Oass_U Book COPYRIGHT DEPOSIT HAND BOOK OF CALCULATIONS FOR ENGINEERS. PRESS OF IylVINGSTON MlDDLEDITCH & CO.. 26 CORTLANDT ST., N. Y. THIS WORK IS DEDICATED TO C. A. H. WITH FILIAL AFFECTION. NEW YORK: Copyright by Theo. Audei. & Co., All rights reserved. 1890. HAND BOOK CALCULATIONS FOR ENGI NEBRS AND FIREMEN. RELATING TO THE STEAM ENGINE, THE STEAM BOILER, PUMPS, SHAFTING, ETC. Comprising the Elements of Mechanical Philosophy, Mensuration, Geometry, Algebra, Arithmetical Signs, and Tables. United States Weights, Measures and Money ; Tables of Wages, with Copious Notes, Explanations and Help Rules Useful for an Engineer. And for Reference, Tables of Squares and Cubes, Square and Cubk Roots, Circumference and Areas of Circles, Tables <>f Weights of Metals and Pipes, Tables of Pressures of Steam, Etc., Etc., Etc. By J^HAWKINS, M. E., Honorary Member National Association of Stationary Engineers ; Editorial Writer, Author Maxims and Instructions for the Boiler Room. THEODORE AUDEL & CO. Publishers, 91 LIBERTY STREET, New York City. . \ V *>0j/. V **>' <* Y \ INTRODUCTION. " I would give a thousand dollars if I knew the principles upon which niy engine works/' This was the remark of a western engineer made to a gentle- man who was admiring the performance of the steam plant under the charge of the former. " I can attend to every nec- essary thing about my whole apparatus; engine, boilers, pumps, pipes, and do all that is expected of an engineer, but 1 don't know why the steam does its work, and I would give a thousand dollars to know." This work is prepared for those who, like the engineer whose words are quoted, wish to know, and are willing to pay the cost, in money and study. Abraham Lincoln once said, in the early days of his opening manhood, with the warm enthusiasm characteristic of his noble mind, " That man who furnishes me with a good book is my best friend;" at the age of 18 he was the proud owner of six volumes. The desire has been strong indeed, in the mind of the author, while compiling this work, upon a single page of which, at times, several days have been spent, that it might come to many aspiring men, with the same potent good, as the few books which Abraham Lincoln had access to in his early strug- gling days. viii INTRODUCTION. In the wide expanse of mathematics it has been a task of the utmost difficulty for the author to lay out a road that would not too soon weary or discourage the student; if he had his wish he would gladly advance step by step with his pupil, and much better explain, by word and gesture and emphasis, the great principles which underlie the operations of mechanics; to do this would be impossible, so he writes his admonition in two short words: In case of obstacles, " go cw." If some rule or process seems too hard to learn, go around the difficulty, always advancing, and, in time, return and conquer. One thing of importance may here be said. The value of a teacher or instructor cannot be overestimated. Men were not made to do their work alone; they are created so that they need assistance and encouragement in every direction except down- wards; to be helped and to help is the universal law. In no profession more than steam engineering does this law hold truer, and while the editor has written down the hard problems he has all the time, while making them as plain as possible to do so, had the secret wish that the learner might have at his side, when the book came to him, a kind and generous tutor, who could, and cheerfully would, go with him over the untravelled road. There is a single unique Book in the world, two thousand years compiling, of which it is said that no person can be called foolish who diligently peruses its pages; so the author's top- most wish has been now to prepare a book so elementary aud yet so wide in its scope that no engineer or fireman could justly be called ignorant who had carefully studied and become familiar with its pages. We quote for a motto— "Education does not consist merely in storing the head with materials; that makes a lumber room of it; but in learning how to turn those materials into useful products; that makes a fac- tory of it; and no man is educated unless his brain is a factory, with storeroom, machinery and material complete." Hence in arranging the materials of this work, the author has aimed to give it a certain completeness and harmony with itself, from beginning to end; to make it " a factory, with INTRODUCTION. ix storeroom, machinery and materials" so abundant in quantity and variety of stores, that it will answer every reasonable require- ment. In excuse for the extreme simplicity of some of the problems presented, the author owns that this feature of the book was incorporated in it through its having come to his knowledge, through a member of the board of U. S. government exami- ners, some years since, that some score of high class marine engineers had come very near losing their positions on account of their ignorance of many of the simplest items of information relating to their duties. It was in consequence of an order received from headquarters at Washington for the re-examination of all the engineers in a certain department, in the Eastern division of bhe marine ser- vice; the order was peremptory, and the examinations to the number of 60 or 70 were held forthwith. And it was a disagreeable fact that while few, or none, were really displaced, the positions of all these really competent engi- neers were in danger of being forfeited, because they had for- gotten the little things they had acquired in earlier days. Hence the truly wise student of this hand-book, even if of •established reputation, will not despise the elementary rules and examples presented. Nor must the humble beginner de- spair of the most difficult. Both extremes will be found in the completed volume. N. H. New York City, June 1st, 1889. PLAN OF THE WORK. The leading idea intended to be illustrated in the following- successive "parts " or chapters is this: that in an informal and not too "dry"' a method, engineers or those aspiring to be such shall be taught to figure the problems relating to the steam engine and boiler; the steam pump; shafting and pulleys; and all other calculations required in the varied duties of steam- engineering in its most intelligent and useful practice. The first four or five parts of the work will be occupied ex- clusively with what may be called the general principles ofmath- e??m^'cs— principles which are used in all times and places and in an infinite variety of machines, and their application to the use of man. Next, these elements will be illustrated by the practice of to-day in steam engineering in its various depart- ments. Eules for calculating horse power of engines and boilers will be given in the plainest manner and fully illustra- ted by diagrams; rules for figuring the safety-valve pressure of boilers, strength of materials, size and capacity of pumps, etc., etc., with help rules, notes and remarks based upon the most approved practical experience. The work will close with valuable and copious tables of roots and powers of numbers, and diameters and circumferences of cir- cles, and all the data commonly found in the most advanced works written for mechanics; hence, the first part of the work, perhaps three quarters of it, will be for instruction and the other part for reference. It must not be forgotten that the elements only of arithmetic, geometry, algebra, mensuration, etc., are to be introduced in the work, but it is upon these elements that the whole structure of mathematics rests, and form the groundwork where the most advanced and the most lowly beginner can meet with mutual respect. PLAN OF THE WORK. XI It is planned that the ultimate result of this publication will be the compiling of a standard and valuable volume, contain- ing all the mathematics relating to steam engineering necessary for an intelligent engineer in his daily practice; hence the author, ere the work proceeds too far, will be pleased to receive the helpful suggestions of his kindly reader as to the most desirable contents for such a comprehensive work. For the space it occupies the explanation of the use of form- tilas or forms will be found to be most useful to the practical man, as it teaches him the school language of expressing calcu- lations. This custom is the same as that followed by the physician in writing aqua pura instead of "pure water"; and the gardener giving Latin names to his plants instead of plain English terms. The use of formulae is so universal that many publications, otherwise of great value to the engineer, are to him as a sealed book; but with the explanations to be found in this work a great part of the difficulty will be obviated. At the issue of Part 1 the whole work is in manuscript, but it will be printed in 10 monthly parts. This is to accommodate the student, to whom a single Paet will be the moderate allow- ance for a month's study, and also to allow snch changes as may seem necessary to perfect the plan of the work before it is advanced to book form. The index of the whole book will be issued with the last number, in convenient shape, and at that time a more formal preface will be written, in which due acknowledgement will be made for assistance from persons and authors whose advice and experience has been drawn upon. One other item may be added, but not enlarged upon: that is the desire to give for a moderate cost, information of large value to the purchaser. An engineer who can figure and do it correctly is of more value than one who cannot, and this esteem is (between the reader and the author) expressed by larger com- pensation and longer service in one position. Hand Book of Calculations. ARITHMETICAL SIGNS. The principal characters or marks used in arithmetical com- putations to denote some of the operations, are as follows : = Equal to. The sign of equality; as 100 cti. = $1— signi- fies that one hundred cents are equal to one dollar. — Minus or Less. The sign of subtraction; as 8-2=6, that is, 8, less 2, is equal to 6. + Plus or More. The sign of addition; as 6+8=14; that is, 6 added to 8, is equal to 14. X Multijiliedby. The sign of multiplication; as 7x7=49; that is, 7 multiplied by 7 is equal to 49. -^ Divided by. The sign of division; as 16-^4 4; that is, 16 divided by 4 is equal to 4. There are still other characters and marks which will be added as needed as the work progresses, but these are the prin- cipal ones. ARITHMETICAL FORMULAS. An arithmetical formula is a general rule of arithmetic expressed by signs. The following 10 formulas include the elementary operations of arithmetic and follow from the succeeding illustrations. 1. The Sum=«£? the parts added. 2. The Differences/^ Minuend — the Subtrahend. 3. The Minuend = the Subtrahend -f- the Difference. 4. The Subtrahend = the Minuend — the Difference. 5. The Product = the Multiplicand x the Multiplier. 6. The Multiplicand = the Product + the Multiplier. 7. The Multiplier = the Product -h the Multiplicand. 8. The Quotient = the Dividend -h the Divisor. 9. The Dividend = the Quotient x the Divisor. 10. The Divisor = the Dividend -^ the Quotient. Formulas or formulas, express the plural of formula — a Latin word which means, simply, a form; hence a formula is a form of stating a problem . Hand Book of Calculations. ARITHMETIC. Arithmetic is the science or orderly arrangement of numbers and their application to the purposes of life. The processes of arithmetic are merely expedients for making easier the discov- ery of results, which every mechanic of ordinary ingenuity would find a means for discovering himself, if really called upon to set about the task, for it is possible for a man to be a good working engineer, and at the same time be quite ignorant of reading, writing or figuring; but experience shows that in order to advance in the confidence of others, it is very necessary to know something of the elements, or first things, of mathemat- ics related directly or indirectly to steam. Arithmetic is the science of numbers, and numbers treat of magnitude or quantity. Whatever is capable of increase or diminution is a magnitude or quantity; a sum of money, a weight, or a surface, is a quantity, being capable of increase or diminution. But as we cannot measure or determine any quan- tity, except by considering some other quantity of the same kind as known, and pointing out their mutual relation, the measurement of quantity or magnitude is reduced to this: Fix at pleasure upon any known kind of magnitude of the same species as that which has to be determined, and consider it as the measure or unit. If, for example, we wish to determine the magnitude of a sum of money we must take some piece of known value, as a dollar, which is the unit of money, and show how many such pieces are contained in the given sum. j 4 Hand Book of Calculations. The foot rule is the unit or measure of length most used for engineering purposes; the foot is divided into twelve inches and the inch is subdivided in half inches, quarter inches, eighths and sixteenths It is plain that into whatever number of parts the inch is divided, wc shall equally have the whole inch if we take the whole of the parts of it; if it were divided into ten equal parts, then ten of these parts would make an inch. The unit of surface in steam engineering is represented by the square inch. The unit of time is in usual practice one minute; thus we say an engine makes so many revolutions per minute, and its per- formance is based upon that. The unit of work is the force required to raise one pound, one foot high from the earth, in the atmosphere, no time being taken in the account; it is known as the foot pound. Atmospheric pressure at the sea level is the unit of pressure. The unit of heat is the amount of heat required to raise one pound of water one degree, usually from 32° to 33° Fahr. The unit of numbers is the figure one (1). These references to the different measures, or units, are made in view of their frequent use in ascertaining duties performed by steam engines and boilers. They enter into all engineering calculations in connection with Tables to be found elsewhere in this volume, and their utility will be clearly explained and readily understood from their combination with practical cal- culations elsewhere found in this volume. Electric units. The unit of electric force is the volt ; the unit of resistance is the ohm; the unit of current strength or volume, is the ampere; the unit of current quantity consid- ered with reference to time is the coulomb; the electric unit of capacity is the farad; the unit of electric power is the watt, etc. The measurement of electricity is one of the newest discover- ies, to which a separate space will herewith be devoted, in which the electric units of force, resistance, etc., will enter into the practical problems relating to electric lighting. Hand Book of Calculations. 15 NOTATION AND NUMERATION. Notation in Arithmetic is the writing down of figures to ex- press a number or numbers, and Numeration is the reading of numbers already written. There are nine figures — 1, 2, 3, 4, 5, 6, 7, 8 and 9 nsed in arithmetic, and the (nanght) to represent nothing. The nnmber 1 is called the unit. The number 9 is a collec- tion of nine of these nnits. By means of these 10 figures we can represent any number. When one of the figures stands by itself, it is called a unit; but if two of them stand together, the right hand one is still called a unit, but the left hand one is called tens; thus, 79 is a collec- tion of 9 units and 7 sets of ten units each, or of 9 units and 70 units, or of 79 units, and is read as seventy-nine. If three of them stand together, then the left hand one is called hundreds; thus 279 is read two hundred and seventy- nine. To express larger numbers other orders of units are formed, the figure in the 4th place denoting thousands; in the 5th place ten thousands; these are called units of the fifth order. The sixth place denotes hundred thousands, the seventh place denotes millions, etc. The French method (which is the same as that used in the IT. S.) of writing and reading large numbers is shown in the following Names of periods. NUMERATION TABLE. Billions. Millions. Thousands. Units. Thousandths. CO Go* 3 Order of B » % g S B . I si Units. 5§ J J I I -3 « & 5 1 t3.S. ^-B« p d3S ^3 ,_, ^3 ^.og 'OdO '■dtSg ns . aj g ,B^2 rHi.rt fl ' 1H c- 11 ^ etc-*-* .,H +j ,-1 3 ? ^ § B :3 § fl 2 3 B a « fl ^ - S H PQ PhH^ Mhh MHh^ ft HWh 87 6, 54 3, 20 1, 28 2, . 489 16 Hand Book of Calculations. The number in the table is read eight hundred and seventy six billion, five hundred and forty-three million, two hundred and one thousand, two hundred and eighty-two, and four hun- dred and eighty-nine thousandths. To express larger numbers other periods are formed in like manner, called Trillions, Quadrillions, Quintiliions, Sextillions, Septillions, Octillions, Nonillions, Decillions. Each of these periods increase the values of all the figures to which it is added 1,000 times. Figures are always read from left to right; thus, one million, one thousand and two is in figures 1,001,002, the figures 1, 1, and 2 occupying the 7th, the 4th and the 1st place, and cyphers the intermediate spaces. The ' ' one million ".at the left.is read first and the unit " two " at the right is read last— and this is the universal rule with the important exception of decimals, hereafter explained. In the table given it will be observed that the long row of figures are divided by commas (,). This is to aid in their ready reading. The first set is called units, the second thousands, the third millions, etc. Beginning at units place, the orders on the right of the deci- mal point, express tenths, hundredths, thousandths, etc. Examples for Peactice. Notation. Write in figures eight million, two hundred fifty- nine thousand eight hundred and ninety-two. Ans. 8,259,892. 2. Write four hundred and sixty-two thousand and nine. Ans. 462,009. 3. Write four billion, four million, four thousand and four. Ans. 4,004,004,004. 4. Write six hundred and two. 5. Write sixteen thousand, seven hundred and ninety two. 6. Write six hundred and eight thousand four hundred and seventy-nine. Hand Book of Calculations. ij Numeration. Read the following numbers: 1. 19. 2. 406. 3. 9,206. 4. 90,009. 5. 896,724. 6. 7,428,940. 7. 63,178,392. This system is called Arabic Notation from the fact that it was introduced into Europe in the 10th century by the Arabs. Its great law is that ten units in any order make one unit of the next order. And the moving a figure one place either increases or dimin- ishes its value by the uniform scale of ten. Hence it is called the Decimal system from the Latin word decern, which means ten. ROMAN NOTATION. This is the method of expressing numbers by letters. I, V, X, L, O, D, M, 1, 5, 10, 50, 100, 500, 1,000 1. Repeating a letter repeats its value, thus: 1=1, 11=2. 2. Placing a letter of less value before one of greater valu^ diminishes the value of the greater by the less; thus, IV=4, IX=9, XL=40. 3. Placing the less after the greater increases the value of the greater by that of the less; thus, YI=6, XI=11, LX=60. 4. Placing a horizontal line over a letter increases its value a thousand times; thus, 1V=4000, M=l, 000,000. i8 Hand Book of Calculations. ADDITION TABLE. 1 1 and 2 and 3 and 4 and 5 and 1 are 2 1 are 3 1 are 4 1 are 5 1 are 6 2 " 3 2 " 4 2 " 5 2 " 6 2 a >y 3 " 4 3 " 5 3 " 6 3 cc 7 3 " 8 4 " 5 4 " 6 4 a w 4 " 8 4 " 9 5 " 6 5 ftf w 5 " 8 5 " 9 5 " 10 6 a >y 6 " 8 6 " 9 6 " 10 6 " 11 7 " 8 7 " 9 7 " 10 7 " 11 7 " 12 8 " 9 8 « 10 8 " 11 8 " 12 8 " 13 9 " 10 9 " 11 9 " 12 9 " 13 9 « !4 10 " 11 10 " 12 10 " 13 10 " 14 10 " 15 6 and 7 and 8 and 9 and 10 and 1 are 7 1 " 8 1 are 9 1 are 10 1 are 11 o " 8 2 " 9 2 " 10 2 " 11 2 " 12 3 " 9 3 " 10 3 " 11 3 " 12 3 " 13 4 " 10 4 " 11 4 " 12 4 " 13 4 « 14 5 « n 5 " 12 5 " 13 5 « 14 5 " 15 6 " 12 6 " 13 6 " 14 6 " 15 6 " 16 7 " 13 7 " 14 7 " 15 7 " 16 7 « 17 8 « 14 8 " 15 8 " 16 8 " 17 8 " 18 9 " 15 9 " 16 9 « 17 9 " 18 9 " 19 10 i( 16 10 « 17 10 " 18 10 " 19 10 " 20 Hand Book of Calculations. ig ADDITION. The first process of arithmetic is Addition; and here the first steps are made by counting upon the fingers as an aid to the perceptions of the total amount of the quantity that has to be -expressed. Persons even of considerable mathemetical experi- ence will often find themselves counting their fingers, or press- ing them down successively on the table in order to assist their memory in performing addition. For example, if we hold up 5 fingers of one hand and 3 of the other and are asked how much 5 and 3 amount to we at once see that the number is 8, as we actually or mentally count the other three fingers from 5. But the best course is to commit very thoroughly to memory an addition table, just as the multiplication table is now com- monly committed to memory by arithmetical students. A ta- ble of this kind is here introduced, and it should be gone over and over again until its indications are as familiar to the mem- ory as the letters of the alphabet, and until the operation of addition can be performed without the necessity of mental effort. The table is so plain as scarcely to require explanation. The sign of addition is -f- It is called plus, or more. The sum or amount of any calculation no matter how small or large contains as many units as all the numbers added. Addition is uniting two or more numbers into one. The result of the addition is called the Sam or Amount. In addi- tion the only thing to be careful about except the correct doing of the sum, is to place the unit figures under the unit figure above it, the tens under the tens, etc. Kule. After writing the figures down so that units are under units, tens under tens, etc. : Begin at the right hand, up and down row, add the column .and write the sum underneath if less than ten. 20 Hand Book of Calculations. If however the sum is ten or more write the right hand figure underneath, and add the number expressed by the other figure or figures with the numbers of the next column. Write the whole of the last column. Examples for Practice. 7,060 248,124 13,579,802 9,420 4,321 83 1,743 889,876 478,652 4,004 457,902 87,547,289 22,227 Ans. Use great care in placing the numbers in vertical lines, as irregularity in writing them down is the cause of mistakes. Rule eor Proving the Correctness of the Sums, Add the columns from the top doivnward, and if the sum is the same as when added up then the answer is right. Add and prove the following numbers : 684 32 257 20. Ans. 993. 42 89 22 99 ? 1006 7008 01 62 ? TABLE OF UMTS. The unit of money in the U. S. is one dollar. The unit of length is one foot. The unit of surface is the square inch. The unit of toork is the foot pound. The unit of heat is one degree, Fahrenheit. The unit of numbers is the figure 1. The unit of electric power is the watt. Hand Book of Calctilations. 21 SUBTRACTION TABLE. 1 from 2 from 3 from 4 from 5 from 1 leaves 2 leaves 3 leaves 4 leaves 5 leaves 2 " 1 3 < ' 1 4 " 1 5 " 1 6 " 1 3 " 2 4 • ' 2 5 " 2 6 " 2 7 " 2 4 " 3 5 ' ' 3 6 " 3 7 " 3 8 " 3 5 " 4 6 ' ' 4 7 " 4 8 « 4 9 " 4 6 " 5 7 ' < 5 8 " 5 9 " 5 10 " 5 7 " 6 8 ' 4 6 9 " 6 10 " 6 11 " 6 8 " 7 9 ■ ' 7 10 " 7 11 " 7 12 « 7 9 " 8 10 < 4 8 11 " 8 12 " 8 13 " 8 10 " 9 11 » • 9 12 " 9 13 " 9 14 " 9 11 " 10 12 ' ' 10 13 " 10 14 " 10 15 " 10 6 from 7 from 8 from 9 from 10 from 6 leaves 7 leaves 8 leaves 9 leaves 10 leaves 7 " 1 8 < • 1 9 « 1 10 < 1 11 " 1 8 " 2 9 ' ' * 10 " 2 11 ■< 2 12 "2 9 " 3 10 ' < 3 11 " 3 12 '< 3 13 " 3 10 "4 11 ' 4 12 " 4 13 '< 4 14 " 4 1 11 " 5 12 ' 13 " 5 14 ' 5 15 « 5 12 " 6 13 ' ' 6 14 " 6 15 < 6 16 " 6 13 " 7 14 ' ■ 7 15 " 7 16 ' 7 17 " 7 14 « 8 15 ' ' 8 16 " 8 17 < 8 18 " 8 15 " 9 1G < ' 9 17 " 9 18 < 9 19 " 9 16 " 10 17 ' ' 10 18 " 10 19 ' < 10 20 " 10 22 Hand Book of Calculations. SUBTRACTION. Subtraction is taking one number from another. As in addition, care must be used in placing the units under the units, the tens under the tens, etc. The answer is called the remainder or the difference. The sign of subtraction is ( — ) Example: 98 — 22=76. Subtraction is the opposite of addition — one " takes from 9> while the other "adds to." Rule. • Write down the sum so that the units stand under the units,, the tens under the tens, etc., etc. Begin with the units, and take the under from the upper fig- ure and put the remainder beneath the line. But if the lower figure is the largest add ten to the upper figure, and then subtract and put the remainder down — this bor- rowed 10 must be deducted from the next column of figures, where it is represented by 1. Examples eoe Pkactice. 892 89,672 89,642,706 46 46,379 48,765,421 846 remainder. Note. In the first example 892—46 the 6 is larger than 2; borrow 10,. which makes it twelve, and then deduct the 6; the answer is 6. The borrowed 10 reduces the 9 to 8, so the next deduction is 4 from 8=4 is the answer. Hand Book of Calcidations. 2j Kule for Proving the Correctness of the Subtraction. Add the remainder, or difference, to the smaller amount of the two sums and if the two are equal to the larger, then the subtraction has been correctly done. Example. 898 246 246 Now then, 652 652 898 correct Ans. Examples, Consisting of Notation, Addition and Sub- traction. 1. Add together twenty-seven thousand four hundred and twenty-eight; ninety-one thousand eight hundred and seventy- nine; sixty-five thousand two hundred and fifty-nine ; and thirty-seven thousand and eight. Ans. 221,574. 2. Add seven hundred billions, nine hundred and one thou- sand; forty millions thirty thousand and ten; five hundred thousand; eight hundred and ninety-one millions and twelve; twenty-four millions two hundred and one thousand and six hundred and forty-four ; and two hundred and ninety-three billions, nine hundred and ninety-two millions, eight hundred and sixty-seven thousand, three hundred and twenty-nine; five billions, fifty millions, five' hundred thousand and five. Ans. 1,000 Billions, or 1 Trillion. Note. — This sum is best done by the aid of the numeration table. It is given for practice to form a habit of accuracy in doing long calculations. 3. From sixty-four thousands two hundred and ten millions nine hundred and twenty thousands six hundred and fifty-one: take twenty-nine thousand five hundred and fifty-four mil- lions, three hundred and seventy-four thousand six hundred and eighty-eight. Ans. 34,656,545,963. 4. From ninety billions, four hundred millions, seven thou- sand and six: take nine billions, one hundred millions, five thousand nine hundred and fifty-six. Ans. 81,300,001,050. 24 Hand Book of Calculations. MULTIPLICATION TABLE. Once 1 is 1 2 " 2 3 " 3 4 " 4 5 " 5 6 " 6 IV « IV 8 " 8 9 " 9 10 " 10 11 "11 12 " 12 2 times 1 are 2 2 " 4 " 6 " 8 "10 "12 "14 "16 "18 "20 "22 "24 3 times 1 are 3 2 " 6 " 9 " 12 "15 "18 "21 "24 "27 "30 "33 "36 3 4 5 6 7 8 9 10 11 12 4 times 1 are 4 2 " 8 3 "12 4 " 16 5 "20 6 "24 7 "28 8 "32 9 "36 10 "40 11 "44 12 "48 5 times 1 are 5 2 " 10 " 15 "20 "25 "30 "35 8 "40 9 "45 50 55 60 7 times 1 are 7 2 " 14 3 "21 4 "28 5 "35 6 "42 7 "49 8 "56 9 "63 10 " 70 11 " 77 12 "84 8 times 1 are 8 2 "16 3 "24 4 "32 5 "40 6 "48 7 "56 8 "64 9 ee iyo 10 "80 11 "88 12 " 96 9 times 1 are 9 2 a 18 3 a 27 4 a 36 5 ee 45 6 ee 54 7 ee 63 8 ee 72 9 ee 81 10 (C 90 11 ee 99 12 "108 10 times 1 are 10 2 i i 20 3 ee 30 4 ee 40 5 ce 50 6 i( 60 7 a 70 8 a 80 9 ee 90 10 a 100 11 ee 110 VI ee 120 11 times 1 are 11 2 " 3 " 4 " 5 " 6 " 7 " 8 " 9 " 10 "110 11 "121 12 "132 22 33 44 55 66 77 88 99 6 times 1 are 6 2 "12 3 "18 4 "24 5 "30 6 "36 7 "42 8 "48 9 "54 10 " 60 11 " 66 12 " 72 12 times 1 are 12 2 " 24 3 " 36 4 " 48 5 " 60 6 " 72 7 " 84 8 " 96 9 "108 10 "120 11 "132 12 "144 Hand Book of Calczdations. 25 MULTIPLICATION. Multiplication is finding the amount of one number increased as many times as there are units in another. The number to be multiplied or increased is called the Mul- tiplicand. The Multiplier is the number by which we multiply,, It shows how many times the multiplicand is to be increased. The answer is called the Product. The multiplier and multiplicand which produce the product are called its Factors. This is a word frequently used in math- ematical works and its meaning should be remembered. The sign of multiplication is X and is read i( times " or mul- tiplied by; thus 6 X 8 is read, 6 times 8 is 48, or, 6 multiplied by 8 is 48. The principle of multiplication is the same as addition, thus 3x8=24 is the same as 8+8+8=24. Eule fob Multiplying. Place the unit figure of the multiplier under the unit figure of the multiplicand and proceed as in the following: Examples. Multiply 846 by 8; and 487,692 by 143. Arrange them thus: 487,692 143 846 ■ 8 1463076 1950768 6,768 487692 69,739,956 26 Hand Book of Calculations. But if the multiplier has ciphers at its end then place it a& in the following: Multiply 83567 by 50; and 898 by 2800. 898 2800 83567 718400 50 1796 4,178,350 2,514,400 Examples foe Practice. 1. Multiply 4,896,780 by 9. 2. " 94,200,642 " 12. 3. " 843,217,896 " 800. 4. " 4,980 " 1,276. 5. " 76 " 7,854. 6. « 34 5 571,248 " 9,876. The product and the multiplicand must be in like numbers. Thus, 10 times 8 gallons of oil must be 80 gallons of oil. 4 times 5 dollars must be 20 dollars; hence the multiplier must be the "number and not the thing to be multiplied. In finding the cost of 6 tons of coal at 7 dollars per ton the 7 dollars are taken 6 times, and not multiplied by 6 tons. When the multiplier is 10, 100, 1000, etc., the product may be obtained at once by annexing to the multiplicand as many ciphers as there are in the multiplier. Example. 1. Multiply 486 by 100. Now 486 with 00 added=48,600. 2. 6,842 X 10,000=how many ? Ans. 68,420,000. To prove the result in multiplication multiply the multiplier by the multiplicand, and if the product is the same in both cases then the answer is right. Hand Book of Calculations. 27 DIVISION TABLE. 1 in 2 in 3 in 4 in 5 in 1 , 1 time 2, 1 time 3, 1 time 4. 1 time 5, 1 time 2, 2 times 4, 2 times 6, 2 times 8, 2 times 10, 2 times 3, 3 " 6, 3 " 0, 3 « 12, 3 " 15, 3 " 4, 4 " 8, 4 « 12, 4 " 16, 4 '< 20, 4 " 5, 5 " 10, 5 " 15, 5 " 20, 5 " 25, 5 " 6, 6 " 12, 6 " 18, 6 " 24, 6 " 30, 6 " 7,7 " 14, 7 « 21, 7 " 28, 7 " 35, 7 " 8, 8 " 16, 8 " 24, 8 " 32, 8 '< 40, 8 " 9, 9 " 18, 9 " 2?, 9 " 36, 9 " 45, 9 (i 10,10 " 20, 10 " 30, 10 " 40, 10 " 50,10 " 6 in 7 in 8 in d in 10 in 6, 1 time 7, 1 time 8, 1 time 0, 1 time 10, 1 time . 12, 2 times 14, 2 times 16, 2 times 18, 2 times 20, 2 times 18, 3 " 21, 3 " 24, 3 " it, 3 " 30, 3 " 24, 4 " 28, 4 " 32, 4 " 36, 4 " 40, 4 " 30, 5 " 35, 5 " 40, 5 " 45, 5 " 50, 5 " ' 36, 6 " 42, 6 " 48, 6 " 54, 6 " 60, 6 " 42, 7 " 49, 7 " 56, 7 " 63, 7 " 70, 7 " 48, 8 « 56, 8 " 64, 8 " Tl, 8 " 80, 8 " : 54, 9 " 63, 9 " 72, 9 '< 81, 9 " 90, 9 " GO, 10 " 70, 10 " 80, 10 " 00, 10 " 100,10 " 28 Hand Book of Calculations. DIVISION. When one number has to be divided by another number the first one is called the dividend, and the second one the divisor, and the result or answer is called the quotient. 1. To divide any number up to 12. Put the dividend down with the divisor to the left of it, with a small curved line sep- arating it, as in the following: Divide by 6)7,865,432 1,310,905—2 Here at the last we have to say 6 into 32 goes 5 times and 2 over; always place the number that is over as above, separated from the quotient by a small line or else pat it as a fraction, thus 2/6, the top figure being the remainder and the bottom figure the divisor, when it should be put close to the quotient; thus— 1,310, 905f. 2. To divide by any number up to 12 with a cipher or ciph- ers after it as 20, 70, 90, 500, 7,000, etc. Place the sum down as in the last example, then mark off from the right of the dividend as many figures as there are ciphers in the divisor; also mark off the cyphers in the divisor; then divide the remaining figures by the number remaining in the divisor; thus: — • Example. Divide 9,876,804 by 40. 40)9,876,804 246,920—4 The 4 cut off from the dividend is put down as a remainder, or it might have been put down as -h or T V. Hand Book of Calculations. 29 Example. Divide 129,876,347 by 1200. 1200)129,876,347 108,230—347 or Aft. Here there is a remainder of 3 and 47 cut off. The three must always be put before the 47 making it a remainder of 347 altogether. 3. To divide by any number that can be broken up into tivo factors as 18, 24, 36, 72, 144, etc. 18 is 3 times 6; then 3 and 6 arc called factors of 18; twice 9 are 18, then 2 and 9 are also factors of 18 Generally any two numbers which when multi- plied together come to the given number, are called factors of that given number. Example. Divide 868.224 by 24. Here 4 times 6=24, therefore 4 and 6 are the factors. Divide first by 4 and then the quotient by 6 as follows : 4)868,224 24 6)217,056 36,176 Example. Divide 9,824.671 by 63. 63=7 times 9. ( 7)9,824,671 >3^ 63 1 y 10 ( 9)1,403,524—3 155,947—1 Here after the division by 7 there are 3 over; and after the division by 9 there is 1 over. What is the full remainder for the sum ? To find the full remainder, multiply the first divisor by the last remainder and add the first remainder. That is 7 multiplied bv 1=7. and 3 added to 7=10. qo Hand Book of Calculations. 4. To divide by any number not included in the last three .cases. This is common long division as it is called. Eule. Write the divisor at the left of the dividend and proceed as in the following: Example. Divide 726,981 by 7,645. 7,645)726981(95 68805 38931 38225 706 Ans. 95y\°A. Examples eor Exercise. 1.— 76,298,764,833 by 9. •2.-120,047,629,817 " 20. 3.— 9,876,548,210 " 48. 4._ 3,247,617,219 " 63. 5._ 7,140,712,614 " 41. 6.— 329,817,298 " 107. 7.-247,698,672,437 " 987. 8.— 2,610,014,723 " 2406. 9.— 10,781,493,987 " 7854. Multiplying the dividend, or dividing the divisor by any number, multiplies the quotient by the same number. Dividing the dividend, or multiplying the divisor by any number, divides the quotient by the same number. Dividing or multiplying both the dividend and divisor by the same number does not change the quotient. Hand Book of Calculations. ji TABLES OF WEIGHTS AND MEASURES REQUIRED BY ENGINEERS. AVOIRDUPOIS, or ORDINARY COMMERCIAL WEIGHT. This table is used for nearly all articles estimated by weight, except gold, silver and jewels. Table. 16 drams (dr.) make 1 ounce, oz. 16 ounces, 1 pound, lb. 25 pounds, 1 quarter, qr. 4 quarters or 100 lbs., 1 hundred-weight, cwfc. 20 hundred-weight, 1 ton, T. LONG- MEASURE, or LINEAR MEASURE. This is used in estimating distances and the length of articles Table. 12 inches (in.) make 1 foot ft. 3 feet, 1 yard, yd. 5 \ yards, 1 rod, rd. 40 rods, 1 furlong, fur. 8 furlongs, 1 common mil , m. SURFACE, or SQUARE MEASURE. This is used in estimating surfaces. Table. 144 square inches (sq. in.) make 1 square foot, sq. ft. 9 square feet, 1 square yard, sq. yd. 30£ square yards, 1 square rod or perch, P. 100 square rods or perches, 1 acre, A. 640 acres, 1 square mile, M. j2 Hand Book of Calculations. MEASURE OF CAPACITY, or LIQUID MEASURE. This is used in measuring all kinds of liquids. Table. 4 gills (gi. ) make 1 pint, pt. 2 pints 1 quart, qt. 4 quarts 1 gallon, gal. DRY MEASURE. This is used in measuring grain, roots, fruit, coal, etc. Table. 2 pints (pt.) make 1 quart, qt. 8 quarts, 1 peck, pk. 4 pecks, 1 bushel, bu. Note. The chaldron, a measure of 36 bushels, formerly employed with some kinds of coal, is now seldom used. CIRCULAR MEASURE. This is used for measuring angles. Table. 60 seconds (") make 1 minute, ' 60 minutes, 1 degree, ° 360 degrees, 1 circum., C. Note. The circumference of every circle, whatever, is supposed to be divided into 360 equal parts, called degrees. A degree is ^ of the circumference of any circle, small or large. A quadrant is a fourth of a circumference, or an arc of 90 degrees. A degree is divided into 60 parts called minutes expressed by sign (') and each minute is divided into 60 seconds expressed by (") so that the circumference of any circle contains 21,600 minutes, or 1,296,000 seconds. TABLE OF WAGES. go a ■ iewot"iooio«oooioocoiooiooo ^tW»»t-050HWMiCi®QOC»H«'^Ot»0 '^T^T^r^T^^T^rHOJ'-4» e^ic^Mr+e *+ e4~;^o c*c.-fcc t-*m WO»Ci00Ortt-05inO00CiHC0iO»Q0©00 CO^^WiCt-QOOOCsOrHWfOOOQOCiTHO: ^tHtH^^tHi-HtHt-HCQCQ s w © IO © IO © © O IO © © O © © © © © © ©'© © CCW-^'^050J>i>OOCtOHCX!Cg'*10COJ>QOO 'tHHtHHHHHt-'HIS DO B •Hie i-fce Hsi e+o r4»H l ^ , 5*« r+oe+o rnte's^co Heoe^eo e+o L-CiCO!>rHOOOW©WCOHOOOOlOCOiHO® ad E i-t»Ot» 1-fcO <*4» r+O •Mter+O fffccHW <*fce HcC r+O OCOCOOWOOCOOCffiCOOSDCOOCCOOM CQCQCQCOCOTt<-lOlO©CO£-OOOOC10©THCQCO 1— 1 tH tH t— I tH CO Pd Wf-oou'joioi'OoowoinoiOOiooo ^H ad e+or+o «*9 >-4w e+o i-fcie+o Hwm»« Hm«*o •sfeo T-niHrtHNWW«»«Om^-1i^LOOlOO© a .- .: rir; -Hfe» rnja * + * e>»»-+o «*?■-♦» *fceH|» h(k> i.tCO^OOOHH«COiOOQOOrHCOO«C»OCO so -*w--fes •*♦» r^o e=t» H»e>(» rHfcoc+o .-fcoe** e^eo CHMLOOOMIO©OM©OM»OCO®0® lC©COi.-THOOOW»lOOTTHOQO®»OC.:rtO® N?}Meo'*OL':oot"QOC500THwa5^oiffl CQ cm®oc:o®5ko®mo«mo®c6oos ox®owc®cmo®coo®xo®c0005 ?( ?; ?7 ?: ?: tj* •* ic o © a t- x xi a o o i-i w co HHrlHH DO -5 ©lO©tOO©©iO ©©©©©©©©©©©© lat-c^ccci-siooiooiooooiooo i-H i-i C< Ci C* CO CO CO "rJH Tt to IO O® t- t- 00 00 OS o TH CO •1 •:*K-*r. :*r: -fee •:*« r-fcc -4« r-fco?*o >-fc0(j*o e*0 c®::cccc:o;c:c®cx®oco®o® © tH co 10 c© © CD IO <© O 00 CO © 55 CO © 0© COO CO «-i i-H i— i— y-t C? Ci CQ C} CO CO CO t* ■**< Tt< IO IO IO CO CO Q l-teofa r*3 Tt>i r*0 1*n+lp> ".lK~+l} l4»HW rH)» Q OQ CO IO CO O CO IO 00 © CO CC O CO CO O CD 00 ©'CO e ic et^oooHOi cc E5 cxonMoaxow rHrHrHrH-HHHCJCJCQWWWMCO For 6 Days © IO O it © © © IO © o o 5 o © o © o © o o co co" •** tj3 id cd i- i~ od © ©' th cJ co -r tri cc 1*1 or © ^t& — — — — — — — — ' — ". I 34 Hand Book of Calculations. SOLID MEASURE, or CUBIC MEASURE. This is used in measuring bodies, or things having length, breadth and height or depth. TABLE. 1728 cubic inches (cu. in.) make 1 cubic foot, cu. ft. 27 cubic feet, 1 cubic yard, cu. yd. 128 cubic feet, 1 cord, C TROY WEIGHT. This is used for weighing gold, silver and jewels. Table. 24 grains (gr. ) make 1 pennyweight, pwt. 20 pennyweights, 1 ounce, oz. 12 ounces, 1 pound, lb. A caret, for gold- weight, is 4 grains; for diamond-weight, is- 3.2 grains. APOTHECARIES WEIGHT. SOLID MEASUEE. FLUID MEASUKE. 20 Grains (gr)=l scruple (sc) 60 minims or drops =1 fluid dram 3 Scruples=:l dram (dr.). 8 Fluid drams=l fluid ounce. 8 Drams = 1 ounce (oz.). 1.6 Fluid ounces =1 pint. 13 Ounces= 1 pound (lb.). 8 Pints=l gallon. Note. Apothecaries, in mixing medicines, usa the pound, ounce (oz.), and grain, of this weight; but divide the ounce into 8 drams (dr.), each equal to three scruples (sc), each scruple being equal to 20 grains. Hand Book of Calculations. 35 TIME MEASURE. Time is used in measuring portions of duration. Table. 60 seconds (sec.) make 1 minute, 60 minutes, 24 hours, 365 days, 366 days, Also, 7 days make 1 100 years 1 century. m. 1 hour, h. 1 day, d. 1 conimo~ vear, c. y. 1 leap yeai, i.y. week, 12 calendar months 1 year, and THE LONG TON FOR WEIGHING COAL. Formerly, 112 pounds, or 4 quarters of 2S pounds each, were reckoned a hundred-weight, and 2240 pounds a ton, now called the long ton. This is now seldom employed in this country, except at the mines for coal, or at the United States Custom- houses for goods imported from Great Britain, in which country such weight continues to be used. CALENDAR OF MONTHS AND DAYS IN A YEAR. January, 1st mo. 31 days. July, 7th mo. 31 days February, 2d " 28 or 29. August, 8th " 31 " March, 3d « 31 days. September, 9th " 30 " April, 4th " 30 " October, 10th " 31 " May, 5 th " 31 " November, 11th « 30 " June, 6th " 30 " December, 12th " 31 " MISCELLANEOUS MEASURES. Counting. Papek. 12 units make 1 dozen. 12 dozen, 1 gross. 20 units, 1 score. 5 scores, 1 hundred. 24 sheets make 1 quire. 20 quires, 1 ream. 2 reams, 1 bundle. 5 bundles, 1 bale. J 6 Maud JJook of Calculations. USEFUL NUMBERS FOR ENGINEERS. 12 inches make one lineal foot. 144 square inches make one square foot. 1728 cubic inches make one cubic foot. 6 feet in length=l fathom. CUBIC INCHES IN BUSHELS AND GALLONS. The standard bushel of the United Slates contains 2150.42 uubic inches ; and the Imperial bushel of Great Britain, .2218.192 cubic inches. The standard liquid gallon of the United States contains 231 cubic inches, and the Imperial gallon of Great Britain, 277.274 cubic inches. The latter is used below. WEIGHTS OF WATER. One gallon of fresh water weighs 10 lbs. One gallon of sea water weighs 10£ lbs. One gallon=To 6 oth of a cubic foot. One cubic foot=6^ gallons. One cubic foot of fresh water weighs 62^ lbs. =1000 ozs. One cubic foot of sea water weighs 64 lbs. A CORD OF WOOD. A pile of wood 8 feet long, 4 feet wide, and 4 feet high, is ar cord. A cord foot (c. f.) is 1 foot in length of this pile, or 16 cubic feet. MEASUREMENTS OF GOVERNMENT LAND. 640 acres or 1 square mile make one section of land; 320 acres=i section; 160 acres=i section. WEIGHTS OF METALS. Wrought Iron, 3dhr cubic inches=l lb., or leu. in.=. 2778 of alb. st Iron, 3 T fo " " =1 " " =.257 " el, soft, 3 T !o " " =1 " « =.2814 iS U 9 t( 3xto a 3tBo a iss, 3roo " " =1 " " =.3 Hand Book of Calculations. 37 UNITED STATES MONEY. Table. 10 mills are 1 cent,, ^ ct. 10 cents are 1 dime, d. 3 dimes or 100 cents are 1 dollar, dol. or $. 10 dollars are 1 eagle, E. The dollar is the unit; hence dollars are written with the sign $ prefixed to them and the decimal point placed after them. Cents occupy hundredths place on the right of the decimal point and occupy two places, hence if the number to be expressed is less than 10 a cipher must be prefixed to the figure denoting them; one dollar and nine cents is written $1.09. Mills occupy the place of thousandths. In business calcula- tions, if the mills in the result are 5 or more, they are consid- ered a cent; if less than 5 they are omitted. STEELING OR ENGLISH MONEY. Table. 4 farthings (qr. or far.) make 1 penny d. 12 pence 1 shilling, s. 20 shillings, 1 pound or sovereign £ 10 florins (fl.) 1 pound, £ FRENCH MONEY. Table. 10 centimes = 1 decime. 10 decimes = 1 franc. The unit of French money is the franc, the value of which in TJ. S. money is 19.3 cents, or about i of a dollar. The Money Unit of the German Empire is the mark, which is divided into 100 pennies. The value of a mark is $0,238, or nearly Si- Canada money is expressed in dollars, cents and mills, which have the nominal value of the corresponding denominations of U. S. money. JS Hand Book of Calcnlatio7is. ROMAN TABLE. I. denotes One. XVII. denotes Seventeen. II. Two. XVIII. Eighteen. III. Three. XIX. Nineteen. IV. Four. XX. Twenty. V. Five. XXX. Thirty. VI. Six. XL. Fortv. VII. Seven. L. Fifty. VIII. Eight. LX. Sixty. IX. Nine. LXX. Seventy. X. Ten. LXXX. Eighty. XI. Eleven. XO. Ninety. XII. Twelve. 0. One hundred. XIII. Thirteen. D. Five hundred, XIV. Fourteen. M. One thousand XV. Fifteen. X. Ten thousand XVI. Sixteen. M. One million. TABLE OF ALIQUOT PARTS. Of a $. Of a Ton. Of a cwt. Of an Acre. Of a Month. cts. % cwt. ton. lb. cwt. id. A. d. m. 50 == i "2" 10 = i 50 = -J 80 = i 18 = i 33* = * 5 = i 25 = i 40 = i 10 — i 25 = i 4 — . i 20 == | 32 = i li — 'i 1» = i H= i m= i 20 = i fi = i 18*- i ' 2 — T V ™ = T v 16 = T V 5 = i 10 = l TIT 1 = 5 = 1 8 = A 3 -A ><"OTE. An Aliquot part of a number is an exact divisor of it; thus, 2, 4 and 8 are exact divisors of 16. Hand Book of Calculations. jp MISCELLANEOUS MEASURES. 3 inches = 1 palm. 3.28 Feet = 1 meter. 4 " = 1 hand. 6 " — 1 fathom. 6 " = 1 span. 830 Fathoms = 1 mile. 18 " =1 cubit. 3 Knots = 1 marine league. 21.8 " =1 Bible cubit. 60 Knots \ 2^ Feet = 1 military pace. 69£ Statute miles > =1 degree. 3 " = 1 common pace. 991.12 Miles ) ¥ V of an inch=a hair's breadth. TABLE. Showing relative value of French and English measures of length. French. English. Milimeter, ... = .. 0.03037 inches. Contimetre, . . . = . . . 0.39371 " Decimetre, ... = .. 3.93710 " Metre, •... = .. .39.37100 <• In the French system of weights and measures, which has been legalized by special act of the U. S. Congress, the metre, litre, gramme, etc., are increased or decreased by the following words prefixed to them: Milli expresses the 1,000th part. Centi " " 100th " Deci " " 10th " Deca " 10 times the value. Hecato " 100 " " Chilio " 1,000 " " " Myrio " 10,000 " " << The following approximate measures, though not strictly ac- curate, are often useful in practical life. 45 drops of water, or a common teaspoonful=l fluid drachm. A common tablespoonful=^ fluid ounce. A small teacupful, or 1 gill =4 fluid ounces. A pint of pure wate = l pound. 4 tablespoon fuls, or a wine glass — -J gill. A common-3ized tumbler =-£ pint. 4 teaspoonfuls = l tablespoonful. 4Q Hand Book of Calculations. REDUCTION. Reduction is changing compound numbers from one denom- ination to another without altering their values. It is of two kinds, Descending and Ascending. Reduction Descending is changing higher denominations to lower, as tons to pounds. Reduction Ascending is changing lower to higher denominations as cents to dollars. To reduce higher denominations to lower. RUL3. Multiply the number of the highest denomination given, by the number required of the next lower denomination to make one of that higher, and to the product add the number, if any, of the lower denomination. Proceed in like manner till the whole is reduced to the re- quired denomination. Example ik Troy Weights. Reduce 63 lb. oz. 10 pwt., to pennyweights. OPERATION. 63 lb. oz. 10 pwt. 12 . ■■ " 12ft 63 Since in 1 pound there are 12 ounces, in 63 pounds there are 63 times 12 ounces, cr 7.6 ounces. 75 g 0z Since in 1 ounce there are 20 penny- 20 weights, in 756 ounces there are 756 times 20 penny-weights: and 10 penny- 15120 weights added, make 15130 penny- 10 weights. 15130 pwt., Ans. Hand Book, of Calculations. dr Example lx Avoirdupois Weight. Reduce six tons, eight hundred weight, three quarters to lbs* 6 T. 8 ewt. 3 20 qrs. 120 8 add above. 128 cwt. 4 512 3 515 qrs. ,. 25 2575 1030 12875 lbs. Answer. Examples for Practice. 1. Reduce 116 tons 68 lbs. to ounces. 2. Reduce 208 tons 42 lbs. to pounds. 3. Reduce 180 degrees of the circle to seconds. 4. Reduce 365 d. 5 h. 48 mi. 50. sec. to seconds. 5. Reduce 75 b. 3 pk. 5 qt; to quarts. ■ >'. Note. Expertness in this rule of arithmetic is of considerable im- portance, as it enters into a vast number of practical questions in every department of manufacturing as well as engineering. 42 Hand Book of Calculations. To reduce lower denominations to higher. Rule. Divide the given number by the number of its denomination required to make one of the next higher, and reserve the re- mainder, if any. Proceed in like manner with the quotient, and so continue uutil the whole is reduced to the required denomination. The number of the required denomination, with the several remainders, if any, will be the answer. Examples. 1. Bring 98,704,623 lbs. to tons and lbs. 2000)98704623 49352 Tons, 62a lbs. Bring 9876 lbs. coal to the long ton, cwt., qrs. and lbs. 2240)987fi(4 tons. 8960 112)916(8 cwt. 896 28)20(0 qrs. Ans. 4 tons 8 cwts. qrs. 20 lbs. Pkoof. Reduction Ascending and Descending prove each other ; for one is the reverse of the other. Notes. A simple number is one which expresses one or more units of the same denomination. A compound number expresses units of two or more de- nominations of the same kind, as 5 yards, 1 foot, 4 inches — or example, page 41, 6 T., 8 cwt, 3 qrs.,— these are compound numbers ; but ten oxen, or five dollars, are simple numbers. Hand Boj/c of Calculations. 4.3 Example. 76,245 gills to gallons, etc. 4)76245 2)190'il— 1 gill. 4)9530—1 pint. 2382—2 quarts. Ans. 2382 gallons, 2 quarts, 1 pint and 1 gill. Examples for Exercise. 1. In 76,298 ounces how many tons, etc. 2. In 648,000 seconds how many degrees? 3. In 15,130 pennyweights how many pounds, etc.? 4. In 3,760,128 cubic inches how many cords? 5. In 785 pints how many gallons ? EXAMPLES IN THE TABLE OF WAGES. 1. What is the amount of 7 weeks, 4-J days work at 7 dollars per week. 7 weeks. 7 dollars. 49=7 weeks pay. 4.6 ; f=4 days per table. 58J=5 hours or i day per table. 54.25 Ans. Fifty-four dollars and 25 cents. 2. What is the amount of one week and $ day extra time at $18.00 per week ? 1 week = 18.00 5 hours= 1.50 19.50 Ans. 3. What do a boy's wages come to, for 5-J days, at #5.00 per week ? 5 days = 4. 16§ per table. 5 hours = 41 J per table. 4.684- Ans. $4.68. 44 Hand Book of Calculations. 4. What do 46 days, 6 hours and a quarter, amount to at $17.00 per week. 1 day per table = 2. 83 J multiply by ( X ) 46 days. 1 5^ amount of fraction. 1698 1 hom r28j * byi 7A, » 1132 divide 13033^=46 days. 170 =6 hours per table. 7iV=i hour. 13210 T 5 2 Ans. $132.10. Examples for Practice. 5. Howm uch 4 days \ 2i hrs. at $ 8. 00 per week. Ans. 15.67 6. 306 a " " 9.00 " " " 459.00 7. 184 a 5 " " 11.00 " « " 33825 8. 11 a " " 4.00 " " ff 9. 39 a 6 " li 15.00 " " ff 10. 1 i( 2 « " 12.00 " " ft 11. Hi" " 14.00 " " a 12. 5 " " 7.00 " " tt Example. In doing the sum for example 9, do it like this: 1 day at $15.00 per week is $2.50 Multiply by 40 days, Deduct 4 hours at 25c., 40 100.00 1.00 Answer, $99.00 There are various " short cuts " in figuring wages, like the last example, which it is well to become familiar with, so that in this important part of mathematics, both quickness and accuracy may be attained. Note. When the fraction is less than i cent it is the gain of the employer by the amount of the fraction — but, if the fraction is more (like f ) it is called a full cent and goes as a full cent to the employee. Hand Book of Calculations. 4.5 NATURAL OR MECHANICAL PHILOSOPHY. Natural philosophy is the science which treats of the laws of the material world; and it is this science, with which the engi- neer has to cooperate, in obtaining the best results from his professional skill. All the calculations relating to steam-engi- neering, are closely connected with the operations set forth in that department of knowledge which is thus termed. u I have learned more about my business/' said a trusted and competent engineer, to the author " from an old work on nat- ural philosophy, which I own, than from all the other books I ever read." Hence it is worth the while, to consider a little, the foundation of this important part of an engineer's education. Natural or Mechanical philosophy is divided into Mechanics, Hydrostics, Pneumatics and Electricity; the engineer in his daily practice is liable to be called upon to deal with one or all of them, for he has to do with machinery, treated under the head first named; with water, treated under the division, Irostatics; air (Pneumatics) and with Electricity; upon analysis it will appear that all the computations in this volume are practically used, in connection with one or more of these divisions. jf6 Hand Book of Calculations. Science shows that there are but few fundamental laws be- neath all creation, and all observation proves that these basis principles are preserved through countless varying forms, therefore, Let it be particularly noted that there are but 68 elemen- tary substances, known at the present day, to exist; these are platinum, gold, silver, copper, iron, lead, tin, sulphur, nickel, mercury, carbon, hydrogen, nitrogen, antimony, arsenic, bis- muth, etc., etc. A substance which cannot be resolved into two or more different substances is called an elementary or sim- ple body; as for example, neither water, coal, nor brass are elementary substances as each can be resolved into other forms, cf matter. Matter is any collection of substance existing by itself in a separate form. Matter appears to us in various shapes, which however can all be reduced to two classes, namely solids or fluids. A Solid offers resistance both to change of shape, and to change of bulk. A Fluid is a body which offers no resistance to change of shape. Fluids again, can be divided into liquids and vapors or gases. Water is the most familiar example of a liquid. A liquid can be poured in drops while a gas or vapor cannot. It is import- ant to note that experiment proves that every vapor becomes a gas at a sufficiently high temperature and low pressure, and, on the other hand, every gas becomes a vapor, at sufficiently low and high pressure. Atoms. An atom is the smallest particle of matter known to exist, they are sometimes called molecules, and are so small that they cannot be divided. Chemistry treats of all which relates to these particles of matter, and to the changes of constitution produced by their action on each other. The combustion of coal is strictly a chemical process, as the mass of fuel is reduced to particles of Hand Book of Calculations. 47 gas and vapor by combination with oxygen, resulting in heat,, which in turn expands water into steam, in the boiler. Now we are brought, with our 68 original elementary sub- stances, to those forces which act upon them, live in number ; these will be explained in the next section, under the title of primary powers. PRIMARY POWERS. The following is a list of all the primary powers which, as yet, have been used by man in accomplishing his purpose in the wide domain of practical life. These are 1. Water power. 2. Wind power. 3. Tide power. lide power. 4. The power of combustion. 5. The power of vital action. To this list may hereafter be added the power of the volcano and the internal heat of the earth; and besides these, science at the present time gives no evidence of any other Gravitation, electricity, galvanism, magnetism and chemical affinity can never be employed as original sources of power. There is no more prevalent and mischievous error than to sup- pose that work can be had from these latter, and no engineer of intelligence will waste his life energy in trying to get "some- thing from nothing " as he will be doing should he attempt the problem. Even in the modern application of electricity it is apparent that it is but the resovoir (a storage battery) or the means of transfer by wires, of the power of combustion, or water, to the work. The same must be said of the elastic force of steam, of air ^S Hand Book of Calculations. and of springs; and also of machinery; they are all bnt the act- ive agents employed between the primary p6 to er and the work. In ail computations of power and the action of machines these first principles should always be borne in mind; it is not the engine which is the source of motion to the machinery, nor yet the steam, but the repulsive energy imparted to the expand- ing water from the burning fuel. THE MECHANICAL POWERS. We now proceed to consider the effect produced, when these forces are made to act by the intervention of other bodies. These intermediate bodies are called machines and by the means of them the effect of a given force may be increased or dimin- ished as desired. Machines ?re divided into simple and compound. The simple machines or what are commonly called Mechanical Powees, are six in number; viz. i 1. The lever. 2. The wheel and axle. 3. The pulley. 4. The inclined plane. 5. The screw. 6. The wedge. These can in turn be reduced to three classes: I. A solid body turning on an axis. II. A flexible cord. . III. A hard and smooth inclined surface. For the mechanism of the wheel and axle and of the pulley, merely combines the principle of the lever with the tension of the cords; the properties of the screw depend entirely on those of the lever and the inclined plane; and the case of the wedge is analogous to that of a body sustained between two inclined planes. Hand Book of Calculations. 49 MACHINERY. Compound machines are formed from two or more simple machines. Tools are the simplest implements of art; these when they become complicated in their structure become ma- chines, and machines when they act with great power, take the name, generally speaking, of engines. The advantage that man has gained by pressing into his ser- vice the great forces of nature, instead of depending on his own feeble arm. is evinced by the fact that aided by the steam engine one man can now accomplish as much labor as 27,000 Egyptians, working at the rate at which they built the pyra- mids (Dapin). The mechanical powers will now be separately considered, it being remembered that none of them create force, but that they only modify and direct it, acting by certain great laws, established by the supreme Creator and generous Giver of the original sources, of both the Primary and Mechanical causes. He will labor most effectively and happily who studies these laws and acts in accordance with their principles, which are those laid down and explained in detail in books relating to N'atural Philosophy. THE LEVER. Lever first kind. jo Hand Book of Calculations. THE LEVEK. The lever is an inflexible bar or rod, some point of which being supported, the rod itself is movable freely about that point as a center of motion. This center of motion is called the Fulcrum or Prop. In the lever three points are to be considered, viz. : the ful- crum or point about which the bar turns, the point where the force is applied, and the point where the weight is applied. There are three varieties of the lever, according as the ful- cram, the weight or the power is placed between the other two, but the action in every case is reducible to the same principle and the same general rule applies to them all. Note. When two forces act on each other by means of any machine,, that which gives it motion is called the power, that which receives it th b weight, hence, In the diagrams the letter P is used to denote the point of application of the forces ; the letter F denotes the fulcrum, or prop, and W the weight. 1st. When the fulcrum (F) is between the force (P) and the weight (W). Fig. 1. Fig. 2. Lever 2d kind. 2d. When the weight (W) is between the fulcrum (F) and the force (P). Fig 2. Hand Book of Calculations. 5* THE LEVER. C w * j^ s \ Fig. 3. Lever 3rd kind. 3rd. When the force (P) is between the fulcrum (F) and the weight (W). Fig. 3. General Eule. The force (P) multiplied by its distance from the fulcrum (F) is equal to the weight (W) multiplied by its distance from the fulcrum. In the following examples the distances are figured in inches and the weight in pounds, the unit of distance in mechanics being one inch, and the unit of loeight being one pound. Note. The following calculations are made on the supposition that the action of the mechanical powers is not impeded by their own weight, or by friction and resistance. Thus, in each cal- culation, in figuring the problems relating to the safety-valve, the weight of the valve, spindle and lever have to be taken into the estimate. A special rule (with illustrations) will be. given in its proper place to show how these are to be provided for. Example. What force applied at three feet from the fulcrum will bal- ance a weight of 112 lbs. applied at 6 inches from the fulcrum (observe diagram of 1st form of lever). Here the leverages are 36 and 6 inches. §2 Hand Book of Calculations. THE LEVER. This is found by dividing 672 by 36. 112 lbs. 6 inches. 36)672(18f 36 Pkoof. 112 lbs. X 6 inches = 672. 18f " X 36 " = 672. 312 288 24 9. 36 That is, 18f lbs. applied at the end of a 3 J foot bar with a fulcrum 6 inches from the point, will lift a box weighing 112 lbs. Example. If 80 lbs. be applied at the extreme end of a 5 foot lever (with prop 1 foot from the point), what force is needed to balance the 80 lbs. The two leverages being 48 inches and 12 inches. Now, multiply the force (P) 80 lbs., by the distance from the fulcrum (F) 48 inches and divide by 12 inches. 48 inches. Proof. 80 lbs. 48x80 lbs. =3840 12X320 "=3840 12 in.)3840 320 lbs. This is an example worked from the lever of the second kind. Hand Book of Calculations, 53 THE LEVER. Under the general rule given, it will be seen that under all circumstances the force multiplied by its distance from the ful- crum, is equal to — or balanced by, the weight multiplied by its distance from the fulcrum; 4 sub-rules are added which will cover all problems where only three of the numbers are known. Fig. 4. Lever 1st kind. To find the power (P) on any lever, ivhen the weight (W) and two distances from the fulcrum (b)are given. Sub-Rule 1. Multiply the weight (W) by its distance from the fulcrum (b) and divide by the distance from P, to b. The quotient is the power. Example. How much to balance 200 lbs., 18 inches from the fulcrum (b) to the end of the lever at (P). The whole length of the lever being 36 inches. 18 in. 200 lbs. 36 in.)3600(100 lbs. Answer. The exMKiple given to illustrate the general rule is similar to this. 54 Hand Book of Calculations. THE LEVER. Fig. 5. Lever of the 2d kind. To find the weight (W) when the power (P) and the two dis- tances from the fulcrum (b) are given. Sub-Rule 2. Multiply the power (P) by its distance from the fulcrum (b) and divide by the distance of the weight (W) from the fulcrum. The quotient is the weight. Example. If 480 lbs. be applied at the end of a lever, 135 inches from the fulcrum, what weight will it lift 45 inches distance from the fulcrum. 480 Prooe. 135 ±440 x 45 = 64,800 480x135 = 64,800. 2400 1440 480 )64800(14 40 lbs. Ans. 45 198 180 180 fp>_ 180 Hand Book of Calculations. 55 THE LEVER. Z 3 Fig. 6. Lever of the 3rd kind. To find the distance of the poiver (P) from the fulcrum (b) the weight and its distance and the power being given. Sub-Rule 3. Multiply the weight (W) by its distance from the fulcrum tad divide by the power. Example. If a weight 900 lbs. be 12 inches from the fulcrum, at what ^stance must 80 lbs. be placed to balance it ? 12 Proof. 900 900 12 10,800. 135 80 10,800. 80)10800(135 inches Ans. 80 280 240 400 To find the distance of the weight from the fulcrum. The power and its distance from the fulcrum and toeight being known. Sub-Rule 4. Multiply the power by its distance from the fulcrum and divide by the weight. $6 Hand Book of Calculations. THE LEVER. Example. (Lever 3d kind.) If the power be 1,000 lbs., 3 inches from the fulcrum, at what distance must the weight (W) 120 lbs. be placed to balance it. 1,000 lbs. power. Proof. 3 in. distance. 1,000 3 = 3,000. 120 25 = 3,000. 120)3000 25 inches. Ans. 25 inches from the fulcrum. The Leverage of the Power. The ratio of the power end of the lever, to the length of the weight end, is called the leverage of the poioer. The three varieties of the lever are shown in Fig. 4, 5 and 6, and in each case the lever is supposed to be seven feet long, and divided into feet. The respective lengths (fig. 4) being 6 feet and 1 foot, the leverage is 6 to 1, or 6. In the second (fig. 5) it is 7 to 1, or 7 ; in the third one-seventh to 1, or 1-7, showing that in the first case the power balances 6 times its own amount ; in the second case 7 times its amount ; in the third case only one-seventh of itself, because it is nearer the fulcrum than the weight. Hand Book of Calculations. 57 THE WHEEL AND AXLE, or PERPETUAL LEVER. Fig. 7. When a lever is applied to raise a weight, or to overcome a resistance, the space through which it acts at one time is small and the work mast be accomplished by a succession of short and intermitting efforts. The common lever is, therefore, used only in cases where weights are required to be raised through short spaces. When a continuous motion is required, as in rais- ing ore from the mine, or in weighing the anchor of a vessel, some contrivance must be adopted to remove the intermitting action of the lever and render- it continuous. The wheel and axle, in its various forms, fully answers this purpose. It may be considered a revolving lever. The wheel and axle may be likened, also, to a couple of pullies of different diameters united together on one axis, of which the larger is the wheel and the smaller the axle, with, a common fulcrum. The power of the wheel and axle is expressed by the number of times the diameter of the axle is contained in that of the wheel, as per the following Rule. Multiply the power at the edge of the wheel by its radius (half its diameter) and divide the product by the radius of the axle. The quotient is the weight that the power will raise. jS Hand Book of Calculations. THE WHEEL AND AXLE. Example. Required the weight that can be raised by a power of 50 lbs. applied at the circumference of a wheel of 5 feet diameter (2^ ft. radius) the weight to be attached to the end of a rope, which is to be wound around a barrel or axle 12 inches in diam- eter. Now then %i feet = 30 inches. 50 lbs power. Radius of axle 6)1500 250 lbs. answer. Note. There are obviously two ways by which the power of the wheel and axle may be increased; either by increasing the di- ameter of the wheel or diminishing that of the axle. The weight to le raised, the diameter of the axle and dameter of the wheel being given, to find the amount of poiver required to raise the weight. Rule. Multiply the weight to he raised by the radius of the axle, and divide the product by the radius of the wheel. Example. Required the power necessary to raise a weight of 400 lbs. by an axle of 10 inches, and wheel of 50 inches in diameter. .Now, then: Weight — 400 ■J diam. of axle 5 i diam. of wheel 25)2000(80 lbs. Ans. 2000 Hand Book of Calculations. 59 THE WHEEL AND AXLE. A ship's capstan is another form of the wheel and axle. Example for Practice. In weighing anchor 6 capstan ^...... „_X^ ^ t~Ls bars are used ; from center of ^^sg^n^^a^ capstan to point of pressure is L [ 6 feet; diameter of axle of cap- W^^S±==^ stan = 24 inches. Now then, /// 1 (^ Oi\\\ if each man exerts 80 lbs. with / 1 1 / 7 ^ ^ a\\\ his bar. //////! II! \W The leverage for force (radius of 12 ft. diam.}=6. Number of men 6 36 Lbs. for each man 80 Divide by radius of axle 1)2880 2880 lbs. Ans. If an allowance of ten per cent, is made for friction and the rigidity of the cord, the answer will be 2592 lbs. Ans. Example for Practice. The diameter of a steering wheel on a ship is 5 feet and the barrel is 15 inches in diameter. If a man appjies a force equal to 200 lbs. what resistance would he overcome? Ans. 800 lbs. THE CHINESE WHEEL AND AXLE. To combine the requisite strength with moderate dimensions and great mechanical powtr has been accomplished by giving different thicknesses to different parts of the axle and carrying 6o Hand Book of Calculations. THE WHEEL AND AXLE. Fig. 9. a rope which is coiled on the inner part through a pulley at- tached to the weight and coiling it in the opposite direction on the thicker part as in fig 9. We see h:re exemplified the principle, that the weight sus- tained by a given power, may be increased as its velocity is di- minished. By inspecting fig. 9 it will be seen that the rope connected with the thinner part of the axle unwinds, while that connected with the thicker part winds up, by which means the ascent of the weight may be rendered slow in any degree, and a> proportionally greater quantity of matter may be added. To find power in this arrangement follow the Rule. The power multiplied by the radius of the wheel, in feet, is- equal to half the weight multiplied by the difference in the= half diameters (radii) of the thicker and thinner parts of the axle. This will be made clear by the following Example. The diameters in fig. 9 are 1 foot and f of a foot; the length of the handle 2 feet 3 inches ; if the exertion put forth is equal to 80 lbs. what weight will be lifted. . Now then to follow the rule. Hand Book of Calculations. 61 THE WHEEL AND AXLE. The length of the handle 2,25 feet. The power exerted 80 lbs. i the difference in the radii .0625)18000(2880 lbs. Ans. 1250 5500 5000 5000 5000 In all these examples the diameter of the rope has been sup- posed to be so small in comparison to that of the drum or bar- rel that it has been neglected; if it is a thick rope, then the leverage must be measured from the center of the barrel to the center of the rope. Example. Wheel and axle, the barrel is 10" in diameter, the rope is 1-j- inches in diameter, the crank handle is 15" radius, and the weight to be lifted is 500. What force must be applied to the handle if 10 per cent, is to be added for friction. Now, then Leverage of weight 5"-f-f " = 575. Being radius of barrel and rope. 500X5,75= 500 lbs. 15)287500(191| lbs. Ans. 15 137 135 25 15 10 » Add for friction 19.17 = 210 ,Vo Ans. 62 Hand Book of Calculations. THE WHEEL AND AXLE. These examples are worked in decimal fractions, the rules and examples of which will be given later. To find the difference in the half diameter of the axle, (Fig. 9.) proceed thus : 1 foot — f = \ foot; \ this for the radius = one-eighth foot, and half this is one-sixteenth, or in deci- mals .0625. (See example.) THE PULLEY. The pulley is a wheel over which a cord, or chain or band is passed, in order to. transmit the force applied to the cord in another direction. The practical effect of the machine depends upon the rope, the wheel being introduced to diminish friction and the effect of imperfect flexibility, but the whole effect of imperfect flexi- bility and friction are not de- stroyed, although in calcula- tions, we proceed as though they were. There is no mechanical advantage gained by a single rope over one or more fixed p allies; but this combination is of the greatest use by enabling us to change the direction of the force. Pulleys are divided into fixed and movable. In the fixed pulley no mechanical advantage is gained, as already explained, but its use is of the greatest importance in accomplishing the work appropriate to the pulley, such as raising water from a well. Thus, it is far more convenient to raise a bucket from a well by drawing downward, as is the case where the rope passes over a fixed pulley above the head, than by drawing upward leaning over the curbing. From its portable form, its cheapness and the facility with which it can be applied, especially in changing or modify- ing the direction of motion, the pulley is one of the most con- venient and useful of the mechanical powers. Fig.. 10. Hand Book of CaI.cn/atio7is. 6j THE PULLEY. It must be observed that in using any system of movable pullies, the whole weight of the pulleys themselves, together with the resistance occasioned by the friction and rigidity of the ropes all act against the power and so far lesseu the weight which it is capable of raising. The moveable pulley by distributing the weights into separate parts, is attended by mechanical advantages proportioned to the number of points of support. Movable pulleys may be arranged according to different system's which increase the efficacy of a given power in different degrees. By means of the pulley great facilities are afforded in raising heavy weights, as boxes of mer- chandise or heavy blocks of stone. Fig. 1 1 represents a con- venient method in building- brick chimneys for steam plants which has been observed by the author, as used by Glasgow, Scotland, masons and builders. The crane at B enables the workmen when the brick and mortar are raised, to swing it around to the point where it is to Fig. 11. be laid or to a platform near it. The lower cord of the rope C D is connected with a wheel and axle ; in the illustration, it may be seen, that instead of the wheel and axle we might fasten a horse to the rope, or attach a sweep to the top of the axis and join a team of horses to the end of it to expedite the work. The employment of this device, in sufficiently large chimneys, enables the builder to dispense with the use of scaffolding, the workmen building into the corners of the chimney, as the work progresses, a ladder of \ or \ inch round iron every fifteen inches, to enable them to go up and down in the interior of the flue. Tli us a large expense is saved in cost of scaffold, and the risk is less for the mason. 6 4 Hand Book of Calculations. THE PULLEY. Fast and loose pullies. These are shown in Fig. 12 where the movable block A car- ries the weight with a fixed counterpart B. Here the rope is attached by one end to the fixed block and is passed over the movable and fixed pullies from one to the other in succession, the power being ap- plied to the other end. This system is known as fast and loose pulley blocks. The fixed end of the rope is sometimes fastened to the movable block. To find the power necessary to balance the weight by the means of a system of fast and loose pulleys. EtJLE 1. Divide the weight by the number of ropes by which it is carried; that is by Fig. 1 2. the number of ropes which proceed from the movable block. The quotient is the power required to bal- ance the weight. Example. A cylinder cover weighing 1200 lbs. is lifted by a pair of blocks of two sheaves each, the rope is fastened to the upper block. Now, then, in two sheaves there are 4 ropes. 4)1200 300 lbs. will balance the weight of the cylinder head. Example. A boiler weighing 6 tons has to be lifted by a pair of treble blocks; how much power must be applied at the end of the rope to balance the weight. 6 tons 2000 6)12000 lbs. 2000 lbs. Hand Book of Calculations. 65 THE PULLEY Sometimes the upper block has 4 sheaves and the lower 3, the rope being- r0 ve and fastened to the lower block, then when the stress comes, there will be 7 singles of the rope holding the weight up. In this case the weight would be divided by 7. In all the above cases, single rope and a single movable block have been used, but we may have several movable blocks each with its own rope. Example. Let a pulley be fastened to a weight of 1200 lbs. and a rope fastened by one end to a beam, brought round the pulley, and the other end fastened to a second pulley; let a second rope be fastened to the beam, brought around this second pulley and fastened to a third pulley; let a third rope be fastened to the beam, brought round the third pulley and then up over a fixed pulley: what weight would put it in balance ? Axswer. In this case the 1200 lbs. is supported by two singles the first rope, hence each single rope bears a weight of 600 lbs. This 600 is the weight the second movable pulley sustains; bence each single of the second rope bears a strain of ™<>=r300 lbs. This 300 lbs. is the weight the third movable pulley sustains; hence each single of the third rope bears '•',:" --150 lbs. Answer. In order to have the rules apply it is accessary to have the cords parallel with each other, as an} other than a "straight pull " altera the mechanical efficiency. 66 Hand Book of Calculations. THE PULLEY. The single fixed pulley as shown in Fig. 10 acts like a lever of the first hind, and simply changes the direction of the forces without modifying the intensity of the power. But the pulley may be employed as a lever of the second hind by suspending" the weight to the axis of the pulley, and fixing one end of the cord to a spot as a fulcrum point X as shown in Fig. 14. Thus the power acts through the diam- eter, A C B, in which B is the fulcrum. In acting as a lever of the third hind, the power is applied to the axis a in Fig. 15, one end of the cord being fixed at h and the weight attached to the other end, c. In the last example the gain is -|. Fig. 15. Hand Book of Calculations. 6j THE INCLINED PLANE. The inclined plane is a slope, or a flat surface inclined to the horizon, on which weights may be raised. By such substitu- tion of a sloping path for a direct upward line of ascent, a given weight can be raised by a power less than itself. The inclined plane becomes a mechanical power in conse- quence of its supporting part of the weight, and of course leaving only a part to be supported by the power. Thus the power has to encounter only a portion of the force of gravity at a time; a portion which is greater or less according as the plane is more or less elevated. The simplest example we have of the application of the inclined plane is that of a plank raised at the hinder end of a cart for the purpose of rolling in heavy articles, as barrels or hogsheads. Again, for another Example. When a horse is (hawing a heavy load on a perfectly hori- zontal plane, his force is spent chiefly in overcoming friction, and the resistance of the air, as the force of gravitation can afford no resistance, in the direction in which the load is moving. lint when the horse is drawing ;i load up a hill he has not only these impediments to overcome hut he lifts a part of the load. If the rise is 1 tool in 20, he lifts one twentieth of the load: if the ascent is one todi in four and the load is two tons, including his own weight, he Lifts 4)4(10(1 lone lbs. 68 Hand Book of Calciilations. THE INCLINED PLANE. Note. The general principle for all calculations relating to the inclined plane may be thus stated. As the length of the plane is to the height or angle of inclination, so is the weight to the power: this principle will be understood by reference to that part of this work relating to Ratio and Proportion. klfr*-* A r^^ r-n ^-2^. """*->. h *'*'••-. 4- ■■■■■■> Fig. 16. There are three elements of calculation in the inclined plane, the plane itself, the base or horizontal length and the height or vertical rise, together forming a right angled triangle. Fig. 16 exhibits an inclined plane. To find the power necessary to raise a given weight, the length and heighth of the inclined plane being known. Rule. Multiply the weight by the height and divide by the length of the plane. Example. Required the power necessary to raise 1280 lbs. up an inclined plane 8 feet long and 5 feet high. Now then : 1280 lbs. 5 feet. 8)6400 800 lbs. Answer. Hand Book of Calculations. 6 9 THE INCLINED PLANE. The length and height of an inclined plane being known* to find the weight that a given power will support upon the plane, EULE. Multiply the power by the length of the plane and divide the product by the height. The quotient is the weight that the power will supp'vfc. Example. The length of an 'nclined plane is 15 feet; the perpendicular height 6 feet: what force will be required to sustain a weignt of 150 lbs.? F f g 16. 150 lbs. 6 feet. 15)900(60 lbs. Answer. 90 00 Fig. 17. The principle of the lever as applied to the inclined plane may be seen illustrated in Fig. 17, where the power is applied at the end of a cord passed round and over the weight (W ). In this case there is the action of a movable pulley, com- bined with an inclined plane, the rolling weight moved by a cord, B P, lapped round it, representing a movable pulley with the weight attached to the axle. Thus the leverage of the power on the inclined plane can be doubled. 7° Hand Book of Calculations. THE SCREW. Fig. 18. The screw is an inclined plane wrapped around a cylinder. Take for example an inclined plane A, B, 0, Fig. 18, and bend it into a circular form resting on its base, so that the ends meet. The incline may be continued winding up- wards rcund the same axis and thus winding inclined planes of any length or height may be constructed. The distance apart of two consecutive coils, measured from centre to centre, or from upper side to upper side, (literally the height of the inclined plane), for one revolution, is "the pitch" of the screw. The screw is generally employed when severe pressure is to be exerted through small spaces; being subject to great loss from friction it usually exerts but a small power of itself, but derives its principal efficacy from the lever or wheel work with which it i-s very easily combined. A screw in one revolution will descend a distance equalto its pitch, or the distance between two threads and the force ap- plied to the screw will move through, in the same time the circumference of a circle whose diameter is twice the length of the lever. Hence the Rule. Hand Book of Calculations. 7' Fig. 19. Kile. The power multiplied by the circumference is equal to the weight multiplied by the pitch. Example. If the distance between the threads be \ inch and the force of 100 lbs. be applied at the end of a lever 3 feet in length; what weight will be moved by the screw ? See the diagram Pig. 19. Twice the length of the lever =6 feet = 72 inches diam. 100 power. 7200 3.14 to get circum. 28800 7200 21000 divide by pitch 5)22608.00 .25 5)452160 00432 AriR. in lbs. J2 Hand Book of Calculations. THE SCREW. If the pitch of screw and length of lever be given, tvhat poiver will be required to move a given weight. Eule. The power multiplied by the circumference is equal to the weight multiplied by the pitch of screw. Example. If the pitch be f of an inch and the lever 2 feet, how much power must be applied at the end of the lever to raise a weight of 6 tons. 6 tons= 12000 lbs. Xf 3 4)36000 2 x 2=diam. X 12=48 inchesX314=15072)90o00(59f lbs. 75360 146400 135648 10752 Note. Out or Fig. 20 gives a view of the winding path of the endless screw. Fig. 20. Hand Book of Calculations. yj THE WEDGE. The wedge is a pair of inclined planes united by their basest or back to back. The wedge has a great advantage oyer all other mechanical powers in consequence of the way in which the power is applied to it, namely, by percussion, or a stroke, so that by the blow of a hammer or sledge almost any constant pressure is overcome. If instead of moving a load on an inclined plane, the plane itself is moved beneath the load, it then becomes a wedge. All cutting and piercing instruments, such as knives, razors,, scissors, chisels, nails, pins, needles, are wedges. The use of the wedge is to separate two bodies by force or to divide into two a single body. In some cases the wedge is moved by blows; in others it is moved by pressure. The action by simple pressure is to be considered. If the weight rests on a horizontal plane and a wedge be forced under it, when the wedge has penetrated its length, the weight will be lifted a height equal to the thickness of the butt end of the wedge, hence the Itule. yd Hand Book of Calculations. THE WEDGE. KlJLE. The power is equal to the weight, multiplied by the thickness of the wedge, divided by the length of the wedge. Example. A wedge 18 inches in length and 3 inches thick, is employed to lift a weight of 100 lbs. ;. what-pressure must be used ? < -- - T^ • - > • Fig. 21. Now then—The weight = 100 lbs. thickness = 3 inches. divided by 18)300(16§- lbs. Ans. 18 120 108 12 18 If a wedge be 12 inches long and 3 inches thick, and the pressure employed be 100 lbs., what weight will be lifted. This is the method of figuring: The power = 100 Thelength= 12 thickness 3)1200 400 lbs. Ans. Hand Book of Calculations. 75 THE WEDGE. The wedge is generally formed of either wood or metal introduced into a cleft already made to receive it, as shown in Fig. 2->. Wlien two bodies are forced from one another by means of a ivedge. Rule. Multiply the resisting power by half the thickness of the head or back of the wedge, and divide the product by the length of one of its inclined sides. Example. Fig. 22. The thickness of the back of a double wedge is 6 inches, and its length, through the middle is 10 inches: what is the power necessary to separate a substance having a resistance of 150 lbs. ? Now then: 150 lbs. to be overcome. -J- thickness 3 10)450(45 lbs. Ans. In many cases, the utility of the wedge depends upon that which is entirely omitted in the theory, viz. the friction which arises between its surface and the substance which it divides — as in the case of nails, etc. The power generally acts by suc- cessive blows, and is therefore'subject to constant intermission, and but for the friction, the wedge would recoil between the intervals of the blows, with as much force as it had been driven forward, and the object of the labor would be constantly frus- trated. The rules for calculation do not apply to instances like the last described. 76 Hand Book of Calculations. SIZES, STRENGTH, ETC., OE ROPE. By reference to page 48 it will be observed that one of the three classes to which the mechanical powers may be reduced is that of & flexible cord; another name for the cord, or rope, is the funicular machine. Hence, the rules and calculations relating to ropes when used for the purpose of producing power, belong with those relating to the inclined plane and wheel and axle. Tic size of a rope is designated by the circumference meas- ured with a thread; thus a three inch rope measures three inches round. Ropes are made of iron, steel, manila and hemp, all of which, even of the same size, vary greatly in strength, durabil- ity and safety. All the tables given for strength of rope must be more or less modified by the time of service, the quality of material and method of manufacture; the strength of pieces from the same coil may vary one-quarter, and a few months service weakens rope from 20 to 50 per cent. A difference in the quality of hemp may also produce a difference of J in the strength of rope of the same size . Table. Shoiving what weight a hemp rope will bear hi safety. Circumfer- ence. Pounds. Circumfer- ence. Pounds. Circumfer- ence. Pounds. 1 in. 200 3| 2450 64 6050 H'" 312 3| 2*12 51 6612 H 612 •4 3200 6 7200 2 800 4i 4512 6i 7812 n , 10L2 U 4050 6* 8450 n 1250 4f 4512 6f 9112 3 1800 5 5000 7 9800 3i 2111 ! 5i 5512 8 12800 The strength of manila is about \ that of hemp. Hand Book of Calculations. 77 SIZES, STRENGTH, ETC., OF ROPE. To find the strength of ropes. Rule. Multiply the square of the circumference by 200, the product will be the weight in pounds the rope will bear with safety. Example. What weight will a L inch rope bear in safety ? 4x4=16 the square of the girth or circumference. Multiply by 200 3200 Ans. See Table for same result. Table showing what weight a good hemp cable will bear in safety. infer- ence. Pounds. Circumfer- ence. rounds. Circumfer- ence . Pounds. 6. 4:320. 9.50 10830. 13. 20280 6.50 5070. 10. 12000. 13.50 21870 7. 5880. 10.50 13230. 14. 23520 i 7.50 6750. 11. 14520. 14.50 25230 8. 7680. 11.50 15870. 15. 27000 8.50 8670. 12. 17280. 15.50 28830 9. 9720. 12.50 18750. j To ascertain the strength of Cables. KULE. Multiply the square of the circumference, in inches, by 120 and the product is the weight the cable will bear, in pounds, with safety. Example. What weight will a 12 inch cable support with safety ? 12X12 = 144 Multiply by 120 2880 144 L7280 Ans. j8 Hand Book of Calctdations. SIZES, STRENGTH, ETC., OF ROPE. Note. Tables for strength of rope are frequently made to show the breaking strain, and then i to ? taken as the safety limit. In the two tables given the allowance is alrerdy ma'de, but for manila rope a further deduction should be made of %. Wet ropes, if small, are a little more flexible than dry; if large a little Ipss flexible. Tarred ropes are stiffer by about i, and in cold weather some- what more so. The stiffness of ropes increases after a little rest. The girth of a rope and its circumference are the same. IRON AND STEEL WIRE ROPE. The use of a round endless wire rope running at a great velocity in a grooved sheave, in place of a flat belt running on a flat-faced pulley, constitutes the transmissio?i of power by wire ropes. The distance to which this can be applied ranges from fifty feet up to about three miles. Ropes of wire — steel and iron — are made up to three inches in diameter, but the ordinary range in the sizes used is small, being from f diameter to 1^ in a range of 3 to 250 horse power. Two kinds of wire rope are manufactured. The most pliable variety contains 19 wires to the strand, and is generally used for hoisting and running rope: ropes with twelve wires and seven wires in the strand are stiff er, and are better adapted for standing rope, guys and rigging. Wire rope is as pliable as new hemp of the the same strength. It is manufactured either with a wire or rope center; the latter is more pliable than the former and will wear better where there is short bending. Hand Book of Calculations. 19 Table of AVike Rope. Hope of 133 Wires (10 wires to a strand.) Diam. Circumf. Ins. Pounds per foot run. Breaking load, lbs. Minimum diam. of drum in feet. Ins. Iron. Cast steel. Iron. Cast steel. H 6* 8.00 148000 310000 8 9 2 6 6.30 130000 250000 7 8 If 5* 5.25 108000 212000 6.5 7.5 If 5 4.10 88000 172000 5 6 4 H 3.65 78000 154000 4.75 5.5 H 41 3.00 66000 126000 4.5 ii 4 2.50 54000 104000 4 5 4 34 2.00 40000 84000 3.5 4.5 l 3* 1.58 32000 66000 3. 4 i 2* 1.20 23000 50000 2,75 3.75 i 2i 0.88 17280 36000 2.5 3.5 t 2 0.60 10260 28000 2 3 A If 0.44 8540 18000 1.75 2.75 1 U 0.35 6960 15000 1.5 2 t li 0.26 5000 1 Hope of 40 Wires (7 ivires to the strand,) Breaking load, lbs. Circumf. Pounds per foot run. Iron. Cast steel. u 4f 3.37 72000 124000 If *i •>.:: 60000 104000 i* 3f 2.28 50000 88000 i* 3f 1.82 40000 72000 l 3 1.50 32000 60000 i 1.12 •^4600 44000 I 21 0.88 17600 34000 l l Iff H 0.70 15200 28000 t n 0.57 11600 22000 a ii 0.41 8200 16000 i if 0.31 5660 12000 1 <: H 0.23 4260 u 0.19 3300 8000 i 0.16 2760 6000 » r i 0.125 2060 .... Note. In the tables given (Jno. A. Roebling's Sons Coy. ) take I to } M the safe working load. So Hand Book of Calculations. SIZES, STRENGTH, ETC., OF ROPE. Example. What is the safe working load of a 2 inch cast steel wire rope. Now then: For breaking weight see Table = 28, 000 lbs. Divide by 7)28,000— for safety. 4,000 lbs. Ans. GENERAL TABLE. Breaking Strain of Hope. 3,000 lbs. per square inch of section for man] la. 6,000 " « " hemp. 12,000 " " " iron wire. 24,000 " . " " steel wire. EULE. Multiply area of the rope in square inches by the figures in the list for kind of rope. Example. What (by the above rule) is the breaking strain of a 5 inch manila rope. Now then: 5 inch rope = 1 A nearly, diam. The area of 1.6 = 2 inches nearly. Multiply by 3 COO per general rule. 6000 lbs. = Breaking strain. CAUTION. The utmost care must be exercised in the use of any tables or rules for strength and safety of rope of wire or hemp and iron chain — a judgment of materials, amount of wear, and finish of manufacture, as well as the known integrity of the makers — all have to be taken into the calculations. Hand Book of Calculations. 81 <3= YV < T1 =D Q== ^— ^T^ q= ^S$y =D Fig. 23. Fig. 24, Fig. 25. IRON CHAINS. Chains are constructed of round rolled iron formed with open links, Figs. 23 & 24, or with stud links, Figs. 25, 26, 27 & 28. The cuts represent different kinds of chain, viz., Fig. 23 the Circular Link; Fig. 24 the Oval Link; Fig. 25 the Oval Stud Link, with pointed stud; Fig. 26 the Oval Link, with broad headed stud; Fig. 27 the obtuse-angled Stud Link, and Fig. 28 the parallel sided Stud Link. The standard proportions of the links of chains, in terms of the diameter of the bar iron from which they are made, are as iollows : Extreme length. Extreme width. Stud-link, 6 Diameters, 3.6 Close-link, 5 " 3.5 Open-link, 6 « 3.5 Middle-link, 5.5 " 3.5 End-link, 6.5 " 4.5 Example. What is the largest chain of the stud link pattern which can be made out of 1 inch iron? Diam. of bar=l inch. Multiply by 6 length of link. 6 in. and for width, multiply 1 in. by 3 T % =3. 6 inches. Answer. — The links should be 6x3A, or less. 82 Hand Book of Calculations, IRON CHAINS. Fig. 26. d ^^ V! u Fig. 27. Fig. 28. Trautwine's Table of Strength of Chains. Chains of superior iron will require i to % more to break them. Diam of rod of which Weight of chain per ft. run. Breaking strain Diam of rod of which Weight of chain per ft. run. Breaking strain of the links are made, of the chain. the links are made. the chain. Ins. Pds. Pds. Tons. Ins. Pds. Pds. Tons. 3-16 .5 1731 .773 1 10.7 49280 22.00 X .8 3069 1.37 1# 12.5 59226 26.44 5-16 1. 4794 2.14 IX 16. 73114 32.64 H 1.7 6922 3.09 IX 18.3 88301 39.42 7-16 2. 9408 4.20 1* 21.7 105280 47.00 l A 2.5 12320 5.50 1H 26. 123514 55.14 9-16 3.2 15590 6.96 IX 28. 143293 63.97 H 4.3 19219 8.58 lji 32. 164505 73.44 11-16 5. 23274 10.39 2 38. 187152 83.55 X 5.8 27687 12.36 2X 54. 224448 100.2 13-16 6.7 32301 14.42 2/ 2 71. 277088 123.7 H 8. 37632 16.80 2X 88. 335328 149.7 1546 9. 43277 19.32 3 105. 398944 178.1 Ton of 2240 lbs. The weight of close link chain is about three times the weight of the bar from which it is made, for equal lengths. Kane von Ott. — An authority comparing the weight, cost and strength of the three materials, hemp, iron wire and chain iron, concludes that the proportion between the cost of hemp rope, wire rope and chain is as 2: 1:3; and that therefore for equal resistance, wire rope is only half the cost of hemp rope, and a third of the cost of chains. Hand Book of Calculations. 83 DECIMAL FRACTIONS. A decimal fraction is one whose denominator is always 10 or 100 or 1000 or some other power as it is called of 10, but its numerator may be any number. For example to, rho, toW are all three decimal fractions. tV is written .1 and is in value one-tenth of a whole number. T V " .7 " " seven-tenths tU " .01 " " one-hundreth " rcW " .001 " " one-thousandth " " So it will be seen that, in decimals, by placing a figure one place to the right makes it a tenth of what it was before, just as in whole numbers. Thus: 1000 is one thousand. 100 is one hundred. 10 is ten. 1 is one. .1 is one-tenth. .01 is one-hundreth. .001 is one-thousandth. If the fraction have a numerator other than 1. Then it is written thus; iV is expressed .5; tVo- is expressed .27; and tWo is expressed .407. The use of the dot (.) is to separate the whole number from the decimal. The first figure after the decimal point is always tenths; the. second figure always hundreths; and the third figure thous- andths, always decreasing towards the left in a tenfold ratio. 84 Hand Book of Calculations. To bring a decimal fraction to a vulgar fraction. From the foregoing it is plain that all we have to do is to put the given decimal down as a numerator; and for a denominator put down the figure 1, with as many cyphers after it as there are figures in the given decimal; then reduce it to its lowest terms. Examples. Bring .25 to a vulgar fraction, ^fr = A = i Answer. Bring .875 to a vulgar fraction. xWir — Mf = If = ■£. Bring .87500 to a vulgar fraction, tVf o%% = t oo 5 o Hence it will be seen by the last example that annexing a cypher to a decimal does not increase its value at all. You add as many naughts to the right as you please without affect- ing the value of the decimal. To bring a wig ar fraction to a decimal. Attach any number of cyphers to the numerator, and divide this by the denominator, being sure to have a figure for each naught attached. Examples. Bring \ to a decimal. 4)100 .25 Answer. Bring if to a decimal. 4)15.0000 16 4)3.7500 .9375 Examples eok Exercise. 1. Reduce \, i, f to decimals. 2. i( -J, f, f and ■$ to decimals. 3. " If, If, H, A, ye, re, Te and T V to decimals. Engineers sometimes find it convenient to reduce a decimal to a particular vulgar fraction, generally quarters, eighths, six- teenths or thirty-seconds. This is done thus: Multiply the given decimal by the denominator you wish to bring it to, mark off as many decimals from right to left as were given, and whatever number is to the left of the decimal point is the required numerator. Hand Book of Calculations. 85 Examples. How many eights are there in .114 ? .114 8 .912 Answer. None, exactly, but nearly -J. How many sixteenths are there in .198 ? .198 16 1188 198 3.138 Answer, a little over T \. Sometimes in reducing a vulgar fraction to a decimal frac- tion the quotient never comes to an end, but the same number keeps on repeating itself as £= 1.66666, etc., without end. This is called a repeating decimal, is written .16. The dot over the 6 represents that it is a repeater. A decimal fraction derives its name from the Latin word decern, ten, which denotes the nature of its numbers. It has for its denominator, a whole thing as a gallon, a pound, a yard, etc., which articles we suppose to be divided in tenths, hundredths; etc. ADDITION OF DECIMALS. Place the quantities down in such a manner that the decimal point of one line shall be exactly under that of every other line; then add up as in simple addition. Example. Thus:— Add together 36.74, 2.98046, 176.4, 31.0071 and .08647. 36.74 2.98046 176.4 31.0071 .08647 247.21403 86 Hand Book of Calculations. Examples for Exercise. 1. Add together 29.0146, 3146.05, 21.09, 6.20471 and 4.075. 2. " " 17.14, 3.9876, 207.10104, 13.1 and 146. 3. Find the sum of 241.01+13.98+1.90246+176.2007+14.- 125. 4. Find thesum of 27.27+1.125+147.5+16.0125+170.9875. SUBTRACTION OF DECIMALS. Place the lines with decimal point under decimal point, as in Addition. If one line has more decimal figures than another, put naughts under the one that is deficient till they are ^qual, then subtract as in simple subtraction. Examples. From 146.2004 take 98.9876. 146.2004 98.9876 47.2128 Answer. From 4.17 take 1.984625. 4.170000 1.984625 2.185375 Answer. Examples for Exercise. 1. From 46.24 take 17.09864. 2. " .2406 " .1400726. 3. Find the value of 240.-27.7065. 4. " " 19.72461-3.9827. MULTIPLICATION OF DECIMALS. Rule. Multiply as in common multiplication without taking notice of the decimal point, add up and so get the product; then count all the figures after or at the right of the decimal points in the multiplier and multiplicand; count from the right towards the left of the product as many figures as the sum of the decimals just counted; put a decimal point before the fig- ures and you have the answer. Hand Book of Calculations. 8j Example. Multiply 27.62 by 5.713. 27.62 5.713 8286 2762 19334 13810 157.79306 The product first stood 15779306, but as there are altogether five decimal figures in the question, we count five beginning at the last or figure 6, and place a decimal point before the figure that stands in the fifth place. The answer is 157.79306. Example. Multiply .00072 by 0.502. .00072 •0502 144 3600 36144 The product is 36144, but as we have nine places of decimals in the example, we must have the same number of decimals in the product. This is done by putting cyphers to the left of the product. The answer is .000036144. Examples for Exercise. 1. Multiply 724.02 by 23.14. 2. a 23.567 by 3.25. 3. t< .3024 by .3055. 4. a .5052 by .0025. 5. a .0002 by .00101. 6. a 176401 by 76.43. 88 Hand Book of Calculations. DIVISION OF DECIMALS. 1. When the divisor is a whole number: divide as in simple division, only when you come to the decimal point place a point under it in the quotient. Divide 763.5676 by 4. Divide 1537.27 by 8. Examples. 4)763.5676 190.8919 8)1537.27 192.15875 After saying 8 into 47 goes 5 times and 7 over, make this 7 into 70; 8 into 70 goes 8 times and 6 ov T er; 8 into 60 goes 7 times and 4 over; 8 into 40 goes 5 times. Divide 72.6432 by 24. 24 is 6 times 4. Divide by 6, and then the quotient by 4. ( 6)72.6432 24-^ ( 4)12.1072 3.0268 Answer. Divide 7196.148 by 1728. 1728)7196.148(4.1644 etc. or 6912 f 12)7196.148 2841 1728 -{ 12)599.679 1728 I t 12)49.7325 11134 10368 4.1644375 7668 6912 7560 6912 648 Hand Book of Caladations. 8? DIVISION OF DECIMALS. Here after the 8 is brought down it goes 4 times, and the remainder is 756; to this attach an 0, and let it go again, and so on as far as it is thought necessarj^. When the number of decimal figures in the divisor is less than that in the dividend, divide without taking notice of the deci- mals; tnen subtract the number- of decimals in the divisor from the number in the dividend; the remainder will be the number to mark off in the quotient. Examples. Divide 172.4025 by .5. .5)172.4025 34.4805 Here we say 1 from 4 leaves 3 : then mark off 3 decimals in answer. Divide .0041275 by .25. 5).0041275 25 5) 8255 1651 Here it is 2 from 7 leaves 5 : mark off 5 in the quotient; we caunot because there are only 4; then attach a cypher to the left and it becomes .01651 Answer. 172.4025 by .5. First shift the decimal back one place and it becomes 1724.- 025 by .5. Then 5)1724.025 3448.05 which is the same as before. Divide .0041275 by .25. First shift the decimal back two places and it becomes 00.41275. 5). 41275 25 5).08255 ,01651 (1651) which is the same as before. i Fig. 38. Let the figure be the trapezoid, the sides 7 and 5 being parallel; and 3 the perpendicular distance between them. Example. Find the area of the above trapezoid, the parallels being 7 feet and 5 feet, and the perpendicular height being 3 feet. 7 5 2)12 6 And 6x3 = 18 square feet. 102 Hand^ Book of Calculations. MENSURATION. To find the Surface or Envelope of a Cylinder. Eule. Multiply 3.1416 by the diameter, to find the circumference; and then by the height. Example. What is the surface of a cylinder whose diameter is 9 inches, and height 15 inches. 3.1416 9 28. 2 7 44= circumference. 15 14l37-<0 282744 424.1160 area of surface in square inches. To find the Surface or Envelope of a Sphere. Note. The surface of a sphere is equal to the convex surface of the circumscribing cylinder; hence the Eule. Multiply 3.1416 by the diameter of the sphere, and: this again by the diameter; because in this case the diameter is the height of the cylinder; Or multiply 3.1416 by the square of the diameter, of the sphere. Example. What is the surface of a sphere whose diameter is 3 feet? 3.1416 9=3 2 28.2744 area cf surface in square feet. Hand Book of Calculations. ioj CONTENTS OF SOLIDS. To find the Contents of a Rectangular Solid, Rule. Multiply the length, breadth, and height together. "What is the content of a rectangular solid whose length is 5 feet, breadth 4 feet, and height 3 feet ? 5 feet 4 feet 20 square feet of base 3 feet 60 cubic feet 104 Hand Book of Calculations, CONTENTS OF SOLIDS. To find the cubic cantents of a Solid Cylinder. Rule. Find the area of the base, and multiply this by the height or length. Fig. 40. Example. "What are the cubic contents of a cylinder whose diameter is 4 feet, and height or length 7£ feet? 4 4 16 .7854 16 47124 7854 12.5664=area of base in square feet 7.5=height or length in feet 628320 879648 Answer, 94. ; 4b00 cubic feet. Hand Book of Calculations. '°5 To find the Cubic contents of a Sphere. Rule. Multiply .7854 by the cube of the diameter, and then take § of the product. Example. Find the cubic contents of a sphere whose diameter is 5 feet* 5 5 .7854 125 25 5 25 = 5 3 39270 15708 7854 98.1750 2 3)196.3500 Answer, 65. 4500 cubic feet. To find the cubic contents of a Frustrum of a Cone. [A frustrum of a cone is the lower portion' of a cone left after the top piece is cut away. Rule. Find the sum of the squares of the two diameters (d, D), add on to this the product of the two diameters multiplied by .7854, and by one- third the height ("h.") Example. Find the cubic contents of a safety valve weight of the fol- lowing dimensions: — 12" large diameter, 6" small diameter, 4" thick. Now: 144+36+72 X. 7854x1.33 252x.7854xl.33x = ^63.23 Ac. cubic inches. lo6 Hand Book of Calculations. VULGAR FRACTIONS. A fraction means a part of anything. If an apple be cut into eight equal parts each part will be called an eighth of the whole apple, and is written -g-. This eighth is a fraction. If we had 3 or 5 or 7 of these pieces of the apple, we would rep- resent it by §, £> or $, as the case might be. All these are fractions. A yulgar fraction is always represented by two numbers (at least), one over the other and separated by a small horizontal line. The one above the line is always called the Numerator, and the one below the line the Denominator. The denominator tells us how many parts the whole thing has been divided into, and the numerator tells us how many of those parts we have. Thus in the fraction f above, the eight is the denominator, and shows that the apple has been divided into eight equal parts; and three is the numerator, and shows that we have three of those pieces or parts of the apple. A proper fraction is one whose numerator is less than the •denominator, as f or f. An improper fraction is one whose numerator is more than its denominator as f or f . J means more than a whole one, because f must be a whole one. Thus f will be 3 thirds -f- 3 thirds -j- 2 thirds or 2f, and this form of fraction is called a mixed number. Hand Book of Calculations. ioj VULGAR FRACTIONS. 1 . To reduce an improper fraction to a mixed number. Divide the numerator by the denominator; the quotient is the whole number part, and the remainder is the numerator of the frac- tional part. Example: V=2f. Example: ¥=5. Example: ¥=3f. 2. To reduce a mixed number to an improper fraction. Multiply the whole number part by the denominator, and add on the numerator; the result is the numerator of the improper fraction. Example: 2f=V-. Example: 5£=V> Example: 3f=V L . 3. To reduce a fraction to its loivest terms. Divide both numerator and denominator by the same number; if by so doing, there is no remainder. Example. Reduce &. Here 4 will divide both top and bottom without a remainder. Divide by 4. 4)A-|. The meaning of this is, that if you divide a thing into 12 equal parts, and take 8 of them, you will have the same as if the thing had been divided into 3 equal parts and you had 2 of them. Example. Reduce tVsVt to its lowest terms. First divide top and bot- tom by 12 and it becomes tW*; then divide top and bottom again by 12 and it becomes H; 12 will again divide them and it becomes ^, which is its lowest term. Examples for Exercise. Reduce to their lowest terms &; H; H; tVA; iff and If Si 4. To reverse the last rule. To bring a fraction of any de- nominator to a fraction having a greater denominator. See how often the less will go into the greater denominator and multiply both numerator and denominator by it. The result is the required fraction. 108 Hand Book of Calculations. Example. Bring \ to a fraction whose denominator is 8. Here 2 goes in 8, 4 times; then multiply the numerator and denominator of i by 4=|, which is the required fraction. Example. Bring f to a fraction whose denominator is 15. Here 3 goes into 15 five times; then f becomes H. 5. If you have a fraction of a fraction, as -J of \, it is called a compound fraction, and should always be reduced to a simple fraction, by multiplying all the numerators together for a new numerator, and all the denominators together for a new denom- inator; then, if necessary, reduce this fraction to its lowest terms. Example. f of f of f. Eeduce this to a single fraction: 3x2x4=24; and 4x3x9=108. Thus tVf is the fraction. Eeduce this 12 ) t %f=I. CANCELLATION. This is a method of shortening problems by rejecting equal factors from the divisor and dividend. The sign of cancellation is an oblique mark drawn across the face of a figure as fi, 0, t- Cancellation means to leave out; if there are the same num- bers in the numerator and the denominator they are to be left out. Example. f of f of f. Here the 3 in the first numerator and the 3 in the 2d denominator are left out; also 4 of the first denomina- tor and the last numerator, thus: Ans.*x!-X^ There is another way of cancellation. Hand Book of Calculations. log Example. — 3 of J of it of T 9 A=by cancellation, thus: gofjqfgof*. = 7 =± $ W in 3X2X34 204 3 A 34 2 The process is as follows: — The first numerator 2 will go into 8 the denominator of the second fraction 4 times; the denomi- nator of the third fraction 18 will go into 90, the numerator of the last quantity 5 times. The numerator of the second frac- tion 3, will go into the denominator of the first fraction, 3 times; 5 will go into 170, 34 times; 2 will go into 4 twice, and 2 into 14, 7 times, and as we cannot find any more figures that can be divided without leaving a remainder we are at the end, and the quantities left must be collected into one expression. On examination we have 7 left on the top row, this is put down at the end as the final numerator; on the bottom we have 3, 2, and 34, these multiplied together give us 204, which is the final denominator. Rules for Cancelling. 1. Any numerator can be divided into any denominator pro- vided no remainder is left, and vice versa, thus: 5 15 *_of*J = ! $ 1$ 2 2 2. Any numerator and denominator may be divided by the same number, provided no remainder is left, and the decreased value of such numerator and denominator be inserted in the place of those cancelled, — 5 Here 8 is divided by 4, and 20 can also be $ of J20 divided by the same number without leaving ^ 31 an y remainder. Answer *V. ;l Example. n n 17 3x2xi7 102 3 t 2 i io Hand Book of Calculations. Examples for Exercise. Eeduce to their simplest form the fractions: 1. I off off of *. This can be done by cancelling. I of * of ? of \ = I Answer. 2. | of if of i By cancelling. * of ■!* of £ = JL = 1 = I Answer. 10 4x3 12 4 4 3 3. f of J of if. By cancelling. % l of 1 of % = JL = 4 Answer. Jfr >< 10 2x3 6 2 3 ADDITION OF FRACTIONS. Add together -J, f and J-. Here it is evident that the sum will be f or 1^. Hence the rule: Bring all the fractions to the same common denominator, add their numerators together for the new numerator, and reduce the resulting fraction to its simplest form. Examples. What is the sum of i+i=i+l=J Ans. What is the sum of f +|+t +l=¥=l|. To bring fractions having different denominators to fractions having one common denominator. Rule. 1. Put all the denominators down in a row; cancel all that are alike except one; also cancel any that will divide into another one without remainder. 2. If there is any number that will divide two or more of those left, then divide by it, putting down those numbers also that cannot be divided. Repeat this till all the numbers are prime numbers. Hand Book of Calculations. 1 1 1 3. Then multiply all these prime numbers together, and their product by all the divisors: the result will be the com- mon denominator for all the fractions. 4. Lastly, divide this common denominator by the denomi- nator of the first fraction, and multiply its quotient by its nu- merator; the product is the new numerator required. Repeat this for each fraction. Example. \y I, v, I, h h tzj t 5 6, £ and f. Bring these fractions to others having a common denominator. •> 6 3 8 3 8 12 16 8 4 There are 2 figures 3, cancel one of them, there are 3 figures 8, cancel 2 of them; next the 2, 8 and 4 will each go into 16, therefore they must be cancelled; the 6 and 3 also, because they will each divide into 12; then there only remain the 12 and 16, place them as below and divide them by 4. See Article 2 of rule 4)12 16 3 4 Then multiply the 3 by the 4 = 12, and this 12 by the divisor 4 = 48, the common denominator. Lastly, bring each fraction to one having the denominator 48 by rule (article 4) heretofore given. HU'liAAU Ans. « V* i I it « U H if A ft Example. — Add together |, |, I, T V and $■ 2) , 3, 4, 10, 3,2, 5 2 10 3 30 2 Divisor 60 Common denominator 48+40+45+42+30 60 • 112 Hand Book of Calciilations. Examples for Exercise. 1. Add together -J, f, \, and f . 2. " « hi, f, andf. 3. " " #, to, t 5 * and |. 4. " " i, i i, h h it and-^. 5. « « |, I, i, I, A and f . SUBTRACTION OF FRACTIONS. Bring the fractions to others having a common denominator, as in addition, and subtract their numerators. Examples. From -J- subtract f = f =•■£. From -J take i. -^ = I = i. 7 _8. 7-6 l T6 £ T6 " lb ' What is the difference between ■£ of f and J of 1-J ? i of f = |; and i of H = i of | = |. Therefore it is f — § = 0. Which is the greater, J of T 9 o or f of A ? ■J of T 9 d = to ; and f of tit =1. Therefore it is tV and |. 27 and 40 90 Then f of A is the greater by $$. Examples for Exercise. l. l-i; t-i; t-A; t-i. 2 5 3.3 2. 7 1 . T 3 ? ^ 7. X6 7. 3. What is the difference between f of f and li of I ? 4. Which is the greatest, 3f of 2f or 8J- of 1| ? MULTIPLICATION OF FRACTIONS. First bring each fraction to its simplest form; then multiply the numerators together for the new numerator, and the denominators together for the new denominator. Reduce the fraction to its simplest form. Hand Book of Calculations. iij Examples 1. Multiply 1 X 1 t\ ; that is |X '. f 5 TT% fi 1> or by canceling 1 3 *X 21 3 t H ' 4 1 4 The A : cancels into the 16 four times, and the 7 into the 21 three times. Hius 1X3 = \ 3, and 1X4 = 4. Answer f . 2. 2tV of 3| X6iof *. 3 5 10 jT 2 1 5 10 7 2 X $ : 1 7 x ^ x T 1 35 ~~ ~2 = 1 u 3 = 17-J- Answer, Examples for Exercise. 1. Multiply ixi; fxt; AxA. 2. " lfxlt; 5|x3A; 4fx2 T V. 3. " }off offxf of | off. DIVISION OF FRACTIONS. Reverse the divisor and proceed as in multiplication. The object of inverting the divisor is convenience in multi- ply i: After inverting the divisor, cancel the common factors. Examples. J-r-li, that is, f-s-i, reverse the f and it becomes |; then the question is fxf = U =J Ans. 4? of tt-irSi of 3i, that is V of »-*■¥ of ¥; cancelling reduces the dividend to J and the divisor to Y and we have |-5- V, that is H-iV= s ^ =*J Ans. Examples for Exercise 1. *-H; »-i; *+i; ^H. 2. 3±-H 5S-r-2i; ^-2f. U4 Hand Book of Calculations. TABLE CONTAINING THE DIAMETERS, CIRCUMFERENCES AND AREAS OF CIRCLES Diam. Area. Circum. Diam. Area. Circum. 0.0 3.0 7.0686 9.4248 .1 .007854 .31416 .1 7.5477 9.7389 «2 .031416 .62832 .2 8.0425 10.0531 £ .070686 .94248 ':s 8 5530 10.3673 A .12566 1.2566 A 9.0792 10.6814 .5 .19735 1.5708 .5 9.6211 10.9956 .6 .28274 1.8850 .6 10.1788 11.3097 .7 .38485 2.1991 .7 10.7521 11.6239 .8 .50266 2.5133 .8 11.3411 11.9381 .9 .63617 2.8274 .9 11.9456 12.2522 1.0 .7854 3.1416 4.0 12.5664 12.5664 .1 .9503 3.4558 .1 13.2025 12.8805 .2 1.1310 3.7699 .2 13.8544 13.1947 .3 1.3273 4.0841 .3 14.5220 13.5088 .4 1.5394 4.3982 .4 15.2053 13.8230 .5 1.7671 4.7124 .5 15.9043 14.1372 .6 2.0106 5.0265 .6 16.6190 14.4513 .7 2.2698 5.3407 .7 17.3494 14.7655 .8 2.5447 5.6549 .8 18.0956 15.0796 .9 2.8353 5.9690 .9 18.8574 15.3938 2.0 3.1416 6.2832 5.0 19.6350 15.7080' .1 3.4636 6.5973 .1 20.4282 16.0221 .2 3.8013 6.9115 .2 21.2372 16.3363 .3 4.1548 7.2257 .3 22.0618 16.6504 A 4.5239 7.5398 .4 22.9022 16.9646 .5 4.9087 7.8540 .5 23.7583 17.2788 .6 5.3093 8.1681 .6 24.6301 17.5929 1.7 5.7256 8.4823 .7 25.5176 17.9071 .8 6.1575 8.7965 .8 26.4208 18.2212 .9 4** 6.6052 9.1106 .9 27.3397 18.5354 Hand Book of Calculations. JI 5 TABLE— {Continued.) CONTAINING THE DIAMETERS, CIRCUMFERENCES AND AREAS OF CIRCLES. Diam. Area. Circuin. 1 Diam. 1 Area. Circum. 6.0 28.3743 18.8496 10.0 78.5398 31.4159 .1 29.2247 19.1637 .1 80.1185 31.7301 .2 30.1907 19.4779 .2 81.7128 32.0442 .3 31.1725 19.7920 .*3 83.3229 32.3584 .4 33.1699 20.1062 .4 84.9487 32.6726 .5 33.1831 20.4204 ..5 86.5901 32.9867 .6 34.2119 20.7345 .6 88.2473 33.3009 .7 35.2565 21.0487 .7 89.9202 33.6150 .8 36.3168 21.3628 .8 91.6088 33.9292 .9 37.3928 21.6770 .9 93.3132 34.2434 7.0 38.4845 21.9911 11.0 95.0332 34.5575 .1 39.5919 22.3053 ,1 96.7689 34.8717 .2 40.7150 22.6195 .2 98.5203 35.1858 .3 41.8539 22.9336 '.S 100.2875 35.5000 .4 43.0084 23.2478 A 102.0703 35.8142 .5 44.1786 23.5619 .5 103.8689 36.1283 .6 45.3616 23.8761 .6 105.6832 36.4425 .7 46.5663 24.1903 .7 107.5132 36.7566 .8 47.7836 24.5044 .8 109.3588 37.0708 .9 49.0167 24.8186 .9 111.2202 37.3850 8.0 50.2655 25.1327 12.0 113.0973 37.6991 .1 51.5300 25.4469 .1 114.9901 38.0133 .2 52.8102 25.7611 .2 116.8987 38.3274 .3 54.1061 26.0752 .3 118.8229 38.6416 .4 55.4177 26.3894 .4 120.7628 38.9557 .8 56.7450 26.7035 .5 122.7185 39.2699 .6 58.0880 27.0177 .6 124.6898 39.5841 .7 59.4468 27.3319 .7 126.6769 39.8982 .8 60.8212 27.6460 .8 128.6796 40.2124 .9 62.2114 27.9602 .9 130.6981 40.5265 9.0 0:5.0173 28.274:5 13.0 132.7323 40.8407 .1 65.0388 28.5885 .1 134.7822 41.1549 .2 66.4761 28.9027 .2 136.847s 41.4690 .3 07.9291 29. 21 OS .3 138.9291 41.7832 .4 69.3978 29.5310 .4 141.0261 42.0973 .5 70.8822 29.8451 .5 1 143.1388 42.4115 .8 72.3823 30.1593 .6 145.2672 42.7257 .7 73.8981 30.4734 .7 147.4114 43.0398 .8 75.4296 30.7870 .8 149.5712 43.3540 .9 76.9709 31.1018 .9 151.7468 43.6681 n6 Hand Book of Calculations. TABLE— {Continued. ) CONTAINING THE DIAMETERS, CIRCUMFERENCES AND AREAS OP CIRCLES. Diam. Area. Circum. Diam. Area. Circum. 14.0 .1 .2 . .3 .4 153.9380 156.1450 158.3677 160.6061 162.8602 43.9823 44.2965 44.6106 44.9248 45.2389 8.0 .1 .2 .3 .4 254.4690 257.3043 280.1553 263.0220 265.9044 56.5486 56.8628 57.1770 57.4911 57.8053 .5 .6 .7 .8 .9 165.1300 167.4155 169.7167 172.0336 174.3662 45.5531 45.8673 46.1814 46.4956 46.8097 .5 .6 .7 .8 .9 268.8025 271.7164 274.6459 277.5911 280.5521 58.1195 58.4336 58.7478 59.0619 59.3761 15.0 .1 .2 .3 .4 176.7146 179.0786 181.4584 183.8539 186.2650 47.1239 47.4380 47.7522 48.0664 48.3805 19.0 .1 .2 .3 .4 283.5287 286.5211 289.5292 292.5530 295.5925 59.6903 60.0044 60.3186 60.6327 60.9469 .5 .6 .7 .8 .9 188.6919 191.1345 193.5928 196.0668 198.5565 48.6947 49.0088 . 49.3230 49.6372 49.9513 .5 .6 .7 .8 .9 298.6477 301.7186 304.8052 307.9075 311.0255 61.2611 61.5752 61.8894 62.2035 62.5177 16.0 .1 .2 '.S .4 201.0619 203.5831 206.1199 208.6724 211.2407 50.2655 50.5796 50.8938 51.2080 51.5221 20.0 .1 .2 !3 .4 314.1593 317.3087 320.4739 323.6547 326.8513 62.8319 63.1460 63.4602 63.7743 64.0885 .5 .6 .7 .8 .9 213.8246 216.4243 219.0397 221.6708 224.3176 51.8363 52.1504 52.4646 52.7788 53.0929 .5 .6 . i .8 .9 330.0636 333.2916 3o6.5353 339.7947 343,0698 64.4026 64.7168 65.0310 65.3451 65.6593 17.0 .1 .2 .3 .4 226.9801 229.6583 232 3522 235.0618 237.7871 53.4071 53.7212 54.0354 54.3496 54.6637 21.0 .1 .2 '.'S A 346.3606 349.6671 352,9894 356.3273 . 359.6809 65.9734 66.2876 66.6018 66.9159 67.2301 .5 .6 .7 .8 .9 240.5282 243.2849 246 0574 248.8456 251.6494 54.9779 55.^920 55.6062 55.9203 56.2345 .5 .6 .7 .8 ,9 363.0503 366.4354 369.8361 373.2526 376.6848 67.5442 67.8584 68.1726 68.4867 68.8009 Hand Book of Calculations. n 7 TABLE— (Continued.) CONTAINING THE DIAMETERS, CIRCUMFERENCES AND AREAS OF CIRCLES, Diam. Area. Clrcum. Diam. Area. Circum. 22.0 380.1827 69.1150 26.0 530.9292 81.6814 .1 383.5963 69.4292 .1 535.0211 81.9956 .2 387.07-6 69.7434 .2 539.1287 82.3097 .3 390.57< 17 70.0575 .3 543.2521 82.6239 .4 394.0814 70.3717 .4 547.3911 82.9380 .5 397.6078 - 70.6858 .5 551.5459 83.2522 .6 401.1500 71.0000 .6 555.7163 83.5664 .7 404.7078 71.3-142 .7 559.9025 83.8805 .8 408.2814 71.6283 .8 564 1044 84.1947 .9 411.6707 71.9425 .9 568.3220 84.5088 23.0 415.4756 72.2566 27.0 572.5553 84.8230 .1 419.0993 72.5708 .1 576.8043 85.1372 .2 422.7327 72.8849 .2 581.0890 85.4513 .3 426.3848 73.1991 .3 585.3494 85.7655 .4 430.0526 73.5133 .4 589.6455 86.0796 .5 433.7361 73.8274 .5 593.9574 86.3938 .6 487.4354 74.1416 .6 598.2849 86.7080 .7 441.1508 74.4557 .7 602.6282 87.0221 .8 444.8809 74.7699 .8 606.9871 87.3363 .9 448.6273 75.0841 .9 611.3618 87.6504 24.0 452.3893 75.3982 28.0 615.7522 87.9646 .1 456.1671 75.7124 .1 620.1582 88.2788 .2 459.9606 76.0265 .2 624.5800 88.5929 .3 463.7698 76.3407 .3 629.0175 88.9071 .4 467.5947 . 76.6549 .4 633.4707 89.2212 .5 471.4352 76.9690 .5 637.9397 89.5354 .6 475.2916 ' 77.2832 .6 642.4243 89.8495 .7 479.1636 77.5973 .7 646.9246 90.1637 .8 483.0513 77.9115 .8 651.4407 90.4779 .9 486.9547 78.2257 .9 655.9724 90.7920 95.0 490.8739 78.5398 29.0 660.5199 91.1063 .1 494 8087 78.8540 .1 665.0830 91.4203 .2 498.7592 79.1681 .2 669.6619 91.7345 .3 603.7255 79.4823 .3 674 2565 92.0487 .4 506.7075 79.7905 .4 67*.8668 92.3628 .5 5 1 0.7052 80,1106 .5 683.4928 92.6770 .6 514.7185 80.4248 .6 688.1345 92.9911 .7 518.7478 HO. 7389 .7 692.7919 93.3053 .8 522 7924 81.0581 .8 697.4650 93.6195 .9 52(5.8529 81.8672 .9 702.1538 93.9336 n8 Hand Book of Calmlations. TA BLE— (Continued.) CONTAINING THE DIAMETERS, CIRCUMFERENCES AND AREAS OIT CHICLES. 706.8583 711.5786 716.3145 721.0662 725.8336 730.6167 735.4154 740.2299 745.0601 749.9060 754.7676 759.6450 764.5380 769.4467 774.3712 779.3113 784.2672 789.2388 794.2260 799.2290 804.2477 809.2821 814 3322 819.39f0 824.47S6 839.5768 834.6898 839.8185 814.9628 850.1229 855.2986 860.4902 865.6973 870.9202 876.15S8 881.4131 886.6831 891.9688 897.2703 902.5874 Circum. 94.2478 94.5619 94.8761 95.1903 95.5044 95.8186 96.1327 96.4469 96.7611 97.0752 97.3894 97.7035 98.0177 98.3319 98.6460 98.9602 99.2743 99.5885 99.9026 100.2168 100.5310 100.8451 101.1593 101.4734 101.7876 102.1018 102.4159 102.7301 103 0442 103 3584 103.6726 103.9867 104.3009 104.6150 104.9292 105.2434 105.5575 105.8717 106.1858 106.5000 Diam. 34.0 .1 .2 .3 .4 .5 .6 .7 .8 .9 35.0 .1 .2 .5 .6 .7 .8 .9 36.0 !£ .3 .4 .5 .6 .7 .8 .9 S7.0 .1 .2 .3 • 4 .5 .6 .7 .8 .9 Area. Circum. 907.9203 106.8142 913.2688 107.1283 918.6331 107.4425 924.0131 107.7566 929.4088 108.0708 334.8202 108.3849 940.2473 108.6991 945.6901 109.0133 951.1486 109.3274 956.6228 109.6416 962.1128 967.6184 973.1397 978.6768 984.2296 989.7980 995.3822 1000.9821 1006.5977 1012.2290 1017.8760 1023.5387 1029.2172 1034.9113 1040.6212 1046.3467 1052. 0880 1057.8449 1063 6 1 76 10*19.4060 1075.2101 1081.0299 1386.8654 1092.7166 1098X835 1104.4662 1110.3645 1116.2786 1122.2083 1128.1538 109.9557 110.2699 110.5841 110.8982 111.2124 111.5265 111.8407 112.1549 112.4690 112.7832 113.0973 113.4115 113.7257 114.0398 114.3540 114.0681 114.9823 115.2965 115.6106 115.9248 11C. 2389 116.5531 116.8672 117.1814 117.4956 117.8097 118.1239 118.4380 118.7522 119.0664 Hand of Calculations. "3 TA BLE— {Continued. ) CONTAINING THE DIAMETERS. CIRCUMFERENCES AND AREAS OF CIRCLES. Diain. Area. Ctrcum. IMam. Area. Circum. 38.0 .1 o A 1134.1149 1140.0918 1146.0844 1152.0927 1158.1167 119.3805 119.6947 120.0088 120.3230 ! 120.6372 42.0 .1 .2 .3 A 1385.4424 1392.0476 1398.6685 1405.3051 1411.9574 131.9469 132.2611 132.5752 132 8894 133.2035 .5 .6 '.8 .9 1164.1564 1170.2118 1170/2830 1182.3698 1188.4724 120.9513 121.2655 121.5796 121.8938 122.2080 .5 .6 .7 .8 .9 1418.6254 1425.3092 1432.0086 1438.7238 1445.4546 133.5177 133.8318 134.1460 134.4602 134.7743 39.0 .1 .2 .3 A 1194.5906 1200.7246 1206.8742 1213.0396 1219.2207 122.5221 122.8363 123.1504 123.4646 123.7788 43.0 .1 .2 .3 .4 1452.2012 1458.9635 1465.7415 1472.5352 1479.3446 135.0885 135.4026 135.7168 136.0310 136.3451 .5 .6 .: .8 ' .9 1225.4175 1231.6300 1237.8582 1244.1021 1250.3617 124.0929 124.4071 124.7212 125.0354 125.3495 .5 .6 .7 .8 .9 1486.1697 1493.0105 1499.8670 1506.7393 1513.6272 136.6593 136.9734 137.2876 137.6018 137.9159 40.0 .1 .2 .3 -4 1256.6371 1262.9281 1269.2348 1375.5573 1281.8955 125.6637 125.9779 126.2920 126.6062 128.9203 44.0 .1 .2 .3 .4 1520.5308 1527.4502 1534.3853 1541.3360 1548.3025 138.2301 138.5442 138.8584 139.1726 139.4867 .5 .6 .7 .8 .9 1288.2493 1294.3189 1301.0042 1307.4052 1313.8219 127.2345 127.5487 127.8628 128.1770 128.4911 .5 .6 .7 .8 .9 1555.2847 1562.2826 1569.2962 1576.3255 1583.3706 139.8009 140.1153 140.4292 140.7434 141.0575 41.0 .1 .2 .3 .4 1320.2548 1326.7084 1333.1603 1339.6458 1346.1410 128.8058 129.1195 129.4336 129.7478 130.0619 43.0 .1 .2 .3 .1 1590.4313 1597.5077 1604.5999 1611.7077 1618.8313 141.3717 141.6858 142.0000 142.3142 142.6283 .5 .6 .7 .8 .9 1352.0520 1359.1786 1365.7210 1372.2701 1378.8529 130.3761 130.0008 131.0044 131. 31 SO 131.6221 ,5 .0 .7 .8 .9 1025.0705 1683.1255 1040/ 002 1647.4826 1054.0847 142.9425 143.2566 143.5708 143.8849 111.1991 120 Hand Book of Calculations, TABLE— (Continued.) CONTAINING THE DIAMETERS, CIRCUMFERENCES AND AREAS OF CIRCLES. Diam. Area. Circum. Diam. Area. Circum. 46.0 .1 .2 .3 .4 1661.9025 1669.1360 1676.3853 1683.6502 1690.9308 144.5133 144.8274 145.1416 145.4557 145.7699 50.0 .1 .2 .3 .4 1963.4954 1971.3572 1979.2348 1987.1280 1995.0370 157.0796 157.3938 157.7080 158.0221 158.3363 .5 .6 .7 .8 .9 1698.2272 1705.5392 1712.8670 1720.2105 1727.5697 146.0841 146.3982 146.7124 147.0265 147.3407 .5 .6 .7 .8 .9 < 2002.9617 2010.9020 2018.8581 2026.8299 2034.8174 158.6504 158.9646 159.2787 159.5929 159.9071 47.0 .1 .2 .3 .4 1734.9445 1742.3351 1749.7414 1757.1635 1764.6012 147.6550 147.9690 148.2832 148.5973 148.9115 51.0 .1 .2 ]3 .4 2042.8206 2050.8395 2058.8742 2066.9245 2074.9905 160.2212 160.5354 160.8495 161.1637 161.4779 .5 .6 .7 .8 .9 1772.0546 1779.5237 1787.0086 1794.5091 1802.0254 149.2257 149.5398 149.8540 150.1681 150.4823 .5 .6 .7' .8 .9 2083.0723 2091.1697 2^99.2829 2107.4118 2115.5563 161.7920- 162.1062 162.4203 162.7345 163.0487 48.0 .1 .2 .3 .4 1809.5574 1817.1050 1824.6684 1832.2475 1839.8423 150.7984 151.1106 151.4248 151.7389 152.0531 52.0 .1 .2 .*3 .4 2123.7166 2131.8926 2140.0843 2148.2917 2156.5149 163.3628 163.6770 163.9911 164.3053 164.6195 .5 .6 .7 .8 .9 1847.4528 1855.0790 1862.7210 1870.3786 1878.0519 152.3672 152.6814 152.9956 153.3097 153.6239 ,5 .6 .7 .8 .9 2164.7537 2173.0082 2181.2785 2189.5644 2197.8661 164.9336 165.2479 165.5619 165.8761 166.190& 49.0 .1 .2 .3 .4 1885.7409 1893.4457 1901.1662 1908.9024 1916.6543 153.9380 154.2522 154.5664 154.8805 155.1947 53.0 .1 .2 .3 .4 2206.1834 2214.5165 2222.8653 2231.2298 2239.6100 166.5044 166.8186 167.1327 167.4469 167.7610 .5 .6 .7 .8 .9 1924.4218 1932.2051 1940.0042 1947. 818 J 1955.6493 155.5088 155.8230 156.1372 156.4513 156.7655 .5' .6 .7 .8 .9 2^48.0059 2256.4175 2264.8448 2273.2879 2281 7466 ! 168.0752 168.3894 168.7035 169.0177 169.3318 Hand Book of Calculations. 121 TABLE— (Continued.) CONTAINING THE DIAMETERS, CIRCUMFERENCES AND AREAS OF CIRCLES. Diain. Area. Circum Diain. Area. 54.0 .1 t 2 ;3 A 2290.2210 2298.7112 2307.2171 2315.7386 23:4.2759 169.6460 169.9602 170.2743 170.5885 170.9026 58.0 .1 .2 .3 .4 2642.0794 2651.1979 2660.3321 2669.4820 2678.6476 .5 .6 .7 .8 .9 2332.8289 2341.3976 2:549.9820 2258.5821 2867.1979 171.2168 171.5ol0 171.8451 172.1593 172.4735 .5 .6 .7 .8 .9 2687.8289 2697.0259 2706.2386 2715.4670 2724.7112 55.0 .1 .2 .3 .4 2375.8294 2384.4767 2393.1396 2401.8183 2410.ol26 172.7876 173.1017 173.4159 173.7301 174.0442 59.0 .1 .2 .3 .4 2733.9710 2743.2466 2752.5378 2761.8448 2771.1675 .5 .6 .7 .8 .9 2419.2227 2427.9485 2436.6899 2445.4471 2454.2200 174.3584 174.6726 174.9867 175.3009 175.6150 .5 .6 .7 .8 .9 2780.5058 2789.8599 2799.2297 2808.6152 2818.0165 56.0 .1 .2 .3 A 2463.0080 2471.8130 2480.6330 2489.4687 2498.3201 175.9292 176.2433 176.5575 176.8717 177.1858 60.0 .1 .2 .3 .4 2827.4334 2836 8660 2846.3144 2855.7784 2865.2582 .5 .6 .7 ,8 .9 2507.1873 2516.0701 2524.9687 2533.8830 2542.8129 177.5000 177.8141 178.1283 178.4125 178.7566 . > .7 .8 .9 2874.7536 2884.2648 2803.7917 2903.3343 2912.8926 57.0 .1 .2 .3 .4 2551.7586 256.). 7200 2569.6071 2578.6809 2587.6085 179.070.8 170.3849 179 6991 180.01.;.; 180.3274 61.0 .1 .2 .3 .4 2922.4666 293^.0563 2941.6617 2951.2828 2960.9197 .5 .<; .7 .8 .9 2590.7227 2605.7626 2614.8183 2623.8896 2632.0767 180.6416 180.9557 1M.2699 181.5841 181.8982 .5 .6 .7 .8 .9 2970.5722 2980.2405 2989.9244 2999.6241 3009.3395 182.2124 182.5265 182.8407 183.1549 183.4690 183.7832 184.0973 184.4115 184.7256 185.0398 185.3540 185.6681 185.9823 186.2964 186.6106 186.9248 187.2389 187.5531 187.8672 188.1814 188.4956 188.8097 189.1239 189.4380 189.7522 190.0664 190.3805 190.6947 191.0088 191.3230 191.6372 191.9513 192.2655 192.5796 192.8938 193.2079 193.5221 193.8363 191.1504 194.4646 122 Hand Book of Calculations, TABLE— (Continued.) CONTAINING THE DIAMETERS, CIRCUMFERENCES AND AREAS OP CIRCLES. Diam. Area. Circum. Diam Area. Circum. 62 .1 .2 .3 .4 3019.0705 3028.8173 3038.5798 3048.3580 3058.1520 194.7787 195.0929 195.4071 195.7212 196.0354 66.0 .1 .2 .3 .4 3421.1944 3431.5695 3441.9603 3452.3669 3462.7891 207.3451 207.6593 207.9734 208.2876 208.6017 .5 .6 .7 .8 .9 3067.9616 3077.7869 3087.6279 3097.4847 3107.3571 196.3495 196.6637 196.9779 197.2920 197.6062 .5 .6 .7 .8 .9 3473.2270 3483.6807 3494.1500 3504.6351 3515.1359 208.9159 209.2301 209.5442 209.8584 210.1725 63.0 .1 .2 .3 .4 3117.2453 3127.1492 3137.0688 3147.0040 3156.9550 197.9203 198.2345 198.5487 198.8628 199.1770 67.0 .1 .2 .3 .4 3525.6524 3536.1845 3546.7324 3557.2960 3567.8754 210.4867 210.8009 211.1150 211.4292 211.7433 .5 .6 .7 .8 .9 3166.9217 3176.9043 3186.9023 3196.9161 3206.9456 199.4911 199.8053 200.1195 200.4336 200.7478 .5 .6 .7 .8 .9 3578.4704 3589.0811 3599.7075 3610.3497 3621.0075 212.0575 212.3717 212 6858 213.0000 213.3141 £4.0 .1 .2 .3 .4 3216.9909 3227.0518 3237.1285 3247.2222 3257.3.89 201.0620 201.3761 201.6902 202.0044 202.3186 68.0 .1 .2 .3 .4 3631.6811 3642.3704 3653.0754 3663.7960 3674.5324 213.6283 213.9425 214.2566 214.5708 214.8849 .5 .6 .7 .8 .9 3267.4527 3277.5922 3287.7474 3^97.9183 3308. 1049 202.6327 202.9469 203.2610 203.5752 203.8894 .5 .6 .7 .8 .9 3685.2845 3696.0523 3708.8359 3717.6351 3728.45U0 215.1991 215.5133 215.8274 216.1416 216.4556 €5.0 .1 .2 .3 .4 3318.3072 3328.5253 3333.7590 3349.0085 3359.2736 204.2035 204.5176 204.8318 205.1460 205.4602 69.0 .1 .2 .3 .4 3739.2807 3750.1270 3760.9891 3771.8668 3782.7603 216.7699 217.0841 217.3982 217.7124 218.0265 .5 .6 .7 .8 .9 3369.5515 3379.8510 3390.1633 3400.4913 3410.8350 205.7743 206.0885 206.4026 206.7168 207.0310 " 8 .5 i .9 3793.6695 3804.5944 3815.5350 3826.4913 3837.4633 218.3407 218.6548 218.9690 219.2832 219.5973 Hand Book of Calculations. 123 TABLE— (Continued.) •CONTAINING THE DIAMETERS. CIRCUMFERENCES AND AREAS OF CIRCLES, Diani. Area. CIrcum. Diani. Area. Circum. 70.0 .1 .2 .3 .4 3848.4510 3.-59.4544 3870.4736 3881.5084 3892.5590 219.9115 220.2256 220.5398 ! 220.8540 221.1681 74.0 .1 .2 .*3 .4 4300.8403 4312.4721 43^4.1195 4335.7827 4347.4616 232.4779 232.7920 233.1062 233.4203 233.7345 .5 .6 .7 .8 .9 3903.6252 3914.7072 3925.8049 3936.9182 3948.0473 221.4823 221.7964 222.1106 222.4248 222.7389 .5 .6 .7 .8 .9 4359.1562 4370.8664 4382.5924 4394.3341 4406.0916 234.0487 234.3628 234.6770 234.9911 235.3053 71.0 .1 .4 3959.1921 3970.3526 3981.5289 3992.7208 4003.9284 223.0531 223.3672 223.6814 223.9956 224.3097 75.0 .1 .2 .3 .4 4417.8647 4429.6535 4441.4580 4453.2783 4465.1142 235.6194 235.9336 236.2478 236.5619 236.8761 .5 .6 .7 .8 .9 4015.1518 4026.3908 4037.6456 4048.9160 4060.2022 224.6239 224.9380 225.2522 225.5664 225.8805 .5 .6 .7 .8 .9 4476 9659 4488.8332 4500.7163 4512.6151 4524.5296 237.1902 237.5044 237.8186 238.1327 238.4469 72.0 .1 .•J .3 .4 4071.5041 4082.8217 4094.1550 4105.5040 4116.8687 226.1947 226.5088 226.8230 227.1371 227.4513 76.0 i i 4536.4598 4548.4057 4560.3673 4572.3446 4584. 3 i77 238.7610 239.0752 239.3894 239.7035 240.0177 .5 .6 .7 .8 .9 4128.2491 4139.0452 4151.0571 4162.4846 4173.9279 227.7655 228.0796 J 228.3938 228.7079 229.0221 .5 .6 .7 .8 .9 4596.3464 4608.3708 4620.4110 4632.4669 4344.5384 240.3318 240.6460 240.9602 241.2743 241.5885 73.0 .1 .2 .3 .4 4185.3868 4196.8615 4208.3519 4219.8579 4281.3797 229.3363 229.6504 229.9646 230.2787 230.5929 77.0 .1 .2 .3 .4 4656.6257 4668.7287 4680.8474 4692.9818 4705.1319 241.9026 242.2168 242.5310 242.H451 243.1592 .5 .6 .7 .8 .9 4242.9172 4254 4701 1266.0394 4277.6240 4289.2243 230.9071 231.2212 231.5354 281.H395 232.1037 .5 .0 .7 .8 .9 4717.2977 4729.4792 4741.6765 475:', 8894 4766.1 LSI 243.4734 243.7876 244.1017 244.4159 244.7301 124 Hand Book of Calculations. TABLE— (Continued.) CONTAINING THE DIAMETERS, CIRCUMFERENCES AND AREAS OF CIRCLES, Dlani. Area. Circum. Diam. Area. Circum. 78.0 .1 .2 .3 .4 4778.3624 4790.6225 4802.8983 4815.1897 4827.4969 245.0442 245.3584 245.6725 245.9867 246.3009 82.0 .1 .2 .'3 .4 5281.0173 5293.9056 5306.8097 5319.7295 5332,6650 257.6106 257.9247 258.2389 258.5531 258.8672 .5 .6 .7 .8 .9 4839.8189 4852.1584 4864.5128 4876.8828 4889.2685 246.6150 246.9292 247.2433 247.5575 247.8717 .5 .6 .7 .8 .9 5345.6162 5358.5832 5371.5658 5384.5641 5397.5782 259.1814 259.4956 259.8097 260. 1239 260.4380 79.0 .1 .2 .3 .4 4901.6699 4914.0871 4926.5199 4938.9685 4951.4328 248.1858 248.5000 248.8141 249.1283 249.4425 83.0 .1 .2 .3 .4 5410.6079 5423.6534 5436.7146 5449.7915 5462.8840 260.7522 261.0663 261.3805 261.6947 262.0088 .5 .6 .7 .8 .9 4963.9127 4976.4084 4988.9198 5001.4469 5013.9897 249.7566 250.0708 250.3850 250.6991 251.0133 .5 .6 .7 .8 .9 5475.9923 5489.1163 5502.2561 5515.4115 5528.5826 262.3230 262.6371 262.9513 263.2655 263.5796 80.0 .1 .2 .3 .4 5026.5482 5039.1225 5051.7124 5064.3180 5076.9394 251.3274 251.6416 251.9557 252.2699 252.5840 84.0 .1 .2 .3 .4 5541.7694 5554.9720 5568.1902 5581.4242 5594.6739 263.893$ 264.2079 264.5221 264.8363 265.1514 .5 .6 .7 .8 .9 5089.5764 5102.2292 5114 8977 5127.5819 5140.2818 252.8982 253.2124 253.5265 253.8407 254.1548 .5 .6 .7 .8 .9 5607.9392 5621.2203 5634.5171 5647.8296 5661.1578 265.4646 265.7787 266.0929 266.4071 266.7212 81.0 .1 .2 .3 .4 5152.9973 5165.7287 5178.4757 5191.2384 5204.0168 254.4690 2*4.7832 255.0973 255.4115 255.7256 85.0 .1 .2 .3 .4 5674.5017 5687.8614 5701.2367 5714.6277 5728.0345 267.0354 267.3495 267.6637 267.9779 268.2920 .5 .6 .7 .8 .9 5216.8110 5229.6208 5242.4463 5255.2876 5268.1446 256.0398 256.3540 256.6681 " 256.9823 257.2966 .5 .6 • .7 .8 .9 5741.4569 5754.8951 5768.3490 5781.8185 5795.3038 268.6062 268.9203 269.2345 269.5486 269.8628 Hand Book of Calculations. ™S TABLE— {Continued. ) CONTAINING THE DIAMETERS, CIRCUMFERENCES AND AREAS OF CIRCLES. Diam. Area. Circum. Diam. Area. Circum. 86.0 .1 .2 .3 .4 5808.8048 5822.3215 5835.8539 5849.4020 5862.9659 270.2770 270.4911 270.8053 271.1194 271.4336 90.0 .1 .2 .3 .4 6361.7251 6375.8701 6390.0309 6404.2073 6418.3995 282.7433 283.0575 283.3717 283.6858 284.0000 .5 .6 .7 .8 .9 5876.5454 5890.1407 5903.7516 5917.3783 5931.0206 271.7478 272.0619 272.3761 272.6902 273.0044 .5 .6 .7 .8 .9 6432.6073 6446.8309 6461.0701 6475.3251 6489.5958 284.3141 284.6283 284.9425 285.2566 285.5708 87.0 .1 .2 .3 .4 5944.6787 5958.3525 5972.0420 5985.7472 5999.4681 273.3186 273.6327 273.9469 274.2610 274.5752 91.0 .1 .2 .3 .4 6503.8822 6518.1843 6532.5021 6546.8356 6561.1848 285.8849 286.1991 286.5133 286.8274 287.1416 .5 .6 .7 .8 .9 6013.2047 6026.9570 6040 72oD 6054.5088 6068.3082 274.8894 275.2035 275.5177 275.8318 276.1460 .5 .6 .7 .8 .9 6575.5498 6589.9304 6604 3268 6618.7388 6633.1666 287.4557 287=7699 288.0840 288.3982 288.7124 88.0 .1 .2 .3 .4 6082.1234 8095.9542 6109.8008 6123.6631 6137.5411 276.4602 276.7743 277.0885 277.4026 277.7168 92.0 .1 .2 .3 .4 6647.6101 6662.0692 6676.5441 6691.0347 6705.5410 289.0265 289.3407 289.6548 289.9690 290.2832 .6 .6 •• .8 .9 6151.4348 6165.3442 6179.2693 6193.2101 6307.1666 278.0309 278.3451 278.6563 278.9740 279.2876 .5 .6 .7 .8 .9 6720.0630 6734.6008 6749.1542 6763.7233 6778.3082 290.5973 290.9115 291.2256 291.5308 201.8540 89.0 .1 .2 .8 .4 6221.1389 6235.1268 6349.1304 6263.1498 6277.1849 270. GO 17 279.9159 280.2301 280 5442 ,8584 93.0 .1 .2 .3 .4 6792.9087 6807.5250 0822.150!) 6836.8046 6851.4680 292.1681 292.4823 292.7964 293.1106 293.4248 .6 i .9 6391.2356 6305.3021 6319.3843 6333. 6347.5958 281.1725 281.4867 281.8009 282.1150 282.4292 .5 ,6 i .9 6866.1471 6880.8419 6895.5524 6910.2786 0025.0205 293.7389 294.0531 294.3072 294. OH 14 294.9956 126 Hand Book of Calculations. TABLE— (Concluded. ) CONTAINING THE DIAMETERS, CIRCUMFERENCES AND AREAS OF CIRCLES* Diam. Area. Circum. Diam. Area. Circum. 94.0 6939.7782 295.3097 97.0 7389.8113 304.7345 .1 6954.5515 295.6239 .1 7405.0559 305.0486 .2 6969.3106 295.9380 .2 7420.3162 305.3628 .3 6984.1453 296.2522 .3 7435.5922 305.6770 .4 6998.9658 296.5663 .4 7450.8839 305 9911 .5 7013.8019 296.8805 .5 7466.1913 306.3053 .6 7028.6538 297.1947 .6 7481.5144 306.6194 .7 7043.5214 297.5088 .7 7496.8532 306.9336 .8 7058.4047 297.8230 .8 7521.2078 307.2478 .9 7073.3033 298.1371 .9 7527.5780 307.5619 95.0 7088.2184 298.4513 98.0 7542.9640 307.8761 .1 7103.1488 298.7655 .1 7558.3656 308.1902 .2 7118.1950 299.0796 .2 7573.7830 308.5044 .3 7133.0568 299.3938 .3 7589.2161 308.8186 .4 7148.0343 299.7079 .4 7604.6648 309.1327 .5 7163.0276 300.0221 .5 7620.1293 309.4469 .6 7178.0366 300.3363 .6 7635.6095 309.7610 .7 7193.0612 300.6504 .7 7651.1054 310.0752 .8 7208.1016 300.9646 .8 7666.6170 310.3894 .9 7223.1577 301.2787 .9 7682.1444 310.7035 96.0 7238.2295 301.5929 99.0 7697.6893 311.0177 .1 7253.3170 301.9071 .1 7713.2461 311.3318 .2 7268.4202 302.2212 .2 7728.8206 311.6460 .3 7283.5391 302.5354 .3 7744.4107 311.9602 .4 7298.6737 302.8405 .4 7760.0166 312.2743 .5 7313.8240 303.1637 .5 7775.6382 312.5885 .6 7328.9901 303.4779 .6 7791.2754 312.9026 .7 7344.1718 303.7920 .7 7806.9284 313.2168 .8 7359.3693 304.1062 .8 7822.5971 313.5309 .9 7374.5824 304.4203 .9 7838.2815 313.8451 100.0 7853.9816 314.1593 Hand Book of Calculations. I2J GEOMETRY. Geometry is one of the oldest and simplest of sciences; it may be defined as the science of measurement; Mensuration as already briefly outlined in this work, belongs properly under this division. Geometry is the root from which all regular mathematical calculations issue. It has claimed the best thought of practi- cal men from the times of the Greeks and Eomans two thous- and years ago; they derived their knowledge of the science from the Egyptians, who in turn were indebted to the Chaldeans and Hindoos in times beyond any authentic history; hence it was under the operations of the laws explained in geometry, that the pyramids of Egypt and the temples of Greece, were con- structed, as well as the engines of war and appliances of peace of ancient times. The elementary conceptions of geometry are few. 1. A point. 2. A line. 3. A surface. 4. A solid, and 5. An angle. From these definitions, as data, a vast number of mathemati- cal calculations have been deduced ; of which a few of the most; elementary will be explained and illustrated in this work; but these few will repay the attention of the student as the mutual relation between practical engineering and geometry is very intimate indeed — as will be apparent. 128 Hand Book of Calculations. GEOMETKICAL DEFINITIONS. A point is mere position, and has no magnitude. A line is that which has extension in length only. The extremities of lines are points. A surface is that which has extension in length and breadth only. A solid is that which has extension in length, breadth and thickness. An angle is the difference in the direction of ^^ two lines proceeding from the same point. ^^ Lines, Surfaces, Angles and Solids constitute the different kinds of quantity called geometrical magnitudes. Parallel lines are lines which have the same — direction; hence parallel lines can never meet, — however far they may be produced; for two lines taking the same direction cannot approach or recede from each other. An Axiom is a self-evident truth, not only too simple to require, but too simple to admit of demonstration, A Proposition is something which is either proposed to be done, or to be demonstrated, and is either a problem or a theorem. A Problem is something proposed to be done. A Theorem is somethi tig proposed to be demonstrated. A Hypothesis is a supposition made with a view to draw from it some consequence which establishes the truth or false- hood of a proposition, or solves a problem. A Lemma is something which is premised, or demonstrated, in order to render what follows more easy. A Corollary is a consequent truth derived immediately from some preceding truth or demonstration. A Scholium is a remark or observation made upon something going before it. A Postulate is a problem, the solution of which is self-evident. Hand Book of Calculations. I2Q Examples of Postulates. Let it be granted — I. That a straight line can be drawn from any one point to any other point: II. That a straight line can be produced to any distance, or terminated at any point; III. That the circumference of a circle can be described about any center, at any distance from that center. ABBREVIATIONS. The common algebraic signs are used in Geometry, and it is necessary that the student in geometry should understand some of the more simple operations of algebra. As the terms circle, angle, triangle, hypothesis, axiom, theorem, corollary, and definition are constantly occurring in a course of geometry, they are abbreviated as shown in the following list: Addition is expressed by . . . . -j- Subtraction - Multiplication " " . . . .X Equality and Equivalency are expressed by = Greater thari. is expressed by . . . , > Less than. st " . . < Thus B is greater than A, is written . . B> A B is less than A " . . . B. -% Fig. 73. Problem V. To draw a parallel line through a given point, Fig. 74-. With a radius equal to the given point from the given line A B, describe the arc D from B taken consider- ably distant from C. Draw L ^ the parallel through <7to touch the arc D. Fig 74, 140 Hand Book of Calculations, Second Method, F-g. 75. From A, the given point describe A/ !» / f Fig. 75. the arc F D, cutting the given line at F ; from F with the same radius, describe the arc E A, and set off F D equal to E A. Draw the parallel through the points A D, Note. When a series of parallels are required perpendicular to a base line A B, they may be drawn as in figure 76 through points in the base line set off at the required distances apart. This method is convenient also where a succession of b parallels are required to a given line C D, for the per- pendicular may be drawn to it, and any number of parallels may be drawn on the perpendicular. Problem VI. To divide a line into a number of equal parts, Kg. 77. To divide the line A B into, say 5 parts. From A and B draw parallels A C, B D on opposite sides; set off any con- venient distance four times (one less than the given number), from A on A C, and on B on B F; join the first on A O to the fourth on B D, and so on. The lines so drawn divide *A B required. Fig. TO. as d\ Second Method, Fig. 78. Draw the * line at A C, at an angle from A, set off say, five equal parts; draw B 5, and draw parallels to it from the other B points of division in A C. These par- allels divide A B as required. Hand Book of Calculations. 141 GEOMETRICAL PROBLEMS. Problem VII. Upon a straight line to draw an angle equal to a given angle, Fig. 79. Let A be the given angle and F G the line. With any radius from the points A and F, describe arcs D E, I H % cutting the sides of the angle A and the line FG. Fig. 79. Set off the arc I H equal to D E and draw F H. /'is equal t-o A as required. Problem VIII. To bisect an angle, Fig. 80. Let A B be the angle; on the center C cut the sides at A B. On % A and B as centers describe arcs cut- ting at D dividing the angle into two equal parts. The angle Fig. 80. Problem IX. To find the center of a circle or of an arc of a circle. First for a circle, . % /' Fig. 81. Draw the chord A B, *\ bisect it by the perpendicu- lar C D, bounded both ways by the circle; and bisect C D for the center G. S' Fig. 81. 142 Hand Book of Calculations. GEOMETRICAL PROBLEMS. Problem X. Through two given points to describe an arc of a circle with a given radius. Fig. 82. On the points A and B as centers, with the given radius, describe arcs cutting at C; and from C, with the same radius, describe an arc A B as re- quired. Fig. 82. Second, for a circle or an arc, Fig. 83. Select three points A B Cm the circum- ference, well apart; with the same radius; describe arcs from these three points cut- ting each other, and draw two lines D E, F G, through their intersections accord- ing to Fig. 68. The point where they cut is the center of the circle or arc. Fig. 83. Problem XL To describe a circle passing through three given points, Fig. 83. Let A B C be the given points and proceed as in last problem to find the center 0, from which the circle may be described. Note. This problem is variously useful; in finding the diameter of a large fly wheel, or any other object of large diameter when ouly a part of the circumference is accessible; in striking out arches when the span and rise are given, etc. Problem XII. To draw a tangent to a circle from a given point in the circumference, Fig. 84. From A set off equal segments A B, A D, join B D and draw A E, parallel to it, for the tan- gent. Fig. 84. Ha mi Book of Calculations. *43 Us Problem XIII. To draw tangents to a circle from points without it, Fig. 85. From A with the radius A C, describe an arc B D, and from with a radius equal to the di- ameter of the circle, cut the arc at B D; join B C, C D y cutting the circle at E F, and draw A E, A F, the tangents. *J> Fig. 85. Problem XIV. Between two inclined lines to draw a series of circles touching these lines and touching each other, Fig. 86. Bisect the inclination of the given lines A B, C D by the line JV" 0. From a point P in this line draw the perpendicular P B to the line A B, and on P describe the circle B D s touching the lines and cutting the center [meat/?. From/? draw E /'perpendicular to the center line, cutting A B at F, and from F de- scribe an arc E G, cut- ting A B at G. Draw G H parallel to B P, giving H, the center of the next circle, to be described with the radius HE, and so on for the next circle I N. Problem XV. To construct a triangle on a given base, the sides being given. First. An equilateral triangle, Fig. 87. On the ends cf a given base A B, with A B as a radius describe arcs cutting at C, and draw A (\ C B. Fig. 86. Pig. 87. j 44 Hand Book of Calculations. Fig. 88. Second. A triangle of un- equal sides, Fig. 88. On either end of the base A D with the side B as a radius, describe an arc; and with the side G as a radius on the other end of the base as a A center describe arcs cutting the arc at E. Join A E, D E. B Note. c This construction may be used for finding the position of a point G or E at given distances from the ends of a base, not necessarily to form a triangle. Pkoblem XVI. To construct a square rectangle on a given straight line. First. A square, Fig. 89. On the ends A B as centers, with the line A B as radius, describe arcs cutting at G; on G describe arcs cutting the others at D E; and on D and E cut these at F G. Draw A F B G and join the in- Fi g . 89. tersections H L Second. A rectangle, Fig. 90. On the base E F draw the perpen- diculars EH, F G, equal to the height of the rectangle and join G H. Fig 90. Problem XVII. To construct a parallelogram of which the sides and one of the angles are given, Fig. 91. Draw the side D E equal to the given length A, and set off the other side D F equal to the other length B, forming the given angle G. From E with D F as radius, describe an arc, and from F, with the radius D E cut the arc at G. Draw F G, E G. Or, the remain- ing sides may be drawn as parallels to D E. D F. -** ! H ■ I \ / \ \ , / c*» -'"' B *Lr "N V '' V v. \ s \ A \ / ' \ \ E ' pi v • A r ^4 c T\ Fig. 91 Ha?id Book of Calculations, W5 GEOMETRICAL PROBLEMS. Problem XYIII. To (/escribe a circle about a triangle, Fig. 92. Bisect two sides A B, A C of the triangle at E F, and from these points draw perpendiculars cutting at K. On the center K, with the radius K A draw the circle .4 B C. Fig. 92. Problem XIX. To describe a circle about a square, and to inscribe a square in a circle. Fig. 94. First. To describe the circle. Draw the diagonals A B, C D of the square, cutting at E; on the center E with the radius E A describe the circle. Second. To inscribe the square. Draw the two diameters A B, G D at right angles and join the points A B, C D to form the square. Note. In the same way a circle may be described about a triangle. Problem XX. To inscribe a circle on a square, and to describe a square about << circle, Fig. 94. First. To inscribe the circle. Draw the diagonals A B, CD of the square, cut- ting at E; draw the perpendicular E Fto one side, and witn the radius E Z 7 describe the circle. Second. To describe the square. Draw two diameters A B, C D at right angles, 1* and produce them; bisect the angle/) ED Fig. M- at the center by the diameter F G, and through Fasx6 G draw perpendiculars A C, B D, and join the points A D and B where they cut the diagonals to complete the square. i^6 Hand Book of Calculations. Problem XXI. To inscribe a circle in a triangle, Fig. 95. Bi- sect two of the angles A G of the triangle by lines cutting at D; from D draw a perpendicular D E to any side, and with D E as radius describe a circle. Problem XXII. To inscribe a pentagon in a circle, Fig 96. Draw two diameters A G, B D at right angles cutting at 0; bisect A at E, and from E with radius E B cut the circumference at G H and with the same radius step round the circle to / and K; join the points to form the pentagon. B k^ ^7- n^ ' / \ *l\ ; 1 / / \ A.r f / r rc^ \ Fig~97. Problem XXIY. To inscribe a hexagon in a circle, Fig. 98. Draw a diameter A G B; from A and B as centers with the radius of the circle A G, cut the circumference at D, E, F, G, and draw A D, D E, etc., to form the hexagon. Note. The points D E, etc., may be found by stepping the radius (with the dividers) six times round the circle. D Fig. 96 Problem XXIII. To construct a hexagon upon a given straight line, Fig. 97. From A and B the ends of the given line describe arcs j D cutting at G; from G with the radius G A describe a circle. With the same radius set oh* the arcs AG, G Fund BD, DE. Join the points so found to form the hexa- gon. Fig. 98. Hand Book of Calculations. 147 Problem XXV. To describe an octagon on a given straight line, Fig. G9. Produce the given line A B both ways and draw perpendiculars ^4 E, B F; bisect the external angles A and B by the lines A H, B C, which make equal to A B. Draw C D and H G parallel to A E and equal to A B; from the center G D, with the radius A B, cut the perpendiculars at E F, and draw E F to complete the hexa- gon. A B Fig. 99. Problem XXYJ. To convert a square into an octagon, Fig. 100. Draw the diagonals of the square cutting at e; from the corners A B C D, with A e as radius, describe arcs cutting the sides at g, h, etc.; and join the points so found to complete the octairon. m i Fig. 100. Problem XXVII. To inscribe an octagon in a circle, Fig. 101. Draw two diameters A C, B D, at right angles; bisect the arcs A B, B C, and C at e, f, etc. , to form the octagon. d Fig. 101. Problem XXVIII. To describe an octagon about a circle, Fig. 102. Describe a square about the given circle A B, draw perpendiculars h, k and 6' to the diagonals, touching the circle, to form the octagon. Or, the points h, h, etc., may be found by cutting the sides from the corners. R t Fig. 102. 14S Hand Book of Calculations. GEOMETRICAL PROBLEMS. Problem XXIX. To describe an ellipse wlien the length and breadth are given, Fig. 103. On the center G, with A E as radius, cut the axis A B at F and G, the foci; fix a couple of pins into the axis at F and 67, and loop on a thread cr cord upon them equal in length to the axi A B, so as when stretched t.> reach the extremity of the conjugate axis, as shown in dot-lining. Place a pencil or drawpcint inside the cord, as at H, and guiding the pencil in this way, keeping the cord equally in tension, carry the pencil round the pins F, G, and so describe the ellipse. Note. The ellipse is an oval figure, like a circle in perspective. The line that divides it equally in the direction of its great length is the transverse axis, and the line which divides the opposite way is the conjugate axis. Second Method. Along the straight edge of a piece of stiff paper mark off a distance a c equal to A C, half the transverse axis; and from the same point a distance a b equal to C D, half the conju- gate axis. Place the slip so as to bring the point b on the line A B of the transverse axis, and the point c on the line D E; Fig. 104. and set off on the drawing the position of the point a. Shifting the slip, so that the point b travels on the transverse axis, and the point c on the conju- gate axis, any number of points in the curve may be found, through which the curve nay be traced. Hand Book of Calculations. /./y PROPORTION, OR THE RULE OF THREE. The Bide of Three, or proportion, is one of the most useful in the whole range of mathematics; a rule by which, when three numbers are given, a fourth number is found, which hears the same relation to the third as the second does to the first ; or a fourth number is found bearing the same relation to the third as the first does to the second. Proportion is the relation of one quantity to another. This relation may be expressed either by the difference of the quantities or by their quotient. In the former case it is called arithmetical relation, in the latter geometrical proportion or simple proportion. Proportion differs from ratio. Ratio is properly the rela- tion of two magnitudes or quantities of one and the same kind ; as the relation of 5 to 10 or 8 to 16. Proportion is the sameness or likeness of two such relations ; thus 5 is to TO, as 8 to 16, or, A is to B as C is to D ; that is, 5 bears the same relation to 10 as 8 does to 16. Hence we say such numbers are in proportion. A proportion is an equality of ratios, and as ratio is the measure of the relations of two like quantities, It is determined by dividing the first quantity by the second. Thus: The ratio of 6 to 3 is 2, or of $8 to $2 is 4. 8 : 2 = 16 : 4, is a proportion. The equality is generally indicated by writing :: between the ratios, thus: 8 : 2 :: 16 : 4 indicates a proportion and is read, eight is to two, as, sixteen is to four. Note. The Bign : is an abbreviated form of -j- and has a like mean- ing. In proportion, three quantities are given, the problem being to rind the fourth, as 2 is to 4 as 6 is to what number — expressed thus: 2 :1::(J: ? Now then: multiply the second term by the third term and divide this product by the first term. _ 4x6 = 24. 24-f-2=12, which is the required number. 150 Hand Book of Calculations. Rule. Of the three given numbers, place that for the third term which is of the same kind with the answer sought. Then consider, from the nature of the question, whether the answer will be greater 01 less than this term. If the answer is to be greater, place the greater of the two numbers for the sec- ond term, and the less number for the first term; but if it is to be less, place the less of the two remaining numbers for the second term, and the greater for the first; and in either case multiply the second and third terms together, and divide the product by the first for the answer,, which will always be of the same denomination as the third term. Note, If the first and second terms contain different denominations, they must both be reduced to the same denomination; and compound numbers to integers of the lowest denomination contained in it. Example. 2. If 40 tons of iron cost , what will 130 tons cost ? Tons. Dor -ons. 40 :4:0:.130 130 13500 450 40)5850|0 1462.5 dollars Ans. The Terms of a ratio are the two numbers compared. The Antecedent is the first term of a ratio, the Consequent is the second term, and the two terms together are called a Couplet. An Inverse Ratio is the ratio formed by inverting the terms of a given ratio. Thus 8:9 is the inverse of 9:8. Each term of a proportion is called a Proportional ; the first and fourth terms are called Extremes ; and the second and third term, Means. When the two means are the same num- ber, that number is a Mean Proportional between the two extremes. Hand Book of Calculations. l 5 l THERMO-DYNAMICS. Heat is treated generally in scientific books under the head- ing of tliermo-clynamics. This term is made from two Greek words which signify, lectively heat-power; i. e., the power which is produced by the combustion or burning of fuel. Without heat there would be no steam engine or steam boiler, and no engineer nor fireman; hence, the consideration of its nature and management and the calculations connected with its employment stand first in the order of subjects, heat, water, steam, now to be explained in their relations to mathematical calculations. HEAT. When two bodies in the neighborhood of each other have un- equal temperatures, there exists between them a transfer of heat from the hotter of the two to the other. The tendency towards an equalization, or towards an equilib- rium of temperatures is universal, and the passage of heat takes place in three ways: 1. By radiation. 2. By conduction. 3. And by convection, or carriage from one place to another by heated currents. RADIATION OF HEAT. Radiant heat traverses air without heating it. By means of a simple apparatus it has been ascertained that the proportion of the total he^>t radiated from different combus- tibles are as follows : Radiated heat from wood, ..... nearly £ 80" 180 100 80 FREEZING >' \r w O 32° - < ( > < s Fig. 107. The on these scales is called Zero, all above the is plus, while all below is minus; thus a temperature of 10° below Zero is written —10° Hand Book of Calculations. 1 57 THE MEASUREMENT OF HEAT. In Fahrenheit's, the space between Freezing and Boiling points is divided into 180 d3grecs, Freez- ing being 32° and Boiling 212°. In Centigrade, Freezing is and Boiling 100°, the space being thus divided into 100 parts; hence its name. In Reaumur's, Freezing is and Boiling is 80°, the space bemg thus divided into 80 parts. In the above diagram it is readily seen that 180° F — 100° C — 80° E; from which we can get the rules for comparing degrees of temperature on one scale with the degrees on. another. In the Cent. grade Thermometer, used in France and in most other countries in Europe, C corresponds to melting ice, and 100° to boiling water. From the freezing to the boiling point there are 100 c In the Reaumur Thermometer, used in Russia, Sweden, Tur- key, and Egypt, 0° corresponds to melting ice, and 80° to boil- ing water. From the freezing to the boiling point there are 80°. Centigrade temperatures are converted into Fahrenheit tem- peratures by multiplying (he former by 9 and dividing by 5, and adding 32° to the quotient; and conversely, Fahrenheit temperatures are converted into Centigrade by deducting 32° and taking Hhs of the remainder. Reaumur degrees are multiplied by § to convert them into the equivalent Centigrade degrees; conversely, libs cf the number of Centigrade degrees give their equivalent in Reaumur degrees. Fahrenheit is converted into Reaumur by deducting 32° and taking l ths of the remainder, and Reaumur into Fahrenheit by multiplying by J, and adding 32° to the product. PYROMETERS. Pyrometers are employed to measure temperatureH above the Innliug point of mercury, about 676° F. I $8 Hand Book of Calculations. THERMODYNAMICS. PYROMETEK. Wedgwood's pyrometer, invented in 1782, was founded on the property possessed by clay of contracting at high tempera- tures. The apparatus consists of a metallic groove, 24 inches long, the sides of which converge, being half an inch wide above and three-tenths below. The clay is made up into little cylinders or truncated cones, which fit the commencement of the groove after having been heated to low redness; their sub- sequent contraction by heat is determined by allowing them to slide from the top of the groove downwards till they arrive at a part of it through which they cannot pass. In Darnell's pyrometer the temperature is measured by the expansion of a metal bar inclosed in a black-lead earthenware case, which is drilled out longitudinally to T 3 o inch in diameter and 7k inches deep. A bar of platinum or soft iron, a little less in diameter, and an inch shorter than the bore, is placed in it and surmounted by a porcelain index 1 J inches long, kept in its place by a strap of platinum and an earthenware wedge. When the instrument is heated, the bar, by its greater rate of expansion compared with the black-lead, presses forward the index, which is kept in its new situation by the strap and wedge until the instrument cools, when the observation can be taken by means of a scale. Another means of estimation, based on the melting points of metals and metallic alloys, is applied simply by suspending in the heated medium a piece of metal or alloy of which the melt- ing point is known, and if necessary, two or more pieces of different melting points, so as to ascertain, according to the pieces which are melted and those which continue in the solid state, within certain limits of temperature, the heat of the fur- nace. A list of melting points of metals and metallic alloys is given in a subsequent chapter. Hand Book of Calculations. i $g THERMODYNAMICS. LUMINOSITY AT HIGH TEMPERATURES. The luminosity or shades of temperature have been observed by M. Pouillet by means of an air-pyrometer to be as follows: — Shade. Temperature. Fahrenheit. Nascent Red. 977° Dark Red 1292 Nascent Cherry Red 1472 ( Jherry Red 1652 Bright Cherry Red 1832 Very Deep Orange 2012 Bright Orange 2192 White 2372 u Sweating" White. . 2552 Dazzling White 2732 A bright bar of iron, slowly heated in contact with air, assumes the following tints at annexed temperatures (Claudel): Fahrenheit. 1 . Cold iron at about 54° •l. Yellow at 437 3. Orange at 473 4. Red at 509 5. Violet at 531 * 6. Indigo at 550 7. Blue at 559 8. Green at 630 9. Oxide Gray (gris d'oxyde) at 753 l6o Maud Book of Calculations. HORSE POWER. A horse power is merely an expression for a certain amount of work and involves three elements — 1 . Force. 2. Space; and 3. Time. If the force be expressed in pounds, and the space passed through in feet, then we have a solution of and the meaning for, the term foot-pound ; from which it will be seen that a foot-pound is a resistance equal to one pound moved upwards one foot. The work done in lifting thirty pounds through a height of fifty feet is fifteen hundred foot-pounds. Now if the foot-pounds required to do a certain amount of work involve a specified amount of time during which the work is performed and if this number of foot-pounds is divided by the equivalent number representing one horse power (which number will be dependent upon the time) then the resulting number will be the horse power developed. Example. Suppose the 1500 foot-pounds just spoken of to have acted in one second. To find the horse power divide by 550, and the result will be the horse power. A horse power is 33,000 foot-pounds, or, in other words 33,000 pounds lifted one foot in one minute, or one pound lifted 3:?,000 feet in one minute, or 550 lbs. lifted one foot in one second. HORSE POWER OF THE STEAM ENGINE. The capacity for work of a steam engine is expressed m the number of horse powers it is capable of developing. Hand Book of Calculations. 161 HORSE POWER. There are three kinds of horse power spoken and written about which engineers should learn to distinguish — these are 1. Xominal, 2. Indicated, and 3. Effective. Engineers and others who have not carefully considered the matter, often use the above as synonymous — or having the same meaning; but in this they are wrong, as the meaning is very far from the same. Nominal horse power is an expression which is gradually going out of use, and is merely a convenient mode of describing the dimensions of a steam engine for convenience of makers and purchasers of steam engines. Indicated horse power is the true measure of the work done within the cylinder of the steam engine and is based upon no assumptions, but is actually calculated. The things necessary to be known m order to make the figures are: 1. The diameter of the cylinder in inches. 2. Length of stroke m feet. 3. The mean effective pressure — that is, the average pressure of the steam on the piston during the full length of the stroke; and 4. The number of revolutions per minute. Effective horse power is the amount of work which an engine is capable of performing, and is the difference between the indicated horse power and horse power required to drive the engine when it is running unloaded. Note. Engine rating, guarantees, etc., are usually based upon tbe indicated horse power, owing to the ease and accuracy with which it can be determined. Rule for calculating horse power. 1. Find the area of the piston. 2. Find the j)ressure in lbs. on the piston, by yrraltiplying the area by the pressure per square inch. r 62 Hand Book of Calculations. HORSE POWER. 3. Find the space in feet travelled through by the piston per minute, by multiplying the length of stroke in feet by twice the revolutions per minute. 4. Find the foot-pounds done by the engine per minute, by multiplying the pressure in lbs. (2) by the travel in feet (3). 5. Find the H. P. by dividing the foot-lbs. (4) by 33000. Example. What is the horse power of an engine, the diameter of the cylinder being 16 inches, length of stroke 24 inches, revolu- tions per minute 120, and the average pressure of steam per square inch on the piston 45 lbs. ? ft. in. Diam. 1 4 .7854 stroke 2 feet. 12 256 No. of strokes 240 16 inches. 47124 480 travel of 16 39270 piston in — 15708 feet* 96 .6 201.0624=area. 45 lbs. pressure. Diam. 256 squared. 10053120 8042496 9047. 8080 ^pressure on piston in lbs* 9047.8 lbs. 480 feet. 7238240 361912 33000 ( 3)43429440 foot pounds. 11)14476480 131.6044 An& 131 T 6 T horse power. Hand Book of Calculations. 163 HORSE POWER. Second Rule. Instead of putting the work down step by step, it is more readily worked as follows: (1) Square the diameter of the piston, (2) multiply it by the length of stroke in feet, (3) by -twice the revolutions, (4) by the pressure per square inch, (5) and by .0000238. Example. 2. What is the horse power of an engine, the diameter of cylinder being 13 inches, length of stroke 12 inches, revolutions per minute 300 and the average pressure per square inch on the piston 67 lbs.? 13 diameter in inches. 13 39 13 169 1 lensrth of stroke in feet. 169 . 600 twice the revolutions, or number of strokes. 101400 67 lbs. pressure per square inch. TO9800 608400 6793800 238 constant multiplier. 54350400 S14 13581 161.0024400 horse power. Ans. Jfiltfj TT. P. J 6^. Hand Book of Calcinations* HORSE POWER. Note. This rule is the same as the first one except that a constant multiplier is used. This is found by dividing .T854 by 33000, which equals .0000238. This very considerably shortens the calculation as will be observed by comparing the two examples given under the rules, i. e., the 16"x24 // and the 13"xl2" engine. Example for Peactice. 3. What is the horse power of an engine, the diameter of cylinder being 24 inches, length of stroke CO inches, revolu- tions per minute GO, and the average pressure being 43to 3 o? Ans. : 4. What is the horse power of an engine, the diameter of cylinder being 6 inches, length of stroke 9 inches, revolutions per minute 400, and the average pressure of steam on the piston being 45 lbs. ? Ans. : 5. What is the horse power of a pair of engines, the diameter of the cylinder being 12 inches, length of stroke 30 inches, rev- olutions per minute 90, and the average pressure of steam 38 lbs. ? Ans. : IMPORTANT. In the rules given for estimating the power of different forms of the steam engine, it will be observed that the area of the piston is a quantity winch is known to a close fraction; the piston speed infect per minute is assumed to be correct, and the reduc- tion of the foot-pounds to horsepower by dividing by 33,000 is the same in all rules; but the average pressure of steam is the doubtful part of the calculation. The reasons f r this are various, such as the varied expan- sion, wire-drawing, steam working against itself in the cylin- der, condensation, cramping of the exhaust, etc., etc. These defects, as well as the average pressure of the steam {and the combined pressure of the steam and the vacuum) are clearly shown by the Indicator; hence, while the rules given are suffi- ciently close for every -day practice, it is important to bear in mind that all questions of the power of engines are much more accurately determined by the Indicator. Hand Book of Calculations. 165 HORSE POWER. Third Rule. To compute horse power of engines by a short rule process. Rule. Multiply the diameter of the cylinder (in inches) by itseK and by the distance in feet travelled by the piston per mmute and divide by 42,000. The quotient gives the horse power for each lb. of mean effective pressure. Example. 6. What is the horse power of a pair of 24x24 horizontal high pressure engines with 120 revolutions per minute. 24x24 = 576 2X120= 240 23040 1152 42,000)138240(3.291 126 2 for 2 cylinders. 122 6.582 84 384 378 60 42 Ans. 6.582 for each lb. of mean effective pressure. To obtain total horse power, multiply by the number of lbs. of steam. Fourth Rule. . To compute horse power of engine, by mean effective press- ure, as shown by indicator. i66 Hand Book of Calculations. HORSE POWER. Rule. Multiply the area of the piston, by the mean effective press- ure per square inch, by the stroke in feet, by the number of strokes per minute (out and back being two strokes), and divide by 33,000. The quotient is the horse power. Example. 7. What is the horse power of an engine, 9 inch bore of cylin- der, 20 inch stroke, and 60 revolutions per minute, with mean effective pressure of 42 lbs. Area of 9 inch per table = 636 M. E. P. 42 1272 2544 33 ,000)534240(16^ H. P. 33 26712 Stroke 20" 204 198 12)534240 ► 62 44540 J revolutions 120 534240 Rule for calculating Horse Power oe Condensing Engines. Proceed as in Rules and Examples on pages 161 and 162, add- ing 10 lbs. to the average pressure on the piston for the increased efficiency — above friction, etc.- for approximate advantage, gained by the use of the condenser. Note. Engine builders add from tV to \ to the nominal horse power of their engines as an approximation of the increased efficiency of their high pressure engines when fitted with condensing apparatus. Hand Book of Calculations. i6j HORSE POWER. Rule for finding the Horse Power of the Compound Engine. Find the horse power of each cylinder separately, then add the two powers together; or in other words treat the two cylin- ders as you would two separate engines. Note. In a compound engine a second cylinder of three or four times the piston area is added, called the low pressure cylinder, into which the exhaust steam of the first or high pressure cyl- inder, instead of being thrown away, is passed and made to yield a further amount of* work. The additional work thus obtained is roughly proportional to the mean effective pressure in the low pressure cylinder multiplied by the difference m area of the two pistons. By this means the power of the engine is increased, and the steam, when finally exhausted, is at a pressure so low that little or no unused work remains in it. Example. 8. What is the H. P. of a compound engine whose high press- are cylinder is 16 inches in diameter, and low pressure 27 inches, with 16 inch stroke, and 250 revolutions per minute. Estimate the mean effective pressure as 70 lbs. for the 16 inch cylinder and 12 lbs. for the 27 inch cylinder. Now: 16" area — 201 27" area = 572.5 inches. 70 pressure 12 lbs. 14070 6870.0 feet traverse 666 feet traverse 666 84420 41220 84420 41220 84420 41220 33,000)9370620(284 nearly. 33,000)4575420(139 nearly. Add 284 high pressure cylinder, non-condensing. 139 low " " " 423 total (newly). 1 68 Hand Book of Calculations. HORSE POWER OF THE COMPOUND ENGINE. Example. 9. What is the horse power of a compound engine, diameter of high pressure cylinder 27-J inches, and mean effective pressure throughout the stroke 36.95 lbs. per square inch. Diameter of low pressure cylinder 48 in., and mean effective pressure 7.35' lbs. per square inch, length of stroke 2 feet 6 inches, and revo- lutions per minute 75 ? Find the H. P. of each cylinder separately, then add the two- powers together. H. P. of high pressure=249.395 &c. do. low do. =151.139 &c. Combined H. P. =400.534 POWER OF THE LOCOMOTIVE. The power of the locomotive is measured at the point where* the wheel touches the rail, and is equal to the load the locomo- tive could lift out of a pit by means of a rope passed over a pulley, and attached to the outside of the tire of one of the driving wheels. The term horse power is not generally used in speaking of the locomotive, as the difference in the work between it and the stationary engine is so great. The power of the locomotive resides in two places, first, the adhesive power which is derived from the weight on the driving wheels, and their friction and adhesion on the rails — it being remembered that the adhesion varies with the weight on the drivers and the state of the rail. Second, the tractive power of the locomotive, which is that derived from the pressure of the steam on the piston applied to the cranks and revolving wheels. Rule to find the House Power of a Locomotive. Multiply the area of the piston in square inches by 2 (there being 2 cylinders to each engine) also by two-thirds the boiler pressure as shown by the gauge; also by the number of revo- lutions per minute; also by the feet traversed by the piston- Divide by 33,000 and the amount will be the horse power. Hand Book of Calculations. i6cy HORSE POWER OF THE LOCOMOTIVE. Example. 10. The locomotive " A. G. Darwin " has 19 inch cylinders, 24 inch stroke, driving wheels 68 inches in diameter. It makes (with ease) 60 miles per hour, with boiler pressure 150 lbs. per square inch. Area of piston 283.5x2 = 567 sq. inches, f boiler pressure = 100 lbs. * To find revolutions per minute. 56700 1 mile in 1 min. = 5280 ft. = 68360 300 revolutions. divide by rim of driving wheel 68" diam. =213.6 inches. Now: 17010000 213.6-^-63.360 = 300 nearly. 4 33,000)68040000 2062 horse power. Note. This engine weighs 120,000 lbs., of which. 72,000 are on the drivers. By actual count it carried its own immense weight added to that of 8 heavy cars a mile in 47 seconds and several miles in 55, 58 and 60 seconds. Example. 11. What isth j power of a locomotive with cylinders 19 inches bore, 30 inch stroke, diameter of drivers 72 inches, running speed 40 miles per hour, boiler pressure 160 lbs. per square inch. Area of piston 19 inches (per table page 116) = 283.5 inches. Steam pressure, say in this example, six-tenths of 160 = 96 lbs. Now tlien, per Rule: 283 ' 5X9 — ° X5X12 -I359f nominal H. P. 33000 3 Note. This must not be taken for the locomotive power, for it is not* This is the power which the engine would develope if the tires on the drivers were as gear wheels fitted to cogs on the track, so that they could not slip, and if the boiler could supply the ■team. I jo Hand Book of Calculations. HORSE POWER OF THE STEAM FIRE ENGINE. Hence the rule given is merely approximate, as nothing can .be told about the internal workings of the engines without a test carefully performed with the indicator — and the results of the latter are modified by the tractive or adhesive power. Note. Colb urn's Rule for Calculating Power of Locomotives takes the full pressure of one cylinder instead of the mean average pressure of two. KlJLE FOR ELNDIKG THE HORSE POWER OF THE STEAM FlRE E^GIKE. Multiply the area of the piston by the average steam press- ure in pounds per square inch; multiply this product by the travel of the piston in feet per minute, divide this product by 33000; seven-tenths of the quotient will be the horse power of the engine. Example. 12. Area of piston 8 inch diameter=50.27 in. (see page 115). Stroke 8 inches, revolutions 150 per minute =200 feet travel. Average steam pressure 100 lbs. Now then: 50.27 area. 100 lbs. steam. 5027.00 200 travel of piston in feet. 33,000 foot-lbs.)100540000(30.4 99 30.4 154 .7 132 • 21to 8 o horse powa*; Hand Book of Calculations. iji THE HORSE POWER OF THE STEAM BOILER. A great deal of trouble lias arisen from the application of this unit, the horse power, to the measurement of the capacity of steam boilers, for the boiler is only one part in the power- producing system. It furnishes the force. It is the magazine where is accumulated and stored the pressure obtained from the combustion of the coal. Xow some engines use steam much more economically than others, and a boiler which could furnish steam to develope power at the rate of 100 horses with the best of these, might not be able to do 40 horse power with the worst. Hence comes the question, what is the horse power of the boiler? To meet the complication which arose from this cause a standard of evaporation of thirty pounds of water per hour, from feed water of 100° Fahrenheit into steam at 70 lbs. gauge pressure, has been adopted as a horse power for steam boilers Some engines can develope a horse power on this number of pounds of steam per hour, others cannot, while many require more hence it is about the present average capacity. Both engineers and steam users have received this standard with unanimity, and so-called " boiler tests," are based upon their evaporative capacity, expressed in lbs. of water per hour. Square feet of heating surface is frequently used to express the horse power. This is figured from the number of square feet tf boiler and tube surface exposed to the action of the fire; but this method is not at all accurate, as the same amount of exposed surface will under some circumstances produce several times as much steam as others, but for the ordinary tubular boiler fifteen square feet of heating surface has been held to be equal to one horse power. The extent of the heating surface of a boiler depends on the length and diameter of the shell and the number and size of the tubes or flues. When setting boilers in brick work, the practice is to rack in the side walls to the shell a few inches belov/ the water line, and thus limit the heating surface. It is customary in calculat- ing the heating surface cf the shell, to consider that two-thirds, of it is exposed to the action of heat. n? Hand Book of Calculations. HORSE POWER OF THE STEAM BOILER. It is also customary to consider that the entire surface of the tubes or flues is exposed to the action of heat. From the table given below, the heating surface of any boiler can be obtained with ease. Table of Heating Surface of Boilers. Diameter of Boil- er Inches. Two thirds heating surface of shell per ft. of length. Diameter of 1 uhe or flue. Inches. Whole external heating surface per It. of length. 24 ■ 4.19 2 .524 26 4.54 *l .589 28 4.89 H .655 32 5.59 3 785 34 5.93 H .850 36 6.28 H .916 40 6.98 4 1.05 42 7.33 « 1.18 44 7.68 5 1.31 48 8.38 7 1.83 50 8.73 8 2.09 54 9.42 10 2.62 56 9.77 11 2.88 60 10.47 13 3.40 66 11.52 16 4.19 72 12,57 20 5.24 Rule for estimating Horse Power of Horizontal Tub- ular Steam Boilers. Mud the square feet of heating surface in the shell, heads and tubes, and divide by 15 for the nominal horse power. Example. What is the heating surface of a boiler having head 72 inches diameter, shell 18 feet long, with 100 tubes, 3 J inches in diam- eter ? Now then: 12)72 inches. 6 6 36 6 feet diameter. .7854 36 47124 23562 28.2744 square feet of surface in one head. Hand Book of Calculations. iyj HORSE POWER OF THE STEAM BOILER. 12)3.50000 .29107 feet diameter of the tube. 3.1416 175002 29167 116663 29167 87501 .916310472 feet circumference of 1 tube. 18 7330483936 916310472 16.4935S8656 square feet surface of 1 tube. 1649.3589 equals square feet surface in all the tubes. 3.1416 6 18.8496 circumference of shell in feet. 18 1507968 188496 339.2928 square feet of surface of shell. 2-thirds of shell 3)678.5856 226.1952 1649.358:) tubes. 15)1875.5541 125. horse power, neglecting the heads. I f 4 Hand Book of Calculations. ARITHMETICAL PROGRESSION. Arithmetical Progression is a series of numbers which succeed each other regularly, increasing or diminishing by a constant number or common difference: As 1, 3, 5, 7, 9, &c. ) increasing series. 15, 12j 9, 6, 3, &c. ) decreasing series. The numbers which form the series are called terms. The f^st and the last term are called the extremes, and the others are called the means. In arithmetical progression, there are five things to be con- sidered, viz.-: 1, The first term. 2, The last term. 3, The common difference. 4, The number of terms. 5, The sam of all the terms. These quantities are so related to each other, that when any three of them are given, the remaining two can be found. Given the first term, the common difference, and the number of terms, to find the last term. Rule. Multiply the number of terms, less one, by the common difference, and to the product add the first term. Example. What is the 20th term of the arithmetical progression, whose first term is one, the common difference -J? 20-1=19 and 19x|=9|; and 9^+1=10^ Ans. Given the number of terms and the extremes, to find the corn-, mon difference. Rule. Divide the difference of the extremes by one less than the number of terms. Hand Book of Calculations. 175 ARITHMETICAL PROGRESSION. Example. The extremes are 3 and 29, and the number of terms 14, required the common difference. 29—3=26; and 26-r- 13=2. Ans. Given the common difference and extremes, to find the number of terms. Rule. Divide the difference of the extremes by the common differ- ence, and to the quotient add one. Example. The first term of an arithmetical progression is 11, the last term 88, and the common difference 7. What is the number of terms? 88 — 11 = 77; and 77^7 = 11; 11+1=12. Ans. Given the extremes and the number of terms, to find the sum of all the terms. Rule. Multiply half the sum of the extremes by the number of terms. Example. How many times does the hammer of a clock strike in 12 hours. 1+12=13 = the sum of extremes. Then 12x(13--2) = 78. Ans. If6 Hand Book of Calculations. WATER. There are some underlying natural laws and other data relating to water which every engineer should thoroughly understand. Heat, ivaier, steam, are the three properties with which he h^s first to deal; like the first rounds of a ladder they lead to higher lessons. The scientific head under which water is treated is Hydros- tatics. Hydraulics is one division of the general subject and means flowing water, not so applicable to steam engineering as the first term, which broadly means the science of fluids, of which water is the principal example. WATER AS A STANDARD. There are four notable temperatures for water, namely, 32° F., or 0° C. = the freezing point under one atmosphere. 39°. 1 or 4° = the point of maximum density. 62° or 16°. 66 = the standard temperature. 212° or 100° = the boiling point, under one atmosphere. The temperature 62° F. is the temperature of water used in calculating the specific gravity of bodies, with respect to the gravity or density of water as a basis, or as unity. Weight of one cubic foot of Pure Water. At 32° F. = 62.418 pounds. At39°.l =62.425 " At 62° (Standard temperature) =62.355 At 212° =59.640 « The weight of a cubic foot of water is, it may be added, about 1000 ounces (exactly 998.8 ounces), at the temperature of maximum density. Ha iid Book of Calculations. if J WATER. The weight of water is usually taken in round numbers, for ordinary calculations, at 62.4 lbs. per cubic foot, which is the weight at 52°.3 F.; or it is taken at 62^- lbs. per cubic foot, where precision is not required, equal to L j™ lbs. The weight of a cylindrical foot of water at 62° F. is 48.973 pounds. Weight of one cubic inch of Pure Water. At 32° F. = .03612 pound, or 0.5779 ounce. At37°.l =.036125 " " 0.5780 " At 62 =.03608 " " 0.5773 " or 252.595 grains. At 212° =.03451 " " 0.5522 " The weight of one cylindrical inch of pure water at 62° F. is • 02833 pounds, or 0.4533 ounce. Volume of one pound of Pure Water. At 32 e F. = .016021 cubic foot, or 27.684 cubic inches. At 39°.l = .016019 " " 27.680 At 02 =.016037 " " 27.712 " At 212° =.016770 " "28.978 The volume of one ounce of pure water at 62° F. is 1.732 cnbic inches. iy8 Hand Book of Calculations. WATER. SEVERAL PRINCIPLES IMPORTANT TO KNOW. Water is practically non-elastic. A pressure lias been applied of 30,000 lbs. to the square inch and the contraction has been found to be less than one-twelfth. Experiment appears to show that for each atmosphere of pressure it is con- densed 47J millionth of its bulk. Water at rest presses equally in all directions. This is a most remarkable property — solids pressing only downward, or in the direction of gravity — the upward direction of the press- ure of water is equal to that pressing downwards, and the side pressure is also equal. A given pressure or bloiv impressed on any portion of a mass of ivater confined in a vessel is distributed equally through all parts of the mass; for example, a plug forced inwards on a square inch of the surface of water, is suddenly communicated to every square inch of the vessel's surface, however large, and to every inch of the surface of any body immersed in it. It is this principle which operates with such astonishing- effect in hydrostatic presses, of which familiar examples are fuund in the hydraulic pumps, by the use of which boilers are tested. By the mere weight of a man's body when leaning on the extremity ci a lever, a pressure may be produced of upwards of 2000 tons; it is the simplest and most easily applicable of all contrivances for increasing human power, and it is only limited by want of materials of sufficient strength to utilize it. The surface of water at rest is horizontal. A familiar exam- ple of this may be noted in the fact that the water in a battery of boilers also seeks a uniform level, no matter how much the cylinders may vary m size. Hand Book of Calculations. fjg WATER. T7ie pressure on any particle of water is proportioned to its depth below the surface, and as the side pressure is equal to the downward pressure, calculations on this principle are easily made. The pressure on a square foot at different depths are approximate, as in the following table. ' Depth in feet Pressure on sq. foot. Depth in feet. ♦ 56 Pressure on sq. foot. 8 500 lbs. 3500 lbs. 16 1000 " 64 4000 " n 1500 " 72 4500 " 3*3 2000 " 80 5000 " 40 2500 " 88 5500 " 48 3000 " 96 6000 <• 1 mile, or 5/280 feet, 330,000 lbs. 5 mf.es, 1,650,000 " This table is based upon an allowance of 62£ lbs. of water to the cubic foot, hence 8 feet X 62 \ = 500, etc. Wafer rises to the same level in the opposite arms of a recurved tube, hence water will rise in pipes as high as its source; this is the principle of carrying the water of an aque- duct through all the undulations of the ground. Any quantity of water, however small, may he made to lal- an e any quant it u, however great. This is called the Hydros- tatic Paradox, and is sometimes exemplified by pouring liquids into cask- through long tubes inserted in the bung holes. As - "ii as i he cask is full and the water rises in the pipe to a cer- tain height the cask bursts witn violence. JN OTE. The in']' regsure, dne to these peculiar natural laws, renders it accessary for the engineer t > make due allowances on the strength of pipes and v< ss< Is used for containing or con- voy i: i8o Hand Book of Calctilations. PUMPS. The action of a pump is as follows: The piston or plunder by moving to one end, or out of the pump cylinder, bares the space it occupied, cr passed through, to be filled by something. As there is little or no air therein a partial vacuum is formed unless the supply to the pump is of sufficient force to follow the piston or plunger of its own accord. If this is not the case, however, as it is where the water level from which the pump obtains its supply is below the pump itself, there being a par- tial vacuum produced, the atmospheric pressure forces the water into the space displaced by the plunger or piston, continuing its flow until the end cf stroke is reached. The water then ceases to flow in, and the suction valve of the pump closes, forbidding the water flowing back the route it came. The piston or plunger then begins to return into the space it has just vacated, and which, has become filled with water, and immediately meets with a resistance which would be insurmountable were the water not allowed to go somewhere. Its only egress is by raising the discharge valve by its own pressure, and passing out through it. This discharge valve is in a pipe leading to the bciler, and in going out of the cylinder by that route the water must overcome boiler pressure and its own friction along the passages. Water is inert and cannot act of itself; so it must derive this power to flow into the feed pipe and boiler from the steam acting upon the steam piston of the pump. The steam piston and pump piston are at the two ends of the same red. Therefore the steam pressure exerted upon the steam piston will ba exerted upon the pump piston direct. Hand Book of Calculations, 181 PUMPS. There being no mechanical purchase in favor of the steam piston, it must have the greater area, otherwise one pressure would balance the other, and the pump would refuse to move. Fcr this reason, all boiler feed pumps have larger steam than water cylinder: generally, at least, 40 per cent, larger. Water will flow into a boiler when the head or height from which it obtains its pressure is greater than the height of a water column represented by the pressure within the bciler, or where the pressure from the water works supply exceeds the pressure of the steam in the boiler. Example. What horse power will be required to deliver 1,000 imperial gallons per minute against a pressure of 80 pounds per square inch, suction lift 20 feet, allowing 20 per cent, friction ? One-pound pressure is eepial to a head of 2.31 feet. Hence the total head to which the water is to be lifted will be 2.31 X 80+20=^)1.8 feet. An imperial gallon weighs ten pounds and the horse power required I raise 1,000 imperial gallons per minute against a ire and suction lift equal to a head of 204.8 feet will be 10X1,000X304.8 ^.^ H p and 33,000 02.00+20^ allowance for friction=74.47 II. P. Rtjxe to find Tin; water capacity of a steam pump per HOUR. 1 . Find the contents of the pump in cubic inches, by multi- plying i la- area by the inches in strokes and by the fraction, it Is tall. 2. Find the cubic inches of water pumped per hour, by mul- tiplying the contents of the pump by the strokes per minute and by 00 minutes. 3. Find the number of cubic feet of water by dividing the cubic inch< s bv 1728. 182 Hand Book of Calculations. PUMPS. Example. How many cubic feet of water will he pumped in an liour by a 'pump 6 inches in diameter and 10 inch stroke, making 60 strokes per minute, the pump being f full each stroke. Now, then: 6X6= 7854 36 diameter squared. 47124 23562 28.2744 10 length of stroke. 282.7440 60 strokes. 16964.6400 60 minutes. 1017^78.4000 3 y i full. 4)3053635. 2000^ 12)783408.8000 1728 <{ 12)63617.4000 12)5301.4500 441.7875 Ans. 441f gallons nearly. To find the pressure in pounds per square inch of a column of tvater. ItULE. Multiply the height of the column of water in feet by .434. Hand Book of Calculations. i8j PUMPS. Example. What is the pressure at the bottom of a column of water 410 feet high ? 440 434 1760 1320 1760 190. tVA Ans. 191 lbs. nearly Xote. The correctness of this calculation is found by multiplying 1 191 by 2.31. To find the height of a column of amter, in feet, the pressure being known. Rule. Multiply pressure by the pressure shown on gauge by 2.31. Example. If pressure shown on the gauge is 95 lbs. to square inch, what is the height of the column of water ? 2.31 95 1155 2079 219.45 Ans. 220 ft, nearly. Note. The correctness of this result is proved by multiplying 220 by .434. To find the horse power necessary to pump water to a given height. Rule. 'Multiply the total weight of the water in pounds, by the height in feet, and divide the product by 33,000. 1 8//. Hand Book of Calculations. PUMPS. Example. What power is required to elevate 90,800 lbs. of water 45 ft. ? 90,800 45 454000 363200 33,000)4086000(12311 horse power. Note. This calculation allows the water to be raised in one minute. To raise the same amount in 60 minutes would require ^Vth the power. Ans. Nearly 3 horse power. To find quantity of ivater pumped in one minute running at 100 feet of piston speed per minute. KULE. Square the diameter of the water cylinder in inches, and multiply by 4. The answer will be in gallons. Example. What quantity of water will be pumped by a 4 inch water cylinder with piston travelling 100 feet per minute. 4 inch diam.=16 inches. 4 64 Ans. in gallons. Note. This is an approximate, not an exact quantity, as will be found by figuring the exact area of the piston 12.566 inches X 100x12 inches^-231=652 6 3 4 T ; but the rule is nearer than the average practice of pumps, owing to leakage of air, etc. To find the capacity of a water cylinder of a steam pump in gallons. EULE. Multiply the area in inches by the length of stroke (this gives the capacity in cubic inches). Next divide by 231 (which is the cubical contents of a U. S. gallon in inches) and the pro- duct is the capacity in gallons. Hand Book of Calculations. ■ 185 PUMPS. Example. What is the capacity of a cylinder 9 inches diameter and 10 inch stroke. 9 inch diameter=see table 63.617 area. 10 Ans. in cubic inches* 231)636.170(2. T V\r gallons. 462 1741 1617 1247 1155 92 To find the steam pressure required when the diameter of steam cylinder, diameter of pump cylinder, and water pressure are given. . Rule. Multiply the area of the pump in inches by the pressure of water in pounds perscpiare inch, and divide the product by the ana of cylinder, plus one-fourth for friction. Example. 6 inches diameter of pump cylinder, 100 pounds pressure per square inch. 70.88 area of steam cylinder. " ' =39.8+i=-±9.7, nearly 50 lbs. pressure per « I Loo square inch. To find the diameter of cylinder required for a direct-acting steam pump. Rule. Multiply the area of pump-bucket or ram in inches by the pressure of water in pounds per square inch, and divide the product by the pressure of steam in pounds per square inch, and add one-fourth to one-half for friction. i86 Hand Book of Calculations. PUMPS. Example. 6 inches diameter of pump, 100 lbs. water pressure per square inch, 50 lbs. steam pressure. 6 inches diameter = 28.27 inches area, Kr . =56.54 inches, area of steam cylinder, 00 add i=56.54-fl4.13=7Gi67=9| inches diameter of steam ■cylinder, nearly. To find the load on a pump: Kule. Multiply the area of pump in inches by the weight of the column of water in pounds per square inch. Example. Pump 3 inches in diameter;, depth of well 30 feet. 3 inches diameter = 7.06 inches area, 30X44 . 1^ — = lo.2 lbs. pressure per square inch, 7.06x13 = 91.78 lbs. total pressure on pump. To find the total amount of pressure that can le exerted in a steam pump. Kule. Multiply the area of the steam piston by the steam pressure. Example. What is the total amount of pressure in a pump cylinder 8-J inches diameter and 80 lbs. of steam ? 8-J inch diam.=per table 56.745 area. 80 lbs. to sq. in. 4539.600 Ans. 4.539 T 6 o lbs. Hand Book of Calculations. 1S7 PUMPS. To find the resistance of the water in the water cylinder. Rule. Multiply the area of the water piston in inches, by the press- ure of water in pounds per square inch. Example. What is the resistance to be overcome in a 7 inch diameter piston, working against a pressure of 110 lbs. 7 in. diam.=(see table) 38.484 square inches. 110 384840 384840 4.233fWV lbs. Ans. To find the number of horse power required to raise a given quantity of water in gallons to a given height in feet. Kule. Multiply the given number of gallons of water ,to be raised per minute by 10 (which is the weight of one gallon) and by the height the water has to be raised in feet, and divide the product by 33,000. There is no account taken of lo*s by leakage or " slip," nor friction in any of these rules; these vary greatly according to the class and condition of pump, if it is working against a high or low lift, and the " slip " depending upon the class ( f valve used for the pump. For well designed direct-acting horizontal pumps, cue-tenth should be enough for " slip," for ordinary purposes, and 25 per cent, tor friction, but when the suction is very long and the height to where the water is raised is great, one-third should be added; but if the pump is old and badly designed, as much as one-half must be added to the total amount required. In an ordinary direct-acting steam pump one-fourth of the power required should be added, but if thedtlivery height is very great and the pipe very long one-half should be added. 1 88 Hand Book of Calculations. PUMPS. Notes. The space betiveen the suction-valve and bucket, plunger or piston, as the case may be, should be as little as possible, con- sistent with ample waterway being given. All passages should be as straight as possible, and when bends are necessary, the radius should be an easy one. Sudden enlargements and contractions in the passages should be avoided, but if an alteration in size or shape of the passages is necessary, it should be made gradually. Care should be taken that the suction-pipe should be the lowest point in the pump. If the pump is required to raise hot water, there should be very little suction; in fact, it is best, if possible, to h aye the water running into the pump. Long suction-pipes should always be provided with a foot- valve just above the windbore or strainer, in the well or }3it. All corners should be well rounded. There should be as few flat surfaces as possible, and where there are any they should be well ribbed. Joints in the suction-pipes and the suction part of the pump must be very carefully made, and perfectly tight. Change of direction in the flow of water should be avoided as much as possible. After a current of water has received an impulse, it is necessary that the motion imparted should be continued with a uniform velocity throughout its whole course. Hand Book oj Calculations, i8g TABLE I. Quantity of Water Discharged per Mixute by Sixgle- Cylindeb P r.M ps, from 2 to 6-inch diameter, at 30 and 40 strokes per minute, 0, 10 and 12 inch stroke. 0-inch Stroke. Diameter Gallons per Minute. of Pomp. :» Strokes. 40 Strokes in. 2 3.0 4.0 2* 4.6 0.25 3 6.7 8.93 3* 8.83 12.2 4 11.96 15.9 44 15.2 20.3 5 18.75 25.0 54 22.69 30.25 6 27.0 3G.0 10-inc.b Stroke. Gallons per Minute. :0 Strokes. 10 Strokes. .1 3.33 5.21 7.44 9.81 13.28 16.88 20.83 25.21 30.0 4.44 6.94 9.92 13.55 17.66 22,55 iC i . i i 33.5 40.0 12-inch Strok \ Gallons per Minute. :>0 Strokes. 40 Strokes. 4.0 6.25 8.93 11.77 15.94 20.26 25.0 30.25 36.0 5.3 8.33 11.9 16.26 21.2 27.6 33.33 40.33 48.0 TABLE IT. Quantity of Water Discharged per Mixute by Double- Cylixder Pumps, from 2 to 6-inch diameter, at 30 and 40 strokes per minute, 9, 10 and 12-inch stroke. 9-inch Stroke. 10-inch Stroke. 13-inch Stroke. Diameter Gallons per Minute. Gallons per Minute. Gallons per Minute. limp. (0 Strokes. 40 Strokes. 30 Strokes. to Strokes. 30 Strokes. K) Strokes. in. 2 0.0 8.0 0.00 8.88 8.0 10.(5 21 9.38 12.5 10.42 13.88 12.5 10 «H 3 13.4 17.86 14.88 19.84 17. .°fi 23.8 H 17.66 24.4 19.02 27.10 23.51 32.52 4 23.92 31.8 26.50 35.32 31.88 42.4 44 30 4 40.6 33.76 45.1 40.52 54.12 5 37.5 50.0 41.66 55.54 50.0 60.66 5 V 45.38 00.5 50.42 67.0 60.5 SO. (]0 (J 54.0 72.0 oo.o 80.0 72.0 96.0 igo Hand Book of Calculations. TABLE III. Pressure of Water at Different Heads in lbs. per Square Inch. c ^~ l 10 20 ao 40 50 60 70 80 90 100 110 120 130 140 3.33 Q.rS 10.0 13.3 16.G 20.0 23.3 26.Q 30.0 33.3 36.6 40.0 43.3 46.6 1.66 3.33 5.0 6.66 8.33 10.0 11.6 13.3 15.0 16.6 18.3 20.0 21.6 23.3 3.041 6.09i 9.14s 12.1 15.2 18.2 21.3 24.3 27.4 30.4 33.5 36.5 39.6 42.6 4.33 S.6G 12.9 17.3 21.6 25.9 30.3 34.6 38.9 43.3 47.6 51.9 56.3 60.6 150 160 170 180 190 200 210 220 230 240 250 260 2T0 280 Of -U CO s s o -^ 53 SI eg a 50.0 25.0 45.7 53.3 26.6 48.7 56.6 28.3 51.8 60.0 30.0 54.8 63.3 31.6 57.9 66.6 33.3 60.9 70.0 35.0 64.0 73.3 36.6 67.0 76.6 38.3 70.1 80.0 40.0 73.1 83.3 41.6 76.2 86.6 43.3 79.2 90.0 45.0 82.2 93.3 46.6 85.3 Ed 64.9 H9.3 73.6 77.D 82.3 86.5 90.9 95.3 99.6 103.9 108.3 112.6 116.9 121.3 TABLE IY. Of the Diameters of Pipes, Sufficient in Size to Dis- charge a Kequired Quantity of Water per Minuie. Cubic Diameter in Cubic Diameter in Cubic Diameter in feet. inches. feet. inches. feet. inches. 1 .96 18 4.07 130 10.94 2 1.36 20 4.29 140 11.35 3 1.66 25 4.80 150 11.75 4 1.92 30 5.25 160 12.14 5 2,15 35 5.67 170 12.51 6 2.35 40 6.07 180 12.67 7 2.60 45 6.53 190 13.23 8 2.72 50 6.80 200 13.57 9 2.88 55 7.12 225 14.40 10 3.04 60 7.43 250 15.17 11 3.18 70 8.03 275 15.91 12 3.33 80 8.60 300 16.62 13 3.46 90 9.10 350 17.95 14 3.60 100 9.<;o 400 19.20 15 3.72 i 110 10.06 500 20.46 13 3.84 ! 120 10.51 <500 23.51 Hand Book of Calculations, iyi EVOLUTION OR SQUARE ROOT. This is one of the most important rules in the whole range of mathematics and well worth the careful attention of the student. Given any power of a number to find its root. To familiarize oneself with the extracting of the square root it is well first to square a number and then work backward according to the Examples here given, and by long and frequent practise become expert in the calculation. But in first working square root it is undoubtedly better to secure the services of a teacher. Example. Find the square root of 186624. Proof 432 18,66,24(432 432 16 83 "266" S64 249 1296 1728 862 1T24 1724 186624 Begin at the last figure 4, count two figures, and mark the second as shown in the Example; count two more, and mark the figure, and so on till there are no more figures; take the figures to the left of the last dot, 18, and find what number multiplied by itself will give 18: there is no number that will do so, for 4x4 = 16, is too small, and 5x5 = 25, is too large; we take the one that is too small, viz., 4, and place it in the quotient, and place its square 16 under the 18, subtract and bring down the next two figures G6. To get the divisor multi- ply the quotient 4 by 2 = 8, place the 8 in the divisor, and say H into :\ times, place the 3 after the 4 in the quotient, and also after the 8 in the divisor; multiply the 83 by the 3 in the quotient, and place the product under the 266 and subtract, then bring down the next two figures 24. To get the next di- vi« r. multiply the quotient 43 by 2=86; see how often 8 goes into 17, twice; place tie- 2 aftei the 43 of the quotient, and the 86 of tie- divisor; multiply the 862 by the 2, and put it under the 1 724, thei] er, 432. IQ2 Hand Book of Calculations. SQUARE ROOT. Example. Find the square root of 735. 47 7 35(27.11 4 335 329 &c. Proof. 2711 2711 2711 541 600 541 2711 18977 5421 5900 5421 5422 &c. 734.9522 We proceed as before till we get the remainder 6, and we see it is notSa perfect .square; we wish the root to be taken to two or three places of decimal; there are no more figures to bring down, therefore, bring down two ciphers and proceed as in the first Example; to the remainder attach two more ciphers and proceed as before; and by attaching two ciphers to the remain- der, you may carry it to any number of decimal places you please. In the above Example the answer is 27.11 &c. Example. Eind the square root of 588.0625. 5,88.06,25(24.25 4 44 188 176 482 1206 964 484^ 5 1 24225 24225 In a decimal quantity like the above, the marking off differs from the former Examples. Instead of counting twos from right to left, we begin at the decimal point and count twos towards the left and towards the right. The rest of the work is similar to the other examples. Notice, that when the .06 is brought down, the figure for a quotient is a decimal. Hand Book of Calculations. ipj SQUARE ROOT. Example. Find the square root of 7986.57246. 7986.57,24 6(89.3676 &c. 64 169 I 1586 1 1521 1783 6557 5349 17866 178727" 1787346 120824 107196 1362860 1251089 11177100 10724076 453024 Notice, the last figure is 6; always bring down two figures at a time, therefore bring down 60. The rest . is similar to the former example. Examples for Exercise. Find the square root of 589824. J- • 2. a 3. (< 4. a 5. a 6. a 7. = 5; &c. Ox Jin: Signs Repbesenjing the Power of Numbers. 6 1 is equal to 6x6 = 36; that is, 36 is the square of 6. is equal to 5x5x5 = 125; that is, 125 is the cube of 5. V is equal to 4x4x4x4 = 256, that is, 256 is the fourth power of 4. be above we have the powers that are most frequently met with; but of course you may have the 5th, 6th, or any r: but whatever tin- power, multiply the given number that number of times by itself, and you will be quite •: for an example, what is the value of ! 7 to ghth power, or ! X7x7x7x7x7x7x7=5764801. ig6 Hand Book of Calculations. The power and the root are often combined, as 4*; this is read as the square root cf 4 cubed. So the numerator figure represents the power, and the denominator figure represents the root. In this case 4 cubed=64, and the square root of 64=8. Answer. Perhaps the most common form that an engineer will meet with this sign is in the following: — 8 f , which is read the cube root of 8 squared. Now 8 squared = 64, and the cube root of 64 is 4. Answer. Find the value of 20*. 20 cubed=8000; and square root of 8000=89.4 &c. Example. 8*+81 What is the value of » ? 3* $*=J/8* = ^/te=4c; 81^=9; 3*= ^¥~=^/2V=5.2 nearly. 4+9 13 t A Hence, ~X~2 == ^¥ = ^'^ or * Answer. ( ) are called brackets, and mean that all the quantities within them are to be put together first; thus, 7 (8 — 6+4x3) means that 6 must be subtracted from 8=2, and 4 times 3=12 added to this 2=14; and then this 14 is to be multiplied by 7=98. CIRCULAR INCHES. A circular inch is a circle whose diameter is one inch; instead of finding diameters in square inches it is frequently conveni- ent to use the circular inch as per the following Rule. To find the circular inches in a circle: square the diameter of the circle in inches. Examples. 1. How many circular inches are there in a safety valve whose diameter is 4J- inches ? 4.5 2 = 4.5 X 4.5 = 20i Answer. HcuiiJ Book of Calculations. 197 CIRCULAR INCHES. '2. How many circular inches in a compound engine whose diameters are 31' and 60' ? Answer. 31 8 and 60 8 = 4561 circular inches. 3. The diameter of a piston is 24 inches, how many circular inches will that give ? Now 24 a = 24x24 = Ans. 576 circular inches. 4. If two pistons of a compound engine are 26" and 50", what ratio will their areas hear to each other ? Instead of finding the areas in square inches find them in circular inches. 26 2 — 676, and 50 2 = 2500. Hence as 676 : 2500:: 1 : Answer, 1 to 3.7 nearly. COMPOSITION OF AIR, ETC. Air is composed of nitrogen and oxygen mixed mechanically and not chemically, (this is unlike water, the parts of which are combined chemically.) Out of every 100 volumes of air 79 parts are composed