ip IN MEMORIAM FLORIAN CAJORi GALLOWAY & PORTER, ^e:^^"^ SLIDE-RULE INSTRUCTOR, FOR INSTRUCTION ON CHADWICK'S IMPROVED SLIDE RULE, CONTAINING NE^\^ AND IMPORTANT RULES UPON THE PRESENT PRACTICE OF ENGINEERING, ETC. FOR THE USE OF EngmeBrs^ MtlliXTnghtS; Mecbaujcs and J-rtisans, Mill ©wners, ^txiitxn Spinners, USalica 'J^vmiBVSf Bleachers and FinisberS; ^allierg i^roprietors, a»d Steam lasers generallu* JOHN CHADWICK, ENGINEER, MANCHESTER. JOHN HErWOOD, Deansqate and Ridqefield, Manchester ; 2, AMEN CORNER, LONDON, E.G. CAJORI ERRATA. Page 9, line 17. After from 4 to 5, 20 a semicolon (;) is omitted. Page 15, fifth line from the bottom, 35 threads upon A, should read : £32 10s. Od. Page 21, line fifteen from the bottom, the * refers to the note in the middle of page 22. Page 32, bottom line, D & inscribed CD should be D & inscribed (Q) and also T> and = Q. Page 32, in table at foot, last line, the triangle a should be inscribed in a circle, thus :© ='866. Page 35, twelve lines from bottom. Should read : 4^ on line D, instead of 4 J. Page 37, Mensuration of Surface, line 13, perimeter x area, should read : length x perimeter -f area of the two ends. Page 65, in the last example under " To Determine Diameter of Pipe to Supply an Engine with Steam," the figures have become crowded in printing, should read : — A 63 18 inches A 63 18 inches. or ^ \/750 7-85 inches B .. 27*4 7*85 inches. Page 69, line 15, R = rate of expansion, the capital R should be printed a small r. j. Page 69, line 17, the capital R should be printed a small r : (1 + log r) Page 73. In the first formula^ the figures have become crowded, should read : — . , u 1 Line C V 3 7*95 Expansions, And as per above rule ^- -^— ^ LineD >■ 16 26 Page 73. In dealing with the Second Formula, the reader will do well to refer to the rule on page 69. Page 73. In dealing with the Third Formula, the reader should con- sult the rule on page 71. 911251 PREFACE. A new edition of a work on the Slide Rule being much inquired for^ the writer, at the request of Mr. John Heywood, publisher, of Man- chester, undertook its editorship. On examining the works extant on the engineer's slide rule , much of the data and methods therein em- ployed are not adapted to give results such as are now required in the present advanced state of engineering science and practice ; therefore, to make the work useful and a text book as it were under such circum- stances, a considerable quantity of fresh subject matter has been added, most of which is entirely new in a work on this subject. The new matter treats on the following subjects : The power trans- mitted through leather belts ; screw-cutting in the slide lathe ; the weight, strength, and safe working load for ropes of hemp, iron and steel wire, also for chains and metal bars ; the maximum speed for turning metals ; rules for bringing English amounts into French, or the reverse, for money, weights or measures. Also rules for nnding the dimensions of the vital parts of a steam engine ; rules for finding the pressure of expanding steam during the stroke of the piston ; also for finding the power developed by a steam engine — single or compound — working under any given conditions as to diameter of cylinder or cylinders, pressure of steam, velocity of piston, rate of steam expansion, condensing or non-condensing as the case may be. The above-named rules on the action of steam are entirely new, and to effect which an additional logarithmic scale has been introduced on the back of the slide which has given to the instrument a new power for dealing with such problems as the above. The gauge points on the lower limb of th einstrument have been speciairy calculated for the work, and every care taken as far as possible to insure accuracy. A table of the temperatures, pressures, and volumes of lib. of saturated steam has also been placed on the same limb of the instrument. Being myself much engaged with business, and having but little spare time at disposal for compiling a work of this class, I have been greatly aided by Mr. Willia m Ashton (inventor of the steam power meter and continuous indicator) "wlTtrhas given me much assistance in ^calcula ting the gauge points, arranging the logarithmic scale, and ^ formulating the new rules for the various subjects introduced. JOHN CHADWICK, engineer. Prince's Bridge Ironworks, Manchester. INTRODUCTION. The Slide Rule is a marvel of ingenuity, utility, and simplicity, its operations not being alone confined to such cases and questions as are usually contained in the small works treating of it, but embracing all matters in which arithmetical calculations are concerned, and by its use the drudgery of computation and the mental labour involved, as well as loss of time occupied therein are greatly abridged ; the advantages derived from its use being in fact similar to those obtained by the application of machinery in obtaining increased efficiency in mechanical operations, and at the same time dispensing with or greatly lessening the manual labour required therein. Indeed, such is the utility of the instrument for perform- ing arithmetical calculations with rapidity and certainty, and so easily is it understood, that if its advantages and simplicity of operation were more generally known, no bank, office, shop, or workshop, where numerical calculations of any description are required, would be without it. We need not be surprised at the public knowing little of its general utility, when a workman who works by the day, hour, or piece, will puzzle and perplex himself at the end of the week to find out the amount of money due to him for the quantity of work he has performed, although he carries a Slide Rule in his pocket and has done for years, which would show him the amount in a moment had he not been too listless and apathetic to look at and understand it during the whole of that time. It is to be hoped that the introduction of the technical school system will make information on the Slide Rule an imperative necessity in order to teach students to dispense with or abridge tedious arithmetical operations. The calculating Slide Rule treated of in the following pages is in the usual form of the workman's jointed two-feet^ ^le, the divisions on the slide andf adjoining^ lines, or the ^^ calculating part of which have been more carefully and __ accurately graduated than is usually the case on such _ instruments, giving it therefore a considerably enhanced value over them. CONTENTS. PAGE Preface . 3 Introduction 5 Explanation of the Lines 9 Numeration 9 Multiplication 11 Division 14 Rule of Three Direct 16 „ Inverse 19 Vulgar and Decimal Fractions 24 Square Root 26 Cube Root 28 Superficial Mensuration 29 Case 1, 2.— To find the Content in Square Feet 29—30 „ 3. „ „ Square Yards 31 ,,4,5,6. „ „ Acres 31—32 ,, 7. „ Area of a Regular Polygon 32 „ 8. ,, Circumference of a Circle 34 ,5 9. „ Area of a Circle 35 „ 10. „ Area from Circumference 36 „ 11. ,, Side of a Square inscribed in a Circle 36 ,, 12. „ Side of a Triangle inscribed in a Circle 36 „ 13. „ Side of a square equal in Area to any given Circle 37 Mensuration of Solids 38 The Weighing of Metals 41 FlatMetals 44 ., Solid Balls and Cylinders 45 „ Piping, Shells, &c. 47 Liquid Measure 48 Tableof Specific Gravities 50 CONTENTS. I'AGE Cask Gauging 50 On the Pitch of Teeth in Wheels 52 Power Transmitted Through Leather Belts 55 Screw-cutting in Slide Lathe 56 Speed of Lathe for Cast Iron Turning 58 English Lineal Measures into French and the Reverse 59 Weight of Hemp Ropes in lbs 60 Strength of Hemp Ropes 61 Weight of Iron Wire Ropes in lbs 61 Strength of Iron Wire Ropes 61 Weight of Steel Wire Ropes in lbs 62 Strength of Steel Wire Ropes 62 Weight of Wrought Iron Chains in lbs : 63 Safe Working Load of Chains in cwts 63 Diameter of Air Pump for a Steam Engine 64 To Determine Diameter of a Pipe to supply an Engine with Steam 64 Area of Steam Ports 65 Pumping Engines 66 The Pressure of Steam in its Relation to its Expansion 67 The Indicated Horse Power of a Steam Engine 71 Compound Engines 72 Cotton Spinning . . ' 73 Block Tackle 78 Miscellaneous Problems , 79 INSTEUCTIOlSrS. EXPLANATION OF THE LINES. At the plain end of the rule, about three-eighths of an inch from the end, it is marked with the first four letters of the Alphabet, A, B, C, D. B, C, are upon the metal slide, and A, D, are upon the wood (which is generally good box w^ood). A line runs through each letter from end to end of the rule, and is called A line, B line, C line, and D line. ^A^^Rj^jxd ^J ^are mar kM..^xactly_ alike, and are comj)osed of, a double radius, and figured from left to rigH^li'om 1 to 10 twice over . The T) line is a single radius, double the length of the others, and figured from 1 to 10. The lines B, C, slide ISetween A, D, and by this operation are all the questions answered in the following work, except square root, which will be explained under that head. Each of the figures are separated by fine lines or divisions, and are divided in the following manner : — Between 1 and 2 are 50 divisions ; from 2 to 3, 20 divisions ; from 3 to 4, 20 divisions ; from 4 to 5, 20 from 5 to 6, from 6 to 7, from 7 to 8, from 8 to 9, from 9 to 10, each have 10 divisions ; the second or right hand radius is a ref etition of the first, and is divided as before. The line D is divided as follows : — From 1 to 2 are 100 divisions; from 2 to 3, 50 divisions; from 3 to 4, 50 divisions ; from 4 to 5, and from 5 to 6, 20 divisions : the rest have 10 divisions each. NUMERATION. Numeration is the art of valuing aright the lines and divisions upon the instrument, for be it known that numeration is the governing rule of instrumental arithmetic, without it nothing can be done, and upon it every other rule is entirely dependent, and in order that the pupil may well understand this rule, observe the following : — 10 First — That all numbers and divisions are arbitrary. Secondly-T^That, a]l the numbers and divisions increase and' decra^'e ibj^a ten-rf'old .proportion. Thirdly — That the lowest numbers must be taken next the joint end of the instrument, increasing towards the right hand. If the pupil attend to the foregoing directions, he will not find any difficulty in pointing out any number upon the radius lines ; they may be best understood in the following order : — If the first 1 next the joint denote 1-tenth, then thev middle 1 will be 1 unit or 1 whole number ; the other figures toward the right are also whole numbers, from the middle 1 to 10 at the plain end. Again, if the first end 1 denote 1 unit or 1 whole number, then the middle 1 will denote 10, and the 10 at the plain end will denote 100. Again, if the first 1 denote 10, the middle 1 will be 100, and the 10 at the plain end 1,000. By repeating again the 10 at the plain end will denote 10,000, &c., &c. The divisions between the figures will also change their value, thus, when the first 1 denotes 1 unit, then the next figure, which is 2, will be 2 units ; and the longer divisions between 1 and 2, being 10 in number, will of course be tenths of a unit, while the 5 shorter sub-divisions of each tenth will each be a fiftieth of a unit. Again the middle 1 will denote 10, and the next figure 2 will be 20; the divisions are parts between 10 and 20 ; each long division will be 1, as 10 and 1 are 11 ; and 5 long divisions following the 10 will be 15, and so on with any number between 10 and 20. We will now proceed with a few^ examples and conclude this rule. Let it be proposed to find 13 on the upper line or line A] Commence at the joint end and say 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and counting 3 long divisions following the 10 is the number 13 required. If a higher value be put upon the first 1, this number may also represent 130 or 1,300, &c. Note in pointing out any number, slide 1 upon B opposite the number required upon A. 11 Point out the number 39|^ upon the line A 1 Commence at the joint as before, and say 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, and 9 long divisions towards 4 is 9, and 1 short division is J ; altogether is 39 J, the number required. This number may also read as 39*5, 395, 3,950, 39,500, &c. According to the value given to the first 1. Find 990 upon the line A? The first 1 will be 10, the middle 1, 100, the next figure, which is 2, will be 200, and the next figure 300, 400, 500, 600, 700, 800, 900, and 9 divisions will be 90, when added together is 990, the number required. This number may also represent 99, 990, 9,900, or 9*9, &c. According to the value given to the first 1. Numerate 188 upon the line A1 The first 1 will denote 10, the middle 100, and 8 long divisions is 80, and 4 short divisions is 8 ; when added is 188, the number required. Numerate 1 9 shillings upon the line A ] The middle 1 will be 10, and each long division between 10 and 20 is of course 1 shilling ; then counting 9 long divisions following the 10 is of course 19 shillings, the number required. This may also read as £19, £190, £1,900, &c. Or calling the 1 or 10 as £1, it may also read as £1 18s. Find £2 15s. upon the line A? The 2 next the joint will be £2, and 7 long divisions is 14s., and 1 short division is 1 shilling, which is £2 15s., the value sought. Find 3,350 upon the line A? The first 1 will denote 100, the middle 1, 1,000, and the 3 towards the right hand is 3,000, and 3 long divisions is 300, and 1 short division is 50 ; when added is 3,350, the number sought. This also may read 33 J or 33*5, 335 or 3*35. According to the value given to the first 1. The foregoing examples being well considered are sufficient for the industrious pupil. MULTIPLICATION. By multiplication we increase or multiply one number by another as often as there are units in either of the numbers. This rule should be well understood by the pupil, for by it many hundreds of questions in many parts of arithmetic are resolved. 12 In multiplication are three principal parts to be well taken notice of : — First — The multiplicand, or number to be multiplied. Secondly — The multiplier, or number by which we multiply. Thirdly — The product, or the number proceeding or pro- duced from both. Rule. — Set the multiplier upon B to 1 or 10 upon A, and opposite any multiplicand upon A will be the product upon B. EXAMPLES. What is the product of 2| multiplied by 3 1 Ans. 7J. Set 3 uponB to 1 upon A, and opposite 2| upon A is 7i upon B, the answer. The slide thus set you have a table of products multiplied by 3, for opposite any multiplicand upon A is the product upon B, as follows : — Multiplicands upon A, 1, 1 J, 2, 2 J, 3, 4, 5, 6, 7. Products upon B, 3, 4J, 6, 7|, 9, 12, 15, 18, 21. And in like manner, at every operation, the lines A and B will form a set of tables. What is the product of 4 multiplied by 5 ? Ans. 20. Set 5 upon B to 1 upon A, and opposite 4 upon A is 20 upon B, the answer. This question is the same as if one had demanded 4 yards of calico at 5d. per yard. What is the product of 12 multiplied by 8 ? Ans. 96. Set 8 upon B to 1 upon A, and opposite 12 upon A is 96 upon B, the answer. What is the product of 14 multiplied by 7 ? Ans. 98. Set 7 upon B to 1 upon A, and opposite 14 upon A is 98 upon B, the answer. What is the product of 84 multiplied by 5 ? Ans. 420. 13 Set 5 upon B to 10 upon A, and opposite 84 upon A is 420 upon B, the answer. What is the product of 18 multiplied by 6J? Ans. 117. Set 6 J upon B to 10 upon A, and opposite 18 upon A is 117 upon B, the answer. What is the product of 325 multiplied by 26^ Ans. 8450. Set 26 upon B to 10 upon A, and opposite 325 A is 8450 upon B, the answer. What is the product of 75 multiplied by 19 ? Ans. 1425 Set 19 upon B to 10 upon A, and opposite 75 upon A is 1425 upon B, the answer. What is the product of 38 multiplied by 23 1 Ans. 874. Set 23 upon B to 10 upon A, and opposite 38 upon A is 874 upon B, the answer. What is the product of 144 multiplied by 12 1 Ans. 1728. Set 12 upon B to 10 upon A, and opposite 144 upon A is 1728 upon B, the answer. What cost 30 yards of cloth at 4s. 6d. per yard 1 Ans. 135 shillings. Set 4s. 6d., or 4 J, upon B, to 10 upon A, and opposite 30 yards upon A is 135 shilliDgs upon B, the answer. What cost 401bs. of coffee at 2s. 9d. per lb. 1 Ans. 110 shillings. Set 2s. 9d., or 2|, upon B to 10 upon A, and opposite 40 upon A is 110 shillings upon B, the answer. What cost 96 bags of nails at 4s. 3d. per bag 1 Ans. 408 shillings. Set 4s. 3d., or 4^, upon B to 10 upon A, and opposite 96 upon A is 408 shillings upon B, the answer. What cost 30 tons of iron at £2 15s. Od. per ton? Ans. £82 10s. Od. Set £2 15s. Od., or 2|, upon B to 1 upon A, and opposite 30 upon A is £82 10s. Od. upon B, the answer. What cost 160 sets of china at £3 5s. Od. per set 1 Ans. £520. 14 Set £3 5s. Od., or 3|, upon B to 10 upon A, and opposite 160 upon A is £520 upon B, the answer. What cost 75,000 bricks at £1 4s. Od. per 1000? Ans. £90. Set £1 4s. Od. upon B to 10 upon A, and opposite 75 upon A is 90 upon B, the answer. What cost 240 umbrellas at 7s. 3d. each? Ans. 1740 shillings. Set 7s. 3d., or 7^, upon B to 100 upon A, and opposite 240 upon A is 1740 shillings upon B, the answer. Note. — The operation may be contracted in the last question ; by setting 7 J upon B to 20 upon A, and opposite 240 upon A is £87 upon B, the answer. What cost 280 gross of tape at 6s 9d. per gross? Ans. £94 10s. Od. Set 6f upon B to 20 upon A, and opposite 280 upon A is £94 10s. Od. upon B, the answer. What cost 144 lamps at 43s. 9d. each. Ans. £315. Set 43s. 9d. upon B to 20 upon A, and opposite 144 upon A is £315 upon B, the answer. What cost 400 cwt. of wrought iron at 28s. per cwt. ? Ans. £560. Set 28 upon B to 20 upon A, and opposite 400 upon A is £560 upon B, the answer. DIVISION. By Division we discover how often one number is contained in another. In division there are three principal parts to be taken. First — the dividend, or number to be divided. Secondly — the divisor, or number by which we divide. Thirdly — the quotient, or number proceeding from the other two. Sometimes there is a fourth number called a remainder. Rule. — Set the divisor upon B to the dividend upon A, and opposite 1 or 10 upon B will be the quotient upon A. 15 EXAMPLES. Divide 42 by 7. Ans, 6 times. Set 7 upon B to 42 upon A, and opposite 1 upon B is 6 times upon A, the answer. This question is the same as if we had demanded in 42 days how many weeks ? Divide 72 by 12. Ans. 6 times. Set 12 upon B to 72 upon A, and opposite 1 upon B is 6 times upon A, the answer. This question is the same as if it had been asked in 72 pence how many shillings ; answer, 6 shillings. Divide 112 by 7. Ans. 16 times. Set 7 upon B to 112 upon A, and opposite 1 upon B is 16 times upon A, the answer. Divide 99 by 11. Ans. 9 times. Set 11 upon B to 99 upon A, and opposite 1 upon B is 9 times upon A, the answer. Divide 364 by 52. Ans. 7 times. Set 52 upon B to 364 upon A, and opposite 10 upon B is 7 times upon A, the answer. Divide 1728 by 72. Ans. 24 times. Set 72 upon B to 1728 upon A, and opposite 10 upon B is 24 upon A, the answer. Divide 6900 by 150. Ans. 46. Set 150 upon B to 6900 upon A, and opposite 10 upon B is 46 upon A, the answer. Divide £7800 amongst 240 men; what is each man's share? Ans. £32 10s. Od. Set 240 upon B to £7800 upon A, and opposite 10 upon B is 35 threads upon A, the answer. In 1890 inches how many threads of 54 inches long? Ans. 35 threads. Set 54 upon B to 1890 upon A, and opposite 10 upon B is 35 threads upon A, the answer. 16 Divide 625 by 25 Ans. 25 times. Set 25 upon B to 625 upon A, and opposite 1 upon B is 25 upon A, the answer. If there are 1000 ounces in a cubical foot of water, how many lbs. of 16oz. each? Ans. 62|lbs. Set 16 upon B to 1000 upon A, and opposite 1 upon B is 62|lbs. upon A, the answer. If 60 minutes of time are equal to 1 hour, during which time the earth passes through 15 degrees of space, how long is the earth in passing through one degree ? Ans. 4 minutes. Set 15 upon B to 60 upon A, and opposite 1 upon B is 4 minutes upon A, the answer. If a set of yarn contains 1920 hanks and weighs 61bs., how many hanks in one pound 1 Ans. 320 hanks. Set 6 upon B to 1920 upon A, and opposite 1 upon B is 320 hanks upon A, the answer. If a set contains 1480 hanks and weighs 51bs., what are the counts of the yarn? Ans. 296 hanks. Set 5 upon B to 1480 hanks upon A, and opposite 1 upon B is 296 hanks upon A, the answer. In 1728 inches how many feet of 12 inches long ? Ans. 144 feet. Set 12 upon B to 1728 upon A, and opposite 1 upon B is 144 feet upon A, the answer. RULE OF THREE DIRECT. This rule teacheth by three numbers given to find a fourth in such proportion to the third as the second is to the first, for which reason it is called the Rule of Three, from its having three numbers given. Rule. — Set the first term upon B to the second term upon A, and opposite the third term upon B is the fourth upon A ; always observing to take the first and third terms on the line B, and the second and fourth upon the line A. 17 EXAMPLES. If 6 yards of cloth cost 2s. 6d., what will 72 yards cost Ans. 30s. Set 6 upon B to 2s. 6d., or 2|, upon A, and opposite 72 upon B is 30 shillings upon A, the answer. If 7 J yards of cloth cost 19s., what will 45 yards cost? Ans. 114 shillings. Set 7 J upon B to 19 upon A, and opposite 45 upon B is 114 shillings upon A, the answer. If I give £6 15s. for 4 packs of wool, how many packs can I buy for £81 ] Ans. 48 packs. Set £6 15s. upon B to 4 upon A, and opposite £81 upon B is 48 packs upon A, the answer. If a man can walk 33 J miles in 4 hours, in what time will he walk 335 miles 1 Ans. 40 hours. Set 33J upon B to 4 upon A, and opposite 335 upon B is 40 hours upon A, the answer. If a man can walk 37 J miles in 5 hours, how long will he be in walking 210 miles'? Ans. 28 hours. Set 37 J upon B to 5 upon A, and opposite 210 upon B, is 28 hours upon A, the answer. If 41bs of candles cost 2s. 6d., what will 721bs. cost 1 Ans. 45 shillings. Set 3 upon B to 2s. 6d. upon A, and opposite 72 upon B, is 45 shillings upon A, the answer. If Icwt. of cast iron cost £1 6s., what will 40cwts. cost? Ans. £52. Set 1 upon B to £1 6s. upon A, and opposite 40 upon B is £52 upon A, the answer. If 24 cost 18s., what will 4 cost? Ans. 3 shillings. Set 24 upon B to 18 upon A, and opposite 4 upon B is 3 shillings upon A, the answer. If 23 cost £1 13s., what will 115 cost? Ans. £8 5s. Set 23 upon B to £1 13s. upon A, and opposite 115 is £8 5s. upon A, the answer. 18 If a carding engine throws off 21bs of cotton in 22 J minutes, in what time will it throw off 801bs. Ans. 900 minutes. Set 2 upon B to 22 J upon A, and opposite 80 upon A is 900 minutes upon A, the answer. If in a wheel of 12 feet circumference there are 72 teeth, how many teeth are there in a wheel of 16 feet circumfer- ence, the pitch being the same ? Ans. 96 teeth. Set 12 upon B to 72 upon A, and opposite 16 upon B is 96 teeth upon A, the answer. A shaft on which is a pully 42 inches diameter drives another shaft 56 revolutions per minute. What should be the diameter of a pully to drive the latter shaft 44 revo- lutions per minute 1 Ans. 33 inches diameter. Set 56 upon B to 42 upon A, and opposite 44 upon B is 33 inches upon A, the answer. If a cistern containing 240 gallons of water, has a cock which discharges 6 gallons in a minute, and another has a cock which discharges 15 gallons per minute, and both cisterns are emptied in the same time, how many gallons does this last cistern contain ? Ans. 600 gallons. Set 6 upon B to 240 upon A, and opposite 15 upon B is 600 gallons upon A, the answer. If 2 men in three days earn 15s., how much will 7 men earn in the same time ? Ans. 528. 6d. Set 2 upon B to 15 upon A, and opposite 7 upon B is 52s. 6d. upon A, the answer. If 3 bundles of cotton cost 18s., what will 17 bundles cost? Ans. 102 shillings. Set 3 upon B to 18 upon A, and opposite 17 upon B is 102 shillings upon A, the answer. If there are 840 yards in one hank, what quantity of hanks will be required to make a warp 1,680 ends broad, and 152 yards long 1 Ans. 304 hanks. Set 840 upon B to 1,680 upon A, and opposite 152 upon B is 304 hanks upon A, the answer. 19 THE RULE OF THREE INVERSE. This Rale teacheth by three numbers given to find a fourth that shall have the same proportion to the second as the first has to the third. Note. — If more requires less, or less requires more, it belongs to this rule, but if more requires more, or less requires less, that question belongs to the Rule of Three Direct. Rule. — Invert the slide and set the first term upon C to the second term upon A, and opposite the third term upon C is the fourth upon A. In inverse proportion with the slide inverted, it may be noted that the products of the vertical numbers on the rule are equal — as 128 64 in the example below. 8 16 1024-1024 EXAMPLES. Suppose I lend my friend £128 for 8 months, hew long ought he to lend me £64 to requite my kindness ? Ans. 16 months. Invert the slide and set £128 upon C to 8 upon A, and opposite £64 upon C is 16 months upon A, the answer. If 64 men perform a piece of work in 70 days, how many men can perform the same in 40 days? Ans. 112 men. Set 64 upon C to 70 upon A, and opposite 40 upon C is 112 men upon A, the answer. If a pasture will feed 32 head of cattle for 6 weeks, how long will it feed 48 head ? Ans. 1 weeks. Set 32 upon C to 6 upon A, and opposite 48 upon C is 4 weeks upon A, the answer. If, when the price of a bushel of wheat is 3s., the penny loaf weighs 18oz., what must the penny loaf weigh when the bushel is worth 4s. 6d. ? Ans. 12oz. 20 Set 3 upon C to 18 upon A, and opposite 4s. 6d. upon C is 12oz. upon A, the answer. If 801bs. be conveyed 54 miles for £1 9s., how far can I have conveyed 3601bs. for the same money? Ans. 12 miles. Set 80 upon C to 54 upon A, and opposite 3601bs. upon C is 12 miles upon A, the answer. If 16 pioneers make a trench in 11 days, how many days will 44 men do the same ? Ans. 4 days. Set 16 upon C to 11 upon A, and opposite 44 upon C is 4 days upon A, the answer. ' If in 15 years £80 gain £50, in what time will £480 gain the same amount 1 Ans. 2| years. Set 15 upon C to 80 upon A, and opposite 480 upon C is 2J years upon A, the answer. If a 15-inch pulley makes 108 revolutions per minute, driving one of 6 inches diameter, required the speed per minute of the latter pulley ? Ans. 270 revolutions. Set 15 upon C to 108 upon A, and opposite 6 upon C is 270 revolutions upon A, the answer. The line shaft in a mill makes 132 revolutions upon its axis per minute, with a pulley 14 inches diameter upon it, driving one of 12 inches diameter placed on the crank of a loom, required the number of picks per minute ? Ans. 154 picks. Set 14 upon C to 132 upon A, and opposite 12 upon C is 154 picks upon A, the answer. The fly wheel shaft of a steam engine makes 33 revo- lutions per minute, having a wheel upon it of 104 teeth, required the number of teeth in the driven wheel that the next shaft may make 49 revolutions per minute 1 Ans. 70 teeth. Set 104 upon C to 33 upon A, and opposite 49 upon C is 70 teeth upon B, the answer. If a 50-inch pulley makes 70 revolutions per minute, driving one of 3| inches diameter, required the speed of the latter pulley 1 Ans. 1,000 revolutions. 21 Set 50 upon C to 70 upon A, and opposite 3| upon C is 1,000 revolutions upon A, the answer. I am spinning 120's with a pinion of 35 teeth, what pinion will 200's require 1 Ans. 21 pinion. Set 120 upon C to 35 upon A, and opposite 200 upon C is 21 pinion upon A, the answer. If a 15 pinion spin ISO's, what counts will 12's pinion spin "? Ans. 225 hanks. Set 15 upon C to 180 upon A, and opposite 12 upon C is 225 hanks upon C, the answer. I am weaving a piece of cloth 16 picks to one inch with a pinion of 28 teeth, what pinion must be put on to give 14 picks to the inch ? Ans. 32 pinion. Set 16 upon C to 28 upon A, and opposite 14 upon C is 32 pinion upon A, the answer. If a 36 pinion gives 93 picks per inch, how many picks will a 22 pinion give? Ans. 152 picks. Set 36 upon C to 93 upon A, and opposite 22 upon C is 152 picks upon A, the answer. If a 19 pinion gives 87 picks per inch, what pinion will give 118 picks per inch 1 Ans. 14 pinion. Set 19 upon C to 87 upon A, and the opposite 118 upon C is 14 pinion upon A, the answer. To this rule belong those questions applicable to the lever,* of which there are three kinds of varieties depending upon the position and the application of a certain fixed point called the fulcrum, on which the lever is supposed to move freely. Those portions of the lever between the fulcrum and the respective weights are called the arms of the lever ; and the length of each of those arms is inversely proportional to the respective weights which they each carry. Hence the product of each arm of the lever multiplied by the weight or force acting upon it is equal to that of the other product of arm by weight. In other words, the product of the weight or power multiplied by the distance it moves through is equal to the weight moved, or resistance overcome, multiplied by the distance it has moved through. The above law is universal, and applicable to all mechanical appliances and engines for raising weights or performing work. 22 General formula for any order of lever, the slide being inverted. ^-.. ^^^ .- 2S> un line A > V V On line C < — : A o bo eg S'S In all cases short arm x great weight — long arm x small weight. A lever of the first kind is shown below, in Fig. I., in which the fulcrum F is situated between the moving power P and the resistance W. Fig. II, is a lever of the second kind, in which the mover P and the resistance W act on the same side of the fulcrum, the load moved being between the fulcrum and the lever. *See Tomlinson's Rudimentary Mechanics, page 32. Fig. III. is a lever of the third kind ; the mover P and the load W also act on the same side of the fulcrum, but the mover P is between the fulcrum F and the load W. ^v A T ^,v H TT T\r IK 23 THE FIRST KIND. The short arm of a lever is 2 inches long, at the end is a weight of 6701bs., the long arm is 33J inches ; what weight must be applied at the extreme end in order to raise the weight? Ans. 401bs. Set 2 upon C to 670 upon A, and opposite 33| upon C is 401bs. upon A, the answer. If 401bs. be suspended on the end of a lever 56 inches from the fulcrum, what weight may be balanced 3^ inches on the other side of the fulcrum ? Ans. 6401bs. Set 40 upon C to 56 upon A, and opposite 3^ upon C is 3401bs, upon A, the answer. If 3601bs. be suspended 3f inches from the fulcrum, at what distance must 181bs. be applied on the other side of the fulcrum to produce an equilibrium? Ans. 75 inches. Set 360 upon C to 3| upon A, and opposite 18 upon C is 75 inches upon A, the answer. THE SECOND KIND. If a lever, having the fulcrum at one end, and 2 J inches from the fulcrum be suspended 781bs., at what distance must 51bs. b3 applied in order to raise the weight 1 Ans. 39 inches. Set 2J upon C to 78 upon A, and opposite 5 upon C is 39 inches upon C, the answer. If a lever, 25 inches long, having a fulcrum at one end, and 5 inches from the fulcrum is suspended 1121bs., how far from the fulcrum must 281bs. be applied in order to raise the weight 1 Ans. 20 inches. Set 5 upon C to 112 upon A, and opposite 28 upon C is 20 inches upon A, the answer. If a lever, 48 inches long, having a fulcrum at one end, and 4 inches from the fulcrum be suspended 551bs., what force must be applied at the end of the lever in order to raise the weight? Ans. 4-581bs. Set 4 upon C to 55 upon A, and opposite 48 inches, the length of the lever upon C, is 4*58 upon A, the answer. 24 THE THIRD KIND. Suppose the lever of a safety valve to be 60 inches long, having a fulcrum at one end, the spindle of the valve is 4 inches from the fulcrum, and at the other end is a weight of 201bs., at what pressure will the steam escape, supposing the area of the valve to be one square inch 1 Ans. 3001bs. pressure. Set 20 upon C to 60 upon A, the length of the longest arm, and opposite 4 upon C is SOOlbs. pressure upon A, the answer. , Suppose the lever of a safety valve to be 48in. long, the fulcrum at one end, and 2 2 Jibs, suspended at the other end ; 6 inches from the fulcrum is the spindle of the valve, and its area equal to 12 square inches ; — at what pressure per square inch will the steam escape? Ans. 151bs. pressure. Set 48 upon C to 22| upon A, and opposite 6 upon C is ISOlbs. upon A, the pressure upon the valve; divide 180 by 12, and it gives 151bs. pressure, the answer. A hydraulic press worked by hand lever, with a force of 2cwts. and a stroke of 1 foot, having by 10 strokes raised the ram 1 inch. Required to find the pressure the ram has applied against its resistance. Here we have 2cwts. moved through 120 inches against resistance moved 1 inch. Set 2 on line C to 120 on line A, and opposite 1 on line A is 240 cwts., or 12 tons, on line C. VULGAR AND DECIMAL FRACTIONS. A fraction is one or more parts of a thing, and supposes the unit divided into a number of parts. It is expressed by two numbers, one above the other, with a line between them, as I, I, f . The upper number is called the numerator, and shows how many of these parts the fraction contains, as |^ is equal to one-eighth part, f is equal to three-seventh parts, &c. The lower number is called the denominator, and shows the number of parts into which the unit is divided. 25 To reduce a vulgar fraction to its decimal expression. Rule. — Set the denomiator upon B to the numerator upon A, and opposite 100 upon B is the decimal required upon A EXAMPLES. Reduce f to its decimal expression. Ans. '75. Set 4 upon B to 3 upon A, and opposite 100 upon B is 75 upon A, the answer. Reduce f to its decimal expression, Ans. '625. Set 8 upon B to 5 upon A, and opposite 100 upon B is '625 upon A^ the answer. Reduce f to its decimal expression. Ans. "375. Set 8 upon B to 3 upon A, and opposite 100 upon B is •375 upon A, the answer Reduce ^ to its equivalent decimal expression. Ans. -125. Set 8 upon B to 1 upon A, and opposite 100 upon B is *125 upon A, the answer. Reduce J to its decimal expression. Ans. '25. Set 4 upon B to 1 upon A, and opposite 100 upon B is •25 upon A, the answer. Reduce y to a decimal. Ans. '571. Set 7 upon B to 4 upon A, and opposite 100 upon B is •571 upon A, the answer. Reduce -{^ to a decimal Ans. -4375. Set 16 upon B to 7 upon A, and oppposite 100 upon B is •4375 upon A, the answer. Reduce —^ to a decimal. Ans. '1 33. Set 15 upon B to 2 upon A, and opposite 100 upon B is '133 upon A, the answer. Reduce -^-^ to a decimal. Ans. '194. Set 36 upon B to 7 upon A, and opposite 100 upon B is '194 upon A, the answer. Reduce if to a decimal. Ans. '321. 26 Set 56 upon B to 18 upon A, and opposite 100 upon B is '321 upon A, the answer. Reduce |^ to a decimal. Ans. '875. Set 8 upon B to 7 upon A, and opposite 100 upon B is •875 upon A, the answer. Reduce -^^ to a decimal. Ans. '75. Set 12 upon B to 9 upon A, and opposite 100 upon B is •75 upon A, the answer. EXTRACTION OF THE SQUARE ROOT. Extraction of the square root is to find out such a num- ber as, being multiplied into itself, the product will be equal to the given number. Rule. — Set the slide even at both ends, then the line C is a table of squares and the line D is a table of roots. For opposite any number upon C is its square root upon D, as in the following table : — Squares upon C... 1 4 9 16 25 36 49 64 81 100 Roots uponD 12345 6789 10 EXAMPLES. Extract the square root of 225. Ans 15. Set the slide even at both ends, as before directed, and opposite 225 upon C is 15 upon D, the answer. Extract the root of 841. Ans. 29. Opposite 841 upon C is 29 upon D, the answer. Extract the square root of 625. Ans. 25. Opposite 625 upon C is 25 upon D, the answer. If the top of a castle from the ground be 45 yards, and surrounded with a river 60 yards broad, what length must the ladder be to reach from the outside of the river to the top ef the castle"? Ans. 75 yards. 27 Opposite 45 upon D is 2025 upon C, the square of the perpendicular, and opposite 60 upon D is 3600 upon C, the square of the base, add the two squares together, and the sum is 5625 ; find 5625 upon C, and opposite is 75 yards, the answer. Case 2. To find a mean proportional between two numbers. • Kule. — Set one of the numbers upon C to the same number upon D, and opposite the other number upon C is the mean proportional sought upon D. EXAMPLES. What is the mean proportional between 16 and 36 1 Ans. 24. Set 16 upon C to 16 upon D, and opposite 36 upon C is 24 upon D, the answer. What is the mean proportional between 16 and 64 ? Ans. 32. Set the slide as before, and opposite 64 upon C is 32 upon D, the mean proportional sought. If a shaft making 16 revolutions per minute gives motion to a third shaft making 121 revolutions per minute, required the revolutions of the intermediate shaft. Ans. 44 revolutions. Set 16 upon C to 16 upon D, and opposite 121 upon C is 44 upon D, the answer. If of three shafts the first makes 18 revolutions and the third 32 revolutions in the same time, how many revolu- tions should the second shaft make ? Ans. 24 revolutions. Set 18 upon C to 18 upon D, and opposite 32 upon C is 24 upon D, the answer. What is the mean square between 3 J broad and | thick ? Ans. 1-43. Set '625 upon C to *625 upon D, and opposite 3 J upon C is 1*43 upon D, the answer. The breadth of a bar of iron is 2|in., and fin. thick, what is the mean square ? Ans 1*015 28 Set '375 upon C to '375 upon D, and opposite 2f upon C is 1*015 upon D, the mean square. To find a mean square between the breadth and depth of a cistern whose breadth is 36in. and depth 24in. Ans. 29*39. Set 36 upon C to 36 upon D, and opposite 24 upon C is 29*39 upon D, the mean square. What is the mean proportion between 9 and 4 1 Ans. 6, Set 9 upon C to 9 upon D, and opposite 4 upon C is 6 upon D, the answer. ^ CUBE ROOT. To extract the cube root is to find out a number, which being multiplied into itself, and then again into itself, the product is equal to the given number. Rule. — Invert the slide, and set the given number upon B to 1 or 10 upon D, and the root will be found where equal value coincides on the two lines B and D. EXAMPLES. Extract the cube root of 216. Ans. 6, Invert the slide, and set 216 upon B to 10 upon D, and opposite 6 upon B is 6 upon D, the root of 216. Extract the cube root of 343. Ans. 7. Set 343 upon B to 10 upon D, and opposite 7 upon B is 7 upon D, the cube root of 343. To cube a given number by one operation. Rule. — Set the given number upon C to 1 or 10 upon D, and opposite the same number upon D, is the cube number upon C. EXAMPLES, What is the cube of 9 1 Ans. 729. Set 9 upon C to 10 upon D, and opposite 9 upon D is 729 upon C, the answer. 29 What is the cube of 8 ? Ans, 512, Set 8 upon C to 10 upon D, and opposite 8 upon D is 512 upon C, the answer SUPERFICIAL MENSURATION. The area of any plane figure is its superficial content, or the measurement of its surface without any regard to thickness. Case 1. Given the length in feet, and breadth in inches to find the content. Rule. — Set the breadth upon B to 12 upon A, and opposite the length upon A is the content upon B. EXAMPLES, What is the superficial content of a board 18 feet long and 11 inches broad 1 Ans. 16| square feet. Set 11 upon B to 12 upon A, and opposite 18 upon A is 16 J square feet upon B, the answer. What is the content of a board 17 inches broad and 30 feet long 1 Ans. 42 J square feet. Set 17 upon B to 12 upon A, and opposite 30 upon A is 42| square feet upon B, the answer. Required the content of a board 5 inches wide and 18 feet long 1 Ans. 7 J square feet. Set 5 upon B to 12 upon A, and opposite 18 upon A is 7^ feet upon B, the answer. What is the content of a board 60 feet long and 11 inches wide 1 Ans. 55 feet. Set 11 upon B to 12 upon A, and opposite 60 upon A is 55 feet upon B, the answer. What is the content of a board 26 feet 8 inches long and 9 inches broad ? Ans. 20 feet. 30 « Set 9 upon B to 12 upon A, and opposite 26 feet 8 inches upon A is 20 feet upon B, the answer. A door is 6 feet 9 inches long and 36 inches wide; required the content. Ans. 20J feet. Set 36 upon B to 12 upon A, and opposite 6 feet 9 inches upon A is 20J feet upon B, the answer. The bottom of a cistern is 5 feet long and 54 inches wide ; required the content. Ans. 22 J feet. Set 54 upon B to 12 upon A, and opposite 5 upon A is 22 J feet upon B, the answer. * Case 2, Given length and breadth in inches to find the content. Rule. — Set the breadth upon B to 144 upon A, and opposite the length upon A is the content upon B in square feet. EXAMPLES, A pane of glass is 27 inches long by 24 inches wide ; required the content. Ans. 4 J feet. Set 24 upon B to 144 upon A, and opposite 27 upon A is 4| feet upon B, the answer. What is the content of a pane of glass 32 inches long bj 18 inches wide? Ans. 4 feet. Set 18 upon B to 144 upon A, and opposite 32 upon A is 4 feet upon B, the answer. What is the content of a board 7J inches wide and 40 inches long ? Ans. 2^2- feet. Set 7 J upon B to 144 upon A, and opposite 40 upon A is 2y^2 ^^®* upon B, the answer. What is the content of a shutter 60 inches long by 54 inches wide 1 Ans. 22J feet. Set 54 upon B to 144 upon A, and opposite 60 upon A is 22^ feet upon B, the answer. 31 Case 3. Given the length and breadth in feet to find the content in square yards. I^ule. — Set the breadth upon B to 9 upon A, and opposite the length upon A is the content in square yards upon B. EXAMPLES. A partition measures 45 feet long and 36 feet wide; how many square yards does it contain 1 Ans. 180 square yards. Set 36 upon B to 9 upon A, and opposite 45 upon A is 180 square yards upon B, the answer. A ceiling measures 15 feet wide by 38 feet long; required the content in square yards. Ans. 63 yards 3 feet Set 15 upon B to 9 upon A, and opposite 38 upon A is 63 J yards upon B, the answer. Case 4. Given the length and breadth, in chains, to find the content in acres. Rule. — Set the length upon B to 1 or 10 upon A, and opposite the breadth upon A is the content in acres and parts upon B. EXAMPLE. How many acres are contained in a plot of land 17 chains 50 links long by 4 chains broad 1 Ans. 7 acres. Set 17J upon B to 10 upon A, and opposite 4 upon A is 7 acres upon B, the answer. Case 5. Given the length and breadth, in perches, to find the content in acres. Rule. — Set the length upon B to 160 upon A, and opposite the breadth upon A is the content upon B in acres. 32 EXAMPLE. What is the content of a field whose length is 72 perches, and its breadth 40 perches'? Ans. 18 acres. Set 72 upon B to 160 upon A, and opposite 40 upon A is 18 acres upon B, the answer. Case 6. Given the length and breadth, in yards, to find the content in acres. Rule. — Set the length upon B to 4840 upon A, ai^d opposite the breadth upon A is the content upon B in acres. EXAMPLE. What is the content of a piece of land 220 yards long by 66 yards wide 1 Ans. 3 acres. Set 220 upon B to 4840 upon A, and opposite 66 upon A is 3 acres upon B, the answer. Case 7. To find the area of a regular polygon. Rule. — Set the tabular number upon C to 1 or 10 upon D, and opposite the length of one of the sides upon D is the area upon C. No. of Slides. 9 10 11 12 Names. Tabular Numbers. Trigon '433 Tetrigon I'OOO Pentagon 1*720 Hexagon 2*598 Heptagon ! 3*634 Octagon I 4-828 Nonagon 6*182 Decagon 7*694 Undecagon 9*366 Duodecagon 11*196 G P S of a Circle D& 0=3*1416 D & inscribed O =*707 D & Area = *7854 D &=[=]=-886 C& Area=*0795 D & inscribed A =-866 33 Examples. Required the area of a trigon, or triangle, whose side is 18 feet? Ans. 140 feet. Set 433 upon C to 10 upon D, and opposite 18 upon D is 140 feet, the area upon C. Required the area of a pentagon whose side is 9 inches ? Ans. 139 inches. Set 1*72 upon C to 10 upon D, and opposite 9 upon D is 139 inches upon C, the area. Required the area of an hexagon whose side is 5 feet? Ans. 64| feet. Set 2*598 upon C to 10 upon D, and opposite 5 upon D is 64f feet upon C, the area. Required the area of an heptagon whose side is 17 feet? Ans. 1050 feet. Set 3 -634 upon C to 10 upon D, and opposite 17 upon D is 1050 feet upon C, the area. Required the area of an octagon whose side is 1 5 inches ? Ans. 1086 inches. Set 4*828 upon C to 10 upon D, and opposite 15 upon D is 1086 inches upon C, the area. Required the area of a nonagon whose side is 3 feet ? Ans. 55| feet. Set 6*182 upon C to 10 upon D, and opposite 13 upon D is 55J feet upon C, the area. Required the area of a decagon whose side is 13 inches? Ans. 1300 inches^ Set 7*694 upon C to 10 upon D, and opposite 13 upon D is 1300 inches upon C, the area. Required the area of an undecagon, whose side is 7 inches ? Ans. 458 inches. Set 9*366 upon C to 10 upon D, and opposite 7 upon D is 458 inches upon C, the area. Required the area of a duodecagon, each side being 5 feet ? Ans. 280 feet, c 34 IS Set 11*196 upon C to 10 uponD, and opposite 5 upon D 280 feet upon C, the area required. Case 8. Given the diameter of a circle to find the circumference, or the circumference to find the diameter. Rule 1. — Set 7 upon B to 22 upon A, and opposite any diameter upon B is the circumference upon A, or vice versa. Rule 2. — Set 1 upon B to 3*1416 upon A, and opposite a^y diameter upon B is the circumference upon A, or vice versa, EXAMPLES. If the diameter of a cylinder be 16 inches, what will be the circumference % Ans. bO\ inches. Set seven upon B to 22 upon A, and opposite 16 upon B is 50J inches upon A, the answer. What is the circumference of a cylinder 21 inches in diameter? Ans. 66 inches. Set 7 upon B to 22 upon A, and opposite 21 upon B is 66 inches upon A, the answer. If the diameter of a doffing cylinder be 1 1 inches, what is the circumference ? Ans. 34 J inches. Set 7 upon B to 22 upon A, and opposite 11 upon B is 34J inches upon A, the answer. The diameter of a carding engine is 51;^ inches ; how many cards 4 inches broad will cover the same ? Ans. 40^. Set 1 upon B to 3*1416 upon A, and opposite 51 J upon B is 161 upon A the circumference, which number divided by 4 and the quotient is 40;| cards, the answer. A drawing frame roller is IJ inches in diameter, required the circumference. Ans. 4*7 inches. Set 1 upon B to 3*1416 upon A, and opposite 1| upon B is 4*7 inches upon A, the circumference. The circumference of a wheel is 245 inches, required the diameter. Ans. 78 inches. 35 Set 7 upon B to 22 upon A, and opposite 245 upon A is 78 inches upon B, the diameter. The circumference of a cylinder is 78-^ inches ; what is the diameter 1 Ans. 25 inches. Set 1 upon B to 3*1416 upon A, and opposite 78 J upon A is 25 inches upon B, the diameter. What is the diameter of a wheel whose circumference is 260| inches? Ans. 83 inches. Set 1 upon B to 3*1416 upon A, and opposite 2 6 Of upon A is 83 inches upon B, the answer. Case 9. Given the diameter to find the area of a circle, or the area to find the diameter. Rule.— Set -7854 upon C to 1 or 10 upon D, and opposite any diameter upon D is the area upon C, or vice versa, EXAMPLES. Required the area of a steam-engine piston whose diameter is 35 inches'? Ans. 962*1 inches. Set *7854 upon C to 10 upon D, and opposite 35 upon D is 962*1 inches upon C, the area required. The diameter of a safety-valve is 4| inches ; required the area. Ans. 14*18 inches. Set *7854 upon C to 10 upon D, and opposite 4| upon D is 14*18 upon C, the area. The area of a piston is 572J inches ; required the diameter. Ans. 27 inches. Set *7854 upon C to 10 upon D, and opposite 572^ upon C is 27 inches upon D, the answer. The area of a circle is 615f inches ; what is the diameter ? Ans. 28 inches. Set *7854 upon C to 10 upon D, and opposite 61 5| upon C is 28 inches upon D, the answer. 36 Case 10. Given the circumference to find the area, or the area given to find the circumference. Rule. — Set '795 upon C to 10 upon D, and opposite any circumference upon D is the area upon C ; or opposite the area upon C is the circumference upon D. EXAMPLES. The circumference of a cylinder is 60 inches ; required the area 1 Ans. 286|- inches. Set -795 upon C to 10 upon D, and opposite 60 upon D is 286^ inches upon C, the answer. The area of a piston contains 420 inches ; what is the circumference? Ans. 72*6 inches. Set -795 upon C to 10 upon D, and opposite 420 upon C is 72-6 upon D, the circumference. Case 11. To find the side of a square inscribed in a circle. Rule. — Set '707 upon B to 1 or 10 upon A, and opposite any diameter upon A is the side of its inscribed square. examples. What is the greatest side of a square inscribed in a circle whose diameter is 8 J inches ? Ans. 6 inches. Set '707 upon B to 1 upon A, and opposite S^ upon A is 6 inches upon B, the answer. If the diameter of a circle be 29 inches, what will be the side of an inscribed square 1 Ans. 20J inches. Set '707 upon B to 10 upon A, and opposite 29 upon A is 20 J inches upon B, the answer. Case 12. To find the side of an equilateral triangle inscribed in a circle. Rule. — Set -866 upon B to 1 or 10 upon A, and opposite any diameter of a circle upon A is the length of one of its sides upon B. 37 EXAMPLES. What is the side of a triangle inscribed in a circle whose diameter is 15 inches? Ans. 13 inches. Set -866 upon B to 1 upon A, and opposite 15 upon A is 13 inches upon B, the answer. If the diameter of a circle be 12 inches, what will be the side of an inscribed triangle? Ans. 10*4 inches. Set -866 upon B to 1 upon A, and opposite 12 upon A is 104 inches upon B, the answer. MENSURATION OF SURFACES. Circumference of a circle = diameter x 3*1416. Area of a rectangle = length x breadth. Area of a triangle = i base x perpendicular height. Area of any right-lined figure of any number of sides is found by dividing it into triangles, finding the area of each and adding them together. Area of any regular polygon is found by inscribing a circle, then I the radius x by length of one side x number of sides = area of polygon. Area of a circle = diameter squared x '7854. Area of an ellipse = long axis x short axis x '7854. Area of a parabola = base x | height. Surface of a prism = length x perimeter x area of the two ends. Surface of a sphere = the convex surface of its circum- scribing cylinder = diameter squared x 3*1416. Area of a sector of a circle = length of arc x | radius. Area of a segment of circle = area of sector - area of triangle. Case 13. Given the diameter of a circle to find the side of a square equal in area. 38 Rule. — Set -886 upon B to 1 or 10 upon A, and opposite any diameter of a circle upon A is the side of a square equal in area upon B. EXAMPLES. If the diameter of a circle be 26 inches, what is the side of a square equal in area 1 Ans. 23 inches. Set -886 upon B to 10 upon A, and opposite 26 upon A is 23 inches upon B, the answer. MENSURATION OF SOLIDS. Solidity or cubic contents of a parallelepiped prism or cylinder = area of base x perpendicular height. Solidity of a cone or pyramid = area of base x ^ height. Solidity of a sphere = I of its circumscribing cylinder = diameter cubed X '5236. Solidity of a spheroid = f of its circumscribing cylinder. Solidity of a paraboloid = 1^ of its circumscribing cylinder. Solid bodies are such as consist of length, breadth, and thickness, as stones, globes, timber, &c., &;c. To ascertain the capacity, content, solidity, or weight of any solid body, the divisors or gauge points marked upon the bottom leg of the rule are always to be made use of, and mast now be explained. The first, second, and third tables are marked Square — Cylinder — Globe. The fourth table is a series of divisors, or gauge points, of polygons from 5 to 1 2 sides, as given in case 7 of superficial mensuration. The fifth table is marked G.P.S. of a circle, which denotes the gauge points of a circle. There are three gauge points under square marked F.F.F., F.T.I., I.I.I. Those figures under F.F.F. are to be used only when the dimensions are all in feet ; as 9 feet long, feet broad, and 5 feet deep. Those figures under F.I.I, are to be used when the dimensions are given in feet and inches ; as 18 feet long, 13 inches broad, and 9 inches deep. 39 Those figures under I.I.I, are to be used when the dimensions are given in inches ; as 60 inches long, 27 inches broad, and 19 inches deep. There are two gauge points for all things having a cylindrical form, marked F.I. and I.I. Those figures under F.I. are to be used when the length is given in feet and the diameter in inches ; as 1 2 feet long and 88 inches diameter. Those figures under I.I. are to be made use of when the length and diameter is given in inches ; as 27 inches long and 13 inches diameter. There are also two gauge points for globular forms, marked F. and I, Those figures under F. are to be made use of when the diameter is given in feet. Those figures under I. are to be used when the diameter is given in inches. In measuring or weighing solid bodies having unequal sides, a mean proportion must be found to arrive at the true square. — See case 2nd in Square Root. Note. — Firstly. That all the gauge points are to be found upon the line A. Secondly. That all the lengths are to be found upon the line B. Thirdly. That all the contents are to be found upon the line C. Fourthly. That all the squares and diameters are to be found upon the line D. Rule. — Set the length upon B. to the gauge point upon A., and opposite the square or diameter upon D. is the content or weight upon*the line C. EXAMPLES. A log of timber is 18 feet long, 3 feet broad, and 2 feet deep ; required the content. Ans. 108 cubic feet. Find a mean square between the breadth and depth by setting 3 upon C to 3 upon D, and opposite 2 upon C is 2*45 upon D, the mean square. Then set 18 upon B to 1 upon A, the gauge point, and opposite 2*45 upon D is 108 feet upon C, the content required. 40 Kequired the solidity of a piece of timber 16 feet long, 9 inches broad, and 4 inches deep ? Ans. 4 cubic feet. Find a mean square. Set 9 upon C to 9 upon D, and opposite 4 upon C is 6 upon D, the mean square. The gauge point under F.I.I, for cubic feet is 144. Set 16 upon B to 144 upon A, and opposite 6 upon D is 4 cubic feet upon C, the answer. What is the solidity of a block of stone 60 inches long, 36 inches broad, and 4 inches thick ? Ans. 5 cubic feet. » Find a mean square. Set 36 upon C to 36 upon D, and opposite 4 upon C is 12 upon D, the mean square. Set 60 •upon B to 1728 upon A, the gauge point, and opposite 12 upon D is 5 feet upon C, the answer. Required the cubical contents of a steam-engine cylinder, the length 4 feet, and diameter 24 inches. Ans. 12 J cubic feet. The gauge point under F.I. for cylinder is 1833. Set 4 upon B to 1833 upon A, and opposite 24 upon D is 12 J feet upon C, the answer. How many cubical inches are there in a cylinder 7 inches long and 9 inches in diameter ? Ans. 445. The gauge point under I.I. for cylinder is 1273. Set 7 upon B to 1273 upon A, and opposite 9 upon D is 445 cubic inches upon C, the answer. Suppose the ball on the top of St. Paul's Cathedral to be 6 feet diameter, how many cubical feet does it contain ? Ans. 113 cubic feet. The globular gauge point under F. is 191. Set 6 upon B to 191 upon A, and opposite 6 upon D is 113 feet upon C, the answer. What is the solidity of a globe whose diameter is 9 inches 1 Ans. 381 solid inches. The gauge point is 191. Set 9 upon B to 191 upon A, and opposite 9 upon D is 381 inches upon C, the answer. What is the solidity of a cone 12 inches long and 12 inches diameter at the base ? Ans. 452 inches. 41 For the contents of a cone take one-third of the perpen- dicular height, equal to 4 inches. Set 4 upon B to 1273, the cylindrical gauge point, upon A, and opposite 12 upon D is 452 inches upon C, the answer. THE WEIGHING OF METALS. This rule is of great use to Engineers, Boiler Makers, Blacksmiths, Mechanics, Moulders, Millwrights, &c., in preparing estimates for shafting, piping, palisades, &c. It is performed in the same manner as the preceding examples, all the answers being avoirdupois weight. Bule. — For square and round metals. Set the length upon B to the gauge point upon A, and opposite the square, or diameter, upon the line D is the weight in pounds and parts upon C. GAUGE POINTS. 1 Cubic inches SQUARE. CYLINDER. GLOBE. FFF. FH. ! m. FI. II. F. I. 578 1 1604 1604 118 194 833 1 1 106 1833 294 294 216 354 1273 22 353 353 260 427 1105 191 307 307 225 369 191 33 530 530 390 640 Cubic feet 144 231 231 170 278 1728 2773 2773 2C4 335 Imperial gallons Water Mercury Brass Copper 180 141 206 222 217 204 260 311 331 257 382 407 396 375 397 310 459 489 476 450 490 132 137 141 345 269 395 424 414 391 425 1143 119 123 697 465 684 733 715 675 Lead 203 296 320 311 294 244 356 384 373 353 Wrought-iron Cast-iron Tin Steel Zinc 222 600 625 638 126 320 860 900 922 182 385 1035 108 111 218 408 110 114 117 231 734 198 206 211 415 Wrought Aluminium Cast Aluminium ^ree Stone Coal 278 240 EXAMPLES. What is the weight of a wrought-iron shaft 8 feet long and 2 inches diameter ? Ans. 841bs. 42 The gauge point under F.I. for cylinders, in a line with wrought-iron, is 382. Set 8 upon B to 382 upon A, and opposite 2 upon D is 841bs. upon C, the answer. Required the weight of a wrought-iron shaft 16 feet long and If inches diameter. Ans. 1281bs. Set 16 upon B to 382 upon A, and opposite If upon D is 1281bs. upon C, the answer. What is the weight of a wrought-iron shaft 18 feet long and 2| inches diameter? Ans. 3911bs. Set 18 upon B to 382 upon A, and opposite 2f upon D»is 39 libs, upon C, the answer. If the fly-wheel shaft of a steam engine is 10 feet long and 6 inches diameter, what will be its weight in wrought-iron 1 Ans. 9451bs. Set 10 upon B to 382 upon A, and opposite 6 upon D is 945 lbs. upon C, the answer. Required the weight of a bar of copper 12 feet long and 1 J inches square *? Ans. 1041bs. Set 12 upon B to 260 upon A, the copper gauge point, and opposite l^ upon D is 1041bs. upon C, the answer. A block of cast-lead is 4 feet long and 6 inches square ; required its weight. Ans. TlOlbs. Set 4 upon B to 203 upon A, the lead gauge point and opposite 6 upon D is 710 lbs. upon C, the answer. What is the weight of a column of Mercury 59 inches long and 1 inch square 1 Ans. 28*81bs. Set 59 upon B to 204 upon A, the gauge point, and opposite 1 upon D is 28*81bs. upon C, the answer. Required the weight of a wrought-iron shaft 5 feet long and 6f inches square. Ans. 7701bs. Set 5 upon B to 296 upon A, and opposite 6f upon D is 770 lbs. upon C, the answer. What is the weight of a brass rod 6 feet long and f inch diameter? Ans. 9 Jibs. Set 6 upon B to 354 upon A, the gauge point, and opposite 75 upon D is 9 Jibs, upon C, the answer. What is the weight of a cast-iron shaft 18 feet long and 3 J inches diameter 1 Ans. 5421bs. 43 Set 18 upon B to 407, the cast-iron gauge point upon A, and opposite 3J upon D is 542lbs. upon C, the answer. Required the weight of a brass roller 40 inches long and 3 inches diameter ? Ans. 84 Jibs. Set 40 upon B to 427 upon A, the gauge point, and opposite 3 upon D is 8 4 Jibs, upon C, the answer. What is the weight of a copper rod 6 feet long and | inch diameter 1 Ans. 71bs. Set 6 upon B to 331 upon A, the gauge point, and opposite 625 upon D is 71bs. upon C, the answer. What is the weight of a bar of steel 10 feet long and IJ inches diameter ? Ans. 601bs. Set 10 upon B to 375 upon A, the gauge point, and opposite Ih upon D is GOlbs. upon C, the answer. Required the weight of a cast-iron shaft 9 feet long and 12 inches diameter ? Ans. 31751bs. Set 9 upon B to 407 upon A, the gauge point, and opposite 12 upon I) is 31751bs. upon C, the answer. What is the weight of a brass roller 8 feet long and 3 inches diameter, having 6 necks upon it, each 4 inches long and 2 inches diameter "? Ans. 174|lbs. First weigh 6 feet of 3 inches round, then weigh 2 feet 2 inches diameter ; add both the weights together for the answer. Set 6 upon B to 354 upon A, the gauge point, and opposite 3 upon D ^re 1521bs. upon C. Then set 2 upon B to 354 upon A, and opposite 2 upon D is 22 J upon C. Add 152 and 22J; the sum is 174Jlbs., the answer. What is the weight of a Wrought-iron shaft 14 feet long and 2| inches diameter, having 4 bosses upon it, each boss 6 inches long and 4 inches diameter] Ans. 3221bs. Set 12 upon B to 382 upon A, and opposite 2f upon D are 2381bs. upon C. Then set 2 upon B to 382 upon A, and opposite 4 upon D are 841bs. upon C ; add 238 and 84 — the sum is 3221bs., the answer. A cast-iron shaft is 5 feet long and 5J inches diameter, having two bosses upon it 9 inches long and 6 J inches dia- meter ; required the w^eight of the shaft. Ans. 3931bs. 44 Set 3| feet upon B to 407 upon A, the gauge point, and opposite 5^1 upon T> are 237 lbs. upon C. Then set 1| feet upon B to 407 upon A, and opposite 6| upon D are 156 lbs. uponC; add 2371bs. to 156, and the sum is 3931bs., the answer. What is the weight of a piece of wrought iron 12 inches long and 2 J inches diameter? Ans. 16*31bs. Set 12 upon B to 459 upon A, the gauge point, and opposite 2 1 upon D are 16-31bs. upon C, the answer. EXAJMPLES IN FLAT METALS. Tn weighing flat metals a mean proportion must be found between the breadth and thickness to arrive at the true square, then proceed by the last rule. What is the weight of a bar of wrought-iron 15 feet long, 3^ inches broad, and | of an inch thick? Ans. 1031bs. Set -625 upon C to '625 upon D, and opposite 3;| upon C is 1*425 upon D, the mean square. Set 15 feet upon B to 296 upon A, the gauge point, and opposite 1*425 upon D are 1031bs. upon C, the answer. What is the weight of a bar of iron 12 feet long, 4f inches broad, and -| of an inch thick 1 Ans. 1201bs. Set *625 upon C to *625 upon D, and opposite 4f upon C is 1*725 upon D, the mean square. Set 12 upon B to 296 upon A, and opposite 1*725 upon D are 1201bs. upon C, the answer. What is the w^eight of a wrought-iron bar 2| inches broad, f of an inch thick, and 18 feet long 1 Ans. 62|lbs. Set *375 upon C to *375 upon D, and opposite 2| upon C is 1*15 upon D, the mean square. Set 18 upon B to 296 upon A, and opposite 1*15 upon D are 62|lbs. upon C, the answer. What is the weight of a bar of wrought-iron 6 inches broad, I inch thick, and 5 feet long 1 Ans. 761bs. Set -75 upon C to *75 upon D, and opposite 6 upon C is 2*12 upon p, the mean square. Set 5 upon B to 297 upon A, and opposite 2*12 upon D are 761bs. upon C, the answer. 45 How many pounds will a bar of brass weigh, 5 inches broad, | of an inch thick, and 6 feet long] Ans. 67|lbs. Set -625 upon C to '625 upon D, and opposite 5 upon C is 1'77 upon D, the mean square. Set 6 upon B to 278 upon A, and opposite 1*77 upon D are 67^1bs. upon C, the answer. What is the weight of a piece of copper 2| inches broad, f of an inch thick, and 28 inches long] Ans. 18 Jibs. Set -75 upon C to -75 upon D, and opposite 2| upon C is 1*44 upon D, the mean square. Set 28 upon B to 311 upon A, and opposite 1*44 upon D are 18 Jibs, upon C, the answer. What is the weight of a bar of steel 5^ inches broad, f of an inch thick, and 9 feet long] Ans. 6 libs. Set -375 upon C to -375 upon D, and opposite 5^ upon C is 1*41 upon D, the mean square. Set 9 upon B to 294 upon A, and opposite 1*41 upon D are 611bs. upon C, the answer. What is the weight of a block of lead 4 inches broad, 1 inch thick, and 32 inches long ? Ans. 52 Jibs. Set 32 upon B to 244 upon A, the gauge point, and opposite 2 upon D, the mean square, are 5 2 Jibs, upon C, the answer. What is the weight of a boiler-plate 12 inches square and I thick] Ans. lOlbs. Set -25 on C to -25 on D, and opposite 12 on C is 1 -73 on D. Set 12 upon B to 356 upon A, and opposite 1*73 upon D, the mean square, are lOlbs. upon C, the answer. What is the weight of a boiler-plate 12 inches square and I thick] Ans. lolbs. Set -375 upon C to -375 upon D, and opposite 12 upon is 2*12 upon D, the mean square. Set 12 upon B to '356 upon A, and opposite 2*12 upon D are 151bs. upon C, the answer. SOLID METAL BALLS AND CYLINDERS. Rule. — For Balls, set the diameter upon B to the gauge point upon A, and opposite the diameter upon D is the weight of the ball upon C. 46 Kule. — For cylinders, set the length upon B to the gauge point upon A, and opposite the diameter upon D is the weight in lbs. upon C. What is the weight of a cast-iron ball 6 inches diameter 1 Ans. 29ilbs. Set 6 upon B to 733 upon A, and opposite 6 upon D is 29|^ lbs. upon C, the answer. What is the weight of a brass ball 7 inches diameter 1 Ans. 541bs. Set 7 upon B to 640 upon A, and opposite 7 upon D is*54 lbs. upon C, the answer. A lead ball is 10 inches diameter, required its weight? Ans. 2151bs. Set 10 upon B to 465 upon A, and opposite 10 upon D are 215 lbs. upon C, the answer. What is the weight of a copper ball 3 inches diameter 1 Ans. 4Jlbs. Set 3 upon B to 597 upon A, and opposite 3 upon D is 4|- lbs. upon C, the answer. What is the weight of a cast-iron governor-ball 9 inches diameter? Ans. 99-41bs. Set 9 upon B to 733 upon A, and opposite 9 upon D is •99*4 lbs. upon C, the answer. What is the weight of a cast-iron cylinder 2 feet long and •9 inches diameter"? Ans. 3971bs. Set 2 upon B to 407 upon A, and opposite 9 upon D are 3971bs. upon C, the answer. What is the weight of a brass cylinder 12 inches long and ^ inches diameter? Ans. 1021bs. Set 12 upon B to 427 upon A, and opposite 6 upon D are 1021bs. upon C, the answer. What is the weight of a solid copper roller 4 feet long and .3 inches diameter? Ans. 1081bs. Set 4 upon B to 331 upon A, and opposite 3 upon D are 1081bs. upon C, the answer. 47 What is the weight of a cylindrical piece of lead 39 inches long and 7^ inches diameter? Ans. 6601bs. Set 39 upon B to 310 upon A, and opposite 7^ upon D is 6601bs. upon C, the answer. WEIGHT OF PIPES. Set the length of the pipes upon B to the gauge point upon A, and from the weight on C shown opposite the external diameter of the pipe on D, deduct that shown on C opposite the internal diameter of the pipe on D, the differ- ence will be the weight of the pipe. For shells or hollow balls, proceed as in last Rule. EXAMPLES. What is the weight of a cast-iron pipe, 9 feet lon^, 4 inches bore, and f-inch thick? Ans. 3151bs. Set 9 upon B to 407 upon A, the gauge point, and opposite 4 upon D are 3551bs. upon C, the weight of a solid cylinder 4 inches diameter and 9 feet long. Then twice f added to the inner diameter is 5| inches, the outside diameter ; allow the slide to remain as above, and opposite 5 J upon D are 6701bs. upon C. Subtract 355 from 670, and it leaves 3151bs. for the weight of the pipe. What is the weight of a cast-iron pipe 5 feet long, 5J inches bore, and |-inch thick ? Ans. 2301bs. Set 5 upon B to 407 upon A, and opposite 5J upon D is 3701bs. upon C : then twice | added to 5| is 7 inches for the outside diameter. Opposite 7 upon D are 6001bs. upon C : subtract 370 from 500, and it leaves 2301bs. for the weight of the pipe. Required the weight of a brass pipe, 3 inches bore, |-inch thick, and 3 feet long ? Ans. 591bs. Set 3 upon B to 354 upon A, the gauge point, and opposite 3 upon D is 771bs. upon C : add twice J-inch to 3, and the sum is four inches for the outside diameter. Opposite 4 upon D is 1361bs. upon C : subtract 77 from 136, and it leaves 591bs. for the weight of the pipe. 48 What is the weight of a copper roller 4 inches bore, 6 feet long, and | thick 1 Ans. 2101bs. Set 6 upon B to 331 upon A, and opposite 4 upon D are 2901bs. upon C : add twice | to 4, and the sum is 5^ for the outside diameter. Opposite 5^ upon D are 5001bs. upon C : subtract 290 from 500, and it leaves 2101bs. for tne weight of the roller. What is the weight of a cast-iron bomb-shell 8 inches diameter and ^ inch thick ? Ans. 23rDS. Set 8 upon B to -733 upon A, and opposite 8 upon D are 69Jlbs. upon C ; then set 7 upon B to -733 upon A, and opposite 7 upon D are 46Jlbs. upon C ; subtract 46|- from 69 i, and it leaves 231bs for the weight of the shell. What is the weight of a copper ball 9 J inches diameter and ^ of an inch thick 1 Ans. 121bs. Set 9 J upon B to '597 upon A, and opposite 9| upon D is 1441bs. upon C ; then set 9^, the inner diameter, upon B to •597 upon A, and opposite 9| upon D, are 1321bs. upon C; subtract 132 from 144, and it leaves 12Ibs. for the weight of the ball. LIQUID MEASURE. To find the quantity of water contained in any given pipes, the length and diameter being given. Rule. — Set the length upon B to the gauge points upon A, and opposite the diameter npon D is the number of gallons upon C. EXAMPLES. A column of water is six feet long and lOi inches diameter; required the content in imperial gallons'? Ans. 22 J gallons. Set 6 upon B to 294 upon A, and opposite 10 J upon D is 22 J gallons upon C, the answer. A column of water is 18 feet high, and 12 inches diameter, how many imperial gallons does it contain ? Ans. 88*3 gallons. Set 18 upon B to 294 upon A, and opposite 12 upon D is 88-3 gallons upon C, the answer. 49 A column of water stands 36 feet high, and is 4 inches in diameter ; required the contents in imperial gallons 1 Ans. 19*6 gallons. Set 36 upon B to 294 upon A, and opposite 4 upon D is 19*6 upon C, the answer. If the cold water-pump of a steam engine be 7J inches diameter, and the length of the stroke 36 inches, how many imperial gallons will be lifted at each stroke 1 Ans. 5*7 gallons. Set 36 upon B to 353 upon A, the gauge-point, and opposite 7^ upon D is 5*7 gallons upon C, the answer. A gallon of water weighs lOlbs.; required the weight of water in a pipe 3 inches bore and 24 feet long? Ans. 73-61bs. Set 24 upon B to 294 upon A, and opposite 3 upon D are 7*36 gallons upon C. Multiply by 10, and the product is 73*61bs., the answer. What is the weight of water in a pipe 7 inches bore and 6 feet high? Ans. 100 lbs. Set 6 upon B to 294 upon A, and opposite 7 upon D are 10 gallons upon C. Multiply by 10, and the product is lOOlbs., the answer. How many gallons per minute will a pump deliver, 9 inches diameter, 3 feet length of stroke, 24 strokes per minute'? Ans. 197 gallons. Set 72 upon B to 294 upon A, and opposite 9 upon D are 197 gallons upon C, the aii^wer. A stone cistern is 3 fe^t square and 3 feet deep ; how many imperial gallons will it contain 1 Ans. 168 J gallons. Set 3 upon B to 1604 upon A, the gauge point, and opposite 3 upon D is 168| gallons upon C, the answer. How many imperial gallons will a cistern contam, the length being 40 inches, the breadth 24 inches, and the depth 16 inches'? Ans. 55 J gallons. Set 40 upon B to 2773 upon A, and opposite 19*6 upon D, the mean square, is 55 J gallons upon C, the answer. D 50 TABLE OF SPECIFIC GRAVITIES. Antimony 6-624 Bismuth Cast 9-823 Brass 8-396 Copper 8*878 Gold 19-258 Iron Cast from 6-900 Do. do. to 7-500 Iron Wrought from. . . . 7-500 Do. do. to .... 7-800 Lead 11-300 Mercury 13-580 Nickel Cast 7*807 Platinum 21-554 Silver 10-511 Steel from 7-730 Do. to 7-900 Tin 7-300 Zinc 7-200 Stones, Ea.rths, &c. Brickwork 1792 Brick Earth 2000 Chalk 2-784 Coal from 1*240 Do. to 1-300 Stones, Earths, &c., Continued. Emery 4-000 Flint 2-582 Glass (Flint) 2-933 Do. White 2-892 Granite 2*643 Grindstone 2-143 Limestone 2*742 Manganese 7-000 Marble 2-7;7 Mill-Stone 2-484 Portland Stone 2*570 Pumice Stone -915 Purbeck Sfcone 2-601 Porphyry 2-754 Pyrites Copper 4*954 Do. Iron 3*900 Salt 2*130 Serpentine 2594 Slate 2*672 Sulphur 2-033 A cubic foot of water is very nearly 1,000 ounces in weight ; the above numbers in the table therefore represent very nearly the weight in ounces of a cubic foot of the material. Hence from the table of specific gravities the weight of any body composed of the materials contained therein may be found, by first finding by the gauge points the weight of an equal volume of water, and multiplying that weight by the tabulated specific gravity. CASK GAUGING In order to perform the difficult part of gauging, the three following dimensions of the cask must be accurately taKcn, viz. : — The bung diameter, The head diameter, V Within the cask. The length of the cask, 51 By these dimensions many persons will imagine the contents of the cask will be perfectly limited ; but it will be easy to conceive that the diameters and length of one cask may be equal to those of another cask, and yet one of those casks may contain several gallons more than the other. Hence it appears that no one general rule can be given by which all sorts of casks can be exactly gauged. And there- fore it is necessary to divide them into four forms of varieties. Thus, if the staves of the cask be very much curved, it belongs to the first variety. If the staves of the cask, between the bung and the head, be something less curved, it belongs to the second variety. If the staves be very little curved, it belongs to the third variety. But if the staves be straight from the bung to the head, it belongs to the fourth variety. Rule. — Multiply the difference between the bung and the head diameter by '7 for the first variety, by '65 for the second variety, by '6 for the third variety, and by '5 for the fourth variety : add the product to the head diameter, and the sum is the mean diameter; and proceed exactly the same as in the last examples. EXAMPLES. In a cask of the first variety^ length 40 inches, bung diameter 32 inches, head diameter 24 inches, how many imperial gallons will it contain 1 Ans. 99 gallons. Here the difference between the bung and head diameters is 8 inches, multiply '7 and the product is 5*6, added to the head diameter is 29*6 for the mean diameter. Then set 40, the length of the cask, upon B to 353 upon A, the gauge- point, and opposite 29*6 upon D is 99 gallons upon C, the answer. In a cask of the second variety, bung diameter 27, head 22, and the length 30 inches ; required the content of the cask 1 Ans. 54 gallons. 52 Multiply 5 by '65, the difference between the head and bung diameters, and the product is 3*25, added to the head diameter the sum is 25*25, or 2 5 J, the mean diameter. Set 30 upon B to 353 upon A, and opposite 25J upon D is 54 gallons upon C. How many imperial gallons are contained in a cask of the third variety, head diameter 18, bung 23, and the length 28 inches? Ans. 35 gallons. Multiply 5, the difference between the head and bung diameter, by '6, and the product is 3*0, added to the head diameter the sum is 21, the mean diameter. Set 28 upon B to 353 upon A, and opposite 21 upon D are 35 gallons upon C, the answer. How many imperial gallons are contained in a cask of the fourth variety, head diameter 25, bung diameter 33, and the length 40 inches'? Ans. 95 gallons. Multiply 8, the difference between the head and bung diameters, by '5, and the product is 4-0. Added to the head diameter, the sum is 29, the mean diameter. Set 40 upon B to 353 upon A, and opposite 29 upon D are 95 gallons upon C, the answer. ON THE PITCH OF TEETH IN WHEELS. Case 1. Given the pitch and diameter to find the number of teeth. Kule. — Set the pitch of the tooth upon B to 3*1416 upon A, and opposite the diameter upon B is the number of teeth upon A. EXAMPLES. How many teeth are there in a wheel If inch pitch, and 22| inches in diameter 1 Ans. 40 teeth. Set If upon B to 3*1416 upon A, and opposite 22 J upon B are^40 teeth upon A, the answer. Required the number of teeth in a wheel 2f inches pitch, and 42 inches diameter ? Ans. 48 teeth. 53 Set 2f upon B to 3*1416 upon A, and opposite 42, upon B are 48 teeth upon A, the answer. How many teeth are there in a wheel 76|^ inches diameter and 2| inches pitch 1 Ans. 102 teeth. Set 2-375 upon B to 3-1415 upon A, and opposite 76 J upon B are 102 teeth upon A, the answer. How many teeth are there in a wheel 83 inches diameter and 4 inches pitch 1 Ans. 65 teeth. Set 4 upon B to 3*1416 upon A, and opposite 83 upon B are 65 teeth upon A, the answer. How many teeth are there in a wheel 24 inches diameter at the pitch line, and 2^ inches pitch ] Ans. 30 teeth. Set 2 J upon B to 3*1416 upon A, and opposite 24 upon B are 30 teeth upon A, the answer. Case 2. Given the diameter at the pitch line, and the number of teeth, to find the pitch of the tooth. Rule. — Set the diameter upon B to the number of teeth upon A, and opposite 3-1416 upon A is the pitch of the tooth upon B. EXAMPLES. A wheel 24 inches diameter has 30 teeth ; what is the pitch of the teeth 1 Ans. 2| inches. Set 24 upon B to 30 upon A, and opposite 3-1416 upon A is 2| inches pitch upon B, the answer. If a wheel is 48| inches diameter, and has 136 teeth, what is the pitch of the teeth 1 Ans. l^ inches. Set 48| upon B to 136 upon A, and opposite 3-1416 upon A is 1^ of an inch pitch upon B, the answer. If a wheel 67 inches diameter has 60 teeth, what is the pitch of the tooth 1 Ans. 3^ inches. Set 67 upon B to 60 upon A, and opposite 3*1416 upon A is 3 J inches pitch upon B, the answer. 54 If a wheel 64 inches diameter has 160 teeth, what is the pitch of the tooth'? Ans. l^ inches. Set 64 upon B to 160 upon A, and opposite 3'1416 upon A is 1*25 equal to IJ inch upon B, the answer. If a wheel 43 inches diameter contains 60 teeth, what is the pitch 1 Ans. 2^ inches. Set 43 upon B to 60 upon A, and opposite 3*1416 upon A is 2^ upon B, the answer. If a wheel 90 inches diameter contains 174 teeth, what is the pitch? Ans. If inches. Set 90 upon B to 174 upon A, and opposite 3*1416 upon A is 1*625 upon B, the answer. Case 3. Give the number of teeth and the pitch to find the diameter. Rule. — Set the pitch upon B to 3*1416 upon A, and opposite the number of teeth upon A is the diameter upon B. EXAMPLES. If a wheel is 1 J inches pitch, and contains 67 teeth, what is the diameter ? Ans. 32 inches. Set 1 J upon B to 3*1416 upon A, and opposite 67 upon A is 32 inches upon B, the answer. If a wheel contains 140 teeth, and is 2^ inches pitch, what is the diameter 1 Ans. 100 inches. Set 2 J upon B to 3*1416 upon A, and opposite 140 upon A is 100 inches diameter upon B, the answer. If a wheel is If inches pitch, and contains 174 teeth, what is the diameter 1 Ans. 90 inches. Set If upon B to 3*1416 upon A, and opposite 174 upon A is 90 inches upon B, the answer. If a wheel has 53 teeth, is If inches pitch, what is the diameter? Ans. 29^ inches. Set If upon B to 3*1416 upon A, and opposite 53 upon A is 29 J inches upon B, the answer. 55 If a wheel is 1 inch pitch, and contains 47 teeth, what is the diameter] Ans. 15 inches. Set 1 upon B to 3-1416 upon A, and opposite 47 upon A is 15 inches upon B, the answer. If a wheel is IJ inches pitch, and contains 160 teeth, what is the diameter at the pitch line 1 Ans. 63*6 inches. Set IJ upon B to 3*1416 upon A, and opposite 160 upon A is 63*6 inches upon B, the answer. Case 4. Given the thickness of a tooth to find the pitch and length. Eule. — Multiply the thickness of the tooth by 2*1 for the pitch. Multiply the same thickness by 2*1, and the product will be the length. EXAMPLES. If the thickness of a tooth is 1| inches, what is the pitch and length? Ans. 3*15 pitch, 1*8 length. Set 2*1 upon B to 1 upon A, and opposite 1 J upon A is 3*15 upon B, the pitch. Set 1*2 upon B to 1 upon A, and opposite 1|- upon A is 1*8, length of the tooth upon B. If a tooth is If inches thick, what is the pitch and length ? Ans. 3*675 pitch, 2*1 length. Set 2*1 upon B to 1 upon A, and opposite If upon A is 3*675 upon B, the pitch. Set 1*2 upon B to 1 upon A, and opposite If upon A is 2*1 upon B, the length. TO FIND THE POWER WHICH MAY BE TRANS- MITTED THROUGH LEATHER BELTS. For Single Belts. To 600 on line A bring the speed of the belt in feet per minute on line B, and opposite breadth of belt in inches on line A is the transmitted horse-power on line B. 56 For Double Belts. To 400 on line A bring the speed of the belt in feet per minute on line B, and opposite breadth of belt in inches on line A is the transmitted horse-power on line B. A single belt 8 inches broad moves at 2100 feet per minute, what power can it transmit ] Ans. 28 horse power. Line A > 600 8 inches. Line B > 2100 28 H.P. A double belt 10 inches broad goes 2500 feet per minUte, what power can it transmit 1 Ans. 62 J horse power. Line A > 400 10 inches. Line B > 2500 62^ H.P. SCREW-CUTTING IN SLIDE LATHE. The number of threads per inch, or in any given length of the leading screw, is to the number of threads per inch, or in the same length of the screw required to be cut, as the number of teeth in the wheel on the leading screw to the number of teeth in the wheel on the lathe spindle. Hence the Rule. — To the number of threads per inehj'^or "^*^^' in any given length on the leading screw found on line A, being the required number of threads per inch or in same ** given length required to be cut, on line B ; then against any number of teeth in the wheel on the lathe spindle shown on line A is the corresponding number of the teeth required in the wheel on the leading screw, shown on line B. (1) Suppose the leading screw of our lathe to have 5 threads per inch, and we require to cut a screw with 8 threads per inch, then to 5 on line A bring 8 on line B. Line A > 5 15 20 25 30 35 40 45 50 55 Teeth in wheel of lathe spind le. Line B — ^> 8 24 82 40 48 56 64 72 SO 88 Teeth in wheel leading screw. Any of the above pairs of wheels will answer the purpose, but if the number of teeth in the change wheels rise by 5 at each step, we have then only 25 with 40, or 50 with 80, or their multiples (75 and 120), (100 and 160), (125 and 200), (150 and 240). 67 The intermediate wheel, if there be one, is only used for transmitting or reversing the motion, and the number of its teeth does not affect the result. (2) Suppose the leading screw to have three threads per inch, and a screw having 14 threads per inch is required. Bring 14 on line B to 3 on line A. Line A > 3 12 15 18 21 24 27 30 33 36 39 42 45 Lathe spindle wheeL LineB >14 56 70 84 98 112 126 140 154 168 182 196 210 Leading screw wheel. Any of which pairs may serve, but if the number of the teeth in the change wheels increase by 5 at each step, we have yf, -Y^Q, J~Q, any of which pairs will answer the purpose. (3) The leading screw having 3 threads to the inch, and a screw having 26 threads to the inch being required. Bring 26 on line B to 3 on line A, and against the number of teeth in the wheel on the lathe spindle shown on line A is the corresponding number of teeth in the wheel on the lead- ing screw, as shown on line B, Line A > 3 12 15 18 21 24 27 30 Teeth in lathe spindle wheel. Line B > 26 104 130 156 182 208 234 260 Teeth in wheel of leading screw. If we suppose the number of the teeth in the change wheels to increase by 5 at each step, then, as above, we have Y^V and rr^^. Suppose we have no 260 wheel, and the wheel 15 is inconveniently small, the pair -^—^ may be replaced by -f^^^ at the same time reducing the speed of the leading screw one half by introducing a pair of intermediate wheels fixed together on the same stud, one having half the nupiber of teeth there is in the other. (4) Suppose the leading screw has 4 threads per inch, and 100 threads per inch are required. Bring 100 on B to 4 on A. Line A > 4 20 30 Teeth in lath e spindle wheel. Line B > 100 500 750 Teeth in wheel of leading screw. Suppose we have wheel 20 on lathe spindle, and having no 500 wheel, we reduce the speed of the screw to one- fourth by a pair of intermediate wheels fixed together on the same stud, having their numbers of teeth respectively in the proportion of 4 to 1, and on leading screw put wheel ^£^ = 125. 58 (5) If the leading screw has 4 threads per inch, and 99 threads per inch be required. Line A > 4 20 Teeth in lathe spindle wheel. Line B > 99 495 Teeth in leading screw wheel. Suppose we have wheel 20 on lathe spindle and have no wheel ^with 495 teeth, we reduce the speed of the leading wheel to one-third by a pair of intermediate wheels fixed together on the same stud, having their numbers of teeth as 3 to 1, then ^§^ = 165 teeth for wheel on leading screw. SPEED OF LATHE FOR CAST-IRON TURNING. Invert the slide and bring 1 on line C to 60 on line A, and opposite the diameter in inches of the article to be turned on line A is the number of revolutions per minute on line C. Line A > 60 V diameter. Line q < 1 A revolutions per minute. A cast-iron pulley, 12 inches diameter, is required to be turned. What should be the number of revolutions per minute 1 Ans. 5 revolutions Line A > 60 12 diameter. Line Q < 1 5 revolutions. For wrong ht-iron, bring 1 on line C of inverted slide to 90 on line A, and opposite diameter in inches on line A is the number of revolutions per minute on line C. Line A > 90 V diameter. Line o < 1 A revolutions. A wrought-iron railway waggon wheel, 30 inches diameter^ is to be turned. What should be the number of revolutions per minute ? Ans. 3 revolutions- Line A > 90 30 diameter. Line q < 1 3 revolutions. 59 ENGLISH LINEAL MEASURES INTO FRENCH, AND THE REVERSE. As 10 English inches are equal to 254 French millimetres, we, therefore, to 10 on line A, bring 254 on line B. Line A then represents a line of inches, while line B represents the corresponding line of millimetres. Line A > 3| 4 5| 7^ 10 12 inches. LineB > 89 lOi-6 139-7 190 254 305 mm. '^^• If the English measure be in eighths of an inch, to 80 on line A we bring 254 on line B. Line A > 28 32 44 60 80 96 ^ths in. . LineB > 89 101*6 139*7 190 254 305 mm. '^^' If the English measure be in tenths of an inch, to 100 on line A we bring 254 on line B. L ine A > 35 40 55 75 100 120 lOths in. LmeB > 89 101-6 139-7 190 254 305 mm. If the English measure be in yards to be compared with French metres, to 100 on line A we bring 91*44 on line B. Line A > 3 4 5 6 7 8 &c. yards. LineB > 274 3*657 4*57 5*48 6*4 7*31 metres. To convert thermometric Centigrade readings into Fahren- heit degrees. o c2 Line A > to 5 V Line B > Set 9 A To Fahrenheit read- ^ ing on line B add 32. To convert Fahrenheit readings into Centigrade, deduct 32 from Fahrenheit. 60 I 1^ From Fahren- |5 heit reading on t^l lineAdeduct32 On line A > to 9 V On line B > Set 5 A P To convert English lbs. into French kilograms. On line A > to 22 v lbs. On line B > Set 10 A kilograms. To convert French kilograms to English lbs. On line A > to 10 V kilograms. On line B > Set 22 A lbs. To convert English into French money of equivalent value. On line A > to 20 V shillings. On line B > Set 25 A francs. To convert French into English money of equivalent value. On line A to 25 V francs. On line B Set 20 A shillings. TO FIND THE WEIGHT OF HEMP ROPES IN LBS. Rule. — To 2 or 20 on line D bring the number of fathoms of rope on line C, and opposite the circumference of the rope in inches on line D is the weight in lbs. on line C. (1) Required the weight of a hemp rope 40 fathoms long and 3 inches in circumference 1 Ans. 901bs. On line C > 40 fathoms. 90 lbs On line D ^^> 2~~ 3 (2) Required the weight of a hemp rope 50 fathoms long and 4 inches in circumference 1 Ans. 2001bs. On line C > 50 fathoms. 200 lbs. On line D > 2 4 61 STRENGTH OF HEMP ROPES. Bring 1 or 10 on line C to 11 on line D, and opposite the circumference in inches on line D is the safe working strain in cwts. on line C ; which x by 6| equals the breaking strain in cwts. ; or, if-rby 3 equals breaking strain in tons. Required the safe working load for a hemp rope 3 inches in circumference] Ans. 7*45 cwts. Oa line C > 1 7*45 cwts. On line D >^~il 3 TO FIND THE WEIGHT OF IRON WIRE ROPES IN LBS. To 1 -1 or 1 1 on line D, bring the number of fathoms of rope on line C, and opposite the circumference of the rope in inches on line D is the weight in lbs. on line C. (1) Required the weight of an iron wire rope 30 fathoms long and 2 inches in circumference 1 Ans. lOOlbs. On line C > 30 fathoms. 100 lbs. OnlineD > M 2 (2) Required the weight of an iron wire rope, 50 fathoms long and 3 inches in circumference ? Ans. 3721bs. On line C > 50 372 l bs. OnlineD > M 3 STRENGTH OF IRON WIRE ROPES. To 5 on line D, bring 13 on line C, and opposite the cir- cumference of the rope in inches on line D is the safe work- ing strain in cwts. on line C. Which x by 6§ = the breaking strain in cwts. Or -r by 3 = breaking strain in tons. (1) Required the safe working load for an iron wire rope 2 inches circumference 1 Ans. 21 cwts. Online C > 13 21 cwts. On line D > 5 2 ins. 62 (2) Kequired the safe working load for an iron wire rope, 3 inches circumference 1 Ans. 46|cwts. On line C > 13 46j^ cwts. On line D > 5 3 ins. WEIGHT OF STEEL WIRE ROPES IN LBS. To I'l or 11 on line D, bring the number of fathom^ of rope on line C, and opposite circumference in inches on line T> is the weight of the rope in lbs. on line C. (1) Required the weight of a steel wire rope 35 fathoms long, 2 inches circumference? Ans. 1161bs. On line C > 35 fa. 116 lbs. On line D > I'l 2 ins. (2) Required weight of a steel wire rope 45 fathoms long, 3| inches circumference 1 Ans. 455lbs. On line C > 45 fa. 455 lbs. On line D > M 3^ STRENGTH OF STEEL WIRE ROPES. Rule. — To 100 on line D, bring 90 on line C, and opposite the circumference in inches on line D, is the safe working load in cwts. on line C, which x by 6§ = breaking strain in cwt. Or -i- by 3 = breaking strain in tons. (1) Required the safe working load on a steel wire rope 3 J inches circumference 1 Ans. 108| cwts On line C > 90 108j cwts On line D > 100 3 J ins. (2) Required the safe working load on a steel wire rope, 2 inches circumference 1 Ans. 35|cwts. On line C > 90 35f cwt. On line D > 100 2 63 TO FIND THE WEIGHT OF WEOUGHT-IRON CHAINS IN LBS. Bring the chain^s length in fathoms on line C to 1 or 10 on line D, and opposite the diameter of the iron in 8th'8 of an inch, of which the links are formed on line D, is the weight of the chain in lbs on line C. (1) Kequired the weight of a J inch chain, 20 fathoms long 1 Ans. 3201bs. O n line C > 20 fa. 320 lbs. Oil line D > i 4 (2) Kequired the weight of a f inch chain, 40 fathoms long? Ans. 14401bs. On line C ^> 40 fa. 1440 lbs. On line D > 1 6 TO FIND THE SAFE WORKING LOAD OF CHAINS IN CWTS. Bring 2 on line C to 1 or 10 on line D, and opposite the diameter of the iron, in 8ths of an inch, of which the links are formed on line D, is the safe working load in cwts. on line C, which multiplied by 4 gives the breaking strain in cwts. Or if divided by 5 gives the breaking weight in tons. (1) Required to find the safe working load for a J inch chain 1 Ans. 32 cwts. On line C > 2 32 cwts . On line D > lO 4 2. Required to find the safe working load for a f inch chain 1 Ans. 72 J cwts. On line C > 2 721 c^ts . On line D > 10 6 64 Average teasional strength of materials in tons, per square inch. Cast Iron 7 Wrought Iron 22 Soft Steel 35 „ (hardened) 54 Cast Steel 54 Copper (Cast) 8 J „ (Wrought) 15 Brass (Cast) 8 Wood. Oak 6| Ash 7J Beech 7 Pine 5 DIAMETER OF AIR PUMP FOR A STEAM ENGINE. Rule. — Bring 7 on line B to 10 on line A, and opposite diameter of cylinder on line A is diameter of air pump on line B. A 10 V diam. cylinder. B 7 A diam. air pump. Required diameter of an air pump for an engine, the cylinder being 25 inches diameter 1 Ans. 17i inches. A 10 25 inches.' ~B 7 17* inches. TO DETERMINE DIAMETER OF PIPE TO SUPPLY AN ENGINE WITH STEAM. To 77 on line A, bring the square root of the piston speed in feet per minute on line B ; and opposite the diameter of the cylinder in inches on line A is the diameter of the steam pipe in inches on line B. 65 An engine with a piston 18 inches diameter going 260 feet per minute, what should be the diameter of its steam supply pipe? Ans. 3-8 inches. A 77 18 inches A 77 18 inches. or B 7260 3-8 inches B 16*15 3-8 inches. An engine with a piston 18 inches diameter going 750 feet per minute, what should be the diameter of its steam supply pipe? Ans. 6*45 inches. A 77 18 inches A 77 18 inches. or B ^750 645 inches B 27'4 6*45 inches. To determine diameter of exhaust pipe. To 63 on line A, bring the square root of the piston speed in feet per minute on line B, and opposite the diameter of the cylinder in inches on line A is the diameter of the exhaust pipe in inches on line B. An engine with a piston 18 inches diameter going 260 feet per minute, what should be the diameter of its exhaust pipe? Ans. 4*65 inches. A 63 18 inches A 63 18 inches. or B ^260 4-65 inches B 16*15 4*65 inches. Suppose the speed of the above piston to be 750 feet per minute, what should be the diameter of the exhaust pipe 1 Ans. 7*85 inches. A 63 18 inches A 63 18 inches. or- B ^750 7-85 inches B 27 •47-85 inches. . AREA OF STEAM PORTS. Let d = diameter of piston in inches. s = speed of piston in feet per minute, a = area of steam port. E 66 When 1 on line C is opposite 1 or 10 on line D, bring the number on line C or B which is opposite the diameter of the piston on line D to 64 on line A, and opposite the piston speed on line A is the area of the steam port on line B. Line A 64_ Line B _^_ Line C ^ Line D ^ Area of exhaust port 1 J to 2 that of steam port ? An engine with piston 20 inches diameter going 450 feet per minute, what should be the area of the steam port 1 Line A ^64 450 Line B 4 28*1 Line C 4- Line D 20- Ans. 28*1 square inches. PUMPING ENGINES. Given the diameter of a pump barrel, the depth of the lift in yards, and the effective pressure of the steam, to find the diameter of the steam cylinder necessary for efficiently working the pump. To the diameter in inches of the pump barrel on line D bring half the effective pressure of the steam in lbs. per square inch on the piston on line C ; then opposite the depth of the lift in yards on line C is the diameter of the steam cylinder in inches on line D. What is the diameter of a steam cylinder to work a pump 12 inches diameter, and 95 yards of a lift, the effective pressure of the steam per square inch being 30 lbs. 1 Ans. 30*2 inches. C 15 lbs. = I pressure. 95 yards. D 12 in. diameter. 30*2 diameter. 67 What is the diameter of a steam cylinder to work a pump 4J inches diameter, the lift being 256 yards, and the effective steam pressure 40 lbs. 1 Ans. 16-1 inches. C 256 yards. 20 lbs. = | pressure. D 16*1 in. diameter. 4 J in. diameter. What is the diameter of a steam cylinder to work a pump 10 inches diameter, the lift being 260 yards, and the effective steam pressure 40 lbs. per square inch *? Ans. 36*2 inches. C 260 yards. 20 lbs. = ^ pressure. D 36-2 in. diameter. 10 in. diameter. What is the [effective pressure of the steam required to work a pump 9 inches diameter, with a 240 yards lift, the steam cylinder being 33 inches diameter. Ans. 35*6 lbs. From our diagram C 240 yards, 17-8 lbs. = J the effective pressure. D 33 in. cylinder. 9 in. pump^ THE PRESSURE OF STEAM IN RELATION TO ITS EXPANSION. As the absolute pressure of a given weight of steam while expanding in the steam engine cylinder is inversely as its volume, we must therefore, to represent such condition, invert the slide, and set the initial absolute pressure of the steam in lbs. per square inch on line C of inverted slide, to the number of inches passed over by the piston to the point at which the steam is cut off on line A ; then opposite every succeeding inch of the stroke of the piston, as shown on line A, is the corresponding absolute pressure on line C. (1) If the absolute initial pressure of the steam be 80 lbs. per square inch on the piston, and the supply be cut off when 10 inches have been passed over by the piston ; what is the absolute pressure at any given amount of expansion 1 Rule. — Bring 80 on line C of inverted slide to 10 on line A. Line A >10 15 20 25 30 35 40 45 50 55 60 inches of stroke. Line C< 80 53^ 40 32 26| 22f 20 17i 16 U{r 13J lbs. pressure. 68 (2) If the absolute initial pressure be 901bs. per square inch on the piston, and the supply be cut olBP when the piston has passed over 9 inches ; what is the pressure at any given position of the piston 1 Line A > 9 10 15 20 25 30 35 40 45 50 55 CO inches of stroke. Line C< 90 81 54 40^ 32r*o 27 23} 20^ 18 16i 14-7 13^ lbs, pressure. (3) If the absolute initial pressure be 601bs. per square inch, and the supply be cut off when the piston has passed 10 inches of the stroke, what is the pressure at any given position of the piston ? ' Line A >10 15 20 25 30 35 40 45 50 55 6 i nches of^ stroke. Line C< 60 40 30 24 20 17; 15 13J 12 10-9 10 lbs. pressure. On the back or underside of the slide are two scales of numbered divisions, those on one edge of the slide being inches, eighths and sixteenths of the inch ; while the other edge is a scale of hyperbolic logarithms which are used in finding the work done by steam during its expansion in the steam engine cylinder. In order to find the hyperbolic logarithm of any number, set 1 on line B to the number on line A, whose logarithm is required, and the required logarithm is found on the scale at the back of the slide, marked by the edge of the end of the rule. The engraved numerals on the scale represent whole numbers, and the 10 intermediate larger divisions between the numerals are therefore decimals in the first place, or tenths, each division counting *!, while the 5 smaller divisions into which the tenths are subdivided are therefore 50ths, and consequently represent decimals in the second place, each division counting '02. Suppose we require the hyperbolic logarithm of 3*5. Set 1 on line B to 3*5 on line A, and on the back of the projecting end of the slide at the edge of the rule we read 1*25, the log. of 3-5. Suppose we require the hyperbolic logarithm of 2*75. Set 1 on line B to 2*75 on line A, and on the back of the slide, as before, we read 1*01, the log. of 2*75. 69 To determine the work which may be done by an engine when steam is admitted on the piston at a given high pressure, and after the piston has moved some given distance, the steam supply be cut off while the pressure of that behind the advancing piston is continually diminishing, we require to know what the average pressure will be 1 The rule for determining the average pressure is, to find as above, the hyperbolic logarithm of the number which expresses the ratio of the steam expansion in the cylinder, to which logarithm add 1 ; and to the number expressing the ratio of the steam expansion on line A, bring the number (1 + log of ratio) on line B, and opposite absolute initial pressure on line A is absolute average pressure on line B. Let P - absolute initial pressure per square inch. K = rate of expansion. P2 = average pressure per square inch. Line A R P Then Line B (1 +log r) , pg (1) Let the absolute initial pressure of the steam on the piston of a steam engine be GOlbs. per square inch, the piston stroke to be 36 inches, and the steam cut off at 12 inches of the stroKe, the steam therefore expanding to 3 times its original volume. Required the average pressure 1 Ans. 42 lbs. Applying 1 on line B to 3 on line A, we find on the back of the slide, at the extremity of the rule, 1 '1 on our scale, which is the hyperbolic logarithm of 3, to which we add 1, making 2'L /-|x mu Line A 3 60 initial pressure. Line B 2*1 42 average pressure. (2) Let the initial absolute pressure of the steam on a square inch of the engine piston be 601bs. and after cut off to expand to 4 volumes. 70 Applying 1 on line B to 4 on line A, we find on the back of the slide, at the extremity of the rule, 1 '38 on our scale, which is the hyperbolic logarithm of 4, to which we add 1. Ans. 35-7 lbs. .t^\ mi^ Line A 4 60 initial pressure. ^ Line B 2*38 35*7 average pressure. (3) Let the initial absolute pressure of the steam on a square inch of the engine piston be 901bs., the piston stroke 45 inches, and the steam cut off at 9 inches, thereby expanding 5 times. Required, the average pressure ? Ans. 46 -QB. Applying 1 on line B to 5 on line A, we find on the back of the slide, at the extremity of the rule, 1*61 on the scale, which is the hyperboh'c logarithm of 5, to which we add 1. making our multiplier 2*61. ,o\ mu Line A 5 90 ir.itial pressure. Line B 2-61 46*98 average pressure. (4) Let the absolute initial pressure of the steam on the piston be 901bs. per square inch, stroke of piston be 50 inches, the cut-off being at 9 inches, the ratio of expansion 50 -T- 9 as per slide rule = 5*55. Required the average pressure 1 Applying 1 on line B to 5*55 on line A, we find on the back of the slide, at the extremity of the rule, 1*71 on our scale, which is the hyperbolic logarithm of 5*55, to which we add 1, making our multiplier 2*71. Then by slide the average pressure throughout the stroke. Ans. 43*9 lbs. ,.. Line A 5*55 90 lbs. initial pressure. Line B 2*71 43*9 lbs. average pressure. The average effective pressure on the piston of a steam engine is the average pressure on the face of the piston urging it forward, less the average pressure on the back of the piston resisting its forward motion ; which in non-con- densing engines is the pressure of the atmosphere — say 151bs. per square inch, together with about 21bs. per square inch of pressure due to the resistance of the steam in the passages on its escaping into the atmosphere, making a total back pressure of about 17 lbs. per square inch, to be 71 deducted from the average absolute pressure on the piston in order to obtain the effective pressure. And in the case of a condensing engine, the back pressure on the piston may be about 3 or 4 lbs. on the square inch (it being the measure of the imperfection of the vacuum), which amount must be deducted from the average absolute pressure on the piston to give effective pressure. TO FIND THE INDICATED HORSE POWER OF A STEAM ENGINE. Bring the average effective pressure of the steam in lbs. per square inch on the piston on line B to the gauge point 42 on line A, and opposite the speed of the piston in feet per minute on line A is a number on line B, which bring to 1 or 10 on line D ; then opposite the diameter of the piston on line D is the horse power on line C. (1) A steam engine, with 15 inches diameter of piston, travelling at 280 feet per minute, the average effective pressure of the steam during the stroke being 521bs. per square inch. Required the indicated horse power 1 Ans. 78 J horse power. Per rule On A > 42 280 feet. On B > 52 lbs. 346 On C > 346 78 j horse power. On D > 1 15 inches. (2) A steam engine, with 7J inches diameter of piston, travelling 320 feet per minute, has an average effective pressure of 55 lbs. on the square inch. Required the indi- cated horse power ? Ans. 23J horse power. Per rule Per rule On A > 42 320 feet. On B > 551bs. 420 On C > 420 231 horse power. On D > 10 7| inches. 72 (3) A steam engine piston, 12 inches diameter, going at 300 feet per minute under an effective pressure of 451bs. on the square inch. Required the indicated horse power? Ans. 46:^ indicated horse power. On A > 42 300 feet. On B > 45 lbs. 321 On C > 321 46| horse power. On D > i 12 inches. (4) A steam piston, 20 inches diameter, going at 600 feet per minute, under an effective pressure of 501bs. p^er square inch. Required the indicated horse power 1 Ans. 284J indicated horse power. On A > 42 600 feet. Per rule ^^ ^ __^ ^^ ^^^^ ^^^ On C > 714 284^ horse power. ^^^^^^^ On D > i 20"" Per rule Per rule COMPOUND ENGINES. To find the total ratio of steam expansion in a Compound Engine, when the ratio of expansion of the steam in the high pressure cylinder, and the diameters of the high and low pressure cylinders, are given. Rule. — Bring the number or rate of expansion in the high pressure cylinder on line C to the diameter of the high pressure cylinder on line D, and opposite the diameter of the low pressure cylinder on line D is the total rate of expansion on line C. C > J* R r = rate of expansion in h. p. cyl. D > A A R = Total rate in all cylinders. Ifi all Having the initial pressure, total number of steam expan- sions, piston speed, and diameter of low pressure piston, we may find the power developed by a Compound Engine by assuming all the work to be done in that cylinder, and com- puting the power by the rules previously j given for such purpose. 73 A Compound Engine, high pressure cyHnder, 16 inches diameter, low pressure cylinder, 26 inches diameter, piston speed 400 feet per minute, steam being cut off at |- stroke in high pressure cylinder. Initial absolute pressure being 145 lbs. Kequired the indicated horse power. Then allow- ing 51bs. of fall in pressure on the steam entering the cylinder, we have 1401b8 on the piston. Ans. 315 horse powder. ^ ^ , ^ Line C — >3 7*95 Expansions. And as per above rule r~ f^ — ^^rr^ — TTa ^ Lme D — >16 26 And log. of 7 '95 as found on back of slide = 2*073, to which add 1 = 3-073. Then Line A 7 '95 140 lbs. initial pressure. Line B 3 073 54 lbs. average pressure, from which, for imperfect vacuum, deduct say 51bs, = 491bs. effective pressure. Then ^ > ^2 ^QQ fe^^- B > 491bs. 466 C >■ 466 315 horse power. D > i 26 inches. Ans. 315 indicated horse power. The mechanical efficiency of a steam engine, or its power of doing external work, may be assumed to be about 80 per cent of its indicated power. Or by slide Line A 10 I.H.P. indicated horse power — if. _ Line B 8 E.H.P. effective horse power. COTTON SPmNING. To find the drafts of Rollers. Eule. — Invert the slide and set the number of teeth in the back roller wheel upon C to 1 upon A, and opposite the number of teeth in the change-pinion upon C is the number of revolutions the change-pinion will make for the back rollers one. Then set the number of teeth in the stud- wheel upon C to the same revolutions upon A, and opposite 74 the number of teeth in the front roller-pinion upon C will be the speed of the front roller upon A. Then set the diameter of the front roller upon C to its revolutions upon A, and opposite the diameter of the back roller upon C is the draft upon A. EXAMPLES. The back-roller wheel contains 44 teeth, change-pinion 22, the stud-wheel 35, and the front roller-wheel 20 teeth; diameter of the front roller l^ inches, back roller 1 inch. Required the draft ? Ans. 5 J draft. Set 44 upon C to 1 upon A, and opposite 22 upon C is 2 upon A. Then set 35 upon C to 2 upon A, and opposite 20 upon C is 3^ upon A. Then set 1 J upon C to 3^ upon A, and opposite 1 upon C is 5J upon A, the draft required. What is the draft of a jack frame, the back roller-wheel having 48 teeth, change pinion 32 teeth, stud wheel 70 teeth, front roller-pinion 21 teeth; diameter of the front roller IJ inches, back roller 1 inch diameter. Ans. 6 J draft. Set 48 upon C to 1 upon A, and opposite 32 upon C is 1^ upon A. Then set 70 upon C to IJ upon A, and opposite 21 upon C is 5 upon A. Then set 1;^ upon C to 5 upon A, and opposite 1 upon C, the diameter of the back roller, is 6 J upon A, the draft. The back roller-wheel contains 60 teeth, change-pinion 24, stud wheel 120, front roller-pinion 40 teeth, diameter of the front roller 1 J inches, back roller f diameter. Required the draft? Ans. 9-6 draft. Set 60 upon C to 1 upon A, and opposite 24 upon C is 2| upon A. Then set 120 upon C to 2 J upon A, and opposite 40 upon C is 7 J upon A. Then set 9 equal l^ inches upon C to 7| upon A, and opposite 7 upon C equal | is 9*6 upon A, the draft. Note. — If an alteration in the draft is to be made this is the rule : Find the draft before the alteration and after ; then set the second draft upon C to the hanks roving of the first draft upon A, and opposite the first draft upon C is the hank roving produced by the alteration upon A. 75 If a stretching frame is producing an 8-hank roving, the back roller-wheel 42, change-pinion 35, stud-wheel, 120, front roller-pinion 24 teeth, diameter of the front roller 1|^ inches, back roller | diameter. Change the 42 to a 54 and 24 for a 34, what hanks roving will be produced by the alteration 1 r First draft 7*71 Ans. < Second draft 7. ( Eoving produced 7 J. Set 42 upon C to 1 upon A, and opposite 35 upon C is 1*2 upon A. Then set 120 upon C to 1*2 upon A, and opposite 24 upon C is 6 upon A. Then set 9 upon C to 6 upon A, and opposite 7 upon C is 7*71 upon A, the first draft. Then set 54 upon C to 1 upon A, and opposite 35 upon C is 1*54 upon A. Then set 120 upon C to 1*54 upon A, and opposite 34 upon C is 5*43 upon A. Then set 9 upon C to 5*43 upon A, and opposite 7 upon C is 7 upon A, the second draft. Then set 7 upon C to 8, the hank roving, upon A, and opposite 7*71 upon C is 7 J upon A, the hank roving produced by the alteration. Drafts required in spinning. Rule. — Invert the slide and set the numbers upon C to 10 upon A, and opposite the single roving, upon C is a divisor upon A. Then set the length delivered from the rollers upon C to the divisor upon A, and opposite the length of the stretch upon C is the draft required upon A. EXAMPLES. Required the draft to spin 156's from a 12 hank roving, the rollers to deliver 40 inches, length of the stretch 52 inches? Ans. 10 draft. Set 156 upon C to 10 upon A, and opposite 12 upon C is 13 upon A, the divisor. Then set 40 upon C to 13 upon A, and opposite 52 upon C is 10 upon A, the draft. What draft will be required to spin 36's from a 2J-hank roving, 42 inches delivered from the rollers, 62 inches the length of the stretch ] Ans. 9| draft. Set 36 upon C to 1 upon A, and opposite 2 J upon C is 14-4 upon A, the divisor. Then set 42 upon C to 14*4 upon A, and opposite 62 upon C is 9f upon A, the draft. Eequired the draft to spin 60's from a 4|-hank roving, 43 inches delivered from the rollers, and a 60 inch stretch 1 Ans. 9*1 draft. Set 60 upon C to 1 upon A, and opposite 4| upon C is 12*7 upon A, the divisor. Then set 43 upon C to 12-7 upon A, and opposite 60 upon C is 9-1 upon A, the draft. If 280's be spun from a 14 single roving, 40 inches delivered from the rollers, 50 inches the length of the stretch, what is the draft 1 Ans. 16 draft. Set 280 upon C to 1 upon A, and opposite 14 upon C is 20 upon A, the divisor. Then set 40 upon C to 20 upon A, and opposite 50 upon C is 16 upon A, the draft. On the Hanks roving. Eule. — Invert the slide and set the counts upon C to 1 upon A, and opposite the draft upon C is a divisor upon A. Then set the lengths delivered from the rollers upon C to the divisor upon A, and opposite the length of the stretch upon C is the hanks roving single upon A. EXAMPLES. I am spinning 190's with 14 draft, 40 inches delivered from the rollers, and a 54 inch stretch. Required the single roving? Ans. 10 single roving. Set 190 upon C to 1 upon A, and opposite 14 upon C is 13*6 upon A, the divisor. Then set 40 upon C to 13*6 upon A, and opposite 54 upon C is 10 upon A, the single roving. If 40's be the counts, 12 the draft, 42 inches delivered from the rollers, stretch 56 inches, what hanks roving will be required 1 Ans. 2 J hanks roving. Set 40 upon C to 1 upon A, and opposite 12 upon C is 3-33 upon A, the divisor. Then set 42 upon C to 3*33 upon A, and opposite 56 upon C is 2J hanks roving upon A. If 270 be the counts, 10 the draft, 40 hanks delivered from the rollers, stretch 54 inches, what is the hanks roving? Ans. 20 hanks roving. 77 Set 270 upon C to 1 upon A, and opposite 10 upon C is 27 upon A, the divisor. Then set 40 upon C to 27 upon A, and opposite 54 upon C is 20 upon A, the hanks roving ? If the counts of the yarn are 120's with a draft of 12 J, the delivery frona the rollers is 40^ inches, length of the stretch, 60 inches, what is the double roving 1 Ans. 13 double rovings^ Set 120 upon C to 1 upon A, and opposite 12| upon C is 9*6 upon A, the divisor. Then set 40 J upon C to 9'6 upon A, and opposite 60 upon C is 6 J upon A, the single roving. Multiply by 2 and the product is 13, the double roving. On the Counts of yarn. Rule. — Multiply the hanks roving, and draft together for a divisor. Then set the length delivered from the rollers upon B to the divisor upon A, and opposite the length of the stretch upon B is the counts of yarn upon A. EXAMPLES. The hanks roving is 13, draft 13, the length delivered from the rollers 39 inches, 50 inches being the stretch. Re- quired the counts of the yarn ? Ans. 200 hanks. By multiplication 12 times 13 are 156. Set 39 upon B to 156 upon A, and opposite 50 upon B is 200 upon A, the answer. The hanks roving is 10, the draft 14, the stretch 54, and the length delivered 40 inches. Required the counts ? Ans. 189 hanks. By multiplication 10 times 14 are 140. Set 40 upon B to 140 upon A, and opposite 54 upon B is 189 upon A, the counts of the yarn. The hanks roving is 2^, the draft 12, the stretch 56 inches, the length delivered 42 inches. Required the counts of the yarn 1 * Ans. 40 hanks. By multiplication 12 times 2 J are 30. Set 42 upon B to 30 upon A, and opposite 56 upon B is 40 upon A, the counts of the yarn. To find the twist per inch of roving or yarn. 78 Rule. — Set 71 upon C to 1 or 10 upon D, and opposite any counts of yarn upon C is the number of turns per inch upon D. EXAMPLES. If the counts spinning be 49's with 16 J turns per inch of yarn, how many turns per inch will 64^8 require*? Ans. 30 turns. Set 71 upon C to 10 upon D, and opposite 64 upon C is 30 turns upon D, the answer. What twist is necessary to spin 144's, 256's, and »36's yarn? Ans. 45, 60, and 22J, turns. Set 71 upon C to 10 upon D, and opposite 144,256, and 36 upon C is 45, 60, and 22 J turns per inch upon D, the answer. BLOCK TACKLE. To find the power to raise a given weight. The number of ropes or chains in tension that shorten between the power applied and the weight moved is the number of times the applied power is multiplied. Whence set the weight to be raised on line B to the number of ropes which shorten between the power and the load moved on line A, and opposite 1 or 10 upon line A is the power required on line B. EXAMPLES. Two blocks having 8 ropes which shorten between power and load. Required the power to raise 4501bs. ? Ans. 56;Jlb8. Set 450 upon B to 8 upon A, and opposite 1 upon A is 56 J lbs. upon B, the power required. Two blocks having 6 ropes which shorten between power and load. Required the power to raise 9841bs. 1 Ans. 1641bs. Set 984 upon B to 6 upon A, and opposite 1 upon A is 1641bs. upon B, the power required. To find the weight to be raised. Rule. — Multiply the power by the number of ropes which shorten between power and weight, and the product is the weight to be raised. 79 BXAMPLB. What weight will a power of 251bs. lift applied to an 8- roped set of blocks *? Ans. 200 lbs. Multiply 25 by 8 and the product is 2001bs., the answer. MISCELLANEOUS PROBLEMS. To find the space through which a body will fall in a given time. Rule. — Set 16*1 upon C to 1 or 10 upon D, and opposite any number of seconds upon D is the space in feet which a body will fall through, EXAMPLES. Through what space will a body fall in 6 seconds ? Ans. 579 feet. Set 16*1 upon C to 1 upon D, and opposite 6 upon D is 579 feet upon C, the answer. Suppose a stone is dropped into a coal pit, and arrives at the bottom in 4 seconds, what is the depth of the pit ? Ans. 257 feet. Set the slide as in the last example, and opposite 4 upon D is 257 feet upon C, the depth of the pit. Problem 2. Given the length of a pendulum to find the number of vibrations per minute. Or given the number of vibrations to find the length of the pendulum. Rule. — Invert the slide and set 39*2 upon B to 60 upon D, and opposite any length of a pendulum upon B is the num- ber of vibrations upon D. Or opposite the number of vibra- tions upon D is the length of a pendulum upon B. examples. Required the length of a pendulum that will vibrate 50 times per minute ? Ans. 56i- inches. 80 Invert the slide and set 39*2 upon B to 60 upon D, and opposite 50 upon D is 56 J inches upon B, the answer. How many vibrations per minute will a pendulum make that is 88 J inches long ? Ans. 40 vibrations. Set 39*2 upon B to 60 upon D, and opposite 88 J upon B is 40 vibrations upon D, the answer. John Heywood, Excelsior Printing and Bookbinding Works, Manchester. THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO 50 CENTS ON THE FOURTH DAY AND TO $1.00 ON THE SEVENTH DAY OVERDUE. AUG 101936 SFP i0 4«kJM ^*^' 16 ?9^ 29Jiil49 GX ^ LD 21-100m-8,'34 911251 ^ ^ THE UNIVERSITY OF CALIFORNIA LIBRARY