Ul DC 1/5 O z 1- (0 DC < I QC U DC 00 0) U. O Id o LJ 5 00 00 > i D ERING o m RADUATI b. bJ O b. O (/) 0) Ul X 1- o K CM CM 0) UJ i5J LJ ^ O Z ^ < -1 o UJ 5 o Z z J Ul (A U v/ Ul LJ DC i X Q. L^^V-^ GIFT OF THE CIVIL ENGINEER'S POCKET-BOOK. MR. TRAUTWINE'S ENGINEERING WORKS. The Field Practice of Laying out Circular Curves for Rail- roads. By JOHN C. TRAUTWINE, Civil Engineer. Eleventh edition, enlarged and rewritten. I2mo, morocco, tuck, gilt edge, $2 50 A New Method of Calculating the Cubic Contents of Exca- vations and Embankments by the aid of Diagrams ; together with Directions for Estimating the Cost of Earthwork. By JOHN C. TRAUTWINE, Civil Engineer. 10 steel plates. Seventh edition, completely revised and enlarged. 8vo, cloth, 2 oo Civil Engineer's Pocket-Book of Mensuration, Trigonometry, Surveying, Hydraulics, Hydrostatics, Instruments and their adjustments, Strength of Materials, Masonry, Principles of Wooden and Iron Roof and Bridge Trusses, Stone Bridges and Culverts, Trestles, Pillars, Suspension Bridges, Dams, Railroads, Turnouts, Turning Platforms, Water Stations, Cost of Earthwork, Foundations, Retaining Walls, etc., etc., etc. In addition to which the elucidation of certain important Principles of Construc- tion is made in a more simple manner than heretofore. By JOHN C. TRAUTWINE, C.E. I2mo, 695 pages, illustrated with about 700 wood-cuts. Morocco, tuck, gilt edge. Twentieth thousand, revised and corrected, 5 Any of the above books will be sent to any part of the United States or Canada on receipt of list price. Send money in Registered Letter, P. O. Order, or Check. E. CLAXTON & CO., PUBLISHERS, No. 930 MARKET STREET, PHILADELPHIA, PA. THE CIVIL ENGINEER'S POCKET-BOOK, OP Mensuration, Trigonometry, Surveying, Hydraulics, Hydrostatics, Instruments and their Adjustments, Strength of Materials, Masonry, Principles of Wooden and Iron Eoof and Bridge Trusses, Stone Bridges and Culverts, Trestles, Pillars, Suspension Bridges, Dams, Railroads, Turnouts, Turning-Plat- forms, "Water Stations, Cost of Earthwork, Foundations, Retaining Walls, Etc,, Etc,, Etc. IN ADDITION TO WHICH THE ELUCIDATION OF CERTAIN IMPORTA-NT PRINCIPLES OF CONSTRUCTION IS MADE IN A MORE SIMPLE MANNER THAN HERETOFORE. BY JOHN C. TRAUTWINE, C.E., AUTHOR OF " A NEW METHOD OF CALCULATING THE CUBIC CONTENTS OF EXCAVATIONS AND EMBANKMENTS," "THE FIELD PRACTICE OF LAYING OUT CIRCULAR CURVES FOR RAILROADS," ETC. ILLUSTRATED WITH 690 ENGRAVINGS FROM ORIGINAL DESIGNS, [wenitdh REVISED AND CORRECTED. PHILADELPHIA: E. CLAXTON & COMPANY. LONDON : TRUBNER & CO. 1883. Entered, according to Act of Congress, in the year 1882, by JOHN C. TRAUTWINE, in tne Office of the Librarian of Congress at Washington. 1LECTROTTPED BY J. FAG AN & SON, PHILADELPHIA. THE AUTHOR i$i TO THE MEMORY OF HIS FRIEND, THE LATE BENJAMIN H. LATROBE, Esq., CIVIL ENGINEER. 50 PUBLISHEES' ANNOUNCEMENT. THE publishers have much pleasure in issuing this twentieth thou- sand of Mr. Trautwine's popular book. Although originally written for civil engineers only, the vast amount of practical information con- densed into it has made it equally a favorite with contractors, builders, and machinists ; besides leading to its adoption as a text-book in many educational institutions. In proof of the estimation in which the work is held by competent professional authorities, we cite the following: Mr. Alfred P. Boiler, C. E., in his " Iron Highway Bridges," says : " For a wonderfully clear and elaborate discussion of force, strains, etc., as well as upon the subject of Trusses arid Strength of Materials, free from all technicalities, the learner is referred to Mr. Trautwine's l En- gineer's Pocket-Book/ a work that should be the corner-stone of every engineer's library." Mr. George L. Vose, C. E., Professor of Civil Engineering in the Massachusetts Institute of Technology, at Boston, in his " Civil En- gineering," says : " Mr. Trautwine's ' Civil Engineer's Pocket-Book ' is, beyond all question, the best practical manual for the engineer that has ever appeared." Mr. Thomas M. Cleemann, C. E., in his " Railroad Engineer's Prac- tice," remarks that "he considers that no other book has appeared which supplies so well a constant want of the engineer, at all stages of his career, as Trautwine's * Engineer's Pocket-Book.' " Many other writers refer to the book in similar terms, beside showing their appreciation of it by extracting freely from it ; and the profes- sional periodicals of the country generally have expressed their ap- proval of it. Although Mr. Trautwine has for twenty years abandoned the active pursuit of his profession, the publishers trust that he will continue to feel sufficient interest in this book to enable them to add to its value in future editions. iv PREFACE TO FIRST EDITION. QHOULD experts in engineering complain that they do not find ^ anything of interest in this volume, the writer would merely remind them that it was not his intention that they should. The book has been prepared for young members of the profession ; and one of the leading objects has been to elucidate in plain English, a few important elementary principles which the savants have envel- oped in such a haze of mystery as to render pursuit hopeless to any but a confirmed mathematician. Comparatively few engineers are good mathematicians ; and in the writer's opinion, it is fortunate that such is the case ; for nature rarely combines high mathematical talent, with that practical tact, and observation of outward things, so essential to a successful engineer. There have been, it is true, brilliant exceptions ; but they are very rare. But few even of those who have been tolerable mathematicians when young, can, as they advance in years, and become engaged in business, spare the time necessary for retaining such accomplish- ments. Nearly all the scientific principles which constitute the founda- tion of civil engineering are susceptible of complete and satisfactory explanation to any person who really possesses only so much element- ary knowledge of arithmetic and natural philosophy as is supposed to be taught to boys of twelve or fourteen in our public schools.* * Let two little boys weigh each other on a platform scale. Then when they balance each other on their board see-saw, let them see (and measure for themselves) that the lighter one is farther from the fence-rail on which their board is placed, in the same proportion as the heavier boy outweighs the lighter one. They will then have learned the grand principle of the lever. Then let them measure and see that the light one see-saws farther than the heavy one, in the same proportion; and they will have acquired the principle of virtual velocities. Explain to them that equatity of moments means nothing more than that when they seat tb.emse.lyes $t their meMK VI PREFACE. The little that is beyond this, might safely be intrusted to the savants. Let them work out the results, and give them to the engi- neer in intelligible language. We could afford to take their words for it, because such things are their specialty ; and because we know that they are the best qualified to investigate them. On the same princi- ple we intrust our lives to our physician, or to the captain of the vessel at sea. Medicine and seamanship are their respective special- ties. If there is any point in which the writer may hope to meet the approbation of proficients, it is in the accuracy of the tables. The pains taken in this respect have been very great. Most of the tables have been entirely recalculated expressly for this book ; and one of the results has been the detection of a great many errors in those in common use. He trusts that none will be found exceeding one, or sometimes two, in the last figure of any table in which great accuracy is required. There are many errors to that amount, especially where ured distances on their see-saw, they balance each other. Let them see that the weight of the heavy boy, when multiplied by his distance in feet from the fence-rail amounts to just as much as the weight of the light one when multiplied by his dis- tance. Explain to them that each of the amounts is in foot-pounds. Tell them that the lightest one, because he see-saws so much faster than the other, will bump against the ground just as hard as the heavy one ; and that this means that their momentum* are equal. The boys may then go in to dinner, and probably puzzle their big lout of a brother who has just passed through college with high honors. They will not forget what they have learned, for they learned it as play, without any ear- pulling, spanking, or keeping in. Let their bats and balls, their marbles, their swings, &c, once become their philosophical apparatus, and children may be taught (really taught) many of the most important principles of engineering before they can read or write. It is the ignorance of these principles, so easily taught even to children, that con- stitutes what is popularly called " THE PRACTICAL ENGINEER ; " which, in the great majority of cases, means simply an ignoramus, who blunders along without knowing any other reason for what he does, than that he has seen it done so before. And it is this same ignorance that causes employers to prefer this practical man to one who ia conversant with principles. They, themselves, were spanked, kept in, &c, when boys, because they could not master leverage, equality of moments, and virtual velo- cities, enveloped in x's, p's, Greek letters, square-roots, cube-roots, &c, and they naturally set down any man as a fool who could. They turn up their noses at science, not dreaming that the word means simply, knowing why. And it must be confessed that they are not altogether without reason ; for the savants appear to prepare their books with the express object of preventing purchasers, (they have but few readers,) from learning why. PREFACE. Vll the recalculation was very tedious, and where, consequently, interpo- lation was resorted to. They are too small to be of practical import- ance. He knows, however, the almost impossibility of avoiding larger errors entirely ; and will be glad to be informed of any that may be detected, except the final ones alluded to, that they may be corrected in case another edition should be called for. Tables which are abso- lutely reliable, possess an intrinsic value that is not to be measured by money alone. With this consideration the volume has been made a trifle larger than would otherwise have been necessary, in order to admit the stereotyped sines and tangents from his book on railroad curves. These have been so thoroughly compared with standards prepared independently of each other, that the writer believes them to be absolutely correct. In order to reduce the volume to pocket-size, smaller type has been used than would otherwise have been desirable. Many abbreviations of common words in frequent use have been introduced, such as abut, cen, diag, hor, vert, pres, &c, instead of abutment, center, diagonal, horizontal, vertical, pressure, &c. They can in no case lead to doubt; while they appreciably reduce the thickness of the volume. Where prices have been added, they are placed in footnotes. They are intended merely to give an approximate or comparative idea of value ; for constant fluctuations prevent anything farther. The addresses of a few manufacturing establishments have also been inserted in notes, in the belief that they might at times be found convenient. They have been given without the knowledge of the proprietors, The writer is frequently asked to name good elementary books on civil engineering ; but regrets to say that there are very few such in our language. " Civil Engineering," by Prof. Mahan of West Point ; " Roads and Railroads," by the late Prof. Gillespie ; and the " Manual for Railroad Engineers," by George L. Vose, Civ. Eng, and Professor of Civil Engineering in Bowdoin College, Brunswick, Maine, are the best. The last, published by Lee & Shepard, Boston, 1873, is the most complete work of its class with which the writer is acquainted. Vlll PREFACE. Many of Weale's series are excellent. Some few of them are behind the times ; but it is to be hoped that this may be rectified in future editions. Among pocket-books, Haswell, Hamilton's Useful Information, Henck, Molesworth, Nystrom, Weale, &c, abound in valuable matter. The writer does not include Rankine, Moseley, and "Weisbach, because, although their books are the productions of master-minds, and exhibit a profundity of knowledge beyond the reach of ordinary men, yet their language also is so profound that very few engineers can read them. The writer himself, having long since forgotten the little higher mathematics he once knew, cannot. To him they are but little more than striking instances of how completely the most simple facts may be buried out of sight under heaps of mathematical rubbish. Where the word " ton " is used in this volume, it always means 2240 Ibs, because that is its meaning in U. S. law. JOHN C. TRAUTWINE. PHILADELPHIA, Nov. 13, 1871. REMAEKS ON THIS EDITION. Q EVERAL slight alterations and additions have been made. Among O the most important are those on the Transit Instrument ; Turnouts ; Mr. Eliot C. Clarke's tables of strengths of Concrete Beams, and of Cement Mortar, both on page 508 ; the Fritz & Say re Splice Plate for rails, on page 395 ; the rate at which rain water reaches a sewer, on page 566 ; Mr. C. L. Gates' table of strength of Built Iron Pil- lars, on page 233 ; comparison of Gordon's and Hodgkinson's strengths of pillars, on page 242 ; and alterations in five of the eight Gordon rules for the strength of Iron Pillars, pages 221 to 223. Hitherto wrong divisors have been used by myself and many others, in some of the Gordon formulas. Thus, in Rule 1, 400 should be 600 ; in Rule 2, 3000 should be 4500 ; in Rule 3, 533.3 should be 800 ; in Rule 5, 200 should be 300 ; in Rule 7, 266.7 should be 400. We have, therefore, altered the RULES and their EXAMPLES in this edition ; but have not been able to alter any of the TABLES in time, except the most important one, on page 232, which has also been enlarged. The reader had better, therefore, insert a caution " not correct " at Tables, pages 224 to 229. The old ones on pages 230, 231, are correct. The SOLID METAL AREAS, however, at the foot of ALL these tables, are correct, and therefore the breaking load for any column con- tained in them, may now be found by multiplying its area by the corresponding number in the Table, page 232. We may add that all the five old rules, now altered, erred on the safe side; the loads given by the new rules being greater in all cases. IT MUST BE BORNE IN MIND that the divisors in the so-called GORDON'S FORMULAS for hollow columns, whether cylindrical or square, are based upon Rankine's rules, p. 235, for finding the square of the Radius of Gyration for THIN columns. WHAT is A THIN COLUMN ? It is usual to apply the rules to columns with thicknesses up to J of the outer diam. ; and in view of this fact, we would respectfully ask the attention of experts to the question WHETHER THIS PRACTICE is SAFE. We have our doubts on the subject, but hope they are unfounded, inasmuch as we believe the rules are so applied even by experts of high mathematical attain- ments, as well as by engineers generally. ERRATA IN THE PRECEDING EDITION (THE 17TH THOUSAND), AND SOME OTHERS, BUT CORRECTED IN THIS. SEE ALSO REMARKS ON THIS EDITION, for others. P. 219. On the 15th and 16th lines of Example 1, for 5 read 2.67. P. 222. Near the middle, for Rule 5, read Rule 1. P. 223. Near the middle, for foregoing wrought read foregoing cast. P. 235. In the small Fig. of a thin hollow rectangle, change c to a, and a to c. P. 236. Upper half, in four places change 3200 to 4800; also in the formula for channel iron, change total area area fl X area web to 4 X sq of total area 4 X sq total area also all the loads for CAST iron in the table are too small. P. 266. 27 lines from bottom, for strain = c h read strain = e h. P. 315. 3d line, for three read two. 8 lines from bottom, omit 1. P. 333. 22d line, for 15 read 5. P. 336. First three letters of 5th line above Rem 1, for com read c m o. P. 417. 10 lines from foot, for prod read quot. P. 437. Near top, for 2.072 and 22.812, read 2.074 and 22.814. P. 495. 21st line, for 8.143 -f read 8.143 X. Also, on the 4th line of table, omit the * after .5, and for the same .5, read .707. P. 590. Art. 7. For Rem 3, p. 463, read Rem 2, p. 462. Art. 8. In rule and formula, for (H defl) 2 read H (defi 2 ). This will make length of chain 588.3 ft, instead of 591.1. P. 637. The tables of loads for CAST iron are too small. CONTENTS. For a full reference to the Contents in detail, see Index, page 677. PAGE MENSURATION 13 PLANE TRIGONOMETRY 39 SQUARES, CUBES, AND KOOTS 48 GEOMETRY 61 ARITHMETIC 69 WEIGHTS AND MEASURES 73 LAND SURVEYING 90 SINES AND TANGENTS, &c 102 CONTOUR LINES 147 DIALLING 150 PAPER 151 THE LEVEL 152 THE ENGINEER'S TRANSIT 157 THE THEODOLITE , 162 THE Box OR POCKET SEXTANT 163 THE COMPASS 164 LEVELLING BY THE BAROMETER 167 PENDULUMS * 172 SOUND 173 STRENGTH OF MATERIALS 174 STRENGTH OF IRON AND WOODEN PILLARS, WITH FULL TABLES. 221 TRUSSES FOR HOOFS AND BRIDGES 243 TRESTLES 307 THERMOMETERS 309 STONEWORK 310 FOUNDATIONS 313 COST OF DREDGING 329 RETAINING- WALLS 331 STONE BRIDGES, CULVERTS, ARCHES 341 BOARD MEASURE .' 357 TABLES OF WEIGHTS OF BARS, BOLTS, PIPES, &c, &c 362 SPECIFIC GRAVITY 383 KAIL JOINTS, CHAIRS, &c .. , 390 xi Xll CONTENTS. PAGE TURNOUTS 397 RAILROADS 409 TABLES OF LEVEL CUTTINGS 420 TURNTABLES 429 WATER STATIONS 432 COST or EARTHWORK 435 CENTRE OF GRAVITY , 442 MECHANICS. FORCE IN RIGID BODIES. ... . 443 CENTRIFUGAL FORCE 494 MORTAR, BRICKS, CEMENT, CONCRETE, &c 496 PLASTERING 509 SLATING .... 510 SHINGLES........ 512 PAINTING .... 512 GLASS, AND GLAZING 514 WATER ... 515 BAIN 518 SNOW , ,.,,... 519 AIR. ATMOSPHERE 519 WIND 520 EVAPORATION, FILTRATION, AND LEAKAGE 521 HYDROSTATICS 521 HYDRAULICS 534 DAMS 583 SUSPENSION BRIDGES 588 FRICTION 597 TRACTION 603 ANIMAL POWER 605 CHORDS TO A KADIUS 1, FOR PROTRACTING 608 GLOSSARY OF TERMS 615 APPENDIX.. 630 SHEARING OF BEAMS 642 OPEN AND CLOSED BEAMS 644 K UTTER' s FORMULA 650 VELOCITIES IN SEWERS 652 RIVETS AND RIVETING 653 CENTERS FOR ARCHES 665 For a full reference to the Contents in detail, see Index, page 677. MENSURATION.- PARALLELOGRAMS. Square. Rectangle. Rhombus. Rhomboid. s s s s A PABAI.LKLOGRAM is any figure of four straight sides, the opposite ones of which are parallel. There are but four, as in the above figs. lu the square aud rhombus all the four sides are equal ; in the rectangle and rhomboid only the opposite ones are equal. In any parallelogram the four angina *miunt to four right angles, or 360; aud any two diagonally opposite angles are equal to each other; hence, having one angle given, the other three can readily be found. In a square, or a rhombus, a diag divides each of two angles into two equal parts; but in the two other parallelograms it does not. To find the area of any parallelogram. Multiply any side, as S, by the perp height, or dist p to the opposite side. Or, multiply together two sides and nat sine of their included angle. The diag a b of any square is equal to one side mult by 1.41421 ; and a side is equal to The side of a square equal in area to a given circle, is equal to diam X .886227. The side of the greatest square, that can be inscribed in a diven circle, is equal to diam X .707107. The side of a square mult by 1.51967 gives the side of an equi- lateral triangle of the same area. All parallelograms as A and C, which have equal bases, a q, aud equal perp heights n c, have also equal areas; aud the area of each is twice that of a tri- angle having the same base, and perp height. The area of a square inscribed in a circle is equal to twice the square of the ra.l. In every parallelogram, the 4 squares drawn on its sides have a united area equal to that of the two squares drawn ou its 2 diags. If a larger square he drawn on the diag a b of a smaller square, its area will be twice that of said smaller square. Either diag of any parallelogram divides it into two equal triangles, and the 2 diags div it into 4 triangles of equal areas. The two diags of any parallelogram divide each other into two equal parts. Any line drawn through the center of a diag divides the parallelogram into two equal parts. Remark 1. The area of any fig whatever as B that is enclosed by four straight lines, may be found thus : Mult together tue two diags a m, n it ; and Mie uat sine of the least angle a o b ; or n o m, formed by their intersection. Div the product by 2. This is useful in land surveying, when obstacles, as is often the case, make it difficult to measure the sides of the fig or field ; while it may be easy to measure the diags ; and after finding their point of intersection o, to measure the re- quired angle. But if the fig is to be drawn, the parts o a, ob, on, o m of the diags must also be measd. Kem. 2. The sides of a parallelogram, triangle, and many other figs may be found, when only the area and angles are given, thus : Assume some particular one of its sides to be ot the length 1 ; aud calculate what its area would be if that were the case. Then as the sq rt of the area thus found is to this side 1, so is the sq rt of the actual given area, to the corre- sponding actual side of the fig. TRIAXGLES. We speak here of plane triangles only ; or those having straight sides. Since the area of any triangle is equal to halt that of a parallelogram which has the same base and perp height, therefore, To find the area of any triangle, having its base and perp height, Mult its base S by its height, or perp dist p to the opposite angle; and div the prod by 2. For an equilateral one, when its perp height is not known; Square one side; and mult the square by the decimal .433013. Any side may be assumed as the base of a triangle; but the perp height must always be measured from the side so assumed ; to do which, the side must sometimes be prolonged, as in Fig E; but the prolongation is not to be considered as a part of the base. Th.e perp height of an equilateral triangle is equal to one side X .866025. Hence one of its sides is equal to the perp height div by .866025 or to perp height X 1.1547. Or. to find a side. mult the s>a rt of its urea by 1.51967. The side of an equilateral triangle, mult bv .65H037 = side of a square of tBP same area; or div by 1.34677 it gives the diam of a circle of same a'rea. To find area, having two sides, and the included angle. Mult together t^ e tw <> sides, and the nat sine of the included angle ; div by 2. Ex. Sides 650 Y l and 980 ft; included angle 69 20'. By the table we find the nat sine, .9356; 13 14 MENSURATION. Having: the three sides. 4<>, or a c, between either pair of opposite angles; and also the two perps, n, n., from the other two angles. Add together these two perps ; mult the sum by the diag; div the prod by 2. Having the fotir sides; and either pair of opposite angles, as n b c 9 noc ; or b a, o, and b c o. Consider the trapezium as divided into two triangles, in each of which are given two sides anfthe included angle. Find the area of each of these triangles as directed under the preceding hea^ Tri- angles," and add them together. Having the four angles, and either pair of opposite sides. Begin with one of the sides, and the two angles at its ends. If the sum of these two n^ie, exceeds 180, subtract each of them from 180. and make use of the rems instead of the angl^ themselves. Then consider this side and its two adjacent angles (or the two rems, as the case ; nav ^ as tbomj of a triangle; and find its area as directed for that case under the preceding he^ .. T r i an gle." Do MENSURATION. .15 the same with the other giveu side, and its two adjacent angles, (or their rems, as the case may be.) Subtract the least of the areas thus found, from the greatest ; the rem will be the reqd area. Having three sides ; and the two included angles. Mult together the middle side, and one of the adjacent sides ; mult the prod by the nat sine of their included angle ; call the result a. Do the same with the middle side and its other adjacent side, and the nat sine of the other included angle ; call the result b. Add the two angles together ; find the diff between their sum and 180, whether greater or less ; find the nat sine of this diflf; mult together the two given sides which are opposite one another ; mult the prod by the nat sine just found ; call the result c. Add together the results a and b ; then, if the sum of the two given angles is lest than 180, subtract c from the sum of a and 6 ; halftbe rem will be the area of the trapezium. But if the sum of the two given angles be greater than 180, add together the three results a, b, and c; half their sum will be the area. Having the two diagonals, and either angle formed by their intersection. See Remarks after Parallelograms, p 13. In railroad measurements Of excavation and embankment, the trapezium I in u o frequently occurs ; as well as the two 5-sided figures I m n o t and Imno s; in all of which m n represents the roadway ; rs, r c, and r t the center- depths or heights ; I u and o v the side-depths or heights, as given by the level ; I m and n o the side- slopes. The same general rule for area applies to all three of these figs ; namely, mult the extreme hor width u v by half the center depth r s, r c, or r t, as the case may be. Also mult one fourth of the width of roadway m n, by the sum of the two side-depths I u and o v. Add the two prods together ; the sum is the reqd area. This rule applies whether the two side- U m r Tl V elopes m I and n o have the same angle of inclination or not. In railroad work, etc., the mid- way hor width, center depth, and side depths (see foot of p 33) of a prismoid are respectively = The half sums of the corresponding end ones, and thus can be found without actual measurement. POLYGONS. b All straight-sided figs of more than 4 sides, are called polygons ; if all the sides and angles are equal, it is a regular polygon ; if not, irregular. Of course their number is infinite ; but those in most common use are the four shown above. Number of Sides. Name of Polygon. Areas. Outer Radii. Sides. Angle con- tained between two sides. Angle at center of circle. 3 4 Equilateral triangle. Square. .4330 1. .5774 .7071 1.0000 1.0000 60 90 120 90 5 6 7 8 9 10 11 12 Pentagon. Hexagon. Heptagon. Octagon. Nonagon. Decagon. Undecagon. Dodecagon. 1.7205 2.5981 3.6339 4.8284 6.1818 7.6942 9.3656 11.1962 .8507 1. 1 .1524 1.3066 1.4619 1.6180 1.7747 1.9319 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 108 120 128 34'.29 135 140 144 147 16'.36 150 72 60 51 25'.71 45 40 36 32 43'.64 30 To find the area of any regular polygon. Mult together one of its sides, a b ; the perp p drawn from the center of the fig to the center of itt side ; and the number of its sides. Div the prod by 2. Or, square one of its sides ; and mult sai4 square by the number in the foregoing table, in the column of areas. Having a side of a regular polygon, to find the rad of a circumscribing circle. Mult the side by the corresponding number in the column of radii. Having the rad of a circumscribing circle, to find the side of the inscribed regular polygon. Divide the given rad by the corresponding rad in the table. Side of an octagon = o o X .41421. Of a hexagon = o o X .57735. 16 MENSURATION. To find the area of any irregular poly goii, a n b c m. Div it into triangles, as a n b, a m c, and a b c ; in each of which find the perp dist o, between its base a I, a c, or I c; and the opposite angle ?i, m, or a; mult each base by its perp dist; add all the prods together ; div by 2. To find the area of a long 1 irregular fig, as a b c d. Between its two ends a 6, and c d, space off equal dists, (the shorter they are the more accurate will be the result,) through which draw the intermediate parallel lines 1, 2, 3, &c, across the breadth of the fig. Measure the lengths of these intermediate lines, and add them together; to the sum add oue half of the two end breadths a b and c d. Mult the entire sum by one of the equal spaces between the parallel lines. The prod will be the area. This rule answers as well if either one or both the ends terminate in points, as at m and n. In the last of these cases, both a b and c d will be included in the intermediate lines ; and half the two end breadths will be 0, or nothing. To find the area of a fig whose outline is extremely irregular. Draw lines around it which shall enclose within them (as nearly as can be judged by eye) as much space not belonging to the fig, as they exclude space belonging to it. The area of the simplified fig thus formed, being in this manner rendered equal to that of the complicated one, may be calculated by dividing it into triangles, &c. By using a piece of fine thread, the proper position for the new boundary lines may be found, before drawing them in. Small irregular \ areas may be found from a drawing, by laying upon * it a piece of transparent paper carefully ruled into small squares, each of a given area, say 10, 20. or 100 sq ft each; and by first counting the whole squares, and theu adding the fractions of squares. CIRCLES. A circle is the area included within a curved line of such a character that every point in it is equally distant from a certain point within it, called its center. The curved line itself is called the circumference, or periphery of the circle ; or very commonly it is called the circle. To find the circumference. Mult diam by 3.1416, which gives too much by only .148 of an inch in a mile. Or, as 113 is to 355 so is diam to circumf; too great 1 inch in 186 miles. Or, mult diam by 3^; too great by about 1 part in 2485. Or, mult area by 12.566, and take sq root of prod. Or, use table p 18 or p 675. The Greek letter IT, also p, is used by writers to denote this 3.1416; and p 2 = 9.86960. To find the diam. Div the oircumf by 3.1416 ; or, as 355 is to 113, so is circumf to diam ; or, mult the circumf. by 7: and div the prod by 22, which gives the diam too small by only about one part in 2485; or, mult the area by 1.2732; and take the sq rt of the prod; or use the following table of circles. The diam is to the circumf more exactly as 1 to 3. 14159265. To find the diam of a circle equal in area to a given square. Mult one side of the square by 1.12838. To find the rad of a circle to circumscribe a given square. Mult one side by .7071 ; or take H the diag. To find the side of a square equal in area to a given circle. Mult the diam by .88623. To find the side of the greatest square in a given circle. Mult diam by .7071. The area of the greatest square that can be inscribed in a circle is equal to twice the square of the rad. ThediamX by 1.3468 gives the side of an equilateral triangle of equal area. Having the chord and rise of an arc, to find radius. Square half the chord. Divide by rise. Add rise. Divide by 2. This rule applies also to area greater than the semicircle ; as does also, Radius = square o'f chord of half arc -- twice rise of whole arc. Having the chord, and rad, to find the rise. Square the rad ; also square half the chord ; take the last square from the first ; take sq rt of the rem ; and subtract it from the rad, if rad is greatest ; but if not, add it to rad. MENSURATION. 17 Having: the racl, and rise, to find the chord. From rad subtract rise (or from rise subtract rad if rise is greatest), square the rem; also square the rad ; from this last square take the first ; take sq rt of rem ; and mult it by 2. Having the rise of arc, and diam of circle, to find the chord. From the diam take the rise ; mult the rem by the rise ; take sq rt of prod and mult it by 2. Above rule applies whether the arc is greater or less than a semicircle. Half the chord of an arc, if div by the rad of the circle, will give the uat sine of half the angle gubtended by the whole chord, or arc. Table of Minutes and Seconds in decimals of a Degree. Min. Deg. Min. Deg. Min. Deg. Sec. Deg. Sec. Deg. Sec. Deg. 1 .016666 21 .350000 41 .683333 1 .0002781 21 .005833 41 .011389 2 .0:{3:333 22 .366666 42 .700000 2 .000556 22 .006111 42 .011667 3 .050000 23 .383333 I 43 .716666 3 .000833 23 .006389 43 .011944 4 .066666 24 .400000 i 44 .733333 4 .001111 24 .006667 44 .012222 5 .083333 25 .416666 45 .750000 5 .001389 25 .006944 45 .012500 6 .100000 26 .433333 ! 46 .766666; 6 .001667 26 .007222 46 .012778 7 . 11(5666 27 .450000 47 .783333; 7 .001944 27 .007500 47 .013056 8 .133333 28 466666 48 .800000 8 .002222 28 .007778 48 .013333 9 .150000 29 .483333 49 .811)666 9 .002500 29 .008056 49 .013611 10 .166666 30 .500000 50 .833333 10 .002778 30 .008333 50 .013889 11 .183333 31 .516666 51 .850000 11 .003056 31 .008611 51 .014167 12 .200000 32 .533333 52 .866666 12 .003333 32 .008889 52 .014444 13 .216666 33 .550000 53 .883333 13 .003611 33 .009167 53 .014722 14 .233333 34 .566666 54 .900000 14 .003889 34 .009444 54 .015000 15 .250000 35 .583333 55 .916666 1 15 .00*167 35 .009722 55 .015278 1ft .266666 36 .600000 56 .933333 16 .004444 36 .010000 56 .015556 17 .283333 37 .616666 57 .950000 17 .004722 37 .010278 57 .015833 18 .300000 38 .633333 58 .966666 18 .005000 38 .010556 58 .016111 19 .316666 39 .650000 59 .9S3333 19 .005278 39 .010833 59 .016389 20 .333333 40 .666666 60 1.000000 20 .005556 40 .011111 60 .016667 of the outer one ; and To find the area of a circle. Square the diam; mult this square by .7854; or more accurately by .78539816; or square the cir- cumf ; mult this square by .07958 ; or more accurately by .07957747 ; or mult half the diam by half the circumf ; or refer to the following table of areas of circles. Also area = sq of rad X 3.1416. The area of a circle is to the area of any circumscribed straight-sided fig, as the circumf of th< circle is to the circumf or periphery of the fig. The area of a square inscribed in a circle, is equal to twice the square of the rad. Of a circle in a square, - square X .7854. It is convenient to remember, in rounding off a square corner a b c, by a quarter of a circle, that the shaded area a b c is equal to about 1. part (correctly .2146) of the whole square abed. To find the breadth of a circular ring-. Having its area, and the diam of its outer circle. Find the area of the whole circle. From it take the area of the ring. Mult the rem by 1.2732. Take the sq rt of the prod. This sq rt will be the diam of the inner circle. Take it from the diam div the rem by 2, for the reqd breadth. To find the length of a circular arc. Find the chord a & of half the arc ; * and mult it by 8. From the prod take the chord a c of the whole arc ; div the rem by 3. This is a close approximation tor flat arcs. It always gives them a little too short. When greater accuracy is required, add as follows, for semicircles, -yx^ P ar t; rise * *i*; rise T 3 irh; rise *> -5-7?; rise i> Twn rise * refer to the two tables of arcs on pages 21 and 23. REMARK. It may frequently be of use to remember, that in any arc bos, not exceeding 29, or in other words, whose chord b s is at least sixteen times its rise, the middle ordinate a o, will be one half of a c, quite near enough for many purposes ; 6 c and s c being tangents to the arc. t And vice versa, if in such an arc we make o c equal ao, then will c be, very nearly, the point at which tangents from the ends of the arc will meet. Also the middle ordinate n, of the half arc o 6. or o *, will be approximately J4 of a o, the mid ord of the whole arc. Indeed, this last observation will apply near enough for many approximate uses even if the arc be as great as 45 ; for if in that case we take %of o a for the ord n, n will then be but 1 part in 103 too small ; and therefore the principle may often be used in drawings, for finding points in a curve of too great rad to be drawn by the dividers ; for in the same manner, y of n will be the mid ord for the arc n b or no; and so on to any extent. On p 434 will be found a table by which the rise or middle ord of a half arc can be obtained with greater accuracy when required for more exact drawings. * To find the chord of half the arc, add together the square of the rise ; and the square of half th pan ; and take the sq root of the sum. t At 29 o c thus found will be but about P art to short. t .0002777778. 18 MENSURATION. TABL.E OF CIRCLES. Circumferences or areas intermediate of those in the table, may be found by simple arithmetics proportion. The diameters, &c, are in inches ; but it is plain that if the diauis are takeu as feet^ yards, &c, the other parts will also be in those same measures. See p 6?7. No errors. Diam. Ins. Circumf. Ins. Area. Sq. Ins. Diam. Ins. Circumf. lus. Area. Sq. Ins. Diam. Ins. Circumf. Area. Sq.Ins. Diam. Ins. Circumf. lus. Area.. Sqlns. 1-64 .049087 .00019 '>. 1 A 10.9956 9.6211 10^ 31.8086 80.516 19 H' 60.4757 291.04 1-32 .098175 .00077 9-16 11.1919 9.967b 34 32.2013 82.516 % 60.8G84 294.83 3-64 .147262 . 00173 % 11.3883 10.321 32.5940 84.541 $6 61.2611 298.65 1-16 .196350 .00307 11-16 11.5846 10.680 V 32.9867 86.590 % 61.6538 302.49 3-32 .291524 .00890 % 11.7810 11.045 % 33.3794 88.664 % (2.04J5.J 306.35 .392699 .01227 13-16 11.9773 1 11.416 33.7721 90.763 % 62.4392 310.24 5-32 .490874 .01917 % 12.1737 11.793 34.1618 92.886 20. 62.8319 314.16 3-16 .589049 .0*761 15-16 12.3700 12.177 11.* 34.5575 95.033 H 63.2246 318.10 7-32 .687223 .03758 4. 12.5664 12.566 i 34.9502 97.205 M 63.6173 322.06 .785398 .6499 1-16 12.7627 12.962 \/ 35.:; 429 99.402 64.0100 320.05 9-32 .883573 .06213 H 12 9591 13.364 % 35.7356 101.62 sj 64.4026 330.06 5-16 .981748 .07670 3-16 13.1554 13.772 36.1283 103.87 g 64.7953 334.10 11-32 1.07992 .09281 H 13.3518 14.186 36.5210 106.14 65.1880 338.16 1.17:410 .11045 5-16 13.5481 14.607 % 36.9137 108.43 % 65.5807 342/25 13-32 1.27627 .12962 H 13.7445 15.033 y 87.306* 110.75 21. 65.9734 346.36 7-16 1.37445 .15033 7-16 13.9408 15.466 12. 37.6991 113.10 Ji 66.3661 350.50 15-32 1.472<>2 .17257 14.1372 15.904 iv 38.0918 115.47 i/ 66.7588 354.66 H 1.57080 .19635 9-16 14.3335 16.349 L/ 38,4845 117.86 % 67.1515 358.84 17-32 1.6SS97 .2-2166 14.5299 16.800 % 38.8772 120.28 67.5442 363.05 9-16 1.76715 .24850 11-16 14.7262 17.257 % 39.2699 122.72 % 67.9309 367/28 19-32 1.86532 .27688 % 14.9226 17.721 % 39.6626 125.19 N 68.3296 371.54 H 1.96350 .3088!) 13-16 15.1189 18.190 % 40.0553 127.68 68.7223 375.83 21-32 2.06167 .8882* y 15.3153 18.665 % 40.4480 130.19 22. 69.1150 380.13 11-16 2.15984 .37122 15-16 15.5116 19.147 13. 40.8407 132.73 i^ 69.5077 384.46 23-32 2.25802 .40574 5. 15.7080 19.635 i/ 41.2334 135.30 i' 69.9004 388.82 H 2.35619 .44179 1.16 15.9043 20.129 34 41.6261 137.89 % 70.2931 393/20 25-32 2.45437 .47937 ^ ! 16.1007 20.629 % 42.0188 140.50 $4 70.6858 397.61 13-16 2.55254 .51849 3-16 16.2970 21.135 34 42.4115 143.14 % 71.0785 402.04 27-32 2.65072 .55914 i/ 16.4934 21.648 % 42.8042 145.80 % 71.4712 406.49 2.74889 .60132 5-16 16.6897 22.166 % 43.1969 14S.19 % 71.8639 410.97 29-32 2.84707 .6 450 4 H 16.8861 22.691 y 43.5896 151.20 23. 72.2506 415.48 15-16 2.94524 .6U029 7-16 17.0824 23.221 14. 43.9823 153.94 3> 72.6493 420.00 31-32 3.04342 '.737J8 K 17.2788 23.758 X 44.3750 156.70 34 73.0420 424.56 3.14159 .78540 9-16 17.4751 24.301 44.7677 159.48 y 73.4347 429.13 1 1-16 3.33794 .88664 17.6715 24.850 % 45.1604 16^2.30 y* 73.8274 43:i.74 K 3.53429 .99402 11-16 17.8678 25.406 X 45.5531 165.13 % 74.2201 438.36 3-16 3.73064 1.1075 H 18.0642 25.967 y* 45.9458 167.99 8 74.6128 443.01 3.92699 1.2272 13-16 18.2005 26.535 H 46.3385 170.87 75.0055 447.69 5-16 4.12334 1.3530 18.45'W 27.109 y 46.7312 17;;. 78 24. 75.3982 452 39 4 31989 1.4849 15-16 18.6.332 27.688 15. 47.1239 176.71 l /i 75.7909 457.11 7-16 4.51604 1.6230 6. 18.8496 28.274 H 47.5166 179.67 34 76.1836 461.86 ^ 4.712W 1.7671 19.2423 29.465 y. 47.9093 182.65 * 765763 466.64 9-16 4.90874 1.9175 \/ 19.6350 30.680 y& 48.3020 1185.66 76.9690 471.44 %' 5.10509 2.0739 y 20.0277 31.919 y* 48.6947 1188.69 % 77.3617 476/26 11-16 5.30144 2.2305 1? 20.4204 33.183 49-0874 191.75 % 77.7544 481.11 % 5.49779 2.4053 5/ 20.8131 34.472 % 49.4801 194.83 'y 78.1471 485.98 13-16 5.69414 2.5832 % 21.2058 35.785 % 49.8728 197.93 25. 78.5398 490.87 % 5.89049 2.7612 % 21.5984 37.122 16. 50.2655 201.06 i^ 78.9325 495.79 15-16 6.08684 2.9483 7. 21.9911 38.485 /^ 50.6582 204.22 3i 79.3252 500. 74 * ! 6.28319 3.1416 22.3838 39.871 H 51.0509 207.39 % 79.7179 505.71 1-16 6.47953 3.3410 i/ 22.7765 41.282 51.4436 210.60 80.1106 510.71 ^ 6.67588 3.546S 2 23.16J2 42.718 14 51.8363 213.82 % 80.5033 515.72 3-16 687223 3.7583 34 23.5619 44.179 % 217 08 H 80.8960 520.77 y 7.06858 3.9761 H 23.9516 45664 H 52l6'217 220.35 % 81.2887 525.S4 5-16 7.28493 4.2000 24.3473 47.173 53.01 14 223.65 26. 81.6814 530.93 % 7.46128 4.4301 IX 24.7400 48.707 17. * 53.4071 226.98 3^ 82.0741 53C.05 7-16 7.65763 4.6684 8. 25.1327 50.265 H 53.7998 230.33 34 82.4668 541.19 fc 7.85398 4.9087 u 25.5254 51.849 X 54.1925 233.71 % 82.8595 546.35 9-16 8.05033 5.1572 25 9181 53.456 54.5K.V2 237.10 % 83.2522 551.55 %\ 8.24668 5.4119 % 26.3108 55.088 % 54.9779 240.53 % 83.6449 556.76 11-16 8.44303 5.6727 1^ 26.7035 56.745 % 55.3706 243.98 3/ 84.0376 562.00 5.9396 % 27.0962 58.426 % 55.7633 247.45 y 84.4303 567/27 13-16' 8! 83573 6.2126 ^ 27.4889 60.132 % 56.1560 _T)0.95 27. 84.8-230 572.56 y 3 9.03208 6.1918 % 27.8816 61.862 18. 56.5487 254.47 H 85.2157 577.87 15- a, 9.22843 6.7771 9. 28.2743 63.617 i^ 56.941 4 >f)K.O'2 H 85.6084 583/21 3. 9.42478 7.0886 28.6670 65.397 34 57.3341 261 .59 86.0011 588.57 1-16 9.62113 7.3662 i/ 29.0597 67.201 y 57.7268 265.18 i,.. 86.3938 593 96 H 9.81748 7.6699 3 i 29.45-24 69.029 Jj 58.1195 268.80 % 86.7*65 5'"9.37 S-16 10.0138 7.9798 U 29.8151 70.882 &2 58.5122 272.45 % 87.1792 604.81 14 10.2102 8.2958 B/ 30.2378 72.760 \ 58.9049 276.12 % 87.5719 610/27 5-16 10.4065 8.6179 3x 30.6305 74662 'y 59.2976 279.81 28. 87.9646 615.75 %' 10.6029 8.9462 y 31.0232 76.589 19. 59.6903 283.53 1 A 88.3573 621/26 M6, 10.7992 9.2806 10. 31.4159 78.540 X 60.0830 287.27 34 88.7500 626.80 1 MENSURATION. 19 TABLE OF CIRCLES (Continued.) Diam. Circumf. Area. Diam. Circumf. Area. Diam. Circumf. Area. Diam. Circumf. Area. Ins. Ins. Sq. lus. lus. IllS. Sq. Ins. Ins. lus. Sq. lus. lus. Ins. Sq.Ins. 28% 89.1427 632.36 38. 119.31 1134.1 K 149.618 1781.4 57^ 179.856 2574.2 89.5354 637.94 X 119.773 1141.6 % 150.011 1790.8 % 180.249 2585.4 % 89.9281 643.55 y* 120.166 1149.1 y 150.404 1800.1 % 180.642 2596.7 34 90.3208 649.18 % 120.559 1156.6 48 150.796 1809.6 y 181.034 2608.0 K 90.7135 654.84 y* 120.951 1164.2 H 151.189 1819.0 H 181.427 2619.4 29. 91.1062 660.52 % 121.344 1171.7 151.582 1828.5 % 181.820 2630.7 H 91.4989 666.23 % 121.737 1179.3 151.975 1837.9 58. 182.212 2642.1 i/ 91.8916 671.96 % 122.129 1186.9 i^ 152.367 1847.5 H 182.605 2653.5 % 92.2843 677.71 39. 122.522 1194.6 M 152.760 1857.0 M 182.998 2664.9 X 92.6770 683.49 X 122.915 1202.3 H 153.153 1866.5 183.390 2676.4 % 93.0697 689.30 H 123.308 1210.0 y 153.545 1876.1 y% 183.783 2687.8 H 93.4624 695.13 % 123.700 1217.7 49. 153.938 1885.7 % 184.176 2699.3 A 93.8551 700.98 X 124.093 1225.4 H 154.331 1895.4 % 184.569 2710.9 30. 94.2478 706.86 % 124.486 1233.2 H 154.723 1905.0 K 184.961 2722.4 U 94.6405 712.76 H 124.878 . 1241.0 % 155.116 1914.7 59. 185.354 2734.0 3 95.0332 718.69 % 125.271 1248.8 % 155.509 1924.4 M 185.747 2745.6 % 95.4259 724.64 40. 125.664 1256.6 155.902 1934.2 h 186.139 2757.2 K 95.8186 730.62 H 126.056 1264.5 % 156.294 1943.9 186.532 2768.8 96.2113 736.62 126.449 1272.4 % 156.687 1953.7 y*. 186.925 2780.5 % 96.6040 742.64 % 126.842 1280.3 50. 157.080 1963.5 % 187.317 2792.2 J* 96.9967 748.69 J^ 127.235 1288.2 H 157.472 1973.3 % 187.710 2803.9 31. 97.3894 754.77 N 127.627 1296.2 157.865 1983.2 % 188.103 2815.7 M 97.7S21 760.87 H 128.020 1304.2 % 158.258 1993.1 60. 188.496 2827.4 y 98.1748 766.99 % 128.413 1312.2 y* 158.650 2003-0 H 188.888 2839.2 8 98.5675 773.14 41. 128.805 1320.3 H 159.043 2012.9 X 189.281 2851.0 M 98.9602 779.31 H 129198 1328.3 159.436 2022.8 H 189.674 2862.9 % 99.3529 785.51 y* 129.591 1336.4 % 159.829 2032.8 % 190.066 2874.8 99.7456 791.73 % 129.983 1344.5 51. 160.221 2042.8 190.459 2**6.6 % 100.138 797.98 y* 130.376 1352.7 M 160.614 2052.8 % 190.852 2898.6 82 100.531 804.25 % 130.769 i:;no.8 M 161.007 2062.9 % 191.244 2910.5 K 100.924 810.54 % 131.161 1369.0 H 161.399 2073.0 61. 191.637 2922.5 M 101.316 816.86 % 131.554 1:57;.'.' U 161.792 2083.1 H 192.030 2934.5 % 101.709 823.21 42. 131.947 1385.4 % 162.185 2093.2 M 192.423 2946.5 8 102.102 829.58 H 132.310 1393.7 H 162.577 2103.3 % 192.815 2958.5 % 102.494 835.97 % 132.732 1402.0 % 162.970 2113.5 % 193.208 2970.6 N 102.887 842.39 h 133.125 1410.3 52. 163.363 2123.7 % 193.601 2982.7 % 103.280 848.83 H 133.518 1418.6 tf 163.756 2133.9 % 193.993 2994.8 83. 103.673 855.30 133.910 1427.0 164.148 2144.2 H 194.386 3006.9 H 104.065 861.79 H 134.303 1435.4 % 164.541 2154.5 62. 194.779 3019.1 N 104.458 868.31 % 134.696 1443.8 y% 164.934 2164.8 K 195.171 3031.3 % 104.851 874.85 43. 135.088 1452.2 % 165.326 2175.1 y* 195.564 3043.5 ^ 105.243 881.41 y 135.481 1460.7 H 165.719 2185.4 % 195.957 3055.7 N 105.636 888.00 % 135.874 1469.1 % 166.112 2195.8 M 196.350 3068.0 ?4' 106.029 894.62 % 136.267 1477.6 53. 166.504 2206.2 % 196.742 30H0.3 Ji 106.421 901.26 H 136.659 1486.2 N 166.897 2216.6 H 197.135 3092.6 B4 106.814 907.92 % 137.052 1494.7 M 167.290 2227. % 197.528 3104.9 K 107.207 914.61 H 137.445 1503.3 8 167.683 2237.5 63 197.920 3117.2 107.600 921.32 % 137.837 1511.9 y 3.097 1017.9 R 143.335 1634.9 M 173.573 2397.5 % 203.81 1 3305.6 H 3.490 1025.0 % 143.728 1643.9 H 173.966 2408.3 65. 204.204 3318.3 J4 13.883 10321 % 144.121 1652.9 y* 174.358 2419.2 M 204.596 3331.1 N 114.275 1039.2 46. 144.513 1661.9 N 174.751 2430.1 N 204.989 3343.9 3^ 114.668 1046.3 H 144.906 1670.9 H 175.144 2441.1 % 205.382 3856.7 % 115.061 1053.5 y*. 145.299 1680.0 y 175.536 2452.0 y* 205.774 3369.6 % 115.454 1060.7 % U5.691 1689.1 56 175.929 2463.0 % 206.167 3382.4 % 115.846 1068.0 N 146.0K4 1698.2 K 176.322 2474.0 M 206.560 3395.3 17 16.2:59 1075.2 N 146.477 1707.4 N 176.715 2485.0 % 206.952 3408.2 ^ 6.632 1082.5 146.869 1716.5 % 177.107 2496.1 66. 207.345 3421.2 H 7.024 1089.8 % 147.262 1725.7 % 177.500 2507.2 fc 207.738 3484.2 % 7.417 10W7.1 47 147.655 734.9 % 177.893 2518.3 y\ 208.131 3447.2 7.810 1104.5 H 148.048 744.2 % 178.285 2529.4 % 208.523 3460.2 8.202 1111.8 M 148.440 753.5 % 178.678 2540.6 y* 208.916 3473.2 % 118.596 1119.2 % 148.833 762.7 57. 179.071 2551.8 209.309 3486.3 % 1.18.988 1126.7 X 149.226 772.1 H 179.463 2563.0 % 209.701 3^99.4 20 MENSURATION. TABLE OF CIRCLES (Continued.) Diam. Circumf. Area. Diam. Circumf. Area. iam. Dircumf. Area. Diam. Circumf. Area. 83^ 262.716 92. 6647.6 210.094 3512.5 75J4 236.405 4447.4 5492.4 289.027 67. 210.487 3525.7 236.798 4462.2 H 263.108 5508.8 289.419 66657 210.879 3538.8 xti 237.190 4477.0 263.501 5525.3 IX 289.812 6683.8 34 211.272 3552.0 K 237.583 4491.8 84. 263.894 5541.8 H 290.205 6701.9 9i 211.665 3565.2 % 237.976 4506.7 H 264.286 5558.3 y% 290.597 6720.1 IX 212.058 3578.5 % 238.368 4521.5 34 264.679 5574.8 Y 290.990 6738.1 % 212.450 3591.7 76. 238.761 4536.5 % 265.072 5591.4 % 291.383 6756.4 x4 212.843 3605.0 239.154 4551.4 y% 265.465 5607.9 K 291.775 6774.7 213.236 3618.3 IX 239.546 4566.4 % 265.857 5624.5 93. 292.168 6792.9 68. 213.628 3631.7 &/ 239.939 4581.3 H 266.250 5641.2 292.561 6811.2 214.021 3645.0 y* 240.332 4596.3 266.643 5657.8 x4 292.954 6829.5 IX 214.414 3658.4 % 240.725 4611.4 85. 267.035 5674.5 % 293.346 6847.8 su 214.806 3671.8 H 241.117 4626.4 267.428 5691.2 y% 293.739 6866.1 1Z 215.199 3685.3 241.510 4641.5 lx 267.821 5707.9 6X 294.132 6884.5 g 215.592 3698.7 77. 241.903 4656.6 zi 268.213 5724.7 H 294.524 6902.9 215.984 3712.2 X 242.295 4671.8 y 268.606 5741.5 % 294.917 6921.3 xl 216.377 3725.7 242.688 4686.9 % 268.999 5758.3 94. 295.310 6939.8 69. 216.770 3739.3 y 243.081 4702.1 % 269.392 5775.1 M 295.702 6958.2 217.163 3752.8 y* 243.473 4717.3 T/ 269.784 5791.9 296.095 6976.7 34 217.555 3766.4 243.866 4732.5 86. 270.177 5808.8 % 296.488 6995.3 217.948 3780.0 ax 244.259 4747.8 270.570 5825.7 296.881 7013.8 TX 218.341 3793.7 % 244.652 4763.1 y 270.962 5842.6 % 297.273 7032.4 K 218.733 3807.3 78. 245.044 4778.4 % 271.355 5859.6 % 297.666 7051.0 219.126 3821.0 r^ 245.437 4793.7 c 271.748 5876.5 % 298.059 7069.6 TX 219.519 3834.7 34 245.830 4809.0 xl 272.140 5893.5 95. 298.451 7088.2 70. 219.911 3848.5 246.222 4824.4 fix 272.533 5910.6 H 298.844 7106.9 220.304 3862.2 H 246.615 4839.8 Jf 272.926 5927.6 M 299.237 7125.6 x4 220.697 3876.0 %\ 247.008 4855.2 87. 273.319 5944.7 g 299.629 7144.3 sz 221.090 3889.8 % 247.400 4870.7 273.711 5961.8 300.022 7163.0 rx 221.482 3903.6 y 247.793 4886.2 ix 274.104 5978.9 xi 300.415 7181.8 &x 221.875 3917.5 79. 248.186 4901.7 274.497 5996.0 a/ 300.807 7200.6 ax 222.268 3931.4 ^ 248.579 4917.2 i< 274.889 6013.2 Tj 301.200 7219.4 tx 222.660 3945.3 % 248.971 4932.7 *y 275.282 6030.4 96. 301.593 7238.2 71. 2-23.053 3959.2 % 249.364 4948.3 X, 275.675 6047.6 i/ 301.986 7257.1 223.446 3973.1 y* 249.757 4963.9 276.067 6064.9 ^ 302.378 7-276 IX 223.838 3987.1 % 250.149 4979.5 88. 276.460 6082.1 xl 302.771 7294.9 SU 224.231 4001.1 %\ 250.542 4995.2 276.853 6099.4 If 303.164 7313.8 IX 224.624 4015.2 % 250.935 5010.9 lx 277.246 6116.7 xi 303.556 7332.8 M 225.017 4029.2 80. 251.327 5026.5 y 277.638 6134.1 S 303.949 7351.8 | 225.409 4043.3 251.720 5042.3 278.031 6151.4 304.342 7370.8 225.802 4057.4 Hi 252.113 5058.0 $ 278.424 6168.8 97.^ 304.734 7389.8 72. 226.195 4071.5 %j 252.506 5073.8 278.816 6186.2 if 305.127 7408.9 226.587 085.7 yz\ 252.898 5089.6 y 279.209 6203.7 Tj 305.520 7428.0 i/ 226.980 099.8 %\ 253.291 5105.4 89. '279.602 6221.1 3x 305.913 7447.1 a/ 227.373 114.0 % 253.684 5121.2 279.994 6238.6 y 306.305 7466.2 i^ 227.765 128.2 % 254.076 5137.1 y 280.387 6256.1 xi 306.698 7485.3 M 228.158 142.5 81. 254.469 5153.0 % 280.780 6273.7 1 307.091 7504.5 i 228.551 156.8 254.862 5168.9 IX 281.173 6291.2 307.483 7523.7 228944 4171.1 Kl 255.254 5184.9 H 281.565 6308.8 98. * 307.876 7543.0 73. 229.336 4185.4 % 255.647 5200.8 * 281.958 6326.4 i/ 308.269 7562.2 229.729 4199.7 3^i 256.040 5216.8 % 282.351 6344.1 Jx 308.661 7581.5 | 230.122 4214 1 S 256.433 5232.8 90. 282.743 6361 .7 309.054 7600.8 230.514 4228.5 1 256.825 5248.9 283.136 6379.4 Xl 309.447 7620.1 !,< 230.907 4242.9 257.218 5264.9 34 283.529 6397.1 309.840 7639.5 xi 2#lv300 4257.4 82. 257.611 5281.0 % 283.921 6414.9 a, 310.232 7658.9 a. 2jf$ . .474 .366711 .327 .223216 .364 .258395 401 .294350 .438 .330858 .475 .367710 .328 .224154 .365 .259358 402 .295330 .439 .331851 .476 .368708 .329 .225094 .366 .260321 403 .296311 .440 .332843 .477 .369707 .330 .226034 .367 .261285 404 .297292 441 .333836 .478 .370706 .331 .226974 .368 .262249 405 .298274 .442 .334829 .479 .371705 .332 .227916 .369 .263214 406 .299256 .443 .335823 .480 .372704 .333 .228858 .370 .264179 407 .300238 .444 .336816 .481 .373704 .334 .229801 .371 .265145 408 .301221 .445 .337810 .482 .374703 .335 .230745 .372 .266111 409 .302204 .446 .338804 .483 .375702 .336 .231689 .373 .267078 410 .303187 .447 .339799 .484 .376702 .337 .232634 .374 .26^046 411 .304171 .448 .340793 .485 .377701 .338 .233580 .375 .269014 412 .305156 .449 .341788 .486 .378701 .339 .234526 .376 .269982 413 .306140 .450 .342783 .487 .379701 .340 .235473 .377 .270951 414 .307125 .451 .343778 .488 .380700 .341 .236421 .378 .271921 415 .308110 .452 .344773 .489 1 .381700 .342 .237369 .379 .272891 416 .309096 .453 .345768 .490 .382700 .343 .238319 .380 .273861 417 .310082 .454 .346764 .491 .383700 .344 .239268 .381 .274832 418 .311068 .455 .347760 .492 .384699 .345 .240219 .382 .275804 419 .312055 .456 .348756 .493 .385699 .346 .241170 .383 .276776 .420 .313042 .457 .349752 .494 .386699 .347 .242122 .384 .277748 .421 .314029 .458 .350749 .495 .387699 .348 .243074 .385 .278721 .422 .315017 .459 .351745 .496 .388699 .349 .244027 .386 .279695 .423 .316005 .460 .352742 .497 .389699 .350 .244980 .387 .280669 .424 .316993 .461 .353739 .498 .390699 .351 .245935 .388 .281643 .425 .317981 .462 .354736 .499 .391699 .352 .246890 .389 .282618 .426 .318970 .463 .355733 .500 .392699 To find the area of a circular zone abed. Knowing the diam of the circle ; the chords a b, and c d ; and the heights o m, and n, of the segments a m b, and end. First find the area of the entire circle; then by means of the preceding table of circular segments, find the areas of the two segments amb, and end; and subtract them from the area of the circle. To find the area of a circular lune abco. A lune is a crescent- shaped fig, comprised between two arcs of circles of diff diams. Having the chord a c, and the heights of the two segments a o c, and a b c, find the areas of those segments ; take the least of these areas from the greatest ; the rem is evidently the area of the lune. THE 26 MENSURATION. An ellipse is a curve, eeee, Pig 1, formed by an oblique section of either a cone or a cylinder, pass- ing through its curved surface, without touching the base. Its nature is such that if two line*, a.* , , . , from any point n in its periphery or circumf, to two certain points/ and g, situated in its long diani c w, (and called the foci of the ellipse,) they will be equal to any other nf and n q. Fig 2, be drawn , d in its long diani c , , two lines, as b /, and b g, drawn from any other point, as 6, in the circumf, to the foci / and g also any Rule. As ?/ m 2 : s a Example, y m ich lines will together be equal to the long diam c w. The line c w dividing the ellipse into tw equal parts lengthwise, is called its transverse, or major axis, or long diam ; and a b, which divides it equally at right-angles to cw, is called the conjugate, or minor axis, or short diam. To find the posi- tion of the foci of au ellipse, from either end, as b, of the short diam, measure off the dists bf and 6 g, Fig 2, each equal to o c, or one-half the long diain. The parameter of an ellipse is a certain length obtained thus ; as the long diam : short diam : : short diam : para- meter; or the short diam 2 parameter is equal to . long diam Any line b a, or c d. Fig 3, drawn from the circumf, to, and at right angles to, either diam. is called an ordirmte ; and the part a y, or c x of that diam, between the ord and the circumf, is called an abscissa, or absciss. To find the length of any ordinal Q a b, drawn to the long diam .ym. Knowing the absciss y a, and the two diams y m, ex. : y a X m ' ab z . s x 3 : y a = 2 ; am = 6. Then as 64 : 9 : : 12_:_ 1.6875. Hence a b = 1/1.6875 = 1.299. Or, mult together the two parts y a and a m, Fig 3, into which the ord divides the long diam ; take the sq rt of the prod; mult this sq rt by the short diam ; div the prod by the long diam. REM. Neither of these rules (nor any other) applies also to ordinates like c d, drawn to the short diam x s. To find the circumf of an ellipse. Mathematicians have furnished practical men with no simple working rule for this purpose. The so-called approximate rules do not deserve the name. They are as follows, D being the long diam : and d the short one.* RULE 1. Circumf = 3.1416 D + <* RULE . 3.1416 / ^P!+^!:\ RULE 8. 2.2 \/ 2 this is the same as Rule 2, but in a diff shape. RULE 4. 2j/ D2-f- 1.4674 d2. Now, in an ellipse whose long and short diams are 10 and 2, the circumf is actually 21, very approximately; but rule 1 gives it = 18.85 ; rule 2, or 3, = 22.65 ; and rule 4, n 20.51. Again, if the diams-be 10 and 6. the cir- cumf actually = 25.59; but rule 4 gives 24.72. These examples show that none of the rules usually given are reliable. The following one by the writer, is sufficiently exact for ordinary purposes; not being in error probably more than 1 part 'in 1000. When D is not more than 5 times as long as d, then, calling the diff between them, Diff, the Circumf = 3. U16 - ^EEi - ? if !- V 2 / 8.8 If B is more than 5 times d, then instead of dividing Diff 2 by 8.8, div it by the number in the following table: D = 6cZ 9 D = 14d 9.6 D = 40 d 9.98 7 9.2 16 9.68 50 10.04 8 9.3 18 9.75 60 10.10 9 9.35 20 9.8 70 10.17 10 9.4 25 9.87 80 10.23 12 9.5 30 9.92 100 10.35 In words, this rule is as follows : Square both D and d ; add these squares together ; div the sum by 2; call the quot A. Next subtract d from D; square the Diff: div this square by 8.8 (or by the proper number from the table) ; subtract the quot from A ; take the sq rt of the reni; mult this sq rt by 3. 1416. The following table of semi-elliptic arcs, has been prepared by this rule. b * The full table in Mr. Haswell's book, Adcock's, and others, is incorrect, esp*. oially OQ the first p*ge. For a more recent rule, see p 680. MENSURATION. 27 TABLE OF &EXGTHS OF SEm-EEMPTIC ARCS. (Original.) Heights. Lengths. Heights. Lengths. Heights. Lengths. Heights. Lengths. .005 1.000 .130 1.079 .255 1.219 .380 1.390 .01 1.001 .135 1.084 .260 1.226 .385 1.397 .015 1.002 .140 1.089 .265 1.233 .390 1.404 .02 1.003 .145 1.094 .270 1.239 .395 1.412 .025 1.004 .150 1.099 .275 1.245 .400 1.419 .03 1.006 .155 1.104 .280 1.252 .405 1.426 .035 1.008 .160 1.109 .285 1.259 .410 1.434 .04 1.011 .165 1.115 .290 1.265 .415 1.441 .045 1.014 .170 1.120 .295 1.272 .420 1.449 .05 1.017 .175 1.125 .300 1.279 .425 1.456 .055 1.020 .180 1.131 .305 1.285 .430 1.464 .06 1.028 .185 1.137 .310 1.292 .435 1.471 .065 1.026 .190 1.142 .315 1.298 .440 1.479 .07 1.029 .195 1147 .320 1.305 .445 1.486 .075 1.032 .200 1.153 .325 1.312 .450 1.494 .08 1.036 .205 1.159 .330 1.319 .455 1.501 .085 1.039 .210 1.165 .335 1.325 .460 1.509 .09 1.043 .215 1.171 .340 1.332 .465 1.517 .095 1.046 .220 1.177 345 1.339 .470 1.524 .100 1.051 .225 1.183 .350 1.346 .475 1.532 .105 1.055 .230 1.189 .355 1.353 ,480 1.540 .110 1.059 .235 1.196 .360 1.361 .485 1.547 .115 1.064 .240 1.202 .365 1.368 .490 1.555 .120 1.069 .245 1.207 .370 1375 .495 1.563 .125 1.074 .250 1.213 .375 1.382 .500 1.571 To find the area of an ellipse. Mult the two diams together; mult the prod by .7854. Ex. D = 10; d = 6. Then 10 X 6 X .7854 47. 124 area. The area of an ellipse is a mean proportional between the areas of two circles, described on its two diams ; therefore it may be found by mult together the areas of those two circles ; and taking the sq rt of the prod. The area of an ellipse is therefore always greater than that of the circular sec- tion of the cylinder from which it may be supposed to be derived. To find the diam of a circle of the same area as a given ellipse. Mult together the 2 diams ; take sq rt of prod. To find the area of an elliptic segment whoso base is parallel to either diam. Div the height of the segment, by that diam of which said height is a part. From the table of cir. cnlar segments take out the tabufar area opposite the quot. Mult together this area, the long diam, and the short diam. To draw an ellipse. Having its long and short diams a I and c d, Fig 4. RULE 1. From either end of the short diam, as c, lay off thedists cf, c/',each equal to e a, or to one-half of the long diam. The - points /, /'are the foci .of the ellipse. Pre- pare a string,/' n f, or /' g f, with a loop at each end ; the total length of string from end to end of loop, being equal to the long diam. Place pins at/ and /' ; and placing the loops ever them, trace the curve by a pencil, which in every position, asatn. or g, keeps the string /' nf, or/' gf, equally stretched all the time. Note. Owing to the difficulty of keeping the string equally stretched, this method is not as satisfactory as the following. RULE 2. On the edge of a strip of paper w s, mark w I equal to half the short diam ; and ws equal half the long diam. Then in whatever position this strip be placed, keep- ing I on the long diam, ami s on the short diam, to will mark a point in the circumf of the ellipse. We may thus obtain as many such points as we please ; and then draw the curve through them by hand. RULE 3. de 28 MENSURATION. To draw a tangent 1 1, at any point n of an ellipse. Draw nf and n/', to the foci ; bisect the angle fnf by the line xp ; draw t n t at right angle* to xp. To draw a joint -n,p, of an elliptic arch, from any point , in the arch. Proceed as in the foregoing rule for a tangent, only omitting tt; np will be the reqd joint. THE PARABOLA. b r F,'4 1 The common or conic parabola, o & c, Fig 1, is a curve formed by cutting a cone in a direction b a, parallel to its side. The curved line o b c itself is called the perimeter of the parabola ; the line o c is called its base ; b a it* height or axis ; b its apex or vertex; any line e , or o a, Fig 2, drawn from the curve, to, and at right angles to, the axis, is an ordinate ; and the part s o, or a b, of the axis, between the ordiuate and the apex b, is an abscissa. The focus of a parabola is that point in the axis, where the abscissa b s, is equal to one-half of the ord e a. The dist from the apex to the focus, is called the focal dist. The focus may be entirely beyond or outside of the curve itself. Its dist from the apex is found thus : square any ord, as o a ; div this square by the abscissa 6 a of that ord ; div the quot by 4. The nature of the parabola is such that its abscissas, as b s, b a, &c, are to each other as, or in proportion to, the squares of their respective ords e s, o a, &c ; that is, as 6 s : b a : : e 2 ; o a 2 ; or b s : e 2 : ; 5 a : o a* . If the square of any ord be divided by its abscissa, the quot will be a constant quantity ; that is, it will be equal to the square of any other ord divided by its abscissa. This quot or constant quan- tity is also equal to a certain quantity called the parameter of the ellipse. Therefore the parameter may be found by squaring e s, or o a, (one-half of the base,) and dividing said square by the height 6 , or b a, as the case may be. If the square of any ord be divided by the parameter, the quot will be the abscissa of that ord. To find the length of a parabolic curve. The so-called approximate rule given by various pocket-books, is, like those for the ellipse, entirely unreliable. It is 2 X V ( 1 A base)2 + 1^ times the (Heights) This rule is, however, close (about 1 per ct in excess) for parabolas, whose height is not more than 1-lOth of the base; and still more so for flatter ones. But in a parabola whose height is 2, and base 1, it gives a curve of 4.73 ; whereas it is actually but about 4.2 ; being an error of nearly 13 per cent, or 1 in 1%. The following by the writer is correct within perhaps 1 part in 00, in all cases ; aud will therefore answer for many purposes. Let adb, Fig 3, or n, a'd. Fig 4, be the parabola, in which are given the base a b or n d; and the height c d or c a. Imagine the complete fig a d b s, or n a d b, to be drawn ; aud in either case, assume its long diarn a b to be the chord or base ; aud one- half the short diam, or c d, to be the height, of a circular arc. Find the length of this circular arc, by means of the rule and table given for that pur- pose. Then div the chord or base a b, or n d of the parabola, by its height c d or c a. Look for the quot in the column of bases in the following table, and take from the table the corresponding multiplier. Mult the length of the circular arc by this; the prod will be the length of arc adb, or n a d, M the case may be. For bases of parabolas less than .05 of the height, or greater than 10 times the height, the multiplier is 1, and is very approx- imate ; or in other words, the parabola will be of almost exactly the same length as the circular arc. To find the area of a parabola m a n &. Mult its base m n, Fig 5, by its height a b ; and take %ds of the prod. The area of any segment, as u b v, whose base M v is parallel to mn, is found in the same way, using u v and a b, instead of mn and a b. To find the area of a parabolic zone, or frus- tum, as vi n u v. RULE 1. First find by the preceding rule the area of the whole parabola m b n ; then that of the segment u b v ; and subtract the last from the first. RULE 2. From the cube of m n, take the cube of u v: call the diff c. ULE . rom e cue o m , a o . From the square of m n, take the square of tt v ; call the diff s. Div c by i. Mult the quot by %ds of the height a . MENSURATION. 29 Original. Base. Mult. Base. Mult. Base. Mult. Base. Mult. .05 1.000 1.10 .999 215 .949 3.20 .983 .10 1.001 1.15 .997 2.20 .951 3.30 .984 .15 1.002 1.20 .995 2.25 .954 3.40 .985 .20 1.004 1.25 .993 2.30 .956 3.50 .986 .25 1.006 1.30 .990 2.35 .958 3.60 .987 .30 1.007 1.35 .987 2.40 .960 3.70 .988 .35 1.007 1.40 .984 2.45 .962 3.80 .989 .40 1.008 1.45 .980 2.50 .963 3.90 .990 .45 1.009 1.50 .977 2.55 .965 4.00 .991 .50 1.010 1.55 .974 2.60 .967 4.25 .992 .55 1.010 1.60 .970 2.65 .969 4.50 .993 .60 1.010 1.65 .966 2.70 .970 4.75 .994 .65 1.011 1.70 .963 2.75 .972 5.00 .995 .70 1.011 1.75 .960 2.80 .973 5.25 .996 .75 1.010 1.80 .957 2.85 .975 5.50 .997 .80 1.009 1.85 .953 2.90 .976 5.75 .998 .85 1.008 1.90 .950 2.95 .978 6.00 .998 .90 1.006 1.95 .946 3.00 .979 7.00 .999 .95 1.004 2.00 .942 305 .980 8.00 1.000 1.00 1.002 2.05 .944 3.10 .981 10.00 1.000 1.05 1.001 2.10 .946 3.15 .982 To draw a parabola, cos, Fig 6. Make of equal to the height eo. Draw ct and t; and divide each of thum into any number of equal parts ; numbering them as in the Fig. Join 1, 1 ; 2, 2 ; 3, 3. &c ; then draw the curve bv hand. It will be observed that the intersections of the lines 1,1; 2, 2, &c, do not give points in , the curve ; but a portion of each of those lines forms a tan- gent to the curve. By increasing the number of divisions on c t and s t, an almost perfect curve is formed, scarcely requiring to be touched up by hand. In practice it is best first to draw only the center portions of the two lines which cross each other just above o ; and from them to work down- ward; actually drawing only that small portion of each successive lower line, which is necessary to indicate the curve. Or the parabola may be drawn thus : Let 6 c. Fig 7, be the base ; and a d the height. Draw the rectangle I nm c ; div each half of the base into any num- ber of equal parts, and number them from the center each way. Div n b, and m c into the same number of equal parts ; and number them from the top, downward. From the points on 6 c draw vert lines ; and from those at the sides draw lines to rf. Then the intersections of lines 1, 1 ; 2. 2, &c, will form points in the parabola. As in the pre- ^ ceding case, it is not necessary to draw the entire lines ; but merely portions of'them, as shown be- tween d and c. Or a parabola may be drawn by first div the height a I, Fig 5, into any number of parts, either equal or unequal; and then calculating the ordi- nates us, &c ; thus, as the height a b : square of half base am:: any absciss 6 s : square of its ord u a. Take the sq rt for us. REM. When the height of a parabola is not greater than l-10th part its base, the curve coin- cides so very closely with that of a circular arc, that in the preparation of drawings for suspen- sion bridges. &c., the circular arc may be em- ployed; or if no great accuracy is reqd,"the circle may be used even when the height is as great as ^ith base. To draw a tangent w r, Fig. 5, to a parabola, from any point tt. Draw t; s perp to axis a b ; prolong a I until 6 w equals s 6. Join w v. The Cycloid, a cT> is the curve described by a point a in the circumf of a circle, a n, during one complete revo- liKion of the circle, rolled along a straight line a 6; which is called the base of the cycloid. Th ; x % 4-32 101 23^ ) c*. i3 7 30 MENSURATION. base a ft is evidently equal to the circumf of th b generating circle an; and the axis, or height d s c, is equal to its diam. The length a c b = 4- times diam of a n. The area a c l> d z= 3 times area of circle a n ; and it.-t ceu of gray is at %ths of c d, measured from the vertex c. To draw a tang co, from any poiut e; draw e s at right angles to the axis d c ; on d c describe the gener- ating circle d t c ; join t c ; from e draw e o parallel to t c. The cycloid is the curve of quickest descent ; so that a body would full from b to c along the curve 6 m c, in less time than along the inclined plane 6 t c, or any other line. PARALLELOPIPEDS. A parallelepiped is any solid contained within six sides, all of which are parallelograms ; and those of each opposite pair, parallel to each other. We show but four of them ; corresponding to the four parallelograms; namely, the cube, Fig 1, which has all its sides equal squares; and all its angles right angles ; the right rectangular prism. Fig 2, has all its angles right angles ; each pair of oppo- site faces equal ; but not all of its faces equal ; the Rhomb, Fig 3, which has a.11 its sides equal rhom- buses ; the Rhombic prism, Fig 4; its faces, rhombuses, or rhomboids; each pair of opposite faces equal ; but not all its faces equal. All parallelepipeds are prisms. To find the solidity of any parallelopiped. Mult the area of any face, as a, b? the perp dist, p, to the opposite face. A cube 18 = 1.90985, its inscribed sphere; or, 1.27324, its inscribed cylinder; or, 3.81972, its inscribed c4 y may be supposed to represent one of these frustums.* If ttie frustum is that of an irreyular 4-sided, or polygonal prism, first div its cross section perp to its sides, into triangles, by lines drawn from ol any one of its angles, as a, 1< ig \Q%. Calculate the area of each of these triangles separately ; then consider the entire frustum to be made up of so many triangular ones; calculate the solidity of each of these by the preceding rule for triangular frustums; and add them together, for the solidity of the entire frustum. Tbe solidity of any frustum whatever, of any prism whatever, Or of a cylinder, may be found thus, find its area. Also find the cen of grai from the base to said ceu of gray. Fig. 10%. Consider either end to be the base; and c of the other end , and the perp dist n c, Mult this dist by the area of the base. To find the surface of any prism ; Figs. 5 to 1O. Whether right or oblique ; regular, or irregular ; mult the circumf of one end, by the perp dist p to the other end ; this gives the surf of the sides ; to which add the surf or area of the two ends, when reqd. Or mult the circumf measd perp to the sides by the actual length a b ; then add the ends. CYL.IICDERS. A cylinder is any solid whose ends are parallel, similar, and equal curved figs ; and whose sections parallel to the ends arc everywhere the same as the ends. Hence there are circular, elliptic, (or cylin- droids), and other cylinders; but when not other- wise expressed, the circular one is understood. A right cylinder is one whose ends are perp to its sides, as Fig" 11 ; when otherwise, it is oblique, as Fig 12. If the ends of a right circular cylinder be cut so as to make it oblique, it becomes an elliptic one ; because then both its ends, and all sections parallel to them, are ellipses. An oblique circular cylinder seldom occurs; it n^ay be conceived of by imagining the two ends of Fig 12 to be c' ' lines forming its curved sides. united by straight To find the solidity of any cylinder. "Whether circular, elliptic, &c ; right or oblique ; as in prisms, mult the area of one end by the perp dist, p, to the other end. Or mult the area measd perp to the sides, by the actual length a 6, Figs 11, 12. The solidity of a cylinder is 3 times that of a cone of the same base and height. To find the surface of any cylinder, As in prisms, mult the circumf of one end by the perp dist p to the other end ; this gives the surf of the sides; to which add that of the ends when reqd. Or mult the circumf measd perp to the sides as at c o, Fig i'2, by the actual length a b, Figs 11 and 12, and add the ends. REM. The solidity of a right cvlinder whose height equals its diam, is to the solidity of its inscribed sphere, as 3 is to 2; the curved surf of the cylinder (that is, its sides,) is equal to the surf of the sphere : and the entire surf of the cylinder, including its ends, is to the surf of the sphere .as 3 to 2; consequently the surf of the two ends of such a cylinder, is equal to half the surf of its sides. Any slant end, c, fig 10%, is au ellipse, of greater area than the circular end. To find the solidity of a cyl- iiidric ungiila, when the cutting' plane does not pass through the base. Figs 13 and 14 : mult the areaof its base, by half the sum of its greatest and least perp lengths on, and c TO. Or mult the area measd perp to the est and least actual lengths o t and g m. To find the surface of a cyl- indric ungiila, when the cutting plane does not pass through the base. Mult the circumf of its base, by the half sum of its greatest and least perp lengths o n, nnd c TO. The prod will be the curved surf of the sides; to which add the ends if reqd. Or mult its cir- cumf measured perp to its sides, as at x, Fig 14, by the half sum of the greatest and least actual lengths o t, gm. Add the ends if reqd. * Our text- hooks on mpnsuration strangely do not give rules for frustums of prisms, although such solids are of frequent occurrence. 32 MENSURATION. n Cylindric uii-n la when the cutting plane passes through the base, making m a, less than m c, or half the diam of, the circle. For the solidity, cube a 6, and take %ds of it ; which call p. Mult the area of the base a d m 6, by a c. Take the prod from p : mult the rem by the height mn; div the prod by a m. See App. p. 630. For the convex surface. Mult the diam m y of the circle, by a b ; call the prod . Mult the IT) length of the arc d m b, by a c ; take the prod from p. Mult the rem by the height mn; div the prod by a m. See App. p. 630 . CIRCULAR RINGS. For the solidity, Mult the area of transverse section of the bar of which the ring is made, by half the sum of the inner and outer diams, a a, and b 6, of the ring; mult the prod by 3.1416. For the surface, Mult together the girt of the bar ; the half sum of the two diams ; and 3.1416. PYRAMIDS AND CONES. d d A pyramid, Figs 1, 2, 3, is any solid which has for its base, a plane fig of any number of sides ; and for its sides, plane triangles, all terminating at one point d, called its apex, or top. When the base is a regular fig, the pyramid is regular; otherwise irregular. For regular figs see Polygons, p. 15. A cone, Figs 4 and 5, is a solid, of which the base is a curved fig ; and which may be considered as made or generated by a line, of which one end is stationary at a certain point d, called the apex or top, while the line is being carried around the circumf of the base, which may be a circle, ellipse, or other curve. The axis of a pyramid, or cone, is a straight line d o in Figs 1, 2, 4 ; and d i in Figs 3 and 5, from the apex d, to the centre of the base. When the axis is perp to the base, as in Figs 1, 2, 4, the solid is said to be a right one; when otherwise, as Figs 3. 5, an oblique one. When the word cone is used alone, the right circular cone, Fig 4, is understood. If snch a cone be cut, as at t. t, obliquely to its base, the new base 1 1 will be an ellipse ; and the cone d 1 1 becomes an oblique elliptic one. Fig 5 will represent either an oblique circular cone, or an oblique elliptic one, according as its base is a circle, or an ellipse. To find the solidity of any pyramid, or cone, Whether regular or irregular; right or oblique; mult the area of its base, by one-third of its perp height d o, Figs 1 to 5. Every pyramid, or cone, has one-third of the solidity of either a prism or a cylinder having the same area of base, and the same perp height; and one-half that of a hemisphere of the same base and height ; in other words, a cone, hemisphere, and cylinder of the same base and height, have solidities as 1, 2. 3. To find the surface of any regular right pyramid, or right cone. Mult the circumf or outline of its base, by the slant height ; take half the prod. This will give the surf of the sides ; to which add that of the base if reqd. In the pyramid, this slant height must be measd from d to the middle of one of the equal sides, and not along one of the edges of the pyramid. Mathematicians have been unable to devise any measurement of the surf of an oblique cone. To find the surface of an irregular pyramid. Whether right or oblique, each side must be calculated as a separate triaugle ; and the several areas added together. Add the area of base if reqd. MENSURATION. 33 To find the solid- ity of any frus- tum of any pyr- amid, or cone, when the base and top are par- allel. RULE 1. Whether regular or irregular, right or oblique, add together the areas of the base, and top, and the area of their mean proportional ; mult the sum by ^d of the perp height o o, Figs 6 and 7, between the base and top. RKM. For the area of the mean proportional, (which is not the area halfway between, and parallel with the ends,) mult together the areas of base and. top ; and take the sq rt of the prod. RULE 2. This applies only to the right circular conical frnst. Add together the squares of the two diams; and the prod of the two diams ; mult the sum by J>d the perp height o o ; mult the prod by. 7864. To find the surface of any frustum of a regular right pyramid, or cone, Figs 6 and 7, when the base and top are parallel. Add together the circumfs of the two ends ; mult the sum by the slant height t ; take half the prod. This gives the surf of the sides ; to which add that of the ends when reqd. In the frustum of the pyramid, the slant height must be measd between the middles s and t of two corresponding sides of the base, and top, Fig. 6. If the pyramid is not regular ; or if it is oblique, then the surf of the sides must be obtained for each aide separately as a trapezoid. WEDGES. aba b a b Fid 9 " Fi^ld" A wedge Is usually defined to be a solid, Figs 8 and 9, generated by a plane triangle, a n c, moving parallel *o Itself, in a straight line. This definition requires that the two triangular ends of the wedge should be parallel ; but a wedge may be shaped as in Fig 10 or 11.. We would therefore propose the following definition, which embraces all the figs ; besides various modifications of them. A solid of five plane faces; one of which is a parallelogram abed, two opposite sides of which, as a c and b d, are united by means of two triangular faces a c n, and & dm, to an edge or line n m, parallel to the other opposite sides a b and c d. The parallelogram abed may be either rectangular, or not ; the two triangular faces may be similar, or not ; and the same with regard to the other two faces. The following rule applies equally to all. To find the solidity of any wedge. Add together the length of the edge m n, and twice the length a b or c d of the back ; mult the su by the perp height p, from the edge to the back ; mult this prod by the breadth of the back, mei perp to its two sides a b and c d ; div this last prod by 6. PRISMOIDS. m eiud A prismoid is Any solid bounded by six plane surfaces, of which but two, as a 6 c d and e f g h, Figs 1 and 2, ar necessarily parallel ; and at least two other opposite ones not parallel. To find the solidity of any prismoid. Add together the areas of the two parallel surfaces ; and four times the area of the section taken halfway between them, and parallel to them ; mult the sum by the perp dist between the two parallel Bides ; and div the prod br 6. In Fig 1, A n is the perp dist ; and in Fig 2. which represents a railroad excavation, it is g c. To find the areas referred to, see Trapezofds and Trapeziums* PP 14< 15. The foregoing rule is the well-known "prismoidal formula;" the very extended application of which to other solids than those which fall strictly within the definition of the prismoid, was first 34 MENSURATION. discovered and made known by Ellwood Morris, civ eng of Philadelphia, in 1840. It embraces <# parallelepipeds, prisms, pyramids, cones, wedges, &c, whether regular or irregular, right or oblique; together with their frustums when cut parallel to their bases; indeed all solids whatever having two parallel faces, or sides, provided these two faces are uuited by surfaces, whether plane or curved, upon which, and through every point of which, a straight line may be drawn from one of the parallel faces to the other. The following six Figs represent a few such solids ; they may be regarded as one-chain lengths of railroad cuttings ; a o being the perp dist between the two parallel ends. o The prismoidal formula applies also to the sphere, hemisphere, and other spherical segments; also to any sections such as abed, and o n i d 6 c, of the cone, in which the sides a d, a c, or od, fc, are straight ; through the apex, or top a. Also to the cylinder when a plane parallel to the sides passes th rough both ends , but not if the plane w x is oblique, as in the fig, though never erring more than 1 in 142. In this last case we must imagine the plane to be extended until it cuts the side of the cylinder likewise extended ; and then by p 32 or 630 find the solidity of the ungula thus formed. Then find the solidity of the small ungula above w, also thus formed, and subtract it from the large one. SPHERES OR GLOSSES. A sphere Is a solid generated by the revolution of a semicircle around its diam. Any line passing entirely through a sphere, and through its center, is called its axis, or diam. Any circle described on the surface of a sphere, from the center of the sphere as the center of the circle, is called a great circle of that sphere; in other words, any entire circumf of a sphere is a great circle. A sphere has a greater content or solidity than any other solid with the same amount of surface; so that if the shape of a sphere be any way changed, its content will be reduced. To find I lie solidity of a sphere.* Cube its diam; mult said cube by .5236. Or cube its circumf; and mult by .01689. Or mult its surface by its diam ; and div by 6. Or refer to the following table of spheres. The solidity of. a sphere is % that of its circumscribing cylinder; or .5236 (hat of its circumscribing cube; or to 4 great circles X % diam; or to cube of rad X 4.1888; or to surface X % diam. To find the surface of a sphere. To find the solidity of a spherical segment. C RULE 1. Square the rad on, of its base; mult ' his square by 3 ; to the prod add the square of its height o s ; mult the sum by the height o s ; and mult this last prod by .5236. RULE 2. Mult the diam aft of the sphere by 3; from the prod take twice the height o s of the segment; mult the rem by the square of the height o s ; and mult this prod by .5236. *In the following table the surfaces and solidities will be inches, feet, yards, 12718 134860 IX 3739.3 21501 TX 6151.5 45367 54. 9160.8 82448 % 12768 135657 RX 3766.5 21736 % 6186.3 45753 IX 9203.3 83021 % J2818 136456 ax; 3793.7 21972 M 6221.2 46141 x4 9246.0 83598 64. 12868 137259 3821.1 22210 & 6256.1 46530 % 9288.5 84177 Ji 12918 JS8065 85. 3848.5 224-49 ax 6291.2 46922 /4 9331.2 84760 /4 12T69 138874 3876.1 22691 K 6926.5 47317 % 9374.1 85344 % 13019 139686 vx 3903.7 22934 45. 6361.7 47713 9417.2 85931 ^ 13070 140501 % 3931.5 23179 6397.2 48112 y* 9460.2 86521 xi 13121 141320 TX 3959.2 23425 IX 6432.7 48513 55. 9.503.2 87114 % 1H172 142142 RX 3987.2 23674 % 6)68.3 48916 u 9546.5 87709 % : 132^2 142966 x4 4015.2 23924 JX 6503.9 49321 9590.0 88307 65. 13273 143794 X 4043.3 24176 6/ 6T39.7 49729 xi 9633.3 88908 % 13324 144625 86. 4071.5 24429 ax 6575.5 50139 9676.8 89511 %' 13376 145460 4099.9 24685 % 6611.6 50551 xi 9720.6 90117 % : 13427 146297 I/ 4128.3 24942 46. 6647.6 50965 K 9764.4 90726 J^ ! 1347-8 147138 ax 41.56.9 25201 6683.7 51382 9808.1 91338 % 13530 147982 H 4185.5 25461 M 6720.0 1 51801 56. 9852.0 91953 H> 13582 148828 MENSURATION. SPHERES (CONTINUED.) 37 I s 1 3 O3 1 5 Surface. 1 I Q < 1 1 S s Ul 1 Ji 13633 149680 34 17437 216505 34 21708 300743 % 26446 404406 66. 13685 150533 % 17496 217597 34 21773 302100 % 26518 406060 13737 151390 % 17554 218693 21839 303463 92. 26590 407721 y4 13789 152251 y 17613 219792 34 21904 304831 34 26663 409384 13841 153114 75. 17672 220894 % 21970 306201 34 26735 411054 34 13893 153980 34 17731 222001 % 22036 307576 % 26808 412726 y 13946 154850 y 17790 223111 % 22102 308957 34 26880 414405 H 13998 155724 a/ 17849 224224 84. 22167 310340 % 26953 416086 14050. 156600 34 17908 225341 34 22234 311728 % 27026 417774 67. 14103 157480 k 17968 226463 34 22300 313118 y 27099 419464 M 14156 158363 % 18027 227588 % 22366 314514 93. 27172 421161 34 14208 159250 18087 228716 22432 315915 34 27245 422862 14261 160139 76. 18146 229848 % 22499 317318 34 27318 424567 14314 161032 34 18206 230984 8^ 22565 318726 % 27391 426277 % 14367 161927 34 18266 232124 k 22632 320140 34 27464 427991 14420 162827 ft 18326 233267 85. 22698 321556 K 27538 429710 v 14474 163731 18386 234414 34 22765 322977 27612 431433 68. 14527 164637 Y 18446 235566 34 22832 324402 K 27686 433160 14580 165547 H 18506 236719 % 22899 325831 94. 27759 434894 34 14634 166460 18566 237879 34 22966 327264 34 27833 43663d H 14688 167376 77. 18626 239041 y* 23(534 328702 34 27907 438373 14741 168295 34 18687 24020<" % 23101 330142 % 27981 440118 K 14795 169218 18748 241376 % 23168 331588 34 28055 441871 14849 170145 18809 242551 86. 23235 333039 28130 44362J jx 14903 171074 34 18869 243728 34 23303 334492 % 28204 445387 69 14957 172007 RZ 18930 244908 34 23371 335951 K 28278 447151 15012 172944 H 18992 246093 % 23439 337414 95. 28353 448920 34 15066 173883 19053 247283 34 23506 338882 J4 28428 45069o % 15120 174828 78. 19111 248475 % 23575 340352 34 28503 452475 34 15175 175774 34 19175 249672 H 23643 341829 H 28577 454259 % 15230 176723 34 19237 250873 23711 343307 y* 28652 456047 N 15284 177677 % 19298 252077 87. * 23779 344792 % 28727 457839 15339 178635 34 19360 253284 34 23847 346281 H 28802 459638 70. 15394 179595 % 19422 254496 34 23916 347772 % 28878 461439 34 15449 180559 * 19483 255713 23984 349269 96. 28953 463248 34 15504 181525 19545 256932 y 2 24053 350771 34 29028 465059 15560 182497 79. 19607 258155 24122 352277 34 29104 466875 34 15615 183471 34 19669 259383 % 24191 353785 29180 468697 % 15670 184449 34 19732 260613 y s 24260 355301 34 29255 470524 % 15726 185430 19794 261848 88. 24328 356819 % 29331 472354 v 15782 186414 34 19856 263088 34 24398 358342 % 29407 474189 71. 15837 187402 y*> 19919 264330 34 24467 359?- 69 y* 29483 476029 34 15893 188394 % 19981 265577 % 24536 361400 97. 29559 477874 34 15949 189389 I/ 20044 266829 24606 362935 34 29636 479725 % 16005 190387 80. 20106 268083 y% 24676 364476 34 29712 481579 34 16061 191389 34 20170 269342 % 24745 366019 % 29788 483438 % 16117 192395 34 20232 270604 % 24815 367568 /4 29865 485302 16174 193104 20296 271S71 89. 24885 369122 29942 487171 Ji 16230 194417 34 20358 273141 24955 870678 ?i 30018 489045 72. 16286 195433 N 20422 274416 34 25025 372240 y 30095 490924 ^ 16343 196453 20485 275694 % 25095 373806 98. 30172 492808 34 16100 197476 V 20549 276977 34 25165 375378 34 30249 494695 16156 198502 81. 20612 278'JM 25236 376954 34 30326 496588 34 16513 199532 0^ 20676 27!>f>f>3 % 25306 378531 % 30404 498486 16570 200566 34 20740 280817 25376 380115 34 30481 500388 % 16628 201604 20804 282145 90. 25447 381704 30558 502296 Ji 16685 202015 34 20867 283447 34 25518 383297 fi 30636 504208 73. 16742 203689 % 20932 284754 34 25589 384894 v 30713 506125 34 16799 204737 % 20996 286061 % 25660 386496 99. 30791 508047 34 16857 205789 21060 287378 34 : 25730 388102 34 30869 509975 % 16914 206844 82. 21124 288696 . Y% 25802 389711 30947 511906 H ! 16972 207903 21 IS!) 290019 H 25873 391327 % 31025 513843 N 17030 208966 34 21253 291345 25944 392945 34 31103 515785 17088 210032 21318 292674 91. 26016 394570 31181 517730 K 17146 211102 i/ 213* 2 294010 26087 396197 % 31259 519682 71. 17204 212175 % 21448 295347 34 26159 % 31338 521638 34 17262 213252 % 21512 296691 % j 26230 399468 100. 31416 523598 34 17320 214333 K 21578 298036 J4 26302 401109 17379 215417 83. 21642 299388 % j 26374 402756 To find the curved surface of a spherical segment. See last Fig. KUT.E 1. Mult the dinm a b of the sphere from which the segment is cut, by 3.1416; mult the prod by the height o s of the seg. Add the area of base reqd. REM. Having the diam n r of the seg, and its height o s, the diam a b of the sphere may be found thus : Div the square of half the diam n r, by its height o s ; to the quot add the height o s. RULE 2. The curved surf of either ft segment, (last Fig,) or of a zouc, (next Fig,) bears the same proportion to the surf of the whole 38 MENSURATION. sphere, that the height of the seg or zone bears to the diam of the sphere. Therefore, first find tke surf of the whole sphere, either by rule or from the preceding table ; mult it by the height of the sez or zone; div the prod by diam of sphere. RULE 3. Mult the circumf of the sphere by the height o t of the seg. To find the solidity of a spherical zone* Add together the square of the rad e d, the square of rad o b and %d of the square of the perp height eo; mult the sum by 1.5708; and mult this prod by the height eo. To find the curved surface of a spher- ical zone. RULE 1. Mult together the diarn m n of the sphere ; the height e o of the zone, and the number 3.1416. Or see preceding Rule 2 for snrf of segments. Rule 2. Mult the circumf of the sphere, by the height of the zone. To find the solidity of a hollow spher- ical shell. Take from the foregoing table the solidities of two spheres having bthe diams a 6, and c d. Subtract the least from the greatest. Here a cor b d is the thickness of the shell. THE REGULAR BODIES. A regular body, or regular polyhedron. Is one which has all its sides, and its solid angles, respectively similar and equal to each other. There are but five such bodies, namely, the Tetraedron, bounded by four equilateral triangles ; the Hexaedron, or cube, bounded by six squares ; the Octaedron, bounded by eight equilateral triangles ; the Dodecaedron, bounded by twelve equilateral pentagons; and the Icosaedron, bounded by twenty equilateral triangles. To find the solidity or the surface of a regular body. For the solidity, cube the length of one of its edges; mult said cube by the correspouding number in the fol- lowing column of solidities. For the surface, square the length of one of its edges ; mult said square by the cor- responding number in the following column of surfs. The solidity of any body as a & cm, generated by a complete revolution of any fig as a b c a, around one of its sides as a c for an axis, may be found thus. Find the area of the generating fig a b c a; also find its cen of grav G, (see top of p 443.) Measure o G perp to a c, and calling o G the rad of a circle find the corresponding circumf. Mult this circumf by the area of the fig a b c a before found. In other words, mult the area of the generating fig by the circumf of the circle described by its cen of grav while revolv- ing. THE ELLIPSOID, OR SPHEROID, Is a solid generated by the revolution of an ellipse around either its long or its short diam. When around the long (or trans verse) diara, as at a, Fig 1, it is an oblong or pro- late spheroid; when around the short (or conjugate) one, as at in, in Fig 2, it is oblate. STames. Surf. Solid. Tetraedron Hexaedi-on Octaedron Dodecaedron .. Icosaedron 1.7320 6. 3.4641 20.6458 8.6602 .1178 1. .4714 7.6631 2.1817 For the solidity in either case, mult the fixed diam or axis by the square of the revolving one ; and mult the prod by .5236. THE PARABOLOID, OR PARABOLIC COXOID, next Fig, is a solid generated by the revolution of a parabola a c b, around its axis, c r. For its solidity mult the area of its base, by half its height, r c. Or inult together the square of the rad a r of the base; the height r c; and the number 1.5708. PLANE TRIGONOMETRY. 39 For the solidity of a frustum, a b g 7i, the ends of which are perp to the axis r c ; add together th squares of the two diams a It and g h ; mult the sum b mult the prod by the decimal .3927. by the height r I; To find the surface of a paraboloid, Mult the rad a r of its base, by 6.2832; div the prod by 12 times the square of the height re: call the quot p. Then add together the square of the rad a r, and 4 times the square of the height r c. Cube the sum ; take the sq rt of this cube ; from the sq rt subtract the cube of the rad a r. Mult the rent by p. Either the solidity, or the surface of a frustum, a b g h, when gh is parallel to a 6, may be found by calculating for the whole paraboloid, and for the upper portion c g h, as two separate paraboloids, and taking their diff. THE CIRCULAR S P I X I, E, Is a solid abny generated by the revolution of a circular segment a b n e a, around its chord a n as an axis. To find its solidity. RULE 1. First find the area of a b e, or half the generating circular segment. Then to the square of a e, add the square of b e; div the sum by 6 y ; from the quot take b e : mult the rein by the area of a e b ; call theprodp. Cubeae; div the cube by 3 ; from the quot take p. Multthe/ rem by 12.5664. RULE 2. When the dist o c is known, from the center of the circle to I the cen of the spindle, then mult that dist o e, by the area of a b e ; call ' the prod p ; cube ae; div the cube by 3 ; from the quot take p ; mult the rem by 12.5664. To find its surface. RULK 1. First find the length of the circular arc a b n ; and mult it hy the dist o e from the center of the circle to the center of the spindle. Call the prod p. Next mult the length an of the spindle, by the rad o b of the circle. From the prod takep; mult the rem by 6.2832. RULE 2. First find the length of the arc a b n. Square a e ; also square b e ; add these squares together; div their sum by b y ; call the quot s; and mult it by a n; call the prod p. Next from stake b e ; mult the rem by the length of the arc a b n. Subtract the prod from p ; mult the rem by 6.2832. To find the solidity of a middle zone of a circular spindle, As h a Jcp I ( a e3 -^jp) X g ) (o e X area of g h I k\ 1 X 6.2832. PLANE TRIGONOMETRY. ; given side : reqd side. PLANS trigonometry teaches how to find certain unknown parts of plane, or straight - sided tri- angles, by means of other parts which are known; and thus enables us to measure inaccessible dis- tances, &c. A triangle consists of six parts, namely, three sides, and three angles; and if we know any three of these, (except the three angles, and in the ambiguous case under " Case 2,") we can find the other three. The following fcur cases include the whole subject; the student should commit them to memory. Case 1. Having? any two angles, and one side, to find the other sides and angle. Add the two angles together; and subtract their sum from 180; the rem will be the third angle. And for the sides, as Sine of the angle . Sine of the angle opp the given side opp the reqd side Use the side thus found, as the given one; and in the same manner find the third side. As a practical example of the use of Case 1, if we measure the line or side a b along one shore of a river; and also the two angles ^ _ a b c, b a c, to an object c on the opposite shore, we can calculate the dist ac, / \ or 6 c, to that object; or by drawing the triangle on paper, to a scale, we can -J * afterward measure a c, or 6 c by the scale. D Case 2. Having two sides, and the angle opposite to one of them, To find the other side, and angles. This is of use only when that gives side which is opp the given angle, is as long, or longer than the other given side. Side opp the . Side opp the . . Sine of angle . Sine of angle given angle reqd angle opp the former opp the latter. Having found the sine, take out the corresponding angle from the table of nat sine". REM. When the given side a c. next Fig, opposite the given angle a b c, is shorter than the other given side a b, then the above proportion alone does not enable us to determine the other parts ; for it is plain that after having; drawn the side a 6. and laid down the given angle a b c, and extended it indefi- ^itely toward n, we cannot tell whether the length of a b is to be drawn from a to c, or from a to d. 40 PLANE TRIGONOMETRY. It a c la as long or longer than a 6, there can be no douot ; for in that case It cannot be drawn toward 6, but only toward n, as in the Fig. Therefore, in taking field-notes, the two sides aud the angle opp one of them, must not be depended upon as sufficient data. Case 3. Having* two sides, and the angle in- cluded between them. Take the angle from 180 ; the rem will be the sum of the two unknown angles. Div this sum by 2 ; and find the uat tang of the quot. Then as The sum of the . rr hp j r ,i:ff . . Tang of half the sum . Tang of half two given sides of the two unknown angles their diff. Take from the table of nat tai>g, the angle opposite this last tang. Add thir angle to the half sum of the two unknown angles, aud it will give the angle opp th( longest given side; and subtract it from the same half sum, for the angle opp tho shortest given side. Having thus found the angles, find the third side by Case 1- As a practical exam pie of theuse^of Ci n - , can ascertain the dist n m across a deep pond, by measuring two lines n o and m o ; and the an- gle 71 o m. From this data we may calculate nm; or by draw- ing the two sides, and the angle on paper, by a scale, we can afterward measure n m on the drawing. Case 4. Having the three sides, To find the three angles : upon one side a 6 as a base, draw (or suppose to be drawn) a perp c g from the opposite angle c. Find the ditf between the other two sides, a c and c 6 : also their sum. Then, aa Th , . Sum of the . . Diff of other . Diff of the tw s other two sides two sides parts ag and half that angle will be equal to ag and bg, of the base. Add half this diff of the parts, to half the base a b ; the sum will be the longest part a g ; which taken from the whole base, gives the shortest part g b. By this means we get in each of the small tri- angles a c ff and cgb, two sides, (namely, a c and ie writer has at times availed himself of the outer door of a house, by opening it until it pointed toward some mountain-peak, the dist of which he knew approximately ; but of the height of 42 PLANE TKIGONOMETRY. Fig. 6. its angle yti of inclination with the horizon found as before; in which case the dist an is calculated. Or if the vert height en is sought, the point c may first be found by sighting upward along a plumb-line held above the head. Ex. 3. To find the approximate height, s x, of a mountain, Of which, perfeaps, only the very summit, z, is visible above interposing forests, or other obstacles ; but the dist, wit, of which is known. In this case, first direct the instrument nor, as m h; and then measure the angle i m x. Then in the triangle i m x we have one side mi; the measd angle imx, and the angle mix (SO ), to find ix by Case 1. But to this i x we must add i o, equal to the height y m of the instrument above the ground: and also o s. Now, o s is apparently due entirely to the curvature of the earth, which is equal to very nearly 8 ins, or .667 ft in one mile ; and increases as the squares of the dists; being 4 times 8 ins in 2 miles ; 9 times 8 ins in 3 miles, &c. But this is somewhat diminished by the refraction of the atmosphere ; which varies with temperature, moisture, &c ; but alwavs tends to make the object x appear higher than it actually is. At an average, this deceptive elevation amounts to about - th part of the curvature of the earth ; and like the latter, it varies with the squares of the dists. Consequently if we subtract part from 8 ins, or .667 ft, we have at once the combined effect of curvature and refraction for one mile, equal to 6.857 ins, or .5714 ft : and for other dists, as shown in the following table, by the use of which we avoid the necessity of making separate allowances for curvature and refraction. Table of allowances to be added for curvature of the earth; and for refraction; combined. Dist. in yards. Allow, feet. Dist. in miles. Allow, feet. Dist. in miles. Allow, feet. Dist. in miles. Allow. feet. 100 .002 % .036 6 20.6 20 229 150 .004 1% .143 7 * 28.0 22 277 200 .007 3/f .321 8 36.6 25 357 300 .017 i .572 9 46.3 30 514 400 .030 H .893 10 57.2 35 700 500 .046 li| 1.29 11 69.2 40 915 600 .066 ] M 1.75 12 82.3 45 1158 700 .090 2 2.29 13 96.6 50 1429 800 .118 2^ 3.57 14 112 55 1729 900 .149 3 5.14 15 129 60 2058 1000 .185 3% 7.00 16 146 70 2SOI 1200 .266 4 -9.15 17 165 80 3659 1500 .415 &A 11.6 18 185 90 4631 3000 .738 5 14.3 19 206 100 5717 Hence, if a person whose eye is 5.14 ft, or 112 ft above the sea, sees an object just at the sea's horizon, that object will be about 3 miles, or 14 miles distant from him. A horizontal line is not a level one, for a straight line cannot be a level one. The curve of the earth, as exemplified in an expanse of quiet water, is level. In Fig 7, if we suppose the curved line t y s g to represent the surface of the sea, then the points t y s and g are on a level with each other. They need not be equidistant from the center of the earth, for the sea at the poles is about 13 miles nearer it than at the equator; yet its surface is everywhere on a level. Up, and down, rei'er to sea level. I^evel means parallel to the curvature of the sea; and horizontal means tangential to a level. Ex. 4. If the inaccessible vert height c d, Fig 8, w"hich he had no idea. For allowance for curvature and refraction see above Table. A triangle whose sides are as 3, 4, and 5, is right angled; and one whose sides are as 7 ; 7; and 9. 9; contains 1 right angle; and 2 angles of 45 each. As it is fre- quently necessary to lay down angles of 45 and 90 on the ground, these proportions may be used for the purpose, by shaping a portion of a tape-line or chain into such a triangle, and driving a stake at each angle. See p. G70. PLANE TRIGONOMETRY. occupied by the top o of the staff; and from o measure the angles iod and doc. This being done, sub, tract the angle toe from 180 ; the rcm will be the angle c o n.* Consequi ly in the triangle noc, have one side no, and two angles, c n o and con, to & find by Case 1 the side o c. Again, take the angle too" 5 from 180 ; the remainder will be the angle n o d, so that in the triangle d n o we have one side n o, and the two angles d n o and nod, to find by Case 1 the side od. Finally, in the triangle cod, we have two sides c o and o d, and their included angle c o d, to find c d, the reqd vert height. REM. If c d were in a valley, or on a hill, and the observations reqd to be made from either higher or lower ground, the operation would be precisely the same. Ex. 5. See Ex 10. To find the dist ao, Fig 9, between two entirely inaccessible objects, Measure a side n TO ; at n measure the angles anm and o n m ; also at TO measure the angles o m n, and o TO n. This being done, we have in the triangle a n TO, one side n m, Fig 9, and the angles anm, aud nma; hence, by Case 1, we can calculate the side an. Again, in the triangle o TO n we have one side n TO, and the two angles o TOW, and TO no; hence, by Case 1, we can calculate the side n o. This being done, we have in the triangle ano, two sides on, and no; and their included angle ano; hence, by Case 3, we can calculate the side oo, which is the reqd dist. It is plain that in this manner we may obtain also the position or direction of the inacces- sible line ao ; for we can calculate the angle n ao ; and can therefrom deduce that of ao; and thus be enabled to run a line parallel to it, if required* By drawing n m on pa- per by a scale, and laying down the four measd angles, the dist oo may be measd upon the drawing by the same scale. If the position of the inaccessible dist c n. Fig 10, be such that we can place a stake p in line with it.we may proceed thus : Place the instrument at any suitable point s, and take the angles psc andean. Also find the angle cpa, and measure the distj*. Then in the triangle p a c find 8 c by Case 1 ; again, the exterior angle n c s, being equal to the two interior and opposite angles c p a, and p a c, we have in the triangle can, one side and two angles to find c n by Case 1. Ex. 6. To find a dist ab, Fig 11. of which the ends only are accessible. From a and ft, measure any two lines o e, 6 c, meeting at c ; also measure the angle a c ft. Then in the triangle aft e we have two sides, and the included angle, to find the third side o ft by Case 3. Ex. 7. To find the vert height o -m, of a hill, above a given point i. Place the instrument atf; measure a TO. Directing the instrument hor, as on, take the angle nam. Then, since a n TO is 90 Fig 12, we have one side a TO, and two angles, nam and an TO, to find nm by Case I. Add no, equal to ai, the height of the instrument. Also, if the hill is a long one, add for curvature of the earth, and for refraction, as explained in Example 3, * ..-' Fig 7. Or the instrument may be placed at the top of the hill ; and an angle of depression measured ; instead of the angle of elevation n am. REM. 1. It is plain, that if the height OTO be previously known, and we wish to ascertain the dist from its sum- mit TO to any point i, the same measurement as before, of the angle n a TO, will enable us to calculate o TO by Case 1. So in Ex. 2, if the height na be known, the angles measd in that example, will enable us to compute the dist a o ; so also in Figs 3, 4, 5, and 7 ; in all of which the process is so plain as to require no further explanation. REM. 2. The height of a vert object by means Of its shadow. Plant one end of a straight stick vert in the ground ; and measure its shadow ; also measure the length of the shadow of the object. Then, as the length of the shadow of the stick is to the length of the stick above * Because, when two straight lines, as o o, and ni, intersect each other at any inclination whatever, the two adjacent angles con and toe amount to 180. Therefore, if one of them is given, we can find *he other by subtractiag ih9 sriven one from 180. 44 PLANE TRIGONOMETRY. ground, so is the length of the shadow of the object, to its height. If the object is inclined, the stick must be equally inclined. Rein. 3. Or the height of a vert object raw, Fig 12^, whose distance r m is known, may be found by its reflection in a vessel of water, or in a piece of looking glass placed perfectly hori: " ; reflectoi tal at r; for as r a is to the height ~isr~ x - made i c is to c o, so is i m to m n. The following examples may be regarded as substitutes for strict trigonome- try : and will at times be useful, iu case a table of sines, &c, is not at hand for making trigonometrical calculations. Ex. 8. To find the dist alt, of which one end only is accessible. Drive a stake at any convenient point a ; from a lay off any angle ft a c. In the line a c, at any convenient point c, drive a stake ; and from c lay off an angle acd, equal to the angle ft a c. In the line c d, at avy convenient point, as d, drive a stake. Then, standing at d, and looking at ft, place a stake o in range with d ft ; and at the same time in the line ac. Measure ao, o c, and erf; then, from the principle of similar triangles, as o c : c d : : a o : ab. Or thus: Fig 14, n h being the dist, place a stake at n ; and lay off the angle h n m 90. At any convenient dist n m, place a stake m. Make the angle h m y 90; a id place a stake at y, in range with h n. Measure, n y and n m ; then, from the principle of similar triangles, as it y : n m : : n in : n h. Or thus, Fig 14. Lay off the angle hnm = 90, placing a stake m, at any convenient dist n m. Measure n m. Also measure the angle n m h. IT- Ji 1 / Find nat tan S of n m A by Table p 103. Mult this nat tang by n m. The prod 1 10 I T- will be n h, Or thus. Lay off angle h n m = 90. From m measure the augle n TO A, and lay off angle nmy equal to it, placing a stake at y in range with h n. Then is n y = n h. Or thus, without' measuring any angle ; t u being the dist. Make u v of any convenient length, in range with t u. Measure any v o ; and o x equal to it. in range. Measure u o ; and oy equal to it in range. Place a stake z in range with both x 11, and t o. Then will y z be both equal to t u, anil parallel to it. Or thus, without measuring any angle. Drive two stakes t and u, in range with the object 8. From t lay off any convenient dist t x, in any direction. From u lay off u w parallel to t x, placing win range with x s. Make u v equal to t x. Measure w v, v x, and x t. Then, as tv v : v x : \xt\ts. Or thus. At a lay off angle o a c = 5 43'. Meas- U ure o c, and mult it by 10 for a o, too long only 1 part iu 935.6. PLANE TRIGONOMETRY. 45 Fi6 17 Ex. 9. To find the dist a ft, of which the ends only are accessible. From a lay off the angle bnc; and from 6, the angle a b d. cnch 90. Make a c and h d equal to each other. Then, cd~a 1>. Or a f) may he considered as the dist across the river in Pigs 15, 13. or 14: and be ascertained in the same way. Or measure any dist. Fig 17, a o; and make a n in line and equal to it,. Also measure ho; and mRke om in Sine and equal to it. Then will ran be both paral- lel to a 6, and equal to it. Ex. 1O. See Ex. 4. To find the entirely inaccessible clist // z, and also its direction. At any two convenient points a and b, from each of which y and z can be seen, drive stakes. Then we have the four corners of a four-sided figure, in which are given the directions of three of its sides, and of its two diags. This data enables us to lay out on the ground, the small four-sided fig a c o t, exactly similar to the large one. Thus, in the Hue a b place a stake c; and make co parallel to fez; o being at the same time in range of the diag a z. Also, from c make c i parallel to 6 y ; i being at the same time in range of a y. Then will i o be in the same direction as y z, or parallel to it. Measure a c, a b, and to ; then evidently, from the principle of similar figures, as a c : a b : : i o : i/ z. If y z were a visible line, such as a fence or road, we could from a divide it into any required portions. Thus, if we wish to place a stake halfway between y and z, first place one half- way between i and o; then standing at a, by means of signals, place a person in range on y z. Or, to find along n b, a point t perp to y z at y, first make ois = 90 ; and measure a . Then, o i : a s : : y z : a t. Ex. 11. To find the position of a point, , Fig? 19, By means of two angles a n b and b n n. taken from it to the three objects a b c, whose positions and dists apart are known. The use of this problem is more frequent in marine than in land surveying. It is chiefly employed for determining the position n of a boat from which soundings are being taken along a coast. As the boat moves from point to point to take fresh soundings, it becomes necessary A^ to make a fresh observation at each point, in order to define its position on the chart. An observation consists in the measurement by a sextant of the two angles an b, b n c. to the sig- nals a 6 c, previously arranged on the shore. When practicable, this method should be rejected; and the observa- tions taken to the boat at the same instant, by two observers on shore, at two of the stations. The boat to show a signal at the proper moment. The most expeditious mode of fixing the point n upon the map, is to draw three lines, forming the two angles, and ex- tended indefinitely, on a piece of trans- parent paper. Place the paper upon the map, and move it about until the three lines pass through the three stations ; then prick through the point n wherever it happens to come. Instead of the transparent paper, an instrument called a station pointer may be used when there are many points to be fixed. But the position of the point n can he found more correctly by describing two circles, as in Fig 19, each of which shall pass through n and two of the station points'. The question is to find the centers oand x of two such circles. This is very simple. We- know that the angle a ob at the center of a circle is twice as great as ary amgle a n b at the circumf of the same circle, when both are subtended by the same chord a h. Consequently , if the angle a n b, observed from the boat, is say. 50, the angle no b must be 100. And, since the three angles of every plane triangle are equal to 180, the two anglf s o ab and o b a are together equal to 180 100'= 80. And, since the two sides a o and b o are equal (being radii of the same circle), therefore, the angles oab and o ba are equal; and each equal to _- rr 40P. Consequently, on the map we have only to lay down at a and b, two angles of 40; the point o of intersection will be the center of the circle a b n. Proceed in the same way with the angle 6 n c, to find the center x. Then the intersection of the two circles at n will be the point sought. 46 CONTENTS OF CYLINDERS, OR PIPES. Contents for one foot in length, in Cub Ft, and in U. S. Gallons of 231 cub ins, or 7.4805 Galls to a Cub Ft. A cub ft of water weighs about 62^ Ibs ; and a gallon about 8% Ibs. Dium* 2, 8, or 1O times as great, give 4, 9, or 100 times the content. For the weight of water in pipes, see Table 2^, page 540. No errors. For 1 ft. in For 1 ft in For 1 ft. in length. length. length. Diam. Diam. Diam. Diam. Diam. in in deci- . a fc, 05 in in deci- . a fc. 00 Diam. in deci- 3 . Ins. mals of 0) g o a Ins. mals of " "^ o a in mals of W rf o a a foot. |i a foot. fc |i Ins. a foot. -' .O a 3* If* i|* 1| 5-16 .0208 .0260 .0003 .0005 .0025 .0040 7. 4 .5625 .5833 .2485 .2673 1.859 1.999 19. 1.583 1.625 1.969 2.074 14.73 15.51 % .0313 .0008 .0057 i/ .6042 .2867 2.145 20. 2 1.667 2.182 16.32 7-16 .0365 .0010 .0078 1/5 .6250 .3068 2.295 17 1.708 2.292 17.15 17J 0417 .0014 .0102 74 .6458 .3276 2.450 21. 1.750 2.405 17.99 9-16 .0469 .0017 .0129 8. .6667 .3491 2.611 17 1.792 2.521 18.86 57 .0521 .0021 .0159 i/ .6875 .3712 2.777 22. 1.833 2.640 1975 11-16 .0573 .0026 .0193 % .7083 .3941 2.948 17 1.875 2.761 20.66 3// .0625 .0031 .0230 3% .7292 .4176 3.125 23.. 1.917 2.885 21.58 13-16 .0677 .0036 .0269 9. .7500 .4418 3.305 \/ 1.958 3.012 22.53 Ys .0729 .0042 .0312 \/ .7708 .4667 3.491 24. 2.000 3.142 23.50 15-16 .0781 .0048 .0359 \/ .7917 .4922 3.682 25. 2.083 3.409 25.50 1. .0833 .0055 .0408 74 .8125 .5185 3.879 26. 2.167 3.687 27.58 i// .1042 .0085 .0638 10. .8333 .5454 4.080 27. 2.250 3.976 29.74 17 .1250 .0123 .0918 I/ .8542 .5730 4.286 28. 2.333 4.276 31.99 74 .1458 .0167 .1249 P .8750 .6013 4.498 29. 2.417 4.587 3431 2. .1667 .0218 .1632 .8958 .6303 4.715 30. 2.500 4.909 36.72 i/^ .1875 .0276 .2066 11. 4 .9167 .6600 4.937 31. 2.583 5.241 39.21 i^J .2083 .0341 .2550 l// .9375 .6903 5.164 32. 2.667 5.585 41.78 74 .2292 .0412 .3085 17 .9583 .7213 5.396 33. 2.750 5.940 44.43 3. .2500 .0491 .3672 74 .9792 .7530 5.633 34. 2.833 6.305 47.13 \/ .2708 .0576 .4309 12. 1 Foot. .7854 5.875 35. 2.917 6.681 49.98 X/ .2917 .0668 .4998 Ml. 042 .8522 6.375 36. 3000 7.069 52.88 74 .3125 .0767 .5738 13. ! 1.083 .9218 6.895 37. 3.083 7.467 55.86 4. .3333 .0873 .6528 17 1.125 .9940 7.43c 38. 3.16T 7.876 58.92 i/^ .3542 .0985 .7369 14. 11.167 1.069 7.997 39. 3.250 8.296 62.06 ITJ .3750 .1104 .8263 Y^ 1.208 1.147 8.578 40. 3.333 8.727 65.28 74 .3958 .1231 .9206 15. 1.250 1.227 9.180 41. 3.417 9.168 68.58 5. .4167 .1364 1.020 if, 1.292 1.310 9.801 42. 3.500 9.621 71.97 i/; .4375 .1503 1.125 16. (1.333 1.396 10.44 43. 3.583 10.085 75.44 ITJ .4583 .1650 1.234 /fj 1.375 1.485 11.11 44. 3.667 10.559 78.99 74 .4792 .1803 1.349 17. il.417 1.576 11.79 45. 3.750 11.045 82.62 6. .5000 .1963 1.469 H! L458 1.670 12.49 46. 3.833 111. 541 86.33 i// .5208 .2131 1.594 18. |1.500 1.767 13.22 47. 3.917 12.048 90.13 /I .5417 .2304 1.724 % 1.542 1.867 13.96 48. 4.000 12.566 94.00 1 Table continued, but with the dianis in feet. Diam. Cub. U. s. Diam. Cub. U. S. Dia. Cub. U.S. Dia. Cub. U. S. Feet. Feet. Galls. Feet. Feet. Galls. Feet. Feet. Galls. Feet. Feet. Galls. 4 12.57 94.0 7 38.49 287.9 12 113.1 846.1 24 452.4 3384 \S 14.19 106.1 X 41.28 308.8 13 132.7 992.8 25 490.9 3672 7% 15.90 119.0 g 44.18 330.5 14 153.9 1152. 26 530.9 3971 % 17.72 132.5 X 47.17 352.9 15 176.7 1322. 27 572.6 4283 5 19.64 146.9 8 50.27 376.0 16 201.1 1504. 28 615.8 4606 74" 21.65 161.9 M 56.75 424.5 17 227.0 1698. 29 660.5 4941 /^ 23.76 177.7 9 63.62 475.9 18 254.5 1904. 30 706.9 5288 % 25.97 194.3 1 A 70.88 530.2 19 283.5 2121. 31 754.8 5646 6 28.27 211.5 10 78.54 587.6 20 314.2 2350. 32 804.3 6017 74" 30.68 229.5 \i 86.59 647.7 21 346.4 2591. 33 855.3 6398 X 33.18 248.2 11 95.03 710.9 22 380.1 2844. 34 907.9 6792 % 35.79 267.7 X 103.90 777.0 23 415.5 3108. 35 962.1 7197 DIGGING, ETC., OF WELLS. 47 DIGGING, &C, OF WELLS. s twice as great as those in the table, for the cub yds of digging, take out those opposite the greater diani ; and mult them by 4. Thus, for the cub yds in each foot of depth of a in diam, first take out from the table those opposite the diani of 153* feet ; namely 6 989 X 4 ~ 27.956 cub yds reqd for the 31 ft diam. But for the stone lining or walling, bricks For diams twice as great as those in the table, for the cub yds of digging, take out those opposite one half of the gre well 31 feet in dia Then 6.989 X 4 ~ 2. . , or plastering, mult the tabular quantity opposite half the greater diam, by 2. Thus, the perches of stone walling for each foot of depth of a well of 31 ft diam, will be 2.073 X 2 ~ 4.146. If the wall is more or less than one foot thick, within usual moderate limits, it will generally be near enough for practice to assume that the number of perches, or of bricks, will increase or decrease in the same pro- portion. The size of the bricks is taken at 8J4 X 4 X 2 inches ; and to be laid dry, or without mortar. In practice an addition of about 5 per cent should be made for waste. The brick lining is supposed to be 1 brick thick, or 8% ins. CAUTIOX. Be careful to observe that the diams to he used for the digging, are greater than those for the walling, bricks, or plastering. No errors. For each foot of depth. For each foot of depth. For this For these three cols use the For this For these three cols use the col use the diam in clear of the "lining. col use the diam in clear of the lining. Diam. Diameter Diam. Diameter in of the in of the Feet. Digging. Cub Yds. of Digging. Stone Lining I ft thick. Perches of 25 Cub Ft. No. of Bricks in a Lining 1 Brick thick. Square Yards of Plaster- ing. Feet. Digging. Stone Lining 1 ft thick. Perches of 25 Cub Ft. No. of Bricks in a Lining 1 Brick thick. Square Yards of Plas- tering. Cub Yds. Digging. 1. .0291 .2513 57 .3491 X 5.107 1.791 750 4.625 34 .0455 .2827 71 .4364 X 5.301 1.822 764 4.713 ^ .0654 .3142 85 .5236 K 5.500 1.854 778 4.800 M .0891 .3456 99 .6109 14. 5.701 1.885 792 4.887 2. .1164 .3770 114 .6982 34 5.907 1.916 806 4.974 34 .1473 .4084 J28 .7855 3l2 6.116 1.948 820 5.062 X. .1818 .4398 142 .8727 % 6.329 1.979 834 5.149 K .2200 .4712 156 .9600 15. 6.545 2.011 849 5.236 3. .2618 .5027 170 1.047 34 6.765 2.042 863 5.323 34 .3073 .5341 184 1.135 % 6.989 2.073 877 5.411 ^ .3563 .5655 198 1.222 % 7.216 2.105 891 5.498 H .4091 .5969 212 1.309 16. 7.447 2.136 905 5.585 4. .4654 .6283 227 1.396 H 7.681 2.168 919 5.673 .5254 .6597 241 1.484 y* 7.919 2.199 933 5.760 i^ .5890 .6912 255 1.571 % 8.161 2.231 948 5.847 H .6563 .7226 269 1.658 17. 8.407 2.262 962 5.934 5. .7272 .7540 283 1.745 34 8.656 2.293 976 6.022 .8018 .7854 297 1.833 J4 8.908 2.325 990 6.109 .8799 .8168 311 1.920 H 9.165 2.356 1004 6.196 % .9617 .8482 326 2.007 18. 9.425 2.388 1018 6.283 6. 1.047 .8796 340 2.095 34 9.688 2.419 1032 6.371 34 1.136 .9111 354 2.182 X 9.956 2.450 1046 6.458 H 1.229 .9425 368 2.269 H 10.23 2.482 1061 6.545 1.325 .9739 382 2.356 19. 10.50 2.513 1075 6.633 7. 1.425 1.005 396 2.444 34 10.78 2.545 1089 6.720 34 1.529 1.037 410 2.531 X 11.06 2.576 1103 6.807 M 1.636 1.068 425 2.618 % 11.35 2.608 1117 6.894 N 1.747 1.100 439 2.705 20. 11.64 2.639 1131 6.982 8. 1.862 1.131 453 2.793 34 11.93 2.670 1145 7-069 34 1.980 1.162 467 2.880 34 12.22 2.702 1160 7.156 k 2.102 1.194 481 2.967 H 12.52 2.733 1174 7.243 % 2.227 1.225 495 3.054 21. 12.83 2.765 1188 7.331 9. 2.356 1.257 509 3.142 34 13.14 2.796 1202 7.418 34 2.489 1.288 523 3.229 34 13.45 2.827 1216 7.505 % 2.625 1.319 538 3.316 H 13.76 2.859 1230 7.593 % 2.765 1.351 552 3.404 14.08 2.890 1244 7.680 10. 2.909 1.382 566 3.491 34 14.40 2.922 1259 7.767 34 3.056 1.414 580 3.578 34 14.73 2.953 1273 7.854 3^ 3.207 1.445 594 3.665 H 15.06 ' 2.985 1287 7.942 X 3.362 1.477 608 3.753 23. 15.39 3.016 1301 8.029 11. 3.520 1.508 622 3.840 34 15.72 3.047 1315 8.116 34 3.682 1 539 637 3.927 34 16.06 3.079 1329 8.203 g 3.847 1.571 651 4.014 H 16.41 3.110 1343 8. ''91 H 4.016 1.602 665 4.102 24. 16.76 3.142 1357 8.378 12. 4.189 1.634 679 4.189 34 17.11 3.173 1372 8.465 34 4.365 1.665 693 4.276 34 17.46 3 204 1386 8.552 J6 4.545 1.696 707 4.364 H 17.82 3.236 1400 8.6*0 4.729 1.728 721 4.451 25. 18.18 3.267 1414 8.727 13. 4.916 1.759 736 4.538 A i sub yd = 2021 r. s. s:a Is. If perches are named in a contract, it is necessary, in order to prevent fraud* to specify the number of cub feet contained in the perch ; for stone-quarriers have one perch, stone- masons another, Ac. Engineers, on this account, contract by the cubic yard. The perch should be done away with entirely ; Perches of 25 cub ft X .926 = cub yds ; and cub yds-j- .926- pers of 25 cub ft. 48 SQUARE AND CUBE HOOTS. Square Roots and Cube Roots of Numbers from .1 to 2$. No errors. No. Square. Cube. Sq. Rt. C. Rt. No. Sq. Rt. C. Rt. No. Sq. Rt. C. Rt. .1 .01 .001 .316 .464 7 2.387 1.786 .4 3.661 2.375 .15 .0225 .0034 .387 .531 8 2.408 1.797 .6 3.688 2.387 .2 .04 .008 .447 .585 9 2.429 1.807 .8 3.715 2.399 .25 .0625 .0156 .500 .630 6 2.449 1.817 14. 3.742 2.410 .3 .09 .027 .548 .669 1 2.470 1.827 .2 3.768 2.422 .35 .1225 .0429 .592 .705 2 2.490 1.837 .4 8.795 2.433 .4 .16 .064 .633 .737 3 2.510 1.847 .6 3.821 2.444 .45 .2025 .0911 .671 .766 4 2.530 1.857 .8 8.847 2.455 .5 .25 .125 .707 .794 .5 2.550 1.866 15. 3.873 2.466 .55 .3025 .1664 .742 .819 .6 2.569 1.876 .2 3.899 2.477 .6 .36 .216 .775 .843 2.588 1.885 .4 3.924 2.488 .65 .4225 .2746 .806 .866 .8 2.608 1.895 .6 3.950 2.499 .7 .49 .343 837 .888 .9 2.627 1.904 .8 3.975 2.509 .75 .5625 .4219 .866 .909 7. 2.646 1.913 16. 4. 2.520 .8. .64 .512 .894 .928 .1 2.665 1.922 .2 4.025 2.530 .85 .7225 .6141 .922 .947 .2 2.683 1.931 .4 4.050 2.541 ,9 .81 .729 .949 .965 .3 2.702 1.940 .6 4.074 2.551 .95 .9025 .8571 .975 .983 .4 2.720 1.949 .8 4.099 2.561 1. 1.000 1.000 1.000 1.000 .5 2.739 1.957 17. 4.123 2.571 .05 1.103 1.158 1.025 1.016 .6 2.757 1.966 .2 4.147 2.581 1.1 1.210 1.331 1.049 1.032 .7 2.775 1975 .4 4.171 2.591 .15 1.323 1.521 1.072 1.018 .8 2.793 1.983 .6 4.195 2.601 1.1 1.440 1.728 1.095 1.063 .9 2.811 1.992 .8 4.219 2.611 .'25 1.563 1.953 .118 1.077 8. 2.828 2.000 18. 4.243 2.621 1.3 1.690 2.197 .140 1.091 .1 2.846 2.008 .2 4.266 2.630 .35 1.823 2.460 .162 1.105 .2 2.864 2.017 .4 4.290 2,640 1.4 1.960 2.714 .183 1.119 .3 2.881 2.025 .6 4.313 2.650 .45 2.103 3.019 .204 1.132 .4 2.898 2.033 .8 4.336 2.659 1.5 2.250 3.375 .225 1.145 .5 2.915 2.041 19. 4.359 2.668 .55 2.403 3.724 .215 1.157 .6 2.933 2.049 .2 4.382 2.678 1.6 2.560 4.096 .265 1.170 .7 2.950 2.057 .4 4.405 2.687 .65 2.723 4.492 .285 1.182 .8 2.966 2.065 ,6 4.427 2.696 1.7 2.890 4.913 .304 1.193 .9 2.983 2.072 .8 4.450 2.705 .75 3.063 5.359 .323 1.205 9. 3. 2.080 20. 4.472 2.714 1.8 3.240 5.832 .342 1.216 .1 3.017 2.088 .2 4.494 2.723 .85 3.423 6.332 .360 1.228 .2 3.033 2.095 .4 4.517 2.732 1.9 3.610 6.859 .378 1.239 .3 3.050 2.103 .6 4.539 2.741 .95 3.803 7.415 .396 1.249 .4 3.066 2.110 .8 4.561 2.750 2. 4.000 8.000 .414 1.260 .5 3.082 2.1J8 21. 4.583 2.759 .1 4.410 9.261 .419 1.281 .6 3.098 2.125 .2 4.604 2.768 .2 4.840 10.65 .483 1.301 .7 3.114 2-133 .4 4.626 2.776 .3 5.290 12.17 .517 1.320 .8 3.130 2.140 .6 4.648 2.785 .4 5.760 13.82 .549 1.339 .9 3.146 2.147 .8 4.669 2.794 .5 6.250 15.63 .581 1.357 10. 3.162 2.154 22. 4.690 2.802 .6 6.760 17.58 .612 1.375 .1 3.178 2.162 .2 4.712 2.810 .7 7.290 19.68 .643 1.392 .2 3.194 2.169 A 4.733 2.819 .8 7.840 21.95 .673 1.409 .3 3.209 2.176 .6 4.754 2.827 .9 8.410 24.39 .703 1.426 .4 3.225 2.183 .8 4.775 2.836 8. 9. 27. .732 1.442 .5 3.240 2.190 23. 4.796 2.844 .1 9.61 29.79 .761 1.458 .6 3.256 2.197 .2 4.817 2.852 .2 10.24 32.77 .789 1.474 3.271 2.204 .4 4.837 2.860 .3 10.89 35.94 .817 1.489 .8 3.286 2.210 .6 4.858 2.868 .4 11.56 39.30 1.844 1.504 .9 3.302 2.217 .8 4.879 2.876 .5 12.25 42.88 1.871 1.518 11. 3.317 2.224 24. 4.899 2.884 .6 12.96 46.66 1.897 1.533 .1 3.332 2.231 .2 4.919 2.892 .7 13.69 50.65 1.924 1.547 .2 3.347 2.237 .4 4.940 2.900 .8 14.44 54.87 1.949 1.560 .3 3.362 2.244 .6 4.960 2.908 .9 15.21 59.32 1.975 1.574 .4 3376 2.251 .8 4.980 2.916 4. 16. 64. 2: 1.587 .5 3.391 2.257 25. 5. 2.924 16.81 68.92 2.025 1.601 .6 3.406 2.264 .2 5.020 2.932 .2 17.64 74.09 2.049 1613 .7 3.421 2.270 .4 5.040 2.940 .3 18.49 79.51 2.074 1.626 .8 3.435 2.277 .6 5.060 2.947 .4 19.36 85.18 2.098 1.639 .9 3.450 2.283 .8 5.079 2.955 .5 20.25 91.13 2.121 1.651 12. 3.464 2289 26. 5.099 2.962 .6 21.16 97.34 2.145 1.663 .1 3.479 2.296 .2 5.119 2.970 J7 22.09 23.04 103.8 110.6 2.168 2.191 1.675 1.687 .2 .3 3.493 3.507 2.302 2.308 .4 .6 5.138 5.158 2.978 2.985 !u 5. ^j 24.01 25. 26.01 117.6 125. 132.7 2.214 2.236 2.258 1.698 1.710 1.721 .4 .5 .6 3.521 3.536 3.55 2.315 2.321 2.327 .8 27. .2 5.177 5.196 5.215 2.993 3.000 3.007 !a .3 .4 .5 .6 27.04 28.09 29.16 30.25 31.36 140.6 148.9 157.5 166.4 175.6 2.280 2.302 2.324 2.345 2.366 1.732 1.744 1.754 1.765 1.776 .7 .8 .9 13. .2 3.564 3.578 3.592 3.606 3.633 2.333 2339 2.345 2.351 2.363 .4 .6 .8 28. .2 5.235 5.254 5.273 5.292 5.310 3.015 3.022 3.029 3.037 3.044 SQUARES, CUBES, AND ROOTS. TABLE of Squares. Cubes. Square Roots, and Cube Roots, of Numbers from 1 to 1OOO. REMARK ON THE FOLLOWING TABLE. add 1 to the fourth and final decimal it Wherever the effect of a fifth decimal in the roots would be to the table, the addition has boen made. No errors. No. Square. Cube. Sq. Rt. C. Rt. No. Square Cube. Sq. Rt. C. Rt. ] 1 l 3.0000 1.0000 61 3721 226981 7.8102 3.9365 2 4 8 1.4142 1.25J9 62 3844 238828 7.8740 3.9579 3 9 27 1.7321 1.4422 63 3969 250047 7.9373 3.9791 4 16 64 2.0000 1.5874 64 4096 262144 8.0000 4. 5 25 123 2.2361 1.7100 65 4225 274625 8.0623 4.020T 6 36 216 2.4495 1.8171 66 4356 287496 8.1240 4.0412 7 49 343 2.6458 1.9129 67 4489 300763 8.1854 4.0615 8 64 512 2.8284 2.0000 68 4624 311432 8.2462 4.0817 9 81 729 3.0000 2.0801 69 4761 328509 8.3066 4.1016 10 100 1000 3.1623 2.1544 70 4900 343000 8.3666 4.1213 11 121 1331 3.3166 2.2240 71 5041 357911 8.4261 4.1408 12 114 1728 3.4641 2.2894 72 5184 373248 8.4853 4.1602 13 169 2197 3.6056 2.3513 73 5329 38S017 8.5440 4.1793 14 196 2744 3.7417 2.4101 74 5*76 405224 8.6023 4.1983 15 225 3375 3.8730 2.4062 75 5625 421875 8.6603 4.2172 16 256- 4096 4.0000 2.5198 76 5776 438976 8.7178 4.2358 17 289 4913 4.1231 2.5713 77 929 456533 8.7750 4.2543 18 324 5832 4.2426 2.6207 78 C084 474552 8.&318 4.2727 19 361 6859 4.3589 2.6681 79 6241 493039 8.8882 4.2908 20 400 8000 4.4721 2.7144 80 6400 512000 8.9443 4.3089 21 441 9261 4.5826 2.7589 81 6561 531441 9. 4.3267 22 484 10648 4.6904 2.8020 82 6724 551368 9.0554 4.3445 23 529 12167 4.7958 2.8439 83 6889 571V87 9.1104 4.3621 24 576 13824 4.8990 2.8845 84 7056 592704 9.1652 4.3795 25 625 15625 5.0000 2.9240 85 7225 614125 9.2195 4.3968 26 676 17576 5.0990 2.9625 86 7396 636056 9.2736 4.4140 27 729 19683 5.1C62 3.0000 87 7569 658503 9.3274 4.4310 28 784 21952 5.2915 3.0366 88 7744 681472 9.3808 4.4480 29 841 24389 5.3852 3.0723 89 7921 704969 9.4340 4.4647 30 900 27000 5.4772 3.1072 90 8100 729000 9.4868 4.4814 31 961 29791 5.5678 3.1414 91 82S1 753571 9.5394 4.4979 32 1024 32768 5.6569 3.1748 92 8464 778688 9.5917 4.5144 33 1039 35937 5.7446 3.2075 93 8G49 804357 9.6437 4.5307 34 1156 39304 5.8310 3.2396 94 8836 830584 9.6954 4.5468 35 1225 42875 5.9161 3.2711 95 9025 857375 9.7468 4.5629 36 1296 46656 6.0000 3.3019 96 9216 884736 9.7980 4.5789 37 1369 50653 6.0828 3.3322 97 9409 912673 9.8489 4.5917 38 1444 54872 6. Hi It 3.3620 98 9H04 941192 9.8995 4.6104 39 1521 59319 6.2450 3.3912 99 9801 970299 9.9499 4.G2G1 40 ItiOO 64000 6.3246 3.4200 100 10000 1000000 10. 4.6416 41 1681 68921 6.4031 3.4482 101 10201 1030301 10.0499 4.6570 42 1764 74088 6.4807 3.4760 102 10404 1061208 10.0995 4.6723 43 1849 79507 6.5574 3.5034 103 1060;) 1092727 10.1489 4.6875 44 1936 85184 6.6332 3.530:5 104- 0816 1124864 10.1980 4.7027 45 2025 91125 6.7082 3.5569 105 1025 1157625 10.2470 4.7177 46 2116 97336 6.7823 3.5830 106 1236 1191016 10.2956 4.7326 47 2209 103823 6.8557 3.0088 107 1449 1225043 10.3441 4.7475 48 21304 110592 6.9282 3.6312 108 1664 1259712" 10.3923 4.7622 49 2401 117019 7.0000 3.6593 1 >9 1881 1295029 10.4403 4.7769 50 2500 125000 7.0711 3.6840 1 2100 1331000 10.4881 4.7914 61 2601 132G51 7.1414 3.7084 1 1 2321 1367631 10.5357 4.8059 52 2704 140608 7.2111 3.7325 1 2 2544 1404928 10.5830 4.8203 53 2809 148877 7.2801 3.7563 1 3 2769 1442897 10.6301 4.a346 54 2<>16 157464 7.34*5 8.T798 1 4 2996 1481544 10.6771 4.8488 55 3025 166375 7.4162 3.8030 115 3225 1520875 10.7238 4.8629 56 3136 175616 7.4833 3.8259 116 3456 1560896 10.7703 4.8770 57 3249 185193 7.5498 3.51485 117 3689 1601613 10.8167 4.8910 58 3364 195112 7.6158 3.870-) 118 3924 1643032 10.8628 4.9049 59 3181 205379 7.6H11 3. 8930 119 14161 1685159 10.9087 4.9181 60 3600 216000 7.7460 3.9149 120 14400 1728000 10.9545 4.9324 50 SQUARES, CUBES, AND ROOTS. TABLE of Squares, Cubes, Square Roots, and Cube Roots, of Numbers from 1 to 1OOO (CONTINUED.) Square. Cube. Sq. Rt. C. lit. No. Square. Cube. Sq. Rt. C.I 14641 1771561 11. 4.9461 186 34596 6434856 13.6382 5.7( 14884 1815848 11.0454 4.9597 187 34969 6539203 13.6748 5.7 15129 1860867 11.0905 4.9732 188 35344 6644672 13.7113 5.7 15376 1906624 11.1355 4.9866 189 35721 6751269 13.7477 5.7, 15625 1953125 11.1803 5. 190 36100 6859000 13.7840 5.7< 15876 2000376 11.2250 5.0133 191 36481 6967871 13.8203 5.7, 16129 2048383 11.2694 5.0265 192 36864 . 7077888 13.8564 5.7< 16384 2097152 11.3137 5.0397 193 37249 7189057 13.8924 5.7 16641 2146689 11.3578 5.0528 194 37636 7301384 13.9284 5.7* 16900 2197000 11.4018 5.0658 195 38025 7414875 13.9642 5.71 17161 2248091 11.4455 5.0788 196 88416 7529536 14. 5. 17424 2299968 11.4891 5.0916 197 38809 7645373 14.0357 5.8 17689 2352637 11.5326 5.1045 198 39204 7762392 14.0712 5.8 17956 2406104 11.5758 5.1172 199 39601 7880599 14.1067 5.8. 18225 2460375 11.6190 5.1299 200 40000 8000000 14.1421 6.* 18496 2515456 11.6619 5.1426 201 40401 8120601 14.1774 5.8 18769 2571353 11.7047 5.1551 202 40804 8242408 14.2127 5.8 19044 2628072 11.7473 5.1676 203 41209 8365427 14.2478 5.8 19321 2685619 11.7898 5.1801 204 41616 8489664 14.2829 5.8 19600 2744000 11.8322 5.1925 205 42025 8615125 14.3178 5.8S 19881 2803221 11.8743 5.2048 206 42436 8741816 14.3527 5.9( 20164 2863288 11.9164 5.2171 207 42849 8869743 14.3875 5.9 20449 2924207 11.9583 5.2293 208 43264 8998912 14.4222 5.9 20736 2985984 12. 5.2415 209 43681 9129329 14.4568 5.9 21025 3048625 12.0416 5.2536 210 44100 9261000 14.4914 5.9 21316 3112136 12.0830 5.2656 211 44521 9393931 14.5258 5.9 21609 3176523 12.1244 5.2776 212 44944 9528128 14.5602 5.9 21904 3241792 12.1055 5.2896 213 45369 9663597 14.5945 5.9 22201 3307949 12.2066 5.3015 214 45796 9800344 14.6287 5.9* 22500 3375000 12.2474 5.3133 215 46225 9938375 14.6629 5.9< 22801 3442951 12.2882 5.3251 216 46656 10077696 14.6969 6. 23104 3511808 12.3288 5.3368 217 47089 10218313 14.7309 6.0 23409 3581577 12.3693 5.3485 218 47524 10360232 14.7648 6.0 23716 3652264 12.4097 5.3601 219 47961 10503459 14.7986 6.0 24025 3723875 12.4499 5.3717 220 48400 10648000 14.8324 6.0. 24336 3796416 12.4900 5.3832 221 48841 10793861 14.8661 6.0< 24619 3869893 12.5300 5.3947 222 49284 10941048 14.8997 6.0, 24964 3944312 12.5698 5.4061 223 49729 11089567 14.9332 6.0( 25281 4019679 12.6095 5.4175 224 50176 11239424 14.9666 6.0 25600 4096000 12.6491 5.4288 225 50625 11390625 15. 6.0. 25921 4173281 12.6886 5.4401 226 51076 11543176 15.0333 6.0J 26244 4251528 12.7279 5.4514 227 '51529 11697083 15.0665 6.K 26569 43IS0747 12.7671 5.4626 228 51984 11852352 15.0997 6.H 26896 4410944 12.8062 5.4737 229 52441 12008989 15.1327 6.11 27225 4492125 12.8452 5.4848 230 52900 12167000 15.1658 6.1' 27556 4574296 12.8841 5.4959 231 53361 12326391 15.1987 6.K 27889 4657463 12.9228 5.5069 232 53824 12487168 15.2315 6.14 28224 4741632 12.9615 5.5178 233 54289 12649337 15.2643 6. If 28561 4826809 13. 5.5288 234 54756 12812904 15.2971 6.1f 28900 4913000 13.0384 5.5397 235 55225 12977875 15.3297 6.11 29241 5000211 13.0767 5.5505 236 55696 13144256 15.3623 6.11 29584 5088448 13.1149 5.5613 237 56169 13312053 15.3948 6.18 29929 5177717 13.1529 5.5721 238 56644 13481272 15.4272 6.1f 30276 5268024 13.1909 5.5828 239 57121 13651919 15.4596 6.2C 30625 5359375 13.2288 5.5934 240 57600 13824000 15.4919 6.21 30976 5451776 13.2665 5.6041 241 58081 13997521 15.5242 6.25 31329 5545233 13.3041 5.6147 242 58564 14172488 15.5563 6.23 31684 5639752 13.3417 5.6252 243 59049 14348907 15.5885 6.24 32041 5735339 13.3791 5.6357 244 59536 14526784 15.6205 6.24 32400 5832000 13.4164 5.6462 245 60025 14706125 15.6525 6.25 32761 5929741 13.4536 5.6567 246 60516 14886936 15.6844 6.26 33124 6028568 13.4907 5.6671 247 61009 15069223 15.7162 6.27 33489 6128487 13.5277 5.6774 248 61504 15252992 15.7480 6.28 33856 6229504 13.5647 5.6877 249 62001 15438249 15.7797 3.28 34225 6331625 13.6015 5.6980 250 62500 15625000 15.8114 6.28 SQUARES, CUBES, AND ROOTS. 51 TABLE of Squares, Cubes, Square Roots, and Cube Roots, of Numbers from 1 to 1OOO (CONTINUED.) No. Square. Cube. Sq. Rt. C. Rt No. Square. Cube. Sq. Rt. C. Rt. 251 63001 15813251 15.8430 6.3080 316 99856 31554496 17.7764 6.8113 252 63504 16003008 118745 6.3164 317 100489 31855013 17.8045 6.8185 253 64009 16194277 15.9060 6.3247 318 101124 32157432 17.8326 6.8256 254 64516 16387064 15.9374 6.3330 319 101761 32461759 17.8606 6.8328 255 65025 16581375 15.9687 6.3413 320 102400 32768000 17.8885 6.8399 256 65536 16777216 16. 6.3496 321 103041 33076161 17.9165 6.8470 257 66049 16974593 16.0312 6.3579 322 103684 33386248 17.9444 6.8541 258 66564 17173512 16.0624 6.3661 323 104329 33698267 17.9722 6.8612 259 67081 17373979 16.0935 6.3743 324 104976 34012224 18. 6.8683 280 67600 17576000 16.1245 6.3825 325 105625 34328125 18.0278 6.8753 261 68121 17779581 16.1555 6.3907 326 106276 34645976 18.0555 6.8824 262 68644 17984728 16.1864 6.3988 327 106929 34965783 18.0831 6.8894 263 264 69696 18399744 16.2481 6.4070 6.4151 328 329 107584 108241 35287552 35611289 18.1108 18.1384 6.8964 6.9034 265 70225 18609625 16.2788 6.4232 330 108900 35937000 18.1659 6.9104 266 70756 18821096 16.3095 6.4312 331 109561 36264691 18.1934 6.9174 267 71289 19034163 16.3401 6.4393 332 110224 36594368 18.2209 6.9244 268 71824 19248832 16.3707 6.4473 333 110889 36926037 18.2483 6.9313 269 72361 19465109 16.4012 6.4553 334 111556 37259704 18.2757 6.9382 270 72900 19683000 16.4317 . 6.4633 335 112225 37595375 18.3030 6.9451 271 73441 19902511 16.4621 6.4713 336 112896 37933056 18.3303 6.9521 272 73984 20123648 16.4924 6.4792 337 113569 38272753 18.3576 6.9589 273 74529 20346417 16.5227 6.4872 338 114244 38614472 18.3848 6.9658 274 75076 20570824 16.5529 6.4951 339 114921 38958219 18.4120 6.9727 275 75625 20796875 16.5831 6.5030 340 115600 39304000 18.4391 6.9795 276 76176 21024576 16.6132 6.5108 341 116281 39651821 18.4662 6.9864 277 76729 21253933 16.6433 6.5187 342 116964 40001688 18.4932 6.9932 278 77284 21484952 16.6733 6.5265 343 117649 40353607 18.5203 7. 279 77841 21717639 16.7033 6.5343 344 118336 40707584 18.5472 7.0068 280 78400 21952000 16.7332 6.5421 345 119025 41063625 18.5742 7.0136 281 78961 22188041 16.7631 6.5499 346 119716 41421736 18.6011 7.0203 282 79524 22425768 16.7929 6.5577 347 120409 41781923 18.6279 7.0271 283 80089 22665187 16.8226 6.5654 348 121104 42144192 18.6548 7.0338 284 80656 22906304 16.8523 6.5731 349 121801 42508549 18.6815 7.0406 285 81225 23149125 16.8819 6.5808 350 122500 42875000 18.7083 7.0473 286 81796 23393656 16.9115 6.5885 351 123201 43243551 18.7350 7.0540 287 82369 2363.9903 16.9411 6.5!lo2 352 123904 43614208 18.7617 7.0607 288 82944 23887872 16.9706 6.6039 353 124609 43986977 18.7883 7.0674 289 83521 24137569 17. 6.6115 354 125316 44361864 18.8149 7.0740 290 84100 24389000 17.0294 6.6191 355 126025 44738875 18.8414 7.0807 291 84681 24642171 17.0587 6.6267 356 126736 45118016 18.8680 7.0873 292 85264 24897088 17.0880 6.6343 357 1274-49 45499293 18.8944 7.0940 293 85849 25153757 17.1172 6.6419 358 128164 45882712 18.9209 7.1006 294 86436 25412184 17.1464 6.6494 359 128881 46268279 18.9473 7.1072 295 87025 25672375 17.1756 6.6569 360 129600 46656000 18.9737 7.1138 2% 87616 25934336 17.2047 6.6644 361 130321 47045881 19. 7.1204 297 88209 26198073 17.2337 6.G719 362 131044 47437928 19.0263 7.1269 298 88804 26463592 17.2627 6.6794 363 131769 47832147 19.0526 7.1335 299 89401 26730899 17.2916 6.6869 364 132496 48228544 19.078 7.1400 300 90000 27000000 17.3205 6.6943 365 133225 48627125 19.1050 7.1466 301 90601 27270901 17.3494 6.7018 366 133956 49027896 19.1311 7.1531 302 91204 27543608 17.3781 6.7092 367 134689 49430863 19.1572 7.1596 303 91809 27818127 17.4069 6.7166 368 135424 49836032 19.1833 7.1661 304 92H6 28094464 17.4356 6.7240 369 136161 50243409 19.2094 7.1726 305 93025 28372625 17.4642 6.7313 370 136900 50653000 19.2354 7.1791 306 93636 28652616 17.4929 6.7387 371 137641 51064811 19.2614 7.1855 307 94249 28934443 17.5214 6.7460 372 138384 51478848 19.2873 7.1920 308 94864 29218112 17.5499 6.7533 373 139129 51895117 19.3132 7.1984 309 95481 29503629 17.5784 6.7606 374 139876 52313624 19.3391 7.2048 310 96100 29791000 17.6068 6.7679 375 140625 52734375 19.3649 7.2112 311 96721 30080231 17.6352 6.7752 376 141376 53157376 19.3907 7.2177 312 97344 30371328 17.6635 6.7H24 377 142129 53582633 19.4165 7.2240 313 97969 30664297 17.6918 6.7897 378 142884 54010152 19.4422 7.2304 314 98596 30959144 17.7200 6.7969 379 143641 54439939 19.4679 7.2368 315 992/0 31255875 17.7482 6.8041 380 144400 54872000 19.4936 7.2432 52 SQUARES, CUBES, AND ROOTS. TABLE of Squares, Cubes, Square Roots, and Cube Roots, of Numbers from 1 to 100O (CONTINUED.) No. Square. Cube. Sq. Rt. C. Ht. No. Square. Cube. Sq. Bt. C.Rt. 381 145161 55306341 19.5192 7.2485 446 198916 88716536 21.1187 7.6403 382 145324 55742968 19.5448 7.2558 447 199809 89314623 21.1424 7.6460 383 146689 56181887 19.5704 7.2622 448 200704 89915392 21.1660 7.6517 384 147456 56623104 19.5959 7.2685 449 201601 90518849 21.1896 7.6574 385 148225 57066625 19.6214 7.2748 450 202500 91125000 21.2132 7.6631 886 148996 57512456 19.6469 7.2811 451 203401 91733851 21.2368 7.6688 387 149769 57960603 19.6723 7.2874 452 204304 92345408 21.2603 7.6744 388 150544 58411072 19.6977 7.2936 453 205209 92959677 21.2838 7.6801 389 151321 58863869 19.7231 7.2999 454 206116 93576G64 21.3073 7.6857 390 152100 59319000 19.7484 7.3061 455 207025 94196375 21.3307 7.6914 391 152881 59776471 19.7737 7.3124 456 207936 94818816 21.3542 7.6970 392 153664 60236288 19.7990 7.3186 457 208849 95443993 21.3776 7.7026 393 154449 60698457 19.8242 7.3248 458 209764 96071912 21.4009 7.7082 394 155236 61162984 19.8494 7.3310 459 210681 96702579 21.4243 7.7138 395 156025 61629875 19.8746 7.3372 460 211600 97336000 21.4476 7.7194 3% 156816 62099136 19.8997 7.3434 461 212521 97972181 21.4709 7.7250 397 157609 62570773 19.9249 7.3196 462 213444 98611128 21.4942 7.7306 398 158404 63044792 19.9499 7.3558 4fi3 214369 99252847 21.5174 7.7362 399 159201 63521199 19.9750 7.3019 464 215296 99897344 21.5407 7.7418 400 160000 64000000 20. 7.3681 465 216225 100544625 21.5639 7.7473 401 160801 6*481201 20.0250 7.3742 466 217156 101194696 21.5870 7.7529 402 161604 64964808 20.0499 7.3803 467 218089 101847563 21.6102 7.7584 403 162409 65450827 20.0749 7.3861 4(58 219024 102503232 21.6333 7.7639 404 163216 65939264 20.0998 7.3925 469 219961 103161709 21.6564 7.7695 405 164025 66430125 20.1246 7.3986 470 220900 103823000 21.6795 . 7.7750 406 164836 66923416 20.1494 7.4047 471 221841 104487111 21.7025 7.7805 407 165849 67419143 20.1742 7.4108 72 222784 105154048 21.7256 7.7860 408 166464 67917312 20.1990 7.4169 73 223729 105823817 21.7486 7.7915 409 167281 68417929 20.2237 7.4229 74 224676 106496424 21.7715 7.7970 410 168100 68921000 20.2485 7.4290 75 225625 107171875 21.7945 7.8025 411 168921 69426531 20.2731 7.4350 76 226576 107850176 21.8174 7.8079 412 169744 69934528 20.2978 7.4410 77 227529 108531333 21.8403 7.8134 413 170569 70444997 20.3224 7.4470 78 228484 109215352 21.8632 7.8188 414 171396 70957944 20.3470 7.4530 479 229441 109902239 21.8861 7.8243 415 172225 71473375 20.3715 7.4590 480 230400 110592000 21.9089 7.8297 416 173056 71991296 20.3961 7.4650 481 231361 111234641 21.9317 7.8352 417 173889 72511713 20.4206 7.4710 482 232324 111980168 21.9545 7.8406 418 174724 73034632 20.4450 7.4770 483 233289 112678587 21.9773 7.8460 419 175561 73560059 20.4695 7.4829 4S4 234256 113379904 22. 7.8514 420 176400 74088000 20.4939 7.4889 485 235225 114084125 22.0227 7.8568 421 177241 74618461 20.51&? 7.4948 486 . 236196 114791256 22.0454 7.8622 422 178084 75151448 20.5426 7.5007 487 237169 115501303 22.0681 7.8676 423 178929 756869<>7 20.5670 7.5067 488 23814 t 116214272 22.0907 7.8730 424 179776 76225024 20.5913 7.5126 48:> 233121 11G9301C9 22.1133 7.8784 425 180625 76765625 20.6155 7.5185 490 240100 117649000 22.1359 7.8837 426 181476 77308776 20.6398 7.5244 431 241081 118370771 22.1585 7.8891 427 182329 77354483 20.6640 7.5302 492 242064 119095488 22.1811 7.8944 428 183184 78402752 20.6882 7.5361 493 243049 11982315 22.2036 7.8998 429 184041 78953589 20.7123 7.5420 494 244036 120553784 22.2261 7.9051 430 184900 79507000 20.7364 7.5478 495 245025 121287375 22.2486 7.9105 431 185761 80062991 20.7605 7.5537 4ns 246016 122023936 22.2711 7.9158 432 186624 80621568 20.7346 7.5595 497 247009 122763473 22.2935 7.9211 433 187489 81182737 20.8087 7.5654 498 24S004 123505992 22.3159 7.9264 434 188356 81746504 20.8327 7.5712 499 249001 124251499 22.3383 7.9317 435 189225 82312875 20.8567 7.5770 500 250000 125000000 22.3607 7.9370 436 190096 82881856 20.8806 7.5828 501 251001 125751501 22.3830 7.9423 437 190969 83453453 20.9045 7.5S86 502 252004 12850300fc 22.4054 7.9476 438 191844 84027672 20.9284 7.5944 503 253009 127263527 22.4277 7.9523 439 192721 84604519 20.9523 7.6001 504 254016 12802KK51 22.4499 7.9581 440 193600 85184000 20.9762 7.6059 505 255025 128787623 22.4722 7.9634 441 194481 85766121 21. 7.6117 506 256036 129554216 22.4944 7.9686 442 1 95364 86350888 21.0238 7.6174 507 257019 1303233*1 22.5167 7.9739 443 196249 86938307 21.0476 7.6232 508 258064 131096515 22.5389 7.9791 444 197136 87528384 21.0713 7.6289 509 259081 1318722X 22.5610 7.9843 445 198025 88121125 21.0950 7.6346 510 260100 13265100C 22.5832 7.9896 SQUARES, CUBES, AND ROOTS. 53 TABLE of Squares, Cubes, Square Roots, and Cube Roots, of Numbers from 1 to 1OOO (CONTINUED.) No. Square. Cube. Sq. Ht. C. lit. No. Square. Cube. Sq. Kt. C. Kt. 511 261121 133432831 22.6053 7.9948 576 331776 191102976 24. 8.3203 512 262144 13A217728 22.6274- 8. 577 332929 192100033 24.0208 8.3251 513 263169 135005697 22.6495 8.0052 578 334084 193100552 24.0416 8.3300 514 264196 135796744 22.6716 8.0104 579 335241 194104539 24.0624 8.3348 5-15 265225 136590875 22.6936 8.0156 580 336400 195112000 24.0832 8.3396 516 266256 137388096 22.7156 8.0208 581 337561 196122941 24.1039 8.3443 517 267289 138188413 22.7376 8.0260 582 338724 197137368 24.1247 8.3491 518 268324 138991832 22.7596 8.0311 583 339889 198155287 24.1454 8.3539 519 269361 139798359 22.7816 8.0363 584 341056 199176704 24.1661 8.3587 520 270400 140608000 22.8035 8.0415 585 342225 200201625 24.1868 8.3634 521 271441 141420761 22.8254 8.0466 586 343396 2012*056 24.2074 8.3682 522 272484 142236648 22.8473 8.0517 587 344569 202262003 24.2281 8.3730 523 273529 143055667 22.8692 8.0569 588 345744 203297472 24.2487 8.3777 524 274576 143877824 22.8910 8.0620 589 346921 204336469 24.2693 8.3825 525 275625 144703125 22.9129 8.0671 590 348100 205379000 24.2899 8.3872 526 276676 145531576 22.9347 8.0723 591 349281 206425071 24.3105 8.3919 527 277729 146363183 22.9565 8.0774 592 350464 207474688 24.3311 8.3967 528 278784 147197952 22.9783 8.0825 593 351649 208527857 24.3516 8.4014 529 279841 148035889 23. 8.0876 594 352836 209584584 24.3721 8.4061 530 280900 148877000 23.0217 8.0927 595 354025 210644875 24.392S 8.4108 531 281961 149721291 23.0434 8.0978 596 355216 211708736 24.4131 8.4155 532 283024 150568768 23.0651 8.1028 597 356409 212776173 24.4336 8.4202 533 284089 151419437 23.0868 8.1079 598 357604 213847192 24.4540 8.4249 534 285156 152273304 23.1084 8.1130 599 358801 214921799 24.4745 8.4296 535 286225 153130375 23.1301 8.1180 600 360000 216000000 24.4949 8.4343 536 287296 153990656 23.1517 8.1231 601 361201 217081801 24.5153 8.4390 537 288369 154854153 23.1733 8.1281 602 362404 218167208 24.5357 8.4437 538 289444 155720872 23.1948 8.1332 603 363609 219256227 24.5561 8.4484 539 290521 156590819 23.2164 8.1382 604 364816 220348864 24.5764 8.4530 540 291600 157464000 23.2379 8.1433 605 366025 221445125 24.5967 8.4577 541 292681 158340421 23.2594 8.1483 606 367236 222545016 24.6171 8.4623 542 293764 159220088 23.2809 8.1533 607 368449 223648543 24.6374 8.4670 543 294849 160103007 23.3024 8.1583 608 369664 224755712 24.6577 8.4716 544 295936 160989184 23.3238 8.1633 609 370881 225866529 24.6779 8.4763 545 297025 161878625 23.3452 8.1683 610 372100 226981000 24.6982 8.4809 546 298116 162771336 23.3666 8.1733 611 373321 228099131 24.7184 8.4856 547 299209 163667323 23.3880 8.1783 612 374544 229220928 24.7386 8.4902 648 300304 164566592 23.4094 8.1833 613 375769 230346397 24.T588 8.4948 549 301401 165469149 23.4307 8.1882 614 376996 231475544 24.7790 8.4994 550 302500 166375000 23.4521 8.1932 615 378225 232608375 24.7992 8.5040 551 303601 167284151 23.4734 8.1982 616 379456 233744896 24.8193 8.5086 552 304704 168196608 23.4947 8.2031 617 380689 234885113 24.8395 8.5132 553 305809 169112377 23.5160 8.2081 618 381924 236029032 24.8596 8.5178 554 306916 170031464 23.5372 8.2130 619 383161 237176659 24.8797 8.5224 555 308025 170953875 23.5584 8.2180 620 384400 238328000 24.8998 8.5270 556 309136 171879616 23.5797 8.2229 621 385641 239483061 24.9199 8.5316 557 310249 172808693 23.6008 8.2278 622 386884 240641848 24.9399 8.5362 558 311364 173741112 23.6220 8.2327 623 388129 241804367 24.9600 8.5408 559 312481 174676879 23.6432 8.2377 624 389376 242970624 24.9800 8.5453 560 313600 175616000 23.6643 8.2426 625 390625 244140625 25. 8.5499 561 314721 176558481 23.6854 8.2475 626 391876 245314376 25.0200 8.5544 562 315844 177504328 23.7065 8.2524 627 393129 246491883 25.0400 8.5590 563 316969 178453547 23.7276 8.2573 628 394384 247673152 25.0599 8.5635 564 318096 179406144 23.7487 8 2621 629 395641 248858189 25.0799 8.5681 565 319225 180362125 23.7697 8.2670 630 396900 250047000 25.0998 8.5726 566 320356 181321496 23.7908 8.2719 631 398161 251239591 25.1197 8.5772 567 321489 182284263 23.8118 8.2768 632 399424 252435968 25.1396 8.5817 568 322624 183250432 23.8328 8.2816 633 400689 253636137 25.1595 8.5862 569 323761 184220009 23.8537 8.2865 634 401956 254840104 25.1794 8.5907 570 324900 185193000 23.8747 8.2913 635 403225 256047875 25.1992 8.5952 571 326041 186169411 23.8956 8.2962 636 404496 257259456 25.2190 8.5997 572 327184 187149248 23.9165 8.3010 637 405769 258474853 25.2389 8.6043 573 328329 188132517 23.9374 8.3059 638 407044 259694072 25.2587 8.6088 574 329476 189119224 23.9583 8.3107 639 408321 260917119 25.2784 8.613'Jt 75 330625 190109375 23.9792 8.3155 640 409600 262144000 25.2982 8.6177 54 SQUARES, CUBES, AND ROOTS. TABLE of Squares, Cnbes, Square Roots, and Cube Roots, of X aim her* from 1 to 10OO (CONTINUED.) I Square. Cube. Sq. Rt. C. Rt. No. Square. Cube. Sq. Rt. C. Rt. 410881 263374721 25.3180 8.6222 706 498436 351895816 26.5707 8.9043 412164 264609288 25.3377 8.6267 707 499849 353393243 26.5895 8.9085 413449 265847707 25.3574 8.6312 708 501264 354894912 26.6083 8.9127 414736 267089984 25.3772 8.6357 709 502681 356400829 26.6271 8.9169 416025 268336125 25.3969 8.6401 710 504100 357911000 26.6458 8.9211 417316 269586136 25.4165 8.6446 711 505521 359425431 26.6646 8.9253 418609 270840023 25.4362 8.6490 712 506944 360944128 26.6833 8.9295 419904 272097792 25.4558 8.6535 713 508369 362467097 26.7021 8.9337 421201 273359449 25.4755 8.6579 714 509796 363994344 26.7208 8,9378 422500 274625000 25.4951 8.6624 715 511225 365525875 26.7395 8.9420 423801 275894451 25.5147 8.6668 716 512656 367061696 26.7582 8.9462 425104 277167808 25.5343 8.6713 717 514089 368601813 26.7769 8.9503 426409 278445077 25.5539 8.6757 718 515524 370146232 26.7955 8.9545 427716 279720264 25.5734 8.6801 719 516961 371694959 26.8142 8.9587 429025 281011375 25.5930 8.6845 720 518400 373248000 26.8328 8.9628 430336 282300416 25.6125 8.6890 721 519841 374805361 26.8514 8.9670 431649 283593393 25.6320 8.6934 722 521284 376367048 26.8701 8.9711 432964 284890312 25.6515 8.6978 723 522729 377933067 26.8887 8.9752 434281 286191179 25.6710 8.7022 724 524176 379503424 26.9072 8.9794 435600 287496000 25.6905 8.70G6 725 525625 381078125 26.9258 8.9835 436921 288804781 25.7099 8.7110 726 527076 382657176 26.9444 8.9876 438244 290117528 25,7294 8.7154 727 528529 384240583 26.9629 8.9918 439569 291434247 25.7488 8.7198 728 529984 385828352 26.9815 8.9959 440896 292754944 25.7682 8.7241 729 531441 387420489 27. 9. 442225 294079625 25.7876 8.7285 730 532900 389017000 27.0185 9.0041 443556 295408296 25.8070 8.7329 731 534361 390617891 27.0370 9.0082 444889 296740963 25.8263 8.7373 732 535824 392223168 27.0555 9.0123 446224 298077632 25.8457 8.7416 733 537289 393832837 27.0740 9.0164 447161 299418309 25.8650 8.7460 734 538756 395446904 27.0924 9.0205 448900 300763000 25.8844 8.7503 735 540225 397065375 27.1109 9.0246 450241 302111711 25.9037 8.7547 736 541696 398688256 27.1293 9.02H7 451584 303464448 25.9230 8.7590 737 543169 400315553 27.1477 9.0328 452929 304821217 25.9422 8.7634 738 544644 401947272 27.1662 9.0369 454276 306182024 25.9615 8.7677 739 546121 403583419 27.1846 9.0410 455625 307546875 25.9808 8.7721 740 547600 405224000 27.2029 9.0450 456976 308915776 26. 8.7764 741 549081 406869021 27.2213 9.0491 458329 310288733 26.0192 8.7807 742 550564 408518488 27.2397 9.0532 459684 311665752 26.0384 8.7850 743 552049 410172407 27.2580 9.0572 461041 313046839 26.0576 8.7893 744 553536 411830784 27.2764 9.0613 462400 314432000 26.0768 8.7937 745 555025 413493625 27.2947 9.0654 463761 315821241 26.0960 8.7980 746 556516 415160936 27.3130 9.0694 465124 317214568 26.1151 8.8023 747 558009 416832723 27.3313 9.0735 466489 318611987 26.1343 8.8066 748 559504 418508992 27.3496 9.0775 467856 320013504 26.1534 8.8109 749 561001 420189749 27.3679 9.0816 469225 321419125 26.1725 8.8152 750 562500 421875000 27.3861 9.0856 470596 322828856 26.1916 8.8194 751 564001 423564751 27.4044 9.0896 471969 324242703 26.2107 8.8237 752 565504 425259008 27.4226 9.0937 473344 325660672 26.2298 8.8280 753 567009 426957777 27.4408 9.0977 474721 327082769 26.2488 8.8323 754 568516 428661064 27.4591 9.1017 476100 328509000 26.2679 8.83S6 755 570025 430368875 27.4773 9.1057 477481 329939371 26.2869 8.8408 756 571536 432081216 27.4955 9.1098 478864 331373888 26.3059 8.8451 757 573049 433798093 27.5136 9.1138 480249 33'/812557 26.3249 8.8493 758 574564 435519512 27.5318 9.1178 481636 334255384 26.3439 8.8536 759 576081 437245479 27.5500 9.1218 483025 335702375 26.3629 8.8578 760 577600 438976000 27.5681 9.1258 484416 337153536 26.3818 8.8621 761 579121 440711081 27.5862 9.1298 485809 338608873 26.4008 8.8663 762 580644 442450728 27.6043 9.1338 487204 340068392 26.4197 8.8706 763 582169 444194947 27.6225 9.1378 488601 341532099 26.4386 8.8748 764 583696 445943744 27.6405 9.1418 490000 343000000 26.4575 8.8790 765 585225 447697125 27.6586 9.1458 491401 344472101 26.4764 8.8833 766 586756 449455096 27.6767 9.1498 492804 345948408 26.4953 8.8875 767 588289 451217663 27.6948 9.1537 494209 347428927 26.5141 8.8917 768 589824 452984832 27.7128 9.1577 495616 348913664 26.5330 8.8959 769 591361 454756609 27.7308 9.1617 487026 360402625 26.6518 8.9001 770 592900 466533006 27.7488 9.1657 SQUARES, CUBES, AND ROOTS. 55 TABLE of Squares, Cubes, Square Roots, and Cube Roots, of Numbers from 1 to 1OOO (CONTINUED.; No. Square. Cube. Sq. St. C. Rt. No. Square Cube. Sq. Rt. C. R*. 771 594441 45831401 27.7669 9.1696 836 698896 58427705 28.9137 9.4204 772 595984 46009964 27.7849 9.1736 837 700569 58637625d 28.9310 9.424! 773 597529 46188991 27.8029 9.1775 838 702244 58848047 28.9482 9.4279 774 599076 46368482 27.8209 9.1815 839 703921 5905897111 28.9655 9.4316 T75 600625 46548437 27.8388 9.1855 840 705600 59270400 28.9828 9.4354 776 602176 467288576 27.8568 9.1894 841 707281 59482332 29. 9.4391 777 603729 469097433 27.8747 9.1933 842 708964 596947688 29.0172 9.4429 778 605284 47091095 27.8927 9.1973 843 710649 599077107 29.0345 9.4466 779 606841 472729139 27.9106 9.2012 844 712336 601211584 29.0517 9.4503 780 608400 474552000 27.9285 9.2052 845 714025 603351120 29.0689 9.4541 781 609961 47637954 27.9464 9.2091 846 715716 605495736 29.0861 9.4578 782 611524 478211768 27.9643 9.2130 817 717409 607645423 29.1033 9.4615 783 613089 480048687 27.9821 9.2170 848 719104 609800192 29.1204 9.4652 784 614656 481890304 28. 9.2209 849 720801 611960049 29.1376 9.4690 785 616225 483736625 ' 28.0179 9.2248 850 722500 614125000 29.1548 9.4727 786 617796 485587656 28.0357 9.2287 851 724201 616295051 29.1719 9.4764 787 619369 487443403 28.0535 9.2326 852 725904 618470208 291890 9.4801 788 620944 489303872 28.0713 9.2365 853 727609 620650477 29.2062 94838 789 622521 491169069 28.0891 9.2404 854 729316 622835864 29.2233 9.4875 790 624100 493039000 28.1069 9.2443 855 731025 625026375 29.2404 9.4912 791 625681 494913671 28.1247 9.2482 856 732736 627222016 29.2575 9.4949 792 627264 496793088 28.1425 9.2521 857 734449 62942279: 29.2746 9.4986 793 628849 498677257 28.1603 9.2560 858 736164 631628712 29.2916 9.5023 794 630136 500566184 28.1780 9.2599 859 737881 633839779 29.3087 9.5060 795 632025 502459875 28.1957 9.2638 860 739600 636056000 29.3258 9.5097 796 633616 504358336 28.2135 9.2677 861 741321 638277381 29.3428 9.5134 797 635209 506261573 28.2312 9.2716 862 743044 640503928 29.3598 9.5171 798 636804 508169592 28.2489 9.2754 863 744769 642735647 29.3769 9.5207 799 638401 510082399 28.2666 9.2793 864 746496 644972544 29.3939 9.5244 800 640000 512000000 28.2843 9.2832 865 748225 647214625 29.4109 9.5281 801 641601 513922401 28.3019 9.2870 866 749956 649461896 29.4279 9.5317 802 643204 515849608 28.3196 9.2909 867 751689 651714363 29.4449 9.5354 803 644809 517781627 28.3373 9.2948 808 753424 653972032 29.4618 9.5391 804 646416 519718464 28.3549 9.2986 869 755161 <;.->62::i9i>9 29.4788 9.5427 805 648025 521660125 28.3725 9.3025 870 756900 658503000 29.4958 9.5464 806 649636 523606616 28.3901 9.3063 871 758641 660776311 29.5127 9.5501 807 651249 525557943 28.4077 9.3102 872 760384 663054848 29.5296 9.5537 808 652864 527514112 28.4253 9.3140 873 762129 665338617 29.5466 9.5574 809 654481 529475129 28.4429 9.3179 874 763876 667627624 29.5635 9.5610 810 656100 531441000 28.4605 9.3217 875 765625 669921875 29.5804 9.5647 811 657721 533411731 28.4781 9.3255 876 767376 672221376 29.5973 9.5683 812 659344 5353*732* 28.4956 9.3294 877 769129 674526133 29.6142 9.5719 813 660969 537367797 28.5132 9.3332 878 770884 676836152 29.6311 9.5756 814 662596 539353144 28.5307 9.3370 879 772641 679151439 29.6479 9.5792 815 664225 541343375 28.5482 9.3408 880 774400 681472000 29.6648 9.5828 816 665856 543338496 28.5657 9.3447 881 776161 683797841 29.6816 9.5865 817 667489 645338513 28.5832 9.3485 882 777924 686128968 29.6985 9.5901 818 669124 547343432 28.6007 9.3523 883 779689 688465387 29.7153 9.5937 819 670761 5W:!.Vi2.V 28.6182 9.3561 884 781456 690807104 29.7321 9.5973 820 672400 551368000 28.6356 9.3599 885 783225 693154125 29.7489 9.6010 821 674041 553387661 28.6531 9.3637 886 784996 695506456 29.7658 9.6046 822 675684 555412248 28.6705 9.3675 887 786769 697864103 29.7825 9.6082 823 677329 557441767 28.6880 9.3713 888 788544 700227072 29.7993 9.6118 824 678976 55947(5224 2S.70.i4 9.3751 889 790321 702595369 29.8161 9.6154 825 680625 561515625 28.7228 9.3789 890 792100 704969000 29.8329 9.6190 826 682276 563559976 28.7402 9.3827 891 793881 707347971 29.8496 9.6226 827 683929 5650923 28.7576 9.3865 892 795664 709732288 29.8664 9.6262 828 685584 567M3552 28.7750 9.3902 893 797449 712121957 29.8831 9.6298 829 687241 569722789 28.7924 9.3940 894 799236 714516984 29.8998 9.6334 830 688900 571787000 28.8097 9.3978 895 801025 716917375 29.9166 9.6370 831 690561 573856191 28.8271 9.4016 896 802816 719323136 29.9333 9.6406 832 692224 575930368 28.8444 9.4053 897 804609 721734273 29.9500 9.6442 833 693889 578009537 28.8617 9.4091 898 806404 724150792 29.9666 9.6477 834 695556 580093704 28.8791 9.4129 899 808201 726572699 29.9833 9.6513 835 697225 682182S75 28.8964 9.4166 J 900 810000 729000000 30. 9.6540 56 SQUARES, CUBES, AND ROOTS. TABLE of Square**, Cubes, Square Roots, and Cube Roots, of Numbers from 1 to 1OOO (CONTINUED.) No. Square. Cube. Sq. Rt. C. Rt. No. Square. Cube. Sq. Rt, C. Rt. 901 902 811801 813604 731432701 733870808 30.0167 30.0333 9.6585 9.6620 951 952 904101 906304 b60085351 862501408 30.8383 30.8545 9.8339 0.8374 903 815409 736314327 30.0500 9.6656 953 908209 8G5523177 30.8707 9.8408 904 817216 738763264 30.0666 9.6692 954 910116 868250664 30.8S69 9.8443 905 819025 741217625 30.0832 9.6727 955 912025 870983875 30.9031 9.8477 906 820836 743677416 30.0998 9.6763 956 913936 873722816 30.9192 9.8511 907 822649 746142643 30.1164 9.6799 957 915849 876467-193 30.9354 9.8546 908 824464 748613312 30.1330 9.6834 958 917764 879217912 30.9516 9.8580 909 826281 75 1081)429 30.1496 9.6870 959 919681 881974079 30.9677 9.8614 910 828100 753571000 30.1662 9.6905 960 921600 884736000 30.9839 9.8648 911 829921 756058031 30.1828 9.6941 961 923521 887503681 31. 9.8683 912 831744 758550528 30.1993 9.6976 962 925444 890277128 31.0161 9.8717 913 833569 761048497 30.2159 9.7012 963 927369 893056347 31.0322 9.8751 914 835396 76:5551944 30.2324 9.7047 964 929296 895841344 31 .0483 9.8785 915 837225 766060875 30.2490 9.7082 965 931225 898632125 31.0644 9.8819 916 839056 768575296 30.2655 9.7118 966 933156 901428696 31.0805 9.8854 917 840889 771095213 30.2820 9.7153 967 935089 904231063 31.0966 9.8888 918 842724 773620632 30.2985 9.7188 968 937024 907039232 31.1127 9.8922 919 844561 776151559 30.3150 9.7224 969 938961 909853209 31.1288 9.8956 920 846400 778688000 30.3315 9.7259 970 940900 912673000 31.1448 9.8990 921 848241 781229961 30.3480 9.7294 971 942841 915498611 31.1609 9.9024 922 850084 783777448 30.3645 9.7329 972 914784 918330048 31.1769 9.i;058 923 851929 786330467 30.3809 9.7364 973 946729 921167317 31.1929 9.9092 924 853776 788889024 30.3974 9.7400 974 948676 924010424 31.2090 9.9126 925 855625 791453125 30.4138 9.7435 975 950625 926859375 31.2250 9.9160 926 857476 791022776 30.4302 9.7470 976 952576 929714176 31.2410 9.9194 927 859329 796597983 30.4467 9.7505 977 954529 932574833 31.2570 9.9227 928 861 184 799178752 30.4631 9,7540 978 956484 935441352 31.2730 9.9261 929 863041 801765089 30.4795 9.7575 979 958441 938313739 31.2890 9.9295 930 864900 804357000 30.4959 9.7610 980 960400 941192000 31.3050 9.9329 931 866761 806954491 30.5123 9.76-15 981 962361 944076141 31.3209 9.9363 932 868624 809557568 30.5287 9.7680 982 964324 946966168 31.3369 9.9396 933 870489 8l2166'j:J7 30.5450 9.7715 983 966289 949862087 31.3528 9.9430 934 872356 814780504 30.5614 9.7750 9S4 968256 952763904 31.3688 9.9464 935 874225 817400375 30.5778 9.7785 985 970225 955671625 31.3847 9.9497 936 876096 820025856 30.5941 9.7819 986 972196 958585256 31.4006 9.9531 937 877969 H22656953 30.6105 9.7854 987 974169 961504803 31.4166 9.9565 938 879844 S2.V293672 30.6268 9.7889 988 976144 964430272 31.4325 9.9598 939 881721 827936019 30.6431 9.7924 989 978121 967361669 31.4484 9.9632 940 883600 830584000 30.6594 9.7959 990 980100 970299000 31.4643 9.9666 941 885481 833237621 30.6757 9.7993 991 982081 973242271 31.4802 9.9699 942 687364 835896888 30.6920 9.8028 992 984064 976191488 SI. 4960 9.9733 943 889249 838561807 30.7083 9.8063 993 986049 979146657 31.5119 9.9766 944 891136 841232384 30.7246 9.8097 994 988036 982107784 81.5278 9.9800 945 893025 843908625 30.7409 9.8132 995 990025 985074875 31.5436 9.9833 946 894916 846590536 30.7571 9.8167 996 992016 988047936 31.5595 9.9866 947 896809 849278123 30.7734 9.8201 997 994009 991026973 31.5753 9.95)00 948 898704 851971392 30.7896 9.8236 998 996004 994011992 81.5911 9.9933 949 900601 854670349 30.8058 9.8270 999 998001 997002999 81.6070 9.9967 950 902500 857375000 30.8221 9.8305 1000 1000000 1000000000 31.6228 10. SQUARE AND CUBE ROOTS. Square Roots and Cube Roots of X ambers from 1000 to 1OOOO. Num. Sq. Rt. Cu.Rt Num. Sq. Rt. Cu.Rt. Num. Sq. Rt. Cu. Rt. Num. Sq. Rt. Cu. Rt. 1005 31.70 10.02 1405 3 .48 11.20 1805 42.49 12.18 2205 46.06 13.02 1010 31.78 10.03 1410 3 .55 11.21 1810 42.54 12.19 2210 47.01 13.03 1015 31.86 10.05 1415 3 .62 11.23 1815 42.60 12.20 2215 47.06 13.04 1020 31.94 10.07 1420 3 .68 11.24 1820 42.66 12.21 2220 47.12 13. 05 1025 32.02 10.08 1425 3 .75 11.25 1825 42.72 12.22 2225 47.17 13.05 1030 32.09 10.10 1430 3 .82 11.27 1830 42.78 12.23 2230 47.22 13.06 1035 32.17 10.12 1435 37.88 -11.28 1835 42.84 12.24 2235 47.28 13.07 1040 32.25 10.13 1440 37.95 11.29 1840 42.90 12.25 2240 47.33 13.08 1045 32.33 10.15 1445 38.01 11.31 1845 42.95 12.26 2245 47.38 13.09 1050 32.40 10.16 1450 38.08 11..-52 1850 43.01 12.28 2250 47.43 13.10 1055 32.48 10.18 1455 38.14 11.33 1855 43.07 12.29 2255 47.49 13.11 1060 32.56 10.20 1460 38.21 11.34 1860 43.13 12.30 2260 47.54 13.12 1065 32.63 10.21 1465 38.28 11.36 1865 43.19 12.31 2265 47.59 13.13 1070 32.71 10.2:5 1470 38.34 11.37 1870 43.24 12.32 2270 47.64 13.14 1075 32.79 10.24 1475 38.41 11.38 1875 43.30 12.33 2275 47.70 13.15 1080 32.86 10.26 1480 38.47 11.40 1880 43.36 12.34 2280 47.75 13.16 1085 32.94 10.28 1485 38.54 11.41 1885 43.42 12.35 2285 47.80 13.17 1090 33.02 10.29 1490 38.60 11.12 1890 4:5.17 12.36 2290 47.85 13.18 1095 33.09 10.31 1495 38.67 11.4:5 1895 43.53 12.37 2295 47.91 13.19 1100 33.17 10.32 1500 38.73 11.15 1900 43.59 12.39 2300 47.96 13.20 1105 33.24 10.34 1505 38.79 11.46 1905 43.65 12.40 2305 48.01 13.21 1110 33.32 10.85 1510 38.86 11.47 1910 43.70 12.41 2310 48.06 13.22 1115 33.39 10.37 1515 38.92 11.49 1915 43.76 12.42 2315 48.11 13.23 1120 33.47 10.38 1520 38.99 11.50 1920 43.82 12.43 2320 48.17 13.24 1125 33.54 10.40 1525 39.05 11.51 1925 43.87 12.44 2325 48.22 13.25 1130 33.62 10.42 1530 39.12 11.52 1930 43.93 12.45 2330 48.27 13.26 1135 33.69 10.43 1535 39.18 11.51 1935 43.99 12.46 2335 48.32 13.27 1140 33.76 10.45 1540 39.24 11.55 1940 41.05 12.47 2340 48-37 13.28 1145 33.84 10.46 1545 39.31 11.56 1915 44.10 12,48 2345 48.43 13.29 1150 33.91 10.48 1550 39.37 11.57 1950 44.16 12.49 2350 48.48 13.30 1155 33.99 10.49 1555 39.43 11. o9 1955 44.22 12.50 2355 48.53 13.30 1160 34.06 10.51 1560 39.50 11.60 1960 44.27 12.51 2360 48.58 13.31 1165 34.13 10.52 1565 39.56 11.61 1965 44.33 12.5:5 2365 48.63 13.32 1170 34.21 10.54 1570 39.62 11.62 1970 44.38 12.54 2370 48.68 13.33 1175 34.28 10.55 1575 39.69 11.63 1975 44.44 12.55 2375 48.73 13.34 1180 34.35 10.57 1580 39.75 11.65 1980 44.50 12.56 2380 48.79 13.35 1185 34.42 10.58 1585 39.81 11.66 1985 44.55 12.57 2385 48.84 13.36 1190 34.50 10.60 1590 39.87 11.67 1990 44.61 12.58 2390 48.89 13.37 1195 34.57 10.61 1595 39.94 11.68 1995 44.67 12.59 2395 48.94 13.38 1200 34.64 10.63 1600 40.00 11.70 2000 44.72 12.60 2400 48.99 13.39 1205 34.71 10.61 1605 40.06 11.71 2005 44.78 12.61 2405 49.04 13.10 1210 34.79 10.66 1610 40.12 11.72 2010 44.83 12.62 2410 49.09 13.41 1215 34.86 10.67 1615 40.19 11.73 2015 44.89 12.63 2415 49.14 13.42 1220 84.93 10.69 1620 40.25 11.74 2020 44.94 12.64 2420 49.19 13.43 1225 35.00 10.70 1625 40.31 11.76 2025 45.00 12.65 2425 49.24 13.43 1230 35.07 10.71 1630 40.37 11.77 2030 45.06 12.66 2430 49.30 13.44 1235 35.14 10.73 1635 40.44 11.78 2035 45.11 12.67 2435 49.35 13.45 1240 35.21 10.71 1640 40.50 11.79 2040 45.17 12.68 2440 49.40 13.46 1245 35.28 10.76 1645 40.56 11. SO 2045 45.22 12.69 2445 49.45 13.47 1250 35.36 10.77 1650 40.62 11.82 2050 45.28 12.70 2450 49.50 13.48 1255 35.43 10.79 1655 40.68 11.83 2055 45.33 12.71 2460 49.60 13.50 1260 35.50 10.80 1660 40.74, 11.84 2060 45.39 12.72 2470 49.70 13.52 1265 35.57 10.82 1665 40.80 11.85 2065 45.44 12.73 2480 49.80 13.54 1270 35.64 10.83 1670 40.87 11.86 2070 45.50 12.74 2490 49.90 13.55 1275 35.71 10.84 1675 40.93 11.88 2075 45.55 12.75 2500 50.00 13.57 1280 35.78 10.86 1680 40.99 11.89 2080 4561 12.77 2510 50.10 13.59 1285 35.85 10.87 1685 41.05 11.90 2085 45.66 12.78 2520 50.20 13.61 1290 35.92 10.89 1690 41.11 11.91 2090 45.72 12.79 2530 50.30 13.63 1295 35.99 10.90 1695 41.17 11.92 2095 45.77 12.80 2540 50.40 13.64 1300 36.06 10.91 1700 41.23 1 1 .9:5 2100 45.83 12.81 2550 50.50 13.66 1305 36.12 10.93 1705 41.29 11.95 2105 45.88 12.82 2560 50.60 13.68 1310 36.19 10.94 1710 41.35 11.96 2110 45.93 12.83 2570 50.70 13.70 1315 86.26 10.96 1715 41.41 11.97 2115 45.99 12.81 2580 50.79 13.72 1320 86.33 10.97 1720 41.47 11.98 2120 46.04 12.85 2590 50.89 13.73 1325 36.40 10.98 1725 41.53 11.99 2125 46.10 12.86 2600 50.99 13.75 1330 36.47 11.00 1730 41.59 12.00 2130 46.15 12.87 2610 51.09 1:5.77 1335 36.54 11.01 1735 41.65 12.02 2135 46.21 12.88 2620 51.19 13.79 1340 36.61 11.02 1740 41.71 12.03 2140 46.26 12.89 2630 51.28 13.80 1345 36.67 11.01 1745 41.77 12.04 2145 46.31 12.SO 2640 51.38 13.82 1350 36.74 11.05 1750 41.83 12.05 2150 46.37 12.91 2650 51.48 13.84 1355 36.81 11.07 1755 41.89 12.06 2155 46.42 12.92 2660 51.58 13.86 1360 36.88 11.08 1760 41.95 12.07 2160 46.48 12.9:5 2670 51.67 13.87 1365 36.95 11.09 1765 42.01 12.09 2165 46.53 12.94 2680 51.77 13.89 1370 37.01 11.11 1770 42.07 12.10 2170 46.58 12.95 2690 51.87 13.91 1375 37.08 11.12 1775 42.13 llll 2175 46.64 12.96 2700 51.96 13.92 1380 37.15 11.13 1780 42.19 12.12 2180 46.69 12.97 2710 52.06 13.94 1385 37.22 11.15 1785 42.25 12.13 2185 46.74 12.98 2720 52.15 13.96 1390 37.28 11.16 1790 42.31 12.14 2190 46.80 12.99 2730 52.25 13.98 1395 37.155 11.17 1795 42.37 12.15 2195 46.85 13.00 2740 52.35 13.99 1400 37.42 11.19 1800 42.43 12.16 2200 46.1W 13.01 2750 62.44 14.01 68 SQUARE AND CUBE ROOTS. Square Roots and Cube Roots of Numbers from 100O tolOOOO (CONTINUED.) Num. Sq. Rt. Cu. Rt. Num. Sq. Rt. Cu. Rt. Num. Sq. Rt. Cu.Rt. Num. Sq. Rt. Cu. Rt. 2760 52.54 14.03 3550 59.58 15.25 4340 65.88 16.31 5130 71.62 17.25 2770 52.63 14.04 3560 59.67 15.27 4350 65.95 16.32 5140 71.69 17.26 2780 52.73 14.06 3570 59.75 15.28 4360 66.03 16.34 5150 71.76 17.27 2790 52.82 14.08 3580 59.83 15.30 4370 66.11 16.35 5160 71.83 17.28 2800 52.92 14.09 3590 59.92 15.31 4380 66.18 16.36 5170 71.90 17.29 2810 53.01 14.11 3600 60.00 15.33 4390 66.26 16.37 5180 71.97 17.30 2820 53.10 14.13 3610 60.08 15.34 4400 66.33 16.39 5190 72.04 17.31 2830 53.20 14.14 3620 60.17 15.35 4410 66.41 16.40 5200 72.11 17.32 2840 53.29 14.16 3630 60.25 15.37 4420 66.48 16.41 5210 72.18 17.34 2850 53.39 14.18 3640 60.33 15.38 4430 66.56 16.42 5220 72.25 17.35 2860 53.48 14.19 3650 60.42 15.40 4440 66.63 16.44 5230 72.32 17.36 2870 53.57 14.21 3660 60.50 15.41 4450 66.71 16.45 5240 72.39 17.37 2880 53.67 14.23 3670 60.58 15.42 4460 66.78 16.46 5250 72.46 17.38 2890 53.76 14.24 3680 60.66 15.44 4470 66-86 16.47 5260 72.53 17.39 2900 53.85 14.26 3690 60.75 15.45 4480 66.93 16.49 5270 72.59 17.40 2910 53.94 14.28 3700 GO. 83 15.47 4490 67.01 16.50 5280 72.66 17.41 2920 54.04 14.29 3710 60.91 15.48 4500 67.08 16.51 5290 72.73 17.42 2930 54.13 14.31 3720 60.99 15.49 4510 67.16 16.52 5300 72.80 17.44 2910 54.22 14.33 3730 61.07 15.51 4520 67.23 16.53 5310 72.87 17.45 2950 54.31 14.34 3740 61.16 15.52 4530 67.31 16.55 5320 72.94 17.46 2960 54.41 14.36 3750 61.24 15.54 4540 67.38 16.56 5330 73.01 17.47 2970 54.50 14.37 3760 61.32 15.55 4550 67.45 16.57 5340 73.08 17.48 2980 54.59 14.39 3770 61.40 15.56 4560 67.53 16.58 5350 73.14 17.49 2990 54.68 14.41 3780 61.48 15.58 4570 67.60 16.59 5360 73.21 17.50 3000 54.77 14.42 3790 61.56 15.59 4580 67.68 16.61 5370 73.28 17.51 3010 54.86 14.44 3800 61 64 15.60 4590 67.75 16.62 5380 73.35 17.52 3020 54.95 14.45 3810 61.73 15.62 4600 67.82 16.63 5390 73.42 17.53 3030 55.05 14.47 3820 61.81 15.63 4610 67.90 16.64 5400 73.48 17.54 3010 55.14 14.49 3830 61.89 15.65 4620 67.97 16.66 5410 73.55 17.55 3050 55.23 14.50 3840 61.97 15.66 4630 68.04 16.67 5420 73.62 17.57 3060 55.32 14.52 3850 62.05 15.67 4640 68.12 16.68 5430 73.69 17.58 3070 55.41 14.53 3860 62.13 15.69 4650 68.19 16.69 5440 73.76 17.59 3080 55.50 14.55 3870 62.21 15.70 4660 68.26 16.70 5450 73.82 17.60 3090 55.59 14.57 3880 62.29 15.71 4670 68.34 16.71 5460 73.89 17.61 3100 55.68 14.58 3890 62.37 15.73 4680 68.41 16.73 5470 73.96 17.62 3110 55.77 14.60 3900 62.45 15.74 4690 68.48 16.74 5480 74.03 17.63 3120 55.86 14.61 3910 62.53 15.75 4700 68.56 16.75 5490 74.09 17.64 3130 55.95 14.63 3920 62.61 15.77 4710 68.63 16.76 5500 74.16 K.65 3140 56.04 14.64 3930 62.69 15.78 4720 68.70 16.77 5510 74.23 17.66 3150 56.12 14.66 3940 62.77 15.79 4730 68.77 . 16.79 5520 74.30 17.67 3160 56.21 14.67 3950 62.85 15.81 4740 68.85 16.80 5530 74.36 17.68 3170 56.30 14.69 3960 62.93 15.82 4750 68.92 16.81 5540 74.43 17.69 3180 56.39 14.71 3970 63.01 15.83 4760 68.99 16.82 5550 74.50 17.71 :*3190 56.48 14.72 3980 63.09 15.85 4770 69.07 16.83 5560 74.57 17.72 3200 56.57 14.74 3990 63.17 15.86 4780 69.14 16.85 5570 74.63 17.73 S210 56.66 14.75 4000 63.25 15.87 4790 69.21 16.86 5580 74.70 17.74 3220 56.75 14.77 4010 63.32 15.89 4800 69.28 16.87 5590 74.77 17.75 3230 56.83 14.78 4020 63.40 15.90 4810 69.35 16.88 5600 74.83 17.76 3240 56.92 14.80 4030 63.48 15.91 4820 69.43 16 89 5610 74.90 17.77 3250 57.01 14.81 4040 63.56 15.93 4830 69.50 16.90 5620 74.97 17.78 3260 57.10 14.83 4050 63.64 15.94 4840 69.57 16.92 5630 75.03 17.79 3270 57.18 14.84 4060 63.72 15.95 4850 VJ.64 16.93 5640 75.10 17.80 3280 57.27 14.86 4070 63.80 15.97 4860 69.71 16.94 5650 75.17 17.81 3290 57.36 14.87 4080 63.87 15.98 4870 69.79 16.95 5660 75.23 17.82 3300 57.45 14.89 4090 63.95 15.99 4880 69.86 16.96 5670 75.30 17.83 3310 57.53 14.90 4100 64.03 16.01 4890 69.93 16.97 5680 75.37 17.84 3320 57.62 14.92 4110 64.11 16.02 4900 70.00 16.98 5690 75.43 17.85 3330 57.71 14.93 4120 64.19 16.03 4910 70.07 17.00 5700 75.50 17.86 3340 57.79 14.95 4130 64.27 16.04 4920 70.14 17.01 5710 75.56 17.87 3350 57.88 14.96 4140 64.34 16.06 4930 70.21 17.02 5720 75.63 17.88 3360 57.97 14.98 4150 64.42 16.07 4940 70.29 17.03 5730 75.70 17.89 3370 58.05 14.99 4160 64.50 16.08 4950 70.36 17.04 5740 75.76 17.90 3380 58.14 15,01 4170 64.58 16.10 4960 70.43 17.05 5750 75.83 17.92 3390 58.22 15.02 4180 64.65 16.11 4970 70.50 17.07 5760 75.89 17.93 3400 58.31 15.04 4190 64.73 16.12 4980 70.57 17.08 5770 75.96 17.94 3410 58.40 15.05 4200 64.81 16.13 4990 70.64 17.09 5780 76.03 17.95 3420 58.48 5.07 4210 64.88 16.15 5000 70.71 17.10 5790 76.09 17.96 3430 58.57 5.08 4220 64.96 16.16 5010 70.78 17.11 5800 76.16 17.97 3440 58.65 5.10 4230 65.04 16.17 5020 70.85 17.12 5810 70. '22 17.98 3450 58.74 5.11 4240 65.12 16.19 5030 70.92 17.13 5820 76.29 17.99 3460 58.82 5.12 4250 65.19 16.20 5040 70.99 17.15 5830 76.35 18.00 3470 58.91 15.14 4260 65.27 16.21 5050 71.06 17.16 5840 76.42 18.01 3480 58.99 15.15 4270 65.35 16.22 5060 71.13 17.17 5850 76.49 18.02 3490 59.08 15.17 4280 65.42 16.24 5070 71.20 17.18 5860 76.55 8.03 3500 59.16 15.18 4290 65.50 16.25 5080 71.27 17.19 5870 76.62 8.04 3510 59.25 15.20 4300 6557 16.26 5090 71.34 17.20 5880 76.68 8.05 3520 59.33 15.21 4310 65.65 16.27 5100 71.41 17.21 5890 76.75 8.06 3530 59.41 15.23 4320 65.73 16.29 5110 71.48 17.22 5900 76.81 8.07 3549 58.50 15.24 4330 65.80 16.30 6120 71.55 17.24 5910 76.88 18.08 SQUARE AND CUBE ROOTS. Otf Square Roots and Cube Roots of tf ambers from 1OOO to 1OOOO (CONTINUED.) Num. Sq. Rt. Cu. Rt. Num. Sq. Rt.JGu.Rt. Num. Sq. Rt. Cu. Rt. Num. Sq. Rt.j Cu.Rt, 5920 76.94 18.09 6710 81.91 18.86 7500 86.60 19.57 8290 91.05 20.24 5930 77.01 18.10 6720 81.98 18.87 7510 8(>.(i6 19.58 8300 91.10 20.25 5940 77.07 18.11 6730 82.04 18.88 7520 86.72 19.59 8310 91.16 20.26 5950 77.14 18.12 6740 82.10 18.89 7530 86.78 19.60 8320 91.21 20.26 5960 77.20 18.13 6750 82.16 18.90 7540 86.83 19.61 8330 91.27 20.27 5970 77.27 18.14 6760 82.22 18.91 7550 86.89 19.62 8340 91.32 20.28 5980 77.33 18.15 6770 82.28 18.92 7560 86.95 19.63 8350 91.38 20.29 5990 77.40 18.16 6780 82.34 18.93 7570 87.01 19.64 8360 91.43 20.30 6000 77.46 18.17 6790 82.40 18.94 7580 87.06 19.64 8370 91.49 20.30 6010 77.52 18.18 6800 82.46 18.95 7590 87.12 19.65 8380 91.54 20.31 6020 77.59 18.19 6810 82.52 18.95 7600 87.18 19.66 8390 91.60 20.32 6030 77.65 18.20 6820 82.58 18.96 7610 87.24 19.67 8400 91.65 20.33 6040 77.72 18.21 6830 82.64 18.97 7620 87.29 19.68 8410 91.71 20.34 6050 77.78 18.22 6840 82.70 18.98 7630 87.35 19.69 8420 91.76 20.34 6060 77.85 18.23 6850 82.76 18.99 7640 87.41 19.70 8430 91.82 20.35 6070 77.91 18.24 6860 82.83 19.00 7650 87.46 19.70 8440 91.87 20.36 6080 77.97 18.25 6870 82.89 19.01 7660 87.52 19.71 8450 91.92 20.37 6090 78.04 18.26 6880 82.95 19.02 7670 87.58 19.72 8460 91.98 20.38 6100 78.10 18.27 6890 83.01 19.03 7680 87.64 19.73 8470 92.03 20.38 6110 78.17 18.28 6900 83.07 19.04 7690 87.69 19.74 8480 92.09 20.39 6120 73.23 18.29 6910 83.13 19.05 7700 87.75 19.75 8490 92.14 20.40 6130 78.29 18.30 6920 83.19 19.06 7710 87.81 19.76 8500 92.20 20.41 6140 78.36 18.31 6930 83.25 19.07 7720 87.8b 19.76 8510 92.25 20.42 6150 78.42 18.32 6940 83.31 19.07 7730 87.92 19.77 8520 92.30 20.42 6160 78.49 18.33 6950 83.37 19.08 7740 87.98 19.78 8530 92.36 20.43 6170 78.55 18.34 6960 83.43 19.09 7750 88.03 19.79 8540 92.41 20.44 6180 78.61 18.35 6970 83.49 19.10 7760 88.09 19.80 8550 92.47 20.45 6190 78.68 18.36 6980 83.55 19.11 7770 88.15 19.81 8560 92.52 20.46 6200 78.74 18.37 6990 83.61 19.12 7780 88.20 19.81 8570 92.57 20.46 6210 78.80 18.38 7000 83.67 19.13 7790 88.26 19.82 8580 92.63 20.47 6220 78.87 18.39 7010 83.73 19.14 7800 88.32 19.83 8590 92.68 20.48 6230 78.93 18.40 7020 83.79 19.15 7810 88.37 19.84 8600 92.74 20.49 6240 78.99 18.41 7030 83.85 19.16 7820 88.43 19.85 8610 92.79 20.50 6250 79.06 18.42 7040 83.90 19.17 7830 88.49 19.86 8620 92.84 20.50 6260 79.12 18.43 7050 83.o 19.17 7840 88.54 19.87 8630 92.90 20.51 6270 79.18 18.44 7060 84.02 19.18 7850 88.60 19.87 8640 92.95 20.52 6280. 79.25 18.45 7070 84.08 19.19 7860 88.66 19.88 8650 93.01 20.53 6290 79.31 18.46 7080 84.14 19.20 7870 88.71 19.89 8660 93.06 20.54 6300 79.37 18.47 7090 84.20 19.21 7880 88.77 19.90 8670 93.11 20.54 6310 79.44 18.48 7100 84.26 19.22 7890 88.83 19.91 8680 93.17 20.55 6320 79.50 18.49 7110 84.32 19.23 7900 88.88 19.92 8690 93.22 20.56 6330 79.56 18.50 7120 84.38 19.24 7910 88.94 19.92 8700 93.27 20.57 6340 79.62 18.51 7130 84.44 19.25 7920 88.99 19.93 8710 93.33 20.57 6350 79.69 18.52 7140 84.50 19.26 7930 89.05 19.94 8720 93.38 20.58 6360 79.75 18.53 7150 84.56 19.26 7940 89.11 19.95 8730 93.43 20.59 6370 79.81 18.54 7160 84.62 19.27 7950 89.16 19.96 8740 93.49 20.60 6380 79.87 18.55 7170 84.68 19.28 7960 89.22 19.97 8750 93.54 20.61 6390 79.94 18.56 7180 84.73 19.29 7970 89.27 19.97 8760 93.59 20.61 6400 80.00 18.57 7190 84.79 19.30 7980 89.33 19.98 8770 93.65 20.62 6410 80.06 18.58 7200 84.85 19.31 7990 89.39 19.99 8780 93.70 20.63 6420 80.12 18.59 7210 84.91 19.32 8000 89.44 20.00 8790 93.75 20.64 6430 80.19 18.60 7220 84.97 19.33 8010 89.50 20.01 8800 93.81 20.65 6440 80.25 18.60 7230 85.03 19.34 8020 89.55 20.02 8810 93.86 20.65 6450 80.31 18.61 7240 85.09 19.35 8030 89.61 20.02 8820 93.91 20.66 6460 80.37 18.62 7250 85.15 19.35 8040 89.67 20.03 8830 93.97 20.67 6470 80.44 18.63 7260 85.21 19.36 8050 89.72 20.04 8840 94.02 20.68 6480 80.50 18.64 7270 85.26 19.37 8060 89.78 20.05 8850 94.07 20.68 6490 80.56 18.65 7280 85.32 19.38 8070 89.83 20.06 8860 94.13 20.69 6500 80.62 18.66 7290 85.38 19.39 8080 89.89 20.07 8870 94.18 20.70 6510 80.68 18.67 7300 85.44 19.40 8090 89.94 20.07 8880 94.23 20.71 6520 80.75 18.68 7310 85.50 19.41 8100 90.00 20.08 8890 94.29 20.72 6530 80.81 18.69 7320 85.56 19.42 8110 90.06 20.09 8900 94.34 20.72 6540 80.87 18.70 7330 85.62 19.43 8120 90.11 20.10 8910 94.39 20.73 6550 80.93 18.71 7340 85.67 19.43 8130 90.17 20.11 8920 94.45 20.74 6560 80.99 18.72 7350 85.73 19.44 8140 90.22 20.12 8930 94.50 20.75 6570 81.06 18.73 7360 85.79 19.45 8150 90.28 20.12 8940 94.55 20.75 6580 81.12 18.74 7370 85.85 19.46 8160 90.33 20.13 8950 94.60 20.76 6590 81.18 18.75 7380 85.91 19.47 8170 90.39 20.14 8960 94.66 20.77 6600 81.24 18.76 7390 85.97 19.48 8180 90.44 20.15 8970 94.71 20.78 6610 81.30 18.77 7400 86.02 19.49 8190 90.50 20.16 8980 94.76 20.79 6620 81.36 18.78 7410 86.08 19.50 8200 90.55 20.17 8990 94.82 20.79 6630 81.42 18.79 7420 86.14 19.50 8210 90.61 20.17 9000 94.87 20.80 6640 81.49 18.80 7430 86.20 19.51 8220 90.66 20.18 9010 94.92 20.81 6650 81.55 18.81 7440 86.26 19.52 8230 90.72 20.19 9020 94.97 20.82 6660 81.61 18.81 7450 86.31 19.53 8240 90.77 20.20 9030 95.03 20.82 6670 81.67 18.82 7460 86.37 19.54 8250 90.83 20.21 9040 95.08 20.83 6680 81.73 18. 83 7470 86.43 19.55 8260 90.88 20.21 9050 95.13 20.84 6690 81.79 18.84 7480 86.49 19.56 8270 90.94 20.22 9060 95.18 20.85 700 81.83 18.85 7490 86.54 19.57 8-280 90.99 20.23 9070 95.24 20.86 60 SQUARE AND CUBE ROOTS. Square Roots and Cube Roots of Numbers from 1000 to 10000 (CONTINUED.) Num. Sq. Rt. Cu. Rt. Num. Sq. Rt. Cu. Rt. Num. Sq. Rt. Cu. Rt. Num. Sq. Rt. Cu. Rt. 9080 95.29 20.86 9320 96.54 21.04 9550 97.72 2 .22 9780 9W.S9 1.39 9090 95.34 20.87 9330 96.59 21.05 9560 97.78 2 .22 9790 98.94 1.39 9100 95.39 20.88 9340 96.64 21.06 9570 97.83 2 .23 9800 98.99 .40 9110 95.45 20.89 9350 96.70 21.07 9580 97.88 2 .24 9810 99.05 1.41 9120 95.50 20.89 9360 96.75 21.07 9590 97.93 2 .25 9820 99.10 .41 9130 95.55 20.90 9370 96.80 21.08 9600 97.98 2 .25 9830 99.15 .42 9HO 95.60 20.91 9380 96.85 21.09 9610 98.03 21.26 9840 99.20 1.43 9150 95.66 20.92 9390 96.90 21.10 9620 98.08 21.27 9850 99.25 1.44 9160 95.71 20.92 9400 96.95 21.10 9630 98.13 21.28 9860 99.30 1.44 9170 95.76 20.93 9410 97.01 21.11 9640 98.18 21.28 9870 99.35 1.45 9180 95.81 20.94 9420 97.06 21.12 9650 98.23 21.29 9880 99.40 1.46 9190 95.86 20.95 9430 97.11 21.1 a 9660 98.29 21.30 9890 99.45 1.47 9200 95.92 20.95 9440 97.16 21.13 9670 98.34 21.30 9900 99.50 21.47 9210 95.97 20.96 9450 97.21 21.14 9680 98.39 21.31 9910 99.55 21.48 9220 96.02 20.97 9460 97.26 21.15 9690 98.44 21.32 9920 99.60 21.49 9230 96.07 20.98 9470 97.31 21.16 9700 98.49 21.33 9930 99.65 21.49 9240 96.12 20.98 9480 9Y.37 21.16 9710 98.54 21.33 9940 99.70 21.50 9250 96.18 20.99 9490 97.42 21.17 9720 98.59 21.34 9950 99.75 21.51 9260 96.23 21.00 9500 97.47 21.18 9730 98.64 21.35 9960 99.80 21.52 9270 96.28 21.01 9510 97.52 21.19 9740 98.69 21.36 9970 99.85 21.52 9280 96.33 21.01 9520 97.57 21.19 9750 98.74 21.36 9980 99.90 21.53 9290 96.38 21.02 9530 97.62 21.20 9760 98.79 21.37 9990 99.95 21.54 9300 96.44 21.03 9540 97.67 21.21 9770 98.84 21.38 10000 100.00 21.54 9310 96.49 21.04 To find Square, or Cube Roots of largre numbers not con* tained in the column of numbers of the table. Such roots may sometimes be taken at once from the table, bj merely regarding the columns of powers as beiug columns of numbers; and those of numbers as being those of roots. Thus, if the sq rt of 25281 is reqd, first find that number in the column of squares; and opposite to it, in the column of numbers, is its sq rt 159. For the cube rt of 857375, tind that number in the column of cubes ; and opposite to it, in the col of numbers, is its cube rt 95. When the exact number is not con- tained in the column of squares, or cubes, as the case may be, we may use instead the number nearest to it, if no great accuracy is reqd. But when a considerable degree of accuracy is necessary, the following very correct methods may be used. For the square root. This rule applies both to whole numbers, and to those which are partly (not wholly) decimal. First, in the foregoing manner, take out the tabular number, which is nearest to the given one ; and also its tabular sq rt. Mult this tabular number by 3 ; to the prod add the given number. Call the sum A. Then mult the given number by 3 ; to the prod add the tabular number. Call the sum B. Then A : B : : Tabular root : Reqd root. Ex. Let the given number be 946.53. Here we find the nearest tabular number to be 947; and its tabular sq rt 30.7734. Hence, 947 = tab num 1 f 946.53 = given num. 3 3 2841 946.53 given num. 3787.53 A. and { 2839.59 947 = tab num. 1.3786.59 - B. A. B. Tab root. Reqd root. Then 3787.53 : 3786.59 : : 30.7734 : 30.7657 +. The root as found by actual mathematical process is also 30.7657 -{-. For the cube root. This rule applies both to whole numbers, and to those which are partly decimal. First take out tTie tabular number which is nearest to the given one ; and also its tabular cube rt. Mult this tabular number by 2 ; and to the prod add the given number. Call the sum A. Then mult the given number by 2; and to the prod add the tabular number. Call the sum B. Then A : B : : Tabular root : Reqd root. Ex. Let the given number be 7368. Here we find the nearest tabular number (in the column of ttfte*) to be 6859 ; and its tabular cube rt 19. Hence, 6859 tab num. ^ f 7368 = given num. 6859 = tab num. "> r 7368 ! _j t and J 14736 b 9 I 21595 13718 7368 = given num. tab num. 21595 = B. 21086 = A. A. B. Then, as 21086 : 21595 The root as found by correct mathematical process is 19.4588. The engineer rarely requires even Tab Root. Reqd Rt. : 19 : 19.4585 GEOMETRY. 61 this degree of accuracy ; for his purposes, therefore, this process is greatly preferable to the ordinary laborious one. To find the square root of a number which is wholly decimal. Very simple, and correct to the third numeral figure inclusive. If the number does not contain at least five figures, counting from the first numeral, and including it, add one or more ciphers to make five. If, after that, the whole number is not separable into twos, add another cipher to make it so. Then beginning at the first numeral figure, and including it, assume the number to be a whole one. In the table find the number nearest to this assumed one ; take out its tabular sq rt ; move the deci- mal point of this tabular root to the left, half MS, many places as the finally modified decimal number has figures. Ex. What is the sq rt of the decimal .002? Here, in order to have at least five decimal figures, counting from the fipst numeral (2), and including it, add ciphers thus, .00,20,00,0. But, as it is not now separable into twos, add another cipher, thus, .00,20,00,00. Then beginning at the first numeral (2), assume this decimal to be the whole number 200000. The nearest to this in the table is 199809; and the sq rt of this is 447. Now, the decimal number as finally modified, namely, .00.20,00,00, has eight figures ; one-half of which is 4; therefore, move the decimal point of the root 447, four places to the left; making it .0447. This is the reqd sq rt of .002, correct to the third numeral 7 included. To find the cube root of a number which is wholly decimal. Very simple, and correct to the third numeral inclusive. If the number does not contain at least five figures, counting from the first numeral, and including it, add one or more ciphers to make five. If, after that, the number is not separable into threes, add one or more ciphers to make it so. Then beginning at the first numeral, and including it, assume the number to be a whole one. In the table find the number nearest to this assumed one, and take out its tabular cub rt. Move the decimal point of this rt to the left, one-third as many places as the fiually modified decimal number has figures. Ex. What is the cube rt of the decimal .002 ? Here, in order to have at least five figures, counting from the first numeral (2), and including it, add ciphers thus, .002.000,0. But as it is not now separ- able into threes, add two more ciphers to make it so ; thus, .002,000,000. Then beginning with the first numeral (2), assume the decimal to be the whole number 2000000. The nearest cube to this in the table in the column of cubes, is 2000376; and its tabular cube rt as found in the col of numbers, is 126. Now, the decimal number as flnnlly modified, namely, .002 000 000, has nine figures ; one-third of which is 3; therefore, move the decimal point of the root 126, three places to the left, making it .126. This la the reqd cube rt of the decimal .002, correct to the third numeral 6 included. To find roots by logarithms, see p. 614. GEOMETKY. Lines, Figures, Solids, defined. Strictly speaking a geometrical line is simply length, or distance. The lines we draw on paper have not only length, but breadth and thickness ; still they are the most convenient symbol we can employ for denoting a geometrical line. Straight lines are also called right lines. A vertical line is one that points toward the center of the earth; and a horizontal one is at right angles to a vert one. A plane figure is merely any flat surface or area entirely enclosed by lines either straight or curved ; which are called its outline, boundary, circumf, or periphery. "We often confound the outline with the fig itself as when we speak of drawing circles, squares, &c ; for we actually draw only their outlines. Geometrically speaking, a fig has length and breadth only ; no thickness. A solid is any body ; it has length, breadth, and thickness. Geometrically similar figs or solids, are not necessarily of the same size ; but only of precisely the same shape. Thus, any two squares are, scien- tifically speaking, similar to each other ; so also any two circles, cubes, &c, no matter how different they may be in size. When they are not only of the same shape, but of the same size, they are said to be similar, and equal. The quantities 01 lines are to each other simply as their lengths; but the quantities, or areas, or surfaces of similar figures, are as, or in proportion to, the squares of any one of the corresponding lines or sides which enclose the figures, or which may be drawn upon them ; and the quantities, or solidities of similar solids, are as the cubes of any of the corresponding lines which form their edges, or the figures by which they are enclosed. Angles. When two straight, or right lines meet each other at any inclina- tion, the inclination is called an angle; and is measured by the degrees con- tained in the arc of a circle described from the point of meeting as a center. Since all circles, whether large or small, are supposed to be divided into 360 degrees, it follows that any number of degrees of a small circle will measure the same degree of inclination as will the same number of a large one. When two straight lines, as o n and a 6, meet in such a manner that the inclination o n a is equal to the inclination o n &, then the two lines are said to be perpendicular to each other; and the angles o n a and o n b, are called right angles ; and are each measd by, or ri /_J__Vu are equal to, 90, or one-fourth part of the circumf of a circle. Any angle, &sced, smaller than a right angle, is called acute or sharp ; and one c ef, larger than a right angle, is called obtuse, or blunt. When one line meets another without crossing it, as in the two foregoing figures, the two angles o n n, and o n 6, are called contiguous, or adjacent; so also c e d, and c ef. Two adjacent angles are together always equal to two right angles; or to 180. There- fore, if we know the number of degrees contained in one of them, and subtract it from 180, we obtain the other. 62 GEOMETRY. When two straight lines cross each other, forming four angles, either pair of those angles which point in exactly opposite direc- tions are called opposite, or improperly, verti- cal angles ; thus, the pair s u t, and v u w are op- posite angles ; also the pair a u v and t u w. The opposite angles of any pair are always equal to each other. "When a straight line a 6 crosses two parallel lines c d, e f, the alternate angles which form a kind of Z, are equal to each other. Thus, the angles don, and o n f, are equal: as are also con, and one. Also the two internal angles on the same side of o b, are equal to tw> right angles, or 180; thus, the two angles c o n, and onf= 18(P ; as do also don and one. An interior angle, In any fig, is any angle formed inside of that fig, by the meet- ing of two of its sides, as the angles c a b, a b c, b c a, of this triangle. All the interior angles of any straight-lined figure of any uumber of sides whatever, are together equal to twice as many right angles minus four, as the figure has sides. Thus, a triangle has 3 sides ; twice that number is 6 ; and 6 right angles, or 6 X 90 = 540 ; from which take 4 right angles, or 360 ; and there remain 180, which is the number of degrees in every plane, or straight lined triangle. This principle furnishes an easy means of testing our measurements of the angles of any fig ; for if the sum of all our measurements does not agree with *he sum given by the rule, it is a proof that we have committed some error. An exterior angle Of any straight-lined figure, is any angle, as a b d, formed OUTSIDE of that fig, by the meeting of any side, as a b, with the prolongation of an adjacent side, as c b ; so likewise the angles c a s, and b cw. All the exterior angles of any straight-lined fig, no matter how many sides it may have, amount to four right angles, or 360. Thus, in he foregoing fig, the three exterior angles a b d, c as, and b cw, amount to 360. In any fig, those angles, as g. h, . and /, which point outward, are called salient ; those whicfc 1 J point inward, as t, are called re-entering*. All angles, as n a m, n o m, at the circumf of a semicircle, and stand- ing on its diam n m, are right angles ; or, as it is usually expressed, all angles in a semicircle are right angles. An angle n s x at the centre of a circle, is twice as great as an angle n m x at the circumf, when both stand upon the same arc n x. All angles, as y dp, y e p, y g p. at the circumf of a circle, and standing upon the same arc. as y p, are equal to each other ; or, as usually expressed, all angles in the same segment of a circle are equal. r The complement of an angle is usually said to S be what it wants of being 90; and its supplement, w r hat it wants of being 180. But in mathematical strictness its complement is its difference from 9O, whether greater or less; and its supplement in like manner js its diff from 180. If any two chords, as a b, o c, cross each other, then as o n : n b : : a n : n c. Hence, nb X an = on'Xnc. That is, the product of the two parts of one of the lines, is = the pro- duct of the two parts of the other line. Sines, Tangents, Ac. Sine, a s, of any angle, a c 6, or which is the same thing, the sine of any circular are, a &, which subtends or measures the angle, is a straight line drawn from one end, as a, of the arc, at right angles to, and terminating at, the rad c b, drawn to the other end b of the arc. It is, therefore, equal to half the chord an, of the arc abn, which is equal to twice the arc ab ; or, the sine of an angle is GEOMETRY. 63 ^tways equal to half the chord of twice that angle; and vice versa, the chord of an angle is always iqual to twice the sine of half the angle. The sine t c of an angle t c b, or of an arc yy t a b, of 90, is equal to the rad of the arc or of the circle ; and this sine of 90 is greater than that of any other angle. Cosine c 5 of an angle a c &, is that part of the rad which lies between the sine and the center of the circle. It is always equal to the sine y a of the c b wants of being 90. The prefix co be- fore sines, &c, means complement ; thus, cosine means sine of the complement. Versed sine s b of any angle a c b, is that part of the rad which lies between the sine, and the outer end 6. It is very common, but erroneous, when speaking of bridges, &c, to call the rise or height s b of a. circular arch a b n, its rersed sine; while it is actually the versed sine of only half the arch. This absurdity should cease ; for the word rise or height is not only more expressive.but is correct. a c 6, is a line drawn from, and at right angles to, the end 6 or a of either rad c 6, or c a, which forms one of the legs of the angle ; and terminating as at w, or c?, in the prolongation of the rad which forms the other leg.* This last rad thus pro- longed, that is, c ic, or c d,as the case may be, is the secant of the angle a c ft. The two tangents are equal; so also are the two secants. The angle t c I being supposed to be 90, the angle t c a becomes the com- plement of the angle a c b, or what a c 6 wants of being 90 ; and the sine y a of tbis complement ; its versed sine t y, its tangent t o ; and its secant c o, are respectively the co-sine, co-versed sine ; co- tangent ; and co- secant, of the angle acb. Or, vice versa, the sine, &c. of a c b, are the cosine. &c. of t c a; because the angle a c 6 is the complement of the angle tea. When the rad c b, c a, or c t, is assumed to be equal to unity, or 1, the corresponding sines, tangents, &c, are called natural ones; and th\ 1O. To find the greatest parallelogram that can be _ Jr \ drawn in any given triangle o n b. Bisect the three aides at a c e, and join Q( \C a c, a e, e c. Then either a e b c, a e c o, or a c e n, each equal to half the /\ /\ triangle, will be the reqd parallelogram. Any of these parallelograms can / \S \ plainly be converted into a rectangle of equal area, and the greatest that can be i in the triangle. !. If a line a c bisects any two sides o &, o n, of a triangle, it will be par- pla dr . allel to the third side n b, and half as long as it. 64 GEOMETRY. 11. To find the greatest square that can be drawn in any triangle axr. From an uugle as a draw a perp a n to the opposite side xr, and find its length. Then o n, or a side v t of the square will = Hem. If the triangle is such that two or three such perps can be drawn, then two or tnree equal squares may be found. Right-angled Triangles. All the foregoing apply also to right-angled triangles; but what follow apply to them only. Call the right angle A, and the others B and C; and call the sides respectively JJ opposite to them a, b, and c. Then is U 6 = a X N. Sine B = a X N. Cos C = c X N. Cot C = c X N. Tang B. c = a X N. Sine C = o X N. Cos B = b X N. Tang C. Also N. Sine of C =; N. Cos C ^/ N. Tang C = ~. And N. Sine of B - 6 ; N. Cos B ~ -; N. Tang B - -. a a c AndN. Sine of A or 9(P = 1. N. Cos A = 0. N. Tang A = infinity. 1. If from the right angle o a line o w be drawn perp to the hypothenuse or long side' h g, then the two small triangles owh, o w g, and the large one otig, will be similar. Also g w : w o :: w o : w h ; and g w X w h w o%. %. A line drawn from the right angle to the center of the long side will be half as long as said side. 8. If on the three sides o h, o g, g h we draw three squares t, u v, or three circles, or triangles, or any other three figs that are similar, then the area of the largest one will be equal to the areas of the two others. 4. In a right-angled triangle whose sides are as 3, 4, and 5 (as is the tri- angle AB C), the angles are very approximately 90: 53 7' 48.38"; and /^ 36 52' 11.62". Their N. Sines, 1. ; .8 ; and .6. Their N. Tangs, infinity ; /ij 1.3333; and .75. \/ &. One whose sides are as 7, 7, and 9.9, has very approx one angle of 90 and two of 45 each, near enough for all practical purposes. PARALLELOGRAMS. A parallelogram is any four-sided straight-lined figure whose opposite sides are equal, as a b c d ; or a square, &c. (See Men- suration.) Any line drawn across a parallelogram from two opposite angles, is called a diagonal, as a c, or b d. A diag divides a parallelogram into two equal parts ; as does also any line mn drawn through the center of either diag ; and moreover, the line TO n itself is div into two equal parts, or is bisected. Twodiags bisect each other; they also divide the parallelogram into four triangles of equal areas. Any two adjacent angles of any paral- lelogram are equal to two right angles, or 180; as the angles dab and a ft c; or a b c and bed; and the four angles are always equal to four right angles, or 360. The sum of the squares of the four sides, is equal to the sum of the squares of the two diags. Rem. Simple a the following operations appear, it is only by care, and good instruments, that they are made to give accurate results. Several of them can be much better performed by means of a metallic triangle having one perfectly accurate right angle. In the field, the tape-line, chain, or a ineasuring-rod will take the place of the dividers and ruler used indoors. To divide a given line, a b, into two equal parts. >From its ends a and b as centers, and with any rad greater than one-half of a b, describe the arcs c and d, and join /. If the line a b is very long, first lay off equal dists a o and bg, each way from thP ends, so as to approach conveniently near to each other ; and then proceed as if o g were the line to be divided. Or measure a 6 by a scale, and thus ascertain its center. To divide a given line, m n, into any given number of equal parts. Prom TO and n draw any two parallel lines TO o and n a, to an indefinite dist; and o'n them, from TO and n step off the reqd number of equal parts of any convenient length : final- ly, join the corresponding points thus stepped off. Or only one line, as TOO. may be drawn and stepped off. as to s; then join sn; and draw the other short Hues parallel to it. To divide a given line, tn n, into two parts which shall have a given proportion to each other. This is done on the same principle as the last ; thus, let the proportion be as 1 to 3. First draw any Hue mo; and with any convenient opening of the dividers, make mx equal to one step ; and xs GEOMETRY. 65 From any s i veil point, p, on a line * t, to draw a perp, j> a. From p, with any convenient opening of the dividers, step off the equals po,pa. Prom o and g as centers, with any opening greater than half o g, describe the two short arcs b and c ; and join a p. Or still better, describe four arcs, and join a y. Or from p with any convenient scale describe two short arcs g and c either one of them with a radius 3, and the other with a rad 4. Then from g with rad 5 describe the arc 6. Join p a. If the point p is at one end of the line, or very near it, Extend the line, if possible, and proceed as above. But if this cannot be done, then from any convenient point, w, open the divid- ers to p, and describe the semicircle, * p o ; through o w draw o w ; join j> t. Or use the last foregoing process with rads 3, 4, and 5. From a given point, o, to let fall a perp o s, to a given line, ni n. . From o, measure to the line m n, any two equal dists, o c, o e ; and from c and e as centers, with any opening greater than half of c e, describe the two arcs a and 6 ; join o t. Or from any point, as d on the line, open the dividers to o, and describe the arc o g ; make i x equal to i o ; and join o x. If the line, a b, is on the gronnd, And a perp is reqd to be drawn from c, first measure off any twc equal dists, c m, c n. At m and n, hold the ends of a piece of string, tape-line, or chain, man; then tighten out the string, &c, as shown by m g n \ being its center. Then will c be the reqd perp. Or if the perp x z is to be drawn from the end of the line w x, first measure xy upon the line, and equal to three feet; then holding the end of a tape- line at x, and its nine feet mark at y, hold the four feet mark at z, keep- ing zx and z y equally stretched. Then zx will be the reqd perp, because 8,4, and 5, make the sides of a right-angled triangle. Instead of 3, 4, and 6, any multiples of those numbers may be used, such as 6, 8, and 10 ; or 9, 12, 15, &c : also instead of feet, we may use yards, chains, &c. y a Through a given point, ft, to draw a line, a c, parallel to another line, ef. "With the perp dist, a e, from any point, n, In ef, describe an arc, t; draw a c just touching the arc. At any point, a, in a line a b, to make an angle ca b, equal to a given angle, m n o. From n and a, with any convenient rad, describe Hie arcs nt.de; measure s t, and make e d equal ' to it; through a d draw a c. 66 GEOMETRY. To bisect, or divide any angle, w x y, into two equal parts. From x set off any two equal dists, x r, x a. From r and with any rad describe two arcs intersecting, as at o ; and join o x. If the two sides of the angle do not meet, as c / and g h, either first extend them until the/ do meet; or else draw lines x w, and xy, parallel to them, and at equal dists from them, so as to meet; then proceed as before. To describe a circle through any three points, a be, not in a straight line. Join the points by the lines a b, be; from the centers of these lines draw the dotted perps meeting, as at o, which will be the center of the circle. Or from b, with any convenient rad, draw the arc m n; and from a and c, with the same rad, draw arcs y and z; then two lines drawn through the Intersections of these arcs, will meet at the center o. To describe a circle to touch the three angles of a triangle is plainly the same as this. To inscribe a circle in a triangle draw two lines bisecting any two of the angles. Where these lines meet is the center of the circle. \m On a given line /-a%to draw a square, w x n, m. From w and x, with rad 10 x, describe the arcs xry and w r e. From their intersection r, and with rad -equal to % of w x, describe s s s. From to and z draw w n and xm tangential to a *, and ending at the other arcs ; join n m. To find the center e, of a given circle. Draw any chord a b ; and from the middle of it o, draw at right angles to it, a diam d g ; find the center c of this diam. To draw a tangent, I e i, to a circle, from any given point, f, in its circumf. Through the center n, and the given point e, draw n o ; make e o equal to e n ; from n and o, with any rad greater than half of o n, describe the two pairs of arc i i; join their intersections i i. . Here, and in the following three figs, the tangents are ordinary or geo- metrical ones ; and may eud where we please. But the trigonometrical tangent of a given angle, must end in a secant, as in the top fig of p 68. Or from e lay off two equal distances e c, e t ; and draw i i parallel to c t. To draw a tang, a * ft, to a circle, from a point, a, which is outside of the circle. Draw a c, and on it describe a semicircle ; through the intersection, a, draw a sb. Here c is the center of the circle. To draw a tang, g 7tt from a circular arc, g a c 9 Of which n a is the rise. With rad g a, describe an arc, a o. Make * a equal to s a. Through t draw g ft. GEOMETRY. 67 To draw a tang to two circles. First draw the line m n, just touching the two sircles ; this gives the direction of the tang. Then from the centers of the circles draw the radii, o o, perp to m n. The points t t are the tang points. If the tang is in the position of the dotted line, sy, the ope- ration is the same. Rein. This empirical method is at least a accurate as the scientific ones, especially if a correct triangular ruler is used for the radii. To draw a hexagon, each side of which shall be equal to a given line, a b. From a and ft, with rad a b, describe the two arcs; from their intersection, t, with the same rad, describe a circle; around the circumf of which, step off the same rad. To draw an octagon, with each side equal to a given line, c e. From c and e draw two perps, cp, ep. Also prolong c e toward / and c with rad c r describe a circle. Hake each of the arcs o t an4 t 1 equal to r o or r i ; and draw ct, cl. Dive*, cl,cr, each into half as many equal parts as thft curve is to be divided into, Draw the lines b 1, b 2, b 3 ; and a 4, a 5, a 6, extended to meet the first ones at e, s, h. Then e, s, h, are points in one-half t'ha curve. Then for the other half, draw similar lines from a to 7, 8, 9 ; and others from b to meet them, as before. Trace the O curve by hand. T Also see Remark in "Men- suration," page 17, after "To 6 Hud the length of a circular arc." To draw an oval, or false ellipse. When only the long diam a ft is given, the following will give agreeable curves, of which the span o b witt not exceed about three times the rise c o. On a ft de- scribe two intersecting circles f any rad; through their intersections 8, v, draw e g; make s g and v each equal to the diam of one of the circles. Through the centers of the circles, draw e y, eh, g d, g t. From e describe h i y ; and from g describe dot. When the span, m n, and the rise, s t, are both given. Make any t w and m r, equal to each other, bin each less than t 8. Drawrw; and through its center o draw the perp t o y. Draw y r z. Make n x equal m r, and draw y x b. From x and r describe n c and m a, ; and from y descri be a t c. By making s d equal to * y, we obtain the center for the other side of the oval. The beauty of the curve will depend upon what portion of t s is taken for m r and t w. When an oval is very fiat, more than three cen- ters are required for drawing a graceful curve; but the finding of these centers is quite as trou- blesome as to draw the correct ellipse. For drawing the ellipse and parabola, see "Mensuration." To reduce any polygon, as abcdefa, to a triangle of the same area. If we produce the side fa toward w; and draw b g parallel to a, c, and join g c, we get equal trl- ingles acb, and a c <7, both on the same base a c; and both of the same perp heij,>.v. inasmuch as jhej are between ths two parallels a c and g b. But the part act forms a portion *f both these trl- ARITHMETIC. 69 p h angles, or in other words, is common to loth. Therefore, if it be taken away from both triangles, the remaining parts, i c 6 of one of them, and i g a of the other, are also equal. Therefore, if the c 6 be left off from the polygon, and the part i g a be taken into it, the polygon g / e'd c i g will the same area as afe d c b a ; but it will have but nve sides, while the other has six. Again, if e s be drawn parallel to d f, and ds joined, we have upon the same base es, and between the same parallels e a and df, the two equal triangles e s d, and e /. with the part cos common to both ; and consequently the remaining part e o d of one, and o sf of the other, *ire equal. Therefore, if oaf be left off from the polygon, and e o d be taken into it, the new polygon y d eg, Fig 2, will have the same area as g f e d c g; but it has but four sides, while the other 'aas five. Finally, if g s, Fig 2, be extended toward n; and d n drawn parallel to c a ; and c n joined, we have on the same base c a, and between the same parallels c and d n, the two equal triangles c s n, and c d, with the part'c 8 t common to both. Therefore, if we leave out c d t, and take in a t , we have the triangle one equal to the polygon g t d c g, Fig 2; or to afedcba, Fig 1. Vhis simple method is applicable to polygons of any number of sides. fo reduce a large fig:, abed e fg, to a smaller similar one. From any interior point o, which had better be near the center, draw lines Jt to all the angles a, b. c, &c. Join these lines by others parallel to the sides 5 of the fig. If it should be reqd to enlarge a small fig, draw, from any point o within it, lines extending beyond its angles ; and join these lines by others parallel to the sides of the small fig. To reduce a map to one on a smaller scale. The best method is by dividing the large map into squares by faint lines, with a very soft lead- pencil; and then drawing the reduced map upon a sheet of smaller ?quares. A pair of proportional dividers will assist much in fixing points intermediate of the sides of the squares. If the large map would be injured by drawing, and rubbing out the squares, threads may be stretched across it to form the squares. Maps, plans, and drawings of all kinds, are now copied, reduced, enlarged, and multiplied, cheaply and expeditiously, by photography. For this purpose they should be prepared only in plain black and white; shading should be done in lines; not in washes;* and care be taken to make the lettering and every part rather more strong and distinct than for ordinary drawings. Ill a rectangular fig, fj h s fl, Representing an open panel, to find the points o o o o in its sides ; and at equal dists from the angles g, and s ; for inserting a diag piece o o o o, of a given width I I, measured at right angles to its length. From g and s as centers, describe several concentric arcs, as in the Fig. Draw upon transparent paper, two parallel lines a o. c c. at a distance apart equal to II; and placing these lines on top of the panel, move them about until it is shown by the arcs that the four dists g o, y o, o, s o, are equal. Instead of the transparent paper, a strip of common paper, of the width 1 1 may be used. RKM. Many problems which would otherwise be very difficult, may be thus solved with an accuracy sufficient for practical purposes, by means of transparent paper. AKITHMETIC, ON this subject we shall merely give a few examples for refreshing the memory of those who for want of constant practice cannot always recall the processes at the moment. Subtraction of Vulgar Fractions. Addition of Vulgar Fractions. 3 4 + 1 f= * Not absolutely necessary. 70 ARITHMETIC, Multiplication of Vulgar Fractions. Division of Vulgar Fractions. l+i=|=>- i*i= = *=***=- = = *= *+*==>& *+'f=V+*=tt=V = f ^1 = 1-1= 4 7 =6 5. To find the greatest common divisor of a Vulgar Fraction. Ex.l. Of T W 70) U2 (2 Ex.2. Off $. >JJ(* 35)70(2 ~4)20(5 70 Ans35. 20 Ans 4. To reduce a Vulgar Fraction to its lowest terms. First find the greatest common divisor ; then divide both the numerator and denominator by it. Thus, in the preceding example ^ffg = -J Ans. And f J = 2 ^ Ans. To reduce a Vulgar Fraction to a decimal form* Divide the numerator by the denominator. Thus, 4 =2 )1. 0(0.5 Ans. V 3 = 4)13(3.25 Ans. 14 = 40)32.0(0.8 Ans. 2 10 . .11 3 20 20 20 Reduce 3 inches to the decimal of a foot. There are 12 ins in a foot; therefore, the question i> to reduce -j 3 ^ to a decimal. Therefore, 12)3.0(0.25 of a foot. Ans. Reduce 2 ft 3 ins to the decimal of a yard. There are 36 ins in a yard; and 27 ins in 2 ft 3 ins; therefore, f^ of a yard = 36)27.0(0.75 of a yd. Ans. 180 180 How many feet and ins are there in .75 of a yard ? Here .75 3 ft in a yd. Ft 2). 25 12 ins in a ft. Ins 3.00 Ans 2 ft 3 ins. How many feet and ins are there in .0625 of a yard ? .0625 3 feet in a yd. No feet, .1875 12 ina in a ft. ft. ins. Ins 2.2500 Ans. 2.25. How many cubic feet are there in .314 of a cub yard ? And cub ins in .46 of a cub ft ? .314 .46 27 cub ft in a yd. 1728 cub ins a cub ft. 2198 ~368 628 92 322 8.478 cub ft. Ans. 46 794.88 cub ins. Ans. ARITHMETIC. 71 Decimals. ADDITION. Add together .25 and .75; also .006, 1.3472, and 43. 25 1.00 Ans. 44.3532 An& SUBTRACTION. Subtract .25 from .75 ; also .0001 from I ; also 6.30 from 9.01. .75 1. 9-01 .25 .0001 6.30 .50 Ans. .9999 Ans. 2.71 Am. MULTIPLICATION. Mult 3 X .3 ; also .3 X .3 ; also .3 X .03; also 4.326 X .003. 3 .3 .3 4.326 .3 .3 .03 .003 .9 Aus. .09 Ans. .009 Ans. .012978 Ans. DIVISION. Divide 3. bj .3 ; also .3 by .3 ; also .3 by .03 ; also 4.326 by .0003. .3)3.0(10. Ans. .3).3(1. Ans. ,03).30(10. Ans. .0003)4.3260(14420. Ana. 3 33 3 5 ~~o Divide 62 by 87.042. 87.042)62.0000(0.712, &c. Ans. 12 4 _60.y294_ "lOTWO 87042 200180 6 6 Divide .006 by 20. 20.000). 0060000<0.0003 Ans. 60000 Duodecimals. Duodecimals refer to square feet of 144 sq ins ; to twelfths of a square or duodecimal foot ; each such twelfth being called an inch; and being equal to 12 square inches ; and to twelfths, each equal to the 12th of a duodecimal inch, or to one square inch. The dimensions of the thing to be measd are supposed to be taken in common feet, ins, and 12ths of an inch ; but as ordinary measuring rules are divided into 8ths of an inch, it is usually guess-work to some extent. Duodecimals are very properly going out of use, in favor of decimals ; we shall therefore give no rule for them. By means of our table of "Inches reduced to Decimals of a Foot," page 75, all dimensions in feet, ins, and 8ths, &c, cau be at once taken out in ft and decimals of a foot. Single Rule of Three ; or, Simple Proportion. If 3 men lay 10000 bricks in a certain time, how many could 6 men lay in the same time ? Thej will evidently lay more ; therefore, the second term of the proportion must be greater than the first. 3:6:: 10000 : 20000 Ans. If 3 men require 10 hours to lay a certain number of bricks, how many hours would 6 men require ? They will evidently require less time ; therefore, the second term of the proportion must be less than the first. 6 : 3 : : 10 : 5 Ans. 3 6)30 5 Ans Double Rule of Three; or, Compound Proportion. If three men can lay 4000 bricks in 2 days, how many men can lay 12000 in 3 days ? Here we se that 4000 bricks require. 3 X 2 = 6 days' work; therefore 12000 will require, 4000 : 12000 : : 6 : 18 days' work. But there are only 3 days to do the 18 days work in ; therefore the number of men must be ^ = 6 men. Ans. A moment's reflection will suffice to reduce any case of double rule of three to this simple form. Arithmetical Progression, In a series of numbers, is a progressive increase or decrease in each successive number, by the addi- tion or subtraction of the same amount at each step; as in 1, 2, 3, 4, 5, Ac., in which 1 is added at each step ; or 10, 8, 6, 4, &c., in which 2 is subtracted at each step; or y< *4, H, 1, 1^. &c. In any such series the numbers are called its terms ; and the equal increase or decrease at each step its com- mon difference. To find the com diff, knowing the first and last terms ; and the number of terms. Find the did between the first and last terms. From the number of terms subtract 1. Div tin diff just found, bj ihe rem. 72 ARITHMETIC. To find the last term, knowing the first term ; the com diff ; and the number of terms. Prom the number of terms take 1. Mult the rem by the com diff. To the prod add the first term. To find the number of terms, having the first and last ones ; and the com diff. Take the diff between the first and last terms. Div this diff by the com diff. To the quot add 1. To find the sum of all the terms, having the first and last ones; and the number of terms. Add together the first and last terms. Div their sum by 2. Mult the quot by the number of terms. Geometrical Progression, In a series of numbers, is a progressive increase or decrease in each successive number, by the same multiplier or divisor at each step ; as 3, 9, 27. 81, &c, where each succeeding term is increased by mult the preceding one by 3. Or 48, 24. 12, 6, &c, or 27, 13}^, 6%, 3%, &c, where each succeeding term is found by dividing the preceding one by 2. The multiplier or divisor is called the common ratio of the series, or progression. To find the last term, knowing the first one ; the ratio ; and the number of terms. Raise the ratio to a power 1 less than the number of terms. Mult this power by the first term. Ex. First term 10 ; ratio 8 ; number of terms 8 ; what is the last term ? Here the number of terms being 8, the ratio 3 must be raised to the 7th power ; thus : 3X3X3X3X3X3X3 = 2187, = 7th power. And 2187 X 10 = 21870 last term. Ans. A man agreed to buy 8 fine horses ; paying $10 for the first; $30 for the second; $90 for the third, &c : how much will the last one cost him? Ans, $21870, as before. To find the sum of all the terms, knowing the first one ; the ratio ; and the number of terms. Raise the ratio to a power equal to the whole number of terms. From this power subtract 1. Div the rem by 1 less than the ratio. Mult the quot by the first term. Ex. As before. What is the sum of all the terms? Here the ratio must be raised to the 8th power ; thus, 3X3X3X3X3X3X3X3 6561 = 8th pow. And 6560 div by 1 less than the ratio * 6560 3, = - = 3280. And 3280 X 10 (or number of terms) 32800 = sum. Ans. In the foregoing case, the 8 horses would cost $32800. Permutation Shows in how many positions any number of things can be arranged in a row. To do this, mult together all the numbers used in counting the things. Thus, in how many positions in a row can 9 things be placed ? Here, 1X2X3X4X5X6X7X8X9 = 362880 positions. Ans. Combination Shows how many combinations of a few things can be made out of a greater number of things. To do this, first set down that number which indicates the greater number of things ; and after it a series of numbers, diminishing by 1, until there are in all as many as the number of the few things that are to form each combination. Then beginning under the last one, set down said number of few things ; and going backward, set down another series, also diminishing by 1, until arriving under the first of the upper numbers. Mult together all the upper numbers to form one prod ; and all the lower ones to form another. Div the upper prod by the lower one. Ex. How many combinations of 4 figures each, can be made from the 9 figs 1, 2, 3, 4, 5, 6, 7, 8, 9; or from 9 any things ? Alligation Shows the value of a mixture of different ingredients, when the quantity and value of each of these last is known. Ex. What is the value of a pound of a mixture of 20 Ibs of sugar worth 15 cts per Ib ; with 30 Ibs worth 25 cts per Ib ? Ihs. cts. cts. S "">* ' =".*. 50 Ibs. 1050 cts. Equation of Payments. A owes B $1200; of which $400 are to be paid in 3 months; $500 In 4 months ; and $300 in 6 months ; all bearing interest until paid ; but it has been agreed to pay all at once. Now, at what time must this payment be made so that neither party shall lose any interest ? $ months. 500 X 4 = 2000 Therefore, = 4.16, &c, months. Ans. 800 X 6 = 1800 1200 5000 A owes B $1000 to be paid in 12 days ; and $500 to be paid in 3 months. What would be the time. tor paying all at once ? $ days. 1000 X 12 = 12000 57000 500 X 90 = 45000 Therefore, -j^ = 88 days. An*. 1500 67000 WEIGHTS AND MEASURES. 73 Simple Interest. What is the simple interest on $865.32 eta for one year, at 6 per ct per annum ? Principal. Interest. Principal. Interest. $100 : $6 :: $865.32 : $51.9192 Sets. 100)5191.92(51.9192 Ans. = 51.91-j What is the interest on $865.32 cts for 1 year, 3 months, and 10 days, at 7 per cent per annum? First calculate the interest for 1 year only ; thus: Prin. Int. Prin. Int. $100 : $7 :: $865.32 : $60.5724 7 100)6057.24(60.5724 Then say, If 1 year or 365 days give $60.5724 int, what will 465 days give? or Days. Int. Days. Int. 365 : $60.5724 : : 465 : $77.16, &c. Ans. At 5 per ct simple interest, money doubles itself in 20 years; at 6 per ct, in 16% years; and at 7 per ct, in 14^ years. Simple Interest is Single Kule of Three. Compound Interest. When money is borrowed for more than a year at compound interest, find the simple interest at the end of the first year, and add it to the principal, for a second principal. Find the simple interest on this second enlarged principal for the next year, and add it to the enlarged principal for a third prin- cipal ; and so on for each successive year. 1 At 5 per ct compound interest, money doubles itself in about 14^- years ; at 6 per ct, in about 11.9 years; and at 7 per ct, in about 10% years. Discount Is a deduction of a part of the interest, when money at interest is paid before it is due. Or it is a deduction. of the whole of the interest in advance, at the time the money is lent. In the first case, if I borrc w $100 for 1 year at 8 per ct, I must at the end of the year pay back $108 ; but if I pay at the end of 3 months, I must add only $2, or the interest for those 3 months, paying back $102; and the diff of $6 is the discount. Therefore, to find the discount in such cases, first find the interest for the full time ; then that for the short time ; and take the diff. In the second case, if I borrow $100 from a bank for one year, at 6 per ct, I receive but 100 6 r: $94; but at the end of the year I must pay back $100. By discounting in this manner, the bank actually gains more than 6 per ct ; for it gains $6 for the use of $94 for 1 year. In the United States, the banks deduct discount for 3 days more than the time stipulated in the note; these are called "days of grace." Commission, or Brokerage, Is a percentage tor so much per each $100) paid to commission merchants for selling our goods; or to brokers, or other kinds of agents, for transacting business for us. It is Single Rule of Three. Ex. If a broker makes purchases for me to the amount of $9362, at 2 per ct, what is his brokerage? Say, as Purchase. Brokerage. Purchase. Brokerage. $100 : $2 : : $9362 : $187.24 Insurance Is a percentage (called a premium) paid to a company for insuring our property against fire, &c. The company, or insurers, (called also underwriters,) deliver to the person insured, a paper bearing their seal, &c, and called the Policy of Insurance ; which contains the conditions of the transaction. Insurance is calculated like Commissions, feet square, by 14 inches"(l^ English bricks) thick = 272^ sq ft of 14 inch wall. It is conven- tionally taken at 272 sq ft; which gives 317 X cub ft. In Brit engineering works the rod is 306 cub ft, or HJi cub yds. The Montreal, (Canada,) toise = 261^ cub ft; or 9.6852 cub yds, or 10.46 perches of 25 cub ft. The Canadian chaldron = 58.64 cub ft. A ton (2240 fts> of Pennsylvania anthracite, when broken for domestic use, occupies from 41 to 43 cub ft of space ; the mean of which is equal to 1.556 cub yds: or a cube of 3.476 ft on each edge. Bituminous coal 44 to .48 cub ft; mean equal to 1.704 cub yd ; or a cube of 3.583 ft on each edge. Coke 80 cub ft. A cubic foot is equal to 1728 cub ins, or 3300.23 spherical ins. .037037 cub yard, or 1.90985 spherical ft. .002832 rayriolitre, or decastere. .028316 kilolitre, or cubic metre, or stere. .283161 hectolitre, or decistere. 2.83161 decalitres, or centisteres. 28.3161 litres, or cub decimetres. 283.161 decilitres. 2831.61 centilitres. 28316.1 millilitres, or cub centimetres. .803564 U. S. struk bushel of 2150.42 cub ins, 1.24445 cub ft. .779013 Brit bushel of 2218.191 cub ins, or 1.28368 cub ft. 3.21426 U. S. pecks. A cubic inch is equal to 16.38663 millilitres; or 1.638663 centilitres; or .1638663 decilitre; or .01638663 HUe; or to .0005787 cub ft; or to .138528 U. S. gill ; or 1.90985 spherical ins. A cubic yard is equal to 3.11605 Brit pecks. 7.48052 U. S. liquid galls of 231 cub ins. 6.42851 U. S. dry galls. 6.23210 Brit galls of 277.274 cub ins. 29.92208 U. S. liquid quarts. 25.71405 U. S. dry quarts. 24.92H42 Brit quarts. 59.84416 U. S. liquid pints. 51.42809 U. S. dry pints. 49.85684 Brit pints. 239.37662 U. S. gills. 199.42737 Brit gills. .26667 flour barrel of 3 struck bushels. .23748 U. S. liquid barrel of 31} galls. 27 cub feet, or to 201.974 U. S. galls. 46656 cub ins. .0764534 myriolitre. .764534 kilolitre, or cub metre. 7.64534 hecatolitres. I 76.4534 decalitres. 764.534 litres, or cub decimetres. 7645.34 decilitres. 21.69623 U. S. bushels (struck). 21.03336 Brit bushels. Liquid Measure. U. 8. only. The basis of this measure in the U. S. is the old Brit wine gallon of 231 cub ins ; or 8.33888 Tbs avoir of pure water, at its max density of about 39. 2 Pahr ; the barom at 30 ins. A cylinder 7 ins diam, aud 6 ins high, contains 230.904" cub ins, or almost precisely a gallon ; as does also a cube of 6.1358 ins on an edge. Also a gallon .13368 of a cub ft ; and a cub ft contains 7.48052 galls ; nearly 7)4 galls. This basis however involves an error of about 1 part in 1362, for the water actu- ally weighs 8.345008 Ibs. cub ins. 4 gills 1 pint =28.875. 2 pints 1 quart 57.750 8 gills. 4 quarts 1 gallon 231 . 8 pints = 32 gills. In the U. S. and Great Brit. 1 barrel of wine r 63 gallons 1 hogshead. 2 hogsheads 1 pipe, or butt. 2 pipes 1 tun. brandy = 31 % galls ; in Pennsylvania, a half barrel, 16 galls; a double barrel, 64 gulls; a puncheon, 84 galls; a tierce, 42 galls. A liquid measure barrel of 31^ galls contains 4.211 cub ft a cube of 1.615 ft on an edge ; or 3.384 U. S. struck bushels. A gill = 7.21875 cub ias. The following cylinders contain some of these measures very approximately. Diam. Height. cub ins. Ins. Ins, Gill (7.21875) ...... 1% ............ 3 ............ Pint .............. 3J* ............ 3 Quart ............. 3% ............ 6 Diam. Height. Ins. Ins. Gallon 7 6 2 gallons 7 12 Sgallons 14 12 lOgallons 14 15 WEIGHTS AND MEASURES. 77 cone is not to be less than 6 struck ones: or to 1.55556 cub ft. To reduce U. &. liquid measures to Brit ones of the same denomina- tion, divide by 1.20032; or near enough for common use, by 1.2 ; or to reduce Brit to U. S. multiply by 1.2. ^ Dry Measure. U. 8. only. The basis of this is the old British Winchester struck bushel of 2150.42 cub Ins ; or 77.627413 pounds avoir of pure water at its max density. Its dimensions by law are 18J^ ins inner diam ; 19.^ ins outer diam ; and 8 ins deep ; and when heaped, the ins high; which makes a heaped bushel equal to Edge of a cube of equal capacity. 2 pints 1 quart, - 67.2006 cub ins - 1.16365 liquid qt ....................... 4.066 ins. 4 quarts 1 gallon, = 8 pints, = 268.8025 cub ins, = 1.16365 liq gal ............. 6.454 "" 2 gallons 1 peck, = 16 pints, = 8 quarts, - 537.6050 cub ins ................. 8.131 " 4 pecks 1 struck bushel, 64 pints, = 32 quarts, rrb gals, -2150.4200 cub ins. 12.908 " A struck bushel = 1.24445 cub ft. A cub ft = .80356 of a struck bushel. The dry flour barrel = 3.75 cub ft; = 3 struck bushels. The dry barrel is not, however, a legalized measure; and no great attention is given to its capacity; consequently, barrels vary considerably. A barrel of flour contains by law, 196 fids. In ordering by the barrel, the amount of its contents should be specified in pounds or galls. To reduce U. S. dry measures to Brit imp ones of the same name, div by 1.031516; and to reduce Brit ones to U. S. mult by 1.031516 ; or for common purposes use 1.032. British Imperial Measure, both liquid and dry. This system is established throughout Great Britain, to the exclusion of the old ones. Its basis is the imperial gallon of 277.274 cub ins, or 10 ft>s avoir of pure water at the temp of 62 Fahr, when the barom is at 30 ins. This basis involves an error of about 1 part in 1836, for 10 as of the water - only 277.123 cub ins. Avoir Ihs. of water. Cub. ins. Cub. ft. Edge of a cube of equal capacity. Inches. 4 gills 1 pint .... 1.25 34.6592 3 2605 2 pints 1 quart 2 50 69 3185 4 1079 2 quarts 1 pottle 5. 138 637 5 1756 10. 277 274 6 5208 2 gallons 1 peck . 20 1 554 548 8 2157 4 pcks 1 bushel 80 I Dry 2218 192 1 2837 13 0417 320 f meas 8872 768 5 1347 2 coombs 1 quarter...... 640. J 17745.536 10.2694 The imp gall = . 16046 cub ft; and 1 cub ft = 6. 23210 galls. The imp gal = 1.20032, or very nearly \\ U. S. liquid galls. A cylinder 1 foot in diameter, and 1 foot high, contains .02909 cub yard. 47.0C16 U. S. liquid pints. .7854 cub foot. 188.0064 U. S. liquid gills. 1357. 1712 cub inches. 4.8947 Brit imp gallons. .63112 U. S. dry bushels. 19.5788 Brit imp quarts. 2.5245 U. S. dry pecks. 39.1575 Brit imp pints. 20.1958 U. S. dry quarts. 156.6302 Brit imp gills. 40.3916 U. S. dry pints. 222.395 Decilitres. 5.8752 U. S. liquid gallons. 22.2395 litres. 23.5008 U. S. liquid quarts. 2.22395 decalitres. .222395 hectolitre. A cylinder 1 inch in diameter, and 1 foot high, contains .2719 Brit imp pint. 1.0877 Brit imp gill. 15.4441 centilitres. 1.54441 decilitres. .154441 litres. .005454 cub foot. 9.4248 cub inches. .2806 U. S. dry pint. .3264 liquid pint. 1.3056 U. S. gill. A sphere 1 foot in diameter, contains .01939 cub yard. 31 .3344 U. S. liquid pints. .5236 cub foot. 125.3376 U. S. liquid gills. 904.781 cub inches. 3.2631 Brit imp gallons. .42075 U. S. bushel. 13.6525 Brit imp quarts. 1.6830 U. S. pecks. 26.1050 Brit imp pints. 13.4639 U. S. dry quarts. 104.4201 Brit imp gills. 26.9278 U. S. dry pints. 14.8263 litres. 8.9168 U. S. liquid gallons. 1.48263 decalitres. 15.6672 U. S. liquid quarts. .148263 hectolitres. A sphere 1 inch in diameter, contains .000303 cub foot. .06043 Brit gill. .5236 cub inch. 8.580 millilitre. .07253 U. S. gill. .8580 centilitre. .08580 deciliu*. 78 WEIGHTS AND MEASURES. French Measures of Length. By U. 8. and British Standard. Ins. Ft. Yds. Miles. Millimetre* .039370 .003281 Centimetre"!" . ... . 39370428 032809 Decimetre 3.9370428 3280869 .1093623 MetreJ 39 370428 3 280809 1 093623 Decametre 1 393 70428 32 80869 10 93623 Hectometre . I Road 328 0869 109 3623 0621375 measures 3280 869 1093.623 .6213750 Myriametre j 32808.69 10936.23 6.213750 * Nearly the ^V part of an inch. t Full % inch. I Very nearly 3 ft, 3% ins, which is too long by only 1 part in 8616. French Square Measure. By U. 8. and British Standard. Sq. Ins. Sq. Feet. Sq. Yds. Acres. Sq Millimetre .001550 .00001076 .0000012 Sq Centimetre 155003 00107641 0001196 So Decimetre . 15 5003 10764101 .0119601 Sq Metre, or Centiare Sq Decametre or Are .. 1550.03 155003 10.764101 10764101 1.19601 1196011 .000247 .024711 10764.101 1196.011 .247110 107641.01 11960.11 2.47110 Sq Kilometre 3861090 sq miles. 10764101 1196011. 247.110 Sq Myriametre 38.61090 " 24711.0 French Cubic, or Solid Measure. According to U. 8. Standard. Only those marked " Brit" are British. Millilitre.orcub Centimetre.... Cub Ins. .0610254 (Liquid. .0084537 gill. \ " .0070428 Brit gill. (Dry. .0018162 dry pint. Centilitre .610254 ( Liquid. .084537 gill. \ " .070428 Brit gill. (Dry. .018162 dry pint. Decilitre 6.10254 (Liquid. .84537 gill = .21134 pint. \ " .70428 Brit gill = .17607 Brit pint. (Dry. .18162 dry pint. Litre, or cubic Decimetre 61.0254 (Liquid. 1.05671 quart = 2.1134 pints. 4 " .88036 Brit quart = .1.7607 Brit pints. (Dry. .11351 peck = .9081 dry qt = 1.8162 dry pt. Decalitre, or Centistere 610.254 Cub Ft. .353156 (Liquid. 2.64179 U. S. liquid gal. -j " 2.20090 Brit gal. (Dry. .283783 bush = 1.1351 peck = 9.081 dry qts. Hectolitre, or Decistere 3.53156 (Liquid. 26.4179 U. S. liquid gal. \ " 22.0090 Brit gal. (Dry. 2.83783 bush. Kilolitre, or Cubic Metre, or Stere 35.3156 ( Liquid. 264.179 U. S. liquid gal.) \ 220.090 Brit gal. V Cub yds, 1.3080. (Dry. 28.3783 bush. j Myriolitre, or Decastere 353.156 { tit*' Sii liquM gia -}cu" 1*. "-"BO. WEIGHTS AND MEASURES. 79 The French Metre.* The French metre was intended to be the one ten-millionth part of the dist 10m either pole of the earth to the equator : but after it had been introduced into use, errors were discovered in the calcu- lations employed for ascertaining that dist ; so that the French metre, like the Brit standard yard, is not what it was intended to be. The U. S. Govt adopts for its length 1.093623 yds = 3.280869 ft = 39.370428 ins U. S. or British measure. But in ordinary business transactions 39.37 ins are a legal metre. At 3 ft 3% ins, the length is but 1 part in 8616 too great. French Weights, reduced to common Commercial or Avoir Weight, of 1 pound = 16 ounces, or 70OO grains. Milligramme Grains. 015432 .15432 Decigramme 15432 Gramme 15432 By law a 5-cent nickel 5 grammes Pounds av .022046 Hectogramme 22046 2.2046 Myriogramme 22046 Quintal 22046 22046 The gramme is the basis of French weights ; and is the weight of a cub centimetre of distilled water at its max density, at sea level, in lat of Paris ; barom 29.922 ins. French Measures of the " Systeine Usuel." This system was in use from about 1812 to 1840, when it was forbidden by law to use even its names. This was done in order to expedite the general use of the tables which we have before given. But as the Systeme Usuel appears in books published during the above interval, we add a table of some of its values, Measures of Length. Ligne usuel x*ouce usuel Pied usuel, Aune usuel, Toise usuel, Yards. Feet. Inches. .09113 1.09362 13.12344 47.245 78.74172 or inch, = 1 ar foot, = 12 2 Hgnes .09113 1.09362 3.93708 6.56181 .36454 1.31236 2.18727 6 pieds Weights, Usuel. Cubic, or Solid, TJsuel. Grain usuel Gros usuel.. .8375 grains. 60.297 1.10258 avoir oz. .55129 avoir Ib. 1.10258 avoir Ib. Litron usuel, or 1 litre Boisseau usuel = 1.7608 British pint. 2.7512 British gala. Once usuel . Marc usuel . Livre usuel, or pound, f 5 Before 1812, or before the "Systeme usuel," the Old System, " Systeme Ancien," was in use. French Measures of the "Systeme Ancien." Lineal. Square. Cubic. Point ancien 0148 ins Sq ins Sq ft Sq. yds C ins C. ft. C.yda. Ligne ancien 0888 ins 00789 .0007 Pouce ancien, 1.06577 ins = .0888 ft Pied ancien 12 7892 ins ~ 1 06577 ft 1.1359 1 1359 1.2106 1 2106 Aune ancien, 46.8939 ins = 3. 90182 ft = 1.30261 yds Toise ancin ~ 6 3946 ft~ 2 1315 yds 40 8908 4.5434 261 482 9 6845 League 2282 toises ~ 2 7637 miles There is, however, much confusion about these old measures. Different measures had the same Dame in different provinces. 5f If the efforts now being made to introduce the metre into general use in the United States should succeed, they will be a source of extreme embarrassment to many millions of persons for many years. 80 WEIGHTS AND MEASURES. Russian. Verst = .6629 U. S. or British mile. Pood = 36.114 ft>s avoir. Spanish. The castellano of Spain and New Granada, for weighing gold, is variously estimated, from 71.07 to 71.04 grains. Ai 71.055 grs, (the mean between the two,) an avoir, or common commercial ounce contains 6.1572 castel ; and a Ib avoir contains 98.515. Also a troy ounce = 6.7553 castel : and a troy ft - 81.01.4 castel. Three U. S. gold-dollars weigh about 1.1 castel. The Spanish mark, or marco, for precious metals, in South America, may be taken in practice, as .5065 of a ft avoir. In Spain, .507(5 ft. In other parts of Europe, it has a great number of val- ues ; most of them, however, being between .5 and .54 of a pound avoir. The .5065 of a ft = 3545> grs : and .5076 ft 3553.2 grs. 1 Marco r: 50 castellanos 400 tomine = 4800 Spanish gold-gr*. The arroba has various values in different parts of Spain. That of Castile, or Madrid, is 25.4025 fts avoir ; the tonelnda of Castile = 2032.2 fts avoir ; the quintal = 101.61 fts avoir ; the libra rr 1.0161 fts avoir; the cantara of wine, &c, of Castile -4.2ftt U. S. galls; that of Havana=r 4.1 galls. The vara of Castile := 32.8748 ins, or almost precisely 32% ins ; or 2 ft 8% ins. The fancgadu of land since 180,1 = 1.5871 acres 69134.08 sq ft. The fanega of corn, kc. ~ 1.59914 U. S. struck bushels. In California, the vara by law = 33.372 U. S. ms ; and the legua = 5000 varas : or 2.6335 U. S. miles. Civil, or Common Clock Time. 60 thirds, marked '" 1 second, marked ". 60 seconds 1 minute '. 60 minutes 1 hour, = 3600 sec. 24 hours 1 civil day, 1440 min. 86400 sec. 7 days 1 week, = 168 hours rr 10080 min. 4 weeks 1 civil month, = 28 days 672 hours. 13 civil months, (or 52 weeks,) 1 day, 5 hours, 48 min, 49 y^ sec ; or 365 days, 5 hours, 48 min, ^y^- Bee, 1 civil year. A solar day is the time between two successive solar noons, or transits of the BUU over the meridian of a place. These intervals are not of equal lengths all the year round. The average length of all the solar days is called the mean solar day; and is the same as the common civil day of 24 hours of clock time. Civil noon is at 12 o'clock ; but solar, or apparent noon, may be about 14}^ min before ; or 16> min after 12 of correct clock time. A sidereal day is the interval between two passages of the same star past the range of two fixed objects; ami is the precise time reqd for one complete rev of the earth on its axis^ The sidereal day never varies : but is always equal to 23 hours, 56 min, 4.09 sec;* so that a star will on any night appear to set, or to pass the range of any two fixed objects, 3 min, 55.91 sec earlier by the clock, than it did on the night before,! so that the number of sidereal days in a civil year is 1 greater than that of the civil days. An astronomical day begins at noon, and its hours are counted from to 24. In comparing it with the civil day, the last is supposed to begin at the midnight before the noon at which the first began. * This gives a means of regulating 1 a watch with much accuracy and by a very simple process. The writer, after having regulated his chronometer watch for a year by this method only, differed but a few seconds from the actual time as deduced from careful solar obser- vations. Even a person not accustomed to ranging objects very accurately, need scarcely err a min- ute in a period of any number of years. It having occurred to him that the motion of a star in a second or two might be visible to the naked eye. he stuck a pin horizontally into a window-jamb ; and placing his eye close to it, sighted along one side of it, at a large star setting behind the top of a roof about 100 feet distant, and found that his conjecture was correct. Those stars which are farthest from the poles appear to move the fastest, and are therefore the best. Those less than of the second mag- nitude are not satisfactory. If the first observations of a given star be made as late as midnight, that same star will answer for about three months, until at last it will begin to pass the range in daylight. Before this happens, the observer must transfer the time to another star which sets later ; if near midnight, the better, as it will serve for a longer time A window looking west is the best. The longer the range, the greater will be the apparent motion of the star; and. consequently, the obser- vations will be more correct. If such a range can be secured as will strike the heavens at an angle of at least 40 above the horizon, the error from refraction will not appreciably affect an observation ; at a much less angle it may do so to the extent of three or four seconds. A caudle must be so placed as to render the pin and the watch visible at the same time. A little practice will render the process very easy, and supersede the necessity for more remarks on the subject. Of course, a memorandum must be made and preserved of. the date, hour, minute, and (approximately) second, at which the first passage of the star took place. Subsequent passages will occur earlier, as shown in the follow- ing table. The watch must be previously known to be right, when taking the first observation, if we require afterward to keep the correct time. Any person who will take the trouble thus to observe, and note down throughout a year, about half a dozen stars following each other at tolerably equal intervals of time, will on almost any clear night afterward be able, after a short calculation, to ascer- tain the correct clock time. The writer observed the passages of two or three stars behind different ranges, on the same nights, in order to obtain a mean of several observations ; his object being to Ascertain how pocket chronometers of the best makers would keep time under the vicissitudes of tem- perature, railroad travelling. Ac, &c, to which they are ordinarily exposed. He used two of the best for this purpose, and the result was that their changes of rate were at times as great as from three to eight seconds per day. For ordinary purposes, therefore, they are of but little, if any, more service than a good common watch, of one-fourth the cost. T More accurately 3 mitt, 55.90944 sec. WEIGHTS AND MEASURES. 81 TABLE showing? bow much earlier a star passes a given range, on each succeeding- night. (Original.) Nights. 1 Min. Sec. 3 55.91 Nights. H. Min. Sec. 43 15.01 Nights. 21 H. Mia. Sec. 1 22 34.11 2 7 51.82 12 47 10.92 22 1 26 30.02 3 11 47.73 13 51 6.83 23 1 30 25.93 4 15 43.64 14 55 2.74 24 1 34 21.84 5 19 39.55 15 58 58.65 25 1 38 17.75 6 23 35.46 16 1 2 54.56 26 1 42 13.66 27 31.37 17 6 50.47 27 1 46 9.57 31 27.28 18 10 46.38 28 1 50 5.48 9 35 23.19 19 14 42.29 29 1 54 1.39 10 39 19.10 20 18 38.20 30 1 57 57.30 31 2 1 53.21 Approximate Tallies of foreign Coins, in 17. S. Currency. It is difficult to obtain positive information respecting these; and it is probable that many of the values in this table are in error from 1 to 5 per cent. Since, however, many of the coins appear in statements of costs of engineering works abroad, it is convenient to have even such an approximation. Dolls. Cts. Dolls. Cts. Augustus. Saxony Carlin. Sardinia. ... 3 g 98 21 Mohur. Bombay " Bengal 7 8 20 15 Carolin. Bavaria 4 93 3 84 Crown. Great Britain 1 13 Ounce Sicily 2 50 " Spain (Half Pistole). 1 95 Pistola Rome 3 37 " Baden Bavaria N Ger- Pistole Spain . 3 90 1 6 10 Sicily 96 Peseta Spain 20 27 Pistareen Spain 20 11 Sweden Norwa. **"* 27 Piastre Turkey Old 42 Copeck Russia. % " " New 4 2^ Dollar. Bolivia, new 96 " *' other authori- TIT " U. S. of Colombia.. .. 93 5 ties 19 cts ; 11 cts, &c " Chili, Peru Ecuador .. 93 1 4 " Liberia 1 5 " Mexico 1 Para 9 " Sandwich Islds .... 1 Pound. Great Britain 4 87 T7 Doubloon. Spain Mexico. 15 65 " Canada, N Scotia New " Central America f 14 < to 50 Bruns, Newfoundland Reale plate Spain 4 10 I 15 65 5 " New Granada 15 34 " Central America average 5*f Drachm. Greece 19 2 Reales Ecuador ' 18% Ducat. Austria, Bohemia, Ham- burg, Hanover 2 28 Rix dollar. Hamburg. Hanover. " " Sweden, Holland... 1 1 10 5 " Sweden 2 l) 1 1 81 " " Bavaria, Austria 1 32 Hungary 97 Franc. France, Belgium, Ac 19.4 " 10 thalers. Prussia 8 5 Franc piece 97 " thaler. Prussia, Poland, N. Florin. Austria, Silesia " Holland, Netherlands, S. Germany 48 38 Germany, Bremen, Saxony, Hanover, " species thaler Saxony 69 98 1 66 45 " (silver! H n 56 " PeC ' Lacof^' . 45000 " P us i 55 Reis (1000) Brazil 1 8 Guilder Netherland 40 Rouble Russia 75 26 3 95 Gulden. Baden 40 Schilling Hamburg 2 Guinea. Great Britain, 21 shil- Shilling. Great Britain 23 V Groschen Poland Prussia. tdk Scudo. Piedmont 1 3*' 4 5 Groschen " " 12 T " Naples Sicily 95 1 " Sardinia 92 Imperial Russia 7 92 1 Kreutzer. Bavaria K ,, 1 28 10. Austria " 60. " or florin.. Livre. France. Sardinia (Franc) " Tuscany, Venice 8 48 18J4 16 Souverain. Austria, Bohemia... Sovereign. Gr Britain, or pound Sous. France very nearly Star Pagoda Madras 3 4 1 57 86 1 81 Lira. Milan .. ... 19 Stiver Holland ...nearly I Marc. Germany. 24 30 Maximilian. Bavaria. 3 30 Testoon Portueal 12 Milrea. Portugal. 1 8 Zecchin Turkey 1 40 Moidore. - 6 50 Zecchino. Rome 2 27 82 TKAVERSE TABLE. Traverse Table for a Distance = 1. Lat. Dep. Lat. Dep. Lat. ^T or or or N. S. E.W. N. S. E.W. N.S. EW. 00' 2 1.0000 1.0000 .0000 .0006 900' 58 20' 2 .9994 .9994 .0349 .0355 880' 58 40' 2 .9976 .9975 .0698 .0703 860' 58 4 1.0000 .0012 56 4 .9993 .0361 56 4 .9975 .0709 56 6 1.0000 .0017 54 6 .9993 .0366 54 6 .9974 .0715 54 8 1.0000 .0023 .52 8 .9993 .0372 52 , 8 .9974 .0721 52 10 1.0000 .0029 50 10 .9993 .0378 50 10 .9974 .0727 50 12 1.0000 .0035 48 12 .9993 .0384 48 12 .9973 .0732 *8 14 1.0000 .0041 46 14 .9992 .0390 46 14 .9973 .0738 46 16 1.0000 .0047 44 16 .9992 .0396 44 16 .9972 .0744 44 18 1.0000 .0052 42 18 .9992 .0401 42 18 .9972 .0750 42 20 1.0000 .0058 40 20 .9992 .0407 40 20 .9971 .0756 40 22 1.0000 .0064 38 22 .9991 .0413 38 22 .9971 .0761 38 24 1.0000 .0070 36 24 .9991 .0419 36 24 .9971 .0767 36 26 1.0000 .0076 34 26 .9991 .0425 34 26 .9970 .0773 34 28 1.0000 .0081 32 28 .9991 .0430 32 28 .9970 .0779 32 30 1 .0000 .0087 30 30 .9990 .0436 30 30 .9969 .0785 30 32 1.0000 .0093 28 32 .9990 .0442 23" 32 .9969 .0790 28 34 1.0000 .0099 26 34 .9990 .0448 26 34 .9968 .0796 36 36 .9999 .0105 24 36 .9990 .0454 24 36 .9968 .0802 24 38 .9999 .0111 22 38 .9989 .0459 22 38 .9967 .0808 22 40 .9999 .0116 20 40 .9989 .0465 20 40 .9967 .0814 20 42 .9999 .0122 18 42 .9989 .0471 18 42 .9966 .0819 18 44 .9999 .0128 16 44 .9989 .0477 16 44 .9966 .0825 16 46 .9999 .0134 14 46 .9988 .0483 u 46 .9965 .0831 14 48 .9999 .0140 12 48 .9988 .0488 12 48 .9965 .0837 12 50 .9999 .0145 10 50 .9988 .0494 10 50 .9964 .0843 10 52 .9999 .0151 8 52 .9987 .0500 8 52 .9964 .0848 8 54 .9999 .0157 6 54 .9987 .0506 6 54 .9963 .0854 6 56 .9999 .0163 4 56 .9987 .0512 4 56 .9963 .0860 4 58 .99*9 .0169 2 58 .9987 .0518 2 58 .9962 0866 2 10' .9998 .0175 890 30' .9986 ,0523 870' 50' .9962 .0872 850' 2 .9998 .0180 58 2 .9986 .0529 58 2 .9961 .0877 58 4 .9998 .0186 56 4 .9986 .0535 56 4 .9961 .0883 56 6 .9998 .0192 54 6 .9985 .0541 54 6 .9960 .0889 54 8 .9998 .0198 52 8 .9985 .0547 52 8 .9960 .0895 52 10 .9998 .0204 50 10 .9985 .0552 50 10 .9959 .0901 50 12 .9998 .0209 48 12 .9984 .0558 48 12 .9959 .0906 48 14 .9998 .0215 46 14 .9984 .0564 46 14 .9958 .0912 46 16 .9998 , .0221 44 16 .9984 .0570 44 16 .9958 .0918 44 18 .9997 .0227 42 18 .9983 .0576 42 18 '.9957 .0924 42 20 .9997 .0233 40 20 .9983 .0581 40 20 .9957 .0929 40 22 .9997 .0239 38 22 .9983 .0587 38 22 .9956 .0935 38 24 .9997 .0244 36 24 .9982 .0593 36 24 .9956 .0941 36 26 .9997 .0250 34 26 .9;)82 .0599 34 26 .9955 .0947 34 28 .9997 .0256 32 28 .9982 .0605 32 28 .9955 .0953 32 30 .9997 .0262 30 30 .9981 .0610 30 30 .9954 .0958 30 32 .9996 .0268 28 32 .9981 .0616 28 32 .9953 .0964 28 34 .9996 .0273 26 34 .9981 .0622 26 34 .9953 .0970 26 36 .9996 .0279 24 36 .9980 .0628 24 36 .9952 .0976 24 38 .9996 .0285 22 38 .9980 .0634 22 38 .9952 .0982 22 40 .9996 .0291 20 40 .9980 .OS40 20 40 .9951 .0987 20 42 9996 .0297 18 42 .9979 .0645 18 42 .9951 .0993 18 44 .9995 .0302 16 44 .9979 .0651 16 44 .9950 .0999 16 46 .9995 .0308 14 46 .9978 .0657 14 46 .9949 .1005 14 48 .9995 .0314 12 48 .9978 .0663 12 48 .9949 .1011 12 50 .9995 .0320 10 50 .9978 .0669 10 50 .9948 .1016 10 52 .9995 .0326 8 52 .9977 .0674 8 52 .9948 .1022 8 54 .9995 .0332 6 54 .9977 .0680 6 54 .9947 .1028 6 56 .9994 .0337 4 56 .9976 .0686 4 56 .9946 .1034 4 58 .9994 .0343 2 58 .9976 .0692 1 58 .9946 .1039 2 20' .9994 .0349 88 D 0' 40' .9976 .0698 860' 60' .9945 .1045 840* Dep. Lat. Dep. Lat. Dep. Lat. or or or E.W. N. S. E.W. N. S. E.W. N.S. TRAVERSE TABLE. 83 Traverse Table for a Distance = 1. (CONTINUED.) Lat. Dep. Lat. Dep. Lat. Dep. or or or or or or N. S. E. W. N. S. E. W. N. S. E.W. .9940 .9945 .1045 .1051 840' 58 80' 2 .9903 .9902 .1392 .1397 820' 58 100' | .9848 .9847 .1736 .1742 80^0' 58 .9944 .1057 56 4 .99U1 .1403 56 4 .9846 .1748 56 .9943 .1063 54 6 .9900 .1409 54 6 .9845 .1754 54 .9943 .1068 52 8 .9899 .1415 52 8 .9844 .1758 52 1 .9942 .1074 50 .9899 .1421 50 10 .9843 . 765 50 1 .9942 .1080 48 1 .9898 .1426 48 12 .9842 . 771 48 1 .9941 .1086 46 4 .9897 .1432 46 14 .9841 . 777 46 1 .9940 .1092 44 6 .9896 .1438 44 16 .9840 . 782 44 18 .9940 .1097 42 8 .9895 .1444 42 18 .9839 . 788 42 20 .9939 .1103 40 20 .9894 .1449 40 20 .9838 .1794 40 22 .9938 .1109 38 22 .9894 .1455 38 22 .9837 .1799 38 24 .9938 .1115 36 24 .9893 .1461 36 24 .9836 .1805 36 26 .9937 .1120 34 26 .9892 .1467 34 26 .9835 .1811 34 28 .9936 .1126 32 28 .9891 .1472 32 28 .9834 .1817 32 30 .9936 .1132 30 30 .9890 .1478 30 30 .9833 .1822 30 32 .9935 .1138 28 32 .9889 .1484 28 32 .9831 .1828 28 34 .9934 .1144 26 34 .9888 .1490 26 34 .9830 .1834 26 36 .9934 .1149 24 36 .9888 .1495 24 36 .9829 .1840 24 38 .9933 .1155 22 38 .9887 .1501 22 38 .9828 .1845 22 40 .9932 .1161 20 40 .9886 .1507 2O 40 .9827 .1851 20 42 .9932 .1167 18 42 .9885 .1513 18 42 .9826 .1857 18 44 .9931 .1172 16 44 .1518 16 44 .9825 .1862 16 46 .9930 .1178 14 46 !9883 .1524 14 46 .9824 .1868 14 48 .9930 .1184 12 48 .9882 .1530 12 48 .9823 .1874 12 60 .9929 .1190 10 50 .9881 .1536 10 50 .9822 J880 10 52 .9928 .1196 8 52 .9880 .1541 8 52 .9821 .1885 8 54 .9928 .1201 6 54 .9880 .1547 6 54 .9820 .1891 6 56 .9927 .1207 4 56 .9879 .1553 4 56 .9818 .1897 4 58 .9926 .1213 2 58 .9878 .1559 2 58 .9817 .1902 2 70' .9925 .1219 83^0' 90 .9877 .1564 81 0' 110' .9816 .1908 79^ I .9925 .1224 58 2 .9876 .1570 58 2 .9815 .1914 58 4 .9924 .1230 56 4 .9875 .1576 56 4 .9814 .1920 56 .9923 .1236 54 6 .9874 .1582 54 6 .9813 .1925 54 8 .9923 .1242 52 8 .9873 .1587 52 8 .9812 .1931 52 10 .9923 .1248 50 10 .9872 .1593 50 10 .9811 .1937 50 12 .9921 .1253 48 12 .9871 .1599 48 12 .9810 .1942 48 14 .9920 .1259 46 14 .9870 .1605 46 14 .9808 .1948 46 16 .9920 .1265 44 16 .9869 .1610 44 16 .9807 1954 44 18 .9919 .1271 42 18 .9869 .1616 42 18 .9806 .1959 42 20 .9918 .1276 40 20 .9868 .1622 40 20 .9805 .1965 40 '22 .9917 .1282 38 22 .9867 .1628 38 22 .9804 .1971 38 24 .9917 .1288 36 24 .9866 .1633 36 24 .9803 .1977 36 26 .9916 .1294 34 26 .9865 .1639 34 26 .9802 .1982 34 28 .9915 .1299 32 28 .9864 .1645 32 28 .9800 .1988 32 30 .9914 .1305 30 30 .9863 .1650 30 30 .9799 .1994 30 M .9914 .1311 28 32 .9862 .1656 28 32 .9798 .1999 28 H .9913 .1317 26 34 .9861 .1662 H 34 .9797 .2005 26 36 .9912 .1323 24 36 .9860 .16(58 24 36 .9796 .2011 24 58 .9911 .1328 22 38 .9859 .1673 22 38 .9795 .2016 22 40 .9911 .1384 20 40 .9858 .1679 20 40 .9793 .2022 20 42 .9910 .1340 18 42 .9857 .1685 18 42 .9792 .2028 18 44 .9909 .1346 16 44 .9856 .1691 16 44 .9791 .2034 16 46 .9908 .1351 14 46 .9855 .1^96 14 46 .9790 .2039 14 48 .9907 .1357 12 48 .9854 .1702 12 48 .9789 .2045 12 50 .9907 .1363 10 50 .9853 .1708 10 50 .9787 .2051 10 52 .9906 .1360 8 52 .9852 .1714 8 52 .9786 .2056 8 54 .9905 .1374 6 54 .9851 .1719 6 54 .9785 .2062 6 56 .9904 .1880 4 56 .9850 .1725 4 56 jtm .2068 4 58 .9903 ,1*86 2 58 .9849 .1731 2 58 .1)783 .2073 2 80' .9903 .1392 820 1000' .9848 .1736 800' 120' .9781 .2079 780' Dep. Lat. Dep. Lnt, Dep. Lat. or or or E. W. N. 8. E. W. N. S. E.W. N.S. 84 TRAVERSE TABLE. Traverse Table for a Distance = 1. (CONTINUED.) Lat. Dep. Lat. Dep. Lat. Dep. or or or or N.S. E. W. 1 N.S. E. W. N.S. E. W. 120' 2 .9781 .9780 .2079 .2084 780' 58 140 2 .9703 .9702 .2419 .2425 760' 58 160' 2 .9613 .9611 .2756 .2762 740' 58 4 .9779 .2090 56 4 .9700 .2431 56 4 .9609 .2768 56 6 .9778 .2096 54 6 .9699 .2436 54 6 .9608 .2773 54 8 .9777 .2102 52 8 .9697 .2442 52 8 .9606 .2779 52 10 .9775 .2108 50 10 .9696 .2447 50 10 .9605 .2784 50 12 .9774 .2113 48 12 .9694 .2453 48 12 .9603 .2790 48 14 9773 .2119 46 14 .9693 .2459 46 14 .9601 .2795 46 16 .9772 .2125 44 16 .9692 .2464 44 16 .9600 .2801 44 18 .9770 .2130 42 18 .9690 .2470 42 18 .9598 .2807 42 20 .9769 .2136 40 20 .9689 .2476 40 20 .9596 .2812 40 22 .9768 .2142 38 22 .9687 .2481 38 22 .9595 .2818 38 24 .9767 .2147 36 24 .9686 .2487 36 24 .9593 .2823 36 26 .9765 .2153 34 26 .9684 .2493 34 26 .9591 .2829 34 28 .9764 .2159 32 28 .9683 .2498 32 28 .9590 .2835 32 30 .9763 .2164 30 30 .9681 .2504 30 30 .9588 .2840 30 32 .9762 .2170 28 32 .9680 .2509 28 32 .9587 .2846 28 34 .9760 .2176 26 34 .9679 .2515 26 34 .9585 .2851 26 36 .9759 .2181 24 36 .9677 .2521 24 36 .9583 .2857 24 38 .9758 .2187 22 38 .9676 .2526 22 38 .9582 .2862 22 40 .9757 .2193 20 40 .9674 .2532 20 40 .9580 .2868 20 42 .9755 .2198 18 42 .9673 .2538 18 42 .9578 .2874 18 44 .9754 .2204 16 44 .9671 .2543 16 44 .9577 .2879 16 46 .9753 .2210 14 46 .9670 .2549 14 46 .9575 .2885 14 48 .9751 .2215 12 48 .9668 .2554 12 48 .9573 .2890 12 50 .9750 .2221 10 50 .9667 .2560 10 50 .9572 .2896 10 52 .9749 .2227 8 52 .9665 .2566 8 52 .9570 .2901 8 54 .9748 .223" 6 54 .9664 .2571 6 54 .9568 .2907 6 56 .9746 ,22.>8 4 56 .9662 .2577 4 56 .9566 .2913 4 58 .9745 .2244 2 58 .9661 .2583 2 58 .9565 .2918 2 130' .9744 .2250 770' 150Q' .9659 .2588 750' L70- .9563 .2924 7300* 2 .9742 .2255 58 2 .9658 .2594 58 2 .9561 .2929 58 4 .9741 .2261 56 4 .9656 .2599 56 4 .9560 .2935 56 6 .9740 .2267 54 6 .9655 .2605 54 6 .9558 .2940 54 8 .9738 .2272 52 8 .9653 .2611 52 8 .9556 .2946 52 10 .9737 .2278 50 10 .9652 .2616 50 10 .9555 .2952 50 12 .9736 .2284 48 12 .9650 .2622 48 12 .9553 .2957 48 14 .9734 .2289 46 14 .9649 .2628 46 14 .9551 .2963 46 16 .9733 .2295 44 16 .9647 .2633 44 16 .9549 .2968 44 18 .9732 .2300 42 18 .9646 .2639 42 18 .9548 .2974 42 20 .9730 .2306 40 20 .9644 .2644 40 20 .9546 .2979 40 22 .9729 .2312 38 22 .9642 .2650 38 22 .9544 .2985 38 24 .9728 .2317 36 24 .9641 .2656 36 24 .9542 .2990 36 26 .9726 .2323 34 26 .9639 .2661 34 26 .9541 .2996 34 28 .9725 .2329 32 28 .9638 .2667 32 28 .9539 .3002 32 30 .9724 .2334 30 30 .9636 .2672 30 30 .9537 .3007 30 32 .9722 .2340 28 32 .9635 .2678 28 32 .9535 .3013 28 34 .9721 .2346 26 34 .9633 .2684 26 34 .9534 .3018 26 36 .9720 .2351 24 36 .9632 .2689 24 36 .9532 .3024 24 38 .9718 .2357 22 38 .9630 .2695 22 38 .9530 .3029 22 40 .9717 .2363 20 40 .9628 .2700 20 40 .9528 .3035 20 42 .9715 .2368 18 42 .9627 .2706 18 42 .9527 .3040 18 44 .9714 .2374 16 44 .9625 .2712 16 44 .9525 .3046 16 46 .9713 .2380 14 46 .9624 .2717 14 46 .9523 .3051 14 48 .9711 .2385 If 48 .9622 .2723 12 48 .9521 .3057 12 50 .9710 .2391 10 50 .9621 .2728 10 50 .9520 .3062 10 52 .9709 .2397 8 52 .9619 .2734 8 52 .9518 .3068 8 54 .9707 .2402 6 54 .9617 .2740 6 54 .9516 .3074 6 56 .9706 .2408 4 56 .9616 .2745 4 56 .9514 .3079 4 58 .9704 .2414 2 58 .9614 .2751 2 58 .9512 .3085 2 140' .9703 .2419 760' 160' .9613 .2756 740' 180' .9511 .3090 720' Dep. Lat. Dep. Lat. Dep. Lat. or or or or or or E. W. N.S. E. W. N.S. E. W. N.S. TRAVEESE TABLE. 85 Traverse Table for a Distance =: 1. (CONTINUED.) Lat. or N. S. Dep. or E. W. Lat. or N. S. Dep. or E. W. t Lat. or N. S. Dep. or E.W. 180' .9511 .3090 720' 20 r O' .9397 .3420 700 220 .9272 .3746 680' 2 .9509 .3096 58 2 .9395 .3426 58 2 .9270 .3751 58 4 .9507 .3101 56 4 .9393 .3431 56 4 .9267 .3757 56 6 .9505 .3107 54 6 .9391 .3437 54 6 .9265 .3762 54 8 .9503 .3112 52 8 .9389 .3442 52 8 .9263 .3768 52 10 .9502 .3118 50 10 .9387 .3448 50 10 .9261 .3773 50 12 .9500 .3123 48 12 .9385 .3453 48 12 .9259 .3778 48 14 .9498 .3129 42^0' .9272 .3746 68^0' 40' .9135 .4067 B6 C Dep. Lat. Dep. Lat. Dep. Lat. or or or or or or E.W: N. S. E. W. N. S. E.W. N. S. 86 TRAVERSE TABLE. Traverse Table for a Distance = 1. (CONTINUED.) Lat. or N.S. Dep. or E.W. Lat. or N.S. Dep. or E.W. Lat. N?S. Dep. or E. W. 24^0' 2 .9135 .9133 .4067 .4073 66^0' 58 260' 2 .8988 .8985 J4384 .4389 640' 58 280' 2 .8829 .8827 .4695 .4700 620 / 58 4 .9131 .4078 56 4 .8983 .4394 56 4 .8824 .4705 56 6 .9128 .4083 54 6 .890 .4399 54 6 .8821 .4710 54 8 .9126 .4089 52 8 .8978 .4405 52 8 .8819 .4715 52 10 .9124 .4094 50 10 .8975 .4410 50 10 .8816 .4720 50 12 .9121 .4099 48 12 .8973 .4415 48 12 .8813 .4726 48 14 16 '.9116 .'4110 44 16 .8967 .4425 44 16 .'8808 .4731 .4736 44 18 .9114 .4115 42 18 .8965 .4431 42 18 .8805 .4741 42 20 .9112 .4120 40 20 .8962 .4436 40 20 .8802 .4746 40 22 .9109 .4126 38 22 .8960 .4441 38 22 .8799 .4751 38 24 .9107 .4131 36 24 .8957 .4446 36 24 .8796 .4756 36 26 .9-504 .4136 34 26 .8955 .4452 34 26 .8794 .4761 34 28 .9102 .4142 32 28 .8952 .4457 32 28 .8791 .4766 32 30 .9100 .4147 30 30 .8949 .4462 30 30 .8788 .4772 30 32 .9097 .4152 28 32 .8947 .4467 28 32 .8785 .4777 28 34 .9095 .4158 26 34 .8944 .4472 26 34 .8783 .4782 26 36 .9092 .4163 24 36 .8942 .4478 24 36 .8780 .4787 24 38 .9090 .4168 22 38 .8939 .4483 22 38 .8777 .4792 22 40 .9088 .4173 20 40 .8936 .4488 20 40 .8774 .4797 20 42 .9085 .4179 18 42 .8934 .4493 18 42 .8771 .4802 18 44 .9083 .4184 16 44 .8931 .4498 16 44 .8769 .4807 16 46 .9080 .4189 14 46 .8928 .4504 14 46 .8766 .4812 14 48 .9078 .4195 12 48 .8926 .4509 12 48 .8763 .4818 12 50 .9075 .4200 10 50 .8923 .4514 10 50 .8760 .4823 10 52 .9073 .4205 8 52 .8921 .4519 8 52 .8757 .4828 8 54 .9070 .4210 6 54 .8918 .4524 6 54 .8755 .4833 6 56 .9068 .4216 4 56 .8915 .4530 4 56 .8752 .4838 4 58 .9066 .4221 2 58 .8913 .4535 2 58 .8749 4843 2 250' .9063 .4226 650' 270 .8910 ,4540 6300' 290' .8746 .4848 61 0' 2 .9061 .4231 58 2 .8907 .4545 58 2 .8743 .4853 58 4 .9058 .4237 56 4 .8905 .4550 56 4 .8741 .4858 56 6 .9056 .4242 54 6 .8902 .4555 54 6 .8738 .4863 54 8 .9053 .4247 52 8 .8899 .4561 52 8 .8735 .4868 52 10 .9051 .4253 50 10 .8897 .4566 50 10 .8732 .4874 50 12 .9048 .4258 48 12 .8894 .4571 48 12 .8729 .4879 48 14 .9046 .4263 46 14 .8892 .4576 46 14 .8726 .4884 46 16 .9043 .4268 44 16 .8889 .4581 44 16 .8724 .4889 44 18 .9041 .4274 42 18 .8886 .4586 42 18 .8721 .4894 42 20 .9038 .4279 40 20 .8884 .4592 40 20 .8718 .4899 40 22 .9036 .4284 38 22 .8881 .4597 38 22 .8715 .4904 38 24 .9033 .4289 36 24 .8878 .4602 36 24 .8712 .4909 36 26 .9031 .4295 34 26 .8875 .4607 34 26 .8709 .4914 34 28 .9028 .4300 32 28 .8873 .4612 32 28 .8706 .4919 32 30 .9026 .4305 30 30 .8870 .4617 30 30 .8704 .4924 30 32 .9023 .4310 28 82 .8867 .4623 28 32 .8701 .4929 28 34 .9021 .4316 26 34 .8865 .4628 26 34 .8698 .4934 26 36 .9018 .4321 24 36 .8862 .4633 24 36 .8695 .4939 24 38 .9016 .4326 22 38 .8859 .4638 22 38 .8692 .4944 22 40 .9013 .4331 20 40 .8857 .4643 20 40 .8689 .4950 20 42 9011 .4337 18 42 .8854 .4648 18 42 .8686 .4955 18 44 .9008 .4342 16 44 .8851 .4654 16 44 .8683 .4960 16 46 .9006 .4347 14 46 .8849 .4659 14 46 .8681 .4965 14 48 .9003 .4352 12 48 .8846 .4664 12 48 .8678 .4970 12 50 .9001 .4358 10 50 .8843 .4669 to 50 .8675 .4975 10 52 .895)8 .4363 8 52 .8840 .4674 8 52 .8672 .4980 8 54 .8996 .436S 6 54 .8838 .4679 6 54 .8669 .49^5 6 56 .81)93 .4373 4 56 .8835 .4684 4 56 .8666 .4990 4 58 .8990 .4378 2 58 .8832 .4690 2 58 .8663 .4995 2 260' .8988 .4384 64 0' .'80' .8829 .4695 620' 300' .8660 .5000 600' Dep. Lat. Dep. Lat. Dep. Lat. or or or or E.W. N.S. E.W. N.S. E.W. N.S. TRAVERSE TABLE. 87 Traverse Table for a Distance = 1. (CONTINUED.) Lat. Dep. Lat. Dep. Lat. Dep. or or or N.S. E.W. N.S. E.W. N.S. E.W. S00' 2 .8660 .8657 .5000 .5005 600' 58 32CQ' 2 8480 8477 .5299 5304 58^0' 58 340' 2 .8290 .8287 .5592 .5597 56^0' 58 4 .8654 .5010 56 4 8474 5309 56 4 .8284* .5602 56 6 .8652 .5015 54 6 8471 5314 54 6 .8281 .5606 54 8 .8649 .5020 52 8 8468 5319 52 8 .8277 .5611 52 10 .8646 .5025 50 10 8465 5324 50 10 8274 .5616 50 12 .8643 .5030 48 12 8462 5329 48 12 .8271 .5621 48 14 .8640 .5035 46 14 8459 5334 46 14 .8268 .5626 46 16 .8637 .5040 44 16 8456 5389 44 16 .8264 .5630 44 18 .8634 .5045 42 18 8453 5344 42 18 .8261 .5635 42 20 .8631 .5050 40 20 8450 5348 40 20 .8258 .5640 40 22 .8628 .5055 38 22 8446 5353 38 22 .8254 .5645 38 24 .8625 .5060 36 24 8443 5358 36 24 .8251 5650 36 28 .8622 .&065 34 26 8440 5363 34 26 .8248 .5654 34 28 .8619 .5070 32 28 8437 5368 32 28 .8245 5659 32 30 .8616 .5075 30 30 8434 5373 30 30 .8241 .5664 30 32 .8613 .5080 28 32 8431 5378 28 32 .8238 .5669 28 34 .8610 .5085 26 34 8428 5383 26 34 .82^5 .5674 26 36 .8607 .5090 24 36 8425 5388 24 36 .8231 .5678 24 38 .8604 .5095 22 38 8421 5393 22 38 .8228 .5683 22 40 .8601 .5100 20 40 8418 5398 20 40 .8225 .5688 20 42 .8599 .5105 18 42 8415 5402 18 42 .8221 .5693 18 44 .8596 .5110 16 44 8412 5407 16 44 .8218 .5698 16 46 .8593 .5115 14 46 8409 5412 14 46 .8215 .5702 14 48 .8590 .5120 12 48 8406 5417 12 48 .8211 5707 12 50 .8587 .5125 10 50 8403 5422 10 50 .8208 5712 10 52 .8584 .5130 8 52 8399 5427 8 52 .8205 .5717 8 54 ,8581 .5135 6 54 8396 5432 6 54 .8202 5721 6 56 .8578 .5140 4 56 8393 5437 4 56 .8198 5726 4 58 .8575 .5145 2 58 8390 5442 2 58 .8195 5731 2 310' .8572 .5150 5900' d30' 8387 5446 570' 350' .8192 .5736 5500' 2 .8569 .5155 58 2 8384 5451 58 2 .8188 .5741 58 4 .8566 .5160 56 4 8380 5456 56 4 .8185 5745 56 6 .8563 .5165 54 6 8377 5461 54 6 .8181 .5750 54 8 .8560 .5170 52 8 8374 5466 52 8 .8178 .5755 52 10 .8557 .5175 50 10 8371 5471 50 10 .8175 .5760 50 12 .8554 .5180 48 12 8368 5476 48 12 .8171 .5764 48 14 .8551 .5185 46 14 8364 5480 46 14 .8168 .5769 46 16 .8548 .5190 44 16 8361 5485 44 16 .8165 .5774 44 18 .8545 .5195 42 18 8358 5490 42 18 .8161 .5779 42 20 .8542 .5200 40 20 8355 5495 40 20 .8158 .5783 40 22 .8539 .5205 38 22 8352 .5500 38 22 .8155 .5788 38 24 .8536 .5210 36 24 8348 .5505 36 24 .8151 .5793 36 26 .8532 .5215 34 26 8345 .5510 34 26 .8148 .5798 34 28 .8529 .5220 32 28 8342 .5515 32 28 .8145 .5802 32 30 .8526 .5225 30 30 8339 .5519 30 30 .8141 .5807 30 32 .8523 .5230 28 32 8336 .5524 28 32 .3138 .5812 28 34 .8520 .5235 26 34 8332 .5529 26 34 .8134 .5816 26 36 .8517 .5240 24 36 8329 .5534 24 36 .8131 .5821 24 38 .8514 .5245 22 38 8326 .5539 22 38 .8128 .5826 22 40 .8511 .5250 20 40 8323 .5544 20 40 .8124 .5831 20 42 .8508 .5255 18 42 8320 .5548 18 42 .8121 .5835 18 44 .8505 .5260 16 44 8316 .5553 16 44 .8117 .5840 16 46 .8502 .5265 14 46 8313 .5558 14 46 .8114 .5845 14 48 .8499 .5270 12 48 8310 .5563 12 48 .8111 .5850 12 50 .8496 .5275 10 50 8307 .5568 10 50 .8107 .5854 10 52 .8493 .5279 8 52 8303 .5573 8 52 .8104 .5859 8 54 .8490 .5284 6 54 8300 .5577 6 54 .8100 .5864 6 56 .8487 .5289 4 56 8297 .5582 4 56 .8097 .5868 4 58 .8484 .5294 2 58 .8294 .5587 2 58 .8094 .5873 2 3200' .8480 .5299 580' 34=0' 8290 .5592 560' 360' .8090 .5878 540' Dep. Lat. Dep. Lat. Dep. Lat. or or or or or or E.W. N.S. E.W. N.S. E. W. N.S. TKAVERSE TABLE. Traverse Table for a Distance = 1. (CONTINUED.) Lat. or N. S. Dep. or E. W. Lat. N. S. Dep. or E. W. Lat. or N. S. Dep. E. W. 360' 2 .8090 .8087 .5878 .5883 54^0' 58 i80' 2 .7880 .7877 .6157 .6161 520' 58 100' 2 .7660 ;7657 .6428 .6432 500' 58 4 .8083 ' .5887 56 4 .7873 .6166 56 4 .7653 .6437 56 6 .8080 .5892 54 6 .769 .6170 54 6 .7649 .6441 54 8 .8076 .5897 52 8 .7866 .6175 52 8 .7645 .6446 52 10 .8073 .5901 50 10 .7862 .6180 50 10 .7642 .6450 50 12 .M)70 .5906 48 12 .7859 .6184 48 12 .7638 .6455 48 14 .8066 .5911 46 14 .7855 .6189 46 14 .7634 .6459 46 16 .80<>3 .5915 44 16 .7851 .6193 44 16 .7630 .6463 44 18 .8059 .5920 42 18 .7848 .6198 42 18 .7627 .6468 42 20 .8056 .5925 40 20 .7844 .6202 40 20 .7623 .6472 40 22 .8052 .59,'iO 38 22 .7841 .6207 38 22 .7619 .6477 38 24 .8049 .5934 36 24 .7837 .621 1 36 24 .7615 .6481 36 26 .8045 .59:59 34 26 .7833 .6216 34 26 .7612 .6486 34 28 .8042 .5944 32 28- .7830 .6221 32 28 .7608 .6490 32 30 .8039 .5948 30 30 .7826 .6225 30 30 .7604 .6494 30 32 .8035 .5953 28 32 .7822 .6230 28 32 .7600 .6499 2b 34 .8032 .5958 26 34 .7819 .6234 26 34 .7596 .6503 26 36 .8028 .5962 24 36 .7815 .6239 24 36 .7593 .6508 24 38 .8025 .5967 22 38 .7812 .6243 22 38 .7589 .6512 22 40 .8021 .5972 20 40 .7808 .6248 20 40 .7585 .6517 20 42 .8018 .5976 18 42 .7804 .6252 18 42 .7581 .6521 18 44 .8014 .5981 16 44 .7801 .6257 16 44 .7578 .6525 16 46 .8011 .5986 14 46 .7797 .6262 14 46 .7574 .6530 14 48 .8007 .5990 12 48 .7793 .62(56 12 48 .7570 .6534 12 50 .8004 .5995 10 50 .7790 .6271 10 50 .7566 .6539 10 52 .8000 .6000 8- 52 .7786 .6275 8 52 .7562 .6543 8 54 .7997 .6004 6 54 .7782 .6280 6 54 .7559 .6547 6 56 .7993 .6009 4 56 .7779 .6284 4 56 .7555 .6552 4 58 .7990 .6014 2 58 .7775 .6289 2 58 .7551 .6556 2 370' .7986 .6018 530' 90 ' .7771 .6293 5iq) n0' .7547 .6561 490' 2 .7983 .6023 58 2 .7768 .6298 A58 2 .7543 .6565 58 4 .7979 .6027 56 4 .7764 .6302 56 4 .7539 .6569 56 6 .7976 .6032 54 6 .7760 .6307 54 6 .7536 .6574 54 8 .7972 .6037 52 8 .7757 .6311 52 8 .7532 .6578 52 10 .7969 .6041 50 10 .7753 .6316 50 10 .7528 .6583 50 12 .7965 .6046 48 12 .7749 .6320 48 12 .7524 .65H7 48 14 .7962 .6051 46 14 .7746 .6325 46 14 .7520 .6591 46 16 .7958 .6055 44 16 .7742 .6329 44 16 .7516 .6596 44 18 .7955 .6060 42 18 .7738 .6334 42 18 .7513 .6600 42 20 .7951 .6065 40 '20 .7735 .6338 40 20 .7509 .6604 40 22 .7948 .6069 38 22 .7731 .6343 38 22 .7505 .6609 38 24 .7944 .6074 36 24 .7727 .6347 36 24 .7501 .6613 36 26 .7941 tows 34 26 .7724 .6352 34 26 .7497 .6617 34 28 .7937 .6083 32 28 .7720 .6356 32 28 .74!)3 .6622 32 30 .7934 .6088 30 30 .7716 .6361 30 30 .7490 .6626 30 32 .7930 .6092 28 32 .7713 .6365 28 32 .7486 .6631 28 34 .7926 .6097 26 34 .7709 .6370 26 34 .7482 .6635 26 36 .7923 .6101 24 36 .7705 .6374 24 36 .7478 .6639 24 38 .7919 .6106 22 38 .7701 .6379 22 38 .7474 .6644 22 40 .7916 .6111 20 40 .7698 .6383 20 40 .7470 .6648 20 42 .7912 .6115 18 42 .7694 .6388 18 42 .7466 .6652 18 44 .7909 .6120 16 44 .7690 .6392 16 44 .7463 .6657 16 46 .7905 .6124 14 46 .7687 .6397 14 46 .7459 .6661 14 48 .7902 .6129 12 48 .7683 .6401 12 48 .7455 .6665 12 50 .7898 .6134 10 50 .7679 .6406 10 50 .7451 .6670 10 52 .7894 .6138 8 52 .7675 .6410 8 52 .7447 .6674 8 54 .7891 .6143 6 54 .7672 .6414 6 54 .7443 .6678 6 56 .7887 .6147 4 56 .7668 .6419 4 56 .7439 .66b3 4 58 .7884 .6152 2 58 .7664 .6423 2 58 .7435 .6687 2 80' .7880 .6157 520' 00' .7660 .6428 500' 20' .7431 .6691 480' Dep. Lat. Dep. Lat. Dep. Lat. or or or or E. W. N. S. E. W. N. S. E. W. N. S. TRAVERSE TABLE. Traverse Table for a Distance = I. (CONCLUDED.) Lat. Dep. Lat. Dep. Lat. Dep. or or or or or or N. S. E. W. N.S. K. W. N.S. E.W. 120 2 .7431 .7428 .6691 .6696 480' 58 W0' 2 .7314 .7310 .6820 .6824 470' 58 440' 2 .7193 .7189 .6947 .6951 460' 58 4 .7424 .6700 56 4 .7306 .6828 56 4 .7185 .6955 56 6 .7420 .6704 54 6 .7302 .6833 54 6 .7181 .6959 54 8 .7416 .6709 52 8 .7298 .6837 52 8 .7177 .6963 52 10 .7412 .6713 50 10 .7294 .6841 50 .7173 .6967 59 12 .7408 .6717 48 12 .7290 .6845 48 2 .7169 .6972 48 14 .7404 .6722 46 14 .7286 .6850 46 4 .7165 .6976 46 16 .7400 .6726 44 16 .7282 .6854 44 6 .7161 .6980 44 18 .7396 .6730 42 18 .7278 .6858 42 8 .7157 .6984 42 20 .7392 .6734 40 20 .7274 .6862 40 20 .7153 .6988 40 22 .7388 .6739 38 22 .7270 .6867 38 22 .7149 .6992 38 24 .7385 .6743 36 24 .7206 .6871 36 24 .7145 .6997 36 26 .7381 .6747 34 26 .7262 .6875 34 26 .7141 .7001 34 28 .7377 .6752 32 28 .7258 .6879 32 28 .7137 .7005 32 30 .7373 .6756 30 30 .7254 .6884 30 30 .7133 .7009 30 32 .73(59 .6760 28 32 .7250 .6888 28 32 .7128 .7013 28 34 .7365 .6764 26 34 .7246 .6892 26 34 .7124 .7017 26 36 .7361 .6769 24 36 .7242 .6896 24 36 .7120 .7021 24 38 .7357 .6773 22 38 .7238 .6900 22 38 .7116 .7026 22 40 .7.353 .6777 20 40 .7234 .6905 20 40 .7112 .7030 20 42 .7349 .6782 18 42 .7230 .6909 18 42 .7108 .7034 18 44 .7345 .6786 16 44 .7226 .6913 16 44 .7104 .7038 16 46 .7341 .6790 14 46 .7222 .6917 14 46 .7100 .7042 14 48 .7337 .6794 12 48 .7218 .6921 12 48 ,7096 .7046 12 50 .7333 .6799 10 50 .7214 .6926 10 50 .7092 .7050 10 52 .7329 .6803 8 52 .7210 .6930 8 52 .7088 .7055 8 54 .7325 .6807 6 54 .7206 .6934 6 54 .7083 .7059 6 56 .7321 .6811 4 56 .7201 .6938 4 56 .7079 .7063 4 58 .7318 .6816 2 58 .7197 .6942 2 58 .7075 .7067 2 t30' .7314 .6820 470' 440' .7193 .6947 460' 450' .7071 .7071 450' Dep. Lat. Dep. Lat. Dep. Lat. or or or or E. W. N.S. E. W. N.S. E. W. N.S. When the anffle exceeds 45, the lats and deps are read upward from the bottom. Rent. Since these lats and deps are for a dist 1, we may proceed as follows for greater dists. Thus, let the dist be 856.1. Add together 800 times, 50 times, 6 times, and ^ time the corresi>ond- ing lats and deps of the table. Ex. What is the lat and dep for 856.1 feet ; the angle being 43 ? Here for 43 we have from the table, lat .7314; dep .6820. Hence, .7314 X 800 = 585.12; and .6820 X 800 =. 545.60 .7314 X 50 r= 36.57 ; and .6820 X 50 = 34.10 .7314 X 6= 4.39 ; and .6820 X 6= 4.09 .7314 X .1 = .07 ; and .6820 X -1 = .07 Lat 626. 15 Dep 583.86 These multiplications may be made mentally. Or we may, with a little more trouble, mult the lat and dep of the table by the given dist. Thus, .7314 X 856.1 = 626.15 lat ; and .6820 X 856.1 = 583.86 dep.* jasmuch as the engineer but rarely needs a traverse table, we have thought it best to give a st one, rather than the common one for y degrees. The first involves more trouble in using it ; ie last is entirely unfit for other than the rude calculations for common surveying with compass courses taken to the nearest y degree. To divide a scale of one mile into feet, first cut off one-sixth of it; then divide the remainder into four equal parts. Each of these parts will be 1100 feet. * Inasmuch correc but the courses 90 LAND SURVEYING. LAND SUKVEYING, IN surveying a tract ef ground, the sides which com* pose its outline are desig. nated by numbers in th order in which they occur. That end of each side whicb first presents itself in the course of the survey, may be called its near end ; aud the other its far end. The num- ber of each side is placed at iti far end. Thus, in Fig 1, the survey being supposed to commence at the corner 6, and to follow the direction is 6, 1 ; and its number is placed at its far end at 1 ; and so of the rest. Let N S be a meridian line, that is, a north and south line; and E W an east and west line. Then in any side which runs northwardly, whether due north, or northeast, as side 2; or northwest, as sides 5 and 1, the dist in a due north direction between its near end and its far end, is called its northing ; thus, a 1 is the northing of side 1 ; 1 ft the northing of side 2; 4 c of side 5. In like manner, if any side runs in a southwardly direction, whether due south, or southeastwardly, as side 3 ; or south westwardly, as sides 4 and 6, the corre- sponding dist in a due south direction between its near end and its far end, is called its southing ; thus, d 3 is the southing of side 3 ; 3 e of side 4 ; / 6 of side 6. Both northings and southings are included in the general term Difference of Latitude of a side ; or more commonly, but erroneously, its latitude. The dist due east, or due west, between the near and the far end of any side, is in like manner called the easting, or westing of that side, as the case may be ; thus, 6 a is the westing of side 1 ; 5/ of side 6 ; c5 of side 5 ; e 4 of side 4 ; and b 2 is the easting of side 2 ; 2 rf of side 3. Both eastings and westings are included in the general term Departure of a side ; implying that the side departs so far from a north or south direction. We may employ the directions (or courses, or bearings, as they are usually called) of sides, as verbs; and say that a side norths, wests, southeasts, &c. We shall call the northings, southings, Ac, the Ns, Ss, Es, and Ws; the latitudes, lats ; and the departures, deps. The preceding Traverse Table consists of the lats, (or Ns and Ss ;) and the deps, (or Es and Ws,) corresponding to diff angles or courses, for a side whose length is 1 ; therefore, to obtain the actual lat and dep of any given side, those taken from the table must be mult by the length of the side. Beyond 44, the lats and deps of this table must be read upward from the bottom of the page. The angles in the table are those whic* the course or bearing of any side would make with a meridian line drawn through either end of said side ; but it is self-evident that what would be N from one end, would be S from the other ; and so of E and W ; in other words, the angle is the same at both ends ; but the direction is reversed. Perfect accuracy is unattainable in any operation involving the measurements of angles and dists. That work is accurate enough, which cannot be made more so without an expenditure more than com- mensurate with the object to be gained. The writer conceives that the accuracy essential to constitute practically fair surveying is purely a matter of dollars and cents. In the purchase and sale of tracts of land, such as farms, &c, an uncertainty of about 1 part in 200 respecting the content, and consequently respecting the price, probably never prevents a transfer ; and on this principle we assume that a survey which proves itself within that limit, may ordinarily be regarded as accurate enough. There is no great difficulty in attaining this limit, which, if exceeded, is the result of bad work. Many circum- stances combine to render trifling errors absolutely unavoidable : * they always become apparent when we come to work out the field notes ; and since the map or plot of the survev, and the calcula- tions for ascertaining the content, should be consistent within themselves, we do what is usually called correcting the errors, but what in fact is simply humoring them in. no matter how scientific the pro- cess may appear. We distribute them all around the survey. Two methods are used for this purpose, both based upon precisely the same principle; one of them mechanical, by means of drawing; the other more exact, but much more troublesome, by calculation. We shall describe both : but will state now, that by proportioning the scale of the plot in the following manner, the mechanical method becomes, in the hands of a correct draftsman, sufficiently exact for all ordinary purposes. Add all the sides in feet together ; and div the sum by their number, for the average length. Div this average by 8; the quot will be the proper scale in feet per inch. In other words, take about 8 ins to represent an average side.t We shall take it for granted, that an engineer does not consider it accurate work to * A 100 ft measuring-chain may vary its length 5 feet per mile, between winter and summer, by mere change of temperature ; and by this alone we shall make a difference of about 1-jl acres in a lot 1 mile square, which contains 640 acres. Even this error amounts to 1 acre in 533. Not one farmer in a hundred would dream of pavine for a scrupulouslv correct survey aitd plot of his property. t It will seldom happen that precisely this quotient will be adopted for a scale. For instance, if the quot should be 738 feet, or 83 feet, per inch, we should adopt the most convenient near number, if smaller the better, as 700, or 80; or rather more than 8 inches to a side. With a scale so proportioned, and with good drawing instruments, the error in protracting (excluding of course errors of the field work) will rarely exceed about ^JJ-Q part of the periphery of the plot; and the area may be found mechanically by dividing the plot into triangles, within ^-i- ff P art * the truth. This remark applies particularly to such plots as may be protracted, and computed within a period so short as not to allow the paper to contract or expand appreciably by atmospheric changes. A larger scale will insure proportionally greater accuracy. The young assistant should practise plotting from perfectly accurate data ; as, for instance, from the example given in the table, p. 97, or, LAND SURVEYING. 91 measure his angles to the nearest quarter of a degree, which is the usual practice among land-survey ors. They can, by means of the engineer's transit, now in universal use on our public works, be readily measured within a miuute or two ; and being thus much more accurate than the compass courses, (which cannot be read off so closely, and which are moreover subject to many sources of error,) they serve to correct the latter in the office. The noting of the courses, however, should not be confined to the nearest quarters of a degree, but should be read as closely as the observer can guess at the minutes. The back courses also should be taken at every corner, as an additional check, and for the detection of local attraction. It is well in taking the com- pass bearings, to adopt as a rule, always to point the north of the compass- box toward the object whose bearing is to be taken, and to read off from the north end of the needle. A person who uses indifferently the N and the S of the box, and of the needle, will be very liable to make mistakes. It is best to measure the least angle (shown by dotted arcs, Fig 2,) at the corners ; whether it be exterior, as that at corner 5; or interior, as all the others ; because it is al- ways less than 180 ; so that there is less danger of reading it off incor- rectly, than if it exceeded 180; taking it for grant- ed that the transit instrument is graduated from the same zero to 180 each way ; if it is graduated from zero to 360 the precaution is useless. When the small angle is exterior, subtract it from 360 for the interior one. Supposing the field work to be finished, and that we require a plot from which the contents may be obtained mechanically, by dividing it into triangles, (the bases and heights of which may be measured by scale, and their areas calculated one by one,) a protraction of it may be made at once from the field notes, either by using the angles, or by first correcting the bearings by means of the angles, and then using them. The last is the best, because in the first the protractor must be moved to each angle ; whereas in the last it will remain stationary while all the bearings are being pricked off. Every movement of it increases the liability to errors. The manner of correcting the bearings is explained on the next page. In either case the protracted plot will certainly not close precisely ; not only in consequence of errors in the field work, but also in the protracting itself. Thus the last side. No 6, Fig 2, instead of closing in at corner 6, will end somewhere else, say, for instance, at ; the dist t 6 being the closing error, which, however, as represented in Fig 2, in more than ten times as great, proportionally to the size of the survey, as would be allowable in practice. Now to humor-in this error, rule through every corner a short line parallel to t 6; and, in all cases, in the direction from t (wherever it may be) to the starting point 6. Add all the sides together ; and measure t 6 by the scale of the plot, then begin- ning at corner 1, at the far end of side 1, say, as the Sum of all . Total closing the sides error 1 6 Lay off this error from 1 to a. Then at corner 2, say, as the Sum of all . Total closing . . Sum of the sides error t 6 sides 1 and 2 Which error lay off from 2 to b ; and so at each of the corners; always using, ns the third term, the sum of the sides between the starting point and the given corner. Finally, join the poiuts a, b, c, d, e, 6; and the plot is finished. The correction has evidently changed the length of every side; lengthening some and shortening others. It has also changed the angles. The new lengths and angles may with tolerable accuracy be found by means of the scale and protractor ; and be marked on the plot instead of the old ones. from those to be found in books on surveying. This is the only way in which he can learn what is meant by accurate work. His semicircular'protractor should be about 9 to 12 ins in diam and gradu- ated to 10 min. His straight edge and triangle should be of metal: w.e prefer German silver, which does not rust as steel does; and thev should be made with scrupulous accuracy by a skilful instru- ment maker. A very fine needle, with a sealing-wax head, should be used for pricking off dists and angles ; it must be held vertically ; and the eye of the draftsman must be directly over it. The lead pencil should be hard (Faber's No. 4 is pood for protracting), and must be kept to a sharp point by rubbing on a fine file, after using a knife for removing the wood. The scale should be at least as long as the longest side of the plot, and should be made at the edge of a strip of the same paper as the plot is drawn on. This will obviate to a considerable extent, errors arising from contraction and expan- sion. Unfortunately, a sheet of paper does not contract and expand in the same proportion length- wise and crosswise, thus preventing the paper scale from being a perfect corrective. In plots of com- mon farm surveys, &c, however, the errors from this source may be neglected. For such plots as may be protracted, divided, and computed within a time loo short to i admit of appreciable change, the ordi- nary scales of wood, ivorv or metal mav be used ; but satisfactory accuracv cannot be obtained with them on plots requiring several days, if the air be meanwhile alternately moist and dry. or subject to considerable variations in tempera'ture. What is called parchment paper is worse in this respect thau good ordinary drawing-_paper. With the foregoing precautions we may work from a drawing, with as much accuracy as is usually attaimed in the field work. Error for side 1. Error for side 2. 92 LAND SURVEYING. When the plot has many sides, this calculating the error for each of them becomes tedious ; and since, in a well-performed survey and protraction, the entire error will be but a very small quantity, (it should not exceed about ^^ part of the periphery,) it may usually be divided among the sides by merely placing about 34, %, and % of it at corners about J4, *4, and % way around the plot; and at n ; . intermediate corners proper- -, "> 1 " 1 u tion it by eye. Or calculation I J? 1 " "I may be avoided entirely by J U drawing a line a 6 of a length equal to the united lengths of all the sides ; dividing it into distances a, 1; 1, 2 ; &c, equal to the respective sides. Make b c equal to the entire closing error ; join a c; and draw 1,1'; 2, 2' , &c, which will give the error at each corner. When the plot is thus completed, it may be divided by fine pencil Hues into triangles, whose bases and heights may be measured by the scale, in order to compute the contents. With care in both the survey and the drawing, the error should not exceed about -R-^TT P ar t of the true area. At least two distinct sets of triangles should be drawn and computed, as a guard against mistakes ; and if the two sets differ in calculated contents more than about -^ ^-^ part, they have not been as carefully prepared as they should have been. The closing error due to imperfect field- work, may be accurately calculated, as we shall show, and laid down on the paper before beginning the plot ; thus furnishing a perfect test of the accuracy of the protraction work, which, if correctly done, will not close at the point of beginning, but at the point which indicates the error. But this calculation of the error, by a little additional trouble, furnishes data also for dividing it by calculation among the diff sides : besides the means of drawing the plot correctly at once, without the use of a protractor ; thus ena- bling us to make the subsequent measurements and computations of the triangles with more cer- tainty. We shall now describe this process, but would recommend that even when it is employed, and especially in complicated surveys, a rough plot should first be made and corrected, by the first of the two mechanical methods already alluded to. It will prove to be of great service in us'ing the method by calculation, inasmuch as it furnishes an eye check to vexatious mistakes which are otherwise apt to occur ; for, although the principles involved are extremely simple, and easily remembered when once understood, yet the continual changes in the directions of the sides will, without great care, cause us to use Ns instead of Ss ; Es instead of Ws, &c. We suppose, then, that such a rough plot has been prepared, and that the angles, bearings, and distances, as taken from the field book, are figured upon it in lead pencil. Add together the interior angles formed at all the corners : call their sum a. Mult the number of sides by 180 ; from the prod subtract 360 : if the remainder is equal to the sum o, it is a proof that the angles have been correctly measured.* This, however, will rarely if ever occur ; there will always be some discrepancy ; but if the field work has been performed with moderate care, this will not exceed about two min for each angle. In this case div it in equal parts among all the angles, adding or subtracting, as the case may be, unless it amounts to less than a min to each angle, when it may be entirely disregarded in common farm surveys. The corrected angles may then be marked on the plot in ink, and the pencilled figures erased. We will suppose the corrected ones to be as shown in Fig 3. Next, by means of these corrected angles, correct the bearings also, thus, Fig 3 ; Select some side (the longer the better) from the two ends of which the bearing and the reverse bearing agreed ; thus bearing fluenced was probably not influ by local attraction. Let side 2'be the one so selected ; as- sume its bearing, N 75 32' E, as taken on the ground, to be correct; through either end of it, as at its far end 2, draw the short meridian line ; par- allel to which draw others through every corner. Now, having the bearing of side 2, N 75 32' E. and requiring that of side 3, it is plain that the reverse bearing from cor- ner 2 is S 75 32' W ; and that therefore the angle 1, 2, m, is 75 32'. Therefore, if we take 75 32' from the entire corrected angle 1 , 2, 3, or 144 57', the rem 69 25' will be the angle m 23 ; consequently the bearing of side 3 must be 8 69 25' E. For finding the bearing of side 4, we now have the angle 23 a of the reverse bearing of Bide 3, also equal to 69 25' ; aud if we add this to the entire corrected angle 234, or to 32 , we ha.\ ' ' ' , the angle a 34=69 25' 4- 69 32' = 138 57' ; which taken from 180, leaves the angle the bearing of side 4 must be S 41 3' W. For the bearing of side owe n ' , . 4: * Because in every straight-lined figure the sum of all its interior angles is equal to twice as many right angles as the figure has sides, minus 4 right angles, or 360.. LAND SURVEYING. careful observation Is necessary to see ho-w the several angles are to be employed at each corner. Rules are sometime* given for this purpose, but unless frequently used, they are soon forgotten. The plot mechanically prepared obviates the necessity for such rules, inasmuch as the principle of proceeding thereby becomes merely a matter of sight, and tends greatly to prevent error from using the wrong bearings ; while the protractor will at once detect any serious mistakes as to the angles, and thus prevent their being carried further along. After having obtained all the corrected bearings, they may be figured on the plot instead of those taken in the field. They will, however, require a still further correction after a while, since they will be affected by the adjustment of the closing error. We now proceed to calculate the closing error tf> of Fig 2, which is done on the principle that in a correct survey the northings will be equal to the southings, and the- eastings to the westings. Pre- pare a table of 7 columns, as below, and in the first 3 cols place the numbers of the sides, and their cor- rected courses ; also the dists or. lengths of the sides, as measured on the rough plot, if such a one has been prepared ; but if not, then as measured on the ground. Let them be aa follows : Side. Bearing. Dist. Ft. Latitudes. Departures. N. S. E. W. 1 2 3 4 5 tt N 16 40' W N 75 32' E S 69 25' E S 41 3' W N 79 40' W S 53 30' W 1060 1202 1110 850 802 705 1015.5 300.3 143.9 390.2 641. 419.3 1163.9 1039.2 304. 558.2 789. 566.7 1459.7 1450.5 1450.5 Error in Lat. 2203.1 Error in Dep. 2217.9 2203.1 9.2 14.8 Now find the N, S, E. W, of the several sides, and place them in the corresponding four columns, thus : By means of the Traverse Table find out the lat and dep for the angle of each course. Mult each of them by the length of the side ; and place the prod in the corresponding col of N, S, E, W. Thus, for side 1, which is 1060 feet long, the latitude from the traverse table for 16 40' is .9580; and the departure is .2868: and .5hO X 1060 = 1015.5 lat; which, since the side norths, we put in the N col. Again, .2868 X 1060 = 30* dep; which, since the side wests, we put in the W col. Proceed thus with all. Add up the four cols ; find the diff between the N and S cols ; and also between the E and W ones. In this instance we find that the Ns are 9.2 feet greater than the Ss ; and that the Ws are 14.8 ft greater than the Es ; in other words, there is a closing error which would cause a correct protraction of our first three cols, to terminate 9.2 feet too far north of the starting point ; and 14.8 feet too far west of it. So that by placing this error upon the paper before beginning to protract, we should have a test for the accuracy of the protracting work ; but, as before remarked, a little more trouble will now enable us to div the error proportionally among all the Ns, Ss, Es, and Ws, and thereby give as data for drawing the plot correctly at once, without using a protractor at all. To divide the errors, prepare a table precisely the same as the foregoing, except that the hor spaces are farther apart; and that the addings-up of the old N, S, E, W columns are omitted. The addition* here noticed are made subsequently. The new table is on the next page. REMARK. The bearing and the reverse bearing: from the two ends of a line will not read precisely the same angle ; and the difference varies with the latitude and with the length of the line, but not in the same proportion with either. It is, however, generally too small to be detected by the needle, being, according to Gummere, only three quarters of a minute in a line one mile long in lat 40. In higher lats it is more, and in lower ones less. It is caused by the fact that meridians or north and south lines are not truly parallel to each other; but would if extended meet at the poles. Hence the only bearing* that can be run in a straight line, with strict accuracy, is a true N and S one ; except on the very equator, where alone a due E and W one will also be straight. But a true curved JE and W line may be found anywhere with sufficient accuracy for the surveyor's purposes thus. Having first by means of the N star (p 99) or otherwise got a true N and S bearing at the starting point, lay off from it 90, for a true E and W bearing at that point. This E and W bearing will be tangent to the true E and W curve. Run this tangent carefully : aud at intervals (say at the end of each mile) lay oft' from it (towards the N if in N lat, or vice versa) an offset whose length in feet is equal to the proper one from the following table, multiplied by the square of the distance in miles from the starting point. These offsets will mark points in the true K and W curve. 15 20 Latitude N or S. 25 30 35 40 45 Offsets in ft one mile from starting point. .058 .118 .179 .243 .311 .385 .467 .559 .667 .795 .952 1.15 1.43 Or, any offset in ft = .6666 X Total Dist in miles* X Nat Tang of Lat. A rhtimb line is any one that crosses a meridian obliquely, that is, is neither due N and S, nor E and W. 94 LAND SURVEYING. Side. Bearing. Dist. Ft. Latitudes. Departures. N. S. E. W. 1 2 3 4 5 N 16 40' W N 75 32' E S G9 25' E S 41 3' W N 79 4(K W S 53 30' W 1060 1202 1110 850 802 705 1015.5 1.7 304.0 2.7 ... 301.3 1013 8 300.3 1.9 1163.9 3.1 558.2 2.2 556 o 298.4 143.9 1.3 ... 1167.0 1039.2 2.9 390.2 1.8 392 ... 641.0 1.3 642.3... ... 1042.1 789.0 2.1 1426 ... 7869 419.3 1.1 566.7 1.8 420.4... 564.9 5729 Sum of Sides. 1454.8 Cor'd Ns. 1454.7 Cor'd Ss. 2209.1 Cor'd Es. 2209.1 Cor'd Ws. Now we have already found by the old table that the Ns and the Wa are too long ; consequently they must be shortened ; while the Ss, and Es, must be lengthened; all in the following proportions: As the Sura of all . Any given .. Total err of . Err of lat, or dep, the sides side lat or dep of given side. Thus, commencing with the lat of side 1, we have, as Sum of all the sides. . Side 1. . . Total lat err. . Lat err of side 1. 5729 1060 9.2 1.7 Now as the lat of side 1 is north, it must be shortened ; hence it becomes = 1015.5 1.7 = 1013.8, as figured out in the new table. Again we have for the departure of side 1, Sum of all the sides. . Sidel. .. Total dep err. . Dep err of side 1. 5729 1060 14.8 2.7 Now as the dep of side 1 is west, it must be shortened ; hence it becomes 304 2.7 = 301.3, as figured out in the uew table. ., Proceeding thus with each N side, we obtain all the corrected lats and deps as shown in the new table ; where they are con- nected with their respective sides by dotted lines; but in practice it is better to cross out the original ones when the cal- culation is finished and proved. If we now add up the 4 cols of corrected N, S, E, W,we find that the Ns = the Ss ; and the Es = theWs; thus proving that the work is right. There is, it is true, a discrepancy of .1 of a ft between the Ns. and the Ss ; but this is owing to our carrying out the corrections to only one decimal place ; and is too small to be regarded. Discrepancies of 3 or 4 tenths of a foot will sometimes occur from this otouse ; but may be neglected. The corrected lats and depa must evidently change the bearing and distance of every irvey by means of the corrected Ftp" 4 ide ; but without knowing either of these, we can now plot the LAND SURVEYING. 95 hits and deps alone. The principle is self-evident, explaining itself. First draw a meridian line N S, Fig 4 ; and upon it fix on a point 1, to represent the extreme west* corner of the survey. Then from the point 1, prick off by scale, northward, the dist 1, 2 =the corrected northing 298.4 of side 2, taken from the last table ; from 2' southward prick off the dist 2', 3', the corrected south- iug 392 of side 3; from 3' southward prick off 3', 4', = southing 642.3 of side 4; from 4' northward prick off 4', 5' = northing 142.6 of side 5; from 5' prick off southward 5', 6' = southing of side 6.t Then from the points 2', 3', 4', 5', 6', draw indefinite lines due eastward, or at right angles to the meridian line. Make by scale, 2', 2 = corrected departure of side 2; and join 1, 2. Make 3', 3 = dep of side 2-j-depof side 3 ; and join 2, 3; make 4', 4 = 3', 3 dep of side 4; and join 3, 4; make 5', 5 = 4', 4 dep of side 5; andjoiu4,5; make 6', 6=5', 5 dep of side 6; and join 5, 6} Finally join 6. 1 ; and the plot is complete. If scrupulous accuracy is not required, the contents may be found by tne mechanical method of triangles; the bearings, by the protractor; and the lengths of the sides, by the scale ; all with an approximation sufficient for ordinary purposes; and perhaps quite as close as by the method by calculation, when, as is customary, the bearings are taken only to the nearest quarter of a degree. We have already said that with a scale of feet per inchr= the error of area need not exceed the YO^th part. But if it is required to calculate the area of the corrected survey with rigorous exactness, it may be done on the following principle, (see Fig 5.) If a ti meridian line N S be sup- N posed to be drawn through the extreme west corner 1 of a survey; and lines (called middle distances) drawn (as the dotted ones in the Fig) at right angles to said me- ridian, from the center of 1 each side of the survey; then if each of th middle dists of such sides as have northings, be mult by the corrected northing of its cor- responding side ; and if each of the middle dists of such sides as have southings, be mult by the corrected south- ing of its corresponding side ; if we add all the north prods into one sum ; and all the south prods into another f these sums from the great- 5st, the rein will be the area Fid 5 * The extreme east corner would answer as well, with a slight change in the subsequent oper- ations, as will become evident. t Instead of pricking off these northings and southings in succession, from each other, it will be more correct in practice to prepare first a table showing how far each of the points 2',3', &c, is north or south from 1. This being done, the points can be pricked off north or south from 1, without mov- ing the tcale each time ; and of course with greater accuracy. Such a table is readily formed. Rule it as below; and in the first three columns place the numbers of the sides (starting with side 2 from point 1 ;) and their respective corrected northings and southings. The formation of the 4th and 5th cols by means of the 3d and 4th ones, explains itself. Its accuracy is proved by the final result being 0. Side. N. lat. S. lat. Dist N or S f N. rom Point 1. S. 2 3 4 5 6 1 298.4 142.6 1013.8 392. 612.3 420.4 298.4 000.0 93.6 735.9 593.3 1013.7 000.0 t A similar table should be prepared beforehand for the dists of the points 2, 3, 4, &c, east from the meridian line. It is done in the same manner, but requires one col less, as all the dists are on the ame side of the mer line. Thus, starting from point 1, with side 2 : Side. E. dep. W. dep. Dist east from meridian line. 2 3 4 5 6 1 1167.0 1042.1 556.0 786.9 564.9 301.3 1167.0 2209.1 1653.1 866.2 301.3 000.0 ' This work likewise proves itself by the final result being 0. 96 LAND SURVEYING. of the survey.* The corrected northings and southings we have already found ; as also the eastings and westings. The middle dists are found by means of the latter, by employing their halves ; adding half eastings, and subtracting half westings. Thus it is evident that the middle dist 2' of side 2, is equal to half the easting of side 2. To this add the other half easting of side 2, aud a half easting of side 3; and the sum is plainly equal to the middle dist 3' of side 3. To this add the other half easting of side 3, and subtract a- half westing of side 4, for the middle di^t 4' of side 4. From this subtract the other half westing of side 4, and a half westing of side 5, for the middle dist 5' of .side 5; and so on. The actual calculation may be made thus : Half easting of side 2 = ^- = 583.5 E = mid dist of side 2. 2 583.5 E 1042.1 1167.0 E Half easting of side 3 = = 521.0 E 2 1688.0 E = mid dist of side 3. 521.0 E 556 2209.0 E Half westing of side 4 = = 278.0 W 1931.0 E = mid dist of side 4. 278.0 W 786.9 1653.0 E Half westing of side 5 = = 393.5 W 2 1259.5 E = mid dist of side 5. 393.5 W 564.9 866.0 E Half westing of side 6 = = 282.4 W 2 583.6 E = mid dist of side 6. 282.4 W _.*- ' 301.3 301.2 E Half westing of side 1 = - = 150.6 W 150.6 E = mid dist of side 1. The work always proves itself by the last two results being equal. Next make a table like the following, in the first 4 cols of which place the numbers of the sidea, the middle dists, the northings, and southings. Mult each middle dist by its corresponding northing or southing, and place the products in their proper col. Add up each col ; subtract the least from the Side. Middle dist. Northing. Southing. North prod. South prod. 1 2 3 4 5 6 150.6 583.5 1688 1931 1259.5 583.6 1013.8 298.4 142.6 392 642.3 420.4 152678 174116 179605 661696 1240281 245345 506399 435 2147322 506399 30)1640923(37.67 Acres. * Proof. To illustrate the principle upon which this rule is based, let ab, be, and c a, Fig 6, represent in order the 3 sides of the triangular plot of a survey, with a meridian line d/drawn through the extreme westcor- ner, a. Let lines b d and c/ be drawn from each corner, perp to the meridian line ; also from the middle of each side draw lines we, mn, so, also perp to meridian ; and representing the middle dists of the sides. Then since the sides are regarded in the order ah, be, ca, it is plain that a d represents the northing of the side aft; fa the northing of ca; and df the southing of & c. Now if we mult the northing ad of the side a b, by its mid dist ew, the prod is the area of the triangle abd. In like manner the northing fa of the side ca, mult by its mid dist s o, gives the area of the triangle a ef. Again, the southing df of the side be, mult by its mid dist mn, gives the area of the entire fig dbcfd. If from this area we subtract the areas of the two triangles ab d, and a cf, the rem is evidently the area of the plot a b c. 80 with any other plot, however complicated. lAND SURVEYING. 97 rrcatest. The rem will be the area of the survey in sq ft ; which, div by 43560, (the number of sq ft ID an acre,) will be the area in acres ; in this instance, 37.67 ac. It now remains only to calculate the corrected bearings and lengths of the sides of the survey, all o"f which are necessarily changed by the adoption of the corrected lats and deps. To find the bearing of any side, div its departure (E or W) by its lat (N or S) ; in the table of nat tang, tiud the quot ; the angle opposite it is the reqd angle of bearing. Thus, for the course of side 2, we have - ^ = .2972 nat tang ; opposite which in the table is the reqd angle, 16 33' ; the bearing, therefore, is N 16 3 33' W. Again : for the dist or length of any side, from the table of nat secants* take the sec opposite to the angle of the corrected bearing ; mult it by the corrected lat (N or fc>) of the side. Thus, for the dist f side 1, we find opposite 16 33', the sec 1.0432. And Sec. Lat. 1.0432 X 1013.8 = 1057.6 the reqd dist. Thfl following table contains all the corrections of the foregoing survey ; consequently, if the bear. Side. Bearing. Dist. Ft. 1 2 3 4 5 6 N 16 33' W N 76 39' E S 69 23' E S 40 53' W N 7944'W S 53 21' W 1057.6 1204.0 1113.3 849.6 800.1 704.3 * , *...* ings and dists are correctly plotted, they will close perfectly. The young assistant is advised to practise doing this, as well as dividing the plot into triangles* and computing the content. In this manner he will soon learn what degree of care is necessary to insure accurate results. Under the heads Mensuration, Geometry, and Trigonometry will be found much pertaining to land surveying. See Remark after Parallelograms. The following hints may often be of service. 1st. Avoid taking bearings and dists along a aircuitous bound- ary line like a b c, Fig 7 ; but run the straight line a c; and at right angles to it, measure off- sets to the crooked line. 2d. Wishing to survey a straight line from a to c, but being una- ble to direct the instrument precisely toward c, on account of intervening woods, or other obstacles ; first run a trial line, as a m, as nearly in the proper direction as can be guessed at. Measure m c, and say, as a m is to w c, so Is 100 ft to ? Lay off a o equal to 100 ft, and o * equal to ? ; and run the final Hne a * c. Or. if m c is quite small, calculate offsets like o g for every 100 ft along a m, and thus avoid the necessity for running a second line. 8d. When c is visible from a, but the intervening ground difficult to measure along, on account of marshes, &c. extend the side y a to good ground at t : then, making the angle y t d equal to y a c, run the line t n to that point d at which the angle n d c is found by trial to be equal to the angle a t d. It will rarely be necessary to make more than one trial for this point d; for, suppose it to be made at x, see where it strikes a c at t; measuret c, and continue from x, making x d =tc. 4th. In case of a very irregular piece of land, or a lake, Fig 8, surround it by straight lines. Survey these, and at right angles to them, measure offsets to the crooked boundary. 5th. Surveying a straight line from w toward y, Fig 9, n Obstacle, o, is met. To pass it, lay off a right angle wtu; measure any t u ; make t u v = 90 ; measure u v ; make u v i ~ 90 ; make v i ~ t u ; make viy = 90. Then is e t ~ v ; and iy u in the straight line. Or, with less trouble, at g make t g n ~ 60 ; measure any g a; make g a =r60 ; and a s ~ g a: make a s i 60. Then is g s r= g a or as; nnd i s, continued toward y, is in the straight line. 6th. Being between two objects, m and n, and wishing to place myself in range with them, I lay a straight rod c b on the ground, and point it to one of the objects m ; then going to the end c, I find that it does not poit)t to the other object. By successive trials, I find the position e d, in which it points to both objects, and consequently is in Vange with them. If no rod * Our table does not con tain nat secants; but the nat sec of any angle is readily found, thus : Di vide 1 by the nat fcosine of the angle. 7 LAND SURVEYING. is at hand, two stones will answer, or two chain-pins. A plumb-line (a pebble tied to a piece of thread) will add to the accuracy of ranging the rod, or stones, &c. THE FOl^OWING TABLE gives deductions or additions to be made every 10O ft as actually chained along sloping ground, iu order to reduce the sloping measurements to Horizontal ones. Even when it is so nearly level that the eye cannot detect the slope, an over-measurement of an inch or two in 100ft may readily occur, it is plain, that, if we measure all the undulations of the ground, we shall get greater totals than if we measure hor, as is supposed always to be done ; but since few surveyors pretend to measure hor until the slope becomes apparent to the eye, their lines are usually -too long by from one to two ins in 100 feet. To counteract this to some extent, chains are frequently made from one to two ins longer than 100 feet; and for ordinary purposes the precaution is a pood one. When greater accuracy is required the chainmen should be attended by a third person, with a rod and slope-level, for taking the inclinations of the ground. These deductions being made, the remain- der will be the actual hor dist. For example, in Fig 10^, each 100 f t a o measured np or down the slo|>e ae plainly corresponds to the shorter horizontal distance a c; the difference or deduction being c n. Taking a o as Rad. then a c is the cosine, and c n the versed sine of the angle e a n of the slope. Thore- fore a o multiplied by the nat. cosine of the ansrte e n n rives the reduced hor dist n r. ; which tak^O from ao gives tho deduction en of oar table. But If while chaining along the slope n we wish to drive stakes that shall correspond with hor dist* u n of 100 ft. it is evident that we must add c n to each 100 ft a o, as shown ;it x e ; and the stake mut be driven at e,. instead of at o. Observe that x e =. C n must be measured horizontally. When the ground is very sloping, alt this calcula-iou may be avoided where great accuracy is not required, by actually holding the chain horizontal, as nearly as can he judged by ey , and finding, by means of a plumb-line, where its raised end would strike the ground. A whole chain at a time cannot be measured in this way; but shorter distances must be taken as the ground requires; at times, on very steep ground, not more than 5 or 10 feet. See note, p 40. Table of Deductions or Additions to be made per 1OO feet, iu chaining over sloping ground. IX ORDER TO REDUCE THE INCLINED MEASUREMENTS TO HORIZONTAL ONKB See p 629 for another table. c /Slope in Deg. Deduct Feet. Rise in 100ft hor. Slope Deg. Deduct Feet. Rise in 100 ft hor. Slope Deg. Deduct Feet. Rise in 100 ft hor. Slope in Deg. Deduct Feet. Rise in . ido ft hor. H .001 .436 K .420 9.189 y 1.596 1808 H 3.521 27.26 9 .004 .873 i^ .460 - 9.629 H 1.675 18.53 H 3.637 27.73 H .009 1.309 H .503 10.07 H 1.755 18.99 3 4 3.754 28.20 i .015 1.746 6 .548 10.51 11 1.837 19.44 16 3.874 28.67 X .024 2.182 % .594 10.95 % 1.921 19.89 H 3.995 29.15 K .034 2.619 % .643 11.39 M 2.008 20.35 X 4.118 29.62 % .047 3.055 H .693 11.84 H 2.095 20.80 % 4.243 30.10 2 .061 3.492 7 .745 12.28 12 2.185 21.26 17 4.370 30.57 /4 .077 3.929 K .800 12.72 M 2.277 21.71 y* 4.498 31.05 y^ .095 4.366 N .856 13.17 *A 2.370 22.17 4.628 31.53 K .115 4.803 % .913 13.61 H 2.466 22.63 *4 4.7HO 32.01 .137 5.241 8 .973 14.05 13 2.563 23.01) 18 4.894 32.49 34 .161 5.678 y* 1.035 14.50 2.662 23 55 H' 5.030 32.98 i^ .187 6.116 y* 1.098 14.95 % 2.763 24.01 X 5.168 33.46 h .214 6.554 H 1.164 15.39 5i 2.86B 24.47 H 5.307 3395 4 .244 6.993 9 1.231 15.84 14 2.970 24.93 19 5.448 34.43 X .275 7.431 H 1 .300 16.29 y* 3.077 25.40 5.591 34.92 % .308 7.870 % 1.371 16.73 y* 3.185 25.86 X 5.736 35.41 H .343 8.309 X 1.444 17.18 H 3.295 26.33 % 5.882 35.90 5 .381 8.749 10 1.519 17.63 15 3 407 26.79 20 I- 6.031 36.40 Chain and Pins. Engineers have abandoned the Gunter's chain of 66 ft, div into 100 links of 7.92 ins in length ; and use one of 100 ft. with links 1 ft long ; and calculate areas in sq ft ; which, div by 43560, reduces them to acres and decimal parts, instead of roods and perches. Both the chain and the v chaiu pins. Fig. 11, hould be of good strong steel; and there should be a stout leather bag for carrying them. To bear ham- mering into hard ground, the pins may be of this shape and size, 11 or 12 ins long, H inch thick, % wide, head '2% wide, with a circular hole of 1 % diam. Each pin should have a strip of bright red flannel tied to its top, th.it it may be readily found among grass, &c. by the hind c'hainman. The length of the chain should be tested every few days; and the target-rod may be used for this purpose. While locating, it is well to have the chain one or two ins longer than 100 feet. Steel wire, No 11 or 12, is a good size for a 100 ft chain. This is scant H inch diam. 41 1 1 ' 1 LAND SURVEYING. 99 Nat Sines of Polar Dists of Polaris or X. Star. Year. 1880 1881 1882 Sine. Year. Sine. Year. 1886 1887 1888 Sine. | Year. Sine. Year. Sine. Year. Sine. .0232 .0231 .0230 1883 1884 1885 .0229 .0229 .0228 .0227 .0226 .0225 1889 1890 1891 .0224 .0223 .0222 1892 1893 1894 .0221 .0220 .0219 1895 1896 1897 .0218 .0217 .0216 Nat Secants of North Latitudes. Lat. Sec. Lat. Sec. Lat. Sec. Lat. Sec. Lat. Sec. Lat. Sec. 1.000 24 1.095 1^0 1.196 /f 1.296 3/0 1.433 52 1.624 2 1.001 72 1.099 IX 1.199 M 1.301 46 1.440 74 1.633 4 1.002 25 1.103 ax 1.203 1.305 IX 1.446 7? 1.643 5 1.004 IX 1.108 34 1.206 i 1.310 17 1.453 7/t 1.652 6 1.006 26 1.113 IX 1.210 IX 1.315 74 1.460 53 1.662 7 1.008 IX, 1.117 IX 1.213 &2 1.320 47 1.466 /4 1.671 8 1.010 27 1.122 74 1.217 41 1.325 IX. 1.473 72 1.681 9 1.013 X2 1.127 35 1.221 ix. 1.330 17 1.480 74 1.691 10 1.016 28 1.133 i 1.225 /I 1.335 74 1.487 54 1.701 11 1.019 IX 1.138 IX 1.228 74 1.340 48 1.495 IX 1.712 12 1.022 29 1.143 37 1.232 42 1.346 1.502 72 1.722 13 1.026 1.149 36 1.236 I/ 1.351 ll 1.509 74 1.733 14 1.031 30 1.155 IX 1.240 IX 1.356 M 1.517 55 1.743 15 1.035 IX 1.158 7^ 1.244 ?i 1.362 49 1.524 i 4 1.754 16 1.040 IX 1.161 xl 1.248 43 1.367 74 1.532 1.766 17 1.046 3X 1.164 37 1.252 74" 1.373 17 1.540 74 1.777 18 1.052 31 1.167 IX 1.256 1.379 % 1.548 56 1.788 19 1.058 1 4 1.170 IX 1.261 74 1.384 50 1.556 74 1.800 20 1.064 ix 1.173 74 1.265 44 1.390 iX^ 1.564 IX 1.812 21 1.071 74 1.176 38 1.269 17 1.396 YL 1.572 74 1.824 1.075 32 1.179 ix/ 1.273 1^ 1.402 74 1.581 57 1.836 1.079 ix_ 1.182 72 1.278 74 1.408 51 1.589 74 1.849 72 1.082 IX 1.186 74 1.282 45 1.414 Yi 1.598 17 1.861 23 1.086 74 1.189 39 1.287 IX 1.420 17 1.606 37 1.874 ^ 1.090 33 1.192 xi 1.291 8 1.427 1.615 58 1.887 To find a Meridian Line (a true North and South line) by means of the North Star. (Polaris.) The north star appears to describe a small circle, n n', &c, Fig 14, around the true north point, OP north pole, as a center. The rad of this circle is estimated bv the angle between the star and the pole, as measured from the earth ; and is called the polar diet of the star. This polar dist be- comes 19 yjy seconds, or very nearly ^ of a minute less every year. On Jan 1, 1880, it is, approxi- mately, 1 19 51". On the first of 1890, it will be about 1 16' 41", &c. When, in its revolution, the star is farthest castor west from the pole, as at n' or n", it is said to be at its greatest E or W elongation. Then its apparent motion for several min is nearly vert, and consequently affords the best opportunity for an observation in the simple manner here described. The arrows in Fig 14 show the direction in which the stars appear to move from east to west when the spectator faces the north. The latitude of the place must be known approximately. Taking it at the closest one in our forego- ing table, the error will not exceed half a min in lat 57, or one-quarter min in lat 40 ; and still less in lower lats. About 3 ft above ground fix firmly, perfectly level, and as nearly east and west as may be, a smooth narrow piece of board, about 3 ft long, to serve as a kind of table. Also prepare another piece a a, about a foot long ; and fasten toil, at right angles, a compass-sight, or astripof thin metal, with a straight slit, < shown by a black line in the fig.) about 6 ins long and JL inch wide. This piece of board is to be slid along the table, as the observer follows the motions of the star toward the east or west: looking at it through the vert slit. Plant a stout pole, about 20 ft long, firmly in the ground, with its top as nearly north as possible from the middle of the table. Its top should tq.12 lean 2 or 3 ft toward either the east or the west ; and a plumb- line must be suspended from its top, with a bob weighing one or two Ibs, -which may swing in a bucket of water plnced orf the ground. This is to prevent the line from being so easily moved by slight currents of air ; and for further steadiness, the pole itself should be well braced from within, a 100 LAND SURVEYING. few feet below Its top. The proper dist a o, of the pole p o, from the table t a, may be found thus : Make an angle n m s, equal approximately to the lat of the place. Open a pair of dividers to equal, by any convenient scale, the height t a of the table ; and draw t a. Then take, by the same scale, the height, p o, of the pole above ground ; and place it upon the sketch, so that the top p shall be by scale a ft 01 two above m n. Then a o, by the same scale, will be about th#- dist reqd ; probably from 3 to 5 yards. A deviation of a ft or so from this will not be important. The correct clock time at any place, for the elongation, may be found within a few min from the following table. Instead of a pole and plumb-line, the writer would suggest a planed, straight-edged board planted vert and braced; its aid* toward the observer. The observer should be at his station at least % of an hour in advance of the time. Placing the board a a, upon the table and in range with the plumb-line and star, he will watch both of them 5 through the vert slit ; sliding the board along the table, so as to keep the slit in the range as long as the star continues to move toward the east or west, as the case may be. An assistant must hold a caudle, or lantern, on a pole near the plumb-line, to enable the observer to see the latter. As the star approaches its elonga- tion, it will appear to move nearly vert for several min. so that it can be seen without moving the slit. When certain by this that the star has reached its elongation, confine the sliding board to the table by sticking a few tacks around its edges. Then let a third person, with another caudle, go off some dist, (a hundred yards or more if convenient,) in a direction toward the star ; and then drive a stake as directed by the observer, who will take care that it is exactly in range with the slit and plumb-line. Another stake must then be driven exactly under either the slit or the plumb-line. Having thus placed the two stakes in the range of the elongation, defer the remainder of the operation ntil morning. From the tables given above take out the sine of the polar dist, and also the secant of the lat. Mult these together. The prod will be the nat sine of an angle called the azimuth of the star. Find the sine in table, p 102, &c. and the angle which corresponds to it. This azimuth angle will be between 1 20' and 2 30', according to lat. Place an instrument over the S stake, sight to the N one, and lav off this angle to the E if the elong was W, or vice versa, and drive a stake to mark it. This last direction is true N and S. It might be supposed that after driving the first two stakes, a true meridian could be had by merely laying off the polar dist, by means of a compass or transit ; but this is not so. Place the compass over the south stake, and take sight to the north one. If, then, the north end of the needle points east of the line, the variation of the compass is east, and rice versa. Times by a correct clock of Elongations of the N. Star. Deduced from U. S. Coast Survey table, calculated for April 1, 1883, to April 1, 1884, and for lat 38 N ; but will answer within about 5 minutes for any lat up to 60 N, and until 1890. Times of Eastern Elongations. Dav of Month. Apr. May. June. July. AUK. Sep. 7 13 19 25 H. M. 6 37 A M 6 14 " 5 50 " 5 26 " 53" H. M. 4 39 A M 4 16 " 3 52 " 3 28 " 35" H. M. 2 37 A M 2 14 " 1 50 " 1 26 " 13" H. M. 12 39 A M 12 16 " 11 52 P M 11 29 " 11 5 " H. M. 10 37 P M 10 14 " 9 50 " 9 27 " 93" H. M. 8 36 P M 8 12 " 7 48 " 7 25 " 7 1 " Times of Western Elongations. Dav of Month. Oct. Nov. Dee. Jan. Feb. Mar. 1 7 13 19 25 H. M. 6 27AM 64" 5 40 " 5 17 " 4 53 " H. M. 4 25AM 4 2 " 3 38 " 3 15 " 2 51 " H. M. 2 28AM 24" 1 40 " 1 17 " 12 53 " H. M. 12 26 A M 12 2 " 11 39 PM 11 15 " 10 51 " H. M. 10 24PM 10 00 " 9 36 " 9 13 " 8 49 " H. M. 8 30 P. M. 86" 7 43 " 7 19 " 6 55 " For days of the month intermediate of those in the table, it will be near enough to make the time 4 min earlier each succeeding day. During nearly all of the four months, March, April. September, and October, the elongations take place in daylight ; so that this method cannot then be used. Nor can it be used at an v time in places south of about 4 north of the equator, because there the north star is not visible. But during that time a meridian may be found by recollecting that when the north star n is on the meridian, or, in other words, is truly north from us, the star Alioth, a, is very nearly vertically above it, if the north star n is on the meridian below the pole: or below it, as at a'", if the north star, n'" is on the meridian above the pole. When the north starn' is at its east elongation, Alioth is horizon- tally west of it. as at a': and when the north star. n". is at its west elongation, Alioth is horizontally- east of it, as at o". All that is necessary, therefore, is to prepare an arrangement of table, pole, plumb line, &c, precisely as before ; except that the plumb-line must be nearer the observer, as he will have to watch Alioth above the north star. Watch through the movable slit until Alioth is on the same vert line with the north star. Then put in two stakes as before, and they will be nearly in a true north and south line.- But to be more exact, either lay off (to the E if Alioth is above LAND SURVEYING, There can be no difficulty in fiud- ing Alioth, as it is one of the 7 bright stars in the fine constel- known as the Greiit Bear, or the Wagon aud Horses. Alioth is the horse nearest tothe fore-wheels of the wagon. The two hind- wheel a t t are known to every schoolboy as the "Pointers," he- cause they point nearly in the di- rection to the North Star. The relative positions of these 7 stars, as shown in Pig 14, are tolerably correct. REMARK OS* THE FOLLOWING TAHI .*? j -I^H^Hr-gl OS _ 66 00 bb - rt ooooooooooooo^ oooooooooooooooooooo q OOOOOOOOOOOOO'-H'-i^'-^'-"'-^ ooooooooooooooooooo o . ' "o os oo i o c c c ooooooooooooooooooooo oooooooooooooooooooo o 000000 000000 0000000000000 0000000000090 : 9 ^ 104 O EH QOQOQOQOQOOT'QOCXJOOOOOOQOQOOOaOQOODQOQOaO i-* ac o us os i-* bo Ir-(^H^-tc^(^ ^^^^^Tj^Tj<-^^TJO lO O O U5 O ppppppppopooooooooo ;sss S c* S i* 55 I i>-*GOOOOOOOO5O5OSO5OSO5OS iC5C5OC^C5C5w5CT5C^C5w5C5C5O5C5CTi ^H^^^i TtT^TjH^'^^Tt^rt'T^^-^T^T^T^T^T^T^T^T^ oooooooooooooooooooo 000000000000000000 oooooooooooooooooooo ' 0< 1 r l 105 ^lOCOl^-OOOSOrHC^^cOCOTt*^^^^^-^!^ {>j>j>^i s t^-t^-i>j>f^j>t>.i>.j>j>i>.j>r>.}>t>. c? o? cp bi lO^cott^-H-HOoojoor-f-cocoio - ~ o O5r- (C5iOi"-C5Nr^COOO OCO^t^-OC^T^CO oooooooooooooooooooo O oooooooooooooooooooo JO -* ** Tf T*^ -^^ ^ rj< ^_ T-H_tO_tn_,_tO tQ i(7. >O 'O iQ to tO c^ 6606600600000000000000060006000000^^-^. ^N C^O5O5^CT5C^C5C>CSC^C>O5C>^O5O5O5OiO5Ci bb ^c^ocftcoiic5 i rtic^o6'-.^^- bb OTO^OSCiCSOSOOoi-i^^-iC^W^WCOCOCO ^ lOiOiOiOiOiOCOCOCOCOCOCOCOCO^COCOCOCOCO r ^ oooooooooooooooooooo O oooooooooooooooooooo O 000000000000000600000000000000060000000000 ^ O Oc^CSO^<^CpCpCpa^OCpCp^C3iO5C>OiC75O5O5Oi j [ ooooooooooooooooocpooo O *6&&&&^^^& ' ooooooooooooooooooooo ^ 106 ri B o O be C rt be -^ o OJ C* US OOGOkOC^OGpiO ^t'O r ^ osgsg^^ci^gs^g^^o^^gi^gigsgigsgi^ ri be q oooooooooooooooooooo iocococot^i>i>i>coG66oc5ca5c>ooO' * ^ t>J>i>l>J>l>i>l>OJ>t^l > 'l>i s -^OOQOGOOOGO ... 00000000000000000000 !O ^.E^1 ^^^^^^oo, QC-^ ^ .o-^-^-N-^-o- (U a J> }> f^ 1 3 CO "o o OJfMOdoOCOCOdsC^iOOO' l Tt*i > -OCOCCQO' '-^ oooooopoooooooooooooo COCOl>l^t^l^l^J>t^GOGOGOG6GOGOGOaOQCGOGO ooooooooooooooooooooo JO j 107 " ' OS O5 "O> OS Oi W Oi OS Oi Oi OS O> o > Oi O5 OS O>'O> O3 00 00 CO O O > i CO 00 r^i iO> 'T^OOCOOCSCS O CO OOOOOOOOOOOOOOOO OOS CO ' C5OSO5OSOOOOOOOOOOOO OOOO Q JS <1 CQ W be S 10 O bb c eS "o bo OOOOOOOOOOOOQOOOOOOO oooooooooooooooooooo oooooooooooooooooooo Tt"O5COOOWCD' 'iO 5C1G5C505O5O5C5OSC5O5O5O5O5C5O505O5O5OSO5 ooooooooooooooooooooo OOOOOOOOOOOOOOQOOOOOO5O5O5OJO5O5OO5O5O5OS ooooooooooooooooooooo 108 OiOSCSOSOiOlCSOiOIOSCSOSOOCSOiOSOiOSOS CO 00 I -^ 'OOOO500 g * tJ 03 3 e r ^ COeOOSCDCOOl^T^ 'OOiC OOiCWOiCDG^J ^T^T^^ T t<-^T^'^^Tt<^^'!*r^^Tl''^CCiCOC7( ^-QOvOCOOQOUDCOOOOiOCCiOOOOOOOQOiOCOO 1 i cc fi OOOOOOOOOOOOOOOOO 1 OOOOOOOOOOOOOOOOOOO*"^ - I r ~R I P i-iOOOOOOOOOOOOOOOO ' OO OiO5 90 L) s ! ^ H O < GO g o3 < K P 5 fc ^cowweocoeowwfittwiNw^r-if-H'-H'.H'-H'-M j^. l^. J> ^. b- j^ }> }> ^- t^. 1> ^- t^-b-t^-t^t^t^l^t^ o co o c* c* a oo ** rt< t- o co COCOOOC'JCC'COCOCOCOOOCOCOCOCOCOCOCOCOCCC^ ** - }>QOQr3QOa5O5O5O5OOO'-' ir-H i ^ ^tOOOOiCtOOtOtQtOr^ -^ r^ ^ r*< T^ rf< ^ T T O O ^ ( ^ ^ 109 i ' CriiOlNOOvO tl-T}O C^WC'*^ '-'OOO O5O50000QO bi I i if it-HOOOOOC5O>C5OSO5QOOOQOOboOOOl>t^ OP opjX) QbgDobobt^t^^-b-^b-^^t^t^b-t^^ 110 r> QOOQOOGOOCOOGOOOQOOOOOOOOOOOOOOOOOODOOOO be ^ iQipioo^^^^^^^^rocoeocococpfoco bb i w OS I OOOOQOOOOOOOaOQOQOQOaOQOQOQOQOQOOOOO COCOCCCOCOCOC7CO(M(N(?OCO Tt^iftiOiO>< I o O C I QOQOOOOOQOOOQOQOOOQOQOQOOOOOOOOOOOQOOOQD p < 50 w 55 bo I-H 0) M Q J ^ s *< R bo O fco I bb o O F I - I J>-t > *t s -t^ t ?^l>l>l s t^l*-t > "t N '^J> >-COCO?DtCtD?O OOOOOCQOXQCOOQOXOOQOQOOOOOariOOOCQOODOOOO COCONMNNNC*C*C*i-iF'-ii-H-H!-'-i i i^OO o 112 g ODOOOOOOOOODOOODOOOD3DGOO5OiO5 i O iQ < ? (N ^ c .? I fi I 0) H . ^ g S o , o fc O I oooooocoooooQOaoixiaDOC Tircaoooaoaoooooao i -i .* QOQOQOOOQOOOGCODOOOOQOOOOOOOOOOOQOQOQOaO Cosin i^t^oioic^wc^ 'Csoo^tnTrNoaoi^i-owo 00000000000000000000(300^00X00000000000000 1 c* CD ' 10 05 ?o r -- iiooo ^ - COCOT^^^OkOiOkOtOCOCOh-^t^l^ 9 fFF" 113 ^~i~g t^- -HiOOOOQOO5COOOO^^C5-^OC5OC^DCO SOOt^J>>OOCD^J>-OOO5O'-iOQir5OC5 "COCO GOO5CiO5OOOt' '" -i-i-4i y S ft p - oooooo T^^ oo ^ oooooooooooo o h I 1 oooa> c XQOQOXXXXXi P 5 OJ H S5 fcb 1 < ^ ^ r ^ r^ rj< ^ ^ r^ rt* O- -H "<^NWCOCCCOrJ<^r^T^OOO) ooooooooooooooooooo WG^dWC^NNC^C^WWC^WWWWOtC^N .J P 5 ^ o c s fc W 05 << H ft fc -^f^i>^i>i>i I- ^ l^ I*- t* ^^T^r^-^^r^r^T^^T^T^ Tf< r^ ^ rt< ^< rH rt< -^ l>i>J>^l^J>l>J>^.J>t^!>3>N.}>I*-l^t^^l> ^r^r^^^r^-^T^ ^^IT^ rj< <* ^j^^ r^ r^ Ty T*> r >co?>i> i-i rt* f- OS W; CD d> W - ^^^T^^^ThT^^^^^T^^T^r^T^rti-^T^^ oooooooo- QQ 5 5 < & o EH <^4^^^^^Tt < ^^'^4hT^^4t < -^^'tt'r^ CC WCOCOCOCOCOCOCOCOCOCOCOT^rtT^TjHr^Tt- CO CO CO .00 CO W CO CO 4< TH 4* 4'^4 ( 4<^-rH-^'^4 J '^-^4h'^4HTH4 | rH -'><-^ O ^^^H^lCl>l-^-l>l-l^t-J>J>3 ii iJ>ri O% O% 0} 0) OS O} O) 0% 0) bb I CpOO(a>i>^!C>^i>i>l>l>(>i>t>i>l>J>t>l> C^CCCOCCOQOOCOCCC^COCOCO C*3C*3COroWCOCT3W .4D CO CD CO CO 00 fcJD 8 E h " bb O5 O5 00 00 00 00 00 GO 000000000000000000000000 ^ 00 I I ^^^^^^^^*a5$*Sftu5*oo*o4oo.o _G o COCOOiCNiOOO- " 117 p-^COTtOpt^GOQOCSOO d o to 1 o r> \ bo QOOOOOQOOOOOOOCBaOOOCBQOOOOCOCOCOOOOOOOO TJH ^ O iO -^ ^ CO -^ fl co r:>t0?>00q:> ^^ r H^Hr-.^- t ^Hp-l>-H^HC< 118 00 s 1 Pi O H Q <$ 00 w fc O(MCCiOJ>QOO' "C . 050000^' W^^COCOCO^Tt^OiOiO 0*0000000000000000000 NCOWCOCOCOCOCOCOCOCOCOCOCQCOCOCOCOC^W r^r^^COCOCOCOCOCOCOWCOCCCOCOCOCOCOCOCO - O 00 SO T*< -W O5 t-*O W - OS t*-O-0 O 00 tO-^.W O iOJ>OCODC' T^ J> O N -O 00 < *< <3 Oi O 1 5O5O5O5OOOOOOOOOOOO O 1 I o I JS ' F-t F-I o O bo H OOOOOOOOOOOOOO' I 120 CB ti s Q o g o ir^^T^Tti^^r^-^^^^T^r^TT^T^^^i" bb (S r->i~i ooo bb 1 C I (MOCOOCO^kOCOCCtOaOCCOOO-HCSCO i O5CDCO'-'a5QOl^i^l^OOa5 iT^^JOCOOOCO QOC5O i'- -' i 5 OS CO 1> i lOOJCOf^-'-HiOO r^^05^^r-HC5COCO^HCX)COOCX)iOCOOOO VfTfiTOC9TOCOWCC<-i-*-r ta pOOQO> OOOOOOQOOOOOOOOOOOOr>OOXOOQOOOOOOOCX)l> WCOCOCOCOCOCOWCOCOC7COCOCOOOCOCOCOCOCO Q 05 MCOJOCOCOJOCCCOCOCOOOCOCOCOCOCOC7COCOCO C g be o o 1 3 3 5 5 5 - OOlOOQOQOCDOOOOOOQOCOQOOOaOQOOOQOOOQOQnOO ^^^^^^^^: o O C osin 122 o be d 0_ bo cpcpcpcocpoocpcpopwopcocpcpcpcpcpcpppcp 00 -~~S . C 00000000001>t^f > -i>l.-J>5CO^COiOiOOiC < <^ ^ O5CiO5O5C5OiO5OSCTJp5 _ J i i i i I bb tO Q WWWC7O:COWWWWTOCOC5O5C7OCPC7C7CO bb I -3 * bb \ ^~>C~QO~O~^ 1 SSSSSSScocpcpcpScococoTOCOcoocccp i O iC*CO^iQ
    OOO5O' iNCO-^iOCOC^QOOiO j^ ... '~''"'" 1 'H *"* "*'"'* **"* *"* ""''""' v .1 . . 123 t*-l^>.l-l>00~l>I>J>J>i>^COCD *> ^ ^ 9? o w a* cp-co - ' C a C 3 o O bo I ^o bi q cocpcpcpfocpcococpcpcococofocpcocpcpcoco r-t r-^ OS 00 CO ^ ^ OS CD C^ O5 lO O CD r ^ r^T^r^T^oooocbcocb^K^^ooaodoocas cpcpcoepcpcpcpcpcpcpcococpccwwcpcpwcp COiOlOOiOiOOtOiO tOtOT^r^Tf-^T^T^T^t-^i-rfi^i COCOCOCOCOCOWWC^fMWC^WNWM Q WCOCOCOCOCOCOCOCOCOCOCOCOWCOWCOCOCOCOCO os to 10 os C^OOOCDCO^HOSt s ^Tt*C^OpO^OCO^-'OSJ>^ l C > JfO bo COOpOpOpQOOOppOpQOOpppOpOpOpQpQOOpOOOOOSOS fpcpcocpcpcococpcpc^cpcocpcpcpcpcpcpcpcocp O^C^CO^tQCD^COOSO^^CO^OCOt^OOOSO | 124 5f?S2 bb a _ ^ . .^ ^- _-i. - rr "w o op r^ c H oS ^ oodoooaoooooQooggo^qpgg^ogaDgpb^OT^^ I C^ CO GO" O^ C* r-H C ^ CO CO CO CO CO C*3 rHO llOr-(r-.COOOCJ>OCOlO^i-OSOO C8 * 000^^^"" ic5c^(^C^W(NCOCOCO^^r^ g COCOC^Csf(^(M(^^^-i^H^^iOOOOOO5OiO5 " ^^^^^^^^^^'5'^^ - 5' - 5' - 7^ < 5<-^^^ cowr^oobobt^io^coc^o^op^ococ^oc^ S^^^^^^^gpopopopopopo^opopoboooo "-H t>- CO 00 _ m^rMll~ltO?^rrirr\rTNr^\r^i(-~i^^ ^, ,, >,-^< < ~ I to q OOOOOOOOOOOOOOOOOO^r-ir-* CO CO CT> SO C< I> CO 00 N ^' ^ " *d* OQ - *^* CO-^iO OOrt*O COiO COOOO- ' - o o c ! W W . OCOC<(OOOCOr^C<(C^C* 127 O K ft O E- W O EH P GO W 00 ^ oooooooooooooooooooo OOOOOOOOOOOOOOOOQCOOQOOOQOOOQOOOOCOCQOX ^^^^^^^'^'^'^^^'^'^^^'^H'vP'^''^ COCOO?^TfJ>J>J>0000 COCOCOCOW-70COCCCOC7COCOCOCOCOCOO700WCO g OD to c O a 1 o I - oooooooooooooooooooo to I 6 to V ft > lOOOooooocsa^oiocsaoGoaoaoooao ^.-H^i^i^^H^HOOOOOOOOOOOOO to i &H I^-i iT^OO Oa5C*CDO5COCDOTft^-i IT^QO^^O - OOOOOOQOOiO5O5O5OOO'- Q 1 f ^ T ' OOQOQOOOOOaOOOQOCX)OOOOOOOOCZ)OOQOCOQOGOaO P fc << 02 w B CQ bo a OOOOOOO5O5O5O5C5O5O5 ooooooooooooocJC5Oic>cpo5q'. J> - J> > ?>l >l> l>i>J> > COCOCDCOCO "< P a r^ GOOOGOGodbooroabaaGOGOGOGCGCGOGOOOGOGOGO U cpqpqpopcpopqpcpopGpGpopopqpopopopopGpop bio O J> CO O ?>^-f > -^i>o^o^Do^o^b^C'iOio r^ GOODaoooaoGoaoooGoaoQOooooGoooGocoooooGO CO w ' qpapopopapopGpapopGpopGpGpopopGpqpGpopGp * bb W C7 ^ o _ CO CO CO ^ ^ I O <(MCOrJiOCOl>aOOiO^iNCO^iOCO^OO OS O o. 1 .,- ^ .-,,-. ^ -. M 130 _J bp * CO lO ^ W -^ C C s ?C < lC v lC < lC < tWS > <' If tr-Hri IH ^ i a> QO O2 p ^,^^^,^^^ l ^^^^^^^^^ l ^^ l ^^ l ^ S ? H ^ ^ _ _ _ (N7^<7<(?*C^^O< ( .-H^-H^r-fOOOOOO r-} 000000X00(30000000000000000000000000000000 ^*^ xxxxxxxxxxxxxxxxxxxxx 'X Xi>-t > -t > "}>'*-i > l>-{>5DCO?OCO?O{OtOiO'OiOOO -^ xopxxopxxopobxxxxxxxxxooxx 1^ COCOCCCOCOCOOOWCOCOCOCOCOCOCOCOCOCOCCCOCO A l| 3 131 o bo * it~COOO^O>OOCD - CO > i CO i ' ID i i CO O O 60QOGOOOOOOOt^'i>J>t^l> l t^~CO?OSOSO5O ; OSO OCOOOQGpGOOOOOOOOQODODGOOOOQOOGOOOGOGpGO I bo I o O CO g 5 <3 o O) EH w o & OSr-HM'^OO5r-HCOOa5 OCOCOiOOCOCOiOOOOCOCOOO^HCOCOGO "COCO ^OOiOiOCOCO^COi t^^i>OOQOQOQOO5O5O5a5 QOOOOOOOQOOOGOQOOOaOOOCXJCXJOOQOaOOOGOQOaOOO _O C 133 B p '-T- '- O500CDOCO'-HOaCi~*-O^W^CiOOCOkO!r'3'MO SOOOOOOOiCiOsasasOiGOaOaOOOaoaOOO Ut)>OlOlOkO^T^Tj<^T^TtlT^T^rt 00 t*- O > o o o < > O O < o ^COCOCOCOCOCOC*CNC-*OOtO OC*U3t^ OOO <"H ^ GOQOOOQOQOOOXQOOOQOQOQOQOaOOOOOOOQOQOaOOO oooooooooooooooooooo a ^ir: 134 b: *H C"2 J> O CO ^r^^-^^oooooodiasasdsasasaodoob CDCpCD^COODODOOQboOCpOOOpOOGpOOOOGpOOOO bC iOOCOCOi>'ODQOO5O^C$COiOCOi> > O' '_- C CO1DCOCO?OCDCOCOl > "i>-i>'f > 'i>'i>"!>J>'OOODOOOO 9.' O bb d H o ^^^fT^Tt | TT'T^Tr < Tr | tOtO>OiOiOtCO>O ^C ^0 g ioaico^(r*co=^^iOQfic*io en Q ^ QOob(^(X)CCQOQOOOOOQOOOQOGOC300000CCWCJbob g ?^t>t*i>-t^^j>^coococococo:ococo?ococo *J >>-t > xif N 't^?>>Z s 3>J>t > CO^O^O?OCOCO?OCOCOCO>O 0_ bb c 3 H ._ CO CO CO CO CO CO CO CO COCOCOCOCOCOCOCOCOCOCOCO CO tO iO O IQ tQ tQ iQ |Q 1Q iQ r - L - ' - ^ ^i^5f^5?^'^5? < !*T?!!'^'^5^'^^^^''!?' c ^?^?^?? < ^ 1 ^T < '~^^ I ??'^^ I ?!?^?^^?^?9P^P^ ) ^P ^p bb o5-^-^coo5'CocoGO- : tcoiioO'-icrcoO' coioob jC G fi W ^^.. ^J_3#_'j*t'_ rfr ^^rt l ^'^tQ>OO iOt6OiOOiOOO < 7 o -I H o ^ (^QpQpOT^Qp^<^Qp6pqpq5QpQpQp ^ bb OGOi>O>O'^COC^i ^pOO>O50pCPGOQOt>t^-C30 P CCOtOOiQiOiOOiO 2 r^ rH-^^'sHTrirH'^ TH.rH ^^^^r^^rH^-^-^^ 2^ *** < H bo c^ ?292S5?5S55S52^^95?2^^^59^?'^?5 < C ^^^? c ?'^' ; ?^?'T t ?*^99'~'? r5 ? c 'P 0( ~ > 9?^ ( ^ i ?! ^ M D H fe o ^ xa^'<^Qpq5qpqpo6qpQpQp bb C 2 ^^^^^^'^^^rH^TH^'^^'^'^'^'^'^TH bb o bb CO O s 137 i ^ oooooDooooooooooooooooooooooQOojDopopojbqo 'i>cooNiOTt7*co.-HOq5T^ S3 -H o 5C oooooooboboboboooooooooododod oooooo6p f-iC^COrfOCOI^Q005Or-i ~C*~CO T^ iO CO l^- '00 cT^ T4H -rfH *+ <-* <<-* ^ ^ j-n _TJ^ ._>O_JO _jO_tO >Q tQ tQ tQ O lQ CO OSCOt^COiO'^COC^' lOCOOOt-COiO^CO^' co co c ^ o c* co o to o i> ^" . _ ^ Qo"co"t- r-H CO --i~CO i CO !N 00 - _ _ . _- _ -. OOOOOOOOOOOOOOCDOOOOOCO 138 QJ OGOOoocscrta^CTiCiOoaoao J < ^ a>opcccopccpopcpQpopcpi^t^t^t^b-i^^-t^ ~" ' " "* SO 00 O -j- ^ cpcpwcpcpepcpcpepcpcocpcococpcpcocpcpcp ^^T^T^^r^rfT^^^^TfiOOkOOiOtOiOtO O l^-b-J-b-}>t^b-I>b-J>-b"^^-}>l>-^-l>'^'J>i>- ^ Oi ,-T< _ _ 7^ _ ( g 00 O5 O ~ iQ 10 CD i O _ s _. _. . . G ^ -. ^ ^ ^T^^COCOCOCOCO(NC oooooooooooooooooooo ^ ocopcp(opopQpQpCipccQpQp(coqpc^qpcQoap g lOOOiOOiOiOiOioirSiOT^^^^^HTf'^^TH ^ .rT COCOCOCpCOCOCOCOCOCOCOCOCOCOCOCOCOCOCOCO COCOCOWOTCOCOCOCOCOr^^r^T^r^Thr^'rt'^'^ O I I o _ ___ _. _ en <> IQ tQ tO tQ tQ tp Cf5 JQ tOkO-'t^r^TfTt 1 '^^* T^ -*t rf " - " ^ g p ooooooooooooooooooooo CO ** i~^}>l>J>.t^^^t^^C^CCiCOCDCD bb c o cS ^ ^ ^ fi "^ *""^ JO - gj ------ J ^WOT^io^^OOct^ '"~ I ilS^^^^^SS^ j ^_| bb 1 'o O5 O5 CD OiOiOSGOGOOQOOOOOOGOOOOOOOOOOOGOl""" 139 be I O p CO O O H DO H 52; H P m ^ ( i 02 iTf*i>OCOCOO5' ' >i i-^41>OCOOOSCOOO5(NOOO< ^< s *-< )^j>t^.J>i>QDOOaOCX)CX)O5OO5O5OOOO' JOOOOOOOOOOOOOO'-H' 4 F^ , H M & o Ibb (?*W(M(NM'^7<(M(^-H'-<'-<'-H^H^-t^^^^H-4r-H cocpcpcpcocpcpwwoowcocpcpcooococococo o 1 5 cc f ^^ t N.^ r ^t^O"*t^OCO50COCOOiO>C*COO.QOQOO5O5O C5OiO5O5O5OSOSO5O5O ho o COOS 'r^J>OCOOOO < ^ tt to to o o co cc co i' OOQOOOOCCXJQOQOOOQOQOO&QOQOOCaOXXQOOOO^ I " 1 142 _I _ OCOOQOO-^'T*^ .s O ^i^^^^t^^t^t^^t^i^t^^ii^i;:^^^^ _j a I bo CJ O O -? ^ d ^ s r .-,..,-- _ - , , , , -_ ._ o ^ K (^NCO^St^ooXojp^^iwccco o " '_ ^H g c^^ii-ir^-Hp-ioooopocftosdsojaidoQOcjo O ^t^i>j>i>i>j>i>.^-j>i>j>i^^-^-^-i>i>^-i^ f bb ^g I ^ _ CO^d*>O5OQOOC < l>Ot'_ . .. be cococo^o^ct^'^^ij09^99^^^?^^'" Hp ~ t a J OOQOQOQOOOQOOOQOOOQpQpQOCjpCXiQOQDQOQOQOQO fiD *** gj iO > O5' iT^tDQOOCxrrCOw' 'COtOi^OS ^ CO 1|! ss r^ ^ (noicioooio ^c us I' i < 5 ^r O O ^^^^j^^^,^.j>^.^,^.i>^.{>.^j>.^^.j>.j> bo a Q bb a XX<^^( oi a tib Jg ------ ^^^^^ o y y y y y y __ 3 ^ | ~ ** m * ^^ ^S^S^^tS^^S^g | \ 143 1 O g " B P o P 05 & w o K <1 p fc << cc ^ hn fc ? B ^ j 5 g < ic o bo I o" I bfi O OOOOOOOOQOOOQOOOOOGOCX)aOODQOOOOOOOQOaO - > r* -^J>J>t^^.l-*.l-j OOOOOOOOOOOOOOOOf TO H B p j>QOooaoaooioio i sa5c35OOooO' <; C GO P w o p fc TO w fc bo a O5COOt30QOaDJ>J>r^^^COO?)O?OtOU5iOiO cocooococococccococococccocoo^oooocococo S? C 'i o ' 'OOSOOi N -COiOTf -t > -i I I O 00 F" tC to ^ CO j^ {S^^^^Z^^ ^.^ WI^C^W^fM'-H , ^- ' '^OOOOOO5O5O^O5 O {^t*}>i>i>i>i>i*.^t-.i>j>j>}>i^^-i'^-i>^- 145 -\$ * Q oooooooooooooooooooo QQ ei) 1 5 * o H )'-it^(M( OO< x -J CD 00 O CO iO t- Oi f-H CO iOl !N w g QC P J ' s D g, ^: be I ooooooooooooooooooo go i c w co co r< iCiOiOiCtCiOiOO OOOOOOOOOOQOODOOaOQOC30QOaOOOQOOOOOO5OiO CO O O lO tO iQ O iQ lOiOtOT^r^^r^rti^T^rt^' - c?c?cococowNw<^OO - OOOOODOOOOODOO^ODOOOOODQOOCOOOOOOOOOOOOOO o -H W 00 -^ Oi iC~^H 0(?t>.QO J>l>J>t>J>OOOOOOQOOOOOOOQOQOOOOOQOQOOOOO bo I o C CiOOSOSO^OOOOOOOOOOOOOOO o OOSOOi^-^OtO-^COC^i (OO500J> ~i~ be c 000000000000000000000 CO<^i>J>l s -J>-i-'i>i>J>J>?>t > - bo B CJ Jij?JLLi- CONTOUR LINES. 147 REM. The table, pp. 608-612, is to furnish the means of laying down angles on paper more accurately than by an ordinary protractor. To do this, after having drawn and measured the first side (say ac) of the figure that is to be plotted ; from its end c as a center, describe an arc - - ny of a circle of sufficient extent to subtend the angle at that point. The rad en with which the arc is described should be as great as convenience will permit ; and it is to be assumed as unity or 1 ; and must be decimally divided, and subdivided, to be used as a scale for laying down the chords taken from the table, in which their lengths are given in parts of said rad 1. Having described the arc, find in the table the length of the chord n t corresponding to the angle act. Let us suppose this angle to be 45; then we find that the tabular chord is .7654 of our rad 1. There- fore from n we lay off the chord n t, equal to .7654 of our radius-scale ; and the lint cs drawn through the point t will form the reqd angle act of 45. And so at each angle. The degree of accuracy attained will evidently depend on the length of the rad, and the neatness of the drafting. The method becomes preferable to the com- mon protractor in proportion as the lengths of the sides of the angles exceed the rad of the protractor. With a protractor of 4 to 6 ins rad, and with sides of angles not much exceeding the same limits, the protractor will usually be preferable. The di- viders in boxes of instruments are rarely fit for accurate arcs of more than about 6 iris diam. In practice it is not necessary to actually describe the whole arc, but merely the portion near t, as well as can be judged by eye. We thus avoid much use of the India-rubber, and dulling of the pencil-point. For larger radii we may dis- pense with the dividers, and use a straight strip of paper with the length of the rad marked on one edge ; and by laying it from c toward s, and at the same time placing another strip (with one edge divided to a radius-scale) from n toward /, we can by trial find their exact point of intersection at the required point t. In such mat- ters, practice and some ingenuity are very essential to satisfactory results. We can- not devote more space to the subject. CONTOUB LINES, A CONTOUR LINE is a curved hor one, every point in which represents the same level ; thus each of the contour lines 88c, 9lc, 94c, &c, Fig 1, indicates that every point in the ground through which it is traced is at the same level ; and that that level or height is everywhere 88, 91 , or 94 ft above a certain other level or height called datum ; to which all others are referred. Frequently the level of the starting point of a survey is taken as being 0, or zero, or datum ; and if we are sure of meeting with no points lower than it, this answers every purpose. But if there is a probability of many Idwer points, it is better to assume the starting point to be so far above a certain supposed datum, that none of these lower points shall become minus quantities, or below said supposed datum or zero. The only object in this is to avoid the liability to error which arises when some of the levels are -{-, or plus ; and some , or minus. Hence we may assume the level of the starting point to be 10, 100, 1000, &c, ft above datum, according to circumstances. The vert diets between each two contour lines are supposed to be equal ; and in railroad surveys through well-known districts, where the engineer knows that his actual line of survey will not require to be much changed, the dist may be 1 or 2 ft only ; and the lines need not be laid down for widths greater than 100 or 200 ft on each side of his center-stakes. But in regions of which the topography is compara- tively unknown ; and where consequently unexpected obstacles may occur which require the line to be materially changed for a considerable dist back, the observa- tions should extend to greater widths ; and for expedition the vertical dists apart may be increased to 3, 5, or even 10 ft, depending on the character of the country, &c. Also, when a survey is made for a topographical map of a State, or of a county, vert dists of 5 or 10 ft will generally suffice. Let the line A B, Fig 1, starting from O, represent three stations (S 1, S 2, S 3,) of the center line of a railroad survey ; and let the numbers 100, 103, 101, 104, along that line denote the heights at the stakes above datum, as determined by levelling. Then the use of the contour lines is to show in the ofnc what would be the effect of changing the surveyed center line A B, by moving any part of it to the right or 148 CONTOUR LINES. left hand.* Thus, if it should be moved 100 ft to the left, the starting point would? be on ground about 6 ft-higher than at present ; inasmuch as its level would then be about 106 ft above datum, instead of 100. Station 1 would be about 7 ft higher or 110 ft instead of 103. Station 2 would be about 7 ft higher, or 108 it instead of 101. If the line be thrown to the right, it will plainly be on lower ground. The field observations for contour lines are sometimes made with the spirit-level ; but more frequently by a slope-man, with a straight 12-ft graduated rod, and a slope instrument, or clinometer. At each station he lays his rod upon the ground, as Rgl. nearly at right angles to the center line A B as he can judge by eye ; and placing the slope instrument upon it, he takes the angle of the slope of the ground to the nearest l / of a degree. He also observes how tar beyond the rod the slope continue* the same ; and with the rod he measures the dist. Then laying down the rod at that point also, he takes the next slope, and measures its length ; and so on as far as may be judged necessary. His notes are entered in his field-book as shown in Fig 2 ; the angles of the slopes being written above the lines, and their lengths below ; and should be accompanied by such remarks as the locality suggests ; such as wood$ rocks, marsh, sand, field, garden, across small run, &c, &c. * In thus using the words right and left we are supposed to have our backs turned to the starting point of the survey. In a river, the rig-lit bank or shore is that which is on the right hand as we descertd it, that is, in speaking of its right or left bank, we are supposed to have our backs turned towards its head, or origin ; and so with a survey. CONTOUR LINES. 149 It Is not absolutely necessary to represent the slopes roughly in the field-book, as in Fig 2; for by using the sign + to signify "up;" "down;" and = "level,'* the slopes may be writ- ten in a straight line, as in Fig 2U. The notes naving been taken, the preparation of the contour lines by means of them, is of course office-work ; and is usually done at the same time as the draw- ing of the map, &c. The v 16+ . 10-t- 20 + 3* |5- Fid a 4-- . 26- 84- ' 70 field observations at each station are then sepa- ( rately drawn by protrac- 91 58 tor and scale, as shown in Fig 3 for the starting point 0. The scale should not be less than about fg inch to a ft, if anything like accuracy is aimed at. Suppose that at said station the slopes to the right, taken in their order, are, as in Fig 2, 15, 4, and 26; and those to the left, 20, 10, and 16 ; and their lengths as in the same Fig. Draw a hor line h o, Fig 3 ; and consider the center of it to be the station-stake. From this point as a center, lay off these angles with a protractor, as shown on tho arcs in Fig 3. Then beginning say on the right hand, with a parallel ruler draw the first dist a c, at its proper slope of 15 ; and of its proper length, 45 ft, by scale. Then the same with c y and y t. Do the same with those on the left hand. We then have a cross-section of the ground at Sta 0. Then on the map, as in Fig 1, draw a line as ra w, or h w, at right angles to the line of road, and passing through tha station-stake. On this line lay down the hor dists ad,ds,sv t a e, e g, g k, marking them with a small star, as is done and lettered in Fig 1, at Sta 0. When extreme accuracy is pretended to, these hor dists must be found by measure on Fig 3 ; but as a general rule it will be near enough, when the slopes do not ex- ceed 10, to assume them to be the same as the sloping dists measured in the field. Next ascertain how high each of the points c y 1 1 n i is above datum. Thus, measura by scale the vert dist d c. Suppose it is found to be 5 ft ; or in other words, that c is 5 ft below station-stake 0. Then since the level at stake is 100 ft above datum, that at c must be 5 ft less, or 100 5 = 95 ft above datum ; which may be marked in light lead-pencil figures on the map, as at d, Fig 1. Next for the point y, suppose we find s y to be 11 ft, or y to be 11 ft below stake O ; then its height above datum must be 100 11 = 89 : which also write in pencil, as at s. Proceed in the same way with t. Next going to the left hand of the station-stake, we find el to be say 2 ft ; but I is above the level of the station-stake, therefore its height above datum is 100 + 2 = 102 ft, as figured at e on the map. Let ng be 5 ft; then is n, 100 -f 5 = 105 ft above datum, as marked at g ; and so on at each station. When this has been done at several stations, we may draw in the contour lines of that portion by hand thus : Suppose they are to represent vert heights of 3 ft. Beginning at Station O (of which the height above datum is 100 ft) to lay down a contour line 103 ft above datum, we see at once that the height of 103 ft must be at t, or at % the dist from tog. Make a light lead-pencil dot at t ; and then go to the next Station 1. Here we see that the height of 103 ft coincides with the station-stake -itself; place a dot there, and go to Sta 2. The level at this stake is 101 ; therefore the contour for 103 150 DIALLING. ft must evidently be 2 ft higher, or at t, % of the dist from Sta 2 to -f 104 ; therefore make a dot at i. Then go to Sta 3. Here the level being 104 above datum, the con- tour of 103 must be at y, or } of the dist from Sta 3 to +99 ; put a dot at y. Finally draw by hand a curving line through t, SI, t, and y- and the contour line of 103 it is done. All the others are prepared in the same way, one by one. The level of each must be figured upon it at short intervals along the map, as at 103 c, 106 c, &c. Or, instead of first placing the + points on the map, to denote the slope dists actu- ally measured upon the ground, we may at once, and with less trouble, find and show those only which represent the points t, S 1, t, y, &c, of the contours themselves. Thus, say that at any given station-stake, Fig 4, the level is 104; that the cross-sec- tion c s of the ground has been prepared as before ; and that we want the hor dists from the stake, to contour lines for 94, 97, 100 ft, Atlas....: 33 X 26 Columbier 3i% X 23 Elephant 28 X 23 24 sheets 1 quire. 20 quires 1 ream. Sizes of drawing? papers. Ins. Tns. Imperial 29 V X 21U Super Royal 2?C| X w/L Royal 23U X 19 Medium 22|| X H^ Demy l9)| X 1^| The English drawing-papers are stronger and superior to the American. Thosa by Whatman have a high reputation ; they are, however, of different qualities. When paper is pasted on muslin, the difference in quality is not so important. Both white ami tinted papers, for the use of engineers, are made in continuous rolls, without seams. Widths 40, 54, and 60 ins ; usual lengths up to 40 yds ; but can be had to order to 400 yds or more. These may also be purchased ready pasted on muslin, in rolls up to 10 yds long. This last, on account of its strength. 152 THE LEVEL. should be used for all drawings which undergo frequent rolling and unrolling; or other hard usage ; particularly working-drawings. For the last purpose, strong car- tridge, or pattern paper answers very well. It is for sale in long rolls; and can be had in lengths of 10 yds, pasted on muslin ; widths 24, 40, 50 ins. Color, a light buff. Tracing 1 paper. Most of that sold, whether domestic or foreign, tears so readily as to be of comparatively little service, except for tracings to be enclosed in letters for mailing. Some of what is called French vegetable tracing-paper, is, how- ever, quite stout and strong, and good for line drawings ; but it shrivels badly under broad washes of color, even when stretched, forming little puddles, which make it difficult to produce a uniform tint. Sizes 18 X 24, and 27 X 40 ins ; also in rolls of 11 and 22 yds. Tracing cloth, usually called tracing muslin, and sometimes vellum cloth, is altogether preferable to tracing paper, on account of its great strength. Widths 18, 30, 33, 36, 40, and 42 ins.; lengths to 24 yds. Common inks dry pale on either tracing muslin or tracing paper; therefore use India ink. Neither the muslin nor the paper takes colors as kindly as drawing paper. Profile paper has hitherto been made in single sheets ; but can now be had in long, continuous rolls; thus avoiding the trouble of pasting separate sheets to- gether. This is a great improvement, and must at once bring the new paper into general use among engineers.* Used for profiles of railroads, &c. Ruled squares. Paper carefully ruled in small squares, so that the divisions answer for a scale for the drawing, is exceedingly useful for sketching out plans, &c. It is sometimes ruled on both sides of the sheet. bl Ro Koman ochre, sap green. Newman s, Ackerm;m s, or Osoorne s colors are among the best in use. Purchase none but the very best India ink. Cakes of colors should always be wiped dry on paper, after bSTng rubbed in water ; and but little water should be used while rubbing; more being added afterward. Lead pencils. Genuine A. W. Faber's Nos. 2, 3, and 4, are very good. The hardness increases with the number. Nos 3 and 4 are good for field-book use : which to prefer, will depend on the character of the paper; No. 3 for smooth, and No. 4 for the coarser or more granular papers. The office draughtsman should have a flat file, upon which to rub his lead to a fine point readily, after using the knife. Hover's patent carbonized writing, letter, and note papers possess the important quality of at once imparting a clear black color to writing done with pale common inks, if the latter contains, as usual, sulphate of iron and gallic acid. These papers do not differ in appearance from ordinary ones ; nor is there much difference in cost. THE LEVEL. ALTHOUGH the levels of different makers vary somewhat in their details, still their principal parts will be understood from the following figure.t The telescope T T rests upon two supports Y Y, called Ys ; out of which it can be lifted, first removing the pins * 5 which confine the semicircular clips n n, and then opening the clips. The pins should be tied to the Ys, by pieces of string, to prevent their being lost. The slide of the object-glass O, is moved backward or forward by a rack and pinion, by means of the milled head A. The slide of the eye-glass E, is moved in the same way by the milled head e. A cylindrical tube of brass, called a shade, is usually furnished with each level. It is intended to be slid on to the object-end O of the telescope, to prevent the glare of the sun upon the object-glass, when the sun is low. At B is an outer ring encircling the telescope, and carrying 4 small capstan- headed sere ws ; two of which, pp, are at top and bottom; while the other two, of which i is one, are at the sides, and at right angles to^ p. Inside of this outer ring is another, inside of the telescope, and which has stretched across it two spider-webs, usually called the CROSS-HAIRS. These are much finer than they ap- pear to be, being considerably magnified. They are at right angles to each other ; and, in levelling, one is kept vert, and the other hor. They are liable at times to be * Made and for sale by James W. Queen & Co, dealers in philosophical apparatus, engineers' sta- tionery, &c, No- 924 Chestnut St, Philadelphia. It can be had also pasted on muslin, in rolls of 10 yds. t The price of a first-class level, by Heller & Brightly, is $145. It is bad economy to buy inferior instruments. THE LEVEL. 153 thrown out of this position by a partial revolution of the telescope, when carrying the level, or when setting the tripod down suddenly upon the ground ; but since, in levelling, the intersection of the hairs is directed to the target-rod, this derangement does not affect the accuracy of the work. Still it is well to keep them nearly vert and hor, by keeping the BUBBLE-TUBE D D as nearly directly over the bar V F as can be judged by eye. This enables the leveller to see that the rod-man holds his rod nearly vert, which is absolutely essential for correct levelling. If perfect verticality is desired, as is sometimes the case, when staking out work, it may be obtained (if the, instrument is in perfect adjustment, and levelled) by sighting at a plumb-line, or other vert object, and then turning the telescope a little in its Ys, so as to bring the hair to correspond. When this is done, a short continuous scratch may be made on the telescope and Y, to save that trouble in future. Heller & Brightly, however, provide their levels with a small projection inside of the Ys, and a corresponding stop on the telescope, the contact of which insures the verticality of the hair. Should the hairs be broken by accident, they may be replaced as directed here- after. The small holes around the heads of the 4 small capstan-screws p, i, just referred to, are for admitting the end of a small steel pin. or leyer, for turning them. If first the upper screw p be loosened, and then the lower one tightened, the interior ring will be lowered, and the horizontal hair with it. But on looking through the tele- scope they will appear to be raised. If first the lower one be loosened, aud the upper one tightened, the hor hair will be actually raised, but apparently lowered. This is because the glasses in the eye-piece E reverse the apparent position of objects inside of the telescope ; which effect is obviated, as regards exterior objects, by means of the object-glass 0. This must be remembered when adjusting the cross-hairs ; for if a hair appears to strike too high, it must be raised still higher; if it appears to be already too far to the right or left, it direction. t must be actually moved still more in the same . This remark, however, does not apply to telescopes which make objects appear inverted. There is no danger of injuring the hairs by these motions, inasmuch as the four screws act against the ring only, and do not come in contact with the hairs them- selves. Under the telescope is the BUBBLE-TUBE D D. One end of this tube can be raised or owered slightly by means of the two capstan-headed nuts n n, one of which must je loosened 'before the other is tightened. On top of the bubble-tube are scratches 154 THE LEVEL. for showing when the bubble is central in the tube. Frequently these scratches, or marks, are made on a strip of brass placed above the tube, as in our fig. There are several of them, to allow for the lengthening or shortening of the bubble by changes of temperatuie. At the other end of the bubble-tube are two small capstan-screws, placed on opposite sides horizontally. The circular head of one of them is shown near t. By moans of these two screws, that end of the tube can be slightly moved hor, or to right or left. Under the bulble-tube is the BAR V F ; at one end of which, as at V,are two large capstan-nuts w w, which operate upon a stout interior screw which forms a prolongation of the Y. The holes in these nuts are larger than the others, as they require a larger lever for turning them. If the lower nut is loosened and the upper one tightened, the Y above is raised ; and that end of the telescope becomes farther removed from the bar; and vice versa. Some makers place a similar screw and nuts under both Ys ; while others dispense with the nuts entirely, and substitute beneath one end of the bar a large circular milled head, to be turned by the fingers. This, however, is exposed to accidental alteration, which should be avoided. When the portions above m are put upon w, and fastened by the screw Y, all the upper part may be swung round hor, in either direction, by loosening the clamp-screw H ; or such motion may be prevented by tightening thatscrew. It frequently happens, after the telescope has been sighted very nearly upon an object, and then clamped by H, that we wish to bring the cross-hairs to coincide more precisely with the object than we can readily do by turning the telescope by hand; and in this case we use the tangent-screw 6, by means of which a slight but steady motion may be given after the instrument is clamped. For fuller remarks on the clamp and tangent-screws, see "Transit." The parallel plates m and S are operated by four levclling-screws; three of which are seen in the figure, at K K. The screws work in sockets K ; which, as well as the screws, extend above the upper plate. When the instrument is placed on the ground for levelling, the lower parallel plate S is never hor, unless by accident; nor is it necessary that it should be. But for levelling, the upper one m must always be made hor (as indicated by the bubble) by the levelling- scrcws. The lower plate S, and the brass parts below it, are together called the tripod-bead ; and, in connection with three wooden legs Q Q Q, constitute the tripod. In the figure are seen the heads of wing-nuts J which confine the legs to tiie tripod-head. Under the center of the tripod-head should always be placed a small ring, from which a plumb-bob may be suspended. This is not needed in ordinary levelling, but becomes useful when ranging center-stakes, &c. To adjust a Level. This is a quite simple operation, but requires a little patience. Be careful to avoid straining any of the screws. The large Y nuts ww sometimes require some force to start them ; but it should be applied by pressure, and not by blows. Before begin- ning to adjust, attend to the object-glass, as directed in the first sentence under "To adjust a plain transit," p. 159. Three adjustments are necessary ; and must be made in the following order: First, that of tile cross-hairs; to secure that their intersection shall continue to strike the same point of a distant object, while the telescope is being turned round a complete revolution in its Ys. This is called adjusting the line Of colliniation, or sometimes, the line of sight; but it is not strictly the line of sight until all the adjustments are finished; for until then, the line of collimation will not serve for taking levelling sights. Second, that of the bubble-tube B D, to place it parallel to the line THE LEVEL. 155 of collimation, previously adjusted ; so that when the bubble stands at the centre of its tube, indicating that it is level, we know that our sight through the telescope is hor. Third, that of the Ys, by which the telescope and bubble-tube are supported ; yo that the bubble-tube, and line of sight, shall be perp to the vert axis of the instru- ment; so as to remain hor while the telescope is pointed to objects in diff directions, as when taking back and fore sights. To make the first adjustment, or that of the cross-hairs, plant the tripod firmly upon the ground. In this adjustment it is not necessary to level the instrument. Open the clips of the Ys; unclamp; draw out the eye-glass E, until the cross-hairs are seen perfectly clear ; sight the telescope toward some clear dis- tant point of an object; or still better, toward some straight line, whether vert or not. Move the object-glass 0, by means of the milled head A, so that the object shall be clearly seen, without parallax, that is, without any apparent dancing about of the cross-hairs, if the eye is moved a little up or down or sideways. To secure this, the object-glass alone is moved to suit different distances ; the eye-glass is not to be changed after it is once properly fixed upon the cross-hairs. The neglect of parallax is a source of frequent errors in levelling. Clamp ; and, by means of the tangent-screw 6, bring either one of the cross-hairs to coincide precisely with the object. Then gently, and without jarring, revolve the telescope half-way round in its Ys. When this is done, if the hair still coincides precisely with the object, it is in adjustment; and we proceed to try the other hair. But if it does not coincide, then by means of the 4 screws jo, t, move the ring which carries the hairs, so as to rectify, as nearly as can be judged by eye, only one-half of the error; remembering that the ring must be moved in the direction opposite to what appears to be the right one ; unless the telescope is an inverting one. Then turn the telescope back again to its former position; and again by the tangent-screw bring the cross-hair to coincide with the object. Then again turn the telescope half-way round as before. The hair will now be found to be more nearly in its right place, but, in all probabil- ity, not precisely so ; inasmuch as it is difficult to estimate one-half the error accu- rately by eye. Therefore a little more alteration of the ring must be made ; and it may be necessary to repeat the operation several times, before the adjustment is perfect. Afterward treat the other hair in precisely the same manner. When both are adjusted, their intersection will strike the same precise spot while the telescope is being turned entirely round in its Ys. This must be tried before the adjustment can be pronounced perfect; because at times the adjustment of the second hair, slightly deranges that of the first one; especially if both Were much out in the be- ginning. To make the second adjustment, or to place the bubble-tube parallel to the line of collimation. This consists of two dis- tinct adjustments, one vert, and one hor. The first of these is effected by means of the two nuts n on the vert screw at one end of the tube ; and the second by the two hor screws at the other end, t, of the tube. Looking at the bubble-tube endwise, from t in the foregoing Fig, its two hor adjusting-screws 1 1 are seen as in this sketch. The larger capstan-headed nut below, has nothing to do with the adjustments; it merely holds the end of the tube in its place. To make the vert adjustment of the bubble-tube, by means of the two nuts nn. Place the telescope over a diagonal pair of the levelling-screws K K ; and clamp it there. Open the clips of the Ys ; and by means of the levelling-screws bring the bubble to the center of its tube. Lift the telescope gently out of the Ys, turn it end for end, and put it back again in its reversed position. This being done, if the bubble still remains at the center of its tube, this adjustment is in order ; but if it moves toward one end, that end is too high, and must be lowered ; or else the other end must be raised. This must be done by means of the two small capstan-headed nuts nn. If the end nn is to be raised, the upper nut must first be loosened, then the lower one tightened, and vice versa ; one-half the error is to be corrected by ihis first process. Then correct the other half by means of the levelling-screws K K. Having thus brought the bub- ble to the middle again, again lift the telescope out of its Ys; turn it end for end, and replace it. The bubble will now settle nearer the center than it did before, but will probably require still further adjustment. If so, correct half the remaining error by the nuts as before : and half by the levelling-screws; and so continue to re- peat the operation until the bubble remains in the center in both positions. That part also of its adjustment will then be complete ; and the bubble will remain at the center (after the instrument is levelled) while the telescope is pointed in any direction. 156 THE LEVEL. To make the hor adjustment of the bubble-tube, place the telescope over two of the levelling-screws K K, which stand diagonally to each other; clamp it; and by the same two levelling-screws bring the bubble to the centre ; at the same time seeing that the bubble-tube appears, as nearly as may be, to be directly under the telescope, or over the bar. Then gently revolve the telescope a little to one side in its Ys, say about y inch, so that the bubble-tube shall stand out on that side, from over the centre of the bar, or from under the telescope. If the bubble then remains at the centre of the tube, the hor adjustment is correct; but if it runs toward one end, that end is too high ; and one-half of the error of the bubble must be corrected by the hor screws 1 1 ; raising the end that is lowest, or lowering that which is highest, as the case may require. This done, turn the telescope back again, until the bubble-tube is over the bar; and bring the bubble again to the center of its tube, by means of the levelling-screws K. Then again turn the telescope as before, so as to make the bubble-tube stand out from over the bar. As it is only by chance that this adjustment is perfected at the first trial, the bubble will most probably again leave the centre of the tube, but not as much as before. If so, again rectify half the error by means of the screws tt\ turn the telescope back again ; re-level the tube ; and so repeat until the bubble remains at the center, both when under the bar, and when standing out from it a short dist. This hor adjustment is usually ready made in new instruments ; and is not at all liable to derangement, except by accidental blows. To make the third adjustment, or to adjust the heights of the Ys, so as to make the line of colliraation parallel to the bar V F, or perp to the vert axis of the instrument. The other adjustments being made, fasten down the clips of the Ys. Make the instrument nearly level by means of all four of the levelling-screws K. Place the telescope over two of the levelling-screws which stand diagonally; and leave it there undamped. Then bring the bubble to the center of its tube, by the two levelling-screws. Swing the upper part of the instrument half-way around, so that the telescope shall ^gain stand over the same two screws; but end for end. This done, if the bubble leaves the center, bring it half-way back by the large cap- stan nuts w, w ; and the other half by the two levelling-screws. Remember that to raise the Y, and the end of the bubble over to, w, the lower w must be loosened ; and the upper one tightened ; and vice versa. Now place the telescope over the other diagonal pair of levelling-screws: and repeat the whole operation with them. Hav- ing completed it, again try with the first pair; and so keep on until the bubble re- mains at the center of its tube, in every position of the telescope. Correct levelling may -be performed even if all the foregoing adjustments are out of order; provided each fore-sight be taken at precisely the same distance frrnn the instrument as the back-sight is. But a good leveller will keep his instrument always in adjustment; and will test the adjustments at least once a day when at work. As much, however, depends upon the rodman, or target-man, as upon the leveller. A rod- man who is careless about holding the rod vert, or about reading the sights correctly, should be discharged without mercy. Forms for level note-books. When the distance is short, so as not to require two sets of books, the following is perhaps as good as any. it best, both with the level and with the transit, to consider the term " TATION to apply to the whole dist between two consecutive stakes; and that its number shall be that written on the last stake. Thus, with the transit, Station 6 means the dist stake 6; that it has a bearing or course of so and so; and its length nd with the level, Station 6 also means the dist from stake 5 to stake - from stake 5 to st the foregoing one as being perfectly simple, and free from liability to mistakes. It does not interfere with designating any stake by its number also, as stake No 6, &c, NOTE. The levelling-screws in manv instruments become very hard to turn if dirty. Clean with water and a tooth-brusn. Don't use oil on neld instruments. Heiier & Brightly's screws are pro- tected against dust. THE ENGINEER'S TRANSIT. 157 THE ENGINEER'S TRANSIT, 158 THE ENGINEER'S TRANSIT. THE details of the transit, like those of the level, are differently arranged by diff makers, and to suit particular purposes. We describe it in its modern form, as made by Heller and Brightly, of Philada. Without the lon bubble- tube F F, Fig 1, under the telescope, and the graduated arc g, it is their plain transit. W T ith these appendages, or rather with a graduated circle in place of the arc, it becomes virtually a Complete Theodolite.* B D D, Fig 2, is the tripod-head. The screw-threads at v receive the screw of a wooden tripod-head-cover when the instrument is out of use. S S A is the lower parallel plate. After the transit has been set very nearly over the center of a stake, the shifting-plate, dd ce, enables us, by slightly loosening the levelling-SCrews K, to shift the upper parts horizontally a trifle, and thus bring the plumb-bob exactly over the center with less trouble than by the usual method of pushing one or two of the legs further into the ground, or spread- ing them more or less. The screws, K, are then tightened, thereby pushing up- ward the upper parallel plate mmmxx, and with it the halt-ball 6, thus pressing c c tightly up against the under side of S. The plumb-line passes through the vert hole in 6. Screw-caps, / g, protect the levelling-screws from dust, &c. The feet, i, of the screws, work in loose sockets, .7, made flat at bottom, to preserve S from being indented. The parts thus far described are generally left attached to the legs at all times. Fig 1 shows the method of attachment. To set the upper parts upon the parallel plates. Place the lower end of U U in x x, holding the instrument so that the three blocks on m m (of which the one shown at F is movable) may enter the three corresponding * The price of a first-class plain transit with shifting-plate and plumb-bob, by Heller & Brightly, in 1882, is $185. One with vert arc g aud long bubble-tube F F, $220. THE ENGINEER'S TRANSIT. 159 recesses in a, thus allowing a to bear fully on m, upon which the upper parts then rest. (The inner end of the spring-catch, I, in the meantime enters a groove around U, just below a, and prevents the upper parts from falling off', if the in- strument is now carried over the shoulder.) Revolve the upper parts horizontally a trifle, in either direction, until they are stopped by the striking of a small lug on a against one of the blocks F. The recesses in a are now clear of the blocks. Tighten int Barom. above in deg ea level ndeg sea level n deg sea level ndeg sea level Fah Ins. Feet. Fah. Ins. Feet. Fah. Ins. Feet. Fah. Ins. Feet. 184 16.79 15221 .3 19.66 11083 .6 22.93 7048 .9 26-59 3164 .1 16.83 15159 .4 19.70 11029 .7 22.98 6991 206 26.64 3115 .2 16.86 15112 .5 19.74 10976 .8 23.02 6945 .1 26-69 3066 .3 16.90 15050 .6 19.78 10923 .9 23.07 6888 .2 26-75 3007 .4 1693 15003 .7 19.82 10870 199 23.11 6843 .3 26-80 2958 .5 1697 14941 .8 19.87 10804 .1 23.16 6786 .4 26.86 2899 .6 17.00 14895 .9 19.92 10738 .2 23.21 6729 .5 26.91 2850 .7 1704 14833 192 19.96 10685 .3 23.26 6673 .6 26.97 2792 .8 17.08 14772 .1 20.00 10633 .4 23.31 6617 .7 27.02 2743 .9 17.12 14710 .2 20.05 10567 .5 23.36 6560 .8 27.08 2685 185 17.16 14649 .3 20.10 10502 .6 23.40 6516 .9 27.13 2637 .1 17.20 14588 .4 20.14 10450 .7 2345 6460 207 27.18 2589 .2 17.23 14543 .5 20.18 10398 .8 23.49 6415 .1 27.23 2540 .3 17.27 14482 .6 20.22 10346 .9 23.54 6359 .2 27.29 2483 .4 17.31 14421 .7 20.27 10281 200 23.59 8M .3 27.34 2435 .5 17.35 14361 .8 20.31 10230 .1 23 64 6248 .4 27.40 2377 .6 17.38 14315 .9 20.35 10178 .2 23.69 6193 .5 27.45 2329 .7 17.42 14255 193 20.39 10127 .3 23.74 6137 .6 27.51 2272 .8 17.46 14195 .1 20.43 10075 .4 23.79 6082 .7 27.56 2224 .9 17.50 14135 .2 20.48 10011 .5 23.84 6027 .8 27.62 2167 186 17.54 14075 .3 20.53 9947 .6 23.89 5972 .9 27.67 2120 .1 17.58 14015 .4 20.57 9896 .7 23.94 5917 208 27.73 2063 .2 17.62 13956 .5 20.61 9845 .8 23.98 5874 .1 27.78 2016 .3 17.66 13896 .6 20.65 9794 .9 24.03 5819 .2 27.84 1959 .4 17.70 13837 .7 20.69 9743 201 24.08 5764 .3 27.89 1912 .5 17.74 13778 .8 20.73 9693 .1 24.13 5710 .4 27.95 1856 .6 17.78 13718 .9 20.77 9642 .2 24.18 5656 .5 28.00 1809 .7 17.82 13660 194 20.82 9579 .3 24.23 5602 .6 28.06 1753 .8 17.86 13601 .1 20 87 9516 .4 24.28 5547 .7 28.11 1706 .9 17.90 13542 .2 2091 9466 .5 24.33 5494 .8 28.17 1650 187 17.93 13498 .3 20.96 9403 .6 24.38 5440 .9 28.23 1595 .1 17.97 13440 .4 21 00 9353 .7 24.43 5386 209 28.29 1539 .2 18.00 13396 .5 21.05 9291 .8 24.48 5332 .1 28.35 1483 .3 18.04 13338 .6 21.09 9241 .9 24.53 5279 .2 28.40 1437 .4 18.08 13280 .7 21.14 9179 202 24.58 5225 .3 28.45 1391 .5 18.12 13222 .8 21.18 9130 .1 24.63 5172 .4 28.51 1336 .6 18.16 13164 .9 21.22 9080 .2 24.68 5119 .5 28.56 1290 .7 18.20 13106 195 21.26 9031 .3 24.73 5066 .6 28.62 1235 .8 18.24 13049 .1 21.31 8969 .4 24.78 5013 .7 28.67 1189 .9 18.28 12991 .2 21.35 8920 .5 24.83 4960 .8 28.73 1134, 188 18.32 12934 .3 21.40 8859 .6 24.88 4907 .9 28.79 1079 .1 18.36 12877 .4 21.44 8810 .7 24.93 4855 210 28.85 1025 .2 18.40 12820 .5 21.49 8749 .8 24.98 4802 .1 2b.9l 970 .3 18.44 12763 .6 21.53 8700 .9 25.03 4750 .2 28.97 916 .4 18.48 12706 .7 21.58 8639 203 25.08 4697 .3 29.03 862 .5 18.52 12649 .8 21.62 8590 .1 25.13 4645 .4 29.09 808 .6 18.56 12593 .9 21.67 8530 .2 25.18 4593 .5 29.15 754 .7 18.60 12536 196 21.71 8481 .3 25.23 4541 .6 29.20 709 .8 18.64 12480 .1 21.76 8421 .4 25.28 4489 .7 29.25 664 .9 18.68 12424 .2 21.81 8361 .5 25.33 4437 .8 29.31 610 189 18.72 12367 .3 21.86 8301 .6 25.38 4386 .9 29.36 565 .1 18.76 12311 .4 21.90 8253 .7 25.43 4334 211 29.42 512 .2 18.80 12256 .5 21.95 8193 .8 25.49 4272 .1 29.48 458 .3 18.84 12200 .6 2199 8145 .9 25.54 4221 .2 29.54 405 .4 18.88 1-2144 .7 22.04 8086 204 25.59 4169 .3 29.60 352 .5 18.92 12089 .8 22.08 8038 .1 25.64 4118 .4 29.65 308 .6 18.96 12033 .9 22.13 7979 .2 25.70 4057 .5 29.71 255 .7 19.00 11978 197 22.17 7932 .3 25.76 3996 .6 29.77 202 .8 19.04 11923 .1 22.22 7873 .4 25.81 3945 .7 29 83 149 .9 19.08 11868 .2 22.27 7814 .5 25.86 3894 .8 29.88 105 190 19.13 11799 .8 22.32 7755 .6 25.91 3844 .9 29.94 52 .1 19.17 11745 .4 22.36 7708 .7 25.96 3793 212 30.00 sealev=0 .2 19.21 11690 .5 22.41 7649 .8 26.01 3742 Below sea level. .3 19.25 11635 .6 22.45 7602 .9 26.06 3892 .1 30.06 52 .4 19.29 11581 .7 22.50 7544 205 26.11 3642 .2 30.12 104 .5 19.33 11527 .8 22.54 7498 .1 26.17 3582 .3 30.18 156 .6 19.37 11472 .9 22.59 7439 .2 26.22 3532 .4 30.24 209 .7 19.41 11418 198 22.64 7381 .3 26.28 3472 .5 30.30 261 .8 19.45 11364 .1 22.69 7324 .4 26.33 3422 .6 30.35 304 .9 19.49 11310 .2 22.74 7266 .5 26.38 8372 .7 30.41 356 191 19.54 11243 .3 22.79 7208 .6 26.43 3322 .8 30.47 408 .1 19.58 11190 .4 22.84 7151 .7 26.48 3273 .9 30.53 459 .2 19.62 11136 .5 22.89 7093 .8 26.54 3213 213 30.59 611 172 PENDULUMS. pl is DESCENT ON INCLINED PLANES,* For more on Inclined Planes, see p 484, &c. IT* ALL BELOW, THE RESISTANCE OF THE AIR IS OMITTED. To find the vel in ft per sec acquired in sliding down an inclined plane a 6. Divide the ver ht b c by the length a 5. The quot will be the nat sine of the angle a. Opposite this sine in the table of nat sines, takeout the angle a, and i* 8 nat cosine. Mult this cosine by the proper coefficient of sliding friction on p 699 '0 or 600, as the case may be. Take the prod from the nat sine. Mult the remainder by 32.2. Mult this last prod by twice the length a 6 in ft. Take the sq rt. If the prod is greater than the nat sine, body will not move down the plane. To find t.lie time of the descent in seconds. First by the foregoing find the acqd vel in ft per sec. Malt the length a 6 by 2. Divide the prod by the acqd vel. The acceleration of gravity on a body falling freely is 32.2 ft per sec; or in other words, a body falling freely from a state of rest, acquires a vel of 32.2 ft per sec for each sec of its descent. But on an inclined* plane this acceleration is less in the same proportion as the actual force down the plane (see pp 485, 486) is less than the wt of the body. Now the actual force down the plane is equal to the theoretical force down the plane, minus the friction of the body on the lane. But the theoretical force down the plane is as the nat sine of a; and the friction of the body s as its pressure on the plane, mult by the coef of friction ; or as the nat cos of a mult by the coef of friction. Hence the acceleration down the plane, or in other words the additional vel in ft per sec that the body acquires in each sec of its descent, becomes reduced to 82.2 X (nat sine of a (nat cos of a X coef of friction.) ) If twice the height in ft fallen freely, or twice the length a b of an inclined plane, be divided by the final acqd vel, the quot will be the time in sec. All the foregoing applies also closely enough to cars or wagons rolling down planes; except that the coef of combined axle and rolling friction, p 602, must then be used instead of that for sliding friction. But it does not apply to cylinders or spheres. The dist in ft to which a body would slide or roll on a liori- _ _ . _ Square of vel at starting, in ft per sec. P Coef of friction (p 599, 600, 602) X 64.4 and the time in sec, _ Twice the dist in ft. before coming to rest. ~~" Vel at starting, in ft per sec. Iii all the above the resistance of the air is omitted. To find the actual force in Ibs which tends to start a bo<3y clown a plane. Mult the wt of the body in Ibs by the nat sine of angle a. Call the prod p. Then mult the wt of the body by the nat cos of angle a ; and mult this last prod by the coef of friction. Take the result from p. The rem is the reqd force. For p is the moving force down the plane ; whereas wt X nat cos X coef is the retarding force of friction, and if it is greater than p, the body will not move down the plane. As the wt of the body, is to this actual force down the plane, so is 32.2 ft per sec, (the accel of a body falling freely,) to the actual accel down the plane, in ft per sec. See Gravity, p 587. PENDULUMS, THE numbers of vibrations which diff pendulums will make in any given place in a given time, are inversely as the square roots of their lengths; thus, if one of them is 4, 9, or 16 times as long as the other, its sq rt will be 2, 3, or 4 times as great ; but its number of vibrations will be but %, %, or % as great. The times in which diff pendulums will make a vibration, are directly as the sq rts of their lengths. Thus, if one be 4, 9, or 16 times as long as the other, its sq rt will be 2, 3, or 4 times as great ; and so also will be the time occupied in one of its vibrations. The length of a pendulum vibrating seconds at the level of the sea, in a vacuum, in the lat of London (51% North) is 39.1393 ins ; and in the lat of N. York (40% North) 39.1013 ins. At the equator about y 1 ^ inch shorter; and at the poles, about T x ff inch longer. Approximately enough for experiments which occupy but a few sec, we may at any place call the length of a seconds pendulum in the open air, 89 ins ; half sec, 9% iiis ; and may assume that long and short vibrations of the same pen- dulum are made in the same time ; which they actually are, very nearly. For meas- uring depths, or dists by sound, a sufficiently good sec pendulum may be made of a pebble (a small piece of metal is better) and a piece of thread, suspended from a common pin. The length of 39 ins should be measured from the centre of the pebble. * Text books generally give merely theoretical rules, which lead only to error in practice. SOUND. 173 In starting the vibrations, the pebble, or 6oft, must not be thrown into motion, but merely let drop, after extending the string at the proper height. To find the length of a pendulum reqd to make a given number of vibrations in a min, divide 375 by said reqd number. The square of the quot will be the length in ina, near enough for such temporary purposes as the foregoing. Thus, for a pendulum to make 100 vibrations per min, we have JJ g- = 3.75 ; and the square of 3.75 = 14.06 ins, the reqd length. To find the number of vibrations per min for a pendulum of given length, in ins, take the sq rt of said length, and div 375 by said sq rt. Thus, for a pendulum 14.06 ins long, the sq rt is 3.75 ; and ^ = 100, the reqd number. REM. 1. By practising before the sec pendulum of a clock, or one prepared as just stated, a person will soon leai n to count 5 in a sec, for a few sec in succession ; and will thus be able to divide a sec into 5 equal parts ; and this may at times be useful for very rough estimating when he has no pendulum. Centre of Oscillation and Percussion. REM. 2. When a pendulum, or any other suspended body, is vibrating or oscillating backward and forward, it is plain that those particles of it which are far from the point of suspension move faster than those which are near it. But there is always a certain point in the body, such that if all the particles were concentrated at it, so thut all should move with the same actual vel, neither the number of oscillations, nor their angular vel,* would be changed. This point is called the center of oscilla- tion. It is not the same as the cen of grav, and is always farther than it from the point of suspension. It is also the centre of percussion of the suspended vibrating body. The dist of this point from the point of susp is found thus : Suppose the body to be divided into many (the more the better) small parts ; the smaller the better. Find the weight of each part. Also find the cen of grav of each part ; also the dist from each such cen of grav to the point of susp. Square each of these dists, and mult each square by the wt of the corresponding small part of the body. Add the products together, and call their sump. Next mult the weight of the entire body by the dist of its cen of grav from the point of susp. Call the prod g. Divide p by^. This/? is the moment of inertia of the body, and if divided by the wtof the body the sq rt of the quotient will be the Radius of Gyration. SOUND. Its velocity, in quiet open air, has been experimentally determined to bo very approximately 1090 ft per sec, when the temp is at freezing point, or 32 Fah ; and that it increases about 1J^ ft per sec for every degree above 32, or, by Boiue authorities, 1 ft for every 2. Tiie first would make it 1090 ft at 32 1150 ft at 80 1100 " 40 1162% " 90 1112% " 500 1175 10 Q 1125 " 60 1187^ " 1100 1137% " 70 1200 " 120 At 32 sound would travel a mile in 4.84 sec ; at 80 e , in 4.6 ; and at 120, in 4.4 sec. If the air iscaltn, fogs, or rain do not appreciably affect the result; but winds do. There is son* reason to believe that very loud sounds travel somewhat faster than low ones. The watchword of sentinels has been heard across still water, on a calm night, 10% miles ; and a cannon 20 miles. Separate sounds, at intervals of -i of a sec, cannot be distinguished, but appear to be connected. The dists at which a speaker can be understood, in front, on one side, and behind him, are about as 4, 3, and 1. Dr. Charles M. Cresson informs the writer that, by repeated trials, he found that in a Philadelphia gas main 20 ins diam, and 16000 ft long, laid and covered in the earth, but empty of gas, and having one horizontal bend of 90, and of 40 ft radius, the sound of a pistol-shot travelled 16000 ft in pre- cisely 16 sec, or 1000 ft per sec. The arrival of the sound was barely audible ; but was rendered very apparent to the eye by its blowing off a diaphragm of tissue-paper placed over th end of the main. Two boats anchored some dist apart may serve as a base line for triang- ulating objects along the coast; the dist between them being first found by firing guns on board one of them. In water the vel is about 4708 ft per sec, or about 4 times that in air. In woods, it is from 10 to 16 times; and in metals, from 4 to 16 times greater than in air, according to some authorities. * For angular vel, see footnote page 447, " Force in Rigid Bodies." 174 STKENGTH OF MATERIALS. STKENGTH OF MATEEIALS, . 1. Endwise* average ultimate crashing loads per re inch, for wood, in pieces whose height does not exceed '2 or 3 times The strengths in all these tables may readily vary as much as one-third part more or less than our average. Weight per' cub. ft. Pounds per sq. inch. Weight per cub. ft. Pounds per sq. inch. Alder 6900 8600 7200 7700 9300 6000 11600 4500 6500 10000 5700 6500 6500 7000 5700 6500 6700 7300 7400 9900 10000 6500 6800 6000 4500 7300 3200 5500 will aver Mahogany Spanish . 8200 4200 6000 6500 9500 7700 6800 5300 5400 5400 7500 7500 3100 5100 3700 9300 7000 6500 6800 6000 12000 3200 5600 Ash ..... 43 to 63 51 53 43 Oak, Quebec, unseas.. " " seas 54 Bay wood . Beech, unseasoned u seasoned Birch Amer unseas . " English 58 " " well seas " Dantzic, very dry Pine pitch " seas English, unseas " " seas 41 " Ameryel'w, uns " * seas " red unseas Box dry Cedar, unseas 56 Pear Crab-tree green Poplar unseas. 48 47 " seas . Deal, red, unseas Plum wet " dry " v .. Sycamore or Planetree Spruce, or Fir, unseas " " " seas . . " " seas 43 37 37 <4 ' " " " Riga .. Elm seas Teak Larch unseas ..... " " " seas " Riga 6000 7200 2900 6100 Hornbeam, unseas " seas Larch green 47 50 Willow unseas " geas " dry age 5500 Ibs. ; hemloi fe, 4500. Seasoned Wh. Pine -K But sldewise the crushing strength Is far less. Thus Hatfleld found for ash only .57 of the endwise strength; live oak .67; oak, maple, hickory, .333; locust, black walnut, cherry, white oak .3; Georgia pine. Ohio pine, whitewood .25; chestnut .'2; spruce, white pine .125; hem- lock .111. He gives as safe equally distributed pressures, which will make but a slight impression, for spruce 250 Ibs per sq inch ; white pine and hemlock 300; Georgia pine 850 ; oak 9oO. But it is well to bear in mind that in practice perfectly equable pressure is rarely secured. The writer found that the resistance was greatest when the position of the annual layers as seen in a cross section of a beam was vertical. But in practice a knowledge of this fact is seldom of use, because in large beams the layers are generally disposed more or less in curves. In a few hur- ried trials on sidewise compression, with fairly seasoned white pine blocks, 6 ius high, 5 ins long, and 2 ins wide, we found that under an equally distributed pressure of 5000 Ibs total or 500 Ibs per sq Inch, they compressed about from ^ to # inch ; which is equal to from % to % inch per foot of height ; or from A to A- of the height ; the mean being about % inch to a foot, or 3*5 of the height. Under 10000 Ibs total, or 1000 Ibs per sq inch, they split badly; and in some cases large pieces flew off. See Bern, p. 193. STRENGTH OF MATERIALS. 175 practice \ve may take its safe strain at from 1000 to 2000 Ibs per sq inch, depending; upon the character of the structure, &c., without regard to the length, except when this is so great that two or more pieces have to be spliced together to make it; thus weakening the piece very much. It is seen from the table that seasoned woods resist crushing much better than green ones; in many cases, twice as well. This must be taken into consideration when building bridges, &c., of timber recently cut. Art. 2. Ultimate average crushing: loads in tons, per square foot, lor stones, &c. The stones are supposed to be ON BKD, and the heights of all to be from. 1.5 to 2 times the least side. Stones generally begin to crack or split under about one-half of their crushing loads. In practice, neither stone nor brickwork should be trusted with more than > to T ^th of the crushing load, ac- cording to circumstances. When thoroughly wet some absorbent sand- stones lose fully half their strength. See head of next page. 4 Tons per sq. ft. 300 to 1200 Mean. Tons. Tons per sq. ft. Mean. Tons. Granites and Syenites. Basalt 750 700 625 175 350 200 170 25 35 60 600 135 70 25 40 Cement, Portland, neat,U. S. or Foreign, 7 days in water Common U.S.cements, neat, 7 days in water CoiicreteofPort. cement, sand, and gravel or brok stone in the proper propor- tions.rammedlmold 6 months old 75 to 150 15 to 30 12 to 18 48 to 72 74 to 120 100 to 150 15 to 35 1300to2300 s that of g | 12 to 18 112.5 22.5 15 60 97 125 25 1800 ranite. 15 Limestones and Mar- blest 250 to 1000 100 to 250 150 to 550 Oolites good Sandstonesfitforbuild- ing Sandstone, red, of Con- necticut and N. Jer- Brick* 40 to 300 20 to 30 30 to 40 50 to 70 400 to 800 70 to 200 Brickwork, ordinary, cracks with 12 months old With good common hyd cements, abt .2 to .25 as much Coignct betoii, 3 months old Rubble masonry, mortar, rough Glass, green,crowu and flint . Brickwork, good, in ce- ment Brickwork, first-rate, in cement Slate Caen Stone " " to crack Chalk hard 20 to 30 Plaster of Paris, i day old or 3 time Ice, firm Crushing height of Brick and Stone. If we assume the wt of ordinary brickwork at 112 Ibs per cub ft, and that it would Crush under 30 tons per sq ft, then a vert uniform column of it 600 ft high, would -.rush at its base, under its own wt. Caen stone, weighing ISO ft>s per cub ft. would /equire a column 1376 ft high to crush it. Average sandstones at 145 fibs per cub ft, would require one of 4158 ft high ; and average granites, at 165 Jbs per ciib ft, one of 8145 feet. But stones begin to crack and splinter at about half their ultimate Crushing load ; and in practice it is not considered expedient to trust them with more than i^th to ^th part of it, especially in important works ; inasmuch as settle- ments, and Imperfect workmanship, often cause undue strains to be thrown on cer- tain parts. The Merchants' shot-tower at Baltimore is 246 ft high ; and its base sustains 6U . tons per sq ft. The base of the granite pier of Saltash bridge, (by Brunei,) of solid masonry to the height of 96 ft, and supporting the ends of two iron spans of 455 ft each, sustains y% tons per sq ft The base of a brick chimney at Glasgow, Scotland, 468 ft high, bears 9 tons per sq ft; and Professor liankine considers that in a high gale of wind, its leeward side may have to bear 15 tons. The highest pier of Rocque- favour stone aqueduct, Marseilles, is 305 ft, and sustains a pressure at base of 13}^ tons per sq ft. For greater pressures on arch stones, see p 342. fTrials at St. Louis bridge by order of Capt James B. Bads, C. E., showed that some magnesian limestone did not yield under less than 1100 tons per sq ft. A column 8 ins high and 2 ins diam shortened ^ inch under pressure ; and recovered when relieved. *Some whole bricks laid flat, crushed by Baldwin Latham, C. E., with 4 to 5 tons per sq inch ; after compressing tn about one- half of their original thickness! B. Latham's "Sanitary Engineering." Can this be correct} 176 STRENGTH OF MATERIALS. Sheet lead is sometimes placed at the .joints of stone col- umns, with a view to equalize the pressure, and thus increase the strength of the column. But experiments have proved that the effect is directly the reverse, and that the column is materially weakened thereby. Does this singular fact apply to cast iron and other materials ? Art. 3. Average crushing- load for Metals. It must be remembered that these are the loads for pieces but two or three times their least side in height. As the height increases, the crushing load diminishes. See " Strength of Pillars." p 221. The crushing load per sq inch, of any material, is frequently called its constant, coefficient, or modulus, of crushing or of com- pression. Pounds per sq. inch. Tons per sq. inch. Cast Iron, usually It is usually assumed at 100000 fts, or say 45 tons per sq inch. Its crushing strength is usually from 6 to 7 times as great as its tensile. Within its average elastic limit of about 15 tons per sq inch, average cast iron shortens about 1 part in 5555; or % inch in 58 ft under each ton per sq inch of load; or about twice as much as average wrought iron. Hence at 15 tons per sq inch it will shorten about 1 part in 370; or full % inch in 4 feet. Different cast irons may however vary 10 to 15 per ct either way from this. U. S. Ordnance, or gun metal ; Some 'Wrong-lit iron, within elastic limit , Its elastic limit under pressure averages about 13 tons per sq inch. It begins to shorten perceptibly under 8 to 10 tons, but recovers when the load is removed. With from 18 to 20 tons, itshortens permanently, about ^th part of its length ; and with from 27 to 30 tons, about y^g- tn part, as averages. The crushing weights therefore in the table are not those which absolutely mash wrought iron entirely out of shape, but merely those at which it yields too much for most practical build- ing purposes. About 4 tons per sq inch is considered its average safe load, in pieces not more than 10 diams long ; and will shorten it J^ inch in 30 ft. average. Brass, reduced T ^th part in length, by 51000; and %by Copper, (cast,) crumbles (wrought) reduced J^th part in length, by 85000 to 125000 38 to 56 175000 22400 to 35840 78.1 10 to 16 165000 ..117000 103000 15500 7350 .102050.. Tin, (cast,) reduced y^th in length, by 8800 ; and % by Lead, (cast,) reduced % of its length, by 7000 to 7700..*. " By writer. A piece 1 inch sq, 2 ins high , at 1200 Ibs the com- pression was 1-200 of the ht; at 2000, 1-29 ; at 3000, 1-8 ; at 5000, 1-3 ; at 7000, 1-2 of the ht. Spelter or Zinc, (cast.) By writer. A piece 1 inch square, 4 ins high, at 2000 fts was compressed 1-400 of its ht ; at 4000, 1-200 ; at 6000, 1-100; at 10000, 1-38 ; at 20000, 1-15 ; at 40000 yielded rapidly, and broke into pieces. Steel, 224000 Ibs or 100 tons shorten it from .2 to .4 part. " American. Black Diamond steel-works, Pittsburg, Penn. experiments by Lieut W. H. Shock. U. S. N., on pieces % in square; and 3}^ ins, or 7 sides long. u TJntempered. 100100 to 104000 " Heated to light cherrv red, then plunged into oil of 82 Fab, 173200 to 199200. . . .* i 11 Heated to light cherry red, then plunged into water of 79! Fah ; then tempered on a heated plate, 325400 to 340800.. .. 333100 " Heated to light cherry red, then plunged into water of 79 j Fah, 275600 to 400000 337800 Elastic limit, 15 to 27 tons 47040 " Compression, within elas limit averages abt 1 part in 13300, or .1 of an inch in 111 ft per ton per sq inch; or .1 of an inch in 5.3 ft under 21 tons per sq inch. Best steel knife edges, of large R E, weigh scales are considered safe with 7000 fts pres per lineal inch of edge ; and solid cylindrical steel rollers under bridges, and rolling on steel, safe with 1/diam in ins X 3 100 000, in Ibs per lineal inch of roller parallel to axis. And per the same, for 73.6 52.2 46.0 6.92 3.28 .186200., 150.8 21 Solid cast Iron wheels rolling on wrought iron, J/Diam ins X 352 000. " " " " " ' cast iron, J/Diam ins X 222 222. " " " steel, ]/ Diam ins XI 300 000. ' " " wrought iron, }/Diamins X 1024000. Solid steel ' " " " " " " cast iron, )/Diam ins X 850 000. From " Specifications for Iron Drawbridge at Milwaukee," by Don J. Wbittemore, C. E. STRENGTH OF MATERIALS. 177 Art. 4. Ultimate average tensile or cohesive strength of Timber, In fts per sq inch ; being the weights which, if attached to the lower end of a vert rod one inch square, firmly upheld at its upper end, would break it by tearing it apart. For large timbers we recommend to reduce these constants % to ^ part. The strengths in all these tables may readily be one-third part more or less than oar averages. Lbs per sq. inch. Lbs per sq. inch. Alder 14000 Mahogany Honduras 8000 Ash English 16000 16000 " American (author) abt Birch 16500 15000 Mangrove, white, Bermuda.... Mulberry 10000 12000 " Amcr'n black .. 7000 Oak Amer'n white "] 12000 " " basket JBeech English 11500 " " red Bamboo 6000 " Dantzic seasoned > 10000 Box 20000 Riga Cedar Bermuda . 7600 " English 9500 Chestnut . . 13000 Pear 10000 10000 Pine Amer'u white red 1 Cyprus 6000 and Pitch Memel Riga 3 10000 Elder 10000 Plane 11000 Elm 6000 Plum .... 11000 " Canada 13000 Poplar 7000 Fir or Spruce 10000 7000 10000 Spruce or Fir..... ... 10000 Hazel 18000 12000 Holly 16000 Teak 15000 20000 Walnut 8000 Hickory, Amer'n 11000 Yew 8000 Lignum Vitfle Amer'n 11000 23000 ^.m^ . flair 2300 Larch Scotch .. . 7000 ff p i ' 1800 Locust 18000 " " " Larch 900 to 1700 Maple 10000 " " Fir, & Pines 650 THESE ARE AVERAGES. The strengths vary much with the age of the tree ; the locality of its growth ; whether the piece is from the center, or from the outer por- tions of the tree; the degree of seasoning ; straightness of grain; knots, &c, Ac. Also, inasmuch as the constants are deduced from experiments with good specimens of small size, whereas large beams are almost invariably more or less defective from knots, crookedness of fibre, &c, it is advisable in practice to reduce these constants as recommended above. The modulus of Elasticity, and its ttse. within the limit of elasticity, uniform rod of given material lengthens or shortens equally under equal additions of load. If thw were also the case beyond said limit, it is plain that there would be some load which would stretch a uniform bar to twice its original length, or shorten it to zero or 0. And this load in Ibs, for a bar of one inch square cross section, is the mod of elas for the given material ; or is the E of authors on Strength of Materials. For example, a one-inch square bar of wrought iron will, within the limit of elas, stretch on an average about 1 part in 12000 of its length under each additional load of 2240 Ibs. Consequently, if the same rate of stretching continued beyond the limit of elas, it is evident that 12000X2240, or 26880000 Ibs, would stretch the bar to twice its original length. Hence these 26880000 Ibs are the mod of elas for average bar iron. And so with any other material. Hence the mod of elas is a load which bears the same proportion to the original length of a uniform bar, as the load which will produce any given amount of stretch, is to the length of said stretch. This fact facilitate* certain calculations ; thus, Load in Ibs reqd ^ Beqd stretch to produce a given I _ lp 1Dches v m< stretch within f ~ orig length * eli elas limit, in ins, j in inches cross MO in sq int Stretch in ins ~i produced by any I _ load within elas f ~ limit, J Load in Ibs X orig length in ins mod of elas X cross sec in sq ins See Table, p 632. E may a . ply auy load within its elas limit, and measure its deflection. Then as expressed by writers E is WP T- 4 A bd 3 . Which means Coef const, or Mod (Load * D fl)s + - 625 wt of clear 8 P ap of bean| ) X cube of span ins, E m fits per sq iucu 4 X Def, ius X breadth, ins X cube of depth, ins. 12 178 STRENGTH OF MATERIALS. Art. 5. Average ultimate tensile or cohesive strength of Metals, per square inch.* The ultimate tensile or pulling load per square inch of any material is frequently called its constant, coejjicient, or modulus of tensioH, or of tensile strength. Pounds Tons per per sq. inch. sq. in. Antimony, cast 1000 Bismuth, cast 3200 Brass, cast 8 to 13 tons, say 18000 to 29000 ibs 23500 " wire, unannealed or hard, 80000. Annealed 49000 Bronze, phosphor wire, hard, 150000. Annealed 63000 Copper, cast 18000 to 30000 24000 " sheet 30000 u bolts, 28000 to 38000 33000 " wire (annealed 16 tons); unannealed 60000 Gold, cast 20000 " wire, 25000 to 30000 27500 Gun metal of copper and tin, 23000 to 55000 39000 " cast iron, U. S. ordnance, 36000 to 40000 38000 Iron, cast, English 13400 to 22400 17900 " " ordinary pig.. 13000 to 16000 14500 American cast iron averages one-fourth more than the above. Average cast iron, when sound, stretches about .00018 ; or 1 part in 5555 of its length ; or % inch in 57.9 ft. for every ton of ten- sile strain per scinch, up to its elastic limit, which is at about Yz its break-strain. The extent of stretching, however, varies much with the quality of the iron ; as in wrought-iron. Cast, malleable, annealed 18 to 25 tons 48160 Iron, wrought, rolled bars, 40000 to 75000, the last exceptional... 57500 " ordinary average. See Rein p 375 44800 " good . " 50400 " superior 60000 " best American, (exceptional) 76100 " " Low Moor, English, average 60000 " plates for boilers, &c, 40000 to 60000 50000 " English rivet iron 55000 to 60000 57500 ** wire, annealed 80000 to 60000 45000 " unannealed, or hard. ..50000 to 100000 76000 " ropes, per sq inch of section of rope 38000 " large forgings, 30000 to 40000 35000 In important practice, good bar iron should not be trusted per- manently with more than about 5 tons per sq inch ; which will stretch it about % inch in from 20 to 25 ft. Good bar iron stretches about 1 part in 12000 of its length ; or about 1 inch in 1000 ft ; or */ inch in 125 ft, for every ton of tensile strain per sq inch of section, up to its elastic limit. This limit usually ranges between 8 and 13 tons per sq inch, or about half the breaking strain, according to quality. The, ultimate stretching of rolled bars is from 5 to 30 per ct of the original length ; usually 15 to 20 per cent. Plates and angle iron 3 to 17 per cent. Heating, even up to 500 Fah, does not weaken bar iron or steel. For stretch by heat see p 310. Lead, cast, 1700 to 2400 by author... 2050 " wire, 1200 to 1600. Pipe 1600 to 1700 " " ... 1650 Platinum wire, annealed, 32000. Unannealed 56000 Steel, plates, range, 60000 to 103000 81500 of Hussey, Wells & Co, Pittsburg, Pa, 91500 to 97400 94450 Bessemer 98600 Bessemer tool 112000 " wire, annealed 30 to 50 tons. Unan, 50 to 90 tons 156800 *Ijf*,rge hars of metal bear less per sq inch than small ones. In cast iron ones 1, 2 and 3 ins sq, the strengths per sq inch were about as 1, .85 and .66 ; and wrought iron prob- ably averages about the same. Iron bars re-rolled cold have tensile strength increased 25 to 50 per ct, with no increase of density. They are said to lose this strength if reheated. STRENGTH OF MATERIALS. 179 Average ultimate tensile, or cohesive strength of Metals per square inch. (CONTINUED.) Capt. James B. Eads foand that forged steel bolts for the St. Louis bridge, 59$ ins diaiu, and 22 to 36 ft. long, broke short with only 30000 tbs per sq inch f while bolts of but % inch section, cut from the large ones, in no instance broke with 8C 1 > mcn - 100000 fts per sq inch, but stretched considerably. Steel, cast, Bessemer ingots, average 63000 " " beat American Bessemer ingots 86600 " " " rolled arid hammered, 120000 to 130000 125000 " " homogeneous, Cammell & Co, England, No 1... 68240 No 2 716SO " " " " " No 3 76160 " puddled bars, rolled and hammered, 65000 to 135000 100000 Steel. Experiments by Lieut. W. S. Shock, U. S N., at Washington, on steel from the Black Diamond Steel- Works, Pittsburg, Pa All the pieces were cut from the same bar, three pieces for each exp. They were turned down to a diani of .62 of an inch at the intended point of fracture, by a groove, in shape of a circular segment, with a chord of about I inch " the bar in its original condition, 109500 to 131900 120700 " heated to light cherry -red, then plunged into oil of 82 Fah, 201 300 to 227500 214400 " heated to light cherry-red, then plunged into water of 79 Fah. Then tempered on aheated plate, 152500 to 176100.. 161300 heated to light cherry-red, then plunged into water of 79 Fah, 132700 to 150500 141600 Tempering in oil usually increases the strength from 40 to 80 per cent. " chrome, made at Brooklyn, N. Y., and tested at West Point Foundry, N.Y , (specific gr 7.816 to 7.956,) 163000 to 199000. Average of 12 specimens 180000 " made from very pure Swedish iron, but containing differ- ent proportions of carbon. The bars were 21% ins long, with 14 ins of this length turned down to a uniform di- am of 1 inch. The breaking wts, however, in the table, are per sq inch : Mark No. 2, carbon .33 per ct, stretched 1.37 ins 68100 " No. 4 " .43 " " 1.37 " 7*5160 " No. 5 " .48 " " 1.25 " 84000 " No. 6 " .53 " " 1.12 " 95200 " No. 7 " .53 " " 0.81 " 92960 " No. 8 " .63 " " 1.00 " 100800 " No. 10 " .74 " " 0.69 " 101920 " No. 12 " .84 " " 1.12 " 123200 " No. 15 1.00 " " 1.00 " 134400 " No. 20 1.25 " " 62 " 154560 With more than about 1.5 per ct of carbon the tensile strength of steel diminishes. A bar of the above No. 15, which broke at 60 tons per sq inch, when turned down for 14 ins of its length ; broke with 79% tons per sq inch when turned down at one point only. This is owing to the fact that the last could not stretch as much as the first, and therefore its diam could not be diminished as much before breaking. All its fibres pulled more unitedly. It will be observed that the steel of greatest strength stretched the least before breaking. This stronger steel would break under a suddenly applied force, or impulse, more easily than a weaker one would ; because the weaker one, by its stretching, gradually breaks the force of the impulse, on the same principle as a spring. Hence the steel, iron, &c, which is strongest against a gradually applied force or strain, may be unfit for uses where the strain comes upon it suddenly. The average ultimate tensile strength of steel is about twice that of wrought iron. Its deflection as a beam within the elastic limits is about $ that of wrought, or % that of cast iron. Its average stretch Pounds 180 STRENGTH OF MATERIALS. is about .1 inch in 111 ft for every ton per sq inch of load, up to its elastic limit, which generally ranges at between % and % of its breaking strength ; the latter being for the harder, stronger, and less stretchy kinds. A uniform bar of rolled steel, gradually loaded, will stretch from -*- to 1- of its length before breaking; or from y^ of an inch to 2.4 ms per foot, according to quality. The mean of these is nearly y 1 ^ of tbe length, or ly^iuch to a foot. When steel, especially if hard, has to be heated to softness in order to give it a required shape, it is thereby weakened. Average ultimate tensile or cohesive strength of Metals per square inch. (CONTINUED.) Pounds Tons per per sq. inch. sq. in. Silver, cast 41000 18 3 Tin, English block ... 4600 2 " wire 7000 3 1 Zinc, cast. ..3000 to 3700 ; (the last by author) 3350 1.5 Art. 6. Average ultimate tensile or cohesive strength of various materials. The strengths in all these tables may readily be one-third part more or less than our averages. Pounds per sq. inch. Tons per sq. ft. Pounds per sq. inch. Tons per sq. ft. Brick 40 to 400 220 141 Marble strong wh Italy * 1034 665 Caen stone, 100 to 200 Cement, hydraulic, Port- 150 9.7 " Champlain, varie- gated * 1666 1071 land, pure, 7 days in water . 300 193 " Glenn's Flls.N.Y, blk * 75()tol034 892 57 4 " 6 months old 450 289 " Montg'y co Pa " 1 year old 550 354 1175 75 6 Common hyd cements average 1-6 as much. The last, neat, adhere to brick and stone with from 15 to 50 Ibs when only 1 month old 32 2 " white*... ** Lee,Mass,white.* " Manchester, Yt,* 550 to 800 " Tennessee, varie- gated* 734 875 675 1034~ 47.2 56.3 43.4 66.5 At end of 1 year, 3 Oolites 100 to 200 150 97 times as much See " Cement," p 506, &c Glass, 2500 to 9000 (p 515) 96 5750 6 385.7 Plaster of Paris, well set. Rope, Manilla, best " hemp best 70 12000 15000 4.5 771 965 Glue holds wood together Sandstone Ohio* 105 6 75 with from 300 to 800... Horn ox 550 9000 35 579 " Pictou, N. S.* " Conn red * 434 590 27.9 37 9 16000 1029 Slate Lehigh * 2475 159 1 Leather belts, 1500 to 5000 Good 3000 193 " Peach bot'm,* 3025 to 4600 r>81 '> 245 1 Mortar, common, 6 mos old, 10 to 20 15 .96 Stone, Ransome's artif.... Whalebone 300 7600 19.3 489 To find the diam in ins of a round rod to bear safely a given pull in fibs. Diam in ins /given pull X coef of = V / ult tensile strength V of material in Ibs pei safetv ,7854. er sq inch ' Iron is weakened by extreme cold. The belief (originating with Styff of Sweden,) is gaining ground that iron and steel are not rendered more brittle by intense cold, but that the great number of * By the author's trials with one of ftiehle's testing machines. Sections broken 1% sq inches. STRENGTH OP MATERIALS. 181 breakages of rails, wheels, axles, s; white pine and hemlock 2500; yellow pine 4300 to 5600 ; white oak 4400. For others see Shearing, Art 2, p 642. Wrought iron is stated at 35000 to 55000 fts per sq inch; cast iron 20000 to 30000; steel 45000 to 75000 fbs; copper 33000. The shearing strength of steel and wrought iron is about % part less than the tensile. The punching of rivet-holes in iron or steel plates, is an example of shearing. The rivets in J tiy tubular bridges are frequently sheared in two, in time, by the motion of the platei through which they are driven. In punching holes, the area of section is evidently found by mult the circumf of the hole by the thickness of the plate in which it ia punched. If a piece of material be supported as shown in Fig -'% its resistance to shearing will be 3 times as great as in Fig 1, where it is sheared across in 2 places only; whereas in Fig 2%, shearing would have to occur at 6 places, as per the 6 dotted lines. Art. 8. Breaking by torsion, or twisting. Let n, Fig 3, be a vert cylindrical rod of any material, 1 inch diam, the lower end of which is immovably fixed ; and let c be a lever L whose leverage (see levers) a b, measd from the axis of the cylindrical rod, is I ft. Suppose that with a spring balance attached to the end b of the lever, we apply force horizontally, and around the axis of the rod as a center, until the rod breaks by being twisted. Then if we mult together the leverage n b in feet, and the amount of force shown by the spring balance in Ibs, and div the prod by the cube of the diam of the rod in ins the quot will be a certain number of foot-pounds ; and will be what is called the constant, or coefficient for torsion, for all cylindrical bars of that material. If we use a square bar, we shall get the coef tor square bars ; and so with any other shape. So that if with any other bar, or shaft, we mult the cube of its diam' in inches by t 182 STRENGTH OF MATERIALS. said constant, and div the prod by the leverage in feet, the quot will be the force in Ibs which will twist the bar in two. In shape of formulas, M^MT ^W^X Constant Brea* g = Constant. And - : = . lor ^ e Also Cube ot diam in ins Leverage in feet m Ibs. Cube of diam ,., ,-. c Leverage v Brkg force __m_ins _ Leverage in feet ^ infos _ Cube of diam Brkg force in ft>s ~ in feet " Constant in ins - Hence we see that the torsional resistance of a cylinder increases directly as the cube of the diam, or cube of one side if square; and diminishes as either the leverage or the force increases. The constant for solid cylinders of average cast iron is about 600; and for wrought iron 800. For puddled steel about 700; cast steel 1000 to 1700. Wrought copper 400. All may vary one fourth part of these more or less. For woods, rough averages. W pine or spruce 20 to 25 ft-lbs. Y pine 35. Ash 40. W oak 50. Locust 75. Hickory 85. To find, by the last formula, the diam of a rod to have a safety of 3, 4, 5, Ac, against a given twisting force in Ibs, first mult said force by 3, 4, 5, Ac, as the case may be, and use the prod as the breaking force. The diam will then be .the safe one. Any angle described by the force at b, when made to revolve around the axis of the rod as a center, during the twisting process, is called the ANGLE OF TORSION. The length of the twisted rod or shaft does not affect the amount of force reqd to produce rupture ; but the longer it is, the greater will be the angle of torsion ; or in other words, the greater will be the dist through which the force must revolve around the axis before fracture takes place. Authorities say that a working shaft should not twist more than 1. We should not expose it to more than .1 of its ult strain. \ If we know the force in ibs per so. inch reqd for shearing any material, see pre- ceding Art, then the force required to break a cylinder of it by torsion, is Tnrsinn One-half the shearing ^ Q \4-ta \/ Cube of rad of force = force in ibs per sq inch X3 ' 141 X cylinder in ins in ibs Leverage in inches. That of a square shaft is about 1^ times that of a round one whose diam is equal to a side of the square ; or about 4- less than that of a round one of the same transverse area. For any solid rectangular shaft . One-third of the shearing v The square of v The square of Breaking force, in fts per so in A one side A the other side. Torsional force = * . ., m T j n jk s Square root of the sum of the ^ Leverage above two squares * in inches. Hollow shafts resist torsion better than solid ones of the same area of metal. Calling the outer and inner diams in ins D and d, then Breaking (D4 _ d t) x Constant Torsional force = \= : -, ,, _.- in Ibs Leverage in ft X D Strength of wrought -iron shafting. The shafting used for the transmission of power to the diff parts of machine-shops, many manufacturing es- tablishments, &c, is subjected to twisting strains. It is usually made cylindrical, and of wrought iron. Experience shows that we may safely use the following for shafts of iron of good quality, bearing but little weight, and well supported at proper intervals, say 8 or 9 ft., by self-adjusting ball and socket hangers. / Horse-powers \/ Number of rev per minute Diam of a wrought _ . / . Horse-power^ iron Shaft in ins. ~ \ / Niimhor nf rvs A A^O. pei Or in words : for the diameter in inches div the number of horse-powers that are to be transmitted along the shaft, by the number of revs which the shaft is reqd to make per min. Mult the quot by 125. Take the cube root of the product. This cube root will be the diameter itself, at the thinnest part, at its bearings. The last formula shows that the faster a shaft revolves under the same number of horse-powers, the less is the torsional strain upon it. This may at first seem strange, but less so when we reflect that a horse-power is made up of pres and dist; therefore, the faster it moves, the less is its pressure. Hence many horse-powers re- STRENGTH OP MATERIALS. 183 i volving rapidly will require a less diam than a small number revolving slower in proportion than its number. Art. 9. Transverse (or across) strength of mate- rials ; or that by which they re- sist breaking, when employed as beams ; as for instance in Fig 4. To find constants. In beams of the same material, and exactly alike, except in their breadths, n d, the strengths vary in the same proportion as those breadths ; that is, if one is 2, 3, or 10 times broader than the other, its strength will be 2, 3, or 10 times as great. If they are alike, except in their clear lengths or spans, a a, between the points of sup- port, their strengths will be inversely as those lengths; that is, if one is 2, 3, or 10 times longer than the other, it will be but %, %, or -J^ part as strong. If they are alike, except in point of depth, o d, measured vert, their strengths will be directly as the squares of their depths; that is, if one is 2, 3, or 10 times as deep as the other, it will also be 4, 9, or 100 times as strong ; or in other words, will require 4, 9, or 100 times as great a load to break it. See Art. 11. It must be remembered that we are now speaking only of strength, or resistance to breaking ; and not of stiffness, or resist- ance to bending, or deflecting. Stiffness follows laws very diff from those of strength. See Art 26, &c. Now, if we combine all the three foregoing elements of size, namely, length, breadth, and depth, we have the fact, that the strength of any beam, of any size, of breadth X the square of its depth. any given material, is in proportion to its - There- its length fore, if WB find by actual trial, what center load will break any beam of known size ; breadth X sq of depth and then find what is the proportion between its : , and its length breakg load, said proportion (or, more strictly speaking, ratio) will also be that which any similar beam has to its breakg load, and will therefore serve to calcu- late the breakg load of any other similar beam of the same material. For instance, if we take any piece of average good white pine, say 6 ins broad, 10 ins deep, and 12 breadth X sq of its depth . 6 X 100 ,feet clear span, we find that its is equal to = 50. length 12 And if we gradually load this at its center until it breaks, we shall find that the breakg load, including half the wt of the clear span of the beam itself, amounts to breadth y sq of depth 22500 Ibs. Therefore, the proportion between the in ins A in ins an a the length in feet breakg load, is as 50 to 22500 ; which is the same as I to 450 ; that is, the breakg load of the beam, including half its own weight, may be found by mult its breadth y sq of depth . * n in3 in ins by 450. And in this same manner may be found the total length in feet center breakg load of any rectangular beam of average quality of white pine. For the neat load one-half the wt of the clear span must be deducted. It is self-evident that the weight of the beam assists to break it, as well as the neat load; and the extent to which it does so, is the same as if one-half of its unsupported wt were concentrated at its center. Hence the rule. On this principle the rule in Art 12 is based. The ratio thus found for any material, is called its coef for ceii breakg loads, or its constant for the same, as it does not vary with the size of the beam. If we take a piece, all of whose dimensions are 1, as 1 breadth v sq of depth inch wide, 1 inch deep and 1 ft span; then the in ins x in ins will be 1 X sq of 1 1 length in feet = 1 ; and the breakg load (including half its own weight) of such a piece is, at once, the constant reqd. See Remarks 1 to 4. In an average piece of white pine, this load will be found to be about 450 Ibs ; or the same as the constant obtained from the large beam. Ho the student may find them for himself; and if he uses materials not included in our table, Art 10, it will be well to supply the deficiency, by inserting his own results. The foregoing directions for finding coefficients may be more briefly expressed 184 STRENGTH OF MATERIALS. by a form., one-half th mla. After finding the neat center breakg load, by experiment, add to It ;he wt of the clear span of the beam, for a total center load; then the Span v Total load Coef for breakg 1 = in j eet __l in lbs strength j Breadth v Square of depth in ins A in inches. Ill a cylinder, as the breadth and the depth are each equal to the diain, it is plain that B X 1^*, amounts to the same thing as diam 3 ; and it is always so exjressed. See Remark, Art 29. p 201. HEM. 1. The variation in strength of equal beams of the same material is so great, that it is necessary to experiment with several pieces, in order to find an average for a constant. The loads given in the preceding tables are also constants, bnt for crush- ing and tension. They are averages of the strengths of the materials, derived from experiment. The actual strength of any particular specimen, if of superior quality, may be considerably greater than the average ; or on the other hand, if of very poor quality, it may fall as much below it. We should always keep this in mind when referring to any table of constants; and if we have doubts as to the quality of the piece of material which we are about to employ, we should make a corresponding deduction from the constant in the table. Reni. 2. If, instead of pine, we had experimented with oak, iron, stone, the process for finding the constant would have been precisely the same. If in- stead of a square beam, we use cylindrical, or triangular ones, or any other shape, such as hollow cylinders, H, T, or U beams, &c, we shall in the same way establish constants for either larger or smaller beams of those shapes, and of precisely the same proportions in every part. See Remark, A rt 29. Or if, iusteadof supjport- ing the beam at both ends, we secure it firmly at one end, and load it at the other end until it breaks, we shall obtain the constant for beams fixed tit one end, and loaded at the other, &c. Remember that the constants are for loads at rest. If they are liable to jars, jolts, vibrations, &c, a large margin must be left for safety. Moreover, the constants given in tables are generally deduced from small specimens free from important defects ; whereas large beams of any kind of material usually contain irregularities, which diminish their strength; and on this account larger allowances for safety should be made as the dimensions of the beam increase. Rein. 3. It is not necessary that & and d be taken in ins, and lengths in ft. They may all be in ins, ft, yds, or any other measure, but since in every-day practice we usually speak of the breadths and depths of beams in ins, and of their lengths in ft, it becomes more convenient so to consider them. If other meas- ures be used, the constant will of course be diff ; but it will still be such that if the same measure be used for calculating the strength of another beam, the final result will be the same as before. In like manner, the loads may all be taken in tons, &c, instead of tbs ; but in giving the rule, it must be stated what measures have been employed. See Remark, Art 29. Rein. 4. There are peculiarities in some materials, which les- sen the reliability of constants derived from experimenting with small pieces. Thus, a large beam of cast iron will break with a less load in proportion than a small one ; because, in the interior of thick masses of that material, more time to cool is required than in the outer surfaces ; in consequence of which, there is a want of uni- formity in the arrangement of the particles of iron, and this conduces to w.eakness. All we can do in such cases, is to exercise judgment and caution in making sufficient allowance for safety. Art. 1O. Table of constants or coefficients for the quiescent breaking? loads of rectangular beams, supported horizon- tally at both ends, and loaded at the center; being the average qui- escent breaking loads in fibs (including one-half the weight of the beams themselves) for beams 1 inch square, and 1 foot clear length between the supports. For safety in practice, not more than about % to % of these constants should be employed ; de- pending upon the importance of the structure, its temporary or permanent charac- ter, and the degree of vibration to which it will be exposed. Thus a roof will prob- ably be as safe at *^, as a bridge at %. Even with a perfectly safe load, a beam may bend too much. See Art -6. p 196. If any of these coefficients be mult by .589 (or say .6) it will give that for a cylin- drical beam whose diam = side of the square. Or if mult by .71 it will give that for a square beam with its diagonal vertical. Any of these constants may vary one-third part either more or less. For any beam, Ceil. Breaks _ Breadth (ins) X Square of depth (ins) Congtant load in lbs. clear span in feet. STRENGTH OF MATERIALS. 185 One Third part of any of these constants (except those for wrought iron and steel), may be taken in ordinary practice as about the average constant for the greatest center load within t3ie elastic limit. The loads here given for wrought iron and steel, are already the greatest within elastic limits. See p 198. The modulus, coef, or constant of rupture or frac- ture of writers on physics is 18 times the loads in this table. See p 195. 1 Transverse Strength. Lbs. Transverse Strength. Lbs. WOODS. Ash English . ... M 650 but at about the average of 2250 Ibs its elas limit is reached " ' Amer White (Author). P " Swamp ^ 650 400 Steel, hammered or rolled ; elas destroyed by 3000 to 7000 5000 Black 300 Under heavy loads hard steel Arbor Vitse Amer O 250 snaps like cast iron and soft Balsam Canada 8* 350 steel bends like wrought iron Beech English M 500 " ' Amer White "* 450 " Amer Red <: ^ 550 STONES, ETC. Birch Amer Black . 450 Blue stone flagging, Hudson River 125 " Amer Yellow ^ 3 450 Brick, common, 10 to 30. .average 20 Cedar Bermuda . . y? S 400 " good Amer pressed 30 to 600 50 average 40 " Amer White 1 ^" P Caen Ston? 25 or Arbor Vit.'."! J -""S Chestnut ** o 250 450 Cement, Hydraulic, English Port- land artificial, Elm English - 350 7 days in water 30 *' Hock Canada . . o> * 600 1 year in water 50 Hemlock .' (by Auth.)*g- Hickory, Amer., " " . ^ g- " " Bitter nut |-S Iron Wood Canada * 400 700 500 600 " " Portland, King- ston, N. Y., 7 days in water. " " Saylor's Port., 7 30 Locust ^ ss 600 days in water. 26 Lignum Vitss * ' 650 " " Common TJ. S. Larch * 400 cements 7 dys Mahogany \ 450 in water 5 650 The followin^ hydraulic ce- " ' Black " 550 ments were made into prisms, in Maple Black . ?~ 550 vertical moulds under a pressure " Soft 550 of 32 fts per sq inch and were Oak English 550 kept in sea water for 1 year " Amer White (by Author). " " Red, Black, Basket... " Live 600 550 600 Portland Cement, English, pure, 1 year old... Roman Cement Scotch pure .... 64 23 Pine, Amer White... ^l>y Author) " " Yellow " " 450 500 American Cements, pure, av about (jrranite 50 to 150 average 25 100 u < pitch " " 550 " Quincy 100 " Memel 450 Glass Millville N Jersey thick Poplar 550 flooring (by Author). 170 P p 185 _ 100X15X45 = 67500 = ^ ^ . the fereakg Clear length in ft 12 ' 12 load reqd. But we must deduct one-half the vrt of the clear length of tbe beam. Now abeam of red sandstone, of 15 ins, by 10 ins, by 12 ft, contains 21600 cub ins - 12% cub tt ; and a cub ft of red sandstone weighs about 140 fts ; therefore the beam weighs 12.5 X 140 = 1750 fts ; one-half of which, or 875 fos, must be taken from the 6626 fts of breakg load, leaving 4750 Ibs as the actual extraneous, or neat breakg load. Rent. 1. If cylindrical, first find the breakg load of a square beam, of which each side is equal to the diam of the cylinder, Mult this load by the dec .6, or more correctly, .589. Hence a square one is 1.7 times as strong as a cylindrical one. See table, p 207. STRENGTH OP MATERIALS. 187 If oval, or elliptic, first find the load for a rectangular beam, whose sides are respectively equal to the two diams, and mult it by .6. If of wood, and triangular, and its base (whether up or down) hor, first find the breakg load for a rectangular beam, whose breadth is equal to the base ; and its depth equal to the perp height of the triangle ; and take one-third of the result as an approximation. When the edge is down, tbe ends must rest in triangular notches in the supports; otherwise, they will be crushed when loaded. For beams of such sections as A to G, the following rude rules of thumb will often be preferred to more intricate ones, being sufficiently approxi- mate for ordinary purposes, and for any material. See near end of Art 24, p 193. For tbe closed Figs A, B, JD, O, (each one supposed to be of equal thickness throughout,) first find the load for a solid beam of the same size and shape. Then find that of a beam of the size of the hollow part. Subtract the last from the first. Take % of the remainder. For C, (its top and bottom being of equal size,) first find for a rectangular beam a a a a. Then for two beams corresponding to the two hollows v v. Subtract these last from the first. Take % of the remainder. For E or F, find for three separate beams r r, i i, t n, and add them together. Take %. T, U, and many other forms of beams in common use, do not admit of any such simple approximate rules. The writer knows of no alternative in such case* except to experiment with a model made of the given material, and thus find the necessary constant, as directed in Art 9. See Remark, Art 29; also Art 24, &c. For I beams, see Arts 37, 38; and for Hodgkiiison beams, see Art SIDE TL. Fi65 35.p 208. Rein. 2. In this case we may remove ^ part of the material of the solid square, or rectangular beam, without diminishing its breaking strength, although it will bend rj^ BL J more. The width may remain uniform, and the depth be re- -; duced either at top or bottom* as shown by the dotted lines |p " 11 at m, Fig 5, strictly two parabolas with bases at load, but the &l- ' u r - straight ones are best in practice. Or the depth may remain uniform, and the breadth be reduced, as shown by the dots at n, which is a top view of the beam. Theoretically, the dotted _ lines in n might meet at the ends of the beam ; but in prac- tice this would not generally leave sufficient material at the ends for the beam to rest upon securely. Such reductions of beams are rarely made when they are of wood ; but in iron ones much expense may be saved thereby. Rem. 3. Load at one end of the beam, Fig 6. the other end fixed, imagine the load to be at the center, and calculate it by the foregoing rule. Then div the result by 4. In this case the lower side of the beam may be cut away in the form of a parabola, as shown by the dots. To draw this curve, see " Parabola." Or the depth may be left uni- form, and the sides be cut away, as shown by the dots at , which is a top view. Art. 13. When the load is equally distributed along? the entire clear length of a horizontal beam. lMI^.!^l^i%IMi^l supported at both ends, as in Fig 7, instead of be- [ i^i^i^u^i^ so__ ing all applied at the center, assume it to be at the center, ..,,.. K' Vj^ and proceed precisely as in the foregoing rule, Art 12. "vty\ -p. - ri W^ Then mult the load by '2. But in this case the wt of the Jt 10 / If entire, clear length of the beam is to be deducted for the ^ neat load. Ex. What will be the equally distributed breakg load of the beam of sandstone In the last example ? Here the center breakg load has already been found to be 188 STRENGTH OF MATERIALS. 6625 Ibs ; and 5625 X 2 = 11250 Ibs, the reqd distributed load. From this snbtrao the wt of the entire 12 feet clear length of beam, or 1750 Ibs ; and the rem,9500 1U is the neat extraneous breakg load. About -J^- part of this, or 950 Ibs, is quite at, much as should be trusted upon so variable and treacherous a material as red sand, stone. REM. 1. A beam requires twice as much breaking load, equally distributed, as it will at its center. In this case the breakg strength of the beam will not be diminished if the top be cut away in the form of a true semi-ellipse, as shown by the dots in Fig 7. Or if the depth must be kept uniform, the sides may be trimmed to two parabolas oc o, o to, Fig 8. The mode of drawing these figs to any span and height will be found under their respective heads; but in practice circular segments will answer. Hem. 2. Load uniformly distributed ALONG THE ENTIRE CLEAR LENGTH, y g, Fig 9, OF A HOR RECTANGULAR BEAM, FIRMLY FIXED AT ONE END ONLY, assume it to be at the center, as in Art 12, and calculate it by the rule in that Art. Then div the result by 2. In this case, theoretically, we may cut off one-half the pro- jecting part y o of the beam, as by the dotted line y o, without diminishing its breakg strength. But in practice it w T ill rarely be advisable to reduce it to a mere thin edge at o. Or the depth c s of the beam may be left uniform, and the sides be cut away, as shown by the two semi-parabolas a c, a c, at t, wftich is the top of the beam. If a c', ac, be even made straight, instead of parabolas, it is plain that there would still be a considerable saving of expense, if the beam is of iron. Art. 14. When the entire breakg 1 load is applied at any point o, Fig 1O, JFt|| not at tBie center; first find by Art 12, what Silt would be the center breakg load ; and deduct half the wt of the beam. Then mult together the two dists o a and o g, in feet. Call the prod a. Also square half the clear length, c g or c a. Call this square b. Then mult the center load by 6, and div the prod by a; the quot will be the reqd load. Such rules do not hold good if the load rests upon the beam for a short distance on either or both sides of the point o, but only when it all rests at that very point alone ; if it does not the load may be increased. REM. 1. The beam will bear less at its center than at any other point; and the breakg load at the center, is to that at any other point, inversely as the square of half the length is to the rectangle of the two parts o a and o g. REM. 2. This beam will bear to be reduced, as at m and n, Fig 5 ; except, that instead of reducing from its center, as in Fig 5, we must do so from where the load is applied. Art. 15. When the beam, instead of being hor, is inclined, as in Fig 11, in any of the foregoing cases, the hor dist o y must be taken as its span, instead of the actual clear length o c ; and s o, s y instead of a o and a c. This applies also to beams fixed at one end, and whether the inclination is upward or downward from the fixed end. NOTE. The quantity of material In inclined beams may be reduced, in the same manner as in hor ones. Art. 16. Triangular beams of wood, according to Barlow's experi- ments with pine, require about % greater breakg loads with the base up, than when it is down. Or with the base down, about ^ less than when up. Tredgold considers them about equally strong in either position ; and that to find the center breakg load, we may first calculate it by Art 12, as if the beam were a rectangular one witk the same base and perp height as the triangle ; and take ^of the result. Hence, the triangle is not an economical shape for a beam; for with only 3^3 the strength of a rectangular one, it has half as much material. Hodgkinson, with cast-iron triangular beams, base up, Fig 10 STRENGTH OF MATERIALS. 189 made the breakg loads equal to % of those of rectangular bars, aa in wood. Rennte's experiments give about the same proportion, with the base up ; but with the base down, he made the strength nearly twice as great, or about -fa that of a rectangular beam of the same width and vertical height. The comparative strengths in the two positions will vary in diff materials, inasmuch as it is affected by the comparative resistances which any given material presents to tension and compression. Within the limit of elasticity the beam will be equally strong, whether the edge or base be up; and will bend equally in either case; so also with the Hodgkinson, or any other form of beam. Art. 17. To find the side of a square nor beam supported at both ends, and reqd to break under a given quiescent center load. RULE. Mult the clear bearing in ft, by the given breakg load in pounds. Div the prod by the corresponding constant p 185. Take the cube root of the quot. This cube root will be the reqd depth or breadth of the beam, approximately, in ins. When the size of the beam is so great that its wt must be taken into consideration, increase either its breadth, as directed, in Remark, Art 20; or its depth, as per Art 21. The breakg, or the sate load of a square beam, if mult by .6, will give that of a Cylinder, whose diam is equal to a side of the square one. Art. 18. When the beam is reqd to bear its center load Safely, mult the given safe load by the number of times it is exceeded by the breakg load. Then find, by Art 17, the side of a square beam to break under this in- creased load. The beam thus found will evidently be approximately the safe one for the actual load ; exclusive, however, of the wt of the beam. When this must be in- cluded, increase the breadth, by Remark, Art 20. If the load is equally distributed, first div it by 2, then proceed precisely as before. Art. 19. When the beam is cylindrical, and reqd to break under its center load, to find its diam, mult the load by 1.7, and by Art 17 find the side of a square beam, to break under this increased load. The side thus found will also be approximately the reqd diam. If to be borne safely, first mult it by the number of times it is to be exceeded by the breakg load. Then mult the prod by 1.7, and proceed precisely as before. REM. 1. In neither case, however, is the wt of the beam itself included. When this is necessary, first find the approximate diam as before. Then calculate the wt of a beam having this diam. Add this wt to the given center load, in either case ; and with this increased center load, repeat the whole calculation. The resulting diam will be the required one very approximately, but stili a mere trifle too small. REM. 2. If the load is equally distributed, first take one-half of it as being a center load, and with this proceed precisely as before. Art. 2O. To find the breadth of a hor rectangular beam, supported at both ends, to break under a g^iveii quiescent center load; mult the center load in ft>s by the span in fret. Mult the square of the depth in ins by the constant p 185. Div the first prod by the last. The quot will be the breadth approximately. Calculate the wt of a beam having this breadth. Then say, as the center load is to half this wt, so is the breadth found, to a new breadth to be added to it. It will still be somewhat too small, owing to the neglect of the wt of the breadth last added. This may readily be found, and its corresponding breadth added. REM. 1. If the load is to be borne safely, (without any regard to the amount of deflection,) first mult it by the number of times it is exceeded by the breakg load. REM. 2. If in either case equally distributed, take half of it as if a center load, and proceed precisely as before. Art. 21. To find the depth, when the breadth is given, mult the load in ft>s by the span in feet. Mult the breadth in ins by the constant p 185. Div the first prod by the last; take the sq rt of the quot for an approximate depth. Calculate the wt of a beam having the depth just found ; add half of it to the given center load, and with this new load repeat the whole calculation; for a more approximate depth, but still somewhat too small, owing to the neglect of the wt of the depth last added. We may find this, and repeat the whole calculation, or wo may merely increase the breadth by Art 20. REM. If the load is to be borne safely, or if It is equally distributed, see Remarks, Art 20. Art. 22. To find the safe dimensions to be griven to a rec- *uiiular beam of given span, supported at both ends, and 190 STRENGTH OF MATERIALS. which BM at the same time exposed both to a transverse strain and to a longitudinal tensile or pulling one, or a longitudinal compressive one. The writer is unable to suggest any better rules than the following, which are at least safe. Namely, when the longi- tudinal strain is tensile, find separately the safe dimensions as if for a beam alone; and as if for a tie alone; and add the two resulting areas together. When the longi- tudinal strain is compressive, find separately the sale dimensions as if for abeam alone ; and as if for a pillar alone ; and add the two resulting areas together. Example 1. A wrought iron rectangular beam of 10 ft span is to sustain with a safety of 6, an equally distx-ibuted transverse load of 100000 Ibs ; and a pulling strain of 200000 Ibs. Of what size must it be? Here the distributed load of 100000 Ibs is equal to a safe center one of 50000 Ibs ; or to a breaking center one of 50000 X 6 = 300000 Ibs. Now first we may assume for the beam some probable approx depth, say 12 ins. Then we find by Art 21 that its breadth as a beam alone will be Breakg load in Ibs X span in ft _ 300000X10 _ 3000000 _ . sq of~defth in ius~X coef, p 185 ~~ ~144 X ^500 ~ 36000 = Again, a bar to bear a pull of 200000 Ibs with a safety of 6, should not break with less than 1200000 Ibs; therefore since fair bar iron breaks with about 50000 Ibs per sq inch, we have 1200000 -=- 50000 = 24 sq ins as the area of bar for the pull alone. We may add all of this to the width of the beam, making it 24 -;- 12 = 2 ins wider ; or 10.33 ins wide in all. Or we may add it all to the depth, thus making the beam 24 -f- 8.33 = 2.88 ins deeper, or 14.88 ins in all. Or part may be added to the breadth, and part to the depth. Example 2. A wrought iron rectangular beam of 10 ft span, is to sustain with a safety of 6, an equally distributed transverse load of 100000 Ibs, and a com-" pressive strain of 200000 Ibs. Of what size must it be? Here first assuming some probable approx depth, say 12 ins, we find as before that its breadth as a beam only will be 8.33 ins. As to the compressive force, it is plain that a pillar for sustaining it should be a hollow one with its sides as wide as possi- ble; and this is to be effected by placing it around the outside of our beam. The pillar will therefore have sides of about 8.33 and 12 ins wide ; and its breaking load must be 200000X6 = 1200000 Ibs, or say 536 tons. Now the length of the pillar measured by its narrowest side is 120-7-8.33 = 14.4 sides; and by table 8, p 232, we find that a hollow square wrought iron pillar 14.4 sides long, breaks with 15.5 tons per sq inch of its metal area. Hence we require 536-7-15.5 = 34.6 sq ins metal area for our pillar. Now the circumf of the pillar is 8.33X2 + 12X2 = 40.66 ins. Hence its thickness must be 34.6 -4- 40.66 = .85 of an inch. Hence both the breadth and the depth of the beam must each be increased twice that much,or 1.7 inch ; thus making it 10.03 ins broad and 13.7 ins deep. It is plain that our pillar is thicker than necessary, because in table 8 the widths are supposed to be by outside measure, whereas our width of 8.33 ins is inside measure. The final outer width of 10.03 ins would make the pillar only 12 sides long; at which it would require 15.7 instead of 15.5 tons per sq inch to break it. Other considerations too abstruse to be explained here, combine to make the resulting dimensions la both examples somewhat in excess. STRENGTH OF MATERIALS. 191 Art. 23. Table of safe quiescent loads for horizontal rec- tangular beams of white pine or spruce, one inch broad, supported at both ends, and loaded at the center; together with their deflections under said loads. Loads applied suddenly will double the deflections in the table ; as when, for instance, if a load is held by hand, just touching a beam, the hold should be suddenly loosed. Caution. Inasmuch as this table was based upon well seasoned, straight grained pieces, free from knots, and other defects, we must not in practice take more than about two-thirds of the loads in the table for a safety of 6 in ordinary building timber of fair quality ; and with these reduced loads should not reduce the deflections. Observe also that our table is for safe center loads, but it is plain that in practice we cannot always apply the term in its utmost strictness ; otherwise the load would have to be sustained by a mere knife-edge, at the very center of the beam. Now, in the instance Rem. p. 192, if we attempted to sustain the center load of 6075 ft>s upon such a knife-edge, it would at once cut the beam in two. If we even applied it along 3 or 4 ins of the length, it would cut into it, and we should not have a safety of 6 against crushing the top of the beam until as in the case of the ends we distributed the load along full 46 ins of length, or about 32 ins for a safety of 4. The safe load is here % of the breakgone; and the last at 450 Ibsatthe center of a beam 1 inch square, and 1 foot clear length between its supports. For mere temporary purposes, ^ part may be added to the loads in the table, thus mak- ing them equal to the \^ of the breakg load. But in important structures, subject to vibration, one ^ P art; should be deducted from the tabular loads, thus reducing them to % of the breaking load. This is especially necessary if the timber is not well seasoued. With the safe loads in this table a beam may bend too much for many practical purposes. When this is the case, we may, by reducing the loads, reduce the deflections in nearly the same proportion ; or see table, p 204. For the neat loads, deduct ^ the wt of the beam itself. The deflections however are the actual ones ; the wts of the beams having been introduced in calcu- lating them, by the rule in art 27, p 199. All the loads in the Table are superabundantly safe against shearing*. Against crushing: at the ends, &c, see "Cautions " below the Table, p 192. Original. Depth of Span 4 ft. Span 6 ft Span 8 ft. Span 10 ft Span 12 ft Span 14 ft Span 16 tt Wt. of 10 ft of beam. load def. !oad| def. load def. load def. load def load def. load def. beam. Ins. fts. ins. ft>s. ins. ft)S. ins. ft>8. ins. fl>8. ins. Bt>8. ins. ft>8. ins. K)8. 1 19 .39 13 92 10 1 8 g 3 6 4.4 2 2 75 .22 50 .45 38 !82 30 1.3 25 1.9 21 2.7 19 3.7 4 3 170 .13 114 .30 85 .53 67 .84 57 1.3 48 1.7 42 2.3 6 4 300 .10 200 .22 150 .39 120 .63 100 .92 86 1.3 75 1.7 8 5 469 .08 312 .18 234 .31 187 .50 156 .72 134 1.0 117 1.3 10 6 675 .06 450 .15 337 .26 270 .41 225 .60 193 .83 168 1.1 12 7 919 .06 612 .12 460 .22 367 .35 306 .51 262 .70 230 .93 14 8 1200 .05 800 .11 600 .19 480 .31 400 .45 343 .61 300 .81 16 9 1520 .04 1014 .10 760 .17 607 .27 507 .40 434 .54 3801 .72 18 10 1875 .04 1-250 .09 937 .16 750 .24 625 .35 536 .49 4681 .64 20 11 2270 1 .04 1514 .08 1135 .14 907 .22 757 .32 648 .44 567 .58 22 12 2700 1 .03 1800 .07 1350 .13 1080 .20 900 .29 772 .40 675 .53 24 14 3675 .03 2450 .06 837 .11 1470 .17 1225 .25 1050 .34 918 .45 28 16 4800 .02 5200 .05 2400 .10 1920 .15 1600 .2-2 1372 .30 1200 .40 32 18 6075 .02 4050 .05 3037 .09 2430 .14 2025 .20 1736 .27 1518 .35 36 20 7500 .02 DOOO .04 3750 .08 3000 .12 2500 .18 2145 .24 1875 .31 40 22 9075 .02 6050 .04 4537 .07 3630 .11 3025 .16 2593 .22 2268 .29 44 24 10800 .02 200 .04 5400 .06 4320 .10 3600 .15 3088 .20 2700| .26 48 (Continued on next page.) 192 STRENGTH OF MATERIALS. Table, continued. (Original.) Depth of Span 18 ft Span 20 ft. Span 25 ft. Span 30 ft. Span 35 ft. Span 40 ft. Wt. of beam load def load def. load def. load def. load def. load def. 10 ft of beam. Ins. Ibs. ins fts. ins. fts. ins. fts. ins. fts. ins. Ibs. ins. fts. 6 150 1.4 135 1.8 108 2.9 90 4.5 77 6.5 67 9.2 12 7 204 1.2 184 1.5 147 2.5 122 3.9 105 5.8 92 7.6 14 8 267 1.0 240 1.3 192 2.1 160 3.2 137 4.6 120 6.4 16 9 338 .92 304 1.2 243 1.9 202 2.8 174 4.0 152 5.5 18 10 417 .82 375 1.0 300 1.7 250 2.5 214 3.5 188 4.9 20 11 605 .74 454 .93 363 1.5 302 2.2 259 3.2* 227 4.3 22 12 600 .68 540 .85 432 1.4 360 2.0 308 2.9 270 3.9 24 14 817 .58 735 .72 588 1.2 490 1.7 420 2.4 367 3.2 28 16 1067 .50 960 .63 768 1.0 640 1.5 548 2.1 480 2.8 32 18 1350 .45 1215 .56 972 .90 810 1.3 694 1.8 607 2.5 36 20 1666 .40 1500 .50 1200 .79 1000 1.2 857 1.6 750 2.2 40 22 2017 .37 1815 .45 1452 '.72 1210 1.1 1037 1.5 907 fO 44 24 2400 .33 2160 .41 1728 .65 1440 .96 1234 1.3 1080 1.8 48 26 2817 .31 2526 .38 2018 .60 1684 .88 1449 1.2 1263 1.6 52 28 3267 .28 2940 .35 2352 .55 1960 .81 1680 1.1 1470 1.5 56 30 3750 .26 3375 .33 2700 .50 2250 .76 1928 1.1 1687 1.4 60 32 4267 .25 3840 .30 3072 .45 2560 .71 2194 1.0 1920 1.3 64 34 4817 .23 4335 .29 3468 .44 2890 .67 2477 .92 2167 1.2 68 36 5400 .22 4860 .27 3888 .43 3240 .63 2777 .86 2430 1.1 72 White oak, and best Southern pitch pine will bear loads % greater. For cast iron, mult the loads in the table by 4.5; and for wrought by 5.3. For these new loads, mult the dels by .4 for cast ; and by .3 for wrought. If the load is equally distributed over the span, it may be twice as freat as the center one, and the defs will be 1^ times those in the table. If the oads in the table be equally distributed along the whole beam, the defs will be but five-eighths as great as those in the table. See Art 2(i, p 196. When more accuracy is reqd, half the wt of the beam itself must be deducted from the center load; and the whole of it from an equally distributed load. The wt of the beam, in the last column, supposes the wood to be but moderately seasoned, and therefore to weigh 28.8 Ibs per cub ft. Uses of the foregoing; table. Ex. 1. What must be the breadth of a hor rect beam of wh pine, 18 ins deep, supported at both ends, and of 20 ft clear length between its supports, to bear safely a load of 5 tons, or 11200 Tbs at its center? Here, opposite the depth of 18 ins in the table, and in the column of 20 feet lengths, we find that a beam 1 inch thick will bear 1215 Ibs ; consequently, 1120 = 9.22 ins, the reqd breadth ; for the strength is in the same proportion as the breadth. Ex. 2. What will be the safe toad at the center of a joist of white pine, 18 ft long, 3 ins broad, and 12 ins deep? Here, in the col for 18 ft, and opposite 12 ins in depth, we find the safe load for a breadth of 1 inch to be 600 Ibs ; consequently, 600 X 3 1800 fts, the load reqd. REM. Cautions in the u#e of the above table. For instance, in placing very heavy loads upon short, but deep and strong beams, we must take care that the beams rest for a sufficient dist on their supports to prevent all danger from crushing at the ends. Thus, if we place a load of b075 fts at the center of a beam of 4 feet span, 18 ins deep, and only 1 inch thick, each end of the beam sustains a yert crushing force of ^^ = 3037 Ibs, and that sidewise of the grain, in which position average white pine, spruce, and hemlock crush under about 800 fts per sq inch, and do not have a safety of 6 until the pressure is reduced to about 133 fts per sq inch. Therefore our beam, in order to have a safety of 6 against crushing at its ends, must rest on each support 3037 -- 133 = 23 sq ins: or for a safety of 4 nearly 16 sq ins. When a pressure is equally distributed side- wise (that is, at right angles to the general direction of the fibres) over the entire pressed surface of a block or beam (to ensure which, the opposite surface must be supported throughout its entire length) the resulting compression might readily escape detection unless actually measured. But when a considerable pressure is applied to only a portion of the surface, as of caps and sills where in contact with the heads and feet of posts, or at the ends of loaded joists or girders, th3 com- pression becomes evident to the eye, because the pressed parts sink below the impressed ones, in conseqr.enceof the bending or breaking of the adjacent fibres What in the first case (especially if slight) would be called compression, would STRENGTH OF MATERIALS. 193 in the second be called crushing 1 ; even when neither might be so great as to be unsafe. Owing to the resistance which said adjacent fibres oppose to being bent or broken, it is plain that a given pressure per sq inch, or per sq foot, &c., will cause somewhat less compression or crushing when applied to only a part of a surface, than when to the whole of it. The writer has seen 40 half seasoned hemlock posts, each 12 ins square, footing at intervals of 5 ft from center to center, upon similar 12 X 12 inch hem- lock sills, to which they were tenoned, and which rested throughout their entire length on stone steps. Each post was gradually loaded with 32 tons, or equal to say 500 Ibs per sq inch ; and their feet all crushed into the sills from J to }% inch. Their heads crushed into the caps to the same extent. In practice the pres- sure at the heads and feet of posts is rarely, if ever, perfectly equable ; and the same remark applies to the ends of loaded joists, girders, &c., in which a slight bending will throw an excess of pressure upon the inner edges of their supports. See other cautions on p. 191. See also footnote, p. 174. Art. 24. Strength of hollow Beams. During the preliminary investigations relative to the construction of the Menai tubular bridge, a few experiments were made on the strength of hollow cast-iron beams of circular, oval, square, and rectan- gular cross-sections, supported at both ends, and loaded at the center. The clear span between the supports was in every fcase 6 ft , the thickness of metal in each beam, % inch ; area of solid cross-section assistant engineer in charge, deduced the following constants, and rules for center breakg loads : Const for circ tubes, .95 ; oval, 1; square, 1.14; rectangle, .91. Then, first finding the area of the solid part of the cross-section in sq ins, Area of solid ., Mean depth, o o, ^, Corresponding Center breaking: in sq ins x in ins constant. load in tons Clear span in feet. Ex. Circular beam, mean depth o o, 3% ins ; area of solid ring, 4.12 sq ins ; clear span, 6 ft. Here, Area. Mean depth. Const. 4.12 X 3.5 X -95 : : 2.28 tons, or 5107 Ibs, breakg load. 6 (length.) The thickness of the cylinder or tube is about ^ of the diam ; and as a mean of 3 trials, it broke with a center load of 2.287 tons, or 5122 Ibs ; span 6 ft. Hence we derive for similar tubes, the constant 475, to be used in the rule, Art 12; that is, center breakg load in Ibs of cast-iron tubes with a thickness of 1 of the outer diam = ~Q tear gpan in feet 5 supposing Mr. Clark's iron to have been of average quality. The average breakg load of 3 square beams was 2.152 tons, or 4820 Ibs; of the rec- tangular ones, 2.3 tons, or 5152 Ibs ; and of the 6 elliptic ones, 3.207 tons, or 7183 Ibs. To all the foregoing extraneous loads must be added half the wt of the beam itself. See Art 9, p 183. .Our rule of thumb on p 187 agrees well with all Mr. Clark's results, except for the oval beams, in which said rule rives a breaks: load of but & that, recorded by him. Hollow beams of thin wrought iron were experimented on at the same time ; and for these Mr. Clark deduced the following constants, to be used with his foregoing rule for cast-iron ones : Constants for thin riveted tubes, circular, 1.74 ; oval, 1.85 ; rectangular, 1.96. welded tubes, " 1.09; " 1.27; 1.61. Art. 25. The following experiments on riveted sheet-iron cylindri- cal beams are by Fairbairn. 1st. Cylinder 18 ft long; 1 ft outer diam; clear span 17 ft ; thickness of iron .037, or ^ of an inch ; wt of tube 107 Ibs. 18 194 STRENGTH OF MATERIALS. Center load. Def. Lbs. IQB. 1360 .32 1920 41 2114 .46 2256 60 Center load. Lbs. 2480 . 2592. 2704 . Def. Ins. .. .60 . .61 . .61 . .65 After bearing 2704 Ibs. for 1% minutes, failed by crushing at top. 2d. Cyl 16 ft 10 ins long ; 12.4 ins outer diarn; clear span 15 ft 7U ins ; thickness of iron .113, or full .J. inch ; wt of tube 392 Ibs. Center load. Lbs. 2000 . 4000 . 6000 Def. Ins. .17 . .34 . .52 Center load. Def. Lbs. Ins. 8000 84 10000 1.06 11440 Broke. With 11440 broke by the tearing of the bottom across the shackle-hole from which the load was suspended. 3d. Cyl 25 ft long; 17.68 ins outer diam ; clear span 23 ft 5 ins; thickness .0631, or full A inch; weight of tube 346 Ibs. Center load. Lbs. 1000 . 2000 . 3000 . 4000 . Def. Ins. . .12 . .21 . .30 . .40 Center load. Lbs. 5000 5280 5840 6120 Def. Ins. 48 .... .51 60 71 With 6400 broke at bottom ; 25 ins from center, by tearing through the rivet-holes. 4tli. Cyl 25 it long; 18.18 ins outer diam; clear span 23 ft 5 ins; thickness .119, or scant }/Q inch ; wt of tube 777 Ibs. Center load. Def. Center load. Def. Lbs. 2000 . 4000 , 6000 . 8000. Ins. . .15 ,. .30 ,. .43 i. .59 Lbs. Ins. 10000 82 KOOO 95 13000 1.04 14240 Broke. Broke through the rivet-holes 3 ft 3 ins from center, after sustaining the load for half a min. The tubes were composed of sheets about 2^ ft wide ; and so long that a single sheet sufficed to form the entire circumf of the tube. They were united by double-riveted lap-joints. The loads were placed on a platform, supported by a rod r, Fig 11, which passed through a hole A in the bottom of the tube s. This rod was attached at its upper end to a block of wood w, rounded at its lower surface, so as to fit the tube. I v -v Circular blocks of wood were fitted into the ends of the tubes, to JtlO 14 "1 k prevent them from crushing at those parts under their loads ; and t the ends rested upon blocks hollowed out to correspond with their cylindrical shape, to a depth equal to about ^ part of their diam. Art. 25%. Breaking load for a beam of any form of cross- section, item. Scientists give the following, which however often differs so much from experiment (see example, p 196 ;* as to prove that the only reliable mode of finding the strengths and deps of beams is by experiment with models of the same form and material ; as has been already done with most of those in common use. If the beam is supported at both eiids, with the load at its center, half its wt must be deducted ; or all of it for an equally distributed load. Moment of inertia of the cross-section of the beam, with respect to a neutral v Constant for rupture of the material axis G G, passing hor through its cen * of which the beam consists, fts. of grav O, in ins. _^__^__ Breakg in Jbt load, _ Dist o g from neutral axis ; (and at ( right angles to it) to the fibre g, the X farthest one from the axis, in ins. r span of ^ i, in ins x m But Prof De Volsoii Wood in his " Resistance of Materials," p 150, says that instead of the farthest fibre g in all cases, we should use that one (either g or z) which is on the side that would yield most readily. If so the disagreement with experiment is still too great ; and moreover the choice of fibre would often * A part of tlie discrepancy is of course due at times to difference of quality of material. STRENGTH OF MATERIALS. 195 be difficult, as it would not always be that one which by trial-calculation would give the least load. On the other hand the formula cannot always be correct as it stands, for then a Hodgkinson beam, p 208, would be equally strong with either flange uppermost. A reliable formula of this kind would be of very great value, but is probably an impossibility. Here in is either 1, %, ^, %, or ^, according to the arrangement of the beam and its load, as referred to in Art 11, p 186. Thus, if supported at both ends and loaded at the center, it will be % or .25. The Constant for Rup- ture is 18 times the constants for center breakg loads given p 185, with the exception (Ran- kine), that for open-work cast- iron beams it is but 9 times the " constant in p 185. Or it is = 3 ( wt of beam + 2 brkg load ) X span 4~X breadth X sq of depth The position of the hor neutral axis G G, may be found by cutting out a correct figure, 4 or 5 ins long, of the sec- tion drawn on thick paper or tin, and balancing it over a straight edge. The line at which it bal- ances is G G. When this has been done, the dist o g, in ins, to the farthest fibre, (which may be either above or below G G, according to the shape of the Fig,) can be measured in ins. *> Moment of inertia of the cross- section, with respect to the neutral axis G G, means the sum which results from adding together the prods found by mult together the infinitely small area of each fibre, as y, of the section, by the square of its dist 6 y from, and at right angles to G G. Such multiplication cannot be performed by ordinary arithmetic, but an approximation, sufficiently close for all practical pur- poses, may readily be made thus : Both above and below G G, and parallel to it, draw lines ,;' Ic, I m, &c, dividing the section into narrow strips. If these lines are equidistant, the subsequent calculations will in some cases be easier; but otherwise it is immaterial whether they are so or not. If they .are drawn no closer together, proportionally to the size of tfie .figure, than in Fig 14%, the approximation will be near enough for practical purposes. The closer they are the more accurate will be the result ; but however close they may be, it will always be a trifle too small. Begin by finding the area in sq ins, of the first strip x xj k, below G G. Mult this area by the square of the dist Or to the cen of grav of the strip. Then proceed to the next strip./ klm; find its area ; and mult it by the square of the dist o s to its center. Do the same with each strip. Add all the prods together, and if the section has the same shape, size, and position above G G as below it, (as would be the case with a square, rectangle, or circle,) mult their sum by 2. The prod will be the reqd moment. But if, as in Fig 14%, the section above G G differs from the portion below it, we must div it also into strips, and proceed as with the lower part. The sum of all the products on both sides of the neutral axis will then be the moment. Moments of Inertia of a few well known figs are given below. Those of similar figs are to each other as their breadths X cubes of depths. Square or rectangle. (Breadth X cube of depth) H- 12, whether any side or diagonal is vertical. * When a beam just breaks under its load, the strain in fbs per sq inch on the fibers g, Fig 14% farthest at right angles from the neutral axis G G, (or according to Prof Wood, the farthest on the side that will yield first, whether those at g or those at z, as the case may be) is called the Mod- ulus, Constant, or Coefficient of Rupture. Writers usually denote it by C ; and its value for hor square or rectangular beams supported at both ends and loaded at the middle i* C = . - - ; where 2 u Q v is the load, and 1 the span. In other words (center load , H wt of clear \ v clear span in tts ' span of bam/ * iu ins Breadth in ins X square of depth in ins. Which gives the same result as the above ; and both of them show how to find C for any given ma* terial by experiment, which ia the only possible way. See Table, p. 135. 196 STRENGTH OF MATERIALS. Hollow square or rectangle. (B X D 3 6 X d 3 ) -* 12. * Circle. Rad< X -7854. Semicircle. Rad* X .1098. Ring. (Outer rad* Inner rad^j X .7854. Ellipse. Long diam vert. Half short diam X half long diam* X .7854. Elliptic ring. Long diam vert. Let L, S, I, s, be half the long and short diams. Then (S X L 8 s X l*> X .7854. .0.1113. j.jit;u ^o A w S A l ) A. .You* Any triangle. (Base X Perp Ht 3 ) -*- 36. 1. (B XV* - 2 6 X &) -*- 12. 2. (B X D 8 + 2 b X d 3 ) -=- 12. 3. (6 X d 3 + &' X ?* - (&' - 6) X d") -*- 3. 4. (6 X d 3 - (fr-jfc) X (d-c? -}- V X d'* - (b'-k) X (*'- c }) -f- 6. Example of Formula 011 page 194. What is the center breaking load of a solid cast-iron beam 4 ins square, and 6 ft, or 72 ins, clear span, supported at both ends? 4* X 4 a 16 V 16 Here the moment of inertia is - " = - ~- = 256 12 = 21.333. The con- stant for rupture, or 18 times our constant for cast iron on p 185 is 2025 X 18 = 36450. Since the beam is supported at both ends, and loaded at the middle, wi, (see Art 11) is i^. The dist off of the farthest fibre from the neutral axis must in a square be equal to % of one side; consequently it is here '2 ins. The clear span is 72 ins. Hence, Breakg _ Mom of In X Con for Rup 71 X Uwf/flbre"" X 8pan 2^-333 X 36450 _ 777588 _ X X 2 X 72 = 21600 fts = 9.64 tons. By our table, p 206, a beam of average cast iron, 4 ins deep, 1 inch broad, and 6 ft span, breaks with 2.41 tons; consequently, four such, or one 4 ins square, would break with 2.41 X 4 9.64 tons: thus confirming the accuracy of the foregoing. Applied in the same way to solid cylinders, the result corresponds equally well with experiment and with our table on p 207. But for Mr Clark's hollow squares, p 193, the formula gives 3.06 tons in- stead of the actual 2.15; and for his hollow cylinders 2.980 instead of 2.287. See footnote, p 194. A true liodgkinson beam, p 208, with top flange of 1 by 3 ins, bottom flange 1.5 by 12 ins, vert web .75 inch thick, total depth 15 ins, clear span 20 ft, has a moment of inertia of 780; dist from neutral axis to upper fibre 10.7 ins, and to the lowest one 4.3 ins. By Hodgkinson it would yield at the lower flange, and by his rule with a center load of 29.24 tons. By the formula it would be 19.8 tons; and by Prof Wood 49.2 tons. Beam I, p 210, actually broke with about 52 tons ; by the formula it would be 40, and by Prof Wood 59 tons. Art. 26. Deflections, or beiidiiigs of beams, under their loads. The foregoing relates to the strength of beams, or their resistance to breakg ; the following to their stiffness, or resistance to bendg. The two follow very diff laws. It is with the defs within the elastic limit that the engineer is chiefly interested. They then are directly as the load and as the cube of the span ; and inversely as the breadth, and as the cube of the depth ; and this, with the following, applies not only to all rectangular beams, but to all others of whatever cross section, provided the sections are similar. See. p 61, llth line of Geometry. With same span, breadth, and load, the deflections within elas limits are in all cases inversely as the cubes of the depths. Hence the depths are inversely as the cube roots of the deflections. With same span, breadth, and deflection, the depths are directly as the cube roots of the loads. Hence the loads for equal deflections are as the cubes of the depths. Under greatest load within limit of elas, the defs are as the square of the span ; and inversely as the depth and breadth. jjc In a rectangle, B and b are respectively the outer and inner dimensions parallel to the neutral axis, whether said axis be lengthwise or crosswise of the figure. STRENGTH OF MATERIALS. 197 If the deflection of a beam supported at both ends and loaded at the center be called 1. Then that of the same beam, with the same load uniformly dis- tributed, will be 625, or % Firmly fixed at both ends, and loaded at the center, by Moseley 2, or $ " uniformly loaded 125, or y% Fixed at one end, and loaded at the other 16. " uniformly loaded 6. The extent to which a beam may bend under even a perfectly safe load, may be too great for mainy purposes in every -day practice. Tredgold and others assume, that in order not to be observed, or that it may not cause the plaster of ceilings to crack, &c, a beam should not deflect at its center more than the z>h part of its span, or ^yth of an inch per ft. Thus, if its span be 20 ft, it should not bend more than f gths, or y> of an inch, which is also x4 th of 20 ft. For such cases see Art 29, p 201. Shafts of* wheels in machinery should not deflect more than half of this, nor a bridge more than about % of it, or say y^ 1 ^- of its span, or y^ inch per loot, under its heaviest loud. We shall allude first to defs within the limits of safety, or of the elasticity of the beam ; and afterward of those not exceeding ^1^. of the span. After the elastic limit is passed, the defs increase irregularly, and more rapidly than before ; and the beam becomes unsafe. As a general rule, the elasticity of a wooden beam is not injured for practical purposes, if the quiescent load does not rxcenl about % of the breaking one. Within the elastic limirs, the defs theoretically vary din'ctly in pro- Sortion to the load, and also to the cube of the span, or clear length; and inversely a proportion to the breadth, and to the cube of the depth. That is, Deflection within > is in proportion to ( Load * cube of 8 P an ^ elastic Hmit3 $ (not equal to) } brea dth X cube of depth. Therefore, constants for the bending* of beams of diff shapes, within the limit of elasticity, may, like those of transverse strength, (see Art 9,) be readily found by experiment. Thus, at the center of any rectangular beam, placed lior upon supports at e-ich end, place any load that is within its elastic limit, and measure the resulting def in ins. Mult the wt of the span of the beam by .625 ; add the prod to the neat load, for a total load. Mult together the total load in Ibs. and the cube of the span in feet. Also mult together the breadth in ins, and the cube of the depth in ins. Div the first prod by the last one. Div the def by the quot. The last quot will be the reqd constant for any rectangular beam of the same kind and quality of material, whether wood, metal, stone, &c. That is, The constant for^i f Total 1< def within 1 Def in ins 1 in ft* elastic, or safe f divided by T R^eaSt! limits of beam. J l 1 *^ Total load v Cube of spaa ' feet Cube of depth in ins. We add to the experimental neat load, the .G.'o, or % of the wt of the clear span of the beam itself, because the wt of the beam equally distributed throughout its span, also aids in producing the def; and it does so to the same extent that %, of it would do, if collected at the center of an imaginary beam having the same strength throughout as the real one, but of no wt except at its very center. Ihereibre, in applying the constants for def to beams intended for actual use, we must not omit to add % of the wt of the span, to the intended center load, for an equivalent total center load, before making the calculations for def. The weights of similar beams (that is, beams proportioned exactly alike in every part, but of diff sizes) increase so much more rapidly than their clear spans, that although a small one may safely bear a load of many times its own wt, a much larger one will break down without any load. Having by experiment found the constant of def for any given material, the def of any similar beam of the same material, whether larger or smaller, and loaded at the center, may be found thus : wl ,,r B , fe _ sMfli Ci c "'" t ' Breadth v Cube of depth limit in ins in ins * in ins. The limit of elasticity of a beam of any particular form, or material, is determined by experiment with a similar beam, as in the case of constants for breakg loads. &c. Thus, load a beam at the center, by the careful gradual addition of small equal loads ; carefully note down the def that takes place within some mins (the more the better) after each load has been applied; in order to ascertain when the defs begin to increase more rapidly than the loads ; for when this takes place, the load for elastic limit has been reached. Sec Remarks, Art 9 and 29. 198 STRENGTH OF MATERIALS. It Is not the defs of the whole beam that are to be noted, but those of its clear span only. Several beams should be tried to get an average constant ; for even in rolled iron beams of the same pattern, and same iron, there is a very appreciable diff of strengths and defs. Then, to get the constant, so as to apply it to similar beams, using the total load applied during the equal defs, including % wt of beam, Constant for greatest ,*, X center loads within = Breadth v Square of depth L X in ins Constants for greatest center loads within limits of elastici- ty, may be had near enough for common practice by taking one-third of the breaking constants in the table on p 185 ; except those for rolled iron and steel.* It is assumed always that the load is not subject to jars or vibrations ; these would increase the defs. REM. Within the limits of elasticity, a beam of irregular shape, such as a T, or a Hodgkinson beam, a triangle, Ac, will bend to the same extent, whether its top or its bottom be uppermost. To find the greatest center load that a given beam, sup- ported at both ends, can sustain without exceeding its elas- tic limit, (beam rectangular.) }/ 3 of the Greatest center load Breadth ^ Square of depth ^, constant on within elas limit of = in ins ^S * n * n8 page 185.* beam in fts Span in feet We will remark that, in practice, it is frequently difficult to ascertain with pre- cision when, or under what load, the defs actually do begin to increase more rapidly than the successive loads. For although theoretically the defs are equal for equal loads, until the elastic limit is reached, yet in practice they are only nearly equal, up to that point This is owing to the fact that no material composing a beam is perfectly uniform throughout in texture and strength; so that instead of perfect equality of defs, we shall have an alternation of larger and smaller ones. Therefore, some judgment is reqd to determine the final point; in doing which, it is better, in case of doubt, to lean to the side of safety. The def of a beam of any form whatever of cross section, if within the limit of elasticity, may be found approximately thus, load v Cube of span v f . Def in = in ft>8 x in inches ' 1DS< mod of elas v moment of inertia p 632 A in ins, p 195 Coef d. Beams supported at both ends ; center load 02083 " " uniform ' 01302 " fixed at one end ; loaded at the other 33333 " " " " ; load uniform 12500 This formula gives the def produced by the load only. To find that arising from the weight of the beam itself, consider said wt as a uniform load ; then find the re- sulting def by the same formula, and add it to that of the load. In a rectangular beam supported hor at both ends and loaded at the middle within elas limit, the def in ins will be / Load , .625 wt of clear span\ y, cube of clear span . \in fos ' of beam in fos / * in ins ~~ 4 X coef elas X breadth in ins X cube of depth in ins And the center load in Ibs, (including .625 wt of clear span of beam) required to produce any given def in ins within elas limit of such a beam will be 4 v pnpf P! v breadth v cube of depth v given def b A in ins A in ins A in ins cube of clear span in ins. * Except for wrought iron and steel ; for which take the whole con- stant, STRENGTH OF MATERIALS. 199 Table of constants for the deflections, within the safe, or elastic limits, of hot rectangular beams, supported at both ends and loaded at the center. The timbers are supposed to be well seasoned; if not, the constant should be increased. 'White oak ...................... 00023* White pine ............................ ] Best southern pitch pine, ) ftnft . 77 # Ordinary yellow pine ............... | and white ash ..... ....... j - 00027 Spruce ................................... \ .00032* Hickory ........................ 00016* Good straight-grained hemlock. Ordinary oaks .......................... J Cast iron ........... 000018 to .000036 .................... ...... ...... Mean .000027* Bar iron ............. 000012 to .000024 ................................. Mean .000018 Stool, rolled ..... 000010 to .000020 ................................. Mean .000015 Full and reliable experiments on the strength and deflections of the various steels are much needed. See table of safe loads and clefs, p 191. It is evident that the stiffer the material is, the smaller will be its constant for bend- ing. All these constants vary somewhat with the quality of the metal. The defs also of timber of the same kind, vary so much with the degree of seasoning, the age of the tree, the part it is cut from, Ac, that the writer considers it mere affectation to pre- tend to assign constants for practical use, more nearly approximate than he has here done. They are averages deduced from his own experiments on good pieces, well seasoned ; and the loads were allowed to remain on for months, instead of minutes, as usual. Every structure is more or less exposed to vibrations and jars, which in time increase the deflections. In several instances, our experimental timbers bore their breakg loads for months before they actually gave way. And in all kinds, less than of the breakg load produced in a few months a permanent set, or def. The following are deduced from single experiments only. An allowance is made ffor the weight of the beam. Rolled iron beams proportioned exactly as the 7-inch Phoenix beam, A,p 210, .0000303f " " " 30 Ibs, 9 inch, " " . ................. 0000321f " " " 50 " heavy 9 inch " " B, p 210, .0000264 41% Ibs, 12 inch " " ................ 0000313f " " " 512J Ibs, 15 inch " " ................. 0000365f 66% Ibs, 15 inch " " ................. 0000438 Art. 27. To find the def in inches, of a hor rectangular beam, supported at both ends, and loaded at its center, with any given load within its elasticity; mult the weight of the clear beam itself, in Ibs, by the decimal .625. Add the prod to the given center load in Ibs. Call the sum the total load. Mult together this total load, the cube of the span in ft, and the constant from the upper table. Also mult together the breadth in ins, and the cube of the depth in ins. Div the first prod by the last one. Ex. What will be the def of such a beam of average white pine, 9 ins broad, 12 ins deep, 21 feet clear span, and weighing 450 Ibs; with a neat center load of 1218.75 Ibs? Here first, 450 X .625 = 281.25 Ibs. And 281.25 -f 1218 75 = 1500 Ibs total load. Hence, 1500 X 21 X -Const. 1500 X 9261 X .00032 4445.2 -- ~ - -OT- ~ 9X1728 - See table of safe loads and deflections, p 191. REM 1. When the load is all at one point not at the center, as at o, Fig 10, mult together the two dists o a, o g, from the load to the points of support. Mult the prod by 4. Div the result by the clear span. Use the quot as if it were the span, in the last rule. The wt of the beam is not here taken into account ; it will of course somewhat increase the def. * Averages near enough for ordinary practice by the writer's own trials. Call- ing- the average elastic dot* of a steel beam, 1, that of a similar average wrought one will be 1.2 : and that of a cast one 1.8. If that of an average cast beam be 1, that of a wrought one will be .67 ; and that of a steel one .56. If that of a wrought one be 1, cast will be 1.5; and steel .83. t We believe that these four beams have the same proportions, as nearly as the process of making them will admit of; so that .000033 may be taken as a near enough avera'ge for all four. As before remarked, extreme accuracy must never be expected in such matters. Two halves of the same iden- tical beam will often give differences greater than this. 200 STRENGTH OF MATERIALS. HEM. 2. When the neat load is equally distributed along" the span, instead of all being at the center, then for an equivalent total center load, add together the neat load, and the entire wt of the clear span of beam ; and mult the sum by the dec .625. With the resulting equivalent center load, proceed precisely as in the foregoing example. Ex. The def of the foregoing beam of white pine, 9 ins broad, 12 ins deep, 21 feet span, weighing 450 Ibs, and bearing an equally distributed load of 1218.75 Bbs? Here first 450 + 1218.75 = 1668.75. And 1668.75 X -625 = 1042.97 Bbs = equivalent center load. Hence 104-2.97 X 21" X -00032 3090.862 9 X 12 - * -T666T ^ - 1987 iD8 ' reqd def ' REM. 3. With an equally distributed load, including the wt of the beam, the def is only %, or the .625 part as great as it would be if the same total load, including the entire wt of the beam, were all applied at the center. RfcM. 4. If the beam in any of these, or the following cases, is inclined, as in Fig 11, use the hor dist o y, instead of the actual span o c. Art. 28. RULE 1. To find the neat center load which will (to- g-ether with the wt of the beam itself) produce any given clef within the elastic limit of the beam ; find the cube of the clear length in feet; mult this cube by the constant from the table on p 199. Also mult the breadth in ins, by the cube of the depth in ins. Div the first prod by the last one. Div the given def in ins, by the quot, for the total reqd load in Ibs. Mult the wt of the clear length of the beam in R>s by .625, and deduct the prod from the load so ob- tained, for the mat load. By formula, Cube of length ., Constant Total load, in feet X in p 199 including = ut iect -^- - = - wt of beam * n * n8 Breadth v Cube of depth in ins A in ins. Ex. What center load in Ibs will (together with the wt of the beam itself) pro- duce a def of .286 of an inch, in a beam of white pine, 21 it span, 9 ins broad, 12 ins deep, and which weighs 450 fibs ? See table, p 191. Cube of 21. Const. Breadth. Cube of 12. Here 9261 X .00032 = 2.9635. And 9 X 1728 = 15552. And -- - 001906 - And - 150 tt For the neat load we must deduct .625 of the wt of the beam ; or 450 Ibs X -625 = 281.25 Ibs; so that the neat load is 1500 281.25 = 1218.75 ft>s, as in Ex 1, Art 27. If the load is uniformly distributed, use precisely the same rule for get- ting the total load. Then mult this load by 1.6. Deduct the entire wt of the clear length of beam. Ex. What equally distributed load will deflect the foregoing beam .1987 ins? Here, proceeding as before, the only diff is that instead of .286 def, we have .1987 def to be div by .0001906. And - 1 = 1042.5 Ibs, as the equivalent center load. And 1042.5 X 1.6 = 1668 Ibs for the total distributed load, including the entire wt of the beam, or 450 Bbs. Hence 1668 450 = 1218 Ibs, the neat distributed load reqd ; agreeing with the preceding example within % of a Ib ; the difif being owing to a neglect of small decimals in the calculation. RULE 2. The length, depth, neat center load, and def being: given, to find the breadth. Neat cen load ,. Cube of length v Constant 1 * in feet X in Art 26 _ , Cube of depth v Def ~ approx. in ins ^ in ins Or sufficient for the neat load alone. Now calculate the wt of a beam with the breadth already found. Mult this wt by .625, then say, as w . Breadth .625 of the Additional Neat center . firgt . , vei ht of breadth found the beam reqd. Add these two breadths together, and their sum will be the total breadth reqd, more approximately ; but still somewhat too small, inasmuch as it provides only for the STRENGTH OF MATERIALS. 201 wt of the beam of the breadth first found, and not for that having the additional breadth. This may readily be calculated and added. See table, p 191. RULE 3. The length, breadth, neat center load, and def, being: given, to find the depth. iii Ibs X iu feet 1 X in Art 26 Cube of depth : in ins approx. Breadth v Def in ins * in ins Take the cube root of this for the depth itself, approximately. REM. This, like the breadth given by the preceding formula, is too small, inasmuch as it does not allow for the wt of the beam. Therefore, when greater accuracy is required, proceed thus: Calculate the wt of a beam having the depth just found. Mult this wt by .625. Add the prod to the neat center load. Consider the sum as a new neat center load; and using it instead of the neat center load first given, go through the whole calculation again, to obtain a new cube of depth. The cube root of this will be more nearly correct ; but still a trifle too small, for the same reason as in the fore- going case. See table, p 191. Art. 29. Deflection not to exceed ^ of an inch for each foot Of clear length, or j|~g- part of the clear length. To obtain constants for such defs, at the center of any hor rectangular beam supported at each end, place any load that is entirely within its elastic limit. Also, measure the def in ins produced by said center load. Then, , ft v Breadth v Cube of depth v Deflection < in ins > in ins x in ins Constant for def of Center load in ft>s Cube of clear length in feet of the span. *To be more exact, .625 of the wt of the beam itself should be added to the actual neat load. See Rem. Art 26. In the same way, constants may be found for beams of any form whatever. If, instead of ^V tne def mns *> n t exceed, ^, -g- 1 ^, Ac, of an inch per foot, then sub- stitute 30, &c, for the 40 of the formula. Table of constants or coefficients for deflections of ? V inch per foot span, of horizontal rectangular beams supported at both ends, and loaded at the center. Hickory 0064 White Oak 0092 Lombardy Poplar 0230 Teak 0080 Horse Chestnut 0170 Average Cast Iron 0011 Average Steel 00044 Best Southern Pitch Pine 0108 White Ash " White Pine, or Common Yel- ") low Spruce, good Hemlock I Red and Black Oak Average Wrought Iron .'.. .00066 .0128 1. Rolled iron beams, proportioned exactly as 7 inch Phoenix beam, A, p 210,... .00121 2. 3. 4. 5. 6. 30 ft, 9 in heavy 9 in 41% Ibs, 12 in 51% Ibs, 15 in 66% fts, 15 in .00129 B, p 210,... .(10106 00125 00131 00175 REM. In experimenting for constants of any kind, with beams of irregular cross-sections, this, for in- stance, it is quite immaterial which breadths and depths are measd; thus, for the breadth we may take a 6, I in, cd. or oe t &c; and for the depth, either we, lr, md, bo, &c. It is only necessary to state what parts actually have been taken, so that the corresponding ones may be measd in any other beam which is to be calculated by the constant derived from the experiment. This remark applies to all constants involving the breadth and the depth. The constant itself will of course vary according to which dimensions are taken in the experiment ; but the results derived from it when applied to other beams of similar forms, will not be affected thereby, if the corre- sponding parts be measd in both cases * * We may ereu take any single oblique measurement, as a b, Im, nc. ad, &c. and call it both the breadth and the depth. This applies to rectangular, or to any other shaped beam*. 202 STRENGTH OF MATERIALS. Art. SO. RULE 1. To find the greatest center load which any given nor rectangular beam, supported at both ends, can sustain without bending more than 4 J o of its span. Breadth v Cube of depth in ins > in ins _ Reqd load * Square of span v Constant in feet X in Art 29 and generally near enough for practice. A To be more accurate, a deduction must be made for the wt of the beam itself. ':o get the neat load. To do this, mult the wt of the span of the beam by .625, and 'de- duct the prod from the approx load just found. REM. 1. The load may also be found by tables in pp 204, 205. RKM. 2. If the load is uniformly distributed, use the same rule or formula; but mult the resulting load by 1.6; and then, if an allowance is reqd for the wt of the beam, deduct the wt of the entire span for the neat load. RULE 2. The neat center load, length, and depth being given. to find the breadth, such that the beam shall not deflect more than | 7 of its span. See tables, pp 204, 205. Square of length ^/ Neat center load ., Constant in feet X in B>s X in Art 29 Cube of depth in ins and generally near enough for practice. Then say, as Centered Add these two breadths together, for the reqd one very approx, but a mere trifle too small. See Remark after Rule 3, Art 28. RULE 3 The neat center load, length, and breadth, being given, to find the depth, such that the beam shall not deflect more than ^fa of its clear length. See tables, pp 204, 205. Square of length v Neat cen load y Constant Cube of depth in feet in ft>s _ ^ in Art 29 _ in ing< Breadth in ins approx. Take the cube root of this for the approx depth itself, generally near enough for practice. Then calculate the wt of the entire clear span of a beam having this depth, mult it by .625, and add the prod to the neat center load. Consider the sum as a new neat center load; and using it instead of the one first given, go through the whole calculation again, for a new cube of depth. The cube root of this will be the reqd depth more approx, but a little too small, for the reason given at Rule 3, p 201. If the neat load is uniformly distributed, first mult it by .625. Use the prod as a center load, and by the foregoing formula find the first approx depth Then calculate the wt of the entire clear length of a beam having that depth. Mult thie wt by .6v?o, and add it to the prod used as a center load. Consider the sum as a new confer load; and using it instead of the one first used, go through the whole calcu- lation again, for a ww cube of depth. The cube root of this will be the reqd depth, approx. but a mere trifle too small, for the reason given at Rule 3, p 201. RULE 4. To find the side of a square beam, when its center load and clear length are given, not to deflect more than ^-5 of its span. Square of length v Center load v Constant _ A certain in feet in fts x in Art 29 ~ fourth power. Next, find the fourth root of this fourth power; th;it is, take its sq rt, then take the sq rt of said sq rt. This last sq rt will be one side of a square beam, generally * We must now employ the square of the length ; not its cube, as in the preceding Arti. STRENGTH OF MATERIALS. 203 near enough for practice ; but which will answer truly only for the given center load, minus .6.5 of the wt of the cle.ir beam; which wt can now be calculated. But if the beam is to answer lor the given center load, plus the wt of the beam, the calculation must now be repeated, thus : Find the wt of the beam just calculated ; mult it by .625; add the prod -to the given center load. Then, with this increased center load, employ the formula as before, obtaining a new fourth power. The sq rt of the sq rt of this will be the reqd side very approx, but a mere trifle too small. See Kern after Rule 3, Art 28, p 201 ; also table, p 205. RULE 5. To find the diam of a solid cylinder, when its cen- ter load and clear length are given. Mult the load by 1.7, and with this increased load proceed, as in the foregoing case of a square beam, to find a cer- tain 4th power. Take the sq rt of the sq rt of this, for a diani sufficient for the given center load, minus .625 of the wt of the beam. For a larger diam, sufficient for the wt of the beam also, first mult said wt just found by .625. Add the prod to the ori- ginal given center load. Mult the sum by 1.7 for a new load; then, with this new load, repeat the formula as before, for a new 4th power. The sq rt of the sq rt of this will be the reqd diam more apprax, but a trifle too small. See Rem, after Rule 3, Art 28. The stiffness of a cyl is to that of a square beam, whose breadth and depth are each equal to the diam of the cyl, as .589 to 1 ; or that of the square one is to that of the cyl as 1 to .589, or as 1.698 to 1 ; in practice we may use .6 and 1.7. Hence, the cylinder will bend 1.7 times as much as a square one, under the same load. When, in any of the foregoing cases, the beam is inclined, as in Fig 11 take the hor dist o y for the span, instead of o c. STONE BEAMS, Table of safe quiescent extraneous loads for beams of g;ood building granite one inch broad, supported at both ends, and loaded at the center; assuming the safe load to be one-tenth of the breaking one; and the latter to be 100 Ibs for a beam 1 inch square, and 1 foot clear span. The half weight of the beams themselves is here already deducted by the rule in Art 12, p 186, at 170 Ibs per cub ft. j CLEAR SPANS IN FEET. a M 1 1 2 3 | 4 5 6 7 8 10 12 15 20 Safe center load ; in pounds. 1 10 5 2 40 20 13 10 3 90 45 29 21 17 4 160 79 52 39 31 26 21 5 250 124 82 61 48 40 34 6 360 179 119 89 70 58 48 42 32 7 490 244 162 120 96 79 67 58 45 36 27 16 8 639 319 212 158 126 104 88 76 59 47 36 22 10 999 499 331 248 197 163 139 120 94 76 58 38 12 1439 718 478 357 284 236 201 174 137 111 85 58 14 1959 978 650 487 388 322 274 238 188 153 118 81 16 2559 1278 850 636 507 421 359 312 246 201 157 109 18 3239 1818 1077 806 643 534 455 396 313 257 200 141 20 3999 1998 1329 995 794 660 563 490 388 319 249 176 22 4839 2417 1609 1205 961 800 682 594 470 387 303 216 24 5758 2877 1916 1434 1145 951 813 708 562 463 362 260 27 7288 3642 2425 1815 1450 1205 1030 898 713 588 462 332 30 8998 4496 2995 2243 1791 1489 1273 1110 882 728 573 415 33 10888 5441 3624 271* 2168 1803 1542 1345 1069 883 666 505 36 12958 6476 4314 3231 2581 2147 1836 1603 1275 1054 832 606 If uniformly distributed over the clear span, the safe extraneous loads will be twice as great as those in the table. For good slate on bed the safe loads may be taken at about 3 times ; for good sandstone on bed at about one-half; and lor g*ood marble or limestone on bed at about the same as those in the table. See table, p 185. 204 STRENGTH OF MATERIALS. STRENGTH OF MATERIALS. 205 Art. 32. Table of greatest center loads of square beams of cast iron, supported at both ends, and reqd not to bend more than . t J of an inch per foot of clear leiig'th, or ? J 5 part of the span. For W. Pine div by 12; or in practice by 18. The loads are about y 1 ^ part greater than would be given by our constant .0011 for average cast iron, Art 29. Wrought iron will bear about -| more than cast, with the same safe deflection. But .625-(or %) of the wt of the beam itself must be deducted from thf-se center loads. If the load is equally distributed, it will be 1.6 times as great as these tabular center loads ; but in this case the wt of the entire clear length of the beam is to be deducted. These deductions are rarely reqd in practice. Greatest Centre Loads, in Pounds. 112 224 336 448 560 672 784 893 J,008 "1,1 20 1,232 1,344 1,456 1,568 1,680 1,792 1,904 2,016 2,128 2,240 2,800 3,360 3,920 4,480 5,600 6,720 7,K40 8,960 10,080 11,200 13,440 15,680 17,920 20,160 22,400 24,640 26,880 29,120 31,360 33,WO 35,840 38,080 40,:i20 42,560 44,800 49,2SO 53,760 58,240 Clear Spans, in Feet. (From Tredgold ) 4 01 A In. 1.2 1.4 1.6 1.7 It8 1.8 1.9 2.0 20 2.1 2.1 2.2 2.2 2.3 2.3 2.4 2.4 2.4 2.5 2.5 2.6 2.8 2.9 2.9 3.1 3.3 3.4 3.5 6 8 .a H, In. 1.7 2.0 2.2 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.0 3.1 3.1 3.2 3.2 3.3 3.4 3.4 3.5 3.5 3.7 3.9 4.0 4.1 4.4 4.6 4.8 4.9 5.1 5.2 6.5 5.7 5.'.) 6.0 10 12 14 16 18 20 22 24 26 28 30 J3 P, & In. 1.4 1.7 1.9 2.0 2/2 2.2 2.3 2.4 2.5 2.6 2.6 2.7 2.7 2.8 2.8 2.9 2.9 3.0 30 3.0 3.2 3.4 3.5 3.5 3.8 4.0 41 4.3 4.4 4.5 p. P In. 1.9 2.2 2.4 2.6 2.8 2. 3.0 3.1 3.2 3.3 3.4 3.5 3.5 3.6 3.6 3.7 3.8 3.8 3.9 3.9 4.1 43 4.5 4.7 4.9 5.1 5.3 5.5 5.7 5.S 6.1 6.:j 6.6 6.8 6.9 1 JL In. 2.0 2.4 2.7 2.9 3.0 3.2 3.3 3.4 3.5 36 3.7 3.8 3.8 3.9 4.0 4.0 41 42 4.2 4.3 4.5 4.7 49 5.1 5.4 5.7 5.8 6.0 62 6.4 6.7 69 7.2 7.4 76 7.8 7.9 8.1 8.3 8.4 8.5 8.7 8.8 8.9 9.0 9.2 9.4 9.6 9.8 ' P In. 2.2 2.6 2.9 3.1 33 3.4 3.6 3.7 3.8 3.9 40 4.1 4.2 4.2 4.3 4.4 44 4.5 4.6 46 4.9 5.1 5.3 5.5 5.8 6.1 6.3 65 6.7 69 7.2 7.5 7,8 8.0 8.2 84 8*6 88 8.9 9.1 92 9.4 9.5 9.6 9.7 10.0 10.2 10.4 10.6 P P In. 2.4 2.8 3.1 3.3 3.5 3.7 3.8 39 4.0 4.2 4.3 4.4 4.4 4.5 4.6 4.7 4.7 4.8 4.9 4.9 5.2 5.5 5.7 59 6.2 6.5 6.7 7.0 7.2 7.4 7.7 8.0 8.3 8.5 8.8 9,Q 9.2 9.4 3 9.8 100 10.1 10.3 10.4 10.7 10.9 11.1 11.4 Ot a> P In. 2.5 3.0 3.3 3.5 3.7 3.9 4.1 4.2 4.3 4.4 4.5 4.7 4.7 4.8 4.9 5.0 5.0 5.1 6.2 5.2 5.5 5.8 6.0 6.2 6.6 6.9 7.1 7.4 7.6 7.8 8.2 8.5 8.8 9.0 9.3 9.5 9.7 9.9 10.1 10.3 10.4 10.6 10.8 10.9 11.0 11.3 11.5 11.8 12.0 P. 3 In. 2.6 3.1 3.4 3.7 3.9 41 4.2 4.4 4.5 4.7 4.8 4.9 4.9 50 5.2 5.2 5.3 5.4 5.4 55 5.8 6.1 6.3 6.5 6.9 7.3 7.5 7.8 8.0 8.2 8.6 8.9 9.3 9.5 9.8 10.0 10.2 10.4 10.6 10.8 11.0 11. '2 11.3 11.5 11.6 11.9 12.2 12.4 12.7 p. P In. 2.7 3.3 3.6 3.9 4.1 43 4.4 4.6 4.7 4.9 6.0 5.1 5.2 6.3 54 5.5 5.5 6.8 5.7 5.8 6.1 6.4 6.7 6.8 7.3 7.6 7.9 8.2 8.4 8.6 9.0 9.4 9.7 10.0 10.3 10.5 10.8 11.0 11.1 11.4 11.5 11.7 119 12.2 12.5 12.8 13.0 13.3 13.5 1 In. 2.9 3.4 3.8 4.0 4.3 4 5 4.6 4.8 4.9 3 5.3 6.4 5.5 6.6 5.7 5.8 5.9 eio 6.0 64 6.7 6.9 7.2 7.6 7.9 82 85 8.8 9.< 9.4 9.8 10.1 10.4 10.7 11.0 11.2 11.5 11.7 ll.f 12.0 12.2 12.4 12. 12.7 13.0 13.4 13.6 13.9 A 1 In 30 3.6 3.9 4.2 4.4 4.6 4.S 5.0 5,1 5.3 5.4 55 5.6 5.7 5.8 6.9 6.0 6.1 6.2 6.3 6.6 7.0 7.2 7.6 7.9 8.3 8.6 89 9.1 94 98 10.2 10.6 10.9 11.2 11.5 11.7 11.9 12.1 12.3 12.5 12.7 12.9 13.1 13.2 136 139 14.2 144 .c p. P In. 3.1 3.7 4.1 43 4.6 4.8 5.0 5.2 6.3 54 5.6 57 5.9 60 6.1 6.2 6.v 6.4 65 6.5 6.9 7.2 7.5 7.7 8.2 8.6 8.9 92 95 97 102 106 10.9 11.3 n.e 11.9 1-2.1 12.4 126 12> 13.0 13/2 13.4 13.6 13.8 14.1 14.4 14.7 15.0 i P, 8 In. 3.2 3.8 42 45 4.8 5.0 5.2 5.4 5.5 5.7 5.8 5.9 6.0 6.1 6.2 6.4 65 6.6 6.7 6.8 7.2 7.5 7.7 8.0 85 89 92 95 9.8 10.1 10.5 11.0 11.3 11.7 12.0 12.3 12.5 128 13.0 132 13.5 13.7 139 141 14.2 146 14.9 1.^.2 15.* .4 .5 .7 .8 .9 8.0 8.1 1 62,720 206 STRENGTH OF MATERIALS. 207 at the center of solid horizontal CYLINDRICAL beams of tie weight of half the beam itself, which must be deducted in order to obtain the lly distributed along the enti( clear length, it will be twice as great as at the center. In this case, rt iron cyls willbear about 1 fourth more than cast. For W. Pine, spruce, ite ash, or best Southern yel pine by 4 or 6. For the breaking load of a square beam, whose breadth loads by 1.7. (ORIGINAL.; sn SlU VO X ^H I J o Ul 3 O SSSSSSSSSSrt-SSSSSSSSSSSSSggSS^ ,, *,. i i i i i i i i i = i : ; : ; i 1 1 ; : ^533^55 r Spans between the End Supports, in Feet d j a innnimmmmF^^iii d H 3 !!Jj!!!!!!!!jM!jj| S5333 5 *> S a innn!nM!8S23SS32=53SS3S23S3, a I ::::::::: :ssgsggSS83^^,w^^c-M :::::;;;: p Nwwei9 *^ ie ' Bao s2a2ss8S8sssss 3 H i i i i i i i i i isSSSSSSSZSS^oo^.-^-^.^- yj|jy :~"^<<^SrtrtSS;8833SS5;!! 2 a c ::::::: :8S3S5!;81SSSS3S^oo^.oo- ii* d M CO a d o ' n rt * tt 00 rt rt OS rt (O rt C ilnf . . ; ; =,*, -g s = 2! !BSgSSSS3 | J 1 ; P5SS5S5S^P 4. Table of breaking loads ii >n, supported at both ends; ii g load. To do this, use the last column. If the of the entire clear length must be dedt aks, divide the loads by 4.5 or bv 6.8; and for w ch equals the diameter of the cylinder, multiply 6 I i "*t-OQC''OOOOCDOCOinrtOOaOO5WCO-H<-*'*u3t~| 1-1 1-1 1- M <*^a e oo a rt ^ j* ^ ^ g jg ^ j*. jg 1 d in i 1 SgSSS^o^ao**..,^*- J: :::::::::: rtrt*^^o 5 *o 5 rt.n^o S 5 . . d a ;. ^-^^^^l^^2^p| OB d a 1 :5|g.^ 1 :: = : i i i i rt^^cc^^^s.rt^^sjll QJJ jjjj d (5 1.T J. rr-r-rr- -7-7^-7-.---.- ~-. ^ tS * o> oe *SfeS sa m qou *!(! i ; : ; : : ; ; : ; ; ; ; : ; ; ; : ; ; ; i : ; i i 208 STRENGTH OF MATERIALS. FiglS Art. 35. Hodgkinsoii's beams have nearly 1% times the strength of a beam of equal wt whose top and bottom flanges ,T are equal. Mr. Hodgkinsori, having found that on an average, cast iron reqd about fii^ times as much force to crush it as it did to pull it apart, contrived the beam (of which Fig 15 is a cross-section at the center) in which the upper or compressed rib or flange, ?/, has but Y\ of the area of the lower or extended one, 6.* The top flange he therefore as- sumed to be safe; inasmuch as its area is some- what greater than the proportion of 1 to 6%: and hence, breaking will take place from the yielding of the bottom flange by extension. As the result of his experiments, he gives the following rule, when the load is applied at top, or equally on both sides of the beam. See Item, next page. Area of bot flange v depth oo Constant = in sq ins ' in ins ' 2.166 Clear length in feet. When the lower flange is as much as about '2*4 inches thick, experiments show that part of the breakg load thus obtained should be deducted; because thick castings are proportionally weaker than thin ones. Half the wt of the beam itself must be deducted,' for the neat breakg load; this, however, is necessary only when the beams are very lori^; for such as are used for ordinary building purposes, it may be ne- glected. If the load is equally distributed, it will be twice as great; but the entire wt of the beam must then be deducted. Ex. The upper rib u = 3 ins X 1 inch = 3 sq ins area; bottom rib 6 = 1% ins X 12 ins = 18 sq ins area; total depth oo, 15 ins ; clear span, 20 ft. Here, 18 X 15 X 2.166 584.82 20 20 = *"*" t ns > t ne reqd load, including % the beam. Now to find the wt of half the beam, we may proceed thus: Mult the entire area of its cross section in sq ins, by the clear span in ins. This gives us the cub ins of iron contained in the beam ; and these div by 8600, give the wt of the beam in tons ; because 8600 cub ins of cast iron weigh about 1 ton ; or near 4 cubic ins 1 ft). Thus, if the vert rib contains 12 sq ins, then since the two flanges con- tain 21, the entire section is 33 sq ins; and the span being 240 ins, we have 33 X 240 = 7920 cub ins of iron. And ~~ = .92 of a ton, the wt of the beam. One-half 8600 of this, or .46 ton, taken from the breakg load 29.241 tons, leaves 28.78 tons as the neat breakg load ; showing that in such cases as this it is scarcely worth while in practice to make the deduction. These beams are not always made of the same section throughout, (see Fig 16,) but diminish toward the ends; this method is therefore not always strictly correct, but no great accuracy is needed in such cases. To find the size of a Hodgkinson beam, reqd to break nnder a given center load, having the depth. Mult the given load in tons by the clear span in feet. Mult the constant 2.166 by the total depth, oo, in ins. Div the first prod by the last; the quot will be the area of the bottom rib in sq ins. This, div by 6, will be the area of the top rib. The bottom rib is usually made from 6 to 8 times as wide as it is thick; and the top one from 3 to 6 times, The thickness of the stem is usually a little greater at bottom than at top; the average thickness being from ^ to -fa of the depth of the beam. To save iron, the width of the bottom flange, and of the top one also if thought proper, may be reduced by curves to about % as great at each end of the beam as at its center; as shown by the middle sketch of Fig 16, of which the upper sketch is a side view. Or, leaving the dimensions of those flanges un- altered, the depth of the vertical rib may be reduced toward the ends, as shown by the loweet sketch. The theoretical curve is here an ellipse. When the width is reduced, the very ends may, for stability, be widened out, as at e, which is a top view. The vert rib is generally strengthened by casting brackets * In practice % is much better and safer than %. FlglG STRENGTH OF MATERIALS. on each side of it, as in the upper sketch. These should not extend entirely to tne upper rib, as they then expose the beam to crack as it cools. To prevent this tend- "eticy, they may be attached alternately to the top and bottom ribs. The upper ones, however, are rarely needed. In designing these beams, as well as in all other castings, it is important to avoid sudden transitions from thin to thick parts ; and to keep all parts as nearly as possi- ble of the same thickness. Otherwise the castings are apt to warp and crack in cooling. Also, bear in mind that the resistance or strength per sq inch is considera- bly less in thick castings than in thin ones. Item. The above rule for breakg loads is safe when the load is equally disposed on top, or on each side of the vert web ; and when said web and the flanges are pro- portioned to each other about the same as those used in Mr. Hodgkinson's experi- ments. But subsequent investigators have found his beams to break with but little more than half the loads given by the rule, when applied to only one side, as bo, or wo, Fig 15, of the top or bottom flange. W. II. Barlow, C. E., London, experimenting since Hodgkinson, finds that when a cast-iron beam is liable to be loaded on only one side of tne flange, the top flange should have an area equal to % that of the entire cross-section of the beam ; and for beams so proportioned, he gives the following : . Constant < 2.333 in tons Clear span in feet. Other experimenters recommend that even for loads pressing vertically through the upright rib, tlio lower flange should have but about 3 instead of 6 times the area of the upper one. Cast beams should always be tested. l^he average ultimate resistance of steel to compression being about twice that to extension, a Hodgkinson beam of that metal should have its lower flange of twice the area of its upper one. Much uncertainty exists in the whole matter. Art. 36. For the purpose of ready reference, we give a few ex- perimental results with cast-iron beams of various shapes : being the actual center breakg loads in tons of sound beams. Some beams of Sterling's toughened cast iron gave results full Y z higher than those of common iron. Actual center breakg; loads in tons, of cast-iron beams. Clear spans in feet. Breadths and depths in inches. 1,5x.5^P K co Span 4U ft. 2.5*. 5 I Br load 2 tons. *., The above inverted. | Br load 2.3 to 2.9 tons. Span 4V ft 325 ton The above inverted. span 41^ ft. ] VO f 3 ' 7 tBB 2 . 27x. fi^aaLa Br load \ ^ I 4.2. * Span 11% ft f Br load 20 tons. Span 18 ft. Br load 22 to 28 tons. 00 Span27%ft.t |#l *~ Br load 2d^ to * As shown by dd, Fig 15. t "After bearing 17 tons, the beam was unloaded, and its elasticity appeared to be but little if at all Injured." Def under 4^ tons. .15 inch ; 8% tons, .3 inch ; 17 tons, l.t inches. ; About two hundred of these beams were tested by center loads of 12 tons. Def & U> ^ inoh. 14 210 STRENGTH OF MATERIALS. Span 15 ft. I j ,. Br load 12^ tons. Br load 10.5 to 11.6 tuns. "l Span 19 ft. 52 Br load 50 to 54 tons. By formula, p 194, it should have been but 40 tons. 15 X 2/j H* Span 30% ft. Br load 58 tons.* m In describing such beams, it is better to give the entire depth of the beam; for when the depth of the wel> is given, doubts arise whether it is meant to include the thicknesses of the two flanges, or not. Every writer, almost, that we have seen, leaves us in this doubt. REM.. In beams either larg-er or smaller than these, but whose cross-sections are proportioned exactly as these are, and whose spans are the same that these have, the center breakg loads will be as the cubes of their cross-section , % , 2, 3, or 10 times as large every lines. Thus, in a beam which is i way, except in span, the breakg load will be TTToTT ' T7 ' ^8 > 8 > 27 or 100 times as great. If the spans also differ, first find the load as above, as if they were the game; then say, as the new span, is. to the span given in our Figs, so is the breakg load thus found, to the actual breakg load for the new beam. Thus, suppose we wish to make a cast-iron beam, 4 times as large every way as the dimensions given in the first of these Figs ; except its span, which is to be, say 10 ft, instead of 4% ft. Here the first breakg load is found tobe4X4X = 64 times as great; or 2 tons X 64 = 128 tons. Next, New Span. Span in Fig. First load. Actual load. 10 : 4.5 : : 128 : 57.6 tons. In such cases we must, however, have regard to Rem, Art 11. The foregoing process applies equally to beams of any other shapes, such as the following ones ; or whether solid or hollow, &c; and of any other materials; so that if we have all the dimensions, and the breakg load of any beam whatever, we may find that for another one of the same material, and of the same proportions of cross-section. It may become advisable in important cases, to even make one or more model beams of some hitherto untried form ; and to break them, in order to find the breakg weight of the actual beam of the same material. In doing this, the defs should abo be measd, in order to see.whether those of the actual beam may not be too great. See Art 26, &c ; and Art 46. Art. 37. Figs 17 show some varieties of the rolled I beam. They have two equal flanges, and a stem or web of uniform thickness. They are 3 v called 6, 9, 15, &c, inch beams according to their lotal G For their strengths as beams see p 212, 213; and as pillars or struts, p 638. '6.25 17 * This iron was " Sterling's toughened," haying about 16 per cent of wrought scrap melted in it. Kaeh of the 89 beams was tested bjr a centr load of 20 tons, which produced deft of from % to ft inca. Entire length, 34^ ft. STRENGTH OF MATERIALS. 211 Table of safe quiescent distributed loads in tons (224O Ibs) of channel bars as beams with the web vertical. The safe loads are here taken at one-third of the ultimate one for iron of superior quality ; but for aver- age iron it will be better in practice to reduce them about one-sixth part before deducting the wt in tons of the span of the beam itself, and which is here in- cluded in the loads. 'If liable to much vibration, as in bridges, deduct one-third to one-half. The beams are supposed to be stayed against bend- ing horizontally. Three stays per beam will suffice for the longest. For a table of these beams as pillars, see p 640. Spn. Hvy wt. 12 in Load. Def. Me wt. 'd 12 Load. in. Def. Hv Wt. y loi Load. in. Def. Hf wt. <1 10 Load. in. Def. Ft. Lbs. Tons. Ins. Lbs. Tons. Ins. Lbs. Tons. IDS. Lbs. Tons. Ins. 10 500 17.82 .13 300 12.32 .13 350 1089 .16 230 8.30 .16 14 700 12.50 .26 420 8.75 .26 490 7.77 .31 322 6.00 .31 18 900 9.64 .43 540 6.84. .43 630 6.07 .52 414 4.64 .52 22 1100 7.87 .65 660 5.62 .65 770 4.91 .78 506 3.75 .78 26 1300 6.66 .90 780 4.73 .90 910 4.10 1.09 598 3.21 1.09 30 1500 5.80 1.20 900 4.10 1.20 1050 3.57 1.45 690 2.77 1.45 II vy 9 in. Med 9 in. Hvy 8 in. Med 8 in. 10 300 8.29 .18 180 6.07 .18 250 6.25 .20 160 4.73 20 14 420 5.90 .35 253 4.34 .35 350 4.46 .39 224 3.48 .39 18 540 4.61 .58 324 3.39 .58 450 3.48 .65 288 268 .65 2* 660 3.77 .86 396 i 277 .86 550 2.86 i .97 352 2.14 .97 26 780 3.19 120 468 2.33 1.20 650 2.4t ! 1.39 416 1.78 1.39 30 900 2.76 1.61 540 2.03 1.61 750 2.05 1.81 480 1.60 1.81 Hvy 7 in. Med 7 in. Hvy 6 in. Hvy 5 in. 10 200 4.55 .23 140 3.75 .23 110 2.32 .27 100 1.70 .32 14 280 3.30 .45 196 2.68 .45 154 1.61 .58 140 1.16 .63 18 360 2.50 .75 252 ! 2.14 .75 198 1.34 .87 180 0.89 1.05 22 440 2.05 1.11 308 1.70 1.11 26 520 1.78 1.56 364 1.43 1.56 Dimensions of the above channel bars in ins, from out to out each way. Depth. wt K Wdth of Flge. Avg Ths of Flge. Ths web. Area. Depth. wt per Ft. Wdth Flge. Avg Ths of Flge. Ths of web. Area. Ins. Lbs. Ins. Ins. Ins. Sq Ins. Ins. Lbs. Ins. Ins. Ins. Sqln. Heavy. ...12 50 3.20 .78 .95 15.01 Heavy.... 7 20 2.55 .53 .55 600 Medium..l2 30 2.70 .78 .45 9.00 Medium.. 7 14 2.29 .53 .30 4.21 Light 12 20% 3.00 .41 .31 6.25 Light 7 10^ 1.98 .44 .22 3.18 Heavy .. 10 35 2.97 .72 .72 10.50 Heavy.... 6 11 2.00 .50 .25 3.30 Medium..! 23 2.61 .72 .36 6.90 Light 6 7^ 1.75 .34 .19 2.25 Light 10 15% 2.50 .34 .31 4.62 Heavy.... 5 10 2.00 .50 .30 3.00 Heavy.... 9 30 2.83 .63 .70 9.00 Light 5 VA 1.55 .37 .18 1.9o Medium.. 9 18 2.43 .63 .30 5.40 Heavy.... 4 73| 1.78 .41 .28 2.33 Light 9 14^ 2.50 .34 .34 4.30 Light 4 6 1.54 .34 .23 1.80 Heavy.... 8 25 2.64 .59 .64 7.50 Heavy.... 3 6 1.71 H4 .40 1.80 Medium.. 8 16 2.30 .59 .30 4.80 Light 3 5 1.50 .34 .20 1.50 Light 8 12M 2.00 .47 .26 3.75 Rein. These tables are condensed from the useful "Tables and Information on Wrought Iron," issued by Messrs Carnegie Brothers & Co, of Pittsburgh, Penna. The o make to order heavier channels without extra charge per ft, by increasing the thick- ness of the web, but not altering the flanges otherwise than that the increased ths of web adds to their out to out width ; their aver- age thickness remaining unaltered. They also make Deck Beams, Fig 18, and all varieties of I, L, -f, T, <&c, bars, segment columns of various kinds, bridge work, roofs, &c. 212 STRENGTH OF MATERIALS. jjfr < ij C3-S s 15 p| Jl - SSS2S22SSSSSEigSCS II -|l j3e - i-, " "^ < 'SI'* 4, * 1~ ** nj S.I ill S fl y *q e E-^ '*" "" 1 ^* SB S -** ^ fl ! - g22538Saasa8SSSSS5cS c a ^s "ti! "* 3 V ^ Q ~^ ||i*l | sss2S2!=ss;r 7 sss;s5?S5a rn 5 1|I| s||*| ^ifi ^1^1 y . IM S2 222555=5555S 91 |jli lllll o 1 |ss8se33:^t:^^ "o * a -^ * g-stl ^|fs c - * ojsll N, H ? S * 2 2 J * S^i3;ggg8822S55?^?5?5 56 Illl cS^ C - 1 g P ^^^.H^^, ^^ N Hf- ll-* Ipfj S iQ - s o c i^3-;ooi^-*MrJN^f^r4oo'666 M H -< 11 If nisi Jh . 2 jf^^^,^--^ !i !li! Itrfii It IS ti'tl a> SsJIS 11 ^^gg SBi^5 s sSSSSSgSgsgsg55tSSSSSSS ~ ^ 3| .| *""a) 5 g J O 1-1 t U< c 'S *1||| B | |S68^R?^SSSS^8^ & *sl* s Vll fe2 d C ^^c.^^^^co^ooooo-.^^^cco ^ ^^ .* e * S ^ ^ C M . _g | g ^ g illijj |52S^ fi C PI 1 l-i "S * i a> e -l ^ ssssssssssssgssssss C-2 |11 *iT"5 'fe * 02 'S *" |IP! in I || .!!! iPi! jrf; 3 w C w ."s ; ^||>- iflU K ? S S5^S--- -- r vl^ e'S 1 1||| STRENGTH OF MATERIALS. 213 Art. 38. Rolled Iron beams of the New Jersey Steel and Iron Company. Made at the Trenton, N. J., Iron Works, Cooper, Hewitt & Co. Morris, Wheeler & Co., Market & Sixteenth Sts., Philadelphia, agents.* Similar to the Phoenix beams, Fig. 17, p 210. J See " Cautions " below preceding table. Depth Thicks. Flange Weight Load Coef. For Area of of beam. of web. width. per yd. use see below. section. Ins. Ins. Ins. Lbs. Lbs. Tons. sq. ins. loi^ .6 5% 200 748000 334 20 16% 1 A 5 150 551000 246 15 12^ .6 51^ 170 511000 228 17 1-/4 .48 4.79 125 377000 168 12.5 10 v^ .47 5 135 360000 161 13.5 10$ az 4.1^ 105 286000 128 10.5 9 .58 4/2 125 268000 120 12.5 9 % 4 85 189000 84.4 8.5 9 .3 '&A 70 152000 67.9 7.0 8 % 4/2 80 168000 75.0 8.0 8 .3 4 65 135000 60.3 6.5 7 % 3/<> 60 1020< X) 45.5 6.0 6 .3 3/^ 50 76800 34.3 6.0 6 \/ 3 40 62600 28.0 4.0 5 5 \ 3 2% 40 30 49100 38700 21.9 17.3 4.0 3.0 4 A 3 37 36800 16.4 3.7 4 g 2% 30 30100 13.4 3.0 Channel iron, upright, thns ]. l ?6 .68 4 140 381000 170 14.0 .42 3 85 238000 106 8.5 9 * .43 3i/j 70 146000 65.2 7.0 9 .33 2 50 104000 46.4 5.0 6 .40 45 58300 26.0 4.5 6 .28 2y<* 33 45700 20.4 3.3 5 .20 1^ 19 22800 10.2 1.9 4 .20 ]1^ 16.5 15700 7.0 1.65 3 .20 1^1 15 10500 4.7 1.5 The above table, as well as that on p 212, is issued by the above named Co. In both, the sate quiescent equally distributed load is taken at l cts per ft). In Belgium I beams are rolled 21.6 ins deep; 334 Ibs per yd. 214 STRENGTH OF MATERIALS. Art. 39. Cooper A Hewitt's box beams, Fig IP, made by the Trenton Iron Co. These beams consist of two channel irons, of 6^ X 2% X \4 inch ; one at top, and one at bot- tom ; and of two vert sides, of ^| inch plate-iron, 18 ins deep. They weigh 69 fbs per foot run. The Co makes also larger and variously modified beams. The following are the results with a beam like the fig ; '^'H ft J n g, 19 ft 5 ins clear span. The ends unconfined further than being steadied sideways. Cen load. ft>s. Def. Ins. Cen load. fts. Def. Ins. H 76862 1H 12990 Interval of 26 days. 19920 1 6 81342 iH 24230 85524 at once a IT! 28744 crackling noise 32284 72 commenced. In 37844 9 10 min, 2 & 42387 78 In 1 hour, 3c 46923 ii 90302 3 51460 % With a side defln of iff 55985 1 3 H This increased 60553 until the side plates gave way 65089 11 at their bottom edges, in an 69954 a hour. Rule for strength of riveted box and I beams. Figs 19 to 25. The greatest safe, uniformly distributed, quiet load in tons (2240 Ibs) for such beams of any size, well made, and so proportioned as to be secure against yielding either sideways or by buckling, may be found near enough for practice by using the "General Rule," p 213; and taking only three-fourths of the resulting load. If the top and bottom flanges diifer in area, use the least one. Art. 41. Fairbairn plate, and box beams, or girders. These are made of rolled plates, and angle iron, riveted together. The plates are usually from /% to % inch thick. The angle-irons from 2^ by "Eft by % to 6 by 6 by 1 inch. The rivets from % to 1% inch diam ; and driven from 3 to 6 ins apart, from center to center. Other shapes of rolled iron are also fre- quently introduced, as the channel-iron, cc; T iron, t, &c, &c. Figs 22 and 23 are common modes of constructing the plate beam for buildings, and for bridges of moderate spans. Fig 25 is an- other form. Frequently several thicknesses of plate-iron are riveted together, to form the top and bottom flanges, to obtain the reqd amount of section. The vertical web, r, or iv, is likewise frequently composed of two thicknesses. In Fig 24 of the box-beam, each of the two flanges con- sists of two thicknesses. This form of beam, when on a large scale, constitutes the tubular bridge. Mr. Fairbairn makes transverse area of the top flange equal to 1% that of the bottom one. He states that with this proportion no separate allowance need be made for the rivet- holes, &c; that large experience sanctions it, and that no gain of strength attends making either one of the flanges to exceed this limit. There is roor * J " rimpntsai The angle- ngth maitiug t.ituei- unu ui uie uauges 10 exceeu uns iimir. A here is room for doubt whether his proportion for the flanges is the best. More experiments are needed on this point. He gives no rules for the thickness of the vert members, irons a a, the channel-irons c c, &c. are to be con ' ' such iron must the a a, the channel-irons cc, <&c. are to be computed as part of the flanges. All rons, as also any others used in the bottom flanges, are subject to tension; and therefore be strongly spliced together at their joints. re ue strongly spiiceu logemer at uieir joinis. Many engineers, however, make both flanges of equal area; conceiving that portion to be safer in consequence of the weakness of the lower joints. The u flanges being compressed, are not appreciably weakened by their joints. pro- upper STRENGTH OF MATERIALS. 215 For the strength of the single-plate beam, Figs 22 and 23, Fair- bairn gives as follows : Tr are ^ of bofc Cen breakg load in tons ") flange in sq ins v Total depth of beam v Constant including y z weight ' \ = in sq ins A in ins A 75 of the beam itself, J Clear span in ins. For the box-beam, Fig 24, precisely the same, except use a constant 80 in- stead of 75. Some very competent authorities, however, regard this as too great; and maintain that the constant should not exceed 75 in either case. Half the wt of the beam must be deducted, to obtain the neat center breakg load; or the whole of it for a uniform load. In beams for buildings, however, or in bridges under 40 to 50 ft span, this deduction will rarely be necessary. As a general approximation, when these beams exceed a very moderate span as bridges, about ^ to ^ part must be added to the wt of wrought iron as deduced from a neat transverse section, such as those in the above Figs, to allow for splicing- plates, stifferiers, chucks, rivets, &c. The following girders have for many years sustained locomotive traffic: 1st, like Fig 23; single track; 2 girders, 15 ft apart from center to center; length of each, 28 ft; clear span, 24 ft; total depth of girders, 28 ins; top and bottom T- irons, t, horizontally 7% X J^; vert^ flange, 3 Xj^j the four longitudinal splicing- prd- plates, * s, each 6 X %'> vert web u, % thick. 2d, like Fig 22 ; single track ; 2 gi arid before the proper proportions were known. 3d, like Fig 23 ; single track ; 2 girders, length of each 68 ft; clear span, 60 ft ; total depth of girders, 4 ft 3 ins; top and bottom T-irons, t, 6 X yV 5 vert flange, 3 X K' tne 4 'longitudinal splicing-plates , , s,Tach 6% X /4 ; vert web, w, % inch thick, luveted to the top and bottom are plates like the dotted ones in Fig 22; each of these is 2 ft wide, % inch thick at the center of the girders, and % at the ends. All have vertical stiffeners. The following are in use on the Charing Cross Railway, England: all like Fig 22. 1st, for a single track, 41 ft clear span, total length 46 ft ; total depth 2^ ft ; top plate, dotted, 16 X %: bottom plate 15 X M ; vert web, w, % thick ; top angle- irons^ X 4 X %; bottom ones 4 X 4 X %\ rivets % diam, 3 ins apart from center to center ; weight of 1 girder, 4 %tons.2d,for a single track, 51 ft clear span, total length 56; total depth 3^ ft ; top plate, dotted, 18 X % ; bottom plate 16 X %; vert web, w, % thick; top and bottom angle-irons, a, 6, 4 X 4 X %\ rivets % diam, 4 inches apart from center to center; weight of each girder, 5^ tons. 3d. For three railway tracks, 2 girders only, 39 ft apart from ceu to cen; clear opan 73; total length 82 ; total depth, 7% ft at center of span ; 6 ft at ends ; top plate, dotted, 2 ft 2 ins wide ; composed of 4 thicknesses of plates, each % thick, at center of span ; and 3 such plates at the ends ; bottom plate 2 ft wide, one thickness of %; and 3 of % each, at the center of the span ; at the ends one of the % is omitted. Vert web, w, 1^ thick at centre of span ; '% for 11 ft at each end; top and bottom angle-irons, 6 X 6 X M; weight of each girder, 25 tons; which would suffice for a single-truck girder 100 ft span. Web stiffened every 3 ft ; as at i ?', Fig 28 ; by Ts of 6 X 3 X % The transverse iron floor-girders, when the main beams are but about 15 ft apart, fora single track, and the girders about 5 to 6 ft apart, are like the shaded parts of Fig 22; that is, without top and bottom plates. Depth about Id iris ; vert web ^ inch thick; 4 angle-irons, each 3^ X 3^ X Y^\ weight about 800 JJbs. In such of the Charing Cross railway bridges as have their main beams 39 ft apart, and where the transverse floor-beams, also 39 ft long, support 3 tracks, the last, 3 ft apart, have the following dimensions: Total depth, 16 ins; top and bottom plates, dotted, 15 X % each ; vert web y thick ; four angle-irons of 5 X 3 X %> Transverse floor-beams may rest upon the top of the main beams ; or upon the inside portion of the bottom flange. The last is not as favorable to the strength of the main beams as the former ; or as when the floor-beams are placed beneath the bottom of the main ones. In all cases the two beams are riveted together. Art. 42. The construction of plate and tnbiilai girders is not as simple as might be supposed from the foregoing Figs. They are composed of separate sheets of iron, not exceeding about 3 X 12 ft in breadth and length in the largest bridges; while in small ones, much smaller dimensions must be employed. "Whenever two plates come into contact, the joint, whether vert or hor, must be strengthened by riveting upon both sides of it at least narrow strips of plate iron 6 to 8 ins wide, called covering-plates, or splicing plates. Or, since the girders re- quire vert stiffenersthe vert joints are frequently covered on both sides of the platei 216 STRENGTH OF MATERIALS. Fig ZJ by vert T-irons, as ww, Fig 26; in which a a represent two plates whose vert joint is to be thus strengthened. When the middle web w, of the T-irons, does not project sufficiently to impart the L , reqd stiffness to the vert . o ^m>^> o web, tt or #, of the beam, broad strips s, Fig 27, of plate iron, may be intro- duced instead of it, in con- nection with 4 angle-irons, as bb. These last are riv- eted together through the vert plates gg of the main beam ; and through the stiff- eners ss- thus protecting the joint of the former, and holding the latter firmly in place. These stiffening-plates are frequently made to project more at bottom than at top; and are at times strengthened by angle-iron riveted along their outer edge and at their base ; thus making them very effective as braces also. The transverse floor-beams or gird- ers, extending from one main beam to the other, are so spaced as to meet the stiffeners, instead of rest- ing upon the weaker intermediate parts of the main girders. These transverse beams themselves generally require to be stiffened in a similar manner, at inter- vals of usually from 3 to 8 It, as in the main girders. The stiffeners near the ends of the main girders, or those resting on the piers and abuts, are (especially in large spans) placed nearer together, and made stouter than the others, because upon them rests the wt of bridge and load. The T and L irons, &c, used to stiffen the sides of the Britannia tubes, 460 ft span, weigh half as much as the plates which compose the The vert T-irons to, or angle-irons &, generally have to be bent both at top and bottom, as shown at i ?', Fig 28, in order to pass the hor angle-irons of the upper and lower flanges. The joints of the c long narrow strips a, Fig 28, which compose the top and bottom plates of the flanges, must also be connected together by riveted splicing-plates. Those at the lower flanges require especial care in this respect, inasmuch as they undergo tension. The same applies to the hor angle-irons h //, Fig 2v), which also must be firmly con- nected at their ends. This is done by pieces cc of bent iron in lengths of 18 to 24 inches. Art. 43. Mr. William Fairbairn, of England, gives the fol- lowing table, calculated for a double-track railway, supported by two box beams, or by two girders He adopts 6 tons per foot run of the span, as the center breakg load of the bridge ; or 12 tons per foot run, uniformly distributed, including the wt of the bridge itself. This 12 tons, he says, is equal to about six times the maximum load that can practically be brought upon the bridge. For spans up to 100 ft, he takes 1 ton per running foot as the wt of the bridge itself, and 2 tons as that of the rolling load, or two trains; and from 100 to 300 ft spans, 1^ tons as the wt of the Bridge; and 1% tons as that of the load; the total being 3 tons per foot in both cases. The depths in the table are equal to ^ of the span, for spans less than 150 ft: for those exceeding 150 ft, they are ^.* The breakg loads include the wt of the bridge itself. The depths are those at the center o*" the span. Except in quite small spans, some saving of material is effected by diminishing the depth from the center to the ends. Single-track bridges will require but about -f s the quantity of material given in the table; the depth of girder remaining the same. If the depth is dimin- ished, the areas of the flanges must of course be increased; or if the depth is increased, the flanges may be reduced; for, as is seen by the foregoing formula, Art 44, the strength varies as the depth, when the flanges remain the same. The propor- tions of the table will serve for plate beams, as well as box; the diff of strength being but as 75 to 80. * These depths are much smaller than is usual in truss bridges In the United States. Here from j* to fy are the common ones ; and are certainly to be preferred for practical reasons of econom?. STRENGTH OF MATERIALS. 217 Table by Mr. Fairbairn, of the Proportions of Tabular Girder Bridges, consisting? of two girders, of from 3O to 3OO ft span ; and intended to sustain two railway tracks. (The sufficiency of these dimensions has been questioned by high authority.) Clear Span. Center Breaking Load of Bridge, or of the two Girders. Sectional Area of bottom of one Girder. Sectional Area of top of one Girder. Depth of a Girder, at the center of the Span. Feet. Tons. Inches. Inches. Feet. In. 30 180 14.63 17.06 2 4 35 210 17.06 19.91 2 8 40 240 19.50 22.75 3 1 45 270 2194 25.59 3 6 50 300 24.38 28.44 3 10 55 330 26.81 31.28 4 3 60 360 29.25 34.13 4 7 65 390 31.69 36.97 5 70 420 34.13 39.81 5 5 75 450 36.56 42.67 5 9 80 480 39.00 45.50 6 2 85 510 41.44 48.34 6 7 90 540 43.88 51.19 6 11 95 570 46.31 54.03 7 4 *100 600 48.75 56.88 7 8 110 600 53.63 6256 8 6 120 720 58.50 68.25 9 3 130 780 63.38 73.94 10 140 840 68.25 79.63 10 9 150 900 73.13 85.31 11 6 160 960 90.00 105.00 10 8 170 1020 95.63 11156 11 4 180 1080 101.25 118.13 12 190 1140 106.88 124.69 12 8 200 1200 112.50 131.25 13 4 210 1260 118.13 137.81 14 220 1320 123.75 14438 14 8 230 1380 129.38 150.94 15 4 240 1440 135.00 157.50 16 250 1500 14063 164.06 16 g 260 1560 146.25 170.63 17 4 270 1620 151.88 177.19 18 280 16SO 15750 183.75 18 8 290 1740 163.13 190.31 19 4 300 1800 168.75 196.88 20 Art. 44. Moments of Rupture and of Resistance. By Moment of Rupture or Breaking Moment is meant the tendency of a load (including the wt of the beam itself or not, as the case may be) to break a beam by a lengthwise pulling apart of some of its fibres and a crushing of others, by the aid of leverage afforded by the beam itself. It is often called simply the moment of the load. It is also this moment that tends to bend the beam as a necessary consequence of the stretching and crushing alluded to; and when considered with reference to its bending effect instead of (as now) its straining and breaking ones it is called the Bending' Moment or Moment of Flexure, or of Deflection, of the load. The load and the wt of the beam together tend also to sever the beam transversely or across its length by a process called shearing. See " Shearing," p 642, but with that we have nothing to do here. The following rules both on Moments of Rupture and on Strains apply to horizontal closed beams (see Open and Closed Beams, p 644) of any form whatever of cross section, whether rectangular, circular or like Figs 22 to 24, p 214, &c. Those on moments apply also to hor open beams like the trusses of a bridge, &c, in which the load is supposed to be concentrated at the panel-points. But in such the shearing tendency vanishes (see Rem 2, Art 5, p 644); and the manner of resisting the moment of rupture, as also the strains caused thereby, differ entirely from those of closed beams. 218 STRENGTH OF MATERIALS, If the beam is inclined, the moment of rupture may still be found by using the hor span and segments instead of the inclined ones; but the resulting longitudinal strains, as well as the shearing forces become changed, involving much complication. We confine ourselves therefore to hor beams. This subject must not be confounded with the principle of Equal- ity of UIoiiieiits, Art 51, p 476, frequently used for finding the strains along the members of a truss. A load causes no moment or strain on any part of a beam that is not between the load and the assumed fulcrum point. Thus in Fig 29 calling c the fulcrum of the load on c x, no load between c and e produces any moment between c and /, because it has no fulcrum and hence no leverage there. The weight of the beam itself is not here included. When required to be so, consider it as a uniform load, and use Case 3 or Case 12, p 220 and 221, and add the result to that obtained for the load. The deflection in ordinary cases may be found by the rule on page 198. Art. 45. Having the moment of rupture of the load, it is necessary to know whether the moment of Resistance, or simply the Resistance of the beam (which term see Art 5 of Open and Closed Beams, p 646) is sufficient to withstand it. This Resistance in any hor solid or closed beam of any form whatever of cross-section may be found thus, subject however to the first paragraph of Art 2.% p 194. Mult its Moment of Inertia in ins by the Constant of Rupture (terms defined on p 195), and divide the prod by the dist in ins of the farthest fibre (or according to Prof Wood, of the farthest fibre on the side which will yield first.) from the Neutral Axis, or cen of grav of the section. Or as the rule is usually expressed by formula, I C -f- t. In a rectangular or circular beam, or in one with equal flanges and uniform web, or in any other in which the parts above and below the neutral axis are similar and equal, the dist of the farthest fibre will in any event be half the depth of the beam. All the dimensions must be in the same measure, that is all in ft, or all in ins. The Constant for Rupture for average rolled iron is about 45000 fcs or say 20 tons per sq inch. Cast iron 36000 ft>s or 16 tons. Good straight-grained, well-seasoned white pine or spruce 8100 fos or 3.6 tons ; yellow pine 9000 or 4 ; good oaks 10000 or nearly 4.5. But as large beams are liable to defects and imperfect seasoning, not more than about two-thirds of tnese constants should be used in practice. See Table, p 185. For either square or rectangular beams it will be easier and as correct to find the mom of res thus. Mult together the sec- tional area in sq ins, the depth in ins, and the constant for rupture per sq inch ; and divide the prod by 6. This is usually expressed by formula, R = - . For a solid flanged beam of equal flanges and uniform web it will be suf- ficiently close to find its resistance thus. Mult the area of only one entire flange by the depth between the centers of grav of the two flanges. Also mult one-sixth of the area of the web (in clear of the flanges) by its depth. Add the two pro- ducts together. Mult the sum by the ult tensile or compressive strength per sq inch (whichever is least) of the material. For average wrought iron this may be taken at 36000 fos or 16 tons. For average cast iron at 18000 ros or 8 tons when the flanges are equal. Rein. 1. Theoretically the webs of closed flanged beams need only be strong enough to bear safely the vert shearing forces of the load, (see "Shear- ing," p 642), and in practice this view may answer for quite short beams; but in long ones there is a tendency to warp or twist sidewise, which must be met by stiffening the web either by an increase of thickness, or by introducing vert stitfeners i ?;, Fig 28, p 216, or in some other way. We cannot pretend to give rules for either of these. Perhaps the best advice is to observe the stiffeners of flanged girders of successful plate-iron bridges, of which a few examples will be found on p 215. The web members of open beams must be calculated like those of truss bridges. Rem. 2. The horizontal strain at any given point of a chord of a hor open beam, is found by dividing the moment of rupture at that point, by the depth of the beam between the centers of grav of the flanges at the same point, see Art 9, p 647. In closed flanged-beams also the same rule is fre- quently used as being safe and sufficiently correct for practice. The span, depth, &c, must all be in the same dimension, that is all in ft, or all in inches, &c. Art. 46. In ordinary practice beams of wood or iron are used either as hor cantilevers, as Fig 29, firmly fixed at one end, or as hor beams supported at each end, -as Fig 31. STRENGTH OF MATERIALS. 219 I General Rule for moments of rapture in fior cantilevers, no matter how irregularly the load or loads may be distributed. Bear in mind that only that part of the load which is beyond (towards the free end from; any as- sumed point tends to break the beam at that point as a fulcrum, and that it does so with a leverage = dist of the cen of grav of that part of the load from the point. The other part of the load has no moment at that point. Thus the whole load o x tends to break the beam at g or I with a leverage = a g or a i as the case may be, a being the cen of grav of the load. And so for the moment at any other point c, Fig 29, as a fulcrum, find the wt of all the load c x between c and the free end / of the beam. Also find the cen of grav * of that part of the load. Mult the weight just found by its leverage c s. Example 1. We use a uniform load in order to illustrate the rule more readily. Let the hor yellow pine beam i 1 be 7 ft long; its breadth and its depth i e each 6 ins ; the whole load o x 4 tons ; and c the point or fulcrum at which the moment of the load is reqd. Then the wt of the load between c and t Is 3 tons ; and its cen of grav s is 1.5 ft or 18 ins from c. Hence the loads moment at c = 3 tons X 18 ins leverage = 54 inch-tons. That is, a load of 3 tons tends with a leverage of 18 ins to rupture the beam at c. Now is the resisting moment or the strength of the beam at c sufficient to withstand this? Using our first rule we have Moment of Inertia in ins (p 195) is = (its area X square of depth) -t- 12 = (36 X 36) -=- 12 = 108 ; the Constant of Rupture for average yellow pine is (p 185) 500 X 18 = 9000 ft>s or 4 tons ; and the dist of the farthest fibre (on either side) from the neutral axis is 3 ins. Consequently we have the resistance of the beani = (108 X 4) -r- 3 = 144 inch-tons, or 2.67 times the moment of the load ; that is the beam has a safety of 2.67. Or by the formula R = ad C - " we have the resistance = (36 X 6 X 4) + 6 = 144 inch- tons as before. - 6 (Art 45,) Example 2. Let Fig 30 be a rolled iron I beam cantilever of the cross section shown in ins at S, projecting hor 10 ft or 120 ins, and bearing a con- centrated load of 2 tons at its free end. Its moment of inertia (p 196) in ins is 184; Con- stant of Rupture p 185 = (2250 X 18) = 40500 ibs = 18.08 tons. Dist of farthest fibre from neutral axis on either side = half depth = 5 ins. Hence its moment of resistance or (I X C) -f- t at any section, = (184 X 18.08 -f- 5)-= 665. 3 inch-tons. The moment of the load at the section i e is = 2 X 120 = 240 inch-tons, therefore the beam has a safety of 2.77 at i e. By the approx process, Art 45. the moment of the beam is (4 X 9) -f (.667 x 8) X 16 = 661.4 inch-tons, or nearly as before. Art. 47. General Rule for M of Hup in hor beams supported at each end, no matter how irregularly the load or loads may be distributed. Let i n, Fig 31, be such a beam of yellow pine of 6 ft or 72 ins span, 6 ins square, and loaded with 3 tons. First find the cen of grav c of the whole load a r, and what portion (1.25 and 1.75 tons) of said load rests on each sup- port i and w, thus, as whole span : whole load : : either arm : portion at other arm. Con- sider the upward reactions thus found (1.25 and 1.75 tons) of the two supports to be two forces acting vert upwards against the ends of the beam at f and n as de- noted by the arrows. Let o be any point whatever in the beam at which as a ful- ,31. 01. t- iqji e a SCO xnjg follow if we use n and the 1.75 tons reaction, but with the load x o. Item. 1. If there is 110 load between i and the fulcrum point, as would be the case if the moment had been reqd at any point between i and a instead of at o, then the above p by itself is the moment. Thus e is 12 ins from t, hence the moment at e of the entire load a x is 1.25 X 12 = 15 inch-tons. 220 STRENGTH OF MATERIALS. Rem. 2. end i or it prod p642. Rem. 3. The resistance of the beam, Fig 31, at any section, or its a d C *- 6 (Art 45), is = (36 X 6 X 4 tons) -=- 6 = 144 inch-tons. Therefore at o it has a safety of 144 -H 36 = 4. Art. 48. Although the foregoing general rules apply to all the following cases, still these last will often expedite calculations. Case 1. Concentrated load at free end. Fig 32. Greatest moment is at o, and = load Xo n. At any other point a it is = load X n. Make o v = greatest moment, join v n; then a cis the moment at any point a. For the strain in tons or Ibs attending the moment in this case or in any of the following, see Art 45. Rem. The dotted line v n of moments, and those in some of the other figs, do not entirely control the modifications of beams, Figs 5 to 9, but assist the shearing forces in doing so. Case 2. Concentrated load at any point a, Fig 33. Greatest moment is at o, and = load X o a. At c it is = load X c a. Make o v = greatest moment, join v a. Then c e is the moment at any point c. The load has no moment be- tween a and n. Case 3. Uniform load throughout, Fig 34. Greatest moment is at o, and = whole load X half o n. At n it is o. At any point a it Qll is = load on a n X half a n. Make o v greatest moment, **C -*" draw the dotted parabola with its vertex at n. Then a c gives fv the moment at any point a. If the load is not uniform the greatest moment is a TV whole load X dist from o to its cen of grav. d and o, i "" ment = (\\-4- e cen of "' 4- Case 4. Load on one part, Fig 35. Greatest moment is at o, and = load X dist from o to cen of grav c of load. At any point t between the load and o, moment = load X tc. At any point a in the load, mor load on a s X dist a e of th grav of load on o * from a. Case 5. Several loads, w x y, Fig 36. Find their centres of grav c, a, s. Greatest moment is at o, and = wXco + xXao + yXso. Or first find the common cen of grav of all the loads, and mult its dist from o by the sum of the three loads. Between the loads the moment at g = y X g s; Case 6. One uniform load and one local one, Fig 37. Greatest moment is at o. Find that of the uniform one by Case 3 ; and that of the local one by case 4, and add them together. No moment between a and n. Case 7. Concentrated S load at center, Fig. 38. Greatest moment is at center, and = half load X half \ \ span. At the supports it is o. Make c s = moment at f - r *\*^ center, join s o, s a ; then n t moment at any point . y (J j \. Or tlie moment at any point n = half load X a n, n being 0*| p M tjir the nearest support. CaseS. Concentrated g load not at center. Fig. x ~ x 39. Greatest moment is at the load, and is = (load X eo A / X ) -T- o a. Make e s = moment at load, join s o, s a ; v /-^ then at any point c the moment is c t. Or at any point /{ ( jp c. moment = (load X a c X a e) -+ span o a ; a being always 7\f~f*~ft~ the support nearest the point. No moment at o or a. Ul L " Jll (ill STRENGTH OF MATERIALS. 220* Case 9. Several concentrated loads x y z, Fig 40. By Case 8 find the greatest moment of each load separately, and for each of them draw its dotted vertical and two inclined lines as in this fig. Then for h x x" ^L the moment at any point whatever ase, measure /' \ the vert dists (in this case e o, e a, e c) to the sloping lines, and add them together. For it is plain that at e we have e o for the moment of the load x at that point ; e a for that of the load y ; and e c for that of the load z ; and so at any other point. Or make en = eo + ea + 6 c ; also make A- i and m h respectively equal to the three dists aboye * and m, and join j hi n k. Then at any point along the beam j k the vert dist to these upper lines gives the moment. Case 1O. Uniform load from end to end, Fig 41. The greatest moment is at the center c, and is = half load X quarter span. At any other point e moment = half load on e o X e a ; or to half load on ea X e o. Make c s = half load X quarter span, and draw a parabola o s a, then at any point e the moment is = e t. The shearing or vertical strain at the center is zero or nothing. See Art 6, p 644. Rem. 1. The weight of the beam itself is usually suchya load, but is frequently so small compared with the load that in this and other cases it may be neglected. Rem. 2. The greatest moment of rupture that can occur at any given point on the span is when the load covers the span from end to end ; and in beams or trusses of uniform depth the hor strains at, any given section are then also greater than under any partial load ; so that if the chords are then strong enough in every part, they will be strong enough for any partial load : which is not the case with web members; any one of which is most strained when the longest segment reaching to it is loaded. See Rem 3, Art 6, p 644. Case 11. Uniform load from a support to part way across, Fig 42. Find the cen of grav g of the load, and by Art 47 what portion of it rests on each support o and x. Then by General Rule, Art 47, the moment at o or x o. At n or at any point a be- tween n and x it is = portion or reaction at xX a; n (or x a as the case may be). At any point c between n and o moment is = reaction at x X x c (load on c n X half c n) or to reaction at o X o c (load on o cX half o c). This plainly applies to unequal loads also, if instead of //s j , 14400 -i- 36 1 -f .5 1.5 800 7^- = 23.8 tons per sq inch; as per Table page 232, for a square hollow cast pillar 20 sides high. A round pillar of the same area, and 20 diams high, was found by Rule 1 to have a breakg load of 235.7 tons. Therefore, when both are 20 diams or sides high, the square one is .' = 1.111 times, or 1J, as strong as the round one. The same pro- portion, however, does not hold good at other equal heights, as may be found by comparing the round and square ones of Table, p. 232. RULE 4. For hollow square wrought iron pillars, the same as for cast iron, except that 36000 is to be used instead of 80000 and 6000 instead of 800; that i8 ' Breakg load = Metal area in sq ins X 36000 in fi>s sq of length in ins -5- sq of one side in ins 6000 For the breakg load in tons use 16.07 instead of 36000. Ex. A square hollow pillar of wrought iron, like the preceding. 6000 hence, 165 -' 3 f tops ) = 15.07 tons per sq inch, as per Table p. 232. ' 11 (ins area) The breakg load of a given hollow square wrought iron pillar may be found by Table p. 232. SOLID CYLINDRICAL IRON PILLARS. Breakg loads of solid cylindrical iron pillars, with flat ends, firmly fixed, and with the loads pressing- upon them equally throughout. Find the area of metal in sq ins contained in the transverse section of the pillar. Square the length of the pillar in ins; also square its diam in ins. Then, STRENGTH OF IRON PILLARS. 223 RULE 5. For solid cylindrical cast iron pillars. Break- load _ Metal area in sq ins X ^0000 in fes sq of length in ins -r- sq of diam in ins 300 For the breakg load in tons use 35.71 instead of 80000. Ex. A solid cylindrical cast-iron pillar 6 ins diam, and 10 ft, or 120 ins long. Here the metal area in sq ins of the transverse section, by Table of Circles, p. 18, is 2a274. The sq of the length in ins, or 1202, is 14400; and the sq of the diam 6 is ^Breakg load _ 28.274 X 80000 ,2261920 __ 2261920 _ in fl>s I"" 14400 -=- 36 ~~ 1 + 1.33 ~ 2.33 " 300 Or to _ 433.4 tons. RULE 6. For solid cylindrical wrought iron pillars. Break" 1 load _ Metal area in sq ins X 36000 in S)s - sq of length in ins -j- sq of diam in ins 2250 For the breakg load in tons use 16.07 instead of 36000. Ex. A solid cylindrical wrought iron pillar 6 ins diam ; and 10 ft, or 120 ins long. Here, as in the preceding case, the metal area is 28.274 sq ins ; the sq of the length in ins =^14400; and the sq of the diam = 36 ins. Hence, Break* load _ 28 - 2 ^ X 36000 _ 1017864 _ 1017864 in fa ~ I 14MO+?6 f+~T8 TIT ' 2250 This pillar and the foregoing cast iron one are of the same dimensions; and each of them is 20 diams long. Hence at 20 diams long a solid wrought iron cylindrical pillar is - ~ = 0.89 times as strong as a cast one. See Table 5, page 230, of such breaking loads. SOLID SQUARE IRON PILLARS. Breakg loads of solid sqnare iron pillars, with flat ends, firmly fixed, and with the loads pressing upon them equally throughout. Find the area of metal in sq ins contained in the transverse section of the pillar. Sq the length of the pillar in ins ; also sq one side of the pillar in ins. If rectangular, square only the least side. Then, RULE 7. For solid square cast iron pillars. Breakg load = Metal area in sq ins X 80000 in tts ^gq of length in ins -f sq of one side in ins ~loo For the breakg load in tons use 35.71 instead of 80000. Ex. A solid square cast iron pillar, 6 ins sqnare, and 10 ft, or 120 ins long. Here the metal area of transverse section is 36 sq ins. The sqof the length in ins, or 1 2 O 2 14400; and the sq of one side in ins, fi2 = 36. Hence, Breakg load = * X 80000 _ 2880000 2880000 _ in Its , ^ 14400 -r 36 ~ 1 + 1 ~~ 2 400 RULE 8. For solid sqnare wrought iron pillars. Breakg load - Metal area in sq ins X 36000 in Bbs sq of length iu ins -f- sq of one side in ins 3000 For the breakg load in tons use 16.07 instead of 36000. Ex. A solid square wrought iron pillar, 6 ins square. 10 ft, or 120 ins long. Here the metal area of transverse section is 36 sq ins. The sq of the length in ins, or 120*- 14400, and the sqof one side iu ins, or 6 2 - 36. Hence, Breakg load - 36 X M000_ 1296000 1296000 _ infts 1 ^ 14400 -36" l-f.133 1.133 ~ 3000 See Table 6, page 231, of such breaking loads. 224 STRENGTH OP IRON PILLARS. Table 1. HOLLOW CYLIND CAST IRON PILLARS. Breaking loads, flat ends, perfectly true, and firmly fixed; and the loads pressing equally on every part of the top. By Gordon's formula. For diams or lengths intermediate of those in the table, the loads may be found near enough by simple proportion. For thicknesses less than those in the table, the breaking loads may safely be assumed to diminish in the same proportion aa the thickness, while the outer diam remains the same. But for greater thicknesses than those in the table, the loads do not increase as rapidly as the new thickness. Still, in practice, they may be assumed to do so approximately, if the new thickness does not exceed about y a part of the outer diam. S CAST IRON. THICKNESS & INCH. (Original.) a "* Outer Diameter in inches. ll (3 2 j 2& 2^ w 3 2X 4 . 4& 5 | 5% 6 3 Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. i 45.2 52.5 59.8 66.8 74.1 88.5 103.0 117.1 131.3 145.5 159.8 1 2 36.2 43.4 51.0 58.6 66.5 81.5 96.7 111.3 125.9 140.5 155.2 2 3 27.2 34.1 41.4 48.8 56.7 71.9 87.6 102.7 117.8 132.9 148.1 3 4 20.2 26.3 32.9 39.7 47.0 61.9 77.5 92.9 108.3 123.7 139.1 4 5 15.1 20.2 25.9 32.0 38.6 62.6 67.4 82.6 97.9 113.4 129.1 5 6 11.6 15.8 20.7 25.9 31.6 44.3 58.2 72.7 87.7 103.0 118.7 6 7 9.1 12.5 16.6 21.0 26.1 37.4 50.1 63.7 78.1 92.9 108.3 7 8 7.3 10.1 13.5 17.3 21.7 31.6 43.2 55.8 69.3 83.4 98.4 8 9 5.9 8.2 11.1 14.3 18.2 26.9 37.3 48.8 61.5 74.9 89.2 9 10 4.9 6.9 9.3 12.0 15.4 23.1 32.4 42.9 54.6 67.1 80.7 10 11 4.1 5.8 7.9 10.3 13.2 20.0 28.3 37.8 48.6 60.3 73.0 11 12 3.5 5.0 6.8 8.9 11.4 17.4 24.9 33.5 43.3 54.2 66.1 12 13 3.0 4.2 5.8 7.6 9.9 152 21.9 29.7 38.8 48.8 60.0 13 14 2.6 3.6 5.1 6.7 8.7 13.5 19.5 26.6 34.9 44.1 54.5 14 15 2.3 3.2 4.5 6.0 7.7 11.9 17.4 23.9 31.4 39.9 49.7 15 16 2.0 2.8 4.0 5.3 6.9 107 15.6 21.5 28.4 36.4 45.6 16 18 1.6 2.3 3.2 4.2 5.5 8.6 12.7 17.5 23.4 30.2 38.0 18 20 1.3 1.8 2.6 3.4 4.5 7.1 10.5 14.7 19.7 25.6 32.3 20 25 1.7 2.3 8.0 4.7 7.0 9.8 13.3 17.6 22.5 25 30 1.2 1.6 2.1 3.2 5-0 7.0 9.4 12.6 16.1 30 35 2.4 3.7 5.2 7.1 M 12.2 35 40 1/2 1.9 2.8 4.0 5.4 7.2 9.5 40 \ 4.31 | 4.91 Veight 5.53 Of 1 f 6.13 50t Of 6.75 Length of pill 7.97 | 9.22 ar in pounds. 10.4 | 11.7 | 12.9 14.1 Area of ring of solid metal in square inches. 1.38 | 1.57 | 1.77 | 1.96 | 2.16 | 2.55 \ 2.95 | 3.34 | 3.73 | 4.12 \ 4.52 a^ J CAST IRON. THICKNESS X INCH. (Original.) a ft 3 Outer Diameter in inches. 5 5^ 6 W 7 1* 8 8}* 9 10 11 12 Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. 2 239 268 297 325 354 383 412 440 469 526 583 640 7 4 205 235 266 296 326 356 387 416 445 504 563 622 4 6 166 196 227 257 288 319 350 380 410 471 532 593 6 8 131 159 188 218 248 279 310 340 371 432 494 557 8 10 103 127 154 182 210 240 270 300 330 391 454 517 10 12 82 103 126 151 177 205 233 262 292 351 413 475 12 14 66 84 104 126 149 174 200 227 255 313 372 434 14 16 54 69 87 106 126 149 173 198 224 277 334 394 16 18 45 58 73 90 108 128 149 172 196 246 300 357 18 20 37 48 62 77 93 111 130 ?5l 173 219 270 323 20 22 32 41 53 66 80 96 113 132 151 194 241 292 22 25 25 34 43 54 65 79 93 109 126 164 206 252 25 30 18 24 81 89 48 59 70 83 96 126 160 199 80 35 13 18 24 30 36 44 53 63 73 98 126 159 35 40 10 14 18 23 28 35 42 50 59 78 102 129 40 45 8 11 15 19 23 28 34 41 48 64 84 107 45 50 6 9 12 15 19 23 28 33 39 53 70 89 50 60 4 6 9 11 14 17 20 24 28 38 50 65 60 70 3 4 6 8 10 12 15 18 21 29 38 50 70 80 3 4 5 6 8 9 11 13 16 22 30 38 80 'Weight of 1 foot of length of pillar, in pounds. 22.1 | 24.5 | 27.0 | 29.4 | 31.9 | 34.4 | 36.9 | 39.4 | 41.9 I 46.6 | 51.6 | 56.6 7.07 | 7.85 | 8.64 Area of ring of solid metal, in square inches. .43 f 10.2 | 11.0 | 11.8 I 12.6 | 18.4 | 14.9 I 16.5 | 18.1 STRENGTH OF IRON PILLARS. 225 Table 1. HOLLOW CYLIXD CAST IROX PILLARS. BREAKING LOADS. (Continued.) BY GORDON'S RULE. Length in feet. CAST IRON. THICKNESS 1 INCH. (Original.) 1 Length in feet. Outer Diameter in Inches. 12 13 14 15 16 18 20 22 24 27 SO 86 4 Tons. 1188 1301 1415 1530 1645 Tons. 1874 Tons. 2103 Tons. 2330 Tons. 2557 Tons. 2*96 Tons. 3236 Tons. 3915 4 6 1138 1253 1368 1484 1601 1833 2066 2295 2525 2866 3208 3890 6 8 1065 1184 1303 1423 1543 1779 2015 2247 2479 2S24 3170 3860 8 10 989 1110 1231 1355 1475 1716 1957 2193 2430 2780 3128 3823 10 12 909 1030 1152 1275 1399 1644 1889 2129 2369 2723 8076 3778 12 14 829 949 1071 1195 1320 1566 1813 2056 2300 2659 3016 3726 14 16 756 873 992 1114 1237 1484 1733 1979 2226 2589 2951 3668 16 18 683 796 913 1034 1155 1401 1651 1899 2147 2515 2879 3604 18 20 618 727 840 958 1077 1320 1568 1817 206(5 2437 2805 3536 20 22 559 663 772 887 1002 1241 1486 1734 1982 2356 2726 3464 22 24 508 606 709 818 929 1163 1404 1650 199 2272 2644 3387 24 26 459 553 651 756 803 1090 1326 1570 1816 2188 2560 3308 26 28 418 506 598 697 800 1020 1250 1489 1733 2103 2475 3226 28 30 380 462 549 644 743 954 1178 1412 1653 2020 2392 3143 30 32 347 424 506 595 689 893 1110 1338 1575 1938 2308 3059 32 34 318 390 467 552 641 836 1046 1268 1500 1857 2225 2974 34 36 592 359 432 511 596 783 984 1199 1427 1779 2143 2889 36 38 SB 331 400 475 556 734 928 1136 1361 1704 2063 2804 38 40 247 305 370 441 518 687 874 1076 1296 1630 984 2720 40 42 229 283 344 411 484 645 825 1019 1232 1560 909 2636 42 44 212 263 321 383 452 605 776 963 1169 1491 834 2.552 44 46 197 246 299 358 423 569 734 915 1114 1428 765 2474 46 48 183 229 280 335 397 536 694 869 1060 1367 696 2396 48 50 170 213 261 314 373 505 656 824 1008 1306 1627 2319 50 55 144 181 222 269 320 438 573 725 893 1172 1474 2135 55 60 124 157 192 233 278 383 503 641 795 1054 1333 1964 60 65 105 134 165 202 242 335 443 568 709 951 1211 J808 65 70 98 118 146 178 213 297 394 507 636 853 1099 1664 70 80 73 92 113 139 168 235 315 408 516 705 914 1414 80 90 58 73 91 112 136 191 257 335 426 586 768 1209 90 100 48 60 75 92 112 157 213 279 356 491 651 1040 100 Weight of one foot of length of pillar, in pounds. 108 | 118 | 128 | 138 | 147 | 167 j 187 j 206 | 226 | 255 | 285 | 344 Area of ring of solid metal, in square inches. 34.6 | 37.7 I 40.8 | 44.0 | 47.1 | 53.4 | 59.7 | 66.0 | 72.2 | 81.7 | 91.1 | 110.0 Si OAOJ. Lxiuj* . j. xii^/jvox Xioo o xi.x uxLj^o. (.unginai.j .2 S in Outer Diameter in Feet. s 3 31* 4 * 1 A 5 5>* 6 7 8 10 12 3 Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. 10 10806 12878 14908 16967 18986 21038 23052 27143 31196 39254 47375 10 15 10453 12564 14628 16712 18754 20825 22855 26972 31045 39133 47273 15 20 9996 12150 14250 16369 18440 20533 22585 26737 80837 38962 47131 20 30 8884 11102 13275 15459 17593 19743 21847 26084 30257 38486 46729 30 40 7688 9907 12113 14344 J6531 18735 20891 25224 29476 87838 46175 40 50 6554 H702 10888 13126 15341 17580 19780 24107 28532 37037 45484 50 60 5553 7575 9690 11892 14100 1K349 18570 23049 27460 86103 44666 60 70 4704 6570 8576 10702 12870 15097 17319 21824 26287 35058 43737 70 80 3998 5699 7570 9595 11692 13873 16070 20566 25048 83924 42712 80 90 3417 4953 6683 8588 10594 12706 14856 19304 23791 32725 41670 90 100 2940 4321 5909 7665 9588 11614 13700 18065 22521 31481 40438 100 110 2547 3788 5238 6888 8677 10606 12613 16869 21270 30213 39220 110 125 1950 2954 4400 5864 7483 9257 11132 14650 19448 27664 36692 125 150 1532 2350 3353 4547 5900 7417 9058 12701 16668 25182 34127 150 Weight of one foot of length of pillar, in pounds. 972 | 1150 | 1325 | 1503 | 1678 | 1856 | 2031 | 2388 | 2740 | 8444 | 4153 Coefficients of safety for hollow cast iron pillars. Mr. Jam^s B. Francis, of Lowell, Mass., a high authority, in his " Tables of Cast Iron Pillars." recommends that in order to allow for unequal loading, imperfect casting, bad end bearings, side blows. &c., we should not take the safe load at more than one-fifteenth of Hodgkinson's breaking load, if the pillars are roughly cast, and the ends not perfectly pinned aud adjusted; and one-fifth when they are so. and the loads about equally distributed. It will be seen by the last table on p 242, how our Gordon's loads differ from Hodgkinson's ; but we think that the same proportions of ours may be taken as safe ; depending on the above con- iitions. 226 STRENGTH OF IRON PILLARS. Or- (t OrJICOC* . - 500t-CO5O-*COCOl rH r-i r-5 "- i-t 00 OS CO CO O 00 t- O5 # i-H 00 O -^ '' ' ' ' J ^ 00 OO" N OJ O i * CO C 1 ! F-! r-5 rH t^t^OOt *-* \g 1-- CO* IM' r-( -H 00 CO W OO CO O OS >O (M -- - OS >O (M 00 CD O -f M I ' ' ' '. rH r- r^ O OO OO I : 1 ! J Sr-icDoicoi--io-fco-M : : : : : : : : : o CO O (N , < rH O O O O O ; H PH '::::::::: *i S *-: e i-j t?. JU ^ F s S V 2- * 3 H< = 1 sg d II STRENGTH OF IRON PILLARS. 227 NCH AR, DE OF SQUARE OJTE r-r- Ot^(M^COCOr* O CN 00 O> O t^ i-J OJ CN O lj r|J a coo^coococoo>os> 5 1 " * ~ til SJ % * 1 ,2 o 6 * -< * a|| 353 ? s = ** I oS 1 (S ; I^-i "2 a a ni a CO 1-- r- I COCOfO rHr-'F-KN MCOCOCO>OO'M i- !<38^3S3S^ ?i^?j^oocco?: ^ (MS> S SI I * . (Ji a tj^siiai I 2 r-IG 9 w t; iO 2 i iO U5rt4 (N ri TH i CO O C^ O CO* rH iOOO>O5H*Oit-COHHrHOOO 00 O CO (M m Cs per sq inch, of the above four pillars, with flat ends, and equally loaded. Coef of Safety = 4 -4- .05 H. By C. L. Gates, C. E. H. A. Square Col. B. Phoanix Col. C. Amer can Col. D. Common Col. Ult. Safe. Ult. Safe. Ult. Safe. Ult. Safe. 15 37067 7822 40476 8521 34434 7249 33693 7093 16 86876 7683 40212 8377 34167 7118 33339 6946 18 36470 7443 39645 8091 33597 6856 32589 6651 20 36024 7205 39030 7806 32982 659(5 31790 I 6358 22 35544 6970 38373 7514 32327 6338 30952 6069 25 34767 6622 37317 7110 31285 5959 29639 5646 30 3:3344 6063 35424 6440 29435 5352 27375 4977 35 31806 5531 33406 1 5810 27512 4789 25108 4367 40 30198 5033 31352 5226 25584 4264 22919 3820 45 28562 4570 29310 4690 23701 3792 20857 3337 50 26932 4143 27321 4203 21900 3369 18952 2916 55 25333 3728 25415 3765 20203 3004 17214 2550 60 23787 3398 23611 3373 18621 2660 15643 2235 234 STRENGTH OF IRON PILLARS. Table of rolled segment-columns of the Phoenix Iron 4 Company, Fig B, p 233. See p 233, for the strength. Four-segment Columns. The least outer diams are those of the circles formed by the segmental pieces; the greatest are from out to out of flats. Four Segments. Four Flats. f Total. External Diam. Thick- ness. Inch. Area. Sq.Ins. of 4. Weight. Lbs.pr ft. of 4. 30.25 37.10 44.00 51.00 Size. Ins. 1 ofl. Area. Sq.Ins. of 4. Weight. Lbs. pr ft. of 4. Area. Sq.Ins. Weight. Lbs. pr ft. Least. Ins. Greatest. Ins. 5-16 8.71 10.79 12.84 14.85 3X9 2 16 4.5 6. 6.75 10. 13.94 20.28 22.S1 33.79 13.21 16.79 19.59 24.85 4419 67.38 66.81 84.79 8 8 /t Sfj Six-segment Columns. Six Segments. Six Flats.f Total. External Diam. Thick- ness. Inch. Area. Sq.Ins. of 6. Weight. Lbs. pr ft. of 6. Size. Ins. ofl. Area. Sq.Ins. of 6. Weight. Lbs. pr ft. of 6. Area. Sq.Ins. Weight. Lbs. pr ft. Least. lus. Greatest. Ins. ~^A" 7-16 23.02 76.56 4X% 15. 50.68 38.02 127.24 13 A Eight-segment Columns. Eight Segments. Eight Flats.f Total. External Diam. Thick- ness. Inch. Area. Sq.Ins. of 8. Weight. Lbs. pr ft. of 8. Size. Ins. ofl. Area. Sq.Ins. of 8. Weight. Lbs. pr ft. of 8. Area. Sq.Ins. Weight. Lbs. pr ft. Least. Ins. Greatest. Ins. 7-16 30.64 102 5X% 24.96 83.2 55.6 185.2 16^ 22% RULE 9. For such forms of transverse section as the follow K Price, Philada, 1880, 5 to 5^ cts per ft. 4- The Fliits (.strengthening strips between the segments') are not now used. STRENGTH OP IRON PILLARS. RULE 10. In such forms of cross-section as Fig's 5 to 14, Ac, in order to attain the greatest accuracy according to present theory, we must introduce in the formulas the square of the least radius of yy ration (see Glos- sary) of the cross-section. Gordon's Rules 1 to 8, pp. 221 to 223, would then assume the following shapes. This however would not alter the results of any of them, in- asmuch as they are based upon this principle. For Cast Iron. For Wrought Iron. toacUn Area f metal in 8q inS X 8000 ' Bre ^ Area of metal in sq ins X 3600 ft3 m 1+ ( When modified b mulas for several sh Solid rec- ^ m tangle " (or square) Q Thin hollow square, _- of uniform U thickness.* Thin hollow rec tan- a uniform """ thickness.* Solid cyl- ^ inder. W Thin hollow cylinder, y^ of uniform ^J thickness.* Cross of J equal JU arms. I O Angle iron, of ribs of * eq u al 1 lengths ~ and thick- * nesses.f Length 2 in ins \ !> i i ( Length 2 in ins ^ 4800 X Rad gyr.2/ y inserting the actual square o apes would read thus, all the di CAST IROX. Metal area X 80000 V36000 X Rad gyr. 2 / " the radius of gyration, the for- mensions being iu inches : WROUGHT IRON. Metal area X 36000 Length 2 in ins ^ Length 9 in ins . 1+1 v least side 2 in ins 1 1 + /%nnnx/ least side 3 in ins J Metal area X 80000 V X 12. / Metal area X 36000 t Length 2 in ins x Length a in ins ^ 1 "*" ( 1800 X ne 8id 2 iD iil8 ) 1+ (36000- Ue8ideainin8 ) * 6. ' Metal area X 80000 yODUUU X g / Metal area X 36000 Length 2 in ins Length a in ins 1 4. 1 f c2 v C ~*" 3 M ) l+( QCftAA fc a ^c-fSa^J V \12c + a/' Metal area X 80000 > ^12 c-fa^ Metal area X 36000 Length 2 in ins / Length 2 in ins x I -f- I Diam 2 in ins 1 1 + ( Diam 2 in ins ) 16. ' Metal area X 80000 \ 36000 X -. g / Metal area X 36000 Length 2 i n ins Length 2 in ins x 1 ~*~ 1 4800 V I)iam 2 in ins 1 1 + ( Diam 2 in ins ) Metal area X 80000 \36000 X g / Metal area X 36000 t Length 2 in ins s Length 2 in ins . 1 + ( 1800 X a 2 iu in8 ) 1 + ( a o 2 in ins ) > ^ 24. 1 Metal area X 80000 \36000 X 24. ' Metal area X 36000 t Length 2 in ins ^ Length a in ins 1 + ( 4800 Y n 2 in ins ) 1 -|- ( n o a in ins ) y 4uu X ^ \3600 24 J & By thin, wo believe, is meant that the single thickness of metal shall not exceed about ^ part of the diam, or side. When the thickness is greater than ^ diam, or side, the loads given by the formulas will be appreciably too great for practical use. See ".Remarks on this edition," p. ix. i This and the next answer also for T iron. 236 STRENGTH OF IKON PILLARS. Angle iron of ribs of unequal 01 lengths, bn but equal C thicknesses. H iron. t C Call the I | area of the H& web W, 1 Iv and that of 11 the two flanges a o, c n, E.* -f CAST IROX. Metal area X 80000 WROUGHT IROX. Metal area X 36000 Length a in ins / Length 3 in ins v 1+ r / o* X c 3 \ ] \180Q X I ) / 1 + ( 36000 y/ a X c2l \j Metal area X 80000 Metal area X 36000 / Length a in ins x , Length a in ins . 1 + UoOX ( m2 X areaF )j 1+ ( 36000 V ( m * v afea * ) ) \i5uu x \i2 F4-W'' *"|ir Channel in the outer end d, o E$sl^ p * nf t.hft t.hiplrne >n. Let d e be the depth from f a flange / o, to the middle line 3s of the web w w o o. Find the that of the two flanges / o, are inches. Then, for cast %4I J IPO total area; also w T^rfK iron, Brkg load = in Ibs Metal area X 80000 Length 2 in inches 1 -f (4800X*X T *ou ^ *"'ea flX area, webNX area / / N \12 X total area 4 X sq of tot a lc / , For wrought iron, instead of 80000, and 4800, use 36000 in both cases. Table of breaking loads in tons (224O Ibs) per square Inch of metal area, of pillars or struts of H section of uniform thickness, by the above formula.f The heights or lengths of the pillars are in flanges a o, or c n. See " Remarks " below. Original. Hts Cast. Wrt. Hts Cast. Wrt. Hts Cast. Wrt. Hts Cast. Wrt. in Figs. Tons. Tons. in Figs. Tons. Tons. in Figs. Tons. Tons. in Figs. Tons. Tons. 1 35.5 16.1 12 20.2 15.0 23 9.38 12.9 38 4.12 9.56 2 34.9 6.0 13 18.8 14.9 24 8.79 12.7 40 3.76 9.15 3 34.0 6.0 14 17.5 14.7 25 8.26 12.4 42 3.45 8.76 4 32.9 6.0 15 16.3 14.5 26 7.77 12.1 45 3.04 8.21 5 31.5 5.9 16 15.1 14.3 27 7.33 11.9 50 2.51 7.37 6 30.0 5.8 17 14.1 14.0 28 6.89 11.7 55 2.09 6.62 7 28.3 5.6 18 13.1 13.8 29 6.53 11.5 60 1.77 5.95 8 26.6 5.5 19 12.2 13.7 30 6.18 11.3 70 1.32 4.85 9 24.9 5.4 20 11.4 13.5 32 5.66 10.9 80 1.02 4.00 10 23.3 15.3 21 10.6 13.3 34 5.02 10.4 90 .81 3.33 11 21.6 15.1 22 10.0 13.1 36 4.54 10.0 100 .66 2.81 The H is stronger than the L, T, -f , or LJ section, the diff increasing with the height. The table may be used for riveted columns with I webs and LJ flanges if properly made. Remarks. This table was calculated for a square H, with a uniform thickness of one-twelfth of its width, but it will answer near enough in practice if the H is longer in either direction than in the other, not exceeding one-fourth part ; and tor any uniform thickness from one-twenty-fourth to one-fifth of the least out to out width of the H. If wider along the flanges than across them (within the above limits) the strength per sq inch becomes somewhat greater than the tabular one, and vice versa, varying with the pro- portions of the section, the height, and whether cast or rolled. Still an allowance of 6 per cent will suffice for either extreme. If the web is thinner than the flanges, the strength per square inch of metal area of the entire H increases somewhat, and vice versa. If the uniform thickness of the H should be as great as one-fifth of its least out to out widt.h, the increase of strength per sq inch will not exceed about 6 per ct in any pillar or strut either wrought or cast; and the diminution of strength in a uniform thickness of only one- twenty-fourth said least width would be at about the same rate. If the flanges taper toward their ends, as is usually the case, it is well for safety to take their metal area at only what it would be if they were uniformly of about their end thickness. * The writer doubts whether this formula applies after a o exceeds about twice a c. f For tables of pillars of L, T, I, +, and LJ sections, see pages 637 to 640. STRENGTH OF IRON PILLARS. 237 In arches of cast iron for bridges, &c, it is usual among English engineers not to allow more than 2^ tons (5600 ft>s) of compression, or thrust, per sq inch. Brunei never subjected cast-iron pillars to more than 1% tons (3360 fos) per sq inch. C. Shaler Smith * employs as maximum working strains, J. of the calcu- lated breaking strain for such hollow chords and posts of bridges as are 1 inch or more in thickness, and not more than 15 diams long. For posts, only ^ ; when not less than % inch thick, nor more than 25 diams long ; or from y 1 ^ to ^y, when % thick or less, and more than 25 diams long. The young engineer must bear in mind that the breakg and the safe loads per sq inch, of pillars of any given material, are not constant quantities ; but diminish as the piece becomes longer in proportion to its diam. If a very long piece or pillar be so braced at intervals as to prevent its bending at those points, then its length becomes virtually diminished, and its strength increased. Thus, if a pillar 100 ft long be sufficiently braced at intervals of 20 ft, then the load sustained may be that due to a pillar only 20 ft long. Therefore, very long pillars used for bridge piers, &c, are thus braced; as are also long horizontal or inclined pieces, exposed to compression in the form of upper chords of bridges; or as struts of any kind in bridges, roofs, or other structures. Mistakes are sometimes made by assuming, say 5 or 6 tons per sq inch, as the safe compressing load for cast iron ; 4 tons for wrought ; 1000 pounds for timber ; without any regard to the length of the piece. But although the final crushing loads, as given in tables of strengths of materials, are usually those for pieces not more than about 2 diams high, they will not be much less for pieces not exceeding 4 or 5 diams. Cautions. Remember a heavily loaded cast-iron pillar is easily broken by a side blow. Cast-iron ones are subject to hidden voids. All are subject to jars and vibrations from moving loads. It very rarely happens that the pres is equally dis- tributed over the whole area of the pillar; or that the top and bottom ends have per- fect bearing at every part, as they had in the experimental pillars.f Cast pillars are seldom perfectly straight, and hence are weakened. Hollow pillars intended to bear heavy loads should not be cast with such mouldings as aa; or with very projecting bases or caps such as g, Fig 19. It is plain that these are weak, and would break off under a much less load than would injure the shaft of the pillar. When such projecting ornaments are required, they should be cast separately, and be at- tached to a prolongation of the shaft, as co", by iron pins or rivets. Ordinarily, it is better to adopt a more simple style of base and cap, which, as at &, can be cast in one piece with the pillar, without weakening it. Hodginson states that while the quantity of material is the same in both pillars, no strength is gained in hollow ones by making the diams greater at the middle than at the ends ; but that in solid ones, with rounded ends, there is a gain of about i.th part ; and in those with flat ends, of about Jth or ^th part, by making the diam at the middle about \% or 2 times that at the ends. Also that a uniform round pillar has the same strength as a moderately tapering one whose diam at half-way up is equal to the uniform diam of the cylindrical one. Also, that when a flat-ended pillar, Fig. 2, is so irregularly fixed, that the pressure upon it passes along its diagonal a a, it loses two-thirds of its strength. Hence the necessity for equalizing, as far as possi- ble, the pressure over every part of the top and bottom of the pillar; a point very difficult to secure in practice. * Of the very skilful, experienced, and intelligent firm of Smith & Latrobe, civ engs, and bridge- builders, Baltimore, Md., The Baltimore Bridge Co. f In important cases, both ends should be planed perfectly true; as is done ill irou bridges, &c. 238 STRENGTH OF WOODEN PILLARS. Steel pillars. Mr. Kirkaldy experimented with a small steel pillar or tube of Shortridge, Howell & Go's homogeneous metal. Its length was 4 ft, or 25.6 diams ; outer diam 1% inch: inner diam 1^; thickness % inch. Area of cross-section 1% sq ins. Flat ends. Under 67300 Ibs, or 30 tons total pressure, or 17.14 tons per sq inch of solid metal section, it bent very slightly. On increasing the pressure con- siderably, the pillar sprang out from under the load. Our preceding table gives 22% tons total, or 13 tons per sq inch, as the ultimate load for a wrought-iron tube of the same size. Mr. M. G. Love, Paris, as the result of a trial with small steel rods, about .4 inch diam, and which had a tensile strength of 48 tons per sq inch, suggests that the comparative strength of pillars of wrought iron, cast-iron, and steel, are probably about as follows : At from 1^ to 5 diarns in length, steel and cast-iron ones have equal strengths ; and either of them is about twice as strong as wrought iron. At 10 diams, steel is 1% times as strong as cast ; and 2.2 times as strong as wrought iron. At 40 diams, steel is 4 times as strong as cast; and 2.7 as strong as wrought. But this needs confirmation. Now that powerful and accurate testing machines are coming more into use, we may hope that the doubts at present existing on such subjects will be set at rest. Mr Stoney advises that until then steel pillars should not be trusted with more than 1.5 the loads of wrought iron ones. WOODEN PILLASS, The strengths of pillars, as well as of beams of timber, depend much on their de- gree of seasoning. Hodgkinson found that perfectly seasoned blocks, 2 diams long, required, in many cases, twice as great a load to crush them as when only moderately dry. This should be borne in mind when building with green timber. In important practice, timber should not be trusted with more than ^ to %of ita calculated crushing load ; and for temporary purposes, not more than % to %. Mr. Charles Shaler Smith. . E., of St. Louis, prepared the following formula for the breaking loads of either square or rectangular pillars or posts, of moderately seasoned white, and common yellow pine, with flat ends, firmly fixed, and equally loaded, based upon experiments by himself. It is Gordon's formula adapted to those woods; and gives results considerably smaller than Hodgkinson's, as is shown on p 242. It is therefore safer. Call either side of the square, or the least side of the rectangle, the breadth. Then, Breakg load in Ibs, per 1 . 5000 t Rule. sq inch of area, of a > == /sq of length in ins v \ pillar of W or Y pine J 1 + ( sq of breadthip ini| X .004 J Or in words, square the length in ins ; square the breadth in ins ; div the first square by the second one; mult the quot by .004 ; to the prod add 1 ; div 5000 by the sum. Ex. Breakg load per sq inch, of a white pine pillar 12 ins square, and 30 ft, or 360 ins long. Here the sq of length in ins is 360 2 = 129600. The square of the breadth is 122 = 144 . and -^- = 900 ; and 900 X -004 = 3.6; and 3.6 + 1 = 4.6. Finally, = 1087 Ibs, the reqd breakg load per sq in. As the area of the pillar is 144 sq ins, the entire breakg load is 10H7 X 1^-4 = 156528 tt>s, or 69.9 tons. See table, p 242. Recent experiments on wooden pillars 20 ft long, and 13 ins square, by Mr. Kirkaldy, of England, confirm the far greater reliability of Mr. Smith's formula. Hence we present the following new set of original tables based upon it. For solid pillars of cast iron and of pine, whose heights range from 5 to tiO times their side or diam, we may say, near enough for practice, that a cast iron one is about 1G% times as strong as a pine one; but no such approximate ratio holds good between wrought iron and pine, or between cast and wrought iron. * Each projects 1 Inch in a 4 inch column ; \% in a 6 inch ; 1% in all larger, t The breaking load in Ibs per sq iuch iu short blocks, by Mr. Smith. STRENGTH OF WOODEN PILLARS. 239 Table of breaking: loads in tons of square pillars of half seasoned white or common yellow pine firmly fixed and equally loaded. By C. Shaler Smith's formula. (Original.) is Side of square pine pillar, in inches. II 1 1 I IK 1 IX 1 IK 1 2 | 2J4 | 2tf UJi | 3 3^ | 3^ 3% | 4 ^f_ BREAKING LOAD. i 1.42 2.54 3.99 5.73 7.80 10.1 12.8 15.7 18.9 22.3 26.1 30.1 34.5 1 M 1.17 2.22 3.59 5.26 7.25 9.6 12.2 15.1 18.3 21.7 25.4 29.2 33.7 Ji .97 1.93 3.19 4.80 6.74 9.0 11.6 14.5 17.6 21.0 24.7 28.6 33.0 }^j % .81 1.66 2.81 4.35 6.19 8.4 10.9 13.7 16.8 20.2 23.9 27.8 32.1 K 2 .68 1.44 2.48 3.92 5.66 7.8 L0.2 12.9 15.9 19.3 23.0 26.9 31.2 2 J4 .57 1-24 2.19 3.53 5.17 7.2 9.6 12.3 15.2 1&.5 22.0 25.8 30.1 B H .49 1.07 1.93 3.16 4.70 6.7 8.9 11.5 14.3 17.6 21.1 24.9 29.1 N .42 .93 1.71 2.85 4.29 6.2 8.3 10.8 13.5 16.7 20.1 23.8 28.0 H 3 .36 .82 1.52 2.55 3.89 5.6 7.6 10.0 12.7 15.8 19.2 22.9 27.0 3 .28 .63 1.21 2.07 3.23 4.8 6.6 8.8 11.3 14.2 17.4 20.9 24.8 4 . .22 .50 .98 1.70 2.70 4.0 5.7 7.7 9.9 12.7 15.7 19.0 22.7 4 J6 .18 40 .81 1.42 2.28 3.4 4.9 6.7 8.8 11.4 14.1 17.2 20.7 /4 5 .15 .34 .68 1.19 1.94 3.0 4.2 5.8 7.7 10.0 12.6 15.5 18.8 5 .12 .28 .57 1.02 1.67 2.6 3.7 5.1 6.8 8.9 11.3 14.0 17.1 X 6 .10 .24 .49 .86 1.44 2.3 3.3 4.6 6.1 8.0 10.2 12.7 15.6 6 .09 .21 .43 .74 1.26 2.0 2.9 4.1 5.4 7.2 9.2 11.6 14.2 7 .08 .18 .37 .66 1.11 1.8 26 3.6 4.9 6.5 8.3 10.5 12.9 7 .07 .16 .33 .59 .98 1.6 2.3 3.3 4.4 5.9 7.6 9.6 11.8 K 8 .06 .14 .29 .52 .87 1.4 2.0 2.9 3.9 5.2 6.8 8.7 10.8 8 /4 .05 .12 .26 .47 .78 1.2 1.8 2.6 3.5 4.8 6.2 7.9 9.9 % 9 .05 .11 .23 .42 .71 1.1 1.6 2.3 3.2 4.3 5.6 7.2 9.1 9 i^ .10 .21 .37 .64 1.0 1.5 2.1 2.9 3.9 5.1 6.6 84 $6 10 .09 .19 .34 .58 .93 1.4 , 2.0 2.7 3.6 4.7 6.1 7.8 10 .17 .31 .53 .86 1.3 1.8 2.5 3.4 4.4 5.7 7.2 11 .16 .28 .48 .79 1.2 1.7 2.3 3.1 4.1 5.3 6.7 11 .14 .26 .44 .72 1.1 1.5 2.1 2.9 3.8 4.9 6.2 12 .13 .24 .41 .65 1.0 1.4 2.0 2.7 3.4 4.5 5.8 12 3 .21 .35 .55 .84 1.2 1.7 2.3 3.1 4.0 5.0 13 4 .18 .31 .46 .70 1.0 1.4 2.0 2.7 3.5 4.4 14 5 .27 .41 .63 .91 1.2 1.7 2.4 3.1 3.9 15 6 .24 .37 .57 .78 1.1 1.5 2.1 2.7 3.5 16 7 .50 .70 1.0 1.4 1.9 2.4 3.1 17 8 .45 .66 .92 1 3 1.7 2.2 2.8 18 20 .76 1.0 1.4 1.8 2.3 20 ll Side of square pine pillar, in inches. II K.S *X 1 *% 1 *% 1 5 | 534 | 5}^ | 5% 6 | 6J4 6}^ | 6% | 7 | 7} K.2 BREAKING LOAD. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. 2 35.8 40.6 45.7 51.1 56.8 62.8 69.0 75.5 82.3 89.4 96.8 104.5 112.4 2 3 31.4 36.1 41.1 46.2 51.8 57.7 63.8 70.2 76.9 84.0 91.3 98.9 106.7 3 4 268 31.2 35.9 , 40.8 46.1 51.7 57.6 63.8 70.4 77.3 84.5 92.1 99.9 4 5 22.6 26.5 30.8 35.4 40.5 45.8 51.5 57.4 63.7 70.3 77.2 84.5 92.1 5 6 18.9 22.5 26.4 30.5 35.2 40.1 45.4 51.0 57.0 63.3 69.9 76.8 84.1 6 7 15.8 19.0 22.6 26.2 30.5 35.0 39.9 45.0 50.5 56.5 62.8 69.4 76.3 7 8 13.3 16.1 19.2 22.5 26.3 30.4 34.9 39.7 45.9 50.4 56.2 62.4 69.0 8 9 11.3 13.7 16.5 19.5 22.9 26.6 30.7 35.0 39.9 44.8 50.2 56.0 62.1 9 10 9.7 11.8 14.2 16.9 19.9 23.2 26.9 30.9 35.2 39.9 44.9 50.3 56.0 10 11 8.3 10.2 12.4 14.8 17.5 20.4 23.8 27.4 31.3 35.6 40.2 45.1 50.4 11 12 7.2 8.8 10.7 12.9 15.4 18.0 21.1 24.3 27.9 31.8 36.0 40.6 45.5 12 13 6.2 7.7 9.4 11.4 13.6 16.0 18.8 21.7 24.9 28.5 32.4 36.6 41.1 13 14 5.5 6.8 8.3 10.1 12.1 14.2 16.7 19.4 22.4 25.7 29.2 33.1 37.3 14 15 4.8 6.0 7.4 9.0 10.8 12.7 15.0 17.5 20.2 23.2 26.4 30.0 33.9 15 16 4.4 5.4 6.7 8.1 9?8 11.5 13.6 15.8 18.3 21.0 24.0 27.3 30.8 16 18 3.6 4.4 5.5 gj 8^0 9^4 12.3 11.2 14.3 13.0 16.6 15.1 19.1 17.4 19.9 24.9 22.7 25^8 18 19 3.3 4.0 5.0 6^0 7.3 8.6 102 11.9 13.8 16.0 18.3 20.9 23.7 19 20 3.0 3.7 4.6 5.5 6.6 7.8 9.3 10.9 12.6 14.6 16.8 19.2 21.8 20 22 2.5 3.0 3.8 4.6 5.6 6.6 7.9 9.2 10.7 12.4 14.3 16.3 186 22 24 . 2.1 2.6 3.2 3?9 4.7 5.6 6.7 7.9 9.1 10.6 12.2 14.1 16.0 24 26 1.8 2.2 2.8 3.4 4.1 4.9 5.8 6.8 7.9 9.2 10.6 12.2 13.9 26 28 1.5 1.9 2.4 2.9 3.5 4.2 5.1 5.9 6.9 8.0 9..? 10.7 12.2 28 30 13 1.7 2.1 2.6 3.1 3.7 4.4 5.2 6.1 7.1 8.2 9.4 10.8 30 32 1.2 1.5 1.9 2.3 2.7 3.2 3.9 4.6 5.4 6.3 7.3 8.4 9.6 32 34 1.1 1.3 1.7 2.0 2.4 2.9 3.5 4.1 4.8 5.6 6.6 7.5 8.6 34 36 1.0 1.2 1.5 1.8 2.2 2.6 3.1 3.7 4.3 5.0 5.8 6.7 7.7 36 38 .9 1.1 1.3 1.6 2.0 2.4 2.8 3.3 3.9 4.5 5.3 6.1 7.0 38 40 .8 1.0 1.2 1.5 1.8 2.1 2.6 8.0 3.5 4.1 4.8 5.5 6.3 40 Continued on next page. 16 240 STRENGTH OF WOODEN PILLARS. Table of breaking: loads in tons of square pine pillars, with flat ends firmly fixed, and equally loaded. (Continued.) Original. .11 Side of square pine pillar, in inches. II S.2 7H 1% \ 8 | 8J4 | 8^ | 8% | 9 I 9H i 9H | 9% | 10 | 10}* | 10^ K.3 BREAKING LOAD. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons Tons. Tons. 2 120.6 129.1 137.9 147.0 156.3 165.9 175.8 186-0 196.4 207.2 218.2 229.5 241.0 2 4 107.9 11H-3 125.0 133.9 143.0 152.6 162.4 172.5 182.8 193.4 204.2 215.3 2?6.6 4 6 91.7 j 99.7 108.0 116.5 125.3 134.$ J43.-5 153.0 162.8 173.5 184.4 195.6 207.0 6 8 75.9 83.2 90.8 98.7 106.8 115.5 ! 124.4 133.6 143.0 153.0 1W3.2 173.7 184.4 8 10 62.0 68.5 75.3 82.4 89.7 97.7 105.8 114.3 123.0 132.3 141.8 151.6 161.6 10 12 50.7 56.5 62.5 68.7 75.1 82.2 89.6 97.2 105.0 1135 122.2 131.2 140.5 12 14 41.8 46.7 51.9 57.3 62.9 69.2 75.7 82.4 89.4 97.1 1 105.0 113.2 121.6 14 16 34.7 38.9 43.4 48.1 53.0 58.5 64.3 70.3 76.5 83.3 90.4 97.7 105.3 16 18 29.1 32.7 3(5.6 40.7 45.0 49.8 549 60.1 65.6 71.7 78.0 84.6 j 91.4 18 20 24 6 >>7 7 31 1 34 7 38 5 42 7 47 2 51 8 56 7 6'> 67 6 7'i 5 79 6 20 23 19.6 22.1 24.8 27.7 30.9 34.4 38.0 41.9 46.0 50.5 55.2 60.* 65.4 23 26 15.8 17.8 20.1 22.5 25.2 28.1 31.1 34.4 379 41.6 45.6 49.8 54.3 26 29 13.1 14.8 16.7 18.7 20.9 23.4 25.9 28.6 31.6 34.9 38.2 41.9 45.6 29 32 10.9 12.3 13.9 15.7 17.6 19.7 21.8 24.2 26.7 29.5 32.3 35.5 38.7 32 35 9.3 10.6 11.9 13.4 15.0 16.8 18.7 20.7 22.8 25.2 27.7 30.5 33.3 35 38 8.0 9.1 10.2 11.5 12.9 14.6 16.3 18.0 19.7 21.8 23.9 26.3 28 8 38 41 6.9 7.9 8.9 10.0 11.2 12.6 14.1 15.6 17.2 19.1 21.0 23.1 25.2 41 44 6.0 6.9 7.8 8.8 9.8 11.0 12.3 13.6 15.0 16.7 18.4 20.2 22.1 44 50 4.7 5.4 6.1 6.9 7.7 8.8 10.0 11.3 12.6 13.7 14.9 16.2 17.5 50 f| 10% | Side of square pine pillar, in inches. II W.2 11 | iiy^ \ \\K \ u% | 12 | | 10% | 11 | lltf | 11& | 11% | 12 BREAKING LOAD. BREAKING LOAD. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. 4 238.9 251.0 263.2 275.9 288.8 302.1 26.5 28.6 31.1 33.8 36.7 39.9 42 6 218.8 230.6 242.7 255.2 267.9 281.0 24.2 26.4 28.7 31.2 33.9 36.8 44 8 195.5 207.0 218.5 230.4 242.5 255.1 22.4 24.3 26.4 28.7 31.2 33.9 46 10 172.2 183.0 194.1 205.6 217.3 229.5 20.7 22.6 24.5 26.7 28.9 31.4 48 12 150.3 160.3 170.7 181.5 192.5 204.0 19.2 20.9 22.7 24.7 26.8 29.2 50 14 130.5 139.8 149 4 159.4 169.6 180.2 17.8 19.5 21.1 23.0 25.0 27.2 52 16 113.2 121.7 1304 139.6 149.0 158.8 16.6 8.2 19.7 21.5 23.3 25.4 54 18 98.7 106.2 114.1 122.5 13KO 140.0 15.5 7.0 18 4 20.1 21.8 23.7 56 20 86.2 92.9 100.0 107.6 115.4 123.6 14.5 5.9 172 .18.8 20.4 22.2 58 22 75.6 81.7 88.1 95.0 102.0 109.5 13.6 4.9 16.2 *17.7 19.2 20.9 60 24 66.7 72.2 77.9 84.1 90.5 97.3 11.8 2.9 13.9 15.2 165 18.0 65 26 59.1 64.0 69.2 74.9 80 6 86.8 10.1 11.1 12.0 13.2 14.3 15.6 70 23 52.6 57.1 61.8 66.9 72.1 77.7 8.9 9.8 10.6 11.6 12.5 13.7 75 30 47.0 51.1 55.3 60.0 64.7 69.9 78 8.6 9.4 10.2 11.1 12.1 80 32 42.1 46.0 49.9 54.0 58.4 63.0 7.0 7.7 8.4 9.1 9.9 10.7 85 34 38.2 41.5 45.0 48.8 52.8 57.1 6.2 6.8 T.4 8.1 8.8 9.6 90 36 34.6 37.7 40.9 44.3 48.0 51.8 5.6 6.1 6.7 7.3 8.0 8.7 95 38 31.5 34.2 37.2 40.4 43.9 47.4 5.1 5.6 6.1 6.6 7.2 7.8 100 40 28.8 31.3 34.0 37.0 40.1 43.5 3.5 ' 3.9 4.2 4.6 5.0 5.5 120 Continued on next page. Remarks. Mr Kirkaldy found for Riga and Bantzic firs, 20 ft long, and 13 ins square, (or 18^ sides high,) 148 and 138 tons total ; or .876 and .817 tons, (1963 and 1829 ft>s,) per sq inch. Mr Smith's rule gives for common pine, 160 tons total ; or .947 ton, or 2121 R>s, per sq inch. Hodgkinson would give for Riga about 297 tons total. Each of Mr Kirkaldy's 20-ft pillars shortened about .6 of an inch total; or .03 inch per ft; or Vg f an inch in 4 ft 2 ins, under a mean of 1900 fts per sq inch. The writer has known 8 unbraced pillars of hemlock, .tolerably seasoned, 12 ins square, and 42 ft high, to be gradually loaded each with 32 tons, or 71680 ibs total ; (or .2222 ton, or 498 ft>s per sq inch) without appreciable yielding. As- suming their strength and stift'ness to be about as for Mr Smith's pine, (as in all our tables,) they should by him yield at 39.9 tons total. With these same data, but with Hodgkinson's formula, they should yield at 69.3 tons; and with Hodgkinson's own data, for seasoned red deal, at 91.6 tons. See Remarks, p 242. STRENGTH OF WOODEN PILLARS. 241 Table of breaking loads in tons of square pillars of half- seasoned white or common yellow pine, with flat ends firmly fixed, and equally loaded. By C. Shaler Smith's formula. (Continued.) As this table was partly made by interpolation, the last figure is not always pre- cisely correct. Original. !!' s.s Side of square pine pillar in inches. |l ffl.2 13 j 14 15 | 16 | 17 | 18 | 19 | 20 21 | 22 | 23 | 24 BREAKING LOAD. Tons. | Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. 4 358 418 482 552 625 703 786 872 964 1060 1161 1265 4 6 335 394 456 526 599 676 760 847 938 1033 1134 1236 6 8 308 367 429 500 572 649 732 818 910 1005 1106 1208 8 10 281 339 400 466 537 612 694 780 870 964 1064 1166 10 12 252 307 365 432 502 576 656 740 829 922 1022 1124 12 14 225 277 333 397 464 536 614 696 784 876 973 1074 14 16 201 250 303 363 428 497 573 652 739 829 925 1024 16 18 179 224 274 331 392 458 531 608 692 780 873 972 18 20 160 201 248 301 359 422 492 566 647 732 822 919 20 22 143 182 224 274 329 388 455 526 604 686 773 866 22 24 127 163 203 249 301 357 421 488 563 642 726 816 24 26 115 148 184 226 275 328 389 453 523 599 680 767 26 28 103 133 167 206 252 302 359 420 490 5SO 638 721 28 30 93 121 152 189 231 278 332 389 453 522 597 677 30 32 84 109 138 173 212 256 307 361 421 487 558 635 32 34 76 99 126 159 196 237 284 335 392 455 523 597 34 36 69 91 116 146 180 219 264 312 366 426 490 560 36 38 63 84 107 134 166 203 245 290 341 397 458 525 33 40 58 77 99 124 154 188 227 270 318 372 429 494 40 42 54 71 91 115 143 175 212 253 298 349 403 465 42 44 50 66 84 107 133 163 198 236 280 328 380 438 44 46 46 61 78 99 123 152 185 221 263 308 358 413 46 48 43 57 73 92 115 142 173 207 247 290 337 389 48 50 40 53 68 86 107 133 162 194 231 272 317 367 50 52 37 50 64 81 101 124 152 182 217 256 300 347 52 54 35 47 60 76 95 117 144 172 205 242 283 328 54 56 33 44 56 71 89 110 135 162 193 228 267 310 56 58 31 41 52 67 84 103 127 153 182 215 253 294 58 60 29 38 49 63 79 98 120 144 172 204 240 280 60 65 25 33 43 55 69 86 105 126 151 179 211 246 65 70 22 29 37 48 60 74 92 111 134 159 187 218 70 75 19 25 33 42 53 66 82 98 118 141 166 195 75 80 16 22 29 37 46 58 72 87 105 125 148 174 80 85 14 19 26 33 41 52 65 78 94 112 132 156 85 90 13 17 23 30 37 46 58 70 85 102 120 141 90 95 12 16 21 27 33 42 53 64 77 93 108 127 95 100 11 14 19 24 30 38 48 58 70 84 99 117 100 110 10 12 16 20 26 33 40 48 58 70 82 97 110 120 9 il 14 17 22 28 34 41 49 60 71 83 120 130 7 9 12 14 18 23 29 36 43 52 61 72 130 140 6 8 10 12 16 20 25 31 37 44 53 62 140 150 5 7 9 11 14 18 22 27 32 38 46 54 150 160 5 6 8 10 13 16 20 24 29 34 41 48 160 170 4 5 7 9 11 14 17 21 25 30 36 43 170 180 4 5 6 8 10 12 15 19 22 27 32 38 180 190 3 4 5 7 9 11 14 17 20 24 29 34 190 200 3 4 5 6 8 10 12 15 18 22 26 31 200 242 STRENGTH OF WOODEN PILLARS. Table of breaking: loads in tons, or in Ibs per sq inch, of cross section of half seasoned square pine pillars, whose heights are measured by one of their sides. By formula, p. 238. 4* 00 &-= II II 8 tt'O s B.fl S .S SQ n. Tons. Lbs. Tons. Lbs. Tons. Lbs. Tons. Lbs. i 2.2232 4980 26 .6027 1350 51 .1960 439 76 .0924 207 2 2.1969 4921 27 .5697 1276 52 .1888 423 77 .0902 202 3 2.1544 4826 28 .5398 1209 53 .1826 409 78 .0879 197 4 2.097S 4699 29 .5116 1146 54 .1763 395 79 .0862 193 5 2.0290 4545 30 .4853 1087 55 .1705 382 80 .0839 188 6 1.9513 4371 31 .4607 1032 56 .1647 369 81 .0821 184 7 1.8665 4181 32 .4379 981 57 .1598 358 82 .0799 179 8 1.7772 3981 33 .4165 933 58 .1545 346 83 .0781 175 9 1.6857 3776 1 34 .3969 889 59 .1496 335 84 .0763 171 10 1.5942 3571 35 .3781 847 60 .1451 325 85 .0746 167 11 1.5040 3369 36 .3599 809 61 .1406 315 86 .0728 163 12 1.4165 3173 37 .3447 772 62 .1362 305 87 .0714 160 13 1.3317 2983 38 .3295 738 63 .1321 296 88 .0696 156 14 1.2513 2803 39 .3152 706 64 .1286 288 89 .0683 153 15 1.1745 2681 40 .3018 676 65 .1250 280 90 .0670 150 16 1.1027 2470 41 .2889 647 66 .1210 271 91 .0656 147 17 1.0353 2319 42 .2772 621 67 .1179 264 92 .0638 143 18 .9723 2178 43 .26ol 696 68 .1147 257 93 .0625 140 19 .9134 2046 44 .2554 572 69 .1112 249 94 .0616 138 20 .8585 1923 45 .2455 550 70 .1085 243 95 .0603 135 21 .8076 1809 46 .2357 528 71 .1054 236 96 .0589 132 22 .7603 1703 47 .2268 508 72 .1027 230 97 .0576 129 23 .7165 1605 48 .2183 489 73 .1000 224 98 .0567 127 24 .6755 1513 49 .2100 472 74 .0973 218 99 .0554 124 25 .6380 1429 50 .2031 455 75 .0951 213 100 .0545 122 Remark. Gordon and Hodgkinson compared. The difference between them is greater in wooden pillars than in hollow cast iron ones. More- over, in the latter, Gordon is sometimes greater, sometimes less, than Hodgkin- son, as seen per lower table. Mr. Smith's assumed strength and stiffness of pine may safely be taken at about one-fourth less than Hodgkinson's for his red deal ; and with this assumption, Hodgkinson's rule would make the strength of Smith's pine (in all our tables for wooden pillars) greater, to the extent shown by the multipliers in the following' table. The truth is probably between the two. See Remark, p. 240. Htin sides. Multr. Htin sides. Multr. Ht in sides. Multr. Htin sides. Multr. Htin sides. Multr. 5 1.04 12.5 1.23 20 ! 1.44 35 1.72 50 1.67 7.5 1.09 15 1.30 25 1 1.54 40 1.76 60 1.63 10 1.15 ' 17.5 1.37 30 ! 1.64 45 1.71 80 159 Gordon's and Hodgkinson's hollow cylindrical cast iron pillars compared. The thickness is usually from ^ to Y 8 of the outer diameter ; and for these limits, the column G, (Gordon), and H, (Hodgkinson), show the proportions of the breaking loads. See " Remarks on this edition," p. ix. Thickness = ^ of the outer diameter. Htin Diams. G. H. 1 Htin Diams. G. 1 1 H. Htin Diams. G. H. Htin Diams. G. 1 H. Htin Diams. G. 1 .90 .97 5 8 1 1 1.25 1.12 10 15 1.11 1.02 20 30 1 1 .97 .941 40 50 .90 .88 70 100 Thickness = y a of the outer diameter. 5 8 1 1 1.23 il 10 1.10 || 15 1 1 1.08 1 1 20 .98 1 1 30 \ .92 1 1 40 .88 1| 50 1 1 .8111 70 Jl .80 1| 100 |l .82 .88 TRUSSES. 243 TKUSSES, Art. 1. When the span of a bridge, roof, &c, becomes so .Treat that single solid beams, supported at their ends, cannot be employed, we resort to compound beams, called trusses, composed of several pieces so arranged and united as to furnish . the reqd strength. The designing, construction, and erection of trusses of great span, especially when of iron, involve such a multiplicity of important detail, that, like the building of locomotives, cars, &c, they have become a specialty, or a dis- tinct branch of business, to which persons confine themselves to the exclusion more or less of other departments ; and thus attain a degree of skill beyond the reach of the general engineer.* The latter, however, should possess a knowledge of the sub- ject sufficient at least to enable him to form a well-grounded opinion of the general merits of a design ; and to guard him against the adoption of one involving serious imperfections. In a volume like this we can aim at nothing more than an attempt to illustrate some few general principles. We shall confine ourselves to such trusses as are in common use; showing first the effects of uniform stationary loads, as in the case of roofs ; and then those of moving loads, such as an engine and train on a bridge. Art. 2. Most of the bridge trusses in common use have two long, straight, par- allel upper and lower members It, ap; and I t, ap, Figs 10, 11, p. 254, called the chords ; or in England, the booms. Vertical pieces placed between, and con- necting the upper and lower chords, are called posts, when they sustain compres- sion; and vertical ties, or suspension rods, &c, when they sustain tension or pull. The oblique pieces seen in these figs are called braces, strut-braces, main-braces, &c, when resisting pres or thrust; or tie-braces, tension- braces, main oblique ties, oblique suspension-rods. &c, when resisting pulls. Sometimes the same piece is adapted to bear both tension and com- pression alternately; and may then be called a tie-strut or a strut-tie. The oblique members alluded to are sometimes called main-braces, whether they are struts or ties ; to distinguish them from counter-braces, or counters. These last are not shown in Figs 10 and 11, but are seen in Figs 28 and 31, crossing the main braces diagonally. These posts, braces, counters, ties, &c, serve not only to keep the two chords asunder, and to prevent them from bending; but to transform the transverse strains produced by the wt of the truss and its load, into other strains, acting longitudinally, or lengthwise, along the diff members ; and to conduct said strains along the truss, to the firm supports of the abuts. A load placed at any one of these members is, of course, partly supported by each abut; one part of it travels up and down alternately between the chords, and along the successive members, until it reaches one abut ; and the other part, in like manner, goes to the other abut. These members, therefore, perform the duty of the vert web of the Hodgkinson beam ; or of the I rolled beams, or of the tubular girder; and on this account are col- lectively called the web members, in contradistinction from the chords. Each portion of any load, while being transferred by the web members, from the spot at which it is placed on the truss, to its final point of support on the abut, produces a strain equal to itself upon every vert web member along which it travels between the parallel chords ; while upon each oblique member encountered on its way, it produces a strain greater than itself, in the same proportion that the oblique member is longer than a vert one. Whether the web members are strained by compression, or by tension; or, in other words, whether they act as struts, or as ties, the amount of strain will be the same. In either case the straining agent is the same identical force, namely the wt, or vert force of gravity of the truss itself, and of its load ; and (Art 18, of Force in Rigid Bodies) whether this force exhibits itself as a pusli, or as a, pull, neither its amount, nor its direction undergoes any change. So far, therefore, as regards the broad principle involved in the duty performed by the web members, they might be divided simply into verticals, and obliques. We shall frequently so desig- nate them. Whatever amount of strain the upper end of an oblique produces in one direction against the upper chord, that same amount will its lower end produce against the parallel lower chord ; but in the opposite direction. That is, if the top or head of any oblique, pushes the upper chord toward the right hand ; its foot will pull the lower chord to the same extent toward the left hand. This, however, is notpre- * The first writer to whom we are indebted for a knowledge of correct principles on this subject is S. WHIPPLK, C. E., the first edition of whose book (beyond all doubt the pioneer one) bears date, Utica, N. York, 1847. He was followed by Bow. of England, and Haupt, of this country, both in 1851. The Murphy-Whipp^e bridge (of which Mr. John W. Murphy, C. E., has built several of the besU owes its name to these two gentlemen. 244 TRUSSES. cisfly correct, inasmuch as when the oblique is a strut, the pres at its foot is some- what greater than at its head, because the foot supports also the wt of the strut itself; or if the oblique is a tie, with its head attached to the upper chord, then the strain is a little greater at the head than at the foot ; because then the head upholds the wt of the oblique, and the foot sustains none of it. This remark applies, of course, to verts also. Another exception is. when the ends of two obliques meet each other; as those at the center of the trusses, in Figs 1U and 11. If, in such cases, the ends of the obliques abut aynnsteacli oilier, instead of being separately attached to the chord, they will at that point exert their strains against each other, instead of against the chord. In any oblique, as c d, Fig 1, the vert dist a c between its ends; and the hor dist a d between the same, are called its vert and hor spreads, or stretches, or readies. Art. 3* There is a great din in princinle between two classes of trusses in common use. In some of them, two chords are absolutely essential, as in the Howe truss, p 283; the Pratt, p 284; the Lattice, p 285; the Warren, p 280; and their various modifications, known as the Murphy-W hippie, the Linville, the Latrop, Ac, o, on, &c, become too long to be safe for upholding their loads of engines, cars, &c, with- out additional precautions. When, therefore, the expense or inconvenience resulting from this would be too great, the verts may, as in Kig 1, be placed so near together as to make half panel* TRUSSES. 245 much higher than they are long; and the obliques (both the main ones, and the dotted counters) then run across one vert, as in the Fig: or across two, if neces- sary. In the Warren girder, the expedient is to introduce verts ; or else a second set of triangles, as in Fig 1, omitting the verts. From 8 to 12 or 15 ft apart are ordinary ' dists for verts in bridges of moderately large spans. Frequently panels are made considerably higher than long; disregarding the economical angle of 4u. In large bridges, the main obliques, instead of being each in one piece, are usually made of two or more parallel pieces, disposed in such a manner as to let the counters pass between them diagonally, without mutual interference. Each lower chord (and fre- quently the upper one also) in large spans, is usually made up of several parallel beams of wood, or bars of iron, side by side. In the Newark Dyke Bridge, England, a transverse section of each lower chord shows 14 iron bars. This enables us to employ smaller beams and bars ; and, moreover, secures greater widtk of truss; thereby diminishing the tendency to lateral or sideways motion. It in no way affects the amount of the strains, or the mode of calculating them ; but is a mere matter of mechanical expediency, or of economy. When the truss is very high, the posts are sometimes made as in Fig 2, where c and c are the upper and lower chords ; and pp a post consisting of two hollow cast-iron pillars, bolted together by their broad flanges at s s. At ss is also placed a hor + shaped casting, of 4 arms ; the outer extremities of which servo to keep in place the iron bars oo, for stiffening the post sideways. Various other ways are in use for building or composing large posts, as well as oblique struts, bv means of T,-f-, U, H, O, or other shaped irons, riveted together. The rolled iron hollow column of the Phoenix Iron Co. of Philadelphia, (p 233, "Strength of Iron Pillars, ") is coming into very common use for such purposes. Lower chords of iron may be made of round or square bars ; or of flat ones joined at their ends by means of splicing-plates, as at B, Figure 3, p 654, or by bolts or pins passing through eyes at the ends of the bars; see Figure 41. Figures 37 to 40 show modes of joining the beams for lower chords of wood so as to resist tension. Upper chords sustain compression only ; and need but a simple butt joint. To allow free expansion and contraction from changes of temperature, when tbe span of an iron bridge truss exceeds about 75 or 100 ft, one end of it should rest upon roll- ers ; or some other device (see p 295) be substituted for the same purpose. If this be neglected, the abuts will be exposed to displace- ment; or if these be sufficient to resist the expansive force, the truss itself will be likely to buckle, if of wrought iron ; or to be frac- tured, if of cast iron, or having a cast-iron upper chord. If in Fig 10 w T e imagine lines crossing the panels diag, as the main obliques shown in the Fig do, but in the opposite direction, as shown in Figs 28 arid 31, they will rep- resent counter-braces, or counters. These, like the main obliques, are in some cases struts, and in others ties. Although important members, they are less so than the main obliques. They are unnecessary when the load is uniform and station- ary, as is usually assumed to be the case in roofs ; )ind are required only when the load is unequal, or a moving one, as in a train crossing a bridge. In this last case they act chiefly while the span is but partially loaded. If the train at any moment covers the entire span, and is of uniform wt, their action ceases for that time. Their office is solely to counteract the deranging tendency of the unequal loading of difl" parts of the truss, as shown in Figs 9%, p 253. In Fig 96, an excess of load along ao would tend to derange the main braces ho and ta; and this would be counteracted by counters across co and ts. The same thing may be effected by arranging the main braces, ho, ta, so as to bear tension as well as compression. The bad effects of une- qual loads must plainly become greater in proportion as the load is heavier than the truss itself; and when the bridge becomes very heavy, so that the load must extend over several panels before its effects become serious, but little couriterbracing is needed; and that at arid near the center only; whereas, in a very light bridge, the counters should extend from the center, where they are most strained; to near the ends, where the strain upon them is least. Inasmuch as we shall first speak of uni- formly loaded trusses, we shall not here say more respecting counters. See Remark, Art 10, p 252. It would at first sight appear that the several parts of abridge truss must be most strained when covered from end to end with its maximum load ; but this is true only of the chords; and of the mam obliques and verts, as la, tp, Fig 10, at the ends of the truss. The other web members are more strained by a part of the load as it passes along the truss ; so that if they be correctly proportioned for a full load, they will 246 TRUSSES. be too weak for a partial one. If all be made as strong as the end ones, they will, it is true, be safe fora passing load; but this would require an expense of material that would be justified only in the case of moderate spans, especially of wood ; in which the additional trouble and expense of getting out and fitting together pieces of many diff sizes, may more than counterbalance the saving in material. Art. 5. Trusses with moving- loads require calculation diff from that for uniform loads. We shall first treat of the latter only ; and in so doing shall not employ the shortest methods, but such as will render the general principles clear to any one acquainted with the simple elements of "Composition and Resolution of Forces." The strains on trusses may be found with all the accuracy needed for prac- tical purposes, by means of diagrams drawn to a scale. The same division of the scale that answers for a foot of length, may also represent a ton, 1000 ibs, or any other convenient wt, load, or strain, and may thus be used for measuring the lines which represent such. The chords, verts, and obliques heretofore mentioned, constitute all the essential elements of a complete truss: but other pieces are necessary for a complete bridge: such as roof and floor beams; transverse bracing for connecting two par- allel trusses with one another, so as better to resist lateral or sidewise motion from winds or lurchings of trains ; bars for tying the truss to the piers and abutments in gome cases, &c. The same may be said of the extension frequently made at the ends of either an upper or a lower chord of a bridge, as shown at n n in the bottom chords of Figs 24 and 26. Here the trusses are perfect without the extensions ; but the bridge requires them, to allow the load to reach and to leave it. They may be needed for the same purpose in an upper chord of a top-road bridge ; or for extending a roof over an entire span, &c. The end vert posts ss, of the same Figs, are not parts of the truss, but supports for upholding it ; also, the posts p and d, Fig 28, are not ea sential to the truss. RKM. Besides the forms of truss already mentioned, there are many others, in some of which arches are introduced either as principal members, or merely aa auxiliaries ; as Town's Lattice* Fig 33 ; the Bow and String, Fig 35; and the Burr, Fig 36, all of much merit. The Lattice and the Burr have both fallen into undeserved disrepute, from the fact that being the first trusses that were extensively introduced upon the railroads in this country, they were built too weak for the heavy engines and trains of the present day, and consequently failed. Art. 6. Chords. When a beam a b, Fig 3, supported at both ends, breaks either und?r its own wt, or under the action of a load placed on top of it, or suspended from it below, it does so be- cause the lower fibres, near its center Z, are pulled asunder ; and its upper ones at u, crushed together to such aa extent as to offer no effective resist- ance. The fig shows this in a some- what exaggerated manner. The ex- treme upper particles at M, and the ex- treme lower ones at /, being the most strained, give way first ; and the strength of the beam being thereby diminished, the adjacent ones give way in rapid succession. The compressed particles of the beam are all atmve a certain point n ; while the extended ones are below it. If we imagine an infinitely fine needle to be held perp to this page, and in that position to be stuck through the point w, passing entirely through the beam, or page, then the infinitely fine hole thus made will pass along what is called the neutral axis of the beam. It is so named because the fibres situated in that line, and which were cut in two by the needle, are neither compressed nor extended, until the strain becomes so great that on its removal the beam will not entirely recover itself ; or. in other words, until the strain exceeds the elastic limit of the beam. Within the limits of elasticity, the neutral axis maybe assumed to pass through the cen of grav of the cross-section of the beam. Thus, if the cross-section be of any of the forms shown in Fig 4, then so long as the beam is safe, or the load within the elastic limits, the line na will pass along its ,. f-w^L v$s$$^$^ cen f grav ; which is at the same time its Eo'4- ^^ ^^ neutral axis. But the chords of a truss S differ essentially in condition from the fibres of the beam, inasmuch as the truss has no neutral axis about which as a center its fibres are extended or compressed to different degrees depending on their dists from it. On the contrary the fibres o'f the two chords react upon each other with nearly uniform intensity, with leverages of so nearly uniform length that their mean or the depth of truss between the centers of grav TRUSSES. 247 of the chords may be taken as their one length. See " Open Beams," p 647. There- fore the same quantity of material, that composes the beam a 6, Fig 3, will present farmore resistance to bending or breaking, if it be cut in two lengthwise, and con- verted into top and bottom chords of a truss ; for the reason that the two chords are then so far removed from each other that all their particles are strained to nearly the same extent at the same time ; so that all the fibres in the upper one must be crushed, and all those in the lower one be pulled apart, at the same instant, rJefore the truss can give way ; whereas, in the single beam a />, the extreme upper and lower fibres break first: then those next to them, and so on, one after the other. They do not all act unitedly, as they do in the chords. It is this principle that gives so much strength to I iron ; to the Hodgkinson beam, &c, (aided by increased leverage.) Art. 7. In the designing of trusses, especially such as may have to bear unequal loads at different parts, as in a bridge, the point chiefly to be aimed at is to dispose its various parts so as to form a series of properly connected tri- angles, because in that shape they present more resistance to derangement of form, than in figs of a greater number of sides. Thus, in the three beams at a, Figs 4^, with a bolt at each junction or joint, the triangular'torm evidently cannot be changed by any but a force sufficient to either bend or break either the beams or the bolts. But in the 4-sided fig t>, the form may readily be changed to that at c, by a force at n entirely too small to injure either the beams or the holts. In the bolts assist to prevent change of form; but in b they are merely pivots, around which great changes may easily take place. Before the strains can be calculated, and the truss propor- tioned to those strains, ITS WEIGHT MUST BE KNOWN; for this tends to break it, as well as the extrane- ous load. But, on the other hand, we cannot learn its wt until we know the size of its iliff members. In this dilemma we must assume for it an approximate wt, based upon our knowledge of somewhat similar trusses already built. This becomes the more necessary as the truss increases in size, so that its own wt becomes greater in proportion to th sit of the load. The table, p 296, gives safe assumed wts for bridge trusses; and p 300 will aid in the case of roofs. In very small spans, especially of bridges, the load is generally so much greater than the wt of the truss, that the latter might almost be neglected entirely. 'Rem. For finding the strains on a paneled truss by means of a drawing, it is best to represent each member by a single line, as in Figs 1, 10, 14, 23, &c. Such is called a skeleton drawing; or diagram of the truss. Each of the parts into which the panel-points divide either chord, or a rafter, is to be regarded as a separate member. As will be shown farther on, a load consisting of some portion of the wt of the truss and its load, is assumed to be supported at each panel-point. All the forces which meet at any panel-point (namely, the aforesaid partial load, and the forces acting lengthwise of the members which meet there) hold each other in equili- brium, or would theoretically keep each other in position without any aid from, pins, rivets, or other fastenings. The forces acting 1 upon a truss (omitting wind) are the downward one of the wt of itself and load ; and the upward one of the reaction of the abut- ments ; and these two forces are equal. They produce all the strains along the members. The figs on pages 292, 294, show some details of modes of uniting the mem- bers of a truss to each other at the panel-points, and elsewhere. Fig- 5 is the most simple form of a roof truss. It consists of two equal rafters o a, o b; and a nor tie-beam a b. Here, as in roofs gen- erally, the entire weight of the truss, and of its load of roof - covering, now, wind, &c, may be assumed to be uniformly distributed across the whole span. A roof con- sists of several trusses, placed usually from 8 to 12 ft apart; but some- times much less, and at others much more. The nal timbers, p, p, called wall -plates, stretching along the top of the wall ; and serving to distribute the wt of the truss and its load over a greater area. On the rafters, and at interval* of a few ft, are fixed pieoes of timber called purling, of 248 TRUSSES. small scantling, running across from truss to truss ; to which the laths or boards are nailed which support the shingles, tin, or slate, &c, which forms the roof-covering. A truss plainly supports all the purlins, roof-covering, snow. &c, &c, which occupy the space half- way on each side of it to the next truss. Thus, suppose a span of 30 ft ; and each ra'fter to be 16.8 ft long ; and if the trusses are say 12 ft apart from center to center, and if we assume (as it is generally well to do,) that the wt of the truss, covering, snow, &c, may amount to 40 fts for every sq ft of area of roof; each truss has to sustain 33.6 X 12 X 40 = 16128 fts, including its own weight. Strictly the wt of the tie-beam shoura be omitted ; because in Fig 5 no part of it is upheld by the rafters. It is very trifling however in comparison with the load. To find the strains upon the different parts of a truss, Fig" 5.f Jb'irst calculate in the manner just shown, the entire wt in K>s of a truss and its load. Through the center U of either rafter draw a vert line Hr. From o draw a hor line H. Join H a. Now on the vert line H r, lay off H I by any convenient scale to represent the entire uniformly distributed wt of one rafter and its load; and draw the hor line IE. Then will IE give by the same scale the hor force at the head of the rafter ; and H E the amount and direction of 'the oblique force * which presses the foot of the rafter ; but which diminishes gradually to nothing at its head.1I The hor force at the foot of the rafter will be equal to that at its head ; t and equal also to the hor pull along the whole length of the tie-beam.t There is no force acting in the direction o a of the actual length of the rafter in such a truss as Fig 5. But the rafters have to be considered in another point of view no less important than their ability to sustain this pres at their feet. Each rafter is an inclined b'eam supported at both ends ; and bearing a heavy load equally distributed along it ; and we must find the dimensions reqd for this purpose also; and add them to those req'l for other purposes. These dimensions may be found by the rules, p 18i*, or by tables, pages 191, or 204, according to the case. The sizes in Fig .1 may be found in the following- manner. Take, for example, a truss of white pine, of 30 t'c span, and 1% ft rise. The wt of the entire roof, ano-r, &s per sq inch ; therefore it will have a safety of 3 under 2000 tts per sq inch ; so that we must provide for the foot of each rafter sq ins of area for resisting this pressure alone. These 5^ sq ins may be ndded either in the breadth . may b of our 5 X 9 rafter, thus making it say 5.6 by 9 ; or to its depth, making it full 5 by 10 at foot. We y Fit, 6 , . may diminish the rafter regularly from the bottom, until it becomes equal to the top: but in practice it is not worth while to do this, unless timbers with the proper taper happen to be at hand. Make the tie-beam about the same size as a rafter. In the next three trusses we shall not enter into this detail of calculation; as we conceive that this example suffices to elucidate its principle. Art. 8. Next to Fig 5, in point of simplicity, is Fig 6 ; which represents a truss for either a bridge or a roof of mode- rate span. It has two equal rafters, and a hor tie-beam a 6 as before; but with the addition of a king- post, king-rod, or suspension-rod o n. Either the tie-beam, or the rafters, or both, may be uniformly loaded. It is immaterial whether the load on the former be equal to that on the latter or not. We shall here consider the truss only as that of a roof. Let y y be points half- way between the king-rod and the abutments. Then will the king-rod sustain all the weight of the portion y y of the tie-beam and its load. The portions of the tie-beam and its load between y, y, and the walls a;, w, are sustained directly by the walls. The entire wt of the truss and its load, it is plain, is sustained ultimately by the ahuts. or walls x w ; but the wt of y y and its load does not reach the walls until after having, as it were, first travelled up the king-rod to o, and from there down the rafters to a and b or, indirectly, by a cir- cuitous route. That the king-rod sustains all between y and y, will be evident when we reflect that a beam a b, when firmly suspended at its center n, may be regarded as two separate beams n b, n a. One-half, of the beam n b. and its load would, in that case, manifestly be borne by the wall x, and the other half by n; and so with n a. Therefore, n upholds one-half of the beam a I and its load ; or, in other words, all between y and y. The king-rod transfer.-- the wt of and on y y, to the heads of the rafters at o. This wt may, therefore, be considered precisely in the light of one resting upon o; and we may proceed to find the strains which it produces upon the rafters and tie beam, by Art 33, p;ige 461, of Force in Rigid Bodies. Namely, on o n make o t, by scale, equal to said total wt of yy and its load, and the wt of the king-rod itself. Complete the parallelogram of forces omtd; and draw its hor diag m d. Then will o m. o d measure the strains produced by said total weight only, along their respective rafters; and cm, c d the pulling forces produced by the same wt only al'ing the tie-beam a b ; causing strain all along it equal to one of them. Art 13, p 449, of Force in Rigid Bodies. It will be observed that in this truss, therefore, unlike Fig 5, there z's a strain o m, or o d, running lengthwise through the rafters from head to foot. To resist this Strain, f It <> rafter nitlSt be regarded as a pillar, with a load on its top equal to the strain ; and the safe area reqd for upholding it may be found by means of the table of strength of wooden pillars, on p ige 239. Call this area when found, a. Next, consider a rafter as a beam supported at both ends. and sustaining a center load equal to half its own wt and actual load; and in that respect proceed precisely as in Fig 5. to find its safe breadth and depth by rules, page 189. To the area of section which results, add area a : and the sum will be the final area reqd for the head o of the rafter. Call it !,. Then measure H K, as iu Fig 5, for the pres caused at the foot only of the rafter by the weight of the rafter and its load. Divide this by the safe amount of 2000 Ibs, as was done in Fig 5; or bv a less number if a safety greater than 3 is reqd. The quot will be the area needed for that purpose alone. Add it to area 6; and the sum will be the total area reqd at the foot. This area may now be regularly diminished upward, bv reducing either the breadth or the depth, until at the head it is equal to area b : but this is rarely done even in iron roofs ; the area at foot being continued to the head. We must not diminish both the breadth and the depth, because we should thereby reduce the pro- portion of area required at every point of the length, to resist this pressure, which, although greatest t the foot, does not disappear entirely until at the very head o. The pull on the tie-beam will be I E added to cm or cd. Find the * This is not a bad proportion of breadth to depth. If we had assumed say 15 ins for the depth, we should have got a rafter so thin as to be laterally weak. Frequently, two or three assumptions and calculations may have to be made before we hit 'upon a satisfactory "proportion- 250 TRUSSES. safe area by dividing their sum by 3333, which is the number of Ibs per sq inch, giving a safety of 3. Then regarding half the length of the tie-beam supported at both euds, aud loaded at its center with or by table, page 191. The resulting area, added to the safe area for the pull just found, will be the eutire section of the tie-beam, uuless some addition be made to the depth, to allow tor what is cut away for the feet of the rafters.* See Rem, p 19'2, also Rem, p 487. As to the vertical king-rod, n o, it must be strong enough to bear safely a pull equal to its own weight, added to the weight of aud upou y y. Jf the rod is of good bar iron, it should have one square inch for a safety of 3, of cross-section for each 20000 tbs of said weight; or see table, page 376. If of wood, it must, for a safety of 3, have at least one sq inch for about each 3333 fts of said weight. A safety of 3 will be enough if the bar is not liable to vibration. When the king-rod is of wood, it is improperly termed a king-post. Since a post is intended to sus- tain a load on its top, the term might lead to the inference that the upper ends of the rafters rested upon, or were upheld by the king post; whereas, as we have seen, they actually uphold it. We add the calculated approximate dimensions for a truss, Fig 1 6, of 30 ft span ; and 7V ft rise. Trusses 12 ft apart cen to cen. Wt of rafters and load on top of them, 40 Ibs per sq ft of area of roof, Wt of and on the tie-beam, includ- ing floor, ceiling, load, and momentum, 100 Ibs per sq ft. Timber white pine. Safety of each piece 3. Rafters 8> ins broad, by 11 deep, at foot, and 7% by 11 at head. Tie-beam 8^ broad, by 11 deep, without any allowance for cutting at feet of rafters. King-rod l^j inch diam if upset; or full 1% if not upset.-|- See note, p 263. With no floor or loading: on the tie-beam, except its own wt, say 1000 Ibs, we have, approximately enough, rafters 6^ ins broad by 9 ins deep, at foot; 6 by 9 at top. Tie-beam, say same as rafter, or 6% X 9. King-rod, ^ inch diam if upset ; % inch if not upaet; but it would be expedient to make it rather more. Trusses 12 ft apart center to center. Art. 9. In Fig 7 we have a truss consisting of two rafters, a 6, a d; a tid-beam, 6 d; a king-rod, a c; and two struts or braces, e c, h c. Either the rafters or the tie-beam, or both, may be supposed to be uniformly loaded. Here, R as in Fig 6, the king-rod a c, upholds the weight of the portion y y . of the tie-beam, and of any load of ' _JH flour, ceiling, people, &c, that may be '*" " placed upon that portion ; together with its own weight. But it also sus- tains, in addition to these, the weight of the two struts e c, he; part of the weight of the portions z r, aud x it, of the rafters ; and part of the weight of the roof-covering, snow, &c, that may rest on said portions. That it up- holds itself, y y, and the struts, is al- most self-evident; but that it upolds part of z r, aud xu, and their loads, is not at first sight so apparent. Such struts are introduced illtO trusses when the rafters become so long as to be in danger of * bending too much, or of breaking un- der their loads; or else requiring the use of inconveniently large timbers to make them of. They act like posts in aff irding partial support to the rafters. They carry a part of the strain upon the rafters down to the foot c. of the king-rod; and the king-rod carries it from there up to the tops, a, of the rafters. From a it passes down through the entire length of the rafters to their feet. Thus, it is seen that the actiou of the struts consists in relieving the rafters from a transverse, or cross-strain which would endanger their safety; and in converting it into a longitudinal strain in the direction of their length, in which they can resist it with less danger. As we proceed with the subject of trusses for bridges as well as roofs, it will be seen that this is the grand duty of such struts and obliques generally. In roofs they thus assist the rafters ; and in bridges, the chords. See Rem, p 375. To find the actual amount of strain in Ibs which the struts thus convey from the rafters to the foot of the king-rod, draw e o and h n verti- cally ; and make each of them, by any convenient scale, equal to the weight in fts of either zr or xu, and its load. From o and n draw the dotted lines, o i. n w, parallel to the struts ; and o fc. n v, par- allel to the rafters ; thus completing the parallelograms of forces, ekoi, and hwnv. Draw the hori- zontal diagonals i k. and v w. Then by Composition and Resolution of Forces, either ek or h v, measured by the same scale as be- fore, will give the longitudinal strain in Ibs upon each one of the struts. This strain presses the struts lengthwise from head to foot. Their feet are pressed in addition by the weight of the struts them- selves; and Mis pressure diminishes regularly toward their heads or 'tops, where it is nothing; as in the case of the rafters in Fig 5. In practice, the wt of the struts is so trifling in comparison with that of the roof portions which they sustain, that it may be neglected, and its safe dimensions may be found by pa*es2:58, 239. Therefore, each strut may be regarded as if a vert pillar, bearing a load equal to^fe or h v. NOW, the strain e k, along the strut :ompounded or composed of the vert strain e s, (which is equal to half c wt of and on zr;) and of the hor strain a k. And the strain h v along the st o. or one-half of the ut h c, is compounded *In cases where no appearance of sagging would be admissible, it is not always enough that the rafters and tie beam be safe ; for they mav be perfectly safe, and yet sag too much for some purposes. When such is the case, refer to table, page 204. tWe have known country road bridges, Fig 6, of 30 ft span, and 7^ ft rise, of two trusses 18 ft apart, in wnicn neither the timbers, nor the probable loads, were larger than in this example. TRUSSES. 251 ef the vert strain h t, (which is equal to half of h n, or one-half of the wt of and on x ;* and of the nor strain t v. These two hor strains s k and tv neutralize or counteract each other, by pressing against each other at the feet of the struts ; and therefore only the vert ones e s and h t pull upon the king-rod ; and they pull it to an extent equal to half the weights of and on z r and xu.* The kiii$?-rod, therefore, upholds in all, 1st, the weight of the two struts ; 2d, the wt of and on y y ; 3d, half the wt of and on z r and x u ;* and 4th, ite own wt. It must, therefore, have sufficient sectional area to safely sustain a pull equal to the sum of these lour. This area may be found by means of the table of bolts on p 376. Make a g by scale equal to the sum of these four wts. Draw am, gl parallel to the rafters; and Imhor. For the dimensions of the rafters, ab,ad, commencing with what they require as beams, supported at the ends, bear in mind that the introduction of the struts ec, h c converts each rafter, as a b, into two shorter ones, a e, e b ; each of which sustains, in the present case, only one-half the load of and on the whole rafter ; or only % of it as a center load. Find the safe dimensions for the short beam, with its smaller center load, by rules, p 189, or by table, p 191. Next, regard the rafter as if a vert pillar, which has to support on its top the pressure indicated by either o m or a L But in doing this, remember that the struts virtually reduce each rafter, as a 6, to two short pillars a e, e b; which require much less area than the whole rafter would. This area may be found by the rule on pag 238 ; or by the table, page 239 ; and added to that required as a beam. The sum will be the total area at the head of the rafter. Next find by means of a line, H E, drawn on the same principle as in Figs 6, 7, and 9, (HI being the entire wt of a rafter and load,) the amount of pres which the/ooe of the rafter sustains from its own wt and load resting on it. Divide it by 2000, (or by whatever other number of tt>s may be con- sidered the safe crushing strength of timber.) The quot will be the safe area in sq ins reqd at the foot for that purpose ; and added to the areas previously found for a beam, and for a pillar, it gives the entire area for the foot. We may diminish either the breadth or the depth, (not both,) of the rafters regularly from foot to head, to accord with the total areas found for those points respectively ; or, which is generally better, may give to them throughout the same area they have at foot. The tie-beam. Here I E added to/ra, or to fl, will give by scale all the pull on the tie-beam. Divide this pull by 3333, the safe pull in ft>9 per sq inch. The quot will be the safe area reqd for that strain. Then consider one-half of the tie-beam to be a uniformly loaded beam sup- ported at each end; and find the safe dimensions by rujes, p 189, or by table, p 191. To these dimen- sions add the area just found for the hor pull ; the sum is the entire area reqd for the tie-beam, un- less some addition be made to compensate for the cutting away at the feet of the rafters.t See Rem, p 192. Below are the calculated dimensions for two trusses, Fig: 7, of 4O ft span ; 10 ft rise ; and 12 ft apart from center to center. In the first of these the tie-beam with its floor, ceiling, and other load, are assumed at the rate of 100 Ibs per sq ft of floor; while, in the second, no specific load is assumed for that member, for reasons before given. In both, the wt of the rafters, with their roof-covering and load of snow, and wind, is taken at 40 Sbs per sq ft of roof surface between the centers of two trusses. The safety of each separate part is taken at 3; except that the unloaded tie-beam is fixed by rule of thumb. Timbers white pine. The great- est dimension in each case is the depth. Dimensions in inches. 1st. Rafters 8 X 10 at foot; and 8 X 9 at head. Tie-beam 8 X 15. Each strut 43^ X 4>. King-rod \% diam if upset ; or 2 ins if not upset. In practice, it is better to make the struts as broad as the rafters. 2d. Rafters 6 X 8 at foot; and 5 by 8 at head. The tie-beam requires, theoretically, only 16 sq ins area; we will make it 6 X 8, like the rafters. Each strut 4>^ X 4^ ; (the same as in the other.) King-rod % diam if upset ; or scant 1 inch if not upset. See Note, p 263. If a tie-rod were used instead of a tie-beam, its diam would be \y inch if upset ; or 1.6 if not. Art. 1O. Fig 9 is a truss with a tie-beam a 6; two rafters w a, z b ; two queen- rods, J or queens, w t, z t, and a hor straining beam d. It may repre- sent a roof uniformly loaded along the rafters and straining beam ; and having a uniform load along the tie- beam. Or only one of these loads may be supposed to exist, as in a bridge with a load along a b ; or a roof with its load along a w z b. The queen w t supports, besides its own weight, all the weight of and on the part y of the tie-beam; and the other one z t, that of and on u y; and r, each being halfway between a queen and an abut. These are the their proper diams can be found by table of bolts, p 376. The parts of * Each strut will thus bear half of the wt of and on z r, or li u, only when, as in Fig 7, the incli- nation of the strut is the same as that of the rafter. If the strut is steeper than the rafter, it will bear more than half; but if it is less steep than the rafter, it will bear less than half; the remainder being in every case borne by the rafter. The parallelogram of forces will of course show all this. When the inclinations of a rafter and strut are not equal, we cannot draw hor diags ik, vw; but from the points i, k, v, w, we must draw hor lines to the vert diags e o. and h n. f The introduction of the struts in Fig 7, renders our process for that form of truss in some meas ure empirical. It is however safe. WHEN A TIE-BEAM is so LONG THAT IT MUST BE SPLICED, allowance must be made for the weakening effect of the splice. For Splices, see p 292 ; and for other joints, p 294. 1 The queens are frequently made of wood. 252 TRUSSES. the tie-beam from and tt to the abuts, or walls, as well as whatever loads those parts may bear, ar sustained directly by the abuts. The queens transfer, as it were, the weights of themselves and of y and u y, with their loads, di- rectly to w and z. To find the strains on the various parts of the truss, first from the center U of a rafter aw, draw a vert line UH; and from w draw a hor line toH to meet it. Join Ho. Make HI by scale equal to the wt of only one rafter and its uniformly distributed load. Also draw og vert, and equal, by the same scale, to the wt upheld by the queen-rod wt, added to one-half the wt of the straining-beam d, and its load ; for it also presses vert at o. Draw g m hor, or parallel to the strain- ing-beam ; and g c parallel to the rafter; thus completing the parallelogram ocgmof forces. The strains on the straining-beam d. The hor line IE and oc to- gether, give all the hor pres against the end w of the straining-beam d ; and it is plain that a similar process on the other side of the truss, would give an equal pres against the end z. These two equal pressures reacting against each other, produce a strain, equal to one of them, throughout the entire length of the straining-beam ; and therefore, the beam must be regarded as a pillar with a load equal to this strain, on its top ; and the dimensions and area of section, for safely supporting it, may be found'by the rule, p 238; or table, p 239. But beside this, the straining-beam, if loaded, must be regarded also as a beam supported at both ends ; and the area necessary for this, as found by tables, page 191, or 204, must be added to that already found. The strains on the rafters. First, consider a rafter to a as an inclined loaded beam supported at both ends ; and find the proper dimensions and area, by the rules on page 188 ; or by the tables, p 191, or 204. Second, consider it as a pillar supporting a load equal to o TO; and by p 238 or 239, find its safe area. Add this area to the one already found ; and their sum will be the area of the rafter at its head. Third, measure by scale the number of Its pres indicated by H E. Divide it by 2000, (the crushing pressure in fts which ordinary average building timber can bear in short blocks with a safety of 3.) The quot will be the area in sq ins reqd at the foot of the rafter, ex- presslv for resisting the crushing effect at that point. Add this area to the two preceding ones ; and the sum will be the total area for the foot. We may diminish to the top ; or may make it of uniform section throughout its length : the last is generally'most convenient. The tie-beam. The hor strain, or pull on the tie-beam, will be equal to the push on the straining-beam ; and is represented by I E and o c together. Find the safe area by table, page 177 : or by dividing the hor strain by 333, which is the pull in fts per sq inch that ordinary building timbeV will bear with a safety of 3. See Rem, p 192, also Rem, p 487. Then, since in this truss the queens divide the tie-beam into three lengths, each of these must be con- sidered as a separate beam, (loaded or unloaded, as the case may be,) supported at each end. Its safe dimensions being found by rules, p 189; or tables, pages 191, or 204. as may be reqd. add the area just found for resisting the pull. Add, if reqd, an allowance for the cutting away at the feet of the rafters. Below are the calculated approximate dimensions for two trusses, Figr 9, of sixty ft span ; 15 ft rise; and 12 ft apart from center to center. All the conditions the same as for the preceding example of Fig 7. 1st. Rafters 12 ins broad, by 14 ins deep at the head ; and 12 X HV> at foot. Straining- beam 12 broad, by 12 deep. Tie-beam 12 broad, by 12 deep. Each queen rod 1ft. ins diarn if upset ; IJf ifnot.t 2d. Rafters 10 X 11% at the head; and 10 X 12}^ at foot. Straining-beam 10 X 11. Tie-beam, say 10 X 12. Each queen-rod y 5 g inch diain if upset ; % if not. Unloaded tie-rod, 1 -=-3_ or 1}$. The proper size for each piece, so tJiat they sJiall all be suitable for tlif truss, cannot be determined at once. We must find any dimensions that will answer for each piece by itself; and afterwards adjust them by recalculation, perhaps 3 or 4 times. Great accuracy is not necessary in doing this. See Note, p 263. For more on roof trusses, see pp 257 to 205 ; 294 ; and 298 to 802. REM. The truss in Figs 9 and 9% affords a good opportunity for alluding, in a general way, to the principle Of COUnterbracing", and to the" necessity (as stated in Art 7) of ad- hering to a triangular arrangement of the parts of a truss. So long as this truss is uniformly loaded throughout its length, it is well arranged for sustaining the resultiug strains; because the strains on each side of the center are equal, and balance each other. But if a heavy load be placed alo ment sho . only, its tendency to depress that portion of the truss will produce the de in Fig 9% ; * A strut or tie cannot be strained along the direction of its length by a force acting at one end, unless there is at the other end an equal force acting in the same straight line but in the opposite direction, and which may be either one single force, or the resultant of two or more forces neither of which acts in that direction. See Strain, p 444. Hence if in Fig 9 we place a load at Z only, a parallelogram v e gn of forces will not give the hor strain v e along the beam Z W, because there is then no equal reacting hor force at the other enrt in the direction from H towards W. In that case a load at Z only, f /^ % x'/ ^ x \.n (represented by z c in Fig X) produces at z the two strains z n, I)/ ^ ^cAI z e; which last pressing towards a tends to make z b revolve around b as a center, thus forcing z down wards, and the joint w upwards, thereby causing the distortions seen in Figs 9^. $%. The force z e therefore evidently tends to break the joint w , and with a moment equal to the force z e. (in tons or Ibs, &c) mult by its leverage w o perp to z a. See Art 46, p 473, and Moments, p 217. If the moment of resistance of the joint can withstand this the truss will remain unchanged ; btu a simple strut from z to a would remove all danger, by sustaining the whole of the force z e effectively, and thus relieving the joiut w entirely. See top of p 462. --,. - y I 1 Ifl \ TKUSSES. 253 because the hor pressure from s toward t, Fig 9J^, will then become greater than that from t toward . The two triangular portions will still retain their original fig; but owing to the ease with which the 4-sided portion, n m c , has been deranged, and changed to 8 t c e, their position becomes altered to tfce dangerous one in Fig 9^. The diag cm s . has been lengthened to ct; while the diag " * TI -m c 5" e n has been shortened to es. Now, if there points, it would have divided the whole truss into triangles ; and then the diag c m could not have become lengthened to c t by any strain less than one sufficient to break this iron bar by pulling it apart; therefore the truss would have remained safe, and unchanged in figure ; for the bar, while preventing cm from lengthening to ct, would, as a consequence, prevent en from shortening to es. Or, omitting the iron bar at cm, suppose a stiff, unbending inclined post to be inserted between e and n. This also will divide the whole truss into triangles ; and it is then plain that en could not be shortened to es by any strain less than one sufficient to break the post by crushing it. Therefore, in this case also, the truss would have remained safe, and unchanged in figure ; for the post, while preventing en from shortening to e s, would, as a consequence, prevent en* from lengthening to ct. Either the bar or the post would be a counterbrace against the effect of un- equal loading. With a uniform load it is not needed. Neither are additional counterbraciug pieces Deeded in bridge trusses of the forms Figs 10, 11, 12, 13, provided each web member is so constructed as to bear alternately compression and extension. See Rem 1, foot of p 306. The next Fig 9 6, shows the bad effect produced in a truss longer than Fig 9^, when the web members are not so constructed. In the Burr bridge, Fig 36, and in, some others, although the truss is divided into trian- gles, yet the inclined braces, ic, Ac, are often impro- perly adapted to bear compression only ; their ends not being firmly attached to the chords. Consequently, with a heavy load at a, the derangement shown in Fig 9 b (analogous to that in Fig 9^) takes place. To pre- vent it, counterbracing must be resorted to, either by inserting struts or ties along the dotted diagonals; or by making the braces capable of resisting tension as well as compression. The last method shows that counterbracing can be performed without the ad- dition of pieces specially called counterbraces, an 1 denoted by the dotted diagonals. All that is re- quired in the principle of counter bracing, is to so arrange and connect the several web members, that the strain produced by unequal loading at any point, as a, between the abuts ; or along any portion, of that distance, shall be properly transferred by them to both abutments. Art. 11. The strains in snch trnsses as Fisp-i 1O and 11, p 254, may be found by three very simple processes when the truss and its load are uniform from the center each way.* When this is the case it is usual and safe to assume that the half load e p. Fig 10 or 11, on the right hand of the center e, rests on the right hand support^ ; and that the half load e. a on the left hand of the center e, rests on the left hand support a.f It is often assumed also, for simplifying the calculations, that the entire weight of the truss and its load is distributed along one chord only. This is plainly incorrect ; but inasmuch as the exfranenus load (such as the covering of slate, snow, etc., on a roof, and the travelling load on a bridge) in many cases ac- tually does rest on one chord only, and is great in comparison with the weight of the truss alone, the error arising from the assumption in such cases is not of practical importance. But in bridges of great span the weight of the truss may bear a large proportion to that of its load ; or there may be an upper and a lower roadway, one resting on each chord; and a roof truss may have to bear not only the covering, snow, &c., on its upper chord or rafters: but a floor with a plastered ceiling beneath it, and all the load incident to any ordinary room, on its lower chord. In such cases the entire weight of the truss and load must be properly distributed along both chords before we can correctly find the strains. But this will in no way affect the principle of the three processes which we are about to explain, and as we proceed we shall give di- rections for both cases. * It is not necessary that the entire load should in itself be uniform ; but merely uniform each way from the center. Thus at e may be say 1 ton ; at a and m each say 5 tons ; at c and n each 2 tons, &c. t This assumption is untrue, and opposed to the unvarying law that every individual portion of the entire weight rests partly on each support. Thus, one portion of the load at o rests partly on p and partly on a; and so with every other portion ; and on this fact depends the difference in the methods of calculating the strains from uniform, and ununiform or moving loads. When, however, the weight of the truss and load is uniform each way from the center we obtain correct results, and more readily by adopting the erroneous assumption. When not thus uniform, the parallelogram of forces is not applicable to a simple uncounterbraced truss. 254 TRUSSES. TRUSSES. 255 Beginning then with uniformly distributed weights of truss and load, and assuming all of said weights to rest on the long chord a p, prepare a correct skeleton diagram of the truss (or at least of one-half of it), such as Figs 10 and 11, in which the height or depth e t, Fig 10, is the vert distance between the centers of the depths of the two horizontal chords. A scale of from /^ to J^ of an inch to a foot will generally be large enough. Then the first process is the very easy one of ascertaining how much of the total uniform weight is to be considered as sustained at each point of support along either the top or the bottom chord, as the case may be; remembering that one half of each end panel is sustained directly by the abut nearest to it, as in the preceding eases. In order more fully to illustrate the following Articles, we shall assume each of the trusses, to Fig 22 inclusive, to be 64 ft long, and 16 ft high. Each truss is as- sumed to be divided into 8 equal panels. Total uniform wt of one truss and its load, 32 tons ; or 4 tons to a panel. Consequently there will be 9 points of support to each truss. Thus, in Figs 14, 15, and 16, in which the load is supposed to rest on top of the truss, and in Figs 10 and 11, in which it rests upon the bottom, the points of support are at a, b, c, d, e, m, n, o, p. Some of these are not shown in the first three Figs. If both chords are loaded, there will be points of support in the short one also. Thus, in Fig 10 there will be 7, and in Fig 11 there will be 8 of them. Now, in Figs 10 and 11, w, x, y, etc., being midway between the points of support, it is plain that (assuming all the weight to be on the lower chord) the point o must sus- tain that portion of it comprised between w and x; nail between y and x ; while the abut p sustains directly the portion from w top. The same principle applies to all the other trusses ; and equally so whether the panels be of the same width or not ; each point of support is assumed to sustain all the uniform wt of truss and load between itself and the two points midway to the adjacent points of support, how- ever unequal the two distances may be. In our Figs 10 to 16, the strong dotted lines of the web members represent ties ; the full lines, struts. The dots intimate that chains may serve as ties. When the panels are of equal length, p o, o w, etc., the dis- tance from p to w will be but half a panel ; consequently then each abut will sustain but half as much wt as each other point. Therefore, to find the amount of wt sus- tained at each of the nine points of support, we have only to div the total wt (32 00 tons) by a number less by 1 than the number of points. The quot = 4 tons, will 8 be a full panel-load, to be at each point, except the two end ones, a and p, at the abuts ; at each of which it will be but half of one of the full panel-loads, or two tons.* The amounts of these panel and half-panel loads should at once be figured on the sketch at their proper points, as is done in our Figs ; a 2 being placed at each end of the truss ; and a 4 at the other points. Each of these panel-loads of course causes a vert strain equal to itself where it rests. As the strains on one half of the truss are the same as those on the other half, the numbers need only be written on one of them ; indeed, the sketch, as a general rule, need show but one-half of the truss. If there is a load on the other chord also, it must be in the same way divided among the points of support of that chord, and be figured as before. The second process. All the panel-loads are of course eventually trans- mitted through the truss to the abuts ; as is manifest from the fact that each abut sustains half the total load. But each panel-load, while travelling, as it were, up and down alternate web members from its original point of support, to the nearest abut, places, so to speak, an additional load, or more correctly produces an addi- tional vert strain equal to itself, at every intervening point of support in each chord.f Our second process consists in finding the amount of this additional vert ttrain at each point of support. * This of course is only when the end panels are of the same length as the others. When not so, the loads at the points of support and on the abutments will plainly vary from the above. f The routes taken by the panel-load strains in Figs 10 and 11, lead at once toward the nearest abut ; but in Figs 14, 15, and 16, they first go forward to the center e of the truss ; and from thence are transmitted backward to the abuts, along the inclined upper member e a, Figs 14 and 16 ; or the inclined lower one i a, of Fig 15. In Fig 16, a peculiar force is generated by the raising of the center of the tie-bar a in; this case will be considered after the others. It is very easy to de- termine, at a glance, whether the panel-load strains travel at once toward the abuts, or toward the center of the truss; and it is essential that this be first done. We have only to assume or suppose, for the moment, that a tie can, like a chain, sustain 17 256 TBUSSES. lu Figs 10 and 11, with parotid horizontal upper and lower chords, the vert strains are very easily found, thus : Remembering that only half of the center panel-load strain at e goes to each abut, begin with the 4 tons at e. In Fig. 10 these 4 tons first go up the tie ei to i, where they produce a vert strain of 4 tons, which figure as in the diagram. But at i these 4 tons separate ; 2 of them going to the abut p, and the other 2 to the abut a. The last 2 first pass down i d to 5, where also they produce a vert strain of 2 tons, which also figure, as must be done with all that follow. At d these 2 tons unite with, or as it were take up and carry along with them the 4 tons already there ; and the entire 6 tons go up the tie d j toj>', where they produce a vert strain of 6 tons. From j these 6 tons go down the strut jc to c, where they also produce a vert strain of 6 tons. At c these 6 tons take up the 4 tons already there, and the entire 10 tons go up ck to k; and thus the process continues until 14 tons find their way to the abut a, where they meet the 2 tons of load already there ; thus making 16 tons, or one-half the wt of the truss and its load; which is a proof that our work is correct so far. In Fig 11, the 4 tons at the center e separate there ; 2 of them going up e i to t', and thence to the abut a as before. In either Fig if there is a uniform load on each chord there is no difference in the second process ; for after having by the " first process " divided each load among the points of support of its own chord, the portion at each point must be taken up as it occurs, and carried on with the others to the abutment as before. The third and last process consists in completing our sketch, or diagram, in such a manner as to enable us to measure by scale the strains produced along every member of the truss, by these vert strains thus accumulated at the difif points of support, a, 6, c, I, k t etc. To do this in Fiy 1O (that is, whenever the web members are alternately vertical and oblique) from each point of support of one chord only, beginning at the center apex i, draw a vert line as iv t jv, etc., to represent by any convenient scale, the vert strain figured at said point; except at the center one t, where the vert line must represent only half the vert strain, inasmuch as that is all that goes to each abut. Draw also the hor lines vu,vu, etc. Then will each oblique line I'M,./ it, etc., give by the same scale the strain (2.2, 6.7, 11.2, 15.7) along its own oblique web mem- ber, as figured. The hor Hues will give the hor strains on the chords at both ends of each oblique. We have figured all these, strains (7, 5, 3, 1) at the head and foot of each oblique. Each of these hor strains extends from the ends of the oblique, to the center of the chord; therefore the end stretches of the chords bear 7 tons hor strain ; the next ones 7 + 5 = 12 tons ; the next ones 7 + 5 + 3 = 15 tons ; and the center one 7-f5-f-3-fl = 16 tons; all of which are figured along the chords.* We have said that the vert, hor, and oblique sides of the triangles give the strains, but it would be more correct to say that each of them gives a force, which being balanced by the other two. thereby causes a strain equal to itself, instead of mo- tion. See " Strains," p 444. The hor strains at the centers of the two chords trill be equal in both Figs 10 and 11, whether one or both chords be uniformly loaded; or if the truss be inverted ; with only the exception in the foot note.* no pressure; and a strut no pull ; then, by looking at any point of support, as at c, Fig 10, we see at once that its strain, being a pull, cannot travel toward the center, along the strut cj, but must go toward the abut a, through the tie c k. Where- as in Fig 14 the pressing strain at c cannot go toward the abut through the tie c A-, but only toward the center, through the strut cj. When, as in Figs 10 and 11, the strains travel at once toward the nearest abut, the web members are slightest at the center of the truss, and stoutest near the abuts ; while the chords are stoutest at the center, and slightest at the abuts. But when the strains first move toward the center of the truss, all this is reversed ; as indicated by the diff thicknesses of the lines in the Figs. * When at the central apex t, Fig 10, the two ends of the obliques td, ira, which meet there, are so arranged as to butt tight against each other, then the center hor strain of 1 ton at that point is not borne by the chords, but by the obliques them- selves; so that there will then be that much less strain at the center point of that chord than along the center stretch d m of the other chord. But if instead of this, they abut against the chord, at some little distance from each other, then the chord also receives the strain; so that the hor strains at the centers of the two chords become equal, as we assume to be the case in all our Figs of uniform trusses uni- formly loaded each way from the center. The same remark applies to the hor strain of .5 of a tou at the center apex e of Fig 11. TRUSSES. 257 The strain along the center vert tie e i of Fig 10, will be equal to the 4 tons at e ; and when the entire wt is assumed to be on the Jong chord, the vert lines at the other points of support will give the vert pulling strains on the other verts, as 6, 10, 14. But with loads upon both chords this last will not be the case; but the strain on each vert tie will then be equal to the vert strain at its foot.* If the loaded truss is inverted, the verts become struts or posts, and the obliques ties ; also the strain on each vert is then the one figured at its top ; but the amount of strain on each part of the entire truss will remain as before. In Fig 10 all the uniform wt is on the long chord, and the resulting strains are all figured on the diagram. We add the strains that would occur in case there were an additional uniform load of 6 tons on the upper chord from I to t. This would give 1 ton at each point of support along that chord, except the two end ones I and f, at each of which it would be but .5 of a ton. All these must be figured on the short chord of the diagram, as were 4, 4, Ac, on the long one. The student may then work out the case for himself. We repeat that uniform trusses and loads require no counter-bracing. For Fig: 1O, but for a load on each chord. e i = 4. tons. dj = 6.5 " c k = 11.5 " b I = 16.5 " i d = 2.8 tons. jc = 8.39 k b = 13.98 I a = 19.01 a 6 or I k = 8.5 tons. 6 cor fc.; = 14.75 " c d or j i = 18.50 " d e or at i = 19.75 " For the '* third process" in Fig 11 (or when all the web members are oblique, whether equally so or not) after having found and figured the loads and vert strains at each point of support precisely as directed for Fig 10, then from every such point in both chords draw a vert line as ev, i u, d r, &c ; and on it lay off sepa- rately by scale both the vert strain that comes to that point through a web member from towards the center of the truss ; and the one that yoesfrom it through another web member towards the abut ; except that at the very starting point e, Fig 11, there is but one vert strain (the one of 2 tons going from it) ; and at the very end a also there is but one, namely that of 14 tons coming to it along the oblique / a ; for the 2 tons at a are not to be included, because they do not roach a by means of a web member. Therefore both at e and at a only a single vert strain is to be laid off.f * It is so in both cases, for in any of these trusses under stationary loads the strain along a web tie whether vert or oblique may be considered to commence at its lower end, that being the end at which the panel loads first act on their route to the abut, and up which they as it were work their way. But under moving loads the same member may have to act both as a tie and a strut ; hence the remark will not apply to such. Referring to what is said above, when the entire wt is on only one chord, tiie vert strains on the two chords are equal at any tie in Fig 10. Hence a line drawn to represent the upper one, may be assumed to repre- sent also the lower one. But when both chords are loaded, the vert strains figured at the two ends of any tie are unequal, and we must then have regard to the true principle. If the verts should be struts or posts (as if Fig 10 should be inverted) then any strain along them must be received from their tops, or the reverse of the case with ties. It will aid the student very much in what follows to familiarize himself with the idea that strains pass only down the struts, and up the ties. Also that vert web members cause no nor strain along; either hor chord ; whereas the two ends of any one oblique web member always cause equal strains on each of two parallel chords; and one of these strains is a push in one direction, while the other is a pull in the opposite di- rection. If the chords are not hor, the strains on them from both verts and obliques will all be more or less oblique. f When all the wt of truss and load is assumed to be on the long chord as in our Fig 11, then the vert strain that comes to any point in the short chord by one web member is plainly the same in amount as that which goes from it by the other web member; and hence only one vert measurement need be laid off for it, as is seen at i v,j v, k v, I v, in the Fig. But at the long chord (or at both chords when there is a load on both) the vert strain that conies to any point is less than the one that goes from it towards the abut, and is evidently the one last figured at that point, as the 2, 6, 10, &c, tons at d, c, b, &c, in Fig 11 ; while the one that goes from that point towards the abut is as evidently equal to the sum of the two strains figured at that point, as the 6, 10, 14, Ac, tons at d, c, 6, &c. When both chords are loaded there will be two vert strains to be figured at each point of support in both chords, except at the very starting point, where will be but one in any case. 258 TRUSSES. Draw also the hor lines, thus forming a series of triangles (as tvu, ivu^jvu^jvu^ &c, of the upper chord ; and a v u, b v M, 6 z w, * center of either hor chord, is not strictly correct except when both chords extend the full length of the span, and are both loaded throughout their entire length ; or in the impossible case of the entire wt being on the long chord. Still in ordinary cases it is a sufficiently close approximation. On this subject see Art 19. p 272. TRUSSES. 259 support in Fig 10 we have one set of three such forces ; and in Pig 11, two sets. In Fig 10 it was not necessary to show those at the long chord. Now each set, or each triangle represents a vert force, a bor one, and an oblique one, keeping each other in equilibrium at the point of support. It is true that there are other forces acting at the same point, but as they hold each other in equilibrium, they do not interfere with the first ones. Thus, both the 7 and the 12 tons hor forces along the chord at k are balanced or held in equilibrium by the equal ones from t and s, on the other half of the truss; without disturbing the forces represented by the sides of the tri- angles. Hence by measuring those sides we obtain the forces and strains themselves. The same principle either by diagram or by calculation applies to Figrs 12 to 13%, when uniform and uniformly loaded. In those Figs all the weight is here assumed to be on the long chord; (but after what we have said, no difficulty can arise when placing loads on the other chord also.) All said wt is first to be properly dis- tributed among the points of support on said long chord, and there figured, as shown by the upper 4s and 2s along that chord in the Figs. This being done, we have figured all the other vert strains, thus providing the data for drawing the vert sides of the triangles ; and these in turn give us the hor and oblique sides which measure the corresponding strains, and all of which are drawn on the Figs. All these trusses being uniform and uniformly loaded from the center each way require no counter bracing. Bear in mind that the vert strains that accumu- late at an abut must equal half the wt of the truss and its load. In Fig: 12, with no oblique at the center, the 4 tons at a having no oblique in contact on either side of them, go to b ; and on their way to b strain ab 4 tons. From b all 4 go along the web members to the nearest abut e as figured. In Figs 12% and 13 the web mem- bers of each are to be regarded in some dogree as belonging to two separate trusses, namely abode and m n op r in Fig 12%; and a b c d e and m n o d e in Fig 13; and the vert strains at their ends are to be found on that assumption, as figured. In Fig 13, o d is a vertical tie. In Fig: 13*4 there being at none of the 4 ton loads on the long chord an ob- lique in contact with them on either side, they (like that at e Fig 10, or at a Fig 12) pass each by itself vertically to the upper chord, where figure them. Of those at 6, 2 go to the abut ?. by way of the oblique b e ; but the other two 4s nil go to e, each by its own oblique. Each of the three hor lines gives the strain produced along the upper chord by an oblique ; but the hor strain along the lower chord is uni- form from end to end, because all the forces that produce it act at its ends only, hor lines. C 4r <2>4 14 6 It is equal to the sum of the three 260 TRUSSES. In Fig: 13%. ftt the 4 ton loads at c and e, there is no oblique In contact with them on either side; therefore they pass at once vertically to t and */, where figure them. Then begin with '2 of the 4 tons at a. All loads that have no oblique in contact oil either side, whether sustained by vert ties or by vert posts, are to be thus transferred at Olice to the opposite chord in order to meet an oblique along which to travel to the nearest abut; and said vert ties or posts will be strained to the amounts of said loads. REM. We call attention to the fact, not sufficiently known, that this paral- lelogram of forces does not always indicate the true strains. Thus in Fig 9, p 251, with a load represented by vg, at z only, and none at w or else- where, v e and v n would not represent the strains along z w and z 6. Nor in Figs 10 and 11, if loaded on only half the truss, would the lines w?>, uv. iu, ju, etc., do so. They do so however in \miform trusses uniformly loaded from the center each way ; also in cases like the Figs on pages 461 to 465 where the strains pass directly from the loads to supports which are in themselves rigid and fixed. See foot note p 252. Art. 13, In Figs 14, 15, and 16, (in which, as before, the span is 64 ft, the rise 16 ft, and the load to a full panel 4 tons,) the second process (or that of finding y t y K J -i bow much additional vert strain is produced at the several points of support by each panel-load, on its way to the nearest abut) differs somewhat from that pursued with Figs 10 and 11, which have parallel upper and lower members. The principle in- volved, however, is precisely the same.* In Fig 14, beginning at the. panel-load b, nearest tin- abut, lay off by scale the vert b r equal to that panel-load. 4 tons. Draw r It'" parallel to the rafter ; and h'" s"' hor. Measure b s"', (2 tons,; and carry forward that amount to c, writing it down over the panel-load 4 ; thus making the vert strain ate 6 tons. (See Remark, Art 15.) Make cr equal to this vert strain; draw rh" parallel to the rafter ; and h" s" hor, as before. Measure cs", (4 tons, ) and carry it forward to d, writing it down, and thus making the vert strain at d 8 tons. Make d r equal to this vert strain; draw r h' parallel to the ratter; and // s r hor, as before. Measure ds', (6 tons,) and carry it forward to the center e, writing it down. Now it is plain that the same process, on the other half of the truss, would bring another 6 tons to e. Write this down also, as in Fig 14. Thus we get for the vert strain at e., 6 -f 4 + 6 = 16 tons, or one-half of the wt of the entire truss and its load. As this must always be the case in such trusses, it proves our work to be correct so far. Third process ase in such trusses, it proves our work to be correct so lar. t. From e draw a vert line e r, by the same scale as before, * When wishing merely to know the amount, without caring to trace the progress of these vert strains at the points of support, in order to calculate the other strains, we may omit part of the fol- lowing, and proceed thus: At the peak e of the truss, the vert strain will always in such trusses he equal to half the entire weight of the truss and its load, (for which, per sq ft. see Table 4, p 301,) and may therefore be written down at once at e. Next count the entire number of points of support a, 6, c, d, e.f, &c, of the truss. From this number, whatever it may be, subtrnct 1. Div the entire weight of the truss and its load by the rera. One half of the quot will be the vert strain at the foot a of the truss, which write down at a. At b the strain will be twice that at a ; ot c three times; at d 4 times ; and so with any number of points of support along a rafter, except at the center one e, where the strain will be twice as great as at the preceding one d. But it has been found already. When this has been done, the vert lines b r, cr, d r, er, &c, may at once be drawn equal by scale to the several vert strains ; then rh,rh,rh, &c, parallel to the rafter ; and t h, a h, Ac, parallel to the tie. Then begin at " Third Process." TRUSSES. 261 equal to the 16 tons of total load at ; draw r h parallel to the rafter ; and k s hor. The fig is now ready for giving by scale the strains along all the ditt' members of the truss. It is by mere chance that the two vert lines b I and e r, representing the loads at 6 and e, happen just to extend to the hor tie a i in our fig. As with the preceding figs, we add the strains in this case also. When as usual, the points of support a, 6, c, d = hr 4- h' r = 28.5 4- 7.13 = 35.63. " bto c = hr 4- h'r 4- h" r - 28.54-7.13 " ato& = Itr + A'r + /"*+ A'" r = 28.5 4-7.13 = 42.76. 4- 7.13 4- 7.13 4- 7.13 = 49.9. * It is probable that the tie-rod is sometimes raised in this manner bj persons ignorant of the fact that they thereby greatly increase the strains on the rafters, &c. All the strains in Figs 14, 15. and 16 may also be found by pre- cisely the same process as that for bowstring and crescent trasses, in Art 19, p 272. TRUSSES. 268 IV 17 Firf/8 RSM. The reason for measuring only parts of the vert lines which, in Figs 14, 16, 16, represent the whole panel-loads, is that the rafter a, Figs 14 and 16; or the tie "at of Fig 15, being inclined, also bear a part of each panel-load ; and since that part does not go forward to the next point of support, but goes backward, along said in- clined member, to the abut at a, it must be omitted in the second process. Thus, in Fig 17, if b a be an inclined rafter resting on an abut a; bg a strut ; and br a vert line repre- senting the load sustained at 6 by b a and bg; if we com- plete the parallelogram bmrn of forces, then will 6m give by scale the strain along the JN strut ; and b n that along the ~ rafter. The strain along the 2 strut is made up or composed of the portion 6 s of vert force ; and the hor force sm. The vert portion bs alone goes to the next point of support; while s m strains the tie ag hor. So also the strain bn is made up of the other portion (bo or sr) of the vert force br; and of the hor force on; which, when the strain bn reaches a, become again resolved into two; one of which, bo, presses vert upon the abut; or, in other words, transfers to the abut the portion bo of the load resting on 6; while the por- tion on, which is equal to sw, strains the tie ag hor. But in Fig 18, where b r also represents a load resting on b, and supported by a strut bg, and by a hor chord b a, if we complete the parallelogram 6 m rn, we have the strain b m along the strut, composed of all the vert force b r, and the hor force NOTE. The following may at times save trouble in designing roof trusses. After the dimensions of all the members of a roof truss of any span have been calculated, then those of any smaller span arranged in the same manner, and having the same rise in proportion to its span : but with the trusses at the same distance apart as in the large one; may be found tafely; and often near enough for practice, thus : Find the are* of cross section of each member of the large truss, in sq ins. Then make the areas of cross section of the corresponding members of the small truss in the same proportion as its span is smaller than that of the large one. The small truss thus obtained will in fact be stronger for its span than the large one. If the total loads sustained by trusses of different spans, including the weight* of the trusses themselves, were in proportion to the spans, then this method would be correct. But, with the trusses at the same distance apart in both oases, only the extraneous total load borne by them is in proportion to the span ; while the total weight of the trusses themselves is as the squares of the spans. In our table of wts of iron roofs on p 300, it will be seen that it is based upon total loads of 40 Ibs per sq ft of ground covered, including the wts of the trusses themselves. Also that the wt of the truss itself of 175 ft span is 8.0a Ibs per sq ft of ground covered ; so that the greatest extraneous load of purlins, slate, snow, &c, tor this span is 40 8.05 = 31.95 Ibs. If now we proportion a roof of 35 ft span by the above mode, or by the table (which is based on that mode), we find that inasmuch as 35 is but i of 175, the short truss will weigh but 396 Ibs, or ivV part of 9379 Ibs, the tabular wt of the 175 ft span. We also see that the 35 ft truss will weigh but 1.61 Ibs per sq ft of ground covered; while the 175 ft one weighs 8.05 Ibs. or 6.44 Ibs more. Therefore the small truss will be as safe under an extraneous load of 40 1.61 = 38.39 Ibs per sq ft, as the large one is under 408.05=31.95 Ibs. Or In other words, if the large one is strong enough, the small one will be about -jt part stronger than necessary. Reductions however will rarely be made to as small as -^ of the original ; and where they o not exceed J^, the method will answer very well in practice. For examples of reducing, see p 301. With the same total load per sq ft, including the wt of the truss itself, and with trusses at the same distance apart in all cases, the strains on the several members of similar trusses, (that is, of trusses precisely alike except in the size of themselves and their parts) will be in the same proportion as the spans ; as will also the areas of cross section, and the wts per foot run of each individual member ; but the total Wts of the trusses will be as the squares of the spans. 264 TRUSSES. r TO. The whole of 6 r is transferred to the next point of support ; while r m and b n produce only hor strains along b a and g y. Art. 16. The roof truss shown by Figs 19, 20, and 21, consists of two complete J'ink trusses, a <' y and ?/ p. Fig 10. It is supposed to be of the same sp;m and height as the others ; and to have the same number ^9) of points of support for the weight Rise 16 ft. TRUSSES. 265 (32 tons,) supposed to be uniformly distributed along its top. Consequently from our first process there will. 'as before, be 2 tons of panel-load at each of the end supports ; and 4 tons at each of the others. Write these down as in Fig 20. Now, at either strut, as dg, draw a vertical line dv in pencil, by any convenient scale, to represent a whole panel- load, (-i tons;) and draw vo parallel to the rafter//*'; and v r parallel to the strut. Measure d r or v o by the scale, and write down the result (1.77 tons) near every strut, as in the Fig. The object of this will be shown hereafter.* The linns d v, v o, v r, may then be rubbed out. Now the strains from the several panel-loads, in passing to their final points of support at the abuts, travel by a route diif from that in either of the preceding cases. The part truss ex a may be regarded, to some extent, as being composed of three separate trusses ; namely, ex a, egc, a cm; as will appear more plainly from en a, eic, and a en, in Fig 21. These may be called first and second secondary trusses. In Fig 19, the half cp y ex- hibits a truss on the same principle, but having a greater number of points of sup- port for the uniform wt. That half truss consists of first, second, and third second- aries, as shown by ey p, tgi, and es h. However far this subdivision may be carried, if the struts occur at equal dists, and if the wt or panel-load supported by each strut is the same, then each panel-load resting on a short strut will travel (one-half of it each way) either to the panel-load of the next longer strut ; or else to one end of the rafter. Thus, with our second process at one of the shortest struts, as 6 m, Fig 20, 2 of its 4 tons go to c at the next longer strut, car; arid 2 of them to the end a of the rafter, as written on the Fig. Then, at the other of the shortest struts, d g, 2 tons go to c ; and 2 to the end e of the rafter. We thus have 4 tons at b ; 4 at d ; and 8 at c. But of these 8 tons at c, 4 travel down the strut c a;, and along the tie x a, to the end a of the rafter ; and 4 along ex and xeto its other end e ; both of these 4 tons are therefore set down as at a and e. When there are more points of support, as along the rafter ep, Fig 19, the process is precisely the same : we first adjust the strains of the four third secondaries, e s h, h r i, i v k, kzp\ placing them at e, //, i, k, and p: then we transfer those thus accumulated at h and k, to e t i, and p ; and finally transfer them from i to e and p, at the ends of the rafter. Now, returning to Fig 20, we see that in addition to the original panel-load of 4 tons at e, we have accumulated 8 tons of vert strain from the other panel-loads ; and it is plain that the same pro- cess, performed along the other half of the truss, would bring 2 -f- 4 = 6 tons more to e, as written in the Fig. Thus it appears that we have 16 tons in all at e ; and this is precisely half the weight (32 tons) of the entire truss and its load; and as this will always be the case in trusses on this principle, it proves our work to be correct thus far. In like manner, the total strains accumulated at the other end a of the rafter; as well as those at its center c, must always each be equal to one quarter of the weight of the entire truss and load. Also the total strains thus accumulated at any longer strut, will be twice as great as that at any next shorter one. See Art 18. Having thus finished our second process of finding the additional strains at tho several points of support produced by the panel-loads on their way to their final points of support, we have only by our third process, to complete the draw- ing, so that we may measure by scale the strains along all the members of the truss. To do this, from the tops of the struts draw vert lines b v, c v, d v to repre- sent the total vert strains accumulated at those respective points; namely, 4 tons at 6, 4 at , d o will give the strains along the struts ; 3.6 tons on b m or d g ; and 7.2 tons on c x. Lay off m *', x i, and g i respectively equal to '6 o, c o, and d o; draw ij, ij, ij\ arid i y, i y, i y, parallel to the ties ; thus completing the parallelograms of forces ij in y, ij x y, and ijg y. Also draw the diags yj, yj, yj; and tne vert lines i w, i w, i ?r, m u, x M.and # u. Also lay off the vert dist e/equal to the total vert strain (1C tons) ate; draw fh (Fig 21) parallel to the rafter a e, and terminating at /tin the other rafter e 1] and h z hor or parallel to the tie a I. Or, which amounts to the same thin.ir, is. Fig 20 make ef equal to the strain (16 tons) at e\ make e z = to half of tf; draw z h hor; and hf. This saves the necessity of drawing more than half the tru>s. Now m y and mj give the strains (4 tons each) along the ties m a, m c, caused by the 4 tons at 6; which strains extend from m to a and c.f In like manner, g y and * When merely wishing to ascertain the strains along the members of such a truss, without caring to trae their progress, Ave may omit part of the following ; and, after having made a correct diagram of the truss ; and found and noted down the force dr or vo mentioned above, we may at once write down the vert strains at the points of support, thus: At e (the apex or peak of the truss) write one half of the entire wt of a truss and its load, (for which, per sqft, see Table 4, p 301,) at the center strut c, one fourth ; at the foot a of the rafter, also one fourth ; at 6 and d, one eighth at each ; and when there are four intermediate subdivisions of the same kind, a- along the rafter ep, Fig 19, one sixteenth of said entire weight at each of such additional points, &c. Then begin at " Having thus finished our second process." t These strains along the ties will b equal to those at the points of support, only where the height of the truss is equal to y of its span ; as in the case before us. When the height is less than V, the trains oa the tie* will be greater than those at the points of support, ; and vice versa. 266 TRUSSES. gj give the strains (4 tons each) extending from g to c and e. In Pig 21, the short ties, o a, o c, i c, i e, show this more distinctly. Next, x y and xj, Fig 20, give the strains (8 tons each*) produced along x a and x e by the 8 tons at c.* This also is shown more plainly in Fig 21, by the ties n a and n e. Again, the nor line It z will give the strain (16 tons) produced along the entire nor tie a /, Fig 21, by the 16 tons at . Fig 20 may be considered one half of Fig 21. In practice, the ties ao,an, &c, Fig 21, of the secondaries, are not always made distinct from that (a I) of the primary truss a e I; but they are so represented in Fig 21, merely to show more plainly that the central portion a; a; of the primary tie a I needs only such dimensions as will enable it to sustain the thrust produced by the 16 ton strain at e: whereas, along its portions x m, x m it must be stout enough to bear, in addition, the thrust along the first secondary ties na,kl\ while at its ends m a, m I it must resist not only the two preceding thrusts, but also those along the second secondary ties o a, o I. Likewise, it is plain that the portion g e of the first secondary tie n e, must be stouter than the portion n g\ because \vhen n eis formed of one bar, its portion g e has to bear also the thrust along the second secondary tie i e. In Fig 20, those portions of the ties which are most strained are shown by stouter lines. We have for the total strains on the ties as follows : See Rem, p 375. Along c m and c g, strain mj or g y 4 tons. A long x e, from x to g. strain ~xjS tons. Along x e, from g to e, strain x j + ffj ~ 8 + 4 = 12 tons. Along t a, from t to x, strain = h z = 16 tons. Along t a, from x to m, strain 7t^4-zy 16 + 8 = 24 tons. Along t a, from m to a, strain = Az + y + 7rty:=16 + 8 + 4 = 28 tons. The strain along a rafter a e would be equal throughout, were it not for the small strains, of 1.77 tons each, at />, c, and d first found.f This will be seen thus : The 8 tons at c, Fig 20, produco/ortv.s at a and e of 5.4 tons each ; which are found by measur- ing uj and w ?/, of the middle parallelogram of forces ij x y. Consequently these two forces acting against each other at the opposite ends of the rafter, produce a uniform pressing strain of 6.4 tons throughout its entire length. Again, the 4 tons at b produce forces at a and c of 2,7 tons each, found by measuring uj and w y of the parallelogram ij m y. Consequently these two forces acting at the opposite ends of the half rafter a c, produce a strain of 2.7 tons along said half. But the 4 tons at d produce in like manner the same amount of strain along the other half, e c. Finally, the 10 tons at e, produce two forces, each equal to e h (17.9 tons;) one of which presses the entire length of each rafter. Consequently we have, as pressing its entire length, the forces e li y uj of the middle parallelogram ; and uj of the parallelogram ij g y ; or 17.9 + 6.4 + 2.7 = 26 tons. Nothing but these 26 tons press it from e to d\ but from d to c we must add 1.77 tons ; from c to & 1.77 more ; and from b to a 1.77 more ; so that we have at last, for tne total strains along 1 a rafter, From e to d, strain = h + uy (of the parallelogram ijgy), and v.j (of the parallelogram ijxy) = 17.9 + 6.44-2.7=26 tons. " dtoc, " -264-1-77 = 27.77. 44 ctofc, " =26 4- 1.77 4- 1.77 = 29.54. " 6 to a, " =26+ 1.77 -fl-77 + 1.77 = 31.31. The center vert e t may be omitted in short spans ; since it sustains noth- ing but the wt of the half (yy) of the central spread xx of the hor tie a I. Thus we have the strains along every member of the truss. See Art 29, p 298. REM. 1. If the main or primary tie is raised either at its center, as p n, Fig 19 ; or if it is raised only as far as y, and is then continued hor, as y o, (as is frequently done,) in either case we must proceed as at Fig 16, p 262 ; and, after hav- ing found the vert strains at all the points of support, as before, we must add to that (16 tons) at e, the amount arising from mult said 16 tons by the vert height tn (Fig 19) to which the tie zn is actually raised above the hor atp, (or to which it would be raised if the inclination of the tie p y were continued to n, instead of being hor like 3/0 ;) and from div the prod by the remaining height ne of the truss. Then, as in Art 15, we must lay off the vert ef, Fig 20, equal to the total vert strain at e, thus found; and, after drawing fh parallel to the rafter, must draw hz parallel to the inclined tie, instead of hor. REM 2. We will explain the reason for using the portions uj and wy, of the diags j y, Fig 20, for measuring strains alon * the rafter a ?. Take, for instance, the strut d g. Here it is evident, that since ig represents by scalo the strain along the strut, the two sides g y and gj of the parallelogram, give the resulting strains along the two * See second note at foot of preceding page. t These small strains become proportionally greater as the rise of the truss increases ; so that when the rise is as great as J of the span, they cause the pressure at the foot of the rafter to be about l.i times that at its bead ; while at ^ rise, it is but about 1.04 times as great. TRUSSES. 267 ties. Now if we take one of these strains, say gj, and on it as a diag dravr the par- allelogram of forces ujng, with two of its sides vert, and two parallel to the rafter, then.; n will give the vert strain which gj produces at ; and ju, the strain which it produces at e, in the direction of the rafter. Then, if we measure the vert strain j w, or g u, we shall find it to consist of the 2 tons which we originally transferred from d to e, by our second process; and, in like manner, iw represents the 2 tons transferred from d to c. Since, then, the strain gj is made up of uj and jn, and since ,/n was transferred to e by calculation during our second process, only j u re- mained to be determined by measurement in the third process. REM 3. It is not necessary actually to draw all three of the parallelograms, as in Fit 20. The large or center one alone will suffice ; for we need only div the several strains measured aloug the strut en, Fig 21 ; and along the ties na, ne, by 2, to get those along the struts 4o and 4f ; and along the ties ao. co, ci, ei. And these, in turn, div by 2, will give those along the smaller subdivisions shown between e andp, Pig 19, if there are such ; and so on with any number of still smaller ones. REM. 4. The student need now have no difficulty in finding the strains produced by a uniform load on a hor Fink Truss, Figs 26, 27 ; the process is more simple than in the roof; for the chord being hor, and the struts vert, there will be no force like the 1.77 tons at 6, c, and d; and the strain along the chord will therefore be uniform from end to end. In Fig 26, nn is not, properly speaking, a truss chord, but merely a beam added only for supporting the roadway. If the truss were in- tended for a hor roof, n n would be omitted. The vert strains at the top of each strut may be at once written down, without tracing their progress to those points. Thus, at the half-way, or center strut, d c, write one half the entire wt of the truss and its load: at each quarter-way strut mm, write one-quarter of the same entire weight ; at each eighth-way strut, hhhh, one-eighth of it, &c. Then, from the feet of the struts, set up vert dists along the struts, by scale, to represent the vert strains just written at the top of each. From the upper ends of these dists draw lines downward parallel to the two ties at the foot of the strut, and ending in said ties ; thus completing a parallelogram of forces at the foot of each strut, as in Fig 20. Draw the hor diag of each of these parallelograms. Measure and add together one- half of the half-way or central diag; one-half of one quarter-way one; one half of one eighth-way one, Ac. The sum will be the uniform hor strain along the entire chord.* The strain along each inclined tie will be found by measuring that side of the parallelogram which is on said tie. For Fink bridge trusses, see p 305. REM. 5. Figs 21% show a few of the many forms of the details of iron roofs. Every maker has his own modifications of them. Most of the figs explain themselves. They will serve as hints. For more on iron roofs, p 298. R and P stand for rafter and purlin. In small roofs, with the trusses only 3 or 4 ft apart, the purlins may, as at 6, be simple % inch or ^ round rods, about 9 ins apart; and the slates may rest immediately on them, being tied to them by iron wire. They may be bent down at their ends, and riveted to the rafters. As the dist between the trusses increases, these purlins may be made of flat iron, from 1 to 3 ins deep, and % inch thick; or of light T iron, &c; and may be trussed, as at 7, so as to admit of being placed several feet apart. When, however, they have to bear great weight, the mode at c, Fig 7, of confining their ends to the rafters, will be too weak. Sometimes they may be arranged as at y. Or the purlins, of either iron or wood, may rest on top of the rafters, as at 1 and 5 ; or their ends may rest in a kind of stirrup, as at t, Fig 2; and at P, Fig 4; in castings placed at the "points of sup- port " of the truss ; or they may be confined to the sides of the rafters by two angle- irons, as at P, Fig 9. Purlins should, when practicable, be sup- ported only at or near the "points of support" of the truss : and as a general rule, it will be expedient to arrange the number of these points with reference to this particular. The rafters are then relieved from transverse strains ; and may be proportioned with regard only to the compressive strain in the direction of their length. Too little attention is sometimes given to this point, and the trans- verse strain is overlooked, to the serious injury of the roof. It is well, however, to bear in mind that thin deep rafters are liable to yield by buckling sideways; and that this tendency is diminished by purlins well secured to them between the " points of support." Sometimes castings similar to 2, are used at the heads; and 3, at the feet, of the struts and vert ties ; which last have their ends cut into right and left hand screws, for insertion into corresponding female screws cut in the castings. At 3, 1 1 is the main tie passing loosely through the lower opening through the cast- ing. Below it, is seen the head of a small set-screw, for tightening together the casting and the tie ; to prevent the former from slipping out of place. There must be different patterns of these castings, tu suit the obliquities of the several obliques; or, in small roofs, the parts a a may be made with hinges, for the same purpose. * If the struts are equidistant, and of equal length, each succeeding one of these half diags will be % part as long as the oue that precedes it. The strains along the oblique ties will not follow the M*tne proportion. 268 TRUSSES. At 4 ami 5 are cast-iron shoes for supporting the ends of the trusses upon the walls. With the exception, perhaps, of these shoes, it is better that the details generally should be of wrought iron. At 8 is a mode of confining- thin metal roof-covering i, to the purlins P, by means of short (about an inch) U-shaped pieces (c c 1 1 is one of them) of the same metal; to which i is riveted by an % inch rivet through each flange 1 1. This may be adopted with corrugated iron covering, which, by its strength, allows the purlins to be placed several ieet apart. See Corrugated Iron. Flat sheets require boards beneath them. At 10 is a mode of confining a wooden purlin P on top of an iron one _p, by means of a crooked spike s s s; which, after being driven from below, is clinched or bent on top. Wooden purlins are sometimes thus required, for nailing elates or plain sheet metal. At 11, c, is a stick of timber inserted between an iron 10 purlin P and the corrugated roof-covering a a. To such sticks plastering-laths may be nailed, when the roof is to be plastered beneath, to avoid condensed moisture. There is room for much ingenuity in all these details. Fig 12 is a rafter made of two channel-bars riveted together ; with a web member c c between them. Two angle- bars are often thus riveted together for a rafter. For more on iron roofs, p 298. Fig: 18 shows a mode of tightening two lengths of a tie-bar by a swivel or turn buckle, t b. The end of one length, n. is cut into a screw ; and the corresponding end, ft, of the turnbuckle is a female screw, into which n fits. The end r of the other length may either be made in the same manner, or, as in the fig, may be plain, and be furnished with a head c. The turnbuckle revolves around both rods, and of course can thus tighten them. Sometimes t b is made of a solid round bar, called a double nut; with a female screw tapped a few inches into each end, right and left. The end of each rod, c and 2, is then cut to a screw. Fig 14 is a mode of tightening four lengths of tie-bars < Fig 14 is a mode of tightening four lengths of tie-bars crossing each other, by means of a ring. The ends inside of the ring are cut into screws, and provided with tightening nuts, as in the fig. The rings are usually % to l 1 ^ inch thick ; 3 to 5 deep ; and 7 to 10 diam. In Art *29 are remarks on the comparative merits of the foregoing plans of roofs, inasmuch as they are the kinds usually employed. In proportioning the sizes of struts for roofs or bridges, bear in mind Hem ''The young engineer," &c, p 237 ; and Bern, p 375, for ties. TRUSSES. 269 a I! J *- Art. 17. Fig- 22 represents a suspension truss on the Boll- man plan ; * the whole weight supposed to be along the top a p. In this, the strain from each panel-load, as for instance that at d, passes down to the foot of its supporting post dj ; and from there is transferred to the two ends a and p of the chord, by- means of two ties, as j a, .; p, upon which the post stands. In this manner the vert strain from each panel -load is separately sustained; and transferred di- rectly as a hor strain to the ends of the chords, by its own post and pair of ties ; without pro- ducing, as in the foregoing cases, an additional vert strain at the points of support of the other panel-loads. So omit our 2nd process ; and having divided the uniform wt of the trues and its load, among the several points of support a, I), c, d, e, &c, as before, we pro- ceed at once to draw the parallel- ograms of forces v u I g, v u If p, Ac, for measuring the strains. To do this we have only to set up the equal vert dists I r, k v, j v, &c, each to represent by scale the 4 ton panel-loads on top of the respective postfc ; then complete each parallelogram by drawing v u, v g parallel to the two ties which support each post. Then the lines I u, I g ; k it, k g, &c. give by scale the strains along the respective ties. The end a of the hor chord is pressed hor by the seven hor forces u o, u o n u o,,, u &c., equal to 1.75 -f 3 + 3.75 -f 4 -f 3.75 + 3 -f 1.75 = 21 tons ; and the other end p is in like man- ner pressed by the seven corres- ponding forces not shown ; and these two sets of equal op- posing forces produce a strain equal to one of them ; or to 21 tons, uniform throughout the entire chord. The tie la car- ries to a so much of the weight of the 4 tons at 6 as is rep- resented by I o, or 3.5 tons ; k a carries to a a weight equal to k o,, or 3 tons ; j a carries j o /y , = 2.5 tons ; i a carries i o in = 2 tons ; w a, w o = 1.5 ; x a, x o = 1; andt/a, yo = .5 ton. All these amount to 14 tons ; which, with the 2 tons of half panel- load at a, give 16 tons ; or half the entire weight (32 tons) of the truss and its load. This is a proof that the strains have been drawn and measured correctly. The other half weight of truss and load is carried in the same way to the other end jp, by means of the ties yp, xp, lp, &c. These wts cause the following strains : 2- -I co * Invented bj Mr. Wendcl Bellman, C. K. 270 TRUSSES. The strain I u = 3.91 tons. The strain I g = 1.82 tons. Ar0r = 3.17 .70 = 4.05 " t g = 4.47 " Each post or vert is of course strained to the amount of a full panel-load, when tlie whole wt is supposed to be on top of the truss. If the load is at the bottom of the truss, see " In the Boll man truss,- 9 near the end of Art 20%, p 282. BOWSTRING, AND CRESCENT TRUSSES, UNIFORMLY 1,OA1>EI>. Art. 18. Before attempting to find the strains on either a uniformly loaded bowstring or a crescent truss, Figs 23, 23 b, 23 c, by means of a diagram, the student should familiar- ize himself with the following remarks : RKM. 1. The basis of the entire process is that at every point of sup- port, beginning at an abut as the first one, we have acting one or more known forces, balanced or held in equilibrium by either one or two unknown ones ; and the object at each point is first, by means of the parallelogram of forces, to find the resultant of the known ones ; and second, by the same principle, to resolve this resultant into two components in the directions of the unknown ones. This is all that is required in either the bowstring or the crescent truss.* REM. 2. While more than two unknown forces exist at any point of support, their amounts cannot be found. If one force is known, and two unknown, the three balancing each other, draw a line by scale to represent the amount and direction of the known one ; and, considering it as one side of a triangle, from its two ends draw lines parallel to the two unknown ones, to meet each other, thus completing the triangle. Then these last two will by the scale give the two unknown ones ; because when three forces meeting at one point, balance each other, three lines representing them both in amount and in direction, will form a triangle. See Rem 2, p 468.t If there are two or more known forces, first find the single re- sultant of them all (Art 36, p 466), and, taking it as one side of the triangle, find the other two sides (that is, the two unknown forces) as before. After a little practice the student will find it unneces- sary to draw more than half the sides of the parallelogram of forces. REM. 3. The bow is to be considered straight from apex to apex. If actually curved it will be much weakened. RKM. 4. At each point of support, or apex, consider every member that meets there, to be a force either pushing towards said point, if along a strut; or pulling from it, if along a tie. This is shown by the arrows in Figs 23, 23 a, X showing their directions be added to the sides of the triangle as in this Fig, they will, ^ ^ as it were, chase each other around the triangle ; that is, the head H " ' ^ of any one of them will touch the butt end of the one next to it ; or no two arrow-heads will meet. But this is not so when three sides of a triangle represent two forces and their resultant, or equivalent in effect ; for the arrow-head of the re- sultant will then meet that of one of the other forces. See Kent. 2, p 468. TRUSSES. 271 J8 272 TRUSSES. from t towards e ; or the other two to- b similarly changed to pulls. The first of course is the easier. According as the known forces (after one or more are so changed, if necessary) are pulls or pushes, the diagonal or resultant will be the same. IU.M". 6. To decide whether an unknown force is a pull or a push ; that is whether the member along which said force acts is a tie or a strut. Having found the resultant of the known forces, add to it an arrow-head to show its direc- tion. Then having on this resultant completed the triangle by means of the two unknown forces, arid arrowheads to them also, placing them so that the three arrows shall chase each other around tlie triangle. Then imagine each arrow of the unknown forces to be placed one at a time without changing its direction, with its head at the apex or point of support under consideration, as if it were pushing against said point. It. in this position, it coincides witl), or covers, the corresponding member, the member is a strut. But if the arrow-head then touches that side of the apex which is opposite to the member, the member is a tie. This simple rule will save much perplexity, and should be thoroughly learned. When there is hut one unknown force, and it is found to form a straight line with the resultant of the known ones, then its arrow must point in the opposite direc- tion from that of the resultant. We may add that of the two unknown forces at any apex, one is al- ways along either the bow or the string ; and must plainly be a strut in th.e first case, and a lie in th last one. For the present we will call it the chord-force. The other unknown force will be along a web member. Now it can always be seen at once that the resultant and the chord force together tend to displace the apex at which they act, by moving it either outwards or inwards, from or towards the truss ; and if we simply consider that it is the duty of the web-member to counteract this displacing tendency, we shall have no trouble in deciding whether it must for that purpose pull or push at the apex, or in other words be a tie or a strut. RKM. 7. After having found the resultant of the known forces, the forces themselves may be con- sidered as tio longer existing. REM. 8. To find the strains correctly requires g-reat care and attention. Plain as the foregoing remarks are, the student will in his first attempts probably com- mit many errors. A little practice however 'will rectify this. A good metallic parallel ruler on rollers, and about 18 inches long, is almost indispensable. Paper ruled in squares facilitates the work. The lead-pencil must be kept sharp, and the lines should be drawn lightly'. . A scale of from % to % of an inch to a foot or ton will generally be found convenient. It will be difficult to find all the strains to within the nearest .1 or .2 of a ton, or even more; be- cause, as the work progresses errors which are inappreciable at the start may insensibly enlarge themselves. It will be seen from our table of bowstrings of 80 ft span, p 274, that the strains on some of the web members are less than .1 of a toil ; so that the diagram may even with great care mislead us to the extent of 100 per cent, or more, in these small strains. Fortunately this is a matter of little importance, for these members are so small that a libeial allowance for errors involves but a trifling waste of material. In the larger strains errors of .1 or .2 of a toft are of no consequence. Never con- aider the work of a diagram complete, however, until after testing it by some of the proofs in Art 19. Art. 19. Example. We will now apply the foregoing; re- marks to the bowstring: truss. Fif 2. Its span is 80 ft ; its rise 10 ft. The bow is divided into 8 equal parts ; and the lower apices are horizontally half-way between the upper ones. The trusses are assunied to be 7 ft apart from center to center. The total wt of the truss and its load is supposed to be equally distributed along the bow only, and to amount to 10.4 tons, which corresponds to 40 tbs per square ft of roof covering. This gives 1:3 tons for a full panel load ; and half as much, or .65 of a ton resting directly (or without passing along the web members) on each abut ; and the finding of these, and figuring them on the diagram as shown, constitute our "first process," Art 11.*? p 255. tfhis done, draw at the abut a a vert line-ow equal by scale to half the entire wt of the truss and load, minus the part panel load which rests directly upon one abut; or to 5.2 .65=4.55 tons. This line represents the vert Upward reaction of the abut against that portion of the wt of the half truss and load that causes the strains which we are about to seek.t This reacting force, which is a known one, balances the two unknown forces, ae along the bow, and ap along the string. To find these we have only ( Rem 2) to consider a v as one side of a triangle, and from its two ends to draw two meet- ing lines on and vn parallel to, or in the same directions as, the unknown forces. Then will v n give by the same scale 9.94 tons strain along the hor string ; and a n 10.93 tons, along the bow. The side an coincides with, or covers, the member ae, which is therefore (Rem 6) a strut. But if vn is placed with its arrow-head at a, without changing its direction, il"will come to the opposite side of a from its corresponding member ap, which is therefore a tie. Now let us go to the apexp. There we find that we have one known force, fthe 9.94 tons along ap) balancing three unknown ones, namely pq,pc, and p e. Hence (Rem 2) we cannot now find these last ; therefore we leave them for the present, and try at the apex e. Here we have two known forces, namely the 10:93 tons pushing along a e, and the 1.3 tons of load which of course push vert downwards. Both these being pushes, neither of them requires to be reversed ; and they balance two unknown forces, namely ec along the bow ; and ep along the oblique. Hence we have only to draw fe (see Fig W) to represent a e ; and x e to represent the 1.3 ton load, and completing the parallelogram fe xd. draw its diagonal de, which is the resultant of/c and xe ; or it alone would balance the two unknown forces. By measuring d e by scale it becomes a known force. Therefore taking it as one side of a triangle, from its two ends draw d s parallel to the bow e c, and e a parallel to the oblique *;Remember that in a truss of any form it is only when the stretches along which a load is uni- forriilv distributed. are equal, that the panel loads are also equal; or that the portion which rests di- rectly on an abut is just half & panel load. t Kach abut of course reacts vert upwards against the entire half wt of the truss and its load : but Inasmuch as that portion of said wt which rests directly on an abut, does not reach the abut by way of the web members, and therefore has nothing to do with their trains, it is omitted from the prwi- ure iet up at the abut for ascertaining said strains. TRUSSES. 273 ep, thus completing the triangle dea. Then da gives by scale the strain 10.56 along ec ; and eg gives .14 of a ton along ep.* Now going again to the apex p, we find that we have two known forces p a, p e, both pulls, balanc- ing two uniinown forces p q, p c. Therefore as at Fig Y. take pf and p a to represent the two known forces ; complete the parallelogram fpsd; draw its diagonal p d ; and taking it as one side of a tri- angle, draw d o parallel to^> q, audp o parallel toj>c. Then is do by scale 10.01 tons strain along pq\ &ndpo is .10 of a ton along pc. Going to c, we have three known forces, ce, cp, and the 1.3 ton panel load, Jill of them pushes, so that none of them need be reversed; and balancing co, cq, both unknown. In this case, as shown at Fig V, we must draw two parallelograms, begin uiug with any two of the known forces ; see Art 36, p 466. Say we begin with ex representing the 1.3 ton load, and ce, representing the 10.56 tons, on these two draw the parallelogram cetx, and its diagonal c t. Then draw ca to represent the third known force cp of .10 of a ton ; and on it and the diagonal ct draw the second parallelogram ctxa. and its di- agonal ca. This last diagonal is the resultant or single force which would balance the two unknown forces co, cq; therefore take it as one side of a triangle, and from its two ends draw two meeting lines parallel to said unknown forces, and measure them by scale to obtain the amounts of those forces. On account of the smallness of our scale we have not shown these two lines. In practice, in first diagonal. In this manner proceed until the final strains on 08, ar, and r t complete the whole. If the entire uniform wt of truss and load is assumed to be on the string or lower chord, as in Fig 23 a, then all the web mem- bers become ties ; but the process remains unchanged. Therefore first distribute the entire wt among the lower apice-* and abuts by our " first process." Then, as in Fig 23, draw the vert line o v ( half the enure wt, minus what rests directly on one abut) and from it find the strains on ae and ap. Then going to e we have one known force ae balancing the two unknown ones ec and ep. After finding these go top. Here we have three known forces, p e, p a, and the panel load ; the first two of which are pulls, while the third (resting on top of the string) is a push vert downwards. Therefore we will reverse this last, and consider it as being a vert pull pi; and draw the first par- allelogram on pi and p a. After finding the resultant of the three pulls, and by it the two forces p q, p c, we go to c. Here of the two known forces, ec, being a push ; and p c a pull, we must reverse one of them, say p c ; representing it by an arrow t c ; and complete the parallelogram on t c and c e. We have now had an instance of all the cases that occur, and have shown how to manage them. Proofs of accuracy of the work. The resultant of the strains on those members (o < and r ), Fig 23, of the half truss that meet at the center *, of the bow, and of half the panel load, 8, at the center, should come out to be a hor line, and equal to the hor .strain along the center stretch, r t, of the string. With all the care, however, that can be taken, it will be very difficult to make the coincidence exact. In some trusses the half truss will have three members meeting at the center of the bow ; one of them (a center vert one) belonging partly to each half of the truss. Then only half the strain on this one, aa well as half the center panel load, is to be used in the proof. All this applies to any form of truss however loaded. Again, if all the uniform wt of the truss and its load is as- sumed to be on the hor string, and if the string * divided into equal, or nearly equal, parts by the web members, the hor strain at the center should be equal, or about equal, to Wt of half truss and load X .25 of the span Depth of truss. But with very unequal divisions of the string, such as will rarely occur, this formula is not even roughly approx. With all the wt uniformly on the bow, unequal divisions of the string have no effect on the center hor strain ; and if the rise does not exceed about one tenth to one eighth of the span, and the bow is about evenly divided, the above formula will be nearly as approx as when the load is on the string. For greater rises, however, multiply the span by the following multipliers ^instead of by the .25 of the above formula), when the load is on the bow ; and the half bow about equally divided into at least two parts. Hise, in Farts of the Span. (Original.; .1 or less | .15 | .2 | .25 | .3 | .333 | .35 | .4 | .45 | .5 Multipliers. .246 | .243 | .239 | .233 | .226 { .222 j .219 | .211 | .202 | .192 RKM. The multipliers for intermediate rises may be taken in simple proportion. When multiplied by the span they give the hor dist from the abut to the center of gravity of one half the loaded bow, (as the .25 of the formula gives that of one half the loaded string,) assuming the load to be concentrated at the points of support on the abut and at the apices. It is this center of gravity of half load so concentrated that must be used for finding the strains in the truss, and not that of half load aa actually evenly distributed; for these two centers of gravity under these two aspects may differ greatly from each other, not only in the evenly loaded bow, but in the string, if divided into very un- equal parts by the web members. Even the number of divisions of the loaded bow makes some dif- ference in this respect, but not so great but that the multipliers in the above table will be correct to within about three per ct at most in any case in which the entire bow has at least four nearly equal divisions. * It is usual, and far more convenient, to draw all such lines at the apices of the diagram itself; but on account of the smallness of the scale of Figs 23, and 23 a, we have drawn them at W, T, and V, to prevent confusion. Also our lines are not drawn to scale because some of the forces are too small to be appreciable with so small a scale. 274 TRUSSES. If both the bow and the string are uniformly loaded, it is plain that the multiplier for any given rise must be somewhere between the .25 of the formula, and the decimal for that rise in our table; and this furnishes an easy method of finding, approx enough for practice, the hor dist from the abut to the cen of grav of a half truss thus loaded. Thus afier allotting to the bow and string their respective proportions of the entire wt of the truss and load, find the hor dist for each of the two separately ; and then combine them ; see p 442. Table of approximate strains in tons on Bowstring trusses of 80 ft span. Trusses 7 ft apart from center to center. Load (including wt of trusses) 40 Ibs per square ft of roof covering; all assumed tj be uniformly distributed on the bow. Bow divided into 8 equal parts; like Fig 23; and straight fiom apex to apex. Lower apices half-way hor between the upper ones. Each column of the table commences near an abut, or end of truss. The first or end web member in the columns is a tie, the next one a strut, and so on alternately towards the center, as in Fig 23. Below the table is given the wt of each entire truss and its load. Rise 20 ft, or y Span. Rise 13J* ft, or K Span. Rise 10 ft, or ft Span. Rise 8 ft, or y 1 ^- Span. Bow. TIB. WEB. Bow. TIE. WEB. Bow. TIB. WEB. Bow. TIE. WKB. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. 7.01 4.85 0.38 8.80 7.50 0.22 10.9 y.4 0.14 ]3.3 . 12.5 0.07 6.02 5.18 0.35 8.30 7 72 0.18 10.6 10.01 0.10 13.0 12.8 0.05 5.56 5.34 0.25 8.03 7.78 0.15 10.4 10.16 0.07 12.8 12.7 0.07 5.34 5.38 0.25 7.90 7.88 0.15 10.3 10.22 0.07 12.7 12.7 0.07 0.10 0.10 0.05 0.03 0.10 0.10 0.04 0.03 (See Note, p. 263.) Total wt = 11.6 tons. Total wt 10.75 tons. Total wt 10.375 tons. Total wt 10.25 tons. Art. 20. The braced arch, uniformly loaded each way from the center, Fig 23 W. In all the preceding trusses, as also in the left hand half Fig 23 Y, there is com- pression in the upper member, whether straight or curved, and tension in the lower one. But, as in either half of Fig 23 W, (supposing m n to be one continuous straight line ; and a gp c one continu- ous unbroken curve;) or in the right band half of Fig 23 Y. (supposing the curve e, s c to be com- plete,) the truss may be so constructed that both members shall be in compression. When this is the case, in order to be able to begin at an abut and work up the strains towards the center, we must have not only as before the vert line c v Cwhich represents the half wt of the entire truss and its load, minus the part panel load that rests directly on the abut,) but also the horizontal line h c, which represents the hor pres at the center of the truss; for when the lower member is not a tie, the abuts take the place of a tie, and react against this hor pres, as well as against the weight of the truss. When only the upper member is compressed, we have seen that it is easy to find this hor pres by either the formula or the table for that purpose in the preceding Art, under the head " Proofs of Accuracy." When both members are compressed, we must, in order to find the hor line h c, assume that all the compression at the centre is borne by only one of them ; and fiud its amount by the formula or the table just al- luded to. TRUSSES. 275 Thus, as on the left hand half of Fig 23 W, we may assume it all to be sustained at d ; or as on the right hand half, at o, between d and the supposed curve of the arch. On the right hand half of Fig 23 Y, it is all supposed to be borne at t. In Fig 23 W, the upper or bor member is then omitted from d to 1; as well us the portion from g top of the arch, unless I o be inclined downward until o coin- cides with the actual arch. In Fig 23 Y also the portion e s of the arch is, or may be, omitted. In either flg sham arches may be used to fill up the intervals, if appearance requires it. In Fig 23 Y, all to the right of y c, and to the left of n a, is considered as forming no part of the truss proper. Having found the hor strain (h c, Figs 23 W, 23 Y) at the center, and knowing the wt (v c) of the half truss and load minus the part panel load that rests directly on the abut, draw them as shown, and find their resultant r c. Then use this resultant precisely as the vert line a v, Figs 23 and 23a, is used: namely, by drawing from its ends two lines for finding the first two unknown forces; then proceed precisely as in those figs. The strains are given in Figs 23 W and 23 Y, to enable the stu- dent to work them out for himself. Proof of accuracy of the 'work. If all has been done correctly, the last resultant near the center of the truss will be in a straight line with the last member; the strain alons; which it represents. Also, the resultant of those members of the half truss which meet at the center of the truss, and of half the center panel load, should come out a hor line equal to the calcu- lated hor strain. But as in the bowstring, &c. it is almost impossible to secure a perfect agreement. It would be well to test the strains near the couter, especially the very small ones, by the principle indicated by the following remark. RKM. It is not necessary to begin at an abut in order to work out the strains: for after having disposed the load properly among the apices, and calcuhited the hor strain at the center of a truss, we may employ Haid strain, and one half of the panel load at the center, as two known forces acting against the half truss at the center; find their resultant; and resolve it along the two members of said half truss that meet there; and thus work on down to an flbut. The line representing the hor strain must evidently be drawn as if pnslii/iy nyaitixt the half truss whose strains are sought. Tllis remark ap- plies also to bowstring trusses, and others, as Figs 10 and 11, (fee., in which we have two or more known forces acting against our half truss ai the center; and not more than two unknown forces of our half truss meeting there also. Cantilevers. Suppose the half o, w, c. of the braced arch Fig 23 W, to be a trussed cur.tilever, with n and c firmly built into, or attached to a wall ; and loaded either along the top o , or otherwise. Then to calculate the strains, begin with only the load concentrated at the outer end or apex o, as a given known force, and resolve said force along o I and op; and so on to n and c taking in on the way the loads at the other apices as before. The same with cantilever d m a, Fig 23 W; or with one of the Warren. &c., truss with parallel chords; or one with both chords curved. The upper chord in all will be in tension; the lower in compression. A revolving truss drawbridge, when open, assimilates to two cantilever* joined back to back. Art. 2OJ4. This fig represents an opened swing-bridge sup- ported on rollers on the pier jj, and by tie-rods a o, at, &c. All the wt of oe is upheld by a n ; that of e s by a i, fec. ; and that of st by the rollers. Draw o ^ / T \ X'i ^^i 6 and ik vertical to represent the wts "^ x sjy^ /x/ ^/^ of of., and e s ; and draw 6 w, k n hori- liil zontaJ. Then will ow,andtn give the '"\ strains along a o, and a i. Also bin will give a hor compressive strain reaching from o to c; and kn one reaching from i to c. If this truss should be suspended from the top of the post ac instead of resting on rollers, then the tie-rod at would uphold all the wt from e. to c ; and the post would sustain the wt of the entire truss. If the truss be inverted the strains will remain the same in amount, but reversed in kind. If one-half of the Fig represent a projecting platform, with c fixed in a wall, the calculation will be as before. Art. 20%. Moving loads, and counterbraciiig. We shall merely glance at this subject in a superficial manner; without pretending to discuss the comparative merits of diff forms of bridge truss. As already stated in Remark, Art 10, counters are not needed n\ a truss uniformly loaded from end to end. The mode which we propose for finding the strains produced in beam trusses by moving loads, differs from that commonly used. It renders the principle more apparent, and is equally reliable and safe. To assist in illustrating the calculations for strains produced by moving loads, we shall suppose the following Fig 23 /, to represent the two trusses, combined into one; of a span m z of 120 feet; divided into 6 panels, each 20 ft long, and 30 ft high. Or, in other words, we (for convenience of calculating) suppose all to be borne by one truss only. The weight of this double truss is 48 tons; or 8 tons per panel ; or A ton per ft run. The floor.'and its several timbers, such a* cross girders, hor bracing, &c, which, although not portions of the truss proper, are essential to the bridge; and to be considered as so much permanent load, equally distributed along the truss, are assumed to weigh an additional '5 tons per ft run : or 33 tons per panel. Having prepared a diagram, begin by finding the strains caused in only the verts and obliques, uy the bridge itself, half the wt of the truss aione being considered to be on the top chord, or 8 tons at each lower apex, and 4 tons at each upper one. Then Oa tp 8 tons. On co, eg each 7.2 tons. " Ao, cq each 14 " "/in, cr " 21.6 " "in, ar " 26 " " im,ax " 36.0 " All of which are set down at their feet. We shall write the moving load strains near their tops ; and the total strains at their centers. We now have the strains on the web members, resulting from the truss itself, and from its uniformly distributed permanent load of floor, &c ; and are prepared to begin with the additional strains resulting from the moving load. We find those on the coun- placed uniformly along y z, its whole wt may be assumed to be concentrated at the middle point r ; and that ^ of it will be borne by the nearest abut x\ and only \ by the farthest abut m. So also with the panel length y w of cars ; its wt being supposed concentrated at q, f of it must be borne by x ; and | of it by m. Of the cars w v, are borne by x ; and -| by m. Of those along u w, f are borne by x ; and % by m ; of those ana is strained by it all ; out wnen mis enure wi readies m top ui me vc.ii, umj the portion f , as above stated, go down to ax to the abut x. The remaining % gooa * Not strictly correct; but a safe and advisable assumption in practice, when the entire wt of tbt engine oan stand on one panel. TRUSSES. 277 to the other abut m. By this process, then, we find that^ of the engine, or 30 tons ; of the cars y u>, or 14 tons ; -| of the cars wr, or 10.5 tons; f- of WM, or 7 tons; % ofws,or3.5 tons; and ^ of s i, or .44 of a ton; making 65.44 tons in all, are borne by the abut x, when the moving load is in the position shown in our fig. This load of 651 tons passes from r to a; and from a to x. The calculation is made, as s.hown-under our fig; below the heading "Engine at r." We next suppose the engine to back into the position y w ; with the train reaching to m\ and with no load between y and x. In this position the strains on q c, and cr, are greater than in any other ; and by a process precisely as before, and as shown under our fig, below the heading " Engine at 7," we find that 45.44 tons of the moving load go to the abut x ; passing from M, and then to u s ; performing each time, calculations precisely similar to the former ones : and as shown under the head- ings " Engine at o," &c. We have stated in a former Art, that a counter is most strained when the moving load extends from it to the nearest abut. Therefore, a counter hp would be most strained with the engine on v u, and the train reaching to m ; and i o, by the engine on u s, with a train to m. s n u o Span 120 ft. With Engine at p, there go to x. Tom. 18 = 3-6 engine. 7 = 2-6 v u. 3.5 =1-6 us. .44 = 1-24 B m. 28 94 ao to x MAIN BRACES. With Engine at q, there go to x. Tons. 24 = 4-6 engine. 10.5 = 3-6 w v. 7 = 2-6 v u. 3.5 = 1-6 u s. .44 = 1-24 s m. ;, With Engine at r, there go to x. Tons. 30 = 5-6 engine. 14 = 4-6 y w. 10.5 =3-6wv. 7 = 2-6 v u. 3.5 =1-6 us. 1 2 nat sec 45 44 go to x 84 728 on e q 65.44 go to x. froTYi load only. 54 528 on c r from load only. 78.528 on a x from load only. 278 TRUSSES. Vertical e p. 28.94 go to X. 18 = remaining 3-6 engine. VERTICALS. Vertical c q. 45.44 go to x. 12 = remaining 2-6 engine. See Rem, p 375. Vertical a r. 65.44 go to x. 6 = remaining 1-6 engine. 46.94 on e p from load only. 57.44 on c q from load only. 71.44 on a r /row Zoad oZy. COUNTERS. Counter i o. Engine at i. 6 =1-6 engine. .44 = 1-24 s m. 6.44 gotnx. 1.2 noi stc. 7J28 on i o Counter h p. Engine at o. 12 = 2-6 engine. 3.5 = 1-6 u s. .44 = 1-24 s m. 15.94 go to x. 1.2 not sec. 19.128 on h p from load only.* Supposing the calculations to have been made as above, we have thereby found the several loads, say 65.4; 45.4; 28.9; 15.9; and (5.4; which go to the abut a?, from the several points r, q, p, o, n, when the engine is at each of those points in succession; with a train reaching in each case to m. Each of these loads, in travelling, as it were, from its particular point, to the abut x, first ascends to the top of its own vert ; and then descends along the adjacent oblique ; producing in said oblique a strain as much greater than the load itself, as the length of the oblique w greater than its vert spread. Now, each of our obliques, as before stated, is 36 ft long, or 1.2 times as long 36 as its vert spread 30 ft, (or the height of truss;) that is, = 1.2; which 1.2 is there- fore the nat sec of the angle which any of our obliques forms with a vert line. There- fore, to find the strain which each of our moving loads produces on the oblique next to it on its way to the abut x, mult each said load by 1.2, as in the above calculations. We thus obtain for these strains, 78.5 tons on a x ; 54.5 on c r ; and 34.7 on e q : all which write near the tops of the obliques, as in our fig, in which we omit some of the decimals, to avoid crowding it. Next, as regards the strains from the moving load alone, upon the verts, we must bear in mind that these last have to sustain not only the loads just considered, which pass up them on their way to the abut x; but also the remaining portion of the wt of the engine; which also passes up them, on its way to th* farthest abut m. Tims, the end vert a r sustains the entire wt of the engine ; whereas our load of 65.4 tons before found, at r, includes only -| of it. The remaining %, or 6 tons, must there- fore be added to it, as is done under the head, " Vertical a r," making 71.4 tons strain on a r, from the moving load only. Write this near the top of a r. For the same reason, at the next vert c q, we add to the load 45.4 before found, that goes from q to a;, tha remaining |-, or 12 tons, of the engine, that go to the abut m, after ascending to c. This we have done under the head " Vertical c g," thus getting f,7.4 tons for the total moving loud-strain on c q. And so at the center vert ep, by adding 18 tons, or the remaining f of the engine, that go to aw, we get 46.9 tons total load-strain on that member; to be written near its top. The strains on the verts of the other half, p m, of the truss, are not to be calculated. They will be the same as those already found. Having now the -loud-strains written near the tops of the verts ; and * Mr De Volson Wood, the accomplished professor of Mathematics and Mechanics in the Stevens Institute of Technology, author of Treatise on Bridges and Roofs, has shown that our method for moving loads is not strictly correct, inasmuch as it makes the strains on the web members somewhat too great. It is therefore at least safe. TRUSSES. 279 the truss strains near their feet ; we have only to add these together to find the total strain on each ; namely, 97.4 tons on a r ; 71.4 on c q ; and 54.9 on ep. We have now done with the main obliques, and the verts. For the counters, we go to the other side, p m, of the truss. Beginning with the panel chop, we examine its two diags /,t r b is thereby relieved from pres to the amount of % eng. From 6 this % eng passes down 6 q as a pre ; but since b q is a tic, it thus becomes relieved of that much pull. From q. this ^ passes up the strut q c, as pull ; thus relieving it from that much pres; an4 so on alternately, until the % eng reaches m. In the same way, when the eng is at q & of it alone pass up q c, down c p, and so on to m, relieving each oblique as before; but to twic-i the extent. Arriving at q, (or in any ease, at the panel next before the center ene p.) and making the allowance for the remainder of the engine, we need go no farther; for the same result will take place on the other half of the truss when the train crosses it in the opposite direction. In practice we should recommend the omission of small considerations like this, when, M here, it conduces to safety to do o. This note is added merely to show the principle. TRUSSES. 281 moving load, is greater than the preservative tensile strain, 4.2 tons, of the truss and floor, acting on it at the same time. Therefore, dp, although a tie, the same as c jo, is liable at times to be compressed rather than pulled. Therefore, it must be so arranged as to act also as a strut ; at least so far as to bear a pres equal to the diff between the 16.73 tons of pres from the load, and the 4.2 tons of tension from the wt of the truss and floor ; or to 12.53 tons. On the next oblique o d, which is a strut, the same as c q, the moving load on o wt produces a pull of 16.73 ; while the truss and floor produce on it a pres of only 8.4 at the same time. Therefore, although it is a strut, it is liable at times to be pulled rather than compressed ; and consequently it must be made able to bear a pull also, equal at least to 16.73 8.4 = 8.33 tons. On the tie e o, the deranging compressive load strain 6.76 is less than the preservative tensile strain 16.8 of the truss and floor acting upon it at the same time. Therefore, it may remain as a tie only ; or in other words, it requires no counterbracing. When this is the case, no other oblique between it and the nearest abutment m needs counterbracing. It is almost needless to remark, that the half xp of the truss requires the same as the half mp, when the engine crosses in the opposite direction. The strains oil the chords are found as directed on p 279. In the Fink truss, Figs 26, 27, the effects of a moving load, may be calculated as for a full uniform max load from end to end. Thus assuming at first, as in the preceding cases, that everything is borne by one truss only; then, when the load is upon the top chord uf the truss, each vert post may in practice be regarded as upholding one-half of that portion of the entire wt of bridge and dis- tributed load which is between the two extreme ends of the two obliques which uphold said post. Thus, in Fig 26, the half-way post dc, bears half of all between a and b. The post m g, half of all between a and d ; the post h o, half of all between m and d.* This is equivalent to saying that the half-way post bears half the entire wt of the bridge and load : each quarter-way post, one-quarter ; each eighth-way post, one-eighth, &c, &c, of this same entire wt of bridge and load; and these consti- tute theoretically the strains on the several posts. But after having got thus far, it is necessary to examine whether some of the smaller ones may not have to be in- creased, for the following plain reason : Suppose we have assumed our max load to be a string of heavy engines, weighing 1 ton per foot run ; or, including the wt of the bridge itself, say 1.4 ton per ft ; and suppose our posts to be as close together as 5 ft ; then the least loaded posts would each bear 5 X 1.4 = 7 tons. But we know that from 16 to 20 tons may n , be concentrated within a length of 5 ft, on four drivers of an engine ; and half of it will have to be supported by each post in succession as a train passes. When we thus find by trial, which posts will be more strained by an engine than by our assumed max per ft run of the whole truss, wo must increase the load first found, correspendingly. In the Fink, and Bollmar: trusses, the verts are always struts or posts. Having fixed upon the load for each post, as p o, Fig 23 h, then for the strain which said load will produce upon each of the obliques, or ties, p c, p A, upholding said post, take any distp d on the post, to represent the load by scale; and draw dw, d r, parallel to the ties; then p w, p n, measured by the same scale, will respectively give the strains on each ; whether they be equally inclined as usual, or not. The two hor lines n a, w a, by the same scale, give the two hor forces which the load at the top o of the post, acting through the ties, produces upon the chord at c and h\ which two equal and op posing forces pro- duce along the intermediate stretch c h of chord, a strain equal to one of them. In other words, either n a, or w a, gives the hor strain produced along c h, by the load at o only. See " Strain," Art 2, of Force in Rigid Bodies, p 444. Strain on the chord. This, from a uniformly distributed load, is the same throughout the entire length of a Fink chord. To find it, observe which obliques, (as m e, g e, ot. Fig 23 1',) of one-half of the truss, terminate, at t,nr. end, e, of the chord. Then, having previously found the loads on the posts, c o, u g, t w, which pertain to * In Fig 26, the load is really on the bottom chord. But in this case also, the same loads art carried to the tops of the posts by the obliques : except at the four posts h e, h o. &c, which have no obliques at their heads. With a bottom load, these may in practice be considered to sustain each only one half-panel wt of trusi alone; but their obiiguet sustain the same load as if the train wer OQ the top chord. 282 TRUSSES. Ro-23.i those obliques, ascertain by the pro- cess in Fig 23 h, the hor forca n a, (in both figs,) which each of those loads produces on one oblique. Add together these forces n a, (there will be but three of them in Fig 23 1, as marked by the dark lines ;) their sum will be the strain along the en- tire chord. The obliques m u, u r, r c, g c, do not terminate at e; and are, therefore, omitted in finding the chord strain. The process is the same whether the verts are all of the same length or not. Or the hor chord-strain produced by each of the loads on the posts c o, u & may be calculated thus, and added together. entire load vy hor dist from post t in, Horizontal strain on post X to end e of chord twice the length of the post. In the Ho II m a ii. Figs 22, 24, 25, for a moving- load, having first pre- pared the working diagram, determine the max weight that can come upon a post. This will be the same for each post. If the moving load is on top of the truss, this load on each post will consist of the greatest wt of engine that can stand upon one panel-length of truss; together with (approximately enough for practice) the wt of one panel-length of truss, floor, &c, if the posts uphold the oblique ties ; (see Boll- man truss, in Art 17;) or one panel-length of floor; and the half of a panel-length of truss, if the posts do not uphold the ties. If the load is at the bottom of the truss, the posts bear no part of either the moving load, or of the floor; but each of them will be strained to the amount of the wt of say one panel of truss, if the posts up- hold the rods ; or of half a panel of truss, if they do not. The loads on the posts may then be written upon the diagram. The obliques or ties, however, when the load is at the bottom, bear (as in the Fink) the same amount of strain from the moving load and floor, as when it is on top. Therefore, when it is at the bottom, each pair of ties sustains not only the load rest- ing on the post which they uphold ; but the wt of one panel-length of floor, and the max panel-w r eight of engine. In other words, lohether the load be on t'>p, or at bottom, the two ties at the foot of each post, sustain a wt equal to a full panel-length of truss and floor; together with the max panel-wt of engine. Having added these wts to- gether, lay off their sum by scale at each post, as shown at 1 i\ k v,j v, i v. Fig 22 ; com- plete I g v u, k g v u, &c ; and measure the strains I u, I g , k u, k g, &c, along the ties. The strains on any pair of ties, may also be calculated thus; having the load they sustain. Strain on short tie i /? N/ hor dist from post to length of "" X farthest end of chord v short tie total length of truss deptfTof truss. 7 , hor dist from post to length of ' x nearest end of chord long tie Strain on ______ long tie. = tutai iength O f trms ' ' * depth of truss. The hor strain on the chord will be uniform throughout, as in the Fink truss ; and will depend upon the max uniform load that can cover the whole bridge; and not, as in the case of the ties, upon the greatest load which each pair of ties may have to sustain in succession; unless we assume our max uniform load to be a string of engines which may bring a max panel-wt of engine upon every pair of ties at once. In that case w r e have only to measure upon one-half of our working diagram, the several hor lines corresponding to MO, &c, in Fig 22 and their sum will be the reqd hor strain on the chord. But if we take our max uniform load on the whole truss, to be a string of cars, we must diminish the chord-strain thus found, in this manner : Add together a full panel-weight of truss, floor, and cars ; then, as the full panel-wt of truss, floor, and engine, (which we before assumed as the straining load of each pair of ties,) is to the panel-wt of truss, floor, arid cars, just found, so is the hor chord-strain before found, to the one reqd. We will repeat, that chords must be strong enough to bear not only the hor pull or push to which they are exposed ; but also to sustain safely, as beams, the trans- Terse strains from the floor, and from the moving load, when these rest upon them. TRUSSES. 282i Art. 21. If a long* beam a ft, Figs 25 and 27, requires to be strengthened, this may be done by the addition of a vert post d c; and two in- clined tie-rods c a, c 6. And if after this, the two halves, d a, d b of the beam still are found to be too weak, additional intermediate posts o, o, may be introduced ; with other ties i,j, to sustain them. In Figs 25 and 27, the roadway is at the chord a 6; no parallel lower chord being necessary, it may be omitted. The inclined ties act as substitutes for it. But if the bridge is so near the water as not to allow the posts and ties to be placed beneath the roadway a 6, we may raise the entire truss upon two posts or piers s s, Figs 24, 26 ; and place the roadway n n, at the lower ends of the posts and ties ; instead of letting it rest on top of the chord, as in Figs 25 and 27. In Figs 24 and 26, the truss and its load do not then rest directly upon the abuts y y, but upon the tops of the posts -i i , ; and the only part that s *t does rest directly on the abuts, is one-half of that small portion of the road- way comprised at each end, between e and n ; in other words, only one half the wt of the roadway of the end panels; the other half being sustained by the inclined ties which meet at e. When the tie-rods all pass from the feet of the posts to the ends of the chords, as in Figs 24 and 25, we have the Boll- man truss. And when, as in Figs 26 and 27, only those which sustain the center post d c, both pass to the ends of the chord, while the others are disposed as in said figs, the Fink truss is the result. BOLLMAN Kg Z5 Kg 27. TRUSSES. 283 Art. 22. Fig 28, shows the general arrangement of a small wooden Howe bridge-truss; Fig 29, some of its details ; and Fig 30. those of an iron truss. High trusses are sometimes made as in Fig 1. The top and bottom chords of the wooden one are each made up of three or more j- 9^ ^_ ^ Ji jvv ^b parallel timbers c c c, placed a email dist apart, to let the vert tie-rods r r pass between them. The main braces, o o, are in pairs or in threes. The pieces com- posing them, abut at top and bot- tom, against triangular angle blocks, s ; which if of hard wood, are solid ; and if of cast- iron, hollow; as shown at T, Figs 30; strengthened by inner ribs. These extend entirely across the three or more chord-pieces. Against their centers, abut also the counterbraces e. These are single pieces in small bridges ; or in pairs, in large ones ; and pass be- tween the pieces which compose a main brace. Where the wooden braces and counters cross each other, they are bolted together. For wooden chords, the angle- blocks are cast,* as at T. The dotted lines show the strengthening ribs; and x serves to keep the block in place. The vert tie-rods r r, of iron, are in pairs, threes, or fours, &c, according to size of bridge ; with a screw and nut at each end. The heads and feet of the braces and counters, butt square against the angle- blocks : and are kept in place only by the tightening of the screws of the vert ties. When the floor is below, as in Fig 28, the end posts p d ; and the ends g i and w 6, of the upper chord, may be omitted ; also i c and b y ; but it is seldom done. E M ~W In Figs 30, of an iron Howe truss, the top chord P, M, and W, is cast in one piece transversely, as at P. Its separate lengths are connected together by flanges and bolts, somewhat as shown at W ; where, a a, are cast longitudinal flanges for strength- ening the transverse bolting-flanges g. Instead of separate angle-blocks at the upper chord, solid ones may be cast in the same piece with the chord itself, as shown at M. The lower chord usually consists, as in other iron bridges, of four or more flat bars of rolled iron, c, placed on edge : and some dist apart, as at R. On top of them rest the lower angle-blocks , which have shallow channels below, for receiving the chord pieces; and thus securing them from lateral motion. A cast washer, a, below the chords, is provided with similar channels on top, for the same purpose. The braces - In large spans, to prevent the pressure of the heads and feet of the obliques from crushing the chords, the angle-blocks are cast with deep projecting Ganges under their bases ; and which, pass- ing between the pieces which compose a chord, extend to the opposite face of the chord. There the flanges bear upon broad washers at the ends of the vert rods. By this means the strains along the obliques are transferred directly to the verts, without at all affecting the chords. Angle block* of curse have openings for the passage of the vert rods. 284 TRUSSES. and counters, o, e, in moderate spans are usually cast in a star-shape, as at.;. The following table gives dimensions sufficient for a strong Howe bridge ; although in wooden bridges it is customary to add arches when the span exceeds about 150 feet. IHmensioiis for each of two trusses of a Howe bridg-e for a single-track railway. Timber not to be strained more than 800 ft>s per sq inch ; nor iron more than 5 tons per sq inch. Iron supposed to be of rather superior quality, requiring from 25 to 27 tons (60450 Ibs) per sq inch to break it. The rods to be upset at their screw-ends. To each of the two sides of each lower chord is sup- posed to be added, and firmly connected, a piece at least half as thick as one of the chord-pieces ; and as long as three panels ; at the center of the span. 9 * 1 es P-i An upper Chord. A lower Chord. An End Brace. A Center Brace. A Counter. End Rod. Center Rod. fe* I s! . o jj o| V 5| 8 SI I A i ij 3 i 525 * OS SK BO Si 02 ** CQ s 02 ** 5 ft Ins. Ins. Ins. Ins. Ins. Ins. 25 6 8 I 4X 5 i 4X10 2 4X 2 4X 5 i 4X 5 2 15-16 2 H 50 9 9 3 6X 7 i 6X10 2 6X 7 2 5X 6 i 5X 6 2 1^6 2 1 1-16 75 12 10 3 6X 9 3 6X11 2 6X 8| 2 6X 6 i 6X 6 2 1 Ji 2 1 3 16 100 13 11 I 6X10 3 6X12 2 8X 9 2 6X 8 i 6X 8 2 2 3-16 2 1 ^16 125 1H 1? 4 6X10 4 6X13 2 9X101 2 6X 9 i 6X 9! 2 2^ 2 \Y 150 21 13 4 8X10 4 8X14 8 8X10! 3 6X 8 2 6X 8 3 2 $ 3 13-16 175 ?4 H 4 10X12 4 10X15 3 8X11 3 8X 8 2 8X 8 3 2^g 3 1J4 200 27 15 4 12X12 4 12X16 3 9X12 3 8X10 2 8X10 3 w 3 1% The rods in onr table are somewhat larger than customary. The same dimensions will serve for a double road for common travel. For bridges of iron, assuming the safe strain for iron to be 5 tons per sq inch, or 14 times as great as the 800 fbs assumed for wood ; the areas of the cross-sections of the individual members will as a general rude approximation, be about one- fourteenth part as great as those of wooden ones. Eq ually strong: wooden, differ very materially in weight. Art. 23. Fig 31, anil iron bridsres of the same span, will not dif T. /SIIXXXIIXIIX PRATT shows in like manner, a wooden Pratt truss : and Fig 32, some details of a small iron one. After the foregoing, they do not need much explanation. Since the angle-blocks have to re- sist tension, instead of thrust, they ai-e placed above the top chord, and below the bottom one. The main obliques are in pairs ; and the smaller single counters pass be- tween them, as in the Howe. In large bridges they are in threes, fours, &c. The vertical posts, which, when of iron, are hollow, are retained in their positions both by the strains on the obliques, which termi- nate above and below their ends ; and by being let into the chords. In large spans, tho details generally vary more or loss from those in the figs. In Fig 31. c c c are the main braces ; and o n o the counters. TRUSSES. 285 When the roadway is below, as in Fig 31, the ends, r 6, y x, of the upper chord ; the end verticals/) and u ; and the two tension obliques in each end panel, may be omit- ted ; and two diagonal struts from 6 and y must then be substituted, extending to the abutments, for upholding the upper chord, &c. Ill tile Pratt the chords may be of the same dimensions as in the foregoing table for Howe's. The posts may have about ^th less area than the main braces of the Howe. The main brace rods, and others, (of the same number as the main brace pieces of the Howe,) may hava the following diams in ins ; allowing the safe strain to be five tons per sq inch. For each Truss of a Pratt Bridge. 25 Ft. 50 Ft. 75 Ft. Spi 100 Ft. ins. 125 Ft. 150 Ft. 175 Ft. 200 Ft. End Main-brace Hods. Ins. 3 of 2% Ins. 3 of 3Ji 2 of 1% 2 of IX 2 of 2^ 2 of 2J^ 2 of 1% 3 of IX Center Main-brace Hods. 2 of 1 2<>flfV -1& 2 of IX 3 of l T 5 ff 3or, 3 of 1% Ins. 1 of ly^" Ins. loflH Cc Ins. >unter Ro< Ins. Iof2 IB at Cent las. 3r. 2 of IX Ins. 2 of 1-j-g- 2 of lyf In Pratt's truss the directions of the main braces and counters are respectively the reverse of the Howe. Many of the remarks in the preceding Art, apply equally in this. Neither the Howe nor the Pratt possesses any special advantage over the other as regards ease of adjustment, &c. In both trusses, arches are frequently added in wooden railroad bridges when the span exceeds about 150 ft.* Art. 24. Town's lattice truss, Fig 33, as originally introduced, and very extensively employed, was of extremely simple construction ; being composed en- tirely of planks from 2 to 3 ins thick; and from 9 to 12 wide; de- t t ll V i pending on the span. Two sets - . ' of these were placed crossing each other at angles of about 90; and were connected to- gether at their intersections by either 2 or 4 treenails of locust, or other hard wood, about 2 ins diam. At the top and bot- tom, similar planks, a a, c c, were treenailed hor, to form the chords ; or in large spans, (many exceeded 150 ft,) there were two upper and two lower chords, (sometimes of timber 6 ins thick,) as shown in the fig, by n w, o o. The transverse section A shows the two upper chords on a larger scale ; each chord consisting of two planks ; one on each side of the lattices. Two trusses of this kind, with a depth equal to % or ^ of any clear span not exceeding about 175 ft ; planks of 3 X 12 white pine ; the open squares 2^ ft on a side, in the clear, were con- sidered sufficient for a common-road bridge 20 ft wide. Many of these bridges warped sidewaya very badly ; and when applied to railroad purposes, failed entirely. In some cases the better mode of three lattices was employed; two of them running in one direction ; and the third in the other direction, passing between them. A funda- mental defect was that the parts were of equal size throughout the span ; whereas the chords should be stoutest at the center ; and the lattices, near the ends. These de- fects caused the lattice to fall into unmerited neglect, in the United States ; whereas * The Baltimore Bridge Co (Smith, Latrobe, Ac) have built a wooden Howe of two trusses, of 300 ft span. 30 high, 26 wide, from center to center of chords, without any arch. It has a wrought-iron lower chord, and is prooortioried for a moving load of but 1000 Ibs per ft run. 19 e x sy LATTICE Pig 33 286 TRUSSES. in Europe it is, when properly proportioned, highly esteemed ; especially for iron bridges ; some of which, on this principle, have been constructed of more than 300 ft span. Another detect was lateral weakness ; or a tendency to warp sideways, owing to the thinness of the trusses. This is obviated in the large bridges referred to, by placing a double truss, Fig 34, (instead of a single one,) at each side of the bridge. The trusses T, D, composing a double one, are placed a foot or more apart ; and are connected to- gether at proper intervals, by short pieces riveted to each one, for stiffening them. At T and D are seen the three lattices or lattice-bars, of each truss ; two of which, on the outside, constitute the main braces; while the center one is thecoun- terbrace. Such a double truss bears some resemblance to the Fairbairn box-girder; the diif being chiefly that in the former the sides are composed of lattice-bars ; and in the latter of solid plates. The Bowstring 1 truss, Fig 35, is an excellent one as regards strength, and economy of construction. It has, however, the disadvantage, in large spans, of a diffi- culty in connecting together overhead the two trusses of a span, so as to be as free from lateral vibration as a bridge with parallel upper and lower chords. In the latter, this con- nection can be made from end to end of the span ; but in the Bow- string it can be done only for some distance each way from the center; from want of headway near the ends. In short spans with low trusses, this defect is not felt. The verts and obliques shown in the fig, may all be struts, or all ties; or either a Howe, a Pratt, or a Warren ar- rangement. The pnll along: the chord y y in a Bowstring truss, is uniform throughout its length ; and is the same in amount as the hor pres at the center or crown, 4, of the arch. Either may be found thus ; remembering that in our calculations of bridge trusses, we assume at first, as in previous exam- ples, that all the weight of the bridge and its load is sustained by one truss only ; so that the resulting strains must finally be divided by 2, to distribute them among the two actual trusses : * BOWSTRING Hor pres at crown; or pull on chord ' half the united wt o/ v ^ bridge and max load x ^ span rise measured between the half-depths of the arch itself, at its foot and crown. Which is the same as for the Howe, and other beam-trusses ; but not for Fink's, and Bollman's ; in both of which it is greater. If the chord has also to act as a beam for supporting cross-floor girders, it must be made stronger. This may be avoided by placing such girders only close to the verts. For the pres along: the direction of the arch, at its foot, o 9 square half the united wt of the bridge and its max load. Also square the hor press at crown just found. Add these two squares together. Take the sq rt of their sum. This is the same process as for a stone arch. The pres does not increase regularly from the crown to the foot ; but in bridge practice it may safely be assumed to do so. Strains on the obliqnes. Whether these (Fig 35) are all struts, or all ties, the strains upon them, according to Prof. Kankine, remain unaltered; and that along any one of them is found as follows, using the numbers of the two verts betAvcen which it is situated, after having first numbered the verts &\ong one-half of the span, as at 1, 2, 3, 4, in Fig 35. The moving load is supposed to be uniform ; and hence the Professor's formulas do not allow for the extra load concentrated upon the driv- ing-wheels of an engine. Strain along any oblique weight of one No of one Jfnnth of ^ ^-tvo oj (me , ?f^ X the oblique\^f __ ^ load J \ twice the 9 'etch of the oblique ^ ^ panels in i JVo. of other- vert vert stretch of the obliq number of one truss. * This and what follows on Bowstring trusses is deduced from Rankine's " Ciy Eng," as the writer understand* it, (edition of 1862, pages 483, 563, to:) not having himself studied this form of truss. TRUSSES. 287 Ex. What is the strain, along the oblique e c, Fig 35, situated between verts No. 2 and No. 3 ; the weight of a panel-length of the moving load alone being 10 tons : the length e c, 12% ft ; and its vert stretch c n, 7.5 ft? There are 8 panels in one truss. Here, (ll^JH) X (^- ? ) - 16.7 X .375 = 6.26 tons, Ans; or ~ = 3.13 tons, for the oblique ec of each of the two actual trusses of a span. Strains oil the verticals. When the arrangement of verts and obliques is as in Fig i>5, then the strains on the verts will depend considerably upon whether the obliques are all struts, or all ties. In either case, however, they will be greatest near the center of the truss, and least near its ends ; resembling in this respect the roofs Figs 14, 15, and 16; but the reverse of the beam truss, such as a Howe, Pratt, or Warren. In the Bowstring, neither the verts nor the obliques sustain any portion of the bow or arch itself; but on the contrary, they are all upheld by it. "When the obliques are all struts, first find the wt of one panel-length of the bridge and its m;tx load ; and from this take the weight of one panel-length of the arch or bow itself. We will call the remainder, the "reduced total panel- weight;" then the No. of vert X next Jess No.\ Strain on ( reduced total \ v .. , any vert = I panel-weight ) X 11+ twice the number of panels * in one truss. .\ \ ) / Ex. What is the greatest strain that can come upon vert No. 3, Fig 35, the weight of a reduced total panel-weight being 13 tons ? Here 13 X l + - = 13 X l + = 13 X 1.375 = 17.875 tons. Or, '- = 8.9375 tons on the vert No. 3 of each of the two actual trusses. But if the obliques in Figr 35 are all ties, the greatest strain upon any one of the verts, is simply equal to one reduced total panel-weight. Proi. Rankine also says that in some cases, under a moving load, if the obliques in Fig 35 are tie;*, some of the verts may have to act at times as struts, as well as ties. To ascertain if this is the case with any given vert, use the following : /No. of vert X next greater No. ^ , w t of one panel of \ per panel * ( twice the number of pandg ) ( bridge only, ) \ in one truss / \without the arch itself./ If the result is minus, the vert will act as a tie only ; but if the result is either or plus, it will show to what extent the vert may havu to act as a strut also. When a Bowstring* truss has no obliques, but only verts, the arch being supposed to be sufficiently stiff in itself to resist change of form by a moving load, the greatest strain on each vert from a uniform moving load, is equal to a re- duced total panel-weight; but in practice we must allow for the greatest load that can come upon a panel. The proper shape for the arch, under a uniform quiescent load, is a parabola; but when the rise does not exceed about % of the span, (which it should not,) a circular segment is sufficiently approximate for common practica Under a uniform quiescent load, obliques are not needed. For an arch trussed as in Fig- 35%, with vertical struts, and with ob* liques all struts, or all ties ; the strains on the arch and obliques may, according 1 2 3 A- "vr to Prof. Rankine, be found by the rules just given for the Bowstring. The upper piece, ow t need only be strong enough to bear the load upon it. When the obliques are all ties, the verts will always be struts or posts; and the strains on any one of them will be given by the foregoing formula for any vert of a Bowstring. When the obliques are all struts, the strain on each vert is equal to the redueed total panel-weight ; and in this case it is possible that some of the verts may at times during the passage of a moving load, have to act as ties, as well as posts. To ascertain if this is the case with any given vert, use the last preceding formula. If th result is plus, it gives the pull or tension which the vert may have to resist as a tie; but if it is either or minus, the vert acts as a post only. Prof. Rankine says that the dotted obliques in the end panels are not absolutely necessary, 288 TRUSSES. as the formula will show ; but that it is still well to insert them, for the sake of greater stiffness. The Lock Ken viaduct, England, of 130 ft clear span, and 18 ft high, has two trusses, 13 ft 8 ins apart clear, for single-track railroad, on the Bowstring principle, Fig 35, omitting only the verts; which, however, affects the strains on the obliques. For convenience of construction, it was built chiefly of rolled channel- iron, (see tt,) of 8 ins by 4, by 4, by % inch. At Fig 35, d shows a transverse section of the arch or bow; which is uniform throughout. That of the uniform chord or string is of the same fig and size as the arch. Each has an area of 33 sq ins in each truss. The top strip, a, a, of rolled iron, is 24 ins by %; and is riveted to the upper flanges of the two channel-irons 1 1 ; which are 8 ins apart, so as j ust to allow the passage between them of the obliques d ; which also are of the same channel-iron. The panels are about 12 ft long near the center of a truss ; and 8 ft at its ends. The wt of the two trusses alone, without the roadway, is very nearly 50 tons. The wt was increased beyond the theoretical requirements, to save the trouble and expense of preparing and fitting together many pieces of diff dimensions; yet, although the bridge is a strong one, the trusses alone, weigh together but .37 of a ton per foot run. Where two obliques cross, one is in two pieces, riveted to the other by straps. In the State of New York are many much -used bridges for common travel, by Mr. Whipple, of 100 feet span, and 12% ft rise ; with two trusses 19 ft apart from center to center ; for two roadways : and having two outside footways, each 6 ft wide. Each truss has 9 panels, braced altogether by vert and oblique tie- rods (no struts) arranged as in Fig 35. The verts next the center of each truss, consist each of two rods of 1% ins diam, welded together at top ; and straddling 2 ft at bottom. The other verts are single, and 2 ins diam. The obliques are all single, and l^ins diam. The arches are of cast-iron. The transverse area of metal of each arch is!8sq ins at the crown ; and 21 sq ins at the spring. The shape of arch transversely resembles a channel- iron with its back upward; the total depth of flange 7 ins ; the width of arch on top, 11 ins at center of span; but increasing uniformly by means of wide open-work on top, to 3 ft at springs. Each consists of 9 straight segments, held together at their but- ting flanges, by the verts themselves ; which pass through them, and have screws and nuts at their ends. The screw-ends are not upset. The thickness of metal in the arches nowhere varies much from % inch. Under the floor, and between the trusses, are horizontal diagonal braces of rods % inch diam ; two of them to each panel ; each of them with a tightening swivel. The chord of each arch consists of 4 rods of 2 ins diam. In the same State, are also many similar common road bridges of 72 ft span. Rise 9 ft ; two trusses, 19 ft apart from center to center ; * and two outside footways of 6 ft each in addition. Each truss has 7 panels, with vert and oblique ties, as in Fig 35. Each cast-iron arch is in 7 straight segments, of the same shape as the foregoing; with a cross-area of metal of about 12 and 15 sq ins. Its width at center of truss 10 ins: at springs, 30 ins. The two verts next the center of each truss, consist each of 2 rods of 1% diam ; the other verts are single, each 1% diam. The obliques are all single, 1 inch diam. The chord or string of each arch, is 4 rods of 1% inch diam. Horizontal diag bracing of 9 inch rods under the floor, as in the foregoing. Some cast-iron bridges of the Severn Valley Railroad, England, of 200 ft clear span, consist of arches rising 20 ft, and supporting the railroad on a level with the tops of the arches, instead of above, them as in Fig 35%. There are no diags between the arches and the roadway, as in that fig; but cast- iron verts only, placed 4 ft apart. The railroad is double track ; and there are four arches, one under each line of rails. The transverse section of an arch is I ; each flange is lr>% ins wide, by 2 ins deep ; the web is 2 ins thick. Total depth at center of span, 4 ft; and at the skewbacks, 4 ft 9 ins Transverse area of each rib at crown, 150 sq iqs. Each arch is cast in 9 segments of equal length. The cast-iron bridge across the Schiiylkill at Chestnut St, Phila. Strickland Kneass, Esq, Engineer, roadway on top, has two arches of 18ift clear *pan each, and 20 ft rise. Clear width, 42 ft. Each arch has fi ribs, about 8 ft apart in the clear; and of the uniform depth of 4 feet, including a hor top rib 8 ins wide ; and a similar one at the bottom. Thickness everywhere 2% ins ; thus giving to each rib a transverse area of 147% sq ins. The standards are vert, with ornamenta- tion. It is a city street bridge. The roadway consists of cast-iron plates, which sup- port a pavement of cubi'-al blocks of granite, laid in gravel. The arches are cast in segments 12 ft 10 ins long; each with end flanges 12 ins wide, for bolting them to- gether with four 1% inch diam screw-bolts at each end. For a change of tempera- ture from 12 to 99 Fah. the crowns of the arches rise 2j^ ins. Under a uniform extraneous load of 100 fos per sq foot, the greatest pres on the arches is but 36UO R>s * With only two trusses, the width between them, in the clear, should not be less than 16 ft. to allow two ordinary vehicles to pass each other readily ; but 18 or 20 ft is still better; more would be unnecessary when there are outside footways. The headway should not be lest than 13 ft. TRUSSES. 289 per sq Inch of their cross-section ; or not more than ^y of the ultimate crushing strength of average cast iron, in short blocks. The Moseley Bridge. Figs 35%, by Thos. W. H. Moseley, of Kentucky, is essentially a wrought-iron Bowstring, with a hollow plate-iron arch of triangular cross-section, apex up; and formed of three plates riveted together; the two side- plates, a 6, ac, having their top and bottom edges bent to form flanges for this purpose. The chords o e ; the verts ; and the two counter- arches tt\ are also of iron. These counter- arches are intended as a substitute for the ob- liques of Fig 35. Each of them consists of two angle-irons, back to back, riveted together, and to the verticals, which pass between them. Each of them has a sectional area equal to half that of the main arch. The verticals are placed about 2 ft apart. They are flat (not square) bars, for convenience of riveting. They pass through holes in the bottom-plate of the main arch, (see dotted line of top Fig,) and are fastened at a by the same rivets which connect the upper flanges of the two side-plates, ab, ac. The chords, oe, are also flat bars ; and have a transverse area half as great as that of the main arches. At their ends they are attached to strong wrought-iron shoes upon which the feet of the main arches rest and abut. The rise of the main arches, measured to the bottom of the arch, is ^ or y 1 ^ of the clear span. The following are the principal dimensions of a single track bridge of 93 ft clear span, (97 ft from out to out of arch,) carrying the " Iron Railroad " at Ironton, Ohio. It was built in I860, and is traversed by heavy engines with trains of pig iron, coal, &c. Rise to bottom of arch 10% ft; to middle of arch 11 ft. Bottom-plate ss of arch, 16% ins, by .44 inch. Side-plates a 6, a c, in clear of flanges, 14% ins, by .29 inch. Top flanges at a. each 3 ins. Bottom flanges b and c, each 1.88 ins. Vert rods, 3 ins, by % inch ; and 2 ft apart from center to center. The chord of each arch consists of two flat bars, each of 4% sq ins of cross-section. The bridge was tested for three weeks by a dead load of y^ a ton ; and a rolling load of 1 ton at the same time, per foot run ; and deflected only % inch. With a load of 1 ton per foot, the pressure at the center of the arches would be about full 4 tons per sq inch of metal ; and a trifle more at the feet. The sectional area of metal in each main arch is 18% S( l ins- These bridges are of easy construction, and consequently cheap. For long spans, vert diagonal bracing (see Fig 35) would probably be essential for preserving the form, of the arches under heavy moving loads.* An iron arch roof in Philadelphia, clear span 80 ft, rise 16ft, con- sists of a uniform arch of single 7-inch Phosnix beam of 6 sq ins sectional area ; weigh- ing '20 Ibs per foot. This rests on cast-iron shoes on the walls. The hor chord or tie at the feet, is of two rods of 1% diam. At the center of this tie is an arrangement similar to No. 15, of Figs 21%, from which diverge upward, to the arch, a central vert rod 1 inch diam ; two struts of 6-inch Phoenix beam, 20 ft apart at the arch ; and two ties each 1% diam, reaching the arch at half-way between the struts and the feet of the arch. There are 10 such trusses, each of which by itself weighs about 3900 fibs; they are placed 16 ft apart; and by means of purlins resting upon them, support the entire weight of the roof, which is of inch boards, covered by thin sheet-iron. The iron roof of a rolling-mill near Boston, Mass, of about 80 ft tpan.and 16 ft rise, has arches of the Moseley section a, />, c. Fig 35%; but without counter arches. The trusses are 12 ft apart. Sides of the arches, clear of the flanges, 7 ins ; upper flanges, 2 ins ; lower ones. 1 inch ; total of each side, 10 ins, by .19 inch thick. Bottom plate 8% inch by^ inch thick. Total sectional area of an arch, 5.925 sq ins. There are besides, a chord; and 24 vert suspending rods ; but no obliques. The roof is covered with corrugated iron, on purlins. When required, the heavy iron rolls of the mill are lifted by tackle supported by a roof-truss. Figs 36, represent the Burr truss; which was formerly more used * Although this bridge seems to hare stood very well for several years, the writer would prefer not to exceed two tons of compressive strain per sq inch on plate-iron in such structures. In several bridges, General Moseley has used a continuous web of % inch iron, instead of the vert suspenders. But in such cases the triangular tube is not applicable for the arch ; and he substitutes two Z bars, riveted together, and to the web, w, which passes between them, as in the foregoing fig. The counter arches being here unnecessary, are omitted. This web would be objectionable in large spans, espe- cially of draw-bridges, on account of the wind. More recently still, he has also introduced lattices, instead of vert bars, in some of his bridges ; together with many innovations on the arrange- ment first described. At 1 ton per ft run, the pull on the chord above, = 8 tons per sq in. 290 TRUSSES. than any other in the United States. It is at present regarded with disfavor by soma, because many early ones tailed under railroad traffic, in consequence of bad propor- tions, and the absence (as in our Fig 3G) of couriterbracing. When properly con- structed it makes an excellent bridge. The common objection to it, and not without reason, is that a truss and an arch cannot be so combined as to act entirely in con- cert; yet, as soon as any ordinary truss begins to fail, the almost invariable remedy is to add an arch. When, however, the two are to be united, it is better to so pro- portion the arch as to be capable by itself of safely sustaining the max load at rest; LJ and to confine the duty of the truss to preventing the arch from changing its form under a moving load. Counterbracing may be effected by strapping the heads and feet of the braces to the chords; or by iron rods parallel to the braces; two to a brace ; with screws and nuts, as at v f. Or by similar rods across the other diags of the panels. The following 1 dimensions answer lor a single-track rail- road bridge of about 150 feet span. Rise from out to out of chords *% of the span ; about fourteen or sixteen panels. Width in clear of arches, 13 ft. Six arch- pieces t, of 8" X 1^" each, to each truss. Upper chord c, 12" X 14". Lower chord a a, two pieces each 8" X !*" Posts ;>, 12" transversely of the bridge, as in the right- hand fig; by 8"; except at the heads and feet, where enlarged to receive the ends of the braces. Braces 8" deep, by 12" wide. Floor girders o, 8" X W \ and 2% to 3 ft apart from center to center. Suspending rods (shown at s; and dotted in x) 1% diam. Counterbrace rods in pairs parallel to braces, about 1V" diam. Bolts for arches, lower chords, &c, \%" diam. Theoretically, the posts, braces, and arches should gradually diminish from the ends, toward the center of the truss ; while the chords should increase ; but in practice, the additional labor of getting out and fitting pieces of diff sizes, frequently makes it more economical to use uniform sizes.* The same amount of arch would answer also, if trussed, as in Kig 35 ; and the arch by itself with a full max load, would be strained less than 800 Ibs per sq inch, at its feet. For a span of 200 ft, with the same proportion of rise, the transverse areas of the several arch and truss pieces should be increased 33 per cent ; and for one of 100 ft, they may be diminished the same. None of these dimensions are the result of close calculation. The dimensions just given will answer for common travel, for a span of 200 ft ; with a depth from out to out of chords, of % the span ; panels 10 to 12 ft long. Many such spans have been built with timbers of about ^ less transverse section; and without counterbracing. The heads of the posts are notched about 2" to 3" into the bottom of the upper chords; and are moreover tenoned into it some ins further,' with two wooden pins through the tenon; see w, Figs 36. Their feet are notched both into and upon the lower chords, so as to leave the two chord-pieces a a only about 2" apart. Through these and the post pass two bolts of about 1" to 1V diam. Since the upper chord resists compression only, its pieces may come together with a plain butt joint, d. To this may be added fishes e d, of stout plank, on the sides of the chord, bolted through by 4 or 8 bolts. The lower chords resist pull; and the pieces composing each lower chord must therefore be joined together somewhat as in Fig 37, &c. These pieces should be as * It will be borne tn mind that our examples are not intended to illnstrate perfectly proportioned structures. None of them would endure strict criticism. There is more waste of timber in an arch built with a uniform transverse section throughout, than in the straight upper chord of a Howe or Pratt similarly built; for both must be proportioned to the greatest strain. This is at the center of the two last; but while that at the center of the arch is as great as ia these, that at its feet ia muoh greater. See Example 2, Art 33, of Force ia Rigid Bodies, p 468. TRUSSES. 291 long as possible, and should never be jointed opposite to each other, but one opposite the middle of the other ; see Fig 39. The braces are merely cut to fit to the heads and feet of the posts, after these last have been fixed in their places ; and usually have no other connection to them than one or two spikes at each end, for small bridges ; or screw-bolts for large ones. The ends of the timbers composing the arches, butt full square against each other; and may also have a wooden dowel. The joints should occur at the posts, as shown at W. The arches are screw-bolted to the posts, as shown in the figs, by bolts of 1 to 1*4 ins diam. Where the arches pass the lower chord, both are notched, and well bolted together. The feet of the arches abut against cast-iron plates. When suspension rods (dotted in Fig 36) are used for assisting to support the road- way, they are placed as shown at s s ; m being a strong block of wood slightly notched on top of the upper arch-pieces. The rods are suspended by a washer and nut on top of the block ; and after passing down between the arches and chord, have a similar arrangement, but inverted, below the last; as shown at g. Floor-girders not exceeding 14 ft clear span, may be 8 ins, by 15 ins deep; and placed not more than about *2% ft apart from center to center. Upon them should be notched and spiked stout string-pieces, say 12" wide, by 9" deep, to carry the rails; and to distribute the pressure of the load. Ilor diagr bracing: for diminishing lateral motion, can be used only under the floors of low bridges ; but in high ones it is introduced also at the top of the trusses. When of timber, these braces are about 4 to b inches thick ; by 6 to 9 deep ; and form a hor cross between each two opposite panels of the two trusses. If the bridge is roofed, and has girders r, Figs 36, 06^, upon and well secured to the upper chords c c, the upper lateral bracing may consist simply of 4 iron rods n w, passing through the chords about midway of their depth ; and having heads and washers on their outer sides. At the center of the cross the rods terminate in an adjusting-ring ; see No 14, of Figs '21%. In a bridge of l. ; >0 ft span, these rods need not exceed % inch diam at the center panel, and 1% at the end ones. If the bridge is high, and not roofed, but open at top, then cross-struts r r, Figs 36%, must be inserted pur- posely, when this rod-bracing is used. If it is also used at the lower chords, the floor- girders perform the duty of these strutsT Iron-bracing is not liable to catch tire from the locomotives. Hor diag bracing is also called sway bracing. A favorite mode of lateral bracing-. W, Figs 3C%. resembles a Howe truss laid flat on its side. In it the diags of the cross are struts of timber; and the pieces r r are round rods. One of the struts is whole, with the exception of a slight mortice on each vert side, at its center, for receiving tenons cut on the inner ends of the two pieces which compose the other diag. At the sides of the chords, the ends of the diags rest upon a ledge, (shown by the dotted line i i,) about 1% ins wide, cast at the bottom of the cast-iron angle-block. The tie-rod r r, passing through the chords of both trusses, being tightened by means of the nut s, holds the diags firmly in place ; and in case of their shrinking a little in time, can be again tightened up by the same means. Various modifications of these methods are in use ; but we cannot afford them space here. The cast angle-block is as deep as a brace ; its thickness need not exceed y 2 inch, in a large bridge. The dark triangle if a top view of it. It has holes for the passage of the rod r r. We have entered somewhat into the details of the Burr truss, because many of them are more or less applicable to others. A good light water-and-fire-proof ma- terial for bridge-roofs ; and one not corroded by the smoke from the engines, is yet a desideratum. Art. 25. Lengthenings-scarfs, splices, or Joints. The lower chords of bridges, being exposed to great pulling strains, require much care in connecting together the ends of the several pieces of which they are composed. There is much uncertainty regarding the strength of the joint-fastenings in common use for this purpose. Experiments on the subject are much needed. When only two pieces, as t and y, Fig 37, or 38, are joined by any of the ordinary methods, it is probably not safe to depend on their possessing more than % of the tensile strength of a s'ingle solid beam of equal cross-section. When the chord, as in Fig 39, is composed of two parallel parts a a, n n, made up of long pieces, breaking joint with each other, as at jjj.eB.ch of the two parts may be made somewhat stronger than either one of them would be by itielf. This is owing to the opportunity afforded of connecting if 292 TRUSSES. them also by bolts b b, and packing-blocks, cc, of wood or iron, intermediate of tha joints jj t &c. By this means the strength of the entire chord may probably be prac- tically rendered equal to one-half of what it would be if solid. If the chord con- sists of 3 or 4 parallel parts, of long pieces, breaking joint, and connected in thtf same way, it will probably have about % of the strength of the solid. Care must- of course be taken that the serviceable area of the pieces shall not be reduced at any intermediate point, to less than it is at the joints. TOP Pig 38 is a simple and efficient form of scarf. Its length i i may be about 3 to 4 times the greatest transverse dimension of the beam. At the center is a block t of hard wood, with a thickness equal to % that of the beam ; a width of 2 or 3 times its thickness; and a length just sufficient to reach entirely through the beam. The beams are connected by 4 screw-bolts nn ; or by 8 of them, if the length requires it. Plates of stout rolled iron, a a, cc, with their ends bent down into the beams,' are occasionally added. These require bolts o o, beyond the ends i i of the scarf. These bolts are not shown in the side view. Fig 37 is another excellent joint with splicing -bJ-ocks e e, instead of the block t of Fig 38. The indentations v v, may each be about Y& as deep as the beam is thick. The length of each splice-block, about 6 times s s. From 4 to 8 screw-bolts, as the case may require. Length of each indent about ^ that of the block itself. Fig 40 is a joint formed by two flat iron links or rings 1 1, let flush into the tim- bers, and retained in place by spikes. The iron may vary from \ to 1 inch in thick- ness ; from 1 to 4 or 5 ins in width ; and 2 to 6 ft in length ; as occasion may require. Fig 40 6, is a joint formed by two blocks c c, of hard wood, passing through the timbers ; and connected by bolts, a a, n n. In Fig 40 a, s s are cast-iron packing-blocks, sometimes used instead of plain wooden ones, at points 6 &, Fig 39, intermediate of the joints jj. The openings in the centers of the blocks are needed only when vertical truss-rods have to pass through those points. At e e, of the same fig, is shown another form, much used in chords composed of two or more parallel strings. Both these are as deep as the chord; and their cross-sections, or end views shown in the fig, may be from 4 to 10 ins long: 2 to 4 ins wide; and from % to 1% ins thick; according to size of bridge, Ac. REM. In selecting hard wood for splicing-blocks, treenails, or for any part of a bridge, it is well to remember that the oaks when in contact with the pines, expedite the decay of the latter; therefore, it is generally better to employ the best southern yellow pine heart wood for such blocks, &c, or interpose sheet iron.* * The tendency of some kinds of timber to produce rapid decay when brought into close contact TRUSSES. 293 Eye-Bars and Pins. The lower chords of iron bridges usually ousist of flat links or bars c and o, W and H, Figs 41, on edge and connected by tight-fitting wrought- iron pins b and P. After deciding on the size of the body W or H of the bars to bear safely the pull upon them, the proper proportioning of their heads or eyes and pins is an abstruse and difficult point upou which much has been written. It was formerly supposed that the diam of the pin should be governed by its resistance to shearing, but experience has shown that this was entirely insufficient. n Pigs 41. J s We give a table of practical conclusions arrived at by that accomplished expert, Chs. Shaler Smith, , . , from 111 experiments by himself on a working scale. The table shews some irregularities, for as Mr. Smith remarks " the bars declined to break by formula." The pin is more strained at the outer links o o than at the inner ones c c c, so that the latter would not require so large a diam, but that this must be uniform throughout in order to secure tight fitting for all of them. When web members as well as chords are held by the same pin the diara and head must be proportioned for that bar of them all which is most strained. When the heads are made by pressure in one piece with the body, the metal v a and u x at the sides of the pin b must be wider than when the heads are first made in sepa- rate pieces by hammering and then welded to the body. But the welded one W requires more iron back of the pin as shown at 1 1. This width 1 1 must be equal to the diam of the pin. The links are supposed to be of uniform thickness. Having drawn a circle b for the pin, lay off on each side of it as at v s, u x, half the width of metal in the table for the head of W or H as the case may be. Then for forming the head of H use only the rad b s as shown. For the head of W lay off also 1 1 = diam of pin. Find by trial the rad g n or g t and use it, except for uniting the head to the body, where use a rad = 1.5 g n as shown. Metal in head Metal in bead Width Thicks. Diam. across pin. Width Thicks. Diam. across pin. of bar. of bar. of pin. W. H. of bar. of bar. of pin. W. H. I. .2 .67 1.33 1.50 1. .55 1.28 1.50 1.60 1. .25 .77 1.33 1.50 1. .60 1.36 1.55 1.72 1. .30 .86 1.40 1.50 1. .65 1.43 1.60 1.76 1. .35 .95 1.50 1.50 1. .70 1.50 1.67 1.85 I. .40 1.04 1.50 1.50 1. .80 1.64 1.67 1.95 1. .45 1.12 1.50 1.53 1. .90 177 1.70 2.05 1. .50 1.20 1.50 1.56 1. 1.00 1.90 1.76 2.21 Art. 26. Figs 42, exhibit joints adapted to most of the cases that occur in practice with wooden beams, &c. They need but little explanation. Fig a is a good mode of splicing a post ; in doing which the line o o should never be in- clined or sloped, but be made vert; otherwise, in case of shrinkage, or of great pressure, the parts on each side of it tend to slide along each other, and thus bring a great strain upon the bolts. When greater strength is reqd, iron hoops may be i used, as at 6, h. and.;, instead of bolts. Fig ft, a post spliced by 4 fishing pieces: ! which may be fastened either by bolts, as in the upper part ; or by hoops, as in the i lower. The hoops may be tightened by flanges and screws, as at s ; or thin iron wedges may be driven between them and the timbers, if necessary. Fig C shows a , good strong arrangement for uniting a straining-beam Ar, a rafter , and a queen-post u ; by letting k and I abut against each other, and confining them between a double queen-post 1 1 ; n n are two blocks through which the bolts pass. A similar arrange- ment is equally good for uniting the tie-beam w, with the foot u, of the queens ; with the addition of a strap, as in the fig. Fig ? is a method of framing one beam into ' another, at right angles to it. An iron stirrup, as at/, may be used for the i same purpose ; and is stronger. Figs g h, ij are built beams. When a beam I or girder of great depth is required, if we obtain it by merely laying one beam flat with other kinds, is a subject of great practical importance; but one which hitherto has received but little attention. Black walnut and cypress are said to cause mutual rot within a year or two. Ob- served cases of this kind should be reported to the leading professional journals. 294 TRUSSES. u TRUSSES. 295 ^ NS then bolt or strap them firmly together to create friction ; we obtain nearly the strength of a solid beam of the tolal depth ; which strength it, as the square ot the depth. Tne strength of a built beam is increased by increasing its depth at its center, where it is most strained ; as in the upper chords of a bridge. This may be done by adding the triangular strip y y between the two beams. Tredgold directs that the combined thicknesses of all the keys be not less than 1.4 times the entire depth of the girder; or when indents are used, as in ij. that their combined depths be at least % that of the girder. If the girder ij be inverted, it will lose mnch of its strength. A piece of plate-iron may be placed at the joints of timbers wneu there is a great pressure ; which is thus more equalized over the entire area of the joint; or cast iron may be used. Frequently a simple strap will not suffice, when it is necessary to draw the two timbers very tightly together. In such cases, one end of each strap may, as at x, terminate as a screw ; and after passing through a cross-bar Z, all may be tightened up by a nut at x. Or the principle of the DOU- BLE KEY, shown at K, may be applied. Sometimes, as at A, the hole for the bolt is first bored ; then a hole is cut in one side of the timber, and reaching to the bolt-hole, large enough to allow the screw nut to be inserted. This being done, the hole is refilled by a wooden plug, which holds the nut in place. Then the screw-bolt is inserted, passing through the nut. By turning the screw the timbers may then be tightened together. Figs 21^, page 268, may be consulted also. When the ends of beams, joists, &c, are inserted into walls in the usual square manner, there is danger that in case of being burnt in two, they may, in falling, overturn the wall. This may be avoided by cutting the ends into the shape shown at TO. When a strap o. Fig R, has to bear a strain so great as to endanger its crushing the timber p, on which it rests, a casting like v may be used under it. The strap will pass around the back r of the casting. The small projections in the bottom being notched into the timber, will prevent the casting from sliding under the oblique strain of the strap. The same may be used for oblique bolts, and below a timber as well as above it. When below, it may become necessary to bolt or spike the casting to the under side of the timber. When the pull on a strap is at right angles to the timber, if then is much strain, a piece of plate iron, instead of a casting, may be inserted between the strap and tht timber, to prevent the latter from being crushed or crippled ; see I and I. Art. 27. Expansion rollers, Fig 43, or planed iron Slides; or rockers, Fig 44 ; or suspension-links. Figs 45, 46; must be provided when an iron span exceeds about t Oft; in order to allow the trusses to contract and expand freely under changes of temperature, without undue strain upon some of its mem- bers. Fig 43 shows the general arrangement of roll- ers; which are cylinders of cast iron or steel, from 3 to 6 ins diam ; and 1 to 4 ft long ; planed smooth. From 4 to 8 or more of these are connected together by a kind of framing, n n; and one such frame is placed under at least one end, of the truss. The rollers rest upon a strong planed cast bed-plate 00; bolted to the masonry below. Under the end of the truss is a sim- ilar plate s s, by which it rests on the rollers. Since a truss of even 200 ft span will scarcely change its length as much as 3 ins by extremes of temperature, the play of the rollers is but small. They are kept in line by flanges cast along the side of the bed-plate. Flanges should also project downward from s s, so as completely to protect the rollers from dust, rain, &c. In Fig 44, r r gives a general idea of a KOCKER ; and Fig 45, , of a SUSPENSION-LINK. U U in each ng is aside view of a cast-iron Fink upper chord, through each end of which passes around pin o, which sustains the entire weight of the truss and its load; and which is sustained by from 4 to 6 rockers or links as the case may be. In a railroad bridge of 205 ft span, across the Mouongahela, the links are 3^ ft long ; and the pins 5 ins diain ; and in others of the same size, over Barren and other rivers, the rockers are a foot wide from r tor; and about 5 ins wide transversely on the curved tread or rim. For the accommodation of these several links and rockers ; as well as of the various bars b b, which constitute the oblique ties of a Fink truss; the ends of the octagonal cast-iron upper chords are widened out, as shown by Fig 46; which is atop view of a longitudinal section of such an end. The rockers, or links, and bars, 6, occupy the spaces n n, &c, between the several partitions of the chord ; and the pin oo passes through them all. except when it is expedient to attach some of the bars to the sides, or to the top of the chord, as at t. These figs are intended merely to illustrate the general principle, without regard to detail of construction. In some English bridges of considerable size, such as the Crumlin Viaduct, of 150 ft spans ; and the Newark Dyke bridge, of 240 span; (both of them Warren girders,) and sustained like the foregoing, by the ends of the upper chords ; no further precaution is taken with regard to expan- sion and contraction, than merely to rest the ends of said chords upon smoothly planed iron plates, upon which they may slide. So also several American bridges. Art. 28. The weight of bridges of the game span, designed by different persons, va- ries considerably, from several causes ; such as the U 296 TRUSSES. form of truss; quality of iron; proportions of wrought and cast iron; coefficient adopted for safety, and for strength of materials ; whether the roadway is on the top chord, or the bottom one, s per sq ft. Mr Nash, architect of Buckingham Palace, experimenting with reference to fire-proof floors for that building, wedged men together as closely as they could pos- 298 TRUSSES. 60 ft span. But in a common bridge also, the greatest load per ft run, on a very short span, will be greater than in a long one ; as in the case of two wheels of a truck hauling a large block of stone, &o ; and this must be taken iuto consideration in building such. It must also be remembered that each transverse floor-girder must bear at least all the weight resting upon two wheels; ho matter how close together the girders may be placed. If they are farther apart than the dist between two axles of a vehicle, they will have to bear more than the load on one pair of wheels. The allowance for safety in a trim* bridge. As the result of a long-continued series of deflections applied to an experimental plate-iron girder of 20 ft span, Mr Fairbairn concludes that a bridge subject to 100 deflections per day, each equal to that produced by % of its extraneous breaking load, would probably break down in about 8 years ; while, with 100 daily deflections equal to that arising from but y^ of its breaking load, it would last fully 300 years. We are of the opinion that a bridge should not have a safety of less than 4 for its max extraneous loud, and Its own weight, combined; nor do we see any use in exceeding 6. From 4 to 5 may be used in temporary structures, or in those rarely exposed to maximum strains ; and 6 in more important ones frequently so exposed. The last will (roughly speaking) generally give a safety of about 2 against reaching the elastic strength, which is the true guide in such matters. But 4, d, &c, usually refer to the ultimate or breaking-down strength; so that a truss with such a safety of 2 would in fact be very unsate. Art. 29. Remarks on king: and queen ; and on Fink trusses, for roofs. The following comparison is founded upon total spans, or lengths of truss, of 151 ft. Rise 30.8 ft ; or i of the total span. Trusses 7 ft apart from cen- ter to center. Each rafter 83 ft long. Total load, including the truss itself, 40 fts per sq ft of roof; or 20.8 tons to each truss. There are seventeen points of sup- 20.8 port in each truss; consequently a full panel-load (Art 11) is == 1.3 tons. Trusses as shown, one-half of each, in Figs 47 and 48. The strain in tons (calculated as if all the weight of truss and load were on the rafters) is marked on each member. The assumed coefficient of safety for ties is 3. Iron is supposed to be used that will not break with a less pull than 20 tons per sq inch ; the assumed safe allow- able pull being therefore here taken at -^ = 6% tons per sq inch. The safe pressure along the rafters is taken at 3^ tons per sq inch. The struts are assumed to be wrought cylindrical tubes, with an outer diam equal to ^ of their length; and of such thickness as will give them a metal area of 1 sq inch for each 2 tons of strain. The rafters are in the present case supposed to be 9-inch rolled Phoenix beams ; 7% sq ins transverse area; weighing 25 fts per foot run. The ends of all ties are supposed to be enlarged, or upset; so that the cutting of the screw-threads shall not diminish their effective area. The purlins are supposed to be at or near the "points of sup- port," so as to produce no cross-strains on the rafters. Table 1. Weight of the Fink truss, of which Fig 47 shows one-half. See also Figs, p 264. Length 154 ft. Rise length. Trusses 7 ft apart. Load 40 fts per sq ft of roof; including trusi. See Note, p 263. (Original.) Name of part. Number of parts. Area of each part, sq ins. Lbs. per foot run each part. Total weight of all the parts. Lbs. Lbs. Rafter 2 7 50 (n 2 1 95 6 5 4150 2 jrfl 2 3 41 ' ' ' ' 979 f 1446 If '.'.'.'.:*.::::: 2 3 66 1 ' 99 (U 4 ' 1 4 n-4 2 2 98 3 27 2 1 47 4 89 us' 676 \k :::::. 2 1 71 5 70 TOO U :.::. g 25 JO } Cj 2 ' Struts < w (i 4 1.20 4.0 136 [ 476 Center vertical y say Joint and splicing-pieces, nuts, &c, &c... say 1 40 400 40 400 Shoes at ends of rafters, say Wt of purlins not included. Total wt of truss 400 =7588 400 7588 erea down from """" amo1 * ' TRUSSES. 299 With the same total load per sq ft, including the trusses, (with trusses 7 ft apart ; rise span) the area, and wt per foot run of each part, as well as the strain upon it, will vary directly as the spans; but the total wts, as the squares of the spans. Hence, it is easy to deduce from the table the areas reqd for smaller spaus. The rafters for small spans are frequently made of round iron rods from \ V A to 1% ins diam ; or of ordinary flat bars. Tubes with the name area of metal, would be better. For trusses also of different spans, and rise of \ the span, 7 ft apart, in which the rafters and struts are of wood, with ties of iron, the strains may be deduced quite closely from those in Figs 47 and 4*. They will, however, be somewhat greater, because wooden struts, not being hollow like our assumed iron ones, must be heavier than the latter to prevent bending. The weight of the load, however, is generally so much greater than that of the truss, that this consideration of the strut is not very material ; so that a roof partly of wood may be assumed in practice to weigh, together with its load, but little more than an iron one ; and the strains on the several parts will be nearly the same in both cases. Table 2. Woijrht of the king and qneen truss, of which Fig: 48 shows one-half. See also Fig 14, p 260. Length 154 ft. Rise i length. Trusses 7 ft apart. Load 40 B>s per sq ft of roof; including truss. See Note, p 263. (Original.) Name of part. Number of parts. Area of each part, sq. ins. Lbs. per foot run of each part. Total weight of all the parts. Lbs. Kafter 2 7.5 25 4150 4150 00 IH . 2 2.2 7 33 146 70 G 2 2.44 8 14 162 80 P 2 2.68 8 94 178 80 i, * 2 2.92 9.74 194.80 1606.30 ... 2 3.41 10.54 11 34 210.80 226 80 B 2 3.65 12 14 242 80 A :.... y :::::::::::: 2 3.65 12 14 242 80 a 2 .0 .0 o ^ \'J 2 33 5 34 \i . : :::::::::: 2 .2 67 1600 Verticals ( I . . 2 .3 1 00 32 00 / 298 66 1 m 2 .4 1 33 53 3S 1 . . 2 .5 1 67 80 00 U .............. 2 .6 2 00 112 00 j Center vertical p 1 1 4 4 67 150 00 150 00 f n 2 88 2 92 64 20^ \r 2 1 05 8 50 91 00 \\ .............. 2 1 28 4 25 136 00 Struts. . < t 2 1 55 5 17 196 37 ^ 1587 50 1 V 2 1 82 6 08 267 63 f 1" ; .:.:;: 2 212 7 08 368 30 ( w 2 2 40 8 00 464 00 J Joint and splicing-pieces, nuts, &c, &c .. say 400 00 400 00 400 00 400 00 Total weight of truss 8592 46 8592 46 Wt of purlins not included. Hence, the wt of the king and queen truss in this instance is equal to -~ 2 6 = 1.132 times (say U times) that of the equally strong Fink; or the Fink is about r upper enns. instead ot being proportioned throughout with reference to the max strain at their feet. If the theoretical diminution toward the tops of the rafters, were made in both cases, the wts of the two forms of truss would be nearly equal. But in practice, on the score of inconvenience, this is rarely done in roofs of moderate span ; say less than about 100 ft. No such diminution, or but very slight, would be admissible even theoretically, when the purlins are not placed at the points of support only. With same total load per sq ft, ineiudiii* trusses themselves at same dist apart, the total wts of trusses are as the squares of their spans; but their wts per ft of span, as well as the cross areas, wts per ft run, and strains along 300 TRUSSES. individual members, are directly as the spans. But see Note. p 263. When the dist apart of the trusses is 7 ft from center to center ; the rise -J- of the span ; assumed load, including the wt of the trusses themselves, 40 Ibs per sq ft of roof covering ; and the various parts proportioned for the several strains per sq inch assumed in Tables 1 and 2; the weight of a properly con- structed Fink truss will be approximately as follows : Total wt in Ibs of _ square of span in ft a Fink roof-truss 3.1 ; and the wt in Ibs per ft of span = s P anin f eet O.I A total K and Q truss, will be about \ part more ; or ^^ ^-L^^L^L^ span in feet Or per foot of span, = : 23716 _ ' ' These rules give 2*?n fi 650 ft>8, for the foregoing Finland ~-^ = 8784 Ibs, for the K and Q truss. From these rules we have drawn up the following Table 3. Approximate weights of roof-trusses of the Fink system. (Original.) Rise 4- span. Trusses 7 ft apart. Load 40 tts per sq ft of roof, including truss. Total Span. Total wt of a Truss. Wt per ft. of Span. Wt per sq ft of ground covered. Total Span. Total wt of a Truss. Wt per ft. of Span. Wt per sq ft of ground covered. Feet. Lbs. Lbs. Lbs. Feet. Lbs. Lbs. Lbs. 20 129 6.46 .92 100 3228 32.3 4.60 25 202 8.08 1.15 105 3557 33.9 4.83 30 290 9.67 1.38 110 3904 35.5 5.06 35 396 11.3 1.61 115 4267 37.1 5.29 40 516 12.9 1.84 120 4640 38.7 552 45 654 14.5 2.07 125 5041 40.4 5.75 50 807 16.1 2.30 130 5452 42.0 5.98 55 976 17.8 2.33 135 5880 43.6 6.21 60 1160 19.4 2.76 140 6336 45.2 6.44 65 1363 21.0 2.99 145 6782 46.8 6.67 70 1584 22.6 3.22 150 7260 48.4 6.90 75 1815 24.2 3.45 155 7750 50.0 7.13 80 2064 25. 3.68 160 8256 51.6 7.36 85 2331 27.^ 3.91 165 8782 53.3 7.59 90 2616 29.1 4.14 170 9324 54.9 7.82 95 2912 80.7 4.37 175 9879 56.5 8.05 For king and queen trusses add ^ part to the tabular wts ; when the rafters are as usual of the same size throughout. See Note, p 263. The wts in the 4th column will remain nearly the same, whatever may be the dist apart. For if this be increased say to 14 ft, each truss will sustain twice as many sq ft of roof; and must itself be at least twice as strong and heavy, in order to do so. We say " at least," because if the dist apart is increased, the wt of thejpwr- lins will generally increase more rupidly than said dist. Thus, if the dist be doubled, the purlins will not only be doubled in length, which alone would double their wt ; but they must also be deeper. In practice, however, long purlins are usually pre- vented from becoming very heavy, by trussing them, as at 7, Figs 21^, page 268. The cost of trusses alone for iron roofs, generally varies between 10 and 12 cts per ft) put up ; depending on the price of iron and labor. The putting up alone from % to 1^ cts per ft). With trusses 7 ft apart, iron purlins will weigh about 2 ft>s per sq ft of ground covered by the roof. Therefore to any wt in the 4th col add 2 Ibs. Mult the sum by from 10 to 12 cts, for the cost of trusses and purlins alone per sq ft of ground. Add for covering with tin or slate on boards, say 15 to 20 cts per sq ft ; or for corrugated iron on the bare purlins say 35 cts ; or if on boards, 38 to 40 cts per sq ft. Bridge trusses, at shop, wrought iron parts 8 to 11 cts per ft> ; cast iron, 5 to 7 cts. HEM. 1. As to the proper total weight, or load, per sq ft of roof, (including snow and wind,) that should be assumed to be sustained by the trusses, engineers differ considerably. The French appear to consider 30 TJS as suflciont; while the English use 40. Since roofs are not subject to violent vibrations like bridges, they do not require so high a coefficient of safety; this should not. however, in our opinion, be taken at leas than 3; and this we consider sufficient. The load is evidently influenced by the character of the roof-covering. Within ordinary limits, for spans not exceeding a'bout 75 ft, and with trusses 7 ft apart, the total load per sq ft, includ- ing the truss itself, purlins, &c, complete, may be safely taken as follows; TRUSSES. 301 Table 4. Span 75 ft or less. and snow. Total. Roof covered with corrugated iron, unbearded, t 8Ibs. 20 Ibs. 28 fts. If plastered below the rafters, 18 " 20 " 38 ' " " corrugated iron, on boards, 11 " 20 " 31 ' If plastered below the ratters, 21" 20" 41 ' ' " " slate, unbearded, or on laths, 13 " 20 " 33 ' 11 " " " on boards, 1J4 ins thick. 16 " 20 " 35 ' " " " " if plastered below the rafters, 26" 20" 46 ' " shingles on laths. 10 " 20 " 30 " If plastered below rafters or below tie beam. 20 " 20 " 40 " For spans from 75 to 150 ft, it will suffice to add 4 fts to each of these totals. Example of use of foregoing tables. A Fink roof 60 feet span ; rise ; trusses 1-4 ft apart ; and to be covered with slate, on boards \% incn thick. Here we see at once from Table 3 that at 7 ft apart, its wt would be about 1160 Ibs ; therefore, at 14 ft apart, it would be 2320 Ibs. But our table is for 40 Ibs per sq ft of roof: while, for slate on boards, 35 Ibs, or ^ part less, is sufficient. Therefore, we may reduce the weight of the truss y 8 part, making it only 2030 fts. See Note, p 263. Ex. 2. Roof as before, 60 span ; trusses only 1 ft apart. Turn to Table 1, where the areas are given for a total length or span of 154 ft. But 60 ft is the = say the .4 part of 154 ft ; therefore, the areas, and the wts per foot run of each member of the 60 ft span, will be .4 of those of the 154 ft one. Thus, the area of a rafter will be 7.5 X -4 = 3. sq ins; which corresponds with a rolled T iron of 3 X 3% ins, and }4 inch average thickness. Its wt per foot run will be 25 X -4 = 10 Ibs. The area of the part n of the main tie will be 1.95 X .4 = .78 sq inch, which we see at once from a table of circular areas, is equal to a round rod very nearly 1 inch diam. Its wt per ft run 6.5 X -4 = 2.6 Ibs ; and so with all the other members. But the total wts will be as the squares of the span. The square of 154 is 23716 ; and that of 60 is 3600. And jj- = .152; therefore, the total wts will be .152 of those in Table 1. Thus, the two rafters will weigh 4150 X -152 = 631 Ibs. The main tie, 1446 X -152 = 220 Ibs, &c. Lastly, if for 35 fibs per sq ft, reduce each area and wt }/ & part. Since the rafters are generally made of T or I iron, a pattern precisely adapted to the calculated strains, will not always be procurable ; and in that. case we may either take the nearest one in excess : or change the dist apart of the truss to suit the pattern on ha'nrt. Owing to the variety of modes of arranging the details of the junctions. Ac. an exact coincidence between the.calculated and the actual wts, is not to be expected ; but we suspect that in properly proportioned roofs, the discrepancy will rarely be found to vary more than about 5 per ct from the results of our rules. It might be supposed that with iron of a tensile quality considerably higher than our assumption of 20 tons per sq inch ; as say of 25 to 30 tons, the truss might be made much lighter. But this is not the case ; because the superiority would affect the ties only ; inasmuch as the compressive strength of iron does not increase with its tensile strength ; but to a certain extent rather the reverse. Now, by Table 1, it appears that the ties in a Fink roof-truss, constitute less than T % of its entire wt. Therefore, iron of even 30 tons, would reduce the weight of the truss less than % part of j 3 ^ part; or y 1 ^ ; and 25 ton iron, about ^V part. Short spans need not have as many subdivisions, or "points of support, ' as a large one; and this will lessen the number of parts of the truss ; but inasmuch as the remaining parts will require to be proportionally stronger, this consideration will not materially affect the wts. While on this subject, we will remark that too few points of support are probably used at times ; owing to either an under- valuation, or an- ignorance of the effect of the transverse strains produced by the load on the parts of the rafter between said points. These parts must be regarded as so many separate beams sup- ported at both ends; or rather, as firmly fixed at both ends, when the pieces composing a rafter are, as usual, strongly connected together ; in which case the beam is about twice as strong as when merely supported. If the separate parts be trussed, like the purlin at 7, Figs 21%, to neutralize this transverse action, it must be remembered that additional compression will be thereby produced lengthwise along the rafter. The best practice is, as far as practicable, to increase the number of points of support, so that the purlins may rest upon them alone, or near them ; and thus relieve Che rafters entirely, or in part, from transverse strain. HEM. 2. As to the effect produced on the weight of a truss, by changing* its rise, no short correct rule can be laid down. Although as a roof becomes flatter, its area becomes less, so that each truss has less total wt of roof- covering, snow, and wind, to sustain, still the strains on most of its members become greater; requiring greater wt of truss. To find this increase with accuracy, it is * See Snow and Wind, p 519, 520. t The corrugated iron itself will weigh from 1 J^ to 2 Tbs per sq ft on the roof. If not plastered under- neath, the condensed moisture of the air, especially from crowded rooms, will fall from the iron into the rooms below. Mere boarding will not prevent this, even if tongued and grooved, unless the circu- lation of air against the under side of the iron is effectually cut off. 20 302 TRUSSES. necessary to make a diagram, and perform all the calculations. The strains on a Fink rafter become more nearly uniform throughout its length, as the pitch of the roof becomes less ; while, with a rise of % span, the strain at its foot is about 1-fy times that at its head. On the contrary, the strains on its struts remain nearly the game in amount for all ordinary rises. In the king and queen truss the strains at the heads and feet of the rafters retain the same pro- portions to each other, at all rises ; the strains on the verticals become less as the roof becomes flatter ; while those on the obliques vary according to their several obliquities. Under these irregularities, which affect the K and Q, much more than the Fink, we can do nothing more than say that when it is merely wished at the moment to form a rough idea of the effect of changing the rise, we may assume the weight of a Fink truss to increase about in the same proportion as we diminish the rise; or to diminish as we increase the rise. Thus, if we increase the rise of the roof in Table I, one-fourth part, so as to make it equal to .25 or } of the span, instead of .2 or-^- of the span, we may diminish its wt y part ; making it about 6000 fts, instead of 8000. Or if we reduce the rise from ^ to y 1 ^, making it only half as great, we shall double its weight, making it 16000 0>s; as rude approximations. Art. 3O. Oil the camber of truss bridges. In practice, the upper and lower chords of bridges are not made perfectly straight, but are curved slightly up- ward ; and this curve is called the camber of the truss or bridge. Its object is to prevent the truss from bending down below a hor line when heavily loaded. A cam- bered chord is of course longer than a straight line uniting its ends ; but in practice the camber is so small that this diflf is inappreciable, and may be entirely neglected. But when the chords are cambered, (see y s and c d, Fig 51,) they become concentric arcs of two large circles, of which the center is at t\ and the upper one plainly be- comes longer than the lower, to an extent which, although much exaggerated in our fig, cannot be overlooked in practice. The verticals, instead of remaining truly vert, become portions of radii of the aforesaid large circles; and although their lengths remain the same, yet their tops become a little farther apart than their feet ; and this renders it necessary to lengthen the obliques or diags a trifle. Therefore, we must find how great is this increase of length of the upper chord beyond the lower one; and divide it equally among all the panels, along said chord; otherwise the several parts of the truss will not fit accurately together. 1st. To find the amount of camber of the lower chord. Di- vide the span in feet, (measured from center to center of the outer panel-points,) by 50. The quot will be a sufficient camber, in inches ; as shown in the following Table of cambers for bridge trusses. Span. Camber. Span. Camber. Span. Camber. Feet. Ins. Feet. Ins. Feet. Ins. 25 0.5 100 2.0 250 5.0 50 1.0 150 3.0 300 6.0 75 1.5 200 4.0 350 7.0 Rem. 1. It is by no means necessary to adhere strictly to this rule ; and the camber by experienced builders of iron bridges is often but one-half the above, or 1 inch per 100 ft of span. Rem. 2. A well built bridge of good design should not, under its greatest load, deflect more than about 1 inch for each 100 feet of its span. The deflec- tion is frequently much less than this. 2d. To find the increase of length in the upper chord. beyond the lower one, having the span ; the depth of truss ; and the camber ; (all in feet, or all in ins.) With any camber not exceeding J^ of the span ; (which, however, is about 7 times as great as is usually given to trusses ;) mult together the depth of truss, the camber, and the number 8 : div the prod t>y the span. The quot will be the increase, in ft, or in ins, as the case may be. Or as a formula, Increase in ft, depth X camber * 8 all in feet; or or in ins, span, all in inches. This rule may be considered practically perfect with any camber not exceeding T^J- of the span. Based upon this principle, we have prepared the following table, which may be used instead of making the above calculations. TRUSSES. 303 Table for finding: increase of length of upper chord beyond lower one. Depth of Mult Camber Depth of Mult Camber Depth Mult Camber Depth of Mult Camber Truss. by Truss. by Truss. by Truss. by y span. 2.00 1.60 H span. 1-9 " 1.00 .888 1-12 span. 1-13 .666 .614 1-16 span. 1 17 " .500 .470 & - 1.33 1.15 1-10 " 1-11 " .800 .727 1-14 M5 " .571 OT 1 1-18 1-20 " .444 .400 O d Ex. How much longer is the upper chord than the lower one, when the depth of the truss is \ of the span ; and the camber 5 ins? Here in the table, and opposite \ span, we find the multiplier 1.15. Therefore, 5 ins X 1.15 = 5.75 ins, Ans. If the truss has say 8 panels, then -^ = .72 inch of this increase must be given to each o panel, along the upper chord. The length of a diagr, or oblique, & c, Fig 50, may readily be found thus : Let a s n c in this fig represent a panel when there is no camber; then o b n c will represent a panel when there is a camber ; and o a and s It together are the portion of the increased length of upper chord given to each panel ; but to an exaggerated scale. Now, to find b c, we have the right-angled triangle a b c, in which we know the side a c, (the depth of truss ;) and the side a b, (equal to the panel width en on the lower chord; added to s 6, or half the portion of the increased length of upper chord given to one panel.) Hence, we have only to square each of those two sides; add the two squares together; and take the sq rt of the sum. This sq rt is b c. Example. Span 200 ft. Height (a c) of truss y s of the span, or 25 ft, or 300 ins. Camber 5 ins ; 10 panels each 20 ft, or 240 ins, (c n,) measured on the lower chord or span. Now, the height being ^ of the span, the increase of length of upper chord will be equal to the camber, 5 ins ; and this divided among 10 panels, will be = .5 inch to each panel ; or o b will be .5 inch longer than en; ind o a and s b will each be .25 inch. Hence, in the right-angled triangle a b c, we Have a 6 = 240.25 ins ; and a c = 300 ins. Hence, Pit/ 50 6 c 57720.0625 + 90000 = ^/ 147720^0625 = 384.344 ins ; or 32 ft, .344 ins. Without any camber, b c would be in the position s c, which is 32 ft, .187 ins long ; or .157 of an inch (about % inch) shorter than b c. An error to this extent would prove seriously inconvenient if the oblique were a cast-iron strut with carefully planed ends, intended to fit closely between planed bearings at the chords ; or a bar with a drilled hole at each end for fitting over pins whose position was fixed and unalterable. In many cases, as when the obliques are merely rods with screw-ends, it is only necessary to be sure that they are long enough; because their exact length can then be adjusted when put into place, by means of the nuts on their ends. So also when the obliques or other pieces are flat bars intended to be bolted or riveted to the sides of the chords; for the final rivet-holes may be made when the pieces come to be finally fitted in place. In the case of wooden obliques, &c, if too long, they can readily be reduced by the saw or chisel. When the panels are all of one size, as is generally the case, it is usual for builders to draw one of them full size on a board platform or floor, to guide in fitting the parts together. In raising a trass, or in other words, when putting its parts together hi their proper position on the abutments and piers, a scaffold or false-works, must first be erected for sustaining the parts until they are joined together so as to form the complete self-sustaining truss. Upon the false works 'the bottom chords are first laid as nearly level as may be ; and the top chords are then raised upon tem- porary supports which foot upon the one that carries the lower chord. The upper chords are at first placed a few inches higher than their final position, or than the true height of the truss, in order that the obliques and verts may be readily slipped into place. After this is done, the top chords are gradually let down until all rests upon the lower chords. The screws are then gradually tightened to bring all the surfaces of the joints into their proper contact ; and by this operation (the upper chord being supposed to have the increased length given by the foregoing rule) the 304 TRUSSES. camber, as it were, forms itself; and lifts the lower chords clear off from their false- works ; leaving the truss resting only upon the abuts or piers, as the case may be. As a support for the falseworks themselves on soft bottoms piles may be driven, to which the uprights of the falseworks may be notched and bolted or banded. In some cases, as of rock bottom in a strong current, it may become expedient to sink cribs filled with stone, as a support for the falseworks. The falseworks should be well protected by fender-piles or otherwise from passing boats, ice and other floating bodies, especially in positions liable to sudden floods : and numerous accidents have shown the expediency of guarding the unfinished truss itself against hig*li winds. This last remark applies as well to roofs as to bridges ; and is too frequently neglected. Art. 31. For very short spans, the rolled iron I beam, page 210, answers every purpose. A single 20 ft>, 7-inch beam under each rail, will suffice for 3 or 4 ft span : and the 52ft), 15-inch one, for 8 to 10 ft. Two 52 ft), 15-inch ones, under each rail, for 15 ft. By employing a greater number of beams, or by intro- ducing a truss-rod, ?*, Fig 52, the spans may be increased; or lighter beams be used. The beams should be fitted at their ends into wide cast-iron shoes, well bolted to the abuts. Care must be taken to insure lateral stability, by means of hor cross-ties and diag bracing. This may generally be secured by notching and bolting the cross-ties to the beams; and by diag rods passing through the chords; and having their inner ends confined by screws to such a ring as is shown at 14, Figs '21%, page 268; or by the diag bracing shown at W, Fig 36%, p 291. To prevent ail overturning tendency in a whole truss when it is not high enough to admit of being horizontally braced overhead, we may introduce wooden knees, or short straight struts or ties, of either wood or iron ; which may foot upon the cross-girders of the floor; and head against either some of the web members, or the upper chord. These braces or ties may be placed either between the two trusses of a span ; or outside of them : or both. When outside, some of the floor-girders may be lengthened out a few feet beyond the lower chord, for receiving the feet of the braces or ties. See Art 41, of Strength of Materials, for other iron beam bridges of still greater spans. Or, a single beam of wood under each rail,* and firmly braced against lateral motion, will suffice. Assuming the weight of entire bridge and load, at two tons per foot, the following dimensions may be used : Span in Ft. Size of Beam. Span in Ft. Size of Beam. 5 10 8X 10 ins. 9 X 12 10 X H " 11 X 16 " 15 20 22}* 12X18 13 X20 14 X 22 16X24 The greatest dimension to be the depth. The ends should be well bolted down to '(jfeiyS^ iV%o * If single beams of sufficient depth cannot be procured, built-beam* may be used ; see g, and ij, Figs 42, p 294. TRUSSES. 305 bolsters. These are long stout sticks of timber, from 10 to 15 ins square, (accord- ing to the span.) laid across the abuts at the bridge-seat, for the chords to rest on. See h, Fig 31. Frequently two are used at each abut, even in small spans; and \ve have seen but one, under railroad spans of 150 feet. Large spans may require three or more. They are not necessarily placed in contact with each other; but may be some feet apart, if required. Or for spans of about 15 to 30 ft, we may use somewhat lighter beams ; and truss each of them as in Fig 52, by an iron bar e s e ; and a center post p. In this case the following dimensions will answer; the total deflection of the rod being 1^ of the clear span. The screw ends of the bars are supposed to be upset; but the areas are given for the body of the rods. For each beam. Span. Ft. Beam. Ins. Section of Rod. Sq Ins. Section of Post. Sq Ins. 15 20 25 30 12 X 15 13 X IT 14 X 18 15 X 20 3^ 4% 6 7 25 33 42 50 It is better to have two rods instead of one under each beam ; each rod being of half the section here given ; and the two placed several ins apart. This affords a better footing for the post. The ends of the beam should be at right angles to the direction of the rod ; and be provided with ample washers e e, of wood or iron, for distributing the pressure from the rod, over the whole area of the ends. The ends of these wash- ers may extend a few ins each way beyond the ddes of the beam, as shown on a larger scale at g. This allows the rods to be outside of the beam ; instead of requiring holes to be bored in the latter, for passing the rods through them. They may be nearer together at the foot of the post. The head of the post may be tenoned into the bottom of the beam ; and be further united to it by ron straps. To prevent the foot from being worn by the rods it should be shod with iron. A cast- ron shoe, as at s, may be bolted to it; having ribs for keeping the bars in place. Or a stout wrought- ron shoe may be well secured to it. In either case the rod at a should be so united to the shoe as to heck any tendency in the'foot to slide toward r or r, under the vibration of passing loads. Perhaps his can "be most conveniently done by making each rod e s e, iu two separate lengths, r, r; and by uniting their lower ends to the shoe at by hooks and eyes : or by eyes and bolts, &c. Various methods are in use for the heads and feet of the posts of large spans ; but we cannot here treat upon details which pertain more to the professional bridge-builder. This mode of trussing is also well adapted to long floor beams; and has been used in long oblique web members ; as well as in long stretches of chords from one point of support to another. Fig. 2 is a vertical, thus trussed in more than one direction. The following dimensions for single-track Fink bridges, with chords and posts of wood; and iron suspension bars; are on the assumption that all the bars deflect Y B of the span ; that the road is on top ; that the bars shall not be strained more than 10000 ft>s, or 4% tons per sq inch, under a weight of bridge and load, amounting in all to two tons per running foot. Assumed wt on each driving-wheel of engine, 5 tons. Screw ends upset. Dimensions for one truss only, of a single-track Fink bridge. The spans are in feet ; the other dimensions are square inches of cross-section of each member. Areas in sq in Areas in sq ins. 21 23 225 280 335 390 We have given the area of the 1st post only. For that of the 2.1, we may take % of the first : for the 3d, % of the second ; and for the 4th. % of the third ; without pretending to any great accuracy. The be best to h iay be flat, square, or round, so that the proper area be maintained. It will usually e them flat. Our assumed total weight of 2 tons per foot of span, for bridge and load together, up to spans of 200 ft, is greater than is usually adopted. Still we 306 TRUSSES. would recommend not to diminish the areas in the last three short tables, or in thb following one, more than % part. They will then be about up to ordinary practice for railroads; and will also be sufficient for bridges for common travel, with a clear width of 1$ ft between the two trusses ; and two outside footpaths, each 5 ft wide, in addition. A width of 18 ft is necessary for allowing two ordinary vehicles to pass each other readily* It should never be less than 16 ft ; and nothing is gained by exceeding 20 ft. It will of course be understood that each member, especially in large spans, will consist of two or more pieces, side by side. Thus, in a span of 200 ft, the 800 sq inches ot each chord, will probably consist of four beams of about 10" X 20", or 12" X 16", placed side by side ; but with sufficient inter- vals between them to allow the several oblique bars to pass. Or it may consist of six beams of smaller size. So also, the 1st, or main rod, of 46% sq ins, will probably be made up of from 4 to 8 bars, of 11.7, or 5.85 sq ins each, placed side by side; occasionally some inches apart. And so with the others. The feet of the opposite posts of the two trusses of a Fink spau, are connected together by ties of wood or iron, to prevent lateral motion; and for the same purpose, diagonals are carried from each upper chord to the foot of the opposite post ; as is usually done in top-road bridges of any kind. These had better be tie-struts. A simple truss like Fig 6, of 30 ft span, and 10 ft high, may have a chord of In" by 18"; rafters 10" by 10"; one rod of 2%" diam; or two rods of l%" diam, and several inches apart transversely of the bridge ; which is far better than one. The Pratt truss, when put together as in Fig 31; each member (except the m-iin obliques x w.yn, &c,) being of one piece, is perhaps as easy of con- struction, and as suitable for small spans, as any other. With the same assumptions as before with regard to total load, upset rods, ~ about 28 cub ins. In ordin tverasrea linary quarrying, a cub yard of solid rock in place, (or about 1.9 cub yds piled up after being quarried.) requires from J^ to % ft. In very refractory rock, lying badly for quarryiug, a solid yard may require from 1 to 2 fts. In some of the most successful great blasts for stone for the Holyhead Breakwater, England, (where several thousands of fts of powder were usually exploded by galvanism at a single Mast,) from 2 to 4 cub yds solid were loosened per ft; but in many instances not more than 1 to 1J4 yds. Tunnels and shafts require 2 to 6 fts per solid yard: usually 3 to 5 fts. Soft, partially decom- posed rock frequently requires more than harder ones. Usually sold in kegs of 25 fts.g The fluid called nitro-glycerine is coming into use instead of powder. Its strength is from 10 to 13 times that of an equal bulk; or from 8 to 10 times that of an equal weight, of powder. Gun-cotton has from 3 to 6 times the strength of an equal weight of powder. * By adding y 1 ^ part to the lengths in the two cols under 180, we get the lengths corresponding to a number of degrees y 1 ^ less than 180; or to 163.63 deg ; which may be taken as about the extremes of temp in the colder portions of the United States. In the Middle States the extremes rarely reach 135, or Yi part less than 180. No dependence whatever is to be placed on results obtained by Wedgewood's pyrometer. f The table shows that the contraction and expansion of stone will cause open joints in winter; and crushing of the mortar in summer, at the ends of long coping-stones. i The melting points are quite uncertain. We give the mean of the best authorities. Assuming that with a change of temp of about 163, wrought iron will alter its length 1 part in 916; this in a mile amounts to 5.764 ft, or about 5 ft 9^ ins ; and in 100 ft to .109 of a foot; or 1% ins; so that a diff of 5 ft, or more, can readily result from measuring a mile in winter and in summer with the same chain ; and a 25 ft rail will change its length full % of an inch. \ Price, 1880, in Atlantic cities, about $3 to $3.25 per 25ft keg ; according to quantity. STONEWORK. 311 Weight of powder in one foot depth of hole. Diameter of hole in inches. 1 I IX 1 1H I 2 | 2^ | 3 | 3^ | 4 | 4^ | 5 | 5J4 | 6 Weight of powder in pounds and ounces avoir. 0"5 | 0"8 I 0"11 | 1"4 | 2 |2"13 1 3"14 | 5"0 | 6"6 | 7"14 | 9"8 | 11"5 holes for blasting*. The holes are generally from 2^ to 4 ft deep; and from 1% to 2 ins diam. Churn-drilling; is much more expeditious and economical than that by jumping, mentioned below. The churn-drill is merely a round iron bar, usually about 1% ins diam, and 6 to 8 ft long; with a steel cutting edge, or bit, (weighing about a tb, and a little wider than the diam of the bar,) welded to its lower end. A man lifts it a few inches; or rather catches it as it rebounds, turns it partially around; and lets it fall again. By this means he drills from 5 to 15 feet of hole, nearly 2 ins diam, in a day of 10 working hours, depending on the character of the rock. From 7 to 8 ft of holes 1% ins diam, is about a fair day's work in hard gneiss, granite, or compact siliceous limestone ; 5 to 7 ft in tough com- pact hornblende ; 3 to 5 in solid quartz ; 8 to 9 in ordinary marble or limestone ; 9 to 10 in sandstone; which, however, may vary within all these limits. When the hole is more than about 4 ft deep, two men are put to the drill. Artesian, and oil wells, in rock, are bored on the principle of the churn-drill. The jumper, as now used, is much shorter than the churn-drill. One mnn (the holder) sitting down, lifts it slightly, and turns it partly around, during the intervals between the blows trom about 8 to 12 ft hammers, wielded by two other laborers, the strikers. It can be used for holes of smaller diameters than can be made by the churn-drill ; because the holder can more readily keep the cutting end at the exact spot required to be drilled. It is also better in conglomerate rock ; the hard siliceous pebbles of which deflect the churn-drill from its vertical direction, so that the hole becomes crooked, and the tool becomes bound in it. The coal conglomerates are by no means hard to drill with a jumper. The jumper was formerly used for large deep holes also, before the superiority of the churn- drill became established. Either tool requires resharpening at about each 6 to 18 inches depth of hole; and the wear of the steel edge requires a new one to be put on every 2 to 4 days. With iron jumpers, the top also be- comes battered away rapidly. As the hole becomes deeper, longer drills are frequently used than at the beginning. .The smaller the diameter of the hole, the greater depth can be drilled in a given time; and the depth will be greater in proportion than the decrease of diam. Under similar circum- stances, three laborers with a jumper will about average as much depth as one with a churn-drill. The hand-drill, in which the same man uses both the hammer and the short drill, is chiefly used for shallow holes of small diam. With it a fair workman will drill about as many feet of hole from 6 to 12 ins deep, and about % inch diam, as one with a churn-drill can do in holes about 3 ft deep, and 2 ins diam, in the same time. Only the jumper or the hand-drill can be used for boring holes which are horizontal, or much inclined. The Bnrleig-h Rock Drill, (Co at Fitchburg, Mass ,) is much more rapid and economical than the foregoing if the work ia so great as to justify the preliminary outfit. It drills at any angle ; and is worked either by steam directly ; or by air compressed by steam into a tank, and thence led to the drills through iron pipes. The air is best for tunnels and shafts, because after leaving the drills it aids ventilation. The cost of each drill and its support is about $450 to $1200 at the shop, depending on the size of the holes, (% to 6 ins diam.) Each compressor, with attached engine, $1100 to $2800. Add boilers, air-tank, air-pipes, house, extra parts, &c. A $2000 compressor will work two drills ; and require 3 inch air-pipes ; and a tank 15 ins diam. by 4 ft long. The boiler and compressor together require 1 man; each drill 2 men, and will drill from one to two holes of about 3 ins diam, and 5 ft deep per hour, depending on the kind of rock ; and about -i more of 2 inch hole. In limestone from 25 to 45 ft of hole with one sharpening. One blacksmith and helper can sharpen for 5 or 6 drills. p r of. Wood's drill also is a first-class machine. Cost of quarrying- stone. After the preliminary expenses of purchasing the site of a good quarry; cleaning off the surface earth and disintegrated ti.p rock ; and providing the necessary tools, trucks, cranes, &c ; the total neat expenses for getting out the rough stone for masonry, per cub yard, ready for delivery, may be roughly approximated thus: Stones of such sizes as two men can readily lift, meas- ured in piles, will cost about as much as from % to % the daily wages of a quarry laborer. Large stones, ranging from % to 1 cub yd each, got out by blasting, from 1 to 2 daily wages per cub yd. Large stones, ranging from 1 to 1% cub yds each, in which most of the work must be done by wedges, in order that the individual stones shall come out in tolerably regular shape, and conform to stipulated dimensions ; from 2 to 4 daily wages per cub yard. The smaller prices are low for sandstone, while the higher ones are high for granite. Under ordinary circumstances, about \}/ A cub yds of good sandstone can be quarried at the same cost as 1 of granite ; or, in other'words, calling the cost of granite 1, that of sandstone will he %: so that the means of the foregoing limits may be regarded as rather full prices for sandstone; rather scant ones for granite : and about fair for limestone or marble. ost of dressing* stone. In the first place, a liberal allowance should be made for waste. Kven when the stone wedges out handsomely on all sides from the quarry, in large blocks of nearly the required shape and size, from % to % of tne rough block will generally not more than cover waste when well dressed. In moderate-sized blocks, (say averaging about % a cub yard each.,) and got out by 312 STONEWORK. blasting, from ^ to % will not be too much for stone of medium character as to straight splitting. About the last allowance should also be made for well-scabbled rubble. The smaller the stones, the greater must be the allowance for waste in dressing. In large operations, it becomes expedient to have the stones dressed, as far as possible, at the quarry ; in order to diminish the cost of transportation, which, when the distance is great, constitutes an important item especially when by land, and on common roads. A Stonecutter will first take out of' wind; and then fairly patent-hammer dress, about 8 to 10 sq ft of plain face in hard granite, in a day of 8 working hours; or twice as much of such infe- rior dressing as is usually bestowed on the beds and joints ; and generally on the faces also of bridge masonry. of the puddle, need not be more than 4 or 5 feet for shallow depths ; or than 5 to 1 ft for great ones: because its use is then merely to prevent leaking. But if there are no 1 races, it must be made wider, so as to resist upsetting bodily; and then, with good puddle, o o may, as a rule of thumb, be % of the vertical depth o / below high water; except when "this gives less than 4 ft; in which case make it 4 ft; unless more should be required for the use of the workmen, fordepositing materials, &c. Or if the excavation for the masonry is sunk deeper than the puddle, the dam must be wider; else it may be upset into the excavated pit. Tlie excavated soil may be raised in buckets by windlasses, or by hand, in successive stages. The pumps may be worked by hniid, or by steam, as the case may require; as also the windlasses generally Deeded for lowering mortar, stone. &c. More or less leak- ing may always be anticipated, notwithstanding every precaution. Where a coffer-dam is exposed to a violent current, and great danger from icp, &c, the ex- pensive mode shown in Figs 10 may become necessarv. The two black rectangles c c. repre- sent two lines of rough cribs filled with stone, and sunk in position; one row being enclosed by the other: with a space several feet wide be- tween them. Sheet-piles p p are then driven around the opposite faces of the two rows of crihs : and the puddle is deposited within the boxing thus provided for it, as shown in the fig. Where the current is not strong enouch to wash away gravel bnckine, we may. on rock especially, enclose the space to he built on. by a siriirle qnadrnngle of cribs sunk by stone; nnd after adopting precautions to prevent the gravel from being pressed in beneath the cribsi apply the backing.* Fi.a;s 10^ show the plan, outside view, and transverse section, to a scale of 20 ft to an inch, of a coffer dam on rock, in 8 to 9 ft water, used successfully on the Schuylkill Navigation. PLAN * A pure clean coarse gravel is entirely unfit for such purposes. A considerable proportion of arth is essftntia) for preventing leaks. For another crib-oonferdam. s*e p 316. 320 FOUNDATIONS. Its construction Is very simple. Uprights 5, about 1 ft square, and 10 ft apart from center to center along the sides of the dam ; and 10 ft in the clear, transversely of the dam, support two lines of hori- zontal stringers, i i; inside of which are the two lines of sheeting- piles, x s, enclosing between them a width of 7 ft of gravel puddle. Two flat iron bars (t t, of the transverse section) tie together each pair of uprights 6 6. These bars are % inch thick, by '2^ ins deep, and 9 It long. Their hooked ends fit into eye-bolts c, which pass turougu the uprights b; outside of which they are fastened by keys, A:, '.see detail sketch.) Between the keys aud 6, were washers. At the corners of the dam (see plan) were additional tie-bars, as shown. A small band of straw, as seen at y, wrapped around the tie- bars just inside of the sheet-piles ; and kept in place by the puddle ; effectually prevented the leaking which generally proves so troublesome ia such cases The stout oblique braces, o o, were merely spiked to the outside faces of the uprights 6. They are not shown in the transverse section. This dam was built on shore; in sections 30 to 40 ft long. These were floated into place, and weitrhteddown, sheet-piled, and puddled with gravel. The dam had sluices by which water was admitted when necessary for preventing the outside head from exceeding 9 ft. The lengths of the uprights 6 b were first found by careful soundings. , TR.SEC b b OUTSIDE. b b PLAN Valuable hints for coffer-dams may be taken from what is said under the head of " Dams," where Fig I affords useful suggestions for coffer-dams also, on rock in shallow water; p 583. The mooring of larjje caissons or cribs, preparatory to sinking them, is sometimes troublesome, especially in strong currents. It may be neces- sary to drive clumps of piles; or to temporarily sink rough cribs filled with stone, to which to attach the long guide-ropes by which the manoeuvring into position, &c, is done. Frequently dams are left standing after the work is done ; if not in the way of navigation, or otherwise objectionable ; inasmuch as the materials are rarely worth the expense of removal. But if removed, the piles should not be drawn out of the ground ; but be cut off close to river bottom ; for if drawn, the water entering their holes may soften the soil under the masonry. It is often expedient to drive two rows of piles from the dam to the shore, for supporting a gangway for the workmen ; or even for horses and carts ; or for a railway for the easy delivery of large stones, &c. Coffer-dams may be sunk through a soft to a firm soil, in shape of a box of cribwork, either rectangular or circular, and without a bottom. This being strongly put together, and provided with proper temporary internal bra -ing, (to be gradually removed as the masonry is built up,) is floated into place; and after being loaded so as to rest on the soft bottom, is sunk by dredging out the soft material from inside. Additional loading will sometimes be required for over- >iiiin^ the friction of the soil against the outside; or it may even become necessary hi drudge away some of the outer material also. On rock it may at times be expedient to drill holes in deep water, for receiving the ends of piles, or of iron rods, &c. This may be done by means of long drill-rods, working in an iron tube or pipe sunk as a guide to the rod ; with its lower end over the spot to be bored Or a diving- bell may be used. Or a cylinder of staves 4 to 12 inches thick, long enough to reach above the surface, and having abroad tarpaulin flap or apron around its lower edge, to be covered with gravel to prevent leaking: maybe sunk, and the water pumped out, to allow a workman to descend, and work in the open air. Piles. When driven in close contact. as on Fig 11, for preventing leakage; for confining puddle in a coffer-dam: or for enclosing a piece of soft or sandy ground, to prevent its spreading when loaded ; or if the outside soil should wash away from COSt Of piles delivered at wharf. Philada. 1873. Hemlock, 6 to 8 ct per foot lineal, Bay yellow pine, 10 to 15 cts, Southern yellow pin,.-, 18 to 25 cts. FOUNDATIONS. 321 around them, &c, they are called sheet-piles. Generally these are thinner than they are wide ; but fre- quently they are square; and as large as bearing piles; and are then called close piles. To make them drive tight to- gether at foot, they are cut obliquely as at /. Occasionally, when driven down to rock through soft soil, their feet are in addition cut to an edge, as at r, so as to become somewhat bruised when they reach the rock, and thus fit closer to its surface. Their heads are kept in line while driv- ing, by means of either one or two longitudinal pieces a and o, called wales or stringers. These wales are supported by gauge-piles, or guide-piles, previously driven 1:1 the required line of the work, and several ft apart, lor this purpose. See Figs 8. A dog-iron d, of round iron, may also be used for keeping the edges of the piles close at top to those previously driven both during and after the driving. Its sharp ends, cc, being driven into the tops of the wales ww, (shown in plan,) it holds the descending pile o firmly m place. A t n, d, p, Fig 11, are other modes occasionally used for keeping the piles in proper line. At^>, the letters s s denote small pieces of iron well screwed to the piles, a little above their feet to act as guides; very rarely used. At m are shown wooden tongues tt, sometimes driven down between the piles after they themselves have been driven ; to assist in preventing leaks. In some cases sheet-piles are employed without being driven. A trench is first dug to their full depth for receiving them ; and the piles are simply placed m these, which are then refilled. Closer joints can be secured in this manner than by driving. ^ When piles are intended to sustain loads on their tops, whether driven all their length into the ground, or only partly so, as in Fig 3, they are called bearing P *?: 1 lhe y i ai ' e generally "round; from 9 to ]8 ins diam at top; and should be, straight, but the bark need not be jemoved. White pine, spruce, or even hem- lock, answer very well in soft soils ; good yellow pine for firmer ones; and hard oaks, elm, beach, &c, for the more compact ones. They are usually driven from about 2% to 4 ft apart each way, from center to center, depending on the character of the soil, and the weight to be sustained. A tread-wheel is more economical than the winch for raising the hammer, when this is done by men. Morin found that the work performed by men working 8 hours per day, was 3900 foot-pounds per man per minute by the tread-wheel ; and only 2600 by a winch. The gunpowder pile driver invented by mechanical engineer of Philada, is a very meritorious machi ' the well-known , .. >rked bv small cartridges of powder, placed one by one in a receptacle on top of the pile; and exploded by the ham- mer itself. It can readily make 30 to 40 blows of 5 to 10 ft, per minute ; and, since the hammer does not come into actual contact with the piles, it does not injure their heads at all ; thus dispensing with iron hoops s ; and g - = 48000 fts, = 21.4 tons safe load by Maj San- ders' rule. The soil was river mud. Our own rule is as follows. Mult together the cube rt of the fall in ft ; the wt of hammer in fts ; and the decimal .023. Divide the prod by the last sinking in ins. -f- 1. The quotient will be the extreme load that will be just, at the point of causing more sinking. For the safe load take from one twelfth to one half of this, according to circumstances. Or, as a formula, Cube rt of v Wt of hammer v 02 , Extreme load __ fall in feet * in pounds i tous Last sinking in inches + 1 Example. The same as the foregoing at Chestnut St Bridge. Here the cube rt of 20 ft fall it 2.714 ft. Hence we have Extreme M = MMXIWXJW = , A in tons .75 -|- 1 1.7o Or say half of tliis, or 21.4 tons, the load for a safety^>f 2. Major Sanders' rule makes the safe load 21.4. The actual one is 18 tons. A safety of '2 is not enough for river mud. See " Proper load for safety," below. But although Major Sanders' rule and our own agree very well in this instance if a safety of 2 Z> taken for each, they differ widely in some others. Thus at Neullly Bridge, France, Perronet's heaviest hammer weighed 2000 fts, fall 5 ft, sinkage .25 of an iucli in the lust 16 blows; or say .016 inch per blow. The piles sustain 47 tons each. Our rule gives 38.8 tons for a safety of 2 ; while San- ders' rule gives 515 tons safe load ! If, as we think probable, there was no actual sinking at the last blow, then onr rule gives 39.3 tons for a safetv of 2 ; while Sanders' gives 0. At the Hull Docks, England, piles 10 ins'square. driven 16 ft into alluvial mud, by a 1500 ft ham- mer. falling 24 ft, sank 2 ins per blow at the end of the driving. They sustain at least 20 tons each, or according to some statements 25 tons. Our rule gives 33.2 tons for the extreme load ; or 16.6 for a safety of only 2. Sanders gives for safety 12.06 tons. As before remarked, 2 is not safetv enough for mud. In mud, it is not primarily the piles, but the piled soil that settles, bodily, for vears. At the Royal Border Bridge, England, piles were very firmly driven from 30 to 40 ft in sand and gravel, iu some cases wet. Pine was first tried, but it split and broomed so badly under the hard driving, that American elm was substituted, with success. They were driven until they sank but .05 inch per blow, under a 1700 Ib monkey, falling 16 ft. They support 70 tons each. Our rule gives 47 tons for a safety of 2 ; while Sanders" gives 364 tons safe load 1 It is the writer's opinion, however, that the piles did not nciually sink, as was (and always is, in such cases) taken for granted by the observers ; but that they were merely compressed or partially crushed by overdriving. Most of the piles were driven until they sank (?) only an inch under 150 blows ; but we doubt whether they were any safer, or farther in the ground, thnn when they had re- ceived only one of them ; and consider such extreme precaution worse than useless. In some experiments (1873) at Philada, a trial pile was driven 15 ft into soft river mud, by a 1600 tt> hammer ; its last sinking being 18 ins under a fall of 36 ft. Only 5 hours after it was driven it was loaded with 65 tons ; which caused a sinking of but a very small fraction of an inch. Our rule gives 6.4 tons as the extreme load. Under 9 tons it sank .75 of an inch ; and under 15 tons, 5 ft. By Miij Sanders' rule its safe load would be 2.14 tons. A U. 8. Govt trial pile, about 12 ins sq, driven 29 ft through layers of silt, sand, and clay, ham- mer y 10 fbs, fall 5 ft, last sinking .375 of an inch, bore 26.6 tons ; but sank slowly under 27.9 tons. Our rule gives 26 tons extreme load. French engineers consider a pile safe for a load of 25 tons, when it is driven to the refusal of 1344 fts, falling 4 ft ; our rule gives 24.2 tons for safety 2. They estimate the refusal by its not sink- ing more than .4 of an inch under 30 blows. In many important bridges &c they drive until there is no sinking under an 800 R> hammer, falling 5 ft. Our rule here gives 31.5 tons extreme load ; or 15.7 for safety 2. As to the proper load for safety, we think that not more than one-half the extreme load given by our rule should be taken for piles thoroughly driven infirm soils; nor more than one-sixth when in liver mud or marsh ; assuming, as we have hitherto done, that their feet do not rest upon rock. If liable to tremors, take only half these load*. FOUNDATIONS. Piles may be made of any required size as regards either length or cross section, by bolt- ing and fishing together side wise and length wise, a number of squared timbers. Piles with blunt ends. At South Street Bridge, Phila, 1200 stout piles of Nova Scotia spruce total cost (piles and driving) of $7 to $8 each. At Wilmington Harbor, Cal, Mr. C. B. Sears, U. S. Army. (Jour. Am. Soc. C. E., Dec 1876) found that m firm compact ^et smud, after the first few blows the piles would not penetrate moie than .5 to 1.5 ins at a blow, no natter how far the 2400 ft hammer fell. The unpointed ones of which there were many thousands, drove quite as readily to aver- age depths of 15 ft in this sand as the pointed ones, and with much less tendency to cant. As a high fall had no farther effect than to batter the heads he reduced it to 10 ft. which drove an average of nbnnt .72 inch to a blow. To insure straight driving, the ends must be at right angles to the length. Instead of driving piles to moderate depths it, mav at times be better to mereh plant them hutt down in holes bored by an auger like Pierce's Well Borer. See p 636. This'will avoid shaking adjacent buildings. See " In Mobile Bay," p 325. The ultimate friction of piles even with the bark on, and driven about 3 ft apart from cen to cen probably never much exceeds about 1 ton per sq ft even when well driven into dense moist sand or loamy gravel ; nor more than .5 to .75 of a ton in common soils and clays; or than .1 to .'2 of a ton in silt or wet river mud depending on the depth and density. The friction of cast iron cylinders seems to be about .3 that of piles. There is a great difference in the penetrability of different ands. Thus, in the Lary bridge, no special uitticulty was found in driving piles ;io ft into deep wet sand : while, in other wet localities, piles of very toi.gh wood, well shod with iron, cannot be driven 6 ft into sand, without being battered to pieces. The same difference has been found in the case of screw-piles. At the Brandywine light-house these could not be forced more than 10 ft into the clean wet sand. Stiff wet clay (and clean gravels) also differ very much in this respect. Generally they are penetrable to any required depth with comparative ease : but. we have seen stout hemlock piles battered to pieces iti driving 6 ft through wet gravel; and Mr. Rendel found that at Plymouth he "could not by any force drive screw-piles more than about 5 ft into the clay, which is not ILK stiff as the London clay," on which the foremen tioned new London and Blackfriars bridges were lounded; and into which'even ordinary wooden piles were driven 20 ft without special difficulty. A mixture of mud with the sand or gravel facilitates driving very much ; but before beginning aii trouble and expense that may be anticipated. Mere boring will often be but a poor substitute for this. As a general rule, a heavy hammer with a low fall, drives more pleasantly than a light one with a high fall. Where a hammer of % ton (1500 fi>s) falling 25 ft, in a very strong ground, shattered the piles; one of 2 tons, (4500 Ibs,) with 7 ft fall, drove them satisfactorily. More blows can be made in the same time with a low fall; and this gives less time for the soil to compact itself around the piles between the bkrtvs. At times a pile may resist the hammer after sinking some distance; but start again after a short rest; or it may refuse a heavy hammer, and start under a lighter one. It may drive slowly at first, and more rapidly afterward, from causes that may be difficult to discover. The driving of one sometimes causes adjacent ones previously driven, to spring upward several feet. A pile is in the most favorable position when its foot rests upon rock, after its entire length has been driven through a firm soil, which affords perfect protection against its bending like an overloaded column; and at the same time creates great friction against its sides; thus as^sting much in sus- taining the load, and thereby relieving the pressure upon the foot. A pile may rest upon rock, and yet be very weak ; for if driven through very soft soil, all the pressure is borne by the sharp point ; and the pile becomes merely a column in a worse condition than a pillar with one rounded end. See Fig I, page 233, Strength of Iron Pillars. In such soils the piles need very little sharpening ; indeed, had better be driven without any ; or even butt end down. The driving of a pile in soft ground or mud will generally cause an adjacent one previously driven, to lean outwards unless means be taken to prevent it. In piling an area of firm soil it is best to begin at its center and work outwards; otherwise the soil may become so consolidated that the central ones can scarcely be driven at all. Elastic reaction of the soil has been known to cause entire piled areas to rise, together with the piles, before they were built upon. In very firm soil, especially if stony; or even in soft soil, if the piles are pointed, and are to be driven to rock ; tlieir feet should be protected by shoes of either wrought iron, as at a, a, and /;, Figs 13 ; spiked to the pile by means of the iron straps r?, forged to them; or of cast iron, as at c, where the shoe is a solid inverted cone, the wide flat base of which affords a good bearing for the flat bottom of the pile-point. The dotted line is a stout wrought-iron spike, well se- cured in the cone, which is cast around it; this holdd the shoe to the pile. Regular Fit/13 The cost of a floating steam pile driver, in Phiiada, scow 2* ft by 50 ft. draft 18 ins, with one engine for driving, and one (to save time) for getting another pile ready ; with one ton hammer, is about $6000; and $500 more will add a circular saw. Ac. for sawing off piles at any reqd depth. Requires engineman, cook, and 4 or 5 others. Will burn about half a ton of coal per day. Driving 20 ft into gravel, and sawing off. will average from 15 to '20 piles per day of 10 hours. In mud about twice as many. Ou land about half as many as in water. A gunpowder driver, scow, Ac, costs about $1500 more; but will do about twice as much work as a steam driver. To drive a pile 20 ft into mud averages about one third ft> of powder: into gravel 4 times as much. One for use on land $1000 to $3000. The U. S. Gunpowder Pile Driving Co. No. 10 S. Delaware Avenue, Pailada, contract for both kinds of machines ; also for dredging and piling. 324 FOUNDATIONS. wrought-iron shoes will generally weigh 18 to 30 l%s ; but sheet iron may be used when the soil Is but moderately compact; plate iron when more so ; and solid iron or steel points, from 2 to 4 ins square at the butt, and 4 to 8 ins long, when very compact and stony. Holes may be drilled in rock for receiving the points of piles, and thus preventing them from slipping ; by first driving down a tube, as a guide to the drill, after the earth is cleaned out of the tube. To preserve the heads to some extent from splitting under the blows of the hammer, they are usually surrounded by a hoop h, Fig d ; from % to 1 inch thick ; and l?i to 3 ins wide. These are, however, sometimes but imperfect aids ; for in hard driving the head will crush, split, and bulge out on all sides, frequently for many feet below the hoop: moreover, the hoops often split open. The heads, therefore, often have to be sawed, or pared off several times before the pile is completely driven; and allowance must be made for this loss in ordering piles for any giveu work; especially in hard soil. Capt Turnbull, U S Top Eng, states that at the Potomac aqueduct, his pileheads were preserved from injury by the simple expedient of dishing them out to a depth of about an inch, and covering them by a loose plate of sheet iron ; as shown in section at c, Figs 13. A verv slight degree of brooming or crushing of the head, materially diminishes the force of the ram. Piles may be driven through small loose rubble without much labor. Shaw's driver does not injure the heads. Piles which foot on sloping rock may slide when loaded. To drive a pile head below water a wooden punch, or follower, as atp, Figs 13, may be used. The foot of this punch fits into the upper part of a casting //. round or square, according to the shape of the pile; and having a transverse partition o o. The lower part f the casting is fitted to the head of the pile t; and the hammer falls on top of the punch. AVhea driving piles vertically in very soft soil, to support retainiug-walls, or other structures exposed to horizontal or inclined forces, care must be taken that these forces do not pu>h over the piles them- selves ; for in such soils piles are adapted to resist vertical forces only, unless they be driven at an inclination corresponding to the oblique force. A broken pile may be drawn out, or at least be started, if not very firmly driven, by attaching scows to it at low water, depending on the rising tide to loosen it. Or a long timber may be used as a lever, with the head of an adjacent pile for its fulcrum. Or a crab worked bv the engine of the pile driver. In very difficult cases the method devised by Mr J. Monroe, C E, may" be used. A 4 inch gas pipe 15 ft long, shod with a solid steel point, and having an outer shoulder for sustaining a circular punch, was thereby driven close to and 2 or 3 ft deeper than two Rilea driven 12 ft, in 37 ft water, and broken off by ice. Four pounds of powder were then deposited i the lower end of the pipe, and exploded, lifting the piles completely out of place. It will often be best to let a broken pile remain, and to drive another close to it. May be drawn by hydraulic press. Ice adheres to piles with a force of about 30 to 40 Iba per sq inch, and iii rising water may lift them out of place if not sufficiently driven. Iron piles and cylinders. Cast iron in various shapes has been much used in Europe for sheet piles; especially when intended to remain as a facing for the protection of concrete work, filled in behind and against them.* Cast-iron cylinders, open at both ends, may be used as bearing piles ; and may be cleaned out, and filled with concrete, if required. The friction in driving is greater than in solid piles, inasmuch as it takes place along both the inner and the outer surfaces. This may be diminished by gradually extracting the inside soil as they go down. Thev require much care, and a lighter hammer, or less fall than wooden ones, to prevent breaking; to which end a piece of wood should be interposed between the hammer and the pile ; or the ram may be of wood. But it is better to use them in the shape of screw cylinders, which, moreover, gives them the advantage of a broad base, as in the following. See foot of p 631. Brunei's process. He experimented with an open cast-iron cylinder, 3 ft outer diarn ; 1% ins thick ; in lengths of 10 ft, connected together by internal socket and joggle joints, secured by pins, and run with lead. It had a sharp-edged hoop or cutter at bottom : and a little above this, one turn of a screw, with a pitch of 7 ins, and projecting one foot all around the outside of the cylinder. By means of capstan bars and wincbe.s, he screwed this down through stiff clay and sand, 58 feet to rock, on the bank of a river. In descending this distance the cylinder made 142 revolutions ; sinking on an average about 5 ins at each. The time occupied in actually screwing was 48J4 hours; or about ly^j- ft per hour. There were, however, many long intervals of rest for clean- ing away the soil in the inside. After resting, there was no great difficulty in restarting. The next fig will give an idea of the arrangement of the screw. The screw-pile of Alex. Mitchell, Belfast, consists usually of a rolled iron shaft A, Figs 14. from 3 to 8 ins diam ; and having at its foot a cast-iron screw S S S. with a blade of from 18 ins to 5 ft diam. The screws used for light-houses, exposed to moderate seas, or heavy ice-fields, are ordinarily about 3 ft diam, have \y% turns or threads, and weigh about 600 Ibs. The round rolled shafts sire from 5 to 8 ins diam. They are screwed down from 10 to 20 ft into clay, sand, or coral, by about 30 to 40 men, pushing with 6 to 8 capstan bars, the ends of which describe a circle of about 30 to 40 ft diam.' For this purpose a platform on piles has frequently to be prepared. In quiet water, this may be supported on scows ; or a raft well moored may be used when the driving is easy ; or the deck of a large scow with a well-hole in the center for the pile to pass through. Roughly made temporary cribs, filled with stone and sunk, might support a platform in some positions. The platform must evidently be able to resist revolving horizontally under the great pushing force of the men at the capstan bars ; and on this account it is difficult to drive screws to a sufficient depth, in clean compact sand, by means of a floating platform. The feet of the piles must be firmly secured to the screws, to prevent * Cast iron, intended to resist sea-water, should be close-grained, hard, white metal. ' In such, the small quantity of contained carbon is chemically combined with the metal; but in the l gravel which form the substratum for a great depth. Into this last they were sunk about 8 or V ft farther, by excavating the inside earth under water, by mea:>8 of an inverted conical screw-pan, or dredger, of % inch plate iron. This was 2 ft greatest diam, and 1 ft deep ; and to its bottom was attached a screw about 1 ft long, for assisting in screwing it down into the soil. Its sides had openings for the entrance of the soil ; and leather flaps, opening inward, to prevent its escape. From opposite sides of the pan, 3 rods of % inch diam projected upward 4 feet, and were there forged together, and connected by an eye-and-bolt joint to a long rod or shaft, at the upper end of which was a four-armed cross-handle, by which the pan was screwed down by 4 men on the staging. " "When a pan was full, a slide which passed over the joint at the bottom -was lifted ; and the pan was raised by a tackle. This pan raised about 1 cub ft at a time. A smaller one of only 1 ft diam, and 1 ft deep, raising about y cub ft, was used when the material was very hard. By this means the cylinders were sunk at the rate of from 2 to 18 ins per day. The slow rate of 2 ins was caused by stones, some of them of 50 H>s. These were first loosened by a screw-pick, which was a bar of iron 3 ft long, with circular arms 12 ins long projecting from the sides. After being loosened by this, the stones were raised by the pan. The expense of all this apparatus was very trifling; and the ex- cavation was done easily and cheaply. After the excavation was finished, aud the cylinder sunk, before pumping out the water, concrete (gravel 2, hydraulic cement 1 measure) was filled in to the depth of 12 feet, by means of a large pan with a movable bottom ; and about 12 days were left it to harden. The water was then pumped out, and the masonry built in open air. In some of the cylin- ders, however, the water rose so fast, notwithstanding the 12 ft of concrete, that the pumps could not keep them clear ; and 6 ft more of concrete had to be added in those. Finally random-stone, or rough dry rubble, was thrown in around the outside* of the cylinders, to preserve them from blows arad undermining." * The masonry extends 20 ft above the cylinders, and above water. The vacuum; and the plenum processes. We can barely allude to the general principles of these two modes of sinking large hollow iron cylinders. In the vacuum invented by Dr. Lawrence Holkor Polls, of London, the cylinder c, Fig 16, while being sunk, is closed air-tight at top, by a trap door, opening upward. A flexible pipe p, of India- rubber, long enough to adapt itself to the sinking of the cylinder, and provided with a stopcock .s on every sq inch ; or nearly 1 ton per sq ft of area of the top. Consequently the cylinder is forced downward in the bed of the river, by this amount of pressure, in addition to its own weight. At the same time, the pressure of the air upon the surface of the water is transmitted through the water to the soil around the open foot of the cylinder: so that if this soil be soft or semi-fluid, it will be pressed up into the nearly void cylinder, in which is no downward pressure to resist it. The descent varies from a few inches, to 4 or 5 ft each time. The process is then repeated, by admitting air again into the cylin- der, opening the trap-door, removing the water and soil, as before, 4 Inch boiler iron. The foot C D, nine timbers high, is continuous, extending eutirely around the caisson; its bottom is shod with cast iron ; its four corners are strengthened by wooden knees 20 ft long. From the bottom, up to the line N, N, 14 ft, the caisson is built of horizontal layers of timbers one foot square; the layers crossing each other at right angles; and the timbers of each layer touching each other well forced and bolted together; and all the joints filled with pitch. To aid in preventing leakage, the nuts and heads of the screws have India-rubber washers ; also all outside seams, as well as all the seams of the layer of timbers N, N, are thoroughly calked; and a layer of tin, enclosed between two layers of fejt, is placed outside of each outer joint; and over the entire top of the layer next helow N, N. When the caisson was built up to N, N, on land, it was launched, floated into position, and anchored ; after which were added for sinking it, fifteen courses of timbers one ft square; and laid one ft apart in the clear ; with the intervals filled with concrete. The top course A B is of solid timber, to serve as a floor for supporting machinery, &c. It was sunk some feet below the very bottom of the river, in order to avoid the teredo. Cribs are sunk outside of the caisson, to form temporary wharves for boats carrving away excavated material ; and for vessels bringing stone, &c. When the caisson was sunk, and the water forced out from the chamber or space CSS D. workmen began to excavate uniformly the enclosed area of river bottom, so as to allow the caisson to descend slowly untilit reached a firm substratum. The space C S S D, as well as the shafts, was then filled up aolid with concrete masonry. A coffer-dam was built on top of the caisson; and in it the regular masonrv of the tower was started. The total height of this tower including the caisson, is about 300 ft. For full details see report. 1873, of W. A. Roebling the chief engineer. A CLUMP OF PILES WELL DRIVEN; and then enclosed by an iron cylinder sunk to a firm bearing, and filled with concrete, is an excellent foundation. The piles may extend to the top of the cylinder, and thus be enclosed in the concrete. Such ail arrangement has been patented by S. B. Gushing, C. K., Providence, R. I. The cyl- inder and concrete serve to protect the piles from sea-worms, and from decay above low water ; and are not intended to support the load above them. ost of a diving dress, with air-pump and tubes, about $1000. Alfred Hale & Co, 332 Washington St, Boston, Mass. Two men can work the air-pump to 50 ft depth. COST OF DEEDGING. DREDGING is generally done by skilled contractors, who own the requisite machines, scows or lighters, &c ; and who make it a specialty. It is necessary to specify whether the dredged material is to be measured in place before it is loosened; or after being deposited in the scow: because it occupies more bulk after being dredged. It was found, in the extensive dredgings for deepening the River St Lawrence through the Lake of St Peter, that on an average a cub yd of tolerably stiff mud in place, makes 1.4 yds in the scow ; or 1 in the scow, makes .715 in place. Also stipulate whether the removal of bowlders, sunken trees, &c, is to constitute an extra. These often require sawing and blasting under water. The-cost per cub yd for dredging varies much with the depth of water: the quantity and character of the material : thedist to which it has to be removed; whether it can be at once discharged from the machine by means of projecting side-shoots or slides ; or must be discharged into scows, to be re- moved to a short dist by poling, or to a greater dist by steam tugs ; whether it can be dropped or dumped into deep water by means of flap or trap doors in the bottom of the hoppers of the scows ; or must be shovelled from the scows into shallow water, (at say 4 to 8 cts per yd ;) or upon land, (at say from 6 to 10 or 20 cts for the shovelling alone, or shovelling and wheeling, as the case may be ;) whether much time must be consumed in moving the machine forward frequently, as when the excavation is narrow, and of but little depth; as in deepening a canal, &c; whether many bowl- ders and sunken trees are to be lifted ; whether interruptions may occur from waves in storms; whether fuel can be readily obtained, Ac, &c. These considerations may make the cost per cub yd in one c?ise from 2 to 4 times as great as in another. The actual cost of deepening a ship-channel through Lake St Peter, to IS ft, from its orig- inal depth of 11 ft, for several miles through moderately stiff mud, was 14 cts per cub yd in place, or 10 cts in the scows; including removing the material by steam tugs to a dist of about V^a mile, and dropping it into deep water. This includes re- pairs of plant of all kinds, but no profit. It was a favorable case. When the buckets work in deep water they do not become so well filled as when the water is shallower, because they have a more vertical movement, and, therefore, do not scrape along as great a distance of the bottom. Hence one reason why deep dredging costs more per yard; in addition to having to bertifted through a greater height. Perhaps the following table is tolerably approximate for large works in ordinary mud, sand, or gravel ; assuming the plant to have been paid for by the company ; and that common labor costs $1 per day. 330 COST OF DREDGING. Table of actual cost of d redgi iig on a large scale; inclnd iiiS dropping tlie material into scows, alongside; or into side-shoots, 011 board. Common labor $1 per day. Repairs of plant are included ; but no pro lit to contractor. (Original.] Cts per Yard, 'Cts per Yard, iu place. j iu scow. Depth Cts per Yard, Cts per Yard, Depth iu Ft. iu place. IU SCOW. iu Ft. Less than 10 8.4 6 25 to 30 10 to 15 9.8 7 80 to 35 15 to 20 11.2 8 35 to 40 20 to 25 14.0 10 18.2 25.2 35.0 13 18 25 For to wing of the scows by steam tugs to a dist of % mik', and dropping the mud into deep water, add 4 cts per yard iu the scow ; for > mile, 6 cts ; for % mil. . b cts ; for 1 mile, 10 cis Add profit to con- tractor. On a small scale work is done to a less advantage; and a corresponding increase must be made in these prices. Also, if the contractor himself furnishes the dredgers and plant, a still further addi- tion must be made. It is evident that the subject admits of no great precision. Small jobs, even iu favorable material, but iu inconvenient positions, may readily cost two or three limes as much per yd as the above: and in very hard material, as iu ceme'iitt-cl gravel and clay, four or five times as much for the dredging. The cost of towing, however, will remain as before, if wages are the same. At present Wages, iSTe. if a contractor provides all the plant for dredging and towing, the total cost, including his profit, will be about 3 time- the above table. The cost of dredgers, tugs. &c, will vary of course with their capabilities, strength of construction, style of finish, whether having accommodations for the men to live on board or not. &c. When for use in salt water, the bottoms of both dredgers and scows should be coppered, to protect them from sea- worms; and if occasionally exposed to high waves, both should be extra strong. The most powerful machines on the St Lawrence cost about $15000 each ; and removed in 10 working hours on an average about 1800 cub vds iu place, or 25'20 in the scows. Good machines, capable, under similar circumstances, of doing as much, may, however, be built for about $25000 to $30000 To remove this quantity to a dist of % tol mile, would require two steam tugs, costing about $000 to $10000 each; and 4 to 6 scows, (some to be loading while otners are awav,) holding from 30 to 60 cub yds each ; and costing from $800 to $1500 each at the shop. Scows with two hoppers are best. Such a dredger would require at least 8 or 10 men, including captain, engineer, fireman, and cook. Each tug 4 or 5 men ; and each scow 2 men. The engineer should be a blacksmith : or a blacksmith should be added. In certain cases a physician, clerk, assistant engineer, &c. mav be needed. Dredgers are often built on the principle of the Yankee Excavator, with but a single bucket or dip- Ser, of from 1 to 2 cub yds capacity. Hull about 25 by 60 ft. Draft 3 ft. Cylinder about 7 or 8 ins iam; 15 to 18 inch stroke; ordinary working pressure 50 to 80 Ibs per sq inch, according to hardness of material. Cost $8000 to $12000. Will raise as an average days' work ^10 hours) from 200 to 500 yds in place, or 280 to 700 in the scow, according to the depth, nature of the material, Ac. Require 5 or 7 men in all aboard, including cook. Burn J^ to 1 ton of coal daily. Tolerably large bowlders, and sunken logs, can be raided by the dipper.* When the material is hard and compacted, the buckets of dredgers should be armed with strong steel teeth projecting from their cutting edge. On arriving at such material, every alternate bucket is sometimes unshipped. By arranging the buckets so as to dredge a few feet in advance of the hull, low tongues of dry land may be cut away ; the machine thus digging its own channel. The dailj work iu such cases will not average half as much as in wet soil. On small operations, dredgers worked by two or more horses, instead of by steam, will answer very well in soft material ; or e.ven in moderately hard, by reducing the size and number of the buckets. " A two-horse machine will raise from 50 to 100 yards of ordinary mud in place, or 70 to 140 in the scow, per day, at from 12 to 15 ft depth. Soft material in small quantity, and at moderate depth, may be removed by the low and expensive mode of the bag-scoop, or bag-spoon. This is simply a bag B, made of canvas or leather, and having Its mouth surrounded by au oval iron ring, the lower part of which ia sharpened to form :i cutting ed?e. It has a fixed handle h. and a swivel handle . One man pushes the bag down into the mud by ft, while another pulls it along by the rope g; and when filled, another raises it bv the rope c. and empties it. If the bag is large, a wind- lass may he used f >r raising it. The men may work from a scow or raft properlv anchored. Or a long-handled metal spoon, shaped like a deeply-dished hoe. mav be used by only one man ; or a larger spoon may be (ruined by a man, and dragged forward and backward by a horse walking in a circle on the scow, &c, &c. The weight of a ciib yd <>f wet dredged mud, pure sand, or gravel, averages about \Y Z tons; say 111 Ths per cub ft: muddy gravel, full l ] / tons; say l'2-i IDS per cub ft. l'ur<; sand or gravel dredge easily; also bed.s of shells Wet dredged clay will slide down a shoot inclined at from 1 to ft. to 1 to 3, according to Us freedom from sand. c ; but wet sand or gravel will not slide down even ii to 1, without a free flow of water to ai-1 it; otherwise it requires much pushing. )f The writer has seen cases in which a circular saw for logs in deep water, would have been a very useful addition to a dredger. It should be worked by steam; and be adjustable to different depths. It would cost but about $500. The American Dredging C'O, No. 10 8. Delaware Avenue, Philada, make both dredgers and pile drivers of many patterns ; and contract for dredging and piling on any scale. RETAINING-WALLS. 331 RETAINIM-WALLS, Art. 1. We here speak only of walls sustaining earth; for those sustaining water, see Hydrostatics. A retaining -wall is one for sustaining the pres of earth, sand, or other filling or backing, deposited behind it after it is built ; in distinction to a face- wall, which is a similar structure for preventing the fall of earth which is in its undisturbed natural position, but in which a vert or inclined face has been excavated. The earth is then in so consolidated a condition as to exert little or no lateral pres, and therefore the wall may generally be thinner than a retaining one. This, however, will depend upon the nature and position of the strata in which the face is cut. If the strata are of rock, with interposed beds of clay, earth, or sand ; and if they dip or incline toward the wall, it may require to be of far greater thickness than any ordinary retaining- wall ; because when the thin seams of earth become softened by infiltrating rain, they act as lubrics, like soap, or tallow, to fa- cilitate the sliding of the rock strata ; and thus bring an enormous pres against the wall. Or the rock may be set in motion by the action of frost upon the clay seams ; or, as sometimes occurs, by the tremor pro- duced by passing trains. Even if there be no rock, still if the strata of soil dip toward the wall, there will always be danger of a similar result; and addi- tional precautions mast be adopted, especially when the strata reach to a much greater height than the wall. A vertical wall has both c o and d vert. Experience, rather than theo- ry, must be our guide in the building of both kinds of wall. We recommend that the hor thickness a 6, Fig 1, at the base of a vert or nearly vert retaining-wall c d b a, which sustains a backing of either s;md, gravel, or earth, level with its top c d, as in the fig, should not be less than the following, in railroad practice, when the foundations are not more than about three feet deep. When the backing? is deposited loosely, as usual, as when dumped from carts, cars, &c. Wall of cut-stone, or of Jirst-class large ranged rubble, in mortar a.b 35 of its entire vert height d b. ; good common xcabbled mortar-rubble, or brick. A " " ' " well-scabbled dry rubble 5 " " " " With good masonry, however, we may take the height d 8 instead of d b, and then the above proportions of d s will give a sufficient thickness at the ground-line o s. See Table, p 338. When the backing is somewhat consolidated in hor layers, each of these thicknesses may be reduced, but no rule can be given for this. The offset o e, in front of the wall, is not included in these thicknesses. When, however, the backing is a pure clean sand, or gravel, we should use only the full dimen- sions; inasmuch as the tremor, caused by passing traius, would neutralize any supposrd advantage from ramming materials so devoid of cohesion. Such sand may be rammed with much advantage for the purpose of compacting it in foundations; but a diff principle is involved in that cuse. When it is done even with cohesive earths, with a view of saving masonry in retaininp-walls. it is probable that the expense will generally be found quite equal to that of the masonry saved. See Rem 4, p 339. The base ah in Fig 1. is J^ of the height bd. In the foregoing thicknesses at base, the back d b of the wall is supposed to be vert; and the face ca either vert, or battered (sloped or inclined back- ward) to an extent not exceeding about 1^ inches to a foot; which limit it is rarely advisable to PX- ceed in practice, owing to the bnd effect of rain, &c. upon the mortar when the batter is great. The base of a vert wall need not in fact be as thick as one with a battered face ; but when the batter does not exceed 1.5 inches to a foot, the diff is very small. See Table, Art 7 p 338. REM. 1. A mixture of sand, or earth, with a larg-e proportion OP ROUND BOWI.DKRS. paving pebbles, Ac, will welch considerably more thnn the material* ordinarily used for backing; and will exert a greater pres aeainst the wall : the thickness of which should be increased, say about J^th to %th part, when such backing has to be used. RKM. 2. The wall will be stronger if all the courses of masonry be laid with an inclination inward, as at oe.b-, especially if of dry masonry, or if time cannot be allowed (as it always should be, when practicable) for the mor- tar to set properly, before the backing is deposited behind it. The object of inolin- 332 RETAINING-WALLS. ing the courses, is to place the joints more nearly at right angles to the direction /P, Figs . 7, and 8, of the pres against the back of the wall ; and thus diminish the tendency of the stones to slide on one another, and cause the wall to bulge. See Art 19 of Force in Rigid Bodies. When the courses are hor, there is nothing to pre- vent this sliding, except the friction of the stones, one upon the other, when of dry masonry ; or friction and the mortar, when the last is used. But if, as is frequently the case, (especially in thick and hastily built walls,) this has not had time to harden properly, it will oppose but little resistance to sliding. But when the courses are inclined, they cannot slide, without at the same time being lifted up the inclined planes formed by themselves. In retaining-walls, as in the abuts of important arches, the engineer should place as little dependence as possible upon moriar; but should rely more upon the position of the joints, fur stability. An objection to this inclining of the joints in dry (without mortar) walls, is that rain-water, falling on the battered face, is thereby carried inward to the earth backing: which thus becomes soft, and settles. This may be in a great measure obviated by laying the outer or face-courses hor; or by using mortar for a depth of ouly about a foot from the face. The top of the wall should be protected by a coping c d, Fig 1, which had better project a few ins in front. After the masonry has been built up to the surface of the ground, the foundation pit should be filled up; and it is well to' con- solidate the filling by ramming, especially in front of the wall. The back d b of the wall should be left rough. In brickwork it would be well to let every third or fourth course project an inch or two. This increases the friction of the eurth against tlie back, and thus causes the resultant of the forces acting behind the wall to become more nearly vert; and to fall farther within the base, giving increased stability. It also con- duces to strength not to make each course of uniform height throughout the thickness of the wall; three courses. By this means the whole masonry becomes niore'efl'tctually interlocked or bonded together as one mass: and therefore less liable to bulge. Very thick walls may consist of a facing of masonry, and a backing of concrete. REM. 3. It is the pres itself of the earth against the back, that creates the friction, which in turn modifies the action of the pres ; as the wt or pres of a body upon an inclined plane produces friction between the body and the plane, sufficient, perhaps, to prevent tlie body from sliding down it. A re- taining- wall is overthrown by being made to revolve around its outer toe or edge e Fig 1. as n. ful- crum, or turning-point; but 'in order thus to revolve, its back must first plainly rise; and in doing so must rul) against the backing, and thus encounter and overcome this triction. The friction exists the same, whether the wall stands firm or not; as in the case of the body on an inclined plane ; the only did' is that in one case it prevents motion ; and in the other ouly retards it. Where deep freezing: occurs the back of the wall should be sloped forwards for 3 or 4 ft below its top as at c o, which should be quite smooth so as to lessen the hold of the frost and prevent displacement. REM. 4. When the wall is too thin, it will generally fail by bulging? outward, at about ^ of its height above the ground, as at a, in Fig - J. A slight bulging in a new wall does not necessarily prove it to be actually unsafe. It is generally due to the newness of the mortar, and to the greater pres exerted by the fresh backing; and will often cease to increase after a few months. It need not excite apprehension if it does not exceed ^ inch for each foot in thickness at a. S^e Remark 3. Art 7, p 339. > The young engineer need not in practice concern himself particularly about the PRECIS- SP GRAY OF HIS BACKING, or about the ANGLE OF SLOPS: at which it will stand ; for the material which he deposits behind his wall one day, may be dry and incoherent, so HS to slope at 1% to 1 ; the next he deposits behind his wall one day, may be dry and incoherent, so HS to slope at 1% to 1 ; the next day rain may convert it into liquid mud. seeking its own level, like water ; the next it may be ice. capable of sustaining a considerable load, as a vert pillar. Moreover, he cannot foretell what may be the nature of his backing; for. as a general rule, this- must consist of whatever the adjacent excavation may produce from time to time : sand to-day, rock to-morrow, &c. Retaining-walls are therefore usually built before the engineer knows the character of their backing; so that in practice, these theoretical considerations have comparatively but little weight. Theory, uncontrolled by observation and common sense, will lead to great errors in every department of engineering ; but, on the other hand, no amount of experience alone will compensate for an ignorance of theory. The two must go hand-in-hand. Again, the settlement of the backing under its own wt, aided by the tremors produced by heavy trains at high speed; its expansion by frost, or by the infiltration of rain; the hydrostatic pressure arising from the admission of the latter through cracks produced in the backing during long droughts ; as well as its lubricating action upon it, (diminishing its friction, and giving it a tendency to slide,) &c, exert at times quite as powerful an overturning tendency as the legitimate theoretical pres does. The action of these agencies is gradual. Careful observation of retaining-walls year after year, will often show that their battered faces are be- coming vertical. Then they will begin to incline outward; and eventually the wall will fail. Theory omits loads that may come on backing increasing its pres. RETAINING- WALLS. 333 Assuming the theoretical views advanced by Professor Moseley to be correct as theories, the thicknesses which we have recommended in Art 1, for mortar walls, correspond to from 7 to 14 times; and for dry walls about 10 to 20 times, the pros' assigned by him ; and we do not consider ours greater than experience has shown to be necessary. See Table 3. Retaining-walls designed by good engineers, but in too close accordance with theory, (which assumes that a resistance equal to twice the theoretical pres is sufficient,) have failed; and the inference is fair that many of those which stand have too small a coefficient of safety. The fact is, (or at least so it appears to us,) there mast be defects in the theoretical assumptions of ome of the most prominent writers who give practical rules on this subject. Thus Poncelet, who certainly is at their head, states that his tables, for practical use, give thicknesses of base for sus- taining 1 j8j. times the theoretrcal pres ; and this he considers amply safe. Yet, for a vert wall of cut granite, his base for sustaining dry sand level with the top, as in Fig 1, is .35 of the vert height- and for brick, .45. But the writer found that when not subject to tremor, a wooden model of a vert wall weighing but 28 fi>s per cub ft, and with a base of .35 of its height, balanced perfectly dry sand sloping at 1% to 1, and weighing 89 fts per cub ft. :n TARIES AS THEIR SPECIFIC GRAVITIES ; and, since granite weighs about 1H5 Ibs per cub foot, or 6 times as much as our model, it follows, we conceive, that a wall of that material, with a base of .35 of its height, must have a resistance of 6 times any true theoretical pres, instead of only 1.8 times; and that his brick wall must have about 5 times the mere" bal- ancing resistance. Our experiments were made in an upper room of a strongly built dwelling ; and we found that the tremor produced by pass- ing vehicles in the street, by the shutting of doors, and walking about the room, sufficed to gradually produce leaning in walls of considerably more than twice the mere balancing stability while quiet : and it appears to us that the injurious effects of a heavy train would be comparatively quite as great upon an actual retaining-wall, supporting so incohesive a material as dry sand. Since, therefore, Poncelet's wall is in this instance sufficiently stable for practice, it seems to us that his theory, which neglects the effect of tremors, &c, must be defective. He also gives 4 of the height as a suf- ficiently safe thickness for a vert granite wall supporting stiff earth; but we suspect that very few engineers would be willing to trust to that pro- portion, when, as usual, the earth is dumped in from carts, or cars; espe- cially during a rainy period. If deposited, and consolidated in layers, theory could scarcely assign any thickness for the wall ; for the backing thus becomes, as it were, mass of unburnt brick, exerting no hor thrust ; and requiring nothing but protection from atmospheric influence, to insure its stability without any retaining-vaill. It is with great diffidence, and distrust in our opinions, that we venture to express doubts respecting the assumptions of so profound an in- vestigator and writer as Poncelet; and we do so only with the hope that the views of more compe- tent persons than ourselves, may be thereby elicited. Our own have no better foundation than ex- periments with wooden and brick models, by ourselves ; combined with observation of actual walls. Art. 3. After a wall a b c o, Fig 3, with a vert back, has been proportioned by our rule in Art 1, it may be converted into one. with an oflfsetted back, as a i n o. This will present greater resistance to overturning; and yet con- tain no more material. Thus, through the center t of the back, draw any line i n ; from n draw n . y P = _ _ In view of the great uncertainty involved in the matter of the actual pressure of earth against retainiug-walls in practice (see Art 2, p 332), arid in order to furnish a simple rule which, although entirely unsupported by theory, is still (in the writer's opinion) sufficiently approximate for ordinary practical purposes, we shall assume that No 1 of the two foregoing formulas applies near enough to walls with in- clined backs c w, also, as Figs 7 and 8, (precisely as they are lettered,) at least until the back of the wall inclines forward as much as tt ins hor, to 1 foot vert, or at an angle cmo of 26 34'. What follows on retaiiiing-walls will involve this incorrect assumption, and must be regarded merely as giving safe approximation. Some appear to assume this perp pres to be the only one acting against the back of the wall; and hence arrive at erroneous practical conclusions. For when, in order to prevent this force from causing the triangle of earth to slide, we place a retaining-wall in front of it, then, instead of motion, the force will produce pres of the earth against the wall ; (see Art 3, of Force, p 445.) But in producing pres, it r 1 > f 336 RETAINING- WALLS. C necessarily produces the new force of friction, between the pressed surfaces of the earth and wall. That is, it' a wall were to begin to overturn around its toe a as a fulcrum, its back c m must of course rise, and iu so doing must rub against the earth filling in contact with it; and this rubbing would evidently act to impede the overturning. So long as the wall does not move, the same friction assists in pre- venting overturning. To ascertain the amount and effect of this friction, let ;/P, Fig 8, represent by scale, the force perp to the back c m; and supposed to have been pre- viously calculated by the foregoing formula No 1. Make the angle yPf equal to the angle of wall friction,* draw ' yf at right angles to y P, or parallel to m c ; make P x equal to yf, and com- Elete the parallelogram P yfx. Then will x P represent y the same scale, the amount of the friction ag-aiiist the back of the wall. Since the fric- tion acts in the direction of the back c m, (see end of Art 62, of Force in Rigid Bodies,) it may be considered as acting at any point P, in that line ; (see Art 18, of Force at page J5-.) Hence we have acting at P, two forces ; namely, the perp force y P, and the friction x l>; conse- quently, by comp and res of force, the diag/P of the parallelogram P yfx, if measured by the same scale, will give us the amount of their resultant ; which is the approx single theoretical force, both in amount and in direction, which the wall has to resist, including the wall friction. o But this force, /P, is also always equal to the perp H O force y P, mult by the nat sec of the angle y Pf of sa the wall friction ; (or divided by its nat cosine) and of \ course may be ascertained thus : vt of triangle v . v nat sec of angle y P f wt of ^ A . Approx theoreti _ c m t * * * of wall friction cmt* l cal pros f P vert depth o r cos y P f X o m Or finally, if it is assumed, as we do throughout, that the earth is perfectly dry (in asmuch as its pressure is then the greatest) and that the angles of nat slope, and of wall friction are then each 33 41' or 1.5 to 1, then in Figs 6, 7 and 8, if the angle cm n between the back c m and the vert o m does not exceed about 26 34' we may assume Approx theoretical pres f P = wt of triangle c m t x .643 which includes the action of the friction of the earth against the back of the wall. REM. 1. When the back of the wall is offset ted or stepped, as in Fig 3, instead of being simply battered, as in Figs 7 and 8, the direction of the pres of the earth will be the same as if the back had the batter i n, on the principle given in Art 34, Fig 17, of Force in Rigid Bodies, p 464. REM. 2. Now to find both the overturning- tendency of the earth, and the resistance of the wall against being overturned around its toe a as a fulcrum, first find the ceu of grav g of the wall (p 442), and through it draw a vert line g h. Prolong /P towards rand draw a v perp to it. By any scale make o = wt of wall, and si = calculated pres /P. Complete the parallelogram sino, and draw its diagonal * n, which will be the resultant of the pres/P and of the wt of the wall ; and should for safety be such that aj be not less than about one-fifth of a m, even with best masont-y and unyielding soil. Otherwise the great pressure so n^ar the toe a may either fracture the wall or compress the soil near that point so that the wall will lean forward. In walls built by our rule, Art 1, or by table, p 338, aj will be more than one-fifth of a m. The pres / P if mult by its leverage a v will give the moment of the pres about a; and the wt of the wall mult by its leverage e a will give that of the wall. The wall is safe from overturning in pro- portion as its moment exceeds that of the pres. It is assumed to be safe against sliding, breaking, or settling into the soil. See Art 15, p 530. *This aiigrle of wall friction is that at which a plane of masonry must b inclitip'1 to the horizontal so that dry sand or earth would slide down it. It is about the same as the oat slope, or 33 41', or 1.5 to 1 ; and its oat secant is 1.202, and its nat cos .832. RKTAINJNG-WALLS. 337 Item. 4. If the earth slopes downward from 0, as at A or B, instead of being hor as in Figs 6, 7, 8, use the wt of the earth c m n instead of c m t, m n being the slope of max pressure. In A the point of application will still be at P (at one third of m C) as in 6, 7, 8 ; but in B it will be a little higher as explained below for Fig 9. Surcharged walls are those in which the earth backing extends above the tops of the walls. According to theory, when as in Fig 9, there is a surcharge v c k of backing, sloping away from c at its natural slope c v, the max pres against the wall is attained when the earth reaches to the level of t a higher point h, tlum one I. P drawn through the cen of grav of the trianirle of earth d r. m, or of t c m: which latter line intersects it at P, at one- third of it< heisrht from the ha-e m n. So that with the small surcharge, we have not only the pres of a greater quantity of parth than t c. m. hut a greater leverage is given to K tor overthiowing the wail. But when 'he snrohavir" i>s made to reach d. then the earth d c m. being like t c m a triangle, the line through it-- ren r.f grav \v II strike the hick'- m at P, at the same height a* that through t cm; or lower down, and with a shorter leverage, thau that through mnr cm. The difi is still greater when the surcharge starts at 6. 338 RETAINING-WALLS. Art. 7. On page 331, Fig 1, we recommend that the base o a at the ground- line of well built vortical walls should not be less than .35, or .4, or .6 of the height ds above said line, depending on the kind of masonry. But a wall with a battered (inclined) front or face as found by Art 8, p 3 >9 (by which the following table was prepared), will be as strong, and at the same time contain less masonry than a vert wall, although the battered one will have the thickest base os. Table 3, of thicknesses at base o s 9 Fig? 1, and at top c ^ to 1 ; sind this we consider the proper one to be used in practical calculations, where safety is the consideration of paramount importance. The less the nat slope, the greater Is the pres : and sinc^ the slope is least when the backing is perfectly dry, (omitting of course its condition when so absolutely \u?i as to become partially fluid,) we have, on the score of safety, confined our tables to dry backing. As stated in Art 1, we cannot recommend dimen- sions less than those there given, when we consider the rough treatment to which masonry is exposed on public works. In carrying: a road along; dang-erons precipices, we should rather be tempted at times to make thicker walls. We imagine, for instance, that the centrifugal force of a heavy train, whirling around a sharp curve, convex on the dangerous side, should not be overlooked in designing walls for such localities. This force is hor; and is applied near the top of the wall : and, consequently, its leverage may be considered as equal to the height: whereas the theoretical pres of the earth is oblique : and is applied at ^ of the height from the bottom ; so that its leverage about the toe of the wall is very short. Moreover, the simple weiffht of the train, pro- duces pres against the wall ; as well as that of the backing. All' such considerations are omitted by theorists. The dangerous pres caused by tremors, Ac, cannot be RETAINING- WALLS. 339 assumed to be applied at % of the height from the bottom ; nor indeed, can it be calculated at all. REM. 2. Wharf walls are an instance where the thickness should be increased, notwithstanding that the pres of the water in front helps to sustain them. The earth behind such walls, is not only liable to be very heavily loaded when vessels are dis- charging; but is apt to become saturated with water, especially below low-water level; and thus to exert a very great pres against the walls. Moreover, the water gets under the wall; and by its upward pressure virtually reduces its weight, and consequently its stability. The same cause of course diminishes the friction of the wall upon its base. Such walls are, therefore, very liable to slide, if the foundation is smooth, and horizontal ; and have done so even when the foundation had a con- siderable inclination backward, as in Fig 1. See Art 9. REM. 3. A retaining-wall is usually in greater danger for a few months after its completion, than after time has been allowed for the mortar to harden perfectly ; and for the backing to settle. When there are suspicions of the safety of a new wall, it would be well to place strong temporary shores against it, at about % to % of its height above ground. In some cases, permanent buttresses of masonrv may be built for the purpose. They should be well bonded into the wall. REM. 4. The pres of the earth backing will be much reduced, if the first few feet of its height be made up in thin hor layers, to be consolidated by being used by the masons instead of scaffolding; as snown at A, Fig 1. Frequently this can be done without inconvenience ; and at very trifling cost. Art. 8. To change a vert retaining-wall, into one with a battered face, which shall present an equal resistance against overturning; although requiring less masonry. This is sometimes termed a transformation of profile. (Original.) Let a b o i, Fig 10, be the vert wall. Mult its bas o t, by 1.225 ; (1.22475 is nearer ;) the prod will be the -i i base o e, of a triangular wall b o e, possessing the _J. a s ft same stability; and yet not requiring much more than half the masonry of the vert one. See Rem 1. This being done, suppose a wall to be desired with a face batter, of say 3 ins to a ft ; or 1 in 4. From the point n, where the face of the triangular wall inter- sects that of the vert one. step off vert any 4 short equal spaces ; and from the upper one m, step off one space hor, to v. Through v and n draw the dotted line s t, which evidently will batter 1 in 4. Then is b s t o approximately the reqd wall ; but a little . thicker than necessary. To reduce it, from t draw the dotted line t b. Mark the point c, where this line intersects the face a i. of the vert wall; and through c draw d I, parallel to s t. Then Is b d I o the reqd wall. Our fig is drawn in an exaggerated manner, so as to avoid confusion in the lines. The base o e of the triangular wall, would not in reality be near so great as it is represented. FiglO i I t It will be observed that as the base increases, the quantity of masonry diminishes. REM. 1. The battered wall will in fact be safer than the vert one, and for this reason ; although the wall after being transformed from a vert one. to one with a battered face, (and consequently with a greater base,) has the same moment of sta- bility as before ; and although the amount of pres of the earth against it. also remains unchanged, still the new wall is better able to resist this pres than before. In other words, the moment or pu- dency of the pres to upset the wall, has become Jess. For let a ?> m i, Fig 11, represent a vertical wall : and fo the amount and direction of pres behind it. Now, the leverage with which this pres tends to overturn t.he wall around its toe m, is the dist m 8, measured from the toe or fulcrum m, and at right angles to the direction fo s c of the pres ; and this leverage mult by the force / o, gives the overturning tendency or moment of naid force. See "Moments and leverage," p 473. Again, let any. represent a triangular wall of the same stability as the other, as found by our rule. Here we still have the same mount fo. and direction fo s c. of pres force against the wall; but it now acts to overturn the wall any around the toe y : and then-fore, with the reduced leverage y ~. Consequently, its overturning tendency is less than before. Therefore, in ordinary language, we may say that the wall is stronger than before, although its moment of stability, or standing tendency, has in itself undergone no change. If the pres/o against the vert back were hor, as in the case of water, then its leverage would evidently be the same in both walls; and the proportion between the overturning moment of the pres, and the moments of stabilitv of the two walls, would be constant. P 528. REM. 2. In attempting to reduce the masonry by adopt- ing a wall, o b e. Fig 10. of a triangular section : or of one nearly approaching a triangle, special attention should be given to the quality of the masonry near the thin toe e; which will otherwise be apt to crack, or fail under the preg. 340 RETAINING- WALLS. Eg 13 MOREOVER, WHEN COMMON MORTAR is useis. WITHOUT AN ADMIXTURE OP CEMENT, which It never should be, in retaiuiug-walls,whereduraOility is au object, a great batter is objec- tionable; inasmuch as the rain, combined with frost, &c, soon destroys the uior- tar. In such cases, therefore, the batter suould not exceed 1 orl^jiustoa ft; and even then, at least the poiiuiug of tne joints, and a few feet in height of both the upper and the lower courses of masonry, should be doue with cement, or cement-mortar. We have observed a most marked diffiu the corrosion of the mor- tar, where, in the same walls, with the same exposure, one portion has been built with a vert face ; and another with a batter of but 1>$ inch to a foot. Common mortar will never set properly, and continue firm, when it is exposed to mois- ture from the earth. This is very observable near the tops and bottoms of abuts, retaining- walls. > of being overturned. This is very apt to occur if it is built upon a hor wooden platform ; or upon a level surf of rock, or clay, without other means than more friction to prevent sliding. This may be obviated by inclining the base, as in Fig I ; by founding the wall at such a depth as to pro- vide a proper resistance from the soil in .front; or in case of a platform, by securing one or more lines of strong beams to its upper surf, across the direction in which sliding would take place. On wet clay, friction may be as low as from .2 to y the weight of tne wall ; on dry earth, it is about ^ to % ; and on sand or gravel, about % to %. The friction of masonry on a wooden fcj platform, is about -6^ o f the wt, if dry ; and % if wet. __~| Counterforts, shown in plan at c c c, Fig 13. consist in an increase of the thickness of the wall, at its back, at regular inter- vals of its length. Wo conceive them to be but little better than a waste of masonry. When a wall of this kind fails, it almost in- variably separates from its counterforts : to which it is connected merely by the adhesion of the mortar ; and to a slight extent, by the bonding of the masonry. The table in Art 7 shows that a very small addition to the base of a wall, is attended by a great increase of its strength; we therefore think that the masonry of counterforts would be much better, and more cheaply employed in giving the wall an additional thickness, along its entire length; and for the lower third of its heignt. Counterforts are very generally used ic retaining-walls by European engineers; but rarely, if ever, by Americans. Buttresses are like counterforts, except that they are placed in front of a wall instead of be- hind it; and that their pronle is generally triangular, or nearly so. They greatly increase its strength; but. being unsightly, are seldom used, except as a remedy when a wall is seen to be failing. Laitd-tieS, or long rods of iron, have been employed as a makeshift for upholding weak re- . taining-wa]ls. Extending through the wall from its face, the laud ends are connected with anchors of masonry, cast-iron or wooden posts; the whole being at some dist below the surface. Retaining walls With curved profiles are mentioned here merely to cau- tion the young engineer against building them. Although sanctioned by the practice of some high, authorities, they really possess no merit sufficient to compensate for the additional expense and trou- ble of their construction. Art. 1O. Among military men, a retaining-wall is called a revetment. When the arth is level with the top, a SCarp revetment; when above it. a counterscarp revetment, or a demi-revetment. When the face of the wall is battered, a sloping ; and when the back is battered, a countersloping revetment. The batter is called the till IIS. Art. 11. Our own experiments lead us to believe that the pressure against a wall, Fig 9, from dry sand, ck, level with the top, is not at all dimin- ished by reducing the width cs of the sand, until it becomes about one-sixth of the dist c, pertaining to the angle cms of natural slope. With less than that limit, the pres was plainly diminished. It would therefore seem to be dangerous to make a retaining-wall thin, merely because the backing does not entend to t.* Table 4. of contents in cub yards for each foot in length of retaining-walls, with a thickness at base equal to .4 of the vert height. Face batter, \% inches to a foot ; or J^th of the height. Back either vert, or stepped according to the rule in Art 3, Fig 3. The strength is very nearly equal to that of avert wall with a base of .4 its height. See table p 338. Experience has proved that such walls, when composed of well- s cabbled mortar rubble, are safe under all ordinary circumstances for earth level with the top. Steps or offsets, oe, at foot, Fig 1, are not here included. *'" A singular fact which the writer believes he was the first to notice in 1859, We cannot under- stand how correct results are to be expected from experiments like those of Gen. Pasley, and others, who confined their bucking in a box. placing their walls in front of it. The friction of the backing against the sides of the box must diminish the pres against the wall, and thus lead to adopting too slight a thickness. We conceive that an experimental wall should diminish in height and thickness, each way from its central portion, (preserving, however, the same proportion between the two,> until it terminates in a point at each end. Ours were made in this way. STONE BRIDGES. 341 TABLE 4. (Original.) Ht. Cub. Ht. Cub. Ht. Cub. Ht. Cub. Ht. Cub. Ht. Cub. Ft. Yds. Ft. Yds. Ft. Yds. Ft. Yds. Ft. Yds. Ft. Yd*. 1 .013 1034 1.38 20 5.00 29 10.9 48 28.8 74 68.5 M .028 11 1.51 X 5.25 30 11.3 49 30.0 76 722 2 .050 % 1.G5 21 5.51 31 12.0 50 31.3 78 761 H .078 12 1.80 K 5.78 32 12.8 51 325 80 80.0 3 .113 X 1.95 22 6.05 33 13.6 52 33.8 82 84.1 X .153 13 2.11 J4 6.33 34 14.5 53 35.1 84 88.4 4 .200 X 2.23 23 661 35 15.3 54 36.5 86 92.5 .253 14 2.45 1 A 6.90 86 16.2 55 37.8 88 96.8 5 .313 H J.63 21 7.20 37 17.1 56 39.2 90 101.3 H .378 15 2.81 J4 7.50 38 18.1 57 40.6 92 105.8 6 .450 X 3.00 7.81 39 190 58 42.1 94 1105 X .528 16 3. '21) >* 8.13 40 20.0 59 43.5 96 115.2 7 .613 Vi 3.40 26 8.45 41 21.0 60 45.0 98 120.1 X 1 ."03 17 3.61 X 8.78 42 22.1 62 48.1 100 125.0 8 .800 M 3.83 27 9.12 43 23.1 64 51.2 102 130.1 X .903 18 4.05 ^ 945 44 24.2 66 54.5 104 135.2 9 1.01 H 4.28 28 9.80 45 25.3 68 57.8 106 140.5 K 1.13 19 4.51 J4 102 46 26.5 70 61.3 10 1.25 X 4.75 29 1 10.5 47 27.6 72 648 See preceding footnote. STONE BEIDGES, Art. 1. In an arch st s, Fig 1, the dist eo is called its span ; t'a its rise; t its Crown ; its lower boundary line, *ao, its soffit, or intrados ; the upper one, rtr, its back, or extrados. The terms soffit and back are also applied to the entire lower and upper curved surfaces of the whole arch. The ends of an arch, or the showing areas comprised between its intrados and extrados, are its faces ; thurf the area st s a is a face. The inclined surfaces or joints, re, r<>, upon which the feet of the arch rest, or from which the arch springs, are the skewbacks. Lines level with e and o, at right angles to the faces of the arch, and forming the lower edges of its fret, (see nn, Fig 2^,) are the spria- in- lines, or springs. The blocks of which the arch itself is composed, are the arcth-stones, or vonssoirs. The center one, la, is the keystone; and the lowest ones, ss. the springers. The term archblock might be substituted for voussoir, and like it would apply to brick or other material, as well as to stone. The parts tr,tr, are the ll aim dies ; and the spaces / r /, t r b, above these, are the spandrels. The material deposited in these spaces is the spandrel filling; ; it is sometimes earth, sometimes ma- soriry ; or partly of each, as in Fig 1. In large arches.it often consists of several parallel SPANDREL-WALLS. II, Fig 2**j, running lengthwise of the rondway, or astraddle of the arch. They are covered at top either by small arches from wall to wall, or by fla't stones, for supporting the material of the roadway. They are also at times connected together by vert cross- walls at intervals, for steadying them laterally, as at tt. Fig 2^. The parts gpen, gpon, Figl, are the ABUTMENTS of the arch; en, on, the 'faces ; gp, gp. the backs; and p n, p n, the bases of the abuts. The bases are usually widened by feet, steps, or offsets, d d. for dis- tributing the wt of the bridge over a greater area of foundation ; thus diminishing the danger of set- tlement. The distance t a in any arch-stone, is called its depth. The only arches in common use ibr bridges, are the circular, (often called segmental); and the elliptic. Art. 2. To find the depth of keystone for cut-stone arches, whether circular or elliptic.* (Original.) Find the rad eo, Fig 1, which will touch the arch at o, a, and e. Add together this rad, and half the span o e. Take the sq rt of the sum. Div this sq rt by 4. To the quot add fa of a ft. Or by formula, * Inasmuch as the rules which we give for arches and abuts are entirely ordinal and novel, it may uot be arnias to state that they are not altogether empirical ; bat are based upon accurate drawing* 342 STONE BRIDGES. Depth of key __ (v ' Rad + half span\ , o /Vw in feet \ 4 / For second-class work, this depth may be increased about i/th part; or for brick or fair rubble, about ^tli. See table of Keystones, p 345. In large arches it is advisable to increase the depth of the archstones toward the springs ; but when the span is as small as about 60 to 80 or 100 feet, this is not at all HKWxxary if the stone is good; although the arch will be stronger if it is done. In practice this increase, even in the largest spans, does not exceed from % to 1 A tny depth of the key ; although theory would require much more in arches of great rise. See Kx 2, p 46S, also Ex 6, p 478, of Force in Rigid Bodies. REM. To find the rad c o, whether the arch be circular or elliptic. Square half the span P, o. Square the whole rise i a. Add these squares together: div the sum by twice the rise i a. Or it may be found near enough for this purpose by the dividers, from a small arch drawn to a scale. Amount of pressure sustained by archstones. IF t>r jges of the same width of roadway, if all the other parts bore to each other the same proportion t*s t je spans, the total pres, as well as the pret per q ft, in arches of the same proportionate rise, wojld increase as the squares of the spans. But in practice the depth of the archstones increases much less rapidly than the span ; while the thickness of the roadway material, and the extraneous load per sq ft, remain the same whether the span be great or small. Hence the pressures increase less rapidly than the squares of the spans. Thus in two bridges of the same width, but with spans of 100 and 200 ft, with depths of archstones taken from our table page 345, and uniform from key to spring; supposed to be filled up solid with masonry of 160 tbs per cub ft, to a level of about 15 inches above the crown, (in- cluding the stone paving of the roadway); with an extraneous load of 100 B>s per sq ft; and the leverages y m, y o, Fig 52, p 479, measured from the center y of the skewback ; the pressures will be approximately as follows : Span 1OO ft. AT KEY. For 1 ft in width of its entire depth. Per sq ft. AT 81 For 1 ft in width of its entire depth. RING. Per sq ft. AT For 1 ft in width of its entire depth. KEY. Per sq ft. AT SP For 1 ft in width of its entire depth. RING. Per sq ft. Rise. w 1 X X Tons. ox MX 31 14 25 18 Tons. 13* 12k 11 9 6% Tons. 58 57 57^ 61^ 67^ Tons. 18^ 19 20 mi 25 Tons. 126 112 97 80 \i 67 fc S UK 24>$ 21 lH Tons. 179 181 188 207 230 Tons. 42 44 47J4 54 y 4 61^ Span 200 ft. Here it is seen that the entire pros of the 200 ft span, with different rises, averages but little more than three times that of the 100 ft one ; while the pres per sq ft of each stone averages but nbout 2^ times that of the 100 ft one. It will also be seen that with the same span, the pres at the key becomes less, while that at the spring becomes greater, as the rise increases. Also that when the archstones are of uniform depth, the pres at either spring of a semicircular arch is about 4 times as great as at the key ; whereas when the rise is but % of the span, the pres at spring averages but about % greater than at the key. These proportions vary somewhat in different spans. The greater pres per sq ft at the springs may be reduced by increasing the depth of the archstones towards the springs. This however is not necessary in moderato spans, inasmuch as good stone will be safe even under this greater pres. By USillg: parallel spandrel Walls, see Fig 2^, p 346, or by partly filling with earth instead of masonry, the pres on the archstones may be diminished, say, as a rough average, about -i- part. and calculations made by the writer, of lines of pres, Ac. of arches from 1 to 300 ft span, and of every STONE BRIDGES. 343 Table 1. Of some existing- arches, with both their actual, and their calculated depths (by our rule) of keystone. Where the two depths are given in the last column, the smallest is for first class cut-stone, and the largest for good rubble, or brick. Those also which are not specified are of first-class cut-stone. C stands for circular, E for elliptic. For 2d class work, add about %th part; and for brick, or fair rubble, about J4th. I a W Sa HI I I 1 SKOJe-a H Q ill o Ammanati. Telford. Telford. Fisk. i hi S S3 02 tf M IB t^Z 533 - S^05W*i (N 53... S3 s i j s2 ; 1 H 4jy*; i g3 s SII SS S 53S 9 & s 5eS eS 5 i 11s I 2 gStS-'tt e^ 8 ^ 3* O i ' i 35 t^ tn 2 OSt^ Jrt t- 2 ... a 83583 5 ^! S ooo iO issss sssse Si S 25S S S 3 5:?;^ OOKSSO O i ; : i i 2 OHO III OOO O Q OOO W KO KO MOO oow e, in cement. Carries ne in Portland cement s a * ; lubble ia cement London Bridge. England Gloucester, across the Severn, England Dora Riparia, Turin, Italy Pout de Alma, Paris, France. Small rough rubbl Soupes, France. An experimental arch of cut-sto Waterloo. Across the Thames at London Tougueland, England. Turnpike s en I 1 .1 d "o 'K I Holy Trinity, Florence. Very slightly pointed... Dean Bridge, Scotland. Turnpike Dunkeld " " Licking Aqueduct. Chesapeake & Ohio Canal ... Royal Border Viaduct, England. Brick in cemen Posen Viaduct, Germany. Brick in cement Allenton. England. Turnpike Falls Bridge, Philadelphia & Reading R R Staines, England. Turnpike Brent R R Viaduct, England. Brick in cement . . d a *5c- S 6. c - 3 H & r; 5| fe cs c a> C22S | M 2 S 1 1 m 3 3 1 O Monocacy Aqueduct, Chesapeake & Ohio Canal... Llaurwast, Wales. Turnpike Avon Viaduct. England. Brick in cement Tonoloway Culvert, under Cbes & Ohio Canal. I Philadelphia & Reading R R " Edinburg & Dalkeith R R. Scotland * The width is not given in any account that we have seen, but probably it was not more than 5 or 6 ft. It. was tested by a distributed load of 360 tons, and by a weight of 5 tons falling 18 ins ; and was uninjured thereby." Total settlement before and after being loaded, about 1^ ins. The depth of arch- atones was increased gradually from 2.67 ft at key, to 3.6 ft at springs. Mortar joints full ^ iunh thick. Rise about y^-th of the span. 344 STONE BRIDGES. The arch on the BOURBONNAIS RAILWAY, is probably the boldest;* and THE CABIN JOHN ARCH, by Oapt, now Gen'l M. 0. Meigs, US Army, the grandest stone one in existence. PONT-Y PRYDD, in Wales, is a common road bridge, of very rude construction ; with a dangerously steep roadway. It was built entirely of rubble, in mortar, by a common country mason, in 1750; and is still in perfect condition. Only the outer, or showing arch-stones, are 2.5 ft deep ; and that depth is made up of two stones. The inner arch-stones are but. 1.5 ft deep ; and but from 6 to 9 inches thick. The stone quar- ried with tolerably fair natural beds; and received little or no dressing in addition. The bridge is a fine example of that ignorance which often passes for boldness. PONT NAPOLEON carries a railroad across the Seine at Paris. The arches are of the uniform depth of 4 ft, from crown to spring. They are composed chiefly of sm>ill rough quarry chips, or spawls ; well washed, to free them from dirt and dust; and then thoroughly bedded in good cement; and grouted with the same. It is in fact an arch of cement-concrete. The' PONT DE ALMA, near it, and built in the same way, has elliptic arches of from 126 to 141 ft span ; with rises of i the span. Key 4.9 ft. These two bridges, considering tb want of precedent in this kind of construction, on so large a scale, must be regarded as verv bold; and as reflecting the highest credit for practical science, upon their engineers, Darcel and Couche. Some trouble arose from the unequal contraction of the different thicknesses of cement. They show what may be readily accomplished in arches of moderate spans, by means of small stone, and good hydraulic cement when large stone fit for arches is not procurable. In Pont Napoleon the depth of arch is barely what our rule gives for second class cut-stone. REM. Our engineers are usually toosparing- of cement. Itshould be freely used, not only in the arches themselves, and in the masonry above them, as a protection from rain-soakage ; but in abuts, wing-walls, retaining-walls, and all other important masonry exposed to dampness. The entire backs of important brick arches should be covered with a layer of good cement, about an inch thick. The want of it can be seen throughout most of our public works. The common mortar will be found to be decayed, and falling down from the soffits of arches; and from the joints of masonry generally, within from 3 to 6 ft of the surface of the ground. The mois- ture rises by capillary attraction, to that dist above the surf of the nat soil ; or descends to it from the artificial surf of embankments, &c ; therefore, cement-mortar should be employed in those portions at least. The mortar in the faces of battered walls, even when the batter is but 1 to !*/ inches per foot, is far more injured by rain and exposure, than in vert ones ; and should therefore be of the best quality. See MORTAR, Ac. We have, however, seen a quite free percolation of surface water through brick arches of nearly 3 ft in depth, even when cement was freely used. In aqueduct bridges, we believe that cement has not been found to prevent leaks, whether the arches were of brick, or even of cut-stone. May not this be the effect of cracks produced by settlement of the arch; or by contraction and expansion under atmos- pheric influence? Cement at any rate prevents the joints from crumbling. Art. 3. The keystones for lararo elliptic arches by the best en- gineers, appear generally to be about % part deeper than our rule requires ; or than is considered necessary for circular ones of the same span and rise. We do not. how- ever, see any sufficient reason for this. The elliptic arch, with its spandrel filling. Jl ^^-TT^^i -It lms slightly less wt; and that wt ha* a trifle less leverage than in a circulai one; and consequently it exerts less pres both at the key, and at the skew- back. See London, Gloucester, and Waterloo bridges, in the preceding table. See upper footnote p 349. RKM. Young engineers arc apt to affect shallow arch-stones; but it would be far better to adop the opposite course: for not only do deep ones make a more stable structure, but a thin arch is ai unsightly an object as too slender a column. According to our own taste, 'arch-stones fullv X .ieepe: than our rule gives for first-class cut stone, are greatly to be preferred when appearance is' consulted Especially when an arch is of rough rubble, which costs about the same whether it is built up ai arch, or as spandrel filling, it is mere folly to make the arches shallow. Stability and durability ohould be the objects aimed at; and when they can be attained even to excess, without increased cost it is best to do so. Rem. Brick arches, from their great number of joints are apt to settle much more than cut stone ont-s when the centers are removed, and thereby tc derange the shape of the arch, and at times, without due care, even to endanger its safety, especially if it b;3 large and flat. When the span exceeds about oO to 3 ft, and particularly if flat, use only brick of superior quality in good cement mortar. With even best materials and work we adviso the young engineer not to attempt brick arches for railroad bridges of greater spans than about the fol- lowing. Considerably larger ones than some of them h:w<> been built, and have stood ; but their coefs of safety are not in all cases satisfactory. In this table the rise is in parts of the span. For more on brick arches, see Art 10, p 674. STONE BRIDGES. 345 Table 2. Depths of keystones for arches of first-class cut stone, by Art 2. For second class add full one-eighth part ; and for superior brick one- fourth to one-third part, if the span exceeds about 15 or 20 ft. Rise, in parts of the span. Original. SPAN. Feet. 1 i I t i i * TV Key. Ft. Key. Ft. Key. Ft. Key. Ft. Key. Ft. Key. Ft. Key. Ft. 2 .55 .56 .58 .60 .61 .64 .68 4 .70 .72 .74 .76 .79 .83 .88 6 .81 .83 .86 .89 .92 .97 1.03 8 .91 .93 .96 1.00 1.03 1.09 1.16 10 .99 KOI 1.04 1.07 1.11 1.18 1.26 15 1.17 1.19 1.22 1.26 1.30 1.40 1.50 20 1.32 1.35 1.3S 1.43 1 48 1.59 1.70 25 1.45 1.4H 1.53 1.58 1.64 1.76 1.88 30 1.57 1.60 1.65 1.71 1.78 1.91 2.04 35 1 68 1.70 1.76 1.83 1.90 2.04 219 40 1.78 1.81 1.88 1.95 2.03 2.18 2.33 50 1.97 200 2.08. 2.16 2.25 2.41 2.58 60 2.14 2.18 2.26 2.35 2.44 2.62 2.80 80 2.44 2.49 2.58 2.68 2.78 298 3.18 100 2.70 2.75 2.86 2.97 3.09 3.32 3.55 120 2.94 2.99 3.10 3.22 8.35 3 61 3.88 140 3.16 3.21 3.33 3.46 3.60 3.87 4.15 160 3.36 3.44 3.58 3.72 3.87 4.17 180 3.56 3.63 3.75 3.90 4.06 4.38 200 3.74 3.81 3.95 4.12 4.29 220 3.91 4.00 4.13 4.30 4.48 240 4.07 4.15 4.30 4.48 260 4.23 4.31 4.47 4.66 280 4.38 4.46 4.63 300 4.53 4.62 4.80 Art. 4. To proportion the abuts for an arch of stone or brick, whether circular or elliptic. (Original.) The writer ventures to offer the following rule, in the belief that it will be found to combine the requirements of theory with those of economy and ease of applica- tion, to perhaps as great an extent as is attainable in an endeavor to reduce so com- plicated a subject, to a simple and reliable working? rule for prac- tical bridge-builders. This is all that he claims for it/Notwithstanding its simplicity, it is the result of much labor on his part. It applies equally to the smallest culvert, and to the largest bridge ; whatever may be the proportions of span and rise ; and to any height of abut whatever. It applies also to all the usual methods of filling above the arch ; whether with solid masonry to the level vf, Fig 2, of the top of the arch; or entirely with earth ; or partly with each, as represented in the fig: or with parallel spandrel-walls extending to the back of the abut, as in Fig 2}^. Although the stability of an abut cannot remain precisely the same under all these conditions, yet the diff of thickness which would follow from a strict investigation of each par- ticular case, is not sufficient to warrant us in embarrassing a rule intended for popu- lar use. by a multitude of exceptions and modifications which would defeat the very object for which it was designed. We shall not touch upon the theory of arches, except in the way of incidental allusion to it. Theories for arches, and their abuts, omit all consideration of passing loads; and consequently are entirely inapplicable in practice when, as is frequently the case, (especially in railroad bridges of moderate spans,) the load bears a large ratio to the wt of the arch itself. Hence the theoretical line of thrust has no place in such cases. Our rule is intended for common practice : and we conceive that no error of practical importance will attend its application to any case whatever ; whether the arch be circular or elliptic. It gives a thickness of abut, which, without any backing of earth behind it, is safe in itself, and in all cases, against the pres. when the bridge is unloaded. Moreover, in very large arches, in which the greatest load likely to come upon them in practice is small in comparison with the wt of the arch itself, and the filling above it, our abuts would also be safe from the loaded bridge, without any dependence upon the earth behind them ; but as the arches become less, and consequently the wt of the load becomes greater in proportion to that of the arch, and of the filling above it, we must depend more and more upon the resistance of the earth behind the abuts, in order to avoid the neces- sity of giving the latter an extravagant thickness. It will therefore be understood throughout, that our rules suppose that after the bridge is finished, earth will be deposited behvtid the abuts, and to the height of the roadway, as usual; except when parallel spandrel walls are used. 346 STONE BRIDGES. To proportion the abutments of a stone bridge. RULE. Find, as in Art 2, the rad c o, Fig 2, in ft, which will touch the arch at o, a, and e. Div this rad by 5. To the quot add y 1 ^ of the rise, and 2 ft. The sum will be the thickness on or ey of each abut at the springing line, or the level from which the arch starts for any abut whose height os does not exceed 1% times the base sp. If of rough rubble add 6 ins to insure full thickness in every part. Thicks o n of abut at spring in ft, when the height o s does not exceed \% times the base 8 p Rad in ft 6 rise in ft 10 . * J Mark the points n and y thus ascertained. Next, from the center t, of the span or chord e o, lay off i />, equal to ^ part of the span. Join a h ; and through n, and parallel to a//, draw the indefinite line gnp of the abut. Do the same with the other abut. Make y m and ng each equal to half the entire height i t of the arch ; and from g draw a straight line g x, touching the back of the arch as high up as pos- sible ; or still better, as shown at t m, with a rad d t or d TO, (to be found by trial,) describe an arc t m. Then gx or t m, will be the top of the masonry filling above the arch.* Fi ;u Now find by trial the point s, Fig 2, at which the thickness sp is equal to two- * EXCEPT WHEN THE RISE IS BUT ABOUT 4- OF THE SPAN, OB LESS; in which case carry the masonry up solid to the level vtf, of the top of the arch. Or if the arch is a large one, ex- ceeding say about 60 ft span ; and especially if its rise is greater than about -^ of its span, it is better to economize masoqry by the use of parallel interior spandrel-walls, 1 1. Fig 2^. carried up to vtf. Fig 2. Indeed, such interior walls may often be advantageously intro- duced in much smaller arches. When high, they are steadied by occasional cross- walls, as tt, Pig 2^. Their feet should be . n Fig 2 the dark part the second cross* wall, similar to tt. STONE BRIDGES. 347 thirds of the corresponding vert height o s, and draw up. Then will the thickness on or ey be that at the springing line of the given circular or elliptic arch of any rise and span; and the lineg'j? will be the back of the abut; provided its height o"s does not exceed 1% times sp; or in other words, provided xp is not less than % of o s. In practice, sp will rarely exceed this limit; and only in arches of considerable rise. But if it should, as for instance at o 7, then make the base q u equal to sp, added to one-fourth of the additional height sg\ and draw the back uw, parallel to g p ; and extending to the same height, &c, as in Fig 2. If, however, this addition of % of * q should in any case give a base q u, less than one-half the total height o q, (which will very rarely happen in practice,) then make qu equal to half said total height ; drawing the back parallel to gp, and extending it to tne same height as before. The additional thicknesses thus found below sp, have reference rather to the pres of the earth behind the abut, than to the thrust of the arch. In a very high abut, the inner line g p would give a thickness too slight to sustain this earth safely. When the height o b, Fig 2, of the abut is less than the thickness on at spring, a small saving of masonry (not worth attending to, except in large flat arches) may be effected by reducing the thickness of the abut throughout, thus: Make ok equal to on, and draw kl. Make <>z equal to 3^ of on, and draw Iz. Then, for any height oh of abut less than on, draw h r, terminating in I z. This bv will be sufficient base, if the foundations are firm. The back of the abut will be drawn upward from v, parallel to g />, and terminating at the same height as g or w. REM. 1. All the abuts thus found will (with the provisions in Art 6) be safe, without any dependence upon the wing-walls ; no matter how high the embkt may extend above the top of the arch. If the bridge is narrow, and the inner faces of the wing-walls are consequently brought so near together as to afford material as- sistance to the abuts, the latter may be made thinner; but to what extent, must depend upon the judgment of the engineer. We, however, caution the young practitioner to be careful how he adopts dimensions less than those given by our rule. There are certain practical considerations, such as carelessness of workmanship; newness of the mortar; danger of undue strains when removing the centers; liability of derange- ment during the process of depositing the earth behind the abuts, and over the arch ; &c, which must not be overlooked; although it is impossible to reduce them to calculation. Whenever it can be done, the centers should remain in place until the embkt is finished; and for some time afterward, to allow the mortar to set well. RKM. '2. A good deal of liberty is sometimes taken, in reducing the quantity of masonry above the springing line of arches of considerable rise, and of moderate spans. When care is taken to leave the centers standing until the earth filling is completed above the arch, and behind its abuts, so that it may not be deranged by accident during that operation ; and when good cement is used instead of common mortar, such experiments may be tried with comparative safety ; especially with culvert arches, in which the depth of arch-stones is great in proportion to the span. They must, however, be left to the judgment of the engineer in charge ; as no specific rules can be laid down for them. They can hardly be regarded as legitimate practice, and we cannot recommend them. We have known nearly semicircular arches, of 30 to 40 ft span, to be thus built successfully, with scarcely a particle of masonry above the springs to back them. Such arches, however, are apt to fall, if at any future period the earth filling is removed, without taking the precaution to first build a center or some other support for them. Even when the embkt can be finished before the centers are removed, we cannot recommend (and that only in small spans) to do less than to make ng. Fig 2, equal to X of the total height it of the arch ; and from g so found, to draw a straight line touching the back of the arch as high up as possible. REM. 3. We have said nothing about battering the faces of the abuts, because in the crossing of streams, the batter either diminishes the water-way; or requires a greater span of arch. Such a batter, however, to the extent of from % to 1^ ins to a ft, is useful, like the offsets, for distributing the wt of the structure, and its embkt, over a greater area of foundation ; especially when the last is not naturally very firm ; or when the embkt extends to a considerable height above the arch. In our tables, Nos 3 and 5, of approximate quantities of masonry in semi- circular bridges of from 2 to 50 ft span, the faces are supposed to be vert. Art. 5. Abutment-piers. When a bridge consists of several arches, sus- tained by piers of only the usual thickness, if one arch should by accident of flood, or otherwise, be destroyed, the adjacent ones would overturn the piers; and arch after arch would then fall. To prevent this, it is usual in important bridges to make some of the piers sufficiently thick to resist the pres of the adjacent arches, in case of such an accident ; and thus preserve at least a portion of the bridge from ruin. Such are called abutment-piers. Our formula of _j_ - --- -j- 2 ft, for the thickness at spring; with the back battering as before, at the rate of ?X- of the span to the rise ; face vert ; will of itself (without any modification for great heights') give a perfectly safe abut-pier, for any unloaded bridge ; and to any height whatever ; due regard being had, however, to the consideration alluded to in the next Art. Thus, for an abut-pier as high as o q, Fig 2 ; or of any greater height ; it is only necessary first to find the thickness o n at spring as before ; and then draw the battered back gnp: extending it down to the base at B ; with- out adding % of the additional height q. This addition is made in the ease of abuts, that they 348 BTONE BRIDGES. may be secure from the pres of the earth behind them ; as well as from the pres of the arch ; a con. sideratiou which does not apply to abut-piers ; in which only tlie pres of the arch is to be resisted. But although the abut-pier tlius found by our formula, would be abundantly safe, yet its shape a b c o. Fig 3, is inadmissible. In practice it would be changed to one somewhat like that shown by the dotted lines ; having an equal degree of batter on both faces. This of course requires more masonry, with but little increase of stability ; but that cannot be avoided. WHEN AN ABUT-PIER is BUILT IN DEEP WATER, or in a shallow stream sub- ject to high freshets, care must be taken that water cannot find its way under the pier, and thus produce an upward pres, which will either diminish, or entirely counteract its efficiency as an abut. See Remark 2, Art 4, of Hy- dros t a lies, p 525. Art. 6. Inclination of the course's of masonry below the spring's of an arch. Although our fore- going rule gives a thickness of abut which cannot be overturned, or upset, by the pres of the arch, yet if the arch be of large span, and small rise, its great hor thrust may produce a sliding out- ward of the masonry near the level of the springs, if the stones are laid in hor courses; especially if the mortar has not set well. This danger, it is true, could be avoided by conflning the courses together by iron bolts and cramps ; or by increasing considerably the thickness of the abuts ; but the expense of doing either of these, leads to the cheaper expedient of inclining the masonry, as shown between o and n, Pig 4 ; the courses near o being steeper; and gradually becoming less steep near n. By this process the arch is virtually prolonged into the body of the abut, so far that when the inclination of the lower masonry ceases, as at n, the direction of the theoretical line of thrust, or of pres of the arch (rudely represented by the dotted curved line o n) is nearly at right angles to the joints of the hor masonry below n; and consequently, said thrust is unable to produce sliding at that point. Be- tween o and n, the line of pres is everywhere so nearly at right angles to the variously inclined joints, as to preclude the possibility of sliding in that interval also. See Art 63 of Force in Rigid Bodies.* The abut being thus safe throughout from both overturning and sliding, can fail only from defective foundations; or from the inferiority of the stone of which it ia built; and which, if soft, may be crushed. This inclination of the masonry is as neces- sary in an elliptic arch, Fig 4%, as in a circular one. The direction of the line of thrust naouniinan elliptic arch, differs but slightly from that in a circular one of the same span and rise. The amount of thrust is also nearly the same in both ; consequently, the same precautions for re- sisting it, must be adopted in each. The dotted line n a o u m, in Fig 4^, gives a tolerable general idea of the theoretical line of thrust in an elliptic arch, when the filling above the arch, and abuts, is either as repre- sented in the fig; or when it is wholly of earth, or * This curved line of pressures is found in the manner directed at Rem 1, p 492. and at Fig 25, p 531. Rankine, Moseley, and others call it the line of resistance, and ap- ply line of pressures to another line which need not be introduced in a practical consideration of abutments, walls, dams, &c. They however call any given point in our line a center of pressure, because at any part of the height of the abut such point shows where all the pressure or thrust may for many purposes be assumed to be concentrated. The perversion of common technical terms is reprehensible. Said other line had better been called the line of resultants. We, both here, and on p 493, 531, and elsewhere, call the one to which we refer the line of pressure, or of thrust, simply because bridge masons have no idea of any other line curving through an abut, and inasmuch as the pressure or thrust is greatest in that line they very properly so term it. Again, we have said above and elsewhere that the bed-joints should be nearly perp to this line of thrust. Theory properly requires them to be at right angles to the resultants which cut the bed-joints at this line of thrust. Still we know that on account of the friction of masonry we may with perfect safety vary as much as about 30 from a right angle to these resultants, without depending at all on the strength of the mortar ; see Art 63, p 487 ; and in using our rule of thumb on the next page for drawing in- clined bed-joints, we shall always be far within the limit of 30; and therefore fully safe from sliding. Oar rules do not call for this line, nor for anything more than the spaa, rise, and radius of the arch. STONE BRIDGES. 349 wholly of solid masonry; although both its amount, and its direction, will differ somewhat according as one or the other of these modes of filling is adopted.* The elliptic form is plainly unfavorable for uniting the arch-stones with the inclined masonry near the springs, so as to receive "the thrust properly ; or about at right angles to its res .Itant. In ordi- nary cases this difficulty may be overcome by making the joints of only the outside or showing arch- tones to conform to the elliptic curve; as between e and a; while the joints of the inner or hidden cues, may have the directions shown between g and u, nearly at right angles to the line of thrust. It will rarely happen, however, that the young engineer will have to construct elliptic arches of suffi- cient magnitude to require either this, or any equivalent expedient. For spans less than 50 ft, with rises not less than about ^ of the span, nothing of the kind is actually necessary, if the mortar is good, and has time to harden. In order to incline the masonry of any abut with sufficient accuracy, it would be necessary first to trace the curved line of pres of the given arch, as directed in Art 72 of Force in Rigid Bodies, so as to arrange the bed joints about at right angles to it at every point of its course; but we offer the following process as sufficing for all ordinary practical purposes; while its simplicity places it within the reach of the com- mon mason. In actual bridges the direction of tne actual thrust changes as the load is passing; therefore, in practice no given degree of Inclination of the abut masonry can conform to it precisely during the entire passage. Consequently, any excess of refinement in this particular, becomes simply ridiculous ; especially in small spans. Rule for inclining: the beds of the masonry in the abnts. Add together the rad cm, Fig 4; and the span of the arch. Div the sum by 5. To the quot add 3 ft. Make o t, on the rad, equal to the last sum. Then is t a central point, toward which to draw the directions of the beds, as in the fig. Draw t s hor, and from t as a center, describe the arc oy\o being the center of the uniform depth of the arch. From y lay off on the arc the dist y n, equal to one-sixth part of t y ; draw t n a. It will never be nec'ssary to C "line the masonry below this t n a. Neither need the inclination extend entirely to the face m i of the abut; but may stop at e, about half-way between i and n. From e upward, the inclination may extend forward to the line e m. See Foot-note p 348. * It will be seen that in consequence of the more sudden curvature of an elliptic arch, near the parts a and . the dotted line of thrust iu Fig 4J^ may pass entirely out from the arch-stones at those points, and enter again below. This causes a tendency in the parts of the arch in that vicinity to rise ; and thus permit the portions near the crown o, to descend. If the arch in Fig 4% were a seg- ment of a circle, the direction of the line of thrust would become but little changed; and the arch would consequently coincide with it more nearly than in the elliptic one; and hence would be more stable than this last. If the arch is a full ".emicircle, with stones of ordinary depth, the line of thrust wil! pass out of it, and into it again, as in the elliptic one ; producing the same deranging ten- dency. See Rem 3, p 493, of Force In Rigid Bodies. So long as theintrados is elliptic, or a full semi- circle, with keystones of the ordinary proportions, no increase in the depth of the arch-stones from the key toward the springs, will prevent the line of thrust from thus falling below the arch : or even materially change its direction in an actual bridge. Thus, if in Fig 4J^, the depth of the arch- stones were increased toward the springs, even as much as is shown by the dotted line w s, the line of thrust in the finished bridge would still be along the dotted line o an; the greater depth serving merely to distribute the thrust over a greater area. If, instead of solid masonry, a hollow vault were constructed at w to support the roadway, the line of thrust at a would rise a little ; and by thus building some parts of the spandrels hollow, and other parts solid, much may be done in the way of keeping tbsline of thrust within the arch-stones. The same could be accomplished by greatly increasing the depth of all the arch-stones; for iu that case a line parallel to o an, and passing through the center of the depth of the new keystone, as well as that of the new springer, would everywhere fall within thearch- Btones ; although not to the same extent in all of them. This may, in fact, be the reason for the great depth of key in elliptic arches, referred to in Article 3. In cases where appearances may be sac- rificed to strength, we might first find (by Rem 3, p 493, of Force in Rigid Bodies,) the line of thrust or pressure of a required arch ; and if, as in Fig 4^ above, U does not fall well within the arch-stones, we mav arrange the latter so as to coincide with it. Sucft a change in the arch-stones would of course in turn slightly change the direction of the line of thrust; but in ordinary practice this might be neglected. As a general rule, the change in the shape of the arch would be detected by but few per- sons; nor indeed, does it follow that the new curve must be at all ungraceful. As a segment arch becomes more flat, the line of thrust coincides more nearly with the arch-stones. The writer draws the following inference, from the results with many of his own diagrams. Front the center c, Fig 1^. from which any segmental arch whose rise does not exceed .4 of its span, IN described, draw two lines c o, c o. forming angles of 25 with c a. Then supposing the depth of the keystone t,o be according to our rule, make the depths of the arch- stones between o and , everywhere equal to 1^ times that of the key. From. the key increase the depth gradually to o and o. Then will the theoretical line of thrust of the arch (supposed to be filled up to the level of n n) everywhere fall within the actual arch-stones ; nowhere tpproachine; their extrados or intrados nearer than about % of the depth of the arch at the same point ; and this will be the case whether the filling be either earth, or solid masonry : or of parallel spandrel walls : or partly of masonry, and partly of earth ; both dis- posed as indicated by the line . t, or i, &c; although both the direction and intensity of the line of thrust will of course vary slightly under these different conditions. This appears to the writer to be an important fact in practice, as regards segmenta! arches of great spans. t The feet of both elliptic and semicircular arches are always made hor; bat it is plain from lig 4}. that this practice is at variance with correct principles of stability in the case of the ellipse. It is the same in the semicircle. In ordinary bridges of the latter form, the vert pres, or weight resting on each skewback. is (roughly speaking) usually about from 3% to 4 times the hor pres on the same ; and the total pres is about 4 times as great as the pres on the keystone. Therefore, theoretically, the skewback should usually be about 4 times as deep as the keystone ; and Us bed, instead of being hor, bould be inclined at the rate of ahont 1 vert to 4 hor. See Example 2, p 468, of Force in Rigid Bodiei. 23 350 STONE BRIDGES. S C When the arch is flat, this inclination may become so steep, especially in the upper parts, that struts, or shores of some kind, must be used for preventing the ma- sonry from sliding down, until the completion of the arch secures it from doing so. The hor courses between the face TO i, and the line o e, will aid somewhat in this respect. The method thus described, should be applied to all very largt arches whose rise is %, or less, of the span. As before remarked, it is not actually necessary in arches not exceeding about 50 ft span, and not flatter than ?- of the span. Indeed, if the earth filling can be deposited before the centers are removed, these limits may be con- siderably extended without danger. Still, since a certain degree of inclination is attended with very little trouble or expense, we would recommend for even such arches, a process somewhat like the follow- ing: From half the span take the rise. Div the rem by 3. Make o t, Fig 5, equal to the quot. Draw t n, and o m, hor. Div the angle t o m into two equal parts, by the line o a. Incline tl.e masonry so as to be parallel to o a, as far down as t n. The inclined course* may extend out to the face o t, or not, at pleasure. REM. 1. The necessity for inclining the courses becomes greater as the span in- creases; because the pres from arches of the same transverse width, but otherwise of the same proportions, and of diff spans, increases nearly aa the squares of the spans; whereas the thickness of the abut at the springs increases only nearly in the same proportion as the span itself. Hence, in the larger span, the abut at spring presents a less proportion of mortar-joint, and of friction, to prevent sliding, than in the smaller one. In practice, the pres of arches does not increase quite as mpidly as the squares of the spans, because the depth of the arch-stones does not increase as fast as the spans; thus reducing the relative wt of the large one considerably. REM. 2. To find the length (ah, Fig 7) . , from face to face of a culvert. From 1 .11. ct the height A t of the embkt, take the above ground height n a of the culvert; the rem will be the height ho of the embkt above the culvert. Then the reqd length a ft is plainly equal to the top width id of the embkt, added to the two dists as, cfc, which correspond to its steepness of side-slopes. Thus, if Jl t ^. the side-slope is, as usual, 1^ to 1, then as and c b will each be equal to 1% times o ft ; or the two together will be 3 times o h. So that if the width i d is 1 4 ft, and A o 5 ft, the length a b will be 14 -f (5 X 3) = 14 + 15 = 29 ft. Art. 7. The following tables, 3, 4, and 5, of quantities, will be found useful for expediting preliminary estimates ; for which purpose chiefly they are intended ; hence no pains have been taken to make them scrupulously correct, but rather a little in excess of the truth. The first column of Table 3 contains the total vert height o c, Fig 6, from the crown n of a semicircular arch, to the foundation or base gm of its abut. The other columns give ap- proximately the number of cub yds contained in each running foot, or foot in length of the culvert, or bridge, measured from end to end (face to face) of the arch proper; and including only the arch and its abuts, as shown in Fig 1 ; or in the half section opmgy in Fig 6; in- cluding footings to the abuts, but omitting the wing-walls (w n), and the spandrel-walls (*), Figs 6 and At the foot of each column is the approximate content in cub yds of the two spandrel- walls by themselves; one over each face of the arch. These spandrel- walls are calculated on the supposition that their thickness at base, at their junc- tion with the wing- walls, where their height is o greatest, is equal to -^ of their height at that point : except where that proportion gives a less thickness at top than 2J^ ft; and that they extend 2 ft (o a) above the top o of the arch. At the top of the arch, they are all supposed to be 2% f't thick at top; that being assumed to be about the least thickness admissible in a rubble wall in such a position. Both the back and the face are supposed to be vert. The contents of these spandrel-walls will vary somewhat, however, even in the same span, with the height of the abut and the arrangement of the wings. They, however, constitute so small a proportion of the entire contents given in Table 5. that this consideration may be neglected in preliminary estimates. They are so firmly bonded into the masonry of the wings at their highest points, arid so strongly connected by mortar with the backing of the arch at their bases, that they require no greater thickness however high the emb may be. The contents of the four wing-walls, of which nj w b, Fig 6, is one, will be found in a table (No. 4) immediately following that for the body of the cul- Vert. We have also added a table (No. 5) for complete semicircular culverts f vari ous lengths, including their spandrel and wing walla. \Fi e -.6 STONE BRIDGES. 351 REM. 1. Although the thickness of wing-walls increases in all parts with their height, they are not made to show thicker at nj than at ti, Fig 6 ; but (as seen in the fig) are ofisetted at their back t n, a little below their slanting upper surf ij, so as to give a uniform width for the steps or flagstones, as the case may be, with which they are covered. In the fig the covering is supposed to be of flagstones ; but steps are preferable, being less liable to derangement. To prevent the flagstones from sliding down the inclined plane jt, the lower stone i should be deep and large, and laid with a hor bed. The flags are sometimes cramped together with iron, and bolted down to the wall. Steps require nothing of that kind, as seen at s, Fig 11, p 355. REM. 2. The tables show the inexpediency of too much con- tracting the width of water-way, with a view to economy, by adopting a small span of arch, when a culvert of greater span can be made, of the same total height. For the wings must be the same, whether the span be great or small, provided the total height it the same fn both cases ; and since the wings constitute a large proportion of the entire quantity of masonry, in culverts of ordinary length, the span itself, within moderate limits, has comparatively little effect upon it. Thus, the total masonry in a semicircular culvert of 3 ft span, 8 ft total height, and 60 ft long between the faces of the arch, is, by Table 5, 151^6 cub yds ; while that of a 5 ft span, of the same height and length, is 152.4. A semicircular bridge of 25 ft span, 24 ft total height, and 40 ft between the faces of the arch, contains 1031 cub yds ; while one of 35 ft span, of the same height and length, contains 1134 yds ; so that in this case we may add nearly 50 per cent to the water-way, by increasing the masonry of the bridge but y^jth part. REM. 3. Partly for the same reason, and partly because the cnlverts for a double-track road are not twice as long; as those for a single- track one, the quantity of culvert masonry for the former will not average more than about from % to % part more than that for the latter; so that it frequently becomes expedient to finish the culverts at once to the full length required for a double track, although the embkts may at first be made wide enough for only a single one, with the intention of increasing them at a future time for a double one. Thus, the average size of culverts for a single track may be roughly taken at 6 ft span, 30 ft lon^ from face to face, and 10 ft total height; and such a one contains, by Table 5, 140 cub yds. For a double track, it would require to be about 12 feet longer; and we see by Table 3 that this will add 2.67 X 12 ~ 32 cub yds ; making a total of 172 yds instead of 140 ; thus adding rather >2ss than y^ part. When the culverts are under very high embkts, and consequently much longer, the addition for a double track becomes comparatively quite trifling. Table 3, of approximate numbers of cnb yds of masonry per foot run, contained in the arches and abutments only, as shown in Fig 1 (omitting wings, and the spandrel-walls over the faces of the arches) of semicircular culverts and bridges, of from 2 to 50 ft span, and of different total heights, ht, Fig 1, or o c, Fig 6. It will be seen that in many cases, a bridge of larger span contains less masonry than one of smaller span, when their total heights are th* same. There is a liberal allowance for footings or offsets at the bases of the abut* TABLE 3. (Original.) Total Height. Span 2ft. Span 3ft. Span 4ft. Span 5ft. Span 6ft. Span 8ft. Span 10ft. Span 12ft. Span 15ft. Feet. 2 Cub. y. 42 Cub. y. Cub. y. Cub. y. Cub. y. Cub. y. Cub. y. Cub. y. Cub. y. 3 60 63 67 4 79 83 87 92 .97 5 99 1 04 1.08 1 15 1.21 g 1 28 1 28 1 28 1 37 1.46 1 58 1.69 7 8 9 10 11 1.62 2.01 2.45 2.94 1.59 1.96 2.38 2.85 3 38 1.55 1.91 2.31 2.76 3 26 1.64 1.95 2.29 2.72 3.19 1.72 1.99 2.27 2.67 3 12 1 85 2.13 2.42 2.77 3.16 1.97 2.26 2.56 2.87 3.19 2J2 2.38 2.65 2.93 3.23 "3.62" 3.34 3 67 12 3.98 3.82 3.72 3.62 3.57 3.52 3.55 4.01 13 4 42 4.29 4.17 4.10 4.02 2.86 4.36 14 5.08 4.90 4.77 4.67 4.57 4.41 4.72 15 5.57 5.42 5.30 5.17 5.01 5.09 16 6 30 6.12 5.97 5.82 5.56 5.69 17 6 87 6 70 6 52 6.26 6.34 18 7 69 7.48 7.27 7.01 7.04 19 8 32 8.07 7.71 7.69 20 9 20 8 92 8.56 8.49 21 9.82 9.46 9.34 22 10.8 10.3 10.2 23 11.3 11.1 24 12.3 12.1 25 13.2 26 14.2 Contents of the two spandrel- walls, over the two ends of the arch, in cub yds. I 2.9 I 3.7 | 4.4 | 5.2 | 5.8 | 7.9 | 9.8 I 12. I 16. 352 STONE BRIDGES. TABLE 3. (Continued.) Total Height. Spaa 20 ft. Spaa 25ft. Total Height. Span 35 ft. Total Height. Span 50ft. Feet. 12 Cub. v. 4.60 Cub. y. Feet. 20 Cub. y. 10.5 Feet. 27 Cub. y. 18.0 13 4.98 21 11.0 28 18.7 14 5.37 6.10 22 11.6 29 19.4 15 5.77 6.41 23 12.2 30 20.1 16 6.18 6.76 24 12.7 31 20.9 17 6.60 7.16 25 13.3 32 21.6 18 7.03 7.61 26 13.8 33 22.4 19 7.47 8.10 27 14.5 34 23.1 20 8.12 8.60 28 15.1 35 23.9 21 8.82 9.02 29 15.7 36 24.7 22 9.57 9.72 30 16.3 37 25.5 23 10.4 10.4 31 17.0 38 26.3 24 11-3 11.1 32 181 39 27.1 25 12.2 12.1 33 19.2 40 28.0 26 13.1 13.0 34 20.4 41 28.8 27 14.1 14.0 35 21.7 42 30.0 28 15.2 15.0 36 23.0 43 31.5 29 i6.3 16.1 37 243 44 33.0 30 17.4 17.2 38 25.7 45 34.6 31 18.6 18.4 39 27.2 46 36.3 32 19.9 19.6 40 28.7 47 38.1 SH 21 2 20.9 41 30.2 48 39.8 5* 22.6 22.2 42 31.8 49 41.6 35 24.0 23.6 43 33.5 50 43.6 36 25.4 25.0 44 35.2 51 45.5 37 26.9 26.5 45 36.9 52 47.4 38 285 28.0 46 38.7 53 49.4 39 30.1 29.5 47 40.6 54 51.6 40 31.7 31.2 48 42.5 55 53.7 41 32 8 49 44.4 56 55.9 42 34.5 50 46.4 57 58.1 43 36.3 58 60.4 44 38.1 59 62.7 45 40.0 60 65.1 Contents of the two spandrel-walls, over th two ends of the arch, in cub yds. 28. | 42. II | 85. II | 195. Art. 8. The following* table of contents of wing-walls, or wings, will, like the preceding one, be useful in making preliminary estimates. The wings no, no, shown in plan at Fig 8, are supposed to form an angle aoc, of 120, with the face, or end o o of the culvert. Their outer or small ends n n, are all assumed to be of the dimensions shown on a larger scale at E. Thickness at base at every part equal to T ^y of the height of the wall at said part; except when that proportion becomes too small to allow the width or thickness at top to be 2.5 ft ; in which case it is en- larged at such parts sufficiently for that purpose. See Remark 2. This happens only 2.5 { m TL 3.125 xTIL when the height m m, Fig E, of the wing, becomes less than 9 ft. Batter of face, 1% ins to a ft ; or 1 in 8. Back vert ; but offsetted, if necessary, for a short dist below the top, so as to give a uniform showing top thickness of 2% ft. The masonry is supposed to be good well-scabbled mortar rubble. The height given in the first column is the greatest one; or that at oo, (or wj, Fig 6,) where the wing joins the face of the culvert. In the table no allowance is made for footings (offsets or steps) at the base of the wings J as these are frequently omitted in wings on good founda- STONE BRIDGES. 353 tiona. In taking out quantities from the table, bear in mind that the height of the wings is usually a little greater than that of the culvert itself. Table 4, of approximate contents, in cub yds, of the four wing- walls of a culvert, or bridge. (Original.) The heights are taken where greatest; as atj w, Fig 6 Height Length of Cub. vds. in Height of Length of Cub. yds. in wing. Feet. Feet. Feet. Feet. 6 1.73 4.04 30 43.S 818 7 3.46 8.85 32 46.8 997 8 5.20 14.6 34 50.3 1192 9 6.U3 21.5 36 53.7 1414 10 8.66 30.2 38 57.2 1661 11 10.4 40.9 40 60.7 1928 12 12.1 53.7 42 64.2 2220 14 15.6 85.2 44 67.6 2552 16 19.1 128 46 71.1 2912 18 2'2.5 183 48 74.6 3306 20 26.0 247 50 78.0 3741 22 29.5 329 55 86.7 4942 24 32.9 426 60 95.3 6404 26 36.4 541 65 104 8131 28 39.8 672 70 113 10155 To reduce cub yds to perches of 25 cub ft, mult by 1.080. To reduce perches to cub yds, mult by .926, or div by 1.08. The contents for heights intermediate of those in the table may be found approximately by simple proportion. RKM. 1. It is not recommended to actually prolong all wings until their dimen- sions become as small as shown at E, in Fig 8. In large ones it will generally be more economical to increase their end height m m, a lew feet. The contents, how- ever, may be readily found by the table in that case also. Thus suppose the height f the wings at one end to be 30 ft, and at the other end 8 ft; we have only to sub- tract the tabular content for S ft high, from that for 30 ft high. Thus, 818 14.6 = 803.4 cub yds required content. HEM. 2. It might be supposed that inasmuch as the wings of arches often have to sustain the pressure from embankments reaching far above their tops, they should, like ordinary retaining-walls, be made much thicker in that case. But the fact that they derive great additional stability from being united at their high ends to the body of the bridge or culvert, renders such increase unnecessary when pioportioned by our rule ; no matter how far the earth may extend above them ; as shown by abundant experience. Relying upon this aid. we may indeed, when the earth does not extend above the top. reduce the the wings, instead of heinc splayed or flared out, as at on. on. merely form straight prolongations of the abutments of the arch, as shown by the dotted lines at og w. In this case the pressure of the earth against the wings is less thnn when they are splayed. We have known the thickness at o to be reduced in such cases to rnther less than % tne height, when the wings were 15 ft high, and the height of the embankment above their tops 16 feet in one case, and 36 ft in another. In another instance, similar wings 25^ ft high, and with 29 ft of embankment above their top. had their bases at o rather less than -^ of the height. In all these cases, the uniform thickness at top was 2.5 feet; backs vertical. We mention them because this particular subject does not seem to be reducible to any practical rule. The last wall appears to us to be too thin ; especially if the earth is not deposited in layers: and after allowing the mortar full time to set. The labor, however, required in compact- ing the earth carefully in layers, may cost more than is thereby saved in the masonry. The young practitioner must bear this in mind when he wishes to economize masonry by such means : and also that the thin wall may bulge, or fail entirely, if the earth backing is deposited while the mortar i* imperfectly set. 354 8TONE BRIDGES. Table 5. Approximate contents in cubic yards, of com- plete semicircular culverts and bridges of from 2 to 50 feet span; including the 2 spandrel walls ; and the 4 wings ; all proportioned by the foregoing directions: and taken from the two preceding tables. The height in the second column, is from the top of the keystone to the bottom of the foundation. The wings are calculated as being 2 ft higher than this, including the thickness of the coping. The wings are frequently carried only to the height of the top of the arch; thus saving a good deal of masonry. Table 4, of wings alone, will serve to make tha proper deduction in this case. The several lengths are from end to end, or from face to face, of the arch proper. The contents for intermediate lengths maybe found exactly; arid those for inter- mediate heights, quite approximately, by simple proportion. In this table, as in No. 3, it will be observed that when the heights are Lite same in both casts, a larger span frequently contains less masonry than a smaller one. A semicircular culvert or bridge contains less masonry than a flatter one, when the total height is the same in both cases; therefore, the first is the most economical as regards cost; but it does uot afford as much area of water-way ; or width of headway. (Original.) a 1 of ^ ~Sb ' 33 2- JS tj J= Length. 30 Ft. as |3 U 1 f g &J *>=* SB JS ^ ,& | ta 56 " g j= >& 3 J= w &*< C g Ft. 2 Ft. 5 6 7 8 10 CubY. 27 37 49 63 101 Cub Y. 32 43 57 7 116 CubY. 42 56 73 93 145 Cub.Y. 52 69 89 113 175 Cub.Y. 72 94 122 153 234 Cub.Y. 92 120 154 193 291 Cub.Y. 112 146 187 233 351 Cub.Y. 132 171 219 273 410 Cub.Y. 152 197 251 313 469 Cub.Y. 172 222 284 353 527 Cub.Y. 192 248 316 393 586 Cub.Y. 212 274 349 433 645 3 5 6 7 8 10 12 28 38 49 63 101 149 34 44 57 73 115 169 44 57 73 93 143 208 54 70 89 112 172 248 75 95 121 152 229 328 96 121 153 191 286 407 117 146 184 230 343 487 138 172 216 269 400 567 158 198 247 308 457 646 179 223 280 348 514 726 200 249 312 387 571 806 221 275 343 426 628 885 4 5 6 7 8 10 12 14 30 38 49 63 100 147 209 35 45 57. 73 114 166 234 46 58 73 92 141 204 285 57 70 88 111 169 243 336 78 96 119 149 224 319 437 100 122 150 188 279 395 539 122 147 181 226 335 472 641 143 173 212 264 390 548 742 165 198 243 302 445 625 844 186 224 274 340 500 701 945 208 250 305 379 555 777 1047 229 275 336 417 611 854 1149 5 6 7 8 10 12 14 41 52 65 100 146 207 47 60 75 114 165 231 61 76 94 141 202 280 75 93 114 168 239 329 102 125 153 223 314 427 130 158 192 277 388 525 157 191 231 331 463 623 184 224 270 386 537 721 212 257 309 440 611 819 239 289 348 495 686 917 267 322 387 549 760 1015 294 355 426 603 835 1113 6 7 8 10 12 14 16 53 66 100 146 206 281 62 76 113 164 219 311 79 96 140 200 277 373 96 116 167 236 325 434 131 156 220 308 420 556 165 196 274 381 516 679 200 236 327 453 611 801 234 276 380 526 706 923 268 316 434 598 802 1046 303 356 487 670 897 1168 337 396 541 743 993 1291 372 436 594 815 1088 1413 8 7 8 10 12 14 16 18 57 70 lot 147 206 . 281 367 67 81 118 165 230 310 405 85 102 145 200 276 370 480 104 124 173 236 323 430 554 141 166 228 308 416 549 704 178 209 284 379 510 669 854 215 251 339 450 603 788 1003 252 294 395 522 696 908 1153 289 337 450 593 790 1027 1302 326 379 505 664 883 1146 1452 363 422 561 736 977 1266 1602 400 464 616 807 1070 1385 1/51 10 8 10 12 14 1(5 18 74 107 148 207 280 366 85 121 166 229 309 402 108 150 201 275 368 475 131 179 236 321 426 548 176 236 306 412 542 693 221 294 377 504 659 839 266 351 447 595 775 984 311 408 518 686 801 1129 357 466 588 778 1008 1275 402 523 658 869 1124 1420 447 581 729 961 1241 1565 492 638 ' 799 1052 1357 1711 12 10 12 14 16 18 20 110 151 206 279 364 470 125 168 228 306 399 512 154 204 272 362 469 598 183 239 317 418 540 684 242 310 405 529 680 855 301 381 493 640 820 1026 359 452 581 751 960 1197 418 523 669 862 1100 1368 476 594 758 974 1241 1540 535 665 846 1085 1381 1711 594 736 934 1196 1521 1882 652 807 1022 1307 1661 2053 STONE BRIDGES. 355 Table 5 (Continued.) (Original.) a SS Wi&< .&, , Sfc, s. * fc * fe w^ 0. OQ g gS gg g$ s |a S gS 8 |g BO g jo J Ft. Ft. Cub.Y. Cub.Y. Cub. Y. Cub.Y. Cub.Y. Cub.Y. Cub.Y. Cub.Y. Cub.Y. Cub.Y. Cub.Y. Cub.Y. 12 162 182 222 262 842 422 502 583 663 743 823 903 14 215 239 286 333 427 522 616 711 805 899 994 1088 1 ^ Id 285 313 370 427 541 654 768 882 996 1110 1223 1337 18 369 404 474 545 686 826 967 1108 1249 1390 1530 1671 20 473 515 600 685 855 1024 1194 1364 1534 1704 1873 2043 22 595 646 748 850 1054 1258 1462 1666 1870 2074 2278 2482 14 237 264 317 371 478 586 693 801 908 1015 1123 1230 1C, 304 335 397 458 582 706 829 953 1076 1200 1324 1447 9n 18 381 416 486 556 697 838 97S 1119 1259 1400 1541 1681 20 479 520 601 682 844 1007 1169 1332 1494 1656 1819 1981 22 598 646 741 837 1028 1220 1411 1603 1794 1985 2177 2368 24 739 795 908 1021 1247 1473 1699 1925 2151 2377 2603 2829 16 327 360 428 496 631 766 901 1036 1172 1307 1442 1577 18 403 441 517 594 746 898 1050 1202 1355 1507 1659 1811 20 500 543 629 715 887 1059 1231 1403 1575 1747 1919 2091 25 22 614 663 760 857 1051 1246 1440 1635 1829 2023 2218 2412 24 751 807 919 1031 1255 1479 1703 1927 2151 2375 2599 2823 26 909 974 1104 1234 1494 1754 2014 2274 2534 2794 3054 3314 28 1085 1160 1310 1460 1760 2060 2360 2660 2960 3260 3560 3860 22 685 743 859 975 1207 1439 1671 1903 2135 2367 2599 2831 24 817 880 1007 1134 1388 1642 1896 2150 2404 2658 2912 3166 26 969 1033 1181 1309 1585 1861 2137 2413 2689 2965 3241 3517 35 28 1130 1205 1356 1507 1809 2111 2413 2715 3017 3319 3621 3923 30 1327 1408 1571 1734 2060 2386 2712 3038 3364 3690 4016 4342 83 1549 1639 1820 2001 2363 2725 3087 3449 3811 4173 4535 4897 35 1946 2054 2271 2488 2922 3356 3790 4224 4658 5092 5526 5960 30 1494 1594 1795 1996 2398 2&00 3202 3604 4006 4408 4810 5212 32 1711 1819 2035 2251 2683 3115 3547 3979 4411 4843 5275 5707 34 1956 2071 2302 2533 2995 3457 3919 4381 4843 5305 5767 6229 ^0 36 38 2228 2519 2350 2650 2597 2913 2844 3J76 3338 3832 4326 4820 5314 5808 6302 6796 40 2835 2975 3255 3535 4095 4655 5215 5775 6335 6895 7455 8015 42 3197 3347 3647 3947 4547 5147 5747 6347 6947 7547 8147 8747 45 3818 3991 4337 4683 5375 6067 6759 7451 8143 8835 9527 10219 50 5063 5281 5717 6153 7025 7897 8769 9641 10513 11385 12257 13129 Art. 9. Especial pains should be taken to secure an unyielding- foun- dation for culverts and drains under high einbkts) otherwise the superincumbent weight, especially under the middle of the embkt, may squeeze them into the soil below, if soft or marshy; and thus diminish the area of water- way, or at least cause an ugly settlement at the midlength of the culvert. Also, in soft ground, the embkt ma,y press the side walls closer together, narrowing the channel. This may be prevented by an inverted arch, or a bed of masonry, between the walls. A stratum from 3 to 6 ft thick, of gravel, sand, or stone broken to turn- pike size, will generally give a sufficient foundation for culverts in treacherous marshy ground ; or quicksand, with but a moderate height of embkt. It should ex- tend a few feet beyond the masonry in every direction, and should be rammed ; the sand or gravel being thoroughly wet, if possible, to assist the consolidation. Piling will sometimes be necessary. If the masonry is built upon timber platforms, or a smooth surface of rock, care must be taken to prevent it from sliding, from the prea of the earth behind it. This same pres may even overthrow the piles, if they are not properly secured against it. Art 1O. I>r; MIS. Brains of the dimen- sions in Fig 11, con- tain 1 perch, of 25 cub ft ; or .926 of a cub yd, per ft run. They are frequently built of dry scabbled rubble, and paved with spawls. When there ia much wash through them, with a consider- able slope, it is better to continue the foundation 356 STONE BRIDGES. solid clear across. This is often done without those causes, inasmuch as the additional masonry ia & mere trifle: and the excavation of a siugle broad foundation-pit is less troublesome than that of two narrow ones. A deep flag-stone /at the entrance, and others at short dists of the length, may be in- troduced in both drains and culverts, to protect from undermining. These drains extend under the entire width of the embkt, from toe to toe; and may terminate in steps, as in the side view at S. They are of course better when built with mortar, with an admixture of cement to prevent the water when full from leaking into and softening the embankment. Sometimes two or three such drains may be placed parallel to each other, instead of a culvert. When two are so placed, they contain only 1>6 times the masonry of one ; still their use will generally involve no saving of masonry over a culvert. A man can crawfthrough Fig 11 to clean it. Art. 11. The drainage of the roadways of stone bridges of several arches, is generally effected by means of open gutters, which descend slightly from the crown of the arches, each way, until they reach to near the ends of the re spective spans. There they discharge into vertical iron pipes built into the masonry. The upper ends of the pipes should be covered by gratings. When inconvenience would result from the water falling upon persons passing under the arches, these pipes may be carried down the entire height of the piers; but when such is not the case, they may extend only to the soffit, or under face of the arch ; allowing the water to fall freely through the air from that height. Table 6, of approximate contents, in cub yds, of a solid pier of masonry, 6 ft by 22 ft on top; and battering 1 inch to a ft on each of its 4 faces. The contents of masonry of such forms must be calculated by the prismoidal formula ; and not by taking the length and breadth of the pier at half its height as an average length and breadth, as is sometimes done. This incorrect method would give only 6492 cub yds as the content of the pier 200 ft high; instead 7178 yds, its true content. High piers may for economy.be built hol- low, with or without interior cross-walls for strengthening them, as the case may require; and the batter is generally reduced to % inch or less to a foot. Hollow piers require good well- bedded ma- * ar J' (Original.) Ht. Ft. Lgth at baso. Bdth at base. Cubic yards Ht. Ft. Lgth at base. Bdth at base. Cubic- yards Ht. Ft, Lgth at base. Bdth at base. Cubic yards. 6 23. 7. 32.5 52 30.67 14.67 537 128 43.33 27.33 2759 7 .17 .17 38.6 54 31. 15. 570 130 .67 .67 2848 8 .33 .33 44.9 56 .33 .33 605 132 44. 28. 2940 9 23.5 7.5 51.3 58 .67 .67 641 134 .33 .33 3032 10 .67 .67 58. 60 32. 16. 679 136 .67 .67 3126 11 .83 .83 64.8 62 .33 .33 717 138 45. 29. 3222 12 24. 8. 71.7 64 .67 .67 757 140 .33 .33 3320 13 .17 .17 79. 66 33. 17. 798 142 .67 .67 34-20 14 .33 33 86.4 68 .33 .33 840 144 46. SO. 3521 15 24.5 8.5 94. 70 .67 .67 884 146 .33 .33 3623 16 .67 .67 102 72 34. 18. 928 148 .67 .67 3728 17 .83 .83 110 74 .33 .33 973 150 47. 31. 3835 18 25. 9. 118 76 .67 .67 1021 152 .33 .33 3944 19 .17 .17 127 78 35. 19. 1070 154 .67 .67 4056 20 .33 .33 135 80 .as .33 1120 156 48. 32. 4168 21 25.5 9.5 144 82 .67 .67 1171 158 .33 .33 4-284 22 .67 .67 153 84 36. 20. 1224 160 .67 .67 4402 23 .83 .83 163 86 ,38 .33 1278 162 49. 33. 4520 21 26. 10. 172 88 .67 .67 1334 164 .33 .33 4640 25 .17 .17 182 90 37. 21. 1392 166 .67 .67 4763 26 .33 .33 192 92 .33 .33 1451 168 50. 34. 4887 27 26.5 10.5 202 94 .67 .67 1510 170 .33 .33 5014 28 .67 .67 212 96 38. 22. 1569 172 .67 .67 5143 29 .83 .83 223 98 .33 .33 1631 174 51. 35. 5275 30 27. 11. 234 100 .67 .67 1695 176 .33 .33 5409 31 .17 .17 245 102 39. 23. 1761 178 .67 .67 5545 32 .33 .33 256 104 .33 .33 1829 180 52. 36. 5680 33 27.5 11.5 268 106 .67 .67 1899 182 .33 .33 6820 34 .67 .67 280 108 40. 24. 1968 184 .67 .67 5962 35 .83 .83 292 110 .33 .33 2041 186 53. 37. 6106 36 28. 12. 304 112 .67 .67 2115 188 ,?3 .33 625? 38 .33 33 329 114 41. 25. 2191 190 .67 .67 640C 40 .67 67 356 116 .33 .33 2269 192 54. 38. 6552 42 29. 13. 383 118 .67 .67 2346 194 .33 .33 6704 44 .33 .33 411 120 42. 26. 2424 196 .67 .67 6859 46 .67 .67 441 122 .33 .33 2504 198 55. 39. 7016 48 30. 14. 472 124 .67 .67 2587 200 .33 .o3 7178 50 .33 .33 504 126 43. 27. 2672 202 .67 .67 7339 BOARD MEASURE. 357 BOAED MEASUKE. Remark on following- table. The table extends to 12 ins by 24 ins, but it is easy to find for greater sizes ; thus, for example, the board measure in a piece of 19 by 22, will be twice that < i a piece of 19 by 11, or 17.42 X 2 = 34.84 ft board meas ; or that of 19*4 by 22, will be that of 10H by 22 added to that of 9 by 22, or 18.79 + 16.50 = 35.29. A foot of board meas is equal to 1 foot square and 1 inch thick, or to 144 cub ins. Hence 1 cab ft = 12 fi board meas. a . Feet of Board Measure conta ned in one running foot of Scantlings a 31 1000 ft board measure = 83tf cub ft. *3 11 THICKNESS 2N" INCHES. Is fen BM ^ 1 1M 1x4 l?i 2 234 2x4 2?* 3 r Ft Bd.M. Ft. Bd.M. Ft. Bd.M. Ft.Bd.M. Ft. Bd.M. Ft. Bd.M. Ft. Bd.M. Ft. Bd.M. Ft. Bd.M. x4 .0208 .0260 .0313 .0365 .0417 .0469 .0521 .0573 .0625 y IX .0417 .0521 .0625 .0729 .0833 .0938 .1042 .1146 .1250 ?x XX .0625 .0781 .0938 .1094 .1250 .1406 .1563 .1719 .1875 3 1. .0833 .1042 .1250 .1458 .1667 .1875 .2083 .2292 .2500 i .1042 .1302 .1563 .1823 .2083 .2344 .2604 .2865 .3125 VX .1250 .1563 .1875 .2188 .2500 .2813 .3125 .3438 .13750 t XX .1458 .1823 .2187 .2552 .2917 .3281 .3646 .4010 .4375 I? 2. .1667 .2083 .2500 .2917 .3333 .3750 .4166 .4583 .5000 2. .1875 .2344 .2813 .3281 .3750 .4219 .4688 .5156 .5625 IX .2083 .2604 .3125 .3646 .4167 .4688 .5208 .5729 .6250 ?5 ax .2292 .2865 .3438 .4010 .4583 .5156 .5729 .6302 .6875 9? 3. .2500 .3125 .3750 .4375 .5000 .5625 .6250 .6875 .7500 3. .2708 .3385 .4063 .4739 .5416 .6094 .6771 .7448 .8125 .2917 .3646 .4375 .5104 .5833 .6563 .7292 .8021 .8750 ^ .3125 .3906 .4689 .5469 .6250 .7031 .7813 .8594 .9375 y 4. .3333 .4167 .5000 .5833 .6667 .7500 .8333 .9167 1.000 4. .3542 .4427 .5312 .6198 .7083 .7969 .8854 .9740 1.063 VX .3750 .4688 .5625 .6503 .7500 .8438 .9375 1.031 1.125 x4 IX- .3958 .4948 .5938 .6927 .7917 .8906 .9896 1.086 1.188 1 5. .4167 .5208 .6250 .7292 .8333 .9375 1.042 1.146 1.250 5 .4375 .5469 .6563 .7656 .8750 .9844 1.094 1.203 1.313 IX .4583 .5729 .6875 .8020 .9167 1.031 1.146 1.260 1.375 ix XX .4792 .5990 .7188 .9583 1.078 1.198 1.318 1 .4:;8 6. .5000 .6250 .7500 '.8750 1.000 1.125 1.250 1.375 1.500 6.* 1 .5208 .6510 .7813 .9115 1.042 1.172 1.302 1.432 1.563 J4 .5417 .6771 .8125 .9479 1.083 1.219 1.354 1.490 1.625 VX XX .5625 .7031 .8438 .9844 1.125 1.266 1.406 1.547 1.688 y 7. .5833 .7292 .8750 1.021 1.167 1.312 1.458 1.604 1.750 7. .6042 .7552 .9063 1.057 1.208 1.359 1 510 1.661 1.813 J4 .6250 .7813 .9375 1.094 1.250 1.406 1.563 1.719 1.875 IX xx .6458 .8073 .9688 .130 1 .292 1.453 1.615 1.776 1.938 8 8. .6667 .8333 1.000 .167 1.333 1.500 1.667 1.833 2.000 8 .6875 .8594 1.031 .203 1.375 1.547 1.719 1.891 2.063 x4 .7083 .8854 1.063 .240 1.417 1694 1.771 1.948 2.125 xtf .7292 .9114 1.094 .276 1.458 1.641 1.823 2.005 2.188 1 9. .7500 .9375 1.125 .313 1.500 1.688 1.875 2.062 2.250 IX .7708 .9635 1.156 .349 1.542 1.734 1.927 2.120 2.313 J4 .7917 .9895 1.188 .385 1.583 1.781 1.979 2.177 2.375 ix xtf .8125 1.016 1.219 .422 1.625 1.828 2.031 2.234 2.438 8 10. .8333 1.042 1.250 .458 1.667 1.875 2.083 2.292 2.500 10. M .8542 1.068 1.281 .495 1.708 1.922 2.135 2.349 2.563 x4 .8750 1.094 1.313 .531 1.750 1.969 2.188 2.406 2.JJ26 xi 9i .8958 1.120 1.344 .568 1.792 2.016 2.240 2.463 2.688 11. .9167 1.146 1.375 .604 1.833 2.063 2.292 2.521 2.750 lit M .9375 1.172 1.406 1.641 1.875 2.109 2.344 2.578 2.813 x4 .9583 1.198 1 438 1.677 1.917 2.156 2.396 2 635 2.875 i xi .9792 1.224 1 469 1.714 1958 2.203 2.448 2.693 2.938 H 12. 1.000 1.250 1.500 1.750 2.000 2.250 2.500 2.750 3.000 12 x4 1.042 1.302 1.563 1.823 2.083 2.344 2.604 2.865 3.125 V* T3. 1.083 1.354 1.625 1.896 2.167 2.438 2.708 2.979 3250 13 J4 1.125 1.406 1.688 1.969 2.250 2.531 2.813 3.094 3.375 x4 14. 1.167 1.458 1.750 2.042 2333 2.625 2.917 3.208 3.500 14 x4 1.208 1.510 1.813 2.115 2.417 2.719 3.021 3.322 3.625 x4 15. 1.250 1.563 1.875 2.188 2.500 2.813 3.125 3.438 3.750 15 1.292 1.615 1.938 2.260 2.583 2.906 3.229 3.552 3.875 x4 16. 1.333 1.667 2.000 2.333 2.667 3.000 3.333 3.667 4.000 16 x4 1.375 1.719 2.063 2.406 2.750 3.094 3.438 3.781 4.125 >4 17. 1.417 1.771 2.125 2.479 2.833 3.188 3.542 3.896 4.250 17 1.458 1.823 2.187 2.552 2.917 3.2H1 3.646 4.010 4.375 18. 1.500 1.875 2.250 2.625 3.000 3.375 3.750 4.125 4.500 18. 19. 1.583 1.979 2.375 2.771 3.167 3563 3.958 4.354 4.750 19. 20. 1.667 2.083 2.500 2.917 3.333 3.750 4.167 4.583 5.000 20. 21. 1.750 2.188 2.625 3.063 8.500 3.9?,8 4.375 4812 5250 21. 22. 1.833 2.292 2.750 3.208 3.667 4.125 4.583 5.042 5.500 22. 23. 1.917 2.396 2.875 3.354 3.833 4.313 4.792 5.270 5.750 23. 24. 2.000 2.500 3.000 3.500 4.000 4.500 5.000 5.500 6.000 24. 358 BOARD MEASURE. Table of Board Measure (Continued.) P" . 51 Feet of Board Measure conta ned in one runn ng foot of Scantlings of different dimensions. (Original.). jj S3 THICKNESS IN INCHES. j>M 3X 3M 3 4 4J4 4>4 4% 5 * Ft.Bd M. Ft.BdM. Ft.Bd.M. Ft.Bd.M. Ft.Bd.M Ft.Bd.M. FtBd.M. Ft. lid. M Ft.Bd.M. . 34 .0677 .0729 .0781 .0833 .0885 .0938 .0990 .1042 .1094 34 iz .1354 .1457 .1562 .1667 .1770 .1875 .1979 .2083 .2188 y>, 9i .2031 .2187 .2344 .2500 .2656 .2813 .2969 ! .3125 .3281 % 1. .2708 .2917 .3125 .3333 .3542 .3750 .3958 .4167 .4375 i. .3385 .3646 .3906 .4167 .4427 .4688 .4948 .5208 .5469 34 IX .4063 .4375 .4688 .5000 .5313 .5625 .5938 .6250 .6563 y% ax .4740 .5104 .5469 .5833 .6198 .6563 .6927 .7292 .7656 % 2. .5417 .5833 .6250 .6667 .7083 .7500 .7917 8333 .8750 2. i/ .6094 .6563 .7031 .7500 .7969 .8438 .8906 .9375 .9844 % jv .6771 .7292 .7813 .8333 .8854 .9375 .9896 1.042 1.094 % 3X .7448 .8021 .8594 .9167 .9740 1.031 1.089 1.146 1.203 % 3. .8125 .8750 .9375 1.000 1.062 1.125 1.188 1.250 1.313 3. .8802 .9479 1.016 1.083 1.151 1.219 1.286 1.354 1.422 34 i^ .9179 1 .021 1.094 1.167 1.240 1.313 1.385 1.458 1.531 H 1.016 1.094 1.172 1.250 1.327 1.406 1.484 1.563 1.641 % 1083 1.167 1.250 1.333 1.416 1.500 1.583 1.667 1.750 4. k 1.151 1 240 1.328 1.417 1.504 1.594 1.682 1.771 1.859 34 | 1.219 1.313 1.406 1.500 1.593 1.688 1.781 1.875 1.969 y* 1.286 1.384 1.484 1.583 1.681 1.781 1.880 1 979 2.078 5.* 1.354 1.457 1.566 1.666 1.770 1.875 1.979 2.083 2188 5. 34 1.422 1.530 1.644 1.750 1.858 1.969 2.078 2.188 2297 34 y* 1.490 1.603 1.722 1.833 1.947 2 063 2.177 2.292 2.406 34 % 1.557 1.676 1.800 1.917 2.035 2.156 2.276 2.396 2.516 % 6. 1.625 1.750 1.875 2.000 2.125 2 250 2.375 2.500 2.625 6. 34 1.693 1.823 1.953 2.083 2.214 2.344 2.474 2604 2.734 34 y* 1.760 1.896 2.031 2.167 2.302 ! 2.438 2.573 2.708 2.843 y% % 1.828 1.969 2.109 2.250 2.391 2.531 2.672 2.813 2953 % 7. 1.896 2.042 2.188 2.333 2.479 2.625 2.771 2.917 3.063 7. v 1 .9(54 2.115 2.266 2.416 2.568 2719 2.870 3021 3.172 34 J4 2.031 2 187 2.344 2.500 2656 2813 2.969 3.125 3.281 y* 2.099 2 260 2422 2.583 2.745 2.906 3.068 3.229 3391 8. 2.167 2333 2.500 2.667 2.833 3000 3.167 3.333 3.500 8. 34 2.234 2.406 2.578 2.750 2 922 3.094 3.266 3.438 3.609 X 34 2.302 2.479 2656 2.833 3.010 3.188 3.365 3.542 3.718 x ?i 2.370 2.552 2 734 2.916 3.099 3.281 3.464 3.646 3.828 9. 2.438 2.625 2.813 3.000 3.187 3.375 3.563 3.750 3.938 9. 4 34 2.505 2.698 i 2.891 3.083 3.276 3.469 3.661 3.854 4.047 34 y^ 2.573 2.771 2.969 3.167 3.365 3.563 3.760 3.958 4.156 H 2.641 2.844 3.047 3.250 3.453 3.656 3.859 4.063 4.266 H 10. 2.708 2.917 3.125 3.333 3.542 3.750 3.958 4.167 4.375 10. 34 2.776 2.990 3.203 3.416 3.630 3.844 4.057 4.271 4.484 x4" Ji 2.844 3.063 3.281 3.500 3.719 3.938 4.156 4.375 ! 4.594 \<, ?4 2.911 3.135 3.359 3.583 3.807 4.031 4.255 4.479 4.703 3^ 11. 2.979 8.208 3.438 i 3.668 3.896 4.125 4.354 4.583 4.813 11. i/. 3.047 3.281 3.516 3.750 3.984 4.219 4.453 4.688 4.922 34 % 3.115 3.354 3.594 3.833 4.073 4.313 4.552 4.792 5.031 y* 3.182 3.427 3.672 3.916 4.161 4.406 4.651 4.896 5.141 | % 12. 3.250 3.500 3.750 4.000 4.250 4.500 4.750 5.000 5.250 12. u 3.385 3.646 3.906 4.167 4.427 4.688 4.948 5.208 5.469 13. 3.521 3.792 4.063 4.333 4.604 4.875 5.146 5.417 5.688 13." V<; 3.656 3.938 4.219 4.500 4.781 5.063 5.344 5.625 5.906 H 14. 3.792 4.083 4.375 4.667 4.958 5.250 5.542 5.833 ! 6.125 14. j 3.927 4.229 4.531 4.833 5.135 5.438 5.740 (5.042 6.344 Jl3 15. 4.063 4.375 4.688 5.000 5.313 5.6'25 5.938 6.250 : 6.563 15. 4 4.198 4.521 4.844 5.166 5.490 5.813 6.135 6.458 : 6.781 % 16. 4.333 4.667 5.000 5.333 5.667 6.000 6.333 6.667 ; 7.000 16. j 4.469 4.813 5.156 5.500 5.844 6.188 6.531 6.875 i 7.219 ^ 17. 4.604 4.958 5.313 5.667 6.021 6.375 6.729 7.083 7.438 17. 4.740 5.104 5.469 5.833 6.198 6.563 6.927 7.292 7.656 18." 4.875 5.250 5.625 6.000 6.375 6.750 7.125 7.500 7.875 18. 19. 5.146 5.542 5.938 6.333 6.729 7.125 7.521 7.917 8.313 19. 20. 5.417 5.833 6.250 6.667 7.083 7.500 7.917 8.333 8.750 20. 21. 5.688 6.125 6.563 7.000 7.438 7.875 8.313 8.750 9.188 21. 22. 5.958 6.417 6.875 7.333 7.792 8.250 8.708 9.167 9.625 22. 23. 6.229 6.708 7.188 7.667 8.145 8.625 9.104 9.583 10.06 23. 34! 6.500 7.000 7.500 8.000 8.500 9.000 9.500 10.00 10.50 24. Creosote or flead oil is by far the best preservative for timber yet known. It is certainly effective against sea worms for at least 25 years, if thoroughly applied. It is a colorless oily fluid distilled chiefly from coal tar. Its peculiar antiseptic quality is due to its carbolic and cresylic acids ; but it also acts by filling the pores of the wood so as to exclude air and moisture. It weighs about 8.8 Ibs per U. S. gkllon. Boils at 500 to 600 Fah. From .5 to 1 gallon, or say 4.5 to 9 Ibs (or 10 to 12 Ibs, if exposed to sea worms) are required per cub ft, depen ing on the nature of the timber It is used at a temp of 200 to 300 F, and under a pressure of 150 to 200 Ibs per *q inch, for BOARD MEASURE. 359 Table of Board Measure (Continued.) .s* si Feet of Board Measure conta ned in one running foot of Scantlings of different dimensions. (Original.) ** OQ n I 1 THICKNESS IN INCHES. 2s H 5& 5% 6 6J4 6M 6% 7 7*4 '* j Ft.Bd.M. Ft.Bd.M. Ft.Bd.M. Ft.Bd.M. Ft.Bd.M- Ft.Bd.M. FuBd.M. FtLd.m. rt.Ld.M. K .1146 .1198 .1250 .1302 .1354 .1406 .1458 .1510 .1563 y*. 14 .2292 .2396 .2500 .2604 .2708 .2813 .2917 .3021 .3125 8 H .3438 .3594 .3750 .3906 .4063 .4219 .4375 .4531 .4688 H I .4583 .4792 .5000 .5208 .5417 .5625 .5833 .6012 .6250 i. X .5729 .5990 .6250 .6510 .6771 .7031 .7292 .7552 .7813 y* M .6875 .7188 .7500 .7812 .8125 .8438 .8750 .9062 .9375 5 X .8021 .8385 .8750 .9115 .9479 .9844 1.020 1.057 1.094 H 2 .9167 .9583 1.000 1.042 1.083 1.125 1.167 1.208 1.X50 2. K 1.031 1.078 1.125 1.172 1.219 1.266 1.313 1.359 1.406 X 1.146 1.198 1.250 1.302 1.354 1.406 1 458 1.510 ' 1.5G3 J4 K 1.260 1.318 1.375 1.432 1.490 1.547 1.604 1.661 1.719 % 3. 1.375 1.438 1.500 1.562 1.625 1.688 1.750 1.813 I.fe75 3. y 1.490 1.557 1 .625 1JJ93 1.760 1.828 1.896 l.%4 2.0ol X 1.604 1.677 1.750 1.823 1.896 1,969 2.042 2.115 2.188 H H 1.719 1.797 1.875 1.953 2.031 2.109 2.188 2 266 2.344 H 4. 1.833 1.917 2.000 2.083 2.167 2.250 2.333 2.417 2500 4. Vi 1.948 2.036 2.125 2.214 2.302 2.391 2.479 2.568 2.56 X 9t 2.063 ! 2.156 2.250 2.344 2.438 2.531 2625 2.719 2.813 X H 2.177 2.276 2.375 2.474 2.573 2.672 2.771 2.870 2.!!G9 H 5 2.292 2.396 2.500 2.604 2.708 2.813 2.i 6.646 6.948 7.250 7.552 7.854 8.156 8.458 8.760 9.063 X 15. 6.8/5 7.188 7.500 7.812 8.125 8.438 8.750 9.063 9.375 15. J* 7.104 7.427 7.750 8.073 8.396 8.719 9.042 9.365 9.688 H 16. 7.333 7.6G7 8.000 8.333 8.667 9.000 9.333 9.667 10.00 16. ^ 7.563 7.906 8.250 8.594 8.938 9.281 9.625 9.969 10.31 X 17. 7.792 8.146 8.500 8.854 9.208 9.563 9.917 10.27 10.63 17. - X 8.021 8.385 8.750 9.115 9.479 9.844 10.21 10.57 10.94 H 18. 8.250 8.625 9.000 9.375 9.750 10.13 10.50 10.88 11.25 18. 19. 8.708 9.104 9.500 9.896 10.29 10.69 11.08 11.48 11.88 19. 20. 9.167 9.583 10.00 10.42 10.83 11.25 11.67 12.08 12.50 20. 21. 9.625 10.06 10.50 10.94 111.38 1181 12.25 12.69 13.13 21. 22. 10.08 10.54 11.00 11.46 111.92 12.38 12.83 13.29 13.75 22. 23. 10.54 11.02 11.50 11.98 112.46 12.94 13.42 13.90 14.38 23. 24. 11.00 11.50 12.00 12.50 13.00 13.50 14.00 14.50 15.00 24. 1 I slightly more brittle and inflammable. In hot weather it exudes to some extent, and discolors the timber. It preserves spikes driven into the timber. Its peculiar smell excludes the timber from dwellings. Ite cost is 12 to 15 cts per gallon ; and that of the process, including oil. will range from 12 to 30 cts per cub ft of timber, according to the quantity of oil, &o; ordinarily perhaps about 15 to 20 cts. At times it may be well for economy to first reduce the timbers to their intended final dimensions. The most perfect process is that patented by I'rof. Charles A. Seely, of New York, under whosp patent Mr W. T. Pelton, N Y, contracts to furnish apparatus and oil, and apply the process; which i* called Seelyizing, or oarboliziug. It may be applied to either green 01 seasoned 360 BOARD MEASURE. Table of Board Measure (Continued.) li Feet of Board Measure contained in one runn ng foot of Scantlings of different dimensions. (Original.) a 2 |>H THICKNESS IN INCHES. SH 79< 8 8x* 8H 8% 9 9M 9H m M Ft.Bd.M. Ft.Bd.M. FtBd.M. Ft.Bd.M FtBd.M Ft.Bd.M Ft.Bd.M FtBd.M. JFt.Bd.M. ' J4 .1615 .1667 .1719 .1771 .182: .1875 .1927 .1979 ! .2031 ^ ^6 .3229 .3333 .3438 .3542 .3646 .3750 .3854 .3958 .4063 X X .4844 .5000 .5156 .5313 .5467 .5625 .5781 .5938 .6094 1. .6458 .6667 .6875 .7083 .7292 .7500 .7708 .7917 .8125 i. % .8073 .8333 .8594 .8854 .9115 .9375 .9635 .9896 1.016 S .9688 1.000 1.031 1.063 1.094 1.125 1.156 1.188 1.219 y* 5i 1.130 1.167 1.203 1 240 1.276 1.313 1.349 1.385 1.422 H 2. 1.292 1.333 1.375 1.417 1.458 1.500 1.542 1.583 1.625 2. xi 1.453, 1.500 1.547 1.594 1.641 1.688 1.734 1.781 1.828 M S 1.615 1.667 1.719 1.771 1.822 1.875 1.927 1.979 2.031 /^J X 1.776 1.833 1.891 1.948 2.005 2.063 .120 2.177 2.234 2 3. 1.938 2.000 2.063 2.125 2.188 2.250 .313 2.375 2.438 3. i^[ 2.099 2. 167 2.234 2.302 2.370 2.438 .505 2.573 2.641 Ji K 2.260 2.333 2.406 2.4*9 2.552 2.625 .698 2.771 2.844 % H 2.422 2.500 2.578 2.656 2.734 2.813 .891 2.969 3.047 X 4. 2.583 2.667 2.750 2.833 2.917 3.000 3.083 3.167 3.250 4. j 2.745 2.833 2.922 3.010 3.099 3.188 3.276 3.365 3.453 ix Ji 2.906 3.000 8.094 8.188 3.281 3.875 3.469 3.563 3.656 L K 3.068 3.167 3.266 3!365 3.464 3.563 3.661 3.760 3.859 H 5. 3.229 3.333 8.438 3.542 3.646 3.750 3.854 3.958 4.063 5. M 3.391 3.500 3.609 3.719 3.828 3.938 4.017 4.156 4.266 24 Q 3.552 3.667 3.781 3.896 4.010 4.125 4.240 4.354 4.469 $6 K 3.714 3.833 3.953 4.073 4.193 4.313 4.432 4.552 4.67'2 H 6. 3.875 4.000 4.125 4.250 4.375 4.500 4.625 4.750 4.875 6. ix 4.036 4.167 4.297 4.427 4.557 4.688 4.818 4.948 5.078 X ^ 4.198 4.333 4.469 4.604 4.740 4.875 5.010 5.146 5.281 j 4.359 4.500 4.641 4.781 4.922 5.063 5.203 5.344 5.484 % 7. 4.521 4.667 4.813 4.958 5.104 5.250 5.360 5.542 5.688 7. U 4.682 4.833 4.984 5.135 5.286 5.438 5.590 5.740 5.891 J4 H 4.844 5.000 5.156 5.313 5.469 5.625 5.782 5.938 6.094 J3 5.005 5.167 5.328 5.490 5.651 5.813 5.975 6.135 6.297 H 8.* 5.167 5.333 5.500 5.667 5.833 6.000 6.167 6.333 6.500 8. i/ 5.328 5.500 5.672 5.844 6.016 6.188 6.359 6.531 6.703 IX M 5.490 5.667 5.844 6.021 6.198 6.375 6.552 6.729 6.906 y* ?i 5.651 5.833 6.016 6.198 6.380 6.563 6.745 6.927 7.109 x 9. 5.818 6.000 6.188 6.375 6.563 6.750 6.938 7.125 7.313 9. ^ 5.974 6.167 6.359 6.552 6.745 6.938 7.130 7.323 7.516 /4 Q 6.135 6.333 6.531 6.729 6.927 7.125 7.323 7.521 7.719 c % 6.297 6.500 6.703 6.906 7.109 7.313 7.516 7.719 7.922 ix 10. 6.458 6.667 6.875 7.083 7.292 7.500 7.708 7.917 8.125 10. ix 6.620 6.833 7.047 7.260 7.474 7.688 7.901 8.115 8.328 ix }* ..781 7.000 7.219 7.438 7.656 7.875 8.094 8.313 8.531 :u % 343 7.167 7.391 7.615 7.839 8.063 8.286 8.510 8.734 H 11. 7.104 7.333 7.563 7.792 8.021 8.250 8.479 8.708 8.938 11. 7.266 7.500 7.735 7.969 8.203 8.438 8.672 8.906 9.141 S 7.427 7.667 7.906 8.146 8.386 8.625 8.865 9.104 9.344 j 2i 7.589 7.833 8.078 8.323 8.568 8.813 9.057 9.302 9.547 9i 12. 7.750 8.000 8.250 8.500 8.750 9.000 9.250 9.500 9.750 12. Ji 8.073 8.333 8.594 8.854 9.115 9.375 9.635 9.896 10.16 % 13. 8.396 8.666 8.938 9.208 9.479 9.750 10.02 10.29 10.56 13. 8.719 9.000 9.281 9.563 9.844 10.13 10.41 10.69 10.97 Js 14. 9.042 9.333 9.625 9.917 10.21 10.50 10.79 11.08 11.38 14. 9.365 9.666 9.969 10.27 10.57 10.88 11.18 11.48 11.78 % 15. 9.688 10.000 0.31 10.63 10.94 11.25 11.56 1.88 12.19 15. J^ 10.01 10.33 0.66 10.98 1.30 11.63 11.95 2.27 12.59 M 16. 10.33 10.67 1.00 11.33 1.67 12.00 12.33 2.67 13.00 16. Ji 10.66 11.00 1.34 11.69 2.03 12.38 12.72 3.06 13.41 w 17. 0.98 11.33 1.69 12.04 2.40 12.75 13.10 3.46 13.81 17. 11.30 11.66 2.03 12.40 2.76 13.13 13.49 3.85 14.22 \i 18. 1.63 12.00 2.38 12.75 3.13 13.50 13.88 4.25 14.63 18. 19. 2.27 12.67 3.06 13.46 3.85 14.25 14.65 15.04 15.44 19. 20. 2.92 13.33 3.75 14.17 4.58 15.00 15.42 15.83 16.25 20. 21. 3.56 14.00 4.44 14.88 5.31 15.75 16.19 16.63 17.06 21. 22. 4.21 14.66 5.13 15.58 16.04 16.50 16.96 17.42 17.88 23. 4.85 15.33 15.81 16.29 16.77 17.25 17.73 18.21 18.69 23. 24. 15.50 16.00 16.50 17.00 17.50 18.00 18.50 19.00 19.50 24. I timber. See admirable paper on this subject by Col. T. J. Cram, U S Engs, Jour Franklin Inst, July. Aug. Sept 1873. Aiso footnote, p 414; also p 362. at top. If the sap of yreen timber be prevented from escaping at fhe ends of the sticks, as in the case of girders, &c, enclosed airtight In brickwork or masonrr. its fermentation will produce dry rot. The painting or varnishing of green timber conduces to the same end. In a free circulation of dry air timber will endure for centuries, if not attacked by worms. Alternate exposure to water and air produces wet rot* BOARD MEASURE. 361 Table of Board Measure (Continued.) -I Feet of Board Measure contained in one running foot of Scantlings of different dimensions. (Original.) a . ij So THICKNESS IN INCHES. M 10 10* 10* 10% 11 11* 11* 11* 12 ; Ft. Bd.M. Ft. Bd.M. FtJBd.M. Ft. Bd.M. Ft.Bd.M.jFt.Bd.M. Ft. Bd.M. Ft. Bd.M. Ft. Bd.M. ,, .2083 .2135 .2188 .2240 .2292 .2344 .2396 .2448 .2500 * u .4167 .427 1 .4375 .4479 .4583 .4688 .4792 .4896 .5000 IX if .6250 .8333 .6406 .8542 .6563 .8750 .6719 .8958 .6875 .9167 .7031 .9375 .7188 .9583 .7344 .9792 .7500 1.000 i* 1.042 1.068 1.094 1.120 1.146 1.172 1.198 1.224 1.250 ?x 1.250 1.821 1.313 1.344 1.375 .406 1.438 1.469 1.500 * X 1.458 1.495 1.531 1.568 1.604 .641 1.677 1.714 1.750 % 2. 1.667 1.708 1.750 1.792 1.833 .875 1.917 1.958 2.000 2. 1.875 1.922 1.969 2.016 2.063 .109 2.156 2.203 2.250 * \f 2.083 2.135 2.188 2.240 2.292 .344 2.396 2.448 2.500 * s 2.292 2.349 2.406 2.464 2.521 .578 2.635 2.693 2.750 9i 3. 2.500 2.563 2.6'25 2.688 2.750 2.813 2.875 2.938 3.000 3. 2.708 2.776 2.844 2.911 2.979 3.047 3.115 3.182 3.250 \6 2.917 2.990 3.063 3.135 3.208 3.281 3.354 3.427 3.500 * H 3.125 3.203 3.281 3.359 3.438 3.516 3.594 3.672 3.750 H 4. 3.333 3.417 3.500 3.583 3.667 3.750 3.833 3.917 4.000 4. 3.542 3.630 3.719 3.807 3.896 3.984 4.073 4.161 4.250 X ?x 3.750 3.844 3.938 4.031 4.125 4.219 4.313 4.406 4.500 x 3.958 4.057 4.156 4.255 4.354 4.453 4.552 4.651 4.750 % 5. 4.167 4.271 4.375 4.479 4.583 4.688 4.791 4.896 5.000 5. 4.375 4.484 4.594 4.703 4.813 4.922 5.031 5.141 5.250 ?x 4.583 4.698 4.813 4.927 5.042 5.156 5.270 5.385 5500 * X 4.792 4.911 5.031 5.151 5.271 5.391 5.510 5.630 5.750 H 6. 5.000 5.125 5.250 5.375 5.500 5.625 5.750 5.875 6.000 6. 5.208 5.339 5.469 5.599 5.729 5.859 5.990 6.120 1 6.250 Hi 5.417 5.552 5.688 5.823 5.958 6.094 6.229 6.365 6.500 * S 6.625 5.766 5.906 6.047 6.188 6.328 6.469 6.609 6.750 9i f. 5.8!<3 5.979 6.125 6.271 6.417 6.563 6.708 6.854 7.000 7. IX 3.042 6.193 d.344 6.495 6.646 6.797 6.948 7.099 7.250 M 6.250 <.406 6.563 6.719 6.875 7.031 7.188 7.344 7.500 * 8 S.458 6620 S.781 6.943 7.104 7.266 7.427 7.589 7.750 K a. 4 6.667 6.833 7.000 7.167 7.333 7500 7.667 7.833 8.000 8 ii 6.875 f.047 1.219 7.391 7.563 7.734 7.906 8.078 8.250 x4 It 7.083 7.260 7.438 7.615 7.792 7.969 8.H6 8-323 8.500 7.292 7.656 r.839 8.021 8.203 8.385 8.568 8.750 5i *.* 7.500 7'.388 7.875 8.063 9 .'250 8.438 8.625 8.813 9.000 9. 7.708 7.901 8.094 3.286 8.479 8.672 8.865 9.057 9.250 * 1 7.917 8.115 8.313 8.510 8.709 . 3.906 9.104 9.302 9.500 i x 8.125 8.323 8.531 8.734 9.141 9.344 9.547 9.750 x5 TO. 8.333 8.542 8.750 8.958 9'.167 9.375 9.583 9.792 10.00 10. 8.542 8.755 8.969 9.182 9.396 9.609 9.823 10.04 10.25 y * 8.750 8.969 9.188 9.406 9.625 9.844 1C. 06 10.28 10.50 3 ^ 8.958 9.182 9.406 9.630 9.854 10.08 10.30 10.53 10.75 Jf- 9.167 9.396 9.625 9.854 10.08 031 10.54 10.77 11.00 a. 9.375 9.609 9.844 1008 10.31 | 0.55 10.78 11.02 11.25 * * 9.583 9.823 10.06 10.30 10.54 1 0.78 11.02 11.26 11.50 * * 9.792 10.04 10.28 10.53 10.77 1.02 11.26 11.51 11.75 H 12. 10.00 10.25 10.50 10.75 11.00 1.25 11.50 11.75 12.00 12. 10.42 10.68 10.94 11.20 11.46 1.72 11.98 12.24 12.50 * 13. 10.83 11.10 11.38 11.65 11.92 2.19 12.46 12.73 13.00 13. 11.25 11.53 11.81 12.09 12.38 2.66 12.94 13 22 13.50 14. 11.67 11.96 12.25 12.54 12.83 3.13 13.42 13.71 14.00 14. 12.08 12.39 12.69 12.99 13.29 3.59 13.90 14.20 14.50 15. 12.50 12.81 13.13 13.44 13.75 4.06 14.38 14.69 15.00 15. 12.92 1324 13.56 13.89 14.21 4.53 14.85 1518 15.50 16. 13.33 13.67 14.00 14.33 14.67 15.00 15.33 15.67 1600 16. 13.75 14.09 . 14.44 14.78 15.13 5.47 15.81 16.16 16.50 17. 14.17 1452 14.88 15.23 15.58 15.94 16.29 16.65 17.00- 17. M 14.58 14.95 15.31 15.77 16.04 16.41 16.77 17.14 17.50 w re 15.00 15.38 15.75 16.13 16.50 16 88 17.25 17.63 18.00 18. 19. 15.83 16.23 16.63 17.02 17.42 17.81 18.21 18.60 1900 19. 20. 16.67 17.08 17.50 17.92 18.33 18.75 19.17 19.58 20.00 20. 21. 17.50 17.94 18.38 18.81 19.25 19.69 20.13 20.56 21.00 21. 22. 18.33 18.79 19.25 19.71 20.17 20.63 21.08 21.54 22.00 22. 23. 19.17 19.65 20.13 20.60 21.08 21.56 22.04 22.52 23.00 23. 24. 20.00 20.50 21.00 21.50 22.00 ^2.50 23.00 23.50 24.00 24. Price of lumber, Phila, 1873: Spruce joists, $28 to $30 per 1000 ft board meas. Hemlock joists, $20 to $23. Yellow pine floor boards, $40 to $60. White pine boards, $30 to $70, according to quality, degree of seasoning, &c. Sawed W pine timbers $40 to $50. Heart Y pine $50 to $55. Hemlock $20 to $25. In 188O prices range from | to J less. 362 WEIGHT OF CAST IRON. The durability of timber in situations either dry or merely damp (not wet), is said to be increased by soaking for a week or two in a solution of either aalt or quicklime, in water. See also " Creosote," p 358. TABLE OF WEIGHT OF CAST IRON.* The weight of a pattern of perfectly dry white pine, if mult by 20, will give approximately the wt of the casting. If well seasoned, but still not perfectly dry, mult by 19, or by 18. Assuming 450 Bbs to a cub ft, a pound contains 3.8400 cubic inches ; a ton 5 cub ft ; and a cubic inch weighs .2604 ft)s. 1 Thickness or Diameter 1 in Inches. Thick- ness or Diam. indeci- mals of a foot. Wt. of a Square Foot. Lbs. Wt. of a Square bar. 1 ft. long. Lbs. Wt. of a Round bar, I ft. Wt. of Balls. Lbs. t 1 Thickness or Diameter in Inches. Thick- ness or Diam. in deci- mals of a foot. Wt. of a Square Foot. Lbs. Wt. of a Square bar. 1 ft. long. Lbs. Wt. of a Round bar, 1 ft. long. Lbs. Wt. of Balls. Lbs. t 1-32 .0026 1.173 .003 .002 3^ .2604 117.3 30.52 23.97 4.162 1-16 .0052 2.344 .012 .010 M .2708 121.8 33.01 25.93 4.681 3-32 .0078 3.516 .027 .021 .0001 % .2813 126.5 35.60 27.95 5.243 M .0104 4.687 .048 .038 .0003 % .2917 131.2 38.28 30.07 5.846 5-32 .0130 5.861 .076 .060 .0005 % .3021 135.9 41.07 32.25 6.498 3-16 .0156 7.032 .110 .086 .0009 H .3125 140.6 43.95 34.51 7.193 7-32 .0182 8.203 .150 .118 .0014 .3229 145.3 46.93 36.85 7.934 x4 .0208 9.375 .195 .154 .0021 4. .3333 150.0 50.01 39.27 8.726 9-32 .0234 10.54 .247 .194 .0030 xi .3438 154.7 53.18 41.77 9.572 5-16 .0260 11.73 .305 .240 .0042 M .3542 159.3 56.46 44.33 10.47 11-32 .0287 12.89 .370 .290 .0056 9i .3646 164.0 59.82 46.99 11.42 .0313 14.06 .440 .346 .0072 .3750 168.7 63.33 49.71 12.43 13-32 .0339 15.24 .516 .400 .0092 xi .3854 173.4 66 >6 52,52 13.49 7-16 .0365 16.41 .598 .470 .0114 % .3958 178.1 70.52 55.39 14.62 15 32 1 .0391 17.56 .687 .540 .0140 y* .4063 182.8 74.28 58.34 15.81 H .0417 18.75 .781 .610 .0170 5. .4167 187.5 78.12 61.37 17.05 916 .0469 21.10 .989 .777 .0243 xi .4271 192.2 82.10 64.47 18.35 K .0521 23.44 1.221 .959 .0334 V* .4375 196.9 86.14 67.65 19.73 n-re .0573 25.79 1.478 1.161 .0444 .4479 201.6 90.29 70.52 21.18 .0625 28.12 1.758 1.381 .0575 xl2 .4583 206.2 94.54 74.26 22.68 13*16 .0677 30.47 2.064 1.621 .0732 % .4688 210.9 98.89 77.66 24.27 % .0729 32.81 2.393 1.880 .0913 % .4792 215.6 103.3 81.16 25.93 15-16 .0781 35.16 2.747 2.158 .1124 % .4896 220.3 107.9 84.72 27.41 1. .0833 37.50 3.125 2.455 .1363 6. .5000 225.0 112.5 88.36 29.44 1-16 .0885 39.84 3.528 2.771 .1636 M .5208 234.4 122.1 95.89 33.28 .0938 42.19 3.955 3.107 .1942 xl2 .5417 243.8 132.0 103.7 37.44 3-16 .0990 44.53 4.407 3.461 .2284 % .5625 253.1 142.4 111.9 41.94 \i .1042 46.87 4.883 3.835 .2664 7. .5833 262.5 153.2 120.2 46.77 f-16 .1094 49.22 5.384 4.229 .3084 X .6042 271.9 164.2 129.0 51.97 N .1146 51.57 5.909 4.640 .3546 .6250 281.3 175.8 138.1 57.54 7-16 .1198 53.91 6.461 5.073 .4058 % .6458 290.7 187.7 147.4 63.47 .1250 56.26 7.033 5.523 .4603 8. .6667 300.0 200.1 157.0 69.82 9-16 .1302 58.60 7.632 5.993 .5204 M .6875 309.4 212.7 167.0 76.58 .1354 60.94 8.253 6.484 .5852 J .7083 318.8 225.8 177.3 83.74 11-16 .1406 63.28 8.900 6.991 .6555 9 .7292 328.2 239.3 187.9 91.35 ax .1458 65.63 9.572 7.518 .7310 9. .7500 337.4 253.1 198.8 99.42 13-16 .1510 67.97 10.27 8.064 .8122 .7708 346.8 267.4 210.0 107.9 % .1563 70.32 10.99 8.630 .8991 .7917 356.2 282.1 221.5 116.8 15-16 .1615 72.66 11.73 9.215 .9920 H .8125 365.6 297.0 233.3 126.3 3. .1667 75.01 12.50 9.821 1.073 10. .8333 375.0 312.5 245.5 136.3 .1771 79.70 14.11 11.09 1.308 .8542 384.4 328.4 257.8 146.8 IX .1875 84.40 15.83 12.43 1.554 M .8750 393.7 344.5 270.6 157.9 % .1979 89.07 17.63 13.85 1.827 % .8958 403.1 361.2 283.7 169.3 ix .2083 93.75 19.54 15.34 2.131 11. .9167 412.5 378.2 297.0 181.5 fcx .2188 98.44 21.54 16.56 2.467 .9375 421.9 395.5 310.6 194.2 ax .2292 103.2 23.64 18.56 2.835 v .9583 431.2 413.3 324.6 207.3 tx .2396 107.8 25.84 20.29 3.241 H .9792 440.6 431.4 338.8 219.2 3. .2500 112.6 28.13 22.10 3.682 12. 1 Foot. 450. 450. 353.4 235.6 t Wts of balls are as the cubes of their diams. To find the weight of a spherical shell. From the weight of a ball which has the outer diam of the shell, take the wt of one which has its inner diam. * For Copper, mult by 1.2 ; Ltad, mult by 1.6; Brass, add 1-7 th ; Zinc, mult by .97. All approximate. WEIGHT OF CAST-IRON PIPES. 363 WEIGHT OF CAST-IRON PIPES per running foot, Assuming the weight of cast-iron at 460 ftn per cub ft, or .2604 ft) per cub inch. No allowance is here made for the spigot and faucet-joints used in water-pipes As the^e are now commonly made, (see Hydraulics, Fig 38,) they add to the weight of each length or section of pipe of any size, about as much as that of 8 inches in length of the plain pipe as given in the table. See Hydraulics, Art 16; also next table. For lead-pipe mult by l.G ; copper, mult by 1.2 ; brass, add l-7th ; welded iron, mult by 1.0667, or add one fifteenth part. iii s^s THICKNESS OP PIPE IN INCHES. S .S H \ % K x K 1 1* 191 IX ! 1H \H \ 2 Wtin Wiia Wtia Wtiu Wt in Wt in Wt in Wt in Wt in Wt in Wt in Wt in Wt in Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. LJbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. 1. 3.07 5.07 7.38 9.93 12.9 16.2 19.7 23.5 27.7 32.1 36.9 47.4 59.1 3.89 6.00 8.61 11.5 14.8 183 22 2 26.3 30.8 35.5 40.6 51.7 64.0 % 4.30 6.92 9.84 13.1 16.6 20.5 24.6 29.1 33.8 38.9 44.3 56.0 68.f % 4 .92 7.84 11.1 14.6 18.5 22.6 27.1 31.8 36.9 42.3 48.0 60.3 73.8 2. 5.53 8.76 12.3 16.2 20.3 24.8 29.5 34.6 40.0 45.7 51.7 64.6 78.7 6.15 9.69 13.5 17.7 222 26.9 32.0 37.4 43.1 49.0 55.4 68.9 86.7 i^ 6.76 10.6 14.8 19.2 24.0 29.1 34.5 40.1 46.1 52.4 59.1 73.2 88.6 H 7.37 11.5 16.0 20.8 25.9 31.2 36.9 42.9 49.2 55.8 62 7 77.5 93.5 3 7.93 12.5 17.2 22 3 27.7 33.4 39.4 457 52.3 59.2 66 4 81.8 98.4 8.60 13.4 18.5 23.8 29.5 35.5 41.8 48.4 554 62.3 70.1 86.1 103. \6 9.21 14.3 19.7 25.4 31.4 37.7 44.3 51.2 58.4 65.9 73.8 90.4 108. % 9.83 15.2 20.9 26.19 33.2 398 46.8 54.0 61.5 69.3 77.5; 94.7 113. 4. 10.3 16.1 22.2 28.5 35.1 42.0 49.2 56.7 64.6 72.7 81.2 99.0 118. y* 11.1 17.1 23.4 30.0 369 44.1 51.7 59.5 67.7 76.1 84.9; 103. 123. X 11.7 18.0 21.6 31.5 38.8 46.3 54.1 62.3 70.7 79.5 88.61 108. 1'28. x 12.3 18.9 25.8 33.1 40.6 48.5 56.6 65.0 73.8 83.9 92.3 112. 133. 5. 12.9 19.8 27.1 34.6 42.5 50.6 59.1 67.8 76.9 87.2 9.0 116. 138. % 13.5 20.8 28.3 36.1 44.3 52.8 54 9 61.5 70.6 73 3 80.0 83.0 90.6 94.0 99.6 103 121. 125. 143. 148. % 14^8 22.6 30 8 39.2 48.0 57^1 66.4 7e!i 86.1 97^4 107.' 129! 153] 6. 15.4 23.5 32.0 40.8 49.8 59.2 68.9 78.9 89.2 99.8 111. 134. 158. X 16.6 25.4 34.5 43.8 53.5 63.5 73.8 84.4 95.3 107. 118. 142. 167. 7. 17.8 27.2 36.9 46.9 57.2 67.8 78.7 89.4 102. 113. 126. 151. 177. X 19.1 29.1 39.4 50.0 60.9 72.1 83.7 955 108. 120. 133. 159. 187. 8. 20.3 30.9 41.8 53.1 64.6 76.4 88.6 101. 114. 127. 140. 168. 197. 21.5 32.8 44.3 56.1 68.3 80.7 93.5 107. 120. 134. 148. 177. 207. 9. 22.8 34.6 46.8 59.2 72.0 85.1 98.4 112. 126. 140. 155. 185. 217. X 240 36.4 49.2 62.3 75.7 89.3 103. 118. 132. 147- 163. 194. 226. 10. 25.1 38.3 51.7 65.3 79.4 93.6 108. 123. 138. 154. 170. 202. 235. h 23.4 401 54.1 68.4 83.0 97.9 113.2 129. 145. 161. 177. 211. 245. 11. 27.6 42.0 566 71.5 86.7 102. 118. 134. 151. 168. 185. 220. 255. X 28.8 43.8 59.1 746 90.4 107. 123. 140. 157. 174. 192. 228. 265. 12. 30.0 45.7 61.5 77.7 94.1 111. 128. 145. 163. 181. 199. 237. 275. 13. 32.5 49.4 66.4 83.8 102. 120. 138. 156. 175. 195. 214. 254. 294. 14. 35.0 5H.1 71.4 89.4 109. 128. 148. 168. 188. 208. 229. 271. 314. 15. 37.4 56.7 76.3 96.1 116. 137. 158. 179. 200. 222. 244. 289. 334. 16. 39.1 60.4 81.2 102. 124. 145. 167. 190. 212. 285. 258. 306. 353. 17. 42.3 61.1 6.1 108. 131. 154. 177. 201. 225. 249. 273. 323. 373. 18. 44.8 67.8 91.0 115. 139. 163. 187. 212. 237. 262. 288. 340. 393. 19. 47.3 71.5 96.0 121. 146. 171. 197. 223. 249. 276. 303. 357. 412. 20. 49.7 75.2 101. 127. 153. 180. 207. 234. 261. 289. 317. 375. 432. 21. 52.2 78.9 106. 133. 161. 188. 217. 245. 274. 303. 332. 392. 452. 22. 54.6 82.6 111. 139. 168. 196. 227. 256. 286. 316. 347. 409. 471. 23. 57.1 86.3 116. 145. 175. 208. 236. 267. 298. 330. 362. 426. 491. 24. 59.6 89.9 121. 152. 183. 214. 246. 278. 311. 343. 375. 441. oil. 25. 62.0 95.6 126. 158. 190. 223. 256. 289. 323. 357. 391. 461. 53t. 26. 61.5 97.3 131. 164. 198. 231. 266. 300. 335. 370. 406. 478. 550. 27. 66.9 101. 135. 170. 205. 240. 276. 311. 348. 384. 421. 495. 570. 28. 69.4 5. 140. 176. Ill 249. 286. 323. 360. 397. 436. 512. 590. 29. 71.8 9. 145. 182. 220. 257. 295. 334. 372. 411. 450. 530. 609. 30. 74.2 2. 150. 188. 227. 266. 305. 345. 384. 424. 465. 547. 629. 31. 76.7 6. 155. 195. 234. 275. 315. 356. 397. 438. 480. 564. 649. 32. 79.1 0. 160. 201. 242. 283. 325. 367. 409. 451. 495. 581. 668. 33. 81.6 Si 165. 207. 249. 292. 335. 378. 421. 465. 509. 698. 688. 34. 84.1 7. 170. 213. 257. 300. 345. 389. 434. 479. 524. 616. 708. 35. 86.5 I. 175. 219. 264. 309. 354. 400. 446. 492. 539. 633. 726. 36. sa.o 4. 180. 225. 271. 318. 364. 411. 458. 506. 554. 650. 746. 42. 104. 6. 210. 262. 315. 370. 423. 478. 532. 588. 644. 753. 864. 48. 119. 8. 239. 298. 359. 422. 482. 544. 605. 669. 733. 856. 982. For warming buildings by steam it usually suffices to allow 1 sq ft of cast or wrought pipe surface for each 120 cub ft of space to be warmed ; and 1 cub ft of boiler for each 2000 cub ft of such space. 364 WEIGHT OF CAST-IRON WATER-PIPES. WEIGHT OF CAST-IROX WATER-PIPES, As used in Phila, and tested by hydraulic press before delivery, to an internal pres of 300 ft>s per sq inch This table includes spigots and faucets, or bells. The pipes are required to be made of remeltcd strong tough gray pig iron, such as may be readily drilled and chipped ; and all of more than 4 ins diam to be cast verti- cally, with the bell end down. Deviations of 5 per cent above or below the theo- retical weights, are allowed for irregularities in casting, which it seems impossi- ble to avoid. Diam. Length. Weight. Thickness of body. Thickness of bell. Thickness ofleadjoint. Ins. Ft. Ins. Lbs. Ins. Ins. Ins. 3 9 135 A K K 4 9 190 % H A i 12 3 220 H it * 8 12 3^ 12 3J4 370 500 I | % % 10 12 3^ 640 1 % 12 12 3}$ 850 ft % A 16 12 3tf 12GO M i $ 20 12 3>* 1815 H >ft A Price, Phila, 1880, about $56 to $67 per ton ; or say 2^ to 3 cents per ft); de- pending on size, and quantity ordered. Elbows, connections, &c. about 6 to 15 per ct more. la ordering anything by the ton, be careful to specify the number of fts (2240). This prevents mis- understandings. Approximate average prices*iii cents per Ib, currency, of iron and steel in Philada, in 1880. Iron bars of ordinary sizes (% inch to 2 ins diam or square,) best Norway, 5^; Swedish, 5; American common, round or square, 3^ to 3%; refined, 3% to 4; increasingly degrees, for both larger and smaller sizes, to about 40 or 50 per ct more for bars of either % inch or of 4 inches diam or square. Extra refined, 1 ct per ft) more. Hoop iron. 4% for \% inch or more wide; to 7 cts for very thin % inch wide. Sheet iron, common, Nos 10 to 16, 4% ; Nos 16 to 25, 7 ; best, about 50 per cent more. Best Russian, 15 cts. Plates, common, 4^; best, 6. Angle, 4. T, 4%. Rolled I beams, 4% to 5. Plioeiiix columns, 5 to 5%. Rails, 2% to 3. Pig iron, common, 1% to 1%; best, 2. Cast iron house fronts, fitted and put up, 6% to 7. Ordinary castings, 2^ to 3% cents. Brass castings, 20 to 22 cts. Steel bars, ordinary sizes, common, 9^ to 10^; best, 13 to 14; machinery, 9. Very small, or large sizes, up to 40 or 50 per ct more. Sheet steel, common, 8 to 10; best, 13 to 14. Plates, homogeneous, 8^. Steel rails, 3*4 to 3%. Cast steel, ordinary, 7 to 9 cts ; tire, 5 to 6 ; spring, 5 ; tool, 12 to 13 cts. Weight per foot run of welded wrought -iron tubes, usually in lengths of 18 ft. Other sizes and lengths made to order. Inner Diam. Outer Diam. Ths. Wt. per Foot. Inner Diam. Outer Diam. Ths. Wt. per Foot. Inner Diam. Outer Diam. Ths. Wt. per Foot. Ins. Ins. Ins. Lbs. Ins. Ins. Ins. Lbs. Ins. Ins. Ins. Lbs. .270 .405 .068 .243 .928 1.660 .366 5.065 2.331 2.855 .262 7.126 .320 .540 .110 .505 .995 1.315 .160 1.976 2.468 2.876 .204 5.773 .320 .840 .260 1.613 1.048 1.315 .134 1.670 3.067 3.500 .217 7.547 .364 .540 .088 .422 1.108 1.900 .396 6.370 3.548 4.000 .226 9.055 .437 .675 .119 .708 1.294 1.660 .183 2.891 4.026 4.500 .237 10.728 .478 1.050 .286 2.338 1.380 1.660 .140 2.258 4.508 5.000 .247 12.492 .494 .675 .091 .561 1.459 2.375 .458 9.390 5.045 5.563 .259 14.564 .580 .840 .130 .987 1.504 1.900 .198 3.604 6.065 6.625 .280 18.767 .623 .840 .109 .845 1.611 1.900 .145 2.694 7.023 7.625 .301 23.410 .675 1.315 .320 3.405 1.807 2.855 .524 13.656 7.982 8.625 .322 28.348 .764 1.050 .143 1.387 1.917 2.375 .229 5.400 9.001 9.688 .344 34.077 .824 1.050 .113 1.126 2.067 2.375 .154 3.667 10.019 10.750 .366 40.641 Prices of welded iron tubes. Philada, 1880, approx; 2 ins diam, 7U cts per ft ; 3 ins diam, 8 cts ; 6 ins diam, 9 cts : 8 ins diam, 1 1 ^ cts. Made on a large scale by W. C Allison ft Co, Junction Car Works, 32d and Walnut Sts, Philada, * In 1882 all about 20 per ct lower. 365 The foregoing well known firm of W. C. Allison Jk Co are builders of R. R. cars of all kinds; and furnish all the appliances for roofs, buildings, and bridges of iron ; and furnish railroad supplies, &c, &c. Seamless I>rawn Brass ami Copper Tubes of the American Tube Works, Boston, Mass. Approximate sizes and weights.* Although tliis is the Go's own table, it is not consistent in itself; for where two thicknesses are given, the wts in some cases correspond with the least, and in others with the greatest ; and there is no way to discriminate between them. Outer Diam. Ins. sr- Ths. Inch. Wt. i per Brs. n Ibs. ft. Cop. Outer Oiam. IDS. LRth. Ft. Ths. In. Wt. i per Brs. u fts. ft. Cop. Outer Diam. Ins. *? Ths. lu. Wt. i per Brs. nibs, ft. Cop. N 10 .049 .375 .375 .083 .095 K 12 .058 .500 .500 \% 13 to 1.75 1.80 2K 12 to 3. 3.13 Yt 10 .058 .625 .625 .109 .120 1 10 .065 .750 .750 1 12 1.88 1.94 2 12 3.13 3.25 IK 10 .065 .875 .875 2 15 2.2 2.25 3 12 3.33 3.5 .083 2K 13 ik 2.25 2.38 3K 10 < 3.5 3.63 IX 15 to 1.25 1.25 J 14 2.38 2.50 3k' 10 < 3.88 4.13 .109 '2H 13 << 2.50 2.67 3^ 10 4.25 4.38 IH 10 " 1.375 1.375 .095 4 10 5. 5.25 w 14 " 1.50 1.60 ZX 13 to 2.75 3. 5 10 M 7. 8. IK 12 " J.631 1.70 .120 These tubes are furnished either plain, or tinned with pure tin. The Co also make the connections, elbows, bends, &c. of the same metals, usually required in plutnbincr. The brass tubes have been used in Boston, &c, as a substitute for lead service-pipes in dwellings for several years ; and nre much approved of. They have also been introduced in the City Hospital, and in some of the hotels. irices, either brass or copper, per foot: Diam %, 25 cts; %, , ,., . _jh. 60 cts ; 1^, 70 cts ; 1%, 80 cts. If tinned, add about one-seventh part to th prices. Liberal discounts on large sales. 1880. 24 * Appro x pri s ; %, 45 cts : 1 inch. 366 WEIGHT OF WROUGHT IRON AND STEEL. Table of Weight of WROUGHT IRON* and STEEL. Wrought iron is here taken at 485 fts per cub f t ; or a sp gr of 7.77. Very pure soft wrought iron weighs from 4&8 to 492 fts per cubic foot. Light weight indicates impurities, and weakness. Steel weighs about 49 J Ibs per cubic toot ; therefore, for steel, an addition must be made to tlie tabular amounts, of about 1 pound in 1OO Ibs. At 485 Ibs per cub ft, a cubic inch weighs .28067 ft ; a ft contains 3.5629 cub ins ; and a ton, 4.6186 cub ft ; and this is about the average of hammered iron The usual assumption is 480 Ibs per cub ft; which is nearer the average of ordinary mild iron. At 480 fts, a cubic inch weighs .2778 of a ft); a ft) contains 3.6 cub ins ; a ton, 4 6bb7 cub ft ; a rod of 1 sq inch area, weighs 10 fts per yard ; or 3% fts per foot, exactly. hiekness Diameter i Inches. Thick- ness or Diani. in deci- mals of Wt. of a Square Foot. Wt. of a Square bar, 1 ft. long. Wt. of a Round bar, 1 ft. long. Wt. of Balls. ill *? = Thick- ness or Diam. in deci- mals of Wt. of a Square Foot. Wt. of a Square bar. 1 ft. long. Wt. of a Round bar. 1 ft. long. Wt. of Ball*. Kfc a foot. Lbs. Lbs. Lbs. Lbs. ~ o'~ a foot. Lbs. Lbs. Lbs. Lbs. 1-32 .0026 1.263 .0033 .0026 *x .2604 126.3 32.89 25.83 4.484 1-16 .0052 2.526 .0132 .0104 .2708 131.4 3557 27.94 5.045 3-32 .0078 3.789 .0296 .0233 .0001 ?'8 .2813 1364 38.37 30.13 5.649 X .0104 5.052 .0526 .0414 .0003 X .2917 141.5 41.26 32.41 6.301 5-32 .0130 6.315 .0823 .0646 .0005 % .3021 146.5 44.26 34.76 7.000 3-16 0156 7.578 .1184 .0930 .0009 K .3125 151.6 47.37 37.20 7.750 7-32 .0182 8.841 .1612 .1266 .0015 % .3229 156.6 50.57 39.72 8.550 X .0208 10.10 .2105 .1653 .0023 4. .3333 161.7 53.89 42.33 9.405 932 .0234 11.37 .2665 .2093 .0033 K .3438 166.7 57.31 45.01 10.32 5-16 .0260 13.63 .3290 .2583 .0045 tt .3542 171.8 60.84 47.78 11.28 11-32 .0287 13.89 .3980 .3126 .0060 % .3046 176.8 64.47 50.63 12.31 M .0313 15.16 .4736 .3720 .0078 X .3750 181.9 68.20 53.57 13.39 13-32 .0339 16.42 .5558 .4365 .0098 >A .3854 Ifc6.9 72.05 56.59 14.54 7-16 .0365 17.68 .6446 .5063 .0123 H .3958 192.0 75.S-.9 5t).9 15.75 15-32 0391 18.95 .7400 .581:} .0151 % .4063 1H7.0 80.05 62.87 17.03 M .0417 20.21 .8420 .6613 .0184 5. .4167 202.1 84.20 (if,. 13 18.37 9-16 .0469 22.73 1.066 .8370 .0262 # .4271 207.1 88.47 69.48 19.78 % .0521 25.26 1.316 1.033 .0359 k .4375 212.2 92.83 72.91 21.26 11-16 .0573 27.79 1.592 1.250 .0478 S /8 .4479 217.2 97.31 76.43 22.82 % .0625 3031 1.895 1.488 .0620 % .4583 222.3 101.9 80.02 24.45 13-16 .0677 32.84 2.223 1.746 .0788 H .4688 227.3 10fi.6 83.70 26.16 X .0729 35.37 2.579 2.025 .0985 H .4792 232.4 111.4 87.46 27.94 15-16 .0781 37.89 2.960 " 2.325 .1211 % .4816 237.5 116.3 91.31 29.80 1. .0833 40.42 3.368 2.645 .1470 6. .5000 242.5 121.3 95.23 31.74 1-16 .0885 42.94 3.803 2.986 .1763 H .51:08 252.6 131.6 103.3 35.88 K .0938 45.47 4.263 3.348 .2093 % .5417 262.7 142.3 111.8 40.36 3-16 .0990 48.00 4.750 3.730 .2461 K .5625 272.8 153.5 S0.5' 45.19 M .1042 50.52 5.263 4.133 .2870 7. .58."3 282.9 165.0 2f!.6 50.40 5*16 .1094 5305 5.802 4.557 .3323 g .6042 293.0 177.0 3Ji.O 56.00 % .1146 55.57 6.368 5.001 .3820 8 .6-;:o 303.1 Ih9.5 48.1 62.00 7-16 .1198 58.10 6.9IJO 5.466 .4365 x .64f,8 313.2 V02.3 5.8 68.40 X .1250 60.63 7.578 5.952 .4960 8. .6fi07 823.3 215.6 C9.3 75.24 9-16 .1302 63.15 8.223 6.458 .5606 M .6875 333.4 229.3 80.1 82.52 N .1354 65.68 8.893 6.985 .6306 /*> .70a3 343.5 243.4 91.1 90.25 11-16 .1406 68.20 9.591 7.533 .7062 % .7292 353.6 247.9 202.5 98.45 % .1458 70.73 10.31 8.101 .7876 9. .7500 363.8 272.8 214.3 107.1 13-16 .1510 73.26 11.07 8.690 .8750 M .7708 373.9 288.2 226.3 116.3 y .1563 75.78 11.84 9.300 .9688 N .7917 384.0 304.0 238.7 126.0 15-16 .1615 78.31 12.64 9.930 1.069 X .8125 394.1 320.2 251.5 136.2 2. .1667 80.83 13.47 10.58 1.176 10. .8333 404.2 336.8 264.5 146.9 x .1771 85.89 15.21 11.95 1.410 X .8542 414.3 353.9 277.9 158.2 .1875 90.94 17.05 13.39 1.674 H .8750 424.4 371.3 291.6 170.1 */ .1979 9599 19.00 14.92 1.969 % .8958 434.5 389.2 305.7 182.6 $4 .2083 101.0 21.05 16.53 2.296 11. .9167 444.6 407.5 320.1 195.6 % .2188 106.1 23.21 18.23 2.658 .9375 454.7 426.3 334.8 209/2 H .2292 111.2 25.47 20.01 3.056 M .9583 464.8 445.4 349.8 223.5 % .2396 116.2 27.84 21.87 3492 H .9792 474.9 465.0 365.2 238.4 3. .2500 121.3 30.31 23.81 3.968 12. 1 Foot. 485. 485. 380.9 253.9 To find the weight of a spherical shell. From the weight of a ball which has the outer diam of the shell, take the wt of one which has its inner diam. * For Copper, add l-7th part. Lead, mult by 1.47. Brass, mult by 1.06. Zinc, mult by ,9. Tin, mult by .95. All approximate. WEIGHT OF SHEET METALS. 367 Weight of one square foot of Rolled, or Sheet Iron, Steel, Copper, or Brass Plates. (From llaswell.) Thickness by the Birmingham gauge for iron wire, sheet iron, and steel. For rolled lead, multiply copper by 1.3 ; and for zinc, multiply wrought iron by the deci- mal .9. Thickness by the American gauge. For rolled lead, multiply copper by 1.3; aud for zinc, multiply wrought irou by the decimal .9. Silver is >j heavier thau steel. *l fco Thick- ness. Iron. Steel. Copper. Brass. o ii> Thick- ness. Iron. Steel. Copper. Brass. Ins. Lbs. Lbs. Lbs. Lbs. Ins. LOB. Lbs. Lbs. Lbs. 0000 .454 18.35 18.54 20.5662 19.4312 0000 .46 18.63 18.87 20.838 19.688 000 .425 17.18 17.35 19.2525 18.19 000 .40964 16.58 16.80 18.5567 17.5328 00 .38 15.36 15.51 17.214 16.264 00 .3648 14.77 14.96 16.5254 15.6134 .34 13.74 13.87 15 402 14.552 .32486 13.15 13.32 14.7162 13.904 1 .3 12.13 12.25 13.59 12.84 .2893 11.70 11.86 13.1053 12382 2 .284 11.48 11.59 12.8652 12.1552 2 .25763 10.43 1057 11.6706 11.0268 3 .259 10.47 10.57 11.7327 11.0852 3 .22942 9.291 9.415 10.3927 9.8192 4 .238 9.619 9.715 10.7814 10.1864 4 .20431 8.273 8.384 9.2552 8.7445 5 .22 8.892 8.981 9.966 9.416 5 .18194 7.366 7.462 8.2419 7.i87 6 .203 8.205 8.287 9.1959 8.6884 6 .16202 6.561 6.648 7.3395 6.9345 7 .18 7.275 7.348 8154 7.704 7 .14428 5.842 5.920 6.5359 6.1752 8 .165 6.669 6.736 7.4745 7.062 8 .12849 5.203 5.272 5.8206 5.4994 9 .148 5.981 6041 6.7044 6.3344 9 .11443 4.633 4.695 5.1837 4.8976 10 .134 5.416 5.470 6.0702 5.7352 10 .10189 4.125 I 4.180 4.6156 4.3609 11 .12 4.850 4.899 5.436 5.136 11 .090742 3.672 I 3.723 4.1106 3.8838 12 .109 4.405 4.449 4.9377 4.6652 12 .08080h 3.272 3.315 3.6606 3.4586 13 .095 3.840 3.878 4.3035 4.066 13 .071961 2.916 2.952 3.2598 30799 14 .083 3.355 3.388 3.7599 3.5524 14 .064084 2.592 2.629 2.903 2 7428 15 .072 2.910 2.939 3.2616 3.0816 15 .05706* 2.311 2.341 2.5852 2.4425 16 .065 2.627 2.653 2.9445 2.782 l(i .05082 2.052 2.085 2.3021 2.1751 17 .058 2.344 2.367 2.6274 2.4824 17 .045257 1.825 1857 2.0501 1.937 18 .049 1.980 1.999 2.2197 2.0972 18 .040303 1.681 1.653 1.8257 1.725 19 .042 1.697 1.714 1.9026 1.7976 19 .03589 1.452 1.468 1.6258 1.5361 20 .035 1.415 1.429 1.5855 1.498 20 .031961 1.293 1.311 1.4478 1.3679 21 .032 1.293 1.305 1.4496 1.369H 21 .028462 1.152 1.166 1.2893 1.2182 22 .028 1.132 1.143 1.2684 1.1984 22 .025847 1.026 1.040 1.1482 1.0849 23 .025 1.010 1.020 1.1325 1.07 23 .022571 .913 .9-25 1.0225 .96604 24 .022 .8892 .8981 .9966 .9413 24 .0201 .814 .824 .91053 .86028 25 .02 .8083 .8164 .906 .856 25 .0179 .724 .734 .81087 .76612 26 .018 .7225 .7348. .8154 .7704 26 .01594 .644 .653 .72208 .68223 27 .016 .6467 .6532 .7428 .6848 271.014195 .574 .582 .64303 .60755 28 i .014 .5658 .5715 .6342 .5992 28 .012641 .511 .518 .57264 .54103 291 .013 .5254 .5307 .5889 .5564 29 .011257 .455 .471 .50994 .4818 30, .012 .4850 .4899 .5436 .5136 30 .010025 .405 .410 .45413 .42907 31 i .010 .4042 .4082 .453 .428 311.008928 .360 .366 .404441 .38212 32 .009 .3638 .3674 .4077 .3852 32L00795 .321 ' .326 .36014! .3402B 33 j .008 .3233 .3265 .3624 .3424 33 .00708 .286 .290 .32072 .30302 34 .007 .2829 .2857 .3171 .2996 34 .006304 .254 .258 .28557 .26981 35 .005 .2021 .2041 .2265 .214 35 .005614 .226 .230 .25431 .24028 36 .001 .1617 .1633 .1812 .1712 36 .005 .202 .205 .2265 .214 37 .004 453 .180 .182 .20172 .19059 38 .003965 .159 .162 .17961 .1697 39 .003531 .142 .144 .15995 .1511H 40 .003144 .127 .128 .14242 .13456 Approximate prices in 1873 for copper or brass sheets, Nos to 25, from 42 to 48 cts per fl>. Ingots, 30 to 35 cts. Castings, 35 to 45 cts. Sheet iron, com- mon, Nos 10 to 16, 514 ; Nos 16 to 25, 8 cts ; best, about 50 per ct more. Sheet steel, common, 8 to 10 cts ; best, 15 to 17 cts. The first two cols vary considerably in different books. Ours are from a table prepared by an Kng- Hsh maker of the g;iuge themselves. See " Gauge," in Tomliuson's Cyclopedia. No trade Stupidity is more thoroughly senseless than the adherence to the various Birmingham, Lancashire, &c, gauges; instead of at once denoting the thickness and diameter of sheets, wire, &c, by the parts of an inch ; as has long been suggested. Thus, No. ^, or No. -^ wire, or sheet-metal of any kind, should be understood to mean ^ or J^ of an inch diaru, or thickness. To avoid mistakes, which are very apt to occur from the number of gauges in use : and from the absurd practice of applying the same No. to different thicknesses of different metals, in different towns, it is best to ignore them all ; and in giving orders, to define the diameter of wire, and the thickness of sheet-metal, by parts of an inch. Or the weight per hundred ft for wire: or per q ft for sheets. may be employed. We believe that the foregoing Birmingham gauge applies to zinc, copper, brass. 368 WEIGHT OF WIRE. and lead; although It ia generally stated to be for iron and steel only. Another Birmingham gauge Is used for sheet brass, gold, silver, and some other metals; but we have ueverseen it stated what those others are. There are different gauges even for wire to be used for different purposes; and various firms have gauges of their own ; not even according among themselves. The American gauge differs from all others, and is equally senseless. We have Stubs (England) gauge: but as Mr. Stubs makes various English gauges, the term by itselj means nothing. Gene- rally, however, in our machine shops, it applies to the Birmingham gauge of the preceding table. Weig-ht of one foot in length of Wire, of Iron, Steel, Copper, or Brass. (From Haswell.) )iameters by the Birmingham gauge for iron >vire, sheet iron, and steel. Diameters by the American gauge. | a Iron. Steel. Copper. Brass. i 1 Iron. Steel. Copper. Brass. C5 5 ^o Q Ins. j Lbs. j Lbs. { Lbs. Lbs. Ins. Lbs. Lbs. Lbs. Lbs. >00 .454 .546207 .55136 .623913 .589286 )000 .46 .56074 .56603 .640513 .605176 K)0 .425 .4786 16 .48317 I .5 46' 52 .516407 000 .40964 .444683 .448879 .507946 .479908 00 .38 .3826 i .38627 .4 37C 99 .41284 00 .3648 .352659 .355986 .40283 .380666 .34 .3063 I .30923 .3 491 21 .3305 .32486 .279665 .282303 .319451 .301816 1 .3 .2385 .24075 .2 72- 3 .25731 1 .2893 .221789 .223891 .253342 .239353 2 .284 .2137 58 .21575 > .2 441 46 .230596 2 .25763 .175888 177548 .200911 .189818 3 .259 .1777 35 .17944 I .2 D3C 54 .19178)5 3 .22942 .13948 .140796 .159323 .150522 4 .238 .1501 )7 .15152 J .1 71 61 .161945 4 .20431 .110616 .11166 126353 .119376 5 .22 .1282 .12947 .1 46: 07 .138376 5 .18194 08772 .088548 .1002 .094666 6 .203 .1092 4 .11023 I .1 241 4 .117817 6 .16202 .069565 .070221 .079462 .075075 7 .18 .0858 .08666 r .0 J8C 75 .092632 7 .14428 055165 .055685 .063013 .059545 8 .165 .0721 ?6 .07282 r .0 324 1 .077836 8 .12849 .043751 .044li4 .049976 .047219 9 .148 .0580 16 .05859, .0 03 .062624 9 .11443 .034699 .035026 .039636 .037437 10 .134 .0475* S3 .04803 i .0 ->4; 53 .051336 10 .10189 .027512 .027772 .031426 .029687 11 .12 .038K .03852 .0 IK 89 04117 11 .090742 .02182 .022026 .024924 .023549 12 .109 .0314* 5 .03178 > .0 64 .033968 12 .080808 .017304 .017468 .019766 .018676 13 .095 .0239 6 .02414 j .0 273 19 .025802 13 .071%! .013722 .013851 .015674 .014809 14 .083 .0182, 6 .01842* $ .0 20? 53 .019696 14 .064084 010886 .010989 .012435 .011746 15 072 .0137;. 8 .01386' .0 I5f 92 .014821 15 .057061- 008631 .008712 .009859 .0093i5 16 .065 .0111J 6 .01130} .0 27 S9 .012079 16|.05082 006845 .006909 .007819 .007587 17 .058 .00891 5 .00899J .0 01 83 .009618 17 .045257 005427 .005478 .006199 .005857 18 .049 .0083C 3 .00642; .0 172 68 .006864 18 .04030: 004304 .004344 .004916 .004645 19 .0*2 .0046' 5 .0047H 4 .005043 19 .03589 003413 .003445 .003899 .003684 20 .035 .0032J 6 .003277 .0 >37 08 .003502 20 .03196! 002708 .002734 .003094 .00292 21 .032 .00271 4 .002739 )31 .002928 21 .02846L 002147 .002167 .002452 002317 22 .028 .00207 8 .002097 )23 73 .002241 22 025347 001703 .001719 .001945 .001838 23 025 .00165 6 .001672 .Of 1- 92 .001787 23 . 02257 J 00135 .001363 .001542 .001457 24 .022 .00128 3 .001295 .0( 14 63 .001384 24 .0201 001071 .001081 .001223 .001 155 25 02 .00106 .001070 .oc 12 11 .001144 25 .0179 0008491 .0008571 .0009699 . 00091 6S 26 .018 .00085 86 .00086S 7 I.OC 09 307 .0009263 26 .01594 0006734 .0006797 .0007692 .0007267 27 .016 .00067 84 .000684 8 I.OC 07 749 .0007319 27 .01419c 000534 .0005391 .0006099 .0005763 28 .014 .00051 94 .000524 3 .01 or. 933 .0005604 28 .012641 0004235 .0004275 .0004837 .000457 29 .013 .00044 79 .000452 1 .Of i).') 16 .0004832 29 .011257 0003358 .0003389 .0003835 .0003624 30 .012 .00038 16 .000385 2 .oc 104 J59 .0004117 30 .010025 0002663 .0002688 .0003042 .0002874 31 .01 .00026 5 .000267 5 .00 327 .0002859 31 .008928 0002113 .0002132 .0002413 .000228 32 .009 .00021 47 .000216 7 .01 02 152 .0002316 32 .00795 0001675 .0001691 .0001913 .0001808 33 .008 .00016 96 .000171 2 .01 01 W .000183 33 .00708 0001328 .0001341 .0001517 .0001434 34 .007 .00012 99 .000131 1 .on 01 183 .0001401 34 006304 0001053 ;. 0001063 .0001204 .0001137 35 .005 .00006 625 .000066 ss .00 00 f568 .00007148 35 .005614 00008366 .00008445 .0000956 .00009015 36 .004 .00004 24 .000042 8 .00 00 1843 .00004574 36 .005 00006625 .00006687 .0000757 .0000715 37 .00445." 00005255 .00005304 .00006003 .00005671 lirmingham g>aus?e for sheet 39 .' 003531 00004166 .00004205 .00004758 uuuu**sfo 00003305 00003336 .00003775 i .00003566 Brass, i Silver, Go If 1, and all 40 .003144 0000262 ; .C0002644 i .00002992; .00002827 metals except iron and steel? ' ' 1 Price of brass and copper wire, approximate for 1880. Alison ia Brass ^ | 1 2 Zi 1 | fc Copper o., Nos 19 & 21 Cliff St, N YorK For 100 ^^ nv rnf ^ v ^ Inch .004 ' .005 8 Inch .015 .016 13 14 Inch .036 19 .041 20 Inc .06 .06 25 26 Inch .095 .103 j Inch 31 .133 32 .143 Nos to 25 Coppe " " " " Highl " " " " Low b r 46 ct< > per ft). arass 36 " rass 40 " .008 J .019 15 .047 '21 .07 11 .113 33 .145 .010 1C .024 16 .051 22 .07 'K .120 34 .148 .012 1] .029 17 .057 23 .077 29 .124 35 .158 .013 IV .034 18 .061 24 .082 30 .126 36 .167 IRON WIRE. 369 Table of Iron wire made by the Trenton Iron Co; Trenton, N. J. Charles Hewitt Esq, President. The numbers of the wires in the first column are those of the Trenton Iron Co ; and correspond to somewhat smaller diaius than those ot the Birmingham gauge. No. by Tr. Ir. CosG. Diam. las. Area. Sq. ius. No. to a sq. in. of metal. Wt. of 100 yds. Lbs. Breakg pull. Lbs. No. by Tr.Ir. CosG. Diam. lus. Area. Sq. ins. No. to a sq.in.of metal. Wt. of 100yds. Lbs. Breakg pull. Lbs. .305 .07306 13.69 73.96 5926 .130 .01327 75.36 13.43 1233 1 .285 .06379 15.68 64.58 5226 1 .1175 .01084 92.25 10.97 1010 2 .265 .05515 18.13 55.83 4570 2 .105 .00866 115.5 8.767 810 3 .245 .04714 21.21 47.72 3948 3 .0925 .00672 148.8 6.804 631 4 .225 .03976 25.15 40.25 3374 4 .080 .00503 198.8 5.092 474 5 .205 .03301 30.29 33.42 2839 .070 .00385 259.7 3.897 372 6 .190 .02835 35.27 28.70 2476 6 .061 .00292 342.5 2.956 292 7 .175 .02405 41.58 24.35 2136 7 .0525 .00216 462.9 2.187 222 8 .160 .02011 49.73 20.36 1813 8 .045 .00159 629.0 1.610 169 S .145 .01651 60.57 16.71 1507 20 .033 .00086 1162.8 .870 96 The wire in this table is supposed to be hard, bright, or unannealed. Annealing renders wire more pliable, but less elastic; and reduces its strength per- haps 20 or 2o per cent. The strength** in the last column are at the rate of 81000 ft>s per sq inch for No. ; 9WOO for No 10; and 111000 for No. 20: and are say from 15 to *25 per ct greater than those of ordinary market wire. The wires in the table are all made of the same specially prepared iron ; and the increased strength per sq in of the smaller ones is due to their more frequent passage through the draw-plates. The choice Swedish and Norway irons would not yield wire of much greater strength ; while the cost would be about 60 per ct more. Hard steel wire averages about twice the strength of iron wire. Unannealed or hard brass wire has about ^ths the strengths of the above table, and about ^ more weight. If annealed, only full half the strength. Hard copper wire may be taken at % of the tabular strengths, and full ^ more weight. To find approximately the number of straight wires that can be got into a cable of given diameter. Divide the diameter of the cable in inches, by the diameter of a wire in inches. Square the quotient. Multiply said square by the decimal .77. The result will be correct within about 4 or 5 per cent at most, in u cylindrical cable. The solidity, or metal area of all the wires in a cable,, will be to the area of the cable itself, about as 1 to 1.3. In other words, the area of the voids is nearly % that of the cable; while that of the wires is fully % that of the cable. All approximate. Prices approx for 1880. I ron W i re of quality of the above table. Best Cast Steel wire. Number of wire by Trenton Iron Co's gauge. to 9 | 10 to 13 | 14 to 17 | 18 to 20 | 23 | 28 Approximate price iu cents per pound. 8 I 10 I 12^ I 14 I 16 I 21J4 I 26^ I 30}$ I 60 | Tinned wire is about one third higher price. The prices of annealed and unannealed wire are the same. The Co make other wire of specified quality to order. BUCKLED PLATES of plate iron are usually 3 or 4 ft square, from 1-20 to 38 ins thick, with a flat rim about 2 ins wide all around, with rivet or bolt holes for holding the plate firmly down to its intended place. The rest of the plate is stamped into the form of a kind of groined arch rising from 1 to 2 ins in the center. They are verv strong, and are used for the floors of tire-proof buildings, and of city iron bridges, covered with asphalt or stone paving. &c. One of 3 ft sq, .25 inch thick, c .rved, 1.75 ins. nnd with a 2 inch rim well bolted down on all sides, required a quiet, equally distributed load of 18 tons to crush it. When unbolted the strength is only half as great. Table of safe, quiet, uniformly distributed loads (.25 of the ultimate ones) for buckled iron plates 3 ft square, arched 1.75 ins, and well bolted down on all sides. Also approx wt iu Ibs per sq yd. B.W.G. lus. Wt. Tons. H>s. Ins. Wt. Tons fts. Ins. Wt. Tons. fts. 18 .018 17 .27 604 X 45 1.0 2240 5-16 113 6.2 13b88 16 .066 24 .43 96 i 3-16 68 2.5 5600 % 135 9.0 20160 12 ! .107 39 .64 1433 y* 90 4.5 10080 Soft puddled steel will bear nearly twice as much. 370 GALVANIZED IRON. Weights in IDS per sq ft, of galvanized sheet iron, both flat and corrugated. The Nos and thicknesses are those of the iron before it is galvanized ; but the weights refer to 'the galvd sheets. When a flat sheet (the ordi- nary size of which is from '2 to "2% ft in width, by 6 to 8 ft in length) is converted into a corrugated one, with corrugations 5 ins wide from center to center, and about an inch deep (the common sizes,) its width is thereby reduced about y^th part, or from 30 to 27 ins; and consequently the weight per sq ft of area covered is increased about ^th part. When the corrugated sheets are laid upon a roof, the overlapping of about 2% ins along their sides, and of 4 ins aiorg their ends, diminishes the covered area about ^th part more ; making their weight per sq ft of roof about %th part greater than before. Or the weight of corrugated iron per sq ft, in place on a roof, is about ^ greater than that of the flat sheets of above sizes of which it is made. GALVANIZED IRON. Weights per square foot. No. by Birming- ham wire Thick- ness in inches. Flat. Lbs. Corru- gated. Lbs. Corrug on roof. Lbs. No. by Birming- ham wire Thick- ness in inches. Flat. Lbs. Corru- gated. Lbs. Corrug. on roof. Lbs. gauge. gauge. 30 .012 .806 .896 1.08 21 .032 l.8 1.81 2.17 29 .013 .857 .952 1.14 20 .035 1.75 1.94 2.33 28 .014 .897 .997 1.20 19 .042 2.03 2.26 2.71 27 .016 .978 1.09 1.30 18 .049 2.32 2.58 3.09 26 .018 1.06 1.18 1.41 17 .058 2.68 2.98 3.57 25 .020 1.14 1.27 152 16 .065 2.96 3.29 3.95 24 .022 1.22 1.36 1.62 15 .072 3.25 3.61 4.33 23 .025 134 1.49 1.79 14 .083 3.69 4.10 4.92 22 .028 1.46 1.62 1.95 13 .095 418 4.64 5.57 N os. 18 to 23 are commonly used for covering roofs. The galvanizing is simply a thin film of zinc on both sides of the sheet, as in what is known as " tinned plates," or " tin ; " which are in reality sheet iron similarly coated with tin. Zinc, like tin, resists corrosion from ordinary atmospheric influences, much better than iron ; and hence the use of these metals as a protection to the iron. A well galvanized roof, of a good pitch, will suffer but little from 5 to 6 years' exposure without being painted. It will then take paint readily, and should be painted. It is better, however, always to paint tin ones at once. Paint does not adhere well to new zinc, and this is the principal reason why new galvanized roofs are not painted: but this may be remedied by first brushing the zinc over with the following: One part of chloride of copper, 1 part nitrate of copper, 1 part of sal- ammoniac. Dissolve in 64 parts of water. Then add 1 part of commercial hydrochloric acid. When brushed with this solution, the zinc turns black ; dries within 12 to 24 hours, and may then be painted. Paint of some mineral oxide of a brown color is generally used : one coat being applied to both sides in the shop ; and the other after being put on the roof. Repainting every 3 or 4 years will suffice afterward. Ungalvanized iron (called BLACK IRON, for distinction) is also very enduring for roofs, if well painted every 1 or 2 years. The chief advantage of galvanized roofing is that it does not require painting so often as the black. The galvanizing adds about % of a ft per square foot of surface, or about % ft per sq ft of sheet as coated on both sides: without regard to the thickness of the sheet. On this principle the above table has been prepared. Those in text- books generally, are incorrect as regards the thinner sheets. Paint for roofs should not have much dryer. See Painting, p 512. The sulphurous fumes from coal are very corrosive of EITHER OALVANIZED OR BLACK IRON ; as may be seen in shops, railroad bridges, or engine houses, roofed with either; if efficient means are not provided for carrying off the smoke: and the same with other metals. THB ACID OP OAK TIMBER is said to destroy the zinc of galvanized iron. See Tin and Zinc. Flat iron is usually nailed upon a sheeting of boards ; but the strength of corrugated iron obviates the necessity for this, and enables it to stretch 5 or 6 ft from purlin to purlin, without inter- mediate support. The corrugated sheets are riveted together on the roof, by rivets of galvanized wire about X inch thick, 300 to a pound, well driven (so as to exclude rain)' at intervals of 3 or 4 inches, all around the edges. The rivet-holes are first punched by machinery, so as to insure coinci- dence in the several sheets: and the rivets are driven hy two men, one above, and one beneath the roof. For black iron, ungalvanized nails, boiled in linseed oil as a partial preservative from rust, are commonly used; as also in shingling or slating. Galvanized ones, however, would be better in nil the*e cases; or even copper ones for slating because good slate endures much longer than either shingles or iron, and therefore it becomes true economy to use durable metals for fastening It. In none of these cases, however, are the nails fully exposed to the weather. The sheets of flat iron are put together by overlapping and FOLDING THE KDOE8. much the same as shown by the fig page 378, head Tin : the joints which run up and down the roof being the same as at s a, and the horizontal ones as at t t; except that inasmuch as these are not soldered in the iron sheets, the joint is made about 5i to 1 inch wide, instead of % inch, the better to provide against leaking. Cleats are used as in tin, with 2 nails to a cleat. The iron plates are best laid on sheeting boards ; but in sheds, &c, are sometimes laid directly on rafters, not more thao about 18 ins apart in the plear ; the plates being allowed to sag a little between CORRUGATED IRON. 371 Ihe rafters, so as to form shallow gutters. In such cases it is well to bevel off the tops of the rafters lightly, as in this fig. A serious objection to iron as a roof covering, is its rapid con- densation of Atmospheric moisture ; which falls from the irou in drops like rain, and may do injury to ceilings, Hours, or articles in the apartments immediately beneath the roof. Painting does not appreciably diminish this ; it may, however, be obviated by plastering, as shown at 11, of Figs 21}^, of Trusses, page 268. Corrugated iron uiakea an excellent permanent street or other awning. Prie Of Corrugated iron in ISSO, 7 to 9 cts per ft ; or 12 to U cts if galvanized, for Noa. 14 to 26. Does not require sheets of best iron, lor roofs. CORRIJOATED IRON. Experiments by the writer, on the strength of corrugated iron. First. A sheet d d, of No. 16 iron, ( - j (about y 1 ^ inch thick,) 27 ins wide, by 4 ft long, with live complete orrugations of 5 ins by 1 inch, was laid on supports o ft 9 ins apart. A block of wood c, 9 ins wide, by 7 ins thick, and 30 ins long, was placed across the center, and gradually loaded with castings weighing ItiuO Ibs. This caused a deflection at the center of precisely J$ an inch. On the removal of the load after an hour, no perma- nent set was appreciable. The severity of the test was pur- posely increased by applying the several castings very roughly, jolting the whole as much as possible.* The sus- pended area of the sheet was 8.44 sq ft ; and since the actual center load of 1600 Its is about equiva- lent to 3000 fts equally distributed, it amounts to = 355 fts per sq ft distributed. But 3000 Ibs distributed would produce a deflection of but about full % of an inch. Again, 355 Ibs per sq ft is about 4 times the weight of the greatest crowd that could well congregate upon a floor. Conse- quently tliis iron, at 3' 9" span, is safe in practice for any ordinary crowd. Moreover, such a crowd would produce a center deflection of only the J^th part of J of an inch ; or yV of an inch ; or TT^-K of the clear span ; which is but two-thirds of Tredgold's limit of -T^-Q of the span. See Art 26 of Strength of Materials, p 196. In one experiment the ends of the sheets rested upon supports dressed so as to present undulations corresponding tolerably closely with the shape of the corrugations; but in the other the supports were flat, and each end of the sheet rested only upon the lower points of the corrugations. No ap- preciable difference was observed in the results Second. An arch of tfo. 18 (^ inch) iron, corrugated like the foregoing, but the depth of corrugation increased to 1*4 ins by the process of arching the sheet ; clear span 6 ft 1 inch ; rise 10 ins ; breadth 27 ins. (of which, however, only 25 ins bore against the abutments ) Each foot o of the arch abutted upon a casting .7', the inner portion t of which was undulated on top, to correspond with the corrugations of the arch, which rested upon it. At j/. (one-fourth of the span.) two wooden blocks were placed, occupying a width of 9 inches, and extending across the arch : on them was piled a load. 1. of castings, to the extent of 4480 Ibs, or 2 tons. Under this load the arch descended about H an inch at y, becoming flatter along that side, and slightly more curved upward along the unloaded side n. Two similar blocks were then placed at n, and two tons of load, a, were piled upon them, in addition to the 2 tons at I ; making a total of 8960 fts.or4tons. Thi brought the arnh more nearly back to its original shape; but still slightly straightened at both n and y. and a little more curved in the center. The load was then increased to 10000 fts. and left standing for several days. Two iron ties, each % by l?i, which were used for pre- venting the abutment castings j from spreading, were found to have stretched nearly % of an inch. Additional ones were inserted, and the load increased to a total of 6 tons, or 134-10 Ibs; parts of it on and 1. and part in the shape of long broad bars of iron at the center of the arch, below the loads * and I. and betwpen n and y. So far as could be judged by eye. the shape of the arch was now almost perfect. The loads s and \ did not touch each other. After standing more than a week, the load was accidentally overturned, crippling the arch. The load was equal to about 1000 fts per sq ft of the arch. Such arches have since come into common use instead of brick, for fireproof floors. Curved roofs of 25 to 3O ft span, rising about i^ span, may be made of ordinary corrugated iron of Nos 16 to 13, riveted as usual ; and having no acces- sories except tie-rods a few feet apart; continuous angle-iron skewbacks; and thin vertical rods to prevent the ties from sagging. Without, however, allowing the deflection to exoeed the ^ Inch ; which was effected by means of stop under the sheet. 372 WEIGHT OF FLAT IRON. Weight of 1 ft in length of FLAT ROLLED IKON, at 4SO Ibs per cubic foot. For cast iron, deduct T V part; for steel, add ^g-; for copper, add j\ for cast brass, add j 1 ^; for lead, add ]^; for zinc, deduct -j^. - a THICKNESS IN INCHES. .2 1-16 H 3-16 ! U 5-16 H 7-16 ! K H K 1 1. .2033 .4166 .6250 .8333 1.012 1.250 1.458 1.666 2.083 2.500 2.916 3.333 % .2344 .4888 .7033 .9375 1.172 1.406 1.640 1.875 2.344 2.812 3.280 3.75 y\ .2603 .5210 .7810 1.042 1 303 1.563 1.823 2.083 2605 3.125 3.646 4.166 X .2335 .5730 .8595 1.146 1.432 1.719 2.006 2.292 2.864 3.438 4.012 4.583 % .3125 .6250 .9375 1.250 1.562 1.875 2.188 2.500 3.125 3.750 4.375 5.000 % .3385 .6771 1.015 1.354 1.692 2.031 2.370 2.708 3.384 4.062 4.740 5.416 H Mi* .7292 1.094 1.458 1.823 2.188 2.550 2.916 3.646 4.375 5.105 5.33 % 3906 .7812 1.172 1.562 1.953 2.344 2.735 3.125 3.906 4.688 5.470 625 2. .4136 .8333 1.25 1.666 2.083 2.500 2.916 3.333 4.166 5.000 5.833 6.666 X .4427 .8855 1.328 1.771 2.214 2.6,>6 3.098 3.542 4.428 5.312 6.196 7.083 .4888 .9375 1.406 1.875 2.344 2.812 3.281 3.750 4.688 5.624 6.562 7.500 % Ml* .9895 1.484 1.979 2.474 2.968 3.463 3.958 4.948 5.936 6.926 7.916 X .5210 1.012 1.562 2.083 2.605 3.125 3.646 4.166 5.210 6.250 7.291 8.333 % .5470 1.094 1641 2.187 2.735 3.282 3.829 4.375 5.470 6.564 7.658 8.750 % .5730 1.146 1.719 2.292 2.865 3.438 4.011 4.583 5.730 6.876 8.022 9.166 % .5390 1.198 1.797 2.396 2.995 3.594 4.193 4.792 5.990 7.188 8.386 9.583 3. .625 1.250 1.875 2500 3.125 3.750 4.375 5.000 6.250 7.500 8.750 10.00 M .6515 1.303 1954 2.605 3.257 3.908 4.560 5.210 6.514 7.816 9.120 10.42 y* .6770 1.354 2.031 2.7')8 3.385 4.062 4.739 5.416 6.770 8.124 9.478 10.83 % .7031 1.406 2.109 2.812 3.516 4.218 4.921 5.625 7.032 8.4o6 9.842 11.25 x .7291 1.458 2.188 2.916 3.646 4.375 5.105 5.833 7.291 8.750 10.21 11.66 X .7555 1.511 2.266 3.021 3.777 4.533 5.288 6.042 7.554 9.066 10.58 12.08 % .7812 1.562 2.343 3.125 3.906 4.686 5.468 6.25 7.812 9.372 10.94 12.50 % .8070 1.614 2.421 3.229 4.035 4.842 5.65 6.458 8.070 9.684 11.30 12.92 A. .8333 1.666 2.500 3.333 4.166 5.000 5.833 6.666 8.333 10.00 11.66 13.23 x .8595 1.719 2.578 3.438 4.297 5.156 6.016 6.875 j 8.594 10.31 12.03 13.75 H .8355 1.771 2.656 3.542 4.427 5.312 6.198 7.083 8.854 10.62 12.40 14.16 % .9115 1.823 2.734 3.616 4.557 5.468 6.380 7.291 9.114 10.94 12.76 14.58 X .9375 1.875 2.812 3.750 4.687 5.624 6.562 7.500 9.374 11.25 13.12 15.00 X .9336 1.927 2.891 3.854 4.818 5.782 6.745 7.708 9.636 11.56 13.49 15.42 H .9895 1.979 2.968 3.958 4.917 5.936 6.926 7.917 9.894 11.87 13.85 15.83 X 1.016 2.031 3.048 4062 5.080 6.096 7.112 8.125 10.16 12.19 14.22 16.25 6. 1.0*2 2.083 3.125 4.166 5.210 6.25 7.291 8333 10.42 12.50 14.58 16.66 X 1.088 2.136 3.204 4.271 5.340 6408 7.476 8.542 10.68 12.81 14.95 17.08 X 1.094 2.188 3.282 4.375 5.470 6.564 7.658 8.750 10.94 13.13 15.31 17.50 K 1.120 2.240 3.360 4.479 5.600 6.720 7.840 8.958 11.20 13.44 15.68 17.92 X 1.146 2.292 3.438 4.584 5.730 6.876 8.022 9.167 11.46 13.75 16.04 18.33 % 1.172 2.344 3.516 4.687 5.860 7.032 8.204 9.375 11.72 14.06 16.40 18.75 H 1.198 2396 3.594 4.791 5.990 7.188 8.386 9.583 11.98 14.37 16.77 19.16 X 1.224 2.448 3.672 4.896 6.120 7.344 8.568 9.792 12.24 14.69 17.13 19.58 6. 1.250 2.500 3.750 5.000 6.250 7.500 8.750 10.00 12.50 15.00 17.50 20.00 H 1.276 2.552 3.828 5.104 6.380 7.656 8.932 10.21 ]2.76 15.31 17.86 20.42 X 1.302 2.604 3.906 5.208 6.510 7.812 9.114 10.42 13.02 15.62 18.23 20.83 % 1.328 2.657 3.984 5.313 6 610 7.968 9.297 10.63 13.28 15.93 18.59 21.25 1.354 2.708 4.063 5.417 6.770 8.126 9.480 10.83 13.54 16.25 18.96 21.66 % 1.381 2.761 4.143 5521 6.906 8.286 9.668 11.04 13.81 16.57 19.33 22.08 H 1.406 2.813 4.218 5.625 7.030 8.436 9.843 11.25 14.06 16.87 19.69 22.50 % 1.432 2.864 4.296 5.729 7.160 8.592 10.02 11.46 14.32 17.18 20.04 22.92 1. 1.458 2.916 4.375 5.833 7.291 8.750 10.20 11.66 14.58 17.50 20.42 23.33 X 1.484 2.939 4.452 5.938 7.420 8.904 10.39 11.87 14.84 17.81 20.78 23.75 M 1.511 3.021 4.533 6.042 7.555 9.066 10.58 12.08 15.11 18.13 21.16 24.16 % 1.536 3.073 4.608 6.146 7.680 9.216 10.75 12.29 15.36 18.43 21.50 24.58 H 1.562 3.125 4.686 6.250 7.810 9.372 10.93 1250 15.62 18.74 21.86 25.00 H 1.588 3.177 4.764 6.354 7.940 9.528 11.12 12.71 15.88 19.05 22.24 25.42 H 1.615 3.229 4.845 6.458 8.075 9.690 11.31 12.92 16.15 19.38 22.62 25.83 x 1.641 3.281 4.923 6.562 8.205 9.846 11.48 13.13 16.41 19.69 22.96 26.25 8. 1.666 3.333 5.000 6.666 8.333 10.00 11.66 13.33 16.66 20.00 23.33 26.66 H 1.693 3.386 5.079 6.771 8.455 10.15 11.85 13.54 6.91 20.30 23.70 27.08 1.719 3.438 5.157 6.875 8.595 10.31 12.03 13.75 7.19 20.61 24.06 27.50 % 1.745 3.489 5.235 6.979 8.725 10.47 12.21 13.96 7.45 20.94 24.42 27.92 X 1.771 3.542 5.313 7.083 8.855 10.63 12.40 14.17 7.71 21.26 24.80 28.33 1.797 3.594 5.391 7.188 8.985 10.78 12.58 14.37 7.97 21.56 25.16 28.75 $i 1.823 3.646 5.469 7.292 9.115 10.94 12.76 14.58 8.23 21.88 25.52 29.17 % 1.849 3.698 5.547 7.3% 9.245 11.09 12.94 14.79 8.49 22.18 25.88 29.58 9. 1.875 3.750 5.625 7.500 9.375 11.25 13.12 15.00 8.75 22.50 26.24 30.00 X 1.901 3.802 5.703 7.604 9.505 11.41 13.31 15.21 9.00 22.81 26.62 30.42 N 1.927 3.&H 5.781 7.708 9.635 11.56 13.49 15.42 9.27 23.12 26.98 30.83 % 1.953 3.906 5.859 7.812 9.765 11.72 13.67 15.62 9.53 23.44 27.34 31.25 ^ 1.979 .958 5.937 7.916 9.895 11.87 13.85 15.84 9.79 23.74 27.70 31.67 H 2.005 .010 6.015 8.021 10.02 12.03 14.04 16.04 20.04 24.06 28.08 32.08 ?* 2.031 .062 6.093 8.125 10.16 12.18 14.21 16.25 20.32 24.36 28.42 32.50 M 2.057 .114 6.171 8.229 10.29 12.34 14.40 16.46 20.58 24.68 28.80 32.92 LO. 2.083 .166 6.250 8.333 10.41 12.50 14.58 16.66 20.82 25.00 29.16 33.33 H 2.109 4.219 6.327 8.438 10.55 12.65 14.76 16.87 21.10 25.30 29.52 33.75 M 2.135 4.270 6.405 8.541 10.67 12.81 14.94 17.08 21.34 25.62 29.88 34.17 WEIGHT OF ANGLE AND T IRON. 373 Weight of 1 ft in length of FLAT ROLLED IROX, at 480 IDS per cubic foot (Continued.) A A S 2 .2 1-16 3-16 T H HICK 5-16 NESS % IN INCHES 7-16 j fc N H % j 10% 2.162 .323 6.486 8.646 10.81 12.97 15.13 1 17.29 2 .62 25.94 30.26 34.58 y* 2.188 .375 6.564 8.750 10.94 13.13 15.31 17.50 2 .88 26.26 30.62 35.00 % 2.214 .427 6.642 8.854 11.07 13.28 15.50 17.71 2 .14 26.56 31.00 35.42 % 2.239 .479 6.717 8.958 11.20 13.43 15.67 17.92 2 .40 26.86 I 31.34 35.83 % 2.?66 .531 6.798 9.062 11.33 13.59 15.86 18.12 2 .66 27.18 31.72 36.25 11. 2.291 .583 6873 9.166 11.46 13.75 16.04 18.33 2 .90 27.50 32.08 36.66 K 2.318 .636 6.954 9.271 1.59 13.91 16.22 18.54 23.18 27.82 32.44 37.08 54 2.344 .688 7.032 9.375 1.72 14.06 16.40 18.75 23.44 28.12 32.80 37.50 X 2.370 .740 7.110 9.479 1.85 14.22 16.59 18.96 23.70 28.44 33.18 37.92 X 2.395 .791 7.185 9.582 1.97 14.37 16.76 19.16 23.94 28.74 33.52 38.33 ^ 2.422 .844 7.266 9.688 2.11 14.53 16.95 19.37 24.22 29.06 33.90 38.75 % 2.-148 .896 7.344 9.792 2.24 14.68 17.13 19.58 24.48 29.36 34.26 39.16 K 2.474 .948 7.422 9.896 2.37 14.84 17.32 19.79 24.74 29.68 34.64 39.58 12. 2.500 5.000 7.500 10.00 2.50 15.00 17.50 20.00 25.00 30.00 35.00 40.00 Plate-iron washers. Standard sizes. Dianas of washers and bolt-holes in ins. Approx thickness by Birmingham wire gauge, p 367. Approx number in one ft). Diams. Ths. I No. Diams. Ths. No. Diams. Ths. No. Ji X 18 543 IM H 14 50 2M 15-16 9 8.7 % 5-16 16 228 m 9-16 12 30 2H 1 1-16 9 6.3 IU 5-16 16 147 m K 12 25.7 3! 1 YA. 9 4.7 t/ X 16 123 1% 11.16 10 17 3 1 % 9 3.7 1 7-16 14 70 2 13-16 10 10.7 3^ 1 H 9 3.0 Price in Philada, 1880, about 16 to 12 cts per ft> for diams from 1.5 to 3 inches. Rolled Star Iron. Standard sizes. Carnegie Bros, Pittsburgh. The thickness is that at center of straight part of one of the four arms, in ins. Rolled in lengths of 20 to 25 ft. Area in sq ins. Wt in tts per ft run. Ins. Ths. Area. Wt. Ins. Ths. Area. Wt. 4X4 h 3.6 12 2.5 X 2.5 % 1.58 5.25 S.5 X 3.5 X 3.3 10 2 X 2 % .80 2.75 3X3 X 24 8 1.5 X 1.5 9-32 .70 2.30 T and Angle Iron. Standard sizes.* Wt per ft. Prices, p 364. Flanges in ins. Thicks. in ins. Weight in B)s. Flanges iuins. Thicks, in ins. Weight inlbs. Flanges in ins. Thicks, in ins. Weight in Ibs. y\ x % X 0.6 2 X 2J4 % 5 3VS X 4 ^ 12. % x y t 3-16 1. 2v/ x 2V 5-16 4.5 39f X 4 12.5 1 X 1 1 X 1 1 A 3-16 3-16 1.2 1.4 2 X 2% 2% X 2 5-16* 5.3 5. 4 X 4 4X4 1 9.6 12.7 \y x \y\ 2 6. 4 X 4 19. y 2.2 2!/2 X 2^ % 6.5 4 X 4 1 25. \$L X \ 1 A 3-16 1.8 2>4 X3 K 6.7 4 X5 M 4.3 \y^ x 1H 2.3 3 X 3 7.3 5 X 3 2.7 14 2.8 N 7.7 5 X 6 M 7.7 1J4 X 2 H 2.8 3 X3H 10.4 5^ X 3 4. m x 2 1% X 2 5-16 3.4 3. 3 X 4 3 X 4 B 8.5 11.3 jj*xJ8 K 4.4 9 2 X 2 H 3.3 3 X 3^ S 8.5 6 X 4J^ 5^ 1 2 X 2 3.9 ^ 10.8 22 2X2 % 4.7 3^X3M 7-16 9.7 THE DIMENSIONS IN THIS TABLE ARE FROM OUT TO OUT EACH WAY, as a a, o c. When these, and thethicknessof, measured at the center of each arm, is the same in M. N, and S. there will scarcely be an appreciable difference in the weights of the three. The dimensions in the table are MARKKT SIZES; that is, they can be bought ready made. The young en- gineer should bear this in mind in such matters when designing; and not introduce sizes that have to be specially made; for this requires the manufac- turer to prepare a* new set of rolls; the expense of which would not be warranted by an order less than some thousands o" dollars. 374 BOLTS, NUTS, WASHERS. Weight of heads and lints of iron bolts.* The weights include both a head, and a nut, which are supposed to be neatly finished off. Rough machine- made ones frequently weigh from J^ to % more. Wt of head about = wt of nut. Diameter of bolt, in inches. K|K|X|K|K|X| i | ix | ** | m | | * | Weight of an hexagonal head and nut in Ibs. .017 I .057 I .128 I .267 I .43 I .73 I 1.10 I 2.14 I 3.78 I 5.6 I 8.75 I 17. I 28.8 .021 .069 .164 Weight of a square head and nut in Ibs. .320 I .55 I .88 I 1.31 I 2.56 I 4.42 I 7.0 10.5 21. 36.4 Au hexagonal head and nut together, weigh about as much as 5 diams of length of the bolt ; or a* 6 diams, if they are square. In machine-made bolts up to 2 ft long, the head is in one piece with tlie shank ; but in very long ones, as in web-members for trusses, it is simply a nut, like the other. Blacksmiths make a head by welding on a ring ; or by upsetting. The thickness y t, of a nut N ; or m g, of a head, II. should be at least equal to the diam co, of the screw. And the width a b, between two parallel sides of the hexagon, is 1>^ times the same diam. Some machinists add % inch to this width, for all diams. If square, each side is made equal to 1^ diams of the screw. The following are the usual nunibers of screw-threads cut per inch of length at the ends of bolts. The depth to which the threads are cut is a little less than their pitch, or distance apart from center to center. I | These may be termed the American stand- _ ard dimensions. "With them a bolt will not yield by tripping off its threads; but will generally break off just below the nut, where the iminution of metal by the cutting of the threads commences. To allow for this drtual reduction of diam, we must first calculate what diam is necessary for sustain- ng the load, or strain, which will come upon it; and then add enough to cover this reduction. That is, we must add twice the depth to which the threads are cut. For this purpose, it will suffice to increase the neat calculated diam about % part, for bolts from 1 to 2% ins diam ; i for 3 ins diam ; Y& f r 6 ins diam.* This of course involves a considerable increase in the quantity of iron ; but is necessary when the bolt is required to fit close in the hole through which it passes. But where this is not the case, the screw end may be UPSET, or made so much thicker than the other part of the shank, that the threads may be cut into it, without injury to the strength of the bolt; as in the next fig. THE FOLLOWING TABLE GIVES THE DIAM FOR EITHER CASE. In carpentry, as well as in ties for masonry, washers, ww, of either cast or wrought iron, are placed between the timber, or stone, and the head and nut; in order to distribute the pressure over a greater surface, and thus prevent crushing ; especially in timber. When much strained against wood, the side of a square wrought-iron washer ; or the diam ww of a circular one, should not be less than 4 diams of the screw, as in the fis? ; and its thickness, t w, % diam at least. Two such square washers will together weigh as much as 18 diams in length of a round rort of the same diam as the screw : or with a square head and nut. as much as 24 diams. Two of same diam as screw; or with a square head and nut. as much as 20 diams. Cast iron washers, being more apt to split under heavy strains, may be made about twice as thick aa wrought ones. "When the strain is very great, the diam of *Price of nuts in Philada, 1880, well finished, square ones of 1.5 to 3 ins on a side, 14 to 12 cts per ft : lare^r si/.^s. 12 to 10 ct->. Hexagon ones about 25 per ct more. Rough bolts and nuts 6 to 7 cts. In 1883 U .ibout 30 per ot lower. BOLT8, NUTS, WASHERS. 375 . 22.0 ' . 35.0 the washer may be 5 or 6 times that of the screw ; and its thickness equal to diam ; but 4 diams will suffice for most practical purposes, or even 2,5 when there ia but little strain, and the thickness may then be but .1 or .2 diam of bolt. See p 373 for such. When the screw end of the bolt is upset as at e, for a length 8 e of 6 diams of the shank g y. we must add to t ,e weight of the entire length of the bolt, an allowance sufficient to cover the addi- tional thickness. This allowance will equal 3J^ diains of length of the body or shank e g of the bolt. But when thus upset, the head, nut, and washers must be enlarged to suit the thicker screw. For bolts of from 1 to 3 ins diam. the total allowance must be increased to the weight of a length equal to 35 diams y g of the shank, or body of the bolt, supposing the washer.s to be round ; and the bead and nut square. When a neat finish is required, the upper edges of the washers are shaped as shown by the dotted lines near w w, and frequently the upset length e need not exceed 2 or 3 diams; both of which slightly diminish the foregoing allowance. When the bolt is not to be much strained, or when the timber is hard, the washers may be but 3 diams of the screw in width, or diam ; and about .4 of a diam in thick- ness; and this will reduce their weight fully one-half. For ease of reference we recapitulate the allowances advisable to be made in preliminary estimates; as meas- ured by the weight of a rod of the same diam as the shank eg\ the length of the rod }>eiug given in its own diameters. BOLT NOT UPSET. Square head and nut together 6 diams. Hexagon" " " " 6 ' 2 round washers : % diam thick; 4 diams in diam; wrought 14 " Square head, and nut; and 2 round washers 20 " BOLT UPSET Upsetting 3.5 " Square head and nut together 9.5 " Hexagon " " " " 7.9 " 2 round washers ; % diam thick: 4 diams of screw in diam.... Upsetting; squarehead; and nut; and 2 round washers If the washers are of the smaller dimensions given above, deduct half their weight. Lock-nut washers. When bolts are subjected to much rough jolting, as at rail-joints, &c, the nuts are liable to unscrew themselves in time. On railroads this is a source of great annoy- ance. SHAW'S LOCK-NUT WASHER is intended to prevent this. It is a simple circular washer made of steel ; with a slit s s cut through it, leaving sharp edges. On one side, a, of the slit, the metal is pressed upward about % inch ; and that on the other side, c, downward, the same distance; so that a perspective view would be somewhat as at*. Now, when the nut is screwed down over the washer, in the direction of the arrow, the slit offers no obstruction ; but if the nut afterward tends to unscrew itself, the sharp upper edge of the slit, along a, presents friction against the bottom of the nut, sufficient, it is said, to prevent it from so doing. ANOTHER DEVICB is to cut at the end of the screw a few threads of a screw of less diam than the main one, and in the opposite direction. The nut is then screwed upon the larger diam ; and after it the lock-nut is screwed in the other direction upon the smaller diam, until it comes into contact with the main nut. At Figs 16 and 17 of rail-joints, other methods will be seen, p 396. THE BILLINGS LOCK-NUT WASHER is also said to be effective. It is a thin hollow cup of tempered spring steel, n n, made by pressure, from a flat circular piece previously heated. It is made by the Wharton Safety Switch Co, of Philada. Diam 2J4 ins; height % inch; thickness % inch; weight 3} ounces.* Table of diameters, weights, and approximate breaking: strains, for round rolled iron bolts, ties, or bars; assuming the breaking strain per square inch of average quality of rolled iron to be as follows: Up to 1 inch square, or 1 inch diam, 20 tons, or 44800 fcs ; from 1 to 2 ins sq or diam, 19 tons : 2 to 3 ins, 18 tons ; 3 to 4 ins, 17 tons ; 4 to 5 ins, 16 tons ; 5 to 6 ins, 15 tons. The first 4 columns of the table are to be used when the screw end of the bolt is enlarged or upset, so that the shank or body of the bolt shall not be weakened by the cutting of the screw threads. But when the shank is so weakened, the diam and weight of the bolt must be taken from the last 2 cols. Rem. But it is very important to know that a long upset rod is no stronger than one not upset, against slowly applied loads or strains. Both will then break at about midlength, under equal pulls. Therefore in such cases the col of greatest diams in the table should be used. EXAMPLE 1. To find the diam of a bolt, that shall just break under a strain, or a load of 52.5 tons, we see by the table and opposite 52.5 ton*, that it will be 1% ins if the screw end is enlarged, and 2.3 ins if it is not. In the first case, the weight of the bolt will be 9.3 B>s per foot run ; and in the second, 13.8 fts. EXAMPLE 2. What must be the diam in order to sustain a strain of 52.5 tons, with a safety of 3? Here 52.5 X 3 ~ 157.5 tons. In the table, the nearest we find to 157.5 tons, is 163.6; opposite which we find the diam 3f^ ins. A diam a trifle less than this will break under a strain of 157.5 tons ; and consequently will have a safety of 3 for 52.5 tons. The breaks: strains in this table will also answer for square BARS, by merely increasing them ^ part; for a round bar has very approximately 4 of the strength of a square one whose side is equal to the diam of the round one ; or the square one has 1 H times the Strength of the round one; or, more correctly, as 1 to .7854. For the Strength Of COPPER bolts, multiply the tabular ones by the decimal .8 ; and for their weight, increase, that of iron ones 4- part. Heads, nuts, and washers, are not included in the table. L me Ji^rWn * Price in 1880, $5 per 100. 376 WEIGHT OF METALS. WEIOHT AJTD STRENGTH OF IRON BOLTS. (Original.) For square ones or for copper see preceding paragraph. Ends enlarged, or upset. Ends not enlarged. Ends enlarged, or upset. Ends not enlarged. Diam. Weight Break- Break- Diam. Weight Diam. Weight Break- Break- Diam. Weight of per foot ing ing of per foot of per foot ing ing of per foot thank run. strain. strain. ihank run. shank run. strain. strain. shank run. Ins. Pds. Tons. Pds. Ins. Pds. Ins. Pds. Tons. Pds. lus. Pds. 1 A .oiu .245 549 1% 8.10 45.7 10236S 2.14 12.0 3-16 .093 .553 1239 13-16 8.69 49.0 1097GO 2.22 12.9 % .165 .983 2202 .35 .321 y& 9.30 52.5 117C03 2.30 13.8 5-16 .258 1.53 34-27 .43 .452 15-16 9.93 56.0 125440 2.38 14.7 3 /s .372 2.21 4950 .50 .654 2. 10.6 59.7 133728 2.45 15.7 7 16 .506 3.00 6720 .58 .897 i/ 12.0 63.8 14291-2 2 59 17.5 1 A .661 3.93 8803 .66 1.14 /4 13.4 71.6 160384 273 19.5 y-16 .837 4.97 11133 .73 1.41 3Z 14.9 79.7 178528 2.88 21.6 % 1.03 6.14 13754 .80 1.67 \/ 16.5 88.4 198016 3.02 23.9 11-16 1.25 7.42 16621 >8 2.03 oZ 18.2 97.4 218176 3.16 26.1 % 1.49 8.83 19779 .96 2.41 3? 20.0 106.9 239456 3.30 28.5 i3-u> 1.75 10.4 2329 fi 1.04 281 H 21.9 116.8 261632 3.45 31.1 % 2.03 12.0 26880 1 12 3.26 3 23.8 127.2 284928 3.60 33.9 15-16 2.33 138 3031-2 1.-20 3.77 1 A 27.9 141.0 315840 3.86 39.1 lin. 265 15.7 35163 1.27 4.27 2 32.4 16X6 366464 4.12 44.4 1-16 295 16.8 37632 1.35 4.77 H 37.2 187.7 420448 4.41 51.0 l /8 3.35 IS.9 42330 1.42 5.28 4 42.3 213.6 478464 4.70 57.8 3-16 373 21.1 47 -2Q i 1.49 5.81 \/ 47.8 227.0 508480 498 65.2 1 A 4.13 23.3 5219-2 1.n5 6.39 I/ 536 l?54.5 570080 5.25 72.9 5-16 4.56 25.7 57568 1.64 70i /4 59.7 283.5 635040 5.53 80.5 % 500 28.2 63168 1.72 7.74 5. 66.1 314.2 703808 5.80 88.1 7-16 5.47 30.8 6899-2 i.*o 8.48 1 A 72.9 324.7 72732 - 6.08 97.0 Y* 5.95 33.6 7526 1.87 9.20 g 80.0 356.4 798336 6.36 106. 9-16 6.46 36.4 81536 1.94 988 % 87.5 389.5 8724SO 6.63 116. % 699 39.4 88250 2.00 10.6 6 95.2 424.1 94998 \- 6.90 126. 11-16 7.53 42.5 95200 2.07 ill. 3 See Rein, p 375. ROLLED LEAD, COPPER, and BRASS: Sheets and Bars. Thickness or LEAD. COPPER. BRASS. Thickness or Diameter, Diameter, or side, Sheets, Square Round Sheets, Square Round Sheets, Square Round or side, in per Jars; Bars; per Bars; Bars; per liars ; Bars; in Inches. Square IFoot 1 Foot Square 1 Foot 1 Foot Square 1 Foot 1 Foot Inches. Foot. long. long. Foot. long. long. Foot. long. long. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. 1-32 1.86 .005 .004 1.44 .004 .003 1.36 .004 .003 1-32 1-16 3.72 .019 .015 2.89 .015 .012 2.71 .014 .011 1-16 3-32 5.58 .044 .034 433 .034 .027 4.06 .032 .025 3-32 U 7.44 .078 .061 5.77 .000 .047 5.42 .053 .044 H 532 9.30 .121 .095 7.20 .094 .074 6.75 .088 .OC9 532 3 16 11.2 .174 .137 8.66 .135 .100 8.13 .127 .100 8 16 7-32 13.0 .237 .187 10.1 .184 .144 9.50 .173 .130 7-32 X 14.9 .310 .244 11.5 .240 .189 10.8 .226 .177 M 516 18.6 .485 .381 14.4 .376 .295 13.5 .353 .277 516 H 22.3 .698 .548 17.3 .541 .425 16.3 .508 .3D9 % 7-16 26.0 .950 .746 20.2 .736 .578 19.0 .691 .543 7-16 H 29.8 1.24 .974 23.1 .962 .755 21.7 .903 .709 H 9 16 33.5 1.57 1.23 26.0 1.22 .955 24.3 1.14 .9CO 9-16 H 37.2 1.94 1.52 28.9 1.50 1.18 27.1 1 41 1.11 K 11 16 40.9 2.34 1.84 31.7 1.82 1.43 298 1.70 1.34 11-16 96 44.6 2.79 2.19 34.6 216 1.70 32.5 2.03 1.60 K 13-16 48.3 3.27 2.57 37.5 255 1.99 35 2 2.38 1.87 13- 16 X 5-2.1 3.80 2.98 40.4 2.94 2.31 37.9 2.76 2.17 % 15-16 5SO 4.37 3.42 41.3 3.38 265 40.6 3.18 2.49 15-16 1. 59.5 4.98 390 46.2 3.85 3.02 4:5.3 3.61 2.81 1. X 66.9 627 4.92 52.0 4.87 3.8-2 48.7 4.57 3.60 H % 744 7.75 6.09 57.7 6.01 4.72 51.2 5.64 4.43 8 i 81.8 9.37 7.37 fi-5.5 7.28 5.7-' 5!) 6 682 537 M 3 83.3 1.2 8.77 09.3 8 65 6.PO 65.0 8.12 6.38 H N 9H.7 3.1 10.3 75.1 10.2 7.98 70.4 9.53 7.19 K K iu4. 5.2 11.9 80.8 11.8 9.25 75.9 11.1 8.G8 M K t. 112. 1 19 75 13.7 86.6 1H.5 10.6 81.3 8fi.7 12.7 14.4 9.97 11.3 % 2. WEIGHT OF METALS. 377 ROOf COpper is usually in sheets of 2^ ft X 5 ft: or 12^ square feet weighing 10 to H fts per sheet ; and is laid on boards * No solder is used in the horizontal joints as it is in tin roofs ; but both the horizontal aud the sloping joints are formed by ouly overlapping and bending the sheets, much as shown by the figs "tinder the head " Tin ; " except that the horizontal joints are bent or locked together, as in this figure ; and then flattened down close. Roof lend generally weighs 4 to 6 tbs persq ft. Its great contraction and expansion, however, render it unfit for this purpose, as it is liable to crack aud leak. It may be used for flashings. Price in 1880 about 9 to 10 cts per Ib. WEIGHT OF BALLS. diameter in Inches. CAST LEAD. CAST COPPER. CAST BRASS. CAST IRON. Diameter in Inches. CAST LEAD. CAST COPPER. CAST BRASS. CAST IRON. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. H .026 .021 .019 .017 5M 30.1 24.1 21.5 19.8 % .088 .070 .063 .058 J 34.7 27.7 24.7 22.7 I. .209 .167 .148 .136 H 39.6 31.7 28.3 25.9 y* .408 .325 .290 .266 6. 45.0 36.0 32.0 29.4 .705 .562 .501 .460 /1j 57.2 45.8 40.8 37.4 H 1.12 .893 .795 .731 7. 71.5 57.2 50.9 46.8 2. 1.67 1.33 1.19 1.07 y% 88.0 70.3 62.6 57.5 y 2.38 1.90 ' 1.69 1.55 8. 106. 85.3 76.0 69.8 y* 3.25 2.60 i 2.32 2.13 y* 127. 102. 91.2 83.7 H 4.34 3.47 i 3.09 2.83 9. 151. 121. 108. 99.4 3. 5.63 4.50 4.01 3.68 y* 178. 143. 127. 117. M 7.15 5.72 5.10 4.68 10. 208. 167. 148. 136. H 8.94 7.14 636 5.85 y* 241. 193. 172. 158. 11. 8.79 7.83 7.19 11 277. 222. 198. 182. 4.* 13.4 10.7 9.50 8.73 y* 317. 253. 226. 207. y 16.0 12.8 11.4 10.5 12. 360. 288. 257. 236. 18.9 15.2 13.5 12.4 ,? 22.7 26.0 17.9 20.8 15.9 18.6 14.6 17.0 The weight! of balls are as the cubes of their diaius. LEAD PIPES; weight of, per Foot ran. Bore, or THICKNESS OF METAL, IN INCHES. inner diam. in inches. rV i A ? A t & i f t J 1 inch. Lbs. Lbs. Lbs. Lbs. Lbs, Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. y .305 .724 1.28 1.95 2.74 3.65 4.53 5.84 8. 2 1.7 15.3 19.5 5-16 .366 .815 1.47 2.20 3.05 4.02 4.96 6.33 9. 4 2.4 16.2 20.5 .427 .967 1.65 2.44 3.35 4.38 5.39 6.82 9. 6 3.2 17.0 21.5 7*16 .488 1.09 1.83 2.69 3.66 4.75 5.82 7.31 0. 3.9 17.9 22.4 .548 1.21 2.01 2.93 3.96 5.11 6.24 7.79 1. 4.6 18.7 23.4 % .670 1.16 2.38 3.42 4.57 5.85 7.10 8.77 2. 6.1 20.4 25.4 % .791 1.70 2.74 3.90 5.18 6.58 7.96 9.75 3. 76 22.1 27.3 TX .911 1 95 3.11 4.39 5.79 7.31 8.82 10.7 4. 9.1 23.9 29.3 1. 1.03 2.19 3.47 4.88 6.40 8.04 967 11.7 5. 20.5 25.6 31.2 .16 2.44 3.84 5.37 7.01 8.77 10.5 127 7! 27.3 33.2 TX 2.69 4.21 5.85 7.62 9.50 11.4 13.7 8. 23.4 29.0 35.1 / AO 2.9t 4.58 6.34 8.23 10.3 12.3 14.7 9.5 24.9 30.7 37.1 y*> .52 3.18 4.94 6.83 8.84 1.0 13.1 15.6 20.7 26.3 32.4 39.0 fix .64 3.43 5.31 7.32 9.47 1.7 14.0 16.6 22.0 27.8 34.1 41.0 H .76 3.67 5.67 7.81 O.I 2.4 14.8 17.6 23.2 29 3 35.8 42.9 % .89 3.92 6.04 8.30 0.7 3.2 15.7 18.6 244 30.8 37.6 44.9 2. 2.01 4.16 6.40 8.78 1.3 3.9 16.5 19.5 25.6 32.2 393 46.8 34 2.25 4.65 7.13 9.76 2.5 5.4 18.2 21.5 28.1 35.1 42.7 50.7 y% 2.49 5 14 7.86 10.7 3.7 6.8 20.0 23.4 30.5 38.0 46.1 54.6 % 2.73 5.83 859 11.7 4 9 8.3 21.7 25.4 32.9 41.0 49.5 58.5 s. 2.98 612 9.32 12.7 6.1 9.7 23.4 27.3 35.4 43.9 52.9 62.4 H 3.22 6.61 10.1 13.7 7.4 1.2 25.1 29.3 87.8 46.8 56.4 66.4 Jrf 3.46 7.10 10.8 14.6 8.6 2.7 268 31.3 40.3 49.7 59.8 70.3 % 3.71 7.59 11.5 156 9.8 4.1 285 33.2 42.7 52.7 63.2 74.2 4. 3.95 8.08 12.2 16.6 21.0 5.6 30.2 35.2 45.2 55.6 66.6 78.1 Lead service pipes for single dwellings in Philadelphia are usually of from Jx inch bore, wt 1 to 2% ft>s; to % inch bore, wt 1% to 3 ft>s per ft run, nppnrninor in Viparl T'ViPv *!** Iv Ikii *!< frnm snrlHpn flnsincr nf sf rmnnnlre hiit . sometimes do so from the freezing of the contained water. See pp 533, 573. Cost of lead pipe in Philada, 1880, about 8% cts per ft>. Tin-lined 15 cts. Made on a large scale by Messrs Tathani Brothers, 226 S Fifth St, Phila. * To which it is held by copper cleats ; as at Fig y, next page. Price of roofing copper in 1880, about 38 cts ; and copper nails 45 cts per tt>. 378 TIN AND ZINC. BRAZED COPPER PIPES j weight of, per Foot run. Thickness INNER DIAMETER IN INCHES. in Inches. 1 1H 1H 1 2 2>4 2X 2J' 3 W 4 ft* 5 6 _: Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. 1-16 .800 .989 1.18 1.37 1.56 1.75 1.93 2.12 2.31 269 3.07 3.45 3.82 4.58 332 1.24 1.52 1.80 2.08 2.36 2.64 2.92 3.20 3.48 4.05 4.61 5.17 5.73 6.85 M 1.70 2.08 2.46 2.84 3.22 2.61 3.99 4.37 4. 75 5.51 6.27 7.03 7.80 9.3'J 3-16 2.70 3.27 3.84 4.41 4.98 5 55 6.12 6.69 7.26 8.40 9.54 10.7 11.8 14.1 H 3.79 4.55 5.IU 6.07 6.83 ?! 5!) 8.35 9.11 9.87 11.4 12.9 14.4 16.0 19.0 516 4.97 5.92 6.86 7 81 8 76 9 70 10 6 11.6 12 5 14.4 16.3 18.2 20.1 r-t a TIN ANI> ZINC. The pure metal is called block tin. When perfectly pure, (which it rarely is, being purposely adulterated, frequently to a large proportion, with the cheaper metals lead or zinc,) its sp grav is 7.29 ; and its weight per cub ft is 455 K>s. It is sufficiently malleable to be beaten into tin foil, only YoVo~ ^ an ^ nc ^ ^" c ^- Its tensile strength is but about 4600 Ibs per sq inch ; or about 7000 ft>s when made into wire. It melts at the moderate temperature of 442 Fah. Pure block tin is not used for common building purposes ; but thin plates of sheet iron, covered with it on both sides, constitute the tinned plates, or, as they are called, the tin, used for covering roofs, rain pipes, and many domestic utensils. For roofs it is laid on boards. The sheets of tin are uni- ted as shown in this fig. First, sev- eral sheets are joined together in the shop, end for end, as at tt\ by being first bent over, then ham- mered flat, and then ^, soldered. These are then formed into a J roll to be carried to the roof; a roll being long enough to reach from the peak to the eaves. Different rolls being spread up and down the roof, are then united along their sides by simply being bent as at a and s, by a tool for that purpose. The roofers call the bending at s a double groove, or iloublf loc . ; and the more simple ones at t, a single groove, or lock. To hold the tin securely to the sheeting boards, pieces of the tin 3 or 4 ins long, by -I ins wide, called cleats, are nailed to the boards at about every 18 ins along the joints of the rolls that are to be united, and are bent over with the double groove 5. This will be understood from y, where the middle piece is the cleat, before being bent over. The nails should be 4-penny slating nails, which have broader heads than common ones. As they are not exposed to the weather, they may be of plain iron. Much use is made of what is called leaded tin, or tarns, for roofing. It is simply sheet-iron coated with lead, instead of the more costly metal tin. It is not as durable as the tinned sheets, but is somewhat cheaper. The best plates, both for tinning and for tarns, are made of charcoal iron : which, being tough, bears bending better. Coke is used for cheaper plates, but inferior as regards bending. In giving orders, it is important to specify whether charcoal plates or coke ones are required ; * also whether tinned plates, or tarns. There are also in use for roofing, certain compound metals which resist tarnish better than either lead, tin, or zinc ; but which are so fusible as to be liable to be melted by large burning cinders fall- ing on the roof frrnn a neighboring conflagration. A roof covered with tin or other i icUl should, if possible, slope not much ZM than five decree*, or about an inch, to a foot; and at the eaves there should be a sudden fall into the rain-gutter, to pre- vent rain from backing up so as to overtop the double-groove joint *, and thus causing leaks. Where eoal is used for fuel, tiu roofs should receive two coats of paint when first put up, and a coat at every * Price, Phila. 1880. Charcoal iron, tinned plates, $10 to $13 per box ; accord- ing to weight and quality, and tarua from $8 to $10. In 1HS2 about 25 per ct lower. TIN AND ZINC. 379 2 or 3 rears after. Where wood only is ased, this is not necessary ; and a tin roof, with l good pitch, will lust 20 or 30 years.J Tinned iron plates are sold by the box. These boxes, unlike glass, have not equal areas of. contents. They may be designated or ordered either by their names or sizes. Many rnaker^, however, have their private brands in addition; and some of these have a much higher reputation than others. Contents of a box of eitber tinned plates or tarns. Names. No. in a box. Sizes. lus. Wt. of a box. Lbs. Marks on the boxes. C 1 N 1 225 13% X 10 iij CI 13M X 9% 105 CIl 12% X 914 98 cm Cross No 1 n 1314 x 10 140 XI Two Cross. No 1 u. 161 XXI u ii 182 XXXI four Cross No 1 VCI \VXI 112 112 14 X 20 14 X 20 112 140* Sheets of larger size may be made to special order ; those of tinned iron, in Enprland ; but leaded tarns are made in Philada also, and elsewhere. Sheets of 5)6 ins by 33, 39. 45, 48, 52, or 58, on hand. Larger ones, up to 3 ftLy 7 ft, and up to No. 26 gauge, (.013 inch.) made to order. A box of '225 sheets of 13% by 10. contains 214.81 sq ft; but, allowing for overlapping, it will cover but about 150 sq ft of roof; even without any allowance for the waste which occurs in cutting away portions in order to fit at angles, &c. To find the area of roof covered by any sheet, first deduct 2 ins from its width, and 1 inch from Its length. ZillC, in sheets, and laid in the same manner as slates, is mnch used in some parts of Knrope for roofing. By exposure to the weather, it soon becomes covered by n thin film of white oxide, which protects it from further injury, and renders the roof very durable.t Corrugated sheet tine is also used. See Galvanized Sheet Iron, page 370. Ziuc sheets are usually about 3 ft by 7 or 8 ft. The gauge differs from that of iron ; thus No 13 is .032 of an inch thick, or 1.22 Ibs per sq ft; No 14, = .035 inch, and 1.35 Ibs; No 15, = .042 inch, and 1.49 Ibs ; No 16, = .049 inch, and 1.62 Ibs per sq ft. Any of these numbers may be used ou root's, for which purpose it should be very pure. WATER KEPT nr ZINC VESSELS is said to become injurious to health : and recently an ontcry has on that account arisen again-a jralvaniz-'d-iron service-pipes in dwellings. Yet such have been in use for many years in New England, Philada. and elsewhere, without as yet any deleterious effects. This is possibly owing to the fact that service-pipes being short, the water is usually all drawn throuch them several times a day ; and hence does not remain in contact with the zinoor lead long enough, to acquire a poisonous character. In taking possession of a house in which the water has remained stagnant in the service-pipes for some considerable time, such water should all be run to waste ; otherwise sickness may ensue from its use. Thre are other sizes, as 10 X 20, 3 a a 8 c "* 1 I 1 CQ ** So i ^ 5 S B 1 1? & 5-5 C li _ 1 a a i a ll III 1 F a 3 S a o III .2 p K M I P M ta 3 c S 6 B i 6% 1 20 74 00 j5^ 11 4% 54 36 00 105i 2 6 1 05 65 00 14^i 12 454 47 30 00 10 * 3 5H 91 54 00 13 13 3% 41 25 00 4 5 78 43 60 12 14 3% 35 20 00 8V 5 4% 65 35 00 10^ 15 3 29 16 00 7^ 6 4 53 27 20 16 23 12 30 7 3^ 41 20 20 8 17 2% 18 8 80 534 8 31^ 34 16 00 7 18 15 7 60 5 9 2% 28 11 40 6 19 \yt 13 5 80 10 25 8 64 5 20 \y 11 4 09 4 * 10% 2 2* 5 13 43^ 21 i 1 9 2 83 10. ^ lf 23 4 27 4 22 ! 1M 8 2 13 %H 10H 1>3 22 8 48 3M 23 24 1 7 6^ 1 65 1 38 2Y Tiller Rope, % in diam, 26 cts. 25 26 K P 1 03 81 1 27 N 5 " 56 i> Ropes from No. 8 to No. 10% are spe- cially adapted for hoisting-rope. 27 J^ H 4 28 2'J 3 2 Large Sash Cord. Small " All kinds of shackles, sockets, swivel books, and fastenings, put on, and splicer made. Copper Rope, corresponding to the above sizes, made to order. Steel ropes are much stronger and more durable than irou. Notes on the use of Wire Rope, by Mr. Roebling. " Two kinds of Wire Rope are manufactured ; the larger sizes, as also the*nost pliable, arecomposei of 133 wires, and are generally used for hoisting or running rope. Those of 49 wires are stiffer. and are better adapted for standing rope, guys, and rigging. Orders should state the use of Rope, and advice will be given. Ropes up to 3 inches diameter are made upon special application. For safe working load, allow 1-5 to 1-7 of ultimate strength, according to speed and vibration. When substituting Wire Rope for hemp rope, it is good economy to allow for the former the same rate per foot run which experience has approved of for the latter. Wire Rope is as pliable as new hemp rope of the same strength ; the former will therefore run over the same sized sheaves and pulleys which are used for the latter. But the greater the diameter of the sheaves, pulleys, or drums, the longer Wire Rope will last. In the construction of machinery for Wire Rope it will 'be found good economy to make the drums and sheaves as large as possible. The size of drum is as follows : The same figure which expresses the circum in inches in the second column of the table is also the minimum diam of drum in feet; doubling that figure will give the max- imum. The diameter of drum should be no less th. r m the minimum, nor is it necessary to exceed the maximum. As an example, take a No. 4 rope, circumference 5 inches; therefore the minimum diam- eter of drum is 5 feet ; and the maximum 10 feet. Or a No. 10}^ rope, circumference 2 inches ; there- fore minimum diam is 2 feet; and maximum 4 feet. A smaller diameter of drum may answer, but the short bending will result in a much more rapid wear. In most cases the Rope will wear twice as long on a max diam as on a minimum. Experience h:s alo demonstrated that the wear increases with the speed. It is better to increase the loHd th:in the speed. Wire Rope is manufactured either with a wire or hemp centre. The latter is more pliable than the former, and will wear hotter where there is short bending. Orders should specify. Wire Rope must nut be coiled or uncoiled like hemp rope. When mounted on a reel the latter should be turned on a spindle to pay off the rope. When forwarded in a coil without reel, roll it over the ground like a wheel, and run off the rope in that way. All untwisting must be avoided. To preserve Wire Rope apply raw linseed oil with a piece of sheepskin, wool inside; or mix the oil with equal parts of Spanish brown and lampblack. To preserve Wire Rope under water or under ground, take mineral or vegetable tar, add 1 bushei of fresh slacked lime to I barrel of tar, (which will neutralize the acid,) and boil it well, then saturate the rope with the boiling tar. The grooves of cast-iron pulleys and sheaves should be filled with well-seasoned blocks of hard wood, set on end. to be reaewed when worn out. This end wood will save the rope and increase ad- hesion. The small pulleys or rollers which support the ropes on inclined planes should be constructed on the same plan. When large sheaves run with a very great velocity, the grooves must be lined eithm- WEIGHT AND STRENGTH OF IRON CHAINS. 381 with leather set on end ; with cork ; or with India rubber. This is done in the case of all sheaves used in the transmission of power between distant points by means of ropes; which frequently run at the rate of 4000 feet per minute. Full information given, if desired, on the size of rope, aud the size and speed of sheaves to be used for transmitting power. Rope % ins diam will transmit 100 horse power to a great dist. Notes on Ki^inj;. " Wire Rope for shrouds and stays is now universally superseding hemp rope, for the following rea- sons : it is much cheaper than hemp rope ; it is more durable, aud it will not stretch permanently under great strains, as is the case with hmp rigging, thus saving much labor in setting up; and it is fully as elastic as hemp rope of equivalent size. Tne rope with 49 wires is best adapted for rigging. The great economy in using wire in place of hemp rigging is the large reduction in size and weight. The bulk of wire rigging is only one-sixth that of hemp, while the weight is only one-half. The ad- vantages of lightness are apparent at a glance to every seaman ; the removal of several tons of weight from the height occupied by the standing rigging must increase both the steadiness and the stability of the ship. Again, on account of its smaller bulk, less resistance is offered to the wind, a fact fully appreciated by captains of steamers, when running against the wind. For protection from rust, wire rigging is galvanized; when not galvanized, apply good paint with a brush or a piece of sheepskin, once or twice a year, and a good set of wire shrouds aud stays will out- last the best built sailing ship or steamer. The best aud cheapest paint is linseed oil mixed with equal parts of lampblack and Spanish brown or Venetian red, or any other preparation of oxide of iron. All vessels in the U S Navy are now rigged with Roebling's Wire Rope exclusively, it having proved the best in the test made by the Government at the Washington Navy Yard." The foregoing is a copy of a circular by Messrs Roebling. On planes in Schuylkill Co. s. No. per 5) Size in ins. Length. Side. No. per keg of 150 Ibs. No. per ft>. -i/ v V 350 2.33 526 3.5 4y X \& 400 2.66 5i/ x T7j 289 1.93 6 X ? 705 4.7 **A X 7 218 1.46 5 X & 5 X *A 48S 390 3.25 2.6 6 X K 6 X A 310 262 2.07 1.75 5 X T 9 7T 295 1.97 6 X ^ 196 1.C.O 5 X % 257 1,71 A size in very common use. is 5^ X tV ; which wei S hs about H ft P er spike. A mile of single-track road, with 2112 cross-ties, 2% feet apart from center to center: and with rails of the ordinary length of 24 feet, or 10 ties to a rail ; thus *~Prlce of spike* and of cut nails, in Philada, in 1880, about 5 cts per ft. Rivets 6 to ej? In 1882, all about 25 per ct less. SPECIFIC GRAVITY. 383 having 440 rail-joints per mile; with 4 spikes to each tie, except at the rail-joints, at each of which there will be 4 spikes ; * will require at a neat calculation 9328 spikes. But an allowance must be made for rail guards at road-crossings, which we mar assume to be '24 ft wide, or the length of a rail. A guard will usually consist of 4 extra rails for protecting the track- rails, and spiked to the 11 ties by which said track rails are sustained. Consequently, such a crossing requires 11. X 8 88 spikes. For turnouts, sidings, loss, &c. we may roughly average 584 t spikes more per mile; thus making in all if we assume one road-crossing per niilej 9328 -f- b8 -f- 584 = 1UOOO spikes per mile; or 5000 Ibs, or 33# kegs of 150 tt>s. Adhesion of spikes. Professor W. R. Johnson found that a plain spike .375, or % inch square, driven 3% ins into seasoned Jersey yellow pine, or unseasoned chestnut, required about 2uOO ft>s force to extract it; from seasoned white oak. about 4000; and from well-seasoned locust, about 6000 ft>s. Bevan found that a t^peuny nail, driven one inch, required the following forces to extract it: Seasoned oeech, b'67 Ibs; oak, 507 ; elm, 327 ; pine, 187. Very careful experiments in Hanover, Germany, BY ENorNKER PUNK, give from 2465 to 3'J40 Ibs, (mean of many experiments, about 3000 fts.) as the force necessary to extract a plain ^ inch square iron spike, 6 ins long, wedge- pointed for one inch, (twice the thickness of the spike,) and driven 4) ins into white or yellow pine. When driven 5 ins, the force reqd was about -fa part greater. Similar spikes, -A- inch square, 7 ins long, driven 6 ins deep, reqd from 3700 to 6745 Ibs to ex- tract them from pine ; the mean of the results being 4873 Ibs. In all cases, about twice at much force wis reqd to extract them from oak. The spikes were all driven across the rain of the wood. Experience shows that wuen driven with the grain, spikes or nails not hold with much more than half as much force. Jagged spikes, or twisted ones, (like an au?er, t or those which were either swelled or diminished near the middle of their length, all proved inferior to plain square ones. When the length of the wedge point was increased to 4 times the thickness of the spike, the resistance to drawing out was a trine less. When the length of the spike is fixed, there is prob-ibly no b -tier shape than the plain square cross- section, with a wedge point twice as long as the width of the spike, as per this tig. Boards of oak or pine, nailed together by from Mo i6tenpermy common cut nails, and then pulled apart in a direction lengthwise of the hoards, and across the nails, tending to break the Utter in two by a shearing action, averaged about 300 to 400 Ibs per nail to sepa- rate them; as the result cf many trials. WEIGHT OF gr do Name. Length. Inches. No. per flX Name. Length. Inches. No. per ft. 3 penny \y 557 10 penny 2% 66 4 " Wg 336 12 " 3% 50 5 " 1% 210 20 " 3% 32 6 " 2 163 30 " 4^z 19 7 " 2 1 4 128 40 " 4% 16 8 " VA 93 50 u 5% 13 The sizes and weights vary considerably with different makers Ours are averages, ^h^ above are machine-made, or cut nails ; in distinction to the wrought nails made by the black sn 5 1 ; . . SPECIFIC GEAVITY. THE sp grav of a body, is its weight as compared with that of an equal bulk of some other body, which is adopted as a standard of comparison. For other substances than air and gases generally, pure water is the usual standard ; and since the weight of a given bulk of water varies somewhat with its temperature ; and also with the state of the air, the former is assumed to be 62 Fah ; and the latter at 30 ins, at sea-level. But where extreme scientific accuracy is not aimed at, all these considerations may be neglected ; and any clear fresh water, at any ordinary temperature, say from (50 to 80 may be used; for if at 70, the resulting sp gr will be but 1 part in 1176 too great; at*75, 1 in 670; at 80, 1 in 454; at 85, 1 in 336. At 62 pure water weighs 62.355 fts avoir per cub ft. To find the sp srrav of a body, heavier than water. Weigh it first in the air ; and then in water ; and find the diff. The diff is what the body loses in water: and is the weight of a bulk of water equal to the bulk of the body. Then say as this, Diff : wt in air : 1 : : sp grav of body. * This suppose** the joint and chair to rest upon a tie: bat when long chairs are ued with a view of placing the rail joint between two ties laid near each other, there will be 8 spikes to a joint : or 1780 per mile more than above; equal to 880 As; making in all, per mile single track, say 12000 pikes, or 000 Ibs, or 40 kegs. t This allows that turnouts and sidings amount to about 1 mile of extra track on 15 miles of road. 4* Price in Philada, 1880, about 5 cts per Q>. Roofing nails of tinned iron, 10 cents. Copper nails, 45 cts. 384 SPECIFIC GRAVITY. The weight of a given bulk of a substance which is either porous, or absorbent of water, cannot be Inferred from its sp gr. Thus pure river aaud, is pure quartz ; and of course has the same sp gr ; yet, a solid cub ft of quartz, weighs nearly twice as much as a cub ft of sand ; on account of the interstices of the latter. A brick, some sandstones, &c. absorb water; so that their sp gr will not furnish the weight of a dry mass of the same. In such cases, the engineer will generally first measure the con- tents of a piece of the substance, if a solid ; and then weigh it; thus ascertaining its weight per cub ft, &c. If it is in grains, or dust, he will measure, and then weigh, a cub ft of it. TO find the Sp grav Of a liquid. First carefully weigh some solid body, as a piece of metal, in the air. Then weigh it in water, and note the loss, sav L. Then weigh it in the other liquid : and note the loss, say /. Then as loss L, is to loss I, so is 1. or the sp gr of water, to the sp gr of the liquid. Or, if the sp gr, and weight of the solid body, are already known, merely weigh it in the liquid. Then as its weight in air, is to its loss in the liquid, so is its sp gr, to that of the liquid. Timber, when first purchased from lumber yards, even under shelter, is rarely, if ever, perfectly dry ; but its weight, if tolerably seasoned, will be about % part greater than given in our tables , or about *4 to % part, if green. Table of specific gravities, and weights. In this table, the sp gr of air, and gases also, are compared with that of water, instead of that of air ; which last is usual. Names of Substances. Average SpGr. Air, atmospheric ; at 60 Fan, and under the pressure of one atmosphere or 14.7 fts per sq inch, weighs g^-j part as much as water at 60 .00123 Alcohol, pure .793 of commerce .834 " proof spirit .916 Ash, perfectly dry. (See footnote, p 386.) average. . .752 1000 ft beard measure weighs 1.748 tons. Ash, American white, dry " .. .61 1000 ft "board measure weighs 1.414 tons. Alabaster, falsely so called ; but really Marbles 2.7 " real; a compact white plaster of Paris average.. 2.31 Aluminium 2.6 Antimony, cast, 6.66 to 6.74 average .. 6.70 native " 6.67 Anthracite, 1.3 to 1.84. Of Penn a, 1.3 to 1.7 usually .. 1.5 A cubic yard solid, averages about 1.75 cub yds, when broken to any mar- ketsize; and loose. Anthracite, broken, of any size. Loose average.. ' " raodorately shaken " ... " heaped bushel, loose, 77 to as. A ton, loose, averages from 40 to 43 cub ft " at 54 fts per cub ft, a cub yard weighs .651 ton. Asphaltum, 1 to 1.8 , " 1.4 Basalt. See Limestones, quarried " 2.9 Bath Stone, Oolite " 2.1 Bismuth, cast. Also native " 9.74 Bitumen, solid. See Asphaltum. Brass, (Copper and Zinc,) cast, 7.8 to 8.4 " 8.1 rolled " 8.4 Bronze. Copper 8 parts; Tin 1. (Gun metal.) 8.4 to 8.6 ' 8.5 Brick, best pressed " common hard " soft, inferior Brickwork. See Masonry. Boxwood, dry Calcite, transparent.. ' 2.722 Carbonic Acid Gas, is 1^ times as heavy as air ' .00187 Charcoal, of pines and oaks ' Chalk, 2.2 to 2.8. See Limestones, quarried ' 2.5 Clay, potter's, dry, 1.8 to 2.1 ' 1.9 dry, in lump, loose Coke, loose, of pood coal ' a heaped bushel, loose, 35 to 42 Ibs " a ton occupies 80 to 97 cub ft In coking, coals swell from 25 to 50 per cent. Equal weights of coke and coal, evaporate about equal wts of water : and each abt twice as much as the same wt of dry wood. Corundum, pure, 3.8 to 4 3.9 Cherry, perfectly dry average .. .672 1000 ft board measure weighs 1.56'2 tons. Coal, bituminous. 1.2 to 1.5 " -. 1.35 " " broken, of any size; loose " 41 " moderately shaken " " " a heaped bushel, loose. 70 to 78 Ibs. " " a ton occupies 43 to 48 cub ft. A cubic yard solid, averages about 1.75 yards when broken to any market size, and )ooe. SPECIFIC GRAVITY. 385 Table of specific gravities, and weights (Continued.) Names of Substances. Average Sp Gr. Chestnut, perfectly dry . (See footnote, p 386.) average . . 1000 board measure weigas I.o25 tons. Cement, hydraulic. American, Rosendale; ground, ktose average .. " U.S. Struck bush,. 70 fts " " Louisville, " 62 " " Copley, " " 67 " English Portland, U.S. struck bush, by Gillmore, 100tol28 " " Various, weighed by writer, 95 to 102.., " " " a barrel 400 to 430 Ibs. " French Boulogne Portland, struck bush, 95 to 110 Differences of 4 or 5 pounds either more or less than we here give per loose struck U.S. bush, often occur in the cement from the same manufactory, owing not only to the difficulty of measuring exactly, but to the want of uniformity in the composition of the stone, de- gree of burning, grinding, dryness, &c. Moreover, the term "loose" is indefinite. We mean by it the average looseness which it has when thrown by a scoop into a half bushel when measuring that quantity for sale. By shaking it may easily be compacted about y part, so a.* to weigh i more per bush, or cub ft. And by ramming, about % part, so as to weigh about % more. So with lime, plas- ter, &c. Copper, cast, 8.6 to 8.8 8.7 rolled 8.8 to 9.0 8.9 Crystal, pure Quartz. See Quartz. Cork 25 Diamond, 3.44 to 3.55 ; usually 3.51 to 3.55 3.53 Earth ; common loam, perfectly dry. loose " " " " shaken " " " " " moderately rammed 41 " slightly moist, loose ." " " more moist, " ' " " " shaken moderately packed 11 " " as a soft flowing mud " " " as a soft mud, well pressed into a box Ether . "6 Elm, perfectly dry. (See footnote, p 386.) average . . .56 1000 ft board measure weighs 1.302 tons. Ebony, dry.., ' 1-22 Emerald, 2.63 to 2.76 l .. 2.7 Fat Flint " 2.6 Feldspar, 2.5 to 2.8 .. 2.65 Garnet, 3.5 to 4.3; Precious, 4.1 to 4.3 " .. 4.2 Glass, 2.5 to 3.45 " common window " .. 2.52 " Millville, New Jersey. Thick flooring glass " -. 2.53 Granite, 2.56 to 2.88. See Limestone, 160 to 180 " 2-72 Gneiss, common, 2.62 to 2.76 " .. 2.69 44 in loose piles " Hornblendic " " quarried, in loose piles Gypsum, Plaster of Paris, 2.24 to 2.30 ' .. 2.27 41 in irregular lumps " ground, loose, per struck bushel, 70 well shaken. " " 80 " " Calcined, loose, per struck bush, 65 to 75 , Greenstone, trap, 2.8 to 3. 2 41 ' 4 quarried, in loose piles Gravel, about the same as sand, which see. Gold, cast, pure, or 24 carat " .. 19.258 " native, pure, 19.3 to 19.34 " .. 19.32 " " frequently containing silver, 15.6 to 19.3 " pure, hammered, 19.4 to 19.6 " .. 19.5 Gutta Percha Hornblende, black, 3.1 to 3.4 Hydrogen Gas, is 14J4 times lighter than air; and 16 times lighter than ' oxygen average.. Hemlock, perfectly dry. (Footnote, p 386.) " .. .4 1000 feet board measure weighs .930 ton. Hickory, perfectly dry. (See footnote, p 386.) 1000 feet board measure weighs 1.971 tons. Iron, cast, 6.9 to 7.4 " .. 7.15 " " usually assumed at " .. 7.21 At 450 ft>s. a cub inch weighs .2604 Tb ; 8601.6 cub inches a ton ; and a ft 3.8400 cub inches ; cast-iron gun metal 7.48 386 SPECIFIC GRAVITY. Table of specific gravities, and weights (Continued.) Names of Substances. Average Sp Gr. Iron, wrought, 7.6 to 7.9; the purest has the greatest sp gr average.. 7.77 " large rolled bars .. 7.6 " sheet . ' At 480 lbn, a cub inch weighs .2778 ft ; and a ft = 3.6000 cub ins. Light iron indicates impurity. Ivory average.. 1.82 Ice, .917 to .922 " .. .92 India rubber " Lignum vitae, dry " .. 1.33 Lard " Lead, of commerce, 11.30 to 11.47; either rolled or cast " .. 11.38 Limestones and Marbles, 2.4 to 2.86, 150 to 178.8 " ordinarily about 2.7 " " " quarried in irregular fragments, 1 cub yard solid, makes about 1.9 cub yds perfectly loose; or about 1% yds piled. In this last case, .571 of the pile is solid; and the remaining .429 part of it is voids piled. . Lime, quick, of ordinary limestone and marbles 2 to 98 fts per cub ft 1.5 ' either in small irregular lumps ; or ground, loose 50 to 58 In either case 1 solid measure makes about 1.8 meas loose; and then .555 of the mass is solid, and .445 is voids. To measure correctly, none of the lumps should exceed about % or Y*(j of the smallest dimension of the vessel used for measuring. Lime, quick, ground, loose, per struck bushel 62 to 70 fts " well shaken, " ' 80 " ' " " thoroughly shaken, " 93% " Mahogany, Spanish , dry * t average . . .85 " Honduras, dry " .. .56 Maple, dry* " .. .79 Marbles, see Limestones. Masonry, of granite or limestones, well dressed throughout " " " well-scabbled mortar rubble. About -^ of the mass will be mortar " " " well-scabbled dry rubble " " " roughly scabbled mortar rubble. About % to X part wiU be mortar " " " roughly scabbled dry rubble At 155 fts per cub ft, a cub yard weighs 1.868 tons ; and 14.45 cub ft, 1 ton. Masonry of sandstone; about ^ part less than the foregoing. " brickwork, pressed brick, fine joints ........ average.. medium quality " " " " coarse ; inferior soft bricks " At 125 fts per cub ft, a cub yard weighs 1.507 tons; and 17.92 cub ft, 1 ton. Mercury, at 32 Pah 13.62 60 " 13.58 " 212 " Mica, 2.75 to 3.1 '. 2.93 Mortar, hardened, 1.4 to 1.9 1.65 Mud, dry, close " wet, moderately pressed ' wet, fluid '. Naphtha .848 Nitrogen Gas is about -^ part lighter than air , Oak, live, perfectly dry, .88 to 1.02 * average. " white, " " .66 to .88 " red, black, &c* " Oils, whale; olive " ..I .92 " of turpentine " . . | .87 Oolites, or Roestones, 1.9 to 2.5 " .. 2.2 Oxygen Gas, a little more than ^ Prt heavier than air .00136 Petroleum .878 Peat, dry, unpressed t Pine, white, perfectly dry, .35 to .45* .40 1000 ft board measure weighs .930 ton.* " yellow, Northern, .48 to .62 ( .55 1000 ft board measure weighs 1.276 tons.* " " Southern, .64 to .80 1000 ft board measure weighs 1.674 tons.* * Oreen timbers usually weigh from one-fifth to nearly one-half more than dry ; and ordinary building timbers when tolerably seasoned about one-sixth more than perfectly* dry. SPECIFIC GRAVITY. 387 Table of specific gravities, and weights (Continued.) Names of Substances. Average SpGr. Pine, heart of long-leafed Southern yellow, unseas. (Footnote, p 386.) ... 1.04 1000 ft board measure weighs 2.418 tons. Pitch 7 Plaster of Paris ; see Gypsum. Powder, slightly shaken 1 Porphyry, 2.66 to 2.8 2.73 Platinum 21 to 22 21.5 native, in grains 16 to 19 ,.... 17.5 Quartz, common, pure 2.64 to 2.67 2,65 finely pulverized, loose " " " " well shaken " " " " well packed '. " quarried, loose. One measure solid, makes full 1% broken and piled Ruby and Sapphire, 3.8 to 4.0 3.9 Rosin 1.1 Salt, coarse, per struck bushel ; Syracuse, N. York 56 Ibs . . " " " " " Turk's Island; Cadiz; Lisbon. 76 to 80 .. " St. Barts 84 to 90 .. " " " " some well-dried West India. ... 90 to 96 .. " " " " Liverpool 50 to 55 . " Liverpool fine, for table use 60 to 62 Sand, of pure quartz , perfectly dried, and loose, usually 112 to 133 Ibs per struck bushel 2.65 At the average of 98 Ibs per cub ft, a struck bushel weighs 122} Ibs j and 18.29 bushels, 1 ton ; a cub yd = 1.181 tons ; 22.86 cub ft, 1 ton. Slight shaking compacts it about 2 to 3 per ct ; and ramming about 12 per ct when dry. " perfectly wet, voids full of water " " " at the mean of 124 Ibs, a cub yard weighs 1.495 tons ; and 18.06 cub 1 ton. " sharp angular sand of pure quartz with very large and very small grains dry may weigh If any ordinary pure natural sand be sifted into 2 or 3 or more parcels of differently sized grains, a measure of any of these parcels will weigh considerably less than an equal measure of the original sand. Thus, a sand weighing 98 Ibs per cub foot, may give others weighing not more than 70 to 80 Ibs. At 98 Ibs per cub ft. 1 bulk of pure quartz, has made 1.68 bulks of sand ; of which the solid occupies .6 ; and the voids .4. But if this same sand be compacted to 110 Ibs per cub ft, then 1 measure of solid quartz makes 1 % measures of sand ; of which % are solid, and % voids. Sand is very retentive of moisture ; and when in large bulks, is rarely as dry as that above in this table. But with its natural moisture, and loose, it is lighter than when dry, its average weight then not exceeding about 85 to 90 Ibs per cub ft ; or 106J4 to 112^ Ibs per struck bushel. See Voids in Sand, p 504. Sandstones, fit for building, dry, 2.1 to 2.73 131 to 171. 2.41 " quarried, and piled. 1 measure solid, makes about \% piled Serpentines, good 2.5 to 2.65 2. Snow, fresh fallen " moistened, and compacted by rain Sycamore, perfectly dry. (See footnote, p 386.) .59 1000 ft board measure weighs 1.376 tons. Shales, red or black 2.4 to 2.8 average.. 2.6 '" quarried, in piles " , Slate 2.7to2.9 " .. 2.8 Silver " .. 10.5 Soapstone, or Steatite 2.65 to 2.8 " .. 2.73 Steel, 7. 7 to 7.9. The heaviest contains least carbon " .. 7.85 Steel is not heavier than the iron from which it is made ; unless the iron had impurities which were expelled during its conversion into steel. Bulphur average. . 2. Spruce, perfectly dry. Footnote, p 386 " .4 1000 ft board measure weighs .930 ton. flpelter, or Zinc 6.8 to 7.2 " .. 7.00 Sapphire ; and Ruby, 3.8 to 4 " .. 3.9 Tallow " .. .94 Tar .. 1. Trap, compact, 2.8 to 3.2 " .. 3. " quarried; in piles " Topaz. 3.45 to 3.65 " .. 8.55 GRADES. Table of specific gravities, and weights (Continued.) Names of Substances. Average Sp Gr. Average Wt of a Cub Ft. Lbs. 7 35 4/>9. 20 to 30 Water pure rain or distilled at 32 Fah Barotn 30 ins 62 417 <. 4, ,< ,< .< g.jO . .4 . 1 62 H55 < .1 t it 212 " " " " . 59 7 At 60, a cub inch weighs .03607 Ib ; or .57712 oz avoir. And a Ib con- tains 27.724 cub ins ; equal to a cube of 3.0263 inches oil each edge. Water sea 1 026 to 1 030 average 1 028 64 08 Although the weight of fresh water is in practice almost invariably assumed as 62^ Ibs per cub ft, yet 62^ would be nearer the truth, at ordinary temperatures of about 70; or a Ib = 27.759 cub ins ; and a cub in zr .5764 oz avoir ; or .4323 oz troy ; or 252.175 grains. The grain is the same in troy, avoir, and apoth. Wax, bees average . . 97 605 Wines, .993 to 1.04 998 62 3 Walnut, black perfectly drv (See footnote p 386 ) " 61 38 1000 ft board measure weighs 1.414 tons. Zinc, or Spelter, 6.8 to 7.2 " 7 00 437.5 Zircon, 4.0 to 4.9 ' 4.45 Table of grades per mile; or per 1OO feet measured hori- zontally. See p 629. Grade in ft. per mile. Grade in ft. per 100 ft. Grade in ft. per mile. Grade in ft. per 100 ft. Grade in ft. per mile. Grade iu ft. per 100 ft. Grade in ft. perniile Grade in ft. per 100 ft. 1 .01894 39 .73S64 77 1.45833 115 2.17803 2 .03788 40 .75758 78 1.47727 116 2.19697 3 .05682 41 .77652 79 1.49621 117 2.21591 4 .07576 42 .79545 80 1.51515 118 2.23485 5 .09470 43 .81439 81 1.53409 119 2.25379 6 -11364 44 .83333 82 1.55303 120 2.27273 7 .13258 45 .85227 83 1.57197 121 2.29167 8 .15152 46 .87121 84 1.59091 122 2.31061 9 .17045 47 .89015 85 1.60985 123 2.32955 10 .18939 48 .90909 86 1.62879 124 2.34848 11 .20833 49 .92803 87 1.64773 125 2.36742 12 .22727 50 .94697 88 1.66666 126 2.38636 13 .24621 51 .96391 89 1.68561 127 2.40530 14 .26515 52 .98485 90 1.70455 128 2 42424 15 .28409 53 1.00379 91 1.72348 129 2.44318 16 .30303 54 1.02273 92 1.74242 130 2.46212 17 .32197 55 1.04167 93 1.76136 131 2.48106 IS .34091 56 1.06061 94 1.78030 132 2.50000 19 .35985 57 1.07955 95 1.79924 133 2.51894 20 .37879 58 1.09848 96 1.81818 134 2.53788 21 .39773 59 1.11742 97 1.83712 135 2.55682 22 .41667 60 1.13636 98 1.85G06 136 2.57576 23 .43561 61 1.15530 99 1.87500 137 2.59470 24 .45455 62 1.17424 100 1.89394 138 2.61364 25 .47348 63 1.19318 101 1.91288 139 2.63258 26 .49242 64 1.21212 102 1.93182 140 2.65152 27 .51136 65 ]. 23106 103 1.95076 141 2.67045 28 .53030 66 1.25000 104 1.96969 142 2.6S939 29 .54924 67 1.26S94 105 1.98864 143 2.70833 30 .56818 68 1.28788 105 2.00758 144 2.72727 31 .58712 69 1.30682 107 2.02652 145 2.74621 32 .60606 70 1.32576 108 2.04545 146 2.76515 33 .62500 71 1.34470 109 2.06439 147 2.78409 34 .64394 72 1.36364 110 2.08333 148 2.80303 35 .66288 73 1.38258 111 2.10227 149 2.82197 36 .68182 74 1.40152 112 2.12121 150 2.84091 37 .70076 75 1.42045 113 2.14015 151 2.85985 38 .71970 76 1.43939 114 2.15909 152 2.87879 GRADES. 389 Remark on preceding: Table. If the grade per mile should consist of feet and tenths, add to the grade per 100 feet in the foregoing table, that corresponding to the number of tenths taken from the table below ; thus, for a grade of 43.7 feet per mile, we have .8H39 -4- .01326= .82765 feet per 100 feet. Ft. per Mile. Per 100 Feet. Ft. per Mile. Per 100 Feet. Ft. per Mile. Per 100 Feet. .05 .1 .15 .2 .25 .3 .35 .00094 .00189 .00283 .00379 .00473 .00568 .00662 .4 .45 .5 .55 .6 .65 .00758 .00852 .00947 .01041 .01136 .01230 .7 .75 .8 .85 .9 .95 .01326 .01420 .01515 .01609 .01705 .01799 Table of grades per mile, and per 100 feet measnred hori- zontally, and corresponding to different angles of inclina- tion. For another table per 100 ft only, see p 629. II Feet per mile. Feet pei 100 ft. H> d Q 3 Feet per mile. Feet pe 100 ft. si> a 3 Feet per mile. Feet pe 100 ft. 8? .2 c S Feet per mile. Feet per 100ft. 1 1.536 .0291 45 69.11 1.3090 1 58 181.3 3.4341 3 26 316.8 5.9994 2 3.072 .0582 46 70.64 1.3381 2 184.4 3.4924 28 319.8 6.0579 3 4.608 .0873 47 72.18 1.3672 2 187.5 3.5506 30 322.9 6.1163 4 6.144 .1164 48 73.72 1.3963 4 190.6 3.6087 82 326.0 6.1747 5 7.680 .1455 49 75.26 1.4254 6 193.6 3.6669 34 329.1 6.2330 6 9.216 .1746 50 76.80 1.4545 8 196.7 8.7250 36 332.2 6.2914 7 10.75 .2037 51 78.33 1.4837 10 199.8 8.7833 38 335.3 6.3498 8 12.29 .2328 52 79.87 1.5128 12 202.8 3.8416 40 338.4 6.4083 9 13.82 .2619 53 81.40 1.5419 14 205.9 8.8999 42 341.4 6.4664 10 15.36 .2909 54 82.94 1.5710 16 208.9 3.9581 44 344.5 6.5246 11 16.90 .3200 65 84.47 1.6000 18 212.0 4.0163 46 347.6 6.5832 1-2 18.43 .3491 56 86.01 1.6291 20 215.1 4.0746 48 350.7 6.6418 13 19.96 .3782 57 87.54 1.6583 22 218.1 4.1329 50 353.8 6.7004 14 21.50 .4073 58 89.08 1.6873 24 221.2 4.1911 52 356.8 6.7583 15 23.04 .4364 59 90.62 1.7164 26 224.3 4.2494 54 359.9 6.8163 16 24.58 .4655 1 92.16 1.7455 28 227.4 4.3076 56 363.0 6.8751 17 26.11 .4946 2 95.23 1.8038 30 230.5 4.3659 58 366.1 6.9339 18 27.64 .5237 4 98.30 1.8620 32 233.5 4.4242 4 369.2 6.9926 19 29.17 .5528 6 101.4 1.9202 34 236.6 4.4826 5 376.9 7.1384 20 30.72 .5818 8 104.5 1.9784 36 239.7 4.5409 10 384.6 7.2842 21 32.26 .6109 10 107.5 2.0366 38 242.8 4.5993 15 392.3 74300 22 33.80 .6400 12 110.6 2.0948 40 245.9 4.6576 20 400.1 7.5767 23 35.33 .6691 14 113.6 2.1530 42 248.9 4.7159 25 407.8 7.7234 24 36.86 .6982 16 116.7 2.2112 44 252.0 4.7742 30 415.5 7.8701 25 38.40 .7273 18 119.8 2.2094 46 255.1 4.8325 35 423.2 8.0163 26 39.94 .7564 20 122.9 2.3277 48 258.2 4.8908 40 431.0 8.1625 27 41.47 .7855 22 126.0 2.3859 50 261.3 4.9492 45 438.7 8.3087 28 43.01 .8146 24 129.1 2.4141 52 264.3 5.0075 50 446.5 8.4554 29 44.54 .8436 26 132.1 2.5023 54 267.4 5.0658 55 454.2 8.6021 30 46.08 .8727 28 135.2 2.5604 56 270.5 5.1241 5 461.9 8.7489 31 47.62 .9018 30 138.3 2.6186 58 273.6 5.1824 5 469.6 8.8951 32 49.16 .9309 32 141.3 2.6768 3 276.7 5.2407 10 477.4 9.0413 33 50.69 .9600 34 144.4 2.7350 2 279.7 5.2990 15 485.1 9.1875 34 52.23 .9891 36 147.4 2.7932 4 282.8 5.3573 20 492.9 9.3347 35 53 76 1.0182 38 150.5 2.8514 6 285.9 5.4158 25 500.6 9.4819 36 55.30 1.0472 40 153.6 2.9097 8 289.0 5.4742 30 508.4 9.6292 37 5683 .0763 42 156.6 2.9679 292.1 5.5326 35 516.1 9.7755 38 58.37 .1054 44 159.7 3.0262 2 295.1 5.5909 40 523.9 9.9218 39 59.90 .1345 46 1(52.8 3.0844 4 298.2 5.6493 45 531.6 10.068 40 61.44 .1636 48 165.9 3.1427 6 301.3 5.7077 50 539.4 10.215 41 62.97 .1927 50 169.0 3.2010 8 304.4 i).7660 55 547.2 10.362 42 61.51 .2218 52 172.0 3.2592 20 307.5 5.8244 6 555. 10.510 43 66.04 1.2509 54 175.1 3.3175 22 310.5 5.8827 44 67.57 1.2800 56 178.2 3.3758 24 313.6 5.9410 On a turnpike road 1 38', or about 1 in 35, or 151 ft per mile, is the greatest slope that should be given to allow horses to trot down rapidly with safety. In crossing mountains, this is often increased to 3, or even to 5. It should never exceed 2%, except when abso- lutely necessary. 390 RAIL- JOINTS, AND CHAIRS. TABLE OF ACRES REQUIRED per mile, and per 1OO feet, for different widths. Width. Feet. Acres per Mile. Acres per 100 Ft. Width. Feet. Acres per Mile. Acres 100 Ft. Width. Feet. Acres per Mile. Acres per 100 Ft. Width. Feet. Acres per Mile. Acres per 100 Ft. 1 .121 .002 26 3.15 .06.) 52 6.30 .119 78 9.45 .179 2 .242 .005 27 3.27 .062 53 6.42 .122 79 9.58 .181 3 .364 .007 28 3.39 .064 64 6.55 .124 80 9.70 .184 4 .485 .009 29 3.52 .067 55 6.67 .126 81 9.82 .186 5 .606 .011 30 3.64 .069 56 6.79 .129 82 9.94 .188 6 .727 .014 31 3.76 .071 57 6.91 .131 Y* 10. .189 7 .848 .016 32 3.88 .073 *A 7. .133 83 10.1 .190 8 .970 .018 33 4.00 .076 58 7.03 .133 84 10.2 .193 Y 1. .019 34 4.12 .078 59 7.15* .135 85 10.3 .195 9 1.09 .021 35 4.24 .080 60 7.27 .138 86 10.4 .197 10 1.21 .023 36 4.36 .083 61 7.39 .140 87 10.5 .200 11 1.33 .025 37 4.48 .085 62 7.52 .142 88 10.7 .202 12 1.46 .028 38 4.61 .087 63 7.64 .145 89 10.8 .204 13 1.58 .030 39 4.73 .090 64 7.76 .147 90 10.9 .207 14 1.70 .032 40 4.85 .092 65 7.88 .149 % 11. .209 15 1.82 .034 41 4.97 .094 66 8. .151 91 11.0 .209 16 1.94 .037 & 5. .094 67 8.12 .154 92 11.2 .211 X 2. .038 42 5.09 .096 68 8.24 .156 93 11.3 .213 17 2.06 .039 43 5.21 .099 69 8.36 .158 94 11.4 .216 18 2.18 .041 44 5.33 .101 70 8.48 .161 95 11.5 .218 19 2.30 .044 45 5.45 .103 71 8.61 .163 96 11.6 .220 20 2.42 .04H 46 5.58 .106 72 8.73 .165 97 11.8 .223 21 2.55 .048 47 5.70 .108 73 8.85 .168 98 11.9 .225 22 2.67 .051 48 5.82 .110 74 8.97 .170 99 12. .227 23 2.79 .053 49 5.94 .112 ^ 9. .170 100 12.1 .230 24 2.91 .055 1 A 6. .114 75 9.09 .172 % 3. .057 50 6.06 .115 76 9.21 .174 25 3.03 .057 51 6.18 .117 77 9.33 .177 RAIL-JOINTS AXI> CHAIRS. A railroad track being weakest at the joints between the rails, where they are deprived of their vertical strength, has of course a greater tendency to bend at those points ; and this bending produces an irregularity in the movement of the train, which is detrimental to both rolling-stock and track. Moreover, that end of a rail upon which a loaded wheel is moving, bends more than the adjacent unloaded end of the next rail ; so that when the wheel arrives at said second rail, it imparts to its end a severe blow, which injures it. Thus, the ends of the rails are exposed to far more injury than its other portions. Numerous devices have been resorted to for strengthening the joints of the rails, with a view of preventing this bending entirely ; or, at least, of causing the two adjacent rail-ends to bend equally, and together; so as to avoid the blows alluded to. None of these joint-fastenings, known as chairs, fish-plates, wooden blocks, &c, have proved entirely satisfactory. Much of the deficiency ascribed to the fastenings, is, however, really due to want of stability in th 'cross-ties at the joints; and more attention must be directed to this latter consideration, before an efficient fastening can be obtained. Observation shows that when the joint-ties are very firmly bedded, almost any of the ordinary fastenings will, (if the joint is placed between two ties, instead of resting upon a tie,)* answer very well ; whereas, when the cross- ties are so insecurely bedded as to play up and down for half an inch or more under the driving-wheels of the engines, the strongest and most effective fastenings soon become comparatively inoperative. All the parts of the best of them will in that case become gradually loosened, warped, bent, or broken. This remark applies to all the fishing-splices, chairs, long wooden blocks, bolts, spikes. &c, in present use. Experience has established the superiority of suspended joints, over supported ones. On a portion of a track carrying a heavy business, with joint-fastenings closely resembling that in Figs 10, (with long wooden blocks B; and long fish-pieces c,) alternate joints were suspended between two ties ; while the intermediate one rested upon a center tie ; the blocks, however, extending over three ties. The ends of the rails were more injured by crushing and brooming in the latter than in the former. And so with a number of other patterns of short joint-fastenings of wrought iron. Long fastenings, perhaps, possess but little superiority over short ones, where the track is not kept in good repair; for the great bearing of the former, although impart- In the first oaae the joint i.s called a SUSPENDED one ; in the last a SUPPORTED one. RAIL-JOINTS, AND CHAIRS- 391 ing increased firm ness on a good track, becomes converted into a powerful leverage, by which it accelerates its own destruction, in a bad one. An element in the injury of joints, is the omission of proper fastenings at the center of the rails. Each rail should be so firmly attached to the cross-ties at and near its center, as to compel the contraction and expansion to take place equally from that point, toward each end. It would probably be somewhat difficult to accomplish this perfectly. The at- tempts hitherto made have failed. One of the earliest suggestions for a joint fastening, was the limti-joint. or fish-splice, or fish-plates, Fig 1, introduced upon the Newcastle and French- town R.R. in Delaware, by Kobt. H. Barr, in 1843.* This consists of two strips of iron, z z. called fishes, rolled to fit the sides of the rail; and bolted together and to the rails, by either keybolts, or screwbolts. Usually 4 bolts are used; sometimes 6. The fishes are usually about % to % inch thick. The curved shape of the inner top and bottom edges of these splices ; and of the top and bottom of the stem of the rai 1, creates a tendency to yield under the bending of the joint under heavy loads, as shown in an exaggerated manner at Figs 2 and 3. Either of these brings a great strain upon the bolt, and is apt to pull it apart. At a later period, Edward Miller, C E, of Philada, in order to remedy these defects, introduced the rolling of the rails with square boulders, as in Fig 4, which however soon fell into disuse. Fish-splices are frequently made with a shallow groove about % inch deep, on their outer side, as at n n, and F, Figs 10. This groove receives either the square head of the bolt, which is then inserted first, and the nut afterwards screwed on ; or else the nut is first placed in the groove, and the bolt afterwards screwed into it. It was supposed that the nut and bolt would thus be prevented from loosening and un- screwing under the jarring of the trains. The result however has not proved satis- factory ; and inasmuch as the groove also weakens the plates, it is falling into disuse. The bolt holes through either the plates or the rail, are made about % inch longer than high, to allow the rails to expand and contract. Such splices, about 18 to 24 ins long, and ^ to % ins thick, with 4 bolts about 4 ins long, by % tc, % inch diarn, and placed between two joint cross-ties about 1 ft apart in the clear, were for many years the most approved in this country; but on some roads of heavy traffic, they are being supplanted by the angle-bar splice, or fish plate, similar to those shown in Fig 11, but usually without the enlargements at cand a. The spikes which confine the angle bars to the cross-ties, serve to counteract the creeping alluded to 3 or 4 lines below, and which the common splice does not prevent. Under the extremes of temperature in the United States, bar iron expands or contracts about 1 part in 916 ; or I inch in 76} feet ; consequently, a rail 30 ft long will vary J-~- inch ; and one 20 feet long fullv x inch. Beside this, the rails are very liable to move or creep bodily in the direction of the heaviest trade; and by this process also the joint-fastenings are exposed to additional strain and derangement. The patent Stop-Chair, Of Mr Jollll A. WilSOn, E, is said to prevent this. It consists simply of a piece of plate-iron, bent to fit to the fish-plate, and to the outer top part of the base of the rail. It is about 3 ins wide, by 7 ins long. Its lower end is held to the tie by two common rail-spikes ; and its upper end is bolted to the outside of a fish-plate by one of the bolts which confine the fish to the rail. Two chairs are used at each joint, one at each end of a fish. Made by the Wharton Switch Co, Phila. .\Vt. I ft> e;ich- All rails appear to l>eoome elongated very slightly at their ends by use ; and this renders a full allowance for contraction and expansion the more necessary. It i a remarRablo fact, not satisfactorily accounted for, that when lengths of from 100 yards, to some miles, of mils hove been perfectly welded, or riveted together tightly, and spiked to the ties as usual, no elongation or contraction by heat or cold could be detected. What is called the compensating fish-joint is in use on many roads. Its novelty consists in a peculiar cup-washer, enclosing a ring of India-rubber 2 inches diam. and ^ inch thick ; and placed under the nut of each of the four bolts. It, is claimed that this prevents the nuts from unscrewing themselves; and diminishes the jar and noise of passing trains. -5f Average price in Philada, in 1880, of wrought-iron chairs, about 4 to 5 cts per Ib. Common fish-plates the same. Bolts and nuts 6 to 7 cts, rough finish. t Price of common fish-plates. Phila, 1882. about 2^ cts per Ib. Angle-bar ones, about 3 cts. Bolts aud nuts 4 cts per Ib. THE I HILA IRON & STEEL Co, 939 North Delaware Avenue, make both kinds. 392 KAIL-JOINTS, AND CHAIRS. Each of the two fish-plates of each joint, is usually from 16 to 24 ins long, fcy % to % ins thick. Th weight of a complete joiut, with its four j^-iuch bolts, cup-washers, &c, from Itt to 24 Ibs. They an generally used without chairs. A new form of fish-plates, contrived by Mr. S, E. Pettier, is shown at Fig 4%. They are about 18 ins long. Near the bottom they are connected by two bolts c; and at top by four bolts v. These last have the billings lock-nut washer a; for which see page 375. This joint seems to be much more effective than the common tish, against both vertical and lateral force. It is of course a suspended joint ; and a quadrant-shaped piece is cut out from the ends of the vertical flanges c c, to prevent interference with the cross-ties. It has been tried on the Reading RR with good results, but not sufficient to displace the common fish-plates. Fig 5 was an early form of supported wrought-iror chair. It is still employed on some roads of light traffic. It is about 7 ins square, % thick, and weight 10K>8. The rolled-iron sleeve -chair, Fig 6, made by the Phoenix Iron Co, of Philadel- phia, and by some other establishments, is in extensive use. It is first rolled in long pieces ; and then sawed into such lengths as may be ordered. When supported, il is usually either 9 or 10 inches long; requiring 4 spikes ; and when suspended, about 2 ft long ; with spikes. It is chiefly used as a supported chair; and as such, is a favorite on many roads of moderate trade. Under heavy traffic, it is deficient in vertical strength, especially when suspended. The curved lips then fre- quently break off at the ends; and occasionally th chair breaks entirely across at the spike-holes. An objection to long chairs of this pattern, is the diffi- culty of sliding them upon the rails ; and the still greater one of sliding them off, when either a chair, or a rail, is to be removed. See Fig 8. Ibs per inch of length; so that ft>s. With rails 20, 24, or 30 feet The sleeve-chair, Fig 1 6, -weighs chairs of 9, 10, or 24 ins long, weigh 13%, 15, or long, there would in one mile of single-track road, be 528, 440, or 352 chairs. The 9-inch ones weigh 7128. 5940, or 4752 R>s ; the 10-inch, weigh 7920, 6600, or 5280 tts ; the 24- inch, weigh 19008, 158W, or 12672 fts.* But an addition should be made to these weights and costs, of say 5 per cent, or -^ part, or % mile-in every 5 miles, for turnouts, sidings, &c. Allowing fom half-pound spikes to a" chair, there are 2112, 1760, 1408 chair-spikes ; or 1056, 880, or 704 fts per mile. Figs 7 show Fisher's wrousrht-iroii rail-Joint, highly esteemed on the Lehigh Valley road ; where it is extensively used under a heavy traffic of coal, mer* * Price io Philada, 1880, 5 cts per tt. RAIL- JOINTS, AND CHAIRS. 393 chandise, and passengers. In this, the disadvantages before alluded to of the long sleeve-chair, are entirely got rid of. At A is shown a transverse section of the fas- tening, with the rail in its place ; while the other portions of Figs 7 show its details separately. It consists of a simple rolled chair c, 6 ins square, and % inch thick ; with two of its sides turned up % of an inch. In its base are 4 bolt-holes, through which are passed from below, two curved double screw-bolts, of round iron 1 inch dium ; one of which is shown at "D. Over the tops of these bolts are let down the two rolled pieces or bars 1 1, each as long as the chair, or t> ins; and shaped to fit the top of the base of the rail. The nuts n n bind firmly together these bars 1 1, the rail s and the chair c. To prevent the nuts n (see Fig B) from unscrewing themselves, a small rod s g, of % inch square iron, notched on the side next to the nut, is inserted between the nut and the rail; one end of it, s, being bent to clasp the nut. In Fig A, the ends g of these rods are shown by 2 small black squares. The whole is sus- pended between two cross-ties, 7 inches apart in the clear; on each of which the rail is simply held by two spikes, as on the other ties ; without any chair. The chairs c, when suspended, very seldom, if ever, break. Those of them first used on this road were but ^ inch thick, and were supported upon the ties; only one double bolt being used instead of two. They frequently broke across the bolt-holes. This excellent fastening was patented by Mr Mark Fisher, of the iron firm of Fisher & Norris, of Trenton, N Jersey. The same firm make also a longer chair, with 3 double bolts. Notwithstandingthe fact that the cross-ties are but from 1% to 8 ft long, (gauge 4 ft 8^,) the road is a very pleasant one to ride upon: there is but little jolting, or rattling at the joints ; and the ends of ' the rails appear to be as well protected as by any joint fastening we have seen. The track is well watched, and carefully kept in line : a precaution especially necessary in a road doing the heavy busi- ness of the Lehigh Valley R R. The success of this joint confirms our opinion that a great length of fastening, or of cross-tie, is not in itself essential to a good track. The weight of this fastening complete is about 13*4 Ibs. With rails 20, 24, or 30 ft long, there would in one mile of single-track road, be 528, 440, or 352 chairs, weighing 7128, 5940, or 4752 Ibs; or, allowing 5 per cent for turnouts and sidings, say 7484, 6237, or 4990 ft>s. A modification of the sleeTe- chair. Fig 6, has been introduced upon the Hudson River R R, involving, to some extent, the same principle as that on the Lehigh Val- ley. It is shown at Fig 8. As but one bar Is here used, and as it is placed on the outside of the rail, its nuts are not in danger of being struck by the flanges of the wheel when a low rail is employed; or when the wheel tires become worn, thus bringing the flanges lower down. It is probable that a suspended chair of this kind, not more than 7 inches long, and with its lip s considerably thicker than they are usually made, would equal the Lehigh Valley one in efficiency. k -j Figs 9 show a remarkable snspended fastening-, called the King-- Joint: highly approved of at one time on the Camden & Amboy road, on which it was employed for many years, under a heavy traffic; to the almost entire exclu- sion of others, except for experimental comparison. K68 It was invented in 1851, by Edwin A Stevens, President of the road. The chief engineer, Col Cook, informed the writer that after several years' trial of a great variety of fastenings, he gave a decided preference to this. It has, however, one very serious defect: namely, that when, in consequence of a bad foundation, the cross-ties play too freely up and down, the ends of the rails frequently split off; either at top, as shown by the line I ; or at bottom, as shown by k. We have never learned, how- ever, that this was productive of accident to trains. When the break occurs above the slot v v the jolting of course gives notice of the fact, and a new rail is put in ; but when below the slot the ring and its wedges continue to uphold the broken piece ; at times probably for days before the frae- tare is observed. Nearly all the fractures are below the slet. 394 RAIL- JOINTS, AND CHAIRS. This fastening consists of a simple welded triangular ring tea, (in the end view ; or m in the aid* Tiew,) % an inch thick, and 3>4 ins wide. This ring passes through a slot vv, (see middle fig,) 4 tag long, cut into the adjacent rail-ends. Two cast-iron wedges w w, 6 to 8 ins long, of a shape to fit the ring and the rail, are inserted between them; and a thinner one S8, of plate iron, below the rail. The first are cast around a cylindrical rod of rolled iron, (w of the end view, ii of the side view,) about % inch diam; aud a little longer than the wedges ; for increasing their strength, and for pre- venting them from falling out from the chair in case they should break, which they sometimes do. The joint is suspended between two cross- ties, 1 ft apart in the clear. The writer examined it carefully for many consecutive years ; and so far as regards protecting the ends of the rails from wear, and freedom from jolting, it certainly appears to him to be fully as ef- fective as any other. He has seen joints laid alternately with it; with heavy cast-iron supported chairs; and with long splices somewhat similar to the one next to be described, but extending over 3 ties. With the exception of the breaking alluded to, the ring appeared to be decidedly superior to the supported chair ; and by no means inferior to the long splice. We have also repeatedly compared it with the heavy rolled joint-fastening Pig 13, which is 5 feet long, and rests on three cross-ties. Under similar conditions of road-bed, whether good or bad, we have been unable to perceive that the ring-joint (when the ties are but a foot apart in the clear) yielded, under heavv trams, any more than it. The weight of a ring itself i? 5> Ibs I tne three wedges about 1% Ibs ; total 13 fts : or nearly the same as the Lehigh Valley fastening, Fig 7. The gauge of the C & A road is 4 ft, 10 ins ; and th cross- ties average about 8^ ft long. Without pretending to advocate this joint in its present shape, we have thought proper to describ* it, as furnishing useful hints on this important subject. It was the first that led us to doubt whether some long fastenings did not furnish the elements of their own destruction, by the long leverage which they afforded for bending and breaking themselves to pieces, when the ties are badly supported. Since the death of Col Cook, a gradual substitution of long combined wood and iron fastenings, in place of the ring-joint, has been commenced ; and is applied wherever the latter is found to have frac- tured a rail. Figs 10 represent the combined suspended joint-fastening introduced upon the Philada & Reading railroad by J. Duttoii Steele, C E. The very heavy tonnage of this road, exceeding perhaps that of any other in existence, demandt particular attention to the joints; and Mr Steele's combination of Trimble's long wooden splice, with a long wrought-iron fish-splice, and a longer rolled-iron chair under the whole, certainly proved itself, for many years, superior to any other of the numerous cast and wrought iron chairs previously tried on that road. The joint is suspended between 2 cross- ties, placed 1 ft apart in the clear. B is a block (known as Trimble's splice) of oak. 3 by 3 ins. and 3 ft long, dressed on one side to fit the out side of the rail : and c is a rolled fish, 17 ins long, (shown more in detail at F.) placed on the insidt of the rail. This fish and the oak block are bolted together, through the rail, by two %-inch screw- bolts a a, 13 inches apart. Under the rail is a rolled chair d. 2 ft, 8 ins long. 5 ins wide, and ^ inch thick ; turned up ^ inch along each of its 2 sides, and fastened to the 2 wooden cross-ties by 4 hook- headed spikes, ^ inch square, by 5V$ ins long. The heads of the two screw-bolts are made somewhat oblong, (about % inch by !>,) for fitting into the groove nn. seen along one side of the fish ; so as to prevent the tendency of the bolts to revolve under the action of the trains, and thus unscrew the nut at the other end. The nuts, however, unscrew themselves, notwithstanding this precaution. We consider 13 ins to be too far apart for the two screw-bolts, inasmuch as it allows the fish in time to become bulged out from the rail quite perceptibly, between the spikes, under the downward pressure on the ends of the rails. This impairs its efficiency. Four bolts would be better. The strain on the screw-bolts is great, both vert and nor; and it becomes greater as the wooden blocks in time lose (as they do) their close fit to the sides of the rails. The blocks then cease to act in perfect unison with the other parts of the fastening, in sustaining passing loads; and when the track is not kept in good order, the various parts mav plainly be seen to yield and move in various directions, independently of each other. The bolt-nuts then loosen : and the fish pieces, long chairs, and long bolts, become bent: and sometimes split, or break entirely. The long wooden blocks B crush and decay soonest near where their tops are in contact with the rail. They, however, have an average life of 6 to 8 years, upon roads kept in tolerable order. These remarks are more or less applicable to all joint-fastenines with long splices, when the ties are allowed to become unstable. With the care taken to prevent this on the Reading mad. Mr Steele's fastening haa sustained the enormous traffic of the line very well for many years. It will, however, be supplanted by better ones more recently introduced. The iron in one of these joints, including bolt* and spikes, weiehs nhnnt 32 ft>< : and with rails ot 20, 24, or 80 ft long, there would be 528,^40, or 352 joints; or 16896, 14080, or 11264 D)8, per mile ol single-track road. RAIL- JOINTS, AND CHAIRS. 395 Fig 11 shows the Fritz and Sayre Splice Plate now (1883) used on the Lehigh Valley R. R. of very heavy traffic ; together with a cross section of the 67 Ibs to a yard steel rail of that road, as designed by Robert H. Sayre. Esq., Chief Eng and Supt; and the standard cast-iron car-wheel tread of the New York Central ; all carefully drawn to scale, and one-third of actual size. These forms of rail and splice are the result of careful study, and each detail has been modified from time to time as experience dictated, until now they are probably the most perfect in this country. Mr. Sayre places the stems of the two plates much farther apart than usual, thus giving the joint greater lateral strength ; at the same time adding to its vertical strength by the support given to the entire lower side of the rail-head by the upper en- largement c ; while the lower one a secures a full bearing on the foot of the rail. The oblong head 6 of the bolts prevents them from unscrewing; and the lock-nut washer v (Shaw's, alias the Verona, p 375) is intended to do the same with the nut. The weight of the two iron splice-plates (each 2 ft long) is 40 Ibs; the 4 bolts (5 ins long, % diam) with their nuts and washers v, 4 Ibs; the 4 spikes (5% ins long, - r 9 g square, two to each plate, their heads shown at o o) 2 Ibs. Total 46 Ibs per joint. The drilled bolt holes in the stem of the rail are 1 inch diam, to allow the rails to contract and expand. These splices are made by the Bethlehem Iron Co, John Fritz, Supt, Bethlehem, Lehigh Co, Penna. 396 RAIL-JOINTS, AND CHAIRS. Fig 13 is Charles E. Smith, Esqr's, inverted T joint-splice, or fastening, of rolled iron, for the U rail of the Camden & Atlantic railroad. It is 5 ft long, and rests on 3 cross- ties, one of which is under the P. 49iPi rail-joint. Its base is 4% ins wide, by ^ inch thick. The vert web is ICTlJ - ! 2 ins high, and 1 inch thick where it joins the base. This joint-piece is riveted loosely to the rails by 8 rivets in pairs ; two pairs being between each two ties. Eight hook headed spikes, 5% inches long, by % inch square, in pairs, (2 pairs on the center tie.) confine both rails and s'plices to the cross-ties. This splice keeps the rails in position very well ; but it would be easier upon the ends of the rails if it rested on four ties, so as to suspend the rail-joint between two of them. The writer has been unable to perceive that the deflection under trains is any less with this splice, than with such as Figs 7, 9, &c, when the ties ara not firm. The weight of iron in one splice, including spikes, is about 70 Bs. With rails 20, 24, or 30 ft long, with 5 per ct allowance for turnouts, &c, there would be 36960, 30800, or 24640 8>s per mile of single track; consequently it is an expensive fastening. The great number of spikes driven into three ties prevents the rails from creeping. A SIMILAR JOINT, but shorter, and inverted, and suspended between ties, has been suggested for edge rails; see Fig 17. Fig 14 is a joint- fastening proposed many years since by Alex. W. Rae, C E, of Penna. It certainly possesses merit. With a length of about 16 ins, and with the addi- tion of a thin wedge, as well as of 2 more bolts, all below the rail, in the positions shown by the dotted lines, it would probably constitute as effective a suspended joint-fastening as has yet been tried. There would be very little strain on the bolts. If 16 inches long, and }4 inch thick, the weight per joint would be about 50 fts, in- cluding 6 bolts, and a wedge beneath the base of the rail. As in all other joints, some device should be used to prevent the nuts from unscrewing themselves by the jolting of passing trains. See Star Washer, Fig 17 ; also, Billings washer, p 375. The star washer being of very thin iron, becomes in time so brittle from rust as to require renewal. The Billings washer occasionally breaks under great strain. A perfect lock-nut-washer of simple construction is still u desideratum. Pig 15 was also one of the numerous joint-fastenings suggested at an early day ; bui. like the foregoing, it never came into use. In lengths of about 8 ins, it would probably make an efficient fastening; especially with the addition of a broad thin wedge between the bottom of the rail and the foot of the chair. This would dimin- ish the difficulty of sliding the chairs on or off of the rail : and would thus make it easy to employ larger ones; besides insuring a firm bearing for the base of the rail upon the fastening. A fastening of tempered steel, like Fig 15, but without any bolt and key, has of late years been us^d on some English roads. Its elasticity permits it to be easily slipped upon the rails. Fig 16 is the Phcenix Iron Co's suspended rail-joint, devised and patented by Sain'l J. Reeves, Esqr, 'President of the Co, of Philada. It consists of three pieces, a, b, c, of rolled iron, each 14 ins long, which are confined to the rail by two bolts ; one of which is shown at nn. The weight, including bolts and nuts, is 35 Ibs. To prevent the six-sided nuts t from loosening themselves, a small strip t t, of thin sheet iron, fits around them; and its ends are bent square over the top and bottom of the piece a: or a star washer is used. . This fastening has a "stop," for preventing the ends of the rails from coming toge- ther; but it is not only ineffective, but decidedly injurious to the e tire fastening. Tig-15 TURNOUTS. 397 TURNOUTS, the switcti-raiis, lonn ine luicrum or center auoui wnicn tuey move; ana are called their heels ; and the ends o and s, at which the motion is greatest, are their toes.f The dist x o reqd for this motion of the toe of each switch- rail, is called its throw, see Fig 5, and x g o is the switch angle. The throw must be equal at least to the width a x of the top of the rail, in addition to a width o , sufficient to allow the flanges of the wheels on the track A B, to pass along readily between the rails b and w. The tops of rails are generally between 2 and 2% ins wide; while about 1% to 2 ins will usually suffice for the flanges. The throw, x o, however, is commonly about 5 ins ; that is. the toes o and s are moved 5 ins from their original position (shown by the dotted lines), when the train is intended to leave the track A B, and to pass along the turnout, to the track C D. The motion is given by means of a switch-lever, which will be described by-and-by. The entire triangular part P, Fig 2, is the tongue of the frog; and its sharp end is the point. Frog-making has become one of the specialties of the day. In ordering them it is not necessary to furnish more than the frog-angle, or num- ber; and an exact cross-section of the rail used on the road. The same frog will answer for turning out either to the right hand or to the left, except the spring- rail frog described farther on. * As the business on the Pennsylvania Central increased, it was found expedient to increase the length of the sidings to one mile. T This applies to the. common or stub switch, Fips 1 and H. In the Wharton, Lorenz and some others, the positions of heel and toe are 'he reverse of this. See rages 407, 408. 398 TURNOUTS. Fig '2 is a plan ; Figs 3 and 4 side views, and Fig Z a cross-section of a frog as frequently made. Tae entire frog was formerly made of cast iron ; hardened by chilling; so as better to resist the action of passing wheels; hut even with this precaution they wore out so much more rapidly than the rails, that the wings and tongue are now capped with the best steel. It is used in thicknesses of from ^ to 1 inch or more: and i.s nrmly bolted down to the cast-iron portion; besides being other wise secured to it in the best makes. It is not necessary to steel the entire length of the wings ; a very trifling economy is gained by leaving their ends m and c unprotected; inasmuch as the wheels do not run upon those parts. All the tongue P, however, should be steeled. The projections t t are merely for bolting the frog down to ths wooden cross-ties. The wings and tongue must evidently be raised above the bottom plate of the frog sufficiently to prevent the flanges of wheels from touching this last. About \% to 2 inches will suffice, as shown at Fig Z ; which is a transverse section of Fig 2, taken across t i w t. The channel is called the MOUTH of the frog at a, Figs 2 and 8; and its THROAT, at the narrowest part w i. Fig 2 ; or z x, Fig 8. That part of the tongue which is back of , Fig 8, or between u and g, ia called its HEEL. Although the frog forms a part of the turnout curve, still its shortness warrants us in making it straight from j to i ; and from s to t, Fig. 8. Fig 'A is a side view of a frog of uniform depth. This depth may be about 3J^ ins ; namely, about 2 ins for the wings and tongue; and 1& for the base plate. In Fig 4, which is also a side view, the projections 1 1, the number of which will vary with the length of the frog, are cast in one piece with the frog ; but are entirely below it. The end one at P. passes also under the end of the adjoining rails ; thus forming a chair for them, as well as for the frog. At the end R is shown a mode of proceeding when the rails are high ; and are to rest upon the ends of the base of the frog, as in the fig. Whichever mode may be employed, it will of course be used at both ends of the frog. Such details are pretty much a matter of whim ; and vary with the notions of the designers. Fig 6 is a frog, as often made, by merely bending two pieces, / g and h t, of ordinary rails, for forming the wings ; and by cutting and bolting together two other pieces, c and c, for the tongue and point. Frogs are now regularly manu- factured on the principle of Fig 6,* and are displacing to a great extent the old cast-iron ones. They are made from 8 to 10 feet long, according to the frog-number, (see Art 2.) At their ends they are drilled for fish-plates, by which they are joined to the track-rails. They thus form, as it were, part of the track, and are therefore less liable than the ordinary frogs to wear loose under the passage of trains. The wing-rails and the point are secured at the proper distance apart by iron pieces, which fit closely to the rails, and are inserted between the wing-rails and the point. They extend beyond, and enclose, the point, thus bracing and staying it. They are, of course, low enough to clear the flanges of the wheels. The parts are bolted to- gether to form one solid frog. To explain the nse of the guide-rails, c C 9 a ft. Fig 1. Suppose wheels to be rolling from A toward B, Fig 1, on the main track : the switch rails being in the dotted positions. On arriving opposite the frog, some irregularity of motion might cause the flanges of the wheels running along the rails n I, to press hard against said rails. Consequently, after passing the throat w i (Fig 2), they would press against the wing i c. ; and passing between c and P, leave the track, or strike the sharp end of P ; breaking it ; and endangering the train. To prevent this, the guide rail c c is placed so near the rails g /t, Fig 1, (generally about \% to 2 ins.) that the flanges at those rails, while passing between them and the guide-rail, not only prevent the flanges at the opposite rails from pressing against the wing t c; but guide them safely along their proper channel from i to m, Fig 2, without striking against the sharp end of the point." * The Pennsylvania Steel Co., office 208 S. 4th St., Phila., manufacture patterns, and made entirely ef steel rails. Prices in 1880 from $35 to $50. such frogs, of different TURNOUTS. 399 In like manner, if the switch-rails be in the positions g o, n s. (or switched,) and if wheels be rolling along the turnout, Fig 1, from A toward D, when they arrive at the frog, the centrifugal force (being of the frog: thus rendering liable the same kind of accident a* in the preceding case. This is pre- vented iu the same manner as before, by the guide-rail a a ; which keeps the flanges in their proper channel from w to c, Fig 2. The narrow flange-way between the guide-rail c c, Fig 1, and the rail b h, need not extend farther than from the end a, Fig 2, of the frog, to about one foot toward g, from the sharp end of the point. In a distance of at least about 2 ft more at each of its ends, the guide-rail should flare out to about 3 ins from the rail b h ; so as to guide the flanges into the narrower part of the flange- way. The same with a a. Guide rails should be very firmly confined to their wooden cross-ties ; inasmuch as they have to resist a strong side pressure. This is usually done by bolting against them two or more stout blocks n n, Fig 5}j ins wide at the head. LENGTH. Ft. Ins. 2 4 2 11 3 6 No. OF FKOG. In loner frogs especially, this reduction is advisable, not only on the score of economy ; but of ease of handling ; and less liability to be broken if the foundation becomes infirm. Art. 4. The laying-out of Turnouts. The words heel and toe are used in this article with reference to the common or stub sivitch, Fig 1, in which the heels are at g and n ; and the toes at o and s. In the Wharton. Lorenz, and some other switches, the positions of heel and toe will be seen to be the reverse of this. The formulas given in our former editions for finding frog dist, rad of turnout, etc, were based upon the old practice of regarding the straight switch-rails go,ns, Fig 1, as forming a tangent to the turnout curve, which last was considered as be- ginning at the toes, o and s, of the switch-rails. The modern practice is to curve the switch-rails so as to form a part of the turnout curve; the latter being supposed to begin at the htels, g and n, of the switch. This view of the case admits of simpler formulas. In each of the Figs 10, 11, and 12, w p z represents the main track. The frog" distance, j?/, is a straight line drawn from the theoretical point of frog, /, to the heel, p, of that switch-rail which, when opened, forms the inner rail of the turn- out. Formerly when the turnout curve was taken as starting at the toe of the switch, the frog dist was a straight line from the theoretical point of frog to the toe, o, Fig 1. of the outer switch-rail, g n, when opened. As already remarked frogs are usually made of Nos. 4 to 12; sometimes with half numbers ; and the turnout radii, etc., are made to conform to them. Scrupulous accuracy is not necessary in these matters. Thus, a deviation, either way, of say 3 per cent in the length of the turnout radius from that given by the table or the formulas, will be almost inappreciable. So too, if a frog number should be used, intermediate of those in the first column of the table, the other dimen- sions may be found, approximately enough, by using quantities similarly interme- diate. A rail almost always has to be cut in two in. order to fill up the said dist; and the exact length of the piece can be found by actual measurement at the time of cutting it. When the turnout is to be traversed by passenger trains the rad should not be less than about 800 feet. Rein. When the turnout leaves a straight track, as in Fig 10, the frog angle is equal to the central angle f c o. When the main track is curved, and the turnout curves in the opposite direction (Fig 11), it is equal to the sum (v f n) of the central angles/c o,fn o ; and when the two curve in the same direction (Fig 32), it is equal to the diff (n f c} of the central angles/c o, fn o. Art. 5. To lay out a turnout, p x. Fig 1O, from a straight track, p z. From the column of radii in the table page 402, select one, c o, suit- 402 TURNOUTS. able for the turnout; together with the corresponding frog number, frog distp/and switch length. Place the frog so that the main-line side of its tongue shall be at / z, precisely in line with the inner edge of the rail, wz: and its theoretical point, J\ at the tabular frog dist pj, from the starting-point p. Stretch a string from q (opposite p) to/; and from it lay off the three ordi- natrs from the table; thus finding three points (in addition to g and j) in the outer curve. Do not, how- ever, drive stakes at these points ; but as each of them is found, measure off from it, inward, half the gauge of the track ; and there drive stakes. Do the same from q and/. The five stakes will all then be in the dotted center line of the turnout, Fig 30; and will q serve as guides to the work, without being liable to be displaced. Remark. The dimensions in the Table below are found by the following formulas; the main track being straight. = Gauge -r- Frog dist. Tangent of hall* frog angle Frog No Or, Frog No Radius c o Or, Radius co Or, Radius co Frog dist p f... Or, Frog dist p f... Or, Frog dist p f... Middle ord Each side ord Switch Length approx enough {f Radius c o -f- Twice the gauge, lalf the cotangent of half the frog .angle. = Twice the gauge X Square of frog number. = (Frog distp / -f- Sine of frog angle) half the gauge. = (Gauge H- versed sine of frog angle) half the gauge. = Frog number X Twice the gauge. = Gauge p q -~- Tangent of half the frog angle. = (Rad c o + half the gauge) X Sine of frog angle. - % gauge, approx enough. " mid ord = -fa (or .188 of the) gauge, approx enough. V Throw in ft X 10000 Tangential diat for chords of 100 ft. for rad c o of turnout curve. See table p 416. TABLE OF TURNOUTS FROM A STRAIGHT TRACK. Fig 10. Gauge 4 ft 8^ ins. Throw of switch 5 ins. For any other gauge, the frog angle for any given frog number remains the same as in the table. The other items may be taken, approx enough, to vary di- rectly as the gauge. Number & Turnout Radius DeflAngof Turnout Curve Frog Dist Pf Middle Ordlnato Side Ords Stub Switch Length o / Feet. o r Feet. Feet. Feet. Feet. 12 4 46 1356 4 14 113.0 1.177 .883 34 UH 4 58 1245 4 36 108.3 1.177 .883 32 11 5 12 1139 5 2 103.6 1.177 .883 31 H>H 5 28 1038 5 31 98.9 1.177 .883 29 10 5 44 942 6 5 94.2 1.177 .883 28 ^A 6 2 850 6 45 89.5 1.177 .883 27 9 6 22 763 7 31 84.7 1.177 .883 25 &A 6 44 680 8 26 80.0 1.177 .883 24 8 7 10 603 9 31 75.3 1.177 .883 22 ^A 7 38 530 10 50 70.6 1.177 .883 21 7 8 10 461 12 27 65.9 1.177 .883 20 &A 8 48 398 14 26 61.2 1.177 .883 18 6 9 32 339 16 58 56.5 1.177 .883 17 &A 10 24 285 20 13 51.8 1.177 .883 15 5 11 26 235 24 32 47.1 1.177 .883 14 V/2 12 40 191 30 24 42.4 1.177 .883 13 4 14 14 151 38 46 37.7 1.177 .883 11 Remark. The switch lengths in the Table merely denote the shortest length of Stub switch that will at the same time form part of the turnout curve, and give 5 ins throw. Pointed, or split- rail switches, like the Lorenz, Ac, require only half this throw ; still, to suit the curve, they should be as long as the Stub, but in practice all kinds seem frequently to be made much shorter than the table requires, thereby sharpening the beginning of the curve. TURNOUTS. 403 Art. 6. To lay out a turnout from a curved main track. There are two Cases : Case 1, Fig 11; when the two curves deflect in op- posite directions. Case 2, Fig 12; when the two curves deflect in the same direction. Exact rules for the di- mensions in these cases would be complicated ; and, moreover, they are not neces- sary, inasmuch as the follow- ing method, using the table in Art 5, is sufficiently close wherever the greater of the two radii is not less than say about 800 feet. For shorter radii (and indeed for all cases) the method by means of a drawing to a large scale (see Art 7) will be found useful. Having determined approx upon a radius for the turnout curve, take from the table p 416 its corresponding deflection angle, and that for the main curve. In Case 1, find the sum of these two angles. In Case 2, find their difference. In the table p 402 find the deflection angle (not the frog angle) nearest to the sum or diff just found, and greater or less than it according as it is preferred to have the turn- out radius c o Jess or greater than the approx one selected. Thus, suppose it is required to lay out a turnout, q a;, Fig 11, Case 1, rad of main curve to be 2865 ft, and that selected approx for the turnout 716.8 ft. The def angles will be, respec- tively, 2 and 8, and their sum 10. The nearest def angles in table p 402 are 10 50' and 9 3V. Here 10 50' will give a turnout rad c o a trifle shorter, and 9 31' one somewhat longer than 716.8ft. From the tabular def angle thus selected; in Case 1, subtract the def angle of main curve; or in Case 2, add them to- gether. The resulting diff or sum is the def angle of the turnout curve; and table p 416 will give its rad c o. Thus, in the above case, tabular angle 10 50' def angle of main curve 2 = 8 50' def angle of turnout curve ; the corresponding rad (c o) of which, by table p. 416 is 649 feet ; or, if 9 31< instead of 10 50' be taken as the tabular angle, we have 9 31' 2 = 7 31', rad c o say 763 ft. With the same data, the operation in Case 2 would be. Turnout defl angle 8 defl angle of main curve 2= 6. Nearest tabular defl angles 6 5' and 5 31'. Here either 6 5' -f 2 = 8 5', rad say 709 ft ; or 50 31 / + 2 o = 7 31', rad say 763 ft. The frog number and switch length in the table, opposite the defl angle thus selected, are the proper ones for the turnout. The frog dist (p f ) is found thus : In Case 1 add to the tabul n ular frog dist half an inch per 100 ft for each degree of defl angle of the easier of the two curves: In Case 2 subtract it. Thus, suppose the frog number found as above, to be 10. The tabular frog dist is 94.2 ft. Let main curve be one of 2. Then in Case 1 add twice % inch, (or 1 inch) per 100 ft, to 94 2 feet ; and in Case 2 sub- tract a like amount. In this example we have (near enough), corrected frog dist for Case 1, 94.3 ft ; and for Case 2, 94.1 ft. TJ 49 I../ / Place the frog with the main-line side of riUl, U// j I it s tongue at/*, in line with the inner edge of the frog-rail, p z, of the main line, and with its theoretical point/ at the distp /(found as above) from the heel,/), of the inner switch rail. Stretch a string from / to the heel q, of the outer switch rail. Measure tho dist, qf, divide it into four equal parts, and lay off three ordinates, found thus: Middle ord = (Square of half qf) -f- twice the rad of turnout curve. Each side ord = three-fourths of middle ord. These three ords, and the points q and /, give us 5 points of the outer rail of the turnout curve ; and from these we measure, inward, half the gauge, and drive 6 center guide stakes, as in Art 5. 404 TURNOUTS. Art. 7. To find frog: divided by 4 4 will give the No of the frog. With care, and a little ingenuity, the %13/a X young student will be able, by similar processes, to solve graphically any turnout case that may pre- sent itself. The method by a drawing has great advantages over the tedious and complicated calcu- lations which otherwise become necessary in cases where curved and straight tracks intersect each other in various directions. The drawing serves as a check against serious errors, which would be detected at once by eye. None of the graphical measurements will be strictly accurate; but with care, none of the errors need be of practical importance. The ordinates for bending rails so as to suit turnout curves can be found from the table, p 418. Art. 8. An experienced track-layer, with a __<^^ bored to admit the cylindrical pin a o, about 2 ins diam ; with a square head s, for the handle or lever h h, about 3 feet long. A wrought collar e e is fastened to 9 o by a pin through both, and turns with s o. The nor piece o t is fast to s o : and near its end t is fastened the short vert round pin i v, about 1 inch diam, which passes loosely through an opening in the end c of the long rod ttn, Fig 14, which opens and shuts the switch. A small pin passes through iujust below c, to prevent the latter from falling. For a throw of 5 ins, the nor dist between the centers of s o and i v must plainly be 2!^ ins, or half the throw. In the fig, o i, Ac, is in the position of only half open. When the switch is a three- throw one, this dist between centers must be five ins: and the length of the piece o i, and the length 1 1 of the stand, must be increased to suit. In a 3 throw, the position of o i c as seen in the fig, is after o i has described a quadrant, thus opening one turnout and closing another. In this position it must be held for the time by a bolt passing through the top of the standard, and through o t near the end i ; otherwise a passing train would be apt to move it, thus partially closing the turnout, and throwing the train off the track. By turn- ing o i around through another quadrant, the remaining third track becomes passable. In a 2 throw, o i is never left in the position shown in Fig 18, butdescribes an entire semicircle each time the turn- out is opened or closed, (t therefore requires no bolt for confining it temporarily. The breadth of the stand t t does not show in the fig : it is usually 6 ins at every part. Art. 1O. Whurton'B patent safety switch. Fig 19 shows the switches set for the main track, and Fig 2'J for the turnout. A and B are the main-track rails. They are continuous, and Fig-18 mcn.es MONKEY SWITCH WHARTON. * mcam 2 P >y a -"^PZ 1 E 9 c z. ^n %20^ K i? V , Itt D 10 n piked to the ties throughout, and are not moved or broken in the working <>f the switch. K and D r the switch-rails, with thuir heti* at n (the place of (he toes in the stub twitch, Art 9.) The 408 TURNOUTS. switch-rajls, like others, are connected together bj clamp-bars g, and slide on the ties. The inntr switch-rail D (called the elevating rail) is blunt ended. At its toe, its top is level with that of A ; but from that point it rises, until at ra, about 4 feet from the toe, its top is about 2} inches (more recently 1% inches) higher than that of A. This enables the flange of the wheel K, in passing to or from the turnout, to pass clear over the rail A. D retains this elevation from m to n. Bevond n it falls gently, until its top is again at the track level. The unbroken main rails, and this carrying of the flanges over the main rail, are special features of the Whartou. The recent reduction from '2% to 1% inches in the rise was made in order to lessen the jar which it gave to trains using the turnout at more than quite moderate speed. The tread of the outer wheel L, in passing to or from the turn- out, runs upon the outer flange (the upper one in the Figs) of the outer switch rail, or grooved rail E. This flange is pointed, so that when E is placed as in Fig '20, its point x fits close up against the stem, and under the head, of the main rail B, so that the wheel L, moving in either direction, can Sass it without jar. The fixed guard-rail tends to draw L away from B, and thus to avoid any anger of its striking the point x. Rail E is elevated like D, to prevent the rocking of cars, etc., which would result if D alone was elevated. Its raistd inner edge z z prevents the flange of wheel K from striking rail A in going up or down the incline at the toe of D in Fig 20. The switch is operated by cranks v w, and a tumbling lever i. This lever, when the switch is set either way, lies so as to bring the orank v a little below its center, so that any lateral pressure on the switch-rails, instead of misplacing the switch, only presses the lever closer to the ground. The weight on the end of the lever tends to bring it clear down to its proper position. Provision against accident In case of misplacement of the switch. Case 1. A train moving in a directiou opposite to the arrow, must ot course go as the switch is set, whether right or wrong. On this account it is usual (when possible) on double-track roads to so place switches that trains not intended for the turnout will approach them only in the directiou of the arrow. For single-track roads the Wharton is so arranged that it will not remain set for the turnout unless actually held so by the switchman. Case 2. If a train comes from the turnout while the switch has been wrongly left set for the main line. Fig 19, the wheels run upon the fixed "safety-castings" H Y, and the flange o o, on H, guides them safely on to the main-track rails A B. Case 8. A train moving along the main track in the direction of the arrow while the switch is wronglv set for the turnout, Fig 20. The cranks v w are so arranged (see Figs) that when the switches are pushed up against the main-track rails, Fig 20, the end c of the movable guard-rail is moved in the opposite direction, and brought close up to the main rail A. In this case the flange of the first wheel K pushes aside c, which, pulling the rod attached to it, throws the crank w (and through it the entire switch) into the proper position, Fig 19, for the main track. A -*- g ^i n D P 9 F,. 21 ->, S r v n E I LORENZ. -, ,, B A,. s """TC n D P 9 Eg. 22 n E Art. 11. The Lorenz Safety Switch* of Wm. Lorenz, Esq., Chief Engineer Reading R. R., in use ou many ot our main roads, is a " split-rail "or " point" one. The movable parts of the point- rails, on main lines, are usually from 15 to 18 feet long. Fig 21 shows this switch set for the main line, Fig 22 for the turnout. The rails A A, B B, are continuous, and spiked to the ties. The switches or point-rails, D E, are spiked to the ties for from 4 to 8 feet back from their heels, n, which, as in the Wharton, are in the places occupied by the toes in the common blunt-ended or stub switch. The point-rail B is so arranged as to form a part of the turnout curve in Fig 22. The switches are oper- ated by any ordinary lever, see p. 406. Short guard-rails, a a, are sometimes added to prevent wheels from running against the points of the switches. Provision against accident In case of misplacement. The remarks on the Wharton switch uu'der Case 1, Art. 10, apply also to the Lorenz on double-track roads. If a train moves in the direction of the arrow, in either Fig. when the switches are misplaced, the wheel flanges push aside the point-rails, D and E, and thus force a passage through the switch. A spring, of rubber or coiled steel, placed on the switch-rod, sometimes at p, sometimes outside of the track, permits this motion of the points ; and after a wheel has thus passed through, the spring returns the rails to their former places. X- Manufactured by the Pennsylvania Steol Co., office 208 S. 4th St., Phila. Price in 1882 of a switch with point rails 18 feet long $125, including rails A and B and guard-rails s s, but exclusive of switch-stand and putting in place. In ordering, give gauge of track, a full-sized section of rail used, directions for drilling fishing-holes, and number of frog. Also state whether the turnout ia right-hand or left, and whether the main line is straight or curved. If curved, state radius, and whether in same direction as turnout (Fig 12), or opposite (Fig. 11). RAILROADS. 409 Platform weigh-scales. Since the making of these is a specialty, we shall not describe their construction. Large ones from 30 to 150 ft long, for weighing from 30 to 150 tons of loaded cars at once, cost at the shop from $25 to $30 per ft of length. The preparation of the pit under the track, with its lining, foundations, &c, and putting the scales into place, will cost about 35 to 40 per ct in addition. Superior ones are made by Messrs Riehle, 9th St above Master, Philada, and by Fairbanks & Co, St Johnsbury, Vermont ; office, 715 Chestnut St, Philada. Testing 1 machines for either tension or compression, up to some hundreds of tons, are also specialties of Messrs Riehle. KAILKOADS. The total annual expenses, on U States railroads, usually range from 4a to 75 per cent of the receipts; the mean being 60 per cent. Few fall below the lower limit ; while few exceed the higher one. The average of all the railroads in England and Wales, previous to 1865, was about 48 per ct ; in 1866, they were 48.8; and in 1867, 60.6 per ct. Table of average per centages of the various general items which constitute the annual expenses of railroads. ITEMS. Approximate Average of U. S. Roads. Average English Re of ads. Mctintencmcp of way cind worJcs* Per ct. 25 Per ct. 19 30f 28 j?t>ctiVs and, rcnevxils of COLTS % 10 9 30 28 (rcneral expenses II 5 6/<2 Damages to persons and goods ^A 8 AfUttJ, .fjwr i , 100 100 In 1869, the proportions which the several items of expense on the Penna Central, between Phila and Pittsburg, bore to the total expense, were very nearly as follows: Conducting transportation, 28.5 per ct; motive power, 30.4; maintaining roadway, 27.3; maintaining cars, 12; general ex- penses, 1.8; total, 100. Each of these items is, however, subject to great variation, not only on diff roads, but on the same road, from year to year. A road with many bridges, deep cuts, high embkts, Ac. to keep in repair, will have heavier maintenance of way than one which has but few ; and this item may be but small one year, and twice as great the next. Fuel may be cheap on one road, and dear on another: thus materially affecting the item of motive power. And so with the other items. Sometimes mainte- nance of way exceeds motive power and cars together; at others, conducting transportation is fully half the total expense. Tiie gross receipts per mile of road in 1868, according to Poor's Rail- road Manual, were, in Massachusetts, $15400; New York, $i;>142; Pennsylvania, $13900; average of the whole United States, about $10000. And the earnings from freight averaged about 2% times that from passengers. The tonnage on the N York roads was 3625 tons per mile ; Massachusetts, 5438 tons; Pennsylvania, 8000 tons; average of U S, about 2500 tons. The total annual expenses on railroads in the United States usually range between 65 and 130 cents per train mile: that is, per mile actually run by trains. Also, between 1 and 2 cents per ton of freight, and per passenger carried one mile. When a road does a very large business, and of such a character that the trains may be heavy, and the cars'full, (as in coal-carrying roads,) the ex- pense per train mile becomes large; but that per ton or passenger, small ; and vice versa, although on coal roads half the train miles are with empty cars. * Including bridges, depots, stations, and other buildings, roadway, superstructure, fences, &c. t Of this 30 per ct of total expenses, about 1 1 (7 to 14) is for fuel ; about 7 (5 to 10) for repairs of engines; and 12 for pay of engine drivers, firemen, cleaners, oil, waste, &c ; as averages of a great number of roads of every character as to length, traffic, &c. Fuel range from 5 to 12 cts per train- mile ; usually 6 to 9 cts. The repairs of engines from 6 to 14 cts : usually about 8 to 12. + The annual repairs of a large 8-whee! passenger car, ranges between about $300 and $000: mail and express cars, from $150 to $300; freight cars, from $75 to $150; coal cars, from $20 to $30, (4 wheels.) Chiefly wages of clerks, ticket-sellers, conductors, brakemen, bridge and switch-tenders, tele- graphers, porters, weighers, &c. Ac; lighting and warming of ears, depots, stations, Ac. II Salaries of the higher officers of the Company ; law, stationery, advertising, office rents, &c, &c. Generally varies from 2^ to 7 per ct. 410 RAILROADS. Table of Annual Expenses of some U S Railroads.* Names of Companies. Lehigh Valley, 1860 1862 " " 1868 and 1869 about. " " 1872 Baltimore & Ohio, main stem, 1859 4434 " " " I860 4254 " 1865 " " ' " 1866 " 1872 ..; East Tennessee A Georgia, 1872 Memphis & Charleston, 1860 2617 Georgia Central, 1872 4180 Penna Central, main line from Phila to Pittsburg, 358 miles, 1859, exclusive of State tonnage tax 7848 " " I860, " " " " " " 1861, " " " " " " 1868, tonnage tax repealed ... " 1869, " " * about... 32000 " 1872, ' Phila & Reading, 1859 1860 " 1868, 365 miles of main road and branches... 17200 " " 1869 1872 North Pennsylvania, 1860, 54 miles long 3213 " " 1862 3240 " " 1867 9534 " " 1868 ... " " 1872. Connecticut; average of all the railroads, 1861 3781 Massachusetts; " " " " 1861 3785 {2700 to 4300 1867 average.. Galena & Chicago, 1859 I860 3102 Phila, Wilmington & Baltimore, main stem, 1859 4586 " " " " " 1860 7100 " " " " " 1861 7785 " " " 1867 17380 New York; all the U R in the State, average,! 1859 4964 " " " " " 1861 5100 " " " " " " 1867 13856 New Jersey R K and Transportation, 1861.. 12213 Louisville & Nashville, 1861. Phila & West Chester, 1861, 27 miles 2274 1862 2282 " 1872 7030 Phila, Germantown & Norristown, 1861, 20 miles 6K)5 " 1862 6405 " " " 1867 18208 New York & Erie, 1861 6461 18 7, with its branches, 784 miles in all 14545 New York Central, 1861 . 8360 ' 1867, with its branches, 696 miles in all.... 15620 English R R, averages for 1856-7-8 Scotch ' ' " " "" Irish " " " " " " * Annu -I t-fpot-t-i ofren omit the lengths of the roads and branches: and as these frequently vary from year to \ ear, it is possible that tne table may ooutain some errors in the first column. f 2528 miles in operation. Total exps equalled 1.56 cts per passenger or ton carried 1 mile. Dead weight of cars, equal to 1.19 tons per passenger ; and to 1.74 tons per ton of freight. Per Per Mile of I Train Road. $ RAILROADS. 411 Fuel for locomotives. As nearly as the writer can judge from a mass of Yery conflicting testimony, a ton of good anthracite, or of bituminous coal, is equal to 1% cords of good dry hard mixed woods, (chiefly white oak ;) or to from t% to '2.% cords * of such soft ones as hemlock, white, and common yellow pine. Much ol the inferior bituminous coal of Illinois is hardly equal to a cord of average wood. On the Illinois 1'eiitral in 1660, the engines ran about 200,000 miles, with average trains of 10% loaded 8-vvheel cars. The wood-burners averaged 39 miles per cord ; or '% cords per 100 uiiles. The coal-burners, with this inferior coal, 36}^ miles per ton ; say 2% tons per 100 miles ; making 1 cord of wood equal to ly 1 ^- tons of coal. On the Chicago, Burlington V Quiiicy K K. a very full trial (about 11000 rniles with each kind) was made with this coal, and with another of good Duality from another locality. The trains averaged 2 % eight-wheel cars, with their aily loads ; and the result was 3.5 tons of the bad coal" and 2.2 of the good, per 100 miles ; making 1 ton of the good coal equal to 1.45 cords mixed woods. On the Great Western R R ol" Mass, the same trains required 2% tons of anthra- cite per 100 miles; or 4 coids of hemlock, (an inferior fuel;) or i ton anth = 1.6 cords hemlock. On the Philada A Reading, passenger trains of 5 large cars, and weighing 54 tons, exclusive of engine and tender, (or about 90 tons with them.) used 2.7 cord> of good mixed woods, or 1.5 tons of anth per 100 miles. Here 1 ton anth~ 1.8 cords of wood. In 1867. passenger trains of 6fe tons, (about 10*2 with engine and tender.) are reported to use 2.22 tons per 100 miles. This is much greater in proportion to the increased weight of train than in the preceding case ; and there would seem to be some discrepancy. The freight trains use 3.45 tons, and the very heavv coal trains 4.75 tons per 100 miles. These last consist of about 96 four-wheel coal-cars, weighing 2}' tons each, or 241 tons in all ; 96 loads of coal, of nearly 4% tons each, or 443 tons in all : and engine and tender about 50 tons; making the entire weight of the train 734 tons. The returning empty trains up the road weigh 291 tons in all. and consume about the same quantity of fuel as the descending loaded ones. The grade of the 95 miles of road in the direction of the loaded trains is descending : most of it al less than 6 feet per mile; and none exceeding 14 ft, except 3 continuous miles of 22.4 ft. and a stretch of less than 2 miles of 35 ft per mile ascending; on which last auxiliary power is used. When the coal trains burned wood they used about Y^ cords to the present ton of coal. The mean weight of the loaded coal trains down, and of the empty ones up, is 512 tons ; and that of the passenger trains each way is 102 tons, or only -JL- as much ; yet the quantity of fuel consumed b< the first is but little more than twice as great as by the last. The first run about 10 miles per hour ; the last, 25 miles. On the Penn'a Central, experiments made 1 ton of Pittsburg bituminous coal equal to 1.49 cords of wood (chiefly white oak) weighing 5215 Ibs, or 2.328 tons ; or 3500 Ibs per cord. Trains of 4 eight-wheel freight cars, weighing 30.8 tons, and loaded with 33 tons; engine and tender 41. 56 tons, making 105.36 tons in all ; in ascending 12 miles of continuous grade of 95 ft per mile, at a speed of 20^ miles per hour, consumed either 1073 fts (.479 ton of coal ; or 2483 Ibs (1.108 ton. or .7094 cord) of wood. This in 100 rniles gives 3.992 tons of coal, or 5.91 cords of wood ; or say 4 tons, or 6 cords ; or 1 ton coal = \% cords wood. And this accords closely with the comparative consumption of an- thracite and wood, in hauling heavy coal trains, on the Reading R R. The passenger engine used in the foregoing experiments weighed "28% tons, of which 18.16 tons rested on 4 drivers, 5% ft diam. It was arranged to be equally well adapted to coal and to wood. The trials show the effect of steep up- grades in increasing the consumption of fuel. Thus, on the Reading road, an engine with 20 tons on with engine and tender about 625 tons ; consuming at the rate of \% tons of anthracite, or 7.1 cords of wood per 100 miles; which in 12 miles would be .57 ton, or .84 cord ; against the .48 ton, or .71 cord, required on the steep grade, with a train weighing but % as much. The run of freight and passenger trains throughout the U States is 20 to 40 miles per cord, and 30 to 60 miles per ton. Much depends upon the adaptation of the engine to the kind of fuel used. A good coal-hnrner may be bad for wood, and vice versa; so that trials with the same engine may give very erroneous results as to the comparative merits of the two kinds of fuel. When wood is used, about ^ cord : or when coal, about % cord of wood, must be used for kindling, and getting up steam rendy for running; and this item is the same for a long run as for a short one : so that long roads have in this respect an advantage over short ones, in economy of fuel. Wood has the disadvantage of emitting parks; and is, moreover, nearly twice as heavy as coal, for the performance of equal duty ; and is, therefore, more expensive to handle. It also occupies 4 or 5 times as much space as coal. traffic woul $18250 per mle. To find the tractive power of a locomotive, mult together the * A oord is 4 X 4 X 8 ft, or 128 cub ft. A cord of good dry white oak. (next to hickory, the best wood for fuel,) weighs 3500 Ibs or 1.563 tons. Dry hemlock, white, or common yellow pine, (all of them inferior for fuel.) about .9 ton. Perfectly green woods generally weigh about to ^ more than when pnrtinlly dried for locomotive use: in other words, a cord of wood, in its partial drving, loses from X to % ton of water, and still contains a large quantity of it. Since this water causes a great waste of heat, green wood should never be used as fuel. The values of woods as fuel are in nearly the same proportion as their weights per cord when perfectly dry. 412 RAILROADS. square of the diam of one piston in ins; the single length of stroke in ins; and the cylinder pressure of the steam in Ibs per sq inch. Divide the prod by the diam of a driver in ins. The quot will be the power in tbs. Whether the engine can really employ that power or not, depends upon the weight on its driving-wheels. And the velocity depends upon tne steam-generating capability of the boiler.* Or ordinarily file tractive power = adhesion = at least, one-fifth of the wt on the drivers. With the rails clean and in good order it may rise even to full one-third of said weight at low speeds. It becomes less at high speeds. Table of appro x greatest wt of train which a good locomotive weighing 27 tons, or 60480 ft>s, all of it on the drivers, can (in addition to the wt of itself and tender, 45 tons,) take up straight grades, at about 8 or 10 miles an hour, when all is in good order, taking the resistance at 8 Ibs per ton on a level ; and the traction at one-fifth the wt on drivers. For other engines the loads will be about as the wts on drivers. With everything in perfect order, friction ut times falls to 5 Ibs, or even to 4 Ibs, per ton wt of train. Grade in Ft. per Mile. Tons. Grade in Ft. per Mile. Tons. Grade in Ft. per Mile. Tons. Grade in Ft. per Mile. Tons. Level. 1458 16 769 60 315 160 113 1 1383 18 725 65 294 not 105 2 1315 20 686 70 275 180 98 3 1253 22 650 75 258 190 91 4 1197 24 617 80 242 200 84 5 1144 26 587 85 228 225 71 6 1096 28 561 90 216 250 60 7 1052 30 536 95 204 275 61 8 1012 32 513 100 194 300 44 9 974 35 482 110 175 350 32 10 939 40 437 120 159 400 22 11 905 45 400 130 146 450 15 12 875 50 367 140 134 500 9 14 819 55 340 150 123 528 7 In practice, trains must not, as a general rule, weigh more than from ^ to % of those in the table. This reduction is necessary in order to have the trains under more complete control; to admit of greater speed in case of detention ; to allow for curves, slippery rails, head winds, &c, &c. The animal expense of running a locomotive and tender, averaging 75 miles a day, for 267 days in the year, or 20000 miles annually, which is common, (have run 60000) may be found in the following manner : Fuel, say 2\ cords per 75 miles, at $3.50 per cord ; $7.87%p?r day $2100 Repairs, at Sets per mile run 1800 Engineer, or driver, 12 months, at $90 1080 Fireman, 12 months, at $50 600 Oil and waste, at 1 ct per mile run 200 Sawing, and loading-up wood; 1% cts per mile run < Supplying water let " " 200 Putting away, cleaning, and getting out, say 120 Locomotive, superintendence " 100 Total $6500 Equal to, say 34 cts per train mile ; or $24.35 per running day; or $17. 81, for every day in the year. This is all 'that is usually stated in annual reports of expenditures; but inasmuch as an engine in active service, even under a judicious system of repairs, generally becomes worthless, (except as old iron.) in say 16 years on an average, an additional allowance of nbout 6 per ct on the first cost, or about $500 to $800, should be made annually for DEPRECIATION OF EACH ENGINE. For Evaporation by Locomotives, see p 434. * The cylinder pres is always less than the boiler pres ; and the disproportion increases with the speed. Thus, at 8 or 10 miles an hour, the boiler pres may be about 110 fts per sqiuch; and the cylinder pres from 90 to 100 fts ; while at a speed of 30 or +0 miles, the proportion may be as 110 to 60 YBaldwin's 8-driver engines of 35 tons, all on the drivers, draw 40 empty coal cars, weighing 100 tons, up a continuous grade of 175 ft per mile for 3^ miles, with many curves and reversed curves of from 450 to 600 ft rad. at 8 miles an hour, as an every -day operation. 7 On the Penn'a Central, engines of 29 ton*, all of it on 8 drivers, take 200 tons of cars and freight up a continuous grade of 95 ft per mile for % miles. The passenger engines take up 3 loaded 8-wheel RAILROADS. 413 The length of Locos out to out is usually about a ft per ton wt. But some very heavy freighters weigh 1.5 tons per ft. Extreme width 9 ft 4 ina for 4 ft 8}$ gauge.* The height Of the pipe, or smoke stack, is usually 13 to 15 ft above the rails. Good StOUt Steel driving-tires,! bearing about 3 to 3^ tons, will run aboutSOOOO or J4 iuch in diaiu ; and will bear 3 or i turnings-down before wearing too thin for safety. Driving xies and journals are about 5> 585 50 44 4 35 335 The weights by other makers do not differ from these materially. The diam of car or engine wheels does not include the flanges ; but is the least diam from tread to tread. Good chilled cast wheels will run about 100,000 miles ; usually however they hardly average 50000 k * Cost of Locomoti ves, Philada, 1880, $250 to $300 per ton of their weight ; the tender being supposed to be thrown in. t Price, in Philada, 1880, 11 to 13 cts per ft. I APPROXIMATE AVERAGE PRICKS OF CARS in the Eastern States in 1880. Passenger-cars for about 50 to 60 persons, $4500 to $5500. Dining-room cars, from $6000 to $8000. Sleeping-cars, $6000 to $10000. Some very extravagant ones, for 60 persons, have cost from $20000 to $25000. Mail and baggage cars, $2800. Freight cars, (house, or box-cars) $750. Gondola, and platform cars, $400 to $500. All the foregoing ou 3 wheels. Coal cars, single, 4 wheels, wooden bodies, $200; with iron bodies, $300. ^ Cost in Philadelphia, in 1880, about 3$ cts per fl>. Wrought-iron axles, hammered, about 5 ct per ft ; and rolled, ift eta. 27 414 RAILROADS. T ? ???i OIMlSI -, 0f Peill yl v rage, about $58000 per mile : aud have about with their equipments, have cost on an ar*. 1 Locomotive to every ......................................... 4 mi i e8 o f road. Engine-house to " ...................... ........... ..... 20 " " 1 Passenger car to " ......................................... 8 " *< 1 Baggage, mail, or express car to every.. .............. 18 " " 2% Freight cars, and trucks, to ......... " ................ 1 6 Coal cars to ............................. " ............... 1 1 Depot, or station.. ................... " ........... ..... 7 " 1 Fuel and water station to ......... " ................ 9 " " The average cost per mile of the railroads of New England has been about $40500; of the Middle States, $55000; of the Southern States, $30000. Miles of railroad in the world, at the close of 1879, about 203000; or I here ''"IT ^ circumf f - the . earth - In Nor ' h America, 90000 ; Europe, and entire Eastern hemi- Standurd size of axles for passenger or freight cars, on a 4 ft 8% track. JLength, total 6 ft 11^ ins ; between hubs 4 ft 0}^ in ; each wheel-seat 7 ins; each journal 7 ins. Diam, at middle Z% ins ; at hubs 4 % ins ; at journals 3% ins. Weight, 347 fts per axle. Table of cubic yards of ballast per mile of road. Side-slope of the ballast 1 to 1. \Vidth in clear between 2 tracks 6 ft. The ties and rails may be laid first, for carrying the ballast along the line; then raised a few ft at a time, and the ballast placed under them. Depth TOP WIDTH, SINGLE TRACK. TOP WIDTH, DOUBLE TRACK. 10 Ft. 11 Ft. 12 Ft. 21 Ft. 22 Ft. 23 Ft. Cub. Y. Cub. Y. Cub. Y. Cub. Y. Cub. Y. Cub. Y. 12 2152 2347 2543 4303 4499 4695 18 8374 3667 3960 6600 6894 7188 24 46f)4 5085 5474 8996 9388 9780 30 6111 6600 7087 11490 11S80 12470 A mail Can break 3 to 4 cub yds per day, of hard quarried .stone, to a size suitable for ballast ; say averaging cubes of 3 ins on au edge. Where other ballast cannot be had, hard-burnt clay is a good substitute. The slag from iron furnaces is excellent. The ties decay more rapidly when gravel or sand is used instead of broken stone, because these do not drain off the rain, but keep the ties damp longer. CrOSS-tieS of 8^ ft, by 9 ins, "by 7 ins, contain 3.719 cub ft each; and if placed 2% ft apart from center to center, there will be 2112 of them per mile, amounting to 291 cub yds. Therefore, if they are completely embedded in the ballast, they will diminish its quantity by that amount. At 2 ft apart there will be 2640 of them, occupying 364 cub yds ; and at 3 ft apart, 1760 of them ; 243 cub yds. Cubic feet contained in cross-ties of different sizes. Dimensions. Dimensions. Dimensions. Ft. Ins. Ins. Cub. Ft. Ft. Ins. Ins. Cub. Ft. Ft. Ins. Ins. Cub. Ft. 8 by 8 by 6 896 2.667 3.000 &H by 8 by 6 Sh 9 6 2.833 3.188 9 by 8 by 6 996 3.oOO 3.375 897 3.500 8^ 9 7 3.719 997 3.938 8 10 6 3.333 8^ 10 6 3.542 9 10 6 3.750 8 10 7 3.889 8^ 10 7 4.132 9 10 7 4.375 8 10 8 4.444 8^ 10 8 4.722 9 10 8 5.000 8 12 8 5.333 8^ 12 8 5.667 9 12 8 6.000 The average life of ties is 6 to 8 years; and the expense for renewing them is serious , amounting probably to not less than ten millions of dollars anuually in the United States alone. The process of Seely of preserving timber by the injection of creosote (see p 358) is altogether the best that has been devised, and there cau be no doubt that its employment will in many cases be true economy. This remark applies as well to timber for bridges, and for engineering and architectural purposes generally. The process is said to preserve timber from the Limnoria, and Teredo Navalit, sea-worms; but in our opinion, there is, as yet, no good foundation for this state- ment. No remedy is, we believe, yet known for this evil, although saturation with creosote certainly delays their attacks for many years if thoroughly done. The limnoria works from near H. W. mark, down to a little below the surface of mud bottom ; the teredo within somewhat less limits. The writer believes that most of the fault usually ascribed to cross-ties, as well as to rail-joints, is in reality due to imperfect drainage of the roadbed. Hence, he does not agree with those who advo- cate VERY LONG TIES ; but considers that with good ballast, on a well-drained roadbed, 8^ ft is as good as more ; and that ^ ft, by 9 ins. by 7 ins : aud 2% ft apart from center to center, is sufficient for the heaviest traffic. On many important roads they are but 8 ft; and on some only 1% ft long; track 4 ft8>. The actual COST OF CUTTING DOWN THE TREES, topping off the branches, and hewing the ties ready for hauling away to be laid, is about 6 to 9 cts per tie, at $1.75 per day per hewer. RAILROADS. 415 The narrow bases of rails resting immediately on the cross-ties, without chairs, frequently produce in time such an amount of crushing in the ties as to injure them materially even before decay begins. Mr P. H. Whitman, of Portland, Maine, has patented a mode of obviating this by letting into the tie a thin block of hard wood under each rail. He sells machines for cutting the grooves for receiving these blocks. Burnetis'id ties rust the spikes away rapidly. Creosoted ones preserve them. Post-and-rail fences, in panels 8% ft long; 5 rails; usually cost between 40 to 100 cents per panel, including the putting up; or from $512 to $1280 per mile of road fenced on both sides, with 1280 panels. Worm fences seven rails high, with two rails on end at each angle, cost about %th less. Labor $1.75 per day. The scarcity or abundance of timber chiefly in- fluences the price ; as is also the case with ties. The Glidden Barbed Steel Wire fence, made by .1. L. Ellwood & Co., at De Kalb, Illinois, has (1882) come into extensive use; many thousands of miles of it having already been put up. Its cost per mile of single row of fence, put up, including the wooden posts and all labor, will usually range from $150 to $250, de- pending on the height of fence, the varying market price of wire, labor, &c. Every sq inch of sectional area of rail, corresponds to 10 Ibs per yard of a single rail ; or to 15.7143 tons per mile of single-track road. Consequently, 15.7143 Thus, a rail of 100 tons per mile of single track, will have a section of 6.364 sq ins ; and will weigh 63.64 Ibs per yd of single rail. Add for turnouts, sidings, road-crossings, and a trifle for waste in cutting. Steel must inevitably take the place of iron for rails, or at least for their tops.* When the ties are in place, and the rails distributed in piles at short intervals, a gang of 6 men can lay % a mile of rails per day, of single track ; or after the ballast is in place, a gang of 15 men will lay about one mile of complete single-track superstructure per week. Approximate average estimate for a mile of single-track railway. Labor $1.75 per day. Grubbing and clearing, (average of entire road,). 3 acres at $50 ........................ $ 150 Grading; 20000 cub yds of earth excavation, at 35 cts .................................. 7000 " 2000 cub yds of rock excavation, at $1.00 ..................................... 2000 Masonry of culverts, drains, abutments of small bridges, retaining-walls, &c; 400 cub yds, at $8, average ................................................................... 3200 Ballast; 3000 cub yds broken stone, at $1.00 ................................................. 3000 Cross-ties; 2112, at 60 cts, delivered ............................................................... 1267 Rails; (60 Ibs to a yard;) 96 tons, at $100, delivered .............. . ..................... 9600 Spikes ................. ................................................................................... 275 Mail-joints, or chairs ................................................................................. 525 Sub-delivery of materials along the line ................................... ..................... 300 Laying track. .................................. .................................................... 600 Fencing ; (average of entire road,) supposing only % of its length to be fenced.. 450 Small wooden bridges, trestles, sidings, road-crossings, cattle guards, 95 14652 47 114709 14767 14S24 14882 14940 14998 15056 15114 15172 152:30 48 15289 15347 15406 i 15465 15524 15583 15642 15701 15761 15826 49 L>880 15939 15999 16059 16119 16179 16239 16300 16360 16421 50 16181 16542 16603 16664 16725 16787 16848 16909 16971 !l7033 51 170J4 17156 17218 17280 17343 17405 17467 117530 17593 117656 52 17719 17782 17845 17908 17971 18035 18098 18162 18226 18-29C 53 18354 18418 18482 18546 18611 18675 18740 18805 18870 18935 n4 19000 19065 19131 19196 19262 19327 19393 19459 19525 19591 55 196YT 19724 19790 19857 19923 19990 20057 20124 20191 20259 56 20326 20393 20461 120529 20596 20664 20732 20800 2()69 20937 57 21005 21074 21143 '21212 21-280 21349 21419 21488 21557 21627 58 21696 21766 21836 J21906 2197 ri 22046 22116 22186 22257 22327 59 22398 22469 22540 22611 22682 22753 22825 22896 22968 23039 60 23111 23183 23255 23327 23399 23472 23544 23617 23689 23762 -y- From the Author'* " Measurement of Excavation and Embankment." RAILROADS. 421 Table 2. Level Cuttings. Roadway 24 feet wide, side-slopes 1% to 1. For double-track embankment, Height in Ft. .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 CuTYds. Cu.Yds. Cu.Yds. Cu.Yds. Cu.Yds. Cu.Yds. Cu.Yds. Cu.Yds. Cu.Yds. Cu.Yds. 8.94 18.0 27.2 36.4 45.8 55.3 64.9 74.7 84.5 1 94.4 104.5 114.7 124.9 135.3 145.8 156.4 167.2 178.0 188.9 2 200.0 211.2 222.4 233.8 245.3 256.9 268.6 280.5 292.4 304.4 3 316.6 328.9 341.2 353.7 366.3 379.0 391.9 4048 417.8 431.0 4 444.4 457-8 471.3 484.9 498.6 512.4 526.4 540.4 554.6 568.8 5 583.3 597.8 612.4 627-1 642.0 656.9 671.9 687.1 702.3 717.7 6 733.3 748.9 764.7 780.5 796.4 812.5 828.7 844.9 861.3 877.8 7 894.4 911.2 928.0 944.9 962.0 979.2 996.4 1014 1031 1049 8 1067 1085 1102 1121 1139 1157 1175 1194 1212 1231 9 1250 1269 1288 1307 1326 1346 1365 1385 1405 1425 10 1444 1465 1485 1505 1525 1546 1566 1587 1608 1629 11 1650 1671 1692 1714 1735 1757 1779 1800 1822 1845 12 1867 1889 1911 1934 1956 1979 2002 2025 2048 2071 13 2094 2118 2141 2165 2189 2213 2236 2261 2285 2309 14 2333 2358 2382 2407 2432 2457 2482 2507 2532 2558 15 2583 2e;o9 2635 2661 2686 2713 2739 2765 2791 2818 16 2844 2871 2898 2925 2952 2979 3006 3034 3061 3089 17 3117 3145 3172 3201 3229 3257 3285 3314 3342 3371 18 3400 3429 3458 3487 3516 3546 3575 3605 3635 3665 19 3694 3725 3755 3785 3815 3846 3876 3907 3938 3969 20 4000 4081 4062 4094 4125 4157 4189 4221 4252 4285 21 4317 4349 4381 4414 4446 4479 4512 4545 4578 4611 22 4644 4678 4711 4745 4779 4813 4846 4881 4915 4949 23 4983 5018 5052 5087 5122 5157 5192 5227 5262 5298 24 5333 5369 5405 5441 5476 5513 5549 5585 5621 5658 25 5694 5731 5768 5805 5842 5879 5916 5954 5991 6029 26 6067 6105 6142 6181 6219 6257 6295 6334 6372 6411 27 6450 6489 6528 6567 6606 6646 6P85 6725 6765 6805 28 6844 6885 . 6925 6965 7005 7046 7086 7127 7168 7209 29 7250 7291 7332 7374 7415 7457 7499 7541 7582 7625 30 7667 7709 7751 7794 7836 7879 7922 7965 8008 8051 31 8094 8138 8181 8225 8269 8313 8356 8401 8445 8489 32 8533 8578 8622 8667 8712 8757 8802 8847 8892 8938 33 8983 9029 9075 9121 9166 9212 9259 9305 9351 9398 34 9444 9491 9538 9585 9632 9679 9726 9774 9821 9869 35 9917 9965 10012 10061 10109 10157 10205 10254 10302 10351 36 1.0400 10449 10498 10547 10596 10646 10695 10745 10795 10845 37 10894 10945 10995 11045 11095 11146 11196 11247 11298 11349 38 11400 11451 11502 11554 11605 11657 11709 11761 11812 11865 39 11917 11969 12021 12074 12126 12179 12232 12285 12338 12391 40 12444 12498 12551 12605 12659 12713 12766 12821 12875 12929 41 12983 13038 13092 13147 13202 13257 13312 13367 13422 13478 42 13533 13589 13645 13701 13756 13813 13869 13925 13981 14038 43 14094 14151 14208 14265 14322 14379 14436 14494 14551 14609 44 14667 14725 14782 14840 14899 14957 15015 15074 15132 15191 45 15250 15309 15368 15427 15486 15546 15605 15665 15725 15785 46 15844 15905 15965 16025 16085 16146 16206 16267 16328 16389 47 16450 16511 16572 16634 16695 16757 16819 16881 16942 17005 48 17067 17129 17191 17254 17316 17379 17442 17505 17568 17631 49 176P4 17758 17821 17885 17949 18013 18076 18141 18205 18269 50 18333 18398 18462 18527 18592 18657 18722 18787 18852 18918 51 18983 19049 19115 19181 19246 19313 19379 19445 19511 19578 52 19644 19711 19778 19845 19912 19979 20046 20114 20181 20249 53 20317 20385 20452 20521 20589 20657 20725 20794 20862 20931 54 21000 21069 21138 21207 21276 21346 21415 21485 21555 21625 55 21694 21765 21835 21905 21975 22046 22116 22187 22258 22329 56 22400 22471 22542 22614 22685 22757 22829 22901 22972 23045 57 23117 23189 23261 23334 2:5406 23479 23552 23625 23698 23771 58 23844 23918 23991 24065 24139 24213 24286 24361 24435 24509 59 24583 24658 24732 24S07 24882 24957 25032 25107 25182 25258 60 25333 25409 25485 25561 25636 25713 25789 25865 25941 26018 For continuation to 100 feet, see TABLE 7. 422 RAILROADS. Table 3. Level Cuttings. Roadway 18 feet wide, side-slopes 1 to 1. For single-track excavation, Depth in Ft. .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 Cu.Yds. Cu.Yds. Cu.Yds. Cu.Yds. Cu.Yds. Cu.Yds. Cu.Yds. Cu.Yds. Cu.Yds. Cu.Yds. 6.70 13.5 203 27.3 34.3 41.3 48.5 55.7 63.0 1 70.4 77.8 85.3 929 100.6 1083 116.1 124.0 132.0 140.0 2 148.1 156.3 164.6 172.9 181.3 189.8 198.4 207.0 215.7 224.5 3 233.3 242.3 251.3 260.3 269.5 278.7 288.0 297.4 306.8 316.3 4 325.9 335.6 345.3 355.1 365.0 375.0 385.0 395.1 405.3 415.6 5 425.9 436.3 446.8 457.4 468.0 478.7 489.5 500.3 511.3 522.3 6 533.3 544.5 555.7 567.0 578.4 589.8 601.3 612.9 624.6 636.3 7 648.1 660.0 672.0 684.0 696.1 708.3 720.6 732.9 745.3 757.8 8 770.4 7830 795.7 808.5 821.3 834.3 847.3 860.3 873.5 886.7 9 900.0 913.4 926.8 940.3 953.9 967.6 981.3 995.1 1009 1023 10 1037 1051 1065 1080 1094 1108 1123 1137 1152 1167 It 1181 1196 1211 1226 1241 1256 1272 1287 1302 1318 12 1333 1349 1365 1380 1396 1412 1428 1444 1460 1476 13 1493 1509 1525 1542 1558 1575 1592 1608 1625 1642 U 1659 1676 1693 1711 1728 1745 1763 1780 1798 1816 15 1833 1851 1869 1887 1905 1923 1941 1960 1978 1996 16 2015 2033 2052 2071 2089 2108 2127 2146 2165 2184 17 2204 2223 2242 2262 2281 2301 2321 2340 2360 2380 18 2100 2420 2440 2460 2481 2501 2521 2542 2562 2583 19 2604 2624 2645 2666 2687 2708 2729 2751 2772 2793 20 2815 2836 2858 2880 2901 2923 2945 2967 2989 3011 21 3053 3056 3078 3100 3123 3145 3168 3191 3213 3236 22 3259 3282 3305 3328 3352 3375 3398 3422 3445 3469 23 3493 3516 3540 3564 3588 3612 3636 3660 3685 3709 24 3733 3758 3782 3807 3832 3856 3881 3906 3931 3956 25 3981 4007 4032 4057 4083 4108 4134 4160 4185 4211 '26 4237 4263 4289 4315 4341 4368 4394 4420 4447 4473 27 4500 4527 4553 4580 4607 4634 4661 4688 4716 4743 28 4770 4798 4825 4853 4881 4908 4936 4964 4992 5020 29 5048 5076 5105 5133 5161 5190 5218 5247 5276 5304 30 5333 5362 5391 5420 5449 5479 5508 5537 5567 5596 31 5626 5656 5685 5715 5745 5775 5805 5835 5865 5896 32 5926 5956 5987 6017 6048 6079 6109 6140 6171 6202 33 6233 6264 6296 6327 6358 6390 6421 6453 6485 6516 34 6548 6580 6612 6644 6676 6708 6741 6773 6S05 6838 35 6870 6903 6936 6968 7001 7034 7067 7100 7133 7167 3tf 7200 7233 7267 7300 7334 7368 7401 7435 7469 7503 37 7537 7571 7605 7640 7674 7708 7743 7777 7812 7847 38 7881 7916 7951 7986 8021 8056 8092 8127 8162 8198 39 8233 S269 8305 8340 8376 8412 8448 8484 8520 8556 10 8593 8629 8665 8702 8738 8775 8812 8848 8885 8922 41 8959 8996 9033 9071 9108 9145 9183 9220 9258 9296 42 9333 9371 9409 9447 9485 9523 9561 9600 9638 9676 43 9715 9753 9792 9831 9869 9908 9947 9986 10025 10064 44 10104 10143 10182 10222 10261 10301 10341 10380 10420 10460 45 10500 10540 10580 10620 10661 10701 10741 10782 10822 10863 46 10304 10944 10985 11026 11067 11108 11149 11191 11-232 11273 47 11315 11356 11398 11440 11481 11523 11565 11607 11649 11691 48 11733 11776 11818 11860 11903 11945 11988 12031 12073 12116 49 1-2159 12202 12245 12288 12332 12375 12418 12462 12505 1-2549 50 12593 12636 12680 12724 12768 12812 12856 12900 12945 12989 51 13033 13078 13122 13167 13212 13256 13301 13346 13391 13436 52 '13481 13527 i 13572 13617 13663 13708 13754 13*00 13845 13891 53 13937 13983 14029 14075 14121 114168 14214 14-260 14307 14353 54 14400 14447 14493 14540 14587 14634 14681 1.47-28 14776 14823 55 14870 14918 14965 15013 15061 15108 15156 15204 15252 15300 56 15348 15396 15445 15493 15541, 15590 15638 15687 15736 15784 57 15833 15882 15931 15980 16029 16079 16128 17177 16227 16276 58 16^26 16376 16425 16475 16525 16575 166-25 16675 16725 16776 59 16826 1*876 16927 16977 17028 17079 17129 17180 17231 17282 60 (17333 17384 17436 17487 17538 17590 17641 17693 17745 17796 For continuation to 100 feet deep, see Table 7. ]fLKO ,KOADS. T*6ie 4. Level Cuttings. dway 18 feet, side-slopes 1^ to 1. For single-track excavation, 423 Depth in Ft. .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 Cu.Yds. Cu.Yds. Cu.Yds. Cu.Yds. Cu.Yds. Cu.Yds. Cu.Yds. Cu.Yds. Cu.Yds. Cu.Yds. 6.72 13.6 20.5 27.6 34.7 42.0 49.4 56.9 64.5 1 72.2 80.1 88.0 96.1 104.2 112.5 120.9 129.4 138.0 146.7 2 155.5 164.5 173.5 182.7 191.9 201.3 210.8 220.4 230.1 240.0 3 249.9 260.0 270.1 280.4 290.8 301.3 311.9 322.6 333.4 344.5 4 355.5 366.7 378.0 389.4 400.9 412.5 424.2 4360 448.0! 460.0 5 472.2 484.5 496.9 509.4 522.0 534.7 547.6 560.5 573.6 586.7 6 600.0 613.4 626.9 640.5 654.2 668.1 682.0 696.1 710.21 724.5 7 738.9 753.4 768.0 782.7 797.6 812.5 827.6 842.7 858.0 873.4 8 888.9 904.5 920.2 936.1 952.0 968.1 984.2 1001 1017 1033 9 1050 1067 1084 1101 1118 1135 1152 1169 1187 1205 10 1222 1240 1258 1276 1294 1313 1331 1349 1368 1387 11 1406 1425 1444 1463 1482 1501 1521 1541 1560 1580 12 1600 1620 1640 1661 1681 1701 1722 1743 1764 1785 13 1806 1^27 1848 1869 1891 1913 1934 1956 1978 2000 14 20-22 2045 2067 2089 2112 2135 2158 2181 2204 2227 15 2250 2273 2-297 2321 2344 2368 2392 2416 2440 2465 16 2489 2513 2538 2563 2588 2613 2638 2663 2688 2713 17 2739 2765 2790 2816 2842 2868 2894 2921 2947 2973 18 3000 3027 3054 3081 3108 3135 3162 3189 3217 3245 19 3272 3300 3328 3356 3384 3413 3441 3469 3498 3527 20 3556 3585 3614 3643 3672 3701 3731 3761 3790 3820 21 3850 3880 3910 3941 3971 4001 4032 4063 4094 4125 22 4156 4187 4218 4249 4281 4313 4344 4376 4408 4440 23 4472 4505 4537 4569 4602 4635 4668 4701 4734 4767 24 4800 4833 4867 4901 4934 4968 5002 5036 5070 5105 25 5139 5173 5208 5243 5278 5'il3 5348 5383 5418 5453 26 5489 5525 5560 5596 5632 5668 5704 5741 5777 5813 27 5850 5887 5924 5961 5998 6035 6072 6109 6147 6185 28 6222 6260 6298 6336 6374 6413 6451 6489 6528 6567 29 6606 6645 6684 6723 6762 6801 6841 6881 6920 6960 30 7000 7040 7080 7121 7161 7201 7242 7283 7324 7365 31 7406 7447 7488 7529 7571 7613 7654 7696 7738 7780 32 7822 7865 7907 7949 7992 8035 8078 8121 8164 8207 33 8250 8293 8337 8381 8424 8468 8512 8556 8600 8645 34 8689 8733 8778 8823 8868 8913 8958 9003 9048 9093 35 9139 9185 9230 9276 9322 9368 9414 9461 9507 9553 36 9600 9647 9694 9741 9788 9835 9882 9929 9977 10025 37 10072 10120 10168 10216 10264 10313 10361 10409 10458 10507 38 10556 10605 10654 10703 10752 10801 10851 10901 10950 11000 39 11050 11100 11150 11200 11251 11301 11352 11403 11454 11505 40 11556 11607 11658 11709 11761 11813 11864 11916 11968 12020 41 12072 12126 12177 122-29 12282 12335 12388 12441 12494 12547 42 12600 12653 12707 12761 12814 12868 12922 12976 13030 13085 43 13139 13193 13248 13303 13358 13413 13468 13523 13578 13fi33 44 13689 13745 13800 13856 13912 13968 14024 14081 14137 114193 45 14250 14307 14364 14421 14478 14535 14592 14C49 14707 14765 46 14822 14880 14938 149P6 15054 15113 15171 15229 152*8 15347 47 15406 15465 15524 15583 15642 15701 15761 15821 15880 15940 48 16000 16060 16120 16181 16241 16301 16362 16423 16484 16545 49 16606 16667 16728 16789 16851 16913 16974 17036 17098 17160 50 17222 17285 17347 17409 17472 17535 17598 17661 17724 I177S7 51 17850 17913 17977 18041 18104 18168 18232 18296 18360 18425 52 18489 18553 18618 18683 18748 18813 18878 18943 19008 119073 53 19139 19205 19270 19336 19402 19468 1U534 19601 19667 119733 54 19800 19867 19934 20000 200(38 20135 20202 20269 20337 20405 55 20472 20540 20608 20676 20744 20813 20881 20949 21018 21087 56 21156 21225 21 '294 21363 21432 21501 21571 21641 21710 21780 57 21850 21920 21990 22061 22131 22201 22272 22343 22414 22485 58 22556 22627 22698 22769 22841 22913 22984 23056 23128 23200 59 23272 23345 23417 23489 23562 23635 23708 23781 23854 23927 60 24000 24073 24147 24221 24294 24368 24442 24516 24590 24665 For continuation to TOO feet deep, see Table 7. 424 RAILROADS. Table 5. Level Cuttings. Roadway 28 feet wide, side-slopes 1 to 1. For double -track excavation, Depth 08 6641 6674 6708 6741 31 6774 6807 6341 6874 6308 6942 6975 7009 7043 7077 32 7111 7145 7179 7214 7248 7282 7317 7351 7386 7421 33 7456 7490 7525 7560 7595 7631 7666 7701 7736 7772 34 7807 7 8 A3 7879 7914 7950 7986 8022 8058 8094 8130 35 8t67 8203 8239 8276 8312 8349 8386 8423 8459 8496 36 8533 8570 8608 8645 8682 8719 8757 8794 8S32 8870 37 8907 8945 8983 9021 90 >9 9097 9135 9174 9212 9250 38 9289 9327 9366 9405 9444 9482 9521 9560 9599 9639 39 9678 9717 9756 9796 9835 9875 9915 9954 9994 10034 40 10074 10114 10154 10194 10235 10275 10315 10356 10396 10437 41 10178 10519 10559 10600 10641 10682 10724 107^5 10806 10847 42 108S9 10330 10972 11014 11055 11097 11139 11181 11223 11265 43 U307 11350 U3J2 11434 11477 11519 11562 11605 11648 11690 44 11733 11776 11819 11863 11906 11949 11992 12036 12079 12123 45 12167 12210 12254 12298 12342 123^6 12430 12474 12519 12563 46 12607 12652 12696 12741 12786 12831 12875 12920 12965 13010 47 13056 13101 13146 13191 13237 13282 13328 13374 13419 13465 48 13511 13587 13603 13649 13695 13742 13788 13834 13S81 13927 49 13974 14021 14068 14114 14161 14208 14255 14303 14350 14397 50 14444 14492 14539 14587 14635 14682 14730 14778 14826 14874 61 14922 14970 15019 15067 15115 15164 15212 15261 15310 15359 52 15407 15456 15505 15554 15804 15653 15702 15751 15801 1 5850 53 15900 15950 15999 16049 16099 16149 16199 16249 1T299 16350 54 16400 16150 16501 16551 16602 1 ! 653 I1H704 16754 16805 [16856 55 16907 16959 17010 17061 17112 17164 17215 17267 17319 i 17370 56 17 422 17474 175'>6 1"578 17630 17682 17735 17787 17839 117892 57 17944 17997 18050 18103 18155 18208 18261 18314 18368 118421 58 18474 18527 18581 186'?4 1^6S8 18742 18795 18S49 18903 18957 59 [19011 19065 19119 19174 19228 19282 19337 19391 19446 19501 60 [19556 19610 19665 197?0 19775 19831 19886 19941 19996 (20052 For continuation to 100 feet, ee Table 7. 425 TaCble 6. Level Cuttings. ^Koadway 28 ft wide, side-slopes 1^ to 1. ^// For double-track excavation, Depth 1 in Ft. .0 .1 .2 .3 ! .4 .5 .6 .7 .8 .9 Cu.Yds. Cu.Yds. Cu.Yds. Cu.Yds. Cu.Yds. Cu.Yds. Cu.Yds. Cu.Yds. Cu.Yds. Cu.Yds. 10.4 21.0 31.6 42.4 53.2 64.2 753 86.5 97.9 1 109.3 120.8 132.5 144.3 156.1 168.1 180.2 192.4 204.8 217.2 2 229.6 242.3 255.0 267.9 280.9 294.0 307.2 320.5 334.0 347.5 3 361.2 374.9 388.8 402.8 416.9 431.1 445.4 459.9 474.4 489.1 4 503.7 618.6 533.6 548.6 5639 579.3 594.7 610.2 625.8 641.6 5 657.5 673.4 689.5 705.7 722.1 738.5 7550 771.7 788.4 805.3 6 822.2 8o9.3 856.5 873.8 891.2 908.8 926.4 944.2 962.0 980.0 7 998.1 1016 1035 1053 1072 1090 1109 1128 1147 1166 8 1186 1204 1224 1243 1263 1283 1303 1322 1343 1863 9 1383 1403 1424 1445 1465 1486 1507 1528 1549 1571 10 1692 1614 1635 1657 1679 1701 1723 1745 1767 1790 11 1812 1836 1858 1881 1904 1927 1950 1973 1997 2020 12 2044 2068 2092 2116 2140 2164 2189 2213 2238 2262 13 2287 2312 2337 2842 2387 2413 2438 2464 2489 2515 14 2541 2567 2593 2619 2645 2672 2698 2725 2752 2779 15 2806 2833 2860 2887 2915 2942 2970 2997 3025 3053 16 3081 3109 3138 3166 3195 3223 3252 8281 3310 3339 17 3368 3397 3427 3456 3486 3516 3546 3576 3606 636 18 3667 3697 3728 3758 3789 3820 3851 3882 3913 3944 19 3976 4007 4039 4070 4102 4134 4166 4198 4231 4'263 20 4296 4328 4361 4394 4427 4460 4493 4527 4560 4594 21 4627 4661 4695 4729 4763 4797 4832 4866 4900 4935 22 4970 6005 5040 5075 5111 5146 5181 5217 5253 6288 23 5324 5360 5396 6432 5469 5505 5542 5578 5615 6652 24 5689 5726 5763 5800 6838 5875 5913 5951 5989 6027 25 6065 6103 6141 6179 6218 6267 6295 6334 6373 6412 26 6461 6491 6530 6570 6609 6649 6689 6729 6769 6809 27 6850 6890 6931 6971 7012 7053 7094 7135 7176 7217 28 7259 7300 7342 7384 7426 7468 7510 7552 7594 7637 29 7680 7722 7765 7808 7851 7894 7937 7981 8024 8067 .30 8111 8155 8199 8243 8287 8331 8375 8420 8464 8509 31 8554 8698 8643 8688 8734 8779 8824 8870 8915 8961 32 9007 9053 9099 9145 9191 9238 9284 9331 9378 9425 33 9472 9519 9566 9613 9661 9708 9756 9804 9851 9900 34 9948 9997 10045 10093 10142 10190 10239 10288 10337 10386 35 10435 10484 10534 10583 10633 10683 10732 10782 10832 10882 36 10933 10983 11034 11084 11135 11186 11237 11288 11339 11391 37 11443 11494 11546 11598 11649 11701 11753 11806 11858 11910 38 11963 12016 12068 12121 12174 12227 12281 12384 12387 12441 39 12494 12548 12602 12656 12710 12764 12819 12873 12928 12982 40 13037 13092 13147 13202 13257 12312 13368 13423 13479 135S5 41 13591 13647 13703 13759 13816 13872 13928 13985 14042 14099 42 14156 14213 14270 14327 14385 14442 14600 14558 14615 14673 43 14731 14790 14848 14906 14965 15024 15082 15141 15200 16259 44 15318 15378 15437 15497 15556 15616 15676 15736 15796 15856 45 15917 15977 16038 16098 16159 16220 16281 16342 16403 16465 46 16526 16587 16649 16711 16773 16835 16897 16969 170-21 17084 47 17146 17209 17272 17335 17o98 17461 17524 17587 17651 17714 48 17778 17842 17905 17969 18033 (18098 18162 182*6 18-291 18356 49 18420 18485 18550 18615 18680 18746 18811 18877 18942 19008 50 19074 19140 19206 19272 19339 19405 19472 19538 16605 19672 51 19739 19806 19873 19940 20008 20075 20143 20211 20279 20347 52 20415 20483 20551 20620 20688 20757 20826 20894 20963 21032 53 21102 21171 21241 21310 213x0 21450 21519 21589 21659 21730 54 21800 21870 21941 22012 22082 22153 22224 22'295 22366 22438 55 22509 22581 22652 22724 22796 22868 22940 23012 23085 23157 56 23230 23302 23375 23448 23521 23594 23667 23741 23814 23888 57 23961 24035 24109 24183 24257 24331 24405 24480 24554 24629 58 24704 24779 24854 24929 25004 25079 25155 25230 25306 25381 59 25457 25533 25609 25686 25762 25838 25915 25992 26068 26145 60 26222 26299 26376 26454 26531 26609 26686 26764 26842 26920 For continuation to 100 fret, see Table 7- 426 RAILROADS. Table 7. Level Cuttings. Continuation of the six foregoing Tables of Cubic Contents, to 100 feet of height or depth. Height or Depth in Feet. Table 1 Table 2 Table 3 Table 4 Table 5 Table 6 Cu. Yds. Cu. Yds. Cu. Yds. Cu. Yds. Cu. Yds. Cu. Yds. 61 23835 26094 17848 24739 20107 26998 .5 24201 26479 18108 25113 20386 27390 62 24570 26S67 18370 25489 20667 27785 .5 24942 27257 18634 25868 20949 28183 63 25317 27650 1S900 26250 21233 28583 .5 25694 28046 19168 26635 21519 28986 64 26074 28444 19437 27022 21807 29393 .5 26457 28846 19708 27413 22097 29801 65 26843 29250 19981 27806 22389 30213 .5 27231 29657 20256 28201 22682 30627 66 27622 30007 20533 2SGOO 22978 31044 .5 28016 30479 20812 29001 23275 31464 67 28413 30894 21093 29406 23574 31887 .5 28812 31313 21375 29813 23875 32312 68 29215 31733 21659 30222 24178 32741 .5 29G20 32157 21945 30635 24482 33172 69 30028 32583 22233 31050 24789 33605 .5 30438 33013 22523 31468 25097 34042 70 30852 33444 22814 31889 25107 34481 .5 31268 33879 23108 33313 25719 34924 71 31687 34317 23404 32739 26033 35369 .5 32108 34757 23701 33168 26349 35816 72 32533 35200 24000 33600 26667 36267 .5 32960 35646 24301 34035 26986 36720 73 33390 36094 24604 34472 27307 37176 .5 33823 36546 24907 34913 27631 37635 74 34259 37000 25214 35356 27956 38096 .5 34697 37457 25522 35801 28282 38561 75 35139 37917 25832 36250 28611 39028 .5 35582 38379 26144 36701 28942 39498 76 36029 38844 26458 37156 29174 39970 .5 36479 39313 26774 37613 29608 40446 77 36931 39783 27092 38072 29944 40924 .5 37386 40257 27411 38535 30282 41405 78 37844 40733 27733 39000 30622 41889 .5 38305 41213 28056 39468 30964 42375 79 38768 41694 28381 39939 31307 42865 .5 39235 42179 28708 40413 31653 43357 80 39704 42667 29037 40889 32000 43852 81 40650 43650 29700 41850 32700 44850 82 41607 44644 30370 42822 33407 45859 83 42576 45650 31048 43806 34122 46880 84 43555 46667 31733 44800 34844 47911 85 44546 47694 32426 45806 35574 48954 86 45548 48733 33126 46822 S6311 60008 87 46561 49783 33833 47850 37056 51072 88 47585 50844 34548 48889 37807 52148 89 48620 51917 35270 49939 38567 53235 90 49667 53000 36000 51000 39333 54333 91 50724 54094 36737 52072 40107 55443 92 51793 55200 37481 53156 40889 56563 93 52872 56317 38233 64250 41678 57694 94 53963 57444 38993 55356 42474 58837 95 55065 58583 39759 56472 43278 59990 96 56178 59733 40533 57600 44089 61155 97 57302 60894 41315 58739 44907 62331 98 58437 62067 42104 59889 45733 63518 99 59583 63250 42900 61050 46567 64716 100 60741 64444 43704 62222 47407 65926 RAILROADS. 427 Table 8, Of Cubic Yards in a 100-foot station of level cutting or filling, to be added to, or sub- tracted from, the quantities in the preceding seven tables, in case the excava- tioua or embankments should be increased or diminished 2 feet in width. Cubic Yards iu a length of 100 feet; breadth 2 feet; and of different depths. Height or Depth in Feet. Cubic Yards. Height or Depth in Feet. Cubic Yards. Height or Depth in Feet. Cubic Yards. Height or Depth in Feet. Cubic Yards. Height or Depth in Feet. Cubic Yards. .5 3.70 .5 152 .5 300 .5 448 .5 696 1 7.41 21 156 41 304 61 452 81 600 .5 11.1 .5 159 .5 307 .5 456 .5 604' 2 14.8 22 163 42 311 62 459 82 607 .5 18.5 .5 167 .5 315 .5 463 .5 611 3 22.2 23 170 43 319 63 467 83 615 .5 25.9 .5 174 .5 322 .5 470 .5 619 4 29.6 24 178 44 326 64 474 84 622 .5 33.3 .5 181 .5 330 .5 478 .5 626 5 37.0 25 185 45 333 65 481 85 630 .5 40.7 .5 189 .5 337 .5 485 .5 633 6 44.4 26 193 46 341 66 489 86 637 .5 48.1 .5 196 .5 344 .5 493 .5 641 7 51.9 27 200 47 348 67 496 87 644 .5 55.6 .5 204 .5 352 .5 500 .5 648 8 5'J.3 28 207 48 356 68 504 88 652 .5 63.0 .5 211 .5 359 .5 507 .5 656 9 66.7 29 215 49 363 69 511 89 659 .5 70.4 .5 219 .5 367 .5 515 .5 663 10 741 30 222 50 370 70 519 90 667 .5 77.8 .5 226 .5 374 .5 522 .5 670 11 81.5 31 230 51 378 71 526 91 674 .5 852 .5 233 .5 381 .5 530 .5 678 12 88.9 32 237 62 385 72 533 92 681 .5 92.6 .5 241 .5 389 .5 537 .5 685 13 96.3 33 244 53 393 73 541 93 689 .5 100 .5 248 .5 396 .5 544 .5 693 14 104 34 252 ,54 400 74 648 94 696 .5 107 .5 256 .5 404 .5 652 .5 700 15 _ 111 35 259 55 407 75 556 95 704 .5 115 .5 263 .5 411 .5 659 .5 707 16 119 36 267 56 415 76 563 96 711 .5 122 .5 270 .5 419 .5 567 .5 715 17 126 37 274 57 422 77 670 97 719 .5 130 .5 278 .5 426 .5 674 .5 722 18 133 38 281 58 430 78 578 98 726 .5 137 .5 285 .5 433 .5 581 .5 730 19 141 39 289 59 437 79 585 99 733 .5 144 .5 293 .5 441 .5 589 .5 737 20 148 40 296 60 444 80 593 100 741 REMARK. The foreg-oinsr tables of level cntting-s may also be used* for widths of roadway greater than those at the heacta Of the tables. Thus, suppose we wish to use Table 1, for a roadbed m n, 16 ft wide, instead of c 6, which is only 14 ft, and for which the table was calculated. It is only necessary first to find the vert dist s a, between these two roadbeds ; and to add it mentally to each height t s, of the given embkt, when taking out from the c a \ 428 RAILROADS. table the numbers of cub yds corresponding to the heights. By this means we obtaim the contents of the embkt c b a p, for any required dist. Next, from these contents subtract that corresponding to the height * a, for the same dist. The remainder will plainly be the embkt m n op. In practice it will be sufficiently correct, to take 8 a to the nearest tenth of a foot, which will save trouble in adding it mentally to the heights in the tables. If the roadbed is narrower than the table, as. for instance, if mnbe the width in the table, but wo wish to flnd the contents for the width c b, then first find s a, and cal- culate the cub yds in 100 ft length of c 6 TO n. Then, in taking out the cub yds from the table, first subtract a mentally from each height; and to the cub yds taken out for each 100 ft, opposite this reduced height, add the cub yds in 100 ft of c 6 m n. To avoid trouble with contractors about the measurement of rock cuts, stipulate in the of say about 2 ft of width of cut will be made, to cover the unavoidable irregularities of the sides. TURNTABLES. 429 TURNTABLES, A turntable is a platform, usually from 40 to 60 ft long, and about 6 to 10 ft wide, (see Fig 1,) upon which a locomotive and its tender may be run, and then be turned around horizontally through any portion of a circle; and thus be transferred from one track to another forming any angle with it. The table is supported by a pivot under its center; and by wheels or rollers under its f.vo ends. Frequently other rollers are added between the center and ends. Beneath the platform is excavated a circular pit about 4 or 5 ft deep, having its circumf lined with a wall of masonry or brick about 2 ft thick, capped with either cut stone or wood. The diam of the pit in clear of this liuing is about 2 ins greater than the length of the turntable. The lining is gen- erally built with a step, as seen in Pig 1, for supporting the circular rail on which the end rollers travel ; or, instead of this step, a detached support may be used for this circular rail, as at u, in Figs 7. At the center of the pit is a solid well-founded mass of masonry or timber, for the pivot to rest on, as seen in Fig 1. This, as well as the step for the end rollers, should be very firm, and perfectly level ; other- wise the platform will be hard to work. The platform is frequently floored across for a width of 6 to 10 ft, to furnish a pathway across the pit, without stepping down into it; especially when under cover of a building. At first they were floored over so as to cover the entire circular pit; but this in- creased not only their cost, but their wt, so as to make them difficult to turn ; besides causing much expense for repairs ; with greater trouble in making them. It is therefore rarely done at present, except where want of space sometimes renders it necessary in indoor turntables. A turntable should be several feet longer than is necessary for merely allowing the engine and tender to stand on it ; for the increased length enables the engine-men to move them a little backward or forward, so as to balance them chiefly upon the central support; and thus relieve the end rollers. By this means the friction while turning is confined as much as possible to the center of motion ; and is therefore more readily overcome than if it were allowed to act at the circumf. The eugine-mea soon learn, by feeling, the proper spot for stopping the engine so as thus to balance the platform. The Sellers cast-iron turntable* of the well-known firm of Wm. Sel- lers & Co, machinists, Philada, shown in Figs 1 to 6. is probably the most perfect one that has been devised. It is expensive in first cost, but economical in the long run. One man can readily turn it without the aid of machinery, when loaded with a heavy engine and tender. It is extensively used ail over the U. S. ; and the principle i's also largely applied to great turning bridges. It consists of two heavy cast-iron girders, one of which is seen in Fig 1 ; and parts of one in Figs 3 and 4. The curved. ~ - tops of these girders are straight, and the bottoms cu; They are perforated by circular openings Scale of Feet forFigs 2. 3. 4 X- Wrought Iron turntables are now coming into coiumou use. 28 430 TURNTABLES. to save metal. A transverse section of one is shown at g, Fig 3. The metal averages abont \% Ins thick. Each of these girders is in two separate pieces, which are fastened to a kind of central hol- low cast-iron boxing, a transverse section of which is shown by A B J ,1 D L, Pig 2 ; and a top view by A B C, Fig 4. This boxing is all cast in one piece. Its four vertical sides, about \% ins 'thick, are solid ; the opening shown in Fig 1 does not exist, but was inserted in the cut in order to show the central pivot. The top and bottom have openings, as in Figs 2 and 4. The girders are fastened to the central boxing, by means of heavy bars 3% ins square, of rolled iron, o, o, Figs 2 and 4, or x, S, Fig 3, tilting iuto sunk recesses on top of the boxing, and tightened in place by wedges j, i, screw- bolted beneath, as shown below S, Fig 3; and by two 2%-iuch key-bolts/. Fig 3, which pass through the bolt holes h, fi, Fig 2; and are confined to the girders by horizontal keys just below R, Fig 3, where the keyhole is seen. The central portion D L, Fig 2, of the boxing, is a hollow cone, open at top and bottom, and loosely surrounding the hollow conical pivot-post P. To save material, two openings II are left in its cir- cumf. The post P is about \% ins thick ; and is firmly bolted down to the large block of cut stone M. which caps the supporting masonry of the post. On top of the post rests loosely a rough cast iron it, without at all ^training the friction rollers. This cap supports the steel box. seen above it in Fig 2, or at tt, Fig 5, or U V, Fig 6, which contains the steel frictioil rollers, r, d. &c, Figs 5 and . There are about 15 of these ; each about 2% ins, both in length, and in greatest diam. They have no axles ; but merely lie loosely in the lower part of the box ; filling its circumf with the exception of about ^ an inch left for play. In the direction of their axes they have but ^ inch play in the box. The lid U, Fig 6, of the roller-box, rests on top of the rollers themselves ; and does not come down to the lower part V of the box by about 5* inch. Both the rollers themselves, and the insides of the box in contact with them, are finished with mathematical accuracy, so as to insure a perfect bearing between them. The rollers are kept constantly well oiled, as much of the ease of turning the platform depends upon it. To oil them, the screws of the cap C are loosened ; and a mix- ture of oil and tallow sufficient for several months is put into the roller-box. On top of the roller- box is the cast cap C C, which is bolted down by 8 screw-bolts of 1^ diam. These bolts sustain all the weight of the entire platform and its load, except what little may rest on the two rollers at each of its ends in case the engine is not perfectly balanced upon the rollers alone. By partly unscrew- ing them, all the platform is lowered around the post; only n, the rollers above it, and the cap C C, remaining unmoved on top of the post. This furnishes the means of adjusting the height of the platform above the bottom of the pit, so as to bring the end rollers to their due bearing upon the cir- cular rail upon which they travel. These end rollers should barely touch lightly on their rail; be- cause the object in the Sellers table, and one of its important features, is to throw all the weight upon the friction rollers. The friction being thus kept near the center of motion, has but little leverage with which to resist the turning of the table. The diam of the roller-box being as great as 15 inches, it is not difficult to balance the engine and tender upon it alone; but this cannot be done upon a central pivot only about 6 ins diam; and more especially because the foot of such a pivot should be convex, as at "f. Figs 7, so as to adjust itself to the slight tilting of the platform when the load enters or leaves it. Under C C, Fig 2, are blocks, ww, of hard wood, which are driven between the eight screws, so as to hold all in place after this adjustment is completed. * All turntables should have the means of making such adjustment. In Figs 7 (of another design of platform) it is attained by placing the central pivot,/, at the foot of a large screw, by turning which the whole platform can be raised or lowered at pleasure. After thus making the adjustment, the screw is keyed fast to the platform, so as to revolve with it without screwing or unscrewing itself. The ends of the two girders of the Sellers are firmly connected transversely by heavy cast-iron beams ; the ends of which project sideways beyond the girders ; and carry the cast iron end rollers, 20 ins diam; two at each end of the platform. Intermediate transverse connection is secured by the wooden cross-ties notched upon the girders to support the rails, and frequently 10 or 12 feet long, for giving a wide footway across the pit. A lever 8 or 10 ft long, fitting iutp a staple, is used for turning these platforms, not on account of friction, but to afford a handhold to the man who turns them. Wooden turntables, with none but two common wheel rollers at each end of the platform are sometimes resorted to from motives of original cost. They are, however, much harder to turn, generally requiring two men, aided by wheelwork : and are more liable to get out of order ; and more expensive to repair. They are made of a great variety of patterns, both as regards the girders and the central pivots, end rollers, &c. Frequently an addition is made of 8 to 12 small rollers travelling on a circular rail of 6 to 12 ft diam, around the pivot as a center. These are in- tended to sustain the whole weight ; the end rollers being so adjusted as to touch their rail only when the platform rocks or tilts as the eneine enters or leaves it. Therefore, there is less resistance from friction than when, as in Figs 7, there are only the end rollers r. In this last case, the engine and tender cannot be balanced so precisely upon the slender central pivot, as to prevent a great part of the wt from being thrown upon the end rollers ; thus materially increasing the frictional resistance. Tn plan, these wooden platforms are generally in shape of a cross ; that is, in i ition to the main v e ,^e platform is intended to carry the wheelwork R x a;, for turning the platform ; and the other arm aerves merely as a balance to it; therefore, neither of them requires to be very strong. It tant to connect the four ends of the two platforms by four beams, as the whole structure is thereby ma- terially stiffened. In the figs the wheelwork Exxis for convenience improperly shown as if it stood upon the main platform. * The price of the Sellers turntable, at the shop in Philada in 1880, is about as follows : 30 ft long, $800: 40 ft, $1200; 45 ft, $1300; 50 ft, $1400; o4 ft, $1800; 60 ft, $1900. A 45-ft one weighs about 23000 B>s ; a 50-ft, 24700 Ks ; a 54-ft, 33000 ; thus making the average in the shop say 5> cts per ft. These prices do not include cross-ties and other woodwork ; nor the circular railway for the end rollers. Machinery for turning, being considered unnecessary, is not attached, unless specially ordered. Its cost is extra. The entire cost of excavating and lining the pit; foundation for pivot; circular rail for end rollers, &c, complete for a 5-ft turntable will vary from $1200 to $2500 in addition, depending on the class of materials and workmanship; and whether the bottom of the pit is paved or not. TURNTABLES. 431 The Figs 7 need butflttle explanation. They represent an actual 45 ft platform, which has been in use for some years./ The convex foot / of the ceutral pivot, about 6 ins diaru, should be faced witr- steel ; and should pest ou a steel step n s. This should be kept well oiled ; and protected from dust by a leather collar ylrouud p, and resting on / inch thick, tor a distance of about 15 ius. It works in a female sorew ia TR.SEC.AT CENTER I ' ' M ' I ' ' I " I or IFOOT the strong cast-Iron nut y y. and serves for raising the whole platform when necessary. When not in use for this purpose, it is keyed tight to the platform, (by a key at its head .) so as to revolve with It. Strong screw bolts t j connect the several timbers at the center of the platform. R is a light cast-iron stand supporting two bevel wheels about 1 foot diam. which give motion by means of an axle d. I % ins diam to two similar ones below, shown more plainly at W and Y. These last give motion by the axle x to the pinion e. (6 ins diam, and '2% ins face,) which turns the platform by work- ing into a circular rack t, (teeth horizontal, 1 inch ntch ; 3^ inches face.) which surrounds the entire pit. This rack is spiked to the under side of a continuous wooden curb H which is upheld bv pieces F, a few feet apart, which are let into the wall J J, which lines the pit. The short beam M N, (about 6 ft,) which carries the lower wheelwork, is suspended stronglv from the beams of the transverse plat- form above it. Instead of the t-vo lower bevel wheels W Y, and the horizontal axle x. a more simple arrangement is to place the pinion at the lower end of the vertical axle d; and let it work into a rack with vertical teeth at u, on the inner face of the stone foundation of the circular rail. For this purpose the stand R should be directly over u. There are two cast-iron rollers r. 2 ft diam, 3 inch face, under each end of the main platform; and one under each end of the secondary one. Although this kind of platform necessarily has much friction, yet one man can generally tnrn a 45- ft or.e by means of the wheelwork, when loaded with a heavy engine and tender. Indoed^ he may do it with some difficulty by hand only, while all is new and in perfect order : but when old. and the'cir- cular railway uneven and dirty, it requires two men at the winches to do it with entire ease. As before remarked, the resistance to turning is di- minished by employing a set of from 8 to 12 rollers or wheels r. Figs 8, about a foot to 15 ins in diam, so ar- ranged as to form a circle 8 to 12 ft diam around the pivot. When this is done, the main girders of the plat- form are placed 8 to 12 ft apart : and long cross-ties are used for supporting the railway track. Also, the main girders are sometimes trussed by iron rods, as in the swing bridge on page 271 : but instead of one post a c. it is best to have two, 6 or 8 ft apart at foot, and meeting at top. The width of platform must then be sufficient to allow the engine to pass the posts ou either side of it. Ten feet will suffice. Fig 8 shows the arrangement of these rollers r, which revolve upon a circular trxck ; while the plat- form rests on their tops hy the track u. The rollers r are held between two wrought-iron rings o, o, about 3 ins deep, by ^ inch thick, which also are carried by the rollers. From each rolle tie-rod t. 1 inch diam, extends to a ring n n. which surrounds the pivot p. closelv, hut not t as to revolve independently of it. These tie-rods keep the rings oo at their proper dist from so that the rollers cannot leave the rails and u. Between each two rollers, the rings o o strengthened by some arrangement like a, to prevent change of shape. The pivot p may be . lal two rollers under each end of the platform, for susta , he the radial ghtly. so he pivot, hould b inFig, ning the There must, of coun the central rollers. Such a platform of 50 ft length can."if"carefulVy made7be"ta'rned.''tnget > ner wi'th an engine and tender, by one man. by means of a wooden lever 12 to 15 ft long, inserted in a staple for that purpose : and therefore may dispense with the transverse platform for sustaining wheelwork. ;ver, is usually added. This, Such rollers as have just been described, in connection with the friction rollers Fig 5 is perhaps the best arrangement for a large turning bridge. At least one end 432 WATER STATIONS. of a platform must be provided with a Catch or stop for arresting its motion at the moment it has reached the proper spot. A common mode is shown at Figs 9. It consists of a wrought-iron bar m n, 4 ft long, 3 ins wide, and % thick ; hinged at its end m, which is confined to the top of the platform. Its outer end 71 is formed into a ring v for lifting it. A strong casting e e, (or in longitudinal section at t t,) about 15 ins long. 3 ins wide, and 1 inch thicK, is also firmly bolted to the top of the plat- form ; and the stop-bar m n rests in its recess r, while the platform is being turned. A similar casting a a is well bolted to the wooden or stone coping c c. which surrounds the top of the lining wall of the pit. When the stop-bar reaches this last casting, as the platform revolves, it rises up one of its little inclined planes t t, and falls into the re- cess of a a, bringing the platform to a stand. When the platform is to be started again, tne bar is lifted out of its recess by the ring v. until it passes the casting; when it is again laid upon the coping c c, and moves with the plat- form ; or, if required, the hinge at m allows it to be turned entirely over on its back. When there is a transverse platform, the proper place for the stop is at thnt end which carries the turning gear; as it is there handy to the men who do the turning. If there is only a main platform, the stop may be placed midway of the rails. Sometimes a Spring Catch is placed at each end of the platform ; and each catch is loosened from its hold at the same instant by a long double-acting lever. All the details of a platform admit of much variety. Sometimes the girders are made of plate iron, like Figs 22 to 24, p 214. Instead of the friction rollers. Fig 5, friction balls 5 or 6 ins diam. of polished steel, are sometimes used. The pivots also are made in many shapes. Plat forms like on, Fig 1O, revolving around one end o as a center of motion, are sometimes useful. The shaded space is the pit. If an engine approaching along the track W, is intended to pass on to any one of the tracks 1, 2, 3, 4. the platform is first put into the required position, and the engine passes at once without de- tention. If the platform is long, it will be necessary to have roller- wheels not only under the moving end a, but at one or two other points, as- indicated by the roller rails cc. 10 WATEE STATIONS, Water stations are points along a railroad, at which the engines stop to take In water. Their distance apart varies (like that of the fuel stations, which accompany them.) from about 6 miles, on roads doing a very large business ; to 15 or 20 miles on those which run but few trains. Much depends, however, upon where water can be had. It has at times to be conducted in pipes for 2 or 3 miles or more. The object in having them near together is to prevent delay from many engines being obliged to use the same station. To prevent interruption to travel, they are frequently placed upon a side track. A supply of water is kept on hand at the station usually in large wooden tubs or tanks, enclosed in frame tank-houses. The tank house stands near the track, leaving only about 2 to 4 ft clearance for the cars. It is two stories high ; the tank being in the upper one ; and having Its bottom about 10 or 1 2 feet above the rails. In the lower story is usually the pump for pumping up the water into the tank; and a stove for preventing the water from freezing in winter.* The tanks are usually circular; and a few inches greater in diam at the bottom than at the top, so that the iron hoops may drive tight. Their capacity generally varies from 6000 to 40000 gallons, (rarely 80000 or more.) depending on the number of engines to be supplied. A tender-tank holds from 1200 to 2400 gallons ; and an onerine evaporates from 20 to 150 gallons per mile, depending on the class of engine; weight of train ; steepness of grade, Ac. Perhaps 40 gallons will be a tolerably full aver- age for passenger, and 80 for freight engines. The following are the contents of tanks of different inner diams, and depths of water. U. S. galls of 231 cub ins ; or 7.4805 galls to a cub ft. See p. 434. Diam. Depth. Gallons. Cub. Ft. Diam. Depth. Gallons. Cub. Ft. Ft. Ft. Ft. Ft. 12 8 6767 905 24 12 40607 5429 14 9 10363 1385 26 13 51628 6902 16 9 13535 1810 28 14 64481 8621 18 10 19034 2545 30 15 79310 10603 20 10 23499 3142 32 16 96253 12868 22 11 31277 4181 34 17 115451 15435 Cypress or any of the pines answer very well for tanks. The staves may be about 2% ins thick for the smaller ones ; to 4 or 5 ins for the largest. The bottoms may be the same. The staves should be planed by machinery to suit the curve precisely. Nothing is then needed between the staves to pro duce tightness. A single wooden dowel is inserted between each two near the top, merely to hold them in place while being put together. The bottom isdowelled together ; and simply inserted into a groove very accurately cut, about an inch deep, around the inner circumf of the tub, at a few inches above the bottoms of the staves. (For iron ones see p 434.) * A frame tank-house. 18 ft square, with stone foundations for both bouse and tank, will by itsel* cost $400 to $600. A brick or stone one somewhat more. ATER STATIONS. 433 One of 2O fi cliaiii, and 12 ft deep, may have 9 hoops of good iron; placed several incjrtfs nearer together at the bottom of the tank than at the top. Their width 3 ins; the thickness of/ftie lower two, y iuch ; thence gradually diminishing until the top one is but half as thick. The^iower two are driven close together. These dimensions will allow for the rivet-holes into place.* Tnree rivets of % iuch diam, and 3 ins apart, in line, are sufficient for a joint of a lower hoop. One of 34 ft diam, 17 deep, may have 12 hoops ; the lower ones 4 ins by ^ ; with three % inch rivets to a lower hoop-joint. The bottom planks of the tank must bear firmly upon their supporting joists, or bearers. A tank must have an inlet-pipe by which the water may enter it; a waste-pipe for preventing overflow; and a discharge or feed-pipe 7 or 8 ias diam. in or near the bottom; through which the water flows out to the tender. The inner end of the discharge-pipe is covered by a valve, to be opened at will by the engine man, by means of an outside cord and lever. To its outer end is generally at- tached a flexible canvas and gum-elastic hose about 7 o: 8 ins diam, and 8 or 10 ft long, through which the water enters the tender-tank. Or, instead of a hose, the feed-pipe may be prolonged by a metallic pipe, or nozzle, sufficiently long to reach the tender; and so jointed as, when not in use, to swing to one side, or to be raised to a vertical position, (in the last case it is called a drop,) so as not to be in the way of passing trains. The same tank may supply two engines on different tracks, at once. The tanks are very durable. The patent frost-proof tank of John Bnriiham, Batavia, Illinois, is simply an ordinary tank, in which the water is prevented from freezing by means, 1st, of a circular roof which protects a ceiling of joists, between which is a layer of sawdust; 2d, by an air-space ob- tained by a ceiling of boards beneath the timbers on which the tank rests. Although the sides are entirely unprotected, no house is necessary : but merely strong posts and beams on a stone founda- tion, for the support of the tauk.t Tanks are frequently made rectangular, with vertical sides of posts lined with plank, and braced across in both directions by iron rods. They are more apt to leak than circular ones. They have been made of iron ; but wood seems to be preferred. The water for supplying the tanks, may be pumped by hand, steam, horse, wind, hydraulic ram, or otherwise, from a running stream ; from a pond made by damming the stream if very small or irregular: from a cistern below the tank ; or from a common well. Many roads doing a business of 10 or 12 engines daily in each direction, depend entirely upon wells ; and pump by hand ; generally two men to a pump. Those doing a very large business, when the supply cannot be obtained by gravity, mostly use steam. The windmill is the most economical power; and when well made, is very little liable to get out of order. Of course it will not work during a calta ; but this objection may be obviated in most cases by having the tanks large enough to hold a supply for several days.i Steam, however, is most reliable. The following* table will give some idea of the power reqd in a steam en- gine for the pumping. In ordering an engine, specify not its number of horse-powers, but the num- ber of gallons it must raise in a given number of hours, to a given height; with a given steam pres- sure, (say about 60 to 80 fts per sq inch.) The pump should be sufficiently powerful not to have to work at night; and should be capable of performing at least 25 per cent more than its reqd duty. A fair average horse should pump in 8 hours the quantities con- tained in the first 3 cols ; to the height in the 4th col ; or sufficient to supply the number of locomo- tives in the 5th col, with about 2000 gals each. Two men should do about ^ as much.** Cub. Ft. Lbs. Gals. Ht. Ft. No. of Locos. Cub. Ft. Lbs. Gals. Ht. Ft. No. of Locos. 1600 2000 2667 3200 3555 4000 100000 125000 166666 200000 222222 250000 11968 14960 19946 23936 26596 29920 100 80 60 50 45 40 6 7J* 10 12 ll y < 4571 5333 64^0 8000 10607 16000 285714 333333 400000 500000 666667 1000000 34194 39893 47872 59840 79787 119680 35 30 25 20 15 10 17 20 24 30 40 60 A reservoir, with a stand-pipe, or water column, is preferable to the ordinary tank, when the locality admits of it ; being less liable than the pump to get out of order ; and being cheaper in the end. The reservoir is supposed to be filled by water flowing into it by * Such a tank, put up in its place, will cost about $400. Greo. J. Burkhardt & Co, 1341 Buttonwood Street, Philada, make tanks their specialty ; and are provided with machinery which secures perfect accuracy of joint in every part. Their work is sought from great distances. i THE " RAILWAY FROST-PROOF TANK Co,' 1 OF BATAVIA, 111, make a specialty of the construction and erection of these tanks, complete in every detail, ready for use. They also make windmills. J ANDREW J. CORCORAN, No 76 John St. N. York, furnishes excellent machines. He also, when de- sired, provides pumps, &c, complete. The cost of windmill alone, for railway stations, varies from about $iOO for 18 ft diam ; to $1400 for 36 ft diara, at the factory. * * The *. or say .8 of a gallon of water to 1 B> of coal, and assume, as on page 432, that a passenger engine evaporntes an average of 40 trallons per mile, and a freight engine 80 galls, we shall have very nearlv 2% tons of coal consumed per 100 miles by the former; and 4^ tons by the latter. The evaporation from a heavily tasked powerful engine may amount to 150 galls or more per mile; but such ia an exceptional case. Thickness near bottom of sheet-iron water tanks, single iveted; safety*; ultima- - x iW punching the rivet holes. riveted ; safety 4; ultimate strength of the iron 40,000 tts per sq inch, but reduced say one-half by "--ugh safe from the water, some are plainly far too thin for handling. 1NNEB DIAMETER IN FEET. Depth in Feet. 5 10 15 20 25 3O 35 40 THICKNESS IN INCHES. Original. 1 5 10 15 20 25 30 .0026 .0130 -0260 .0391 .0521 .0651 .0781 .0052 .0260 0521 .0781 .1042 .1302 .1562 .0078 .0391 .0781 1172 .1562 .1953 .2344 .0104 .0520 .1042 .1562 2084 .2604 .3124 .0130 .0651 .1302 .1953 .2604 3255 .3906 .0156 .0781 .1562 .2344 .3125 .3906 .4687 .0182 .0911 .1823 .2734 .3645 .4557 .5470 .0208 .1042 .2083 .3125 .4166 .5208 .6250 Within the limit* of the table we may assume both the pres and the thickness to vary as either diam or depth. See Table, p 532. CIRCULAR ARCS IN FREQUENT USE. The fifth column is of use for finding points for drawing arcs too large for the beam-compass, on the principle given near foot of p 17. In even the largest office drawings it will not be necessary to use more than the first three decimals of the fifth column : and after the arc is subdivided into parts smaller than about 35 each, the first two decimals .25 will generally suffice. Original. Rise For For rise Rise For For in Degin For rad length of of half in Degin For rad length of rise of of arc. by chord mult rise of arc. mu^rise chord mult chord. by by chord. by rise by 1-60 o 9 9.75 313. 1 .00107 .2501 X ' 56 8.70 8.5 1.04116 .2538 1-45 10 10.75 253.625 1.00132 .2501 1-7 6346.90 6.625 1.05356 .2549 1-40 11 26.98 200.5 1.00167 .2502 .155 68 53.63 5.70291 1.06288 .2557 1-35 13 4.92 153.625 1.00219 .2502 1-6 73 44.39 5. 1.07250 .2566 1-30 15 15.38 113. 1.00296 .2503 .18 7911.73 4.35803 1.08428 .2576 1-25 1817.74 78.625 1.00426 .2504 1-5 87 12.34 3.625 1.10347 .2593 1-20 22 50.54 505 1.00665 .2506 .207107 90 3.41422 1.11072 .2599 1-19 24 2.16 45.625 1.0073T .2507 .225 96 54.67 2.96913 1.12997 .2615 1-18 25 21.65 41. 1.00821 .2508 X 106 15.61 2.5 1.15912 .2639 117 26 50.36 36625 1.00920 .2509 .275 11514.59 2.15289 1.19082 .2665 1-16 28 30.00 32.5 1.01038 .2510 .3 12351.30 1.88889 1.22495 .2692 1-15 30 22.71 28.625 1.01181 .2511 K 134 45.62 1.625 1.27401 .2729 1-14 32 31.22 25. 1.01355 .2513 .365 144 30 98 1.43827 1.32413 .2766 1-13 34 59.08 21.625 1.01571 .2515 .4 154 38.35 1.28125 1.38322 .2808 1-12 37 50.96 18.5 1.01812 .2517 .425 161 27.52 1.19204 1.42764 .2838 1-11 41 13.16 15.625 1.02189 .2520 .45 167 5633 1.11728 1.47377 .2868 1-10 45 14.38 13. 1.02646 .2525 .475 174 7.49 1.05402 1.52152 .2699 1-9 50 6.91 10.625 1.03260 .2530 .5 180 1. 1.57080 .2929 * AN UNCOVERED RESERVOIR 50 ft diam by 12 ft deep, lined with, brick or masonry, will usually cost from $2500 to $3500, according to circumstances. t THE PRICE OP A CAST-IRON WATER-COLUMN, of 6-inch bore; with bed-plate; holding-down bolts, and washers ; connecting-pipes ; swing-joint with copper arm 9 ft long ; valve ; hand- wheel ; &c ; com- plete, ready to set up, (by the Pascal Iron Works, Phtlada, in 1871,) is $475 at the shop. COS'I OF EARTHWORK. 435 COST OF EARTHWORK. THE following is takeu from the last edition of the writer's volume on the Measurement of Excava- tions and Embankments. Al*t. 1. It is advisable to pay for this kind of work by the cubic yard of excavation only ; in- utead of allowing separate prices for excavation and embankment. By this means we get rid of th difficulty of measurements, as well as the controversies and lawsuits which often attend the deter- mination of the allowance to be made for the settlement or subsidence of the embankments. It is, moreover, our opinion that justice to the contractor should lead to the Ellg'lisll prac- tice of paying 1 the laborers by the cubic yard, instead of by the day. Kxperience fully proves that when laborers are scarce and wages high, men can scarcely be depended upon to do three-fourths of the work which they readily accomplish when wages are low, and when fresh hands are waiting to be hired in case any are discharged. The contractor is thus placed at the mercy of his men. The writer has known the most satisfactory results to attend a system of task- work, accompanied by liberal premiums for all overwork. By this means the interests of the laborers are identified with that of the contractor ; and every man takes care that the others shall do their fair share of the task. Ellwood Morris. C E, of Philadelphia. Was, we believe, the first person who properly investigated the elements of cost of earthwork, and reduced them to such a form as to enable us to calculate the total with a considerable degree of accuracy. He published his results in the Journal of the Franklin Institute in 1841. His paper forms the basis on which, with some variations, we shall consider the matter ; and on which we shall extend it to wheelbarrows, as well as to carts. Throughout this paper we speak of a cubic yard considered only as solid in its place, or before it is loosened for removal. It Is scarcely necessary to add that the various items can of course only be regarded as tolerably close approximations, or averages. As before stated, the men do less work" when wages are high ; and more when they are low. A great deal besides depends on the skill, observation, and energy of the con- tractor and his superintendents. It is no unusual thing to see two contrp^tors working at the same prices, in precisely similar material, where one is making money, and the other losing it, from a want of tact in the proper distribution of his forces, keeping his road's in order, having his carts and bar- rows well filled, &c. &c. Uncommonly long spells of wet weather may seriously affect the cost of exe- cuting earthwork, by making it more diflicult to loosen, load, or empty ; beside's keeping the roads in bad order for hauling. The aggregate cost of excavating and removing earth is made up by the following items, namely : 1st. Loosening the earth ready for the nhoreUers. 2d. Loading it by shovels into the carts or barrowi. 3d. Hauling, or wheeling it away, including emptying and returning. 4th. Spreading it out into successive layers on the embankment. 5th. Keeping the hauling -road for carts, or the plank gangways for barrows, in good order. 6th. Wear, sharpening, depreciation, and interest on cost of tools. 7th. Superintendence, and water carriers. 8th. Profit to the contractor. We will consider these items a little in detail, baaing our calculations on the assumption that com- mon labor costs $1 per day. of 10 working hours. The results in our tables must therefore be in- creased or diminished in about the same proportion as common labor costs more or less than this. Art. 2. loosen i n:r the earth ready for the shovellers. This is generally done either by ploughs or l>y picks ; more cheaply by the first. A plough with two horses, and two'men to manage them, at $1 'per day for labor, 75 cents per day for each horse and 87 cents per day for plough, including harness, wear, repairs. &c. or a total of $3 87, will loosen, of strong heavy soils, from 200 to 300 oubic yards a day, at from 1.93 to 1/29 cents per yard ; or of ordinary loam, from 400 to 600 cubic yards a day, at from .97 to .64 of a cent per yard. Therefore, as an ordi- nary average, we may assume the actual cost to the contractor for loosening by the plough, as fol- lows: strong heavy soils, 1.6 cents ; common loam, .8 cei.t : light sandy soils, .4 cent. Very stiff pure olav, or obstinate cemented gravel, may be set down at 2.5 cents : they require three or four horses. By the pick, a fair day's work is about 14 yards of stiff pure clay, or of cemented gravel ; 25 yards of strong heavy soils; 40 yards of common" loam ; 60 yards of light sandy soils all measured in place; which, at $1 per day for labor, gives, for stiff clny. 7 cents; heavy soils. 4 cents; loam, 2.5 cents; light sandy soil, 1.668 cents. Pure sand requires but very little labor for loosening; .5 of a cent will cover it. Art. 3. Shovelling the loosened earth into carts. The amount shovelled per day depends partly upon the weight of the material, but more upon so proportioning the number of pickers and of carts to that of shovellers, as not to keep the latter waiting for either material or carts. Tn fairlv regulated gangs, the shovellers into carts are not actually engaged in shovelling for more than six-tenths of their time, thus being unoccupied but four- tenths of it: while, under bad management, they lose considerably more than one-half of it. A shoveller can readily load into a cart one-third of a cubic yard measured in place (and which is an average working cart- load), of sandy soil, in five minutes : "of loam, in six minutes: and of any of the heavy soils, in seven minutes. This would give, for a day of 10 working hours, ]'20 loads, or 40 cubic yards of light sandy soil ; 100 loads, or 33^ cubic yards of loam : or 86 loads, or 28.7 yards of the heavy soils. But from these amounts we must deduct four tenths for time necessarily lost: thus reducing the actual work- ing quantities to 24 yards of light sandy soil, 20 yards of loam, 17.2 yards of the heavy soils. When the shovellers do less than this, there is some mismanagement. Assuming these as fair quantities, then, at $1 per day for labor, the actual cost to the contractor for shovelling per cubic yard measured in place, will be, for sandy soils, 4.167 cents; loam, 5 cents; heavy soils, clays. Ac, 5.81 cents. In practice, the carts are not usually loaded to any leas extent with the heavier soils than with the lighter ones. Nor. indeed, is there any necessity for so doing, inasmuch as the difference of weight of a cart and one third of a cubic yard of the various soils is too slight to need any attention ; espe- cially when the cart road is kept in good order, as it will be by any contractor who understands hi* 436 COST OF EARTHWORK. own interest. Neither is it necessary to modify the load on account of any slight inclinations which may occur iu the grading of roads. An earth-cart weighs by itself about % a ton. Art. 4. Hauling away the earth; dumping-, or emptying; and returning to reload. The average speed of horses iu hauling is about 2^ miles per hour, or 200 feet per minute ; which is equal to 100 feet of trip each way ; or to 100 feet of lead, as the distance to which the earth is hauled is technically called.* Besides this, there is a. loss of about four minutes in every trip, whether long or short, in waiting to load, dumping, turning, &c. Hence, every trip will occupy as rnauy minutes as there are lengths of 100 feet each in the lead ; aud four minutes besides. Therefore, to find the number of trips per day over any given average lead, we divide the number of minutes in a working day by the sum of 4 added to the number of 100 feet lengths contained in the distance to which the earth has to be removed ; that is, The number (600) of minutes in a working day _ the number of trips, or loads 4 -f- the number of 100- feet lengths in the lead ~~ removed per day, per cart. And since % of a cubic yard measured before being loosened, makes an average cart-load, the num- ber of loads, divided by 3, will give the number of cubic yards removed per day by each cart; aud the cubic yards divided into the total expense of a cart per day, will give the cost per cubic yard !V>r hauling. In leads of ordinary length one driver can attend to 4 carts ; which, at $1 per day, is 25 cents per cart. When labor is"$l per dav. the expense of a horse is usually about 75 cents; and that of tiie cart, including harness, tar, repairs, &c, 25 cents, making the total daily cost per care $1.25. The expense of the horse is the same on Sundays and on rainy days, as when at work : and this consid- eration is included iu the 75 cents. Some contractors employ a greater number of drivers, who also help to load the carts, so that the expense is about the same in either case. EXAMPLE. How many cubic yards of loam, measured in the cut. can be hauled by a horse and cart in a day of 10 working hours, (600 minutes,) the lead, or length of haul of earth being 1000 feet, (or 10 lengths of 100 feet,) and what will be the expense to the contractor for hauling, per cubic yard, assuming the total cost of cart, horse, and driver, at $1.25? 600 minutes 600 , 43 loads Here, ; = = 43 loads. And =: 14.3 cubic yards. 4+ W lengths of 100 feet, 14 3 And , f- - = 8.74 cents per cubic yard. 14.3 cub yds In this manner the 2d and 3d columns of the following tables have been calculated. Art. 5. Spreading, or levelling off* the earth into regular thin layers Oil the embankment. A bankraan will spread from 50 to 100 cubic yards of either common loam, or any of the heavier soils, clays, &c. depending on their dryness. This, at $1 per day, is 1 to 2 cents per cubic yard; and we may assume 13^ cents as a fair average for such soils : while I cent will suffice for light sandy soils. This expense for spreading is saved when the ear*h is either dumped over the end of the embank- ment, or is wasted; still, about y cent per yard should be allowed iu either case for keeping the dumping-places clear and in order. REMARK. When removing loose rock, which requires more time for loading, say, No. of minutes (600) >>i a working day __ No. of loads removed, 6 + No. of 100-/eet lengths of load ~ per day, per cart. Art. 6. Keeping the cart-road in good order for hauling* No ruts or puddles should be allowed to remain unfilled: rain should at once be led off by shallow ditches ; and the road be carefully kept in good order; otherwise the labor of the horses, and the wear of carts, will be very greatly increased. It is usual to allow so much per cubic yard for road repairs ; but we suggest so much per cubic yard, per 100 feet of load ; say -^ of a cent. Art. 7. Wear, sharpening, and depreciation of picks and Shovels. Experience shows that about % of a cent per cubic yard will cover this item. Superintendence and water-carriers. These expenses win vary with lorsal circumstances ; but we agree with Mr. Morris, that 1 % cents per cubic varrt will, underordinary circumstances, cover both of them. An allowance of about y cent mav in justice be added for extra trouble in digging the side-ditches ; levelling off the bottom of the cut to {trade ; and general trimming up. In very light cuttings this may be increased to J^ cent per every yard. At y cent, all the items in this article amount to 2 cents per cubic yard of cut. Art. 8. Profit tO the Contractor. This may generally be set down at from 6 to 15 per cent, according to the magnitude of the work, the risks incurred, and various incidental cir- cumstances. Out of this item the contractor generally has to pay clerks, storekeepers, and other agents, as well as the expenses of shanties. &c ; although these are in most cases repnid by the profits of the stores; and by the rates of boarding and lodging paid to the contractors by the laborers. Art. 9. A knowledge of the foregoing items enables its to calculate with tolerable accuracy the cost of removing earth. For example, let it be required to ascertain the cost per cubic yard of excavating common loam, meas- ured in place; and of removing it into embankment, with n aroraee haul or l^ad of 1000 feet: the wages of laborers being $1 per day of 10 working hours ; a horse 75 cts a day ; and a cart 25 cts. One driver to four carts. Jf When an entire cut is made into an embankment, the mean haul is the dist between centers of gravity of the cut aud embkt. EARTHWORK. 437 Cents. Here we hayrtiost of loosening, say by pick, Art 2, per cubic yard, say, 2.50 Loadinajfito carts, Art. 3, " " 5.00 Hauling 1000 feet, as calculated previously in example, Art. 4, " 8-74 Spr/ading into layers. Art. 5, " 1.50 Keeping cart-road in repair, Art. 6, 10 lengths of 100 ft, 1.00 Various items in Art. 7, 2.00 Total cost to contractor, 20.74 Add contractor's profit, say 10 per cent, 2.074 Total cost per cubic yard to the company, 22.814 It is easy to construct a table like the following, of costs per cubic yard, for different lengths of lead. Columns 2 and 3 are first obtained by the Rule in Article 4 ; then to each amount in column 3 is added it particular circumstances, in tnis manner the tables have been prepared. By Carts. Labor .SI per day, of 1O working 1 hours. 1 t s Common Loam, Strong Heavy Soils, II \\ If K -si a.2 TOTAL COST PER CUBIC TOTAL COST PER CUBIC 32~ >, "2 ft YARD, EXCLUSIVE OF YARD, EXCLUSIVE OF js it Is PROFIT TO CONTRACTOR. PROFIT TO CONTRACTOR. ^5 a> |5 a rl c s l M i-1 3 i !lj 1*1 iji o fi! 1 | 3 i * OJ b Feet. Cu.Yds. Cts. Cts. Cts. Cts. Cts. Cts. Cts. Cts. Cts. 25 470 2.66 13.69 1244 11.99 10.74 6.00 14.75 13.50 12.25 50 44.4 2.81 13.86 12.61 12.16 10.91 6.17 14.92 13.67 12.42 75 42.1 2.97 14.05 12.80 12.35 11.10 6.36 15.11 13.86 12.61 100 40.0 3.12 14.22 12.97 12.52 11.27 6.53 15.28 14.03 12.78 150 36.4 343 14.58 13.33 12.88 11.63 6.89 1564 14.39 13.14 200 33.3 3.75 14.95 13.70 13.25 12.00 7.26 16.01 14.76 13.51 300 28.6 4.37 15.67 14.42 13.97 12.72 7.98 16.73 15.48 14.23 400 25 5.00 16.40 15.15 14.70 13.45 18.71 17.46 16.21 14.96 500 22.2 5.63 17.13 1588 . 15.43 14.18 19.44 18.19 16.94 15.69 600 700 20.0 18.2 6.25 17.85 16.60 16.15 14.90 20.16 18.91 17.66 16.41 800 16^7 7.48 19.28 18.03 17.58 16.33 20.88 21.59 19.63 20.34 19.09 17.13 17.84 900 15.4 8 12 19.92 18.67 18.22 16.97 22.23 20.98 19.73 18.48 1000 14.3 8.74 20.74 19.49 19.04 17.79 23.05 21 80 20.55 19.30 1100 13.3 9.40 21.50 20.25 1980 18.55 23.81 22.56 21.31 20.06 1200 12.5 10.0 22.20 20.95 20.50 19 25 24.51 23.26 22.01 20.76 1300 11.8 10.6 22.90 21.65 21.20 19.95 25.21 23.96 22.71 21.46 1400 11.1 11.2 23.60 22.35 21.90 20.65 25.91 24.66 23.41 22.16 1500 10.5 11.9 24.40 23.15 23.70 21.45 26.71 25.46 24.21 22.96 1600 10.0 12.5 25.10 23.85 23.40 22.15 27.41 26.16 24.91 23.66 1700 .52 13.1 25.80 24.55 24.10 22.85 28.11 26.86 2561 24.36 1800 .09 13.7 26.50 25.25 24.80 23.55 28.81 27.56 26.31 25.06 1900 .70 14.4 27.30 2605 25.60 24.35 29.61 28.36 27.11 25.86 2000 .33 15.0 28.00 26.75 26.30 25.05 30.31 29.06 27.81 26.56 2250 54 16.6 29.85 28.60 28.15 26.90 32.16 30.91 29.66 28.41 2500 90 18.1 31.60 30.35 29.90 28.65 33.91 32.66 31.41 30.16 M mile .58 19.0 32.64 31.39 30.94 29.69 34.95 33.70 32.45 31.20 3000 .88 21.2 35.20 33.95 33.50 32.25 37.51 36.26 35.01 33.76 3250 .48 22.8 37.05 35.80 35.35 34.10 39.36 38.11 36.86 35.61 3500 .13 24.3 38.80 37.55 37.10 35.85 41.11 39.86 38.61 37.36 3750 .82 259 40.65 39.40 38.95 37.70 42.96 41.71 40.46 39.21 4000 .54 27.5 42.50 41.25 40.80 39.55 44.81 43.56 42.31 41.06 4250 .30 29.1 44.35 43.10 42.65 41.40 46.66 45.41 44.16 42.91 4500 .08 30.6 46.10 44.85 44.40 43.15 48.41 47.16 45.91 44.66 4750 3.88 32.2 47.95 46.70 46.25 45.00 50.26 49.01 ! 47.76 46:51 5000 3.70 33.8 49.80 48.55 48.10 46.85 52.11 5086 49.61 48.36 1 mile 352 35.5 51.78 50.53 50.08 48.83 54.09 52.84 51.59 50.34 134m. 2.86 43.8 61.40 60.15 59.70 58.45 63.71 62.46 61.21 59.% \Yi m. 2.40 52.1 71.02 69.77 69.32 68.07 73.33 72.08 70.83 69.58 1% DV. 2.07 60.4 80.64 1 79.39 78.94 77.69 82.95 81.70 80.45 79.20 2 m. 1.82 68.7 90.26 89.01 88.56 87.31 92.57 91.32 ! 90.07 88.82 438 COST OF EAKTHWORK. By Carts. Labor $1 per clay, of 1O working hours. ! II 47.0 44.4 42.1 40.0 36.4 33.3 28.6 25.0 22.2 20.0 18.2 16.7 15.4 14.3 13.3 12.5 11.8 11.1 10.5 10.0 9.52 9.09 8.70 8.33 7.54 6.90 6.58 5.88 5.48 5.13 4.82 454 4.30 4.08 3.88 3.70 3.52 2.86 2.40T 2.07 1.82 2 .11 e-M _a.S if 2.66 2.81 2.97 3.12 3.43 375 4.37 5.00' 5.63 6'.'87 7.48 8.12 874 9.40 10.0 10.6 11.2 11.9 12.5 13.1 13 7 14.4 15.0 16.6 18.1 19.0 21.2 22.8 24.3 25.9 27.5 29.1 30.6 32.2 33.8 35.5 43.8 52.1 Pure stiff Clay, or cemented Gravel, Light Sandy Soils, TOTAL COST PER CUBIC TOTAL COST PER CUBIC YARD, EXCLUSIVE OF YARD, EXCLUSIVE OF PROFIT TO CONTRACTOR. PROFIT TO CONTRACTOR. lit H 1 ' *u 111 It! |,| L| 1*1 fr< oo s" J2 i** * s-g ** - * Cts. 1900 Cts. 17.75 Cts. 14.50 Cts. 13.25 Cw. 11.52 Cts. 10.77 Cts. 10.25 Cts. 9.50 19.17 17.92 14.67 13.42 11.69 10.94 10.42 9.67 19.36 18.11 14.86 13.61 11.88 11.13 10.61 9.86 19.53 18.28 15.03 13.78 12.05 11.30 10.78 10.03 20'26 19^01 15'.76 14.51 12.78 12.03 11.51 10.76 20^98 19.73 15.48 15.23 13.50 12.75 12.23 11.48 21.71 20.46 17.21 15.96 14.23 13.48 12.46 12.21 22.44 21.19 17.94 16.69 14.96 14.21 13.69 12.94 23.16 21.91 18.66 17.41 15.68 14.93 14.41 13.66 23.88 22.63 19.38 18.13 16.40 15.65 15.13 14.38 24.59 23.34 20.09 18.84 17.11 16.36 15.84 1509 25.23 23.98 20.73 19.48 17.75 17.00 16.48 15.73 .26.05 24.80 21.55 20.30 18.57 17.82 17.30 16.55 26.81 25.56 22.31 21.06 19.33 18.58 18.06 17.31 27.51 26.26 23.01 21.76 20.03 19.28 18.76 18.01 28 21 26.96 23.71 22.46 20.73 19.98 19.46 18.71 28.91 27.66 24.41 23.16 21.43 - 20.68 20.16 19.41 29.71 28.46 25.21 23.96 22.23 21.48 20.96 20.21 30.41 29.16 25.91 24.66 22.93 22.18 21.66 ! 20.91 31.11 29.86 26.61 25.36 23.63 22.88 22.36 21.61 31.81 30.56 2731 26.06 24.33 23.58 23.06 22.31 32.61 31.36 28.11 26.86 25.13 24.38 23.86 23.11 33.31 32.06 28.81 27.56 25-83 25.08 24.56 23.81 35.16 33.91 30.66 29.41 27.68 26.93 2641 25.66 36.91 35.66 32.41 31.16 29.43 28.68 28.16 27.41 37.95 36.70 33.45 32.20 30.47 29.72 29.20 28.45 40.51 39.26 36.01 34.76 33.03 32.28 31.76 31.01 42.36 41.11 37.86 36.61 34.88 34.13 33.61 32.86 44.11 42.86 39.61 38.36 36.63 35.88 35.36 34.61 45.96 44.71 41.46 40.21 38.48 37.73 37.21 36.46 47.81 46.56 43.31 42.06 40.33 39.58 39.06 38.31 49.66 48.41 45.16 43.91 42.18 41.45 40.93 4018 51.41 50.16 46.91 45.66 43.93 43.18 42.66 41.91 53.26 52.01 48.76 47.51 45.78 45.03 4451 4376 55.U 53.86 50.61 49.36 47.63 46.88 46.36 45.61 57.09 55 84 52.59 51.34 4961 48.86 4834 47.59 66.91 65.46 62.21 60.96 59.23 58.48 57.96 57.21 7633 75.08 71.83 70.58 68.85 68.10 67.58 66.83 85.95 84.70 81.45 80.20 7847 77.72 77.20 76.45 95.57 94.32 91.07 89.82 88.09 87.34 86.82 86.07 Art. 1O. By WtieelbarrOWS. The cost by barrows may be estimated In the same manner as by carts. See Articles 1, Ac. Men in wheeling move at about the same average rate as horses do in hauling, that is, 2# miles an hour, or 200 feet per minute, or 1 minute per every 100-feet length of lead. The time occupied in loading, emptying, &c (when, as is usual, the wheeler loads his own barrow,) is about 1.25 minutes, without regard" to length of lead; besides which, the time lost in occasional short rests, in adjusting the wheeling plank, and in other incidental causes, amounts to about 1 part of his whole time; so that we must in practice consider him as actually working but 9 hours out of his 10 working o^es. at the rate of 2.25 minutes per 100 feet of lead. To find. then, the number of Barrow-loads which he can remove in a day, multiply the number of minutes (600) in a working day by .9: and divide the product by the sum of 1.25, added to the number of 100-feet lengths in the lead ; that is, The number of minutes in a working day X .9 _ the number of trips or of loads 1 .25 + the number of 100- feet lengths of lea d ~ removed per day per barrow. Se* Remark, next page. The number of loads divided by 14 will rrive the number of cub yards, since a cub yard, measured in place, averages about 14 loads. And the cost of a tvheelrr and barrow per day. (say $1 per man and 5 cents per barrow.) divided by the number of oub yards, will give the cost pt>r yard fjr leading, wheeling, and emptying. COST OF EARTHWORK. 439 _ nmanir cubic yards of common loam, measured in place, will one man load, wheel, and empty, per day of 10 working hours, (or 600 minutes :) the lead, or distance to which the earth is removed being 1000 feet, (or 10 lengths of 100 feet ;) and what will be the expense per yard, supposing the laborer and barrow to cost SI. 05 per day ? u<\ r loads per day. And = 3.43 cub yd* per day. And - 30.6 cent, per cub yard for loading, wheeling away, emptying, and returning. This would be increased almost inappreciably by the cost of the shovel, which, in the following tables, however, is included in the cose of tools. Item. For rock, which requires more time for loading, say No of minutes in a working day X .9 _ X o o f loads removed 1.6 -4- JVo o/ 100-/ee lengths of lead ~ per day, per barrow. Al*t. 11. The following tables are calculated as in the case of carts, by first finding columns 2 and 3 by means of the Rule iu Art 4, and then adding to each sum in column 3, the variable quantity of .1 of a cent per cubic yard per 100 feet of lead for keeping the wheeling-planks in order; and the prices of loosening, spreading, superintendence, water-carrying, &c, per cubic yard, as given in th preceding Articles 2 to 7. By Wheelbarrows. Labor $1 per day, of 1O working hours, 3 !l ft S* Common Loam, Strong, Heavy Soils, a sl If li "Sw TOTAL COST PER CUBIC TOTAL COST PER CUBIC fcj "E * YARD, EXCLUSIVE OF YARD, EXCLUSIVE OF c| f* >! PROFIT TO CONTRACTOR. PROFIT TO CONTRACTOR. sf 1* 3s *s$ tl *;] 1^1 ii if Ill |,1 33 111 y Ill ill u e * c. 1 1 111 B j pi 11 u &< OQ * EC E ** CO * E M E - * ^Feet. Cu.Yds. ~~Ct" Cts. Cts. Cts. Cts. Cts. Cts. Cts. Cts. 25 25.7 4.09 10.12 8.87 8.42 7.17 11.62 10.37 9.12 7.87 50 22.1 4.75 10.80 9.55 9.10 7.85 12.30 11.05 9.80 8.55 75 19.3 5.44 1.52 10.27 9.82 8.57 13.02 11 77 10.52 9.27 100 17.1 6.14 2.24 10.99 10.54 9.29 13.74 12.49 11.24 9.99 150 14.0 7.50 3.65 12.40 11.95 10.70 15.15 13.90 12.65 11.40 200 11.9 8.82 5.02 13.77 13.32 12.07 16.52 15.27 14.02 12.77 250 10.3 10.2 6.45 15.20 14.75 13.50 17.95 16.70 1545 14.20 300 9.07 11.6 7.90 16.65 16.20 14.95 19.40 18.15 16.90 156S 350 8.14 12.9 19.25 18.00 17.55 16.30 20.75 19.50 1825 17.00 400 7.36 14.3 20.70 19.45 19.00 17.75 22.20 20.95 19.70 18.45 450 6.71 15.6 22.05 20.80 20.35 19.10 23.55 22.30 21.05 19.80 500 6.17 17.0 23.50 22.25 21.80 20.55 25.00 23.75 22.50 21.25 600 5.32 19.7 26.30 25.05 24.60 23.35 27.80 26.55 25.30 24.05 700 4.67 22.5 29.20 27.95 27.50 26.25 30.70 29.45 28.20 26.95 800 4.17 25.2 32.00 30.75 30.30 29.05 33.50 32 25 31.00 29.75 900 3.76 27.9 34.80 33.55 33.10 31.85 36.30 35.05 33.80 32.55 1000 3.43 30.6 37.60 36.35 35.90 34.65 39.10 87.85 36.60 35.35 1200 2.91 36.1 43.30 42.05 41.60 40.35 44.80 43.55 42.30 41.05 1400 2.53 41.5 48.90 47.65 47.20 45.95 50.40 49.15 47.90 46.65 1600 2.24 46.9 54.50 53.45 52.80 51.55 56.00 54.75 53.50 52.25 1800 2.00 52.5 60.30 59.05 58.60 57.35 61.80 60.55 59.30 58.05 2000 1.81 58.0 6600 64.75 64.30 63.05 67.50 66.25 65.00 63.75 2200 1.66 63.3 71.50 70.25 69.80 68.55 73.00 71.75 70.50 69.25 2400 1.53 686 77.00 75.75 75.30 74.05 78.50 77.25 76.00 74.75 K mile. 1.39 75.5 84.14 82.89 82.44 81.19 85.64 84.39 83.14 81.89 440 COST OF EARTHWORK. By Wheelbarrows. Labor $1 per day, of 1O working boars. 3 I'S P If Pure Stiff Clay, or Ce- mented Gravel, Light Sandy Soils, If 2 I !! 5 ^ ^ if 2 a TOTAL COST PER CUBIC TOTAL COST PER CUBIC o' 2 ; o * YARD, EXCLUSIVE OF YARD, EXCLUSIVE OF o^ " .2 * Nf PROFIT TO CONTRACTOR. PROFIT TO CONTRACTOR. 2 J .3 ii to* . Ifj * if Ill 2-gl u O 88 D. III Ill III U M 5 S5~ I 5 75 fc S ;* S * * *g BTP Feet. Cu.Yds. Cts. Cts. Cts. Cts. Cts. Cts. Cts. Cts. Cts. 25 25.7 4.09 14.62 13.37 10.12 8.87 8.79 8.04 7.52 6.77 50 22.1 4.75 15.30 14.05 10.80 9.55 9.47, 8.72 8.20 7.45 75 19.3 544 16.02 14.77 11.52 10.27 10.19 9.44 8.92 8.17 100 17.1 6.14 16.74 15.49 12.24 10.99 10.91 10.16 9.64 8.89 150 14.0 7.50 18.15 16.90 13.65 12.40 12.32 11.57 11.05 10.30 200 11.9 8.82 19.52 18.27 15.02 13.77 13.69 12.94 12.42 11.67 250 10.3 10.2 20.95 19.70 16.45 15.20 15.12 14.37 13.85 13.10 300 9.07 11.6 22.40 21.15 17.90 16.65 16.57 15.82 15.30 14.55 350 8.14 12.9 23.75 22.50 19.25 18.00 17.92 17.17 16.65 15.90 400 7.36 14.3 25.20 23.95 ?070 19.45 19.37 18.62 18.10 17.35 450 6.71 15.6 26.55 25.30 22.05 20.80 20.72 19.97 19.45 18.70 500 6.17 17.0 28.00 26.75 23.50 22.25 22.17 21.42 20.90 20.15 600 5.32 19.7 30.80 29.55 26.30 25.05 24.97 24.22 23.70 22.95 700 4.67 22.5 33.70 32.45 29.20 27.95 27.87 27.12 26.60 25.85 800 4.17 25.2 36.50 35.25 32.00 30.75 30.67 29.92 29.40 28.65 900 3.76 27.9 39.30 38.05 34.80 33.55 33.47 32.72 32.20 31.45 1000 3.43 30.6 42.10 40.85 37.60 36.35 36.27 35.52 35.00 34.25 1200 2.91 36.1 47.80 46.55 43.30 4205 41.97 41.22 40.70 39.90 1400 2.53 41.5 53.40 52, 15 48.90 47.65 4757 46.82 46.30 45.55 1600 2.24 46.9 59.00 57.75 54.50 53.25 53.17 52.42 51.90 51.15 1800 2.00 52.5 6480 63.55 60.30 59.05 58.97 58.22 57.70 56.95 2000 1.81 58.0 70.50 69.25 66.00 64.75 64.67 63.92 63.40 62.65 2200 1.66 63.3 76.00 74.75 71.50 70.25 70 17 69.42 68.90 68.15 2400 1.53 68.6 81.50 80.25 77.00 75.75 75-67 74.92 74.40 73.65 % mile. 1.39 75.5 88.64 87.39 84.14 82.89 82.81 82.06 81.54 80.79 Art. 14. Removing 1 rock excavation by wheelbarrows. A cubic yard of hard rock, in place, or before being blasted, will weigh about 1.8 tons, if sandstone or conglomerate, (150 Ibs per cubic foot;) or 2 tons if good compact granite, gneiss, limestone, or marble, (168 ft>s per cubic foot.) So that, near enough for practice in the case before us, we may as- sume the weight of any of them to be about 1.9 tons, or 4256 Ibs per cubic yard, in place ; or 158 Ibs per cubic foot. Now, a solid cubic yard, when broken up by blasting for removal by wheelbarrows or carts, will occupy a space of about 1.8, or l4 cubic yards ; whereas average earth, when loosened, swells to but about 1.2. or li of its original bulk in place; although, after being made into embank- ment, it eventually shrinks into less than its original bulk. In estimating for enrth. it is Resumed that yV cubic yard, in place, is a fair load for a wheelbarrow. Such a cubic yard will weigh on an average 2430 B>s, or 1.09 tons ; therefore, 174 Ibs, is the weight of a barrow-load, of 2.31 cubic feet of loose earth. Assuming that a barrow of loose rock should weigh about the same as one of earth, we may take it at -^ of a cubic yard; which gives ^- = 177 Ibs per load of loose rock, occupying 2 cubic feet of space. In the following table, columns 2 and 3 are prepared on the same principle as for earth, as directed in Article 4. Column 4 is made up by adding to each amount in column 3, .2 of a cent for each 100 feet length of lead, for keeping the wheeling-planks in order; and 45 cents per cubic yard, in place, as the actual cost for loosening, including tools, drilling, powder. Ac: as well as moderate drainage, and everv ordinary contingency not embraced in column 3. Contractor's profits, of course, are not here included. Ample experience shows that when labor is at $1 per day. the foregoing 45 cents per cubic yard, in place, is a sufficiently liberal allowance for loosening hard rock under all ordinary circumstances. In practice it will generally range between 30 and 60 cents ; depending on the position of the strata, hardness, toughness, water, and other considerations. Soft shales, and other allied rocks, may fre- quently be loosened by pick and plough, as low as 15 to 20 cents ; while, on the other hand, shallow cuttings of very tough rock, with an unfavorable position of strata, especially in the bottoms of ex- cavations, may cost $1, or even considerably more. These, however, are exceptional cases, of com- paratively rare occurrence. The quarrying of average hard rock requires about ^ to % Ib of powder per cubic yard, in place; but the nature of the rook, the position of the strata. &c. may increase it >F EARTHWORK. 441 to H B>. or more. Soft rockCrquently requires more powder than hard. A good churn-driller will drill 8 to 10 feet in depth, of holes about 2*4 feet deep, and 2 inches diameter, per day, in average hard rook, at from 12 to le cents per toot. Drillers receive higher wages than common laborers. Hard Rock, by Wheelbarrows. Labor $1 per day, of 10 working hours. Length of Lead, or dis- tance to which the rock is wheeled. Number of cubic yards, in place, wheeled per day by each barrow. Cost per cubic yard, in place, for loading, wheeling, and emptying. Total cost per cubic yard, iu place, ex- clusive of profit to contractor. Length of Lead, or dis- tance to which the rock is wheeled. Number of cubic yards, in place, wheeled per day by each barrow. Cost per cubic yard, in pface, for loading, wheeling, emptying. Total cost per cubic yard, in place, eX' elusive of profit to contractor. Feet. Cubic Yds. Cents. Cents. Feet. Cubic Yds. Cents. Cents. 25 12.2 8.64 53.7 600 2.96 35.5 81.7 50 10.7 9.81 54.9 700 2.62 40.1 86.5 75 9.58 11.0 56.2 800 2.34 44.8 91.4 100 8.66 12.1 57.3 , 900 2.12 49.5 96.3 150 7.26 14.5 59.8 1000 1.94 54.1 101. 1 200 6.25 16.8 62.2 1200 1.65 63.6 115.0 250 5.49 19.1 64.6 1400 1.44 72.9 120.7 300 4.89 21 5 67.1 1600 1.28 82.2 130.4 350 4.41 23.8 69.5 1800 1.15 91.5 140.1 400 4.02 26.1 71.9 2000 1.04 100.8 149.8 450 3.69 28.5 71 4 2200 .953 110.2 159.6 500 3.41 308 76.8 2400 .879 119.5 169.3 Art. 15. Removing rock excavation by carts. A cart-load of I rock may be taken ati of a cubic yard, in place. This will weigh, on an average, 851 S>s ; or but 41 i Tbs more than a cart-load of average soil. Since the cart itself will weigh about % a ton, the total ; loads are very nearly equal in both cases. Columns 2 and 3 of the following table are prepared on the i same principle as for earth, as directed in Art. 4. Column 4 is made up by adding to each amount in | column 3, the following items : For blasting, (and for everything except those in column 3; loading, and repairs of cart-road,) 45 cents per cubic yard, in place; for loading, 8 cents, per cubic yard, in place; and for repairs of road, .2, or i of a cent for each 100-feet length of lead. Contractor's profit not included. Hard Rock, by Carts. Labor $1 per day, of 10 working hours. Length of Lead, or dis- tance to which the rock is hauled. Number of cubic yards, in place, hauled per day, bveach cart. Cost per cubic yard, in place, for hauling, and emptying. Total cost per cubic yard, in place, ex- clusive of profit to contractor. Length of Lead.ordis tance to which the rock is hauled. Number of cubic vards, in place, hauled per day, by each cart. Cost pet cubic vard. in place, for hauling, and emptying. Total cost per cubic yard, in place, ex- clusive of profit to contractor. Feet. Cubic Yds. Cents. Cents. Feet. Cubic Yds. Cents. Cents. 25 19.2 6.51 59.6 1800 5.00 250 81.6 50 18.5 677 59.9 1900 4.80 26.0 82.8 75 17.8 7.03 60.2 2000 4.62 27.1 84.1 100 17.1 7.29 60.5 2250 4.21 29.7 87.2 150 16.0 7.81 61.1 2500 3-87 i 32.3 90.3 200 15.0 8.33 61.7 ^ mile 3.70 33.7 92.0 300 13.3 9.37 63.0 3000 3.33 37.5 96.5 400 12.0 10.4 64. 3250 3.12 40.1 99.6 500 16.9 11.5 65. 3500 2.92 42.8 1028 600 100 12.5 66 3750 2.76 45.3 105.8 700 9.23 13.6 68. 4000 2.61 47.9 108.9 800 8.57 14.6 69. 4250 2.47 50.6 112.1 900 8.00 15.6 70. 4500 2.35 53.2 115.2 1000 7.50 16.7 71. 4750 2.24 55.8 118.3 1100 7.06 17.7 72.9 5000 2.14 58.4 121.4 1200 , 6.67 18.7 74.1 Imile 2.04 61.2 124.8 1300 6.32 19.8 75.4 lYt " 1.67 75.0 141.2 1400 6.00 20.8 76.6 I}*" 1.41 88.8 157.6 1500 5.71 21.9 77.9 1% " 1.22 102.5 174.0 1600 5.45 22.9 79.1 2 " 1.08 116.3 190.4 1700 5.22 24.0 80.4 2K" .962 130.0 206.8 " I^oose rock " will cost about 30 cts per yd less; and even solid rock will average about 10 cts less than the tables. 442 CENTER OF GRAVITY. CENTEE OF GEAVITT, The cen of 4; rav of a square, rectangle, rhombus, or rhom- boid, is at the intersection of its two diagonals. Of a circle, ellipse, or regular polygon, in the center of the figure. Of a triangle, at the intersection of lines drawn from any two angles, to the middles of the sides respectively opposite said angles. Or, draw a line from any one of the angles, to the middle of the side opposite said angle ; the cen of grav is in this line at ^ of its length from the side which it bisects. Of either a trapezium, or a trapezoid, draw the two diags a c and b d. Div either of them, say a c, into two equal parts as at in. Take the longest part d s, of the other diag d. b, atid set it off from 6 to n. Join M m, and div it into 3 equal parts. The cen of grav will be at o, the first of these divisions from m. Of a trapezoid only. Prolong either parallel side, as 6 a, in eitiier direction, say toward g ; and make a g equal to the opposite side d c. Then prolong the other parallel side d c, in the opposite direction ; make ing c k equal to the side 6 . Join y h. Find the centre e of a 6; and the center / of d c. Join ef. Then o is the ceo of grav dc + 2aft e/ Or lo = X -5 dc + a b Of a semicircle. Mult the height or rad a b by .4244; the prod will be a c; and c is the cen of grav. Of a sector of a circle, adbc. Mult twice the chord a 6, by the rad a c. Div the prod by 3 times the length a d b of the arc of the sector ; (sec Lengths of Arcs, p 21, 23.) The quot is c o; and o is the cen of grav. Of a quadrant, c o = c d X .6002. Of a segment of a circle, nob. Cube the chord a 6. Div this cube by 12 times the area of the segment ; (see Areas of Segments, page 24.) The quot will been; and n is the cen of grav. Of a circular arc n o 6, (the line alone,) not exceed- ing a semicircle. As the length of the arc is to its chord, so is rad c o to c n ; n being the cen of gr. Or, quite approx, mult the rise o by .65 for n. Or more correctlv, if the rise s o is .01 of the chord a b, or less, mult it by .666 for aw; if '.1, mult by .665; if 15 by .663: if .2. by .660; if .25 by .657 ; if .3, by .653 ; if .35, by .649 ; if .4, by .645 ; if .45, by .641 ; if .5, by .637. Of a parabola a b <% at o in the axis x 6, $ths of its length from x. Of a semiparabola a b x. Atn; on being 3-eighths of the half base a x, and o x being fths of z 6. The common center of gravity g, of two figs a and is fauna u.t.s : First hud tueti separate cen of grav and I; of the two tiga combiued. The longest division must evidently be next to the smallest body b. If the common cen of grav of the three bodies a. 6, and c. is required, begin with any two of them as has just now been done. Then draw gc, and measure it. Then suppose the united areas of a-and 6 to be placed at a, as if one fig at that point; and call it Fig g. Add the aroaof this fig to that of Fig c; and, as before, say as a of c (the greatest of them) to the longest di solid bodies having wt, tne process 1 , C 1 V V~. x.,in U instead of areas. In either case it is immaterial which two w begin wit MECHANICS. .FdRCE IN RIGID BODIES. 443 IK AN IRREGULAR PIG whicb^Hiay be divided into triangles, trapeziums, trapezoids, Ac, the process In the same as with the figj^dT &. c, and d. The ceil of Jfrav of any plane fig may be found by drawing it to a scale on pasteboariKtneti cut out the figure ; balance it in two or more positions over the edge of a table, or on a shdrp knife-edge ; and mark on it the directions of the edge. Where these directions intersect each faher, will be the reqd point, near enough for most practical purposes. The paper on which the fig is prepared, must be so stiff that the fig will not bend when balanced. Of a cube, parallelepiped, cylinder, prism, sphere, sphe- roid, ellipsoid; the cen of grav is in the cen of the body. Of any frustum of either a rig-lit or oblique cylinder; or of a right or oblique prism; whether cut parallel to its base, or obliquely. In the center of the axis of the frustum. Of a r igli t pyramid, or cone ; in its axis, at i^ of its length from the base. Of any pyramid, or cone ; whether right, or oblique. In a line from its Vertex, or top, to the cen of grav of its base ; and at 34 the length of this line from the base. Of a frustum of a right cone, cut parallel to its base. Call the rad of the larger end R; and that of the smaller end r; and the height of the frustum nieasd on its axis, h. Then. R2 4- 2 R r X .- --- --- = dist on axis from greater end, or base. Of a frustum of a right pyramid, cut parallel to its base. Call the area of the large end A ; that of the small end a ; and the height measd on the axis, h ; then, i axis from greater end, or base. A + V\a + a Of any frustum of any pyramid, or cone; whether right, or ob- lique: or whether the ends of the frustum are parallel to the base or not; it is in a line to, drawn between the centers of grav of the two ends of the frustum j but its height y c above the lower end an, and perp to it, must first be found upon the line t y, drawn from the cen of grav of the small end, to, and at right angles with, thelarge end an. Having first found the line t y, whether for a pyramid or a cone, use it instead of the height ft. in the preceding formula, for a frus- tum of a right pyramid. The result will be y c ; and by drawing c a parallel to a n, we find the reqd point , in the line t o. Of the curved or slanting surface only, of a right cone; in the axis ; and at % of its length from the base. Of a hemisphere ; in its axis, at % of its length from the base. Of a segment of a sphere. From twice the rad of the sphere, take the height of the segment. Square the remainder. Then from 3 times the rad of the sphere, take the height of the segment. Div the square just found by this last rem ; and take % of the quot, for the dist from the cen of the sphere to the cen of grav of the segment. Of the curved surface only of a hemisphere; of a spherical seg- ment; or of a zone: at the middle of its axis, or height. Of a paraboloid : in its axis, and at % of its length from the base. REM, "We must not confound cen of grav with cen of weiqht. It does not follow when a body bal- ances on a knife-edee, that ther? is equal wt on both sides of it; but merely that the wts of the several particles on one side, when mult by their respective leverages, or dists. at right angles from the knife-edge, have a united moment about the knife-edge, equal to that of the particles on the other side of it, when mult by their leverages. See Leverage, ate, in Force in Rigid Bodies, p 473, &c. MECHANICS. FOECE IN RIGID BODIES. Art. 1. Mechanics (the very foundation of civil engineering) is that brunch of science which treats of the effects of force upon matter. This broad application of the term necessarily includes hydrostatics, pneumatics, &c; but ordinarily it is restricted to the effects produced upon matter by the application of outward, extrane- ous, or mechanical force, from whatever source it maybe derived; whether from steam, water, air, wind, gravity, animals, &c. Some of these effects consist in either deranging, or separating the particles of matter which comprse a body, by pushing them close together : or by pulling them farther apart; and in nil such cases the extraneous force is to be considered only as acting against the natural inherent forces, or strength of the materials of which the bodies consist. This class of effects 444 FORCE IN RIGID BODIES. is comprised under the distinct head of Strength of Materials. Another and very important class refers only to the action of force upon entire bodies ; either to move them ; or to keep them stationary. Force, when it moves bodies, is called Dyna- mic force; and when it keeps them stationary, Static. Hence, we have the sciences of Statics and Dynamics ; which imply merely the effects of force as to giving motion to bodies ; or to keeping them at rest. See Note p. 459. In examining the abstract static and dynamic effects of force thus applied to entire bodies, we of Bourse have to assume that it does not break, bend, penetrate, or alter in any way, the shape of the body. The body i,s to be regarded merely as something that receives the force ; but is in no way affected thereby, further than that it moves, or stands (or rests) when the force does so. That is, the forces are assumed to act upon each other only, when there are more than one ; the body being only the field of their action. This assumption is of course not strictly true in any case, because no body is so perfectly rigid as not to have its shape somewhat changed by the application of force ; still, it will be seen, as we proceed, that it is not liable to produce error. Thus, it will be evident that the illustrations of force in the following pages would not apply to walls, arches, beams, &c, of snow, cotton, loose sand, or soft clay ; and in the same manner, although to a less extent, would they be influenced by the assump- tion of any yielding whatever of the particles composing bodies of metal, stone, or wood. Such yielding has of course to be taken into consideration in nearly all cases in practice ; but it must be done by a totally distinct process, under the head of Strength of Materials. The two processes do not interfere with each other. It is so absolutely essential in the study of Statics and Dynamics, and in reading the following articles, to keep this assumption constantly in mind, that we repeat it; namely, that the force is to be con- sidered only as acting upon entire bodies; or upon bodies as a whole; incapable of being broken, bent, or affected by it in any manner whatever, othec than in merely beiug moved: or kept station- ary by it. For instance, if we wish to ascertain the effects of a given force /A to overturn or upset a stone. S, Fig 25, around one of its edges, n, we suppose the stone to remain whole. But if we wish to know whether the edge n will be in danger of being fractured when the whole weight of the stone comes upon it during the process of upsetting, we resort to the crushing strength of stone. It is plain, that a body when pushed, or pulled in several directions at once, may not as a whole, have the slightest tendency to move in one direction rather thiui another; yet some of its particles must tend to move in one direction ; and others in others. Statics and Dynamics regard the body only in the first point of view. RRM 1. When not otherwise stated, or apparent, the force in many of the following examples, is supposed to be imparted to the body in a direction passing through its cen of grav : so as to move, pull, or press it, without tending to make it revolve, or upset. By examining first the action of forces imparted in that manner, certain elementary principles become much more easy of comprehension. Rein. 2. If the direction of tlie imparted force does not pass through the t^en of j; rav of a body free to move, the body will still move forward in that direction just as far as before; but while so doing, it will also revolve around its ceu of grav to the same extent as if it did not move forward at all, and as it would around a fixed axis, and under the full action of the force. REM .1 Lastly, when not otherwise stated, the force is supposed to be applied to the rigid body in a direction at right angles to its surf at the point where it isaj)- plied ; otherwise (except as per Art 19) only a part of the force will be imparted to the body ; that is, will be put into it, or enter it; and produce an effect upon it. Art 2. Matter is any substance whatever, as metal, stone, wood, water, air, steam, gas, &c. A body is any quantity ot matter, as a piece, a pound, or a. cub ft, &c, of it. The weig-ht of a body is the amount of vert pres, or pull, which the force of gravity in that body exerts while the body is at rest. In ordinary practical mechanics, the quantity of matter in a body is measured by its wt ; for a method which is the only theoretically correct one, but which is not adapted to popular use, see note to p. 455. Motion is change of place, or of position. Since all bodies are constantly in motion in consequence of the revolution of the earth, the ordinary use of the word here refers to the motion produced by the extraneous force under consideration at the time. The base of a body, as ordinarily understood, is that part of it which is underneath, and upon which it stands when acted upon by its own wt, or by the wt of other bodies also, which may be placed upon it. But in fact, a base may be at any part whatever of a body ; at its top, on one side, &c ; as when we press a block of wood upward against the ceiling of a room ; or horizon- tally against a wall ; the part which touches the ceiling, or wall, is the base. Strain is the effect produced by the action of tqual forces or of equal parts of unequal forces, against each other ; or in opposite directions. It may consequently be applied to the effect which these equal parts or wholes produce among the par- tides of the body upon which they act ; as tending either to push them closer to- gether ; or to pull them farther apart ; in other words, to crush, or to rend the body. But as before stated, this view of strain belongs, not to pure Statics and Dynamics, but to Strength of Materials. Again, it may refer to the action of the forces upon each other; or, as usually stated, upon the body as a whole. IN RIGID BODIES. 445 Thns, if two men pu)K>r push, with equal forces, a body that is between them, their forces merely react against, count*fDalance, or destroy (see Rem, Art 13) each other; but they produce no effect whatever upon the'cody as a whole ; it remains at rest ; and has no more tendency to move in any direction, than if both forces were absent. The forces then merely strain (pull or push against, without moviuaf each other, and keep each other at rest; but no practical error will arise from adopting the common phraseology, and saying they act upon the body to keep it at rest. Strain is measured by weight; as by pounds, tons, &c. Its amount or quantity is equal to that of only one of the two equal opposing forces. Thus, if two men pull against each other at two ends of a rope, each with a force of 30 ft>s. the strain on the rope is but 30 fts ; and it is equal throughout the length of the rope. The two 30 R> forces strain each other, 30 Ibs ; as is made manifest if a spring-balance is inserted at any part of the rope. If a rope passes over a pulley, and equal \vts be suspended at each end of it, then the two equal forces of gravity of the two wts strain against each other; and also strain the rope ; to an amount equal to one of them. For more on Strain, see Art 13, p 449. Two equal opposing forces produce strain and a tendency to motion, among the particles which compose a body to which they are imparted ; but exert no tendency to motion upon the body as a whole; because the particles are held together by their internal cohesive force ; and this cohesive force reacts against our extraneous imparted forces. If it is not sufficient to do so completely, the body is broken ; and the remaining portion of the imparted forces, becoming motion, scatters the fragments in all directions. But the body as a whole, is isolated, detached matter, which is not adhering to anything by any kind of force. It in itself opposes no resistance to force ; it is inert. Therefore, so far as the imparted forces are concerned with it, as a ivhole, to give it motion in any direction, it is merely a corpse, over which two contending forces are destroying each other, in their struggles to get possession of it. If the contending forces are equal, they will strain against each other, until they are mutually destroyed ; but the body will remain unmoved by either. If they are unequal, the surviving portion of the greater one will move on at a slackened pace, in its former course; moving the unresisting body along with it. See Rem 3, &c, of Art 28, p 458. Art. 3. Force is that principle of which, considered simply as a mechanical agent, we know but little more than that when it is imparted, that is, put into, a body, it produces either motion alone ; or strain, with or without motion. This is all that force, mechanically considered, can do under any circumstances whatever. When it produces motion alone, it is called moving force. When it produces strain alone, or without motion, straining force ; or simply strain, pull, push, tension, or pres,as the case may be. As, for instance, when a body rests quietly on a table, or on a post, Ac; or is suspended by a chain, rope, &c ; its effect is simply strain ; a push or pres on the post or table, and a pull or tension, on the rope. There can be no strain except where there are at least two forces to strain against each other. When it produces strain and motion at the same time ; or in other words, when a part of it produces strain ; and a part of it, motion ; it is called working force ; and its effect is work. Thus, when a man is lifting a wt, a part of his force is straining against the equal force of gravity of the wt; while the other part of his force is giving motion to the wt ; so that all his force combined, is at work ; or is working force ; or performs work. See Art 11, p 448. A straining force is often called a stress. These are not different kinds of force ; but diff effects of the same force ; for there are no diff kinds of mechanical force; it is all the same: no matter from what source it may be derived. Indeed, we might generally, and without fear of being misunderstood, call it simply motion, work, or strain, according to which of the three effects it is producing at the time; and we shall frequently do so in the following pages. The only means we have of measuring it, is by measuring the quantity of these its three effects. Not hint? font force can resist force. When all the force imparted, or put into, a body, is unresisted by opposing force ; it produces motion only ; when it is all resisted, it produces strain only ; when a part of it is unresisted, and a part resisted, the first part produces motion; and the last part strain ; which two combined constitute work. A single force can produce motion ; but at least two are required to produce strain. See Art. 13, p 449. Force is sometimes defined to be that principle which either does produce, or tends to produce motion; or which either does prevent, or tends to prevent it. Friction, however, which is classed among forces, always tends to prevent motion ; therefore, when the word force is used alone, it must be remembered that friction is not included. REM 1. We infer from both reason and observation, that force, when once imparted, or put into a body, would, if not resisted by other force, remain in it, and move it forever, in a straight line; in the direction in which the force was imparted; and at a uniform velocity, or rate of speed. For we observe in practice, that motion continues longer in proportion as we remove resisting forces ; hence we infer, that if all resistances could be removed, it would continue forever. But in practice we cannot remove all resisting forces, such as those of gravity, wt, friction, the air, Ac. These strain against, or resist the forces which we may impart; so that with the exception of a body falling by the force of gravity, in a vacuum, it is difficult to mention a case of motion alone, without strain. REM 2. Matter, in itself, cannot resist force. See Arts 5 and 11. When force produces motion alone, without strain, it is often called Dynamic force ; when 29 446 FORCE IN RIGID BODIES. continuous strain without motion. Static ; when a sudden strain for an instant only, as a blow of a hammer, Impulsive; and its action, an impulse. See Art 9. REM 3. The Living Force, or vis viva of scientific writers, is nothing more than an expression referring to the quantity of work (motion and strain combined) which the force in a body at any given instant, could perform, if left to itself, with- out afterward receiving anj r additional force. See Art. 24, p 455. Art. 4. When force is imparted in any direction which passes through the cen of grav of a homogeneous rigid body, perfectly free to move, it quickly diffuses itself equally through every part ; and gives to every atom composing the body, the same tendency to move onward in the same direction. But although this diffusion takes place rapidly, especially in compact bodies, still it requires some time ; generally so short as not to be appreciable without close ob- servation. If we put the body B, Fig 1, into motion by striking it a blow near the top D or bottom, the struck end will for an instant move faster than the other; after which D both will move with equal velocity ; because the great force which for au infinitely short time acted upon the top only, becomes equally diffused throughout the body. We are not, however, now treating of cases of this kind, in which the direction of the force does not pass through the cen of grav of the body ; for the body may then ro- tate, or whirl around as it moves forward. 7. This equal diffusion of force is owing to the inherent cohesive force of bodies, which -I- 111 1 holds their atoms together, somewhat as lime holds together the grains of sand in a of wt lifted one foot high ; or one ft> of any kind of resistance, overcome in any direction whatever, for the dist of one foot ; and is called a, font-pound. Or, when more conve- nient, we may use foot-tons, &c, &c ; as we use a two-ft rule, a yardstick, or a tape- line, as may best suit our purpose. The unit of KATE of work, or the quan- tity done in a given time, is oneft-lb per sec. The same quantity of force which will overcome a given resistance through a given dist. in a given time, will also overcome any other resistance through any other dist, in that same time, provided the resistance and dist when mult together give the same amount as in the first case. Thus, the force that will lift 50 fts through 10 ft in a sec, will lift 500 fts, 1 ft; or 25 fts, 20 ft; or 5COO fts j- 1 ^- of a ft in a sec ; and in all these cases the amount of force expended, as well as of work done, is precisely the same. In practice, the adjustment of the speed to suit diff amounts of resistance, is usually effected by the medium of cog-wheels, belts, or levers. By means of these, the engine, water-wheel, horse, or other motive power, may be made to overcome small resistances rapidly ; or great ones slowly, by the same working force. For more on working force, see Art 17, 18, 19, &c. Also Note, p 459. Art. 12. When vel undergoes no change, it is said to be uniform ; so with force, motion, strain, and work. \Vhen any of them becornei grachially greater, it is said to be accelerated; when gradually less, retarded. If the acceleration, or retarda- tion is in exact proportion to the time ; that is, when during any and every equal interval of time, the same degree of change takes place, it is uniformly accelerated, or retarded. \Vhen otherwise, the words variable, and variably are used. Confusion arises from the frequent use by even the best writers, of the words " constant," and " uniform," instead of uniformly accelerating force. Although the expressions are strictly correct, still they are inexpedient ; for when causes and effects are so intimately connected, as moving force and motion, it is desirable that the same adjectives should be equally applicable to both ; and we should not in the same sentence read of constant or uniform force, producing inconstant/or uuuniform motion. Gravity is a uniformly accelerating force when it acts upon a body falling freely ; for it then increases the vel at the uniform rate of .322 of a foot during every hundredtlTpart of a sec ;' or 32.2 ft in every sec. Also when it acts upon a body moving down an inclined plane ; although in this case the increase is not so rapid, because it is caused by only a part of the gravity ; while another part presses the body to the plane; and a third part overcomes the friction. It is a uniformly retarding force, upon a body thrown vert upward; for no matter what may be the vel of the body when projected upward it will be diminished .322 of a foot in each hundredth part of a sec during its rise ; or 32 2 ft during each entire sec. At least, such would be the case were it not for the varying resistance of the nir at diff vels. It is a uniformly straining force when it causes a body at rest, to press upon another body ; or to pull upon a string by which it is suspended. The fore'going expressions, like those of momentum, strain, push. pull, lift, work, Ac, do not indicate diff kinds of force; but merely diff kinds of effects produced by the one grand principle, force. See footnote, p 455; also p 587. The above 32.2 ft per sec is called the acceleration of gravity ; and by scientific writers is conventionally denoted by a small g 1 : or, more correctly speak- ing, since the acceleration is not precisely the same at all parts of the earth, g de- notes the acceleration per sec, whatever it may be, at any particular place. See note to Art 25, p 455. Art. 13. Reaction. Strain. Strain, as before said, in Art 2, is either a pull or a push. The term may be applied equally to the act. or to its fffrctit; that is, it may be said to be either the action of opposing forces against each other: or tbe effects which that action produces upon the particles of the body in which they act. A single force cannot produce strain : for since nothing but force can oppose, resist, pull, or push, against force, there must he at least two, to strain by pulling or push- ing against each other. This mutual opposition or straining is called also the war- on of the forces. It can take place only between equal forces, or equal portions of unequal ones. When the forces are equal, and meet from diametrically opposite direc- tions ; that is, in the same straight line, but in opposite directions along it, they become entirely converted into reaction, or strain. If they are unequal ; or are not 450 FORCE IN RIGID BODIES. in the same straight line ; but meet obliquely ; then only equal portions of them will re.ict against each other; while the remainder will continue as motion ; unless soiae third force is present to prevent it; as when friction is generated. The reaction of the equal wholes or parts consists in their mutually resisting, opposing, arresting, balancing, equilibrating, straining, pulling, pushing, counteracting, or destroying eacli other. All these words are equally applicable. As a matter of convenience only, we often say that the bodies themselves react. If a cauuou-ball in its flight cuts a leaf from a tree, we say that the leaf has reacted against the ball with precisely the same degree of force that the ball acted against the leaf. That degree of force was sufficient to cut off a leaf, but not to arrest the ball ; for, after a smalt portion of the moving force of the ball had been converted into straining force to react against the resistance of the leaf, the remainder was sufficient to carry it onward in its course. It, has. how- ever, lost precisely as much force as that which the leaf opposed to it. A ship of war, in running against a canoe, receives as violent a blow as it gives ; but the same blow that will upset or sink a canoe, will not appreciably affect the motion of a ship. The fist of a pugilist striking his opponent in the face receives as severe a blow as it gives; but the blow which may seriously damage a nose, mouth, or eyes, may have no such effect upon hard knuckles. The paiu received in the one case, and not in the other, is of course no measure for force. BEM. We have just said that strain is the mutual destruction of two equal amounts of force. This may readily be conceived in cases where two bodies come into sudden contact, and arrest each other's progress by a mutual blow; for we then see that their forces are lost, inasmuch aa they no longer produce either motion, strain, or work. But in continuous strains (pulls, or pushes) the principle is not so evident, at first. For instance, when a wt rests upon a table, and continues to strain it day after day, it may be asked where is the loss of gravity force in the weight; inasmuch as it weighs aa much after pressing for a long time, as it did at the beginning; or where is the loss of inherent cohe- sive force in the material of the table, which is as strong aa at first? The reply is that gravity, and the natural strengths, or inherent forces of matter, are being inces- santly maintained or renewed, by unceasing streams, as it were, of those forces. The gravity of a wt which presses a table, or pulls on a rope, at one moment, is not identically the same that pressed or pulled it the moment before; and so with the cohesive force of the material of the table. If it were not so, a post or a rope which would be broken by a single force of 10 tons, would also be broken by sustaining one ton 10 consecutive times ; for the one ton would each time react against, or destroy one ton of the inherent resisting force of the post or rope; and in ten applications would destroy it all. We must therefore conclude that these natural forces are being unceasingly supplied from that inexhaustible source of power " by which all things are upheld." When we lean forward against a strong wind, we are continuously exerting new force against the continuous stream of force which assails us ; and in the same way does a post or a chain sustain its load. Continuous strains produced by the force of water, steam, animals, &c, we well know can only be maintained by a continuous sup- ply of said force : to be as continuously reacted against, or destroyed, by whatever opposing force of grinding, pumping, Ac, it is directed against. The strain of an impulse, blow, or stroke, lasts only for an instant, because new force is not supplied to make it continuous. Two equal forces, straining against each other, do not even keep a body at rest; but the body rests merely because the two forces destroy each other, and therefore cannot prevent it from resting. As a matter of convenience only, we may, however, say they keep it at rest. We might even assume strictly that force produces motion only ; and that the destruction of force produces strain, and thus restores rest. Art. 14. While a horse is hauling a boat on a canal, it is not the boat and ita load which react against his force ; -because matter cannot react against force; it is the force of friction of the boat against the water ; and the resistance of the wa- ter in front of the boat, produced by its cohesive force. So with an engine and train on a level railroad ; the only resistance to the steam force of the engine, is the force of friction at the axles and tires of the wheels, and the pres force of the air in front. But on an up grade, the engine has also to partly lift the train ; or, in other words, to react against its forces of friction and gravity combined. Neither the horse nor the engine exerts any more force upon the opposing forces, than these last exert upon them ; but both the horse and the engine possess a surplus of power beyond what is ne- cessary to strain against or destroy the forces opposed to them : and this surplus, as moving force, enables them in addition to move forward, as in the case of the cannon-ball just alluded to; and since the unresisting matter of the boat, load, and train, is attached to them by the tow-rope and coupling- links, they of course must follow. The resistance which an abutment opposes to the pres of an arch ; or a retaining-wall to the pres of the earth behind it, is no greater than those pres themselves ; but the abut and wall are, for the sake of safety, made capable of sustaining much greater pressures, in case accidental circumstances kould produce such. Art. 15. The mere fact that a body is subjected to g^reat Strains from equal forces reacting upon it in opposite directions, does not of it- self render the body more difficult to move than if it were free from strains ; but in the cases which usually present themselves in practice the straining forces gener- ate friction, which does oppose motion. Thus let B, Fig 8, be a block resting on a nor support, and acted upon by a downward force d of say 100 tons, produced say by an immense block of granite resting upon B. Now it is plain that this 100 tons downward force will be met and balanced by a 100 tons upward force u, being the resistance of the hor support. Heuce these two equal reacting forces produce in the body B a strain of 100 tons ; but evidently do not impart to it as a whole any tendency to move in any direction whatever ; nor do they tend to prevent it from being moved in any direction. The body therefore remains as be- fore a mere inert mass incapable of resisting the slightest moving force. Now suppose no friction to exist at either the base or the top of B. Then the slightest hor force A, a mere breath, would slide B along the hor support, moving it from DRCE IN RIGID BODIES. 451 under the 100 top^olock on its top. No matter how heavy B might be, the same smallest force wfculd slide it, the only difference being that the heavier it was the less would beXts velocity. The quantity of motion (Art 9) will be the same for any wt. The heaviest bodies resting upon the surface of the earth, as well as ourselves, would be swept along by the slightest breeze if it were not for friction. If the screw of a vise be worked until it produces a great strain in the jaws of the vise, the vise is not thereby rendered more difficult to move. Again, if a strain of thousands of tons were produced by the jaws of a vise, in a body weighing an ounce, this immense strain would not prevent, nor even tend in the smallest degree to prevent, the ounce body from falling down from the jaws of the vise. It is prevented by the third force, friction, which compels the one ounce of gravity force to become vert strain, instead of motion. The two forces of thousands of tons each, which produce the strain of the vise, are thereby entirely destroyed, aa regards their action upon the body as a whole. Hence they could not prevent the one ounce from pro- ducing motion in it; nor could they affect it as a whole in any way ; for all their action is actually against each other. It is on this principle alone that strains'do not interfere with motions. If a body H, Fig 4, of 10 tons weight, is suspended from a long rope, its reaction against the equal opposing force at the other end of the rope, produces a continuous strain among all the particles which compose the rope ; but which does not in the least affect the rope considered as a whole ; inasmuch as it does not tend to move it in any direction. Now, in this case, there is no friction to be overcome; and we know from daily experience that it is therefore easy to move the unresisting body a little dist, by applying a very small hor force/. We cannot move it far, as, for instance, to m, because we then have not only to move, but to Lift it, or overcome its gravity force, through the vert height vc. In doing this, it is true our force does not have to sustain the entire wt of the body ; because most of it is sustained by the rope. Still, if we move it at all, we have to overcome some of its weight ; otherwise, a mere breath would move it, although very slowly. If we attempt to move it by an upward force w, we shall have still more of its wt to re- sist us ; and if by a downward one d, we shall be resisted by the cohesive force of the rope. Therefore, in this case, we can move it more readily by the hor force /. Art. 16. If two unequal forces, which for illustration we will call 10 and 12, are imparted, either as pulls or as pushes, in precisely opposite directions, to a rigid body on which no other force is acting, then the small force 10, and 10 parts of the large one, will destroy each other as strain ; after which, of course, they can produce no effect of any kind ; but the remaining two parts of the large one, meeting with no opposing force to react against, will continue onward as motion, in the same di- rection as before; taking the unresisting body along with them. In such a case, the large force is said to overcome the small one ; and such an expression is very con- venient, in reference to the entire original forces. But in a strictly scientific sense, one force cannot overcome another. Thus, in the foregoing case, so far as the strain Is concerned, the large force must be considered as separated into 10 straining, and 2 moving parts. Neither of the two 10 forces which strained against each other overcame, or gained any advantage whatever over the other ; for the two reacted equally on each other, and mutually arrested", equilibrated, and destroyed each other. And this is plainly the most that one force can do to another. The 2 force of the large body took no part whatever in the conflict; but, on the contrary, moved out of the way of it. As it had lost 10 twelfths of its previous moving force, it now moves the body only by virtue of the remaining 2 twelfths ; and consequently with but 2 twelfths of its former vel. Since two equal' opposing forces, or equal portions of unequal ones, thus bring each other to a stand-still, or equilibrate each other, they are called Static; from the Latin "Sto, I stand ;" and that branch of the science of force which treats only of cases in which all the applied forces keep each other at rest, is called " Statics," or " Equilibrium." Art. 17. When force has once been put into a body, it can only be taken out again by the reaction of some opposing force. The same identical portion of any force cannot produce both motion and strain at the same time. When continuous force is applied, as in mills, Ac, to do both, (or, in other words, to work,) it must be considered to divide itself into two parts for those separate purposes. If, while at work, the resistances to be overcome become less, the strain also becomes less, and the motion becomes greater; and vice versa. Motion is diminished by converting part of it into strain ; and strain, by converting it into motion. Thus, the motion or moving force in a cannon-ball is gradually converted into strain, by working against the resisting force of the successive strata of air through which it passes. These gradually destroy all its force by this process, and thus permit it to rest. The moving force which remains in % railroad train after steam is shut off, is gradually destroyed by working against the resistance of 452 FORCE IN RIGID BODIES. the air ; of friction at the axles and rims of the wheels ; up-grades, curves. Ac. The strain produced in a, rope by two men pulling against each other at its opposite ends, is converted into motion if the rope breaks ; throwing both men backward. So with the strain on a bent bow, if the string is cut- er if we cut the rope that holds a balloon, part of the force with which the balloou before strained the rope becomes moving force, and by it the balloon ascends. The other part balances gravity. Art. 18. If force / be imparted to any rigid body, as N, Fig 5, at any point c ; and if/o represent the direction in which it was imparted, whether as a pull or a push, then the force would produce the same effect upon the body considered as an entire mass, as if it had been im- parted as either a pull, or a push, in the same direction, at any other point of the body in said line ; as at t, /, $, o, &c. Under Composition and Resolution of Forces, it will be seen to be sometimes IV 5 necessary to consider a push fc, to be changed to a pull oh\ and vice versa; when we wish to ascertain the joint effect or resultant of a pull and push imparted to a body at the same time. See Remark 1, Art 29, p 459. The foregoing important principle holds good, no matter how many diff forces may be acting upoo the body at the same time, in diff directions ; or how much the direction of their joint effect, or re- sultant, may differ from that of any one of them; the action of each force, considered separately, may be regarded as just stated. The tendencies of several forces, acting at the same moment, may therefore frequently be first investigated one by one; and these tendencies then combined into one; or the forces themselves may first be combined into one or more resultants, as directed under Com- position and Resolution of Forces, and the effect of these resultants considered. The engineer has generally to divide all the forces actipg upon his structures into two classes; namely, those whose tendency is to secure the stability he requires; and those which tend to impair that stability. He therefore first finds the resultant, or joint effort of each class separately ; and then compares these two resultants with each other. The mode of doing this will be shown further on. See Arts 35, 72. and e, will all enter B, if the resistance ofR is as great or greater than they ; if not, they will move B. But the force F# is riot at right angles torthe surf at its point of applica- tion g; therefore, only a portion of it will (by the theory) be imparted to the body B, however great may be the resistance of B. Under the head Composition and Resolution of Forces, it will be seeu that when a force is thus applied obliquely to a surf, its action at the point g is precisely the same as that of two entirely separate forces ; one of which, v g, is at right angles to the surf at g ; and the other. s g, par- allel to the same surf. Only vg will enter the body H It 1 ; A h while ag will remain in the body F g, which carried the - 1 - * V. v entire force to B ; and (if F g also is rigid) will (by theory) "T cause it to move in the direction g t, unless some third force be opposed a.t right angles to it also. The quantity of each of the forces v g and a g, is very readilv found thus : On the line F g measure off by any con- venient scale a dist gi, to represent the amount in fts. tons, s, and is overcome through a dist of 1000 ft per min, the rate of the work is in each Ibs vel tts vel case the same, namely, 33000 ft-tts per min, or one horse-power; for 3300 X 10 H3 X 1000 33000 f't-ft>s per min. The quantity of motion of a body (Art 9) is also estimated in ft-tbs ; and under the head Levers, it will be seen that the tendencies (called moments) which the power and the weight re- spectively have to commence motion about the fulcrum as a center, are measured in the same term. Work, mere motion, and moments, are, however, effects of force so diff from each other, that confu- sion is no more likely to occur, than in applying the same measure, one foot, to materials as diff ad cloth, bar iron, hoards, &c. In scientific phraseology, work is either useful or prejudicial, the latter being the quantity of force lost by friction, by the resistance of the air, &c. Thus, in pump- ing water, part of the applied force or power is lost in the friction of the diff parts of the pump: so that a steam or water power of 100 tbs, moving 6 ft per sec, cannot raise 100 tts of water to a height of 6 ft per sec. Therefore machines, so far from paining power, according to the popular idea, ac- tually lose it, in one sense of the word. In the practical application of all machinery, the object is twofold; namelv. to enable us conveniently to apply straining force, to balance, react against, or destroy, the resisting forces of friction, and the cohesive forces of the material which is to be operated operated on, after the resisting forces which had acted upon them have thus been rendered ineffectiva. Art. 23. The total quantity of work that will be performed by the moving force that is in a body at any given moment, provided that after changing from mere moving force into working force, it is left to expend itself in uniformly retarding work, without receiving any additional force to aid it, (as in the case of the moving force in the locomotive in Art 20, after steam is shut off; and when said force begins to work against the resistances of the road,) is as, or in proportion to, FORCE IN RIGID BODIES. 455 (not equal to,) th/wt of the body, mult by the square of its vel at the moment it begins to work.' For example, if a train at the time steam is shut off, has in it an accumulated or stored-up moving force of 10 miles an hour ; and if that force will by itself work the train against the resistances of the road for a dist of one-quarter of a mile, before coming to a stop; then, if steam is shut off while the train is moving at 2, 3, or 4 times that vel, and consequently with 2, 3, or 4 times the moving force, it will work through 4, 9, or 16 times the dist of the first case, before coming to rest. If bullets of equal wt be fired with vels proportioned to each other as 1, 2, 8, they will respectively penetrate a plank to depths as 1, 4, 9. If an engine, water- wheel, &c, works steadily in a mill, grinding at 2, 3, or 4 revolutions per min, it per- forms only 2, 3, or 4 times as much uniform work per min, as when at but 1 rev per min. But if steam or the water be suddenly shut off at 2, 3, or 4 revs per mm, then the 2, 3, or 4 times quantity of moving force accumulated in the machinery at that moment, will, as ivorking force, run the mill through 4, 9, or 16 times as many revs before stopping, as if shut off at 1 rev. If a rolling ball, started against a row of bricks, will overcome their resistances, and knock them down for a distance of 4 ft; then, if it be started at a vel 3, 4, or 5 times as great, it will overcome and knock them down for dists of 9, 16, and 25 times 4 ft ; and in but 3, 4, or 5 times the time. But in all these cases the rate of the work done, that is, the quantity done t'n any given time, as one sec, is directly as the vels. Thus, the locomotive whose steam Is shut off at 20, 30, or 40 miles an hour, will require but 2, 3, or 4 times as many seconds for running its 4, 9, or 16 dist before it comes to a stop ; in other words, when its moving force is 2, 3, or 4 times as great, it will overcome but 2, 3, or 4 times the resistances in the same time , although the total amount of resistances over. come will be as 4, 9, and 16. And so with the other examples. Rem. We know that the dist through which a body must fall by the unif ac- cel force of gravity in order to acquire any given vel and moving force, is as the square of said vel ; but directly as the time of falling. Also if a body is thrown vert upwards with any given vel or force, grav will retard it unif, and the height to which it will rise by the time that grav destroys all the force with which it was thrown, will be as the square of said vel ; but the time will be directly as said vel. And so with anybody moving in any direction, and acted upon by any unif accel or retard force whatever. It will either acquire or part with its moving force within dists proportionate to the square of its vel, and in times proportionate to its simple vel. See Caution* Gravity, p 587. Art. 24. Vis viva, or living* force. The preceding article serves as an introduction to this subject ; of which we shall endeavor to give some idea in plain language. The term itself is merely one of those absurdities to which savants re- sort, in order to impart an air of mystery to their writings. We might with the same propriety speak of a brickbat viva, or a living hod of mortar. We have seen in Art 23, that if that portion of force in a body which is occupied in giving motion alone to the body, be suddenly converted into working force, the quantity of work which it would perform against uniformly retarding resistances, before being entirely destroyed, or coming to rest, would be in proportion to the square of its vel at the time of beginning to work. Now, if " vis viva," or " living force," were merely the name given to this force; or to the quantity of work done by it, (as measured in ft-Ibs, by mult the resistances in Ibs, by the dist in ft through which they were over- come,) the expressions, although silly, would still convey an idea readily understood by practical men. We could then say, for instance, of a moving body, that its vis viva was 100 foot-fibs ; mean- ing thereby that it would overcome a uniform resistance of I ft through a dist of 100 ft; or a resist- ance of 100 Ibs through a dist of 1 ft, &c. But scientiflc writers apply the terms to a purely imagi- nary quantity, equal to twice this : and which does not exist in any body, under any circumstances. The reason they do so is that it facilitates some of their calculations. But the practical engineer need not concern himself with either this reason, or vis viva itself; the simple statement of facts contained in the preceding and following articles, probably contains all that it is essential for him to know on the subject of moving force, converted into uniformly retarded working force. Art. 25. The actual total amount of worlc, in ft-lbs, that can be accomplished by a given moving force, when converted into unaided working force, is found by div the square of the vel of the body in ft per sec, by 64.4 ; and then mult the quot by the wt of the body in pounds ; or, in shape of a formula, Uniformly retarded _ wt of tJie working v S 1 of it* vel infi P er sec working force body, in Ibs * 64.4. Or to wt X fall in ft reqd to give the vel. See " Falling Bodies," p 587. An imaginary force equal to double this, will be the vis viva, or living force of the savants : or Vis viva = Wei 9 f 't f body v s? /*'<* wl ft per sec* in Ibs 32.2. * For the purposes of abstract science, it is not sufficiently exact to measure the quantity of matter by its wt; because the wt or gravity of a body, as shown by a spring balance, varies somewhat in diff latitudes, and at diff heights above the fevel of the sea. Therefore, the vel with wnich it will fall by that gravity, of course becomes a proper measure of the gravity ; because it also varies in the same proportion; all moving force being in proportion to the vel it "imparts. Therefore, if the wt of a body at any place, be div by the vel imparted by gravity in one sec at the snme place, (and called the acceleration of gravity of that place.) the quot will be the same at nil places. And ?ince the quantity of matter undergoes no change at diff places, the measure of that quantity should likewise 456 FORCE IN RIGID BODIES. By way of practical application of the first formula, snppose that a railroad train of 500 tons, is moving at the rate of 15 miles an hour ; and that steam is suddenly shut off; into how much working force will this moving force be converted? Here, 500 tons = 1,120,000 ft>s ; and 15 miles per hour, = 22 feet per sec; and the square of 22 = 484 ; Hence, by the formula, we have, Uniformly .,.;.* r>d* 484 retarded = JJjjS; X ,-7 = 1,120,000 X -7- = 1,120,000 X 7.5 = 8,400,000 ft-ft>s. working force J body b4 - 4 &A Now, how far will this working force work the train before its coming to rest, sup- posing the resistance of friction, air, &c, to be uniform, and to amount to 10 Jbs per ton wt of train ; or to a total resistance of 500 X 10 = 5000 Ibs. This resistance being assumed to present itself equally at every point along the road, the reqd dist evi- 8400000 ft H* 8 dently becomes = 1680 feet; for 1680 X 5000 = 8,400,000 ft-Ibs. In practice, the resistance of the air certainly would not be uniform ; but if we were to introduce its variableness into the question, the solution would become very difficult. The unif retard force div by the dist run gives the total fric. Art. 26. If an engine has not sufficient power to overcome the friction of a train, and to impart some (no matter how little) motion, by the first stroke of the piston, it will not be able to do so by any greater number of strokes. For the motion of the first stroke was entirely converted into strain by the resisting force of friction, &c, of the train; and by the time a second stroke can apply a second instalment of force, renewed friction is also ready to react against it, and thus destroy it ; and so with any number of strokes. The friction of the train is caused by its gravity or wt: and, since gravity acts as an unceasing stream of force, continually pouring into every body ; and being continually destroyed by the reaction of whatever the body is resting upon ; so, in like manner, does it maintain a constant stream of fric- tion between the body itself, and what it rests upon. It is plain that the rolling friction of the wheels, and the sliding friction of their journals, along one mile of road, are not identically the same as that on the next mile ; although it may be precisely the same in amount. So with quiescent friction; that of one instant is not identically the same as that of the next instant; for if it were, then the strain from the first stroke of the piston would destroy at least a portion of it; and a few more strokes the whole of it. Art. 27. If a body were perfectly rigid, any force imparted to it at one end would at the same instant reach the other end, no matter how long it might be ; and moreover, no amount of force could break it, no matter how thin it might be. But no bodies are perfectly rigid; and hence their inherent tensile or compressive forces, or strengths, will yield to any extraneous force applied in excess. Therefore, if we wish to transmit a great amount of force through a body which would otherwise crush, or pull apart under it we must take time, and transmit it by degrees. Thus, the coupling links of a long train of cars, would snap instantly under a pull sufficient to throw back into the train, at one effort, a moving force of 20 miles an hour; if an engine of such power existed. In practice we are therefore compelled to transmit the force from the engine to the train in instalments so small as not to ex- ceed the tensile force of the links. Rapid speeds are produced by the force thus accumulated in the train by degrees. See Art 20, p 453. The cogs of wheels which transmit force from the motive power, to the working points in ma- chinery, are frequently broken if the power is applied too suddenly ; that is, too much of it at once. If a pistol -ball be thrown with very little vel, against a pane of glass, its force will have time to dif- fuse itself ovar the whole pane, and will probably crack and shatter it in every direction ; but if it be shot with great vel from a pistol, it will frequently pass through the glass so quickly as not to give its force time to spread over the whole pane ; but it will merely allow it time to act upon the small circular piece which comes into immediate contact with it; and it will therefore cut out this small piece neatly, and carry it away. A person may safely skate across a thin piece of ice, which would break under his wt at rest ; for, before the ice has time to bend sufficiently to break, the load is removed from it. undergo none. Therefore scientific men adopt - body ; and they call the resulting quot the "mass " of the body, to distinguish it from mere wt. Thus at any place where the acceleration of gravity is 32.2 feet per sec, and where a body weighs 20 Ibs by 20 a spring balance, the body s mass, or scientific quantity, is equal to rr .621. To the prac- tical man, this mass or quantity conveys no idea whatever: but it is plain that the ordinary measure by weight cannot be perfectly correct, because the weight changes at diff places, while the quantity remains the same; and the measure by size would be equally incorrect, because the size varies with the temperature. Since, therefore, the only absolutely correct measure of quantity in a body is the scientific mass ; and since the imaginary vis viva is the quantity, mult hv the square of the vel. we have vis viva represented strictly by. Mass X Vel2; or the M.V2 of scientific writers. In science, the mass of 100 Ibs of iron is equal to that of 100 Ibs of cotton. The greatest discrepancy that can occur at various heights and latitudes, by adopting wt as the measure of quantity, would not be likely to exceed 1 in 300; or, under ordinary circumstances, 1 in 1000. FORCI RIGID BODIES. 457 A string may safelv^dstain a wt of 1 ft) suspended from our hand ; and if we wisb to impart a great u^rward vel to the wt, we evidently can do so only by imparting t$ it a great force ixand we may do this by jerking the string violently upward. But if it has not tensile force, or strength, sufficient to transmit this force all at once from our hand to the wt, it will break. It plainly is not broken by the wt, but by the excessive force which we endeavored to pass along it. We might safely give the wt all the vel or force we desire, by simply raising the string slowly at first, and more and more rapidly by degrees ; thus putting the force into the body gradually, in instalments too small to break the string. Some imagine that the string is broken by the so-called inertia of the wt, which they say causes it to resist moving force; and that we actually feel its resistance; and that it is made apparent by a spring balance, if we hold the balance in our hand, with the string and wt attached to it. Tnere is no doubt that when we jerk the string upward, the balance will indicate that there is an increased strain upon its spring. But this strain arises not from resistance of the wt, but from the direct action of the force which, as motion only, we have imparted to the string, to be by it conveyed, as work, to the wt. Work, because, when it reaches the wt, it has not only to impart motion to it ; but also to strain against its gravity force. The wt cannot receive great vel, unless we impart to it great force; this force is plainly imparted through the medium of the string: and if we attempt to impart too much at once, the string must break. The breaking of the string by the pull, is the same as the breaking or bending of a nail under too heavy a blow of a hammer: in both c;ses the failure is caused by the attempt to transmit at once a quantity of force which the inherent strength of the body is insufficient to sustain. Whether that force is motion or strain makes no difference. Springs ease the force of blows because their elasticity gives them time, by gradually yielding, to receive the whole action of the force, and react against, or destroy it, by degrees, or part at a time. The imparting, and the receiving, of force are often attended" by the same effects. If we hold our hand still, and let a hard play-ball strike it, the hand will experience the same sensation as if we first throw the ball upward ; and" afterward strike it with our open hand, at the moment it is turning to fall, and is consequently still, or at rest. In the first case the ball im- parts its moving force, as strain, to the hand at rest; in the second, the hand imparts its moving force as strain, to the ball at rest. In the first, we received ; in the second, we gave away force ; and in both cases, the effect on the hand was the same. Art. 28. Composition and resolution of forces. We hnve already said that when diff forces are imparted * (whether so applied, or not) in the same direction to a rigid body free to move unresistedly, they all act as motion alone, in that same direction. If two equal forces are imparted in diametrically opposite directions, they mutually destroy each other entirely, as strain (pull or push) against each other r thereby producing strain among the particles of the body; but having no tendency to move the body, as a whole, in either direction. If unequal, and in diametrically opposite directions, the whole of the small one, and a part of the large one, equal to the small one, destroy each other as strain; while their diff remains as motion, (or a moving of the whole body,) in its original di- rection. But if t w< "*m d m. - . n forces, a o and b o, Figs 9, whether equal or un- equal, are imparted at the same time to an un- resisting rigid body o, in directions either con- verging toward ; or di- verging from, the same point o, at any angle whatever ; then the body o cannot possibly be kept at rest by them ; or in other words, equilibrium cannot exist between them ; or they cannot balance, or completely react against each other ; the body must move. Equal parts of each of the two forces will mutually destroy each other as strain among the par- ticles of the body; while the remaining portions will unite to constitute a single force r o, which will move the whole body in a direction o d, in the line r o extended : and which direction o d will always be somewhere bftvjeen those in which the separate forces would have moved it. If we lay off c o and t o by any convenient scale, to represent respectively the amount of the forces a o and b o. and then complete the parallelogram o c r t ; the diag r o, measured by the same scale, will represent both the direction and the amount, of the single remaining force. * Tt is absolutely necessary to keep distinctly in mind the diff between applied and imparted force. Writers carelessly confound the two very frequently. See Art 19. 458 FORCE IN RIGID BODIES. The same procesa will answer also for forces which instead of motion, produce strain, not only in the particles of the body, but in the body itself considered as a whole ; or, in other words, a tendency to press or pull the entire body in a certain direction. Thus, suppose that two men were either pull- Ing or pushing with the forces c o and t o ; trying in vain to detach a piece o of rock, from a cliff of which it forms a portion ; and which, by its inherent force of cohesion to the cliff, defies their efforts. Here we have a case of extraneous forces, resisted, or reacted against, or balanced, by strength o/ material. As in the case of motion, the two forces partly destroy each other as strain among the particles of the body; and the remainders combine to forni the single force ro, which tends to move the whole body toward d. The rock resists this single force, by a cohesive force precisely equal, and diametri- cally opposite to it; and so long as it does so, there is strain but no motion. The piece of rock may have strength enough to oppose a much greater resistance ; but cannot actually exert it unless the men also exert more force. In the matter of comp and res of forces, it must be remembered that when force i applied to a body in order to produce motion, care must be taken that there is no other force to prevent it ; but when the force is intended to produce strain, it is equally necessary that other force should be present to oppose it ; for strain is the opposition of forces. The fig ocrt, Figs 9, is called the parallelogram of forces. The two original forces co, to, are called the components of the force ro; which results from their joint action ; and the force r o is called the resultant of the original ones which compose it. The principle of the parallelogram of forces, than which there is none more important in the whole range of mechanical science, may be expressed thus : If any two forces, (both motions, or both strains,) whose directions either converge toward, or diverge from, the same point, be represented both in quantity and in di- rection by two adjacent sides of a parallelogram : then will their resultant be simi- larly represented by the diag of the parallelogram,* REM. 1. If one of the forces, as c, upper Fig !%, is a pull, and the other a push, then to find their result- ant o t we must, before drawing the parallelogram of forces, move (or imagine to be moved) one of the forces to the opposite side of the point o, so as to cbange it from a pull to a push, or vice versa, so that both shall be pulls, or both pushes, as shown by the two lower figs. Otherwise we should obtain a wrong resultant no of the top fig. Either a push or a pull equal to ot, if applied at o, would be equal in effect to the push a and the pull c. The remark is of frequent use when finding strains in bowstring and crescent trusses ; as in many other cases. Item. 2. When any three forces as a, b, c, form- ing only two angles axb and bxc, balance each other at any point x, then a straight line as oe can be drawn through that point so that all three forces shall be on one side from it ; then also a parallelogram x n can be drawn on the three lines a, 6, c, having the middle line b for its diagonal ; and this diagonal will be of a different character from the two outer forces a and c; that is, if they are pulls, it will be a push, and vice versa. But if as in the tnree balancing forces /, i, s, three angles as s x t, txi, sx i, are formed, neither such a line, nor such a parallelogram can be drawn ; and the three forces will all be alike, all pulls or all pushes. All this is evident from the two figures. REM. 3. We have alluded to equal parts of each component as being lost, or de- stroyed, by reacting against each other; thus producing within the body a straining of its particles ; and therefore having no tendency to move, push, or pull, the body os a whole, in any direction. Let 6 a and c a be any two components, and na their resultant. From the two angles b and c, opposite to the diagonal, draw bo and ct at right angles to the diagonal; or to the diagonal extended, if necessary, as in Fig 9%. These two lines, bo, tc, will always be equal to one another; whatever may be the lengths and directions of the components b a, ca. When two forces, as 5 a, ca, are imparted at a. there occurs a loss offeree equal to what would result from the reaction of two forces equal to ft o and c i. It is lost by becoming strain against the cohesive forces of the parti- cles which compose the body a. In anticipation of what is said in Art 31, we will state that the force 6 a may be regarded as made up of the forces 60, oa; and the force ca, of ct, ta; which act also in those directions, when ft a and c a converge toward a, as in Fig 9V6 ; or in the directions a o, o 5, and a i, i c, when the forces diverge from a, as in Fig 9%. In either case, however, these forces, ft o, ao, ct, a, &c, must be considered as being imparted at a. This being supposed, It be- comes plain that when 6 a and c a meet at a, inasmuch as ft o and ct destroy each other as strain against the internal cohesive forces of the body, there remains nothing to act upon the body consid- ered as a whole, except oa and ta; which, being together equal to na, (as seen in the fig.) are, in other words, equal to, or actually compose, the resultant n a of the two components 6 a, ca. See RetnS. * Components and Resultants may be calculated by the form- ulas in Art 45, p 472, when a diagram is not considered sufficiently accurate. ^Qfim ] RIGID BODIES. 459 We conceive that eafiof the original forces endeavors as it were to compel the other to leave its own course, aud fpWow that of its antagonist; and the struggle continues until they have succeeded in forcing each other into the same direction. This is of course effected by their reactions against each other; and, as occurs in all cases of reaction, they expend equal parts of their forces on each other. When the two forces act in diametrically opposite directions, where there is no neutral diag direction that can be adopted, there is no alternative but for the larger force to react against or de- stroy the smaller one entirely ; thereby losing an equal amount of its own force. Its remains totter n slowly in their former unchanged direction. The writer can see no difference of principle between the reaction of opposite forces; that of oblique ones; and that of those at right angles to each other. REM. 5. When the direction a&. Fig 9%, of one of the forces, forms an angle ft a n, greater than 90, with the diagonal, the shape of the parallelogram of forces becomes such that the two equal lines bo and ct, cannot be drawn at right angles to the diag a n itself; or within the parallelogram ; in which case the diag must be extended each way, as to o and t; and the lines bo, ct, must be drawn at right angles to the extensions. . . When this occurs, the component forces a o, a t, cannot as in Fig 9J^ be measured on the diag a n of the parallelogram ; because they will be greater than it ; but must, like bo, ci, be measured outside of the fig. And here it must be remembered that oo and ai no longer measure forces acting (like those in Fig9^) in the same di- rection. Thus the strain along a b may be considered (see Comp and Res of Forces. Art 31) to be made up of two forces imparted at a ; namely, a hor force equal to o b, and a vert one equal to a o, acting upward. And the strain along a c, as made up of one hor force equal to i c, and a vert one ai. (greater than the whole diag,) acting downward ; both of them imparted at o. Hence, the re- sultant on we find is equal to the diff between the two vert compo- nents a o and a i. Thus it is seen that this shape of the parallelo- gram in no way affects the principle laid down in Remark 3. Art. 29. According to Art 18, the force w e, Fig 10, may be considered as im- parted to the rigid body B at any point whatever in its line of direction we; also, the force xi, at any point in its direction x d ; conse- quently, both of them may be considered as imparted at the same point a; inasmuch as it is situated in both these lines. Hence, it is immaterial, so far as regards the effect of those two converging forces upon the body considered as one entire rigid mass, whether they are actually im- parted like zo and yn, at the same point o; or like we and 25 1, at diff points i and e. For in either case their result- ant, or joint effect upon the body as a whole, is precisely i the same ; namely, a tendency to move the body in the i same line of direction oat. This tendency will actually _ /( \ produce motion if no opposing force prevents; otherwise JmlOL 1 it will produce strain in the body. J ~ REM. 1. Hence the resultant R, of two converging forces F/, Fig 10J4; or of two diverging ones F/, Fig 10%, acting in the same plane, but imparted at diff points . i NOTE. The savants now call "Dynamics" Kinetics; "Motion" they call ^Kinematics; and u Working Force," Energy. They subdivide Energy into Potential and Kinetic. Potential energy is that whose work is measured by Resistance X Distance, as in Arts 11 and 22. Kinetic Energy performs work meus- . wt X vel^in ft per sec uredby _ - ; as j n Arts 23 and 25. The employment of such terms in the study of mechanics merely muddles the mind and memory of the young engine^,-. 460 FORCE IN RIGID BODIES. \ Fiq 11 of a rigid body W, may be found as readily as when imparted at the same point; as at o, Figs 9, or Fig 10. Thus, produce their lines of direction, either forward as in Fig 10> : or backward as in Fig 10% ; as the case may require, until they meet, as at b. Make b a by any scale, equal to the force /; and b c equal to the force F. From a and c, draw lines respectively parallel to 6 c and b a ; thus complet- ing the parallelogram of forces, baic. The diag 6 i of this'parallelogram, measured by the same scale, will represent the reqd resultant R both in quantity, and in direction. It is thus seen that it is not necessary that the point b shall be in the body itself. REM. 2. It is perhaps almost useless to again remind the young student that the bodies are all along assumed to be rigid ; or inelastic, and incapable of being broken or bent by the imparted forces. For otherwise the force/, in Fig 10>$, might split off the top of the body ; or F might crush to dust its toe t; or both might penetrate it. But, assuming that the material is sufficiently strong to resist such splitting, crushing, and penetration, we at present confine ourselves to the effect of the forces, whether as motion, push, or pull, upon the body as a whole. The splitting, crushing. &c, is a mat- ter that must be considered under the head of Strength of Materials. It is of course quite as neces- sary in practice to pay attention to these effects as to the others, but it must be done by a separate process. .T1. Art. 3O. Since the effect produced upon a rigid body (con- sidered as a whole) by the resultant (ac, Fig 11) of any two forces (6 c, dc) tending to or from the same point, is the same as the joint effect of those two forces themselves, it follows that if we oppose to those two forces a third one (nc) equal to the resultant (ac), and diametrically opposite to it, that this third force will com- pletely react against, balance, or destroy said two forces; or rather their remains. It is frequently necessary to consider such a third force, (n c,) equal and opposite to a resultant (a c); and inasmuch as we do not know that any specific name has been applied to it, although one is needed, we suggest anti-resultant. Re- sultant (a c) may be defined to be a single force which will pro- duce upon a body considered as a whole, the same result that its components (6 c, dc) produce. Or as a force which, if its direction were reversed, (thus making an anti-resultant,) would balance its components. In the preceding Figs, the arrows represent pressures; if all the arrows be reversed, thus indi- cating pulls, the^principle and processes remain precisely the same; for force is still only force ; and its effect upon a rigid body, considered as a whole, is the same whether it act as a pull, or as a push. Bee Art 18. When the forces diverge from the same point, their strain is a pull, or a tension ; when they con- verge toward it, a push, or pres, or compression. Art. 31. By a process the reverse of that in Art 28, any single force, od, Fig 12, may be resolved into two component ones, n d, m d, one on each side of it, and in the same plane with it ; which would produce the same effect as it upon a rigid body, d, (considered as a whole,) by merely drawing from d, 2 lines dg, dt, showing the di- rections of the two forces; and then, drawing from o two other lines o n, o m, respectively parallel to dg, d t ; thus completing the parallelogram (dnom) of forces, upon od as its diag. Then measure dn, and dm, by the same scale as od; and they will give the amount of each of those forces. See Rem, p 260. It is plain that an infinite number of differently proportioned parallelograms, such as d n o m, d $ o a, Ac, may be drawn upon any line o d as a diag; and in any one of them, two adjacent sides will rep- resent components equal in effect to the single force o d, represented by the diag. Thus the forces nd,md, are equal to o d, as regards their effect upon a rigid body d, as a whole. So are also the forces s d and a d ; consequently the effect of s d, and a d, is equal to that of n d, and m d. It will be observed that the longer any two components on the same diag are, (as nd,md, longer than s d, a d,) the more nearly in a straight line, and more directly opposed to each other, do they become : and consequentlv the more nearly do they mutually destroy each other ; leaving smaller portions of each to act upon the body. Thus the portion of the great forces nd. md, left to act upon the body d, is no greater than that of the small forces ad, a d ; this remainder being in both cases represented by the resultant o d. RKM. Hence, if we have two forces, as the two pulls a b, ac, Fig 12^, whose amounts and directions both are given; and which are counteracted, or held in equilibrium, by two otner forces such as the two pulls a/, ae, whose directions alone are known, it becomes easy to find the amounts a d and a o of these last, thus : Complete the .- ,. parallelogram b a c t ; and draw its diag a t. Make a t fl; rt 1Q 1 / \ equal to at, and in a line with it, Complete the paral- -TICJ J,C~9~ / lelogram adio; then plainly a d will be the amount of ** ** f the force in the direction a/; and ao that in the direc- tion ae. FORCE IN SID BODIES. 461 Art. 32. It follows from the fefregoing articles, that a single force cannot be resolved into two components, onp^of which only is in the same direction as that Jorce itself; for if a line representing that force be taken as a diag, it is self-evident that no parallelogram can he drawn upon it which shall have any of its sides par- allel to said diag. Therefore a rope, as a b. F* 15, sustaining a wt to, so long as it remains perfectly vert, that is, pre- cisely in the direction of the force of gravity of the wt, will receive no assistance in upholding the wt by having added to it a single rope as 06, or by; or one extending from the wt itself in any in- clined direction. In other words, a perfectly vert rope cannot sustain one part of a load, and one in- clined rope another part. All this, indeed, is a result of the fact stated in Art 15; that any force, however great, (as the vert force of an immense suspended weight w, Fig 15.) will be turned out of its direction by any other force, however small, (as a slight pull from a rope 06, or by,) unless there i>e some third force Co prevent it. In the present instance, this third force might be a third rope ; for the rope a b will be relieved, and still remain vert, if we employ two oblique ones to assist it, pro- vided they be exactly opposite each other; or, in other words, that all three ropes, or forces, be iu %ue plane. W So also in the case of a vert post sustaining a load ; the pres from the load cannot pass vert through the axis of the post, if the load at the same time is partly sustained by a single oblique brace pressing against the post. Indeed, such a brace, by turning away the direction of the strain from the axis of the post, may very materially diminish the power of the latter to sustain the load; for it will be found under Strength of Materials, that if the strain along a post or column does not pass directly through its axis, the column may in some cases lose two-thirds of its strength. The principle of course applies to force in any other direction, as well as vert. A resultant may be greater or less than either one of its two oblique components: bnt it can never be greater, or even quite equal, to both of them : on the plain prin- ciple that any two sides of a triangle are greater than the third side. If the com- ponents are equal, and inclined to each other at an angle of 120, the resultant will be equal to one of them; therefore, the same weight that would bn j ak a single vert rope, or post, would break two ropes each of the same strength as the single one. or two posts, inclined 120 to each other. If the angle o a b, or y a fc, which either of the forces form with the diag a 6, exceeds 90, see Rems 5, of pp 459, 465. Art. 33. The principle of the parallelogram of forces is of constant applica- tion in constructions of every kind; for instance, bridges, centers, roofs, retaining- walls, &c. Figs 13, 14, 15, 16, show a few of the most simple cases of force (the load w) applied to produce strain ; by reacting against opposing forces ya, oa, presented by the walls. In all these, the load w, applied at a, is a single force of gravity ; and consequently acts, in a vert direction downward. It is to be resolved into two com- ponent forces in the direction am, an, in order that we may find the strains which it produces (according to the ordinary phraseology) along the pieces am, an, so that we may proportion their dimensions to resist those strains ; which strains are in fact produced by the reactions of the three, forces, of the load, and the two walls. To do this, in all the figs, from a draw a vert line a 6, to represent the direction of grav, or of the force in the load w. On this line, lay off by any convenient scale, the dist a b to represent the amount in fbs, tons, &c, of the load w. Also, from a draw the two lines am, an, in the directions of the reqd component forces. Then complete the parallelogram of forces, by drawing lines b o, b y, from b, respectively parallel to am, a n. Then will a o, measd by the same scale as a b, give the amount of strain, whether push or pull, which the load w produces along the piece am; and in like mariner will ay give the amount which it produces along the piece a n. It must be especially borne in mind, that we here speak only of the amounts and directions of the strains produced by the extraneous load w alone ; without reference to those produced by the weight of the pieces themselves. If the force acting at a is not vert, but oblique, then the direction of b must of course be drawn oblique ; but if the force at a is gravity or wt, it mutt be vert. 30 462 FORCE IN RIGID BODIES. m-- This mode of finding- strains does not apply unless a w, a n pass straight to supports m, n able to react against them in the same straight lines. Thus in Fig 9, p 251, with a load at Z only, the parallelogram would not give the strain along Z W because at W there is no reacting force in a direction from H to W. Footnote, p 252. REM 1. Fig 16^ will explain what we mean in saying that the strains e t, e s, are in reality produced by the walls; although they are usually ascribed to the load l t which is represented by the diag e i. We have said that a force cannot produce strain unless there is opposing force to strain against. Now, when we place the force of the load I at the point e, it is evident that it is upheld by the walls at A and B; or in other words, that it reacts against these walls ; and the walls against it. The wall A furnishes the force indicated by the arrow A ; and which may be considered as the resultant of the hor force c; and of the vert one o. So also the force B; as the re- sultant of m and n. Now these forces A and B are ap- plied at the point , just as well as the load I is ; for they pass up as pushes, along the rafters; as the force of I passes up as a pull, along the rope. The rafters and rope are mere- ly the mediums through which the three forces reach e ; and the forces in passing through them from end to end, of course produce in them strains respectively proportionate to the forces. Now, the forces e t and e. s, which are usually said to be produced by the load, are nothing more or less than the two forces A and B, produced by reaction of the walls; and which, for convenience of drawing the parallelogram of forces in practice, are laid off each way from e. We have then three forces t e, s e, and e i, all acting at e, to produce strain alone; and this they must do by straining against each other. The following is the manner in which they do so. The two hor components m and c, (which will always be equal to each other; no matter how ditfereut the slopes of the two ratters may be,) being dia- metrically opposite in direction, react or strain against, or balance, each other ; thereby producing a hor strain, equal to one of them, throughout every part of each rafter. The two vert components o and , (however unequal they may be,) will together be equal to the load I; or to its representative e i ; and having a direction exactly opposed to it, they react against, or balance it ; thereby producing in every part of the rafter e 8, a vert strain equal to n; and in the rafter e t, one equal to o. There- fore, since n, is here greater than o, the rafter e s bears more of the load I, than the rafter e t does; and in the same proportion. Thus, we see that every part of each of the three forces e i, et,es, produces strain, by balancing an equal part of one of the others. The walls really oppose to the load no force greater than its own ; namely, o and n. against e i. With the hor components TO and c, the walls react only against each other. Hence is seen the error of saying that the load /. produces the forces e s, e t. As it is difficult, however, to introduce a new phraseology, in place of one which, although errone- ous, is in universal use, we also shall speak of component strains like e t, c s, as if they were really pro- duced by the resultant, or load, e i. And in alluding to resultant motion, we shall probably often say they are the effects of components, instead of effects of their remainders, after the components have Sartially destroyed each other's moving forces by straining against each other to produce change of irection. BBM 2. The truth of such examples as Fig 14, with a rope or string, may easily be shown by means of two spring balances, to which the ends m and n of the string may be fastened. Suspend a weight w from the string, and the balances will show the strains along a m and a n. The balances must be held in inclined positions. The student should try all such experiments. This one will show that in proportion as the two parts am, an, of the rope, approach nearer to one straight line, the greater will be the strain produced upon them by any given load, or force w; and so great will this be, that if the weight w be only one pound, two of the strongest men cannot strain the rope perfectly straight be- tween them. Or if they stretch the rope alone to as nearly a straight line as possible, and if then a weight of a few IDS be suspended from it, this small weight will pull the men closer together. Or if the rope be stretched nearly straight .between the two spikes so firmly driven as to require a great force to draw them, it will be found that a much smaller force applied as at w, will draw them readily. In ether words, a rope so situated, and with force, or power w, applied to it in this manner, between its ends, and oblique to its di- rection, becomes a machine; for by it power may, (to use the *P 16J 5CE IN RIGID BODIES. 463 ordinary incorrect expression,) be gained. It is called the fuilidllar Iliactlilie ; or some- times simply the COTCi. Fig 16>6 shows the principle on which this machine is frequently employed for overcomingj/great resistance, r, througa a short distance, by a small power p. One end c, of a rope c d r. is -orrnly fixed. The rope passes over a pulley d ; and its other end is tied to the resist- ance, or load r. By applying a small downward force p, at the center of the rope, drawing it down to s, the load r is thereby raised a short dist ; for the same great strain which the small force p pro- duces from to d, extends also down the rope, from d to r ; except a slight loss produced by the friction of the pulley. Thus, the strain along the back-stays of a suspension bridge, is equal to that on the main chains just inside of the suspension piers; supposing the cables to rest upon rollers. In the theoretical consideration of ropes and chains, they are in most cases assumed not to stretch ; to be perfectly flexible ; without weight; and infinitely thin. In such a machine the two parts s c, s d, Fig 16}^, are to be considered as two entirely distinct ties , in the same manner as a m and a n, Figs 13 and 16, are two distinct struts Each of 'these ties may have to sustain a different amount of strain, depending on their respective inclinations to s p. Thus if the load p. Fig 16>6, be suspended from a perfectly frictiooless pulley or slip knot resting on the perfectly flexible cord c s d r, and if this pulley or knot be at first placed near c or d, it, with its load p will descend by gravity along the cord until it comes to rest at a. which is the lowest point that the cord admits of its attaining and at which alone the angles of Inclination of 8 C and s d to s P become equal; and the strains on the two parts will then ue equal. But it us in Fig 14 the short string which sustains W is tied fast to the cord (so as not to move as the pulley did) at any point a, such that the angles of inclination of a m and a n to the diagonal a b shall be different, then the strains or pulls along a m and a 11 will also bp different. It is Immaterial whether m and n, Fig 14, or c and d. Fig 16 i, arc at the same height or not. For more on the funicular machine see p 662 Let the end g of the rope g c on be fixed ; a power of 9 tons at n ; the rope passing over a pulley at P ; and bent out of line at c by a fixed pin. Make c g and c o by scale each equal to the power 9 atn; and complete the parallelogram ; the diagonal ex of which is then found to be. say 6; or a result- ant of 6 tons. Now, in this case, theoretically the strain lengthwise of the rope is everywhere equal to the power n, or 9 tons ; and we have found that it produces also a strain ex, against the pin at c, of 6 tons. It also produces a pushing strain on the pulley P. Its amount may be found in the same way, by measuring 9 tons by scale each way from o toward c and n ; completing the parallelogram; and measuring its diagonal resultant. But now let us use this rope as a funicular machine ; and apply a power x c of 6 tons at c. We find that this 6 tons produces a strain eg or c o, of 9 tons along the rope; and this strain along co will pass along to n; and thus the power of 6 balances a resistance of 9 tons acting at n in the direction n o. The diagonal c x or any other will plainly be vertical only when the angles of Inclination of c g and c o, with the horizon are equal. If they differ, both the di- rection and the length of the diagonal will change. All will remain the same if the end g instead of being fixed, is passed over a pulley as at P, and a load or a pull equal to that at the other end is applied to it. Rem. 3. The surfaces of contact of pieces used in construction, are called joints. When a piece is intended to resist compression, or push, it is called a strut; or if inclined, it is often called a brace; or if vertical, a post, pillar, or column. When to resist tension or pull, a tie. When to resist both tension and pull alternately, a tie-strut, or a strut-tie. A strut should be stiff or inflexible ; but a rope, chain, or thin rod, may answer for a tie. REM 4. To distinguish a tie from a strut at a glance is sometimes difficult; but it may be done thus. From the point a, Figs 1*%, at which the force acts, draw a line a c, in the direction in which the force, if at liberty, would move away from that point. On any part, a o, of that line as a diag, draw a paral- lelogram offerees. Through the point a draw a line i i, parallel to the other diag 1 1. Then all the pieces which are on the same side of that line, that a c is, are struts ; while those on the opposite side, are ties. We may also frequently determine, by imagining the piece to be a rope or chain, in- stead of a beam ; and seeing whether it would then bear the strain. If it would it is a tie ; if not, a strut. When a piece of material is used to resist forces which tend to bend or break it crosswise or trans- versely of its length, as in Figs 47. 48, 49, 50, it is called a beam; such as joists, girders, &c. The beam transversely ; but in our present illustrations of comp and res of forces, this strain, although frequently the most important one, could not be well considered at the same time. 464 FORCE IN RIGID BODIES. R Art. 34. Since any single force may be resolved into two oblique ones in the same plane with it, and which shall produce the same effect upon a rigid body con- sidered as a whole, it follows that the single strain along any piece a m or a w, of the four figs on p 461, may be thus resolved. Jn practice, it 13 frequently necessary to do this ; and especially so for finding components at right angles to each other, in hor and vert directions. For instance, the joint o d. Fig 17, at the foot of the beam A, if made at right angles to the resultant r r of all the pressures along the beam, of course receives the whole of these pressures; which consequently are all imparted to the abutment ; leaving no portion unresisted, so as to pro- duce sliding; or even a tendency to slide along the joint o d. Conse- quently, this joint is perfectly adapted to its duty. See Art 19. But a joint of the form of 6 i c, which is equally effective, is sometimes reqd for re- ceiving a single strain (like that along A) along a piece E ; and in order to properly proportion the vert and hor faces 6 i. and c t, of the joint, we must flnd the proportion existing between the vert, and the hor compo- nents equal to the single strain r r along E. To do this is very easy ; foi we have only to lay off by scale, any length e n along r r, to represent the single strain in that direction; and on it as a diag, from e and n draw vert and hor lines e t, n t, meeting in t. Then e t measured by the same scale, will give the vert strain ; while n t will give the hor one. The parts 6 i, i c of ttie joint, must consequently have the same proportion as p. *,| these two components have to each other ; bearing in mind, however, that JjlCI _lf since joints should be at right angles to the forces they have to sustain, * the vert part b i must bear the hor strain ; and the hor part i c, the vert one. When, by Art 33, we are finding, by means of the parallelogram of forces o n y g. Fig 18, the total strains o n. o g, which an extraneous load F produces along two beams. FR, Fg.it is easy at the same time to find the vert and hor compo- nents also ; by drawing the two hor lines n t, gj, and measuring them by the same scale used for the diag oy. Likewise measure o t. and oj, for the correspond- ing vert forces at the joints; because when n t and (jj may be drawn inside of the parallelogram, (which is not always the case; as see Fig 18^.) the component forces in the direction of any diag, \vhethervertornot. are measured respectively from the point o, where the extraneous force F is imparted to the beams ; to those points t and j, where the diag is met by the equal lilies n t, gj. Sot R-'rn, p 2(50. REM. 1. It ia an important fact that however diff may be either the inclinations, or the lengths of the two beams; or how diff the total strains in the directions of their respective lengths; the hor s; ruins, caused both by the extraneous load and by the weights of the beams themselves, will always be equal on both of them. Thus, in Fig 18, n t is equal to gj ; and in Figs 13 to 10, if hor lines be drawn from o and y, to a 6, those in any one fig will be equal to each other. HEM. 2. The beam o R, Fig 18, is not to be considered as acted upon at the same time by three distinct forces on, ot, and t n ; nor the beam og by three forces og, oj t gj\ but each is acted upon by one force ; thus o R is acted upon by o n ; which pro- duces upon it precisely the same effect as would be produced by its two components o t, n t. And so with the other beam. Each beam may be considered as receiving from the load F, either one force or its two components. The vert component oj, of the triangle o gj, being longer than o t, of the triangle o tn, shows that the beam o y pg. The beam and its load may be regarded as a single body, acted upon, and kept at rest, by three forces ; namely, its owa gravitv or wt; the force ft, at c; and the force/, at a. No other forces act on it. Now, gravity acts vert only ; and in the case before us it may all be re- garded as acting in the line p g. The force at c can act only at right angles to the surf or joint at that place, (see Art 19;) and since the joint is vert, the force A must be hor, or along ft p. The question now is, how to find the direction of the third force /. To do this we must avail ourselves of the principle that when three forces, not parallel to ea^h other, hold a body at rest, or in equilibrium, as these three forces hold the beam a c, their directions all tend to or from one point; which is either at the cen of grav of the body ; or in a vert line passing through said cen. Hence, since the vert direction p g of the force of gravity of the body ; and the direction ft p of the force ft, meet at p, therefore, the direction fp, of the force /, must also meet there. Hence we have only to draw a line fp, in order to find the reqd direction. A post intended to support the end a of the beam, should have the position fa; and the joint o i should be at right angles to fa; and not to a c, as mfekt at first be supposed from Figs 13 and 16, of Art 32; in which the wt of the beams is not considered. Having found the directions of the three forces in Fig 22^, it only remains to find their amounts. To do this, we already have one of them given, namely, gravity, or the wt of the beam and its load ; and we know that they must be in proportion to the sides of the triangle drawn parallel to their di- rections. Consequently, if on the vert direction p g, we lay off by scale any portion whatever, as p d, to represent the force of gravity, then will the hor side of the triangle, pdb, represent by the same scale the hor pres at c : and the side b p, the oblique pres at a. The hor pres at the foot is equal to that at the head of the beam. It is of course included in the oblique pres /; which is compounded of said hor force, and of the vert force at a. The vert force is equal to the weight of the beam and its load; none of which is sustained at c ; nor can be, so long as the joint at the head and wall is vert. Ex. 2. This is very similar to the preceding. Let a 6 ij\ Fig 22%, be the half of any arch bridge, loaded or unloaded equally throughout; and of which we know in either case the total wt ; and that the cen of grav of said wt is somewhere in the vert line gg. Now this half bridge is, like the preceding beam, kept in equilibrium, or at rest, by three forces only; namely, the wt; a hor pres A, at the crown, arising from the other half of the arch; and an oblique pres o, at the springing, or skew- back j i. To find the directions and amounts of these forces, from a draw a hor line, meeting the vert gg at c. From c draw a line to the center of ./ 1 ; this is the direc- tion of the oblique force o. From c measure down by scale any dist cs, on the vert direction g q, to represent the weight ; and from *, draw s t hor. Then s t, measd by the same scale, will be the hor pres 7i; and c t, the oblique one o. The joint j i, at the spring of the arch, bears all of c s ; that is, all the wt of the half arch, and half FO IN RIGID BODIES. 469 load. No vert pres or wtfis sustained at the center In of the arch ; nothing but the hor pres. Butj> i also sustains this hot pres, for c f is composed ol c s and s t. The oblique force ct constitutes the total thrust exerted by the entire arch, against each of its two abuts; and the line ct shows the direction in which this thrust enters the abut at the skewbackji. Alter entering at that point, it begins to curve downward, on the principle explained in Art 72. Since ctia the hypothenuse of a right-angled triangle, of which the leg cs represents the half wt; and st the hor pres of the arch, it follows that the total thrust c t of an arch may be found thus : add together the square of half its wt ; and the square of the hor pres ; and take the sq rt of the sum. This applies also to arches of iron or wood. The joints of any arch which is a portion of a circle, are usually drawn toward the center of the circle; and, practi- cally, this answers every purpose; but it is plain that strict theory would require the joint ij to be at right angles to co. Bo also the other joints of the archstones would be reqd to be perp to the pres which they have to sustain. Moreover, since the pres ct, upon the joint ji, is much greater than the pres st, upon the joint In, theory would require the joint \ji to be proportionally deeper than In; whereas in practice they are usually made the Mine, except in very large arches. See Stone Bridges. P 349, footnote. The last few lines of Ex 1, respecting the her pres at the foot, and at the head, apply equally here. The young stu- dent should familiarize himself thoroughly with the princi- ple illustrated by these two examples, as it is one of very frequent application in practice ; as in retaining- walls, abut- meats, &c. Here, as in the preceding example of the beam, we do not consider the strains produced along the length of the arch Inji itself ; but merely the two forces which, acting at its center In, and at its foot ji, keep each other, and the wt of the half arch and its load, in equilibrium. For the others, see Stone Bridges. Art. 39. A third mode of finding 1 the resultant R, Figs 23, of any number of forces E, F, G, in one plane; and acting: through one point, x. Draw two lines, H H, and V V, at right angles to each other. From their point of intersection o, draw lines by any convenient scale, to represent the directions and amounts of the forces. By Art 34, resolve each of these forces into two component ones parallel to H H and V V. Thus F o is re- solved into o and eo ; Go, into m o and to; E o, into to and no. Then measure by the scale, and add together, those components io and to, parallel to H H, and which tend to move the point o toward the left hand. Also add together those (in this case only one) o, which tend to move o toward the right hand. Subtract the least snm from the greatest; their diff, equal to s o, will be the resultant of the two sets of forces which respectively tend to move o to the right, and to the left. In this instance, this so must evidently be placed to the right of o, because the components on that side give the greatest sum. Next, add together those components eo, no, parallel to W, which tend to move the point o up- ward. In like manner add together those (in this case only one) components mo, parallel to V V, which tend to move o downward,. Subtract, as before, the least sum from the greatest; their diff, equal to ao, will be the resultant of the two set* of forces which respectively tend to move o upward and downward. In this instance, a o must be measd off below o, because the upward tendency Is the greatest. By this process, then, we first reduce all the original forces to t wo components, so and ao. This being done, we have only to complete the parallelogram of forces osca, and draw its diag co; which will be the flnal single resultant of all the original forces. From x draw xy parallel to co, and make by equal to co ; then is by. or R, th reqd resultant; and 6 the point fnr imparting it to ;he body P, ao that its effect may be equal to that of the three original forces combined. 470 FORCE IN RIGID BODIES. Art. 4O. Even when any number of forces in the same plane do NOT tend to or from the same point, the principle of the polygon offerees, or of Art 39, may be used in precisely the same manner as at Figs 21 and 23, for finding the length and direction of their resultant. Or if they are in equilibrium, and hence can have no resultant, they will still form a closed figure as A Fig 21, or N Fig 22, as well as if they acted through one point. There will however be this difference, that when all the forces, as a, 6, and c. Fig 21, tend to or from one point o, we know that their resultant, as R, must be applied parallel to A, and must tend to or from that same point o. In other words, -we know where its point of application must be. And so with the resultant co, Art 39. But when the forces do not tend to or from one point, and we find their resultant by Art 38 or 39, we know only its amount and direction ; but do not know where to apply it. In such cases we may use Art 36, p 466, Fig 19^. Art. 41. Forces in different planes ; but tending: to or from the same point. Such forces cannot, like those in one plane, be correctly rep- resented together on one flat surf, such as a sheet of paper. Thus, let Fig 27 be a cube; and tx, ex, ix, three forces acting in the directions of its edges ; and all tending to the same point x. It is plain that the relative positions of these forces are not correctly represented; for txc,txi, and csct, are in reality right angles; whereas, in the fig, t x c appears to be an acute one ; c x t, a right angle ; and t x i an obtuse one. On this account the resultant of such forces cannot be had by measurement from a drawing. Recourse must therefore be had to calculation ; which, however, will be facilitated by a drawing. The theoretical principle is very simple ; being, in fact, the same as when the forces are all in one 'plane; namely, first find by Art 28 the re- sultant of any two of them, (for any two are really in one plane;) then find the resultant of this resultant and the third force; and 80 on to the end. It is easy to find the first resultant; but the others are more troublesome. Instances are comparatively rare, in which the resultants of such forces are reqd to be found: the attention of the engineer being gen- erally confined to those in one plane ; as when proportioning bridges, roofs, retaining- walls, &c. FORCE IN^IGID BODIES. 471 Art. 42. To find but all tendin ult ant of forces in different planes, rough one point. In cases where -mathematical accuracy is not necessary, and the number of forces only three, or four, the writer will venture to propose a method by models ; which, if open to the objection of empiricism, has the ad vantage of requiring less time than other processes ; is sufficiently correct for most practical purposes; and shows the resultant in its actual position, which ie done by no method of calculation. Let ao, bo, co, Fig 30, be the three forces, meeting at o ; their angles with each other, a o b, 6 o c, Co a, (which alone are necessary in this method,) being of course known. Draw on pasteboard the JTi30 three forces ao,bo,co, as in Fig 31, with their actual angles ao b, b o c. c o a. By Art 28, draw the parallelogram of forces for the middle pair bo, co; and draw its diag w o, which will be the re- sultant of those two; leaving the resultant of it, and a o. yet to be found. Cut away neatly the whole fig, aoacwb a. Make deep knife-scratches along ob, o c, so that the two outer triangles may be more readily turned at angles to the middle one. Turn them until the two edges o a, o a, meet; and then paste a piece of thin paper along the meeting joint, to keep them in place. Stand the model upon its side o b w c as a base ; and we shall have the slipper shape aobw, Fig 32 ; o w being the sole, and a o ft the hollow foot. We new have the first resultant w o, and the third remaining force a o, in their actual relative, po- sitions. Now, to find their resultant, also in its actual position, cut a separate triangular piece of paste- board of the size and shape of w a o. Find the center i, of the edge w a, and draw a line i o on each side of it. Finally, by means of tbe edges ao, wo, paste this piece to the inside of the model, along its center-Hue wo. This done, io represents one half of the reqd resultant, in its actual position. The reason why it represents but one-half of it is plain ; for, as be- fore stated, we now have a o and w o in their actual positions in the model; consequently, if we complete the parallelogram of forces wo an, and draw its diagonal no, this last will be their resultant. But since the two diags of every parallelogram divide each other into two equal parts, the diag aw; thus divides the resultant : consequently t o is one-half the resultant. If there be four forces, as an, bn, en, dn, Fig 34, draw them as in the fig, with their actual angles anb, bnc, &c. Draw also the re- sultants n v, of an and 6 n ; and n w, of n c and n d. Then cut out the entire fig, as before; and paste together the two edges an, an. Then we have the two resultants av, aw, Fig 35, forming two simple forces, in their actual relative positions ; and we have only to measure their dist apart from v to w; and thence find their resultant ar, which will evidently be that of the four original forces. Or, as in the preceding case, cut out a separate piece of pasteboard, avw. Fig 35, and having drawn on each side of it a line from a to the center o of vw, paste it inside of the model. Then will ao represent one-half of the resultant of the four forces, in its actual position. Should the model be exposed to hard usage by workmen, it should be made of wood ; the triangles anb, bnc, _. Rem. To find the portions of z borne by a and by * p say, as the whole span s a is to the whole load z, so is n a to the load on s ; and as 8 a is to z so is n to the load on a. Ex. 3. If the beam sustains several loads at diff points, as in Fig 49, calculate for each of them separately, using the leverages a f, a c, a o, &c ; and add all together for total F. For portions of each of these loads borne by a and v see above Rem. For more on this subject see p 218. Ex. 4. If the beam in any such case, in inclined, as in Fig 50, the hor dist a o, a g, &c, must be taken as measured from the fulcrum a, instead of at, a i, &c 5 because, since all the forces are vert in direction, only a hor line can be at right angles to them , and serve to measure their leverages from th fulcrum a. If the beam be rigid, and its ends cut hor, as shown in this fig, it will have no tendency to slide; because all the forces which through it are applied to the bodies m and p are vert; and since the joints are at right angles to those bodies at those points, the entire forces will be imparted also ; no por- tion of them remaining unresisted, to act as motion, so long as the beam remains rigid, and consequently straight. But if it bends un- der either its own wt, or that of its load, new forces come into action, which will tend to push the supports outward from each other; so also in the foregoing cases. It is only where we may practically regard a beam as rigid, or unchangeable under the forces, that the foregoing concentration of entire weights or forces at the cen of grav, can be safely assumed. It will not apply when we are investigating the strength, and deflections of beams ; see Art 58. After having thus obtained F, in any of these cases, or in other words, having found how much f the entire wt of beam and load bears upon one support, we have only to subtract it from the eutire wt, to obtain that on the other support. It is plainly immaterial which end of the beam is assumed to be the fulcrum in any of these cases. Ex. 5. Let a o, Figs 51, be a hor beam 10 ft long, projecting from a vert wall a c ; and resting at one end on a step a; the other end being sustained by either a strut, or a tie p c, 12& ft long. The beam, and its uni- form load, weighing together 3 tons, what will be the push- ing strain along the direction of the strut; or the pulling strain along the tie ? Draw the Fig to scale ; and meas- ure a i (which will be found to be 6 ft) at right angles to c o. Now, the weight of a rigid body, when considered only with regard to its effect in moving the entire un- altered body, or in straining it bodily against another body, acts the same as if it were all concentrated at its cen of grav; and since we are now about to consider it in that light, and not as tending to lend or break the beam a o (in which case only half its uniform load, and wt must be assumed to be concentrated at its cen of grav ;) we consider the 3 tons wt to act at g, 5 ft, or half the length of the beam from a. Now, the 3 tons, being a force of grav, will act in a vert direction ; and since the beam is hor, a g is at right angles to this direction of the force exerted by the beam and its load. Consequently, if we assume the beam to be a lever, movable about a, as a fulcrum, a g is the leverage of that force of grav ; and the moment of that force about a, consequently, is 3 X 5 15 foot-tons. But this moment is reacted against by that of another force in the direction c o ; which acts at the point o of the lever o o, to uphold the beam and its load. The leverage a i of this force, that is, the dist from the fulcrum a, and at right angles to the direction c o of the force, has already been found to be 6 ft ; consequently, the force itself, in order to have a moment of 15 ft- tons alxmt a (as the beam and its load have) must evidently be = 2.5 tons, the reqd strain along the strut, or along the tie, o c; for 2.5 X 6 15. Ex. 6. The following- is very important in its application to arches of any material. Let end rj, Fig 52, represent one half of a bridge arch. If this half were not prevented by the hor pres, ha, of the opposite half, it would evidently fall, as in the shaded fig. by turning about the point r as a fulcrum. (See Rem 1.) Let us find what this hor pres amounts to in any case. To do this, we may consider the half bridge cndrj to be a lever. Suppose its wt to be 80 tons ; and t be concentrated at its cen of grav g ; 'the vert line g s being of course the direction in which it would act. And let its leverage r t, about r, be \ ft; r t of course being at right angles to the direction 5RCE IN RIGID BODIES. 479 g a. TheLKisits moment about r equal to 80 X 6 = 480 ft-tous. (Art 46.) Now, whatever may b< the anvsruiu of the hor force h a, which acts at the end a of this lever, to counteract this moment of 480 ft-tons, its leverage (Art 46) is plainly equal to re, measured from the fulcrum r, and at right angles to the direction A i of said force. Suppose we find by mea- surement from the drawing that r e is 8 feet. Then the force itself must necessarily be = 60 tons which is the hor pres which the opposite half of the bridge exerts against the keystone a, of the arch ; for 60 X 8 480 ft-tons of moment. KKM. 1. But so long as an arch is not deranged, but remains firmly in position, the half arch, in- stead of tending to revolve about the point r, presses equally over the entire surf r of of its skewback. Therefore, the leverage with which the hor force A acts upon the skewback, is actually y o, measured from the center of rd, and in practice it must be used instead of r e. In the same manner, ym be- comes the leverage for the wt, instead of r t. HEM. 2. The cen of g-rav of the half arch, can be found by making a drawing end rj, about 4 to 6 ins long, on pasteboard, or on a stiff drawing-paper, to a scale. Cut out the fig; and balance it flatways on a sharp straight-edge, or over the edge of a table, in two directions or positions. Where these two directions intersect each other is the cen of grav. It is not indeed this cen itself that is needed, but the line g s, of its direction ; which may be found t once by taking care that the straight-edge is parallel to the back n d, while balancing the fig. RKM. 3. Under the head leverage, may be classed the tread-wheel; windlass and lever; capstan and lever; and all axles turned by a winch or by a crank ; such as the drum and winch with which a water-bucket is raised from a well, &c. They are all merely continuous simple levers, of which the axis is the fulcrum ; the rad of the circle described by the power is one arm, and the rad of that de- scribed by the shaft, drum, , there is no gain of power ; for here the diam a b is a lever of two equal arms, revolving around its fulcrum at the center of the pulley. Consequently, the wt and the power have equal leverages; 480 FORCE IN RIGID BODIES. Fio53 o each equai to the rad of the circle ; aud in order to balance a wt W of say 1 ton. the power F must also be 1 ton ; for if one of them moves, the other must plainly move with the same rel. To raise the wt, the power must exceed the wt; because it has also to overcome the friction of the axle around which the pulley revolves, and the friction of the rope in the groove around its circumf. These frictions become so great when many pulleys are combined, that theoretical cal- culations of the power are of little value. Although a fixed Sulley gives no gain of power, t is very convenient for al- lowing change of direction in applying the power; so that by pulling downward, or hor, &c, we can cause the wt to rise vert. It is plain that the rope in this pulley is equally strained at all points. Theo- retically, this is the case with any one single rope, as rcdf ge. Fig 52%, passing around any numberof pulleys, whether fixed, as A or D, or movable, as B ; and all the theoretical calculations of the power may be based upon this principle T/iT// alone. They will, however, be incorrect in practice, on account of the friction just alluded to. In Fig 52%, where only one rope is used, the lower pulley-block B , to which the wt W is attached, is directly upheld by the two parts df and eg of the rope. Consequently each of these parts is equally strained by a force equal to one half the wt ; and since the whole single rope is theo- retically strained to the same extent, that part of it to which the power is applied must be strained equal to half the wt W ; or, in other words, the power itself must be equal to half the wt, and will move twice as fast, aud twice as far. In Fig 53, the lower pulley-block ty is sustained directly by the 4 parts cccc of the single rope; therefore, each part of the rope, and consequently the whole o'f it, is equally strained by a force equal to % of the wt W ; and the power P must be equal to the same }^, and will move 4 times as fast, and 4 times as far, as the wt. It is immaterial whether the two pulleys in the lower block of Fig 53, be placed one above the other, as shown, or (as usual, and more convenient) side by side; so also with those in the upper block. If there were 3, 4, or 5. &c, pulleys in each block, then there would be 6, 8, or 10 sustaining parts c c, &c, of rope, each stretched equal to -^, |-, or y 1 ^- of the wt W ; and the power would also be in the same proportions to the wt; in other words, to find the theoretical pro- portion of the power to the wt, divide 1 by the number of parts c ccc of rope which directly sustain the lower block. In our fig it is %. The same rule applies to Fig 52%. When more than one rope is used in a system of pulleys, the strains become diff from the foregoing, on the principle illustrated by Fig 53J4. Here the lower block y b, with its attached wt of say 4 tons, is directly sustained by the two parts a and c of one rope. Consequently, each part has a strain of % the wt, or '2 tons ; which is uniform throughout that rope. But all the 2 tons strain on the part c is sustained by the hook s ; while that on the part a is sustained bv-the two parts n and m of the other rope : each of which plainly sustains one half of it. or 1 ton, which is uniform throughout this second rope to its very end. ir @fo \ Therefore the power also is 1 ton, or y of the wt W. The mode of V . / / proceeding is the same, whatever may be the number of movable pul- VI I/ leys. To find the theoretical proportion of the power to the wt. mult together continuously as many 2s as there are movable pulleys, and "P| div 1 by the prod. Thus, here we have two movable pulleys, and 2 X 2 n 4 ; and % = the answer. If there were 4 movable pulleys, we should have 2X2X2X2 = 16; and y 1 ^ = answer. In all our figs, that end of the rope to which the power P 1 is applied is represented as hanging vertically, and parallel to the other parts of the rope ; but this was done merely because the power is supposed to be a weight, and of course acting vert. But if the power is muscular force, or any other kind that may act in any direction whatever, then the power end of the rope, as mn. Fig 53, may have that direction in Tfhich it is most convenient. The amount of power required will not be thereby changed ; for it is plain that leverage from the center of the pulley o s to m, when the power is at n. and acting in the inclined di- rection of the rope, is equal to that from the same center to o, when the power is at P, and acting vert. The parts of the ropes, except the power-ends, (as m n, Fig 53) are sup- tL Us RIGID BODIES. 481 posed to be 53) it will ' _.flel; for if a part be inclined (as is the end attached to the fixed pulleys in Fig trained more than the other parts, See Art 4, p 668. Tlie Wedg'C is a kind of double inclined plane, (Art 60.) and is a very powerful machine But inasmuch as it is usually worked by blows, the effect of which cannot be calculated ; and in*<- much as its theoretical action is in practice totally changed by friction, no serviceable rules can be given respecting it. Art. 55. Parallel forces are those whose directions, as in Fig 53*^, (whether opposite to one another, or not; or whether in the same plane, or not,) are parallel. In Fig 53*/, the forces, although acting upon one plane, oooo, are not in one plane, but in several. It is a peculiarity of parallel forces in one plane, that all their arms, or leverages with respect to any given point in the same plane are in the same straight line. Thus, if /, TO, n, o, Fig 54, he in the same plane, then their leverages p q, p r, p a. about the point p. are all in the same straight line p a. The pointy is supposed to be in the same plane as the forces. Two parallel forces are evidently always in the same plane: that is, the same flat surf could coincide with 'both of them ; and their re- to saying:, in other words, that if 3 parallel forces hold each other in equilibrium, they are in the same plane. See Rem 2, page 468. Fitf 54 Art. 56. The resultant of any number of parallel forces, whether in the same plane, or in the same direction, or not, is always parallel to them. If they all act in the same direction, whether in the same plane or not, their resultant is equal in amount to their sum : or, in other words, an autiresultant force sufficient to balance them, must be equal to all the forces added together. But if they are in oppo- site directions, their resultant will be equal to the diff between those which act in one direction, and those which act in the opposite one ; and its direction will be that of the greater sum. Thus, in Fig 53^, if the forces pointing to the left amount to 10 tons, arid those to the right 4 tons; then the resultant will be 10 4 = 6 tons ; and it will point to the left. The parallel vert downward forces of gravity, upon the innu- merable separate particles, situated in the infinite number of imaginary vert planes, in any body, as W, Fig 55, is an illustra- tion of this. If any such body be suspended by a string from a spring-balance B, the vert upward pull of the string will balance or equilibrate all these innumerable forces. Consequently, the string represents their antiresultant. which is equal to their re- sultant. We know that the vert pull on the string, as shown by the spring-balance, is equal to the wt of the body ; which wt is made up of the innumerable parallel vert forces alluded to. Thus we see that when any number of parallel forces, whether in the same plane or not, net in the same direction, their antiresultant is parallel to them, and equal to their sum ; consequently their resultant must be so also. The same principle applies to parallel forces in any direction whatever. When a body thus acted on by gravity is kept at rest, or balanced, as in the fig, then the direction of the resultant or antiresultant, or of the string in the fig, passes through a certain point, called the center of gravity of the body. This is a certain point, upon which when acted upon by gravity only, the body will balance itself, in whatever position it may be placed : and if the entire wt or grav of the body could be concentrated into that single point, its effect, whether regarded as moving the entire rigid body, or as producing strain (pnll or push) between it and another rigid body, would remain precisely the same as it actually is with the grav diffused throughout the entire mass.* * In some bodies the cen of grav is also the center of the wt of the body ; but very frequently this is not the case. Thus, in a body a b c. Fig 55>," with its cen of grav at c. there is more wt on the side a c. than on the side c 6. If a body W, Fig 55, suspended freely from any point n, is at rest, its cen of grav is directly under said point. If the body W be pushed a little to one side, and then left to itself, it will plainly tend of itself to swing back to its first position : and when this is the case, it is said to be in stable equilibrium. But if the body, instead of being suspended, be balanced on top of a slim rod, and if we then push it a little to one side, it will not tend to return, but will fall over ; and therefore the equilibrium of a body so balanced is said to be unstable. Also 482 FORCE IN RIGID BODIES. K 3 56 Art. 57. That point through which the direction of a single antiresultant force must pass, in order to balance several other ibices acting at diff points ; or, in other words, that point through which the direction of the resultant of those forces must pass, is called the center of pressure, or of force, or of strain, of those forces, as the case may be. For instance, let S, Fig 56, be a common wooden box; but having one side, as o o, looselv fitted, so as barely to allow of pushing it back- ward and forward. Fill the box with dry sand, (clean small gravel will be better,) aud it will be found that there is but one single point, i, at which we can. by holding to it a thin rod r i, balance the pres of the gravel against the opposite side of o o. If we apply the rod at any other point, o o will give way before the sand ; if the rod is held above , the bottom of o o will be pushed outward ; if held below i, the top of oo will move outward. This point i is dist above the bottom of the sand % of the depth of the sand ; in other words, the cen of pres of sand of any depth is, like that of water, at % of that depth from the bottom. In the case before us. the depth is supposed to be uniform, so that the cen of pres is at the same height above the bottom, clear across the box. Now the balancing force applied through the rod at i, is the antiresultant of all the pressures re- sulting from the several particles of gravel against the opposite side of oo; and its effect upon the rigid body o o, (omitting of course any tendency to bend or break it, which comes under the head of Strength of Materials,) is precisely the same as that of all those forces combined; except that it is in the opposite direction. Its tendency to push oo bodily, or as an entire mass, toward the right hand, is precisely the same as that of the gravel to push it to the left hand; or it is the same as would result were we to heap up sand in front of oo, so as to balance the sand behind it. RKM. It is this important principle of the cen of pres, that enables us to adopt the convenient practice of representing, by a single line, the effect of force actually distributed over a considerable surf. Thus, in Fig 52, the hor force ha, by which each half of the arch mutually prevents the other half from falling, is actually distributed over an area whose depth is the depth cj of the keystone; and its breadth, that of the whole bridge, as measd acres* the roadway. Yet the arrow ft a, when drawn to a scale, perfectly represents the effects of this distributed force in upholding the half arch, considered as an entire rigid mast. So far as regards splitting or cracking the stone immediately at a, the effects would of course be diff; but as the whole force is only supposed, for convenience, to be applied at a, this diff is merely ideal in this instance. See Arts 5 and 8, p 525, 526. It is evident from the foregoing that "cen of grav" means nothing more than "cen of force; " ex- cept only that the former is a convenient term for denoting that the force is that of grav alone. Art. 58. It may be well here to direct particular attention to the fact alluded to in Ex 4, Art 54, that when either grav or other forces are to be considered with regard to their eifects in bending, or breaking bodies; instead of moving or straining them as entire masses, supposed to be rigid, or incapable of change of form, they cannot be assumed to be concentrated at either the cen of force, or the cen of grav. In the last case our applied extraneous forces are supposed to be brought to bear only upon other extianeous forces acting upon the bodies at the same time ; the bodies themselves being regarded mere- ly as unalterable mediums through which said forces are enabled to act upon each other; but in bending, breaking, twisting, shearing, &c, our extraneous or mechanical forces must be considered to act against the inherent cohesive forces of the bodies themselves; therefore these forces, which before were entirely neglected, now acquire a primary importance ; the question of strength of materials comes in, and the assumption of perfect rigidity must be altogether discarded. Thus, in Fig 56*4, so far as regards either moving a rigid body c o, or straining it against the force n, it is immaterial whether we employ the two equal parallel forces a, ft, or a single force m, equal to both of them, and acting at their center of force. But since no bodies are absolutely rigid, but may all be bent or broken, it is plain that the two forces a, b, straining the body c o against the force n. would bend or break it much more readily than the force m would. An absolutely rigid hor beam * s, would sustain any amount of load I, without bending; and consequently would always press vert upon its upright supports u, u, without any tendency to press them sideways. But an actual beam n , if overloaded, will bend ; thereby generating at its ends forces which are not vert, but which will tend to overthrow the supports * t. Art. 59. To find the point of impartation of the resultant Of parallel forces. Case 1. Two parallel forces, 1 ton, and 3 tons, Fig 57, in in such cases as that of a grindstone supported by its hor axis passing through its cen of grav, if we cause it to revolve a short dist, and then leave it to itself, it will have no tendency either to re- turn, or to keep on revolving ; and its equilibrium is called indifferent. Bee p 635. FORCE RIGID BODIES. 483 -4TONS 3 TONS Fig 57 the same direction ; and cprtiequently in the same plane. Draw any straight line a o, uniting their direckms 1 w, 3 n. Measure this Hue, and/div it into two cl x~ "\-&. parts, io, ia, proportioned like the TTt '^"^~"~'~"j^T. '1TON forces; hut plaora inversely to the forces ; that is, place the longest part near the smallest force, and vice versa. Through the point i draw the direc- tion of the resultant R, parallel to the forces. Then is t the point of imparta- tion. Make the resultant, 4, equal to the sum of the two forces. Ex. Let one force be 1 ton; and the other, 3 tons; and let ao be 8 ft. Then, as the sum, 4 tons, of the two forces ; is to the length, 8 ft, of a o ; so is the large force, 3 tons ; to the long part, i cr, of a o. Or, 4 : 8 : : 3 : 6 = t a. Consequently, i o = 8 6 = 2 ft ; as shown in the fig. The foregoing, as well as some other facts connected with parallel forces, will be more easily recalled to mind, by associating them with the idea of the common steel-yard, Pig 58. Her* the two forces in Pig 57 are represented by the wt 1 ft at c, and the wt 3 tts at d ; one of them 3 times as great as the other. We know that these weights, when sus- pended at dists fa,fb, one of them 3 times as great as the other, from the fulcrum/, are balanced by their antiresultant//, which is equal to 3 + l~4:fl>s, or the sum of the two forces. This antiresultaut has precisely the same tendency to pull the steel-yard upward, that the two weights have to pull it downward. Also/*, equal to fh, but acting in the opposite direction, is the resultant of the two forces; its effect to pull the steel-yard bodily downward, when applied to it at/, is precisely equal to that of the two wts applied at a and 6. The effect of the two wts at a and 5, to bend or break the steel-yard, would plainly be very diff from that of their resultant */, applied at/. See Art 58. Case 2. Two unequal parallel forces, da of 3 tons, and hf of 4 tons, Fig 59. imparted to a rigid body in opposite directions, but not in the same straight line. Starting from the line of direction dm, of the small force, draw any line mn, passing through and beyond the direction hf of the large one. Now find the amount of the resultant R. This, by Art 56, is equal to the diff between hf and da, since they are in opposite directions ; and has the same direc- tion as the large one ; therefore it is equal to 4 3 = 1 ton; and its direction is the same as that of hf. Then say, as the resultant is to the small force d a, so is the dist om, to the dist o, along the line m n. Through draw c, parallel to the two forces ; and c will be the reqd point of impartation of the resultant R ; the tendency of which to move the entire rigid body, will be equal to the joint tendency of d o and hf. An antiresultant at t would of course balance the two forces da, hf. This case, like the preceding one, is illustrated by the steel-yard, Fig 58; where ad, and/ft, repre- sent the two forces in the last fig ; while 6 c represents their balancing antiresultant, corresponding to t. Case 3. Couples. Two equal parallel forces, a, and &, Fig 59 a, imparted to a rigid body in opposite directions, but not in the same straight line, are called a couple. The force of a couple, means simply one of the forces ; the perp dist c, between the directions of the two forces, is called the arm, or leverage of the couple. If one of the forces be mult by this arm, the prod is the moment of the couple, in foot-Ibs, Ac. A couple has no tendency to move the entire body forward in the direction of either force ; but merely to make it rotate around a point o, half-way between the points at which the two forces are imparted. A couple (as is also the case with two equal opposing forces in the same straight line) has no single resultant ; only another couple can hold it in equilibrium. Case 4. Any number of parallel forces, whether in the same plane, or in the same direction, or not. The process in this case consists in a mere repetition of that in Case 1, Fig 57, as follows. Namely, commencing with those forces which point in the same direction, as a, b, c. Fig 60, which all point downward ; between the directions of any two of them, as a and 6, draw any straight line oi, anddiv it into two parts jo, ji, proportioned like the forces, but placed inversely to the forces. Through ./draw ms, parallel and equal to the forces a and b. Then is ma the resultant of those two forces. Next find the resultant of this resultant m and any third force, as c, in the same manner That is, draw any line t w, uniting their directions; div it into two parts It and Iw, proportioned like the forces, but placed inversely; Through I draw ng, and make it equal to m , and c. Then is Pi.,59 484 FORCE IN RIGID BODIES. n g the resultant of the three forces a, b, c; and g is the cen of force of those forces, aud consequently is the point of application of their resultant. So do with any number in the same direction. Then proceed in the same manner with those which point in the opposite direction, as d, e, x, y; and having found their resultant, find by Case 2, the resultant of the two resultant forces, now obtained in opposite directions. It is not at all necessary that the forces be supposed to act upon a plane surf, as in Fig 60; the pro- plane. This is 'a consequence of' the principle laid down in Art 18, Fig 5 ; namely, that the effect pro- duced upon a rigid body by an imparted force, remains the same, no matter at what point of the uody it be imparted, so long as that point is in the line of the direction of that force. I e ErfGO ] at Although Figs 60 and 61 serve to illustrate the principle, they plainly do not give the actual posi- tions of the forces and resultant; because they are necessarily drawn in a kind of perspective, so that all the parts cannot be measd by a scale. The amounts of the resultants are easily fouud by calculation ; inasmuch as they are equal to the sums of the forces. The points for imparting them can be found correctly from a drawing in plan, like Fig 62 : where the stars represent by scale the actual dists apart of the directions of the forces. The quantities of the forces, instead of being shown by lines, are to be written in figures, as shown in the fig. This being done, it is easy to tiiid the points a, g, &c, of the resultants. Art. 60. The Inclined Plane is a rigid straight plane surf, as a &, Fig 63, not hor. If a vert line b c be drawn from the top b of the plane, to meet a hor line ttc, drawn from its bottom a, then be is called the height of the plane; ac its fca.sv ; and a b its length. The angle b ac, which the plane forms with the hor line a c, is called its inclination, slope, or steepness ; which, however, is frequently ex- pressed also by the proportion which th* base bears to the height ; thus, if the length ac of the base be 1, 1%, 2, &c, times that of the height be, the inclination or slope is said to be 1 to 1, 1% to 1, 2 to 1, &c. The angle 6 ac is the angle of inclination of the plane. It follows from Arts 56, 57, that when one rigid body as N or M, Fig 63, is placed loosely upon an- other, as upon the rigid plane a b, the effect produced \ by its wt is the same as if all that wt were concen- \"1 trated at its cen of grav g, and acted in the direction iV of a vert line git; drawn through said center. When we assume the wt to be thus concentrated at the point g, we must remember that all other parts of the body must be considered to be without weight ; although still retaining their inherent cohesive force, or strength. If, as in N, this vert line gv, which now represents the direction of the entire wt of the body, passes beyond, or outside of the base, the body must fall; because this wt meets with no opposing force, in the direction vg r to react against it; and thus prevent it from producing motion. See Re- mark, Art 65. But if, as in the body M, the line g v falls within the base r 8, the body will not upset : but we shall have (Art 19) a force gv equal to the wt of the body, and applied obliquely to a rigid surface aft, *t 1W63 IN RIGID BODIES. 485 the point vjaarfconsequently resolvable into two components; namely, iv, perp to the surf ao, and therefore istfp arted to it as a pressure ; and x v, parallel to the surf, and consequently not imparted to it. All these lines may be drawn by scale, to represent their respective forces. When we consider a single force as y v to be thus resolved into two components, with a view to ascertaining their effects, it is plain that said single force must then be considered as no longer existing; but as being replaced by its components. Now the component force xv being parallel to the plane, it follows (Art 15) that thepreasure or strain i v, no matter how great it may be, cannot in the slightest degree oppose the cross action of the moving force x v, no matter how small it may be ; and x v must therefore produce motion in the body, causing it to slide down the plane ; unless some third force, not yet spoken of, shall present itself, opposed to xv. But any forces which press bodies together, always produce a new force, fric- tion, at the joint, or surfs of contact of the bodies ; and this friction acts in direct opposition to any force, in any direction whatever that is in the plane of that joint or surfs. See Friction. Therefore the force i v produces fric between the surfs, r s, of the body M and of the inclined plane ; and this fric acts in the direction r a, or diametrically opposite to that of the force xv. The amount of fric depends upon that of the pres ; as also upon "the nature of the bodies at whose surfs it is produced, upon the degree of smoothness of those surfs, and upon whether they are lubricated or not. If the fric is greater than the force xv in the opposite direction, the body of course cannot move ; but if less, it will move, under the action of a force equal to the excess of xv over the fric. It must be remem- bered that the pres component t v, which produces fric on an inclined plane, is not equal to the wt of the body, but is less than it. It is equal only when the surf is hor, so that the vert force g v, rep- resenting the entire wt of the body, is at right angles to the joint, and when, consequently, it all acts as pres. Therefore, the steeper the plane becomes, the less is the fric; because then less of the wt of the body acts as pres, and more of it as moving force. Hence, a locomotive has less adhesion on an inclined grade, than on a level ; for the so called adhesion is in reality nothing but fric. But although both the perp pres and the fric become less in amount as the plane becomes steeper, yet they constantly retain the same proportion to each other; until pressures become so great that abrasion of the surfs of contact takes place; the proportion of the fric to the pres then increases. REM. It is evident that when we wish to push a body up an inclined plane, we must overcome both the Trie, and the parallel force x v ; but in pushing it down, we are opposed only by the fric ; for the parallel force assists us. Art. 61. Experiment has determined the amount of fric which takes place be- tween the surfs of such materials as are employed in construction ; that is, it has determined the proportion (or, more correctly, the ratio) between the pres and the fric. Any person may easily do this for himself, thus: A body r.s M, Fig 63, is placed upon the plane surf ab; of which one end, as fc, is gradually raised until the body is barely about to begin to slide. When this takes place, we know that the force, x v has become barely equal to the fric ; and the angle b a c, which the plane then makes with the hor a c, is called the angle of friction, or limiting angle of resistance, or angle of repose, for the particular kind of surf experimented on. Now, a little reflection will show that whatever may be this angle, feac, of fric. the line xv, which measures not only the parallel force, but also the fric existing at that moment, (and at no other one,) is to the line i v, which measures the perp pres, (not the wt of the body ;) as the vertical height ft c of the plane at the same moment, is to its hor base a c. That is, at the point of sliding, as Fric- tion : Perp Pressure : : Height : Base; or, as Base : Ht : : Perp Pres : Friction. Therefore, when a body barely begins to slide, measure a c hor. and b c vert; div the last by the first, and the quot will be the proportion which the fric of the bodies experimented upon, bears to the pres which causes it. Or, measure the angle b ac in degrees. &c ; the nat tang of this angle will be that same propor- tion. This proportion is called the coefficient of friction for those bodies; a table of which will be found under Friction. A hor line dg. drawn from g, and terminating in vi extended, will, when measd by the same scale as gv.iv, xv, give a hor force which, without the aid of friction, would react against the force xv, and prevent it from moving the body down the plane. Or if the length a b of the plane be taken by scale to represent the wt of a body, then 6 I, perp to a 5, to meet a c pro- duced at I, will give that same hor force. Art. 62. If the length m n, Fig 63%, of an inclined plane, be taken by a scale, to represent the wt in Ibs, tons, &c, of any body placed upon it ; then the base o n will, by the same scale, give the perp pres in fts, tons, &c, which the body imparts to the surf of the plane ; and the height m o will give the amount of force parallel to m n, and which tends to move the body down the plane, either by sliding or rolling. If the pres on be mult by the proper coeflf of fric, the prod will plainly be the actual amount of fric in ft>s, &c. If the fric ~~ ; - thus obtained proves to be greater than the I res. perp to plane, sliding force m o, then the body will remain at - j rest on the plane; but if less, then sliding or .tlCI 6 3 IT rolling down the plane will be the result ; and J & the amount of force which starts or begins the motion, will be equal to the excess of mo over the fric. As the motion continues, it will be accelerated by the accumulation of gravity. See p 172, p 449. When a body is placed upon an inclined plane, whether it slides or not, the pres which it pro- duces at right angles to the surf of the plane, is equal to the wt X nat cosine of angle of slope; the sliding force parallel to the surface of the plane, = wt X nat sine of angle of slope; the actual amount of fric = wt X nat cosine angle of slope X coeff of fric. For the Trie does not vary as the angle of slope of the plane, but as the cosine of that angle ; in the same manner as the perp pres varies. Coeffs of fric are given on pp 599, 600, 602. Ex. 1. Suppose we wish to slide a wooden box M, Fig 63, filled with stone, and weighing in all 486 FORCE IN RIGID BODIES. 1200 9>s, up the iron rails of aa inclined plane, sloping 5 ; what force must we use, parallel to to* plane; assuming the coeff of wood on iron to be .4, or ^^ of the perp pres? Here we have to over- come the parallel force x v, and the fric. Now, as just stated, this parallel force x v is equal to, wt X nat sine of slope, = 1200 X .087 104.4 Ibs. The fric is equal to, wt X nat cos of slope X coeff of frm; = 1200 X .996 X .4 = 478. Consequently. 104.4 -}- 478 582.4 Ibs, is the force reqd. In fact, however, this force merely balances the downward tendency of the box, together with its fric ; thus rendering them incapable of resisting any additional upward force; but it is plain that we must apply some additional force, in order to impart motion to the now unresisting box. Now, suppose we wish to slide the box down the plane, what force must we use? Here nothing re- sists us but the fric, just found to be 478 Ibs. The parallel force helps us to the amount of 104.4 Ibs; therefore we need only to add 478 104.4 = 373.6 S>3. For acceleration on inclined planes see p 172. The following table will facilitate calculations respecting the draft required on grades, inclined planes, Ac. In practice, allowance for friction must be made ill tne last 2 cols ; see p 172, and near foot of 485. Original. Pres. on Tendency Inclination or Slope of the Plane. For the nat sine of slope, divide the vert height by the sloping length. Plane, in parts of the wt. Or, nat. cos. of angle of Plane. Pres. on Plane, in fts per ton. down the Plane, in parts of the wt. Or. nat. sine of angle of Plane. Tendency down the Plane, in Ibs per ton. Ft. per mile. Deg. Min. 1 n 3. 1760.00 18 26 .9437 2125 .3162 708. n 4. 1320.00 14 2 .9702 2173 .2425 543. a 5. 1056.00 11 19 .9806 2196 .1962 439. a 6. 880.00 9 28 .9864 2210 .1645 368. n 8. 660.00 7 8 .9923 2223 .1242 278. u 9. 588.66 6 20 .9939 2226 .1103 247. a 10. 528.00 5 43 .9950 2229 .0996 223. n 11.4 461.94 5 00 .9962 2231 .0872 195. n 12. 440.00 4 46 .9965 2232 .0831 186. n 14.3 369.23 4 00 .9976 2232 .0698 156. n 15. 352.00 3 49 .9978 2233 .0666 149. n 19.1 276.73 3 00 .9986 2237 .0523 117. n 20. 264.00 2 52 .9987 .0500 112. n 23.1 229.04 2 30 .9990 i .0436 97.7 n 25. 211.20 2 17 .9992 2238 .0398 89.2 u 28.6 184.36 2 00 .9994 ii .0349 78.2 n 30. 176.00 1 55 < .0334 74.8 n 32.7 161.47 1 45 .9995 2239 .0305 68.4 n 35. 150.86 1 38 .9996 .0285 63.8 n 38.2 138.22 1 30 .9997 2240 .0262 58.6 n 40. 132.00 1 26 .0250 56.0 n 45.8 115.29 1 15 it ti .0218 488 n 50. 105.60 1 9 .9998 " .0201 45.0 n 57.3 92.16 1 'i .0175 39.1 Q 60. 88.00 57>$ .9999 it .0167 37.4 n 70. 75.43 49 .0143 32.0 n 76.4 69.12 45 ii ii .0131 29.3 n 80. 66.00 43 ii ii .0125 28.0 n 90. 58.67 38 .0111 24.9 n 100. 52.80 34 1.0000* i .0100 22.4 n 114.6 46.07 80 I M .0087 19.6 n 125. 42.24 27% u .0080 17.9 n 150. 35.20 23 ei n .0067 15.0 n 175. 30.17 19% i< 14 .0057 12.8 n 200. 26.40 17 il .0050 11.2 n 229.2 23.04 15 i .0044 9.77 n 250. 21.12 14 n .0041 9.18 n 300. 17.60 11^ i l .0033 7.39 n 343.9 15.35 10 u , p, p, p, lines at right angles to them ; we may press a piece of cut stone against them with any force whatever, applied in the direction of the stone itself, without danger of its sliding ; provided only that the di- rection of the pres along s does not form with the perp p an angle exceeding 32. But sliding will take place, whether the pres be great or small, if, as at o, o, o, o, said angle exceeds 32. The angle of fric is, by some writers, called, in such cases, the limiting: angle of resistance. Kem. The friction at the feet of rafters when highly inclined diminishes very much their horizontal pressure and tendency to split off the ends of the tie-beams. The angle of fric of oak endwise against hard limestone, is, according to Moriu, 20%; therefore, if the walls, Ac, of a room consisted of such lime- stone, we could not press a piece of oak endwise against it without sliding, if the angle withp exceeded 209; and the legs of a wooden trestle, Fig 66, would not spread, on the level surf of such limestone, under any wt w, if the angle abcbe less than 20% ; but certainly would if it be greater, unless other preventives besides fric at the feet be depended on. In this case the fric amounts to very nearly -j&y of the pres : that being the proportion cor- responding to '20%. These two illustrations show how wide is the applica- tion of this principle : for the announcement of which we are (the writer believes) indebted to Moseley. EigGG Art. 64. To find the effect of an extraneous force (fg, Fig- 67,) imparted in any direction, to a rigid body (B) on an inclined plane, ip ; when we know the angle of fric, and the wt of the body. The prin- ciple laid down in the preceding Art } enables us to do this. Through the cen of grav c, of the fl-i 3* body, draw avert line a w ; and extend j /' ! the 'direction fg of the force, to meet this line, as at o. Make o a by scale, to represent the wt of the body ; and o z to represent the amount of the force fg. Then is o a point at which we may assume both these forces to be im- parted to the body. (Art 29.) Complete the parallelogram of forces a x z o, by _^ drawing a x, and z x, parallel and " "C"^ f*rt equal to o z, and o a. Draw the diag tlO b I x o, and extend it to meet the plane, as at t. Make the line t v perp to the surf of the plane. This done, we have a single force x o, equal in effect upon the rigid body, to its wt, and fg combined. This single force may (Art 18) be considered as imparted to the body at any point that lies in its line of direction x t ; therefore, we will assume it to be imported at t, where it encounters the force of fric acting in the direction 8 e, of the joint formed between the body, and the plane. Now, if t strikes within the base te,tv being at right angles to this joint, it follows from the last Art, that if the angle xtv is less than the angle of fric corresponding to the nature of the materials which 488 FORCE IN RIGID BODIES. compose the body and plane, then the body will remain at rest on the plane. But if the angle * t 9 be greater than said angle of fric, the body will slide up or down the plane, (according to circum- stances, stated in the next paragraph ;) if the angles be equal, the body will be just on the point of beginning to slide either up or down. When the angle x t v is on the down hill side of v t, as in the fig, the tendency of the body will evi- dently be to move up the plane ; but if, in consequence of a diff direction of the force / g, (and conse- quently of the resultant x o,) the angle x t v is on the up hill side of v t, then the tendency will be down the plane. REM. 1. If the direction of the resultant x o, or the point t, falls outside of its base s e, the body, instead of sliding, will upset. It will fall up hill, if t strikes p i up hill from the base; and down hill, if t strikes down hill from the base. See Remark, Art 65. REM. 2. In order to draw the parallelogram of forces a xz o, and its resultant diag x o, the line a, o, which represents the wt, may sometimes have to be regarded as pulling instead of pushing down- ward at the point o, where the other force meets it. See Art 28, Fig 9J4 ; and Fig 69, Art 65. , 68 REM. 3. It follows from the foregoing, that when at the joints p q, r *, Fig 68, of a mass of masonry ; or at the joints of timbers in carpentry, iron work, &c, the fric alone is depended on to prevent sliding, the re- sultant AS me, co, o n, &c, of all the forces acting at any joint, must not form an angle m c i, c o a, o n e, with a perp c i, o a, n e, to the joint, greater than the angle of fric corresponding to the nature of the ma- terials whose surfaces constitute the joint. REM. 4. The extraneous force reqd to move a body up a plane, will be the least when its direction, i n. Fig 67. makes with surf ip, of the plane, an angle, nip, equal to the angle of fric. Art. 65, To find the force required to prevent a body S, Fig 69, from falling; when the direction, o w, of its wt, strikes outside of its base 1 1, Thus, suppose we wish to impart a pulling force at e, and in the direction e a, to prevent the body from upsetting down the plane. Through the cen of gray c, draw a vert line x w; and continue the line of direction of a e to meet it at o. From o draw oy at right angles to the plane t p. By scale make o w equal to the wt of the body ; and from w draw w y par- allel to o a. Make e a equal to w y ; then is e a the reqd force, which will resist all tendency of the body to fall. For in the par- allelogram of forces o w y z, we have the force o w tending to make the body fall; and the force o z (equal to e a) tending to prevent it from falling; and the resultant o y, of these two forces, equal to ir joint effect, is at right angles to the surf of the plane ; and their is consequently (Art 19) all imparted to it as pres ; no part being left unresisted, to produce motion in any direction. For as before said, when two forces, as o w, o z, are compounded into one result- ant force o y, those two forces must be considered as no longer ex- isting; thus, in this case, so long as we regard the joint effect of ow and o z as being concentrated in their resultant o y, we cannot of course, consider them as acting in other directions at the same time ; so that there is, as it were, no longer any wt, o w, tending to make the body fall ; nor any force c a, tending to uphold it; but only the single force o y, which presses the inert body against, and at right angles to, the surf i p ; imparting to it a tendency to move only in the direction o y ; which ten- dency is reacted against by the inherent cohesive force, or strength, of the plane. REM. If the resultant of all the forces of any kind, acting upon a body, does not strike inside of its base, no matter whether the base be inclined or hor, the body must evidently move ; on the same prin- ciple as when grav is the only force acting upon it, in a direction which strikes outside of the base. See Art 35, Fig 19. Or, in other words, in all cases, if the direction of the resultant of all the forces acting upon a body, meets with no resisting force as it passes out of the body, the body must move in the direction of that resultant. Thus, suppose that the only forces acting upon S, were three, in the directions ao,yo. and wo, or the reverse of what they are in the II fig; and that y b was their resultant. Then, as this resultant meets with no opposing force at v, the body must move in the direction v b. Or, the body A, Fig 70, rests on a hor base. Being acted upon only by grav, the direction of the resultant is gj ; which, as it leaves the body at 3, encounters an equal opposing force from the resistance of the ground ; and consequently, no motion takes place. But now suppose an upward force tj, to be imparted to the body ; and let it be four tin;es as great as the grav. Then the resultant will be a single force, three times as great as grav ; and acting in the direction t n. As this direction leaves the body at o, it meets no opposing force; hence the body must move in the direction o n, since unopposed force always produces motion. Art. 66. Stability. The stability of a structure, or of any body, is, strictly speaking, that resistance which its wt alone enables it to oppose against forces tend- ing to change its position. Such resistance may be assisted by extraneous wts, or by other forces properly applied; but such must be distinguished from the stability inherent in the structure, or body itself. To insure the stability of a structure, the disposition of its parts, as well as that of the entire mass, tfmst be such that neither of them shall move, either by sliding, or by overturning, under the action of the im- parted forces. Stability is therefore a branch of Statics; or of forces at rest, or in equilibrium with each other ; Art 16, p 451. RCE IN RIGID BODIES. 489 Stability ijtfiist not be confounded wi tit strength. A structure may be very strong; and yet very unstable. A block of stone is quite as strong while sliding down a smooth plane, or rolling down a steep bank, as when resting on a firm hor base ; but it has stability only in the last case. A pyramid of weak chalk may have great stability : while a globe of granite or cast iron, has very little". We generally have to examine into the strength, as well as the stability of our structures : but it must be done by diff processes. The stability has reference to the structure considered as consisting of one or more rigid bodies, which may be moved as entire masses, but not broken, or changed in form, by the applied forces. See Remark 2, Art 29, p 460. Those forces which tend to impair the stability of a structure, are called acting ones ; and those which tend to maintain it. resisting ones. This distinction is merely a matter of convenience ; for all the forces act, and resist. The forces which affect the stability of a rigid structure considered as one mass, are its wt ; extra- neous wts, or strains . and the foundation, or support; which last reacts as an antiresultant (Art 30; against the others. When these three balance each other, the structure is stable. When the struc- ture is to be considered as composed of several rigid bodies, then the joints or surfaces of contact be- tween these bodies must also be regarded as so many secondary foundations, and these also must re- spectively balance the forces acting upon them ; otherwise these parts may slide, or overturn, while other parts may remain firm. Art. 67. In order to guard against accidents, a structure must generally be so designed as to be capable of resisting much greater forces than those which it sustains under ordinary circum- stances. The proportion which, with this object, we give to the resisting forces, in excess of the act- ing ones, is called the coefficient of stability ; or simply the stability, or the safety, of the structure. Thus, if we make it capable of resisting 2, 3, or 6 times the amount of the ordinary acting forces, we say it has a stability, or a coeff of stability, or a safety, of 2, 3, or 6. Art. 68. Since the stability of a structure, considered apart from its founda- tion, consists entirely in the resistance which its several parts, as well as the entire mass, can present against both sliding and overturning, it follows that two precau- tions, already adverted to in previous articles, must be resorted to. Namely, 1st, against sliding, take care that the resultant of all the forces acting upon any joint, (including that between the base and the foundation,) shall act either at right angles to said joint ; so as to be entirely imparted to it as strain, (press or pull,) leaving no part unresisted to tend to produce motion ; or else that it shall not de- viate from a right angle, to a greater extent than the angle of fric corresponding to the materials which compose the joint; so that the portion of it which is not im- parted at right angles, shall be resisted by friction ; and thus be prevented from producing a motion of sliding. Otherwise, instead of relying upon the position of the joints, resort must be had to the cohesive strength of joint-fastenings, such as bolts, spikes, cramps, joggles, mortises and tenons, mortar, cement, &c, to prevent sliding. As to mortar and cement, however, it is important to remember that frequently, and especially in very massive work, they have not time to harden, or acquire their full strength, before the acting forces are brought to bear upon them : therefore, great care is necessary, when we use them as substitutes for position. On this account we frequently cannot consider a mass of masonry to be a single rigid body, but must regard it as composed of several detached rigid bodies ; the stability of each of which must be separately provided for, before we can secure that of the whole. Therefore, in large massive structures of importance, we should, as far as possible, omit all consideration of the strength of the mortar, and rely for stability chiefly upon placing the joints at or nearly at right angles to the forces acting upon them. In the 2d precaution, against overturning ; we must take equal care that the resultant of all the forces acting upon any single part, or upon the whole structure, shall fall within the base-joint of that part, or whole. See Remark, Art 35; Art 60; Remark 1, Art 64; Remark, Art 65; Remark 2, Art 72. Art. 69. and 49) thai overturned about any given point a, is equal to that which would be produced if the entire wt of the body were concentrated at its cen of grav g ; and acted at the end i of a straight lever a i, of which a is the fulcrum ; or at the end o, s, or , of any straight lever (Art 49) ao, ax. an: or of any bent lever ai n, a i s, at o, a so, provided that in every case there is the same leverage at, measured from the fulcrum a, and at right angles to the direction mn of the force of grav of the body. So far as regards tendency to resist overturning, it is immaterial (Art 18) at what point of the body, in this line of direction m n, we conceive the grav to act : or whether as a push at o, or a pull at t, as denoted by the arrows. We have also said that the tendency, or moment, of this force . Moment of stability. We have already stated (see Arts 46 t the resistance which any rigid body as B, Fig 71, opposes against being Fig 71 TTL , , of grav, or wt, to produce or to resist motion about the fulcrum a, through the me- dium of any of these levers, is found by mult the force or wt in fbs, by the leverage ai in feet. The prod in ft-fbs is generally called the moment of the force about the point a ; but in cases like that before us, in which this moment becomes the measure of the stability of the body, it is called the moment of stability (or simply the star 490 FORCE IN RIGID BODIES. bility) of the. body, about that point. Therefore, if bodies of the same size and shape have diff wts, or sp gravities, their respective stabilities will be in proportion to their wts, or sp gr. A body may have diff moments of stability, about diff points. Thus it would be far more difficult to overturn B about the point 6, than about a ; because the lever- age bi is 2> times as great as at ; and since the wt and the point of the cen of grav remain unchanged, the moment about 6 is 2% times as great as about a. BBM. 1. Let a b c o, Fig 72, be a squared block of stone 6 feet long ; on a nor base ;. and weighing 12 tons ; and h, a force applied to overturn it about the toe c. Since its A Ti length o c, is 6 feet, its cen of grav t, will be dist o g, or 3 ft back from T~ o. Consequently, the moment with which the block resists being overturned, is 12 (tons) X 3 (ft leverage) = 36 ft-tous. Now. suppose the upper half a ft o, to be removed ; the remainder o b c will weigh but 6 tons. But its cen of grav , is farther from o, than that of the whole block was. Being now triangular in shape, the dist o y will be % of o c; or will be 4 ft. Consequently, the resisting moment will be 6 (tons) X 4 (ft leverage) '24 foot-tons. So that although the block has but half the wt of the first one, it has % as great resisting power. It is on this principle, that in order to save masonrv, the faces of retaining- walls, Ac, are sloped, or battered back. REM. 2. Of two bodies, as A and B, Fig 72^, of precisely the same size, wt, and position; and having the same moment of stability ; one may re- quire the expenditure of a greater amount of the same degree of force, than the other, to overturn it. Thus, let the upper part en, of A ; and the lower part o y, of B, be made of lead ; and the remain- ing part of each, of cork. Then the cen of grav of the body A, will be near the dot t ; and that of B, near the dot . The wt of both bodies being the eame; and the cen of grav of both, being at the same hor dist, a o, from the fulcrums o, o, around which the bodies are to be overturned ; their mo- ments of stability must also be the same. Conse- quently, both will require, in order to begin to overturn them, precisely the same amount of force, applied in any same given direction ; and at any same given point; as, for example, the equal forces , /and g, applied in a hor direction, at the points i and i. But the body B will plainly require this force . requi to be continued for a longer time, (or in other words, will require the expenditure of a larger amount of force,) in order to actually overthrow it, than the body A will ; for the body A will be overthrown when the force has acted for only the short time necessary to move it into the position of the dotted lines J. The cen of grav being then carried to e, which is beyond the base at o, the body must necessarily fall. But in the body B, the force must act at least long enough to move it into the position N ; for not until then will its cen of grav , be moved to the position I, so as to be beyond the base. When the stability of a body is considered only with regard to the degree of force necessary for its resistance to beginning to move, (which degree is the same in both A ad B : and is the force with which engineers are most concerned,) it is called the static stability of the body ; and when considered with regard to the total expenditure of that same de- gree of force, necessary to complete the overthrow of the body, it is called its dynamic, or moving sta- bility. The engineer rarely need concern himself about the last; his object being to secure his structures against beginning to move. RKM. 3. It is not alone the wt of the body itself, which contributes to its stability in all cases ; for this may be assisted by extraneous wts or loads. Thus, the wt of a pier P, Fig 73, gives it in itself a certain degree of stability ; but when we add the wt of the two equal arches, its stability is thereby increased, supposing the foundation to be secure. And a passing load, when it is directly over the pier, increases it still more. It is true that the wt of the arches might crush the pier to fragments, if the stone be soft; but this is a matter of Strength of Materials ; not of stability ; and must be examined into by itself. If the two arches be of unequal sizes, or if there be but one arch, the stability of the pier may become either increased or diminished, according to circumstances; as will appear farther on. Whether the force acting for or against the stability of a structure, be gravity, or pushes, or pulls, produced from other sources, is, as in other cases, a matter of no importance; for force is simply force, no matter whence de- rived. We have, therefore, only to look at the diff forces acting upon our structures, as so many tendencies to produce motions in certain directions. If these tendencies are reacted against, or destroyed by others, they will not produce it ; but if they meet no resistance, mo- tion must take place. The only peculiarity we need assign to grav, is that the direction of its actio* i* always vert downward ; while other forces may be imparted in either that, or any other direction. The resultant of grav combined with other force or forces, may be in any direction. Art. 7O. Fig 73%. In the principal cases of stability that present themselves to the civil engineer, both the acting and resisting forces w x,f,f,f, &c, may all be considered to be imparted and acting in the same plane ; which is a vert one, el o c, passing through the cen of grav, t?, of the structure, m np q r s t u; and of course, coinciding with the line of direction w x, of its wt, or force of gravity. The plane f. I o c, and all the forces, therefore, may be considered as coinciding with a leaf of paper standing vert on one edge. This renders the calculations much more simple .] NMPZ cannot upset, no matter how great may be the pres cy ; see Remark 2. From i 492 FORCE IN RIGID BODIES. draw if, at right angles to the line P Z ; and measure the angle cit, which the resultant cy forma with it. This, we find, is greater than 32; -that is, it exceeds the angl; of fric between surfaces of dressed stone. Therefore, the part NMPZ must slide along the joint PZ. This might possibly be prevented by good mortar, if time be allowed it to solidify properly, before the centers are eased so M to bring the pres of the arch upon the abut; or by iron cramps, stone joggles, &c; but these are expensive. The most obvious remedy, as well as the least expensive, is simply to incline the joint PZ into a direction somewhat like from R to Z : so as to receive the pres of the resultant cy more nearly at right augles ; at least so nearly as to be fairly within the limits of the angle of fric. If this is done, stability is secured ; for the part NMPZ, being now safe against both sliding and overturn- ing, can move in no other way ; unless the strength of the stone composing the masonry is insufficient to bear the pres, and may therefore crush to pieces under it. But this is a question of Strength of Materials. See Remark 2. Art 35. The point i, of Fig 75, comes much nearer to Z than would be desirable in practice ; for it might cause crushing at Z. See Rem 2, following. Having thus provided for both the sliding and the overturning stability of the abut as far down as the joint P Z, we will now examine as far down as the joint F h. Taking the entire part N M F L of the abut, we first find its weight, say 25 tons ; and this we assume to act at its cen of grav K, and in the vert direction v K I. The amount and direction of the thrust of the arch at o, of course, remain as before. Therefore, from the point v, where the two directions meet, lay off ve to represent as before the 30 tons thrust of the arch ; and v I, the 25 tons wt of the abut. Complete the parallelogram of forces v e s I, and draw its diag v s ; which, measd by the same scale, will give the resultant of all the forces acting upon the part N M F L. Now we see that the direction of this resultant does not fall within the base F L ; but, on the contrary, passes out of the body atj ; outside of which it meets no force to resist it. Consequently, (Remark, Art 65,) since this resultant must be considered as an only force acting upon an inert body or abut, NMF L, without wt, (Art 35,) that body must upset around the point L ; or around the nearest joint in the masonry between L and./; and cannot con- tinue to stand of itself, unless its base be above j. It is true that by placing earth behind it. espe- cially if well compacted by ramming, the abut of a small arch might be made to stand safely even upon the base W X ; and in the case of arches of moderate spans, this aid may be resorted to for strengthening the abuts, when there is no danger that the earth may be washed away by floods or rains, and thus expose them to ruin ; and this is generally and properly done. RKM. 1. If in the same manner that the point i was found in the joint P Z. others between P Z and W X be determined also : then a curve, commencing at the akewback o, and drawn through them, will represent the line of pressures or of resistance, or of thrust, (see footnote, p 348) through the abut. At any point whatever in this line, say at i, the entire pres above said point may be supposed to be concentrated ; while the entire length, as cy, of that resultant which cuts said point, gives the amount of said pres at that point ; and the direction, as c i, of the same resultant ia also the direc- tion in which said pres acts upon said point. See Art 15 of Hydrostatics. REM. 2. The line of pres enables us to determine another very important point connected with the stability of a structure. It is not sufficient in practice that this line should strike merely within the base; it must strike at a considerable dist within. If the structure and its foundation were ofeso- lutely rigid, so that no conceivable force could bend or break them, this would not be necessary ; but all materials are more or less weak, so that if great pressures come too near to their edges, there ia danger of splitting or crushing at those points ; or if near the edge of a base, an unequal settlement -f the soil beneath may take place. Therefore, even in structures of but small size, the dist iZ, Fig 75, of the line ot pres, from the outer point Z. should never be less at any joint than y of the width f that joint ; except, perhaps, in a case like that of a small arch in which the earth filling is depoa- FORCE IN RIGID BODIES. 493 iited behind the abuts before the centers are removed. In important works, it should not be less than about % of the width of the joint; and it is still better, when possible, at H ; or, in other words, at the center of each joint. When, as at Q, a footing U is added at a base, W X should be taken as the joint; not W Q. REM. 3. The line of pressure in an arch itself, as H J N T, Fig 75, may also be found much in the same way, thus : First divide the half arch H J N T, and the filling above it, by vert lines ru, wx, &c, which need not be at equal dists apart. Four such lines will suffice for a flat arch, and about six for a semicircular one. We then consider in turn, and separately, each part, as r u H J, w x H J, N D T H J, &e, which extends from these lines to the center H J of the half arch ; the last of these- being the entire half arch. The cen of grav, and the wt, of each of these parts, must be found ; also, (Ex 6, p 478,) the hor pres at the keystone. Now each of these parts, like the beam in Ex 1, or the half arch in Ex 2, p 469, is acted upon, and kept in equilibrium, by three forces ; namely, the hor pres at the keystone, (see Ex 6, p 478 ;) its own wt, acting vert; and the reacting force of the part next behind it. We proceed with each part sepa- rately, as we did with the entire beam alluded to, thus : Beginning with the part r u H J, from its cen of grav, ra, draw a vert line TO/. From the center E of the keystone, draw a hor line E n, to meet mf. From n lay off nf by scale, to represent the wt of the part r u H J ; and from / lay off fg, hor by scale, to represent the hor pres at the key. Draw the diag ng; which will give, by scale, the re- sultant of these two forces. The point b, at which the diag ng intersects the vert ru, is a point in the line of pres reqd. Next go to the part tvxlIJ; and in the same manner find another point in the line w x, using the cen of grav and the wt of that part. The hor pres will be the Same in each part.* Finally, treat the entire half arch N D T H J in the same way. The resultant diag of this last will pass through o, the center of the skewback, if the archstones have the same depth throughout. A curve drawn by hand through the points thus found, will be the reqd line of pres. These points will not all fall equally well within the thickness of the archstones. In- deed, if the intrados of the arch is a full semicircle, or semi-ellipse, some of them will even fall entirely below the archstones, as shown at a and u, in the dotted line of pres in Fig 4>, p 348. When this is the case, the tendency of the line of pres is to bend the arch still more upward at a and u; and thus allow the parts about the crown o to descend. Each half, o e and o g, of the arch is then in the same condition that a crooked pillar or column would be if a heavy load were placed on top of it. The line of pres of the load would then not paas through the axis of the pillar, but outside of it, or as the string of a bow; so that instead of being borne in safety, the load would bend the pillar still more; as the tightening of the string would bend the bow. If the concave side of the pillar, or of the bow, should crush under the undue strain upon it, the whole would fail. In like manner, if the stone of the arch along the concave intrados near a and u, is not strong enough to resist the undue crushing strain thus brought against it, it will crumble, and the arch will fail, by rising at its haunches, and falling at its crown. See footnotes to Art 6 of Stone Bridges, p 349. Art. 73. The stability of bodies on inclined planes, as regards overturning, is measd in the same way as when the base is hor ; namely, by mult their wt, by the perp dist (ao, or c (, at A, B, and D, Fig 76,) from the fulcrum, or turning-point a or c, to the vert line of direction (g o) drawn from the cen of grav of the body. Hence, it is evident that the body B has less overturning stability about its toe a, than the similar body A has, when the force, w, tends to upset it down hill. But it has more than A, when the force tends to upset it up hill, or about the toe c; for the leverage t c of B is greater than that, o c, of A. The body C, which would overturn upon a level base, because the line g o strikes outside of th base ; would be stable against overturning, if placed as D upon an inclination, where the vert g o # That the hor pres throughout every part of the arch is equal to that at its center, may be under- stood by supposing one half of the arch to be changed into a bent column having the skewback for its base : and the kevstone for its top ; and to be loaded on its top by a wt equal to the hor pres at the center of the arch. Now it is plain that every part of this column has equally to bear the vert pres of the load ; on the same principle that every part of a long bent hook has equally to bear the vert pull of any load upheld by it. And on the same principle every part of the arch sustains a bori- ,.! ^ u~r pres at the same poin 1 . , thus producing the curved line of pres or thrust of the arch. In other words, the hor pres, although equal at every part of the arch, is, as it were, pushed downward as we approach the skewback, by the increasing vert pres of the greater height of masonry and earth, 32 494 CENTRIFUGAL FORCE. strikes within the base. Inasmuch as the leverage a o, is greater than the one c t, D would present more resistance to a force tending to upset it down hill, than up hill. Structures built upon slopes are, however, liable to slide t that is, they are deficient in frictional stability. In practice this is remedied by cutting the slope into hor steps, as at E. But works so constructed are not as strong as if the base were a continuous hor line; because the vert faces of the steps break the bond of the masonry; and because the mortar in the higher portions a d, being in greater quantity than that in the lower portions e y, necessarily allows more settlement of the masonry in the former ; and thus renders the work liable to crack, or split open vertically. The case is analogous to that of a foundation, firm in some parts, and com- pressible in others. Therefore, when circumstances permit, the foundation should be levelled off as at d v, or if the masonry has to sustain down-hillward pressures, v should be lower than d; and the courses of masonry be laid with a corresponding inclination. CENTBIFTJGAL FOECE, Art* 1. CENTRIFUGAL force is that with which a body while revolving around a cen- ter, tends to fly off in a radius, or direct line drawn through said center and the body : and is entirely diff and distinct from the force which the body has in the curved direction in which it is moving. Centripetal force, on the contrary, (ahvays precisely equal to the centrifugal, but in an opposite direc- tion,) prevents the body from so flying off; and keeps it in the circle.* They are both called central farces. It is centrifugal force that pulls the string with which we whirl a stone : or which tends to burst a millstone, or a fly-wheel when revolving rapidly ; or which presses a railway train against the outer rails in curves ; -while the cohesive force of the string, millstone, or fly-wheel ; or the spikes at the rails, furnish the centripetal force. Both forces vary directly as the wt of the body ; and also as the square of its vel ; and inversely as the rad of the circle in which it moves. Let Fig 1 represent a uniform homogeneous circular ring like a fly-wheel, revolving by means of its arms, around its cen- ter of motion, c; and Fig 2, any isolated body n, revolving by means of a string, around its center of motion, c. In either case, if the di- meusion a t of the body, in the direction of a rad a c, does not exceed about % t part of the entire rad a c, we may near enough, for ordi- nary practice, consider en as the rad infect to be used in calculating the centrifugal force ; n being the cen of grav of the entire body in Fig 2 ; and the cen of grav of its cross-section at t a in Fig 1. In either case, if we know the wt of the body in ft>s or tons, &c ; the rad c n, in feet; and either the vel in ft per sec with which the point n revolves; or else the number of revs which the body makes per rain; then The centrif force, in Wt of ^ Square of vel at n, Ibs or tons, Ac, as the _ the body * in ft per sec, case may be, jj^ c n in ft x 32 . 2 . The centrif force _ Wt of v Square of number v R , . f . in Iba or tons, Ac. ~ the body X of revs per min. X .Barf c n in ftl Ex. Suppose the body in either Fig 1, or Fig 2, to weigh 1.2 tons ; the rad c n to be 2.5 ft; (in which case the circumf will be 15.7 ft;) and suppose the body to make 100 revs per min. In this case its vel will plainly be 15.7 X 100 = 1570 ft per min ; or ^ = 26.17 ft per sec. Then by the first for- aiula, Centrif force Centrif force L, x 10000 x .s = .oog =10LgtBBB . * Centrifugal and centripetal force being always exactly equal to each other, it may be asked why do rapidly revolving bodies sometimes break; and the fragments fly off? Does not this prove that the centrifugal force has become greater than the centripetal one? The answer is, No. When the centrifu- gal force becomes precisely equal to the ultimate strength of the string, or of the millstone. &c. so that the string, stone, &c, can furnish no greater resistance, then that force which is moving the body for- ward, causes the breakage ; and at that instant both the centrifugal and centripetal forces ce.ase en- tirely ; and the fragments of the body do not fly off in the direction which the centrifugal force had ; but in a tangent, or at right angles to it : being carried in that direction by said other force. t When a t is equal to % of a c, the rad thus found will be but -fa part too short ; when % of a c, about J part too short ; and when % of a c, about % part too short ; and the resulting centrif forces will consequently be too small in the same proportion. But when, as is usually the case, in fly-wheels, &o, a t is much less than even X of a c, the error is not worth notice in prnctice. t These are not strictly correct, not only because n is not in reality at the cen of grav of t a ; but because the ring, Fig I, or the body, Fig 2, must be united to the center c, either by arms, or by a string, or wire, Ac- and the weight of these arms, &c. will slightly shorten the rad of gyration. Se Art 2. But in practice this effect is usaally too small to be regarded ; or a trifling allowance is made for it by guess. IFUGAL FORCE. 495 Al*t 2. But in the cas^of a solid homogeneous uniform circular tody, like a millstone of uni- form thickness, Fig 3, rejurfving around its center c either vert or hor, the'rad to be used is what is called the rarijsla of gyration, c h. In such a body the parts nearer.Ane circumf move with greater vel and force than those nearer thecentelC But there is, in such cases, an imaginary interior circle ~pj , Q h m s y, caned the circle of gry ration ; such that if all ^^^^^ ' the weight of the entire body be supposed to be concentrated at the cit cunif of said circle, then th force of the revolving body would produce the same effect as it actually does while distributed over the whole body Any point as h, in said circle, where it is cut by any rad c w, is the cen- ter of g-y ration of that rad. In a circular body of uni- form thickness, like a millstone, it is very easy to find the rad c A of gyra- tion ; for we have only to mult the actual ouf-to-out rad c w, by the deci- mal .707. This being done, we use the resulting rad c A of gyr/instead of the rad c n of Figs 1 and 2; and in precisely the same manner; using also the vel at A, instead of that at n in Figs 1 and 2. See " center of gyration," p 617 pe: rad of the circle of gyr is 1.296 X 2 = 2.592 ft ; and its circumf A m s y is 2.592 X 3.1416 = 8.143 ft. And since the stone makes- 326 revs per miu, the vel of the circle of gyr is 8.143 X 326 = 2655 ft per min ; or, = 44.25 ft per sec. Hence the Wt of v Square of vel at h Ceil trif forces the body A in ft per sec Ex. Let Fig 3 be a millstone, 3 ft 8 ins, or 3.666 ft diam ; its wt .5 of a ton ; and making 326 revs T min. Its circumf will then be 11.52, and will move 3755.5 ft per min ; or 62.592 ft per sec. Its 6 = 1.833 ft; consequently, its rad of gyr cA is 1.833 X .707 = 1.296 ft; and the diam .5 (on X 44.252 rad ch, i X 1958 i ft X 32.2; 979 Ceiitrif force = Wt of ~, Sq of number v the body X of revs per min X .5 ton X 3262 X 1 .296 .5 X 106276 X 1.296 68867 Al*t 3. When, in a homogeneous revolving ring, Fig 4, of uniform thickness, the dimension a t becomes so great, that it is not sufficiently ac- curate to assume the point h, of the rad of gyr ch, to be at the cen of grav of at, as was done in Figs 1 and 2, the true rad c/t may be found thus : Add the square of the inner rad c t, to the square of the outer rad c a. Div their sum by 2. Take the su rt of the quot. Ex. Let c t be 3 ft ; and c a, 7 ft. Then, Bad of *yr, ch- /S2 + 72 _ /9 + 49 _ / ~V"~ V" 1 " V' - ~ 5.38 ft. The rad of gyr, or dist from the cen c be found thus : olution, to the circle of gyr, of a few other bodies, may solid globe thin circular ring thin hollow globe solid cone thin straight rod arund its centre. Rad X -707 its diameter, Rad X .5 " its diameter, Rad X .633 " its diameter, Rad X .707 " its diameter, Rad X .817* " its axis, Rad X .548 1 (nd, Rad X .577 " " " revolving around any point) / AS _i_ between its two ends. Call the length of > . / . T g ., its two arms A and a ^\/ 3A + 3 - It is plain from the last example, that any body may have any number of radii of gyration ; de- pending upon the position of the cen of rev." Coal used by engines per horse-power per hour. Condensing engines, !K to '^A **; non-condensing, 3 to 7 fcs ; depending on the quality of the coal, perfection of engines, &c. * Perfectly correct, only when infinitely thin. When the thickness is appreciable, first measure the rad to the center of the thickness ; then mult by .817 for a practical approximation. t Here rad means the rad of the circular base of the cone. 496 MORTAR, BRICKS, ETC. MOETAR, BRICKS, &c. Art, 1. Mortar. The proportion of 1 measure of quicklime, either in ir- regular lumps, or ground^* and 5 measures of sand, is about the average used for common mortar, by good builders in our principal Atlantic cities; and if both materials are good, and well mixed (or tempered) with clean water, the mortar is certainly as good as can be desired for such ordinary purposes as require no addi- tion of hydraulic cement. The bulk of the mixed mortar will usually exceed that of the dry loose sand alone about }/% part. Quantity required. 20 cub ft, or 16 struck bushels of sand, and 4 cub ft, or 3. 2 struck bushels of quicklime, the measures slightly shaken iu both cases, will make abt 2'2^ cub ft of mortar; sufficient to lay 1000 bricks oftho ordinary average size of 814 by 4 by 2 ins, with the coarse mortar joints usual in interior house-walls, varying say from % to % inch. With such joints, 1000 such bricks will make 2 cubic yards of massive work ; and nearly % of the mass will be mortar. For outside or showing joints, where a whiter and neater-looking mortar is required, house-builders in- crease the proportion of lime to 1 in 4, or 1 in 3. For mortar of fine screened gravel, for cellar-walls of stone rubble, or coarse brickwork, 1 measure of lime to 6 or 8 of gravel, is asual ; and the mortar is good. In average rough massive rubble, as in the foregoing brickwork, about % of the mass is mortar: consequently a cubic yard will require about as much as 500 such bricks ; or 10 cubic feet, (8 struck bushels) of sand ; and 2 cub ft, or 1.6 bushels of quicklime. Superior, well-scabbled rubble, carefully laid, will contain but about TJ- of its bulk of mortar ; or 5} cub ft sand, and 1.1 cub ft lime, per cub yard. For public engineering works, especially in massive ones, or where exposed to dampness, an addi- tion should be made in either of the foregoing mortars, of a quantity of good hyd cement, equal to about % of the lime; or still better, % of the lime should be omitted, and an equal measure of cement be substituted for it. If exposed to water while quite new, use little or no lime outside. With bricks of 8^ by 4 by 2 ins, the following are the quantities of mor- tar ami of bricks for a cubic yard of massive work. Thickness of Joints. Proportion of Mortar in the whole mass. No. of Bricks per cub yard. No. of Bricks per cub toot. 574 21.26 1 " f 522 19.33 : ?::. 475 , 17.60 - 1 .. 433 .. .. 16.04 In estimating for bricks in massive work, allow 2 or 3 per ct for waste ; and in common buildings, 5 per ct. or more. Much of the waste is incurred in cutting bricks to fit angles, Ac. In Philadelphia a barrel of lump lime is allowed for 1000 bricks ; or for 2 perches (25 cub ft each) of rough cellar-wall rubble. Somewhat less mortar per 1000 is contained in thin walls, than in massive engineering structures ; because the former have proportionally more outside face, which does not require to be covered with mortar; but thin walls involve more waste while building; so that both require about the same quantity of materials to be provided. Careful experiments show that mortar becomes harder, and more adhesive to brick or stone, if the proportion of lime is increased. Hence, on our public works the proportion of one measure of quicklime to 3 of sand, is usually spec- ified, but probably never used. Lime is usually sold in lump, by the barrel,* of about 230 Ibs net, or 250 fts gross. A heaped bushel of lump lime averages about 75 fts. Ground quicklime, loose, averages about 70 fts per struck bushel ; and 3 bushels loose just fill a common Hour barrel ; but from 3.5 to 3.75 bushels, or 245 to 280 fts can readily be compacted into a barrel. General remarks on mortar and lime. On too great a pro- portion of our public works, the common lime mortar may be seen to be rotten and useless, where it has been exposed to moisture ; which will be carried by the capillary action of earth to several feet above the natural surface; or as far below the artificial surface of embankments deposited behind abutments, retaining-walls, &c. The same will frequently be seen in the soffits of arches under em- bankments. Common lime mortar, thus exposed to constant moisture, will never harden properly. Even when very old and hard, it absorbs water freely. Cement also does so, but hardens. Brickdust, or burnt clay, improves common mortar ; and makes it hydraulic. In localities where sand cannot be obtained, burnt clay, ground, may be substituted; and will gen- erally give a better mortar. Protection of quicklime from moisture, even that of the air, is absolutely essential, otherwise it undergoes the process of air-slacking, or * Price of quicklime in lump in Philadelphia, about $1.20 to $1.30 per barrel. By the bushel, without barrel, about 35 cts. Ground lime, about 25 cents more per barrel. Sand, sea, per cart load of 20 struck bushels, $2 to $2.50. Bar sand $1 to $1.25 (from the river at the city). In 188O. TAR, BRICKS, ETC. 497 spontaneous slacking Jby'which it becomes reduced to powder as when slacked by water as usual, but without heatiiyjK'and with but little swelling. As this air slacking requires from a few months to a year or more.jipending on quality and exposure, it gives the lime time to absorb sufficient carbonic acid from me air to injure or destroy its efficacy. But quicklime will keep good for a long- time if first ground, and then well packed in air-tight barrels. The grinding also breaks down refractory particles found in all limes, and which injure the mortar by not slacking until it has been made and used. For the same reason it is better that lime should not be made into mortar as soon as it is slacked, but be allowed to remain slacked for a day or two (or even several) protected from rain, sun, and dust. Lime slacked in great bulk may char or even set fire to wood. Lime paste and mortar will keep for years, and improve, if well buried in the earth.- Also for months if merely covered In heaps under shelter, with a thick layer of sand. The paste shrinks and cracks in drying ; but the sand in mortar prevents this. As approximate averages varying much according to the character and degree of burning of the limestone; ana to the fineness or coarseness of the sand, one measure of good quicklime, either in lump, or ground ; if wet with about % a measure of water, will within less than an hour, slack to about 2 measures of dry powder. And if to this powder there be added about % more measures of water, and 3 measures of dry sand, and the whole thoroughly mixed, the result will be about 3^ measures of mortar. Or the same slacked dry powder, with about 1 measure of water, and 5 measures of sand, will make about 5% measures of mortar. In both cases the bulk of the mortar will be about % part greater than that of the dry sand alone. If % of a measure of water be used for slackiug, the result, instead of a dry powder, will be about l l /i measures of stiff paste ; or with 1 whole measure of water for slacking, the result will be about 1% measures of thin paste, of about the proper consistence for mixing with the sand. Very pure, fat limes, slack quickly, and make about from 2 to 3 measures of powder ; while poor, meagre ones, require more time, and swell less. Slow slacking, and small swelling, in case the lime has been properly burnt, are not in general bad properties ; but on the contrary, usually indicate that it is to some extent hydraulic. In this case it makes a better mortar ; especially for works exposed to moisture, or to the weather. Very pure limes are the worst of all for such exposures; or are bad weather-limes ; and in important works, should never be used without cement. Shell lime appears to be about the same as that from the purest limestones; but that from chalk is still more inferior, and will not bear more than about 1^ measures of sand; its mortar never becomes very hard. Madrepores (commonly called coral) appear to furnish a lime intermediate between those of chalk and limestone. They require to be but moderately burnt. The average weight; of common hardened mortar is about 105 to 115 fts per cub ft. Grout is merely common mortar made so thin as to flow almost like cream. It is intended to fill interstices left in the mortar -joints of rough masonry; but unless it contains a large amount of cement, it is probably entirely worthless; since the great quantity of water injures the properties of lime; and moreover, its ingredients separate from each other; the sand settling be- low the lime. Besides this, it will never harden thoroughly in the interior of thick masses of ma- sonry ; indeed, the same may probably be said of any common lime mortar. In such positions, it has been found to be perfectly soft, after the lapse of many years. Both the sand and the water for lime mortar, should be free from clay and salt. The clay may be removed by thorough washing; but it is extremely dif- ficult to get rid of the salt from seashore sand, even by repeated washings. Enough will generally remain to keep the work damp, and to produce efflorescences of nitre on the surface; whether with lime, or with cement mortar. Slacking by salt water gives less paste than fresh. Mortar should not be mixed upon the surface of clayey ground ; but a rough board, brick, or stone platform should be interposed. Pit sand sifted from decomposed gneiss, and other allied rocks, is ex- cellent for mortar; its sharp angles making with the lime a more coherent mass than the rounded grains r.f river or sea sand. Mortar should be applied wetter in hot than in cold weather; especially in brickwork ; otherwise the water is too much absorbed by the masonry, and the mortar is thereby injured. The tenacity, or cohesive strength, that is, the resistance to a pull of good common lime mortar of the usual proportions of lime and sand, and 6 months old, is about ? rom 15 to 30 fts per sq inch ; or .96 to 1.9 tons per sq ft. With less sand, or with greater age, it will be stronger. The crushing strength of good common mortar 6 months old is from 150 to 300 ft>s per sq inch, or 9.7 to 19.3 tons per sq foot. The sliding resistance, or that which common mortar opposes to any force tending to make one course of masonry slide upon another, is stated by Roudelet, to be but 5 ft>s per sq inch ; or about Jtf ton per sq ft, in mortar 6 months old. Transverse strength of good common mortar 6 months old. A bar 1 inch square and 12 ins clear span, breaks with a center load of 4 to 8 Ebs. The lime in mortar decays wood rapidly, especially in close, damp situations. Still the soaking of timber for a week or two in a solution of quicklime in water appears to act as a preservative. Iron* so completely embedded in mortar as to exclude air and moisture, has been found perfect after 1400 years ; but if the mortar admits moisture the iron decays. So, probably, with other metals. The adhesion to common bricks, or to rough nibble at any age will average about % of the cohesive strength at the same age; or say 12 to 24 fbsper sq inch, or .75 to 1.5 ton per sq ft at 6 months old. If care be taken to exclude dust entirely, by dipping each brick into water before laying it, or by sprinkling the stone by a hose, &c, the adhesion will be in- creased. On the other hand, much dust may almost prevent any adhesion at all. The precaution of wetting is especially necessary in very hot weather, to prevent the warm bricks or stone from kill- ing the mortar by the rapid absorption and evaporation of its water. The adhesion to very smooth hard pressed bricks, or to smoothly dressed or sawed stone is considerably less. 498 MORTAR, BRICKS, ETC. Art. 2. Bricks, size, wt. Ac.* A common size in our eastern cities is about 8.25 X 4 X 2 ins ; which is equal to 66 cub ins ; or 26.2 bricks to a cub ft ; or 707 bricks to a cub yard. For the number required with mortar, see table, p 496. In ordering a large number a minimum limit of dimension should be specified in order to prevent fraud. A brick X inch le-s each way than the above, contains but-52.5 cub ins ; thus requiring full 25 per cent more bricks to do the same work ; in addition to 25 per ct more cost for laying, which is generally paid for by the 1000. The weig-ht of a grood common brick of 8.25 X 4 X 2 ins, will aver- age about 4.5 tts ; or 118 tts per cub ft 3186 Ibs or 1.42 tons per cub yard ; or 2.01 tong per 10OO. A j; 4 X 4 X 2, required per sq foot of wall is as follows : Thickness of Wall. No. of Bricks. S^ius, or 1 brick 14 12?i " orl^" 21 17 " or 2 " 28 21^" or2^" 35 25%" or3 42 Laying per day* a bricklayer, with a laborer to keep him supplied with materials, will in common house walls, lay on an average about 1500 bricks per day of 10 working hours, lu tiie neater outer faces of back buildings, from 1000 to 1200 ; in good ordinary street fronts, 800 to 1000 ; or of the very finest lower story faces used in street fronts, from 150 to 300, depending on the number of angles, &c. In plain massive engineering work, he should average about 2000 per day, or 4 cub yds; und in large arches, about 1500, or 3 cub yds.t Since bricks shrink about yj part of each dimension in drying and burning, the moulds should be about y 1 ,- P art Iar 8 er every way than the burnt brick is intended to be. Good well-burnt bricks will ring when two are struck together. At the brick-yards about Philadelphia, a brick-moulder's work is 2333 bricks per day ; or 14000 per week. He is assisted by two boys, one of whom supplies the prepared clay, moulding sand, and water ; while the other carries away the bricks as they are moulded. A fourth person arranges them in rows for drying. About % of a cord, or 96 cub ft of wood, is allowed per 1000 for burning. Paving? Witll brick. In our cities this is done over a 6-inch layer of gravel, which should be free from clay, and well consolidated. With bricks of 8)4X4X2 ins, with joints of from X to 14 inch wide, a sq yard requires, flatwise, as is usual in streets, 38 bricks; edgewise, 73; endwise, ' ' iks and gravel, will in 10 hours pave about Lt-ii done, suiid is brushed into the joints. Art. 3. The Crushing 1 Strength Of bricks of course varies greatly. A rather soft one will crush under from 450 to 600 Ibs per sq inch ; or about 30 to 40 tons per sq ft ; while a first-rate machine-pressed one will require about 200 to 400 tons per square foot. This last is about the crushing limit of the best sandstone; % as much as the best marbles or limestones ; and % as much as the best granites, or roofing slates. But masses of brickwork crush under much smaller loads than single bricks. In some English experiments, small cubical masses only 9 inches on each edge, laid in cement, crushed under 27 to 40 tons per sq ft. Others, with piers 9 ins square, and 2 ft 3 ins high, in cement, only two days after being built, required 44 to 62 tons per sq ft to crush them. Another, of pressed brick, in best Portland cement, is said to have withstood 202 tons per sq ft; and with common lime mortar only y as much. It must, however, be remembered, that cracking and splitting usually commence under about one- half the crushing loads. To be safe, the load should not exceed % or fa of the crushing one ; and o with stone. Moreover, these experiments were made upon low masses ; but the strength decreases with the* proportion of the height to the thickness. The pressure at the base of a brick shot-tower in Baltimore, 246 feet high, is estimated at 6> tons The Peerless Brick CO, office 208 South Seventh St, Philada. make superb smooth (not shining) bricks of various shapes and colors (as white, black, gray, butt, brown, red, &c) (or ornamental architectural purposes. Their standard size is 8% X 4^ X 2% = 82 cub ins, or H larger than the above 8% X 4 X 2 ins. For either plain rectangular or voussoir shapes their price in 1880 is $4 per hundred if red ; and $5.50 if of other colors. For simple curved mouldings $4.50 to $5.50 if red ; and $6 to $7 of other colors. The color extends throughout the body of the brick. With a few of these judiciously distributed among common bricks, beautiful architectural eflects may be pro duced, both indoors and out. at far less cost than in stone. The same Co will, if a sufficient order is given, furnish voussoir bricks for specified radii, but of the quality and finish of ordinary good hard brick, at from 25 to 50 per ct advance on prevailing market rates of common plain ones. * Prices in Philada in 1873. Bricks alone; Salmon, or soft, $8 per 1000. Hard brick, $10. Back stretchers (generally used for the facings of back buildings, &c,) $16. Paving brick, $15. Pressed, (for lower stories'of first-class fronts,) $30. t Bricklaying ; including mortar and scaffolding : averaging an entire dwelling. $8 per 1000. Best pressed bricks in first-class fronts, $15 to $20. In 188O all average about 2O per ct lea. 149. An average workman, with a laborer to supplv the bricks and gravel, will in 10 hours pave about 2000 bricks ; or 53 sq yds flat, 27 edgewise, 13 endwise. When ' , ETC. 499 per sq ft; and In a briok chimne^r'at Glasgow, Scotland, 468 feet high, at 9 tons. Professor Rankine calculates that in heavy gale^Chis is increased to 15 tons, on the leeward side. The walls of both are of course much thicker ^MJottom than at top. With walls 100 feet high, of uniform thickness, the pressure at base wouhi^Se 5.4 tons per sq ft. With our present^mperfect knowledge ou this subject, it cannot be considered safe to expose even first-class pressed brickwork, in cement, to more than 12 or 15 tons per sq ft ; or good hand-moulded, to more than-two-thirds as much. Tensile strength 01 brick, 40 to 400 fbs per sq inch ; or 2.6 to 26 tons per sq ft. The English rol of brickwork is 306 cub feet, or l\% cub yards; and requires about 4500 bricks of the English standard size ; with about 75 cub ft of mortar. The English hundred of lime, is a cub yd. Frozen mortsir There is risk in using common mortar in cold weather. If the cold should continue lorig euough to allow the frozen mortar to set well, the work may remain safe , but if a warm day should occur between the freezing and the setting of the mortar, the sun shining on owe side of the wall may melt the mortar on that side, while that on the other side may remain frozen hard. In that case, the wall will be apt to fall ; or if it does not, it will at least always be weak ; for mortar that has partially set while frozen, if then melted, will never regain its strength. By the writer's own trials hydraulic cements seemed not to be injured by freezing. Experiments for rendering brick masonry impervious to Water. Abstract of a paper read before the American Society of Civil Engineers, May 4, 1870, by William L. Dearborn, Civil Engineer, member of the Society. The face walls of the Back Bays of the Gate- houses of the new Croton Eeservoir, located north of Eighty-sixth Street, in Central Park, were built of the best quality of hard-burnt brick; laid in mortar composed of hydraulic cement of New York, and sand mixed in the proportion of one measure of cement to two of sand. The space between the walls is 4 ft ; and was filled with concrete. The face walls were laid up with great care, and every precaution was taken to have the joints well filled and /nsure good work. They are 12 ins thick, and 40 ft high ; and the Bays when full generally have 36 ft of water in them. When the reservoir was first filled, and the water was let into the Gate-houses, it was found to filter through these walls to a considerable amount. As soon as this was discovered, the water was drawn wt of the Bays, with the intention of attempting to remedy or prevent this infiltration. After care- fully considering several modes of accomplishing the object desired, I came to the conclusion to try " Svlvester's Process for Repelling Moisture from External Walls." The process consists in using two washes or solutions for covering the surface of brick walls : one composed of Castile soap and water ; and one of alum and water. The proportions are : three-quar- "rs of a pound of soap to one gallon of water; and half a pound of alum to four gallons of water; both substances to be perfectly dissolved in the water before being used. The walls should be perfectly clean and dry ; and the temperature of the air should not be below 50 degrees Fahrenheit, when the'compositions are applied. The first, or soap wash, should be laid on when at boiling heat, with a flat brush, taking care not to form a froth on the brickwork. This wash should remain twenty-four hours ; so as to become dry and hard before the second or alum wash is applied ; which should be done in the same manner as She first. The temperature of this wash when applied may be 60 or 70 ; and it should also remain twenty-four hours before a second coat of the soap wash is put on ; and these coats are to be repeated alternately until the walls are made impervious to water. The alum and soap thus combined form an Insoluble compound, filling the pores of the masonry, and entirely preventing the water from penetrating the walls. Before applying these compositions to the walls of the Bays, some experiments were made to test Che absorption of water by bricks under pressure after being covered with these washes, in order to determine how many coats the wall would require to render them impervious to water. To do this, a strong wooden box was made, put together with screws, large enough to hold 2 bricks ; and on the top was inserted an inch pine forty feet long. In this box were placed two bricks after being made perfectly dry. and then covered with a coat of each of the washes, as before directed, and weighed. They were then subjected to the pressure of a column of water 40 ft high ; and. after remaining a sufficient length of time, they were taken out and weighed again, to ascertain the amount of water they had absorbed. The bricks were then dried, and again coated with the washes and weighed, and subjected to press- ure as before; and this operation was repeated until the bricks were found not to absorb any water. Four coatings rendered the bricks impenetrable under the pressure of 40 ft head. The mean weight of the bricks (dry) before being coated, was &% Ibs; the mean absorption wai one-half pound of water. An hydrometer was used in testing the solutions, As this experiment was made in the fall and winter, (1863,) after the temporary roofs were put on to the Gate house, artificial heat had to be resorted to, to dry the walls and keep the air at a proper temperature. The cost was 10.06 cts per sq ft. As soon as the last coat had become hard, the water fras let into the Bays, and the walls were found to be perfectly impervious to water ; and they still re- main so in 1870, after about 6^ years. BRICK ARCH (FOOTWAY OP HIGH BRinoK). The brick arch of the footway of High Bridge is the arc of a circle 29 ft 6 in radius ; and is 12 in thick ; the width on top is 17 ft; and the length covered was 1381 ft. The first two courses of the brick of the arch are composed of the best hard-burnt brick, laid edge- wise in mortar composed of one part, by measure, of hydraulic cement of New York, and two parts of sand. The top of these bricks, and the inside of the granite coping against which the two top courses of brick rest, was, when they were perfectly dry, covered with a coat of asphalt one-half an inch thick, laid on when the asphalt was heated to a temperature of from 360 to 518 Fahrenheit. On top of this was laid a course of brick flatwise, dipped in asphalt, and laid when the asphalt was hot ; and the joints were run full of hot asphalt. On top of this a course of pressed brick was laid flatwise in hydraulic cement mortar, forming the paving and floor of the bridge. This asphalt was the Trinidad variety ; and was mixed with 10 per cent, by measure, of coal tar; and 25 per cent of sand. A few experiments for testing the strength of this asphalt, when used to cement bricks together, were made, and two of them are given below. Six bricks, pressed together flatwise, with asphalt joints, were, after lying six mouths, broken. The distance between the supports was 12 ins ; breaking weight, 900 tts ; area of single joint, 28J$ sq ins. The asphalt adhered so strongly to the brick as to tear away the surface in many places. 500 CEMENT, CONCRETE, ETC. Two bricks pressed together end to end, cemented with asphalt, were, after lying 6 months, broken. The distance between the supports was 10 ins; area of joint, 8> sq ins; breaking weight, 150 tt>s. The area of the bridge covered with asphalted brick, was 23065 sq ft. There was used 94200 Bs of asphalt, 33 barrels of coal tar, 10 cub yds of sand, 93HOO bricks. The time occupied was 109 days of masons, aud 148 days of laborers. Two masons and two labor- ers will melt and spread, of the' first coat, 1650 sq ft per day. The total cost of this coat was 5.25 cents per sq ft, exclusive of duty on asphalt. There were three grooves, 2 ins wide by 4 ins deep, made entirely across the brick arch, and immediately under the first coat of asphalt, dividing the arch into four equal parts. These grooves were filled with elastic paint cement. This arrangement was intended to guard against the evil eflects of the contraction of the arch in winter; as it was expected to yield slightly at these points, and at no other point; and then the elastic cement would prevent any leakage there. The entire experiment has proved a very successful one, and the arch has remained perfectly tight. In proposing the above plan for working the asphalt with the brickwork, the object was to avoid depending on a large aontiuued surface of asphalt, as is usual in covering arches, which very fre- quently cracks from the greater contraction of the asphalt than that of the masonry with which it is in contact ; the extent of the asphalt on this work being only about one-quarter of an inch to each brick. This is deemed to be an essential element in the success of the impervious covering." A cheap and effective process for preventing the percolation of water through the arches of aque- ducts, and even of bridges, is a great desideratum. Many expensive trials with resinous compounds have proved failures. Hydraulic cement appears to merely diminish the evil. Much of the trouble is probably due to cracks produced by changes of temperature. Art. 4. Hydraulic Cements.* Certain limestones, when burnt, will not slack with water; but when the burnt stone is finelv ground, and made into a paste, it possesses the pro- perty of hardening under water; and is therefore called hydraulic cement. So long as the propor- tion of those ingredients which impart hydraulicity, is so small that the burnt stone will slack ; bat still make a paste or a mortar, which will harden under water, it is called hydraulic lime. This does not harden so promptly, or to so great a degree, as the cements. Hydraulic limes slack more slowly, and swell less, in proportion to their hydraulicity ; some requiring many hours. Artificial hydraulic limes and cements, of excellent quality, may be made by mixing lime and clay thoroughly together; then moulding the mixture into blocks like bricks; which are first dried, then burnt, aud finely ground. The celebrated artificial English Portland cement, is made by grinding together in water chalk and clay. The fine particles are floated away to other vessels, and allowed to settle as a paste ; which is then collected, moulded, dried, burnt, and ground. Natural Port- land is that made from limestone, or other material of very rare occurrence, which combines naturally that proportion of lime and clay which gives the above artificial Portlands their pre-eminence. This alone constitutes its difference from our common natural hyd cements. The weig-ht of the Rosendale, and other American cements, together with some foreign ones, will be found on page 385. Saylor's Port* land weighs about 120 ft>s per struck bushel. The writer found by 10 years' trial that if, after setting, dampness is absolutely excluded, Cements preserve iron, lead, zinc, copper, and brass ; and that Plaster of Paris preserves all except iron, which it rusts somewhat unless galvanized. Lime-mortar probably preserves all of them, if kept free from damp. Protection from moisture, even that of the air, is very essential for the preservation of cements, as well as of quicklime. On this account the barrels are generally lined with stout paper. With this precaution, aided by keeping the barrels stored in a dry place, raised above the ground, the cement, although it may require more time to set, will not otherwise very appreciably deteriorate for six months; but after H or 16 months, Gillmore says it is unfit for use m important works. But in lumps, kept dry, it will remain good for 2 or 3 years; and may be ground as required for use. Good Portland cement is stated by good authority rather to improve by free exposure to the air under cover ; but whether this is correct or not, we cannot say. Restoration by rebarning- may be effected. If the injured ground ce- ment is spread in a thin layer, on a red-hot iron plate, for about 15 minutes, its good qualities will be in a great measure restored. The time should be ascertained by trial. If it has been actually wet, and lumpy, or cemented into a mass, it should first be broken into small pieces, and then ground. Or these pieces may be first kiln-burnt at a bright red-heat for about 1>$ hours ; and then ground. Art. 5. For roughcasting 1 , or stuccoing the outside of walls, very few hydraulic cements are fit. Mr. Downing, in his work on " Country Houses," excepts that from Berlin, Connecticut, as the only one within his extended knowledge, that is suitable. Portland cement ia said to be good for that purpose. A wall with a northern exposure in Philada was coated with it in i860 ; and appears to be in perfect condition in 1880. Quantity required. A barrel of cement, 300 ft>s ; and 2 barrels of sand, (6 bushels, or 1% cub ft;) mixed with about % a barrel of water, will make about 8 cub ft of mortar, 192 sq ft of mortar-joint X inch thick = 21 # sq yarda. 288'' " K " " = 32 " " 384 ' " X " " -*2 " " 768 " " " H " " =85* " * Prices of hyd cements in Philada, 1880, by the large importing firm of French, Richards & Co, corner of York Avenue and Callowhill St. English Portland, $3.25 to $4 per barrel of about 400 fts gross; according to quality and quantity. Saylor's Portland, per barrel of 400 fts gross, $3 to $3.50. Rosendale, per barrel of about 300 fts net, Si. '25 to 1.50. Other C. 8 cements, per barrel of 800 fts net, $1.15 to $1.50. Ground calcined plaster of Paris, selected, barrel of 300 fts net, $2 to $2.25. Commercial, barrel of about 250 tts net, $1.50 to $1.60. In 1882 English and Saylor's Portland about 10 per ct less. CEJJHEOT, CONCRETE, ETC. 501 Or, to lay 1 cubic yard^tff-522 bricks of 8^ by 4, by 2 ins, with joints % inch thick; or a cubic yard of roughly scrabbletf-fubble stonework. The quantity of sand may be increased, however, to 3 or 4 measures for ordinary work. JPointiiig mortar. Gen Gillmore recommends " 1 part by weight of good cement powder, to 3 or 3% parts of sand. To be mixed under shelter, and in quantities of only 2 or 3 pints at a time, using very little water, so that the mortar, when ready for use, shall appear rather incoherent, and quite deficient in plasticity. The joints being previously scraped out to a depth of at least % an inch, the mortar is put in by the trowel ; a straight-edge being held just below the joint, if straight, as an auxiliary. The mortar is then to be well calked into the joint by a calking- iron and hammer; then more mortar is put in, and calked, until the joint is full. It is then rubbed and polished under as great pressure as the mason can exert. If the joints are very One they should be enlarged by a stonecutter, to about -A- inch, to receive the pointing. The wall should be well wet before the pointing is put in, and kept in such condition as neither to give water to, nor take it from the mortar. In hot weather, the pointing should be kept sheltered for some days from the sun, so as not to dry too quickly." Why not finish joints at once, without subsequent pointing ? Author. Art. 6. Color is no indication of strength in hyd cements. The finer they are ground the better. At least 90 per ct should pass through a sieve of 50 meshes per lineal inch, of Wire No 35 Amer wire gauge (.0056 inch thick) ; or 2500 meshes per sq inch. Weight is a good indication when equally well ground. A flat cake of good cement paste placed in water as soon as it admits of so doing safely, and left in it for a week, should show no cracks. New cement is not as good as when a few weeks old. The term Setting does not imply that the cement has hardened to any great extent, but merely that it has ceased to be pasty and has become brittle. Quick setting cements may do this sufficiently to allow small experimental samples to be lifted and handled care- fully within flve to thirty minutes ; while others may require from one to eight or more hours. Slow setting does not indicate inferiority, for many of the very best are the slowest setting. A layer of very quick setting cement may partially set, especially in warm weather, before the masonry is properly lowered and adjusted upon it, and any disturbance after setting has commenced is prejudicial. Such are to be regarded with suspicion, and sub- mitted to longer tests than slow ones. Still, quick setting ones are best in certain cases, as when exposed to running water, &c. They may be rendered slower by adding a bulk of lime paste equal to 5 or 15 per ct of the cement paste, without weakening them seriously. As a general rule cements set and harden better in water than in air, especially in warm weather. If, however, the temp for the first few days does not exceed 55 to 65 Fah, there seems to be no appreciable difference in this respect; but in warm air cement dries instead of setting, and thus loses most of its strength. In hot weather every precaution should be used against this. The time reqd to attain the greatest hardness is many years, but after about a year the increase is usually very small and slow, especially with neat cement. More- over, auy subsequent increase is a matter of little importance, because generally by that time, and often much sooner, the work is completed and exposed to its maximum strains. and retards setting, and weakens the cement paste. But although with sand the strength of the mortar may never attain to that of the neat paste, yet it increases with age in a greater proportion ; so that a neat paste which at the end of a year would be but twice as strong as in 7 days, may with sand yield a mortar which at the end of a year will be 8, 4, or 5 times as strong as it was in 7 days. Good Port- lands neat usually have at the end of a year from 1.5 to 2 times their strength at the end of 7 days ; and the American natural cements, Rosendale, Louisville, Cumberland, &c, from 2.5 to 3.5 times; but inasmuch as Portlands average (roughly speaking) about 5 or 6 times the strength of the others in 7 days, they still average about 2.5 to 3 times as strong in a year or longer. Cements of the same class differ much in their rapidity of hardening. One may at the end of a month gain nearly one-half, and another not more than one-sixth of its increase at the end of a ear, at which time both may have about the same strength. Hence, tests for 1 week or 1 month are y no means conclusive as to their final comparative merits. There seems to be a period occurring from a few weeks to several months after having been laid, at which cement and its mortars for a short time not only cease from hardening, but actually lose Strength. They then recover, and the hardening goes on as before. It has been suggested that this opinion has originated in some oversight of the experimenters, but the writer believes it to be founded on fact. In his expts with various hvd cements of the consistence of mortar, even without sand, the writer detected no change of bulk in setting. Art. 7. Mr. Wm. W. Ulaclay, . E. (see his very instructive paper in Trans. Am. Soc. C. E., Dec, 1877), found that in the testing of cements the temperature of the air and water had far more influence than bad before been suspected. Thus neat Portland moulded in air at 30 Fah, and kept 6 days in water at 40, had a tensile strength of but 156 ft>s per sq inch, while that kept in water of 70 had 299 Ibs, or nearly twice as much. Other bars moulded in air at 60, after 6 days in water of 40, broke with 113 fbs tensile per sq inch, while those in water at 70 required 254 Ibs, or about 2.25 times as much. But at the end of only 20 days the strengths of these last were as 212 to 336 fts, or as 1 to 1 .6 ; the weaker one having in that time gained rapidly on the stronger. As longer time would of course bring them still nearer to an equality, the ultimate effects of temperature within certain limits are fortunately not so important in actual prac- tice as the first expts might lead us to infer. Work must go on notwithstanding changes of tempera- ture, but we must take care that our mortar shall at all times be strong enough even under their most injurious influences. Cements in open air are certainly more or less injured by drying instead of setting, as the temp exceeds about 65 to 70. But if mixed only ia small quantities at a time, and quickly laid in masonry of dampened stone, so as to be sheltered from the air, the injury is much reduce'd. The sand and stone should both be damp, not wet, in hot weather, and a little more water may be used in the cement paste: also if possible not only the mortar while being mixed, but the masonry also should then be shaded Mr Maclay found that 6 day specimens of neat Portland broken direct from the water were much stronger than if first left 24 hours to dry in the shade at tolerably high temps. But the reverse occurs with such U. S. natural cements as Rosendale. &c, the strength of some being largely increased by such drying. Experiments in Europe with Portlands kept 3 months y b 502 CEMENT, CONCRETE, ETC. in water, seemed to show the weakest period for such to be at 2 days' exposure, when the strength was but half as great as when first taken from the water. But on the 4th day they were even stronger than at first; and the strength then increased with time as if there had been no interruption. The effects of colcl, although it retards the setting, do not appear to be serious otherwise. If the cement mortar even freezes almost as quickly as the masonry is laid with it, it does not seem to depreciate appreciably. The writer has found this to be the case also with lime mortar, even when a few hours after freezing the temp became so high as to soften the frozen mortar again. But although the mortar of either lime or cement may not be thereby injured, the work, espe- cially in thin brick walls, may be ruined and overthrown. Thus if soon after the mortar through th entire thickness of such a wall be frozen, the sun shines on one face of it so as to soften the mortar of that face, while the mortar behind it remains hard, it is plain that the wall will be liable to settle at the heated face, and at least bend outward if it does not fall. The writer has observed that coat- ings of cement applied to the backs of arches on the approach of winter, aud left unprotected, were entirely broken up and worthless on resuming work the next spring. The heating 1 of sand and cement in freezing weather seems to be a bad prac- tice, especially if to be placed in cold water. But for use out of water Mr Maclay says they may be heated to 50 or 60. Cold water for mixing is probably no farther inju- rious than that it retards the setting. All cements when mixed with sand to a proper consistence for mortar will fall to pieces if placed in water before setting has commenced. Portlands do so even without sand ; but U. S. natural cements of good quality do not. Art. 8. Strength of cements. The strength as before stated is much affected by the temp of the air and water, as also by the degree of force with which the cement is pressed into the moulds ; by the extent of setting before being put into the water, and of drying when taken out; and still more by the consideration of whether or not it sets while under the influence of pressure, which increases the strength materially. On this account cements in actual masonry may under ordinary circumstances give better results than in-door expts. These causes, together with tbe degree of thoroughness of the mixing- or gang- ing-, the proportion of water used, and other considerations may easily affect the results 100 per ct, or even much more. Hence, the discrepancies in the reports of different experimenters. Rein. Portlands require more water than the common U. S. cements, and shrink less in mixing. See next Art. Also, mortars require more than concrete, espe- cially when the last is to be well rammed, in which case it should be merely moist, so aa barely to cohere when pressed into a ball by hand. If more water is present, the consoli- dation by ramming is proportionally imperfect. To assure himself that the quality of cement furnished is equal to that contracted for, the engineer should reserve the right to bore with a long auger into any part of each barrel, and to reject every barrel of which the sample drawn out does not satisfy the stipulations. On works using large quan- tities, there should be one person specially detailed to this duty. One advantage of very strong cements is their economy, even at a higher cost, in allowing the use of a larger proportion of the cheaper ingredients, sand, gravel, and broken stone. Almost any common U. S. cement, if of good quality, will with 1.5 or 2 measures of sand give a mortar strong enough for most engineering pur- poses : but a good Portland will give one equally strong with 3 or 4 meas of sand ; and will, therefore, be equally cheap at twice the price; beside requiring the handling, storing, and testing of only half the number of barrels. After what has been said it is plain that great latitude must be allowed in attempting to prepare a table of approximate average strengths. The writer can pretend to nothing more than the following, which is deduced from reliable reports, aided by a few experiments of his own on transverse strengths, a summary of which last forms the last column. It is singular that most experimenters test only the tensile strength, the coefficient of which is seldom wanted in practice. If one measure of cement slightly shaken be mixed to a paste with about .35 meas of water if a common U. S. cement, or about .40 meas if Portland, in the shade, and in a temp of from 60 to 90, this paste will occupj^bout .7 meas if common, and about 86 if Portland, when well pressed into wooden moulds by the fingers (protected from corrosion by gloves of rubber or buckskin). If then allowed from 80 minutes to some hours (according to its setting properties) to set ; then removed from the moulds, and at the end of '24 hours total, placed in water of the above limits of temp for 7 days, and brokeu at once when taken from the water, the samples will generally exhibit about the following strengths. Those for compression are supposed to be cubes ; and those for transverse strength in the table were beams 1 inch square, and 12 ins clear span, loaded at the center. Table A. Average Strengths of neat Cements after 6 days in water, and broken directly from the water. Tensile, fts 1" X per sqin. per sqin. per sq ft. i" x 12 '.fts. Portlands, artificial, either foreign, or the " National " of Kingston, N.Y. 200 to 350 1400 to 2400 90 to 154 25 to 45 " Saylor's natural, Coplay, Penn.. 170 to 370 40 to 70 1100 to 1700 250 to 450 71 to 109 16 to 29 26 3 to 7 All below the lowest of these should be rejected ; the average of the table may be considered fair; and all above the highest superior. After only 24 hours in water the strength of the common ^U. S. cements averages about half that for 6 days, but with considerable variations both ways. In like manner at the end of a year neat Portlands average from 1.5 to 2 times as strong as in 6 days ; and our common cements from 2.5 to 3.5 times. The London board of works require that Portlands after 7 days in water shall have at least 35 Ibs transverse, and 350 Ibs tensile strength. Some have reqd 500 or more ten- sile to break them. For Portlands the writer found the transverse strengths of several well known English brands moulded as before described, to be 26 to 40 Bbs after 7 days in water; National Portland of Kingston, N. Y., 40 and 46; Baylor's Portland (only 2 trials) 26 f, CONCRETE, ETC. 503 fts. ToepfTer, G-^awitz, fc Co, of Stettin, Germany, warrant all their Portland (known &aJA& " Stera " brand) equal to 500 fts tensile after 7 days in water. Some of it has borne 760 fts. ./ Mr. J. Ilerbert Shedd, as Engineer of the Water Works and Sewers of Providence-fTl. I., rejected all Rosendale which when mixed to a stiff paste, and allowed 30 min in air to set, and then put into water for only 24 hours, broke with less tension than 70 fts. At first he found some that broke with 10 to 15 fts; some that would not set at all in water; and but little that bore 30 fts. Now samples frequently bear 100 fts or more; but that usually sold still rarely exceeds 40 to 50, and frequently scarce half as much. The Sewer Department of St. Louis. Missouri, requires all Louisville, Kentucky, cement to bear at least 40 fts tensile after 24 hours in water. Some of it now shows as high as 100 or more; and 60 or 70 would have heen adopted as the mininuin, but for the fear that it would have encouraged the making of too quick setting cement. Most of that sold will probably not exceed 30 fts. Art. 9. Cement mortar is cement mixed with water and sand only. The writer found that for making cement pastes of about equal consistency and fit for mortar by themselves, the English Portlands, slightly shaken in the measure, required an average of about .4 of their own bulk of water; and the D. S. common cements about .35. The Portland pastes when thoroughly mixed and slightly pressed by hand into a box shrank about one-eighth of their bulk as dry shaken cement; and the others about one-fourth ; or in other words the common U. S. cements shrink about twice as much as the Portlands ; and these are about the proper proportions to assume in estimating the quantity of cement for theoretically filling the voids in sand. But when sand is added, more water is reqd. It is impossible to lay down rules for all cases, but as a very rough average, mortar will require an addition equal to about .2 of the bulk of dry sand ; varying of course with the weather, &c. Trial on the work in hand is better than rules. Any addition of sand weakens cement, especially as regards ten- sion; as it does also lime mortar. But economy requires its use. Sand also retards the setting, so that cement which by itself would set in half an hour, may not do so for some days if mixed with a large proportion of sand. This weakening effect will of course vary with different cements, and with many circumstances inferable from Art. 7, &c. As a rough average the following is perhaps not far from the truth as regards either tensile or transverse strength when not rammed. Sand. M i VA 2 3 4 5 6 7 8 Strength. 1 % y* .4 1 A .3 % t Ye * Y* The crushing- strength does not diminish so rapidly ; but for each pro- portion of sand we may take the strength preceding it in the table, as an approximation. Moreover the crushing strength with sand increases with age much more rapidly than the tensile ; and the more EO the greater the proportion of sand. As a general rule with cements of good quality we shall have mortars fit for most engineering pur- poses if we do not exceed from 1 to 1.5 measures of dry sand to 1 of the common cements ; or from 2 to 3 of sand to I of Portland. The shearing strength of neat cements averages about one-fourth of the tensile. The adhesion of cements to bricks or rough rubble, at dif- ferent ages, and whether neat or with sand, may probably be taken at an average of about three- fourths of the cohesive or tensile strength of the cement or mortar at the same age. If the bricks and stoae are moist and entirely free from dust when laid, the adhesion is increased ; whereas if very dry and dusty, especially in hot weather, it may be reduced almost to 0. The adhesion to very hard, smooth bricks, or to finely dressed or sawed masonry is less. The voids in sand of pure quartz like that found on most of our sea- shores, when perfectly dry and loose, occupy from .303 of the mass in sand weighing 1 15 fts per cub ft, to .515 in that.weighing 80 fts. Usually, however, such dry sand weighs say from 105 fts with voids of .364 : to 95 fts, with voids .424 ; the mean being 100 fts, with voids .394.* But the wet sand in mortar occupies about from 5 to 7 per cent less space than when dry ; the shrinkage averaging say 6 per ct or ^ part ; thus making the voids .323 of the 105 ft sand when wet ; and .387 of the 95 ft ; the mean of which is .355. But to allow for imperfect mixing, &c, it is better to assume the voids at .4 of the * If greater accuracy is desired pour into a graduated cylindrical measuring-glass 100 measures of dry sand. Pour this out, and fill the glass up to 60 measures with water. Into this sprinkle slowly the same 100 measures of dry sand. These will now be found to fill the glass only to say 94 measures, having shrunk say 6 perct; while the water will reach to say 121 measures ; of which 1219427 measures will be above the sand : leaving 6027 33 measures filling the voids in 94 measures of wet sand; showing the voids in the wet sand to be H .351 of the wet mass. If the sand is poured into the water hastily, air is carried in with it, the voids will not be filled, and the result will be quite different. Since a cubic foot of pure quartz weighs 165 Ibs, it follows that if we weigh a cubic foot of pure dry sand either loose or rammed, then as 165 is to the wt found, so is 1 to the solid part of the sand. And if this solid part be subtracted from 1, the remainder will be the voids, as below. Wt in Ibs per cub ft dry 80 85 90 95 100 105 110 115 Proportion of solid .485 .515 .546 .576 .606 .636 .667 .697 Proportion of voids .515 .485 .454 .424 .394 .364 .333 .303 504 CEMENT, CONCRETE, ETC. dry sand. Moreover, since the cements, as before stated, shrink more or leas when mixed with water, and worked up into mortar, it would be as well to assume that to make sufficient paste to thoroughly fill the voids, we should not use a less volume of dry common cement, slightly shaken, than half the bulk of the dry sand ; or than .45 of the bulk if Portland. The bulk of the mixed mortar will then be about equal to or a trifle less than that of the dry sand alone. The best sand is that with grains of very uneven sizes, and sharp. The more uneven the sizes the smaller are the voids, and the heavier is the sand. It should be well washed if it contains clay or mud, for these are very injurious to mortar or concrete. Art. 1O. Cement concrete or Be ton, is the foregoing cement mortar mixed with gravel or broken stone, brick, oyster shells, &c, or with all together. In concrete as in mortar, it is advisable on the score of strength that all the voids be filled or more than filled. Those of broken stone of tolerably uniform size and shape are about .5 of the mass ; with more irregularity of size and shape they may decrease to .4. Those of gravel vary like those of sand, and had like it better be taken at .5 when estimating the dry cement. We shall then have as follows. . For 1 cub yd of concrete of stone, gravel, and sand, without voids. 1 cub yd broken stone with .5 of its bulk voids, requiring .5 cub yd gravel. 0.5 cub yd gravel " .5 of 0.25 cub yd sand " .5 .25 cub yd sand. .1'25 (or %) cub yd dry cement. It Is probable that mistakes have occurred from inadvertently assuming that because the voids in a broken mass, constitute a certain proportion of the bulk of said mass ; therefore, the original solid has swelled in only that same proportion. Thus, if a solid cubic yard of stone be broken into small irregular pieces, which have among themselves about the same proportions of large and small cues, as usually occurs in quarrying, or in railroad rock-cuttings ; and if these be loosely thrown into a heap, the .47 of this heap, or rather less than half of it, will be voids. But it does not follow, therefore, that the original solid cub yd has swelled only .47, or nearly one-half, or makes only 1J cub yds of broken stone ; although many young engineers would probably consider this a very full allowance ; and would suppose that they were quite just to the company, if they counted for the contractor one solid yard of excavation for every 1% yds of fragments. Now, it is plain that if .47 of the broken heap are voids, the remaining .53 must be stone. But these .53 constituted the original solid cubic yard : and they still remain equal to it in actual solidity. Hence we must say as follows : If .53 of the broken mass occupies oue cub yd of actual space, how much space will the whole mass occupy ; or, Of the Cub yd Entire Cub yds broken mass. of space. broken mass. of space. Hence, we see that a solid cub yd of stone, when so broken, swells to 1.9, or nearly 2 cub yds ; and hence a proper allowance to a contractor, would be 1 cub yd solid, for every 1.9 cub yds of pieces ; or the yds of pfeces must be divided by 1.9 for the solid yards. If we know that a cubic yard of any stone, break: ,1 T C w ~ vu- u , , ~. jaks to, say 1.9 yds, then to find the proportions of voids, and solid, in the broken mass, proceed thus : The solid part of the broken mass must occupy 1 cub yd of space; and the question is what part of 1.9 yds does this 1 yd constitute. The answer is 53. therefore 53 hundredths of the broken mass is solid; and of course the remaining 47 bun- 1.9 dredths are voids. If a cubic foot solid weighs N Ibs ; but when broken up, or ground, only n Ibs per cub ft, then n divided by N, will be the proportion of solid in the broken mass. If the broken stone is loosely piled up, it will occupy a little less space, say about 1.8 cub yds ; in which case the voids will be .44 ; and the solid, .56 of the mass. We will here venture to express our doubts whether hard rock when blasted and made into embankment, settles to less than 1% yds for every solid yd. Mr Ellwood Morris gives as the result of certain embankment of hard sandstone, made under his supervision, an increase of bulk of y 5 ^; or in other words, that 1 cub yd of rock in place, made lj 5 ^, or 1.417 yds of embankment This corresponds to very nearly .7 solid; and .3 voids ; while 1% yds to 1 solid, corresponds to .6 solid; and .4 voids. The rough sides of rock excava- tions, make it difficult to measure them with accuracy; and we cannot but suspect, that something of this kind has interfered with the results obtained by Mr Morris. He, however, may be right, and By some careful experiments of our own, an ordinary pure sand from the sea shore, perfectly dry, and loose, weighed 97 Ibs per cub ft; and its voids were .41, and the solid .59 of the mass. By thorough shaking, and jarring, it could be settled the .1333 part, (halfway between ^, and ^,) and no more. It then weighed 112 Ibs per cub ft; and its voids were then .32; and the solid, .68 of the mass. Another pure quartz sand, of much finer grain, perfectly dry and loose, weighed but 88 fts per cub ft ; the voids were .466 ; and the solid .534 of the mass. By thorough shaking and jarring it could be reduced; like the former, onlv the .1333 part: it then weighed 101.6 Ibs per cub ft; and its voids were .384 ; and the solid .616. Another, consisting of the finest sifted grains, of the last, weighed 82 Ibs per cub ft ; so that its voids, and solid, each were very nearly .5 of the mass. This could be com- pacted about % part; and then weighed 98J^ 8>s per cub ft. The first, or coarsest of these sands, when quite moist, but not wet, perfectly loose, weighed but 86 Ibs per cub ff or 11 Ibs less than when dry. It could be rammed in thin layers, until it settled one- fifth part ; and then weighed 107^ Ibs per cub ft. Voids .348. t solid .652. The second sand, similarlv moist and loose, weighed but 69 Ibs per cub ft ; or 19 Ibs less than when dry. It could be rammed in thin layers, until it settled % part ; and then weighed 103^ Ibs per cub None olf 'these sands when dry, and loose, if poured gently into water to a depth of 15 inches, set- tled more than about one-fifteenth part; the coarsest one, considerably less. NCRETE, ETC. 505 Here the .125 cub yd of drv>6ment constitutes one-eighth of the single mass ; or one-fourteenth of all the dry ingredients as^aleasured separately. For 1 ciib jpcTof concrete of broken stone and sand without X^ voids. 1 cub yd broken stone, with .5 of its bulk voids requiring | .5 cub yd sand. .5 cub yard sand, " .5 " " " " | .25 cub yd dry cement. The strength of concrete is affected by the quality of the broken stone, as well as by that of the cement, the degree of ramming, &c. Cubes of either of the above with Port- land, as well as one composed of 1 meas of good Portland to 5 of sand only, well made, and rammed, should either in air or in water require to crush them at different ages, not less than about as follows. Agre in months 1 36 9 12 Tons per sq ft 15 40 65 85 100 Under favorable conditions of materials, workmanship and weather, the strengths may be from 50 to 100 per ct greater. For transverse strength as beams see p 507. If not rammed the strength will average about one-third part less. With common U. S. cements, if of good quality from .2 to .3 of the strength of Portland concrete may be had. Slow setting cements are best for concrete, especially when to be rammed. It may not be amiss to state here that when masonry is backed by concrete the two are liable in time to crack apart from unequal settlement, especially if the ramming has not been thorough; also that in variable climates cast iron cylinders filled with concrete are frequently split horizon- tally by unequal expansion and contraction. In such structures it is safest to consider the cyls as mere moulds for the concrete: and to depend upon the last only for sustaining the load. The concrete for the New York City docks consists of 1 measure of either English or Saylor's Portland, 2 of sand, 5 of broken stone (hard trap). That made of Eng- lish Portland, after drying a few days, and then being immersed 6 weeks, required about 30 tons per sq ft to crush it. Saylor's would probably require the same. At the Missis- sippi Jetties, (see "South Pass Jetties" by Max E. Schmidt, C. E., Trans Am Soc C E, Aug 1879) Saylor's Portland 1 ; sand 2.76; gravel 1.46; broken stone 5. In the foundations of the Washington Monument at Washington, D.C., (1880) English Portland (J. B. White & Bros) 1 ; sand 2 ; gravel 3 ; broken stone 4 ; and according to a Govt. Report, has a crushing strength of 155 tons per sq ft when 7.5 months old. At Croton Dam, N. Y., (1870; Kosendale 1 ; sand 2; broken stone 4.5. Some at the same work, and deposited under water, had 6 meas of stone; and at the end of a year had be- come so hard that it was found necessary to drill and blast a portion that had to be removed. I, ime with cement weakens all of them, but General Q. A. Gillmore, our best authority, repeatedly states that even in important concrete work in either the air or water, (pro- vided the water does not come into contact with it until setting takes place), from .25 to even .5 of the neat cement paste of the U. S. common cements may be replaced by lime paste without serious dimi- nution of either strength or hydraulicity ; and with decided economy. It retards the setting which is often of great advantage, especially with quick setting cements which at times cannot on that ac- count be advantageously used without some lime. Moulded blocks of Portland concrete of even 50 tons wt can generally be handled and removed to their places in from 1 to 2 weeks. Ramming of concrete, when properly done, consolidates the mass about 6 or 6 per ct, rendering it less porous, and very materially stronger. The rammers are like those used in street paving, of wood, about 4 ft long, 6 to 8 ins diam at foot with a lifting handle, and shod with iron; weight about 35 ftts. They are let fall 6 or 8 ins. The men using them, if standing on the concrete, should wear india-rubber boots to preserve their feet from corrosion by the cement. Ramming- cannot be done under water, except partially, when the concrete is enclosed in bags. A rake may, however, be used gently for levelling concrete under water. Blake's Stone Crusher (Co, New Haven, Connecticut), is useful for breaking the stone more cheaply than by hand on a large work. The two sizes best adapted to this purpose cost about $900 and $1300 ; break 6 to 7 cub yds per hour ; and require steam-engines of about 8 to 10 horse power to run them properly. According to Mr. J. J. R. Croes, C. E. (see " Construction of Croton Dam," Trans. Am. Soc. C. E., Feb. 1875), a machine will require about as follows: 1 engine man, 1 or 2 men to break the larger stones to a size that will enter the machine, 1 driver to horse-cart, 1 man to feed the stone into the machine, 2 to keep him supplied with stone, 1 at the screen, 2 wheeling away the broken stone to the stone-heap, 1 or 2 to receive it at the heap. Say 10 or 12 men in all. The size of the broken stone for concrete is gen- erally specified not to exceed about 2 ins on any edge ; but if it is well freed from dust by screening or washing, all sizes from .5 to 4 ins on any edge may be used, care being taken that the other ingredi- ents completely fill the voids. The common (not Portland) cements, when used as mortar for brickwork, often disfigure it, especi- ally near sea coasts, and in damp climates, by white efflorescences which sometimes spread over the entire exposed face of the work, and also injure the bricks. This also occurs in stone masonry, but to a much less extent, and is confined to the mortar joints; and injures only porous stone. It is usually a hydrous carbonate of soda or of potash often containing other salts. Gen'l Gillmore recommends as a preventive to add to every 300 Its (1 barrel) of the cement powder, 100 fts of quicklime, and from 8 to 12 Tbs of any cheap animal fat. The fat to be well incorporated with the quicklime before slacking it preparatory to adding it to the cement. This addition will re- tard the setting, and somewhat diminish the strength of the cement. It is also said by others that linseed oil at the rate of 2 gallons to 300 fts of dry cement, either with or without lime, will in all exposures prevent efflorescence , but like the fat it greatly retards setting, and weakens. 506 CEMENTS, CONCRETE, ETC. Concrete is good for bringing; up an uneven foundation to a level before starting the masonry. By this means the number of horizontal joints in the masonry is equalized, and unequal settlement is thereby prevented. Concrete may readily be deposited under water in the usual way of lowering it, soon after it is mixed, in a V shaped box of wood or plate-iron, with a lid that may be closed while the box descends. The lid however is often omitted. This box is BO arranged that on reaching bottom a pin may be drawn out by a string reaching to the surface, thus permitting one of the sloping sides to swing open below, and allow the concrete to fall out. The box is then raised to be refilled. In large works the box may contain a cubic yard or more, and should be sus- pended from a travelling crane, by which it can readily be brought over any required spot in the work. The concrete may if necessary be genily levelled by a rake soon after it leaves the box. Its consistency and strength will of course be impaired bv falling through the water from the box : and moreover it cannot be rammed under water without still greater injury. Still, if good it will in due time become sufficiently strong for all engineering purposes. Concrete has been safely deposited in the above manner in depths of 50 ft. The Treniie, sometimes used for depositing concrete under water, is a box of wood or of plate iron, round or square, and open at top and bottom ; and of a length suited to the depth of water. It may be about 18 ins diam. Its top, which is always kept above water, is hopper- shaped, for receiving the concrete more readily. It is moved laterally and vertically by a travelling crane or other device suited to the case. Its lower end rests on the river bottom, or on the deposited concrete. In commencing operations, its lower end resting on the river bottom, it is first entirely filled with concrete, which (to prevent its being washed to pieces by falling through the water in the tremie) is lowered in a cylindrical tub, with a bottom somewhat like the box before described, which can be opened when it arrives at its proper place. After being filled it in kept so by throwing fresh concrete into the hopper to supply the place of that which gradually falls out below, 'as the tremie is lifted a little to allow it to do so. The wt of the filled tremie compacts the concrete as it is deposited. A tremie had better widen out downwards, to allow the concrete to fall ont more readily. See " Gill- more on Cements." The area upon which it is deposited must previously be surrounded by some kind of enclosure, to prevent the concrete from spreading beyond its proper limits ; and to serve as a mould to give it its intended shape. This enclosure must be so strong that its sides may not be bulged outwards by the weight of the concrete. It will usually be a close crib of timber oV plate-iron without a bottom : and will remain after the work is done. If of timber it may require an outer row of cells, to be filled with stone or gravel for sinking it into place. Care must be taken to prevent the escape of the concrete through open spaces under the sides of the crib or enclosure. To this end the crib may be scribed to suit the inequalities of the bottom when the latter cannot readily be levelled off. Or iuside sheet piles will be better in some cases; or an outer or inner broad flap of tarpaulin may be fastened all around the lower edge of the crib, and be weighted with stone or gravel to keep it in place on the bottom. Broken stone or gravel or even earth (the last two where there is no current) heaped up outside of a weak crib will prevent the bulging outwards of its sides by the pressure of the concrete. After the concrete has been carried up to within some feet of low water, and levelled off, the masonry may be started upon it by means of a caisson (page 316) ; or by men in diving dresses. Or if the concrete reaches very nearly to low water, a first deep course of stone may be laid and the work thus brought at once above low water without any such aids. The concrete should extend out from 2 to 5 feet (according to the case) beyond the base of the masonry. All soft mild should be removed before depositing concrete. BjlJJS partly 11 lied With Concrete, and merely thrown into the water may be useful in certain cases. If the texture of the bags is slightly open, a portion of the cement will ooze out, and bind the whole into a tolerably compact mass. Such bags, by the aid of divers, may be employed for stopping leaks, underpinning, and various other purposes, that may suggest themselves. Such bags may be rammed to some extent. Tarpaulin may be spread over deep seams in rock to prevent the loss of concrete ; and in some cases, to prevent it from being washed away by springs. There is much room for judgment in the various applications of concrete; especially under water. Concrete has been used in very large masses; as in the founda- tion of a graving dock at Toulon, France; where it was deposited to a thickness of 15 feet, over an area of 400 ft by 100 ft ; forming, as it were, a single artificial stone of that size. It was deposited under water; an immense mould having first been prepared for its reception, by enclosing the area with close piling, 1 ned inside with tarpaulin. On top of this foundation were similarly built, likewise under water, the sides of the dock ; inside of great boxes, or enclosures of timber, conforming to the shape. The last deposits of concrete were then faced with masonry. Walls of buildings are also fre- quently built of cement concrete deposited between planks as a mould ; and which are moved upward as the building goes on. Flues may be made in these walls by ramming concrete around a tube, which can afterwards be lifted out; and be used for the next course above. The dome of the Pantheon, at Rome, 142 ft diam, and now nearly 2000 years old, is of concrete. The R. R. viaducts Pont Napoleon, and Pont d'Alma, at Paris, have arches of 115 and 141 feet span, of concrete. With regard to the mixing of concrete, Gen Gillrnore gives the method pursued and described by Lieut Wright. The gravel and pebbles being first separated by screening, into different sizes, " the concrete was prepared by spreading out the gravel on a platform of rough boards, in a layer from 8 to 12 ins thick ; the smaller pebbles at the bottom, and the larger ones on top. The mortar was then spread over the gravel as uniformly as possible. The materials were theu mixed by 4 men : 2 with shovels, and 2 with hoes ; the former facing each other, and always working from the outside of the heap, to the center. They then went back to the outside, and re- peated this operation, until the whole mass was turned. The men with hoes worked each in conjunc- tion with a shoveller, and were required to rub each shovelful well into the mortar, as it was turned and spread, or rather scattered on the platform by a jerking motion. The heap was turned over a second time, in the same manner, but in the opposite .direction ; and the ingredients were thus thor- oughly incorporated ; the surface of every pebble being well covered with mortar. Two turnings usually sufficed, and the concrete was then carried to the foundation in which it was to be used. The uccess of the operation depends, however, entirely upon the proper management of the hoe and CONCRETE, ETC. 507 shovel ; and althougte'fhig may be easily learned by the laborer, yet he seldom acquires it without the particular attention of the overseer." It is bard work. Or simple machinery is sometimes employed for incorporat- ing the ingredients of concrete, when large quantities are required. A machine that has been much used successfully in Germany, consists simply of a cylinder about 13 ft long, and 4 ft diam, open at both euds ; and lined on the inside, which is perfectly smooth, with sheet iron. It is inclined 6 or 8 degrees with the horizon. This cylinder is made to revolve 15 or 20 times per min, by means of a simple leather strap or band arouud its outside; and to which motion is given by a locomotive, which at the same time worked a heavy mill for mixing the mortar. This simple machine easily turns out from 105 to 130 cub yds of concrete in 10 hours ; and when worked in connection with a mortar mill, at a trifling expense." " When concrete is deposited in water, especially in the sea, a pulpy gelatinous fluid exudes from the cement, and rises to the surface. This causes the water to assume a milky hue; hence the term laltance, which Preach engineers apply to this substance. As it sets very imperfectly, and with some varieties* of cements scarcely at all, its interposition between the layers of concrete, even in moderate quantities, will have a tendency to lessen, more or less sensibly, the continuity and strength of the mass. It is usually removed from the enclosed space by pumps. Its proportion is greatly diminished by reducing the area of concrete exposed to the water, by using large, boxes, say from 1 to \}4 cub yds capacity, for immersing the concrete." Weight of good concrete 130 to 160 Ibs per cub ft, dry. Cost of concrete $5 to $9 per cub yard if roughly deposited ; and $9 to $15 if first made into blocks; depending on size, cement, locality, wages, &c. M. F. Colonel's beton. The artificial stone which bears this engineer's name has for several years been used in France with perfect success, not only for dwellings, depots, large city sewers, &c, but for the piers and arches of bridges, light-houses, &c. Bridge arches of 116 ft span, and of low rise, have been built of it. It is composed of 5 measures of sand, 1 of sifted dry-slaked lime, and from ^ to % measure of ground Portland cement. Or of sand 6, cement 1, lime %; &c. These are first well mixed together dry, and then placed in a mixing-mill ; at the same time sprinkling them with .3 to A measure of water, so as to moisten them slightly, without wetting them. They are then thoroughly incorporated by mixing, until they form a stiff pasty mass, slightly coherent. This is then placed in a mould, in successive thin layers, each of which is well compacted by blows of a 16 B) rammer. The top of each layer may be scored or cross-cut, to make the next one unite better with it. Owing to the small proportion of water, it sets soon; and may generally be taken from the mould in from a few hours to a few days, depending on the size of the block ; and left to harden. River sand is the best, inasmuch as it requires less lime and cement than.pit sand, to make equally good stone.(?) The cement should be a rather slow-setting one; and both it and the lime should be screened, to exclude lumps. About \% bushels, or \% cub ft of the dry materials, make 1 cubic foot of finished stone, weighing about 140 Ibs; resisting 100 to 150 tons per square foot at o months old. 250 to 400 in 2 years. Arches of it are made no thicker than brick ones. An arch, pier, wall, foundation, &c, may be built of it, as one stone, instead of in separate blocks. In sewers the centers may be struck within 10 to 15 hours after the arch is finished ; and the water may be admitted within a week or less. The distinctive features of Coignet's beton are: the very small proportion of water; the thorough incorporation of the ingredients; and the consolidation of the separate layers by ramming. It is difficult for a person who has never seen the process, to credit the rapidity, facility, and economy with which blocks of good stone can be made by it. Its cost, as compared with perfectly plain dressed granite, does not exceed one-half; while for ornamental work it compares even far more favorably. Hence the Coignet beton, or artifi- cial stone, is nothing more than good, well-prepared mortar, mixed with very little water : and well rammed into moulds, in successive layers. A mixture of 1 measure of hydraulic cement, and 3 measures of sand, similarly treated, has been successfully used in the U/S., for some years, in building* of all kinds. Ornamental work can be furnished at J4 the price of stone ; and will answer equally well. F'or full information, see Gillmore's " Coignet Beton." Both Ransomes, (England,) and the Sorel (Boston, Mass,) artificial stones are too expensive for general engineering or architectural purposes. Transverse Strength of Concrete Beams.* Average results of 24 beams, 10 ins square, made of good Rosendale and English Portland cements, pit sand and screened pebbles, few exceeding 1 inch diam. The beams were buried for 6 months, in a pit 4 ft deep, in gravelly soil, exposed to the rain, snow, &c. A first set of beams all broke on being taken from the moulds after 7 or 8 days, although carefully handled. To avoid this, the bottom of the pit itself was rammed to a smooth, hard surface : immediately upon which a new set was made by ramming the concrete into 2 inch planed plank moulds without bottoms. The moulds were removed after 24 hours, and when all were done the earth was filled in over the undisturbed beams. Very little of the soil adhered to them. Their wt in all cases when tested was about 150 Ibs per cub ft, or 520 Ibs wt of 5 ft clear span of beam; one half of which, or 260 Ibs, must be deducted from the cen breakg loads of the 5 ft spans below; and 124 Ibs from the 2 ft 4.5 ins ones, as explained p 183. The coefficient or Constant C is the cen breakg load in Ibs for a beam 1 inch square, and 1 ft clear span, like those in table p 185 ; and like them is found by the formula at top of p 184. Its use is shown by the formula at foot of p 184. -v- Both these useful tables (the only ones we know of on the subject) were kindly furnished us by Eliot O. Clarke, Esq.. Principal Assistant iu charge of the Improved Sewerage Works of Boston, Mass. , for which the experiments were made. 508 CEMENT, CONCRETE, ETC. Proportions of mate- rials by measure. Center Breaking- load in Ibs, including half wt of beam. Constant c. Cement. Sand. Pebbles. Span 2 ft 4.5 ins. Span 5 ft. Rosendale 1 2 5 1782 690 3.7 " 1 3 7 all broke in handling Portland 1 3 7 3926 1995 9,8 1 4 9 3648 8.1 *' 1 6 11 2822 1190 6.2 Tensile Strength of Cement Mortars, of medium coarse sea-beach sand, and good Rosendale, and English Portland ce- ments; being averages of about 25000 experiments in the years 1878 to 1882. The area of breaking section was 2.25 sq ins. The proportions of sand and cement were by measure. The mortar was rammed into the moulds, and the specimens were immersed in water as soon as they would bear handling, and so remained for 1 day, or 1 week, or for 1, 6, or 12 months. The strengths are in Ibs per sq iiicn. Rosendale. Neat. Cement 1. Sand 1. Cement 1. Sand 1.5. ID. 1W. 1M. 6M. 1Y. 1W. IMi 6M. 1 Y. 1W. 1M. 6M. 1Y. 71 92 145 282 290 56 116 180 236 41 90 135 210 Cement 1. Sand 2. Cement 1. Sand 3. Cement 1. Sand 5. | 22 49 105 1 169 12 25 65 121 10 36 80 Portland. Neat. Cement 1. Sand 1. Cement 1. Sand 1.5. 102 303 412 | 468 494 160 225 347 387 Cement 1. Sand 2. Cement 1. Sand 3. Cement 1. Sand 5. 126 163 279 ooo oZo 95 130 198 257 55 78 116 145 'PLASTERING. 509 PLASTERING. THE plastering of the inside walls of buildings, whether done on laths, bricks, or stone, generally consists of three separate coats of mortar. The first of these is called by workmen the rough or scratch coat; arid consists of about 1 measure of quicklime, to 4 of sand ; (which latter need not be of the purest kind;) and % measure of bul- lock or horse hair; the last of which is for making the mortar more cohesive, and less liable to split off in spots. This coat is about % to ^ inch thick ; is put on roughly ; and should be pressed by the trowel with sufficient force to enter perfectly between and behind the laths; which for facilitating this should not be nailed nearer together than ^ an inch. In rude buildings, or in cellars, Ac, this is often the only coat used. When this first coat has been left for one or more days, accord- ing to the dry ness of the air, to dry slightly, it is roughly scored, or scratched, (hence its name,) with a pointed stick, or a lath, nearly through its thickness, by lines run- ning diagonally across each other, and about 2 to 4 ins apart. This gives a better hold to the second coat, which might otherwise peel off. If the first coat has be- come too dry, it is well also to dampen it slightly as the second one is put on. The second coat is put on about % to % inch thick, of the same hair mortar, or coarse, stuff. Before it becomes hard, it is roughed over by a hickory broom, or some substitute, to make the third coat adhere to it better. The third coat, about % inch thick, contains no hair; and forgiving it a still whiter and neater appearance, more lime is used, say 1 of lime, to 2 of sand ; and the purest sand is used. This mortar is by plasterers called stucco; a name also applied to mortar when used for plastering the outsides of buildings. Or in- stead of stucco, the third coat may be, and usually is, of hard finish, or gauge stuff; which consists of 1 measure of ground plaster of Paris, to about 2 of quicklime, without sand. Hard finish works easier; but is not as good as stucco, for walls in- tended to be painted in oil. The plaster of Paris is for hastening the hardening. Either of these third coats is smoothed or polished to a greater or less extent, according to whether it is to show, or to be papered, painted, &c. The polishing tools are merely, the trowel ; the hand- lioat, (a kind of wooden trowel ;) and the water-brush, (a short-handled brush for wetting the surface part at a time with water, in order to polish more freely.) For finer polishing, a float made of cork is used. The smooth piece of board about 10 to 12 ins square, with a handle beneath, on which the plasterer holds his mortar until he pUts it on to the wall with his trowel, is called a hawk. The more thoroughly each coat is gone over with the water-brush and trowel, (which process is called hand-floating,) the firmer and stronger will it be. Frequently only two coats of plastering are put on in inferior rooms ; or where great neatness of appearance is not needed. The first is of hair mortar, or coarse stuff; this is scratched with the broom, and then covered by the finishing coat of finer mortar, (stucco.) If this last is nearly all lime, or with but very little sand, to make it work easier, it is called a slipped coat. Without any sand it is called ftne stuff. Neither is as good as stucco, if the wall is to be papered. When this is the case, the third coat also may have a little hair, to give it more strength ; but this is not absolutely necessary. A very good effect may be produced in station-houses, churches, &c, by only two coats of plaster in which fine clean screened gravel is used instead of sand. When lined into regular courses, it resem- bles a buff-colored sandstone, very agreeable to the eye. In purchasing plastering hair, care must be taken that it has not been taken from salted hides; inasmuch as the salt will make the walls damp. For the same cause sea-shore sand should not be used. It is almost impossible to wash it entirely free from salt. In brick walls intended to be plastered, the mortar joints should be left very rough, to let the plas- ter adhere. If it is put on smooth walls, without first raking out the mortar to the depth of nearly an inch, it is very apt to fall off; especially from outside walls ; as can be seen daily in any of our cities. As this raking out of brick joints is tedious and expensive, it would generally be better to use paint rather than plaster. The walls should also be washed clean from all dust; and should be slightly dampened as the plaster is put on. To imitate granite on outer walls : after the second or smooth coat of plaster is dry, it receives a coat of lime wash, slightly tinted by a little umber, or ochre, Ac. After this is dry, in case it appears too dark, or too light, another may be applied with more or less of the coloring matter in it. Finally, a wash of lime and mineral-black is sprinkled on from a flat brush, to imitate the black specks of granite. By this simple means, a skilful workman can produce excellent imitations. The horizontal and vertical joints of the imitation masonry, may be ruled in by a small brush, using the same black wash, and a long straight-edge. The rough surfaces of all walls are more or less warped, or out of line ; and it is not possible for the plasterer to rectify this perfectly by eye, as may be seen in almost every house. Even in what are called first-class ones, a quick eye can generally detect unsightly undulations of the plastered urfaces. To prevent this, the process of screed ing: is resorted to. Screeds are a kind of gauge or guide, formed by applying to the first rough coat, when partly dried, horizontal strips of the plastering mortar, about 8 ins wide, and from 2 to 4 ft apart all around the room- These are made to project from the first coat, out to the intended face of the second one; anrtarwhile soft are carefully made perfectly straight, and out of wind with each other, by means of the plumb-line, straight-edge. &c. When they become dry, the second coat is put on, filling up the broad horizontal spaces between them ; and is readily brought to a perfectly flat surface, corresponding with that of the screeds, by means of long straight-edges extending over two or more of the latter. A day's work at plastering:. A plasterer, aided by one or two laborers to mix his mortar, and to keep his hawk supplied, can average from 100 to 200 sanare yards a day, of first coat ; about % as much of second ; and half as 33 510 SLATING. msh of third, which requires more care. The amount will depend upon the number of angles, sir* of rooms, whether on ceilings or on walls, &c, &c. Gen Gillmorc's estimate of cost of plastering-* 100 square yards with 2 or with 3 coats. Common labor $1 per day. Materials. Three Coats. Hard finished work. Two Coats.. Slipped coat finish. Quicklime 4 casks. 8 - 2000 4 bushels. 7 loads.* 2^ bushels. 13 Ibs. 4 days. 3 days. $4.00 .85 .70 4.00 .80 2.00 .25 .90 7.00 3.00 2.00 3Ji casks. 2000. 3 bushels. 6 loads. 13 Ibs. 3J^ days. 2 days. $3.33 4.00 .60 1.80 .90 6.1'2 2.00 1.20 $19.95 " for fine stuff Plaster of Paris Laths Hair Common Sand White Sand Nails Mason's labor . . Cartage Cost of 100 square vards $25.50 This amounts to 25 V6 cts per sq yd for 3 coats ; and say 20 cts for 2 coats, gee Art 5, P 500. Plastering laths are usually of split white or yellow pine, in lengths of about 3 to 4 ft long ; and hence called 3. or 4 ft laths. They are about 1^ ins wide, by % inch thick. They are nailed up horizontally, about % inch apart. The upright studs of partitions are spaced a* such distances apart, (generally about 15 ins from center to center,) that the ends of the laths may be nailed to them. Laths are sold by the bundle of 1000 each. A square foot of surface requires 1 & four feet laths ; or 1000 such laths will cover 666 sq ft. Sawed laths may be had to order, ot any re- quired length. A carpenter can nail up the laths for from 40 to 60 q yds of plastering in a day of 10 hours ; depending on the number of angles in the rooms, &c. * A load (one-horse), both in the U. S. and in England, usually means a cub yd; but many dealers adopt 20 struck bushels 25 cub ft fully a ton. SLATING. ROOFING slates are usually from ^ to *< inch thick; about -^ being a commo* average. They may be nailed either to a sheeting of rough boards (c, #, in the fig) from % to 1J4 inch thick, (which should be, but rarely are, tongued and grooved,) * Average prices of plastering in Philada, 1873, in cts per sq yard. Three coats, including laths, scaffold &c, 50 to 55 cts. Two coats 35 to 40. Three coats on brick or stone (no laths reqd,) 50 to 55. Outside plastering, 60; or if to imitate marble, 75. Simple plaster cornices, 1 to 2 cts per inch of girth, per ft run. Plaster center flowers for parlors, $5 to $15 each, put up. The plastering of a 20 ft front, 3 story dwelling, with large 3 story back buildings, 81000 to $1300. Stipulate expressly to pay only for surfaces actually plastered : and thu.- avoid extras, even if vou have to pay a few cts more per yard. SLATING. 511 laid horizontally x firt5m rafter to rafter; or sloping, from purlin to purlin, as tho case may be ; or to^stout laths t tt about 2 to 3 ins wide, and from 1 to 1^ thick, nailed to the rafters at distances apart to suit the gauge of the slates. Two nails are used to each sb*e ; one near each upper corner. They may be either of copper, (which is the most durable, but most expensive,) of zinc, or of either galvanized or tinned iron. The last two are generally used ; or in inferior work, merely plain iron ones, pre- viously boiled in linseed oil, as a partial preservative from rust. Rust, however, sometimes weakens them so much that they break ; and the slates are blown off in high winds, to the danger of passers by. Since good slate endures for along series of years, it is true economy to use nails that are equally durable. In iron roofs, the slates, instead .of being nailed to boards, are sometimes tied directly to the iron purlins, by wire. A square of slating, shingling, &c, is 100 sq ft. In laboratories, chemical factories, Ac, subject to acid fumes, it is difficult to provide a metal fastening that will not be eaten away. In such cases it is best to depend chiefly upon a layer of mortar between the slates. This will harden before the metal fastenings give way ; and will hold the slates in place, while new fastenings are being inserted. The least pitch considered advisable for a roof, to prevent rain or snow from being driven through the interstices between the slates, is about 26^ ; or 1 vert to 2 hor ; which corresponds to a rise of y the span in a common double pitched roof. Bat even at steeper pitches, rain, and more particularly snow, will be forced through the roof by violent winds ; especially if laths alone be used, or even boarding alone. To avoid this, a layer of mortar about y inch thick, may be spread over the touching surfaces of the slates if on laths. If on boards, the same process may be adopted ; or the.more common one of first covering the boards with a layer of what is called slating felt; hut which in reality is merely thick brown paper, soaked in tar. This is sold in long continuous rolls, 28 ins wide, and weighing from 40 to 50 fts. A 50 ft roll will cover about 300 sq ft of roof. With proper precautions against the admission of rain and snow, a pitch as flat as 1 in 2^, or even 1 in 8, may be adopted. The thickness of slate on a roof Is double; except at the laps is, is, &c, where it is triple. The lap is measured from the nail hole (under t) of the lower slate, to the lower edge or tail, s, of the pper one ; and is usually about 3 ins. In order that the showing lower edges of the slates shall, when laid, form regular straight lines along the roof, the nail holes are made at equal distances from said lower edges ; so that any irregularity of length is concealed from view at the hidden heads of the slates. The slater estimates the length of his slate from the nail hole to the tail discarding the narrow strip between the nail hole and the head. If from this reduced length the lap be deducted, then one-half of the remainder will be the gauge, weathering, or margin, of the slating; or, in other words, the showing or exposed width of the courses of Blates. The gauge in ins multiplied by the width of a slate in ins, gives the area in sq ins of finished roof covered by a single slate ; and if 144 (the sq ins in a sq footl be divided by this area, the quotient will be the number of slates required per sq ft of roof. The upper side of a shite is called its back; the lower one, its bed. Slating, like shingling, must evidently be commenced at the eaves, and extended upward. Since the beds of the slates are not exactly parallel to the boarding, and consequently do not rest flat upon It, those at the lower edge 10 would easily be broken. To prevent this, a tilting strip (a stout wide lath, with its upper side planed a little bevelling, to suit the slope of the slates) is first nailed around near the eaves, for the tails of the lowest course of slates to rest on. This is shown on a larger scale atT. Slate of the best quality has a glistening semi-metallic appearance, somewhat like that of a surface of paper rubbed with black-lead pencil. That of a dull earthy aspect, is softer, more absorbent, and consequently more liable to yield to atmospheric influences, rain, frost, &c. Iron pyrites frequently occurs in slate; and since it always decomposes and leaves holes, should never be ad'mitted on a roof. Of two qualities of slate, that which absorbs the least weight of water, when pieces of equal size are soaked for an hour or two, is eenerallv the best; being least liable to split by frost, and become weather-worn. This tst is easily applied. In England the different sizes are distinguished by absurd names of no meaning. In the United States they are called 6 by 12's ; 16 by 24's, &c, according to their measures in inches. They may be cut to order, of almost any prescribed dimensions, or shape. Those in common use vary from about 7 by 14, to 12 by 18. The first forms about 5 to 6 inch courses; and the last about 7 to 8 inch; depending upon how" far from the head the nail holes are pierced. The farther this is, the firmer will the slating be. Slate roofs, like iron ones, heat the rooms immediately below them very much. This is somewhat diminished when the slates are on boards, instead of laths; and still niore bv a coat of plaster be- neath. They are also liable to break when walked on ; less so when bedded in mortar. Weight of slate roofs. Slate weighs about 175 fts per cub foot; therefore, a sq ft, H inch thick, weighs about 1.8 Ibs; y 3 ^, 2.7 fts; and % thick, 3.6 fts. But owing to the overlapping, a square foot of roof requires about 2^ sq ft of slate of ordinary sizes ; and if the slate is laid on boards an inch thick, tae weight per sq ft of roof will be increased about 2% fts ; or with \y 4 inch boards, 2.8 fts. Laths will weigh about % ft per sq ft of roof. Hence, Appro* Weight of one Rq ft of Slating, in Ibs. Slate Hinch. thick on laths 4.75 " " on 1 inch boards 6.75 . " 3-16 " on laths 7.00 on 1 inch boards 9.00 " " " onlVf" " 9.55 " J4 " on laths 9.25 " " " on 1 inch boards 11.25 onlJ4" " 11.80 If slating felt is used, add ^ ft ; or if the slates are bedded in } inch of mortar, add 3 tbs. 512 SHINGLES. For the total weight borne by the roof trusses, that of the purlins also must be added. This wil) not vary much from the limits of 1^' to 3 Ibs per sq ft in roofs of moderate span. Add tor wind and snow, say 20 fts per sq ft; * and finally add the weight of the truss itself. For stopping* the joints between slates (or shingles, &c) and chimneys, dormer windows, &c, a mixture of stirt' white lead paint, as sold by the keg, with sand enough to pre- vent it from running, is very good; especially if protected by a covering of strips of lead, or copper, tin, &c. nailed to the mortar-joints of the chimneys, after being bent m, as to enter said joints ; which should be scraped out for an inch in depth, and afterward refilled. Mortar protected in the same way, or even unprotected, is often used for the purpose ; but is not equal to the paint and sand. Mor- tar a few days old, (to allow refractory particles cf lime to alack,) mixed with blacksmith's cinder* and molasses, is much used for this purpose; and becomes very hard, and effective. SHINGLES, WHITE cedar shingles are the best in use ; and when of good quality will last 40 or 50 years in our Northern States. They are usually 27 ins long; by from 6 to 7 ins wide ; about ^ inch thick at upper end ; and about % at lower end or butt ; and are laid in courses about 8% ins wide ; so that not quite ^ of a shingle is exposed to the weather. They ar usually laid in three thicknesses ; except for an inch or two at the upper ends, where there are four. Thev are nailed to sawed shiugling-laths of oak or yellow pine; about 16 ft long; 2^ ins wide, and 1 inch thick ; placed in horizontal rows about 8% ins apart. These are nailed to the raft- ers, or purlins ; which, for laths of the foregoing size, should not be more than 2 ft apart from center to center. Two nails are used to each shingle, near its upper end. They should not be of less size than 400 to a ft. Those of wrought iron being the strongest, are the best; cut ones are apt to break by the warping of the shingles. Two pounds of such nails will suffice for 100 sq ft of roof, including w'aste. An average shingle 7^ ins wide, in 8}4 inch courses, exposes 63% sq ins ; making '2% shingles to a sq ft of roof; but to allow for waste, and narrow shingles, it is better in practice to allow about 3 shingles to a sq ft. Shingling, like slating, must plainly be begun at the eaves : and extended upward. For closing the joints between the shingles, and chimneys, dormer windows, &c, see at end of Slating. Cypress and white pine are also much used for shingles, being much cheaper, hut scarcely half as durable, t All shingles wear quite thin in time by rain and exposure. In warm damp climates they all decay within 6 to 12 years. PAINTING. THE principal material used in house-painting, is either white lead, or oxide of Bine, ground in raw (unboiled) linseed oil, by a mill, to the consistency of a thick paste. In this condition, it is sold by the manufacturers in kegs of 25, 50, and 100 ros. To prepare it for actual use, merely requires the addition of more linseed oil, say 3 or 4 pints to 10 Ibs of the keg paint, for thinning it sufficiently to flow readily under the brush. Good painting requires 4 or 5 coats ; but usually only 4 are used in principal rooms ; and 8 in inferior ones. Each coat must be allowed to dry perfectly before the next one is put on. One ft of the keg paint will, after being thinned, cover about 2 sq yds of first coat; 3 yds of second; and 4 yds of each subsequent coat ; or 1 sq yd of 3 coats will require in all, 1.08 fts ; of 4 coats, 1% fts ; of 5 coats, 1.58 fts. The reason why the first coats require so much more than the subsequent ones, is that the bare surface of the wood absorbs it more. When, as is usual, raw or unboiled oil is used for thinning, dryers must be added to it; otherwise the paint might require several weeks to harden; whereas, with dryers, from 1 to 3 days, according to the weather, suffice for each coat to become hard enough to receive the next one. The dryers most commonly used, are powdered litharge, in the proportion of one heaped teaspoonful ; or Japan var- nish, 1 table-spoonful, to 10 fts of the keg paint. Either sugar of lead, or sulphate of zinc, may also be used instead of litharge ; and in the same proportion. Although both litharge and Japan varnish are dark-colored, yet the quantity is so small as not to appreciably affect the whiteness of the paint. If the varnish is used in excess, as is often done in the hurry to have work finished, it produces cracks all over the surface. No dryer is necessary if painters' boiled oil be used for thinning. Mere boiling will not cause oil to harden more rapidly"; but that intended for painters, has litharge added to it previously to boiling , in the proportion of 1*4 fts to each 10 gallons of raw oil. In some works written for the use of house painters, it is asserted that boiling renders the oil too thick for any but coarse outdoor work. But this is entirely a mistake; for if the boiling be properly done, the oil will be quite thin enough tor the best inside work ; and will moreover be clearer than while raw ; and * Price of slate, felt, and slating in Philada in 1873, is from 12 to 14 cts per sq ft, according to quality of slate; kind of nails. &c ; but exclusive of boarding. With copper nails add 2 cts per sq ft. The slate from Peach Bottom, York Co, Penna, is the best in the State. It commands 2 or 3 cts per sq ft more than the others. A roof of leaded tin, will cost about the same as one of slate; and not. much more than half as much as good cedar shingles, (in Philada.) Felt about 4 cts per ft. f Price of shingles in Philada. in 1873 : Best cedar (a*out 7 to 8 ins wide, by 27 ins long,) $50 to $60. White pine or cypress, $40 to $50. Shingling laths, $3 to $4 per 1000. Cedar shingles, laths, nails, and shingling complete, 30 cts per sq ft of roof; or about twice aa much as slate or leaded tin roofing, -""-"" '-> -J-tijrt' 'PAINTING. 513 Will impart to the palufttffsui face a more shilling appearance. The heat should be barely sufficient to produce boilmglxor about t>UO Fan. The boiliug should continue about 1^ hours; the oil being thoroughly stirnwai short intervals, to prevent the litharge from settling at the bottom. The fire may then beutfiowed to subside; when the operation will be completed. A sedimeut will then form at the bottsfu ; which must be left behind when the oil is poured off. Although no dryer is necessary ' il, still a little litharge may be added when great expedition demands it. Painters rarely use tuis oil. on account of its trifling increase of cost. Another substance much used with the thinning oil, (except for the first coat,) is spirits of turpen- tine ; called " turp" by the workmen. The quantity of oil may be diminished, to the extent of the added turp. This being more fluid than oil, causes the paint to work more pleasantly under the brush. It moreover diminishes the tendency of the paint to become yellow ; especially in rooms kept closed for some time. It is also much cheaper than oil. It should not be used, or but sparingly, for exposed outdoor work; inasmuch as its tendency is to impair the tirmuess of the paint; and although its effects are scarcely appreciable indoors, they are quite apparent when the work has to resist the weather. As the fashions change in house-painting, the surface is at times required to present a shining or glossy finish : at other times a dead one is in vogue. Tbe glossy one is that which the paint will uuturaily have, provided that no more turp than oil be used in the thinning. The dead finish is obtained 'by using no oil, but turp alone, for the last coat; which in that case is called a flatting coat. Although turp is not properly a dryer, still, as it evaporates quickly, it facilitates the hardening of the paint. In outdoor work it is usually advisable to use more dryer than inside, so that the paint may sooner become hard enough not to be injured by dust or rain. Otherwise less would be better. When, instead of a white finish, one of seme other color is required, the coloring ingredient is mixed with the white paint to be used in the last coat only ; although two coloring coats are some- times found to be necessary before a satisfactory effect is produced. The coloring ingredients may be indigo, lampblack, terra sienna, umber, ochre, chrome yellow, Venetian red, red lead, &c, &c; which ara ground in oil, ready for sale, by the manufacturers of the white-lead and zinc paints. They are simply well stirred iuto the white paint. All surfaces to be painted, should first be thoroughly dry, and free from dust. If on wood, all plane-marks, and other slight irregularities, should first be smoothed off by sand-paper, when the neatest finish is required. Also, all heads of nails must be punched to about % inh below the sur- face. To prevent knots from showing through the finished work, (as those in white or yellow pine would do, on account of the contained turpentine,) thev must first be killed, as it is termed. A usual and effective way of doing this, is by covering them with two coats of shellac varnish ; which, when dry, should be smoothed by sand-paper. Another mode, not quite so certain, is by one or two coats of white lead mixed with thin glue- water, or size, as it is called. After these preparations, the first, or priming coat, is put on ; in which there should be no turp; because it would sink at once into the bare wood, leaving the white lead behind it, in a nearly dry friable condition. After this the nail holes, cracks, &c, must be filled with common glaziers' putty, made of whiting (fine clean washed chalk) and raw linseed oil ; boiled oil will not answer ; the putty would be friable. The puny would be apt to fall out, if put in before priming, because the wood would absorb the oil, and the putty would then shrink. After the first coat is perfectly dry, the second one is put on ; and for it about 1 measure of turp may be mixed with 3 measures of the thin- ning oil. In the third, and any subsequent coats, equal measures of turp and oil, may be used for thinning, if the work is required to dry with a gloss ; but if it is to finish dead, the last coat must be a flatting one; or one in which the "thinning oil is entirely omitted, and turp alone substituted for it. Painters generally clean their brushes by merely pressing out most of the paint with a knife ; and then keep them in water until further use. If to be put away for some time, they may be thoroughly cleaned by turp; or by soap and water. To prevent a hard skin from forming on the top of their paint when not used for some days, they pour on a little oil. The best paints for preserving? iron exposed to the weather, appear to be pulverized oxides of iron, such as yellow and red iron ochres; or brown hematite iron ores finely ground; and simply mixed with linseed oil. and a dryer. White lead applied directly to the iron, requires incessant renewal : and indeed probably exerts a corrosive effect. It may. how- ever, be applied over the more durable colors, when appearance requires it. Red lead is said to be very durable, when pure. An instance is recorded of pump-rods, in a well 200 ft deep, near London, which, having first been thus painted, were in use for 45 years; and at the expiration of that time, their weight was found to be precisely the same as when new; thus showing that rust had not affected them. When the size of the exposed iron admits of it, its freedom from rust may be very much promoted by first heating it thoroughly; and then dipping it into, or washing it well with, hot linseed oil; which will then penetrate into the interior of the iron. For tinned iron exposed to the weather, on roofs, rain pipes, Ac, Spanish brown is a very durable color. The tin is frequently found perfectly bright and protected, when this color has been used, after an exposure of 40 or 50 years. White paint washes off in a few years by rain. Plastered walls should if possible be all wed to dry for at least a year, before being painted in oil ; otherwise the paint will be liable to blister. They may, if preferred, be frescoed (water-colors, mixed with size) to the desired tint during the interval. The painting of unseasoned wood hastens its decay. If the surface to be painted is greasy, the grease must first be removed by water in which is dissolved some lime. Washes for outside work. Downing, in his work on country houses, recommends the following: For wood-work; in a tight bushel, slack half a bushel of fresh lime, by pouring over it boiling water sufficient to cover it 4 or 5 ins deep; stirring it until slacked. Add 2 tts of sulphate of zinc (white vitriol) dissolved in water. Add water enough to bring all to the con- sistence of thick whitewash. Apply with a whitewash brush. This wash is white; but it may b colored by adding powdered ochre Indian rod, umber, &c. If lampblack is added to water colors, U * Average cost of Painting* in Philada, 1873, including scaffold &c, per square yard. Four coats in plain colors 40 cts ; 3 coats, 35. Graining in imitation of oak, walnut &c, 90. White lead ground in oil, in kees 15 cts per Ib. The cost of painting and glazing a 20 ft front, 3 story dwelling, with large 3 story back buildings, $600 to $700. A church of 60 by 80 ft, with base- meat story, and galleries, $2500 to $3000. Avoid extras ; or stipulate for them in advance. 514 GLASS, AND GLAZING. should first be thoroughly dissolved in alcohol. The sulphate of zin iu a tew weeks. lauses the wash to become hard For brick, masonry, or rougrh-cast. Slack ^ a bushel of lime as before; then flli the barret % full of water, and add a bushel of hydraulic- cement. Add 3 fts of sul- phate of ziuc. previously dissolved in water. The whole should be of the thickness of paint; and may be put on with a whitewash brush. The wash is improved by stirring iu a peck of white sand, just before using it. It may be colored, if desired, like the preceding. He also gives the following cheap oil-paint for outside work on wood, brick, stone, &c ; and says it becomes far harder and more durable than common paint: One measure of ground fresh quicklime; add the same quantity of fine white sand, or fine coal ashes: and twice as much fresh wood ashes; all the foregoing to be passed through a tine sieve. Mix well together dry. Mix with as much raw linseed oil as will make the mixture as thin as paint. Apply with a painter's brush. It may be col- ored like the foregoing, taking care to mix the colors well with oil before adding them. It is best t Also, another, said to stand 15 to '20 years : 50 fts best white lead : 10 quarts raw linseed oil : % fl> dryer; 50 ftis finely sifted sharp clean sand; 2 tt>s raw umber. Add very little, say % pint of tur- pentine. Apply with a large brush. Coment for stopping* joints, such as around chimneys, &c, &c. White lead ground in oil, as sold by the keg;" mixed with enough pure sand to make a stiff paste that will not run. It grows hard by exposure, and resists heat, cold, and water. Pieces of stone may be strongly cemented together by it, allowing a few mouths for proper hardening. Whitewash lor inside work., according to Mr. Downing, "is made more fixed atid permanent, by adding 2 quarts of thin size to a pailful of the wash, just before using. The best size for this purpose is made of shreds of glove leather; but any clean size of good quality will answer." as thin glue-water. We will add, that the common practice of mixing salt with white- wash, should not be permitted. Paper pasted on a wall which has previously been covered with salt whitewash, is very apt to become wet. and loose, and to fall off during damp weather. The white- wash shnuld he scraped off. and the wall or partition covered with a coat or two of thin size, to pro- tect the paper from the effect of the salt that may still adhere to the plaster. GLASS, AND GLAZING. WINDOW glass is sold by the box ; and whatever may be the size of the panes, a box contains as nearly 50 sq feet of glass as the dimensions of the panes will admit of In the following table, those numbers which have no + after them, denote that they amount to precisely 50 sq ft; while in the others a part of a pane would have to be added to make up the 60 ft. Panes of any size may be made to order by the manufacturers. The sizes given In the following table, as well as many others, are generally to be had ready made. Ordinary window glass of all the sizes iu the table, is about one-sixteenth of an inch thick ; and this is the thickness supposed to be intended when a greater one is not specified. Double-thick glass is nearly ^ inch ; and its price is 50 per ct more than the single thick. It is of course much stronger than'the single. ** The panes are confined to the sash by glaziers' putty, made of whiting (powdered chalk) and raw linseed oil ; and by small triangular pieces of thin tin. about % inch on a side, which uphold the glass while the putty is being put on ; and are allowed to remain afterward, as a protection while the putty continues soft. TABLE OF XFMBERS OF PAXES IX A BOX. Size in ins. Panes to a 'box. Size in ins. Panes to a box. Size in ins. Panes to a box. Size in ins. Pane to a bo 6X8 6 X 10 6 X 12 150 120 100 11 X 12 11 X 14 11 X 16 54 4 46 - 40 - 14 X 22 14 X 24 14 X 26 23 + 21 + 19 + 18 X 28 18 X 30 18 X 33 14 4- 13 4 12 4 7X8 128 + 11 X 18 36 - 14 X 28 18 + 18 X 36 11 4 7 X 10 7 X 12 7 X 14 102 + 85 + 73 + 11 X 20 11 X 22 12 X 14 32 - 29 - 42 - 15 X 16 15 X 18 15 X 20 30 26 + 24 20 X 24 20 X 26 20 X 28 15 134 12 4 8 X 10 90 12 X 16 37 - 15 X 22 21 + 20 X 30 12 8 X 12 75 12 X 18 33 - 15 X 24 20 20 X 33 10 4; 8 X H 8 X 16 644- 53 + 12 X 20 12 X 22 30 27 - 15 X 26 15 X 28 18 + 17 + 20 X 36 20 X 40 10 9 9 X 10 80 12 X 24 25 - 15 X 30 16 22 X 26 12 - 9 X 12 (H + 13 X 14 39 - 16 X 18 25 22 X 30 10 - 9 X U 57 + 13 X 16 3* - 16 X 20 22 + 22 X 36 9 - 9 < n 50 13 X 18 30 - 16 X 22 20 + 22 X 40 8 - 9 X 18 4t + :j X 20 27 - 16 X 24 18 + 22 X 44 7 - T) X 12 60 : X 22 25 - 1H X 28 16 + 24 X 27 n - li) X It 51 + 3 X 24 23 - 16 X 32 14 + 24 X 30 10 1') X 15 48 :', X 26 21 - 18 X 20 20 2t X 34 8 - li) X 18 45 4 X 16 32 - 18 X 22 18 + 24 X 38 7 - 10 X 13 40 4 X 18 28 - 18 X 24 16 + 24 X 42 ; 7 - 1J X 20 36 4 X 20 25 - 18 X 26 15 + 24 X 43 6 - WATER. 515 The best qualities ofXmerican glass made in the vicinity of Philadelphia, Boston, Plttsbnrg, Ac, ar for most purelyjrfse/wZ purposes, as good as those from foreign countries: but when the highest degree of &euus per sq inch. Boston rods by author, 3500 to 5200. Crushing strength, 6000 to 10000 fts per sq inch. Transversely, (by the writer's trials,) Millville, N. J. r flooring glass, 1 inch square, and 1 foot between the end supports, breaks under a center load of about 170 fts ; consequent.y, it is considerably stronger than granite, except as regards crushing; in which the two are about equal. REMARK. Windo.w and other glass which contains an excess of potash or of soda is very liable to become dull ia time, owing to the decomposition of those ingredients by atmospheric Influences. WATEE, PURE water, as boiled and distilled, is composed of the two gases, hydrogen and oxygen ; in the proportions of 2 measures hyd, to 1 of ox; or 1 weight of hyd, to 8 of ox. Ordinarily, however, it contains several foreign ingredients, as carbonic, and other acids; and soluble mineral, or organic substances. "When it contains much lime, it is said to be hard ; and will not make a good lather with soap. The air in its ordinary state contains about 4 grains of water per cub ft. The average pressure of the air at sea level, will balance a column of water of 34 ft In vert height, or about 30 ins of mercury. At its boiling point of 212 Fan, its bulk Is about ^ 3 greater than at 70. Us weight per cub ft is taken at 62% fts, or 1000 ounces avoir; bat 62% fts would be nearer the truth, as per table below. It is about 815 times heavier than air, when both are at the temp of 62 ; and the barom at 30 ins. With barom at 30 ins the wt of perfectly pure water is as follows. At max density 62.425 fts. Lbs per Cub Ft. Temp in Deg of Fah. 32^ 6-2.417 40 62.423 50 62 409 60 62.367 Temp in Deg of Fah. Lbs per Cub Ft. 70 62.302 80 62.218 90 62.119 212 59.7 *The prices for American single-thick glass, per box, in small orders are (in Philadelphia, in 1873,) approximately as below. Double-thick 50 per ct more. Liberal discounts are made on heavy orders. For ground glass add about $2.50 per box. Size in Inches. 1st Quality. 2d Quality. 3d Quality. 4th Qnality. From to 6 by 8 8 by 10 $ 5.00 $ 4.50 $4.25 $ 4.00 8 by 11 10 bv 15 5.25 4.75 4.40 4.20 11 by 14 12 bv 18 6.00 5.50 5.00 4.65 14 by 16 16 bv 24 6.25 6.75 5.25 4.90 18 by 22 18 by 30 7.50 6.75 5.75 5.25 20 by 30 24 by 30 9.25 8.25 6.50 6.00 24 by 31 24 by 36 10.00 8.75 7.00 6.25 28 by 46 30 by 48 11.50 10.50 8.50 30 bv 50 82 bv 52 12.25 11.25 9.00 34 by 58 34 by 60 15.00 14.00 11.00 The charge by glaziers for putting the glass into new windows, including putty, tins, and two coats of paint to the sash, (one of which is a priming coat,) is (1873) equal to the co'st of the glass or the above prices. See footnote, p 513. For reglazing old sash, removing the broken panes, the charge is about twice as great. In small quantities, the following are also approximate prices for American glass : Large plates of U inch thick, 75 cts to $1 per sq ft. One inch thick. Si .40 to $1.80; if either is ground. 10 to 15 cts addi- tional per sq ft. Ribbed glass, ^ inch thick, 50 cts ; Klnch,60ctspersqft. Stained glass, single thick- ness, (^ inch,) or figured white enamelled glass, (single thickness,) 60 to 75 cts per sq ft. Superior thicker strong figured glass, first ground, and the transparent figures then formed by polishing awar portions of the ground surface, $1.25 to $2.00 per sq ft. Mufted glass is an inferior article of fanciful colored patterns, attached by some Imperfect process which allows them to peel off after a year or two of exposure to the weather. 516 WATER. The weight of water affords an easy way to find the cubic contents of a vessel. First weigh the ves- sel by itself; and then full of water. The diff will be the weight of the water ; and this divided by 62.3 or by the number in the table opp the temp of water, will be the contents in cub ft. To obtain the size of commercial measures by means of the weight of water. At the common temperature of from 70 to 75 Fan, a cub foot of fresh water weighs very approxi- mately 62 L 1 B)d avoir. A cubic half foot, (6 ius on each edge,) 7.78125 fi>s. A cub quarter foot, (3 ina on each edge,) .97268 B>. A cub yard, 1680.75 Ibs; or .75034 ton. A cub half yd, (18 ins on each edge,) A:).094 fos: or .0938 ton. A cub'iuch, .036024 ft ; or .576384 ounce; or 9.2222 drams ; or 252. 170 grains. An inch square, and one foot long, .432292 B>. Also 1 ft> = 27.75903 cub ins, or a cube of 3.028 ins on au edge. An ounce, 1.735 cub ins ; a ton, 35.984 cub ft, all near enough for common use. Original. Liquid Measures. r. s. cm U S Pint Lbs Avoir, of "Water. .26005* 1.0402 Liquid and I>ry. Lbs Avoir. of Water. British Imp Gill 31211* " " Pint , 1.248:iR V S Quart 2.0804 8.3216 262.1310 1 2104 " " Quart... 2 49715 U. S. Gallon 8 Ibs 5^ oz U. S. Wine Barrel, 31^ Gall I>ry Measures. U S Pint. " " Gallon.. 9 9SS6 " " Peck 19 9772 " " Bushel.. .79 9088 * 4.9942 ; or ve French Centilitre y nearly 5 ounces. Measures. 021981 U S Quart . 2.4208 . 9.6834 U. S. Gallon U S Peck 19 3668 Decilitre 2198J * Or 4 ounces ; 2 drams ; 15.6625 grs. Litre 2.1981 Decalitre, or Ceutist ere 21.9808 2198 0786 t Or 5.6271 drai J 3.5169 ounces. ns ; or 153.866 gra. Its max density is when its temp is a little more than 39 Fah ; or about 7 warmer than the freezing point. By best authorities 39.2. From about 39 it expands either by cold, or by heat. When the temp of 32 reduces it to Ice, its wt is but about 57.2 Ibs per cub ft : aud its sp gr about .9175, according to the latest determination by L. Dutour. Hence, as ice, it haa expanded J^- part of its original bulk as water ; and the sudden expansive force exerted at the mo- ment of freezing, is sufficiently great to split iron water-pipes ; being probably not less than 30000 Ibs per sq inch. Instances have occurred of its splitting cast tubular posts of iron bridges, aud of ordi- nary buildings, when full of rain water from exposure. It also loosens and throws down masses of rock, through the joints of which rain or spring water has found its way. Retaining- walls also are some- times overthrown, or at least bulged, by the freezing of water which has settled between their backs and the earth filling which they sustain ; and walls which are not founded at a sufficient depth, are often lifted upward by the same process. It is said that in a glass tube }\ inch in diarn, water will not freeze until the temp is reduced to 23; and in tubes of less than -^Q- inch, to 3 or 4. Neither will it freeze until considerably colder than 32 in rapid running streams. Anchor ice, sometimes found at depths as great as 25 ft. consists of an aggregation of small crystals or needles of ice frozen at the surface of rapid open water ; and probably carried below by the force of the stream. It does not form under frozen water. Since ice floats in -water; and a floating body displaces a wt of the liquid equal to its own wt, it follows that a cub ft of floating ice weighing 57.2 fts, must displace 57.2 R>-* of water. But 57.2 Ibs of water, one ft square, is 11 ius deep; therefore, floating ice of a cubical or par- allelopipedal shape, will have y^- of its volume under water; and only y 1 ^ above; and a sq ft of ice of any thickness, will require a wt equal to y*y of its own wt to sink it to~the surf of the water. In practice, however, this must be regarded merely as a close approximation, since the wt of ice is some- what affected by enclosed air- bubbles. Pure water is usually assumed to boil at 212 F, in the open air, at the level of the sea ; the barom being at 30 ins ; and at about 1 less for every 520 ft above sea at from 212 to 220; and it is stated that if all air be previously extracted, it requires 275. It evaporates at all temps; dissolves more substances than any other agent ; and has a greater capacity for heat than any other known substance. It is compressed at the rate of about the ^ T ^ ?TT part, (or about T ^ of an inch in ISj 1 ^ ft,) by each atmosphere or pressure of 15 Ibs per sq inch. Wheu the pres is removed, its elasticity restores its original bulk. Effect oil metals. The lime contained in many waters, forms deposits in metallic water-pipes , and in channels of earthenware, or of masonry ; especially if the current b WATER. 517 low. Some other^tffTstances do the same; obstructing the flow of the water to such an extent, that it is always expedient to us'e pipes of diameters larger than would otherwise be necessary. See Hy- draulics, Ar>tc, for a long time. The sulphuric acid contained in the water from coal mines corrodes iron pipes rapidly. In the fresh water of canals, iron boats have continued in service from 20 to 40 years. Wood remains sound for centuries under either fresh or salt water, if not exposed to be worn away by the action of currents ; or to be destroyed by marine insects. Sea water differs a little in weight, at diff places ; but at the same place it is appreciably the same at all depths; and may be generally assumed at about 64 Ibs; or 1% per cub ft more than fresh. The additional \% fts, or -$\-% part of its entire weight, is chiefly common salt. Sea water freezes at 27 Fah ; the ice is fresh.' A teaspoonful of powdered alum, well stirred into a bucket of dirty water, will generally purify it sufficiently within a few hours to be drinkable. If a hole 3 or 4 ft deep be dug in the sand of the sea- shore, the infiltrating water will usually be sufficiently fresh for washing with soap ; or even for drinking. It is also stated that water may be preserved sweet for many years by placing in the con- taining vessel 1 ounce of black oxide of manganese for each gallon of water. It Is said that water kept ill zsinc tanks; or flowing through iron tubes galvanized inside, rapidly becomes poisoned by soluble salts of zinc formed thereby ; and it is recommended to coat zinc surfaces with asphalt varnish to prevent this. Yft. in the city of Hartford, Conn, service pipes of iron galvanized inside and out, were adopted in 1855. fit the recommendation of the water commissioners; and have been in use ever since. They are likewise need in Philadel- phia and other cities to a considerable extent. In manv hotels and other buildincs in Boston, the " Seamless Drawn Brass Tube " of the American Tubes Works at Boston, has for many years been in use for service pipe; and has given great satisfaction. It is stated that the softest water may be kept In brass vessels for years without any deleterious result. The action of lead upon some waters (even pure ones) is highly poisonous. The subject, however, 518 RAIN. is a complicated one. An Injurious ingredient may be attended by another which neutralizes its action. Organic mutter, whether vegetable or animal, is injurious. Garbouic acid, when uotiu excess, is harmless, bee near bottom of page 379. Ice may be so impure that its water is dangerous to drink. RAIN. The quantity that falls annually in any one place, varies greatly from year to year ; the extremes being frequently greater than 2 to 1. In making calculations for collecting water in reservoirs, whether for feeding canals, or for supplying cities, we cannot safely assume more than the minimum fall observed for many years ; or rather, somewhat less. And from even this must be deducted the amount (a quite considerable one) lost by evaporation and leakage after it has been collected. The following table shows in some cases, the average annual falls j and in others, the least and the greatest ones observed at several places ; including snow -water. It ia highly probable that most of the results are merely approximate. See Evaporation, p. 521. Inches per au. Augusta, Georgia 23 Albany, N.York 31 to 51 Arkansas 41 Bath, Maine 30 to 50 Inches per an. Port Laramie, Nebraska ............... 20 Fort Worth, Texas .................... 41 FortMcIntosh, " , .................... Fort Dallas, Oregon ................... 30 to 39 34 to 45 35 , Key West, Flor Lebanon, Pe Michigan Monterey, Cal .............. . .......... 12^ Marietta, Ohio .............. . ......... 35 to 54 New Orleans, Louisiana ............... 51 New Fane, Vermont .................. 36 to 74 1 New England ............... average. . 47 Natchez, Miss ......................... 37 to 58 New York State ............ average.. 36J4 Ohio ....................... " .. 36 Philadelphia, Penna .................. 23 to 59 av for 32 years, to 1870. . . 45.8 * Pennsylvania .............. average.. 41 Savannah, Georgia ................... 30 to 60 Stow, Mass ........................... 33 to 49 St.Louis, Mo .......................... 42 Washington, D. C ................ . ---- 41 West Chester, Penna .................. 39 to 54 Williamstown, Mass ................... 26 to 39 Baltimore, Md 40 Boston, Mass 25 to 46 Charleston, S. C 40 to 76 ! Canada 36 Carlisle, Penna 34 Detroit, Michigan 30 Frankford, Peuna 33 to 54 Fort Gaston, California, in 9 months. 129 Fort Yuma, Cal 3} Port (not Fort) Orford, Oregon 69 Fort Pike, Louisiana 72 Fort Pierce, E. Florida 63 Fort Conrad. New Mexico 6% Fort Kent, Maine 364 Fort Preble, " 45^ Fort Constitution, N. Hamp 35^ Fort Adams, Rhode Island 52>$ Fort Hamilton, N. York Harbor 43% Fort Niagara, N.Y 3H{ Fort Monroe, Virg 51 Fort Kearney, Nebraska 28 The greatest fall recorded in one day in Philada, was 6 ins, which fell iu 2 hours, in July, 1842. The greatest in any month, was 12 ins, in the same month. It has not reached 9 ins per month, more than 5 or 6 times, in 25 years. During a tremendous rain at Norristown, Ponna, in 1865, the writer saw evidence that at least 9 ins fell within 5 hours. At Ge- noa, Italy, on one occasion, 32 ins fell in 24 hours ; at Geneva. Switzerland, H ins in 3 hours ; at Mar- seilles, France, 13 ins iu 14 hours; in Chicago, Sept. 1878, .97 inch in 7 minutes. Near London, England, the mean total fall for many years is 23 ins. On one occasion, 6 ins fell in 1% hour! In the mountain districts of the English lakes, the fail is enor- mous ; reaching in some years to 180, or 240 ins ; or from 15 to 20 ft ! while, in the adjacent neighbor- hood, it is but 40 to 60 ins. At Liverpool, the average is 34 ins; at Edinburg, 30; Glasgow, 22; Ire- 1'iiirt. 36; Madras, 47; Calcutta, 60; maximum for 16 years, 82; Delhi, 21; Gibraltar, 30; Adelaide, Australia, 23 : West Indies. 36 to 9(i ; Rome, 39. On the Khassya hills north of Calcutta, 5CO ins, or 41 ft 8 in. have fallen in the 6 rainy mouths I la other mountainous districts of India, annual falls of 10 to 20 ft are common. It requires a quite heavy rain, for 24 hours, to yield the depth of an inch; still, inasmuch as at rare iutervals falls of as much as from 1 to 3, or even 6 ins per hour occur, these latter depths must be considered in planning sewers, culverts, etc. See Remark, p 566. As a general rule, more rain falls in warm than in cold countries; and more in elevated regions than in low ones. Local peculiarities, however, sometimes reverse this; and also cause great diffs in the amounts in places quite near each other; as in the English lake districts just alluded to. It is sometimes difficult to account for these variations. In some lagoons in New Granada, South America, the writer has known three or lour heavy rains to occur weekly for some months, during which not a drop fell on hills about 1000 feet hk'h. \\irhin 10 miles' distance, and within full sight. At another locality, almost a dead-level plain, fullv % <>f the rains that fell for 2 years, at a spot 2 miles from his residence, occurred in the morning ; while those which fell about 3 miles from it, in an opposite direction, were in the afternoon. " The returns of several rain gauges in the Longdendale district, England, for 1847, gave the rain- falls at diff altitudes above the sea, as follows: " At 1750 ft altitude ........... 56.5 ins. 1800 " ........... 62.1 " At 500 ft altitude .............. 4fi.fi ins. 809 " .............. 50.5 " 1700 " .............. 52.1 " " The annual average fall at Edinhurp, 200 ft above the sea. In three successive years, -was 30 ins. In the Pent-land hills, a few miles south, and 700 ft above sea, 37.4 ins ; and at Carlops, similarly situ- ated near the last, but 900 ft above sea, 49.2 ins." * In 1869. during which occurred the greatest drought known in Philada for at least 50 years, ii was 4S.84 ius. -AIR. 519 There ai*Trobably but few places in the United States, where an annual fall of 2 feet may not be safely relied on; and since, as an ordinary average, cue- half of it majrbe collected into reservoirs, we should have for a square mile of drainage, one foot deep, 2787S4GO cift ft ; equal tt 7s per cub ft. It was very dry and incoherent. A cub ft of heavy snow may, by a gentle sprinkling of water, be converted into ano'ut half a cub foot of slush, weighing 20 Ibs ; 'which will not slide or run off from a shingled roof sloping 30, if the weather is cold. A cub block of snow saturated with water until it weighed 45 H>s per cub fr, just sl'd on a rough board inclined at 45; on a smoothly planed one at 30 j and on slate at 18; all approxi- mate. A prism of snow, saturated to 52 E>s per cub ft; one inch square, and 4 ins high, here a wt of 7 Ibs ; which at first compressed it about } part of its length. European engineers consider 6 fts per sq ft of roof, to be sufficient allowance for the wt of snow ; and 8 Ibs for the pressure of wind ; total, 14 Ibs. The writer thinks that in the U. S. the allowance for snow should not be taken at las than 12 EH ; or t'.)e total for snow and wind, at 20 Bs. There is no dancer that snow on a roof will become saturated to the extent just alluded to; because a rain that would supply the necessary quantitv of water, would also by its violence wash away the snow; but we entertain no doubt whatever that the united pressures from snow and wind, in o'ur Northern States, do actually at, times reach, and even surpass, 20 Ibs per pq ft of roof. See Table 4, p 301, of Trusses. The limit of pprpetnnl snow at, the equator is nt the height of about 1GOOO ft, or *ij 3 miles above sea-lfvel; in lat 45 north or south, it is about half that height; while near the poles it is about at sea level. AIE.-ATMOSPHEEE, The atmosphere is known to extend to at least 45 miles above the earth. Its composition is about .79 measures of nitrogen gas, and .21 of oxygen gas ; or about .77 nit, .23 ox, by weight. It generally contains, however, a trace of water; carbonic acid, aud carburetted hydrogen gases; and still less ammonia. Vhen the barom is at 30 ins, aud the temperature 60 Fah, air weighs about ^ ^ part as much as water; or 535 grains- 1.224 commercial ounces .0765 commercial ft, per cub ft. Or 13.072 cub ft weigh 1 ft ; uutJ a cub yd, 2.0(>6 B>s. Or a cube of 30.82 ft on each edge, 1 ton. When colder it weighs more per cub ft, and vice versa, at the rate of about a grain per deg of Fah. The average weight of the entire atmospheric column, (at least 45 miles high,) at sea level, is 14% fts avoir per sq inch ; or 2124 fts per sq f t ; = weight of a column of water 34 ft. or of mercury 30 inches nigjj. , ins u what is usually called the " pressure of the air." At % mile above sea level it is but 14.02 fts per sq inch; at % mile 13.33; at % mile, 12.66; at 1 mile, 12.02; &tl% mile, 11.42; at 1^ mile 10 88- aud at 2 miles, 9.80 Bxs. Therefore, a pump in a high region, will not lift water to as g'reat a height as in a low one. The pres of air, like that of water, is, at any given point, equal in all directions Tt is often stated that the temperature of the atmosphere lowers or becomes colder, at the rate of 1 Fah for each 300 ft of ascent above the enrth's surface ; but this is liable to many exceptions, and varies much with local causes. Actual observation in balloons seems to snow that up to the first 1000 ft, about 200 ft to 1, is nearer the truth ; at 2000 ft, 250 ; at 4000 ft, 300 ; and at a mile. 350. In breathing, a grown person at rest requires from .25 to .35 of a cub ft of air per minute; which when breathed vitiates from 3^ to 5 cub ft. When walking or hard at work, he breathes and vitiates two or three times as much. About 5 cub ft of fresh JMI- per i>ei>ou per minute is reqd for the perfect ventilation of rooms in winter; 8 in summer. Hospitals 40 to 80. Beneath the general level of the surface of the earth in temperate re- 5 ions, a tolerably uniform temp of about 50 to 60 Fah exists at the depth of about 50 to 60 ft ; and acreases about 1 for each additional 50 to 60 ft; all subject, however, to considerable deviations from n.any lociil causes. In the Rose Bridge colliery. Kngland, at the depth of 2424 ft. the tempera ture of the coal is 93.4 Fah; and at the bottom of a' boring 4lfi9 ft deep, near Berlin, the temp is 119 The air is a very slow oomlnetor of heat; hence hollow walls serve to retain the heat in dwellings; besides keeping them dry. It rushes into a vacuum nenr sea level with a vel of about 1157ft por sec: or JS^ miles per minute; or about as fast as sound ordinarily travels through quiet air. See Sound, p 173. Ulie all other elastic fluids, it expands equally with equal increases of temperatnre. Every increase of 5 Fah, expands tho luilk of any of them slieht.lv more than 1 per cent of that which it has at Fah ; or 500 about doubles its bulk at zero. The bulk of any of them diminishes inversely in proportion to the total pressure to 520 WIND. which it is subjected. Thus, if we have a cylinder open at top, aud 1 ft deep, full of air at it natural Srea of about 15 tts per sq inch ; if by means of a piston we apply an additional pres of 15 fts per aq ich, making 30 fts in all, or twice as much as the nut pres, theu the uir will be compressed into tt ins of depth of the cylinder, or cue-half of what it occupied before. Or it we apply 45 fts additional, making 60 fts ia all, or 4 times the natural pres, then the air will be compressed into y^ of the depth of the cylinder. Experiment shows that this holds good with air at least up to pressures of about 750 Ibs per sq inch, or 50 times its uat pres; the air in this case occupying the -^ part of its natural bulk. In like manner the bulk will increase as the total pres is diminished ; so that if we remove our additional 45 fts per sq inch, the air in the cylinder will regain its original bulk, aud will precisely fill the cylinder. Substances which follow these laws, are said to be perfectly tUstic. Under a pres of about 5><| tons per sq inch, air would become as dense, or would weigh as much per cub ft, as water. Since the air at the surface of the earth is pressed 14% fts per sq inch by the atmosphere above it, and since this is equal to the wt of a column of water 1 inch sq, and 34 ft high, it follows that at depths of 34, 68, 102 ft, &c. below water, air will be compressed into %, %, y. &c, of its bulk at the surf; because at those depths it is exposed to pressures equal to 2, 3, 4, &c, times 14% fts per sq inch, inasmuch as the pres of the atmosphere on the surf, is in each case to be added to that of the water. The pres of the water alone at those depths, would be but 1, 2, 3, &c, times 14% Ibs per sq inch. Tile greatest heat of the air in the sun probably never exceeds 145 Pah ; nor the greatest cold 74 at night. About 130 above, aud 40 below zero, are the extremes in the U. S. east of the Mississippi : and 65 below in the N. W. ; all at common ground level. It ia stated, however, that 81 has been observed in N.E.Siberia: and -j- 101 Fah in the shade in Paris; and + 153 in the suu at Greenwich Observatory, both in July, 1881. Ill a diving-bell, men, after some experience, can readily work for several hours at a depth of 51 feet; or under a pressure of 2^ atmospheres ; or 37^ Ibs per sq inch. But at 90 ft deep ; or under 3.64 atmospheres ; or nearly 55 fts per sq inch, they can work for but about an hour, without serious suffering from paralysis ; or even danger of death. 'Still at the St. Louis bridge some work was done at a depth of 110 % ft; pres 63.7 fts per sq inch. The dew point is that temp f varying) at which the air deposits its vapor. WIND. by Smeaton, is'prepared. On Mt. Washington, N. H., 180 miles per hour has been observed. Vel. in Miles per Hour. Vel. in Ft. per Sec. Pres. in Lbs. per Sq. Ft. Remarks. 1 1.467 2 933 .005 .020 Hardly perceptible. 3 4'.400 .045 ' ct> ^_^/ 4 5.867 .080 5 7.33 .125 n ."."."".- nil 10 14.67 .5 lt/\l ^ 18.33 .781 Fresh breeze. o 15 22. 1.125 20 25 29.33 36.67 2. 3.125 Brisk wind. The pres against a semicylindrical 30 44. 4.5 Strong wind. surface a c b nom 40 58.67 8. High wind. is about ha If that 50 73.33 12.5 Storm. against th e fiat 60 88. 18. Violent storm. aurf abnm 80 117.3 32. Hurricane. 100 146.7 50. Violent hurricane, uprooting large trees. ' Tredgold recommends to allow 4O Ibs per sq ft of roof for the pres of wind against it; but as roofs are constructed with a slope, and consequently do not receive the full force of the wind, this is plainly too much.* Moreover, only one-half of a roof is usually ex- posed, even thus partially, to the wind. Probably the force in such cases varies approximately as the sines of the angles of slopes. According to observations in Liverpool, in 1860, a wind of 38 miles per hour, produced a pres of 14 Ibs per sq ft against an object perp to it; and on of 70 miles per hour, (the severest i?ale on record at that city.) 42 fts per sq foot. These would make the pres per sq ft, requ treat as given in Smeaton's table. We should ourselves give the preference to the Livprpool ol ft. " It is stated that as high as 55 fts has been observed at Glasgow. High winds often lift roofs. The gauge at Oirard College. Philada, broke under a strain of 42 fts per sq ft: a tornado passing at the moment, within y mile. By inversion of Smeaton's rule, if the force in fts per sq ft, be mult by 200. the sq rt of the prod will give the vel in miles per hour. Smeaton's rule is used by the U. S Signal Service. * The writer thinks 8 fts per sq foot of ordinary double-sloping roofs, or 16 ft for thed-rooft, Bnffl- tent allowance for pres of wind. Sett Table 4, p 301 ; also Snow, p 519. HYDROSTATICS. 521 EVAPORATION, FILTRATION, AND LEAKAGE. The amount of evaporation from surfaces of water exposed to the natural effects of the open air, is of course greater in summer thau in winter; although it is quite perceptible in even tbe coldest weather. It is greater iu shallow water thaii in deep, inasmuch as the bottom also becomes heated by the sun. It is greater in running, than in standing water; on much the same principle that it is greater during winds than calms. It is probable that the average daily loss from a reservoir of moderate depth, from evaporation alone, throughout the 3 warmer moLths of the year, (June, July, August,) rarely exceeds about y^W inch, in any part of the United States. Or y 1 - inch during the 9 colder months ; except in the Southern States. These two averages would give a daily one of .15 inch ; or a total annual loss of 55 ins, or 4 ft 7 ins. It probably is 3.5 to 4 ft. By some trials by the writer, in the tropics, ponds of pure water 8 ft deep, in a stiff retentive clay, and fully exposed to a very hot sun all day, lost during the dry sea- son, precisely 2 ins in 16 days ; or % inch per day ; while the evaporation from a glass tumbler WHS y^ inch per day. The air in that region is highly charged with moisture; and the dews are heavy. Every day during the trial the thermometer reached from 115 to 125 in the sun. The total annual evaporation in several parts of England and Scotland is stated to average from 22 to 38 ins ; at Paris, 34 ; Boston, Mass, 32 ; many places in the U. S., 30 to 36 ins. This last would give a daily average of y 1 ^ inch for the whole year. Such statements, however, are of very little value, unless accompanied by memoranda of the circumstances of the case; such as the depth, exposure, size and nature of the vessel, pond. &c. which contains the water. &c. Sometimes the total annual evaporation from a district of country exceeds the rain fall ; and vice versa. On canals, reservoirs, &c, it is usual to combine the loss by evaporation, with that by filtration. The last is that which soaks into the earth; and of which some portion passes entirely through the banks, (when in embankt;) and if in very small quantity, may be dried up by the sun and air as fast as it reaches the outside; so as not to exhibit itself as water; but if in greater quantity, it becomes apparent, as leakage. E. H. Cxi 11. E, states the average evaporation and filtra- tion on the Sanely and Beaver canal. Ohio, (38 ft wide at water sur- face; 26 ft at bottom ; and 4 ft deep,) to be but 13 cub ft per mile per minute, in a dry season. Here the exposed water surf in one mile is 200640 sq ft; and in order, with this surf, to loje 13 cub ft per min, or 18720 cub ft per day of 24 hours, the quantity lost must be ^^^ - 0933 ft - l ^ inch in depth per day. Moreover, one mile of the canal contains 675840 cub ft ; therefore, the number of days reqd for the combined evaporation and filtration to amount to as much as all the water in the canal, is *LZJ? ?JL?_ 36 davs. Observations In warm weather on a 22-mile reach of the Chenango canal, N 18720 York, (40; 28; and 4 ft,) gave 65^ cub ft per mile per min ; or 5 times as much as in the preceding case. This rate would empty the canal in about 8 days. Besides this there was an excessive leakage at the gates of a lock, (of only 5^ ft lift.) of 479 cub ft per min. 22 cub ft per mile per min ; and at aqueducts*and waste- weirs, others amounting to 19 cub ft per mile per min. The leakage at other locks with lifts of 8 ft, or less, did not exceed about 350 cub ft per miu. at each. On otjier canals, it has been found to be from 50, to 500 ft per min. On the Chesapeake and Ohio canal, (where 50, 32, and 6 ft,) Mr. Fisk, C E. estimated the loss bv evap and filtration in 2 weeks of warm wenther, to be 3 uai to all the water in the canal. Professor Kaitkiiio assumes 2 ins per ay, for leakage of canal bed, and evaporation, on English gates, on the original Erie canal, (40, 28. and 4 ft,; at 100 cub ft per m'ile per min : or 144000 cub ft per day. The water surf in a mile is 211200 sq ft ; therefore, the daily loss would be equal to a depth of 211 ~ >682 ft> ~ 8Hy ^ ln8 ' SeC Cnd f RalU> P 519 ' On the Delaware division of the Pennsylvania canals, when the supply is temporarily shut off from any long reach, the water falls from 4 to 8 ins per day. The filtration will of course be much greater on embankts, than in cuts. In some of our canals, the depth at high embankts becomes quite considerable; the earth, from motives of economy, not being filled in level under the bottom of the canal; hut merely left to form its own natural slopes. At one spot at least, on the Ches and Ohio canal, where one side is a natural face of vertical rock, this depth is 40 ft. Such depths increase the leakage very greatly ; especially when, as is frequently the case the em- bankts are not puddled ; and the practice is not to be commended, for other reasons also. The total average loss from reservoirs of moderate depths, In case the earthen dams be constructed with proper care, and well settled by time, will not exceed about from ^ to 1 inch per day ; but in new ones, it will usually be considerably greater. The loss from ditches, or channels of small area, i* much jrreatpr than that from navigable canals; so that long canal feeders usually deliver but a small pro- portion of the water which enters them at their heads. HYDKOSTATICS. Art. 1. Hydrostatics treats of the pressure of quiet water ; and other liquids. The pros of liquids against any point of any surf upon which they act, whether said surf be curved or plain, is always at right angles to that 522 HYDROSTATICS. Soint. At any given depth, the pres of water is equal in every direction ; and is in i i ect proportion to the v?rt depth below the surf. In all cases whatever, the total pres of quiet water against, and perp to any surf, is equal to the wt of a uniform column of water, the area of whose cross-section parallel to its base, is everywhere equal to the area of the surf pressed ; and whose height is equal to the vurt depth of the cen of grav of the surf pressed, beluw the hor surf of the water. This fact is one of those important ones of frequent application, which the young student should impress firmly upon his memory. The wt of a cub ft of fresh water is usually assumed to be ti:i^ ft>s avoir; which is sufficiently correct for ordinary engineering purposes;, although 62^ is nearer the truth for ordinary temperatures of about 70 i'ah. Hence, To find the total pres of quiet water against, and perp to any surf whatever, as a dam, embkt, lock-gate, per sq ft at diff vert depths ; and also the total pi es against a plane one ^ foot wide extending vert from the surface to those depths. The first in- creases as the depths ; the last as the squares of the depths. For the pres in Ibs per sq inch at any given depth, mult the depth in ft by .434. For tbs per sq ft, mult by 62.5. For tons per so ft, mult by .0279. For the depth in 1't at which any given pres exists, divide the ft>s per sq inch by .434; or the fts per sq ft by 62.5: or the tons per sq ft by .0279. D in Ft. Per a Tot P. D Ft. Per a Tot P. I) in Ft. Per K Tot P. D in Ft. Per F?. Tot P. D in Ft. Per il Tot P. 1 62. 31 11 687. 3781 21 1312. 13781 31 1937. 30031 41 2562. 52531 2 125. 125 12 750. 4500 22 1375. 15125 32 2000. 32000 42 2625. 55125 3 187. 281 13 812. 5281 23 1437. 16531 33 2062. 34031 43 2687. 57781 4 250. 500 14 875. 6125 24 1500. 18000 34 2125. 36125 44 2750. 60500 5 312. 781 15 937. 7031 25 1562. 19531 35 2187. 38281 45 2812. 63281 6 375. 1125 16 1000. 8000 26 1625. 21125 36 2250. 40500 46 2875. 66125 7 437. 1531 17 1062. 9031 27 1687. 22781 37 2312. 42781 47 2937. 69031 8 500 2000 18 1125. 10125 28 1750. 24500 38 2375. 45125 48 3000. 72000 9 562. 2531 19 1187. 11281 29 1812. 26281 39 2437. 47531 49 3062. 75031 10 625. 3125 20 1250. 12500 30 1875. 28125 40 2500. 50000 50 3125. 78125 Thus we see that at the depth of 36 ft, the pres of water against a single sq ft of surf, whether hor, rert, or oblique, is fully 1 ton ; requiring great precaution to prevent leakage, or breaking. At 72 ft, it would be 2 tons. &c. A pres of 62^ fts per sq ft gives a pres of .434 fts per sq inch. Further; let a 6, Fig 3>^, be a tube of 36 ft vert height ; full of water ; with a bore so small that the tube would contain say only one pound of water; and let this tube open at its lower end into a vessel also full of water; tlie top and bottom of which are 8 ft apart. Then the 1 Tb of water in the tube, will cause each sq ft of the top of the vessel, (which is 36 ft below the surf of the water in the tube) to be pressed upward with a force of 2250 fts, as per table. Each sq ft of the bottom of the vessel (which is 44 ft below the- surf of the water in the tube) will be pressed downward with a force of 2750 Ibs ; and any par- ticular sq ft of the sides of the vessel, will be pressed hor outward, with the force given in the table, opposite to the depth of the cen of grav of said sq ft below the same water surf of the top of the tube, whatever said depth may happen to be. Or, suppose, first only th3 ?ower vessel to be filled with water, and its inner surf to be sustaining the pres arising therefrom ; if we then flil the 36 ft tube with its 1 Ib of water, this 1 ft will create an additional pres of 2250 tbs against every sq ft of said inner surf; so that if each of the 6 sides of the vessel be 8ft square; or contain in all 384 sq ft of inner surf, this 1 ft of water will produce addi- tional pres of 864000 Ibs. or full 385 tons, against them. If we then press upon the top of the water with our thumb to the extent of 1 ft, we shall thereby redouble this enormous pres. This fact, how- ever, belongs to Art. 7, p 526. Art. 2. Surfaces pressed on both sides; and immersed. When two bodies of water of diff depths, press against two oppo- site sides of a plane which is completely immersed, whether vert or eloping; as, for instance, against the two sides a 6, no, Fig 4; or the two sides d e. c r, then, the total pres against i b, i e, a b, n o, or c r, &c, may still be found by the foregoing rule, in Art 1 : but the XXCESS of pres against the part a b, or d e, of the immersed plane, beyond the counier-pres against the opposite part n o, or c r, will be equal to the wt of a column of water whose section is equal to the area of the part a b, or d e. (as the case may be;) and whose vert height is equal to m n. or xp. the vert diff of level of the two bodies of water. Consequently, this excess of outward pres is found by mult together, the area of a 6 or d e, in sq ft; the vert height m n or xp, in ft ; and the constant 62.5 fts wt of a cub ft of water. Thus, if a 6 is 10 ft high, and 20 ft long : and the vert height mn, 12 ft; then the excess of pres against a 6, over that against no, will be 10 X 20 X 12 X 62.5=150000 fts. The excess will be greater on d e, than on a b, although both are exposed to the same vert depths mn, xp ; because the area of d e is greater than that of a b. Moreover, this excess of outward pres is equally distributed over the entire area of a b or d e ; being no greater at b and e, than at a or d; in other words, every sq ft of area of a 6 or deis pressed outward at right angles to its surf, by an excess of force equal to the wt of a column of water 1 ft sq ; and of a height equal to m n, or xp. this pres of the air may. and should be omitted ; because it is counterbalanced by an equal pres of air against the opposite side, face, or surf of the pressed body. It becomes necessary, therefore, to take it into consideration only when the opposite face of the body is not exposed to a counterbalancing atmospheric pressure; as when there is a vacuum on that side. 524 HYDROSTATICS. JfilK This will be understood by means of Pig 5. which may represent five Slauks, 1, 2, 3, 4, and 5, forming a dam, and seen endwise; each one 1 ft i depth, and say 20 ft long hor ; making the area of each surf pressed, equal to 20 sq ft. The pres in tts against each separate 20 sq ft of area, calculated by the rule in Art 1, is shown in the tig. Now, the outward pres against the upper immersed '20 ft area, or that of plank 3, is 2125 ttis ; while the counter-pres against it trom the other side is 625 tts ; making the excess of outward pres equal to 3125 625 = 2500 B>s. Again, at the lowest plank, number 5, the outward pres exceeds the inward one by 5625 3125 =r 2500 tts, the same as in the upper one. And so of any other equal area of surf, at any depth whatever ; the excess depending upon the vert height of m /t, will be equally distributed over a b. it only remains to show that the total excess of outward pres against a b, is equal in amount to the wt of a uniform column of water with a base equal in area to a 6. and with a height equal to m n. Thus, we have seen that in the instance before us, the excess amounts to 3 times 2500 tt>3, or to 7500 tt>s. Now, the wt of the column of water will be 60 (or area of a fc) X wt n (or 2 ft) X 62.5 B)s 7500 Bt>s ; or the same as the excess pres on a 6. The excess of pres against the entire side s b, over that against n o, is evidently the diff between those two pressures calculated respectively by the rule in Art 1. Art. 3. Surfaces of equal widths, commencing- at the level of the water, but extending to diff depths, measured vert; and having 1 the same inclination to the surf of the water; sustain total pressures proportional to the squares of those depths. In Fig 6, let the two vert sides, a no t, and b m c s, of a vessel, 4, 5, &c, times greater than the depth n o, the pres against the surf b mcs, will be 4, 9, 16. 25, &c, times greater than that against ano t. This will be seen by referring to the pressures figured on the left side of Fig 5, where, as stated in Art 2, the surf of plank 1, exposed to the prs on the left side, is 20 sq ft ; that of planks 1 and 2, 40 sq ft ; that of planks 1, 2, and 3, 60 sq ft, &c. All these surfs commence at the level of the water; and all of them being vert, are of course at the same inclination with the water surf; but their depths are re- spectively 1, 2, and 3 ft. The pres against the surf of 1, is 625 tts; that against the surf of 1, 2, is 625 -f- 1875 2500 ; and that against the surf of 1, 2. 3, is 625 -j- 1875+ 31 25 = 5625. But 2500 is four times 625; and 5625 is nine times 625. And the pres against the entire surf s &, (which is 5 times as deep as plank 1,) is 25 times as great as that against plank 1 ; or 625 X 25 15625 fts - the sum of all the pressures marked on the left side of Fig 5. This follows, from the Rule in Art 1 ; for twice the area of surf, mult by twice the vert depth of the oen of grav below the surf, must give 4 times the pres : three times the area, by three times the depth, must give 9 times the pres, &c. See third columns of table, p 523. It follows, also, that at any particular point, or against any given area placed at various depths, the pres will increase simply as the vert depth : thus, if there be three areas, each one sq ft, placed in the same positions, but with their centers of grav respectively 8, 16, and 24 ft below the surf, the pres against them will he respectively as 8, 16, and 24; or as}, 2, and 3. See se -on ' crlumns table, p 523. Art. 4. The pressure of quiet water, in any one given di- rection, against any given surf, whether vert, hor, inclined, flat, or curved, is equal to the wt of a uniform column of water, the area of whose section, parallel to its base, is everywhere equal to the area of the projection* of the pressed surf taken perp to the given direction; and the height of the column equal to the vert depth of the cen of grav of the pressed surf below the upper surf of the water. Hence the RULE. To find the pres in Ibs, mult together the area in sq ft of the projection taken at right angles to the given direction; the vert depth in ft of the cen of grav of the pressed surf below the upper surf of the water ; and the constant 62.5 B>s wt of a cub ft of water. Ex. Let m cs n, Fig 7, be an inclined surf, sustaining the pres of water which is level with its top m c. Then the total pres against me s n, and at right angles to it, as found by the rule in Art 1, is an illustration of the pres- ent rule ; because the projection of mean, taken at right angles to the given direction, or parallel tomcsn, is in fact me s n itself, or equal to it. Hence the rule in Art 1 is merely a simple modification of the present one, appli- cable to the case of total pres against any surf. But if it be reqd to find only the vert or downward pres against m c s n, in pounds, mult together the area of the hor projection aocm in sq ft; the vert depth in ft of the cen of grav of m c s n below the surf; and 62.5, Or if only the hor pres against m c s n be sought, mult together the area of the vert projection a o s n; the vert depth of the cen of grav of me an; and 62.5. In Fig 8 also, the total pres against efg h is found by rule in Art 1 : while the hor and vert pressures against it are found as in Fig 7, by using the projec- tions efk i, and k i g h. In Fig 7 the vert pres is downward; while in Fig 8 it is upward ; but this circumstance in no respect affects the rule. RKM. I. It will be observed in both figs, that the vert projections, aosn, efki, will remain the same, no matter what may be the inclination of the pressed purfs me s n, and efg h; the degree of inclination therefore does not influence the hor pres, but only the total, and the vert ones. a * See Projection, in our glossary. fDROSTATICS. 525 IW10 ; with the same depth : and 62.5 Again, let Fig 9pej5resent a conical vessel full of water; is base 6 c, 2 ft d^eem; its vert height a n, 3 ft ; then the circumf of the base will be li.2832 ft; the apaof the base 3.1416 sq ft; the length of its slant side a 6 or a c, 3.16 (I; the areaf'of its curved slanting sides will be 6 ' 2832 X 3 - 16 y 9a sq ft; and the vert depth of the cen of grt height a n from the apex a, < Here, to tiud the total pres against the base, we have by rule in Art 1, 3.1416 X 3 X 62.5 589.05 tt>s. For the total pres against the slant sides, by the same rule, 9.93 X 2 X 62.5 1241.25 tt>s. For the vert pres upward against the entire area of the ilaut sides, we have given the area of the base (which is here the hor projection of the slant sides) ~ 3.H16; and the vert depth of theceu of gray of the slant sides, 2 ft. Therefore, 3.1416 X 2 X 62.5 =: 35*2.7 fts, the upward vert pres. Finally, for the hor pres in any given direction against the slant sides of one half of the cone, we have the vert projection of that half, represented by the triangle ale, with its base 2 ft, and its perp height 3 ft ; and consequently , with an area of 8 sq ft. The depth of its ceu of grav is 2 ft : therefore, 3 X 2 X 62.5 = 375 fts, the'reqd hor pres.* In Fig 10, widen represents a vessel full of water, the total pres against the semi-cylindrical surf avemdk. and perp to it, must be also hor. because the surf is vert; but inasmuch as the surf is curved, this total pres, as found by rule in Art 1, acts against it in many di- rections, wuich might be represented by an infinite number of radii drawn from o as a center. But let it be reqd to find the hor pres in ft>s, in one direction only, say parallel to o .e, or perp to a d; which would be the force tending to tear the curved surf away from th flat sides a 6 v, and desk, by producing fractures along the lines a v and d k ; or which would tend to burst a pipe or other cylinder. Ju this case, mult together the area of the vert projection a d k v in sq ft; the depth of the cen of grav of the curved surf in ft; (which, in the semi-cyliuder would be half of e m, or of o i;) and 62.5. Since the resulting pres is resisted equally by the strength of the vessel along the two lines a v and d k. it is "plain that each single thickness along those lines need only be sufficient to resist safely one half of it; and so in the case of pipes, or other cylinders, such as hooped cisterns or tanks. See Art 16, p 531. Should the pres against only one half of the curved surf, as edmk be sought, and in a direction parallel to o d, tending to produce frac- tures along the lines e m, and d k, then use the vert projection oen ' as before. It follows, that if the face of a metallic piston be made concave or convex, no more pres will be reqd to force the piston through any dist, than if it were fiat; lor the pres against the face of the piston, in the direction in which it moves, must be measured by the area of a projection of that face, taken at right angles to said direction ; and the area of said projection will be the same in all three cases. HEM. 2. If a bridge pier, or other construction, Fi;;- 1O V. be founded on sand or gravel, or on any kind of foundation through which water may find its way underneath, even in a very thin sheet, then the upward pres of the water will take'effect upon the pier ; and win tend to lift it, with a force equal to the wt, of the water displaced by the pier ; (see Arts 17 and 18 ;) or in other words, the effective wt of the submerged portion of the pier, will be reduced 62^ fts per cub ft; or nearly the half of the ordinary wt of masonry. But if the foundation be on rock, covered with a layer of cement to prevent the infiltration of water beneath the masonry, no such effect will be produced; but on the contrary, the vert pres downward, afforded by the bat- tering sides of the pier, and bv its offsets, will tend to hold it down, and thus increase its stability ; which, in quiet water, will then actually be greater than on land. Art. 5. To divide a rectangular surf, whether vert as ft b c d, or inclined as m nop, Fig 11, whose top a b or m n is level with the surf of the water, by a hor line .r 2, such that the total pres against the part above said hor line, shall equal that against the part be- low it. RULE. Mult one half of the length of 6 c, or m p, as the case may be, by the constant number 1.4142; the prod will be b 2, Ex. ' Let ft c= 12 ft. Then 6 X 1.4142 = 8.4852 ft ; or 6 2. Let TO p 16 ft. Then 8 X 1.4142 = 11.3136 ft, or m x. REM. The line x 2, thus found, must not be confounded with the cen of pres, which is entirely diff. See Art 8. Art. 6. In a rectangular surf, whether vert as a b c d f or in- clined as m n op, Fig 11, whose top a b or m n coincides with the surf of the water, to find any number of points, as 1,2, Ac. through which if hor lines, as 1 ./ r 528 HYDROStATICS. 8. To find the cen of pres against either a circular, or an elliptic surf, pressed on one side only* whether vert, or inclined ; and having its top either coinciding with the surf of the water, or below i( , Call the vert depth of the cen of pres below the water surf, h. The vert (or inclined, as the case may be) se/wi-diam of the surf, r. The vert dist of the cen of the pressed surf, below the water surf, d. Then, h f- d. In a vert circle with top at surf, h = 1% rad. Art. 1O. Walls for resisting- the pres of quiet water. A study of what we have said on retaining- walls for earth, will be of service in this connection. It is of course assumed that the water does not find its way under the wall ; and that the wall cannot slide. In making calculations for walls to resist the pres be but one foot in Ir.nyth; (not height, or thickness;) for then the number of cub ft contained in it, is equal to that of the s^ ft of area of its cross-section, or profile; so that these sq ft, when mult by the wt of a cub ft of the masonry, give the wt of tiie wall. In ordinary cases, it is well for safety to assume that the water extends down to the very bottom line of the wall. Now. by Art 1, the total pres of quiet water, against the rec- tilineal back of a wall, whether vert or sloping, is found in tts, by mult together the area in sq ft of the part actually pressed, Cor in contact with the water;) half the vert depth of the water, in ft, (being the vert depth of the cen of grav of a rectilineal back, below the surf;) and the constant 62.5 tts; and this total pres is always perp to the pressed area. When the lack of the watt is vert, as in Fig 20^. this pres p is of course less than when it is bat- tered; and is also hor ; and it tends to overthrow the wall, by making it revolve around its outer toe, or edge t. Tbe cen of pres is at c ; c being % the vert depth on; in other words, the entire pres of the water, so far as regards overthrowing the wall as one mass, (see Art 1, of Force in Rigid Bodies,) may be considered as concentrated at the point c : where it acts with an overthrowing lever- age tl, (see Art* 41, 49, 50, Force in Rigid Bodies.) The pres in Ibs, mult by this leverage in feet, gives the moment in ft-fts of the overturning force ; (see Art 49, Force in Rigid Bodies.) The wall, on the other hand, resists in a vert di- rection g a, with a moment equal to its wt, (supposed to be concen- trated at its cen of grav g,) mult by the hor dist at, which consti- tutes the leverage of the wt with respect to the point t as a fulcrum. If the moment of the water is greater thnn that of the wall, the lat- Fl ter will be overthrown ; but if less, it will stand. *-^_ / 9 q REM. I. Art 49 of Force in Rigid Bodies, will sufficiently explain the subjects of moments and leverage; and make it evident that the same prinjiple applies also to sloping backs, as in Fig 21. Here the overturning moment of the water is equal to its calculated pres p X its leverage tl; while the moment of stability of the wall is equal to its wt X its leverage at. By aid of a drawing to a scale, we may on this principle ascertain whether any proposed wall will stand. For we have only to calculate the pres p; then apply it at c. and at right angles to the hack; prolong it ~to 1; 'measure'? 2 by the same scale. Then calculate the wt of wall ; find its cen of grav "g : draw g a vert, and measure the leverage a t. We then have the data for calculating the two moments. For finding the cen of grav, see Cen of Grav, Trapezoid, p 442. RIM. 2. If the water, instead of being quiet, Is liable to waves, the wall should be made thicker. Art. 11. To find the thickness, a c, of a vert wall. Vis 22 sustaining quiet wa*er level with its top, and as deep as the wall is high; so as to resist being: overturned. RULE. Divthe number 1, orl 1 ^. 2, 3, or Ac, (according as the resistance of the wall is reqd to be either just equal to, or 1 ^. 2, 3. or Ac, times as great as the overturning notion of the water.) by 3 times the sp grav of the mate- rial of which the wall is built. Take the sq rt of the quot. Mult this sq rt bv the vert depth of the water in ft. The prod will be the reqd thickness, The sp gr of a dressed granite wall may be taken at 2.5 ; of dressed sand- stone, 2.2; common mortar rubble. 2: brickwork. 1.8. T^ j* O O Ex. What must he the thickness of a vert wall, built of mortar rubble of In 1 & Lj a so gr of 2 ; and reqd to present a resistance equal to 1 .5 time* the pres of \i? the water ; the depth of water, or height of wall a n, each being 20 ft? Here, 1.5-r6 = .25; and the sq rt of .25= .5; and .5X 20 = 10 ft = ac. Or mult the ht by the proper decimal below. Sp. Gr. Lbs per Cub Ft. Resist = 1.5 pres. Resist = 2 pres. Resist = 3 pres. Dressed Granite... Dressed Sandstone Mortar Rubble Brickwork 2.5 t 156 137 125 112 .447 .477 .500 .527 .516 .550 .578 .609 .633 .674 .707 .746 To change a vert wall into a battered one. see Art 8. p 389. Art. 12. To find the thickness, w n, at the base of a rigrht- angled triangular wall wmn, Fig 23. sustaining at its vert back wm, the pressure of quit water 1..1 arith it. ton ./* > r)n i\ the wall ii hiffh. in a la rsier baimr overturned. fDKOSTATICS. 529 RULE. DiT the nupafcer 1, 1 V$, or Ac, (according as the resistance of the wall it reqd to be jus^etfual to, or 1&, 2, or &c, times as great as the overturning ~" action of thejirftter,) by twice the sp gr of the material of which the wall is built. TaJe the sq rt of the quot. Mult this sq rt by the depth of water, or height of wail in ft. The prod will be the required thickness, m n, in feet. Or, (original ;) mult the thickness, m o, of a vert wall by 1.225. Ex. As before ; wall of rubble, of sp gr of 2 ; resistance to be 1.5 times the pros of the water, depth 20 ft. Reqd the thickness, mn. Here, 1.5-r 4=r.375. The sq rt of .375- .6124 ; aud .6124 X 20-12.25 ft ~m n. Or mull the tit by tlie proper decimal be- _ low. REM. The pressure against a wall sustaining water is not increased bv a de- posit of earth on the same side at its natural slope, if the earth is imne'rvious to water and in sufficient quantity to prevent the water from reaching the wall. Sp. Or. Lbs. Resist =1. 5 pres. Resist 2 pres. Resist = 3 pres. Dressed Granite... 2.5 156 .548 .633 .775 Dressed Sandstone. 2.2 137 .584 .675 826 Mortar Rubble 2. 125 .613 .707 .866 Brickwork. . 1 8 112 646 746 Notwithstanding their greater thickness at base, such triangular walls contain, as seen by the fig, not much more than half the quantity of masonry reqd for vert ones of equal strength. This is owing to the fact that their cen of grav is thrown farther back ; thus increasing the leverage by which the wt of the wall resists overthrow. Art. 13. To find the thickness, a b, at the base of a wall. Fig 24, with a vert back, ft a f and with a face, it ft, having any given batter; to sustain the pres of quiet water level with its top. and \,. r as deep as the wall is high; so as to resist being overturned. RULE. Square the vert depth of the water, or wall, in feet. Mult this square by 1. or by \%, 2. 3, or. Ac. (according as the resistance of the wall is reqd to be just equal to. or 1%. 2, 3, or, Ac. times as great as the overturning action of the water.) Call the prod a. Square the entire amount of batter, h n, in feet. Mult this square by the sp gr of the masonry of which the wall the sum by 3 times the sp pr of the masonry. Take the sq rt of the quot This sq rt will be the reqd base, a b. Ex. Rubble wall 20 ft hi^h, sp gr 2 ; ba-ter of face 4 inches to a ft ; or total batter hn, = 6.666 ft. What must be the thickness a b, at base, that the resistance of the wall shall be 1.5 times the pres of the water? Here, the square of 20 400; and 400 X 1.5 600, or prod a. Again, the square of 6.666 = 44.44; and 44.44 X 2 = 88.88, or prod b. Now, 600 -4- 88.88 = 688.88 ; aud 6 ^- = 114.81 ; aud )/lU.bl =: I 10.715, ab, the base reqd. The following table is drawn up from this rule : Or mult the lit sa by the proper decimal below. Dressed Granite.. Dressed Sandstone Mortar Rubble Brickwork 3 & 25 2.2 1 1.8 Wt.of a cub. ft. of Wall. Lbs. 156 137 125 112 The Resist of the Wall to be equal o 1> times the pres of the Water. The Resist of the Wall to be equal to twice the pres of the Water. Batter 1 in to a foot. .449 .480 .502 .530 Batter 2 ins. to a foot. .458 .488 .510 .539 Batter 4 ins. to a foot. Batter 6 ins. to a foot. Batter 1 in. to a foot. Batter 2 ins. to a foot. Batter 4 ins. to a foot. Batter 6 ins. to a foot. .593 .622 .646 .674 .487 .515 .536 .5(52 .532 .558 .578 .602 .519 .552 .571 .610 .526 .560 .586 .618 .551 .583 .609 .640 The greater the batter, and consequently the greater the base, the less masonry is reqd to secure the same strength. Art. 14. The following: table (although not scrupulously correct) shows how greatly the safety of a wail sustaining water, is aifected by its form; the quantity of masonry re- maining the same. It will be observed that the vert wall is the least safe of all in the table. The overturning tendency of the water is here taken as 1. Iii France are dam walls from 4O to 7O ft high, with bases of -fjj the height; the face battering y 4 ^ of the height; the back, against which the water presses, being vert. These correspond to No. 12 of the following table, for which the resist is 2.6. But if the water pressed against the battered side, the resist would be 4.9 : as at No. 6, of the table ; but on this point see the next Art, inasmuch as theory and practice differ here. 530 HYDROSTATICS. All these walls contain precisely the same quantity of masonry. The masonry is supposed to be mortar rubble, weighing 125 fbs per cubic foot ; or twice as much Base in parts of height. Appro* resist ot wall. safety also will be greater or less, in precisely the same proportion. 1 Vertical wall 5 1 5 2 Face vertical ; back batters one-tenth height 55 1 8 3 " " " " onc-flfth " 6 2 2 4 625 2 6 5 ' one-third " 667 8 5 ' ' " " four-tenths " 7 4 9 7 " " ' " one-half " 75 14 8 Back vertical ; face batters one-tenth height .55 1.8 9 10 " " " " one-fourth " .6 625 2.1 2 2 H " * " one- third " 667 2 4 12 " " " " four-tenths " .7 2.6 13 14 Back and face each batter one- tenth height 'e 2.9 2 2 15 " " " < one-fifth " 7 3 4 16 75 4 6 17 " " " " one third " 833 9 18 " " " " four tenths " | 36 0) rbx When the base of a triangular wall, of sp grav 2, is less than % the height, the wall is theoretically safest when the water presses the vert side ; but if the base is greater than } the height, it is safest with the water on the battered side ; but for the practical view, see Art. 15. Art. 15. Our statement that walls are stronger with the water pressing? ag'aiiist the sloping* back rather than against the vert one, holds good so long as both the wall and the foundation may be considered unyielding, as has been the assumption of writers until within a short period. But of late years the erection of many enor- mous reservoir walls, some exceeding 160 ft iu ht, (the chief danger in which arises from their own wt.) has induced European scientists, Rankine among others, to investigate the subject very thor- oughly. They agree that the water should press against the vert back, because the pressure is then less ; and the resultaat falls farther back from the toe, thus distributing the pressure more equally not only through the masonry, but also along the foundation, thereby lessening the liability of either to yield. These practical considerations, they maintain, far outweigh that of the higher theoretical safety when the water presses on the slant back. To make this clearer, Fig 25, drawn carefully to scale, represents a dam wall at Poona. Hindoostan, de- signed by Mr. Fife, C. E. of England. It is built of heavy mortar rubble of 150 fts per cub ft. Its total vert ht is 100 ft ; thickness at base 60 ft 9 ins ; and at top 13 ft 9 ins. The front slopes 42 ft. and the back 5 ft, in 100. Its cen of grav is at G ; and 1 ft in length of the wall weighs 249.4 tons. Its foundation is 7 ft deep ; but we shall here assume the water to press against its entire back xv. This pres would be 139.6 tons. On ru it would be 151.4 tons. Through G draw a vert line G s ; and from c, where the direction of the pres P of the water strikes G s, lay off en by scale to represent the 139.6 tons water pres against x v ; and c t the 249.4 tons wt of 1 ft length of wall. Com plete the parallelogram of forces cnmt, and draw its diagonal c m. Then cm represents the resultant of all the pressures upon the base u v, and at I it cuts the base 20 ft back from the toe u. Doing the same with the 151.4 tons pressure .p against the back r u, we get the result- ant o y, which cuts the base at i, only 12.7 ft back from the toe v ; or 7.3 feet less than I is from u. The points I and tare called centers of resistance of the base; or centers of pres upon the base. See Rem 1, page 492. Now if the back xv were truly vert, the theoretical safety against overturning around u as a fulcrum the writer finds would be 2 ; with the actual xv, battered 5 ft, it is 2.2; and with the water pressing against ru it would be 3, around the other toe v; or 50 per ct greater than with the rert back ; or 36 per ct greater than with the actual back xv. If we treat a portion of the wall (as r xhf) as if it were an entire wall, with the water first against one back, and then against the other, we shall obtain the centers of resistance d and z in the bed-joint/ A. If in the same manner we as- ume three or four such walls, each having the top rx, we may find the centers r of resistance at other bed-joints. Through these centers draw the slightly curved dotted lines b, d, etc, I : and 6, z, etc, t; called lines of resistance, or lines of pressure;* which at any bed- joint in the ht of the wall show the point at which the pres upon it may be assumed to be concen- trated. It does not thow in what direction the pres comes to that point. Said direction is that of the resultant that cuts at that point. In Fig 25, it is seen that when the water presses the back xv, the line bdl of pres falls farther within the wall than when the water is against ru, in which case 6 z i would be the line. Hence both the wall and its foundation are less liable to derangement by frac- ture or crushing when the water presses against xv; and the earth foundation u v is then more evenly loaded, and hence less liable to yield unequally so as to cause cracks in the wall ; and * Moseley unfortunately applies this last term to another line seldom if ever used by engineers nd 'or which " Hue of resultants " would probably have answered as well. STATICS. 531 on this account x v is made theJjSck of the wall, notwithstanding that the theoretical safety against overturning would be 36 pejxa greater if the water pressed against r u. According to Rankine, the dist e I , or K i, from theartuer e of the base u v, to the point where the resultant c m or o y (as the case may be) cuts tbe ba^e-fshould not exceed .25 of the base u v, unless the foundation is unusually firm ; and on soft foundations it should cut at or near tbe center e. In tbePoona wall it is .173 of tbe base. He says, howler, that tbe common practice among British engineers is to make si or ei from .3 to .375, or from three-tenths to three-eighths o the base. The first would bring tbe resultant as near. and tbe last considerably nearer to the toe than i in the Poona dam would be to the toe v if the water pressed against ru. If from the end m or y of the resultant, we draw m 2 or y a horizontal, then c 2 or oa (as the case may be) will measure the entire vert pres on the base u v ; and m 2, or y a will measure the hor pres against the back of the wall, which tends to make the courses of masonry slide on each other as well as to make tbe whole wall slide on its foundation. Having this hor pres*, and knowing that the fric- tion of masonry on masonry is about .6 of the wt, or of the vert pres upon it, it is easy to ascertain if the wall is safe in this respect or not. Also, there can be no sliding at any bed-joint at which the resultant makes an angle not greater than about 30 or 32 with a line at right angles to said joint. See Art 63, p 486; and table, page 599. Such sliding never occurs in walls of ordinary forms. Good mortar well set aids against sliding ; but it is better not to rely upon it. Entire walls have slidden on slippery foundations. See Rem 2, p 331 : Rem 2, p 339; and Art 9, p 340, for preventives. When the resultant cuts at the center e of the base, the pres is distributed equally over the entire base; and its amount per sq ft of base will be = to ft ' which is the m a pres. But when as at I or i it cuts areMMM to q ft ' nearer to a toe, the pres will be greatest at that toe. and will diminish regularly to the other end of the base. When el or ei (as the case may be) is not greater than one-sixth of uv, then the max pres persq ft at the nearest toe is = mean presx(l + (6 - )j. When el or ei is more thaa / 2 v I3(.S eloref) \ * u ' one-sixth of u v, then max pres per sq ft at nearest toe mean pres X The max pres allowed should evidently not exceed the aafe strength per sq ft of the masonry or soil. To tind the least pres or that at the toe farthest from the resultant, subtract the mean pres from the max pres ; mult the remainder by 2 ; take the prod from the max pres. It may happen that the prod is greater than the max pres ; and when this is the case it shows that there is tension at the farthest toe; that is, that the wall at that toe does not pres at all upon the base, but has a tendency to rise, or to pull away from it. Thin may he exemplified by H, Fig 25, thus, draw a hor line ai for the base : at its center c draw the vert c v, equal to the mean vert pres ; and at i draw it for the max pres. Also draw tvn; then an give* the least vert pres on the base, being that at a. And any vert line drawn from t n to any point on a i will give the vert pres at said point. But if tbe mean pres should be as small as c e, then when we draw teo we find that a portion g o of it falls below the base a i ; showing that at g pres ceases, while tension begins, and increases to ao at a. In order that there shall be no tension, the resultant cm or oy must not cut the base uv nearer to a toe than one-third of u v, which is the dist in the Poona wall. Rankine states that in order that the masonry shall be secure against crushing?, the dist e /, or e i from the resultant to the center e of the / Twice the total vert pres on the base uv \ base must not exceed .5 - / 3 time8 the safe crushing^nguTof masonry^ ) . The vert P re sq ft X area of base uv in sq ft. ' and the safe strength both to be in the same terms, tons or Jbs. First-class rubble in cement mortar, or good cement concrete, should be safe with 8 tons pres per sq ft, which limit will rarely be reached ; and sound earth or gravel foundations sunk to a depth sufficient to protect them from frost, rain, sliding, s per sq inch, by the number denoting the reqd degree of safety. The quotient is the aafe cohesion. Divide the given interior pressure in B>s per sq inch by this safe cohesion. (Indeed this interior pres might first be reduced 15 fts per sq inch, on account of the outer pres of the air, which assists the pipe; but it is so small that this is rarely done.) Call the quot m. To half of m add 1. Mult the sum by m. Mult the prod by the inner rad in ins. Rent. 1. It is often known beforehand (as in large steam boilers,) that the safe thickness will he less than one-thirtieth of the rad; and in that case we may find the thickness by merely mult TO by inner rad in ins. Example. How thick should be the cast iron cyl of a hyd press of 14 ins bore, or 7 ins inner rad, to resist with a safety of 4 an internal pres of 2000 tts per sq inch; taking the ult cohesion of the iron at 18000 tts per sq inch ? 532 HYDROSTATICS. .2222 ; to which adding 1 we have 1.2222. And finally 1.2222 X .4*44 X 7 =: 8.8 ins, the reqd thickness. But to allow for irregular casting, air-bubbles, &c, make it .5 in more, or 4.3 ins. Rom. 2. Want of uniformity in the cooling: of thick castings makes them proportionally weaker thau thin ones, so that in order to reduce thickness in important cases we should use only best iron remelted 3 or 4 times, by which means an ult cohesion of about 30000 tbs per sq inch may be secured. But even with this precaution no rule will Pl>ly safely in practice to cast cylinders whose thickness exceeds either about 8 to 10 ins, or the inner rad however small. Under a pres of 8000 ft>s per sq inch, water will ooze through cast iron 8 or 1O ins thick ; and under but 250 Jbs per sq inch, through .5 inch. Table of thicknesses of single-riveted wrought iron pipes, tanks, standpipes. Ac, by the above rule, to bear with a safety of 6 a quiet pressure of 1000 ft head of water, or 434 Ibs \-e: sq inch ; the ult cob. of fair quality plate iron being taken at 480uO ft>s per sq inch, or at 8000 fts for a safety of 6 ; which is farther reduced to 8000 X .56 4480 ft>s, to allow tor weakening by rivet holes; for single-riveted cyls have but about .56 of the strength of the solid sheet; and double- riveted ones about .7. With the above pres and other data, the rule here leads to thickness .1016 X iuuer rad in ins. For a similar table for tanks, see p 484; and for cast iron and lead pipes, loot of this, and top of next page. (Original.) Di. Ins. Ths. Ins. Di. Ins. Ths. Ins. Di. Ths. Di. Ths. Di. Ths. Di. Di. Ths. .5 1.0 1.5 2.0 3.0 4.0 .025 .051 .076 .102 .152 .203 5 8 8 10 12 14 .254 .305 .406 .508 .609 .711 16 18 20 22 24 27 .813 .914 1 016 1.117 1.219 1.371 30 33 36 42 48 54 1.52 1 68 1.83 2.13 2.44 2.74 60 66 72 8i 96 108 3.05 3.35 366 4.27 4.88 5.49 120 132 144 19'2 240 288 10 11 12 16 20 24 6.09 6.70 7.31 9.75 12.19 14.63 For a less liead or pressure, or for any safety less than 6, it is safe and near enough in practice, to reduce the thickness of wrought iron cyls in the same proportion as said head, pres, or safety is less than - the tabular one. l>oiible-riveted cylinders. Fairbairn says, are about 1.25 times as strong as single-riveted. Hence they may be one-fifth part thinner. Lap-welded ones are nearly 1.8 times as strong as single-riveted; and hence may be- only .56 as thick. Many continuous miles of double- riveted pipes in California have beeu in use for years with safetys of but 2 to 2.6. In one case the head is 1720 ft, with a pres of 746 fts per sq inch ; diam 11.5 ins ; thickness, .34 inch ; safety, 2.6 by rule p 531 for such iron as in our table, Cast iron city water pipes require a somewhat greater thickness than that given by the rule, p 531, to enable those of small bore to endure the necessary handling, and to Srovide against irregular casting, and the air- bubbles or voids to which all castings are more or less able. In the writer's opinion experience has shown that if we employ one-eighth of the ultimate co- hesion of the iron in using the rule p 531, and then add .3 inch to every resulting thickness, weshall obtain satisfactory practical thicknesses. In preparing the following table the ult con of the cast iron is taken at 18000 fbs per sq inch ; and in the table .3 inch is added to each result of the rule. Table of Practical Thicknesses for Cast Iron Water-pipes. Heads of Water, in Feet. 1 50 | 75 100 125 150 200 250 300 400 500 600 700 | 800 1000 a Pressures in Pounds, per sq inch. 21.7 32.6 43.4 54.3 65.1 86.8 109 130 174 217 i 260 304 1 347 434 Thickness of Pipe in Inches. Original. 2 .31 .32 .32 .32 .33 .34 .35 ,36 .38 .40 .42 .45 ' .47 ; .51 4 .32 .33 .34 .35 .36 .38 .40 .42 .46 .50 .54 .59 ' .63 j .72 6 .33 .35 .36 .37 .39 .42 .45 .48 .54 .60 .67 .73 .80 .94 9 .34 .37 .39 .41 .43 .48 .52 .57 .66 .76 .85 .95 1.05 ! 1.25 12 .36 .39 .42 .45 .48 .54 .60 .66 .78 .91 1.03 1.17 1.30 , 1.57 15 .37 .41 .45 .48 .52 .60 .67 .75 .90 1.06 1 22 1.38 1 55 1.89 18 .39 .43 .48 .52 .56 .65 .75 .84 1.02 1.21 1.40 1 60 1.80 2.20 24 .42 .48 .53 .59 .65 .77 .90 1.01 1.26 1.51 1.77 2.03 2.29 2.84 30 .45 .52 .59 .67 .74 .89 1.04 1.19 1.51 1.82 2.13 2.46 2.79 3.47 36 .47 .56 .65 .74 .83 1.01 1.19 1.37 1.75 2.12 2.50 2.90 3.29 4.11 48 .53 .65 .77 .89 1.00 1.24 1.49 1.73 2.23 2.73 3.23 3.76 4.29 5.38 60 .59 .74 .88 1.03 1.18 1.48 1.79 2.08 2.71 3 33 3.97 4.63 528 6.65 72 .65 .83 1.00 1.18 1.36 1.72 2.09 2.14 3.19 3.94 4.70 5.49 6.28 7.91 84 .71 .91 1.12 1.33 1.53 1.95 2.38 2.80 3.67 4.55 5.43 6.36 7.28 9.18 96 .77 1.00 1.24 1.47 1.71 2.19 2.68 3.15 4.16 5.15 6.17 7.22 8.27 10.5 OSTATICS. 533 Table of thicfcrtess of lend I pipe to bear internal pressures with a afety of 6; taking UWlmimate cohesion of lead at 1400 Ibs per sq inch. By rule on p 531. Item. Although these thicknesses are safe againstquiet pressures.they might not resist shockaprfused by too sudden closing of stop-cocks against running water. See Service pipes, p 377. 1 1 1 I H 100 Hea 200 da in 300 Feet. 400 500 1 1 a 1 IK l^ IX 2 100 Hea 200 dsin 800 Feet. 400 500 Pr 43.4 esin] '86.8 bs per sq ir 130 1 174 ch. 217 Pr 43.4 esinl 86.8 bspe 130 p sq in 174 oh. 217 T .026 .038 .051 .064 .076 .089 hickn .055 .083 .111 .138 .166 .193 ess in .089 .134 .179 .223 .268 .313 Inchc .128 .192 .256 .320 .383 .447 58. .171 .256 .341 .427 .512 .597 1 .102 .127 .153 .178 .204 hickn .221 .276 .332 .387 .442 ess in .357 .447 .536 .626 .714 Inch* .511 .639 .767 .895 1.02 Mi .682 .853 1.02 1.20 1.36 Rein. The valves or stop-grates of water-pipes must be closed slowly, and this precaution increases with their diams. Otherwise the sud- den arresting of the momentum of the running water will create a great pressure against the pipes in all directions, and throughout their entire length behind the gate, even if it be many miles ; thus endangering their bursting at any point. Hence stop-gates are shut by screws, (p 573) which pre- vent any very sudden closing ; but in large diams even the screws must be worked very slowly to avoid bursting. Art. 17. The buoyancy of liquids. When a body is placed in a liquid, whether it float or sink, it evidently displaces a bulk of the liquid equal to the bulk of the immersed portion of the body ; and the body in both cases, and at any depth, aud in any position whatever, is buoyed up by the liquid with a force equal to the wt of the liquid so displaced. Thus, if we immerse entirely in water a piece of cork c, c, Fig 26, or any body of less sp gr than water, the cork will by its wt, or force of gravity, tend to descend still deeper; but the upward buoyant force of the water, being greater than the downward force of gravity of the cork, will compel the latter to rise with a force equal to theditf between the two. In this case, the cork receives a total downward pres equal to the wt of the vert column of water above it, shown by the vert Hues in vessel 1 ; and a total upward pres equal to the wt of the column shown in vessel 2. The diff be- tween these two columns is evidently (from the figs) equal to the bulk of the cork itself; therefore the diff between their wts or pressures, (or, in other words, the buoyancy of the water,) is equal to the wt or pres of the water which would have occupied the place of the cork : or, in other words, of the water which is displaced by the cork. This diff, or buoyancy, will plainly be the same at any depth whatever of entire immersion. Now the cork, if left to itself, will continue to rise until a por- tion of it reaches above the surf, as in vessel 3 ; so that the downward pressing column ceases to exist; aud the cork is then pressed downward only by its own wt. But as it now remains station- ary, we know (from the fact that when two opposite forces keep a body at rest, they must be equal to one another) that the upward pres of the water must be equal to the wt of the cork. But the upward pres of the water arises only from the shaded column shown in vessel 3; and this column is tas in the case of total immersion) equal to the bulk of water displaced. Therefore, in all cases, the buoy- ancy is equal to the wt of water displaced ; and when the body floats on the surf, the buoyancy, or the wt of water displaced, is also equal to the wt of the body itself. If the immersed body <% c, be of iron, or any other substance spe- cifically heavier than water, the diff between the upward and downward pres wilt of course remain the same ; or equal to the wt of water displaced. But the wt of the body is now greater than that of the water which it displaces; or, in other words, the downward force of gravity of the body is greater than the upward buovant force of the displaced water; and therefore the body descends, or sinks, with a force equal to the diff between the two. Thus, if the body be a cub ft of cast iron, weighing 450 Ibs, while a cub ft of fresh water weighs 62J^ Ibs, the iron will descend with an effective force of only 450 62^ = 387.5 fts. If the immersed body has the same sp j?r as the fluid, it will neither rise nor sink ; but will remain wherever it is placed ; because then the wt of the body, and the buoyancy of the water, are equal. For floating bodies, see p 635. The air also buoys bodies upward to an extent equal to the wt of air displaced; therefore, although a pound of iron, and a pound of feathers, weighed in the air, will balance each other yet in the exhausted bell-glass of an air-pump the feathers will outweigh the iron, by as much as the bulk of air which they displaced outweighs the bulk of air displaced by the iron. A balloon rises in the air on the same principle that corlt rises in water. Its ascending force is equal to the diff between its wt when full of ea<<. and the wt of the bulk of air which it displaces. The balloon does not actually tend to rise, but to descend; but the air being, bulk for bulk, heavier than the balloon, pushes the latter upward with more force than the gravity, or the wt of the balloon, exerts to bring it down. So also warm smoke has no tendency in itself to rise. It is pushed up by the heavier cold air. No substance tends to rise: but all tend downward teward the center of th earth. 534 HYDRAULICS. m Art. 18. A body lighter than water, if placed at tbe bottom of a vessel containing: water, will not rise unless the water can get under it, to buoy it, or press it upward, as the air presses a balloon or smoke upward. Thus, if one side of a block of light wood, perfectly flat and smooth, be placed upon the similarly flat and smooth bottom of a vessel, and held there until the vessel is Oiled with water, the downward pres will keep it in its place, until water insinuates itself beneath through the pores of the wood. But if the wood be smoothly varnished, to exclude water from its pores, it will remain at the bottom. Fig 28 On the other hand, a piece of metal may be pre- vented from sinking: in water, by subjecting it to a suffi- cient upward pres only, while the downward pres is excluded. Thus, if the bottom TV ft-* ^ an P en glass tube, t, Fig 27, and a plate of iron m, be made smooth enough to be |5 m , / water-tight when placed as in the fig ; and if in this position they be placed in a ,J vessel of water to a depth greater than about 8 times the thickness of the iron. th upward pres of the water will hold the iron in its place, and prevent its sinking,- because it is pressed upward by a column of water heavier than both the column of air, and its own weight, which press it downward. On this principle iron ships float. HEM. 1. A retaining-- wall, as in Fig: 28, founded on piles, may be strong enough to re- sist the pres of the earth e behind it, in case water does not find its way underneath; and yet may be overthrown if it does; or even if the earth around the heads of the piles becomes satu- rated with water so as to form a fluid mud. In either case, the upward pres of the water pgainst the bottom of the wall will vir- tually reduce the wt of all such parts as are below the water surf, to the extent of 62> H>s per cub ft; or nearly one-half of the or- dinary wt of rubble masonry in mortar. RKM. 2. Although the piles under a wall, as in Fig 28, may be abundantly sufficient to sustain the wt of the wall ; and the wall equally strong in itself to resist the pres of the backing e; yet if the soil as around tbe piles be soft, both they and the wall may be pushed outward, and the latter overthrown by the pres of the backing e. From this cause the wing-walls of bridges, when built on piles in very soft soil, are frequently bulged outward and disfigured. In such cases, the piling, and the wooden platform on top of it, should extend over the whole space between the walls; or else some other remedy be applied. Art. 19. Draught of vessels. Sincea floating body displaces a wt of liquid equal to the wt of the body, we may determine the wt of a vessel and its cargo, by ascertaining how many cub ft of water they displace. The cub ft, mult by 62V, will give the reqd wt in fl>s. Suppose, for instance, a flat-boat, with vert sides, 60 ft long. 15 ft wide, and drawing unloaded 6 ins, or .5 of a ft. In this case it displaces 60 X 15 X .5 = 450 cub ft of water ; which weighs 450 X 62J^ = 28125 Tbs ; which consequently is the wt of the boat also. If the cargo then be put in. and found to sink the boat 2 ft more, we have for the wt of water displaced by the cargo alone, 60 X 15 X '^ X 62>^ = 112500 Its ; which is also the wt of the cargo. So also, knowing beforehand the wt of the boat and cargo, and the dimensions of the boat, we can find what the draught will be. Thus, if the wt as before 140625 be 140625 fts. and the boat 60 X 15, we have 60 X 15 X 62^ = 56250; and = 2 5ft tbe required 56250 draught. In vessels of more complex shapes, aa in ordinary sailing vessels, the calculation of the amount of displacement becomes more tedious ; but the principle remains the same. Art. 2O. Compressibility of liquids. Liquids are not entirely in- compressible ; but for most engineering purposes they may be so considered. The bulk of water is diminished but about one-thousandth part by a pres of 324 fts per sq inch, or 22 atmospheres ; vary, ing very slightly with its temperature. It is perfectly elastic ; regaining its original bulk when the pres is removed. HYDRAULICS, A rt. 1. Hydraulics treats of the flow or motion of water through pipes, aqueducts, rivers, and other channels; also- through orifices or openings of various kinds : of machinery for raising water ; as well as that in which water furnishes the moving power. The science of hydraulics, in many of its departments, is but imperfectly understood ; therefore, some of the rules given on the subject are to be regarded merely as furnishing close approximations to the truth. On the flow of water through pipes. See Caution, p 566. Inasmuch as the experiments on which the following rules are based, were made with pipes care- fully laid in straight lines ; and perfectly free from all obstructions to the flow of the water, some allowance must in practice be made for this circumstance. Workmen do not lay long lines of pipes in perfectly straight lines ; it is almost impossible to avoid very numerous, although slight devia- tions, both vert and hor ; the soil itself, in which the pipes are imbedded, especially when in embkt, will settle unequally ; especially in streets liable to heavy traffic, which not only frequently deranges, but occasional! v breaks water pipes whose tops are 3 or 4 ft below the surf. The material used for calking the joints, may be carelessly left projecting into the interior, and thus cause obstructions; the water is frequently muddy, or is impregnated with certain salts, or gases, which form deposits, or incrustations, wh ch materially impede, the flow. See Art 27. Moreover, the pipes themselves are not cast perfectly straight, or smooth, or of uniform diam ; and irregular swellings, by producing eddies, HYDRAULICS. 535 retard the flow as wirtfas contractions ; and accumulations of air do the same. Under the most favor- able circuustaric*r therefore, it is expedient to make the diams of pipes, even lor temporary pur- poses, sufficiency large to discharge at leant 20 per ct more than the quantity actually needed ; and if there is qnseasion to anticipate deposit, or incrustation, a still larger allowance should be made in permanent pipes, especially iu those of small diam ; because in them, the same thickness of incrusta- tion occupies a greater comparative portion of the area. Perhaps it would be best to allow an equal increase, of say from J^ to 1% inch, to each diam, whether great or small ; inasmuch as the thickness of incrustation will be the same for all diams, or nearly so. The cost of pipes does not increase as rapidly as their discharging capacities ; thus, if the diam be Increased only -fa part, the disch will be increased about 25 per cent; if % part, nearly 50 per cent,' if J^ part, the disch will be doubled. See Table 2. Within these limits, the increase of thickness for the larger diams, and the increased expense of laying, will add but little to the cost; vhich will therefore augment only a little more rapidly than the diahis. The increased diam involves no waste of water ; since the disch may be regulated by stopcocks. LEVEL eL-li The term HEAD or TOTAI* HEAD of water, as applied to the flowage of water through canals, pipes, or openings in reservoirs, &c, means the vert dist i v or p o, Fig 1, from the level surf, mi, of the water in the reservoir, or source of supply, to the center (or more properly to the cen of grav) o, of the orifice (whether the end of a pipe, r o, t o, v o,zo. lo; or any other kind of opening) through which the disch takes place freely, into the air ; or the vert dist a u, or /.a, from the same surf, m i, to the level surf, g u, of the water in the lower reservoir; when the disch takes place under water. Thus, in the case of disch into the air, the vert dist i v orpo, is the total head for either of the pipes ro, t o, v o, zo, or I o; and i k is the head for the orifice, k, in the side of the reservoir. And for disch under water, au. or / g, is the head for either the pipe j, or the opening n; without any regard whatever to their depths below the surf of the lower water; which, according to the older authorities, do not at all afiect their disch. Weisbach, however, a more recent experimenter, says the disch will be about -fa part less under water, than into the air. In the case of circular pipes or openings ; or of rectangular, or many other shaped openings, the center, and the cen of grav coincide: but iu triangular or trapezoidal ones, they do not; hence in all cases of disch into air the head must be measured totheceu of grav of the disch'opening ; as it is this head alone that causes the flow. The pres which it imparts to the water, overcomes the friction op- posed by thu sides and bottom of the channel, and thus enables the water to advance. After water has descended along any channel, to any given dist, that head by which it was driven through that dist, is sometimes called lost head; or the water is said to have lost so much head in that dist. A portion of a pipe mav have a head greater than the total head of the entire pipe. Thus the point 6 in the pipe lo, has a head 6 1 ; while the entire pipe has only the head p o. Both in theory and in practice it is immaterial as regards the vel, and tSie quantity of water discharged, whether the pipe is inclined downward, as ro. Fig* 1; or lior. as r o: or in- clined upward, as lo\ provided the total head po, and also the length of the pipe, remain nn changed. If one pipe is longer than another, its sides will evidently present more friction against the water, and thus diminish the vel and the quantity of disch. The inclined pipes, r o, I o, being of course a little longer than the nor one uo, will therefore each disch a trifle less water: but if the hor one were extended slightly beyond o, so as to give it the same length as the others, then each of the three would disch the same quantity in the same time. Rem. 1. It is evidently necessary that the upper or entry endr, v or Z of the pipe be so far at least below the surface m f of the water as to enable the water first to overcome the resistance which the sharp edges of the end of the pipe oppose to its entrance ; and then to force the water into the pipe with the same vel it i* intended to have through it. The total head p o must be considered as di- vided into three parts, of which either p w or i may represent the two devoted to the two above dis- tinct duties ; while the third part or the remainder w o or s v overcomes the resistanoe of friction, &e, inside of the pipe, and is hence called the friction head, or the resist- ance head ; while the first two parts are called the entry head, and the velocity head. The vel head is the same as the theoretical head in Table 10, p 552; and experiment shows that with the usual sharp edged entry, the entry head is (near enough for practice) half as great. If the entry is shaped like Pig 7, p 554, the entry head disappears almost entirely. This, however, has hut little effect on the vel or discharge, except in pipes shorter than 1000 diamg. It becomes more apparent as they shorten. The friction head may be all above, or all 536 HYDRAULICS. below the entrance into the pipe, or part above aud part below, without affecting the vel or disahare* of the water. Since the friction, in pipes of the same diam, increases in amount in the same proportion as their lengths, the water when it first enters the pipe encounters but little friction, and has great vel ; but this gradually decreases as the advancing water encounters the friction along increased lengths of the pipe; and finally becomes slowest when the water fills the whole length, and begins to flow from the disch end o. The vel then becomes uniform along the pipe so long as the entry and vel heads (is) are sufficient to allow the water of the reservoir to enter the pipe with that name veL If even much more th;in these two heads is left above the entrance into the pipe, the effect of the surplus is not at all diminished thereby as regards overcoming the friction along the inside of the pipe; and con- sequently the vel of 'the disch will undergo no change. Thus, if i g he sufficient head for a pipe laid direct from to o, and if the pipe be afterward changed to the position I o, the vel and disch will be the same in both positions. Therefore nothing more is necessary than to be certain that the depth below the surface m i of the reservoir shall not be less than t *. Theoretically, is at the center (or rather, at the cen of grav) of the entry opening of the pipe, or aperture ; and the head i t. above it. is equal to J.5 times that found in Table 10. opposite to the vel in the pipe. Thus, if we have found by calculation (by Rule 1. following) that the total head p o will produce along the pipe a vel of 6 ft per sec, we find in Table 10, opposite the vel of 6 ft. the vel head .56 ft. Therefore f .56 4- .28 = .84 ft is the least dist that the center of the end of the pipe must be placed below the surf m i. But the end of the pipe should in practice always be entirely below water; otherwise air and floating im- purities will be drawn into it, and cause obstructions ; therefore> if the pipe is large, say in this case 3 feet in diam, it is evident that the center of its upper end cannot be placed lees than \% ft under water; or nearly 3 times as deep as the true theoretical vel head. Moreover, the water surface of reservoirs is always liable to considerable changes of height; so that the end of the pipe must be placed at such a depth that the water can flow into it at the lowest stages. As before stated, this will be attended by no diminution of disch. The above vel of 6 ft per sec, is full 4 miles an hour, and is one seldom reached in water pipes in practice ; hence we see that in ordinary cases the vel aud entry heads together need in theory rarely exceed .56 -f- -28 = .84 of a foot, which is usually but a small part of the total head, p o. A straight line a o, drawn from the above S to the cen of grav o of the disch end, is called the tllllic grade-lilt of any pipe so, r o, t o, vo, z o, or I o, commencing at the ir, and ending at o; and any such pipe may be bent into easy curves, (having radii not less than 5 diaras of the pipe.) as t o, zo; and still deliver as much water a* a straight one of the same actuil length, provided the tops of its highest bends are kept below this hydraulic grade-line; and provided that arrangements be made for the escape of any air that may accumulate at the tops of the bends. See Art 5, and Fig 42. If the hydraulic grade-line so be divided into any number of equal parts, as c, ex, kc; and if the actual" length of any pipe terminating at o, be afso divided into the same number of equal parts, then will b c be the friction head of the first division of the pipe ; zx that for the lir.st two divisions, Ac. If the pipe commences vertically under s, like all those in Fig 1 : and is straight, like r o, vo, or lo, the equal divisions of the pipe will be vert under those of the line so. But if the pipe be curved either hor, or vert, like to and z o, this of course will not be the case. REM. 2. A great diff exists between the condition of a hor pipe, and one inclining- downward from the reservoir, in case it should become necessary to prolong* them in their original directions. Thus, if we extend the length of the hor pipe r o be- yond o, it is plain that we do not thereby increase the total head of water; for the vert dist from the surf of the reservoir, to the di*ch end of the pipe, will still remain equal to po. Consequently the additional friction along the sides of the extended pipe, will cause the water to flow more slowly than before; or, in other words, the disch will be less. Now suppose an inclined pipe so, atfirtinp from the exact, vel head s. It is plain that every equal dist, s c, ex, &c. along this pipe, has its equal fric- tion head be, d x, &c; and however far the pipe may be extended beyond o. with the same degree of inclination, its friction head will be extended in the same proportion; and consequently the friction along the additional length will be thereby counteracted, so that no chance will take place in the vel of the flow; nor, consequently 1 , in the quantity discharged. The vel head will remain the same as before, for it still has merely to supply the same vel of water as at first. Such a pipe is said to be in train. This will not be the case to the same extent if the entry end of the pipe, as a. Fig 1%, be placed be- low a; for it is evident that if ao be the original length of the pipe, and wo its friction hejvd ; then if we double this length, by extending it to n, we do not thereby double the friction head ; for the new friction head m n is not equal to twice wo. and the disch will of course be slower. Still, if s a is (as will often be the case in practice) very small in proportion to the friction head wo of the original pipe, then mn will be so nearly twice w o, that the diminution of dich from extending the pipe to any reqd dist, will be but slight. REM. 3. Of the outward, or bursting* pressure, of water in pipes. When any pipe, as, for instance, any of those in Fig 1, is full of water at rest, this pres is greater than when the water is flowing through it; and is that due to the total head above the point pressed. Thus, at the point 4 on the pipe r o, it is that due to the head 4, 1 : at the point fi in the pipe I o, that due to the head 6, 1 : at the point o in any of the pipes, that due to o p, Ac. Therefore it may always be readilv calculated in Ibs. by the Rule p 522, of Hydrostatics; namely, mult together the area in sq ft, of the portion pressed ; the total head, in ft. above its cen of grav ; 'and the constant number 62.5. If the discharge end of the pine be partially opened, the water will move slowly, and the prea will become less, and when the water ii flowing freely through any Fi HYDRAULICS. 537 full pipe of iMrttorm diam, the ends of which are entirely open, it becomes still more reduced; and ff the plp^has its entry end precisely at the vel head s, Fig 1, aud itself lying upon the hydraulic grade-line s o. (a case which never occurs iu practice,) there will be uo bursting pres ; and the pipe will experience uo pres of any kiud. except that of the weight of the ruuuiug water upon its lower side. But if any part of the ruuuiug pipe, or all of it, be below the hyd grade-line, as iu all the pipes in Fig I. then every point of such pipe will be subject to a bursting pres due to a head equal to the vert dist of said point, below the yradv-line ; aud which consequently may be calculated in Ibs, in Precisely the same way as in the preceding case of water at rest. Thus the point 4, in the pipe ro, 'i* I, will be pressed outward by the head 4, 3 ; the point 5, in the pipe to, by the head 5, 3; the point 6, iu lo, by 6. 3; aud the point 7 will be pressed outward by the head 7, 8; &c. If at any point iu any pipe thus below the hydraulic grade-line, aud discharging freely, au open vert commu- nicating pipe be inserted, the flowing water will rise in it to the level of said grade-line. Thus, it will rise iu a pipe from 6 to 3, from 5 to 3, from 4 to 3, from 7 to 8, &c. Advantage may be taken of this, to supply water, or establish fountains, at intervals along a great line of pipes. Small vert pipes of this kiud, called piezometers, (pressure measurers,) either made of glass, or else furuished with a floating index, are sometimes used for detecting the position of accidental obstruc- tions iu a line of pipes. If the water in these is found at any time to fall below its proper level, it shows that the pres upon it has become diminished by some obstruction in the interval between it and the outlet, or disc becomes a maximum ; and the water some obstruction, or partial closing of the pipe, between the pie: end. If the outlet be entirely closed, the pres on the piezometer 1 will rise in it to a level with the surf mi of the water in the reservoir. By having several piezome- ters, the point at which an obstruction has taken place can be approximately ascertained; aud thus much labor saved in searching for it. If the lower end o. Fig 1. of any of the pipes, instead of discharging freely into the air, discharges under water of which t.h represents the surf, then the hydraulic grade-line must be drawn from s to e, instead of to o ; and p e becomes the head, instead of p o. RKM. 4 When the pipe rises above the hydraulic grade-line in any part, an entire change of condition takes place throughout. In Fig 1^>, the pipe agno rises above the grade-line so; which, however, can no longer be properly so called; for the pipe must now be considered as divided into two sections, agn, and nyo; each having its own grade-line, as t n, no. In a long undulating line, B m "\ "T-- _ *- -- i n .._..! n it may thus become necessary to consider many separate sec- tions, with their respective grade-lines. In our fig, the sec- tion agn has the friction head TIT; only ; and the shorter section ny o has the greater friction head t o ; consequently the water would move more slowly in agn than in nyo; and would re- quire a greater diam than it, in order to carrv the same quan- tity of water." Or, if the diam tire pipe, then the quantity of water delivered at o. will be that due to the small head n b only. It will then flow from a to nulling that portion of the pipe; but from n to o the pipe will not be full, but will carry otf the water as iu an iron gutter. The bursting pres at any point, as g, of the section agn, will be measured by its corresponding vert line, as g c ; and if the sections have different diams, (proportioned to their vels,) the pres at any point, as ?/, in section nyo, will be measured by its corresponding vert, as ?/ x. But if both sections be of the same diam, then, since section nyo becomes virtually an open gutter, it can experience no burst- ing pres. If a pipe be closed suddenly, the arrested momentum of the flowing water will exert a great pres, sufficient in most cases to burst with ease any ordinary street pipe. These are therefore closed very slowly by valves moved by screws; see Figs 35 to 36 j. Leaden service-pipes in dwellings are fre- quently burst by closing the stopcock too quickly. RKM. 5. Resistance to flow in pipes of diff materials. It was formerly supposed that the material of which the pipe was made, exerted no influence on the flow, provided the insideswere equally smooth; but later observations show that this is not the case. Weis- bach states that in wooden pipes of from 2V to 4> ins diam. he found the frictional resistance to be as much as ]% times as great as in equally smooth cast-iron ones. If this be correct, we infer that the friction heads only, in table p 544, Ac, should be multiplied by 1.75 to adapt them to wooden pipes. Moreover it is said that Darcy found the friction in corroded iron pipes to be twice as great as in new smooth ones. If so the friction heads in the table should be multiplied by 2 for such pipes. Darcy found that the usual formulas (those here given) agreed sufficiently closely with the ac- tual results with perfectly clean smooth cant-iron pipes ; except that they made the disch rather too large in small pipes ; and rather too small in lare;e ones ; hot that when the insides of the pipes were smoothly covered with pitch, the disch was increased about % part; and became about equal to that through tubes made of glass. The vel hea.d/or any given vel, of course, remains the same for all ma- terials and diams of pipe ; only \\\e friction head will vary. See Remark 2, Art 4. The later researches of Ganguillet and lint tor on this subject are very im- portant; see p 651. Art. 2. To find the velocity, and the quantity discharged through a straight, smooth, cylindrical cast-iron pipe, r o, v o f or I 0^ Fig 1, whose length is not shorter than 4 times its diam; knowing its total head po; its length; and its diam, or bore. 538 HYDRAULICS. Rule. Mult thediam in feet, by the total head in feet. Call the prod a. Add Approx vel in ft per sec Total length in ft + 54 diams in ft. For any head not less than at the rate of 4 ft per mile,* multiply theapprox vel thus found by the number corresponding to the diam in ft in the table below. If the pipe is m good order, this last vel will probably be within 5 to 10 per ct of the truth, inasmuch as it corresponds with Kutter's. Diam in Ft. No. Diam in Ft. No. Diam in Ft. No. Diam in Ft, No. .1 .48 .6 .87 1.5 1.10 4 1.37 .2 .63 .7 .91 2. 1.18 5 1.42 .3 .71 .8 .95 2.5 1.24 6 1.46 .4 .77 .9 .98 3. 1.30 7 1.50 .5 .82 1.0 1.00 35 1.34 10 1.60 Then to find the discharge in cub ft per see, mult the vel last found, by the area of circular transverse section of the pipe in sq ft. Said area may be taken from Table 3, p 541. If the disch end of the pipe is under water, the total head is the vert dist between the surfs of the water of the two reservoirs. Ex. A straight pipe a mile, or 5280 ft long ; with a diam of 1 foot, has a total fall of 12 ft, measured from the water surf in the reservoir, to the cen of grav of its lower end or opening. With what vel will the water flow through it ; and how much will be dischd per sec? Here the diam in ft, mult by the total head in ft = 1 X 12 = 12 = a. Again, the length in ft is 5280 ; and 54 times the diam in ft is 54 ; and these two added 5334 = .00225. The sq rt of together = 5334. And the prod a, div by 5334, = .00225 is .04743. And .01743 X constant 48 = 2.27 ft per sec approx vel. The num- ber in the above table, for 1 ft diam is also 1 ; therefore, 2.27 X 1 = 2.27 ft per sec the reqd vel. Dischargee. The area of cross section of a pipe 1 ft diam is .7854 of a sq ft. Hence, 2.27 vel X -7854 area = 1.782 cub ft per sec disch. Item. 1. Table Hfo. 1, calculated by this rule, shows the vels and dischgs through a pipe, one mile long, and 1 ft diam, under different heads. But they will be very nearly the same for any greater lengths; and also quite approximate for shorter ones not less than 1000 or even 500 diams long, provided that in all cases they have the same RATE O/HKAD ; that is, if the given pipe of 1 it diam, is 2 or 3 miles long, it must have 2 or 3 times as much head as the pipe in the table which is 1 mile long; or if the given pipe of 1 ft diam, is %, %, or %, &c, of a mile long, it must have but y, ^, ^ as much total head as the 1 mile one in the table, in order to have very nearly the same vel and disch. Special Note. The above rule and formula for finding the first or approx vel, are modifications by Poncelet of the original by Eytelwein ; and were until within a few years, quite generally accepted by engineers as correctly applicable by themselves to all diams. without the use of such a Table as the above. But later experiments have proved that such was not the case, and therefore the writer has, by the aid of Kutter's formula, added the above table of corrections. See " antion," p 566 ; and Matter's formula, p 650. Rem. 2. When we speak of the vel of water in a pipe, river, &c, we allude to the mean vel of the entire cross section of the water. As in a river the vel is usually greater half way across it, and at the surface, than it is at the bot- tom and sides, so in a pipe the vel is actually greater at the center of its cross sec- tion than at its circumf. The mean vel referred to in our rules is an assumed uni- form one which would give the same discharge that the actual ununiform one does. * About .9 of an Inch per 1OO ft. fDRAULIOS. 539 TABLE 1. Oftlie actual velocities and discharges through a pipe, 1 ft fii diams 1 mile, or 528O diams in length; and of cast iron ; smooth, and straight. Head is the vert dist from the surf of the water in the reservoir, to the ce'n of grav of the lower end of the pipe, when the disch is into the air; or to the level surface of the lower reservoir, when the disch is under water. Head in Ft. per 100 Ft. Head in Ft. per Mile. Velocity in Ft. per Second. Discharge in Cub. Ft. per Second. Discharge in Cub. Ft. per 24 Hours. Head in Ft. per 100 Ft. Head in Ft. per Mile. Velocity ii Ft. per Second. Discharge in Cub. Ft. per Second. Discharge in Cub. Ft. per 24 Hours. .0019 .1 .208 .1633 14114 1.515 80. 5.85 4.602 397613 .0038 .2 .293 .2301 19880 1.704 90. 6.23 4.900 423435 .0057 .3 .359 .2819 24360 1.894 100. 6.56 5.144 444312 .0076 .4 .415 .3267 28229 2.083 110. 6.87 5.395 466128 .0095 .5 .464 .3638 31435 2.272 120. 7.18 5.639 487209 .0114 .6 .508 .b989 34464 2.462 130. 7.47 5.866 506822 .0132 .7 .549 .4311 37217 2.652 140. 7.76 6.094 526521 .0151 .8 .585 .4602 39760 2.841 150. 8.05 6.322 546048 .0170 .9 .623 .4901 42343 3.030 160. 8.30 6.534 564576 .0189 1. .656 .5144 44431 3.219 170. 8.55 6.715 580176 .0237 .25 .735 .5753 49701 3.408 180. 8.80 6.903 596418 .0284 .5 .805 .6322 54604 3.596 190. 9.04 7.100 613440 .0331 .75 .871 .6832 59011 3.788 200. 9.28 7.276 628704 .0379 2. .928 .7276 62870 4.261 225. 9.84 7.696 664848 .0426 .25 .984 .7696 66484 4.735 250. 10.4 8.168 705728 .0473 .5 1.04 .8168 70572 5.208 275. 10.8 8.482 732844 .0521 .75 1.08 .8482 73284 5.682 300. 11.3 8.914 769824 .0568 3. 1.13 .8914 76982 6.629 350. 12.3 9.621 831168 .0758 4. 1.31 1.028 88862 7.576 400. 13.1 10.28 888624 .0947 5. 1.47 1.150 99403 8.532 450. 13.9 10.91 943056 .1136 6. 1.61 1.264 109209 9.47 500. 4. 11.50 994032 .1325 7. 1.74 1.366 118022 10.41 550. 5. 12.09 1044576 ,1514 8. 1.86 1.455 125740 11.36 600. 6. 12.64 1092096 -1703 9. 1.96 1.539 132969 12.30 650. 6. 13.11 1132704 4894 to. 2.08 1.633 141145 13.25 700. 7. 13.66 1180224 .2273 12. 2.27 1.782 153964 14.20 750. 18.0 14.13 1220832 .2652 14. 2.45 1.924 166233 15.15 800. 18.6 14.55 1257408 .3030 16. 2.62 2.057 177724 16.09 850. 19.1 15.00 1296000 .3409 18. 2.78 2.183 188611 17.04 900. 19.6 15.39 1329696 3788 20. 2.93 2-301 198806 17.99 950. 20.3 15.94 1377216 .4735 25. 3.28 2.572 222156 18.94 1000. 20.8 16.33 1411456 .5682 30. 3.59 2.819 243604 22.73 1200. 22.7 17.82 1539648 .6629 35. 3.88 3.047 263260 26.52 1400. 24.5 19.24 1662M6 .7576 40. 4.15 3.267 282288 30.30 j 1600. 26.2 20.57 1777248 .8523 45. 4.40 3.451 298209 34.08 : 1800. 27.8 21.83 1886112 .9470 50. 4.64 3.638 314352 37.87 1 2000. 29.3 23.01 1988064 1.136 60. 5.08 3.989 344649 47.35 i 2500. 32.8 25.72 2221560 1.326 70. 5.49 4.311 372470 56.81 3000. 35.9 28.19 2436040 To reduce cub ft to U. S. gallons, mult by 7.48. Since, therefore, 8 cub ft are equal to 60 gals, (very nearly,) if we divide the cub ft per 24 hours, by 8, we get the number of persons that may be daily supplied with 60 gals each, by a pipe constantly running full, and at the vel given in the third col. This condition does not exist in city water-pipes; the water in them being comparatively stag- nant. Therefore, the results of the rule and table do not at all apply to them. See Art 33, p 580. KKM. If the pipe, instead of being straight, has easy carves. (say with radii not less than 5 diams of the pipe,) either hor or vert, the disch will not be materifllly diminished, so long as the total heads, and total actual lengths of pipes remain the same; provided be made for the ecape of air accumulating at the tops of the curves. See Fig 42, p 579. Notwithstanding what is said about bends on pages 549, 550, we advise to make the radius as much more than 5 diams as can conveniently be done. To find either the area of pipe, opening, or channel-way ; or the mean vel; or the quantity discharged, when the other two are given. This applies to openings in the sides of vessels, to rivers, and to all other channels aa well as to pipes. Disch in cub ft Disch in cub ft Area in _ per sec Dd Mean vel _ per secoud S( * feet " mean vel in in ft per sec &rfia , n feet per sec. sq feet. Disch in cub ft _ area in x mean vel in per second sq feet ft per second. Or all the terms may be in inches instead of feet ; and minutes or hoars instead of second*. 540 HYDRAULICS. TABLE 2. (Original.) Of the rliani of long pipe reqcl to deliver either more or lea* water than that of 1 ft diam in Table 1, under the same rate of incli- nation, or of head in ft per mile. To Hud the disch (but uot the vel) through another pipe, not less than about 1000. or at least 500, of its own diams in length ; first take out the disch through the 1 ft one, from Table 1. Div the reqd disch by this tabular one. Look for the quot in the third column of Table '2, of proportions of disch ; and opposite to it, in columns 1 and 2, will be found the reqd diarn. See Rem 2, p 543. From this table we see that a 13-inch pipe will deliver nearly 1% time* as much as a 1 ft one ; a 14-inch one, nearly 1% times; a 15-inch oue, nearly 1% times; and a 16- inch one, fully twice as much as the 1 ft one; &c, of the same length and bead. This use of this table is not sufficiently correct for pipes less than about 1OOO (or at furthest 5OO) diams long-; therefore, for shorter ones, use preceding Rule, p 538. For more on finding diams required for a giveu dis- charge, see Art 4. See Caution, p 566. Diam. Diam. Proportion of Disch. to that Diam. Diam. Proportion of Disch. to that Diam. Diam. Proportion of Disch. to that of long of long through a 1 ft. of long of long through a 1 ft. of long of long through a 1 ft. Pipe, In Inches. Pipe, in Feet. lung Pipe, with the same Pipe, in Inches. Pipe, in Feet. lona Pipe, with the same Pipe, in Inches. Pipe, in Feet. long Pipe, with the same head per mile. head p-.-r mile. head per mile. 1 .0833 .0020 10^ .8750 .7157 28 2.333 8.319 13* .1250 .0055 11 .9167 .8044 30 2.5 9.822 2 .1667 .0113 .9583 .8987 30J4 2.521 10.0 .2083 .0198 12 32 2.667 11.6 3 .2500 .0310 ISM 1.'042 !l06 34 2.833 13.5 3J* .2917 .0458 13 1.083 .221 36 3. 15.5 4 .3333 .0643 14 1.167 .470 38 3.167 17.8 4/* .3750 .0857 15 1.250 .746 40 3.333 20.2 5 .4167 .1119 16 1.333 2.053 42 3.5 22.9 .4583 .1422 17 1.417 2.388 44 3.667 25.7 6 .5 .1767 18 1.5 2.754 48 4. 32.0 6Va .5417 .2159 19 1.583 3.153 54 4.5 42.0 7 .5815 .2600 20 1.667 3.585 60 5. 54.9 .6250 .3090 21 1.75 4.051 66 5.5 69.8 8 .66S7 .3631 22 1.833 4.551 72 6. 85.8 .7083 .4220 23 1.917 5.084 78 6.5 104.8 9 .75 .4871 24 2. 5.649 84 7. 126.1 .7917 .5575 2.052 6.000 96 176. 10 .8333 .6337 26 2.167 6.912 120 10'. 302. Examples of the use of Tables 1. and 2. Having a head from a reservoir to a certain point of delivery, of 20 ft, in a dist of 1860 ft, and wishing to receive 6 cub ft of water per sec; what must be tiie diam of a pipe to accomplish this? In the first place, we find that a fall of 20 ft in 1860, is equal to a fall of 1.075 ft in 100 ft. Then we see by Table 1, that with a fall of 1.075 ft in 100, a long pipe of 1 ft diam yields about 3.8 cub ft per sec. But we want -- = 1.58 times as much as the 1 ft pipe can deliver ; and by Table 2, we see that the pipe, to do this, under the same rate of head, must be about 14}^ ins in diam. In practice, we should adopt at least 15 ins. Near enough, we may say that double the diam gives 5% times the disch. TABLE 2^. Weight of water (at 62V Ibs per cub foot) con- tained in one foot length of pipes of different bores. (Original.) Bore. Ins. Water. Lbs. Bore. | Water. Ins. I Lbs. Bore. Water. Lbs. Bore. Water. Lbs. Bore. Water. Lbs. Bore. Ins. Water. Lbs. K .00531 2. 1.3581 3% 5.0980 T% 19.098 13V$ 61.877 22 164.33 K .02122 % 1.5331 4. 5.4323 % 20.392 14. 66.545 23 179.60 .04775 14 1.7188 6.1325 8. 21.729 71.384 24 195.56 H .08488 5 1.9150 y* 6.8750 y 23.109 15. 76.392 25 212.20 .13263 3 2.1220 H 7.6601 y* 24.530 J* 81.568 26 229.51 % .19098 M 2.3395 5. 8.4880 % \ 25.993 16. 86.916 27 247.51 .25994 X 2.5676 \/ 9.3580 9. 27.501 y^ 92.434 28 266.18 I. .33952 y 2.8063 X 10.270 H 30.641 17. 98.121 29 285.53 If .42969 3 3.0557 11.225 10. 33.952 H 103.97 30 305.57 y .53050 % 3.3156 6. 12.223 ^ 37-432 18. 110.00 31 326.27 y .64190 3.5862 13.262 11. 41.082 VJ 116.20 3'J 347.66 % .76392 iU 3.8673 ix 14.345 y* 44.901 19. | 122.56 33 369.74 RZ .89654 1Z 4.1591 8/ 15.469 1? 48.891 129.10 34 392.48 % 1.0398 RX 4.4615 7. 16.636 H 53.04!) 20. 135.81 35 415.90 % 1.1936 % 4.774f> 17.846 13. 57.379 21. 149.73 36 440.00 And in larger pipes, as the squares of their bores. Thus a pipe of 40 or 60 ins bore, will contain 4 times ag much as one of 20 or 30 ins bore ; and one of -j&r, % as much at one of % inch. At 62J/4 Ibs per cub ft, a sq inch of water 1 ft high weighs .432292 of a fl). HYDRAULICS. 541 TABLE reas and Contents of Pipes ; and square roots of IHams. (Original ) Correct. Diarn. iu lus. s Diam. in Feet. Area in sq ft, also cub ft, iu 1 foot length of Pipe. Sq. rt. of Diam. iu Ft. Diam. iu lus. Diam. iu Feet. Area in sq ft, also cub ft, iu 1 foot length of Pipe. Sq. rt. of Diam. iu Ft. Diam. lus. Diam. iu Feet. Area in sq ft, also cub ft, in 1 foot length of Pipe. Sq. rt. of Diam. in Ft. U .0208 ,0003 .145 4. .3333 .0873 .579 15. 1.250 1.227 1.118 516 .0260 .0005 .161 % .3438 .0928 .588 24 1.271 1.268 1.127 H .0313 .0008 .177 l i .3542 .0985 .590 /^ 1.292 1.310 1.136 716 .0305 .0010 .191 y .3646 .1040 .604 K 1.313 1 353 1.146 }i .OUT .0014 .2C4 x .3750 .1104 .612 16. 1.333 1.396 1.155 916 .0469 .0017 .217 % .3854 .1167 .621 M 1.354 1.440 1.163 y .0521 .0021 .228 H .3958 .1231 .629 1.375 1.485 1.172 1116 .0573 .0026 .239 y .4063 .1296 .637 H 1.396 1.530 1.181 % .0025 .0031 .250 5 .4167 .1363 .645 17. 1.417 1.&74 1.180 1816 .0677 .0036 .2GO K .4271 .1433 .653 i/ 1.437 1.623 1.199 H .0729 .0042 .270 X .4375 .1503 .6CO % 1.458 1.670 1.207 15-16 .0781 .0048 .280 % .4479 .1576 jm X 1.479 1.718 1.216 1. .0803 .0055 ,28i) X .4583 .1650 .677 18. 1.5 1.767 1.224 1-16 .0885 .0002 .297 N .4688 .1725 .(JS5 y* 1.542 1.867 1.241 M .C938 .0009 .305 a i .4792 .1803 .(il)3 19. 1.583 1 969 1.258 316 .o;)90 .0077 .314 y .4896 .1878 .700 M 1.625 2.074 1.274 X .1042 .0085 .322 6. .5 .n:64 .707 20. 1 667 2.18'! 1.291 516 .1094 .009* .330 ft .5208 .2131 .722 1 A 1.708 2 292 1.307 % .1146 .0103 .338 S .5417 .2304 .736 21. 1.750 2.405 1.323 7-16 .1198 .0113 JU4 H .5625 .2485 .750 y* 1.791 2.521 1.339 K .1250 .0123 .35 1 7. .5833 .2673 .764 22. 1.833 2.640 1.354 9-16 .1302 .0133 .3 42. 3.500 .621 1.871 .2292 .0412 .478 12. .7854 ' 1.000 4. 3.666 10.56 1.914 13-16 .2344 .432 .484 v\ 1 .021 .8184 1 010 48. 4.000 12.57 2.000 1*16 .23!>6 .2448 .0451 .0471 .489 .495 j 1.042 1 063 .8522 .8866 1.020 1.031 60! 4.500 5.000 15.90 1963 2.121 2.236 3. .2500 .0491 .500 13. 1.083 .9218 1.041 66. 5.500 23.76 2.345 U .2604 .0532 .510 H .104 .9576 1.051 72. 6.000 28.27 2.449 .2708 .0576 .520 M .125 .9940 1.060 78. 6.500 33.18 2.550 y\ .2813 .0621 .530 K .146 1.031 1.070 84. 7.000 38.48 2.646 $4 .2917 .0668 .540 14. .167 1.069 1.080 90. 7.500 44.18 2.739 K .3021 .0716 .550 y*. .187 1.108 1.090 96. 8.000 60.27 2.828 K .3125 .0767 .560 * .208 1.147 1.099 3 * .32-29 .0819 .570 H 1.229 1.187 1.110 For con tents in gallons, see p 46. 35 542 HYDRAULICS. Art. 3. To find the total head in feet, that must be given to a straight. smooth. cylindrical cast-iron pipe, not less than 4 diams long, to enable it to disch agiveu read suau- tity per sec ; knowing its diam in feet. See Rem 2, p 543. Also, Rem p 539. Rur,K. Square the given disch in cub ft per sec. Add together the total length of the pipe in feet, and 54 times its diam in ft. Mult the sum by the square of the disch just found. Call the prod p. Next, div the diam in ft by the dec .235. Take, from Table 5, the fifth power of the quot. Divp by this 5th power. The quot will be the reqd head in ft. See Caution, p 566. /Disch in cu&y / Length hi feet \ Total ftea J !23d~ ' Ex. A straight, clean, cast iron pipe 6 ins, or .5 ft diam. and 20 ft long, is reqd tn disch 3.066 cub ft per sec. What total head must it have? Here the square of the disch, or 3.0662, is 9.4. The length in ft, added to 54 diam in ft 20 + 27 47 ; and 9.4 X 47 = 441.8, or p. Next, the diam in ft, div by .235, or = 2.128. The 5th power of 2.128, by Table 5, is 43.64.* Hence the total head REM. The following 1 rule is more simple; but is applicable only when the pipe is so long that the 54 diams may be omitted from the calculation, without affecting the result to a practical degree ; in which case writers call it a long* pipe. If, however, we take it as low as 1000 diams, the resulting head will be but about 6 per ct, or y 1 ^ part too small ; at 2000 diams, -Ar too small, CTule and formula give the diams of such pipes too small. To obtain correct results, in suchca^es, mult the diam as found by the rule or formula, by the corresponding multiplier in the foU0wing table. Table Qfniultipliers for the till 1Xd = .oX .2012 X .135 = .0136 ft, or about % inch only, the additional head reqd. See next table. No. 8. Du Btiat's rule for the additional head required to over- come the resistance of circular bends in water pipes. Having given the diam of the pipe, in ft ; the rad of the bend, in ft; and the vel in ft per sec.t Div tr. Fig 2, or half the diam of the pipe, by the rad sw of the outer side of the bend. The quot will be th nbt versed sine of Du Buat's angle of reflexion. Take this versed sine from unity, or 1. The rem will be the nat cosine of the same angle. From the Table of Nat Sin and Tang, take both the angle and the nat sine corresponding to this nat cosine. Call the angli- R. Also, square the nat sine, and call this square S. Take the angle w s o from 180. Div the rem by twice the angle R *>f reflexion just found. Call the quot T. Finally, mult together the constant dec .00375. the square of the vel in ft per sec, the quot T, and the square S. The prod will be the reqd extra head in feet. * This central angle, when the bend is less than a semicircle, will always be equal to either one of the two exterior angles, a A c, or x b d. Fig 2, formed by the tangents a d and x c. Either of these angles is called the angle Off deflection of the bend. t When we have given the diam of the pipe in ft. and the quantity of water required in cub ft per sec, the vel in ft psr sec will be found by dividing the quantity by the area of the pipe. This area may be taken from Table 3, Art 2, p 541. 550 HYDRAULICS. Ex. The same as the foregoing one for Weiabach's rule ; that is, a pipe of 18 ins, or 1.5 ft diaia; rad sw of outer side of beud, 5 Jo ft; vel, 3.6 ft per sec. What extra head will the beud require, in order that this vel may not be diminished? Here, ~ = .13043 = nat versed sine of angle of reflexion. And 1 .13043 = .86957 = nat cos of same angle. In the Table of Nat Sin and Tang, we find, opposite the nat cos .86957, the angle R 29 35 ; and its nat sine .4937. The square of .4937 .2437 = S. Again, 180 90 =: 90. And 90 _ 5400_mm __ Qf Finallv tne 8qU are of the vel is 12.96: hence, we have .00875 X 59 10' ~~ 3550 min 12 96 X 1.52 X .2437 = .0172 ft, the reqd extra head : or about 3- of an inch. Weisbach's rule gave .0136 ft, or about % of an inch. Hence we see that the resistance produced by well-rounded beuda is not great. When the rad r s. Fig 2, of the bend, is less than about two diams of the pipe, which will rarely hap- pen, the resistance to the flow of the water iucreases very rapidly; while, on the other hand, by Weisbach's rule, as we understand it, no advautage ap- pears to be gained by using a rad greater than 5 diauirf of the pipe.* Employ- ing Weisbach's formula, the writer has drawn up the following table of head* reqd to overcome the resistance of one bend of 90. for diff vels iu ft per sec ; and for any diam whatever. This table extends from a rad of 5 diams down to one of % diam ; which is the smallest possible, inasmuch as it leads to a bend like Fig 3. A vel of 12 ft per sec is equal to 8.18 miles per hour; one which will rarely occur, inasmuch as it requires a head of about 330 feet per mile. TABLE 8. Heads required to overcome the resistance in circular bends of 9O. Original. Fin 3 Velocity in feet per Second. 1 ft. 1 2 ft. 1 3 ft. I 4 ft. 1 5 ft. j 6 ft. 7 ft. I 8 ft. I 9 ft. 10 ft. 12ft. HEADS IN FEET. Rad 5 diams of . the pipe. .001 .004 .009 .016 .025 | .036 .050 .065 .082 .101 .145 Rad 3 diams. . .001 .004 .010 .017 ,027 .038 .052 .069 .086 .106 153 .019 .029 .074 .094 .167 Rad = 2 diams. . . .001 .005 .011 .012 .057 .116 Rad= 1M diams... .001 .005 .012 .021 .033 .048 .066 .086 .108 .134 .192 Rad -114 diam... .002 .007 .015 .026 .041 .059 .080 .104 .132 .163 .235 Rad = l diam... .002 .009 .020 .036 .056 .081 .110 .144 .182 .225 .324 Rad= *fdiam... .005 .018 .041 .072 .113 .162 . .221 .288 .365 .450 .649 Rad = ft diam. . . .016 .062 .140 .248 .388 .559 .761 .994 1.26 1.55 2.24 If the central angle r s o, Fig 2, should be either greater, or less than 9O, then the heads given in the table, must be increased, or dimin- ished directly in the same proportion. Experiments by Ronnie, with a pipe 15 ft long ; and 1 4 ft head, gave the following disch iu cub ft per sec : inch bore ; with t vertical bends near discharge end 85.00 sec. 4 vertical bends near supply end... 84.00 " Straight 00699 cub ft. I One bend at right angles near end 00556 15 semicircular bends 00617 " | 24 bends at right angles 00253 The mean of many careful experiments tried at Liverpool, England, with a leaden pipe, 75 ft long, % inch bore, under 8 ft head, gave the following number of sees to discharge one gallon of water : 75 ft pipe, straight and horizontal.. 81.56 sec. 2 hor bends near dischai-ge end ... 83.33 " 2 hor bends near supply end 81.80 " The rad of the bends is not stated. See Minutes Trans Inst Civ Eng, vol 12, page 501. 7fi K .flfl 1.48 77 .8 20.9 27 41.7 75 69.5 HYDRAULICS. 553 Art. ,*. On the flow of water through vertical openings fur- nished with short tubes. When water flows from a reservoir, Fig 6, through a vert partition mm a a, the thickness a TO of which is about 2^ or 3 times the least transverse dimension of the opening, (whether that dimension be its breadth, or its height;} or when, if the partition be very thin, as n n, the water Hows through a tube, as at t, the length of which is about 2 or 3 times its least transverse dimension, then tbe effluent stream wll entirely fill the opening, or the tube, as shown in Fig 6 ; or, in technical language, will run with a full flow ; or a full bore; aud will disch more water in a given time, than if ^ ^~ the tube were either materially longer or shorter. For if 77^. /> JL longer than 3 times the least transverse dimension, the Jj \\\ |) How will be impeded by the increased friction against the " v sides of the tube ; and if shorter than about twice the least transverse dimension, the water will not flow in a full stream, but in a contracted one, as Fig 11, p 554. This will be the case whether the tube be circular, or rectilinear, in its cross shown by section. To find approximately the actual vel. and disch into the air, through a tube, or opening, either circular or recti- linear in its outline, or cross-section ; and whose length c i, or c e+ in the direction of the flow, is about 2]4 or * times its least transverse dimension ; when the surface-level. >-, Fig 6, remains constantly at the same height; and which height must not be below the upper edge of the tube, or opening. RULE 1. Take out the theoretical vel from Table 10, p 552. corresponding to the bead measured vert from the center (or more properlv, the cen of grav) c, of the opening, to the level water surf . Mult it by the coeff of disch .81. The prod will be the reqd vel, in ft per sec. Mult this actual vel by the transverse area of the opening, in sq ft. If circular, knowing its diam. this area will be found in Table 3, p 541. The prod will be the quantity of water dischd, in cub ft per sec ; within, probably, 3 or 4 per cent. RULE 2. Find the sq rt of the head in ft. Mult this sq rt by 6.5. The prod will be the actual vel in ft per sec. Ex. An opening c o ; or box-shaped tube c f, Fig 6, is 3 feet wide, by .25 of a ft high ; and its length in the direction cior e e in which the water flows is about .62 of a ft, or about 2^ time.* its least transverse dimension, or its height. The head from the cen of grav c, of the opening, to the constant surf-level *, is 4 feet. What will be the vel of the water; and how much will be dischd per sec? By Ride 1. The theoretical vel (Table 10. p 552,) corresponding to a head of 4 ft is 16 ft per sec. And 16 X .81 = 12.96 ft per sec, the actual vel reqd. Again, the transverse area of the opening, or of the tube, is 3 ft X .25 ft = .75 sq ft. And .75 X 12.96 = 9.72 cub ft ; the quantity dischd per sec. By Rule 2. The sq rt of 4 is 2. And 2 X 6.5 = 13 ft per sec, the reqd vel, as before ; the very slight diflf being owing to the omission of small decimals in the coeffs. of the vert t case, use .71 or .7 instead of the .81 of Rule 1 ; or 5.7 instead of the 6.5 of Rule 2. REM. 2. When the thickness a TO of the vert partition m m a a ; or the length c e of the tube t. Fig 6, is increased to about 4 times the least transverse dimension of the opening ; or of the diam. when circular; then the additional friction against its sides begins appreciablv to lessen the vel and disch. In that case, or for still greater lengths, up to 100 diams, they may be found approximately, by using instead of the coeff of disch .81 in Rule 1, the following coeffs, by which to mult the theoretical veil of Table 10, p 552. Or use Rule, p 538. TABLE 11. See Caution, p 566. REM. 1. If the short tube f projects partly inside of partition n n, the disch will be diminished about y s part. In that ( Length of Pipe in Diams. Coeff. Length of Pipe in Diams. Coeff. 4 .80 40 .62 6... ... .76 60... ... .60 10 .74 60 .57 15... ... .71 70... ... .55 20 .69 80 .52 25... ... .67 90... ... .50 30 .65 100 .48 tube, in the direction in which the water flows, becomes icnsion, the disch is diminished ; so that for lengths from plate, we may use .61, instead of the .81 of Rule 1. For RKM. 3. When the length of the opening less than about twice its least transverse d IJiJ times, down to openings in a very thi such openings, however, see Arts 9 and 10. REM. 4. But on the other hand, the disch through such short openings and tubes as are shown In Fig 6. may be increased to nearly the theoretical ones of Table 10, by merely rounding off neatly the edges of the entrance end or mouth, as iu Fig 7; which is the shape, and half actual size of one with which Weisbach obtained .975 of the theoretical vel and discharge, when the head was 10 ft; and .95* 554 HYDRAULICS. with a head of oue foot; so that in similar cases, .975, and .958 may be used instead of the coeff .81 in Rule 1. As much as .92 to .94 may be obtained by widening the opening, m n, toward its outer mouth, o , Fig, 8, making the divergence, or angle a. about 5: or by widening it toward its inner mouth, as at t c. Fig 9; but increasing the angle of divergence, at b, to from 11 to 16. In all cases, we consider the small end as being the opening whose area must be multiplied by the vel to get the discharge. In some experiments made with large pyramidal wooden troughs 9.5 ft long, with an inner mouth of 3.2 X 2.-* ft, and a discharging one of fi'2 X .4* ft; and under a head of 9J feet, the discharge was .98 of the theoretical oue. due to the smaller end. Therefore, .98 may be used in such cases, instead of the .81 of Rule 1. KKM. 5. The discharge through a short opening of small transverse section may even be made 5O per cent greater than the theoretical one, by adopting the shape, Fig 10; where m n is sup- posed to be the diameter of the opening. _ The best proportions appear to be about as . follows: oy 9 inches; mn~linch; be K -m ____ J)_ =1.8 inch; o*=X inch ;/< d~2 ins; the 1\. PltsT" curves, a m, and d n, beiug quadrants; DC -s!;:;f; :::: ----4H--J-S [V^^ the angle, z, of divergence, about 5 6' ; ' ""IX*- F==^ and the tube of polished metal. In this t \ V IL ~ 5:5. case use 1.55, or more safelv, 1.5. instead rl6 10 v 7 *^ of the .81 of Rule 1. The only experi- -vlgj lvr merits with this form have been on a very small scale. To what extent it may be applicable is unknown. So far as regards the ordinary operations of the engineer, this subject is perhaps more curious than useful ; for he will rarely have any difficulty in making his openings large enough, without resorting to such aids ; except, perhaps, that of a rounding off the inner edges, as in Fig 1 ; whicii is usually done. Art. 9. On the disch of water through openings in thin vert partitions, with plane or flat faces, e e, or n n, Fig 11.* If the face ee, or n n, instead of being plane, and vert, should be curved, or inclining in diff directions toward the opening, then the disch will be altered. When water flows from a reservoir. Fig 11, through a vert plane plate or partition nn, which is not thicker than about the least transversedimension of the opening, whether thatdimension be its breadth, or its height o o; t or when, if the partition e e itself is much thicker, we give the opening the shape shown at b. (which evidently amounts to the same thing.) then the effluent stream will not pass out with & full flow, as in Fig 6, but will assume the shape shown in Fig 11; forming, just outside of the opening, what is called the vena, contracta, or contracted vein. In order that this contraction may take place to its fullest extent, or become complete, G H the inner sharp edges of the opening must not approach either the .__ . . surf of the water, or the bottom or sides of the reservoir, nearer \\lfi 11 than about 1^ times the least transverse dimension of the opening. The contracted vein occurs at a dist of about half the smallest di- mension of the orifice, from the orifice itself. In a circular orifice. at about half the diam dist; and ordinarily its area is about .62 or nenrly % that of the orilice itself. At this point the actual mean vel of the stream is verv nearly (abont .97) the theoretical vel given by Table 10, p 552 ; and hence the actual dischs are but ".62, or nearly % of the theoretical ones. Case 1. To find the actual disch into air.J through either a circular or rectilinear^ opening in a thin vert plane parti- *We believe that these rules for thin plate are also sufficiently approximate for most practical purposes, if the opening be in the bottom of the reservoir; or in an inclined, instead of a vert side. t When the side of a reservoir, or the edge of a plank. &c. over which water flows, has no greater thickness than this, the water is said to flow through, or over, thin i>lato. or thin partition. J Should the disch take place under water, as in Fig 12. both surf -levels re- levels. After making the calculation with this head, we should, according to Weisbach, deduct the 7-3- part; inasmuch as he states that the disch is that much less when under water, than when it takes place freely into the air. Other experimenters, however, assert that it is precisely the same in both cases. If the shape of the opening is oval, triangular, or irregular, the head must be measured vert from its cen of gray. HYDRAULICS. 555 fioii, when the contraction is complete; and when the surf- level, , remains constantly at the same height; water being- supplied to the reservoir as fast as it runs out at the open- in^.* RULE 1. When the head, measured vert from the center (or rather from the cen of grav) c, of the opeuiug, to the surf level a of the reservoir, is not less than 1 ft. nor more than 10 ft ; and when the least transverse dimension of the opening is not less than an inch, mult the theoretical vel in ft per sec due to the head, (Table 10, p 552,) by the coeff of disch. 62. (See Kern 2.) The prod will be the actual mean vel of the water through the opening. Mult this vel by the area of the opening in sq ft; the prod will be the discb in cub ft per sec, approximately. When the head is greater than 10 ft, use .6, instead of .62. RULE 2. Fiud the sq rt of the bead in ft. Mult this sq rt by 5 ; the prod will be the vel in ft per sec ; which mult by the area as before for the disch. Ex. What will be the disch through an opening in complete contraction, whose dimensions are 6 ins, or .5 ft vert ; and 4 ft hor ; the vert head above the ceu of grav of the opening- being constantly By Rule 1. The theoretical vel (Table 10, p 552) corresponding to 6 ft head, is 19.7 ft per sec. And 19.7 X .62 12.214 ft, the reqd vel. Again, the area of the opening .5 X 4 2 sq ft; and 12 214 X 2 - 24.428 cub ft per sec ; the disch. By Rule 2. The sq rt of 6 = 2.45 ; and 2.45 X 5 - 12.25 ft per sec, the reqd vel ; and 12.25 X 2 = 24.5 cub ft per sec, the disch. Both very approx even if the orifice reaches to the surface of the issuing water. Rem. 1. The coef .62 is a mean of results of many old experimenters. In 1874 Genl. T. G. Ellis of Massachusetts conducted an elaborate series (Trans Am Soc C E, Feb 1876) on a large scale, the general results of which, within less than 1 per ct, are given in the follow- ing table. See also Rem 3. The sharp edged orifices were in iron elates .25 to .5 inch thick. Orifice. 2 ft sq. 2 "lon K , 1 ft high 2 " long, .5 high 2 " diam. Head above Center. 2. to 3.5 ft. 1.8 to 11.3" 1.4 to 17.0 " 1.8 to 9.6" Coef. .60 to .61 .60 to .61 .61 to .60 .59 to .61 Rem. 2. Extreme care is reqd to obtain correct results; but for many purposes of the engineer an error of 5 to 10 per ct is unimportant. REM, 2. It will rarely happen that greater accuracy Is reqd than that obtained by the foregoing rules; but when such does occur, aid may be derived from the following table deduced from the experiments of Lesbros and Poncelet, on openings 8 ins wide, of diff heights, and with diff heads. Use that coeff in the table which applies to the case, in- stead of the .62 of Rule 1. In some of the cases in this table, the upper edge of the opening is nearer the surf-level of the reservoir than 1> times its least transverse dimension. TABLE 12. Coefficients for rectangular openings in thin vertical partitions in full contraction.* Head Head The breadth in all the openings rr 8 inches. above cen. of grav. of opening above cen. of grav. of opening Ins. H Ins. EIGHT OF OPENIN Ins. 1 Ins. GK Ins. I .Ins. in Feet. in Inches. 8 6 4 3 2 1 1 .4 033 4 70 0666 8 65 69 0833 64 68 125 \y>. 61 64 68 1666 2 60 62 64 68 2083 2i 59 .61 62 64 67 250 8 60 61 62 64 67 2917 31^ 57 60 61 62 64 66 3333 PI 58 60 61 63 64 66 .3750 4% .56 .59 .60 .61 .63 .64 .66 .4167 5 .57 .59 .61 .62 .63 .64 .66 .6666 8 .59 .60 .61 .2 .63 .64 .65 1 12 .60 .60 .61 .62 .63 .63 .64 1 **et the dimensions of the opening o, be 2 ft by 3 ft; making its area 6 sq ft. Now, c being above the top of the opening; and cd being 16 ft; the sq rt of which is 4; how long will the reservoir m be in filling from c to of? Here we have p= 200 X 2 X 4 = 1600 ; and y = 6 X -62 X 8.02 = 29.83. And : = 53.6 sec, the time reqd. REM. I. If it should be reqd to find the time of filling m, from its bottom c, up to rf. we may do so very approximately by calculating by the first rule in Art 9. the time reqd from e to the center of the opening o, as if all that portion of the disch took place into air; and afterward, from the center of the opening to d. by the rule just given. This case is similar to that of filliug a lock from the canal reach above, in which the surf- level mav be considered constant. REM. 1 If the bottom of the opening o. should coincide with the bottom of the reservoir, then the coeff will become greater than .62. See Art 11, for obtaining coeffs for imperfect contraction. R,;M. 3. If the opening, instead of being in complete con- traction, ia of any of the shapes Figs 6 to 9, then a reference to Art 8 will show what coeff must be substituted for .62. Gisr. 3. Uisch from one prismatic reservoir, Fig 15, W, into another, X, of any comparative sizes whatever, through an opening o, in a plane thin vert partition, and in complete contraction; when the water rises in X. while it falls in W. To find the time in which the water, flowing from W into X, through o, will fall through the dist a s, so as to stand at the same level a c. in both reservoirs. In this case, the water reqd to fill X from e to d, (d being the bottom of the opening o.) flows out into the air; and the time necessary for it to do so, must be calculated separately from that reqd above d, which flows into water. RULE. First from e to d. Find the hor area of each reservoir, in sq ft. Mult the hor area of X, by the vert depth de in ft, for the cub ft contained in that portion. Div these cub ft by the hor area of W. The quot will be the dist am, in feet, through which the water in W must descend, in order to fill X to d. Now calculate by Rule 1, Case 1 , the number of sees in which W would empty itself into the air, under a constant head equal to an; also the time" in which it would empty itself into air, under a constant head equal to mn. Take the diff be- tween these two times. This diff will be the time in which the water in n. , Mf W would fall from a to m ; and that in X rise from to d ; under a con- _TlU 1 J * t(ini hea(1 el"* 1 to an. Mult thi time by 2. for the time rcqd in the O actual eas bfere us, uder a haad varying from a w to TO n. T thii W X HYDRAULICS. 557 time we mast add that still reqd for the water to fall from m to a; filling from d to c. To find tbti time, take the square root of the remaining head, inn, in feet. Mult together, this sq rt ; the hor area of W; the hor area of X; and the constant number 2. Call the prod p. Next, add together the hor areas of W and X. Mult together, the sum of these areas ; the constant number 8.02 ; the area of the opening o, in sq ft; and the coett' of disch of the opening o (.which coelf tor opening in com- plete contraction will be about .62). Call the prod y. Divpbyy. The quot will be the additional time reqd in sees, very approximately. Ex. Let the hor area of W be 100 sq ft : and that of X, 60 sq ft. Let an be 20 ft ; and TO n 16 ft ; and the area of the opening o, '6 sq ft. In what time will the water descend from a to , and rise from e to c ? Inasmuch as the method of finding the time for filling from e to d, by the water falling from a to m, requires no further exemplification, we will confine ourselves to the additional time necessary for filling from if to c, by the water falling from m to 8. To find this, we have, the sq rt of the head ' Fig 16 20.1 sec; the additional time reqd, very approximately. NOTE 1. If the opening, as , coincide with, or form portions of, the sides s. s. of the reservoir : in which case contraction takes place only along the upper edge a, a, of the weir, Fig 22, as shown at a in Fig 20; but is suppressed entirely at the ends. so that the water flows out as shown in plan by Fig 23. And Case 2d, in which, as in Fig 22, the vert ends ah, ah. as well as the crest a, a, are formed with a sharp corner in thin plate; and are, moreover, removed from the sides v. t>, of the reservoir, a dist equal at least to the head am; so that contraction takes place at the ends of the weir, as well as along its crest; and less water flows out, s shown in plan at a, a, Fig 24. __ JLlfl 19 HYDKAULICS. 559 To find the Htlisch over a weir in thin plate, when, as in Figs 21 and 23, tKere is no contraction at its two ends. a a RULE. Find the cube of the head am. Fig 20. in ft. Take the sq rt of this cube. Mult together this sq rt; the length a, a. of the wtir, in ft; and the constant uuuiber 3.33. The prod will be the quantity disch, in cub ft per sec. Or, in shape of a formula, The disch, in _ the, sq rt of the cube of v the length a a. of v const number cubftpersec, ~ the head am, in ft A the over/all, in ft A 3.33. Ex. How many cub ft per sec will flow over such a weir in thin plate. 200 ft long: having a head, a TO. of 1.5 ft, measured to the level suit o w*.of the reservoir; and with no contraction at either end? Here, the cube of 1.5 = 3 375. Aud the sq rt of 3.375 = 1.837. And 1.837 X 200 X 3.33 = 1223.4 cub ft per sec, the reqd disch.* This rule will also be very approximate even when there is contraction at both ends, provided the length of the w r eir is at 'least 1O times as great as the head a >n and provided the head is not less than 2 or 3 ins in depth. Indeed, it will be within about 6 per cent of the truth, for weirs with contraction at the ends, and whose lengths sire but 4 times the head ; and for the mnny cases in which no closer approximation is reqd, the disch may betaken at once from the following table. To find the discharge over a weir in thin plate; when contract ion takes place at both ends, as shown by Figs 22 and 24. RULE. Proceed precisely the same as in Rule for Case 1 ; except that when there is contrac- tion at both ends, i part of the head am is to be taken from the length, before using it as a multi- plier ; and when there is contraction at one end only, A, part of a TO must first be taken from the length a a. Ex. How many cub ft per sec will flow over such a weir in thin plate, 200 ft long; having a head cm, of 1.5 ft; with contraction at both ends? Here, the cube of 1.5 = 3.375. And the sq rt of 3.375= 1.837. Again, ^ of a m = .3 of a ft. And 200 .3 = 199-7. Therefore we have 1.837 X 199.7 X 3.33 = 1221.6 cub ft per sec, the reqd disch ; or practically the same as when there is no contraction, in this case. REM. If instead of 3.33, we use 3.41, the two foregoing rules will apply to heads n m. as small as H HI> inch ; and coeffs between 3.33 and 3.41 may be used for heads between about 5 inches, and % an inch, where more than common accuracy is aimed at. We may also use 3.3 instead of 3.33, for heads greater than 2 feet. * Eytelwein's rule for weirs, over a thin edge: and extending Mult togothor. the sq rt of the head : the h^nd iMM' : the leneth of overfall : (all in feet ;) and the constant, number 3.4. The prod will be tho disoh in cnh ft ppr PC. It givps 1249. instead of the above 1223 cub ft, The sq rt of the hed is 1.226; and 1.225 X 1.5 X 200 X 3.4 = 1249. The rules are in fact identical, except in the coefficients. 560 HYDRAULICS. TABLE 13. Of actual discharges in ciib ft per see, for each foot in length of weir in thin plate: and without contraction at either end ; a />, Fig: 2O. being vert, and not less than twice the head am. Very approximate also, when there is con- traction at both ends, provided the length be at least 1O times the head. And but about 6 per cent in excess of the truth, if the length be but 4 times the head. (Original.) The decimal .01 of a foot, is precisely .12 of an inch : or scant % inch. Head, am, in Feet. Cub. ft. per Second. Head, am, in Feet. Cub. ft. per Second. Head, a m, in Feet. Cub. ft. per Second. Head, am, in Feet. Cub. ft. per Second. Head, am, in Feet. Cub. ft. pt^r Second. .03 .OH .22 .351 .58 1.47 .94 3.04 2.4 12.2 .04 .OJ7 .24 .401 .60 1.54 .96 3.14 2.5 13.0 .05 .038 .26 .452 .62 1.62 .98 3.21 2.6 13.8 .06 .050 .2S .505 .64 1.70 1. 3.^3 2.7 14.6 .07 .063 .30 .560 .66 1.78 1.1 3.85 2.8 15.4 .08 .077 .32 .603 .68 1.86 1.2 4.38 2.9 16.2 .09 .092 .34 .659 .70 1.95 1.3 4.94 3. 17.1 .10 .108 .36 .719 .72 2.03 1.4 5.51 3.1 18.0 .11 .124 .38 .78J .74 2.12 1.5 6.11 3.2 18.9 .12 .142 .40 .842 .76 2.21 1.6 6.73 3.3 19.8 .13 .160 .42 .907 .78 2.30 1.7 7.37 3.4 20.7 .14 .178 .44 .972 .80 238 . 1.8 8.04 3.5 21.6 .15 .198 .46 1.04 .82 2.47 1.9 8.72 3.6 22.5 .16 .218 .48 Lit .84 2.56 2. 9.42 3.7 23.5 .17 .239 .50 1.18 .86 2.65 2.1 10.0 3.8 24.4 .18 .260 .52 1.25 .88 2.74 2.2 10.8 3.9 25.4 .19 .282 .54 1.32 .90 2.84 2.3 11.5 4. 26.4 .2 .305 .56 1.40 .92 2.94 Iu calculating this table, the coeff 3.41 was used for beads from .03 ft to .3 ft; then 3.33 to 2 ft; then 3.3 to the end. From the Lowell experiments, by Mr Francis, it appears that when the depth, m a, is 1 foot, and the entire sheet of water, after passing over the weir, strikes a bor solid floor placed only about 6 ins below the crest o, <>f the weir, the disch is thereby diminished bur. about the YTf^yTT P art ' an< * tnat wnen tne nea d amia about 10 ins, and falls into water of con- siderable depth, no diff whatever is perceptible in the disch, whether the surf of that water be about 4. or about 13 in* below the crest a ; and that a fall below the crest a, equal to one-half of the head m, is quite sufficient. If the water in the reservoir, or in the feeding canal, in- stead of being stagnant, has a slight current toward the weir, the dis< % h will be but very little increased thereby when the head a m is several ins. Mr Francis observed that a current of I foot per sec, or nearly .7 of a mile per hour in- creased the disch but about 2 per cent, when the head was 13 ins; and one of 6 ins per sec. about 1 per ct, when the head was 8 ins. Whenever the effeot of the current, however, is so great as to require notice, proceed as follows : Find in Table 10, p552, the theoretical head A which corresponds to the observed vel of the approaching current. Add this head to the head o TO. Fig 20. Cubethesum. Take the sq rt of this cube, calling it a. Also cube the theoretical head A. Take the sq rt of this cube. Call it ft. From i take b. Find the square of the remainder. Take the cube rt of this square, for a new head H', to be employed in Cases 1 and 2, instead of H, or o TO ; or in shape of a formula, /i2~)3 r=H'; the new head. Art. 15. If the inner face of the weir and dam, instead of being vert, as a b, Fig 2O. is sloped, as a or t>. Fig 25; tho contraction on the crest will be diminished ; and consequently the dioh will be in- creased. This will also be the case if the inner corner oredce of the crest be rounded off, instead of being left sharp ; or if the .sides of the reservoir c 'nverge more or les^as they approach the weir: so as to form wings for guiding the water more directly to it: or if a fc. Fig '20. be less than twice a m. Indeed, so many modifying circumstances exist to embarrass experi- ments on this, and similar subjects, that some of those which have hoen made with great care, are rendered inapplicable as other than tolerable ~I7* 9 *\ approximations, in consequence of the neglect to take into consideration & some local peculiarity, which was not at the time regarded as exerting an appreciable effect. Unless, therefore, circumstances admit of our com- bining all the conditions mentioned in the first, part of this article ; and [thereby securing very ap- nroximate results, we must either resort to an actual measurement of the disch in a vessel of known capacity; or else he contented with rules which may lead t-o errors of 5, 10. or more per cent, in pro- portion as we deviate from those conditions. Frequently, even 10 per ct of error may be of little real importance. The following rule for finding the discharge over an over- HYDRAULICS. 561 fall, or weir, approximately, has been prepared by the author from various data. Mult together the length, a, a, of the weir. Figs 21 and 22, in feet: the head, am. Fig 20. measured to the lentil surf of the reservoir, in ft; the theoretical vel in ft per sec, corresponding to the head a m. (Table 10. p 552 :) and that coeff from the following table, which agrees most nearly with the case. The prod will be the reqd disch in cub ft per sec, near enough for ordinary purposes ; and probably quite as close us can be arrived at without actual measurement in each case that presents itself. Ex. How much water will be dischd over a weir 60 ft long; the crest of which is level, smooth, and 3 ft wide, or thick ; and over which the head am, Fig 20, is 8 ins, or .6666 ft thick ? Here the theo- retical vel for a head of 8 ins. (Table 10, p 552) is 6.55 ft per sec. The coeff for a weir whose crest is level, u(i 3 ft wide, with a head of 8 ins. is by the following table .31. Consequently, 60X.6666 X 6.56 X .31 = 81.21 cub ft pe,r sec ; the reqd disch approximately. TABLE 14, Of coefficients of approximate discharge over weirs of different thicknesses, varying- from a sharp edire to Sleet. (Original.) 3 Ft Thick ; smooth ; Head a m in Feet. Head a m in Inches. Sharp Edge. 2 Inches Thick. sloping out- ward: and downward. 3 Ft Thick ; smooth, and level. from 1 in 12 to 1 in 18. .0833 1 .41 1 .32 .27 .1666 2 .40 .38 .34 .30 .'25 3 .40 .39 .34 .31 .3333 4 .40 .41 .35 .31 .4166 5 .40 .41 .35 .32 .5 6 .39 .41 .35 .33 .5833 7 .39 .41 .35 .32 .6666 8 .39 .41 .34 .31 .8333 10 .38 .40 .34 .31 1. 12 .38 .40 .33 .31 2. 24 .37 .39 .32 .30 3. 36 .37 .39 .32 .30 i inches. ^; and if 1 inch, fa or RKM. 1. When the water, after pass- in- over a weir. Fig 26, instead of falling freely into the air, is carried away by a slightly inclined apron or trough, T, the floor of which coincides with the crest, a. of the weir, then the disch is not appre- ciably diminished thereby when the head a m is 15 ins or more. But if the head a m is but 1 ft, then the calculated disch must be reduced about -^ part; if 6 inches, y^ ; if ! one-half, as approximations. RF.M. 2. Professor Thomson, of Dublin, proposed the use of triangular notches, or weirs, for measuring the disch ; inasmuch as then the periphery always bears the same ratio to the area of other form. Experimenting with a right-angled triangu- lar notch in thin sheet iron. Fig 26^ ; with heads of from 2 to 7 ins, measured vert, from the bottom of the notch, to the level surfoftftf quiet water, he found the disch in cub ft per sec to be as follows : Find the fifth power of the head in inche*, (Table 5, pages 546-7.) Take the sq rt of this 5th powr, (Table 6, page 548.) Mult this sq rt by .0051. Or by formula, eubfTpe^sec ~ Head in inche * $ * -O 051 - RKM. 3. Fig 36V shows a singular effect observed at Clegg's dam. across Cape Fear River, N C. It is from measurements made by Ellwood Morris, C E ; by whom they were communicated to the writer. The dam is of wooden cribwork ; and its level crest, 8 ft 5 ins wide, is covered with plank ; along which the water glides in a smooth sheet, 6 ins deep, (nt the time of measurement.) At the upper end of this sheet, and in a dist of about 2 ft, a head of 9 ins forms itself, as in the fig. We have no comments to offer ; but consider the fact to be of sufficient interest to he made more widely known. Art. 16. On the flow of water throngh open channels. The following rules, to the end of Art '20, must be regarded merely as approximations. The subject Is in- rolved in much uncertainty. See Re m, p 562, and " Caution," p 566. 562 HYDRAULICS. * 2' * FtjtX.fi i To ascertain approximately the mean vel of all the water in any given cross-section of a river, canal, or other water- course; having given the greatest surf vel only.* When no great accuracy is reqd, this mean vel may be deduced from the greatest surf vel. Thus, select a place where the stream is for some dist (the longer the better) of tolerably uniform cross- section; and free from counter-currents, slackwater, eddies, rapids, &c. Observe, bv a seconds- watch, or pendulum, how long a time a float (such as a small block of wood) placed iu" the swiftest part of the current, occupies in passing through some previously measured dist. From 50 feet for slow streams, to 150 ft for rapid ones, will answer very well. This dist in ft, or ins, div by the entire number of seconds reqd by the float to traverse it, will give the greatest surf vel iu ft or ins per sec. Take 4 of this vel; or, iu other words, mult it by the dec .8. The prod will be approximately tb mean vel of the entire body of water. The result will be somewhat more accurate, if, instead of mult by .8 for every surf ve"l, we mult by the following decimals in the 4th col.* Ex. What will be the mean vel of all the water moving through a uniform channel, the greatest surf vel of which is found to be 60 ins per sec? Here, 60 X .83 = 49.8 ins per sec ; the reqd vel. TABLE 15. See Caution, p 566. SURFACE VELOCITY. Mult, greatest Surface Vel., by SURFACE VELOCITY. Mult. freatest urface Vel., by Inches per Sec Feet per Sec. Miles per Hr. Inches per Sec. Feet per Sec. Miles per Hr. 4 12 20 40 60 .333 1. 1.667 3.333 5. .227 .6S2 1.136 2273 3.409 .76 .77 .79 .81 .83 80 100 120 140 160 6.667 8.333 10. 11.7 13.3 4.515 5.682 6.818 7.955 9.091 .85 .86 .87 .88 .89 The surf vel should be measd in perfectly calm weather, so that the float may not be disturbed by wind ; ad. for the same reason, the float should not projoct much above the water. The measurement should be repeated several times to insure accuracy. In very small streams, the banks and bed may be trimmed for a short dist, so as to present a uniform channel- way. The float should be placed in the water a little dist above the point for commencing the observation ; so that it may acquire the full vel of the water, before reaching that point. To tin l the discharge in the foregoing case. Measure the breadth of the stream at the surface ; and also a sufficient number of depths, taken in a straight line across ; in order to calculate the area of its cross-section. This area in sq ft, mult by the mean vel in ft per sec, will evidently give the disch in cub ft per sec. Rent. If the channel is in common earth, especially if sandy, the loss by soakage into the soil, and by evaporation, will frequently abstract so much water that the disch will gradually become less and less, the farther down stream it is measured. Long canal feeders thus generally deliver into the canal but a small proportion ot the water that enters their upper ends Art. 17. To find approximately the vel at the bottom of a stream ; at any part of its cross-section.* RULE. First measure the surf vel over the same part. Mult it by the corresponding decimal \n the The remainder will be approximately the reqd bottom vel. Ex. Surf vol 100 ins per sec. And 100 X .86 = 86 ins per sec, mean vel ; and 86 X 2 = 172 ; and 172 100 = 72 ins per sec, bottom vel. * As we have already stated, these rules give in many cases very rude results ; that in Art 17 par- ticularly. With the same surface vel, a wide and deep stream will have greater mean and bottom vels than a small shallow one. There Is no reliable rule. Art. 18. TABLE Incites per sec ; DRAULICS. 563 Of approximate velocities of streams in the foregoing rules. See Caution, p 566. ** 2 MeaikVel. of all the Stream, in Ins. per Sec. Greatest Bot- tom Vel., in Ins. per SeCvj Greatest Sur- face Vel.. in Ins. per Sec. o eg | Is, S 02 S Greatest Bot- tom Vel.. in Ins. per Sec. lift! Greatest Bot- tom Vel., in Ins per Sec. i& V s ! "si| lit 3 2.28 1.54 23 18.2 13.3 48 394 30.8 86 73.4 60.9 4 3.04 2.08 24 19. 14. 50 41.2 32.3 88 75.3 62.7 5 381 2.62 25 19.8 14.6 52 42.9 33.8 90 77.1 64.3** 6 4.58 3.16* 26 20.6 15.2 54 44.7 35.3 92 78.9 65.9 7 5.36 3.72 27 21.4 15.9 56 46.4 36.8 94 80.8 67.7 8 6.14 4.2S 28 22.3 16.6 58 48.2 38.4 96 82.7 69.3 9 6.92 4.84 29 23.1 17.2 60 49.9 39.8f 98 84.5 71. 10 7.70 5.40 30 23.9 17.9 62 517 41.4 100 86.3 72.6 11 8.49 5.98| 31 24.8 18.5 64 535 43. 105 90.9 7.9 12 9.29 6.58 32 25.6 19.3 66 55.3 446 110 95.6 81.2 13 10.1 7.16 33 26.5 19.9 68 57.1 46.1 115 100.2 85.3 14 10.9 7.72; 34 27.3 20.6 70 58.9 47.7 120 104.9 89.8 15 11.7 8.34 35 28.1 21.3 72 60.7 49.4 125 109.6 94.3 16 12.5 8.i62 06 29. 22. 74 62.5 50.9 130 114.3 98.5 17 13.3 9*.5S 38 30.7 23.5|| 76 64.3 526 135 119.1 103.-H- 18 14.1 10.2 40 32.6 25. 78 66.2 54.3 140 123.9 108. 19 14.9 10.8 42 34.2 26.5 80 67.9 55.8 145 128.8 112. 20 15.7 11.4 44 36. 2$. 82 69.8 57.6 150 133.7 117. 21 16.5 12.1 46 37.7 29.3 84 71.7 59.3 160 142.4 125. 22 17.3 12.7 * .18 of a mile per hour; and begins to scour fine clay. This, and the follow- ing, are mere approximations, chiefly deduced by Du Buat from experiments on a trifling scale. More observations on this subject are needed. There is reason to believe that the scouring vel is consider* ably too small for clay. The subject is very intricate. T Ji of a mile per hour; and just lifts fine sand. tt 5.85 miles per hour; (surface vel J .44 of a mile per hour ; lifts sand as coarse as linseed. 7.67 mile.-* per hour) will roll stones 5 .51 of a mile per hour ; moves One gravel. of a foot diam. II 1 Jf miles per hour ; moves pebbles about an inch in diam. II 2. '26 miles per hour; moves pebbles as large as an egg. ** 3J$ miles per hour; begins to wear away soft shistus. (Scouring: action is supposed to be as square of vel. According to Smeaton, a vel of 8 miles an hour will not derange quarry rubble stones, not exceed- ing half a cub ft, deposited around piers, Arc ; except by washing the soil from under them, i inch per sec, - 5 ft per min. - .056818 of a mile, or 300 ft per hour. 1 foot per sec. = 60 ft per min, = .681816 of a mile, or 3600 ft per hour. To reduce inches per sec, to feet per minute, multiply by 5. " hour, " " 800. " " " " " to miles per hour, divide by 17.6. One mile per hour 88 ft per min = 1.4667 ft, or 17.6 ins per sec. Art. 19. The simplicity, and easy application of the fore- going rules frequently lead to their adoption for the graug-- injj: of streams; but it must be stated that this method is involved in much uncertainty. The experiments of Mr Francis, at Lowell, have shown its liability to err at least 15 per cent in deficiency ; while observations by others would seem to indicate that it may, under diff circumstances, be as much (if not more) in excess of the truth. Mr Francis found that surface floats of wax, 2 ins diam. floating down the center of a rec- tangular flume 10 ft wide, and 8 ft deep, actually moved about 6 per cent slower than a tin tube 2 ins diam, reaching from a few ins above the surf, down to within \% ins of the bottom of the flume: and loaded at bottom with lead, to insure its maintaining a nearly vert position. While the wax surf float moved at the rate of 3.73 ft per sec, the rate of the tube (which was evidently very nearly the same as that of the center vert thread of water) was 3.98 ft per sec. Also, that in the same flume, with vels of the center tube varying from 1.55, to 4 ft per sec, the vel of the tube was less than that of the mean vel of the entire cross section of water in the flume, about as .96 to 1, for the lesser vel ; and .93 to. 1 for the greater ve!. While. In another rectangular flnme 20 ft wide and 8 ft deep, with vels varying from 1.16 to 1.84 ft per sec. that of the tubes was greater than that of the entire mass of water, about as 1 .04 to 1. In a flnme 29 ft wide, by 8. 1 ft deep, with vels of about 3 ft per sec. it was as 1 to .9 ; and in a flume 36^ ft wide, by 8.4 ft deep, with vels of aboirt 3^ ft per sec, as 1 to .97. C'harles Filet. Jr, C E. found in the Mississippi "at diff points on the river, in depths varying from 54 to 100 ft: nnd in currents varying from 3 to 7 miles an hour that the speed of a float supporting a line 50 ft long, is almost alwnvs greater thnn that of the surf float alone " The same results were obtained with lines 25 and 75'ft long: the excess of the speed of the line floats being about 2 per cent over that of the simple floats : and Mr Ellet concludes, therefore, that the mean vel of the entire cross-section of the Mississippi, instead of being less, is absolutely greater 564 HYDRAULICS. by about 2 per cent, than the MEAN surf vel. He, however, employed .8 of the greatest surf rel at representing approximately, in his opinion, the mean vel of the entire cross-section of water. In shallow streams, he always found the surf float to travel more rapidly than a line float. European trials of the mean vel of separate single verticals, in tolerably deep rivers, have resulted in from .85 to .96 of the surf vel at each vertical. The mean of all may be taken at .9. Art. 2O. To grange a stream by means of its velocity, more accurately than by the preceding- methods. The following is perhaps the most accurate means that can be adopted for gaujrin^ a stream by means of direct measurement of its vel. la case of a large river it will involve some trouble, time, and expense; but where accuracy is neces- sary, those considerations must yield. Select a Slace where the cross-section remains for a short ist, tolerably uniform, and free from counter-cur- rents, eddies, still water, or other irregularities. Prepare a careful cross-section, as Fig 27. By means of poles, or buoys, n, n, divide the stream into sections, a. b, c, Ac. Plant two range-poles, R, R, at the upper end; and two others at the lower end of the dist through which the floats are to pass ; for observing the time, by a seconds watch, or a pendulum, which they occupy in the passage. Then measure the mean vel of each section a, o, c, &c, separately, and directly, by means of long floats, as F L, reaching to near the bottom ; and projecting a little above the surf. The floats may be tin tubes, or wooden rods; weighted in either case, at the lower end, until they will float nearly vert. They must be of diff lengths, to suit the depths of the diff sections. For this purpose the float may be made in pieces, with screw-joints. The area of each separate section of the stream in sq ft. being mult by the observed mean vel of its water in ft per sec, will give the disch of that section in cub ft per sec. And the discharges of all the sepa- rate sections thus obtained, when added together, will give the total disch of the stream. And thia total disch, div by the entire area of cross-section of the stream in sq ft, gives the mean vel of all the water of the stream, in ft per sec. Art. 21. To find, approximately; the mean vel, and disch in an open canal, aqueduct, river, A-c, of uniform cross-sec- tion, and fall throughout; knowing: its dimensions, and rate of fall, or descent in feet per foot. See Caution, p 566. Also Art 16. Fiq28 Rule 1. Find area of the water-way, abcoa, Figs 28 29, 30, in *q ft. Find the dist aft co, in feet ; being that portion of the cross-section of the aides and bottom of the channel bed which is in actual contact with the water. This dist fa called the WET PERIMETER, or WET BORDER of the channel. Mult together the area abcoa; the fall in feet per foot ; and the constant number 8975 ; (in practice we may use 9000.) Div the prod by the wet perimeter. Take the sq rt of the quot. From this sq rt subtract the constant dec .1085). The rem will be the mean vel in ft per sec of all the water in the entire cross-section of the stream. Or by formula, Mean vel __ /Area of water-way in sq ft X fall, in ft per ft X 8975 V 5 \/ ' wet perimeter in feet. ~ 1 - .1089. Ex. A canal of rectangular cross-section, as Fig 29, is to be 10 ft wide ; and to have 3.5 ft depth of water. Its bottom is to have a uniform fall, slope, or descent, at the rate .246 of a ft in a length of 492 ft. "What will be the mean vel of all the water ; and how many cub ft of water will it disch per sec ? Here, the area of water- way = 10 X 3.5 = 35 sq ft. The fall, '.'246 ft, div by 492 ft (the dist in which it occurs) gives .0005 of a ft fall for every foot of length. The wet perimeter, = a 6 -f- 6 c -j- c o = 3.5 -f 10 + 3.5 = 17 ft. Hence we have --- 005 X 897 = 9.239. And the sq rt of 9.239 = 3.04. And 3.04 .1089 = 2.9311 ft per sec, the reqd vel. The disch will evidently be, this vel mult by the area of water-way in sqft; or 2.9311 X 35 = 102.588 cub ft per sec. See Remark, p 562: also, Remark of next Art. _ Rule 2. Vel in ft _ /Area of water-way, in sq ft Twice the fall per sec 4 / ^e7^erim*e^~in~ft * in ft per mile. Vel in ft __ /Area of water-way, in sq ft per sec 4 / wet perimeter in ~j t For the foregoing example it gives a vel of 3.3 ft. instead of 2.93 ft; and some experiments with sewers would indicate that 3.3 is nearer the truth; but as before remarked, perfect accuracy in these matters must not be looked for. The degree of smoothness of the channel, and even the muddiness of the water, will affect the result appreciably. Rule 3. Mr. Poole, C E. England, tested the following on open canal portions of the Rocquefavonr aqueduct, France, and found it to accord closely with fact. inft 4 /!< - ec=\ / Jl Vel in ft . / innn _ total fall in ft X area of water-way, in sq ft per tec =\ / JUUUU total length in ft X wetperim in ft. Our above Rale 1 gives in the example thereto, a vel of 2.93 ft per sec ; aud Mr. Poole's rule gives 8.21 feet. Rule 2 give* 3.3 ft. See Kutter'g formula, page 6Qi HYDRAULICS. 565 Art. 22. To/ft iid, approximately, what fall must be given to every foot A n length of a uniform canal, or other water- course. the dimensions of which are known; to enable it to discharge a reqd quantity, in a given time. HULK 1. Piv the number of cub ft of water reqd per sec, by the area in sqftof the water-way a b coa, Figs 28, 29, 30, &c. The quot will be the mean vel which the water must have, iu ft per sec, in order to disch the reqd quantity. Next find the square of this vel. Also find the length of wet perimeter, abco, in ft. Mult together the square of the vel ; the wet perim ; and the constant dec .000111*. Div the prod by the area of water-way, abco a. Call the quot M. Next mult together, the simple vel itself; the wet perim; and the constant dec .00002426. Div the prod by the area of water-way, a t> c b a. Add the quot to M. The sum will bo the reqd fall, iu decimals of a foot, which must be given to every foot in length of the canal. Or, in shape of a formula, See Caution, p 566. Fall fn every _ Veil X wet perim X .0001114 Vel Xwet perim X .00002426 foot of length area o f wat er . wa y in sq ft "*" area of water-way in #q ft.' Ex. The same as the foregoing. What fall per foot must be given to a rectangular canal, or water- oourse, Fig 29, 10 ft wide, and having 3.5 ft depth of water, to enable it to disch 102.55 cub ft per sec ? Here, the area of water-way, a 6 c o a, = 35 sq ft. And - = 2.93 ft per sec, the vel. And 2.93 = 8.5849. Also, the wet perim a 6 co =3. 5 + 10 + 3.5 = 17 feet. Hence, 8.5849 X 17 X .0001114^ .016258 _ Q^^ _. M 35 ~~ = 35 . 2.93 X 17 X .00002426 .0012084 Again; - ^ - = = .000035. Finally, .000465 -j- .000035 = .0005 ft ; the reqd fall ia every foot of length of the canal ; the same as in the preceding example. See Remark, Art 16, p 562. Rnle 2. Or the following formula,, based on that in Rule 2, Art 21, may be used. The aquare of the vel ) .. . . . , ( ^rea of water-way *" edb For the foregoing example it gives 2.085 ft per mile ; or very nearly .0004 ft per ft. Instead of .0005. Rule 3. For the total fall in feet in the whole length of the canal or other channel, the following is approximate. , length of canal v Wet perim \ /Sqof vel in \ Total fall _ / (W7 . in feet. * in feet. ) x / ft per sec. ) infect ~\ Tr. area in sq feet. 7 \ 6iT~ / This rule gives for the above canal, assuming it to be one mile or 5280 ft long, a total fall of 2.564 ft, or 31.68 ins ; or .000486 ft for each ft of length ; or nearly as Rule 1. Rule 4. Mr. Poole says the fall per mile in ins, reqd in a canal, to impart any given mean vel, in ins per sec, may be found thus : Div the area of water-way in sq ft, by the wet perim in ft. The quot is what is called the hydraulic mean depth, or mean radius, of the canal.* Mult this by 1'2, to reduce it to ins ; and call the prod D. Then div the given mean vel, by the dec .909. Square the quot. Div said square by twice D. By formula, The read fall /given mean vel in ins per see\ 2 . (n in- permit^ ( - co , ut number ^ - ) ^ For the example at the first part of thin Art, Mr. Poole's rule gives a fall of 30.3 ins per mile : while the rule at the head of the Art, gives a fall of .0005 of a ft per ft in length ; which is equal to 31.68 ins per mile. The two rules, therefore, agree as closely as can be expected in cases of this kind. REM. In designing; watercourses, it is important to bear in mind that when a canal, as a b, Fig 31, either of uniform cross-section, and descent, or otherwise, is carried directly from a reservoir, without any enlargement of the canal at the point of junc- tion, a slight fall iorms itself at a, somewhat of the shape shown on an ^T"~-~ -~ b exaggerated scale in Fig 31. This fall will not only produce an excess of vel at that point, beyond that tor which the canal was intended; but will also di- minish the calculated depth of water in the canal, throughout its entire length, to an amount equal to the fall. Thus, if b m were the depth calculated by "O- O 4 our rules, the actual depth would be n m. In large JjlCl U J- oanals, as in those intended for navigation, this fall J at a will be approximately equal, iu feet, to * Or of anv stream ; or of a pipe. In a cylindrical pipe running full; or in any open channel of lemicircular cross-section, running full, the hyd mean depth thus found ia equal to X of the diam of the circle. 566 HYDRAULICS. The square of the calculated mean veil, in ft per tec X .0155, .9 and in smaller canals, such as mill courses, to The square of the calculated mean vel, in ft per tec, X .0155. m At the great m^an vel of 3 miles an hour, or 4.4 ft per sec, this fall at a -would be as much as .333 of a foot, or 4 ins, in a large canal: and .4 of a foot, or 4 8, in an ordinary mill-course. At 1^ miles per hour, or 2.2 ft per sec. it would be .083 of a ft. or 1 inch, in the canal: and .1 ft, or 1.2 ins. in the mill course. Without this fall the quiescent water of the reservoir would not flow out suffi- ciently fast to maintain the calculated vel in the canal. To destroy this fall, or rather, to reduce it to an almost inappreciable quantitv. and thus secure the calculated denth and disch ; at the same time providing a substitute for the fall, it is merely necessary to enlarge the canal at its junction with the reservoir: giving it at that point a shape corresponding to some deeree to that of the contracted vein. Thus, if a b. Fig 32, be the width of the canal, make c d equal to .7 of that width : at the reservoir, make e f 2 cd\ and describe two arcs of a circle a c and bf, with a rad am \% a b. See Remark. Art 16, p 562. Remark. The rate at which rain water reaches a sewer or culvert. &c, may. according to Uie admirable Report ou European Sewerage Systems" by Mr Rudolph Hering, Civ. and San. Eng. of Philada, be found approximately by the follow- ing formula by Mr Burkli-Ziegler. See Trans. Am. Soc. C. E, Nov 1881. Canal Cub. ft. per EFlmi= accordi g A f A cub ft of ra!nfall X Per second per acre, X to judgment during heaviest tall. * I Av. slope of ground I in feet per 1000 ft ^ No. of acres drained Ills coefficient for paved streets i .75: for ordinary cases ,fi?5 : and for suburbs with gardens, lawu-, au-i macadamized streets .31. His average heaviest fall is from 1% to 2% ins per hour. To this the writer will add that each Inch of rainfall per hour, corresponds closely enougu to 1 cub ft per SCO per acre; so that if \%e liberally allow for 3 or 4, &c, ins per hour of average heavi- est rainfall, the third term of the above equation also becomes simply 3 or 4, &c. Example. If an area of 3100 acres (nearly 5 sq miles), with an average slope of 5 ft per 1000 ft, receives a rainfall averaging 3 ius per hour when heaviest, then, assuming a coefficient of .5, the rate at which the water would reach the mouth of a sewer at the lower end of the 3100 acres would be 5 X 3 X ' 5 X 3 X .203 = .305 cub ft per sec per acre ; or .305 X 3100 = 945.5 cub ft per sec, total. Now suppo.se the fall of the intended sewer to be say 4 ft per mile ; and that for fear of the too rapid wearing away of its brickwork by debris swept along by the water, we limit its vel to 6.3 ft per sec, which may be permitted on occasions as rare as rains of 3 ins per hour, although for tolerably constant flow, where liable to debris, it should not exceed about 5 ft per sec. To find the diameter, look in the Table of Vels in Sewers, p 652, for a diam corresponding as near as may be, to a vel of 6.3, and to a tall of 4 ft per mile. We find this diam to be 14 ft, the area of which is 154 sq ft. Hence, 154 X 6.3 = 970 cub ft per sec = capacity of sewer. This is a trifle more than our 945.5 cub ft per sec of rainfall ; nevertheless, to allow for deposits in the sewer, it would be advisable to increase the diam say to 14.5 or 15 ft. See Caution, below. Caution. (1880.) It has for some time been known that tlie Rules given in this volume for the velocity and discharge through pipes and channels are only approximate. In cases of common occurrence with pipes of 6 to 24 ins diam they are liable to vary as much as f om 5 to 15 per ct from the truth, sometimes in exces^, and sometimes in deficiency: and in unusual cases much more. Hence also the Rules for diams and falls are similarly defective. These remarks do not apply to the Rules for orifices and weirs in thin plate. At- tempts have been made by eminent men. suoh as Weisbach, Darcy, Kutter and others to devise rules that should apply equally well to all diams, heads, and falls; but they have not succeeded perfectly, although Kutter's formula (on p 650) is a great advance on all preceding ones. Mr. J. T. Fanning, Civ. Eng. of Manchester, New I lamp- shire, in his " Practical Treatise on Water Supply Engineering," (D. Van Nostrand, N. Y., 1882, Publisher) appears to the writer to have found the source of error of his prede- cessors, and has given rules which, although much more complicated than those in this book, are perhaps as simple and as correct as the nature of the subject admits of. Art. 23. The writer has prepared Tables 17 and 18 for the convenience of forming, at the instant, an approximate idea of the requisite dimensions of channels tor fulfilling certain given conditions, but no such tables are perfectly reliable. Stut- ter's formula, p 650, may be used for such cases. HYDRAULICS. 567 TABLE 17. -Of mean velocities in feet per second; and of discharges/411 cubic feet per second, in a rectangular chan- nel 1O 1't wi. Obstructions by piers. When the area of section of a channel of running water, is diminished at any point by narrowing the stream; or by placing in it piers, such as those of a bridge. &c, then the same quantity of water which formerly flowed through the wider channel, has to force its way through the narrower one. This it can do only bv, as it were, heaping itself up. at a short dist above the obstruc- tion, so as there to form a head, as co. Fig 34, suffi- ciently great to overcome the increased resistance; and thereby f >rce along the same quantity of water as before; but of course at an increased vel. If the straight Hue ab represent the original water surf, then in ordinary oases, the curved line a * c b will approximately represent the new one, (but la an exaggerated manner,) as produced when the ends of the pier are properly rounded. ends or tne pier are properly rounded. REM. 1. It is important to bear in mind, that before the erection of an obstruction, the vel of the original stream is greatest at the surf; aud diminishes gradually toward the bottom; but when a pier. &c, is built, the increased vel produced by the head thereby created, is actually much greater we the surf; but is nearly uniform throughout the action upon the bottom than before existed, or tha Architecture ; and must be looked upon merely as probable approximations. They suppose the pier, &c, to be properly rounded or pointed at their upstream ends, so as to give as free a passage as pos- TABLE 19. Of heads produced by obstructions to streams. Original Vel. of Stream.* Kind of Bottom which begins to wear away under Bottom Vel. equal to those in the first three cols. I TV Vopoi iV tion o occnp i f Are ed ty i a Of 0] ftheO >igina bstrtu i I Water-w jtions, i| t T' f *"- *. Ooze, and Mud... Clay He .0003 .0011 .0045 .0182 .0409 .0728 .1137 .1638 .4550 id of .0004 .0014 .0056 .0225 .0507 .0902 .1410 .2030 .56*0 Water .0004 .0017 .0039 .0276 .0621 .1104 .1725 .2484 .6901 prod .0006 .OSI23 .0091 .0364 .0819 .1456 .2275 .3276 .9100 nc^d Fee .001 .0 Obsti .0033 .0133 .0532 .2128 .4788 .8412 1.320 1.915 5.280 notion .0067 .0267 .1069 .4276 .9621 1.710 2.672 3.848 10.69 sj in .0162 .06*6 .2584 1.036 2.326 4.144 64T5 9.304 25.9 Ins. 3 6 12 24 36 48 60 72 120 Ft. M K 1 2 3 4 5 6 10 Miles. .170 .341 .681 1.36 2.04 2.72 3.41 4.09 6.81 Sand Gravel Small Shingle.... Large " .... Soft Shistiis Stratified Rocks.. Hard Rocks See Table, Art 18. TABLE 2O. Increased velocities produced at and by rounded, or pointed obstructions. If square, these vels must, accord- ing to Nicholson, be increased % part. of Stream.* Proportion of Area of Watei A 1 A 1 i 1 * '-way, occnpied by the Obstructions, i i i I i I f I f Per Sec. Per Hour. V .28 .56 1.13 2.27 3.39 4.54 5.60 6.78 11.3 Blocity .29 .58 1.16 2.33 3.48 4.66 5.80 6.96 11.6 - jroduce .30 .60 1.20 2.40 3.60 4.80 6.00 7.20 12.0 d at the .32 .64 1.26 2.52 3.78 5.04 640 7.56 12.6 Obstrn .35 .70 1.40 2.80 4.20 5.60 7.00 8.40 14.0 ction in .394 .788 1.58 3.16 4.74 6.32 7.88 9.48 15.8 Feet p< .52 1.05 2.1 4.2 .3 8.4 10.5 12.6 21.0 jr Fecon .7 1.4 2.8 5.6 8.4 11.2 14.0 16.8 28.0 d, 1.05 2.1 4.2 8.4 12.6 1*5.8 21.0 25.2 42.0 Ins. 3 6 12 24 36 48 60 72 120 Ft. u i 6 10 Miles. .170 .3(1 .681 1 36 2.04 2.72 3 41 4.09 6.81 * A very vagne expression. Does it refer to the greatest surface vel at mid-channel ; or to the me*a rel of the entire cross section? HYDRAULICS. 571 Art. Sft^l'he resistance of water against a flat surface mov- ing- ttfrrousrli it at rig-tit angles, is nearly as the squares of the vel : and, according to Hutton, its amount in tbs per sq ft approx = Square of vel in ft per sec. Or like the pres of a running' stream against a perp tixed fiat surface, it is = wt of a col of water whose base pressed surf, and whose ht head due to the vel as per table p 552. The resist of a sphere is to that of its great circle about as 1 to 2.9. When the moving surf, instead of being at right angles to the direction in -which it moves, forms another angle with it, the resistance becomes less in about the following proportions. Therefore, when the surf is inclined, first calculate the resistance as if at right angles: and then mult by the following decimals opposite the angle of inclination : 90.. ..1.00 60... . .88 40. . . . .58 20. . . .16 80 .. .. .98 55 ... . .83 35 ... . .46 15 ... . .10 70 .. .. .95 50 ... . .76 30 ... . .34 10 .. . .06 65 .. .. .92 45 ... . .68 25 ... . .24 5 .. . .02 The scour, or abrading- power of moving water is considered to be as the square of its vel. Art. 27. To calculate the horse-power of falling* water, on the ordinary assumption that a horse-power is equal to 33000 Bbs lifted 1 foot vert per min. That of average horses is really but about % as much, or '22000 R>s, 1 foot high per min. Mult together the number of cub ft of water which fall per min ; the vert height or head in feet, through which it falls ; and tbe number 62.3, (the wt of a cub ft of water in Bbs ;) and div the prod by 33000. Or, by formula, cub ft v vert v lls 9 The number of __ per min * height in ft A 62.3 horae-poivera ' 33000 Ex. Over a fall 16 ft in vert height, 800 cub ft of water are disckd per min. How many horse- powers does the fall afford ? cub ft ft Ibs 800 X 16 X 2. 3 797440 HerC ' 33000 = 33000- vFater-wheelS do not realize all the power inherent in the water, as found by our rule. Thus, undershots realize but from y to % ; breast-wheels. ^ ; overshots, from % to % ; tur- bines, % to .85 of it ; according to the skill of design, and the perfection of workmanship. Even when the wheel revolves in a close-fitting casing, or breast, elbow buckets give considerably more power than plain radial or center-buckets. Of the power actually received by a wheel, part is expended in friction, ft per sec ; and in the column of heads in Table 10, opposite to 3.5 vel per sec, we fiurl the reqd head .190 of a ft. Having thus found the head, we must now h'nd the quantity of water which passes any given area of the stream in a aiin. Thus, suppose that the immersed part of a float when vert is 5 ft Ion?, and 1 ft wide or deep ; then the area of this part which receives the force of the current, is 5 X 1 : ~ 5 square feet. Hence, area vel 5 sq ft X 210 1050 cub ft per min. Having now the cub ft per min, and the vert height or head, the number of horse- powers of the stream of the given area, is found by the foregoing rule, or formula. * A committee of the Franklin Institute, in ISflO. gave .71 as the coefficient for a ram at the Girard College, in which the diam of drive-pipe was 2^ ins : its length, 160 ft; fall. 14 ft. Delivery-pipe, 1 inch diam ; 2260 ft long: vert rise, or height to which the water was raised. 93 ft. No details of the experiment are given. Some large rams in France give it usrful effect of from .6 to .65 of the whole power expended. It is an excellent machine for many purposes; and is somettrnes used for filling railway tanks at water stations. t Such wheels, for floating- mills, in Europe, rarely exceed 15 ft diam. Whatever the diam, they may have about 18 to 20 floats. The floats are from 8 to 16 ft long; and about ^ to ^ as deep as the diam of the wheel. They should not dip their entire depth into the water, but nearly so. They should not be in the same straight line with the radii ; but should incline from them 30 up stream, to produce their full effect. All these remarks applv to wheels moving freely in a wide or indefinite channel ; as in the case of a floating mill, built on a scow, and anchored out in a stream : but not to wheels for which the water is dammed up, and acts with a prac- tical fall. No great exactness is to he expected in rules on this subject. The best vel for tht wheel if about .4 that of the stream. 572 HYDRAULICS. cub ft per min 1050 ert ht in ft Ibt 62.3 33000 ~ 33000 ~ ' But in practice the wheels actually realize but about -j^j of this power of the stream, when working In an open channel ; and still less when the water flows with the same vet through a narrow artificial chanuel, but little wider than the wheel. Therefore, the actual power of our wheel will be but .377 X .4 r: .1508; or about y of a horse-power; or 33000 X .1508 = 4976 ft-fts per min. Making a rough allowauce for the friction of the machine at its journals, &c, we should have say about 4400 ft-fts of useful power ; that is, the wheel would actually raise about 440 B)s 10 ft high ; or 44 fts 100 ft high, &c, per uiiu. The vel of the stream must not be measured at the surface ; but at about % of the depth m which the floats are to dip, or be immersed. This, however, is chiefly necessary in shallow streams, in which the depth of the float bears a considerable ratio to that of the water. This power of a running; stream, (for any given area of transverse section,) increases as the cubes of the vels; for, as we have seen, the power in ft-Bbs per min is found by mult together the weight of water which passes through the section in a miu, and the virtual head in ft; and since this weight increases as the vel, and this head as the square of the vel, the prod of the two (or the power) must be as the cube of the vel. Therefore, if the vel in the foregoing case had been 10.5 ft per sec, or 3 times 3.5 ft, the power of the wheel would have been 27 times as great, or .1508 X 27 4.07 horse-powers. Art. 28. Miscellaneous on city water-pipes. The pipes are laid to conform to tue vert uudulatiou.i of the street surfaces. In Philadelphia there is about one mile of street pipe to each 1300 in- habitants. The tops of tiie pipes are laid not less than 3% feet below the surface of tA street; but in 3-inch pipes the water has at times been frozen at that depth. 8tOp- ValveS, or yateS, opening vertically in grooves, are placed across the street pipes at intervals of from 100 to 300 yds. Their use is to shut off the water from any section dur- ing repairs ; the water of such sections being allowed to run to waste, and to soak into the ground. Figs 35, 36, 36>, and 36^, will ex- C plain the general principle of these valves. J The details are much varied by diff makers. Figs 36J4 and 36^ are a form adopted by the Chapman Valve Co, of Boston, Mass.* In it the valve itself is hollow. Fig-s 35 and 36 are two styles used in Philada. The valve itself, v. Fig 36, is a circular plate of cast iron ; which, when down, as in the fig, closes the pipe. As in other styles, it opens vertically by means of a screw D; the valve rising into the cast-iron case or box B B. The screw is turned by a handle fitting on th square-head A. Whatever the style of the valve, It Is, when in place, protected by a surrounding box, ii, Fig 35, generally of plank or cast iron, with four vert sides; a movable iron top, k, level with the street; and no bottom. This top is taken off when the valve needs inspection. Two of the opposite sides of the box of course have openings for the passage of the pipes to or from the valve. Since the tops of the pipes when laid are at least 3 l <4 ft below ground, the tops of the valve-cases for small pipes. Fig 35. are also considerably below it; and for such pipes, the lifting-screw, D, is usually attached to the top of the valve itself; and extends above the case, as in the fig; being sus- tained by standards II. connected at top by cross-pieces. On these cross-pieces, is a nut at D, so confined' that it cannot move vertically; and by turning which the screw is raised, without itself turning. In Fig 35 the valve is down, or closed; when open, the top of the screw will reach nearly to k: hut still will allow the cover k to be in place. In large valves. Fig 36. which is for a 30-inch pipe, the screw D is so long, that if it rose when the gate v is lifted or open, its top h would project above the street paving. On this account it is not allowed to rise, but merely to turn ; and in so doing it raises the gate v by means of the projection t, in which is fitted a brass interior screw, or nut, through which the lifting-screw D works. The 1'ft- iug-screw foots into a socket above c ; and passes through the top of the case at o: resting upon it by means of the small enlargement or collar shown in the fig. Immediately above this collar, the screw- stem is surrounded br a loose ring, kept in place by the screws ; and thus preventing the lifting- screw from rising when it is turned by means of a key or lever, applied to its square-head ft. When so turned, the valve v rises up the screw ; and enters the top d D o of the cover. In a front view, the top casting gEg of the cover, would present the upper half of a circle. The three principal castings which compose a valve box or cover, are bolted together by means or continuous fl-inges, as at q g g. The ioint faces of these castings are carefully smoothed ; and a thin strip of lead is inserted between them, as a precaution asain^t leaks. The gates, as well as the grooves in which they slide, are usually made very slightly tapering toward the bottom, so as to in * No. 79 Water St. Boston. The Ludlow Valve Manufacturing Co, 193 River St. Troy. New York: and the Boston Machine Co. office 20 State St, Boston, are also noted for their excellent valves. The latter make the Coffin's patent double disk valve, besides other patterns; and many styles of hy- drants, or fireplugs. &c; also, locomotive aud stationary engines, dredging machines, iron bridge*, and heavy tools and machinery in general. Pig35 HYDKAULIOS. 573 gure closing tightly. The small oblong recss seen below the gate, admits small particles of foreign matter which might otherwise prevent the gate from closing perfectly. The grooves a a a a, ID which Fig 36J the gate slides, as well as those parts of the gate which come into contact with them, are carefully faced with brass, to diminish friction. At the top of the cover, the screw-stem passes through a stuffing-box, which prevents leaking at that point. Very careful workmanship is required throughout. The gates, especially of large mains, must be closed very slowly ; otherwise, the too sudden arrest- ing of the momentum of the flowing water would be apt to break either them or the covers ; or burst the pipes. As a precaution against this, the covers for very large valves are cast with outside about 50 ft. Fire-plug's, Figs 40, are placed as much as possible at summits, so as to serve also for washing the streets ; and for the escape of accumulated air. They average about 8 in num- ber to each mile of pipe ; or 1 to each block of buildings. No g-al VRIliC action has been observed where lead pipes or brass unite with cast-iron ones. Rust has not very ma- terially affected the pipes in 30 to 35 years ; even without any preservative meas- ures. See Art 34. A coat of whitewash on the outside, impedes rust quite appreciably. Some 22- inch ones which have laid in Philada for 50 years, and 4-inch ones for 65 years, are still in use, in fair condition.! No pipe as small as 3 ins diam is carried farther thail 4OO feet, without being connected with larger ones at both ends. Even this dist is at times too great for so small a pipe. None less than 4 inches diam should \>e laid in cities ; and even they only for lengths of a few hundred feet. Their insufficiency is chiefly felt in case of fire. Six ins would be a better minimum. No more leakage occurs in winter than in summer; except from the bursting of private service-pipes by freezing. The service-pipes for supplying: single dwellings, are of lead; * The following table gives a tolerable average of the weights of valves, for pipes of,various bores, by different makers ; with an approximation to average retail prices in 1873. Bore. Ins. Wt. Lbs. 72 105 173 Price. Bore. Ins. Wt. Lbs. Price. $ Bore. Ins. Wt. Lbs. Price. $ Bore. Ins. Wt. Lbs. Price. 400 450 950 3 4 6 21 28 42 8 10 12 310 575 800 56 70 96 14 16 20 1200 1600 2100 175 230 350 22 24 30 2200 2500 5200 t To compact the earth thoroughly against the pipes excludes air, and greatly impedes rust. Pipes lay be corroded by the leakage of gas through the body as well as the joints of adjacent gas pipes. 37 574 HYDRAULICS. and of % to % inch bore. They are connected with the street mains n, Fig 37, by a brass ferrule/, which is here shown at ^ real size. The dotted lines show its ^ inch bore. The ferrule is merely hard driven into a slightly conical hole reamed out of the main, as at . The lead pipe o is attached to the other end of the ferrule ; overlapping it about 1% ins; and the joint soldered, t. The extra thickness near /, is for giving v^ proper shape and strength for hammering the ferrule into the fC-\ ft O7 y\ main. The pipe and solder are shown iu section. Besides the stopcocks attached to each service pipe, and to its branches through the house, there is an underground one by which the city authorities can stop off the water in case ot delinquency iu payment of dues : and another by which the plumber can stop it off when so required during indoor repairs. Of late years galvanized ll'OIl tubes are being much used for service-pipes, especially for hot water; being less subject to contraction and expansion, which produce leaks. See near bottom of page 517, for such water pipes. Brass service pipes are now much used in Boston. See bottom of page 517 ; also table, page 3(55, and foot of 377. In IMiilada, the faucets e d o, Fig 38. of the city pipes, laid until of late years, are of the following dimensions. The width of the joint, or clear distance between the spigot and the faucet, is the same in all ; namely, % inch. Bo 3 8 Inner Diam. Inner Diam. Dpth.mnJ Thickness QO, of Pipe. ii.ofFaucet. of Faucet.] ate. i at d. ate. Ins. Ins. Ins. jlns. Ins. Ins. 30 32?^ 6 1 l>a 2 20 22* 6 H 1 IK 16 18 6 % H IK 11 14 5 3 4 9-16 X *K 10 11M 5H H % i55 8 9K 5tf H i 6 IX 7-16 % IM 4 5^ ^A 7-16 % IK 3 4^ 4 H % 1*4 The small beads at and m, on the spigot end of the pipe, project about }4 i nCQ : and are to prevent tae calking material from entering the pipe. The calking consists of about 1 to 2 ins in depth of well- rammed untarred gasket, or ropeyarn : above which is poured melted lead, confined from spreading by means of clay plastered around the joint. The lead is afterward compacted by a calking hammer. The weight of lead used per joint, averaged about as follows : 30 ins diam, 107 tbs ; 20 ins, 60 fts ; 18 ins, 50 fts ; 16 ins, 35 Its ; 12 ins, 24 fts ; 10 ins, 16 fts ; 8 ins, 13 fts ; 6 ins, 10 fts; 4 ins, 8 fts ; 3 ins, 5 fts. Hydraulic engineers are, however, now pretty generally re- ducing the depth of faucets about % part; and the depth of the lead joint to 2 or 2^ inches, and its wt to 2 Bis per inch of diam for pipes of any diam under 30 ins. Faucets are now frequently made as shown at W ; leaving for the lead joint a thickness of % inch at top, and -j-^g inch below. At the average f .4 inch, and a depth of 2^ ins, we have about 1}^ fts for every inch of outer diam of pipe; and for an entire joint as follows, the diams being the inner ones: 2 ins, 3.67 fts: 3 ins, 5 fts; 4 ins, 6.7 fts; 6 ins, 9.3 fts ; 8 ins, 12 fts ; 10 ins, 15 fts ; 12 ins. 17.7 fts ; 14 ins, 20.7 fts ; 16 ins. 23.3 fts ; 18 ins, 26 fts ; 20 ins, 28.6 fts ; 24 ins, 34.3 fts ; 30 ins, 43 fts. The form of joint at W is said to prevent to some extent the tendency of the lead to creep, or work loose by contraction and expansion. Pipes are now often made 12 ft long, instead of 9 ft as formerly ; thus requiring only about 452 joints 11 ft 8 ins apart per mile, instead of about 610 joints 8 ft 8 ins apart ; thereby reducing the expense for lead very materially. Mr Thomas Wicksteed, engineer of the East London water-works, England, says that more than 50 years' experience proves that slightly tapering wedges of pine, about 4 ins long, 2 ins wide, and S| ins thick at the butt, carefully shaped to suit the curve of the pipe, and well driven, answer all the purposes of lead joints, at considerably less cost. Cost of water pipes 9 ft long*, per mile, laid ; assuming the pipes to cost 3 cts per ft, or $67.20 per ton, delivered along the streets; lead for the joiuts, of the weights formerly used in Philada, at 10 cents per ft ; and the laying, (including digging and refilling the trenches, in earth; making the joints, including the gasket yarn for that purpose.) according to rates which experience has shown to be fair when common labor costs $1 per day. A small addition must be made if unpaving and repaving are to be done. Owing to the constant fluctuations in prices, such tables answer but an imperfect purpose. In 1873 the laying would be nearly twice as great. Diam. Thick's. Tons Cost per Lead. Lead. Lnvini? Laving. Total. Ins. Ins. per Mile. Mile. $ Lbs. $ .erf, Cts. per" Mile. $ 30 1 860 57792 66500 66oO 100 5280 69722 24 y 570 38804 45000 4500 65 3432 46236 20 H 400 26880 37300 3730 45 2376 32986 18 y\ 358 24058 31100 3110 40 2112 29280 16 325 21840 21800 2180 36 1901 25921 14 H 267 17943 18000 1800 32 1690 21433 12 % 205 13776 15000 1500 28 1478 16754 10 % 171 11491 10000 1000 25 1320 13811 8 H 111 7459 8100 810 22 1162 9431 6 % 84 5645 6300 630 20 1056 7331 4 % 58 3898 5000 500 18 951 5349 3 H 33 2218 3200 320 16 845 ssas 2 H 24 HM3 1500 150 15 792 2555 For thickness of heads, see p 532 of HydrostaJ 300 fibs per sq iuch. ULICS. 575 etal pipes to resist safely the pressures of various Those iu the last table have been tested to about 700 feet head, or The following- are the contract prices for laying- pipes in the city oc Brooklyn. N. York, in 186y- raulic or aspbaltic cement. When exposed to pressure from great heads, they are spirally wrapped or banded with sheet or plate iron. A 10-inch pipe thus banded with iron of 1 X % inch ; spirals 4 ins apart; sustained a test pressure of 406 Ibs per sq inch ; equal to a head of water of 935 ft. Their cost is but about half that of cast-iron pipes. They are made by J. A. Woodward, Williamsport, Penna; and have been quite extensively and successfully used for both water and gas. Water pipes of bored yellow pine logs, laid in Philada 50 to 60 years ago. are frequently quite sound, and still fit for use, except where outer sap wood is decayed. When this is removed, many of these old pipes have been relaid in factories, to l.# A top width of 15 ft to 20 ft, and inside slopes of 3 to 1, are adopted in some important cases; with outer slopes of 2 to 1. Both slopes, however, are at times made only \% to 1. The level water surf should be kept at least 3 or 4 feet below the top of the embkt ; or more, if liable to waves. In a large reservoir, a quite moderate breeze will raise waves that will run 3 ft (measured vert) up the inner slope. A low wall, or close fence, ?. Fig 41, is s times used as a defence against them. The top and the outer slopes should be protected at least by sod or by grass. To assist in keeping the top dry, it should be either a little rounding, or else sloped toward the outside. The soft soil and vegetable matter should be carefully removed from under the that leakage may not take place under them. To aid in this, a double row of sheet piles, or a sunk wall of cement masonry, carried to a suitable depth below the bottom, may be placed along the inner toe in bad cases. If there are springs beneath the base, they must either be stopped, or led away by pipes. The embkt should be carried up in layers, slightly hollowing toward the center, and not ex- ceeding a foot in thickness ; and all stones, stumps, and other foreign material, such as clean gravel, sand, and decomposed mica shists, Ac, that may produce leakage, carefully excluded. These layers shriuld be well consolidated by the carts; and the easier the slopes are, the more effectively can this be done. The layers, however, should not be distinct, and separated by actual plane surfaces; but each succeeding one should be well incorporated with the one below. This has sometimes been done by driving a drove of oxen, or even sheep, repeatedly over each layer: in addition to the carting. Rollers are not to be recommended, as they tend to produce seams between the layers. This might possibly be obviated by projections on the circumf of the roller. Gravelly earth is an excellent material, perhaps the best. The choicest material should be placed In the slope next to the reservoir ; and should be deposited and compacted with special care in that portion, so as to prevent the water from leaking into the main body of the dam, and thus weakening It. It is not amiss to introduce a bench fc, Fig 41, in the outer slope, to diminish danger from rain- wash by breaking the rapidity of its descent. If the bottom of the reservoir itself is on a leaky soil, er on fissured rock, through the seams of which water may escape, it must be carefully covered with from 1 J^ to 3 feet of good puddle: which, in turn should be protected from abrasion and disturbance, by a layer of gravel; or of concrete, either paved or not. according to circumstances. Reservoirs constructed with the foregoing dimensions, and with care, may remain safe for an in- definite period ; but where serious damage would result from failure, the following additional pre- cautions should he taken. The inner slopes should be carefully faced up to the very top. with at least a close dry rubble-stone pitching*, not less than about 15 to 18 ins thick; as a protection against wash, and against muskrats. These animals, we believe, always commence to burrow under water. If the slopes are * The writer suggests that a top width equal to 2 ft + twice the sq rt of the height in feet, will b 3fe for any height whatever of reservoir properly constructed in other respects. 578 HYDRAULICS. Fig 41 much steeper than 2 to 1, this dry pitching will be apt to be overthrown by the sliding down of the softened earth behind it, if tue water in the reservoir should for any cau.-,e be drawn tiuwu rather suddenly. It will be much more effective, but of course more costly, if laid in hydraulic ctrneut ; aud still more so if laid upon a layer a few ins thick of cemeut-and-gravel concrete; especially if this last be underlaid by a layer about 1}^ to 3 ft thick of good puddle, spread over the face of the slope ; tha great object being to protect the inner elope from actual contact with the water. If this can be ef- fectually accomplished, slopes as steep as 1^ to 1 will be perfectly secure; for the danger does not arise from any want of weight of the earth for resisting overthrow. Special care should be bestowed upon the inner toe of the slope, to prevent water from finding its way beneath it, and softening the earth so as to undermine the stone pitching. Near the top, reference should be had to danger of de- rangement by ice, frost, rain, and waves. Flat inner slopes tend not only to prevent the displace- ment of the pitching; but increase the stability of the embankment, by causing the pressure of the water (which is always at right angles to the slope) to become more nearly vertical ; and thus to hold the embankment more firmly to its base than if there were no water behind it. Sometimes the toes of both the inner and outer slopes abut against low retaining- walls in cement. This gives a neat finish, and tends to preservation from injury. Many engineers, in order to prevent leaking, either through or beneath the embankment, construct a putldle- Wall, jt>, Fig 41, of well- rammed impervious soil, (gravelly clay is the best,) reaching from the top to several feet below the base. This wall should not be less than 6 or 8 ft thick on top, for a deep reservoir; and should in- crease downward by offsets (and not by slopes, or batters) at the rate of about 1 in total thickness, to 3 or 4 in depth. Other engineers object to these puddle-walls ; and contend that leakage should be prevented by making both the inner slopes, and the bottom of the reservoir, water-tight, by means of puddle, concrete, and stone facing in cement, as just alluded to. They argue that if the embankment is well constructed, it is itself a puddle-wall throughout. Near Sail Francisco, Cal, are two earthen reservoir clams built about 1864, one 95 ft high, 26 on top, inner slope 2.75 to 1, outer 2.5 to 1. The other 93 high, 25 on top, inner slope 3.5 to 1, outer 3 to 1. In each the puddle- wall is carried 47 ft deeper than the base. No stone facing. It is difficult to prevent water under higli pressure from finding* its way through considerable distances along- seams iron pipes laid under reservoir embankments ; or along the tie-rods sometimes used through the pud- dle of coffer-dams ; and the same is apt to occur under the bases of embankments which rest on smooth rock. Special care should be taken that the earth used in such positions is not of a porous nature; ani that it is thoroughly compacted all along the seam; and the straight continuity of the sen;a should be interrupted or broken as frequently as possible by projections. Faucets or flanges do this to a limited extent in the case of iron pipes; and something similar, but on a larger scale, should at short intervals be constructed in the shape of collars or yokes of cement stonework, in the case of rock or masonry. For more on dams, see Dams, p 583, also 528 &c. It is usually advisable to divide reservoirs into two parts, so that while the water in one part is being drawn off for use, that in the other may purify itself by settling its sediment. Also, one part may remain in use, while the other is being cleaned or repaired. Many days, or even two or three weeks, sometimes, are required for the complete settlement of the very fine clayey particles in muddy water: depending on the depth of the reservoir. One or more flights of steps to the bottom of the reservoir, should be provided. Mud in Reservoirs. The reservoirs of the New River Water Co, London, England, were uncleaned for 100 years, during which mud 8 ft deep was deposited, or about an inch annually. At Philadelphia it is about .25 inch per annum from the Schuylkill, and 1 inch from the Delaware Riv. At St Louis, Missouri, about 3 to 4 ft per year! In shallow reservoirs vegetation takes place, and injures the water. A depth of about 20 ft appears to prevent this. Water flowing through marsh lands is sometimes unfit for arinking purposes. That, for instance, in some sections of the Concord River, Massachusetts, was reported by the eminent hydraulic engi- neer, Loammi Baldwin, of Boston, to be absolutely poisonous from this cause. The construction of a large deep reservoir is not only a very costly, but a very hazardous under- taking. With every watchfulness and care, it is almost impossible entirely to prevent leaking ; although this may not manifest itself for months, or even years. Should a break occur, especially near a city, it would probably be attended by great loss of life and property. If the water once finds its way in a stream, either across the unpaved top, or through the body of the embankment, the rapid destruction of the whole becomes almost certain. Art. 3O. STORING RESERVOIRS. The entire annual yield of a stream mav be much more than sufficient for supplying a certain population with water; and yet in its natu- ral condition the stream may not be available for this purpose, because it becomes nearly dry in sum- mer, when water is most needed; while, at other seasons, the rains and melted snows produce floods which supply vastly more than is reqd ; and which must be allowed to run to waste. A storing reser- voir is intended to collect and store up this excess of water, so that it may be drawn off as reqd during the droughts of summer, and thus equalize the supply throughout the entire year. This, when the lo- cality permits, is effected by building a dam across the stream, to form one side of the reservoir ; while the hill-slopes of the vallev of the stream form the other sides. The stream itself flows into this roservoir at its up-stream end. "When the stream is liable to become nearly dry during long summer droughts experience shows that the capacity of the reservoir should be equa". to "from 4 to 6 months' supplv. ac- cording to circumstances. During the construction of the dam. a free channel must he provided, to pass the stream without allowing it to do injury to the work. If the dam were built precisely Mke Fig 41. entirely of earth, it would plainly be liable to destruction by being washed away in case the reser- vo'r sbou'.t Become SG f&'.l that the water would begin to flow over its top. To provide against this we may, by means of masonry, or of cribs filled with broken stone, or otherwise, construct either th HYDRAUllCS. 579 (rhole, or a part of the dam, to serve as ap/bverfall, or a Waste-weir. Or a side chan- nel (an open cut, pipes, or a cul\vrt,J&)ma,y be provided at one or both ends of the dam, arid in the natural soil, at such a level as to cajaryaway the surplus flood water before it cau rise h.gh enough to overtop the eartheu dam. BesidjjfThese, and the pipes for carrying the water to the town, there should be a order that. if necessary "for repairs, on/fur cleaning by scouring, all the water may be drawn off. The entrances to the city pipes shouldjeprotected by gratings, to exclude fish, &c. To facilitajre repairs or renewals of all valves, fcc, which are under Jwater, the reservoir ends of the pipes or culverts to which they are attached, maybe surrounded by a water-tight box or chamber, which will usually be left open to tho reservoir ; but may be closed when repairs are required. Access may then be had to them by entering at the outer end, after the water has flowed away from inside. In case the outlet is through a long line of pipes which cannot thus be entered, a special entry for this purpose may be cast in the pipe itself, near the outer toe of the embkt; to be kept closed except in cas of repairs. Sometimes a bet. ter, but more expensive means of access to such valves, is secured by enclosing them in a Valve- tO WCr of masonry. This is a hollow vert water-tight chamber, like a well ; but near the toe of the inner slope; having its foundation at the bottom of the reservoir; whence the tower rises through the water to above its surf. This chamber is provided with valves or gates usually left open to the reser- voir ; but which may be closed when repairs are needed ; and the water in the tower allowed to escape from it through the open valves of the outlets. This done, workmen can descend through the tower by ladders from the aperture at its top. At times the outlets for the discharge of surplus flood water are, like those for scouring, placed at, or just above, the level of the bottom of the reservoir. In order that these may work in case of a sud- den flood at night, &c, they must be furnished with self-acting valves, which will open of their own accord when the flood is about to rise too high. This may be effected by attaching them to floats, the rising of which, when the water is high, wiM pull them open. All such outlets should be large enough to let men enter them for repairs. They should by no means be laid through the artificial earthen body of the dam itself, without being supported upon masonry reaching down to a firm natural foundation ; otherwise they are very apt to be broken by the subsidence of the embankment. It is usually safer to carry them through the firm natural soil near one end of the dam. Their valves, if only single, should be at their inner or reservoir end, so as to leave the outlets themselves usually empty, for inspection ; but it is better to have two valves, so that one may be used when the other needs repair ; and in this case one may be placed at each end. Reservoirs which are supplied by pumps, need no precautions against overflow; because the pumping is stopped when they are tilled to the proper height. Large storing reservoirs necessarily submerge more or less land, which has therefore to be purchased. They frequently prevent spring floods from injuring low lands farther down stream ; inasmuch as they in- tercept the descending water. In case there are mills down stream from the reservoir, they would evidently be deprived of water for driving them, unless a portion of that stored in the reservoir be devoted to that purpose. Water thus applied to compensate for the loss of the natural stream, is called Compensation water ; and the reservoir, a compensating one. Art- 31. Distributing: reservoirs. Frequently a valley fit for a storing reservoir can be found only at a long dist (sometimes many miles) from the town; and it then be- comes expedient to construct also an additional one of smaller size than the storing one, near the town ; and at as great an elevation above it as circumstances will permit ; but lower than the storing one. This is called, by way of distinction, a distributing reservoir, because from it the water, after having flowed into it from the storing reservoir, through the long supply pipe which connects them, is distributed in various directions through the town, by means of the street mains, or pipes. This small reservoir should hold a supply sufficient at least for a few days ; a few weeks would be better; and the end of the supply pipe which terminates in it, should be p'rovided with a valve for shutting off the supply from the storing reservoir. These precautions permit repairs to be made along the line of supply pipe without depriving the town of water in the mean time. With a view to such repairs ; as well as to scouring out sediment from the supply pipe, this last should be provided with Outlet Valves at various low points along the entire interval between the two reservoirs ; especially at those at which the valves may disch into natural watercourses. On opening these valves, the out- rush of the water carries away sediment; and leaves the pipe empty for inspection. Art. 32. Air valves. Air is apt to collect gradually at the high points of vert curves along the supply pipes; and, unless removed, produces more or less obstruction to the flow. This may be prevented by air valves, see Fig 42, which is % of the full size of those once used in Philada. This simple device consists of a cast-iron box, ccdd. confined to the main pipe m m, by screw-bolts passing through its flange dd. It has a cover gng, confined to it by screws tt; and at the top -f which is an opening n, for the escape of air from within. In this box is a float /, which may be a close tin or copper vessel, or of layers of cork, as supposed in the fig; or &c. This float has a spindle or stem 8 s. fast to it; which passes through openings in the bridge-bars a a. and o ; thereby allowing the float to rise and fall freely, but prevent- cg it from moving sideways. When the pipe mm is empty, the float i* down; its base y resting on the cross-bar a a. The stem ss has fixed to it avalve , which rises and falls with it and the float. Suppose the pipe mm to be empty, and consequently the float, and the valve v , down. Then, if water be admitted into the pipe, it will rise and fill also the box as far up as e; and in doing so will lift the float/, and the valve v, lo the position in the fig ; thus preventing 580 HYDRAULICS. egress to the outer air, by closing the opening at v. Now, air carried along by the water, will, en account of its lightness, ascend to the highest points it meets with. Hence, when such air arrives under the opening at a a, it will rise through it, and ascend to e; the closed valve preventing it from going farther. Thus successive portions of air uscend, and in time accumulate to such an extent as gradually to force much of the water downward out of the box. When this takes place, the float, which is held up only by the water, of course descends also; and iu doing so, pulls down with it the valve v. The accumulated air then instantly escapes through the openings at v and n, into the atmosphere ; and the water in the pipe mm, immediately ascends again into the box, carrying with it the float ; and thus again closing the valve v. The valve, and the valve- seat e, are faced with brass, to avoid rust, and consequent bad tit. The whole is protected bv au iron or wooden cover, reaching to the level of the street, somewhat as in Fig 35, p 572. Air valves are no longer used in city pipes; their place being supplied by the fireplugs at average distances of about 150 yds apart. These, being placed as much as possible at the summits of undulations in the lines of pipes, for convenience of washing the streets, and being frequently opened for that purpose, permit also the escape of accumulated air. The escape of compressed air through an air valve, or other opening, has been known to produce bursting of the main pipes; for the escape is instantaneous, and permits the columns of water in the pipes on both sides of the valve, to rush together with great forces, which arrest each other, and react against the pipes. Water for city use should not be drawn from the very bottom of the reservoir, because it will the* be apt to carry along the sediment; which not only injures the water, but creates deposits within the pipes ; thus obstructing the flow. In fixing upon the necessary capacity of a reservoir, this must be taken into consideration ; inasmuch as all the water below the leve! for drawing oft", must be re- garded as lost. When circumstances justify the expense, it is well to curve up the reservoir end of the service main, so as to provide it with valves at ditf heights; for drawing off only the purest stratum that may be in the reservoir. With this view, the valve-tower before spoken of generally has such valves communicating with the water in the reservoir ; and by this means only the purest is admitted into the tower; and from it, into the city pipes. This refinement, however, is rarely practicable. Such valves must of course be worked by watchmen. The quantity of water reqd in cities, has been found by experience to increase faster than the population. About 60 gallons, or 8 cub ft, per day, to each inhabitant, is usually considered a fair ample allowance. Many European cities have not half as much ; while New York, and some others, use and waste half as much more. With efficient means for preventing waste, 60 gallons would probably suffice for any commercial city ; but inasmuch as cleanliness and health are promoted by its free use, as few restrictions as possible should be introduced. Ill fixing upon the diams of pipes for supplying cities, it is necessary to bear in mind, that by far the greater portion of the 24 hour*' yield is actually drawn from them during only 8 to 12 hours of daylight; and therefore the capacity of the pipes must be sufficient to furnish the daily supply in much less than 24 hours. Again, during the hot summer months, much more water is used than during the winter ones ; and this consideration necessitates a still larger diam. Art. 33. Systems of street pipes for supplying cities. The writer knows of no practical rules for proportioning the diams for such systems. The various com- plications involved, render a purely scientific investigation of little or no service. With much hesi- tation, he ventures the following purely empirical rules of his own ; based on such limited observa- tions as have casually fallen under his notice. RULE I. When, at no point in a system of city pipes, is the head, or vert dist below the surface of the reservoir, compared with the hor dist from the reservoir, less than at the rate of 50 ft per mile, then the population in the last column of the following Table A, may be abundantly supplied, for all city purposes, by either one pipe of the inner diam or bore in the 1st col; or by 2, 3, &c, pipes of the diams in the other cols. These diams are given to the nearest safe % inch. The supply is assumed to be about 60 gallons per day to each inhabitant. TABLE A. (Original.) 1 2 BTTIMBEB, 3 | 4 OF PIP! 6 is. 8 12 24 Population Diam. Ins. Diam. Ins. Diam. Ins. Diam. Ins. Diam. Ins. Diara. Ins. Diam. Ins. Diam. Ins. 6 4% 3% 3J^ 3 2%~ 2% 1% 1647 8 6% 4% 4 8* 3% 2% 3465 10 7% 6* 5% 5 4% 3% 3 5908 11 7% 7 9324 14 10% 9% 8% 7 6% 5% 4% 1370* 16 18 u* 13% 10% 11% 8% 6 6% 4% 5% 19141 25677 20 r *y* 13 11% 9% 8% 5% 33426 22 14J4 12% 10% 9^6 8$ 6M 42433 24 1HV 15 H 18 H 11% 10* j 9 6% 52671 26 19% 1H% 15 12% 11* 9% 64447 28 30 i* 22% 19% 16* 17% 13% 14% 13% MM 11* 8 8% 77565 91580 32 24% 20% 18* 15% 14 n% 9 108160 34 36 25% 22 27% 23 19% 20% 16% 17% 15 15% 12% 18* 10* 125840 144480 40 30% 25% 23% 19% 17% 15 uS 188320 44 33J4 28% 25% 12% 239600 48 36% 31 27% 23% 21% If? 13% 297600 54 60 66 72 80 41 1 34% 45% j 38% 50% | 42% 54% | 46^ 60% 51% 81* 34% 38% 41% 29% 32% 35% 39* 23% 26% 29% 31% 85* 20% , 22% 24% 26% 29% 15% 18% 22% 391200 511200 650400 800000 106400G HYDKAULICS. 581 It is well to ahow in addition from ^ inch to 1 inch, or more, (depending on the character of the water,) to each diam ; for deposits and concretions. The water, after reaching the city through one or more large main pipes from the reservoir, must be distributed through the streets by means of smaller mains branching from the large ones. The diams of these smaller ones also may be found by Table A. Thus, if a street, with its alleys, &c, contains about 6000 persons, (the rate of head being, as before, not less than 50 ft to a mile at any point of the system,) then we see ly the table that a 10- inch pipe will answer. It would be well to lay BO city street pipes of less than 6 ins diam. Mains which cross each other should be connected at some Of their intersections, to allow the water a more free circulation through- out the entire system; so that if the supply at any point is temporarily cut off from one direction by closing the valves for repairs, or is diminished by excessive demand, it may be maintained by the flow from other directions. Avoid dead ends when possible, as the water in them becomes foul and unwholesome. RULE 2. With the same diams, different rates of head will supply the proportionate populations in cof. 3 of Table B. Or, to find the diams which at different rates of head will supply the same popula- tions given in the last col of Table A, mult the diam given in Table A, by the corresponding number in col 4 of Table B ; or (approximately) do as directed in col 5. TABLE B. (Original.) COL. 1. COL. 2. COL. 3. COL. 4. COL. 5. Rate of Head, in Feet per Mile. Rate of Head, compared with that in Table A. Proportionate Populations. Proportionate Diam. to supply the Populations in Table A. Remarks. 5 .1 .32 1.58 10 .2 .45 1.37 12}$ .25 .50 1.32 Add one-third. 15 .3 .55 1.27 Add full one-fourth. 20 .4 .64 1.20 Add one-fifth. 25 .5 .71 1.14 Add one-seventh. 30 .6 .78 1.11 Add one-ninth. 35 .7 .84 1.07 Add one- fourteenth. 40** .75 .87 1.06 Add one-sixteenth. 45 J .'95 1.05 1.02 Add one-fiftieth. 50 1.0 1.00 1.00 75 1.5 1.23 .92 Deduct one-thirteenth. 100 2.0 1.41 .88 Deduct one eighth. 125 2.5 1.59 .83 Deduct full one-sixth. 150 3.0 1.73 .80 Deduct one-fifth. 200 250 4.0 5.0 2.00 2.25 .76 .73 Deduct nearly one-fourth. Deduct nearly two-sevenths. 300 60 2.46 .69 Deduct three-tenths. 400 8.0 2.83 .66 Deduct full one-third. 500 10.0 3.18 .63 Examp. By Table A we see that with the rate of head of 50 feet per mile, a 30-inch pipe will supply a population of 91580 ; but with three times that rate of head, or 150 ft per mile, we see by col 3, Table B, that the same pipe will supply 1.73 times as many persons, or 91580 X 1.73 = 158433 persons. But if, at this greater rate of head, we still wish to supply only 91580 persons, then we find in col 4, Table B, that we may diminish the diam of the pipe from 30, down to 30 X .80 = 24 ins; or, by col 5, we have 30 6 = 24 ins. Again, after the water has reached the city by the 30-inch pipe of Table A, if we wish to distribute it through the city by say eight branches or smaller mains, we see by col 6, Table A, that each of them must have at least 13}^ ins diam. From these eight, other smaller ones may branch off into the cross streets, alleys, Ac; and in estimating the supply required for any particular street main, we must evidently add what is reqd also for such cross streets, &c, &c, as are to be fed from said main. If certain limited parts of a city pipe system have considerably less rates of head than most of the remainder, it may become expedient to supply the former by a special separate main of Inrger diam j which may start either directly from the reservoir ; or as a branch from the grand leading main which feeds the lower parts, according to circumstances. It must be remembered, that although by increasing the diams, an abundant supply may be ob- tained under a small rate of head, as well as under a great one. yet the water will not rise to as great a height in the service pipes for supplying the different stones of dwellings, &c. Even with the diams in Table A, the water, under ordinary use, will not rise in these pipes to the full height of the surface of the reservoir; and if an unusual'dra wing-off is going on at the same time at many parts of the system, as in case of an extensive fire, or frequently during the hot summer months, it may ot rise to even one-half of that height. Art. 84. The following: has been found very effective for preventing concretions in water pipes. Formerly in Boston, cast* iron city pipes, 4 ins diam, became closed up in 7 years : and those of larger diam became seriously reduced in the same time. But during the last 8 years, in which this varnish has been used, no con* eretions have formed. Coal-pitch varnish to be applied to pipes and castings, 582 HYDRAULICS. made for the Water Department of Philadelphia, under the following: conditions: ;arth or sand which to remove the loose First. Every pipe must be thoroughly dressed and made clean, free frc eliugs to the irou iu the moulds ; hard brushes to be used in finishing the dust. Second. Every pipe must be entirely free from rust when the varnish is applied. If the pipe can- not be dipped immediately alter being cleansed, the surface must be oiled witu huseed oil to preserve it until it is ready to be dipped : no pipe to be dipped after rust has set in. Third. The coal tar pitch is made from coal tar, distilled until the naphtha is entirely removed, and the material deodorized. It should be distilled until it is about the consistency of wax. The mixture of five or six per cent of linseed oil is recommended. Pitch which becomes hard and brittle when cold, will not answer for this use. Fourth. Pitch of the proper quality having been obtained, it must be carefully heated in a suit- able vessel to a temperature of 300 degrees Fahrenheit, and must be maintained at not less than this temperature during the time of dipping. The material will thicken and deteriorate after a number of pipes have been dipped; fresh pitch must therefore be frequently added; and occasionally the vessel must be entirely emptied of its old contents, and refilled with fresh pitch; the refuse will be hard and brittle like common pitch. Fifth. Every pipe must attain a temperature of 300 degrees Fahrenheit, before it is removed from the vessel of hot pitch. It may then be slowly removed and laid upon skids to drip. AH pipes of 20 inches diameter and upward, will require to remain at least thirty minutes in the hot fluid, to attain this temperature; probably more in cold weather. Sixth. The application must be made to the satisfaction of the Chief Engineer of the Water De- partment; and the material be subject at all times to his examination, inspection, and rejection. Seventh. Payment for coating the pipes will only be made on such pipes as are sound and suffi- cient according to the specifications, and are acceptable independent of the coating. Eighth. No pipe to be dipped until the authorized inspector has examined it as to cleaning and rust; and subjected it thoroughly to the hammer proof. It may then be dipped, after which, it will be passed to the hydraulic press to meet the required water proof. Ninth. The proper coaling will be tough and tenacious when cold on the pipes, and not brittle or with any tendency to scale off. When the coating of any pipe has not been properly applied, and does not give satisfaction, whether from defect in material, tools, or manipulations, it shall not be paid for; if it scales off or shows a. tendency that way, the pipe shall be cleansed inside before it can be recoated or be receivable as an ordinary pipe.* FREDERIC GRAFF, Chief Eng Water Department. Art. 35. The syphon, or siphon f If one leg a b of a bent tube or pipe a be, Fig 43, of any diam, filled with water, and with both its ends stopped, be placed in a reservoir of water, as in the fig; and if the stoppers be then removed, the water in the reservoir will begin to flow out at c, and will continue to do so until its level is reduced to t, which is the same aa that of the highest end c of the pipe or syphon. The flow will then stop. The parts a b and 6 c are called the legs of the syphon, 6 being its high- est point ; and this is correct so far as relates to it merely as a piece of tube ; but considering it purely with regard to its character as a hydrau- lic machine, the part t a below the level of the highest end c, may be en- tirely neglected; for the water in the reservoir will not be draw'n down below the level of the highest end, whether that be the inner or the outer one. Therefore, if the disch end be above the water in the reservoir, as, for instance, at w, no flow will take place. The vert height 6 o, from the highest part of the syphon, to the lowest level t, to which the reservoir is to be drawn down, must not, theoretically, exceed about 33 or 34 ft; or that at which the pres of the air will sustain a column of water. Practically it must be less, to allow for the friction of the flowing water, and for air which forces its way in. And still less at places far above sea level; for at such the reduced weight of the atmospheric column will not balance so great a height of water. In order readily to understand, or at any time to recall the principle on which the syphon acts, bear in mind that we may theoretically consider the end of the inner leg to be not actually immersed below the water surf, but only to be kept precisely at it, as the surf descends while the water is flowing out: but may re- gard the vert dist 6 o as the length of the outer leg ; and a varying dist, which at first is b s, and finally b o (as the surf of the reservoir descends) as the length of the inner leg; and that the flow continues only while this outer leg is longer than this inner one. The books are wrong in saying that the outer leg 6 c mnst be longer than the inner one b a, in order that the water may run at all. The principle then is simply this: that both these legs 6 c, and bi, being first filled with water, (the part i a being considered at first as a portion of the reservoir, and not of the syphon,) it follows that when the stop- pers are removed from the ends c and a, the air presses equally'against these ends ; but the great vert head of water b o iu the outer leg b c, presses against the air at c, with more force than the small head of water 6s in the inner leg bi, does against the air at a or i.l Consequently, the water in be will tend to fall out more rapidly than that iu ft t ; and as it commences to fall, would produce a vacuum at 6, were it not that the pres of the air against the other end a or t, forces the water up i b, to supply the place of that which flows out at c. In this manner the flow continues until the surf of the water in the reservoir descends to t. on the same level as c. The pressures of the vert heads bo, bo, in the two legs be, bt, being then equal, it ceases. The syphon principle may be employed for draining ponds into lower ground at a considerable dist, eveia though an elevation of several feet (in practice perhaps not exceeding about 28 ft above the level to which the pond is to be reduced) may intervene. In such a case an escape must be provided at ttie summit (or summits, if there are niore than one) of the bends, for the disch of free air, which will inevitably enter, and soon stop the flow, unless this precaution be taken. The air-valve Fig 42 * Such coating is said by D^rcy to increase the discharge materially. tThis subject belongs more properly to Pneumatics. } Said pres of the air at a or i, is of course not direct ; but is transmitted through the water to a; ftfld thence upward through the syphon to i. [>AMS. 583 ,/ill not answer for this, because as soon as the valve v opens, the syphon becomes n effect two separate tubes i#pen at top; and the water will fall in both. An ori- fice at the escape will be weeded for filling the syphon at the start; and to pre- sent the water thus intnwluced, from running out. stopcocks must be provided at ;he ends, and kept closed until the filling is completed. The greatest pains must be taken to make all the joints perfectly air-tight. The motive power or head which causes the flow in a syphon, is the ng this head, the entire length a b c ot the syphon, and its diam, all in It, the disch may be found approximately by either of the rules given in Art 2 for straight pipes. These rules f :>y Col Crozet's syph approximately uy eii/iier ui me i iues givt:u in /vri & 101 siraigiit give 55% galls per min, instead of the 43% galls actually discnd Ion, with a head of 20 ft, as stated oil p 661, which see. DAMS. WE can devote but little space to this subject, in addition to what Is said on earthen dams for re- servoirs p 577; and on stone ones, p 528 &c. Those we shall BOW describe will also answer for uch reservoirs, when the perishable nature of timber is not an objection. Primary objects in the erection of dams, are, a foundation suffi- jiently firm to prevent them from settling, and thus leaking; the prevention of leaks through their backs, or under their bases ; and the prevention of wear of the bottom of the stream in front of the dam, by the action of the falling water. For the first purpose, hard level rock bottom is of course ' best; and should be chosen, if possible. In that case, thick planks, tt. Fig 6, (single or double, ihe case may be,) closely jointed, and reaching from the crest c, to the back lower edge w, (where hey should be scribed down to the rock;) with a good backing, b, of gravel, will suffice to prevent :aks. Gravel, or rather very gravelly soil, is far better than earth for this purpose ; for if the water bould chance to form a void" in it, the gravel falls and stops it. To prevent this backing from being disturbed near the crest of the dam, by floating bodies swept along by freshets, a rough pavement of stones, ahout 15 to 18 ins deep, as shown in Fig 7, should be added for a width of about 10 to 20 ft; or until its top becomes 3 to 5 ft below the crest c of the dam, according to circumstances. Rgl ROCK In Fig 1, (a dam on the Schuylkill navigation,) the upper timbers, e, are all close jointed, and laid teaching, so as not to require planking in addition. But if the bottom of the stream is gravel or earth, there must in addition to these be used twc thicknesses of sheet piles, p, Fig 2, &c, close driven, breaking joint, to a depth of several ft. to pre- vent leaking through the soil beneath the base of the dam. Frequently but one thickness is used. If the bottom is soft or open for a depth of only a few ft, it is at times better to remove it. and base the dam on the firmer stratum below; still, however, using the sheet piles. Old decayed timber and other rubbish should be removed from the base. In very bad soils of greater depth, it may be neces- iry to support the dam entirely upon a platform resting on bearing piles. Here great p'recautions re necessary against leaks ; but the case occurs so rarely, that we shall not stop to consider it. As to the wearing away of the bottom of the stream by the water falling over the front of the dam, precautions should be used in all cases except that of very hard rock, or of medium rock protected C GRAVE L by a considerable depth of water. The dam, Fig 1, was built upon a tolerably firm micaceous gneiss in nearly vert strata, covered by about 2 feet of water in ordinary stages. lii 39 years the rock was 584 DAMS. Rq3 worn away in front of the dam, as shown in the fig, to the average depth of 3 feet; or very nearly 1 inch per year. The depth of water on the crest c, was usually from 6 to 18 ins ; rarely 5 or 6 ft dur- ing freshets; and but a few times during the whole period, 8 or 9 ft. At Jones's dam, on Cape Fear River; height of dam, 16 ft; front vert; fall, usually 10 ft, into 6 ft depth of water; the soft shale rock, in vert strata, was, in the course of a few years, worn away 16 ft ; and the dam was undermined to such an extent as to fall into the cavity. In another case, dam 36 ft high ; front vert ; the water falling upon nearly vert strata of hard shale rock, usually covered by but about 2 ft of water ; in about 20 years wore it to an irregular depth of from 10 to 20 ft ; and extending from the very face of the dam', to 70 or 80 ft in front of it. In Fig 2, upon a stream subject to very violent freshets, the gravel was washed awav for a consid- erable width and depth beyond the apron, as at A. To prevent a repetition, the cavity was tilled with cribwork full of stone, clear across the river. A deposit of blocks of loose stone, of even a ton weight or more, will not serve as a pro- tection in front of a darn exposed to high freshets; but will soon be swept away. A common precaution against this wear, in low dams, is an apron, a a, Fig 2 ; or d d, Fig 3 ; of either rough round tree trunks, or of hewn timber, laid close together ; extending under the entire base of the dam, and from 15 to 30 ft in front of its face. These are sometimes bolted to pieces, ss, Fig 2 ; or yy, Fig 3 ; laid under them across the stream. In Fig 3, with very soft bottom, these pieces yy are supposed to be bolted to short piles 1 1, driven " for that purpose. At times a distinct wide low timber crib, filled with stone, and covered on top with stout plank, has been placed in front of the dam, to receive the fall of the water; and is effective in protecting the bottom. Also, in some cases, a dam of less height, and of cheap character, has been built at a short distance down stream from the main one, in order to secure at all times a deep pool in front of the latter for breaking the force. Another precaution Is to substitute a sloping front like cl, Fig 4, or such as Figa 1 and 2 would form if reversed, for the nearly vert one of the other figs ; thus to some extent reducing the force of the wa- ter. This, however, is but a partial remedy, especially for soft bottoms in shallow water; for the sliding sheet still descends with great force. The best form of dam, per- haps, in such cases, is that shown in Fig 5, in which the front consists of a series of steps of about 1 vert, to 3 or 4 hor. These ef- fectually break the force of the water; and, with the addition of an apron o o, secure a satisfactory result. It is ob- jected against this form, as also against Figs 4 and 6, that their fronts are liable to he torn by descending trees, ice, and other bodies swept but little weight; for when such the front timbers. On the Sea along during freshets ; but experience shows that this objection ha bodies pass, the sheet of water is thicker than usual ; and protects Nav, the timbers cl, Fig 6, scarcely wear thin at the rate of an inch in 10 to 15 years. Fig 6 ROCK The forms of wooden dams are many; (see the figs, which show those most used :) varying with the circumstances of the case, and with the fancy of the designer. In the United States they are usually of cribwork. of either rough round logs with the bark on, or of hewn timber; in either case about a foot through. These timbers are merely laid on top of each DAMS. 585 ther, forming in plan a series of rectangles with sides of about 7 to 12 ft. They are not notched together, but simply bolted by 1 inch square bolts (often ragged or jagged) about 2 to 2Jf> feet long, through every timber at every intersection. These are not found to rust or wear seriously, even when exposed to a current. Square bolts hold best. Round logs are flattened where they lie upon ench other. Experience shows that firmer but more expensive connections are entirely unnecessary. The sribs are usually, but not always, tilled with ough stone. In triangular dams, disposed is in Figs 1, '2, and 7, this stoue filling is jot so essential as in other forms ; because the weight of the water, and of the gravel backing, tends to hold the dam down on its base. Still, even in these, when the lower ibers are not bolted to a rock bottom, or otherwise secured in place, some stone may necessary to prevent the timbers from Lting away while the work is unfinished, and the gravel not yet deposited behind it. ra 7 On rock, the lowest timbers are often bolted -FIQ I to it, to prevent them from floating away during construction; and when the water Is some feet deep, this requires coffer-dams. Or. the cribs may be built at first only a few feet high ; then floated into place, and sunk by loading them with stoue; for the reception of which a rough platform or flooring will be reqd in the cribs, a little above their lowest timbers. The bolting to the rock may then be dispensed with. The water may flow through the open cribwork as the building higher goes on ; attention being paid to adding stone enough to prevent it floating away if a freshet should happen. Or, cribs shown in plan at cc, Fig 8, loaded with stone, may be sunk, leaving one or more intervals, like that at o o o o, between them, for the free escape of the water. These openings to be finally closed by floating into them closing-cribs shaped like . The workmanship of a dam in deep water can of course be much better executed in coffer-dams, than by merely sinking cribs. The joints can be made tighter: the stone filling better packed ; the sheet piling more closely fitted, &c. When a very uneven rock bottom in deep water, or the introduc- tion of sluices in the dam, or any other considerations, make it ex- pedient to build dams within coffer-dams, both should be carried on in sections ; so as to leave part of the channel-way open for the es- cape of the water. Commencing at one or both shores, the first section of the coffer-dam may reach say quarter way or more across the stream. In the section of The dam itself built within this enclos- ing coffer-dam, ample sluices should be left for the water to flow through when we come to build the closing section of the coffer-dam. When the dam has been finished, these sluices may be closed by drop-timbers*. Before removing one section of coffer-dam, the outer end of the enclosed section of dam itself must be firmly finished in such a manner as to constitute a part of the inner end of the next section of coffer-dam. It is impossible to give details for every contingency ; the en- gineer must rely upon his own ingenuity to meet the peculiarities of the case before him. In some cases of shallow water, mere mounds of' earth may answer for coffer-dams; or rough stone mounds, backed with earth or gravel. After the water has passed beyond the crest, c in the figs, there is no necessity for preventing its leaking down among the crib timbers: on the contrary, the thick sheeting planks, (or squared tim- bers, as occasion may require.) ci. Figs 4 and 6, which form the slopes along which the water then flows in some dams, are usually not laid close together, but with open joints of about ^> inch wide be- tween them, for the express purpose of allowing part of the water to fall through them, so as to keep the timbers beneath them partially wet: which, to some extent, renders them more durable. In Figs 1, 4, 6, and 7, the water of the lower pool flows freely back among the crib timbers, and rough quarry stones with which the cribs are filled either partly or entirely. In Figs 4 and 6, these stones are not shown. In the dam, Fig 1, none were used. In Fig 2, they were as shown. A substantial, and not very expensive dam of the form of Fig 7. may be built of rough stone in cement. Some hewn timbers should be firmly built horizontally into the masonry of the sloping back c n w, at a few feet apart, with their tops level with the surf of the masonry. To these must be well spiked close-jointed sheeting-plank cn>, for protecting the masonry from the action of the water, and of floating bodies. The gravel backing b, may be omitted ; but the sheet piles p, and an apron in front of the dam, will be as indispensable in yielding soils, as if the dam were of timber. Figs 1. 2, 4, 6. and 7, are sections drawn to a scale, of existing dams in Pennsylvania that have- stood successfully the force of heavy freshets for a long series of years. t These fres'hets at times carry along large bodies of ice, trees, houses, bridges, &c. ; and have risen to 11 ft above the crests. Fig 1. on the Sch Nav, was built in 1819, and served perfectly for 39 years, until in 1858 the decay of much of its timber, especially of the close-laid top ones, e, rendered it necessary to build a new one just in front of it. It was of extremely simple construction ; and was never filled w'ith stone. The bottom tim- bers, o o, 10 ft apart, were bolted to the rock ; and immediately over each of them, was such a series of inclined timbers a,s is shown in the fig. The top ones, e. however, were close jointed, and laid touching, so as to form the top sheeting, instead of thinner planks. The short pieces at t were lai-) in the same way. No coffer-dam was used : but the bottom pieces were first bolted to the rock ; 10 ft apart ; then the stringers and the sloping pieces were added. The close covering (e) was carried forward from each end of the dam, until at last a space of only about 60 ft was left in the center, for the water to pass. The close covering for this space heing'then all got ready, a strong force of men was set to work, and the space was covered so rapidly that the river had not time to rise sufficiently high to impede the operation. * Timbers ready prppared fnr closing an opening through which water is flowing; and suddenly dropped into place by means of grooves or guides of some kind for retaining them in position. Sev- eral such timbers may at times be firmly framed together, and then be all dropped at once; closing the opening or sluice at one operation ; especially when it is of small size. In some cases, a crib may be sunk on the up-stream side of such an opening, for closing it. t Those on the Schuylkill Navigation were obligingly furnished by James F. Smith, Esq. chief engineer and superintendent of that work. Other valuable information from the same source will b found in different parts of this volume. 586 DAMS. Fig 2 is a canal feeder dam on the Juniata. Here s s are timbers stretching clear across the stream, (about 300 ft,) and sustaining the apron aa, of stout hewn timbers laid touching. This lam was filled with stone, for the retention of which the front sheeting planks were added. Fig 6 is on the Sch Nav ; was built in 1855. It is a form much approved of on that work, for such situations; namely, firm rock foundation, with a considerable depth of water in front. The highest dam (32 ft) on the Sch Nav, is very similar to it; built in 1851. All the dams on this work are of hewn timber, chiefly white and yellow pine. The water occasionally runs from 8 to 12 ;eet deep over their crests; and then overflows and surrounds many of the abuts. The vertical back allows the overflowing water to leak down among all the lower timbers of the dam, and thus tend to their preservation. Fig 4 shows the dams on the Monongahela slackwater navigation ; W. Milnor Roberts, eng. They are of round logs, with the bark on j flattened at crossings. The longest ones in the fig are 10 feet apart along the length of the dam. Experience shows that such dams possess all the strength neces- sary for violent streams. On rock, the lowest timbers are bolted to it. Fig 7 has been successfully used to heights of 40 ft.* Fig 3 is intended merely as a hint for a very low dam on yielding bottom. Its main supports are piles ii, from 4 to 8 ft apart, according to the height of the dam; and other circumstances ; and tt are short piles for sustaining the apron dd. It may be extended to greater heights by adding braces in front ; which may be covered by stout planks, to form an inclined slide for the overfalliug water. Many effective arrangements of piles, and sloping timbers for dams on soft ground, will suggest them- selves to the engineer. Thus, at intervals of several feet, rows of 3 or more piles may be dnveu trans- versely of the dam ; the top of the outer pile of each row being left at the intended height of the crest, while those behind are successively driven lower and lower; so that when all are afterward con- nected by transverse and longitudinal timbers, and covered by stout planking, and gravel, they will form a dam somewhat of the triangular form of Fig 7. It would be well to drive the piles with an inclination of their tops up stream. There is much scope for ingenuity both in designing, and in constructing dams under various cir- cumstances ; and in turning tiie course of the water from one channel to another, by means of ditches, pipes, or troughs, &c., at diff heights; aided at times by low temporary dams or mounds of earth ; or f sheet piles, &c ; or by coffer-dams ; so as to keep it away from the part being built, Each locality will have its peculiar features ; and the engineer must depend on his judgment to make the most of them. Abutments of clams as a general rule should not contract the natural? width of the stream ; or, if they must do so, as little as possible ; for contractions increase the height, and violence of the overflowing water in time of freshets ; during which a great length of overfall is especially desirable. They should be very firmly connected with the ends of the dams; and should, if the section of the valley admits of it, be so high : and carried so far inland, that the high water of freshets will not sweep either over them, or around their extremities; and thus endanger under- mining, and destruction. In wide, flat valleys they cannot be so extended without too much ex- pense ; and the only alternative is to found them so deeply and securely as to withstand such action ; making their height such that they will, at least, be overflowed but seldom. Their enda adjacent to the dam, should be rounded off, so as to facilitate the flow of the water over the crest. They are best built of large stone in cement; for although sufficient strength may be secured by timber, that material decays rapidly in such exposures. If of earth only, they are very apt to be carried away if a freshet should overtop them. Sluices should be placed iu every important clam, in order that all the water may be drawn off, if necessary, for the purpose of repairs ; or of removing mud deposits; or finding lost articles of importance, &c. They may be merely strong boxings, with floor, sides, and top of squared timbers ; and passing through the breadth of the dam, just above the bottom. To pre- vent trees, &c, from entering and sticking fast in them, some kind of strong screen is expedient. In common cases a sluice should not exceed about 3> ft by 5 ft in cross-section; otherwise it becomes hard to work. Two or more such openings may be used when much water is to be voided. They should be near the abutments. The gates or valves for opening and shutting them, should be at the up-stream end; for if at the lower one, accumulations of mud, ams are sometimes, but rarely, built in the form of an arch ; convex up stream. This form is strong; and when the shores are of rock it may be expedient to use it ; but if the banks are soft, they will be exposed to wear by the curreni thrown against them at the abuts of the arch. At times dams are built obliquely across the stream, with the object of increasing the length, and consequently reducing the depth of water over the crest in times of freshets. The argument, however, appears to the writer to be of but little weight, inasmuch as the reduction of depth would extend but a trifling distance up stream from the dam; and would therefore scarcely have an appreciable effect in diminishing the injury to the overflowed district above. Moreover, the increased expense is probably always more than commensurate with any advantage gained. Some dams are subject to "tremblings," which have not been satisfactorily accounted for. They exhibit themselves chiefly as undulations of the air, produced by the falling water; and which occasionally cause a rattling of windows within a distance of ^ a mile or more. We have known this to be stopped unintentionally in one case, by building a well-covered * Cost of crib dams. Formerly, with common labor at $1 per dry ; plank $20, and other timber $10 per 1000 ft board measure, delivered: stone for filling. $1 per cub yard ; gravel 40 cts per cub yd; iron for bolts, n, against the front of the dam, for preventing the abrasion tf the bot- tom. In other cases a secies of oblique timbers placed against the front of the dam, and part way up it, at a slope of abouL*5$to 1, and covered with plauk, has been perfectly effective in stopping it. The proper time for building- clams is of course at the longest period of low stage'of water. To ascertain in advance approximately, the height to which the water will rise above the crest of a dam : or rather, a little back from it ; the crest being above the level of the original water. This will vary with the shape of the crest, as may be seen by reference to Fig 26 Vo of Hydraulics, which, however, is a very peculiar case. Still, until we have more experiments, appreciable deviations from the results of such rules must, be expected in practice. Square the disch of the stream in cub ft per sec. Call this square, s. Square the length of the overfall in ft. Mult this square by 7. Call the prod p. Divides by p. Take the cube rt of the quot.* Thvs cube rt will be the reqd approximate height of rise in ft. M'hen, in times of freshets, the -water rises above the crest to a height equal to that of the dam itself, there is no perceptible fall at the dam ; aud boats may pass in safety over the crest. For measuring the disch over dams, see arts 14 and 15 of Hydraulics. See also Art 1 of Hydro- statics. In shape of a formula, the foregoing rule will be, Rise _ cube n * /discharge in cub ft per sec^\ inft ' J \i~(i e ngth of overfall in Jt^. ) When the dam is originally a Q,... __ ____ submerged, or drowned one, as 1), Fig 9 ; the following is a rough approximation, probably O: _ : ; _ _ ^- _^2^ ... - ~. n somewhat in excess; off being the natural level of the |______ _ _ ______ ~ --- J water previous to building the dam; and oc the natural depth in ft, of the -water above the intended crest. Then the required depth ac, of the up-stream water, above the crest when built, will be, approximately, ^^^t^ ac = oc + cuberootof /^^ 2 \ Fit) Q V7X length 2/ J J Having a c, deduct o c ; and the rem will be a o, or the required rise produced by the dam.* The rnles given for the varying rise of surface for consid- erable distances lip stream from dams ; as well as for some allied sub- jects in hydraulics ; are extremely complicated ; and require much greater knowledge of mathematics than is usually found among civil engineers ; and so far as regards their application to the actualities of common occurrence, they are probably no less useless than complicated. The short table on page 317 will at times be of use in finding the di- mensions of timbers for sustaining the pressure of different heads of water. GKAVITY, FALLING BODIES, Caution. Owing to the resistance of the air none of the follow- ing rules give perfectly accurate results in practice, especially at great vels. The greater the sp gr of the body the better will be the result. The latitude, and the height above sea level also cause a slight difference. The air resists both rising and falling bodies. The dist through which a body falls freely in the first second is very nearly 16 1 ft; and it increases as the squares of the times. Therefore to find the number of feet fallen in any case, find the square of the number of sec; and mult it by 16.1. The n t in new wooden ones. Particular care should be bestowe*! upon the strength of the foints of the side parapets; for the uttdulations and lateral motions of the "bridge expose them to violent deranging forces in every direction. The parapets should be high and stout: and not restricted to mere service as hand-rnils. or guards. As a ru'e of thumb, one-half the sq rt of the spnn will be about a good height for them in ordinary cases, provided it is not less than a hand-rail requires. 594 SUSPENSION BRIDGES. "We do not think that diagonal horizontal bracing should, as is usual, be omitted under the floir. It may readily be efioeiml by iron rods. All the cables need not be at the sides of the bridge. One or more of them may be over its axis; especially in a wide bridge. One wide footpath iu the center luay be used, instead of two narrow cues at the sides. The platform or roadway should be slightly cambered, or curved upward, to the extent say of about 2-jg. of the span. Art. 11. The Niagara suspension bridge, built in 1852-3, John A Eoebling, engineer, consists of a single span of b_'l^ ft measured straight from center to center of towers ; and 800 ft of clear suspended length of roadway. It has two floors or roadways : the upper one, for a single-track railway, is 25 > ft; and the lower one, for common tcavel, 24>g ft wide, out to out of everything. The lower one is 11) ft wide in the clear of everything. They are 17 ft apart verti- cally. The trusses are 18 ft total height, throughout. They are on the Pratt arrangement; see p 284 ; with verticals 5 ft apart from cen to cen; and single oblique iron rods, 1 inch square, running in each direction across four of the 5 ft panels. Where these rods pass each other, they are tiad together by 10 or 12 turns of -fa inch wire. Each vertical consists 'of two pieces of 4)4 by 6> timber, placed 4}^ ins apart, to allow the oblique rods to pass between them. Both upper and lower floor girders are in two pieces, of 4 by 16 ins each. Pairs 5 ft apart. The tops and bottoms of the verticals pass be- tween the two pieces which form each floor girder. No tenons or mortises are used in the framing. There are four cables of iron wire ; two on each side of the bridge. Each cable is 10 ins diarn. The wire is scant No. 9 of the Birmingham wire gauge, or scant ,14b inch diam. Sixty wires have a united transverse section equal to one square inch of solid iron. Each of the four cables contains 3(UO wires, with a united cross-section of 60.4 sq ins of solid metal. Therefore, the area of solid metal in a section of all the four cables together is 241.6 sq ins, or 1.678 sq ft ; weighing 814 Ibs per ft of span. The wires of each cable are first made up, in place, into 7 small strands ; and these are firmly bound together throughout by a continuous close wrapping of wire. The strength of each individual" wire is 1640 fts. or .73214 of a ton. This is equal to 98400 fts, or 43.93 tons per sq inch of solid metal ; or to 5943360 fts, or 2653.3 tons per cable ; or to 10613.2 tons ultimate strength of the four cables together. One cable on each, side of the bridge deflects 54 ft ; and the other 64 ft; average deflection 59 ft, or about -j*r of the span. With this av defl the tension on the cables at the tops of the towers averages 1.82 times the total suspended wt of the span and its load. See table, Art 1. The wt of the suspended span itself is about 900 tons ; and if the greatest extraneous load on the two floors together be taken at 1J4 tons per ft run, we have the total suspended wt 900-4- (800 X 1 J4) = 1900 tons. And 1900 X 1.82 = 3458 tons tension at towers; or very nearly ^ of the ultimate strength of the cables, without any allowance for momentum, or wind. But such loads, although possible, are not permitted to come upon the bridge; and moreover a part of the strain is borne by the upper stays. The wires were perfectly oiled before being made into strands; and when the strands were being bound together to form a cable, the whole was again thoroughly saturated with oil and paint. The cables do not hang vertically; but the two upper ones are about 37 ft apart from center to cen- ter, where they rest upon the towers, (where all four are on the same level ;) and are drawn to within 13 ft of each other at the center of the span ; and at the level of the railway track on top of the bridge : while the two lower ones are about 39 ft apart at the towers, and 25 ft at the center of the span, and at the level of about halfway between the two floors. This drawing in of the cables contributes much to lateral stability ; as do also the upper and lower floor of stout plank. There is no horizontal diagonal floor bracing. There are 624 suspenders of wire rope, 1% ins diam, and 5 ft apart, or corresponding with the floor girders, which they uphold; and with the wooden verticals of the trusses. They do not hang verti- cally ; but incline inward. The masonry towers are all founded on rock. They are 78J4 ft high above the bottom of the bridge; and 60,4 ft above the upper floor. The two at each end of the span are 39 ft apart from center to cen- ter. At the level of the lower floor they are 19 X 20 ft ; and 21 ft apart iu the clear. At the level of the upper floor they are 15 ft square; and 24 ft apart in the clear. From there they taper regularly to the top, where they are 8 ft square. They are built of limestone, in heavy dressed hor courses ; laid in cement ; vertical joints grouted. The upper courses are dowelled. On top of each tower is a cast-iron plate, 8 ft sq, and 2}^ ins thick, bedded in cement. Part of the top of this plate is planed, as upon it move the rollers which support the cast-iron saddles on which the cables rest. At each tower, each cable has its separate saddle and rollers. Each saddle rests on 10 cast-iron rollers 25^ ins long, and 5 ins diam/carefully planed. They lie loosely, and close together; and are kept in place by side flanges on the bed-plate. The cast saddles are each 5 ft long, by 25^ ins wide. Their bottoms, which rest on the rollers, are flat, and planed. Their tops are curved to a rad of 6J4 ft ; to suit the bend of the cables over the piers ; and each saddle has a longitudinal groove, in which the cable lies. The passage of the heaviest trains produces less than % an inch of movement in a saddle. The floors have a camber of 5 feet. A change of 100 Fah of temperature causes an average variation of about 2^ ft in the deflection of the cables, or in the camber of the roadways; and one of 150, (about the extreme to which the bridge is exposed,) about 3% ft. The passage of a train weighing 291 tons, and covering the entire length of the span, caused a deflection of 10 ins ; and an ordinary train deflects it only from 3 to 5 inches. This bridge has, since the year 1853, demonstrated the applicability of the suspension principle to large span railway bridges. Its entire cost was not quite $400,000. For more, see top of p 589. Art. 12. The wire suspension bridg-e near Freybnrg 1 , Swit- zerland, finished in 1834, Mr. Chaley, engineer, and still in full service, is of very simple construction, and has served as the prototype for several in this country. It is for common travel only ; and is narrow: its entire width of platform being but 21% ft; and its clear available width but 19 ft. The dist from cen to cen of its towers is 889 feet; and its clear spnn be- tween abutments 800 ft ; or the same as the Niagara. There are 4 cables, each 5 ins diam. Each of them consists of 1056 wires of No. 10, or full ^ inch diam, (or 71 wires to the sq inch of solid mptnl ;) arranged in 20 strands of about 53 wires each. The four cables, therefore, have a united area of but 40 sq. ins of solid metal ; weighing 202 Ibs or .09 of a ton, per ft run of spaa. All its suspenders are SUi 'SIGN BRIDGES. 595 Tertlcal ; about 5 ft anaKf and each upholds one end of a transverse floor girder. It hai no std trussing except th^yfight one of the wooden baud-railing, which is about 6 feet high ; and conse- quently, with Ujf'great span it is quite flexible. The deflection of the cables is ^ of the span ; hence the strain upon them at the top of the towers at either end, is 1.82 times (see table, Art 1,) the wt of the suspended span itself, and its extraneous load; and supposing the wire to be as good as that of the Niagara, the breaking strain of the four cables would be 60 X 44 2640 tons ; and their safe strain cannot be taken at more than % as much, or 880 tons. The suspended weight reqd to produce this safe strain would of course be - := 481 tons. The suspended weight of the span itself cannot well be less than .3 of a ton per ft run : or 240 tons in all ; * thus, leaving 484 240 244 tons To.* the maximum safe extraneous load. This amounts to .305 of a ton per ft run of span ; or 36 fts per sq ft of its platform, 19 ft wide in the clear. The French allowance is 41 fts per sq ft; t and since no allowance is here made for momentum or wind, it is plain that this celebrated bridge, on account of its slight cables, and its flexibility, is by no means a strong one. In that respect, as well as steadi- ness, it is much inferior to the one next spoken of. It is said, however, to have withstood very severe tempests; and also to have been occasionally completely covered by crowds of people. If so, their lives were not very secure. Art. 13. The wire suspension bridge across the Schnylkill at Philada, finished in 1842, Chas Ellet, Jr, engineer, is somewhat similar in character, and in the dimensions of its details, to the preceding; but being of much less span, is much stronger. Its span from cen to cen of towers is 358 ft; suspended platform between abuts 342 ft. It has ten cables of 3 ins diam ; five on each side. Their united sections present 55 sq ins of solid iron ; or nearly as much as the preceding bridge of 800 ft clear span. The five cables on either side have different deflections, ranging between the y\j- and the -Jj of the span from tower to tower. The dist from cen to cen of towers at either end of the span is 35^ ft; and on top of each tower the cables (considerably flattened at that point) lie side by side on a single roller about 30 ins long, and 6 ins smallest diam, which has 5 grooves, for their reception. Each cable is drawn- in about 3^ ft at the center of the span. At in- tervals of 20 ins the parallel wires of the cable have a close wrapping of finer wire for a distance of 3 ins. The suspenders are of wire; and are % inch diam; and 4 ft apart. On any one cable they are 20 n apart. They all incline slightly inward. The width of the platform from out to out is 27 ft; and in clear of hand-rails 25 ft. Inside of the hand-rail is a footway, 4 ft 4 ins wide, on each side of the bridge. The remaining 16 ft 4 ins serves for a double carriage way, or double-track street railway. Figs 5 show the arrangement of the wood- work, on a scale of ^ inch to a ft. The trussing of the parapets is on the Howe system. (see p 283.) which does not appear to be as well adapted as the Pratt, to suspension bridges. The diagonals in the Fairmount bridge work themselves out of place laterally, by the vibrations of the bridge ; ly seen several of them almost on the point of falling out entirely. Being under municipal charge, it is of course neglected. The upper chords u, are 12 ins wide by 6 ins deep; the lower ones I, and the stringer c, below them, are each 12 wide, by 7 deep. The diagonals i are all 4 ins wide, by 5 deep. The angle-blocks nt their ends are of cast iron, hollow, and about % inch thick. The vert iron rods v, (in pairs,) are % inch diam near the center of the span; and lj.g at its ends. The top chords are spliced on each vert face by an iron bar. of 5 ft by 3 ins, by y 2 inch; with 4 bolts passing through them. The splice of the bottom chord has merely 2 bolts, side by side ; (see Figs 5 ;) which (except S) are to a scale of % inch to a ft. The floor girders g, 4k ft apart from cen to cen, are 6 by 14 ins at their ends ; and 6 by 16 at center. The floor is of two thicknesses of 2-inch plank; except the footpaths, which are single thickness. The wires were well oiled when the cables were made ; and afterward painted. At S is shown the mode of uniting a suspender with a cable, o, bv means of a small cast-iron yoke g, which straddles the cable ; and on the back of which is a groove % of an inch wide, in which the suspender rests. The metal of the yoke is about ) inch thick. Since the lower ends of the wires which compose a suspender cannot themselves be formed into a screw-bolt, for upholding the floor girders, they are passed through the eye of a screw-bolt of bar iron ; then doubled on themselves, and held by a wrapping of wire. It is well to introduce a yoke here also, to prevent the wear of the wire by friction. The small fig on the right of S is an edge view of the yoke ' g. TR SEC. SIDE. * This is probably nearly its actual weieht, as obtained by comparing it with the Fairmount bridge, which, by a careful estimate by the writer, weighs .375 of a ton per ft run ; but is considerably wider than the Freyburg ; and carries four lines of light street-rails. But if the Frey burg has longitudinal joists, it will weigh about .03 ton more per ft run. t The greatest load that can come upon an ordinary bridge, is a dense crowd of people; and this the French pncineers estimate at 41 fts per sq ft of platform. This is certainly as great as can well occur under ordinary circumstances ; but it may be considerably exceeded, the French estimate, moreover, includes no allowance for wind, or for the crowd being in motinn. Including these, the writer thinks that no suspension bridge should have a less safety than 3, against 100 fts per sq ft; added to the weight of the bridge itself. A less coeff of safety is admissible in a wire bridge than in an iron trussed one, on accoun; of the greater reliability of the material. See foot-note p 297. 596 SUSPENSION BRIDGES. There is no transverse bracing under the floor ; nor are there longitudinal floor joists resting on the girders. Owing to the waut of the distributing effect of these ; and to the use of so many small cables instead of but 2 or 4 larger ones ; as well as to the inefficient trussing of the hand-railing or para- pets, the bridge is much less steady than it would otherwise be. Still it is very rarely (as during the trotting of a herd of cattle) that the trembling or undulating of the bridge becomes seriously objectionable. Ordinarily it cannot be said to be at all so ; and but few persons would notice it. It must be greatly less than on the Freyburg bridge. With wire of the same quality as the Niagara, (or 44 tons per sq inch breaking strength,) the Fair- mount bridge would, with a safety of 3, (omitting momentum and wind.) sustain an extraneous load of 31:6 tons; which is equal to 1.01 ton per ft run of span ; or 90 Ibs per sq ft of its clear platform. This last is 2.5 times as great as the strength of the Freyburg, with the same quality of wire. The Fairmount is, however, we believe, built with wire of but 36 tons per sq irich ultimate strength. If so, its greatest extraneous load becomes reduced to 260 tons ; or .76 ton per ft run ; or 68 Bbs per sq ft of platform, or nearly twice that of the Freyburg. The towers are of cut granite, in heavy courses. They are 8*4 ft square at the ground line, or level of the floor; about 5 ft sq at the top; and about 30 ft high. The backstays have the same angle of direction as the main cables. Art. 14. The Wheeling: bridge across the Ohio at Wheeling, Vir- ginia, also by Mr Ellet. had a span of 1010 ft between the towers ; and 960 feet clear span between the abuts ; and was 26 ft wide from out to out. Its mode of construction was much the same even in de- tail as that of the Fairmount bridge; except in having 12 cables instead of 10. The 12 cables con- sisted of 6600 wires of No. 10 Birmingham gauge, presenting a sectional area of 93 sq ins of solid metal, weighing 313 fts, or .14 of a ton. per foot of span. The weight of the woodwork was about the same per foot run of span as in the Fairmount. Although its clear span was 2.8 times as great as the Fairmount, yet its cables had but 1.7 times as great area of solid metal. The entire suspended wt between towers, is stated at but 440 tons; therefore, with an average deflection of yV of the span, for a safety of 3 against 100 Tbs per sq ft of platform of 24 ft clear width ; or 1.07 tons per ft run of span, the area of solid metal in the cables should have been 173 sq ins, with 44 ton wire like that of the Ni- agara : or 214 sq ins, with 36 ton wire, which we believe was the quality actually used. Art. 15. The suspension canal aqueduct at Pittsburgh Perm, built in 1845, John A Roebling, Esq, engineer, has seven spans of 160 fc each. Deflection 14)6 ft; or about yy- of the span. It has but two cables, each 7 ins diam. The two together contain 3800 No. 10 wires, making 53 sq ins of solid metal section. Ultimate strength of each wire 1100 B>s ; equal to 35.2 tons per sq inch of solid metal; and making the ultimate strength of the two cables together 1866 tons. The prism of water in the wooden aqueduct is 4 ft deep ; by 14% ft average width ; and weighs 265 tons per span. The wt of one span of the structure itself is about 111 tons ; making the total sus- pended wt at each span 376 tons. The tension on the two cables at either end of a span, with a dett of y^, is 1.46 times the total suspended weight ; see table, p 588. Hence it is in this case 376 X 1.46 = 549 tons ; and the strength of the cables is = 3.4 times the constant strain upon them. On one side of the water is a towpath for horses ; and on the other a footpath ; each 7 ft clear width. With these occupied by horses and people, the foregoing safety would be reduced to about 3. The loaded boats do not add materially to the weight, inasmuch as they displace a bulk of water equal to their own wt; and but little of the displaced water remains on a span at the same time with the boat. The great wt of the water prevents undulations ; and the aqueduct is therefore very steady. On this account a less coeff of safety is admissible than on a common bridge.. The aqueduct leaked badly along its lower corners. Art. 16. In 1796. Mr James Finley. of Fayette County, Penn, introduced suspension bridges in the U. S. ; and built several with spans of 200 feet and less. Many of them were very primitive structures ; but answered sufficiently well for the times. They had usually either two or four chains, composed of links from 7 to 10 feet long, formed by bending about 1^-inch square bars of iron, and welding their ends together. At each link-end, "was a vertical suspender rod of 2 ins by fo inch iron; which, at its lower end, was bent and welded into a stirrup for upholding one end of a transverse floor beam. On these beams rested longitudinal joists supporting the floor plank. Finley used deflections as great as 7, or even % of the span ; and his piers were frequently single wooden posts ; the two at each end being braced together at top. Such were used in a span of 151 J^ ft clear, across Will's Creek, Alleghany Co, Penn. It had two chains. The defl was % of the span. The double links of 1% inch sq iron, were 10 feet long. The center link was horizontal, and at the level of the floor; and at its ends were stirrnped the two central transverse girders. From the ends of this central link, the chains were carried in straight lines to the tops of the single posts. 25 ft high, which served as piers or towers. The back- stays' were carried away straight, at the same angle as the cables ; and each end was confined to four buried stones of about ' % a cub yard each. The floor was only wide enough for a single line of ve- hicles. All the transverse girders were 10 ft apart; and supported longitudinal joists, to which the floor was spiked. There were no restrictions as to travel ; but lines of carts and wagons in close MIC- cession, and heavily loaded with coal, stone, iron. &c, crossed it almost daily : together with droves of cattle in full run. The slight hand-railing of iron was tinged, so as not to he be.nt J>y the undu- lation* of the bridge. Six-horse wagons were frequently driven across in a rapid trot. It was built in 1820; ami an observant engineer friend, who in 1838 took the sketch and measurements upon which this description is based, informed the writer that the iron was as perfect, and as sharp on all its edges, as on the day it was built. The iron was the old-fashioned charcoal, of full 30 tons per sq inch ultimate strength. The united cross-section of the two double links was 7.56 sq ins ; which, at 30 tons per sq inch, gives 227 tons for their ultimate strength : or say 76 tons, with a safety of 3. Now, with a defl of % span, the tension on the cables (see table, p588) is but .9 of the suspended total wt. The two chains in plane would therefore sustain, with a safety of 3. a quiet suspended load of 76 X .9 = 68 tons ; and as the span itself did not weigh more, than 15 tons, we have 53 tons for the safe ex- traneous weight, omitting all consideration of wind and momentum. This is equal to .35 ton per ft of span ; equal to .7 ton for a bridge wide enough for two vehicles to pass. This primitive FRICTION. 597 bridge wopJUTflierefore safely sustain a greater load per foot run of span, than the Freyburg. These otdoridges frequently failed by the rotting of the end posts ; or were carried away by fresh- ets ; but^we have never heard of a failure from the breaking of the chains. Many of them were built in a much more perfect style than the one just described ; and on the most used roads in the Union. Art. 17. As it is sometimes convenient to form a rough idea at the moment, of the size of cables required for a bridge, we suggest the following rule for finding approximately the area in sq ins of solid iron iu the wire required to sustain, with a safety of 3,* the weight of the bridge itself, together with an extraneous load of 1.205 tons per foot run of span ; which corresponds to 100 E>s per sq ft of platform of 27 ft clear available width. This suffices for a double carriage-way, and two footways. The deflection is assumed at yV ^ tne 8 P an J atl( i tne w i re to have an ultimate strength of 36 tons per solid square inch, as per table, page 369; and which can be procured without difficulty. For spans of 1OO ft or more, RULE. Mult the span in feet, by the square root of the span. Divide the prod by 100. To tha quot add the sq rt of the span. Or, as a formula, span X sq rt of span -f- sqrt of span. Area of SOLID metal of all the cables ; in square ins ; for spans over 100 feet 100 For spans less than 100 feet, proportion the area to that at 100 ft. If a defl of j\j- is adopted instead of ^, the. area of the cables may be reduced very nearly ^- part- The following table is drawn up from this rule. The 3d col gives the united areas of all the actual wire cables, when made up, including voids. (Original.) Span Feet. Solid Iron in all the Cables. Areas of all the Finished Cables. Span in Feet. Solid Iron in all the Cables. Areas of all the Finished Cables. Span Feet. Solid Iron in all the Cables. Areas of all the Finished Cables. Sq. Ins. Sq. Ins. Sq. Ins. Sq. Ins. Sq. Ins. Sq. Ins. 1000 348 446 400 100 128 150 30.6 39.2 900 300 385 350 84 108 125 25.2 32.3 800 254 326 300 69 89 100 20 25.6 700 212 272 250 55 71 75 15 19.2 600 171 219 200 42 54 50 10 12.8 500 134 172 175 36.4 46.7 25 5 6.4 Having the areas of all the actual cables, we can readily find their cliam. Thus, suppose with a gpan of 500 ft, we intend to use four cables. Then the area of each of them will be =43 sq ins ; and from the table of circles, page 18, we see that the corresponding diam is full 7% ins. The above areas are supposed to allow for the increased wt of a depth of truss, and other additions necessary to secure the bridge from violent winds, and from undue vibrations from passing loads. When these considerations are neglected, and a less maximum load assumed, the foregoing descrip- tions of the Wheeling and Freyburg bridges show what reductions arc practicable. Weight, suffi- ciently provided for, is of great service in reducing undulation. FRICTION. Art. 1. 'See Arts 15, 60, 61, 62, &c, of Force in Rigid Bodies.) Friction is that resistance (produced bv an interlocking of the roughnesses of surfaces in contact with each other) which opposes the sliding of one body along another. It occurs in all machinery : in water, gas. or air, moving along pipes, or other channels; and in all cases whatever where one body mores along another in contact with it. It is the principal cause of what is called '* loss Of power." in all these cases ; which means simply that the whole of any moving force (such as steam, gravity, moving water, wind, muscular force, &c) cannot be directly applied through machinery, to give motion to other bodies, because a portion of it is lost in overcoming friction in the friction were not first overcome, there could be no motion at all. In hydraulics, this loss of power is usually termed "loss of head," meaning that portion of the head of water, which by its pres counter- acts the fric of the water in the pipes, &c, and thus permits the other portion to produce motion or flow. In nearly all the cases which come under the notice of the engineer, the surfaces in contact are rendered more or less smooth by art; and the smoother they are made, the less is the friction, because the roughnesses are diminished ; and of course take less hold on each other. But in even the most highly polished surfaces, there is sufficient roughness to produce some friction. Although the * The writer must not be understood to advocate a safety of 3 against 100 fts per sq ft, in addition fo the weight of the bridge, in all cases. He believes that limit to be about a sufficient one for a pro- perly designed wire suspension bridge for ordinary travel ; but for an important railroad bridge, he would (according to position, exposure, ratio, to the pressing 1 force ; and is entirely independent of the Size Of tlie Surfaces themselves. A brick will slide just as reauily when lying on its broad side, as wneu on a narrow one. This proportion, however, is airi" for diff materials; and for diff degrees of smoothness; and moreover, is less when the surfs are clean and well lubricated, than when dirty aud dry. Whatever this proportion may be in any particular case, it is called the coefficient of friction for that case. Thus, if we wish" to slide hor a block of dry- cut stone weighing 1 ton, upon a surf of the same kind, we shall have to exert a force of about % of a ton ; or if we Grst place a weight of 2 tons upon the moving stone, making the total pres upon its base 3 tons, we must exert a sliding force of % of 3 tons. Because the fric of dry cut stoue upou dry cut stone averages about % of the force which presses the two surfs together; or in other woius, the coeff of fric of dry cut stone on dry cut stone, is about %, or .66, But if the pres upon the sliding surfs is sufficient to produce abrasion (indeed, while it is much less,) the fric becomes greater: but no precise law has yet been discovered for estimating it. Tne fric between surfs actually in motion, is less than when they have been for some time quiescent or stationary. Hence it is usual to consider it under the heads of moving, and quiescent fric. Expe- rience, however, has shown that slight jarring suffices to remove this diff ; aud since all structures, even the heaviest, are subject to occasional jarring, (as a bridge, or a neighboring building, or even a hill, during the passage of a train : or a large factory, by the motion of its machinery ; or in num- berless cases, by the action of the wind,) it is considered expedient, in construction, not to rely on fric for stability, any further than the coeff for moving friction may justify. Beside the foregoing subdivisions of sliding fric, we have rolling, aud journal fric. At present we shall confine ourselves to sliding. The coeff of moving* fric is the same at all vels;* and this, in connection with the foregoing statements that friction increases as the pres, so long as this is con- siderably below the abrading point; and that it is independent of the amount of surf ; are considered the three grand laws of fric. Unfortunately, the experiments on this subject by Morin, which are those by which practical men are chiefly guided; were made with slight pressures; not exceeding about 30 Ibs per sq inch of contact-area of the bodies, even with the strongest bodies, such as iron, &c ; which, in machinery very often have to move under far greater pres. It is well known that fric, under much heavier pres, increases very considerably beyond the extent assigned by Morin ; as will be seen by our table of results found by Rennie ; Art 3. Locomotives would frequently be unable to draw the loads they do, if their fric (miscalled adhesion) on the rails did not greatly exceed the limits of iron on iron assigned by Morin. The fact is, that although it is customary to consider our knowledge of the laws of fric to be very complete, it is extremely defective. The common assertion that "fric is the same at all vels," may lead to mistakes, unless we clearly distinguish between the coeff, and the amount. If one man drags a sled one mile in one hour ; and another man drags it one mile in % of an hour; then, inasmuch as both the coeff, and the dist, are the same in both cases: each man has overcome the same amount of fric. But if one drags it 4 miles, in the same time that the other drags it 1 mile, then, although the coeff, and the time, remain unchanged, the amount of fric is 4 times as great in the first case as in the last ; and has reqd the man to expend 4 times as much force to overcome it alone; without any regard to the force expended in moving himself so rapidly. In other words, the quantity, or amount of fric; and the total amount of power reqd to overcome it, is in proportion to the space passed over; without any reference to the time reqd. So a train moving at 60 miles per hour, overcomes just as great an amount of fric in 1 mile, as does a train moving at 10 miles per hour; but the fast one overcomes 6 times as much per hour, min, or sec, as the slow one. So with pivots, journals, &c, in machinery ; the more rapidly they revolve, thu greater is the amount of friction they must overcome in a given time; although the degree, or coeff, or intensity of the fric remains the same at all vels.* Art. 2. To find the proportion, or ratio, of fric. to the pres which produces it, (or the coeffof fric.) for cliff materials. This may be ascertained by making of one of them an inclined plane; and finding by trial, what slope it must have, to enable the other just to begin to slide down it. Div the vert height of this slope, by its hor base ; aud the quot will be the reqd coeff. It will also be the nat tang of the angle which the slope makes with the hor; so that by looking for the coeff in a table of nat tane, the angle itself will be found opposite to it. If the weight of the sliding body be mult by the coeff, the quot will be the force reqd to drag it horizontally along such a surf as that of the inclined plane; or more correctly, the force required to barely balance the fric ; still more force must be added to produce motion. The angle of slope is usually called the angle of fric ; sometimes the angle of re- pose ; and the limiting angle of resistance. See Force in Rigid Bodies, Art 63, p 486. Experiments on fric with unguents cannot well be made on a small ncale bv means of an inclined plane; on account of the cohesion, or the stickiness of the ungnent; which when the sliding body is very light, requires a proportionally greater slope thnn when it is heavy. In trying experiments on fric for himself, the student must not expect by any means to arrive at the same results as the following; because the slightest deviation in point of hardness, smoothness, dryness. dust, inovin^ friction, of perfectly smooth, clean, and ry, plane surfaces, chiefly from Morin. Materials Experimented with. Coeffof Fric : or Propor- tion of Fric to the Pres. Oak ou oak ; all the fibers parallel to the motion .48 " " moving fibres at right angles to the others; and to the motion... .32 " ' all the fibres at right angles to the motion. .34 " moving fibres on end ; resting fibres parallel to the motion .19 41 cast iron, fibres at right angles to motion .37 Elm on oak, fibres all parallel to motion .43 Oak ou elm, " " " ., 25 Elm ou oak, moving fibres at right angles to the others, and to motion .45 Ash on oak, fibres all parallel to motion .40 Fir on oak, ' " " " .36 Beech ou oak l4 " 4t l( .36 Wrought iron od oak, fibres parallel to motion .62 Wrought iron on elm, " " " " ' .25 Wrought iron on cast iron, fibres parallel to motion .19 " " on wrought iron, fibres all parallel to motion .14 Wrought iron on brass .17 Wrought iron on soft limestone, well dressed .49 " " hard " " " 24 44 " ' 4 " 4< l4 " wet 30 44 4 ' or steel on hard marble, sawed. By the writer about.. .17 " 44 4< " 41 smoothly planed, and rubbed mahogany, fibres par- allel to motion *. .18 44 4I 4< " " smoothly planed wh pine .16 Cast iron on oak, fibres parallel to motion .49 44 " " elm, " " " " 20 44 44 4< cast iron .15 44 ' ll brass ,15 Steel on cast iron .20 Steel on steel. By the writer .14 Steel on brass 15 Steel on polished glass. By the writer about.. .11 ' 4 quite smooth, but not polished; ou perfectly dry planed wh pine, fibres parallel to motion about.. .16 44 quite smooth, but not polished; on perfectly dry planed and smoothed mahogany, fibres parallel to motion about.. .18 Yellow copper on cast iron 19 onoak 62 Brass on cast iron.. .22 44 on wrought iron, fibres parallel to motion .16 " on brass 20 44 on perfectly dry pluued wh pine, fibres parallel to motion about.. .19 41 4l " " 4< and smoothed mahogany, fibres parallel to mo- tion about .. .24 Polished marble on polished marble. By the writer average 16 44 " ou common brick 44 .44 Common brick on common brick 44 .64 Soft limestone well dressed, on the same .64 Common brick, on well-dressed soft limestone .65 " " " hard " 60 Oak across the grain, on soft limestone, well dressed .38 " " " hard 38 Hard limestone on hard limestone, both 4I ' 4 .38 '- 4l " soft " " 44 4< 67 Soft 4< 4< hard " 4< 4< " 65 Wood on metal, generally, .2 to .62 mean.. .41 Wood, very smooth, on the same, generally, .25 to .5 44 .. .38 Wood, " " on metal, " .2 to .62 4< .. .41 Metal on metal, very smooth, dry 44 .15 to .22 44 .. .18 Masonry and brickwork, dry " " .6 to .7 " .. .65 44 " with wet mortar about.. .47 44 <4 4I " slightly damp mortar " .. .74 44 on dry clay .* 4< .. .51 41 44 moist" ' 4 .. .33 Marble, sawed ; on the same ; both dry. By the writer.* average ' .. A " ll " " " both damp " ..* 44 4 .. .55 41 " on perfectly dry planed wh pine. 4l ..* 4< 4 .. .45 44 4< on damp planed wh pine 44 ..* 4 .. .6 44 polished, on perfectly dry planed wh pine 4 * ' .. .26 White pine, perfectly dry : planed; on the same; all the fibres parallel to motion about.. .4 <4 4I damp, pinned ; on the same " .. .6 * But after a few trials the surfaces become so much smoother aa to reduce the angles as much as from i Q to 5 ; tlie sliding -blocks weighing about 30 fts each. 600 FRICTION. Art. 3. Table of coefficients of moving- friction of smooth plane surfaces, when kept perfectly lubricated. (Morin.) Substances. Dry Soap. Olive Oil. Tal- low. Lard. LardA Plum- bago. Oak on oak, fibres parallel to motion 164 075 067 "" " " " fibres perpendicular to motion 083 072 '* on elm, fibres parallel to motion 136 073 066 " on cast iron, fibres parallel to motion 080 " on wrought iron, " " " . . 098 Beech on oak, fibres " ' " 055 Elm on oak, " " " " 137 070 nun " on elm, " " " " 139 .060 " cast iron, " " " " 066 Wrought iron on oak, fibres parallel, greased and wet, .256. " " " " fibres parallel to motion 214 085 " " on elm, " '' " " 055 078 076 " " on cast iron " " " 066 103 076 070 082 081 " " on brass fibres '" " " 078 103 Cast iron on oak, fibres parallel to motion .189 . 10 ' " on elm, " " " " .075 .061 .078 .077 .075 091 " '* on cast iron, with water 314 197 064 100 070 " " on brass 078 103 075 .Uo Copper on oak. fibres parallel to motion 069 Yellow copper on cast iron 066 072 Brass on cast iron. 077 086 ,uo8 " on wrought iron 072 081 089 " on brass 058 Steel ou cast iron 079 105 081 " on wrought iron 093 076 ' ' on brass .053 .056 .067 .133 191 .159 241 ' ' on oak, with water, .29 g- friction of the wooden frigate Princeton was found by lin Institute in 1844. to average about .067 or one-fifteenth of the pressure The launching a committee of the Frankli during the first .75 of a sec, and .022 or one forty-fifth for the next 4 sees of her motion. The slope of the ways was 1 in 13, or 4 dear, 24 mins. TheV were heavily coated with tallow. Pressure on them = 15.84 fts per sq inch, or 2280 5>s per sq ft. In the first .75 of a sec the vessel slid 2.5 ins j in the 4 next sees 15 ft, 6.5 ius ; total for 4.75 sec 15.75 ft. See footnote, p 598. Coeffs of friction of dry surfaces, under pressures grad- ually increased up to the limits of abrasion. (By G. Rennie, C E ) Pres. in Los. per Square Inch. Wrought Iron Wrought Iron. Wrought Iron on Cast Iron. Steel on Cast Iron. Brass on Cast Iron 32.5 186 224 336 448 560 672 709 784 .140 .250 .271 .312 .376 .409 .174 !292 .333 .365 .367 .376 .434 .166 .300 .333 .347 .354 .358 .403 .157 .225 .219 .215 .208 .233 .233 .234 232 The irregular- ities in this last column are re- markable. 821 .273 Art. 4. Pivot friction. To find the amount of power consumed by the fric of pivots. The base of the pivot is supposed to be flat; that being the best form. The moment of pivot fric, or. in other words, its tendency to prevent the pivot from revolving ; is found by mult the amount of the fric itself, by its leverage ; which last is equal to % (two thirds) of the rad of the pivot. Hence, first find the fric, by mult the entire pres on the pivot in Ibs. by the corresponding coeff taken from one of the foregoing tables. Mult the fric so obtained, by % of the circumf of the pivot, in feet.* The prod will be the^amount of power consumed by fric at each turn of the pivot, ex- pressed in foot pounds. And these ft-fts. mult by the number of turns made per min, will give the power expended per min in overcoming the fric. Ex. A pres of 22400 Ibs is supported by a steel pivot 6 ins diam, revolving on a cast-iron step ; and kept well lubricated with oil. What power must the prime mover expend in overcoming tho fric? Here, the diam of the pivot being 6 inches, its circumf is 18.84 ins: and two-thirds of 18. 84 is 12.56 ins, or 1.05 ft. Also, the corresponding number for steel on cast iron in the table, is about .08. Con- sequently, 22400 X .08 X 1.05 - 1881.6 ft-fts of fric per revolution. * Kqual to the whole circumf described by ^ of the rad. 601 Now, If the pivot makes say 60 tu/HSper min, fric will consume 1881.6 X 60 112896 ft-fts of power powers of thi NOTE. The>Hlim ot the pivot should be as small as considerations of strength will admit; for to motion Increases with the greater leverage of the larger one. See Pivot, in Glossary. \Vtioii the pres on a pivot floes not exceed 15O It** per s| inch lor east iron, or 30U Ibs for steel, with good lubrication, it will revolve rapidly in machinery for a long time, without sensible wear. When the motion is very slow and intermittent, as in lock-gates, draw-bridges, turntables, &c, from 1 to even 2.5 tons per sq inch for cast iron ; and from 2 to 4 tons for steel, are used. A flat base* or foot, is best for a pivot ; and should be kept well lubricated, and free from dust or grit. Art. 5. The friction of the journals of axles, gudgeons, or trunnions, in their boxes or bearings, is a species of sliding fric; yet somewhat distinct from the foregoing. It bears a less proportion to the pres than in the case of flat surfaces. Neither the fric itself, uor the power reqd to overcome it, is affected by the length of the journal in its bearings; but its resistance to motion increases with its diam : and in the same proportion; be- cause the leverage with which the fric resists motion, is as the diam. Therefore, it becomes impor- tant to employ strong materials for journals, as well as tor pivots, in order to reduce the diam as much as possible. To find the amount of power consumed by the fric of jour- nals : Mult together, the weight or the pres sustained by the journals, in pounds; the corresponding number taken from the following table, and the circumf of the journal in feet. The prod will be the loss of power at each rev, expressed in ft-B>s. Ex. A pres of 22400 pounds is sustained by two journals of cast iron, 6 ins diam, and running iu cast-iron bearings well lubricated with lard. How much power of the prime mover is consumed by fric, at each rev ? Here, we have the pres on the journals, 22400 Ibs ; the corresponding number from the following table, .054 ; and the circumf of the journals 18.84 ins, or 1.57 ft; consequently, Power consumed = 22400 X -054 X 1.57 = 1899 ft-fts per rev. Table of journal, or axle friction.* For the boxes of jour- nals, cast iron perfectly smoothed, is as good a material for light pressure as any, if kept constantly well be Q c O i 9 1 O OIL, TALLOW, OR LARD. Very soft, purified carriage grease. Continuously. the journal rapidly. Brass or bronze is much used. Being softer than cast iron it does not cut so much when badly oiled ; but is less durable. Babbitt or other soft metals are useful under great pressures. S.cg |KJ Bell-metal on bell-metal .097 .075 .075 .075 .075 .125 .100 .116 .03 to .054 .03 to .054 .03 to .054 .03 to .054 .092 .070 .03 to ,054f .03 to .054 .03 to .05 .065 .090 Cast-iron on bell-metal .194 .251 .161 .189 .079" Wrought-iron on bell-metal Wrought-iron on cast-iron Cast-iron on cast-iron .137 Wrought-iron on lignum vitae.... Cast-iron on ligriumvitte Lignumvitse on cast-iron 188 .185 Lignumvitse on lignumvitse Cast-iron on brass Wrought-iron on brass .190 .250 .160 .190 .075 .075 .100 Brass on cast-iron.... lard mbag .111 .109 * Prof B. H. Thurston of the Stevens Inst,, see Jour Franklin Inst, Nov 1878. proves that contrary to common opinion the coef of journal fric diminishes with increase of pres up to about 600 Ibs per sq inch, which is seldom reached. It then increases. Also that the coef ia much affected by vel, and by the temp of the journal. Much depends on smoothness of journal and bearing. t On railroads this is certainly at times as low as .02; and probably even .015 at high vels; and much less in well made machinery, with polished journals and bearings. So with the other metals in the table. 602 FRICTION. Art. 6. Friction rollers. If a journal J, instead of revolving on ordi- narv bearings, be supported on friction rollers R, R, its fric will be reduced in nearly the same pro- portion that the diam ot the axle o or o of the rollers, is less than the diam of the rollers them- Thus, if the fric of J when in ordinary bearings be 1000 pounds, it will, if placed on rollers 12 ins diam, revolving on axles 3 ins diam, be reduced almost to 250 pounds, or to one-fourth, or as 12 ia to 3. Art. 7. Rolling friction is that which takes place where the circumf of a wheel, or of any rolling body, comes in contact with the surf on which it rolls. It appears to fol- low the same law that applies'to sliding friction, viz, that so long as abrasion does not take place, it increases as the pres. It is also inversely as the diams. In practice there are usually two ways of applying the force reqd to overcome rolling fric. The first is at the axis of the rolling body ; as the force of a horse is applied at the axis of a wheel in a wagon ; or of a man, at that of a wheelbarrow. The second is at the circumf of the roller; as when workmen push along a heavy timber laid on top of two or more rollers ; or as the ends of an iron bridge play backward and forward by contraction and expansion, on top of rollers, or balls of metal. The fric is much less in the second case, than in the first; for although in the second there are two rolling fries to be overcome, (one at the top, and one at the bottom of the roller,) yet these two are much less than the one rolling, and the one axle- fric of the first case. The few experiments that have been made on the coeffs of rolling fric, discon- nected from axle fric, have been too incomplete to serve as a basis for practical rules. Art. 8. Rolling, and axle friction combined ; as in railroad cars. A body will not slide, or roll down an inclined plane, unless the plane is so steep that the sliding force of grav is sufficient to overcome the sliding or rolling fric (as the case may be) of the body. See Art 62, of Force in Kigid Bodies. The fric is in proportion to the weight, or rather to the pres, of the body ; and the sliding force of gravity is in proportion to the height of the plane as compared with its hor length or base. Therefore, if the height of the plane has to be i, y 1 ^, J^Q. &c, of its base, before the body begins to slide, or to roll down it, we know that the fric of the body is 4, y 1 ^, 1 & c , of its weight; or, more correctly speaking, of its pres on the plane. This pres is equal to Ihe weight only when the plane is level, or hor ; and becomes less than the weight, as the plane be- eomes steeper; but the diff is so slight on moderate railroad grades, that it may be neglected in such eases as that now before us. See Table, p 486, Force in Rigid Bodies. It is found that railroad cars with wheels and journals of the ordinary diams, (about 28 to 32 ins for wheels, and about 3 ins for journals.) begin to roll down a grade when it is as steep as from 16 to 24 ft per mile. This diff is owing to a varying condition of the rails as to smoothness, dust, and irregu- larities; and to diff proportions between" the diams of the wheels and journals; the kind of springs; khe kind and quantity of oil used in lubricating; and to other minor considerations. There are 5280 Ft in a mile ; therefore, 16 and 24 ft per mile give for the height of the plane in proportion to its base, 1 to 330; and ^ = 1 to 220. Therefore, the combined rolling and axle fric of the car vary 16 24 2240 from -a-A-TT to -T-i^ part of its weight; or since there are 2240 fts in a ton, it varies from to 330 220 6oO 2240 = 6.8 to 10.2 B)s per ton weight of car ; and the coeff of its fric varies from -^\-^ or .00303, to ^Q or .00454. Perhaps 8% H>s per ton, or ^y of the weight, (or .00379,) which corresponds to = 20 ft grade per mile, (or to an inclination of 13 mius,) may in practice be assumed as the 264 average on American railroads : and that about 1 tb of this mav be ascribed to rolling; and 7J^ to axle or journal friction, but much uncertainty exists on this subject. Under the same circumstances, this fric, and its coeff, remain unchanged at all vels: that is, if the fric is 8% tbs per ton at a vel of 1 mile per hour, it will be 8> B>s per ton at 60 miles per hour ; and precisely the same amount of it will be generated at each rev of the wheel in either case ; and the same amount of motive power will in either case be required to overcome it during one rev. This being the case, it may at first sight appear paradoxical that the motive power or traction reqd to overcome the fric, must increase with the vel. But it is evident that this follows from the fact that the fric at each rev requires the same amount of power to overcome it, at whatever vel ; consequently, since at 60 miles an hour there are 60 revs in the same time that there is 1 at 1 mile an hour, there must be 60 times more power continuously applied during the high vel than during the low one. Still the total amount of fric, and also the total amount of power called into action in travelling any given dist. will be the same at all vels; for whether the car travels 60 miles in 1 hour, or in 60 hours, its wheels make the same number of revs, and generate the same total amount of friction. At 1 mile per hour, we apply our continuous power in a small stream, but for a long time; and at 60 miles per hour, we apply it in a stream 60 times larger, but for only one 60th as long a time. The engine will expend the same total amount of power against fric during a 60-mile trip at 1 mile an hour, as during the same 60 miles at 60 miles an hour; but still, an engine to perform the 60 miles in 1 hour, must be 60 times as powerful as one that can barely do 1 mile per hour ; in other words, the fast engine must be able to apply 60 times as much force at any one instant. The same principle applies to the fric of pivots, bands. &c, at diff vels. The effect of grades is not here considered. In them all, the coe.ff of fric remains unchanged at all vels ; but the <|Iiantity of fric to be over- come ill a giveil time, of course varies as the vels; and, therefore, as in the case of the cars, greater vel requires greater power during the time of motion.* When a train moves rapidly, other resistances are generated ; such as that of the air, (which is almost inappreciable at low speeds;) jolts against irregularities of the rails; fric against the rails, caused by the cars swaying from side to side of the track, &c. These, however, do not affect the frio just spoken of. A double purchase crane, with a weight of 7000 Ibs suspended from it, showed a fric Of ^ the weight; *The young student should reflect well on this subject, and familiarize himself with the broad dis- tinction between the unvarying coeff of fric, and the varying amount of frio at diff vels. See footnote, p 598. j?*< TKACTIQiW; \ ' VJ 603 9r nearly 800 fts. One ton suspended a, a*SH end of a chain passing over 2 cast-iron sheaves of 2 feet diam; with wrought-iron jouruaii*<^'forkiug iu brass bearings, well oiled, gave y^- of the weight; or er ton. The fric of au unloaded locomotive, is about 12 fts per ton of -'120 fts; its weight; with a tram attached, this is increased about 1 ft per ton of train. Moriu says the fric of a sled on dry ground is % of the pres. Babbage states that a block of stone floor, GO per cent ; with both wooden surfaces greased, only 6 per cent; and with the block on top of wooden rollers 3 ins diam, only 2.6 per cent. Rubble masonry on wet clay .2 to .35. For the friction of Hydraulic presses see p 632. TEACTION. Traction oil common roads, and canals; or the power reqd to draw vehicles and boats along them. In connection with this subject read the preceding and the following The following table coach and passengers, as ascertained by me results are given per s shows tolerable approximations to the force in fts per ton, reqd to draw a stage up ascents on the Holyhead turnpike road iu England, (a fine road,) by horses ; ns of a dynamometer. The entire weight was 1% tons; but in the table, the ingle ton. From the nature of such cases, no great accuracy is attainable. Proportional Ascent. Ascent iu Ft. per Mile. At 4 Miles per Hour. At 6 Miles per Hour. At 8 Miles per Hour. At 10 Miles per Hour. Lbs. Lbs. Lbs. Lbs. 1 in 15J4 340.7 210 216 225 240 1 " 20 264. 196 202 212 229 1 " 26 203.1 155 160 166 175 1 " 30 176. 137 142 147 154 1 " 40 132. 114 120 124 130 1 " 64 82.5 109 115 120 126 1 " 118 44.7 102 107 113 120 1 " 138 38.3 99 103 109 117 1 " 156 33.9 98 101 106 112 1 " 245 21.6 93 96 101 107 1 " 600 8.8 81 85 91 96 Level. 0. 76 80 85 91 Miles per hour. y. 1 . The following results, raost of them with the same instrument, are also in fts per ton ; with a four- wheeled wagon, at a slow pace, on a level; and the roads iu fair condition. On a cubical block pavement ........................... 32 fts per ton ........... to 50. " McAdam road, of small broken stone .............. 6'2 " " probably to 75. " gravel road ....................................... 140 " " " " Telford road, of small stone on a paving of spawls 46 " " 75. " broken stone, on a bed of cement concrete ........ 46" " ' "75. " common earth roads .............................. 200 to 300. On a plank road 30, to 50 fts. The tractive power of a horse diminishes as his speed in- creases; and perhaps, within certain limits, say from % to four miles per hour, nearly in inverse proportion to it. Thus, the average traction of a horse, on a level, and actually pulling for 10 hours in the day, may be assumed approximately as follows: Lbs. Traction. Miles per hour. Lbs. Traction. 333.33 iy .............. 111.11 ... 250. 2^ .............. 100. 200. 2% .............. 90.91 13^ ............. 166.66 3 .............. 83.33 1?? .............. 142.86 3^ .............. 71.43 2 .............. 125. 4 .............. 62.50 If he works for a smaller number of hours, his traction may increase as the hours diminish ; down to about 5 hours per day and for speeds of about from \y to 3 miles per hour. Thus, for 5 hours per day his traction at 2^ miles per hour will be 200 fts, Ac. When ascending a hill, his power dimin- ishes so rapidly, from having partially to raise his own weight, (which averages about 1000 to 1100 fts,) that up a slope of 5 to 1. he can barely struggle along without any load. On such an ascent, (see table, p 486, of Force in Rigid Bodies,) he must exert a force equal to 439 fts per ton. or of 15)6 fts for the 1000 fts of his own weight. Assuming that on a level piece of good turnpike, he would when h;iul- ing a cart and load, together weighing 1 ton, have to exert a traction of 60 fts; then on ascending a hill of 4 inclination, (or 1 in 14.3 ; or 36M ft per mile,) he would have to exert 156 fts. against the gravity of the 1 ton : and 67 fts. against, that of his own weight: or 223 fts in all. He may, for a few inins, exert without injury, about twice his regular traction. This calculation shows that up a hill of 4, an average horse is fully tasked in drawing a total load of one ton ; and should, therefore, be allowed, in such a case, to choose his own gait ; and to rest at short intervals. A fair load for a single road in good order, is about half a ton, in addition to the cart, which will be about half a ton more. With two horses to this same cart, the load alone may be about 1 % tons. REM. Since the action of gravity is the same on good roads and bad ones, it follows that ascents become more objectionable the better the road is. Thus, on an ascent of 2, or 184.4 ft pe'r mile, gravity alone requires a traction of 78 fts per ton ; 604 TRACTION. which is about 10 times that ou a level railroad at 6 miles an hour ; but only about equal to that ou & level common turnpike road, at the same speed. Therefore, (to speak somewhat at random,) it would require 10 locomotives instead of 1 ; but only 2 horses instead of 1. A grade of I in 35 ; or 150 ft to a mile; or 1 'M' , is about the steepest that permits horses to be driven down a hard smooth road, in a fast trot, without danger. It should, therefore, not be exceeded except when absolutely necessary, especially ou turnpikes. On eajtuis) ami other waters, the liquid is the resisting medium that takes the place of friction on level roads. But unlike friction, its resistance varies as the squares ol' the vels; (see Art '26 of page 571.) at least from the vel of 2 ft per sec, or 1.361 miles per hour; u that of 11 ^ ft per sec, or 7.81 m per h. As the speed falls below 1 % in per h, the resistance varies less and less rapidly; and this is tue case whether the moved body floats partly above the surface; or is entirely immersed. In towing along stagnant canals, m per h; for freight most frequently from 1J4 to 2. Less force is required to tow a boat at say 2 m per h, where there is no current, than at say \% m per h, against a current of % m per h. because in the last case the boat has to be lifted up the very gradual inclined plane or slope which produces the current. Therefore, a steamboat which has its speed increased say 3 or 4 m per h by a descending current, will have it retarded more than 'A or 4 m per h when returning against the current. The force required to tow a boat along a canal depends greatly upon the comparative transverse sectional areas of the channel, and of the immersed portion of the bout. When the width of a canal at water-line is at least 4 times that of the boat; and the areaof its transverse section asgreatas at least 6}^ times that of the immersed transverse section of the boat, the towing at usual canal vels will be about as easy as in wider and deeper water. With less dimensions, it becomes more difficult. (D'Au- buisson.) Much also depends on the shape of the bow and other parts of the boat ; and on the propor- tion of its length to its breadth and depth. Hence it is seen that the mere weight of the load is by no means so controlling an element as it is ou laud. The whole subject, however, is too intricate to be treated of here. Morin states that naval constructors estimate the resistance to sailing and steam vessels at sea, at but from about .5 to .7 of a R> for every sq ft of immersed transverse section, when the vel is 3 ft per sec, or 2.046 miles per hour. It is far greater on canals. On the Sell uy lit ill Ufavig-ation of Pennsylvania, of mixed canal and slackwater, for 108 miles, the regular load for 3 horses or mules, is a bout of very full build ; and no keel : 100 ft long, 17^ ft beam ; and 8 ft depth of hold ; drawing ;VA ft when loaded.* Weight of boat about 65 tons; load 175 tons of coal, (2240 Us:) total weight 240 tons, or 80 tons per horse or mule. On the down trip with the loaded boats, for 4 days, the animals are at work, actually towing, (except at the locks.) for 18 hours out of the 24; thus exceeding by far the limits of time usually allowed for Ou the canal sections, (which have 60 ft water-line ; and 6 ft depth,) thespeed is 1% miles per hour ; and on the deep wide pools, 2 miles. On the up trip with the empty 65-ton boats, the average speed is about 2U miles per hour. The empty boats draw 16 to 18 ins water ; and frequently keep on without stopping to rest day or night through the entire distance cs per ton. The intelligent engineer and superintendent of the Sch Nav, Jumes F Smith, gives as the results of his own extensive observation, that one of these large boats loaded (240 tons in all) may, without distressing the animals, be drawn along the canal sections, for 10 hours per day, as fellow's : By one average horse or mule, at the rate of 1 mile; by two animals, at 13^ miles; and "by three, atl% miles per hour. When four animals are used the gain of time is very trifling. At a time of rivalry among the boatmen, one of them used 8 horses ; but with these could not exceed 2J6 miles per hour in the canal portions. Two or more horses together cannot for hours pull as much as when working sepa- rately. If our preceding short table of the traction of a horse at diff vels for 10 hours is correct, then the traction of the above loaded conl boats (240 tons) on the canal sections of the navigation, is as follows : The last column shows the traction in fts per sq ft of area of immersed transverse section where largest ; viz, about 95 sq ft. Horses. Miles per Hour. Lbs. per Ton. Lbs. per Sq Ft. 3 \% 42* ^ . 1 78 4 50 3 on pools 2 3 75 1.56 3 95 8.... ....2K... 8 0. ...3.33.... ...8.42 3up-trip 2^ %Y 4.61 12.50 Laehiiie Canal, Canada, 120 ft wide at water-line ; 80 ft at bottom ; depth on mitre sills 9 ft ; 6 horses tow loaded schooners with ease. Before the enlargement of the Erie <'MUll,f its dimensions were 40 ft water-line ; 28 ft bottom ; 4 ft depth of water. The average weight of the boats was about 30 tons. With 75 tons of load, or 105 tons total, they were towed by 2 horses, at the rate of about 2 miles per hour ; which by our table gives a traction of nearly 2.4 fts per ton. The boats were about 80 ft long ; 14 ft beam ; full 83/4 ft draught loaded ; hence the traction by our table >vould be about 5.7 fts per sq ft of immersed transverse section. * Cost of boats in 1872 averaged about $2500 ; arid their repairs about $150 per annum. They last from 9 to 12 years. Before the canal was enlarged, boats weighing about 22 tons, and carrying 60 tons of coal, were used. Length 70 ft; beam 13; draft empty, about 1 foot; when loaded, 4 ft ; cost $700. These are approximately the dimensions of the freight boats, on our ordinary canals. With 2 horses, speed \% miles. t Length 363 miles ; cost $19680 per mile. The enlarged canal has 70 ft ; 42 ft ; and 7 ft of water ; and cost $90800 per mile for the enlargement only. The cost of the several canals in Pennsylvania has ranged between $23000 and $50000 per mile. 5WER. 605 While, for the 82-ton loaded boatajp*?fie footnote, on the smaller canal, (the boats nearly touch- ing bottom,) the tractiona-ifjSes, would be 3% fts per ton ; or about twice as great as the above 1.78 fts. It also wouJA^SirjJreper sq ft of immersed section. Traction on^c'fevel straight railroad, at speeds not exceeding about 12 miles per hour^Wrrom 5 to 10 fts per ton, depending on diam of wheels and journals ; lubrication, condition of t ANIMAL POWEE, Art. 1. So far as regards horses, this subject has been partially considered under the preceding head, Traction. All estimates on this subject must to a certain extent be vague, owing to the diff strengths aud speeds of animals of the same kind ; as well as to the extent of their training to any particular kind of work. Authorities on the subject differ widely ; and sometime!* express themselves iu a loose manner that throws doubt on their meaning. We believe, however, that the following will be found to be as close approximation to practical averages as the nature of the case admits of with our present imperfect knowledge. We suppose a good average trained horse, weighing not less than about J^ a ton, well fed aud treated. Such a one, when actually walking for 10 hours a day, at the rate of 2 J^ miles per hour, on a good level road, such as the tow-path of a canal, or & circular horse -path, * can exert a continuous pull, draught, power, or traction, of 1OO Ibs. Now, 2Ji miles per hour, is 220 ft per min. or 3% ft per sec; and since 10 hours contain 600 min, his day's work of actual hauling on a level, at that speed, amounts to min ft Ibs 600 X 220 X 100 13200000ft-ftsperday. Or, 22000 ft-Ibs per min. or 366% ft-Ibs per sec.t Which means that he exerts force enough during the day to lift 13 200000 fts 1 foot high; or 1 320000 Rs 10 feet high; or 132 000 fts 100 ft high, &c. He may exert this force either in traction (hauling) or in lifting loads. If he has to raise a small load to a great height, the machinery through which he does it must be so geared as to gain speed, at the losa (commonly but improperly so expressed) of power. Whether he lifts the great weight through a small height, or the small weight through a great height, he exerts precisely the same amount of force or power. In connection with this subject, the student should read Arts 5, 9, 11, &c, of Force in Rigid Bodies. Also, see Hauling by Horses and Carts. Experience shows that within the limits of 5 and 10 hours per day, (the speed remaining the same,/ the draft of a horse may be increased in about the same pro- portion as the time is diminished ; so that when working from 5 to 10 hour! per day, it will be about as shown in the following table. Hence, the total amount of 13 200 000 ft-ftf per day may be accomplished, whether the horse is at work 5, 6, or 8, &c, hours per day.+ This, of course, supposes him to be actually lifting or hauling all the time; and makes no allowance for stop- pages for any purpose. Table of draft of a horse, at 2^ miles per hour, on a level. Hours per day. Lbs. Hours per day. Lbs. 10 ....'. 100 7 142J 9 lllj 6 166% 8 125 5 200 Experience also shows that at speeds between % and 4 miles an hour, his force or draught will be inversely in pro* portion to his speed. Thus, at 2 miles an hour, for 10 hours of the day, hif draught will be miles miles ftg 9>s 2 : iy z : : 100 : 125 draught. At m miles, it would be 166% fts ; at 3 miles, 83 & fts; and at 4 miles, 62^ B>s ; as per table i Traction. Therefore, in this case also, the entire amount of his day's work remains the same ; and within * To enable a horse to work with ease in a circular horse- walk, its diam should not be less than 25 ft; 30 or 35 would be still better. f A nominal horse-power is 33000 ft-fts per minute; this being the rate assumed by Boulton and Watt in selling their engines ; so that purchasers wishing to substitute steam for horses, should not be disappointed. Their assumption can be carried out by a very Btrong horse day after day for 8 or 10 hours ; but as the engine can work day and night for months without stopping, which a horse cannot, it is plain that a one-horse engine can do much more work than any one such horse. Hence many object to the term horse-power as applied to engines ; but since every- body understands its plain meaning, and such a term is convenient, it is not in fact objectionable. Boulton and Watt meant that a one-horse engine would at any moment perform the work of a very strong horse. An average horse will do but 22000 ft-fts per min. + It is plain that although the day's labor will be the same, that of an hour, or of a min, will vary with the number of hours taken as a day's work. It must be remembered that a working day of a given number of hours, by no means implies, in every case, that number of hours of actual work; but includes intermissions and rests. \ This remark about speed will not apply to loads towed through the water. Thus, if his draught at 2 miles an hour be 125 Ibs ; and at 4 miles, "62 H fts : he will on land draw loads in these proportions ; but in hauling a boat through the water at the greater speed, he has to encounter the increased resistance of the water itself ; which resistance at 4 miles is much more than twice as great as at 2 miles ; probably 4 times as great. Therefore, at 4 miles on a canal, his draught of 62>$ fts would not suffice for a load half as great as he could tow with his draft of 125 fts at 2 miles. 39 606 ANIMAL POWER. all the foregoing limits of hours and speed, may be practically taken to be about 13200000 ft-Ibs per day ; or 22000 ft-B>s per min of a day of 10 hours. But it does uot follow that the horse can always in practice actually lift loads at that rate; because generally a part of his power is expended in overcoming the friction of the machinery which he puts in motion ; aud moreover, the nature of the work may require him to stop frequently ; so that in a working day of 8 or 10 hours, the horse may not actually be at work more than 5, 6, or 7 hours. As a rough approximation, to allow for the waste of force in overcoming the friction of hoisting machinery, and the weight of the hoisting chains, buckets, &c, we may say that the Useful or paying daily net work of a horse, in hoisting- by a com- mon gill, is about 10000000 t't-fts. That is, he will raise equivalent to 10000000 tt>s net of water, or ore, &c, 1 foot. The load which he can raise at once, including chains, bucket, aud aa allowance for friction, will be as much greater than his own direct force,, as the diam of the horse - walk is greater than that of the winding drum; aud it wiii jiove that much slower than he does. His own direct force will vary according to the number of hours per day that he may be required to work, as io the foregoing table. With these data, the size of the buckets can be decided on ; and of these there should be at least two, so that the empty one at the bottom may be filled while the full one at top is being emptied ; so as to save time. The same when the work is done by men. Art. 2. A practised laborer hauling along: level road, by a rope over his shoulders; or in a circular path, pushing before him a hor lever, at a speed of from 1 ^ to 3 miles per hour, exerts about % fl> part as much force as a horse; or 2 200000 ft-B>s per day ; or 3666% ft-Ibs per min of a day of 10 hours of actual hauling or pushing. But laborers frequently have to work under circumstances less advantageous for the exertion of their force than when haulingor pushingin the manner just alluded to ; and in such cases they cannot do as much per day. Thus in turning a winch or crank like that of a grindstone, or of a crane, the continual benditig of the body, and motion of the arms, is more fatiguing. The size of a WillCll Should not exceed 18 illS, or the rad of acircle of 3 ft diam; and against it a laborer can exert a force of about 16 fts, at a vel of '2% ft per sec, or 150 ft per min, making very nearly 16 turns per min ; for 8 hours per day. To these 8 hours an addition must be made of about % part, for short rests. Or if a working day is taken at 8, or 10, &c, hours, -^ part must generally be taken from it for such rests. On the foregoing data an hour's work of 60 min of actual hoisting would be Hw ft min 16 X 150 X 60 = 144000 ft-ftg; or, deducting ^ part for rests, 115200 ft-fts per hour of time, including rests. In practice, however, a further deduction must be made for the fric of the machine, and for the wtof the hoisting chains j and in case of raising water, stone, ore, &c, from pits, for the wt of the buckets also. As a rough average we may assume that these will leave but 100000 ft-lb.s of paying, or useful work per hour; that is, that a man at a winch will actually lift equivalent to 100OOO Ibs of water, ore, fcc, 1 foot high per hour's time, in- cluding rests. This is equal to 1666% ft-Tbs per min of a day of 10 hours, including rests. Therefore, in a day of 10 working hours he would raise 1 000000 fts net. 1 foot high ; Or jllSt ^ part Of What a horse WOllld do With a gin in the same time. We have before seen that in hauling along a level road, he can at a slow pace perform about % of the daily duty of a horse. He may also work the winch with greater force, say up to 30 or even 40 fts; but he will do it at a proportionately slower rate; thus, accomplishing only the same daily duty. With a gin, like tKtise for horses, but lighter, with 2 or more buckets, a prac- tised laborer will in a working day of 10 hours, raise from 1 200000 to 1 400000 ft-fts net of water, ore, &c. With a shallow well or pit, more time is lost in emptying buckets than in a deep one; but the deep one will require a.greater wtof rope. To save time in all such operations on a large scale, there should be at least two buckets; the empty one to be fi.led while the-full one is being emptied. It is also best to employ 2 or more men to hoist at the same time, by winches, at both ends of the axis ; and the men will work with more ease if the winches are at right angles to.each other. Each winch handle may be long enough for 2 or 3 men. An extra man should be employed to emptv the buckets. He may take turns with the hoisters. The same remarks apply in some of the following cases. On a treadwheel a practised laborer will do about 40 per cent more daily duty than at a winch ; or in a working day * of 10 hours, including rests, he will do about 1 400000 ft- Ibs. And he can do this whether he works at the outer circumf of the wheel, stepping upon foot- boards, or tread-boards, on a level with its axis ; or walks inside of it, near its bottom. In both cases he acts by his wt, usually about 130 to 140 Ibs ; and not by the muscular strength of his arms. When at the level of the axis, his wt acts more directly than when he walks on the bottom of the wheel; but in the first case he has to perform a slow and fatiguing duty resembling that of walking up a continuous flight of steps ; while in the second he has as it were "merely to ascend a very slightly in- clined plane; which he can do much more rapidly for hours, with comparatively little fatigue: and this rapidity compensates for the less direct action of his wt. Therefore, in either case, as experience has shown, he accomplishes about the same amount of daily duty. Treadwheels may be from 5 to 25 ft in diam, according to the nature of the work. They are 'generally worked by several men at once , and may at times be advantageously used in pile-driving, as well as in hoisting water, stone, &c. By a good common pump, properly proportioned, a. practised laborer will in a day of 10 working hours, raise about 1000000 ft-'fts of water, net.t Bailing with a light bucket or scoop, he can accomplish about 200000 ft- Ibs net of water. By a bucket and SWape, (a long lever rocking vertically ; and weighted at one end so as to balance the full bucket hung from the other; often seen at country *The working day must be understood to include necpssary rests, and such intermissions as th nature of the work demands : but does not include time lost at meals. A working <#ay of 10 hours may, therefore, have but 8, 7, or 6, s suspended from a pulley, was raised by 10 to 40 men pulling at separate cords, from 35 to 40 fts of the ram were allotted to each man, to be lifted from 12 to 18 times per min, to a height of 3% to 4% feet each time, for about 3 min at a spell, and then 3 min rest. It was very laborious ; and the gangs had to be changed about hourly, after performing but K an hour's actual labor. Hauling by horses. See Traction. When working all day, say 10 working hours, the average rate at which a horse walks while hauling a full load, and while returning with the empty vehicle, is about 2 to 2^ miles per hour; but to allow for stoppages to rest, &c, it is safest to take it at but about 1.8 miles ;>er hour, or 160 ft per miu. The time lost on each trip, in loading and unloading, may usually be taken at about 15 min. Therefore, to find the number of loads that can be hauled to any given dist in a day, first find the time in min reqd in hauling one load, and return- ing empty. Thus : div twice the dist in ft to which the load is to be hauled ; or in other words, div the length in ft, of the round trip, by 160 ft. The quot is the number of min that the horse is in mo- tion during each round trip. To this quot add 15 min lost each trip while loading and unloading ; the sum is the total time in min occupied by each round trip. Div the number of min in a working day (600 min in a day of 10 working hours) by this number of min reqd for each trip ; the quot will be the number of trips, or of loads hauled per day. Ex. How many loads will a horse haul to a dist of 960 ft, in a day of 10 working hours, or 600 min? Here, 960 X 2 = 1920 ft of round trip at each load. And -'-- = 12 min, occupied in walking. And Jb() goo min in 10 hours 12 + 15 in loading, &c)r:,27 min reqd for each load. Finally, = - - 22.2, or 27 mm per trip Bay 22 trips; or loads hauled per day. Table of number of loads hauled per day of 1O working; hours. The first col is the distance to which the load is actually hauled ; or half the length of the round trip. The cost of hauling per load, is supposed to be for one-horse carts ; the driver doing the loading and unloading; rating the expense of horse, cart, and driver at $2 per day. See Cost of Earthwork, page 437. * The tympan revolves on a hoi- shaft: and is a kind of large wheel, the spokes, arms, or radii of which are gutters, troughs, or pipe?, which at their outer ends terminate in scoops, which dip into the water. As the water is gradually raised, it flows along the arms of the wheel to its axis, where it is dischd. The scoop wheel is a modification of it. It is an admirable machine for raising large quantities of water to moderate heights. We cannot go into any detail respecting this and other hydraulic machines. t A kind of large wheel with buckets or pots at the ends of its radiating arms ; revolves on a hor axis ; discharges at top. The buckets are attached loosely, so as to hang vert, and thus avoid spill- ing until they arrive at the proper point, where they come into contact with a contrivance for tilting and emptying them. The noria is similar, except that the buckets are firmly held in place, and thus spill much water. It is therefore inferior to the Persian wheel. 1 An endless revolving vert chain of buckets. D'Aubuissou and some others erroneously call thii tic noria. It is an effective machine. 608 ANIMAL POWEU. Dist. Feet. No. of Loads. Cost per Load. Dist. Feet. No. of Loads. Cost per Load. Dist. Miles. No. of Loads. Cost per Load. Cts. Cts. Cts. 50 38 5.26 1500 18 11.11 1 7 28.57 100 37 5.H 2000 15 13.33 IK 6 33.33 2'K) 34 5.88 2500 13 15.39 1% 5 40.00 300 32 6.23 3000 11 18.18 2 4 50.00 400 30 6.67 3500 10. 20.00 3 3 66.67 600 27 7.41 4000 9 22.22 4 2 100.00 1000 22 9.09 5000 7 28.57 9 1 I 200.00 i If the loading and unloading is such as cannot be done by the driver alone ; but requires the help of cranes, or other machinery, an addition of from 10 to 50 cts per load may become necessary. Haul- ing can generally be more cheaply done by using 2 or 3 horses, and one driver, to a vehicle. The neat load per horse, in addition to the vehicle, will usually be from J^ to 1 ton, depending on the condition, and grades of the road. From 13 to 15 cub ft of solid stone ; or from 23 to 27 cub feet of broken stone, make i ton. In estimating* for hauling rough quarry stone for drains, CUlvertS, feC, bear in mind that each cub yard of common scabbled rubble masonry, requires the hauling of about 1.2 cub yds of the stone as usually piled up for sale in the quarry ; or about % of a cub yd of the original rock in place. A Cllb yd Of Solid Stone, when broken into pieces, usually occupies about 1.9 cub yt9s perfectly lOOSe t or about 1% when piled up. A strong cart for stone hauling, will weigh about % ton ; or 1500 fts ; and will hold stone enough for a perch of rubble masonry ; or say 1.2 pers of the rough stone in piles. The average weight of a good working horse is about % a ton. Mori 11 gives the following results from careful experiments made by him for the French Government. The draft of the same wheeled vehicle on a road, may in practice be considered to be, 1st. On hard turnpikes, and pavements; in proportion to tho loads ; Inversely as the diams of the wheels ; and nearly independent of the width of tire. It increases to uncertain extents with the inequalities of the road ; the stiffness (want of spring) of the vehicle; and the speed; (considerably less than as the square roots of the last.) 2d. On soft roads, the tlraft is less with wide tires than with narrower ones; and for farming purposes he recommends a width of 4 ins. With speeds from a walk to a fast trot, the draft does uot vary sensibly. CHORDS TO A RADIUS 1, M. 1 2 3 4 6 5 6 7 8 9 1O M. 0' .0000 .0175 .03*9 .0524 .0698 .0872 .1047 .1221 .1395 .1569 .1743 0' 2 .0006 .0180 .0355 .0529 .0704 .0878 .1053 .1227 .1401 .1575 .1749 2 4 .0012 .018!) .0361 .0535 .0710 .0884 .1058 .1233 .1407 .1581 .1755 4 6 .0017 .0192 .0566 .0541 .0715 .0890 .1064 .1238 .1413 .1587 .1761 6 8 .0023 .0198 .0372 .0547 .0721 .0896 .1070 .1244 .1418 .1592 .1766 8 10 .0029 .0204 .0378 .0553 .0727 .0901 .1076 .1250 .1424 .1598 .1772 10 12 .00:55 .0209 .0384 .0558 .0733 .0907 .1082 .1256 .1430 .1604 .1778 12 14 .0041 .0215 .0390 .0564 .0739 .0913 .1087 .1262 .1436 .1610 .1784 14 1 .0047 .0221 .0396 .0570 .0745 .0919 .1093 .1267 .1442 .1616 .1789 16 18 .0052 .0227 .0401 .0576 .0750 .0925 .1099 .1273 .1447 .1621 .1795 18 20 .0058 .0233 .0407 .0582 .0756 .0931 .1105 .1279 .1453 .1627 .1801 20 22 .0064 .0239 .0413 .0588 .0762 .0936 .1111 .1285 .1459 .1633 .1807 22 24 .0070 .0244 .0419 .0593 .0768 .0942 .1116 .1291 .1465 .1639 .1813 24 2fi .0076 .0250 .0425 .0599 .0774 .0948 .1122 .1296 .1471 .1645 .1818 26 28 .0081 .0256 .0430 .0605 .0779 .0954 .1128 .1302 .1476 .1650 .1824 28 30 .0087 .0262 .0436 .0611 .0785 .0960 .1134 .1308 .1482 .1656 .1830 30 32 .0093 .0-268 .0442 .0617 .0791 .0965 .1140 .1314 .1488 .1662 .1836 32 34 .0099 .0273 .0448 .0622 .0797 .0971 .1145 .1320 .1494 .1668 .1842 34 36 .0105 .0279 .0454 .0628 .0803 .0977 .1151 .1325 .1500 .1674 .1847 36 38 .0111 .0285 .0460 .0634 .0808 .0983 .1157 .1331 .1505 .1679 .1853 38 40 .0116 .0291 .0465 .0640 .0814 .0989 .1163 .1337 .1511 .1685 .1859 40 42 .0122 .0297 .0471 .0646 .0820 .0994 .1169 .1343 .1517 .1691 .1865 42 44 .0128 .0303 .0477 .0651 .0826 .1000 .1175 .1349 .1523 .1697 .1871 44 46 .0134 .0308 .0483 .0657 .0832 .1006 .1180 .1355 .1529 .1703 .1876 46 48 .0140 .0314 .0489 .0663 .0838 .1012 .1186 .1360 .1534 .1708 .1882 48 50 .0145 .0320 .0494 .0669 .0843 .1018 .1192 .1366 .1540 .1714 .1888 50 52 .0151 .0326 .0500 .0675 .0849 .1023 .1198 .1372 .1546 .1720 .1894 52 54 .0157 .0332 .0506 .0681 .0855 .1029 .1204 .1378 .1552 .1726 .1900 54 56 .0163 .0337 .0512 .0686 .0861 .1035 .1209 .1384 .1558 .1732 .1905 56 58 .0169 .0343 .0518 .0692 .0867 .1041 .1215 .1389 .1513 .1737 .1911 58 60 .0175 .0349 .0524 .0698 .0872 .1047 .1221 .1395 .1569 .1743 .1917 60 TABLE OP CHORDS. 609 Table of 0iorcls, in parts of a ratl 1; for protracting Continued. M. 11 12 13 14 15 16 17 18 19 20 M. 0' 2 4 6 8 10 .1917 .1923 .1928 .1934 .1940 .1946 .2091 .2096 .2102 .2108 .'2114 .2119 .'22G4 .2270 .2276 .2281 .2287 .2293 .2437 .2443 .2449 .2455 .2460 .2466 .2611 .2616 .2622 .2628 .2634 .2639 .2783 .2789 .2795 .2801 .2807 .2812 .2956 .2962 .2968 .2973 .2979 .2985 .3129 .3134 .3140 .3146 .3152 .3157 .3301 .3307 .3312 .3318 .3324 .3330 .3473 .3479 .3484 .3490 .3496 .3502 0' 2 4 6 8 10 12 14 16 18 20 .1952 .1957 .1963 .1969 .1975 .2125 .2131 .2137 .2143 .2148 .2299 .2305 .2310 .2316 .2322 .2472 .2478 .2484 .2489 .2495 .2045 .2651 .2G57 .2662 .2668 .2818 .2824 .2830 .2835 .2841 .2991 .2996 .3002 .3008 .3014 .3163 .3169 .3175 .3180 .3186 .3335 .3341 .3347 .3353 .3358 .3507 .3513 .3519 .3525 .3530 12 14 16 18 20 'M: 26 28 30 .1981 .1986 .1992 .1998 .2004 .2154 .2160 .2166 .2172 .2177 .2328 .2333 .2339 .2345 .2351 .2501 .2507 .2512 .2518 .2524 .2674 .2680 .2685 .2691 .'2697 .2847 .2853 .2858 .2864 .2870 .3019 .3025 .3031 .3037 .3042 .3192 .3198 .3203 .3209 .3215 .3364 .3370 .3376 .3381 .3387 .3536 .3542 .3547 .3553 .3559 22 24 26 28 30 32 34 36 38 40 .2010 .2015 .2021 .2027 .2033 .2183 .2189 .2195 .2200 .2206 .2357 .2362 .2368 .2374 .2380 .2530 .2536 .2541 .2547 .2553 .2703 .2709 .2714 . .2720 .2726 .2876 .2881 .2887 .2893 .2899 .3048 .3054 .3060 .3065 .3071 .3221 .3226 .3232 .3238 .3244 .3393 .3398 .3404 .3410 .3416 .3565 .3570 .3576 .3582 .3587 32 34 36 38 40 42 44 46 48 50 .2038 .2044 .2050 .2056 .2062 .2212 .2218 .2224 .2229 .2235 .2385 .2391 .2397 .2403 .2409 .2559 .2564 .2570 .2576 .2582 .2732 .2737 .2743 .2749 .2755 .2904 .2910 .2916 .2922 .2927 .3077 .3083 .3088 .3094 .3100 .3249 .3255 .3261 .3267 .3272 .3421 .3427 .3433 .3439 .3444 .3593 .3599 .3605 .3610 .3616 42 44 46 48 50 52 54 56 58 60 .2067 .2073 .2079 .2085 .2091 .2241 .2247 .2253 .2258 .2264 .2414 .2420 .2426 .2432 .2437 .2587 .2593 .2599 .2605 .2611 .2760 .2766 .2772 .2778 .2783 .2933 .2939 .2945 .2950 .2956 .3106 .3111 .3117 .3123 .3129 .3'278 .3284 .3289 .3295 .3301 .3450 .3456 .3462 .3467 .3473 .3622 .3628 .3633 .3639 .3645 52 54 56 58 60 M. 21 22 23 24 25 26 27 28 29 30 M. 0' 2 4 6 8 10 .3645 .3650 .3656 .3662 .3668 .3673 .3816 .3822 .3828 .3833 .3839 .3845 .3987 .3993 .3999 .4004 .4010 .4016 .4158 .4164 .4170 .4175 .4181 .4187 .4329 .4334 .4340 .4346 .4352 .4357 .4499 .4505 .4510 .4516 .4522 .45'27 .4669 .4675 .4680 .4686 .4692 .4697 .4838 .4844 .4850 .4855 .4861 .4867 .5008 .5013 .5019 .5024 .5030 .5036 .5176 .5182 .5188 .5193 .5199 .5204 0' 2 4 6 8 10 12 14 16 18 20 .3679 .3685 .3690 .3696 .3702 .3850 .3856 .3862 .3868 .3873 .4022 .4027 .4033 .4039 .4044 .4192 .4198 .4204 .4209 .4215 .4363 .4369 .4374 .4380 .4386 .4533 .4539 .4544 .4550 .4556 .4703 .4708 .4714 .4720 .4725 .4872 .4878 .4884 .4889 .4895 .5041 .5047 .5053 .5C58 .5064 .5210 .5216 .5221 .5227 .5233 12 14 16 18 20 22 24 26 28 30 .3708 .3713 3719 .3725 .3730 .3879 .3885 .3890 .3896 .3902 .4050 .4056 .4061 .4067 .4073 .4221 .4226 .4232 .4238 .4244 .4391 .4397 .4403 .4408 .4414 .4561 .4567 .4573 .4578 .4584 .4731 .4737 .4742 .4748 .4754 .4901 .4906 .4912 .4917 .4923 .5070 .5075 .5081 .5086 .0092 .5238 .5244 .5249 .5255 .5261 22 24 26 28 30 32 34 36 88 40 .3736 .3742 .3748 .3753 .3759 .3908 .3913 .3919 .3925 .3930 .4079 .4084 .4090 .4096 .4101 .4249 .4255 .4261 .4266 .4272 .4420 .4 H>5 .44,",! .4437 .4442 .4590 .1 -)!.> .4601 .4607 .4612 .4759 .4765 .4771 .4776 .4782 .4929 .4934 .4940 .4946 .4951 .5098 .5103 .5109 .5115 .5120 .5266 .5272 .5277 .5283 .5289 32 34 36 38 40 42 44 46 48 50 .3765 .3770 .3776 .3782 .3788 .3936 .3942 .3947 .3953 .3959 .4107 .4113 .4118 .4124 .4130 .4278 .4283 .4289 .4295 .4300 .4448 .4i:> .4459 -.4*65 .4471 .4618 .4(i:?4 .4629 .4635 .4641 .4788 .4793 .47C9 .4805 .4810 .4957 <49flS .4968 .4974 .4979 .51-26 .5131 .5137 .5143 .5148 .5294 .5300 .5306 .5311 .5317 42 44 46 48 50 52 54 56 58 60 .3793 .3799 .3805 .3810 .3816 .3965 .3970 .3976 .3982 .3987 .4135 .4141 .4147 .4153 .4158 .4306 .4312 .4317 .4323 .4329 .4476 .4482 .4488 .4493 .4499 .4646 .4652 .4658 .4663 .4669 .4816 .4822 .48-27 .4833 .4838 .4985 .4991 .4996 .5002 .5008 .5154 .5160 .5165 .5171 .5176 .5322 .53-28 .5334 .5339 .5345 52 54 56 58 60 610 TABLE OF CHORDS. Table of chords, in parts of a rad 1; for protracting: Continued M. 31 32 33 34 35 36 37 38 39 40 M. 0' 2 4 6 8 10 .5345 .5350 .5356 .5362 .5367 .5373 .5513 .5518 .5524 .5530 .5535 .5541 .5680 .5686 .5691 .5697 .5703 .5708 .547 .5853 .5859 .5864 .5870 .5875 .6014 .6020 .6025 .60151 .6036 .6042 .6180 .6186 .6191 .6197 .6202 .6208 .6:546 .6352 .6357 .6363 .6368 .6374 .6511 .6517 .6522 .6528 .6533 .6539 .6676 .66S2 .6687 .6693 .6698 .6704 .6840 .6846 .6851 .6857 .6862 .6868 0' 2 4 6 8 10 12 14 16 18 20 .5378 .5384 .5390 .5395 .5401 .5546 .5552 .5557 .5563 .5569 .5714 .5719 .5725 .5730 .5736 .5881 .5886 .5892 .5897 .5903 .6047 .6053 .6058 .6064 .6070 .6214 .6219 .6225 .6230 .6236 .6379 .6385 .6390 .6396 .6401 .6544 .6550 .6555 .6561 .6566 .6709 .6715 .6720 .6725 .6731 .6873 .6879 .6884 .6890 .6895 12 14 16 18 20 22 24 26 28 30 .5406 .5412 .5418 .5423 .5429 .5574 .5580 .5585 .5591 .5597 .5742 .5747 .5753 .5758 .5764 .5909 .5914 .5920 .5925 .5931 .6075 .6081 .6086 .6092 .6097 .6241 .6247 .6252 .6258 .6263 .6407 .6412 .6418 .6423 .6429 .6572 .6577 .6583 .6588 .6594 .6736 .6742 .6747 .6753 .6758 .6901 .6906 .6911 .6917 .6922 22 24 26 28 30 32 34 36 38 40 .5434 .5440 .5*46 .5451 .5457 .5602 .5608 .5613 .5619 .5625 .5769 .5775 .5781 .5786 .5792 .5936 .5942 .5947 .5953 .5959 .6103 .6108 .6114 .6119 .6125 .6269 .6274 .6280 .6285 .6291 .6434 .6440 .6445 .6451 .6456 .6599 .6605 6610 .6616 .6621 .6764 .6769 .6775 .6780 .6786 .6928 .6933 .6939 .6944 .6950 32 34 36 38 40 42 44 46 48 50 .5462 .5468 .5174 .5479 .5485 .5630 .5636 .5641 .5647 .5652 .5797 .5803 .5808 .5814 .5820 .5964 .5970 .5975 .5981 .5986 .6130 .6136 .6142 .6147 .6153 .6296 .630*2 .6307 .6313 .6318 .6462 .6467 .6473 .6478 .6484 .6627 .6632 .6638 .6643 .6649 .6791 .6797 .6802 .6808 .6813 .6955 .6961 .6966 .6971 .6977 42 44 46 48 50 52 54 56 58 60 .5490 .5496 .5502 .5507 .5513 .5658 .5664 .5669 .5675 .5680 .5825 .5831 .5836 .5842 .5847 .5992 .5997 .6003 .6009 .6014 .6153 .6164 .6169 .6175 .6180 .6324 .6330 .6335 .6341 .6346 .6489 .6495 .6500 .6506 .6511 .6654 .6660 .6665 .6671 .6676 .6819 .6824 .6829 .6835 .6840 .6982 .6988 .6993 .6999 .7004 52 54 56 58 60 M. 41 42 43 44 45 46 47 48 49 50 M. 2 4 6 8 10 .7004 .7010 .7015 .7020 .7026 .7031 .7167 .7173 .7178 .7184 .7189 .7195 .7330 .7335 .7341 .7346 .7352 .7357 .7492 .7498 .7503 .7508 .7514 .7519 .7654 .7659 .7664 .7670 .7675 .7681 .7815 .7820 .7825 .7831 .7836 .7841 .7975 .7980 .7986 .7991 .7996 .8002 .8135 .8140 .8145 .8151 .8156 .8161 .8294 .8299 .8304 .8310 .8315 .8320 .8452 .8458 .8463 .8468 .8473 .8479 0' 2 4 6 8 10 12 14 16 18 20 .7037 .7042 .7048 .7053 .7059 .7200 .7205 .7211 .7216 .7222 .7362 .7368 .7373 .7379 .7384 .7524 .7530 .7535 .7541 .7546 .7686 .7691 .7697 .7702 .7707 .7847 .7852 .7857 .7863 .7868 .8007 .8012 .8018 .8023 .8028 .8167 .8172 .8177 .8183 .8188 .8326 .8331 .8336 .8341 .8347 .8484 .8489 .8495 .8500 .8505 12 14 16 18 20 22 24 26 28 30 .7064 .7069 .7075 .7080 .7086 .7227 .7232 .7238 .7243 .7249 .7390 .7395 .7400 .7406 .7411 .7551 .7557 .7562 .7568 .7573 .7713 .7718 .7723 .7729 .7734 .7873 .7879 .7884 .7890 .7895 .8034 .8039 .8044 .8050 .8055 .8193 .8198 .8204 .8209 .8214 .8352 .8357 .8363 .8368 .8373 .8510 .8516 .8521 .8526 .8531 22 24 26 28 30 32 34 86 38 40 .7091 .7097 .7102 .7108 .7113 .7254 .7260 .7265 .7270 .7276 .7417 .7422 .7427 .7433 .7438 .7578 .7584 .7589 .7595 .7600 .7740 .7745 .7750 .7756 .7761 .7900 .7906 .7911 .7916 .7922 .8060 .8066 .8071 .8076 .8082 .8220 .8225 .8230 .8236 .8241 .8378 .8384 .8389 .8394 .8400 .8537 .8542 .8547 .8552 .8558 32 34 36 3& 40 42 44 46 48 50 .7118 .7124 .7129 .7135 .7140 .7281 .7287 .7292 .7298 .7303 .7443 .7449 .7454 .7460 .7465 .7605 .7611 .7616 .7621 .7627 .7766 .7772 .7777 .7782 .7788 .7927 .7932 .7938 .7943 .7948 .8087 .8092 .8098 .8103 .8108 .8246 .8251 .8257 .8262 .8267 .8405 .8410 .8415 .8421 .8426 .8563 .8568 .8573 .8579 .8584 42 44 46 48 50 52 54 56 58 60 .7146 .7151 .7156 .7162 .7167 .7308 .7314 .7319 .7325 .7330 .7471 .7476 .7481 .7487 .7492 .7632 .7638 .7643 .7648 .7654 .7793 .7799 .7804 .7809 .7815 .7954 .7959 .7964 .7970 .7975 .8113 .8119 .8124 .8129 .8135 .8273 .8278 .8283 .8289 .8294 .8431 .8437 .8442 .8447 .8452 .8589 .8594 .8600 .8605 .8610 52 54 56 58 60 TABLE OF CHORDS. 611 Table jf chords, in parts of a rad 1 ; for protracting 1 Continued. M. 51 52 53 54 55 56 57 58 59 60 M. 0' 1 4 6 8 10 .8610 .8615 .8621 .8626 .8631 .8636 .8767 .8773 .8778 .8783 .8788 .879* .8924 .8929 .8934 .8940 .8945 .8950 .9080 .9085 .9oyo .9095 .9101 .9106 .9235 .9240 .9245 .9250 .9256 .9261 .9389 .9395 .9400 .9405 .9410 .9415 .9543 .9548 .9553 .9559 .9564 .9569 .9696 .9701 .9706 .9711 .9717 .9722 .9848 .9854 .9859 .9864 .9869 .9874 1.0000 1.1005 1.0010 1.0015 1.0020 1.00* 0' 2 4 6 8 10 12 14 16 18 20 .8642 .8647 .8652 .8657 .8663 8799 .8804 .8809 .8814 .8820 .8955 .8960 .8966 .8971 .8976 .9111 .9116 .9121 .9126 .9132 .9266 .9271 .9276 .9281 .9287 .9420 .9425 .9430 .9436 .9441 .9574 .9579 .9584 .9589 .9594 .9727 .9732 .9737 .9742 .9747 .9879 .9884 .9889 .9894 .9899 1.0030 1.0035 1.0040 1.0045 1.0050 12 14 16 18 20 22 24 26 28 30 .8668 .8673 .8678 .8684 .8689 .8825 .8830 .8835 .8841 .8846 .8981 .8986 .8992 .8S97 .9002 .9137 .9142 .9147 .9152 .9157 .9292 .9297 .9302 .9307 .9312 .9446 .9451 .9456 .9461 .9466 .9599 .9604 .9610 .9615 .9620 .9752 .9757 .9762 .9767 .9772 .9904 .9909 .9914 .9919 .9924 1.0055 1.0060 1.0065 1.0070 1.0075 22 24 26 28 30 32 34 36 38 40 .8694 .8699 .8705 .8710 .8715 .8851 .8856 .8861 .8867 .8872 .9007 .9012 .9018 .9023 .9028 .9163 .9168 .9173 .9178 .9183 .9317 .9323 .9328 .9333 .9388 .9472 .9477 .9482 .9487 .9492 .9625 .9630 .9635 .9640 .9645 .9778 .9783 .9788 .9793 .9798 .9929 .9934 .9939 .9945 .9950 1.0080 1.0086 1.0091 1.0096 1.0101 32 34 36 38 40 42 44 46 48 50 .8720 .8726 .8731 .8736 .8741 .8877 .8882 .8887 .8893 -.8898 .9033 .9038 .9044 .9049 .9054 .9188 .9194 .9199 .9204 .9209 .9343 .9348 .9353 .9359 .9364 .9497 .9502 .9507 .9512 .9518 .9650 .9655 .9661 .9666 .9671 .9803 .9808 .9813 .9818 .9823 .9955 .9960 .9965 .9970 .9975 1.0106 1.0111 1.0116 1.0121 1.0126 42 44 46 48 50 52 54 56 58 60 .8747 .8752 .8757 .8762 .8767 .8903 .8908 .8914 .8919 .8924 .9059 .9064 .9069 .9075 .9080 .9214 .9219 .9225 .9230 .9235 .9369 .9374 .9379 .9384 .9389 .9523 .9528 .9533 .9538 .9543 .9676 .9681 .9686 .9691 .9696 .9828 .9833 .9838 .9843 .9848 .9980 .9985 .9990 .9995 1.0000 1.0131 1.0136 1.0141 1.0146 1.0151 52 54 56 58 60 M. 61 62 63 64 65 66 67 68 69 7O M. 2 4 6 8 10 1.0151 1.0156 1.0161 1.0166 1.0171 1.0176 1 .0301 i.o;;o6 1.0311 1.0316 1.0321 1.0326 1.0450 1.0455 1.0460 1.0465 1.0470 1.0475 1.0598 1.0603 1 .0608 1.0C13 1.0618 1.0623 1.0746 1.0751 1.0756 1.0761 1.0766 1.0771 1.0893 1.0898 1.0903 1.0907 1.0912 1.0917 1.1039 1.1044 1.1048 1.1053 1.1058 1.1063 1.1184 1.1189 1.1194 1.1198 1.1203 1.1208 1.1328 1.1333 1.1338 1.1342 1.1347 1.1352 .1472 .1476 .1481 .1486 .1491 .1495 0' 2 4 6 8 10 12 14 16 18 20 1.0181 1.0186 1.0191 1.0196 1.0201 1.0331 1.0336 1.0341 1.0346 1.0351 1.0480 1.0485 1.0490 1.0495 1.0500 1.0628 1.0633 1.0638 1.0643 10648 1.0775 1.0780 1.0785 1.0790 1.0795 .0922 .0927 .0932 .0937 .0942 1.1068 1.1073 1.1078 1.1082 1.1087 1.1213 1.1218 1.1222 1.1227 1.1232 1.1357 1.1362 1.1366 1.1371 1.1376 .1500 .1505 .1510 .1514 .1519 12 14 16 18 20 22 24 26 28 30 1.0206 1.0211 1.0216 1 .0221 1.0226 1.0356 1 .0361 1.0366 1.0370 1.0375 1.05t4 1.0509 1.0514 1.0519 1.0524 1.0653 1.0658 1.0662 1.0667 1.0672 1.0800 1.0805 1.0810 1.0815 1.0820 .0946 .0951 .0956 .0961 .0966 1. 092 1. 097 1. 102 1. 107 1. Ill 1.1237 1.1242 1.1246 1.1251 1.1256 1.1381 1.1386 1.1390 1.1395 1.1400 .1524 .1529 .1533 .1538 .1543 22 24 26 28 30 32 34 36 38 40 1.0231 1.0236 1.0241 1.0246 1.0251 1.0380 1.0385 1.0390 1.0395 1.0400 1.0529 1.0534 1.0539 1.0544 1.0549 1.0677 1.0682 1.0687 1.0692 1.0697 1.0824 1.0829 1.083* 1.0839 1.0844 1.0971 1.0976 1.0980 1.0985 1.0990 1.1116 1.1121 1.1126 1.1131 1.1136 1.1261 1.1266 1.1271 1.1275 1.1280 1.1405 1.1409 1.1414 1.1419 1.1424 .1548 .1552 .1557 .1562 .1567 32 34 36 38 40 42 44 46 48 50 1.0256 1.0261 1.0266 1.0271 1.0276 1.0405 1.0410 1.0415 1 .0420 1.0425 1.0554 1 .0559 1 0564 .0569 .0574 1.0702 1 .0707 1.0712 1.0717 1 .0721 1.0849 1.0854 1 .0859 1.0863 1 .0868 1.0995 1.1000 1.1005 .1010 .1014 1.1140 1. 145 1. 150 1. 155 1. 160 1.1285 1.1290 1.1295 1.1299 1.1304 1.1429 1.1433 1.1438 1.1443 1.1448 .1571 .1576 .1581 .1586 .1590 42 44 46 48 50 52 54 56 58 60 1 .0281 1.0286 1.0291 1.0296 1.0301 1 .0430 1.0435 1.0440 1.0445 1.0450 .0579 .0584 .0589 .0593 .0598 1.0726 1.0731 1.0736 1.0741 1.0746 1.0873 1.0878 1.0883 1.0888 1.0893 .1019 .1024 .1029 .1034 .1039 1. 165 1. 169 1. 174 1. 179 1. 184 1.1309 1.1314 1.1319 1.1323 1.1328 1.1452 1.1457 1.1462 1.1467 1.1472 .1595 .1600 .1605 .1609 1.1614 52 54 56 58 60 612 TABLE OF CHORDS. Table of Chords, in parts of a rad 1 ; for protracting Continued. M. 71 72 73 74 75 76 77 78 79 80 M. 2 4 6 8 10 12 14 16 18 20 1.161* 1.1619 1.1324 1.1628 1.1333 1.1638 1.16*2 1.16A7 1.1652 1.1657 1.1661 1.1756 1.1760 1.1765 1.1770 1.1775 1.1779 1.1784 1.1789 1.1793 1.1798 1.1808 1.1896 1.1901 1.1906 1.1910 1.1915 1.1920 1.1924 1.1929 1.1934 1.1938 1.1943 1.2036 1.2041 1.2046 1.2050 1.2055 1.2060 1.2064 1.2069 1.2073 1.2078 1.2083 1.2175 1.2180 1.2184 1.2189 1.2194 1.2198 1.2313 .2318 .2322 .2327 .2332 .2336 .2341 .2345 .2350 .2354 .2359 1.2450 1.2455 1.2459 1.2464 1.2468 1.2473 1.2586 1.2591 1.2595 1.2600 1.2604 1.2609 1.2722 1.2726 1.2731 1.2735 1.2740 1.2744 1.2748 1.2753 1.2757 1.2762 1.2766 1.2856 1.2860 1.2865 1.2869 1.2874 1.2878 1.2882 1.2887 1.2891 1.2896 1.2900 0' 2 4 6 8 10 12 14 16 18 20 1.2203 1.2208 1.2212 1.2217 1.2221 1.2478 1.2482 1.2487 1.2491 1.2496 1.2614 1.2618 1.2623 1.2627 1.2632 22 24 26 28 30 1.1668 1.1671 1.1676 1.1680 1.1685 1.1690 1.1694 1.1699 1.1704 1.1709 1.1807 1.1812 1.1817 1-1821 1.1826 1.1831 1.1836 1.1840 1.1845 1.1850 1.1948 1.1952 1.1957 1 1962 1.1966 1.1971 1.1976 1.1980 1.1985 1.1990 1.2087 1.2092 1.2097 1.2101 1.2106 1.2226 1.2231 1.2235 1.2240 1.2244 .2364 .2368 .2373 .2377 .2382 1.2500 1.2505 1.2509 1.2514 1.2518 1.2636 1.2641 1.2645 1.2650 1.2654 1.2771 1.2775 1 2780 1.2784 1.2789 1.2905 1.2909 1.2914 1.2918 1.2922 22 24 26 28 30 32 34 36 38 40 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 1.2111 1.2115 1.2120 1.2124 1.2129 1.2249 1.2254 1.2258 1.2263 1.2267 .2386 .2391 .2396 .2400 .2405 1.2523 1.2528 1.2532 1.2537 1.2541 1.2659 1.2663 1.2668 1.2672 1.2677 1.2793 1.2798 1.2802 1.2807 1.2811 1.2927 1.2931 1.2936 1.2940 1.2945 1.1713 1.1718 1.1723 1.1727 1.1732 1.1854 1.1859 1.1864 1.1868 1.1873 1.1994 1.1999 1.2004 1.2008 1.2013 1.2134 1.2138 1.2143 1.2148 1.2152 1.2272 1.2277 3.2281 1.2286 1.2290 1.2295 1.2299 1.2304 1.2309 1.2313 .2409 .2414 .2418 .2423 .2428 .2432 .2437 .2441 .2446 .2450 1.2546 1.2550 1.2555 1.2559 1.2564 1.2568 1.2573 1.2577 1.2582 1.2586 1.2681 1.2686 1.2690 1.2695 1.2699 1.2704 1.2-08 1.2713 1.2717 1.2722 1.2816 1.2820 1.2825 1.2829 1.2833 1.2838 1.2842 1.2847 1.2851 1.2856 1.2949 1.2954 1.2958 1.2962 1.2967 1.2971 1.2976 1.2980 1.2985 1.2989 42 4* 46 48 50 52 54 56 58 60 1.1737 1.1742 1.1746 1.1751 1.1758 1.1878 1.1882 i.issr 1.1892 1.1896 1.2018 1.2022 1.2027 1.2032 1.2036 1.2157 1.2161 1.2166 1.2171 1.2175 M. 81 82 1 83 84 85 86 | 87 88 89 | M. 0' 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 SO la" 54 56 58 60 1.2989 1.2993 1.2998 1.3002 1.3007 1.3011 1.3121 1.3126 1.3130 1.3134 1.3139 1.3143 1.3252 1.3-257 1.3261 1.3265 1.3270 1.3274 1.3383 1.3387 1.3391 1-3396 1.3400 1.3404 1.3409 1.3413 1.3417 1.8421 1.3426 1.3430 1.8434 1.3439 1.3443 1.3447 1.3512 1.3516 ] .3520 1.3525 1.3529 1.3533 1.3640 1.3644 1.3648 1.3653 1.3657 1.3661 1.3767 1.3771 1.3776 1.3780 1.3784 1.3788 1.3792 1.3797 1.3301 1.3805 1.3809 1.3893 1.3897 1.3902 1.3906 1.3910 1.3914 1.4018 1.4022 1.4026 1.4031 1 4035 1.4039 O v 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 43 50 ~52~ 54 56 58 60 1.3015 1.3020 1.3024 1.3029 1.3033 1.3038 1.3012 1.3016 1.3051 1.3055 1.3147 1.3152 1.3156 1.3161 1.3165 1.3279 1.3283 1.3287 1.3292 1.3296 1.3538 1.35f2 1.3546 1.3550 1.3555 1.3559 1.3563 1.3567 1.3572 1.3576 1.3665 1.3670 1.3674 1.3078 1.3682 1.3687 1.3691 1.3695 1 .3(599 1.3704 1.3918 1.3922 1.3927 1.3931 1.3935 1 .4043 1.4047 1.4051 1.4055 1.4060 1.3169 1.3174 1.3178 1.3183 1.3187 1.3300 1.3305 1.3309 1.3313 1.3318 1.3813 1.3818 1.3822 1.3H26 1.3830 1.3834 1.3839 1.3843 1.3847 1.3851 1.3939 1.3943 1.3947 1.3952 1.3956 1.4064 1.4068 1.4072 1.4076 1 .4080 1.3060 1.3064 1.3068 1.3073 1.3077 1.3191 1.3196 1.3200 1.3204 1.3209 1.3322 1.3326 1.3331 1.3335 1.3339 1.3152 1.3456 1.3460 1.3465 1.3469 1.3580 1.35S5 1.3589 113593 1.3597 t.3708 1.3712 1.3716 1 .3721 1.3725 1.3960 1.3964 1.3968 1.3972 1.3977 1.4084 1.4089 1.4093 1.4097 1.4101 1.3086 1.3090 1.3035 1.3099 1.3218 1.3222 1.3226 1.3231 1.3348 1.3352 1.3357 1.3361 1.3365 1.3370 1.3374 1.3378 1.3383 1.3477 1.3482 1.3486 1.3490 1.3495 1.3499 1.3503 1.3508 1.3512 1.3606 1.3610 1.3614 1.3619 1.3623 1.3627 1.3631 1.3636 1.3640 1.3729 1.3733 1 .3738 1.3742 1.3746 1.3750 1.3764 1.3759 1.3763 1.3767 1.3855 1.3860 1.3864 1 .3868 1.3872 1 .3876 ] .3881 1.3885 1.3889 1.3893 1.3981 1.3985 1.3989 1.3993 1.3997 1.4105 1.4109 1.4113 1.4117 1.4122 1.3104 1.3103 1.3112 1.3117 1.3121 1 .3235 1 .3239 1.3244 1.3248 1.3252 1.4002 1.4006 1.4010 1 .4014 1.4018 1.4126 1.4130 1.4134 1.4138 1.4142 BLE OF LOGARITHMS. 613 Mims of Numbers, from O to 1OOO.* No. A - 2 3 4 5 6 7 8 Prop. 00000 30103 47712 60206 69897 77815 84510 90309 95424 10 00000 00432 00860 01283 01703 02118 02530 02938 03342 03742 415 11 04139 04532 04921 05307 05690 06069 06445 06818 07188 07554 379 12 07918 08278 08636 08990 09342 09691 10037 10380 10721 11059 349 13 11394 11727 12057 12385 12710 13033 13353 13672 13987 14301 323 14 14613 14921 15228 15533 15836 16136 16435 16731 17026 17318 300 15 17609 17897 18184 18469 18752 19033 19312 19590 19865 20139 281 16 20412 20682 20951 21218 21484 21748 22010 22271 22530 22788 264 17 23045 23299 23552 23804 24054 24303 24551 24797 25042 25285 249 18 25527 25767 26007 26245 26481 26717 26951 27184 27415 27646 236 19 27875 28103 28330 28555 28780 29003 29225 29446 29666 29885 223 20 30103 30319 30535 30749 30963 31175 31386 31597 31806 32014 212 21 32222 32428 32633 32838 33041 33243 33445 33646 33845 34044 202 22 34242 34439 34635 34830 35024 35218 35410 35602 35793 35983 194 23 36173 36361 36548 36735 36921 37106 37291 37474 37657 37839 185 24 38021 38201 38381 38560 38739 38916 39093 39269 39445 39619 17T 25 39794 39367 40140 40312 40483 40654 40824 40993 41162 41330 171 26 41497 41664 41830 41995 42160 42324 4248S 42651 42813 42975 164 27 43136 43206 43156 43616 43775 43933 44090 44248 44404 44560 158 28 44716 44870 45024 45178 45331 45484 45636 45788 45939 46089 153 29 46240 46389 46538 46686 46834 46982 47129 47275 47421 47567 148 30 47712 47856 48000 48144 48287 48430 48572 48713 48855 48995 143 31 49136 49276 49415 49554 49693 49831 49968 50105 50242 50379 138 32 50515 50650 50785 50920 51054 51188 51321 51454 51587 51719 134 33 51851 51982 52113 52244 52374 52504 52633 52703 52891 53020 130 34 53148 53275 53402 53529 53655 53781 53907 54033 54157 54282 126 35 54407 54530 54654 54777 54900 55022 55145 55266 55388 55509 122 36 55630 55750 55870 55990 56110 56229 56348 56466 56584 56702 119 37 56S20 569371 57054 57170 57287 57403 57518 57634 57749 57863 116 38 57978 58092 58206 58319 58433 58546 58658 58771 58883 58995 113 39 59106 59217 59328 59439 59549 59659 59769 59879 59988 60097 110 40 60206 60314 60422 60530 60638 60745 60852 60959 61066 61172 107 41 61278' 61384 61489 61595 61700 61804 61909 62013 62117 62221 104 42 62325 62428 62531 62634 62736 62838 H2941 63042 63144 63245 102 43 63347 63447 63548 63648 63749 63S48 63948 64048 64147 64246 99 44 64345 64443 64542 64640 64738 64836 64933 65030 65127 65224 98 45 65321 65417 65513 65609 65705 65801 65896 65991 660*6 66181 96 46 66276 66370 66464 66558 666ol 06745 66838 66931 67024 67117 94 47 67210 67302 67394 67486 67577 67669 67760 67851 67942 68033 92 48 68124 68214 68304 68394 68484 68574 6S663 68752 68842 68930 90 49 69020 69108 69196 69284 69372 69460 69548 69635 69722 69810 88 50 69897 69983 70070 70156 70243 70329 70415 70500 70586 70671 86 51 70757 70842 70927 71011 71096 71180 71265 71349 71433 71516 84 52 71600 71683 71767 71850 71933 72015 72098 72181 72263 72345 82 53 72428 72509 72591 72672 72754 72835 72916 72997 73078 73158 81 54 73239 73319 73399 73480 73559 73639 73719 73798 73878 73957 80 55 74036 74115 74193 74272 74351 74429 74507 74585 74663 74741 78 56 74818 74896 74973 75050 75127 75204 752S1 75358 75434 75511 77 57 75587 75663 75739 75815 J75891 75966 76042 76117 76192 76267 75 58 76342 76417 76492 76566 76641 76715 76789 76863 76937 77011 74 59 77085 77158 77232 77305 77378 77451 77524 77597 77670 77742 73 60 77815 77887 77959 78031 78103 78175 78247 78318 78390 78461 72 61 78533 78604 78675 78746 78816 78887 78958 79028 79098 79169 71 62 79239 79309 79379 79448 79518 79588 79657 79726 79796 79865 70 63 79934 80002 80071 80140 80208 80277 80345 80413 80482 80550 69 64 80618 80685 80753 80821 80888 80956 81023 81090 81157 81224 68 65 81291 81358 81424 81491 81557 81624 81690 81756 81822 81888 67 *Each log is supposed to have the decimal sign . before it. 614 TABLE OF LOGARITHMS. Logarithms of Numbers, from O to 10OO* (Continued.) So. f 1 2 3 4 5 6 7 8 9 Prop. 66 81954 82020 82085 82151 82216 82282 82347 82412 82477 82542 66 67 82607 82672 82736 82801 82866 82930 82994 83058 83123 83187 65 68 83250 83314 8337S 83442 83505 83569 83632 83695 83758 83821 64 69 83884 83947 84010 84073 84136 84198 84260 84323 84385 84447 63 70 84509 84571 84633 84695 84757 84818 84880 84941 85003 850ti4 62 71 85125 85187 85248 85309 85369 85430 85491 85551 85612 85672 61 72 85733 85793 85853 85913 1 8597 3 86033 86093 86153 86213 86272 60 73 86332 86391 86451 86510 86569 86628 86687 86746 86805 86864 59 74 86923 86981 87040 87098 87157 87215 87273 87332 87390 87448 58 75 87506 87564 87621 87679 87737 87794 87852 87909 87966 88024 57 76 88081 88138 88195 88252 88309 88366 88422 88479 88536 88592 56 77 88649 88705 88761 88818 88874 88930 88986 89042 89098 89153 56 78 89209 89265 89320 89376 89431 89487 89542 89597 89652 89707 55 79 89762 89817 89872 89927 89982 90036 90091 90145 90200 90254 54 80 90309 90363 90417 90471 90525 90579 90633 90687 90741 90794 54 81 90848 90902 90955 91009 91062 91115 91169 91222 91275 91328 53 82 91381 91434 91487 91540 91592 91645 91698 91750 91803 91855 53 83 91907 91960 92012 92064 92116 92168 92220 92272 92324 92376 52 84 92427 92479 92531 92582 92634 92685 92737 92788 92839 92890 51 85 92941 92993 93044 93095 93146 93196 93247 93298 93348 93399 51 86 93449 93500 93550 93601 93651 93701 93751 93802 93852 93902 50 87 93951 94001 94051 94101 94151 94200 94250 94300 94349 94398 49 88 94448 94497 94546 94596 94d45 94694 94743 94792 94841 94890 49 89 94939 94987 95036 95085 95133 95182 95230 95279 95327 95376 48 90 95424 95472 95520 95568 95616 95664 95712 95760 9580S 95856 48 91 95904 95951 95999 96047 96094 96142 96189 96236 96284 96331 48 92 96378 96426 96473 96520 96507 96614 96661 96708 96754 96801 47 93 96848 96895 96941 96988 97034 97081 97127 97174 97220 97266 47 94 97312 97359 97405 97451 97497 97543 97589 97635 97680 97726 46 95 97772 97818 97863 97909 97954 98000 98045 98091 98136 98181 46 96 98227 98272 98317 98362 98407 98452 98497 98542 98587 98632 45 97 98677 98721 98766 98811 98855 98900 98945 98989 99033 99078 45 98 99122 99166 99211 99255 99299 99343 99387 99431 99475 99519 44 99 99563 99607 99651 99694 99738 99782 99825 99869 99913 99956 44 * Each log is supposed to have the decimal sign . before it. The log of 2870 is 3.45788 " " " 287 is 2.45788 " " " 28.7 is 1.45788 " " " 2.87 is 0.45788 The log of .287 is 1.45788 " " " .028 is 2.44716 " " " .002 is 3.30103 " .0002 is 4.30103 What is the log of 2873 ? Here, log of 2870 = 3.45788 And prop 153 X 3 = 459 3.458339 To find roots divide the log (with its index) of the given number, by that number which expresses the kind of root. The quotient will be the log of the required root. Example. What is the cube root of 2870? Here, the log of 2870, with its index, is 3.45788. And '- = 1.15263. Hence the cube root is 14.1 The Hyperbolic, or Napierian logarithm is the common log of the table multiplied by 2.3025851. GLOSSARY OF TERMS. 615 GLOSSAEY OF TEEMS. Abacus ; the flat square member on top of a column. Absciss or abscissa ; any portion of the axis of a curve, from the vertex to any point from which a Hue leaves the axis at right angles, and extends to meet the curve itself; said line being called an. ordinate. An absciss and ordinate together are called co-ordinates. Acclivity ; an upward slope, or ascent of ground, &c. Adit ; a Horizontal passage into a mine, &c. Adze; a well-known curved cutting instrument, for dressing or chipping horizontal surfaces. Air-vessel. Motion is imparted to the water in a line of pipes, by the forward stroke of the piston of a single-acting pump; but during the backward stroke, this motion is stopped; and the water in the pipes comes to rest. Therefore, at the next forward stroke, all the water has to be again set in motion ; and the force that must be exerted by the pump to do this is much greater than would be required if the motion previously imparted had been maintained during the time of the backstroke. The addition of an air-vessel secures this maintenance of motion, and thus effects a great saving of power; besides diminishing the danger of bursting the pipes at each forward stroke. It is merely a tall and strong air-tight iron box, usually cylindrical, strongly bolted on top of the pipes just beyond the pump, and communicating freelv with them through an opening in its base. It is full of air. The forward stroke of the piston then forces water not only along the pipes, but also into the lower part of the air-vessel, through the opening in its base ; thus compress. ng its contained air. But during the backstroke, this compressed air, being relieved from the pressure of the pump, expands; and in so doing presses upon the water in the pipes, and thus keeps it in motion until the next for- ward stroke; and so on. An air-vessel also acts as an air-cusliion ; permitting the piston to apply its force to the water in the pipes gradually ; thus preserving both the pipes and the pump from vio- lent shocks. The air in the vessel, however, becomes by degrees absorbed and taken away by the water; and its action as a regulator then ceases. To prevent this, fresh air must be forced into the vessel from time to time by a condenser, or forcing air-pump. A double-acting pump does not so much need an air-vessel. There is no particular rule for the size or capacity of air-vessels. In prac- tice it appears to vary from about 5 to 50 times that of the pump ; with a height equal to two or more times the diam. A stand-pipe, which see, is sometimes used instead of an air-vessel. Alternating motion; up and down, or backward and forward, instead of revolving, &c. Angle-bead, or plaster bead; a bead nailed to projecting angles in rooms, to protect the plaster on their edges from injury. Angle-block; a triangular block against which the ends of the braces and counters abut in a Howe bridge. Angular velocity. See footnote to Art 6 of Force in Rigid Bodies. Page 447. Anneal; to toughen some of the metals, glass, Ac, by first heating them, and then causing them to cool very slowly. This process however lessens the tensile strength. Anticlinal axis ; in geology ; a line from which the strata of rocks slope away downward in oppo- site directions, like the slates on the roof of a house ; the ridge of the roof representing the axis. Apron; a covering of timber, stone, or metal, to protect a surface against the action of water flow- ing over it. Has many other meanings. Arbor. See Journal. Architrave ; that part of an entablature which is next above the columns. Applies also when there are no columns. Also, the mouldings around the sides and tops of doors and windows, attached to either the inner or outer face of the wall. Arris ; a sharp edge formed by any two surfaces which meet at an angle. The edges of a brick are arrises. Ashler ; a facing of cut stone, applied to a backing of rubble or rough masonry, or brickwork, Astragal; a small moulding, about semi-circular or semi-elliptic, and either plain or ornamented by carving. Axis ; an imaginary line passing through a body, which may be supposed to revolve around it; as the diam of a sphere. Any piece that passes through and supports a body which revolves ; in which case it is called an axle, or shaft. Axle-box. See Journal-box. Axletree; an axle which remains fixed while the wheel revolves around it, as in wagons, &c. Azimuth. The azimuth of a body is that arc of the horizon that is included between the meridian circle at the given place, and another great circle passing through the body. Backing ; the rough masonry of a wall faced with finer work. Earth deposited behind a retainlng- wall, &c. Balance-beams; the long top beams of lock-gates, by which they are pushed open or shut s Balk ; a large beam of timber. Baliast; broken stone, sand or gravel, &c, on which railroad cross-ties are laid. Ball-cock; a cistern valve at one end of a lever, at the other end of which is a floating ball. Th ball rises and falls with the water in the cistern ; and thus opens or shuts the valve. Ball-value. See Valve. Bargeboards ; boards nailed against the outer face of a wall, along the slopes of a gable end of a house, to hide the rafters, &c ; and to make a neat finish. Batter, (sometimes affectedly batir.) or talus ; the sloping backward of a face of masonry. Bay ; on bridges, &c, sometimes a panel ; sometimes a span. Bead ; an ornamenfeither composed of a straight cylindrical rod ; or carved or cast in that shape on any surface. Bearing ; the course by a compass. The span or length in the clear between the points of support of a beam, &e. The points of support themselves of a beam, shaft, axle, pivot, &c. Bed- moulding ; ornamental mouldings on the lower face of a projecting cornice, &c. Bed-plate ; a large plate of iron laid as a foundation for something to rest on. Beetle ; a heavy wooden rammer, such as pavers use. Bell-crank. See Crank. Bench-mark; a level mark cut at the foot of a tree for future reference, as being more permanent than a stake. Herm,, or berme; a horizontal surface, as if for a pathway, and forming a kind of step along the face 616 GLOSSARY OF TEEMS. of sloping ground. In canals, the level top of the embankment opposite and corresponding to the towpath is called the berm. Bessemer steel is formed by forcing air into a mass of melted cast iron ; by which means the excess of carbon in the iron is separated from it, until only enough remains to constitute cast steel. The carbon is chemically united with the steel, but mechanically with the iron. Beton; concrete of hydraulic cement, with broken stone and bricks, gravel, &c. Bevel; the slope formed by trimming away a sharp edge, as of a board, &c. Edges of common drawing rulers aud scales are usually bevelled. See 13, Figs 42, of Trusses, p 294. Bevel gear; cog-wheels witn teeth so formed that the wheels can work into each other at an angle. Bilge; the nearly flat part of the bottom of a ship on each side of the keel. Also, the swelled part of a barrel, &c. To bilge is to spring a leak in the bilge, or to be broken there. Bitts ; the small boring points used with a brace. Blast-pipes; in a locomotive; those through which the waste steam passes from the cylinder into the smoke-pipe, and thus creates an artificial draft in the chimney, or smoke-pipe. Boasting ; dressing stone with a broad chisel called a boaster, aud mallet. The boaster gives a gmoother surface after the use of the point, or the narrow chisel called a tool. Bond; the disposing of the blocks of stone or brickwork so as to form the whole into a firm struc- ture, by a judicious overlapping of each other, so as to break joint. Applies also to timber, &c, in various ways. Bonnet; a cap over the end of a pipe, &c. A cast-iron plate bolted down as a covering over an aperture. Bore ; inner diameter of a hollow cylinder. Boss ; an increase of the diameter at any part of a shaft for any purpose. A projection in shape of a segment of a sphere, or somewhat so, whether for use or for ornament; often carved, or cast. Box-drain; & square or rectangular drain of masonry or timber, under a railroad, &c. Brace ; a kind of curved handle used for boring holes with bitts. The head of the brace remains stationary, being pressed against by the body of the person using it, while the other part with the bitt is turned round by his haud. Also, an inclined beam, bar, or strut, for sustaining compression. Bracket; a projecting piece of board, &c, frequently triangular, the vertical leg attached to the face of a wall, and the horizontal one supporting a shelf, &c. Often made in ornamental shapes for supporting busts, clocks, &c. Also, the supports for shafting ; as pendent, wall, and pedestal brackets. Brake; an arrangement for preventing or diminishing motion by means of friction. The friction is usually applied at the circumference of a revolving wheel, by means of levers. On railroads, the car-brakes should be worked by steam, as those of Loughridge, Westiughouse, and Creamer. Also, such a handle as that of a common pump. Brass is composed of copper and zinc. Brasses; fittings of brass in many plummer-blocks, and in other positions, for diminishing the friction of revolving journals which rest upon them. Braze; to unite pieces of iron, copper, or brass, by means of a hard solder, called spelter solder, and composed, like brass, of copper aud zinc, but in other proportions. Break joint; to so overlap pieces that the joints shall not occur at the same place, and thus pro- duce a bad bond. Breast-summer; a beam of wood, iron, or stone, supporting a wall over a door or other opening; a kind of lintel. Breast-wall; one built to prevent the falling of a vertical face cut into the natural soil; in dis- tinction to a retaining-wall or revetment, which is built to sustain earth deposited behind it. Breech; the hind part of a cannon, &c. Bridge,, or Iridge-piece, or bridge bar ; a narrow strip placed across an opening, for supporting something without closing too much of the opening. Bronze is composed of copper and tin. Bulkhead ; on ships, &c, the timber partitions across them. Also, a long face of wharf parallel Buoy ; a floating body, fastened by a chain or rope to some sunk body, as a guide for finding the latter, 'sometimes also used to indicate channels, shoals, rocks, &c. Burnish ; to polish by rubbing; chiefly applies to metals. Bush to line a circular hole by a ring of metal, to prevent the hole from wearing larger. Also, when a piece is cut out, and another piece neatly inserted into the cavity, the last piece is sometimes said to be bushed in ; sometimes it is called a plug. Buttress ; a vertical projecting piece of brickwork or masonry, built in tront or a wall to * cSLmVa large wooden box with sides that may be detached and floated away. Caliber; the inner diameter, or bore. Calipers compasses or dividers with curved legs, for measuring outside diameters. Calk, or caulk; to fill seams or joints with something to prevent leaking. Calking iron ; a tool for forcing calking into a joint. Camb. or cam, or wiper; a piece fixed upon a revolving shaft in such a manner as to produce an alternating or reciprocation motion in something in contact with the cam. An eccentric. Camber a slight upward curve given to a beam or truss, to allow for settling. Camel; a kind of barges or hollow floating vessels, which, when filled whir water, are fastened to the sides of a ship ; and the water being then pumped out, they rise by their buoyancy ; and lift the ship so that she can float in shallower water. Cantilevers ; projecting pieces for supporting an upper balcony, &c Cants, rims, or shroudings; the pieces forming the ends of the buckets of water-wheels, to prevent t ^nns^n "ing 'hollow rope^drum surrounding a strong vertical pivot, upon the head of which it rests and around which it turns. Its top is a thick projecting circular piece, having holes around its outer edge or circumference, for the insertion of the ends of levers ; or capstan-bars. It is a kind oi Ve eL-artaftrto convert the outer surface of wrought iron into steel, by heating it while in contact w'th charcoal. Casemate; in fortification ; the small apartment m which a cannon stands. Castors ; rollers usually combined with swivels; as those used under heavy furniture, &c. Causeway ; a raised footway or roadway. Cavetto ; "a moulding consisting of a receding quadrant of a circle. Cementation the process of converting wrought iron into steel, by heating it in contact with cnar- 'GLOSSARY OF TERMS. 617 eoal. This process produces blisters on the steel bars ; hence blister steel. These are removed, and the steel compacted, by reheatiug it, aud then subjecting it to a tilt-hamiuer. It is then tilted steel, or shear aM&l. Or if the blister steel is broken up; remelted; and then run into ingots or blocks; it is called cast, or ingot steel; which is harder aud closer-grained than tilted steel. It may be softened, and thus become less brittle, by annealing. The ingots may be converted into bars by either rolling or hammering, the same as shear and blister. Center ; the supports of an arch while being built. Canter of gravity. See Art 56, of Force, p 481. Also see Cen of Grav, p 442. Center of gyration. Suppose a body free to revolve around an axis whicu passes through it in any direction; or to oscillate like a pendulum hung from a point of suspension. Then suppose in either case, a certain given amount of force to be applied to the body, at a certain given dist from the axis, or from the point of suspension, so as to impart to the body an angular vel; or in other words, to came it to describe a number of degrees per sec. Now, there will be a certain point in the body, such that if the entire wt of the body were there concentrated, then the same force as before, applied at the same dist from the axis, or from the point of su.speusion as before, would impart to the body the .same angular motion as before. This point is the center of gyration ; and its dist from the axis, or from the point of su.spension, is the Radius of gyration, of the body. To find the position of this center, or the length of this rad, see Moment of Iuertia,,iu this glossary. One use of the center of gyration is to enable us to calculate the momentum or moving force in a revolving or oscillating body. This force in foot pounds is found by mult the wt of the body in tts, by the vert height in ft through which it would have to fall, in order to acquire the vel which its center of gyration has. This height may be found in Table 10, p 552, of Hydraulics. Also p 495. Center of oscillation, or of vibration. See Rem 2, of Pendulums, p 173. Center of percussion, in a moving body, is that point which would strike an opposing body with greater force than any other point would. If the opposing body is immovable, it will receive all the force of a rigid moving body which strikes with its center of percussion. See Pendulum, page 173. Cesspool; & shallow well for receiving waste water, filth, &c. Chamfer; means much the same as bevel ; but applies more especially when two edges are cut away 30 as to form either a chamfer-groove, (see 14, p 294, of Trusses,) or a projecting sharp edge. Cheeks ; two flat parallel pieces confining something between tnem. See w, at 15, of Figs 21^, of Trusses, p 265. Chilling, chill- hardening, or chill -cast ing ; giving great hardness to the outside of cast-iron, by pouring it into a mould made of iron instead of wood. The iron mould causes the outside or skin of the casting to cool very rapidly ; and this for some unknown reason increases its hardness. This pro- cess is frequently confounded with case-hardening. Chock ; any piece used for filling up a chance hole, or vacancy. Chuck; the arrangement attached to the revolving shaft, arbor, or mandril of a lathe, for holding the thing to be turned. Churn-drill; a long iron bar. with a cutting end of steel ; much used in quarrying, and worked by raising it aud letting it fall When worked by blows of a hammer or sledge it is called a jumper. Cima, or cymn ; a moulding nearly in shape of an S. When the upper part is concave, it is called a cirna recta ; when convex, a cima reversa. See page 67. Clack valve. See Valve. Clamp; a piece fastened by tongue and groove, transversely along the end of others, to keep them from warping. A kind of open collar, which, being closed by a clam-screw, holds tight what it sur rounds. See Cramp. Clap-boards; short thin boards, shingle-shaped, and used instead of shingles. Claw ; a split provided at the end of an iron bar, or of a hammer, &c, to take hold of the heads ot nails or spikes for drawing them out ; as in a common claw-hammer. Cleat ; a piece merely bolted to another to serve as a support for something else; as at 7, 8. 9, 10, &c, p 294, of Trusses. Often used on shipboard for fastening ropes to, as at 11. Also a piece of board uailed across two or more other boards, for holding them together, as is often done in temporary doors, &c. Clewi*. See Shackle. Click. See Ratchet. Clip ; a fastening like that on the tops of the Y's of a spirit level ; being a kind of half collar opening by a hinge. Clutch ; applied to various arrangements at the ends of separate shafts, and which by clutching or catching into each other cause both shafts to revolve together. A kind of coupling. Cock; a kind of valve for the discharge of liquids, air, steam, &c. Coefficient; or a Constant of friction, safety, or strength, Ac, may usually be taken to be a num- ber which shows the proportion (or rather the ratio) which friction, "safety, tensile strength, &c, bear to a certain something else which is not generally expressed at the time, but is well understood. Thus, when we say that the coeff of friction of one body upon another is y 1 ^-, Ac, it is understood that the friction is in the proportion of y^-th of the pressure which produces it. A coeff of safety of 3, ni^ans that the safety has a proportion or ratio of 3 to I to the theoretical breaking loud. A coeff of 500 fts, or of '20 tons, Ac, of tensile strength of any material, denotes that said strength is in the proportion of 500 KM. or of 20 tous. &c, to each square inch of transverse section. 6 times their diam. In lines of shafting 4 diams. To find the diam, see Gudgeon. Jumper ; a drill used for boring holes in stone by aid of blows of a sledge-hammer. Kedge ; a small anchor. Keepers ; the pieces of metal or wood which keep a sliding bolt In its place, and guide it in sliding. Kerf; the opening or narrow slit made in sawing. Key-bolt. See Cotter-bolt. Keystone ; the center stone of an arch. Kibble; the bucket used for raising earth, stone, &c, from shafts or mines. King-post, king-rod ; the center post, vertical piece, or rod, in a truss ; all those on each side of it are queen-posts, or queen-rods. Frequently called simply kings and queens. Knee ; a piece of metal or wood bent at an angle ; to serve as a bracket, or as a means of uniting two surfaces which form with each other a similar angle. Lagging, or sheeting ; a covering of loose plank ; as that placed upon centers, and supporting the archstones. Also, an outer wooden casing to locomotive boilers and others. Landing ; the resting-place at the end of a flight of stairs. Lantern wheel. See Trundle. Lap; to place one piece upon another, with the edge of one reaching beyond that of the other. Lap-welding; welding together pieces that have first been lapped; in distinction to butt- welding. Lead, (pronounced leed ;) in locomotives, a certain amount of opening of the port-valve before each stroke of the piston begins. The distance to which earth is hauled or wheeled. Leading-beam; leading-pili; one placed as a guide for placing others. Leading-wheels; iu a locomotive, those frequently placed in front of the driving-wheels. Leaves ; the cogs of pinions. Ledge; a part projecting over like a shelf; a rock so projecting. A narrow strip of board nailed across other boards, to hold them together, as in temporary ledge-doors. Lewis; an arrangement composed of 2 or 3 pieces of metal let into a wedge-shaped hole in a block of stone, by which to raise the block. Lighter ; a scow, raft, or other vessel, used for unloading vessels out from the shore, Linch-pin; a pin near the end of an axle, to hold the wheel on. Link ; one of the divisions of a chain : or a piece shaped like one. Link-motion; a device for regulating the movement of the main or port valve in a locomotive. Lintel; a horizontal beam across an opening in a wall, as seen in windows, doors, &c. When of wide span, and supporting heavy brickwork or masonry, it is called a breast-summer, or bressummer. Lock; those common door locks which are entirely concealed within the thickness of the door, are called mortice locks ; those which are screwed against the face of a door, rim locks. It must be remem- bered that locks are " right and left." Louvre ; a kind of vertical window, frequently at the tops of roofs of depots, &c, provided with hor- izontal slats, which permit ventilation, and exclude rain. Lozenqe; the shape of a rhomb ; often called diamond-shaped. Luq in casting, small projections from the general surface, and for various purposes, such as for lifting the body ; or for a flange for joining it to another ; or for a support for something else. Mallet ; the wooden, hammer used by stonecutters. 40 622 GLOSSARY OP TERMS. Mandrel ; an iron rod used as a core around which a flat piece may be bent into a cylindrical shape Also the shaft that carries the chuck of a lathe. Manhole; an opening by which a man can enter a boiler, culvert, c. to clean or repair it. Mattock; a kind of pick with broad edges for digging Maul; a heavy wooden hammer. Mean, arithmetical; half the sum of two numbers. ' , geometrical; the sq rt of the product of two numbers. Mean-proportional ; the same as the geometrical mean. Meridian ; a north and south line. Noon. Mitre-joint; a joint formed along the diagonal line where the ends of two pieces are united at an angle with each other. Mitre-sill; the sill against which the lock gates of a canal shut. Modulus ; a datum serving as a means of comparison. Same as constant or coefficient. MoUulus of elasticity ; see pp 177 :iud 632. Modulus of Rupture pp 185, 195. Moment ; tendency of force acting with leverage. See p 475. Moment of inertia, of a body either revolving (or imagined to revolve) around an axis, as a grind, tone; or oscillating, like a pendulum. Suppose that the shortest dist from the axis (or from tne point of suspension, as the case may be,) to each single individual particle of the body, has been meas- ured; also that each of these dists has been squared: and that each square has been mult by the weight of that particle to which the dist was measured; also that all these last products are added into one sum. That sum is the moment of inertia of the body; and is the I of scientific writers. In practice we may suppose the body to be divided into portions of a cub inch, or some other size ; and use these instead of the theoretical infinitely small particles. When the moment of inertia of a mere surface is wanted, (instead of that of a solid,) the surf must be supposed to be div into an infinite number of small areas, which must be used instead of the weights of the particles of the solid body. For an example, see page 195, Art 25, of Strength of Materials. If the moment of inertia is div by the wt of the entire solid body ; or by the area of the entire surf, as the case may be ; the square root of the quot will be the radius of gyration, of the body or surf. If the length of the rad of gyration be laid off perp from the axis, or from the point of suspension, it will reach to the center of gyration. See Cen of Gyration, in this Glossary. The moment of inertia of a body is ~ its wt X sq of rad of gyr. Moment of rupture, or of bending : the tendency which any load or force exerts to break or bend & body by the aid of leverage. Its amount is found in foot-pounds by multiplying the force in fts, by the length of leverage in feet between it and that part of the body upon which the tendency is exerted. Moment of stability. See Art 69, of Force in Rigid Bodies, page 489. Momentum; moving force. See page 4W. Force in Rigid Bodies. Monkey ; the hammer or ram of a pile-driver. Monkey-wrench, or screw-wrench; a spanner, the gripping end of which can be adjusted by means f a screw to fit objects of different sizes. Moorings; fixtures to which ships, &c, can make fast. Mortise; a hole cut in one piece, for receiving the tenon which projects from another piece. Muck; soft surface soil containing much vegetable matter. Muntins, or mullions ; the vertical pieces which separate the panes in a window-sash. Nailing-blocks ; blocks of wood inserted in walls of stone or brick, for nailing washboards, &c, to. Nave; the main body of a building, having connecting wings or aisles on each side of it. The hub of a wheel. Neioel; the open space surrounded by a stairway. Newel-post; a vertical post sometimes used for 'sustaining the outer ends of steps. Also the large baluster often placed at the foot of a stairway. Nippers ; pincers. An arrangement of two curved arms for catching hold of anything. Normal; perpendicular to. According to rule, or to correct principles. Nosing ; the slight projection often given to the front edge of the tread ot a step ; usually rounded. Nut, or burr ; the short piece with a central female screw, used on the end of a screw-bolt, &c, for keeping it in place. Ogee; a moulding in shape of an S. the same as a cima. Ordinate ; a line drawn at right angles from the axis of a curve, and extending to the curve. Oscillate ; to swing backward and forward like a pendulum. Out of wind, pronounced wynd ; perfectly straight or flat. Ovolo ; a projecting convex moulding of quarter of a circle ; when it is concave it is a cavetto, or hollow. Packing ; the material placed in a stuffing-box, &c, to prevent leaks. Packing-pieces ; short pieces inserted between two others which are to be riveted or bolted together, to prevent their coming in contact with each other. Pall, or pawl. See Ratchet. Parapet; a wall or any kind of fence or railing to prevent persons from falling off. Parcel; to wrap canvas or rags round a ropp. Parge ; to make the inside of a flue smooth by plastering it. Patent hammer ; a hammer with several parallel sharp edges for dressing stone. Pay. To cover a surface with tar, pitch, &c. A ship word. Pay out. To slacken, or let out rope. Pediment; the triangular space in the face of a wall that is included between the two sloping sides f the roof and a line joining the eaves. Penstock. See Forebay. Pier; the support of two adjacent arches. The wall space between windows, &c. A structure built ut into the water. Pierre perdue ; lost stone ; random stone, or rough stones thrown into the water, and let find their own slope. Pilaster ; a thin flat projection from the' face of a wall, as a kind of ornamental substitute for a column. Pile-planks; planks driven like piles. Pillow-block, or plummer- block; a kind of metal chair or support, upon which the journals of hor- izontal shafts are generally made to rest, and on which they revolve. Pinion; a small cog wheel which gives motion to a larger one. Pintle ; a vertical projecting pin like that often placed at the tops of crane-posts, and over which the holding rings at the tops of the wooden guys fit. Also, such as is used for the hinges of rudders or of window-shutters to turo around. 5ARY OF TERMS. 623 Pitch ; the slope of a roof^/. The distance from center to center of the teeth of a cog-wheel, or the threads of a screw^-'BoilqA tar. Also the dist apart of rivets, &c. Pitman; a conjaCung-rod /or transmitting motion from a prime mover to machinery at a distance, moved by Mu^ Pit-s^w ; a large saw worked vertically by two men, one of whom (the pitman) stands in a pit. Pivot; the lower end of a vertical revolving shaft, whether a part of the shaft hsr.lf, or attached to it. It should be tlat; and both ic aud the step or socket upon which it rests should be of hard steel. If a steel pivot has to revolve rapidly aud continuously, it is well to proportion its diaui, so as not to have to sustain more than 250 fts per sq inch; otherwise it will wear quicklv. Dust and grit should for the same reason be carefully guarded against. Pivots which revolve but seldom, and slowly, as those of a railroad turntable, may be trusted with half a ton. or even a whole ton per sq inch. As a rude rule, cast-iron pivots should not be loaded with more than half as much as steel ones. A steel one may be welded to the foot of its cast-iron shaft; or may be inserted part way into it; and the whole strengthened by iron bands shrunk on. Planish; to polish metals by rubbing with a hard smooth tool. Plant; the outfit of machinery, &c, necessary for carrying on any kind of work. Plaster-bead; a small vertical strip of iron or wood nailed along projecting angles in rooms, to protect the plaster at those parts. Platband; a plain, flat, wide, slightly projecting strip, generally for ornament. When narrow, It is called a fillet. Pliers ; a kind of pincers. Plinth ; the square lowest member of the base of a column or pillar. Plug ; a piece inserted to stop a hole. Screw-plug, a plug that is screwed into a hole. Plumb ; vertical. Plummet, or plumb-bob ; a weight at the lower end of a string, for testing verticality. Plunger ; a kind of solid piston, or one without a valve. Point; a kind of pointed chisel for dressing stone. To put a finish to masonry by touching up the outer mortar joints. To dress stone with a point and mallet. Pole-plate ; a longitudinal timber resting on the ends of tie-beams of roofs ; and for supporting the feet of the common or jack rafters, when such are used. Port; the opening or passage controlled by a valve. Prime ; to put on the first coat of paint. Priming also is when water passes into a steam cylinder along with the steam. Projection. If parallel straight lines be imagined to be drawn in any one given direction, from every point in any surface s. whether flat, curved, or irregular, then if all these lines be supposed to be intersected or cut by a plane, either at right angles to their direction, or obliquely, the figure which their cross-section thus made would form upon said plane, is called the projection of the sur- face s. If such lines be supposed to be drawn from a person's face, in a direction in front of him, and to be cut by a plane at right angles to their direction, their projection on the plane would be the person's full-face portrait. If the lines be drawn sideways from his face, the projection will be his profile. The projection of a globe upon a flat plane, will evidently be a circle if the plane cuts the lines at right angles ; and an ellipse if it cuts them obliquely. Shadows cast by the sun are projec- tions. Proportion. See Ratio. Puddle; earth well rammed into a trench, &c, to prevent leaking. A process for converting cast iron into wrought by a puddling furnace. rufi-mill ; a mill for tempering clay for bricks or pottery, &c. Pulley ; a circular hoop which carries a belt in machinery. Puppet; in machinery, a small short pedestal or stand. Puppet-valve. See Valves. Purlins; the horizontal pieces placed on rafters, for supporting the roof covering. Piit-loys, or put-lockft ; horizontal pieces supporting the floor of a scaffold; one end being inserted into put-log hole* left for that purpose in the masonry. Quay; a wharf. Quoin; the hollow into which a quoin-post of a canal lock-gate fits. Stones, usually dressed, placed along the vertical angles of buildings, chiefly for ornament. Quoin-pout ; the vertical post on which a lock-gate turns. The heel-pflst. Rabbet, or rebate ; a half groove along the edge of a board, &c. See 16, p 294, of Trusses, where two rabbets are shown overlapping each other. Race; the channel whioh conducts water either to or from n water-wheel : the first is a head race ; the last a tail race. The waves produced by the meeting of strong opposing currents ; also, a rapid tidewav, or roost. Rtirk and pinion ; the rack is a straight row of cogs on a bar, and called a rack-bar ; and the pinion is a small cog-wheel working into u. /moment of inertia Radius of gyration. See Center of gyration. Kad of gyr is ~ / Rag-bolt. See Jag-spike. N weight of body Raft-wheel ; one with teeth or pins which catch into the links of a chain. Rails ; the horizontal pieces in a door. Ram ; the hammer of a pile-driver. Random stone ; rip rap, or rough stones thrown promiscuously into the water, to form a founda- tion, Ac. Rasp ; a coarse file. Ratchet and pall ; the former is sometimes a straight bar, at others a wheel : in either case it ii furnished with teeth between which the pall drops aud prevents backward motion. Used for safety in hoisting machinery, &c. The pall is sometimes called a click. Ratio. Simple ratio is a number denoting how often one quantity is contained in another. Thus, the ratio of 5 to 10 is J>^, or i ; and the ratio of 10 to 5 is \P , or 2. When, of four numbers, two have to each other the same ratio that the other two have, the numbers are said to be in proportion to each other. Thus, 6 has the same ratio (2) to 3. as 100 has to 50 ; therefore, 6, 3, 100, and 50, are said to be in proportion ; or, as 6 : 3 : : 100 : 50. In other words, an equality of ratios ia called pro- portion. Ratio and proportion are often confounded with one -another; but the error is one of no importance. Duplicate ratio is that of the squares of numbers. Ream. A hole, wider at top than at bottom, (see 19. p '294, of Trusses,) through which a screw, bolt, the cylinders Steam-pipe ; the one which leads steam from a boiler to the steam-chest. Step ; a cavity in a piece for receiving the pivot of an upright shaft ; or the end of any upright piece. Stiles ; the flat vertical pieces between and at the sides of the panels in doors. &c. Stork; the eye with handles for turning it, in which the dies for the cutting of screws are held. Stone.-itp, or stoved. or upset; when a rod of iron is heated at one end. and then hammered end- wise so that that part becomes of greater diameter or stouter than the remainder. The heads of bolts are frequently made in one piece with the shank in this way ; and the screw ends of long screw-rods are often upset, so that the cutting of the threads of the screw mny not reduce the strength of the bar. Strap; a long thin narrow piece of metal bolted to two bodies' to hold them together. A strap- hinge is a strap fastened to a shutter. &c, and having an eye or a pin at one end for fitting it to the jther part of the hinge which is attached to the wall. Stratum ; a layer, or bed ; as the natural ones in rocks, &c. Stretcher ; a brick, or a block of masonry laid lengthwise of a wall. A frame for stretching any )hing upon. Stretcher-course ; a course of masonry all of stretchers, without any headers. Strike; an imaginary horizontal line drawn upon the inclined face of a stratum of rocks. Thus, If the slates or shingles on a roof represent inclined strata of rocks, then either the ridge or the eaves jf the roof, or any horizoutal line between them, will represent their strike. The inclination ia walled the dip of the strata; and the strike is always at right angles to it by compass. String ; variously applied to longitudinal pieces. String board; the boarding (often ornamented) at the outer ends of steps in staircases. It hides the horses, as the inclined timbers which carry the steps are called. String-course; a long horizontal course of brick or masonry projecting a little beyond the others ; and often introduced for ornament. 626 GLOSSARY OF TERMS. Strut ; a prop. A piece that sustains compression, whether vertical or inclined. Strut-tie, or tie-strut ; a piece adapted to sustain both tension and compression. Stub-end; a blunt end. Stud ; a short stout projecting pin. A prop. The vertical pieces in a stud partition. Stud-bolt. See Standing-bolt. Stuffing-box; a small boxing on the end of a steam cylinder, and surrounding the piston-rod like a collar ; or in other positions where a rod is required to move backward and forward, or to revolve, In an opening through any kind of partition, without allowing the escape of steam, air, or water, &c, as the case may be. The box is filled with greased hemp or other packing, which is kept pressed close around the moving rod by means of a top-piece or kind of cover called the gland, which may be screwed down more or less tightly upon it at pleasure. The rod passes through the gland also. Se Figs 40 of Hydraulics, where b b is the gland of the stuffing-box just below it between taud t ; and tha small circle shows the cavity for the packing or stuffing. Sumpt, or sump; a draining well into which rain or other water may be led by little ditches from different parts of a work to which it would do injury. Surbase; the inside horizontal mouldings just under a window-sill. Also those around the top of a pedestal, or of wainscoting, &c. Swage, or swedge; a kind of hammer, on the face of which is a semi-cylindrical, or other shaped S:oove or indentation ; and which, beiug held upon a piece of hot iron and struck by a heavy hammer, ave.s the shape of the indentation upon the iron. Switch, ; the movable tongue or rail by which a train is directed from one track to another. Swivels; devices for permitting one piece to turn readily in various directions upon another, with- out danger of entanglement or separation. At 13, p 265, of Trusses, is a tightening swivel; the castors under the legs of heavy furniture are swivelled rollers. Synclinal axis ; in geology, a valley axis, or one toward which the strata of rocks slope downward from opposite directions. The line of the gutter in a valley roof may represent such an axis. Ts ; pieces of metal in that shape, whether to serve aa straps, or for other purposes. So also with L's. S's, Ws, -f-'s, &c. See figs to Welded Iron Tubes, p 365. Tackle; a combination of ropes and pulleys. Talus ; the same as batter. Tamp; to fill up with sand or earth, &c, the remainder of the hole in which the powder has been poured for blasting rock. Tap; a kind of screw made of hard steel, and having a square head which may be grasped by a wrench for turning it around, and thus forcing it through a hole around the inside of which it cuts an interior screw. To strike with moderate force. To make an opening in the side of any vessel. Tappet; a pin or short arm projecting from a revolving shaft; or from an alternating bar, and in- tended to come into contact with, or tap, something at each revolution or stroke. Teeth ; or cogs of wheels. Temper ; to change the hardness of metals by first heating, and then plunging them into water, oil, &c. To mix mortar, or to prepare clay for bricks, &c. Templet; the outline of a moulding or other article, cut out of sheet metal or thin wood, to serve as a pattern for stonecutters, carpenters, &c. Tenon; a projecting tongue fitting into a corresponding cavity called a mortise. Terracotta; baked clay. Brick is a coarse kind. Thimble ; an iron ring with its outer face curved into a continuous groove. A rope being doubled around this and tied, the thimble acts as an eye for it, and prevents that part of the rope from wear- ing. Also, a short piece of tube slid over another piece, or over a rod, &c, to strengthen a joint, &c. Thread ; the continuous spiral projection or woi-m of a screw. Through- stone; a stone that extends entirely through a wall. Throw; the radius, or distance to which a crank " throws out " its arm. Applies in the same way to lathes. Some use it to express the diameter instead of the radius. To avoid mistakes, the terms "single" and "double" throw might be used. Tides; those well-known rises and falls of the surface of the sea and of some rivers, caused by the attraction of the sun and moon. There are two rises, floods, or, high tides ; and two falls, ebbs, or low tides, every 24 hours and 50 minutes ; a lunar day j making the average of 6 hours 12V6 minutes be- tween high and low water. These intervals are, however, subject to great variations; as are also the heights of the tides ; and this not only at diif^rent places, but at the same place. These irregularities are owing to the shape of the coast line, the depth of water, winds, and other causes. Usually at new and full moon, or rather a day or two after, (or twice in each lunar month, at intervals of two weeks,) the tides rise higher, and fall lower than at other times; and these are called spring tides. Also, one or two days after the moon is in her quarters, twice in a lunar mouth, they both rise andfall lass than at other times; and are then called neap tides. From neap to spring they rise and fall more daily; and vice versa. The time of high water at any place, is generally two or three hours after the moon has passed over either the upper or lower meridian ; and is called the establishment of that place ; because, when this time is established, the time of high water on any other day may be found from it in most cases. The total height of spring tides is generally from 1^ to 2 times as great as that of neaps. The great tidal wave is merely an undulation, unattended by any current, or progressive motion of the particles of water. Each successive high tide occurs about 24 minutes later thau the preceding one; and so with the low tides. Tie; any piece that sustains tension or pull. Tightning-ring. See 14, of Figs 21^, of Trusses, p 265. Tightning- screw. See Set-screw. Tire ; the iron ring placed around the outer circumference of the felloe of a wheel. Toggle joint. In Fig 16, of Force in page 461, suppose a TO and a n to be two stiff bars, hinged together at a. It is plain that if we press downward at a, the result will be a great pushing force against any bodies placed at the ends m and n. Such an arrangement of two bars for producing such pressure is a toggle-joint. Tongue ; a long slightly projecting strip to be inserted into aoorresponding groove, as in tongued and grooved floors. Tooling ; dressing stone by means of a tool and mallet; the tool being a chisel with a cutting edge of I to 2 inches wide. Tooling is generally done in parallel stripes across the stone. Torus; a projecting semi-circular, or semi-elliptic moulding; often used in the bases of columns, It is the reverse of a scotia. Trailing -wheels ; in a locomotive, those sometimes placed behind the driving-wheels. Train; a number of cog-wheels working into each other. Tramway ; any two smooth parallel tracks upon which wheels without flanges may run. In rail OP TERMS. 627 tramways the rails thamselve^ htfve flanges ; but ia wide stone ones for common vehicles, none are required. S Transom ; a beam acre^s the opening for a door, &c. Also, a horizontal piece dividing a high win- dow into two storiesyfce, &c. Tread', the boriai miles long, and has no shafts, although it grades up from each end, which is the most unfavor- able of all conditions for ventilation without shafts. It was made so for facilitating drainage. Its ven- tilation is maintained by air forced in from the ends. The Hoosick tunnel, Mass, 4^ miles long, has shafts ; one of them 1030 ft deep ; but they are for expediting the work. Shafts generally cost from 1% to 3 times as much per cub yd as the main tunnel, owing to the greater difficulty of excavating and removing the material ; and getting rid of the water, all of which must be done by hoisting, When through earth, they must be lined as well as the tunnel ; and the lining must usually be an , under-pinning process. Or the lining may first be built over the intended shaft, and then sunk by page 327. Their sectional area commonly varies from about 40 to 100 . undermining it gradually ; see p . sq ft. They have the great advantage of expediting the work by increasing the number of points at which it can be carried on ; but if placed too close together, their cost more than compensates for this. The air in some tunnels, while being constructed, is much more foul than in others; so that after the work has been commenced, shafts with forced air may be expedient where they were not anticipated. In excavating the tunnel itself, a heading or passage-way, 5 or 8 ft high, and 3 to 12 ft wide, is driven and maintained a short distance (10 to 100 ft, or more, according to the firmness of the material) in advance of the main work. In rock, the heading is just below the top of the tunnel, so that the men can conveniently drill holes in its floor for blasting; but in earth, the heading ia driven along the bottom of the tunnel, that being the most convenient for enlarging the aperture to the full tunnel size, by undermining the earth, and letting it fall. In earth, the top and sides of the heading, as well as of the tunnel, must be carefully prevented from caving in before the lining is built; and this is done by means of rows of vertical rough timber props, and horizontal caps or over- head pieces, between which and the earth rous;h boards are placed to form temporary supporting sides and ceiling to the excavation. The props and caps are placed first : and the boards are then driven in between them and the earthen sides of the excavation. These are gradually removed as the lining is carried forward. The lining, when of brick, is usually from 2 to 3 bricks thick (17 to 26 inches) at bottom, and from 1^ to 2} less. Loose vegetable surface soil " 15 per ct: or 1 in 6% less. Puddled clay " 25 per cub yds of very carelessly scabbled rubble; or 1^ yds of somewhat carefully scab- bled. II Seep 504. * Solid part .526 ; voids .474 of the entire bulk. t " " .570; " .430 " " " J " " .630; " .370 " " " " " .670; " .320 " t4 I " " .800; " .200 " " " 630 APPENDIX. 631 Approximate c4tst of buildings per cubic foot, at Philada pric n 1873 : iucluerfng every cub ft of space from roof to cellar floor. Plaiu brick dwellings, such ;is mo __ m , u _ rices In 187*3 rinclu*rfng every cub ft of space from roof to cellar floor. Plain brick dwellings, such as most of those in Poilada. 12 to 15 cts. Better class, highly finished throughout, 15 to 18 cts. First class, with cut stone fronts, 20 to 30 cts. Plain brick churches, public schools, court-houses, theaters, &c, 12 to 16 cu. Ornate Gothic churches with much cut stone facing, 30 to 45 cts, exclusive of spires. Large plain brick or rubble RR shops, depots, station-houses, &c, 9 to 12 cts; or with ornamental finish and best materials, 15 to 20 cts. First class city stores, marble fronts, high stories, fire-proof, (so called,) throughout to roof; best materials and workmanship, 18 to 25 cts. Small buildings cost more per cub ft than large ones of the same finish. Also isolated or corner buildings, cost more than those which have two party-walls. In Philada dwellings of brick, the carpentry and lumber usually cost each about one-fourth of the entire building. Memorial Hall of the Centen- nial Buildings cost 68 cts per cub ft, exclusive of iron dome. P 327. The plenum process as applied at the South St bridge, Philada, by Mr. John W. Murphy, contracting engineer, differs materially from that described on p 327 ; and moreover deserves notice on account of the great simplicity and efficacy of his plant.* This consisted partly of two canal boats, decked, each 100 ft long, by 17}^ ft wide, and 8 ft depth of hold. They were 'anchored parallel to each other, 15 ft apart. Supported by the boats, and over the space between them, was a strong four-legged shears about 50 ft high ; at the lop of which was attached tackle for handling the cast iron cylinders. In the hold of one of the boats was a Burleigh Compressor having two pistons of 10 ins diam, and 9 ins stroke; together with its boiler. On the deck of the same boat stooda vertical air-tank or regulator, 22 ft long, by 2 ft diam, made of quarter inch boiler iron. This served to maintain a supply of com- pressed air in the submerged cylinder in case of an accidental stopping of the compressor; which otherwise would probably be fatal to the laborers in the cylinder. The condensed air flowed from this air-tank to the air-lock of the cylinder through a hose 4 ins diam, made of gum elastic and can- vas, and so long, and so placed, as to extend itself as the cylinder went down, thus maintaining the communication at all times. Entirely across both boats, and across the interval between them, ex- tended two heavy wooden clamps, each 3 ft wide by 18 ins high; each composed of three pieces of 12 X 18 inch timber strongly bolted together. At the centers of these clamps the two inner vertical sides which faced each other were hollowed out to the depth of a foot by concavi- ties corresponding to the curve of the cylinders. The distance apart of the clamps was regulated by two strong iron rods, having screws and nuts at their ends for that purpose. Thus when a section of a cylinder was hoisted by means of the shears into its position over the space between the two boats, the two concavities of the clamps wre brought into contact with it, and the nuts being then screwed up, the cylinder was firmly held in place by the clamps. The shears could then be used to raise another section of the cylinder to its place upon the first one. that the two might be bolted to- gether. By repeating this process the height of the cylinder would soon become too great to allow the shears to place another section upon it; in which case the nuts of the screws were slightly loosened, and the cylinder was allowed to slip down slowly into the water until its top was but a little above the surface. The screws were then again tightened, and the cylinder again held fast until other sections were added and bolted to it. When there was danger that the upward pressure of the condensed air might lift a cylinder, the clamps were raised by the shears clear of the boats; then tightened to the cylinder, and a platform of plauks laid upon them, and loaded with stone. The air-lock was so arranged as not to require to be removed when a new sec- tion was to be bolted on. This was effected as follows. . Sections of the cylinder were bolted together in the manner just described, until its foot rested on the bottom, with its top a few feet above high water. A heavy cast iron diaphragm 1^ inches thick, to form the floor of the air-lock, was then placed on top. Then was added another 10 ft high section of the cvlinder, to form the chamber of the air-lock. These were bolted together ; and then another diaphragm was added at top to form the roof of the air-lock. These diaphragms were furnished with openings, and with doors and valves corresponding with those shown in Fig 17. p 327. and remained permanently in the cylinders when the work was finished. If the depth of soil to be passed through before reaching rock is so great as to require other sections of cvlinder to be bolted on above the top of the air-lock this may be done to any extent, inasmuch as it is immaterial whether the air-lock is under water or not. To keep the cylinder both air- and water-tight the faces of the Hanges before being bolted together were smeared with a mixture of red and white lead and cot- ton fiber. At the South St bridge the cylinders were 4. 6, and 8 ft diam ; in lengths or sections 10 ft long. They were all \y inch thick. Inside flnnges 2% ins wide, 1} thick, with bolt- holes 1J4 inch diam, by 5 ins apart from center to center. The bottom edge has no flange. A 10 ft section of an 8 ft Cylinder weighs 14600 fts ; of a 6 ft one, 10800 ; of a 4 ft one, 6800. An 8 ft dia- phragm, 2800 fts ; 6 ft, 1600; 4 ft, 783. The rock under the soil was quite uneven in places ; but was levelled off as the cylinders went down. These were then bolted to it by cast iron brackets. The work went on, day and night, summer and winter: with no inter- ruption from the tides, floods, or floating ice; and the thirteen columns were sunk, filled with con- crete, and completed in 11 months ; much of which was consumed in levelling off the rock, and bolt- ing the brackets. The want of guides caused much tilting, trouble, and delay. Rise and fall of tide about 7 ft. Greatest depth of soil, gravel, Ac. passed through. 30 ft : least, 6 ft. Depth of water about 25 ft. The work was under charge of John Anderson, a very skillful and energetic superintendent of such matters. The entire neat cost of the cylin- ders in place, and filled with hydraulic concrete was probably not far from $120 per ft of total length for the 8 ft ones ; $80 for ttie 6 ft ; and $50 for the 4 ft diams. There were three gangs of workmen and each gang worked 4 hours at a time. Such cylinders have cracked through, around their entire circum- ference, in many parts of the U. S. in very cold weather; owing to the diff of contraction of the iron, and of the concrete filling. Ignorant use of them may be attended by great danger. * See 'a full and very instructive description with engraving*, by D. M. Stauffer, Superintending Engineer for the city, in the Journal of the Franklin lost, Nov. 1872. From it the above few item* 632 APPENDIX. P 177. Moduliises of Elasticity in Ibs per sq incb. For the meaning of the term see p 177. Authors differ considerably in their data on this subject. Where it was possible to adopt an average without error of practical importance, we have done so ; but where the differences were too great for this, we have omitted the material entirely. Fortunately it is a matter of but little importance to the engineer ; for, with the exception of iron and steel, he rarely need care to know the exact degree to which his materials are lengthened or shortened by their loads. It must be carefully remembered that in practice the principle of mod of elas applies only within the elas limit of materials. This limit may ordinarily be safely taken at about one-third of the breaking tensile strengths given on pages 177 to 180. This is done in our 5th column. It is, how- ever, but a tolerable approximation. There is reason to doubt the accuracy of many of the items ic this table. Round and square bare do not stretch the same. MATERIAL. Modulus or Coeff of Elasticity. Stretch or C in a lengtl under a 1000 Ibs per sq iu. ompression i of 10 ft, load of 1 ton per sq in. Approx elas limit. Ash Ibs per sq in. 1 600 000 IDS. 075 Ins. 168 Ibs per sqin. 4500 Beech 1 HOO 000 092 207 4000 Birch 1 400 000 086 192 5000 Brass, cast 9 200 000 .013 .029 6000 wire 14 200 000 .009 .019 16000 Chestnut. 1 000 000 .120 269 4500 18 000 000 .007 015 6300 < ' wire 18 000 000 .007 015 10000 Elm ... 1 000 000 120 269 2000 Glass . 8 000 000 015 034 3200 12 000 000 010 022 4500 " " average to 23 000 000 17 500 000 to .005 .007 to .012 .015 to 8000 6250 " wrought, in either < 18 000 000 to .006 (to .015 to 13000 " " " average 40 000 000 29 000 000 .003 .004 .007 .009 26000 19500 ' " wire, hard 26 000 000 .005 .010 27000 15 000 000 .008 .018 13000 Larch 1 100 000 .109 .244 2300 I ead sheet 720 000 .167 1100 < w i re 1 000 000 .120 1100 I 400 000 .086 192 2700 0ak 1 000 000 120 269 to 2 000 000 to 060 to 134 Bveraee 1 500 000 .080 .179 3300 Pine white or yellow 1 600 000 .075 .168 3300 glate 14 500 000 .008 .018 3700 1 600 000 .075 .168 S.iOO oPppi K ar a 29 000 000 .004 .009 34000 to 42 000 000 35 500 000 to .003 003 to .006 007 to 44000 39000 Sycamore 1 000 000 .120 .269 4000 Teak 2 000 000 .060 .134 5000 Tin. cast 4 600 000 .026 1500 P 603. Friction of Hydraulic Presses. Is not certainly known. Mr. John Hick, C E, England, says that it is independent both of the width of the leather collar, and of the length of the ram in" the cylinder ; and that in a press in good order it may be found very approximately thus, Rule. Mult the diam of the ram in inches by 25. Divide the total pressure against the bottom of the ram by the product. Example. In a press of 18 ins diam the total pressure is 680000 Ibs; how much must be deducted from this for friction ? Here, 18 X 25 = 450. And - = 1511 Tbs, answer. Hence iu an 18 inch press the ratio of the friction to the total pressure is ^ ftfkv r .00222 ; or s _ L_ =: 450, the fric = of the pressure. ; 15 ins, ; 12 ins, In a press 24 ins diam it will be ^-^; 18 ins, 6 ins, y4-7y ; 3 ins, -J-r part of the entire pressure. But a small press by Tang-ey of Birmingham, under a pressure of but 360 Ibs per sq inch, showed a friction of 1 per ct, or 4}^ times greater than the above; while other experiments make it equal to the pressure in fts per sq inch X .4 of the depth of collar in ius X cir- cumf in ins. which is much greater than either. Recent expts in this country make ifr iu some cases vastly greater still; while others by Gen Q,. A. Gillmore con- firm Mr. Hicks. BLE OF ORDINATES. 633 of Orclinates 5 ft apart. Chord 1OO ft. For Railroad Curves, For the radii corresponding to the angles of deflection, see page 416. Ordinates for angles intermediate of those in the table can at once be found by simple proportion. This table was calculated by rule p 20. Distances of the Ordinates from the end of the 100 feet Chord. Ang. of Defl. Mid. 50ft. 45ft. 40ft. 35ft. 30ft. 25ft. | 20ft. 15ft. 10 ft. 5ft. ' i 4 .014 .014 .014 .013 .012 .010 .008 .008 .005 .003 8 .029 .029 .028 .026 .024 .022 .018 .015 .010 .005 12 .043 .043 .041 .038 .037 .033 .028 .022 .015 .008 16 .058 .058 .056 .052 .049 .044 .037 ,030 .020 .011 20 .073 .072 .070 .066 .061 .055 .047 .037 .026 .014 24 .087 .086 .083 .077 .074 .066 .056 .045 .031 .017 28 .102 .101 .098 .092 .086 .077 .065 .052 .036 .019 32 .116 .115 .112 .106 .098 .088 .075 .058 .042 .022 36 .181 .130 .126 .119 .110 .099 .084 .066 .047 .024 40 .145 .144 .140 .133 .123 .110 .093 .074 .052 .027 44 .160 .158 .153 .145 .135 .121 .103 .081 .057 .030 48 .174 .172 .167 .158 .147 .132 .112 .088 .062 .033 52 .189 .187 .181 .171 .159 .143 .122 .095 .068 .035 56 .204 .202 .195 .185 .171 .154 .131 .103 .073 .038 1 .218 .216 .209 .198 .183 .164 .140 .111 .078 .041 4 .233 .231 .223 .211 .196 .175 .150 .118 " .DBS .043 8 .247 .245 .237 .224 .208 .186 .159 .125 .088 .046 12 .262 .260 .252 .237 .220 .196 .168 .133 .094 .049 16 .276 .274 .265 .251 .232 .207 .177 .140 .099 .052 20 .291 .288 .279 .264 .244 .218 .187 .148 .104 .055 24 .306 .303 .293 .277 .256 .229 .197 .155 .109 .057 28 .320 .317 .307 .291 .269 .240 .206 .163 .114 .060 32 .334 .331 .321 .304 .281 .251 .215 .171 .120 .063 86 .349 .345 .335 .317 .293 .262 .224 .178 .125 .066 40 .364 .360 .349 .330 .305 .273 .233 .185 .130 .069 44 .378 .374 .363 .343 .318 .284 .242 .192 .135 .072 48 .393 .389 .377 .356 .330 .295 .251 .200 .141 .075 52 .407 .403 .391 .370 .342 .305 .261 .208 .147 .077 56 .422 .418 .405 .383 .354 .316 .270 .215 .152 .080 2 .436 .432 .419 .397 .366 .327 .280 .222 .157 .083 4 .451 .446 .433 .409 .379 .338 .289 .230 .162 .086 8 .465 .461 .447 .425 .391 .349 .298 .237 .167 .088 12 .480 .475 .461 .437 .403 .360 .308 .245 .173 .090 16 .495 .490 .475 .450 .415 .371 .317 .252 .178 .093 20 .509 .504 .489 .463 .428 .382 .326 .260 .183 .096 24 .523 .518 .503 .476 .440 .393 .334 .267 .188 .099 28 .538 .533 .517 .489 .452 .404 .346 .275 .194 .102 32 .552 .547 .531 .503 .465 .415 .355 .282 .199 .104 36 .567 .562 .545 .516 .477 .425 .364 .289 .204 .107 40 .582 .576 .559 .529 .489 .436 .373 .297 .209 .110 44 .596 .590 .573 .542 .501 .447 .382 .304 .214 .113 48 .611 .605 .587 .555 .513 .458 .391 .312 .219 .116 52 .625 .619 .601 .569 .526 .469 .401 .319 .225 .118 56 .640 .634 .615 .582 .538 .480 .410 .326 .230 .121 3 .654 .648 .629 .595 .550 .491 .419 .334 .235 .124 4 .669 .662 .643 .608 .562 .502 .428 .341 .240 .127 8 .683 .677 .657 .621 .574 .512 .438 .349 .246 .130 12 .698 .691 .671 .635 .587 .523 .448 .357 .251 .132 16 .713 .705 .685 .649 .599 .534 .457 .364 .257 .135 20 .727 .720 .699 .662 .611 .545 .466 .371 .262 .138 24 .742 .734 .713 .675 .623 .556 .475 .378 .267 .141 28 .756 .749 .727 .688 .635 .567 .485 .386 .272 .144 32 .771 -763 .741 .702 .648 .578 .494 .394 .278 .146 86 .786 .777 .755 .715 660 589 503 401 .283 149 40 .800 .792 .769 .728 .673 .600 .512 .408 .288 .152 44 .814 .806 .783 .741 .685 .611 521 .415 .293 .155 48 .829 .821 .797 .754 .697 .621 .531 .423 .298 .158 52 .843 .835 .811 .768 .709 .632 .541 .431 .304 .160 56 .858 .850 .825 .781 .721 .643 .550 .438 .309 .163 4 .873 .864 .839 .794 .734 .655 .559 .445 .314 .166 10 .909 .900 .874 .827 .764 .682 .582 .464 .327 .173 20 .945 .936 .909 .860 .795 .709 .606 .482 .340 .179 30 .981 .972 .944 .893 .825 .736 .629 .501 .354 .186 40 1.017 1.008 .979 .926 .a55 .764 .652 .519 .367 .193 50 1.054 1.044 1.014 .959 .886 .791 .676 -.538 .380 .199 5 1.091 1.080 1.048 .993 .917 .818 .699 .557 .393 .207 10 1.127 1.116 1.083 1.026 .947 .845 .722 .576 .406 .214 20 1.164 1.152 1.118 1.058 .978 .872 .746 .594 .419 .220 30 1.200 1.188 1.153 1.092 1.009 .900 .769 .613 .432 .228 634 TABLE OF ORDINATES. Table of Ordinates 5 ft apart. (Continued.) Distances of the Ordinates from the end of the 100 feet Chord. Ang. of Deft. Mid. 50ft. 45 ft. 40ft. 35ft. 30ft. 25ft. 20ft. 15ft. 10ft. 5ft. 540 1.236 1.224 1.188 1.124 1.039 .927 .792 .631 .445 .235 50 1.273 1.2HO 1.223 1.157 1.070 .954 .816 .649 .458 .241 6 1.309 1.296 1.258 1.191 1.100 .982 .839 .668 .472 .243 10 1.345 1.332 1.293 1.224 1.130 1.009 .862 % .686 .485 .255 20 1.382 1.368 1.328 1.256 1.161 1.036 .886 .705 .498 .262 30 1.419 1.404 1.362 1.290 1.192 1.064 .909 .724 .511 .269 40 1.455 1-440 1.397 1.323 1.222 1.091 .932 .742 .524 .276 50 1.491 1.476 1.432 1 .355 1.253 1.118 .956 .761 .537 .283 7 1.528 1.512 1.467 1.389 1.284 1.146 .979 .779 .551 .290 10 1.564 1.548 1.502 1.422 1.314 1.173 1.002 .798 .564 .297 20 1.600 1.584 1.537 1.454 1.345 1.200 1.026 .816 .576 .304 30 1.637 1.620 1.572 1.488 1.375 1.228 1.048 .835 .590 .311 40 1.073 1.656 1.607 1.521 1.405 1.255 1.071 .854 .603 .318 50 1.710 1.692 1.641 1.553 1.436 1.282 1.095 .872 .616 .324 8 1.746 1.728 1.677 1.587 1.467 1.810 1.118 .891 .629 .332 80 1.855 1.836 1.782 1.687 1.559 1.392. 1.188 .946 .669 .353 e 1.965 1.944 1.886 1.787 1.651 1.474 1.258 1.002 .708 .873 30 2.074 2.052 1.991 1.887 1.742 1.556 1.328 1.057 .748 .394 10 2.183 2.161 2.096 1.987 1.834 1.637 1.398 1.114 .787 .415 30 2.292 2.269 2.201 2.087 1.926 1.719 1.468 1.170 .827 .436 11 2.401 2.377 2.306 2.186 2.018 1.802 1.538 1.226 .866 .457 30 2.511 2.486 2.411 2.286 2.110 1.884 1.609 1.282 .906 .478 12 2.620 2.594 2.516 2.386 2.203 1.967 1.680 1.339 .946 .499 30 2.730 2.703 2.621 2.485 2.295 2.049 1.750 1.395 .985 .520 13 2.889 2.811 2.726 2.585 2.387 2.132 1.820 1.451 1 .025 .541 30 2.949 2.920 2.832 2.685 2.479 2.214 1.891 1.507 1.065 .562 14 3.058 3.028 2.937 2.785 2.571 2.297 1.961 1.564 1-105 .583 30 3.168 3.136 3.042 2.884 2664 2.379 2.031 1.620 1.144 .604 15 3.277 3.245 3.147 2.984 2.756 2.462 2.102 1.676 1.184 .625 30 3.387 3.354 3.252 3.084 2.848 2.544 2.172 1.732 1.224 .646 16 3.496 3.462 3.358 3.184 2.941 2.627 2.243 1.789 1.264 .667 17 3.716 3.680 3.569 3.384 3.125 2.792 2.384 1.902 1.344 .709 18 3.935 3.897 3.779 3.584 8.310 2.958 2.525 2.014 1.424 .751 19 4.155 4.115 3.990 3.784 3.495 3.123 2.666 2.127 1.504 .793 20 4.375 4.332 4.201 3.984 3.680 3.288 2.808 2.240 1.583 .836 22 4.815 4.768 4.624 4.386 4.050 3.620 3.093 2.467 1.744 .922 24 5.255 5.204 5.048 4.789 4.423 3.952 3.379 2.695 1.905 1.008 26 5.697 5.642 5.473 5.192 4.798 4.286 3.665 2.924 2.068 1.094 28 6.139 6079 5.898 5.595 5-171 4.622 3.952 3.154 2.232 1.181 30 6.582 6.517 6.323 5.999 5.544 4.958 4.239 3.385 2.396 1.268 32 7-027 6.957 6.751 6.406 5.922 5.297 4.530 3.619 2.565 1.356 34 7.472 7.398 7.179 6.813 6.300 5.637 4.822 3.854 2.733 1.445 36 7.918 7.841 7.609 7.2'22 6.679 5.978 5.115 4.090 2.901 1.535 38 8.367 8.286 8.041 7.633 7.060 6.320 5.410 4.327 3.069 1.626 40 8.816 8.731 8.474 8.044 7.442 6.663 5.705 4.565 3.238 1.718 GEILLAGE, After piles have been driven, and their heads carefully sawed off to a level, if not under water, the spaces between them are in important cases filled up level with their tops with well rammed gravel, stone spawls, or concrete, in order to impart some sustaining power to the soil between the piles. Two courses of stout timbers, ("from 8 to 12 ins square, according to the weight to be carried) are then bolted or treeuailed to the tops of the piles and to each other, as shown in the Fig, forming what is called a grillage. On top of these is bolted a floor or plat- form of thick plank for the support of the masonry ; or the timbers of the upper course of the grillage may be laid close together to form the floor. The space below the floor should also, in important cases, be well packed with gravel, spawls, or concrete. If under water, the piles are sawed off by a diver, or by a circular saw driven by the engine of the pile-driver, and the grillage is omitted. Instead of it the masonry or concrete may be built in the open air in a caiasoti, p 316 ; which gradually sinks as it becomes filled ; or on a strong platform which is lowered upon the piles by screws as the work progresses, p 328. Or a strong caisson may first be sunk entirely under water, and then be filled with concrete, p 507, up to near low water; the caisson being allowed to remain. Or the caisson may form a cofferdam, to be first sunk, and then pumped dut. If the ground is liable to wash away from around the piles, as in the case of bridge piers, &c, defend it toy sheet-piles, or Tip- rap, or both ; p 314. BUOYANCY, ATION, METACENTER, ETC. 635 or O, either solid or a vessel of any shape, at rest, as L, *" buoyancy and flotation) be considered as acted upon only by two A floating bodv may (so far as equal vert for shown by Jtf, equal arrows.* One of them is the wt of the body itself, al- ways acting vert downwards, and as if concentrated at the oen of grav G of the body. The other is the "buoyancy or upward pres of the water; and is equal to the wtof the water displaced by the body : or to the wt of the "floating body it- self. It always acts vert upwards, and as If concentrated at the cen of grav W of the displaced water.f W is also called the center of pressure, or of buoyancy of the water; and a vert line drawn through it is called the axis or vertical of buoyancy, or of flotation. This W of course shifts its position with every change in that of the body. Thus in L it is at the cen of grav of the rectangle o o b b : and in N, at that of the tri- angle a a v. When a floating body L or P is at rest or undisturbed by any third force, then G and W will be in the same vert line t t&g L; or e e fig P ; which line is called the axis, or vertical of equilibrium. If (T fig L is then above W, the body will be in unstable equilibrium ; that is, if any third force, as the wind, F fig N, causes the axis of equilibrium to lean, the body will upset; for the forces G and W then no longer act in the same straight line, but in two parallel lines; thus forming a coil pie (Case 3, p 483): and instead of holding each other, and the body, in equilibrium, they cause it to rotate around a point half way between them. Thus the force G fig N impels the upper part of the body downwards, while W impels its lower part upwards. Hence the body must upset, even if F ceases. J But if the upright vessel L be so loaded that G shall be vert below W, then it will be in stable equilib; so that if a third force F, causes the axis of equilib tt to lean as in fig O, the couple or forces G and W will tend to prevent it from upsetting ; since G impels the lower part downward, while W impels the upper part upward ; so that if F ceases to act, the vessel will right itself. It is true however that even in this case the thiid force F may be so great as to en- tirely overpower the combined forces G and W, so that a vessel may upset in a hurricane, although judiciously loaded and ballasted for ordinary winds. The tendency of the leaning bouy either to upset or to right itself increases with its inclination, and is readily found in foot-pounds as stated in Case 3, p 483. by mult one of the forces by the perp dist in ft existing between the two forces at any given inclination. Uneven loading 1 , instead of a third force, may cause a vessel at rest to incline as in fig F ; and yet the vessel so leaning will be in stable equilibrium, G being below W ; for its axis e e of equilibrium is vert, although not coinciding with the axis of symmetry of the vessel, as it does at t tin L. Hence if a third force, as wind, should cause e e to lean, G and W will both tend to bring it back to a vert position. It is plain however that a vessel thus unevenly loaded would more easily upset towards the right hand than towards the left; hence it should be so loaded as to float upright when at rest. Also the heaviest articles of the cargo should be placed lowest in the hold, in order to keep G as far below W as pos- sible. Persons in a boat in danger of upsetting should squat down instead of standing up. When a third force causes the axis of equilibrium 1 1 fig L of a floating body to lean as in figs N and O, then if a vert line be drawn upwards from the center W of flotation, the point M at which said line cuts said axis is called the metaceiiter of the body. This metacenter shifts its position according to the inclination of the axis of equilib ; but so long as it is higher than the cen of nv G of the body, as in fig O, the body will remain in stable equilib, and will restore itself to its iginal position as soon as the disturbing force ceases to act. But if M is below G as in fig N, the equilib will be unstable, and the body must upset even if the third force ceases to act. A hor section of a body at water-line is called its plane of flotation. * The body is in fact acted upon by other forces, such as the hor pressures of the water against its immersed portions ; but as all of these in any one given direction are balanced by equal ones in the opposite direction, they have no effect upon the forces G and W. It is also acted upon by the air, which presses it downwards with a force of 14.75 Ibs per sq inch ; but this is balanced by an equal pres of the surrounding air upon the surface of the water, and which is nsnaitted 'art 7, p 526) vert upwards against the immersed bottom of the floating body. t This buoyancy is made lip of the parallel upward pressures of the innumerable vert filaments of the displaced water as shown by Fig 26, p 533 ; and ;he axis of flotation is their resultant, as in the case of parallel forces Fig 55, p 481. t When a body is held in eqiiilib by two forces, one of which is its >wn wt, it follows, that inasmuch as gravity or wt acts vert downwards only, the two forces must be in the same vert line. If a disturbing force should raise the cen of grav of the body, then on the removal of said force this cen of grav will naturally descend to its original position, and thus restore the equilibrium, which in such cases is called stable. But if the disturbing force lowers the cen of grav of the body, said cen will not return to its original position ; but will continue to descend, and the body will" fall ; its equilib having been unstable. If the cen of j; rav of the body neither rises nor falls, its equilibrium is called indifferent; and the body will remain in any position in which a third force may place it, as a ball on a level table, or a grindstone on its axis. 636 TEST BORINGS. Pierce's well borer is an excellent tool for boring into soils, clay, sand or gravel, even when quite indurated. It removes pebbles and stones smaller than the bore. The augers are made from 6 to 18 ins diain, according to the required purpose; and if the hole should re- quire to be enlarged for the insertion of tubing or curbing, a rimmer is at- tached. If loose running sand, slush, &c, are met with, the sand-sides and valves are put on ; but for these materials, (which require tubing even for test holes) Pierce's sand-auger is better. A light derrick about 25 or 30 feet high, with winch, drum, simple cog-gear, and two-block tackle are required for raising the auger iiud its rods at short intervals for emptying. The square socket-jointed rods are 10 to 14 ft long, of 1.5 inch sq iron. Where no tubing is needed the auger is screwed down at the rate of from 3 to 20 ft per hour by hook- wrench levers 6 ft long, worked by '2 to 4 men, or by a horse, according to depth, hardness, diam, &c. Another man attends the winch by which the auger is fed and raised. Where tubing is required the progress is of course much slower. In dry soils a bucket of water is thrown down the hole whenever the auger is raised. In boring many shallow holes near each other the moving of the derrick consumes much of the time. The Pierce borer may be ad- vantageously used for sand-piles, p 328; and at times instead of driving wooden piles, it may be better to plant them (perhaps butt down) in holes bored by this auger, ramming the earth well around them afterwards. This will save adjacent build- ings from the jarring and injury done by a pile-driver. -X- In testing; through hard gravel mixed with cobbles, or even through rock, a welded iron tube about 4 to 6 ins diam, in screw jointed lengths of about 8 ft, and with a steel cutting-edge ring screwed around its foot, may be held over the spot, and if possible be driven down so far as to stand by itself; or if this cannot be done it must be held in place at first by other means. Inside of this tube the boring is done by a boring-bit weighing about 150 Ibs, of about 3 inch diam iron, 3 ft long~ with a steel chisel-edge foot about 4 ins wide , and with an eye at top for a derrick rope. It is expedient to have at bund an extra boring bit of about 400 Ibs, in case the hardness of the material should require it. To vork the bit, a few turns of the i ope are taken around the drum, and a man pulls at the slack end of the rcpe, while the bit is being raised a foot or two by the crab. When so raised this oian lets go his end of the rope, thus loosening the turns around the drum, and the bit falls. Previous to each stroke of the bit the rope must be slightly twisted, in order to change the position of the chisel at each stroke, so that the hole may be round. At intervals of about a foot in depth, as the boring goes on, the bit must (as in other drilling opera- tions) be lifted out by the crab to allow the debris which has accumulated in the well tube to be re- moved. Except for very shallow holes this is best done by a sand-pump, a simple form of which is a welded iron tube for the pump barrel, about 4 ft long, and with a diam say about an inch less than that of the well tube into which it is to be lowered. At its foot is a leathe'r up- valve ; and at top is a falling or bucket handle for lowering it into, or lifting it out of the well-tube by the derrick rope and crab. In this pump-barrel works a close-fitting sucker or piston of wood and leather, with a handle or pump-rod about 6 ft long, with an eye at top for a rope. By this rope the sucker-rod is jerked suddenly upwards a foot or two. by hand a few times, de scending each time by its own wt. Water having first been poured into the well, this process pumps the debris (including fragments of egg size) from the bottom of the well tube into the pump barrel, which is then lifted out of the well and emptied. The well tube is then driven down a little farther, the boring bit is again lowered into it, and the boring is continued until enough more debris accumu- lates to again require the pump; and so on alternately. Frequently something will stick in the sucker, and keep it open so as to prevent the pump from working. The pump must then be lifted out from the well, and the obstacle removed. An ordinary clay's work with this tool in hard gravel mixed with cobbles, will be but from 2 to 4 ft of depth. Pierce's hand-drill is much more rapid. To avoid bruising the top of the well-tube by hard driving, and thus destroying the fit of the screw joints of its separate lengths, there must, while driving, be placed upon it some attachment of hard wood or iron to receive the blows of the heavy maul. For boring: under water the derrick may be placed at the center of a raft or scow of about 15 by 30 ft, with an opening about a foot square for working the tools. If in a tideway there will be trouble in keeping a raft or scow constantly in position ; and some arrangement must be made for a fixed platform to work from. * The Pierce Well Excavator Co, No 29 Rose St, New York, furnish these tools complete, as well as all those for artesian and other boring, together with earth screw- augers, windmills, pumps, tubings, &c. Or they will contract to do the work itself, using their own tools. Their charges, in 1878, were about $1 per ft of depth for a well 17 ins diam, lined with wood; and from $3 to $5 per foot for holes 5 ins diam in solid rock, to a depth of 1000 ft or more. Their price for the above well-borer of any one given diam from 6 to 18 ins, and with rods for a depth of 100 ft, including derrick, tackle, levers, Ac, complete, is about $150, including rimmer, $5, and sand-sides and valves, $10. The sand-auger $35 extra. They furnish well-tubing of No 16 to 18 galvanized iron, 1 foot diam, at 50 to 75 cts per ft run. A lighter tool for holes about 6 ins diam, and 10 to 20 ft deep, and which would require no derrick, but would be raised by hand, can be furnished for about $40, exclusive of rimmer, and sand-sides. The same Co furnish their Portable Hand Rock ]>rill including drill sharpener, 6 drills, &c, at $230. With it one man can drill 3 inch holes to a depth of 50 or more feet, at the rate of from 1 to 5 ft per hour, according to hard ness, depth, &c. It requires, of course, a sand-pump. A screw-auger of steel about 15 ins long, and 1.75 ins diarn, either single or (far better) double twist, with jointed boring rods of 1.5 inch square iron or (much better) steel, each joint about 10 ft long, may be u.sed for test boring in clay, sand, or fine gravel, of all which it will bring up samples. It will not bore through pebbles ; but at moderate depths these may be broken up, or penetrated by a bar with a cutting edge. Loose sand or slush will of course require gas-pipe tub ing. It may be worked to a depth of 100 ft in a day or two, by '2 to 4 men according to hardness and depth , levers 3 to 4 ft long. It must have a derrick, &c, like the well-borer. The above Co can fur- AND + PILLARS. 637 Table A. Br<*lLRing: loads in tons (224O Ibs) per square inch of metal area, of pillars or struts of L, T, or + section of equal arms, and of uniform thickness; by formulas on p 235. The heights or lengths of the pillars are in out to out arms of the sec- tion. See " Remarks " below. For table of pillars of H section see pp 236 and 638. (Original.) Hts Cast. Wrt. Hts Cast. Wrt. Hts Cast. Wrt. Hts Cast. Wrt. Arras. Tons. Tons. in Arms. Tons. Tons. in Arms. Tons. Tons. in Arms. Tons. Tons. 1 35.5 16.1 12 17.2 14.7 23 7.20 11.9 38 3.02 8.19 2 34.7 16.0 13 15.8 14.4 24 6.72 11.6 40 2.75 7.78 3 33.5 16.0 14 14.5 14.2 25 6.28 11.3 42 2 52 7.39 4 32.0 15.9 15 13.3 14.0 26 5.88 11.0 45 2.21 6.84 5 30.1 15.8 16 12.2 13.7 27 5.52 10.8 50 1.81 6.03 6 28.1 15.7 17 11.3 13.5 28 5.19 106 55 1.51 5.34 7 26.1 15.5 18 10.4 13.2 29 4.88 10.3 60 1.28 4.73 8 24.1 15.4 19 9.6 13.0 30 4.61 10.0 70 .95 3.77 9 22.2 15.2 20 8.9 12.7 32 4.11 9.6 80 .73 3.05 10 20.4 15.1 21 8.3 12.4 34 3.69 9.1 90 .58 2.51 11 18.7 14.9 22 7.7 12.1 36 3.33 8.6 100 .47 2.10 Remarks. If the arms or members taper towards their ends, as they nearly always do in practice, instead of being of uniform thickness, the strength is thereby lessened to an extent that varies with the proportions of the section, the height, and whether of cast or of wrought iron ; and although in common practice this taper is not great, still for safety it is well to assume the metal area at only what it would be if the tapering members had a uniform thickness about equal to that at their ends. If one of the two members is shorter than the other the height must be measured by the short one ; and we shall in that case err on the safe side by using the loads in the table as the breaking ones, for these last will in fact be greater at the reduced height. Still if lengths of arms differ more than one-sixth part use the formula. The above table may also be used for this channel form LJ , thus, If the out to out length of a flange is just one-half the out to out length of web, and if the uni- form thickness is not greater than one-sixth, nor less than one-eighteenth of the out to out length of web, th ;n the breaking load per sq inch for any length or height measured in flanges may be taken at once from the table, sufficiently correct for practice. But if the out to out length of a flange is either greater or less than half that of the web of the uniformly thick channel iron, but not shorter than one-fifth of the out to out length of web, (or than two thicknesses of web, if they amount to more than the other) tiien multiply the load in the above table by the corresponding multiplier in the table below. See another table of channel-bar pillars, p 640. (Original.) a la Length of flange in parts of length of web ; both from oat to out. si .2 | .25 .3 .35 .4 .45 .5 .6 .7 .8 .9 l.O CAST. Multipliers for the co umns of cast iron in table A. 5 .85 .90 .93 .95 .97 .99 i. 1.01 1.02 1.03 1.03 1.04 10 .67 .77 .85 .90 .94 .97 I. 1.04 1.07 1.09 1.11 1.13 15 .58 .69 .78 .85 .91 .96 1.06 1.10 1.14 1.17 1.20 20 .54 .66 .75 .83 .89 .95 1.07 1.13 1.17 1.21 1.24 30 .50 .62 .72 .81 .88 .95 1.08 1.15 1.20 1.25 1.29 40 .49 .61 .71 .80 .87 .94 1.09 1.16 1.21 1.27 1.32 50 .48 .60 .70 .79 .87 .94 1.09 1.16 1.22 1.28 1.33 100 .47 .59 .70 .79 .86 .94 1.09 1.18 1.24 1.30 1.35 ROLLED. Multipliers for the co umns of wrought iron in table A. 5 .98 .99 .99 .99 .99 .99 1. I, 1. 1. j 10 .93 .96 .97 .98 .99 .99 1. 1. 1.01 1.01 L01 15 .87 .92 .94 .96 .98 .99 1.01 1.01 1.02 1.03 1.04 20 .81 .87 .91 .94 .97 .99 1.02 1.02 1.04 1.05 1.06 30 .70 .80 .86 .90 .95 .98 1.03 1.06 1.09 1.11 1.12 40 .63 .74 .82 .87 .92 .97 1.04 1.08 1.11 1.14 1.16 50 .59 .70 .78 .85 .91 .96 i! 1.05 1.10 1.14 1.17 1.19 100 .50 .62 .72 .81 .88 .95 i. 1.08 1.15 1.20 1.25 1.29 If the flanges are thinner than the web the strength per square inch of the whole metal area becomes less ;. and if thicker than the web it becomes greater; both to an extent varying with every different proportion of the parts, and with the height or length of strut or pillar as measured by flanges, and also with whether it is of cast or of wrought iron. If the difference in thickness is not more than one-eighth part either way it may be disregarded without serious error ; but if more, use the formula (Raukine's) p 236. nish it complete for about $125. In non-running earths and sands considerable depths may be reached without tubing, as the sides will not cave. For work in loam, clay, or non-running sand, an effective screw-auger can be made by any good blacksmith, by merely forming a one inch sq bar of iron or steel into cork- screw shape about 2 ft long, with 6 complete turns 6 ins iu diam ; its lower end sharpened to form a vertical cutting edge, which should project say .5 of an inch beyond the spiral of the screw, in order to diminish friction. It will hring up full samples. Requires a derrick, or some other simple mode of lifting, when the screw is full. 41 638 ROLLED I BEAMS AS PILLARS. 2 .a .23 - S Sag 15 ft 5 ^S'S^-e' ^,- j^HT lOOit Ort^C^lOtr-'-fT-iOOlOC C CO 10 1C 5 10 Tf -f rf CO JC '^ CO >O rji CO C CO CO O <^ " "t'i7 '7 i~7 1 7 ift 1C ^ Tt* 't CO CO CO C^ O^ C1 ^J COCOCOCOCOCOCOCOCOCOCOCOCOCOOOfOCOCOCCCQCT 238888833$ LLED I BEAMS AS PILLARS. \ CO (N O 00 O ^ ft ^i-H > ft ft r* S.fc^S r-OOOiOTtirJHCOCOC^iM(MC^r i-r ;OC5OSO5OiOOOOOOCO ii ^ ^ . ^ . "5 . *^ . ^ . '. . '. . ". . ^ . ^ r-5 co ^' co t>- oi o w" cc 10 to oo ** c 28 21. 14 32 239 15 45 20 159 12 107 30 22.5 15 28 235 13 43 17 156 11 105 32 24. 16 25 230 11 40 15 153 10 103 34 25.5 17 23 226 10 38 14 149 9 100 36 27. 18 21 222 9 36 13 146 8 98 38 28.5 19 19 218 8 34 11 143 7 96 40 30. 20 17 213 7 32 10 140 6 94 9 Hy. 9 Mm. 8 Hy. 8 Mm. 1 160 162 96 97 134 135 85 86 2 1.5 1 154 161 93 97 130 134 83 85 4 3. 134 160 80 96 112 133 70 85 6 45 3 110 159 65 96 91 132 57 84 8 6. 4 90 158 53 95 72 131 45 84 10 7.5 5 72 157 42 95 58 130 36 83 12 9. 6 58 154 33 94 46 127 28 83 14 10.5 7 47 150 27 93 30 125 23 82 16 12. 8 38 147 22 92 25 123 19 81 18 13.5 9 32 143 18 91 21 120 15 80 20 15. 10 27 140 15 90 19 118 13 78 22 16.5 11 23 138 13 88 17 115 11 76 24 18. 12 20 136 11 86 15 112 10 74 26 19.5 13 17 134 9 84 13 109 8 72 28 21. 14 15 131 8 82 12 105 7 70 30 22.5 15 13 129 7.3 80 10 102 6 68 32 24. 16 12 126 6.6 78 9 99 5.5 66 34 25.5 17 11 122 5.9 76 8 96 5 64 36 27. 18 10 118 5.2 74 7.4 93 4.5 62 38 28.5 19 9 115 4.5 72 6.7 89 4 60 40 30. 20 8 111 3.8 70 6.0 86 3.5 58 7Hy. 7 Mm. 6 Hy. 5 Hy. 1 105 106 75 76 54 55 53 54 2 1.5 1 103 106 72 75 51 54 50 54 4 3. 2 87 105 61 75 43 53 41 53 6 4.5 3 70 104 49 74 33 52 31 52 8 6. 4 55 104 39 74 25 51 24 51 10 7.5 5 43 103 30 73 20 50 18 50 12 9. 6 34 100 24 72 16 49 14 49 14 10.5 7 27 98 19 71 13 48 11 47 16 12. 8 23 96 16 70 10 47 8.8 45 18 13.5 9 18.2 94 13 68 8 46 7.2 43 20 15. 10 15.5 " 92 11 66 7 45 6 41 22 16.5 11 13.5 89 9 64 6 44 5 40 24 18. 12 12 86 8 62 5 42 4.3 38 26 19.5 13 10.5 83 7 60 4.5 40 3.7 36 28 21. 14 9 80 6 58 4. 38 3.2 34 30 22.5 15 7.5 77 5 56 3.6 37 2.8 32 32 24. 16 6.9 75 4.6 54 3. 36 2.5 31 34 25.5 17 6.2 72 4.2 52 2.6 35 2.2 30 36 27. 18 5.5 69 3.7 50 2.3 34 2.0 28 38 28.5 19 5.0 66 3.3 48 2.1 33 1.8 26 40 30. 20 4.5 63 3.1 46 1.9 30 1.6 24 3NTINUOUS BEAMS. MODELS. 641 Copffnuous beams. When a single beam, as a b, Fig 40, is supported not only-lit its two ends, but at one or more intermediate points, it is said to be con- tinuous. It is stronger than if it were cut into two parts, a c, b c, each supported at both ends ; because the tensile strength of the particles at o (lower Pig) assists in counteracting the bendg or breakg tendency of loads on the in- termediate parts o ra, on, of the lower Fig. These particles at o must be torn asunder before the beam (if properly proportioned) can fail. Such a beam, rara, if very long and flexible, will, under its own wt, assume the shape of the reversed curve m so s n ; or if it be stiff, and heavily loaded, the same effect will follow. The points ss, at which the curves reverse, are called tlie points of contrary flexure; and the spans are virtually reduced from mo and n o, to ms and ns. When the beam is supported at only 3 points, as in the Fig, and uniformly loaded, the point of contrary flexure is dist from the central support % of the span ; so that each span, om, on, becomes virtually reduced about ^ part; and the defs will be but about y% as great as if there were two separate beams. The sections of the beam at s and s will then experience no hor strain ; but merely the vert one arising from half the wt between m and s, and n and s. The position of the point of contrary flexure varies with the number of interme- diate supports, and with the manner of loading ; and in bridges, &c, where the load moves along the beam, it changes its place during the transit, so as to bring the points s s considerably nearer to the central support o; thus reducing materially the ad- vantage commonly supposed to arise from connecting together the ends of adjacent bridge-trusses ; if indeed there is any advantage in so doing, which is doubtful. The principle, however, becomes very useful in the case of long rafters or girders, stretch- ing over several points of support, especially when uniformly loaded. Each interval, except the two end ones, will have two points of contrary flexure ; and will then have nearly twice as much strength, under an equally distributed load, as a single beam no longer than said interval. Comparison between models and actual structures. Many practical men imagine that if a model is strong, an actual bridge, roof, Ac, con- structed with precisely the same proportions, must be equally strong in propor- tion to its size. This arises from their ignorance of the fact that the strength of similar beams, trusses, &c, increases only in proportion to the squares of their spans ; while their weight increases as the cubes of the spans ; so that a model 5 or 10 ft long may show a great surplus of strength; while the roof or bridge of 50 or 100 ft span, constructed like it in every respect, may break down under its own weight. We may compare the two in the following manner: Let us suppose a model 4 feet long of a bridge truss, its wt 6 ft>s, and the extraneous center load reqd to break it 120 fbs, or 20 times its own wt. Then its entire center breakg load, including half its own wt, is 120 + 3 = 12-i ft>s. Now suppose we are going to build a bridge truss of 200 ft, or 50 times the span of the model. The strength of the truss will be 50 a , or 2500 times that of the model; that is, it will require f000 times that of the model, or 125000 X 6 = 750000 ft>s ; and one-half of this weight, or 375000 fbs, must be deducted from its entire center strength, in order to find its ex- traneous center load. But in this case the half weight is greater than the entire center strength ; consequently the truss would break under its own wt. If, instead of a center load in the model, we had broken it by an equally distributed one, the calculation would plainly be the same, except that in the model the entire weight, instead of % of it, would be added to the extraneous load for the entire distributed breakg load; and in the truss, its whole weight must be deducted from its breaking strength, to get the extraneous distributed load. If the breaking load of a model is 2, 3 or 4, &c, times as great as its weight, then a similar structure 2, 3 or 4, s, inch-lbs, &c, by mult the load in tons or fts, &c, by its leverage, or the shortest or perp dist h e or / s of its line of direction a m from the fulcrum in ft or ins. See " Moments," p 217 and 473. If the load instead of being* concen- trated like I is distributed in any way along the whole or a part of the beam, its leverage is measured from the fulcrum perp to the line of direction of its cen of grav ; which is plainly the case also with a concentrated load, because its line of direction also passes through its cen of grav. Before the beam bends, its leverage is evidently greater than afterwards, and it becomes less as the bending increases; but as very little bending is allowed in practical cases the leverage may generally be assumed not to change, but to remain as when the beam is hor. The load evidently tends also to strain, or break the beam at any point what- ever as t, Fig 1, between itself and the fulcrum /, and is assisted in so doing by the wt of beam between t and o. Therefore any such point t may also be assumed to be a fulcrum. The moment of the load will of course be less at such point than at /because its leverage t s will be shorter. Art. 2. In the closed beam i a e o, Fig 2, the load tends to revolve about the neutral axis n as the supporting fulcrum of its lever the beam, as shown by the dotted lines, and thereby to strain all the fibres from top to bottom of the beam at the section i n e, by stretching lengthwise those above w, and compressing lengthwise those below n. The greatest strain is at the top and bottom fibres ; and from them both ways it diminishes until at n it is nothing. The load also stretches or compresses the fibres length wise at every vert section along the entire length of the beam, more or less according to its lever- age and moment at said section ; most near the fixed end and least near the free end; so that the extent of stretch indicated by s i is the total accumulated stretchings that have taken place in the top fibres at every point from i to a. The same is the case with the stretches and compressions of the fibres anywhere be- tween i e and a o, as indicated by the varying hor dists between n i and n s, or between n v and n e. The compressed fibres below n, and comprised between n v and n e would as it were vanish, being crushed or mashed flat against the face of the wall. Art. 3. A closed beam a a. Fig- 3, supported at only one point whether at the center or not, and balanced by two either equal or un- equal loads, may plainly be regarded as two levers each of which is essentially in the same condition as Fig 2. Whether the loads are con- centrated or distributed their leverages n e, n e are as before to be measured from n and perp to the lines of direction v o, v o of their centers of grav as in Fig 2. Both the 5 ton loads are mani- festly upheld by the support, which of course reacts vert upwards against them in a vert line with their common cen of grav n, with a force of 10 tons as per the central arrow. Item. 1. Each end load in Fig 3 being 5 tons, suppose each lever n e to be 4 ft. Then the moment of each load about the fulcrum n would be = 5 X 4 = 20 ft-tons. Hence it might seem that over the support the fibres of the beam near n would have to resist a combined moment of 40 ft-tons. Hut they have actually to present a resistance of but 20 ft-tons, on the same principle that if two men pull against each other at two ends of a rope, each with a force of say 30 fbs, the strain or pull on the rope is not 60 but only 30 fts, because strain is the reaction, (pressure or pull) against each other of two equal opposing forces, and is equal to only one of them. *The two above equal moments are merely two forces acting through leverages. See "Strain," Art 2, p 444. Art. 4. A closed beam, Fig 1 4, supported at both ends, and loaded at only one point, whether at the center or not, with a concentrated load, : o 646 OPEN AND CLOSED BEAMS. may also like Fig 3 be regarded as two levers with their common fulcrum at n in a vert line with the cen of grav of the load. This however is by no means so manifest at first sight as in Fig 3, but needs a little explanation. Let the beam bear 10 tons concentrated at its center, then evidently 5 tons of it will rest pressing down- wards on each end support; and each support will therefore press upward or react against an end of the beam with a force of 5 tons as per the arrows. Now these two 5-ton reactions of the sup- ports in Fig 4 are to be considered as taking the place of the two 5-ton end loads in Fig 3 ; while the 10-ton load in Fig 4 takes the place of the 10-ton reaction of the support in Fig 3, and hence in this view of the case is no longer to be considered at all as load, but merely as a fixture for holding the common ful- crum n of the two levers in place, or in equilibrium with the upward end reactions. Being no longer regarded as load, it of course cannot in such cases be assumed to have any moment of rupture; that property being now transferred, to the end reactions. Still, to avoid awkwardness of expression we always speak of the mo- ment of the load even in such cases, rather than of the moments of the reactions of the load. In both Figs 3 and 4 the forces at work are the same in amount, but plainly reversed in direction. Rent. If the load Is diatribnted as the 6 tons in Fig 5, instead of concentrated as in Fig 4, we still consider the beam as consisting of two levers with their common fulcrum n in a vert line with the cen of grav c of the load. But to find the mo- ment of the load (or more correctly, the moment of the reactions of the supports) about n we must proceed a little differently. Thus let the beam be 3 ft span, and the load uniform, weighing 6 tons, 6tons c 4. X Rq5. and being 1 ft long. Find by Art 47, p 219, how much of this load rests on each support, (4 tons on * a, and 2 tons on o.) The upward reactions of the supports will therefore also be 4 and 2 tons. Then first find the moment about n of either one of the reac- tions, say of the 4-ton one at a. This moment will plainly be (4 tons X 1 ft) = 4 ft-tons. Then find the moment about n of that part (3 tons) of the load that is between n and a, by mult the wt (3 tons) of that part by the hor dist (en = .25 of a ft) between its cen of grav and n. This last moment (3 tons X -25 of a ft) = .75 of a ft-ton, being downward, evidently diminishes or counteracts the upward moment of the 4-ton reaction at a about the same fulcrum n to the same extent, and is therefore to be subtracted from it, thus leaving 4 .75 = 3.25 ft-ton for the mo- ment of the 6-ton load about n. The same result will follow if we use the 2-ton reaction of o, with the hor lever- age o w, and the part of the load between o and n. To find the moment for any other point than n see Moments, Case 11, p 220^ Art. 5. In a closed beam, say Fig 2, each of the fibres throughout the entire depth of the yielding section i n e opposes the breaking moment of the load by a Resisting Moment or Moment of Resistance of its own. As the breaking moment about n of the load is made up of its gravity-force or weight mult by its leverage or perp distance n c from the fulcrum or neutral axis n, so the resisting moment about n of each separate fibre, say for instance the one at *, is made up of its natural longitudinal tensile or compressive force or strength mult by its leverage or perp dist n i above or below the same fulcrum n. We have already said (and it is self-evident from the fig) that the extent to which any fibre is stretched or crushed lengthwise is in proportion to its dist above or below n; hence on the principle that action and reaction are equal and in opposite directions, (Arts 13 and 14, p 449) and that within the elastic limit '/ tensio sic vis" (as is the stretch so is the force) the length wise force which stretches or com- presses any fibre, and with which that fibre reacts to resist being stretched or compressed is also as its dist or leverage from n. Now suppose three fibres to be at the dists 1, 2, 3 iris from n, or in other words let their leverages about n be 1, 2 and 3 ins. Then as just said their lengthwise resisting forces must also be as 1, 2, and 3 ; and hence (since the moment of a force is the force mult by its leverage) the resisting moments about n of these three fibres are as 1 X lj 2 X 2, and 3X3, or as 1, 4 and 9 ; that is as the square.? of their dists from the fulcrum n. Thus we see that in a closed beam the lengthwise resistance in ft>s or tons, &c, of each fibre is as its dist from the fulcrum ; while its resisting moment about that fulcrum is as the square of said dist. CLOSED BEAMS. 647 Art. 6. Thte^ft fact as will be shown in Art 8 constitutes the great differ- ence betweeni^sed beams and open ones. It also explains why the strengths of closed^beams are as the squares of their depths, while in open ones such as the trusses of bridges, roofs, &c, they are simply as their depths. It also shows that the strongest form of beam is that in which as much of the material as possible is taken from near the neutral axis where it has but little resistance, and placed at the top and bottom of the beam where it may exert great resistance, as in the common rolled I beam, and the Hodgkinson cast ones, p 208. Art. 7. The Moment of Inertia (which may be found by the ap- proximate method on p 195; at any section of a closed beam, when multiplied by the Constant of Rupture for the material of which the beam is made (the strain in Ibs per sq inch borne by the extreme fibres when on the point of yield- ing, see p 195) and the product divided by the dist in ins of the farthest fibre of the section (or according to Prof De Volson Wood the farthest fibre on the side thai, will yield first, see Art 25V, p 194) from the neutral axis n, gives the Moment ol Resistance of the closed beam at that section. This is usually expressed by the formula R = i-. t In order that the beam shall not fail at that section, its moment of resistance OT its 1 C -t must there be at least equal to the load's Moment of Rupture; and for a safety of 3, 4 or 6, &c, it must be 3, 4 or 6, &c, times as great. The modes of finding- the moments of rupture and of resistance in many cases are given under that head, p 217, &c. Art. 8. But in an open beam i x e a, Fig 6, there is no neutral axis to act as a supporting fulcrum for the beam and its load at n, but the inner end e of the lower chord e a now becomes the fulcrum ; and the beam with its load now tends to revolve about it as per the dotted lines, stretching lengthwise the fibres of the upper chord or flange i z, and crushing in like manner those of the lower one e a. It is plain that the breaking moment of the load is the same as in the closed beam, Fig 2, being still the load X its leverage ea orix', but the resisting moment of the beam consists now of the longitudinal tensile and crushing strengths of the fibres of the two chords or flanges only, X by their leverage, the depth of the beam ; while the web members resist only the vert or shearing force of the load, which in an open beam does not tend to shear the web members, but as end loads to press or pull them in the direction of their lengths. In practice the depth of any open beam or truss is measured from i to e, the centers of grav of the cross sections of the chords, which are as- sumed to be so far apart and so thin that we may do so without sensible error. In doing this we of course thereby assume what is impossible, namely, that all the fibres of one chord at any vert section of the truss are equally distant from all those of the other chord, and hence that all of them have the same leverage, namely the depth of truss measured between the centers of grav of the chords. Although this cannot be strictly correct under any circumstances, still in beams whose chords or flanges are thin in proportion to the aforesaid depth it is suffi- ciently so for practice. This is much more simple than the case of closed beams, and greatly facilitates the finding of the moments and hor strains in the chords of open ones, as follows. Art. 9. Let Fig 6 be a hor open beam 1.5 ft deep from i to e, and projecting 6 ft from a wall into which it is firmly fixed by its flanges or chords i and e; and let the concentrated load L< at its outer end be 1 ton. This load tends to pull the beam into the dotted position by stretching or tearing apart the fibres of the upper chord at i. Now with how great a moment of rupture does it tend to do this, and how strong must the fibres of the chord be at i in order that their mo- ment of resistance may oppose it safely ? We shall here leave the wt of the beam itself out of consideration. When required to be included see Case 10, p 220^- Regard the lines i e and e a as the two arms of a bent lever resting on its ful- crum e. This lever is plainly acted upon and balanced by two equal moments, one at each end a and i\ namely at a the moment of rupture of the load, equal to (1 ton X 6 ft leverage a e} = 6 ft tons ; and at i the resisting moment of the beam, equal to the hor pull or strain on the fibres at the chord i X 1-5 ft, leverage *' e. But we do not yet know what amount of hor pull by the fibres at i is required to 648 OPEN AND CLOSED BEAMS. balance the moment of the load. It is however very easily found by merely divid- ing the 6 ft tons moment of the load by the 1.5 ft leverage of the fibres, that is, by the depth of the beam. Thus we get (6 -f- 1.5) = 4 tons pall at ?' ; and we then have the 6 X 1 = 6 ft tons moment of the load, balanced by the 1.5 X 4 = 6 ft tons mo- ment of the fibres. Therefore in order just to balance the moment of the load, thecho ' of 3,4 lower one upon which it acts as an equal hor compressing one. In shape of a formula the above stands thus. Hor strain at any Moment of load Load X its leverage point in a hor flange = at that point _ at that point of an open cantilever Depth of beam Depth of beam Hence if we know the size and of course the ultimate longitudinal tensile and corupressive strength of the flange or chord, we have by transposition the ulti- mate or breaking load of the hor open beam, thus, Breaking load at any _ Ultimate strength o f flange X Depth of beam. point of a hor open cantilever Leverage of load at that point. And for a safety of 3, 4 or 6, &c, we have Safe load = ^ &c * the ult stren g th of flange X Depth of beam. Levefage of load at that point. Art. 1O. Also in a hor open beam or truss supported at both ends, after having found the moment of the load at any point (by "moments," p 217, &c) the strain on the beam as also its load in Ibs or tons are found in the same way or by the same formulas. Rein. 1. The longitudinal strains on the flanges of hor elosed beams with thin webs such as common rolled I beams, or Figs 22 to 24, p 214, as well as their loads, are also frequently computed in this same ready way, instead of the more troublesome one at foot of p 194, or in " Moments," p 217, &c. The webs are then left entirely out of consideration as regards the hor strains. Although not strictly correct, it is sufficiently so for ordinary practice, and is safe. With these assumptions the dimensions or sectional areas of the top and bottom flanges are proportioned to the safe unit strains of the material. Thus Hodgkinson having found that the ultimate coinpressive strength of cast-iron averaged about 6 times as great as its tensile one, gave his upper flange only one-sixth the area of the lower one, in order that both should be equally strong. In wrought-iron the ten- sile strength is somewhat the greatest, which would lead to making the lower flange the smallest, but here this consideration is outweighed by the practical ones of greater ease of manufacture and of handling or placing which require equal flanges. Item. 2. If the flanges are not horizontal, although the beam or truss itself may be so, the longitudinal strains on the flanges will be increased; and the transverse or shearing strains on the webs will also be changed as stated in Art 12. If the beams are inclined, modifications arise which we shall not treat of. Strangely, most of our standard authorities on bridge building do not even allude to them. Rein. 3. The principle of the bent lever in open beams explains why the strength of a truss is as its depth, (the length of its vert lever-arm) instead of as the square of its depth as in closed beams. The strength however is inversely as the length in both kinds. Art. 11. The web members of an open beam or common truss like Figs 10 and 11, p 254, uniformly loaded, carry the vert or shearing forces of the load and beam from the center each way, up and down alternately from one chord to the other, until finally the end ones deposit it as load on the supports or abut- ments. For each member receives and carries its share of the shearing force in the shape of an end load, thus changing the shearing tendency into an alternately pulling and compressing one according as the alternate members are ties or struts. In doing this any web member that is oblique is (on account of its obliquity) strained to an extent that exceeds its load in the same proportion that the oblique length of the member exceeds the length it would have had if it had been vert, as explained in Art 11, p 253, &c. This excess of strain over the load on the obliques f I exhibits itself at th^fr ends as hor pull along one chord, and hor compression along the other j^trnd th/se hor strains on the chords are the same as those found e Case 10, p 220%. Thus it is seen in Figs 10 and 11 that the hor chord (as there found by tracing up the dift'oreri incuts, v ertiua.1 mciiiucia incicij' ^uuvc^ HJCJLJ lutiuo veil* uu ui vtuvvn iiuiii unc chord to the other, at which last they transfer them to oblique members which cnora to me ouier, ai wiucii iasi iney uaiisier mem 10 oouque memuers wnica can convey them laterally. If both a pull and a push act at once in opposite directions on a web member, their din is the actual strain. Rem. As a matter of economy in small spans it is often better not to , proportion the sizes of the individual members to the strains they have to bear ; but to give to the flanges throughout their entire length the same dimensions as are required at their most strained part, namely, at the center ; and to make all the web members as strong as the most strained or end ones. This avoids the extra trouble and expense of getting out and fitting together many pieces of yarious sizes. Art. 12. Oblique or curved flanges. We have hitherto supposed the beams and their flanges to be horizontal; but a beam may be hor, and yet have one or both of its flanges oblique or curved as at A and B. In such cases the longitudinal strains along the flanges become greater; and the vert or shearing strains across the web in most cases less. See Bern at end of Art. It is plain that such flanges must as it were intercept to some extent (depending on their inclination) the vert force at any point, and convert it into an oblique one along the flanges, somewhat as the oblique web members of an open beam do. To find these new strains at any point o, Figs A, B, of either an upper or lower oblique or curved flange, first ascertain by "Moments," the hor strain at that point for a beam with the depth o e; and by "Shearing," p 642, &c, the shear also. Then from that point o draw a hor line h equal by scale to the hor strain ; and from its end draw v vert and ending either at the flange (produced if necessary) if straight as in A ; or at a tangent / from o if the flange is curved as at B. Then will / in either fig give by the same scale the longitudinal strain along the flange at o ; and h and v are the components of that strain. As a formula, the Rule reads thus, o being the angle formed by h and I at o. web at o. For exceptions, see Rein. The foregoing applies also to oblique flanges of open hor beams. In the hor triangular flanged beam D with a concentrated load at its free end, draw a o vert and equal by scale to the load, and draw o c hor. Now here the whole load rests upon the upper end a of the oblique flange a n, which therefore sus- tains all of it as an end load, which it deposits as vert press- shear: , at ?i, and thus entirely prevents it from exerting any aring force whatever upon any part of the beam. The searng orce waever upon any par o e eam. e shaded web is therefore of no use here. The line a c meas- ures the strain along the oblique flange ; a o the vert pressure at n ; and o c the hor pull of the load all along the upper flange a e. Also a o and o c are the components of a c. 650 KUTTER S FORMULA. a the abutments or supports these pulls along ca and cb become converted into ver- tical pressures, together equal to the load I ; and into hor pressures compressing a b. Here also the shaded web is unnecessary ; as would likewise be the case if the load were transferred to e, and a single vert post (shown by the dark line) provided to carry it down to c, as the string before carried it up to c - If there is no such post the web acts, and the strain on either oblique flange is found as for A and B. But it is only in a few similar cases that the oblique flange entirely supplants the continuous web. If umber give* for finding the strain at any point of an oblique or curved web as follows. First find the shear as before as for a horizontal flange. Then If the coiiipressed./frm<7e is inclined down to the nearest suppwt, or Jfthe stretched flange is inclined down from " take the diff between the vert component v and the shearing force. But Jfthe compressed flange is inclined down from the nearest support, or Jfthe stretched flange is inclined down to take the sum of the vert component v and the shearing force. Rein. Hence in these last two cases (which do not include any of our above figs) the vertical force on the web is increased. As Humber remarks, in girders or beams with curved or oblique flanges the greatest strain in the web is not always where the greatest shearing strain is produced. Art. 13. The moment of rupture at any point t in a bent piece R with a load upon or suspended from c, is equal to the load X its leverage 1 1, perp to c w. This moment tends to break the piece R at its cross section at t by tearing apart the fibres to the right of its neutral axis, and by compressing those to the left of it ; and to this mo- ment the piece R opposes the moment of resistance of that section as in the case of a beam. In an arelieil piece as S loaded at any one point o, draw on,om, also o w vertical and equal by scale to the load, and complete the parallelogram o e w c of forces. Then will o e and o c by the same scale give two forces into which the load is resolved, and acting in the directions o TO, on, much as the two strings of two bows o a w, o u m. The force o e tends to break the bow o u m at any section u with a moment = the force X its leverage u v drawn from the point, and perp to o ra; and the force o c tends to break o a n at any point in the same way with its leverage. The section at u or elsewhere resists as in R. The weights of the pieces R and S themselves have not been taken into con- sideration. KUTTER'S FOEMULA, Kiit tor's general formula for the mean velocity in feet per second in pipes, aqueducts, canals, rivers, &c. This formula is the joint produc- tion of two eminent German engineers, Ganguillet and Kutter, but for conven- ience is usually called by the name of the last. Here 11 is the coef for roughness of sides of pipe or channel as given by the table below. The slope is the quo- tient arising from dividing the fall in any portion of the length by the length of that portion. The wet perimeter'is the length in ft found by measuring across the channel such parts of its sides and bottom as are in contact with the water; see Art 21, p 564. The Mean Radius is the quotient arising from the area of cross section of the water divided by the wet perimeter. In pipes running full it is always equal to one-fourth of the bore. All the dimen- sions must be in ft. Then by Kutter's formula FORMULA. 651 V/Mean Had X Slope. I/ Mean Rad Table of n, or eoeffs of Roughness of Wet Perimeter. .009 for well planed timber. .010 " plastered with neat cement; also for glazed pipes. .011 " plastered with 1 measure of sand to 3 of cement. .012 " unplaned timber, or unlined cast-iron pipes. If the pipes are very smooth, .011 may be used. .013 " Ashlar or Brickwork. .017 " Rubble. .020 u Canals in very firm gravel. .025 " Rivers and Canals in moderately good order and regimen, and perfectly free from stones and weeds. .030 " Rivers and Canals in moderately good order and regimen, having stones and weeds occasionally. .035 " Rivers and Canals in bad order and regimen, overgrown with vegetation, and strewn with stones or detritus of any sort. Rem. To avoid t lie use of this troublesome formula, at least in the case of clean cast-iron pipes, the writer would remark that he has found that for any head not less than at the rate of 4 ft per mile, or say .9 inch in 100 ft, (which is of very rare occurrence for pipes) it gives results for a pipe of 1 ft diam so nearly identical with our Table on p 539, that said Table (alter 4 ft per mile) may be considered as drawn up from Kutter's formula, using .012 for n. For diams greater than 1 ft his formula gives greater vels, and for smaller diams smaller vels than that by Poncelet, near top of p 538, from which our Table p 539 was prepared. But the writer fortunately also finds that when the head is not less than at the rate of 4 ft per mile, we may use the simple formula near top of p 538, and multiply the vel thus found by the corresponding number in the Table below the Rule for vels on said page. The product will be Kutter's vel. For the same reason we add the following table of vels in sewers. Remarks on Kutter's Formula. . - also shows that the foundations of the clusive, but cannot be given here. H ves w proay seom er more an rom whereas in quite possible cases the old formulas would err 50 or even 100 per cent. As we have before remarked, extreme accuracy is not to be expected in such mat- ters ; but we almost always may and should ensure that the error shall at least be be necessar. ters ; but we almost always may and should ensure tnat tne error snail at jeasi oe on the safe side by making n a little larger than may be supposed to be necessary. If in that case the roughness of the channel afterwards proves to be less than we had supposed it might be, and the vel and discharge therefore greater than we need, the supply may be regulated by stop-gates. Inasmuch as the same degree of roughness, or n, diminishes the vel and disch to a greater proportional extent in small pipes or channels, especially in high vels, than in large ones, care should be taken that their sides be made as smooth as possible. 652 VELOCITIES IN SEWERS. VELOCITIES IN SEWEES, Table of vels in Circular Brick Sewers when running full, by Kutter's formula, p 650, but taking n at .015 instead of his .013, in consideration of the rough character of sewer brickwork generally. When running: only half full the vel will be the same as when full, but this is not the case at, any other depth whether greater or less. At greater ones it increases until the depth equals very nearly .9 of the diam, when it is about 10 per cent greater than when either full or half full. From depth of .9 of the diatn the vel decreases whether the depth becomes greater or less. At depth of .25 diam the vel is about .78 of that when full ; and then diminishes much more rapidly for less depths. All this applies also to pipes. The vel for any fall or diam intermediate of those in the table can be found by simple proportion. Original. Fall in ft per mile. 2 3 ] Mametei 6 rg in feel 8 12 16 20 Fall in ft per 100 ft. Velocities in feet per second. .1 .19 .27 .35 .50 .64 .89 1.10 1.34 .0019 .2 .30 .42 .53 .74 .93 1.26 1.56 1.84 .0038 .4 .46 .65 .80 1.08 1.39 1.81 2.20 2.60 .0076 .6 .59 .81 1.00 1.35 1.70 2.22 2.70 3.18 .0114 .8 .69 .95 1.17 1.57 1.94 2.56 3.08 3.60 .0151 1.0 .79 1.07 1.32 1.77 2.16 2.84 3.43 3.96 .0189 1.25 .89 1.21 1.49 1.98 2.42 3.17 3.8 4.5 .0237 1.50 .98 1.33 1.64 2.18 2.64 3.5 4.2 4.9 .0284 1.75 1.06 1.44 1.78 2.34 2.85 3.8 4.5 5.3 .0331 2.0 1.15 1.55 1.91 2.53 3.1 4.0 4.8 5.6 .0379 2.5 1.32 1.78 2.18 2.85 3.5 4.5 5.4 6.3 .0473 3.0 1.44 1.94 2.38 3.2 3.8 5.0 6.0 6.9 .0568 3.5 1.58 2.10 2.58 3.4 4.1 5.3 6.5 7.4 .0662 4. 1.68 2.2 2.7 3.6 4.4 5.7 6.9 7.9 .0758 5. 1.90 2.5 3.1 4.1 4.9 6.3 7.6 8.7 .0947 6. 2.06 2.7 3.3 4.4 5.4 6.9 8.3 9.6 .1136 7. 2.2 3.0 3.6 4.8 5.8 7.5 9.0 10.4 .1325 8. 2.4 3.2 3.8 5.1 6.2 8.0 9.7 11.1 .1514 9. 2.5 3.4 4.1 5.4 6.6 8.5 10.3 11.8 .1703 10. 2.7 3.5 4.3 5.7 6.9 9.0 10.8 12.5 .1894 12. 2.9 3.9 4.8 6.3 7.6 9.9 11.9 13.6 .2273 15. 3.3 4.4 5.4 7.1 8.5 11.0 13.3 15.3 .2841 18. 3.6 4.8 5.9 7.7 9.3 12.1 14.5 16.7 .3409 21. 3.9 5.1 6.3 8.4 10.0 13.0 15.7 17.9 .3975 24. 4.2 5.5 6.8 8.9 10.8 13.9 16.8 19.2 .4546 27. 4.5 5.9 7.2 9.5 11.4 14.8 17.9 20.4 .5109 30. 4.7 6.2 7.5 9.9 12.0 15.6 18.8 21.5 .5682 35. 5.0 6.7 8.2 10.8 13.0 16.8 204 23.2 .6629 40. 5.4 7.1 8.7 11.5 13.9 18.0 21.7 24.8 .7576 45. 5.6 7.5 9.2 12.2 14.8 19.1 23.0 26.3 .8523 50. 5.9 8.0 9.7 12.8 15.5 20.1 24.2 27.7 .9470 60. 6.5 8.7 10.7 14.1 17.0 22.1 26.5 30.3 1.136 70. 7.0 9.4 11.5 15.2 18.4 23.9 28.5 32.8 1.326 80. 7.4 10.1 12.3 16.2 19.7 25.5 31.0 35.0 1.515 90. 7.9 10.7 13.1 17.2 20.9 27.0 32.3 37.1 1.705 100. 8.4 11.3 13.8 18.2 22.0 28.5 34.1 39.1 1.894 A vel of 1O ft per sec = 600 ft per minute = 36000 ft, or 6.818 miles per hour. About 5 ft per sec is as great as can be adopted in practice to prevent the lower parts of the sewers from wearing away too rapidly by the debris carried along by the water. RIV, 653 R, Figs 3 p 654, shows the /urual shapes of rivets as sold.* The weights in the following table of course include the head ; hut the lengths, as usual, are taken " under the head/' or are those of the shanks only. In practice, discrepancies of 5 or 6 per ct in wt may be expected. From Carnegie Bros. & Go's "Useful Information," by C. L. Strobel, C E. E AND RIVETING. AND RIVETING, Length of Shank. Ins. / % | K IHnmct % erg of Rl M vets in ii % iches. 1 IK Vi , 3.0 8.5 Weight of 100 B [vets, in pounds. i/ 3.8 9.9 17.3 i 4 4.6 11.2 19 4 25.6 389 \/ 5.4 12.6 21.5 28.7 43.1 65.3 91.5 123 iz 6.2 13.9 23.7 31.8 47.3 70.7 98.4 133 37 6.9 15.3 25.8 34.9 51.4 76.2 105 142 2 1.1 16.6 27.9 37.9 55.6 81.6 112 150 K 8.5 18.0 30.0 41.0 59.8 87.1 119 159 s or tons. Or by a formula, l>iaiu ill ins =\ /~ Shearing force >Qcoef of safety \/ Ult shearing strength per sq inch X .7854 If the rivet is to be double-sheared, first mult only half the shearing force by the coef of safety. Then proceed as before. Or, near enough for practice, mult the diam in single shear by the decimal .7. The ultimate shearing- mi it for average rivet-iron may be taken at about 45000 tt>s, or 20.1 tons per sq inch of circular sheared section. Table of ultimate single shearing 1 strength of rivets. (market sizes), in single shear ; at 45000 Bbs or 20.1 tons per sq inch. This table is not to be used when as in our " Example," Art 5, the crippling strength of the rivet governs the strength of the joint. If the rivet is in double shear it will have twice the strength in the table. For the cliam in double shear to equal the strength in the table, mult the diam in the table by the decimal .7 ; near enough for practice ; strictly, .707. Diam. Ins. Diam. ins. fts. Tons. Diam. Ins. Diam. Ins. as. Tons. Diam. Ins. Diam. Ins. fts. Tons. H .125 552 .246 .562 11183 4.99 1 1.000 35343 15.8 .187 1242 .554 % .625 13806 6.16 1.062 39899 17.8 K .250 2209 .986 .687 16705 7.46 1V& 1.125 44731 20.0 .312 3452 1.54 ' A 4 .750 19880 8.88 1.187 49838 22.2 *4 .375 4970 2.22 .812 23332 10.4 IK 1.250 55224 24.6 .437 6765 3.02 % .875 27060 12.1 1.312 60885 27.2 1 A .500 8836 3.94 .937 31064 13.9 m 1.375 66820 29.8 RIVETS/AND RIVETING. 657 The tensile stren&m. of a properly proportioned joint is equally as either the sessional area of the net plate (not covers) across the cen- ters of only one rowjpi rivets j or as the shearing or the crippling (as the case may be) areas of alL-die rivets in a lap, or of all the rivets on one side of the joint-line in a DUK The tensile strength of fair quality of plate iron, before the rivet holes are s^nlade, averages about 45000 fts, or 20.1 tons per sq inch ; but we shall for safety assume, as stated in Art 2, that the making of the holes reduces the strength of the net iron that is left about one-seventh part, or to 38500 ft>s, or 17.2 tons per sq inch. Rent. Even this is considerably too great for laps, or for butts with one cover, owing to the weakening of the iron in such by the bending shown at W, Figs 3. But we are not speaking of such. See Art 7. Art. 4. The friction between the plates in a lap, or between the plates and the covers in a butt, produced by their being pressed tightly together by the contraction of the rivets in cooling, adds much to the strength of a joint while new, perhaps as much as 1.5 to 3 tons per sq inch of circ section of all the rivets in a lap, or of all on one side of a single-cover butt ; or 3 to 6 tons of all on one side of a double-cover butt. In quiet structures, this friction might continue to exist, either wholly or in part, for an indefinite period ; but in bridges, &c, sub- ject to incessant and violent jarring and tremor, it is probably soon diminished, or entirely dissipated. Hence good authorities recommend not to rely on it, and it is, therefore, omitted in what follows. Art. 5. We now give rules for finding the number of rivets required for a double coyer butt-joint (the only kind of which we shall treat), and their clear or net distance apart. This dist -f one diam is the pitch of the rivets, or their dist from center to center. The principle of the rule will be explained further on, at Art 7. First, select a diam of rivet either equal to or greater than .85 times the thickness of the plate. In practice they are generally 1.5 times for plates % inch or more thick ; and 2 for thinner than % in. Second, mult the greatest pull that can come upon the joint by the coef (3, 4, or 6, &c) of safety, and call the product p. Third, multiply the crippling area of the rivet (that is, its diam X the thick- ness of plate) by 60000. The prod is the ult crippling strength of a rivet. Call it m. Fourth, divide p by m. The quotient will be the number of rivets to sustain the given pull with the reqd degree of safety. Then, the clear distance apart will be Diam X thickness of plate X 60000 Thickness of plate X 38500 Example. A double-cover butt-joint in .5 inch thick plate is to bear an actual pull of 33750 ft>s, with a safety of 4 ; or not to break with less than 33750 X 4 = 135000 ft)s. How many rivets must it have ; and how far apart must they be? First, Here .85 times the thickness of the plate is .5 X .85 = .425 inch ; there- fore, our rivets must not be less than .425 inch in diam ; but we will take .75 inch diam. Second, The greatest pull X coef of safety = 33750 X 4 = 135000 Ibs = p. Third, The crippling area of a rivet X 60000 = .75 X -5 X 60000 = 22500 = m. Fourth, -*-- = TTo^T^r = 6 rivets required on each side of the joint-line. And the clear space or net width between them will be Diam X thickness of plate X 60000 _ 22500 _ Thickness of plate X 38500 = 19250 . ' And the pitch = net space + diam = 1.1688 + .75 = 1.9188 ins, = 2.56 diams. , ** In practice, to avoid troublesome decimals, we might make the net space 1.2 ins; and the pitch 1.95; but to show farther on the working of the rule, we ad- here to the more exact ones. The entire width of net iron, if all 6 rivets are in one row, is equal to one clear space X number of rivets, = 1.1688 X 6= 7.0128 ins; and the entire width of plate is equal to one pitch X number of rivets, = 1.9188 X 6 = 11.5128 ins. The net width X by the thickness of plate (7.0128 X .5) gives 3.5064 sq ins area of net iron ; and this area X by our tensile unit, 38500 fas, gives 135000 fts as the ultimate pull required to break it, as in the beginning of the example. 658 RIVET8 AND RIVETING. The area of the entire unholed plate is 11.5128 X .5 = 5.7564 sq ins; and its tensile strength before the Holes are made is 5.7564 X 45000 = 259038 fts. 1 85000 The strength of our joint, omitting friction, is therefore ^TTTTT: = .52 of that of the original unholed plate. If we place the 6 rivets ill 2 rows of 3 each, we shall have hut 3 rivet-holes in a straight line across the plate ; and as the united diams of the 3 holes thus dispensed with is equal to (3 X '75) = 2.25 ins, the width of net iron in a row is increased that much. This of course increases the strength of the net iron, but not that of the joint; because the latter (as proportioned by the rule) depends precisely as much on the crippling strength" of the rivets as on the tensile one of the net iron ; and inasmuch as the 6 rivets will now yield under the same pull as before, it is plain that the joint is no stronger than before. But the 2 rows are the most economical, because we may reduce the width of the joined plates 2.25 ins, or from 11.5128 to 9.2628 ins, without diminishing either the number of rivets, or the calculated necessary net iron in one row. The ult strength of this net iron in one row is still 135000 ; but that of the unholed plate of reduced width is now (9.2628 X .5 X 45000; = 208413 Ibs; so that the strength of the joint is now -- 5 = .65 of that of the reduced unholed plate. If we place the 6 rivets in 3 rows of 2 each, we may reduce the entire width of joint 3 ins, by saving 4 holes in the first row ; thus making it only 11.5128 3 = 8.5128 ins, still retaining the first required width (7.0128 ins) of net iron in that row. The ult strength of the entire unholed twice-reduced plate is then 8.5128 X -5 X 45000 = 191538 fts ; and that of our joint without friction is ~ = .7 of it. Art. 6. The distance apart of the rows, from center to center of rivets, should not be less than two diameters of a rivet-hole. Rem. 1. With our constants for tension, shearing, and compression, the rivets will not yield first by shearing in a double-cover butt (and of course in double shear), except when the diam is either equal to or less than .85 of the thickness of the plate, which will rarely happen. At .85 the crippling and shearing strength of a rivet are equal when using our assumed coeffs of crip- pling, shearing, and tension. Item. 2. Our example was chosen to illustrate the rule. It will rarely hap- pen in practice that the rule will give a number of rivets without a fraction ; or that may be divided by 2 and by 3 without a remainder. In case of a fraction, it is plainly best to call it a whole* rivet ; although the joint thereby becomes a trifle stronger than necessary. Or rivets of a slightly dirf diam may be used. If the number of rivets comes out say 7 or 9, we may make 2 rows of 3 and 4, or of 4 and 5, &c. Moreover, the width of the plate is frequently fixed beforehand by some requirement of the structure, and we must arrange the rivets to suit, taking care in all cases to maintain the calculated area of net iron in one row, &c. Rem. 3. We have (as we at first said we should do) confined ourselves to the simple butt-joint with 2 covers, and with the .JC _ 5 _ . rivets in either 1, or in 2 or more parallel rows on each side of the joint-line ; this being the strongest and the one in most common use in engineering structures. Necessity at times calls for less simple arrangements, for which we cannot afford space, and the strength of which is not so readily calculated. These sometimes yield results which appear strange to the uninitiated ; thus, this lap-joint breaks across the net iron of one plate, along either c c or o o, where there is most of it, and where, therefore, it might be supposed to be the strongest. Rem. 4. The following 1 table shows approximately the comparative strengths of the common forms of joints when properly proportioned ; varying with quality of sheets, and of rivets : With friction. The original uuholed plate 1.00 Double-riveted butt with two covers 80 Double-riveted butt with one cover 65 Single-riveted butt with one cover 50 Double-riveted lap 65 Single-riveted lap 50 Without friction. 1.00 .64 .52 .40 .52 .40 RIVI RIVETING. 659 Rein. 5. The irnately attain* double- abular strengths for the lap-joints will be approx- ng the following proportions, according as the joint is ^ Calling thickness of plate Double rlv In thicknesses. zigzag. In diams. Single In thicknesses. rlv. In diams. 1. 1.67 9.0 7.0 2.0 3.33 .6 1.0 5.4 4.2 1.2 2.0 1. 1.67 5.67 4.5 2.0 .6 1.0 3.4 2.7 1.2 Then make diani of rivet " " breadth of lap " " pitch from cen to cen " " dist from end of plate to edge of holes . " " dist apart of rows from cen to cen Item. 6. If two or more plates on top of each other, as the four in A B or M H, are to be jointed together so as to act as one plate of the thickness c e, the diams of the rivets, and the thickness of the covers cc.ee will depend upon whether the junctions of the plates are all in one line with each other as at c c, in A B, or whether they break joint with each other as at 0, 1, 2, 3 in M H. It is plain that the two covers c c by means of their connecting rivets convey from A to B, across the joint c c, all the strength that partly compensates for the severance of the four plates at that joint ; whereas the two covers e e, e e, and their rivets in like manner convey from n of one single plate, to o of the adjoining one, across the joint between those two letters, only the strength that partly com- pensates for the severance of that single plate ; and so with the joints at 1, 2, and 3. Therefore the covers c c, and their rivets, must be four times as strong as those at any one of the four joints 0, 1, 2, 3. The first, c c, are to be regarded as joining two solid plates A and B, each of the fourfold thickness c c ; and the others as joining two of the single thickness. The covers c c will, therefore, each be about two-thirds of the thickness c c ; and the others each about two-thirds as thick as a single plate. Hence, when any number of superimposed plates break joint, the covers and diams of rivets may be of the same as when only two plates of single thickness are jointed. Since the weakest point of a structure is the measure of its effective strength, the tie A B is weaker than M H ; for although M H has four weak cross-sections, namely, through all four plates at each of the joints 0, 1, 2, 3, yet each section is weakened only by the loss of part of the strength of a single plate; whereas the section c c is weakened by the loss of an equal part of the strength of each of the four plates; and is, therefore, weaker than any other sec- tion entirely across either A B or M H ; and consequently A B is weaker than M H. Art. 7. Principle of the Rule in Art 5. Omitting friction, and having the proper coefts of tension, shearing, and crippling, it is plain that when we equalize the tensile strength of the net plate across one row of rivets, with the single or double shearing 1 strength of all the rivets in a lap, or of all on one side of the joint-line in a butt, we have for the net width or space between twe rivets, Net space againstvy' Thicknessv/ Tension _ Once or twice the cir-vy Shearing shearing /\ of plate S\ unit cular area of a rivet /\ unit. From this follows, by transposing, _ Once or twice the circ area of a rivet X Shearing unit R 1 1 shearing- Thickness of plate X Tension unit. And, also, that when we equalize the tensile strength of net plate with the 660 RIVETS AND RIVETING. crippling strength of the rivets, we have, calling the diain of a rivet X thickness of plate, its crippling* area. Net space against \s Thickness \/ Tension Crippling area \s Crippling crippling S\ of plate s\ unit of a rivet /\ unit. From this follows, by transposing, Rule 2. ye ag^inc 8 t Ce ii: Cripplipg area of a rivet X Cri PP lin g imit crippling* Thickness of plate X Tension unit. Now take any diam of rivet, and find its circular area. Mult this circ area by the shearing unit; divide the prod by the prod of diam X crippling unit. The quotient will evidently be the thickness or plate (not cover) which, with the assumed units, will make the shearing and crippling strengths equal to each other Assumed diameter in single shear. And then - .-. - r - will give the fixed proportion in Thickness of plate single shear which (with said units) any diam must bear to any given thickness of plate whatever, to ensure equality between the crippling and shearing strengths. With our assumed units of 45000 Ibs per sq inch for shearing, and 60000 fts per sq inch for crippling, the diameters for equality will be found to be 1.7 times the thickness of the plate when in single shear ; and .85 when in double shear. If the diam is less than 1.7 or .85 of the thickness of the plate, as the case may be, the foregoing calculation will show that the shearing strength of the rivet is less than its crippling strength, and that, therefore, it will give way by shearing. Hence, we equalize the net strength of the plate with the shearing strength of the rivet by using Rule 1. If, on the other hand, the diam is more than 1.7 or .85 of the thickness of the plate, as the case may be, the calculation will show the crippling strength of the rivet to be the least ; aftd then we use Rule 2. Rem. 1. Butt joints in double shear, or with 2 covers, being the only ones here considered, and inasmuch as rivets may always be used with a diam greater than .85 of the thickness of the plate, we may in practice always use Rule 2 for such joints ; and, therefore, we gave it alone. Item. 2. When using these rules for other kinds of joint, such as laps, or butts with single covers, remember that the rivets in such are in single shear; and, therefore, we can use Rule 2 above (for crippling) only when the diam is either 1.7 or more times the thickness of plate. If less, use Rule 1 above for shearing*; all on the assumption that our foregoing coefs of crippling and shearing are used. But the coef for tension must be changed for each kind of these other joints, to allow for the weakening effects of the bending shown at W, Figs 3, as deduced approximately from experiment. The writer believes that the fol- lowing tension units will give safe approximate results without friction. For double-cover butts, double-riveted, 38500 Ibs per sq inch, as adopted above. For double-riveted laps, or one-cover butts, 28000. For single-riveted laps, or one-cover butts, 24000. But, as before remarked, no great certainty is attainable in riveting. Rein. 3. A joint may fail by crippling without the facts being known or even suspected, for it does not imply that anything breaks, but merely that the joint has stretched ; and this might not be detected even on a slight inspection of it. Still it might, and probably often has sufficed to endanger, and even destroy both bridges and roofs by generating strains where none were provided for. LOADS/ON PROPS. 661 LOADS ON PROPS. A load resting on two props either at its ends or otherwise. When a load c of any shape whatever, rests in any position upon two props x and z, the portions of its wt borne by the respective props will be to each other inversely as the horizontal distances ox, oz, from the cen of grav o of the load, to the props. Thus if o z is two, three I or four times as great as o x, then will x bear two, three, or four times as much of the entire wt of the load c as z does. There- fore to find bow much each prop bears, first find the cen of grav c of the entire load ; and its hor dist (say o x) from either one oT the props, (say x.) rph The entire hor dist x z . Entire wt . ru t . Wt borne by the ** between the two props of load * * * * other prop z. And this wt taken from the entire load leaves the wt borne by x. This all amounts to the same as if we consider x z to be the clear span of a beam without wt, and supporting a load equal to c, concentrated at the cen of grav of c. See Bern, p 478. Conversely, to place two props x and z so that each may bear a given por- tion of the entire load c, take any two hor dists o x and o z from the cen of grav c, inversely as the two portions of the load to be borne by each. Thus if x is to bear two-thirds of the wt, make o z equal to two-thirds of x z. SYPHON ; continued from p 583. At Blue Ridge Tunnel, Virginia, Col. C. Crozet constructed a drainage syphon 1792 ft long of cast iron faucet pipes 3 ins bore, 9 ft long. Its summit was 9 ft above the surface of the water to be drained ; and its discharge end was 20 ft below said surface, thus giving it a head of 20 ft. At the summit 570 ft from the inlet, was an ordinary cast iron air-vessel with a chamber 3 ft high and 15 ins inner diam. In the stem connecting it with the syphon was a cut-off stop- cock ; and at its top was an opening 6 ins diam, closed by an air tight screw lid. At each end of the syphon was a stopcock. To start the flow these end cocks are closed, and the entire syphon and air-vessel are filled with water through the opening at top of air-vessel. This opening is then closed airtight, and the two end cocks afterwards opened ; the cut-off' cock remaining open. The flow then begins, and theoretically it should continue without. diminution, except so far as the head diminishes by the lowering of the surface level of the pond, lint in practice with very long syphons this is not the case, for air begins at once to disengage itself from the water, and to travel up the syphon to the summit, where it enters the air-vessel, and rising to the top of the chamber gradually drives out the water. If this is allowed to continue the air would first fill the en- tire chamber, and then the summit of the syphon itself, where it would act as a wad completely stopping the flow. The water-level in the air chamber can be detected by the sound made by tapping against the outside with a hammer. To prevent this stoppage, the cut-off at the foot of the chamber is closed before the water is all driven out ; and the lid on top being removed the chamber is refilled with water, the lid replaced, and the cut-off again opened. The flow in the meantime continues uninterrupted, but still gradually diminish- ing notwithstanding the refilling of the chamber; and after a number of refill- ings it will cease altogether, and the whole operation must then be repeated by filling the whole syphon and air chamber with water as at the start. At Col. Crozet's syphon at first owing to the porosity of the joint-caulking, which was nothing but oakum and pitch, air entered the pipes so rapidly as to drive all the water from the chamber and thus require it to be refilled every 5 or 10 minutes; but still in two hours the syphon would run dry. The joints were then thoroughly recaulked with lead, and protected by a covering of white and red lead made into a putty with Japan varnish and boiled linseed oil. But even then the chamber had to be refilled with water about every two hours ; and after six hours the syphon ran dry, and the whole had to be refilled. In this way it continued to work. In the writer's opinion an inside, and probably an outside coating of the pipes and air-vessel with the coal pitch varnish, Art 34, p 581, would effect a great im- provement. 662 THE CORD OR FUNICULAR MACHINE. THE COED OE FUNICULAE MACHINE, Art. 1. Some allusion to this subject has already been made on ,_ which see. Theory requires that the cord, rope or string, &c, shall be ab- solutely flexible, inextensible, frictionless, without weight, and infinitely thin ; and that such pulleys, posts, pins or pegs, loops or rings as may be used with the cord shall also be absolutely frictionless ; and at times deyoid of weight unless said wt is included in one of the acting forces. These assumptions cannot of course be realized in practice, which however will agree with theory in propor- tion as we approximate to them ; and this we can frequently do so far as to render the theory of great use. We know that all cords have wt and thickness, and can be stretched ; and that so far from being flexible, they may require very con- siderable force to bend them around pulleys, posts, pins, &c. They also possess friction. Also that all pulleys, pins, sliding rings or loops &c, have more or less friction ; which when there are many of them, may entirely vitiate all calcu- lations. A pulley has, in itself, no advantage over a smooth cylindrical pin, or post (as the case may be) except that its friction being a rolling* one, is less than the sliding* friction which takes place between a cord and a pin, s, and holding m a, g a together at a with both hands ; or let m a and g a both hang vertically from the pulley, and let him hold their lower ends apart by one hand at eacli end. Now if each part of the rope will bear but a little more than 100 fts, and z v a little more than 200 tbs, he will be safe from falling. But if he lets go one end of the rope, putting it into the hands of an- 664 THE CORD OR FUNICULAR MACHINE. other man standing by to hold it ; or if he hooks one end to a projecting spike in a wall close at hand, the rope will break, and he will fall, because he at once doubles the strain along both the ropes a m n g a, and z v. Rem. 2. Fig 7 shows a device by which boatmen sometimes haul a boat out of the water, and up on to the beach, at a landing, when it is too heavy for their unaided efforts. The rope e m x n b is to be considered as horizontal in this case. One end b being fastened to the bow of the boat, the rope is carried past one smooth post n, to another at m, around which it makes a whole turn ; and a man stands at that end e to take in the slack while the others, taking hold of the rope midway between m and n, pull it into a position m a n in which, if the angle man exceeds 12O, (see Rem 3, Art 3) each component an, am of their force will exceed said force itself; and a strain equal to one of these components (except so far as it may be reduced by the friction of the rope against the post n) will be transmitted uniformly to the boat at b, drawing it a short distance up the beach. The rope is then straightened again from m to n by taking in the slack at e, and the operation is repeated as often as necessary. To find the strain a m or a n, divide the force by twice the nat cosine of the angle x a n, or x a m. Or to find how many times the strain exceeds the force divide the distance a n by twice the distance <|H y JV a x - Here a x represents only half the force, or half sr- ' j ^--'"'^1 the diagonal or resultant of a m and a n. / "~j| The force of a man pulling by jerks at a; or a W will average between 30 and 80 ibs. ._ _ Art. 5. We will now dispense with the fric- rlU I \A tionless pulley required by the principle of the * Uf| cord, and which would of itself roll along the cord until it came to rest at a point which would cause equal angles and consequently uniform strain throughout. We will substitute for it a tight knot which cannot slide, at a point a, Fig 8, which causes unequal angles bam,ba w, be- tween the direction a & of the force /, and the two parts a m, a n, of the cord. Here, completing the parallelogram 6 ca o, we find that the part a n of the cord is strained to an amount denoted by ac; and that same strain would affect any extension of the cord on that side, like that to v, Fig 4. The part a m would be strained equal to a o ; and that T} Q v |\4 strain would continue to the end m, or to the end i-in O, f pi of any extension of that part. Hence the strain is not uniform from end to end of the cord, as it would have been if the knot had been frictionless; in which case it would have slid along the cord until the angles would be equal. Rem. 1. But even in this case of a tight knot at a, there is always one direction, as as, in which the force can be imparted so as to cause equal an- gles sam,san, with the direction a s of the force t, and then the strain will be uniform from end to end, as if a frictionless pulley had been used. Rem. 2. With a tight knot at a it is plain that a force may be imparted there from any direction as/a, ta &c. If the direction coincides with either part a m, or a n of the rope, that part will bear all the strain ; the other part remaining entirely free. Rem. 3. From Rule 1, p 27, for drawing an Ellipse, it will be seen that at whatever point as a, Fig 8, we apply force to an in extensible cord nam with fixed ends, that point will be in the circumference of an ellipse, the foci of which are at the ends m t n. CENTERS FOR/ARCHES. 665 CENTEM-FOB AKCHES, Art. 1. A center'IsXiemporary wooden structure (built lying flat, on a full size drawing, on a fixejr platform, under cover or not) for supporting an arch while it is being builtX^It consists of a number of trusses or frames,/, f, Fig. 1. placed from 1 to 6Jf\ apart from cen to cen, and covered with a flooring /, I, of rough boards or planks, usually laid close, and called the sheeting or lag- ging, immediately upon which the archstones are laid. In Fig 3, the lag- ging is not laid close. There is no great economy in placing the frames very far apart, on account of the greater required amount of lagging, the thickness of which increases rapidly. For the thickness of lagging see Rein. 9, Art. 8. The frames are of many designs. Thus Figs? and 9, pp 250, 251, are often used for small spans (say 15 to 25 ft), their upper timbers supporting throughout their length planks on edge, with their upper edges trimmed to conform to the curve of the arch. Fig 14, p 260, cov- ered in the same way, is some- times used for still longer spans, say 25 to 40 feet ; also Fig 28, p 283; Fig 31, p 284; and Fig 35, p 286, for still longer ones. InFig 1, /, /, are these frames as seen in place transversely of the arch a a. They rest by the ends of their chords, c, upon wooden striking wedges w, Fig 1, supported by standards com- posed of posts p, whose tops are connected by cap-pieces o; and whose feet rest on string- ers s; the whole being braced diagonally as shown. If the ground is very firm, and the arch light, the standards may rest on it, with the interposition of adjusting-blocks, n, be- low the stringer, to accommodate irregularities of the surface of the ground, as in the Fig. These blocks should be somewhat double- wedge-shaped, so that by driving them the standard may be raised at any point in case it should settle a little into the ground. But for heavy arches the standards must rest on a much firmer foundation, such as short blocks of brickwork sunk a few feet into the ground, or some other device adapted to the case. Frequently projecting offsets or footings, or at times re- cesses, are provided in the masonry of the abutments and piers for this express purpose ; and with a view to this it is well to design the center at the same time as the arch. Knowing the wt of the arch the proper dimensions of the posts may readily be found by the table p 239. Up to spans of 50 or 60 ft a single row of rjpsts (one under each end of each frame) will suffice ; but for much larger ones two or three rows, 2 or more feet apart may be- come expedient, as in the lower Fig 2. The striking or lowering-wedges before alluded to are for striking or lowering the center after the completion of the arch. They consist of pairs of wedge-shaped blocks, w w, at A, Figs 2, of hard wood, from 1 to 2 ft long, about half as wide, and a quarter or more as thick, (sufficient to lower the center from say 2 to 6 or more inches, according to span and other circumstances,) rest- ing on the cap o, of the standard, while the chord c of the frame rests on them. When the end of a frame is supported by two or more posts p, as at B, Fig 2, instead of upon one, the striking-wedges are sometimes made as there shown ; and where B v is one long wedge at right angles to the abutment, and acting as four wedges which may all be low- ered together by blows against the end B. Up to spans of 60 or 80 ft, all the frames may rest on but two wedges like B t>, Figsl. f o 666 CENTERS FOR ARCHES. each so long as to reach transversely across the entire arch. Then all the frames can be lowered at one operation, as described near end of Art 9. If we had to consider only the friction of dry wood against dry wood, the taper of these wedges might be as steep as 1 vert to 3 hor, without any danger of their sliding upon each other of their own accord ; and they would then require very moderate blows to start them, or even to entirely separate them, when the center had finally to be lowered. But it is of the utmost importance, especially in large arches, that the centers should be lowered very slowly, otherwise the momentum acquired by so heavy a body as the arcli in descending suddenly even but 2 or 3 ins, might possibly affect its shape, or even its safety. Therefore the wedges should not have a taper steeper than about 1 in 6 or 8 for arches of less than about 50 ft span ; or than 1 in 8 or 10 for larger spans. Vertical lines at equal dists apart should be drawn on the long sides of the wedges as a guide for lowering them all to the same extent at a time ; and this should not ex- ceed in all about half an inch a day in intervals of about an eighth of an inch, for 50 ft spans ; or about .1 to .25 of an inch per day in all, for spans over 100 ft. Slowness is especially to be recommended in brick arches, not only because their greater number of joints exposes them to greater derangement of shape, but because even good brick has much less than the average crushing strength of good granite, limestone, or sandstone, and therefore is far more liable than they to crack, or even to crush (as the writer has seen) when the strains are thrown almost entirely upon their edges, as described, in Art 3. For more on brick arches, see Art 9. At Gloucester Bridge, England, of first class cut stone, span 150 ft, rise 35 ft, the centers were entirely struck within the very short space of 3 hours ; and the crown of the arch descended 10 ins! At Crrosveiior Bridge, England, of first class cut stone, span 200 ft, rise 42 ft, such care was taken in easing the centers that the crown of the arch settled but 2.5 ins. This case however was sagging or settlement, consisted essentially of vertical and in- clined posts or struts, see Fig 3, footing on four temporary piers of masonry, 7 or 8 feet thick, built in the river, parallel to the abut- ments, and as long as they. These piers supported six frames (or rather six series) about 7 ft apart cen to cen, of such struts, footing on cast iron shoes. Fig 3 shows half of one series. Each frame or series consisted of four fan-like sets of posts, all in the same ver- tical plane. The long horizontal pieces seen extending from side to side of the arch were bolted to the struts to increase their stiffness ; and other pieces for the same purpose united the six series transversely. Here each strut sustains its own share of the weight of the archstones, and transfers it directly to the unyielding foundation of the pier ; whereas in the usual trussed centers, the entire load rests upon the frames, and is finally transferred to the comparatively unstable support of the posts at their ends. The tops p of the posts of a series varied about from 5 to 8 ft apart cen to cen ; and were connected by a continuous curved rib, rr, of two thicknesses of 4 inch plank, bent to conform approximately to the curve of the arch. On this rib were placed pairs of striking-wedges w like Fig 2, about 16 ins long, 10 to 12 ins wide, and tapering 1.5 ins, so near together (varying about from 2.5 to 3.5 ft cen to cen) that there was a pair under each joint of the archstones, a a. On these wedges, and ex- tending over all six of the frames, were the lagging pieces /, 4.5 ins thick. This peculiar arrangement of the striking-wedges and Ia ging has, in large spans, great advantages over the usual one of placing them only at the ends of the frames. In the last the entire center and the entire arch are lowered together, without giving an opportunity to rectify any slight derange- ments of shape or inequality of bearing that may have occurred in the arch during its construction, This center, designed by Mr. Trubshaw, admits of lowering either the whole equally, or any one part a little more or less than the others. He had much experience in large arches, and stated that during the striking he found that he had an arch under better control, or could humor it better, by keep- ing the haunches a little down, and the crown a little up, until near the end of the operation. CENTERS FPK7ARCHES. 667 atom. 1. Instead of piers ofinasonry for supporting the feet of the posts, wooden cribs or pihj*iiiay often be used if the arch is over water. The priii cip I ejw suitor ting; even trussed frames by struts at points of the chi>ra as far rom the abutments as circumstances will admit of (in addition to thQse at the/very ends) should always be applied when possible, in order to reduce their/sagging to a minimum. Stops or offsets in the masonry of the abujbments and piers may be provided for receiving the feet of such struts, when tHey are inclined. Rem. 2. Screws may be used instead of wedges for lowering centers. At the Pont d' Alma^Paris, ellipse of 141.4 ft span, and 28.2 ft rise, the frames were sup- ported by woodfen pistons or plungers, the feet of which rested on sand con- fined in pi sc to- iron cylinders 1 ft in diam and height, and having near the bottom of each a plug which could be withdrawn and replaced at pleasure, thus regulating the outflow of the sand and the descent of the center. This de- vice succeeded perfectly, and is well worthy of adoption under arches exceeding about 60 ft span. When much larger than this the driving of the wedges on striking requires heavy blows, and becomes a somewhat awkward operation, re- quiring at times a battering-ram, even when the wedges are lubricated. In rail- road cuttings crossed by bridges, the earth under the arch has been made to serve as a center, by dressing its surface to the proper curve, and then embedding in it curved timbers a few feet apart, and extending from abut to abut, for supporting the close plank lagging. Rem. 3. All centers must yield or settle more or less under the wt of the arch, especially when supported only near their ends; and since the arch itself also settles somewhat not only when the centers are struck, but for some time after, it is advisable to make them at first a little higher than the finished arch is intended to be. This extra height, when the supports are at the ends, may be from 2 to 4 ins per 100 ft of span for cut stone arches (according to time of striking, character of masonry, workmanship, etc.), and about twice as much in brick ones. Rem. 4. The proper time for striking centers is a disputed point among engineers, some contending that it should be done as soon as the arch is finished and sufficiently backed up; and others that the mortar should first be given time to harden. It is the writer's opinion that inasmuch as in cut-stone arches the mortar joints should be very thin ; and in which, in fact, the mortar is at best of very little service, it is of no importance when they are struck ; but that in brick or rubble, the numerous joints of both of which require much mortar, (which for hardness should consist largely of cement,) 3 or 4 mouths, or longer, if possible, should be allowed it to harden sufficiently to prevent undue compression and consequent settlement when the centers are struck. The con- tinuance of the centers need not interfere with traffic over the bridge. Art. 2. The pressure of archstones against a center is very trifling until after the arch is built up so far on each side that the joints form angles of 25 or 30 with the horizontal. Theoretical discussions on this pressure make no allowance for accidental jarrings in laying the archstones, or by the accumulation of material ready for use, laborers working on it, &c. Without going into any detail, we merely advise on the score of safety not to assume it at less than about the following pro- portions or ratios to the weight of the entire arch, namely, in a semicircular arch .47; rise .35 span, .61; rise .25 span, .79; rise .2 span, .86; rise .167 span, or less, 1, or equal to the wt of the arch. This gives the pressure of a semicircular arch upon its 'centers rather less than half its wt. Ihe wt of the centers themselves when supported only near the ends must be considered as part of the load borne by them. Art. 3. We have seen that as an arch a a a is being gradually built upward on both sides, after passing the points e, e, Fig 4, where its joints form angles a s e, of about 30 with the horizontal a a, the arch begins to press more and more upon the centers ; thereby tending to flatten them at the haunches, as shown at h in the dotted line ; and consequently to raise them at the crown, as shown at c. But as the building goes on still higher, the added stones press much more heavily upon the centers than those below had done, and thereby tend to a final derange- ment of the centers just the reverse of that caused by the lower ones ; namely to depress them at the crown a, as at o; and consequently to raise the haunches as at n; and this the more because the upper stones actually tend to lift or ease the lower ones from the lagging. In some cases where this tendency has been in- creased by forcing* the keystones into place by too hard driving, the lagging under the haunches could be drawn out without any trouble before the centers were eased at all. On striking the centers this tendency to sink at crown and 668 CENTERS FOR ARCHES. rise at haunches is very apt to exhibit itself more or less dangerously in the arch- stones themselves, as in Fig 5, causing those near the crown to press very hard together at the extrados, and to separate from each other at the intrados ;" while near the haunches the reverse takes place. Hence the angles of the stones are frequently split and spawled off near c and h by this unequal pressure. These Rg 4. derangements are of course much more likely to be serious in high arches than in flat ones, especially if their spandrels are not sufficiently built up befoee lowering the centers. In the Grosvenor bridge, before alluded to, of 200 ft span, this dangerous excess of pressure near c and h was prevented by covering the skewback joint of the springing course at each abutment with a wedge of lead 1.5 ins thick at the iu- trados of the arch, and running out to nothing at the extrados. Beside this a strip 9 ins wide of sheet lead was laid along the intrados edy;e of every joint until reaching that point at which it was judged that the line of pressure would pass from the intrados to the extrados ; after which similar strips were laid along the extrados edges of the joints, up to the crown. Hence when the centers were struck, this excess of pressure merely compressed the lead, and was thus enabled to distribute itself more evenly over the entire depth of the joints. See Trans Inst Civ Eng London, vol i. At the bridge at Wen illy, France (of 5 elliptic arches of 120 ft span, and 30 ft rise), the centers were so radically defective in design that the arches sank 13.25 ins at crown during the time of building; and 10.5 ins more during and immediately after the striking ; or say 2 ft in all. Their construction made the striking very tedious and hazardous ; greatly endangering the lives of the workmen and the existence of the arches. Some of the joints at the extrados at the haunches opened an inch each ; and those at the intrados of the crown .25 of an inch. By the exercise of great care and humoring in lowering the centers, these openings were much reduced. Rem. 1. Chamfering the edges of the archstones diminishes the danger of their spawling off from unequal pressure ; as does also the scrap- ing out of the mortar of the Joints for an inch or two in depth be- fore striking the centers. Rem. 2. It is evident that in order to prevent, or at least to diminish the alternate derangements of the center, those of its web members which at first acted as struts near the haunches, Fig. 4, to prevent them from sinking as at A, must afterwards act as ties to prevent them from rising as at n; while those which at first acted as ties near the crown a, to prevent it from rising as at c, must afterwards act as struts to prevent it from sinking as at o. In other words, the principle of counter- bracing must be attended to as well in a frame or truss for a center, as in one for a bridge. If the web members are on the Warren or simple triangle system, as in Fig 23, p 271, this may be effected by making each member a tie- strut; or the Pratt, or the Howe system, Fig 35, p 286, may be used. Art. 4. From the foregoing it is plain that a simple unbraced wooden R ARCHES. 669 arcli. or curvedrife'Jg<t to bend Ins. nore than Ins. y 8 inch. Ins. 6 A 3% 4K 4% 5 5% 5 2 H m 3% 3J^ 4 4 12 2y 2V-4 2% JKg 3 3 jlx w| 1% 1% 2 2 2 5 4 1 V/8 m 1$ With thicknesses three quarters as great as these, the bending may reach a full quarter inch ; which may be allowed in dists apart of 3 or more ft. Rem. 1O. Centers are framed, or put together, (like iron bridges) on a firm, level temporary floor or platform, on which a full-size drawing of a frame is 674 CENTERS FOR ARCHES. first made. As eac'a frame is finished, it is removed to its place on the piers or abuts. from 35 to 50 ft high. It contains about 15400 cub yds of masonry.* Each center consisted of 7 frames or trusses of hemlock timber, of the Bowstring pattern, with lattice (Fig 33, p 285) web-members; and as nearly as may be, of the same span arid rise as the arches. They were placed 4.5 ft apart from center to center; and were supported near each end /, Fig 13 (a transverse section to scale) by a hemlock post p, 12 ins square. The bow was of two thicknesses bb of hemlock plank, 6 ins apart clear, in lengths of 6 ft, with their upper edges cut to suit the curve of the arch. Each piece was 4 ins thick, by 13.5 ins deep at its middle, and 12 ins at its ends. These pieces did not break joint; but at each joint were four % inch bolts, with nuts and washers, uniting them with chocks or filling-in pieces. The bow, bb, footed on top of the ends of the chords /; and the angle formed by their meeting (seen only in a side view) was (for about 2.5 ft horizontal and 5.5 ft vertical) filled up solid with vertical pieces, to afford a firmer base for resting the frame on n ; beyond which it extends (in a side view) about 18 ins. The chords /were of two thicknesses of 4X 12 hemlock plank, 6 ins apart clear, and most of them in two or three lengths; breaking joint, and with two % inch bolts, with nuts and washers, at each joint, for bolting them together, and to filling-in pieces. The wel> members of each frame were 26 lattices, o, of 3 X 12 inch hemlock, crossing each other about at right angles, at intervals of about 3.5 ft from center to center, and passing between the two thicknesses b b of the bow, and// of the chords. A few of the lattices were in two lengths, and the joints were not at the crossings. The lattices were connected at each crossing by two hard wood treenails 9 ins long, and 2 ins diam ; and one such, 18 ins long, passed through the intersection of each end of a lattice with a bow or chord. The first lattice foots aboi^t 4 ft from the end of a chord. They do not extend above the top of the bow. All the spaces between the two thicknesses of bow or chord, where not occupied by the ends of lattices, were completely filled by chocks, well spiked. , Each frame contained about 360 cub ft of timber; and weighed about 5 tons. They were very flexible laterally until in place, and braced together by 4 transverse horizontal planks spiked to their chords ; and by 5 others above them, spiked to the lattices. Until the keystones were placed, all the joints of the frames continued tight, under the pressure from the arch, and from the unfinished backing to the height of about 14 ft above the springing line ; but after the keystones were set, all the joints of the chords alone opened from .25 to .75 of an inch ; and at the same time the lagging un- der the haunches of the arches became slightly separated from the soffit of the masonry. Each center sank but a full inch at the middle, under the pressure from the arch and 14 ft of backing. The portion of the bridge above the piers was about two thirds completed before the centers were struck. There was one wedge w, w, (32.5 ft long, of 12 X 12 inch oafr) under each end of a center. It was trimmed to form 7 smaller ones w, w, each 4.5 ft long, and tapering 7 ins ; one under each end of each frame /. They played between tapered blocks a, a, of oak, 2 ft long, 1 ft wide, let 1 inch into the cap c, or into the piece , on which last the frames /,/, rested. The sliding surfaces were well lubricated with tallow when put in place. The weflgfes were struck with ease, at one end of a center at a time, by an oak log battering-ram 18 ft long, and nearly a ft in diam, suspended by ropes, and swung and guided by 4 men. They generally yielded and moved several inches at the second blow with a 3 or 4 ft swing. Although each wedge was loosened entirely within 2 or 3 minutes, thus lowering the centers very suddenly, yet on account of the X- This bridge, finished without accident, at the end of 1882, reflects much credit on William Lorenz Esq, Ch. Eng; on Mr. Charles W. Buchholz, Assistant in Charge ; and on the skilful and energetic contractors, William & James Nolan, of Reading, Penna. These last most cordially assisted the writer in making observations during the entire progress of the work. FOR ARCHES. 675 , U good charact^of the rodsonry, not the slightest crack of a mortar joint could after- wards be^fected iaXny part of the work. After three days the average sinking of the kp5mones waMmly .35 of an inch ; the least was 1^; and the greatest % of an inch. The heads and feet of the posts p compressed the hemlock caps c, and the sills, about % of an inch each, showing that for arches of this size the caps and sills had better be of some harder wood, as yellow pine or oak; although probably the compression was facilitated by the large mortices, 3 by 12 ins, and 6 ins deep. Art. IO. Brick Arches. Since even good brick fit for large arches has far less crushing strength than good granite or limestone, and is inferior even to good sandstone, while its weight does not differ very materially from stone, it is plain that it cannot be used in arches of as great span as stone can. Some of those already built, and which have stood for many years, have a theoretical co- efficient of safety of but about 3 ; whereas the authorities direct us not to trust even stone with more than one-twentieth of its crushing load. This last, however, ap- pears to the writer to be one of those hasty assumptions which, when once ad- mitted into professional books, are difficult to be got rid of. It is his opinion that with good cement, and proper care in striking the centers, one-tenth of the ulti- mate strength is sufficiently secure against even the abnormal strains caused by the settling at crown, and rising at the haunches when the centers are struck. It is useless to attempt to fix limits of safety for bad materials poorly put together. Item. 1. The common practice of building brick arches in a series of con- centric rings, as at a c e e, Fig 12, with no other bond between them than that afforded by the mortar, is censured by authorities, on the ground that the line of pressure in passing from the extrados to the intrados tends to separate the rings, and thus weaken the arch by, as it were, splitting it longitudinally. The reason for using these rings, instead of making the radial joints continuous throughout the depth m n of the arch, as at b, is to avoid the thick mortar-joints at the back of the arch, and shown in the Fig. If the center of an arch built as at b be struck too soon, the soft mortar in these thick joints will be so much compressed as to cause great settlement at the crown, throwing the arch out of shape, and creating such inequality of pressure as might even lead to its fall, especially if flat. As a compromise between rings and continuous joints, they are sometimes employed together, so as to get rid of some of the long radial joints ; and at the same time to break at intervals the continuity of the rings. Thus in Fig 12. which is supposed to be brick-and- a-half deep, beginning at the abutment a, we may lay half-brick rings as far as say to e o e; then cutting 1 away the brick o to the line e e, we may lay from. e e to m n a block of bricks with continuous radial joints, the same as at b; and then start again with three rings; and so on alternately. A still better, but more expensive, mode would be to fill ee,mn with a regular cut-stone voussoir. The proper intervals for changing from rings to blocks will depend upon the number of the rings and the depth c a of the arch ; reference being also had to reducing the amount of brick cutting as much as possible. These points can be best decided on from a drawing of a portion of the arch on a scale of 3 or 4 ins to a foot. Generally the rings are made only half-brick, or about 4 to 4.5 ins thick, as at a c; and in Brunei's Maidenhead viaduct of two ellip- tic brick arches of 128 ft span, and 24.25 ft rise ; the boldest brick arches yet at- tempted ; but which have been estimated to have a co-efficient of safety of but three against crushing at the crown. So many othersof from 70 to 100 ft span have been successfully built entirely in rings of either half or whole brick thick, as to justify us in attaching but little weight to the above theoretical objection, provided first class cement be used, and time allowed it to become nearly or quite as hard as the bricks themselves, before striking the centers. Under such circumstances we should not object to a series of rings even 1.5 bricks thick, laid alternately header and stretcher, as at 6. If the bricks were voussoir-shaped, that is, a little thicker at one end than the other, then rings a whole-brick thick could be used without any in- crease in thickness of mortar-joint at the back of each ring. Still with more than one ring, the radial joints would not be continuous, as at b, but broken as at ac. Such bricks however would be more expensive to make; and moreover, in order fully to answer the intended purpose, they would have to be made of many patterns, so as to conform to the many radii used in arches; and even to the radii of the different rings, when the depth of the arch required several of them. 676 TO MEASURE ANGLES BY A 2-FOOT RULE. To make voussoir-shaped bricks that shall insure continuity of radial joints with uniform thickness of mortar, in deep arches, is therefore (commercially speaking) impossible; and we must depend on good cement to overcome the difficulty. Item. 2. Wet the bricks before laying. See last paragraph of p 497. Item. 3. When the ends or faces of a brick arch are to be finished with cnt- Stone voussoirs, these had better not be inserted until some time after the completion of the brickwork, the hardening of the mortar, and a partial easing of the centers ; lest they be cracked or spawled by the unequal settlements of them- selves and the bricks. For more on brick arches see p 344. To Measure Angles by a 2 Ft Kule, etc, The four fingers of the hand, held at right angles to the arm, and at arras-length from the eye, cover about 7 degrees, Aud 7 corresponds to about 12. 2 ft in 100 ft; or to 36.6 ft iu 100 yds ; or to 645 ft in a mile ; or in the same proportion as the distance. The following: Table may sometimes be found useful for the rough measurement of angles, either on a drawing, or be- tween distant objects in the field. If the inner edges of a common two-foot rule be opened to the ex- tent shown in the column of inches, its edges will be inclined at the angles shown in the columns of angles. Since an opening of % of an inch up to 19 inches or about 105, corresponds to from about % to 1, no great accuracy is to be expected ; and beyond 105 still less ; the liability to error in- creasing very rapidly as the opening becomes greater. Thus, the last J^ inch corresponds to about 12. As to the table itself, angles for openings intermediate of those therein given, may be calculated to the nearest minute or two, by simple proportion, up to 23 inches of opening, or about 147. Table of Angles corresponding: to openings of a 2-foot rule. (Original.) D, degrees ; M, minutes. Correct. Ins. D. M. Ins. D. M. Ins. D. M. Ins. D. M. Ins. D. M. Ins. D. M. M 1 12 4>4 20 24 8^ 40 13 l2Ji 61 23 16^ 85 14 MX 115 5 1 48 21 40 51 62 5 86 3 116 12 M 2 24 X 21 37 M 41 29 % 62 47 H 86 52 % 117 20 3 00 22 13 42 7 63 28 87 41 118 30 % 3 36 K 22 50 % 42 46 *A 64 11 % 88 31 H 119 40 4 11 23 27 43 24 64 53 89 21 120 52 i 4 47 5 24 3 9 44 3 L 3 65 35 17 90 12 21 122 6 5 2:3 24 39 44 42 66 18 91 3 123 20 M 5 58 34 25 16 K 45 21 y* 67 1 34 91 54 34 124 36 6 34 25 53 45 59 67 44 92 46 125 54 M 7 10 M 26 30 % 46 38 M 68 28 34 93 38 34 127 14 7 46 - 27 7 47 17 69 12 94 31 128 35 H 8 22 H 27 44 K 47 56 H 69 55 M 95 24 H 129 59 8 58 28 21 48 35 70 38 96 17 131 25 2 9 34 6 28 58 10 49 15 14 71 22 18 97 11 22 132 53 10 10 29 35 49 54 72 6 98 5 134 24 v\ 10 46 H 30 11 M 50 34 34 72 51 H 99 00 H 135 58 11 22 30 49 51 13 73 36 99 55 137 35 % 11 58 H 31 26 H 51 53 y* 74 21 y* 100 51 H 139 16 12 34 32 3 52 33 75 6 101 48 141 1 H 13 10 M 32 40 % 53 13 H 75 51 *A 102 45 H 142 51 13 46 33 17 53 53 76 36 103 43 1M 46 3 14 22 7 33 54 il 54 34 15 77 22 19 104 41 23 146 48 14 58 34 33 55 14 78 8 105 40 148 58 34 15 34 M 35 10 J* 55 55 M 78 54 34 06 39 X 151 17 16 10 35 47 56 35 79 40 07 40 153 48 X 16 46 X 36 25 % 57 16 M 80 27 x 08 4] M 156 34 17 22 37 3 57 57 81 14 09 43 159 43 *A 17 59 M 37 41 % 58 38 H 82 2 H 10 46 H 163 27 18 35 38 19 59 19 82 49 11 49 168 18 4 19 12 8 38 57 12 60 00 L6 83 37 20 112 53 24 180 00 19 48 39 85 60 41 84 26 113 58 Or this table may be used thus. From any point measure 12 ft towards each object, and place marks. Measure the dist in ft between these marks. Suppose the first cols in the table to be ft instead of ins ; then opposite the dist in ft will be the angle. One-eighth of a ft is 1.5 ins. The following is a good way to measure an angle. Measure 100 or any other number of ft towards each object, and place marks. Measure the dist between the marks. Then As dist measured . 1 . . Half the dist . nat sine of toward one object -*-.. between marks Half the angle. Find this nat sine in the table of nat sines, take out the corresponding angle, and multiply it by 2. See near foot of p 41. TO FIND To FERENCES OF CIRCLES. 677 Ircuinljjrof circles when the diam contains decimals. Role. Find the/dircumf for the whole number by table, p. 18. Then for the decimal paryuse the following table : Diam. Circ. / Diam. Circ. Diam. Circ. i Diam. Circ. Diam. Circ. .1 .314159 .01 .031416 .001 .003142 .0001 .000314 .00001 .000031 .2 .628319 .02 .062832 .002 .006283 .0002 .000628 .00002 .000063 .3 .942478 .03 .094248 .003 .009425 .0003 .000942 .00003 .000094 .4 1.256637 .04 .125664 .004 .012566 .0004 .001257 .00004 .000126 .5 1.570796 .05 .157080 .005 .015708 .0005 .001571 .00005 .000157 .6 1.884956 .06 .188496 .006 .018850 .0006 .001885 .00006 .000188 .7 2.199115 .07 .219911 .007 .021991 .0007 .002199 .00007 .000220 .8 2.513274 .08 .251327 .008 .025133 .0008 .002513 .00008 .000251 .9 2.827433 .09 .282743 .009 .028274 .0009 .002827 .00009 .000283 Example. What is the circumf of a circle whose diam is 67.35824 ins, or feet, Ac. ? Here, first by table, p 18, we have for diam 67. Circumf = 210.487 Then by above table .3 .05 .008 .0002 .00004 .942478 = .157080 = .025133 = .000628 = .000126 Circumf required = 211.612445 Rem. This mode is correct within an error of less than 1 in the third, fourth, or fifth decimal, according to the number of decimals in the first circumf, taken from table, p 18. Thus the true circumf (found approximately to be 211.612) will not be as great as 211.613, nor as small as 211.611. Many things Abrasiortoy streams, 563, 570. Abutment-piers, 347. Abutments, to proportion, 345. Acceleration of gravity, 172, 449, 587. Acre, square and circular, 76. Acres drained by pipes, 569. required per mile for R R, 390. Action and reaction, 449. Adhesion of cement, 503. of glue, 620. of mortar, 497. of nails and spikes, 383. Adjustment of compass, 164. of hand-level, 167. of plumb level, 167. of slope instrument, 167. of spirit level, 154. of the box-sextant, 164. of theodolite, 162. of transit, 160. Air, 519. buoyancy of, 033. compressed, in cyls, 327, 631. lock, 327, 631. pressure of, in a diving-bell, 520. quantity of, for ventilation, 519. valve for water-pipes, 579. vessel, 615. Alioth, (star,) 100. Alligation, 72. Angle iron, 373. Angle-blocks of Howe truss, 283. complement and supplement of, 62. exterior and interior, Ac, 62. limiting of resistance, 487. of friction or repose, 453,'* 1 485, 598. of maximum pressure, 335. on sloping ground, 40, 41. salient, and re-entering, 62. to meas by a tape line, footnote, 41, 42. " " by a 2 ft rule, 676. " " by hand, 67G. " " by sextant, 41. ' without any instrument, 41. Angles of deflection, &c, 416. of depression, 40. Angular velocity, 447. Animal power, 605. INDEX. contained in the Index may be found in the Glossary. Anthracite, wt of, 76, 384. Apothecaries' weight, 74. Appendix, 630. Application, point of, 447. Aqueducts, flow in, 562. Kutter, 650. Pittsburg, 596. Arches, braced, 274, 287. BrieJe, 675. cast-iron, Chesinut St bridge, 288. " " Severn Valley R R, 288. " " Whipple's, 288. existing, 343. Centers for, 665. hor. pres. equal throughout, 493. iron, in roofs, two examples of, 289. large, rubble, 344, 506. line of pressure or thrust, 348, 493. " " " to find, 493. of corrugated iron, 371. pressures on key, 342, 468, 491. principles of, 341, 479, 491. J675. stone and brick for culverts, &c, 341, " " " tables of, 345, 351, &c. wooden, rule for, 307. Archstones, tables of, 343, 345. to find depth of, 341. Arcs, cen of grav of, 442. circular, tables of length, 21, 23. having span and rise, to find rad, 16. in common use, table, 434. large, to draw, 17, 67. Arithmetic, 69. decimal, 71. Artificial stone, Coignet's, 507. Ashlar masonry, cost of, 312, 630. Atmosphere, 519. Avoirdupois or commercial wt, 74. Axis of flotation, 635. Bag-scoop, or bag-spoon, 330. Bailing by bucket, day's work at, 606, Ballast for railroads, 414. Balloon, principle of, 533. Balls, weight of, 362, 366, 377. Barometer, levelling by, 167. levelling, tables for, 169, 171, Batter, 347. Beams) box, Fairbairn's, 214, 217. 679 680 INDEX. Beams, channel, 211, 213, 640. closed and open, 644. Cooper & Hewitt's, 213, 214. constants for breaking loads, 185. continuous, 641. Of concrete, 507. cylindrical, 186, 193. deck, 211 . deflections of, 191, 196. deflections of ^-Q of the span, 201. examined on the principle of the lever, 477. exposed to both transverse and lon- gitudinal strains, 190. general facts respecting strengths of, 186. Hodgkinson's, 208. hollow, 193. inclined, 188. limit of elasticity, 197. loads within elastic limit, 185, 198. moments of rupture and resistance, 217. parts may be cut away without loss of strength, 187. plate, 214. rolled I, 210, 212, 213. " as pillars, 638. " as short bridges, 304. shearing of, 181, 642. Btone, 185, 203. strength of, 183, &c. " of some experimental beams, 209. table of constants for safe deflec- tions, 199. " of loads not to bend more than -$1 -Q of the span, 204, 205. " of safe loads, 191, 203. tables of breaking loads, 206, 207. to find break'g load at any point, 188. " " center breaking load, 186. " " safe dimensions, 189. " " the breadth of a beam, 189. " " the depth " " 189. " " uniform load, 187. to splice, 291. Trenton, 213. triangular, 187, 188. Searing and reverse bearing not alike, 93. Bearing power of soils, 314. Bell-joints or faucets of pipes, 574. Bends in water-pipes, effects of, 539, 548. Beton, 504. Coignet's, 507. Blake's stone-crusher, 505. Board measure, table of, 357. Boats, cost, 604. Bodies, falling, 587. Table of vels, 552. floating, 533, 635. mass of, 456. regular, the solidities, &c, of, 38. Body, defined, 444. Boiling ^vater to meas hts by, 170. Bollman truss, 269, 282. Bolts and nuts, 374; to find diam, 180; copper, 375. strength of, upset and not upset, 375. Boring in soils by augur, 313, 636. test-holes, wells, 313, 636. Borrotv-pits,te measure beforehand,30. Bottoms to bear diff vels, 563, 570. Bowstring truss, 270, 286. Box beams, Fairbairn's, 214. drains, 355. sextant, 163. Bracing, horizontal diagonal, 291, 304. Brass, rolled, weight of, 376. sheets, weight of, 367. wire, " " 368. Brick Arches, 344, 675. Bricklaying, a day's work at, 498. Bricks, and mortar, 496. crushing strength, 175, 498. English rod of, 499. number in a sq ft of wall, 498. paving with, 498. prices in Philada, 498. proportions of brick and mortar, 496. tensile strength, 499. to render impervious to water, 499. weight of, 384, 498. Bridges, arches of existing, 343. Bollman, 269, 282. Burr, 289. bowstring, 270, 286. braced arch, 274, 287. coefs of safety, 298. camber, 302. Centers for, 665. cast-iron arches, 288. Chestnut St, Philada, 288. cub yds of masonry in ; tables, 351, 353, 354. - deflections, 302. depth of archstones, 341, 345. Fink, 281, 305. floor girders, 215, 291, 296. greatest load on, 297. Howe, 283, 284. iron, cost of, 300. lattice, 285. Bridges, Moseley, Pratt, 284, raising of, 303. spandrel walls of, 346. stone, 341. stone, drainage o^foadways of, 356. suspension, swing, strain's on, 275. to proportion stone abutments, 345. Town, 285. turning, friction rollers for, 431. Warren, 254, 279. weight of, 295. WissahicJton, 674. Whipple's cast-iron, N. York, 288. width and headway, 288, 306, 307. British imperial measures, 77. Brokerage, 73. Brunlee's iron piles, 326. Babble-glass, to replace, 162. Btickled plates, 369. Buildings, cost per cub ft, 631. Buoyancy of air, 533. of liquids, 533, 635. Buttresses, 340. Cables, number of wires in, 369. suspension, 588, 597. Caissons, 316. for East River susp. bridge, 328. Camber of trusses, 302. Canals, boats, cost, 604. flow in, 564. leakage of, 521. traction on, 604. Cantilevers, 219, 275, 643. Cars, axles of, 414. weight of, 413. Wheels, 413. Cart, weight of, 436, 608. Castings, to judge the wt of by the patterns, 362. Cast-iron, weight of, 362. pi pes, 363, 364. Ceilings, weight of, 248. Cement, adhesion of, 503. brick-dust, 496. effect of freezing on, 502. " in preserving metals, 500. for pointing, 501. for rough casting, 500. for stopping crtcks around chim- neys, 512, 514. hydraulic, 500. " cost, 500. mortar, 503, 508. Portland, how made, 500. 681 Cement, protection from moisture, 500. quantity reqd for mortar, 500. restoration by reburuing, 500. setting of, 501. strength of, 502, 503. Wt. 385, 500. Centers lor arches, 665. Center of buoyancy, 635. of force or pressure, 333, 482. of force of water, 526, 635. of gravity, 442, 481. of gyration, 495, 617, 622. of oscillation, 173. of percussion, 173. Centrifugal and centripetal force, 494. Chain., Gunter's, 74, 98. Chaining, deductions on sloping ground, 40, 98. Chains, wt and strength of, 381. Chairs, railroad, 390. sleeve, 392. " modified, 393. Wilson's, stop, 391. Channel-iron, as pillars, 236, G37, 640. wtof and strength as beams, 211, 213. Channels, flow in, 562, &c. Charing Cross Railw'y, girders on,215. Chord of 1 of earth's great circle, 22. Chords, long, 419. of a truss defined, 243. of flat iron bars, 293. principle of, 246. table of for protracting, 608. Circles, 16, 62, 66. By decimals, 677. earth's great, 22. table of, 18. to describe large ones, 17, 67, 434. Circular arcs, 17, 21, 23, 434. inch, 76. lunes, 25. ordi nates, to calculate, 20. rings, 17, 22, 32. sectors, 22. segments, areas of, 24. spindle, solidity of, 39. tables, of, 21, 23, 434. " in common use, 434. xouss, areas of, 25. Cisterns, 432, 434, 532. Civil, or clock time, 80. Clay, swelling of, 314. Clinometer, or slope instrument, 167. Cloth, tracing, 152. Coal and coke, wts of, 76, 384. corrosive fumes from, 370, 511. per H. P., 495. 682 INDEX. Coffer-dams, 317. earthen, 317. of cribs, 318, 319. of two enclosures of cribs, 319. Cog-wheels, power of a train of, 479. Cohesion, tables of, 177, 179, 180. Coiff net's beton, 507. Coins, foreign, value of, 81. Cold, effect of, on iron, 180. Colors, 152. Columns. See Pillars. Combination, 72. Commission, 73. Compass, to adjust, 164. variation of, 165. Compensation -water, 579. Composition and res of forces, 457. Compound levers, 479. Compressed air in cyls, 327, 631. Concrete, 504. As Beams, 507. at Croton darn, &c, 505. cost, 507. large arches built of it, 344, 506. mixing of, 506. modes of using under water, 506. proportion of void and solid, 504. ramming of, 505. strength of, 505, 507 weight of concrete, 507. Cones, 32. Conical screw-pan, for excavating, 326. Conoid, solidity of, 38. Constants for deflections of -%\^j of the span, 201. of rupture, 185, 195. Of Elas., 177. to find for beams, 183, Ac. Continuous beams, 641. Contour lines, 147. Contracted vein, 552, 554. Contrary flexure, point of, 641. Cooper and Hewitt beams, 213, 214. Copper, cost, 367, 368. for roofs, 377. pipes, weight of, 365, 378. weight of, 367, 376. Cord, or funicular machine, 463, 662. Corrugated iron, 370, 371. cost of, 371. Cost of buildings, 631. cements, 500. Of concrete, 507. of dredging, 330. earthwork, 435. hauling, 607. iron, 364. lumber, 361. Of R R, 416, Cost of masonry, 312, 630. Shops, 415. Counterbracing, 245, 252, 275, 306 Rm. Counterforts, 340. Couples, 483. Couplings for tubes, 365. Creeping of rails, 391. Creosote, 358. Crescent truss, 270. Crib dams, cost of, 586. foundations, 315. Cribs, 315. Cross-hairs, to replace, 162. Cross-ties, 414. Crowds, weight of, 297, 595, footnote. Crusher, stone, 505. Crushing loads, tablesof, 174,175, 176. Cube roots, rules for, 60. Table of, 48. Cubic foot, what equal to, 76. inch, what equal to, 76. or solid measure, 76. yards in a cutting 2 ft wide, 427. Culverts, tables of cub yds in, 351, 353, 354. to find lengths of, 350. Curvature of the earth, table of, 42. Curves, railroad, table of, 416, 633. Cuttings, level, table of, 420. to prepare a table of, 418. Cycloid, 29. Cylinders, 31. brick, 327. for foundations, 324, 327, 631. friction of, 323. riveted, strength as beams. 193. screw, 324. strength of, 193, 531. table of contents of, 46, 47, 77. with piles inside, 329. Cylindric beams, hollow, strength of, 193. cast-iron beams, table of loads, 207. ungulas, 31, 630. Cyma, to draw, 67. Dams, 583, 528. coffer, 318. discharge over, 558, 561. earthen, 317, 578. in California, 531, 578. one at Poona, &30. rise of water produced by, 587. sluices in, 686. stone, 528, 529, Ac. tremblings in, 586. Dam walls, high, in France, 529. SDEX. 683 rJtJ&srtwvet^tl) bailing, 606. Day's wort at bricklayi at dresfjitfgstone, 312. at-dflfling, 311. at excavating, 435. at hauling by a rope, 606. at plastering, 509. At tin roof, 379. at treadwbeel, tympan, 606. with a gin, winch, &c, 606. Decagons, 15. Decimals, 71. Inflection angles, fcc, 416. Deflections of beams, 191, 196. " bridges, 302. Degree, length of, in a mile, 22. of long and lat, 74. Dew point, 520. Diagonal horizontal bracing, 291. of a truss, to find length of, 69, 303. Dialling, 150. Discharge, canals, 561. pipes, 538. In sewers, 566, 652. under water, 554, note. Discount, 73. Distances by sound, 173. Distributing reservoirs, 579. Diving-bell, pressure in, 520. Diving-dress, cost of, 329. Dodecaedrons, 38. Dodecagons, 15. Dollar, 74. Draft, of vessels. 534, 635. on roads and canals, 603, 607. Drainage of roadways, of bridges, 356. Drains, box, 355. pipes, discharge by, 568, 569. should be well fouhded under high embankt, 355. terra cotta, 569. Drawbridge, strains on, 275. Drawing materials, 152. Dredged material, wt of, 330, 385, 386. Dredging, 329. by bag-spoon, 330. by screw-pan, 326. cost, 330. Drilling in rock, 311, 324, 636. Drop timbers, 585. Dry drains, 355. measure. 77. Dynamics defined, 444, 459, note. E and Wline, to run, 93. Earth, bearing power of, 314. embankment, settlement of, 630. friction on, 340, 603. [&c. natural slope of, 338. Pressure of, 331, Earth, table of curvature, 42. Earthwork, cost of, with tables, 435. Elasticity, limit of, in beams, 197. limit of in steel and iron, 176, 185. modulus of, defined, 177. " " table of, 632. Elevation of outer rail, 419. Ellipse, 25, 630; to draw, 27. Ellipsoid, solidity of, 38. Elliptic arcs, table of, 27. segments, areas of, 27. Elongation of north star, 99. Embankments, settlement of, 630. Energy, 459. Equality of moments, 476. Equation of payments, 72. Equilibrium, stable, &c. Note, 481, 635. Establishment of port. See Tides, 626. Evaporation, 521. of locomotives, 434. Excavation, cost of, 435. level cuttings, tables of, 420. 100 ft long, arid 2 ft wide, table, 427. Expansion links, rockers, rollers, 295. of iron bridges, 245. of solids by heat, 310. Eye-bars and pins, 293. Fairbairn beams, 214. on the safety of iron bridges, 298. table of box girders, 217. Falling bodies, 587. Vels of, 552. False-works, defined, 303, 619. Fascines, 328. Faucet, or bell-joints of pipes, 574. Fellowship, 73. Fences, 415. Ferrules for service pipes, 574. Figure, defined, 61 . irregular, to find the areas of, 16. or maps, to enlarge or reduce, 69. Filtration, leakage, &c, 521. Fink truss, 264, 281, 297, 305. roof, weight of, 298. 300. Fireplugs, or hydrants, 576. Fisher's rail-joint, 392. Fifih-j>lates, 391. Flexure, point of, contrary, 641. moating bodies, 533, 635. Floor girders for bridges, 215, 291, 296. Foot, and decimals of inches, table, 75. cubic, of substances, wt of, 384. " what equal to, 76. spherical, what equal to, 76, 77. 684 INDEX. Force, can be fully imparted only at right angles, 452. centre of, 333, 482, 526, 635. centrifugal, 494. conip and res of, 457. couples, 483. denned, 445. equality of moments, 476. great, is imparted gradually, 453, 456. impartation of, 447. in rigid bodies, 443. living, or vis viva, 446, 455. moving, 448. nothing can destroy but other force, 445. of a pile driver, 321. parallel, 481. parallelogram of, 458. parallelepiped of, 471. point of application, 447. polygon of, 467. " " resultant of, 466, &c. working, defined, 448, 454. Forces in different planes, 470. Foreign coins, value of, 81. Foundations, 313. by plenum process, 327, 631. by vacuum process, 326. in caissons, 316. on artificial islands, 328. on cribs, 315. on cylinders of iron, brick, &c, 327. on fascines, 328. on grillage, 634. on piles enclosed in a cylinder, 329. on random stone, 314. . on random stone and piles, 315. on sand piles, 328. safe loads on, 314. testing for, 313. Fractions, vulgar, 69. Francis, 7.^B.,hydraulic expts, 558, 560. French weights and measures, 78. Friction, 172 ; 456, near top ; 597. angle of, 453, 485, 598. at backs of walls. Kern 3, 332. head, 535, 543. hydraulic press, 632. in pipes, 551. journal, 601. masonry on wood and earth, 340, 603. of piles, 323. Parry's friction rollers, 429. pivot, 600. power consumed by, 601. Friction rollers, 176, 295, 429, 602. rollers for drawbridges, 431. rolling and axle, 602. tables of, 599, 600. Wall, 336. Frictional stability, 486, 494. Frogs, switches, &c, 397. J. Wood's self-acting, 400. Pennsylvania Steel Go's, 398, 400. tables of, 401, 402, spring-rail frog, 400. Frustums of prisms, solidity, 30. Fuel for locomotives, 411. per H. P., 495. Fumes, corrosive, 370, 511. Funicular machine, 463, 662. g, 449. Gain of power, 474. Galvanized iron, 370. Galvanizing, 370. Gates for water pipes, 572. Gauge, Am and Birm, 367. Stubs, 368. Geometry, practical, or drawing of figures, 61. Gin, day's work with, 606. Girders, for bridge floors, 215, 291, 296. Hodgkinson's, 208. plate and box, 214, 217. rolled I, 210. Glass and glazing, 514. strength of, 175, 180, 185, 515. Glossary of terms, 615. Glue, adhesion of, 620. Gold and silver, 74. Gordon's rules for iro"n pillars, 221, &c. table of pillars by his rules, 232, &c. Grades, tables of, 388, 389, 629. hydraulic, 536. wt of trains on, 412. Gravity, acceleration of, 172, 449, 587. center of, 442, 481. " to find mechanically, 443. on inclined planes, 172, 486. specific, 383. table of, 384. Great Bear, 101. Grillage, 634. Grout, 497. Gudgeons, to find diams, 620. Guide rails, 398. Gunter's chain, 74, 98. [622. Gyration, rad. and cen. of, 173, 495, 617, Hand level, Locke's, 166. 685 Hand) to meas Hauling by hoj by men, 606. effect of width of ti^s, 608. Head of water, 535VVirtual, 571. theoretical, table', 552. Heads and nut^oT bolts, 374, 375. Headway, ofyforidges, 288 note. 307. Heat) expansion of solids by, 310. Heights, to find by barometer, 167. to find by boiling water, 170. " " reflection, 44. " " shadow, 43. " " trigonometry, 40, &c. Hemp ropes, 382. Heptagons, 15. Hexaedrons, 38. Hexagon, area of, 15. to draw, 67. Hodgkinson on pillars, 221. beams, 208. Hollow beams, strength of, 187, 193. Horse, traction of, 603, 605. amount of work by a gin, 606. " " " in pumping, 606. power, 571, 605. " coal per, 495. " of a running stream, 571. " of falling water, 571. walks, diameter of, 605. weight of, 605. Howe truss, 283. " table of, 284. Hydrants, or fireplugs, 576. Hydraulics, 534. acres drained by pipes, 569. adjutages, 554. air valves, 579. bends, resistance of, 539, 548. " " table of, 550. bursting pressure in pipes. 531, 536. cases of incomplete contraction, 557. city pipe systems, 580. Clegg's dam, singular effect at, 561. coeffs of Lesbros and Poncelet, 555. disch in open channels, 561. 564. " from one reservoir into another, 556. " openings, 551. " over weirs, 558. " through pipes, 537. " " " tables of, 539, 544. " " short tubes. 543. 553. " thin partition, 554. " " triangular notches,561. Hydraulics, floating mills, 571. flow through pipes, 534. " affected by material of pipes, 537. " Weisbach's rule for. 543. Francis', Mr., experiments, 558, 560. friction head, 535. Fireplugs, 576. friction in pipes in pumping, 551. head, defined,- 535. ' required for a pipe, 542. " " for bends ; table, 550. horse-powers of falling water, 571. ' " of running water, 571. hydraulic grade line, 536. " mean depth, 565. " radius, 565, 650. Kutter's formula, 650. obstructions by piers, 570. piezometers, 537. pressure of running water, 571. [544. pipes, discharge, tables of, 539, 540, " " " Weisbach's, 544. " " through, 537. " general laws of flow, 534. " old formulas said to be defect- ive, 543. " resistance to pumping, 551. " to find diam, 542. Rennie's expts with bends, 550. reservoirs, 577. square roots of. 5th powers, 548. velocity head, 5^5. " in regular channels, 562. " in rivers, 563. " in sewers and drain pipes, 568, 569, 652. " of falling bodies,table of, 552. of flow in pipes, 537, 544. vena contracta, 552, 554. Hydraulic cements, 500. grade, 536. press, 632. radius, 565, 650. ram, 571. Hydrostatics, 521. buoyancy of liquids, 533, 635. centre of pressure, 482, 526, 635. compressibility of liquids, 516, 534. draught of vessels, 534, 635 pressure against various surfaces,522. ** in pipes, 532, 536. " independentofquantity,522. " of water, 521. " table of, 523. '." transmission of, 526. " walls to resist, 528. 44 686 INDEX. Hydrostatics, pressure, walls to resist, tables of, 528, 530. I beams, 210. as pillars, tables of, 638, 639. safe loads, tables of, 212, 213. Ice, 516. may lift piles, 324. strength of, 175. Icosaedron^ 38. Imperial British measures, 77. Impulse or impact defined, 448. experiments on, 448. Inch, circular, 76. cubic, what equal to, 76. of rain, 519. spherical, what equal to, 77. Inches in decimals of a foot, 75. Inclined plane, 172, 484. acceleration, 172. ropes for, 381. stability on, 493. table of pres on, 486. Inertia, 446. moment of, 173, 195. " in beams, 195; 647. Insurance, 73. Interest, 73. Iron buckled 'plates, 369. Angle, 373. cast, wt of square and round, 362. corroded by coal fumes, 370. cost, 364. Chains, 381. effect of cold on, 180. " " mortar on, 500, 592. " " water on, 324, note, 517. elastic limit of, 176, 178. paints for, 513. porosity, 532. proper kind for sea water, 324. rolled, wt of square and round, 355. ropes of, 380. Star, 373. sheet, how put on roofs, 370. strength, bolts, 375. " crushing, 176. " of beams. See Beams. " pillars, 221, 638. " shearing, 181. " " table of, 656. " tensile, 178. " torsional, 182. " transverse, 185. stretching of, 178. weight of angle, T, star, &c, 373. " of bolts, 376. " of chains, 381. Iron, weight of corrugated, 370. weight of flat bars, 372. " of pillars, 224, 234, &c. " of sheet, 367. " of wire, 368, 369. Jet, water, 325. Joints, Alex. W. Rae's, 396. bell, or faucet, in pipes, 574. C. E. Smith's inverted j., 396. figures of iron, 268. " " wooden, 294. Fisher's, 392. fish-plates, 391, &c. " " compensating, 391. flexible, for water pipes, 575. for rails, 390. J. Button Steele's, 394. Fritz & Sayre's, 395. Pettie's, 392. Phoenix sleeve, 392. " suspended, 396. ring, of C. & Amboy R R, 393. riveted, 653. Wilson's stop-chair, 391. Journal friction, 601. Journals, 621. Jumper and hand-drill, 311. Keystones, depth of, 341. table of existing, 343. pressure on, 342, 468, 491. Knees in water pipes, effect of, 550. Knife edges, strength of, 176. Knot, 74. Kinetics, 459. Kutter's formula, 650. Land measure, 76. numb of acres drained by pipes, 569. reqd per mile for R. R., 390. surveying, 90. Laths, plastering, 510. shingling, 512. slating, 511. Latitude, 74. Lattice truss, 285. Lead, balls, 377. for roofs, 377. pipes, wt, 377. " strength of, 533. req in laying pipes, 574. strength of, 176-178. white, 512. wt of bars and sheets, 376. Leakage of canals, &c, 521. SDEX. 687 Leakage of reseprtfirs, 578, 521. Level, book i^rm. for, 156. cuttiagfMables of, 420. hand, Locke's, 166. meaning of the word, 42. plumb, to adjust, 167. the spirit, or engineer's, 152. Levelling, by barometer, 167. by boiling water, 170. leverage and moments, 473, 645. Levers, 473. beams regarded as, 477. compound, 479. Lime, 496. Effect on timber, 497. air-slacking, 496. from coral, 497. weight, 386, 496. Limit of elasticity in beams, 185, 197. Line of pressure, or thrust of an arch, 348, 493. Lines, 61. Link, expansion, 295. of Gunter's chain, 74. Liquid measure, 76. Liquids, buoyancy of, 533, 635. compressibility of, 516, 534. transmission of pressure, 526. Load, on a bridge or floor, 297, 595. on a roof, 300. On props, 661. safe, on earth, 314. Of sand, 510. Locke's hand-level, 166. Lock Ken viaduct, 288, 326. Lock-nut, washers, 375, Locomotives, annual expense of run- ning, 412. cost, 413. dimensions of, 413. evaporation, 434. fuel, 411. tractive power of, 411. wt of trains on grades, 412. Logarithms, 613. Long Meas, 74. Longitude, length of degrees of, 75. Lorenz safety stvitch, 408. Lumber, board measure, 357. price in Philada, 361. Lune, circular, area of, 25. Man, weight of, 297, 595. work by pump, treadwheel, &c, 606. work of hauling by a rope, 606. Maps, to reduce or enlarge, 69. Masonry, cost of, 312, 630. cub yds in culverts, 351 to 355. cub yds in wing walls, 353. Masonry, weight of, 386, 529. Mass, defined, 456. Materials, crushing strength, table, 174, &c. shearing and torsion, 181. strength of, 174. " " tensile, 177 to 180. " " transverse, 185. weight of, 384, &c. See Weight. Matter, defined, 444. Quantity, 455. Measures, 73, &c. to find size of commercial by the weight of water, 516. Measuring on sloping ground, 40, 98. Mechanics, 443. Melting point, 310. Mensuration, 13. Meridian line, to find, 99. Metacenter f 635. Metal roofing, 268, 377, 378. Metals, crushing strength, 176". expansion and melting by heat, 310, shearing strength, 181. sheet, wt of, 367-376. tensile strengths, 178. transverse strengths, 185. Metre, 79. Mile, sea, and land, 74. scale, 89; Mills, floating, 571. Minutes and sec, in dec of a deg, 17. Mitchell's screw-pile, 324. Models, compared with structures, 641. Modulus of Rup., 195. Of Bias., 177. Moment of inertia, 173, 195, 647. of resistance, 217, 646. of rupture, 217, 645. Of stability, 489 Moments, equality of, 476. and leverages, 473. Momentum, 448. Money, 74-81. Monkey-switch, 407. Mortar, bricks, 496. Mortar, adhesion of, 497. cement-mortar, 503, 508. frozen, 502. Grout, 497. with brickdust, 496. Pointing, 501. proportion of, in brickwork, 496. " " rubble, 386. protects iron, 592. Decays wood, 497. should not be depended on against sliding, 332, 453. strength, 497. Moseley on strength of beams, 19U. Moseley's, W. H., bridges, 289. 688 INDEX. Motion defined, 444. accelerated, retarded, &c, 449. quantity of, 448. Mud, weight of, 330, 385, 386. accumulation in reservoirs, 578. MuskratS) 577. Nails, weight of, 383. adhesion and shearing strength of, 383. shingling, 512. slating, 511. Needle of compass, 164. Neutral axis, 195, 246, 645. Nonagons, 15. North and South line, to find, 99. North star, 100. diagram of its positions, 101. Nuts, weight of, 374. Obliques in a truss, defined, 243. " to find length of, 69, 303. Obstaeles 9 to pass, 97. Obstructions by piers, 570. Octaedrons, 38. Octagons, area of, 15 ; to draw, 67. Ordinates for bending rails, 418. of R. R. curves, 20, 416, 633. Oscillation, center of, 173. Ovals, to draw, 68. Overfalls, or weirs, discharge by, 559. tables of discharge, 560, 561. fainting house, 512. cost, 513. Panel of a truss, defined, 244. " " length of, 244. panel-point defined, 244. to find diagonal for, 69, 303. Paper, 151. profile and tracing, 152. Parabola, 28. to draw, 29. Paraboloid, solidity, &c. of, 38. Parallel forces, 481. Parallelograms, 13, 64. of forces, 458. Parallelopipeds, 30. of forces, 471. Patterns, 362. Paving, with bricks, 498. Belgian, cost of, 313. Pencils, 152. Pendulums, 172. Pentagons, 15., Percussion, center of, 173. Perimeter, wet, Rule 1, 564, 650. Permutation, 72. Perpendiculars, to draw, 65. Phcenix I beams, 199, 201, 210. " as pillars, 638. pillars for trestles, 308. rolled segment pillars, 234. Pierce's Well JSorer, 636. Pierre perdue, or random stone, 314. Piers, abutment, 347. cub yds, in, 356. obstructions by, 570. Piezometers, 537. Piles, 315, 320, 323, 534. blunt ended, 323. Brunei's, 324. Brunlee's, 326. driver, Shaw's gunpowder, 321. driving by short ropes, 607. " by treadwheel, 321. French rule for safety, 322. friction of, 323. hollow, 326. ice around, 324. inside of a cylinder, 329. iron, 324. Mitchell's screw, 324. preserving heads from splitting, 324. sand, 328. Sanders' rule for driving, 322. sheet, 321. shoes, for, 323. Trautwine's rule for driving, 322. to withdraw, 324. wooden, cost, 320. Pillars, iron, strength of, 221. table of hollow cast round, 224, 232. " " " square, 222, 232. " " wrt round, 228, 232. " " wrt square, 232. " solid cast round, 226, 232. solid cast square, 227, 232. " solid wrt round, 230, 232. " solid wrt square, 231, 232. L, -h H , &c, 235 to 236, 637 to 840. Phoenix rolled segment, 234. " iron, various ; strength of, 233. pinned, hinged, or jointed, 233. safety, coefficient of, 225. swelled at center, 237. Steel, 238. wooden, strength of, 238. wrt I beams, as pillars, 638. Pins for eyebars, 293. Pipes, air-valves for, 579. INDEX, 689 Pipes, Ball's ipaffaud cement, 577. t of, 539, 548. branches, 575. contents, 46, 540, 541. copper and brass, wt of, '365. cost of laying, 574. cub ft in 1 ft length, 541. flow affected by material, 537. flow through. See Hydraulics. gallons in 1 ft length, 46. giitta percha, 577. iron, wt of cast, 363, 364. " " wrought, 364. knees, effect of, 550. miles of, in Philada, 577. of bituminized paper, 577. of bored logs, 577. of lead, strength, 533. wt, 377. pressure in, 531, 536. resistance to pumping, 551. service, 533, 573. ferrules for, 574. steam warming, 363. swellings in, effect of, 551. terra cotta, 569. thickness to resist pressure, 531. to mend, 575. to prevent concretions in, 581. valves or gates for, 572. Ward's flexible joint, 575. water, must be closed slowly, 533, 573. wedges for, instead of lead, 574. wt of lead for laying, 574. wt of water in 1 ft lengths, 540. WyckofTs patent, 577. Pipe-drains, 568, 569. Pivot friction, 600. Plane, inclined, 172, 484. " ropes for, 381. stability on, 488, 493. " table of pres on, 486. Plane trigonometry, 39. Planck, thickness to bear pres of water, 317. Plastering, 509. Plate-iron beams, 214. Plates, buckled, 369. Plenum process for foundations, 327,631. Plumb-level, to adjust, 167. Point of contrary flexure, 641. Pointing mortar, 501. Polygons, 15. To draw, 67. of forces, 467. to reduce to a triangle, 68. Poona dam, 530. Porosity of cast iron, 532. Port, establishment of. Tides, 626. Portland cement, 385, 500. Potts, Dr f process, 326, 631. Powder, 310. quantity in blasting, 311. Power, animal, 605. defined, 449, 474. fifth, 546. gain of, 474. locomotive, 411. Water, 571. Pratt, truss, 284, 285, 306. Preserving timber, 358, 362, 497. Press, hyd, friction of, 632. Pressure, center of, 333, 482, 526, 635. in pipes, 531, 536. line of, in arches, 348, 493. of air in a diving-bell, 520. of water, 522, 526. Of earth, 331. table of, 523. Of wind, 520. of running water, 571. Prismoidal formula, 33. Prismoids, solidity of, 33. Prisms, and frustums of, 30. Profile paper, 152. Progression, 71, 72. Proportion, 71. Props, 661. Protracting by chords, 147. table of chords for, 608. Pulleys, 479, 662. Pumping, by hand, &c, 433, 606. engines, cost of, 433. Purlins, 247, 267, 294, 300. Pyramids, 32. frustums of, 33. Quarrying, cost of, 311, 440. Radii and ordinates of curves, 416. Radius, hyd mean, 565, 650. of gyrations, 173, 495, 617, 622. Railroads, 409. annual expenses, &c, 409. ballast for, 414. cars, 413. cost of, 414, 415. locomotives, 411. shops, cost of, 415. Ties, 414. water-stations, 432. Kails, 415. creeping of, 391. elevation of outer, 419. guide, 398. joints, chairs, &c. See Joints. 690 INDEX. Jin Us, ordinates for bending, 418. welded together, do not creep, 391. Main, 518. Reaching sewers, 566. Raised tie-bar, strains by, 262. Raising of bridges, 303. Ram, hydraulic, 571. Random stone foundations, 314. Reaction, strain, 444, 449. Reflection, to meas heights by, 44. Refraction and curvature, table, 42. Regular bodies, the, 38. Reservoirs, 577. Evap from, 521. compensating, 5T9. discharge from one into another, 556. distributing, 579. Storing, 578. leakage of, 578. 3fud in, 578. Resistance, moment of, 217, 646. " angle of, 485, 487, 598. Retaining-walls, bulging of, 332. offsetted, 333. On Piles, 534. surcharged, 334, 337. table of contents, 341. tables of, 334, 338, 528 to 530. theory of, 334. transformation of, 339. with curved profiles, 340. with offsetted, or stepped backs, 333. Revetments, 340. Rhumb line, 93. Right angle, to lay off by a triangle, 42. Rings, area of, 22. Solidity of, 32. joint for rails, 393. tightening, 268. to find breadth, 17. Riprap, 314. Rise of roofs, effect of, on their wt, 301. Rivers, flow in, 563. Riveted joints and rivets, 653, Rivets, No. in 100 Ibs, 653. shearing strengths, table, 656. Roads, greatest slope for, 389. traction on, 603, 605, 607. Rock, excavation, cost of, 440. increase of bulk when quarried, 440, 630. quarrying, 311, 440. Rocker, expansion, 295. Rod of brickwork, English, 499. Rollers, expansion, 295. " loads on, 176. friction, 176, 43*1, 602. Parry's, 429. Rolling friction, 602. Roofs, arched iron, 289. Bowstring, 270. cost of, 300. Copper, 377. Roofs, coverings, wt of, 301. crescent, 270. Fink, 264. weight of, 298, 300. how covered with sheet iron, 370. " " " corrugated iron, 370. iron, details of, 268. Lead, 377. least pitch for metal coverings, 378. " " " slate " 511. simple, 247, &c. Tin, 378. strains on trusses, 247 to 268, 298. weight allowed on, 300. wind on, Tredgold's allowance for, 520. wooden, details of, 294. wt of, 263, 298 to 301. wt of, affected by rise, 301. Roots, fifth, 546. square and cube, table, 48. square of 5th powers, 548. to calculate, 60. Ropes, 382. wire, 380. wire, Roebling's notes on, 380. Rot, timber, 360. Rubble masonry, cost, 312, 630. proportion ot mortar in, 386. quarry, loose, 440, 608. Rule of three, 71. 2 foot, to measure angles by, 676. Rupture, constant for, 185, 195. moment of, 217, 645. modulus of, 195. Russian weights and measures, 80. Safety, in pillars, 225 ; in bridges, 298. Sand, singular fact in pres of, 340. voids in, 503. Wt of, 387, 503. piles, 328. Load of, 510. pump, 328, 636. Scales, platform, 409. Scarfs for timbers, 291. Schuylltill River bridge of cast-iron arches, 288. Scour of streams, 563, 570, 571. Screw,' American standard proportions, 374. The Screw, 479. cylinders, 324. piles, 324. wrench, 629. Seamless brass and copper tubes, 365. Seaworms, 414. Secants, cosecants, &c, to find, 101. Sector of a circle, 22. C of Gr, 442. Sediment in reservoirs, 578. INDEX. 691 Segment, circulas^area of, 22. e of, 24. spherical, curved surf of, 37. " solidity of, 34. Sellers 9 turntable, 429. Service pipes, 377, 533, 573. Settlement of embankment, 630. Severn Valley R. R. arch bridges, 288. Sewers, tables of, 568, 569, 652. velocities in, 568, 652. Sextant, box or pocket, 163. Shafting, iron, strength of, 182. for tunnels, 627. Shearing strength of woods, 181, 642. of nails, 383. Of iron &c, 181. of rivets, table of, 656. Sheet-piles, 321. Sheet-metals, wt of, 367. Shell, spherical, 38. weight of, 362, 366. Shingling, 512. Shops, railroad, cost of, 415. Shrinkage of embankment, 630. Similar figures, &c, defined, 61. Sines and tangents explained, 62. table of, 102. Single rule of three, 71. Skeleton diagram, 247. Sleeve-chair, 392. Slating, 510. Slope instrument, 167. natural, of earth, 338. of maximum pres,335. per 100 ft, 98, 388, 389, 629. Sluices in dams, 586. Snow, 519. Soils, or earths, safe loads for, 314. weight of, 385. Solids, expansion by heat, 310. Sound, 173. Spandrel walls, 341, 346. Spanish measures and weights, 80. Specific gravity, 383. " table of, 384. Spheres, or globes, contents of one 1 ft diam, 77. contents of one 1 inch diam, 77. segments of, 34, 37. table of, 35. zones of, 38. Spherical shells, 38. weight of, 362, 366. Spheroids, 3.8. Spikes, weight of, 382. adhesion of, and nails, 383. Spindle, circular, solidity of, 39. Splicing of timbers, 291. Sqtiare hollow iron beams, 193. Squares and cubes, and roots, 48. Stability, 488. diff measures of, 490. frictional, 486. moment of, 475, 489. of an arch, to find, 491. on inclined planes, 488, 493. Stand-pipes, for waterworks, 625. for water stations, 433. Star, Nortli, 100. Alioth, 100. Star-iron, 373. Stars, to regulate a watch by, 80. table of times, 80. Statics defined, 444. Station house, cost of, 415. water, 432. Steam-pipes, 363. Steel, cost of, 364. elastic, limit of, 176, 179. pillars, 238. strength of, 176, 178, 185. wt of, 366, 367. Stone, artificial, 507. beams, 203. breaking, 414, 505. bridges, 341. broken, swelling of, 440, 608, 630. " voids in, 504, 630. cost of dressing, 311. " quarrying, 311, 440. dams, 528, 530. expansion of, 310. random, or pierre perdue, 314. strength of, 175, 180, 185, 203. Stonework, 310. cost of, 312, 630. " " Bunker Hill Mont, 312. weight of, 386, 528. Stop-valves, or gates, 572. Storing reservoirs, 578. Strain defined, 444, 449. on trusses, 243, &c. three simple processes for finding, 253. Streams^ bottoms, to bear diff vels, 570, 563. flow in, to measure, 562. horse-power of falling, 571. " " running, 571. scour in, 563, 570, 571. virtual head of, 571. 692 INDEX. Street pipes. See Pipes. Strength of beams. See Beams. crushing, 174. of cylinders, 193, 531. of glass, 175, 180, 185, 515. of iron pillars, 221. of iron rods, 376. of materials, 174. of shafting, 182. of stones, 175, 180, 185, 203. of wooden pillars, 238. shearing, 181, 642. tensile, 177, &c. torsional, 131. transverse, 183, &c. Stress, 445. Struts to distinguish from ties, 463. Stucco, 509. Surcharged retaining-walls, 334, 337. Surveying, 90. Suspension bridges, 588. aqueduct at Pittsburgh, 596. data for cables, table, 588. Fairmount, Philada, 595. Finley's, 596. Freyburg, 594. length of cables, to find, 590. " of suspending rods, 591. links, 295. Niagara, 588, 594. strains on cables, 588, 589. " on piers, 591. table of wire in cables, 369, 597. Wheeling, 596. Sway bracing, 291. Swing-bridge, strains on, 275. Switches, frogs, &c, 397. Lorenz, 408. monkey, 407. tumbling, 405. Stub, 405. Wharton's safety, 407. Swivel, tightening, 268. Syphon, 582, 661. T-iron, 373. Table of acres reqd per mile, 390. angles by a 2-ft rule, 676. arcs in common use, table, 434. balls, wt of, 362, 366, 377. board meas, 357. Bolts, 376. camber, 302. chains, 381. chords for protracting, 608. ; circles, 18. By decimals, 677. circular arcs, 21, 23, 434. Table of circular segments, 24. constants for defl 3-^ span, 201. " for safe defl, 199. cpntents of cylinders or pipes, 46, 47, 540, 541. contents of retaining-walls, 341. Cooper & Hewitt's beams, 213. copper pipes, wt of, 365, 378. cubic foot and inch, 76. cub yds in a pier, 356. " ' culverts, 351, 354. " " retaining-walls, 341. " " wing walls, 353. curvature of the earth, 42. curves for railroads, 416. deductions in -chaining on sloping ground, 98. diams, &c, of circles, 18. elliptic arcs, 27. Fairbairn girders, 217. Fink trusses, 298, 300, 305. friction, 599. grades, 388, 389, 629. heads and nuts of bolts, 374. heads, theoretical, 552. Howe bridges, 284. inches in decimals of a ft, 75. iron pillars, Gordon's rule, 224, &c. iron rods, strength of, 376. joints, riveted, 654. lead pipes, wt of, 377. level cuttings, 420. levelling by barom or boiling water, 171. loads on grades, 412. logarithms, 613. min and sec in decimals of a deg, 17. motion of stars, 81. polygons, 15. Pratt truss, 285, 306. pressure of water, 523. rivets, 653. ropes, 380, 382. safe loads of wooden beams, 191. sines, &c, 102. slopes, 98, 388, 629. spheres, 35. square roots of 5th powers, 548. squares and cubes, and roots, 48. tin, 379. Turnouts, 402. traverse, 82. Trenton beams, 213. value of foreign money, 81. vel of falling bodies, table, 552. walls to resist earth, 334, 338. INDEX. 693 Table of walla to resist water, 528 to 530. weight of flat aiyi corrug iron, 370. " iron/nd steel bars, 366. " me^al sheets, 367. " wire, 368, 369. weights and measures, 74 to 81. weights of bridges, 296. Tangents explained. 63, 66, 101. to draw, 67. Tanks, 432, 434, 532. Tarns, wt per box, 379. Tenders of locomotives, 413. Tensile strength of materials, 177, &c. Tension, to find diams to resist, 180. Terms, glossary of, 615. Teredo, 414. Terra cotta pipes, 569. Tetraedrons, 38. Theodolite, 162. Thermometers, 309. TJiin partition explained, 554. Tides, 626. Tie-rod raised, strains by, 262. Ties, railroad, 414. to distinguish from struts, 463. Tightening swivel, 268. ring, 268. Timber, cost of, 361. crushing strength, 174. preserving, 358, 362, 497. shearing, 181, 642. tensile strength, 177. transverse strength, 185, 317. weight, 384, &c. Time, by a dial, 150. by stars, 80. civil or clock, 80. reqd to transmit force, 456. Tin, 378. contents and wt per box, 379. Tires, wagon, 608. Of locos, 413. Torsion, 181. Town's lattice truss, 285. Tracing paper and cloth, 152. Traction, on roads, &c, 411, 603, 605. Train of cog-wheels, power of, 479. weight of, on grades, 412. Transit, the engineer's, 157, Transverse strength of materials, 185. Trapeziums, 14. in earthwork measurements, 15. Trapezoin, 14. Traverse table, 82. Treadwheel, day's work on, 606. Tremblings in dams, 586. Tremie, 506. Trenton beams, table of, 213. Trestles, 307. of Phoenix segment pillars, 308. Triangles, 13, 63. to measure by trigonometry, 39. Triangular walls to resist water, 529. Trigonometry, 39. Troy weight, 74. Trusses, 243. beam, 244. Bollman, 269, 282. Bowstring, 270, 286. braced arch, 274, 287. Burr, 289. camber of, 302. cantilevers, 219, 275, 643. cost, 300. counterbracing, 245, 252, 275, 306. crescent, 270. Fink bridge, 281, 305. Fink roof, 264, 297, 300. for roofs, 247, 298 to 302. for short bridges, 304. horizontal diag bracing, 291. Howe, 283. king and queen roof, 297 to 299. lattice, or Town's, 285. parts of, denned, 243. j Pratt, 284, 306. raising of, 303. roof, wt as affected by rise, 301. strains, 3 simple processes for find* ing, 253. suspension, 244. Warren, 254, 279. weight of bridge, 295. weight of roof, 263, 298 to 300. Tubes, cast-iron, table of, 363, 364. couplings, 365. gutta percha, 577. lead, 377, 533. seamless brass and copper, 365. various kinds, 577. welded, 364. wrought-iron, table of, 364. Tumbling switch, 405. Tunnels, 627. Turnbuckle, 268, 628. Turnouts, frogs, switches, Ac, 397. table of, 402. Laying out of, 401. Turnpikes, max slope, 389. Turntables, cast-iron, Sellers', 429. other kinds, 430. Undecagons, 15. 694 INDEX. Vngulas, 31. Upset rods, strength of, 375. Vacuum process, 326. Valve towers, 579. Valves, air, 579. defined, 628. for water pipes, 572. Variation of the compass, 165. Velocity , angular, 447 defined, 447,449. in pipes, 534. Kutter's formula, 650. of sound, 173. of water in rivers, tables, 562, 563. " in regular channels, tables, 567, 568. " in sewers, 568, 652. of wind, 520. theoretical of falling bodies, table, 552. to abrade soils, 563. vel head, 535. virtual, 477. Vena contracta, 552, 554. Ventilation f air required, 519. Vernier > 162. Vertical, meaning of, 61. Verticals, in a truss, defined, 243. Vessels, draft of, 534, 635. Viaduct, Loch Ken, 288, 326. Virtual velocity, 477. head, 571. Vis viva, 446, 455. Voids in broken stone, 504, 630. in rubble, 496, 630. in sand, 503. Vulgar fractions, 69. Walls, high, for dams, in France, 529. intended to resist water, 528, 534. " " ." tables, 528,530. " table of contents, 341. number of bricks reqd, 496, 498. retaining, 331. On piles, 534. " several forms with the same quantity of mason- ry, 530. [&c. " tables of, 334, 338, 341, 528, " transformation of, 339. " triangular, 529. spandrel, 341, 346. wharf, 339, 534. wing, 350. wing, table of contents, 353. Ward's flexible pipe-joint, 575. Warren truss, 254, 279, &c. Washers, size, and weight of, 373, 374. lock-nut, 375. Wash'n Monument concrete, 505. Washes for outdoor's work, 513. for brickwork, 514. white, 514. Watch, to regulate by stars, 80, note. Water, 515. boiling, to measure heights by, 170. buoyancy of, 533, 635. cisterns, 432, 434, 532. column or stand-pipe, 433, 625. compensation, 579. cub ft, and gallons in pipes, 46, 541. effects on iron, 324, 516. horse-power of, 571. pressure, 522. Table of, 523. " of running water, 571. " in pipes, 531, 536. quantity reqd in cities, 580. resistance to moving bodies, 571. stations on railroads, 432. under pres, oozes through iron, 532. walls, to resist, 528, 534. wheels, 571. wt of, for finding the size of commer- cial measures, 516. wt of, in 1 ft length of pipes, 540. Water-jet, 325. Web members of a truss, 243. Wedges, 33, 481. instead of lead in pipe laying, 574. Weight allowed on a bridge, 297. allowed on a roof, 300. cars, 413. carts, 436, 608. cement, 385, 500. crowds, 297, 595, footnote. defined, 444. iron, angle, T, channel, star, 373. " corrugated, 370. " flat bars, 372. " pillars, 224, &c. " square and round, 366. lime, 496. masonry, 386, 529. nails, 383. of a cub ft of substances, 384. of a horse, 605. of a roof truss, as affected by its rise, 301. of balls, 362, 366, 377. of bolts, nuts, washers, 374. of bricks, 384, 498. 695 Weight of bridj of ceilings and floors, 248. of dredged materials, : of earths, 385. of patterns, 362. of pipes, brass anXcopper, 365. , 364. " lead, 377. roof covering, 301, 511. Rails, 415. roofs, iron, 263, 298, 299, 300. sheet metals, 367, 376. shells, 362, 366. snow, 519. Sand, 387, 503. spikes, 382. tenders, 413. timber, 384, &c. water, 515. " in 1 ft length of pipes, 540. wire, 368, 369. Weights and measures, 73. Weirs, 558. Weisbach's rule for flow in pipes, 543. Wells, contents of, 47. boring, 636. lining of, 47. Wet perimeter, rule 1, p 564, 650. Wharf walls, 339, 534. Wharton's safety switch, 407. Wheelbarrow ft, in earthwork, 438. Wlieels, cog, power of a train of, 479. of cars, 413. tires, 608. water, 571. wheel and axle, 477. Whipple, Mr., 243, note. Whipple bridges, arched, 288. Whitewash, 514. Winch, day's work at, 606. Wind, 520. mills for water stations, 433. Wing-walls, cub yds in, 353. Wire gauges, 367. Fences, 415. rope, wt and strength, 380. " Mr. Roebling's notes on, 380. strength of, 369. Rigging, 381. table of weight, 368, 369. Wires , to find the number of in a cable, 369. Wood, crushing strength, 174. shearing, " 181. table of, 642. tensile, " 177. transverse, " 185, 191. Wooden jnllars, rules for, 238. table of, 239 to 242. Work defined, 448. quantity of, 454, 455. unit of, 449. unit of rate of, 449. Zinc or spelter, 378, 388. in sheets for roofing, 379. paint, 512. paint does not adhere well to, 370. price of, 379, note. strength of, 176, 180. vessels for water, said to be un- healthy, 379. Zones, circular, 25. spherical, 38. Jtemarh. Many things not mentioned in this Index will be found in the Glossary. THE END. INDEX TO ADVERTISEMENTS. PAGE HELLER & BRIGHTLY, 699 CAMBRIA IRON AND STEEL WORKS, 700 J. G. BRILL & Co., . 700 D. P. DIETERICH, 701 GEO. J. BURKHARDT'S SONS, . 702 WILSON BROTHERS & Co., 702 WM. SELLERS & Co., 703 CARNEGIE BROS. & Co., LIMITED, 704 JAMES BEGGS & Co., 704 TATHAM & BROTHERS, 705 VANDERBILT & HOPKINS, 706 H. B. SMITH MACHINE Co., 706 PENNSYLVANIA STEEL Co., 707 SAMUEL J. CRESWELL, 708 FRENCH'S PAINT, PLASTER, AND CEMENT DEPOT, . . 708 VERONA TOOL WORKS,. METCALF, PAUL & Co., . . . 709 CRESCENT STEEL WORKS, MILLER, METCALF & PARKIN, 709 THE PHILADELPHIA BRIDGE WORKS, COFRODE & SAYLOR, 710 BORGNER & O'BRIEN, 7ia COOPER, HEWITT & Co., 711 WM. B. SCAIFE & SONS, 712 CHESTER STEEL CASTINGS Co., 712 ELBA IRON & BOLT Co., LIMITED, 713 THE PEERLESS BRICK Co., 714 THE PHCENIX IRON Co., 714 697 RUMENTS OF PRECISION.) -fe O 7C Z w ?s h-l PJ 5 2 y E o" c/: > s o I Messrs. HELLER & BRIGHTLY publish a book containing tables and maps useful to Civil Engineers and Surveyors ; also contains much valua- ble information respecting the selection, proper care, and use of field in- struments. A copy of this book they send by mail, post-paid, to any Civil Engineer or Surveyor in any part of the world on receipt of a postal card containing the name and post-office address of the applicant. (699) Cambria Iron and Steel Works. OFFICE, WORKS, 2 1 8 S. Fourth St., Philadelphia, Johnstown, Penna, RAILS Of all Weights, including Light Rails for Narrow-Gauge Eoads and Street Eailways. SPLICE BARS, BOLTS, AND NUTS. CAMBRIA PATENT LOCK-NUT JOINT. J. G. BRILL & CO., PHILADELPHIA, BUILDEBS OF RAILWAY & TRAMWAY PASSENGER CABS, ALSO, Small Freight Cars, Hand Cars, and Cane Cars. SPECIAL ATTENTION TO WORK BUILT IN SECTIONS FOB FOREIGN COUNTRIES. 700 GOODYEAR'S< RUBBER WAREHOUSE HAVE A LARGE STOCK CONSTANTLY ON HAND OF ALL KINDS OF Vulcanized Unite, aianted to Mechanical and other purposes, PQ CQ D en W IX Q O o O Machine Belting W 1'1'tt Smooth Metallic Rubber Surface. This Company has manu- factured the largest Belts [made in the world for the Principal Elevators at Chi- cago, Buffalo and New York. Steal and later Hose, PLAIN AND RUBBER LINED. RUBBER "TEST" HOSE, made of Vul- canized Para Rubber and Carbolized Duck ; Cotton "CABLE "HOSE, Circular, Wov- en, Seamless, Antiseptic, for the use of Steam and Hand Fire Engines, Force Pumps, Mills, Factories, Steamers, aud Brewers' use. O o o o >< w > ?c CAR SPRINGS OF A Superior Quality, ' And of all the va- rious Sizes used. CORRUGATED Rubber Mats Slatting For Halls, Flooring, Stone and Iron Stairways, etc. NEW YORK Belting & Facial Co., Waterproofs, Rubber Clothing, Boots, Shoes, etc. SUITABLE FOE MEN, WOMEN, AND CHILDREN, D, P, DIETEBICH, 308 Chestnut Street, PHILADELPHIA. 701 ESTABLISHED 1840. GEO. J. BDEKHAEDTS SONS, (Successors to GEO. J. BURKHARDT & CO.) Factory, N. Broad, below Cambria St. Down Town Office, No. 215 Chestnut St. CEDAR WATER TANKS. From 1,OOO to 1OO,OOO Gallons Capacity. Cedar is well kfiown for its great durability. Tanks made of this material will last twice as long as Pine, and can b'e furnished by us at about the same cost. JOHN A. WILSON, FRED.. G THORN, JOSEPH M. WILSON, Civil Engineer. Architect. Civil Engineer and Architect . WILSON BROTHERS & Co. Civil Supers an! Architects, 115 BROADWAY, 435 CHESTNUT ST., NEW YORK. PHILADELPHIA. SURVEYS MADE FOR RAILWAY LINES. PLANS AND SPECIFICATIONS FURNISHED FOR BUILDINGS, BRIDGES, WATER WORKS, SEWERAGE SYSTEMS, HARBOR IMPROVE- MENTS, AND ALL CLASSES OF ENGINEERING AND ARCHITECTURAL WORK. CONSTRUCTION OF WORK ATTENDED TO. EXAMINATIONS MADE OF RAILWAY, MINING AND OTHER PROPERTIES. 702 WM.3ELLERS & CO. Main Wee aid f iris, ffiaieljlia. MACHINE TOOLS FOR RAILWAYS AND MACHINE SHOPS. Turning-Tables for Locomotives. Shafts, Pulleys, Hangers, and Mill Gearing. SELLERS' 1876 INJECTOR FOR FEEDING STEAM BOILERS WITH WATER. NEW YORK OFFICE, No. 79 LIBERTY ST. 703 CARNEGIE BROS, & CO,-LlM, PKOPRIBTOKS OF THE UNION IRON MILLS ITHIRTY-THIRD ST, PITTSBURGH PENNSYLVANIA, MANUFACTURERS OF STRUCTURAL IRON Bridge Iron, Iron Beams, Channel Bars, Car Truck Channels, Angles, Tees, Universal Mill Plates, Bar Iron, Light Steel and Iron Rails. SPECIAL ATTENTION GIVEN UNUSUAL SHAPES AND SIZES. Lithographs of sections and book of detailed information giving calculation of strain, etc., furnished to Engineers and Architects on application. NEW YORK OFFICE: Room 32, No. 55 Broadway, N. Y. JAMES BEGGS & Co.. PRACTICAL MECHANICAL ENGINEERS, MACHINISTS AND DRAUGHTSMEN. MANUFACTURERS AND DEALERS Machinery, Tools and Supplies, c* AGENTS *^ P. H. & F. M. ROOTS' ROTARY BLOWERS, GARDNER GOVERNOR, KEYSTONE INJECTOR, ROWLAND ENGINE, DETROIT LUBRICATOR, BUFFALO FORGES, ETC. 9 DEY STREET, NEW YORK. 704 TATHAM No. 226 South Fifth Street, n Sheet Lead, Lead Pi Tin Pi THERS, No. 82 Beebnan Street, New 7ork, OF Lead Pipe, Tin-lined !ron Pipe, Block- Shot, Chilled Shot, and Buck Shot, ' WEIGHTS OF SHEET LEAD. Weight per sq are foot. X| Weight per Thick- Weight per Thick- ! Weight per Th ck- ness. 3 " 4 " .042 in. .051 " .068 " 5 fts. 6 " 7 " .085 in. .102 ' .119 " 8 fts. 9 " 10 " .136 in. .153 " .170 " i**'; 16 " .jo:; in. .2:57 ' .271 " WEIGHTS OF LEAD PIPE. 2i a if I a i s a . i Weight per fot aud rod. a ff | "3 Weight per foot and rud. I Weight per foot aud rod. 11 u." % in. 7 fts. per,rod. 10 oz. per foot. 6 8 % i -, 2% fts 3"^ '" per foot. 22 25 l^in. 3 Its. per foot. 3% fts. " 14 16 " 1 tl " 12 % 11. 16 " per rod. * " 4% 19 " 1/4 fts. " 16 1 M " per foot. II 4 6 " 25 19 r. l^in. 14 fcin- 9 " per rod. 7 2)4 " ' K 4}^ ' 17 % ft . per foot. 9 3 " 21 5 ' 19 1 " 11 v: 23 fts. " M 16 I in. 4% " 21% fts. per rod. 'K 1%'in. H 4 27 13 " 2 41 19 2 " per foot. 11 5 17 3 25 H 21 Kin. 12ft <(. per rod. 8 < 3J4 ' < 17 " 8J<2 ' 27 " 1 ' per foot. 9 4 ' " 2 1 2 in. 4% 15 1 V<5 ' 13 4% ' 24 6 " 18 2 ' ' 16 IVfi n. 2 ' 10 7 " 22 2,^' 20 2^ ' 12 9 " 27 LEAD PIPE OF LARGER CALIBRE. Indies. I Thick. ^ Thick. Thick. & Thick. 3 ft. 17 20 22 25 31 oz, ft. oz. 14 16 18 8 21 ft. oz. 11 12 15 16 18 20 ft, oz. 8 9 9 8 12 8 14 \^i inch, 2 fts. per foot. 2 " 3 fts. % inch, 4J^. 6J^, and 8oz ^ " 6. 7H, and 10 <>/ ^ " 8 and 10 oz. LEAD WASTE PIPE. 3 inch, 3 1 A and 5 ft<. per foot. 4^ inch, < and 8 fts. per foot. 4 " 5, 6. and 8 IDs. per foot. 5 " 8, 10, and 12 fts. per ft. BLOCK-TIN PIPE. per ft. *4 inch, 10 and 12 oz. per foot 1J inch, 2 and '2 l /> fts. per foot. 1 " 15 aud 18 oz. " 2 " 2^ and 3 fts. Itf " 1H and 1'4 ft. ' SHOT-AMERICAN STANDARD SIZES. No. Diameter iu inches. CHILLED DROP SHOT. DROP SHOT. No. Diameter iu inches. CHILLED DROP SHOT. Duoi- SHOT No. of shot ! No. of shot No. of shot No. of shot to the oz. to tfce oz. to the oz. to theoz. n 05 2385 2326 2 15 88 H; 11 06 1380 1346 1 16 73 71 10 07 868 848 B 17 61 59 9 08 585 568 BB 18 52 50 8 09 409 399 BUR 19 43 42 7 10 299 291 T 20 36 6 u 223 218 TT 21 31 5 12 172 18 F 22 27 4 13 136 132 FF 23 24 3 14 109 106 COMPRESSED BUCK SHOT. No. | Diameter in inches. No. of balk to the ft. | No. Diameter in inches. No. of balls to the Ib. 3 2 1 25 27 30 82 288 225 172 140 | 00 000 Balls " 34 36 38 44 113 100 83 50 E. W. VANDERBILT. E . M. HOPKINS. VANDERBILT & HOPKINS, RAILROAD TIES, CAR AND RAILROAD LUMBER, White anl Yellow Pine, M, Gnm anf Cypress, 120 LIBERTY STREET, NEW YORK. Boards, Plank and Dimension Lumber Sawed to Order. General Railroad Supplies, and Roofing Slate. [OUR NEW STYLB IO-INCH MOULDER.] WOOD-CUTTING MACHINERY OF SUPERIOR QUALITY, For Car Shops, Planing Mills, and Sash and Door Factories. Correspondence Solicited. Cable Address, "Machine, Phila." H. B. SMITH MACHINE CO., 926 MARKET ST., PHILADELPHIA, PA., U. S. A. 706 STEEL CO,, /No. ^208 South Fourth Street, PHILADELPHIA, PA., Manufacturers of STEEL RAILS AND STEEL FORCINGS. STEEL RAILROAD FROGrS In Several Improved Patterns. IMPROVED SAFETY SWITCHES, RAILROAD-CROSSING FROGS, SWITCH STANDS AND SIGNALS, INTERLOCKING APPARATUS, by which Switches are operated from convenient stations with a great saving of expense, and securing Freedom from Accidents. WORKS AT STEELTON, PENNA. NEW YOMK AGENT, STEPHEN W. BALDWIN, 160 Broadway, N. T. 707 SAMUEL J. CRESWELL, ARCHITECTURAL AND ORNAMENTAL -*IRON WORKS*- Twenty-Third and Cherry Sts,, PHILADELPHIA. Fronts for Buildings, Girders, Columns, "Wrought Iron Beams. Sidewalk Lights, Stairs, Railings, Orestings, Stable Fixtures, and Lamp Posts. FRENCH'S PAINT, PLASTER, AND CEMENT -^ DEPOT, gh York Avenue and Callowhill Street, PHILADELPHIA, PA. S* MANUFACTURERS AND IMPORTERS.- FOREIGN AND DOMESTIC PAINTS, PLASTERS, AND CEMENTS OF ALL GRADES. 708 WORKS, MtETCALF, PAUL & co. McCance's Block, Serena Aye, & Liberty St., PittsWi, Pa. MAKE A SPECIALTY OF SOLID STEEL RAILROAD TRACK TOOLS. ALSO, SOLE MANUFACTURERS OF THE PATENT VERONA NUT LOCK, BRANCH HOUSE, No. 22 West Lake Street, Chicago. ST-2 SEND FOB CATALOGUE. =^ CRESCENT STEEL WORKS. ESTABLISHED 1865. MILLER,METCALF& PARKIN OFFICE, No. 81 WOOD STREET, Works, 49th and 50th Streets and A, V, E, E,, PITTSBURGH, PA. Fine Tool Steel, Drill Rods, Needle Wire, Cold Rolled Strips, Clock Spring- Sheets, etc. etc. PHILADELPHIA, .... 1232 MARKET STKEET. NEW YOKK, ...... 178J WATEK STKEET. CHICAGO, .... 22 & 24 WEST LAKE STREET. 709 DIE-FORGED EYE-BARS. MACHINE RIVETING. THE PHILADELPHIA BRIDGE WORKS, SHOPS AT POTTSTOWN, PA. JOSEPH H. COFRODK. FRANCIS H. SAYLOR. COFRODE & SAYLOR, CIVIL ENGINEERS AND BRIDGE BUILDERS, OFFPCE, No, 257 SOUTH FOURTH STREET, PHILADELPHIA. Contractors for the Construction and Erection of IRON or WOODEN BRIDGES, VIADUCTS, TURN-TABLES, ROOFS, and BUILDINGS. Plans and Prices Furnished on Application, Specifications Solicited, BRIDGE RODS WITH UPSET ENDS. CASTINGS FOR WOODEN BRIDGES. Superior Quality. Best Workmanship. BORGNER & O'BRIEN, Twenty -Third Street, above Race Street, PHILADELPHIA, PA, U.S. A, MANUFACTURERS OP FIRE BRICK AND CLAY RETORTS For Heating and Melting Furnaces of Every Description, Particnlar Attention Given to Soecial Snanes. 710 EDWARD CpsffR,^ JAMES HALL, t65 ' EDWIN F. BEDELL. COOPER, HEWITT & CO., No. 17 BURLING SLIP, NEW YORK. TRENTON IRON WORKS, TRENTON, N. J. PEQUEST IRON WORKS, OXFORD, N. J. RINGWOOD IRON WORKS, RINGWOOD, N. J. DURHAM IRON WORKS, RIEGELSVILLE, PA. IRON ORE, PIG IRON, ROLLED IRON BEAMS, CHANNELS, ANGLES, AND TEES, WELDLESS, DIE-FORGED EYE-BARS, RAILS, MERCHANT IRON, BRAZIER AND WIRE RODS, STAPLES, RIVETS, CHAINS, BRIDGES, ROOFS, AND OTHER IRON STRUCTURES. IRON AND STEEL WIRE OF ALL KINDS. A Specialty is made of Superior Qualities of Wire straightened and cut to lengths. 711 WM. B. SCAIFE & SONS, OFFICE, NO. 119 FIRST AVENUE, PITTSBURGH, PA., DESIGN, MANUFACTURE AA T D ERECT IRON MILL BUILDINGS, AND ALL KINDS OF IRON ROOF FRAMES, CORRUGATED IRON FOR ROOFS AND SIDING, SHEET AND PLATE IRON-WORK, CaldwelPs Patent Hollow-Shaft Spiral Conveyer, For Grain, Goal, Sawdust, Cottonseed, Flaxseed, etc. <^ SEND FOR CATALOGUES AND PRICE-LISTS. < *-^* From 1-4 to 15,000 Ibs. Weight. True 'to pattern, sound and solid, of unequalled strength, toughness and durability. An invaluable substitute for forg- ings, or for cast-iron requiring three-fold strength. Gearing of all kinds, Shoes, Dies, Hammer- Heads, Cross- Heads for Locomotives, etc. 15,000 Crank Shafts and 10,000 Gear Wheels of this steel now running prove its superiority over other Steel Castings. SPECIALTIES: CRANK SHAFTS, CROSS-HEADS, AND GEARINGS. STEEL CASTINGS STEEL CASTINGS OP EVERY DESCRIPTION, Please send for Circulars. Address CHESTER STEEL CASTINGS CO., Works, Chester, Pa. OFFICE, 407 LIBRARY ST., PHILADELPHIA. 712 ELBA iHOlrt BOLT CO., Limited, V^ MANUFACTURERS OF MERCHANT BAR IRON, SKELP IRON, SPLICE BARS, RAILWAY TRACK BOLTS, CAR, BRIDGE, AND MACHINERY BOLTS, NUTS, ETC. We invite the attention of RAILROAD MEN especially to our make of Splice Bars and Track Bolts. Using the best brands of REFINED IRON, and paying close at- tention to the finish of our manufactures, we are enabled to offer our patrons BOLTS, NUTS, SPLICE BARS, ETC., of excellent quality. Our works have been enlarged within a few years; all orders are now executed with promptness; all our work guaranteed. SEND FOR PRICE-LISTS AND INFORMATION TO ELBA IRON & BOLT Co., LIMITED, PITTSBURGH, PA. LOVEJOY & DRAKE, Agents, 49 Cortlandt Street, Netv 713 THE PEERLESS BRICK COMPANY, OP PHILADELPHIA, MANUFACTURE AND KEEP IN STOCK ARCHITECTURAL SHAPES, 300 KINDS. ALSO BED Pressed Fronts. Extra fine in color and quality. BUFF, solid rich color beautiful. One of the finest bricks made. DRAB, handsomer and more durable than stone. BROWN, very strong and superior to brown stone. GRAY, a very desirable shade. All the above are solid colors throughout. Special Shapes made to order. Particular attention paid to making and fitting ARCHES of all kinds and sizes (from plain and moulded bricks) from drawings furnished. BLACK, Velvety jet face. The only black brick fit for a fine building, producing a beautiful effect, and free from the glossy and greasy look of other black or dipped bricks. DIAPERING AND ORNAMENTAL Bricks made in the above colors. For further descriptions see pages 466 to 469 of this book. Illustrated Catalogue and Price-List sent free on application. Office, 208 S, Seventh St, (W.Washington Square.) SAMUEL HART, Pres. JOSEPH WOOD, JR., Treas. THE PH(ENIX IRON COMPANY No. 41O WALNUT STREET, PHILADELPHIA, PA. Manufacture Rolled Beams, Channels, Angles, Tee, Shape, and Bar Iron of all sizes. Roof Trusses, Girders, and Joists for Fire-Proof Buildings framed and fitted as per plans. PHCENIX Wrought-Iron Columns of all sizes. DIE-FORGED WELDLESS EYE-BARS a specialty. Designs and Estimates furnished upon application. 714 SCffiNTIFIf^D MECHANICAL BOOKS, AUCHIWCI.O8S. The Practical Application of toe Slide- J\_ Valve and Link-Motion to Stationary, Portable, Locomotive, and Marine Engines, with New and Simple Methods for proportioning the Parts. By WILLIAM S. AUCHINCLOSS, C.E. Seventh Edition. Revised and Enlarged. 8vo, cloth. $3.00. BIL.GRAM. Slide- Valve Gears. A new graphical method for Analyzing the Action of Slide-Valves, moved by eccentrics, link- motion, and cnt-oft gears, By HUGO BILGRAM, M.E. 16mo, cloth. $1.OO. /COOPER. A Treatise on the Use of Belting for the Traiis- vy mission of Power. With numerous illustrations of approved and actual methods of arranging Main Driving and Quarter Twist Belts, and of Belt Fastenings. Examples and Rules in great number for exhibiting and calcu- lating the size and driving power of Belts. Plain, Particular, and Practical Directions for the Treatment, Care, and Management of Belts. Descriptions of many varieties of Beltings, together with chapters on the Transmission of ower by Ropes ; by Iron and Wood Fractional Gearing; on the Strength of Belting Leather ; and on the Experimental Investigations of Morin, Briggs. s. By JOHN H. COOPER, M. E. 1 vol., demi octavo, cloth. S3.5O. Po Belt and others. DANBY. Practical Guide to the Determination of Miner- als by the Blowpipe. By DR. C. W. C. FUCHS. Translated and edited by T. W. DANBY, M.A., F.G.S. $2.5O. DRAKE. A Systematic Treatise, Historical, Etiological, and Practical, of the Principal Diseases of the Interior Valley of North America, as they appear in the Caucasian, African, Indian, and Esquimaux varieties of its population. By DANIEL DRAKE, M.D. 8vo, sheep. $5.OO. DWYER. The Immigrant Builder; or, Practical Hints to Handy-Men. Showing clearly how to plan and construct dwellings in the bush, on the prairie, or elsewhere, cheaply and well with Wood, Earth, or Gravel. Copiously illustrated. By C. P. DWYER, Architect. Tenth edi- tion. Demy 8vo, cloth. $1.5O. /"I ENTRY. The House Sparrow at Home and Abroad.. V7" By THOMAS G. GENTRY. Demy 8vo, cloth. $2.0O. jO IRARO. Herpetologry of the United States Exploring VJT Expedition of the years 1838, '39, '40, '41, and '42, under the command of Commodore Wilkes. With a folio Atlas of over Thirty Plates elegantly col- ored from nature, executed under the supervision of Dr. Charles Girard of the Smithsonian Institution. 4to, cloth. $3O.OO. tion ; n Saws. History, Development, and Ac- ion and Comparison ; Manufacture, Care, and Use of all kinds of Saws. By ROBERT GIUMSHAW. Large Octavo. 234 ILLUS- TRATIONS. This thorough work, impartially written in a clear, simple and practical style, treats the Saw scientifically, analyzing its action and work and describing, under the leading classes of Reciprocating and Continuous Acting Saws, the various kinds of large and small Hand, Sash Mnlay Ji- Drag, Circular, Cylinder, and Band Saws, as now and formerly used for Cross- Cutting, Ripping, Scroll-Cutting, and all other sawing operations in Wood Stone, and Metal, Ice, Ivory, etc., in this country and abroad. With Appen- dices concerning the details of Manufacture, Setting, Swaging, Gumming 'ihng. etc. : Tables of Gauges, Log Measurements from 10 to 24 feet, and from 12 to 96 inches; Lists of all U. S. Patents on Saws from 1790 to 1880, and other valuable information. Elegantly printed on extra heavy paper. Copiously indexed. Of immense practical value to every Saw user. $2.5O. 1 PUBLICATIONS OF E. CLAXTOK & CO., PHILADELPHIA. I < KITISH \\v. V Supplement to Qrimshaw on Saws, coii- \JT taining additional practical matter more especially relating to the forms of saw teeth for special material and conditions, and to the behavior of sa\vs under particular conditions. One hundred and twenty illustrations. By ROBERT GRIMSHAW. Quarto, cloth. $2.OO. f 1 RIMSMAW. Saws. The History, development . Action, \JT Classification, and Comparison of Saws of all kinds. With Copious Ap- pendices. Giving the details of manufacture, filing, setting, gumming, etc. Care and use of saws ; table of gauges ; capacities of saw-mills ; list of saw patents ; and other valuable information. By ROBERT GRIMSHAW, author of " Modern Milling," etc., etc. Second and greatly enlarged edition, with Sup- plement, 354 illustrations. Quarto, cloth. $4.0O. -Modern Milling. Being the substance of \JT two addresses delivered at the Franklin Institute, outlining the modern methods of Flour-Milling by " New Process " and Rollers. By ROBERT GRIM- SHAW. 8vo, 28 illustrations. Sl.OO. HARTMAN and MECHENER'S Coiichology.-oiichologia Cestrica. The Molluscous Animals and their Shells of Chester County, Pa. With numerous illustrations. By WILLIAM D. HARTMAN, M.D.. and EZRA MECHENER, M.D. 12mo., cloth. $1.OO* HOBSON. The Amateur Mechanic's Practical Hand- book. Describing the different tools required in the workshop, the use of them, and how to use them ; also, examples of different kinds of work, etc., with descriptions and drawings. By ARTHUR JI.G. HOBSON. 12mo, cloth, $1.25. LONCi and BIT ELL. The C'adet Engineer, or Steam for the Student. By JOHN H. LONG, Chief Engineer. (J. S v Navy, and R. H. BUEL, Assistant Engineer, U. S. Navy, Demy 8vo. cloth. $2.25. MORTON. The System of < 1 alciilatiiig Diameter, Circum- ference, Area, and Squaring the Circle. By JAMES MORTON. 12mo. cloth. $1.00. MOORE. The Universal Assistant anil Complete Mechanic. A Hand-Book of One Million Industrial Facts, Processes, Rules, For- mulae, Receipts. Business Forms. Tables, etc., in over Two Hundred Trades and Occupations Together with full directions for the Cure of Disease and the Maintenance of Health. By R. MOORK. A new revised edition, illustrated by 500 Engravings. T2mo, cloth. $2.5O. ATIHOL,S. The Theoretical and Practical Boiler-Maker JM and Engineer's Reference Book By SAMUKL NICHOLS, Foreman Boiler- Maker. 12mo, cloth, extra. $2.5O. XTYSTROM. A New Treatise 011 Elements of Mechanics, il establishing strict precision in the meaning of Dynamical Terms, accom- panied with an Appendix on Duodenal Arithmetic and Metrology. By JOHN W. NYSTROM. C.E. 8vo, cloth. Price reduced to 5=2.00. ATYSTROM. A New Treatise on Steam Engineering, Physi- 1\ cal Properties of Permanent Gases, and of Different Kinds of \apor. By JOHN W. NYSTROM. C.E. Svo, cloth. $1.5O. OVERMAN. -Mechanics for the Millwright. Engineer, Machinist. Civil Engineer, and Architect. By FREDERICK OVERMAN. 12mo, cloth. l.")0 illustrations. $1.50. OWEN. Report of a Geological Survey of Wisconsin, Iowa, and Minnesota. By DANIKI. DALK OWKN. 300 Illustrations. '2 \ ols. 4to, Cloth. $7.50. 2 PUBLICATIONS OF E. CLAXTONXt CO., PHILADELPHIA. RIDDRtL.. The Carpenteraiid Joiner Modernized. Third editigai, revised and corpeffted, containing new matter of interest to the Carpenter, Stair-Builder, Carriage-Builder, Cabinet- Maker, Joiner, and .Mason ; also explaining^ne utility of the Slide Rule, lucid examples of its accuracy in calculation; showing it to be indispensable to every workman in giving the mensurajami of surfaces and solids, the division of lines into equal parts, circumferences of circles, length of rafters and braces, board measure, etc. The whole illustrated with numerous engravings. By ROBERT RIDDELL. 4to, cloth. S7.5O. J )II>I>EL,L,. The ITew Element* of Hand Railing. Revised XX edition, containing forty-one plates, thirteen of which are now for the first time presented, together with the accompanying letter-press description. The whole giving a complete elucidation of the Art of Stair-Building. By ROBERT RIDDELL, author of " The Carpenter and Joiner Modernized," etc. One volume, folio. $7 .00. RIDDELL. Mechanic's Geometry; plainly Teaching the Carpenter, Joiner, Mason, Metal-plate Worker in fact, the artisan in any and every branch of industry whatsoever the constructive principles of his calling. Illustrated by accurate explanatory card-board Models and Diagrams. By ROBERT RIDDELL. Quarto, cloth. Fully illustrated bv fifty large plates. S5.OO. 1>IDDELL. Lessons on Hand-Railing for Learners. By V ROBERT RIDDELL, author of "New Elements of Hand-Railing," "The Carpenter and Joiner Modernized," etc. 4to, cloth. Third edition. $5.0O. 1)IDDELL. The Artisan. Illustrated by forty plates of 1\ Geometric drawings, showing the most practical methods that may be applied to works of building and other constructions. The whole is intended to advance the learner by teaching him in a plain and simple manner the utility of lines, and their application in producing results which are indis- pensable in all works of art. By ROBERT RIDDELL, late teacher of the artisan class in the Philadelphia High School, etc. $5.OO. E OPER. A Catechism of High-Pressure, or Non-Coiideiis- ing Steam-Engines; including the Modelling, Constructing, and Man- agement of Steam-Engines and Steam-Boilers. With valuable illustrations. By STEPHEN ROPER, Engineer. Sixteenth Thousand, revised and enlarged IHino. tucks, gilt edge. $2.OO. KOPER. Handbook of the Locomotive, including the Construction of Engines and Boilers, and the Construction, Manage- ment, and Running of Locomotives. Bv STEPHEN ROPER. Eleventh edition LSmo, tucks, gilt edge. S2.5O. I) OPER. Handbook of Land and Marine Engines: iiiclud- 1 ing the Modelling, Construction, Running, and Management of Land and Marine Engines md Boilers. With illustrations. By STEPHEN ROPER, Engi- neer. Tenth edition. 12mo, tucks, gilt edge. $3.5O. I ) OPER. Handbook of Modern Steam Fire-Eiigiiies. With II illustrations. By STEPHEN ROPER, Engineer. 12mo, tucks, gilt edge. $3.5O. 1) OPER. Engineer's Handy-Book. Containing a Full Ex- 1 plunation of the Steam-Engine Indicator, and its Use and Advantage-!* to Engineers and Steam Users: with Formulae for Estimating the Power of ah Classes of Steam-Engines: also Facts, Figures, Questions, and Tables for By STEPHEN ROPER, Engineer. 1 vol., 16mo, 675"pa^tUCks, 01^6^6. fr^SO. 1)OPER. Questions and Answers for Engineers. This little 1 book contains all the Questions that Engineers will be asked when un- dergoing an examination for the purpose of procuring licenses, and they are so plain that any engineer or fireman of ordinary intelligence may commit them to memory in a short time. By STEPHEN ROPER, Engineer. $3.OO. PUBLICATIONS OF E. i LAXToN | . .... PHILADELPHIA. 11OPER, I ne and Abn*e of the Steam-Boiler. By Stephen 11 frith illustrations. 18mo. tucks, gilt ^l.OA>.-tity uit fQeOMpanied ':>' spec in eat! oils ii::: r:- ... ..": ,x- planatory text. New edition, revised. Imperial 4to. Cloth. S1O.OO. ^ I. O \ >. Constructive Arehiteetnre. A Guide for the O Ider and Carpenter: - for the construc- tion of Roofs. Domes, ai iples of the fttf Orders ef Arckiltdbtrt : selected from Grecian and Roman art. with the figured dimensions of tht :0. SPAXi. A Practical Treatise 011 I.i^lun in- Protection. ^. By HENRY \v ih illustrations.. 12m. i.50. . A New Method of Calculating the Cubic _L Oo^ents of Excavations and Embankmt aid of Diagrams: together with Directions for estimating the cost of Earthwork. By J TRAVTWINE. CJ1 -. edition, completely revised ami enlarged, bvo, cloth. 2.OO. L'TWIXE. The Field Practice of Laying Ont Circular i'.roiids. By JOHN C. TRAUTWIXE . C.E. Eleventh-.. 18S2, revised and enlarged. 12mo, tuck. $2.50. r r"RAFTWIXE*S Civil Engineer's Pocket- Book of 3Iensnra- I n. Trigonometry Ilydrauli '' P ? n - C1 ? and Iron Roof and E "idges and Cnily ^nsion Br: ads. Tun: lonns. Water Stations, Cost of Earthwork, Fowndati JOH?* C. TRAUTWT > - .no, 696 pages, mor. tncks. gilt edges. Twentieth Thousand. Revised and corrected. 5.OO. ^IT^EBBER. 4 ifannal of Power for Machines, Shafts, and M Belts. With the History of Cotton Manufacture in the United - Bv S \ y :k contains the Power required by Cotton. Woollen. Worsted, and Flax Machinery, Shaft- ing, and Tools, with Summaries of the Machines and Po\ver used in a number of Cotton Mills on various f& strength and speed of Shafting and Belting : Corrected Tables of the Centennial Turbine Tests, I niilaMphi* UBS fcfcaki: - .:,\ TV- >: Fa le.s :- : Yarn and Historical Sketch of the growth : n manufacture in the States. The whole, with an explanatory prefece, forming an octavo volume of 296 pages, neatly bound in clot ^\^HITE.-Tli Elements of Theoretical and Oescriptive \ > .mv for the ose of Colleges and Academies. By CHARLES J. WRITE, A M Xumerous illustrations. 1 vol., demi 8vo. Fourth edition, revised. "$2.00. ^ll^HITXEY. Hetallie Wealth of the United State- \\ "ribedVnd compared with that of other countries. By J. D. WHITNEY. S3.50. Any of the fqjegoing books wiU be sent by mail, post-paid, on receipt of price, by E. CLAXTON