Boats. 1826, Sun 12 hours 16 min. 1826, North America*. 10 u 20 " 1841, Troy t 8 u 10 " 1841, South America J. 7 * 28 " 1852, Fr. Skiddy 6 hours 24 min. 1860, Armenia II 7 " 22 " 1864, Daniel Drew$.... 6 " 51 " 1864, CWncey Vibbardl. 6 " 42 ' * 7 landings. f 4 landings. $ 9 landings. 6 landings. || u landings. Timing Distance. From i4th St., Hudson River, N. Y., to College at Mount St. Vincent, 13 mile*. NOTE. Where landings have been made, and the river crossed, the distance between the pointa given is correspondingly increased. 1870, R. E. Lee, of St. Louis, (non-condensing), New Orleans to St. Louis, Mo., 1180 miles (without passengers or freight), 4 to 5 miles per hour adverse current; Vicks- burg, i day 38 mm.; Memphis, 2 days 6 hours g min.; Cairo, 3 days i hour.; and to St. Louis, 3 days 18 hours 14 mm., inclusive of all stoppages. 1870, Natchez, of Cincinnati, Ohio, from New Orleans to Baton Rouge, 120 miles, in 7 hours 40 min. 42 sec. Runs from New Orleans to Natchez, 295 miles, by different Boats. 1814, Orleans, 6 days 6 hours 40 mm. I 1856, New Princess, 17 hours 30 min. 1840, Edward Shippen, idayS hours. \ 1870, R. E. Lee, 16 hours 36 min. 47 sec. Ice-"boats. Distances in Statute Miles. 1872, Haze, of Poughkeepsie, N. Y., to buoy off Milton, 4 miles, in 4 min. 1872, Whiz, of Poughkeepsie, N. Y., to New Hamburg, 8.375 miles, in 8 min. PASSAGES OF STEAMERS AND SAILING VESSELS. 897 PASSAGES OF STEAMERS AND SAILING VESSELS. Distances in Geographical Miles or Knots. Steamers. Side-wheels. 1807, Phoenix, of Hoboken, N. J. (John Stevens), New York, N.Y., to Philadelphia, Penu. First passage of a steam vessel at sea. 1814, Morning Star, of Eng., River Clyde to London, Eng. First passage of an English steamer at sea. 1817, Caledonia, of Eng., Margate, Eng., to Cassel, Germ., 180 miles, in 24 hours. 1819, Savannah, of N.Y., about 340 tons 0. M., Tybee Light, Savannah River, Ga.. to Rock Light, Liverpool, Eng., 3640 miles, in 25 days 14 hours; 6 days 21 hours or which were under steam. 1825, Enterprise, of Eng., 500 tons, Falmouth, Eng., to Table Bay, Africa, in 57 days; and to Calcutta, India, in 113 days. First passage of a steamer to India. 1830, Hugh Lindsay, 411 tons, 80 BP, Bombay, India, to Suez, Egypt, 3103 miles, in 31 days running time. 1837, Atlanta, of Eng., 650 tons, Falmouth, Eng., to Calcutta, in 91 days. 1839, Great Western, of Eng., Liverpool to New York, N. Y., 3017 miles, in 12 days 1 8 hours. 1870, Scotia, of Eng., Queenstown, Ireland, to Sandy Hook, N. J., 2780 miles, in 8 days 7 hours 31 min. 1866, New York to Queenstown, 2798 miles, in 8 days 2 hours 48 min.; thence to Liverpool, Eng., 270 miles, in 14 hours 59 min.; total, 8 days 17 hours 47 min. Screw. 1874, India Government Boat, Steel, length 87 feet, beam 12 feet, draught of water 3.75 feet, mean speed for one mile 20.77 miles per hour, and maintained a speed, of 18.92 miles in i hour. 1877, Lusitania, of Eng., London to Melbourne, Australia, via Cape, u 445 miles, in 38 days 23 hours 40 min. Sailing "Vessels. 1851, Chrysolite (clipper ship), of Eng., Liverpool, Eng., to Anjer, Java, 13000 miles, in 88 days. The Oriental, of N. Y., ran the same course in 89 days. 1853, Trade Wind (clipper ship), of N. Y., San Francisco, CaL, to New York, N. Y., 13610 miles, in 75 days. 1854, Lightning (clipper ship), of Boston, Mass., Melbourne, Australia, to Liver- pool, Eng., 12 190 miles, in 64 days. 1854, Comet (clipper ship), of N. Y., Liverpool, Eng., to Hong Kong, China, 13040 miles, in 84 days. 1854, Sierra Nevada (schooner), of N. H., Hong Kong, China, to San Francisco, Cal., 6000 miles, in 34 days. 1854, Red Jacket (clipper ship), of N. Y., Sandy Hook, N. J., to Melbourne, Aus- tralia, 12 720 miles, in 69 days n hours i min. 1855, Euterpe (half-clipper ship) of Rockland, Me., New York to Calcutta, India, 12 500 miles, in 78 days. 1860, Andrew Jackson (clipper ship), of Boston, New York, N. Y., to San Fran- cisco, Cal., 13610 miles, in 80 days 4 hours. 1865, Dreadnought (clipper ship), of Boston, Honolulu, Sandwich Islands, to New Bedford, Mass., 13470 miles, in 82 days; and 1859, Sandy Hook, N. J., to Rock Light, Liverpool, Eng., 3000 miles, in 13 days 8 hours. 1865, Sovereign of the Seas (medium ship), of Boston, Mass., in 22 days sailed 5391 miles = 245 miles per day. For 4 days sailed 341.78 miles per day, and for i day 375 miles. 1866, Henrietta (schooner yacht), of N. Y., Sandy Hook, N. J., to the Needles, Eng., 3053 miles, in 13 days 21 hours 55 min. 16 sec. 1866, Ariel and Serica (clipper ships), of England, Foo-chou foo Bar, China, to the Downs, Eng., 13 500 miles, in 98 days. 1869, Sappho (schooner yacht), of N. Y., Light-ship off Sandy Hook, N. J., to Queenstown, Ireland, 2857 miles, in 12 days 9 hours 34 min. ELEMENTS OF MACHINES AND ENGINES. ELEMENTS OF MACHINES AND ENGINES. BLOWING ENGINES. Furnaces. Two. Fineries. Two. {England.) 240 Tons Forge Pig Iron per Week. Engine (non-condensing). Cylinder, 20 ins. in diam. by 8 feet stroke of piston. Boilers. Six (plain cylindrical), 36 ins. in diam. and 28 feet* in length. Grates, 100 sq. feet. Blowing Cylinders. Two, 62 ins. in diam. by 8 feet stroke of piston. Pressure, 2.17 Ibs. per sq. inch. Revolutions, 22 per minute. Pipes, 3 feet in diam.= 168 area of cylinder. Tuyere*. Each Furnace, 2 of 3 ins. in diam. ; i of 3.25 ins. ; and i, 3 of 3 ins. Each Finery, 6 of 1.33 ins. ; and i, 4 of 1.125 in s. Temperature of Blast, 600. Ore, 40 to 45 per cent, of iron. Furnaces. Eight, diam. 16 to iSfeet. Dowlais Iron Works (England). 1300 Tons Forge Iron per Week; discharging 44000 Cube Feet of Air per Minute. Engine (non-condensing). Cylinder, 55 ins. in diam. by 13 feet stroke of piston. Pressure of Steam. 60 Ibs. per sq. inch, cut off at . 33 the stroke of piston. Valves, 120 ins. in area. Boilers. Eight (cylindrical flued, internal furnace), 7 feet in diam. and 42 feet in length ; one flue 4 feet in diam. Grates, 288 sq. feet. Fly Wheel. Diam., 22 feet; weight, 25 tons. Blowing Cylinder, 144 ins. in diam. by 12 feet stroke of piston. Revolutions, 20 per minute. Blast, 3.25 Ibs. per sq. inch. Discharge pipe, diam. 5 feet, and 420 feet in length. Valves. Exhaust, 56 sq. feet; Delivery, 16 sq. feet. Furnaces. LacJceriby (England). 800 Tons Iron per Week. Engine (horizontal, compound condensing). 32 and 60 ins. in diam. by 4.5 feet stroke of piston. Blowing Cylinders. Two, 80 ins. in diam. by 4.5 feet stroke of piston. Pressure, 4.5 Ibs. per sq. inch. Revolutions, 24 per minute. Pipe, 30 ins. in diam. ; volume, 12.25 times that of blowing cylinders. IP. Engine, 290 Ibs. ; Blowing cylinders, 258; efficiency, 89 per cent. Valves. Area of admission, . 16 of area of piston ; of exit, . 125. Volume. 190000 cube feet of air are supplied per ton of iron. Blcrwer and. Exhausting ITan. The Huyett & Smith Manufacturing Co., Detroit, Mich. Blower. Grate Surface. Outlet. Diam. Pulleyi. Face Pulleys. Revolu- tions at 3 oz. Air at 302. H?at 3oz. Revolu- tions at 6 oz. Air at 6 oz. IP at 6 oz. No. Sq. Ft. Sq. Ins. Ina. Ins. Per Min. Cube Ft. No. Per Min. Cube Ft. No. s 4 10 15 3 2 3 I i-5 5500 3500 930 1870 .38 .76 7500 5000 1330 2670 I.I 2.16 } 14 50 4 2-5 2700 3 120 1.27 4000 4440 3.63 4 18 75 5 3-25 2OOO 4680 I.QI 3000 6670 5-5 5 26 "5 6 4-25 1500 7830 3-2 2300 II 100 9.1 6 36 '75 7 5-25 1300 10900 4-47 1800 15600 12.8 7 56 280 9 6.25 1000 1750 7-iS 1400 24900 20.4 * 40 feet would have afforded economy in fael. ELEMENTS OF MACHINES AND ENGINES. 899 COTTON FACTORIES. (English.) For driving 22060 Hand-mule Spindles, with Preparation, and 260 Looms, with common Sizing. Engine (condensing). Cinder, 37 ins. in diam. by 7 feet stroke of piston; volume of piston space, 53.6 cube feet. Pressure of Steam. (Indicated average) 16.73 Ibs. per sq. inch. Revolutions, 17 per minute. Friction of Engine and Shafting. (Indicated) 4.75 Ibs. per sq. inch of piston. IIP, 125. Total power = i. Available, deducting friction = . 717. {305 hand mule spindles, vrith preparation, S*3SSf? or 10.5 looms, with common sizing. Including preparation : i throstle spindle = 3 hand-mule, or 2.25 self-acting spindlea x self-acting spindle = 1.2 hand-mule spindles. DREDGING MACHINES. Dredging 20 Feet from Water-line, or 180 Tons of Mud or Silt per Hour ii Feet from Water-line. Length upon deck, 123 feet ; beam, 26 feet. Breadth over all, 41 feet. Immersed section at load-line, 60 sq. feet. Displacement, 141 tons, at load-draught of '2-83 feet. Engine (non -condensing). Cylinders, two, 12.125 ms - in diam. by 4 feet stroke of piston. Boilers. Two (cylindrical flue), diam. 40.5 ins., and length, 20 feet 3 ins.; two flues, 14.625 ins. in diam. Heating surface, 617 sq. feet. Grates, 37 sq. feet. Pressure of Steam, 25 Ibs. per sq. inch; throttle .25 open, cut off at .5 the stroke of piston. Revolutions, 42 per minute. Buckets. Two sets of 12, 2.5 feet in length by 15 ins. at top and 2 feet deep; vol- ume, 6.25 cube feet. Chain Links, 8 ins. in length by .5 inch diam. Scows or Camels. Four, of 40 tons capacity each. STEAM HOPPER DREDGER. (Wm. Simons $ Co.) "NEPTUNE" (English). Length, 1 50 feet; breadth, 32 feet. Dredge 'from 6 Ins. to 25 Feet. Capacity of Hopper, 500 to 600 Tons. Engines. Two (compound), 375 IP, for dredging and propulsion, and one for raising bucket-frame and anchor-posts. A like designed dredger ot 1000 tons' capacity has dredged 25 ooo tons silt per week and transported it 4 miles. Dredging 1000 Tons of Mud or Silt per Hour, 5 to 35 Feet in Depth. Capacity of Hopper, 1000 Ton*. Engines. Two (compound), IP 1000. Speed. 9 knots per hoar. Steam IDredging Crane, (English.) Lift, 30 Feet per Hour. 21 280 24640 Volu of Buck LbB. II2O 1680 Tons. 25 37-5 and d. 32 III w o C.YdB. Weight f Crane Lbs. 18000 33480 Tons. 5 7 lume of cket. Lbs. 2240 3360 Tons. 50 60 an d. 30 40 9OO ELEMENTS OF MACHINES AND ENGINES. Electric Lanncli. Steel. "HILDA," "MARY," "FLO," and "THEO." Length, 40 feet; Beam, 6.5; Hold, 3. i. Load-draught, 40 passengers, 1.66 feet. Motor, B? 3.5. Revolutions, 700 psi minute. Speed, 6 miles per hour. Accumulators, under the seats, and when fully charged, capacity for 8 hours at full speed. Charging is effected at landings at termination of route. Builders. J. B. Seath & Co., Glasgow, Scotland. HOPPER DREDGER "BELFAST No. 3." IRON AND STEEL. Length over all, 190 feet,, on deck, 189; between perpendiculars and for tonnage, 185; Beam, -$S.$feet; Hold,, 14.1 feet; Tonnage, Gross, 760 tons; Net, 372; Mean draught, g. 5 feet, loaded, 12.5. Displacement, 1860 tons. Immersed Section, 490 dfeet. Freeboard, 2.75 feet. Dredging Capacity, 1000 tons per hour. Cylinders. Two of 20 ins. in diam. and two of 38. 5 ins. Stroke of piston 30 ins. Pressure of Steam, go Ibs. per ninch. Revolutions per minute, 80. IP 850. Boilers, two. Grate surface, 81 nfeet. Heating surface 2120, and Condensing 1150. Propeller, 9 feet in diameter. Fuel, capacity 50 tons. Crew, 13. Weight, Hull, 500 tons. Speed, 8. 5 knots per hour. Builders. Wm. Simons & Co., Renfrew, Scotland. "HERCULES," Panama Canal Length on deck, 100 feet ; beams, 40, 60, and 45 feet; depth of hold, 12 feet. Slot, 36 feet in length by 6 feet 7 ins. in width. Ways. Two, one 40 feet and one 60 feet, by 5 feet in width. Buckets. 38; volume, 1.33 cube yards. Spuds, 2 feet in diam. and 60 in length. Engines. Two of 100 BE* each, and two of 40 B? each. Boilers. Three (horizontal tubular), 16 feet in length. Elevator and Discharge. Maximum, 24 cube yards per minute. Crane. CWood.) Hull Length on deck, loofeet; beam, wfeet; load-draught, 4. 5 feet. Radius of crane, 46 feet; height, 70 feet; counter-balance, 70 tons. Boiler. Heating surface, 500 sq. feet. Pressure of Steam, 80 Ibs. per sq. inch, IB?, 150. Propellers. Two, 4.25 feet in diam. Speed, 5 miles per hour. Engine to operate crane. Cylinder. 10 ins. in diam. by 12 ins. stroke of piston. FLOUR MILLS. 30 Barrels of Flour per Hour. Water- wheels, Overshot. 5, diam. 18 feet by 14.5 feet face. Buckets, 15 ins. in depth. Water. Head, 2.5 feet. Opening, 2.5 ins. by 14 feet in length ovei each wheel. 5 Barrels of Flour per Hour, and Elevating 400 Bushels of Grain 36 Feet. Water- wheel, Overshot. Diam. 22 feet by 8 feet face. Buckets, 52 of i foot in depth. Water. Head, from centre of opening, 25 ins. Opening, 1.75 ins. by 80 ins. in length. Revolutions, 3.5 per minute. Stones, three of 4.5 feet; revolutions, 130. Three Run of Stones, Diameter 4 Feet. Water-wheel, Overshot. Diam. 19 feet by 8 feet face. Buckets, 14 ins. ifi depth. Or, Steam-engine (non-condensing). Cylinder, 13 ins. in diam. by 4 feet stroke, Boiler (cylindrical flued). Diam. 5 feet by 30 in length; two flues 20 ins. in diain. ELEMENTS OP MACHINES AND ENGINES. 901 HOISTING ENGINES. ITor 3Pile Driving, Hoisting, Mining, etc. Uidgerwood ]Vtanuf g Co., Ne\v York. SINGLE CYLINDERS. DOUBLE CYLINDERS. H Cylinder. Capacity. Cost, with Boiler.* H> Cylinder. Capacity. Cost, with Boiler.* H7T Ins. Lbs. $ No. Ins. Lbs. $ 4 5X5 IOOO 600 8 5X8 2OOO 950 6 6X8 I25C > * 67i > 12 6X8 2500 I 050 10 7X 10 1800 825 2O 7 X 10 3500 1350 15 8 X 10 2800 1050 30 8 X 10 6000 550 20 9 X 12 4000 1275 40 9X12 8000 2000 25 10 X 12 5000 '375 50 10 X 12 9000 2350 * Complete. Details and. Operation. Boiler. Leaders. Lift. Blows Piles Fuel Engine. Drum. Dimen- sions. Tubes. Ram. Hoist. Ram. per Minute. per 10 Hours. H?ir. H> Ins. Ins. No. Lbs. Feet. Feet. No. No. Lbs. 10* 20 12 X 24 14 X 26 32X75 40X84 48 of 2 in. 80 of 2 in. 1953 2700 40 75 8 to 12 8 to 12 25 29 50 100 * Weight complete, 8500 Ibs. Mining Engines and Boilers. ( Various Capacities. ) Engine, Boiler, etc., as given for Pile Driving, page 902. Operation. 250 to 300 tons of coal in 10 hours. Fuel, 40 IDS. coal per hour. Water, 20 gallons per hour. Weight of Engine and Boiler, 4500 Ibs. The Hancock Inspirator. For a Lift of Water oj 25 Feet No. Diair Steam-pipe. eter. Suction. Discharge at Pressure of 60 Lbs. No. Diam Steam-pipe. eter. Suction. Discharge at Pressure of 60 Lbs. Ins. Ins. G'lls.perh'r. Ins. Ins. G'lls.perh'r. 10 375 5 120 30 1.25 i-5 1260 12.5 5 75 220 35 1.25 '5 1740 15 5 75 300 40 1-5 2 2230 20 75 i 540 45 i'5 2 2820 25 i 1.25 900 50 2 2-5 3480 Temperature of feed water at 20 feet lift, 100 ; and on 3 feet lift, 145. HYDROSTATIC PRESS. (Cotton.) 30 Bales of Cotton per Hour. Engine (non-condensing). Cylinder, 10 ins. in diam. by 3 feet stroke of piston. Pressure of Steam, 50 Ibs. per sq. inch, full stroke. Revolutions, 45 to 60 per minute. Presses. Two, with J2 -inch rams; stroke, 4.5 feet. Pumps. Two, diam. 2 ins. ; stroke, 6 ins. For 83 Bales per Hour. Engine (non-condensing). Cylinder, 14 ins. in diam. by 4 feet stroke of piston. Boilers. Three (plain cylindrical). 30 ins. in diam. and 26 feet in length. Gratet, 32 sq. feet. Pressure of Steam, 40 Ibs. per sq. inch. Revolutions, 60 per minute. Presses. Four, geared 6 to i, with two screws, each of 7.5 ins. in diam. by 1.625 to pitch. Shaft (wrought iron).- Journal, 8.5 ins. fly Wheel, 16 feet in diam. : weight, 8960 Ibs. QO2 ELEMENTS OF MACHINES AND ENGINES. LOCOMOTIVE. "EXPERIMENT" (Compound)* Cylinders, one each, 12 and 26 ing. in diam., and lie 26 ins. by 2 feet stroke of piston. Boiler. Heating surface, 1083.5 sq. feet. Grate, 17.1 sq. feet. Pressure of Steam, 150 Ibs. per sq. inch, cut off at .35. Speed, 50 miles per hour. Weight. Empty, 34.75 tons. Street Railroad or Tramway Engine. Cylinder, 7 ins. in diam. by n ins. stroke of piston. Boiler, 78 tubes 1.75 ins. in diam. by 4 feet in length. Heating surface, 160 sq. feet. Grate, 4.25 sq. feet. Wheels, 2.33 feet in diam. Base, 4.5 feet. Gauge, 4 feet 8.5 ins. Cost. Average per mile in England, 2.52 pence sterling = 4. 48 cents. PILE-DRIVING. Driving One Pile. Engine (non-condensing) Cylinder, 6 ins. in diam. by i foot stroke of piston. Boiler (vertical tubular). 32 ins. in diam., and 6.166 feet in height. Grates, 3.7 sq. feet. Furnace, 20 ins. in height Tubes, 35, 2 ins. in diam., 4.5 feet in length. Revolutions, 150 per minute. Drum, 12 ins. in diam., geared 4 to i. Leader, 40 feet in height. Ram. 2000 Ibs., 2 blows per minute. Fuel, 30 Ibs. coal per hour. Driving Two Piles. Engine (non-condensing). Cylinders, two, 6 ins. in diam. by 18 ins. stroke of piston. Boiler (horizontal tubular). Shell, diam. 3 feet, and 6 feet in length. Furnace end 3.75 feet in width, 3.5 feet in length, and 6 feet in height. Pressure of Steam, 60 Ibs. per sq. inch. Revolutions, 60 to 80 per minute. Frame, 8.5 feet in width by 26 feet in length. Leaders, 3 feet in width by 24 feet in height. Rams. Two, 1000 Ibs. each, 5 blows per minute. PUMPING ENGINES. CORLISS STEAM-ENGINE Co., Providence, R. 7. VERTICAL -BEAM ENGINE (Com- pound). Cylinders. 1 8 and 36 ins. in diam. by 6 feet stroke of piston. Pumps. Four plunger, 19 ins. in diam. hy 3 feet stroke of piston. Displacement per revolution of engine, 84.96 cube feet. Boilers. Three, vertical fire tubular. Grate. 93 sq. feet. Heating surface, 1680 sq. feet. Pressure of Steam, 125 Ibs. per sq. inch, cut off at .22 feet. Revolutions, 36 per minute. IIP 313. Fly-wheel. 25 feet in diam., weight 62000 Ibs. Fuel. Cumberland coal, 486 Ibs. per hour, inclusive of kindling and raising steam. Ash and Clinkers, 9.4 per cent. Duty for one week, 113 271 ooofoot-lbs. Water delivered, 17 621 gallons per minute, against head of 180 feet. Duty, average for 1883, per 100 Ibs. anthracite coal, 106 048 ooofoot-lbs. For Elevating 200000 Gallons of Water per Hour. LYNN, Mass. ENGINE (Compound). Cylinders, 17.5 and 36 ins. in diam. by 7 feet stroke of piston; volume of piston space, 61.2 cube feet. Air Pump (double act- ing), 11.25 ins - in diam. by 49.5 ins. stroke of piston. Pump Plunger, 18.5 ins. in diam. by 7 feet stroke. Boilers. Two (return flued), horizontal tubular; diam. of shell, 5 feet; drum, 3 feet ; tubes, 3 ins. Length of shell, 16 feet. Grates, 27.5 sq. feet. Pressure of Steam, 90.5 Ibs. ; average in high-pressure cylinder, 86 Ibs., cut off at i foot, or to an average of 44.5 Ibs. ; average in low-pressure cylinder, 27 Ibs., cut off at 6 ins. , or to an average of 10. 8 Ibs. Revolutions, 18.3 per minute. Fly Wheel. Weight, 24000 Ibs. Evaporation of Water, 4644 Ibs. per hour. Loss of action by Pump, 4 per cent. Consumption of Coal. Lackawanna, 291 Ibs. per hour. Duty, 205772 gallons of water per hour, under a load and frictional resistance of 73.41 Ibs. per square inch, equal to 103923217 foot-lbs. for each 100 Ibs. of coal. ELEMENTS OF MACHINES, MILLS, ETC. 903 u Ga8lcitt"at Saratoga, N.T. Engine (Horizontal Compound). Cylinders. High pressure, 2 of 21 ins. diam. Low pressure, 2 of 42 ins. uiani., all 3 feet stroke of piston Pumps. Two of 20 ins. diam. by 3 feet stroke of piston. Fly Wheel, 12.33 f 66 * 1 ' n diam. ; weight, 1-2000 Ibs. Boilers (horizontal tubular). Two of 5.5 feet in diam. by 18 feet in length. Heat- ing surface, 2957 sq. feet. Grates, 51 sq. feet of grate; to heating surface, i to 58, and to transverse section of tubes, i to 7. Chimneys, 75 feet. Pressure of Steam. Mean of 20 hours, 74.25 Ibs. per sq. inch. Revolutions, 17.87 per minute. IIP. High : pressure cylinders, 109.2; low-pressure, 76 65. Total, 185.8. Fuel Anthracite, 6.9 Ibs. per sq. foot of grate per hour. Evaporation, per sq. foot of heating surface per hour, 1.175 Ibs. ; per Ib. of coal, 9.25 Ibs. ; per cent, of non-combustible, 3.2. Duty, 112 899993 foot-lbs. per 100 Ibs. coal. Heating surface per IIP, 14.9. Steam per sq. foot of surface per hour, 1. 19 Ibs. ; per sq. foot of surface per Ib. of coal per hour from 212, 11.28 Ibs. Ericsson's Caloric. For an Elevation of 50 Feet. Dimen- Space occupied. Volume Pipes, Suction and Fuel per Hour. Furnace. C U S T Deep Well Pump. Extra. sions. Floor. (Height. Sur. Dis- charge. Nut Anthr. Gas. Gas. Coal. Pump. Pipes P Plain. r Foot. Galvan. Ins. Ins. Ins. Gall. Ins. Lbs. Cub. ft. $ * $ . $ T~ 5 34Xi8 48 ISO 75 15 150 6 39X20 200 75 2-5 18 200 210 8 48X21 63 350 i 25 235 250 10 .64 .86 12 54X27 6 3 800 1.5 6 320 J5 .80 IeI 5 12* 42X52 65 1600 2 12 450 .92 1.25 * Over 90 feet, 92 cents. t Duplex. Including engine and pump, oil-can and wrench, complete in all but suction and discharge-pipe. SUGAR MILLS. Expressing 40 ooo Ibs. Cane-juice per day, or for a Crop of 5000 Boxes of 450 Ibs. each in four Months 1 Grinding. Engine (non-condensing). Cylinder, 18 ins. in diam. by 4 feet stroke of piston. Boiler (cylindrical flued). 64 ins. in diam. and 36 feet in length ; two return flues, 20 ins. in diam. Heating surface, 660 sq. feet. Grates, 30 sq. feet. Pressure of Steam, 60 Ibs. per sq. inch, cut off at . 5 the stroke of piston. Revolu tions, 40 per minute. Rolls. One set of 3, 28 ins. in diam. by 6 feet in length; geared i to 14. Shafts, ii and 12 ins. in diam. Spur Wheel, 20 feet in diam. by i foot in width. Fly Wheel, 1 8 feet in diam. ; weight, 17 400 Ibs. Weights. Engine, 61 460 Ibs. ; Sugar Mill, 65730 Ibs. ; Spur Wheel and Connect ing Machinery to Mill, 28 680 Ibs. ; Boiler, 18 520 Ibs. ; Appendages, 6730 Ibs. Total, 181 120 IbS. STONE AND ORE BREAKERS. (See p 957-) No. Re- ceiver. Pul D'm. ey- Face. > &> cr Weight. No. Re- ceiver. Ins. Feet. Ins. Feet. I-P. Lbs. Ins. A I 4X10 5X10 1.66 2-75 6 6 250 iSo 4 5 4000 6700 1 9X15 IlXlS 3 7Xio 2 7-5 250 6 8000 7 13X15 3 5Xi5 2-33 8 180 9 9 loo 8 15X20 4 7X15 2-33 9 1 80 9 10490 9 18X24 Face. In7 > s Feet. 8 1 80 150 125 Feet. '5 '33 ' 33 M NOTK. Amount of product depends on distance jaws are set apart, and speed. Product given in Table is due when jaws are set 1.5 ins. open at bottom, and ma- chine is run at its proper speed and diligently fed. It will also vary somewhat with character of stone. Hard stone or ore will crush faster than sandstone. A cube yard of stone is about one and one third tons. Weight. 904 ELEMENTS OF MACHINES. CHIMNEYS. STEAM FIRE-ENGINE. Airioslieag, N". H. 1st Class* Steam Cylinder. Two of 7.625 ins. in diara. by 8 ins. stroke of piston. Water Cylinder. Two of 4. 5 ins. in diam. Boiler (vertical tubular). Heating surface, 175 sq. feet. Grates, 4.75 sq. feet; Pressure of Steam. 100 Ibs. per sq. inch. Revolutions, 200 per minute. Discharges. Two gates of 2.5 ins., through hose, one of 1.25 ins. and two of i inclk Projection. Horizontal, 1.25 ins. stream, 311 feet; two i inch streams, 256 feet Vertical, 1.25 ins. stream, 200 feet. Water Pressure. With 1.125 ins. nozzle, 200 Iba Time of Raising Steam. From cold water, 25 Ibs., 4 rain. 45 sec. Weights. Engine complete, 6000 Ibs. ; water, 300 Ibs. SAW-MILL. Two Vertical Saws, 34 Ins. Stroke, Lathes, etc. Engine (non-condensing). Cylinder. 10 ins. in diam. by 4 feet stroke of piston. Boilers. Three (plain cylindrical), 30 ins. in diam. by 20 feet in length. Pressure of Steam. go Ibs. per sq. inch. Revolutions, 35 per minute. NOTE. This engine has cut, of yellow-pine timber, 30 feet by 18 ins. in i minute. STONE SAWING. Emerson Stone Saw Co. (Diamond Stone Saw, Pittsburgh, Penn.). to IP, 150 sq. feet of Berea sandstone, inclusive of both sides of cut, in i hour. CHIMNEYS. LAWRENCE, Mass. Octagonal, 222 Feet above Ground, and 19 Feet below. Foundation^ 35 Feet square and of Concrete 7 Feet deep. (Hiram F. Mills.) Shaft. 234 feet in height, 20 feet at base, and 11.5 at top; 28 ins. thick at base and 8 at top. Core. 2 feet thick for 27 feet, and i foot for 154. Horizontal Flues. 7.5 feet square, and Vertical flue or cylinder of 8.5 feet, 234 high, with walls 20 ins. thick for 20 feet, 16 for 17 feet, 12 for 52 feet, and 8 for 145 feet. Purpose. For 700 sq. feet grate surface. Weight. 2250 tons. Bricks, 550000. NEW YORK STEAM HEATING Co. Quadrilateral, 220 Feet above Ground and i Foot below. (Chas. E. Emery, Ph.D.) Shaft. 220 feet in height, and 27 feet 10 ins. by 8 feet 4 ins. in the clear inside. Foundation. i foot below high water. Capacity. Boilers of 16000 H>. Cost of* Steam-Engines and. Boilers complete, and of Operation per Day of 1O Hours, inclusive of .Labor, Fuel, and Repairs. (Chas. E. Emery, Ph.D.) IFF. Engine. Water orate< IHPper Hour. Evap- per Lb.of Coal. Coa IIP. Iper Day. Labor. Sup- plies and Re- pairs. Cost of Coal.* Total Cost of Operafn, including Coal. Lbs. Lbs. Lbs. Lbs. $ * $ $ 6.25 Portable Vertical ( & * 42 7-5 56 394 33 73 2.86 12.5 " " ) 1: 38 717 1.75 .41 3.56 29 Horizontal ( ^ 8 32 8 40 1308 2.25 .60 2-43 5-45 112 Single Condensing. . . 23 8.8 26.1 3-75 1.17 6.14 11.66 2 7 6 552 ci ei 22.2 22.2 8.8 8.8 25.2 25.2 15663 r 5 2.12 4.02 14.58 29.16 22.27 4'- 5* $ 4.42 per ton (2240 Ibs.), including cartage. GEAPHIC OPERATION. 90S GRAPHIC OPERATION. Solutions of* Questions "by a Grraphic Operation. 1. If a man walks 5 miles in i hour, how far win he walk in 4 hours? Operation. Draw horizontal line, divide it into equal parts, as i, 2, 3, and 4, representing hours. From each of these points let fall vertical lines A C, i i, etc., and divide A C into miles, as 5, 10, 15, and 20, and from these points draw equi- distant lines parallel to the horizontal. Hence, the horizontal lines represent time or hours, and the vertical, distance or miles. Therefore, as any inclined line in diagram represents both time and distance, course of man walking 5 miles in an hour 2 3 4 is represented by diagonal Ae; and if he walks for 4 hours, continue the time to 4, and read off from vertical line A C the distance = 20 miles. 2. How far will a man walk in 2 hours at rate of 10 miles in i hour ? His course is shown by the line A o, representing 20 miles. 3. If two men start from a point at the same time, one walking at the rate of 5 miles in an hour and the other at ro miles, how far apart will they be at the end of 2 hours? . ^ ,| Their courses being shown by the lines A r and A o, the distance r o represents the difference of their distances, 10 *> 20= 10 miles. 4. How long have they been walking? Their courses are now shown by the lines A o and A 4, the distance 2 4 represents the difference of their times, or 2 *\ 4 = 2 hours. 5. When they are 10 miles apart, how long have they been walking ? Their courses are again shown by the lines A r and A o, the distance r o repre- sents the difference of their distances of 10 miles, and A '2, 2 hours. 6. If a man walks a given distance at rate of 3.5 miles per hour, and then runs part of distance back at rate of 7 miles, and walks remainder of dis- tance in 5 minutes, occupying 25 minutes of time in all, how far did he run ? Operation. Draw horizontal line, as A C, representing whole time of 25 minutes; set off point e representing a convenient fraction of an hour (as 10 minutes), and a i equal to corresponding fraction of 3.5 miles (or .5833); draw diagonal A n, produced indefinitely to 0, and it will represent the rate of 3. 5 miles per hour. Set off C r equal to 5 minutes, upon same scale as that of A C; let fall vertical r s, and draw diagonal C u at same angle of inclination as that of A n; then from point u draw diagonal u O, inclined at such a rate as to represent 7 miles per hour; thus, if i n represents rate of 3.5 miles, s 0, being one half of the distance, will represent 7 miles. The whole distance between the two points fe thus determined by C x, and dis- tance ran by u s, measured by scale of miles employed. Verification. The distances A e and A i are respectively 10 minutes rr.i66 of an hour, and .5833 mile = .i66 of 3.5 miles. Hence, C x = . 875 mile, and us = .5833 mile. Consequently, the man walked A 0=1.875 mile = 15 minutes, ran Qu= .5833 mile = 5 minutes, and walked u C = .2916 mile. 7. If a second man were to set out from C at same time the man referred to in preceding question started from A, and to walk to A and return to C, at a uniform rate of speed and occupying same time of 25 minutes, at which points and times will he meet the first man ? Operation. As A C represents whole time, and Cx distance between the two points, v z and t x will represent course of second man walking at a uniform rate, and he will meet the first man, on his outward course, at a distance from his start- ing-point of A, represented by A o, and at the time A a; and on his return course at distance A v. x m, and at the time A c. 906 MISCELLANEOUS. MISCELLANEOUS. No., Diameter, and Number of Shot. (American Standard.} Compressed Buck Shot. No. Diam. Shot per Lb. No. Diam. Shot per Lb. No. Diam. Shot per Lb. 3 2 Inch. 25 .27 No. 284 232 o Inch. 3 32 No. 173 140 00 ooo Inch. % No. US 98 Balls, .38 Inch, 85 No. per Ib. ; .44 Inch, 50 No. per Ib. Chilled Shot. Diam. Shot perOz. No. Diam. Shot perOz. No. Diam. Shot perOz. No. Diam. Shot per Oz. Inch. No. Inch. No Inch. No. Inch. No. 05 2385 9 .08 585 6 .11 223 i .16 73 .06 1380 8 Trap 495 5 .12 172 B 17 61 Trap .07 1130 868 8 7 "Trap 409 345 4 3 13 .14 I 3 6 109 BB BBB .18 .19 52 43 Trap 7,6 7 .i 299 2 15 88 Drop Shot. No. Diam. Pellets per Oz. No. Diam. Pellets per Oz. No. Diam. Pellets perOz. No. Diam. Pellets perOz. Inch. No. Inch. No. Inch. No. Inch. No. Extra Fine Dust .015 84021 9 Trap 688 5 .12 168 BBB .19 42 Fine Dust Dust 03 .04 10784 4565 .08 Trap 568 472 4 3 13 .14 132 106 T .2 36 12 05 2 326 8 .09 399 2 15 86 TT .21 3 1 ii .06 1346 7 Trap 338 I .16 7i p 10 Trap 1056 7 .1 291 B 17 59 .22 2 7 10 .07 8 4 8 6 .11 218 BB .18 50 FF 23 24 The scale of the Le Roy standard (adopted by the Sportsman's Convention) com- mences with .21 inch for TT shot, and reduces .01 inch for each size to .05 inch for No. 12. The number of pellets per oz. being the actual number in perfect shot. The number of pellets by this standard is nearly identical with that of the Amer- ican Standard. Tatham's scale is same as Le Roy's, but number of pellets is deduced mathemat- ically, by computing them from the specific gravity of the lead. Drains, Diameter and Q-rade of, to Discharge Rainfall. Diam. Diam. Grade one in. Acres. Diam. Grade one in. Acres. Ins. Ins. 4 30 5 40 .2 20 .6 20 5 5 80 5 7 20 .2 60 .6 60 5 'JO i 8 120 5 6 60 i 80 .8 Grade one in. Acres. Diam. Grade one in. Acres. Ins. 60 2.1 80 5-8 I2O 2.1 15 240 7-8 80 2-5 1 20 7.8 60 2-75 80 9 120 4-5 60 10 80 5-3 18 240 10 British and Metric Measures, Commercial Equivalents of. ( Johnstont Stones, F. R. S.) Length. Millimeters. Yard 914-4 Foot 3*- 8 Inch 25.4 Weight. Grammes. Pound 453-6 Ounce 28.35 Grain. . Volume. Cube Centimeter. Gallon 4554 Quart 1136 .0648 | Ounce 28.4 MEMORANDA. 907 MEMORANDA. Physical and Mechanical Elements, Constructions, and. Results. Belting. Double. 600 IP (to be transmitted) -=- velocity of belt in feet per minute, or 191 IP-:- number of revolutions per minute-i- diameter of pulley in feet = width in ins. Machine Belts. 1500 to 2000 HP -4- velocity of belt in feet per minute = width in ins. (Edward Sawyer.) Blast Pipe of* a Locomotive. Best height is from 6 to 8 diameters of pipe, and best effect when expanded to full diam. of pipe at 2 diameters from base. Boiler Riveting. A riveting gang (2 riveters and i boy) will drive in shell, furnace, etc., a mean of 12.5 rivets per hour. Brick or Compressed Fuel is composed of coal dust agglomerated by pitchy matter, compressed in molds, and subjected to a high temperature in an oven, in order to expel the moisture or volatile portion of the pitch and any fire- damp that may exist in the cells of the coal. Bridge, Highest. At Garabil, France, 413 feet from floor to surface of water, and 1800 feet in length. Bronze, Malleable. P. Dronier, in Paris, makes alloys of copper and tin malleable by adding from .5 per cent, to 2 per cent, quicksilver. Building Department, Requirements of. (New York.) Furnace Flues of Dwelling Homes hereafter constructed at least 8-inch walls on each side. The inner 4 ins. of which, from bottom of flue to a point two feet above 2d story floor, built of fire-brick laid with fire-clay mortar; and least dimensions of furnace flue 8 ins. square, or 4 ins. wide and 16 ins. long, inside measure; and when furnace flues are located in the usual stacks, side of flue inside of house to which it belongs may be 4 ins. thick. If preferred, furnace flues may be made of fire clay pipe of proper size, built in the walls, with an air space of i inch between them, and 4 ins. of brick wall on outside. Boiler Flues to be lined with fire-brick at least 25 feet in height from bottom, and in no case walls of said flues to be less than 8 ins. thick. All flues not built for furnaces or boilers must be altered to conform to the above requirements before they are used as such. Steam Pipes not to be laid within two inches of any timber or woodwork, unless it is protected by a metal shield, and then the distance not to exceed one inch. All floors, ceilings, and partitions to be protected from heat by a metal tube one inch in diameter in excess of the pipe, and the intervening space filled with mineral wool, asbestos, or other incombustible material Horizontal and Hot-Air Pipes in stud partitions to be double, with an interven- ing space between them of at least half an inch, and a space of three inches around -a pipe: the inner face of the partition to be lined with tin plate and the outer faces with iron lath or slate. Hot-air pipes not to be permitted in any stud partition un- less it shall be at least eight feet distant in a horizontal line from the furnace To shield the effect of their heat in wood or stud partitions, to have a double metal collar, with two inches of air space between them and holes for ventilation, or to be enclosed in brick masonry at least four inches in thickness. Cement. Iron to Stone. Fine iron filings, 20 parts, Plaster of Paris, 60, and Sal Ammoniac, i ; mixed fluid with vinegar, and applied forthwith. Chimney Draught. W w h = D. W and w representing weights of a cube foot of air at external, and internal temperatures, h height of chimney or pipe in feet, and D value of draught. See Weight of Air, page 521. Chinese or India Ink improves with age, should be kept in dry air, and in rubbing it down the movement should be in a right line and with very little pressure. MEMORANDA. Coal, Effective "Value of. Theoretical quantity of heat per IP is 2564 units per hour, and average quantity of heat in a Ib. of coal that is utilized in the generation of steam in a boiler is 8500 units; hence, theoretical quantity of coal required per IP per hour = ~-^ = .3 Ibs., after the water has been heated 8500 into atmospheric steam, being theoretically nearly 7.5 per cent, of total heat re- quired to change 30 Ibs. water at 60 into steam of 60 Ibs. effective pressure. The total heat developed by the combustion of coal, when utilized evaporatively, ranges from .55 to .8, but in practice it does not exceed 65 per cent. Coast and Bay Service. A velocity of current of 2.5 feet per second will scour and transport silt, and 5 to 6.5 feet sand. For river scour the velocities are very much less. Cold, Greatest. liquid Nitrous Acid. -220, produced by a bath of Carbon, Bisulphide, and Corrosion of* Iron and Steel. The corrosion of steel over iron is, as a mean, fully one third greater. Cost of Family of Mechanics in TTrance ranges from $220 to $600 per annum, of which clothing costs 16 parts, food 61, rent 15, and mis- cellaneous 8. Crushing Resistance of Brick. A pressed brick of Philadelphia clay withstood a pressure of 500000 Ibs. for a period of 5 minutes. Earth -work. Shovelling. Horizontal, 12 feet. Vertical, 6 feet. When thrown horizontal, 12 to 20 feet, i stage is required, and from 20 to 30, 2 stages. When vertical, 6 to 10 feet, i stage is required. Wheelbarrow. Proper distance up to 200 feet. Number of Loads and "Volume of Earth per Day. One Laborer. (C. Herschell, C. E.) Distance. Trips. Volume. Distance. Trips. Volume. || Distance. Trips. Vo ume. Feet. 20 5 70 100 No. 120 110 IOO 9 8 Cub. Yds. 23-5 16.9 14.4 13-8 Feet. ISO 200 250 300 No. 96 94 92 90 Cub. Yds. Feet. 13-3 350 12.8 400 12.4 450 12 II 500 No. 88 86 84 82 Cu .Y.ds. 1.6 1.2 0.9 0.5 Volume of a barrow load, 2.5 cube feet. Portable Railroad and Hand Cars. For a distance of 550 feet, 60 cube yards can be transported per day. Horse Cart. Volume of Earth transported per Day. One Laborer. Distance. Trips. Volume. Distance. Trips. Volume. Distance. Trips. Volui Feet, joo 500 No. 86 67 Cub. Yds. 13-6 Feet. 1000 1500 43 Cub. Yds. 8.6 6.4 Feet. 2000 2500 No. 25 Cub. Yds. 5 4-3 Volume of each load, 8 cube feet. Ox Cart is less in cost at expense of time. Electric Light, Candle IPower of. Maxim Incandescent Lamp.-* Current with 30 Faure cells, 74 volts, 1.81 Amperes, 16 standard candles. With 50 like cells, 124 volts, and 3.2 Amperes, 333 candles. (Paget Hills, LL. D.) The elavated electric lights at Los Angeles, Cal., are distinctly visible at sea for a distance of 80 miles. Engine and Sugar IVlill, "Weights of. ENGINE (now- condensing). Cylinder. 30 ins. in diam. by 5 feet stroke of piston. Boilers (cylindrical flue). 70 ins. in diam. by 40 feet in length. Weights. Engine, 105000 Ibs. ; Boilers, com< plete, 75000 Ibs. ; Sugar-mill, 40 ins. by 8 feet, 220050 Ibs. ; Connecting Machinery, 137 179 Ibs. Cane carriers, etc. , 46 787 Ibs. MEMORANDA. 909 Filtering Stone. Artificial Clay, 15 parts; Levigated Chalk, 1.5; and Glass Sand, coarse, 83.5. Mixed in water, molded, and hard burned. Fire-engine, Steam. Relative effect for equal cost compared with a hand engine, as i to 113. Each IIP requires about 112 weight of engine. Floating Bodies, Velocities of. At low speeds resistance increases somewhat less than square of velocity. In a Canal, at a speed of 5 miles per hour, a large wave is raised, which at a speed of 9 miles disappears, and when speed is superior to that of the wave, resistance of boat is less in proportion to velocity, and immersion is reduced. Length of Vessel. The proper length for a vessel in feet (upon the wave-line theory) is fifteen sixteenths of square of her speed in knots per hour. Flow of Air. 67 -^/h = Velocity per second X C. h representing column of water in ins., and C a coefficient ranging from 56 to 100. Circular orifices, thin plate 56 to .79 Cylindrical mouth-pieces, short 81 " .84 do. da rounded at inner end 92 " .93 Conical converging mouth-pieces 9 " i Conoidal mouth-piece, alike to contracted vein 97 " i Fl vies, Corrugated. (Wm. Parker.) - M * ~ 2) = Working stress in Ibs. per sq. inch. T representing thickness in i6ths of an inch, and D diameter in ins. Steel, corrugations 1.5 ins. deep. Experiments upon a furnace 31.875 ins. in diam., 6.75 feet in length, and with 13 corrugations. Foundation Files. When piles are driven to a solid foundation, they act as columns of support, and are designated Columns, and when they derive their supporting power from the friction of the soil alone, they are termed Piles. Authorities differ greatly as to the factor of safety for Piles, varying .1 to .01 of impact of ram. ( Weisbach. ) As columns, their safe load may.be taken at from 750 to 900 Ibs. per sq. inch. Authorities give a higher value (Rankine and Mahon, 1000); but it is to be borne in mind that when piles are driven to a solid resistance, they are frequently split, and consequently their resistance is much decreased. As a rule, the following coefficients for ordinary structures are submitted: When the piles are wholly free from vibration consequent upon external impulse, .35 to .4, and when the structures are heavy and exposed to irregular loading, as storehouses, etc., .15 to .2. Ordinarily, the bearing of a properly driven pile not less than 10 ins. in diam. may be taken at 10 tons. Friction of* Bottoms of "Vessels. At a velocity of 7 knots per hour, a foul bottom requires 2.42 IP over that for a clean bottom. Friction of Planed Brass Surfaces in muddy water is .4 pressure. Q-as, Steam, and Hot-air Engines. Relative costs of gas, steam, and air engines per IP: Otto Gas engine, 8.75; Steam engine, 3.5; and Hot-air engine, 4. Heat. Available heat) 16431535 __ expended per IIP per hour} ~ Total heat of combustion x Coefficient for fuel ~" consumption of coal per IIP. Coal 14000 X 772 units = 10 808 ooo. Theoretical evaporative power = 15 Ibs. water. Efficiency of furnace = . 5 ; then 10 808 ooo x . 5 = 5 404 ooo, and 43' 535 5404000 = 3.04 Ibs. per IIP per hour. Ice Boats, Speed of. Maj.-Gen. Z. B. Tower, U. S. A.; assigns the speed of Ice boats at twice that of the wind, and the angle of sail, to attain greatest speed, to be less than 90. Japan Coal. Analysis of Bituminous. Specific Gravity, 1.231. Carbon, 77.59. Hydrogen, 5.28. Oxygen, 3.26. Nitrogen, 2.75. Sulphur, 1.65 Ash, 8.4^ and loss, 98. Its evaporative effect = 4. 16 Ibs. water per Ib. of coal 910 MEMORANDA. Lee- way. A full modelled vessel, with an immersed section of i to 6 of her longitudinal section, and with an area of 36 sq. feet of sails to i of immersed sec- tion, will drift to leeward i mile in 6. A medium modelled vessel, with an im- mersed section of i to 8, and with like areas of sail and section, will drift i in 9. Light, Standard, of. Photometric, English. Spermaceti candles, 6 per Ib. ; 120 grains per hour. Carcel burner = 9. 5 candles. Locomotive Axles, Friction of. .016 of weight. Hence, if radius of wheel = .1, axle friction at periphery ^-j- 10 = 3.73 at periphery. Mercurial Gauge. To prevent freezing, apply or introduce Glycerine on top of column. Metal Products of TJ. S., 18SS. Value, $222000000. Mississippi River, Silt in. Near St. Charles the volume of silt borne per day in 1879 was 475 457 cube yards, and on one day, July 3, it was 4 113600. At times the volume equals 3 ozs. per cube foot of water. Motive Power. A sailing vessel having a length 6 times that of her breadth, requires, for a speed of 10 knots per hour, an impelling force of 48 Ibs. per sq. foot of immersed section. Mowing Machine. Kirby^s (Auburn, N. Y.) 670 Ibs., 2 horses, i acre heavy clover in 46 min. Ordnance, Energy of. In a competitive test ot a 9 -inch Woolwich gun, and a 5. 75-inch Krupp, the energy per inch of circumference of bore was re- spectively 118 and 123 foot-tons; their penetration therefore by the wrought iron standard being about the same, but their total energies were respectively 16400 and 5800 foot-tons. At Mepper a shot of no Ibs., with a velocity of 1749 feet per second, and a strik- ing energy of 2300 foot-tons, passed through a target composed of two plates of soft wrought iron 7 ins. thick, with 10 ins. of wood between them, and passed 800 yards beyond. Petroleum. One Ib. crude oil heated i Ib. water 315.75 28.21 Ibs. water at 60 converted to steam at 212. Relative evaporative effects of Oil and Anthra- cite coal as i to 3.45. Population, Comparative Density of, and N"um"ber of Persons living in a House in different Cities. Chicago, 4 ; Baltimore and Naples, 4. 5 ; Philadelphia, 6 ; London, Boston, and Cairo, 8; Marseilles, 9; Pekin, 10 ; Amsterdam, n ; New York, 13.5; Hamburg, 17.07; Rome and Munich, 27; Paris, 29; Buda Pesth, 34.2; Madrid, 40; St. Peters- burg, 43.9; Vienna, 60.5 ; and in Berlin, 63. Power of a 'Volcano. An eruption of that of Cotopaxi has projected a mass of rock of a volume of 100 cube yards a distance of 9 miles. Power Required to Draw a 'Vessel or Load up an In- clined Hydrostatic Rail or Slip Way. ( Wm. Boyd, Eng. ) W I = R ; C d W -i- D = F ; and P d' c =f. W representing weight of vessel, or load and cradle, I inclination of ways, as length -r- rise, R resistance of vessel or load, Y friction of cradle and rollers, and f 'friction of plunger in stuffing-box, all in tons, C and c coefficients of friction of cradle and stuffing-box, d diameter of axle of rollers, d' product of circumference of plunger and depth of collar or stuffing, all in ins., and f pressure per sq. inch on plunger, in Ibs. Hence, W ^ -=- = I, and R -}- F -}-/= power in tons. ILLUSTRATION. Assume weight of a vessel and cradle 2000 tons, pressure on plunger 2500 Ibs. per sq. inch, inclination of ways i in 20, diameters of axle of roll- ers and of rollers 3 and 10 ins., depth of collar 2 ins., and circumference of plunger 50 ; what would be the power required ? C = . 2, and c = .6. 2000 .2 X 3 X 2000 2500 X 2 X 50 X .6 Then = 100 tons ; = 1 20 tons ; = = 67 tons ; 20 xo 2240 and too -f- i2o-f 67 = 287 tons. MEMORANDA. 911 Propeller Steamer, Ordinary ^Distribution of !Po\ver in a. Power developed by engine, 88 IIP; Power expended in its operation, 12. Per cent. Friction of load 7. 5 " of propeller 7.5 Per cent Power expended by slip of propeller. ... 14 " in propulsion 71 I*ump, Centrifugal, has lifted water 28 to 29 feet, drawn it horizontally 800 feet, and then lifted it 15 feet. Also drawn it 24 feet, and projected it 50 feet Railway 'Trains. Power and Resistance. A railway train running at rate of 60 miles per hour = 88 feet per second, and velocity a body would acquire in felling from 88 feet = 88-^8.02 = 120.3 feet Consequently, in addition to power expended in frictional and atmospheric resistance to train, as much power must be expended to put it in motion at this speed, as would lift it in mass to a height of i2i feet in a second. If the train weighed 100 tons = 224000 Ibs., then 224000 X 120.3 26747200 foot Ibs., and if this result was obtained in a period of 5 minutes, it would require 120.3-7-5 X 224000-^-33000=163.3 IP in addition to that required for frictioual resistances. To raise the speed of a train from 40 (58.66 feet per second) to 45 (66 feet per sec- ond) miles per hour, the power required in addition to that of friction would be as 58. 66 -f- 8. 02 = 53. 44 feet is to 66 -f- 8. 02 = 67. 57 feet = 67. 57 53. 44 = 14. 1 3 feet. Assume a train of 100 tons, running at rate of 60 miles per hour, and total retard- ing power at . i its weight ioo-=- 10 = 10. Then 224 ooo x 10 X 120.3 = 26 947 200-:- 22400 = 1203 feet, which train would run before stopping. If, however, train was ascending a grade of i in 100, the retarding force = .n (n -f- 100) of weight = 24640, distance in which train would come to rest would be 26 947 200 -4- 24640 = 1093. 6 feet Relative Non-conductit>ility of Materials. MATERIAL. Per cent. MATERIAL. Per cent. MATERIAL. [Percent. Hair felt Mineral wool No i 67 e T ' It! i .0 Mineral wool, No. 2 83.2 Charcoal 0/-5 63.2 Asbestos 36 ? " " and tar 71 ? Pine wood Coal ashes 68 J Loam . . . qs Air snace. 2 ins. . 1 34 'I . it. 6 Resistance to a Steam-vessel in Air and "Water. In air 10 per cent, of IIP, and in water, at a speed of 20 miles per hour, 90 per cent, or 8 IIP per sq. foot of immersed amidship section. Saws, Circular. 30 ins. in diameter, are run at 2000 revolutions per minute = 3.57 miles. Spur G-ear has been driven at a velocity of i mile per minute. Sugar Mill Rollers. 5 feet by 28 ins., at 2.5 revolutions per minute requires 20 IP, and 18 feet per minute is proper speed of such rolls. Surface Condensation, Experiments on. (B. G. Nichol.) Tube of Brass, .75 Inch External Diameter. No. 18 B W G. Surface = 1.0656 sq.feet. Duration of Experiment, 20 Minutes. STEAM. Vertical. Horizontal. Temperature 255 ,K 17-75 Ibs. 18.5835 " 52.32 19.0625 * 256 18.25 Ibs. 29-9585 " 84-34 " 30-4375 " 253 1 6. 75 Ibs. 24.0835 " 67-8 24-5625 " 254 ,K 17.25 Ibs. 43-0835 " 121.29 ( 43-5625" Pressure per sq. inch per gauge. Condensation by tube surface " persq. ft. of " per hour Condensed during experiment . . . Steamers' Engines, Weights of. Engine, Boiler. Water, and all Fittings ready for Service per IH. Mercantile steamer 480 Ibs. I Light draught. . . . . 280 Iba English Naval 3 6o | Torpedoes 60 " Ordinary Marine Boiler with Water 196 Ibs. Wind, Pressure of. Estimate of, upon Structures. 30 Ibs. per sq. foot Per lineal foot of a locomotive train = 10 feet in height, 300 Ibs. per sq. foot A Tornado has developed a pressure of 93 Ibs. per lineal foot MEMORANDA. Vi a S xi ez C an al . Passages by Steamers. 1882," Stirling Castle, ' ' Shang. hai to Gravesend, in 29 days 22 hours and 15 wu'w., including i day 22 Aours and 30 wiw. in coaling and detentions. " Glenarc," Amoy to New York, N. Y., in 44 days and 12 Aowrs, including deten- tion at Suez. From Gibraltar in n days. Zino Foil in Steam-boilers. Zinc in an iron steam-boiler consti- tutes a voltaic element, which decomposes the water, liberating oxygen and hydro- gen. The oxygen combines with fatty acids and makes soap, which, coating the tubes, prevents the adhesion of the salts left by evaporation. The mealy deposit ean then be readily removed. Piles. To Compute Extreme Load a Foundation Pile will Sustain. = L. R representing weight of ram, P weight of pile, and L extreme load, all in Ms.; h height of fall of ram, and s distance of depression of pile with last blows, both in feet. ILLUSTRATION. Assume a ram 1000 Ibs. to fall 20 feet upon a pile of 400 Ibs., what resistance will the earth bear, or what weight will the pile sustain when driven by the last blow, from a height of 20 feet, .5 inch ? s = . 5 of 12 ins. = .0416. 10002X20 20000000 Then - == = 343 406 Ibs. (400 -f- 1000) X 0416 58. 24 Perimeter. The limits or bounds of a figure, or sum of all its sides. Of a canal it is the length of the bottom and wet sides of its transverse section. Flood Wave. The flood wave of the Ohio River in March (1884) was 71 feet i inch at Cincinnati, being higher than that of any previous record. Ice. Crushing Strength of, as determined by U. S. testing machine, ranged from 327 to looo Ibs. per sq. inch Atmosphere. If pure air is exhausted of 2. 5 per cent, of its oxygen, it will not support the combustion of a candle. Blasting Paper. Unsized paper coated with a hot mixture of yellow prussiate of potash and charcoal, each 17 parts; refined saltpetre, 35; potassium chlorate, 70; wheat starch, 10, and water, 1500. Dry, cut into strips, and roll into cartridges Circular Saws. Speed, 9000 feet per minute. Thus, for an 8 ins., 4500 revolutions, and progressively up to a 72 ins. , 500 revolutions. (Emerson.) Foods, Relative Value of, compared with 1OO I_/bs. of Q-ood Hay. Additional to page 203. Lbs. Acorns 68 Barley and Rye, mix'd 179 Barley straw . . 180 Buckwheat 64 Buckwheat straw. ... 200 LlM. Linseed 59 Mangel-wurzel 339 Pease and Beans 45 Pea-straw 153 Potatoes 175 Lttf. Rye 54 Turnips 504 Wheat 46 Wheat, Pea, and Oat- chaff 167 Depth of the Ocean. Mean depth is estimated by Dr. Krummel at 1877 fathoms = 1.85 geographical miles. Q-a houi 3-as-engine. A gas-engine 1.5 actual IP will cost, with gas at 8 cents per ir, 10 cents per hour for 10 hours. (Am. Engineer.) Locomotive. Average daily run 100 miles at a cost of $ 12.80 for driver, fireman, fuel, and repairs. (A'. J. Central R. R. Co.) Consumption of Fuel per Milt. Passenger, 25 to 30 Ibs. coal. Freight, 45 to 55 Ibs., or one cord wood per 40 miles. MEMORANDA. 913 Masonry. In laying stones in mortar or cement, they should rest upon the course beneath them, more than upon the material of joint. Steel Q-vua (Krnpp's). Bore, 15.75 ins.; length of bore, 28.5 feet; of gun, 32.66 feet. Weight, 72 tons. Charge, 385 Ibs. prismatic powder; projectile, chilled iron, 1660 Ibs., with an explosive charge of 22 Ibs. of powder. Moment of shot at muzzle, estimated at 31 ooo foot-tons, and range 15 miles. Saw-Mill. 7722 feet of i inch Poplar boards in One Hour. Engine (Non-condensing). Cylinder. 12 by 24 ins. stroke of piston. Boilers. Two (cylindrical flued), 38 ins. in diam. by 26 feet in length, two 14 ing. return flues in each. Heating Surface. 780 sq. feet. Grates. 42.5 sq. feet. Pressure of Steam. 125 Ibs. per sq. inch, cut off at 16.5 ins. Revolutions. 250 to 350 per minute. Saws. Two circular, 60 and 66 ins. in diam. NOTE. Grates set 28 ins. from under side of boilers, without bridge- wall, and a combustion chamber under boilers, 4 feet in depth. Fuel, sawdust. Steatn Keating. 62 50x5 cube feet of space requires 6000 sq. feet of heat- ing surface to attain a temperature of 70 in the vicinity of the city of New York in its coldest weather. Or, One sq. foot of iron pipe will heat 10.5 cube feet of space in an ordinary build- ing, temperature of exterior air 70. (Felix Campbell ) "Velocity of* Steam. Steam at a pressure of 60 Ibs. -f- atmosphere has a velocity of efflux of 890 feet per second, and as expanded, a velocity of 1445 feet. Blasting. In small blasts i Ib. powder will detach 4.5 tons material, and iu large blasts 2.75 tons. (See page 443.) Delta Metal (Iron and Bronze). Specific gravity 8.4. Melting point 1800. (See page 384.) Jarrah Wood of Axistralia. Impervious to insects and the Teredo Pfavalis. Natural and Artificial O-as. Relative water evaporating powers differ in localities, but are assumed at 900 and 600 heat units (B. T. U.) Nat- ural compared 'with. Bitviminovis Coal is effective in the ratio of 2.38 to i. Free Board of Vessels. For each foot of depth of hold (from ceiling to under side of main deck), . i inch added to i. 5 ins. for a depth of 8 feet. Thus, for 24 feet depth i. 5 -f- . i x 8 0024 = 3. i ins. (American. ) Or, 2 ins. for 8 feet depth and . i for each foot in addition thereto. (Lloyd' 9.) Colors for "Working Drawings. Brass Gamboge. Bricks Carmine. Clay Burnt Umber. Concrete Sepia with dark markings. Copper Lake and Burnt Sienna. Granite India Ink, light Iron, cast . . .Neutral tint. " wrought . Prussian Blue. Lead Ind. Ink tinged with P. Blue. Steel Neutral tint, Water Cobalt. Wood Burnt Sienna. Burnt Umber. Stones and Earths . Yellow Ochre. and Black. " " and B't Umber. Red and Indigo. Burnt Sienna and Indigo. Stowage of Chain Cat>le. Square of diameter of chain in ins. mul- tiplied by .35 will give volume of space required to stow i fathom. Asphalt Mortar. Asphaltum i part, powdered asphaltic limestone 7.5 parts, residuum oil .28 parts, sand .6 parts. Melt asphaltum and add the rest in order named. Asphalt Concrete. Asphalt mortar n parts and broken stone 9 parts. Asbestos is a fibrous variety of Actinolite orTremolite, composed of silica, alumina, magnesia, oxide of iron, and water. It resists heat, moisture, and many acids. 914 MEMORANDA. Daily IToocl of an :Ksqrzima\a. Flesh ot a sea-horse 8.5 and Bread 1.75 Ibs., Soup 1.25, Spirits i, and Water .9 pint. (Sir W. E. Parry.) Ooignet'a Concrete. For walls that resist moisture. Sand, Gravel, and Pebbles, 7 parts; Argillaceous Earth 3 parts, and Quicklime i part. Hard and quick setting. Sand, Gravel, and Pebbles, 8 parts; Earth, burned and powdered Cinders, each i part, and Unslacked hydraulic Lime i 5 parts. For a very hard mixture, add cement i part. Transmission or Conductivity of* Temperature in the Earth.. At Edinburgh thermometers set at a depth of 16 feet in the earth at- tained their maximum and minimum at about six months after the corresponding maximum and minimum of the surface, being lowest or coldest in July. The average rate of transmission of heat, as observed at Schenectady, N. Y., was, downwards, 2.9 feet per month, and upwards 3.4 feet. (Olin H. Landreth.) Shafts. When loaded transversely, the diameters of the journal should first be determined, its dimensions then at any other point can be deduced from those diameters. It being observed that the diameters at any two points should be pro- portional to the cube roots of the stress at those points. Journals. For operation at high speed a greater length is required than for low speed. The less their length, the less may be its diameter for a given stress, and consequently the friction will be less. When in constant operation, a large surface is required to reduce heating, and as friction increases with diameter, not with length, for like stress, it is best to lengthen. Wrought Iron. For 50 revolutions length to diameter as 1.2 to i, and for every 50 revolutions additional .2 should be added. Thus, for 1000 revolutions the length to diameter should be 5 times. Cast Iron. Length to diameter as .9, and Steel as 1.25 of above value. (W. C. Unwin.) Non-conducting Materials. By the investigations of Prof. J. M. Ordway of New Orleans, he determined the relative non-conducting values of the following materials, compared with a naked pipe, to be: Hair-felt, burlap i Asbestos paper, hair-felt, duck 1.18 Pine charcoal 1.26 Air space 4 (Engineering, vol. 39, page 206.) Cork in strips 2 Rice-chaff. a.a Clay and vegetable fibre 2.8 Naked pipe 31 Marine Transportation of Troops. Height between decks (deck to under side of beam), men 6 feet, horses 7 feet. Hatchways. Horses at least 10 by 10 feet. Vessels. Horses, beam not less than 30 feet. Men, all ranks, 2 to 2.5 tons capacity; horses, 10 tons. Rations. If biscuit in bags, 10000 require 950 cube feet of volume; if it is in barrels, 1350 cube feet. Cabins. Officers, 30 sq. feet and 195 cube feet of volume, two men 42 sq. feet, and 270 cube feet of volume, and for each additional man 10 sq. feet, exclusive of bed space of 6 by 2 feet. Hammocks. To compute number that can be swung under a deck. -Hi x = I representing length under deck in feet, and b breadth in int. 6 16 (Sir G. Wolseley.) Horse -TPower of Boilers. 30 Ibs. water evaporated into dry steam, from feed at 100, under a pressure of 70 Ibs. per sq. inch mercurial gauge per hour. (Centennial Exhibition, 1876.) 34.5 Ibs. water as above from feed at 212 into steam at 212. (Am. Soc. Mechanical Engineers.) MEMOB^DOA, 915 Penetration of Light in \Vater. Mediterranean, clear sunlight in March, at a depth of 1200 feet; in winter, 600 feet. (M. M. Fol and Sararin.) Railroad. Horse. First in operation in 1826-7. fins. First in use in England about 14501 Iron Steamers. First build in 1830. Lxicifer Match.. First made in 1819. Watches. First constructed in 1476. Xjoad. on Stone per sq. foot. Church of All-Saints at Angers, 86000 IDS. Pantheon at Rome, 60000 Ibs. Flexible Faint for Canvas. Water i. Grind while hot with .83 parts oil paint. Fuel. Evaporation of 9 Ibs. water from 212: 1 Ib. good coal. 2 Ibs. dry peat 3.25 " cotton stalks. 3.75 " wheat straw. Yellow soap 1.66 parts. Boiling .75 Ib. petroleum. 2.5 Ibs. dry wood. 3.5 " brush wood. 4 " megass, or cane refuse. Tramways or Street Tfcailroacls. Resistance on straight and level tracks 15 to 40 Ibs. per ton, or an average of 30 Ibs. Power required on a good track to start a car, as determined by A. W. Wright, M.W.S.E., 116.5 Ibs., and to maintain it in motion 17.2 Ibs. C. E. Emery, Ph. D., rnade it 13 Ibs. On a bad track, the power is 134.6 Ibs. to start, and 35 to maintain it in motion. Power required, as determined by Mr. Wright, to start a car is 33.53 IP, with an average load and day's work, and 133.22 to maintain it in motion. Average work of a car-horse 5.75 hours per day for a term of service of 6 yeara Strong draught-horses will exert a power of 143 Ibs. @ 2.75 miles per hour for 22 miles, and an ordinary one 121 Ibs. for 25 miles. (G^yffier.) Cable Railway. Mr. Wright gives the power required per ton * at 1.92 E?. * All tons here and elsewhere are given at 2240 Ibs. Result of Experiments on Motors for Street Railroads. (1885.) At Antwerp, by Capt. D. Gallon, FR.S., etc. i. Locomotive Engine and Car. Ordinary type of steam-engine, surface condenser (Krauss). a. Surface condenser, vertical boiler, escape super- heated (Black and Hawthorn). 3. ** Compound engine, compressed air, water - tube boiler (Beaumont). 4. " and car combined. Ordinary type of steam-engine, water- tube boiler (Rowan). ' $. " " " combined. Electric Fausse Batteries. Weight of Train per Passenger. Fuel con Per Mile of Course. gamed Per Seat per Mile of Course. Oil, Tallow, etc. Water per Mile of Course. Lbs. 5. Electric. ...1.78 4. Steam 2.3 3. Comp'dair r 2.55 Lbs. 4. Rowan 5.22 5. Electric 6.16 2. Black and ) Q Q Hawthorn. . J b ' B2 i Krauss 9.1 Lbs. .1 23 23 .25 .66 Lbs. .038 .038 073 .101 255 Gallons. Rowan 75 Comp'd air. i. 06 Black and > Rg Hawthorn P' by Krauss 6.52 3. Comp'd air. .39.48 NOTE. The economy of the Rowan motor occurred mainly from the extent of its condensing fewer, by which warm water was supplied to the boiler. 916 MEMORANDA. Corrosive Effects of Salt-water on Steel or Iron. ( J. Farquharson. ) Loss of Plates Submerged for Six Months. Area 12 Sq. Feet. ( .07 Ib. " I -445 " Fractional Resistance of a Railway Train. (C. H. Hudson.] Resistance per ton, due to atmosphere at maximum speed, .132 Ib. ; to start, 17.27 Ibs. ; and to maintain in motion, 5.1 Ibs. Blasting Q-elatine. (G McRoberts, F.C.S.) Is composed ( Nitro glycerine 93 parts ) FffppHvp nnwfir r ft0f i ha by weight. ... 1 Nitro-cotton 7 } M ve P wer - J 4 fo( * It freezes hard at a low temperature (35 to 40). At ordinary temperature above freezing, it does not explode by shock, but when frozen it readily explodes. It is insoluble in water Specific gravity i 55 to i 59. Effective Power of some other Explosives. Nitro-glycerine, 1270 foot-lbs. ; Dynamite, No i, 900; Gun-powder, extra strong. as Curtis and Harvey's, 272; Dynamite, No. 2, of 18 nitro-glycerine, 71 nitrate or potash, 10 of charcoal,, and i of paraffin, 531, and Fulminate of Mercury, 367. Bolts of Wrought Iron as Affected, by the Thread. (D. K. Clark.) Strength per Square Inch of Metal. Diam. of Bolt. Tool. Strength when cut. Loss. Diam. of Bolt. TooL Strength when cut. Lbs. 44845 51005 43 6l 3 41888 Loss. Per cent. 28 H 26 33 Ins. 1.25 1.25 I X Dies.* Chaser. Old Dies. New Dies. Lbs. 40812 38528 55H9 42650 Per cent. 25 2 9 ii 30 Ins. '.625 .625 .625 Chaser. Old Dies. New Dies. Chaser. * Die not giren, evidently new. Approximate Bottom "Velocities of Flow- of "Water in Channels, at which following Materials begin to Move. (Haupt.) Feet. Miles. Feet. Miles. | Sc. Hour. Sec. Hour. 25 .5 17 34 Microscopic sand and clay. Fine sand. 3 2.04 {Small stones, 1.75 inch in diam. I .68 Coarse sand and fine gravel. ( Flint stones, size of 1.75 1.19 Pea gravel. 3-33 2 -3 | hen's eggs. ( Rounded pebbles, i inch in {2-inch square brick- 9 1.39 \ diam. 5 3-4 1 bats. Scouring force of the current is proportioned to the square of its velocity. Transporting capacity varies as sixth power of the velocity. Hence the impor- tance of Increasing lottom velocities, both to effect a scour and to prevent deposits. Chimney. (MetternicJc Lead Mining Co.) Foundation 36 feet square by 11.5 in height; base circular 24.6 feet by 39.37 in height; shaft, 397.5 feet in height, 24.6 feet at base, and 12.48 at top; flue 12.48 and 9.84 feet in diameter. Total height 441.6 feet. Evaporation of "Water. Ins. , Ins. January 9 I April 3.1 February. i.a I May 4.61 . 5.86 Mean, as observed at Boston, Mass. March i . 8 I June . July 6.28 August 5.49 September... 4.09 October November . . December. . . Ins. *:tl Total 39.11 ins. MEMORANDA. 917 Central Widtn of a Roadway in. a Cut. Feet. Feet. Railway, single line ........... 18 to 20 I Public road ................... 281030 " double line .......... 30 " 33 | Turnpike road ................ 38 " 40 Hydraulic Ram. Efficiency under Heads of Supply from 2 to 24 Feet, and Delivery of Discharge at Elevations from 15 to loofeet. Measurements from Valves of Ram. To Compute I?er Cent, of* Total "Volume of Water Ex- pended. . C = Per cent. H representing head of supply, and E elevation of discharge, E both in feet, and C = .8. ILLUSTRATION. What is volume of water delivered with a head of 21 feet to an elevation of 60 feet? X .8 = .28 per cent. Hence, if the volume of discharge is 100 cube feet, vol- 60 ume elevated is 100 X .28 = 28 cube feet. Inversely. By formula of E. B. Weston, M. Am. Soc. C. E. - = V. S representing number of cube feet expended in ram per minute, h dif- 100 h ference in elevations of ram and delivery in feet, and V volume raised in cube feet. C = 65 to 70. Assume as preceding, H = 21 feet, E = 60 feet, and S = 100 cube feet 100X65X21 136500 Then, -- = = 35 cube feet 100X60-21 3900 Norm. To conform to the preceding formula C should be 52. To Compute Elements of a Screw Propeller. and = = p p R 33 ooo p R P representing mean pressure on piston per sq. inch in Ibs. , a area of piston in sq. ins. , p pitch of propeller and I length of stroke, both in feet, R number of revolutions per minute, and T thrust of propeller in Ibs. ILLUSTRATION. The elements of operation of a steam-engine are: Mean pressure on piston, having an area of 1000 sq. ins., is 30 Ibs. ; length of stroke 2 feet; revolu- tions of engine 130 per minute; and pitch of propeller 12 feet. What is the thrust of the propeller, and what the power of the engine? 30X2 X2Xxooo_ i _ 7 ^ Qn ,3oX2X2XioooXi 3 o_ 15600 OOP 12 33000 33000 Centrifugal Pump. (Southwark Foundry and Machine Co. ^on-condensing.) ' Pumps. two of 42 ins., with runners 68 ins. in diameter; discharge pipe 42 ins. Engines. Two of 28 ins. in diameter of cylinder, and 24 ins. stroke of piston. Boilers. 12 Horizontal tubular. Heating surface, 8568 sq feet; Grate, 330 sq. feet; Combustion natural. Pressure of Steam 70 Ibs. per sq. inch, cut off at .625. Revolutions, 130 to 160 per minute. Height of Delivery, o to 36 feet. Weight. Pumping plant exclusive of boilers 300000 Ibs. Discharge. From Dry-dock from a depth of water of o to 36 feet, mean per min- ite 1 12 92 2 gallons, 4H* 91 8 MEMOBANDA. FViction of a Non-condensing Engine. (Prof. R. H. Thurston.] Friction of a non condensing engine is given at from 2 to 4.75 Ibs. per sq. inch of piston, being least at low pressure. The conclusions drawn from a series of exper- iments are as follows: 1. It is sensibly constant at any given speed of engine at all loads. 2. It is variable with variation of speed of engine, increasing with the speed. 3. It increases with increase of pressure of steam. NOTE. This per cent, of friction is somewhat less than that given ante at p. 733. Visibility of" "Vessel's Sidelights. The minimum distance of visibility assigned by the International regulations for green and red lights is 2 nautical miles. "Weight of Anvils. The weight of an anvil for forging iron should be 8 times that of the hammer, and for steel 12 times. (Prof. Friedrick Rich.) Temperature of Mines. Temperature of copper-mines of Lake Su- perior increases i for every 100.8 feet of depth. The usual gradient is from 50 to 55 feet. (H. A. Wheeler.) Horse. In transportation by sea occupies the space of 10 tons measurement, and requires that of 300 cube feet of air. Stalls 6 feet in length in the clear of padding and haunch piece, 2 feet 2 ins. in clear width between padding, 10 per cent, of this width 2 ins. narrower, and 5 per cent, of it 6 ins. longer. Mule. A pair will draw, including cart, 1500 to 2000 Ibs. Aes. Will carry 100 to 200 Ibs. 15 miles per day. Camel. The Arabian, or Dromedary, has one hump on back, the Bactrian has two. Large animals will carry 1500 Ibs. for 3 or 4 days, or 1000 Ibs. for several days, and 450 to 600 Ibs. for a long march. One has travelled 115 miles in n hours. EJlephant. Weight, 3 to 5 tons; weight one can carry about 1450 Ibs. ; 2000 Ibs. have been carried. Occupies 55 sq. feet; will travel on a good road at a rate of 2.5 miles per hour for 6 hours. "Whales. Greenland Right, length 50 to 60 feet. Finner, 80 feet. Speed, 10 io 12 miles per hour. Extreme weight, 74 tons. IP estimated at 145. Chimneys. Late experiments as to the draught of chimneys have developed the result that an increase of its area near to the top increases the draught. Cost of Maintenance of Street Railroads, 1876. Average of 16 roads, 102 miles in length, with 1297 cars and 10300 horses. (H. Haupt.) Cost per horse, and average number to a car eight. Repairs of harness $ 4.06 Shoeing $22.77 Feed 124.39 Stall expenses. 42.13 Replacing horses 22.1 Total $215.45 Cost per month of each horse, $ 18. On one of the longest railroads in the City of New York, on the least populous route, the daily cost per passenger, exclusive of general expenses, was 2.88 cents, and inclusive of general expenses 4.1 cents. Magnesia Covering for Steam-pipes and Boilers. Experiments made by Bureau of Steam Engineering, U.S.N., developed the fol- lowing comparative results: Felt (as standard) ... 100 | Sectional magnesia.... 103.07 | Sawdust 90.5 APPENDIX. 919 APPENDIX. River Steamboat. Wood Side 'Wheels. freight and ^Passenger. " BOSTONA. " HORIZONTAL LEVER ENGINES (Non-condensing). Length on deck, 302 feet 10 ins.; beam, 43 feet 4 ins.; hold, 6 feet. Tons, 993.52. Immersed section of light draught of 26 ins., 83 sq.feet. Capacity for freight, 120* tons (2000 Ibs. ). Cylinders. Two of 25 ins. in diam. by 8 feet stroke of piston. Boilers. Four of steel, 47 ins. in diam. by 30 feet in length, 6 flues in each. Heating surface, 903 sq. feet. Grate surface, 98 sq. feet. Pressure of Steam, 154 Ibs. per sq. inch, cut off at .625. Revolutions, per minute. Speed, 10 miles per hour against current of upper Ohio, 3 to 5 miles. To Compute Meta-centre of Hull of a Vessel. Operation of Formula in Naval Architecture, page 660. Assume a sharp- modelled yacht, 45 feet in length, 13.5 feet beam, and 9.5 feet hold, with an immersed amidship section of 42 sq. feet, and a displacement of 900 cube feet at a mean draught of water of 6 feet lf^~D^ = Meta-centre. . See pages 650, 659, Ordinates (dx) taken at intervals of 2.5 feet are as follows: y y 2 = o = .0 = .63= .216 = i.3 3 = 2.197 = 2 3 = 8 = 2.8 3 = 21.952 = 3. 6 3 =: 46.656 y8 y9 ylO y" y 12 y* yi5 f cubes = 6. s 8 = 287.496 = 6.73 = 300.763 = 6.75 = 307.547 = 6.5 =287.496 = 6.25 = 244.14 = 5.8 = 195.112 ~ 5 ~~ *^5 y I7 3 = 2. 4 =13.824 y 18 =i-5 = 3-375 y'9 a = .8 = .512 y 3 = o = .0 2272.814 2-5 = 5 8 3 195.112 imation of function o of ordinates for value 5082.035 of /y3 d x = 5682.035. And 1 of 5682.035 = ^ of 6 = feet 3 900 3 NOTE. The other elements of this vessel are: Area of load-line, 401. 12 sq.feet ; Displacement in iveight, 27.974 tons ; do. at load- draught, .955 tons per inch; 'Depth of centre of gravity of displacement below load- line, i. 49 feet; Volume of displacement, fa volume of 'immersed dimensions, 26.8 per cent. To Compute Height of Jet in a Conduit 3?ipe from a Constant Head. (Weisbach.) 7 /\ /d'\4 ~ = *'* and ~~ = k" * *' an ^ ^" re P resen ^ n ff heights due to velocity of efflux, loss of head and of ascent, I length of pipe or conduit, and d and d' diameters of pipe and jet, all in feet, v velocity of efflux in feet per second, G and C' coefficients of friction of inlet of pipe and outlet, and z a divisor determined by experiment with diameters of .5 to 1.25 ins., ranging from 1.06 to 1.08. ILLUSTRATION. If conduit pipe for a fountain is 350 feet in length, and 2 ins. in diameter, to what height will a jet of .5 inch ascend under a head of 40 feet? Assume C and 0'. 8 and .5, h = 2sfeet, d z tns. = .i66, and .5 = .5 12 = .0416. Then 2 J 02O APPENDIX. To Compute Head and Discharge of Water in Pipes of Gfreat Length. It becomes necessary first to determine the velocity of the flow, which is =* 4 V V 6d2 = = 1.273 , independent of friction. V representing volume of water in cube feet, and d diameter of pipe in inf. When head, length, and diameter of pipe are given, Coefficients of friction C, for velocity of flow, range from .0234 to .0191 for veloci- ties from 3 to 13 feet per second, and c that for the pipe as a mean at .5. See Weis- bach's Mechanics, Vol. i., page 431. ILLUSTRATION. What head must be given to a pipe 150 feet in length and 5 ins. in diameter, to discharge 25 cube feet of water per minute, and what velocity will it attain at that head? C = .o2 4 and c = . S . 2 c X I2 2 Then 1.273 6oXs a = x - 2 73 X 2.4 = 3.055/6^ velocity per second, and viP /TV' Or, 4. 72 = V in cube feet per minute, and . 538 | / -= = d in ins. yl-T-fl V H> ILLUSTRATION. Assume elements of preceding case. f 150-7- 1.42 p= -538 X -^69607 = .538 X 9.301 = 5 ins. To Compute Fall of a Canal or Open Conduit to Con- duct and Discharge a G-iven "Volume of ^Water per Second. Coefficient of friction in suck case is assumed by Du Buat and others at .007565. C _? x = h. h representing height of fall, I length of canal, and p net perime- ter, all in feet; A area of section of canal in sq. feet, and v velocity of flow in feet per second. ILLUSTRATION i. What fall should be given to a canal with a section of 3 feet at bottom, 7 at top, and 3 in depth, and a length of 2600 feet, to conduct 40 cube feet of water per second ? C = .oo 7 6, p = 3 + (V3 2 -f 2 2 X 2) = 10.21 feet, A= 7 -^-~-2.= 15 sq. feet, and v = = 2.66 feet. , 2600X10.21 2.66 2 oft Then .0076 X ^ = J3-45 X .xx = 1.48 feet. 2 . What is volume of water conducted by a canal, with a section of 4 feet at bottom, 12 at top, and 5 in depth, with a fall of 3 feet, and a length of 5800 feet? i6.8/ee. "* 40 X 3.23 feet velocity 129.2 cube feet. For Dimensions of transverse profile of a canal, see Weisbach, page 492, vot i. APPENDIX. 9 2I MAGNESIA COVERING FOR STEAM BOILERS, HEATED PIPES, ETC. Robert A.. Keas"bey 9 Jersey City and. New York. This covering is devoid of organic matter, hence it possesses great rapacity to resist a high temperature, combined with high rank in the order of noi> conductors. It is furnished for pipes in the form of hollow cylinders divided longitu- dinally, and covered with canvas ; for boilers, in blocks ; and for covering odd fittings, filling floors, etc., in dry mass in bags. Relative Valne of Noi\ - Conducting Coverings on \Vromgnt -Iron Steam I*ipe. Determined toy Tests at St. I^oxiis Water- Works. Condensation in Cube Centimeters per Foot per Hour. (John A. Laird, M. E.) Material. Analysis. C. C. Magnesia, Sectional ( Carbonate of Magnesia. . . 92 . 20 No. Magnesia, Plastic ( Carbonate of Magnesia.. . 92 . 20 33-53 (Asbestos . 82 oo 33-4 Asbestos Fire Felt, Sectional . . . ( Carbonaceous . 18 oo 100 36.75 Asbesto-Sponge Molded j Plaster of Paris . 92.80 | Fibrous Asbestos . 4.20 100 37-13 NOTE. The test at the New York Post-Office gave Fire-felt superior to Magnesia DISTANCES, VELOCITIES, AND ACCELERATION. To Compvite Velocities of an. Accelerated Body. ^/ V 2 _j_ ( 2 v ' S), Or, v + T v' = V. v and v' representing original and accelerated velocities, and V final velocity, all in feet per second; S distance or space passed over in feet, and t time in seconds. -- = V. V representing average velocity in feet per second. V t S, and 2 V V = v. ILLUSTRATION i. A body moving with a velocity of 10 feet per second, is acceler- ated at rate of 4 feet per second, per second, for a period of 6 seconds; what are its different velocities? v = 10, v' = 4, t 6. Then, 10 + 6X4 = 34 feet final velocity. *- = 22 feel average velocity. 22 X 6 = 132 feet distance passed over. Vio 2 -f-( 2 X 4 X 132) = V II 5 6 = 34 f eet i and 2 X 22 34 = lofeet original velocity. . V v . V + v ... V2 ~~ v2 "1=:t>'S, t>2-}-2t>'S = V 2 , ai V-v = t, and 2. A body is projected vertically with a velocity of 200 feet per second, and is retarded at tbe rate of 30 feet per second, per second; wbat height will it have passed through when its velocity is reduced to 80 feet per second, and in what time? v = 200, v' 30, and V = 80. Then 2 ~ 8 = 4 seconds. 3 ._A vehicle being drawn with a velocity of 25 feet per second, is accelerated 5 feet per second, per second; what is its velocity and time of operation at the end of ioo feet ? v = 25, v' =. 5, and V = ioo. Then = 15 seconds. Q22 APPENDIX. 4. A stream of water, after flowing a distance of 120 feet, is ascertained to have a velocity of 40 feet per second, with an accelerating velocity of 2 feet per second, per second; what was its primitive velocity and time of flow? S = 120, V = 40, v' = 2. Then \/4o 2 2 x 2 x 120 = 33. 47 feet. 33 ' 47 = 3. 26 seconds. Delivery and. Friction in Hose. (B. F. Hartford, Am. Soc. C. E.) Hose 2. 5 ins. in diameter. Nozzles not exceeding i. 5 ins. Rubber or Leather. .0408 vd 2 and .497 cd 2 VP = G; / 24 ' 51 G and and ^-1^ = v .003 175 6 c 2 d 4 P I and .000021 6 I v 2 d* =P\ P p = P'; 2.3o6(P p) and 1.123X20 bc*d* bv 2 d* _ . -p- H, i .003 175 6 c 2 d* I and x. G representing gallons dis- charged per second, v velocity in feet per second, P pressure of stream at hydrant or source of supply, p pressure lost in hose, and P' pressure at nozzle, all in Ibs. per sq. foot, d diameter of nozzle in ins., H head of supply at hydrant, h head at nozzle, and I length of hose, all in feet, x fraction ofP at nozzle, b coefficient of material of hose, and cfor nozzle. b = i for rubber hose and 1.167 for leather. c = .82 for smooth nozzle and .64 for ring. ILLUSTRATION. Assume length of a rubber hose 200 feet, pressure at hydrant 100 Ibs., diameter of ring nozzle 1.25 ins., and volume of discharge 4.97 gallons per second; what are the other elements to be obtained by preceding formulas? 497 X .64 X 1.252 X V' = 4-97 gallons. 2 4- 5^X4- 97 = J7 ^feet M.5I X 4.97 and /2.0I2 x 4-97 = . ns 4^484l _97 2 == -o = V v V -64-^/100 .64 2 Xi.25 4 i c oi3 857 X i X 4. 97 2 X 200 = 63. 52 Ibs. 100 63. 52 = 36. 48 Ibs. looX .3648 = 35.48 Ibs. 2.306 (100 63. 52) = 84. i2feet. ^ // 9 =94.12 feet, i .003175 X i X-64 2 X i. 25* X 200= i .6352 = .3648 = 0;. 314.96(1 .3648) _ 200 46750.82X100(1 .3648) _ f . i X .64* X 1.25* ~ i iX 77 96 2 X 1.25* ' /C6t For vertical Jets, see page 549. G-anging of Weirs. When there is an Initial Velocity. (H~+ft f h f ) = H'. H and H' represent- ing depth of water on weir, and when corrected to include effect of initial velocity of approaching water, and h head to which this velocity is due, all in feet. Velocity in Pipes. C V** I = V. r representing mean radius or hydraulic mean depth,* I sine of angle of inclination equal to loss of head per unit of length, V velocity in feet per second, and C a mean coefficient 0/142. In small Channels. C = 30 to 50. NOTE. Sectional area of a pipe or conduit, divided by perimeter, Is termed mean rarft,and whef the pipe, conduit, or channel is but partially filled, the area is termed hydraulic mean depth. * See also page 553. APPENDIX. 923 Metric Factors. In addition to pp. 27-37. By Act of Congress, July, 1866. By French Metric Computation Measures. x Liter per cube meter = .007 48 gallons per cube foot . . . | .007 48 gallons. \Veights and Pressures. i Centimeter of mercury per sq. inch = . 192 91 Ib. per ) sq. inch j i Atmosphere (14.7 Ibs.) = 6.6679 kilograms i Inch of mercury per sq. inch = 2. 54 centimeters i Pound per sq. inch = 453. 6029 grams i Cube foot per ton = .0275 cube meter .1929117 Ib. 6.6678 kilogram*. 2. 54 centimetres. 453- 59 2 6 grammes. .0279 cubic metre. Heat. i Caloric per Kilogram = 1.8 heat units per Ib | 1.8 heat units. Velocity. i Meter per second = 3. 280 833 feet per second | 3. 280 869 feet. I*o\ver and "Work. i Kilogrammeter (k X m) = 2.2046 x 3.28083. . i Foot-pound = . 138 26 kilogrammeters i Kilogram per cheval = 2.2352 Ibs. per EP. . . . i Sq. foot per IP = .091 63 sq. meter per cheval'. j.233 . 138 25 kilogrametre. 2- 2353 pounds. .091 63 sq. metre. Miscellaneous. i Avoirs Lb.= '453 6 kilogram. i Ton = 1.016057 tonne. i Sq. Inch = 645. 161 29 sq. miWrs. i Sq. Foot = .092903 sq. meter. i Cube Foot = .028 317 cube meter. [ Cube Yard = .764559 cube meter. i Mile per hour = 26.8225 meters per minute. i Knot " " (6086.44 feet) = 30.9192 " " " i Cube Meter per minute = 7.848 cube yards per hour. i " Yard " " =45.8718 " meters " " v 2 V 2 tiooomotive Brakes. and -= = distance in which a train is 64- 4 / 3/ Stopped, v and V representing velocity in feet per second, and miles per hour, and f proportion of resistance of brakes to weight of train. Brakes, self-acting, on all wheels,/ . 14. Ordinary hand,/= .023 to .031. As- cending i in .5 resistance is/-f 2 ; descending i in .5 / 2 '. Hydraulic Rams. Efficiency decreases rapidly as height to which wate? is to be raised increases above the fall or head. Number of times the height to which the water is raised exceeds that of the head of the supply and efficiency per cent. ( Walter S. Button, C. and M. E.) Number ... 4 5 6 7 8 9 10 n 12 13 14 15 16 18 19 20 25 Efficiency . . 75 72 68 62 57 53 48 43 38 35 32 28 23 17 15 12 a Speed of water in pumps, 200 feet per minute. To Compute Weignt of Water at any Temperature. - = W. W and w representing weights of water per cube T + 4 6i.2 3W 500 ' 1 + 461. 2 foot at temperature T, and at maximum density 0/39. 2 = 62. 425 Ibs., and 46i.a c equal absolute temperature. ILLUSTRATION. Required weight of a cube foot of water at temperature of 60. 60 + 461.2 t =62.37 E* 500 60-4- 461.3 924 APPENDIX. Results of Experiinents or Performances Steam-engines and. Boilers. of Cylinders, Cut-off, Vacuum, and Diameters in Inches, Revolutions per Minute^ Pressure, Water, and Coal in Lbs., and Surfaces and Areas in Sq. Ins. ELEMENTS OF ENGINE. HARRIS. Non-con- densing. COR Con- densing. LISS. Con- densing. BOILERS. Cylinder 18X42 74.29 58.5 4-74 26.93 105-47 12.64 92.83 l8. 59 2.34 2.07 >oo Ibs. 18X42 73-6 7 6 '37 7-94 29.47 "5-43 13.07 102.36 25.39 3-18 2.82 1.83 753 t Steam p 24X60* 59-62 92.88 18.02 89.38 270.58 12.55 1.98 26.4 1.83 er Ib. of cot Number . . 2 60 12 1536.92 51-75 1256.64 29.7 5-93 "4-3 8.85 8.21 10.3 9.64 to i. Revolutions Pressure in Pipe Diameter Length Cut off Tubes 50 Mean effective Pressure IIP Heating Surface Grate " Friction EP Calorimeter Net IP Heating to Grate Grate to Calorimeter. . Temperature of Feed . Steam per Lb. of ) Combustiblet . . j ' ' Steam per Lb. of) Pnal 1 Water per net IP ) per hour j " * * * Coal per IIP per) Vacuum Combustible per) IIP per hour . J Relative efficiency .... * Weight of engine, 40 c Coal per Sq. Foot) of Grate per hour J " Steam per Temp. 212 1 8.21 Ibt., and evaporation 9 WINDMILLS. (Andrew J. Corcoran, New York.) (Improved. Patented June and August, 1888; March and June, 1889.) "Volume of" "Water "Pumped, per ^Minute. From 10 to 200 Feet. Diameter of Wheel. V 10 BRTICAL Dl 15 STANCE FRO 25 M WATER i 50 ro POINT 01 75 DELIVERY 100 IN TEST. 'SO 200 Feet. Gallons. Gallons. Gallons. Gallons. Gallons. Gallons. Gallons. Gallons. 8-5 15.242 10.162 6.162 3.016 10 48.262 3 2 - I 75 19.179 9-563 6.638 4.25 12 86.708 57-805 33-941 17.952 11.851 8.485 5-68 '4 111.665 74-443 45-139 22.569 15.304 11.246 7.807 4.998 16 155.982 103.988 64.6 31.654 19.542 16.15 9.771 8.075 18 20 249-93 309.604 159-954 206.403 97.682 124.95 52.165 63.75 32.513 40.8 24.421 31.248 17-485 19.284 12. 211 15.938 25 3 532.517 1080.112 355-012 728.828 212.381 430-848 106.964 216.172 71.604 146.608 49-725 107.712 37-349 74-8 26.741 54-043 Factory in Jersey City. Velocity of Wind. The average over the United States, as determined by the Signal Service of the U S. Army, is 5769 miles per month, or about 8 miles per hour. Experience has determined that, to operate a windmill, there is required an average velocity of wind of six miles per hour. - - - = pressure oj mind per sq.foot of surface in Ibs. Or, r a and u' a . v representing velocity of air in feet per second, and ' 400 200 in miles per hour. None. For useful tables and formulas see " Windmills as a Prime Mover." by A. R. Wolff, J. Wiley & Sons, New York, 188$. APPENDIX. To Compute Head in l^bs. per Sq. Inch to Resist Fric- tion, of Air in Long and. Rectilineal IPipes, etc. V 1728 V'L - = H; H( 3 . 7 d)S8 3 .i V3 /H( 3 . 7 d)8 3 .i \ L~~ V representing volume discharged a 60" " (3.7 d) 5 83.i " s /- - =. -T- 3.7 = d, and >T ~^ = IP. y 83.1 H 12" x 33000 in cube feet per minute, L length of pipe in feet, d diameter of pipe in ins. , H head and P pressure, both in Ibs. and per sq inch, v velocity of discharge in feet per sec- ond, a area of discharge in sq. ins., and IP horse-power of friction of air alone. ILLUSTRATION. Assume volume of air discharged 44000 cube feet per minute, diameter of discharge pipe 40.54- ins. (say 1280 sq. ins. net), length of pipe 1000 feet, and pressure at discharge 3.5 Ibs. per sq inch. Then 44 ooo X 1728 K 1280X60 44 ooo 9 X looo __ 6310406250000" /i 936 ooo ooo X looo V 83. i X. 3068 74i 5 IP- = 990 feet, and (3.7 X 40.5)* X 83.3=6 310406250000. .3068 Ibs.; /. 3068 X 6 310 406 250 ooo = 44000 cube feet; 3.7 = 40.5 in*., and 1280X60X990X3.5+ 3068 12 X 33000 Volume of Enclosed. Air at O that may t>e Heated, One Sq.u.are Foot of Iron Heating Surface. ENCLOSURH. He in Cellar. iter in Room. ENCLOSURE. H Celfar iter ta rioom. Dwellings Cub. ft. 70 Cub. ft. 50 1 So Large stores, average Hotels Cub. ft. 90 IOO mo Cuh ft no 125 200 Offices Close stores . . , Churches .. Commercial tf* of Chimney for a G-iven Diam, of Flue. Height of Chimney in Feet Diam. ol SUue. 50 60 70 80 90 IOO no 125 135 'So 175 200 5 250 i 300 Ins. H? H> H> IP IP H? IP FP K> HP IP HP H> IP i IP 16 18 | 30 84 P2 f 9 too 107 113 IIQ 124 45 250 270 288 30.S 321 345 353 35 i 55 330 370 4 os 438 46S 4QO S30 Sbo i 75 . 860 000 PIS IOOO 1036 1090 1185 1270 '34.S I4l5il48o 90 - I3SO 1410 isoo 1556 1640 1770 1800 2OOO 2IOOI2I9O IOO TT ;..*<> . f ,,oJ!o *~ 1665 1725 1820 188011970 2115 , ^j i 2255 2390 2520(2645 Fr intermediate Diameters and Powers, take proportionate Diameters and Powers. E - Square Chimney deduct oneninth to one twelfth of Diameter of Round, for Side Friction of Water in Iipes. (Weisbach.) - * C = h. I representing length of pipes in feet, v = ^- or velocity in feet per second, V volume of water in cube feet elevated per second, d diameter oj pipes in ins. . and C a coefficient, ranging from .069 when velocity .ifoot, .0387^07 -5 foot, .0375 for i foot, .0265 for zfeet, 023 for 4 feet, .02 14 for 6 feet, .0205 for 8 feet, .0193 for 12 feet, and .01 82 for 20 feet. ILLUSTRATION. Assume volume 125 cube feet, raised 25 feet per hour, through a pipe 2 ins. in diameter and 500 feet in length ; how many feet of vertical head will the friction in the pipe be equal to ? Then 83 6 ^ XI 2 25 = 3- 1 velocity, and C = .028. Hence, .1865X500X3.18* X .028 = 14.6 feet, and 25 -f 14.6 = 39.6/efc 926 APPENDIX Marine Boiler. Tests of the United States Government on a Boiler built for the U. S. Cruise} 1 Alert." Conducted by a Board of Naval Engineers, April, 1899. Heating Surface, 2125 sq. ft. Grate Surface, 48 sq. ft. Ratio, 44 : i. Tlie Batococ Elements k and "Wil Cumberland Min 8th nth cox C e Run i 3 th Anthra- cite, egg i4th any. Cumb 20th erland 2ist Cardiff 2 4 th Moisture in coal per cent 5.25 4.09 2-77 .0 2 1.63 i Refuse in dry coal per cent. . . . 7-39 IO -5 12.33 ii. 6 IO 7-88 Boiler pressure Ibs 218 Temperature of feed-water 3 157-2 219 93-4 ~.y 9 1 110.5 152.6 160 204 160.5 Draught at base of pipe .61 .26 Draught in furnace 43 ''.26 ' < 5 *3 55 Blast pressure ash pit 3 -1- e _|_ tjj .07 4" -53 J 5 .27 J 4 Temperature of gases in flue . . 49 8~ I ' J 595 567 520 410 470 433 (% C0 2 . . 11.9 10.9 ii. i IO.2 9-7 10.8 Analyses of flue gases <% 7-8 .5 8.2 .06 8.7 .0 8.8 9-3 .2 8.1 Moisture in steam, decimals of one per cent OQ Dry coal per sq. ft. grate per '^y ' .0 .0 .0 hour Ibs 28.8 Water evap. per hr. IV. & at 212, Ibs. : 4O '4- 41 .y 15.4 ' 10.2 Per Ib dry coal 9.41 10.66 Per Ib combustible . TO '93 1 T Rr "33 Per sq. ft. heating surface . . Per sq. ft. grate surface 12. 13 5.23 231.9 9- 6 5' 9-Si 427.4 421.3 6.94 307.2 12.30 167.8 11.89 5-34 236.3 13.8 4-13 183-1 NOTE. The test of April i3th was made with air heated by the flue gases to 168. Proportions of G-rat* and. Heating Surfaces of \Vater- r Pvit>e Boilers, as Determined by Tests of I3at>cocli and "Wilcox Boilers from ISrS to 1884. (Committee of U. S. Centennial Exhibition and Individuals.) Water evaporated from and at 212. Duration of Test. S Grate. ur f ace Heating. s. Ratio. Combus sun Per Grate. bible con- icd. Per Heating S f Coal per Grate Hour. Evapoi by Com- bustible. ation Coal. Ash per cent. Hours. Sq.Ft. Sq Feet. Sq. Feet. Sq. Feet. Sq. Feet. Lbs. Lbs 8 44-5 1676 37-7 8.88 .256 12.131 120 50.7 1980 39i II. 21 .26 12.99 11.62 9.71 13-7 216 54-7 2148 39.1 12.22 .292 11.982 24 61.9 2760 44-6 8.22 .198 11.626 10.09 13.2 22 59-5 2757 46.3 14.25 .307 9-93 "43 9.96 12.9 13-5 39-7 1680 42.3 5-8 6.26 12.495 ".53 4 25 1403 56.1 12.41 ! 27 6 13.44 12.38 11.52 7 >5 10.25 70 3126 44-7 18.15 .406 20 12.42 11.32 8.8* Coals : * Anthracite, American, f Bituminous, Welsh, t Bituminous, Scotch. Bituminous, Powelton. f Test in London. A Galloway boiler of standard efficiency, at this exposition, having a ratio of heat- ing surface to grate of 25 to i, and feed water at a temperature of 56, gave the fol- lowing results: Consumption of coal, 8.87 Ibs. per hour per sq. foot of grate. Pressure of steam, 70 Ibs. per sq. inch. Water evaporated per hour per sq. foot of heating surface, 3 Ibs. ; water evaporated per Ib. of combustible, 9.68 Ibs., and per Ib. of coal, 8.63 Ibs. APPENDIX. 927 To Compute Area of Cylinder of* a Steam-engine and. G-rate and Heating Surfaces of a Boiler. When Required Power is Given. II is assumed that IP of a steam engine is at- tained by evaporation of 33.6 Ibs. water per hour, at a temperature of 212 from feed water at 100. NOTK. This Is a deduction from the elements of the estimate as given by the Am. Soc. of Mech'l Engineers, in order to put temperature of the feed at 100 instead of 212. Non condensing (Single Cylinder}. ^-^ 5- X 1728 = area of cylinder DO X 2 K X v> in sq. ins. V representing volume of i Ib. of water at terminal pressure of steam in cube feet, R number of revolutions per minute, and S stroke of piston in feet. ILLUSTRATION. Required EP of an engine is 300, initial pressure of steam 70 Ibs. mercurial gauge, cut off at .5 stroke of piston of 4 feet, and number of revolutions 60 per minute. What should be areas of cylinder of engine and grate and heating surfaces of boiler? Clearance in cylinder and steam passages = 1.8 ins. =.15 foot, point of cutting off =4 -=-.5 = 2 feet. Then (formula p. 711), 70 X (2 + .15 -=-4 + .15) = 36. 26 Ibs. terminal pressure, and steam at this pressure has a density or volume, which is its reciprocal (formula table p. 708) of 11.26 cube feet for each Ib of water contained in it. Hence, ii 26 X 36.26 X 300 ^ 122 486 = 851X1728=14705 cube 60X60X2X2*' 14400 which -=-48 ins. stroke = 306. 35 sq. ins., to which is to be added for friction of en- gine and load and waste of steam 15 per cent. 45-95 -f- 36*35 35 2 -3 ins- Grate Surface Evaporation of fresh water in an efficient marine boiler, from a temperature of feed of 100, is assumed, with a proportion of heating surface to grate of 30 to i, to be, with a combustion of 20 Ibs. coal per sq. foot of grate per hour, 213 Ibs. per sq. foot of grate, and 10.3 Ibs. per Ib. of coal. E T-P Hence, =. area. L representing evaporation per sq. foot of grate per hour. ILLUSTRATION. Assume elements of preceding, with evaporation as above. 33.6X300. 213 Heating Surface. -Then 47.32 X 30= 1419 sq.feet area. For the several types of boilers the following units should be used: Ratio of Heating Surface to Gratt - = 47.32 sq.fset. 30 to i II 50 to i Coal consumed per Sq. Foot of Grate per Hoar inLbs. is 2O 30 5 20 30 Marine 164 214 314. l8* 242 3-3Q ICQ 2O7 2OQ 182 241 333 Portable I slate black, iron is present. (2.) Dissolve a little prussiate of potash, and mix it with the water; if iron is present, it will turn blue. For Lime. Into a glass of water put two drops of oxalic acid and blow upon it. If it becomes milky, lime is present. For Acid. Immerse a piece of litmus paper in it. If it turns red, it is acid. If it precipitates on adding lime-water, it is carbonic acid. If a blue paper is turned red, it is a mineral acid. APPENDIX. 929 TOBIN BRONZE. (Trade-mark registered.) The Aiisonia Brass and. Copper Co., Ne-w York, N. Y. Sole Manufacturer. Specific gravity, 8.379. Weight of a cube inch, .3021 of a Ib. Tensile strength i-inch round rod, 79 600 Ibs. per sq. inch. Elastic limit, 54 257 Ibs. per. sq. inch. Elongation in a rod i inch in diameter and 8 ins. in length, 15.4 per cent. Reduction, 37.26 per cent. Fairbanks. Is readily forged into bolts and nuts at a dark-red heat, Torsional strength and Elastic limit equal to machinery steel. Torsional Strength. Bolt .5 inch in diameter and i inch in length, load at end of lever ifoot. Torsion, 2.67. Elastic limit, 328 Ibs. Rupture, 633 Ibs. Torsion point of rupture, 92.2. J. E. Denton, Stevens' s Institute. Crushing Strength, maximum, 181 ooo Ibs. per sq. inch. Fairbanks. Plates. Weight per Square Foot. Thickness. Weight. Thickness. Weight. Thickness. Weight. Thickness. Weight. Ins. .0625 .125 .1875 25 .3125 375 Lbs. 2.72 5-44 8.16 10.88 13-59 16.31 Ins. 4375 5 5625 .625 6875 75 * Lbs. 19.03 21-75 24.47 27.19 29.91 32.63 Ins. .8125 .875 9375 i 1.0625 1.125 Lbs. 35-35 38.06 40.78 43-5 46.22 48.94 Ins. 1.1875 1.25 I.3I25 1-375 1-4375 1-5 Lbs. 51-66 54.38 57-1 59.82 62.53 65.25 Bolts and Rods. Weight per Lineal Foot. Diameter. Weight. Diameter. Weight. Diameter. Weight. Diameter. Weigbt. Diam. Weight. Ins. Lbs. Ins, Lbs. Ins. Lbs. Ins. Lbs. Ins. Lbs. .25 .177 75 1.6 1-5 6.42 2-5 I 7 .8 4 45.57 3 J 25 .279 .8125 1.88 1.625 7.5 2.025 19.6 4.25 51-44 375 399 .875 2.18 1-75 8.7 2-75 21-53 4-5 57.64 .4378 544 9375 2.5 1.875 10 2.875 23.52 4-75 64.24 5 .711 i 2.84 2 11.38 3 25-53 5 71.16 5625 .899 1.125 3-6 2.125 12.87 3-25 30.05 5.25 78.46 .625 I. II 1.25 4-46 2.25 14.43 3-5 34-86 5-5 86.11 6875 1.34 1-375 5-36 2-375 16.06 3-75 40.01 6 102.45 Owing to its great strength and non-corrosive properties the rods are ex- tensively used for bolts, forgings, etc., Marine and Naval Machinery, Sugar- houses, Breweries, Pump Piston-Rods, and Yacht Shafting. The plates are used for Pump Linings, Condenser Heads, Hulls of Yachts, Centreboards, and Rudders. Drop Forgings and Nails of every description can be made of it. Weights of Steam-engines and Boilers with Water. Per Indicated IP in Lbs. Merchant Steamer 480 Royal Navy " 360 Steamboats 280 4 i* Torpedo Boats 60 Marine Boilers 196 Locomotive" 60 930 APPENDIX. Weight and Strength of Ordinary Stud - Link Chain Catole. Dime Diana. isions of 1 Length. .ink. Width. Weight per Fathom. Admiralty Proof- stress.* Dimei Diam. isioij of 1 Length. ink. Width. Weight per Fathom. Admiralty Proof- stress.* Ins. Ins. Ins. LbB. Lbs. Ins. Ins. Ins. Lbs. Lba. 375 2.25 1-35 7-55 375 8.25 4-95 101.6 76160 4375 5 2.625 3 ;:l 75 "3 13-4 7840 10080 '.625 9 9-75 & 121 I 4 2 90720 96400 5625 3-375 2.025 17.2 12320 75 10.5 6-3 164.6 1*4320 .625 3-75 2.25 21 15680 .875 11.25 6-75 l8 9 141 680 .6875 4-125 2-475 25.4 19040 12 7.2 215 161 280 75 4-5 2.7 30.2 22680 .125 12.75 7-65 242.8 182000 &75 5-25 3-15 41.2 30800 25 13-5 8.1 276.2 204 1 20 6 3-6 53-8 40320 375 I4-25 8-55 303-2 227 360 1.125 6-75 4-05 69 50960 5 15 9 336 252000 25 7-5 4-5 84 63000 75 l6. 5 9-9 406.6 304 940 * Adopted by Lloyds. NOTE i. Safe Working- stress is taken at half the Proof-stress. 2. Proof-stress and Safe Working- stress for close-link chains are respectively two thirds of those of stud-link chains. 3. Average Proof-stress is 72 per cent, of ultimate strength, or 17000 Ibs. per sq. inch of section of both sides. Safe working-stress is half the proof-stress, or 8500 Ibs. per sq. inch of section. Weight of close-link chain is about three times the weight of the bar from which it is made, for equal lengths. 4. Ultimate Strength per sq. inch of section of metal is 35 ooo Ibs. Comparing the weight, cost, and strength of the three materials, hemp, iron wire, and chain iron, the proportion between the cost of hemp rope, wire rope, and chain is as 2 : i : 3 ; and, therefore, for equal resistances, wire rope is only half the cost of hemp rope, and a third of the cost of chains. (Karl von Ott.) Height and Retrocession of Niagara Falls. (J. Pohlman.) 1842. Height. American ........... 167 feet. Horseshoe... ....... 158 ' Jas HalL Width. American ........... 600 ' Horseshoe ........... 1800 1886. Average retrocession 2.5 feet per annum ( Woodward). 200 feet in ii years, and 9 feet per year in 42 years. Descent of the river below 15 feet per mile. Bridge. Over Oxus on Caspian sea, 6230 feet in length. LENGTHS OF ENGLISH RACE-COURSES. Course. Miles. Course. Miles. Course. Miles, NEWMARKET. Across the Flat 1.292 DONCASTER. Circular '9*5 GOODWOOD. Cup Course 2.5 4.206 Fitzwilliam Cambridgeshire 1.136 2.266 Red House St. Leger 'I" 1.825 New Course i-5 Round Rowley Mile 3-579 I. OOQ Cup Course 2.634 NEW CASTLE 1.796 2 Summer Course .... 2 Craven I. 2? YORK. Two-year old, new . . yearling..... .702 ,277 Derby and Oaks. . . . Metropolitan i-S 2.25 Stakes Course Two-mile..,.. i-75 1.923 Railway Speed in England. 1887. North Western Railway. To Crewe, 158.5 miles in 178 minutes without a stop. Caledonian. Carlisle to Edinburgh, 100. 75 miles, including 10 consecutive miles of elevation of i in 80, in 104 minutes. APPENDIX. 931 f - W 0\ CO* OOCOMSOOOOh* * CO\O M U"l Is O\ f ^ o ! * s 8 Si, Jo '* m mvo voixooONOw row wmtxin og 88 g g jig a 1^ m mm m j mmc4inc5vovovotxooo\ONO do ON NN win Ninmmmo OO 2 H NOW ? Y! a o ^ g oo tx -^-m COM rx tx w vo --m rxin in oo N TJ- oo M g 3 - * o 3 05 H 8 i fad 0-8 5 '5>* ,Q 03 ^S g^fe oo" OO OO inmmo mo. mvo mm mm ^ ro ^- ^- m mvo votxooONOONOot N m in ^ in m in N N >2 mvo vo O txoo co ON ON 6 6 w o'cl N m m wmmmmo mo fc 1 v ^ - O ** ^ QD (g 1 8 ^ tx i- - ^o^.. o. M ro M ro ON tx M fl^fe-'l I H X CO p S 5 fe H K i o o o OO omminmo mo mmvom ^ ro ro <* m mm mvo votxcoONON ONN mm mo OO OO MM Mm mo mO S ^ "jl| tJC I^M m mr? r? m ? Q^ 0. p r ~ ffi 5 - m vo vo vo tx txoo OOO\ONO o cj Mm mm mm mm MM MM MM MM k * r H i \j. S gvoONOOOONC* tx roONOv i ONVO M H m vo 3 1 S o S c-^5 S ro ro ro * * m mvo vo tx txoo ON 1 ON O o m m 1 mi o I 'S ^^S rt e> j m m m ^ Tt- m m m vo vo txoo oo ON ON os o 1 do m m B 1 |77777777 7777 932 APPENDIX. Niagara River The canal through which the water is to be drawn commences at a distance of 1.5 miles above the Falls. The water-storage of the river is computed at 328855 square miles, viz. , 87 620 of lake and 241 235 of shed. The annual rainfall being 37 inches. Assuming the rainfall to be but 3^ inches, the flow over the Falls would be 213000 cube feet per second. The Lake survey computes it at 265 ooo cube feet. The BP designed to be used by the company constructing the canal is 120000. A late and corrected determination of levels gives Lake Ontario 246 and Lake Erie 578 feet above mean tide at oity of Nevr York. Height of* Towers, Spires, etc. (Additional to page 180.) Eiffel Tower, Paris .......... 984.3 feet. I Cathedral, Strasburg .......... 465 feet Cathedral, Rouen ........... 492 " | City Hall, Philadelphia ........ 535 " Zenith and Meridian Distance at New York. C. H. and A-ltitn.de (Lat. 40 42' 44".) of Sun June 2ist, Zenith distance. . . 17 15' 44" I Dec. 2ist, Zenith distance. . . 64 9' 44" Meridian altitude. 72 44' 16" | Meridian altitude . 25 50' 16" "Water-pump. First in use 283 years B.C. Rotating introduced in i7th century. Plunger pistons invented by Morland (England), 1674. Double acting by De la Hire (France). Symbolic Hatching and Designations. As adopted by Engineer Department, U. S. Navy. The folio-wing are designed and added "by the A-uthor BABBITT. NICKEL. See Colors for Drawings, p. IQ& GUM. ZINO. CONCRETE. APPENDIX. 933 .Results of Experiments on Operation of Steam- Engine. (C. E. Emery, M. E.) Condensing. Cylinder t . . . 16 X 42 ins. Non-Condensing. 18 X 42 ins. 73-37 IDS. 29-47 73-6 Condensing. Cutoff. 189 IP 78 79 Non-Condensing. .189 "5-43 I3-07 102. 36 29.231 3-25 3-66 .8685 .8676 Pressure, 81.69 Ibs. " mean effective. 31.06 " Revolutions per min. . . 60.3 Water per IIP per hour in Ibs Friction H? 10. 09 Net IP.... 68. 7 Coal " .".".".""." Coal per net BP per hour in Ibs. Relative efficiency per steam Relative efficiency per coal. . . Safety Valves of Steam-Boilers. Boilers operated at a low pressure of steam require proportionately larger safety valves than when operated at a high pressure. Thus: If steam at 20 Ibs. pressure per sq. inch is raised to 30 Ibs., a valve nearly one half more capacity is required; but if raised from 100 Ibs. to no Ibs., a valve of nearly one tenth more capacity is required. Belting. Double belts will transmit one and one half times the power of single. Wide belts are less effective per unit of area than narrow. Long belts are more effective than short. Driving belts may be driven at a velocity of 3500 feet per min. ute. Lathe belts from 1500 to 2000 feet. Economy of wear requires less velocities. Non-Conductors of Temperature. Their Comparative Efficiency. Materials in Italics are whotty free from Carbonization or Ignition from slow con- tact with Boilers or Steam-pipes. The following Materials were used, as Covering to a Steam-pipe 2 Ins. in Diameter. Pounds of Water Heated 10 per Hour through One Square Foot of the Material (J. M. Ordway.) Material i Inch in Thickness. Lbs. Relative Solidity. Material i Inch in Thickness. Lbs. Relative Solidity. i Wool loose 8.x .56 13. Anthracite coalpow- \ 2 Feathers of live geese . . Q.6 . e der j 35-7 5.o6 3. Lamp black, loose 4 Felt of hair fa 10. 3 .56 1.85 14. Magnesia, calcined) and compressed ) 42.6 2.8 5 48 6. Lamp black, compressed 10.6 3.44 16. Asbestos, fine 4Q 81 7. Charcoal of cork 8. Magnesia, calcined) and loose ) 11.9 12.4 53 23 17. Sand 18. Stag wool, best (fine) threads of brittle glass) J 62.1 13 5-27 9 Magnesia, carbonate ) 19. Paper 14 of and light . . j 13-7 .6 20 Rice-chaff .... 18 7 to. Charcoal of white pine. ii. Magnesia, carbonate) of, and compressed ) 12. Plaster of Paris... 13-9 iS-4 30. a 1.19 i-5 7.68 21. Bit. coal-ashes, loose. . 22. Asbestos paper, tight. 23. Anth. coal-ashes, loose 24. Clav and veg'ble fibre. 21 21.7 27 10. - Flow of Several Rivers. Minimum Dry Weather. In Cube Feet per Minute. St. lAwrence, at BrockviJ'.e, Ont., 18000000. Mississippi, at St. Pauls, Minn 2000000. Connecticut, at Holyoke, Mass. ... 300 ooo. Okio, at Pittsburgh, Penn. . . Illinois, at La Salle, III 36 ooo. Seine, at Paris, France looooo. Mohawk, at Cohoes, N. Y 58 800. Thames, at London, England. . 360001 Chicago, at Chicago, 111 36 ooo, 934 APPENDIX. Standard TJ. S. ^Weights and Measures. (U. S. Coasi Survey.) Lineal. Inch to Millimetres. Foot to Metre. Yard to Metre. Mile to II Metre to Kilometres. Inches. Metre to Feet. Metre to Yards. Kilometre to Miles. 25.4 Chain = .304801 20.1169111 .9144 etres. P I 1-60935 || 39.37 athom = 1.829 metre .not = 1853.27 metres 3.28083 s. So., no 1.093611 ile = 259 h .62137 ectares. Inches to Feet to Ceutimetre. Decimetre. Yard to Metre. Acre to Hectare. Centimetre to Inch. Metre to Feet. Metre to Yards. Hectare to Acres. 6.452 9.29 .836 .4047 155 10.764 1.196 2.471 Volume. (Fluid.) Dram to Milli- litres.* Ounce to Milli- litres. Quart to Litre. Gallon to Litres. Milli- litref to Dram. Millilitre to Ounce Litres to Quarts. Deca- litre to Gallons. Hecto- litre to Bushels. 3-7 29.57 .94636 3- 7 8 544 .27 .338 1.0567 2.6417 2-8375 Cube. Inch to Centimetres. Foot to Metre. Yard to Metre. Bushel to Hectoliter. Centimetre to Inch. Decimetre to Inches. Metre to Feet. Metre to Yards. 16.38^ f .02832 .765 35242 .061 61.023 35-3I4 1.308 Weight. Grain to Milligrams. Av. Ounce to Grains. Av. Pound to Kilogram. Tr. Ounce to Grains. Milligram to Grain. Kilogram to Grains. Hectogram J to Av. Ounces. Kilogram to Pounds 64.7989 28.3495 45359 31.10348 01543 15432.36 3-5274 2 . 204 62 Quintal to Av. Pounds, 220.46. Tonnes to Av. Pounds, 2204.6. Grams to Tr. Ounce, .03215. Av. Pound = 453. 592 427 7 grams. Kilogram 15 432. 356 39 grains. NOTE. The U. S. yard is equal to the British yard. British gallon = 4. 543 46 litres. Bushel = 36.3477 litres. Value of tlie .Metre in terms of the British Imperial Yard, and of the Committee Metre (C.M.) of the U. S. Coast and Geodetic Survey. (0. H. Tittman.) Authority. Hassler 39.380917 Kater 370 79 Bailey 369 678 Clarke 370 432 Comstock 369 85 39.36994 39-3699 39-36973 39-369 39 -3697 -36984 Mean.... 39.3698 Dead Sea and Valley of* tlie Jordan. Portions of these are 1300 feet below the level of the sea. (R. E. Peary.) Value of* GJ-old. From 1501 to 1889 the ratio of gold and silver varied from 1 1. 1 to 22. Dnr ability of* Woods. Wood columns or posts, set in earth opposite to course of its growth, are more durable than when set with it. * Cube centimetres. f Cube centilitre. too Grams. $ Millien. APPENDIX. 935 Miscellaneous Operations. To Remove Paint. Apply chloroform. To Restore Color of a Fabric. When destroyed by an acid ap- ply ammonia to neutralize it, and then chloroform. Silverware. Warm, and cover with a mild solution of collodion in alcohol, applying it with a soft brush. Grilt Frames. To restore, rub with a sponge moistened with spirits of turpentine. Egg Stain. On silver, rub with salt. Iron Rust. To remove from white fabrics, saturate the spots with lemon-juice and salt, and expose to the sun. Ink: Stains. Wash with pure fresh water, and apply oxalic acid. If this changes the stain to a red color, apply ammonia. Clinkers on Brick:. Apply oyster shells on the top of a clear fire. Antidotes for Poisons. Additional to page 185. Antimonial Wine or Tartar Emetic. A A r arm water to induce vomiting. Arsenic or Fowler's Solution. Emetic of mustard and salt, a tablespoonfuL Then, butter, sweet-oil, or milk. Bed Bug. Oil of vitriol, corrosive sublimate, sugar of lead. Caustic Soda or Potash, and Volatile Alkali. Drink freely of lemon -juice or vinegar in water. Carbolic Acid. Flour and water, and glutinous drinks. Carbonate of Soda, Copperas, or Cobalt. Administer emetic; soap or mucilagi- nous drinks. Chloroform. -Apply cold water to head and face, artificial respiration, and gal- vanic battery. Laudanum, Morphine, or Opium. Administer strong coffee, mustard flour, butter or oils in warm water, and exercise. Muriatic or Oxalic Acid. Give magnesia mixed, and soap dissolved with fresh water. Nitrite of Silver. Bali in water. Sulphate of Zinc or Red Precipitate. Give milk or white of eggs copiously. Sulphuric Acid. Aqua fortis. Strychnine. Emetic of mustard or sulphate of zinc, aided by warm water. Motive iPower of the "World. Steam-engines. In Horse-Power. United States. ... 7 500 ooo I Germany 4 500000 I Austria 1 500000 England 7000000 | France 3000000 | Other countries. 19000000 Steam-boilers in Foreign Countries, France, including Locomotive. . 51 390 | Germany 60700 | Austria 12000 Locomotives in Foreign Countries. France 7000 | Germany. . . 10000 | Austria 2800 | Other countries. . . 85 200 The steam-engines of the world represent the power or work of i ooo ooo ooo men. (Bureau of Statistics, Berlin, 1887.) Destructive Stress of Belting. (Horace B. Gale.) In Lbs. per Sq. Inch. Material. Maxi- mum. Minimum. Exten- sion.* Material. Maxi- mum. Minimum. Exten- sion.* Best Leather. Raw hide Lba. 8000 6750 Lbs. 2850 3000 Inch. .018 .18 Rubber Cotton belt'g Lba. 3888 2913 Lba. 3000 2000 Inch. .059 .037 * At 400 Ibs. per eq. inch. APPENDIX. Largest Constructions and. INTatural FormationSo New Opera-House, Paris. Covers 3 acres, and has a volume of 4287000 feet. Popocatapetl, Highest active Volcano, Mexico. Has a crater one mile in diametei and looo feet in depth. (See p. 182.) Telegraph Wire over river Kistnah, India. 6000 feet in length and 1000 feet in elevation. (See p. 179.) Chinese Wall, Built 220 B.C. (See p. 179.) Lambert Coal Mine, Belgium. 3490 feet in depth. Mammoth Cave, Kentucky. Some of its chambers are traversed by navigable branches of the subterranean river Echo. St. Gothard Tunnel Its summit is 900 feet below the surface at Andermatt, and 6600 feet below the peak of Kastlehorn. (See p. 179.) Bibliotheque Nationals, Paris. Founded by Louis XIV., contains 1400000 vol- umes, 300000 pamphlets, 175000 MSS., 300000 maps and charts, and 150000 coins and medals. Engravings i 300000, contained in 1000 volumes, and 100000 portraits. Desert of Sahara, Africa. Length 3000 miles, average breadth 900 miles, and area 2000000 sq. miles. Pyramid of Cheops, Egypt. Volume of masonry 89028000 cube feet; weight of stone computed at 6 316 ooo tons. (See p. 174. ) Bell, Moscow. Circumference at base 68 feet, height 21 feet. (See p. 181.) Bridges.* Rialto, Venice. A single arch of marble, 98.5 feet in length. Clifton Suspension, Bristol, Eng. Span 703 feet, elevation 245 feet. Niagara Suspension, U. S. Cantilevers, of steel, length 810 feet. Elevation above the rapids 245 feet. Britannia, England. 1512 feet in length, and elevation 103 feet. Forth, Frith of Forth, Scotland. Length 8098.5 feet, exclusive of approaches of 5349.5 feet. Two Cantilever spans of 1710 feet each. Piers 360 feet above water. Roadway 150 feet in the clear above water. Iron and steel 54000 tons. Masonry 250040 tons. Tay, Scotland. Length 2 miles, 85 piers, and elevation 77 feet. Colnmns or !Pillars. When a column or pillar is without its vertical line; one with slightly rounded ends becomes capable of greater resistance than one with square ends. Experiments at the U. S. Arsenal at Watertown, Mass., developed that the ver- tical resistance of timber, to transverse compression or crushing, was about one third of its resistance to longitudinal compression, and hence, that the area of the cap or the head of a timber column, should proportionately exceed that of the column. Steam-engine Notes. Horse- power, t Nominal. Is usually computed from the volume of steam dis- charged from the cylinder. Its measure for an ordinary non-condensing engine is about .4 of its actual power. It refers more to the dimensions of an engine than its capacity. Indicated. Is the measure of the force exerted by an engine, and from this is to be deducted for leaks, friction of its parts!, and of its connecting parts, about 10 per cent. Feed Water. Ordinarily 2 to 3.5 gallons or 17 to 30 Ibs. of water are required for each IIP. Fuel The ordinary consumption of fuel may be taken at 3 Ibs. per IIP for a non-condensing engine, and 2 Ibs. for a condensing. Boilers. 12 to 15 sq. feet of heating surface, or .4 to .5 of grate surface, with natural draught, will give one IIP. Flow of Steam. The velocity of it in feet per second, may be determined by the formula, 6oVT -f- 460 = V; or, 60 times the square root of the sum of the tem- perature of it in degrees, and 460. Thus for a pressure of 100 Ibs. per sq. inch a velocity of 900 feet may be obtained. ( John Richards, Phila. ) * Additional to page 181. t See also pp. 733, 734. APPENDIX. 937 Atlantic and Pacific Oceans. There is not any difference in the mean levels of these Oceans at Aspinwall and Panama, as determined by Geo. M. Totten, who constructed the Panama Railroad. Origin and. IPeriocl of Grreat Inventions. See also Chronology, pp. 71, 72, 915. Air-engine. Amonton, 1699. Stirling, 1827. Ericsson, 1855. Air-pump. Otto Gueriche, 1650. Anemometer. Walflus, 1709. Balloon. First, Lyons, France, 1783. Barometer.* Torricella, 1643. Battery. Electric, 1745; claimed by Kleist, Cunseus, and Muschenbroch. Bridges (Suspension). Of chains, China, 100 B. C. Bayonets. At Bayonne, 1670. Socket bayonet, 1699. Bells. In Christian church, 400; in France, 550. Bellows. Egypt, 1490 B.CX Bessemer Steel Sir Henry Bessemer, 1856. Blankets.* England, 1340. Blasting. Germany, 1620. Bullets. Of stone, 1418; of iron, 1550. Calico Printing. Egypt; introduced in England 1696. Camera Obscura. Roger Bacon, 1214; Newton, 1700; Daguerre, 1839. Candles. Of tallow, 1290. Cannon. m8; England, 1521. Carriages Vienna, 1515; England, 1580. Clocks.* To strike, by Arabians, 800; by Italians, 1200. Com.* 1184 B.C. ; China, 1200 B.C. ; Rome, 576; England, uoi. Compass.* China, 2634 B.C. Cotton Gin. Whitney, 1793. Dyeing. 1490 B.C. Prussian Blue, Berlin, 1710. Dynamite. Sobrero, 1846; Nobel, 1867. Electric Discoveries.* Leyden Jar, Cunaeus, 1746 ; Electric Light, Davy, 1800; first patent of it, Greene & Staite, 1846. Electro- Magnetism. Oersted, Copenhagen, 1819. Electrotyping. Jacobi of Russia and Spencer of England, 1837. Engraving. China, 1000 B.C. ; on metal, 1423; line or steel, 1450; etching, 1512. Gas. Murdock Cornwall, 1792; Meter, Clegg, 1807; Dry meter, Malam, 1820. Glass.* Egypt, 1740 B.C. Windows, France, i2th century. Gold Leaf. Egypt, 1700 B.C. Gunpowder. Unknown; rediscovered 1324. Horseshoes. 300; 'of iron, 480. Hydraulic Press. Bramah, 1796. Hydraulic Ram. Whitehurst, 1772. Hydrogen. Isolated by Cavendish, 1766. Iron Vessels. J. Wilkinson, England, 1787; Ship, 1821; Steam-boat, 1830; Shipbuilding, 1833. Kaleidoscope. Sir Daniel Brewster, 1814-17. Knives. Table, England, 1550. Life-boat. 1817. Lithography. Senefelder, about 1796. Locomotive. Watt, 1769 and 1784. Cngnot, 1769. Matches. Friction, 1829. Medicine. From Greece, in Rome 200 B. C. Mirrors. Glass, Venice, i3th century. Newspaper. First authentic, 1494. Omnibus Paris, 1827. Organs. 755. England, 951. Oxygen. Priestley, 1774. Paper. From silk, China, 120 B.C. ; from rags, Egypt, 1085. Pens. Of steel, 1803; gold, 1825. Pencils. Of lead, 50. England, 1565. Pianoforte. Italy, 1710. Phonograph. Edison, 1877. Photograph. England, 1802; perfected, 1841. Pottery. Oldest, Egypt, 20006. CL Post-Office. Vienna and Brussels, 1516. Stamps. England, 1840. Printing.* Types, L. Coster, 1423. Railroad. * Passenger, England, Sept. 27, 1825. Sewing-machine. Patented. England, 1755. Sleeping-car. 1858; Pullman, 1864. Soap. England, i6th century. Spectacles. Italy, isth century. Telephone. A. G. Bell and C. J. Blake, Boston, 1874. Torpedo. Credited to D. Bushnell, 1777. Indicates that the subject it also given at pp. 71, 72. 4 K 938 Cobalt.., G9ld.... Iridium . APPENDIX. Values of some Precious Metals. Per Pound, Troy. 250 295 Osmium Platinum Potassium ^Variable. 102 25 Rhodium . . $ 415 Ruthenium 075 Silver* ia Missions Tea, Coffee, etc . 20 Sugar Fuel for Households \\ Milk Linen and Cotton. . . . 20 Butter and Cheese. . 35 Expenditure in. England, for Various Purposes and of Articles Compared with, that of Spirituous Liquors- In Millions of Pound Sterling. Woollen Goods 46 Bread 70 Rents 130 Liquors 136 Aluminum.. Elastic limit of bars in tension 14000 Ibs. per sq. inch. Specific heat .2185. Melt at 1400. Malleable at from 200 to 300. Tensile strength, ultimate, 26000 Ibs. Modulus of elasticity, 12000000. Shrinkage .022 per linear foot. It is comparatively unaffected by exposure to air or water. Cube inch weighs .0926 Ib. A cube foot weighs 160.013 * ( Continued on page 976.) Bushels of Seed Required per Acre. Barley Beans .1.5 t t -75 : 7 4 .16 i 25 02.5 2 33 33 i In Bushels per A Flax 5 t( ere. ) 2 875 2.25 5 i Oats 2 t 5 2.5 5 2 .06 i-5 04 7 ' 3-5 ' 10 1 2.5 1 2 " .16 ' 2 Grass, blue. . .625 " orchard. i. 5 " Herds 1 .. .375 " Timothy .5 Hemp i Parsnips.. Pease Potatoes.. Rice . Buckwheat. . . Clover, red... white. Corn, brown . " Indian... Rye Millet i '.'fes Turnips... Wheat.... Mustard 25 See also page 193. Domestic Remedials. Colors. Discharged by an acid, can be restored by Ammonia. Flies. Carbolic Acid (20 drops), evaporated on a hot surface, as a shovel, will drive them from a room. Ink. To remove stains from a white fabric, wet with Milk and cover with Salt, Mildew stains. May be discharged by Buttermilk. Mosquito. Camphor Gum, vaporized over the chimney of a gas-burner or lamp, will drive them from a room. Mats. To drive them off, apply Chloride of Lime to their locality. Sewer Gas. The noxious effects removed by Chloride of Lime. Snnstroke. Remove patient to a cool place, administer water freely, and Quinine or Salicate of Soda. Comparative "Values of UTood. for Sheep. Wool and Tallow Produced. FOOD. Wool. Tallow. FOOD. Wool. Tallow. Wheat x Oats 94 Barley . . .. .89 Lb8. 97 7 7 R Lbs. 99 7 i Corn-meal, wet 83 Buckwheat 79 Rye, without salt 58 Lbs. 93 7 97 Lbs. .29 55 .71 Pease 88 I .7 Potatoes, with salt .... 3 45 .3 Rye, with salt... .87 Q7 S8 " without salt. .28 45 .10 APPENDIX. 939 Croton .A-queduct. New York, 1890. Dimensions, Length, and. Capacity. ^h.::::::::::::: *!:S ra ' les } 30.75 m.* m umgth. Pipes to Central Park reservoir, 2.37 miles in length. Tunnel under Harlem river, 307 feet below tide-water level Course. From Croton Lake, 350 feet above the Dam, and runs generally Southerly, through Westchester Co. and the 24th Ward of New York, to a point 7000 feet N. on Jerome Park, with a uniform inclination of .7 feet per mile; its general form, that of a horse shoe with curved invert ; being 13.33 feet in height and 13.6 feet in width ; having a computed capacity of 318 millions of gallons per day. From thence, where it is contemplated to construct a large reservoir, for the supply of the annexed districts of the city, to its termination at issth Street and xoth Avenue, its capacity is reduced to 250 millions of gallons per day, and the Aqueduct which from there is to be operated under pressure, is circular in its section, 12.3 feet in diameter, with varying inclinations, the portion under the Harlem river being 10.5 feet From 135th Street it is connected to 12 cast-iron pipes, 48 ins. in diameter, 4 of which connect with the old Aqueduct, 4 with the present City distribution, and 4 leading through Convent, (gih) and 8th Avenues to the Reservoir in Central Park. The operating capacity of all being equal to that of the Aqueduct, 250 millions of gallons. The Aqueduct is for the greater portion of its length a tunnel, it raising to the surface but at four points, from which it can be emptied through gates into the adjacent rivers. Capacity. The water-shed ef the Croton, in extreme dry weather, with sterage, is 250 million gallons per day. The present storage system includes Croton Lake, Reservoir at Boyd's Corners, the middle branch Reservoir of the Croton valley, and several lakes, with a total capacity of 10000 million gallons: three dams being in progress of construction and others contemplated, viz., one at Carmel and one at Quaker Bridge. The Capacity of the Reservoir in Central Park is computed at 1000 million gallons. Ice. Additional to p. 195. 1.5 ins. thick will support a man; 5 ins., an 84-lbs. cannon; 10 ins., a body of men; 18 ins., a railroad train. Yield of Oil in Seeds. Per Cent. Mustard, white 37 Hemp 19 Linseed 17 Corn, Indian 7 Oats 6.5 Clover-hay 5 Flour- wheat 3 Barley a. 5 Rape 55 Almond, sweet 47 bitter 37 Turnip 45 Additional to page 189. Historical Events and N"otatole Facts. Australia. Discovered 1622. Banana. Produce per acre 44 times greater than potato, and 131 times greater than wheat. Camels. Some can travel 800 miles in 8 days. Catacombs. Of Rome, remajns of 6000000 bodies. China. Authentic history of it, 3000 B.C. Crucifixion. 37. Library of Alexandria 47 B. C. contained 400000 books. Pens. Steel, consumption 4000000 per day. Slavery. Abolished in Eng. West Indies, 1834; Russia, 1861. N". Latitude reached toy Explorers. 1884. Adolphus W. Greely, U. S. Army, 83 24'. The distance from this to the Pole is 456 02 miles. 940 APPENDIX. Drilling. Rand Drill Co., New York. Drills. Cylinder. Diain. Usual Depth Drilled. Diam. of Bottom of Hole. Depth Drilled in 10 Hour*. Diam. of Hose. Diam. of Steel. Steam Boiler. Steam Pipe. No. Kid Ins. 1.875 Fest. 1.5 lus. Feet. Ins. 75 Ins. .625 If 3 Ins. 75 2 .25 4 i .0625 CO . ye 75 5 i 2 and 2 A 3 and 3 A 3.25 and 3. 25 A 4 and 4 A 2-75 3-125 3-25 3.625 4. 5 6 to 10 10 to 15 15 20 20 to 30 i-5 i-75 i-75 2 2.25 60 70 7 70 70 75 25 . 5 .125 .125 to 1.25 375 . 5 7 10 10 12 15 1.25 i-5 i-5 2 2 7 5.5 '& .75 20 tO 21 2-5 Raiid. Air Compressors. Rand- Corliss Class " B B*." Compound Steam Condensing. Compound Air. Steam Pressure 125 Ibs. Terminal Capacity in Free Air Cylinder Steam. Diameters. Air. Stroke. Revolutions per Air Pressure per Minute. High. Low. High. Low Minute. at 80 Ibs. Cube Feet. Ins. Ins. Ins. Ins Ins. No. IH 5 . 670 10 18 10.5 17 30 85 102 1196 12 22 13 21 36 83 l82 1562 '4 26 15 24 36 83 238 1650 '4 26 15 24 42 75 252 1920 16 30 17-5 28 36 75 293 2242 16 30 17-5 28 42 75 342 2395 16 30 17.5 28 48 70 365 2520 18 34 20 32 36 75 384 2897 18 34 2O 32 4 2 75 442 3128 18 34 20 32 48 70 475 3960 20 38 22.5 36 48 70 604 4100 22 40 24 38 48 65 625 453 22 42 25 40 48 65 690 5000 24 44 26.5 42 48 65 7 6 3 6000 26 48 29 46 48 65 9*5 6820 28 52 30 48 48 65 1040 Rand "Imperial" Type X. Duplex Steam Non- Condensing. Compound Air. Steam Pressure 80 to 100 Ibs. gteftm Capacity In Free Air per Min. Duplex Steam Cylinders. Dian AirC High. icter of Revolutions and Air Blinders. Stroke. per Pressure, Low. Minute. * 100 Ibs. Cube Feet. Ins. Ins. [ns. Ins. No. IH>. 145 6 6 -5 10 8 200 25 245 7 7-5 12 10 190 4* 37 8 9 14 12 175 63 535 10 10 16 14 165 91 75 12 ii 18 16 150 120 1050 J 4 *3 22 l6 150 178 Rand "Imperial" Type XI. Duplex Air Cylinders. Belt Driven. Capacity in Free Air per Min. Air Cylinders. Diam of gt k each. per Min. Air Pressure 60 Ibs. per Sq. Inch. 100 Ibs. Cube Feet. Ins. Ins. No. IH>. Iff. 11.7 4 4 200 1 .7 2-3 22.7 5 5 2OO 3-3 4-5 38 6 6 200 5-5 7-5 62 7 7 2OO 9 12 93 8 8 2OO 13-5 lg-5 163 10 10 1 80 24 30 275 12 12 175 4 53 APPENDIX} 941 Suspension. Furnaces IVTorison, The Continental Iron Works, Brooklyn, N. Y. La for Corrugated. Furnaces. Board of U. S. Supervising Engineers, October ioth, 1891. P X B =T. P = working pressure in Ibs. per sq. inch. D mean diameter of fur- 15 600 nace=inside diameter+1, and T thickness of metal, both in ins. Corrugated not less than 1.5 inches in depth, and flat surface of ends not exceed- ing 6 inches in length. Thickness of Metal in Suspension Furnaces for dif- ferent Diameters and "Working Pressures in Lt>s. Per Sq.. Inch. As Determined by the Formula in the Rules and Regulations of the U. S. Board. Inside Diam. Ins. A H 1 Itt iV H i if A 1 U 1 ff H H 1 28 162 178 i95 211 227 243 260 276 292 308 325 34i 357 390 29 157 172 1 88 2O4 220 235 251 267 283 298 314 330 345 377 30 152 167 182 198 213 228 243 258 274 289 304 319 335 365 31 14? 162 177 192 2O6 221 236 251 265 280 295 310 325 354 32 143 157 172 1 86 2OO 215 229 243 258 272 286 301 315 344 33 139 153 167 181 195 208 222 236 250 264 278 292 306 334 34 135 i 4 8 162 176 189 20.J 216 230 243 257 270 284 297 325 35 131 144 158 171 184 197 210 223 237 250 263 276 289 316 36 128 141 153 166 179 192 205 218 230 243 256 269 282 307 37 125 137 150 162 175 187 200 212 225 237 250 262 275 300 38 121 134 146 158 170 182 195 207 219 231 243 255 268 292 39 118 130 142 154 166 I 7 8 190 202 214 225 237 249 261 285 40 116 127 139 150 162 174 185 197 208 220 232 243 255 278 41 113" 124 136 147 158 170 181 192 204 215 226 238 249 272 42 no 121 132 144 155 166 177 188 199 210 221 232 243 265 43 108 IIQ 130 140 151 162 i73 184 195 205 216 227 238 260 44 105 116 127 137 148 158 169 1 80 190 201 211 222 233 254 45 103 114 124 134 i45 155 165 176 1 86 197 207 217 228 248 46 101 in 121 132 142 152 162 172 182 192 203 213 223 243 47 99 109 119 129 i39 149 159 169 179 189 198 208 218 238 48 97 107 117 126 136 146 156 165 175 185 195 204 214 234 49 95 105 114 124 133 143 152 162 172 181 191 200 210 229 50 93 103 112 121 131 140 150 159 168 178 187 196 206 225 Si 9i IOI no 119 128 137 147 156 165 174 183 193 2O2 220 52 90 99 108 117 126 135 144 153 162 171 I 80 l8 9 I 9 8 216 53 88 97 1 06 US 124 132 141 150 159 168 177 186 195 212 54 87 95 104 H3 121 130 139 147 156 165 174 182 191 208 55 85 94 102 III 119 128 136 145 i53 162 171 179 1 88 205 56 84 92 IOO IO9 117 126 134 142 151 159 168 176 184 201 57 82 90 99 107 US 123 132 140 148 156 165 173 181 198 58 81 89 97 105 H3 121 130 138 146 154 162 170 178 195 59 79 87 95 103 III 119 127 135 143 151 159 167 175 191 60 78 86 94 102 no 117 125 133 141 149 157 165 172 188 942 APPENDIX. Influence of the Rotation of* the Earth on IMoving Bodies. The Rotation of the Earth on its axis effects an appreciable displacement of the rails in a line of railroad. In the case of an express train weighing 400 tons, running N. at the rate of 50 miles per hour, the pressure on the right hand or Eastern rail is computed at 501 IDS., and with a steamer, alike to the Inman Line "City of New York," the press- ure is computed at 936 Ibs. This lateral force increases to the Poles. (T. Von Bavier.j Bacteria in Earth-soil. In Virgin soil; soil from beneath Roadways; from Gardens; adjacent to Factories; from Courtyards and Cemeteries. -a a pli* COc $ No. 124800 til &* *!: No. Meters 750 * .061 022 cube inches. til * CO c No. 64200 No. 590 The number very rapidly decreases in the deeper layers of the earth, both in virgin soil and in that which has been polluted. (John Reimers.) "Water-meters. "Worthington's. New York. Diam. of Re- ceiving Pip.. Volume delivered per Minute. Diam. of Re- ceiving Pipe. Volume delivered per Minute. Diam. of Re- ceiving Pipe. Volume delivered per Minute. Diam. of Re- ceiving Pipe. Volume delivered per Minute. Ins. .625 75 Cube ft. 3 S Galls. 11.25 22.5 Ins. i i-5 Cube ft. 6 Galls. 37-5 45 Ins. 2 3 Cube ft. 8 23 Galls. 60 172 Ins. 6 Cube ft. 58 120 Galls, 435 900 NOTE i. .The volume of delivery here given, for each meter, can be exceeded. 2. Extreme velocity of a meter produces incessant and improper resistances; hence, in order that the instrument may operate only within a perceptible reduction of the head of the supply, it should be of a capacity to effect its duty at a moderate velocity of operation. Telescopes. Galileo's first telescope magnified but three times; but by the addition of a con- cave eye and convex object glass he attained a magnifying power of 30 times. The construction of large lenses is at present limited by the chromatic aberration, or separation of light in a telescope. Euler was the first to discover the principle governing this aberration and the method of abolishing it. Diameters of the Principal Objective Grlasses. United States. Location. Diameter. Focal Length. Ins. 12 Feet. Wesleyan University . . 12 I2.S 15 Madison. Wia . . 12.56 20.2 Location. Diameter. Focal Length. Ins. Feet. Washington . . 26 32 472 University, Va Lick Observatory. . . . 26 36 ft University of Southern California contemplates the construction of one of 40 ins The largest telescopes outside the U. S. are, Gates Head, England, 24 ina ; Vienna Austria, 27 ins. ; Nice, France, 28 ins. ; Pulkowa, Russia, 30 ina * To have four lenses of 24 inches. APPENDIX. 943 Manufacture of Ice. Machinery and. Apparatus. Produc- tion of Ice in 24 Hours, Steam-Engine. Com- pressors. Blocks of Ice. Water required M^ute. Coal. Op Engi- eratora. Fire- men. Lab- Weight of Engine and Plant. Tons. Ins. Rev. Ins. Ins. Gallons. T's.* Lbs. I 7X 9 00 SXiot 8X 8X28 5 5 2 2 2OOOO 3 8X16 80 5X15 8X15X28 IS i 2 2 2 58000 5 10X20 75 6X18 8X15X28 20 1-5 2 2 2 69000 10 12X30 70 8X20 j 11X22X28 ( 1 11X11X28) 30 2 2 2 3 101 000 12.5 14X30 65 8X25 j 11X22X28 ( | 11X11X28 j 35 2-5 2 2 3 129000 15 14X30 65 10X20 j 11X22X28 | 1 11X11X28) 40 3 2 2 4 167 ooo 20 16X30 55 10X30 ) 11X22X28 1 } 11X11X28 ) So 4 2 2 5 190000 30 - - - j 11X22X28 ( | 11X11X28) 60 5 2 2 6 225000 4 18X36 So 12X30 11X11X28 90 6-5 2 2 7 260000 45 20X36 5o 15X30 11X11X28 2 2 8 60 24X36 45 I2X30J 11X11X28 2 2 9 80 26X48 45 20X36 11X22X28 100 13 2 2 10 360000 * 2000 Ibs. f One compressor. J And one 16x36 ins. additional. All others two compressors, and all single acting. Pressure of Steam. For all 75 Ibs. per square inch. Out - off. For the three first, which are slide valves, three eighths. For the others, as Corliss engines, one fifth. The volumes of ice above given cover that lost in thawing the molds to release it. The coal given as that required is inclusive of that required to distil water from which to make the ice. NOTE. In order that the proper dimensions of engine and plant may be arrived at for a required volume of ice, it is necessary that the quantity and temperature of the water supply should be furnished. 2. The ice is produced from water of distillation; hence, it is clear and trans- parent. "When a Machine is operated, "by "Water-power. As the water from which the ice is made is not distilled from steam, as in the case where steam is the motive power, the ice produced is less clear or transparent, and is known as " white ice. " Refrigerating. Engines for Refrigerating are in all respects alike to those for Ice-making, with two thirds more capacity. As distilled water is not required in refrigerating, the saving of fuel in consequence is fully thirty per cent. Refrigerating by compression involves a much less expenditure of water than vhen it is attained by absorption. Elements of a Test of Operation and. Capacity of n Refrigerating Machine. Ice Liquefied in 24 Consecutive Hours, 78.41 Tons of 2000 Ibs. Steam-engine. Non-condensing, 18X36 ins. Compressors 12.375X30 ins. Pressure of Steam, 86 Ibs. Revolutions per minute, 56.5. IHP. Steam-cylinder, 84.3. Of compressors, 67.78. Temperature of condensing water, 76.2. Of condenser room, 62.5. Volume of condensing water per minute, 21.19 gallons. Of brine per meter, 35 670 cube feet = 2 496 900 Ibs. Evaporating pressure, 25.22 Ibs. Condensing pressure, 157.12 Ibs. Anthracite coal, consumed, 6108 Ibs. Conibvsfible, 83.63 per cent. Coal per IHP per hour, 3.02 Ibs. Consumption equivalent to the liquefaction of one ton of ice. 77.88 Ibs. 944 APPENDIX. and. .Asplialt lavement. Barber Asphalt Paving Co., New York. Rock Asphalt is amorphous limestone impregnated with asphaltum, whereas Trinidad asphalt pavement is a mixture of sand, pulverized limestone, aud asphal- tic cement. The asphaltic cement is composed of refined Trinidad asphalt, with a little residuum oil of petroleum, the pavement being an artificial asphaltic sand- stone. In the cities of Europe, where asphalt pavement has been laid, the practice is to spread from 1.5 to 2.5 inches of it on a bed of concrete. The process of preparing the material for use is to crush the rock to powder, heat it to about 280, spread it on the concrete, and then compress it by rammers. The use of natural asphaltum, found in the United States, as Albertite and Grahamite, was resorted to, but without success, when the pitch or asphalt lake in Trinidad, W. I., was discovered; by combining this material with highly refined pe- troleum, a satisfactory cement was produced, which being mixed with a sharp sili- cious sand and powdered limestone, a desired sandstone was formed; a compound possessing the necessary firmness and resistance to the changes of temperature and durability, under the wear of loaded vehicles, combined with smoothness, cleanli- ness, and comparative freedom from noise; without danger from the slipping of horses' feet, usual with pavements with smooth surface. So evident was the useful application of this construction, that in 1870 an essay of its merits was made in Newark and New York, and in 1876 it was further essayed on an extended scale in Washington, its merits being evidenced by a Board of U. S. Engineers. Since which time it has been laid in over 100 other cities in the U. S. to an extent of about 20,000,000 square yards. The advantages of such a pavement are the reduction of the resistance to traction, economy of transportation, and freedom from jolting in travel, added to cleanliness and public health, as it is without seams or joints wherein filth may be collected. Its durability in wear is less than granite, and greater than sandstone, wood, or macadam. As regards the cost of its maintenance, it is less than that of any other material maintained in like condition of repair. Origin and. Development. The utility of asphalt for covering of a road was not discovered until 1849. As- phalt rock, broken up, was laid in the manner of a macadamized road, aud the re- sult was such that in 1854 a street in Paris was laid with compressed asphalt on a foundation bed of concrete. In 1869 it was first laid in London, and is now extensively laid in the cities of Europe to an extent in excess of 3,000,000 square yards. SyiTostitntes. Tar. As a substitute for it it was essayed to use the inex- pensive tar, obtained from gas-works; but as it is deficient in the required cement- ing qualities, susceptible of being rendered viscid by the heat of summer, and brittle by the cold o f winter, the use of it was abandoned. Wood. Wood-pavement is laid in London and Paris on a foundation of concrete, and it lasts from 4 to 6 years. Stone-blocks filled in with asphaltum water-proof filling has been practised with success. In some of the principal cities of Europe, the uniformity in the dimen- sions and shape of the blocks contribute to their durability. The cost of such a pavement is in excess of all others. Macadam. Macadam pavement is unsuited for cities from the wear of heavy Tehicles, and the great cost of maintenance. Brick. Brick, hard burned, laid in two courses on 6 inches of sand, the first course on its face, and the second on its longitudinal edge, has been used in Holland, Ohio, and Illinois. The duration of such a pavement depends wholly on the uni- formity of the material and its burning. In general practice it was found to be neither enduring nor economical. Gen'l Gillmore, U. S. engineer, in his report (1879) submits the following: Requisite of a Good Pavement. A good pavement must be smooth, aud to promote easy draught must give a firm and safe foothold for animals, and not polish or become slippery under wear; must be, as nearly as possible, noiseless and free from dust or mud, and made of durable material, laid upon a firm foundation, and be susceptible of repairs at moderate cost at all seasons of the year. APPENDIX. 945 Suitable Foundations for Pavements. A firm and unyielding foundation is quite as necessary for stability and endurance of a pavement as for any other structure. Following are suitable foundations for street-pavements, in order of value, pro- vided their thickness is adapted to character of subsoil and nature of traffic, viz. : i. hydraulic concrete 5 to 8 ins. in thickness; 2. rubble-stone set on edge side by side, but not in close contact, with interstices filled in with hydraulic concrete ; 3. an old coal-tar pavement properly brought to slope and grade; 4. rubble-stone set on edge and wedged closely in contact like sub-pavement of a Telford road; 5. an old pave- ment of stone-blocks, cobble, or rubble stone ; and 6. an old Macadamized or gravel road, or a compost layer of broken stone or gravel, 8 or 10 inches thick. The best pavements now prominently before the public, classified with respect to the materials of which they are made, are Asphalt, Stone block, Wooden block, and Coal-tar pavements. The wooden-block pavement is not entitled to a place in the list. Stone Pavements. The best is formed with rectangular blocks from 3. 5 to 4. 5 ins. thick, 10 to 13 in length on wearing surface, and 8 to 9 inches deep, set upon frheir longest edge across the street, upon a foundation of hydraulic concrete. Asphalt Pavements. Best asphalt is one having for a foundation a bed of hy- draulic cement, or something equivalent thereto in firmness and durability, and for its wearing surface either the natural bituminous limestone known as asphalt rock, derived from the Jurassic region on the confines of Switzerland, or, preferable thereto, an artificially compounded mixture of refined asphaltum and silico- calcareous sand, in which the calcareous ingredient is finely pulverized limestone. As the material for first-named pavement comes principally from vicinity of Neufchatel, the pave- ment is known as the Neufchatel. Asphaltum for the other pavements referred to comes from Island of Trinidad, and the pavement is sometimes called the Trinidad asphalt. Neufchatel pavement. Has been extensively laid in London, Paris, and other Eu- ropean cities. Although these two pavements represent the best type of street surface, there is a characteristic and somewhat important difference between them, due to the fact that the Trinidad contains nearly 75 per cent, of sharp silicious sand, and does not, therefore, become polished and slippery by wear; while the Neufchatel, being composed entirely of bituminous limestone (a species of amorphous pulverulent chalk, without grit, impregnated with bitumen), is by no means free from this fault. A variety of asphalt pavement adapted to streets of exceptionally steep grade, is one Sbrmed with rectangular blocks of compressed asphalt concrete. Comparative Merits of* the Several Pavements. 1. Their First Cost. In cost of construction, wood is the cheapest; Coal tar com- position second; Sheet asphalt like the Trinidad third; Stone-blocks fourth, and Asphalt blocks fifth. 2. Their Durability. Assuming each of the four pavements named to be the bes* of its kind, stone and asphalt will possess the longest life, and wood and coal-tar very much the shortest. Between the first two and the last two there is a wide gap. Unless the stone be of good quality, asphalt will take first place and stone second. 3. Cost of Maintenance. Order of merit under this head would place stone and asphalt first, and wood and coal-tar last. If the asphalt is good, well mixed and laid, the stone must be both tough and hard in order to maintain the first place. Relative Loads for Roadways and Pavements. At Low Speed. (J. W. Howard, C. E.) Loads which a Horse can draw on a level, each day of 10 hours, on following roads. Roadway. Lbs. Resistance of Load. Roadway. Lbs. Resistanc in Term of Load. Asphalt 6oCK Hard Earth Stone Block uoy^ 3006 U 37 .076 Worn Stone Block .191 Ordinary Stone Block. 1828 . 124 Cobble Stone .2 Hard Macadam Hard Gravel I39 1 I27Q .164 .178 Ordinary Earth Sand...' 456 228 5 i. 946 APPENDIX. Sn"b-M:arin.e Torpedoes. Formula for Determination of Pressure per Square Inch of Various Explosives at Different Distances. 3 //66 3 6(A+E)C\ 2 V\ (D + .oi) 2 -' / = P. A representing angle with the vertical passing through the centre of the charge, made by a line drawn from it to the surface exposed to the shock, determined from the nadir,* in degrees; E a constant for the explosive, as de- termined by experiment; C weight of the explosive in Ibs.; D distance from centre of the explosive to the surface exposed, in feet ; and P the mean pressure, corresponding to that which would be transmitted to a disc of copper, by a Rodman indenting -tool, per square inch of surface exposed to the shock, in Ibs. (Brev. Brig. General H. L. Abbott, U.S.A.,i88i.) Value ofE,or Relative Strength of Explosives Fired under Water. 4 I'o K ln < o S | S ' =.. i.- Explosive. 4 W i 11 P 5 oo l! Explosive. S bi w fc 11 l-r 1 ii X ^ < Dualin 232 116 III 108 Forcite No. i . . 333 Dynamite No. 1 1 186 118 No. 2 . 75 36 1 20 75 83 88 Rackarock . . . 220 _ Explosive Gelat. 89 259 125 117 "3 Nitro glyc'ne . 100 in 71 81 86 Gun-cotton 81 8 7 Rendrock... . 20 IOI 67 78 84 Electric No. i . . . 33 67 Si 69 77 " ... . 40 1 60 9 1 94 95 u No. 2... Hercules No. i . . 28 77 43 211 38 109 62 106 72 105 Vulcan No. i . 60 30 1 66 99 95 78 f 3 " No. 2 . . 42 118 74 83 8 7 " No. 2 . 35 114 72 82 86 ILLUSTRATION. Assume the distance between the line of the centre of a charge of dynamite No. i and the bottom of a vessel to be 5 feet, the angle between the line of centre of the distance and the bottom, measured from the nadir, to be 180, the constant for the charge 186, and its weight 100 Ibs. What would be the mean pressure on the object in Ibs. per sq. inch? A = 180, E = i86, C = 100, and D = 5. 3 // V\ 66 3 6 (180+186) i 2428 736 Xioo\ _ ; ) 29.489 * A point of the globe directly under our feet, or that opposite the zenith. f Standard of comparison. J For (5 + .oi) 2 - 1 , see p. 310. Thus, ' = X log. 5.01 = 2.1 X .699 837 = Numbtr 29.489 When the Object is not in a Vertical line with the Explosion. ILLUSTRATION. Assume a charge of gun-cotton weighing 882 Ibs., set in water, at a horizontal distance of 24, and a vertical of 86 feet from the object; what would be the effect? To obtain A, or angle of divergence, 180 Tan. = 15 25', and 180 86 i 5 o 35' = 164 35' = 164.58. D = \/24 2 -f86 2 = 89, and E = 135. Hence, p / /66 3 6(i6 4 . 5 8 + 135) 88 2 \ Log. of 6636 = 3. 821 906 " " 164.58 + 135 = 2.476513 " " 88a =2.945469 Product =9.243888 4.093822 Quotient =5.150066 Log. of 89 -f oi = 1.949439 2.1 '949439 3898878 Log. of 89.01 2>x = 4.0938219 3110.300132 Log cnbe root of Quotient = 3.433 37 8 = Number 2712.5 Z&x.ss APPENDIX. 947 Efficiency of Water-Tribe Steam-B oilers. In a late test by J. J. Thorneycroft of his patented boiler, the following elements and results are reported to the Institute of Civil Engineers. See Vol. XCIX., 1889. Engine. Triple expansion, Cylinders 14, 20, and 31.5, by 16 ins. stroke of piston, and jacketed. Independent engines for Circulating pump, Blower, Donkey, and Sheering. All exhausting into engine condenser. Results of* Trials. Furnace. Elements and Dimensions. Natural Draught. Blast Draught. 26.2 26.2 Heating " " " surface to grate 3 1837 61.2 1837 1837 61.2 1837 6l. 2 1837 Pressure of steam in boiler, p'r sq. in. 200.8 70.1 196.3 1 86 164.2 7O.I 194.9 " blast in fire- room in ins. 27 49 2 Revolutions of engine per minute. . 192.8 165.2 234.2 268.7 318.4 Coal per sq. foot of grate per hour. . Water evaporated from and at 2 1 2 ) per Ib. of coal, ash utilized J II. I 7-74 11.22 18.6 10.48 29.8 10.2 66.8 8.89 Do. do. per Ib. of carbon.. 13.08 12. 18 II.7 10.04 Do. do. per sq. foot of) heating surface per hour j 1.24 3-2 4-7 8.05 Temperature of gases in chimney.. 44 of air in fire-room.... 474 4210 69.30 540 71.4 610 ^0-3 ? ?'6 Fuel per IIP per hour. 2.28 1.981 2.28 1.99 ' jjp " " 150.3 89^1 2.03 282.1 2.04 AAQ 2 2.32 Efficiency of boiler per cent i 86.8 81.4 .84 % .42 .- 3 8 Water used for jacket per IIP per) hour in Ibs. . . ... 1 Fuel. Calorific value of 14900 thermal units per Ib., equal to 1.025 of a Ib. of car- bon. Each Ib. of coal, if completely consumed, is capable of evaporating 15.41 Ibs. water from and at 212. Barbed. Steel-wire JETencing. (Galvanized or painted.) J. A. Roebling's Sons Co. , New York. Four points, barbs 6 inches apart, 15 feet = i Ib. 4< 44 44 2 44 44 ,2 44 __ j t( On Spools. 15 feet in length of the regular measures and 12 feet of the thickset, weigh each one Ib. Spool, about 18 x 18 X 17 ins., measuring 3.5 cube feet, weighing from 60 to 100 Ibs., and length of wire ordinarily 1500 feet. Thickset or Hog weighs .2 more. To Compute Volume of* Boards that can toe Sawed out of a Round Log. (M. J. Butler, C.E.) RULE. From diameter of log in inches subtract 4, multiply remainder by one half of it, multiply proceed by length of log in feet, and divide product by 8 ; result will give number in feet. d~^4 X X Z -r- 8 = V. d representing least diameter in inches, I length of log itfeet, and V volume in feet of board measure. ILLUSTRATION. Assume a log 30 ins. in diameter and 15 feet in length. 30 4 X 26 -T- 2 X 15 -5- 8 = 633.75 feet B.M. Foot-Pound When for Unit of Work Is i Ib. lifted, thrust, or projected through i foot, against gravity or inertia, and is expressed in pounds or tons, with- out regard to the period of its action. When for Unit of Rate of Work Is i Ib. liftod etc., as above, i foot In a given period, as in i second or minute. 948 APPENDIX. TVire Rope.* Galvanizing decreases strength of unannealed wire 5 per cent, and its ductility 15 per cent. Breaking Weight of No. 20, B W G (.035 in.) crucible steel rope of 6 strands, 1.75 ins. in circumference: Wires, 78 to 102 tons per square inch, and Ropes 5.75 to 10.47 tons. Annealed Wire is not affected by galvanizing, but its ductility is reduced from 179 twists to 58, = 68 per cent. Annealing Wire reduces its strength 45 per cent., but increases its elasticity 77 per cent. Tensile Strength of crucible steel wire averages 85 tons (80 to 90) per sq. inch. Permanent set, Bessemer iron wire 12 tons per sq. in., or .25 of ultimate tenacity. Variation of tensile strength of like pieces of steel wire, galvanized or plain, is but 3 per cent, for the former and 8 for the latter Modulus of Elasticity (ME). Iron wire 22400000, Steel 35000000, and crucible Steel 33 ooo ooo. Bending. Stress due to it, in a wire of the material and dimensions given. ME = 32 ooo ooo. Diam. of pulley 10.5 13-125 16.875 18.75 24 ins. Stress per sq. inch 50 40 31.4 28 . 2 22 tons. Durability. Life of steel wire ropes over iron pulleys, of material and dimen- sions above. Number of times rope passed over the pulleys without Breaking. Load 1568 Ibs. Ins. Ins. Ins. Ins. Ins. Ins. Ins. Diam. of pulley ... 5.25 7.875 10.5 13.125 16.875 18.75 24 Number of times. . 6075 10300 16000 23400 46800 72700 74100 53 ioot 85 2oof 392 soot 336 600 Over Pulleys 24 Inches in Diameter. Load 1568 Lbs. Manufacture of T. & W. Smith. BW G Diameter. Number of Bends be! Wire and strands laid in opposite direction. 1-24 Inch. 3-24 Inch. ore Breaking. Wire and strandu laid in same direction. 3-24 Inch. Ordinary crucible steel Patent improved steel Plough steel No. 20 2O 20 19 18 22 Ins. 035 035 035 .042 .049 .028 No. 74100 96000 109000 66000 87000 III OOO No. 51000 57000 54000 32000 47400 J.8 700 No. 26000 42800 34400 79000 17 zoo 2O7OO Iron wire Crucible steel Crucible steel... NOTE. By author: diameter of pulleys should be = 10 circumferences of rope. Tenacity of* Dovetails. White Pine, 6 inches square. Notch in Length equal to Depth of Timber. S and D each representing proportion or depth of cuts to width of S Destruction. .25 3. 9 tons. 33 5-75 " .41 5.1 Destruction. D . 125 6 tons. .167 6 " .208 6 " Greatest strength in a double dovetail is attained when D = . 167, and in a single, when S = .33. (Gen? I O. M. Poe, U. S. E. ) Shafting for Lathes and Mills. Diameter. Should be given in inches or quarters only. Length. Not to exceed 20 feet. Velocity. Machinery, 125 to 150 revolutions per minute; Woods, 200 to 300. Power. Applied at middle of length of shaft whenever practicable. Hangers. With adjustable boxes, in order the easier to maintain a shaft in line. * From a ppr by A. S. Biggart. Ins'n C.E. f Long's patent lay. APPENDIX. 949 Cost of Sawing and. Dressing Stone. Sa-wing. Per Cube Foot. Bedford Stone. 20 cents'. At Chicago, Soft, medium, 8 to 10 cents; Limestone, Magnesian, and Oolites. Medium, 13 to 17 cents; Marble and Grranite, Hard, 25 to 30 cents. Rate in 10 Hours. Ins. Granite Bluestone 8 36 to 40 Marble, Tenn. . . " Vermont. Brownstone I Limestone 20 tO 25 I magnesia, oolite Ins. 10 to 15 36 40 to 70 NOTE. Depth of cut without reference to its length or number of saws. (R. J. Cooke.) Dressing. Per Square Foot. Labor $ 3 per Day. Hard Limestone. Bush hammered, rough, 25 cents; Medium work, 30 cents; Fine work, 35 cents. Cost of Raising "Water. 1OOOOOO Imperial or 1 SOO OOO TJ. S. Grallons 1 Foot. Average 0/15 Years. Low Service. I High Service. By water, 1.23 cents. By steam, 13.2 cents. | By steam ____ 25 cents. of Drifted Bolts. Steel, One Inch in Diameter. Hole Six Inches in Depth. Mean Holding Resistance per Lineal Inch. Wood. Hoi* i5-i6tha. Hole i4-i6ths. Hole i3-i6ths. Hole i2-i6ths. Hole i5-i6ths. Ra Hole i4-i6ths. ios. Hole i3-i6ths. Hole i2-i6ths. Yellow pine White oak... Lbs. 3 6i 1 300 Lbs. 616 1778 Lbs. 761 2AQQ Lbs. 400 im Lbs. 47 .^2 Lbs. .8 .71 Lbs. i Lba. 53 .As Hemlock in i5~i6ths hole 415 Ibs. per lineal foot to withdraw it, and White or Norway Pine i2-i6ths hole 830 Ibs. To obtain maximum holding resistance of timber, diam.of hole to bolt as 13 to 16. Relative holding resistance between driving parallel or perpendicular to the fibre is as i to 2. (J. B. Tschamer.) Resistance of tlie Air to Falling Bodies. Falling Body Lead Ball, 2 ins. in Diameter, Weight i Ib. Body Falling Horizontally. Weight i Ib. In Vacuo. In Air. One Foot Square. Two Feet Square. Final Retar- Final Retar- Final Retar- Veloc- ity. Fall. Veloc- ity. Fall. dation per Sec. Veloc- ity. Fall. dation per Sec. Veloc- ity. Fall. dation per Sec. Feet. Feet. Feet. Feet. Feet. Feet. Feet. Feet. Feet. Feet. Feet. 16 32 30 15-5 5 28 14-33 1.66 13-33 13-33 2.66 48 64 55 43-5 4-5 35 33 15 24 37-3 24 80 96 77 6 7 .5 12.5 38 38.5 41-5 61 66.66 (P. H. Van Der Weyde.) Retardation is Inversily as Density of Body. Velocity after fall of one second becomes measureably uniform; the increased velocity being balanced by the in- creased resistance. Resistance of the air at moderate velocities, to the velocity of a fall ing body, is as the square of its velocity. Thus, when the velocity is doubled, the resistance is quadrupled, and when trip- led, nine times greater. Applicable alike to a cannon ball in air or a body in water. 4 950 APPENDIX. Cost of a Horse-l?ower "by Steam.* Joule's equivalent (p. 504) = 772 units = heat required to raise i Ib. water i, = elevation of i Ib., i foot high. Unit of evaporation, to evaporate i Ib. water to steam at the pressure of the at- mosphere = 966. i British thermal units. Horsepower 33 ooo Ibs. *? 42.75 heat units =. i IP i , and 42.75 x 60 min. =. 2565 per IP per hour. 25 5 r= 2.655 Ibs. water required to generate i EP. : \\ *? "^ } * other fueU see p. 4 86. Then I4 * = 15-01 I 900.1 . water evaporated per Ib. of coal. Hence, = .1769 Ibs. coal per IP per hour, or ' = 5.65 IP per hour per Ib. of coal. Assuming in all of the above, the normal condition, that there is neither expen- diture of water or temperature in the operation. Operalively. From elements furnished, in part by Thos. Pray, C.E., the cost of a IIP at the pressures and expansions given is as follows: Coal at $ 3 per 2000 Ibs. Engine. Non-condensing. i8"X42" 58.5 Ibs. 4. 74 ins. 8. 12 Ibs. 9.99 " 18.59 " 2.07 " 6. 21 cents. Initial pressure of steam ..................... 83.6 Ibs. Cut off ...................................... ins. Terminal Pressure ........................... 8.71 Ibs. Evaporation per Ib. of coal ................... 10.31 " *' " IP .......................... 16.84 " Coal per IIP per hour ........................ 2.28 " Cost per IIP per 10 hours .................... 2.6 cents. A Condensing Pumping Engine has been operated at a cost of 2.28 cents. * For Horse-powtr tee pp. 441, 733, 758, and 014. Cost of Water IPower on Driving Shaft. Per IP. Power is variable, depending upon variation in head of water, as when it is de- In order Jlvereu ill a river nuujeui. i/u noc uy ncoucto, uu&i/ ui watei, Him ui piauu. iu uiuer to attain an average daily power, the power must be increased to meet the loss of head by back water in freshets. * a ~* & Cost ijs-s o?T5 Cost LOCATION. j.p 3 a >J E 03 per LOCATION. H? 5 ri >> Ja per si" 3 * IK fr 0^^ ^ = H' No. Feet. Feet. $ No. Feet. Feet. $ Manchester. N. Y. . 890 30 30 44 Lowell, Mass. . . . IOOO 575 13 IOO Lawrence, Mass. . . IOOO 490 28 42 " " . . IOOO 290 18 57 Cost of a looo IP Plant independent of cost of water about $ 45 per H?. Head. Co From TOO at O] Supply f a 1 to Disci 3o ike large in 400 Pla Feet. 500 txt n 600 nd.e Head. r di] From IOO ffere Supply 200 nt I to Disci 300 leac large in 400 Is. Feet. 500 600 Feet. 10 20 $ 9 I 38 1 no 46 125 54 140 62 155 70 170 77 30 40 26 20 32 25 39 3i 45 37 5i 42 11 At Lawrence and Lowell, a Mill Power 30 cube feet of water per second, with a head of 25 feet At Manchester it is 38 cube feet, with a head of 20 feet. (Chas. T. Main, M.E.} For Power of a Mill Wheel see pp. 565, 566. APPENDIX. 951 Steam Plant. Daily and. Yearly Cost of Coal and. Hia"bor in Operating a Plant of 1OOO H>. Year of* 308 days of 10 . 35 hours, coal at &3 per ton of 2OOO Ibs. Deduced front Reports ofChas. T. Main, M.E. ENGINB. Exhaust steam used. Coal per HP per hour. Attenc per Boiler. ance and HPper I Engine. Stores >ay. Stores. Coal per IP per day. fDaily per HP fDaily for looo H? Yearly. Per cent. Lbs. c c e 9 9 9 9 ( 1-75 53 .60 25 2.14 3-52 3520 10841,60 Compound.. 25 1-5 45 .60 25 1.84 3-16 9732.80 1.25 38 .60 25 2.76 2760 8500.80 2-5 75 .40 .22 3.06 4-43 4430 13644.40 Condensing . (25 2.06 .62 .40 .22 2.52 3-76 3760 11580.80 (So 1.63 49 .40 .22 2 3-" 3110 9578.80 Non-Con- $ 3 .90 35 .20 3- 6 7 5-12 5 120 15769.60 densing. . . 25 (50 i.'88 73 .56 35 35 .20 .20 2-99 2-3 4.27 3-47 4270 347 13151.60 10687.60 * For heating. f Including coal. Yearly Cost of 1OOO EP and of a EP. Year of 308 days of 1O.2S hovirs. Coal at $3 per ton of SOOO IVbs. Deduced from Reports of Chas. T. Main, M.E. ENGINE. *Exhaust Steam used. Engine and House. fOperat- ing Ex- penses. Boiler- house and Shed. fOpera- tor's Expense. tCoal and Labor and Stores. STotal per IP Compound... Condensing . . . Non-Condens- ing. . . Per cent. ( r 5 (so r 5 (50 {25 * 40 40 40 33 33 3i 29.50 29.50 29.50 1 able. $ 5-02 5.02 5-02 4.14 4.14 3-95 3-7 3-70 3-7 Injector, r Not inc 18.36 16.16 13.90 24.80 21.12 17-33 24.28 19.46 Depreciation uding Cost I 2-50 2.20 1.89 Hi 2.36 3-95 33' 2.65- , Taxes, Int of Plant in 10841.60 9732.80 8500.80 13644-4 11588.80 9578.8o 15769.60 13151.60 10687.60 erest, and Insu column 3 and 18361.60 16952.80 15410.80 21 164.40 18608.80 16888.80 23 419- 6 20161.60 17037.60 ranee. 5- (50 * For Heating. t As per previous T Sugar in Mortar. It has been demonstrated that the addition of saccharine matter to lime-mortar is very beneficial, as it enables it to be laid in frosty weather. It is claimed also that it causes the mortar to set very soon and strengthens it, and that it can be laid with dry bricks. As sugared water dissolves lime, it is necessary to dissolve the sugar first, and then add the water to the lime slowly and cautiously. The mortar should be very stiff. Proportions. For mortar, coarse brown sugar, 2 Ibs.; lime, i bushel; sand, e bushels. If sugar is added to mixed mortar, it renders it too thin. (Manufacturer and Builder.) Belting. Speed of belts, single and double, i inch in width, should not ex- ceed for the first, 800 feet per minute, and for the second 500 feet, each = one H*. Railroad. Speed.. London, North Western, and Caledonian. London to Edinburgh, 400 miles Speed, 55.4 miles per hour ; 3 stops 50.9 miles. Engine, tender, and car?, 348 ooo Ibs. Chicago, Burlington, and Quincy.i+.8 miles in 9 minutes. 952 APPENDIX. Cost of Irrigation, per Acre. California. From $7.18 to $53.33. Colorado. $3.7510 $10.80. Utah France. Average of several, $ 58. India. Average of several, $ i. 75 to $ 10. Alloy That expands in cooling: Lead 9 parts, Antimony 2, Bismuth i. Extremes of Temperature. Artificial, 135 (Faraday}. Atmosphere, 77 (Back). Extension of Woods "by Water, (de Volson Wood.) Elongation. Pine 065 Lateral. Pine 2.6 Oak 085 Chestnut.. .165 Oak 3.5 Chestnut.. 3.65 Smokeless Powder. Gun 6 ins. in diam. Charge 17.64 Ibs. Energy at muzzle, 4609 foot-tons. Per Ib. of powder 139.7, and per weight of gun 720. "Volume of "Water Flo\ving over Niagara Falls. 270000 cube feet per second. Since 1842, Horseshoe Fall has receded 140.5 feet, and American 36.5 feet. (J. Bogart, S. E.) ROOFS. To Compute Stress on Roofs. Velocity and Pressure of Wind. RULE. Multiply square ofvelocity of wind in feet per second by .0023, or square of its velocity in miles per hour by .005, and product will give pressure in pounds per sq. foot. Or, v 2 X .0023 = P, and V 2 x .005. Also, .0023 v 2 sin. x = P. P representing pressure per sq. foot in Ibs., x angle of incidence of wind with plane of surface in degrees, V velocity of wind in miles per hour, and v velocity in feet per second. Direction of wind usually makes an angle of 10 with the horizon, hence 10 is to be added to horizontal plane of direction of the wind. ILLUSTRATION i. Assume wind with a velocity of 100 feet per second to impinge upon a plane roof set at an angle of 45; what would be the pressure per sq. foot? Sin. 45+ 10 = .819. .0023 X ioo 2 X .819 = 18.837 #> 2. Assume the wind to have a velocity of 150 feet per second, and angle of roof 60; what would be the pressure per sq. foot? Sin. 60 + 10 = .94. .0023 x 150 x .94 = 48.75 Ibs. Pressure of Snow. This pressure decreases per square foot in Ratio of half space, to length of rafters, or height divided by space. Pressures for "Various .A.ngles or Ratios. At 15 Pounds Weight per Square Foot. h-4-8 Degrees. Lbs. h-i-S Degrees. Lbs. h-i-s Degrees. LU. 5 33 25 45 33 40 26 34' 10. 6 12.6 i3-4 .2 .14 17 45' 15 39' 13-9 14-3 14.4 125 .11 .10 14 2' 12 31' 11 19' 14-5 14.6 14.7 "Weights on Roofs. Single tiles 20 Slates, ordinary 15 Asphalt on slabs 20 Paper, tarred 6 Per Sq [ron, shee Zinc, shee Slates on Iron, shee jtare Foot in Lb t 8 t 8 y. Iron, corrugated, on iron 4.3 Zinc " " 4.7 Snow 20 Wind 10 ron 10 t on iron . . 5 APPENDIX. 953 Comparative Operations of a Simple and a Compound Locomotive. Brooklyn and Union Elevated Railway of Brooklyn, N. T. Forney Type. ELEMENTS. Simple. Compound. ELEMENTS. Simple. Compound. Cylinders, ins Drivers diam . 11X16 42 ins 11.5 18X16 42 ins Coal per car mile. Water 11.05 IDS - 26 070 lbs. 6.88 lbs. 19 862 lbs. u revolu- 1 Gain in fuel 07 7^ tions per mile } Boiler diam 480 42 ins. 480 42 inc. Evaporation ) from 212) 8.09 lbs. 9. 97 lbs. Flues O D i 5 ins Gain in water 23 8 Number .... 124. 124 Water per car ) Exhaust tip, diam. Grates, water Area, sq. feet. . . Heating surface, ) sq feet J 3. 25 ins. 15^6 289.46 289.46 mile} Pressure of ) steam, ave' J Revolu's per min. Miles per hour. . . 73.85 lbs. 136 lbs. 56.27 lbs. 136 lbs. 222 27.73 Ratio of do. to) IP 223.6 grate J 18.5 18.5 Weight loaded . 45 35 45 850 Coal . . . 1800 lbs. 24^0 lbs. Miles run.. . 122 122 High. Explosives. Firing l*oint and Relative Strength. DESIGNATION. Firing Point. Order of Strength. DESIGNATION. Firing Point. Order of Strength. Expl Gelat (Vou^e's) Degree, jfic 106 IT Tonite Degrees. 68.24 Helluofrite 106. 17 Bellite 65.7 Nitre-glycerine (old) -,6c IOO Rack- a- rock . 61.71 ' ' fresh Q2-37 Atlas powder. 60.43 " French . Sm 'less Powder (Nobel) Gun-cotton 1889 346 81.85 92.38 83.12 Ammonia, dynamite. . Volney's powder No. i " " No. 2 60.25 58.44 53- 18 " laboratory Si.qi Melinite 50.82 Dynamite No. i Emmensite No. i Oxinite fr. Pieric acid. Amide powder 301 81.31 77.86 %% Fulminate, silver. I 4*f , mercury. . Mortar powd. , Dupont Forcite No i. 315 500 30Q 50.27 49-9* 23- *3 (Lieut. W. Walt cc, U. S < A.rmy.) Centrifugal IPuimp. To Compute tlie Required Velocity of tlie Outer ICdge of the Blades. When the Height of the Required Lift of Water is Given. The edge of the blades must have a velocity at least equal to that acquired by body falling from the given height. Then, to lift water | and sand 20 i water ) , / > feet. ) V 2 g h = V 64. 4 X 20 = 35. 89 feet. Comparison of Operation and Cost of a Q-as and, Steam lEngine. (In addition to page 587. ) Elements. Gas.* Steam. Brake IP 76 75 ( Generator, 70.5$ Boiler, 72^ " " or per cent, of heat in) power, to total heat generated., j Mechanical efficiency of motor Power to operate engine (do. and engine, 12.7^ 18* 6 9* and engine, 7^ 9-75# 75* os-a; Coal for B BP per hour 3 1 /* i 34 lbs. 2 6 lbs ^pace occupied, including gen- ( erator or boiler ) 470 sq. feet. 360 sq. feet Professor YVitz. Bryan Don, 954 APPENDIX. STEAM-ENGINES. Compound. Duration of Operation 2 Hours. Cylinders. 5.5, 9, and 15.5 ins. in diameter. Stroke of piston Revolutions. 150 per minute IIP 40. Boilers. Fire tubular. Tubes, 38 of 2 ins. ; 6.25 feet in length. Heating Surface. 158 sq. feet. Grates. 5.7 sq. feet. Pressure of Steam. 175 Ibs. per sq. inch. Water. Weight consumed, 1 140 Ibs. Evaporation per Ib. of coal, 9. 8 Ibs. Drawn from jackets, 84 Ibs. Consumption per IP per hour, 1.425 Ibs. Temperature of feed, 55. Consumed 116 Ibs. per IIP per hour 1.45 Ibs. ; per sq. foot of grate 10.2 Ibs. Indicator Diagrams. Mean IP 54=13.31 IIP; Intermediate 18= 12 IIP; Con- densing 7. 5 = 14. 7 IIP = 40. Fly-wheel. 5.5 feet in diameter and 10.5 ins. in width. Weight of Engine and Boilers, without water, 14 560 Ibs. Builders. Marshall & Co., Kreigly, Eng. PUMPING ENGINE. Vertical Compound. Cylinders. 34 and 66 ins. in diameter by 60 ins. stroke of piston. Pressure of Steam. 74.81 Ibs. per sq. inch. Vacuum, 26.25 ins. Revolutions. 25. 51 per minute. Grate Surface. 70 sq. feet. Pressure of Water by Gauge. 62.02 Ibs. Head, including lift, 155. 17 feet = 67.62 Ibs. Fuel. 675 Ibs. per hour. Duty. 104 820 431. Stack, in height, 125 feet. Constructors. The Edward P. Allis Co., Milwaukee, Wis. ELECTRIC DYNAMO ENGINE. Triple Expansion. Arc Lights. 500. Water entrained in steam 7.39$. Cylinders. 14, 25, and 33 ins. in diam. by 48 ins. stroke of piston. Condenser. Separate. Circulating Pump, 16 X 16 ins. ; Air-pump, single-acting, 24 X 16 ins. Cylinders, 12 x 16 ins., operating both pumps. Revolutions, 61.29. IIP 16.4. Pressure of Steam. 125 Ibs. per sq. inch; Revolutions, engine, 99.12; Steam per IIP per hour, 12.94 lbs - Iff 5 l6 - Injection Water. 72. Reservoir, 90. Constructors. The Edward P. Allis Co., Milwaukee, Wis. Railroad. Signals and. Significations. Stop," one pull of bell-cord. Go ahead," Two pulls. Back up," three pulls. Down breaks," one whistle. " Off' breaks," two whistles. "Back up," three whistles. "Danger," continued whistles. "A cattle alarm," rapid short whistles. Go ahead," a sweeping parting of the hands, on level with the eyes. Back slowly," a slowly sweeping meeting of fhe hands, over the head. Stop," downward motion of the hands with extended arms. 'Back," beckoning motion of a hand. ' Danger," a red flag or light waved up the track. 'Stop," red flag raised at a station. 'Start," lantern at night raised and lowered vertically. 'Stop," lantern swung at right angels across the track. Back the train," lantern swung in a circle. Metropolitan Opera House, Ne-w York. Capacity. Seating, 3600. Standing, 400. If the saloons attached to the private boxes were removed, the total capacity would be 5000. APPENDIX. 955 Distillation, of Fresh. TVater. Process of G. W. Baird, U. S. Navy, New York. Marine Steamers for long voyages, operated under a high pressure of steam, are necessarily provided with Evaporators, to replace the water expended in leaks and vents, and to provide for the ordinary requirements for fresh water. This process is an improvement upon existing methods, inasmuch as it furnishes the water potable, and it is as follows: The Evaporator contains a series of tinned metallic coils and a volume of sea- water; which is designed to be evaporated by the passage of steam from the engine boilers through the coils. The water condensed in them is returned to the boilers; the water vaporized from the sea- water, external to the coils, is either led to the Engine condenser, to replenish that lost by leaks and vents, as from gauge cocks, etc. ; or if required for potable purposes, is led to a Distiller, where it is aerated, condensed, and filtered, from which it is drawn for use. As the sea- water is evaporated in vacuo, vaporization occurs at a temperature below that at which much scale is precipitated. Hence the shell and coils are both measurably free from it. Results of* tn Experiment. Pressure in coils, 20 Ibs. above atmosphere; temperature of steam in coils, 259.3; temperature of feed water, 131.66; temperature of the water vaporized, 212; water vaporized per hour, 103.33 Ibs. ; water condensed in the coils per hour, 112.12 Ibs.; total heat in the steam, 1193.7, and in the water "vaporized, 1178.6. Capacities of* Kvaporators and. Uistillera. Gallons per day 0/24 hours. No. i 2 Evapo- rator. Dis- II tiller. No - Evapo rator. Dis- tiller. No. 4 4-5 Evapo- rator. Dis- tiller. Gallons. 600 I2OO Gallons. 600 3 1200 U 3.5 Gallons. 2OOO 2000 Galloni. 1600 1600 Gallons. 3000 3000 Gallons. 2OOO 2500 I Evapo- Dis- rator. tiller. Gallons. 4000 6000 Gallons. 2500 3000 Coal iProdiaotion and. Consumption Of the World Per Diem. Production. Estimated at 3 360000000 to 3 696000000 Ibs. ooo ooo Ibs. ; Smelting 20000000 Ibs. ; Do- Consumption. Generation of steam, Land and Marine, 62400 Iron Ore, 28800000 Ibs. ; other metals, 23000000 Ibs. ; Forges, mestic use, 57 600000 Ibs. Total, 2 700000000 Ibs. Corrosion of* "Wrought Iron. The purer the water, the more active it is in corroding and pitting Wrought iron plates. This arises from the greater presence of air in pure water, and hence a greater proportion of Oxygen. (Locomotive. ) Earth. Boring and. Heat of Mines. Sperentoerg, near Berlin. Bore, 4172 feet in depth, about 1000 feet in ex- cess of Artesian well at St. Louis. In lower levels of some of the shafts in the Omstock mines, prior to the draining into the Sutro Tunnel, the water was at a temperature of 120. Preservatives of Iron. Pitch, Black Varnish, Asphalt and Mineral waxes are among the best, provided the acid and ammonia salts, which frequently occur in tar and tar products, are removed. If in addition these substances are applied hot to warm iron, the bituminous and asphaltic substances form on the surface of the iron an enamel, which, unlike to other coatings, is not microscopically porous, and consequently it is impervious to water Spirits and Naptha varaishe* are injurious. (Prof. Lewis.) 956 LIGHTNING-CONDUCTORS. Code of R-ules for the Erection, of Lightning-Con- d. victors. Lightning-rod Conference. Points. Point of terminal should not be sharp not sharper than a cone of which the height is equal to radius of its base. A foot lower down a copper ring should be screwed and soldered on to the upper terminal, in which ring should be fixed three or four sharp copper points, each about 6 inches in length. It is desirable that these points be platinized, gilded, or nickel-plated. Upper Terminals. Number of conductors or points to be specified will depend upon size of the building, material of which it is constructed, and comparative height of the several parts. No general rule can be given for this. Ordinary chimney-stacks, when exposed, should be protected by short terminals connected to the nearest rod. Insulators. Rod is not to be set off from building by glass or other insulators, but attached to it by metal fastenings. Attachment. Rods should be led down the side of building which is most ex- posed to rain. They should be secured firmly, but the holdfasts should not pinch the rod, or prevent contraction and expansion. Factory Chimneys. Should have a copper band around the top, and stout, sharp copper points, each about i foot in length, at intervals of 2 or 3 feet throughout the circumference, and the rod should be connected with all bands and metallic masses in or near the chimney. Ornamental Iron-work. All vanes, ridge-work, etc., should be connected with conductor, and it is not absolutely necessary to use any other point than that afforded by such ornamental iron- work, provided the connection be perfect and the mass of iron considerable. Material. Copper, weighing not less than 6 ozs. per foot in length, and the con- ductivity of which is not less than 90 per cent, of that of pure copper, either in the form of tape or rope of stout wires, no one wire being less than No. 12 B.W.G. Iron may be used, but should not weigh less than 2.25 Ibs. per foot in length. Joints. Bad joints diminish the efficacy of the conductor; therefore every joint, besides being well cleaned, screwed, scarfed, or riveted, should be thoroughly sol- dered. Protection. Copper rods to the height of 10 feet above the ground should be protected from injury and theft by being enclosed in an iron pipe reaching some distance into the ground. Painting. Iron rods, whether galvanized or not, should be painted; copper ones may be painted or not. Curvature. Rods should not be bent abruptly. In no case should the length of it between two joints be more than half as long again as the line joining them. When a string-course or other projecting stone- work will admit of it, the rod should be carried through, instead of around, the projection. In such a case the hole should be large enough to allow for expansion, etc. Masses of Metal. As far as practicable it is desirable that the conductor be con- nected to extensive masses of metal, such as hot- water pipes, etc., both internal and external; but it should be kept away from all soft metal pipes, and from in- ternal gas-pipes. Bells inside well protected spires need not be connected. Earth Connection. It is essential that the lower extremity of the conductor be buried in permanently damp soil ; hence proximity to rain-water pipes and to drains is desirable. It is a very good plan to bifurcate the conductor close below surface of the ground, and adopt two of following methods for securing escape of the lightning to earth. A strip of copper tape may be led from the bottom of the rod to t?he nearest gas or water main not merely to a lead pipe and be soldered to it; or a tape may be soldered to a sheet of copper 3 feet x 3 feet and .0625 inch thick, buried in permanently wet earth, and surrounded by cinders or coke; or many yards of the tape may be laid on a trench filled with coke, taking care that the surfaces of copper are, as in previous cases, not less than 18 square feet. Where iron is used for the rod, a galvanized iron plate of similar dimensions should be employed. Inspection. The conductor should be satisfactorily examined and tested by a qualified person, as injury to it often occurs up to the latest period of the works from accidental causes and carelessness. Collieries. The head-gear of all shafts should be protected by proper lightning- conductors to prevent explosion of fire damp by sparks from atmospheric elec- tricity being led to the mine by the wire ropes of the shaft and iron rails of the galleries. STONE BREAKER. CRUSHER. STEAM HEATING. 95 f Stone Breaker and. Ore Cruslier. Stone Breakers and Ore Crushers are used in mak- ing Macadam for construction of roads ; material for concrete; ballasting railroads, crushing ores, quartz, corundum, and all brittle substances ; they can be ad- justed to pass a mass from the size of a pea to larger - diameters, depending upon the capacity of the machine. Crushed to Cubes 0/2 Inches. Per Hour. No. Receiver. Volume. Extreme Weight of Stone. Weight Produced. ] Length. )imension Breadth. c ' Height. Pulley. Speed. IP Ins. Cub. yds. Lbs. Lbs. Ft. ins. Ft. ins. Ft. ins. Ins. No. i 3X 1.5 40 IOO ! I . 6 .10 5X i 250 5 2 6X 2 I 5 6o I 200 2.10 2. I 2- 3 nX 5 250 4 3 ioX 4 3 1800 4900 4 3- 3 3- 9 2oX 6 250 6 4 ioX 7 5 3800 7800 5- i 3- 9 4- 5 2 4 X 7-5 250 8 5 i5X 9 8 7400 15500 6. 6 5 5-ii 3oX 9 250 15 6 15X10 9 7800 16000 6. 6 5- 5 5-" 30X10 250 i5 7 2oX 6 10 53 II 200 5- 3 2. II 4. 6 30X10 250 15 8 20X10 10 8100 18 300 6.10 5 -.9 5." 36X12 250 20 9 12X30 16 14200 33000 7.10 8. 4 6. 4 36X12 250 30 10 15X30 20 14200 35000 7.10 8. 4 6. 4 36X12 250 3 NOTE. The 30X15 and the 36X24 are preparatory Crushers, the former breaking 500 cube yards in 10 hours to 4 ins., and the latter 800 cube yards to 8 ins. Crusher -with. Revolving Screen. Dimen- Volume. Extreme Weight of Stone. Weight Produced. 'J'l Length. Mrnensions Breadth. U"' '_ Depth. Pulley. Speed per Min. IP Ins. ioX 7 i5X 9 15X10 20X10 Cub. yds. 5 8 9 10 Lbs. 3800 6800 7300 7700 Lbs. 10200 17700 18 loo 21 500 Ft. ins. 5. i 6. 6 6. 6 6.10 Ft. ins. 3-9 5 5-5 5-9 Ft ins. 4- 5 5-ii 5-n 5-n Ins. 2 X 7-5 2.6X 9 2.6X10 3 X i Rev. 250 250 250 250 No. 8 i5 12 M Steam Heating and. Boilers. Steam Heating. Is effected Directly or Indirectly. In the first case, the steam is conveyed through a pipe, or to a cluster of them, at whatever point they are required, termed a Radiator ; air being heated by contact with the exterior sur- face of the pipes, and the water of the condensed steam flows back (by gravity) through the return pipes discharging into the boiler. In the second case, steam is conveyed in like manner to a cluster of pipes enclosed in a chamber, in the lowest part of the building, usually the cellar, the air within the chamber, upon being heated, ascends by its rarefaction, and is led to the space or apartment required to be heated. Hot-water -Heating. This system consists of circulating hot water in the radiators instead of steam. The boiler, pipes, and radiators are fully filled with water the flow or circulation pipes attached to the top of the boiler and the return pipes to the bottom. The water in the boiler, when heated, rises and circulates through the pipes and radiators, and parting with a portion of its heat it becomes denser, and gravitates through the return pipe to the boiler, where it is again heated. This system requires a much greater proportion of radiating surface than that of steam. 958 BOILEES. HYEDAULIC CEMENT. ELEMENTS. No> B< 2 >ile: Wrou rs. j ght-iror 4 r n continual Water Legs. 5 | 6 ion. 7 Cast-Iro O I u Legs. 2 | 3 Shell di;un Ins oe 28 1 " over jacket. .. " 3* 35 si 4 1 45 J! 5 1 55 54 57 24 30 3i 32 35 38 " height " " extreme " j/ 72 80 s' 43 9 45 92 2 64 33 67 33 69 A/ 72 Furnace, diam " 21 24 3 3 2 38 40 18 J 9 21 24 Tubes, No., do. 2 ... u 44 56 84 9 1 124 160 30 36 44 56 u length " QQ 34 34 42 4 2 42 27 OQ QO 34 Steam-outl'ts 2, diam. " j" 2 2 2-5 2-5 3 3 *5 J" i-5 o" 2 2 Chimney flue, diam. " 8 8 10 10 12 12 7 7 8 8 Water- line from base " 55 59 63 70 73 74 5' 54 55 59 Heating surface.. Qfeet 75 105 140 185 260 320 45 60 75 i5 Direct radiating ) u surface supplied j 450 630 830 1050 1500 i goo 260 350 450 630 For Direct radiation, each nfoot of radiating surface will heat from 50 to loocube feet of air space, and for Indirect, from 25 to 50 cube feet; the range depending upon the conditions of construction of building and its exposure to external air. HYDKAULIC CEMENT. IPortland. In addition to pp. 589-590. Some limestones when burned, ground finely, and made into paste, attain the element of hardening in water, and are termed Hydraulic. Cements are classed as Natural and Artificial. The stone from which Portland or Hydraulic Cement is made in the United States is found in stratified beds of aqueous deposits, which in extent cover about one-third of the area of the State of New York, the western part of Vermont, and also in New Jersey, Penn- sylvania, Maryland, Virginia, and East Tennessee. Analysis of Glens Falls and best German cement are nearly identical, both In their quality and volumes, and all advantage claimed for the former is that it is finer grained, and that, in common with this, that it sets slowly, usually requiring from 4 to 5 hours. Consequently, the mixture can be made in a larger volume without being rendered useless by setting before all of mass is required. It is only in the stopping of joints leaking water under a pressure that the quick setting of cement is better. Tensile Strength.. Cement 1, Sand 3. Per square inch. Sieve. Sets. 17 Days' Test. 28 Days' Test. No.50. No. 100. tial. Hard. Max. Min. age. Max. Min. age. Min- Min. Lbs. Lbs. Lbs. Lbs. Lbs Lbs. Aqueduct No. 3 Vertical Wall. . Swing Bridge. . Vertical Wall. . Swing Bridge. . Vertical Wall. . Fort Miller Glens Falls Waterford . Glens Falls Waterford . Glens Falls ioo T IOO ioo T IOO ioo T ioo T 98.25 98875 99- 125 99 99 50 5? 140 120 230 230 780 347 404 369 379 382 346 254 305 302 ?8o 309 337 342 350 331 321 394 442 385 398 407 37 2 300 320 330 326 295 350 373 350 341 343 Swing Bridge. . Vertical Wall. . Waterford . Glens Falls ioo T IOO 99.125 99 155 160 300 255 368 ^ 323 273 343 337 404 442 322 326 Vertical Wall. . Glens Falls ioo T 99.125 105 190 374 313 332 440 374 405 Wm. P. Judson, Deputy State Engineer, Cru.sh.ing Strength. Per Square Inch. Tests of strength made at New York and Brooklyn Bridge. . T. Day. Lbs. 490 In Air. Lbs. 948 Water. Period. Air. In Water. Period. h, Air. Water. libsT 1408 Lbs. 653 Weeks. 2 Lbs. 750 Lbs. 515 Weeks. 3 Lbs. ELECTRIC MOTOE 959 Klectrio Motor. The Crochtr- Wheeler , New York. This Motor has been designed to remove difficulties which experience has developed to be attendant upon other instruments of like purpose. Care has been taken in its design and construction. The bearings are oiled automatically, and magnetic circuit is made as perfect as practicable. Its centre of gravity is low, machine strongly built, weight of it comparatively low, and its efficiency high. Designed to run at low speed, in order to reduce wear, heating journals, etc. EP Weight Veloc- ity. Pulle Diam. y. Face. D Length. intension Breadth s. Height. Betwec ho Length. n Bolt- 68. Breadth Sh Diam. ifts. From Base of Motor. No. ~LbI~ Ins. Ins. Ins. Ins. Ins. Ins. Ins. Ins. Ins. Via 18 2IOO I? 1 .:! 375 i 7-375 5-5 7-875 4.625 2-75 25 3-375 .125 26 2000 {?.:? 375 i 9-75 7-5 8-5 6-375 3-375 375 3-6875 .166 26 l8oO 2-5 1-25 9-75 7-5 8-5 6-375 3-375 375 3-6875 25 65 1500 3 2 14-75 9-5 io-75 % 9-75 4-3125 5625 4-75 5 100 1350 3-5 3 18.25 i 13 ' i-5 5-5 6875 5-75 i *57 1050 4 3-5 !9 3-25 15-5 2.25 7 .875 7.0625 2 290 1050 6 3 25 2.625 i8.375 8.25 9-25 I 8.25 3 300 IOOO 8 4 26.25 5-625 i8.375 8.25 9-25 i 8.25 5 485 IOOO 7-5 4-5 28 8-75 21 8.75 9-75 1.125 9-25 G Grooved. F Flat. Application of the Motor. For Printing-rooms and mechanical Shops of medium capacity and Elevators, one of 5 IP is sufficient. To Compute 3?o>ver required, for Elevators. RULE. Multiply twice* product of weight to be raised in Ibs., and height of as- cent in feet per minute; divide by 33 ooo, and the quotient will give the number of IP required. Small Motors, of . 166, . 125, and .0833 IP are adapted for operating Fans, Blowers, Sewing-machines, Small Lathes, Presses, Tools, Models in operation, Ro- tating Advertisements, Organ blowing, Raffing wheels, Knife sharpeners, Cloth and Paper cutting, Experimental models, etc., etc. Electric Fans. For Ventilation of Offices, Restaurants, Kitchens, Sick-rooms, etc., etc. Constructed in various styles, Plain and Nickel-plated. Fans 12 Inches in Diameter. Regular, .0833 IP motor. Fast, .125 IP motor. Double (a Fan at each side), .1666 IP motor, or one i6-inch Fan. La Rue Construction. Variable speed, Fan 24 ins.in diam. 20 Inch, . 125 IP motor. Electric 3?nmps. Pump, .166 IP, will elevate 500 gall, water per day ot 10 hours 100 feet in height. These pumps are arranged to operate automatically, so that when a receiving tank is tilled, the pump is arrested. Capacity of I*vimp per Hovir. FP Gallons. || IP Gallons. FP Gallons. .166 25 IOO 250 5 i 37 750 3 5 1670 2600 -A.ro Circuit Motors. Arc Motors' totter from all others. The manner of connecting the circuit and of their operation varies from that of other motors. They are furnished from .125 to 5 IP. They should always be connected to the arc circuit by a competent lineman. * The twice is taken to covr loss of power by friction of all the part*. 960 DYNAMO LEATHER BELTS. WIRES AND CABLES. IDynamo Leather Belts. Belting, For Dynamos and Electric Light Machinery should be double and endless, and* not over .33 inch thick, when run at a velocity of 4000 feet or more per minute ; should be perforated to prevent air-cushion- ing, the perforations may be .093 75 inch in width, .281 25 inch in length, and placed 1.5 inches apart; furnishing about 50 openings per sq. foot of belt, without material injury to the tensile or operating strength of it. In order to protect the surface of a Dynamo Belt it should be rendered impervious to the mineral oil used on it, which is destructive to the fibre of the leather. LEATHER LINK BELTS. Where Belts are run at right angles and at short distances apart, Leather Link Belts are recommended, as they are very pliable and have uniform oscillation. Link Belts made .6875 inch thick, forming when combined two full cir- cles, assure the required uniformity of oscillation. WIRES AND CABLES. Telegraph, Telephone, and Electric JL.io.ht Wires and. Cables. For Aerial, Su.lb-marine, arid. "Underground.. The Okonite Company, Ltd., New York. Insulation. In consequence of the decomposing fnfluence of the elements upon insulated wires exposed to them, it is necessary that their insulation should be as perfect in construction and enduring in material as it is practi- cable to attain. In order to effect an enduring insulation this Company uses a compound, termed Okonite, a material possessing both tenacity and resist- ance to abrasion, while it is equally unaffected by extremes of temperature, with insulation of a high order. Telegraph and Telephone AVire. Size of Insulation and Diameter per B W G; External Diameter of Insulation in 32<* ; Weights per Mile in Lbs. , and Insulation Resistance per Mile in Megohm*. Insulation. Insulation. Insulation. Insulation.. ^ ^ PLAIN. BKAIDKD L," PLAIN. BBAIOKD No. 1 1 If If |j io Is No. 1 1 |l .M. J. ' S ii .2 OS 9 5 la '1 Ji 5 O Ss fc I S 1 JI 40 24 I4 29 IOOO 43 IOOO 55 ~i6 6 l6 4 1600 1600 4i 24 13 3 33 I2OO 48 1200 56 16 6 6-5 I 7 8 2OOO 198 2OOO 42 22 13 3 45 IOOO 60 IOOO 57 16 5 7 200 2400 223 2400 43 2O 12 3-5 56 IOOO 7 1 IOOO 58 1 6 3 8 240 2000 267 2600 44 2O II 4 65 I2OO 80 I2OO 59 16 2 9 288 2800 3i8 2800 45 18 II 4 81 IOOO 96 IOOO 60 *4 8 5 I 7 6 IOOO 194 IOOO 46 18 10 9 1 I2OO 107 I2OO 61 14 7 5-5 192 I2OO 211 I2OO 47 18 8 5 119 1600 I600 62 H - 6 2OI 1400 221 I4OO 48 18 6 2OOO 175 2OOO 63 14 5 7 2 3 6 l6oO 239 IOOO 49 18 5 7 197 2400 220 2400 64 T 4 3 8 273 I800 300 1800 50 18 3 8 244 2600 271 2600 65 M 2 9 3 2I 2OOO 351 2OOO 51 18 2 9 299 3000 329 5000 66 12 6 6-5 275 IOOO 295 IOOO 52 io 9 4-5 126 IOOO 143 .lOOO 67 i 12 5 7 2OO l6oO 313 j 1600 53 1.6 8 5 140 1400 158 1400 68 IO 5 |7 369 IOOO 304 i looo 54 16 7 i i57 1500 I 7 6 1500 [', 69 10 4 '7 5 387 1200 ii 413 1200 VOLTMETERS AND AMMETERS. 961 "Voltmeters and Ammeters. \Veston Electrical Instrument Co., Newark, N". J. Instruments are Direct Reading. A multiplying constant being unnecessary, except with voltmeters of high range, as a simple inspection of position of pointer on scale indicates value in amperes or volts, and as pointer immediately becomes fixed, or "dead beat," it reduces period for reading, added to which there does not exi.st the "magnetic lag" which induces different deflection with like current or potential, as in instruments in which moving parts are of iron. All readings of the scale commence at o, and the uniformity of the divisions facilitates the visual subdivision of them. Correction* for Temperature are unnecessary, being not in excess of .25 percent, for a range of 35 above or below 70, and for the ammeter less than i per cent, for a like range of temperature. Circuit. These instruments can be retained in circuit without injury, as the beating effect, except in the high ranges, is inappreciable. Calibrating Coil. In Voltmeters provided with one. changes in the scale value from accidental injury are readily checked. All the respective parts are made to a uniform gauge, and are interchangeable. Standard Voltmeters. No. Scale in Volte, o up to Single- Scale Division Volte. Readable to Volte. Remarks. ( No. Scale in j S ^jJ- ta / 1 DiviEion "P * i Volte. Readable to Volte. Remarks. i i 150 i O.I 7 600 5 0.5 1 s 150 I O.I C. K. C. C. 8 600 5 0-5 C. K, C. C. * 3 I 'J O.K. 9 (600 4 0-4} O.K. f ' l /30 Vsoo) i 0.1 J !*- {150 ( 3 Yso 0.1 i VMO) O.K. 9# J750 (150 5 X 0.1 f C. K. I 4 (15 Vio 0.1 i ViooJ O.K. 10 (600 4 0-4} 0.2 J C. K. M 300 2 450 i 3 0.2 o-3 C. K. C. C. C. K. C. C. ii 12 750 5 1500 1 10 0-5 X NOTE i. Calibrating coil (C. C.) and contact key (C. K.) are attached to instru ments only when so stated. 2. Instruments up to No. 6, provided with contact key, can be kept in circuit by depressing it and giving it a quarter turn. 3. Volt- meters of other ranges furnished as required. High-Range "Voltmeters No. 1. In these instruments part of the resistance is contained in a separate box, de- signed to obtain high insulation. Without calibrating coil or contact key. Scale from o to 150 volts aud a resistance box, multiplying valve of scale divisions. No. Volte to 150. f Scale. 13 '4 2250 3000 By 15 " 20 Volte to 150. Scale. J| Ho. Volte to 150. Scale. 3750 4500 By 25 " 30 j| 18 5250 6000 By 35 " 49 \l NOTB i. Above are graduated to read in lamps in addition to graduation in volts, indicating by simple inspection number of arc lamps open on the circuit. 2. Other multipliers giving intermediate ranges, applicable to these high-range voltmeters, are furnished. 3. Reversing Key. When many and rapid tests of distribution of po- tential or electric currents of unknown polarity are required, this key enables oper- ator to open the circuit by turning the milled head through an angle of 45, or to reverse direction of current in the instrument by an additional movement of 45. Milli- Voltmeters. Scale in Single- Readable Scale in S [g le - ; Readable Volte. Scale Division to Remarks. . No. Volte. DiSrioni Vo ? u Remarks. o up to Volte. VlltB. o up to .02 0.0002 O.OOOO2 fo.oi (O.I 0.0002 < 0.00002) O.OO2 I O.OOO2 ) Zero in centre. Contact key redur. sensib. (Zero in cn- i i 10 times. .01 0.0002 0.00002 1 tre. Right ( and left. (0.02 i i (0-2 O. OOO2 O. OOOO2 ) O.OO2 j O.OOO3 f Higher range compensated for 1 1 V temperat:ue. 962 VOLTMETERS AND AMMETERS. RAILROAD CRANE. No. x. 150 volts in single volts readable to tenths. 1.5 amp. in o.oi amperes readable to tenths. "Volt- Ammeters. Scale. No. 2. 150 volts in single volts readable to tenths. 3 amp. in 0.02 amperes readable to tenths. JVXilli- Ammeters No. Scale in Milli-ampere. o up to Single-Scale Division Milli-ampere. Readable to No. Scale in Milli-ampere. o up to Single-Scale Division Milli-ampere. Readable to o J50 I O.I 500 5 0-5 X 300 2 0.2 5 50 0-5 0.05 2 600 5 -5 6 500 5 o-5 3 IOOO 10 IO O.I O.OI 4 1500 10 I 7 500) 10 j with resistance box to give readings to 10 and 100 volts. Ammeters. No. Scale in Ampere. Single-Scale Division Readable to No. Scale in Ampere. Single-Scale Division Readable o up to Ampere. up to Ampere. i 5 0.05 0.005 7 200 2 O. 2 2 15 O.I O.OI 8 250 2 0.2 3 25 0.2 02 9 300 2 0.2 4 50 0.5 O.O5 10 400 5 0-5 5 100 I O. I ii 500 5 o-5 6 150 I O.I Portable Direct- Reading Voltmeters and ^Vattmeters for Alternating and. Direct- Current Circuits. Voltmeters, 22 ranges, from 7. 5 to 3000 volts. Wattmeters, 12 ranges, from 150 to 30000 watts. S-witchtooard Ammeters and Voltmeters for Central Stations and Isolated Plants. Illuminated Dial Instruments, " Round Pattern " Instruments, have substantially the same characteristics as the Portable Standard Instruments. Are " dead beat," have uniform scales, can be kept in circuit continuously. Railroad. Crane. The Farrell Foundry and Machine Co., A nsonia, Conn. Post. Of cast iron, in one piece, fitted to deck-plate, with faced joints and secured by bolts running through a stone foun- dation, set up on anchor plates on its under side. Jib. Of two wrought-iron beams, bolted at head and foot to a bonnet and shoe, with tie bolts between them, and secured to the post by bolts which lead from its head to a yoke, which turns on a pin in the hub. Hub. With a pin is fitted into head of post, on which the jib turns. Yoke. Is secured by two bolts, which lead down through and are secured at the deck-plate on the foundation. Gearing. Double and set for both fast and slow motions, and detachable, to admit of lowering load by a brake. Chain. Triple " B Crane," and all sheaves have roller bushes. Capacity. Radius. Capacity. Weight. Radius. Capacity Weight. Radius. ~Feet. 16 20 structLug ca Capacity. Weight. Feet. 12.5 10 Tons. 2* 4 \ Lbs. 5500 6500 Designed f Feet. 15 15 or operation Tons. 6 10 on Wrecki Lba. 10400 14600 ng and Cou Tons. 15 20 r. Lbs. 17400 23800 VACUUM PUMPS. "Vacuum IPumps. acuum Pumps. Air pumps are so termed when they are used in connection with vacuum pans, multiple effects, or niters. It is impracticable to define a general rule for their capacity, as the cir- cumstances of their operation vary in different cases. Vacuum Evaporators. Their dimensions depend upon the temperature to which they are submitted, the evaporation, character of the liquid concentrated, vacuum desired, and type and efficiency of the condenser. Dry Exhaustion. When air alone is withdrawn. f j M = Q. V and V representing volumes Oj cylinder and receiver, M volume of air in receiver at commencement of operation, both in cube feet, n number of strokes of piston, and Q volume of air remaining after n strokes of piston. Condensation. There are two systems in operation for vacuum pans and multiple effects, viz. Dry System. Where the condenser is fitted with a leg pipe or barometric tube, through which the injected water passes off by gravitation. Wet System. When the pump receives and discharges the condensing water, in addition to its maintaining a vacuum. In either system the pump is required to discharge: ist. The air contained in the injection water, in the liquid, and in the pan, pipes, and condenser. 2d. The incon- densable gases evolved from the liquid in operation. Notes. The Pan and its immediate connections are made of iron, copper, bronze, or alloys. In designating the design and construction of pump required, the liquor, the volume, the degree of concentration required, and the time in which the operation must be completed, should be furnished. To facilitate transportation, the bed plates of the large sizes are cast in two parts and bolted together. An order for a pump should state : ist. What liquor, and volume of it, is to be evap- orated in a given period, as an hour? 2d. What the diameter of pan or evaporating vessel, and what that of vapor pipe when it enters condenser? 3d. What the heating surface of pan, and has it a steam jacket and coils, and if coils, what is their diameter and length ? 4th. If heating surface is of iron, brass, or copper ? sth. What the average temperature of condensing water, and what the volume of it ? Duplex "Vacuum IPumps. Fly-wh.ee! Type for "Dry" or "Wet' System. Di Diameter of Vacuum Cylinders. MKNSIONS. Diameter of Steam Cylinders. Stroke. Volume per Rev- olution. Displace Feet Pist< Mi Per Min. rnent, at 75 m Speed per nute. Per Hour. Dia Suction and Discharge. meter of Pi Steam. pes. Exhauit. Ins. Ins. Ins. Cub. feet. Cub. feet. Cub. feet. Ins. Ins. nt. 6 5 6 589 29.45 1767 .25 8 6 6 1.047 52-35 3Mi t -25 5 10 7 6 I-635 81.82 4909 g -25 10 8 6 1-635 81.82 4909 1 -5 12 8 9 2-356 117.81 7068 "3 5 12 9 9 2-356 117.81 7068 5 14 9 9 3.207 160.35 9621 g 5 in 10 10 9 9 3.207 4.188 160.35 209.4 9 621 12564 | 5 5 -5 5 16 12 9 4.188 209.4 12564 \ -5 18 12 9 5.301 265 15898 "C 5 18 14 9 5.301 265 15898 1 5 20 12 9 6-545 327- 2 5 19635 2 5 20 H 9 6-545 327-25 19635 5 22 f i 9 7.919 395-95 23757 5 5 22 16 9 7.919 395-95 23757 -5 3 24 16 9 9.424 471-25 28275 < 5 3 4 18 9 9.424 471.25 28275 5 3 964 DRAWING, TRACING, SECTION, ETC., PAPER. Drawing, Tracing, IProfile, Cross-section, IPlioto- printing ^Papers ancl Clotlis. Keuffel & Esser Co., New York, Chicago, St. Louis, San Francisco In Sheets. Whatman's Hand-made in all sizes, H P, C P, and R. Universal, For general drawing and water-colors, six sizes, 14X17 ins. to 27X40 ins. Normal, Not Hand-made, but very similar to the Not Hot Pressed, in Royal, Imperial, and Double Elephant. Duplex, Cream color, for fine detail and general drawings, in Royal, Imperial, and Double Elephant. Duplex, Drab color, heavy, Double Elephant only. Paragon, Medium rough, in Royal, Imperial, and Double Elephant smooth in Double Elephant only. Bristol- Board (Reynolds's), tive sizes, 12.5X15.25 ins. to 21.5X28.75 ins., 2, 3, or 4 sheets in thickness. K. & E. Patent Office Bristol- Board, 10X15 ins. and 15X20 ins. K. &. E. Bond Paper, light and very tough, three sizes, 19X24 to 27X4 ins - Tracing Papers. Vegetable (French), five sizes, 13X17 to 29X42 ins. Cupola, very tough and transparent, 28X39 ins. Hermes (slight grain), 20X30 and 30X40 ins. Ceres, tough, 20X27 and 27X40 ins. Corona, thick, 27X40 ins. Of these the Vegetable, Ceres, and Corona are natural Tracing paper (not prepared). In rolls: Parchment, Thick Parchment, Abacus, Patera, Colonna, thin and medium, 30, 36, 42 ins. in width (can often be substituted for tracing cloth), Corinthian, Gothic, Doric, Alba (for transferring), Lotus, and Libra. Drawing Papers in Rolls. Duplex, medium, cream color, 30, 36, 42, 56, and 62 ins. in width. Do., thick, drab color, 36 and 56 ins. in width. Universal, for general drawing, water-colors, etc., 36, 42, 56, and 62 ins. in width. Lava, similar to Universal, pearl gray. Anvil, medium and thick, surface and appearance similar to Whatman's Not Hot Pressed, medium, 36, 42, and 62 ins. in width ; thick, 62 and 72 ins. in width. Paragon, pebbled surface (similar to egg shells), thin, medium, thick, and extra thick. All 58 ins. in width, except medium rough, which is also 36 and 42 ins. With smooth surface (similar to What- man's N. H. P. on one side, smooth on the other), medium and thick, both 58 ins. in width, except medium, also 36 and 72 ins. All can be had by the yard, in ro-yard lengths, or in rolls of about 35 Ibs. Detail. Economy, medium and light, 5o-yard rolls, 36 and 60 ins. in width. Simplex, light, medium, and heavy (Manila), 36, 42, 48, and 54 ins. in width, in 50 and 100 yard or ioo-lb. rolls. ATonnted. Universal, Duplex, Lava, Anvil, and all the Paragon Papers are mounted on muslin, in all the widths; by the yard, or in 10, 20, or 30 yard rolls. All the sheet papers are also to be obtained mounted up to 20X30 feet. Photo-printing Papers. Helios Blue Print, medium and thick, pre- pared (sensitized), 24 to 54 ins. in width. E. T. Paper, thin, for mailing, prepared, 2 4 3 36, and 42 ins. in width. Columbia, Blue Print Papers, medium, thick, and thin (mailing), 24 to 42 ins. in width. Prepared paper is in 10 or 50 yard rolls; unprepared in 50 - yard rolls. Blue Process Cloth, prepared and unprepared, in lo-yard rolls, 30, 36, and 42 ins. in width. Nigrosine (Positive Black Process), 10- yard rolls, 30, 36, and 42 ins. in width, prepared only. Umbra (Positive Black Process) requires no developing bath. Maduro (Negative Brown Process) Paper and Cloth, requiring only a flxing-bath; 30, 36, and 42 ins. in width, prepared only. From Maduro prints on thin paper positive blue or brown prints of the original can be taken. Maduro is the latest. Profile Cross- Section and Tracing Papers. Tracing and Drawing Cloths in red, green, orange, or blue. NOTB. A complete catalogue of Drawing Materials and Surveying Instruments, 500 pp., mailed on application. REFRIGERATION. 965 Mechanical Refrigeration. The De La Vergne Refrigerating Machine (7o., New York. Mechanical Refrigeration is effected by Compression, Condensa- tion, and Expansion of a liquefiable gas. The Refrigerating or Heat-absorbing agents are Ammonia, Ether, Sul- phurous Oxide, Carbonic Acid, etc., which undergo the operations above given. The De La Vergne Machine is operated with Ammonia. Compression. The gaseous agent is compressed if Ammonia is used to from 125 to 175 Ibs. per sq. inch ; during which operation heat is devel- oped in proportion to the pressure exerted upon the gas, or the relative vol- ume to which it has been reduced. Condensation. The heat developed in the operation of compression is withdrawn from the compressed gas, which is forced through coils of metal pipe, surrounded with cold water. As soon as the condition of satura- tion is reached, the gas assumes a liquid state. Expansion. The liquefied gas is also passed through coils of metal pipe, suspended or seated in a space where the substance to be cooled, as air, water, brine, beer, etc., is introduced ; the pressure in the interior of the coils being at a lower point than that required for the maintenance of the gas in the liquid state. The liquefied gas, upon entering these coils', again expands, and extracts from them and the substance around them the same quantity of heat that was previously given up by the gas to the water of condensation. The gas, having passed through this routine of operation of refrigerating, is now in a condition to be used in a repetition of it. The gas is forced through these coils by the pressure in the condenser, which, in the use of Ammonia, is generally from 125 to 175 Ibs. per sq. inch. Under this pressure and the cooling action of the water, liquefaction occurs, and the resulting liquefied gas flows to a stop-cock, having a minute opening, by which the pressure is reduced from 10 to 30 Ibs. per sq. inch in the expansion coils, and where the liquid through reduction in pressure is again transformed into a gas. By the ex- hausting operation of a gas pump, this pressure is maintained, and then the gas is forced by compression into the condenser again. Thus the expansion coils, although similar to those for condensation, are operated for the reverse, which is the absorption of heat by the liquefied gas, instead of the extraction of heat from it. In Operation, heat is transmitted from the outside through the walls of the ex- pansion or cooling c/>ils, and is absorbed by the expanding liquefied gas within such coils. This heat is borne by the gas through the pump into the condenser, where it is in turn transferred to the cooling water through the walls of the condenser coils, and ultimately carried away by this water. NOTE. Liquefied ammonia in a gaseous condition at atmospheric pressure and temperature of 60, expands about 1000 times, and upon its expansion re-absorbs a quantity of heat equal in amount to that originally held and evolved from it during liquefaction. The liquefied gas, entering the coils through the minute opening in stop-cock, is immediately relieved of a pressure of 125 to 175 Ibs., that requisite to maintain it in a liquid state, when it boils and expaLds into gas. To obtain this, heat is re- quired, and which alone can be supplied from the substance surrounding the coils, such as air, brine, water, etc. As a result, the surrounding substance is reduced in temperature, the quantity of heat withdrawn by the gas being the same as that which was withdrawn from it during its liquefaction in the condenser. Consequently, if the expansion coils are set in an insulated space, it will be re- frigerated; and if brine or any liquid surrounds the coils, it will be reduced in tem- perature, and brine, in this condition led into a space through a pipe or open con- duit, will refrigerate it. 966 REFRIGERATION. FORCITE POWDER. Results of Operation of Refrigerating Machines of SOO* Tons. At Lion Brewery, New York. Duration of Test n h zomin. Steam Cylinders. Diameter, 36 ins. ; Stroke of Piston, 36 ins. Pressures of steam (mean), 48.4 Ibs. Gas Compressors. Two double acting; diam. 18 ins. ; Stroke of Piston, 36 ins. ; back-pressure, 28.22 Ibs. ; condenser, 180.78 Ibs. persq. inch; Revolutions, 39.55 per minute. Test for cooling made by running water of a mean temperature of 100.95 over wort, Baudelot Cooler, and cooling same to a mean temperature of 50.77. Refrigeration, equal to melting of 210 tons Ice per day of 24 hours. Horse Power. IBP = 313, and assuming consumption of coal at 3 Ibs. per hour per IB?, ratio of refrigeration =: 20. 84 Ibs. ice per Ib. of coal. If operated under ordinary condensing pressure of 156 Ibs., the IB? would be 278, and ratio 23. 47 Ibs. ice per Ib. of coal ; IB? per ton of ice per day = 1. 183. Of a 26-Ton Machine. At Bohlen-Huse Machine and Lake Ice Co., Memphis, Tenn. Duration of Operation 20 Days. Steam Cylinder: Diameter, 22 ins. ; Stroke of Piston, 28 ins. ; Steam, 93.49 Ibs. per sq. inch. Gas Compressors, Two single-acting: diam., 14 ins. ; Stroke of Piston, 28 ins. Revolutions, 40 13 per min. Temperatures : Cooling water 63, brine 18.62; coal consumed, 180597 Ibs. ; Ice produced, i 221 172 Ibs. Ice-making, 26.83 tons per day of 24 hours. Steam-boiler evaporated 5.5 Ibs. water per Ib. coal. * All tons are given at 2240 Ibs. See foot-note, p. xxvi. FORCITE POWDER. American For cite Powder M'fg Co., New York. Foroite. Is an improvement in Nitro-glycerine compounds, and it presents the following elements : It is less sensitive to shock than other explosives. Assuming Dynamite No. i as the Standard 100. Forcite No. X, 95 per cent, Nitro-glycerine, 133 per cent, intensity. i*75 " " 125 3 t 4 o " " 95 5 per cent, stronger than Dynamite No. i. f Within 5 per cent, the strength of No. i, 75 per cent. It is more powerful than any other known explosive in our market. See Report of Henry L. Abbott, Lieut. -Col. E. U. S. A. It is safe in handling and transportation, quintuple force-caps being ap- plied to explode it, and free from noxious fumes. Water-proof, free from the absorption of moisture, and is not injured by submersion in water. Directions in Use. In Blasting, fill the hole, and thoroughly tamp the charge. Thaw it, if frozen, as frozen powder will not explode with its proper effect. Exploder or caps should be maintained dry, and are not to be stored in same buildings as the powder. Powder, ignited by weak caps, instead of being exploded, emits noxious vapors. I*er Cent, of Nitre-glycerine in Brands of Foroite. Gelatine 95 I No. i 75 I No. 2 50 I No. 3 40 I No. 3 B 2 a No.iX 8o| " 2 X....6o| 3 X.... 45 | "3A.... 35 | " 3 C 30 SURFACE CONDENSATION. REFRIGERATING. 967 SURFACE CONDENSATION. Wheeler Condenser & Engineering Works, New York. Construction. The Wheeler Condenser, alike to others for the same purpose, is an elongated vessel, cylindrical or cubical, with the necessary at- tachments for Steam and water connections. Its distinguishing features are : The exhausted steam, upon entering the condenser, impinges upon a perforated scattering plate, which distributes it generally over the tubes and thus diverts the deteriorating effect of the direct impingement of it upon one portion of the tubes ; the steam, expanding in a void above the tubes, is reduced in pressure, and consequent temperature, before it flows into contact with the surfaces of the tubes. Each pair of tubes is composed of an external and internal tube, set hori- zontally, the inner tube having an open end, the other end being screwed into a removable head or vertical diaphragm, which is set at a space of a few inches from a like head, into which one end of this large tube is screwed, the other end being closed by a screw cap. This design permits the tubes to expand or contract, without the use of tube packings or ferrules of any kind, as only orfe end of each tube is fixed. The tubes are tinned both externally and internally, and can be readily withdrawn for cleaning, etc. Operation. Tftie tubes are divided into two distinct tiers ; the condens- ing water flowing through the small tubes in the lower division passes out of then* open ends and through the annular space between their external sur- faces and the internal surfaces of the larger tubes, and from thence into the upper division, and through its tubes in like manner to the space between the two heads referred to, and finally out through the discharge pipe. The circulation of the condensing water is by this manner of flowing ren- dered very active, and consequently a less volume of it is required, and there is less tube surface needed for a required volume of condensation. Results of* an Operation to Determine tne Efficiency of tliis Condenser, -with. and. -without a Vacuum. Steam Condensed per Hour per Sq. Foot of Condensing Surface. Condenser. Vac- uum. Te jection Water. mperatut Dis- charge WateV. es. Reser- voir. Steam Con- densed. Condenser. Te In- jection Water. mperatui Dis- charge Water. es. Reser- voir. Steam Con- densed. Lbs. 204.2 With Vacuum Ins. }H.S Deg's. 56.5 Deg's. 98 Deg's. 138 Lbs. 1 01. 8 Without Vacuum* Deg's. J78-5 Deg's. 139 Deg's. 201 * As a simple surface condenser without air pump attached. REFRIGERATING AND ICE-MAKING. A Refrigerating Machine is one that produces as low a temperature as a given volume of ice, at the temperature attained, would in melting from the temperature of the air, or void to be refrigerated = 142 (142.6) of temper- ature are required to transfer one Ib. ice at 32 to one Ib. water at 32, which difference represents the Latent heat. In order to operate such a machine for the formation of ice, there will be required, instead of 142, about 236. Thus, Assume the water from which the ice is to be formed to be of an average temperature of 72; then to reduce it to 32, before ice can be formed, 40 or 40 thermal units are to be abstracted from each Ib. of water; then 142 are to be ab- stracted from the Ib. of water of 32 to reduce it to one Ib. ice at 32. 968 EEFRIGEKATIXG AND JOE-MAKING, ETC., ETC. If the ice is produced at the general temperature of 18, and the Specific heat of it is taken at .5; then, 32 18 X-5 = 7. To reduce this water from 72 to 32, there is a reduction of 40 or thermal units from each Ib. of water. If ice is produced at 18, Then 7 additional, as deduced above, are required. In practice it is observed that the average loss of temperature by radiation of it from the freezing tank, melting the external surface of the ice, to withdraw it from the molds, etc., is fully 20 per cent, of the total capacity of the machine. Hence, of the 236 which are to be abstracted from the water per Ib. of ice, in order to reduce it to ice, 47.2 are lost by radiation. And 40+ i4 2 H~7-h47 2 36 are to be ab stracted from each Ib. of water of 72, in order to produce i Ib. ice at 18. Consequently, If 142 are required in Refrigerating machine and 236 in Ice- making, the relative requirements are as i to i .66 or as 6 to 10. Refrigerating Capacity. Of a machine is designated by the number of Ibs., or tons of Ice, which it is capable of producing. One Ib. of ice at 32 absorbs 142 or thermal units in melting. Hence, one ton of ice absorbs 142 X 2240^318000, and a machine of 50 tons' capacity absorbs 318000 x 50 15900000 every 24 hours of its operation. Ice-malting Capacity. Of a machine is also designated by the number of Ibs., or tons of Ice, which it is capable of producing. To freeze one Ib. of water at 72 to ice at 18, it requires the absorption of 236, viz., To reduce one Ib. of water at 72 to 32, it requires the absorption of 40, to freeze it requires 142; to reduce ice from 32 to 18 requires 14 x .5 7 (Specific heat of ice = .5). Reduction of temperature from surface of freezing tank and withdrawing the ice from its molds by the application of heat, about 20$ of total capacity of machine = 20$ of 236 = 47. Hence, Total heat to be absorbed per Ib. of ice = 40 -f 142 -f- 7 -f 47 = 236 . Ratio of Capacity of Refrigerating to Ice-making. As 142 : 236 : : 6 : 10, as pre- ceding, or a Refrigerating machine of 9.97 tons capacity will produce about 6 tons of ice in the same period. Higliest Klevation of a l^alre. Colorado. " Green Lake" is 10252 feet above level of the sea and 300 feet in depth. Magnifying. Bavaria, Munich, possesses a microscope that magnifies 16000 diameters. fower of Scre-vv Bolts. Results of an Experiment. Wrought-iron. Diameter, 2 ins. Thread, V. Pitch, .22 ins. Mean Power applied at a circumference of 78.85 ins., 213 Ibs. Loss by friction, 10. 19 per cent. (Jas. McBride, M. Am. Soc. M. E.) Duration, of Railroad. Cross-ties. IDnration of Following \Voods. Wood. Years. Wood. Years. Wood. Years. White Cedar 8 7< Chestnut 7 5 Yellow Pine 6 White Oak R Red Spruce Hemlock 5 5 Black Cypress 8 Red Oak 5-5 Tamarack 4 The elements of durability are Resistance to decay and to wear. White Oak com- bines both qualities to the highest degree. Yellow Pine resists wear, but not decay. Red Cedar and Black Cypress resist decay, but not wear. Ties should not be cut when the tree is in leaf, and should be well seasoned or preserved by some antiseptic process before being laid. Proper draining of a road-bed will add to the duration of ties, and all indentations of their surface by tools, etc., should be avoided, and all spike-holes plugged to avoid the absorption of water. (H. W. Real.) GAS AND ELECTRIC LIGHTING. RAILROAD SPEED. 969 GAS AND ELECTRIC LIGHTING. (In Addition to pp. 583-587). Gras. Candle 3?oxver and. Consumption of Different Burners. Candle Power. Cons ump- Candle Power. Consump- tic tion Burner. No. Per Foot per Hour. per Hour per Lamp. Burner. No Per Foot per Hour. per Hour per Lamp. No. Fe et. | No. Fact. Batswing 10 2 . 3-3 Flat ) In ( 60 JC Flat ) from . . Flame j to "5 13-8 2. 3 JO 5 4 4 6 8 | Clus-*j 90 Flame) ters. ( 150 4 5-5 5 20 3<> Electric. A.rc Lamps. Relative Current. C Horiz'tal. Angle 7. andle Pow Angle 10. r. Angle 20 a . Angle 40. Watte Required. Units per Cotts* of Gas. Elec- trici. Ampert. No. No. No. No. No. No. Hour. 6 92 175 207 322 460 300 3 2.6 7 8 156 300 350 546 7 8o 400 4 3-77 10 22O 420 495 77 IIOO 500 5 4-83 * Per Candle Power for Batswing Burner. Arc Lights should be set high and for the following causes: 1. Their high candle power and distance apart being in excess of gaslights. 2. Light radiating at a depressed angle is greater than when cast horizontally. 3. Horizontal rays are not as steady as angular. NOTE. The greatest intensity with continuous currents is at an angle of 40 be- low a horizontal line. To Determine the Coefficient of Minimum Lighting I?ower in Streets. L H -r- D 3 = Co. L representing candle power of lamps. D maximum distance from lamp, and H height of lamp, both in feet, and Co, coefficient. Usual standard for Gaslighting is assumed for a unit of pavement 50 feet dis- tant for a lamp of 12 candle power 9 feet in height. Hence, 12 X 9 -T- so 3 = .000864. Adopting this coefficient, the following capacities of arc lights will give the same standard of light at the following freight and distance. A minimum standard would increase the coefficient to .001 728. NOTE. One arc light can replace from 3 to 6 gas-lamps, according to locality and standard of light adopted. 2. Arc lighting, based on the substitution of one light for 3.5 to 4 gas-lamps, would double the minimum standard of light; while the average standard would be increased from 10 to 12 times. (Eliminated, etc., from Papers of Henry Robinson, M.I.C.E.) Railroad Speed. 1891, Sept. 14. N. Y. Central and Hudson River R. R. From Grand Central Station to East Buffalo, N. Y. 436 miles in 426 minutes, actual running time = 61 . 40+ miles per hour. Weight of Train 230 tons. From Station to Fairport, 361 miles in 360 minutes, there delayed by a hot journal. 1891. Philadelphia and Reading R. R. One mile in 39.75 seconds=the rate of 90 . 54 miles per hour. "Flying Scotchman," London to Edinburgh, 400 miles; stops, 44 minutes ex- cluded, in 8 . 5 hours = 47 . 05 miles per hour. Weight of Train, excluding locomotive, 80 tons. 970 TENACITY AND RESISTANCE OF BOLTS. Tenacity of Round, and Sqxiare "Wrougnt-Iron JBolts, Holes of Different Diameters. Round. .75-inch, driven into a bole of .625 inch, in White Pine, for 12 ins., required 6875 Ibs. to withdraw it. i-inch, driven into a hole ot .75 inch, in White Pine, for 12 ins., required 10612 IDS. to withdraw it; and in Norway Yellow Pine, 10830 Ibs. i-inch, screwed, 8-threads per inch into a hole .8125 inch, in White Pine, for 12 ins., required 15 125 Ibs. to withdraw it, and one of 12 threads required 15 250 Ibs. i.i25-ins., driven into a hole of .875 inch, in Hemlock, for 12 ins., required 8875 Ibs. to withdraw it. Square The difference between that and Round, under like conditions, was essentially different, and when a hole was bored 10 ins. in depth, the difference was not essential. Railway Spilses. Length iu Tie. Chestnut. T Y. Pine. o Withdra W. Cedar. w W. Oak. Hemlock. Remarks. Ins. 4.6 Lbs. 3264 Lbs. 3198 Lbs. 2305 Lbs. 4330 Lbs. 3485 In solid wood, sharp pointed. Ship Spikes. . 375 inch square and 7 ins. in depth, driven 3 ins. in White Pine and drawn back, required 1617 Ibs. , their edge with the grain of the wood, and 1317 Ibs. with it across. Note. The above are deduced from Experiments of Gen. Weitzel, U. S. E., 1874-77. Resistance of Bolls, after being 7 months driven = 10 per cent, greater than im. mediately after, and when driven through in direction of fibre it is but 60 per cent, of that of being withdrawn. Smooth bolts have greater retention than ragged, either driven or withdrawn. Moderate " ragging " reduces their power 25 per cent., and extreme 50 per cent. Relation between diameters of bolt and hole showed that the resistance of a bolt of i inch in a .6875-inch hole was greater than in one of .75 or .8125 inch. With a -75-inch bolt the resistance was greater in a hole of .625 inch, and was one quarter greater than in one of a sixteenth greater or less. One-inch square bolt in a .875-inch hole was the same as a round bolt in a ,6875- inch hole. Screw-bolts are about 50 per cent, more effective than plain round. Long pointed blunt bolts are more effective than short pointed. Experiments of Mr. F. Collingwood and Wm. H. Paine, made in connection with construction of the New York and Brooklyn Bridge, gave for a i-inch round bolt, driven in a .9375-inch hole, in best Georgia Pine, a resistance of 15 ooo Ibs. per lineal foot, and in a .875-inch hole 12 ooo Ibs. In lighter woods the tenacity was less. Mr. J. B. Tscharner, in the laboratory of the University of Illinois, determined that a like bolt (i-inch round), under like conditions in White Pine, was 6000 Ibs., and that a bolt driven parallel to the grain of the wood has but half of the resist ance of that driven perpendicular to it. Further, that assuming a bolt of i inch in a .0375-hole as i, that if driven in a .75-inch hole it would be 1.69, and in a .8125- inch hole 2.13. Relative Driving Resistance of" Roxind and Sq.ixare Steel Bolts. One Inch in Diameter. Drive into Pine Wood. Six Inches in Depth. Square. Round. i 3972 662 9375 4260 710 875 4660 777 .8125 4050 675 9375 2250 375 875 379 8 633 .8125 4728 788 Power applied in Lbs . ... Tenacity per Inch of Depth (J. H. Powell and A, E. Harvey). NOTE. Inasmuch as the amount of metal in the Round bolts is but .7854 that of the square, Round drift bolts are the least expensive. MORTAR. SPEED OF VESSELS. MTC. 971 Mortar. Brick. Clean and Sharp Sand, 3 parts; Lime, i part; laid in a bed suffi- ciently large to admit of the composition being in a thin layer. In slaking the lime, apply sufficient water to prevent its burning. Stir rapidly and thoroughly, in order to enable the water to cover each lump of lime as it deli- quesces; and when this operation is fully effected, stir the substance into a condi- tion alike to milk, and then mix it with 'the sand in the bed, with water sufficient to render the mass semi-fluid. In this condition it should remain for a period of at least 24 hours a longer pe- riod is preferable. When required for use, add and thoroughly mix with it another part of sand. Hair Mortar. Lay the hair on a floor and beat it, in order to break the bunches and remove foreign substances. Then soak and wash it in water for 24 hours, to remove all glutinous matter. Spread it on a layer of sand in a bed, add lime, and proceed as directed for Brick Mortar. J^arge Trees in. Australia. In Victoria, Eucalyptas. One 435 and one 450 feet in height. Speed, of Vessels. To Determine the True Speed of a Vessel by Consecutive and. Alternate Runs over a Measured Distance. Assume the Runs as follows: Run. Miles or Knots. IBt Result. ad Remit. Result. 4th Result. Mean of Result*. I I 5 .6 12.9 2 IO.2 ' ' ' VJ 12.6 ' 12.3 >."' g , 12.55 3 14.4 12.7 12.5 \ 12.5 } 12-45 { 12.45 = True Speed. 4 II : v -' 12.4 j } 12.4 12. 1 J 12.35 1 5 13.2 r ' i^i 12.3 12.5 6 ii. 8 62.s-f-? = i2.; Ordinary mean Sneed. NOTE. The mean of second result is sufficiently accurate for ordinary determi- nations. Velocity of tlie Current. To Determine tne Velocity of* tne Current in Line of* tne Vessel's Course. From the observed speed of the vessel deduct her true speed, and the difference is the velocity of the current. ILLUSTRATION. Assume preceding runs. Runs. SJH Observed. ed. True. Difference. Miles or Knots. X 5-6 3-*5 With the vessel 2 3 4 5 O.2 4-4 i 12.45 2.25 1-95 75 Against With Against With do. do. do. do. 6 I'.B 65 Against da Relative Corrosion of Wrought Iron in Sea Water. In Air i. In contact with brass 3-4 I In contact with lead s . 5 ' copper 4.9) " " gun-metal 6.5 In contact with tin 8.7. 972 PILE-DRIVING. RINGING ENGINE. PILE-DRIVING. (Continued from page 672.) To Compute \Veiglit of Ram. (Molesworth.) (hP \ -_- i j = R. P representing weight of pile in Ibs., h height of fall of ram, and L length of pile, both in feet, and A area of section of pile in sq. ins. Piles are distinguished according to their position and purpose : thus, Gauge Piles are driven to define limit of area to be enclosed, or as guides to the permanent piling. Sheet or Close Piles are driven between gauge piles to form a compact and continuous enclosure of the work, and are driven as close and uniform to each other as practicable of attainment, and the intervening space or joint, however close, is made water-tight by the introduction of a " feather " driven in a groove on the sides of the piles. Crushing. Crushing resistance of a pile, unless of very hard wood, should not be estimated to exceed a range of from 500 to 1000 Ibs. per sq. inch. Refusal of a pile intended to support a weight of 13.5 tons can be safely taken with a ram of 1350 Ibs., falling 12 feet, and depressing the pile .8 of an inch at final stroke. Pneumatic Piles. A hollow pile of cast iron, 2.5 feet in diameter, was depressed into the Goodwin Sands 33 feet 7 ins. in 5. 5 hours. Water Jets. A. stream of water is ejected under pressure at the point of a pile, and, rising around it, removes the end and surface resistance, so that it will be more easily driven. Suited for sand or fine soil. Nasmyth's Steam Pile-hammer has driven a pile 14 ins. square, and 18 feet in length, 15 feet into a coarse ground, imbedded in a strong clay, in 17 seconds, with 20 blows of ram, making 70 strokes per minute. Shaw's Gunpowder Pile-driver is operated by cartridges of powder on head of pile, which are ignited by fall of the ram. 30 to 40 blows per minute have been made under a fall of 5 and 10 feet. Sheet filing. Bevelling 120 | Shoeing 25 To Compute Coefficient of Resistance of the Eartl*. - = C. R representing resistance of the earth, h height of fall of ram, and d ftnal depression, both in feet. Ringing Kngine Requires i man to each 40 Ibs. of ram, which varies from 450 to 900 Ibs. To Color Brass (Copper and. Zinc) Blu.e. Mix in a close vessel 100 grains = 6. 5 oz. Troy, of Carbonate of Copper and 750 grains 4.06 Ibs. Troy, of Ammonia; shake until solution is effected and then add distilled water; shake, and the solution is ready for use. Keep it cool and effectively stopped. If deteriorated, add a little Ammonia. Articles to be colored, to be perfectly clean, suspended in motion in the solution; remove therefrom in from 2 to 3 minutes, wash in pure water, and dry in sawdust or like effective material. Expose during the operation as little to the air as practicable. Other alloys, as copper and tin and argentine, are not available. (Chemical Journal.) STEEL SPEINGS. 973 STEEL SPRINGS. (Additional to page 779.) To Compute Safe Elements of Springs.* D representing dejlection and t thickness of plates, both in i&h* of an inch; I length of span or bearings when weighted, and b breadth of plates of springs, both in ins.; n number of plates, and L load or stress in 1000 Ibs. NOTE. The plates are assumed to be similar and regularly formed. ILLUSTRATION. Assume a spring of the following elements : I = 20 and 6 = 3 in*., t = 4 i6*A, n = 5, and L 2400 Ibs. = 6 -^ = 6.66 ,6*: 3 /6.66x 3 xT3xl_,/6 4 oo__, 3 / V 6. . 8 X 2o3 _ q 76400 _ .8X2o3 _6400_ ^ * ' A f.f. XX -, \/ A ? ,r,8^ ^' _ 6.66X3X5 V ioo ' 6.66X3X4 3 1280 8 X zo3 6400 3 X 4 2 X 5 240 - - 5 = - = 3 -j- ins. - - - - - = = 2. 4 1000 Ibs. 6.66X4 3 X5 2133 5X20 100 NOTE. When back or short plates are added, they are to be added to the number of plates if of the ruling breadth and thickness. When extra thick back or short plates are adde.d, they are to be represented by plates of ruling thickness having an equivalent resistance, prior to computation by formulas for D and L, and are thus ascertained: multiply number of additional plates by cube of their thickness, and divide product by cube of ruling thickness. ILLUSTRATION. Assume as preceding, thickness of plates = 4 16^ number of them 5, and 3 extra plates of 5 i6ths to be added. Then, L^_ = 21* = 5 .86 = no. of plates, and 5 + 5.86 = 10.86, the no. of plates 0/4 i6. ik ~A 3 I 4 or 6 6 4-5 3 I 4 or 6 6.625 5 3 4 i 6 7.625 6 3 4 I 8 7.625 6 3 4 i 8 or 10 9-75 8 6 Eddy Valves and Hydrants are adopted by Jf ire Insurance Companies. MEMORANDA. .A.ruminnm. (Continued from page 938.) The available properties of Aluminum are its relative lightness, freedom from tarnish, not being affected by sulphurous fumes and being slowly oxidized by a moist atmosphere, its extreme malleability, its facility of being cast, its high speci- fic heat and electrical and heat conductivity, and its extreme ductility. Its transverse and torsional resistances are very low, its maximum shearing re sistance for castings 12000 Ibs., and forgings 16000 Ibs. per square inch. It is adapted for structures under water, can be welded by electricity and an- nealed if heated and gradually cooled just below a red heat. The tensile strength of its wire is greater than that of its rolled metal. Its properties are materially changed and impaired by alloying it with small per- centages of other metals, and its tensile resistance, relative to its weight, is in plates as strong as steel at 80000 Ibs. per square inch t and in cold drawn wire as strong as it is at 180000 Ibs. (Alfred E. Hunt.) ^Magnesium. Specific gravity 1.74, is .33 lighter than Aluminium; is harder, tougher, and denser; less affected by alkalies, and takes a higher polish. Staff. Staff is composed of Plaster of Paris, water, and hemp fibre, the latter used to bind the mass. For ornamental pieces, matrices of hardened gelatine are used. It resists the weather and even frost after being saturated. Boiler Setting. The fire-brick should be laid with very thin joints, and set in Kaolin* or pre- pared fire-clay, so thin that it is necessary to lay it with a spoon instead of a trowel. Every fifth course should be a header course. (" The Locomotive. 11 ) GUue. Its tenacity varies from 500 to 700 Ibs. per square inch. ITriotion of Engines and. Grearing. (In addition to pages 469-478, etc.) Deduced from Experiments of Alfred Saxton, Manchester Assn. of Engineers. Spur Gearing. 25.9 per cent. I Belt Driving 28.6 per cent. Rope Driving 29.6 " | Direct Acting 23.8 " Engines 6 and 10.3 per cent. Spur gearing gave the best result when not complicated with rope driving. Rope driving gave best results at high speeds. Belt driving for developing large power is only equal to an average rope-driving engine. Relative Value of various "Woods, their Crushing Strength and Stiffness feeing Combined. Mahogany 3.7 Yellow pine. .. 3 Spruce 3.6 Sycamore 2.6 Walnut 3.4 Cedar i Comparative "Value of Long Solid Columns of various Materials. (Hodgkinson.) Cast Iron 1000 | Cast Steel. . . . 2518 | Oak 108.8 | Pine 78.5 Hence, To compute destructive weight of an Oak or Pine column, take weight for one of Cast iron of like dimensions, and if for Oak divide by 9, and for Pine by 12.7. * A variety of clay, one of the two ingredients in Oriental porcelain ; the other ia termed in China fttunts. Teak 9.4 I Elm 5 English oak ... 5.8 Beech 4.4 Ash 5. i I Quebec oak. ... 4. i SPIRALLY RIVETED IRON OR STEEL PIPE. 977 Spirally Riveted Iron, or Steel 3?ipe. Abendroth & Root Mf'g Co., Neioburgh, N. Y. Spirally Riveted !3VEetal I?ipe. Compared with Wrought or Cast iron Pipe, has the advantage of low original cost and expense of transportation, maintaining a nearly equal bursting pressure with that made of heavier material. It is made of Sheet Iron or Sheet Steel, varying in thickness from No. 20 to No. 12 B.W.G., ac- cording to diameter and pressure. The rivets in the seam are set by compression, while the laps are thoroughly coated with hydraulic cement to make it water tight. Connections. When a moderate pressure is maintained, these pipes, their ends being crimped, are usually connected by a cement joint, as shown in the annexed cut. When the pressure is excessive, a bolted, joint is resorted to, as also shown, and which is in effect a stuffing box or sleeve joint, dispensing with lead calking, and admitting of a slight flex- ure of the pipe. j For service connections the collar may be tapped. When lead calking is required, the inner ends of the pipe are reinforced by an iron collar. Bursting Fressnre. ll Jl-a ll ll *_ 3 * I! Per Sq. Inch. g Per Sq. Inch. 1! Per 8q. Inch. Per Sq. Inch. II Per Sq. Incb. 55 5 5.S 5~ 5*9 Ins. Lbs. Ins. Lbs. Ins. Lb. Ins. Lbs. Ins. Lbs. 3 900 to 1300 6 350 to 800 10 275 to 650 16 190 to 400 22 125 to 300 4 700 " looo 8 350 " 825 12 225 " 550 18 150 " 375 2 4 no " 275 5 550 " 800 9 300 " 750 14 200 " 470 20 140 " 325 In order to enable an estimate of the relative cost of these pipes, com- pared with cast and ordinary wrought-iron pipes, the weights of each are submitted. 'Weights. Heavy Spiral. U ID*. L+- a a 5 a Heavy Spiral. i l*n ^ Heavy Spiral. . fe ^ He Spi a < ih ^,4- i o i I P ~ . 22 Volt, -s Vo Day, -s Ds Month -s Mo Atmosphere, -s At Barometric Be Breadth -s b b' Evaporation, -ive.... Evp Foot pound, -s, tons Fp.Ft Force F Year, -s Ys Triangle, -s A A' Centrifugal force Cf Triple Tpl Centre of gravity Cg Friction Fn Unit -s Heat Hu Circumference, -s..C.c.c' Coefficient or Factor. . Co. Compound . .... Cpd Gravity ,..*g Height, -s H . h . h' Horse-power B? Calorific or French . . Co Vacuum Vm Velocity V v v' Cube Cub or |5] Effective EH* Versed sine v-sin Cylindrical Cyl Indicated IBP Vertical Vt Nominal NB? Volume -s Vol vol Depth, -s dp . dp' Departure Dpt Inclination In Joule's Equivalent jE Ijjiti tilde ....... Lat Chaldron, -s Ch Chord, -s Co Diameter, -s. ... D . d . d,' Distances. Inch, -es. ins. Feet ft Bushel, -s Bl Length -s L 1 1' Cube foot feet Cf Barrel, -s bbl Yard s Yds Hyperbolic . . . Hyp. log. Mercurial gauge Mg Gallon, -s gl Chain, -s Chn Rood, -s Rd Microliter (Greek lambda) X Milliliter, -s ml Knot, -s K Mile, -s .Ms Millimeter, -s mm Centimeter, -s cm Meridian M Modulus of Elasticity . ME Moment s Mt Deciliter, -s dl Liter, -s 1 Number, -s No Ordinate, -s O . o . o' Perpendicular Pr Pitch -s Ph Ph' Dekaliter, -s dal Hektoliter -s .... hi Kiloliter or Stere, -s. Kr Water-line Wl Meter, -s m Dekameter -s dA Hektameter, a hk Kilometer, s km Pressure, -s P . p . p' Quadruple Qpl Radius, -ii R . r . r' Revolution, -s.. Rev. rev. Secant Sec Weight, -s W . w . w' Pound, -s Ib. . Ibs. Ton, -s (2240) Tons " (2000) ... Tons Milligram -s mg Centare orsq. meter, -s Ce A re - s a Hectare -s Ha Cosecant Cosec Draught of water Dw Sine Sin. Centigram -s eg Cosine Cosin. Decigram, -s dg Slip Sp Gram, -s g Equivalent, U. S. or ) French ) Eq< Solid Sd Dekagram, -s dgra Hektogram, -s hgm Kilogram or Kilo, s. Kg Myriagram -s y Specific gravity Sg Span Sn Electric. Ampere, -s. Am Farad, -S. (Greek cap. phi) 4> Microfarad, -s (Greek phi) Stability St Steam .... . Stm Quintal -s . . . q Stroke S s Miller or Tonneau, -8 Mr i TT = 3 1416 982 MEMORANDA. Relative Efficiency of a Non-condensing Steam-Engine and a Ki-Sulphide of Carbon (C 82) Engine. With like Engine, Boiler, and Fuel, as developed by competitive tests of both at Riverdale, near Chicago, 1892-94. Cylinder, 16 x 42 ins. Jacketed and with automatic Cut-off. Boiler, horizontal cylindrical fire tubular. Grates, 21.6 sq. feet, and Heating surface, 1028 sq. feet. Fuel, anthracite; Combustion, natural draught. Steam. Coal consumed per i IP per hour, 4.299 Ibs. C S 3 . " " 2.49 " =42.08 per cent., or as i to 1.72 -}-, or 942.6 Ibs. coal in each ton. Mortar for Masonry loelcrw Freezing-point. Anhydrous Carbonate of Soda, u Sodium carbonate" (Na2CO3), 2.3 Ibs. per gallon, dissolved in water, maintained at 86, mixed with equal volume of water. Mix the mortar with 25 per cent, more of this solution than if pure water was used. Hands of operatives should be protected, as with India-rubber gloves. Extra cost 35 cents per cube yard of masonry. Setting of mortar accelerated. It will set at 5 below freezing point as readily as at 10 above. (Caen and Vive- Saint Lo R'y.) Mons. Rabat. Freestone. Result of a Series of Tests of Connecticut Brownstone to Resist Crushing. Per Sq. Inch of Cross Section. Portland stone 6222 to 10928 Ibs Colt's Fire arms Mfg. Co. New England Brownstone Co. . 7843 " 13 297 '* U. S. Ordnance Dept. Portland Shaler and Hall 9330 " 13 980 " u " Highest Rail-way in Europe. Brienzer Rothhornbahn. Alps, 7886 feet at summit level. Rack and Pinion. Greatest grade, i in 4. Elevation in Feet of Localities in the Upper Missis- sippi and \Vest of L.ake Michigan. (In addition to page 582.) Davenport, la 615 Dubuque, " 665 Ft. Madison, la 600 Independence, la 850 Keokuk, " 618 Monticello, 880 Ashland, Bay City,Wis. 610 La Crosse, " 744 Milwaukee, " 697 Ft. Ridgely, Minn 1230 Ft. Snelling, " 820 Minneapolis, " .... 856 St. Paul, " .... 831 Age of Trees. Lime i too years Ft. Ripley, Minn Chicago, 111 .1130 Galesburg 111 Ottawa, " .... Peoria, " Rockford, " . 800 St. Louis Mo Oriental Plane. 1000 yeara Spruce 1200 " Oak 1500 " Walnut ooo " Olive 800 u Yew 3200 Orange 670 " Appleton, Wis 653 Cedar 2000 years. Cyprus 800 " Elm 300 " Ivy 335 " Larch 576 ' Monolith. (In addition to page 179.) Wisconsin. At World's Fair, 10 feet square at base, 115 in height, and 4 at apex. Angle of Repose of Earth. Clay, dry 29 damp, well drained 45 wet !6 Earth, vegetable dry.. 29 moist. Shingle. . , 39 Gravel, clean ......... 48 with sand. ... 26 Sand, wet ............ 26 34 R. E. Aide-Memorie. The co-efficient f friction = tan. of degrees, as co-efficient of shingle == tan. of 39 --Si- MEMORANDA. 983 To Compute the H* of a "W>ought-iron Shaft. RULE. Multiply cube of diameter of shaft in its journal in inches, by number of its revolutions per minute and divide product by 80. EXAMPLE. Diameter of journal 17 ins., and revolutions of engine 20. What is its IP? 17 3 x 20 = 98 260, which -4- 80 = 1228. 25 H?. To Determine the South toy the Hour-hand of a Watch, between the Hours of 8 A.3VL. and 4 P.M. When the Sun is Visible. OPERATION. Point the hour hand to the sun, and half the distance between thai point and the figure 12 is the South. XOTK. The greatest error is in latitude 38, and is about 15* too far Eat at 8 A.M. and 15* too far West at 4 P.M. Hence allowances are to be correspondingly made. Q-reatest Depths and Heights. (In addition to pp. 179-184.) Greatest depth of Ocean, Pacific 28 ooo feet. Deepest Well in North America, Wheeling, Va 4 560 44 Mine in Comstock, Nev 3000 Highest Mountain in North America, St. Elias. . : 18000 " Structure, Washington Monument 550 " Tide, Bay of Fundy '. .T. .'KV. . . . i * 50 Tests for \Vater. (In addition to page 851.) If Hard When mixed with a solution of soap, it will be rendered milky. If Carbonic Acid is present. When mixed with lime water it will be rendered milky. If Sulphate of Lime (Gypsum) is present. Mix with a little chloride of barium, and if a white precipitate is formed, which will not dissolve when nitric acid is added. Comparative Tenacity of Cut and "Wire Iron Nails. In Lengths from 1.125 t 6 ins., and Driven in Spruce and Pine Timber. In Spruce. The tenacity of ordinary cut nails exceeded that of wire 47.51 per cent.; of finishing, 72.22; and of box, 50.88. In Pine. Box nails taper perpendicular to grain of wood 135.2 per cent; paral- lel to it, 100.23; a d driven in end of wood, 64.38. Average of 58 series of tests, with 40 sizes of nails, 72.74 per cent Maj. J. W. Reilly, U. S. O. D. By Wm. H. Burr, C. E. Hydro-GJ-eology. U. S. Census Report, Vol. XVII. Upper Missouri. The average annual Discharge of the principal tribu- taries of it, to the precipitation on the basin = 30. 8 per cent. Upper Mississippi. Average annual Rainfall in it and in the valley west of Lake Michigan, 33.7 inches. Average flow per second per square mile of drainage area, .703 cube feet. Elevation of ordinary low water at its extreme source above the Ocean, 1680 feet Drainage area above mouth of Missouri river, 169000 square miles. J. L. Greenleaf, C. E. Cost of an Electrical EP. Available IP, 84 per cent, of Indicated. Coal at $3.75 per Ton. Coal, 64 cents. Wages, water, etc., 16 cents. Alex. Siemens, M. I. C. E. 984 MEMORANDA. Capacity of Gri.rd.ers and. Floor Beams. Loaded in their Centre. Girders of single span of Georgia or Yellow Pine, 10 ins. in breadth, 12 ins. in depth, and 10 feet in length between its sup- ports at both ends. The mean capacities of the woods in ordinary use in the floors of buildings for one inch square and one foot between supports are as follows: Lbs. Georgia or Yel. Pine. . 850 Hemlock 450 Chestnut 480 White Pine. White Oak 650 Lbs- Spruce 550 Canada Oak 560 N.E.Pine 5 < Ash .................. 900 In the computation of the strength of posts, girders, and floor beams in build- ings, it is impracticable of assured safety at all times to assume their strength at their mean value as determined by experiment, inasmuch as allowance is to be made for defects, as knots and shakes, fungus growth and dry rot. Hence a Factor of safety or a deduction of capacity in each case must be resorted to. In the case above given, the strength of the pine is assumed at 850 and coeffi- cient of capacity at 4. Hence, 10 X I = 3 o6oo Ibs. and - = 3060 Ibs. 4 X 10 10 If Uniformly Loaded, this result would be doubled. Estimate of ."Paint Required, for Open. Iron "Work on Bridges, etc. First coat, .625 gallon per ton; second coat, .375 gallon. (71 J. Swift, C. E.} To Compute T--P of a Stream of Water. When Maintained at a Uniform Height. ILLUSTRATION. Assume section 22 sq. feet, velocity of flow 5 feet per second, and fall 25 feet. 22X5 = 110 cube feet water in volume per second. 1 10 X 60 x 62. 5 = 41 2 500 Ibs. water per minute. Then, ^-^ - = weight oj water in Ibs. -7-33000 = 312.5 theoretical IP per minute, from which is to be deducted loss of efficiency of instrument of applica- tion, and which are given. (See pp. 561-580.) Assume a Turbine wheel at. 7 of efficiency. Then, 312.5 X .7 = 218. 75 effective IP. If the surface is maintained (impounded) at a uniform height, the power of it. for the period of working hours, will be 218.75 x 60 X N. N representing the num- ber of hours. The result, then, for a period of 10 hours would be 218.75 X 60 X 10 = 131 250 IP. If a stream has a supply equal to the expenditure of it, it will overflow in the intervals between working hours; but if it is unequal to the operating volume or consumption required, there must be a storage reservoir, and the greater the area of it the less the decrease of the head or level of it, when being drawn from. To Compvite Elements of a Flume. The Volume of Water. Area and Height being Given. / V - A v zghG=. volume. I 2 p) -*- 2 = height. V representing volume in cube \3 I feet, A area in sq.feet, C coefficient of discharge, h height, and I length in feet. Assume Volume 36.29 cube feet, Area 8 sq. feet, and C .6. 36 .29 or, ~ = ! = MEMORANDA. 985 Effect of Tapping of T^.ong-leaf Pine. Late tests, conducted by the U. S. Dep't of Agriculture, of 32 trees, have conclu- sively evidenced that the timber of the Long leaf Pine is in no wise affected by the tapping of it for turpentine. See Circular No. 9, Forestry Division. Comparative Strength of Tapped and Untapped Long-leaf Pine. In Pounds per Sq. Inch. Condition. Specific Gravity. Tensile. Transverse. Crushing. Detrusive. Tapped. 25 pieces green . . .... Lbs. TC 448 Lbs.* 176 Lbs. 4755 Lbs. dry 687 14 7S7 177 6627 648 115 tests mean ?6 re 081; eil8 6ofi Untapped. m tests, mean .71 1 6 A.2Q i^6 *66i * One inch square and one foot in length, weight supported from one end. See Circular No. 8. Tapped and untapped is known as "boxed" and " unboxed." The pores of wood leading upward, or in the direction of its growth, facilitate the flow or passage of moisture in that direction. Hence, timber set inclined or vertical, with the abut end uppermost and exposed to moisture, will decay at the top more readily than if set with the abut down. The effect of varying the set of wood is frequently observed in the staves of a cistern or tub, etc., some of them being saturated with moisture and others quite drv. To Compute Weight of Flue and Tvitmlar Marine Steam Boiler. To weight of the metal plates, as determined by their area and thickness, add as follows : For Laps and Rivets. One Ib. per sq. foot for each . 125 ins. in thickness of plate. " Bolts, Stays, and Braces. 20 per cent, of total weight of the plates in Ibs. " Mean and Handhole Plates. 750 to 1000 Ibs. Notes on. Portland. Cernent and. Cement Mortars. (In addition to pp. 515, 589, 871, 907, 958.) Cements that harden rapidly produce a brittle texture ; they should increase in strength with uniformity. The strength at termination of one day should not ex- ceed 45 per cent, that of the seventh day. The addition of Sulphate of Lime and Gypsum to American Portland cement in- creases the strength of the cement; with 3 per cent, of the former it increases it 64 per cent. ; from that it diminishes the effect; and with 5 per cent, of the latter it increases it 33 per cent. To English Portland cement 2 per cent, of Sulphate of Lime increased it 60 per cent. Dry Sands. Standard. Weight, 92 Ibs. per bushel, and its voids are 47. 5 per cent. Natural or Bar. 103 Ibs., and 41.25 per cent. Requirements and Specifications. Tensile Tests. An average of 5 briquettes in each case. Fineness. 97. 5 per cent, through No. 50 sieve, and 87. 5 through No. 100 sieve. Specific Gravity. To exceed 3. Homogeneous. Discs, 3.5 ins. in diameter, and .375 thick at centre, tapering to a sharp edge at the circumference; they should not crack or warp. Samples. Ten per cent, of the quantity selected at random. Tensile Strength. Limit of results, 10 per cent. For Exposure in Salt-water. Initial set with fresh-water not to exceed ten min- utes. 4 986 MEMORANDA. Absorptive Power of* Charcoal. Of Fine Boxwood. By Volumes. Ammonia 30 Carbonic acid 35 Carbonic oxide 9. 42 Cavburetted hydrogen 5 Hydrogen 1.75 Nitrogen. 6.5 Nitrous oxide Oxygen Sulp'd hydrogen. . 40 9-25 55 3?op Safety and. Relief "Valves. Crane Co., Chicago. BRASS. IRON BODY. BRASS SEAT WT u^ r\j j J \ \ 2 t I 1 2. 8 s "* H. P. tio of Valve to Grate. Centre of alve to end of Outlet. & > Ins. No. Sq. Ft. Ins. 75 3 t 6 1.32 i 6 10 1.25 10 20 3-68 *-5 20 30 5-3 2 30 40 9.42 2-5 4 75 14.72 3-75 H. P. Ratio of Valve to Grate. Centre of Valve to end of Outlet. No. 40 to 75 75 loo 100 125 125 150 150 175 175 200 250 300 Sq. Ft. 14.72 21.2 28.86 47-7 84.82 Ins. 3-75 4-375 4-75 6-375 6-375 7- I2 5 Approved by U. S. Board of Supervising Inspectors of Steam Vessels, and will be approved by all Local Inspectors, on a basis of One square inch of area to three square feet of grate surface. -A.meri.oan. "Woods. With the Order of their Strength. Order. Ash, mountain, Pyrus Americana.. Fraxinus pistacicefolia ....... 234 Oregon, Fraxinus Oregana... 210 red, " pubescens. . 105 white, " Americanum 29 prickly, ^fanthoxylum Ameri- canum ......................... Basswood, Linden, Tilia Americana 249 Beech, Fagus ferruginea .......... 24 Butternut, Juglans cinerea ........ 205 Button-wood, Conocarpus erecta. .. 76 Cedar, white, Libocedrus decurrens. 200 " red, canoe, Thuya gigantea. Cherry, wild red, Prunus Pennsyl- vania ......................... Cherry, wild black, Prunus serotina 119 Chestnut, Castanea vulgaris ....... Cotton wood, Populus monilifera. .. Cucumber, mountain, Magnolia acu- minata ................. ...... 208 Elm, slippery, Ulmusfulva ........ " white, " Americana. . 114 Fir, white, Abies grandis .......... 280 Gum, sweet, Liquidambar styraci- flua ............................ 222 Hemlock, Tsuga Mertensiana ...... 87 Hickory, shell-bark, Carya alba. . . 12 41 nutmeg, " myristi- cceformis ....................... x 150 Hickory, pignut, Carya porcina Iron wood, Cyrilla racemiftora Larch, hackmatack, Larix Ameri- cana Laurel, big, Magnolia grandiflora. . " white " glauca Lignumvitae, Guaiacum sanctum. . . Lime, wild, JTanthoxylum Pterota.. Locust, Robinia pseudacacia Maple, mountain, Acer spicatum. . . " sugar, hard, " saccharinum " silver, soft, " dasycarpum. " swamp, " rubrum Oak, black, Quercus tinctoria " live, u virens " white, " alba Pine, white, Pinus strobus " yellow, " Arizonica " pitch, " rigida " scrub, " inops Poplar, white wood, Liriodendron tulipifera Redwood, Sequoia sempermrens Satin wood. Jfanthoxylum caribaum Spruce, white, Picea alba Sycamore, Platanus occidentalis . . . Tulip, yellow Walnut, black, Juglans nigra Willow, Salix Icevigata Order. 44 305 94 139 170 143 56 126 1 68 214 215 246 '57 163 231 ELECTRICAL. 987 Electrical. Compiled by Prof. A. E. Kennelly. Units in Electrical Engineering. The following units have been legally adopted by the U. S. Government, 53^ Con- gress, 1894: Unit of Resistance. The International Ohm, represented by the re- sistance offered to an unvarying electric current by a column of mercury at the temperature of melting ice, 14.4521 grammes in mass, of a constant cross-sec- tional area, and of the length one hundred and six and three-tenths centimetres. Unit of Cvirrent. The International Ampere, which is the one tenth of the unit of current of the centimetre-gramme second system of electro magnetic units, and is the practical equivalent of the unvarying current which, when passed through a solution of ni Irate of silver in water, in accordance with standard specifications, deposits silver at the rate of .001 118 gramme per second. Unit of Electromotive Force. The International Volt, which is the electromotive force that, steadily applied to a conductor whose resistance is one international ohm, will produce a current of an international ampere, and is practically equal to 1.434 times the electromotive force between the poles or elec- trodes of the voltaic cell known as Clark's cell, at a temperature of 15 C., and pre- pared in the manner described in the standard specifications. Unit of Quantity. The International Coulomb, which is the quantity of electricity transferred by a current of one international ampere in one second. Unit of Capacity. The International Farad, which is the capacity of a condenser charged to a potential of one international volt by one international coulomb of electricity. Unit of \Vork. The Joule, which is equal to io7 units of work in the C.-G.-S. system, and which is practically equivalent to the energy expended in one second by an international ampere in an international ohm. Unit of ]Power. The Watt, which is equal to 10? units of power in the C.-G.-S. system, and equivalent to work done at the rate of one joule per second. Unit of Induction. The Henry, which is the induction in a circuit when the electromotive force induced in circuit is one international volt, while the in- ducing current varies at one ampere per second. The following list presents these units with their derivatives, and also other elec- tro-magnetic units which are in use: Resistance, O hm. Megohm, one million ohms; Begohm, one billion ohms; Tregohrn, one trillion ohms; Microhm, one millionth ohm; Bicrohm, one billionth ohm. Current, Ampere. Bicro-ampere, one billionth ampere; Micro ampere, one millionth ampere; Milli-ampere, one thousandth ampere; Centi ampere, one hun- dredth ampere; Deci-ampere, one tenth ampere; Deka-ampere, ten amperes; Heclo- ampere, one hundred amperes; Kilo-ampere, one thousand amperes. E. M. IT., Tort. Microvolt, one millionth volt; Kilovolt, one thousand volts. Capacity, Farad. Bicrofarad, one billionth farad; Microfarad, one mill- ionth farad. Worlz, Joule. Kilojoule, one thousand joules; Megajoule, one million joules. IPower, Watt. Kilowatt, one thousand watts. Induction, Henry. Microhenry, one millionth henry; Millihenry, one thousandth henry. Magnetic Flux, Weber. Kiloweber, one thousand webers; Megaweber, one million webers. Magnetic Relnctance, Oersted Millioersted, one thousandth oersted. . '* Intensity, Gauss. Kilogauss, one thousand gausses. Magnetomotive Force, Gilbert. The Q-ilbert is the M. M. F. produced by .7958 ampere-turn. The Oersted is the reluctance of a cubic centimetre of air measured between opposed parallel faces. The Weber is the flux produced by a M. M. F. of one gilbert through a mag netic circuit in which the reluctance is one oersted. The Q-anss is an intensity of one weber per normal sq. centimetre. 988 ELECTRICAL. Electrical. (British Association.) Resistance. Unit of resistance is termed an Ohm, which represents resist- ance of a column of mercury of i sq. millimeter in section and ^.0486 meters in length, at temperature o C. i eoo ooo Microhms = i Ohm. i Microhm = 1000 absolute electro-magnetic units. i Ohm = i ooo ooo ooo i oooooo Ohms = i Megohm or io J 5 " Kleotro-xnotive Force. Unit of tension or difference of potentials is termed a Volt. i oooooo Microvolts. . = i Volt. i Volt = 100000000 absolute electro-magnetic units. i Megavolt . . . = i oooooo Volts. Current. Unit of current is equal to i Ampere, or the current in a circuit which has an electro motive force of i Volt and a resistance of one Ohm. Capacity. Unit of capacity is termed a Farad. i oooooo Microfarads or io~ 9 absolute units of capacity = i Farad. Heat. Unit of heat is quantity required to raise one gramme of water from 00 C. to i C. of temperature. Quantity. Unit of Quantity, one Coulomb, and is the quantity of Electricity transferred by one ampere during one second. r j?o Determine the East and. "West ^Meridian. Set up a rod vertically on a level area or plane in the approximating meridian and describe arcs, with a radius of about twice the height of the rod.* At any time before M. mark the point in the arc where the shadow of the rod touches it, and in the P. M., at the same length of time of before M., mark the point of the shadow on the arc; remove the rod, and a line drawn through these points will give the true bearing of E. and W. To Compute the Increase of -A^rea of a Circle or "Volnme of a Ctiloe. By Differential Calculus. CIRCLE, n x 2 = u and 2 -n x d x=du. u representing area, x radius of circle, and du increase of area. ILLUSTRATIONS. i. Assume x 10 ins., and d x, or difference of radius, .05 inch. Then, 2X3. 1416 X 10 X .05 = 3. 1416 sq. ins. Circle of 10 ins. radius = 3. 1416 sq. ins. Hence, / - = V 404 = 20.0997 ins. , the increased diameter. CUBE. x3 = u and 3 x 2 d x = du. 2. Assume x side of a cube 12 ins. and d x increase of side .05 inch. x representing side of cube, and du increase of volume. Then, 3 x i2 2 X .05 = 21.6 cube ins. Hence, -^123-1-21.6 12.05 ins. the increased side. du -idx .15 , dx .05 , .0125 = - = ^. .0125 and = ^-=.004166 and- 77 = 3- Hence, the u x 12 x 12 .004166 cubical expansion is three times that of the linear. * Varying with the latitude of the location, a the more vertical the sun the greater the height of thu rod. ENERGY AND MOTION KINETICS. 989 Energy and. ^Motion. The science of Motion is included in Mathematics, and is termed Kine- matics ; the science of Force, Dynamics or Kinetics ; and the investigation or operation of forces in equilibrium, Statics. All standards of Energy and Motion are Units, as the unit of Length may be an inch, foot, yard, or mile, but usually a foot ; that of Time a second, minute, hour, or day, usually a second; and \ r elocity by the number of units of lengths or opera- tion in a unit of time. Uniform Acceleration is the uniform increase or decrease of velocity per unit of time or distance, but this increase or decrease of velocity is that which the force produces in a unit of time ; hence =F. m representing unit of force, a unit of acceleration, t the time, and F the force. ILLUSTRATION. Assume a body moving at the rate of 50 feet per second, and at the end of 10 seconds it has acquired a velocity of 75 feet per second, the increase of velocity is 25 feet in 10 seconds, equal to 2.5 feet per second in each second, or 2. 5 feet per second per second. * 2. Given a uniform acceleration of velocity of 40 feet per second per second; what is the acceleration in yards per minute per minute? 40 feet per) _4oX 60 feet per min. per sec. ) _4QX6o 2 = sec. per sec. J ~ 40 X 6o 2 " ' min. ( 3 min. per min. 3. A train of cars 2 minutes after starting attains a uniform velocity of 15 miles per hour ; what is the acceleration in miles per minute per minute? 15 miles per hour =. .25 mile per minute, increase of velocity = . 25 mile per minute which occurs in 2 minutes. Hence, acceleration = . 5 of . 25 mile per minute = . 1 25 mile per min. per min. Kinetics. Force and Mass. If two bodies of equal dimensions and unequal weights be simultaneously pro- jected with like Velocity, the heavier one will go farther than the lighter; but if these bodies were projected with like Force, the lighter one would go the farthest. The difference is in consequence of the difference of the Mass or Matter, and as a result it requires more force to stop the heavier body when started than the light one. In operation, there are two common Units of Mass, as there are two of Force. The Poundal, or British absolute unit of force, is that which the action on a mass- pound for one second produces in it a velocity of one foot per second. The Dyne is that force which, acting on a mass-gram for one second, produces in it a velocity of one centimeter per second. OPERATION. If 15 poundals bear upon a mass of 70 Ibs., in what time will it pro- duce a velocity of 60 feet per minute? i poundal = a velocity of i foot per sec. in i Ib. in i second. Hence, 15 poundals == a velocity of i foot per sec. in i Ib. in second. 15 poundals = a velocity of i foot per sec. in 70 Ibs. in seconds, and 15 poundals = a velocity of 60 feet per sec. in 70 Ibs. in = 280 seconds. Hence, ^^ = P. m representing unit of force, v unit of velocity, t unit of time, and F the force. * Second per second, Minute per minute, etc., although unusual, is proper. Thus, the expression, " The train went with a velocity of 60 miles per hour,'' is indefinite, as it may have gone at that rate but for a period of one minute, or 10 minutes ; whereas, 60 miles per hour per 'hour indicates both the rate of the velocity and of that per hour. 990 KINETICS. GAS ENGINES. Impulse Is when a force acts during a given time. Thus, if a force of 5 Ibs. bears upon an object during 3 seconds, 3 X 5 = 15 units of effect, and = F, representing the number of units of impulse, Momentum, or Moment Is the product of the number of units of velocity with which the mass is moving. ILLUSTRATION. A mass of 200 Ibs. is moved with a uniform velocity of 75 yards per minute ; during what time is a force of 80 Ibs. required to arrest the motion? 75 3 = 3. 75 feet per second, and 2 ^_ 75 = 750, which -4- 80 = 9. 375 seconds. 2. A mass of 6.75 Ibs. is acted upon by a force of .5 Ib. during 5 minutes ; what is the velocity acquired? 5 minutes=3oo seconds, and 300-:-. 5 (half pound) = 150 units of impulse, and ^- 22. 2 feet per second. 6-75 3. A rod 8 feet in length, weighing 8 Ibs., has a weight of 205 Ibs. suspended from one end and 60 Ibs. from the other ; at what point in the bar will the effect of the weights be equalized? By rule To Compute Position of Fulcrum, p. 624, 8 -r- ^ 4- i = - = 1.8113 feet= distance of 205 Ibs. from its end. and 8 60 4-4167 i. 8113=: 6. 1887 feet = distance of 60 Ibs. from its end. Then, to include weight of rod 205-]- 1. 8113-7-60-1-6.18874- 1 =4.1246, and 8-7-4.1246 = 1.9396 feet = distance of 205 Ibs. from its end, including its weight of the rod. Hence, 8 1-9396 = 6. 0604 -\-feet = distance of 60 Ibs. from its end. Verification. 205 -}- 1.9396 X 1-9396 = 401. 367 Ibs, , and 60 -f- 6.064 -f- x 6.064 ~f* = 401.376 Ibs. GAS ENGINES. Grcts Engines. Are divided into three types. T yP es ' Theoretical. Efficiency. 1. Engines igniting at constant volume, without previous) compression ........................................ J 2. Igniting at constant pressure, without previous compres- ) R sion ................................................ J 2 3. Igniting at constant volume, with previous compression. .. 3 = .34 In the first two types the cylindrical conditions are most favorable to cooling, and a practical efficiency of .06 is attained. In the third the conditions for loss by cooling are very favorable, and an actual efficiency of .17 is obtained. The ordinary heat efficiency is 17 per cent, of all the heat expended in an en- gine, and the highest obtained 25 per cent. For powers up to 20 IIP, when gas is cheap, as in towns and cities, it competes with steam, as it is more economical and more convenient, and is most usually re- sorted to for a power of from 4 to 6 horses. Some engines have been constructed and are in use of 100 IP. Non-Compression Kngines. Are principally used for small power, as up to .5 IP. The pressure is applied only during a portion of the stroke. In the Leiioir the piston is moved only for about. 5 its stroke, when it re- ceives a mixed volume of gas and atmospheric air, which is ignited by an electric spark, the pressure rising to about 45 Ibs. per sq. inch above the atmosphere. The piston then is driven through the remaining portion of the stroke, and at the end of it the pressure falls to about 3 Ibs. The mean effective pressure being usually 8.5 Ibs. per sq. inch. GAS ENGINES. 991 M.ean rtesxilts of Operation of* Thirteen 10ngin.es of Five Different Constructions. Remits. Iff BH? Gas I per BHP Revolution Mnfute. Heat Con- verted into Work. Mean. Least. Extreme. No. 15-35 3-42 336 No. 12-55 2-7 27.75 Cube feet. 22.21 18.92 3-9 Cube feet. 28.05 23-58 33-4 No. 180.7 132 223.8 Per cent. '7-43 10.5 21.2 This Type of engine is held to be wasteful of gas, as it consumes over 90 cube feet of 16 candle-power gas per IIP per hour. The Otto fc JLanger is a free-piston or atmospheric engine, admitting of high piston speed and great expansion, hence it is more economical. An explosion of gas drives the piston upward, and by the projectile force of it and the reduction of the temperature of the gas under it, a partial vacuum is formed, the piston returning under atmospheric pressure. In an engine with a cylinder of 12.5 ins. diameter and an observed stroke of pis- ton of 40 ins., 25 cube feet of gas gave a maximum gauge pressure of 54 Ibs. per sq. inch, and a result of 2.9 per IIP per hour. Compression Engines possess the advantage, of furnishing greater power With less volume and weight, as well as economy of operation. The Ot to. It is a single-acting-piston engine, serving alternately as a pump and a motor, and one explosion of gas is given for every two complete revolutions. OPERATION. The piston receives a volume or charge of gas and air, then returns, compressing the volume into a space at end of its stroke, which mixture is ignited, and the pressure therefore forces up the piston, when it returns with an exhaust valve open to free it from the force and products of combustion ; at the termination of the stroke it is in position to receive a new charge. Thus, one driving stroke of the piston is given for two revolutions of the engine. The Mean effective pressure, with gas of 16 candle-power, is about 55 to 60 IbB. per sq. inch, the maximum pressure of the explosion being from 140 to 160 Ibs., and even up to 180 Ibs. In an engine rated at 6 IP the consumption of gas was 21 cube feet per IIP, and for brake IP 29 cube feet. The Clerk has a second cylinder, termed the charging, the function of which is to receive a charge of gas and air at each stroke and deliver it into the motor cylinder. The charge dispels the consumed gas of a previous operation and fills it with mixture, to be compressed by the return stroke of the piston and ignited at each revolution. The Mean effective pressure in an engine of 12 IP is 65 Ibs. per sq. inch, pressure of compression 57 Ibs., and maximum pressure of explosion 238 Ibs. Gas consumed 24 cube feet of 24 candle power per IIP per hour. One of 9-inch cylinder and 20 ins. stroke, at 132 revolutions per minute, developed 27.5 IIP per hour, or 23.2 IP at the brake. The CampToell and 2VdLid.lan.xl are of this type, and are alike to it in the use of a charging cylinder. The Stodspnt resembles it also, but the operation differs in the passing of each charge from the gas pump, which is a combination of the motor piston, into an intermediate reservoir, whence it blows out into the motor cylinder and dis- charges the burned gas. Three-Cycle engines are like the Otto in their operations, but give only one impulse for every three revolutions, one extra double stroke being used in re- ceiving a charge of air and expelling it at the exhaust port, so that the compres- sion space is cleared at each operation. The earliest of these engines was known as the Linford ; those now in use are the Griffin and the Barker, 992 GAS ENGINES. The Atkinson Cycle gives an impulse at each revolution of the crank shaft, and the piston, by a system of links, is connected so that it makes two out and two in strokes for each revolution of the crank-shaft, and one explosion is given for each cycle of four strokes. It resembles the Otto in using the same piston alternately for pump and motor operations, but differs from it in making unequal strokes. This arrangement en- ables the exhaust gases to be swept out of the cylinder at every operation and great expansion is obtained. This engine is held to be very economical, as in recent trials it consumed but 22 cube feet of gas per brake IP per hour. (The Practical Engineer.) The Crossley is a horizontal engine, with a single cylinder, and of nominal powers from . 5 to 30 IP, indicating from 2 to 85 IP : with double cylinders, from 16 to 170 IIP. A 12 IP engine has developed 28 IIP and 23 at the brake = 82^ of the indicated, consuming 20 cube feet of gas per IIP per hour. Results of Trials of Q-as Engines. Type. IFF. Gas per IH*. Heat converted into Work Revolu- tions per Minute. Type. IFF. Gas per IFF. Heat converted into Work Revolu- tions per Minute. Otto. Clerk. Beck. No. 22.56 3-42 27.46 6.12 Cube ft. 23-6 30-9 20.39 20.67 Per cent. 17-5 14.46 15-5 21. 1 No. 158-7 160.3 132 168.9 Griffin. Forward. Fawcett. Atkinson. No. 17.46 5-54 11.49 11.15 Cube ft. 18.92 20.79 18.4 19.22 Per cent. 21.2 I 9 .2 19.6 22.8 No. 223.8 (T. L. Miller.) Results of Trials of Crossley Engine -with London. Coal Q-as.* Pressures and Revolutions per minute, Pressures and Water in Ibs., Gas in cube feet, and Efficiency and Heat per cent. Power. Full. Half. Power. Full. Half. Revolutions per minute. . . Explosions " u Mean initial pressure Mean effective " Brake IP 1 60. i 78.4 196.9 67.9 14.74 158.8 41.1 196.2 73-4 Water for cooling per hour Mean pressure during ^ working stroke equiv- ! alent to work in pump- [ ing stroke . . J 7i3 2.19 480 Indicated IP 17.12 9 - 73 Corresponding IIP .58 Mechanical efficiency 86.1 76.2 205 8 Heat converted to work.. . Heat rejected in jacket) 22.1 20.9 Gas per IIP per hour .... 355-3 20 76 water j 43-2 41.1 Gas per brake IP per hour. 24.1 27.77 Heat rejected in exhaust. . 35-5 38 (D. K. Clark.) Pressures Produced l>y the Explosion of Q-aseous Mixtures in a Closed "Vessel. Mixture of Air and Coal- Gas, Temperature 64 Mixture. Pressure! per Mixture. *y I! MM. Pressure per Gas. Air. Sq.Vh. Gas. Air. Sq. Inch. || Gas. Air. Sq. Inch. Volume. Volumes. Lbs. Volume. Volumes. Lbs. 1 Volume. Volumes. Lbs. I 5 96 I 9 69 I 13 52 I 7 89 I II 63 II (D. Clerk.) * See Report on Trials of Motors for Electric Lighting for Society of Art*, 1889. f Maximum above the atmosphere. DIMENSIONS OF BOLTS, NUTS. TENACITY OF NAILS. 993 Standard. Dimensions of Iron and Copper Bolts and !N"\rts, TJ. S. Navy. Square and Hexagonal Heads and Nuts. Finished. From .25 Inch to 6 Inches in Diameter. Dl AM Bolt. HTER. Effec- tive. Effective Area. DlAM Head a Hexagonal. ETKB. nd Nut. Square. WIDTH. Head & Nut. Hexagonal Square. DK Head. Hex. and Square. PTH. Nut. Hex. and Square. Threads. Ins. Ins Sq. Ins. Ins. Ins. Ins. Ins. Ins. No. 25 .185 .026 9-16 23-32 5 25 25 o 3125 24 045 11-16 27-32 19-32 19-64 3125 8 375 .294 .067 25-32 31-32 11-16 n-32 375 6 4375 345 093 29-32 3-32 25-32 25-64 4375 4 5 4 .125 i 25 -875 4375 5 3 5625 454 .162 1-25 -625 31-32 31-64 9-16 2 .625 507 .202 i. 7-32 5 i. 1-16 17-32 .625 I 75 .62 .302 i. 716 75 1.25 .1625 75 O 875 731 .419 1.21-32 2. 1-32 i. 7-16 23-32 -875 i x -837 1-875 2. s-,6 i. 5-8 13-16 i 1.125 .94 694 2. 3-32 2. 9-16 1.13-16 29-32 1.125 7 1-25 1.065 .891 2. 5-l6 2.27-32 2 i 1-25 7 '375 1.16 1-057 2.17-32 3- 3-32 2. 3-l6 i. 3-32 i-375 6 1.284 1.294 2-75 3-11-32 2-375 i. 3-16 6 1.625 1.389 I-5I5 2.31-32 3-625 2. 916 i. 9-32 1.625 5-5 1.75 1.491 1.746 3- 3-i6 3.875 2-75 J.375 1.75 5 '875 1.616 2.051 3-I3-32 4- 5-32 2.I5-I6 1.15-32 1-875 5 2 1.712 2.302 3-19-32 4-I3-32 3-125 i. 9-16 2 4-5 2.25 1.962 3-023 4- i-S 2 4.15-16 3-5 i-75 2.25 4-5 2-5 2.176 3-7I9 4-I5-32 5-I5-32 3-875 1.15-16 2-5 4 2-75 2.426 4.622 4-29-32 6 4-25 2.125 2-75 4 3 2.629 5-428 5.11-32 6.17-32 4-625 2. 5-l6 3 3-5 3-25 2.879 6. 5 I 5-25-32 7. 1-16 5 2-5 3-25 3-5 3-5 3- l 7- 547 6- 7-32 7-19-32 5-375 2. II-l6 3-5 3-25 3-75 3-3I7 8.641 6.625 8125 5-75 2.875 3-75 3 4 9-99 7. 1-16 8.21-32 6.125 3- 1-16 4 3 4-25 3-798 11.329 7-5 9. 3-16 6-5 3-25 4-25 2.875 4-5 4.028 12.743 7.15-16 9- 2 5-32 6.875 3- 7->6 4-5 2-75 4-75 4-256 14.226 8-375 0.25 7 25 3-625 4-75 2.625 5 4.48 15-763 8.13-16 o. 25-32 7.625 3.13-16 5 2-5 525 4-73 I7-572 9-25 i. 5-16 8 4 5-25 2-5 55 4953 19.267 9.11-16 1.27-32 8-375 4- 3-i6 5-5 2-375 5-75 6 5.203 21.262 ! 10. 3-32 23.098 10.17-32 2-375 2.29-32 8-75 9-125 4-375 4. 9-16 I' 75 2-375 2.25 For Rough Bolts and Nuts, add .066 to above dimensions, and for other notes see PP- 156-159- Relative Tenacity of Wrougnt-Iron Cut and Wire Nails. Per Cent, of Cut and Wire Nails. Dimen- sions. SPRUCE. Designa- tion. Per Cent. Dimen- sions. Designa- tion. PINK. Per Cent. Direction of Penetration. Ins. 1.125X6 Ordinary. 47-5 Ins. 1.25X4 Box. 135-2 {Taper perpendic ular to grain. 1.125X4 Finish. 72.2 1.25X4 Box. 100.2 (Taper parallel to I grain. 1.125 X 4 Box. 509 i 25X4 Box. 66.4 In end. 2X4 Floor. 80 1.25X4 Box. 99-9 lln the three ( ways. In Spruce 40 tests of cut nails averaged 60 per cent. In Spruce and Pine combined the average was 72.7 per cent. t:ao n ^~^ i U See p. 970. . ( Wm. H. Burr, C E.) 994 COMPRESSION OF AIR. Compressed, and Compression, of Atm.osph.eric Computations of Flow, Operation, Kfiect, .Po\ver, etc. For fuller information, see "Compressed Air," by Fredk. C. Weber, M.E., before the Engineering Society of Columbia College, April 22d, 1896 ; Wm. L. Saunders, N. Y. ; also by Frank Richards, Mem. A.S.M.E. ; a lecture by R. A. Parke ; I). K. Clark's Pocket- Book: a treatise by W. C. Unwin, in Vol. CV. of Proceedings of In- stitution of C. E. of Great Britain, and a treatise of The Norwalk Iron Works Co., etc. Pressure and. Temperature. Under constant pressure the volume of air varies directly as the Absolute temperature. For constant volume the pressure is in direct proportion to an increase in temperature. Compression. Heat and Mechanical energy are mutually convertible ; when, therefore, the piston of an air-compressing engine is in operation, heat is evolved (theoretically) in exact proportion to the work performed, in the ratio of one British thermal unit (B T U) for every 772* foot pounds expended. When atmospheric air is compressed, the degree of its compression may be indicated by a pressure gauge. The heat evolved by the compression of air generates by expanding it an increased resistance, and involves increased power to compress it. This loss of power consequent upon the expansion of the air by the heat of com- pression is so great that it is necessarily essayed to reduce the heat, and a cooling medium is resorted to, to abstract it in the operation of compression. The rate of increase of temperature of air during compression is not uni- form, as the temperature rises faster during the primitive stages of compres- sion than the later. Thus, in compressing from i to 2 atmospheres, the increase of temperature will be greater than in compressing from 2 to 3 atmospheres, and in like ratios. The rate of increase also varies with the initial temperature, as the higher it is the greater will be the rate of increase at any point of the compression. When air at atmospheric pressure and o is compressed to 15 Ibs. gauge, the final temperature will be 100, or an increase of 100. If at 60, it will be 175, an increase of 115, and at 90 it will be 210, an increase of 120. The rise in temperature due to the compression of atmospheric air at 32, when it is reduced to one-fourth its volume, is given by Kimball at 344. The great reduction of the temperature of compressed air when it is dis- charged from the compressing cylinder, against a resistance, as the cylinder piston of an engine, precludes the economical operation of using it expan- sively alike to steam or any similar vapor. The available energy of com- pressed air is that which it exerts against a resisting medium, in its increase of volume by expansion. When air is compressed, if it neither gains or loses temperature by com- munication with any other body, the heat generated by compression, re- maining and adding to it, the operation is termed Adiabatic compression. When pressure is removed from compressed air and it expands without re- ceiving heat externally, the air is termed to have expanded A diabatically. If during the compression of air, it is maintained at a uniform tempera- ture by the reduction of it, coeval with its generation, the compression is termed Isothermal. Hence when the air remains at a uniform tempera- ture throughout the operation of compression or expansion, it is designated as Isothermal. * Joule's. Later experiments put it at 778. COMPRESSION OP AIR. 995 The specific heat of atmospheric air at constant pressure is .2 377, hence the unit of heat that would" raise the temperature of i Ib. of water i would raise the temperature of i Ib. of air (i -f- .2377) = 4.207. 13.141 cube feet of air at 62 (table, p. 521) weigh i Ib., and air at 60 compressed to half its volume evolves 116 heat, and the specific heat of air under constant pressure is .2377, which x 116 = 27.573 heat units, pro- duced by the compression of i Ib. or 13.141 cube feet of free air into one- half its volume : Hence, 27.573 x 778 = 21 452 foot Ibs., and as heat and mechanical energy are held to be convertible terms, = .65 H* pro- Tiooo duced or lost by the compression of i Ib. of air. 33000 "Volume, Mean Pressure, and Temperature of Com- pressed. Air. From i to 200 Lbs. and from 60 to 672. Air assumed at 14.7 Ibs. and Temperature 60. Pressure Lba. VOLUME Constant Temperature Isothermal. OF AlB. Not Cooled. Adiabatic. MEAN Constant Temperature Isothermal. PRKSHURB Not Cooled. Adiabatic. AIR PBB STR During Co onl Constant Temperature OKB mpreasion Y- Not Cooled. final Tem- perature,* Air not Cooled. I I o 60 i ?363 95 .96 975 43 44 71 2 . 8803 .91 1.87 1.91 95 .96 80.4 3 8305 .876 2.72 2.8 1.41 88.9 7861 .84 3-53 3-67 1.84 1.86 98 5 7462 .81 4-3 4-5 2.22 2.26 106 10 5952 .69 7.62 8.27 4.14 4.26 '45 15 495 .606 10.33 11.51 5-77 5.99 178 20 4237 543 12.62 14.4 7-2 7.58 207 25 3703 494 '4-59 17.01 8-49 9-5 234 3 3289 .464 16.34 19.4 9.66 10.39 255 35 2957 42 17.92 21.6 10.72 ".59 281 4 45 2462 393 37 19.32 20.52 23.66 25-59 12.62 12.8 13-95 302 321 So 2272 35 21 79 27.39 13.48 15-05 339 i 65 1844 33' .301 22.77 23.84 24.77 29.11 30-75 31.69 15-05 I5-7 6 15.98 16.89 17.88 357 i 70 1735 .288 26 33-73 16.43 18.74 405 i 5 1639 ,276 26.65 35.23 17.09 19.54 420 So 1552 .267 27-33 36-6 17.7 20.5 432 85 1474 257 28.05 37-94 18.3 21.22 447 90 1404 .248 28.78 39.18 18^87 22 459 95 134 24 29-53 40.4 19.4 22.77 472 100 105 1281 1228 .232 .225 30.07 30.81 41.6 42.78 19.92 20.43 23 43 24.17 I no 1178 .219 3i.39 43-9 1 20.9 24.85 57 "5 "33 .213 31.98 44.98 21.39 25-54 518 1 20 1091 .207 32.54 46.04 21.84 26.2 529 125 1052 .202 33-07 47.06 22.26 26.81 540 130 1015 .197 33-57 48.1 22.69 27.42 55 135 0981 34-05 49.1 23.08 28 -5 560 140 095 .'188 34-57 50.02 23- 4 1 28.66 570 '45 .0921 .184 35-09 51 23-97 29.26 580 I 5 .0892 .18 35-48 5189 24. 28 29.82 589 1 60 .0841 .172 36.29 24.97 30.91 607 170 .0796 .166 37-2 55-39 25-7 1 32.03 624 180 0755 .16 37- 9 6 57-01 26.36 33-04 640 190 .0718 154 38.68 58.57 27.02 34-o6 657 200 .0685 .149 39-42 60.14 27.71 35-02 672 * Produced by compression. (Frank Richardt.* 99 COMPRESSION OF AIR. For determination of absolute pressure add 14.7 Ibs. to gauge pressure. Column 2 gives the volume of air (initial i), assuming that its temperature has not risen during the compression, or that if the air has not been wholly cooled during the compression, it has been cooled to the initial temperature after the compression. Or volume of one cube foot of free air at given pressure. Absolute isothermal compression is not attainable, as it is impracticable in the compression of air, simultaneously to abstract all the heat evolved in the compres- sion. This column, however, does give the volume of air that will be realized, if it is transmitted to such a distance from the compressor or in any manner that the heat is abstracted before it is used. Air radiates its heat very rapidly, and this column may be taken to represent the volume of available air after compression. Column 3 gives the volume of air at completion of the compression, assuming that the air has neither lost nor gained during the compression, and that all the heat developed by the compression remains in the air. The condition represented by this column adiabatic compression is alike to that of isothermal compression, never actually attained. In any compression, the air will lose some of its heat, and consequently the air is not as heated at any period of the compression to the ex- tent that theory assigns to it. Physically, the theory is correct, but practically it fails. The slower a compressor is operated, the more readily will the air radiate some of its heat, and as a result, the less will be its volume and less the power rc~ quired for compression. Column 4 gives the mean effective resistance to the piston of the air-compressor cylinder in the stroke of compression, assuming that the air throughout the stroke remains uniformly at its initial temperature isothermal compression but as the air does not remain at constant temperature during compression, the results in this column are to be essayed to be attained in economical compression. Column 5 gives the mean effective resistance to be overcome by the piston, as- suming there is not any cooling of the air during compression adiabatic compres- sion inasmuch as there is always some cooling of the air during compression, the actual mean effective result will be somewhat less than that given in the column. For the computation of power required for the operation of the air-compressor cylinder, the results given may be taken, with a per cent, added for friction* o to 10 per cent. and the result will very nearly give the power required to operate the compression. Column 6 gives the mean effective resistance for the compression of the stroke of the piston in compressing air isothermally from that of 14.7 Ibs. to any given pressure. ILLUSTRATION. Assume an air-compressing cylinder 20 ins. in diameter by 2 feet stroke of piston, making 75 revolutions per minute, with an adiabatic pressure of 75 Ibs. 2o 2 X-7854X 35-23 (columns) X 75 X 2 X 2 -=- 33 ooo = 100. 6 IP. ILLUSTRATION. Assume an adiabatic pressure of 50 Ibs. by gauge, the volume ot air compressed and delivered will be (column 3) ,35 for each stroke of the piston in a cylinder full of free air; while for the compressing part of the stroke i 35 .65, the mean resistance will be 15.05 Ibs. (column 7). Thus, 15.05 x .65 + 50 X -35 = 27.28; corresponding very nearly with 27.39 (column 5) for the whole stroke. Comparing isothermal compression with adiabatic, to 50 Ibs. as above, in column 6 is 13.48 which x i 2272 = .7 728 (column 2) -j- 50 X -2 272 = 21.78 or 21.79, as given in column 4. Columns 6 and 7 are useful in the computation of power in the first operation ot compression, as the function of the first cylinder is that of compression only. The results given in columns 7 and 8 are elements of computation for the IP of the compressing engine, and a like computation applied to the result in the air engine will give the power attained in the compression of the air. Column 7 gives also the mean effective resistance for the compression of the stroke in compressing air isothermally from a pressure of 14.7 Ibs. to any given pressure, and column 8 gives the theoretic temperature of the air after compression adiabatic to the given pressure. * In some operations the air will become so cooled that it will, by the resulting decrease of require- ment of power of operation, fully compensate for the friction of the compressing machine. COMPRESSION OF AIB. 997 To Compute IH? "with, the Elements of the Preced- ing Table. Assume a cylinder 40 ins. in diameter, with 4 feet stroke of piston, in which air is compressed by 75 revolutions at 75 IDS. pressure per sq. inch. Area of cylinder, less .5 that of piston rod, 1250. sq. ins. and mean pressure per stroke of piston as per table (column 5) 35.23. Then 1250 X 35.23 X 75 X 4 X 2-7-33000 = 800.6 IP. Efficiency of Engine of Operation. The efficiency of an engine is the per cent, of power developed by it, that it bears to that required to compress the air, the loss by leaks, friction in pipes, of parts and heated air from the en- gine-room (varying with the weather and the season), including that of the driving engine. Compressed air can be transmitted with great facility, provided the trans- verse area of the conduit is proportioned to the volume and pressure of the flow, and the suitability of the interior surface of it for its transmission. Under such conditions, the volume of the external flow or discharge of air may be computed by the volume of the cylinder of the air engine and the number of strokes of its piston, less the loss and friction of the flow, which may be estimated at 5 per cent. Theoretical Efficiency of the compression and delivery pf air T-f-<=zE. T and t representing the absolute temperatures of the air at its entrance into the operating cylinder and its flow from the compressor. In order, then, to increase the efficiency, the heat evolved during compression of the air must be abstracted, or by operating at a lower pressure. Practical Efficiency is the difference between the power developed by the dis- charged air and that expended in its compression, and in operation at a low speed of compressing engine and under a pressure of but from 60 to 75 Ibs. an efficiency of .9 has been attained. Spray injection of cold water into a cylinder Is more effective than a water jacket, and by compressing the air in two or more cylinders, and cooling it between them, the work lost or expended in the heating of the air by its compression is much reduced. Hence compound compression with inter-coolers has been intro- duced with advantage.* If air is flowing with uniformity, a like weight of it flows through each trans- verse section per section. Hence, G a V = W; G representing weight of a cube foot of air in Ibs. ; a, area of transverse section in sq.feet; V, velocity in feet per second; and W, weight of air in Ibs. per second. Friction of Air in Long Pipes. _V ! L /.coco d . C * _VM. = d . MM , d > = ^ y iooood*C V ^ 10 ooo h representing volume of air delivered in cube feet per minute ; L = length of pipe in feet ; d== diameter of pipe in inches ; and C = coefficient as per following table : " -SSli-S" .5* 2-5" .6513.5" -78715" .934! 8" I-I25JI2'' x. 2 6|2o" 1.4 1.25 .42|2' .5653" .7314' -84l6' i. |io 1.2 |i6 i.34l24 MS For fifth power of d, see pp. 303, 304. ILLUSTRATION. It is required to transmit 1200 cube feet of free air per minute, at 75 Ibs. gauge pressure, through a pipe 4 ins. in diameter and 1000 feet in length; what is the additional pressure required to overcome the friction in the pipe? 1200 X- 1639 (col. 2, Table, p. 995) = 196.68 cube feet. '96.682x1000 = lu looooX 1024 (45) x.84 Mr. Unwin gives the following : .0027 i -j- 3 -i- 10 d = C. d representing diameter of pipe in feet, and C a constant, due to diameter of the pipe. For pipes less than one foot in diameter, .5. C = . 00435, .656 = .00393, and for .98 feet = .oo 351. * x88z. Norwalk Iron Works Co. claim to have first constructed Compound Compressors. COMPRESSION OF AIR. 'J?o Compute Loss of Head in Flow of A.ir in Long JPipes. V 2 4. I C X ^- = h. V representing velocity of air in feet per second, C as above, I length, d diameter of pipe, and h head, all in feet. Assume a pipe having a diameter of .5 foot and a length of 1000 and the velocity of the air 10 feet per second. C = . 0027 ( i -f" ) =.00432. Then, x .00432 x -- ' = 53-71 feet. Assume the transmission of 1200 cube feet of free atmospheric air per minute, through a pipe 4 inches in diameter and 1000 feet in length, under a gauge press- ure of 73.5 Ibs. persq. inch; what will be the additional pressure or head required? 1200 cube feet of free air -r- 73 ' ' 4 ' 7 = 1200-^-6 = 200 feet at 73.5 Ibs. 4 6 = 1024 and C for 4 ins. = .84. 2oo 2 Xiooo /looooX 1024 X .84 X 4-65 Then = 4.65 Ibs. head, and / - - = 10 ooo X 1024 X .84 V 1000 200 cube feet. If, however, this volume of free air was under a pressure, the volume of free during its transmission would be due to the pressure. Thus, if it was 58.8 Ibs. gauge, the volume would be -^- 7 - =i 5, and 200 X 5 = 1000 cube feet. Loss of Pressure per Mile of Pipe. II V 2 \ P 1 /[i , ) = P- P 1 = conventional pressures as given below ; V repre- V \ 14 072 d/ senting initial velocity in feet per second, d diameter of pipe in feet, and P terminal pressure in Ibs. per square inch. Assuming initial velocities of 25, 50, and 100 feet per second and initial pressures of 50, loo, and 200 Ibs. absolute. e air 58.8 Ibs. per Diameter of Pipe, One Foot. Diameter of Pipe, Two Feet. Initial Velocity. Terminal Pressure = P. Initial Pressure lost in Initial Velocity. Terminal Pressure = P. Initial Pressure lost in ' P^So. Pi = 100. P J =200. One Mile. P I = 5o. P* = 100. P I = 200. One Mile. Feet. Lbs. Lbs. Lbs. Per cent. Feet. Lba. Lbs. Lbs. Per cent. 25 48.8 97-7 195-4 2.4 25 49-4 98.9 197.8 1.2 50 100 45-3 26.9 90.0 53-8 181.2 107.6 9.4 46.2 50 100 47-7 40.1 95-4 80.3 190.8 160.6 4-6 19.8 ILLUSTRATION. Assume initial pressure 50 Ibs. per sq. inch, velocity 100 feet per second, diameter of pipe or conduit one foot. 50 / f i J = 50 x V - 2 9 = 26.92 Ibs. terminal pressure. Hence, if 50 26.92 = 23.08, 100 = 46.2 per cent, loss in one mile. The per cent, loss in one mile is the same, whatever the initial pressure, the velocity increasing and the density decreasing with the length of the pipe. Results observed by Prof. A. B. W. Kennedy, M.Inst.C.E., in the operation of a plant of six Compound cylinder engines, each operating two compressors, having a combined capacity of 2000 IP. For a distance 0/3.1 miles, through a pipe n.8 inches in diameter. At the termination of the flow of air as it was about to enter the motor, it was heated from a coke-burning stove. Compression of the air 88.2 (73 .5-1-14.7) Ibs. per sq. in. at a temperature of 150 reduced to 66.15 I DS -> and delivery of the com- pression cylinders 348 cube feet of air at atmospheric pressure and 70 tempera- ture per IIP per hour. The average loss was 3 per cent, velocity of air 1550 feet per minute, with an IIP of 1250. COMPRESSION OF AIE. 999 Summary of Results of two experiments, each with cold and heated air, in the Transmission of Compressed air at Paris, 1889, for a distance of 4 miles. Motor 10 IP and pressure of air reduced to 66 Ibs. One IIP gave .845 IIP in compression, or 348 cube feet of air per hour from atmospheric pressure of 88.2 Ibs. A summary of other results showed that a small motor at a distance of 4 miles from the compressor indicated i IP for 2 IP at the motor, or 2.5 IP when the air was not heated before entering the motor. Heating the air caused a saving of 225 cube feet of it per IIP, at a cost of 4 cents per IIP. The exhausted air from a motor, when that in the pipe is even but slightly heated, will be so much reduced in temperature as to be available for cooling and even freezing application, so great is the effect of instantaneous expansion of the air when exhausted that ice is formed in the air-ports of the cylinder, and hence the operation of a plant at high pressures or above 90 Ibs. is held to be objectionable. By operating at full pressure, the high velocity of the flow mechanically restricts the deposit of ice crystals, but inasmuch as the useful effect decreases with an in- crease of pressure, it is held by Robert Zahner and others that 60 Ibs. is the limit unless the operating air is reheated. When air is operated expansively at half-stroke, the temperature falls 160, and at one-fourth stroke 284. Compressed air is the only power of general application, as it can be applied, extended, and distributed without restriction to distance, course, elevation, and depression, and under ground or water, and under some of these conditions the only power at all practicable of operation. Alike to water it can be stored, which condition is unattainable with steam. Heating Compressed Air. When compressed air has been transmitted to the point where it is to be employed, an increase of power is attainable by the addition of heat to it, before it is applied. Absolute temperature is 461.2. Hence when the air is 60, the absolute temperature is 461.2 + 60 = 521.2, and when it is 30, it is 431.2 abso- lute. Loss of Efficiency. Initial Temperature of Air 62. Pressure. Final Tem- perature. Efficiency. Reduced. I Loss of. Pressure. Final Tem- perature. Effic Reduced. cncy. Loss of. Lba. 29.4 44.1 58.8 Degrees. 178 2 5 8 321 Per cent. 82 73 6 7 Per cent. 18 27 33 Lbs. 73-5 147 Degrees. 373 559 Per cent. 6 3 51 Per cent. 37 49 Assuming efficiency of Compression and also that of the Engine at 80 per cent, the resultant efficiency of the combination at 147 Ibs. pressure = X 51 = 32.6 per cent. At 44. i Ibs. the efficiency : 80X80 X 82 = 52. 5 per cent. (D.K.Clark.) Air expands at constant pressure from 32 to 212 .002036 per degree of tem- perature. Efficiency" of Cooling. Cooling of compressed air effects a saving of power required for its compression, and aids in the lubrication of the piston. It is most effective at low pressures. Thus at 15 Ibs. pressure the temperature consequent upon compression is raised from 60 to 177 and from 75 to 90, but 39. When air is heated by compression and water is introduced it becomes saturated, and when after performing its work it is exhausted into the open air, it expands so rapidly that its temperature is frequently reduced below zero, and, as a result, the moisture in the air gravitates as ice in the exhaust passage of the engine, and its capacity is choked and even closed. Hence it is imperative that the air of com- pression should be maintained as dry as practicable. IOOO COMPRESSION OF AIR. Air Receivers. The operation of a Receiver, if of sufficient volume, is to reduce the effect of the pulsations consequent upon the stroke of the compressor, for without it the press- ure of the air at its delivery from the compressor to a pipe would be momentarily in excess of the average pressure of operation. This effect may be reduced by in- creasing the length of the pipe, also by the attachment of a second Receiver at the termination of a long pipe. As the presence of a Receiver checks the flow of the compressed air, some of the water which is in the air, which otherwise would be borne with the current, is precipitated. Efficiency of Compressed Air Engines. At the ordinary pressure of 60 Ibs. per sq. inch, the decrease in resistance effected by the cooling of the air is held to be equal to the friction of the compressor. This effect is greater with high than low temperatures of the air, in consequence of the higher temperature at the higher pressures of the air. Adiabatic Expansion. The more air is in compression and the friction of its passage in the pipe in- creased, the efficiency of compression is increased. The following table gives the Lowest Pressures which should be operated in the Compressor, with varying amounts of friction in the pipe: & Lbs. 20.5 29.4 70.9 64-5 it | II if 4 4e 3 n if c .2 s| o ^ i >* || Lbs. 38-2 47 P'r ct 60.6 57-9 Lbs. 14.7 17.6 Lbs. 52.8 61.7 P'r ct. 55-7 539 Lbs. 20.5 235 Lbs. 7-5 76.4 P'rct. 52-5 51-3 Lbs. 26.4 29.4 Lbs. 82.3 88.2 P'r ct. 50.2 49 Operation and Mean. Results of a Hardie Motor at Rome, 1ST. Y., 1895. Elements. One Run. Mean of Screw. Elements. One Run. Mean of Screw. Pressure persq. inch. Distance run ... . 1.41 -3 eg 1 01 Ibs. 2. 6 1 miles Difference in tem- perature in heater Temperature of air entering heater 6"? 2 68.2 at start and fin- ish . 40 29 8 Temperature of air IIP leaving heater Temperature of air at exhaust 240.3 130 7 219.6 123.5 Water supplied Air per IIP per min- ute 29-37 6 21.46 Ibs. 6 5 cube ft Heat absorbed in heater... I7S.I 117.1 Power from heater.. 43- 2 ^o.ipercwt. The power obtained from the Reheater was about 45 per cent. (Frederick C. Weber.) Io\ver Required to Compress Air at tlie Uniform Temperature of 62. Pressure per Sq. Inch. HP per Cube foot of Com- pressed Air Volume of Compres'd Air per inin.perH? Pressure per Sq. Inch. H?per Cube foot of Com- pressed Air Volume of Compres'd Air per min. per H? Pressure per Sq. Inch. H?per Cube foot of Com- pressed Air Volume of Compres'd Air per Lbs. No. Cube feet. Lbs. No. Cube feet. Lbs. No. Cube feet. 30 .089 11.25 120 1.07 938 210 2-37 .422 45 .211 4-73 135 1.27 .788 225 2.61 384 60 .356 2.88 ISO 1.48 .667 240 2.84 352 75 .516 1.94 165 1.69 591 255 3-9 324 90 .69 i-45 1 80 X. 9 I 523 270 3-34 3 105 .874 1.14 195 2.14 .468 300 3-84 .26 At the Mont Cenis tunnel, 64 cube feet of compressed air per minute through a cast-iron pipe 7.625 ins. in diameter, 5325 feet in length, and under a pressure of 838 Ibs. : the loss of the head including leaks and friction was but 3.5 per cent., and in a length of pipe of 20 ooo feet the loss was but 5 per cent, of the initial pressure. (D. K. Clark.) COMPRESSION OF AIR. 1001 Gauge Press- ure at en- trance to the Pipe. H 5' 2- VI OO OOvp VQ *. ON N 00 &- ra OOVI ON*. N vi ON 00 M 58 *. M ON H *. 8 M VI M ON 8 OJ M VO VI OJ 8 OJ W 10 M O ON M ON TO OJ ONOJvp^ 8 ***h 1 IFf* 5 w r Gauge Press- ure at en- trance to the Pipe, ^ s 1 Diameter Pipe in Ir *. ON TO M JP* FR s s s o Oovi vi vi vi TO ONOI M Q vl ui vO N Oi OJ OJ M W W M O VO TOvi 8 TOV| VI ONO-I HI *. p ON 00 On (0 ON ONVI 1 M O vO TO ON - 01 VI VI VO ON N M 00 OJ M 00 ON*. M M 00 00 w to M ON^ O^ N VIVIMVOOJ K3 vO On M ON O^'ONbivO *^ vO OJ vo ^ \O 1002 COMPRESSION OF AIR. Volume of* Free Air in Cvttoe Feet RecfuiredL in. Motor per ItP per ]Miiixite.* Without Reheating. Gauge Pressure in Pounds at 60. Point of Cut-off. I 75 .66 5 -33 25 * 30 40 50 60 70 80 90 IOO no 125 150 17-05 I3 'I 12.6 10.85 9-5 9.1 31.2 25-6 24.8 25-8 37 23-3 18.7 17-85 16.4 20.6 21.3 17.1 16.2 14-5 15-2 15.6 2O. 2 16.1 15-2 12.9 13-4 19.4 15-47 12.8 11.85 13-3 18.8 14.2 12.3 11.26 11.4 18.42 14.6 13-75 "93 10.8 10.72 18.1 14-35 13-47 11.7 10.5 10.31 17.8 14-15 13-28 11.48 IO.2I 10 17.62 13.98 13.08 "3 IO.O2 9-75 17.4 13-78 12.9 n. i 9.78 9.42 To these results is to be added the per cent, of clearance as determined in each case. If the air is reheated, the volume in the table will be decreased, depending upon the temperature of the air at admission, and it is proportional to T-r-T", T repre- senting absolute temperature at 60, and T' 460 -f- temperature of air at admission to motor. Hence, if the air is reheated to 300, the volume in the table is to be multiplied by 460+60 _ 520^_ 460 + 300 760 To Ascertain the Economical point of Cut-off for the Gauge Pressures in the Table. An inspection of it will show. Thus, at 60 Ibs. the least volume of free air per IIP is at .33 cut-off, and at 80 Ibs. at ,25. (Frederick C. Weber.) Loss of Pressure toy Friction of* Compressed. Air in Pipes, In Pounds per Square Inch for 1000 Feet of Pipe. Volume of Free Air, Compressed to a Gauge Pressure of 60 Ibs. per Square Inch, Delivered per Minute. Diana, of Pipe. So 75 IOO 125 ISO 200 250 300 400 600 Ins. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. I 10.4 1.25 2.63 5-9 1.5 1.22 2-75 4-59 7-65 ii 3 35 79 1.41 2.2 3-17 5-64 8.78 2-5 .14 32 57 9 1.29 2-3 3.58 5^8 9.2 3 .11 .2 31 44 .78 1.2 3 1.77 3-H 7-05 3-5 15 .21 38 59 85 3-4 4 .2 3i 45 '.Bo 1.81 5 .1 i5 .26 59 Cube Feet. Diam. f Pipe. 800 1000 I2OO 1500 1800 2000 2500 3000 4000 5000 Ins. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. 3-5 6.03 4 3.22 5.02 7-23 "3 5 1.04 1.63 2-35 3-66 5-28 6-5 10.2 6 8 .41 .1 .64 .16 93 23 1.46 37 2.09 53 2.59 65 4 .06 1.02 1.47 10.3 2.61 4 ~o8 10 .13 17 .21 33 47 .84 1 -3 12 13 .19 34 53 * Copyrighted. (Rand Drill Co., F. A. Halsey.} COMPRESSION OF AIR. 1003 Dimensions and Elements of Air Compression, Operated by Steam. DlAMK Air. TEH OF CYL Compres- sions. INDKR. Steam. Stroke of Piston. Revolu- tions per Minute. Volume dis- charged. Steam. DlAMBTEB Exhaust. OF PlPl Air. 8. Water. IP Ins. Ins. Ins Ins. No. Cube ft. Ins. Ins. Ins. Ins. No. 8 5 8 10 200 116 2 2-5 2 5 18 IO 6.75 IO 12 180 195 2-5 3 2-5 75 30 J 4 9-5 14 16 150 427 3 4 4 57 16 9-5 16 16 I 5 558 3 4 4 82 20 i3-5 20 24 no 960 5 6 5 25 145 22 22 24 no 1160 5 6 5 .25 175 26* '7-5 24 30 90 1659 6 8 6 25 215 28 7-5 28 30 9 1924 8 10 6 25 300 32 21.5 30 36 80 2686 8 10 8 5 350 Horse-P ower Required to Compress One Cn"be IToOt of Free A_ir per Minute to a, GHven Pressure, and. the Power Required to JJeliver One Cu/be Foot of A.ir at a Given Pressure. Compressing to Given Pressure. Delivering to Given Pressure. Compressing to Given Pressure. Delivering to Given Pressure. Gauge Pressure. Constant Temper- ature. Without Cooling. Constant Temper- ature. Without Cooling. Gauge Pressure. Constant Temper- ature. Without Cooling. Constant Temper- ature. Without Cooling. Lbs. H? IP H? IP Lbs. H? H? H? H? 5 .0188 .0196 .0251 .0263 55 .0994 .127 .4711 .6023 IO .0332 .0361 0559 .064 60 .104 .1342 .5285 .6818 15 045 .0502 .091 .1014 65 .1081 .1403 .5861 .7608 20 -055 .0628 .1299 .1483 70 .1124 .1472 .6481 .8483 25 .0637 .0742 .1719 .2004 75 .1163 '537 .7095 938 30 .0713 .0846 .2168 2573 80 .1193 1597 .7684 .0291 35 .0782 .0942 .2644 .3189 85 .1224 1655 .8304 1231 4 -0843 , 1032 -3137 .3842 90 .1256 .171 .8944 .2176 45 .0895 .1117 3 6 37 -4535 95 .1289 .1763 .9616 .3148 So .0951 1195 .4185 .526 100 .1312 .1815 1.0243 .4171 To these must be added a per cent, due to the estimated friction of the com- pressor. Mean and Terminal Pressures of Compressed Air at Several Points of .Expansion and at Given Grange Pressures. When the Pressure is Less than Atmosphere it is Given Absolute. Cut Pressure 50. Pressure 60. Pressure 70. Pressure 80. Pressure 90. Pressure 100. off at Mean. Termi- nal. Mean. Termi- nal. Mean. Termi- nal. Mean. Termi- nal. Mean. Termi- nal. Mean. Termi- nal. Point. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. .125 4-5i 3-47 7-51 4.01 10.51 4-54 I3-5I 5-o8 16.52 5.61 I9-5I 6.15 .2 13.84 6.74 17.7 7-77 22.06 8.81 26.6 9-85 30.78 10.88 35 H 11.92 25 18.45 9-23 23-6 10.65 28.74 12.07 33-89 13-49 39- 4 14.91 44.19 i-33 33 25.84 13-83 32.13 .96 38.41 3-09 44.69 5-22 50.98 7-35 57.26 9.48 375 29.07 i-34 35-85 3-85 42.63 6.36 49.41 7 .88 56.2 11.39 62.98 13.89 .5 37- J 2 7-49 45-14 13.26 53- 16 17 61.18 20. 8 1 69.19 24.56 77-21 28.33 .625 42.99 i8.53 51.92 23.69 60.84 28.85 69.76 34.01 78.69 39.16 87.61 44-32 75 46.98 28.34 56-52 35-01 66.05 41.68 75-59 48.35 85.12 55-02 94.66 61.69 (Frank Richards. ) For Volume of Air Transmitted in Cube Feet per Minute in Pipes of Diameters from i to 24 inches, see Frank Richards, p. 109. * At elevations of 2000, 6000, and 10000 feet above the sea-level the capacity and IP of this enj vrould be reduced respectively to 1560, 1373, and 1200 feet capacity and 207, 195, and 182 HP. 1004 COMPRESSION OF AIR. Heat Prodnoed toy Compression of Dry .Air. Without Cooling. Pressure above Atmos- phere. Volume. Tempera- ture of the Air Pressure above Atmos- phere. Volume. Tempera- ture of the Air. Pressure above Atmos- phere. Volume. Tempera- ture of the Air. Lbs. Cube feet. O Lbs. Cube feet. Lbs. Cube feet. o i. 60 22 .5221 218.3 88.2 2516 454-5 1.47 .9346 74-6 29.4 .4588 255.1 102.9 .2288 490.6 3-67 8536 94.8 36.7 .4113 287.8 117.0 .2105 523.7 7-35 7501 124.9 44-1 3741 3I7.4 132.3 1953 554 n. ii .6724 151.6 58.8 '3194 369-4 205.8 .1465 681 14.7 .6117 175-8 73-5 .2806 4I4.5 279.3 "95 781 The presence of moisture will increase these results as it increases both the specific heat and the heat-conductive capacity of the air. ( W. L. Saunders.) Kffioienoy of an Engine. With perfect expansion, without the air receiving any increase of temperature, the efficiency at pressures above the at- mosphere and friction in pipes are estimated as follows : Per Cent. Friction estimated. 14.7 29.4 44.1 58.8 73-5 88.3 Lbs. 2.0 5-8 14.7 Per cent. 70.44 57- *4 Per cent. 68. 81 64-49 48.53 Per cent. 64.87 62.71 55-13 Per cent. 61.48 60. 12 55.64 Per cent. 58.62 57-73 54-74 Per cent. 66.23 56.59 53-44 As friction increases, the most efficient and economical pressures increase. Lowest Pressures at Compression. Friction. Compres- sion. Efficiency. Friction. Compres- sion. Efficiency. Friction. Compres- sion. Efficiency. Lbs. tl 8.8 Lbs. 20.5 29.4 38.2 Per cent. 70.92 64.49 60.64 Lbs. 11.7 & Lbs. , 61.7 Per cent. 57.87 55-73 53.98 Lbs. 20.5 23-5 26.4 Lbs. 70.5 76.4 82.3 Per cent. 53.52 51.26 50.17 13.134 cube feet of air at 62 (table, p. 521) weigh i lb., and air at 60 compressed to half its volume evolves 116 heat, and as the specific heat of air under constant pressure is .2377, which x 116 = 27.573 heat units, produced by the compression of i lb. or 13.134 cube feet of free air into one-half its volume : Hence, 27.573 X 778 = 21452 foot-lbs., and as heat and mechanical energy are held to be convertible terms, " =.65 IP produced or lost by the compression of i lb. of air. Inas- 33000 much, then, as the compression of air develops heat, and if the temperature of the compressed air is reduced to that of the atmosphere from which it is drawn before being used, the mechanical effect of this difference in heat is work lost. "Work Lost toy Heat of Compression. Air assumed to be cooled to temperature of atmosphere between stages of com- pression and without effect of jacket cooling. Gauge Pressure. First Stage. Second Stage. Third Stage. Fourth Stage. Gauge Pressure. First Stage. Second Stage. Third Stage. Fourth Stage. Lbs. Per cent. Per cent. Per cent. Per cent. Lbs. Per cent. Per cent. Per cent. Per cent. 60 23 n.8 4-45 800 47-4 26.3 14-3 80 25-3 I3-I 4.8 IOOO 49-2 28.1 14.4 100 27.6 14.6 7.41 1200 Si.6 28.6 14.8 200 34-4 18.9 8.27 I4OO 52 29.4 15 400 40.7 22.9 u 1600 53-3 3 15.5 600 44.6 24.6 13 i I800 54 30.6 16.1 The power of compressing at high pressure is not proportional to the pressure. (Frederick C. Weber.) COMPRESSION OF AIR. 1005 Loss of IPressure throngli Friction of -A.ir in. I^ipes. Per 100 Feet of Length (Initial Gauge Pressure 80 Lbs. at Receiver). Equivalent Volume of Free Air Discharged. X 1-25 Ins. .12 45 i 2 2.5 DIJ 3 LMKTR 4 a OF I 5 >IPK. 6 7 8 10 12 14 Per minute. 25 50 75 zoo 200 300 400 500 750 1000 I 500 2OOO 3000 4000 5000 6000 7500 10000 Ins. .24 i 2.4 Ins. Ins. Ins. Ins. Ins. Ins. Ins. Ins. Ins. Ins. Ins. Ins. .18 4 7 13 - - - - - - - - - - 2.15 3-3 .67 2-5 27 4 4 .06 .1 .22 4 i i. 60 3-70 .07 .12 3 5 I .2 2 .012 03 05 .12 .2 i-3 3 .013 .023 052 .22 1.4 2-5 .012 .027 .048 US .2 3 1.25 .017 .036 .07 .1 15 .22 4 .015 .026 4 I .06 .09 '7 .012 .018 .028 .04 75 = 5 - i - - - (Frederick C. Weber.) ILLUSTRATION. An air compressor furnishes 500 cube feet of free air per minute at a pressure of 80 Ibs. per square inch in the receiver. If this air is used at the end of a 3-inch pipe 1000 feet in length, the loss due to friction will be 10 X 4 = 4 Ibs. If a like volume of air were supplied by the same compressor at a like press- ure and passed through a s-inch pipe 1000 feet in length, the loss would be only ,03 x 10 = .3 Ibs. ; thus illustrating the importance of using pipes of large diameter. Strictly, the loss of pressure is not directly proportional to the length of the pipe, but for all practical purposes it may be taken. Elbows and irregularities in pipes increase the friction in excess of the figures here given. The results in the table represent the loss by friction in the pipes. There is also a slight loss due to friction of the air with itself at the mouth of a pipe when it leaves the Receiver. Leakage. All leaks in compressors or valves, air receivers or pipes, should be strictly guarded against for economy, as they are fully as expensive as steam leaks. When air, at 60 Ibs. pressure, issues from a leaky joint in a pipe at a velocity of over 500 feet per second the waste of it will become apparent. IVlean Effective Pressures in the Compressing and. Delivery of* Free Air to a Griven <3-axige Pressure in a Single Cylinder. Gauge Press- ure. Compr Adia- batic. ession. Isother- Gauge Press- Compr Adia- ession. Isother- Gauge ' Compr Press- Adia- ession. Isother- Gauge Press- Compr Adia- batic. ession. Isother- mal. Lbs. i 2 3 4 5 10 Lbs. 1.41 1.86 2.26 4.26 Lbs. 43 95 2.22 4.14 Lbs. '5 20 25 30 35 40 Lbs. 5-99 7-58 9-5 10.39 "59 12.8 Lbs. 5-77 7-2 10.72 Lbs. Lbs. 45 | 13-95 50 I5-05 55 i 15-98 60 16.89 65 i X7 .88 70 | 18.74 Lbs. 12.62 13.48 14-3 I5-05 I5-76 16.43 Lbs. 85 90 95 IOO Lbs. 19-54 20.05 21.22 22 22.77 23-43 Lbs. 17.09 III 18.87 19.4 19.92 (Frank Richards.) IOO6 COMPRESSION OF AIR. To Compute tlie Steam fressure and." IPoint of Cutting off for a GUven .A-ir Compressor. Assume steam and air cylinders each 22 X 24 ins. and temperature of initial air 62 Area: 378 ' 23 * x 24 5.26 cube feet per stroke. 1720 - = .4 Ibs., and, if compressed adiabatically, 68755 (see ante)x .4 = 27502 foot-lbs., and assuming friction of operation at 12 per cent. ^-^ 31252 foot- .88 Z&s. resistance to be overcome by steam pressure. Hence, 3I ^-=41.2 Z&s., which corresponds with .2 cut-off at initial press- ure of 80 Ibs. gauge pressure. (F. C. Weber, M. E. ) Xo Compute Volume of One Found of Dry Air in Culoe Ifeet and Weight of One Cxitoe Foot of it in Pounds, At Various Temperatures and at Atmospheric Pressure. T + T'-r- 39. 819 = volume. T representing temperature of air and T' absolute tem- perature in degrees. // r T = 62. 62 + 461 -=-39. 819= 13.134 cube feet Inversely. 39. 819 -f- 62 + 461 = . 076 097 Ibs. NOTE. For Table of Volumes, Pressures, and Density at 62 = i, and for Computation of Volume, Weight, Pressure, Density, and Elasticity at other Temperatures, see pp. 521, 522. When the Pressure and Temperature of Air both vary. 2- 7093 X P -i- T = cube feet in Ibs. T representing absolute temperature and press- ure in Ibs. per sq. inch. ILLUSTRATION. What is the weight of a cube foot of air at 60 Ibs. pressure and ioo? . 2.7093 X 60 +14. 7 -=-461 +100 = .3607 Ibs. 461 + 100 -4- 60 -f- 14. 7 Inversely. - - =2.771 volume. 2.7093 Comparison of Single and. Compound Compression. Assume areas of cylinders for Single and Compound compression respectively loo and 33.33 sq. ins. , and pressure of compression 100 Ibs. per sq. inch. Resistance to cylinder of single compression = 100 x 100= 10000 Ibs., and to second cylinder of compound compression = 33. 33 x 100 = 3333 Ibs. The resistance upon the large piston is its area multiplied by the pressure re- quired to force the air from its cylinder into the less. In this case it is 30 Ibs. per sq. inch ; but inasmuch as this 30 Ibs. presses upon the reverse side of the less pis- ton, and thus assists the operation, the net resistance to forcing the air from the large into the less cylinder is equal to the difference of the area of the two pistons, X the 30 Ibs. pressure, 66.66 x 30 2000 Ibs. Hence, the resistance to forcing the air from the larger into the less cylinder is 2000 Ibs., and the resistance in the small cylinder to the compression of it to 100 Ibs. = 3333 Ibs., the sum of the resistance = 5333 Ibs. (The Norwalk Iron Works Co.) The compression of air develops heat, and if the temperature of the compressed air is reduced to that of the atmosphere from which it is drawn before being used, the mechanical effect of this difference in heat is work lost. * Deducting area of piston-rod. f 2 feet stroke. COMPRESSION OF AIR. Isothermal Compression. P V hyp. log. - = F. P representing atmospheric pressure in Ibs. per sq. foot-=. 14.7 X 144 = 2116.8, V volume of i lb. air at atmospheric pressure (62) = 13. 141 cube feet, p andp' terminal and atmospheric pressures absolute in Ibs. per sq. inch, and F foot Ibs. per lb. of air. Assume p = 80 Ibs. per gauge. Then, 2116.8 x 13-141 X hyp. log. ^=27 814.7 x 1.8625 = 51 804 /ooMte. Acliat>atic Compression. One Cylinder. P V ^-^ (j^\ ~S~ i = F. n, omitting cooling of jacket, = i . 408. Then, as preceding, 27 814.7 X 3-45 X (^\ i == 95960 X 6.44.291=95.960 K.7i65 = 68 7 55/oo-H*. Compound Air Cylinders. Two Cylinders. Air cooled to atmospheric temperature before admission to second cylinder. P V ^^ ( ~-\ n + f j n 2 = F. p 2 andp 3 representing terminal press- es in ist and \d cylinder** P V - = 95 9 6o, as preceding, and p 2 = 3 = 37 25- '* Then, 95 960 X f 5ZJ_J + /^ilLj 2 = 95960 X 1.30964 -1.3096-2 = . 6192 X 95 960 = 59 418 foot-lbs. For N Cylinders, 95 960 X N X R t29 i = F. N representing number of cylinders and R ratio of compression, equal in each cylinder. NOTB. Initial pressure in ist cylinder = 14.7 Ibt.; terminal 37.25 Ibs. absolute; initial in ad cylinder 37.25, same as terminal in ist, and at terminal in 21! cylinder = 94.7 Ibs. absolute. To Compute \Vorlz per Pound of Air in Compressing it to SOO Lt>s., Gfauge Pressure, from an Initial Temperature of 62 in First Cylinder. Cooling to Atmospheric Temperature before Air is admitted to next Cylinder, and Jacket Cooling not considered, hence n= 1.408. F = 959 6oXNxR- 29 i =^ ~= = = R. p representing atmosphere in Ibs. per sq. lnch= 14.7, p l terminal pressure (absolute) in ist cylinder and admission to zd=\ = Vi4-7 X 86.8 = 35.7, jp 2 terminal in 2d cylinder and admission to %d = > V'4-7 X 514-7 =86.8, jp 3 terminal pressure in $d cylinder and admission to ^th = Vpz XJ>4 = V86.8 X SM-y^sn,!^ terminal pressure = 514.7 Ibs., and N = 4- Hence, = 2.43, = 2.43, ^^=2.43, and ^ = 2.44. 95 960 X 4 X 2.43'2 i = 95 960 X 4 X .2937 = 112 734 foot-pounds. k Po Compute the Steam Pressure Required in tlie Steam Cylinder of a Sixnple -A.ir Compressor. When the Air Pressure and Diameter of Both Cylinders are Given. x ( ~\ 2 = PI- P and P t representing mean effective air and steam pressures in Ibs. per sq. inch, E, mechanical efficiency of Compressor, and d and di diameter of air and steam cylinders. IOO8 COMPRESSION OF AIR. ILLUSTRATION. Assume pressure of air 60 Ibs., diameter of steam and air cylin- ders respectively 12 and 14 ins., and mechanical efficiency .85. Mean eflective air pressure of air for adiabatic compression at 60 = 30.75 Ibs. (see table, p. 995). 3^Z5 x (iiV:= 36. 18 X 1.36 = 49-2 Ibs. per sq. inch. Corresponding to a steam pressure of 70 Ibs. gauge, at .375 cut-off. Temperature is a direct function of the pressure, hence it is apparent that in the multiple stage compression, where the temperature, by the application of inter- coolers, is reduced back to that of the atmosphere before admission to each cylin- der, that the loss in radiation is reduced. In compound compression, in order to divide the work equally, the ratio of compression should be the same. The temperature of the air (theoretical) in the single stage compression her given is about 400, and that at the end of each compression in the compound case is about 200. The mean effective pressure or resistance of the air of compression in a single C! 804 cylinder, and for the given pressure and temperature, is Isothermally 68 7 '144X13-141 = 27.38, and Adiabatically = 36.33, and 51 804 -5- 68 755 = 75.35. Hence, 144 X 13- *4 r Adiabatic compression is but 75.35 per cent, as effective as Isothermal.* For the heat evolved and given to the air by Adiabatic compression is diffused to the surrounding media before the air is admitted to the Motor cylinder of an engine, the extra work in compression is lost, and in the case here referred to, the loss is 100 75.35 = 24. 65 per cent. In a water-jacketed cylinder, the loss is not so much, as the heat of compression does not rise so high. (Frederick C. Weber.) To Compute tHe Weight of Air used in a Motor per Minute for a Griven -A.mou.nt of "Work. ^~- = = cube feet. N representing number of IP. U = P V ~ P T i I M . p initial and P t exhaust pressure in Ibs. per sq. inch, V volume P 1 1 of air in cube feet, n 1.408, and T and Ti absolute temperatures at admission and atmospheric temperature ; W weight of air per minute to deliver, N IP per minute, and w weight per cube foot at atmospheric pressure in Ibs. N assumed 12, w .076, T and T x 63 -}- 460 = 523, and 300 -f- 460 = 761, P 80 Ibs. per gauge, and volume at 62 = 13. 141 cube feet. 33 X 12 X 523 .408 14. 7 X 144 X 13- 141 X X i -- MOS X 761 X .0761 1.400 i 94.7 - 2 7*8ooo - - - = 89. 2 cube feet per minute. 95 960 X (i .i552' 2fl ) . 4174X76! X. 076! = 2 319510 89.2-^.686 129.9 cube feet without reheating and 129.9 X .686 = 89. l cube feei when reheated. 1.408-1 _ t Log. 14.7 X. 29 = 1.167317 X- 29 .338521 i. 1 552 . 94. 7 X. 29 = 1.97635 X .29 = . 573141 94-7 .234620 i .23462=1.76538 and number of .76538 i =.5826 i = .4i74. By Logarithms. 959601=4.98209 33000 = 4.51851 .4174=762055 12 = 1.07918 761=^88138 523 = 2.71850 .0761=2.88138 8.31619 6.36540 Log. of 1.95079 = 89.28 cube feet. 1.95079 * For an illustration of the curves of pressure, see Frank Richards. Frontispiece and p. 43. t By Logarithm*. COMPRESSION OF AIR. Dimensions of Valves, Pipes, and. Clearance of Air Cylinders. Pressure of Air, 75 Lbs. Cylinder. Area. Free Air. Pressure, 75 Lbs. INLET Diameter. PIP. Area. DlSCHARG Number. VALVKS. Area. . Ins. Sq. ins. Per cent. Per cent. Ins. Sq. ins. No. Sq.ins. 10.25 X 12 78 .0098 .047 2 3-'4 2 5-4 12.25 X H "3 .0086 043 2-5 4-9 2 8.8 14.25 X 18 154 .0066 033 3 7 3 13.2 i6.25X 18 201 .0066 .023 3-5 9.6 13.2 18.25 X 24 255 .0049 .0225 4 12.5 8 35-2 20.25 X 24 3 o 4 .0049 .0225 4-5 '5-9 8 35-2 22.25 X 24 380 .0049 .0225 5 19.6 10 44 30.25 X 60 707 .002 .01 6 28.2 18 79 .2 36.25X48 1018 .002 .01 7 38.5 20 88 Clearance, .0625 inch at each end of cylinder. The area of the discharge depends upon the speed of the compressor; for a speed of 300 feet per minute, ten per cent, of area of cylinder; for a speed of 450 to 500, fifteen per cent. (W. L. Saundcrs.) 1010 TIDAL OR FLUVIAL EFFECT. Tidal or Fluvial Effect on Speed of a Steam or Lilie Propelled Vessel. Deduced from the Experiments and Notes of Edwin A. Stevens, Associate, and C. P. Paulding, Junior, Members N. A. and M. E. To Compute Velocity of the Tide or Current in Feet per Minute. TJ C = V, representing the velocity in feet per minute ; C, length of course R t -\- r T in feet ; R and r, T and t, respectively, whole number of revolutions of engine, and times of run in minutes, both against and with a tide or current ILLUSTRATION Assume C one mile = $ 280 feet, R and r 970 and 548 number of revolutions, and T and t 8.45 and 4.77 times. What is velocity of tide or current? 970 548 5280 970X4-77+548X8.45 =5280 422 4626.9-1-4630.6 = 240.7 feet velocity. To Compute Advance per Revolution of Engine in Feet per Minute. or Current. S+ A. 5 280 + 240.7 X 8.45 = 73H R 970 970 NOTE. If distance is given in knot of 6080 feet, add 15. 151 per cent. To Compute Speed of Vessel in Feet per Minute. rT + R* =& 5*80 548X8.454-970X4-77 9257-5 To Compute Number of Revolutions to Run one Mile in Still Water and the Slip. _ N 548X8.45 + 970X4.77 = 9257.5 = 6 revolutio T+t 8.45 + 4.77 13-22 700 . 37= U>li 700.26 = 58 .74 lost in slip = 7 . 7 5 per cent. NOTE. In applying these formulae, the number of revolutions in the run should be as uniform as practicable. Between runs, a variation of 5 per cent, will not materially affect the result. STEAM SIPHON. A.n Independent Lifting Pximp. Capacity for a Discharge Pipe 2 Ins. in Diameter, per Minute. Discharge. Gallons. 119.68 Water raised. . Feet. 14 13 13 Ins. 6 ressure. Discharge. Water raised. Pressure. Lbs. 3 40 50 Gallons. 63.54 85-71 100 Feet. 13 13 13 Ins. 2 2 2 Lbs. 60 70 80 157.57 .Friction. Losses and. Distribution, of* 3?ower in Machinery. From 8 to 4OO IP. Losses Range from 55 to 65 per Cent. Per Cent. Friction of Engine 10 to 1 1. 8 " of Shafting 15 "17.7 " of Belts and Gearing. 15 "17.7 Per Cent. Friction of Lathes and Ma- chinery 15 to 17.7 Effective* Operation 45 "35.1 CAST IKON, DEW-POINT, AND COLUMNS. IOI I Strength of Cast Iron. As determined by Tests on a Riehle Instrument at Lexington, Ky. Average of 16 Tests. Tensile, per Sq. Inch. Elastic Limit. Modulus of Elasticity. Transverse, per Sq. Inch. Elastic Limit. Modulus of Elasticity. Lbs. 24436 Malleable. 41582 Lbs. 21 469* 31042 Lbs. 28240000 13000000 Lbs. Annealed. 4425 Refined. 2435 Lbs. 2508 (Jame, Lbs. 21 OOOOOO 19300000 5 H. Wtlls.) To Ascertain the Degree of AJbsolnte Dryness in the Air and. the Dew-:Point. Mason" 1 s Hygrometer. ll 11 1 Excess of | Dryness. Absolute Dryness. Dryness Observed. If 8 - Is 11 .2> ' 5 2.92 40.83 3 5 7 8 33 18.67 13 2.17 30-33 18 3 42 3-5 .58 8.17 8-5 .42 19.83 '3>5 2.25 31-5 18.5 3-o8 43-17 4 .67 9-33 9 5 21 14 2.33 32.67 *9 3-i7 44-33 45 75 10.5 9-5 58 22.17 i4-5 2. 4 2 33-83 i9-5 3-25 45-5 5 83 11.67 10 .67 23-33 15 2-5 35 20 3-33 46.67 To Ascertain the Drynm. OPERATION. From temperature of the air subtract that of the wet thermometer, add excess of dryness from the table for the differ- ence, multiply sum by 2, and the result will give absolute dryness in degrees. ILLUSTRATION. Temperature of air, 57; wet thermometer, 54. Hence, 5754 = 3. Add .5, from table, = 3. 5 which X 2 = 7 degrees. To Ascertain the Dew- Point. From temperature of the air subtract Absolute Dryness and result will give the Dew-Point in degrees. ILLUSTRATION. Temperature of air, 57 ; Absolute Dryness = 7. Hence, 57 7 = 50 Dew- Point. Safe Gnashing Strength of Columns of a Height not ex- ceeding IS times their Diameter. In Pounds per Square Inch of Transverse Section. Material. Lbs. Material. Lbs. Material. Lbs. Basalt 2 875 Iron, wrought 14 400 Mortar, common 36 Brick hard Limestone hard 7 20 Oak white 4^2 " common Granite hard 58 " common Marble hard 432 i S '- a * -^ ^ _ " 'i * p I ' c? N" c?^ cT'w cT 1J W ^llll *s ET* 2"8 a H ^a MMMWMM a'^ CO T- IOVO t^OO ^ g 1 1 w Q PQ <5 1 M cs co ^ ^vg g. 8 .2 <* VOVO t^OO ON O rt rt "" 7? ^ - w O - 2 ^ : s ^ "^ S EH ^ EH fe 00 no* p , 8 8 at gj oo w Q P3 f h s , 4* fl Q O CQ 1- fe * Ed W H Ill"ll fl N 1 3 8 8 a s- O PQ - - H fi Q| g^ M |g|= g | PQ fe - Q gf| 8 8 vo fe w c O : oo ON O M : ^ 5 Years. From One to Ninety -nine. 1 ^ 1 e> 1 'ON 1 & \ t H S m ? 10 vo t^ ^ ^-'S 8,1 ON N| 8 s;i s b 2" J?^ ET" 2" S a a a 00 oo* oo 00 s IS lo? S rxoo ONO M co g ^ SI eg eS g 00 H w co ^*- invo a) * r< o K 1 ! ^ : ^ c? cT c?^ pT'S c? %s It C IS \ & S. g. ft VO t-xOO ON O M N 3 VO vo vo VO^ 1 ^0 vo' ^ ^SMME?;?!? ^SO -^ gj Ivg vS < m ^ mvo txoo *| rt - a & s? 1 5 s en M "^ "".c ^ * 5 ^ ON o ^ d ^cT ( NN N Coto QCC,2 * "> vo | t^ oo ON" 2S CN CO Tt- VOVO t^OO .; t^ * c?l 8> & (N CEMENT, HUMIDITY, METRIC MEASURES. IOI3 Tire Cement. Mix bisulphide of carbon, 160 parts; gutta-percha, 29 parts; caoutchouc, 40 parts; and isinglass, 10 parts. Pour the mass into the crevices of a rupture after they have been properly cleaned. If the rent is large, apply the cement in layers. Bind up the tire lightly, let the cement dry for twenty-four to thirty-six hours, remove the binding and the protruding cement with a sharp knife, which must previously have been dipped in water. German practice. Relative Humidity and Dew !Point of the Air. As Determined by a Dry and Wet Thermometer. Difference of Temperature between the Two Thermometers and Degrees of Hu- midity. Saturation being 100. *tf MB* sjj 32 42 s 52 62 72 82 9 2 C DiflF. of Tempera- ture of Air. 4- 52 62 72 82 ^ Diff. of Tempera- ture of Air. 42 C 52 62 72 C .2 92 I 2 3 4 5 6 87 75 92 5 78 72 66 60 93 86 So | 4 8 82 77 % 94 89 84 79 74 69 95 90 8 5 80 76 72 95 90 85 81 77 73 O 9 10 ii 12 54 49 44 40 36 33 59 54 sj 46 *a 39 62 58 54 5 r -> 47 44 65 6 1 57 54 8 68 64 60 57 54 5 1 70 ^8 62 53 13 H 15 * 16 g 3 27 3* 33 30 27 25 4i 38 35 32 30 28 45 4* 39 36 34 32 4 8 45 4-? 4'-' 3^ 35 50 47 45 43 4i 33 OPERATION. If temperature of air is 72 and difference of temperature between the thermometers is 7 The humidity or dew point is 65 degrees. (Greenwich Observatory. ) Reduction, of Metric IMeasures. As enacted by the Congress of the United States and in United States Measures. In addition to pp. 27-33, &, 44, 47, 923, 934- Caloric X 3- 968 = B.T.U. (Centigrade x 1.8) + 32 degrees. * Centimeters X .3937 = inches. Centimeters -r- 2. 54 = inches. Cheval vapeur x .9863 = IP. Cube Centimeters-r- 16. 383 cube inches. Cube Centimeters -T- 3.69 =: fl. drachms. Cube Centimeters 4- 29. 57 = fluid oz. Cube Meters X 35-315 cube feet. Cube Meters X 1.308^1 cube yards. Cube Meters X 264.2 = gallons. Grams X 15-432 grains. Grams -=- 981. = dynes. Grams (water) -4- 29.57 = fluid ounces. Grams-:- 28.35 = ounces avoirdupois. Grams per cube centimeter -=- 27.7 = Ibs. per cube inch. Gravity Paris = 980. 94 centimeters per second. Hectare x 2.471 = acres. Hectoliters x 3.531 = cube feet. Hectoliters X 2.84 = bushels. Hectoliters x -131 =cube yards. Hectoliters -f- 26. 42 = gallons. Joule X .7373 = f ot pounds. Kilo per Meter x .672 = Ibs. per foot. Kilo per Cheval x 2. 235 = Ibs. per BP Kilo per Cu. Meter x .026= Ibs. per cu ft. Kilogram-meters x 7.233 = foot Ibs. Kilogram per sq. cent, x 14- 223 Ibs. per sq. inch. Kilograms X 2. 2046 = pounds. Kilograms x 35- 3 = ounces avoirdupois. Kilograms -r- 1 102. 3 = tons, t Kilometers x .621 = miles. Kilometers -r- 1.6093 = miles. Kilometers X 3280. 7 = feet Kilo Watts X i 34 = IP. Liters x 61.022 = cube inch. Liters x 33-84 = fluid ounces. Liters X .2642 == gallons. Liters -f- 3. 78 ! = gallons. Liters 4- 28.316 = cube feet. Meters x 39.37 = inches. Meters x 3-281 = feet. Meters X 1-094 = yards. Millimeters x .03937 = inches. Millimeters -r- 25.4 = inches. Square Centimeters x .i55 = sq. inches. Square Centimeters-:- 6. 451 = sq. inches. Square Kilometers x 247.1 = acres. Square Meters X 10.764 = sq. feet. Square Millimeters x -0155 = sq. inches. Square Millimeters -r- 645. i = sq. inches. Watts -r- 746. = IP. Watts -r- .7373 = foot pounds per second. > All degrees are given Fahrenheit. t Tons in this item are computed at 2000 Ibs. IOI4 CHIMNEY DRAUGHT, STEAM VESSELS, ETC. To Ascertain the Height of* a Chimney for a Re- quired. Draught. Divide 7.6 by the absolute temperature of the externnl air, and 7.9 by the like temperature of the gases in the chimney at the point of their delivery into it; sub- tract this quotient from the former, divide the required draught by the difference, and the quotient will give the height of the chimney in feet. Or, - = h. D representing the draught in inches of water, T temperature 7-6 7-9 T t of air -f- 460, t temperature of gases -f- 460, and h height of chimney in feet. ILLUSTRATION. Assume temperature of air 20, and that of the gases 600, and required draught .6 inch .6 .6 . , . - = - = ?i.6feet. 7.6 7.9 .00838 20 -f- 460 600 4- 460 To Ascertain the Draught of* a Chimney. In Inches of Water. Proceed as above to determine the difference of temperature, subtract the latter from the former, and multiply the remainder by the height of the chimney in feet. ILLUSTRATION. Assume like temperature and height of chimney as abpve. 7 ,.6 = .6 ic7, Resistance of Steam Vessels. : ' The thrust of a propeller on the resisting collars of a propeller shaft is the meas- ure of the power applied to the propulsion of the vessel. .66X33ooo_ p PXS TO> PX22 3 2_ TTr> D * X S* 3 n1i~ 33000 -"* C ~^K~ IIP. P representing the thrust of the propeller in Ibs., S and S l the speed of the ves- sel in feet per minute and knots per hour, D displacement in tons (2240), A area of immersed amidship section in square feet, and C and C t constants. Assume the following elements of the steamer "El Sol": Length between per- pendiculars, 377. 2 feet ; amidship section, 934 square feet ; displacement, 6760 tons ; S and Sj, 14.75 feet and 14.5 knots ; C and C\, 310 and 813; and IIP 3500. .66 X 33 coo = 6 p 5i 695 X 1475 = 61g p. 5 *95 X 2232 = M75X3500 33000 33000 6760* X.4- 53 ^357-5 X3Q48 93^X^3 310 310 813 W V S3 D. K. Clark gives = EIP. W representing wetted surface in square feet. Coefficients of* Radiation of Heat. For a Period of One Hour from 10. 76 Square Feet * of Surface. Silver, polished 16 Copper, red 20 Brass, polished ->^- 32 Sheet iron, polished. . . 56 Sheet-iron, leaded 81 Sheet-iron, black 345 Glass, polished 373 Cast-iron, rusted 419 Paper 470 Stone, building 499 Soot 500 Water 662 (^ Home Study.") Heat Radiated, per Sq.u.are Foot per Hour. From a Temperature of 180 to 159 in Units. Tin-plate 1.37 | Sheet-iron 2.24 | Glass 2.18 (Tredgeld.) * One square Metre. FORCED DRAUGHT, GEOLOGICAL STRATA, ETC. Forced. Draught in Marine Boiler. Compressed Air Exhausting Blast in the S. S. "Resolute." Blowing .Engine. Engine. Coal per hour. Coal* per IIP per hour. Water evapo- rated per Ib. of coal Blowing Engine. Engine. Coal per hour. Coal perlH? per hour. Water rated per Ib. of coal Iff IIP Lbs. Lbs. Lbs. IH? IH? Lbs. Lbs. Lbs. Natural ) draught. j 57-5 213 3-72 10.77 4.2 118.8 348 2-93 782 .96 88.8 289 3.26 8.82 5 119.8 374 3-12 7-53 2 100.5 315 3.12 8 6 127.9 ! 400 3 .12 7 3 106.1 321 3-04 7.82 7-4 135-7 1 420 3.10 7-03 (D. K. Clark.) Absorption of Q-eological Strata. Per Cent, by Volume. Formation. Location. Water in loo parts. Authority. Dolomite Joliet, 111 Lemont 111 1. 06 1. 12 4.76 2-5 5-55 .29 .42 29 4-4 4.2 2.1 12.15 .08 { } 25 i '3 i5/ 29 23-95 n.6 4.81 6.25 12 /33 } \4o J Daniel W. 3S. Metal. G. P. Merrill. M, Delessee. D. W. Mead. E. Wetherel. M Delessee. G. P. Merrill. M. Delessee= E. Wetherel. G. P. Merrill D. W. Mead. R. J. Hinton. Mead,C.E.) ii Winona, Minn ii Red Wing Minn ii Mantorville " Gabbro Duluth u Granite Hornblende E St Cloud " Freestone Calcareous Grand Beauchamp, France Bedford Ind Limestone " Galena Rockfor'd 111 " Trenton u Oolite Cheltenham Eng " Devonian .... Boulogne t ranee Quincy 111 u Big Sturgeons Bay, Wis. . . Grand Beauchamp, France it Cheltenham, Eng Gloucestershire, Eng , Fond du Lac Wis Sandstone u Quartzose " Oolite " Old red tt Fort Snelling, Minn. Jordan, " Berea Ohio u 11 Clay dry Sand and Gravel *H Cast-iron. Abater I?ip< To Compxite Thickness of (- . 25 = T, and \- .25 = T. H representing head of pressure of water in 9600 4250 feet, d internal diameter of pipe, and T thickness, both in inches, and p interior press- ure in Ibs. per sq. inch. ILLUSTRATION. Assume head of water 200 feet, diameter of pipe 8 ins., and in- terior pressure 86.83 Ibs. per sq. inch. 200 X 8 , j 86.83 , ^-.25 =.417 ins., and -\- .25 =.4134 ins. 9600 4250 For faucet ends, the equivalent length of pipe, equal in weight to that of the faucet, 7 -j- d -4- 15 = ins. See ante, p. 147. (D. K. Clark.) * Anzin briguettes. The fuel consumed and the power were doubled, but the evaporative efficiency was reduced. IOl6 FRICTION AND FLOW OF WATER IN METAL PIPES. Friction of Flow of Water in Smooth Metal Pipes, From .5 to 3.5 Inches in Diameter. To Compute tlie JLioss of Head. Per ioo Feet. .0126 + 5i ZJ - x x Ji__ __ H d internal diameter, H, Zoss o/ head due V o a 2 p which X 40000 (volume cf water) = 45 560 Ibs. equivalent evapora- tion at 212 ; and 45 560 -f- 4000 350 (Ibs. of combustible) = 12.48 Ibs. water evapo- rated per Ib. of combustible from and at 212. If 34.5 Ibs. water evaporated from and at 212 = one IIP, a boiler or boilers, oper- ating with the given elements, will have developed 45 560-^-34.5 x 10= 132.1 H*. * Am. Soc. M. E t See p. 478. $ See also Am. Soc. M. E., 1884. APPENDIX. 1025 Heating Surface. Of a Steam Boiler, etc. Heat is communicated to the transmitting surfaces of a steam boiler in the fol- towing order of eft'ect viz., incandescence, flame and gases of combustion ; and that cransmitted by radiation of it, from one surface to another, is reduced, in the ratio as the square of the distance between the surfaces, and it is also reduced by a de- pressed inclination of the surface upon which the current of the heat impinges, and contrariwise increased by a raised inclination. Evaporative Efficiency. The evaporative efficiency of a boiler, or of an assigned area of heating surface, as one sq. foot, depends so entirely upon the thickness, position, and condition of it that it is wholly impracticable to assign a determinate value to it. It is also measurably affected by the duration of the time of the trans- mission of the gases of combustion over it. Theoretical and Attainable Evaporation. If all the heat of the combustion of coal in the furnace of a steam boiler was utilized, the evaporation from one pound of best anthracite would be from and at 212 about 15 Ibs. of water, but only 80 per cent, of that has been attained. At the Centennial Exhibition in Philadelphia in 1876, the average evaporation from 15 boilers of different types, with grate area as 35 to i, was 10.27 Ibs. of water; and the averages of evaporation per sq. foot of heating surface per hour was 2.99 Ibs., varying from 1.75, to 9 ; and of the temperature of the escaping gases, 410. Experiments with locomotive boilers by D. K. Clark, having from 52 to 90 sq. feet of heating surface per sq. foot of grate, gave with coke* an average evaporation, at the ordinary temperature and pressures. 9 Ibs. of water per Ib. of fuel. In horizontal tubular boilers, with heating to grate surface as 25 to i, the vol- ume of water evaporated per Ib. of fuel decreased as the fuel consumed per sq. foot of grate area increased. Fuel Water evaporated from 212 Tempera- Fuel Water evaporated from 212 Tempera- per hour persq. foot persq. foot of heating per Ib. of coal. persq. foot of heating ture of escaping gases. per hour persq. foot persq. foot of heating per Ib. of coal. persq. foot of heating ture of escaping gases. of grate. surface. surface. of grate. surface. surface. Lbs. Lbs. Lba. Lba. Deg. Lbs. Lbs. Lbs. Lbs. Deg. 6 .24 10.49 2.52 444 16 .64 8.21 5-25 897 8 32 10.35 3-31 472 18 .72 7-7 5-54 999 IO 4 10.05 4.02 532 20 .8 7.3 2 5-85 1074 12 .48 9-53 4-57 685 22 .88 7.04 6.19 1130 14 .56 8.87 4.96 766 24 .96 6.82 E 7!,., 6-54 "74 r c -\r Benj. F. Isherwood, U. The efficiency is also dependent upon the area of it for the contact of furnace heat, flame, the gases, and the period of the application or transmission of the heat over it ; and inasmuch as flame imparts more heat than the inflammable gases, the diameter of tubes should be, so far as practicable, of a capacity to admit of the flow of it. Radiation of Heat to Surfaces. If the thickness and surfaces of boiler plates and tubes were uniform, or progressively reduced in thickness and resulting capacity of radiation, their progressive effect and the proper temperature of the gases at the point of delivery, as into a smoke-pipe or chimney-stack, could be readily obtained by a dividend, of the difference of temperatures between that of the entrance of the gases of combustion into the flues or tubes, and that of 212, the divisor being that of any assumed number. Thus, assume the gases at the bridge wall at 1700, which is the temperature assigned by Chief Engineer Benj. F. Isherwood, U. S. N., being the result of his observation and extended experiments. Then, at six locations or divisions of the temperature, 1700" - = 248 and 1700 2480=1452 and 1245 , 212 = 207 ; 1452 207 = 1245 and '3 = 1072 and I0720 ~ 2I20 = i4 4 o ; I072 o_ I44 o = 928 and 9280-2,20 6 o _ I20 o = 8o8 o and 8o8 ~ 2I2 = 990. IO26 APPENDIX. Thus, at termination of 6th location, 99 are radiated in its passage from the sth, and the temperature of exits 808 212 = 596. Professor RanTdne asserts that when the difference of temperature between the water of evaporation and the gases of combustion is very great, that the rate of conduction increases faster than the ratio of the difference, and is nearly propor- tional to the square of the difference of temperature, and which may be thus ex- pressed. T Z -i- C = R. T and t representing the temperatures of the gases and the water, C a constant derived from experience, 160 to 200, and R ratio of conduction in ther- mal units. Assume temperature of gases and water 1500 and 212 and C = 180. 1500 212 , _,, , =0216 thermal units. 1 80 Robert Wilson, London, 1896. Gives for multitubular and other boilers with heating surface to grate area from 30 to 40 to i : 9 sq. feet heating surface to evap- orate i cube foot fresh water, or 4.5 sq. feet of total heating surface per H*. In locomotive and like boilers with a blast draugni, with heating surface to grate from 60 to 80 to i : 6 sq. feet heating surface to evaporate i cube foot fresh water, or 3 sq. feet of total heating surface per IP ; and the highest average efficiency, i sq. foot of heating surface for 13.5 Ibs. water, or 4.66 sq. feet for i cube foot of water. And in ordinary externally fired boilers, with heating surface to grate 10 to 16 to i : 18 sq. feet of heating surface to evaporate i cube foot watei, or 9 feet per IP. Vertical Boilers. Usually, are wasteful of fuel, but when in good condition and the tubes properly spaced, 16 sq. feet of heating surface have evaporated i cube foot of water or 8 sq. feet per IP, with a consumption of i Ib. coal to evaporate 8 Ibs. water = 7.75 Ibs. per cube foot. Usually 10 to 12 sq. feet of heating surface are required per IP. Tubes. The number and sectional area of tubes in a boiler (horizontal multi- tubular) and the spaces between them should be determined by the transverse area over the bridge wall or the grate surface, and the required facility of the ascending current of steam from over furnace, and in all cases should be set in direct vertical lines, and, in consideration of their efficiency, their lower or inner surface kept free from accumulation of the mechanical deposit of ashes and soot. Inasmuch as the surface of a tube increases directly with its diameter and its sectional area, or capacity as the square of it, it becomes necessary in order to at- tain like economy of evaporation by them, when of different diameters, as i and 2 ins., that the length of the greater diameter must be twice that of the less. Thus, the 2-inch tube having four times the sectional area or capacity for the passage of flame or gases than the less, and but twice its heating surface, the length of the greater must be twice that of the less.* Length of Flues and Tubes. The greater the velocity of the current of the gases in a flue or tube, the greater will be its temperature at their exit, and consequently the greater the waste of it, unless the length of them is proportional to the velocity of the current. The absorption of the heat of the gases requires time, and hence the longer the course of them, if duly proportioned, the greater their effect.! Peclet assigns the proportion of radiating heat from coal in its condition of per- fect combustion at .5 of its latent heat. Length and Area of Tubes. With ordinary smoke-pipe or chimney draught, the length of a tube should not exceed 40 times its diameter. Experiment with a tube 2.5 ins. in diameter, the temperature of the gases at their delivery being 500, the length of it was 100 ins., or 40 times its diameter, and with a blast draught, as in a locomotive boiler, or blower draught ; the length may be much increased. By late experiments on a railway in France it was found that greater economy was attained by increasing the length of locomotive tubes above 12 feet. In conse- quence of which the Baldwin Locomotive Works, of Philadelphia, assign a length of 14 and 14.5 feet for a 2-inch tube = 84 and 87 times the diameter. Prior to this 2-inch tubes were usually but 10 and 12 feet in length. In all cases, however, the length should be in direct proportion to the diameter. Vertical Fire Tubes. Are not as effective as horizontal or inclined, as the gases * Se$ P- 742- t For the evaporating capacity of tubes of different lengths, see also p. 742. APPENDIX. IO27 which lose some of their temperature by contact with the surface of the tubes are not replaced by the centre current, and the additional temperature of it imparted. Water Tubes. Whether vertical or inclined, enable the steam which is gener- ated at their inner surface to rise as fast as it is generated, and, as a consequence, the velocity of the current of the water is increased. Water or Masonry Bridge Walls. At the termination of the grates, by diverting the current of the gases of combustion, enable them to be better commixed, and effect a more effective combustion and consequent economy. In the computation of the area of heating surface, the areas of the furnace above the grates, bridge wall if water, combustion chamber, flues, tubes (fire or water), and connections to water-line are to be taken, and also two-thirds of all the gas surfaces above it, inasmuch as they, as in the case of a steam chimney, are sur- rounded by steam at a high temperature. Steam Heating. (In addition to p. 527. ) To Compute Area of Plate or 3?ipe Surface of Cast- Iron, to Compensate the Reduction of Temperature in an Enclosed. Space by the Exposure of GJ-lass to the External A-ir, or its Equivalent in Exposed Walls. fp g = A. T and t representing degrees of required temperature of space and of external air, H temperature of heating surface, and A area of radiating plate or pipe surface in sq. feet. ILLUSTRATION. Assume temperature of external air 70, of heating surface 160, and required temperature of space 70. 7O o 2O o JL =,667 sq.feet of heating surface for each sq.foot of glass. Hence, if the area of the glass in windows or lights of a room is 80 sq. feet, then .667 X 80 = 53.36 sq.feet additional radiating surface. Radiation. One square foot of Direct Radiating surface will heat Cube Feet. In an ordinary domestic room with glass windows , 35 to 45 In an ordinary public room. ... 45 "55 u small dormitories 50 "60 " large dormitories 45 "55 " hallway and passages 60 "80 " school and low-ceiled lecture- rooms and offices 60 "75 Cube Feet. In churches and high -ceiled halls 65 to 95 " small low -ceiled factories and workshops 50 " 60 " small high -ceiled factories and workshops 60 " 70 " large high -ceiled factories and workshops 75 " 140 Steam Exhaust Heating. Exhaust steam, according to the condition in which it flows from an engine, contains water and oil, varying from 10 to 20 per cent, of both combined. Conse- quently they should be arrested before entering a heating plant ; and as any restric- tion to the flow of the steam involves a resistance to it, or that which is termed back pressure, the receiving and distributing pipes should have the greatest prac- ticable capacity and least restriction to the flow of it by curves and angles. To meet an emergent requirement for heat, a direct connection to the boiler, through an automatic reducing pressure valve, must be furnished, and, contrari- wise, a relief valve should be furnished, in order that when a reduced temperature or volume of steam is required in heating, it may escape into the air. To Design and Proportion an Exhaust Stenm Plant. It is necessary first to ascertain the area of radiating surface required to condense the exhaust steam, and to obtain this the volume of the steam which the engine would exhaust must be ascertained. To determine which (see table, p. 708), give weight of water evaporated from 212 per sq. foot of heating surface. 1028 APPENDIX. ILLUSTRATION. Assume the area of the heating surface is 9000 sq. feet, of grate 30, and fuel consumed 16 Ibs. per sq. foot of grate per hour. The water evaporated per hour per sq. foot of grate = 8. 21 (see table, p. 1025) X 16 = 131.36 Ibs. water, which x 26.36 (the volume of i Ib. steam at 212) = 3 462. 65 cube feet of steam. Water evaporated per hour per IIP in a non-condensing engine ranges from 25 to 40 Ibs., from which, for condensation and leaks, 10 per cent, should be deducted to obtain the volume of steam available for heating. One square foot of Direct Radiating surface will heat Cube Feet, i Cube Feet. In dwelling-houses 45 to 55 In factories, stores, and shops 90 to 100 " offices 65 " 75 I " churches, auditoriums, etc. 150 " 200 For Indirect Radiation deduct 20 per cent., and when the heat is transmitted by blast from a Blower add from 4 to 6 times, in accordance with its volume and con- sequent velocity. Hot- Water Heating. The sectional area of the main pipe should, in all cases, exceed that of its branches, and for each sq. inch of its section, if short, indirect, and at a slight in- clination, 50 sq. feet, and if long, direct, and vertical, 100 sq. feet. One square foot of Direct Radiating surface will heat, the average temperature of the water 160, from 80 to 100 per cent, more surface than by Steam. Horse-Power Required, to Drive Machinery. In addition to Frictional Resistances, etc., pp. 475-478. See a very full table of HP required in American Machinist, April 12, 1894, and February 6, 1896. Referring to the following table, it will be noticed that the loss of power varies between wide limits, but in all cases the mechanical loss is large, averaging over 41 per cent. Machinery. Work. Total H?. Horse Shafting. Power to Machin- ery. Drive Shafting. per cent. Union Iron Works Frontier I. & B. Works. . Baldwin Loc. Works W Sellers & Co Engines and Machinery. Marine Engines, etc Locomotives Heavy Machinery Machine Tools Cranes and Locks Presses and Dies 406 25 2 5OO 102.45 1 80 135-5 ' 35 150 400 9 I 2000 40.89 66.11 II 75 100 35 i? 500 61.56 'OS 68.24 24 75 qoo 23 32 .80 .40 .41 49 3i So .25 Pond Machine Tool Co. . . Yale & Towne Co Ferracute Machine Co. . . Bridgeport Forge Co Hartford Mch. Screw Co. Heavy Forgings Machine Screws.... (Prof. J. J. Flather.) Refrigerating Machinery. For the cooling of Brine and other liquids by the alternate compression and ex- pansion of air. , T t P T P representing power required in foot-pounds, T absolute maximum temperature of the air in the hot or compressive end of the refrigerator, t absolute minimum tem- perature of the air in the cold or expansion end, and C cooling work in thermal units. (David Thomson.) ILLUSTRATION. Assume T = 80, t = 30, and C = 80 30 = 50. 80 772X5QX~ 8o J = 31.25 X 1.6 = 50. Hence, the most economical results, as regards power used, are obtained when the machine is operated within a small range of temperature, as in a brewery, where the temperature of the water is frequently reduced to but 10. These formula are applicable to all refrigerating machines, whether operated by f ?2. 3, = 3 s 600 X .625 = 24 125 foot -pounds, and ^-^ X- _ 3O APPENDIX. IO29 air, ether, ammonia, or any other liquid. In an ammonia machine, or any other operated on the same principle, in which mechanical power is applied, the value of P is the heat theoretically required, at the rate of i heat-unit for 772 foot-pounds or power, and the formula i becomes (ammonia): Heat required for the operation, r -'r r _ c=c. The ammonia machine is, theoretically, economically superior, as heat is less ex- pensive than its equivalent in mechanical power. The nature of the vapor operated controls the capacity of the machine. Relative Capacities of Cylinder Required. Ammonia i Carbonic acid 16 Methyl Chloride 1.8 Methyl ether ...................... 1.8 Sulphuric acid ..................... 2.6 Ether ............................. 15. i (D. K. Clark.) Sewerage. In order that an estimate of the volume of excessive rainfalls may be com- puted, the following data are derived from the valuable report of the Sewerage Com- mission of Baltimore, 1897 : Philadelphia, July 23, 1887 ..................... 4.23 inches in 13 minutes. Chestertown, Md., Aug. 15, 1894 ............... 3.64 " "30 *' Washington, B.C., June 30, 1895 ............... 6.27 " " 10 " The average of 26 falls was 3 inches in 10 minutes. . In the city of New York a fall of i inch in 10 minutes has frequently occurred 6 inches per hour. Flags. Safe Transverse Strength. Loaded in Middle. Supported at Both Ends. c. Freestone, Little Falls .............. 121 " Belleville, N. J .......... 101 " Connecticut... .......... 65 " Dorchester, Mass ........ 50 c. Slate, mean of 242 and 537 .......... 390 Glass ........................ . ..... 210 Bluestone ......................... 178 Granite, Quincy ................... 131 bd 2 To Compute Safe Load, -y- X C = Load in Ibs. C representing one-tenth of breaking weight. Assume a flag or block of Quincy Granite, 6 feet in width, 6 ins. in depth, and 3 feet in length between its supports. 6 X 12 X 6 2 X 131 = 72 X 131 = 9 432 Ibs. 3X 12 Average weight of 17 different Sandstones, a ascertained by Lieut.-Col. Gilmore, U.S.A., 143 Ibs. CAST-STEEL FLAT ROPES. John A. Roebling's Sons Co., New York. Dimensions. Weight per Foot. Strength. D imeMi on.. | Jtftl Strength. Ins. Lbs. Lbs. Ins. Lbs. Lbs. 375 X 2 1.19 35700 5X3 2.38 71400 375 X 2.5 1.86 55^00 5X3-5 2.97 89000 375X3 2 60000 5X4 3-3 99000 375X3-5 2-5 75000 5X4-5 4 I2OOOO 375X4 2.86 85800 5X5 4.27 128000 375X4-5 3.12 93600 5X5-5 4.82 144600 375 X 5 3-4 100 000 .5X6 5-i 153000 375X5-5 3-9 IIOOOO 5X 7 5-9 177000 Steel Wire Flat Ropes are composed of a number of strands, alternately twisted right and left, laid aside of each other and sewed together with soft iron wires. They are used sometimes in place of round ropes in shafts of mines: wound upon a narrow drum, requiring less space than a round rope. Soft iron sewing-wires wear out sooner than the steel strands, and then it is necessary to replace them with new iron wires. 1030 APPENDIX. Illustrations in. Logarithms. To Compute the Length of an Arc of a Circle to Radius 1. RULE. To log. of degrees in the arc, add 2.241 877, and sum is log. of length. NOTE. When the arc is in minutes, seconds, etc., take their decimal equivalents. ILLUSTRATION. An arc of a circle is 57 17' 44" 48" -\- ; what is its length? 17 44" 48" = .2957. Log. 57 -2957=_i-75 8l2 3 Log. 3. 1416 -r- 180 2.241877 Log. i = . ooo ooo To Compute the Degrees in. an Arc of a Circle -when the Length is Griven. RULE. To log. of length of arc, add 1.758 123, and from the sum subtract log. of its radius, and remainder will give log. of degrees. ILLUSTRATION. How many degrees are there in an arc when the length is 2 and the radius i ? Log. 2 = .301 030 Log. 180 -r- 3. 1416 = 1.758123 2.059153 Log. i = .000000 " 2.059 J 53 = 114- 59 l66 = "4 35' 3 ' To Compute the Angles of a Triangle, the Length of the Sides feeing GHven. RULE i. To the logs, of the differences between any two sides and half the sum of the sides, add the arithmetical complements of the logs, of half the sum of the sides, and the difference between it and the remaining side, and divide the sum by 2, the quotient is the logarithmic tangent of half the angle opposite to the latter side. RULE 2. To the logs, of half the sum of the sides, and the difference between it and any side, add the arithmetical complements of the logs, of the two other sides, divide the sum by 2, and the quotient is the logarithmic cosine of half the angle opposite the former side. RULE 3. To the logs, of the differences between any two sides and half the sum of the sides, add the arithmetical complements of the logs, of three sides, divide the sum by 2, and the quotient is the logarithmic side of half the angle opposite to the remaining side. ILLUSTRATION. The sides of a triangle are A 679, B 537, and C 429. What are the angles? = 822.5 and 822.5 679 = 143.5 = difference between two sides and half sum of sides. 822. 5 429 = 393. 5 and 822. 5 537 285. 5 = differences as above obtained. Log. 143.5 2.156852 " 393-5 = 2.594945 Co. Log. 822. 5 10 = 7.084 864 '' " 285.5 10 = 7.544394 2)19.381055 9.690528 Log. 143.5 = 2.156852 " 285.5 = 2.455606 Co. Log. 679 10 = 7. 1 68 130 " " 537 10 = 7.270026 2)19.050614 9-525307 2. Log. 822. 5 = 2.915 136 " 143-5 = 2.156852 Co. Log. 537 10 = 7.270026 " " 429 10 = 7.367543 2)19.709557 9- 8 54779 1. 9. 690 528 = . 5 Log. e Sin. = 19 35" 2. 9. 854 779 = .5 " Cos. = 44 17' 30" 3- 9-525 307 = -5 " Tan. = 26 7' 30 90 o' o" APPENDIX. 1031 To Compute tlie Volume of a Pyramid or Cone. RULE. To logs, of area of base and height add 7.522 879, and the sum is the log. of the volume. ILLUSTRATION. The largest Pyramid of Egypt has a base of 700 feet square, a height of 500 feet, aud assuming its faces to be triangular, and to be constructed solid of granite weighing 2654 oz. per cube foot, what is its volume and weight? Log. 700 = 5.690196 Log. volume = 7.41 2 045 " 500 = 2^698970 Log. 2054 = 3. 423 901 " 1-7-3 = 1.522879 Log. 2240 -f- 16 = 5. 445 632 Log. volume 7.912045 Log. weight = 6.781 578 Log. 7.912 045 = 81 666 667 cube feet " 6.781578= 6.047 528 tows. Centrifugal Pumps. Morris Machine Worlcs, Baldwinsville, N. T. Centrifugal Pumps. Are simple in construction, and for the raising of great vol- umes of water, to or from a low elevation, are superior to a piston pump, in their dispensing of valves, piston and its packing, etc. ; and as a result they will effectively raise gravel, sand, paper pulp, sewage, silt, and like material : all of which a piston pump is wholly impractica- ble of raising. The efficiency of one when properly propor- tioned and constructed is fully 65 per cent, of the power expended to operate it, and they are also applicable to furnish surface-water for the condensers of marine and other engines, and brine in ice -making machines, in dredging, wrecking, etc., etc., at a less cost of operation than any other pump of like capacity. Standard Horizontal Pump. No. Capacity per Minute. Minimum Power for each Foot of Lift. Diameter and Face of Pulley. Weight. Galls. IP. Ina. Lbs. J-5 50 to 70 .024 6X 6 168 '75 75 to loo 037 7X 8 232 2 no to 150 054 8X 8 306 2-5 175 to 250 .086 8X 8 348 3 2 5 35 .124 8X 8 400 4 450 .0 600 .223 10 X 10 545 5 750 to 900 372 15 X 12 826 6 i xx) to T 400 .496 15 X 12 965 8 I 700 tO 2 2OO .844 20 X 12 i 500 10 2 2OO tO 3 OOO 1.093 24 X 12 2170 12 3 ooo to 4 ooo 1.49 30X14 3050 15 4 800 to 6 ooo 2.38 40 X 15 7 zoo *i5 4 800 to 6 ooo 2.38 30X15 3i5o 18 7 500 to 10 ooo 3-73 40 X 15 9000 *i8 7 500 to loooo 3-73 30 X 16 350 22 12000 tO 14000 5-96 48 X 20 12000 * Refers to low-lift pump. The number of pump is also diameter of discharge opening in inches. Where more than 25 feet of discharge pipe is attached to pump, one or two sizes larger than the pump discharge is recommended. Railway. Speed. Chicago, Burlington & Quincy, passenger train, Denver to Eckley, 14.8 miles in 9 minutes = 98. 66 miles per hour. IO32 ASBESTOS FELTINGS, CEMENTS, ETC. Asbestos Fabrics, Felts, Cements, Locomotive Lagging, 1C to. H. W. Johns ManvUle Co.. New York. Steam -Pipe and. Boiler Coverings, Paelzings, Etc. In the protection of surfaces from loss of heat, Hair-Felt and other organic materials, in consequence of their destructibility at high temperatures, have been very generally superseded by Felts made from Asbestos. Numerous tests of the relative non-conductivity of materials published by au- thorities have given an impression that Asbestos is an inferior non conductor of heat. This, however, is an error, as these tests are made with the dense or crude forms of Asbestos, while in its fibrous state it contains numerous air cells. The best insulator known is air confined in minute cells, so that heat cannot be re- moved by convection, and the value of insulating substances depends upon the power of holding minute volumes of air in a manner that precludes circulation. Asbestos Fibrous Fabrics are claimed, therefore, to be the very best and most durable non-conductors of heat. Asbestos Fire-Felt Is a fabric "felted" from Asbestos fibres. As its air-cells are innumerable and microscopic in size, Fire-Felt is a successful application of the air-space principle. In addition to its superior insulating properties, it is fire-proof, flexible, light in weight, susceptible of any desired mechanical arrangement, and indestructible. It is particularly adapted for Marine, Mine, and Railway work, as moisture and vibra- tion will not disintegrate it, and it will withstand much rough usage. It is sup- plied in cylindrical sections for pipes, in sheets for boilers, drums, flues, etc., in rolls for grouped pipes, cylinders, hot-air pipes, etc., and in blocks for Locomotive and Boiler Lagging, etc. Asbestos Fire-Felt, Asbesto-Sponge Felted, and As- besto- Sponge Molded Sectional Pipe Covering. These are formed into cylinders, cut lengthwise, in order that they may be laid over pipes, and are furnished with a canvas jacket, secured by metal bands. They are suitable for both high and low steam pressures. The Fire-Felt, being com- posed wholly of Asbestos, is especially adapted for highest pressures and super- heated steam. Champion, Zero, Brine, and Ammonia Sectional Pipe Covering. "Champion" is an economical covering for low-pressure steam and hot- water pipes. "Zero" effectually prevents water and gas in pipes from freezing. Brine and Ammonia Pipe Coverings prevent the formation of ice on the line of pipe, and produce important economies in refrigerating and ice plants. Asbestos Cement Felting. Composed of Asbestos fibre, infusorial earth, and a cementing compound, ap- plied to pipes, boilers, etc., while heated. Furnished in bags or barrels. One bag contains sufficient material to cover about 40 square feet of surface i in. in thickness, and weighs about 120 Ibs. net. A,sbestos Lagging for Locomotives. Composed wholly of pure Asbestos, suitable for all styles of locomotives. In slabs, 6 ins. in width by 36 ins. in length, from .5 in. to 2 ins. in thickness. Asbesto-Sponge Hair-Felt Is very elastic, and, in consequence of the large proportion of Asbestos in it, it is not liable to injury from steam heat. In rolls of about 300 square feet, 6 ft. in width, and .375 in. in thickness. ASBESTOS FELTINGS, CEMENTS, ETC. 1033 Hair-Felt. Of various thicknesses. In bales of 300 square feet, 72 ins. in width. Asbestos Cloth. Pure fibres of Asbestos spun into threads and woven into cloths. Produced in various weights and widths. Fine cloth, 36 ins. in width, weighs 3.33 oz. per square foot. Medium cloth, 36 ins in width, weighs 4.66 oz. per square foot. Heavy cloth, 36 ins. in width, weighs 6.25 oz. per square foot. Asbestos Packings and Asbesto-Metallio Packings. These are especially adapted for the extreme high-pressure and high-speed en- gines of modern times. They are supplied in flat, round, and special shapes to meet all requirements. Asbestos Mill-Board. Composed of pure Asbestos fibres. Valuable for sheet packing and general joint-work, for gas fire-backs, screens, partitions, and general fire-proofing pur- In sheets 40 by 40 ins., from .03125 to .5 in. in thickness. "Asbestos Non-Burn Paper or Building Felt." Composed of pure Asbestos fibres. Used as fire-proof lining between floors, side- walls, etc., of frame and other structures ; also for railroad-car partitions. It is vermin, acid, and fire proof, and is also made damp-proof. Supplied in rolls weighing about 80 Ibs., 36 ins. in width. Three weights thin, medium, and heavy. Nickel Steel and Shafting. Nickel Steel is well adapted for shafting, as it has greater elasticity and tensile strength than steel, it being fully 30 per cent, greater, the latter being 20 per cent. With 4.7 per cent, of nickel in the composition of steel, the elastic strength has been increased from 36000 to 41000 Ibs. per Q inch, and the transverse from 67 ooo to 89 ooo Ibs. Electrical Expressions and. Equivalents. Rate of Operation. One Watt. i Ampere per sec. at one volt. 7373 foot-lbs. per sec. 44.238 " " min. 2654.28 " *' hour .50:* mile-lbs. " .00134 = -^ 'ff. One tP. 550 foot-lbs. per sec. 33000 " " min. 375 mile-lbs. " hoir 746 Watts .746 Kilowatt One Kilo-watt. 737.3 foot-lbs. per sec. ^yf- " " min. 502.7 " " hour -34 H. Quantity of Operation. One Watt-Hour. 2654.28 foot-lbs. .503 mile-lbs. i ampere - hour X one volt. IP-hour. 74 6 Quantity of Operation. One IP-Hour. 1980000. foot Ibs. 375. mile-lbs. 746. Watt-hour. .746 Kilowatt-hour Quantity of Current. ' One Ampere- Hour. One Ampere flowing for one hour, irrespective ol the voltage. Watt hour-:- volts. Force Moving in a Circle. Torq.ue. One pounj at a radius of one foot. 1034 CHAINS. Chains. ITor Cables, Cranes, etc., of AVrought Iron or Steel. Cable chains are designated as Open or Stud link, and Crane, Sus- pension, and Hauling as Short or Open link. Tensile strength of fibrous Wrought Iron and of soft Steel is assumed at 56 ooo Ibs. per square inch. Short-link.* The average ultimate tensile strength of the link of a chain is ascertained to be 1.625 times that of the rod or bar from which it is forged, and to avoid injury to a chain in testing it should not be subjected to a stress in excess of one half of its tensile strength ; nor, in consequence of the disastrous results of the rupture of a cable or crane chain, should it be subjected in operation to a stress in excess of one half of its testing strength. The average tensile strength of i inch round and chain-rolled wrought iron and steel, is further assumed at 44 ooo Ibs., and a link of such chain at 71 500 Ibs., or 1.625 times greater than that of a rod or bar. Hence, a chain of i inch may be tested to 35 750 Ibs., f and submitted to a working stress of 17 875 Ibs. The Pencoyd Iron Works gives 17920 Ibs., the Pennsylvania Railroad 15000, and Molesworth and D. K. Clark, both of England, give respectively 15 680 and 13 440 Ibs. When the lead of a chain is inclined to the stress, as when it encompasses a weight in mass, or is applied to cant-hooks, the greater the angle, the greater the stress with a given load, as the stress on each chain will be in the same ratio to half the load that the length of one half or side of the chain bears to the vertical distance in a line between the point of suspension of the chain and the load. Thus, Multiply half the load by the length of one lead of the chain, divide the product by the vertical distance and the result will give the capacity of the chain. Or, X sec. .5 angle of spread of the chain = Stress. ILLUSTRATION. Assume the load to be i ooo Ibs., the length of one side of the chain to be 5 ins., the vertical distance 4 ins., and the spread of the chains 6 ins. Then i ooo -4-2X5-^-4 = 625 Ibs. on each chain. Stud-link. Authorities on the relative strength of this and a short or open link are very materially divided. D. K. Clark gives the safe working load of the two links approximately as: Short-link D* -f- 10.7, Stud-link D -f- 7.07. An excess of strength for the stud link. D representing diameter of rod in eighths of an inch. Again, he gives the ultimate safe working stress of a stud-link chain at 9 tons (20 160 Ibs.) and of a short-link at 6 tons (13440 Ibs.). This is wholly at variance with the preceding rule for the determination of the stress on a chain, when it is spread or diverging outward from the vertical; for as the stress of the load increases in the proportion that the length of one lead of a chain is to the vertical distance, as here illustrated, the length of the stud not only spreads the span of the link, but subjects it to a severe combined tensile and transverse stress on its outer surface, in the direction of the central line of the stud, as the tensile and transverse strength of wrought iron or steel being less than that of their crushing strength, the neutral axis is lowered and the stress on the outer surface correspondingly increased. Chaining over Inclined Surface. 1 1 cosin. A = l. L representing length of line on surface, A angle of inclination, and I length of line reduced to the horizontal. * Crane chains are usually of this consiruction. fThe table on p. 457 is for English iron and is for 31 360 Ibs. APPENDIX. 1035 Hydraulics of* a Fire - Engine. With a ring nozzle the Coefficient of discharge is about .74. Loss of Pressure by Friction in Hose. Loss of head varied as the square of the velocity ot the flow and nearly as the length of the hose. The effect of a difference in diameter of hose, even of .125 inch for 2.5 ins., may cause a loss of 25 per cent. Wind. Pressure. Normal pressure is estimated at 15 Ibs. per sq. foot and maximum at 30 Ibs. ; but on elevated structures, in consequence of the partial vacuum or minus pressure in their rear, the effect of the wind is much increased. At the summit of the Eiffel Tower, 1097 feet, the pressure has been observed to be 5 times that at the Central Meteorological Bureau, at its height of 70 feet be- low. R. Kohfahl. From observations of the St. Louis tornado in 1896, the pressure was computed to vary from 45 to 90 Ibs. per sq. foot; and from experiments of C. F. Martin at Mt. Washington, U. S., it was shown that rapid and intense fluctuation occurs ; and by Kernot, that a marked difference results from the presence of other buildings. T. Bates. Effect of a Low Temperature on Iron and Steel. In Tons per Sq. Inch. Metal, Temperature. Elastic Limit. Breaking Stress. Teneile Strength. Ratio of Elastic and Breaking. Wrought- ( Iron Bar. ) - 18.2 18.5 25.2 26.5 100 .72 7 ( '9-3 27.8 107-5 7 Steel, ( 64 15.2 25-7 100 59 Siemens' ] -4 15.7 27.7 102.9 Angle. ( 112 18.9 28.7 123.8 .66 Malleable ( Iron Bar, j -Jo 19.6 20 25.5 26.4 100 102.3 $ ( 112 20.3 27.4 103.2 74 The conclusions from these results are: 1. Elastic and ultimate limits are raised by low temperatures. 2. The variation in mechanical properties by a reduction of temperature is great- er in steel and least in malleable iron. The variations between the extremes at temperatures given are, in per cent. : Metal, Elastic Limit. Increase. Tensile Strength. Increase. Elongation. Decrease. Siemens' steel 2^ 8 Wrought-iron bar Malleable iron 5-4 3-2 7-5 7-5 14.1 5-2 3. The compression by impact diminishes with a reduction of temperature, in like manner with elongation under tension. 4. The loss of malleability is 8 per cent, at 4, and 23 at 112. The change being least in hammered iron, and greatest in rivet iron. 5. The flexibility of iron is slightly changed in soft rivet at 4, and in rolled bar iron at 112; but all other qualities were more or less injuriously affected at the lowest temperature. M. Rudeloff. Relative Hardening of* Cement and Mortar in Fresh, and Salt Water. From experiments with cement with varying proportions of sand, it was shown that, when it was mixed with and submitted to fresh water, it became harder. N. M. Koning and L. Bienfort. Evaporative I?o-wers of* Coke and Coal. From experiments at Colmar. The calorific values were i and .8933.^, Weber. 1036 APPENDIX. The Hiightest Known. Su.tostari.oe Is the pith of the Sunflower; its specific gravity .028. Elder pith, hitherto held to be the lightest, is .09 ; Reindeer's hair .1, and Cork .24. Hence, Reindeer's hair has a buoyancy of i to 10, and Sunflower pith i to ^.Froitzheitn. Ratio between Surface arid. Mean Velocity of the Wet Section of* a 3Vtill-Raoe, As determined by a series of experiments, is .60 to .65, being less than that of .80, usually taken, and that of .71 to .72 in channels with earth banks. R. P. T- Tutein Northenius. Testing of* Stones. Granite, Marble, and Sandstone lose strength by saturation with water, and Sandstone and Granite are most aflected by frosts __ M. Gary. Resistance of Wronght-Iron. and Steel Rivets in. a Lap of not I^ess than. Three. Per Sq. Inch. Elasticity per sq. inc^h. How Made Tempera- ture. Resistance to Shearing. Elasticity per sq. inch. How Made. Tempera- ture. Resistance to Shearing. Lbs. Lbs. Lbs. Lbs. IRON. STEEL. . 25536 31360 Hand. ) Bright red heat r 5800* i 6720 31360 32704 Hand.) Bright red heat. 6384 7168 25 536 } Hydrau- White j 7 i68 31 360 Hydrau- White j 8512 31 360 J lic. heat, j 8288 3 2 74 lic. heat. \ 9408 The Iron submitted to an extension of 12 per cent, before fracture, and the Steel 18 per cent.Dupuy. IVEuzzle Velocity- of the G-errnan. Infantry Rifle. A series of experiments gave the following results : Muzzle velocity 2070 feet per second, and the maximum at 10 feet from the muz- zle 2130 feet. InsVn. C. E. Effect of a Diamond- Edged Saw. Result of its operation at the Paris Exposition. In semi -hard and soft stone, n.8 ins. per minute. Cost a cents per sq. foot; cost by hand sawing. 15 cents. /. Laftargue. Forced Draught. For non-caking coal it is necessary to reduce the width of the air space between the grates of a furnace to .125 inch. The greater the force of the blast, the less is the evaporative effect of the fuel, as illustrated in a steam-boiler plant, where the evaporation with natural draught was from 7 to 8 Ibs. of water per Ib. of coal; it fell to 4 Ibs. upon the introduction of a blast draught, and although there was a length of flue of 400 feet, and a chimney 130 feet in height, flame was generated at the top of the chimney, evidencing that carbonic oxide left the furnace unconsumed. D. K. C. Efficiency of liand Uralses. From a series of experiments on the tender of a locomotive on the Northern Railway of France, it was deduced that the frictional resistance absorbed 82.3 per cent of the power applied. In general it is assumed that the efficacy of hand brakes does not exceed 20 pe* cent. D. K. C. A.cetylene Formula. C 2 H 2 , and a Specific Gravity .91. M. Hempel. APPENDIX. Effect of Kiln IDrying on. Pine and. Hemlock. White Pine. Weight of a cube foot, 36.4 Ibs. ; dried at 212, 22 Ibs. RedPine.~ 32.3 Ibs. ; at 212, 31 Ibs. Hemlock. 53 Ibs. ; at 212, 31.3 Ibs. Prof. H. T. Bovey, LL.D. Test of an Iron \Vire Rope 3.S ins. in Diameter. Construction. Six strands on a core of hemp, and each of six other strands on a central core containing 108 wires, .058 inch in diameter. Tensile strength of wire 260 ooo Ibs. per sq. inch, and united strength of all 740 ooo Ibs. Reduction of di- ameter with a stress ot 150 tons 1.4 inch, and ultimate strength 560000 Ibs., a coeffi- cient of 75 percent. A. Martens. Consolidation of Loose or Made Q-ronnd May be successfully attained by the driving of piles as close together as the earth will admit withdrawing of them, and filling their holes with a weak concrete. In an instance recited, 8 piles, 30 inches apart, driven to a depth of 23 feet in made ground, supported a brick chimney 213 feet in height on a base of 43 feet square. Hoffman. For the foundation of the buildings of the Paris Exposition, the ground was rammed by conical and suitable monkeys from a pile-driver, and the holes filled with hard substances rammed down. Dular. A. New <3-eneral Formula for Train Resistance. 4-f-S t.2-\ -- ~=J = R. R representing resistance in Ibs. per ton (2000), S ve- locity in miles per hour, and T weight of train in tons.H. L. J. Resistance of Fast Passenger Trains on Straight Road. Experiments on the Northern Railway of France at velocities varying 15 to 35 miles per hour. i. 45 -f- . ooo8V a = R. De Laboriette. M! agnail vi in. A new alloy of aluminum. Spec, gravity, 2-3; Melting-point, noo ; Tensile strength with 5 per cent, of magnesium, 30000 Ibs., and, with an addition of from 5 to 20 per cent, of it, the alloy becomes similar to brass and bronze, and, with 50 per cent, it loses its hardness and ceases to be useful for mechanical purposes; but as it is capable of receiving a very high polish, it is eminently suited for op- tical and like instruments. Mielke. IMaximite. its susceptibil- of charge of projectiles, viz., insensibility to heat and shock. To test its suscepti ity to chemical change, it is maintained at a temperature of 165 for a period 15 minutes. as ugh a it ex- , 00 ragmens, an a r 2 -nc se, smary exploded, burst into 7000 fragments. It freezes below the boiling-point of water, and possesses the advantage of expanding in passing from a fluid to a solid. A. P. H. Cylinder Ratios for Compound and Triple Expansion Steam - Engines. By experiments of Mr. Greacen. of Perth Amboy, N. J., under different initial pressures and the bushing of the high-pressure cylinder, he determined the ratios to be: With 60 Ibs. pressure per sq. inch, i to 4; with 85 Ibs., i to 5; with no Ibs., i to 6; with 135 Ibs., i to 7; and with 160 Ibs., i to 8. B. C. Ball. 1038 APPENDIX. B el t - D ri vi ng. Belts for high speed, running over 4000 feet per minute, should be of single, thin, pliable, and tough leather; and if singly compounded, they may be run at 9000 feet per minute, with less loss from slipping. Narrow pulleys are more effective than wide; thus, two belts of 20 ins. will transmit more power than one of 40 ins. over a wide pulley. Great convexity of pulley increases wear of belt, and induces loss of power. .0625 inch in convexity is sufficient for a pulley of 6 ins., and a less driven pulley may be flat on its face. /. Tullis. Texxi.peratu.re in IVlines. From observations made in Australia, the mean result in rock was an increase of i to each 137 feet f descent. /. Sterling. At Lake Superior, U. S., at 105 feet, 59, and at 4580 feet, 79, a difference of i for 223.7 feet - At st - Gothard Tunnel it was j for 60 feet. A. Agassiz. Relative Efficiency of a Reciprocating Piston Pump, a Rotary ifump, and. a Steam. Sipnon. Water raised to an Elevation of 17.66 feet, and Pounds oj Water raised per Pound of Steam. Reciprocating Pump, 135.6 Ibs. ; Rotary, 108.6 Ibs. ; and Steam Siphon (Giffard's), 37.4 Ibs. B. F. Isherwood, U. S. N. Tenacity of Nails and Drift Bolts. Experiments made at Sibley College furnish the following results : Cut Nails are superior to Wire in all positions ; and, as the pointing of a nail in- creases its efficiency, the pointing of a cut nail would increase its tenacity about 30 per cent. Barbing decreases their tenacity about 32 per cent. Wire Nails. Their tenacity decreases with time of service. Surface of a nail should be slightly rough. Nails should be wedge-shaped in both directions, where there are not special dangers of the splitting of the wood. Nails are 50 per cent, more effective when driven perpendicular to the grain of the wood than with it, and most effective when driven perpendicular to the surface, and when submitted to impact they hold less than .083 the stress they can withstand when it is grad- ually applied. Drift Bolts, when round, are superior to square, and the holes into which they are to be driven should be respectively, .8125 and .875 of their diameter. Relative Tenacity of Woods. White pine, i; Basswood, 1.2; Yellow pine, 1.5; Chestnut, 1.6; Elm and Sycamore, 2; Beech, 3.2; and White oak, 3. F. W. Clay. Lubrication of Metal Bearings. From results of extended experiments on the Paris- Lyons- Mediterranean Railway, extending from 1871 to 1890, it was shown that: Lubricating Wicks of Wool have a delivery of oil over that of cotton of from 50 to ioo per cent. ; that 'their renewals were but as 68 to 100 of the cotton, and that they were less liable to firing. Bearings. The wear of White Metal was 50 per cent, less than that of Bronze, and bearings of it diminished the resistance of trains of 300 tons, running from 16 to 26 miles per hour, 20 per cent. ; but as the speed was increased, this gain was diminished, but it remained always at 5 per cent. E. Chabal. Execution of Masonry or Brickwork during Severe Frost. A committee of the Austrian Union of Engineers and Architects, after extended experiments, submitted that Portland Cement with 7 per cent, of common salt in cold water, and the stone or brick dry, was the most effective, and that Lime mortar was useless. Alfred Greil APPENDIX. 1039 Talole for Reducing Observed Daily Variation of Needle to Mean Variation of the Day. U. S. Coast and Geodetic Survey, 1878. SEASON. Nee A.M. iloEa netic A.M. at of Mean 2 Meridian. A.M. A.M. iag- A.M. A.M. Need NOON. eWei P.M. it of IV Meridi P.M. [ean& an. P.M. [agnet P.M. ic P.M. P M, Spring 6 / 3 4 2 I 7 4 5 3 i 8 / 4 5 3 2 9 t 3 4 2 2 IO f I I I A. II t I 2 2 Noon. A. I h. 2 h. 3 A. 4 h. 5 A. 6 4 4 3 2 5 6 4 3 5 5 3 5 4 4 2 2 3 3 i i 2 2 I I i i Summer Autumn Winter... Elevations, at Various Locations, of Bench-Mar Its .AJoove Mean Level of the Ocean at Sandy Hook, N. J. See Treasury Annual Report for 1899 of Superintendent U. S. Coast and Geodetic Survey. Location. Elevation in Metres. Bench-Marks. Location. Elevation in Metres. Bench- Marks. Albany, N. Y... Alex'dria B.,N.Y. Altoona, Pa 5-013 78.9705 354-0357 1.268 6.7175 20.0484 162.0168 96.8627 187.595 181.4580 180.8124 168.4273 1.822.9081 95-9903 190.0727 202.0164 226.5461 163.3446 27.8699 1.565.1693 183.2189 184.589 191 .0642 110.8262 209.0303 98.6150 l3 6 -399 3.6951 2.6313 112.0816 97.5692 58.5092 120.8052 191.4484 225.6832 227 . 5684 198.0135 BM 2 PBMA PRRNo.i24 a Ii A PBM 12 PBM i C BMi BMIX City BM PBM 7 PBM 182 City BM TBM 176 No. 215 N 2 PBM 275 BMI. (USE) PBM 8 H 685 No. V TBM 87 T L PRR No. i BM i No. XXIII TBM 197 TBM 484 BM 245 PBM 194 Lake Michigan, 111. . Leavenw'th, Kan. . . Little Rock, Ark... Memphis, Tenn Minneapolis, Minn. . Mobile, Ala 179.1269 238.4976 71-9737 80.5465 256.1462 3.7448 24.9946 6.8144 1.6034 4.59 2 7 2.2991 299.5705 80.3769 76.7016 248.2064 187.718 18-5763 196.0783 11.7284 7.7678 10.8961 .2659 253.9426 129.9745 216.854 7L5558 1780.1063 337.5317 183-5342 271.0406 131.7546 126.3358 3L3525 132.024 2.936 1013.8567 PBM 100 PBM 251 BMi PBM TBM 13 A BMPolka S H'way H. E.42d St. No. 5 PBM 345 BMA BMA 8 'o 5A SGS PBM 231 E City BM No. XIII TBM PBM 284 PBM 14 TBM 193 BM 16 Ov PBM 394 Pt. -Office M BM 44 No. XL PBM i A 3 BMCNo.4 M 2 Annapolis, Md... Biloxi, Miss Brooklyn, N. Y. . . Burlington, Iowa. Cairo, 111 Cheyenne, Wyo. . Chicago, 111 Cincinnati, O Colorado Sp's, Col. Columbus, Ky Cumberland, Md. Dakota, Minn Dayton, O Natchez, Miss Newport News, Va.. New Orleans, La. . . New York, N.Y.... Omaha Neb Ontario, Pt. Dal, Can. Oswego, N. Y Owego, " Parkersburg, W.Va. Perth Amboy, N. J. Pr'rieduChien,Wis. Red Bank, N. J Richmond, Va Round Brook, N. J. St. Augustine, Fla. St. Joseph, Mo St. Louis, Mo St. Paul, Minn Schenectady, N. Y. . Sedalia Col Decatur, Ala Delta, La Denver (nr.), Col.. Detroit, Mich.... Dubuque, Iowa... Duluth, Minn Easton, Penn .... Erie, Pa ... Fort Jefferson, Ky Florence, Ala Gov'nor's I.,N.Y. Harrisburg, Pa... Helena, Ark. . . ". ! '. Jackson, Tenn Jefferson City, Mo. Kansas City, Kan. " Mo.. La Crosse, Wis.. . Sioux Iowa Topeka Kan .... Utica N Y Van Buren, Ark. ... Vicksburg, Miss. . . . Vincennes, Ind.... Washington, D.C... Winona, Kan For exact location and description of Bench- Marks, see Report as above, pp. 548-549. IO4O APPENDIX. Insulation of Steam Boilers and IMpes. From the experiments of several parties in England, St. Petersburg, and Canada, the following results were obtained: With steam at pressures ranging from 3 to 150 Ibs. per sq. inch, and averaging 75 Ibs., the condensation of steam in pipes, per sq. foot per hour, was: Uncovered, .60 Ibs. ; with mica insulation, with steam from 47 to 244 Ibs., averaging 150 Ibs., .143 Ibs. With steam at 150 Ibs. permanent in the pipes, it was computed that each sq. foot of uncovered surface involved an annual loss of $2.11; with ordinary and good bagging, 55 cents; and with mica insulation, 28 cents. From experiments on the Canadian Pacific Railway, the rate of cooling of water- tanks from the boiling point in 5 hours, the loss of temperature varying from 84 in the uncovered to 20 in the one covered with mica; and from other experiments on the Grand Junction Railway, with 5 locomotives, with steam at from 140 to 150 Ibs. pressure per sq. inch, the observed effects, after the fires were drawn, were: The uncovered boiler lost 56 Ibs. pressure in one hour, while the covered lost, re- spectively, 24, 20, 13, and 6 Ibs., the last with mica covering. Engineering, 1901, p. 234. Hdq.uid Fuel. From experiments made with crude Borneo oil, of the composition: Carbon, 87.9 per cent; hydrogen, 10.78; oxygen, 1.24; flash-point, 211; boiling-point, 395; and caloric value, 18.831 B. T. U. The constituents of fuel oils give off vapor at temperatures from 100 up to boil- ing-point of the oil, near which point a residuum of dense carbon is precipitated, tending to choke pipes and to accumulate in the furnace. The following methods of using it are: t. Injecting it into the furnace under pressure, as spray; 2. Spraying it by air or steam ; 3. Vaporizing it. The evaporative efficiency under the first was 12 Ibs. water from ; and at 212 per Ib. of oil, an excess of air and a large furnace being required for combustion. Under the second, 13 to 14 Ibs., less air being required. Under the third, 15 to 16 Ibs., a minimum of air being required. The conclusions deduced were: ist. A reduction in consumption of fuel with the oil, compared with coal, of about 40 per cent. 2d. A reduction in bunker space of about 15 per cent, for equal weights of fuel. 3d. A reduction in furnace labor of at least 50 per cent. The oil should be filtered before being used. E. L. Orde. Effects of Repeated Stress on tlie Strength of Wrought Iron. From Tests Made on a Bridge that had been 24 Years in Service. The maximum stress being 6.64 tons per square inch, no reduction in strength or durability from its service was observed. Zimmermann. Safe Static .Load on Ordinary Foundations. In Tons per Square Foot. Alluvial soil 5 Clay and sand, moist 1.33 Clay, hard and dry 1.5 to 3 Earth, firm x to 1.5 Sand, sharp and clean i to 1.5 Gravel, dry 2. 25 Stones, broken and concrete. . 3 Aide Memoir e and Rankine APPENDIX. IO4I Air-I?ixmps of Condensing Steam-Eiigines. Are more effective when operated independent of the engine, in consequence of possessing the advantage of their operation being varied to meet their require- ments; and vertical single-acting, at a velocity of piston not exceeding 400 feet per minute, are more effective than double-acting. The required dimensions and resulting capacity of full flowing pumps may be computed from the table of H. R. Worthington on p. 738. A displacement of pump cylinder of one-fifth of a cube foot per minute is held to be proper for one IIP. Effect of the TJse of Oil or Tallow in a Steam-Boiler. Oil or grease introduced in a steam boiler, combining with alluvial or calcareous sediments from the feed water, if not held in suspension by rapid circulation over the heated surfaces, as the crowns of the furnaces and tubes, and withdrawn by pump or blown out as it subsides, will settle upon the upper surfaces of the fur- nace and tubes, involving the burning of the metal and their consequent disruption under pressure. Stress on Trussed. Beam or Rods. In addition to pp. 621-623, 823. King Truss. To Compute Stress on Beam. . *" "" "'*: ~~,..-> _^ gec t -_ _ cosec t s. W repre- 4 i j I \ senting weight uniformly distributed, I length ' of beam between supports, d depth of truss, both in feet, and i angle. ILLUSTRATION. Assume W = 2ooo Ibs., Zc= 20 , d z.o^ feet. t= xi 42', and cotan. 1 = 5.1. 2000X20 2000 8x = 2 495 Ibs., = X 5 i = 2 550 & To Compute Stress on Rods. wi . w w v - - sec. i = cosec. t. In absence of angle put . Bd 4 4 2 d ILLUSTRATION. Assume as preceding. Sec. t = 1.02, cosec. i = 5.1. X 1.02 = 2 445.4 Ibs., = X = 5- = 2 550 Ibs. o x 2.03 4 To Compute Stress on Centre. ILLUSTRATION. - W= - x 2000= i 250 Ibs. 8 8 Queen Truss. To Compute Stress on Beam. / . -yy j ,, sec. t = | W. cotan. t, c .-= 6.67 feet. o d 8 -^i _ A_L^-^ r>~ ILLUSTRATION. Assume as preceding. 1 d= 2 feet, and angle i=i6 42', sec. i = 1.044, and cotan. i = 3.42. I044=26loW , s . ja To Compute Stress on Rods. ILLUSTRATION. Assume as preceding, cosec. i = 3.56. 2000 X2 X 1.044^2610^., and 3 8X2 _ 3 W In absence of angle put ^- e 3 a To Compute Stress on Centre. IO42 ORTHOGRAPHY OF TECHNICAL WORDS AND TERMS. ORTHOGRAPHY OF TECHNICAL WORDS AND TERMS. Orthography in ordinary use of following words and terms is so varied, that they are here given for the purpose of aiding in the establishment oi d uniformity of expression. Abut. To meet, to adjoin to at the end, to border upon. Abut end of a log, etc., is that having the greatest diameter or side. But and Butt end, when applied in this manner, are corruptions. Adit, in Mining, the opening into a mine. Amidships. The middle or centre of a vessel, either fore and aft or athwartships. The amidship frame of a vessel is at g& and is termed dead flat. Arabesque. Applied to painted and carved or sculptured ornaments of imaginary foliage and animals, in which there are no perfect figures of either. Synonymous with Moresque. Arbor. The principal axis or spindle of a machine of revolution Arris. A term in Mechanics, the line in which the two straight or curved sur- faces of a body, forming an exterior angle, meet each other. The edges of a body, as a brick, are arrises. Ashlar. In Masonry, stones roughly squared, or when faced. Athwart. Across, from side to side, transverse, across the line of a vessel's sourse. Athwartships, reaching across a vessel, from side to side. Bagasse. Sugar-cane in its crushed state, as delivered from the rollers of a mill. Balk. In Carpentry, a piece of timber from 4 to 10 ins. square. Baluster. A small column or pilaster ; a collection of them, joined by a rail, forms a balustrade. Banister is a corruption of balustrade. Bark. A ship without a mizzen-topsail, and formerly a small ship. Bateau. A light boat, with great length proportionate to its beam, and wider at its centre than at its ends. Batten. In Carpentry, a piece of wood from i to 2. 5 ins. thick, and from i to 7 ins. in breadth. When less than 6 feet in length, it is termed a deal-end. Berme. In Fortifications and Engineering, a space of ground between a rampart and a moat or fosse, to arrest the ruins of a rampart. The level top of the embank- ment of a canal, opposite to and alike to the towpath. Bevel. A term for a plane having any other angle than 45 or 90. Binnacle. The case in which the compass, or compasses (when two are used), ia set on board of a vessel. Bit. The part of a bridle which is put into an animal's mouth. In Carpentry, a boring instrument. Bitter End. The inboard end of a vessel's cable abaft the bitts. Bitts. A vertical frame upon a deck of a vessel, around or upon which is secured cables, hawsers, sheets, etc. Bogie. Pivoted truck, to ease the running of an engine or car around a curve. Boomkin. A short spar projecting from the bow or quarter of a vessel, to extend the tack of a sail to windward. Bowlder. A stone rounded by natural attrition ; a rounded mass of rock trans- ported from its original bed. Buhr-stone. A stone which IB nearly pure silex, full of pores and cavities, and used for Mill* Bunting. Woolen texture of which colors and flags are made. Burden. A load. The quantity that a ship will carry. Hence burdensome. Cog. A small cask, differing from a barrel only in size. Commonly written Keg. ORTHOGRAPHY OF TECHNICAL WORDS AND TERMS. 1 043 Caliber. An instrument with semi-circular legs, to measure diameters of spheres, or exterior and interior diameters of cylinders, bores, etc. A pair of Calibtrs is superfluous and improper. Calk. To stop seams and pay them with pitch, etc. To point an iron shoe so as to prevent its slipping. Cam. An irregular curved instrument, having its axis eccentric to the shaft upon which it is fixed. Camber. To camber is to cut a beam or mold a structure archwise, as deck- beams of a vessel. Camboose. The stove or range in which the cooking in a vessel is effected. The cooking- room of a vessel; this term is usually confined to merchant vessels: in vessels of war it is termed Galley. Camel. In Engineering, a decked vessel, having great stability, designed for use in the lifting of sunken vessels or structures. Also to transport loads of great weight or bulk. A Scow is open decked. Cantle. A fragment; a piece; the raised portion of the hind part of a saddle. Cantline. The space between the sides of two casks stowed aside of each other. When a cask is laid in the cantline of two others, it is said to be stowed bilge and cantline. Capstan. A vertical windlass. Caravel. A small vessel (of 25 or 30 tons' burden) used upon the coast of France in herring fisheries. Carlings. Pieces of timber set fore and aft from the deck beams of a vessel, to receive the ends of the ledges in framing a deck. Carvel built. A term applied to the manner of construction of small boats, to signify that the edges of their bottom planks are laid to each other like to the matt- ner of planking vessels. Opposed to the term Clincher. Caster. A small phial or bottle for the table. Casters. Small wheels placed upon the legs of tables, etc., to allow them to be moved with facility. Catamaran. A small raft of logs, usually consisting of three, the centre one be- ing longer and wider than the others, and designed for use in an open roadstead and upon a sea-coast. Chamfer. A slope, groove, or small gutter cut in wood, metal, or stone. Chapetting. Wearing a ship around without bracing her fore yards. Chimney. The flue of a fireplace or furnace, constructed of masonry in houses and furnaces, and of metal, as in a steam boiler. See Pipe. Ckinse. To chinse is to calk slightly with a knife or chisel. Chock. In Naval Architecture, small pieces of wood used to make good any de- ficiency in a piece of timber, frame, etc. See Furrings. Choke. To stop, to obstruct, to block up, to hinder, etc. Cleats. Pieces of wood or metal of various shapes, according to their uses, either to belay ropes upon, to resist or support weights or strains, as sheet, shoar, beam cleats, etc. Clincher built. A term applied to the construction of vessels' bottoms, when the lower edges of the planks overlay the next under them. Coak. A cylinder, cube, or triangle of hard wood let into the ends or faces of two pieces of timber to be secured together. The metallic eyes in a sheave through which the pin runs. In Naval Architecture, the oblong ridges banded on the mastfl of ships. Coamingt. Raised borders around the edges of hatches. Coble. A small fishing boat Cocoon. The case which certain insects make for a covering during *he period of their metamorphosis to the pupa state. IO44 ORTHOGRAPHY OF TECHNICAL WORDS AND TERMS. Cog. In Mechanics, a short piece of wood or other material let into the faces of a body to impart motion to another. A term applied to a tooth in a wheel when it is made of a different material than that of the wheel. In Mining, an intrusion of matter into fissures of rocks, as when a mass of unstratitied rocks appears to be in- jected into a rent in the stratified rocks. Cogging. In Carpentry, the cutting of a piece of timber so as to leave a part alike to a cog, and the notching of the upper piece so as to conform to and receive it. Alike to indenting or tabling. Colter. The fore iron of a plough that cuts earth or sod. Compass. In Geometry, an instrument for describing circles, measuring figures, etc. A pair of Compasses is superfluous and improper. Connecting Rod. In Mechanics, the connection between a prime and secondary mover, as between the piston-rod of a steam-engine and the crank of a water- wheel or fly-wheel shaft. The term Pitman is local, and altogether inapplicable. Contrariwise. Conversely, opposite. Orotsways is a corruption. Corridor. A gallery or passage in or around a building, connected with various departments, sometimes running within a quadrangle ; it may be opened or enclosed. In Fortifications, a covert way. Cyma. A molding in a cornice. Damasquinerie. Inlaying in metal. Davit. A short boom fitted to hoist an anchor or boat. Deals. In Carpentry, the pieces of timber into which a log is cut or sawed up. Their usual thickness is 3 by 9 ins. and exceeding 6 feet in length. Improperly restricted to the wood of fir-trees. Dike. In Engineering, an embankment of greater length than breadth, imper- vious to water, and designed as a wall to a reservoir, a drain, or to resist the influx of a river or sea. Dingey (Nautical). A ship or vessel's small boat. Dock. In Marine Architecture, an enclosure in a harbor or shore of a river, for ihe reception, repair, or security of vessels or timber. It may be wholly or only partially enclosed. See Pier. When applied to a single pier or jetty, it is a misapplication. Dowel. A pin of wood or metal inserted in the edge or face of two boards or pieces, so as to secure them together. This is very similar to coaking, but is used in a diminutive sense. An illustration of it is had in the manner a cooper secures two or more pieces in the head of a cask. Draught. A representation by delineation. The depth which a vessel or any floating body sinks into water. The act of drawing. A detachment of men from the main body, etc. Ordinarily written draft. Dutchman. In Mechanics, a piece of like material with the structure, let into a slack place, to cover slack or bad work. See Shim. Edgewise. An edge put into a particular direction. Hence endwise and sidewist have similar significations with reference to an end and a side. Edgtvays is a corruption. Euphroe. A piece of wood by which the crowfoot of an awning is extended. Fault. In Mining, a break of strata, with displacement, which interrupts opera- tions. Also, fissures traversing the strata. Felloe, Felloes. The pieces of wood which form the rim of a wheel. Fetch. Length of a reservoir, pond, etc. , along which the wind may blow towards the embankment or dam. Flange. A projection from an end or from the body of an instrument, or any part composing it, for the purpose of receiving, confining, or of securing it to a suf> port or to a second piece. Flier. In Carpentry, a straight line of steps in a stairway. Frap. To bind together with a rope, as tofrap a fall, etc. ORTHOGRAPHY OF TECHNICAL WORDS AND TERMS. 1 045 Frieze. In Architecture, the part of the entablature of a column which is between the architrave and the cornice. Frustum. The part of a solid next the base, left by the removal of the top or segment. Frustrum, although used by some lexicographers, is erroneous. Furrings. Strips of timber or boards fastened to frames, joists, etc., in order to bring their faces to the required shape or level. Galeting. Putting galets into pointing-mortar or cement. Galets. Pieces of stone chipped off by the stroke of a chisel. See Spall. Galiot. A small galley built for speed, having one mast, and from 16 to 20 thwarts for rowers. A Dutch-constructed brigantine. Gate. In Mechanics, the hole through which molten metal is poured into a mold for casting. Geat aud Gett are corruptions. Gearing. A series of teeth or cogged wheels for transmitting motion. To gear a machine is to prepare to connect its parts as by an articulation. Gingle. To shake so as to produce a sharp, clattering noise, commonly Jingle. Girt. The circumference of a tree or piece of timber. Girth. The band or strap by which a saddle or burden is secured upon the back of an animal, by passing around his belly. In Printing, the bands of a press. Gnarled. Knotty. Grave. In Nautical language, to clean a vessel's bottom by burning. Graving. Burning off grass, shells, etc., from a ship's bottom. Synonymous with Breaming. Grommet. A wreath or ring of rope. Gymbal Ring. A circular rynd for the connection of the upper mill-stone to the spindle by which the stone is suspended, so that it may vibrate upon all sides. Harpings. The fore part of *,he wales of a vessel which encompass her bows, and are fastened to the stem. Cat harpings, ropes which brace in the shrouds of the lower masts of a vessel. Hogging. A term applied to the hull of a vessel when her ends drop below her centre. See Sagging. Horsing. In Naval Architecture, calking with a large maul or beetle. Jam. To press, to crowd, to wedge in. In Nautical language, to squeeze tight. Jamb. A pier; the sides of an opening in a wall. Jib. The projecting beam of a crane from which the pulleys and weight are sus- pended. A sail in a vessel. Jibe. To shift a boom-sail from one tack to another; hence Jibing, the shifting of a boom. Jigging. Washing minerals in a sieve. Keelson. The timber within a vessel laid upon the middle of the floor timbers, and exactly over the keel. When located on the floors or at the sides, it is termed a sisters or a side keelson. Kerf. Slit made by cut of a saw. Kevel. Large wooden cleats to belay hawsers and ropes to, commonly Cavil. Lacquer. A spirituous solution of lac. To varnish with lacquer. Lagan. Articles sunk in the water with a buoy attached. Laitance. A pulpy, gelatinous fluid washed from the cement of concrete depos- ited in water. Lap-sided. A term expressive of the condition of a vessel or any body when it will not float or sit upright. Lay-to. To arrest headway of a vessel, without anchoring or securing her to a buoy, etc., as by counterbracing her yards, or stopping her engine. Leaf. A trench to conduct water to or from a mill-wheel. Leech. In Nautical language, the perpendicular or slanting edge of a sail wher not secured to a spar or stay. ORTHOGRAPHY OF TECHNICAL WORDS AND TERMS. Luf. The fullest part of the bow of a vessel. Mad. A large double-headed wooden hammer. Mantle. To expand, to spread. Mantelpiece. The shelf over a fireplace in front of a chimney. Marquetry. Checkered or inlaid work in wood. Matrass. A chemical vessel with a body alike to an egg, and a tapering neck. Mattress. A quilted bed; a bed stuffed with hair, moss, etc., and quilted. Mitred. In Mechanics, cut to an angle of 45, or two pieces joined so as to make a right angle. Mizzen-mast. The aftermost mast in a three-masted vessel. Mold. In Mechanics, a matrix in which a casting is formed. A number of pieces of vellum or like substance, between which gold and silver are laid for the purpose of being beaten. Thin pieces of materials cut to curves or any required figure. In Naval Architecture, pieces of thin board cut to the lines of a vessel's timbers, etc. Fine earth, such as constitutes soil. A substance which forms upon bodies in warm and confined damp air. This orthography is by analogy, as gold, told, old, bold, cold, fold, etc. Molding. In Architecture, a projection beyond a wall, from a column, wainscot, etc, Moresque. See Arabesque. Mortise. A hole cut in any material to receive the end or tenon of another piece, Muck. A mass of dung in a moist state, or of dung and putrefied vegetable matter. Mul I ion. A vertical bar dividing the lights in a window ; the horizontal are termed transoms. Net. Clear of deductions, as net weight Newel. An upright post, around which winding stairs turn. Nigged. Stone hewed with a pick or pointed hammer instead of a chisel. Ogee. A molding with a concave and convex outline, like to an S. See Cymo and Talon. Paillasse. Masonry raised upon a floor. A bed. Pargeting. In Architecture, rough plastering, alike to that upon chimney* Parquetry. Inlaying of wood in figures. See Marquetry. Parral. The rope by which a yard is secured to a mast at its centre. Pawl. The catch which stops, or holds, or falls on to a ratchet wheel. Peek. The upper or pointed corner of a sail extended by a gaff, or a yard set ob- liquely to a mast. To peek a yard is to point it perpendicularly to a mast. Pendant. A short rope over the head of a mast for the attachment of tackles thereto; a tackle, etc. Pennant. A small pointed flag. Pier. In Marine Architecture, a mole or jetty, projecting into a river or sea, to protect vessels from the sea, or for convenience of their lading. See Dock. Erroneously termed a Dock. Pile. In Engineering, spars pointed at one end and driven into soil to support a Superstructure or holdfast. Spile is a corruption. Pipe. In Mechanics, a metallic tube. The flue of a fireplace or furnace when constructed of metal ; usually of a cylindrical form. The term or application of Stack (which refers solely to masonry) to metallic pipe is a misappli- cation. Piragua. A small vessel with two masts and two boom-sails. Commonly termed Perry-augur. Pirogue. A canoe formed from a single log, propelled by paddles or by a sail, with the aid of an outrigger. Plastering. In Architecture, covering with plaster cement or mortar upon walla or laths. In England, termed laying, if in one or two coat work; and pricking up, If in three-coat work. , . Plumber block. A bearing to receive and support the journal of a shaft* JWocre. Maata of one piece, without tope. ORTHOGRAPHY OF TECHNICAL WORDS AND TERMS. Poppets. In Naval Architecture, pieces of timber set perpendicular to a vessel's bilge ways, and extending to her bottom, to support her in launching. Porch. An arched vestibule at the entrance of a building. A vestibule supported by columns. A portico. Portico. A gallery near to the ground, the sides being open. A piazza encom- passed with arches supported by columns, where persons may walk; the roof may be flat or vaulted. Pozzuolana. A loose, porous, volcanic substance, composed of silicious, argilla- ceous, and calcareous earths and iron. Prize. In Mechanics, to raise With a lever. To pry and a pry are corruption* Proa, Flying. A narrow canoe, the outer or lee side being nearly flat. A frame- work, projecting several feet to the windward side, supports a solid bearing, in the form of a canoe. Used in the Ladrone Islands. Purlin. In Carpentry, a piece of timber laid horizontal upon the rafters of a roof, to support the covering. Bump. In Architecture, a flight of steps on a line tangential to the steps. A concave sweep connecting a higher and lower portion of a railing, wall, etc. A loping line of a surface, as an inclined platform. Rarefaction. The act or process of distending bodies, by separating their parts and rendering them more rare or porous. It is opposed to Condensation. Rebate. In Mechanics, to pare down an edge of a board or a plate for the purpose of receiving another board or plate by lapping. To lap and unite edges of boards and plates. In Xaval Architecture, the grooves in the side of the keel for receiving the garboard strake of plank. Commonly written Rabbet. Remou. Eddy water without progressive action, in bed of a river; a return of water against direction of flow of a river. Rendering. In Architecture, laying plaster or mortar upon mortar or walls. Rendered and Set refers to two coats or layers, and Rendered, floated, and Set, to three coats or layers. Reniform. Kidney-shaped. Resin. The residuum of the distillation of turpentine, itorin u a corruption. Riband. In Xaval Architecture, a long, narrow, flexible piece of timber. Rimer. A bit or boring tool for making a tapering hole. In Mechanics, to Rime in to bevel out a hole. Riming. The opening of the seams between the planks of a vessel for the purpose of calking them. Rotary. Turning upon an axis, as a wheel. Rynd. The metallic collar in the upper mill-stone by which it is connected to the spindle. Sagging. A term applied to the hull of a vessel when her centre drops below her ends. The converse of Hogging. Scallop. To mark or cut an edge into segments of circles. Scarcement. A set back in the face of a wall or in a bank of earth. A footing. Scarf. To join; to piece; to unite two pieces of timber at their ends by running the end of one over and upon the other, and bolting or securing them together. Scend. The settling of a vessel below the level of her keel Selvagee. A strap made of rope-yarns, without being twisted or laid up, and re- tained in form by knotting it at intervals. Sennit. Braided cordage. Sewage. The matter borne off by a sewer. Sewed. In nautical language, the condition of a vessel aground ; she is said to be tewed by as much as the difference in depth of water around her and her floating depth. Sewerage. The system of sewers. Shaky. Cracked or split, or as timber loosely put together. Shammy. Leather prepared from the skin of a chamois goat IO48 ORTHOGRAPHY OF TECHNICAL WORDS AND TERMS. Sheer. In Naval Architecture, the curve or bend of a ship's deck or sides. To she.!r, to slip or move aside. Sheers. Elevated spars connected at the upper ends, and used to elevate heavy bodies, as masts, etc. Shim. In Naval Architecture, a piece of wood or iron let into a slack place in s frame, plank, or plate to fill out to a fair surface or line. Shoal. A great multitude; a crowd; a multitude of fish. School is a corruption. Shoar. An oblique brace, the upper end resting against the substance to be sup- ported. Sholes. Pieces of plank under the heels of shears, etc. Shoot. A passage way on the side of a steep hill, down which wood, coal, etc., are thrown or slid. The artificial or natural contraction of a river. A young pig. Sidewise. See Edgewise. Signalled. Communicated by signals. Signalized, when applied to signals, is a misapplication of words. Sill. A piece of timber upon which a building rests ; the horizontal piece of tim- ber or stone at the bottom of a framed case. Siphon. A curved tube or pipe designed to draw fluids out of vessels. Skeg. The extreme after-part of the keel of a vessel; the portion that supports the rudder-post. Slantwise. Oblique; not perpendicular. Sleek. To make smooth. Refuse ; small coal. Sleeker. A spherical-shaped, curved, or plane-surfaced instrument with which to smooth surfaces. Slue. The turning of a substance upon an axis within its figure. Snying. A term applied to planks when their edges at their ends are curved or rounded upward, as a strake at the ends of a full-modelled vessel. Spall. A piece of stone, etc., chipped off by the stroke of a hammer or the force of a blow. Spoiling, breaking up of ore into small pieces. Spandrel In Architecture, the irregular triangular space between the outer lines' or extrados of an arch, a horizontal line drawn from its apex, and a vertical line from its springing. Sponson. An addition to the outer side of the hull of a steam vessel, commencing near the light water-line and running up to the wheel guards; applied for the pur- pose of shielding the deck-beams from the shock of a sea. Spnnson- sided. The hull of a vessel is so termed when her frames have the out- line of a sponson, and the spaee afforded by the curvature is included in the hold. Spending, Sponsing, etc., are corruptions. Squilgee. A wooden instrument, alike to a hoe, its edge faced with leather or vulcanized rubber, used to facilitate the drying of wet floors, or decks of a vessel. Stack. In Masonry, a number of chimneys or pipes standing together. The chimney of a blast furnace. The application of this word to the smoke-pipe of a steam-boiler is wholly erroneous. Stage. In Engineering, the interval or distance between two elevations, in shovel- ling, throwing, or lifting. Steeving. The elevation of a vessel's bowsprit } cathead, etc. Strake. A breadth of plank. Strut. An oblique brace to support a rafter. Style. The gnomon of a sun-dial. Sump. In Mining, a pit or well into which water may be led from a mine or work. Surcingle. A belt, band, or girth, which passes over a saddle or blanket upon a horse's back. Stooge. To bear or force down. An instrument having a groove on its under ide for the purpose of giving shape to any piece mbjected to it when receiving a blow from a hammer. ORTHOGRAPHY OF TECHNICAL \VORDS AND TERMS. IO49 i\ Sypkered. Overlapping the chamfered edge of one plank upon the chamfered edge of another in such a manner that the joint shall be a plane surface. Talus. In Architecture, the slope or batter of a wall, parapet, etc. In Geology, a sloping heap of rubble at foot of a cliff. Template. In Architecture, a wooden bearing to receive the end of a girder to distribute its weight. Templet. A mold cut to an exact section of any piece or structure. Tenon. The end of a piece of wood, cut into the form of a rectangular prism, de- signed to be set into a cavity of a like form in another piece, which is termed the mortise. Terring. The earth overlying a quarry. Tester. The top covering of a bedstead. Tholes. The pins in the gunwale of a boat which are used as rowlocka Thwarts. The athwartship seats in a boat. Tide-rode. The situation of a vessel at anchor, when she rides in direction of tht current instead of the wind. Tire. The metal hoop that binds the felloes of a wheel Tompion. The stopper of a piece of ordnance. The iron bottom to which grape- shot are secured. Treenails. Wooden pins employed to secure the planking of a vessel to the frames. Trepan. In Mining, the instrument used in the comminution of rock in earth- boring at great depths. Trestle. The frame of a table ; a movable form of support. In Mast-making, two pieces of timber set horizontally upon opposite sides of a mast-head. Trice. In Seamanship, to haul or tie up by means of a rope or tricing-line. Tue-iron or Tuyere. The nozzle of a bellows or blast-pipe in a forge or smelting. furnaca Vice. In Mechanics, a press to hold fast anything to be worked upon. Voyal. In Seamanship, a purchase applied to the weighing of an anchor, leading to a capstan. Wagon. An open or partially enclosed four-wheeled vehicle, adapted for the transportation of persons, goods, etc. Wear. I* nautical language, to put a vessel upon a contrary tack by turning her around stern to the wind. Weir. A dam across a river or stream to arrest the water; a fence of twigs or stakes in a stream to divert the run of fish. Whipple-tree. The bar to which the traces of harness are fastened. Wind-rode. The situation of a vessel at anchor, when she rides in direction ol the wind instead of the current. Windrow. A row or line of hay, etc., raked together. Withe. An instrument fitted to the end of a boom or mast, with a ring, through which a boom is rigged out or mast set up. Woold. To wind; particularly to bind a rope around a spar, etc. Astragal* In Architecture, a round molding, surrounding the head or base of a colunm. In Gunnery, a like molding on cannon near the mouth. Creusote. An oily colorless liquid, procured from coal-tar. Flume, a channel for conducting water, as that by which the surplus water of a canal is led to a lower level. Forebay. The part of a Mill-race or Penstock, from which water flows upon a water-wheel. Grillage. A frame, constructed of beams laid in parallel rows, and crossed at right angles, with others notched over them. Designed to uniformly distribute or extend the area of a foundation. IO5O ORTHOGRAPHY OF TECHNICAL WORDS AND TERMS. Hypotenuse. Commonly, but incorrectly, hypothenuse. ArcMtectUre > a pier that juts out or P"***" into a river or a Kibble. In Mining, a metallic bucket in which ore is drawn up to the surface Lewis One or two frustums of a right-angled metallic wedge set inverted or W*^ * MraSU; orTfaduct. ^ En9ineerin ^ a c y lindrj cal pillar terminating a wing wall of a bridge Nautical Wra ?P ed with canvas or tarred rope, to resist wear from Payed. Nautical. Painted, tarred, or greased, to resist moisture and wear. floodgate*' An artificial conduit for water to a water-wheel, and furnished with a Ravel. To disentangle, untwist, or unweave. The usual prefix Un i. wholly superfluous. Roil. To render turbid, to stir or mix. Scabble. The dressing of the faces of rough stones, as with a broad chisel. Served, Service. Nautical. The layer of wrapping, as spun yarn, lines, etc. around a stay or rope, to resist friction and wear. Shackle, or Clevis. An open link set in a chain, secured by a pin running through eyes m its ends, which, when withdrawn, admits chain to be parted at that point. Soffit. In Architecture, the under side of an opening ; the lower surface of a vault or arch; also the under surface of an arch between columns * le with Strike, in Geology, is the compass direction of the intersection of the plane of stratified rock with the plane of the horizon. Altars. In Naval Architecture, the steps on the sides and end of a marine dock. Gin. An instrument operated by men or animals for the raising or drawing of heavy bodies; usually a vertical revolving windlass and lever. Sump. In Salt-works, a pond in which the sea or saline water is retained for use in the future. Skeet. Nautical. A scoop with a long handle, for use in wetting the sails or the sides of a vessel. Wyes. The vertical standards on which the telescope of a Theodolite or Level is supported, and which admits of their being reversed by a reversal of its ends When the telescope is reversed by rotation on its trunnions, the instrument is termed a Transit. Cantalcver. An angular lever, as a projecting bracket under a balcony, the eaves of a building or the span of a bridge, when the intrados is defined by lines from the abutments, at an angle elevating from the horizon. Camel. In Naval Architecture. A decked and flat-bottomed vessel, alike to a scow; adapted for transportation of heavy material, in the raising of sunken ves- vels, etc., and for the transportation of heavy materials, as armaments from vessels. to a shore, etc., commonly, but erroneously, a scow. Scow. An open and flat-bottomed vessel, adapted for operation in shallow water. Sprocket. A radial projection on the circumference of a wheel; to engage the links of a chain, as those on the wheel base of a capstan. Spud. In Mechanics. A spade-like instrument for recovering a tool in a tuba well. In Surveying. A nail driven in a monument or stake, to designate a line or point. Beam. In Mechanics. When vibrating, as in a Vertical or Side-lever Steam or other Engine, it is simply a Beam, as Main, Overhead, Side-lever, Air-pump, etc. Working is superfluous, and Walking is a local vulgarism. Size. In Mining. A separation of coarse and fine grains or parts. In Mechanics or Arts. A weak, viscous, and glutinous substance or varnish. In Geometry or Volume, the application of it to areas, structures, etc. , is objectionable. Corruption of Assize, a Statute of measure and price, ORTHOGRAPHY OP TECHNICAL WORDS AND TERMS. 10$ I Adjutage. An opening in a vessel for tne efflux of a fluid. Archean. Oldest period of geological time. A term given to crystalline schists and massive rocks underneath the oldest fossiliferous stratified rocks. Bascule Bridge. A bridge structure for the passage of vessels in a river; by which a single or divided floor is counterpoised by the weight of the inner end or ends. The whole of the movable floor or floors resting on a transverse shaft. Beton. Artificial stone made by the admixture of broken stone, shingle, gravel, etc., with hydraulic cement and water. When mixed with lime it is termed Con- crete. Breast- summer. A beam of metal, stone, or wood, designed to sustain a wall over a doorway or like opening or floor; a lintel. Chaplet. A metallic support or Stud, set in a mold to sustain a core against the pressure of the metal when fluid. Crab. A shaft, vertical or horizontal, constructed as a rope drum; for the draw- ing or raising of heavy bodies, and operated by a winch or handspikes in the manner of a windlass or capstan. Dolmen or Tolmen. (Celtic.) A Druidical monument consisting of a large stone set horizontal on two vertical stones at a short distance apart, and a few feet in height. (Breton.) An excavated stone containing human remains. Also Cromleeh. A large flat and crooked stone, set horizontal upon four others set vertical, alike to a table. Firmer Tools. Short chisels and gouges, as distinguished from ordinary long- bladed, and usually operated by hand. The gouge is ground upon its outer side; the ordinary upon its inner. Flitch. In Construction. The combination of wood with iron, either in plates or a flanged beam. Gantry. A frame of posts and header, to support a travelling winch, wherewith heavy weights, as stone for foundation walls, or machines, may be raised and trans ported. Jag. A rough point or barb on the projecting surfaces of metal; produced by nicking it, as with a chisel, or by casting. Jaggers. The rough projections. Jagging. The insertion of a jagged or serrated bar, shaft, or eye-bolt in a casting, . to prevent its being easily withdrawn. Key. In Mechanics, a tapered wedge used in connection with a gib and strap, and also with brasses; to adjust the length of the rod to which they are attached. A Cottar. Lacustrine. Pertaining to a lake, and applied to deposits which are present in lake basins. Lewis. A combination of one or two dovetailed iron pieces, with a shackle eye and bolt; set into a dovetailed under cut in a body of stone, marble, or cement block, and set out and secured by the insertion of a wedge. Lewis Bolt. A bolt with a jagged end, for insertion in a block of stone, etc., and leaded in. Luting. The laying or insertion of a paste, cement, or adhesive material of plas- tic consistency, in or over a crevice or between joints of a pipe, etc. Mandrel. A metal spindle for chucking lathe work. A tapered metal rod or former on which nuts, etc., etc., are dressed to shape. Moraine. Material as rocks, earth, etc., pushed or deposited by glaciers. Pein. The lesser head of a hammer, and is termed Ball when it is spherical; Cross when in the form of a round-edged ridge, at right angles to the axis of the handle; and Straight, when a like ridge is in the plane of the handle. Pierre Perdue. Lost or random stone projected in water, usually for a base to a superstructure, or to construct a Breakwater. Riprap. Scrim. A screen or shade; a thin, coarse cloth, used for temporary windows or doors in a building in progress of completion. Seepage. Percolation ; oozing fluid or moisture; also, the volume of a fluid that percolates. Slope. A step, an excavation in a mine to enable ore to be rendered accessible by a shaft or drift. To remove the contents of a vein. Sullage. Scoriae, cinder, scurf, etc., which floats on the surface of molten metal. Tackle. The connection of two or more blocks and a rope. THE END. YA UNIVERSITY OF CALIFORNIA LIBRARY i