LIBRARY 
 
 UNIVERSITY OF CALIFORNIA 
 
 GIFT OF 
 
 Class 
 


 
 
 
 

 
 i 
 
MECHANIC'S CALCULATOR; 
 
 COMPREHENDING 
 
 PRINCIPLES, RULES, AND TABLES 
 
 IN THE VARIOUS DEPARTMENTS OP 
 
 MATHEMATICS AND MECHANICS; 
 
 MILLWRIGHTS, ENGINEERS, AND ARTISANS IN GENERAL. 
 
 PROM THE FIFTH GLASGOW EDITION 
 
 How have we obtained this great superiority over these poor savages ? Because science has been t 
 work, for many centuries, to diminish the amount of our mental labour by teaching us the easiest mode 
 of calculation *tuBi of Machinery. 
 
 PHILADELPHIA : 
 THOMAS WARDLE, 15 MINOR STREET. 
 
 Stereotyped by L. Johnson. 
 
 1842. 
 

 INTRODUCTION. 
 
 IT is our intention, in these introductory pages, to make a few 
 observations on the nature of scientific knowledge, which may be 
 useful to the young reader in enabling him to understand more 
 clearly the subjects contained in the volume, and in guarding him 
 against the adoption of false theory, or the wasting of his time 
 in inquiries which can terminate in no useful result. Such intro- 
 ductory observations are rendered the more necessary, as a correct 
 knowledge of the subjects to which they relate, is the only sure 
 foundation on which there can be raised a solid superstructure 
 of science. 
 
 It is a general opinion that scientific knowledge is entirely dif- 
 ferent from all other kinds of knowledge ; or that it requires for 
 its cultivation a constitution of mind only to be met with here 
 and there in the great family of mankind ; and what is said of the 
 poet is also thought of the philosopher that he is born, not made. 
 All men are certainly not equally endowed with capacities for the 
 acquisition of scientific knowledge, but there are few men indeed 
 who are totally unprivileged. The man who would relinquish 
 scientific pursuits merely because he had no hope of reaching 
 the eminence of a Newton, a Watt, or a Davy, is no better than 
 him. who, in despair of ever obtaining a share of wealth equal 
 to that of the rich inheritor of the land, would cease to make any 
 honest exertion to raise himself from a state of the most squalid 
 wretchedness. We would not be understood by this to bring the 
 acquisition of knowledge into invidious comparison with the 
 acquisition of wealth the one is in every case a godlike employ- 
 ment, but the other is often the concomitant of vice. 
 
 The young mechanic should be made well aware that the 
 knowledge of the man of science differs from the knowledge of 
 
 1 5 
 
6 INTRODUCTION. 
 
 ordinary men, not so much in kind as in degree ; and the know- 
 ledge which guides the little boy in the erection of his summer- 
 house, constitutes a part of that knowledge which guides the best 
 architect in the erection of the most splendid edifice. The boy 
 raises his paper kite in the air, with no other end in view save 
 his own amusement he has learned to do so by seeing other 
 boys do the same, and by trials he finds that the kite will fly 
 better in a moderate wind than in a perfect calm, and that the 
 weight at the tail may be too heavy or too light, and he regulates 
 his actions accordingly : so far he is a little philosopher. A man 
 raises a kite knowing all that the boy knew, but he raises it with 
 a view of determining the state of the atmosphere so far as 
 electricity is concerned, for which purpose, instead of employing 
 the hempen cord, which was sufficient for the purpose of the boy, 
 he employs a metallic wire, which he knows by experience will 
 conduct the electricity from the clouds to the earth, and thus 
 effects his design. In this respect the knowledge of the man is 
 more extensive than that of the boy, but this additional knowledge 
 has been obtained exactly in the same way as the knowledge of 
 the boy, that is to say, by experience. Even the Indian, unlearned 
 as he seems to be, is in some respects a philosopher. He sees 
 daily that the paddle of his canoe is to appearance broken when 
 he puts it into the water; but it is only to appearance, for by re- 
 peated trials, he finds that the paddle is as whole when in the 
 water as when out of it. He knows also, by repeated trials, 
 that the fish, while it shoots along through the clear flood, does 
 not appear to be where it really is ; for though the most unerring of 
 marksmen, yet if he throws his dart directly at the point where 
 the fish appears, he will certainly miss it. In vain will he try to 
 strike the fish on the same principles as he strikes the bird flying 
 in the air ; but he finds, that when he directs his dart to a line which 
 is nearer to him than that in which the fish seems to move, he will 
 strike the fish. The Indian remembers the circumstance of his 
 paddle, and other circumstances of a like kind, and concludes that, 
 when bodies are viewed through water, they do not seem to be in 
 the place in which they really are. When he knows and acts upon 
 this principle, he is a man of science so far as this is concerned. 
 The man of science, indeed, as we commonly understand that 
 appellation, knows much more than this : he knows that many 
 other substances have a like effect in changing the apparent 
 

 
 INTRODUCTION. 
 
 position of objects when seen through them ; that one produces a 
 greater and another a less change, and by repeated trials he ascer- 
 tains the actual amount of their changes by measurement, and 
 can subject them to the most rigid calculation; all of which 
 knowledge is obtained in the same way as that of the Indian, but 
 is more extensive. 
 
 An examination of facts is the foundation of all true science ; 
 but science does not consist in a mere examination of facts. 
 They must be compared with each other, and the general circum- 
 stance of their agreement carefully marked. "When we have 
 compared several facts together, and find that there is one general 
 circumstance in which they agree, this one circumstance becomes, 
 as it were, a chain by which they are all linked together. This 
 general circumstance of agreement, when expressed in language, 
 is what is called a law. For instance, it is a law that all bodies, 
 when left to fall freely, will tend to the earth ; and this law has 
 been framed by us, because in all cases which we have examined 
 this has been the case ; and the term gravity, by which this law 
 is designated, is nothing else than a name invented to express a 
 circumstance in which we have found innumerable facts to agree. 
 It was known for a very long time that water would not rise in a 
 sucking pump to a height of more than thirty-two feet, and this 
 was said to take place because nature abhorred a vacuum. The ' 
 reason given was afterwards found to be false, yet the knowledge 
 of the fact was exceedingly useful in the construction of pumps 
 for lifting water. About the middle of the seventeenth century, 
 Toricelli, the pupil of Galileo, made experiments on the subject, 
 and found that fluids would rise in tubes or in sucking pumps 
 higher in proportion as they were lighter; and collecting all the 
 facts together, he concluded that the fluids were forced up by the 
 pressure of the atmosphere, and thus laid down one of the most 
 important laws of physical science. A collection of such laws 
 which refers to some particular class of objects, when properly 
 arranged, becomes what is called a theory. Thus we see that a 
 theory, properly so called, is founded on an examination of par- 
 ticular facts, and of course cannot refer to any other but those 
 facts which have been examined ; or, if it is attempted so to do, 
 it is no . longer a theory, but an hypothesis or supposition. 
 Hypotheses, although they ought not to be relied upon, are never- 
 theless useful, as in our endeavours to discover whether they be 
 
8 INTRODUCTION. 
 
 true or false, we may at last ascertain the class of facts to which 
 they belong, and thus arrive at the true theory. 
 
 In the examination of facts, it is to be observed, that we must 
 depend on the information derived through the medium of the five 
 senses, that, is, the senses of seeing hearing touching tasting 
 and smelling ; for it is only by bodies affecting these organs 
 that the properties of matter become known to us ; and all that 
 the mind does is to compare and classify the information thus 
 derived. 
 
 It is a common error to suppose that many of our greatest 
 inventions and discoveries were made by accident. Many 
 wonderful anecdotes are told in support of this assertion ; but 
 the very circumstance of their exciting our wonder is sufficient 
 to show that they are out of the common course of our experience, 
 and that, therefore, before they are received, they ought to undergo 
 a careful examination. A multitude of facts might be adduced 
 to prove that knowledge is more regularly progressive than is 
 commonly imagined. Far be it from us to detract from the 
 merits of those great men who have, from time to time, benefited 
 mankind by their important discoveries ; but from a survey of 
 the history of science, we are led to the conviction, that where- 
 ever a new path has been struck out in the great field of truth, 
 that path has been previously prepared by former inquirers. Had 
 Kepler not discovered the three fundamental laws of the planetary 
 motions, it is highly probable that the Principia of Newton never 
 would have issued from the pen of that illustrious man ; and 
 had it not been for the brilliant discoveries of Dr. Black on the 
 subject of heat, it is probable that Watt never would have nrade 
 his improvements on the steam engine, that invaluable distributer 
 of power. It is not unlikely, however, from the state of know- 
 ledge in the days of Newton, that, independent of the exertions 
 of his mighty mind, the knowledge contained in the Principia 
 would soon after have been given to the world by some one or 
 more individuals and the like may be said of the inventions of 
 James Watt. 
 
 The great lesson which we would wish the young mechanic to 
 learn from these observations is that great discoveries are never 
 made without preparation that previous knowledge is necessary 
 to turn what are called accidental occurrences to good account. 
 And when he is told that the law of gravitation was suggested 
 
INTRODUCTION. 9 
 
 to Newton by the falling of an apple from a tree in his garden ; 
 or that the invention of the cotton jenny was suggested to Har- 
 greave by the circumstance of a common spinning-wheel conti- 
 nuing in its ordinary motion while in a state of falling to the 
 ground let him be well assured, that, had the minds of Newton 
 and Hargreave not been previously stored with knowledge, these 
 discoveries never would have been made by them. Apples and 
 spinning wheels had fallen a thousand and a thousand times, but 
 the knowledge necessary to turn these circumstances to good 
 account was first concentrated in the minds of these two illus- 
 trious benefactors of mankind. 
 
 In Smith's Wealth of Nations it is related that the ingenious 
 apparatus for opening and shutting the valves of the steam engine 
 was introduced by the accident of an idle boy having fastened a 
 brick as a counterweight to the handles which opened and shut 
 the valves, and thus allowed him time to leave the machine and 
 go to play. This simple trick of an idle boy, it is said, gave rise 
 to the apparatus which superseded the constant attendance of a 
 person while the engine was at work. This, however romantic, 
 is not the fact the invention originated in necessity, no doubt, 
 but it was begun and perfected by a thorough mechanic, Mr. H. 
 Brighton, about the year 1717. 
 
 While we are on this subject we cannot pass over another very 
 common prejudice, which we conceive has a very hurtful tendency 
 on the progress of the young mechanic. We allude to the pride 
 that some men take in boasting that all their knowledge is ori- 
 ginal ; or that they are self-taught. This is, in other words, 
 stating, that no assistance has been taken either from teachers or 
 books; and goes only to prove, that the knowledge of the indivi- 
 dual so circumstanced must be very limited indeed. The unas- 
 sisted exertions of one man must be very feeble, when compared 
 with the collected exertions of the many who have gone before 
 him in the career of discovery. That man must know little of 
 geometry who has not availed himself of the use of Euclid's 
 Elements, or some work of a similar nature ; and the Elements 
 of Euclid would have been meager and confined, had he not 
 availed himself of the discoveries of his contemporaries and pre- 
 decessors. A like remark may be made on the cultivation of 
 every department of knowledge ; and to those whom we are now 
 
:'; 
 
 10 INTRODUCTION. 
 
 addressing we say learn from others all that you possibly can, 
 and when you have done so, try to correct and improve what you 
 have obtained. We know of no dishonourable means of acquiring 
 knowledge, and therefore wherever we meet it we are disposed 
 to respect it, even though it should not contain one particle of 
 originality, if such be possible ; for it is not easy to conceive 
 how any man should be in possession of useful knowledge, and 
 not make some new application of it ; and a new application of 
 an old principle is certainly one constituent of originality. With 
 a knowledge of what others have done, that workman will be 
 less likely to waste his time in enterprises which may ruin him 
 by their failure, or in speculations which are unsupported by the 
 principles of science. 
 
 In the museum of the mechanics' class of the university founded 
 by the venerable Anderson of Glasgow, there is preserved the 
 model of a machine to procure a perpetual motion. For the con- 
 trivance and execution of this beautiful specimen of workmanship, 
 we are, we believe, indebted to an ingenious clock-maker of Dun- 
 dee, who has proven himself a master in the use of his tools. 
 But had he been acquainted with the first principles of mechanics, 
 or with the nature and failure of the various attempts which had 
 been made before his time for the same purpose, he would have 
 seen the utter folly of his enterprise, and would have spent the 
 seven years which he occupied in the construction of this truly 
 beautiful model in some more useful employment. These seven 
 years might have been devoted to the construction of timepieces 
 which would have been of infinite service to the commerce and 
 navigation of his country in guiding the lonely mariner when 
 far away on the billow in determining the exact distance and 
 direction of the part for which he is bound whereas, the model 
 of his perpetual motion is preserved in the museum as a lasting 
 monument of this clock-maker's ignorance, perseverance, and 
 handicraft. 
 
 It is another common error to suppose that genius alone can 
 make a man a great mechanic, a great chemist, or a great any 
 thing. Some one makes the remark, that every man is more 
 than half humanity ; and we do believe that the differences of 
 the degrees of knowledge of different men arise more from their 
 difference of application than from original differences of capa- 
 
INTRODUCTION. 11 
 
 city. Let, therefore, the young workman earnestly try to learn, 
 and we do assure him that he will make advances which will be 
 proportional to his application. 
 
 This book has been written with the view of assisting the 
 young workman in obtaining a knowledge of the calculations 
 connected with machinery. The first part is devoted to such 
 parts of arithmetic as workmen generally require, and in which 
 they are most commonly deficient. Nor is this deficiency to be 
 wondered at, since the school books in our language contain, 
 generally speaking, no explanation of the nature of the rules 
 which they give, and are, moreover, embarrassed with so many 
 divisions and subdivisions, that the mind of the scholar is per- 
 fectly perplexed, nor can it lay hold of the great leading principles 
 which pervade the whole system. As this is the great instrument 
 used throughout the book, we have endeavoured to make its use 
 and management easily understood. The examples which we 
 have given are indeed few and simple ; but, if carefully consi- 
 dered, they will be found sufficient to establish the principle. 
 The mere habit of calculation cannot be said to constitute a 
 knowledge of arithmetic ; it is easily obtained, but is of no avail 
 without the principles. This is well illustrated by an occurrence 
 of but recent date. To construct a set of mathematical tables 
 requires, not only a knowledge of principles, but also immense 
 calculation. M. De Pronney was desired by the government of 
 France to construct a very large set of such tables ; a task which 
 would require the labour of a mathematician for many years. 
 But Pronney fell upon an expedient which was every way worthy 
 of a man of science. A change in the fashions of the Parisians 
 had thrown about five hundered wig-makers idle, and Pronney 
 contrived at once to give employment to these barbers, and at 
 the same time to serve the purposes of science. He digested the 
 principles of the calculation of these tables into short and simple 
 rules, and printed forms of them, which he gave into the hands 
 of these workmen, who, in a few months, produced a set of tables, 
 the most correct and extensive that ever has been made. The 
 peruke-makers may, so far as the construction of the tables was 
 concerned, be regarded as mere machines, under the guidance of 
 M. de Pronney. The same principle has been of late years car- 
 ried to a far greater extent by our countryman, Professor Babbage, 
 who has invented a machine by which logarithms and astrono- 
 
12 INTRODUCTION. 
 
 mical tables may be calculated and printed with the most unerring 
 certainty, thus obviating the necessity of employing either calcu- 
 lators or compositors. Let not these statements induce you, 
 however, to neglect the practice of calculation ; on the contrary, 
 improve yourself in it wherever you can, but be also careful to 
 learn the principle. 
 
 In that part devoted to geometry, we have given such informa- 
 tion without demonstration as was necessary to the right under- 
 standing of the rest of the book ; and the like may be said of the 
 conic sections, mensuration, and useful curves. Thus far the 
 book may be said to be a compend of certain branches of the 
 mathematics. It is hoped that the reader, to whom such studies 
 are new, will not be contented to stop here ; but will be induced 
 to investigate these subjects in theory ; and for such as may be 
 desirous of entering on such a course of study, where there is 
 nothing to be met with but unsophisticated truths connected 
 together by a chain of the most beautiful relations, we intend to 
 offer a few words of well-meant advice as to the order and means 
 of prosecuting such studies.* 
 
 In the first place, let the Elements of Euclid be studied so far 
 as the end of the first book, in the course of which it should be 
 borne in mind, that there is nothing really difficult to be met with. 
 The greatest difficulty is, we believe, this, that, to a proposition 
 which is so simple as to be almost self-evident, there is often 
 
 * In a very creditable work, recently published, " Stuart's History of 
 the Steam Engine, " it is stated that mathematics is not necessary to 
 make a great mechanic, and Watt is cited as an instance. The instance 
 chosen is most unfortunate for the author's assertion. Watt was de- 
 scended from a family of mathematicians, and inherited in the highest 
 degree the genius of his ancestors. One instance will sufficiently prove 
 this. With a desire to determine what relation the boiling point bore to 
 the pressure of the atmosphere on the surface of the water, he made 
 several experiments with apothecaries' phials, and having found the rela- 
 tion between the pressure and temperature of ebullition, under different 
 circumstances, he laid the temperatures down as abscissae, and the pres- 
 sures as ordinates, and thus found a curve whose equation gave that well 
 known formula, the equation of the boiling point. No man but a mathe- 
 matician of high attainments would have thought of such a method of 
 proceeding. To this we may add, that mechanics is a branch of mathe- 
 matics ; tor, as Sir Isaac Newton has denned it, " mechanics is the geo- 
 metry of motion." 
 
INTRODUCTION. 13 
 
 attached a long demonstration, which is apt to lead the reader to 
 suppose that there is really something mysterious in it, which he 
 does not understand. This proceeds from the fact, that it often 
 requires a greater deal of circumlocution to show the connection 
 of simple propositions with first principles, compared with propo- 
 sitions which are more complex ; but we have no hesitation in 
 saying, that if the steps of the propositions are carefully consi- 
 dered, one by one, they will be easily understood, and will lead 
 at last to perfect conviction ; for, as Lord Brougham has well 
 observed, " Mathematical language is not only the simplest and 
 most easily understood of any, but the shortest also ;" and Euclid 
 has transmitted to posterity a specimen of the purest mathemati- 
 cal language. Of Euclid's Elements, there are various editions. 
 Those of Simpson and Playfair are generally used in this country, 
 and are deservedly popular. That of Dr. Thomson is a very 
 valuable work, and very correct. But we beg to recommend to 
 the workman the edition of Mr. Robert Wallace, of Glasgow, 
 both for its execution and cheapness. The demonstrations are 
 clear and short ; many new propositions are added, and the con- 
 nection of theory with practice is never omitted where it can be 
 introduced. 
 
 When the first book of Euclid has been read, the study of al- 
 gebra should be commenced, on which subject there are few good 
 treatises to be found. That which we think best is the treatise 
 of Euler, a book which has come from the hand of a master, and 
 is therefore characterized by great simplicity. Another good 
 book is the treatise of Saunderson. Let either of these works, 
 or others if they cannot be had, be read carefully so far as to 
 equations of the second degree. If any one part of this depart- 
 ment can be said to be difficult, it is that of powers and roots, 
 which is a subject of the greatest importance ; and should, 6*h 
 that account, receive the most careful attention ; and, if the trea- 
 tise of Euler be used, we have no hesitation in saying, that little 
 difficulty will be experienced. It may be necessary to observe, 
 that attention should be paid all along to the intimate connection 
 of arithmetic and algebra, which will tend to the better under- 
 standing of them both. Having advanced thus far, Euclid must 
 again be returned to ; and, after revising the first book, read on 
 to the sixth inclusive. Occasional revision of the algebra is 
 recommended, and an advancement as far as equations of the 
 
14 INTRODUCTION. 
 
 third degree ; after which Euclid may be read to the termination. 
 The study of trigonometry may then be introduced ; on which 
 subject we have various works of various merits. The treatise 
 prefixed to Brown's Logarithmic Tables may be employed ; and 
 when it is understood, and the management of the logarithmic 
 tables acquired, the works of Gregory, Lardncr, or Thomson may 
 be consulted ; the last is the most simple. After the study of 
 trigonometry, Simpson's conic sections may be read with advan- 
 tage. 
 
 Perhaps it may be a kind of relief at this stage, to see some- 
 thing of the application of mathematics to mechanics, and, for 
 this purpose, the work of Keil on Physics, or the article Mecha- 
 nics, Mutton's Mathematics, Tegg's edition. The neat little 
 treatise of Mr. Hay of Edinburgh will answer the same purpose 
 exceedingly well. But for the purpose of obtaining a good know- 
 ledge of theoretical mechanics, a more extensive knowledge of 
 mathematics than we have hitherto supposed becomes absolutely 
 necessary. A knowledge of the method of fluxions and fluents, 
 or the differential and integral calculus, which bear a strong ana- 
 logy to each other, and which have been employed for similar 
 purposes. The simplest work on fluxions, and we believe the 
 best, is the treatise of Simpson ; and this may be followed by a 
 perusal of Thomson's differential and integral calculus. With 
 this preparation the student may now go on to read the first vo- 
 lume of Gregory's Mechanics, a book in which, we believe, he 
 will find ample satisfaction. The second volume of this excellent 
 work is almost entirely popular, and can cause no difficulty what- 
 ever. Another work, well worthy of a perusal, is that of Sir 
 John Leslie : we allude to his Natural Philosophy ; a book which, 
 though neither strictly mathematical, nor strictly popular, yet 
 contains much valuable information communicated in both ways. 
 Indeed all the works of this great man, although much has been 
 said against them as to the floridness of their style, will, never- 
 theless, be found to amply repay the trouble of a perusal. 
 
 We will not lengthen out these directions, as we conceive that 
 when the student has advanced thus far he will be possessed of 
 much valuable information, and will have a sufficient knowledge 
 of both books and things to guide himself in his future inquiries. 
 We say future inquiries, as it is our firm conviction that he who 
 has advanced to the point we have considered, will be too deeply 
 
INTRODUCTION. 15 
 
 imbued with a love of science, even for its own sake, ever to 
 cease from its cultivation, so long as his mind is capable of 
 cherishing one ray of its benign influence. 
 
 As the library of the workman cannot be very extensive, the 
 few books which it contains should be well chosen. The treatises 
 published by the Society for the Diffusion of Useful Knowledge 
 cannot be too warmly recommended ; and are easy of access from 
 their cheapness and mode of publication. Indeed, the foundation 
 of this society forms a most important era in the history of man- 
 kind ; and we fondly hope, as we firmly believe, that the benevo- 
 lent exertions of its talented members will be crowned with suc- 
 cess. 
 
 In recommending this course of reading we do not mean to 
 Insist that the mechanic should leave unopened works in the 
 lighter walks of literature. In the volumes we have mentioned 
 he will find directions for the construction and management of 
 the steam engine, and other powerful and complicated machines ; 
 but it should not be forgotten that in the dramas of Shakspeare, 
 and the novels of Fielding, Smollett, and Scott, he will find 
 illustrations of the structure and economy of the human mind, 
 the most powerful machine of all. These, and the poetry and 
 the periodical literature of the day, together with historical and 
 biographical works, will often afford agreeable and instructive 
 relaxation from severer studies. 
 
 The author of the following pages hopes that his work will be 
 found useful to workmen in general ; and though no book was 
 ever written so as to meet the views of every man, yet he trusts 
 that the artisan will find much information in it, which he daily 
 requires, collected and compressed within a smaller compass than 
 in any work of a similar nature. Should this book be deemed a 
 failure, it must at least be acknowledged that its aim has been 
 utility ; and that to a class of men on whose' intelligence, exertions, 
 and welfare, the prosperity of the nation depends. There is, 
 indeed, a strong competition between this and other kingdoms in 
 the improvement of the arts and manufactures ; and, although 
 Britain still stands pre-eminent among the nations in this respect, 
 yet she must not, on that account, relax her endeavours toward 
 improvement, otherwise she will be seen left lagging behind. 
 When we reflect on the circumstance, that it is to workmen them- 
 selves that we have ever been indebted for improvements in the 
 
16 INTRODUCTION. 
 
 arts, it is reasonable to expect that this is the source from whence 
 such improvements will continue to flow; and among workmen 
 it may be safely affirmed, that he who is the most intelligent will 
 be the most likely to make improvements. 
 
 Add to these considerations the fact, that there is a pleasure 
 inseparable from the study of science, which is perfectly inde- 
 pendent of all its other advantages, and that the poor man as well 
 as the rich has a right to partake in its enjoyment. The diffusion 
 of scientific knowledge among the working classes becomes thus 
 not only a duty which every man owes to his country, but, besides 
 this, it is an act of benevolence, as it tends to administer pleasure 
 to a class of most useful men, who, in a multitude of cases, 
 suffer grievous privations. The workingman, however, should 
 be made well aware, that no exertion of any individual, or society 
 of individuals, can be of any avail in the diffusion of knowledge, 
 unless the workingman shall make an earnest exertion in tho 
 pursuit of science. To the young mechanic we then say ear- 
 nestly endeavour to improve your mind ; and enliven your spare 
 hours by the cultivation of science; and should this little volume 
 facilitate your progress in that manly employment, the desire of 
 the author shall be fulfilled. 
 
 GLASGOW, 21th August, 1832. 
 

 THE 
 
 MECHANIC'S CALCULATOR. 
 
 ARITHMETIC. 
 
 VULGAR FRACTIONS. 
 
 1. IN many cases of division after the quotient is ob- 
 tained, there is a remainder, which is placed at the end of 
 the quotient, above a small line with the divisor under it : 
 thus- 88 divided by 12 gives the quotient 7 and remainder 4, 
 which is written 12) 88 (7 T 4 f . Now, this T \ is called a frac- 
 tion ; and it is written in this way to show that 4 ought to 
 be divided by 12 ; and in all cases where we meet with num- 
 bers written in this form, we conclude that the number above 
 the line is to be divided by that under the line. This 
 should be well borne in mind, as it is of the greatest use in 
 obtaining a clear notion of fractions. 
 
 2. A fraction is said to express any number of the equal 
 parts into which one whole is divided. It consists of two num- 
 bersone placed above and the other below a small line. 
 The upper number is called the Numerator, because it 
 numerates how many parts the fraction expresses; and the 
 under number is called the Denominator, because it ex- 
 presses or denominates of what kind these parts are ; or, 
 in other words, the denominator shows into how many parts 
 one inch, foot, yard, mile one whole any thing is sup- 
 posed to be divided ; and the numerator shows how many 
 of these parts are taken : as T 4 j of a foot. The denominator 
 shows that the foot is here divided into 12 equal pails 
 (inches ;) and the numerator 4, shows that four of these 
 parts are taken (4 inches.) 
 
 2* 17 
 
18 
 
 3. If the numerator had been equal to the denominator, 
 as |f, then the value of the fraction would have been one 
 whole (foot;) and the numerator, being divided by the 
 denominator, gives 1 as a quotient. In the fraction }* of a 
 foot, the numerator is greater than the denominator, and 
 the value of the fraction is greater than one : for the foot 
 being divided into twelve equal parts, (inches,) and fourteen 
 such parts (inches) being expressed by this fraction, its 
 value is more than one foot; and the numerator being 
 divided by the denominator, gives 1-^. Again, T 6 2 f a f ot 
 is just six inches, or one-half foot ; and had the foot been 
 divided into two equal parts, one of these parts would have 
 been equal to T 6 j, or 5 is equal to T V. From this we may 
 conclude, that when the numerator is equal to, less, or 
 greater than the denominator, the value of the fraction is 
 equal to, less, or greater than one whole. It is, then, not the 
 numbers which express the numerator and denominator of 
 a fraction, but the relation they bear to each other, that 
 determines the real value of a fraction. , J J iV are a ^ 
 equal, although expressed by different numbers, the deno- 
 minators of all the fractions being respectively doubles of 
 their numerators. 
 
 4. From what has been said, it will easily be seen, that, 
 if we multiply or divide both terms of any fraction by the 
 same number, a new fraction will be found, equal to the 
 first; thus, *; multiply both terms by 2, we get T 8 f , or 
 divide them by 2, J, and these again by 2, 3. All who 
 know any thing of a common foot-rule will understand this, 
 at sight. 
 
 5. The first use which we shall make of the principle last 
 stated, is to bring two or more fractions to the same deno- 
 minator, and that without altering their real values. For 
 example, take and | of a foot. Multiply both terms of the 
 first fraction by the denominator of the second, 4 : we get 
 fy. Next multiply both terms of the second fraction by the 
 denominator of the first fraction, that is, by 3 : the result 
 is ,'j. Now it will be seen (from No. 4) that these two 
 fractions, T 8 ^ and -, 9 j, are equal to the two -J and |, with 
 this additional advantage, however, that they have the same 
 denominator, 12 : the great use of which will be seen here- 
 after. A like process is employed in the case of three or 
 more fractions: thus, 3, ! y multiply the terms of the 
 first fraction by 4 and 5, the denominators of the second 
 
LGA 
 
 VULGAR FRACTIONS. 19 
 
 and third, we get | ; next multiply the second I by 3 and 
 5, the denominators of the first and third, we next get -Jf- ; 
 lastly, multiply the third by the denominators of the first and 
 second, 3 and 4, we get |. It will be useful to look over 
 what we have done. In obtaining the numerators of the 
 new fractions, we have multiplied each numerator in the 
 former fractions by all the denominators except its own ; 
 and so also for the denominators. But 3 multiplied by 4, A 
 and 4 multiplied by 3, are the same thing, viz. 12 : so, like- 
 wise, 3 multiplied by 4 multiplied by 5 is 60, and will be 60 in 
 whatever order we take theih 3 by 4 by 5, or 4 by 3 by 5, or 
 5 by 3 by 4 ; when, therefore, we have obtained one deno- 
 minator, it is sufficient. Hence the usual rule to reduce 
 fractions to a common denominator: Multiply each nume- 
 rator by all the denominators except its own for new nume- 
 rators, and all the denominators together for the common 
 denominator. 
 
 6. We are now prepared to add two or more fractions 
 together. It is very easy to see how we may add f and $ 
 of an inch, and that their sum is f ; but it is not quite so 
 evident how we are to add and | of a foot. If we had 
 them, however, of one denomination, the difficulty would 
 vanish. By No. 5, bring them to a common denominator 
 they stand thus : T \ and T \, or 8 and 9 inches ; add the 
 numerators, and under their sum place the denominator, j-J ; 
 divide the numerator by the denominator, (No 1,) the quo- 
 tient is 1 T 5 3, or one foot five inches. The reason of bring- 
 ing them to a common denominator is, that we cannot add 
 unlike quantities together : and we do not add the denomi- 
 nators, their only use being to show of what kind the quan- 
 tities are. The rule, then, is bring the fractions to a 
 common denominator, add the numerators together, and 
 under their sum place the common denominator. 
 
 7. In subtraction we bring the fractions to a common 
 denominator, and taking the lesser from the greater of the 
 two numerators, place under their difference the common 
 denominator. The reason of this may be easily inferred 
 from (No. 6) f subtracted from 5, when brought to a com- 
 mon denominator, T 6 ? from T 8 ? the difference is $, equal to 
 i, by No. 4. 
 
 8. To take one number as often as there are units in 
 another, is to multiply the one number by the other. To 
 multiply 4 by 2, is to take the number four two times, as 
 
20 ARITHMETIC. 
 
 there are two units in 2 ; and to multiply 4 by 5, is to take 
 four one-half times, or the half of four, as there is only half 
 a unit in the fraction . This may be thought so simple, 
 that it need not be stated ; but, let it be observed, that it 
 explains a fact in the multiplication of fractions, which 
 many excellent practical arithmeticians do not understand ; 
 viz. how that, when we multiply by a fraction, the product 
 is less than "the number multiplied. If the fraction is to 
 be multiplied by |, (let the fractions both refer to an inch,) 
 this is taking (inch) | times, or taking the one-fourth part 
 of one-half inch, which is one-eighth. The product is 
 obtained by this simple process : multiply the numerators 
 together for a new numerator, and the denominators to- 
 gether for a new denominator ; the new fraction will be the 
 product. That this is true in general may be shown by 
 taking other fractions, thus : f of f, the product by the 
 rule is ^ 4 T , which may be simplified by dividing the nume- 
 rator and denominator by the same number, on the principle 
 of No. 4 ; if 4 be the divisor, the result is J, which is the 
 same as ^ 4 T . Now, that is the real product of f by f , may 
 be shown thus : divide a line AB 
 into six equal parts ; take two of c ' j ' D 
 these parts, and join them by A ; : : : : , B 
 CD. Divide CD into four parts, 
 
 and it will be seen that the two parts of this line CD are just 
 equal to one division on the line AB ; or f of CD is equal 
 to % of AB ; so that | of f is i. The rule, then, is general. 
 
 9. Division is the reverse of multiplication ; hence, to 
 divide in fractions, invert the divisor, and proceed as in 
 multiplication. Thus, to divide by |, invert the divisor 
 |, it becomes f , which, multiplied by gives multiplied by 
 A, equal to f ; and by dividing, to make the fraction less, 
 we obtain f , which, by No. 1, is just 2 or twice. This is 
 the quotient ; and it is easily seen, if these fractions relate 
 to a foot, that there are 2 quarters or twice | of a foot, in 
 one-half foot, or 5. 
 
 10. We have now endeavoured to explain the nature of 
 the fundamental rules of vulgar fractions, as simply as pos- 
 sible ; but instances often occur, where it is necessary to 
 prepare for these operations ; first, where whole numbers 
 are concerned ; and secondly, where the fractions are large, 
 and, consequently, not so easily managed. 
 
 11. As to the first, where whole numbers are concerned, 
 
.GAR 
 
 VULGAR FRACTIONS. 
 
 it is to be observed, that when unit, or 1, is used, either to 
 multiply or divide a number, it does not change the value 
 of that number. Thus, 6 multiplied by 1 is 6, and 6 divided 
 by 1 is 6. According to the principle shown in No. 1, we 
 may write the number 6 in this way, f , without altering 
 its real value with this advantage, that we have it now in 
 the form of a fraction. We shall illustrate this by a few 
 examples, and show that numbers, whether whole or frac- 
 tional, are in this department of arithmetic managed by the 
 same rules. 
 
 Add 8 to |, here we write them f and |, which, brought 
 to a common denominator, are, \ 2 , | their sum is y; then 
 by No. 1, divide the numerator by the denominator, we get 
 8|, the number we set out from. 7|, which is read seven 
 and a third, may on the same principle be put in the form 
 of a common fraction : for it is 7 wholes added to part of 
 a whole, and may be thus written, \ and 5, equal to 2 f l and 
 whose sum is */; divide the 22 by the 3, the result is 7|, 
 the first number. This very simple principle is often used, 
 and is embraced in the following rule multiply the whole 
 number by the denominator of the fraction, add the nume- 
 rator, and under the sum place the denominator. 
 
 12. When the fractions are very large, it becomes neces- 
 sary to bring them to a simple form, not only that we may 
 rnqre easily see their value, but that they may be more 
 readily operated upon. Thus, ^ is not so simple nor so 
 easily managed as y 1 ^, and the one fraction is just equal in 
 value to the other; for, by No. 4, the numerator and deno- 
 minator of 7 6 j being both divided by 6, gives -jL. Also, 
 2\nnr when l is use d as a divisor, gives --$. Whenever 
 we can find a number which will divide both terms of the 
 fraction without remainders, we ought to employ it, and thus 
 make the fraction simpler in form, though of exactly the 
 same value. The divisor thus used to simplify fractions, is 
 usually called the common measure, and may frequently be 
 found at sight, although sometimes there is no such number 
 at all. Thus, in f , it is seen at once that 2 is the common 
 measure ; but in the fraction | no such common measure 
 can be found : consequently, the fraction cannot be made 
 more simple. Sometimes, also, two or more numbers will 
 divide the fraction ; thus, f- may be divided by 4 or by 2 
 the greatest is preferred, because it brings the fraction to 
 the lowest terms at once. When this cannot be obtained at 
 

 22 ARITHMETIC. 
 
 sight, the following rule may be employed : Divide the 
 greater term by the less; if these leave any remainder, 
 divide the last term by it ; and thus go on dividing the last 
 divisor by the last remainder, and that divisor which leaves 
 no remainder is the greatest common measure. This rule 
 may be applied to the following example : 
 
 1470 By the above rule. 
 
 "2205 1470 ) 2205 ( 1 
 
 1470 
 
 735 ) 1470 ( 2 
 1470 
 
 735 is the common measure ; therefore, 
 
 735 ) ( |, the simple form of the fraction. 
 
 2205 
 
 DECIMAL FRACTIONS. 
 
 13. LET us examine the number 3333, (three thousand, 
 three hundred, thirty and three.) The same figure is us^d, 
 out for every place it is removed towards the left, its value 
 is increased ten times ; and consequently, if we begin at 
 the left hand side, and go on towards the right, we see that 
 every figure has a value ten times less than the same figure 
 placed one place nearer the left, each number expressing 
 tenth parts of the number next it to the left. Hundreds are 
 just tenth parts of thousands ; tens are tenth parts of hun- 
 dreds ; and units are tenth parts of tens, &c. Now. the 
 same 3333, with a point placed before any of its figures, 
 would still have the same property of each figure towards 
 the right, having a tenth part of the value it would have 
 had in the next place towards the left : that is to say, the point 
 has no effect in altering the relative value of the figures ; but 
 it has this effect, that the figure which stands at its right 
 hand would signify units : thus, 33'33, where we have 
 the same figures as before, with a point placed betwixt the 
 middle two, and from what has been said, we conclude that 
 
*' 
 
 DECIMAL FRACTIONS. 23 
 
 the 3 to the left of the point is units. From this it follows 
 that the next 3 on the right of the point is tenth parts of 
 unity, and the 3 following that again tenth parts of a tenth 
 part of unity, or hundredth parts. Had it been written 
 thus : 3.333, the last three to the right of the point would 
 have been a tenth less again, &c ; so that all the figures 
 that follow the point to the right are less than units, conse- 
 quently, they are fractional ; and from their decreasing by 
 tenths each place, they are called Decimal fractions from 
 the Latin word decem, ten. Thus, then, -5% may be written *3. 
 
 14. It is to be observed here, that the use of the cipher 
 (0) is in decimals quite similar to what it is in whole 
 numbers, where its only use is to remove some figure from 
 the units' place, and therefore alter its value tenfold. Thus, 
 in the number 40, the cipher of itself signifies nothing, but 
 serves to remove the 4 to the tens' place. Had it been 04 
 here the cipher is of no use, because there is no figure to 
 remove beyond it from the units' place. The same is true of 
 any number of units. Now, we have seen that '3 is just T ^- 
 and, from what has been said, it will follow, that .03 is three 
 hundredth parts, or T ^, as the cipher in *03 removes the *3 
 a place farther from the units' place towards the right, and 
 (No. 13) makes it ten times less in value than it would have 
 been had it been one place nearer the left ; or, it is now 
 tenth parts of a tenth part. For the same reason "003 is 
 the same as T7 \ 7 . 
 
 15. The number 33 is read thirty and three, and *33 
 is read three tenths and three hundredths, or sometimes 
 thirty-three hundreds. Now, -f^ added to T f -$ give (No. 
 6) y^V which, simplified, is T \%, (No. 4.) If we wished to 
 write T ^Vo * n tf 16 other form, it is done simply thus : point 
 in tenth's place, in hundredth's place, and 3 in thousandth's 
 place ; that is, '003. Take, now, T % and T | ; adding, then, 
 
 by No. 6, we gety 4 ^ ^, simplified T % 6 Q , which, written with the 
 point, is simply *46. We may now see, that any number 
 placed after the decimal point is a fraction ; which may be 
 expressed by a numerator which is that number, and a de- 
 nominator consisting of 1, with as many ciphers annexed 
 as there are figures in the numerator : thus, *3034 is the 
 same thing as T ^ 3 oV 
 
 16. These simple statements being understood, all that 
 follows will be easy. The principle being kept in rnind, 
 that the numbers to the one side of the point have the same 
 

 4K 
 
 24 ARITHMETIC. 
 
 relation to one another as those on the other, every figure 
 on the one side of the point as well as on the other, being 
 ten times greater than it would have been in the next place 
 to the right, and ten times less than in that to the left. 
 
 17. To add decimal fractions, we proceed just as in whole 
 ' numbers, placing units under units, and consequently points 
 
 under points, and carrying to each new column to the left, 
 by 1 for every ten in the column already added. As \ may 
 be written T 5 ^ or -5 ; 7 may, therefore, be written 7*5 ; 4| 
 may be written 4*5. Now, add 7*5 and 4-5 by 
 the rule we have given, and we will obtain a 7*5 
 result which must be correct, as may be 4-5 
 proved by principles laid down in the former 12-Q 
 chapter. Here we have kept the points under 
 each other, and put a point in the answer just under the 
 others, and the sum is 12, with no decimal fraction. Take 7-j- 
 and bring it to the form of a common vulgar fraction, by the 
 principle, No. 11, and it will be y ; do so like wise with 4-}- and 
 we get | ; they have a common denominator, and add them 
 by No. 6, we have 2 ^ 4 , now, this fraction, by No. 4, is equal 
 to V 2 , or 12, the same as before. Take now 135*7, and 1-23, 
 and -764, and 9-102, and 8-003, and -035 ; to find their 
 sum. Here we place, as before, all the points under each 
 other, and proceed as in addition of whole numbers, carry- 
 ing by tens and pointing the sum in the line under the other 
 points : 
 
 135-7 
 
 1-23 
 764 
 
 9-102 
 
 8-003 
 035 
 
 154-834 
 
 18. Subtraction is managed in like manner as in common 
 numbers, the same attention being paid to the points. 
 Thus, subtract 33*785 from 1967-32; 
 
 they are placed thus, and subtracted as 1967*320 
 in whole numbers, the point in the 33*785 
 
 answer being placed in a line with the 1933-535 
 others. It is to be observed, that there 
 are more decimal places in the under number than in the 
 
:Y 
 
 DECIMAL FRACTIONS. 25 
 
 upper, and the deficiency may be supplied by adding ciphers 
 to the upper line, which, as there is no significant figure 
 beyond, does not alter the value of the number. 
 
 19. Multiplication of decimal fractions is performed as in 
 whole numbers, paying no attention to the points until the 
 product is obtained, when we point off as many places from 
 the right hand side of the product, as there are decimal 
 places in both the multiplicand and the number which mul- 
 tiplies, or multiplier. Thus, multiply 
 
 36-42 by 4-7. Here -174 are pointed 
 off as decimals, as there are two deci- 
 mal places in the multiplicand and one 
 in the multiplier in all three. That 
 this rule is correct, may be inferred 
 from the results of a former example in 
 No. 8. Here we multiplied 4 by J, and found the pro- 
 duct to be 2 : now, -| is equal to T 5 ^, which may be written 
 5 ; then let us multiply 4 by '5, as directed 
 above, and we will find the same result, 2 ; 
 where, by principle of No. 14, the cipher 
 being pointed off, there remains 2 a whole 2*0 
 number. 
 
 20. Division may be properly defined, the finding of one 
 number (the quotient), such, that when multiplied by 
 another (the divisor), will give a product equal to a third 
 (the dividend). The dividend may thus be viewed as the 
 product of the quotient and divisor ; hence, the quotient and 
 divisor should, together, contain as many decimal places as 
 the dividend. This being observed, the rule will be easily 
 followed : Divide as in whole numbers, and when the quo- 
 tient is obtained, point off from the right as many places 
 for decimals as those of the divisor want of those in the 
 dividend. Divide 22-578 by 48-6, 
 
 the quotient 4'6ff J- is obtained by 48-6)22-578(4-6ff.f 
 common division, and pointed thus, 
 
 because the divisor wants only one decimal place to have 
 as many as the dividend. In many cases, when the quo- 
 tient is obtained, there will not be as many figures as make 
 up the number of decimal places required ; here we must 
 place one or more ciphers betwixt the point and the quo- 
 tient figures, so as to make up the number required. Thus, 
 divide 1-0384 by 236, the quotient is 44 only two places, 
 whereas there should be four decimals in the quotient ; 
 
 3 
 

 
 26 ARITHMETIC. 
 
 because there are four in the dividend and none in the 
 divisor. We, therefore, place the quotient thus, -0044 , 
 and to prove that this is the true quotient, we have only to 
 multiply it by the divisor, and the product being the same 
 as the dividend, the operation must be correct. 
 
 21. From the great facility with which decimal fractions 
 may be managed, it is very desirable that we could bring 
 vulgar fractions to the same form, in order that they might 
 more easily be wrought with. Now, this may be done on 
 the principles already laid down : take the fraction |, and, 
 on the principle of No. 4, multiply both terms by 1000, it 
 then becomes -J, which is equal to I ; divide (No. 4) both 
 numerator and denominator by 8 ; then 8) | (iVYn which 
 last fraction is expressed in the decimal notation thus, (on 
 the principle of No. 15,) -125, which, from the way it has 
 been derived, must be equal to . This may, however, be 
 found more immediately thus : add as many ciphers to the 
 numerator as you find necessary, and divide by the denomi- 
 nator thus, 8)1000(-125. If we have only to add one 
 cipher before we get a quotient figure, we put a point in 
 the quotient ; but if more, then we put as many ciphers in 
 the quotient after the point. Thus, & ; 25)100(-04, and 
 ^V is just T ^, or -04. 
 
 22. In many cases the quotient would go on without end ; 
 but it is to be observed, that it is not necessary to continue 
 any operation in decimals, at least in mechanical calcula- 
 tions, beyond three or four places, as ten thousandth parts 
 are seldom necessary to be considered in practice. For 
 similar reasons, it is unnecessary to give rules for repeating 
 and circulating decimals : i. e. decimal numbers, when the 
 same figures recur in some order thus, '3333, or, 142142, 
 &c., carry them to four places, and it is all that is neces- 
 sary. 
 
 Other applications of these principles will be found in the 
 next chapter, on compound numbers. 
 
 COMPOUND NUMBERS. 
 
 23. IN mechanical calculations, we are often concerned 
 with weights and measures, and it is necessary to know how 
 to operate with the numbers which express these. The rules 
 
 
COMPOUND NUMBERS. 27 
 
 given in books of arithmetic are generally very long, and, 
 therefore, not very easily understood ; yet the steps of the 
 operation are simple. We shall therefore show the mode of 
 procedure, in some very easy examples, and the reader will 
 find no difficulty in applying the principles he may thus im- 
 bibe to cases more complex. 
 
 24. If we have to add 9 yards 2 feet 6 y ds - feet inch, 
 inches, to 2 yards 1 foot 3 inches, 8 yards 
 
 feet 11 inches, long measure. Then we ^ 
 must in this, as in all other cases of com- 
 pound addition, arrange them in order, 20 1 8 
 the greater towards the left hand, and the lesser towards 
 the right ; and there must be a column for every denomina- 
 tion of weight or measure, in which column the respective 
 quantities must stand, so that feet will stand under feet, inch- 
 es under inches, pounds under pounds, and ounces under 
 ounces, &c. Add now the column toward the right, which 
 in this example amounts to 20 inches, or 1 foot 8 inches, we 
 therefore put down the 8 inches under the column of inches, 
 and add the 1 foot to the column of feet, which comes to 4 
 feet ; that is, 1 yard and 1 foot. The 1 foot is put down un- 
 der the column of feet, and the 1 yard is added, or carried, as 
 it is usually called, to the column of yards, whose sum is 20. 
 
 If we have to add 2 tons tons cwt quar< lbs> oz> 
 
 2 cwt. 1 quar. 17 Ibs. 10 2 2 1 17 10 
 
 oz. avoirdupois, to 12 tons 12 10 2 2 
 
 10 cwt. 2 Ibs. 2 oz., 2 cwt. 2 1 18 3 
 
 1 quar. 18 Ibs. 3 oz., and Q o Q 911 
 
 9 Ibs. 11 oz.; then, from ~^g ^ g j-g JQ~ 
 
 what was remarked above, 
 
 they will be put down as in the margin. Then the sum of 
 the right hand column is 26 oz., which is 1 Ib. 10 oz., we 
 put down the 10 in the column of oz., and carry the 1 Ib. to 
 the column of Ibs. ,which is next ; and this when added 
 comes to 47 Ibs., that is, 1 quar. and 19 Ibs. ; the 19 is put 
 in the column of Ibs. and the 1 is carried to that of quars., 
 which comes to 3, which not amounting to 1 cwt. we put 
 down the 3 in the column of quars. and carry nothing to 
 the column of cwts., which, when added, amounts to 14, this 
 we put down, and, as it does not amount to 20 cwt. or 1 ton, 
 we carry nothing to the column of tons ; and when this co- 
 lumn is added, its sum is 14. 
 
 25. In Subtraction the same principle of arrangement is to 
 
 

 28 ARITHMETIC. 
 
 be observed, and the lesser quantity is to be put under the 
 greater. If we have to subtract 1 ton 13 cwt. 2 quars. 17 Ibs. 
 12 oz., from 9 tons 8 cwt. tons cwt quars . lbs< oz> 
 1 quar. 4 Ibs 7 oz. avoirdu- 98147 
 pois, they are arranged as j 13 2 17 12 
 
 in the margin. We begin -^ ^g g j^ jj-j- 
 
 to subtract at the lowest 
 
 denomination, viz. oz. 12 oz. from 7 oz. we cannot, but 
 we add a Ib. or 16 oz. to the 7, which is supposed to be 
 borrowed from the column of Ibs. which stands next it, 
 towards the left ; now 16 added to 7 makes 23, and 12 from 
 23 leaves 11, which is put down in the column of oz. Now 
 we must pay back to the column of Ibs. the pound or 16 oz. 
 which we borrowed, therefore, it is 18 from 4. Here we 
 have to borrow from the column of quars., and 1 quar. 
 being 28 Ibs. we borrow 28, then 28 and 4 are 32, there- 
 fore 18 from 32 leaves 14, which is put down, and the 1 
 quar. paid back to the column of quars.; 3 from 1, we 
 cannot, and must borrow 1 cwt. or 4 quars., therefore 3 
 from 5 and 2 remains, which is put down. Add 1 to 13 for the 
 1 cwt. that was borrowed, then 14 from 8, we cannot, but 
 borrow 20 from the next column, then 14 from 28 and 14 
 remains. Pay back to the column of tons the 1 ton, or 20 
 cwt. which we borrowed, then 2 from 9 and 7 remains, which 
 is put down. 
 
 The same principle holds in other examples, the only va- 
 riation being that the numbers to be borrowed from the next 
 higher column, will depend upon the relative values of these 
 columns, which may be known by examining a table of the 
 particular weight or measure to which the example may refer. 
 
 26. In Mu] tiplication, which is only a short way of perform- 
 ing addition in particular cases ; the principles are nearly 
 similar : thus, to multiply 3 tons 2 cwt. 2 quars. 6 Ibs. 10 
 oz. by 3; they are arranged tons cwt quarg lbs ^ 
 as in margin. 1 hen the first 3 2 2 6 10 
 product is 30 oz. or 1 Ib. 3 
 
 which is carried to the co- ^ ~ HF TT" 
 
 lumn of lbs., and 14 oz., 
 
 which is put down in the column of oz. The product of 
 the lbs. is 18, and the one Ib. carried is 19, which not 
 amounting to 28 lbs. or 1 quar., nothing is to be carried to 
 the column of quars. The product of the quars. is 6, 
 which is 1 cwt. to be carried and 2 quars. to be put down. 
 
COMPOUND NUMBERS. 
 
 29 
 
 The product of cwts. is 6, and the one carried from the 
 former column makes 7, nothing being carried ; the co- 
 lumn of tons is 9. By examining the following examples, 
 and referring to the tables of weights and measures, the 
 general application may be easily inferred. See Appendix 
 to Arithmetic. 
 
 Degrees, min. seconds. yds. feet. inch. 8th parts. 
 
 23 
 
 14 
 
 17 
 6 
 
 17 
 
 2 
 
 9 
 
 6 
 5 
 
 139 25 42 89 J2 _0 6 
 
 Carry by ~60 60 ~3 12 ~8 
 
 27. It may not be out of place here to notice, Duodeci- 
 mal, or whutis commonly called Cross Multiplication ; which 
 is very useful to artificers in general, in measuring timber, &c. 
 The foot is divided into 12 inches, each inch into 12 parts, 
 and each part again into 12 seconds ; these last, however, 
 are so small, that they are generally neglected in calculation. 
 If we wish to find the surface of a plank, whose breadth 
 
 8 5 
 
 1 
 
 5 
 
 10 
 
 11 
 
 is I foot 7 inches, and length 8 feet 5 
 
 inches, we place the one under the other, 
 
 feet under feet, inches under inches, &c., 
 
 as in the margin. Multiply the inches 
 
 and feet in the upper line, by the feet 
 
 in the under line, placing the product ^ 
 
 of the inches, under the inches, and that of the feet, under 
 
 the feet. Then multiply the inches and feet, of the upper 
 
 line, by the inches in the under line, placing the product 
 
 one place further towards the right, and carry by twelves 
 
 where necessary ; as in this example, 7 times 5 is 35, that 
 
 is, two twelves and 11 over; the 11 is put down, and the 
 
 2 added to the product of the next column, 7 times 8 is 
 
 56, and the 2 carried makes 58, that is four twelves and 
 
 10 over ; the 10 is put down, and the 4 carried to the next 
 
 column. These are now added, observing again to carry 
 
 by twelves. 
 
 feet. inch. feet. inch, parts. 
 
 47 35 4 6 
 
 84 12 3 4 
 
 36 
 I 
 
 "38" 
 
 424 
 
 8 
 
 434~ 
 
 6 
 10 
 11 
 
 3 11 
 
30 ARITHMETIC. 
 
 The feet in the example are square feet, but the inches 
 are not square, as might be thought at first sight, but 12th 
 parts of a square foot ; and also the numbers standing in 
 the third place, are 12th parts of these 12 parts of a foot, 
 and so on. 
 
 28. Before we consider the Division of compound num- 
 bers, it will be necessary to attend a little to the nature of 
 reduction. This is usually thought by beginners to be very 
 perplexing, but a little attention to the principle, will ob- 
 viate all this apparent difficulty. 
 
 In every lineal foot there are 12 inches, and therefore 
 there will be 12 times as many inches, in any number of 
 feet, as there are feet ; thus, in 8 feet there are 8 times 12, 
 that is, 96 inches. In every Ib. avoirdupois there are 16 
 ounces, therefore in 18 Ibs. there are 18 times 16, that is, 
 288 ounces. So that we multiply the higher denomina- 
 tion, by that number of the lower which makes one of the 
 higher, and the product is the number of the lower contained 
 in the number of the higher, which we multiply. In the pre- 
 vious examples, feet and pounds are the higher denomina- 
 tions, and inches and ounces are the lower. From these 
 remarks it will be easy to see, how we proceed in finding 
 the number of & parts of an inch contained in 3 yards 2 
 feet 7 inches, and $ parts, long measure. Bring the yards 
 to feet, 3 multiplied by 3 are 9, to which we add the 2 
 feet, which make 11. This brought to inches, is 11 mul- 
 tiplied by 12, or 132, to which we add the 7 inches, making 
 139. This brought to parts gives 139, multiplied by 8, 
 that is, 1112, to which we add the 5 eighth parts, making 
 1117 the answer. 
 
 The examples subjoined are managed in a like manner ; 
 the multipliers varying with the kind of weight or mea- 
 sure. 
 
 cwt. quar. Jbs. acres. roods. poles. 
 
 27 1 22 22 3 24 
 
 4 mult. 4 mult. 
 
 108 quars. 88 roods 
 
 1 add 3 add 
 
 109 quars. 91 roods 
 
 28 mult. 40 mult. 
 
 3052 Ibs. 3640 poles 
 
 22 add 24 add 
 
 3074 Ibs. 3664 poles. 
 
COMPOUND NUMBERS. 31 
 
 The work is reversed, when we wish to ascertain how 
 many of a higher denomination are contained in any num- 
 ber of a lower. Thus, in 1440 inches, long measure, there 
 will be one foot for every 12 inches, we therefore divide 
 1440 by 12, and the quotient will be the number of feet, 
 that is, 120 feet. Then there is no remainder, but if there 
 had, it would have been of the same kind with the dividend, 
 that is, inches. In the same way tind how many tons, cwts.^> 
 quars. and Ibs., are contained in 12345678 oz. 
 
 oz. in 1 Ib. 16 ) 12345678 ounces. 
 Ibs. in 1 quar. 28 ""7771604 Ibs. 14 oz. 
 quars. in cwt. 4 )27557 quars. 8 Ibs. 
 
 cwt. in 1 ton 20 )6889 cwt. 1 quar. 
 
 344 tons 9 cwt. 
 
 The answer therefore is 344 tons 9 cwt. 1 quar. 8 Ib. 14 
 oz. which may be proved by reducing the work to ounces 
 by the method given above. 
 
 29. It is frequently of great use, to express compound 
 numbers fractionally ; thus, so many feet and inches as the 
 fraction of a yard. What fraction of a yard is 2 feet 8 
 inches ? Now, from what has been said on vulgar fractions, 
 it will be easily seen that one yard is here the unit, or de- 
 nominator of the fraction, which must of course be brought 
 to inches. Now there are 36 inches in one yard, which 
 must be the denominator of the fraction, and the numera- 
 tor will be the quantity taken ; that is, 2 feet 8 inches re- 
 duced to inches, or 32 inches. The fraction therefore is 
 |f, or simplified f , which, turned into a decimal, is 0'8888, 
 one yard being 1. So likewise, what fraction of a cwt. is 
 2 qrs. 14 Ibs. 3 oz.? This last reduced to ounces is 1123, 
 which is the numerator of the fraction, and the denomina- 
 tor is 1 cwt. reduced to oz., or 1792 oz.; the fraction is 
 therefore j^f f , which is expressed decimally 0'6264. We 
 think that these examples will be sufficient to show the 
 mode of procedure, and it remains for us to consider the 
 reverse of this ; to estimate the value of such fractions in 
 terms of the weight or measure to which they refer. 
 
 30. It will be easily seen, that one-half of a foot is twelve 
 times greater than one-half of an inch, or that any given 
 part of a foot, is a twelve times greater part of an inch ; thus, 
 5 of a foot is y of an inch ; so that to bring any fraction of 
 
32 ARITHMETIC. 
 
 a foot to the fraction of an inch, we have only to multiply the 
 numerator by 12. So likewise * of a pound avoirdupois, is 
 Y , of an ounce, and | of a yard is | of a foot, or 3 T 6 of an 
 inch; and if we divide the numerator by the denominator, 
 we get in the last example | of a yard, equivalent to 7} 
 inches. 
 
 What is the value of of 1 cwt. ? By applying the fore- 
 going principle it will be found that 5 of 1 cwt. is of a 
 quar., or a 28 times greater part of 1 lb., that is l ^ a ; that is 
 37 j Ibs. also j of 1 lb. is 16 times 3 of an ounce, or y, 
 equal to 5| ounces. 
 
 31. It will generally be found best to express these deci- 
 mally, thus, the last example will be | of a cwt. or 0.333 
 of a cwt., or 1.333 of a quar., or 37.666 of a pound. Thus 
 it appears that any fraction of a cwt. is 4 times greater than 
 a like fraction of a quarter, and any fraction of a quarter 
 is 28 times greater than a similar fraction of a pound ; 
 hence, to reduce a fraction of a higher to its value in a 
 lower denomination, we multiply the numerator of the frac- 
 tion, by that number which expresses how many of the lower 
 are contained in one of the higher, while the denominator 
 remains unaltered. On the other hand, to bring a fraction 
 from a lower to a higher denomination, the numerator re- 
 mains the same ; but we multiply the denominator by that 
 number which expresses how many of the lower is contained 
 in one of the higher. Thus J of an inch is J v of a foot, or 
 j^y of a yard ; or expressed in decimals 0.3333 of an inch, 
 or 0.0277 of a foot, or 0.00924 of a yard. 
 
 32. On a like principle the value of a decimal expressing 
 weight or measure, may be determined, simply by multiply- 
 ing the decimal by that number of the next lower denomi- 
 nation, which is contained in one of the higher, and cutting 
 off the proper number of decimals in the product, thus : 
 
 37689 of a cwt. 
 4 
 
 1.50756 quarters. 
 
 28 
 14.21168 pounds. 
 
 
 3.38688 
 Here it will be observed, that the integers or whole num 
 
POWERS AND ROOTS. 33 
 
 bers cut off are not multiplied, and the value of .37689 of 
 a cwt. is 1 quar. 14 Ibs. 3.386 oz. 
 
 We will conclude this chapter on compound numbers, 
 with some remarks on Division. The same arrangement is 
 to be observed here as in addition ; the greater quantity be- 
 ing towards the left of the lesser. 
 
 Let it be required to divide 13 yards 2 feet 8 inches by 
 4. We say 4 in 13, 3 times and 1 over, that is one yard, 
 which must be reduced to feet, the next lower denomination; 
 that is 3 feet, and the 2 feet are five feet now 4 in 5, 1 and 
 1 over, which last being a foot, must be reduced to inches ; 
 it is therefore 12 inches, and the 8 make 20; then 4 in 20, 
 5 times ; the answer therefore is 3 yards, 1 foot, 5 inches. 
 
 yards. feet. inch. yards. feet. inch. 
 3) 16 2 9(5 1 11 
 
 15 
 
 POWERS AND ROOTS. 
 
 
 
 32. THE square of any number is the product of that 
 number multiplied by itself: thus, the square of 2 is 4, the 
 square of 4 is 16, the square of 5 is 25, <fec. The cube of 
 any number is the product of that number multiplied twice 
 by itself : thus, the cube of 2 is 8, the cube of 3 is 27, the 
 cube of 4 is 64, &c. On the other hand, when we talk <K 
 
34 ARITHMETIC. 
 
 the square and cube roots of any numbers, we mean such 
 numbers that, when squared or cubed, will produce these 
 numbers : thus, 2 is the square root of 4, 3 is the square root 
 of 9, and 4 is the square root of 16, &c. In like manner, 3 
 is the cube root of 27, 4 the cube root of 64, 5 the cube root 
 of 125, <fec. The cube and cube root are said to be of higher 
 order than the square and square root; and there are higher 
 orders than these, with which we shall not concern ourselves, 
 as they will not occur in our calculations. The method of 
 raising any number to the square and cube powers, will be 
 sufficiently obvious from what has been said above ; but the 
 method of extracting the square and cube roots is not by any 
 means so easy. We shall give the rules for the extraction 
 of these roots ; and as they are long, we would recommend 
 the beginner to compare carefully each step in the example, 
 with that part of the rule to which it refers ; and by doing 
 so attentively, he will find that the greater part of the diffi- 
 culty will vanish. 
 
 33. The rule for extracting the square root is this : 
 
 First Commencing at the unit figure, point off periods 
 of two figures each, till all the figures in the given number 
 are exhausted. The second point will be above hundreds 
 in whole numbers, and hundredths in decimals. 
 
 Second Tf the first period towards the left be a complete 
 square, then put its square root at the end of the given num- 
 ber, by way of quotient ; and if the first period is not a com- 
 plete square, take the square root of the next less square. 
 
 Third Square this root now found, and subtract the 
 square from the first period ; to the remainder annex the 
 next period for a dividend, and for part of a divisor double 
 the root already obtained. 
 
 Fourth Try how often this part of the divisor now found 
 is contained in the dividend, omitting the last figure, and 
 annex the quotient thus found, not only to the root last 
 found, but also to the divisor, last used. 
 
 Fifth Then multiply and subtract, as in division ; to the 
 remainder bring down the next period, and, adding to the 
 divisor the figure of the root last found, proceed as before. 
 
 Sixth Continue this process till all the figures in the 
 given number have been used ; and if any thing remain, 
 proceed in the same manner to find decimals adding two 
 ciphers to find each figure. 
 
 * ff f f 
 
POWERS AND ROOTS. 
 
 35 
 
 The square root of 365 is required. 
 
 365(19-1049 
 1 
 
 29 
 9 
 
 265 
 261 
 
 381 
 1 
 
 400 
 381 
 
 38204 
 4 
 
 190000 
 152816 
 
 382089 
 9 
 
 3718400 
 3438801 
 
 382098 | 279599 
 
 The square root of 2 to six places of decimals is required. 
 2(1-414213 
 
 281 
 1 
 
 400 
 281 
 
 2824 
 4 
 
 11900 
 11296 
 
 28282 
 2 
 
 60400 
 56564 
 
 282841 I 383600 
 
 1 I 282841 
 
 2828423 | 100759 
 
 34. The easiest rule for the extraction of the cube root 
 is this : 
 
 By trials, take the nearest cube to the given number, 
 whether it be greater or less, and call it the assumed cube : 
 thus, if 29 was the given cube whose root was to be ex- 
 tracted, then, 3 times 3 times 3, or 27, is the nearest less 
 cube, and 4 times 4 times 4, or 64, is the nearest greatest 
 cube ; 27 is the nearer of the two, therefore, 27 is the as- 
 sumed cube. 
 
 Add double the given cube to the assumed cube, and 
 multiply this sum by the root of the assumed cube, and this 
 product divided by the given cube, added to twice the 
 
30 ARITHMETIC. 
 
 assumed cube, will give a quotient which will be the re- 
 quired root, nearly. 
 
 By using, in like manner, the cube of the last answer, as 
 an assumed root, and proceeding in the same manner, we 
 will get a second answer nearer the truth than the first, and 
 so on. 
 
 Find the cube root of 21035-8. 
 
 If 20 is assumed, its cube is 8000 ; if 30, its cube is 27000, 
 the one a great deal too small and the other too great : let us 
 therefore try some number between them, as 27 ; the cube 
 of this is 19683, which we shall call the assumed cube ; then, 
 twice the assumed cube is 39366 twice the given cube is 
 42071-6. 
 
 Therefore, the sum of the given cube and twice the as- 
 sumed cube is 60401-8, and the sum of the assumed cube 
 and twice the given cube is 61754-6. 
 
 Wherefore, by the rule, 
 
 61754-6 
 27 
 
 4322822 
 1235092 
 
 60401-8 ) 1667374-2 ( 27-6047 
 
 This quotient is the root nearly ; and by using 27*6047 in 
 the same way that we used 27, we will get an answer still 
 nearer the true root. For a Table of Powers and Roots, 
 see Grier's Mech. Diet. 
 
 THE SLIDING RULE. 
 
 35. We are indebted for the invention of this useful in- 
 strument to Edmond Gunter. It is a kind of logarithmic 
 table, whose great use is to obtain the solution of arithme- 
 tical questions by inspection, in the multiplication, division, 
 and extraction of the roots of numbers. It consists of two 
 equal pieces of boxwood, each 12 inches long, joined toge- 
 ther by a brass folding joint. In one of those pieces there 
 is a brass slider. On the face of this instrument, there are 
 engraven four lines, marked by the letters A, B, C, and D : 
 at the beginning of each line, the lines A and D being 
 
THE SLIDING RULE. 37 
 
 marked on the wood part of the rule, and B and C on the 
 brass slider. 
 
 36. Before the use of the sliding rule can be explained, 
 it is necessary that a correct idea should be formed of the 
 method of estimating the values of the several divisions on 
 these lines. Let it be observed, then, that whatever value 
 is given to the first 1 from the left, the numbers following, 
 viz. 2, 3, 4, 5, &c., will represent twice, thrice, four times, 
 &c., that value. If 1 is reckoned 1 or unity, then 2, 3, 4, &c., 
 will just signify two, three, four, <fcc. ; but if 1 is reckoned 
 10, then 2, 3, 4, &c., will represent 20, 30, 40, &c. If the 
 first 1 is reckoned 100, then 2, 3, 4, &c., will represent 200, 
 300, 400, <fec. The value of the 1 in the middle of the line 
 is always ten times that of the first 1 ; the value of the 
 second 2 is ten times that of the first 2 : so that if the value 
 of the first 1 be 10, that of the second 1 will be 100 ; the 
 first 2 will be 20, and the second 2 will be 200, &c. The 
 value of these divisions being understood, we may now at- 
 tend to the minute divisions between these. Now, on the 
 lines A, B, and C, there are 50 small divisions betwixt 1 and 
 
 2, 2 and 3, 3 and 4, <fec. ; and it follows, from the nature of 
 the larger divisions, that if the first 1 be reckoned 1, or 
 unity, each of these small divisions between 1 and 2, 2 and 3, 
 &c., will be -jJy, or *02 ; and supposing still the first 1 to be 
 unity, then the small divisions from thie second 1 to 2, 2 to 
 
 3, <fcc., will each be ten times greater than a-^, or *02, that 
 is, each of them will be -}, or -, or *2. In the same way, 
 if the first 1 represents 100, the first 2 will be 200 ; if the 
 second 1 be 1000, the second 2 will be 2000, &c. ; and on 
 the same principle as above the small divisions or 50th parts 
 will represent each -f v of 100, or 2, in the first half, or from 
 the first 1 to 2, 2 to 3, &c., and T V of 1000 r 20, in the second 
 half; or from the second 1 to the second 2,2 to 3, &c. 
 
 37. These divisions- being understood, we may proceed to 
 show the method of using this rule in the solution of arith- 
 metical questions. 
 
 38. To find the product of two numbers : 
 
 Move the slider, so that 1 on B stands against one of the 
 factors on A ; then the product will be found on the line A, 
 against the other factor on the line B. 
 
 Thus, to find the product of 3 by 8 : 
 
 Set 1 on B to 3 on A ; then against 8 on B will be found 
 the product 24 on A. 
 
 4 
 
38 ARITHMETIC. 
 
 For the product of 34 by 16 : 
 
 Set 1 on B against 16 on A, then look on B for 34, and 
 against it on the line A will be found the product 544. 
 
 39. To find the quotient of two numbers : 
 
 This may be done in two ways, either set 1 on the slider 
 B against the divisor on A, then against the dividend on A 
 the quotient will be found on B. Or, set the divisor on B 
 against 1 on A, then the quotient will be found on A against 
 the dividend on B ; therefore, in general, it is to be remem- 
 bered, that the quotient must always be found on the same 
 line on which 1 was taken, and the divisor and dividend on 
 the other line. 
 
 Thus, to find the quotient of 96 divided by 6 : 
 
 Move the slider till 1 on B stands against 6 on A ; then 
 the quotient 16 will be found on B against the dividend 
 96 on A. 
 
 In like manner, to find the quotient of 108 divided by 12, 
 we may take the latter form of the rule, thus : 
 
 Set 12 on B against 1 on A ; then on the line A will be 
 found the quotient 9 against 96 on B. 
 
 40. To solve questions in the rule of three or simple pro- 
 portion : 
 
 Set the first term on the slider B to the second on A ; 
 then on the line A will be found the fourth term, standing 
 against the third term on B. 
 
 If 4 Ibs. of brass cost 36 pence, what will 12 Ibs. cost ? 
 
 Move the slider so, that 4 on B will stand against 12 on 
 A ; then against 36 on B will be found the fourth term 108 
 on A. 
 
 41. To extract the square root : 
 
 Move the slider so, that the middle division on C, which 
 is marked 1, stands against 10 on the line D, then against the 
 given number on C the square root will be found on D. 
 
 It is to be observed before applying this rule, that if the 
 given number consists of an even number of places of figures, 
 ^as two, four, six, <fcc., it is to be found on the left hand part 
 of the line C ; but if it consists of any odd number of places, 
 as three, five, seven, &c., it is to be found on the right hand 
 side of C, 1 being the middle point of the line. 
 
 To find the square root of 81 : 
 
 Here the number of places are even, being two ; therefore, 
 the number 81 is sought for on the left hand side of the 
 line C. 
 
MARKS OP CONTRACTION. 39 
 
 Set 1 on C against 10 on D ; then against 81 on C will 
 be found 9, the square root on D. 
 
 For the square root of 144 : 
 
 Set 1 on C to 10 on D ; then against 144 on C will be 
 found the square root 12 on D. 
 
 42. To find the area of a board or plank : 
 
 The rule is, to multiply the length by the breadth, the 
 product will be the area ; therefore, by the sliding rule, 
 
 Set 12 on B against the breadth in inches on A ; then on 
 the line A will be found the surface in square feet, against 
 the length in feet on the line B. 
 
 To find the area of a plank 18 inches broad and 10 feet 
 3 inches long : 
 
 Move the slider so that 12 on B stands against 18 on A ; 
 then will 10^ on B stand against 15| on A, which 15| is 
 square feet. 
 
 This may be proved by cross multiplication. 
 10 3 
 1 6 
 10 3 
 516 
 
 15 4 6 
 
 43. For the solid content of timber. 
 
 The rule is to multiply length, breadth, and thickness all 
 together. 
 
 Set the length in feet on C to 12 on D ; then on C will 
 be found the content in feet against the square root of the 
 product of the depth and breadth in inches on D. 
 
 What is the content of a square log of timber, the length 
 of which is ten feet, and the side of its square base is 15 
 inches. 
 
 Set 10 on C against 12 on D; then will 15 on D stand 
 against the content 15| on C. 
 
 44. Other particulars on the measurement of timber will 
 be given hereafter, when we come to Mensuration. 
 
 MARKS OF CONTRACTION. 
 
 45. WE earnestly request that particular attention be paid 
 to this chapter, not because it is difficult, but because it is of 
 the greatest importance to the clear understanding of what 
 
40 ARITHMETIC. 
 
 follows in this book, and contributes greatly towards its 
 shortness and simplicity. 
 
 46. When we mean to say that one thing is equal to an- 
 other, we use this mark = thus, 3 added to 5=8, is read 
 thus, 3 added to 5 is equal to 8. 
 
 47. But the words, added to, may also be represented by 
 the mark -f thus, 3 + 5 = 8, is read, 3 added to, or plus, 5 is 
 equal to 8. 
 
 48. So likewise the difference of two numbers may be 
 represented by the mark , which is a short way of express- 
 ing the word subtract, thus, 5 3=2, is read from 5 sub- 
 tract 3 the difference is equal to 2 ; and thus, 3 + 62=7 
 is a short way of writing, to 3 add 6 and subtract 2, the result 
 is equal to 7. 
 
 49. After the same manner the mark x is used instead 
 of the words multiply by, thus, 3x2=6, is read 3 multi- 
 plied by 2 is equal to 6. 
 
 50. To show that the operation of division is to be per- 
 formed this mark is sometimes used, viz. -T-, which is a short 
 way of writing the words, divided by, thus, 1 5 -j- 3 = 5, is read 
 15 divided by 3 is equal to 5 : but we will in general place 
 the divisor below a line with the dividend above it, on the 
 principle stated in vulgar fractions, thus, 1 / =5 the same 
 as 15-i-3 = 5. 
 
 51. The square of any number or quantity is marked by a 
 small 2 placed at its upper right hand corner, thus, 3 2 =9 is 
 read, the square of 3 is 9. The cube is marked by a 3 placed 
 in the same way, as 3 3 =27, that is, the cube of 3 is 27. 
 
 The square root is noted in a similar manner by the frac- 
 tion * placed in the same way, as 9 i =3, and so likewise the 
 cube root, as 27* = 3 ; but the square root is often denoted 
 by^/placed before the number or quantity, thus, x/9=9^ = 3, 
 and the cube root, in like manner, by ^/, thus, <$/27=27*=3. 
 
 52. Parentheses ( ) are used to show that all the numbers 
 within them are to be operated upon as if they were only one ; 
 thus, 3+2x5, means that 3 is to be added to the product of 
 2 and 5, that is, the amount of this is 13 ; but (3+2) x5, 
 means that 3 and 2, that is, 5, is to be multiplied by 5, and 
 the result will be 25 ; a very different thing from what it 
 was before, which arises entirely from the use of paren- 
 theses. In like manner 3 + 2 3 =7, but (3 + 2) a =25 ; here, 
 as in every other case, the whole of the numbers within 
 the parentheses are taken as one whole, and as such **e 
 
PROPORTION. 41 
 
 affected by whatever is without the parentheses. The 
 same thing is often marked by drawing a line over all the 
 numbers or quantities to be taken as one whole ; thus, instead 
 of (3+2)x5, we may write 3+2x5; also (6x4) 3x2 
 is the same as 6x4 3 X 2, both being equal to 42. 
 
 53. The rule for the measurement of the surface of timber, 
 given in our remarks on the sliding rule, may be expressed 
 thus, length X breadth =area; and the rule for simple pro- 
 portion, to be given in the next chapter, may also be written 
 thus : 
 
 Second term X third term, 
 
 = fourth term. 
 
 first term, 
 
 54. It is obvious that this is merely a kind of short hand 
 which might be carried still farther ; for instance, in the 
 last example we might make F stand for the first term, 
 S for the second, T for the third, and for the last, and 
 the rule would then be 
 
 L f z = 
 
 55. We again insist that the young reader will read this 
 chapter carefully over. 
 
 PROPORTION. 
 
 56. When four numbers following each other are such 
 that the first is as many times greater or less than the 
 second, as the third is greater or less than the fourth, they 
 are said to be in proportion ; thus, 2, 4, 3, 6, usually written 
 thus, 2 : 4 : : 3 : 6 ; the mark : being put for the words, is 
 to, and : : for, as, so that this would be read, 2 is to 4 as 3 
 is to 6. Here the first is half the second, and the third is 
 half the fourth, and they are therefore in proportion ; but 
 they may be arranged otherwise and yet be in proportion, 
 thus, 4 : 2 : : 6 : 3, where the first is twice as large as the 
 second, and the third is twice as large as the fourth. In all 
 cases the two middle terms are called the means, and the 
 two outer terms are called the extremes. The product of 
 the two means is equal to that of the two extremes, thus in 
 the last example, 2x6=12, and 4x3 = 12. Now, if we 
 wanted the last term, to wit, 3, it could easily be found by 
 
 4* 
 
42 ARITHMETIC. 
 
 means of this property of numbers in proportion. If we had 
 only three terms given, as 4 : 2 : : 6, to find the fourth in 
 proportion, which is the last extreme, and 4 is the first 
 extreme. Now, we must find such a number, that, when 
 multiplied by 4, the product will be equal to the product of 
 the means; 2x6=12, to find such a number we have only, 
 by the definition of division, to divide the product of the 
 two means, viz. 12 by the first extreme 4, and the quotient 
 3 will be the answer. So universally 6:9:: 12 : where 
 the last term will be found, as before, by multiplying the 
 twa means 12x9 = 108, and dividing the product 108 by 
 the first extreme 6, the quotient will be the last extreme 18, 
 hence 6 : 9 : : 12 : 18. The rule may be expressed simply 
 thus : let F stand for the first term, S the second, T the 
 
 S x T 
 
 third, and the last, then we have - = , and this 
 
 r 
 
 rule holds true whether the numbers be whole or fractional ; 
 and here it may be observed, that it will in most, if not in 
 all cases, be best to turn all vulgar fractions, when they 
 occur, into decimals ; thus, 2 : 3f : : 6| : or f : y : : \ 5 *: 
 
 2 = f = 2-5 1 
 
 3| = U = 3-666 V 2-5 : 3-666 :: 6-25 : 
 
 6 = *f = 6-25 J 
 
 Here the mode of determining the fourth term is the 
 same in all ; the two means being X, and their product -v-, 
 by the first term. This is usually called the rule of three, 
 and is of the utmost utility in practical arithmetic. We 
 shall now show how it is to be applied. 
 
 If we pay 40 pence for 2 feet of wood, how much will we 
 pay for 6 feet at the same rate ? Here it is clear we will 
 pay in proportion to the quantity of wood ; for as many 
 times as we have 2 feet, we will pay so many times 40 pence ; 
 that is, the price will be in proportion to the quantity of 
 wood. So that we may say, as the one quantity of wood is 
 to another quantity, so will be the price of the first quantity 
 to the price of the second. Hence the terms in the question 
 will stand arranged thus : 2 : 6 :: 40 : 120, which term 120 
 is the price of 6 feet, and is found by the rule given above ; 
 
 57. In every question in simple proportion, there will 
 always be three terms, one of which is of the same kind 
 
COMPOUND PIIOPO-RTION. 43 
 
 with the answer sought, whether it be money, measure, 
 time, force, or any tiling, which term in the question we 
 put in the third place ; as in the last question the answer 
 was to be money, and therefore the money in the question, 
 40 pence, was placed as the third term. When this is done, 
 we next consider whether the answer will be greater or less 
 than the third term, and place the greater or less of the 
 other two terms next it in the second place, and the other 
 one first, as the answer may require ; after which, employ 
 the rule given above to find the answer. 
 
 58. As, for example, 40 men will do a piece of work in 
 15 days, in how many days will 20 men do the same? 
 Here the answer must be days ; consequently, 15 goes in 
 the third term, and 20 men will take more time than 40 to 
 do it, therefore we must put the greatest in the second place, 
 and the least in the first ; and it therefore stands thus : 
 20 : 40 : : 15 : the answer 30, which is found by the rule. 
 
 COMPOUND PROPORTION. 
 
 59. COMPOUND PROPORTION depends entirely on the same 
 principles as simple proportion. For instance, if 2 feet of 
 fir cost 40 pence, what will 6 feet of mahogany cost, 3 feet 
 of mahogany being equal in value to 9 of fir. Here we 
 may find the price of the 6 feet of mahogany as if they 
 were fir, and it comes out, by the last article, 120 pence, 
 but 3 is to 9 as the price of fir is to that of mahogany ; 
 therefore we put the 120, the price of 6 feet of fir, in the 
 third term, and state the proportion, 3:9:: 120 : 360, the 
 price of 6 feet of mahogany. The same would have been 
 more easily found by stating it thus : 
 
 2 : 6 : : ? , n 
 
 3 : 9 :: $ 4 
 
 6 : 54 : : 4<f7 360. Ans. 
 
 where the proportions are stated under each other, and 
 multiplied together, which produces 3x2 = 6 and 6x9 
 = 54, two terms of a new proportion, in the simple rule, 
 where 40 is the third term ; and this is only the particular 
 example of a general rule, where we may have as many 
 
44 ARITHMETIC. 
 
 proportions as we please reduced to the form of a simple 
 question in the rule of three. As, therefore, that quantity 
 which is of the same kind with the required answer is put 
 in the third term, the rest will be found to go in pairs ; 
 two expressing relation of price, two relation of quality, 
 two relation of time, which must be put in proper order in 
 the first and second terms, as directed for simple propor- 
 tion. When this is done, all the first terms of these several 
 proportions are to be multiplied together for a new first 
 term, all the second terms together for a new second term, 
 which being placed with the third, in the form of simple 
 proportion, and operated upon as there directed, will give 
 the answer. 
 
 Forty boys are set to dig a trench in summer; 14 spade- 
 fuls can be dug in summer for 12 in winter; 6 men can do 
 as much as 13 boys ; and 16 men can do it in 104 days in 
 winter : how long will the boys take ? Here the answer 
 is to l?e, how many days ? We have in the question 104 
 days ; the third term, relative of difficulty, 14 spadefuls 
 and 12 spadefuls ; of strength, 6 men to 13 boys ; relation 
 of numbers, 16 to 40 ; which will be stated thus : 
 Relation of number, 40 : 16"} makes the time less. 
 
 Relation of difficulty, 14 : 12 v :: 104 makes the time less. 
 Relation of strength, 6:13J makes the time greater. 
 
 Product, 3360 : 2496 : : 104 : 77 T V F days, Ans. 
 
 ARITHMETICAL AND GEOMETRICAL PROPORTIONS 
 AND PROGRESSIONS. 
 
 60. THE subject of this chapter is often referred to in 
 elementary books on mechanical science ; and for this rea- 
 son, we shall draw the attention of the reader, for a little 
 while, to the subject. 
 
 61. When we inquire as to the difference of two num- 
 bers, we inquire for their arithmetical ratio ; but when we 
 inquire as to the quotient of two numbers, we inquire for 
 their geometrical ratio. Thus, 12 3 = 9 and 12 -~- 3 = 
 4 ; here 9 is the arithmetical ratio of 12 and 3, and 4 is the 
 geometrical ratio of the same numbers. From this it will 
 be seen, that ratio and relation are terms which have the 
 same signification. 
 
PROPORTIONS AND PROGRESSIONS. 45 
 
 62. When four numbers follow each other, and are such 
 that the difference of the first two is the same as, or equal 
 to, the difference of the last two, these numbers are said to 
 be in arithmetical proportion; thus the numbers 12, 7, 9, 4, 
 form an arithmetical proportion, because the difference of 
 12 and 7 is the same as the difference of 9 and 4, both being 
 5. The numbers in an arithmetical proportion may be 
 varied in their position, but still the result will be an arith- 
 metical proportion ; for instance, 12, 7, 9, 4, may be written 
 12, 9, 7, 4, or 9, 12, 4, 7 : but the most remarkable pro- 
 perty of arithmetical proportion is this, that the sum of the 
 first and last terms is always equal to the sum of the second 
 and third ; thus, 12 + 4 = 16 and 9 + 7 = 16; and from 
 this it evidently follows, that to find the fourth term, we 
 add the second and third terms together, and from their 
 sum subtract the first ; the remainder is the fourth term. 
 
 63. An arithmetical progression is a series of numbers 
 such, that, in taking any three numbers in succession, the 
 difference of the first and second is the same as the differ- 
 ence of the second and third ; thus, 1, 2, 3, 4, 5, 6, 7, 8, or 
 14, 12, 10, 8, 6, 4, 2, where the difference of the succeeding 
 numbers in the first is 1, and in the second 2. As the num- 
 bers in the first increase from the beginning, it is called an 
 increasing arithmetical series, or progression, and as they 
 decrease, from the beginning, in the second example, it is 
 called a decreasing arithmetical progression, or series. 
 
 64. Let us place any one of these progressions above 
 itself, in this manner : 
 
 2 
 14 
 
 4 
 12 
 
 6 
 10 
 
 8 
 8 
 
 10 
 6 
 
 12 
 4 
 
 14 
 2 
 
 16 16 16 16 16 16 16 
 
 writing the same progression as increasing and decreasing, 
 the respective terms of the one being directly under the re- 
 spective terms of the other in columns, as above, the lowest 
 line of the three being the sums of the several columns, 
 which are all seen to be 16. Now, it will be obvious, that 
 the first column consists of the first and last terms of the 
 series, 2, 4, 6, &c., with their sum, which is 16 ; the second 
 column consists of the first but one and the last but one of 
 the terms of the same series, together with their sum, which 
 is likewise 16. The third column consists of the first but 
 two and the last but two terms, with their sum, which again 
 
46 ARITHMETIC. 
 
 is 16. We may therefore infer, that, in an arithmetical 
 progression, the sum of any two terms, equally distant from 
 the first and last, is equal to the sum of any other two terms 
 which are equally distant from the first and last, or equal to 
 the sum of the first and last. It will also be seen, that the 
 under line, or sum of the two series, is therefore equal to 
 twice the sum of one of the progressions. Now, there are 
 seven sixteens, or 112, which is twice the sum of the pro- 
 gression, therefore 2)112(56 is the sum of the progression. 
 
 65. It is also apparent, that if any term be wanting, that 
 term may be found by adding the common difference, or 
 arithmetical ratio, of the progression, to the term going be- 
 fore the term sought, or subtracting it from the term which 
 follows, if the series is increasing, but the reverse if decreas- 
 ing. Thus, 2, 4, 8, the term awanting between the 4 and 8, 
 may be supplied, either by adding the common difference, 
 2, to the 4, or subtracting it from the 8, and we thus get 6. 
 The same may be found by taking the sum of the terms on 
 each side of the term sought, and dividing by 2 ; thus, 4 + 8 
 = 12, then 2)12(6, the same as before ; so, likewise, 3, 5, 
 7, 9, 13. To fill up the term awanting between 9 and 13, 
 we have 9 + 13 = 22, therefore 2)22(11, which is the 
 number sought, and it is called the arithmetical mean. 
 
 66. The quotient of two numbers is their geometrical 
 ratio, and thus a fraction, as T 6 , expresses the ratio of 6 to 
 12, and therefore 1 : 2 : : 6 : 12 is the same thing as = ^. 
 We thus get another view of the rule of three, and it is use- 
 ful to view any subject of this kind in different ways, as by 
 so doing we acquire a more accurate and extensive know- 
 ledge of its nature and application. The limits of this book 
 will not permit us to dwell on this subject, as we have dis- 
 cussed the subject of proportion in a former chapter. 
 
 67. In a series of progression of numbers, as 2, 4, 8, 16, 
 32, 64, where the quotient of any term, and that which fol- 
 >ows it, is equal to the quotient of any other term, and that 
 which follows it, such progression is said to be geometrical. 
 
 68. Let us take the geometrical progression, 2, 6, 18, 54, 
 162, and write it as we did the arithmetical, both as in in- 
 creasing and decreasing series, thus : 
 
 2 6 18 54 162 
 
 162 54 18 6 2 
 
 324 324 324 324 324 
 
PROPORTIONS AND PROGRESSIONS. 
 
 47 
 
 Here we observe that the product of the terms of each 
 column is the same, whatever column we take ; and we 
 arrive at a knowledge of the fact, that the product of the 
 first and last terms is the same as the product of any other 
 two terms, one of which is as many places distant from the 
 first as the other is distant from the last term. 
 
 69. If one term in the above series were wanting, for in- 
 stance the second, that is, 6, take the terms on each side of 
 it, and find their product, 2x18 = 36, now the square root 
 .of this, or 6, will be the number sought, which is called 
 the geometrical mean. In like manner we might find the 
 geometrical mean between 18 and 162 ; thus, 18 x 162 = 
 2916, the square root of which is 29161 = 54, the number 
 sought. The geometrical mean is sometimes called the 
 mean proportional. 
 
 70. The sum of any geometrical series may be found 
 thus : 
 
 (The greater extreme X ratio) less extreme, 
 
 i 5 ratio i ! =the sum of series. 
 
 thus the sum of the last series is 
 (162 x 3) 2_ 486 2 _ 484 _ 
 
 31 O O 
 
 J & & 
 
 71. Terms relating to proportion often occur in books 
 read by mechanics, of which it would be useful to know 
 the signification; and, to prevent their being misapplied, 
 we give the following illustration. If there be four num- 
 bers in proportion, as 4 : 16 : : 3 : 12, then, 
 
 Directly, 4 
 
 Alternately, 4 
 
 Inversely, 16 
 
 C Compounded, ...4 + 16 
 
 \ That is, ;20 
 
 C Divided, 4 - 
 
 I That is, 12 
 
 ("Converted,. ...4 
 \ That is, 4 
 
 1 Also, 4 
 
 LThat is, 4 
 
 5 Mixed, 4 -h 16 
 
 2 That is, 20 
 
 To these may be added, duplicate ratio, or ratio of the 
 
 
 16 
 
 
 
 3 
 
 12 
 
 
 3 
 
 
 1 
 
 5 
 
 12 
 
 
 K . 4 
 
 
 1 
 
 2 
 
 3 
 
 16 
 
 16 
 
 
 3-f 
 
 12 
 
 12 
 
 
 16 
 
 
 1 
 
 5 
 
 12 
 
 -16 
 
 16 
 
 
 3 
 
 12 
 
 12 
 
 
 16 
 
 
 :' 
 
 9 
 
 12 
 
 16 
 
 + 4 
 
 
 3 
 
 12 
 
 + 3 
 
 S 
 
 50 
 
 
 3 
 
 15 
 
 
 16 
 
 4 
 
 
 3 
 
 12 
 
 3 
 
 1 
 
 2 
 
 
 3 
 
 9 
 
 
 4- 
 
 -16 
 
 3 - 
 
 f 12 
 
 3- 
 
 -12 
 
 ] 
 
 2 
 
 
 15 
 
 9 
 
 
 
48 ARITHMETIC. 
 
 squares ; triplicate ratio, or ratio of the cubes ; sub-dupli- 
 cate ratio, or ratio of the square roots ; and sub-triplicate 
 ratio, or ratio of the cube roots. 
 
 POSITION. 
 
 72. POSITION is a rule in which, from the assumption of 
 one or more false answers to a problem, the true one is 
 obtained. 
 
 73. It admits of two varieties, single position and double 
 position. 
 
 74. In single position the answer is obtained by one as- 
 sumption ; in double position it is obtained by two. 
 
 75. Single position maybe applied in resolving problems, 
 in which the required number is any how increased or dimi- 
 nished in any given ratio ; such as when it is increased or 
 diminished by any part of itself, or when it is multiplied 
 or divided by any number. 
 
 76. Double position is used, when the result obtained by 
 increasing or diminishing the required number in a given 
 ratio, is increased or diminished by some number which is 
 no known part of the required number ; or when any root 
 or power of the required number, is either directly or in- 
 directly contained in the result given in the question. 
 
 SINGLE POSITION. 
 
 77. Pule. Assume any number, and perform on it the 
 operations mentioned in the question as being performed on 
 the required number. Then, as the result thus obtained is 
 to the assumed number, so is the result given in the ques- 
 tion to the number required. 
 
 Exam. Required a number to which if one half, one- 
 third, one-fourth, and one-fifth of itself be added, the sum 
 may be 1644. 
 
 Suppose the number to be 60 : then, if to 60 one- 
 half, one-third, one-fourth, and one-fifth of itself be 
 added, the sum is 137. Hence, according to the rule, as 
 137 : 1644 : : 60 : 720, the number required. The truth 
 of the result is proved by adding to 720 one-half, one -third, 
 <fcc., of itself, and the sum is found to be 1644. The num- 
 ber 60 was here assumed, not as being near the truth, but 
 
POSITION. 49 
 
 as being a multiple of 2, 3, 4, and 5 ; and in this way the 
 operation was kept free from fractions. By the assumption 
 of any other number, however, the answer would have 
 been found correctly, but not so easily. The reason of the 
 operation is so obvious as not to require illustration. 
 
 DOUBLE POSITION. 
 
 78. Rule. Assume two different numbers, and perform 
 on them separately the operations indicated in the question. 
 Then, as the difference of the results thus obtained is to 
 the difference of the assumed numbers, so is the difference 
 between the true result and either of the others to the cor- 
 rection to be applied to the assumed number which gave 
 this result. Add the correction to this number, if the cor- 
 responding result was too small ; otherwise, subtract it. 
 
 79. A more general rule is this. Having assumed two 
 different numbers, perform on them separately the opera- 
 tions indicated in the question, and find the errors of the 
 results. Then, as the difference of the errors, if both results 
 be too great or both too little, or as the sum of the errors, 
 if one result be too great and the other too small, is to the 
 difference of the assumed numbers, so is either error to the 
 corrections-to be applied to the number that produced that 
 error. 
 
 80. When any root or power of the required number is 
 in any way contained in the result given in the question, 
 the preceding rules will only give an approximation to the 
 required number. In this case the assumed numbers should 
 be taken as near the true answer as possible. Then, to ap- 
 proximate the required number still more nearly, assume 
 for a second operation the number found by the first, and 
 that one of the two first assumptions which was nearer the 
 true answer, or any other number that may appear to be 
 nearer it still. In this way, by repeating the operation as 
 often as may be necessary, the true result may be approxi- 
 mated to any assigned degree of accuracy. 
 
 81. It may be further observed also, that the method of 
 double position, besides its use in common arithmetic, is of 
 much utility in algebra, affording in many cases a very con- 
 venient mode of approximating the roots of equations, and 
 finding the value of unknown quantities in very compli- 
 cated expressions, without the usual reductions. 
 
 82. Exam. 1. Required a number, from which, if 2 be 
 
50 
 
 
 ARITHMETIC. 
 
 subtracted, one-third of the remainder will be 5 less than 
 half the required number. 
 
 Here, suppose the required number to be 8, from which 
 take 2, and one-third of the remainder is two: This being 
 taken from one-half of 8, the remainder is 2, the first result. 
 Suppose, again, the number to be 32, and from it take 2 : 
 one-third of the remainder is 10, which being taken from 
 the half of 32, the remainder is 6, the second result. Then, 
 the difference of the results being 4, the difference of the 
 assumed numbers 24, and the difference between 5, the true 
 result, and 6, the result nearest it, being 1 ; as 4 : 24 : : 1 : 6, 
 the correction to be subtracted from 32, since the result 6 
 was too great. Hence, the required number is 26. 
 
 83. Exam. 2. -If one person's age be now only four 
 times as great as another person's, though 7 years ago it 
 was six times as great ; what is the age of each ? 
 
 Here, suppose the age of the younger to be 12 years ; then 
 will the age of the older be 48. Take 7 from each of these, 
 and there will remain 5 and 41, their ages 7 years ago. 
 Now, 6 times 5 is 30, which, taken from 41, leaves an error 
 of 1 1 years. By supposing the age of the younger to be 15, 
 and proceeding in a similar manner, the error is found to 
 be 5 years. Hence, as 6, the difference of the errors, (both 
 results being too small,) is to 3, the difference of the as- 
 sumed numbers, so is 5, the less error, to 2, the correction ; 
 which, being added to 15, the sum, 17, is the age of the 
 younger, and consequently that of the older must be 70. 
 
 Both the rules above given for double position depend 
 on the principle, that the differences between the true and 
 the assumed numbers, are proportional to the differences 
 between the result given in the question and the results 
 arising from the assumed numbers. This principle is quite 
 correct in relation to all questions which in algebra would 
 be resolved by simple equations, but not in relation to any 
 others ; and hence, when applied to others, it gives only 
 approximations to the true results. The subject is of too 
 little importance to claim further illustration in this place. 
 
 84. Exam. 3. Required a number to which, if twice 
 its square be added, the sum will be 100. 
 
 It is easy to see that this number must be between 6 and 
 7. These numbers being assumed, therefore, the sum of 6 
 and twice its square is 78, and the sum of 7 and twice its 
 square is 105. Then, as 105 78 : 7- 6 : : 105 100: -18; 
 
 

 WEIGHTS AND MEASURES. 51 
 
 which, being taken from 7, the remainder, 6-82, is the 
 required number, nearly. To this let twice its square be 
 added, and the result is 99-8448. Then, as 105 99-8448 : 
 7 6-82 : : 105 100 : -1746 ; which, being taken from 7, 
 the remainder is 6*8254, the required number still more 
 nearly ; and if the operation were repeated with this and 
 the former approximate answer, the required number would 
 be found true for seven or eight figures. 
 
 APPENDIX TO ARITHMETIC, 
 
 COHTTAISriBTG 
 
 TABLES OF WEIGHTS AND MEASURES. 
 
 ENGLISH. 
 AVOIRDUPOIS WEIGHT. $ 
 
 Drachms. 
 
 16 = 1 Ounce. 
 
 256 = 16 = 1 Pound. 
 7168 = 448 = 28 = 1 Quarter. 
 286782 = 1792 = 112 = 4 = 1 Cwt. 
 573440 = 35840 = 2240 = 80 = 20 = 1 Ton. 
 Tons are marked t. ; hundred weights, cwt. ; quarters, 
 qr. ; pounds, Ib. ; ounces, oz. ; and drachms, dr. 
 
 TROY WEIGHT. 
 
 Grains. 
 
 24 = 1 Pennyweight. 
 480 = 20 = 1 Ounce. 
 5760 = 240 = 12 = 1 Pound. 
 Pounds are marked, Ib.; ounces, oz.; pennyweights, 
 dwt. ; and grains, gr. 
 
 LONG MEASURE. 
 
 Barley corns. r ? , 
 
 3 = 1 Inch. 
 
 36 = 12 = 1 Foot. 
 108 = 36 = 3 = 1 Yard. 
 594 = 198 = 16-5= 5-5= 1 Pole. SH**; 
 23760 = 7920 = 660 =220 = 40 = 1 Furlong. 
 190080 = 63360 =5280 =1760 =320 = 8 = 1 Mile. 
 

 ^ 
 
 52 ARITHMETIC. 
 
 
 SQUARE MEASURE. 
 
 Inches. 
 
 144 = 1 Foot. 
 1296 = 9=1 Yard. 
 39204= 272*= 30| = 1 Pole. Per*. 
 1568160 = 10890 = 1210 = 40 = 1 Rood. 
 6272640 = 43560 = 4840 = 160 = 4 = 1 Acre. 
 
 * ' k* .> -^ v 
 
 SOLID MEASURE. 
 
 Inches. 
 
 1728 = 1 Foot. 
 46656 = 27 = 1 Yard. 
 
 WINE MEASURE. 
 
 Pints. 
 
 2 = 1 Quart. 
 
 8 = 4=1 Gallon. 
 
 336 = 168 = 42 = 1 Tierce. 
 
 504 = 257 = 63 = 1-5= 1 Hogshead. 
 
 672 = 336 = 84 = 2 = 1-5= 1 Puncheon. 
 
 1008 = 504 = 126 = 3 =2 = 1-5= 1 Pipe. 
 
 2016 = 1008 =252 = 6 =4 =3 =2 = 1 Tun 
 
 ALE AND BEER MEASURE. 
 
 Pints. 
 
 2 = 1 Quart. 
 8 = 4 = 1 Gallon. 
 72 = 36 = 2=1 Firkin. 
 144 = 72 = 18 = 2 = 1 Kilderkin. 
 288 = 144 = 36 = 4=2 = 1 Barrel. 
 432=216= 54= 6 = 3 = 1-5= 1 Hogshead. 
 576 = 288 = 72 = 8 = 4=2 = 1-5= 1 Puncheon. 
 864 = 432 = 108 = 12 = 6 = 3 =2 = 1-5=1 Butt 
 
 DRY MEASURE. 
 
 Pints. 
 
 8 = 1 Gallon. 
 
 16 = 2 = 1 Peck. 
 
 64 = 8 = 4=1 Bushel. 
 
 256 = 32 = 16 = 4 = 1 Coom. 
 
 512 =64 =32 =8=2=1 Quarter. 
 
 2560 = 320 = 160 = 40 = 10 = 5=1 Wey. 
 
 5120 = 640 = 320 = 80 = 20 = 10 = 2 = 1 Last 
 
WEIGHTS AND MEASURES. 53 
 
 TIME. 
 
 60 seconds = 1 minute, 60 minutes = 1 hour, 
 
 24 hours = 1 day, 365 days = 1 year, nearly. 
 
 THE CIRCLE. 
 
 m 
 
 The circle is divided into 360 equal parts, called degrees. 
 Seconds. 
 
 60 = 1 Minute. 
 
 360 = 60 = 1 Degree. 
 32400 = 5400 = 90 = 1 Quadrant. 
 129600 = 21600 = 360 = 4 = 1 Circumference. 
 Degrees, minutes, and seconds, are marked 
 4 5' 6" 4 degrees, 5 minutes, 6 seconds. 
 
 REMARKS ON ENGLISH WEIGHTS AND MEASURES. 
 
 Troy weight is used frequently by chemists, and also in 
 weighing gold, silver, and jewels ; but all metals, except 
 gold and silver, are weighed by avoirdupois weight. 
 
 175 troy pounds are equal to 144 avoirdupois pounds. 
 
 175 troy ounces = 192 avoirdupois ounces. 
 
 14 oz., 11 dwt., 15 grs. troy = 1 Ib. avoirdupois. 
 
 18 dwt., 5 gr. troy = 1 oz. avoirdupois. 
 
 3 miles long measure = 1 league. 
 
 69 T ' T English miles = 60 geographical miles. 
 
 1089 Scottish acres = 1369 English acres. 
 
 A chaldron of coals in London = 36 bushels, and weighs 
 3136 Ibs. avoirdupois, or nearly 1 ton, 8 cwt. 
 
 The ale gallon contains 282 cubic inches, and the wine 
 gallon contains 231 cubic inches the wine gallon being to 
 the ale gallon nearly as 1 Ib. avoirdupois to 1 Ib. troy. 
 
 By an Act of Parliament passed in 1824, and carried 
 into execution in 1826, imperial weights and measures were 
 introduced by this. 
 
 The pound troy contains 5760 grains. 
 
 The pound avoirdupois contains 7000 grains. 
 
 The imperial gallon contains 277*274 cubic inches. 
 
 The bushel (dry measure) contains 2218*192 cubic 
 inches. 
 
 To find the value of the old in terms of the new, or the 
 reverse, the following table of multipliers is given. 
 
 5* 
 

 54 ARITHMETIC. 
 
 Dry. Wine. Ale. 
 
 To convert the old into new x 0-96943 0-83311 1-01704. 
 To convert new into old x 1-03153 1-20032 0-98324. 
 
 Examples. What is the value in imperial measure, of 
 32 wine gallons old measure ? 
 
 83311 x 32 = 26-65952 imperial gallons. 
 
 In like manner 4 bushels imperial measure = 1-03153 X 
 4 == 4-12612 old or Winchester bushels. 
 
 FRENCH WEIGHTS AND MEASURES. 
 
 Old System. 
 
 English Troy Grains. 
 The Paris Pound = 7561 
 
 Ounce == 472-5625 
 Gros = 59-0703 
 Grain = -8204 
 
 * -* 
 
 Eng. Inches. 
 
 The Paris Royal Foot of 12 Inches = 12-7977 
 
 The Inch = 1-0659 
 
 The Line, or one-twelfth of an inch = -0074 
 
 Eng. Cubical Feet. ' 
 
 The Paris Cubic Foot = 1-211273 
 
 The Cubic Inch = -000700 
 
 MEASURE OF CAPACITY. 
 
 The Paris pint contains 58-145 English cubical inches, 
 and the English wine pint contains 28*875 cubical inches ; 
 or the Paris pint contains 2-0171082 English pints ; there- 
 fore to reduce the Paris pint to the English, multiply by 
 2-0171082. 
 
 New System. 
 
 MEASURES OF LENGTH. 
 
 English Inches. 
 
 Millimetre = -03937 
 
 Centimetre == -39370 
 
 ,,, Decimetre = 3-93702 
 
 Metre = 39-37022 
 
 Decametre = 393-70226 
 
FRENCH WEIGHTS AND MEASURES 
 
 55 
 
 Hecatometre = 3937-02260 
 
 Chiliometre = 39370-22601 
 
 Myriometre = 393702-26014 
 
 M. P. Y. Ft. In. 
 
 A Decametre is = 00 10 2 9-7 
 
 A Hecatometre = 00 109 1 -1 
 
 A Chiliometre = 04 213 1 10-2 
 
 A Myriometre = 61 156 -6 
 
 Eight Chiliometres are nearly 5 English miles. 
 
 MEASURES OF CAPACITY. 
 
 English Cubic Inches. 
 
 Millilitre = -06102 
 
 Centilitre = -61024 
 
 Decilitre = 6-10244 
 
 Litre = 61-02442 
 
 Decalitre = 610-24429 
 
 Hecatolitre = 6102-44288 
 
 Chiliolitre = 61024-42878 
 
 Myriolitre = 610244-28778 
 
 A Litre is nearly 2- wine pints. 
 
 14 Decilitres are nearly 3 wine pints. 
 
 A Chiliolitre is a tun, 12 '75 wine gallons. 
 
 WEIGHTS. 
 
 Milligram me 
 
 Centigramme 
 
 Decigramme 
 
 Gramme 
 
 Decagramme ,- 
 
 Hecatogramme 
 
 Chiliogramme (Kilogram)' 
 Myriogramme < 
 
 English Grains. 
 
 0154 
 
 1544 
 
 1-5444 
 
 15-4440 
 
 154-4402 
 
 1544-4023 
 
 15444-0234 
 
 154440-2344 
 
 A Decagramme is 6 dwts. 10-44 gr. troy; or 5-65 dr. 
 avoirdupois. 
 
 A Hecatogramme is 3 oz. 8*5 dr. avoirdupois. 
 A Chiliogramme is 2 Ibs. 3 oz. 5 dr. avoirdupois. 
 A Myriogramme is 22 1*15 oz. avoirdupois. 
 100 Myriogrammes are 1 ton, wanting 32-8 Ibs. 
 
56 
 
 
 ARITHMETIC. 
 
 AGRARIAN MEASURES. 
 
 Are, 1 square Decametre* = 3*95 Perches. 
 
 Hecatare = 2 Acres, 1 Rood, 30*1 
 
 Perches. 
 
 FIR WOOD. 
 
 Decistre, 1-1 Oth Stere 
 Stere, 1 Cubic Metre* 
 
 = 3-5315 cub. ft. Eng. 
 = 35-3150 cub. ft. 
 
 DIVISION OF THE CIRCLE. 
 
 100 seconds 
 100 minutes 
 100 degrees 
 4 quadrants 
 
 1 minute. 
 1 degree. 
 1 quadrant. 
 1 circle. 
 
 THE ENGLISH DIVISION. 
 
 60 seconds 
 
 60 minutes 
 
 360 degrees 
 
 1 minute. 
 1 degree. 
 1 circle. 
 
 DIMENSIONS OF DRAWING PAPER IN FEET AND INCHES. 
 
 Ft. In. Ft In. 
 
 1 7 X 
 
 6 
 
 7 
 7 
 
 Demy 
 
 Medium 1 10 X 
 
 Royal 2 X 
 
 Super royal 2 3 X 
 
 Imperial 2 5 X 
 
 Elephant 2 3| X 
 
 Columbier 2 9| X 11 
 
 Atlas 29x22 
 
 Double elephant 34x22 
 
 Wove antiquarian 44x27 
 
GEOMETRY. 
 
 DEFINITIONS. 
 
 1. A POINT is that which has position, 
 but no magnitude, nor dimensions ; neither 
 length, breadth, nor thickness. 
 
 2. A Line is length, without breadth or 
 thickness. 
 
 3. A Surface or Superficies, is an exten- 
 sion or a figure of two dimensions, length 
 and breadth ; but without thickness. 
 
 4. A Body or Solid, is a figure of three 
 dimensions, namely, length, breadth, and 
 depth or thickness. 
 
 5. Lines are either Right, or Curved, or mixed of these 
 two. 
 
 6. A Right Line, or Straight Line, lies all in the same 
 direction, between its extremities ; and is the shortest dis- 
 tance between two points. 
 
 When aline is mentioned simply, it means a Right Line. 
 
 7. A Curve continually changes its di- 
 rection between its extreme points. 
 
 8. Lines are either Parallel, Oblique, 
 Perpendicular, or Tangential. 
 
 9. Parallel lines are always at the same 
 perpendicular distance ; and they never 
 meet, though ever so far produced. 
 
 10. Oblique lines change their distance, 
 and would meet, if produced on the side 
 of the least distance. 
 
 11. One line is Perpendicular to an- 
 other, when it inclines not more on the 
 one side than the other, or when the angles 
 on both sides of it are equal. 
 
 12. A line or circle is Tangential, or 
 is a tangent to a circle or other curve, 
 when it touches it without cutting, when 
 both are produced. 
 
 57 
 
 
 
58 ARITHMETIC. 
 
 13. An Angle is the inclination or open- 
 ing of two lines, having different direc- 
 tions, and meeting in a point. 
 
 14. Angles are Right or Oblique, Acute or Obtuse. 
 
 15. A Right Angle is that which is made 
 by one line perpendicular to another. Or 
 when the angles on each side are equal to 
 one another, they are right angles. 
 
 16. An Oblique Angle is that which is 
 made by two oblique lines ; and is either 
 less or greater than a right angle. 
 
 17. An Acute Angle is less than a righ't 
 angle. 
 
 18. An Obtuse Angle is greater than a 
 right angle. 
 
 19. Superficies are either Plane or Curved. 
 
 20. A Plane Superficies, or a Plane, is that with which 
 a right line may, every way, coincide. Or, if the line 
 touch the plane in two points, it will touch it in every point. 
 But, if not, it is curved. 
 
 21. Plane Figures are bounded either by right lines 01 
 curves. 
 
 22. Plajie figures that are bounded by right lines have 
 names according to the number of their sides, or of their 
 angles ; for they have as many sides as angles ; the least 
 number being three. 
 
 23. A figure of three sides and angles is called a Tri- 
 angle. And it receives particular denominations from the 
 relations of its sides and angles. 
 
 24. An Equilateral Triangle is that 
 whose three sides are all equal. 
 
 25. An Isosceles Triangle is that which 
 has two sides equal. 
 
 26. A Scalene Triangle is that whose 
 three sides are all unequal. 
 
 27. A Right-angled Triangle is that >Xj 
 which has one right angle. / I 
 
DEFINITIONS. 
 
 59 
 
 28. Other triangles are Oblique-angled, 
 and are either obtuse or acute. 
 
 29. An Obtuse- Angled Triangle has one 
 obtuse angle. 
 
 30. An Acute-angled Triangle has all 
 its three angles acute. 
 
 31. A figure of four sides and angles is called a Quad- 
 rangle, or a Quadrilateral. 
 
 32. A Parallelogram is a quadrilateral which has both 
 its pairs of opposite sides parallel. And it takes the fol- 
 lowing particular names, viz. Rectangle, Square, Rhombus, 
 Rhomboid. 
 
 33. A Rectangle is a parallelogram, i 1 
 
 having right angles. I I 
 
 34. A Square is an equilateral rec- 
 tangle ; having its length and breadth 
 equal. 
 
 35. A Rhomboid is an oblique-angled 
 parallelogram. 
 
 36 A Rhombus is an equilateral rhom- 
 boid ; having all its sides equal, but its 
 angles oblique. 
 
 37. A Trapezium is a quadrilateral 
 which hath not its opposite sides parallel. 
 
 38. A Trapezoid has only one pair of 
 opposite sides parallel. 
 
 39. A Diagonal is a line joining any 
 two opposite angles of a quadrilateral. 
 
 40. Plane figures that have more than four sides are, in 
 general, called Polygons ; and they receive other particular 
 names, according to the number of their sides or angles. 
 Thus, 
 
 41. A Pentagon is a polygon of five sides ; a Hexagon 
 of six sides; a Heptagon, seven; an Octagon, eight; a 
 Nonagon, nine ; a Decagon, ten ; an Undecagon, eleven ; 
 and a Dodecagon, twelve sides. 
 
 42. A Regular Polygon has all its sides and all its 
 angles equal. If they are not both equal, the polygon is 
 irregular. 
 
 43. An Equilateral Triangle is also a regular figure of 
 
60 
 
 GEOMETRY. 
 
 three sides, and the Square is one of four : the former being 
 also called a Trigon, and the latter a Tetragon. 
 
 44. Any figure is equilateral, when all its sides are 
 equal : and it is equiangular when all its angles are equal. 
 When both these are equal, it is a regular figure. 
 
 45. A Circle is a plane figure bounded 
 by a curve line, called the Circumference, 
 which is everywhere equidistant from a 
 certain point within, called its Centre. 
 
 The circumference itself is often called a Circle, and 
 also the Periphery. 
 
 46. The Radius of acircle is aline drawn 
 from the centre to the circumference. 
 
 47. The Diameter of a circle is a line 
 drawn through the centre, and terminating 
 at the circumference on both sides. 
 
 48. An Arc of a circle is any part of the 
 circumference. 
 
 49. A Chord is a right line joining the 
 extremities of an arc. 
 
 50. A Segment is any part of a circle 
 bounded by an arc and its chord. 
 
 51. A Semicircle is half the circle, or a 
 segment cut off by a diameter. 
 
 The half circumference is sometimes 
 called the Semicircle. 
 
 52. A Sector is any part of a circle 
 which is bounded by an arc, and two radii 
 drawn to its extremities. 
 
 53. A Quadrant, or Quarter of a circle, 
 is a sector having a quarter of the circum- 
 ference for its arc, and its two radii are 
 perpendicular to each other. A quarter 
 of the circumference is sometimes called 
 a Quadrant. 
 
 e 
 
DEFINITIONS. 
 
 61 
 
 54. The Height or Altitude 'of a figure 
 is a perpendicular let fall from an angle, 
 or its vertex, to the opposite side, called 
 the base. 
 
 * 55. In a right-angled triangle, the side 
 opposite the right angle is called the Hy- 
 potenuse ; and the other two sides are 
 called the Legs, and sometimes the Base 
 and Perpendicular. 
 
 56. When an angle is denoted by three 
 letters, of which one stands at the angular 
 
 point, and the other two on the two sides, 5 A O 
 that which stands at the angular point is 
 read in the middle. Thus, DAE. 
 
 57. The circumference of every circle is supposed to be 
 divided into 360 equal parts, called degrees ; and each de- 
 gree into 60 minutes, each minute into 60 seconds, and so 
 on. Hence a semicircle contains 180 degrees, and a quad- 
 rant 90 degrees.^ 
 
 58. The measure of an angle, is an arc 
 of any circle contained between the two 
 lines which form that angle, the angular 
 point being the centre ; and it is estimated 
 by the number of degrees contained in 
 that arc. 
 
 59. Lines, or chords, are said to be 
 Equidistant from the centre of a circle, 
 when perpendiculars drawn to them from 
 the centre are equal. 
 
 60. And the right line on which the 
 Greater Perpendicular falls, is said to be 
 farther from the centre. 
 
 61. An Angle in a segment is that 
 which is contained by two lines, drawn 
 from any point in the arc of the segment, 
 to the two extremities of that arc. 
 
 62. An Angle on a segment, or an arc, is that which is 
 contained by two lines, drawn from any point in the oppo- 
 
 
 site or 
 mities 
 
 supplemental part of the circumference, to the extre- 
 of the arc, and < 
 
 containing the arc between them. 
 
62 
 
 GEOMETRY. 
 
 63. An Angle at the circumference, is 
 that whose angular point or summit is 
 anywhere in the circumference. And an 
 angle at the centre, is that whose angular 
 point is at the centre. 
 
 64. A right-lined figure is Inscribed in a 
 circle, or the circle Circumscribes it, when 
 all the angular points of the figure are in 
 the circumference of the circle. 
 
 06. A right-lined figure Circumscribes a 
 circle, or the circle is Inscribed in it, when 
 all the sides of the figure touch the circum- 
 ference of the circle. 
 
 66. One right-lined figure is Inscribed in 
 another, or the latter Circumscribes the 
 former, when all the angular points of the 
 former are placed in the sides of the latter. 
 
 67. A Secant is a line that cuts a circle, 
 lying partly within and partly without it. 
 
 68. Two triangles, or other right-lined figures, are said 
 to be mutually equilateral, when all the sides of the one are 
 equal to the corresponding sides of the other, each to each : 
 and they are said to be mutually equiangular, when the 
 angles of the one are respectively equal to those of the 
 other. 
 
 69. Identical figures, are such as are both mutually equi- 
 lateral and equiangular ; or that have all the sides and all 
 the angles of the one, respectively equal to all the sides and 
 all the angles of the other, each to each ; so that, if the one 
 figure were applied to, or laid upon the other, all the sides 
 of the one would exactly fall upon and cover all the sides 
 of the other ; the two becoming as it were but one and the 
 
 fsame figure. 
 
 70. Similar figures, are those that have all the angles of 
 the one equal to all the angles of the other, each to each, 
 and the sides about the equal angles proportional. 
 
 71. The Perimeter of a figure, is the sum of all its sides 
 taken together. 
 
 72. A Proposition, is something which is either proposed 
 
THEOREMS. 
 
 63 
 
 to be done, or to be demonstrated, and is either a problem 
 or a theorem. 
 
 73. A Problem, is something proposed to be done. 
 
 74. ATheorem, is something proposed to be demonstrated. 
 
 75. A Lemma, is something which is premised, or de- 
 monstrated, in order to render what follows more easy. 
 
 76. A Corollary, is a consequent truth, gained imme- 
 diately from some preceding truth, or demonstration. 
 
 77. A Scholium, is a remark or observation made upon 
 something going before it. 
 
 THEOREMS. 
 
 1. In the two triangles ABC, DEF, if 
 the side AC be equal to the side DF, and 
 the side BC equal to the side EF, and the 
 angle C equal to the angle F ; then will 
 the two triangles be identical, or equal in 
 all respects. 
 
 2. Let the two triangles ABC, DEF, have the angle A 
 equal to the angle D, the angle B equal to the angle E, and 
 the side AB equal to the side DE ; then these two triangles 
 will be identical. 
 
 3. If the triangle ABC have the side 
 AC equal to the side BC ; then will the 
 angle B be equal to the angle A. 
 
 The line which bisects the vertical angle 
 of an isosceles triangle, bisects the base, 
 and is also perpendicular to it. 
 
 Every equilateral triangle, is also equiangular, or has all 
 its angles equal. 
 
 4. If the triangle ABC, have the angle 
 A equal to the angle B, it will also have 
 the side AC equal to the side BC. 
 
 Every equiangular triangle is also equi- 
 lateral. 
 
 5. Let the two triangles ABC, ABD, 
 have their three sides respectively equal, 
 viz. the side AB equal to AB, AC to AD, 
 and BC to BD ; then shall the two triangles 
 be identical, or have their angles equal, 
 
64 GEOMETRY. 
 
 viz. those angles that are opposite to the equal sides ; viz. 
 the angle BAG to the angle BAD, the angle ABC to the 
 angle ABD, and the angle C to the angle D. 
 
 6. Let the line AB meet the line CD ; 
 then will the two angles ABC, ABD, taken 
 together, be equal to two right angles. _-. 
 
 7. Let the two lines AB, CD, intersect 
 in the point E ; then will the angle AEC 
 be equal to the angle BED, and the angle 
 AED equal to the angle CEB. 
 
 8. Let ABC be a triangle, having the 
 side AB produced to D ; then will the out- 
 ward angle CBD be greater than either 
 of the inward opposite angles A or C. 
 
 9. Let ABC be a triangle, having the 
 side AB greater than the side AC ; then 
 will the angle ACB, opposite the greater 
 side AB, be greater than the angle B, op- 
 posite the less side AC. 
 
 10. Let ABC be a triangle ; then will 
 the sum of any two of its sides be greater 
 than the third side ; as, for instance, AC 
 -}- CB greater than AB. 
 
 1 1 . Let ABC be a triangle ; then will the 
 difference of any two sides, as AB AC, 
 be less than the third side BC. 
 
 12. Let the line EF cut the two parallel 
 lines AB, CD ; then will the angle AEF 
 be equal to the alternate angle EFD. 
 
 13. Let the line EF, cutting the two 
 lines AB, CD, make the alternate angles 
 AEF, DFE, equal to each other; then 
 will AB be parallel to CD. 
 
 
THEOREMS. 65 
 
 14. Let the line EF cut the two paral- 
 lel lines AB, CD ; then will the outward 
 angle EGB be equal to the inward opposite 
 angle GHD, on the same side of the line 
 EF; and the two inward angles BGH, 
 GHD, taken together, will be equal to two 
 right angles. 
 
 15. Let the lines AB, CD, be each of 
 them parallel to the line EF ; then shall 
 the lines AB, CD, be parallel to each 
 other. 
 
 16. Let the side AB, of the triangle 
 ABC, be produced to D ; then will the out- 
 ward angle CBD be equal to the sum of A^\. 
 the two inward opposite angles A and C. 3 *iJ' 
 
 17. Let ABC be any plane triangle; 
 
 then the sum of the three angles A-f B-{-C r""~~7 C 
 
 is equal to two right angles. * ' 
 
 If two angles in one triangle, be equal 
 to two angles in another triangle, the third 
 angles will also be equal, and the two triangles equiangu- 
 lar. 
 
 If one angle in one triangle, be equal to one angle in 
 another, the sums of the remaining angles will also be 
 equal. 
 
 If one angle of a triangle be right, the sum of the other 
 two will also be equal to a right angle, and each of them 
 singly will be acute, or less than a right angle. 
 
 The two least angles of every triangle are acute, or each 
 less than a right angle. 
 
 18. Let ABCD be a quadrangle ; then 
 the sum of the four inward angles, A-f B 
 -f-C+D is equal to four right angles. 
 
 19. Let ABCDE be any figure; then 
 the sum of all its inward angles, A + B + 
 
 C-fD + E, is equal to twice as many ^u/ 
 
 right angles, wanting four, as the figure 
 has sides. 
 
 6* 
 
 

 66 GEOMETRY. 
 
 20. Let A, B, C, &c., be the outward 
 angles of any polygon, made by producing 
 all the sides; then will the sum A+B + 
 C-f D+E, of all those outward angles, be 
 equal to four right angles. 
 
 21. If AB, AC, AD, &c., be lines drawn 
 from the given point A, to the indefinite 
 line BE, of which AB is perpendicular; 
 then shall the perpendicular AB be less 
 than AC, and AC less than AD, &c. 
 
 22. Let ABCD be a parallelogram, of 
 which the diagonal is BD ; then will its 
 opposite sides and angles be equal to each 
 other, and the diagonal BD will divide it 
 into two equal parts, or triangles. 
 
 If one angle of a parallelogram be a right angle, all the 
 other three will also be right angles, and the parallelogram 
 a rectangle. 
 
 The sum of any two adjacent angles of a parallelogram 
 is equal to two right angles. 
 
 23. Let ABCD be a quadrangle, having the opposite sides 
 equal, namely, the side AB equal to DC, and AD equal to 
 BC ; then shall these equal sides be also parallel, and the 
 figure a parallelogram. 
 
 24. Let AB, DC, be two equal and parallel lines; then 
 will the lines AD, BC, which join their extremes, be also 
 equal and parallel. 
 
 25. Let ABCD, ABEF, be two paral- 
 lelograms, and ABC, ABF, two triangles, 
 standing on the same base AB, and between 
 the same parallels AB, DE ; then will the 
 parallelogram ABCD be equal to the paral- 
 lelogram ABEF, and the triangle ABC 
 equal to the triangle ABF. 
 
 Parallelograms, or triangles, having the same base and 
 altitude, are equal. For the altitude is the same as the 
 perpendicular or distance between the two parallels, which 
 is everywhere equal, by the definition of parallels. 
 
 Parallelograms, or triangles, having equal bases and al- 
 titudes, are equal. For if the one figure be applied with 
 its base on the other, the bases will coincide or be the same, 
 because they are equal: and so the two figures, having the 
 same base and altitude, are equal. 
 

 THEOREMS. 
 
 
 67 
 
 26. Let ABCD be a parallelogram, and 
 ABE a triangle, on the same base AB, and 
 between .the same parallels AB, DE ; then 
 will the parallelogram ABCD be double 
 the triangle ABE, or the triangle half the 
 parallelogram. 
 
 A triangle is equal to half a parallelogram of the same 
 base and altitude, because the altitude is the perpendicular 
 distance between the parallels, which is everywhere equal, 
 by the definition of parallels. 
 
 If the base of a parallelogram be half that of a triangle, 
 of the same altitude, or the base of a triangle be double that 
 of the parallelogram, the two figures will be equal to each 
 other. 
 
 27. Let BD, FH, be two rectangles, 
 having the sides AB, BC, equal to the 
 sides EF, FG, each to each; then will the 
 rectangle BD be equal to the rectangle 
 FH. 
 
 28. Let AC be a parallelogram, BD a 
 diagonal, EIF parallel to AB or DC, and 
 GIH parallel to AD or BC, making AI, 
 1C, complements to the parallelograms 
 EG, HF, which are about the diagonal DB: 
 
 then will the complement AI be equal to the complement 
 1C. 
 
 29. Let AD be the one line, and AB 
 the other, divided into the parts AE, EF, 
 FB ; then will the rectangle contained by 
 AD and AB, be equal to the sum of the 
 rectangles of AD and AE, and AD and 
 
 EF, and AD and FB : thus expressed, AD . AB=AD . AE 
 +AD . EF+AD . FB.* 
 
 If a right line be divided into any two parts, the square 
 of the whole line, is equal to both the rectangles of the 
 whole line and each of the parts. 
 
 * Instead of the mark X > a point is often used; thus, length X 
 breadth = area, is the same as length . breadth = area. Instead of the 
 parenthesis, a stroke is often used; thus, (first-}- last) - 2= arith- 
 metical mean, is the same thing as first -f- last -f- 2 = arithmetical 
 mean. For the square root this mark y/ is sometimes used, and for the 
 cube root /. &c. 
 
b* Hk * *W:-' 
 
 68 GEOMETRY. 
 
 30. Let the line AB be the sum of any 
 two lines AC, CB ; then will the square 
 of AB be equal to the squares of AC, CB, 
 together with twice the rectangle of AC 
 . CB. That is, AB 3 =AC 3 -fCB 2 -f 2AC 
 . CB. 
 
 If a line be divided into two equal parts ; the square of 
 the whole line will be equal to four times the square of half 
 the line. 
 
 31. Let AC, BC, be any two lines, and 
 AB their difference ; then will the square 
 of AB be less than the squares of AC, BC, 
 by twice the rectangle of AC and BC. 
 Or, AB 2 = AC 3 + BC 3 2AC . BC. 
 
 32. Let AB, AC, be any two unequal 
 lines; then will the difference of the 
 squares of AB, AC, be equal to a rect- 
 angle under their sum and difference. 
 That is, AB a AC 3 = AB -f AC~ 
 AB AC. 
 
 33. Let ABC be a right-angled tri- 
 angle, having the right angle at C ; then 
 will the square of the hypotenuse AB, be 
 equal to the sum of the squares of the 
 other two sides AC, CB. Or AB 3 =AC 3 
 +BC*. 
 
 34. Let ABC be any triangle, having CD 
 perpendicular to AB ; then will the differ- 
 ence of the squares of AC, BC, be equal 
 
 to the difference of the squares of AD, 
 
 BD ; that is, AC 3 BC 3 =AD 3 BD 3 . A D B D A B 
 
 35. Let ABC be a triangle, obtuse-angled at B, and CD 
 perpendicular to AB ; then will the square of AC be greater 
 than the squares of AB, BC, by twice the rectangle of AB, 
 BD. That is, AC 3 =AB 3 +BC 3 -j-2AB . BD. 
 
 36. Let ABC be a triangle, having the angle A acute, and 
 CD perpendicular to AB ; then will the square of BC, be 
 less than the squares of AB, AC, by twice the rectangle of 
 AB, AD. That is, BC 3 =AB 3 +AC 2 - 2AD . AB. 
 
THEOREMS. 
 
 37. Let ABC be a triangle, and CD the 
 line drawn from the vertex to the middle 
 of the base AB, bisecting it into the two 
 equal parts AD, DB ; then will the sum 
 of the squares of AC, CB, be equal to twice 
 the sum of the square of CD, AD ; or AC 3 
 -fCB 2 =2CD a +2AD 3 . 
 
 38. Let ABC be an isosceles triangle, 
 and CD a line drawn from the vertex to 
 any point D in the base : then will the 
 square of AC, be equal to the square of 
 CD, together with the rectangle of AD and 
 
 DB. That is AC a =CD 3 +AD . DB. 
 
 39. Let ABC D be a parallelogram, 
 whose diagonals intersect each other in E : 
 then will AE be equal to EC, and BE to 
 ED; and the sum of the squares of AC, 
 BD, will be equal to the sum of the squares 
 of AB, BC, CD, DA. That is, 
 
 AE=EC, and BE=ED, 
 and AC 2 +BD 2 =AB 2 +BC s -f CD 3 +DA a 
 
 40. Let AB be any chord in a circle, 
 and CD a line drawn from the centre C to 
 the chord. Then, if the chord be bisected 
 in the point D, CD will be perpendicular 
 to AB. 
 
 41. Let ABC be a circle, and D a point 
 within it ; then if any threeliines, DA, DB, 
 
 DC, drawn from the point D to the cir- 
 cumference, be equal to each other, the 
 point D will be the centre. 
 
 42. Let two circles touch one another internally in the 
 point ; then will the pqint and the centres of those circles 
 be all in the same right line. 
 
 43. Let two circles touch one another externally at the 
 point ; then will the point of contact and the centres of the 
 two circles be all in the same right line. 
 
 44. Let AB, CD, be any two chords at 
 equal distances from the centre G ; then 
 will these two chords AB, CD, be equal 
 to each other. 
 

 70 
 
 
 GEOMETRY. 
 
 45. Let the line ADB be perpendicular 
 to the radius CB of a circle ; then shall 
 AB touch the circle in the point D only. 
 
 46. Let AB be a tangent to a circle, 
 and CD a chord drawn from the point of 
 contact C ; then is the angle BCD mea- 
 sured by half the arc CFD, and the angle 
 ACD measured by half the arc CGD. 
 
 47. Let BAC be an angle at the circum- 
 ference ; it has for its measure, half the 
 arc BC which subtends it. 
 
 48. Let C and D be two angles in the 
 same segment ACDB, or, which is the 
 same thing, standing on the supplemental 
 arc AEB ; then will the angle C be equal 
 to the angle D. 
 
 49. Let C be an angle at the centre C, 
 and D an angle at the circumference, both 
 standing on the same arc or same chord 
 AB ; then will the angle C be double of 
 the angle D, or the angle D equal to half 
 the angle C. 
 
 50. If ABC or ADC be a semicircle, 
 then any angle D in that semicircle, is a 
 right angle. 
 
 51. If AB be a tangent, and AC a chord, 
 and D any angle in the alternate segment 
 ADC ; then will the angle D be equal to 
 the angle BAC made by the tangent and 
 chord of the arc AEC. 
 
 52. Let ABCD be any quadrilateral in- 
 scribed in a circle ; then shall the sum of 
 the two opposite angles A and C, or B 
 and D, be equal to two right angles. 
 
THEOREMS. 
 
 71 
 
 53. If the side AB, of the quadrilateral 
 ABCD, inscribed in a circle, be produced 
 to E ; the outward angle DBE will be 
 equal to the inward opposite angle C. 
 
 54. Let the two chords AB, CD be 
 parallel ; then will the arcs AC, BD, be 
 equal; or AC=BD. 
 
 55. Let the tangent ABC be parallel to 
 the chord DF ; then are the arcs BD, BF, 
 equal; that is, BD=BF. 
 
 56. Let the two chords AB, CD, inter- 
 sect at the point E ; then the angle AEC, 
 or DEB, is measured by half the sum of 
 the two arcs AC, DB. 
 
 57. Let the angle E be formed by two 
 secants EAB and ECD ; this angle is 
 measured by half the difference of the two 
 arcs AC, DB, intercepted by the two se- 
 cants. 
 
 58. Let EB, ED, be two tangents to a 
 circle at the points A, C ; then the angle 
 E is measured by half the difference of the 
 two arcs CFA, CGA. 
 
 59. Let the two lines AB, CD, meet 
 each other in E ; then the rectangle of AE, 
 EB, will be equal to the rectangle of CE, 
 ED. Or, AE . EB = CE . ED. 
 
 When one of the lines in the se- 
 cond case, as DE, by revolving about 
 the point E, comes into the position 
 of the tangent EC or ED, the two 
 points C and D running into one ; 
 

 72 
 
 GEOMETRY. 
 
 then the rectangle of CE, ED, becomes the square of CE, 
 because CE and DE are then equal. Consequently, the 
 rectangle of the parts of the secant, AE . EB, is equal to 
 the square of the tangent, CE a . 
 
 60. Let CD be the perpendicular, and 
 CE the diameter of the circle about the 
 triangle ABC ; then the rectangle CA, 
 CB is = the rectangle CD . CE. 
 
 61. Let CD bisect the angle C of the 
 triangle ABC ; then the square CD 2 -f the 
 rectangle AD . DB is = the rectangle 
 AC . CB. 
 
 * 
 
 62. Let ABCD be any quadrilateral in- 
 scribed in a circle, and AC, BD, its two 
 diagonals ; then the rectangle AC . BD is 
 = the rectangle AB . DC -f the rectangle 
 AD . BC. 
 
 63. Let the two triangles ADC, DEF, 
 have the same altitude, or be between the 
 same parallels AE, CF ; then is the sur- 
 face of the triangle ADC, to the surface 
 of the triangle DEF, as the base AD is 
 to the base DE. Or, AD : DE : : the tri- 
 angle ADC : the triangle DEF. 
 
 64. Let ABC, BEF, be two triangles 
 having the equal bases AB, BE, and whose 
 altitudes are the perpendiculars CG, FH ; 
 then will the triangle ABC : the triangle 
 BEF : : CG : FH. 
 
 Triangles and parallelograms, when their bases are equal, 
 are to each other as their altitudes ; and by the foregoing 
 one, when their altitudes are equal, they are to each other 
 as their bases ; therefore, universally, when neither are 
 equal, they are to each other in the compound ratio, or as 
 the rectangle or product of their bases and altitudes. 
 
 65. Let the four lines A, B, C, D, be A 
 proportionals, or A : B : : C : D ; then 
 
 will the rectangle of A and D be equal to 
 the rectangle of B and C ; or the rectangle 
 A . D = B . C. 
 
 G B H E 
 
THEOREMS. 
 
 73 
 
 /-A 
 
 66. Let DE.be parallel to the side BC 
 of the triangle ABC ; then will AD : DB 
 : : AE : EC. 
 
 AB : AC : : AD : AE, 
 AB : AC : : BD : CE. 
 
 67. Let the angle ACB, of the triangle 
 ABC, be bisected by the line CD, making 
 the angle r equal to the angle s : then will 
 the segment AD be to the segment DB, 
 as the side AC is to the side CB. Or, AD 
 : DB : : AC : CB. 
 
 68. In the two triangles ABC, DEF, if 
 AB : DE : : AC : DF : : BC : EF ; the 
 
 two triangles will have their correspond- 
 ing angles equal. 
 
 69. Let ABC, DEF, be two triangles, 
 having the angle A = the angle D, and 
 the sides AB, AC, proportional to the sides 
 DE, DF; then will the triangle ABC be 
 equiangular with the triangle DEF. 
 
 70. Let ABC be a right-angled tri- 
 angle, and CD a perpendicular from 
 the right angle C to the hypotenuse 
 AB ; then will 
 
 CD be a mean proportional between AD and DB ; 
 AC a mean proportional between AB and AD ; 
 BC a mean proportional between AB and BD. 
 
 71. All similar figures are to each other, as the squares 
 of their like sides. 
 
 72. Similar figures inscribed in circles, have their like 
 sides, and also their whole perimeters, in the same ratio 
 as the diameters of the circles in which they are inscribed. 
 
 73. Similar figures inscribed in circles are to each other 
 as the squares of the diameters of those circles. 
 
 74. The circumferences of all circles are to each other 
 as their diameters. 
 
 75. The areas or spaces of circles, are to each other as 
 the squares of their diameters, or of their radii. 
 
 76. The area of any circle, is equal to the rectangle of 
 half its circumference and half its diameter. 
 
 7 
 
 
PROBLEMo. 
 
 1. To bisect a line AB ; that is, to divide it into two 
 equal parts. 
 
 c 
 
 From the two centres A and B, with 
 any equal radii, describe arcs of circles, 
 intersecting each other in C and D ; 
 and draw the line CD, which will bi- 
 sect the given line AB in the point E. 
 
 2. To bisect an angle BAC. 
 
 From the centre A, with any radius, A 
 
 describe an arc cutting off the equal J\ 
 
 lines AD, AE ; and from the two centres -&/ \E 
 
 D, E, with the same radius, describe /\ /\ 
 
 arcs intersecting in F ; then draw AF, B ^ / C 
 
 which will bisect the angle A as re- * 
 quired. 
 
 3. At a given point C, in a line AB, to erect a per- 
 pendicular. 
 
 From the given point C, with any ra- 
 dius, ttit off any equal parts CD, CE, of 
 the given line ; and, from the two centres 
 D and E, with any one radius, describe 
 arcs intersecting in F ; then join CF, 
 which will be perpendicular as required. 
 
 OTHERWISE. 
 
 When the given point C is near the end of the line. 
 
 From any point D, assumed above 
 the line, as a centre, through the given 
 point C describe a circle, cutting the 
 given line at E ; and through E and 
 the centre D, draw the diameter EDF; 
 then join CF, which will be the per- ] 
 pendicular required. 
 
 C B 
 
 74 
 
PROBLEMS. 75 
 
 4. JVom /&e given point A, to /e /// a perpendicular on 
 a given line BC. 
 
 From the given point A as a centre, 
 with any convenient radius, describe an 
 arc, cutting the given line at the two 
 points D and E ; and from the two cen- 
 
 \ 
 
 f 
 
 tres D, E, with any radius, describe two B \ 
 arcs, intersecting at F ; then draw AGF, 
 which will be perpendicular to BC as re- 
 quired. 
 
 OTHERWISE. 
 
 When the, given point is nearly opposite the end of the 
 line. 
 
 From any point D, in the given line 
 BC, as a centre, describe the arc of a cir- 
 cle through the given point A, cutting 
 BC in E : and from the centre E, with 
 the radius EA, describe another arc, cut- 
 ting the former in F; then draw AGF, 
 which will be perpendicular to BC as re- 
 quired. 
 
 5. Jit a given point A, in a line AB, to make an angle 
 equal to a given angle C. 
 
 From the centres A and C, with any 
 one radius, describe the arcs DE, FG. 
 Then, with radius DE, and centre F, de- 
 scribe an arc, cutting FG in G. Through 
 G draw the line AG, and it will form the 
 angle required. 
 
 A F B 
 
 6. Through a given point A, to draw a line parallel to a 
 given line BC. 
 
 From the given point A draw a line AD 
 
 to any point in the given line BC. Then E A . F 
 
 draw the line EAF, making the angle at A ^X. 
 
 equal to the angle at D (by prob. 5) ; so B D~~C 
 
 shall EF be parallel to BC as required. 
 
76 
 
 GEOMETRY. 
 
 7. To divide a line AB into any proposed number of 
 equal parts. 
 
 Draw any other line AC, forming any 
 angle with the given line AB ; on which 
 set off as many of any equal parts AD, 
 DE, EG, FC, as the line AB is to be 
 divided into. Join BC ; parallel to which 
 draw the other lines FG, EH, DI ; then 
 these will divide AB in the manner required. For those 
 parallel lines divide both the sides AB, AC, proportionally. 
 
 8. To find a third proportional to two given lines AB, AC. 
 
 Place the two given lines AB, AC, A B 
 
 forming any angle at A ; and in AB take 
 also AD equal to AC. Join BC, and 
 draw DE parallel to it ; so will AE be 
 the third proportional sought. 
 
 9. To find a fourth proportional to three lines, AB, AC, 
 
 AD. 
 
 Place two of the given lines AB, A B 
 
 AC, making any angle at A; also A C 
 
 place AD on AB. Join BC ; and pa- A 
 rallel to it draw DE ; so shall AE be 
 the fourth proportional as required. 
 
 10. To find a mean proportional between two lines, AB, 
 
 BC. 
 
 Place AB, BC, joined in one straight 
 line AC : on which, as a diameter, de- 
 scribe the semicircle ADC ; to meet which 
 erect the perpendicular BD ; and it will 
 
 be the mean proportional sought, between 
 
 ABandBC. A o B c 
 
 11. To find the centre of a circle. 
 
 Draw any chord AB ; and bisect it 
 perpendicularly with the line CD, which 
 will be a diameter. Therefore CD bi- 
 sected in O, will give the centre, as re- 
 quired. 
 
 
PROBLEMS. 
 
 77 
 
 12. To describe the circumference of a circle through 
 three given points, A, B, C. 
 
 From the middle point B draw chords 
 BA, BC, to the two other points, and bi- 
 sect these chords perpendicularly by lines 
 meeting in O, which will be the centre. 
 Then from the centre O, at the distance 
 of any one of the points, as OA, describe 
 a circle, and it will pass through the two 
 other points B, C, as required. 
 
 o 
 
 13. To draw a tangent to a circle, through a given 
 point A. 
 
 B A 
 
 When the given point A is in the cir- 
 cumference of the circle : join A and the 
 centre O ; perpendicular to which draw 
 BAG, and it will be the tangent. 
 
 14. On a given line B to describe a segment of a circle, 
 to contain a given angle C. 
 
 At the ends of the given line make 
 angles DAB, DBA, each equal to the 
 given angle C. Then draw AE, BE per- 
 pendicular to AD, BD ; and with the 
 centre E, and radius EA or EB, describe 
 a circle ; so shall AFB be the segment 
 required, as any angle F made in it will 
 be equal to the given angle C. 
 
 15. To cut off a segment from a circle, that shall contain 
 a given angle C. 
 
 Draw any tangent AB to the given cir- 
 cle ; and a chord AD to make the angle 
 DAB equal to the given angle C ; then 
 DEA will be the segment required, any 
 angle E made in it being equal to the 
 given angle C. 
 
 7 * 
 
78 
 
 GEOMETRY. 
 
 16. To make a triangle with three given lines, AB, AC, 
 BC. 
 
 With the centre A, and distance AC, 
 describe an arc. With the centre B, and 
 distance BC, describe another arc, cutting 
 the former in C. Draw AB, BC, and 
 ABC will be the triangle required. A 
 
 B- 
 
 17. To inscribe a circle in a given triangle ABC, 
 
 Bisect any two angles A and B, with 
 the two lines AD, BD. From the inter- 
 section D, which will be the centre of the 
 circle, draw the perpendiculars DB, DF, 
 DG, and they will be the radii of the cir- 
 cle required. 
 
 18. To describe a circle about a given 
 triangle ABC. 
 
 Bisect any two sides with two per- 
 pendiculars DE, DF, and their inter- 
 section D will be the centre. 
 
 19. To inscribe an equilateral triangle in a given circk. 
 
 Through the centre C draw any di- 
 ameter AB. From the point B as a 
 centre, with the radius BC of the < 
 given circle, describe an arc DCE. 
 Join AD, AE, DE, and ADE is the 
 equilateral triangle sought. 
 
 20. To inscribe a square in a given circle. 
 
 Draw two diameters AC, BD, cross- 
 ing at right angles in the centre E. 
 Then join the four extremities A, B, 
 C, D, with right lines, and these will 
 form the inscribed square ABCD. 
 
 
PROBLEMS. 
 
 21. To describe a square about a given circle. 
 
 Draw two diameters AC, BD, cross- 
 ing at right angles in the centre E. 
 Then through their four extremities 
 draw FG, IH, parallel to AC, and FI, 
 GH, parallel to BD, and they will 
 form the square FGHI. 
 
 79 
 
 D H 
 
 22. To inscribe a circle in a regular polygon. 
 
 A 
 
 Bisect any two sides of the poly- 
 gon by the perpendiculars GO, FO, 
 and their intersection O will be the 
 centre of the inscribed circle, and OG 
 or OF will be the radius. 
 
 23. To describe a circle about a regular polygon. 
 
 Bisect any two of the angles, C 
 and D, with the lines CO, DO ; then 
 their intersection O will be the centre 
 of the circumscribing circle ; and OC, 
 or OD, will be the radius. 
 
 24. To make a square equal to the sum of two or more 
 
 given squares. 
 
 Let AB and AC be the sides of two 
 given squares. Draw two indefinite 
 lines AP, AQ, at right angles to each 
 other ; in which place the sides AB, 
 AC, of the given squares ; join BC ; 
 then a square described on BC will be 
 equal to the sum of the two squares 
 described on AB and AC. 
 
 p B 
 
 25. To make a square equal to the difference of two given 
 
 squares. 
 
 Let AB and AC, taken in the same 
 straight line, be equal to the sides of 
 the two given squares. From the 
 centre A, with the distance AB, de- 
 scribe a circle ; and make CD perpen- 
 dicular to AB, meeting the circumference in D ; so shall a 
 
 C B 
 

 80 
 
 GEOMETRY. 
 
 
 square described on CD be equal to AD 2 AC 2 , or AB 3 - 
 AC 2 , as required. 
 
 26. To make a triangle equal to a given pentagon 
 ABCDE. 
 
 D 
 
 Draw DA and DB, and also EF, 
 CG, parallel to them, meeting AB pro- 
 duced at F and G : then draw DF and 
 DG ; so shall the triangle DFG be 
 equal to the given pentagon ABCDE. 
 
 27. To make a square equal to a given rectangle A BCD. 
 
 Produce one side AB, till BE be 
 equal to the other side BC. On AE 
 as a diameter describe a circle, meet- 
 ing BC produced at F ; then will BF 
 be the side of a square BFGH, equal 
 to the given rectangle BD, as required. 
 
 r> 
 
 A H 
 
 APPENDIX TO GEOMETRY. 
 
 6 a 
 
 INSTRUMENTS. 
 
 28. To facilitate the construction of geometrical figures, 
 we add a short description of a few useful instruments 
 which do not belong to the common pocket-case. 
 
 29. Let there be a flat ruler, AB, from 
 one to two feet in length, for which the 
 common Gunter's scale may be substi- 
 tuted ; and, secondly, a triangular piece 
 of wood, , 6, c, flat, and about the same 
 thickness as the ruler : the sides ab and 
 
 be of which are equal to one another, and form a right 
 angle at b. For the convenience of sliding, there is usually 
 a hole in the middle of the triangle, as may be seen in the 
 figure. 
 
 30. By means of these simple instruments many very 
 useful geometrical problems may be performed. Thus, to 
 draw a line through a given point parallel to a given line. 
 
 

 INSTRUMENTS. 81 
 
 Lay the triangle on the paper so that one of its sides will 
 coincide with the given line to which the parallel is to be 
 drawn ; then, keeping the triangle steady, lay the ruler on 
 the paper, with its edge applied to either of the other sides 
 of the triangle ; then, keeping the ruler firm, move the tri- 
 angle along its edge, up or down, to the given point ; the 
 side of the triangle which was placed on the given line will 
 always keep parallel to itself, and hence a parallel may be 
 drawn through the given point. 
 
 31. To erect a perpendicular on a given line, and from 
 any given point in that line, we have only to apply the 
 ruler to the given line, and place the triangle so, that its 
 right angle shall touch the given point in the line, and one 
 of the sides about the right angle, placed to the edge of the 
 ruler the other side will give the perpendicular required. 
 
 32. If the given point be either above or below the line, 
 the process is equally easy. Place one of the sides of the 
 triangle about the right angle on the given line, and the 
 ruler on the side opposite the right angle, then slide the tri- 
 angle on the edge of the ruler till the given point from 
 which the perpendicular is to be drawn is on the other side, 
 then this side will give the perpendicular. 
 
 33. Other problems may be performed with these instru- 
 ments, the method of doing which it will be easy for the 
 reader to contrive for himself. 
 
 34. When arcs of circles of great diameter are to be 
 drawn, the use of a compass may be substituted by a very 
 simple contrivance. Draw the chord of the arc to be de- 
 scribed, and place a pin at, each ex- 
 tremity, A and B, then place two 
 
 rulers jointed at C, and forming an 
 angle, ACB = the supplement of half 
 the given number of degrees ; that is 
 to say, the number of degrees which the arc whose chord 
 given is to contain, is to be halved, and this half being sub- 
 tracted from 180 degrees, will give the degrees which form 
 the angle at which the rulers are placed, that is, the angle 
 A.CB. This being done, the edges of the rulers are moved 
 along against the pins, and a pencil at C will describe the 
 arc required. 
 
 35. Large circles may be described by a contrivance 
 equally simple. On an axle, a foot or a foot and a half 
 
GEOMETRY. 
 
 long, there are placed two wheels, M and 
 F, of which one is fixed to the axle, name- 
 ly, M, and the other is capable of being 
 shifted to different parts of the axle, and, 
 by means of a thumb-screw, made capable 
 of being fixed at any point on the axle. 
 These wheels are of different diameters, say of 3 and 6 
 inches, the fixed wheel F being the largest. This instru- 
 ment being moved on the paper, the circles M and F will 
 roll, and describe circles of different radii : the axle will al- 
 ways point to the centre of these circles, and there will be 
 this proportion : 
 
 As the diameter of the large wheel 
 
 Is to the difference of the diameters of the two wheels, 
 
 So is the radius of the circle to be described by the 
 
 large wheel 
 To the distance of the two wheels on the axle. 
 
 36. If the diameters of the wheels are as above stated, 
 and it is required to describe a circle of 3 feet radius, then 
 from the above proportion we have 6:6 3 : : 3 feet or 
 36 inches : 18 inches = the distance of the two wheels, to 
 describe a circle 6 feet in diameter. 
 
 37. It may be observed, that it will be best to make the 
 difference of the wheels greater if large circles are to be de- 
 scribed, as then a shorter instrument will serve the purpose. 
 
 38. We will conclude this appendix, by making a few 
 remarks on the Diagonal Scale and Sector, the great use 
 of the latter of which, especially, is seldom explained to 
 the young mechanic. 
 
 39. The diagonal scale to be found on the plain scale in 
 common pocket-cases of instruments, is a contrivance for 
 measuring very small divisions of lines ; as, for instance, 
 hundredth parts of an inch. 
 
 40. Suppose the accompanying cut to B E A 
 represent an enlarged view of two di- 
 visions of the diagonal scale, and the 
 
 bottom and top lines to be divided into 
 two parts, each representing the tenth 
 part of an inch. Now, the perpendicular 
 lines BC, AD, are each divided into ten 
 equal parts, which are joined by the 
 crossing lines, 1, 2, 3, 4, &c., and the diagonals BF, DE, 
 
INSTRUMENTS. 83 
 
 are drawn as in the figure. Now, as the division FC is 
 the tenth part of an inch, and as the line FB continually 
 approaches nearer and nearer to BC, till it meets it in B, 
 it will follow, that the part of the line 1 cut off by this 
 diagonal will be a tenth part of FC, because Bl is only one- 
 tenth part of BC ; so, likewise, 2 will represent two-tenth 
 parts, 3, three-tenth parts, and so on to 9, which is nine- 
 tenth parts, and 10, ten-tenth parts, or the whole tenth of 
 an inch; so that, by means of this diagonal, we arrive at 
 divisions equal to tenth parts of tenth parts of an inch, or 
 hundredths of an inch. With this consideration, an exami- 
 nation of the scale itself will easily show the whole matter. 
 It may be observed, that if half an inch and the quarter of 
 an inch be divided, in the same manner, into tenths and 
 tenths of tenths, we may get thus two-hundredth and four- 
 hundredth parts of an inch. 
 
 THE SECTOR. 
 
 41. THIS very useful instrument consists of two equal 
 rulers each six inches long, joined together by a brass 
 folding joint. These rulers are generally made of boxwood 
 or ivory ; and on the face of the instrument, several lines 
 or scales are engraven. Some of these lines or scales pro- 
 ceed from the centre of the joint, and are called sectorial 
 lines, to distinguish them from others which are drawn 
 parallel to the edge of the instrument, similar to those on 
 the common Gunter's scale. 
 
 42. The sectorial lines are drawn twice on the same face 
 of the instrument; that is to say, each line is drawn on both 
 legs. Those on each face are, 
 
 A scale of equal parts, marked L, 
 A line of chords, marked C, 
 
 A line of secants, marked S, 
 
 A line of polygons, marked P, or Pol. 
 These sectorial lines are marked on one face of the instru- 
 ment; and on the other there are the following: 
 A line of sines, marked S, 
 
 A line of tangents, marked T, 
 A line of tangents to a less radius, marked t. 
 
 
84 GEOMETRY. 
 
 This last line is intended to supply the defect of the former, 
 and extends from about 45 to 75 degrees,. 
 
 43. The lines of chords, sines, tangents, and secants, but 
 not the line of polygons, are numbered from the centre, 
 and are so disposed as to form equal angles at the centre; 
 and it follows from this, that at whatever distance the sec- 
 tor is opened, the angles which the lines form, will always 
 be respectively equal. The distance, therefore, between 10 
 and 10, on the two lines marked L, will be equal to the 
 distance of 60 and 60 on the two lines of chords, and also 
 to 90 and 90 on the two lines of sines, &c., at any particu- 
 lar opening of the sector. 
 
 44. Any extent measured with a pair of compasses, from 
 the centre of the joint to any division on the sectorial lines, 
 is called a lateral distance; and any extent taken from a 
 point in a line on the one leg, to the like point on the 
 similar line on the other leg, is called a transverse or 
 parallel distance. 
 
 With these remarks, we shall now proceed to explain the 
 use of the sector, in so far as it is likely to be serviceable to 
 mechanics. 
 
 USE OF THE LINE OF LINES. 
 
 45. This line, as was before observed, is marked L, and 
 its uses are, 
 
 To Divide a line into any number of equal parts : Take 
 the length of the line by the compasses, and placing one of 
 the points on that number in the line of lines which denotes 
 the number of parts into which the given line is to be 
 divided, open the sector till the other point of the com- 
 passes touches the same division on the line of lines marked 
 on the other leg ; then, the sector being kept at the same 
 width, the distance from 1 on the line L on the one leg, 
 to 1 on the line L on the other, will give the length of one 
 of the equal divisions of the given line to be divided. 
 Thus, to divide a given line into seven equal parts : take 
 the length of the given line with the compasses, and setting 
 one point on 7, on the line L of one of the legs, move the 
 other leg out until the other point of the compasses touch 
 7 on the line L of that leg ; this may be called the trans- 
 verse distance of 7 on the line of lines. Now, keeping the 
 sector at the same opening, the transverse distance of 1 
 will be the length of one of the 7 equal divisions of the 
 
INSTRUMENTS. 85 
 
 given line ; the transverse distance of 2 will be two of these 
 divisions, &c. 
 
 46. It will sometimes happen, that the line to be divided 
 will be too long for the largest opening of the sector ; and 
 in this case we take the half, or third, or fourth of the line, 
 as the case may be ; then the transverse distance of 1 to 1, 
 will be a half, a third, or a fourth of the required equal part. 
 
 47. To divide a given line into any number of parts that 
 shall have a certain relation or proportion to each other: 
 Take the length of the whole line to be divided, and placing 
 one point of the compasses at that division on the line 
 of lines on one leg of the instrument which expresses 
 the sum of all the parts into which the given line is to be 
 divided, and open the sector till the other point of the 
 compasses is on the corresponding division on the line of 
 lines of the other leg. This is evidently making the sum 
 of the parts into which the given line is to be divided a 
 transverse distance ; and when this is done, the proportional 
 parts will be found by taking, with the same opening of 
 the sector, the transverse distances of the parts required. 
 To divide a given line into three parts, in the proportion 
 of 2, 3, 4: The sum of these is 9; make the given line a 
 transverse distance between 9 and 9 on the two lines of 
 lines ; then the transverse distances of the several numbers 
 2, 3, 4, will give the proportional parts required. 
 
 48. To find a fourth proportional to three given lines : 
 Take the lateral distance of the second, and make it the 
 transverse distance of the first, then will the transverse dis- 
 tance of the third be the lateral distance of the fourth ; then, 
 let there be given 6:3:: 8, make the lateral distance of 
 3 the transverse distance of 6 ; then will the transverse 
 distance of 8 be the lateral distance of 4, the fourth propor- 
 tional required. 
 
 49. This sector will be found highly serviceable in draw- 
 ing plans. For instance, if it is wished to reduce the draw- 
 ing of a steam engine from a scale of Is inches to the foot, 
 to another of f to the foot. Now, in 1| inches there are ^/ 
 parts ; so that the drawing will be reduced in the propor- 
 tion of 12 to 5. Take the lateral distance of 5, and keep 
 the compasses at this opening; then open the sector till the 
 points of the compasses mark the transverse distance of 12 ; 
 keep now the sector at this opening, and any measure taken 
 on the drawing, to be copied and laid off on the sector as a 
 
 8 
 
86 MECHANICAL DRAWING 
 
 lateral distance, the transverse distance taken from that 
 point will give the corresponding measure to be laid down 
 in the new drawing. 
 
 50. If the length of the side of a triangle, of which we 
 have the drawing, is to be reckoned 45 ; what are the 
 lengths of the other two sides ? Take the length of the 
 side given, by the compasses, and open the sector till the 
 measure be the transverse distance of 45 to 45 ; then the 
 lengths of the other sides being applied transversely, will 
 give their numerical lengths. 
 
 USE OF THE LINE OF CHORDS. 
 
 51. By means of the sector, we may dispense with the 
 protractor. Thus, to lay down an angle of any number of 
 degrees : take the radius of the circle on the compasses, 
 and open the sector till this becomes the transverse distance 
 of 60 on the line of chords ; then take the transverse dis- 
 tance of the required number of degrees, keeping the sec- 
 tor at the same opening; and this transverse distance being 
 marked off on an arc of the circle whose radius was taken, 
 will be the required number of degrees. 
 
 We will not enter farther on the use of the sectorial lines, 
 as what we have said will, we hope, be found sufficient for 
 the purposes of the practical mechanic. 
 
 MECHANICAL DRAWING AND PERSPECTIVE. 
 
 52. A FLAT rectangular board is first to be provided, of 
 any convenient size, as from 18 to 30 inches long, and from 
 16 to 24 inches broad. It may be made of fir, plane tree, 
 or mahogany ; its face must be planed smooth and flat, and 
 the sides and ends as nearly as possible at right angles to 
 each other the bottom of the board and the left side should 
 be made perfectly so ; and this corner should be marked, 
 so that the stock of the square may be always applied to 
 the bottom and left hand side of the board. To prevent 
 the board from casting, it is usual to pannel it on the back 
 or on the sides. 
 
 53. A T square must also be provided, such that by 
 
AND PERSPECTIVE. 87 
 
 means of a thumb-screw fixed in the stock, it may be made 
 to answer either the purposes of a common square, or bevel, 
 the one-half of the stock being movable about the screw, 
 and the other fixed at right angles on the blade. The blade 
 ought to be somewhat flexible, and equal in length to the 
 length of the board. 
 
 54. Besides these, there will be required a case of mathe- 
 matical instruments ; in the selection of which it should be 
 observed, that the bow compass is more frequently defect- 
 ive than any of the other instruments. After using any of 
 the ink feet, they should be dried ; and if they do not draw 
 properly, they ought to be sharpened and brought to an 
 equal length in the blade, by grinding on a hone. 
 
 55. The colours most useful are, Indian ink, gamboge, 
 Prussian blue, vermilion, and lake. With these, all colours 
 necessary for drawing machinery or buildings may be 
 made ; so that, instead of purchasing a box of colours, we 
 would advise that those for whom this book is intended 
 should procure these cakes separately : the gamboge may 
 be bought from an apothecary a pennyworth will serve 
 a lifetime. In choosing the rest, they should be rubbed 
 against the teeth, and those which feel smoothest are of the 
 best quality. 
 
 56. Hair pencils will also be necessary, made of camel's 
 hair, and of various sizes. They ought to taper gradually 
 to a point when wet in the mouth, and should, after being 
 pressed against the finger, spring back. 
 
 57. Black-lead pencils will also be necessary. They 
 ought not to be very soft, nor so hard that their traces can- 
 not be easily erased by the Indian rubber. In choosing 
 paper, that which will best suit this kind of drawing is 
 thick, and has a hardish feel, not very smooth on the sur- 
 face, yet free from knots. 
 
 58. The paper on -which the drawing is to be made, must 
 be chosen of a good quality and convenient size. It is then 
 to be wet with a sponge and clean water, on the opposite 
 side from that on which the drawing is to be made. When 
 the paper absorbs the water, which may be seen by the 
 wetted side becoming dim, as its surface is viewed slantwise 
 against the light, it is to be laid on the drawing board 
 with the wetted side next the board. About half an inch 
 must be turned up on a straight edge all round the paper, 
 
88 MECHANICAL DRAWING 
 
 and then fastened on the board. This is done because the 
 paper when wet is enlarged, and the edges being fixed on 
 the board, act as stretchers when the paper contracts by 
 drying. To prevent the paper from contracting before the 
 paste has been sufficiently fastened by drying, the paper is 
 usually wet on the upper surface, to within half an inch of 
 the paste mark. When the paper is thoroughly dried, it 
 will be found to lie firmly and equally on the board, and is 
 then fit for use. 
 
 59. If the drawing is to be made from a copy, we ought 
 first to consider what scale it is to be drawn to. If it is to 
 be equal in size to, or larger than the copy ; and a scale 
 should be made accordingly, by which the dimensions of 
 the several parts of the drawing are to be regulated. The 
 diagonal scale, a simple and beautiful contrivance, will be 
 here found of great use for the more minute divisions ; and 
 whenever the drawing is to be made to a scale of 1 inch, 
 5 inch, | inch to the foot, a scale should be drawn of 20 or 
 30 equal parts ; the last of which should be subdivided into 
 12, and a diagonal scale formed on the same principles as 
 the common one, but with eight parallels and 12 diagonals, 
 to express inches and eighths of an inch. For making such 
 scales to any proportion, the line L on the sector will be 
 found very convenient. 
 
 GO. Great care should be taken in the penciling, that an 
 accurate outline be drawn, for on this much of the value 
 of the picture will depend. The pencil marks should be 
 distinct, yet not heavy, and the use of the rubber should be 
 avoided as much as possible, as its frequent application 
 ruffles the surface of the paper. The methods already 
 given for constructing geometrical figures will be here 
 found applicable, and the use of the T square, parallel 
 ruler, <fcc., will suggest themselves whenever they require 
 to be employed. 
 
 61. The drawing thus made of any machine or building 
 is called a plan. Plans are of three kinds a ground plan, 
 or bird's-eye view, an elevation or front view, and a per- 
 spective plan. 
 
 62. When a view is taken of the teeth of a wheel, with 
 the circumference towards the eye, the teeth appear to be 
 nearer as they are removed from the middle point of the 
 circumference opposite the eye, and it may not be out of 
 
AND PERSPECTIVE. 89 
 
 place here to give the method of representing them on 
 
 paper : If AB be the circumference of 
 
 a wheel as viewed by the eye, and it is > 
 
 required to represent the teeth as they A\ 
 
 appear on it. Only half of the circum- A ||j 
 
 ference can be seen in this way at one 
 
 time, consequently we can only represent the half of the 
 
 teeth. On AB describe a semicircle, which divide into 
 
 half as many equal parts as the wheel has teeth ; then from 
 
 each of these points of division draw perpendiculars to the 
 
 wheel AB, then will these perpendiculars mark the relative 
 
 places of the teeth. 
 
 63. When the outline is completed in pencil, it is next 
 to be carefully gone over with Indian ink, which is to be 
 rubbed down with a little water, on a plate of glass or 
 earthenware so as to be sufficiently fluid to flow easily out 
 of the pen, and at the same time have a sufficient body of 
 colour. While drawing the ink lines, the measurement 
 should all be repeated, so as to correct any error that may 
 have slipped during the penciling. The screw in the 
 drawing pen will regulate the breadth of the strokes ; which 
 should not be alike heavy ; those strokes being the heaviest 
 which bound the dark part of the shades. Should any line 
 chance to be wrong drawn with the ink, it may be taken 
 out by means of a sponge and water, which could not be 
 done if common writing ink were employed. 
 
 65. In preparing for colouring it is to be observed, that 
 a hair pencil is to be fixed at each end of a small piece of 
 wood, made in the form of a common pencil, one of which 
 is to be used with colour, and the other with water only. 
 If the colour is to be laid on, so as to represent a flat sur- 
 i'ace, it ought to be spread on equally, and there is here no 
 use for the water brush ; but if it is to represent a curved 
 surface, then the colour is to be laid on the part intended 
 to be shaded, and softened towards the light by washing 
 with the water brush. In all cases it should be borne in 
 mind, that the colour ought to be laid on very thin, other- 
 wise it will be more difficult to manage, and will never 
 make so fine a drawing. 
 
 66. In colours even of the best quality, we sometimes 
 meet with gritty particles, which it is desirable to avoid. 
 Instead of rubbing the colour on a plate with a little water, 
 
 8 * 
 
90 MECHANICAL DRAWING 
 
 as is usual, it will be better to wet the colour, and rub it on 
 the point of the forefinger, letting the dissolved part drop 
 off the finger on to the plate. 
 
 67. In using the Indian ink, it will be found advanta- 
 geous to mix it with a little blue and a small quantity of lake, 
 which renders h much more easily wrought with, and this 
 is the more desirable as it is the most frequently used of all 
 the other colours in Mechanical Drawing, the shades being 
 all made with this colour. 
 
 The depth and extent of the shades will depend on 
 various circumstances on the figure of the object to be 
 shaded, the position of the eye of the observer, and the 
 direction in which the light comes, <fcc. The position of 
 the eye will vary the proportionate size of any object in a 
 picture when drawn in perspective. Thus, if a perspective 
 view of a steam engine is given, the eye being supposed to 
 be placed opposite the end nearest the nozzles, an inch of 
 the nozzle rod will appear much larger than an inch of the 
 pump rod which feeds the cistern ; but if the eye is sup- 
 posed to be placed opposite the other end of the engine, 
 the reverse will be the case. But in drawing elevations 
 and ground plans of machinery, every part of the machine 
 is drawn to the proper scale an inch or foot in one part 
 of the machine, being just the same size as an inch or foot 
 in any other part of the machine. So that by measuring 
 the dimensions of any part of the drawing, and then apply- 
 ing the compass to the scale, we determine the real size of 
 the part so measured. Whereas, if the view were given in 
 perspective, we would be obliged to make allowance for the 
 effect of distance, &c. 
 
 68. The light is always supposed to fall on the picture 
 at an angle of forty-five degrees, from which it follows, that 
 the shade of any object, which is intended to rise from the 
 plane of the picture, or appear prominent, will just be equal 
 in length to the prominence of the object. 
 
 69. The shades, therefore, should be as exactly measured 
 as any other part of the drawing, and care should be taken 
 that they all fall in the proper direction, as the light is sup- 
 posed to come from one point only. 
 
 70. It is frequently of great use for the mechanic to take 
 a hasty copy of a drawing, and many methods have been 
 given for this purpose by machines, tracing, &c. We 
 give the following as easy, accurate, and convenient. 
 
91 
 
 Mix equal parts of turpentine and drying oil, and with a 
 rag lay it on a sheet of good silk paper, allowing the paper 
 to lie by for two or three days to dry, and when it is so it will 
 be fit for use. To use it, lay it on the drawing to be copied, 
 and the prepared paper being nearly transparent, the lines 
 of the drawing will be seen through it, and may be easily 
 traced with a black-lead pencil. The lines on the oiled 
 paper will be quite distinct when it is laid on white paper. 
 Thus, if the mechanic has little time to spare, he may take 
 a copy" and lay it past to be recopied at his leisure. 
 
 Care and perseverance are the chief requisites for attain- 
 ing perfection in this species of drawing. Every mechanic 
 should know something of it, so that he may the better un- 
 derstand how to execute plans that may be submitted to 
 him, or make intelligible to others any invention he him- 
 self may make. 
 
CONIC SECTIONS. 
 
 
 DEFINITIONS. 
 
 A CONE is a solid figure having a circle for its base and 
 terminated in a vertex ; it may be conceived to be formed 
 by the revolution of a triangle about one of its sides. 
 
 Conic Sections are the figures made by a plane cutting a 
 cone. According to the different positions of the cutting 
 plane there arise five different figures or sections, namely, 
 a triangle, a circle, an ellipse, an hyperbola, and a parabola: 
 the last three of which only are peculiarly called Conic 
 
 Sections. If the cutting plane pass through the vertex of 
 the cone, and any part of the base, the section will be a 
 triangle ; as VAB, fig. 1. If the plane cut the cone parallel 
 to the base, or make no angle with it, the section will be a 
 circle ; as fig. 2. The section DAB is an ellipse when the 
 cone is cut obliquely through both sides, or when the plane 
 is inclined to the base in a less angle than the side of the 
 cone is, fig. 3. The section is a parabola, when the cone 
 is cut by a plane parallel to the side, or when the cutting 
 plane and the side of the cone make equal angles with the 
 base, fig. 4. The section is an hyperbola, when the cutting 
 plane makes a greater angle with the base than the side of 
 the cone makes, fig. 5. And if all the sides of the cone be 
 continued through the vertex, forming an opposite equal 
 cone, and the plane be also continued to cut the opposite 
 cone, this latter section will be the opposite hyperbola to 
 the former. 
 
 92 
 
DEFINITIONS. 93 
 
 The Vertices of any section, are the points where the 
 cutting plane meets the opposite sides of the cone, or the 
 sides of the vertical triangular section. 
 
 Hence the ellipse and the opposite hyperbolas, have each 
 two vertices ; but the parabola only one ; unless we consider 
 the other as at an infinite distance. 
 
 The Axis, or Transverse Diameter, of a conic section, is 
 the line or distance between the vertices. 
 
 Hence the axis of a parabola is infinite in length. 
 
 The centre is the middle of the axis. 
 
 Hence the centre of a parabola is infinitely distant from 
 the vertex. And of an ellipse, the axis and centre lie 
 within the curve ; but of an hyperbola, the axis and centre 
 lie without it. 
 
 A Diameter is any right line drawn through the centre, 
 and terminated on each side by the curve ; and the extremi- 
 ties of the diameter, or its intersections with the curve, are 
 its vertices. 
 
 Hence all the diameters of a parabola are parallel to the 
 axis, and infinite in length. Hence also every diameter of 
 the ellipse and hyperbola has two vertices ; but of the para- 
 bola, only one ; unless we consider the other as at an infi- 
 nite distance. 
 
 The Conjugate to any diameter, is the line drawn through 
 the centre, and parallel to the tangent of the curve at the 
 vertex of the diameter. 
 
 Hence the conjugate of the axis is perpendicular to ft. 
 
 An Ordinate to any diameter, is a line parallel to its con- 
 jugate, or to the tangent at its vertex, and terminated by 
 the diameter and curve. 
 
 Hence the ordinates of the axis are perpendicular to it. 
 
 An Absciss is a part of any diameter contained between 
 its vertex and an ordinate to it. 
 
 Hence, in the ellipse and hyperbola, every ordinate has 
 two determinate abscisses ; but in the parabola only one ; 
 the other vertex of the diameter being infinitely distant. 
 
 The Parameter of any diameter is a third proportional 
 to that diameter and its conjugate, in the ellipse and hyper- 
 bola, and to one absciss and its ordinate in the parabola. 
 
 The Focus is the point in the axis where the ordinate is 
 equal to half the parameter. 
 
 The ellipse and hyperbola have each two foci ; but the 
 parabola only one. 
 
** 
 
 &4 CONIC SECTIONS. 
 
 PROBLEMS FOR THE CONIC SECTIONS. 
 THE PARABOLA. 
 
 1. Given two abscisses A and B, together with the ordi- 
 nate of A, to find the ordinate of B. 
 
 v/ absciss B x ordinate A 
 
 = ordinate B. 
 
 */ absciss A 
 
 Ex. An absciss is 9, and its ordinate is 16, it is required 
 to find the ordinate of another absciss 36. 
 
 ^/ 36 X 16 6 x 16 
 
 = = 32, the required ordinate. 
 
 \/ 9 
 
 2. Given the ordinate and absciss, required the para- 
 meter. 
 
 ordinate * 
 
 : = parameter. 
 
 absciss 
 
 Ex. The ordinate being 12 and absciss 6, then, 
 
 12 9 144 
 
 = = 24 = the parameter required. 
 
 3. To find the length of the curve of a parabola, cut off 
 by a double ordinate to the axis. 
 
 <S (ordin. 9 + f abs. 8 ) x 2 = the length of the curve. 
 
 Ex. The length of the double ordinate being 12 and the 
 absciss 2, then, 
 s/(6 9 -f f 2 9 ) x 2 = 12-858 = the length of the curve. 
 
 NOTE. This rule is sufficiently correct for practice, but 
 will not apply when the absciss is greater than the half 
 ordinate. 
 
 THE ELLIPSE. 
 
 1 . To find an ordinate, we have the proportion : 
 
 As the transverse axis is to the conjugate, so is the square 
 root of the product of the two abscisses, to the ordinate. 
 
 Ex. The transverse axis being 60, the conjugate 45, the 
 one absciss 12, and the other 48, then, 
 
 60 : 45 : : v/ (48 X 12) : 18 = the ordinate required. 
 
PROBLEMS. 95 
 
 2. To find the absciss. 
 
 v/ (the half conju. 2 ordin. 2 ) x trans, axis 
 
 conjugate axis. 
 
 distance between the ordinate and centre of the axis, which 
 being added to the half axis, will give the greater absciss, 
 or being subtracted, will give the shorter absciss. 
 
 Ex. One axis being 20 and the other 15, what are the 
 abscisses to the ordinate whose length is 6. 
 
 </ f7*5 3 6 a ^ X 20 
 
 Z- - ; = 6 = the distance from the centre, 
 1 ) 
 
 wherefore 10 + 6=16 = the longer absciss, and 10 
 6 = 4 = the shorter. 
 
 3. To find the conjugate axis. 
 
 As x/(one absciss x other absciss) is to their ordinate, 
 so is the transverse axis to the conjugate. 
 
 Ex. The transverse axis being 200, the ordinate 60, 
 one absciss is 40 and the other 160, then, 
 
 x/(160 x 40) : 60 : : 200 : 150 = the conjugate axis. 
 
 4. To find the transverse axis. 
 
 Take the square root of the difference of the squares of 
 the ordinate and half conjugate, and add to this the half 
 conjugate if the lesser absciss is used, but subtract the half 
 conjugate if the greater absciss is used. In either case call 
 the result of this part of the operation M, then, 
 
 conjugate x absciss x M 
 
 = transverse axis. 
 ordinate a 
 
 Ex. If the ordinate be 20, the lesser absciss 14, and the 
 conjugate 50, then by the above, 
 
 v/(25 3 20 2 ) -f 25 == 40 = M. 
 
 50 X 14 X 40 
 
 = 70 = the transverse axis. 
 
 Jt 
 J 
 
 5. To find the circumference of an ellipse. 
 
 sum of the sq. of the two axes\ c 
 
 - - - ) X 3-1416 = circumfer. 
 
 Ex. The one axis being 24 and the other 18, then, 
 
 24 s 4- IR 3 \ 
 /- ) X 3-1416 = 66-643 = circumference. 
 
96 CONIC SECTIONS. 
 
 THE HYPERBOLA. 
 
 1. To find the ordinate. 
 
 As the transverse axis is to the conjugate ; so is the square 
 root of the product of the two abscisses, to the ordinate. 
 
 Ex. The transverse axis being 24, the conjugate 21, and 
 the absciss 8 ; then, 
 
 24 : 21 :: v/(32 x 8) : 14 = the ordinate. 
 
 2. To find the abscisses. 
 
 -v/(ord. 2 -f half conju. 2 )x trans, axis 
 
 = distance between 
 conjugate 
 
 the ordin. and centre. Then this distance, added to the 
 half transverse, gives the greater absciss; or, subtracted 
 from it, the less. 
 
 Ex. The transverse axis being 40, the conjugate 32, 
 and the ordinate 12 ; then, 
 
 /(12 3 + 16 J )X40 = 25 = distance from the middle Qf 
 o 
 
 the transverse. Wherefore, 25 -f 20 = 45 = the greater 
 absciss ; and 25 20 = 5 = the lesser. 
 
 3. To find the conjugate. 
 ordinate x transverse axis 
 -^/(product of the abscisses) ~~~" 
 
 Ex. The transverse axis being 144, the lesser absciss 
 48, and its ordinate 84 ; then, 
 
 84 x 144 
 --- 126 = the conjugate required. 
 
 4. To find the transverse. 
 
 Take the half conjugate, and, according as the lesser or 
 greater absciss is used, add it to, or subtract it from, the 
 square root of the sum of the squares of the half conjugate 
 and of the ordinate, and call this result m ; then, 
 
 abscissa x conjugate x m 
 
 = the transverse axis. 
 ordmate- 5 
 
 Ex. The conjugate being 18, the lesser absciss 10, and 
 its ordinate 12; then, 
 
 9 + ^/(0 + 12 2 ) = 9 -f 15 = 24 = m ; 
 
 10 X 18 x 24 QA 
 
 = 30 = the transverse axis. 
 

 PROBLEMS. 
 
 97 
 
 Descriptions of Conic Sections on a Plane 
 
 1. Parabola. Let AB be a v 
 right line and C a point with- 
 out it, and DKF a ruler in the 
 
 same plane with the line and 
 point, so that one side, as DK, 
 be applied to AB, and KF 
 coincide with the point C ; 
 on F, fix one end of the 
 thread FNC, and the other at 
 the point C ; and let part of 
 the thread, as FN, be brought to the side KF by a pin N ; 
 then let the square DKF, be removed from B to A, applying 
 its side DK dose to BA; and in the mean time the thread 
 will be always applied to the side KF ; and by the motion 
 of the pin N there will be described a curve called a semi- 
 parabola. Then bringing the square to its first position 
 moving from B to H the other semi-parabola will be 
 described. 
 
 2. Ellipse. If two points, as 
 A and B, be taken in any plane, 
 and in them is fixed a thread 
 longer than the distance between 
 them, and this be extended by 
 means of a pin C ; and the pin 
 
 be moved round from any point till it return back again 
 to the same place, the thread being extended all the while, 
 the figure described is an ellipse. 
 
 3. Hyperbola. If to the point A, one end of the ruler 
 AB be placed, so that about that point as a centre it may 
 freely move ; and if to the other end B is fixed the ex- 
 tremity of the thread 
 
 BDC shorter than the 
 ruler AB, and the other 
 end of the thread fixed 
 in the point C, coincid- 
 ing with the side of the 
 ruler AB in the same 
 place with the given 
 point A ; let part of the 
 thread BD be brought 
 to the side of the ruler 
 
 9 
 
98 CONIC SECTIONS. 
 
 AB by the pin D ; then let the ruler be moved about the 
 point A from C to T, the thread extended, and the re- 
 maining part coinciding with the side of the ruler ; by the 
 motion of the pin D a semi-hyperbola will be described. 
 The ellipse returns into itself: but the parabola and hyper- 
 bola are unlimited. 
 
 USEFUL CURVES. 
 
 THE Cycloid is a very useful curve ; and may be defined, 
 the curve formed by a nail in the rim of a wheel, while it 
 moves along a level road. The cycloid may be described 
 on paper, thus: If the circumference of a circle be rolled 
 
 on a right line, beginning at any point A, and continued 
 till the same point A arrive at the line again, making just 
 one revolution, and thereby measuring out a straight line 
 ABA equal to the circumference of the circle, while the 
 point A in the circumference traces out a curve line 
 AGA6A : then this curve is called a cycloid ; and some 
 of its properties are contained in the following lemma: 
 
 If the generating or revolving circle be placed in the 
 middle of the cycloid, its diameter coinciding with the axis 
 AB, and from any point there be drawn the tangent CF, 
 the ordinate CDE perpendicular to the axis, and the chord 
 of the circle AD ; then the chief properties are these : 
 
 The right line CD = the circular arc AD ; 
 
 The cycloidal arc AC = double the chord AD ; 
 
 The semi-cycloid ACA= double the diameter AB, and 
 
 The tangent CF is parallel to the chord AD. 
 
 If the ball of a pendulum be made to move in a cycloid, 
 its vibrations will be isochronous, or, they will all be per- 
 formed in the same time. The cycloid is also the line of 
 swiftest descent, or, a body will fall through the arc of this 
 curve, from one given point to another, in less time than 
 through any other path. See Centre of Oscillation. 
 
Tl 
 
 CONIC SECTIONS. 99 
 
 The Catenary is that curve which is formed by a chain 
 or chord of uniform texture, when it is hung upon two 
 points, and left to hang freely, without any restraint. It 
 matters not whether these points of suspension be in the 
 same horizontal line or not, or whether the chain be slack 
 or tight ; still the curve will be a catenary. A knowledge 
 of this curve is very useful in the construction of suspension 
 bridges. See the chapter on Strength of Materials. 
 
 
 
L m - : iM* 
 
 ^^NSURATION, 
 
 DEFINITIONS. 
 
 
 To the definitions in geometry the following are added, 
 in order to make the subject of mensuration understood. 
 
 1. A. prism is a solid, of which the sides are parallelo- 
 grams, and the ends equal, similar, and parallel plane 
 figures. The figure of the ends gives the name to the 
 prism ; if the ends are triangular, the prism is triangular, 
 &c. If the sides and ends of a prism be all equal squares, 
 the prism is called a cube ; and if the base or ends be a 
 parallelogram, the prism is called a parallelopipedon. The 
 cylinder is a round prism, having circular ends. 
 
 2. The pyramid has any plane figure for its base, and 
 its sides triangles, of which all the vertices meet in a point 
 at the top, called the vertex of the pyramid. 
 
 3. A sphere or globe is a solid bounded by one continued 
 surface, every point of which surface is equally distant from 
 a point within the sphere, called the centre. The diameter 
 or axis of a sphere, is any line which passes through its 
 centre, and is terminated at both ends by the circumference. 
 
 4. A prismoid has its two ends as any unlike parallel 
 plane figures of the same number of sides ; the upright 
 sides being trapezoids. 
 
 5. A spheroid is a solid resembling the figure of a sphere, 
 but not exactly round one of its diameters being longer 
 than the other ; and, likewise, a conoid is like a cone, but 
 has not its sides straight lines but curved. 
 
 6. A spindle is a solid formed by the revolution of some 
 curve round its base. 
 
 7. The axis of a solid is a straight line drawn through 
 the solid, from the middle of one end to the middle of the 
 opposite end. 
 
 8. The height of a solid is a line drawn from the vertex, 
 perpendicular to the base, or the plane on which the base rests. 
 
 9. The segment of a solid is a part cut off by a plane, 
 parallel to the base ; and the frustum is the part remaining 
 after the segment is cut off. 
 
 100 
 

 MENSURATION. 
 
 101 
 
 SURFACES. 
 1. For the area of a square, rhombus, or rhomboid. 
 
 Base x height = area. 
 
 Ex. The base of a rhombus is 16, the height 9 ; there- 
 fore, 16x9 = 144= area. 
 
 2. For the area of a triangle. 
 
 5 (base X height) = area. 
 
 Ex. The base of a triangle is 2|, and height 7 5 ; there- 
 fore, g (2-25 X 7-5) = 8-437, the area. 
 
 3. For the area of a trapezoid. 
 5 (sum of the two parallel sides) X height = area. 
 Ex. In a trapezoid one of the parallel sides is 16&, the 
 other is 14, and the height or perpendicular distance be- 
 tween them is 7 ; therefore, 
 
 (16-125 -f 14-25) X 7 = 106-3125, the area. 
 
 4. For any right-lined figure of four or more unequal 
 
 sides. 
 
 Divide it into triangles, by lines drawn from various 
 angles ; find the area of each ; then, the sum of these areas 
 will be the area of the whole figure. 
 
 5. For a regular polygon. 
 
 Inscribe a circle ; then, 5 (radius of insc. circle x length 
 of one side x number of sides) = area. 
 
 Ex. In a polygon of 8 sides, the length of a side is 16, 
 and radius of inscribed circle 19-2 ; then 5 (3 X 16 x 8) = 
 1230, the area. 
 
 The following table will greatly facilitate the solution of 
 questions connected with polygons. 
 
 No. of 
 sides. 
 
 Name of 
 Polygon. 
 
 Ang. F at 
 cent. 
 
 Ang. C of 
 Polygon. 
 
 Area. 
 
 A. 
 
 B. 
 
 c. 
 
 3 
 
 Trigon 
 
 120 
 
 60 
 
 0-4330127 
 
 2- 
 
 1-73 
 
 579 
 
 4 
 
 Tetragon 
 
 90 
 
 90 
 
 1-0000000 
 
 1-41 
 
 1-412 
 
 705 
 
 5 
 
 Pentagon 
 
 72 
 
 108 
 
 1-7204774 
 
 1-238 
 
 1-174 
 
 852 
 
 6 
 
 Hexagon 
 
 60 
 
 120 
 
 2-5980762 
 
 1-156 
 
 = Radius 
 
 = Length 
 
 
 
 
 
 
 
 
 of side. 
 
 7 
 
 Heptagon 
 
 5|f 
 
 128f 
 
 3-6339124 
 
 1-11 
 
 867 
 
 1-16 
 
 8 
 
 Octagon 
 
 45 
 
 135 
 
 4-8284271 
 
 1-08 
 
 765 
 
 1-307 
 
 9 
 
 Nonagon 
 
 40 
 
 140 
 
 6-1818242 
 
 1-062 
 
 681 
 
 1-47 
 
 10 
 
 Decagon 
 
 36 
 
 144 
 
 7-6942088 
 
 1-05 
 
 616 
 
 1-625 
 
 11 
 
 Undecagon 
 
 32 FT 
 
 147 TT 
 
 9-3656405 
 
 1-04 
 
 561 
 
 1-777 
 
 12 
 
 Dodecagon 
 
 30 
 
 150 
 
 11-1961524 
 
 1-037 
 
 515625 
 
 1-94 
 
102 MENSURATION. 
 
 The first column of this table gives the number" of sides 
 of the polygon ; the second, the name ; the uses of the 
 third and fourth will be explained in the note at the bottom 
 of the page,* and the uses of the rest will appear by the fol- 
 lowing rules and examples. The answers found are only 
 approximate, but come sufficiently near the truth for all 
 practical purposes. 
 
 Side of polygon a x No. column AREA = area. 
 
 Ex. In a figure of 10 equal sides, the length of one 
 side being 8, we have 8 $ = 8 x 8 = 64 ; hence 64 X 
 7-6942088 = 492-4293632 = the area. 
 
 Take the length of a perpendicular, drawn from the 
 centre to one of the sides of a polygon, and multiply this 
 by the numbers in column A, the product will be the radius 
 of the circle thai contains the polygon. 
 
 Ex. If the length of a perpendicular drawn from the 
 centre to one of the sides of an octagon be 12, then 12 x 
 1-08 = 12-96 = radius of circumscribing circle. 
 
 ^The radius of a circle multiplied by the number in 
 column B, will give the length of the side of the correspond- 
 ing polygon which that circle will contain. Suppose, for 
 an octagon, the radius of a circle ^p be 12*96, then 12*96 
 X '765 = 9-9144 = the length of one side of th<> inscribed 
 polygon of 8 sides. 
 
 The length of the side of a polygon multiplied by the 
 corresponding number in the column U, will give the radius 
 of circumscribing circle. Thus the length of one side of a 
 decagon being 10; then 10 x 1'625 = 16*25 = radius 
 of circumscribing circle. 
 
 6. For the circle. 
 1st, diametet x 3-1416 = circumference; 
 
 * The third and fourth columns of the table of polygons will greatly 
 facilitate the construction of these figures by the aid of the Hector. Thus, 
 if it he required to describe a polygon of eight si.l.-s th.-n U,k in column 
 Angle F, opposite Octagon, and you find 45. With the chord of 60 on 
 the sector as radius describe a circle, then taking the length 45 on the 
 same line of the sector, mark this distance off on the circunifi-rrncr, 
 which being repeated round the circle, will give the points of junction 
 of the sides of the octagon. The fourth column of the table gives the 
 angle in degrees, which any two adjoining sides of the respective figures 
 make with each other. 
 

 MENSURATION. 
 
 circumference 
 
 103 
 
 = diameter ; 
 
 3-1416 
 
 3rf, circumference x radius = area. 
 Ex. In a circle whose diameter is 14 inches, we have, 
 1st, 14 x 3-1416 = 43-9824, the circumference; 
 
 J, 43-9824 
 
 14, the diameter; 
 
 3-1416 
 
 14 
 
 3d, diameter -r- 2 = radius , so 
 
 2 
 
 (43-9824) X 7 
 
 7 = radius. Then, 
 153-9384, the area. 
 
 7. For the length of the arc of a circle. 
 
 Radius x '079577 x number of degrees = length of arc. 
 Ex. If the radius be 12, and number of degrees 22, then, 
 12 x '079577 X 22 = 21-008328, the length. 
 
 8. For the area of a circular sector. 
 
 Radius X 5 length of arc. 
 
 Ex. The radius being 12, and length of arc 2 1-008328 ; 
 then, 12 x 10-504164 =a 126-049968, the area. 
 
 9. For the area of a circular segment. 
 .rtar.' J 
 
 TABLE OF THE AREAS OF CIRCULAR SEGMENTS. 
 
 H. 
 
 Area. 
 
 H. 
 
 Area. 
 
 H. Area. 
 
 11 
 
 Area. 
 
 01 
 
 001329 
 
 14 
 
 066833 
 
 27 -171089 
 
 40 
 
 293369 
 
 02 
 
 003748 
 
 15 
 
 073874 
 
 28 
 
 180019 
 
 41 
 
 303187 
 
 03 
 
 006865 
 
 16 
 
 081112 
 
 29 
 
 189047 
 
 42 
 
 313041 
 
 04 
 
 010537 
 
 17 
 
 088535 
 
 30 
 
 198168 
 
 43 
 
 322928 
 
 05 
 
 014681 
 
 18 
 
 096134 
 
 31 -207376 
 
 44 
 
 332843 
 
 06 
 
 019239 
 
 19 
 
 103900 
 
 32 | -216666 
 
 45 
 
 342782 
 
 07 
 
 024168 
 
 20 
 
 111823 
 
 33 
 
 226033 
 
 46 
 
 352742 
 
 08 
 
 029435 
 
 21 
 
 119897 
 
 34 
 
 235473 
 
 47 
 
 362717 
 
 09 
 
 035011 
 
 23 
 
 128113 
 
 35 
 
 244980 
 
 48 
 
 372704 
 
 10 
 
 040875 
 
 23 
 
 136465 
 
 36 
 
 254550 
 
 49 
 
 382699 
 
 11 
 
 047005 
 
 24 
 
 .144944 
 
 37 
 
 264178 
 
 50 
 
 392699 
 
 12 
 
 053385 
 
 25 
 
 153546 
 
 38 
 
 273861 
 
 001 
 
 000042 
 
 13 
 
 059999 
 
 26 
 
 162263 
 
 39 
 
 283592 
 
 002 
 
 000119 
 
 This may be done easily by the help of the preceding 
 table ; to use which, divide the height of the segment by 
 the diameter of the circle, and look for the quotient in the 
 
104 MENSURATION. 
 
 column H, opposite to which will be found a number in 
 column AREA, which multiplied by the square of the dia- 
 meter will give the area of the segment. Should the height 
 of the segment be greater than the diameter, find by the 
 foregoing rule the area of the remaining segment, and by 
 subtracting this from the area of the whole circle, the area 
 of the greater segment will be found. 
 
 18 
 Ex. Let the height be 18 and diameter 48, then =-37; 
 
 which, in the column marked H in col. AREA, corresponds 
 to -264178 ; hence 48 a x -264178 = 608-6661 = the area. 
 
 10. For the area of a cycloid. 
 Area of generating circle X 3 = area of cycloid. 
 Ex. The diameter of generating circle being 10, then 
 (10 X 3-1416) x V X 3 = 235-619, the area of cycloid. 
 
 11. For the area of a parabola. 
 (Base x height) x f = the area. 
 Ex. The base being 20, and height 6 ; then, 
 20 X 6 x f = 80, the area. 
 
 12. For the area of an ellipse. 
 (Long axis x short axis) X '7854 = area. 
 Ex. The greater axis being 300, and lesser 200 ; then, 
 300 X 200 X -7854 = 47124, the area. 
 
 SOLIDS. 
 1. For the surface and content of a prism or cylinder. 
 
 1st. Area of two ends + length x perimeter = surface. 
 
 2d. Area of base X height = content. 
 
 The circumference of a cylinder is 6, and its length 9 
 inches ; what is the surface and content ? 
 
 The area of each end is 2-85 ; therefore 2 x 2-85 = 
 5-7 = the area of the two ends, and then 5-7 + (6x9) 
 = 59-7 = the area of the whole cylinder. Also, 2-85 X 
 9 = 25-65 = content. 
 
 2. For a cone or pyramid. 
 
 1st. 5 (slant height x perimeter of base) -f area of base 
 = surface. 
 
MENSURATION. 
 
 105 
 
 2d. 3 (area of base X perpend, height) = content. 
 
 Ex. Slant height is 10, perimeter of base 16; then, 
 (10 X 16 = 80 -f 16 = 96, surface of a four-sided pyra- 
 mid, whose side at the base is 4. 
 
 The area of the base of a cone being 147*68, and per- 
 pendicular height 14, 
 
 Then (14 X 147-68)^= 689-17, content. 
 
 3. For a cube or parallelepiped. 
 
 1st. The 'sum of the areas of all the sides = surface. 
 
 2d. Length X breadth x depth = content. 
 
 Ex. In a parallelepiped the length 30, breadth 6, and 
 depth 4. 
 
 30 x 6 x 4 = 720, content, and 648 = the surface. 
 
 It is worthy of remembrance that one cubic foot contains 
 1728 cubic inches, 22,000 cylindric, 3300 spherical inches, 
 and 66 conical. The cone, sphere, and cylinder, are as 1, 
 2, and 3. 
 
 4. For regular or platonic bodies, or bodies of equal sides. 
 
 1st. Linear edge 3 x tabular number of figures for sur- 
 face = surface. 
 
 2d. Linear edge s x tabular number of figures for soli- 
 dity = content. 
 
 No. of Sides. 
 
 Name. 
 
 Multiplier for 
 Surface. 
 
 Multiplier for 
 Solidity. 
 
 4 
 6 
 8 
 12 
 20 
 
 Tetrahedron, 
 Hexahedron, 
 Octahedron, 
 Dodecahedron, 
 Icosahedron, 
 
 1-7320508 
 6-0000000 
 3-4641016 
 20-6457288 
 8-6602540 
 
 0-1178513 
 1-00000 
 0-4714045 
 7-6631189 
 2-181695 
 
 Ex. In an Octahedron the length of the ridge of a side 
 is 5, therefore 5 a X 3-4641016 = 86-6025 = surface, and 
 53 x .4714045 = 58-9255, the solidity. 
 
 5. For the surface of\ sphere and segment. 
 
 Diameter 2 X 3-1416 = surface of the whole sphere. 
 
 Ex. If the diameter be 36, then 36 a x3-1416=4071-504 
 square inches = surface. 
 
 Height of segment x diameter of sphere X 3*1416 = 
 surface of segment. 
 

 106 MENSURATION. 
 
 Ex. The diameter of the sphere being 12, and the 
 height of segment 6, then 
 6 Xl2 x 3-1416 = 226-1952 = surface of spheric segment. 
 
 6. For the content of a sphere and spheric segment. 
 
 Diameter 3 x 0-5236 = content. 
 
 Ex. If the diameter of a sphere be 2 inches, then 2 3 
 X 0-5236 = 4-1888 = the content. 
 
 (radius of segment's base 2 x 3 + height of segment 2 ) X 
 height x '5236 = content of segment. 
 
 Ex. If the height of a spheric segment be 2, and radius 
 of base 6, then 
 
 (6 8 X 3 -f 2 3 ) X 2 x '5236 = 117-2864 = content. 
 
 7. For the solidity of a spheroid. 
 Revolving axis 2 X fixed axis x "5236 = content. 
 NOTE. If the spheroid revolve round the greater axis, 
 it is said to be oblate ; if round the lesser, oblong. 
 
 Ex. The two axes of a spheroid are 24 and 18; there- 
 fore, 
 
 24 2 X 18 X -5236 = 5428-56 = content if it be oblate. 
 18 3 X 24 x '5236 = 4071-5 = content if it be oblong. 
 
 8. For the solidity of a parabolic conoid. 
 
 Area of base X half the height = content. 
 Ex. The height being 18, and the diameter of base 24, 
 then the area of the base therefore is 452-39 ; hence 
 452-39 X 9 = 4071-51 the content. 
 
 9. For the frustum of a cone or pyramid. 
 (perim. of one end + perim. of the other end) x slant height 
 
 ~~2~ 
 = surface. 
 
 Ex. In the frustum of a triangular pyramid the peri- 
 meter of one end is 25, that of the other 36, and the slant 
 height is 10 ; therefore, 
 
 (25 + 36) X 10 
 
 L 305 = the surface. 
 
 & 
 
 ^/(area of one end + ar. of other) -J- area of one end + ar. of other 
 
 3 
 
 X height = content. 
 Ex. A log of wood is 20 feet long ; its ends are squares, 
 
MENSURATION. 107 
 
 of which the sides are respectively 12 and 16 inches ; there- 
 fore, 
 
 33600 inches _ 
 
 TIMBER MEASURE. 
 
 EXAMPLES of timber measuring have already been given in 
 the department allotted to arithmetic, but it is necessary to 
 be here somewhat more particular. The surface of a plank 
 is found : 
 
 1st. By multiplying the length by the breadth. When 
 the board tapers gradually, add the breadth at both ends 
 together, and take the half of this sum for the mean 
 breadth. 
 
 2d. By the sliding rule. Set the length in inches on 
 B to 12 on A, and against the length in feet on B will be 
 the area in square feet and decimals on A. 
 
 Ex. A board is 12 feet 6 inches long and 1 foot 3 
 inches broad ; hence, 
 
 12 : 6 
 1 : 3 
 12 : 6 
 3:1:6 
 
 15 : 7 : 6 
 
 1st. For the content of squared timber, length X mean 
 breadth x mean thickness = content. 
 
 2d. By the sliding rule. Find the mean proportional 
 between the breadth and thickness, then set the length on 
 C to 12 on D, and against the mean proportional on D the 
 solid content on C. If the mean proportional be in feet, 
 reduce to inches. 
 
 Ex. A log is 24 feet long, the mean depth and breadth 
 being each 13 inches. 
 
 1 : 1 
 1 : 1 
 
 1:2:1 
 24 
 
 28 : 2 : 
 
108 MENSURATION. 
 
 For round timber. 1st. Take one-fourth of the mean 
 girth and square it, this multiplied by the length will give 
 the content. 
 
 2d. By the sliding rule. Set the length in feet on C 
 to 12 on D, then against the quarter girth in inches on D, 
 will be the content on C. 
 
 This gives no allowance for bark, but there is usually a 
 deduction made of about an inch to the foot of quarter girth. 
 The rule given above gives the customary, but not the true 
 content ; the following gives the true content. 
 
 One-fifth of the girth squared and multiplied by twice the 
 length = content. 
 
 Ex. The mean girth of a tree being 5 feet 8 inches, and 
 its length 18 feet, the two rules will apply as below: 
 4) 5 : 8 (1 : 5 5) 5 : 8 (1 : 1 : 7 
 1 ; 5 1:1:7 
 2:0:1 1:3:4:6 
 IS 36 
 
 36 : 1 : 6 46 : 1 : 6 
 
 Trees very seldom have an equal girth throughout, one 
 end being generally much smaller than the other : the girth 
 taken above is the mean girth ; that is to say, the girths of 
 both ends added together, and their sum halved for the 
 mean girth. It is to be observed, however, that, if the 
 difference of the girths is great, it will be best to find the 
 content of the tree as if it were a conic frustum. The 
 method of using the sliding rule in the measurement of tim- 
 ber has been given before. 
 
 ARTIFICERS' WORK. 
 
 ARTIFICERS compute the contents of their works by 
 several different measures ; as, glazing and masonry by the 
 foot; painting, plastering, paving, &c., by the yard of 9 
 square feet; flooring, partitioning, roofing, tiling, &c., by 
 the square of 100 square feet; and brickwork, either by 
 a yard of 9 square feet, or by the perch, or square rod or 
 pole, containing 272$ square feet, or 30| square yards, 
 being the square of the rod or pole of 16 feet of 5 yards 
 long. As this number 272| is troublesome to divide by, 
 the is often omitted in practice, and the content in feet 
 divided only by the 272. But when the exact divisor 
 
 
 
ARTIFICERS' WORK. 109 
 
 272| is to be used, it will be easier to multiply the feet by 
 4, and then divide successively by 9, 11, and 11. Also to 
 divide square yards by 30|, first multiply them by 4, and 
 then divide twice by 11. 
 
 BRICKLAYERS' WORK. Brickwork is estimated at the 
 rate of a brick and a half thick. So that, if a wall be more 
 or less than this standard thickness, it must be reduced to 
 it, as follows : Multiply the superficial content of the wall 
 by the number of half bricks in the thickness, and divide 
 the product by 3. The dimensions of a building are usu- 
 ally taken by measuring half round on the outside, and half 
 round on the inside ; the sum of these two gives the com- 
 pass of the wall, to be multiplied by the height, for the 
 content of the materials. Chimneys are by some measured 
 as if they were solid, deducting only the vacuity from the 
 hearth to the mantel, on account of the trouble of them. 
 And by others they are girt or measured round for their 
 breadth, and the height of the story is their height, taking 
 the depth of the jambs for their thickness. And in this 
 case, no deduction is made for the vacuity from the floor 
 to the mantel-tree, because of the gathering of the breast 
 and wings, to make room for the hearth in the next story. 
 To measure the chimney shafts, which appear above the 
 building, gird them about with a line for the breadth, to 
 multiply by their height. And account their thickness 
 half a brick more than it really is, in consideration of the 
 plastering and scaffolding. All windows, doors, <fcc., are 
 to be deducted out of the contents of the walls in which 
 they are placed. But this deduction is made only with 
 regard to materials ; for the whole measure is taken for 
 workmanship, and that all outside measure too, namely, 
 measuring quite round the outside of the building, being 
 in consideration of the trouble of the returns or angles. 
 There are also some other allowances, such as double 
 measure for feathered gable ends, &c. 
 
 Ex. The end wall of a house is 28 feet 10 inches long, 
 and 55 feet 8 inches high, to the eaves ; 20 feet high is 2 
 bricks thick, other 20 feet high is 2 bricks thick, and the 
 remaining 15 feet 8 inches is 1^ brick thick ; above which 
 is a triangular gable, 1 brick thick, and which rises 42 
 courses of bricks, of which every 4 courses make a foot. 
 What is the whole content in standard measure ? 
 
 Ans. 253-626 yards. 
 10 
 
110 MENSURATION. 
 
 MASONS' WORK. To masonry belong all sorts of stone- 
 work ; and the measure made use of is a foot, either super- 
 ficial or solid. Walls, columns, blocks of stone or marble, 
 &c., are measured by the cubic foot ; and pavements, 
 slabs, chimney-pieces, &c., by the superficial or square 
 foot. Cubic or solid "measure is used for the materials, 
 and square measure for the workmanship. In the solid 
 measure, the true length, breadth and thickness, are taken, 
 and multiplied continually together. In the superficial, 
 there must be taken the length and breadth of every part 
 of the projection, which is seen without the general upright 
 face of the building. 
 
 Ex. In a chimney-piece, suppose the 
 Length of the mantel and slab, each 4 feet 6 inches ; 
 
 Breadth of both together, 3 2 
 
 Length of each jamb, 4 4 
 
 Breadth of both together, 1 9 
 
 Required the superficial content. Ans. 21 feet, 10 inch. 
 
 CARPENTERS' AND JOINERS' WORK. To this branch 
 belongs all the wood-work of a house, such as flooring, 
 partitioning, roofing, &c. Large and plain articles are 
 usually measured by the square foot or yard, &c., but 
 enriched mouldings, and some other articles, are often 
 estimated by running or lineal measures, and some things 
 are rated by the piece. 
 
 In measuring of joists, it is to be observed, that only one 
 of their dimensions is the same with that of the floor ; for 
 the other exceeds the length of the room by the thickness 
 of the wall and of the same, because each end is let into 
 the wall about of its thickness. 
 
 No deductions are made for hearths, on account of the 
 additional trouble and waste of materials. 
 
 Partitions are measured from wall to wall for one dimen- 
 sion, and from floor to floor, as far as they extend, for the other. 
 
 No deduction is made for door-ways, on account of the 
 f trouble of framing them. 
 
 In measuring of joiners' work, the string is made to ply 
 close to every part of the work over which it passes. 
 
 The measure for centering for cellars is found by making 
 a string pass over the surface of the arch for the breadth, 
 and taking the length of the cellar for the length ; but in 
 groin centering, it is usual to allow double measure, on 
 account of their extraordinary trouble. 
 
 

 Ill 
 
 In roofing, the length of the house in the inside, to- 
 gether with of the thickness of one gable, is to be con- 
 sidered as the length ; and the breadth is equal to double 
 the length of a string which is stretched from the ridge 
 down the rafter, and along the eaves-board, till it meets with 
 the top of the wall. 
 
 For staircases, take the breadth of all the steps, by making 
 a line ply close over them, from the top to the bottom, and 
 multiply the length of this line by the length of a step, for 
 the whole area. By the length of a step is meant the 
 length of the front and the returns at the two ends ; and 
 by the breadth, is to be understood the girth of its two outer 
 surfaces, or the tread and riser. 
 
 For the balustrade, take the whole length of the upper 
 part of the hand-rail, and girt over its end till it meet the 
 top of the newel post, for the length ; and twice the length 
 of the baluster upon the landing, with the girth of the hand- 
 rail, for the breadth. 
 
 For wainscoting, take the compass of the room for the 
 length ; and the height from the floor to the ceiling, making 
 the string ply close into all the mouldings for the breadth. 
 Out of this must be made deductions for windows, doors, 
 and chimneys, &c., but workmanship is counted for the 
 whole, on account of the extraordinary trouble. 
 
 For doors, it is usual to allow for their thickness, by add- 
 ing it unto both the dimensions of length and breadth, and 
 then to multiply them together for the area. If the door 
 be paneled on both sides, take double its measure for the 
 workmanship ; but if the one side only be paneled, take 
 the area and its half for the workmanship. For the sur- 
 rounding architrave, gird it about the outermost parts for 
 its length ; and measure over it, as far as it can be seen 
 when the door is open, for the breadth. 
 
 Window-shutters, bases, <fec., are measured in the same 
 manner. 
 
 In the measuring of roofing for workmanship alone, 
 holes for chimney-shafts and skylights are generally de- 
 ducted. But in measuring for work and materials, they 
 commonly measure in all skylights, luthern-lights, and 
 holes for the chimney-shafts, on account of their trouble 
 and waste of materials. 
 
 Ex. To how much, at 65. per square yard, amounts the 
 wainscoting of a room ; the height, taking in the cornice 
 
 
112 MENSURATION. 
 
 and mouldings, being 12 feet 6 inches, and the whole com- 
 pass 83 feet 8 inches ; also three window-shutters are each 
 7 feet 8 inches by 3 feet 6 inches, and the door 7 feet by 3 
 feet 6 inches ; the door and shutters, being worked on both 
 sides, are reckoned work and half work ? 
 
 Ans. d636, 12s. 2%d. 
 
 SLATERS' AND TILERS' WORK. In these articles, the 
 content of a roof is found by multiplying the length of the 
 ridge by the girth over from eaves to eaves ; making allow- 
 ance in this girth for the double row of slates at the bottom, 
 or for how much one row of slates or tiles is laid over an- 
 other. When the roof is of a- true pitch, that is, forming 
 a right angle at top, then the breadth of the building with 
 its half added, is the girth over both sides. In angles 
 formed in a roof, running from the ridge to the eaves, when 
 the angle bends inwards, it is called a valley ; but when 
 outwards, it is called a hip. Deductions are made for 
 chimney-shafts or window-holes. 
 
 Ex. To how much amounts the tiling of a house, at 
 255. 6(/. per square ; the length being 43 feet 10 inches, 
 and the breadth on the flat 27 feet 5 inches, also the eaves 
 projecting 16 inches on each side, and the roof of a true 
 pitch ? 24, 9s. 5|rf. 
 
 PLASTERERS' WORK. Plasterers' work is of two kinds, 
 namely, ceiling which is plastering upon laths and ren- 
 dering, which is plastering upon walls ; which are measured 
 separately. 
 
 The contents are estimated either by the foot or yard, or 
 square of 100 feet. Enriched mouldings, &c., are rated by 
 running or lineal measure. 
 
 Deductions are to be made for chimneys, doors, win- 
 dows, &c. But the windows are seldom deducted, as the 
 plastered returns at the top and sides are allowed to com- 
 pensate for the window opening. 
 
 Ex. Required the quantity of plastering in a room, the 
 length being 14 feet 5 inches, breadth 13 feet 2 inches, and 
 height 9 feet 3 inches to the under side of the cornice, 
 which girts 85 inches, and projects 5 inches from the wall 
 on the upper part next the ceiling deducting only for a 
 door 7 feet by 4. 
 
 Ans. 53 yards 5 feet 3 inches of rendering, 
 18 5 6 of ceiling, 
 
 39 Oji of cornice. 
 
ARTIFICERS' WORK. 113 
 
 PAINTERS' WORK. Painters' work is computed in square 
 yards. Every part is measured where the colour lies ; and 
 the measuring line is forced into all the mouldings and 
 corners. 
 
 Windows are done at so much apiece. And it is usual 
 to allow double measure for carved mouldings, &c. 
 
 Ex. What costs the painting of a room at Gd. per yard ; 
 its length being 24 feet 6 inches, its breadth 16 feet 3 
 inches, and height 12 feet 9 inches ; also the door is 7 feet 
 by 3 feet 6 inches, and the window-shutters to two windows 
 each 7 feet 9 inches by 3 feet 6 inches, but the breaks of 
 the windows themselves are 8 feet 6 inches high, and 1 foot 
 3 inches deep deducting the fire-place of 5 feet by 5 feet 
 6 inches ? Ans. 3, 3s. W^d. 
 
 GLAZIERS' WORK. Glaziers take their dimensions either 
 in feet, inches, and parts ; or feet, tenths, and hundredths. 
 And they compute their work in square feet. 
 
 In taking the length and breadth of a window, the cross 
 bars between the squares are included. Also, windows of 
 round or oval forms are measured as square, measuring 
 them to their greatest length and breadth, on account of the 
 waste in cutting the glass. 
 
 Ex. Required the expense of glazing the windows of a 
 house at 13d. a foot; there being three stories, and three 
 windows in each story. 
 
 The height of the lower tier is 7 feet 9 inches, 
 
 of the middle 6 6 
 
 of the upper 5 3| 
 
 and of an oval window over the door 1 10% 
 
 the common breadth of all the windows being 3 feet 9 
 
 inches. Ans. 12, Is. S^d. 
 
 PAVERS' WORK. Pavers' work is done by the square 
 yard. And the content is found by multiplying the length 
 by the breadth. 
 
 Ex. What will be the expense of paving a rectangular 
 courtyard, whose length is 63 feet, and breadth 45 feet ; 
 in which there is laid a footpath of 5 feet 3 inches broad, 
 running the whole length, with broad stones, at 35. a yard 
 the rest being paved with pebbles, at 2s. 6d. per yard ? 
 
 Ans. 40, 5s.W%d. 
 
 PLUMBERS' WORK. Plumbers' work is rated at so much 
 a pound, or else by the hundred weight, of 112 pounds. 
 Sheet lead used in roofing, guttering, &c., is from 7 to 12 
 
 10* 
 
114 MENSURATION. 
 
 lb. to the square foot. And a pipe of an inch bore is com- 
 monly 13 to 14 Ib. to the yard in length. 
 
 Ex. What cost the covering and guttering a roof with 
 lead, at 19s. the cwt. ; the length of the roof being 43 feet, 
 and breadth or girth over it 32 feet the guttering 60 feet 
 long, and 2 feet wide, the former 9 lb., and the latter 8 lb. 
 to the square foot ? Ans. 113, 3s. 
 
MECHANICS. 
 
 DEFINITIONS. 
 
 1. A BODY is any quantity of matter collected together. 
 
 2. Whatever communicates, or has a tendency to com- 
 municate motion to a body, is called a force. 
 
 3. That department of knowledge which comprehends a 
 statement of the effects of forces on bodies, is called Mecha 
 nics. If a body be put in motion by the action of one or 
 more forces, the consideration of the circumstances of this 
 body belongs to that branch of Mechanics called Dynamics ; 
 but if two or more forces act on a body in such a way that 
 they destroy each other's effects, and the body remains at 
 rest, or in equilibrium, the consideration of the circum- 
 stances of a body, in this case, belongs to that department 
 of Mechanics called Statics. 
 
 4. The density of matter, is the quantity of matter con- 
 tained in any body compared with its bulk. Thus lead is 
 denser than cork. 
 
 5. The weight of a body, is its quantity of matter, with- 
 out regard to its bulk. 
 
 6. When we speak of some given space, which a moving 
 body passes over in a given time, we speak of the velocity 
 of the body. If a body moves over one foot of space in one 
 second of time, it is said to have a velocity of one foot in 
 the second ; and its velocity would be increased to the 
 double, if it passed over two feet in one second of time. 
 
 7. If, while the body is in motion, the velocity continues 
 the same, the body is said to have a uniform motion ; but 
 if, while the body moves onward, the velocity continually 
 increases, it is said to have an accelerated motion ; and, on 
 the other hand, if during the progress of the body in motion, 
 the velocity continually decreases, the body is said to have 
 a retarded motion. 
 
 8. The quantity of matter in a moving body, multiplied 
 by the velocity with which it moves, is called the quantity 
 of motion, or momentum of the body. 
 
 115 
 
116 
 
 MECHANICS. 
 
 9. Gravity is that force by which all bodies endeavour to 
 descend towards the centre of the earth. 
 
 AXIOMS, OR PLAIN TRUTHS. 
 
 IF a body be at rest, it will remain at rest ; and if in mo- 
 tion, it will continue that motion, uniformly in a straight 
 line, if it be not disturbed by the action of some external 
 cause. 
 
 The change of motion takes place in the direction in 
 which the moving force acts, and is proportional to it. 
 
 The action and reaction of bodies upon one another, are 
 equal. 
 
 LAWS OF MOTION. 
 
 Uniform motion is caused by the action of some force, 
 
 by one impulse, on the body : and if 
 
 b signify the quantity of matter to be moved, 
 / the force which caused the body's motion, 
 v the velocity with which the body moves, 
 m the momentum of the body in motion, 
 8 the space passed over by the moving body, 
 t the time of describing that space ; 
 
 and if b = 3, m = 6, v = 2,/ = 6, * = 4, t = 2 : then 
 
 the figures in the examples will show the application of the 
 
 theorems. 
 
 THEOREMS. 
 
 EXAMPLES. 
 
 6. 
 
 m 
 
 L, 
 
 in 
 
 xt 
 
 .fxt 
 
 3. 
 
 6 
 
 6 
 
 
 6 X2 
 
 6x2 
 
 . 
 
 v 
 
 V 
 
 
 8 
 
 8 
 
 . 
 
 2 
 
 * 2 
 
 * 
 
 4 
 
 4 
 
 f 
 
 m : 
 
 b X 
 
 v 
 
 b 
 
 X 8 
 
 6: 
 
 6 
 
 : 3 X2 
 
 3x4 
 
 
 
 
 
 * 
 
 
 
 
 
 t 
 
 
 
 
 
 2 
 
 
 
 
 h v 
 
 M 
 
 b 
 
 X 8 
 
 6. 
 
 
 . q v 9 
 
 3 X4 
 
 
 
 J - 
 
 f XN 
 
 c 
 
 
 t 
 
 . 
 
 
 . u A 
 
 2 
 
 
 9 : 
 
 t X 
 
 t 
 
 v : 
 
 xm 
 
 t_xf 
 b 
 
 4: 
 
 2 
 
 X 2 
 
 2 
 
 X 6 2 
 
 X6 
 
 b 
 
 * 
 
 
 3 
 
 3 
 
 
 m 
 
 8 
 
 f 
 
 
 
 
 6 
 
 4 
 
 
 6 
 
 
 r : 
 
 T 
 
 7 
 
 '77 
 
 
 
 2: 
 
 3 
 
 1 2 
 
 : 
 
 
 
 
 
 s 
 
 s xb 
 
 
 s x b 
 
 
 4 
 
 4 
 
 X3 4x3 
 
 / : 
 
 V 
 
 m 
 
 : 
 
 f 
 
 i 
 
 '2 : 
 
 2" 
 
 : 
 
 "6~ 
 
 i 
 
 
MOTION. 117 
 
 * ^^ 
 
 OF ACCELERATED MOTION. 
 
 If the moving force continues to act all the while that the 
 body is in motion, then that motion will be uniformly ac- 
 celerated : such is the case with bodies falling to the earth, 
 as the force of gravity acts constantly. Now, it has been 
 found by experiment, that a body falling through free space, 
 in the latitude of London, will, by the force of gravity, fall 
 through 16*095 feet in the first second of time ; and as forces 
 are measured by the effects they produce, this 16*095 may 
 be taken as the measure of the force of gravity ; and as this 
 quantity does not differ materially from 16 feet, we shall 
 neglect the fraction -095 in our calculation of the circum- 
 stances of falling bodies. 
 
 The subjects of consideration here are, the time that the 
 falling body is in motion, the space it falls through in that 
 time, and the velocity which it has acquired in falling 
 through that space, or that velocity with which it would 
 continue to move, supposing gravity to cease its action, and 
 the motion of the body becoming uniform. 
 
 The time is always supposed to be taken in seconds, and 
 the space in feet. 
 
 The velocity acquired = 32 x time of falling, 
 
 or =v/(64 x space fallen through) 
 
 the velocity acquired 
 The time of falling =- - ~^ 
 
 32 
 
 f/the space fallen through\ 
 
 the velocity acquired * 
 The space fallen through = ^ 
 
 or = time, a X 16. 
 
 Ex. If a body falls through 100 feet, then 
 -s/(64 x 100) = 80 = the velocity acquired, 
 
 80 
 
 = 2f = 2-5 = the time of falling. 
 
 32 
 
 If the space described be 64 feet, then 
 
 = 2 = the time of falling, 
 
 32 X 2 = 64 = the velocity acquired. 
 If the space descended be 400, then 
 
118 MECHANICS. 
 
 x 64) = 160 = the velocity acquired, 
 
 1155! = 5 = the time of falling. 
 o2 
 
 If the times be as 1, 2, 3, 4, 5, &c. 
 
 The velocities will be as 1, 2, 3, 4, 5, &c. 
 And the spaces as 1,4, 9, 16, 25, &c. 
 
 The space for each time as 1, 3, 5, 7, 9, &c. 
 
 COLLISION OF BODIES. 
 
 IF two bodies, A and B, in motion, weigh respectively 5 
 and 3 Ibs., and their velocities respect- 
 ively 3 and 2 before they strike, - - ? - 
 then will 3 X 5 be the momentum of 
 A, and 2 X 3 that of B, before the stroke ; also, 5 + 3 = 
 8 is the sum of their weights ; then, 1st. If the bodies move 
 'the same way, the quotient arising from the division of the 
 sum of the momentum* of the two bodies, by the sum of 
 their weights, will give the common velocity of the two 
 bodies after the stroke. 2rf. If the bodies move contrary 
 ways, then the quotient arising from the division of the 
 difference of their momentums, by the sum of their 
 weights, will give the common velocity after the stroke. 
 3d. If one of the bodies be at rest, then the quotient of the 
 momentum of the other body, divided by the sum of the 
 weights of the two bodies, will give the common velocity 
 after the stroke. Hence, assuming the numbers given above, 
 
 1 |E | *S 
 
 we have, in the first case, - = 2 ; in the second 
 
 o 
 
 H = 1| ; and in the third - = 1?, as the common 
 
 8 o 
 
 velocity after the stroke. 
 
 When the bodies are perfectly elastic, the theorems be- 
 come more complicated. 
 
 If the weight of the one body be A, and the velocity V ; 
 the weight of the other body B, and its velocity v : then, 
 
 1st. If the bodies move in the same direction before the 
 stroke, _ 
 (2BxoMA-BxV) = t 
 
 -BXP) = the velocity of B after the stroke. 
 A-flS 
 
PARALLELOGRAM OP FORCES. 119 
 
 2d. If B move in the contrary direction to A before the 
 stroke, 
 
 (A B)xV 2xBxV 
 
 i - i - - - = velocity of A after the stroke. 
 A + B 
 
 (A ~ B)X r + n + AXY = Cecity of B after the stroke. 
 
 A -j- D 
 
 3rf. If the body B had been at rest before it was struck 
 by A, then 
 A TC 
 
 X V = the velocity of A after the stroke. 
 
 = the velocity of B after the stroke. 
 
 Ex. If the weight of an elastic body A be 6 Ibs., and 
 its velocity 4, and the weight of another body B be 4 Ibs., 
 and its velocity 2 ; then we have these results : in the first 
 case, 
 
 The sum of these two velocities, viz. 5*2 and -8 = 6, which 
 was the sum of the velocities 2 and 4 before the stroke; 
 and this is a general law. The reader may exercise him- 
 self with the rules for the other cases. 
 
 It is to be observed, that when non-elastic bodies, that is, 
 bodies which have no spring, strike, they will both move 
 in the direction of the motion of that body which has the 
 greater momentum ; but if they are elastic, they will recoil 
 after the stroke, and move contrary ways. 
 
 THE COMPOSITION AND RESOLUTION OF FORCES. 
 
 ' IF a body be acted upon 
 by two forces, one of which 
 would cause it to move from 
 A to B in any given time, 
 and the other would cause 
 it to move from A to C in 
 
120 MECHANICS. 
 
 the same time ; then if these forces act upon the body at 
 one instant, it will move in neither of the lines AB, AC, 
 but in the line AD, which is the diagonal of the parallelo- 
 gram of which the two lines AB and AC are containing 
 sides ; and by the action of the two forces, the body will be 
 found at D, at the end of the time that it would have been 
 found at B or C, by the action of either of the forces singly. 
 This important fact in mechanical science, is usually called 
 the parallelogram of forces. From this statement it will 
 be seen, that if we have the quantity and direction of any 
 two forces urging a body at the same instant, we can find 
 the resulting motion, both in quantity and direction. 
 
 It will not be difficult to understand, that if the two forces 
 which act upon a body, act not at an angle, but in the same 
 straight line, and in contrary directions, the resulting 
 motion will be in that straight line, and in the direction of 
 the greater force ; but if the forces be equal, the body will 
 remain at rest. If, while a body A is urged by a force in 
 the direction AB, which would carry it to A, it be acted 
 on by another force in the direction AC which would carry 
 it to C, and a third force in the direction DA, which would 
 carry it over a space as great as that from D to A, these 
 being the sides and diagonals of a parallelogram, the body 
 A will remain at rest. Also, if a body A has a tendency to 
 move in the direction AB, but is counteracted by a force 
 DA, and if we wish to keep the body A from moving, 
 altogether, we must apply another force AC, forming the 
 other side of the parallelogram of which AB is one side and 
 AD the diagonal. 
 
 If there be three forces acting on a body at the same time, 
 make the sides of a parallelogram represent any two of 
 them ; then the diagonal of this parallelogram, together 
 with the third force as the two sides of another parallelo- 
 gram, will give a diagonal which will be the result of the 
 three forces acting at once on the body. 
 
 If the two forces which urge the body, both produce a 
 uniform motion, the resulting motion will be in a straight 
 line ; but if one of them act by impulse, which would pro- 
 duce a uniform motion, and the other act constantly so as 
 to produce an accelerated motion, the resulting motion will 
 be in a curve. Thus, if the ball of a cannon were sent in 
 a horizontal direction, it would never deviate from this 
 straight line unless acted on by some external force. The 
 
THE LEVER. 121 
 
 force of gravity acts on the body constantly, so as to draw 
 it to the earth, by a uniformly accelerated motion ; and the 
 result is, that the ball will move in a curve, and this curve 
 may be easily shown to be that of the parabola. The re- 
 sistance of the air being taken into account together with 
 these circumstances, constitute the basis of the science of 
 gunnery. 
 
 We shall give a simple example, to show the application 
 of the former part of this subject. One force will cause 
 the body A to move 20 miles in a day, and another, acting 
 at right angles, will cause it to move 18 miles a day; draw 
 these lines 20 and 18 from the line of lines on the sector, 
 as the sides AB, AC, of a parallelogram, and complete it : 
 draw the diagonal, then measure it, and it will be found 
 to be 26*9, the resulting motion ; and the angle being 
 measured, will give the direction. There are other methods 
 of doing this by calculation, but this is simple, and is suffi- 
 cient to show the principle. 
 
 MECHANICAL POWERS. 
 
 1. A MACHINE is any instrument employed to regulate 
 motion, so as to save either time or force. No instrument 
 can be employed by man so as to save both time and force ; 
 for it is a maxim in mechanics, that whatever we gain in 
 the one of these two, must be at the expense of the other. 
 
 2. The simple machines, or those of which all others are 
 constructed, are usually reckoned six : the lever, the wheel 
 and axle, the pulley, the inclined plane, the wedge, and 
 the screw. To these the funicular machine is sometimes 
 added. 
 
 3. The weight signifies the body to be moved, or the 
 resistance to be overcome ; and the power is the force em- 
 ployed to overcome that resistance, or move that body. 
 They are frequently represented by the first letters of their 
 names, W and P. 
 
 THE LEVER. 
 
 4. A LEVER is an inflexible bar, either straight or bent, 
 supposed capable of turning round a fixed point, called the 
 fulcrum. 
 
 11 
 
122 MECHANICS. 
 
 According to the relative positions of the weight, power, 
 and fulcrum, on the lever, it is said to be of three kinds, 
 viz. when the fulcrum is somewhere betwixt the weight and 
 power, it is of the first kind ; when the weight is between 
 the power and the fulcrum, it is of the second kind ; and 
 when the power is between the weight and the fulcrum, it 
 is of the third kind: thus, 
 
 5- I* * ; ' 
 
 6. Zd. = = -' 
 
 7. 3d. , '- 
 
 8. In the first and second kinds there is an advantage of 
 power, but a proportionate loss of velocity ; and in the third 
 kind, there is an advantage in velocity, but a loss of power. 
 
 9. When the weight x its distance from the fulcrum = 
 the power x its distance from the fulcrum, then the lever 
 will be at rest, or in equilibrio ; but if one of these pro- 
 ducts be greater than the other, the lever will turn round 
 the fulcrum in the direction of that side whose product is 
 the greater. 
 
 10. In all the three kinds of levers, any of these quanti- 
 ties, the weight or its distance from the fulcrum, or, the 
 power or its distance from the fulcrum, may be found from 
 the rest, such, that when applied to the lever, it will remain 
 at rest, or the weight and power will balance each other. 
 
 weight X its dist. from fulc. 
 
 11. v^ ? ;r-r = power. 
 
 dist. of power from fulc. 
 
 power X its dist. from fulc. 
 
 12. ~ j . . f-= = weight. 
 
 dist. of weight from fulc. 
 
 weight x dist. weight from fulc. 
 
 13. - =dist. power from ful. 
 
 power 
 
 powerxdist.powerfrom fulc. 
 
 14. 5 ___ =dist. weight from fulc. 
 
 weight. 
 
 15. In the first kind of lever, the pressure upon the ful- 
 crum = the sum of weight and power ; in the second and 
 third = their difference. 
 
 16. If there be several weights on both sides of the ful- 
 crum, they may be reckoned powers on the one side of the 
 fulcrum, and weights on the other. Then, if the sum of 
 the product of all the weights x their distances from the 
 
THE LEVER. 123 
 
 fulcrum be = to the sum of the products of all the powers 
 X their distances from the fulcrum, the lever will be at rest, 
 if not, it will turn round the fulcrum in the direction of that 
 side whose products are greatest. 
 
 17. In these calculations, the weight of the lever is not 
 taken into account ; but if it is, it is just reckoned like any 
 other weight or power acting at the centre of gravity. 
 
 18. When two, three, or more levers act upon each other 
 in succession, then the entire mechanical advantage which 
 they give, is found by taking the product of their separate 
 advantages. 
 
 19. It is to be observed, in general, before applying these 
 observations to practice, that if the lever be bent, the dis- 
 tances from the fulcrum must be taken, as perpendiculars 
 drawn from the lines of direction of the weight and power 
 to the fulcrum. 
 
 Ex. In a lever of the first kind, the weight is 16, its 
 distance from the fulcrum 12, and the power is 8 ; there* 
 fore, by No. 13 of this chapter, 
 
 = 24, the distance of power from the fulcrum. 
 
 8 
 
 In a lever of the second kind, a power of 3 acts at a distance 
 of 12 ; what weight can be balanced at a distance of 4 
 from the fulcrum ? Here, by No. 12, 
 
 3 * 12 = 9, weight. 
 
 In a lever of the third kind, the weight is 60, and its dis- 
 tance 12, and the power acts at a distance of 9 from the 
 fulcrum; therefore, by No. 11, 
 
 60 x 12 . , 
 = 80, the power required. 
 
 If there be a lever of the first kind, having three weights, 
 7, 8, and 9, at the respective distances of 6, 15, and 29, 
 from the fulcrum on one side, and a power of 17 at the dis- 
 tance of 9 on the other side of the fulcrum ; then a power is 
 to be applied at the distance of 12 from the fulcrum, on the 
 last mentioned side : what must that power be to keep the 
 lever in balance ? 
 
 Here (6x7) + (15 x 8) + (29 X 9) = 423 = the 
 effect of the three weights on the one side of the ful- 
 crum ; and 17 X 9 = 153 = the effect of the power on the 
 other side. Now, it is clear that the effect of the weight is 
 
124 MECHANICS. 
 
 far greater than the effect of the power ; and the difference 
 423 153 = 270 requires to be balanced by a power ap- 
 plied at the distance of 12, which will evidently be found 
 by dividing 270 by 12, which gives 22'5, the weight re- 
 quired. 
 
 20. The Roman steel-yard is a lever of the first kind, so 
 contrived that only one movable weight is employed. 
 
 The common weighing balance is also a lever of the first 
 kind. The requisites of a good balance are: that the points 
 of suspension of the scales and the centre of motion, or ful- 
 crum of the beam, be all in one straight line that the arms 
 of the beam be equal to each other in every respect that 
 they be as long as possible that the centre of gravity of 
 the beam be a very little below the centre of motion that 
 the beam be balanced when the scales are empty, &c. 
 But we may ascertain the true weight of any body even by 
 a false balance, thus : weigh the body first in one scale, 
 then in the other, and multiply their weights together; 
 then the square root of this product will be the true weight 
 
 THE WHEEL AND AXLE. 
 
 21. THE wheel and axle is. a kind of lever, so contrived 
 as to have a continued motion about its fulcrum, or centre 
 of motion, where the power acts at the circumference of the 
 wheel, whose radius may be reckoned one arm of the lever, 
 the length of the other arm being the radius of the axle, 
 on which the weight acts. If the power acts at the end 
 of a handspike fixed in the rim of the wheel, then this in- 
 creases the leverage of the power, by the length of the 
 handspike. 
 
 The wheel and axle consists of a wheel 
 having a cylindric axis passing through its 
 centre. The power is applied to the cir- R 
 cumference of the wheel, and the weight 
 to the circumference of the axle. 
 
 In the wheel and axle, an equilibrium 
 takes place when the power multiplied by 
 the radius of the wheel, is equal to the 
 weight multiplied by the radius of the 
 axle ; or, P : W : : CA : CB. 
 
 For the wheel and axle being nothing else but a lever 
 so contrived as to have a continued motion about its ful- 
 
THE WHEEL AND AXLE. 125 
 
 crum C, the arms of which may be represented by AC and 
 BC, therefore, by the property of the lever, P : W : : CA 
 :CB. 
 
 If the power does not act at right angles to CB, but 
 obliquely, draw CD perpendicular to the direction of the 
 power, then, by the property of the lever, P : W : : CA : 
 CD. 
 
 22. It will be easily seen, that if two wheels fastened 
 together and turning round the same centre, be so adjusted, 
 that while they turn round they will coil on their cir- 
 cumferences strings, to which weights are suspended ; 
 one of those wheels being larger than the other, the larger 
 wheel will coil up a greater length of the string than the 
 smaller one will do in the same time, and this will depend 
 either on the radii or circumferences of the two wheels. 
 The velocity of the weight will be in proportion to the 
 length of string coiled in a given time ; therefore, the ve- 
 locity of the weight will be greater as the wheel is larger. 
 Now, as in the lever we saw that a small weight required a 
 great velocity to balance a large weight with a small velo- 
 city, we may infer, that the rules given for levers will also 
 apply to the wheel and axle ; since the velocity of any body 
 on a lever depends upon its distance from the fulcrum. 
 
 Ex. A weight of 13 Ibs. is to be raised at a velocity 
 of 14 feet per second, by a power whose velocity is 20 feet 
 per second ; how great must that power be ? 
 
 13 X 14 
 
 -- = 9*1, the power required. 
 
 If the velocity of the weight, be to that of the power, as 
 14 to 20, and the radius of the axle on which the weight is 
 coiled be 7 ; then, 
 
 20 x 7 
 
 =10, radius of wheel on which the power acts. 
 
 If a Weight of 36 Ibs, is to be raised by an axle 3 inches 
 diameter ; what must be the power applied at the end of a 
 handspike 4 inches long, fixed in the rim of the wheel con- 
 nected with the axle, the wheel being 6 inches diameter ? 
 
 Here the handspike will increase the distance of the 
 power from the fulcrum, and will add to the diameter of the 
 wheel twice its own length ; therefore, 8 -f- 6 = 14 ; 
 hence, 14 : 3 : : 36 : 7'77, the power required to keep the 
 weight in equilibrio. 
 
 11* 
 
126 MECHANICS. 
 
 23. Wheels acting oh each other by teeth or bands, may 
 be easily calculated in the same way. Thus, if a wheel 
 which has 30 teeth, drives another of 10 teeth, it is evident, 
 that as the larger wheel has three times as many teeth as 
 the smaller, the smaller wheel will be turned round three 
 times for once that the larger one is turned round ; so that 
 the velocities of the wheels will be inversely as their num- 
 ber of teeth. In like manner, it is clear, that if the larger 
 wheel drives the smaller not by teeth but by a band, their 
 revolutions will be inversely as their circumferences. 
 
 Ex. The number of teeth in one wheel are 160, and 
 in another driven by it are 20, and the larger wheel makes 
 12 revolutions in a minute ; how many does the smaller one 
 make ? 
 
 20 : 160 : : 12 : 96 = the number ojf turns which the 
 smaller wheel makes in a minute. 
 
 24. The larger wheel is usually called the wheel, driver, 
 or leader, and the smaller one is called the pinion, driven 
 wheel, or follower. 
 
 25. Let us now see what would be the action of two 
 wheels and a pinion. If the first wheel contains 80 teeth, 
 the pinion 12 teeth, and second wheel 36 teeth. Place the 
 first wheel and the pinion on the same axis, so that they 
 move together, one revolution of the one being in the same 
 time as a revolution of the other, and the pinion drives 
 the second wheel. ^If the first wheel makes 16 revolutions 
 in a minute, the pinion will do the same, and the pinion 
 drives the second wheel ; therefore, 36 : 12 :: 16 : 5$ 
 = the velocity of the second wheel. Place these so, that 
 the teeth of the first wheel act in the teeth of the pinion, 
 and those again act in the teeth of the second wheel. If 
 the first wheel make, as before, 16 turns in a minute, then 
 the pinion will make 12 : 80 : : 16 : 106 r ' f = in a minute ; 
 consequently, the revolutions .of the second wheel will be 
 36 : 12 : : 106 T *j : 35-55 = turns of the second wheel in 
 a minute. 
 
 26. When there are a number of 
 wheels A, B, C, D, E, acting on 
 the respective pinions a, 6, c, rf, e, 
 as then the effect of the whole may 
 be found thus : if the letters which 
 represent the wheels and pinions be 
 understood to signify the number of teeth of each, 
 
THE WHEEL AND AXLE. 127 
 
 power xAxBxCxDxE 
 
 - = weight. 
 axbxcxdxe 
 
 If the velocity of the first wheel be used instead of the 
 power applied, then this rule will give the resulting velo- 
 city instead of the weight. 
 
 Ex. If the numbers of the teeth of the wheels are 9, 6 
 9, 10, 12, and those of the pinions 6, 6, 6, 6 ; then if the 
 power applied be 14 Ibs., we have 
 
 14 X 9 X 6 X 9 X 10 X 12 
 
 = 105 Ibs., the weight. 
 6x6x6x6x6 
 
 And, by the remark under the rule, if the first make 14 
 revolutions in the minute, the speed of the last will be 105 
 in the same time. 
 
 The same rule will apply to the case where the wheels 
 act on each other by ropes or straps, if the circumferences 
 of the wheels and pinions are used for the number of teeth. 
 
 27. It often happens, in the construction of machinery, 
 that two shafts must be connected by means of toothed 
 wheels, in such a way, that the one shaft's velocity shall 
 bear a certain proportion to that of the other shaft ; and we 
 must determine the numbers of teeth in each of the con- 
 necting wheels and pinions. 
 
 Take the respective numbers of teeth in the pinions at 
 pleasure, and multiply all these together, and their product 
 again by the number of turns that the one shaft is to make 
 for one turn of the other shaft. Take, now, this product, 
 and find all the .numbers which will divide it without a re- 
 mainder, or divide its divisors without a remainder always 
 excepting the number 1. Arrange all these in one line, and 
 separate them into parcels or bands, each containing as many 
 numbers, or factors (as they are called) as you please ; but 
 observing, that there must be as many bands as there are 
 wheels required ; then the product of the numbers in each 
 band will give the number of teeth in the respective wheels. 
 Thus, if one shaft is to turn 720 times for another shaft's 
 once, and there be interposed 4 pinions, one of which is 
 fixed to the end of the one shaft, each pinion having 6 teeth 
 or leaves : then, 6x6x6x6x 720 ; all the divisors or 
 factors of which are 3, 2, 3, 2, 3, 2, 3, 2, 2, 2, 3, 5, 2, 2, 3 ; 
 these divided into 4 bands at pleasure, give the number of 
 teeth in the wheels. Thus, 
 
 *,. 
 
128 MECHANICS. 
 
 f2x3x5 =30, T3X3X5 =45, 
 
 F'th J 2x2x2x3 = 24, n I 3x2x2x2x2 = 48, 
 12x2x3x3 = 36, Jr '] 3X3X2 =18, 
 
 L2x2x3x3 = 36, L3x2x2x2 =24. 
 
 The application of what we laid down may be thus illus- 
 trated. In finding the number of teeth in the wheels of an 
 orrery, we extract from Marrat's Mechanical Philosophy. 
 " There is considerable difficulty in proportioning the num- 
 ber of teeth in wheels for clocks, orreries, &c. the problem 
 indeed is indeterminate ; we shall, however, give an ex- 
 ample, that will point out a method by which any ingenious 
 mechanic may complete a piece of machinery, such as an 
 orrery, so as to show, at all times, in what part of its orbit 
 any planet is. The following example is for Mercury; 
 this planet goes round the sun in 87d. 23h.; now, as the 
 hour hand of a clock goes round twice in 24 hours, 
 it will make 175 1- revolutions in 87d. 23h. For the 
 fraction -J-J, take any multiple of the denominator plus 
 or minus unity, and make it the third term of the propor- 
 
 472 
 
 tion; thus say, as!2:ll::515: 472 nearly; for 
 
 515 
 
 473 
 
 is one unit less in each than a multiple of {* by 43 = - 
 
 516 
 
 hence the revolutions become 175 = Now the 
 
 515 515 
 
 only difficulty remaining, is to find proper factors or divi- 
 sors that will divide the numerator and denominator 
 without a remainder, in order to determine the number of 
 teeth and leaves in the wheels and pinions. For the 
 numerator, the best method I have found is to make trial 
 of the numbers 2 X 5 or 10, as often as we can, and if we 
 do not succeed, to try successively the prime numbers 3, 
 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, &c. I find by 
 trial the numerator will break into the factors 101 X 39 X 
 23 =90597, I conclude then that these numbers 101, 39, 
 23, may be the number of teeth in three wheels. I can 
 easily break the denominator into the numbers 103 and 5 ; 
 but as 103 is too large for the teeth in a pinion, and being 
 a prime number, another number must be sought for that 
 will answer the purpose better. Again say, as 12 : 11 : : 
 
 1825 : 1673, the revolutions now become 175 or 
 
THE WHEEL AND AXLE. 129 
 
 OO1 fMQ 
 
 . Hence I find by trial that the numerator (321048) 
 
 can be broken into the factors 91 X 72 X 49 = 321048, 
 which may be three wheels having that number of teeth in 
 each. Again, the denominator of the fraction, or 1825, is 
 capable of being broken into the factors 73 X 5 X 5 = 
 1825. Now the product of the number of teeth in all the 
 wheels, divided by the product of the number of teeth in 
 all the pinions, will give the revolutions. For example, 
 32104 -f- 1825 = 175 revolutions, llh. Om. Is. 58 thirds, 
 which does not exceed the 87d. 23h. (or 175 ]-J revolutions) 
 by two seconds. The numbers last found for the wheels 
 and pinions, may be transformed by multiplication into 
 
 98X91X72 144X98X91 
 more convenient numbers, ^3 xlo x 5 = 73 x 10 X10 
 
 = 175r. llh. Om. Is. 58th. either of which will be a train 
 of wheel-work proper for such a motion, and this train may 
 be conveniently attached to the pinion of the hour-wheel 
 of a clock. The reason for finding a new fraction, will| 
 appear evident; for *if we take the original number 
 175j| = 2 j|- 1 , we shall find it impossible to break the 
 numerator into factors without leaving a fraction, which is 
 inconsistent with wheel-work, as nothing but whole num- 
 bers Vvill answer the purpose. It is obvious that the higher 
 we take a multiple of }} the nearer we approach to the true 
 time of revolution, provided we can break the numerator 
 and denominator into proper numbers for the teeth and 
 leaves of the wheels and pinions. It is necessary to ob- 
 serve, that there must be either three wheels and three 
 pinions, or, if the numbers when broken be too large, if we 
 can break them into five wheels and five pinions, it will be 
 the same thing ; because as the hands of a clock go round 
 with the sun, that motion would make two wheels and two 
 pinions (attached to the pinion on the hour wheel) go round 
 the contrary way to what they ought; but three or five will 
 answer the intended purpose." 
 
 28. As the subject of wheel- work is of the greatest im- 
 portance to mechanics, we shall resume it in a more 
 advanced part of this work, where it may be more properly 
 introduced. 
 
130 
 
 MECHANICS. 
 
 THE PULLEY. 
 
 29. IF a rope or string pass round the groove or rim of a 
 wheel, movable round an axle, with a power at the one end 
 of the String or rope, and a weight at the other, such a 
 machine is called a Pulley. The axis of the pulley may be 
 either fixed or movable. If the axis of the pulley be 
 fixed, it only serves to change the direction of the power's 
 action ; but if it be movable, the power acts with an ad- 
 vantage of two to one. 
 
 The accompanying engraving exhibits various forms of 
 the pulley. AB is a beam from which they are suspended. 
 
 1234 
 
 \Vi. 
 
 No. 1, is the fixed pulley in which there is no other ad- 
 vantage gained than that the power P and weight W move 
 in a contrary direction. No. 2, is a movable pulley, in 
 which the power P by moving upwards raises the pulley, 
 to the block of which the weight W is attached ; but the 
 one end of the string being attached to the beam AB, the 
 power must move twice as fast as the weight, and there 
 will be a gain of power proportional. No. 3, is a combi- 
 nation of two movable pulleys, in which the gain of power 
 will be four ; and No. 4 is a combination of two fixed and 
 two movable pulleys, in which the gain of power will be 
 the same as in No. 3. 
 
 30. If in a system of pulleys, where each pulley is em- 
 braced by a cord, attached at one end to a fixed point, and 
 at the other to the centre of the movable pulley next above 
 it, and the weight is hung to the lowest pulley ; then, the 
 effect of the whole will be = the number 2 multiplied by 
 itself, as many times as there are movable pulleys in the 
 system : thus, if there be 4 movable pulleys, then 2x2 
 
THE PULLEY. 131 
 
 . < 2 X2 =16: wherefore, if the weight be one lb., it will 
 oe sustained by a power of one oz. avoirdupois. 
 
 31. When there are any number of movable pulleys on 
 one block, and as many on a fixed block, the pulleys are 
 called Sheeves, and the system is called a Muffle ; and the 
 weight is to the power inversely as one is to twice the 
 number of movable pulleys in the system, or 
 
 the weight to be raised 
 
 twice the number of mov. pulleys = e power ' 
 Ex. In a muffle where each block has 4 sheeves, one 
 block being fixed and the other movable, a weight of 112 
 Ibs. is to be raised ; how great must be the power ? 
 
 112 
 
 -g =14 Ibs., the power required. 
 
 If a power of 236 Ibs. is to be applied to a tackle con- 
 nected with two blocks of pulleys, one fixed, consisting of 
 6, and another movable, of 5 pulleys ; what weight can be 
 raised ? (Here the rule above must be reversed.) 
 
 Therefore 236 X 10 = 2360 Ibs., the weight. 
 
 REMARK. In all the above cases of the pulley, the strings, 
 cords, or ropes, are supposed to act parallel to each other ; 
 when this is not the case, the relation of power and weight 
 may be found by applying the principle of the parallelo- 
 gram of forces ; thus, draw ab in the direction of the 
 power's action and of that length, taken from 
 a scale of equal parts, which expresses the 
 quantity of that power ; next, draw bd a per- 
 pendicular to the horizon, and from a draw 
 ad parallel to 6c, the direction of the string, 
 which is fastened at c: then the power is to 
 the weight, as ba is to bd; and the strain on the hook at c, 
 is as ad to db, these lines being all measured on the same 
 scale of equal parts. 
 
 It may be further observed, that the pulley is a species 
 of lever of the second kind ; where the point at which the 
 string is fastened may be called the fulcrum ; the axis of 
 the pulley the place of the weight, and the place of the 
 power the other end of the lever ; or, the diameter of the 
 pulley may be reckoned the length of the lever, the weight 
 being in the middle. 
 
 
132 MECHANICS. 
 
 THE INCLINED PLANE. 
 
 32. WHEN a power acts on a body, 
 on an inclined plane, so as to keep that 
 body at rest; then the weight, the 
 power, and the pressure on the plane, 
 will be as the length, the height, and L 
 
 the base of the plane, when the power acts parallel to the 
 
 plane ; that is, 
 
 The weight f 1 AC, 
 
 The power < will be as >BC, 
 
 The pressure on the plane (^ J AB. 
 
 These properties give rise to the following rules : 
 
 weight x height of plane 
 
 power = 2, . 
 
 length of plane 
 
 . , power X length of 
 height of plane 
 
 weight x base of plane 
 
 pressure on the plane = -: = ; 
 
 length of plane 
 
 33. The force with which a body endeavours to descend 
 down an inclined plane, is as the height of the plane. 
 
 When the power does not act parallel 
 to the plane, then from the angle C of 
 the plane, draw a line perpendicular 
 to the direction of the power'^ action ; 
 then the weight, the power, and the 
 pressure on the plane, will be as AC, CB, AB. 
 
 When the line of direction of the power is parallel to the 
 plane, the power is least. 
 
 34. If two bodies, on two inclined planes, sustain each 
 other, by means of a string over a pulley, their weights 
 will be inversely as the lengths of the planes. 
 
 35. In the exercises on inclined planes, it is often neces- 
 sary to find the length of the base, and height, or length of 
 the plane. Any two of these being given, the third may 
 be found and this is done on the principle stated in Geo- 
 metry, that the hypotenuse 2 of a right-angled triangle (the 
 length of the plane) is equal to the base 3 + height 3 . 
 
 Ex. The height of an inclined plane is 20 feet, and its 
 length 100 ; what is the pressure on the plane of a weight 
 of 1000 Ibs.? Here we must first ascertain the base, 
 (100 3 20 a ) = 97-98 = the base of the plane ; and from 
 
THE INCLINED PLANE. 133 
 
 what has been said above, 100 : 1000 : : 97'98 : 979-8 the 
 pressure upon the plane ; also 100 : 20 : : 1000 : 200, the 
 power necessary to keep the body from rolling down the 
 plane. 
 
 If a wagon of 3 cwt. on an inclined railway of 10 feet to 
 the 100, be sustained by another on an opposite railway of 
 10 feet to 90 of an incline ; what is the weight of the second 
 wagon? Here 100 : 90 : : 3 : 2'7 cwt. = the weight of 
 the second wagon. 
 
 36. The- space which' a body describes upon an inclined 
 plane, when descending on .the plane by the force of 
 gravity, is to the space which it would fall freely in the 
 same time, as the height is to the length of the plane ; and 
 the spaces being the same, the times will be inversely in 
 this proportion. 
 
 Ex. If a body roll down an inclined plane 320 feet long, 
 and 26 feet in height ; what space will it pass down the 
 plane in one second, by the force of gravity alone ? 
 320 : 26 : : 16 : 1-3 foot = the answer. 
 
 This subject, as connected with railways, will be resumed 
 when we come to treat of friction and railways. 
 
 THE WEDGE. 
 
 37. THE wedge is a triangular prism, formed either of 
 wood or metal, whose great use is to split or raise timber, 
 stones, &c. 
 
 The circumstances in which it is applied are such that 
 it is not easy to devise a general rule to determine the 
 amount of its action. The wedge has a great advantage 
 over all the other mechanical powers, in consequence of the 
 way in which the power is applied to it, namely, by per- 
 cussion, or a stroke,' so that by the blow of a hammer, 
 almost any constant pressure may be overcome. 
 
 THE SCREW. 
 
 38. THE screw is a kind of continued inclined plane, 
 being an inclined plane rolled about a cylinder the 
 height of the plane being the distance between the centres 
 of two threads, and its length the circumference ; hence, 
 
 12 
 
 '* * 
 
134 MECHANICAL CENTRES. 
 
 the rule to find the power of a screw pressing either up- 
 wards or downwards, is as the distance between two threads 
 of the screw is to the circumference where the power is 
 applied : thus, if the distance of the centres of two threads 
 of the screw be | of an inch, and the radius of the hand- 
 spike attached to the screw be 24 inches ; the circumference 
 of the screw will be 150f inches, nearly: therefore, 
 * : 150| : : 1 : 603| ; and if the power applied be 150 
 Ibs., the force of the screw will therefore be 603 X 150 
 = 90480 Ibs. 
 
 39. REMARKS ON THE MECHANICAL POWERS. The me- 
 chanical powers may be variously modified and applied, 
 but still they form the elements of all machinery. In our 
 calculations of their effects, we have not made allowance 
 for friction, or the resistance arising from one body rubbing 
 against another a subject which will be discussed hereafter. 
 The justice of the remark made before, will now be seen 
 to hold generally, that of the two velocity and power 
 whatever we gain in the one, we lose in the other ; or, as 
 power and weight are opposed to each other, there will 
 always be an equilibrium between them, when the power 
 X its velocity = the weight x its velocity, that is, when 
 the momentum of the one is equal to the momentum of the 
 other. 
 
 All the advantage that we can obtain from the mechanical 
 powers, or their combinations, is to raise great weights, or 
 overcome great resistances, and this must be done at the 
 expense of time; or, to generate rapid velocities, as in 
 turning-lathes, or cotton-spinning machinery, and this if 
 done at the expense of power. 
 
 MECHANICAL CENTRES. 
 
 1. THESE are the centres of gravity, oscillation, percus- 
 sion, and gyration. 
 
 THE CENTRE OF GRAVITY. 
 
 2. THERE is a certain point in every body, or system of 
 bodies connected together ; which point, if suspended, the 
 
 

 THE CENTRE OF GRAVITY. 135 
 
 body or system of bodies will remain at rest when acted 
 upon by the force of gravity alone ; this point is called 
 the Centre of Gravity. If a body or system of bodies be 
 suspended by any other point than the centre of gravity, 
 such body or system of bodies will move round that point, 
 until the centre of gravity be in a vertical line with the 
 point of suspension. If a body be sustained from falling 
 by two forces, the lines of direction in which these two 
 forces act, will meet in the centre of gravity of the body, 
 or, in the vertical line which passes through it. 
 
 3. It is often useful in calculation to consider the whole 
 weight of a body as placed in its centre of gravity, but it 
 is to be remembered, that gravity and weight do not 
 signify the same thing gravity is the force by means of 
 which bodies, if left to themselves, fall to the earth in 
 directions perpendicular to the earth's surface ; weight, on 
 the other hand, is the resistance or force which must be 
 exerted, to prevent a given body from obeying the law of 
 gravity. 
 
 4. To find the centre of gravity of any plane figure, me- 
 chanically : Suspend the figure by any point near its edge, 
 and mark the direction of a plumb-line hung from that 
 point, then suspend it from some other point, and mark 
 the direction of the plumb-line in like manner. The 
 centre of gravity of the figure will be in that point where 
 the marks of the plumb-line cross each other. For instance, 
 if we wish to find the centre of gravity of the arch of a 
 bridge, we draw the plan upon paper to a certain scale, 
 cut out the figure, and proceed with it as above directed ; 
 and by means of the plumb-line from the points of sus- 
 pension, its centre of gravity will be found ; whence, by 
 measuring the relative position of this centre in the plan 
 by the scale, we may determine by comparison its position 
 in the structure itself. 
 
 5. We can find the centre of gravity of many figures by 
 calculation. 
 
 6. The centre of gravity of a line, parallelogram, prism, 
 cylinder, circle, circumference of a circle, sphere, and 
 regular polygon, is the geometrical centre of these figures 
 respectively. 
 
 7. To find the centre of gravity of a triangle draw a line 
 from any angle to the middle of the opposite side, then f 
 
 

 136 MECHANICAL CENTRES. 
 
 of this line from the angle will be the position of the centre 
 of gravity. 
 
 8. For a trapezium, draw the two diagonals, and find 
 the centres of gravity of each of the four triangles thus 
 formed, then join each opposite pair of these centres of 
 gravity, and the two joining lines will cut each other in 
 the centre of gravity of the figure. 
 
 9. For the cone and pyramid, the centre of gravity is in 
 the axis, at the distance of of the axis from the vertex. 
 
 10. For the arc of a circle, 
 
 radius of circle x chord of arc 
 
 - = distance of the 
 length of arc 
 
 centre of gravity from the centre of the circle. 
 
 11. For the sector of a circle, 
 
 2 x chord of arc x radius of circle f 
 
 : ^ = distance of 
 
 3 X length oi arc 
 
 the centre of gravity from the centre of the circle. 
 
 12. For a parabolic space, the distance of the centre of 
 gravity from the vertex is | of the axis. " 
 
 13. For a paraboloid, the centre of gravity is f of the 
 axis from the vertex. 
 
 14. For two bodies, if at each end of a bar a weight be 
 hung, the common centre of gravity will be in that point 
 which divides the bar, in the same ratio that the weights 
 of the bodies bear to each other, and this point will be 
 nearest the heavier body. 
 
 Examples. If the line drawn from the middle of the base 
 of a triangle to the opposite angle be 15, then we have 
 
 - X 2 = 10 = the distance of the centre of gravity from 
 3 
 
 the vertical angle. 
 
 If the height of a cone be 24 inches, then we have 
 
 24 
 
 x 3 = 18 = the distance of the centre of gravity from 
 
 the vertex. 
 
 If the length of the arc of a circle be 157*07, and the 
 chord 153-07, and radius 200 ; then, 
 
 200 x 153-07 
 
 r^r^= = 194-9 = distance of the centre of 
 
 1 oY'U7 
 
 gravity from the centre of the circle. 
 
 If there be the sector of a circle of which the chord, 
 

 OSCILLATION AND PERCUSSION. 137 
 
 radius, and length of arc, are the same as in the last ex- 
 ample, we have 
 
 2 X 153-07 X 200 ,. , 
 
 f the 
 
 3 x 157-07 
 
 centre of gravity from the centre of the circle. 
 
 In a parabolic space, if the axis be 25 inches long, then 
 25 
 - x 3 =s 15 = the distance of the centre of gravity from 
 
 the centre. 
 
 30 
 In a paraboloid, if the axis be 30, then we have x 2 
 
 8 
 
 = 20 = the distance of the centre of gravity from the vertex. 
 A bar of wood, 24 feet long, has a weight suspended at 
 each end, that at one end being 16 Ibs., and the other 4 : 
 then, we have 20 : 24 : : 16 : 19-2 
 and 20 : 24 : : 4 : 4-8 
 
 the distances of the weights from the common centre of 
 gravity, the greater weight being least distant. Hence we 
 see, that 19-2 -f 4-8 = 24, the whole length of the bar; 
 and also 4 x 19-2 = 16 x 4-8 = 76'8 ; so that the prin- 
 ciple of virtual velocities, stated before, holds good here also ; 
 and here it may be observed, that it is of the greatest im- 
 portance to trace any leading principle of this kind through 
 its various applications, as it serves to link together and 
 harmonize the whole, and enables us to apply and remember 
 it with greater facility. 
 
 It is often necessary to determine the centre of gravit)*- 
 experimentally, as in many cases it cannot be conveniently 
 done by calculation. To maintain the firmness of any 
 body resting on a base, it is necessary that the perpendicu- 
 lar drawn from the centre of gravity of the body, to the 
 base on which it rests, be- within that base ; and the body 
 will be the more difficult to overset, the nearer that per- 
 pendicular is to the centre of the base, and the more ex- 
 tensive the base is, compared to the height of the centre of 
 gravity. 
 
 THE CENTRE OF OSCILLATION. THE PENDULUM, AND 
 CENTRE OF PERCUSSION. 
 
 1. THE centre of oscillation in a vibrating body, is that 
 point in the axis of vibration, in which, if the whole matter 
 12* 
 
138 MECHANICAL CENTRES. 
 
 contained in the body were collected, and acted upon by 
 the same force, it would, if attached to the same axis of 
 motion, perform its vibrations in the same time. The 
 centre of oscillation is always situated in the straight line 
 which passes through the centre of gravity, and is perpen- 
 dicular to the axis of motion. It will be seen by these 
 remarks, that the subject of pendulums must be considered 
 here. 
 
 2. In theory, a simple pendulum is a single weight, con- 
 sidered as a point, hanging at the lower extremity of an 
 inflexible right line, having no weight, and suspended from 
 a fixed point or centre, about which it vibrates, or oscil- 
 lates ; a compound pendulum, on the other hand, consists 
 of several weights, so connected with the centre of suspen- 
 sion, or motion, as to retain always the same distance from 
 it, and from each other. 
 
 3. If the pendulum be inverted, so that the centre of 
 oscillation shall become the centre of suspension, then the 
 former centre of suspension will become the centre of os- 
 cillation, and the pendulum will vibrate in the same time : 
 this is called the reciprocity of the pendulum ; and it is a 
 fact of the greatest utility, in experimenting on the lengths 
 of pendulums. 
 
 4. Of the simple pendulum we may observe, that its 
 length, when vibrating seconds, must in the first place be 
 determined by experiment, as it vibrates by the action of 
 gravity, which force differs at different distances from the 
 pole of the earth. By the latest experiments, the length of 
 the seconds' pendulum in the latitude of London, has been 
 found to be 39-1393 inches, or 3-2616 feet; the length at 
 the equator is nearly 39-027, and at the pole 39*197 inches. 
 The length for the latitude of London may be taken for all 
 places in Britain, without any material error. 
 
 5. The times of vibration of two pendulums, are directly 
 proportional to the square roots of the lengths of these pen- 
 dulums. 
 
 6. Thus : what will be the time of one vibration of a 
 pendulum of 12 inches long at London? 
 
 v/39-1393 : v/ 12 :: I : 0-5537 = time of one vibration. 
 
 If the pendulum be 36 inches long, 
 
 ^/39-1393 : v/ 36 : : 1 : 0-9599 = time of one vibration. 
 
 7. The lengths of the pendulums are to each other in- 
 

 OSCILLATION AND PERCUSSION. 139 
 
 versely as the squares of the numbers of vibrations made in 
 a given time. 
 
 What is the length of a pendulum vibrating half-seconds, 
 or making 30 vibrations in a minute ? 
 
 (60) 2 : (30) a :: 39-1393 : 9-7848 = length in inches. 
 The length of a pendulum to make any given number of 
 vibrations in a minute, may be easily found by the following 
 short rule : 
 
 140850 = 
 
 number of vibrations 8 " 
 
 Thus a pendulum to make 50 vibrations in a minute, will 
 be 
 
 140850 140850 
 
 -SO*- = -250CT = 56 ' 34 mches m len * th - 
 8. All the rules for simple pendulums may be expressed 
 as follows : 
 
 The time of one vibration in seconds of any pendulum is 
 
 number of vibrations in one second 
 the length of the pendulunA 
 
 QQ.1 QQQ / 
 
 39-1393 
 
 Exam. If the number of vibrations of a pendulum be 
 6256, then 
 
 = 1- 598 = the time of one vibration. 
 6256 
 
 Or, if the length of the pendulum be 100 inches, then 
 
 rSr,) -"- 
 
 The length of a pendulum in inches is 
 
 = 39-1393 x time of one vibration 3 ; 
 
 39-1393 
 
 number of vibrations 3 
 
 Exam. If the time of one vibration be 1'598 ; find the 
 length. 39-1393 x 1'598 3 = 100, length of pend. 
 
 Or, if the number of vibrations in a second be as above, 
 6256, then we have 
 on. i QOQ 
 
 = 100, length of pendulum. 
 
140 MECHANICAL CENTRES. 
 
 The number of vibrations in a second may be found thus : 
 
 I 39-1393 , .. . 
 
 r = j-: =s number of vibrations ; 
 
 \length of pendulum 
 
 or, the number of vibrations in a second is 
 
 _ 1 
 
 ~" time of one vibration 
 If the time of one vibration be, as above, 1-598 ; then 
 
 = -6256, number of vibrations ; 
 I*o98 
 
 or, if the length of 100, we have 
 
 39-1393x 
 \ = -6256. 
 
 When a clock goes too fast or too slow, so that it shall 
 lose or gain in twenty-four hours, it is desirable to regulate 
 the length of the pendulum so that it shall go right. The 
 pendulum bob is made capable of being moved up or down 
 on the rod by means of the screw. If the clock goes too 
 fast, the bob must be lowered, and if too slow, it must be 
 raised ; and we have this rule : number of threads in an 
 inch .of the screw X the time in minutes that the clock 
 loses or gains in 24 hours ; this product divided by 37 will 
 give the number of threads that the bob must be screwed 
 up or down, so that the clock shall go right. 
 
 Ex. If the rod have a screw 70 threads in the inch, and 
 the pendulum is too long, so that the clock is 12 minutes 
 slow in 24 hours ; then we have 
 
 2 X 70 X 12 
 
 = 45if = threads we must raise the bob, 
 
 37 
 
 so that the clock shall go right. 
 
 9. It is often desirable that a pendulum should vibrate 
 seconds, and yet be much shorter than 39'1393 inches; 
 which may be done by placing one bob on the rod above 
 the centre of suspension, and another below it : then, having 
 the distances oj[the weights from the centre of suspension, 
 we may find the ratio which the weights should bear to 
 each other by this rule. Call D the distance of the lower, 
 and d the distance of the upper weight, from the centre of 
 suspension ; then, 
 
 39-1393 x D D 2 ^ 
 39-1393 X d -f d 3 
 
OSCILLATION AND PERCUSSION. 141 
 
 a number which, when multiplied by the lower weight, 
 will give the higher. D and d are taken in inches. 
 
 Ex. In a pendulum having two bobs, the one 12 inches 
 below the centre of suspension, and the other 9*6 inches 
 above the same centre, the lower weight being 8 ounces ; 
 what is the upper weight ? 
 
 39-1393 X 1212* 
 39-1393 X 9-6 -f 9'6 a 
 
 then, 0-696 X 8 = 5-568 ounces = the weight of the 
 upper bob. 
 
 10. If a common walking-stick be held in the hand, and 
 struck against a stone, at different points of its length, it 
 will be found that the hand receives a shock when it is 
 struck at any part of the stick, but at one particular point, 
 at which, if the stick be struck, the hand will receive no 
 shock. This point is called the centre of Percussion, and is 
 usually defined thus : The centre of percussion is that point 
 in a body revolving about an axis, at which, if it struck an 
 immovable obstacle, all the motion of the body would be 
 destroyed, so that it would incline neither way after the 
 stroke. 
 
 11. The distance of the centre of percussion from the 
 axis of motion, is the same as the distance of the centre of 
 oscillation from the centre of suspension ; and the same 
 rules serve for both centres. See Oscillation. 
 
 12. The distance of either of these centres from the axis 
 of motion, is found thus : 
 
 13. If the axis of motion be in the vertex of the figure, 
 and the motion be flatwise ; then, 
 
 14. In a right line, it is = f of its length ; 
 
 In an isosceles triangle = | of its height ; 
 In a circle = of its radius ; 
 In a parabola = | of its height. 
 
 15. But if the bodies move sidewise, we have it 
 In a circle = | of the diameter ; 
 
 In a rectangle suspended by one angle = f of the 
 diagonal. 
 
 16. In a parabola suspended by its vertex, 
 
 = 4 axis + i parameter ; 
 but if suspended by the middle of its base, 
 = $ axis -f- i parameter. 
 
142 MECHANICAL CENTRES. 
 
 17. In the sector of a circle = 3 X arc X radius 
 
 4 x chord 
 
 1Q T (radius of base) 9 
 
 18. In a cone = f axis 4- * . 
 
 d F\ \S O VIQ 
 
 
 5 x axis 
 
 2 X radius 3 
 
 19. In a sphere = - -f radius -f d, where 
 
 5 (d -f- radius) 
 
 d is the length of the thread by which it is suspended. 
 
 20. We have given these rules for the sake of reference, 
 but we shall illustrate by examples the most useful. 
 
 Examples. What must be the length of a rod without a 
 weight, so that when hung by one end it shall vibrate 
 seconds ? 
 
 To vibrate seconds, the centre of oscillation must be 
 39*1393 inches from that of suspension; hence, as this 
 must be f of the rod, 2:3:: 39-1393 : 58-7089 inches = 
 the length of the rod. 
 
 What is the centre of percussion of a rod 46 inches long? 
 f X 46 = 30 1 inches from the axis of motion. 
 
 In an isosceles triangle, suspended by one angle, and 
 oscillating flatwise, the height is 24 feet ; what is the dis- 
 tance of the centre of percussion from the axis of motion ? 
 | X 24 = 18 feet. 
 
 In a sphere the diameter is 14, and the string by which 
 the sphere is suspended is 20 inches ; therefore, 
 
 n V "7* Qft 
 
 20=- +27 = 27-725; 
 
 5 (20 + 7) ' 135 
 
 so that the centre of oscillation or percussion is farther from 
 the axis of motion than the centre of the sphere, by 7-725 
 inches. 
 
 THE CENTRE OF GYRATION AND ROTATION. 
 
 21. IT will be seen, that the last two centres refer to 
 bodies in motion round a fixed axis, and belonging to the 
 same class : there is yet another centre to be considered, 
 of the utmost importance to the practical mechanic. We 
 saw, in determining the centre of oscillation, that we were 
 finding a point in which, if all the matter of the body were 
 collected, the motion would be the same as that of the body 
 which motion was caused by the action of gravity ; but 
 
GYRATION AND ROTATION. 143 
 
 when the body is put in motion by some other force than 
 gravity, the point in question becomes the centre of Gyra- 
 tion. The centre of gyration may therefore be defined, 
 that point in a body or system of bodies revolving round 
 an axis, in which point, if all the matter in the body or 
 system of bodies were collected, the same number of revo- 
 lutions in a given time would be generated by the applica- 
 tion of a given force, as would be generated by the same 
 force applied to the body or system of bodies itself. 
 
 22. The position of the centre of gyration is a mean pro- 
 portional between the centres of oscillation and gravity. 
 
 23. The centre of gyration of the following bodies may 
 be found by these rules : 
 
 24. For a straight line or cylinder, whose axis of motion 
 is in one end, = length x 0*5775. 
 
 25. For a cylinder or plane of a circle, revolving about 
 the axis, or the circumference about the diameter, = radius 
 X 0-7071. 
 
 26. For the plane of a circle about its diameter = 5 
 radius. 
 
 27. For the surface of a sphere about its diameter = 
 radius X -8165. 
 
 28. For a solid sphere or globe, about its diameter = 
 radius X '6324. 
 
 29. For the circumference of a circle upon a perpendicu- 
 lar axis passing through the centre = radius* 
 
 Ex. What is the distance of the centre of gyration 
 from the centre of motion, of a rod 58-7089 inches long? 
 Here 58-7089 X -5775 = 33-9044. 
 
 In a wheel of uniform thickness, revolving about its axis, 
 the diameter is 36 inches ; hence 18 X -7071 = 12 = dis- 
 tance of the centre of gyration from the axis. 
 
 In a solid globe revolving about its diameter, which is 
 2 feet, the distance of the centre of gyration is = 12 X 
 6324 = 7-5888 inches. 
 
 30. Effects are proportional to their causes ; the motion 
 generated in any body is proportional to the force which 
 produces that motion ; hence we see, that all constant forces 
 may be compared to the force of gravity. And it is often 
 useful to know the time in which a revolving body of a 
 certain weight, acted upon by a known constant force, will 
 acquire a given velocity. The principles we have laid 
 
144 MECHANICAL CENTRES. 
 
 down in discussing the inclined plane, will here be found 
 serviceable. 
 
 As the weight of the body moved, 
 
 Is to the weight or force causing it to move, 
 
 So is the length of an inclined plane, such that the 
 
 given force would just support the body upon it, 
 To the height of the plane. 
 
 Now, if in a wheel 6 feet diameter, whose weight, 400 Ibs., 
 is turned by a force of 56 Ibs., acting at the distance of 18 
 inches from its centre of motion, its centre of gyration being 
 5 feet from the same centre ; what will be the time required 
 to give by this force a velocity of 20 feet per second at the 
 centre of gyration. Here, by the lever, 
 
 the force exerted at the centre of gyration. We now wish 
 to know the length of time in which a body would acquire 
 a velocity of 20 feet per second, on an inclined plane, 
 whose length is to its height as 400 is to 16f % ; wherefore, 
 by the laws of falling bodies, we have 
 16 16-8 
 -32~ : = -32~ : = 525 ' 
 
 the time required to fall perpendicularly ; therefore, by the 
 inclined plane, we have, 20 : 400 : : -525 : 10-5 = the 
 time required. 
 
 31. All the circumstances comprehended under this kind 
 of rotatory motion, may be expressed by the following rules : 
 
 Let W express the weight of a wheel, 
 
 F, the force acting upon the wheel, 
 
 D, the distance of the force from the axis of motion, 
 
 G, the distance of the centre of gyration from the 
 axis of motion, 
 
 tj the time the force acts, 
 
 v, the velocity acquired by the revolving body in that 
 
 time. 
 G X W X ^ __ G X W X v _ 
 
 D X t X 32 F X t X 32 
 
 F X P X t X 32 _ F X D X t X 32 
 
 W X v G X v 
 
 G X W X v _ F X D X X 32 
 
 = W 
 
 FXDX32 GXW 
 
GYRATION AND ROTATION. 145 
 
 It is to be observed, before applying these rules, that the 
 number of turns of a revolving body in a minute are often 
 given, and it is required to find the velocity of feet per 
 second. A wheel of 8 feet diameter, for instance, makes 12 
 revolutions in a minute ; how many feet does a nail in its 
 circumference pass over in a second? Here, 8 x 3*1416 
 = 25-1328 feet the nail passes through in one revolution, 
 but 25-1328 x 12 = 301-5936 = the feet it passes through 
 in a minute; hence, 60)301-5936(5-0265, the velocity in 
 ft. per second. The whole may be expressed shortly thus : 
 
 8 X 3 ' 14 ' 6 X 12 = 5-0265. 
 60 
 
 Ex. What must be the weight of a fly-wheel that makes 
 12 revolutions in a minute, whose diameter is 8 feet, urged 
 by a force of 84 Ibs. at its rim, acting for 6 seconds, the 
 distance of the centre of gyration being 3 feet 6 inches ? 
 84 X 4 x 6 x 32 
 
 Ibs. 
 
 3-5X5-0265 
 
 In a wheel which is 2 tons weight, and 12 feet diameter, 
 the centre of gyration is 6 feet from the centre of rotation, 
 the velocity with which this wheel moves is 10 feet per 
 second ; what force must be applied for 8 seconds, at the 
 distance of 3 feet from the centre, to generate that velocity ? 
 
 = " 1562 of a ton " 3 cwt 1 ' 496 
 
 What is the distance of the centre of gyration from the 
 centre of motion of a fly-wheel, the force which moves the 
 wheel being 2 cwt., acting at the distance of 7 feet from 
 the centre of motion, and for 10 seconds, the weight of the 
 wheel being 2 tons, and its velocity 8 feet per second ? 
 Here 2^ tons = 50 cwt. 
 
 2X7X10X 12 ,1.- " j. , f 
 
 - 50 x 8 -- = MI f eet distance of centre of gyration. 
 
 What is the velocity acquired by a fly-wheel acted upon 
 by a force of 84 Ibs., at the distance of 4 feet from the axis, 
 the time in which the force has been acting is 7 seconds, the 
 weight of the wheel l tons, and the distance of the centre 
 of gyration 5 feet from the centre of motion ? Here l 
 ton == 30 cwt. = 3360 Ibs. ; therefore, 
 
 84 X 4 X 7 X 32 
 
 - - -- -- = 4-4 feet per second, the velocity 
 5 X 
 
 acquired by the wheel. 
 
 13 
 
146 CENTRAL FORCES. 
 
 
 CENTRAL FORCES. 
 
 1. INTIMATELY connected with the foregoing subject is 
 that of central forces, the nature of which may be illus- 
 trated by a very simple instance. When a boy causes a stone 
 in a sling to revolve round his hand, the stone is kept from 
 flying off by the strength of the string, which continually 
 draws the stone, as it were, to the hand or centre of motion ; 
 but if the string is let go, or breaks, then the stone will fly 
 off in a straight line, by means of its centrifugal force ; 
 the strength of the string, while it restrains this tendency, 
 is called the centripetal force : when both forces are spoken 
 of they are jointly called central forces. 
 
 2. When a body revolves round a fixed centre, the cen- 
 tripetal force may sometimes be the cohesion of the par- 
 ticles of which the body is composed, or sometimes it may 
 be the power of some attracting body such as gravity in 
 the case of the planets. 
 
 3. In talking of the angular velocity of a revolving body, 
 we mean not the space which is passed over in a given time, 
 but the number of degrees, minutes, <fcc., that the body de- 
 scribes in a certain time, whether the circle be large or 
 small. Thus, a body moving in a circle of 10 feet diameter, 
 may have an angular velocity of 15 in a second, so may 
 also another body moving in a circle of three feet diameter ; 
 they will complete their respective circles in the same time, 
 but the actual spaces they pass through are very different ; 
 that is, their angular velocities are the same, but their actual 
 velocities are not. 
 
 4. The central forces are as the radii of the circles 
 directly, and the squares of the times inversely, also the 
 squares of the times are as the cubes of the distances. 
 When a body revolves in a circle by means of central forces, 
 its actual velocity is the same as it would acquire by falling 
 through half the radius by the constant action of the centri- 
 petal force. From these facts the following rules for cen- 
 tral forces are derived. 
 
 . veloc. of rev. body 3 X weight of body 
 
 5. _ ' _L_ _. centnf. force. 
 
 radius of circle of revolution x 32 
 
 velocity of revol. body 8 X weight of body __ .. 
 
 centrifugal force X 32 
 the circle of revolution. 
 
CENTRAL FORCES. 147 
 
 centrif. force X 32 x rad. circle . , f , 
 
 7. - = - ? -r. - , , , = weight of the re- 
 veloc. ot revolving body 2 
 
 volving body. 
 
 rad. circle X 32 X centrifugal forceN 
 - -T-- - ) = velocity. 
 
 weight 
 
 9. There will be no difficulty in applying what has been 
 said to practice. 
 
 There are two fly-wheels of the same weight, one of 
 which is 10 feet diameter, and makes 6 revolutions in a 
 minute ; what must the diameter of the other be to revolve 
 3 times in a minute? Here 6 a : 3 a : : 10 : 2-5 = the 
 diameter of the second. 
 
 What is the centrifugal force of the rim of a fly-wheel, 
 its diameter being 12 feet, and the weight of the rim 1 ton, 
 making 65 turns in a minute ? 
 
 8 >< 8 ^ 16 X 65 = 40-84 = 
 
 the velocity in feet per second ; hence, 
 
 the tendency to burst. 
 
 Let us employ the centre of gyration. If the fly above 
 mentioned is in two halves, which are joined together by 
 bolts capable of supporting 4 tons in all their positions, the 
 whole weight of the wheel is l tons, the circle of gyration 
 is 5-5 feet from the axis of motion ; what must be its velocity 
 so that its two halves may fly asunder ? The force tending 
 to separate the two halves will be 5 of the whole force ; 
 wherefore, by the rule, 
 88X4X8-5X8 
 
 ^ 
 
 
 11 x 3-1416 == 34-5576 = circumference of circle of gy- 
 ration, wherefore, 34-5576 : 30-636 : : 60 : 53-191 revolu- 
 tions in a minute. 
 
 10. The steam engine governor, or conical pendulum, 
 acts on the principle of central forces. It is so constructed, 
 that when the balls diverge, or fly outwards, the ring on 
 the upright shaft is raised, and that in proportion to the in- 
 crease of the velocity, squared ; or, the square roots of the 
 
148 CENTRAJL FORCES. 
 
 distances of the ring from the top, corresponding to two 
 velocities, will be as these velocities. 
 
 Ex. If a governor makes 6 revolutions in a second, 
 when the ring is 16 inches from the top ; what will be the 
 distance of the ring when the speed is increased to 10 revo- 
 lutions in the same time ? The balls will diverge more, 
 consequently the ring will rise and the distance from the 
 top become less ; therefore, we have 
 
 10 : 6 :: v/ 16 or 4 : 2-4; 
 
 which, squared, gives 5'76 inches, the second distance of 
 the ring from the top. See Steam Engine. 
 
 11. We shall elsewhere introduce other particulars on 
 rotation and central forces. 
 
 STRENGTH OF MATERIALS, MACHINES, 
 MODELS, &c. 
 
 MATERIALS are exposed to four different kinds of strain : 
 
 1st. They may be torn asunder, as in the case of ropes 
 and stretchers. The strength of a body to resist this kind 
 of strain is called its Resistance to Tension, or Absolute 
 strength. 
 
 2d. They may be crushed or compressed in the direction 
 of their length, as in the case of columns, truss beams, &c. 
 
 3d. They may be broken across, as in the case of joists, 
 rafters, <fec. The strength of a body to resist this kind of 
 strain is called its Lateral strength. 
 
 4th. They may be twisted or wrenched, as in the case 
 of axles, screws, &c. 
 
 Extensive and accurate experiments are necessary to 
 determine the several measures of these strengths in the 
 different materials ; and when this is done, the subsequent 
 calculations become comparatively easy. We shall, there- 
 fore, in the first place, lay down the results of the experi- 
 ments of practical men. 
 

 STRENGTH OF MATERIALS. 
 
 A. 
 
 149 
 
 TABLE OF THE FLEXIBILITY AND STRENGTH OF TIMBER. 
 
 Name of the Wood. 
 
 U 
 
 E 
 
 s 
 
 c 
 
 Teak, 
 
 818 
 
 9657802 
 
 2462 
 
 15555 
 
 Poon, 
 
 596 
 
 6759200 
 
 2221 
 
 14787 
 
 English oak, 
 
 598 
 
 3494730 
 
 1181 
 
 9836 
 
 Do. 
 
 435 
 
 5806200 
 
 1672 
 
 10853 
 
 Canada oak, 
 
 588 
 
 8595864 
 
 1766 
 
 11428 
 
 Dantzic oak, 
 
 724 
 
 4765750 
 
 1457 
 
 7386 
 
 Adriatic oak, 
 
 610 
 
 3885700 
 
 1583 
 
 8808 
 
 Ash, 
 
 395 
 
 6580750 
 
 2026 
 
 17337 
 
 Beech, 
 
 615 
 
 5417266 
 
 1556 
 
 9912 
 
 Elm, 
 
 509 
 
 2799347 
 
 1013 
 
 5767 
 
 Pitch pine, 
 
 588 
 
 4900466 
 
 1632 
 
 10415 
 
 Red pine, 
 
 605 
 
 7359700 
 
 1341 
 
 10000 
 
 New English fir, 
 
 757 
 
 5967400 
 
 1102 
 
 9947 
 
 Riga fir, 
 
 588 
 
 5314570 
 
 1108 
 
 10707 
 
 Do. 
 
 
 3962800 
 
 1051 
 
 
 Mar forest fir, 
 
 588 
 
 2581400 
 
 1144 
 
 9539 
 
 Do. 
 
 403 
 
 3478328 
 
 1262 
 
 10691 
 
 Larch, 
 
 411 
 
 2465433 
 
 653 
 
 
 Do. 
 
 518 
 
 3591133 
 
 832 
 
 
 Do. 
 
 518 
 
 4210830 
 
 1127 
 
 7655 
 
 Do. 
 
 518 
 
 4210830 
 
 1149 
 
 7352 
 
 Norway spar, 
 
 648 
 
 5832000 
 
 1474 
 
 12180 
 
 NOTE. The extensive use of the above table will be 
 shown hereafter. 
 
 U. The ultimate strength. E. Lateral strength. S 
 Transverse strength. C. Cohesion. 
 
 B. 
 
 Table showing the weight that will pull asunder a prism 
 one inch square. 
 
 Ibs. Ibs. 
 
 Cast gold, 22000 I Bismuth, 29000 
 
 Cast silver, 41000 I Good brass, 51000 
 
 13* 
 
150 
 
 STRENGTH OF MATERIALS. 
 
 It*. 
 Anglesea copper, 34000 
 Swedish copper,---- 37000 
 
 f*nat Ii-rn ^flftOft 
 
 
 It*. 
 
 . . 1 l'>7O 
 
 
 . 7^0 
 
 
 
 Bar iron, ordinary,- 68000 
 Do. Swedish, 84000 
 
 Tlar etf>l erft . . . 1 *> ( II II )( > 
 
 COMPOSITIONS 
 
 Gold 5, copper 1, 
 Silver 5, copper !, 
 S wed. copper 6, tin 
 Block tin 3, lead 1, 
 Tin 4, lead 1, zinc 
 T^ead 8. zinc 1. 
 
 OF 
 
 ---50000 
 .-48500 
 1,. 64000 
 -10200 
 1,. 13000 
 ..-45000 
 
 Do. razor temper, 15000Q 
 Cast tin, Eng. block, 5200 
 
 
 
 y.inn ., 9fiOO 
 
 c. 
 
 The same from Rennie : 
 
 Weight that would tear 
 it asunder in Ibs. 
 
 Cast steel, 
 
 Swedish iron, 
 
 English iron, 
 
 Cast iron, 
 
 Cast copper, 
 
 Yellow brass, 
 
 Cast tin, 
 
 Cast lead, 
 
 Good hemp rope,.*. 
 Do. one inch diam. 
 
 Length in feet that would 
 break with its own weight 
 
 134256 39455 
 
 72064 19740 
 
 55872 16938 
 
 19096 6110 
 
 19072 5092 
 
 17958 5180 
 
 4736 1496 
 
 1824 306 
 
 6400 18790 
 
 5026 18790 
 
 D. 
 
 The cohesive force of a square inch of iron ; from dif- 
 ferent experimentists. 
 
 
 IDS. 
 
 ..11 1O77 
 
 English iron, 
 
 IDS. 
 
 61600 
 
 
 
 > 1* I. 
 
 
 TV 1 U ' 
 
 . . !i 1 'If'ill 
 
 Tin 
 
 79ORJ. 
 
 T\n 
 
 
 
 . '. MK-..I 
 
 
 . .<; I nn 1 
 
 T\~ f ^QO4/< 
 
 Russian iron, 
 
 ..^0479 
 
 
 AQ1 QQ 
 
 
 
 r.r.i) in 
 
 Fin 
 
 
 T)f. 
 
 . 55000 
 
 Welsh do. 
 
 16255 
 
STRENGTH OF MATERIALS. 151 
 
 E. 
 
 Table of the lateral strength of the following materials, 
 one foot long, and one inch square. 
 
 * Weight that will Weight which they can 
 
 break tUetn. bear with safety. 
 
 Cast iron, 3270 Ibs. 1090 Ibs. 
 
 Oak, 627 209 
 
 Memelfir, 390 -r 130 -r- 
 
 American white pine,* 206 69 
 
 F. 
 The force necessary to crush one cubic inch. 
 
 Aberdeen granite, blue, 24556 
 
 Very hard freestone, 21254 
 
 Black Limerick limestone, 19924 
 
 Compact limestone, 17354 
 
 Craigleith stone, 15568 
 
 Dundee sandstone, 14919 
 
 Yorkshire paving stone, 15856 
 
 Redbrick, 1817 
 
 Pale red brick, 1265 
 
 Chalk, 1127 
 
 Cubes of one-fourth of an inch. 
 
 Iron cast vertically, 11140 
 
 horizontally, 10110 
 
 Cast copper, 7318 
 
 Cast tin, 966 
 
 Cast lead, 483 
 
 Having made these statements, we shall proceed to show 
 how, by the assistance of theoretical results, they may be 
 applied to the wants of the practical engineer. 
 
 The absolute strength of ropes or bars, pulled length- 
 wise, is in proportion to the squares of their diameters. 
 All cylindrical or prismatic rods are equally strong in every 
 part, if they are equally thick, but if not, they will break 
 where the thickness is least. 
 
 The lateral strength of any beam or bar of wood, stone, 
 metal, &c., is in proportion to its breadth X its depth 2 . 
 In square beams the lateral strengths' are in proportion to 
 the cubes of the sides, and in general of like-sided beams 
 as the cubes of the similar sides of the section. 
 
152 STRENGTH OF MATERIALS. 
 
 The lateral strength of any beam or bar, one end being 
 fixed in the wall and the other projecting, is inversely as 
 the distance of the weight from the section acted upon ; 
 and the strain upon any section is directly as the distance 
 of the weight from that section. 
 
 If a projecting beam be fixed in a wall at one end, and 
 a weight be hung at the other, then the strain at the end in 
 the wall, is the same as the strain upon a beam of twice 
 the length, supported at both ends and with twice the 
 weight acting on its middle. The strength of a projecting 
 beam is only half of what it would be, if supported at both 
 ends. 
 
 If a beam be supported at both ends, and a weight act 
 upon it, the strain is greatest when the weight is in the 
 middle ; and the strain, when the weight is not in the 
 middle, will be to the strain when it is in the middle, as 
 the product of the weight's distances from both ends, is to 
 the square of half the length of the beam. Take any two 
 points in a beam supported at both ends ; call one of these 
 points a and the other b ; then a weight hung at a will 
 produce a strain at 6, the same as it would do at a if hung 
 at b. 
 
 In a beam supported at the ends p 
 
 A and B ; the strain at C, with the 
 
 whole weight placed there, is to the strain at C with the 
 whole weight placed equally between C and P, as AC is 
 to AP x 5 PC ; and the strain at C by a weight placed 
 equally along AP, is to the strain at C by the same weight 
 placed on C, as AP is to AC. 
 
 If beams bear weights in proportion to their lengths, 
 either equally distributed over the beams or placed in similar 
 points, the strains upon the beams will be as their lengths 2 . 
 
 If a beam rest upon two supports, and at the same time 
 be firmly fixed in a wall at each end, it will bear twice as 
 much weight as if it had lain loosely upon the supports ; 
 and the strain will be everywhere equal between the 
 supports. 
 
 In any beam standing obliquely, or in a sloping direction, 
 its strength or strain will be equal to that of a beam of the 
 same breadth, thickness, and material, but only of the 
 length of the horizontal distance between the points of 
 support. 
 
 Similar plates of the same thickness, either supported aC 
 
STRENGTH OF MATERIALS. 153 
 
 ihe ends or all round, will carry the same weight either 
 uniformly distributed or laid on similar points, whatever be 
 their extent. 
 
 The strength of a hollow cylinder, is to that of a solid 
 cylinder of the same length and the same quantity of mat- 
 ter, as the greater diameter of the hollow cylinder is to the 
 diameter of the solid cylinder ; and the strength of hollow 
 cylinders of the same length, weight, and material, are as 
 their greater diameters. 
 
 The lateral strength of beams, posts, or pillars, are dimi- 
 nished the more they are compressed lengthwise. 
 
 The strength of a column to resist being crushed is 
 directly as the square of the diameter, provided it is not so 
 long as to have a chance of bending. This is true in metals 
 or stone, but in timber the proportion is rather greater than 
 the square. 
 
 The strength of homogeneous cylinders to resist being 
 twisted round their axes, is as the cubes of their diameters ; 
 and this holds true of hollow cylinders, if their quantities 
 of matter be the same. 
 
 PROBLEMS. 
 
 To find the strength of direct cohesion : 
 
 Area of transverse section in inches X measure of cohe- 
 sion = strength in Ibs. to resist being pulled asunder. 
 
 Ex. In a square bar of beech, 3 inches in the side, we 
 have 3 X 3 X 9912 = 89208 Ibs. 
 
 NOTE. The measure of cohesion for timber is taken 
 from col. C, table A, and for other materials, from tables B 
 or C. 
 
 In a beam of English oak, having four equal sides, each 
 side being four inches, we have 
 
 4 x 4 X 9836 = 157376 Ibs., the strength. 
 
 In a rod of cast steel', 2 inches broad and 1 inch thick, 
 we have 2 X Is X 134256 = 402768 Ibs., the strength. 
 
 What is the greatest weight which an iron wire -jL- of an 
 inch thick will bear ? 
 
 The area of the cross section of such wire will be '007854, 
 hence we have -007854 X 84000 = 659-736 Ibs. 
 
 To find the ultimate transverse strength of any beam : 
 When the beam is fixed at one end and loaded at the 
 other, then the dimensions being in inches, 
 
154 STRENGTH OF MATERIALS. 
 
 breadth x depth* x transverse strength , . . 
 
 - = the ultimate 
 length of beam 
 
 transverse strength. 
 
 NOTE. In column S, Table A, will be found the trans- 
 verse strength of timber, and in table E, that of iron, &c. ; 
 and let it be observed, that when the beam is loaded uni- 
 formly, the result of the last rule must be doubled. 
 
 What weight will break a beam of Riga fir, fixed at one 
 end and loaded at the other, the breadth being 3, depth 4, 
 and length 60 inches ? 
 
 * ^ "! 886Mb.. 
 
 What weight uniformly distributed over a beam of 
 English oak would break it, the breadth being 6, depth 9, 
 and its length 12 feet? 
 
 If the number be taken from table F, we must use the 
 length in feet. 
 
 When the beam is supported at both ends, and loaded in 
 the centre, 
 
 tabular value of S. tab. A X depth* X breadth x 4 _ 
 
 length 
 the weight in pounds. 
 
 NOTE. When the beam is fixed at one end and loaded 
 in the middle, the result obtained by the rule must be in- 
 creased by its half. When the beam is loaded uniformly 
 throughout, the result must be doubled. When the beam 
 is fixed at both ends and loaded uniformly, the result must 
 be multiplied by three. 
 
 Ex. What weight will it require to break a beam of 
 English oak, supported at both ends and loaded in the 
 middle, the breadth being 6, and depth 8 inches, and length 
 12 feet? 
 
 IgTCjLJL' X 6 X 4 = 17834. 
 144 
 
 By using table E : 
 
 depth 9 X breadth X tabular number 
 length in ieet 
 
PROBLEMS. 155 
 
 Ex. What weight will a cast iron bar bear, 10 feet 
 long, 10 inches deep, and 2 inches thick, laid on its edge? 
 
 10 
 
 The same on its broad side : 
 2 9 X 10 X 1090 
 
 4360 Ibs. 
 
 10 
 
 To find the breadth to bear a given weight. 
 length X weight 
 
 number in table E x depth 8 
 What must be the breadth of an oak beam, 20 feet long 
 and 14 inches deep, to sustain a weight of 10000 Ibs. ? 
 
 20 X 10000 
 ua x 2Q9 = 4-85 inches = the breadth. 
 
 To find the length : 
 
 depth 8 x breadth x tabular number . 
 
 r-r - = length. 
 
 weight 
 
 In a beam 1 ft. deep and 4 in. broad, the weight being 
 5000 Ibs. ; then we have, if the beam be made of Memel fir, 
 
 = 14-97 feet, length required. 
 
 5000 
 To find the depth : 
 
 length x weight N = d 
 .tabular number x breadth/ 
 
 Jfc 
 
 We wish to support a weight of 2000 Ibs. by a beam of 
 American pine ; what is its depth, its length being- 20 feet 
 and breadth 4 inches ? 
 
 To find the deflection of a beam fixed at one end, and 
 loaded at the other : 
 
 length of beam in inches 3 X 32 x weight , 
 
 tab. numb. E (in table A) X breadth x depth 3 
 flection in inches. 
 
 NOTE. If the beam be loaded uniformly, use 12 instead 
 of 32 in the rule. 
 
 If a weight of 300 be hung at the end of an ash bar fixed 
 
156- STRENGTH OF MATERIALS. 
 
 in a wall at one end, and five feet long, it being 4 inches 
 square : what is its deflection ? 
 60 3 x 32 X 300 
 
 1<28 mchei = 
 
 6580750 x 4 x 4 
 
 If the beam be supported at both ends and loaded in the 
 middle : 
 
 _ length (in inches) 8 x weight _ . 
 
 tab. numb. (E, table A) X breadth X depth 8 ~" 
 
 NOTE. When the beam is firmly fixed at both ends, the 
 deflection will be f of that given by the rule. 
 
 Ex. If a beam of pitch pine, 8 inches broad, 3 inches 
 thick, and thirty feet long, is supported at both ends and 
 loaded in the centre with a weight of 100 Ibs. ; what is its 
 deflection ? 
 
 " 
 
 4900466X8 8" 
 
 If the beam had been firmly fixed at both ends, the de- 
 flection would have been 
 
 4-408 x | = 2-938 inches. 
 
 If the beam had been supported at both ends, and loaded 
 uniformly throughout, the deflection would have been 
 
 4-408 X = 2-754. 
 
 To find the ultimate deflection of a beam of timber 
 before it breaks : 
 
 length (in inches) 9 
 
 - = ultimate deflection. 
 tab. numb. U (table A) x depth 
 
 What is the ultimate deflection of a beam of ash, 1 foot 
 broad, 8 inches deep, and 40 feet long ? 
 
 480* 
 
 - = 72-72 inches, the ultimate deflection. 
 
 ivO X O 
 
 To find the weight under which a column placed verti- 
 cally will begin to bend, when it supports that weight : 
 tab. numb. E (table A) x least thickness 3 x greatest x -2056 
 
 length (in inches) 8 
 
 3= weight in pounds. It will be found by the application 
 of this rule, that it will require 40289-22 Ibs. to bend a 
 beam of English oak 20 ft. long, 6 in. thick, and 9 in. broad. 
 
 BEAMS. 
 
 WE take the liberty here of introducing a short extract 
 from Messrs. Hann and Dodds' Mechanics, on the subject 
 
BEAMS. 
 
 157 
 
 of beams. " In the construction of beams, it is necessary 
 that their form should be such that they will be equally 
 strong throughout. If a beam be fixed at one end, and 
 loaded at the other, and the breadth uniform throughout its 
 length, then, that the beam may be equally strong through- 
 out, its form must be that of a parabola. This form is 
 generally used in the beams of steam engines." 
 
 Dr. Young and Mr. Tredgold have considered that it will 
 answer better, in practice, to have some straight-lined 
 figure to include the parabolic form ; and the form which 
 they propose is to draw a tangent to the point A of the 
 parabola ACB. 
 
 To draw a parabola. 
 Let CB represent the 
 length of the beam, and 
 AB the semi-ordinate, or 
 half the base ; then, by 
 the property of the para- 
 b61a, the squares of all 
 ordinates to the same 
 diameter are to one an- 
 other as their respective abscisses. Now, if we take CB 
 = 4 feet, and AB = 1 foot, we may proceed to apply this 
 property to determine the length of the semi-ordinates 
 corresponding to every foot in the length of the beam, as 
 follow : CB : AB 8 : : CF : EF 3 ; 
 
 that is, 48 : 12 a :: 36 : 108 = EF S ; 
 the square root of which is 10'4 nearly = EF. 
 AndCB : AB a : : CG : GH 2 ; 
 
 48 : 12 a : : 24 : 72 
 the square root of which is 8-5 nearly = GH. 
 CB: AB 3 :: CI : IK 2 ; 
 48 : 12 a : : 12 : 36 = IK 3 ; 
 the square root of which is 6 inches = IK. 
 Now, if we take CL = 6 inches, 
 thenCB : BB a : : CL : LM a ; 
 
 48 : 12 a :: 6 : 18 = LM 3 ; 
 
 the square root of which is 4'24, which is very near 4| 
 inches s= LM. Now, if any flexible rod be bent so as just 
 to touch the tops A, E, H, K, M, of the ordinates, and the 
 vertex C, then the form of this rod is a parabola. 
 To draw a tangent to any point A of a parabola: 
 From the vertex C of the parabola draw CD perpendicu- 
 
 14 
 
 GH 8 ; 
 
158 STRENGTH OP MATERIALS. 
 
 lar to CB, and make it equal to AB; then join AD, and 
 the right line AD will be a tangent to the parabola at the 
 point A ; that is, it touches the parabola at that point. In 
 the same manner, we may draw a tangent to the parabola 
 at any other point, by erecting a perpendicular at the vertex 
 equal to half the semi-ordinate at that point. 
 
 When a beam is regularly diminished towards the points 
 that are least strained, so that all the sections are similar 
 figures, whether it be supported at each end and loaded in 
 the middle, or supported in the middle and loaded at each 
 end, the outline should be a cubic parabola. 
 
 When a beam is supported at both ends, and is of the 
 same breadth throughout, then, if the load be uniformly 
 distributed throughout the length of the beam, the line 
 bounding the compressed side should be a semi-ellipse. 
 
 The same form should be made use of for the rails of a 
 wagon-way, where they have to resist the pressure of a 
 load rolling over them. 
 
 MODELS. The relation of models to machines, as to 
 strength, deserves the particular attention of the mechanic. 
 A model may be perfectly proportioned in all its parts as a 
 model, yet the machine, if constructed in the same propor- 
 tion, will not be sufficiently strong in every part ; hence, 
 particular attention should be paid to the kind of strain the 
 different parts are exposed to; and from the statements 
 which follow, the proper dimensions of the structure may 
 be determined. 
 
 If the strain to draw asunder in the model be 1, and if 
 the structure is 8 times larger than the model, then the 
 stress in the structure will be 8 3 = 512. If the structure is 
 6 times as large as the model, then the stress on the struc- 
 ture will be 6* = 216, and so on ; therefore, the structure 
 will be much less firm than the model ; and this the more, 
 as the structure is cube times greater than the model. If 
 we wish to determine the greatest size we can make a ma- 
 chine of which we have a model, we have, 
 
 The greatest weight which the beam of the model can 
 bear, divided by the weight which it actually sustains = a 
 quotient which, when multiplied by the size of the beam in 
 the model, will give the greatest possible size of the same 
 beam in the structure. 
 
 Ex. If a beam in the model be 7 inches long, and bear 
 a weight of 4 Ibs., but is capable of bearing a weight of 2G 
 
SHAFTS. 159 
 
 Ibs. ; what is the greatest length which we can make the 
 corresponding beam in the structure ? Here 
 
 therefore, 6' 5 X 7 = 45' 5 inches. 
 
 The strength to resist crushing, increases from a model 
 to a structure in proportion to their size, but, as above, the 
 strain increases as the cubes ; wherefore, in this case also, 
 the model will be stronger than the machine, and the 
 greatest size of the structure will be found by employing 
 the square root of the quotient in the last rule, instead of 
 the quotient itself; thus, 
 
 If the greatest weight which the column in a model can 
 bear is 3 cwt., and if it actually bears 28 Ibs., then, if the 
 column be 18 inches high, we have 
 
 (12) = 3-464; 
 
 wherefore, 3'464 x 18 = 62-352 inches, the length of the 
 column in the structure. 
 
 SHAFTS. 
 
 THE strength of shafts deserves particular attention ; 
 wherefore, instead of incorporating it with the general sub- 
 ject, strength of materials, we have allotted to it % separate 
 chapter under that head. 
 
 Whenthe weight is in the middle of the shaft, the rule is 
 
 in Ibs. x length in feet\ 
 - - - I = diameter m inches. 
 oOO 
 
 This is to be understood as the journal of the shaft, the 
 body being usually square. 
 
 What is the diameter of a shaft 12 feet long, bearing a 
 weight of 6 cwts., the weight acting at the middle ? 
 
 inches. 
 
 If the weight be equally diffused, we have, the weight in 
 Ibs. x length ; extract the cube root and divide by 10; the 
 quotient is the diameter. 
 
 Thus, take the last example, then 672 x 12 = 8064 ; 
 the cube root of which is 20-05, which divided by 10 gives 
 2-005, the diameter of the shaft. 
 
 3 1/ weight 
 
160 STRENGTH OP MATERIALS. 
 
 If a cylindrical shaft have no other weight to sustain be- 
 sides its own, the rule is, ^(-007 x length 3 ) = diameter; 
 thus, if a shaft having only the stress of its own- weight be 
 10 feet long; 
 v/(-007xl0 3 ) = 2-645 the diameter of the shaft in inches. 
 
 For a hollow shaft supporting so many times its own 
 weight, we have 
 
 012 x length 3 x No. times its own weighty 
 1 + inner diameter* /*" 
 
 outer diameter in inches. 
 
 For wrought iron shafts find the diameter by the forego- 
 ing rules, which apply to cast iron, then multiply by -935, 
 and for oak shafts the multiplier is 1-83, and for fir 1-716. 
 
 Ex. What is the diameter of a cast iron shaft 12 feet 
 long, and the stress it bears being twice its own weight ? 
 Here we have, 
 
 v' (-012 X 12" X 2) = 6-44 inches. 
 For wrought iron, using the multiplier, 
 6-44 x -935 = 6-0215, 
 and for oak, using the multiplier, 
 
 6-44 x 1-83 = 11-3852, 
 and for fir, we have 
 
 .^ 6-44 X 1-716 = 11-05104. 
 
 A nnPbften used in practice, though by no means a cor- 
 rect one, for determining the diameter of shafts is this. 
 The* cube root of the weight which the shaft bears taken in 
 cwts. is nearly the diameter of the shaft in inches. It will 
 be found safe in practice, to add one-third more to this 
 result. 
 
 If a cast metal shaft has to bear a weight of U ton, that 
 is, 30 cwts., then we have, 
 
 </ 30 = 3-107 inches by this rule ; 
 
 and supposing it 12 feet long, we will apply the other rule, 
 we have, 
 
 ' 3360 X 12' 
 
 We have now considered the strength of shafts, so far as 
 regards their power to resist lateral pressure by weight act- 
 ing on them; we have now to consider their power to 
 resist torsion or twisting. 
 
SHAFTS. 161 
 
 For cylindrical shafts, we have, 
 
 240 X No. of horses* power \ 
 >No. of revolutions in a minute/ ~ 
 the diameter of the shaft in inches. 
 
 This rule is for cast iron ; and it may be used for 
 wrought iron by multiplying the result by -963, or for oak 
 by 2-238, or for fir by 2-06. 
 
 If the shaft belong to a 7 horse power engine, and the 
 strap turns ll times in a minute, 
 
 ?|(- n>5 ) = 5-267 inches diameter for cast iron. 
 
 For fir, 5-267 X 2-06 = 10-85. 
 
 For oak, 5-267 X 2'38 = 12-535. 
 
 And for wrought iron, 5-267 X -963 = 5-0719. 
 
 NOTE. This rule comes from the best authority, and 
 gives perfectly safe results, though some employ 340, in- 
 stead of 240, as a multiplier, which gives a greater diameter 
 to the shaft. We may compare the two : 
 
 ' 34 X ^ = 5-916, 
 
 11-5 
 
 whereas the other was 5*267 something more than half an 
 inch of difference. 
 
 It is to be remembered, that these rules relate to the 
 shafts of first movers, or the shafts immediately connected 
 with the moving power. But these shafts may communi- 
 cate motion to other shafts, called second movers, and these 
 again to others, called third movers, and so on. The dia- 
 meters of the second movers may be found from those of 
 the first, by multiplying by -8, and those of the third 
 movers, by multiplying by -793, thus, if the diameter of 
 the first mover be 5-267, then that of the second will be 
 5-267 X -8 = 4-2136, and that of the third mover will be 
 5-267 X -793 = 4-1767. 
 
 One material may resist, much better than another, one 
 kind of strain ; but expose both to a different kind of strain, 
 and that which was weakest before may now be the strong- 
 est. This may be illustrated in the case of cast and wrought 
 iron. The cast iron is stronger than the wrought iron when 
 exposed to twisting or torsional strain, but the malleable 
 iron is the stronger of the two when they are exposed to 
 
 14* 
 
162 
 
 STRENGTH OF MATERIALS. 
 
 lateral pressure. We shall subjoin a few results of experi- 
 ments on the weight which was necessary to twist bare | 
 fclose to the bearings. 
 
 Cast metal,., 
 
 lb. oz. 
 
 , 9 17 
 
 lb. ox. 
 
 English iron wrought,. 10 2 
 
 Do. vertical 
 Cast steel,.., 
 
 cast, 10 10 
 17 9 
 
 Swedish iron wrought, 9 8 
 Hard gun metal, 5 
 
 Shear steel,, 
 
 17 l 
 
 Brass bent, 4 11 
 
 Blister steel 
 
 16 11 
 
 Copper cast, 4 5 
 
 It would appear that the strength of bodies to resist torsion 
 is nearly as the cubes of their diameters. 
 
 REMARKS. The rules and statements we have now given 
 will often find their application in the practice of the engi- 
 neer. On the proper proportioning of the magnitude of 
 materials to the stress they have to bear, depends much of 
 the beauty of any mechanical structure ; and, what is of fai 
 greater moment, its absolute security. We will, in the Ap- 
 pendix to this book, give some examples of the application 
 of these principles to practice. 
 
 TABLE OF THE DIAMETERS OF SHAFT JOURNALS. 
 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 60 
 
 70 
 
 80 
 
 90 
 
 100 
 
 5 
 
 5-9 
 
 4-7 
 
 4-1 
 
 3-7 
 
 3-5 
 
 3-3 
 
 3-1 
 
 3-0 
 
 2-9 
 
 2-7 
 
 6 
 
 6-3 
 
 5*0 
 
 4-4 
 
 4 
 
 3-7 
 
 3-5 
 
 3-4 
 
 3-2 
 
 3 
 
 2-9 
 
 7 
 
 6-6 
 
 5-2 
 
 4-6 
 
 4-2 
 
 3-9 
 
 3-6 
 
 3-5 
 
 3-4 
 
 3-3 
 
 3-1 
 
 8 
 
 6-9 
 
 5-5 
 
 4-8 
 
 4-4 
 
 4-1 
 
 3-9 
 
 3-7 
 
 3-5 
 
 3-4 
 
 3-3 
 
 9 
 
 7-2 
 
 5-7 
 
 5 
 
 4-5 
 
 4-2 
 
 4 
 
 3-7 
 
 3-6 
 
 3-5 
 
 3-4 
 
 10 
 
 7-4 
 
 5-9 
 
 5-2 
 
 4-7 
 
 4-4 
 
 4-1 
 
 3-9 
 
 3-7 
 
 3-6 
 
 :{-.'> 
 
 15 
 
 8-5 
 
 7-0 
 
 6-0 
 
 5-5 
 
 5-1 
 
 4-6 
 
 4-5 
 
 4-3 
 
 4-2 
 
 4-0 
 
 20 
 
 9-3 
 
 7-4 
 
 6-6 
 
 5-9 
 
 5-6 
 
 5-2 
 
 5-0 
 
 4-6 
 
 4-5 
 
 4.4 
 
 30 
 
 10-7 
 
 8-4 
 
 7-4 
 
 6-9 
 
 6-5 
 
 5-9 
 
 5-7 
 
 5-5 
 
 5-2 
 
 5-0 
 
 40 
 
 11-7 9-5 
 
 8-3 
 
 7*4 
 
 6-9 
 
 6-6 
 
 6-2 
 
 5-9 
 
 5-7 
 
 5-li 
 
 50 
 
 12-6 
 
 10-0 9-0 
 
 8-0 
 
 7*4 7-2 
 
 6-8 
 
 6-5 
 
 6-2 
 
 .-)! 
 
 60 
 
 13-6 
 
 10-8 9-3 
 
 8-6 
 
 7-7 7-4 
 
 7-2 
 
 6-8 
 
 6-7 
 
 6-4 
 
 In the preceding table of the diameters of the shafts of first 
 movers, the number of horses' power of the engine is given 
 in the left-hand column, and the number of revolutions the 
 shaft makes in a minute is given in the top column. Then, 
 to use the table, we have only to look for the power of the 
 engine in the side column, and the number of turns the shaft 
 
JOISTS AND ROOFS. 163 
 
 makes in a minute in the line which runs across the top, 
 and where these columns meet will be found the diameter 
 of the shaft in inches. The table is constructed for cast' 
 iron, and first movers ; the rules for finding the second and 
 third have been given above, as also for finding equally 
 strong shafts of other materials. 
 
 This table answers for first movers only. It may, how- 
 ever, be made to give results for second and third movers, 
 by using the multipliers for that purpose, formerly given. 
 
 What is the diameter of the journal of the shaft of the 
 first mover in a 30 horse power engine, the shaft making 
 40 revolutions in a minute? Here, by looking in the table, 
 in the side column of horses' power, we find 30, and in the 
 top column of revolutions, we find 40, and where these 
 columns meet, we find 6*9 = the diameter of the first 
 mover, in inches ; wherefore, the second mover of this 
 power and velocity will be = 6'9 X -8 = 5-52 inches ; and, 
 in like manner, the third mover will be = 6*9 x "64 = 
 4*416 inches = the diameter of the third mover to the same 
 power and speed. 
 
 JOISTS AND ROOFS. 
 
 JOISTS should increase in strength in proportion to the 
 squares of their lengths ; for instance, a joist 16 feet long 
 should be four times as strong as another joist 8 feet long, 
 similarly situated; because 8* : 16*: : 1 : 4. From what 
 has been previously stated, it will easily appear, that the 
 stress on a beam or joist supported at both ends, increases 
 towards the middle, where it is greatest ; it therefore fol- 
 lows, that a beam should be strengthened in proportion to 
 the increasing strain ; and, as it would not be easy to add 
 to the thickness of a beam towards the middle, which would 
 destroy the levelness of the floor, a good substitute may be 
 to fasten pieces to the sides of the joist, and thus increase 
 its breadth ; thus causing the beam to taper, in breadth, 
 from the centre to the ends. In this way joists may be 
 made much stronger than they usually are of the same 
 length, and out of the same quantity of timber. It may 
 also be observed, that joists are twice as strong when firmly 
 fixed in the wall, as when loose ; but it is to be remarked, 
 that they have, when fixed, a far greater tendency to shake 
 the wall. It is also to be remarked, that a joist is four times 
 stronger when supported in the middle. 
 
164 JOISTS AND ROOFS. 
 
 If the letter L represent the length of some known joist, 
 whose strength has been tried, and D its depth, and T its 
 thickness ; and if another joist is required of equal strength 
 with the former, when similarly situated ; whose length is 
 represented by /, its, depth by d, and its thickness by t; we 
 have the following rules : 
 
 x /\ D- x T x l 
 
 d 
 
 D 3 x I 2 X T ... d* xtxL 
 
 \ 
 
 ) 
 
 - rxlT DxT 
 
 If a joist 30 feet long, 1 foot deep, and 3 inches thick, be 
 sufficient in one case, what must the depth of a beam be, 
 similarly placed, whose length is 15 feet, its depth and 
 thickness bearing the same proportion to each other, as in 
 the former beam ? Here, by the first theorem, we have, 
 
 r/1 3 X^ 158 \ _ ^/(-25) = -6298 feet = 7-55 inches 
 
 the depth ; and therefore 12 depth : 3 thickness : : 7-55 : 
 1-88 the breadth. 
 
 If the given beam be, as in the last example, 12 inches 
 deep, 3 thick, and 30 feet long, and the required beam, of 
 the same strength, is 8 inches deep, and 6 inches thick, then 
 by the 4th we have, 
 
 If a joist, whose length is 30 feet, depth 12 inches, and 
 thickness 8, is given, to find the depth of another of equal 
 strength, only 6 inches thick, and 28-28 feet long? Here, 
 by the 2d, we have, 
 
 |12 a x 3 x 28-28 a 
 \ -- 6 x 30* " = mches tne depth. 
 
 To find the thickness from the same circumstances, we 
 have by the 3rf, 
 
 12 x 28-2S 3 x 3 
 
 Q8 ^ QAa - = 6 inches, the thickness. 
 
 o X >U 
 
 The same remarks hold true to a certain extent in roof- 
 ing. A high roof is both heavier and more expensive than 
 a low roof, as they will always be as the squares of the 
 lengths of the couple-legs, so far as the scantling is con- 
 cerned ; but the slates and other materials increase in weight 
 
WHEELS. 165 
 
 and expense as the length of the couple-legs simply. High 
 roofs have, however, the advantage of being less severe 
 upon the walls than low ones, that is to say, so far as a 
 tendency to push out the walls is concerned. To obtain 
 the length of the rafter from that of the span, a common 
 rule is to multiply the span by '66, which gives the length 
 of the rafter; thus, 14 feet of span gives 14 x "66 = 9'24 
 feet, the length of the rafter. 
 
 NOTE. The numbers in the tables of the strength of 
 materials are such as will break the bodies in a very short 
 time ; the prudent artist, therefore, will do well to trust no 
 more than about one-third of these weights ; also great 
 allowance must be made for knotty timber, and such as is 
 sawn in any part across or obliquely to the fibres. 
 
 WHEELS. 
 
 IN page 136 we promised again to resume the subject of 
 wheel-work ; and we now proceed to consider, in the first 
 place, the formation of the teeth of wheels. 
 
 A Cog-wheel is the general name for any wheel which 
 has a number of teeth or cogs placed round its circumference. 
 
 A Pinion is a small wheel which has, in general, not 
 more than 12 teeth ; though, when two toothed wheels act 
 upon one another, the smallest is generally called the pinion ; 
 so is also the trundle, lantern, or wallower. 
 
 When the teeth of a wheel are made of the same material 
 and formed of the same piece as the body of the wheel, they 
 are called teeth; when they are made of wood or some other 
 material, and fixed to the circumference of the wheel, they 
 are called cogs ; in a pinion they are called leaves ; in a 
 trundle, staves. 
 
 The wheel which acts is called a leader, or driver ; and 
 the wheel which is acted upon by the former is called a 
 follower, or the driven. 
 
 When a wheel and pinion are to be so formed that the 
 pinion shall revolve four times for the wheel's once, then 
 they must be represented by two circles, whose diameters 
 are to one another, as 4 to 1. When these two circles are 
 so placed that they touch each other at the circumferences, 
 then the line drawn joining their centres, is called the line 
 
166 WHEELS. 
 
 of centres, and the radii of the two circles, the proportional 
 radii. 
 
 These circles are called, by mill-wrights in general, 
 pitch-lines. 
 
 The distances from the centres of two circles to the ex- 
 tremities of their respective teeth, are called the real radii, 
 and the distances between the centres of two contiguous teeth 
 measured upon the pitch-line, is called the pitch of the wheel. 
 
 Two wheels acting upon one another in the same plane, 
 are called spur geer. When they act at an angle, they are 
 called bevel geer. 
 
 Teeth of wheels and leaves of pinions require great care 
 and judgment in their formation, so that they neither clog 
 the machinery with unnecessary friction, nor act so irre- 
 gularly as to produce any inequalities in the motion, and a 
 consequent wearing away of one part before another. Much 
 has been written on this subject by mathematicians, who 
 seem to agree that the epicycloid is the best of all curves 
 for the teeth of wheels. The epicycloid is a curve differing 
 from the cycloid formerly described, in this, that the gene- 
 rating circle instead of moving along a straight edge, moves 
 on the circumference of another circle. 
 
 The teeth of one wheel should press in a direction per- 
 pendicular to the radius of the wheel which it drives. As 
 many teeth as possible should be in contact at the same 
 time, in order to distribute the strain amongst them ; by 
 this means the chance of breaking the teeth will be dimi- 
 nished. During the action of one tooth upon another, the 
 direction of the pressure should remain the same, so that 
 the effect may be uniform. The surfaces of the teeth in 
 working should not rub one against another, and should 
 suffer no jolt either at the commencement or the termination 
 of their mutual contact. The form of the epicycloid satis- 
 fies all these conditions ; but it is intricate, and the involute 
 of the circle is here substituted, as satisfying equally these 
 conditions, and as being much more easily described. 
 
 Take the circumference ABC of the 
 wheel on which it is proposed to raise 
 the teeth, and let a be a point from 
 which one surface of one tooth is to 
 spring, then fasten a string at H, such 
 that when stretched and lying on the 
 circumference shall reach to a; fix a 
 
WHEELS. 167 
 
 pencil at a, and keeping the string equally tense, move 
 the pencil outwards, and it will describe the involute of the 
 circle which will form the curve for one side of the tooth. 
 Fasten the string at B so that its end, to which the pencil 
 is fixed, be at the point from which the other face of the 
 tooth is to spring and proceed as above ; then the curve 
 of the other side of the tooth will be formed ; and the figure 
 of one of the teeth being determined, the rest may be traced 
 from it. 
 
 The teeth of the pinion are formed in like manner. 
 
 The observation of practical men has furnished us with 
 a method of forming teeth of wheels, which is found to an- 
 swer fully as well in practice as any of the geometrical 
 curves of the mathematician. 
 
 We have the pattern here of 
 the segment of a wheel with 
 cogs fixed on in their rough 
 state, and it is required to bring j 
 them to their proper figure : * 
 
 they are consequently understood 10 be much larger than 
 they are intended to be when dressed. The arc 6, 6, is the 
 circumference of the wheel on which the bottoms of the 
 teeth and cogs rest. Draw an arc a, a, on the face of the 
 teeth for the pitch line of their point of action ; draw also 
 </, rf, for their extremities or tops. When this is done, the 
 pitch circle is correctly divided into as many equal parts as 
 there are to be teeth. The compasses are then to be opened 
 to an extent of one and a quarter of those divisions, and with 
 this radius arcs are described on each side of every division 
 on the pitch line a, a, from that line to the line d, d. One 
 point of the compasses being set on c, the curve /, #, on 
 one side of one tooth, and 0, n, on the oilier sides of the 
 other are described. Then the point of the compasses being 
 set on the adjacent division A:, the curve /, 7/1, will be de- 
 scribed : this completes the curved portion of the tooth e. 
 The remaining portion of the tooth within the circle a, a, 
 is bounded by two straight lines drawn from g and m to- 
 wards the centre. The same being done to the teeth all 
 round, the mark is finished, and the cogs only require to be 
 dressed down to the lines thus drawn. 
 
 It will be easy to determine the diameter of any wheel 
 having the pitch and number of teeth in that wheel given. 
 Thus, a wheel of 54 teeth having a pitch of 3 inches, we 
 
168 
 
 WHEELS. 
 
 51 '5 inches diameter, nearly. 
 
 have 54 x 3 = 162 inches, the circumference, conse- 
 quently, 
 _. ^ 162 
 
 3-1416 
 or about 4 feet 3| inches. 
 
 In the following table we have given the radii of wheels 
 of various numbers of teeth, the pitch being one inch. To 
 find the radius for any other pitch, we have only to multiply 
 the radius in the table by the pitch in inches, the product 
 is the answer. Thus for 30 teeth at a pitch of 3 5 inches, 
 we have 4-783 x 3-5 = 16'74 inches, the radius. 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 1-668 
 
 1-774 
 
 1-932 
 
 2-089 
 
 2-247 
 
 2-405 
 
 2-563 
 
 2-721 
 
 2-879 
 
 3-038 
 
 20 
 
 3-196 
 
 3355 
 
 3-513 
 
 3-672 
 
 3-830 
 
 3-989 
 
 4 148 
 
 4-307 
 
 4465 
 
 4624 
 
 30 
 
 4-783 
 
 4-942 
 
 6-101 
 
 5-260 
 
 5-419 
 
 5-578 
 
 5-737 
 
 5-896 
 
 6-055 
 
 6214 
 
 40 
 
 6-373 
 
 6-532 6-643 
 
 6-850 
 
 7-009 
 
 7-168 
 
 7-327 
 
 7-486 
 
 7-695 
 
 7-804 
 
 60 
 
 7-963 
 
 8-122 
 
 8-231 
 
 8-440 
 
 8-599 
 
 8-753 
 
 8-962 
 
 9-076 
 
 9-235 
 
 9-399 
 
 60 
 
 9-553 
 
 9-712 
 
 9872 
 
 10031 
 
 10-190 
 
 10-349 
 
 10-508 
 
 10662 
 
 10-826 
 
 10-935 
 
 70 
 
 11-144 
 
 11-303 
 
 11-463 
 
 11-622 
 
 11-731 
 
 11-940 
 
 12-099 
 
 12-758 
 
 12-417 
 
 12-676 
 
 80 12-735 
 
 12 895 13-054 
 
 13213 
 
 13370 
 
 13531 
 
 13690 
 
 13-849 
 
 14-008 
 
 14-168 
 
 90 14-327 
 
 14-436 14-645 
 
 14-804 
 
 14-963 
 
 15-122 
 
 15281 
 
 15-441 
 
 15-600 
 
 15-759 
 
 100 
 
 15-918 
 
 16-072 
 
 mo*; 
 
 16-395 
 
 16-564 
 
 16713 
 
 16873 
 
 17-032 
 
 17-191 
 
 17-350 
 
 110 
 
 17-504 
 
 17-668 
 
 17-987 
 
 17-827 
 
 18-146 
 
 18305 
 
 18464 
 
 18-623 
 
 18-782 
 
 18-941 
 
 120 
 
 19-101 
 
 19260 
 
 19-419 
 
 19-578 
 
 19-737 
 
 19-896 
 
 20-055 
 
 20-214 
 
 20-374 
 
 20-633 
 
 130 
 
 20-692 
 
 20-851 21-010 
 
 21-169 
 
 21-328 
 
 21-488 
 
 21-647 
 
 21-806 
 
 21-460 
 
 22-124 
 
 140 
 
 22-283 
 
 22-442 22-602 
 
 22-761 
 
 22-920 
 
 23074 
 
 23-238 
 
 23397 
 
 23556 
 
 23716 
 
 150 
 
 23875 
 
 24-03424-193 
 
 24-352 
 
 24511 
 
 24620 
 
 24-830 
 
 24-989 
 
 25-148 
 
 25307 
 
 160 
 
 25-466 
 
 25-625 
 
 25-784 
 
 25-944 
 
 26-103 
 
 26-262 
 
 26-421 
 
 26-580 
 
 26-739 
 
 26-894 
 
 170 
 
 27-058 
 
 27-217 
 
 27-376 
 
 27-535 
 
 27-694 
 
 27853 
 
 27-931 
 
 28-172 
 
 28331 
 
 28 190 
 
 180 28-699 
 
 28-808:28-967 
 
 29-126 29 286 
 
 29-445 
 
 29604 
 
 29-763 
 
 29-922 
 
 30-086 
 
 19030241 
 
 30-40030-559 
 
 30-718 30 877 
 
 31-036 
 
 31-196 
 
 31-355 
 
 31-514 
 
 31-673 
 
 200 
 
 31-832 
 
 31-99232-150 
 
 32-310 
 
 32-469 
 
 32-628 
 
 32-787 
 
 32-846 
 
 33-105 
 
 33-264 
 
 210 33-42-1 
 
 33 583 33-742 
 
 33-901 
 
 34060 
 
 34219 
 
 34-278 
 
 34-537 
 
 34-697 
 
 34 B66 
 
 22035-015 
 
 35-17435333 
 
 35-492 
 
 35-652 
 
 35-811 
 
 36-970 
 
 36-129 
 
 :-;i;--jxx 
 
 36-447 
 
 230,36-60? 
 
 36-766 36-925 
 
 37-084 
 
 37-243 
 
 37-402 
 
 37561 
 
 37-720 
 
 37-880 
 
 38-039 
 
 240]38-198 
 25039-790 
 
 38-357,38-516 
 39-94940 1 0.x 
 
 38-725 
 40-262 
 
 38-835 
 40-426 
 
 38-994 
 40-585 
 
 39-153 
 40-744 
 
 39312 
 40-904 
 
 39-471 
 41-063 
 
 39-631 
 41-222 
 
 260 41-381 
 
 41-541, 
 
 41-699 
 
 41-858 
 
 42019 
 
 42-177 
 
 13-336 
 
 42-495 
 
 42-654 
 
 42-813 
 
 270 42-973 
 
 43-132:43291 
 
 43-450 
 
 43-609 
 
 43-768 
 
 43-927 
 
 44-087 
 
 44231 
 
 44-405 
 
 280 44-564 
 
 44-723 ! 44-882 
 
 45-042 
 
 45-201 
 
 45360 
 
 45519 
 
 45-678 
 
 45-837 
 
 45-996 
 
 290 46-156 
 
 46-315 
 
 46-474 
 
 46633 
 
 46-792 
 
 46-751 
 
 47-111 
 
 47-270 
 
 47-429 
 
 47-588 
 
 This will be found a very useful table in abridging calcu- 
 lation, for instance, if we wish to find the radius of a wheel 
 having 132 teeth, we look for 130 at the left-hand side 
 column, and 2 at the top, and where these columns meet, 
 we find the number 21-010, which, if the pitch of the 
 wheel be 2 5 inches, we multiply by 2. 
 
WHEELS. 169 
 
 21*010 X 2*5 == 52*525 inches, radius of required wheel. 
 
 An easy practical rule for the same purpose is the fol- 
 lowing : 
 
 Take the pitch by a pair of compasses, and lay it off on 
 a straight line, seven times, divide this line into eleven 
 equal parts ; each will be equal to four of the radius, which 
 is supposed to consist of as many parts as the wheel has 
 teeth. 
 
 Let the pitch be two inches, and the number of teeth 60, 
 then the diagram will show how to lay it down. 
 
 A l 2 3 * 5 6 7 R 
 123456789 10 11 
 C D 
 
 4, 8, 12, 16, <fec. 
 
 The upper line is the pitch laid off seven times, and 
 forming AB, which is divided into 11 equal parts, one 
 of which, CD, being repeated for every four teeth in 
 the wheel, that is, in this case, fifteen times, will give the 
 radius. 
 
 The same may be done by calculation, going by the 
 principles of the rule, thus, 
 
 2x7 = 14, then = 1-272, which divided by 4 
 
 gives 0-318 = the value of of the radius; 
 
 wherefore, -318 X 60 = 19-08. 
 By the table we have, 
 
 9-552 X 2 = 19-104, 
 the difference in the two results being 
 
 19-104 19-08 = -024, or twenty-four thousandth 
 parts of an inch. 
 
 Reversing the operation, let it be required to find the 
 pitch, the radius of the wheel being 19-104, and number 
 of teeth 60. 
 
 We have ^^ - = -318, then -318 X 4 = 1-272, 
 
 and 1-272 X 11 = 13-992. Now this is the whole line AB, 
 
 13*992 
 and therefore, = 1*998, which is so very nearly 
 
 two inches, the difference being 2 1*998 = *002 of an 
 inch, we ought in practice to take two as the pitch. 
 
 15 
 
170 WHEELS. 
 
 A little reflection on the part of the reader will show 
 that since ~ = -636, and = 1-571, and - .[ 
 
 we have, 
 
 (1) pitch x -159 x number of teeth = radius. 
 
 r^x radius 
 
 ( ' number of teeth x -159 = P 
 
 radius 
 
 (*) pitch X '159 = nUmberofteeth * 
 Thus, 
 (1) 2 X '159 x 60 = 19-08 = radius, 
 
 ^ 60^159 - 8 
 
 (3) 2 x \ 159 = 60 = number of teeth. 
 
 NOTE. The number *16 maybe employed instead of 
 159, being easily remembered. These rules are approxi- 
 mate, and the error diminishes as the number of teeth in- 
 creases. The true pitch is a straight line, but these rules 
 give it an arc of the circle, which passes through the centre 
 of the teeth, whereas it should be the chord of the arc. 
 
 An eminent writer on clock-work gives the following 
 rules regarding wheels and pinions : 
 
 (A) As the number of teeth in the wheel -f 2-25, 
 Is to the diameter of the wheel, 
 
 So is the number of teeth in the pinion -f- 1'5, 
 To the diameter of the pinion. 
 
 A wheel being 12 inches diameter, having 120 teeth, 
 drives a pinion of 20 leaves ; wherefore, 
 
 120 + 2-25 = 122-25 and 20 -f 1-5 = 21-5, 
 Then 122-25 : 12 : : 21-5 : 2-1 104 = the diameter of the 
 pinion. 
 
 (B) As the number of teeth in the wheel -f 2-25, 
 Is to the wheel's diameter, 
 
 So is 5 (teeth in wheel -f- leaves in pinion) 
 To the distance of their centres. 
 
 A wheel's diameter being 3*2 indies, number of teeth 96, 
 the leaves in the pinion being 8, then, 
 
 104 
 96 + 2-25, = 98-25 and (96 + 8) = - - = 52. 
 
WHEELS. 171 
 
 Hence, 98-25 : 3-2 : : 52 : 1-6936 = the distance which 
 the centres ought to have. 
 
 The strength of wheels is a subject which has occupied 
 the attention of the most eminent practical engineers, but 
 the rules they have given us are entirely empirical, that is 
 to say, the result of experiment. 
 
 The strength of the teeth will vary with the velocity of 
 the wheel, the strength in horses' power at a velocity of 
 2-27 feet per second, will be 
 
 breadth of the tooth x its thickness 8 _ 
 length of tooth"" 
 
 Required the strength in horses' power of a tooth 4 inches 
 broad, 1-3 inches thick, and 1-6 inches long, at a velocity 
 of 2'27 feet per second, here we have 
 
 . = 4-225, the horses' power at a velocity of 2*27. 
 
 The power at any other velocity may be found by pro- 
 portion, thus the same at 6 feet per second. 
 
 2-27 : 6 :: 4-225 : 11-1 = horses' power at a velocity 
 of 6 feet per second. 
 
 The thickness of a tooth x 2-1 = the pitch. 
 
 The thickness of a tooth x 1'2 = length. 
 
 Ex. The thickness of a tooth being 1 inches, then we 
 have 
 
 1-5 X 2-1 = 3-15 == the pitch. 
 1-5 X 1-2 = 1-8 = the length. 
 
 The breadth in practice is usually 2-5 times the pitch. 
 
 The arms of wheels generally taper from the axle to the 
 rim, because they sustain the greatest stress towards the 
 axle. It is obvious, that the more numerous the arms of a 
 wheel are, they each suffer a proportionately less strain, as 
 the resistance will be diffused over a greater number. 
 The power acting at the rim X length of arm 8 
 
 number of arms X 2656 X O'l 
 and cube of depth. 
 
 Ex. If the force acting at the extremity of the arm of a 
 wheel be 16 cwt.; the radius of the wheel being 5 feet, and 
 the number of arms 6, then we have 16 X 112 = 1792 Ibs. 
 = the force ; wherefore, 
 
 1792 X 5 3 224000 
 
 6 X 2656 X 0-1 = T59F6 = U ' breadth and CUbe f 
 depth. 
 
172 WHEELS. 
 
 Now, let us suppose that the breadth is two inches, we 
 must divide this 140 by it, whence, 
 
 140 
 
 - = 70, the cube of the depth, 
 2 
 
 and the cube root of 70 will be found = 4-121, which is 
 the depth of each arm. 
 
 When the depth at the axis is intended to be double of 
 the depth at the rim, the number 1640 is to be used in the 
 rule instead of 2656. 
 
 The tables which follow will be found in the highest 
 degree useful to the practical mechanic. 
 
WHEELS. 
 
 - - 
 
 173 
 
 TABLE OF PITCHES OF WHEELS IN ACTUAL USE IN MILL 
 WORK. 
 
 oo 
 
 
 
 GO 
 o o 
 
 c 
 
 c 
 
 
 
 _ 
 
 ?9 
 
 Water-wheel, 3 
 Steam-engine, 4 * 
 
 P BS 
 
 s s 
 
 7 7 
 
 =r ET 
 
 
 
 Cfc O 
 
 1 
 
 ~- 
 
 (1 
 2_ 
 
 Horse-mill, 
 Horse-mill, 
 
 Nature of the machinery. 
 
 v 
 
 w 
 
 fc> 
 
 "? 
 
 - 
 
 pp 
 
 . . 
 
 ** 
 
 a oi 
 
 
 
 
 
 *> CO 
 O5 tO 
 
 to to 
 
 - o 
 
 r 
 
 to to 
 
 c 
 
 O5 
 
 CO 
 
 
 
 Oi O 
 
 Vt 
 
 ^*^- 
 
 Horses' power. 
 
 CO CO 
 
 co to 
 
 i: 
 
 *0- 
 
 
 
 to to 
 
 to co 
 
 CO rft 
 
 u 
 
 to to 
 
 Pitch in inches. 
 
 *-W~ 
 
 
 
 
 
 
 
 00 05 
 
 O5 Oi 
 
 ,, 
 
 CO 
 w~ 
 
 r: 
 
 4us 
 
 . 
 
 *Hu^- 
 
 O5 01 
 
 w~ 
 
 ~ 
 
 ^ 4^ 
 
 Breadth of teeth in inches. 
 
 tO 05 
 
 ~ 5 
 
 
 
 <"i ri 
 
 :| 
 
 3 
 
 CO 
 
 ~ o 
 
 00 *. 
 
 to 
 
 2 
 
 H 
 
 CO CO 
 
 Teeth. 
 
 ' 
 
 CJ 
 
 - 
 
 00 
 
 tc 
 
 Ol 
 
 ^ 
 
 (C 
 
 v 
 
 to to 
 
 Oi GO 
 
 CO 
 
 to u 
 
 ->l- 
 
 **- 
 
 M- 
 
 
 CO CO 
 
 Revolves per 
 minute. 
 
 00 
 
 
 
 o 
 
 GO 
 
 o ^- 
 
 
 
 co to 
 
 X 
 
 c 
 
 CO 
 cot* 
 
 to 
 
 to 
 
 
 
 O5 
 
 to 
 
 
 
 O5 05 
 00 
 
 Diameter. 
 
 0< 
 
 *. 
 
 - CO 
 
 to oo 
 
 <i 
 
 c 
 
 t 
 00 
 
 ^ 
 
 
 rft tO 
 ^ 
 
 to 
 
 Ol CO 
 
 ^ 
 
 g 
 
 to to 
 
 o to 
 
 Teeth. 
 
 1 
 
 8 
 
 co to 
 
 CO O^ 
 <C X 
 
 Vi 
 
 ~l 
 
 O5 
 
 
 
 
 
 X 
 
 O5 
 GO tO 
 
 CO 
 
 2 
 
 to 
 
 
 
 
 co to 
 
 hi CO 
 CO 
 
 Revolves per 
 
 minute. 
 
 Ol 
 
 VI 
 
 Ol O5 
 
 ^s 
 
 
 co to 
 
 05 <! 
 
 <c 
 
 CO 
 
 
 cotvcotcj 
 
 O5 
 
 CO 
 O5 
 
 
 
 eetoi 
 
 *- Ol 
 
 Diameter. 
 
 
 
 to to 
 
 ic 
 
 co to 
 
 -. 
 
 Oi 00 
 
 ^ CO 
 
 
 
 ^, 
 
 Oi O 
 
 Breadth proportional to 10 
 horses' power,and present 
 velocity. 
 
 <l 00 
 
 Oi Oi 
 
 wl 
 
 ^ Oi 
 
 
 S3 
 
 00 
 
 
 5 
 
 O 
 
 h- 00 
 
 4 01 
 
 c: 
 
 Oi Ol 
 
 Vi 
 
 O5 Ol 
 
 rft CO 
 
 CO 
 
 CO 
 
 CO CO 
 
 
 
 CO CO 
 
 Present velocity per second, 
 
 T* 4 
 
 CO 
 
 CO 01 
 Oi -^ 
 
 ~ 
 
 Vi 
 
 00 
 CO 
 
 
 to to 
 
 Ol 
 
 GO CO 
 Ol 
 
 GO 
 
 O5 O 
 
 05 4^ 
 
 ., 
 
 
 
 C: 
 
 H- Ol 
 
 1 ' 
 GO ^ 
 
 01 Oi 
 
 ^ 
 
 to 
 
 Breadth in inches propor- 
 tional to 10 horses' power, 
 at 3 {. p. second, that is, 
 reducicg all the examples 
 to the same denom. 
 
 to -1 
 *J 
 
 05 C5 
 
 f~ 
 
 CD 
 
 CO 
 01 
 
 
 GO CO 
 
 00 H- 
 
 co - 
 
 CO 00 
 CO 
 
 O Oi 
 
 05 
 
 i^ 
 
 10 05 
 CO Ol 
 
 
174 WHEELS. 
 
 EXPLANATION OF REFERENCES, &c., IN THE FOREGO- 
 ING TABLE. 
 
 ' The only defect in this geering, which has been 16 years at work, 
 is the want of breadth in the spur-wheel and pinion : they ought to have 
 been 6 inches or more, as they will not last half so long as the bevel- 
 wheels and pinions connected with them. 
 
 8 Has been 16 years at work. The teeth are much worn. 
 
 3 Has been 16 years at work. This geering is found rather too nar- 
 row for the strain, as it is wearing much faster than the rest of the wheels 
 in the same mill. 
 
 4 and 5 This wheel has wooden teeth, and has been working for three 
 years. 
 
 6 This is a better pitch for the power than the following. 
 
 7 This pitch has been found to be too fine. 
 
 In the foregoing table the wheels are all reduced to what 
 may be called one denomination. 1st. By proportioning 
 all their breadths to what they should be, to have the same 
 strength, if the resistance were equal to the work of a steam 
 engine of ten horses' power. 2rf. By supposing their pitch- 
 lines all brought to the same velocity of three feet per se- 
 cond, and proportioning their breadth accordingly. This 
 particular velocity of three feet per second has been chosen, 
 because it is the velocity very common for overshot wheels. 
 Such cases as appear to have worn too rapidly, are marked, 
 which may tend to discover the limit in point of breadth. 
 
 TABLE OF PITCHES. 
 
 THE succeeding table of pitches of wheels was drawn up 
 in the following manner : The thickness of the teeth in 
 each of the lines is varied one-tenth of an inch. The 
 breadth of the teeth is always four times as much as their 
 thickness. The strength of the teeth is ascertained by 
 multiplying the square of their thickness into their breadth, 
 taken in inches and tenths, <fcc. The pitch is found by 
 multiplying the thickness of the teeth by 2*1. The num- 
 ber that represents the strength of the teeth, will also repre- 
 sent the number of horses' power, at a velocity of about 
 four feet per second. Thus, in the table where the pitch 
 is 3*15 inches, the thickness of the teeth 1*5 inches, and 
 the breadth 6 inches, the strength is valued at 13 horses' 
 power, with a velocity of four feet per second at the pitch- 
 line. 
 
WHEELS. 
 
 175 
 
 Table of Pitches of Wheels, with the breadth and thick- 
 ness of the teeth, and the corresponding number of 
 horses 1 power, moving at the pitch-line at the rate of 
 three, four, six, and eight feet, per second. 
 
 Pitch in 
 inches. 
 
 Thick- 
 ness of 
 teeth in 
 inches. 
 
 Breadth 
 of teeth 
 in inches. 
 
 Strength of 
 teeth, or no. 
 of horses' 
 power, at 4 
 feet per se- 
 cond. 
 
 Horses' 
 power at 3 
 feet per 
 second. 
 
 Horses' ' 
 power at 6 
 feet per 
 second. 
 
 Horses' 
 power at 8 
 feet per 
 second. 
 
 3-99 
 
 9 
 
 7-6 
 
 27-43 
 
 20-57 
 
 41-14 
 
 54-85 
 
 3-78 
 
 8 
 
 7-2 
 
 23-32 
 
 17-49 
 
 34-98 
 
 46-64 
 
 3-57 
 
 7 
 
 6-8 
 
 19-65 
 
 14-73 
 
 29-46 
 
 39-28 
 
 3-36 
 
 6 
 
 6-4 
 
 16-38 
 
 12-28 
 
 24-56 
 
 32-74 
 
 3-15 
 
 5 
 
 6- 
 
 13-5 
 
 10-12 
 
 20-24 
 
 26-98 
 
 2-94 
 
 4 
 
 5-6 
 
 10-97 
 
 8-22 
 
 16-44 
 
 21-92 
 
 2-73 
 
 3 
 
 5-2 
 
 8-78 
 
 6-58 
 
 13-16 
 
 17-34 
 
 2-52 
 
 2 
 
 4-8 
 
 6-91 
 
 5-18 
 
 10-36 
 
 13-81 
 
 2-31 
 
 1 
 
 4-4 
 
 5-32 
 
 3-99 
 
 7-98 
 
 10-64 
 
 2-1 
 
 1-0 
 
 4- 
 
 4-0 
 
 3-0 
 
 6-0 
 
 8-0 
 
 1-89 
 
 9 
 
 3-6 
 
 2-91 
 
 2-18 
 
 4-36 
 
 5-81 
 
 1-68 
 
 8 
 
 3-2 
 
 2-04 
 
 1-53 
 
 3-06 
 
 3-08 
 
 1-47 
 
 7 
 
 2-8 
 
 1-37 
 
 1-027 
 
 2-04 
 
 2-72 
 
 1-26 
 1-05 
 
 6 
 5 
 
 2-4 
 2- 
 
 86 
 5 
 
 64 
 375 
 
 1-38 
 75 
 
 1-84 
 
 1- 
 
HYDROSTATICS. 
 
 HYDROSTATICS comprehends all the circumstances of the 
 pressure of non-elastic fluids, as water, mercury, &c., and 
 of the weight and pressure of solids in them, when these 
 fluids are at rest. Hydrodynamics, on the other hand, refers 
 to the like circumstances of fluids in motion. 
 
 The particles of fluids are small and easily moved among 
 themselves. 
 
 Motion or pressure in a fluid is not in one straight line 
 in the direction of the moving force, but is propagated in 
 every direction, upwards, downwards, sidewise, and oblique. 
 
 From this property it is, that water will always tend to 
 come to a level, for if two cisterns be filled with water, the 
 one 10 feet deep, and the other 6, there will be more pres- 
 sure on the bottom of the 10 feet, than the 6 feet cistern ; 
 and, if the bottoms of both cisterns be on a level, and a 
 pipe be made to communicate between them, then the water 
 in the deep cistern will exert a greater pressure than that 
 in the other, and will cause the other to rise till their pres- 
 sures become equal, that is, when their surfaces come to a 
 level ; and this will hold true, however different the sur- 
 faces of the two cisterns may be in area. Hence, if water 
 be communicated through pipes between any number of 
 places, it will rise to the same level in all the places, whe- 
 ther the pipes be straight or bent, wide or narrow ; and any 
 fluid surface will rest only when that surface is level. 
 
 If a vessel contain water, the pressure on any point in 
 the sides or bottom, is proportional to the perpendicular 
 height of the fluid, above that point, in the side or bottom. 
 
 The pressure of a fluid upon a horizontal base, is equal 
 to the weight of a column of the fluid, of the area of the 
 base multiplied by the perpendicular height of the fluid, 
 whatever be the shape of the containing vessel : so that by 
 a long and very small pipe, the strongest casks or vessel* 
 
 176 
 
HYDROSTATICS. 177 
 
 may be burst asunder by the pressure of a very small quan- 
 tity of water. 
 
 Ex. Into a square box a tube is fixed, so that it shall 
 stand perpendicularly ; the area of the bottom of the box 
 is 9 square feet, and the height of the top of the tube above 
 the bottom of the box is 5 feet, and therefore the pressure 
 on the bottom is 5 X 9 = 45 cubic feet of water. Now 
 the weight of one cubic foot of water is found to be very 
 nearly 1000 ounces avoir., therefore, 45 X 1000 = 45,000 
 ounces, = 1 ton, 5 cwt. qrs. 12 Ibs. 8 oz. 
 
 The content in imperial gallons of any rectangular cis- 
 tern may be found thus, 
 
 cistern's content in cubic feet X 6*232, 
 or cistern's content in inches x "003607, 
 cistern's content in cubic inches _ 
 
 277-274 
 content in imperial gallons. 
 
 From these rules, which are approximate, it is easy to 
 see that of the three, the length, breadth, and depth of a 
 cistern, any two being given the third may be found, so 
 that the vessel shall contain any given number of gallons, 
 thus, 
 
 number of gallons 
 
 = the third 
 any two dimensions in feet X 6*232 
 
 dimension in feet. 
 
 For the content in gallons of a cylindric vessel, 
 
 diameter 8 X length X 4-895, 
 
 if the dimensions are in feet, but if the diameter be in 
 inches, use -034 instead of 4-895, and should both dimen- 
 sions be in inches, use -002832, or divide by 352-0362. 
 Also when the length and diameter are in feet, 
 
 number of gallons 
 
 - = diameter 
 length x 4-895 
 
 number of gallons __ 
 diameter 2 x 4-895 ~ 
 
 For a sphere we have diameter 3 X 3*263 content in 
 gallons, the diameter being in feet, but when the diameter 
 is in inches, use the number -001888. These rules may be 
 illustrated by the following examples. 
 
 The length of a cistern being 8 feet, its breadth 4-5, and 
 depth 3, then will its content be 8 X 4-5 X 3 = 108 cubic 
 
 , 
 
178 HYDROSTATICS. 
 
 feet, hence 108 X 6-232 = 673-06 gallons may be con- 
 tained in it. 
 
 It is required that a cistern should contain 1000 gallons, 
 but must not exceed 10 feet in length and 5 in breadth, 
 wherefore, 
 
 1000 1000 
 
 10 x 5 x 6-232 = liTFe = 
 
 A cylinder is 6-5 feet long and 3 inches diameter, there- 
 fore 6-5 X 3 a X -034 = 1-989 gallons that it will contain. 
 
 A pipe is to be made 20 inches in length, what must be 
 its diameter so that it shall contain 5 gallons ? 
 
 5 X 354 
 
 = 9-4 inches. 
 
 The quantity of pressure upon any plane surface on which 
 a fluid rests, is equal to the pressure upon the same plane 
 placed horizontally at the depth of its centre of gravity. 
 
 If any plane surface, either vertical or inclined, be piaced 
 in a fluid, the centre of pressure of the fluid on the plane 
 is at the centre of percussion, the surface of the fluid being 
 supposed the centre of motion. Thus it will be found that 
 in a cistern whose sides are vertical, the centre of pressure 
 on the sides is two-thirds from the top, which is also the 
 centre of percussion. 
 
 To ascertain the whole pressure on a flood-gate, or other 
 surface exposed to the pressure of water, a very near ap- 
 proach to the truth may be made by these rules the breadth 
 and depth being taken in feet. 
 
 31-25 X breadth X depth 8 = pressure in Ibs. 
 
 -2727 X breadth X depth 9 = pressure in cwts. 
 
 If the gate be wider at the top than bottom, 
 
 /breadth at top breadth at bottomx 
 31-25 X (- -) + breadth 
 
 \ o / 
 
 at bottom X depth* = pressure in Ibs. ; and "2727, used in- 
 stead of 31-25, will give the pressure in cwts., nearly. 
 
 Exam. What is the pressure upon a rectangular flood- 
 gate, whose breadth is 25 feet, and depth below the surface 
 of the water 12 feet? 
 
 31-25 X 25 X 12 3 = 112500 Ibs. pressure. 
 
 If the breadth at top be 28 feet, that at bottom 22, and 
 the height 12, as before, then, 
 
HYDROSTATICS. 179 
 
 OQ OO 
 
 31-25 X L_ 4. 22 X 12 a = 108000 Ibs. pressure. 
 
 3 - > 
 
 The weight of a cubic foot of river water is about J T of 
 a cwt. The pressure at the depth of 30 feet is about 13 
 Ibs. to the square inch. And at the depth of 36 feet the 
 pressure is about 1 ton to the square foot. The weight of 
 an imperial gallon of water is about 10 Ibs. 
 
 Ex. What is the pressure at the depth of 120 feet on a 
 square inch? 
 
 30 : 120 : : 13 : 52 = the pressure, and at the same 
 depth, 36 : 120 : : 1 : 3j tons on the square foot. 
 
 It is not difficult to see that the strength of the vessels 
 or pipes which contain or convey water must be regulated 
 according to the pressure. 
 
 The thickness of pipes to convey water must vary in pro- 
 portion to the height of the head of water X diameter of 
 pipe -T- the cohesion of one square inch of the material of 
 which the pipe is composed. 
 
 By experiment it has been found that a cast iron pipe 1 5 
 inches diameter and | of an inch thick of metal, will be 
 sufficiently strong for a head 600 feet high. A pipe of oak 
 15 inches diameter and 2 inches thick, is sufficient for a 
 head of 180 feet. When the material is the same, the 
 thickness of the material varies with the height of head X 
 diameter of pipe. 
 
 Ex. What must be the thickness of a cast iron pipe 10 
 inches diameter for a head of 360 feet ? 
 
 360 X 10 X I 3 , . . A . i 
 ~600 X 15 = * f an mch thlckness - 
 
 If the same pipe is to be made of oak, then 
 
 360 x 10 X 2 
 
 180X15 = 2f thickness m inches. 
 
 When conduit pipes are horizontal and made of lead, 
 their thicknesses should be 2, 3, 4, 5, 6, 7, 8 lines, when 
 the diameters are 1, 1|, 2, 3, 4|, 6, 7 inches and when 
 the pipes are made of iron, their thickness should be 1,2, 
 3, 4, 5, 6, 7, 8 lines, when their diameters are 1, 2, 4, 6, 
 8, 10, 12. 
 
 The plumber should be aware that the tenacity of lead is 
 increased four times, by adding 1 part of zinc to 8 of lead. 
 
 When the vessel which contains the water has, besides the 
 

 180 HYDROSTATICS. 
 
 pressure arising from the weight of the water, to resist an 
 additional pressure exerted by some force on the water, as 
 in Bramah's press, where the pressure exerted by means of 
 a force pump on the water in a small tube, which commu- 
 nicates with a large cylinder, is, by the principles stated 
 before in this chapter, multiplied on the piston of the 
 cylinder as often as the area of the tube is contained in the 
 area of the piston of the cylinder. If the area of the tube 
 be one inch, the area of the piston 92 inches, and if the 
 pressure on the water in the tube be 16 Ibs., then the pres- 
 sure on the piston will be 16 X 92 = 1472 Ibs. 
 
 The annexed figure and description taken from the 
 Popular Encyclopedia, will give a clearer idea of the 
 operation of this press. " Here AB is the bottom of a 
 hollow cylinder, into which a piston 
 P is accurately fitted. Into the bot- 
 tom of this cylinder there is intro- 
 duced a pipe C leading from the 
 forcing pump D ; water is supplied 
 to this pump by a cistern below, from 
 which the pipe E is led, being fur- 
 nished with a valve opening upwards 
 where it is joined to the pump barrel. 
 Where the pipe C enters into the pump barrel there is also 
 a valve opening outwards into the pipe; consequently, 
 when the piston D rises, this valve shuts, and the valve at 
 the cistern pipe opens, and the fluid rises into the pump 
 barrel. The top of the piston rod, P, is fixed in the bottom 
 of the board on which the goods are laid, and when the 
 piston rises the goods are pressed against the top of the 
 framing of the machine. When the piston begins to de- 
 scend, the cistern valve shuts, and the water is forced 
 through the pipe C into the large cylinder AB ; and by the 
 law of fluids before alluded to, whatever pressure be exerted 
 by the piston D on the surface of the water in the pump, 
 will be repeated on the piston of the large cylinder AB as 
 many times as the area of the small piston D is contained 
 in the area of the large piston AB ; that is, if the area of 
 the pump-piston were one square inch, and that of the 
 cylinder 100 inches, and if the piston were forced down 
 with a pressure of 10 Ibs., then the whole pressure on the 
 bottom of the piston AB will be 10 times 100, that is, 1000 
 Ibs. When the page which is now before the reader was taken 
 
HYDROSTATICS. 181 
 
 wet off the types, it was all deeply indented in consequence 
 of the pressure of the printing press ; but after being dried, it 
 was subjected to the action of Bramah's press, by which 
 process, as will be seen, these indentations have been nearly 
 obliterated. In the press by which this has been accom- 
 plished, the pump has a bore of three-fourths of an inch in 
 diameter, and the cylinder one of eight inches, their areas 
 are therefore to one another, as 9- 16th to 64, (the squares 
 of the diameters,) that is, as 1 is to 113; hence if the 
 pressure upon the pump-cylinder be 56 Ibs., (which can be 
 easily effected by boys,) the pressure upon the piston of 
 the large cylinder will be 56 X 113, that is, 6'328 Ibs. 
 This astonishing power has also been employed in the 
 construction of cranes." 
 
 To ascertain the thickness of metal necessary for the 
 cylinder of such presses, this rule will serve : 
 
 pressure per square inch X radius of cylinder 
 cohesion of the metal per square in. pressure 
 thickness of metal necessary for the cylinder to sustain the 
 pressure. The pressure being in Ibs. 
 
 NOTE. The cohesive force of a square inch of cast iron 
 is 18,000 Ibs. 
 
 What is the thickness of metal in a cast iron press whose 
 cylinder is 12 inches diameter, the pressure being 1*5 tons 
 on the circular inch ? 
 
 A circular inch is to a square inch as 0-7854 to 1, there- 
 fore 1'5 tons per circular inch =1*9 per square inch = 
 4256 Ibs. 
 
 Here we have 
 
 4256 X 6 
 18000 4256 ~ 
 
 What is the thickness of metal in a press of yellow brass, 
 whose cylinder is 10 inches in diameter, and which is in- 
 tended for a pressure of 2 tons to the square inch ? 
 
 The cohesive force of yellow brass being 17958, we have 
 by the same rule, 
 
 2 tons = 4480 Ibs. 
 
 = 1'66 inches, the thickness of the 
 
 17958 4480 
 
 metal. 
 
 16 
 
182 HYDROSTATICS. 
 
 When the diameter remains the same, the thickness ap- 
 pears to increase with the increase of pressure. 
 
 FLOATING BODIES. 
 
 WHEN any body is immersed in water, it will, if it be of 
 the same density of the water, remain suspended in any 
 place ; but if it be more dense than the water it will sink, 
 and if less dense it will float. 
 
 Bodies immersed and suspended in a fluid lose the weight 
 of an equal bulk of the fluid, and the fluid acquires the 
 weight that the body loses : also, bodies floating on a fluid 
 lose weight in proportion to the quantity of fluid they dis- 
 place. 
 
 When a body floats upon the water, it will sink in the 
 water till the water which is displaced be equal in weight 
 to the weight of the body. 
 
 When a body floats on a fluid, it will only be at rest 
 when the centre of gravity of the body and the centre of 
 gravity of the displaced fluid are in the same vertical line ; 
 and the lower the centre of gravity is, the more stable will 
 the body be. 
 
 The buoyancy of casks, or the load which they will carry 
 without sinking, may be estimated at about 10 Ibs. to the 
 ale gallon, or 282 cubic inches of the content of the cask. 
 
 SPECIFIC GRAVITY. 
 
 SPECIFIC gravity is the relative weight of any body of a 
 certain bulk, compared with the weight of some body taken 
 as a standard of the same bulk. The standard of compari- 
 son is water ; one cubic foot of which is found to weigh 
 1000 ounces avoir, at a temperature of 60 of Fahrenheit, 
 so that the weight expressed in ounces of a cubic foot of any 
 body, will be its specific gravity, that of water being 1000. 
 
 To determine the specific gravity. 
 
 Jf a body be a solid heavier than water Weigh it first 
 carefully in air, and note this weight; then immerse it in 
 water, and in this state note its weight. Then divide the 
 body's weight in air by the difference of the weights in air 
 and water, the quotient is the specific gravity. 
 
SPECIFIC GRAVITY. 
 
 183 
 
 If a body be a solid lighter than water Tie a piece of 
 metal to it, so that the compound may sink in water then 
 to the weight of the solid itself in air, add the weight of the 
 metal in water, and from this sum subtract the weight of the 
 compound in water, which difference makes a divisor to a 
 dividend, which is the weight of the solid in air, then the 
 quotient will be the specific gravity. 
 
 If the body be a fluid Take a solid, whose specific 
 gravity is known, and that will sink in the fluid ; then take 
 the difference of the weights of the solid in and out of the 
 fluid, and multiply this difference by the specific gravity 
 of the solid ; then, this product divided by the weight of 
 the body in air, will give the specific gravity of the fluid. 
 
 On these principles there has been constructed tables of 
 specific gravities, one of which we insert. The column, 
 specific gravity, may be taken to represent the weight of 
 a cubic foot. 
 
 TABLE OF SPECIFIC GRAVITIES. 
 
 METALS. 
 
 Specific Graity. 
 
 Arsenic, 5763 
 
 Cast antimony, 6702 
 
 Cast zinc, 7190 
 
 Castiron, 7207 
 
 Cast tin, 7291 
 
 Bar iron 7788 
 
 Cast nickel, 7807 
 
 Cast cobalt, 7811 
 
 Hard steel, 7816 
 
 Soft steel, 7833 
 
 Cast brass, 8395 
 
 Cast copper 8788 
 
 Specific Gravity. 
 
 Cast bismuth, 9822 
 
 Cast silver, 10474 
 
 Hammered silver, ... 10510 
 
 Cast lead, 11352 
 
 Mercury, 13568 
 
 Jewellers' gold, 15709 
 
 Gold coin, 17647 
 
 Cast gold, pure, 19258 
 
 Pure gold, hammered, 19361 
 
 Platinum, pure, 19500 
 
 Platinum, hammered, 20336 
 Platinum wire,.. ,. 21041 
 
 STONES, EARTHS, ETC. 
 
 Brick, 2000 
 
 Sulphur, 2033 
 
 Stone, paving, 2416 
 
 Stone, common, 2520 
 
 Granite, red, 2654 
 
 Glass, green, 2642 
 
 Glass, white, 2892 
 
 Glass, bottle, 2733 
 
 Pebble, 2664 
 
 Slate, 2672 
 
 Marble, 2742 
 
 Chalk, 2784 
 
 Basalt, 2864 
 
 Hone, white razor, . . . 2876 
 Limestone, .. .. 3179 
 
184 
 
 HYDROSTATICS. 
 
 RESINS, ETC. 
 
 Specific Gravity. Specific Gravity 
 
 Wax, 897 1 Bone of an ox, 1659 
 
 Tallow, 945 | Ivory, 1822 
 
 sur- 
 
 Air at the earth's 
 
 face, If 
 
 Oil of turpentine, 870 
 
 Olive oil, 915 
 
 LIQUIDS. 
 
 Distilled water, 1000 
 
 Sea water, 1028 
 
 Nitric acid, 1218 
 
 Vitriol.. 1841 
 
 WOODS. 
 
 Cork, 246 
 
 Poplar 383 
 
 Larch, 544 
 
 Elm and new English fir, 556 
 Mahogany, Honduras,- -560 
 
 Willow, 585 
 
 Cedar, 596 
 
 Pitch pine, 560 
 
 Pear tree, 661 
 
 Walnut, 671 
 
 Fir, forest, 694 
 
 Elder, 695 
 
 Beech, 696 
 
 Cherry tree, 715 
 
 Teak, 745 
 
 Maple and Riga fir, 750 
 Ash and Dantzic oak, -*760 
 
 Yew r Dutch, 788 
 
 Apple tree, 793 
 
 Alder, 800 
 
 Yew, Spanish, 807 
 
 Mahogany, Spanish, 852 
 
 Oak, American, 872 
 
 Boxwood, French, 912 
 
 Logwood, 913 
 
 Oak, English, 970 
 
 Do. sixty years cut," 1170 
 
 Ebony, 1331 
 
 Lignumvitae, 1333 
 
 Specific gravity of gases, that of atmospheric air being 
 
 : 1. 
 
 Hydrogen, 0-0694 
 
 Carbon, 04166 
 
 Steam of water, 0-481 
 
 Ammonia, 0-5902 
 
 Carburetted hydrog., 0-9722 
 
 Azote, 0-9723 
 
 Oxygen, 1-1111 
 
 Muriatic acid, 1-2840 
 
 NOTE. The specific gravity of atmospheric air at a tem- 
 perature of 60 Fah. and barometric column 30 inches is 
 1"^ according to M. Arago, and in round numbers we may 
 regard water as 825 times heavier than air. 
 
 Carbonic acid, 1-5277 
 
 Alcohol vapour, 1-6133 
 
 Chlorine, 2-500 
 
 Nitrous acid, 2-638 
 
 Sulphuric acid, 2*777 
 
 Nitric acid, 3-75 
 
 Oil of turpentine, 5'013 
 
SPECIFIC GRAVITY. 185 
 
 The preceding table will be found of the utmost use in 
 determining the weight and magnitude of bodies. 
 To find the magnitude of a body from its weight : 
 
 weight of body in ounces ' , . f 
 
 : = content m cubic feet. 
 its specific grav. in table 
 
 How many cubic feet are in one ton of mahogany? 
 Here 20 x 112 x 16 = 35840 ounces in a ton ; therefore, 
 
 35840 .. , 
 
 = 64 cubic feet. 
 5 60 
 
 Had the timber been fir, then 
 
 ?|^-= 64-46 cubic feet. 
 556 
 
 Or English oak : 
 
 35840 
 
 - = 36-94 cubic feet. 
 970 
 
 To find the weight of a body from its bulk : 
 
 cubic feet X specific gravity = weight in ounces. 
 What is the weight of a log of larch, 14 feet long, 2j 
 broad, and 1* thick? 
 
 Here 2-5 X 1'25 X 14 = 43-75 ; then, 
 43-75 x 544 = 23800 ounces = 13 cwt. 1 qr. 3 Ibs. 8 oz. 
 What is the weight of a cast iron ball, 2 inches diameter? 
 Here the content of the globe will be 2 8 X '5236 = 4-1888 
 cubic inches = -00242 feet, and then -00242 X 7271 = 
 17-29 ounces = 1-08 Ibs. 
 
 A bullet of lead of the same magnitude would be *00242 
 X 11344 = 27-44 ounces = 1'71 Ibs. 
 
 If we wish to determine the quantity of two ingredients 
 in a compound which they form, 
 
 Let H be the weight of the heavy body. 
 A, its specific gravity. 
 L, the weight of the lighter body. 
 /, its specific gravity. 
 C, the weight of the compound. 
 c, its specific gravity. 
 Then, 
 
 (A_ /) X c 
 also, 
 
 (h c) x / r 
 
 77 - 77 - X U 
 
 (h /) X c 
 
 16* 
 
 
186 
 
 HYDROSTATICS. 
 
 Ex. A mixture of gold and silver weighed 170 Ibs. and 
 its specific gravity was 15630 ; hence 
 
 h (by the table) = 19326. / = 10744 
 c = 15630 C = 170 Ibs. wherefore, by the rule, 
 
 (19326 15630) X 10744 39709824 
 
 (19326 10744) X 15630 > "134136660 
 
 = -296 x 170 = 50-32 Ibs. of gold ; 
 consequently there will be 170 50-32 = 119-68 Ibs. of 
 silver. 
 
 The weight of bodies their magnitudes and also their 
 quantities in a compound, may thus be found by means of 
 a table of specific gravities ; and for the more expeditious 
 calculation in practice we add the following memoranda : 
 
 430-25 cubic inches of cast iron weigh one cwt., as also 
 397-60 of bar iron, 368-88 of cast brass, 352-41 of cast 
 copper, and 372*8 of cast lead. 
 
 14-835 cubic feet of common paving stone weigh one ton, 
 as also 14-222 of common stone, 13-505 of granite, 13-070 
 of marble, 64-46 of elm, 64 of Honduras mahogany, 51-65 
 of fir, 51-494 of beech, 42-066 of Spanish mahogany, and 
 36-205 of English oak. 
 
 For wrought iron square bars, allow 100 inches in length 
 of an inch square to a quarter of a cwt. 
 
 A similar cast iron bar would require 9 feet in length for 
 a quarter of a cwt. One foot in length of an inch square 
 bar weighs 3^ Ibs. also the breadth and thickness being 
 taken in, th of an inch, and the length in feet. 
 
 length x breadth X thickness X 7 
 
 -- - = the weigh! 
 
 in avoirdupois pounds. 
 
 Ex. An iron bar 10 feet long, 3 inches broad, and 2 
 thick. Here 3 inches = 24, and 2 5 = 20-8ths ; therefore, 
 
 233 lb8 ' 
 
 For the weight of a cast iron pipe : 
 
 The length being taken in feet, the diameter and thick 
 ness of metal in inches, then we have 
 
 0-0876 X length x thickness x (inner diameter 4 
 thickness) = the weight in cwts. 
 
 For a leaden pipe the rule is, 
 
 0-1382 X length x thickness x (inner diameter -j 
 thickness) = the weight in cwts. 
 
SPECIFIC GRAVITY. 187 
 
 NOTE. The weight of a cast iron pipe is to a leaden 
 pipe of the same dimensions nearly as 7 is to 11. 
 
 Ex. If the inner diameter or bore of a cast iron pipe 
 be 3 inches, and its thickness | of an inch ; what is the 
 weight of 14 feet of it? 
 
 0876 X 14 X * X (3 + i) - '99645 cwt. = 3 qrs. 
 27 Ibs. 9 oz. 
 
 A leaden pipe is 12 feet long, the bore is 4 inches, and 
 thickness of metal of an inch ; therefore, 
 1382 X 12 X I X (4 + $) = 1-762 cwt. = Icwt. 3 qrs. 1 Ib. 
 
 For the weight of the rim of a fly-wheel. Let D be the 
 diameter of the fly, exclusive of the rim, taken in inches ; 
 then take the difference of this and the diameter of the fly, 
 including the rim, and call this difference d, T being the 
 thickness of the rim of the fly, from side to side, then we 
 have 
 0073 x T x d X (D -|- d) = the weight of the rim in cwts. 
 
 Ex. If the interior diameter of the fly be 100 inches 
 = D, half the difference of the exterior and interior dia- 
 meter 5 = rf, hence if the rim is 10 inches broad, as the 
 exterior diameter will then be 110, and let the thickness 
 of the rim be 4 inches = T, then, 
 
 0073 X 4 X 5 X (100 -f 5) = 15-33 cwts. 
 
188 
 
 HYDROSTATICS. 
 
 TABLE A. 
 
 Of the weight of I lineal foot of Swedish iron, of all breadths and 
 thicknesses i from 1 tenth of an inch to 1 inch, in pounds and deci- 
 mal parti. 
 
 1 
 
 wi 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 *0 
 304 
 
 1-0 
 
 lOUnof 
 iKta. 
 
 -068 
 
 101 
 
 135 
 
 169 
 
 203 
 
 237 
 
 270 
 
 338 
 
 1 
 
 
 135 
 
 203 
 
 270 
 
 338 
 
 406 
 
 473 
 
 541 
 
 608 
 
 676 
 
 2 
 
 
 304 
 
 406 
 
 507 
 
 609 
 
 710 
 
 811 
 
 913 
 
 1-014 
 
 3 
 4 
 
 
 541 
 
 676 
 
 811 
 
 947 
 
 1*069 
 
 1-217 
 
 1-352 
 
 
 845 
 
 1-014 
 
 1-183 
 
 1-352 
 
 1-521 
 1-826 
 SHISO 
 
 1-690 
 
 5 
 
 
 1-217 
 
 1-490 
 1-657 
 
 1-623 
 H393 
 
 9-099 
 
 2-367 
 
 6 
 
 7 
 
 
 2-164 
 
 2-434 
 
 2-657 
 
 8 
 
 
 2-739 
 
 3-043 
 
 9 
 
 
 3-381 
 
 1-0 
 
 TABLE B. 
 
 Of the weight of 1 lineal foot of Swedish iron, of all breadths and 
 thicknesses, from 1 inch to 6 inches, in pounds and decimal parts. 
 
 1 li 1J Ij 
 
 2 
 
 i 
 
 3 
 
 N 
 
 4 
 
 6 
 
 6 
 
 IB. 
 
 3-38 4-23 5-07 5-91 
 
 6-76 
 
 8-45 
 
 10-14 
 
 11-83 
 
 13-52 
 
 16-91 
 
 20-29 
 
 1 
 
 6-296-34 7-40 
 
 8-45 
 
 10-56 
 
 12-68 
 
 14-79 
 
 16-91 
 
 21-13 
 
 25-36 
 
 U 
 
 7-60 8-87 
 
 10-14 
 
 12-67 
 
 15-21 
 
 17-75 
 
 20-29 
 
 25-36 
 
 30-43 
 
 U 
 
 1 10-35 
 
 11-83 
 
 14-78 
 
 17-75 
 
 20-71 
 
 23-67 
 
 29-58 
 
 35-50 
 
 1| 
 
 
 13-52 
 
 16-91 
 
 20-29 
 
 23-67 
 
 27-05 
 
 33-81 
 
 40-51 
 
 2 
 
 
 
 21-13 
 
 25-36 
 
 29-58 
 
 33-81 
 
 42-26 
 
 50-72 
 
 2* 
 
 
 
 
 30-43 
 
 35-50 
 
 40-57 
 
 50-72 
 
 60-86 
 
 3 
 
 
 
 
 
 41-42 
 
 I,-:'! 
 
 59-lfi 
 
 71-00 
 
 3* 
 
 
 
 
 
 
 54-10 
 
 67-62 
 
 81-14 
 
 4 
 
 
 
 
 
 
 
 84-52 
 
 101-44 
 
 5 
 
 
 
 
 
 
 
 
 12172 
 
 6 
 
SPECIFIC GRAVITY. 
 
 189 
 
 TABLE C. 
 
 Of the weight of 1 superficial foot of Swedish iron plate from 100/A 
 part of an inch thick to one inch. 
 
 Thickness in 
 parts of an inch. 
 
 Weight in Ibe. 
 
 Thickness in 
 parts of an inch. 
 
 Weight in Iba. 
 
 01 
 
 406 
 
 1 
 
 4-057 
 
 02 
 
 811 
 
 2 
 
 8-114 
 
 03 
 
 1-217 
 
 3 
 
 12-172 
 
 04 
 
 1-623 
 
 4 
 
 16-232 
 
 05 
 
 2-029 
 
 5 
 
 20-286 
 
 06 
 
 2-434 
 
 -6 
 
 24-344 
 
 07 
 
 2-840 
 
 7 
 
 28-401 
 
 08 
 
 3-246 
 
 8 
 
 32-458 
 
 09 
 
 3-651 
 
 9 
 
 36-516 
 
 10 
 
 4-057 
 
 1- 
 
 40-573 
 
 TABLE D. 
 
 Of Multipliers for the other Metals, wherebv their weights may be 
 found from the above Tables. 
 
 Metal., 
 
 Multi- 
 pi UTS. 
 
 Metals. 
 
 Multi- 
 pliers. 
 
 Platinum, laminated 
 - nurifipd 
 
 2-846 
 2-503 
 2-486 
 2-47 
 1-457 
 1-350 
 1-344 
 1-136 
 1-132 
 
 Copper, cast 
 
 1-128 
 1-096 
 1-080 
 1-003 
 1- 
 980 
 925 
 960 
 937 
 
 Brass wire 
 
 Pure gold, hammered 
 oust 
 
 in t cast 
 
 Steel 
 
 
 Iron, Swedish 
 
 Pure silver, hammered 
 cast * * 
 
 , British 
 
 - - cast 
 
 
 Pewter 
 
 , hammered . . 
 
 Tin, cast 
 
 
190 
 
 HYDROSTATICS. 
 
 TABLE E. 
 
 Table of the weight of one square foot of different metals in variant 
 thicknesses, in pounds ana decimal parts. 
 
 Tbickne 
 in MOH f 
 an inch. 
 
 Mai. Iron. 
 Swed. 
 
 Mai. Iron, 
 EngUaT 
 
 Cut Iron. 
 
 Copper. 
 
 Bra*. 
 
 Lad. 
 
 1 
 
 2-535 
 
 2-486 
 
 2-345 
 
 2-860 
 
 2-738 
 
 3-693 
 
 2 
 
 5-070 
 
 4-972 
 
 4-690 
 
 5-720 
 
 5-476 
 
 7-386 
 
 3 
 
 7-605 
 
 7-458 
 
 7-035 
 
 8-580 
 
 8-214 
 
 11-079 
 
 4 
 
 10-140 
 
 9-944 
 
 9-380 
 
 11-440 
 
 10-952 
 
 14-772 
 
 5 
 
 12-675 
 
 12-130 
 
 11-725 
 
 14-300 
 
 13-690 
 
 18-465 
 
 6 
 
 15-216 
 
 14-916 
 
 14-670 
 
 17-160 
 
 16-428 
 
 22-158 
 
 7 
 
 17-851 
 
 17-402 
 
 16-415 
 
 20-020 
 
 19-166 
 
 25-851 
 
 8 
 
 20-280 
 
 19-888 
 
 18-760 
 
 22-880 
 
 21-904 
 
 29-544 
 
 9 
 
 22-815 
 
 22-774 
 
 21-105 
 
 25-740 
 
 24-642 
 
 33-237 
 
 10 
 
 25-350 
 
 24-260 
 
 23-450 
 
 28-600 
 
 27-380 
 
 36-930 
 
 11 
 
 27-885 
 
 26-746 
 
 25-795 
 
 31-460 
 
 30-118 
 
 40-623 
 
 12 
 
 30-410 
 
 29-232 
 
 28-140 
 
 34-320 
 
 32-856 
 
 44-316 
 
 13 
 
 32-945 
 
 31-718 
 
 30-485 
 
 37-180 
 
 35-594 
 
 48-009 
 
 14 
 
 35-480 
 
 34-204 
 
 32-880 
 
 40-040 
 
 38-332 
 
 51-702 
 
 15 
 
 38-015 
 
 36-690 
 
 35-225 
 
 42-900 
 
 41-170 
 
 55-405 
 
 16 
 
 40-550 
 
 39-176 
 
 37-570 
 
 45-760 
 
 43-908 
 
 59-098 
 
 TABLE F. 
 
 Table of the iveight of 1 foot in length of malleable Iron rod, from 
 one-fourth to 6 inches diameter. 
 
 Diam. 
 
 Wrifht 
 
 Diam. 
 
 Weight 
 
 Diam. 
 
 Weight. 
 
 Dum. 
 
 Wright 
 
 Inch. 
 
 IU. 
 
 Inch. 
 
 Ite. 
 
 loch. 
 
 Ibt. 
 
 Inch. 
 
 Its. 
 
 * 
 
 -163 
 
 U 
 
 8-01 
 
 3* 
 
 27-65 
 
 42 
 
 59-06 
 
 1 
 
 368 
 
 u 
 
 9-2 
 
 31 
 
 29-82 
 
 41 
 
 62-21 
 
 * 
 
 654 
 
 2 
 
 10-47 
 
 3* 
 
 32-07 
 
 5 
 
 65-45 
 
 I 
 
 1-02 
 
 i 
 
 11-82 
 
 31 
 
 34-4 
 
 5* 
 
 HS-7C, 
 
 n 
 
 4 
 
 1-47 
 
 2* 
 
 13-25 
 
 31 
 
 36-81 
 
 6* 
 
 72-16 
 
 I 
 
 2 
 
 21 
 
 14-76 
 
 3| 
 
 39-31 
 
 51 
 
 75-63 
 
 1 
 
 2-61 
 
 81 
 
 16-36 
 
 4 
 
 41-89 
 
 5* 
 
 79-19 
 
 u 
 
 3-31 
 
 21 
 
 18-03 
 
 4* 
 
 44-54 
 
 51 
 
 82-83 
 
 H 
 
 4-09 
 
 21 
 
 19-79 
 
 4| 
 
 47-28 
 
 51 
 
 86-56 
 
 11 ' 
 
 4-94 
 
 22 
 
 21-63 
 
 41 
 
 50-11 
 
 51 
 
 90-36 
 
 1ft 
 
 5-89 
 
 3 23-56 
 
 4i 
 
 53-01 
 
 6 
 
 94-25 
 
 if 
 
 6-91 
 
 3$ I 25-56 
 
 41 
 
 56 
 
 
 
SPECIFIC GRAVITY. 
 
 191 
 
 TABLE G. 
 
 Table of the weight of cast iron Pipes, 1 foot long, and of different 
 thicknesses. 
 
 Diam. o 
 bore. 
 
 * 
 
 Inch. 
 
 1 
 
 Inch. 1 
 
 i 
 
 Inch. 
 
 i 
 
 Inch. 
 
 1 
 
 Inch. 
 
 i 
 
 Inch. 
 
 1 
 
 Inch. 
 
 Inch. 
 
 Ib. 
 
 Ib. 
 
 Ib. 
 
 Ib. 
 
 Ib. 
 
 Ib. 
 
 Ib. 
 
 1 
 
 3-06 
 
 5-06 
 
 7-36 
 
 9-97 
 
 12-89 
 
 16-11 
 
 19-63 
 
 ti 
 
 3-68 
 
 5-98 
 
 8-59 
 
 11-51 
 
 14-73 
 
 18-25 
 
 22-09 
 
 H 
 
 4-29 
 
 6-9 
 
 9-82 
 
 13-04 
 
 16-56 
 
 20-4 
 
 24-54 
 
 il 
 
 4-91 
 
 7-83 
 
 11-05 
 
 14-57 
 
 18-41 
 
 22-55 
 
 27- 
 
 2 
 
 5-53 
 
 8-75 
 
 12-27 
 
 16-11 
 
 20-25 
 
 24-7 
 
 29-45 
 
 2| 
 
 6-14 
 
 9-66 
 
 13-5 
 
 17-64 
 
 22-09 
 
 26-84 
 
 31-85 
 
 2 
 
 6-74 
 
 10-58 
 
 14-72 
 
 19-17 
 
 23-92 
 
 28-93 
 
 34-36 
 
 21 
 
 7-36 
 
 11-5 
 
 15-95 
 
 20-7 
 
 25-71 
 
 31-14 
 
 36-81 
 
 3 
 
 7-98 
 
 12-43 
 
 17-18 
 
 22-19 
 
 27-62 33-29 
 
 39-28 
 
 3* 
 
 8-59 
 
 13-34 
 
 18-35 
 
 23-78 
 
 29-45 
 
 35-44 
 
 41-72 
 
 3d 
 
 9-2 
 
 14-21 
 
 19-64 
 
 25-31 
 
 31-3 
 
 37-58 
 
 44-18 
 
 31 
 
 9-76 
 
 15-19 
 
 20-86 
 
 26-85 
 
 33-13 
 
 39-73 
 
 46-63 
 
 4 
 
 10-44 
 
 16-11 
 
 22-1 
 
 28-38 
 
 34-98 
 
 41-88 
 
 49-1 
 
 4* 
 
 11-1 
 
 17-08 
 
 23-37 
 
 29-97 
 
 36-87 
 
 44-08 
 
 51-6 
 
 4 
 
 11-66 
 
 17-94 
 
 24-54 
 
 31-44 
 
 38-65 
 
 46-17 
 
 54- 
 
 41 
 
 12-27 
 
 18-87 
 
 25-77 
 
 32-98 
 
 40-5 
 
 48-32 
 
 56-45 
 
 5 
 
 12-80 
 
 19-78 
 
 26-99 
 
 34-51 
 
 42-33 
 
 50-46 
 
 59- 
 
 51 
 
 13-5 
 
 20-71 
 
 28-23 
 
 36-05 
 
 44-18 
 
 52-62 
 
 61-36 
 
 5 
 
 14-11 
 
 21-63 
 
 29-45 
 
 37-58 
 
 46-02 
 
 54-76 
 
 63-81 
 
 51 
 
 14-73 
 
 22-55 
 
 30-68 
 
 39-12 
 
 47-86 
 
 56-91 
 
 66-27 
 
 6 
 
 15-34 
 
 23-47 
 
 31-91 
 
 40-65 
 
 49-7 
 
 59-06 
 
 68-73 
 
 61 
 
 15-95 
 
 24-39 
 
 33-13 
 
 42-18 
 
 51-54 
 
 61-21 
 
 72- 
 
 6 
 
 16-57 
 
 25-31 
 
 34-36 
 
 43-72 
 
 53-39 
 
 63-36 
 
 73-41 
 
 61 
 
 17-18 
 
 26-2S 
 
 35-59 
 
 45-26 
 
 55-23 
 
 65-28 
 
 76-1 
 
 7 
 
 17-79 
 
 27-15 
 
 36-82 
 
 46-79 
 
 56-84 
 
 67-65 
 
 78-53 
 
 7| 
 
 18-41 
 
 28-08 
 
 38-05 
 
 48-1 
 
 58-91 
 
 69-79 
 
 81' 
 
 71 
 
 19-03 
 
 29- 
 
 39-05 
 
 49-86 
 
 60-74 
 
 71-95 
 
 83-45 
 
 71 
 
 19-64 
 
 29-69 
 
 40-5 
 
 51-38 
 
 62-59 
 
 74-09 
 
 86- 
 
 8 
 
 20-02 
 
 30-83 
 
 41-71 
 
 52-92 
 
 64-42 
 
 76-23 
 
 88-35 
 
 8* 
 
 20-86 
 
 31-74 
 
 42-95 
 
 54-45 
 
 66-26 
 
 78-38 
 
 90-81 
 
 81 
 
 21-69 
 
 32-9 
 
 44-4 
 
 56-21 
 
 68-33 
 
 80-76 
 
 93-49 
 
 81 
 
 22-09 
 
 33-59 
 
 45-4 
 
 57-52 
 
 69-95 
 
 82-68 
 
 95-72 
 
 9 
 
 22-71 
 
 34-52 
 
 46-64 
 
 59-07 
 
 71-8 
 
 84-84 
 
 98-18 
 
192 
 
 HYDROSTATICS. 
 
 Diim. 
 of bore. 
 
 ,L 
 
 Inch. 
 
 JL 
 
 li 
 
 ,JL 
 
 iL 
 
 1 
 
 Inch. 
 
 Inch. 
 
 Ib. 
 
 Ib. 
 
 Ib. 
 
 Ib. 
 
 Ib. 
 
 Ib. 
 
 Ib. 
 
 * 
 
 23-31 
 
 35-43 
 
 47-86 
 
 60-59 
 
 73-63 
 
 86-97 
 
 100-63 
 
 9* 
 
 23-93 
 
 36-36 
 
 49-09 
 
 62-13 
 
 75-47 
 
 89-13 
 
 103-1 
 
 9f 
 
 24-55 
 
 37-28 
 
 50-32 
 
 63-66 
 
 77-32 
 
 91-28 
 
 105-54 
 
 10 
 
 25-16 
 
 38-2 
 
 51-54 
 
 65-2 
 
 79-16 
 
 93-42 
 
 108 
 
 10* 
 
 25-77 
 
 39-11 
 
 52-77 
 
 66-73 
 
 80-99 
 
 95-57 
 
 110-44 
 
 10* 
 
 26-38 
 
 40-04 
 
 54 
 
 68-26 
 
 82-84 
 
 97-71 
 
 113 
 
 10| 
 
 27 
 
 40-96 
 
 55-22 
 
 69-8 
 
 84-67 
 
 99-86 
 
 115-35 
 
 11 
 
 27-62 
 
 41-88 
 
 56-46 
 
 71-33 
 
 86-52 
 
 102-01 
 
 117-81 
 
 Hi 
 
 28-22 
 
 42-8 
 
 57-67 
 
 72-86 
 
 88-35 
 
 104-15 
 
 120-26 
 
 ill 
 
 28-84 
 
 43-71 
 
 58-9 
 
 74-39 
 
 90-19 
 
 106-3 
 
 122-71 
 
 11* 
 
 29-45 
 
 44-64 
 
 60-13 
 
 75-93 
 
 92-04 
 
 108-45 
 
 125-18 
 
 12 
 
 30-06 
 
 45-55 
 
 61-35 
 
 77-46 
 
 93-6 
 
 110-6 
 
 127-6 
 
 MM 
 
 i 
 
 1 
 
 i 
 
 1 
 
 1} 
 
 
 H 
 
 If 
 
 1 
 
 of bora. 
 
 Inch. 
 
 teh. 
 
 tech. 
 
 tech. 
 
 lock 
 
 lack. 
 
 
 loch. 
 
 Inch. 
 
 Inch. 
 
 Ib. 
 
 Ib. 
 
 Ib. 
 
 Ib. 
 
 Ib. 
 
 Ib. 
 
 Ib. 
 
 Ib. 
 
 Ib. 
 
 12* 
 
 63-5 
 
 97-3 
 
 114 
 
 132 
 
 149 
 
 167 
 
 205 
 
 243 
 
 285 
 
 13 
 
 66 
 
 101 
 
 118 
 
 137 
 
 154 
 
 173-6 
 
 212 
 
 252 
 
 294 
 
 13* 
 
 68-4 
 
 104-8 
 
 122 
 
 141-6 
 
 160 
 
 179 
 
 219 
 
 260 
 
 304 
 
 14 
 
 76 
 
 108-2 
 
 126 
 
 146 
 
 165 
 
 185 
 
 227 
 
 269 
 
 314 
 
 14* 
 
 73-4 
 
 112-3 
 
 130 
 
 151 
 
 170 
 
 192 
 
 234 
 
 277 
 
 324 
 
 15 
 
 76-8 
 
 115-7 
 
 135 
 
 156 
 
 176 
 
 198 
 
 242 
 
 286 
 
 334 
 
 15* 
 
 78-1 
 
 119 
 
 139 
 
 161 
 
 181 
 
 204 
 
 250 
 
 295 
 
 344 
 
 16 
 
 80-7 
 
 123 
 
 143 
 
 166 
 
 187 
 
 211 
 
 257 
 
 303 
 
 355 
 
 16* 
 
 83-1 
 
 126-5 
 
 147 
 
 170-1 
 
 192 
 
 217 
 
 264 
 
 312 
 
 363 
 
 17 
 
 85-5 
 
 130 
 
 152 
 
 178-5 
 
 198 
 
 223 
 
 271 
 
 322 
 
 376 
 
 17* 
 
 87-8 
 
 133-5 
 
 157 
 
 180-5 
 
 203 
 
 229 
 
 278 
 
 330 
 
 383 
 
 18 
 
 90-5 
 
 137 
 
 161 
 
 185 
 
 209 
 
 236 
 
 285 
 
 338 
 
 393 
 
 18* 
 
 93 
 
 140-5 
 
 166 
 
 190 
 
 217 
 
 241 
 
 293 
 
 347 
 
 402 
 
 19 
 
 95-5 
 
 144-8 
 
 169 
 
 195 
 
 222 
 
 247 
 
 300 
 
 354 
 
 412 
 
 19* 
 
 97-8 
 
 148-5 
 
 174 
 
 200 
 
 227 
 
 253 
 
 307 
 
 363 
 
 422 
 
 20 
 
 100 
 
 162 
 
 178 
 
 205 
 
 233 
 
 259 
 
 315 
 
 372 
 
 432 
 
 20* 
 
 102-5 
 
 156 183 
 
 210 
 
 238 
 
 26 
 
 323 
 
 381 
 
 442 
 
 The following TABLE pf the weight of different sub- 
 stances used in building and engineering requires no ex- 
 planation. 
 
SPECIFIC GRAVITY. 
 
 193 
 
 Names of Bodiw. 
 
 eight of a 
 bic foot m| 
 ounces. 
 
 Weight of a 
 :ubic foot in 
 pounds. 
 
 Weight of a Weight of a I 
 cubic inch in 'cubic inch in 
 ounces. pounds. 
 
 Nuniber of 
 ubic inches in 
 a pound. 
 
 Copper, cast 
 Copper, sheet 
 
 8788 
 8915 
 8396 
 3271 
 7.631 
 1344 
 7833 
 7816 
 7190 
 7292 
 9880 
 8784 
 1520 
 1250 
 2000 
 2416 
 2672 
 2742 
 3160 
 2880 
 945 
 240 
 544 
 556 
 660 
 96 
 745 
 760 
 852 
 970 
 870 
 915 
 932 
 927 
 1000 
 1028 
 1015 
 1026 
 13568 
 
 549-25 
 557-18 
 524-75 
 454-43 
 476-93 
 09-00 
 89-56 
 88-50 
 449-37 
 455-75 
 619-50 
 649-00 
 95-00 
 78-12 
 125-00 
 151-00 
 167-00 
 171-37 
 197-50 
 180-00 
 59-06 
 15-00 
 34-00 
 34-75 
 41-25 
 43-50 
 46-56 
 47-50 
 53-25 
 60-62 
 54-37 
 57-18 
 58-25 
 57-93 
 62-50 
 
 63-43 
 64-12 
 
 848-00 
 
 086 
 159 
 852 
 203 
 410 
 6-456 
 4-527 
 4-517 
 4-156 
 
 3178 
 3225 
 3037 
 263 
 276 
 4103 
 2833 
 2827 
 26 
 2636 
 3585 
 3177 
 055 
 0452 
 0723 
 0873 
 0967 
 0991 
 1143 
 1042 
 0342 
 0087 
 0197 
 0201 
 024 
 0252 
 027 
 0275 
 0308 
 0351 
 0315 
 0331 
 0337 
 03352 
 03617 
 0372 
 -0367 
 037 
 4908 
 
 3-146 
 3-103 
 3-293 
 3-802 
 3-623 
 2-437 
 3-530 
 3-537 
 3-845 
 3-790 
 2-789 
 3-147 
 18-190 
 22-120 
 13-824 
 11-443 
 10-347 
 10-083 
 8-750 
 9-600 
 29-258 
 115-200 
 50-823 
 49-726 
 41-890 
 39-724 
 37-113 
 36-370 
 32-449 
 28-505 
 31-771 
 30-220 
 29-665 
 29-288 
 27-648 
 26-894 
 27-242 
 26-949 
 2-037 
 
 
 
 
 
 Iteel, hard 
 
 
 
 4215 
 5-710 
 5-0775 
 8787 
 7225 
 1-156 
 1*396 
 1-544 
 1-585 
 1-826 
 1-664 
 5462 
 138 
 315 
 321 
 382 
 
 431 
 440 
 493 
 561 
 503 
 529 
 539 
 536 
 578 
 594 
 587 
 593 
 7-851 
 
 
 
 
 
 Stone, paving - 
 
 QlotP . , 
 
 
 White lead 
 
 fjlooq . . 
 
 
 
 
 Flm . 
 
 Pine, pitch 
 
 Tpak . 
 
 A c>i 
 
 Mahogany 
 
 Out . 
 
 Oil of turpentin 
 
 Linseed oil ... 
 Spirits, proof . 
 Water, distilled 
 
 Tar . 
 
 
 
 
 
 17 
 
194 HYDROSTATICS. 
 
 The foregoing tables and rules will be found of the ut- 
 most service, in the ready calculation of the weight of 
 materials commonly used in engineering. 
 
 What is the weight of a bar of Swedish iron 16 feet long, 
 3 inches broad, and 1*1 inch thick? 
 
 By table B, 3*38 is the weight o a piece of Swedish 
 iron, of one foot long and one inch square, wherefore, 
 
 3-38 X 16 X 3 = 162-24; and then for the fraction -1, 
 in table A, we have for the weight of 1 foot by 1 of an 
 inch square = -034; hence, -034 X 3 X 16 = 16-32; 
 wherefore the sum of the two = 162-24 + 16-32 = 178-56 
 Ibs., the weight. 
 
 If we wish the weight of an equal bar of cast iron, we 
 must employ the multipliers in table D ; hence, 
 178-56 X -925 = 165-168. 
 
 If we wished it for lead, the multiplier from the same 
 table being 1-457, we have, 
 
 178-56 X 1-457 = 260-1619 Ibs., &c., &c. 
 
 Then if lead were 1 penny per pound, the price of such 
 a bar would be 
 
 The following practical rules are often useful and may 
 be easily remembered. 
 
 For round bars of iron, 
 
 diameter (m) a x length in ft. X 2-6 = weight of 
 wrought iron in Ibs. 
 
 diameter (m) 9 x length in ft. X 2-48 = weight of 
 cast iron bars in Ibs. 
 
 A cylindrical bar is 2 inches diameter and 29 inches long, 
 therefore, 2 a x 2-5 x 2-6 = 26 Ibs. if it be wrought iron, 
 but if cast, 2 s X 2-5 X 2-48 = 24-8 Ibs. 
 
 Multiply the sum of the exterior and interior diameters 
 of a cast iron ring by the breadth and thickness of the rim, 
 and also by 0-0074, the results will be the weight in cwts. 
 
HYDRODYNAMICS. 
 
 As hydrostatics embraces the consideration of fluids at 
 rest, so hydrodynamics or hydraulics comprehends the cir- 
 cumstances of fluids in motion. Of this science, little, 
 comparatively speaking, is yet known ; but as it is of the 
 utmost importance to man, we will endeavour to lay before 
 our readers a statement of the more important results of 
 recent inquiry into it. 
 
 If a fluid move through a pipe, canal, or river, of various 
 breadths, always filling it, the velocity of the fluid at dif- 
 ferent parts will be inversely as the transverse sections of 
 these parts. 
 
 Thus let there be a canal, AB, of various breadths at 
 different places, then will the velocity in the portion ab be 
 to that of the velocity in cd, as the area of the cross section 
 at cd is to that at 06, and the velocity at ef will be to that 
 at cd as the area at cd is to the area at e/", being always in 
 inverse proportion. 
 
 Suppose the velocity at ab 10 feet per second, and the 
 area there 100 feet, then if the area at cd be 25 feet, we 
 have 25 : 100 : : 10 : 40 feet, the velocity of the water 
 at cd; and if the area at efbe 50 feet, then 50 : 25 : : 40 
 : 20 feet, the velocity at e/*, the canal being kept con- 
 tinually full. 
 
 The quantity of water that flows through a pipe, or in a 
 canal or river, at any part, is in proportion to the area 
 multiplied by the velocity at that part. 
 
 The calculation of the motion of rivers is often of the 
 highest utility to the engineer. This is sometimes done by 
 the employment of very intricate formulas, but such 
 methods, if easier could be found, would evidently be in- 
 
 195 
 
196 HYDRODYNAMICS. 
 
 consistent with the nature of this work. The method which 
 we shall give is simple, and will be found to answer all the 
 purposes of the practical man. 
 
 In a river, the greatest velocity is at the surface and in 
 the middle of the stream ; from which it diminishes toward 
 the bottom and sides, where it is least. 
 
 The velocity at the middle of the stream may be ascer- 
 tained, by observing how many inches a body floating with 
 the current passes over in a second of time. Gooseberries 
 will fit this purpose exceedingly well; if they are not at 
 hand, a cork may be employed. 
 
 Take the number of inches that the floating body passes 
 over in one second, and extract its square root ; double this 
 square root, subtract it from the velocity at top, and add 
 one, the result will be the velocity of the stream at the 
 bottom. 
 
 And these velocities being ascertained, the mean velocity, 
 or that with which if the stream moved in every part, 
 it would produce the same discharge, may be found = 
 the velocity at top \/ve\oc\iy at top -f- *5. 
 
 Exam. If the velocity at the top and in the middle of 
 the stream, be 36 inches per second, then, 36 (2 X \/36) 
 -f 1 = 36 12 -j- 1 = 25 = the least velocity, or the 
 velocity at bottom. And the mean velocity will be = 
 36 v/36 + -5 = 36 6 -f -5 = 30-5. 
 
 When the water in a river receives a permanent increase 
 from the junction of some other river, the velocity of the 
 water is increased. This increase in velocity causes an in- 
 crease of the action of the water on the sides and bottom, 
 from which circumstance the width of the river will always 
 be increased, and sometimes, though not so frequently, the 
 depth also. By the reason of this increased action of the 
 water on the bottom, the velocity is diminished until the 
 tenacity of the soil or the hardness of the rock afford a 
 sufficient resistance to the force of the water. The bed of 
 the river then changes only by very slow degrees, but the 
 bed of no river is stationary. 
 
 It is of the greatest use to know the amount of the action 
 of any stream on its bed, and for this purpose a knowledge 
 of the nature of the bed and of the velocity at bottom, are 
 absolutely necessary. 
 
 Every kind of soil has a certain velocity which will insure 
 the stability of the bed. A less velocity would allow the 
 
HYDRODYNAMICS. 197 
 
 waters to deposit more of the matter which is carried with 
 the current, and a greater velocity would tear up the chan- 
 nel. From extensive experiments it has been found, that 
 a velocity of 3 inches per second at the bottom, will just 
 begin to work upon the fine clay used for pottery, and, 
 however firm and compact it may be, it will tear it up. 
 A velocity of 6 inches will lift fine sand 8 inches, will lift 
 coarse sand (the size of linseed) 12 inches, will sweep 
 along gravel 24, will roll along pebbles an inch diameter 
 and 3 feet at bottom, will sweep along shivery stones the 
 size of an egg. 
 
 When water issues through a hole in the bottom or side 
 of a vessel, its velocity is the same as that acquired by a 
 body falling through free space from a height equal to that 
 of the surface of the water above the hole. 
 
 The most correct rule for ascertaining the velocity of 
 water running through pipes and canals is this : 
 
 57 X height of head X diam. of pipex , 
 
 .length of pipe X 57 X diam. of pipe/- 
 the velocity in inches with which the water will issue from 
 the orifice. All the measures are understood to be taken in 
 inches. 
 
 Exam. If there be a reservoir of water whose depth is 
 6 feet, having a tube 1 foot long and 2 inches bore, open 
 so as to let the water escape at a distance of 18 inches from 
 the bottom, then we have, 6 X 12 = 72 = whole depth 
 of water on the reservoir, and 72 18 = 54, the height 
 of the head of the fluid above the orifice, wherefore by the 
 rule, 
 
 57 X 54 X 2-5x l/7695 
 
 _ l/7695x 
 - \|Vl7lo; > 
 
 x 57 X 
 
 v/ (4-5) X 23 = 2-121 X 23 = 49-49 inches per second, 
 the velocity of the water. And, by multiplying this result 
 by the area of the orifice, we get the quantity discharged in 
 one second hence, if the pipe be circular, we have, 
 
 2-5 2-5 X 3-1416 
 
 = r25 = radius, and - - - = half 
 2 2 
 
 circumference = 1-9635 = area of orifice, hence, 49*49 x 
 1-9635 = 97-173 cubic inches. 
 
 The quantity of water that flows out of a vertical rectan- 
 gular aperture, that reaches as high as the surface, is | of 
 
 17* 
 
198 HYDRODYNAMICS. 
 
 the quantity that would flow out of the same aperture, 
 placed horizontally at the depth of the base. 
 
 When water issues out of a circular aperture in a thin 
 plate placed on the bottom or side of a reservoir, the stream 
 is contracted into a smaller diameter, to a certain distance 
 from the orifice. The vein is smaller at the distance of half 
 the diameter of the orifice where the area of the section of 
 the vein is -} of that of the orifice, and at the above point 
 the stream has the velocity given by theory, so that to ob- 
 tain the quantity of water discharged, we multiply the 
 Telocity by the area of the orifice, and J-J of this will be 
 the true result. When the water issues through a short 
 tube, the vein of the stream will be less contracted than in 
 the former case, in the proportion of 16 to 13. But when 
 the water issues through an aperture which is the frustum 
 of a cone, whose greater base is the aperture, the height of 
 the conic frustum one half the diameter of the aperture 
 and the area of the small end to that of the large end, as 
 10 : 16; then, in this case, there will be no contraction of 
 the vein ; and from this it may be inferred, that, when a 
 supply of water is required, the greatest possible from a 
 given orifice, this form should be employed. 
 
 To determine the quantity of water discharged by a 
 small vertical or horizontal orifice, the time of discharge, 
 and the height of the fluid in the vessel, when any two 
 of these quantities are known. 
 
 Let A represent the area of the small orifice, W the 
 quantity of water discharged ; T the time of discharge, H 
 the height of fluid in the vessel, and g = 16*087 feet, the 
 space described by gravity in a second. 
 
 Then we have, 
 
 W 
 
 2 X t X \/ g X H 
 W 
 
 4 X g X t 2 X A a 
 
 By means of these formulae we may determine the quan- 
 tity of water W which is discharged in the same time T, 
 from any other vessel in which A' is the area of the orifice, 
 
HYDRODYNAMICS. 
 
 199 
 
 and H the altitude of the fluid ; for since t and g are con- 
 stant, we shall have 
 
 W : W = A v/ H : A' v/ H'. 
 
 Table showing the quantity of Water discharged in one 
 Minute by Orifices differing in form and position. 
 
 Constant 
 Height of the 
 Fluid above 
 the centre of 
 the orifice. 
 
 Form and position of the Orifice. 
 
 Diameter of 
 the orifice. 
 
 No. of cubic 
 inches dis- 
 charged in a 
 minute. 
 
 Ft. in. lin. 
 
 
 Lines. 
 
 
 11 8 10 
 
 Circular and Horizontal, 
 
 6 
 
 2311 
 
 
 Circular and Horizontal, 
 
 12 
 
 9281 
 
 
 Circular and Horizontal, 
 
 24 
 
 37203 
 
 
 Rectangular and Hori- 
 
 
 
 
 zontal, 
 
 12 by 3 
 
 2933 
 
 
 Horizontal and Square, 
 
 12 side 
 
 11817 
 
 
 Horizontal and Square, 
 
 24 side 
 
 47361 
 
 900 
 
 Vertical and Circular, 
 
 6 
 
 2018 
 
 
 Vertical and Circular, 
 
 12 
 
 8135 
 
 400 
 
 Vertical and Circular, 
 
 6 
 
 1353 
 
 
 Vertical and Circular, 
 
 12 
 
 5436 
 
 507 
 
 Vertical and Circular, 
 
 12 
 
 628 
 
 From these results we may conclude, 
 
 1. That the quantities of water discharged in equal times 
 by the same orifice from the same head of water, are very 
 nearly as the areas of the orifices ; and, 
 
 2. That the quantities of water discharged in equal times 
 by the same orifices under different heads of water, are 
 nearly as the square roots of the corresponding heights of 
 the water in the reservoir above the centres of the orifices. 
 
 3. The quantities of water discharged during the same 
 time by different apertures under different heights of water 
 in the reservoir, are to one another in the compound ratio 
 of the areas of the apertures, and of the square roots of 
 the heights in the reservoirs. 
 
 This general rule may be considered as sufficiently cor- 
 rect for ordinary purposes ; but, in order to obtain a great 
 degree of accuracy, Bossut recommends an attention to 
 the three following rules. 
 
 1. Friction is the cause, that, of several similar orifices 
 
200 
 
 HYDRODYNAMICS. 
 
 the smallest discharges less water in proportion than those 
 which are greater, under the same altitudes of water in the 
 reservoir. 
 
 2. Of several orifices of equal surface, that which has the 
 smallest perimeter ought, on account of the friction, to give 
 more water than the rest, under the same altitude of water 
 in the reservoir. 
 
 3. That, in consequence of a slight augmentation which 
 the contraction of the fluid vein undergoes, in proportion 
 as the height of fluid in the reservoir increases, the expense 
 ought to be a little diminished. 
 
 Table of Comparison of the Theoretic with the Real dis- 
 charges from an orifice one inch in diameter. 
 
 height of the 
 water in the 
 reservoir 
 above the 
 
 ;/: I '.I.. 
 
 Theoretical dis- 
 
 . li.-irv - '! r 
 acircular orifice 
 one inch in dia- 
 
 .... ' r 
 
 Real discharges 
 
 iu the same time 
 through the 
 same orifice. 
 
 Ratio of the theoretical 
 to the real discharge!. 
 
 orifice. 
 
 
 
 
 Paris feet. 
 
 Cubic inches. 
 
 Cubic inches. 
 
 
 1 
 
 i:isi 
 
 2722 
 
 to 0-62133 
 
 2 
 
 6196 
 
 3846 
 
 to 0-62073 
 
 3 
 
 7589 
 
 4710 
 
 to 0-62064 
 
 4 
 
 8763 
 
 5436 
 
 to 0-62034 
 
 5 
 
 9797 
 
 6075 
 
 to 0-62010 
 
 6 
 
 10732 
 
 t;r,;> i 
 
 to 0-62000 
 
 7 
 
 11592 
 
 7183 
 
 to 0-61965 
 
 8 
 
 12392 
 
 7672 
 
 to 0-61911 
 
 9 
 
 13144 
 
 8135 
 
 to 0-61 892 
 
 10 
 
 13855 
 
 8574 
 
 to 0-61883 
 
 11 
 
 14530 
 
 8990 
 
 to 0-61873 
 
 12 
 
 15180 
 
 9384 
 
 to 0-61819 
 
 13 
 
 15797 
 
 9764 
 
 to 0-61810 
 
 14 
 
 16393 
 
 10130 
 
 to 0-61 795 
 
 15 
 
 16968 
 
 10472 
 
 to 0-61716 
 
 1 
 
 2 
 
 3 
 
 4 
 
 It appears from this table, that the real as well as the 
 theoretical discharges are nearly proportional to the square 
 roots of the heights of the fluid in the reservoir. Thus 
 for the heights 1 and 4, whose square roots are as 1 to 2 
 feet, the real discharges are 2722 and 5436, which are to 
 one another as 1 to 1-997, very nearly as 1 to 2. 
 
HYDRODYNAMICS. 201 
 
 Let it be required to determine the quantity of water dis- 
 charged from an orifice of 3 inches in diameter, under an 
 altitude of 30 feet. Then, since the real quantities dis- 
 charged are in the compound ratio of the orifices, and the 
 square roots of the altitudes of the water, and since the 
 theoretical discharge by an orifice 1 inch in diameter, under 
 an altitude of 15 feet is 16968 cubical inches in a minute, 
 we have 1 v' 15 : 9 ^ 30 = 16968 : 215961, the theoreti- 
 cal discharge. But the theoretical is to the real discharge 
 as 1 to *62, the above value being diminished in that ratio, 
 gives 133309 cubic inches for the real quantity of water 
 discharged by the orifice. 
 
 The following formulae have been given by M. Prony 
 as deduced from the preceding experiments of Bossut, 
 
 Q = 0-61938 AT v' 2 g H, 
 
 A being the area of the orifice in square feet, H the altitude 
 of the fluid in feet, T the time, g the force of gravity at 
 the end of a second, and Q the quantity of water in cubic 
 feet. As x/ 2 g is a constant quantity, and is equal to 
 7-77125, we have 
 
 Q = 4-818 AT ^ H for orifices of any form. 
 If the orifices are circular, and if d represents their dia- 
 meter, then 
 
 Q = 3-7842 d* T ^/ H. 
 From the second of these equations we obtain 
 
 A Q 
 
 3X 4-818 T v/ H 
 
 Q 
 
 II 
 
 4-818 A v/ H 
 Q 
 
 (4-818 AT) 8 
 
 These formulae will be found to give very accurate re- 
 sults ; but if we wish to obtain a still higher degree of ac- 
 curacy, we must not use the mean co-efficient 0*6194, but 
 the one in the table which comes nearest to the circum- 
 stances of the case. Thus if the head of water happens to 
 be small, such as 1 foot, then we must take the co-efficient 
 0-62133, and if it happens to be great, we must use the least 
 co-efficient 0-61716. 
 
202 
 
 HYDRODYNAMICS. 
 
 Table containing the quantity of Water discharged over a weir. 
 
 Depth of the up 
 peTMe of the 
 wMtetmd balow 
 
 *E^ E "r 
 
 1 Cubic tat of water 
 dachargrdinainiaute 
 by every inch of the 
 Wartctm^aceortfcqt 
 tote Built fanautaT 
 
 Cubic feet of water 4V 
 charged in a mint* by 
 .rrrfmcho/ tbewJZ. 
 board accordJoc to up* 
 
 1 
 
 0,403 
 
 0,428 
 
 2 
 
 1,140 
 
 1,211 
 
 - 3 
 
 2,095 
 
 2,226 
 
 4 
 
 3,225 
 
 3,427 
 
 5 
 
 4,507 
 
 4,789 
 
 6 
 
 5,925 
 
 6,295 
 
 7 
 
 7,466 
 
 7,933 
 
 8 
 
 9,122 
 
 9,692 
 
 9 
 
 10,884 
 
 11,564 
 
 10 
 
 12,748 
 
 13,535 
 
 11 
 
 14,707 
 
 15,633 
 
 12 
 
 16,758 
 
 17,805 
 
 13 
 
 18,895 
 
 20,076 
 
 14 
 
 21,117 
 
 22,437 
 
 15 
 
 23,419 
 
 24,883 
 
 16 
 
 25,800 
 
 27,413 
 
 17 
 
 28,258 
 
 30,024 
 
 18 
 
 30,786 
 
 32,710 
 
 Table containing the quantities of Water discharged by Cylindrical 
 Tubes one inch in diameter and of different lengths, whether the 
 Tubes were inserted in the bottom or in the sides of the vessel, 
 
 Len<tbi of the Tube* .xprcMd in 
 
 The tube filled with the 
 issuing fluid 
 
 The tube not filled with 
 the issuing fluid 
 
 18 
 
 Cub.ol iodM d- 
 
 12274 
 12188 
 12168 
 
 9282 
 
HYDRODYNAMICS. 
 
 203 
 
 Table of comparison of the Theoretical with the Real Discharges from 
 an additional Tube of a cylindrical form, one inch in diameter and 
 two inches long. 
 
 Constant altitude 
 
 Theoretical dis- 
 
 Real discharges in the 
 
 
 of the Water in 
 the reservoir 
 above the centre 
 of the orifice. 
 
 charges through a 
 circular orifice onn 
 inch in diameter. 
 
 same time by a cy- 
 lindrical tube one inch 
 in diameter and two 
 inches long. 
 
 Ratio of the theoretical 
 to the real discharges. 
 
 Paris feet. 
 
 Cubic inches. 
 
 Cubic inches. 
 
 
 1 
 
 4381 
 
 3539 
 
 1 to 0-81781 
 
 2 
 
 6196 
 
 5002 
 
 to 0-80729 
 
 3 
 
 7589 
 
 6126 
 
 to 0-80724 
 
 4 
 
 8763 
 
 7070 
 
 to 0-80681 
 
 5 
 
 9797 
 
 7900 
 
 to 0-80638 
 
 6 
 
 10732 
 
 8654 
 
 to 0-80638 
 
 7 
 
 11592 
 
 9340 
 
 to 0-80573 
 
 8 
 
 12392 
 
 9975 
 
 to 0-80496 
 
 9 
 
 13144 
 
 10579 
 
 to 0-80485 
 
 10 
 
 13855 
 
 11151 
 
 to 0-80483 
 
 11 
 
 14530 
 
 11693 
 
 to 0-80477 
 
 12 
 
 15180 
 
 12205 
 
 to 0-80403 
 
 13 
 
 15797 
 
 12699 
 
 to 0-80390 
 
 14 
 
 16393 
 
 13177 
 
 to 0-80382 
 
 15 
 
 16968 
 
 13620 
 
 to 0-80270 
 
 1 
 
 2 
 
 3 
 
 4 
 
 Hence it follows, that the velocity in English inches will 
 be V = 22-47 v/ H for additional tubes. 
 
 M. Prony has given the following formulae, as deduced 
 from the preceding table. 
 
 T= 
 
 II: 
 
 4-9438 T 
 Q 
 
 H 
 
 (4-9438 d* T) 3 
 
 The resistance that a body sustains in moving through a 
 fluid is in proportion to the square of the velocity. 
 
 The resistance that any plane surface encounters in mov- 
 ing through a fluid with any velocity, is equal to the weight 
 of a column whose height is the space a body would have 
 
204 HYDRODYNAMICS. 
 
 to fall through in free space to acquire that velocity, and 
 whose base is the surface of the plane. 
 
 Ex. If a plane 16 inches square, move through water at 
 the rate of 13 feet per second ; then, 
 13 3 
 
 64-= 2 ' 6 = 
 
 the space a body would require to fall through free space to 
 acquire a velocity of 13 per second, wherefore, as 2-6 feet 
 =31*2 inches, we have 16 X 31-2 = 499-2 cubic inches= 
 the column of matter whose height and base are required ; 
 therefore, since 1728 cubic inches = 1 cubic foot of water 
 weighs 1000 ounces, we have 1728 : 499-2 : : 1000 : 288 
 ounces = 18 Ibs. which is the amount of resistance met 
 with by the plane at the above velocity. 
 
 As action and reaction are equal and contrary, it is the 
 same thing whether the plane moves against the fluid, or 
 the fluid against the plane. 
 
 WATER WHEELS. 
 
 MOTION is generally obtained from water, either by ex- 
 posing obstacles to the action of its current, or by arresting 
 its progress during part of its descent, by movable buckets. 
 
 Water-wheels have three denominations depending on 
 their particular construction, undershot, breast, and over- 
 shot. If the water is to act on the wheel by its weight, it 
 is delivered from the spout as high on the wheel as possible, 
 that it may continue the longer to press the buckets down ; 
 but when it acts on the wheel by the velocity of the stream, 
 it is made to act on the float-boards at as low a point as 
 possible, that it may have acquired previously the greatest 
 velocity. In the first case, the wheel is said to be overshot, 
 in the second, undershot. The overshot wheel is the most 
 advantageous, as from the same quantity of water it gives a 
 greater power, but it is not always that we can employ an 
 overshot wheel from the smallness of the fall. When this 
 is the case, we must deliver the water farther down than 
 the top of the wheel, and, in this case, it becomes a breast- 
 wheel, and partakes in some degrees of the properties of 
 the overshot. When we cannot employ a breast wheel, 
 we must have recourse to the undershot, which is the least 
 
\T K \{ 
 
 Hate I. 
 
 Fiu. 2. 
 
 f-U T 'n 
 
 Published by Thomas Wardle Philadelphia. 
 

WATER W II EELS. 205 
 
 powerful of all. The force of a stream of water against 
 the floats of an undershot wheel is equal to a column of 
 water, whose base is the section of the stream in that place, 
 and height the perpendicular height of the water to the 
 surface. Where the quantity of water is given, its force 
 against the floats of the wheel is directly proportional to 
 the velocity of the stream, or the square root of the height 
 of the surface. These remarks hold true only when the 
 water is allowed to escape from the float boards, after it 
 has struck them. For if the floats be too near each other, 
 then the water from one float will be sent back and obstruct 
 the progress of the next float. 
 
 Engraved representations of trie three forms of the water 
 wheel are given in plate 1st. Fig. 1 is a representation 
 of the undershot; fig. 2 of the breast; and fig. 3 of the 
 overshot water wheel. The floats of the undershot as 
 likewise of the breast wheel are flat, those of the latter 
 being fitted so nearly to the water way that little of the 
 fluid is allowed to escape between their edges and the stone 
 or brick work, as may be seen in the figure. The over- 
 shot wheel is furnished with buckets instead of floats, so 
 constructed that they shall retain as much as possible of 
 the water from the time they receive it until they arrive at 
 the lowest point, where each bucket should be emptied, 
 since if any water be carried by the bucket in its ascent it 
 will be just so much unnecessary weight that the wheel has 
 to lift. The following geometrical construction will show 
 the method of forming the buckets so that there shall be the 
 greatest possible advantage derived from the overshot wheel. 
 
 This bucket is formed of three planes ; 
 AB is in the direction of the radius of 
 the wheel, and is called the start, or 
 shoulder ; BC is called the arm, and CH 
 the wrist. These buckets are so con- 
 structed, that when AB makes an angle 
 of 35 with the vertical diameter of the 
 wheel, the line AD is horizontal ; and the 
 area of the figure ADCB is equal to that M \ \ 
 of FOB A ; so that as much water is re- \ \ 
 tained in the bucket in this position as would fill FCBA ; 
 the whole of the water is not discharged until CD becomes 
 horizontal, which takes place when the line AB is very 
 near the lowest point. 
 
 18 
 
206 HYDRODYNAMICS. 
 
 To find the velocity of the water acting upon the wheel, 
 ^.(height of the fall x 64'38) = the velocity in feet per 
 second. 
 
 Ex. If the height of the fall be 14 feet, then we have 
 v '(14x64-38) = % /901-32=:30-02 feet per second, nearly. 
 
 To find the area of the section of the stream, 
 
 The number of feet flowing in 1 second 
 
 velocity in feet per second 
 the section of the stream in square feet. 
 
 Ex. If there be 40 feet flowing in a second, and the 
 velocity of the stream is 5 feet per second, then, 
 
 i=8 = 
 5 
 
 the area of the section of the stream in square feet. 
 
 To calculate the power of the fall : 
 
 Area of section of stream where it acts upon the wheel x 
 height of fall x 62 = the number of Ibs. avoir, the wheel 
 can sustain, acting perpendicularly at its circumference, so 
 as to be in equilibrium. If this number of Ibs. which keeps 
 the wheel at rest be diminished, the wheel will move. 
 
 If the wheel move as fast as the stream, it is clear that 
 the water would have no effect in moving it, if the wheel 
 were to move faster than the stream, the water would be a 
 positive hindrance to its motion ; and it can only be ad- 
 vantageous when the velocity of the stream is greater than 
 that of the wheel. There is a certain relation between the 
 velocity of the wheel and that of the stream, at which the 
 effect will be the greatest possible or a maximum. 
 
 The effect of an undershot wheel is a maximum when 
 the velocity of the wheel is 3 of the velocity of the stream. 
 
 Ex. If the area of the cross section of a stream be 6 
 feet, and its velocity 4 feet per second, and a fall of 16 feet 
 can be procured, then we have 4x6 =24, the number of 
 cubic feet flowing per second : 
 
 v/ (16x64'38)=32, the velocity of the water at the end 
 of the fall : 
 
 S)A 
 
 = I, the section of the stream at the end of the fall in 
 
 square feet : 
 
 | X 16 X 62* = 750 lbs.= the weight which the wheel 
 will sustain in equilibrium. 
 
WATER WHEELS. 207 
 
 Now, the effective velocity of the stream is the difference 
 between the velocities of the stream and wheel, and the 
 wheel's velocity being 5 of that of the stream, the difference 
 or effective velocity will be ; now, the power of the 
 stream is as the square of the effective velocity, and the 
 square off is -*. We must multiply the power of the fall 
 as above calculated by this , and also by 1, in order Hhat 
 the wheel may move with the proper velocity ; hence, 750 
 X$X5 = Hl,j Ibs. raised through 10^ feet per second, the 
 velocity of the wheel, which is 3 of 32 the velocity of the 
 stream. An undershot water wheel is capable only of rais- 
 ing ^ 4 7 of the weight of the water to the height of the fall. 
 From numerous experiments on water wheels, it has been 
 found, that in practice the water not being allowed to es- 
 cape from the floats immediately after it has impinged upon 
 them, the maximum effect is, when the velocity varies be- 
 tween 3 and 2, that of the water being nearly ^ 2 T . There 
 is another deviation from theoretical result, in consequence 
 of the water not being allowed to escape immediately from 
 the float-boards, as the water is heaped up to about 2k times 
 its natural height, and thus acts partly by its weight, and 
 partly by its force in consequence of which it happens, 
 that a well-constructed undershot water wheel, instead of 
 raising fa of the weight of the water expended on the 
 height of the fall, will raise . 
 
 The effective head being the same, the effect of the wheel 
 will depend on the quantity of water expended ; and the 
 quantity of water being the same, the effect of the wheel 
 depends on the height of the head of the fall. 
 
 The section of the stream being the same, the effect will 
 be nearly as the cube of the velocity. 
 
 Overshot water wheel. If the water in the buckets of an 
 overshot wheel be supposed to be equally diffused over half 
 the circumference of the wheel, then the whole weight of 
 the water in the buckets is to its power to turn the wheel 
 as 11 to 7. 
 
 An overshot water wheel will raise nearly as much water 
 to the height of the fall, as is expended in driving the 
 wheel : if the height of the fall be reckoned from the bucket 
 that receives the water to the bucket that discharges it. 
 According to the last experiments, the velocity of an over- 
 shot wheel should be between 2 and 4 feet per second for 
 
208 HYDRODYNAMICS. 
 
 all diameters of wheels. A breast wheel partakes of the 
 properties of the two foregoing, as part of its action de 
 pends on the velocity, and part on the weight of the water 
 which moves it. 
 
 Circumstances will regulate which of these three species 
 of water wheels is to be employed. For a large supply of 
 water with a small fall, the undershot wheel is the most ap- 
 propriate. For a small supply of water with a large fall, the 
 overshot ought to be employed. Where both the quantity 
 of water and height of fall are moderate, the breast wheel 
 must be used. 
 
 Before erecting a water wheel, all the circumstances must 
 be taken into account, and our calculations made accordingly. 
 We must measure the height of head velocity, and area of 
 stream, &c., to do which a slight knowledge of levelling will 
 be required. What follows will make this subject suffi- 
 ciently plain. 
 
 Levelling. A pole about 10 feet long must be procured, 
 and also a staff about five feet long, on the top of which is 
 fixed a spirit level with small sight holes at the ends, so 
 that when the spirit level is perfectly horizontal, the eye 
 may view any object before it through the sights in a per- 
 fectly horizontal line. If you have to measure the perpen- 
 dicular distance between the bottom and top of a hill, for 
 instance ; place the level staff on the side of the hill in such 
 a way that when the level is truly set, the top of the hill 
 may be seen through the sights. Keep the level in this 
 position and look the contrary way, then cause some person 
 to place the 10 feet staff before the sight further down the 
 hill, and looking through the sights to the staff, cause the 
 person to move his finger up or down the staff until the 
 finger be seen through the sights, and mark the position of 
 the finger on the staff. Keep your ten feet staff in the 
 same place, and carry your level staff down the hill to a 
 convenient distance, then fix it in the same way as before; 
 and looking through the sights at the ten feet staff, cause 
 the person to bring his finger towards the bottom of the 
 staff, and move his finger up or down the staff in the same 
 way until it be seen through the sights, and mark the place 
 of the finder. Then the distance between the two finger 
 marks added to the height of the level staff, will be the 
 perpendicular distance between the place where the level 
 staff now stands and the top of the hill. The process Vs 
 
WATER WHEELS. 
 
 209 
 
 perfectly simple, and it will not be difficult to repeat it 
 oftener if the height of the hill requires it. 
 
 This process will give what is called the apparent level, 
 which however is not the true level. Two stations are on 
 the same true level when they are equally distant from the 
 centre of the earth. The apparent level gives the objects 
 in the same straight line, but the true level gives the line 
 which joins them as part of a circle whose centre is the 
 centre of the earth. In small distances there is no sensible 
 difference between the true and apparent level of any two 
 objects. When the distance is one mile, the true level will 
 be about 8 inches different from the apparent level. This 
 will serve well enough to remember, but more correctly 
 speaking it is 7'962 inches for one mile, and for all other 
 distances the difference of the two levels will be as the 
 square of the distance. Thus at the distance of two miles 
 it will be, 
 
 l a : 2 3 : : 8 : 32 inches, or 2 feet 8 inches nearly. 
 
 These circumstances must be strictly observed in the 
 formation of canals, railways, &c., &c. 
 
 The following table will save the trouble, of calculation. 
 The distances are measured on the earth's surface. 
 
 Distance 
 measured in 
 yards. 
 
 Allowance 
 inches. 
 
 Distance 
 measured in 
 miles. 
 
 Allowance in 
 f.-et and 
 inches. 
 
 100 
 
 0-026 
 
 1 
 
 * 
 
 
 
 200 
 
 0-103 
 
 * 
 
 2 
 
 300 
 
 0-231 
 
 1 
 
 4 
 
 400 
 
 0-411 
 
 1 
 
 8 
 
 500 
 
 0-643 
 
 2 
 
 2 8 
 
 600 
 
 0-925 
 
 3 
 
 6 
 
 700 
 
 1-260 
 
 4 
 
 10 7 
 
 800 
 
 1-645 
 
 5 
 
 16 7 
 
 900 
 
 2-081 
 
 6 
 
 23 11 
 
 1000 
 
 2-570 
 
 7 
 
 32 6 
 
 1100 
 
 3-110 
 
 8 
 
 42 6 
 
 1200 
 
 3-701 
 
 9 
 
 53 9 
 
 1300 
 
 4-344 
 
 10 
 
 66 4 
 
 1400 
 
 5-038 
 
 11 
 
 80 3 
 
 1500 
 
 5-784 
 
 12 
 
 95 7 
 
 1600 
 
 6-580 
 
 13 
 
 112 2 
 
 1700 
 
 7-425 
 
 14 
 
 130 1 
 
 18* 
 
210 HYDRODYNAMICS. 
 
 Construction of a water wheel. To find the centre of 
 gyration of a water wheel, take the radius of the wheel 
 and the weight of its arms, rim, shrouding, and float 
 boards. Then call the weight of the rim R, which must 
 be multiplied by the square of the radius, and the pro- 
 duct be doubled and then carried out. Next the weight 
 of the arms called A must be multiplied by the square 
 of the radius, and be doubled and carried out as before. 
 Then the weight of the water in action called W must 
 be multiplied by the square of the radius and carried 
 out. If these products be added together into one sum 
 they will form a dividend. For a divisor, double the sum 
 of the weights of the rim and the arms, and add the weight 
 of the water to them. Divide the dividend by the divisor, 
 and the square root of the quotient will be the radius of 
 gyration. 
 
 Ex. In a wheel 24 feet diameter The weight of the 
 arms is 2 tons, the shrouding and rims 4 tons, and the 
 water in action 2 tons ; hence, by the above, 
 
 R = 4 tons x 12 8 x 2 = 1152 
 A = 2 tons x 12 a X 2 = 576 
 W=2tons x 12 8 = 288 
 
 Their sum 2016 dividend, and 
 2 x (4 + 2 -f 2) = 16, the divisor. 
 
 The answer, J^ = N/ J 26 = 11-225. 
 
 Tables for the more ready performance of calculations 
 for water wheels are usually given in books of Mechanics ; 
 the construction and use of which we shall now proceed to 
 explain. 
 
 1. Find, by measuring and levelling, the height of the 
 fall of water which is reckoned from its upper surface to 
 the middle of the depth of the stream, where it acts upon 
 the float-boards. 
 
 2. Find the velocity acquired by the water in falling 
 through that height, which is done thus : multiply the 
 height of the fall by 64-38, extract the square root of the 
 product which would be the velocity of the stream if there 
 were no friction, but to allow for friction take away ~ v of 
 this result for the true velocity. 
 
WATER WHEELS. 211 
 
 3. Find the velocity that ought to be given to the float- 
 boards, by taking 4 of the velocity of the water, which 
 product will be the number of feet the float-boards have to 
 pass through in one second of time to produce the maxi- 
 mum effect. 
 
 circumference of wheel 
 
 velocity of the float-boards 
 
 the number of seconds that the wheel takes to make one 
 turn. 
 
 4. Divide 60 by the last number. The quotient is the 
 number of revolutions the wheel makes in one minute. 
 
 5. Divide 90 by the last quotient, the new quotient is the 
 number of turns of the millstone for one of the wheel : 90 
 being the number of turns that a millstone of five feet dia- 
 meter ought to make in a minute. 
 
 6. As the number of turns of the wheel in a minute 
 
 Is to the number of turns of the millstone in a minute, 
 So is the number of staves in the trundle 
 To the number of teeth in the spur-wheel, avoiding 
 fractions. 
 
 7. The number of turns of the wheel in a minute X 
 the number of turns of the millstone for one turn of 
 the wheel = the number of turns of the millstone per 
 minute. 
 
 Or, by a different method, multiply the number of teeth 
 in the spur-wheel by the number of turns of the water- 
 wheel per minute, and divide this product by the number 
 of staves in the trundle, the quotient is the number of turns 
 of the millstone per minute. 
 
 In this way has the following table been constructed for 
 a water-wheel of 15 feet diameter, the millstone being 5 
 feet diameter and making 90 tipns in one minute. 
 
212 
 
 HYDRODYNAMICS. 
 
 A MILLWRIGHT'S TABLE, 
 
 In which the Velocity of the Wheel is Three-Sevenths of the Velocity 
 of the Water, allowance being made fur the Effect* of Friction on 
 the Velocity of tie Stream for a Wheel of Fifteen Feet diameter. 
 
 Height of 
 the fill of 
 water. 
 
 Velocify of 
 the water 
 per second. 
 
 Velocity of 
 
 :,; j-r 
 second, 
 bein< 3-7ths 
 of that of 
 the water. 
 
 Revolutions 
 of the 
 
 w r u r 
 
 NumtxT of 
 revolutions of 
 the millstone 
 for one of 
 the 
 wheel 
 
 Teeth in the 
 wheel, and 
 staves in the 
 trundle. 
 
 Revolutions 
 of the mill- 
 stone per 
 minute by 
 these (laves 
 and teeth. 
 
 1 
 
 
 
 iii 
 
 8 
 
 ijl 
 
 Revolution*. 
 100 parts ofa 
 resolution. 
 
 Revolution*. 
 100 parts of a 
 
 revolution. 
 
 11 
 
 Revolutions. 
 100 parts of a 
 revolution. 
 
 1 
 
 7-62 
 
 3-27 
 
 4-16 
 
 21-63 
 
 130 6 
 
 90-07 
 
 2 
 
 10-77 
 
 4-62 
 
 5-88 
 
 15-31 
 
 92 6 
 
 90-16 
 
 3 
 
 13-20 
 
 5-66 
 
 7-20 
 
 12-50 
 
 100 8 
 
 90-00 
 
 4 
 
 15-24 
 
 6-53 
 
 8-32 
 
 10-81 
 
 97 9 
 
 89-67 
 
 5 
 
 17-04 
 
 7-30 
 
 9-28 
 
 9-70 
 
 97 10 
 
 90-02 
 
 6 
 
 18-67 
 
 8-00 
 
 10-19 
 
 8-83 
 
 97 11 
 
 89-86 
 
 7 
 8 
 
 20-15 
 21-56 
 
 8-64 
 9-24 
 
 10-99 
 11-76 
 
 8-19 
 7-65 
 
 90 11 
 84 11 
 
 89-92 
 89-80 
 
 9 
 
 22-86 
 
 9-80 
 
 12-47 
 
 7-22 
 
 72 10 
 
 89-68 
 
 10 
 
 24-10 
 
 10-33 
 
 13-15 
 
 6-84 
 
 82 12 
 
 89-86 
 
 11 
 
 25-27 
 
 10-83 
 
 13-79 
 
 6-53 
 
 85 13 
 
 90-16 
 
 12 
 
 26-40 
 
 11-31 
 
 14-40 
 
 6-25 
 
 75 12 
 
 90-00 
 
 13 
 
 27-47 
 
 11-77 
 
 14-99 
 
 6-00 
 
 72 12 
 
 89-94 
 
 14 
 
 28-51 
 
 12-22 
 
 15-56 
 
 5-78 
 
 75 13 
 
 89-77 
 
 15 
 
 29-52 
 
 12-65 
 
 16-13 
 
 5-58 
 
 67 12 
 
 90-06 
 
 16 
 
 30-48 
 
 13-06 
 
 16-63 
 
 5-41 
 
 65 12 
 
 90-06 
 
 17 
 
 31-42 
 
 13-46 
 
 17-14 
 
 5-25 
 
 63 12 
 
 89-99 
 
 18 
 
 32-33 
 
 13-86 
 
 17-65 
 
 5-10 
 
 61 12 
 
 89-72 
 
 19 
 
 33-22 
 
 14-24 
 
 18-13 
 
 4-96 
 
 60 12 
 
 90-65 
 
 20 
 
 34-17 
 
 14-64 
 
 18-64 
 
 4-83 
 
 58 12 
 
 90-09 
 
 It is desirable that the millwright should possess short 
 easy rules, which would answer the purposes of practice 
 rather than the conditions of mere theory. The following 
 will be found useful, as they give the power with allowance 
 for friction and waste of water. 
 
WATER WHEELS. 213 
 
 For an undershot : 
 
 Height of fall x quantity of water flowing per minute _ 
 
 5000 
 the number of horses' power which the effect is equal to. 
 
 For an overshot : 
 
 Power of an undershot X 2<| = horses' power. 
 
 For a breast-wheel : 
 
 Find the power of an undershot from the top of the fall 
 to where the water enters the bucket ; then for an overshot 
 for the rest of the fall the sum of these two is the power 
 of the breast wheel. 
 
 NOTE. The quantity of water flowing per minute, ana 
 the height of the fall are both taken in feet. 
 
 Ex. What power can be obtained from an undershot 
 wheel the fall being 25 feet, the section of the stream 
 being 9 feet, and the velocity of the water 18 feet per 
 minute ? 
 
 9 X 18 X 25 4050 , , 
 
 -5000- = 5000 " * 81 f a horSe P Wer ' 
 one horse power being unit. 
 
 And an overshot in the same situation would be "81 X 
 2-5 = 2-025 horses' power. 
 
 And if, in a breast wheel, the water enters the bucket 10 
 feet from the top of the fall, then we have, 
 
 10 X 8 X 9 720 , _ 1800-0 
 
 ' X2 ! ~~ : 
 
 5000 
 
 for an overshot, and for the undershot we found it before 
 81 ; hence, *36 -j- '81 = 1-17 horses' power for the breast 
 wheel. 
 
 IN plate 1st, fig. 4, we have given a view of Barker's 
 mill, where CD is a vertical axis, moving on a pivot at D, 
 and carrying the upper millstone w, after passing through 
 an opening in the fixed millstone C. Upon this axis is 
 fixed a vertical tube TT communicating with a horizontal 
 tube AB, at the extremities of which A, B, are two aper- 
 tures in opposite directions. When water from the mill- 
 course MN is introduced into the tube TT, it flows out of 
 the apertures A, B, and by the reaction or counterpressure 
 of the issuing water, the arm AB, and consequently the 
 
214 HYDRODYNAMICS. 
 
 whole machine, is put in motion. The bridge-tree db is 
 elevated or depressed by turning the nut c at the end of 
 the lever cb. In order to understand how this motion is 
 produced, let us suppose both the apertures shut, and the 
 tube TT filled with water up to T. The apertures A, B, 
 which are shut up, will be pressed outwards by a force 
 equal to the weight of a column of water whose height is 
 TT, and whose area is the area of the apertures. Every 
 part of the tube AB sustains a similar pressure ; but as 
 these pressures are balanced by equal and opposite pres- 
 sures, the arm AB is at rest. By opening the aperture at 
 A, however, the pressure at that place is removed, and 
 consequently the arm is carried round by a pressure equal 
 to that of a column TT, acting upon an area equal to that 
 of the aperture A. The same thing happens on the arm 
 TB ; and these two pressures drive the arm AB round in 
 the same direction. This machine may evidently be ap- 
 plied to dfive any kind of machinery, by fixing a wheel 
 upon the vertical axis CD. 
 
 This ingenious machine has not been much employed, 
 even in those situations for which it is best adapted ; partly, 
 we suspect, from the millwright's not having in his posses- 
 sion sufficiently simple rules for its construction ; as the 
 theory of Barker's mill, simple as its construction and action 
 may appear, is not by any means well developed. In the 
 mean time the following directions may be found useful to 
 the mechanic. 
 
 1. Make each arm of the horizontal tube, from the centre 
 of motion to the centre of the aperture of any convenient 
 length, not less than of the perpendicular height of the 
 water's surface above these centres. 
 
 2. Multiply the length of the arm in feet by -61365, and 
 the square root of this product will be the proper time for a 
 revolution, in seconds ; then adapt the other parts of the 
 machinery to this velocity ; or, 
 
 If the time of a revolution be given, multiply the square 
 of this time by 1-6296 for the proportional length of the 
 arm in feet. 
 
 Multiply together the breadth, depth, and velocity per 
 second of the race, and divide the las], product, 14-27 X 
 the square root of the height; the result is the area of 
 either aperture ; or, multiply the continual product of the 
 breadth, depth, and velocity of the race, by the square root 
 
215 
 
 of the height, and by the decimal '07, the last product 
 divided by the height will give the area of the aperture. 
 
 Multiply the area of either aperture by the height of the 
 head of water, and this product by 55*795 (or, in round 
 numbers, 56) for the moving force, estimated at the centres 
 of the apertures in Ibs. avoirdupois. 
 
 Ex. If the fall be 18 feet from the head to the centre 
 of the apertures, then the arm must not be less than 2 feet, 
 as i of 18 = 2, and v/(2 x '61365) = x /(l-22730) = 1-107 
 = the time of a revolution in seconds ; also, the breadth 
 of the race being 17 inches, and depth 9, and the velocity 
 of the water 6 feet per second, here we have, 
 
 17 in. = 1-41 feet, and 9 in. = -75 feet, then 
 
 1-41 x '75 x 6 = 6-34 = the area of section of the 
 race x velocity of water ; hence, 
 
 6-39 X v/ 18 x '07 == 1-896 = the area of the aperture 
 in inches ; and, 
 
 1-876 X 18 X 56 = 1909 Ibs. the moving force. 
 
 The following dimensions have been employed in prac- 
 tice with success. The length of arm from the centre pivot 
 to the centre of the discharging hole, 46 inches ; inside 
 diameter of the arm, 13 inches ; diameter of the supplying 
 pipe, 2 inches ; height of the working head of water 21 
 feet above the level of the discharge. When a machine 
 of these dimensions, and in such circumstances, was not 
 loaded and had one orifice open, it made 115 turns in a 
 minute. 
 
 
PNEUMATICS. 
 
 PNEUMATICS comprehends the knowledge of the proper- 
 ties of common air and elastic fluids in general. 
 
 Air is capable of being compressed to almost any degree, 
 that is, may be forced into a space infinitely smaller than 
 the space which it commonly occupies, and this is effected 
 by additional pressure. When this additional pressure is 
 .taken away, the air will regain, by its elasticity, its former 
 magnitude. Were it not for this circumstance, the subject 
 of this chapter might have been introduced when we dis- 
 cussed the equilibrium and motion of water and fluids, 
 which are non-elastic or incompressible, as their fundamen- 
 tal laws are the same. It has, indeed, been found by recent 
 experimenters, that water, mercury, <fcc., are compressible, 
 but to a very limited degree ; so that although the distinction 
 of elastic and non-elastic fluids is not absolutely correct, it 
 is yet sufficiently so to retain Pneumatics, in elementary 
 arrangement, as a distinct branch of science. 
 
 The air or atmosphere is a fluid body which surrounds 
 the earth, and gravitates on all parts of its surface. 
 
 The mechanical properties of air are the same as other 
 elastic fluids, and being the most common, inquiries in 
 pneumatics are generally confined to this fluid. 
 
 The air has weight. A cubic foot of it weighs 1-2857 
 ounces at the surface of the earth, or, as some state it, 1'222. 
 
 The air being an elastic fluid, it is compressible and ex- 
 pansible, and its degrees of compression and expansion are 
 proportional to the forces or weights which compress it. 
 
 All the air near the earth's surface is in a state of com- 
 pression, in consequence of the weight of the atmosphere 
 which is above it. 
 
 As the less weight that presses the air compresses it the 
 less, or causes it to be less dense, and as the higher we 
 rise in the atmosphere there will be the less weight, so the 
 higher we go in the atmosphere the air will be the less 
 dense. 
 
 216 
 
PRESSURE OP THE ATMOSPHERE. 217 
 
 The spring or elasticity of the air is equal to the weight 
 of the atmosphere above it, and they will produce the same 
 effects since they always sustain and balance each other. 
 
 If the density of the air be increased by compression, its 
 spring or elasticity is also increased, and in the same pro- 
 portion. 
 
 By the pressure and gravity of the atmosphere on the 
 surface of fluids such as water, they are made to rise in 
 pipes or vessels, where the spring or pressure within is 
 taken off or diminished. This fact, a knowledge of which 
 is applied to a multitude of useful purposes, will not be 
 difficult of explanation. LU a tube 3 feet long be filled 
 with water, the tube being open at one end and close at the 
 other; one unacquainted witn the subject might naturally 
 expect that if this tube were held perpendicularly with the 
 open end downmost, the water would flow out of the tube 
 by reason of its weight. But if we consider all the circum- 
 stances, we will see that this can only happen- on certain 
 conditions. The water has a tendency to fall to the earth 
 in consequence of its weight, but then the air of the atmo- 
 sphere, which we have stated before as also possessed of 
 weight, presses upon the surface of the water at the open 
 end of the tube ; and as the pressure of fluids of all kinds 
 is exerted in every direction, it follows, that the air will 
 have a tendency to force the water up the tube. Now the 
 pressure of the atmosphere at the surface of the earth is 
 about 15 Ibs. for every square inch, which is therefore the 
 force by which the water will be pressed up the tube by 
 the action of the air. A column of water 3 feet high does 
 not exert such a pressure on the base ; wherefore, as the 
 pressure upwards is greater than the pressure downwards, 
 the water will remain suspended in the tube. 
 
 Let us now take a tube 36 feet long, similar to the 
 former, filled with water and inverted in the same way as 
 before, it will now be found that a part of the water will 
 flow out of the tube, the reason of which will be easily seen. 
 It was stated under Hydrostatics, that the pressure of a 
 column of water 30 feet high was equal to 13 Ibs. on the 
 square inch. So that we see, that the pressure of the air 
 will keep 30 feet of the water in the tube, but it will keep 
 more, for the pressure of the air is 15, and that of 30 feet 
 of water is only 13 ; and as the pressure of the water will 
 be as its depth, we say, 13 : 15 : : 30 : 34, which, there- 
 
 19 
 
218 PNEUMATICS. 
 
 fore, is the greatest height at which the water will be sup- 
 ported by the pressure of the atmosphere. 
 
 For the purpose of arriving at this conclusion of the 
 effect of the pressure of the atmosphere, we might have 
 employed a much shorter tube if we had used a heavier 
 fluid than water, for instance, mercury. Now the cubic 
 foot of mercury weighs 13600 ounces, and a cubic inch will 
 be found, 13600 
 
 -J728- = 7-866 ounces, 
 
 or nearly 8 ounces, that is about half a pound avoirdupois; 
 therefore 30 inches will weigh 15 Ibs., hence, the atmo- 
 sphere will balance by its pressure 30 inches of mercury. 
 Thus we have arrived at the principle of the barometer, or 
 weather glass, as it is commonly called. The pressure of 
 the air at the surface of the earth is not always constant, 
 but varies within certain limits. The mean pressure is 
 about 14 Ibs. to the square inch, and the corresponding 
 height of the mercury in the barometer will therefore be 
 15 : 14 : : 30 : 28 inches. 
 
 It will appear evident, from what has been said before, 
 that as the higher we ascend in the atmosphere there will 
 be less pressure, and therefore the mercury in the barometer 
 will fall, and this fact has been used as a means of measur- 
 ing heights by the barometer. If the air were of the same 
 uniform density up to the top of the atmosphere as it is at 
 the earth's surface, we might very easily determine its 
 height, for the specific gravity of air being to that of water 
 as 1-222 to 1000, nearly, we have this proportion, 1-222 : 
 1000 : : 33-25, (the mean height of a water barometer in 
 feet,) : 27200 feet, which is very nearly 5| miles ; but by 
 a process which proceeds on correct principles, the height 
 of the atmosphere has been estimated at about 50 miles. 
 The law of the diminution of density at different heights in 
 the atmosphere is this, that if the heights increase in arith- 
 metical progression, the densities will decrease in geometri- 
 cal progression ; for instance, if the density at the surface 
 of the earth be called 1, and if at the height of 7 miles it 
 be called 4 times rarer than at 
 
 14 16, 
 
 21 it will be 64 times rarer, 
 
 28 256, 
 
 35 1024, 
 
PRESSURE OF THE ATMOSPHERE. 219 
 
 and in this way it might be shown, that at the height of one- 
 half the diameter of the earth, one cubic inch of atmospheric 
 air of the density which we breathe, would expand so much 
 as to fill the bounds of the solar system. 
 
 Many eminent men have investigated this subject, and 
 derived theorems of great use for determining altitudes by 
 the barometer. Some of these are exceedingly complex 
 and unfitted for a work of this nature : that of Sir J. Leslie 
 is the most simple, and gives results sufficiently near the 
 truth for all ordinary purposes. 
 
 As the sum of the heights of the mercury at the bottom 
 and top of the mountain is to the difference of the heights, 
 so is 52000 to the altitude of the mountain in feet. 
 
 At the bottom of a hill the barometer stood at 29-8, and 
 at the top 27-2, wherefore, 
 
 29-8 + 27-2 = 57 = the sum, 
 and 29-8 27-2 = 2'6 = the difference ; 
 hence, 57 : 2-6 : : 52000 : '2372 feet, the height of the 
 mountain nearly. 
 
 When air becomes denser, its elastic force is increased, 
 and that in proportion. Thus, when air is compressed into 
 half its bulk, its elastic force will be double of what it was 
 before. 
 
 It will, therefore, be easy to calculate the elastic force 
 of air compressed any number of times ; thus, if, by any 
 means, we condense the air in a vessel into 5 of the space 
 which it occupied when not confined, it will press on the 
 inside of the vessel with a force of 1.5 x 3 = 45 Ibs. on 
 every square inch. It must be remembered, however, that 
 the atmosphere presses with a force of 1 5 Ibs. on each square 
 inch of the outside of the vessel, which therefore counter- 
 acts so much of the force of the condensed air within the 
 real pressure, therefore, is 45 15 = 30 Ibs. It is clear, 
 then, that whatever be the degree of condensation of the 
 enclosed air, we must always deduct the pressure of the at- 
 mosphere to ascertain its true effect. The young mechanic 
 will easily understand what is meant by the phrase a 
 pressure of 2, 3, 4, or any number of atmospheres, one 
 atmosphere being understood as exerting a pressure of 15 
 Ibs. on the square inch, two atmospheres 30, and three 45, 
 <fec. When the air is by any means entirely taken out of 
 any vessel, there is said to be a vacuum in that vessel. 
 
 What is the whole amount of pressure on the inside sur- 
 
220 PNEUMATICS. 
 
 face of a sphere, which contains air condensed to | of its 
 natural bulk, and is 6 inches in diameter within. Here, 
 by mensuration, we have, 6- x 3-1416 = 113-0976 = the 
 surface of the inside of the sphere and 15x4 15 = 
 45 = the pressure on a square inch, therefore, 113*0976 X 
 45 = 5089-3920 Ibs. on the inner surface of the globe. 
 Here the globe is supposed to be in a vacuum. 
 
 In a cylinder 6 feet long, and closed at the bottom, a pis- 
 ton is thrust down to the distance of one foot from the bot- 
 tom, the cylinder being 24 inches in diameter, then, by the 
 rules in mensuration, the area of the piston will be found to 
 be 452-4 inches, the diameter of the piston being 24 inches, 
 and the cylinder being 6 feet long, and the piston being 
 pressed down to 1 foot from the bottom, the air will be com- 
 pressed into of its former bulk, and its elastic force will 
 be 6 times greater than it was before. At first it was 15 Ibs. 
 to the square inch, but now it will be 15 x 6 = 90 on the 
 square inch, and one atmosphere being deducted for the 
 contrary pressure of the atmosphere above the piston, the 
 pressure is 90 15 = 75 Ibs. to the square inch, where- 
 fore, 452-4 x 75 = 33930 Ibs., the force by which the 
 
 piston will be pressed upwards. 
 
 * 
 
 THE SYPHON. 
 
 A SYPHON, or, as it is frequently written, siphon, is any 
 bent tube. 
 
 If a syphon be filled with water and inverted, so that the 
 bend shall be uppermost, then if the legs be of equal 
 length, and it be held so that the two lower ends of the 
 syphon are on a level, then we will find that if the perpen- 
 dicular height of the bend of the tube above the level of the 
 two ends be not more than 32 or 33 feet, the water will re- 
 main suspended in the tube. It will not be difficult to see 
 how this happens, for the atmosphere pressing on the water 
 at the orifice of the tube at each extremity, presses the 
 water up the tube with a force capable of raising it 33 feet ; 
 but in the case supposed, the orifices and the legs are equal, 
 and do not exceed the limit of 32 or 33 feet, therefore, 
 since the pressure on one orifice is the same as the pressure 
 on the other, there will be an equilibrium and the water 
 in the one leg has no more power to move than that in the 
 other. 
 
 If we now suppose the syphon to be inclined a little, so 
 
PUMPS. 221 
 
 that the two orifices shall not be on a level, or what is the 
 same thing, if we suppose the length of the one leg to be 
 greater than that of the other, we will find that the equi- 
 librium will be no longer maintained ; and the water will 
 flow out of the orifice which is lowest. For although the 
 air presses equally on both orifices with -a force of 15 Ibs. 
 to the square inch, yet the contrary pressures downwards 
 by the weight of the water are not equal, therefore motion 
 will ensue where the power of the water is greatest. If 
 the shorter leg be immersed in a vessel of water, and the 
 syphon be set a running, the water will flow out of the 
 lower end of the syphon, until the other end be no longer 
 supplied. Instead of filling the syphon with water, as has 
 been supposed above, a common practice is to apply the 
 mouth to the lower orifice, and by sucking, exhaust the air 
 in the tube, which diminishes the pressure at the other 
 orifice, and consequently the action of the atmosphere will 
 force the water in the vessel up the tube of the syphon and 
 fill it, and it will continue to act in the same way as before. 
 
 PUMPS. 
 
 A PUMP is a machine used for exhausting vessels con- 
 taining air, or for raising water, sometimes by means of 
 the pressure of the atmosphere, sometimes by the condensa- 
 tion of air, and sometimes by a combination of both. 
 
 It may be necessary here to explain what is meant by the 
 term valve, that our remarks on the action of the pump may 
 be rendered more intelligible. 
 
 A valve is usually defined to be a close lid affixed to a 
 tube or opening in a vessel, by means of a hinge or some 
 sort of movable joint, .and which can be opened only in one 
 direction. There are various kinds of valves. The clack 
 valve consists merely of a circular piece of leather covering 
 the hole or bore of the pipe which it is intended to stop, 
 and moving on a hinge, sometimes a part of itself, and 
 sometimes made of metal. The butterfly valve, which is 
 superior to the clack valve, consists of two pieces of leather 
 each formed into the shape of a half circle ; they are at- 
 tached by hinges on their diameters, or straight parts, to a 
 bar that crosses the centre of the orifice to be closed. The 
 button or conical valve consists of a plate of brass ground 
 in such a way as exactly to fit the conical cavity in which it 
 lies. Sometimes valves are made in the form of pyramids 
 
 19* 
 
222 
 
 PNEUMATICS. 
 
 consisting of four triangular flaps which form the sides of 
 the pyramid, and move upon hinges which are placed round 
 the edge of the orifice to be closed. The tops of these 
 flaps must all meet accurately in the middle of the orifice, 
 and are supported by four bars which meet in the centre. 
 
 The action of the air pump may be thus explained. Lei 
 R be the section of a glass 
 bell, called a receiver, closed 
 at the top T, but open at the 
 bottom, and having its lower 
 edge ground smooth, so as 
 to rest in close contact with 
 a smooth brass plate, of 
 which SS is a section. In 
 the middle is an opening A, 
 which communicates by a 
 tube AB with a hollow cy- 
 linder or barrel, in which a 
 solid piston P is movi -d. 
 
 I 
 
 The piston rod C moves in 
 an air-tight collar D, and at 
 the bottom of the cylinder a valve V is placed, opening 
 freely outward, but immediately closed by any pressure 
 from without. There is thus a free comqiunication between 
 the receiver R, the tube AB, and the exhausting barrel BV. 
 When the piston CP is pressed down, and has passed the 
 opemng at B, the air in the barrel BV will be enclosed, and 
 will be compressed by the piston. As it will thus be made 
 to occupy a smaller space than before, its density, and con- 
 sequently its elasticity, will be increased. It will therefore 
 press downwards upon the valve V with a greater force 
 than that by which the valve is pressed upwards by the 
 external air. This superior elastic force will open the 
 valve, through which, as the piston descends, the air in the 
 barrel will be driven into the atmosphere. If the piston be 
 pushed quite to the bottom, the whole air in the barrel will 
 be thus expelled. The moment the piston begins to ascend, 
 the pressure of the air from without closes the valve V com- 
 pletely, and, as the piston ascends, a vacuum is left beneath 
 it ; but, when it rises beyond the opening B, the air in the 
 receiver R and the tube AB expands, by its elasticity, so 
 as to fill the barrel BV. A second depression of the piston 
 will expel the air contained in the barrel, and the process 
 
PUMPS. 223 
 
 may be continued at pleasure. The communication be- 
 tween the barrels and the receiver may be closed by a stop- 
 cock at G. In consequence of the elasticity of the air it 
 expands and fills the barrel, diffusing itself equally through- 
 out the cavity in which it is contained. The degree of 
 rarefaction produced by the machine may, however, be 
 easily calculated. Suppose that the barrel contains one- 
 third as much as the receiver and tube together, and, there- 
 fore, that it contains one-fourth of the whole air within the 
 valve V. Upon one depression of the piston, this fourth 
 part will be expelled, and three-fourths of the original quan- 
 tity will remain. One-fourth of this remaining quantity 
 will in like manner be expelled by the second depression 
 of the piston, which is equal to three-sixteenths of the ori- 
 ginal quantity. By calculating in this way, it will be found 
 that after thirty depressions of the piston, only one 3096th 
 part of the original quantity will be left in the receiver. 
 Rarefaction may thus be carried so far that the elasticity of 
 the air pressed down by the piston shall not be sufficient to 
 force open the valve. 
 
 We now proceed to the consideration of the common 
 suction pump. This pump consists of a hollow cylinder A, 
 of wood or metal, which contains a piston B, 
 stuffed so as to move up or down in the cy- 
 linder easily, and yet be air tight: to this 
 piston there is attached a rod which will reach I TJ~j i 
 at least to the top of the cylinder when the ~pl~ 
 piston is at the bottom. In the piston there 
 is a valve which opens upwards, and at the 
 bottom of the cylinder there is another valve 
 C also rising upwards, and which covers the - 
 orifice of a tube fixed to the bottom of the cy- 
 linder, and reaching to the well from whence 
 the. water is to be drawn. This tube is commonly called 
 the suction tube, and the cylinder, the body of the pump. 
 
 From what has been said of the pressure of the atmo- 
 sphere, it will not be difficult to understand how this ma- 
 chine operates. For when the piston is at the bottom of the 
 cylinder, there can be no air, or at least very little, between 
 it and the valve C, for as the piston was pushed down, the 
 valve in it would allow the air to escape instead of being 
 condensed, and when it is drawn up, the pressuie of the air 
 would shut the valve, and there would be a vacuum produced 
 
224 PNEUMATICS. 
 
 in the body of the cylinder when the piston arrived at the 
 top. But the air in the cylinder being very much rarified, 
 the pressure of the valve C on the water at the bottom will 
 be greatly less than that of the external atmosphere on the 
 surface of the water in the well ; therefore, the water will 
 be pressed up the pump to a height not exceeding 32 or 
 33 feet. As the valves shut downwards, the water is pre- 
 vented from returning, and the same operation being 
 repeated, the water may be raised to any height, not 
 exceeding the above limit, in any quantity. 
 
 The quantity of water discharged in a given time, is de- 
 termined by considering that at each stroke of the piston a 
 quantity is discharged equal to a cylinder whose base is the 
 area of a cross section of the body of the pump, and height 
 the play of the piston. Thus, if the diameter of the cylin- 
 der of the pump be 4 inches, and the play of the piston 3 
 feet, then, by mensuration, we have to find the content of a 
 cylinder 4 inches diameter, and 3 feet high now, 4 inches 
 is the j of a foot, or '333, hence, -333* x '7854 = -110999 
 X -7854 = -08796 = the area of the cross section of the 
 cylinder in square feet; hence, '08796 X 3 = -2639 the 
 content of the cylinder in cubic feet = the quantity in 
 cubic feet of water discharged by one stroke of the piston. 
 Now, a cubic foot of water weighs about 63*5 Ibs. avoirdu- 
 pois, wherefore, '2639 x 63-5 = 16-756 Ibs. avoirdupois, 
 and an imperial gallon is equal to 10 Ibs. of water ; whence, 
 dividing the above number 16-756 by 10, we get the num- 
 ber of ale gallons = 1-6756. The piston, throughout its 
 ascent, has to overcome a resistance equal to the weight of 
 a column of water, having the same base as the area of the 
 piston, and a height equal to the height of the water in the 
 body of the pump above the water in the well. 
 
 In our calculations of the effects of the pump, it will be 
 necessary to determine the contents of pipes, for which 
 purpose the following simple rules will serve. 
 
 Diameter of pipe in inches 9 = number of avoirdupois 
 poumls contained in 3 feet length of the pipe. 
 
 If the last figure of this be pointed oft as a decimal, the 
 result will be the number of ale gallons, and if there be 
 only one figure this is to be considered as so many tenths 
 of an ale gallon : ale gallons x 282 = the number of cubic 
 inches. 
 
 Thus, in a pipe 5 inches diameter, we have, 
 
PUMPS. 225 
 
 5 9 = 25 = number of avoirdupois pounds contained in 3 
 feet of the pipe 2-5 = the number of ale gallons and 2'5 X 
 282 = 705 cubic inches. 
 
 These rules find the content for three feet in length of 
 the pipe, the content for any other length may be found by 
 proportion ; thus, for a pipe 6 inches in diameter, and 12 
 feet long ; we have, 6 s = 36 = pounds of water avoir, con- 
 tained in the pipe to the length of 3 feet ; therefore, 
 
 3 : 12 : : 36 : 144 = the number of pounds in 12 feet 
 length, and, 
 
 14-4 = ale gallons, and 14-4 x 282 = 4060*8 = the 
 cubic inches in 12 feet length. 
 
 The resistance which is opposed to a pump rod in raising 
 water, is equal to the weight of a column of water whose 
 base is the area of the piston, and height the height of the 
 surface of the water in the body of the pump above the 
 surface of the water in the well, together with the friction 
 and the piston and piston rod. 
 
 Suppose the body of the pump to be 6 inches in diameter, 
 and the height to which the water is raised be 30 feet, and 
 also the weight of the piston and rod is 10 Ibs., and the 
 friction is \ of the whole weight of the water. 
 
 Now, 6 a = 36 = the Ibs. avoirdupois of 3 feet of the 
 column of water, but the column is 30 feet, therefore, 3 : 
 30 : : 36 : 360 Ibs., the weight of the whole column. To 
 this we must add the effect of friction, which we have sup- 
 posed to be 4 of the weight of the water ; hence, 
 9 An 
 
 - = 72 Ibs., and this must be added to the weight of 
 
 
 the column of water, which gives 360 -f 72 = 432 Ibs. the 
 whole amount of resistance arising from the weight of the 
 water and friction ; to this must be added the weight of the 
 piston and pump rod, therefore, 432 -f 10 = 442 = the 
 whole resistance opposed to the rising of the piston, any 
 thing greater than this will raise it. 
 
 In the construction of pumps it is usual to employ a lever 
 to work the piston, which gives an advantage in power ; 
 and if in the case estimated above, we employ a lever whose 
 arms are in the proportion of 10 to 1, the pump might be 
 wrought with a force of 44'2 Ibs., or we may say 45 Ibs. 
 
 For the convenience of workmen we insert the following 
 table. It has been calculated on the supposition that the 
 handle of the pump is a lever which gives an advantage on 
 
226 
 
 PNEUMATICS. 
 
 the piston rod of 5 to 1, and that a man can, with a pump 
 30 feet long, and a 4 inch bore, discharge 27'5 wine gallons 
 (oil measure) in a minute. And if it be required to find 
 the diameter of a pump that a man could work with the 
 same ease as the above pump at any required height above 
 the surface of the well, this table will give the diameter of 
 bore, and the quantity of water discharged in a minute. . 
 
 Height of ebeownp 
 
 towtewfiiMor 
 
 the mill in (eet. 
 
 Diameter of the Inn 
 whm the pbtoo 
 work, in (S3 
 
 WUardkeharpdper 
 minute in wine iiUMtol, 
 plloo.aadp.nta. 
 
 10 
 
 6*93 
 
 81 6 
 
 15 
 
 5-66 
 
 54 4 
 
 20 
 
 4-90 
 
 40 7 
 
 25 
 
 4-38 
 
 32 6 
 
 30 
 
 4- 
 
 27 2 
 
 35 
 
 3-70 
 
 23 3 
 
 40 
 
 3-46 
 
 20 3 
 
 45 
 
 3-27 
 
 18 1 
 
 50 
 
 3-10 
 
 16 3 
 
 55 
 
 2-95 
 
 14 7 
 
 60 
 
 2-84 
 
 13 5 
 
 65 
 
 2-72 
 
 12 4 
 
 70 
 
 2-62 
 
 11 5 
 
 75 
 
 2-53 
 
 10 7 
 
 80 
 
 2-45 
 
 10 2 
 
 85 
 
 2-3$ 
 
 9 5 
 
 90 
 
 2-31 
 
 9 1 
 
 05 
 
 2-25 
 
 8 5 
 
 100 
 
 2-19 
 
 8 -1 
 
 We stated before that water could not be raised to a 
 greater height than 32 feet by means of the kind of pump 
 we have described, and it may seem strange that this table 
 extends to 100 ; but these are pumps acting on a different 
 principle, by means of which water may be raised to any 
 height, and whose action will be considered before we 
 leave this subject. 
 
 The lifting pump. This pump, like the suction pump, has 
 two valves and a piston, both opening upwards ; but the 
 valve in the cylinder, instead of being placed at the bottom 
 of the cylinder, is placed in the body of it, and at the height 
 
PUMPS. 227 
 
 where the water is intended to be delivered. The bottom 
 of the pump is thrust into the well a considerable way, and 
 if the piston be supposed to be at the bottom, it is plain, 
 that as its valve opens upwards, there will be no obstruction 
 to the water rising in the cylinder to the height which it is 
 in the well ; for, by the principles of Hydrostatics, water 
 will always endeavour to come to a level. Now when the 
 piston is drawn up, the valve in it will shut, and the water 
 in the cylinder will be lifted up ; the valve in the barrel 
 will be opened, and the water will pass through it, and can- 
 not return, as the valve opens upwards ; another stroke 
 of the piston repeats the same process, and in this way the 
 water is raised from the well : but the height to which it 
 may be raised, is not in this, as in the suction pump, limited 
 to 32 or 33 feet. To ascertain the force necessary to work 
 this pump, we are to consider that the piston lifts a column 
 of water whose base is the area of the piston, and height 
 the distance between the level of the water in the well and 
 the spout, at which the water is delivered. Thus, if the 
 diameter of the pump's bore be 4 inches, and the height 
 of the spout above the level of the well = 40 feet, then we 
 have 4 3 == 16 Ibs. in three feet of the barrel ; wherefore, 
 
 3 : 40 : : 16 : 213| Ibs. the weight of the water, and the 
 friction and weight of the piston and rod must be added to 
 this, to find the whole force necessary. If the friction be 
 reckoned, as it usually is, |, then we have, 
 
 wherefore, 213 -|- 42 = 255; as we have neglected frac- 
 tions we may reckon it 256, and if the weight of the piston 
 and rod be 20 Ibs. the whole will be 256 -f 20 = 276 Ibs., 
 the whole force necessary to balance the piston ; any thing 
 greater than this will raise it. 
 
 The forcing pump remains to be consi- 
 dered. The piston of this pump has no 
 valve, but there is a valve at the bottom of 
 the cylinder, the same as seen at A. In the 
 aide of the cylinder, and immediately above 
 the valve B, there is another valve A open- 
 ing outwards into a tube which is bent up- 
 wards to the height H at which the water 
 is to be delivered. When the piston is 
 raised, the valve in the bottom of the pump 
 
228 PNEUMATICS. 
 
 opens, and a vacuum being produced, the water is pressed 
 up into the pump on the principle of the sucking pump. 
 But when the piston is pressed down, the valve A at the 
 bottom shuts, and the valve B at the side which leads into 
 the ejection pipe opens, and the water is forced up the tube. 
 When the piston is raised again the valve B shuts, and the 
 valve A opens. The same process is repeated, and the 
 water is thrown out at every descent of the piston, the dis- 
 charge therefore is not constant. 
 
 It is frequently required that the dis- 
 charge from the pump should be continu- 
 ous, and this is effected by fixing to the 
 top of the eduction pipe an air vessel. 
 This air vessel consists of a box AB, in 
 the bottom of which there is a valve C 
 opening upwards into the box. This valve 
 covers the top of the eduction pipe D. A 
 tube, , is fastened into the top of the box, 
 which reaches nearly to the bottom of the box, it rises out 
 of the box, and is furnished with a stop cock. If the stop 
 cock be shut, and the water be sent by the action of the 
 pump into the air vessel, it cannot return because of the 
 shutting of the valve at the bottom of the box ; and be- 
 cause of the space occupied by the water, the air in the box 
 is condensed, and will consequently exert a pressure on the 
 water in the air vessel. If the water fill three-fourths of 
 the box, then the air will be compressed so as to exert four 
 times its original force ; and the stop cock being opened, 
 the water will be forced up the tube, with a force which 
 will send it one less than as many times 32 feet as the sir 
 is compressed, that is, in the case supposed 3 x 32 = 96 
 feet. On this principle it is that jets of fountains act. 
 
 The air vessel may therefore be considered as a magazine 
 of power, and so long as there is as much water forced into 
 the air vessel by pumping, as there is forced out by the 
 pressure of the air, there will be a constant jet of water. 
 
 The force necessary to raise the piston in this pump, is 
 found exactly in the same way as for the suction pump. 
 And the force necessary to depress the piston, is found by 
 taking the weight of a column of water, whose height is 
 equal to the height of the spout where the water is de- 
 livered above the level of the piston, before it begins to 
 descend. Thus, if the piston when raised is 26 feet above 
 
WINDMILLS. 229 
 
 the level of the well, and the spout is 63 feet above the same 
 level, therefore, the height of the column is 63 26 = 37 
 feet; and supposing the diameter of the ejection pipe to 
 be 5 inches, we have for 3 feet of the pipe 5 3 = 25 Ibs., 
 wherefore for 37 feet we have, 
 
 3 : 37 :: 25 : 308 Ibs. 
 
 The weight of the piston and its rods oppose the raising of 
 the piston, but assist in depressing it. 
 
 The power applied to the piston rod of a suction pump 
 should be an intermitting power, otherwise there will be a 
 waste of power ; but in a forcing pump the power must be 
 continued throughout equable. A single stroke steam en- 
 gine will be best to raise water by the sucking, and a 
 double stroke by a forcing pump. The piston rod of a 
 forcing pump should be loaded with a weight sufficient to 
 balance a column of water, whose base is the section of the 
 piston, and whose height is the excess of the height of the 
 spout from the level of the water in the cistern above 68 
 feet. This will cause a regular application of power when 
 this pump is wrought with a steam engine. 
 
 WIND AND WINDMILLS. 
 
 WE have seen the effect of the pressure of air, arising 
 from its weight and elasticity when at rest ; it now remains 
 for us to consider its effects when put in motion, as in the 
 case of wind. 
 
 Were it not for the irregularity in direction and force of 
 the wind, it would be the most convenient of all the first 
 movers of machinery, but even as it is, its efficacy may be 
 taken advantage of, and deserves our consideration. 
 
 The force with which wind strikes against a surface, is 
 as the square of the velocity of the wind. This simple 
 theorem is so nearly true that it may be employed without 
 fear of error. 
 
 The force in avoirdupois pounds with which the wind 
 strikes on any surface on which it acts perpendicularly may 
 be found by using the rule, 
 
 surface struck X velocity of wind 9 X "002288 ; 
 where the surface and velocity of wind are taken in feet, 
 and the time 1 second. If the wind moves at the rate of 
 30 feet per second, and the surface exposed to its action be 
 14 feet square, then, 14 X 30 9 x '002288 = 28-8288. 
 
 From this statement it might appear at first sight, that in 
 20 
 
230 
 
 PNEUMATICS. 
 
 the case of mills which act by the impulse of wind on re- 
 volving surfaces called sails it might appear, we say, that 
 the greater quantity of sail exposed to the action of the 
 wind, the greater would be the effect of the machine. But 
 this has been found not to hold : it would appear that the 
 wind requires space to escape. The sails of the windmill 
 may be supposed to intercept a cylinder of wind ; and it 
 would seem, that when the whole cylinder is intercepted, 
 the effect of the machine is diminished ; and it is concluded 
 from experiments, that the sails should not intercept above 
 seven-eighths of the cylinder of the wind. 
 
 We here subjoin a tabular view of the effects of wind at 
 different velocities. 
 
 Tdbk showing the pressure of the Wind for the following Velocities. 
 
 Velocity of the Wind. 
 
 Force upon 1 square foot In 
 pound* avoir. 
 
 Mile* in 1 hour. 
 
 Feet In 1 aecond. 
 
 1 
 
 1-47 
 
 005 
 
 2 
 
 2-93 
 
 020 
 
 3 
 
 4-40 
 
 044 
 
 4 
 
 5-87 
 
 079 
 
 5 
 
 7-33 
 
 123 
 
 10 
 
 14-67 
 
 492 
 
 15 
 
 22-00 
 
 1-107 
 
 20 
 
 29-34 
 
 1-968 
 
 25 
 
 36-67 
 
 3-075 
 
 30 
 
 44-01 
 
 4-429 
 
 35 
 
 51-34 
 
 6-027 
 
 40 
 
 58-68 
 
 7-873 
 
 45 
 
 66-01 
 
 9-963 
 
 50 
 
 73-35 
 
 12-300 
 
 60 
 
 88-02 
 
 17-715 
 
 80 
 
 117-36 
 
 31-490 
 
 100 
 
 146-70 
 
 49-200 
 
 Windmills are constructed either so that the sails shall 
 move in a horizontal plane, or in a plane nearly vertical ; 
 their former are called horizontal, and the latter vertical 
 windmills. In plate 2, fig. 1 and 2, we have given a plan 
 and section of a horizontal windmill, on an improved con- 
 
AII hi, 
 
 Published by Thonias V&rile Philadelphia . 
 

WINDMILLS. 231 
 
 struction. HH are the side walls of an octagonal building 
 which contains the machinery. These walls are surmounted 
 by a strong limber framing GG, of the same "form as the 
 building, and connected at top by cross-framing to support 
 the roof, and also the upper pivot of the main vertical 
 shaft AA, which has three sets of arms, BB, CC, DD, 
 framed upon it at that part which rises above the height 
 of the walls. The arms are strengthened and supported 
 by diagonal braces, and their extremities are bolted to 
 octagonal wood frames, round which the vanes or floats EE 
 are fixed, as seen in outline in fig. 2, so as to form a large 
 wheel, resembling a water wheel, which is less than the 
 size of the house by about 18 inches all round. This space 
 is occupied by a number of vertical boards or blinds FF, 
 turning on pivots at top and bottom, and placed obliquely, 
 so as to overlap each other, and completely shut out the 
 wind, and stop the mill, by forming a close case surround- 
 ing the wheel ; but they can be moved altogether upon 
 their pivots to allow the wind to blow in the direction of a 
 tangenj, upon the vanes on one side of the wheel, at the 
 time the other side is completely shaded or defended by 
 the boarding. The position of the blinds is clearly shown 
 at FF, fig. 2. At the lower end of the vertical shaft AA, 
 a large spur-wheel aa is fixed, which gives motion to a 
 pinion c, upon a small vertical axis d, whose upper pivot 
 turns in a bearing bolted to a girder of the floor n. Above 
 the pinion c, a spur-wheel e is placed, to give motion to 
 two small pinions /", on the upper ends of the spindles g, 
 of the millstone h. Another pinion is situated at the 
 opposite side of the great spur-wheel aa, to give motion to 
 a third pair of millstones, which are used when the wind 
 is very strong ; and then the wheel turns so quick as not 
 to need the extra wheel e to give the requisite velocity to 
 the stones. The weight of the main vertical shaft is borne 
 by a strong timber 6, having a brass box placed on it to 
 receive the lower pivot of the shaft. It is supported at its 
 ends by cross-beams mortised into the upright posts 66, as 
 shown in the plan, fig. 2. A floor or roof 1 1 is- thrown 
 across the top of the brick building to protect the machinery 
 from the weather, and to prevent the rain blowing down 
 the opening through which the shaft descends, a broad cir- 
 cular hoop K is fixed to the floor, and is surrounded by 
 another hoop or case L, which is fixed to the arms DD of 
 
232 PNEUMATICS. 
 
 the wheel. This last is of such a size, as exactly to go 
 over the hoop K, without touching it when ihe wheel turns 
 round. By this means, the rain is completely excluded 
 from the upper room M, which serves as a granary, being 
 fitted up with the bins mm, to contain the different sorts 
 of grain which is raised up by the sack-tackle. A wheel i 
 is fixed on the main shaft, having cogs projecting from both 
 sides. Those at the under side work into a pinion on the 
 end of the roller K, which is for the purpose of drawing 
 up sacks. Another pinion is situated above the wheel t, 
 which has a roller projecting out over the flap-doors seen 
 at />, in fig. 2, to land the sacks upon. The two pinions 
 7nm, fig. 2, are turned by the great wheel aa, and are for 
 giving motion to the dressing and bolting machines, \vhirh 
 are placed upon the floor N, but are not shown in the 
 drawing, being exactly similar to the dressing machines 
 used in all flour-mills. The cogs upon the great wheel a 
 are not so broad as the rim itself, leaving a plain rim about 
 three inches broad. This is encompassed by a broad iron 
 hoop, which is made fast at one end to the upright.post b ; 
 the other being jointed to a strong lever n, to the extn m< 
 end of which a purchase o is attached, and the fall is radfc 
 fast to iron pins on the top of a frame fixed to the ground. 
 This apparatus answers the purpose of the brake or gripe 
 used in common windmills to stop their motion. By pull- 
 ing the fall of the purchase o, it causes the iron strap to em- 
 brace the great wheel, and produces a resistance sufficient 
 to stop the wheel. The mill can be regulated in its motion, 
 or stopped entirely, by opening or shutting the blinds F, 
 which surround the fan-wheel. They are all moved at 
 once by a circular ring of wood situated just beneath the 
 lower ends of the blinds upon the floor 1 1, being connected 
 with each blind by a short iron link. The ring is moved 
 round by a rack and spindle which descend into the mill 
 room below, for the convenience of the miller. The mode 
 of bringing the sails back against the wind, which Mr. 
 Beatson invented, is, perhaps, the simplest and best for 
 that end. He makes each sail AI, fig. 3, to consist of six 
 or eight flaps or vanes, AP 61,61 c 2, &c., moving upon 
 hinges represented by the dark lines, AP b 1, c 2, <fec., so 
 that the lower side b 1 of the first flap wraps over the hinge 
 or higher side of the second flap, and so on. When the 
 wind, therefore, acts upon the sail AI, each flap will press 
 
WINDMILLS. 233 
 
 upon the hinge of the one immediately below it, and the 
 whole surface of the sail will be exposed to its action. 
 But when the sail AI returns against the wind, the flaps 
 will revolve round upon their hinges, and present only 
 their edges to the wind, as is represented at EG, so that 
 the resistance occasioned by the return of the sail must be 
 greatly diminished, and the motion will be continued by 
 the great superiority of force exerted upon the sails in the 
 position AI. In computing the force of the wind upon the 
 sail AI, and the resistance opposed to it by the edges of the 
 flaps in EG, Mr. Beatson finds, that when the pressure 
 upon the former is 1872 pounds, the resistance opposed by 
 the latter is only about 36 pounds, or T l j part of the whole 
 force ; but he neglects the action of the wind upon the 
 arms, CA, &c., and the frames which carry the sails, be- 
 cause they expose the same surface in the position AI, as 
 in the position EG. This omission, however, has a ten- 
 dency to mislead us in the present case, as we shall now 
 see ; for we ought to compare the whole force exerted 
 upon the arms, as well as the sail, with the whole resistance 
 which these arms and the edges of the flaps oppose to the 
 motion of the windmill. By inspecting the figure it will 
 appear, that if the force upon the edges of the flaps, which 
 Mr. Beatson supposed to be 12 in number, amounts to 36 
 pounds, the force spent upon the bars CD, DG, GF, FE, 
 &c., cannot be less than 60 pounds. Now, since these bars 
 are acted upon with an equal force, when the sails have the 
 position AI, 1872 -f 60 = 1932 will be the force exerted 
 upon the sail AI, and its appendages, while the opposite 
 force upon the bars and edges of the flaps when returning 
 against the wind will be 36 -f- 60 = 96 pounds, which is 
 nearly ^ of 1932, instead of /^ as computed by Mr. Beat- 
 son. Hence we may see the advantages which will pro- 
 bably arise from using a screen for the returning sail instead 
 of movable flaps, as it will preserve not only the sails, but 
 the .arms and the frame which supports it. from the action 
 of the wind. 
 
 Figures 4 and 5, plate 2d, represent the most improved 
 form of the vertical windmill ; aaaa, are the vanes or sails 
 of the mill, which communicate motion to the wind-shaft b 
 and the crown wheel c ; rf, the centre wheel which conveys 
 this motion along the shaft e to the spur-wheel f; g, a 
 wheel, or trundle, on the end of the spindle of the upper 
 20* 
 
834 PNEUMATICS. 
 
 or turning millstone ; f, the case in which the millstones 
 are placed ; k, the bridge-tree which supports the spindle 
 of the turning-stone ; /, another wheel, or trundle, on the 
 end of the shaft m, which conveys the motion lower down 
 the building to another spur-wheel n; this spur-wheel puts 
 other two millstones in motion at pleasure, in the same 
 manner as the former ; 0, the brake, or rubber, for stopping 
 the mill, it operates by friction ; />, the governor for regu- 
 lating the motion, by opening or shutting the wind-boards 
 on the vanes ; q, the director which carries round the roof 
 with the wind, by keeping the vanes always at right angles 
 to it. On the spindle of this director is placed an endless 
 screw, working into a wheel which turns a shaft having a 
 pinion fixed at the other end of it. This pinion works into 
 another wheel connected with the rack pinion, which pin- 
 the whole roof in motion. 
 
 The wind does not act perpendicularly on the sails of a 
 wind-mill, but at a certain angle, and the sail varies in the 
 degree of its inclination at different distances from the 
 centre of motion, in resemblance to the wing of a bir.l : 
 this is called the weathering of the sail. The angles of 
 weathering have been found by Smeaton as follows. The 
 radius being divided into 6 equal parts, and the first part 
 from the centre being called 1, the last 6. 
 
 Distance from Angle with Angle with the 
 
 the centre. the axis. plane of motion. 
 
 1 72 18 
 
 2 71 19 
 
 3 72 18 
 
 4 74 16 
 
 5 77| 12i 
 
 6 83 7 
 
 Smeaton gives the following maxims for the construction 
 of windmills. 
 
 1. The velocity of the windmill sails, whether unloaded 
 or loaded, so as to produce a maximum, is nearly as the 
 velocity of the wind, their shape and motion being the same. 
 2. The load at the maximum is nearly but somewhat less 
 than, as the square of the velocity of the wind, the shape 
 and position of the sails being the same. 3. The effects of 
 the same sails at a maximum are nearly but somewhat less 
 than, as the cubes of the velocity of the wind. 4. The load 
 
WINDMILLS. 
 
 235 
 
 of the same sails at the maximum is nearly as the squares, 
 and their effects as the cubes of their number of turns in a 
 given time. 5. When the sails are loaded so as to produce 
 a maximum at a given velocity, and the velocity of the 
 wind increases, the load continuing the same, then, when 
 the increase of the velocity of the wind is small, the effect 
 will be nearly as the squares of the velocities ; but when 
 the velocity of the wind is double, the effects will be nearly 
 as 10 to 27] ; and when the velocities compared are more 
 than double of that where the given load produces a maxi- 
 mum, the effect increases only as the increase of the velo- 
 city of the wind. 6. If sails are of a similar figure and 
 position, the number of turns in a given time will be in- 
 versely as the radius of length of the sail. 7. The load at 
 a maximum that sails of a similar figure and position will 
 overcome, at a given distance from the centre of motion, 
 will be as the cube of the radius. 8. The effect of sails of 
 similar figure and position are as the square of the radius. 
 
 Rules for modelling the sails of Windmills. 
 
 The accompanying cut is the 
 front view of one sail of a wind- 
 mill. The length of the arm 
 AA, called by workmen the 
 whip, is measured from the 
 centre of the great shaft B, to 
 the outermost bar 19. The breadth of the face of the whip 
 A next the centre, is ^ of the length of the whip, and its 
 thickness at the same end is | of the breadth; and the back 
 side is made parallel to the face for half the length of the 
 whip : the small end of the whip is square, and at its end 
 is 1-1 6th of the length of the whip, or half the breadth at 
 the great end. 
 
 From the centre of the shaft B, to the nearest bar 1 of 
 the lattice is l-7th of the whip, the remaining space of 
 6-7ths of the whip is equally divided into 19 spaces ; l-9th 
 of one of these spaces gives the size of the mortice, which 
 must be made square. 
 
 To prepare the whip for mortising, strike a gauge score at 
 about three-quarters of an inch from the face on each side, 
 and the gauge score on the leading side, 4, 5, will give the 
 face of all the bars on each side ; but on the other side the 
 faces of all the bars will fall deeper than the gauge score, 
 
236 PNEUMATICS. 
 
 according to a certain rule, which is this : Extend the 
 compasses to any distance at pleasure, so that 6 times that 
 extent may be greater than the breadtii of the whip at the 
 seventh bar. Set off these six spaces upon a straight line 
 for a base, at the end of which raise a perpendicular ; set 
 off the same six spaces on the perpendicular, and divide the 
 two spaces on the perpendicular which are farthest from 
 the base, each into 6 equal parts, so that these two spaces 
 will contain 13 points. Join each of these 13 points with 
 the end of the base farthest from the perpendicular. 
 
 To apply this scale to any given case, set off the breadtii 
 of the whip at the last bar (that is the bar at the extremity 
 of the sail) from the centre of the scale, along the base to- 
 wards the perpendicular, and at this point raise a perpen- 
 dicular to cut the oblique line nearest the base ; also set off 
 the breadth at the seventh bar in the same manner, and at 
 this point raise a perpendicular to cut off the thirteenth 
 oblique line. Now, from the point where the first of these 
 two perpendiculars cuts the first oblique line from the base, 
 to the intersection of the second perpendicular with the 
 thirteenth oblique line, there is drawn a line joining the 
 two points of intersection ; and perpendiculars being drawn 
 from the points where this joining line cuts the oblique 
 lines to the base, will be the several distances of the face 
 of each bar from the gauge line. These distances give a 
 difference, set off for each bar to the seventh, which must 
 be set off for all the rest to the first. The length of the 
 longest bar is of the whip. 
 
 We now proceed to show the method of weathering the 
 sails. Draw AB = the length of the vane, BC its breadth, 
 and BCD the angle of the weather at the extremity of the 
 vane, equal to 20 degrees. With the length of the vane 
 AB, and breadth BC, construct the isosceles triangle ABC ; 
 and from the point B, draw BD perpendicular to CB, then 
 BD is the proper depth of the vane. 
 
 Divide the line AB into any number of parts, say four, 
 at these divisions draw the lines 1 E, 2 F, 3 G, &c. 
 
WINDMILLS. 237 
 
 parallel to the line BC. Also, from the points of division, 
 1, 2, 3, &c., draw the lines 1 i, 2 k, 3 1, &c. perpendicular 
 to 1 E, 2 F, 3 G, &c., all of them equal in length to BD. 
 Join Ei, Fk, Gl, &c., then the angles 1 Ei, 2 Fk, 3 Gl, 
 <fcc., are the angles of the weather for these divisions of the 
 vane ; and if the triangles be conceived to stand perpen- 
 dicular to the paper, the angles i, k, 1, and D, denoting the 
 vertical angles, the hypotenuses of these triangles will 
 
 Eive a perfect idea of the weathering of the vane as it recedes 
 oin the centre. 
 
HEAT, STEAM, &c. 
 
 IT would be out of place in a work of this nature to 
 enter into a minute detail respecting the nature of heat ; in 
 this section, therefore, we shall confine ourselves to a de- 
 scription of the more important of its mechanical properties. 
 
 Heat expands bodies, that is, increases their dimensions. 
 Different bodies expand differently by the application of 
 the same quantity of heat. With the same degree of heat, 
 solids expand less than liquids, and liquids less than gases. 
 
 On the principle that bodies expand by heat, is con- 
 structed the Thermometer. The action of this instrument 
 is very simple. It consists of a small glass tube with a 
 hollow bulb at one end, and at the other end it is closed. 
 The bulb is filled with mercury, as likewise a part of the 
 tube, the other portion of the tube being entirely deprived 
 of air. When heat is applied to the bulb of the thermome- 
 ter, the mercury expands and rises in the tube, and accord- 
 ing to the degree of heat applied to it, so will the mercury 
 rise. To the tube there is attached a divided scale, to de- 
 note the degrees of heat by the rising of the mercury, which 
 scale is thus formed. The bulb of the thermometer is put 
 into melting ice, and the height of the mercury is marked 
 on the scale ; this is called the freezing point, and num- 
 bered 32. The bulb is then put into boiling water, and the 
 height of the mercury in the tube is marked upon the scale 
 and numbered 212 this is called the boiling point. The 
 space betwixt these two points on the scale is divided into 
 180 equal parts, called degrees, and the scale is then ex- 
 tended both above and below these points. This is the 
 scale commonly used in this country, and is known by the 
 name of its inventor, Fahrenheit. But the French and many 
 philosophers in Britain use a thermometer having a scale 
 of much more simple construction, called, from the nature 
 of its divisions, the Centigrade scale. The freezing point, 
 which in Fahrenheit is marked 32, is in the Centigrade 
 
 238 
 
HEAT. 239 
 
 marked or zero ; and the boiling point, in Fahrenheit 
 marked 212, is in the Centigrade marked 100. In Reaumur's 
 thermometer the freezing point is marked 0, and the boiling 
 point 80. 
 
 Let F represent Fahrenheit, R Reaumur, and C Centi- 
 grade, then we have the following rules for converting the 
 degrees of any one of these thermometers into the corres- 
 ponding temperature, as marked in the others : 
 
 (1.) F = C xl-8 -f 32. 
 
 (2.) F =-^5 + 32. 
 
 w -^- , V^ ^ ' 
 
 < 4 -) c =iS- v< iV' r - * . 
 
 (5.) B-lffz^). 
 
 (6.) R = C x 0-8. 
 
 Thus 185 Fahrenheit's will be found to correspond to 
 85 of the Centigrade, and 68 of Reaumur's thermometer. 
 (1.) 85 x 1-8 -f 32 = 185. 
 
 (3.) = = 85. 
 
 <> - >' 
 
 (5.) * X (185 -32) = 6g 
 
 ML 
 
 (6.) 85 X 0-8 = 68. 
 
 There are many other particulars regarding the thermo- 
 meter which it would be inconsistent with the design of 
 these pages to consider : what we have said will be sufficient 
 for the understanding of what is hereafter to follow on the 
 subject of steam, &c. 
 
 Before we introduced the subject of the thermometer, we 
 stated the fact of the expansion of bodies by heat. Bars 
 of the following substances, whose length at a temperature 
 
240 HEAT. 
 
 of 32 was 1, were heated to 212 Fahrenheit, and expanded 
 so as to become, 
 
 Cast iron, 1-00110940 
 
 Steel, 1-00118990 
 
 Copper, 1-00191880 
 
 Brass, 1-00188971 
 
 This is the expansion in length ; the expansion in length, 
 breadth, and thickness, will be found by multiplying the 
 above numbers by 3. 
 
 The effects of different degrees of heat on different bodies, 
 according to Fahrenheit's scale, are shown below. 
 
 Cast iron thoroughly melted, 20577 
 
 Cast iron begins to melt, 17977 
 
 Greatest heat of a common smith's forge, 17327 
 
 Flint glass furnace, strongest heat, 1 5897 
 
 Welding heat of iron, (greatest) 1 3427 
 
 Swedish copper melts, 4587 
 
 Brass melts, 3807 
 
 Iron red hot in the twilight, 884 
 
 Heat of a common fire, 790 
 
 Iron bright red in the dark, 752 
 
 Zinc melts, 700 
 
 Mercury boils, 672 
 
 Lead melts, 594 
 
 The surface of polished steel becomes uniformly 
 
 deep blue, 580 
 
 The surface of polished steel becomes a pale 
 
 straw colour, 460 
 
 Tin melts, 442 
 
 A mixture of 3 tin and 2 lead melts, 332 
 
 Heat passes through different bodies with very different 
 degrees of velocity, and according to the rapidity or slow- 
 ness with which heat passes through any body, it is said to 
 be a good or a bad conductor of heat. The conducting 
 power of copper being 1, that of brass will be 1, iron, 1-1, 
 tin, 1*7, lead, 2-5. The densest bodies are generally the 
 best conductors of heat ; but this is not universal, as platina, 
 one of the densest of all metals, is one of the worst con- 
 ductors. Earthy substances are much inferior to metals in 
 their conducting power, and the worst conductors of all are 
 the coverings of animals. 
 
 When heated bodies are exposed to the air they lose por- 
 tions of their heat by projection in right lines into space 
 from all parts of their surface. This is called the radiation 
 
HEAT. 241 
 
 of heat. Bodies which radiate heat best have the power of 
 absorbing it in the same proportion, and the least power 
 of reflecting it; hence, in leading steam through a room, 
 it would be absurd to use black pipes, because, in that case, 
 much of the heat would escape by radiation before the 
 steam would be carried to the place where it was to be used. 
 If the steam is used to heat the apartment, black pipes are 
 the best. Hence the cylinder of a steam engine ought to 
 be polished, but the condenser should not. Vessels in- 
 tended to receive heat should be black. 
 
 The comparative quantities of heat existing in different 
 bodies may be ascertained by marking the time which equal 
 quantities of them require to cool a certain number of de- 
 grees, reckoning their capacities for heat to be as these 
 times estimated by the volume ; or, if divided by the spe- 
 cific gravity of the substance, by the weight. 
 
 It is necessary here to distinguish carefully between what 
 is called the specific heat of a body, and its capacity for heat, 
 these two terms being often confounded. If we take two 
 bodies at the same temperature, and expose them to the 
 action of a greater heat, it will be found that one body 
 will have absorbed a greater quantity of heat than the 
 other, by the time that they have acquired an equal tem- 
 perature ; and the amount of this additional heat, referred 
 to some standard, is denominated the specific heat of the 
 body. Thus if it be found that it requires 1 degree of heat 
 to raise water from one temperature, T, to another tempera- 
 ture, /, and if to produce the same change of temperature 
 in steam it requires 0-847 degrees, then is 0-847 the specific 
 heat of steam, water, as the standard, being I'OOO. The 
 capacity of one body for heat compared to another is not 
 the relative quantities of heat required to raise them a 
 certain number of degrees, but the absolute quantities con- 
 tained in them at the same temperature. 
 
 CAPACITIES OF BODIES FOR HEAT. 
 
 OASES. 
 
 Atmospheric air, 1 -7900 
 
 Aqueous vapour, 1 '5500 
 
 Carbonic acid gas, 1-0454 
 
 LIQUIDS. 
 
 Alcohol, 1-0860 
 
 Water, 1-0000 
 
 21 
 
242 
 
 HEAT. 
 
 Solution of muriate of soda, 1 in 10 of water, -9360 
 
 Sulphuric acid, diluted with 10 parts water, -9250 
 
 Solution of muriate of soda in 6*4 of water, -9050 
 
 Olive oil, -7100 
 
 Nitric acid, specific gravity 1-29895, '6613 
 
 Sulphuric acid, with an equal weight of water, '6050 
 
 Nitrous acid, specific gravity 1 -354, -5760 
 
 Linseed oil, '5280 
 
 Oil of turpentine, '4720 
 
 Quicksilver, specific gravity 13*30, '0330 
 
 SOLIDS. 
 
 Ice, -9000 
 
 White wax, -4500 
 
 Quicklime, with water, in the proportion of 16 to 9, *4391 
 
 Quicklime, -3000 
 
 Quicklime saturated with water, and dried, -2800 
 
 Pit coal, -2800 
 
 Pit coal, '2777 
 
 Rust of iron, '2500 
 
 Flint glass, specific gravity 287, '1900 
 
 Iron, '1300 
 
 Hardened steel, -1230 
 
 Soft bar iron, specific gravity 7-724, -1190 
 
 Brass, specific gravity 8-356 -1160 
 
 Copper, specific gravity 8-785, -1140 
 
 Sheet iron, -1099 
 
 Zinc, specific gravity 8-154, -1020 
 
 White lead, -0670 
 
 Lead, -0352 
 
 Specific beau. Specific heal*. 
 
 Specific heat of water equal 1. Specific heat of water equal 1. 
 
 Bismuth, 0-0288 Tellurium, .0-09 12 
 
 Lead .0*0293 Copper 0-0949 
 
 Gold, 0-0298 I Nickel 0-1035 
 
 Platinum, 0-0314 Iron 0- 1 100 
 
 Tin, 0-0514 ! Cobalt, 0-1498 
 
 Silver, 0-0557 ', Sulphur, 0-1880 
 
 Zinc,., ..0-0927 
 
 
HEAT. 243 
 
 Large quantities of heat must enter into bodies, and be 
 concealed, to enable them to pass from the solid to the fluid 
 state, or from the fluid state to that of vapour. Thus the 
 quantity of heat necessary to convert any given weight of 
 ice into water, would raise the same weight of water 140 
 degrees of Fahrenheit. This quantity of heat is not sensi- 
 ble, but is, as it were, kept hid or latent ; nor can it be de- 
 tected by the touch, or by application of the thermometer. 
 
 Every addition of heat applied to water in a fluid state, 
 raises the temperature until it arrives at the boiling point ; 
 but however violently the fluid may boil, it does not become 
 hotter, nor does the steam that arises from it indicate a 
 greater degree of heat than the water ; hence, a large pro- 
 portion of the heat must enter into the steam and become 
 latent. The quantity of heat that becomes latent in steam, 
 was found by Dr. Black to be 810 degrees of Fahrenheit. 
 
 Under the common pressure of the atmosphere at the 
 surface of the earth, (15 Ibs. on the square inch,) water 
 cannot be raised above a temperature of 212 Fahr. ; but 
 when exposed to greater pressure, by being confined in a 
 vessel, the water may be raised to a much higher degree of 
 heat, and if, in this state of confinement, the heat applied 
 be insufficient to cause the water to boil : if the vessel should 
 be open, steam will rush out, and the water which remains 
 will fall in temperature to 212. On the contrary, water 
 boils at very low temperatures when the pressure is dimi- 
 nished ; as in an exhausted receiver, or at the tops of 
 mountains. 
 
 When the temperature of steam is reduced, it assumes 
 again the fluid form, and the quantity of latent heat set free 
 by steam in passing to. the state of water, has been found, 
 by Mr. Watt, to be 945 degrees. He also found that a 
 cubic inch of water may be converted into a cubic foot of 
 steam ; and that when this steam is condensed, by injecting 
 cold water, the latent heat which the steam gives out in 
 passing to the fluid state, would be sufficient to heat 6 cubic 
 inches of water to the temperature of 212, or the boiling 
 point. It is generally considered that steam raised from 
 boiling water occupies 18 hundred times as much space as 
 the water did from which it was raised, and instead of 
 making the latent heat of steam 810, as Dr. Black found it, 
 more correct experiments show it to be 1000, at the com- 
 mon pressures of the atmosphere ; but the latent heat of 
 
244 HEAT. 
 
 steam is inversely proportional to the degree of pressure 
 under which it is produced ; that is, the latent heat is 
 greatest where the pressure is least, and least where the 
 pressure is greatest. 
 
 It has lately been discovered that the sensible heat and 
 latent heat of steam at any one temperature added together, 
 give a sum which is constant ; that is to say, which is the 
 sum of the sensible and latent heat of any other tempera- 
 ture, or under any other pressure. Now, the sensible heat 
 of steam at the ordinary pressure of the atmosphere is 
 212 32 = 180 ; and the latent heat has been found to be 
 1000, their sum is 1180, which is the constant sum of the 
 latent and sensible heats of steam under any other pressure. 
 Thus, at the temperature of 248, where the elastic force of 
 the steam is equal to two atmospheres, or a pressure of 30 
 Ibs. on the square inch, the sensible heat will be 248 32 
 = 216, wherefore the latent heat is 1180 216 = 964, 
 and so of the other temperatures. 
 
 It has also been found that while the elasticity of steam 
 increases in geometrical progression, with a ratio of 2, the 
 latent heat diminishes with a ratio of 1*0306, differing not 
 very materially from a unit. 
 
 Many experiments have been made to ascertain the 
 elastic force of steam of various temperatures. The most 
 valuable of them are those recently made by the French 
 academicians, the results of which are given below in a 
 tabular form ; and the practical man will duly estimate the 
 value of this gift of science. 
 
 The following simple rule is easily remembered and ap- 
 plied, and comes near enough to the truth for all practical 
 uses. 
 
 (temperature + 100y = 
 
 steam in inches of mercury. Thus if the temperature be 
 307, then, 
 
 307 + 100 
 
 ~~m~ 
 
 then 2-3 X 2*3 X 2-3 X 2'3 x 2-3 X 2-3 = 148-0359, this 
 divided by 30, gives the atmospheres, 
 
 148-0359 
 
 = 4-93 atmospheres. 
 
 30 
 
HEAT. 
 
 245 
 
 TABLE OF THE ELASTICITY OF STEAM, 
 
 BY M. ARAGO AND OTHERS. 
 
 Elasticity of 
 iteam, the 
 pres. of the 
 atmosphere 
 being!. 
 
 Corresponding 
 
 ta s5iS of 
 
 Elasticity of 
 (team, the 
 pres. of the 
 atmosphere 
 being 1. 
 
 Corresponding 
 temp, in deg.ot 
 Fahrenheit. 
 
 1 
 
 212 
 
 13 
 
 380-66 
 
 ii 
 
 234 
 
 14 
 
 386-94 
 
 2 
 
 250-5 
 
 15 
 
 392-86 
 
 2 
 
 263-8 
 
 16 
 
 398-48 
 
 3 
 
 275-2 
 
 17 
 
 403-83 
 
 3 
 
 285 
 
 18 
 
 408-92 
 
 4 
 
 293-7 
 
 19 
 
 413-78 
 
 4 
 
 300-3 
 
 20 
 
 418-46 
 
 5 
 
 307-5 
 
 21 
 
 422-96 
 
 5d 
 
 314-24 
 
 22 
 
 427-28 
 
 6 
 
 320-36 
 
 23 
 
 431-42 
 
 6 
 
 326-26 
 
 24 
 
 435-56 
 
 7 
 
 331-7 
 
 25 
 
 439-34 
 
 7 
 
 336-86 
 
 30 
 
 457-16 
 
 8 
 
 341-78 
 
 35 
 
 472-73 
 
 9 
 
 350-78 
 
 40 
 
 486-59 
 
 10 
 
 358-78 
 
 45 
 
 499-24 
 
 11 
 
 366-85 
 
 50 
 
 510-6 
 
 12 
 
 374 
 
 
 
 In constructing this table the results were derived from 
 experiments up to 24 atmospheres, after which the formula 
 which follows was employed. 
 
 E = (l +T + 0-7153)* 
 
 Where E represents the elasticity, and T the temperature, 
 by the centigrade thermometer, regarding 100 as unity, 
 and T the excess of temperature above 100. It maybe 
 observed that this formula is more accurate in very high 
 temperatures than for low. 
 
 21* 
 
246 
 
 HEAT. 
 
 ELASTIC FORCE OF STEAM, BY DR. URE. 
 
 Temp. 
 
 Elastic I 
 force. 
 
 Temp. 
 
 Elastic I 
 force. 1 
 
 Temp. 
 
 Elastic 
 force. 
 
 Temp. 
 
 Eltic 
 force. 
 
 24 
 
 0-170, 
 
 155 
 
 8-500 
 
 53-600 
 
 281-8 
 
 104-400 
 
 32 
 
 0-200, 
 
 160 
 
 9-600 245 
 
 56-340 
 
 283-8 
 
 107-700 
 
 40 
 
 0-250 
 
 165 
 
 10-800 J245-8 
 
 57-100 
 
 285-2 
 
 112-200 
 
 50 
 
 0-360' 
 
 170 
 
 12-050 848-5 
 
 60-400 
 
 287-2 
 
 114-800 
 
 55 
 
 0-416 
 
 175 
 
 13-550;250 
 
 61-900! 
 
 289 
 
 118-200 
 
 60 
 
 0-516 
 
 180 
 
 15-160 
 
 251-6 
 
 63-500! 
 
 290 
 
 120-150 
 
 65 
 
 0-630| 
 
 185 
 
 16-900 
 
 254-5 
 
 66-700 
 
 292-3 
 
 123-100 
 
 70 
 
 0-726i 
 
 190 
 
 19-000 
 
 255 
 
 67-250 
 
 294 
 
 126-700 
 
 75 
 
 0-860: 
 
 195 
 
 21-100 257-5 
 
 69-800 
 
 295 
 
 129-000 
 
 80 
 
 oioi 
 
 200 
 
 23-600 
 
 260 
 
 72-300 
 
 295-6 
 
 130-400 
 
 85 
 
 no; 
 
 205 
 
 25-900 
 
 260-4 
 
 72-800 
 
 297-1 
 
 133-900 
 
 90 
 
 360, 
 
 210 
 
 28-880 
 
 262-8 
 
 75-900 
 
 298-8 
 
 137-400 
 
 95 
 
 640 ! 
 
 213 
 
 30-000 
 
 264-9 
 
 77-900 
 
 300 
 
 139-700 
 
 100 
 
 860! 
 
 216-6 
 
 33-400 
 
 265 
 
 78-040 
 
 300-6 
 
 140-900 
 
 105 
 
 2-100 1 
 
 220 
 
 35-510 
 
 267 
 
 81-900! 
 
 302 
 
 144-300 
 
 110 !2-456i 
 
 221-6 
 
 36-700 
 
 269 
 
 84-900 
 
 303-8 
 
 147-700 
 
 115 
 
 2-820 
 
 225 
 
 39-110 
 
 270 
 
 86-300 
 
 305 
 
 150-560 
 
 120 
 
 3-300 
 
 226-3 
 
 40-100 -J71-J 
 
 89-000 
 
 306-8 
 
 155-400 
 
 125 J3-830 
 
 230 
 
 43-100;273'7 
 
 91-200 
 
 303 
 
 157-700 
 
 130 4-366 
 
 230-5 
 
 43-500 j 275 
 
 93-480 
 
 310 
 
 161-300 
 
 135 5-070 
 
 231-5 
 
 46-800:275-7 
 
 94-600 
 
 311-4 
 
 164-800 
 
 140 
 
 5-770 
 
 235 47-920 977-9 
 
 97-800 
 
 312 
 
 167-000 
 
 145 
 
 6-600 
 
 238-5 50-300 979-5 
 
 101-600 
 
 312 
 
 165-5 
 
 150 J7-530 
 
 240 51-700|]280 
 
 101-900 
 
 
 
 Before we describe the application of steam in the 
 steam engine, we shall briefly allude to some other useful 
 purposes to which it has been subjected. It has been as- 
 certained that one cubic foot of boiler will heat about 2000 
 feet of space, in a cotton mill, to an average heat of about 
 70 or 80 Fahr. It has also been proved that one square 
 foot of surface of steam pipe is adequate to the warming of 
 200 cubic feet of space. This quantity is adapted to a well 
 finished, ordinary brick or stone building. Cast iron pipes 
 are preferable to all others for the diffusion of heat, the pipes 
 being distributed within a few inches of the floor. Steam 
 is also used extensively for drying muslin and calicoes. 
 Large cylinders are filled with it, which, diffusing in the 
 apartment a temperature of 100 or 130, rapidly dry the 
 suspended cloth. Experience has shown that bright dyed 
 yarns, like scarlet, dried in a common stove heat of 128, 
 have their colour darkened, and acquire a harsh feel ; 
 while similar hanks, laid on a steam pipe heated up to 165, 
 
HEAT. 
 
 247 
 
 retain the shade and lustre they possessed in the moist state. 
 Besides, the people who work in steam drying rooms are 
 healthy, while those who were formerly employed in '.he 
 stove heated apartments, became, in a short time, sickly 
 and emaciated. The heating, by steam, of large quantities 
 of water or other liquids, either for baths or manufactures, 
 may be effected in two ways : The steam pipe may be 
 plunged, with an open end, into the water cistern ; or the 
 steam may be diffused around the liquid in the interval be- 
 tween the wooden vessel and the interior metallic case. 
 
 Elastic force of vapour of alcohol of a specific gravity of 
 0-813, water being 1. 
 
 Alcohol of S. G. 0-813. 
 
 Temp. 
 
 Force of vap. 
 
 Temp. 
 
 Force of vap. 
 
 32 
 
 0-40 
 
 180-0 
 
 34-73 
 
 40-0 
 
 0-56 
 
 182-3 
 
 36-40 
 
 45-0 
 
 0-70 
 
 185-3 
 
 39-90 
 
 50-0 
 
 0-86 
 
 190-0 
 
 43-20 
 
 55-0 
 
 1-00 
 
 193-3 
 
 46-60 
 
 60-0 
 
 1-23 
 
 196-3 
 
 50-10 
 
 65-0 
 
 1-49 
 
 200- 
 
 53-00 
 
 70-0 
 
 1-76 
 
 206-0 
 
 60-10 
 
 75-0 
 
 2-10 
 
 210-0 
 
 65-00 
 
 80-0 
 
 2-45 
 
 214-0 
 
 69-36 
 
 85-0 
 
 2-93 
 
 216-0 
 
 72-20 
 
 90-0 
 
 3-40 
 
 220-0 
 
 78-50 
 
 95-0 
 
 3-90 
 
 225-0 
 
 87-50 
 
 100-0 
 
 4-50 
 
 230-0 
 
 94-10 
 
 105-0 
 
 5-20 
 
 232-0 
 
 97-10 
 
 110-0 
 
 6-00 
 
 236-0 
 
 103-60 
 
 115-0 
 
 7-10 
 
 238-0 
 
 106-90 
 
 120-0 
 
 8-10 
 
 240-0 
 
 111-24 
 
 125-0 
 
 9-25 
 
 244- 
 
 118-20 
 
 130-0 
 
 10-60 
 
 247-0 
 
 122-10 
 
 135-0 
 
 12-15 
 
 248-0 
 
 126-10 
 
 140-0 
 
 13-90 
 
 249-7 
 
 131-40 
 
 145-0 
 
 15-95 
 
 250-0 
 
 132-30 
 
 150-0 
 
 18-00 
 
 252-0 
 
 138-60 
 
 155-0 
 
 20-30 
 
 254-3 
 
 143-70 
 
 160-0 
 
 22-60 
 
 258-6 
 
 151-60 
 
 165-0 
 
 25-40 
 
 260-0 
 
 155-20 
 
 170-0 
 
 28-30 
 
 262-0 
 
 161-40 
 
 173-0 
 
 30-00 
 
 264-0 
 
 166-10 
 
 178-3 
 
 33-50 
 
 
 
 
248 STEAM ENGINE. 
 
 THE STEAM ENGINE. 
 
 IT is not consistent with the plan of this book, that we 
 should enter into minute details as to all the modifications 
 and departments of the steam engine; a subject which 
 would of itself occupy a large volume. We shall, however, 
 attempt to explain the leading principles on which this in- 
 valuable machine operates, so that the mode of calculating 
 its effects may be the more clearly comprehended. 
 
 The engine of Newcomen consists of a hollow cylinder 
 furnished with a solid piston. This piston is attached 
 to a rod, the top of which is connected with a large beam, 
 resting upon a fulcrum in the centre. To the other end 
 of this large beam, called the working beam, the pump rod 
 is attached. When steam is admitted into the bottom of 
 the cylinder, it will, by the superiority of its elastic force 
 above the pressure of the atmosphere, assisted by the coun- 
 teraction of the weight of the pump rod, cause the piston 
 to rise to the top of the cylinder. But when the piston 
 arrives at this point, cold water is injected into the cylinder, 
 by which the steam is condensed, and a vacuum formed, 
 then the pressure of the air on the top of the piston will 
 cause it to descend to the bottom of the cylinder. The 
 steam is again injected and again condensed, and thus the 
 operation of the machine is continued. This is called the 
 atmospheric engine. It is liable to this objection, that 
 there is a great waste of steam, and consequently of fuel 
 incurred in consequence of the steam being condensed in 
 the cylinder, since the cylinder must be heated to a certain 
 temperature, before the steam which it contains can exert 
 a sufficient elastic force, and the admission of cold water 
 cooling it down below this temperature, a considerable 
 quantity of steam is employed in again raising its heat to 
 the proper point. 
 
 In order to obviate this defect, the illustrious WATT 
 made such arrangements as enabled him to condense the 
 steam in a separate vessel, and thus to maintain a uniform 
 temperature in the cylinder. By this great improvement 
 the effect of the same quantity of steam was increased in 
 about the proportion of 12 to 7. Such was the principle 
 of Watt's single-acting engine ; but he afterwards so ar- 
 ranged the structure of the machine as to admit the steam 
 
STEAM ENGINE. 249 
 
 alternately above and below the piston, and still to con- 
 dense it in a separate vessel, as will be understood from the 
 description of the engraving, plate III, which will be given 
 a little farther on. This form of the steam engine is called 
 the double-acting low-pressure engine. 
 
 The steam engine was further improved by Mr. Watt, 
 by his shutting off the steam when the piston had passed 
 through a portion of its stroke, by which means the acce- 
 lerated motion of the piston is counteracted, from the elastic 
 force of the steam diminishing during its expansion. This 
 is the principle of what is called the expansive engine. 
 
 In the high-pressure steam engine, the steam, of high 
 temperature, is admitted into the cylinder alternately above 
 and below the piston ; but instead of being condensed, it is 
 allowed to escape into the atmosphere. In this engine, 
 which is the most simple in its construction, the steam acts 
 by its elastic force alone. 
 
 The construction of the low-pressure double-acting steam 
 engine, will be understood in its more minute details, from 
 the following description. 
 
 Plate III is a side elevation of a low-pressure portable 
 double-acting steam engine, in which the boiler and the 
 other principal parts are drawn in section. 
 
 After the flame from the furnace A passes under the 
 whole bottom surface of the boiler, it enters the flue C, 
 from which it escapes into a flue running up one side of the 
 boiler ; from this side flue it passes into the end flue D, 
 which carries it into a flue running along the other side of 
 the boiler; and from this last the smoke is conducted into 
 the chimney E. The bridge B helps to spread the flame 
 over the bottom of the boiler. When the furnace is cleaned, 
 the plate between the end of the furnace bars and the bridge 
 can be drawn forward by means of two handles, (one of 
 which only is shown,) in order that the cinders may be 
 pushed over the end of the furnace bars into the ashpit. 
 
 If one of the gauge cocks, FF, is opened, it will emit 
 steam ; and the other cock if opened will blow out water, 
 if the boiler be just as full of water as it ought to be. As 
 these cocks stop up sometimes, a wire may be passed down 
 through them, if the part above the key is not bent over. 
 The water should always stand somewhere between the 
 dotted lines passing below the ends of the gauge cocks. 
 G is a small valve opening inwards, placed in the man-hole 
 
J. 
 
 250 STEAM ENGINE. 
 
 door, to keep the sides of the boiler from being pressed 
 together by the force of the atmosphere, if the steam should 
 happen to be suddenly condensed by the water that feeds 
 the boiler. HH is the feed pipe, and the small valve sus- 
 pended from the point O, of the lever K, regulates the 
 quantity of water passing into the boiler ; the lever which 
 works the feed valve is connected by means of a rod to the 
 float I, which rises or falls along with the water in the 
 boiler, and this opens or shuts the valve, according as the 
 water stands low or high in the boiler. The pipe L con- 
 ducts the water into the feed pipe from a cistern fixed 
 above the boiler house, which is kept full by means of the 
 hot water pump, which takes in water from the hot well. 
 The cistern on the top of the boiler house should be large 
 enough to fill the boiler, as also the large cistern on which 
 the engine stands, if they should happen to be empty at any 
 time. The pipe M carries away any overplus water from 
 the feed pipe. NN is the pipe which conveys the steam 
 from the boiler into the nozles, and the safety valve is 
 placed above the bend in it. Q is a section of the cylinder, 
 showing also the outside of the metallic piston. The oblong 
 opening, near the top of the condenser R, admits a jet of 
 cold water to condense the steam after it has acted in the 
 cylinder. The injection cock is bolted to the outside of 
 the oblong opening, and the water which is forced through 
 it into the condenser by the pressure of the atmosphere is 
 taken from the large cistern on which the engine stands; 
 this cistern is always kept nearly full by the cold water 
 pump. The hot and cold water pumps are both wrought 
 off the same spindle P, fixed in the working beam, a pump 
 being attached to each end of the spindle. The foot valve 
 S is placed between the condenser R, and the air pump T. 
 The bucket shown in the air pump is not sectioned. The 
 valve in the air pump bucket, and the discharging valve, 
 which opens into the hot well on the top of the air pump, 
 have each a shallow flat-bottomed recess turned on the top, 
 so as to fit nicely the flat-bottomed disks X and W ; the 
 one disk is keyed on the air pump rod, and the other is 
 fixed by means of studs and nuts to the hot well ; as it 
 gives more water way, it is an improvement to have the 
 recess in the valve, rather than in the disk. If each valve 
 had not a recess turned in it to contain a quantity of water, 
 which, as it is forced out by the disk, reduces the momen- 
 
STEAM ENGINE. 251 
 
 turn of the valve by degrees ; the stroke of the valve on its 
 disk or guard would be very great, and the parts would 
 soon work out of order. The pipe which carries away the 
 water that is pumped out of the condenser by the air pump, 
 is shown near the top of the hot well, on the side farthest 
 from the cylinder. 
 
 The way in which the tire is regulated, is as follows :- 
 When the steam gets too strong, the water in the boiler 
 rises in the feed pipe, and carries up the float W ; and as 
 the float is connected by a chain and a pulley with the 
 damper V, the damper descends into the flue, and reduces 
 the draught in the furnace, and the force of the steam. 
 Again, if the steam gets too low, the float falls and raises 
 the damper, to increase the draught. The two pulleys 
 which form the connexion between the damper and the 
 float are both fixed on one shaft ; on account of the one 
 being placed exactly behind the other, one of them only 
 can be seen. 
 
 As the balls YY are carried round along with the rod Z, 
 when the engine is going too quick, the balls by their cen- 
 trifugal force fly out, and the rods and levers in connexion 
 with them shut more or less a valve at A', in the steam 
 pipe ; if the engine goes too slow, the balls fall down, and 
 open this valve to give the engine more steam to bring up 
 its motion. The rod B', and the lever C', form part of the 
 connexion with the valve in the steam pipe and the go- 
 vernor. 
 
 It is clear, that the power of the steam engine will de- 
 pend upon the energy of the steam, 1st. Steam of two 
 atmospheres will, other things being equal, produce double 
 the effect of steam of one atmosphere. 2d. the force of the 
 steam remaining the same, the power of the engine will 
 depend on the extent of surface acted upon, that is, on the 
 area of the piston. 3d. these two circumstances remaining 
 the same, the power of the engine will depend on the 
 velocity with which the piston moves. 
 
 For the sake of illustration, let us suppose that steam is 
 admitted into the cylinder, so as to press down the piston 
 with the force of one hundred pounds, and that the length 
 of the stroke is five feet; and suppose that the end of the 
 piston rod is attached to a beam whose fulcrum is in the 
 centre, and that to the other end of the beam there is 
 attached a weight of any thing less than one hundred 
 
252 STEAM ENGINE. 
 
 pounds, there being no friction. By the descent of the 
 piston, the weight at the end of the beam will be raised 5 
 feet; therefore it follows, that 100 pounds raised 5 feet 
 during one descent of the piston, will express the mechanic- 
 al effect of the engine. The reader will easily perceive 
 that the weight at the end of the beam must be somewhat 
 less than 100 pounds, for as it acts contrary to the power 
 of the piston, if they were equal the machine would be at 
 rest. If we suppose the area of the piston double of what 
 it was before, other things being the same, the engine 
 would raise 200 pounds through the same space of 5 feet 
 in the same time : and the same effect would evidently 
 ensue if we supposed the area of the piston to remain as it 
 was at first, but the force of the steam to be doubled. If 
 the area of the piston and force of steam be the same as at 
 first, but the length of stroke doubled, then the mechanical 
 effect of the engine will be 100 Ibs. raised 10 feet high 
 during one descent of the piston ; and if the descents be 
 performed in the same time, this engine will be double the 
 power of the first. 
 
 Let us proceed now to actual cases. In the common 
 low-pressure steam engine of Watt, steam is admitted into 
 the cylinder whose elastic force is somewhere about that 
 of the atmosphere, which we have all along supposed to be 
 15 Ibs. to the square inch; but friction and imperfect 
 vacuums tend to diminish this pressure, and the effective 
 pressure may be reckoned only four-fifths of this. If the 
 pressure of the steam is diminished by its one-fifth part, 
 which is 3 Ibs. to the square inch, then will the effective 
 pressure be 12 Ibs. to the square inch. The working 
 pressure is generally reckoned at 10 Ibs. to the circular 
 inch, and Smeaton only makes it 7 Ibs. The effective 
 pressure we have taken is between these extremes, being 
 equivalent to 9'42 Ibs. to the circular inch. 
 
 Mr. Tredgold gives the following table, which will show 
 how the power of the steam, as it issues from the boiler, is 
 distributed. In an engine which has no condenser: 
 
 The pressure on the boiler being 1 0'OOO 
 
 1. The force necessary for producing 
 motion of the steam in the cylinder- *0069 
 
 2. By cooling in the cylinder and pipes '0160 
 
 3. Friction of piston and waste -2000 
 
STEAM ENGINE. 253 
 
 4. The force required to expel the steam 
 
 into the atmosphere '0069 
 
 5. The force expended in opening the 
 valves, and friction of the parts of an 
 engine -0622 
 
 6. By the steam being cut off before the 
 
 end of the stroke '1000 
 
 Amount of deductions 3920 
 
 Effective pressure- 6080 
 
 In one which has a condenser : 
 The pressure on the boiler being 1000 
 
 1. By the force required to produce motion 
 
 of the steam into the cylinder 007 
 
 2. By the cooling in the cylinder and pipes 016 
 
 3. By the friction of the piston and loss 125 
 
 4. By the force required to expel the steam 
 through the passages 007 
 
 5. By the force required to open and close 
 the valves, raise the injection water, and 
 overcome the friction of the axes 063 
 
 6. By the steam being cut off before the end 
 
 of the stroke 100 
 
 7. By the power required to work the air- 
 pump 050 
 
 368 
 
 632 
 
 If we now suppose a cylinder whose diameter is 24 inches, 
 the area of this cylinder, and consequently the area of the 
 piston in square inches, will be, 
 
 24 3 x '7854 = 452-39. 
 
 Let us also make the supposition that steam is admitted 
 into the cylinder of such power as exerts an effective pres- 
 sure on the piston of 12 Ibs. to the square inch ; therefore, 
 452-39 X 12 == 5428-68 Ibs., the whole force with which 
 the piston is pressed. If we now suppose that the length 
 of the stroke is five feet, and the engine makes 44 single 
 or 22 double strokes in a minute, then the piston will move 
 through a space of 22 x 5 x 2 = 220 feet in a minute ; 
 and from what has been said before, it will not be difficult 
 to see, that the power of the engine will be equivalent to a 
 weight of 5428 Ibs. raised through 220 feet in a minute. 
 22 
 
254 STEAM ENGINE. 
 
 This is the most certain measure of the power of a steam 
 engine. It is usual, however, to estimate the effect as equi- 
 valent to the power of so many horses. This method, 
 however simple and natural it may appear, is yet, from 
 differences of opinion as to the power of a horse, not very 
 accurate ; and its employment in calculation can only be 
 accounted for on the ground, that when steam engines were 
 first employed to drive machinery, they were substituted 
 instead of horses ; and it became thus necessary to estimate 
 what size of a steam engine would give a power equal to 
 so many horses. 
 
 There are various opinions as to the power of a horse. 
 According to Smeaton, a horse will raise 22,916 Ibs. one 
 foot high in a minute. Desaguliers makes the number 
 27,500 ; and Watt makes it larger still, that is, 33,000. 
 There is reason to believe that even this number is too 
 small, and that we may add at least 11,000 to it, which gives 
 44,000 Ibs. raised one foot high per minute. 
 
 Now, in the case above, we found that the engine of 24 
 inch cylinder, would raise 5428 Ibs. through the space of 
 220 feet in one minute ; and it is easily seen that it could 
 raise 220 x 5428 Ibs. through one foot in the same time, 
 therefore, 220 x 5428 = 1194160 Ibs. raised through one 
 foot in one minute, is the effective power of the engine ; 
 and from these considerations it will be easy to find the 
 power according to the different estimates of a horse's 
 power. For, 
 
 1194160 
 22916 = 52 hor8es P wer ' 
 
 according to Smeaton. 
 
 1194160 
 
 T7500- = 4 
 according to Desaguliers. 
 
 1194160 
 
 -33000- = 3 
 .according to Watt. 
 
 1194160 
 44000" = 2 
 according to the usual estimate. 
 
 The reader will have no difficulty in forming a general 
 rule for estimating the power of a steam engine. (The 
 
STEAM ENGINE. 
 
 255 
 
 effective pressure on each square inch x the area of piston 
 in square inches x length of stroke in feet X number of 
 strokes per minute) -7- 44000 = the number of horses' 
 power of the engine. 
 
 What is the power of a low-pressure engine, whose 
 cylinder is 30 inches diameter, length of stroke 6 feet, 
 making 16 double strokes in the minute? 
 
 NOTE. An easy rule to find the area of the piston in 
 square inches, is this, 
 
 The diameter x circumference 
 = area. 
 
 Here we have, 
 
 30 x (30 X 3-1416) 
 4 
 
 equa the area of the piston in square inches ; and 12 the 
 effective pressure, 6 the length of stroke, 16 the number of 
 double strokes in a minute ? 
 
 706-86 X 12 x 6 X 16 X 2 1628605-44 = 
 
 44000 44000 
 
 horses' power. 
 
 If the cylinder of a high-pressure steam engine has a 
 piston of 5 inches diameter, with a twelve inch stroke, 
 making 32 double strokes in a minute ; steam being ad- 
 mitted of an elastic force equivalent to 7 atmospheres on the 
 inside of the cylinder. Its effective pressure will be 7 X 
 15 = 105 Ibs. to the square inch without friction ; but al- 
 lowing one-fifth for friction, the effective pressure will be 
 105 21 = 84 Ibs. to the square inch. 
 
 here 5 X (3>1 * 1( ^ = 19-63 the area of the piston : 
 
 19-63 X 84 X 1 X 32 X 2 105530-88 
 44000 ~T4000T~ 
 
 horses' power. 
 
 A convenient rule for finding the power of a high-pres- 
 sure engine, is let P be the force of the steam in the 
 boiler, A the area of the piston, and V the velocity of the 
 piston in feet per minute, then, 
 
 0-9 P 6 x A x V 
 
 = horses' power. 
 
 44000 
 
 The pressure of the steam in a boiler is 30 Ibs. per 
 square inch, the diameter of cylinder 12 inches, length of 
 
 
256 STEAM ENGINE. 
 
 stroke 3 feet, and the engine making 30 double strokes per 
 minute. Here the area of piston will be 113*097, the 
 velocity of piston = 3 x 30 X 2 = 180 feet per minute, 
 and since 0*9 x 30 6 = 21, then, 
 
 0-9 x 30 6 x 113-097 x 180 _ 427506*66 _ 
 
 44000 44000 
 
 9'7 horses' power. 
 
 We might simplify this rule still farther on the consi- 
 deration, that the divisor 44000 may be viewed as the de- 
 nominator of a fraction whose numerator is one, and by 
 converting this into a decimal, and multiplying by it, we 
 might avoid the necessity of division. 
 
 Since - = '0000227, hence we may devise the rule. 
 44000 
 
 Effective pressure of steam x area of piston in square 
 inches x length of stroke in feet x number of strokes per 
 minute x 227 ; and from the product cutting off seven 
 places as decimals ; = the horses' power of the engine. 
 
 This is for a single stroke engine for a double stroke 
 engine the multiplier is 227 X 2 = 454. 
 
 If the cylinder be 42 inches diameter, and the piston 
 moves 210 feet per minute, then the engine being low 
 pressure, we have, 
 
 area of cylinder equal 1385-44 ; hence 227 x 1385*44 x 
 210 x 12 = 792527097: 
 
 and the seven figures cut off as decimals, leave 79 horses' 
 power. 
 
 These are at best but approximations, and for safety it 
 might be advisable that a lower number than 12 should be 
 employed, as the^effective pressure of the steam ; the num- 
 ber 10 may be used as being easily managed, and coming 
 near the truth ; and thus the above rule may be simplified 
 by neglecting the pressure of the steam, and cutting off six 
 places for decimals instead of seven, as there is reason to 
 believe that the above results will answer only ponies in- 
 stead of strong horses. 
 
 The stroke of an engine is commonly reckoned equal to 
 one complete revolution of the crank shaft, and therefore 
 double the length of the cylinder, and it has been stated 
 by Mr. Thomas Tredgold, that to ascertain the velocity of 
 the piston when the engine performs at its maximum, we 
 may employ the rule, 
 
STEAM ENGINE. 257 
 
 120 x v/ length of stroke = velocity. 
 If an engine has a two feet stroke, then, 
 
 120 X V 2 = 120 x 1-4142 = 169-704, 
 or we may say 170, as the velocity of the piston per minute 
 in feet ; wherefore as the engine has a single stroke of 2 
 feet we have, 
 
 170 
 
 = 42^ strokes in the minute. 
 4 
 
 If an engine have a four feet stroke, then we have, 
 
 120 X x/ 4 = 120 x 2 = 240 = 
 the velocity of the piston per minute ; and, 
 
 240 
 
 = 30, equal the number of strokes per minute. 
 8 
 
 The safety valves of most of the steam engines in this 
 part of the country, are generally loaded with a weight of 
 from 3 to 4 Ibs. to the square inch of their area; let us 
 take 3-1 Ibs. in the present instance. The temperature of 
 steam necessary to balance this pressure, is, according to the 
 best experiments, 223 degrees of Fahrenheit's thermometer. 
 But besides this sensible heat, there is a quantity of latent 
 heat not indicated by the thermometer, and which can only 
 be detected when the steam passes, by condensation, into 
 the fluid state ; as the latent heat is then given out. Now, 
 if the latent heat of the steam at the above temperature, be 
 found on the principle stated in our remarks on heat, that the 
 sensible and latent heats of steam at all temperatures, when 
 added together, make a constant quantity ; we will find that 
 the latent heat of steam at this temperature is 989. The real 
 quantity of heat then in the steam is 223 + 989 = 1212 
 degrees. We will not be far from the truth in supposing, 
 that one cubic foot of this steam will, when condensed into 
 water, measure one cubic inch ; and the steam is supposed 
 to be condensed by the injection of cold water. Now it is 
 evident, that the temperature of the water formed by the 
 condensation of the steam, will be somewhere between the 
 temperature of cold water and the boiling point. Say that 
 the temperature of the injected water is 50 degrees, and that 
 the temperature of the water arising from the condensation 
 of the steam is 100. We must deduct the 100 degrees from 
 the heat of the uncondensed steam, that is, 1212 100 = 
 1112, which is left to be communicated to the injection 
 22* 
 
258 STEAM ENGINE. 
 
 water ; and since each cubic inch of the cold water requires 
 50 of heat to raise it to the temperature of the water found 
 after the condensation of the steam, therefore, 
 
 1112 
 
 =r 22 T 3 3 cubic inches 
 50 
 
 of water necessary to condense one cubic foot of steam to 
 the temperature of 100, the injected water being 50. 
 
 From these considerations may be derived a rule for de- 
 termining the quantity of water necessary to condense any 
 quantity of steam, at any given temperature. 
 Total heat of the steam temperature of warm water 
 
 temp, of warm water temp, of cold water 
 quantity of steam in cubic feet = the quantity of cold water 
 in cubic inches necessary to produce the effect. 
 
 Let us illustrate this by an example. What quantity of 
 cold water will it require of the temperature of 60, to con- 
 dense 8 cubic feet of steam, of the temperature of 223, to 
 water at 90 ? The whole heat is as before, 989 + 223 
 1212, wherefore by the rule, 
 
 1212 90 
 
 > > -MiM)-^ cubic m. -ins r= 
 
 200*2 
 
 = 1 7 of a cubic foot of water. 
 
 1 < '. ^ 
 
 From this it will be easy to determine how much water 
 must be discharged by the pump which feeds the condenser, 
 in order that a proper vacuum may be formed. 
 
 From practice it would appear that about 26 cubic inches 
 of cold water for condensing should be used for each cubic 
 foot of the capacity of the cylinder. 
 
 We may infer from observation, that the engines com- 
 monly in use require betwixt 3 and 4 gallons of cold water 
 per minute for each horse's power. If the water is return- 
 ed as it is in some engines, then a greater quantity will be 
 necessary. Now, in the usual construction of engines, the 
 pump rod which supplies the condenser with cold water, 
 is lixed halfway between the end of the beam and the cen- 
 tre ; hence, the length of its stroke is one-half that of the 
 piston in the large cylinder: therefore, if there be a 40 
 horse power engine, the length of whose stroke is 6 feet, 
 the length of the stroke of the pump will be 3 feet. 
 
 Now an imperial wine gallon occupies a space of 277*274 
 
STEAM ENGINE. 259 
 
 cubic inches, and 7% gallons will occupy a space of 277*274 
 X 7*5 = 2079-555 cubic inches ; and as the engine is 40 
 horses' power, there must be discharged in one minute, 
 
 2079-555 X 40 = 83182-2 cubic inches, 
 and if the engine makes 30 strokes per minute, then 
 
 oo 1 QOO 
 
 = 277-274 cubic inches 
 
 30 
 
 discharged at one stroke : but the stroke is 3 feet long, and 
 it remains only to find what must be the diameter of a 
 pump's bore, whose length is 36 inches, so that its capacity 
 shall be 2772 ; hence we find that, 
 
 2772 
 
 - - = 77 inches, 
 30 
 
 nearly equal to the area of the pump's bore ; now the area 
 of circles are to each other as the squares of their diame- 
 ters, and the area of a circle whose diameter is 9, is 63-6 ; 
 therefore, 
 
 63-6 : 77 : : 9 3 : 98, 
 
 the square root of which will be the diameter of the pump, 
 and will be found = 9*9 inches. 
 
 With respect to the fly wheel, 
 
 Horses' power of engine X 2000 
 
 Velocity of circumfer. wheel in feet per second* 
 the weight of the fly wheel in cwts. 
 
 If the diameter of the fly of a 30 horse power engine be 
 20 feet, and make 18 revolutions per minute, then, 
 
 20 X 3-1416 = 62-832 = 
 
 circumference in feet, and 62-832 X 18 = 1128-97 feet, 
 the space which the circumference moves through in one 
 minute ; hence, 
 
 1128-97 
 
 =18-81 feet per second; 
 oO 
 
 30 X 2000 60000 
 henCE ' 18-81' -863?" 
 =3 8 tons 9 cwts. the weight of the fly. 
 
 In the working of the valve of a steam engine, an eccentric 
 wheel is often employed, and it becomes necessary to cal- 
 culate the degree of eccentricity necessary to give a certain 
 length of stroke. The eccentric wheel's radius may be 
 
260 STEAM ENGINE. 
 
 easily found; thus, suppose the length of stroke required is 
 20 inches, and the diameter of the shaft on which the wheel 
 is screwed is 5 inches, and the thickness of metal required 
 to key on the wheel 2 5 inches. Take the half of the re- 
 quired stroke, that is, 10 inches, as the distance of the cen- 
 tre of the shaft from the centre of required wheel, and add- 
 ing to this the half thickness of the shaft = 2 inches, as 
 likewise the thickness of metal necessary for keying =25, 
 then 10 + 2s + 2$ = 15 inches, the radius of the wheel. 
 Now let E be = the radius of the eccentric wheel L = the 
 length of the eccentric rod, and / = the length of the bar 
 between the other end of this rod and the slide ; and let e 
 = the length of slide ; then, 
 
 _ Lxe /xE 
 
 j = JL = 
 
 / e 
 
 /X E Lxe 
 
 Suppose the length of the stroke of the slide e = 6 inches, 
 the length of the slide rod / = 5 inches, and the radius of 
 the eccentric = 24 inches = E, then the length of the rod 
 
 5 x 24 
 
 L = = 20 inches. 
 
 o 
 
 The other rules are wrought on the same principle. 
 
 We have before spoken of the governor while treating of 
 central forces and rotation. It remains for us here only 
 to observe, that the governor performs in one minute half 
 as many revolutions as a pendulum, whose length is the 
 perpendicular distance between the plane in which the balls 
 move and the centre of suspension. Thus, if the distance 
 between the point of suspension and the plane in which the 
 balls move be 28 inches : 
 
 I/ 39-1386 \ 
 .K ) = 1*182 vibrations in a second from the 
 
 nature of the pendulum ; hence, 
 
 I . I QO 
 
 = 0-591, the revolutions of the governor in a se- 
 
 
 cond, or 0-591 x 60 = 35-46 in one minute. 
 
 The piston rod of a steam engine may be made to move 
 up and down in a right line in various ways. The rod may 
 be made to terminate in a rack, the teeth of which act in 
 
STEAM ENGINE. 261 
 
 the teeth of an arched head of the long lever, called the 
 working beam : but the most efficacious of all contrivances 
 of this kind, is that of Watt, commonly called the parallel 
 motion. This contrivance is founded on geometrical prin- 
 ciples, which it would be inconsistent with the plan of this 
 work to consider ; we shall therefore simply describe the 
 contrivance of this illustrious mechanic. 
 
 The working beam has an alternating circular motion 
 round its centre A, and it is clear that the points B and G 
 will have a circular motion round the common centre A. 
 Let the point B be exactly in the middle, between the 
 centre and end of the beam. Let there be a bar or rod 
 CD, of the same length as AB, capable of moving round 
 the centre C, by means of a pivot. The other end of this 
 rod is attached by means of a pivot, to the rod DB. Now, 
 by the alternate rising and falling of the beam, the points 
 
 B and D will move in circular arches, but the middle point 
 P, of the connecting rod BD, will move upwards and 
 downwards in a vertical straight line, or at least so very 
 nearly so, as the difference cannot be perceived. Now, to 
 this point P, there is attached the end of the pump rod, 
 which will, of course, follow the direction of the impelling 
 point, and move in a straight line. For the purpose of 
 communicating a similar motion to the other piston rod, 
 conceive another rod CP' introduced, of the same length as 
 BD, and its extremities moving likewise on pivots. The 
 piston rod of the cylinder is attached to the point P', and 
 this point moves quite in the same way as the point P. 
 The only difference in the motion of these two points will 
 be, that the point P' will move twice as fast as the point P, 
 or will, in the same time, move twice as far. 
 
262 STEAM ENGINE. 
 
 The length of the links are made = 4 to 5, the length of 
 the stroke being 1, according to circumstances, the longer 
 link being preferred when practicable. From the length 
 of the links must be determined the position of the radius 
 bar, for the vertical distance between the centres of motion 
 of the working beam and the radius bar must be equal to 
 the length of a link. 
 
 When the parallel bar is not more than one-half of the 
 working beam's radius, then, 
 
 Let B = radius of the beam, 
 P = length of parallel bar, 
 S = length of stroke, 
 R = length of radius bar; we have 
 
 B 2Px'(|S) R 
 
 B v/(B [S] 8 ) x 2 P ^ 
 
 Suppose the length of the beam from the centre = 12 
 feet, the length of stroke 6, and of parallel bar 5 feet, 
 that is, B = 12, S = 6, and P = 5, then, 
 
 B 2 P X (58)" = 12 10 X (*6) = 18 
 = the dividend ; then, B v/(B fl (IS] 9 ) x 2 P = 12 
 % /(12 9 [6] 8 ) x 10 = 1211-62 X 10 = 0-38 
 X 10 = 3-8 the divisor, wherefore, 
 
 18 
 
 - -f 5 = 9-74 = the length in feet of the radius bar. 
 3*8 
 
 When the parallel bar is more than half the length of the 
 radius of the beam, the rule is, 
 
 2P Bx(*S)' 
 B (B 9 [S] 8 ) x 2 P " 
 
 by which rule it will be found that when the length of 
 stroke and radius bar are each 6, and the radius of beam 10 
 feet, the length of radius bar will be 2-75 feet. 
 
 Many rules have been given for the quantity of fuel 
 necessary for the production of steam, but they cannot be 
 depended on, so many circumstances must be taken under 
 consideration the quality of material used for fuel and 
 the mode of constructing the fireplace. 
 
 It has been found that 3 cwt. of Newcastle coals are 
 equivalent to 4 cwt. of Glasgow coals, or 9 cwt. of wood, 
 or 7 cwt. of culm. A chaldron of coals in London contains 
 36 bushels, and weighs 3136 Ibs., or nearly 1 ton, 8 cwt. 
 
: . 
 
 STEAM ENGINE. 263 
 
 It would appear, that in the common low-pressure steam 
 engines, the consumpt of coal per hour for 1 horse power, 
 is about 16 Ibs., of wood 56 Ibs., and of culm 35 Ibs. 
 These statements are given somewhat large, and by proper 
 regulation much less fuel might serve. 
 
 In the boiler there are certain proportions generally ob- 
 served. The width, depth, and length, are as the numbers 
 1, !!, 2-5. So that if the width be 5 feet, then the depth 
 will be 1-1 x 5 = 5 feet 6 inches ; and the length 5 x 2'5 = 
 12 ft. 6 in. ; and the whole content of the boiler will be, 
 
 5 X 5-5 x 12-5 = 343-75 cubic feet. 
 Now Boulton and Watt allow 25 cubic feet of space in the 
 boiler for each horse power ; and according to this estimate, 
 
 =13 and a fraction, the number of horses' power 
 25 
 
 of this engine for which this boiler would be fitted. Some, 
 instead of computing the size of boiler in this way, allow 
 5 square feet of surface of water for each horse's power; 
 but in all cases, it is common to make the boiler of a size 
 fitted for an engine of at least 2 horses' power more than 
 that to which it is applied. 
 
 There are two ways of loading the safety valve of a 
 boiler ; the one by placing a weight on the top of it, and 
 the other by causing the weight to act on the valve by a 
 lever. 
 
 When the weight is placed upon the valve ; area of 
 valve X pressure per square inch = whole weight, and also 
 
 whole weight 
 
 2 =: pressure per square inch. 
 
 area of valve 
 
 Thus, if a weight of 50 Ibs. be placed upon a valve whose 
 area is 10 inches, then the pressure per square inch is 
 
 = 5 Ibs. pressure per square inch. 
 
 When the weight acts by a lever, it is placed at one end, 
 the fulcrum being at the other, and the valve connected 
 with the lever somewhere between them ; this, then, is a 
 simple case of the lever. Hence, if the length of the lever 
 be 24 inches, the diameter of the valve 3 inches, (its area 
 will be 7,) the distance between the fulcrum and the valve 
 3 inches, then to give 60 Ibs. pressure per square inch on 
 the valve 60 X 7 = 420 Ibs. the whole pressure on the 
 valve, and 
 
264 
 
 STEAM ENGINE. 
 
 420 X 3 
 
 = 60 Ibs. will be the weight hung at the 
 
 end of the lever to give the required pressure. 
 
 To find the action of the weight of the lever divide its 
 whole length by the distance of the valve from the fulcrum, 
 and multiply the quotient by half the weight of the lever. 
 
 The following rules for calculations connected with the 
 steam engine are extracted from a useful little compendium 
 lately published by Mr. Templeton, of Liverpool. These 
 rules we have inserted here, not so much for their superior 
 accuracy, as from a desire to present our readers with 
 methods by which they may approximate to the true results 
 by means of the sliding rule. It is to be observed that the 
 term gauge point is used to denote the number to be taken 
 on the line stated in the rule. 
 
 Length of 
 stroke in 
 ft and in. 
 
 Gauge point. 
 
 Length of 
 stroke in 
 ft. and in. 
 
 Gauge point. 
 
 2 
 
 2 6 
 2 9 
 3 
 3 6 
 4 
 4 6 
 5 
 
 295 
 318 
 322 
 33 
 335 
 343 
 355 
 385 
 
 6 
 7 
 8 
 
 392 
 41 
 414 
 
 M AIM' 
 
 3 
 3 6 
 4 
 4 6 
 
 rx ENGINES. 
 
 3 
 31 
 
 317 
 326 
 
 RULE. Set the gauge point upon C to 1 upon D, and 
 against the number of horses* power upon C, is the diame- 
 ter in inches upon D ; or, against the diameter in inches 
 upon D, is the number of horses' power upon C. 
 
 Ex. 1. What diameter must a cylinder be with a 4 feet 
 stroke, to be equal to 20 horses' power ? 
 
 Set 343 upon C to 1 upon D ; and against 20 upon C is 
 24'2 inches diameter upon D.. 
 
 Ex. 2. What number of horses' power will an engine 
 be equal to, when the cylinder's diameter is 19 inches and 
 stroke 3 feet ? 
 
 19 a X '7854 X 7-25 x 192 394672-7328 
 
 33000 33000 
 
 11*96 or 12 horses' power nearly. 
 
 
STEAM ENGINE. 265 
 
 77ie proportion of parts of a high-pressure steam en- 
 gine. The length of the stroke should, if possible, be 
 twice its diameter. The velocity in feet per minute should 
 be 103 times the square root of the length of the stroke in 
 feet. And, as 4800 is to the velocity thus found, so is the 
 area of the cylinder to the area of the steam passages. 
 
 The proportions of the parts of an atmospheric en- 
 gine. The length of the cylinder should be twice the dia- 
 meter. The velocity in feet per minute should be ninety- 
 eight times the square root of the length of the stroke in 
 feet. The area of the steam passages will be as 4800 is to 
 the velocity in feet per minute, so is the area of the cylin- ' 
 der to the area of the steam passage. Tf the area of the 
 cylinder in feet be multiplied by half the velocity in feet, 
 and that product by 1-23 added to 1-4 divided by the dia- 
 meter in feet, the result divided by 1480 will give the cubic 
 feet of water required for steam per minute. If the num- 
 ber of times the quantity of water required for injection 
 must be greater than that required for steam, in general it 
 will be about twelve times the quantity, but it had better be 
 a little in defect than excess. The aperture for the injec- 
 tion must be such that the above quantity of water will be 
 injected during the time of the stroke. In order that the 
 injection be sufficiently powerful at first, the head should 
 be about three times the height of the cylinder ; and making 
 the jet apertures square, the area should be the 850th part 
 of the area of the cylinder. The conducting pipe should 
 be about four times the diameter of the jet. 
 
 The proportions of the parts of a single-acting low- 
 pressure engine. The length of the cylinder should be 
 twice its diameter. The velocity of the piston in feet per 
 minute should be ninety-eight times the square root of the 
 length of the stroke. The area of the steam passages should 
 be equal to the area of the cylinder, multiplied by the velo- ^ 
 city of the piston in feet per minute, and divided by 4800;'*' 
 The air pump should be one-eighth of the capacity of the 
 cylinder, or half the diameter and half the length of the 
 stroke of the cylinder, and the condenser should be of 
 the same capacity. The quantity of steam will be found 
 by multiplying the area of the cylinder in feet by half the 
 velocity in feet ; with an addition of one-tenth for cooling 
 and waste, and this divided by the volume of the steam 
 corresponding to its force in the boiler, gives the quantity 
 
 23 
 
266 RAILWAYS. 
 
 of water required for steam per minute, from whence the 
 proportions of the boiler may be determined.* At the com- 
 mon pressure of two pounds per circular inch on the valve, 
 the divisor will be 1497. The quantity of injection water 
 should be twenty-four times that required for steam, and 
 -the diameter of the injection pipe one-thirty-sixth of the 
 diameter of the cylinder. The valves in the air pump bucket 
 should be as large as they can be made, and the discharge 
 and foot valves not less than the same area. 
 
 Summary of proportions of a double engine, working 
 at full pressure. The length of a cylinder should be twice 
 its diameter ; for a cylinder having this proportion exposes 
 less surface to condensation than any other enclosing the 
 same quantity of steam. The area of the steam passages 
 should be about one-fifth of the diameter of the cylinder ; 
 or their area should be equal to the area of the cylinder, 
 multiplied by the velocity of the piston in feet per minute, 
 and divided by 4800. The diameter of the air pump should 
 be about two-thirds of the diameter of the cylinder, and 
 half the length of stroke ; and the larger the passages 
 through the air bucket and the discharging flap are, the 
 better. The quantity of water for injection should be about 
 23 times that required for steam, or about 26 cubic inches 
 to each cubic foot of the contents of the stroke of the pis- 
 ton. Watt considered a wine pint, or 28g cubic inches, 
 quite sufficient. There should be 62 times as much water 
 in the boiler as is introduced at one feed. 
 
 These proportions are taken from Tredgold's valuable 
 treatise on the steam engine. 
 
 RAILWAY^ STEAMBOATS, &c. 
 
 IT has been deduced from very extensive experiments or 
 the Liverpool and Manchester railways, that the effective 
 power of a locomotive engine is about *3 of the pressure of 
 the steam on the piston, on the calculated power of th* 
 engine being 1. In one case, for instance, a cylinder 21 
 inches diameter was used, the elasticity of steam in the 
 boiler was 30 Ibs. to the square inch, above the pressure 
 
 * To 459 add the temperature in degrees, and multiply the sum by 
 76-5. Divide the product by the force of the steam in inches of mer- 
 cury, and the result will be the space in feet the steam of a cubic foot 
 of water will occupy. 
 
RAILWAYS. 267 
 
 of the atmosphere. The length of the rail, which was in- 
 clined, was 3165 feet, and the height 24 feet. The time 
 of drawing 6 loaded wagons, each weighing 9010 Ibs. up the 
 rail, was 570 seconds, during which time the engine made 
 444 single strokes, each 5 feet long. Now, 
 
 21 3 x *7854 = 346-36 = the area of the piston in square 
 inches, wherefore, 346'36 X 30 = 10390 Ibs. = the pres- 
 sure of steam upon the piston, whose stroke was 5 feet, 
 and number of strokes in the given time 444 ; hence 444 x 
 5 = 2220 feet = the space through which the power 10390 
 has traversed; therefore, 10390 x 2220 = 23065800 Ibs. 
 = the impelling power of the engine. Now, it was found 
 that the actual weight including resistance moved, was 
 7124415 Ibs.; then, 
 
 which will give the effect about 30-9 per 
 ioOoooOO 
 
 cent., but the foregoing number may be taken as a safe me- 
 dium, that is, 30 per cent or *3. 
 
 The amount of retardation, arising from steam carriages 
 moving on railways, has been estimated thus ; 
 
 Loaded carriages weighing altogether 8522 Ibs. the fric- 
 tion amounted to 50 Ibs., or the y^ part of the weight. In 
 empty carriages weighing 2586 Ibs., the friction amounted 
 to 10 Ibs., or the g-gr P art f tne weight; and the friction 
 may be regarded as a constant retarding force. Wrought 
 iron rails seem from a multitude of experiments to be much 
 better than those of cast-iron, as they are more durable and 
 cause less friction. 
 
 The Rocket was tried, weighing 4 tons and 5 cwt., to it 
 there was attached a tender with water and coals, weighing 
 3 tons, 2 cwt. quar. 2 Ibs. ; and two carriages loaded 
 with stones, weighing 9 tons, 10 cwt. 3 qr. 26 Ibs., making 
 in all 17 tons. At full speed she moved at the rate of 30 
 miles in 2 hours, 6 minutes, 9 seconds, or 14} per hour 
 at the end of stage, about 6 miles ; and the greatest velocity 
 was 29 miles per hour. The quantity of water used 92'6 
 cubic feet, and it required 11 T 7 ^ Ibs. of coke for each cubic 
 foot of steam. 
 
 In the Rocket the boiler is cylindrical, with flat ends 6 
 feet long, and 3 feet 4 inches in diameter. To one end of 
 the boiler there is attached a square box as a furnace, 3 feet 
 long by 2 feet broad, and about 3 feet deep at the bottom 
 of this box five bars are placed, and the box is entirely 
 surrounded with a casting, except at the bottom and th^ 
 
268 RAILWAYS. 
 
 side next the boiler. Betwixt the casting and the box 
 there is left a space of about 3 inches, which is kept con- 
 stantly filled with water. The upper half of the boiler is 
 used as a reservoir for steam ; the under half being kept 
 filled with water, and through this part copper tubes reach 
 from one end to the other of the boiler, being open to the 
 fire box at one end, to the chimney at the other; these 
 tubes are 25 in number, each being 3 inches in diameter. 
 The cylinders were each 8 inches in diameter, and one was 
 at each side of the boiler; the piston had a stroke of 16$ 
 inches. The diameter of the large wheels was 4 feet 8| 
 inches. The area of the surface of water, exposed to the 
 radiant heat of the fire, was 20 square feet, being that sur 
 rounding the fire box or furnace ; and the surface exposed 
 to the heated air or fiame from the furnace, or what may 
 be called communicative heat, is 117*8 square feet. 
 
 The average velocity of the Rocket may be stated at 14 
 miles per hour, and during one hour she evaporates 18*24 
 cubic feet of steam, with a consumpt of about 17*7 Ibs. of 
 coke for each cubic foot of water. 
 
 An empirical rule has been given for the ascertaining of 
 the quantity of fuel necessary for steam carriages, which 
 may be useful. 
 
 The weight of the load X 61*55 -f weight of carriages 
 
 898 
 
 the quantity of coals required to carry one mile, but a near 
 approximation to the truth may be to allow 2 Ibs. for every 
 ton for one mile. 
 
 Iron railroads are of two descriptions. The flat rat'/, or 
 tram road, consists of cast iron plates about 3 feet long, 
 4 inches broad, and 5 an inch or 1 inch thick, with a 
 flaunch, or turned up edge, on the inside, to guide the 
 wheels of the carriage. The plates rest at each end on 
 stone sleepers of 3 or 4 cwt. sunk into the earth, and they 
 are joined to each other so as to form a continuous horizon- 
 tal pathway. They are, of course, double ; and the distance 
 between the opposite rails is from 3 to 4s feet, according 
 to the breadth of the carriage or wagon to be employed. 
 The edge rail, which is found to be superior to the tram 
 rail, is made either of wrought or cast iron ; if the latter be 
 used, the rails are about 3 feet long, 3 or 4 inches broad, 
 and from 1 to 2 inches thick, being joined at the ends by 
 cast metal sockets attached to the sleepers. The upper 
 edge of the rail is generally made with a convex surface 
 
RAILWAYS. 
 
 269 
 
 to which the wheel of the carriage is attached by a groove 
 made somewhat wider. When wrought iron is used, which 
 is in many respects preferable, the bars are made of a 
 smaller size, of a wedge shape, and from 12 to 18 feet 
 long; but they are supported by sleepers, at the distance 
 of every 3 feet. In the Liverpool railroad the bars are 15 
 feet long, and weigh 35 Ibs. per lineal yard. The wagons 
 in common use run upon 4 wheels of from 2 to 3 feet in 
 diameter. Railroads are either made double, 1 for going 
 and 1 for returning ; or they are made with sidings, where 
 the carriages may pass each other. See M* Culloch* s Diet. 
 
 Table showing the effects of a force of traction of 100 pounds, at 
 different velocities, on canals, railroads, and turnpike roads.* 
 
 Velocity of motion. 
 
 Load moved by a power of 100 lb. 
 
 
 
 On a Canal. 
 
 On a level Railway. 
 
 On a level 
 Turnpike Road. 
 
 Milf* per 
 boar. 
 
 r ^r 
 
 Total ma 
 
 Uteful ef- 
 
 Total mats 
 
 Uwful ef- 
 
 Total man 
 
 Uieful ef- 
 
 
 
 moved. 
 
 fect. 
 
 moved. 
 
 fect. 
 
 
 fect. 
 
 
 
 Ib.. 
 
 Ib. 
 
 Ib.. 
 
 Ibs. 
 
 Ib*. 
 
 Ira. 
 
 2* 
 
 3-66 
 
 55,500 
 
 39,400 
 
 14,400 
 
 10,800 
 
 ,800 
 
 ,350 
 
 3 
 
 4-40 
 
 38,542 
 
 27,361 
 
 14,400 
 
 10,800 
 
 ,800 
 
 ,350 
 
 3* 
 
 5-13 
 
 28,316 
 
 20,100 
 
 14,400 
 
 10,800 
 
 ,800 
 
 ,350 
 
 4 
 
 5-86 
 
 21,680 
 
 15,390 
 
 14,400 
 
 10,800 
 
 ,800 
 
 ,350 
 
 5 
 
 7-33 
 
 13,875 
 
 9,850 
 
 14,400 
 
 10,800 
 
 ,800 
 
 ,350 
 
 6 
 
 8-80 
 
 9,635 
 
 6,840 
 
 14,400 
 
 10,800 
 
 ,800 
 
 ,350 
 
 7 
 
 10-26 
 
 7,080 
 
 5,026 
 
 14,400 
 
 10,800 
 
 ,800 
 
 ,350 
 
 8 
 
 11-73 
 
 5,420 
 
 3,848 
 
 14,400 
 
 10,800 
 
 ,800 
 
 ,350 
 
 9 
 
 13-20 
 
 4,282 
 
 3,040 
 
 14,400 
 
 10,800 
 
 ,800 
 
 ,350 
 
 10 
 
 14-66 
 
 3,468 
 
 2,462 
 
 14,400 
 
 10,800 
 
 ,800 
 
 ,350 
 
 13-5 
 
 19-9 
 
 1,900 
 
 1,350 
 
 14,400 
 
 10,800 
 
 ,800 
 
 ,350 
 
 The subject of steam vessels has been investigated by 
 different engineers, on mathematical principles, but the 
 calculations which their rules direct are by far too intricate 
 for a work of this nature. We will, however, insert a state- 
 ment of the proportions, <fcc., of several steamboats already 
 made, which will doubtless be acceptable to the practical 
 man, and those who wish to investigate the theory will find 
 ample material in the work of Tredgold. 
 
 * The force of traction on a canal varies as the square of the velocity ; 
 but the mechanical power necessary to move the boat is usually reckoned 
 to increase as the cube of the velocity. On a railroad or turnpike, the 
 force of traction is constant; but the mechanical power necessary to 
 move the carriage, increases as the velocity. 
 
 23* 
 
270 
 
 STEAMBOATS. 
 
 Sc.Sc A 
 
 O >- ^* tf5 O 
 tC tfj Ci O 
 
 J 
 
 B 
 
 0. 
 
 Jl 
 
 -~ 
 
 y *- C - C * .. 
 
 c - c- . T j a 8 * 
 
 -o'3fc:2&g * fe 
 
 1*^5^2 if 1^^*^ s 5 
 
 f.o^oll Icclp . J 
 
 ^<2.^:22 S '^ 'J .* 1 1 J g 
 
 oo - cc *n c* 
 
 I ll 
 
 H.s 
 
 e!35SlS 
 
 (S, 
 
 * 
 
 2 2 
 d. 
 
 j 
 
 .5 
 
 .s.s 
 
 O 
 
 ' TW -3 
 
 flC- 
 
 I ; ; 
 
 It illl- 
 '1 litn 
 
 il 
 
 III! 
 
STEAMBOATS. 
 
 271 
 
 e-S.g 
 
 A 
 
 i 
 
 til 
 
 
 I !! 
 
 Ji 
 
 - C 00 ^ 00 
 
 I 
 
 !,; 
 
 !! 
 
 3 TT< 
 
 .s.s.s 
 
 .5 
 
 ii 
 
 c - 
 
 - 
 
 i 
 
 '-*.^ 00 
 
 * 1 
 
 i 
 
 5 d 
 2S i 2^S2 
 
 o e 
 
 3 .S.S^.S i 
 
 a 
 
 * N 00 O CO _- w 
 
 C* ^ ^ W W P l-H 
 
 i?S=S 
 I" 
 
 II urf 
 
 a; ill 
 j^&ifc- 
 
 <y g^ 
 
 O4 
 
 : ;l 
 
 11.11-s 
 
 -88-^ 
 
 II 
 
 UHQAQQ^ 
 
272 
 
 STEAMBOATS. 
 
 
 .s-^ 
 
 to J 
 
 dd s 
 
 U 
 
 .S.2 
 
 d, 
 
 i 
 
 |Jj= 
 
 *i 
 
 li 
 
 ss 
 
 I -^ 
 
 I 4~,> 
 
 *g* 
 
 I! 
 
 .s.s 
 ?-: = 
 
 i 
 
 a 
 
 ..2 
 
 ^. 00 
 
 dS 
 
 
 fe 
 
 : - 
 
 d. 
 I 
 
 H d 
 
 ss 
 
 ^ ^ 
 
 s l-fl 
 
 ii 
 
 * _<i 
 
 * i 
 
 ii= 
 
 i si 
 
 i i 
 
 ~; 
 
 h 
 
 -. 
 
 w -!: 
 
 in? 
 
 t"3 
 
 & s 
 
 illi- 
 
 11M 1 
 
 ,3 *S. 
 
 iiin 
 
 Jill 
 
 a s a 
 
 i=|r. 
 iJHJi 
 
 r- -3 * -S -3 
 
 I s o 6 d d S 
 UQCOQ^ 
 
 ^J 
 
 |li 
 
 s||l 
 
 J| Sa> 
 g-a-o 
 
ANIMAL STRENGTH. 273 
 
 The rule for determining the tonnage is according to law, 
 but by no means according to correct principles. It is as 
 follows : 
 
 Take the length = L from the back of the main stern 
 post to the fore part of the main stem, beneath the bow- 
 sprit, and subtract from it the length of the engine room 
 = E, and from the remainder subtract three-fifths of B = 
 the breadth of the vessel taken from outside to outside of 
 the planks at the widest part of the vessel, whether it be 
 above or below the wales, and divide this last remainder by 
 188 ; the quotient multiplied by the square of B will give 
 the register tonnage, or, 
 
 x B> = tonnn e e - 
 
 Wherefore the length being 162 feet, the length of engine 
 room 47, and the breadth of the vessel 32, then, 
 
 tonnage. 
 
 ANIMAL STRENGTH. 
 
 THERE is a certain load which an animal can just bear 
 but cannot move with it, and there is a certain velocity with 
 which an animal can move but cannot carry any load. In 
 these two circumstances it is clear, that the exertion of the 
 animal can be of no avail as a mover of machinery. These 
 are, as it were, the extremes of the animal's exertion, 
 where its effect is nothing ; but between these two extremes, 
 there must be weights and velocities with which the animal 
 can move, and be more or less efficient. 
 
 If one man travel at the rate of three miles an hour, and 
 carry a load of 56 Ibs., and another move at the rate of 4 
 miles an hour and carry a load of 42 Ibs., the speed of 
 the first is 3, and the load 56, the useful effect may there- 
 fore be estimated as the momentum = 168. The other 
 carries only 42 Ibs., but travels at the rate of 4 miles an 
 hour ; therefore, in the same way, his useful effect will be 
 4 x 42 = 168, the same as before: hence the effect of 
 
274 ANIMAL STRENGTH. 
 
 these two men are the same. It will not be difficult to 
 show, that in the same time they perform the same quantity 
 of work. For the first will in six hours carry 56 Ibs. 3x6 
 = 18 miles, aa he travels at the rate of 3 miles an hour ; 
 and if he be supposed to carry a different load, but of the 
 same weight every mile, he will in the six hours have car- 
 ried altogether 18 x 56 = 1008 Ibs. ; but the other carries 
 in the same way, 4 times 42 Ibs. every hour, that is 168 
 Ibs. in one hour therefore in 6 hours he will have carried 
 168 x 6 = 1008 Ibs., the same as the other. 
 
 It will now be seen what is meant by the phrase useful 
 effect, and from what has been observed above, we will be 
 led to conclude, that when the load is the greatest which 
 the animal can possibly bear ; the useful effect is nothing, 
 because the animal cannot move ; and when the animal 
 moves with its greatest possible speed, the useful effect will 
 also be nothing, for then the animal can carry no load ; 
 and it becomes a very useful problem to determine where 
 between these two limits, the load and speed are so related 
 that the useful effect of the animal will be the greatest. By 
 investigation it has been found that the maximum effect of 
 an animal will be when it moves with j of its greatest 
 speed, and carries ths of the greatest load it can bear. 
 
 Thus, if the greatest speed at which a man could travel 
 or run, without a load, be 6 miles per hour; and if the 
 greatest load which he can bear, without moving, be 2 
 cwt., then this reduced to Ibs. is 280 Ibs., hence, 
 
 280 x 4 ( 
 
 = 124-4 Ibs. = the load, and =2 miles, the 
 
 9 3 
 
 speed with which a man will produce the greatest useful 
 effect. 
 
 Sir John Leslie gives a formula for a horse's power, in 
 traction, in which he denotes the velocity in miles per 
 hour, | (12 V) 9 by which it will be found that if a 
 horse begins this pull with a force =144 Ibs., he would 
 draw 100 at the rate of 2 miles, 64 at 4, and 36 at 6 ; the 
 greatest effect being at 4 miles per hour. 
 
 The French employ a measure of animal action which 
 they denominate a Dynamical unit, which is a cubic metre 
 of water raised to the height of a metre. 
 
 There are so many causes operating to produce variations 
 in animated beings even of the same kind, that it is difficult, 
 if not impossible, to form a correct estimate of the amount 
 
, t - 
 
 FRICTION. 275 
 
 of any one particular class, or the comparative strength of 
 different classes, hence we find great differences in the 
 results of different experimenters. 
 
 Gregory has estimated the average force of a man at rest 
 to be 70 Ibs., and his utmost walking velocity, when un- 
 loaded, to be 6 feet per second ; and that a man will pro- 
 duce the greatest mechanical effect in drawing, when the 
 weight was 31 Ibs., with a velocity of 2 feet per second. 
 But this is not the most advantageous way of applying the 
 strength of men, although it has been found to be the best 
 way of employing the strength of horses. Robertson Bu- 
 chanan states, that the mechanical effects of men in work- 
 ing a pump, in turning a winch, in ringing a bell, and 
 rowing a boat, are as the numbers 100, 167, 227, and 248. 
 According to Hatchette, of a man working at the cord of a 
 pulley to raise the ram of a pile engine = 50 dynamical 
 units. A man drawing water from a well by means of a 
 cord 71 ; a man working at a capstan 116. The dynami- 
 cal unit being, as stated before, equivalent in English mea- 
 sure to 2208 Ibs., or 4 hogsheads of water raised to the 
 height of 3 '281 feet in a minute ; these things being con- 
 sidered, the above results will give the average strength of 
 men per day. 
 
 We meet with similar difficulties in estimating the 
 strength of horses. According to Desaguliers and Smea- 
 ton, 1 horse equal to 5 men. According to Bossut, 1 horse 
 equal to 7 men. Schulze makes it 14 men ; and Bossut 
 states, that I ass is equivalent to 2 men. It is also stated 
 by Amontons, that 2 horses yoked in a plough exert a 
 power of 150 Ibs. See the section on the Steam Engine. 
 
 FRICTION. 
 
 WE have considered the effects of the first movers of 
 machinery, and we must now direct our attention to the 
 subject of Friction, which, as we have frequently noticed, 
 tends to diminish these effects. On this subject it is not 
 our intention to dwell long, as all the researches that have 
 been hitherto made in this branch of mechanical science, 
 are not of such a nature as to furnish means of deducing 
 satisfactory laws. The resistance arising from one surface 
 
276 FRICTION. 
 
 rubbing against another is denominated friction ; and it is 
 the only force in nature which is perfectly inert its ten- 
 dency always being to destroy motion. Friction may thus 
 be viewed as an obstruction to the power of man in the 
 construction of machinery ; but, like all the other forces 
 in nature, it may, when properly understood, be turned to 
 his advantage, for friction is the chief cause of the stability 
 of buildings or machinery, and without it animals could not 
 exert their strength. 
 
 The friction of planed woods and polished metals, with- 
 out grease, on one another, is about one-fourth of the pres- 
 sure. 
 
 The friction does not increase on the increase of the rub- 
 bing surfaces. 
 
 The friction of metals is nearly constant. 
 
 The friction of woods seems to increase after they are 
 some time in action. 
 
 The friction of a cylinder rolling down a plane, is in- 
 versely as the diameter of the cylinder. 
 
 The friction of wheels is as the diameter of the axle di- 
 rectly, and as the diameter of the wheel inversely. The 
 following hints may be of use for the purpose of diminish- 
 ing friction. 
 
 The gudgeons of pivots and wheels should be made of 
 polished iron ; and the bushes or collars in which they 
 move should be made of polished brass. In small and 
 delicate machines, the pivots or knife edges should rest on 
 garnet. Oily substances diminish friction swine's grease 
 and tallow should be used for wood, but oil for metal. 
 Black lead powder has been used with effect for wooden 
 gudgeons. The ropes of pulleys should be rubbed with 
 tallow. 
 
 As to the friction of the mechanic powers. The simple 
 lever has no such resistance, unless the place of the ful- 
 crum be moved during the operation. In the wheel and 
 axle the friction on the axis is nearly as the weight, the 
 diameter of the axis, and the angular velocity it is, how- 
 ever, very small. When the sheaves rub against the blocks 
 the friction of the pulley is very great. In most, if not in 
 all screws, the friction of the screw is equal to the pres- 
 sure the square threaded screw is the best. 
 
 In the inclined plane, the friction of a rolling body is 
 far less than that of a sliding one. 
 
FRICTION. 277 
 
 To estimate the amount of the friction of a carriage upon 
 a railway, we have, 
 
 P x T 
 
 p == friction, 
 
 in which rule P signifies the power that will keep the 
 wagon on the plane, independent of friction, T the time of 
 descent without friction, both of which are to be deter- 
 mined by the laws of the inclined plane given before : and 
 t is the time of actual descent of the wagon or carriage. 
 
 There is a loaded carriage on a railroad 120 feet in 
 length, having an inclination of one foot to the hundred. 
 The carriage, together with its load, weighs 4500 Ibs. 
 Now, the height of the plane may be found by the princi- 
 ples of geometry, from the proportion of similar triangles. 
 
 100 : 120 : : 1 : 1-2 = the height of the plane ; and by 
 the laws of falling bodies, and the properties of the inclined 
 plane, 
 
 |- X 120 = -2731 x 120 = 32-772 = the time 
 
 in seconds in which the carriage would descend the plane 
 without friction and by the properties of the inclined 
 plane, 100 : 1 : : 4500 : 45 = the force that sustains the 
 carriage, without friction, from rolling down the plane ; 
 wherefore, by the rule, 
 
 45 - 20-421 = the friction in pounds, 
 
 which retards the carriage in rolling down the railway. 
 
 24 
 
 
OF MACHINES IN GENERAL, 
 
 THEIR REGULATION AND COMPARATIVE EFFECTS. 
 
 A MACHINE, howsoever complicated it may be, is nothing 
 else than an organ or instrument placed between the work- 
 men, or source of force or power, whatever it may be, and 
 the work to be done. Machines are used chiefly for three 
 reasons. 1. To accommodate the direction of the moving 
 force to that of the resistance which is to be overcome^ 2. 
 To render a power, which has a fixed and certain velocity, 
 effective in performing work with a different velocity. 3. 
 To make a moving power, with a certain intensity, capable 
 of balancing. or overcoming a resistance of a greater in- 
 tensity. 
 
 These objects may be accomplished in different ways, 
 either by using machines which have a motion round some 
 fixed point, as the three first mechanic powers ; or by those 
 which furnish, to the resistance to be moved, a solid path 
 along which it may be impelled, as is the case in the last 
 three mechanic powers : hence some authors have reduced 
 the simple machines to two the lever and inclined plane. 
 Simplicity in the construction of machines cannot be too 
 warmly recommended to the young engineer; for com- 
 plexity increases the friction and expense, and endangers 
 the chance of success from the derangement of the parts. 
 In consequence of friction, it is well known, that no 
 machine can overcome a resistance without an expense of 
 the power of the first mover, and as the more complicated 
 f the machine is, the greater will the friction be ; so also will 
 the machine be less powerful. If two machines be con- 
 structed, the one simple and the other complex, and be 
 such, that the velocity of the impelled point is to the 
 velocity of the working point in the same proportion in 
 both ; then will the simple machine be the most powerful. 
 
 The methods of communicating motion from one point 
 to another are infinitely diversified ; and we, in the last 
 
 278 
 
MACHINERY. 279 
 
 chapter, gave an account of the best of these which have 
 hitherto been invented. We confine ourselves in the 
 mean time to a few general remarks on the construction of 
 machinery. 
 
 When heavy stampers are to be raised in order to drop 
 on matter to be pounded, the wipers by which they are 
 raised should be of such a form, that the stampers may be 
 raised by a uniform pressure, or with a motion as nearly as 
 possible uniform. If this is not the case, and the wiper is 
 merely a pin sticking out of the axis, the stamper will be 
 forced into motion at once, which will occasion violent jolts 
 in the machine, together with great strains on its moving 
 parts and points of support. But if gradually lifted, no 
 inequality will be felt at the impelled point of the machine. 
 The judicious engineer will therefore avoid, as much as 
 possible, all sudden changes of motion, especially in any 
 ponderous part of a machine. 
 
 When several stampers, pistons, or other reciprocal 
 movers are to be raised and depressed, common sense 
 teaches us to distribute their times of action in a uniform 
 manner, so that the machine may be always equally-loaded 
 with work. When this is done, and the observations in 
 the foregoing paragraph attended to, the machine may be 
 made to move almost as smoothly as if there were no re- 
 ciprocations in it. Nothing shows the ingenuity or skill 
 of the contriver more than the simple yet effectual con- 
 trivances for obviating those difficulties which are unavoid- 
 able, from the nature of the work to be done by the 
 machine, or of the power applied. There is also much 
 ingenuity required in the management of the moving 
 power, when it is such as does not answer the kind of mo- 
 tion required ; for instance, in employing a power which 
 necessarily reciprocates to produce a motion which shall be 
 uniform, as in the employment of a steam engine to drive 
 a cotton mill. The necessity of reciprocation of the first 
 mover causes a waste of much power. The impelling 
 power is wasted first in imparting, and then in destroying 
 a vast quantity of motion in the working beam. The en- 
 gineer will see the necessity of erecting the mover in a 
 separate building from the machinery moved, which pre- 
 vents the great shaking and speedy destruction of the 
 buildings. 
 
 The gudgeons of a water wheel should never rest on the 
 
280 MACHINERY. 
 
 building, but should be placed on a separate erection ; and 
 if this is not practicable, blocks of oak should be placed 
 below them, which tend to soften all tremors, like the 
 springs of a carriage. 
 
 It will often conduce to the equality of motion of ma- 
 chinery, to make the resistance unequal, to accommodate 
 the inequalities of the moving power. There are some 
 beautiful specimens of this kind in the mechanism of the 
 human body. 
 
 It is always desirable, that the motion of a machine should 
 be regular, when this can be effected ; and we now pro- 
 ceed to state the various methods that have heretofore been 
 employed for producing regularity in the motion of the 
 machine, both as regards the reception and distribution of 
 power. 
 
 Even supposing that the first mover is perfectly constant 
 and equable in its action, the machine may not be regular 
 in its movement, from the irregularity of the resistance to 
 be overcome. Hut still, if botli the power and the resist- 
 ance were perfectly regular, the machine would not be 
 perfectly uniform in its motion ; for there are particular 
 positions in which the moving parts of a machine are more 
 efficacious than in others, as in the crank for instance : 
 Ii^ncc the energy of the first mover will be unequally trans- 
 mitted, and irregularity in the motion of the machine will 
 consequently follow. The motion of some machines bears 
 a constant tendency to accelerate, others to retard ; and 
 others alternately to accelerate and retard ; and perhaps 
 in no ease whatever can the motion of a machine be said to 
 be perfectly uniform. Hut of this we will speak more at 
 large when we come to treat of the maximum eflect of 
 machines. 
 
 We intend to confine our attention chiefly to the regula- 
 tors of machinery employed in the steam engine, making 
 occasional remarks on others as we go along. 
 
 For the purpose of regulating the moving power, the 
 conical pendulum or governor is commonly employed. 
 The nature of this beautiful contrivance lias been described 
 under central forces, and alluded to in our remarks on the 
 steam engine. The ring on the shaft act* upon a lever of 
 the first kind, whose other end opens or shuts a valve, which 
 is fixed in the pipe that admits the steam from the boiler to 
 the cylinder ; and according to the degree of opening or 
 
MACHINERY. 281 
 
 shutting of this valve, and consequently the divergence or 
 convergence of the balls, or the velocity of the shaft, will 
 be the quantity of steam admitted to the cylinder. The 
 governor is frequently applied to the water wheel, and acts 
 in a similar way by a board or valve in the shuttle, which 
 delivers the water to the wheel. So likewise in the wind- 
 mill, it is employed to furl or unfurl more or less sail. 
 
 Sometimes the governor is found inadequate to the regu- 
 lation of the machine, and another contrivance of great 
 power and simplicity is introduced. The machine is made 
 to work a pump, which tends continually to fill a cistern 
 with water. From this cistern there proceeds an eduction 
 pipe, leading to the reservoir, from which the water is 
 drawn by the pump. This simple contrivance is so regu- 
 lated, that when the machine goes with its proper velocity, 
 the pump throws just as much water into the cistern as the 
 ejection pipe draws from it; consequently, the water in the 
 cistern remains at the same level. But if the machine goes 
 too fast, then the pump will throw in more water than is 
 let out by the ejection pipe, wherefore the level of the 
 water will rise in the cistern. If the machine goes too 
 slow, the level of the water will in like manner fall. Now, 
 on the surface of the water in the cistern, there is a float 
 which rises or falls with the surface of the water ; and is 
 thus made to answer the same purpose as the ring of the 
 governor. It may be observed, that the delicacy of this 
 kind of regulator will depend, in a great measure, upon 
 the smallness of the surface of the water which supports the 
 float ; for then a small difference between the supply and 
 discharge, will cause a greater difference in the elevation 
 or depression of the surface, than if the surface were large. 
 
 To procure a constant supply of steam in the steam en- 
 gfne, it is necessary that the water in the boiler be always 
 at the same level. To effect this purpose, there is a lever 
 fixed on a support, on the top of the boiler, to one end of 
 which lever there is attached a slender rod, which descends 
 into the boiler, and is terminated by a float, which rests 
 oti the surface of the water in the boiler. To the other 
 end of the lever, there is attached another rod, to the end 
 of which is affixed a valve, opening and shutting the orifice 
 of a pipe which leads into the boiler. The top of the pipe, 
 where the valve is placed, opens into a cistern of water, 
 which is supplied by a pump driven by the engine itself. 
 
 24* 
 
282 MACHINERY. 
 
 When the water in the boiler falls below its common level, 
 in consequence of the formation of steam, the float falls 
 with it, and consequently depresses that side of the lever 
 to which the float rod is attached ; the other arm rises and 
 opens the valve at the top of the pipe, which leads from 
 the cistern into the boiler, and thus admits water until the 
 float rises to the proper height, and then the valve is closed. 
 In this beautiful contrivance, the water is not supplied to 
 the boiler in jolts, but the float and valve continuing in a 
 state of constant and quick vibration, the supply is rendered 
 quite constant. 
 
 There is a very ingenious contrivance called the Tacho- 
 meter, from its use as a measure of small variations in 
 velocity, which is often employed in the steam engine and 
 other machinery. The simplicity of this contrivance will 
 render its action easily understood. If a cup with any 
 fluid, as mercury, be placed on a spindle, so that the brim 
 of the cup shall revolve horizontally round its centre, then 
 the mercury in the cup will assume a concave form, that is, 
 the mercury will rise on the sides of the cup, and be de- 
 pressed in the middle ; and the more rapid the motion of 
 the cup is, the more will the surface of the mercury differ 
 from a plane. Now, if the mouth of this cup be closed, and 
 a tube inserted in it, terminated in the cup by a ball-shaped 
 end, and half filled with some coloured fluid, as spirits of 
 wine and dragon's blood ; then it is clear, that the more 
 the surface of the mercury is depressed, the more the fluid 
 in the tube will fall, and vice versa: consequently, the 
 rapidity or slowness of the motion of the cup, will be indi- 
 cated by the height of the coloured fluid in the tube ; and 
 thus it becomes a measure of small variations in velocity. 
 
 In the steam engine, we also find an apparatus for regu- 
 lating the strength of the fire of the boiler, which apparatus 
 is called the self-acting damper. There is a tube inserted 
 inlo the boiler, reaching nearly to the bottom, which tube is 
 open at both ends. Now, from the principles of Pneuma- 
 tics, it is plain, that the greater the pressure of the steam in 
 tUe boiler is, the water will be pressed to the grenter height 
 in this tube. The water in the tube supports a weight, to 
 which there is attached a chain going over two wheels ; and 
 to the other end of the chain is attnched a plate, which 
 slides over the mouth of the flue which leads into the fire. 
 These things are so formed, that the rising of the weight 
 
MACHINERY. 283 
 
 in the tube will cause more or less of the flue to be covered 
 by the plate ; and thus increase or diminish the quantity 
 of air which feeds the fire. Now, if there is too much steam 
 produced, there will be a greater pressure on the surface 
 of the water in the boiler, and it will be forced up the tube 
 the weight in the tube will be raised, and consequently the 
 plate at the other end of the chain will fall, and cover more 
 of the mouth of the flue, and thus diminish the quantity 
 of air which feeds the fire ; and there will consequently be 
 generated in the boiler a less quantity of steam. 
 
 We come now to speak of the nature and use of the fly 
 wheel. A fly in mechanics may be defined to be a heavy 
 wheel or cylinder, which moves rapidly upon its axis, and 
 is applied to a machine for the purpose of regulating its 
 motion. 
 
 We have already stated that there are many circum- 
 stances which tend to render the motion of a machine ir- 
 regularvariation in the energy of the first mover, whether 
 it be water, wind, steam, or animal strength variation in 
 the resistance or work to be done and variations in the 
 efficacy of the machine itself, arising from the nature of its 
 construction, whereby it is of necessity more effective in 
 6ne position than in another. We have already seen how 
 many of these irregularities are compensated, and we are 
 now come to speak of the fly, which is the simplest and 
 most effective of them all. The principle on which the fly 
 acts is that of inertia, one of the most important of the first 
 principles of mechanical science. At any one given time, 
 a body must be in one or other of these two states rest or 
 motion. And let any body be in one or other of these two 
 states, it has no power within itself to change it, if it be 
 at rest, it has no power to put itself in motion and if in 
 motion, it has no power in itself either to increase, diminish, 
 or destroy that motion. From a knowledge of this fact, 
 and from what was stated before on the momentum, or 
 moving force of a body, that it is the quantity of matter 
 multiplied by the velocity of the moving body the nature 
 of the operation of the fly will be easily understood. 
 
 As the fly wheel, to do its office effectually, must have 
 a considerable velocity, it is clear that its rim, which has a 
 considerable weight, must also have a considerable momen- 
 tum, and consequently a considerable power to overcome 
 any tendency either to increase or retard its motion. 
 
284 MACHINERY. 
 
 To apply these observations to actual cases, let us sup- 
 pose that a single horse drives a gin. When the gin has been 
 set in motion, the animal cannot exert a uniform strength 
 there will be occasional increases and relaxations in the 
 velocity of the gin ; but suppose a fly wheel to be added, 
 then, whenever the animal slackened its exertions, the 
 machine would have a tendency to move slower, but the 
 momentum which the fly had acquired, would overcome this 
 tendency to retardation, and continue the motion of the 
 machine at the same rate as before, until the animal had 
 recovered itself so as to exert the same strength as before. 
 So, likewise, if the animal exerted an extraordinary pull, 
 the inertia of the wheel would oppose a resistance which 
 would check the tendency to increase in the velocity of the 
 gin. In this way the fly wheel regulates the motion of the. 
 gin, whether the animal takes occasional rests, or makes 
 occasional extraordinary exertions. It is evident that the 
 fly would operate in the same way, if the first mover were 
 steam, water, or wind, and that the other regulators which 
 we have described, are merely assistants to the fly wheel. 
 
 Variations in the resistance, or work to be performed, 
 are also compensated by the fly wheel. For instance, in a 
 small thrashing mill without a fly. When the machine is 
 not regularly fed with the corn, there will be an occasional 
 resistance, which will have a sensible effect on the whole 
 train of the machinery, even the water wheel itself; which 
 irregularity, may, however, be avoided by the introduction 
 of a fly, as its inertia will procure equality of motion : but 
 it may be observed, that when the machine is large, there 
 will be less necessity for a fly, as the inertia of the machine 
 itself will then effect the same purposes. 
 
 It was before stated, that even supposing the first mover 
 and resistance to be perfectly uniform, the machine itself is 
 liable to variations in energy at different positions. It was 
 seen, for instance, that a crank is more effective in one 
 position than another ; but the momentum communicated 
 to the fly, when the crank is in the most effective position, 
 will carry the crank past its least effective position. There 
 are many cases, however, where there are irregularities of 
 motion proceeding from the nature of the machinery, which 
 could be compensated better than with a fly. Thus, if a 
 bucket is to be drawn from the bottom of a coal pit, which 
 is 60 fathoms in depth: the weight of bucket being 14 
 
MACHINERY. 
 
 285 
 
 cwt., and the chain by which it is coiled up round the cylin- 
 der weighing 8 Ibs. to every fathom, it is plain, that when 
 the bucket is at the bottom, not only the weight of the 
 bucket, but also the weight of the chain, will require to be 
 overcome in the raising of the bucket. Now the weight of 
 the chain is 60 + 8 = 480 Ibs., and the amount of the 
 weight of the bucket is 14 cwt. or 1568 Ibs. ; hence 1 568 + 
 480 == 2048 Ibs. ; but the weight of the chain will always 
 be getting less as it is coiled round the cylinder, until the 
 bucket comes to the cylinder, when the chain will be all 
 coiled, and there will remain only the weight of the bucket. 
 Now, the use of a fly may be advantageously dispensed 
 with, if the barrel on which the chain is coiled is formed 
 like a cone ; the diameter of the barrel thus increasing with 
 the uniform diminution of the weight. 
 
 The effect of the fly wheel in accumulating force, has led 
 some to suppose that there is, positively, a creation of force 
 in the fly ; but this is a mistake, for it is only, as it were, a 
 magazine of power, where there is no force but what has 
 been delivered to it. The great use of the fly wheel is thus 
 to deliver out at proper intervals, that force which has been 
 previously communicated to it ; and although there is ab- 
 solutely a small loss of power by the use of the fly, yet this 
 is more than compensated by its utility as a regulator. 
 
 The motion of machines may, as stated before, be re- 
 duced to three kinds. That which is gradually accelerated, 
 which generally takes place at the commencement of a 
 machine's action: that which is entirely uniform: that 
 which is alternately accelerated and retarded. The nearer 
 that the motion of a machine approaches to uniformity, the 
 greater will be the quantity of work done. 
 
 In order that the few remarks, which we intend to make 
 on the effect of machines, may be clearly understood, we 
 desire the reader to attend to the following definitions. 
 
 The impelled point of any machine, is that point at which 
 the force which moves the machine, may be considered as 
 applied as the piston of a steam engine, or the float board 
 of a water wheel. 
 
 The working point, on the contrary, is that point where 
 the resistance may be supposed to act. 
 
 The velocity of the moving power is the same as the ve- 
 locity of the impelled point, the velocity of the resistance 
 is the same as the velocity of the working point. 
 
286 MACHINERY. 
 
 The performance or effect of a machine is measured by 
 the resistance or work performed, (calculated by weight,) 
 multiplied by its velocity, which is, in other words, the mo- 
 mentum of the working point. The momentum of impulse, 
 on the other hand, is measured by the energy of the first 
 mover, (also estimated by weight,) multiplied by the ve- 
 locity of the impelled point. 
 
 These definitions being understood, we proceed to a 
 simple statement of principles. 
 
 When any power is made to act in opposition to a resist- 
 ance, by means either of a simple or compound machine ; 
 which machine will be in a state of rest, when the velocity 
 of the power is to that of the resistance as the weight of 
 the resistance is to that of the power. In this state of things 
 the machine can do no work, because it has no motion ; but 
 if the power is increased, so as to overcome the resistance, 
 the machine will have an accelerated motion so long as 
 the power exceeds the resistance. If the power should 
 diminish, the machine would accelerate less and less, until 
 its motion became uniform. The same effect would ne- 
 cessarily follow, if the resistance increased, a circumstance 
 which may arise from various causes. From the resistance 
 of the air, which increases with an increase of velocity ; 
 and also from friction, which often increases with the in- 
 crease of velocity. Hence we find, that machines have 
 commonly a tendency to become uniform in their motion. 
 
 We have seen before, while treating of the water wheel, 
 that the velocity of the floats of the undershot wheel, must 
 be less than the velocity of the stream. For, when the 
 float board is at rest, the water will impinge on it with the 
 greatest possible effect ; but so soon as the float begins to 
 move, then it leaves the water, as it were, and does not re- 
 ceive the whole impetus of the stream ; and if the velocity 
 of the float were equal to that of the stream, it is clear that 
 the water would have no effect upon it at all ; and, as was 
 stated before, there is a certain relation between the velocity 
 of the wheel and that of the stream, at which the effect will 
 be a maximum. This is not confined to the water wheel, 
 but is common to all machines, as we have seen illustrated 
 in the steam engine. 
 
 We have seen before, that the maximum effect of an 
 animal was, when its velocity was one-third of its greatest 
 possible speed, and the load which it bore, or the resistance 
 
MACHINERY. 287 
 
 which it overcame, was equal to four-ninths of its greatest 
 possible load. 
 
 The following tables (A and B) constructed from the re- 
 sults of Dr. Robison, will be useful to the mechanic. 
 
 Table A contains the least proportions between the vela- 
 cities of the impelled and working points of a machine ; or 
 between the levers by which the power and resistance act. 
 
 The use of this table is very simple, for suppose we 
 wished to raise 3 cubic feet of water per second, by means 
 of a water wheel, whose radius was 8 feet, (= the length 
 of the lever by which the power acts,) and the power which 
 moves the wheel being 6 cubic feet of water per second. 
 
 Employ this rule : 
 
 Power, 
 
 _- X 10 = a number, 
 
 Resistance, 
 
 which look for in column M, and against it in column N, 
 will be found a number which, when multiplied by the 
 length of lever at which the power acts, will give the length 
 of lever at which the resistance should act. 
 
 Wherefore, in the above case, 
 / 
 
 X 10 = 20, the number corresponding to which is 
 3 
 
 0-732051, hence 0-732051 X 8 = 5-856408 = the radius 
 of the axle at which the resistance or work to be done acts. 
 This table will be found very useful in the construction 
 of machines ; but they are frequently already constructed, 
 and it becomes then necessary for us to regulate the power 
 and resistance in order to produce a maximum effect, with- 
 out making any alteration in the machine. For this pur- 
 pose we employ table B, in order to show the use of which 
 we give the following rule and example : 
 
 Length of lever of resistance. 
 
 - = a number, which, when 
 Length of lever ot power, 
 
 found in column O, will stand against a number in column 
 P : such, when multiplied by the energy of power, will give 
 the proper energy of resistance. Thus, if a man exerts a 
 constant force of 56 Ibs. on the handle of a capstan, whose 
 leverage is 4 feet, and the barrel is one foot in radius, then 
 we have, 
 
 = a number, which will be found in column O, coi 
 4 4 
 
288 
 
 MACHINERY. 
 
 responding to which will be found, in column P, the num- 
 ber 1*8885; wherefore, by the rule, 
 
 1-8885 X 56 =105-756 = the resistance which the man, 
 in these circumstances, can overcome with the greatest ad- 
 vantage, or with the maximum mechanical effect. 
 
 TABLE A. 
 
 M 
 
 N 
 
 M 
 
 N 
 
 1 
 
 0-048809 
 
 20 
 
 0-732051 
 
 2 
 
 0-095445 
 
 21 
 
 0-760682 
 
 3 
 
 0-140175 
 
 22 
 
 0-788854 
 
 4 
 
 0-183216 
 
 23 
 
 0-816590 
 
 5 
 
 0-224745 
 
 24 
 
 0-843900 
 
 6 
 
 0-264911 
 
 25 
 
 0-870800 
 
 7 
 
 0-303841 
 
 26 ' 
 
 0-897300 
 
 8 
 
 0-341641 
 
 27 
 
 0-923500 
 
 9 
 
 0-378405 
 
 28 
 
 0-949400 
 
 10 
 
 0-414211 
 
 29 
 
 0-974800 
 
 11 
 
 0-449138 
 
 30 
 
 1-000000 
 
 12 
 
 0-483240 
 
 40 
 
 1-236200 
 
 13 
 
 0-516575 
 
 50 
 
 1-449500 
 
 14 
 
 0-549193 
 
 60 
 
 1-645600 
 
 15 
 
 0-581139 
 
 70 
 
 1-828400 
 
 16 
 
 0-612451 
 
 80 
 
 2-200000 
 
 17 
 
 0-643168 
 
 90 
 
 2-162300 
 
 18 
 
 0-673320 
 
 100 
 
 2-316600 
 
 19 
 
 0-702938 
 
 
 
 TABLE B. 
 
 
 
 P 
 
 
 
 P 
 
 * 
 
 1-8885 
 
 7 
 
 0-03731 
 
 1 
 
 1-3928 
 
 8 
 
 0-03125 
 
 * 
 
 0-8986 
 
 9 
 
 0-02669 
 
 1 
 
 0-4142 
 
 10 
 
 0-02317 
 
 2 
 
 0-1830 
 
 11 
 
 0-02037 
 
 3 
 
 0-1111 
 
 12 
 
 0-01809 
 
 4 
 
 0-0772 
 
 13 
 
 0-01622 
 
 5 
 
 0-0587 
 
 14 
 
 0-01466 
 
 6 
 
 0-0457 
 
 15 
 
 0-01333 
 
MACHINERY. 
 
 289 
 
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290 COTTON SPINNING. 
 
 It is not by any means an easy matter to estimate the 
 relative quantities of work done by different machines. 
 Their effects are generally stated as equivalent to so many 
 horses' power, and the following data are commonly given : 
 One horse's power, at a maximum, is equivalent to the 
 raising of 1000 Ibs. 13 feet high in one minute. In cotton 
 factories, 100 spindles, with preparation, are allowed to each 
 horse power for spinning cotton yarn twist, or five times 
 that number of spindles, with preparation, for mule yarn, 
 No. 48 ; and if it be No. 110, ten times, that number of 
 spindles, with preparation and the power-loom factories 
 12 beams with subservient machinery. 
 
 Thus a steam engine on Watt's principle, having a cylin- 
 der of 30 inches diameter, and a stroke of 6 feet, making 
 21 double strokes per minute, will give, by the usual cal- 
 culation, 
 
 7854 X 30 a X 10 X 6 X 21 X 2 
 
 44000 
 
 40 horses' power. Hence such an engine will drive 4000 
 spindles cotton yarn twist, or 20,000 spindles mule twist, 
 No. 48, or 40,000 mule twist spindles, No. 110, or 480 
 power-looms in each of which cases subservient or pre- 
 paratory machinery is included. 
 
 RULES FOR COTTON SPINNERS. 
 
 IN the following calculations the reader is supposed to be 
 acquainted with the construction of the various machines 
 employed in the cotton manufacture, so that the rules are 
 only intended to assist the memory of the practical man in 
 cases of particular difficulty. The effects of shafts, belts, 
 drums, pulleys, pinions, and wheels, in varying velocity, 
 depend upon the principles established when treating of the 
 mechanical powers, and the calculations connected with 
 them may be easily performed by the rules given in that 
 section. 
 
 To find the draught on the spreading machine, count the 
 number of teeth of the wheel on the end of the feeding roller 
 shaft, calling it the first leader, and also the number of teeth 
 on the pinion which it drives, calling it the first follower, 
 and in like manner reckon all the leaders and followers on 
 
COTTON SPINNING. 291 
 
 to the last follower i. e. the wheel on the calender roller 
 shaft, omitting all intermediate wheels, then, 
 
 product of leaders x diam. calender roller 
 product of followers x feeding roller 
 
 If the teeth of the leaders be 160, 22, and 20, and those 
 of the followers 90, 22 and 40 ; the diameter of calender 
 roller 5, and feeding roller 2 inches ; then, 
 
 160 X 22 X 20 X 5 
 
 - - == 2-26 = the draught. 
 90 X 22 X 40 X 2 
 
 The reader will have no difficulty in applying the prin- 
 ciple of this rule to the calculation of the draught of other 
 machines in cotton manufacture. 
 
 To find the number of twists per inch given to the rove 
 by the fly frame : 
 
 Turns of front roller per minute x its circumference = 
 length of rove produced in one minute, dividing the turns 
 of the spindle per minute by that product, gives the number 
 of twists on the rove per inch. 
 
 Let the revolutions of the front roller per minute be 100, 
 and the circumference 4 inches, then 100 X 4 = 400 inches 
 = 33 feet 4 inches of rove produced in a minute, where- 
 fore, if the spindle revolve 600 times in a minute, then, 
 
 A A ft 
 
 - = 1*5 twists per inch. 
 
 The proper diameter of the taking-out pulley, or men- 
 doza pulley of the stretching frame that shall regulate the 
 motion of the carriage to the delivery of the rove, may be 
 found by taking the product of the diameter of the front 
 roller x the number of teeth in the mendoza wheel, and 
 dividing by the number of teeth in the front roller pinion, 
 and subtracting from the quotient the diameter of the men- 
 doza bond. Thus if the diameter of the front roller be 1| 
 inches, the diameter of the mendoza bond 5 inch, the teeth 
 in front roller pinion 20, and in mendoza wheel 110, then, 
 
 = the diameter of mendoza pulley. 
 
 The revolutions of the spindle of the throstle may be 
 found thus : 
 
 * 
 
 ** 
 
292 COTTON SPINNING. 
 
 turns of cylinder per minute X its diameter 
 diameter of wharve 
 
 A cylinder of 7'5 inches diameter makes 450 revolutions 
 per minute, and the diameter of the wharve is 1 inch, 
 hence, 
 
 450 X 7*5 
 
 - - - = 3375 =s turns of the spindle per minute. 
 
 To find the draught of the roller of the jenny, take the 
 product of the teeth of the front roller pinion X the grist 
 pinion X diameter of back roller for a divisor, and take 
 the product of the diameter of front roller x the number of 
 teeth of the crown wheel x those of the back roller wheel 
 for a dividend, then the dividend divided by the divisor will 
 give the draught. Thus if the teeth of the crown wheel be 
 72, back roller wheel 56, front roller pinion 18, and grist 
 pinion 24, the diameter of front roller 1 inch, and of back 
 roller f , then, 
 
 10-66= 
 
 In order to determine the size of yarn from hank rove, 
 we must first find the quantity of rove given out by the 
 roller during one stretch, which is =* the whole length of 
 stretch the inches gained, and calling this the divisor, 
 the dividend will be found by taking the product of the 
 number of hank rove x the length of the stretch x the 
 draught, the quotient will be the size of yarn produced. 
 Thus, if the draught be as found above 10-666, the stretch 
 56, the gaining of carriage 5 inches, and the rove 5 hank, 
 then, 
 
 10-66 X 5 X 56 
 
 - - = 58-52 = size of yarn. 
 5o D 
 
 To find the effect of a change of the grist pinion on the 
 jenny. 
 
 Take the product of the pinion producing a known size 
 of yarn, and call it the dividend, if this be divided by any 
 other number of yarn, the quotient will be the correspond- 
 ing grist pinion ; or if another grist pinion be used as a 
 divisor, the quotient will be the corresponding size of yarn 
 produced. Thus if No. 70 yarn be produced by a pinion 
 of 24 teeth, then, 
 
COTTON SPINNING. 293 
 
 24 X 70 
 
 = 28 = the number of teeth in a grist pinion 
 
 that shall produce yarn No. 60 ; and also 
 
 24 X 70 
 
 - == 42 = the number of yarn that shall be pro- 
 duced by a grist pinion of 42 teeth. 
 
 Take the product of the diameter of the front roller x 
 the teeth of the mendoza wheel, and divide by the teeth of 
 the pinion on the front roller that drives the mendoza 
 wheel. From the quotient thus found, subtract the diame- 
 ter of the mendoza band, and the remainder is the diameter 
 of a pulley that will move the carriage out with the same 
 speed as the yarn passes through the front rollers. When 
 this is found, the diameter of such a pinion as will give a 
 certain gain on the stretch may be found by multiplying 
 the last result by the full length of the stretch, and divide 
 the product by the difference of the length of the stretch 
 and the gaining required. Thus, if the length of stretch 
 be 56 inches, the gain upon stretch 5 inches, the diameter 
 of the front roller 1 inch, and of the mendoza band jj- of an 
 inch, the number of teeth on the mendoza wheel 112, and 
 on the front roller pinion 18, then, 
 
 i- = 6-22 -625 = 5-595 = the 
 
 diameter of mendoza pulley, to move the carriage uniformly 
 with the delivery of the front roller, and 
 
 56x5-595 313-32 
 
 - - = = 6-14 = the diameter of men- 
 
 56 5 51 
 
 doza pulley to move the carriage with a gain of five inches 
 on the stretch. 
 
 The number of twists given to cotton yarn varies with 
 the quality of the fibre of the wool, the fineness of the yarn, 
 and whether it be intended for warp or weft. But omitting 
 the variation necessary for differences in the length of fibre, 
 which is comparatively trifling, the number of twists in the 
 inch will vary with the square root of the No. of the yarn, 
 or a good practical rule is this, 
 
 v/No. of yarn X 3-75 for the twists per inch of warp 
 yarn, and 
 
 x/No. of yarn X 3-25 for wefts. 
 25* 
 
294 COTTON SPINNING. 
 
 Thus for No. 36 warps, we have, 
 -v/36 x 3-75 = 6 x 3-75 == 22-5 twists per inch. 
 And for No. 64 wefts, 
 
 -v/64 x 3-25 = 8 x 3-25 = 26 twists per inch. 
 When cotton yarn is put up in hanks or spindles, it is 
 coiled upon a reel, one revolution of which takes up 54 
 inches of thread, and this length of yarn is denominated a 
 thread. 
 
 54 in. = l yards = 1 thread or round of the reel. 
 
 120 = 80 = 1 skein or ley. 
 
 840 = 560 = 7 = 1 hank or No. 
 
 15120 = 10080 = 126 = 18 = 1 spindle. 
 
 Cotton yarn is sold by weight, and its fineness is esti- 
 mated by the No. of hanks in a pound. Thus, No. 20 
 yarn contains 20 hanks, or 20 X 840 yards = 16800 yards 
 in one pound ; No. 64 contains 64 hanks or 64 x 840 = 
 53760 yards of thread in a pound ; consequently the 
 diameter of the thread of No. 64 must be much less than 
 the diameter of the thread of No. 20. 
 
 When the yarn is in cops the fineness is determined by 
 reeling a few hanks, and by finding their weight, the No. 
 of the yarn may be found by proportion ; thus if a spindle 
 be reeled, and its weight found to be 4 ounces 8 drachms, 
 then by proportion, since there are 18 hanks in a spindle, 
 and 16 ounces in a pound, and 16 drachms in an ounce, 
 we have, 
 
 4 : 16 : : 18 : 64 = the number of the yarn ; or, 
 
 288 __ , 
 
 - . . = No. of yarn ; 
 
 weight of a spindle in oz. 
 
 and, 
 
 -^ ? - = weight of a spindle in ounces. 
 
 No. of yarn 
 
 
SQUARE AND CUBE ROOTS. 
 
 295 
 
 No. 
 
 Square root 
 
 Cube root 
 
 No. 
 
 Square roo 
 
 Cube roo 
 
 No. 
 
 Square root 
 
 Cube root. 
 
 1 
 
 1- 
 
 1- 
 
 49 
 
 7- 
 
 3-659 
 
 97 
 
 9-8488 
 
 4.594 
 
 2 
 
 1-4142 
 
 1-259 
 
 50 
 
 7-0710 
 
 3-684 
 
 98 
 
 9-8994 
 
 4-610 
 
 
 
 1-7320 
 
 1-442 
 
 51 
 
 7-1414 
 
 3-708 
 
 99 
 
 9-9498 
 
 4-626 
 
 4 
 
 2- 
 
 1-587 
 
 52 
 
 7-2111 
 
 3-732 
 
 100 
 
 10- 
 
 4-641 
 
 5 
 
 2-2360 
 
 1-709 
 
 53 
 
 7-2801 
 
 3-756 
 
 101 
 
 10-0498 
 
 4-657 
 
 6 
 
 2-4494 
 
 1-817 
 
 54 
 
 7-3484 
 
 3-779 
 
 102 
 
 10-0995 
 
 4-672 
 
 7 
 
 2-6457 
 
 1-912 
 
 55 
 
 7-4161 
 
 3-802 
 
 103 
 
 10-1488 
 
 4-687 
 
 8 
 
 2-8284 
 
 2- 
 
 56 
 
 7-4833 
 
 3-825 
 
 104 
 
 10-1980 
 
 4-702 
 
 9 
 
 3- 
 
 2-080 
 
 57 
 
 7-5498 
 
 3-848 
 
 105 
 
 10-2469 
 
 4-717 
 
 10 
 
 3-1622 
 
 2-154 
 
 58 
 
 7-6157 
 
 3-870 
 
 106 
 
 10-2956 
 
 4-732 
 
 11 
 
 3-3166 
 
 2-223 
 
 59 
 
 7-6811 
 
 3-892 
 
 107 
 
 10-3440 
 
 4-747 
 
 12 
 
 3-4641 
 
 2-289 
 
 60 
 
 7-7459 
 
 3-914 
 
 108 
 
 10-3923 
 
 4-762 
 
 13 
 
 3-6055 
 
 2-351 
 
 61 
 
 7-8102 
 
 3-936 
 
 109 
 
 10-4403 
 
 4-776 
 
 14 
 
 3-7416 
 
 2-410 
 
 62 
 
 7-8740 
 
 3-957 
 
 110 
 
 10-4880 
 
 4-791 
 
 15 
 
 3-8729 
 
 2-466 
 
 63 
 
 7-9372 
 
 3-979 
 
 111 
 
 10-5356 
 
 4-805 
 
 16 
 
 4- 
 
 2-519 
 
 64 
 
 8- 
 
 4- 
 
 112 
 
 10-5830 
 
 4-820 
 
 17 
 
 4-1231 
 
 2-571 
 
 65 
 
 8-0622 
 
 4-020 
 
 113 
 
 10-6301 
 
 4-834 
 
 18 
 
 4-2426 
 
 2-620 
 
 66 
 
 8-1240 
 
 4-041 
 
 114 
 
 10-6770 
 
 4-848 
 
 19 
 
 4-3588 
 
 2-668 
 
 67 
 
 8-1853 
 
 4-061 
 
 115 
 
 10-7238 
 
 4-862 
 
 20 
 
 4-4721 
 
 2-714 
 
 68 
 
 8-2462 
 
 4-081 
 
 116 
 
 10-7703 
 
 4-876 
 
 21 
 
 4-5825 
 
 2-758 
 
 69 
 
 8-3066 
 
 4-101 
 
 117 
 
 10-8166 
 
 4-890 
 
 22 
 
 4-6904 
 
 2-802 
 
 70 
 
 8-3666 
 
 4-121 
 
 118 
 
 10-8627 
 
 4-904 
 
 23 
 
 4-7958 
 
 2-843 
 
 71 
 
 8-4261 
 
 4-140 
 
 119 
 
 10-9087 
 
 4-918 
 
 24 
 
 4-8989 
 
 2-884 
 
 72 
 
 8-4852 
 
 4-160 
 
 120 
 
 10-9544 
 
 4-932 
 
 25 
 
 5- 
 
 2-924 
 
 73 
 
 8-5440 
 
 4-179 
 
 121 
 
 11- 
 
 4-946 
 
 i 26 
 
 5-0990 
 
 2-962 
 
 74 
 
 8-6023 
 
 4-198 
 
 122 
 
 11-0453 
 
 4-959 
 
 27 
 
 5-1961 
 
 3- 
 
 75 
 
 8-6602 
 
 4-217 
 
 123 
 
 11-0905 
 
 4-973 
 
 28 
 
 5-2915 
 
 3-036 
 
 76 
 
 8-7177 
 
 4-235 
 
 124 
 
 11-1355 
 
 4-986 
 
 29 
 
 5-3851 
 
 3-072 
 
 77 
 
 8-7749 
 
 4-254 
 
 125 
 
 11-1803 
 
 5- 
 
 30 
 
 5-4772 
 
 3-107 
 
 78 
 
 8-8317 
 
 4-272 
 
 126 
 
 11-2249 
 
 5-013 
 
 31 
 
 5-5677 
 
 3-141 
 
 79 
 
 8-8881 
 
 4-290 
 
 127 
 
 11-2694 
 
 5-026 
 
 32 
 
 5-6568 
 
 3-174 
 
 80 
 
 8-9442 
 
 4-308 
 
 128 
 
 11-3137 
 
 5-039 
 
 33 
 
 5-7445 
 
 3-207 
 
 81 
 
 9- 
 
 4-326 
 
 129 
 
 11-3578 
 
 5-052 
 
 34 
 
 5-8309 
 
 3-239 
 
 82 
 
 9-0553 
 
 4-344 
 
 130 
 
 11-4017 
 
 5-065 
 
 35 
 
 5-9160 
 
 3-271 
 
 83 
 
 9-1104 
 
 4-362 
 
 131 
 
 11-4455 
 
 5-078 
 
 36 
 
 > 
 
 3-301 
 
 84 
 
 9-1651 
 
 4-379 
 
 132 
 
 11-4891 
 
 5-091 
 
 37 
 
 6-0827 
 
 3-332 
 
 85 
 
 9-2195 
 
 4-396 
 
 133 
 
 11-5325 
 
 5-104 
 
 38 
 
 6-1644 
 
 3-361 
 
 86 
 
 9-2736 
 
 4-414 
 
 134 
 
 11-5758 
 
 5-117 
 
 39 
 
 6-2449 
 
 3-391 
 
 87 
 
 9-3273 
 
 4-431 
 
 135 
 
 11-6189 
 
 5-129 
 
 40 
 
 6-3245 
 
 3-419 
 
 88 
 
 9-3808 
 
 4-447 
 
 136 
 
 11-6619 
 
 5-142 
 
 41 
 
 6-4031 
 
 3-448 
 
 89 
 
 9-4339 
 
 4-464 
 
 137 
 
 11-7046 
 
 5-155 
 
 42 
 
 6-4807 
 
 3-476 
 
 90 
 
 9-4868 
 
 4-481 
 
 138 
 
 11-7473 
 
 5-167 
 
 43 
 
 6-5574 
 
 3-503 
 
 91 
 
 9-5393 
 
 4-497 
 
 139 
 
 11-7898 
 
 5-180 
 
 44 
 
 6-6332 
 
 3-530 
 
 92 
 
 9-5916 
 
 4-514 
 
 140 
 
 11-8321 
 
 5-192 
 
 45 
 
 6-7082 
 
 3-556 
 
 93 
 
 9-6436 
 
 4-530 
 
 141 
 
 11-8743 
 
 5-204 
 
 46 
 
 6-7823 
 
 3-583 
 
 94 
 
 9-6953 
 
 4-546 
 
 142 
 
 11-9163 
 
 5-217 
 
 47 
 
 6-8556 
 
 3-608 
 
 95 
 
 9-7467 
 
 4-562 
 
 143 
 
 11-9582 
 
 5-229 
 
 48 
 
 6-9282 
 
 3-634 
 
 96 
 
 9-7979 
 
 4-578 
 
 144 
 
 12- 
 
 5-241 
 
 25* 
 
296 
 
 SQUARE AND CUBE ROOTS. 
 
 No. 
 
 Square root. 
 
 Cube root. 
 
 No. 
 
 Square root. 
 
 Cube root. 
 
 No. 
 
 Square root. 
 
 Cube root 
 
 145 
 
 12-0415 
 
 5-253 
 
 193 
 
 13-8924 
 
 5-778 
 
 241 
 
 15-5241 
 
 6-223 
 
 146 
 
 12-0830 
 
 5-265 
 
 194 
 
 13-9283 
 
 5-788 
 
 242 
 
 15-5563 
 
 6-231 
 
 147 
 
 12-1243 
 
 5-277 
 
 195 
 
 13-9642 
 
 5-798 
 
 243 
 
 15-5884 
 
 6-240 
 
 148 
 
 12-1655 
 
 5-289 
 
 196 
 
 14 
 
 5-808 
 
 244 
 
 15-6204 
 
 6-248 
 
 149 
 
 12-2065 
 
 5-301 
 
 197 
 
 14-0356 
 
 5-818 
 
 246 
 
 15-6524 
 
 6-257 
 
 150 
 
 12-2474 
 
 5-313 
 
 198 
 
 14-0712 
 
 5-828 
 
 246 
 
 15-6843 
 
 6-265 
 
 151 
 
 12-2882 
 
 5-325 
 
 199 
 
 14-1067 
 
 5-838 
 
 247 
 
 15-7162 
 
 6-274 
 
 152 
 
 12-3288 
 
 5-336 
 
 200 
 
 14-1421 
 
 5-848 
 
 248 
 
 15-7480 
 
 6-282 
 
 153 
 
 12-3693 
 
 5-348 
 
 201 
 
 14-1774 
 
 5-857 
 
 249 
 
 15-7797 
 
 6-291 
 
 154 
 
 12-4096 
 
 5-360 
 
 202 
 
 14-2126 
 
 5-867 
 
 250 
 
 15-8113 
 
 6-299 
 
 155 
 
 12-4498 
 
 5-371 
 
 203 
 
 14-2478 
 
 5-877 
 
 251 
 
 15-8429 
 
 6-307 
 
 156 
 
 12-4899 
 
 5-383 
 
 204 
 
 14-2828 
 
 5-886 
 
 252 
 
 15-8745 
 
 6-316 
 
 157 
 
 12-5299 
 
 5-394 
 
 205 
 
 14-3178 
 
 5-896 
 
 253 
 
 15-9059 
 
 6-324 
 
 158 
 
 12-5698 
 
 5-406 
 
 206 
 
 14-3527 
 
 5-905 
 
 254 
 
 15-9373 
 
 6-333 
 
 159 
 
 12-6095 
 
 5-417 
 
 207 
 
 14-3874 
 
 5-915 
 
 255 
 
 15-9687 
 
 6-341 
 
 160 
 
 12-6491 
 
 5-428 
 
 208 
 
 14-4222 
 
 5-924 
 
 256 
 
 16- 
 
 6-349 
 
 161 
 
 12-6885 
 
 5-440 
 
 209 
 
 14-4568 
 
 5-934 
 
 257 
 
 16-0312 
 
 6-357 
 
 162 
 
 12-7279 
 
 5-451 
 
 210 
 
 14-4913 
 
 5-943 
 
 258 
 
 16-0623 
 
 6-366 
 
 163 
 
 12-7671 
 
 5-462 
 
 211 
 
 14-5258 
 
 5-953 
 
 259 
 
 16-0934 
 
 6-374 
 
 164 
 
 12-8062 
 
 5-473 
 
 212 
 
 14-5602 
 
 5-962 
 
 260 
 
 16-1245 
 
 6-382 
 
 165 
 
 12-8452 
 
 5-484 
 
 213 
 
 14-5945 
 
 5-972 
 
 261 
 
 16-1554 
 
 6-390 
 
 166 
 
 12-8840 
 
 5-495 
 
 214 
 
 14-6287 
 
 5-981 
 
 262 
 
 16-1864 
 
 6-398 
 
 167 
 
 12-9228 
 
 5-506 
 
 215 
 
 14-6628 
 
 5-990 
 
 263 
 
 16-2172 
 
 6-406 
 
 168 
 
 12-9614 
 
 5-517 
 
 216 
 
 14-6969 
 
 6- 
 
 264 
 
 16-2480 
 
 6-415 
 
 169 
 
 13- 
 
 5-528 
 
 217 
 
 14-7309 
 
 6-009 
 
 265 
 
 16-2788 
 
 6-423 
 
 170 
 
 13-0384 
 
 5-539 
 
 218 
 
 14-7648 
 
 6-018 
 
 266 
 
 16-3095 
 
 6-431 
 
 171 
 
 13-0766 
 
 5-550 
 
 219 
 
 14-7986 
 
 6-027 
 
 267 
 
 16-3401 
 
 6-439 
 
 172 
 
 13-1148 
 
 5-561 
 
 220 
 
 14-8323 
 
 6-036 
 
 268 
 
 16-3707 
 
 6-447 
 
 173 
 
 13-1529 
 
 5-572 
 
 221 
 
 14-8660 
 
 6-045 
 
 269 
 
 16-4012 
 
 6-455 
 
 174 
 
 13-1909 
 
 5-582 
 
 222 
 
 14-8996 
 
 6-055 
 
 270 
 
 16-4316 
 
 6-463 
 
 175 
 
 13-2287 
 
 5-593 
 
 223 
 
 14-9331 
 
 6-064 
 
 271 
 
 16-4620 
 
 6-471 
 
 176 
 
 13-2664 
 
 6-604 
 
 224 
 
 14-9666 
 
 6-073 
 
 272 
 
 16-4924 
 
 6-479 
 
 177 
 
 13-3041 
 
 6-614 
 
 225 
 
 16- 
 
 6-082 
 
 273 
 
 16-5227 
 
 6-487 
 
 178 
 
 13-3416 
 
 5-625 
 
 226 
 
 15-0332 
 
 6-091 
 
 274 
 
 16-5529 
 
 6-495 
 
 179 
 
 13-3790 
 
 5-635 
 
 227 
 
 15-0665 
 
 6-100 
 
 275 
 
 16-5831 
 
 6-502 
 
 180 
 
 13-4164 
 
 5-646 
 
 228 
 
 15-0996 
 
 6-109 
 
 276 
 
 16-6132 
 
 6-510 
 
 181 
 
 13-4536 
 
 5-656 
 
 229 
 
 15-1327 
 
 6-118 
 
 277 
 
 16-6433 
 
 6-518 
 
 182 
 
 13-4907 
 
 5-667 
 
 230 
 
 15-1657 
 
 6-126 
 
 278 
 
 16-6733 
 
 6-526 
 
 183 
 
 13-5277 
 
 5-677 
 
 231 
 
 15-1986 
 
 6-135 
 
 279 
 
 16-7032 
 
 6-534 
 
 184 
 
 13-5646 
 
 5-687 
 
 232 
 
 15-2315 
 
 6-144 
 
 280 
 
 16-7332 
 
 6-542 
 
 185 
 
 13-6014 
 
 5-698 
 
 233 
 
 15-2643 
 
 6-153 
 
 281 
 
 16-7630 
 
 6-549 
 
 186 
 
 13-6381 
 
 5-708 
 
 234 
 
 15-2970 
 
 6-162 
 
 282 
 
 16-7928 
 
 6-557 
 
 187 
 
 13-6747 
 
 5-718 
 
 235 
 
 15-3297 
 
 6-171 
 
 283 
 
 16-8226 
 
 6-565 
 
 188 
 
 13-7113 
 
 5-728 
 
 236 
 
 15-3622 
 
 6-179 
 
 284 
 
 16-8522 
 
 6-573 
 
 189 
 
 13-7477 
 
 5-738 
 
 237 
 
 15-3948 
 
 6-188 
 
 285 
 
 16-8819 
 
 6-580 
 
 190 
 
 13-7840 
 
 5-748 
 
 238 
 
 15-4272 
 
 6-197 
 
 286 
 
 16-9115 
 
 6-588 
 
 191 
 
 13-8202 
 
 5-758 
 
 239 
 
 15-4596 
 
 6-205 
 
 287 
 
 16-9410 
 
 6-596 
 
 192 
 
 13-8564 
 
 5-768 
 
 240 
 
 15-4919 
 
 6-214 
 
 288 
 
 16-9705 
 
 6-603 
 
SQUARE AND CUBE ROOTS. 
 
 297 
 
 No. 
 
 Square root. 
 
 Cube root. 
 
 No. 
 
 Square root. 
 
 Cube root. 
 
 No. 
 
 Square root. 
 
 Cube root. 
 
 289 
 
 17- 
 
 6-611 
 
 337 
 
 18-3575 
 
 6-958 
 
 385 
 
 19-6214 
 
 7-274 
 
 290 
 
 17-0293 
 
 6-619 
 
 338 
 
 18-3847 
 
 6-965 
 
 386 
 
 19-6468 
 
 7-281 
 
 291 
 
 17-0587 
 
 6-626 
 
 339 
 
 18-4119 
 
 6-972 
 
 387 
 
 19-6723 
 
 7-287 
 
 292 
 
 17-0880 
 
 6-634 
 
 340 
 
 18-4390 
 
 6-979 
 
 388 
 
 19-6977 
 
 7-293 
 
 293 
 
 17-1172 
 
 6-641 
 
 341 
 
 18-4661 
 
 6-986 
 
 389 
 
 19-7230 
 
 7-299 
 
 294 
 
 17-1464 
 
 6-649 
 
 342 
 
 18-4932 
 
 6-993 
 
 390 
 
 19-7484 
 
 7-306 
 
 295 
 
 17-1755 
 
 6-656 
 
 343 
 
 18-5202 
 
 7- 
 
 391 
 
 19-7737 
 
 7-312 
 
 296 
 
 17-2046 
 
 6-664 
 
 344 
 
 18-5472 
 
 7-006 
 
 392 
 
 19-7989 
 
 7-318 
 
 297 
 
 17-2336 
 
 6-671 
 
 345 
 
 18-5741 
 
 7-013 
 
 393 
 
 19-8242 
 
 7-324 
 
 298 
 
 17-2626 
 
 6-679 
 
 346 
 
 18-6010 
 
 7-020 
 
 394 
 
 19-8494 
 
 7-331 
 
 299 
 
 17-2916 
 
 6-686 
 
 347 
 
 18-6279 
 
 7-027 
 
 395 
 
 19-8746 
 
 7-337 
 
 300 
 
 17-3205 
 
 6-694 
 
 348 
 
 18-6547 
 
 7-033 
 
 396 
 
 19-8997 
 
 7-343 
 
 301 
 
 17-3493 
 
 6-701 
 
 349 
 
 18-6815 
 
 7-040 
 
 397 
 
 19-9248 
 
 7-349 
 
 302 
 
 17-3781 
 
 6-709 
 
 350 
 
 18-7082 
 
 7-047 
 
 398 
 
 19-9499 
 
 7-355 
 
 303 
 
 17-4068 
 
 6-716 
 
 351 
 
 18-7349 
 
 7-054 
 
 399 
 
 19-9749 
 
 7-361 
 
 304 
 
 17-4'355 
 
 6-723 
 
 352 
 
 18-7616 
 
 7-060 
 
 400 
 
 20- 
 
 7-368 
 
 305 
 
 17-4642 
 
 6-731 
 
 353 
 
 18-7882 
 
 7-067 
 
 401 
 
 20-0249 
 
 7-374 
 
 306 
 
 17-4928 
 
 6-738 
 
 354 
 
 18-8148 
 
 7-074 
 
 402 
 
 20-0499 
 
 7-380 
 
 307 
 
 17-5214 
 
 6-745 
 
 355 
 
 18-8414 
 
 7-080 
 
 403 
 
 20-0748 
 
 7-386 
 
 308 
 
 17-5499 
 
 6-753 
 
 356 
 
 18-8679 
 
 7-087 
 
 404 
 
 20-0997 
 
 7-392 
 
 309 
 
 17-5783 
 
 6-760 
 
 357 
 
 18-8944 
 
 7-093 
 
 405 
 
 20-1246 
 
 7-398 
 
 310 
 
 17-6068 
 
 6-767 
 
 358 
 
 18-9208 
 
 7-100 
 
 406 
 
 20-1494 
 
 7-404 
 
 311 
 
 17-6351 
 
 6-775 
 
 359 
 
 18-9472 
 
 7-107 
 
 407 
 
 20-1742 
 
 7-410 
 
 312 
 
 17-6635 
 
 6-782 
 
 360 
 
 18-9736 
 
 7-113 
 
 408 
 
 20-1990 
 
 7-416 
 
 313 
 
 17-6918 
 
 6-789 
 
 361 
 
 19- 
 
 7-120 
 
 409 
 
 20-2237 
 
 7-422 
 
 314 
 
 17-7200 
 
 6-796 
 
 362 
 
 19-0262 
 
 7-126 
 
 410 
 
 20-2484 
 
 7-428 
 
 315 
 
 17-7482 
 
 6-804 
 
 363 
 
 19-0525 
 
 7-133 
 
 411 
 
 20-2731 
 
 7-434 
 
 316 
 
 17-7763 
 
 6-811 
 
 364 
 
 19-0787 
 
 7-140 
 
 412 
 
 20-2977 
 
 7-441 
 
 317 
 
 17-8044 
 
 6-818 
 
 365 
 
 19-1049 
 
 7-146 
 
 413 
 
 20-3224 
 
 7-447 
 
 318 
 
 17-8325 
 
 6-825 
 
 366 
 
 19-1311 
 
 7-153 
 
 414 
 
 20-3469 
 
 7-453 
 
 319 
 
 17-8605 
 
 6-832 
 
 367 
 
 19-1572 
 
 7-159 
 
 415 
 
 20-3715 
 
 7-459 
 
 320 
 
 17-8885 
 
 6-839 
 
 368 
 
 19-1833 
 
 7-166 
 
 416 
 
 20-3960 
 
 7-465 
 
 321 
 
 17-9164 
 
 6-847 
 
 369 
 
 19-2093 
 
 7-172 
 
 417 
 
 20-4205 
 
 7-470 
 
 322 
 
 17-9443 
 
 6-854 
 
 370 
 
 19-2353 
 
 7-179 
 
 418 
 
 20-4450 
 
 7-476 
 
 323 
 
 17-9722 
 
 6-861 
 
 371 
 
 19-2613 
 
 7-185 
 
 419 
 
 20-4694 
 
 7-482 
 
 324 
 
 18- 
 
 6-868 
 
 372 
 
 19-2873 
 
 7-191 
 
 420 
 
 20-4939 
 
 7-488 
 
 325 
 
 18-0277 
 
 6-875 
 
 373 
 
 19-3132 
 
 7-198 
 
 421 
 
 20-5182 
 
 7-494 
 
 326 
 
 18-0554 
 
 6-882 
 
 374 
 
 19-3390 
 
 7-204 
 
 422 
 
 20-5426 
 
 7-500 
 
 327 
 
 18-0831 
 
 6-889 
 
 375 
 
 19-3649 
 
 7-211 
 
 423 
 
 20-5669 
 
 7-506 
 
 328 
 
 18-1107 
 
 6-896 
 
 376 
 
 19-3907 
 
 7-217 
 
 424 
 
 20-5912 
 
 7-512 
 
 329 
 
 18-1383 
 
 6-903 
 
 377 
 
 19-4164 
 
 7-224 
 
 425 
 
 20-6155 
 
 7-518 
 
 330 
 
 18-1659 
 
 6-910 
 
 378 
 
 19-4422 
 
 7-230 
 
 426 
 
 20-6397 
 
 7-524 
 
 331 
 
 18-1934 
 
 6-917 
 
 379 
 
 19-4679 
 
 7-236 
 
 427 
 
 20-6639 
 
 7-530 
 
 332 
 
 18-2208 
 
 6-924 
 
 380 
 
 19-4935 
 
 7-243 
 
 428 
 
 20-6881 
 
 7-536 
 
 333 
 
 18-2482 
 
 6-931 
 
 381 
 
 19-5192 
 
 7-249 
 
 429 
 
 20-7123 
 
 7-541 
 
 334 
 
 18-2756 
 
 6-938 
 
 382 
 
 19-54*8 
 
 7-255 
 
 430 
 
 20-7364 
 
 7-547 
 
 335 
 
 18-3030 
 
 6-945 
 
 383 
 
 19-5703 
 
 7-262 
 
 431 
 
 20-7605 
 
 7-553 
 
 336 
 
 18-3303 6-952 
 
 384 
 
 19-5959 
 
 7-268 
 
 432 
 
 20-7846 
 
 7-559 
 
298 
 
 SQUARE AND CUBE ROOTS. 
 
 No. 
 
 Square root. 
 
 Cube root. 
 
 No. 
 
 Square root. 
 
 Cube root. 
 
 No. 
 
 Square root. 
 
 Cuberoot. 
 
 433 
 
 20-8086 
 
 7-565 
 
 481 
 
 21-9317 
 
 7-835 
 
 529 
 
 23- 
 
 8-087 
 
 434 
 
 20-8326 
 
 7-571 
 
 482 
 
 21-9544 
 
 7-840 
 
 530 
 
 23-0217 
 
 8-092 
 
 435 
 
 20-8566 
 
 7-576 
 
 483 
 
 21-9772 
 
 7-846 
 
 531 
 
 23-0434 
 
 8-097 
 
 436 
 
 20-8806 
 
 7-582 
 
 484 
 
 22- 
 
 7-851 
 
 532 
 
 23-0651 
 
 8-102 
 
 437 
 
 20-9045 
 
 7-588 
 
 485 
 
 22-0227 
 
 7-856 
 
 533 
 
 23-0867 
 
 8-107 
 
 438 
 
 20-9284 
 
 7-594 
 
 486 
 
 22-0454 
 
 7-862 
 
 534 
 
 23-1084 
 
 8-112 
 
 439 
 
 20-9523 
 
 7-600 
 
 487 
 
 22-0680 
 
 7-867 
 
 535 
 
 23-1300 
 
 8-118 
 
 440 
 
 20-9761 
 
 7-605 
 
 488 
 
 22-0907 
 
 7-872 
 
 536 
 
 23-1516 
 
 8-123 
 
 441 
 
 21- 
 
 7-611 
 
 489 
 
 22-1133 
 
 7-878 
 
 537 
 
 23-1732 
 
 8-128 
 
 442 
 
 21-0237 
 
 7-617 
 
 490 
 
 22-1359 
 
 7-883 
 
 538 
 
 23-1948 
 
 8-133 
 
 443 
 
 21-0475 
 
 7-623 
 
 491 
 
 22-1585 
 
 7-889 
 
 539 
 
 23-2163 
 
 8-138 
 
 444 
 
 21-0713 
 
 7-628 
 
 492 
 
 22-1810 
 
 7-894 
 
 540 
 
 23-2379 
 
 8-M3 
 
 445 
 
 21-0950 
 
 7-634 
 
 493 
 
 22-2036 
 
 7-899 
 
 541 
 
 23-2594 
 
 8-148 
 
 446 
 
 21-1187 
 
 7-640 
 
 494 
 
 22-2261 
 
 7-905 
 
 542 
 
 23-2808 
 
 8-153 
 
 447 
 
 21-1423 
 
 7-646 
 
 495 
 
 22-2485 
 
 7-910 
 
 543 
 
 23-3023 
 
 8-158 
 
 448 
 
 21-1660 
 
 7-651 
 
 496 
 
 22-2710 
 
 7-915 
 
 544 
 
 23-3238 
 
 8-163 
 
 449 
 
 21-1896 
 
 7-657 
 
 497 
 
 22-2934 
 
 7-921 
 
 545 
 
 23-3452 
 
 8-168 
 
 450 
 
 21-2132 
 
 7-663 
 
 498 
 
 22-3159 
 
 7-926 
 
 546 
 
 23-3666 
 
 8-173 
 
 451 
 
 21-2367 
 
 7-668 
 
 499 
 
 22-3383 
 
 7-931 
 
 547 
 
 23-3880 
 
 8-178 
 
 452 
 
 21-2602 
 
 7-674 
 
 500 
 
 22-3606 
 
 7-937 
 
 -548 
 
 23-4093 
 
 8-183 
 
 453 
 
 21-2837 
 
 7-680 
 
 501 
 
 22-3830 
 
 7-942 
 
 549 
 
 23-4307 
 
 8-188 
 
 454 
 
 21-3072 
 
 7-685 
 
 502 
 
 22-4053 
 
 7-947 
 
 550 
 
 23-4520 
 
 8-193 
 
 455 
 
 21-3307 
 
 7-691 
 
 503 
 
 22-4276 
 
 7-952 
 
 551 
 
 23-4733 
 
 8-198 
 
 456 
 
 21-3541 
 
 7-697 
 
 504 
 
 22-4499 
 
 7-958 
 
 552 
 
 23-4946 
 
 8-203 
 
 457 
 
 21-3775 
 
 7-702 
 
 505 
 
 22-4722 
 
 7-963 
 
 553 
 
 23-5159 
 
 8-208 
 
 458 
 
 21-4009 
 
 7-708 
 
 506 
 
 22-4944 
 
 7-968 
 
 554 
 
 23-5372 
 
 8-213 
 
 459 
 
 21-4242 
 
 7-713 
 
 507 
 
 22-5166 
 
 7-973 
 
 555 
 
 23-5584 
 
 8-217 
 
 460 
 
 21-4476 
 
 7-719 
 
 508 
 
 22-5388 
 
 7-979 
 
 556 
 
 23-5796 
 
 8-222 
 
 461 
 
 21-4709 
 
 7-725 
 
 509 
 
 22-5610 
 
 7-984 
 
 557 
 
 23-6008 
 
 8-227 
 
 462 
 
 21-4941 
 
 7-730 
 
 510 
 
 22-5831 
 
 7-989 
 
 558 
 
 23-6220 
 
 8-232 
 
 463 
 
 21-5174 
 
 7-736 
 
 511 
 
 22-6053 
 
 7-994 
 
 559 
 
 23-6431 
 
 8-237 
 
 464 
 
 21-5406 
 
 7-741 
 
 512 
 
 22-6274 
 
 8- 
 
 560 
 
 23-6643 
 
 8-242 
 
 465 
 
 21-5638 
 
 7-747 
 
 513 
 
 22-6495 
 
 8-005 
 
 561 
 
 23-6854 
 
 8-247 
 
 466 
 
 21-5870 
 
 7-752 
 
 514 
 
 22-6715 
 
 8-010 
 
 562 
 
 23-7065 
 
 8-252 
 
 467 
 
 21-6101 
 
 7-758 
 
 515 
 
 22-6936 
 
 8-015 
 
 563 
 
 23-7276 
 
 8-257 
 
 468 
 
 21-6333 
 
 7-763 
 
 516 
 
 22-7156 
 
 8-020 
 
 564 
 
 23-7486 
 
 8-262 
 
 469 
 
 21-6564 
 
 7-769 
 
 517 
 
 22-7376 
 
 8-025 
 
 565 
 
 23-7697 
 
 8-267 
 
 470 
 
 21-6794 
 
 7-774 
 
 518 
 
 22-7596 
 
 8-031 
 
 566 
 
 23-7907 
 
 8-271 
 
 471 
 
 21-7025 
 
 7-780 
 
 519 
 
 22-7815 
 
 8-036 
 
 567 
 
 23-8117 
 
 8-276 
 
 472 
 
 21-7255 
 
 7-785 
 
 520 
 
 22-8035 
 
 8-041 
 
 568 
 
 23-8327 
 
 8-281 
 
 473 
 
 21-7485 
 
 7-791 
 
 521 
 
 22-8254 
 
 8-046 
 
 569 
 
 23-8537 
 
 8-286 
 
 474 
 
 21-7715 
 
 7-796 
 
 522 
 
 22-8473 
 
 8-051 
 
 570 
 
 23-8746 
 
 8-291 
 
 475 
 
 21-7944 
 
 7-802 
 
 523 
 
 22-8691 
 
 8-056 
 
 571 
 
 23-8956 
 
 8-296 
 
 476 
 
 21-8174 
 
 7-807 
 
 524 
 
 22-8910 
 
 8-062 
 
 572 
 
 23-9165 
 
 8-301 
 
 477 
 
 21-8403 
 
 7-813 
 
 525 
 
 22-9128 
 
 8-067 
 
 573 
 
 23-9374 
 
 8-305 
 
 478 
 
 21-8632 
 
 7-818 
 
 526 
 
 22-9846 
 
 8-072 
 
 574 
 
 23-9582 
 
 8-310 
 
 479 
 
 21-8860 
 
 7-824 
 
 527 
 
 22-9564 
 
 8-077 
 
 575 
 
 23-9791 
 
 8-315 
 
 -180 
 
 21-9089 
 
 7-829 
 
 528 
 
 22-9782 
 
 8-082 
 
 576 
 
 24* 
 
 8-320 
 
SQUARE AND CUBE ROOTS. 
 
 299 
 
 No. 
 
 Square root. 
 
 Cube root. 
 
 No. 
 
 Square root. 
 
 Cube root. 
 
 No. 
 
 Square root. 
 
 Cube root. 
 
 577 
 
 24 0208 
 
 8-325 
 
 625 
 
 25- 
 
 8-549 
 
 673 
 
 25-9422 
 
 8-763 
 
 578 
 
 24-0416 
 
 8-329 
 
 626 
 
 25-0199 
 
 8-554 
 
 674 
 
 25-9615 
 
 8-767 
 
 579 
 
 24-0624 
 
 8-334 
 
 627 
 
 25-0399 
 
 8-558 
 
 675 
 
 25-9807 
 
 8-772 
 
 580 
 
 24-0831 
 
 8-339 
 
 628 
 
 25-0599 
 
 8-563 
 
 676 
 
 26- 
 
 8-776 
 
 581 
 
 24-1039 
 
 8-344 
 
 629 
 
 25-0798 
 
 8-568 
 
 677 
 
 26-0192 
 
 8-780 
 
 582 
 
 24-1246 
 
 8-349 
 
 630 
 
 25-0998 
 
 8-572 
 
 678 
 
 26-0384 
 
 8-785 
 
 583 
 
 24-1453 
 
 8-353 
 
 631 
 
 25-1197 
 
 8-577 
 
 679 
 
 26-0576 
 
 8-789 
 
 584 
 
 24-1660 
 
 8-358 
 
 632 
 
 25-1396 
 
 8-581 
 
 680 
 
 26-0768 
 
 8-793 
 
 585 
 
 24-1867 
 
 8-363 
 
 633 
 
 25-1594 
 
 8-586 
 
 681 
 
 26-0959 
 
 8-797 
 
 586 
 
 24-2074 
 
 8-368 
 
 634 
 
 25-1793 
 
 8-590 
 
 682 
 
 26-1151 
 
 8-802 
 
 587 
 
 24-2280 
 
 8-372 
 
 635 
 
 25-1992 
 
 8-595 
 
 683 
 
 26-1342 
 
 8-806 
 
 588 
 
 24-2487 
 
 8-377 
 
 636 
 
 25-2190 
 
 8-599 
 
 684 
 
 26-1533 
 
 8-810 
 
 589 
 
 24-2693 
 
 8-382 
 
 637 
 
 25-2388 
 
 8-604 
 
 685 
 
 26-1725 
 
 8-815 
 
 590 
 
 24-2899 
 
 8-387 
 
 638 
 
 25-2586 
 
 8-608 
 
 686 
 
 26-1916 
 
 8-819 
 
 591 
 
 24-3104 
 
 8-391 
 
 639 
 
 25-2784 
 
 8-613 
 
 687 
 
 26-2106 
 
 8-823 
 
 592 
 
 24-3310 
 
 8-396 
 
 640 
 
 25-2982 
 
 8-617 
 
 688 
 
 26-2297 
 
 8-828 
 
 593 
 
 24-3515 
 
 8-401 
 
 641 
 
 25-3179 
 
 8-622 
 
 689 
 
 26-2488 
 
 8-832 
 
 594 
 
 24-3721 
 
 8-406 
 
 642 
 
 25-3377 
 
 8-626 
 
 690 
 
 26-2678 
 
 8-836 
 
 595 
 
 24-3926 
 
 8-410 
 
 643 
 
 25-3574 
 
 8-631 
 
 691 
 
 26-2868 
 
 8-840 
 
 596 
 
 24-4131 
 
 8-415 
 
 644 
 
 25-3771 
 
 8-635 
 
 692 
 
 26-3058 
 
 8-845 
 
 597 
 
 24-4335 
 
 8-420 
 
 645 
 
 25-3968 
 
 8-640 
 
 693 
 
 26-3248 
 
 8-849 
 
 598 
 
 24-4540 
 
 8-424 
 
 646 
 
 25-4165 
 
 8-644 
 
 694 
 
 26-3438 
 
 8-853 
 
 599 
 
 24-4744 
 
 8-429 
 
 647 
 
 25-4361 
 
 8-649 
 
 695 
 
 26-3628 
 
 8-857 
 
 600 
 
 24-4948 
 
 8-434 
 
 648 
 
 25-4558 
 
 8-653 
 
 696 
 
 26-3818 
 
 8-862 
 
 601 
 
 24-5153 
 
 8-439 
 
 649 
 
 25-4754 
 
 8-657 
 
 697 
 
 26-4007 
 
 8-866 
 
 602 
 
 24-5356 
 
 8-443 
 
 650 
 
 25-4950 
 
 8-662 
 
 698 
 
 26-4196 
 
 8-870 
 
 603 
 
 24-5560 
 
 8-448 
 
 651 
 
 25-5147 
 
 8-666 
 
 699 
 
 26-4386 
 
 8-874 
 
 604 
 
 24-5764 
 
 8-453 
 
 652 
 
 25-5342 
 
 8-671 
 
 700 
 
 26-4575 
 
 8-879 
 
 605 
 
 24-5967 
 
 8-457 
 
 653 
 
 25-5538 
 
 8-675 
 
 701 
 
 26-4764 
 
 8-883 
 
 606 
 
 24-6170 
 
 8-462 
 
 654 
 
 25-5734 
 
 8-680 
 
 702 
 
 26-4952 
 
 8-887 
 
 607 
 
 24-6373 
 
 8-466 
 
 655 
 
 25-5929 
 
 8-684 
 
 703 
 
 26-5141 
 
 8-891 
 
 608 
 
 24-6576 
 
 8-471 
 
 656 
 
 25-6124 
 
 8-688 
 
 704 
 
 26-5329 
 
 8-895 
 
 609 
 
 24-6779 
 
 8-476 
 
 657 
 
 25-6320 
 
 8-693 
 
 705 
 
 26-5518 
 
 8-900 
 
 610 
 
 24-6981 
 
 8-480 
 
 658 
 
 25-6515 
 
 8-697 
 
 706 
 
 26-5706 
 
 8-904 
 
 611 
 
 24-7184 
 
 8-485 
 
 659 
 
 25-6709 
 
 8-702 
 
 707 
 
 26-5894 
 
 8-908 
 
 612 
 
 24-7386 
 
 8-490 
 
 660 
 
 25-6904 
 
 8-706 
 
 708 
 
 26-6082 
 
 8-912 
 
 613 
 
 24-7588 
 
 8-494 
 
 661 
 
 25-7099 
 
 8-710 
 
 709 
 
 26-6270 
 
 8-916 
 
 614 
 
 24-7790 
 
 8-499 
 
 662 
 
 25-7293 
 
 8-715 
 
 710 
 
 26-6458 
 
 8-921 
 
 615 
 
 24-7991 
 
 8-504 
 
 663 
 
 25-7487 
 
 8-719 
 
 711 
 
 26-6645 
 
 8-925 
 
 616 
 
 24-8193 
 
 8-508 
 
 664 
 
 25-7681 
 
 8-724 
 
 712 
 
 26-6833 
 
 8-929 
 
 617 
 
 24-8394 
 
 8-513 
 
 665 
 
 25-7875 
 
 8-728 
 
 713 
 
 26-7020 
 
 8-933 
 
 618 
 
 24-8596 
 
 8-517 
 
 666 
 
 25-8069 
 
 8-732 
 
 714 
 
 26-7207 
 
 8-937 
 
 619 
 
 24-8797 
 
 8-522 
 
 667 
 
 25-8263 
 
 8-737 
 
 715 
 
 26-7394 
 
 8-942 
 
 620 
 
 24-8997 
 
 8-527 
 
 668 
 
 25-8456 
 
 8-741 
 
 716 
 
 26-7581 
 
 8-945 
 
 621 
 
 24-9198 
 
 8-531 
 
 669 
 
 25-8650 
 
 8-745 
 
 717 
 
 26-7768 
 
 8-950 
 
 622 
 
 24-9399 
 
 8-536 
 
 670 
 
 25-8843 
 
 8-750 
 
 718 
 
 26-7955 
 
 8-954 
 
 623 
 
 24-9599 
 
 8-540 
 
 671 
 
 25-9036 
 
 8-754 
 
 719 
 
 26-8141 
 
 8-958 
 
 624 
 
 24-9799 
 
 8-545 
 
 672 
 
 25-9229 
 
 8-759 
 
 720 
 
 26-8328 
 
 8-962 
 
300 
 
 SQUARE AND CUBE ROOTS. 
 
 No. 
 
 Square root. 
 
 Cube root 
 
 No. 
 
 Square root. 
 
 Cube root. 
 
 No. 
 
 Square root. 
 
 Cube root. 
 
 721 
 
 26-8514 
 
 8-966 
 
 769 
 
 27-7308 
 
 9-161 
 
 817 
 
 28-5832 
 
 9-348 
 
 722 
 
 26-8700 
 
 8-971 
 
 770 
 
 27-7488 
 
 9-165 
 
 818 
 
 28-6006 
 
 9-352 
 
 723 
 
 26-8886 
 
 8-975 
 
 771 
 
 27-7668 
 
 9-169 
 
 819 
 
 28-6181 
 
 9-356 
 
 724 
 
 26-9072 
 
 8-979 
 
 772 
 
 27-7848 
 
 9-173 
 
 820 
 
 28-6356 
 
 9-359 
 
 725 
 
 26-9258 
 
 8-983 
 
 773 27-8028 
 
 9-177 
 
 821 
 
 28-6530 
 
 9-363 
 
 726 
 
 26-9443 
 
 8-987 
 
 774 
 
 27-8208 
 
 9-181 
 
 822 
 
 28-6705 
 
 9-367 
 
 727 
 
 26-9629 
 
 8-991 
 
 775 
 
 27-8388 
 
 9-185 
 
 823 
 
 28-6879 
 
 9-371 
 
 728 
 
 26-9814 
 
 8-995 
 
 776 
 
 27-8567 
 
 9-189 
 
 824 
 
 28-7054 
 
 9-375 
 
 729 
 
 27- 
 
 9- 
 
 777 
 
 27-8747 
 
 9-193 
 
 825 
 
 28-7228 
 
 9-378 
 
 730 
 
 27-0185 
 
 9-004 
 
 778 
 
 27-8926 
 
 9-197 
 
 826 
 
 28-7402 
 
 9-382 
 
 731 
 
 27-0370 
 
 9-008 
 
 779 
 
 27-9105 
 
 9-201 
 
 827 
 
 28-7576 
 
 9-386 
 
 732 
 
 27-0554 
 
 9-012 
 
 780 
 
 27-9284 
 
 9-205 
 
 828 
 
 28-7749 
 
 9-390 
 
 733 
 
 27-0739 
 
 9-016 
 
 781 
 
 27-9463 
 
 9-209 
 
 829 
 
 28-7923 
 
 9-394 
 
 734 
 
 27-0924 
 
 9-020 
 
 782 
 
 27-9642 
 
 9-213 
 
 830 
 
 28-8097 
 
 9-397 
 
 735 
 
 27-1108 
 
 9-024 
 
 783 
 
 27-9821 
 
 9-216 
 
 831 
 
 28-8270 
 
 9-401 
 
 736 
 
 27-1293 
 
 9-028 
 
 784 
 
 28- 
 
 9-220 
 
 832 
 
 28-8444 
 
 9-405 
 
 737 
 
 27-1477 
 
 9-032 
 
 785 
 
 28-0178 
 
 9-224 
 
 833 
 
 28-8617 
 
 9-409 
 
 738 
 
 27-1661 
 
 9-036 
 
 786 
 
 28-0356 
 
 9-228 
 
 834 
 
 28-8790 
 
 9-412 
 
 739 
 
 27-1845 
 
 9-040 
 
 787 
 
 28-0535 
 
 9-232 
 
 835 
 
 28-8963 
 
 9-416 
 
 740 
 
 27-20'^ 
 
 9-045 
 
 788 
 
 28-0713 
 
 9-236 
 
 836 
 
 28-9136 
 
 9-420 
 
 741 
 
 27-2213 
 
 9-049 
 
 789 
 
 28-0891 
 
 9-240 
 
 837 
 
 28-9309 
 
 9-424 
 
 742 
 
 27-2396 
 
 9-053 
 
 790 
 
 28-1069 
 
 9-244 
 
 838 
 
 28-9482 
 
 9-427 
 
 743 
 
 27-2580 
 
 9-057 
 
 791 
 
 28-1247 
 
 9-248 
 
 839 
 
 28-9654 
 
 9-431 
 
 744 
 
 27-2763 
 
 9-061 
 
 792 
 
 28-1424 
 
 9-252 
 
 940 
 
 28-9827 
 
 9-135 
 
 745 
 
 27-2946 
 
 9-065 
 
 793 
 
 28-1602 
 
 9-256 
 
 841 
 
 29. 
 
 9-439 
 
 746 
 
 27-3130 
 
 9-069 
 
 794 
 
 28-1780 
 
 9-259 
 
 842 
 
 29-0172 
 
 9-442 
 
 747 
 
 27-3313 
 
 9-073 
 
 795 
 
 28-1957 
 
 9-263 
 
 843 
 
 29-0344 
 
 9-446 
 
 748 
 
 27-3495 
 
 9-077 
 
 796 
 
 28-2134 
 
 9-267 
 
 844 
 
 29-0516 
 
 9-450 
 
 749 
 
 27-3678 
 
 9-081 
 
 797 
 
 28-2311 
 
 9-271 
 
 845 
 
 29-0688 
 
 9-454 
 
 750 
 
 27-3861 
 
 9-085 
 
 798 
 
 28-2488 
 
 9-275 
 
 846 
 
 29-0860 
 
 9-457 
 
 751 
 
 27-4043 
 
 9-089 
 
 H0 
 
 28-2665 
 
 9-279 
 
 J847 
 
 29-1032 
 
 9-461 
 
 752 
 
 27-4226 
 
 9-093 
 
 800 
 
 28-2842 
 
 9-283 
 
 1848 
 
 29-1204 
 
 9-465 
 
 753 
 
 27-4408 
 
 9-097 
 
 801 
 
 28-3019 
 
 9-287 
 
 849 
 
 29-1376 
 
 9-468 
 
 754 
 
 27-4590 
 
 9-101 
 
 802 
 
 28-3196 
 
 9-290 
 
 850 
 
 29-1547 
 
 9-472 
 
 755 
 
 27-4772 
 
 9-105 
 
 803 
 
 28-3372 
 
 9-294 
 
 1851 
 
 29-1719 
 
 9-476 
 
 756 
 
 27-4954 
 
 9-109 
 
 804 
 
 28-3548 
 
 9-298 
 
 1852 
 
 29-1890 
 
 9-480 
 
 757 
 
 27-5136 
 
 9-113 
 
 805 
 
 28-3725 
 
 9-302 
 
 853 
 
 29-2061 
 
 9-483 
 
 758 
 
 27-5317 
 
 9-117 
 
 806 
 
 28-3901 
 
 9-306 
 
 854 
 
 29-2232 
 
 9-487 
 
 759 
 
 27-5499 
 
 9-121 
 
 807 
 
 28-4077 
 
 9-310 
 
 855 
 
 29-2403 
 
 9-491 
 
 760 
 
 27-5680 
 
 9-125 
 
 808 
 
 28-4253 
 
 9-314 
 
 856 
 
 29-2574 
 
 9-494 
 
 761 
 
 27-5862 
 
 9-129 
 
 809 
 
 28-4429 
 
 9-317 
 
 857 
 
 29-2745 
 
 9-498 
 
 762 
 
 27-6043 
 
 9-133 
 
 810 
 
 28-4604 
 
 9-321 
 
 858 
 
 29-2916 
 
 9-502 
 
 763 
 
 27-6224 
 
 9-137 
 
 811 
 
 28-4780 
 
 9-325 
 
 859 
 
 29-3087 
 
 9-505 
 
 764 
 
 27-6405 
 
 9-141 
 
 812 
 
 28-4956 
 
 9-329 
 
 860 
 
 29-3257 
 
 9-509 
 
 765 
 
 27-6536 
 
 9-145 
 
 813 
 
 28-5131 
 
 9-333 
 
 861 
 
 29-3428 
 
 9-513 
 
 766 
 
 27-6767 
 
 9-149 
 
 814 
 
 28-5306 
 
 9-337 862 
 
 29-3598 
 
 9-517 
 
 i 767 
 
 27-6947 
 
 9-153 
 
 815 
 
 28-5482 
 
 9-340 863 
 
 29-3768 
 
 9-520 
 
 : 768 
 
 27-7128 
 
 9-157 
 
 816 
 
 28-5653 
 
 9-344 || 864 
 
 29-3938 
 
 9-524 
 
SQUARE AND CUBE ROOTS. 
 
 301 
 
 No. 
 
 Square root. 
 
 Cube root. 
 
 No. 
 
 Square root. 
 
 Cube root. 
 
 No. 
 
 Square root. 
 
 Cube root. 
 
 865 
 
 29-4108 
 
 9-528 
 
 910 
 
 30-1662 
 
 9-690 
 
 955 
 
 30-9030 
 
 9-847 
 
 866 
 
 29-4278 
 
 9-531 
 
 911 
 
 30-1827 
 
 9-694 
 
 956 
 
 30-9192 
 
 9-851 
 
 867 
 
 29-4448 
 
 9-535 
 
 912 
 
 30-1993 
 
 9-697 
 
 957 
 
 30-9354 
 
 9-854 
 
 868 
 
 29-4618 
 
 9-539 
 
 913 
 
 30-2158 
 
 9-701 
 
 958 
 
 30-9515 
 
 9-857 
 
 869 
 
 29-4788 
 
 9-542 
 
 914 
 
 30-2324 
 
 9-704 
 
 959 
 
 30-9677 
 
 9-861 
 
 870 
 
 29-4957 
 
 9-546 
 
 915 
 
 30-2489 
 
 9-708 
 
 960 
 
 30-9838 
 
 9-864 
 
 871 
 
 29-5127 
 
 9-550 
 
 916 
 
 30-2654 
 
 9-711 
 
 961 
 
 31- 
 
 9-868 
 
 872 
 
 29-5296 
 
 9-553 
 
 917 
 
 30-2820 
 
 9-715 
 
 962 
 
 31-0161 
 
 9-871 
 
 873 
 
 29-5465 
 
 9-557 
 
 918 
 
 30-2985 
 
 9-718 
 
 963 
 
 31-0322 
 
 9-875 
 
 874 
 
 29-5634 
 
 9-561 
 
 919 
 
 30-3150 
 
 9-722 
 
 964 
 
 31-0483 
 
 9-878 
 
 875 
 
 29-5803 
 
 9-564 
 
 920 
 
 30-3315 
 
 9-725 
 
 965 
 
 31-0644 
 
 9-881 
 
 876 
 
 29-5972 
 
 9-568 
 
 921 
 
 30-3479 
 
 9-729 
 
 966 
 
 31-0805 
 
 9-885 
 
 877 
 
 29-6141 
 
 9-571 
 
 922 
 
 30-3644 
 
 9-732 
 
 967 
 
 31-0966 
 
 9-888 
 
 878 
 
 29-6310 
 
 9-575 
 
 923 
 
 30-3809 
 
 9-736 
 
 968 
 
 31-1126 
 
 9-892 
 
 879 
 
 29-6479 
 
 9-579 
 
 924 
 
 30-3973 
 
 9-739 
 
 969 
 
 31-1287 
 
 9-895 
 
 880 
 
 29-6647 
 
 9-582 
 
 925 
 
 30-4138 
 
 9-743 
 
 970 
 
 31-1448 
 
 9-898 
 
 881 
 
 29-6816 
 
 9-586 
 
 926 
 
 30-4302 
 
 9-746 
 
 971 
 
 31-1608 
 
 9-902 
 
 882 
 
 29-6984 
 
 9-590 
 
 927 
 
 30-4466 
 
 9-750 
 
 972 
 
 31-1769 
 
 9-905 
 
 883 
 
 29-7153 
 
 9-593 
 
 928 
 
 30-4630 
 
 9-753 
 
 973 
 
 31-1929 
 
 9-909 
 
 884 
 
 29-7321 
 
 9-597 
 
 929 
 
 30-4795 
 
 9-757 
 
 974 
 
 31-2089 
 
 9-912 
 
 885 
 
 29-7489 
 
 9-600 
 
 930 
 
 30-4959 
 
 9-761 
 
 975 
 
 31-2249 
 
 9-915 
 
 886 
 
 29-7657 
 
 9-604 
 
 931 
 
 30-5122 
 
 9-764 
 
 976 
 
 31-2409 
 
 9-919 
 
 887 
 
 29-7825 
 
 9-608 
 
 932 
 
 30-5286 
 
 9-767 
 
 977 
 
 31-2569 
 
 9-922 
 
 888 
 
 29-7993 
 
 9-611 
 
 933 
 
 30-5450 
 
 9-771 
 
 978 
 
 31-2729 
 
 9-926 
 
 889 
 
 29-8161 
 
 9-615 
 
 934 
 
 30-5614 
 
 9-774 
 
 979 
 
 31-2889 
 
 9-929 
 
 890 
 
 29-8328 
 
 9-619 
 
 935 
 
 3t)-5777 
 
 9-778 
 
 980 
 
 31-3049 
 
 9-932 
 
 891 
 
 29-8496 
 
 9-622 
 
 936 
 
 30-5941 
 
 9-782 
 
 981 
 
 31-3209 
 
 9-936 
 
 892 
 
 29-8663 
 
 9-626 
 
 937 
 
 30-6104 
 
 9-785 
 
 982 
 
 31-3368 
 
 9-939 
 
 893 
 
 29-8831 
 
 9-629 
 
 938 
 
 30-6267 
 
 9-788 
 
 983 
 
 31-3528 
 
 9-943 
 
 894 
 
 29-8998 
 
 9-633 
 
 939 
 
 30-6431 
 
 9-792 
 
 984 
 
 31-3687 
 
 9-946 
 
 895 
 
 29-9165 
 
 9-636 
 
 940 
 
 30-6594 
 
 9-795 
 
 985 
 
 31-3847 
 
 9-949 
 
 896 
 
 29-9332 
 
 9-640 
 
 941 
 
 30-6757 
 
 9-799 
 
 986 
 
 31-4006 
 
 9-953 
 
 897 
 
 29-9499 
 
 9-644 
 
 942 
 
 30-6920 
 
 9-802 
 
 987 
 
 31-4165 
 
 9-956 
 
 898 
 
 29-9 GO 6 
 
 9-647 
 
 943 
 
 30-7083 
 
 9-806 
 
 988 
 
 31-4324 
 
 9-959 
 
 899 
 
 29-9833 
 
 9-651 
 
 944 
 
 30-7245 
 
 9-809 
 
 989 
 
 31-4483 
 
 9-963 
 
 900 
 
 30- 
 
 9-654 
 
 945 
 
 30-7408 
 
 9-813 
 
 990 
 
 31-4642 
 
 9-966 
 
 901 
 
 30-0166 
 
 9-658 
 
 946 
 
 30-7571 
 
 9-816 
 
 991 
 
 31-4801 
 
 9-969 
 
 902 
 
 30-0333 
 
 9-662 
 
 947 
 
 30-7733 
 
 9-820 
 
 992 
 
 31-4960 
 
 9-973 
 
 903 
 
 30-0499 
 
 9-665 
 
 948 
 
 30-7896 
 
 9-823 
 
 993 
 
 31-5119 
 
 9-976 
 
 904 
 
 30-0665 
 
 9-669 
 
 949 
 
 30-8058 
 
 9-827 
 
 994 
 
 31-5277 
 
 9-979 
 
 905 
 
 30-0832 
 
 9-672 
 
 950 
 
 30-8220 
 
 9-830 
 
 995 
 
 31-5436 
 
 9-983 
 
 906 
 
 30-0998 
 
 9-676 
 
 951 
 
 30-8382 
 
 9-833 
 
 996 
 
 31-5594 
 
 9-986 
 
 907 
 
 30-1164 
 
 9-679 
 
 952 
 
 30-8544 
 
 9-837 
 
 997 
 
 31-5753 
 
 9-989 
 
 908 
 
 30-1330 
 
 9-683 
 
 953 
 
 30-8706 
 
 9-840 
 
 998 
 
 31-5911 
 
 9-993 
 
 909 , 30-1496 
 
 9-686 
 
 954 
 
 30-8868 
 
 9-844 
 
 999 
 
 31-6069 
 
 9-996 
 
 26 
 
302 RECIPES. 
 
 USEFUL RECIPES FOR WORKMEN. 
 
 SOLDERS. 
 
 For Lead. Melt one part of block tin, and when in a 
 
 state of fusion add two parts of lead. If a small quantity 
 
 'of this, when melted, is poured out upon the table, there 
 
 will, if it be good, arise little bright stars upon it. Resin 
 
 should be used with this solder. 
 
 For Tin. Take four parts of pewter, one of tin, and one 
 of bismuth ; melt them together, and run them into thin 
 slips. Resin is also used with this solder. 
 
 For Iron. Good tough brass, with a little borax. 
 
 CEMENTS. 
 
 A very strong glue is made by adding some powdered 
 chalk to common glue when melted ; and a glue which will 
 resist the action of water may be formed by boiling one 
 pound of common glue in two quarts (English measure) of 
 skimmed milk. 
 
 Turkey Cement.'' Dissolve five or six bits of mastich, as 
 large as peas, in as much spirit of wine as will dissolve it. 
 In another vessel dissolve as much isinglass, (which has 
 been previously soaked in water till it is softened and 
 swelled,) in one glass of strong whisky ; add two small bits 
 of gum galbanum, or ammoniacum, which must be rubbed 
 or ground till dissolved, then mix the whole by the assist- 
 ance of heat. It must be kept in a stopped phial, which 
 should be set in hot water when the cement is to be used. 
 
 For turners, an excellent cement is made by melting in 
 a pan over the fire one pound of resin, and when melted 
 add a quarter of a pound of pitch : while these are boiling 
 add brick dust, until, by dropping a little upon a cold stone, 
 you think it hard enough. In winter it is sometimes found 
 necessary to add a little tallow. 
 
 ^ In joining the flanches of iron cylinders or pipes, to 
 withstand the action of boiling water and steam, great in- 
 convenience is often felt by the workmen for want of a 
 durable cement. The following will be found to answer : 
 Boiled linseed oil, litharge, and white lead, mixed up to a 
 proper consistence, and applied to each side of a piece of 
 flannel, linen, or even pasteboard, and then placed between 
 the pieces before they are brought home, as it is called, or 
 joined. 
 
RECIPES. 303 
 
 For Steam Engines an excellent cement is as follows : 
 Take of sal ammoniac two ounces, sublimed sulphur one 
 ounce, and cast iron filings or fine turnings one pound ; mix 
 them in a mortar, and keep the powder dry. When it is 
 to be used mix it with twenty times its quantity of clean 
 iron turnings, or filings, and grind the whole in a mortar, 
 then wet it with water, until it becomes of a convenient 
 consistence, when it is to be applied to the joint ; after a 
 time it becomes as hard and strong as any other part of the 
 metal. V^ 
 
 LACQUERS AND VARNISHES. 
 
 Old Varnish is made by pouring, by little and little, half 
 a pound of drying oil on a pound of melted copal, constantly 
 stirring with a piece of wood. x When the copal is melted, 
 take the mixture off the fire and add a pound of Venice 
 turpentine ; then pass the whole through a linen cloth. 
 When the varnish gets thick by keeping, add a little Venice 
 turpentine; and if it be wished of a dark colour, amber 
 should be used instead of copal. 
 
 Black varnish for iron is made of twelve parts of amber, 
 twelve of turpentine, two of resin, two of asphaltum, and 
 six of drying oil. 
 
 For cabinet work and musical instruments a varnish may 
 be made thus : Take four ounces of gum sandarack, two 
 ounces of lack, the same of gum mastich, and an ounce of 
 gum elemi ; dissolve them in a quart of the best whisky ; 
 the whole being kept warm when they are dissolved, add 
 half a gill of turpentine. 
 
 Lacquer is a varnish to be laid on metal, for the purpose 
 of improving its appearance or preserving its polish. The 
 lacquer is laid on the surface of the metal with a brush : the 
 metal must be warm, otherwise the lacquer will not spread. 
 
 For brass a good lacquer may be made thus : Take one 
 ounce of turmeric root ground, and half a drachm of the best 
 dragon's blood; put them in a pint of spirits of wine, 
 (English measure,) and place them in a moderate heat, 
 shaking them for several days. It must then be strained 
 through a linen cloth, and being put back into the bottle, 
 three ounces of good seed-lack, powdered, must be added. 
 The mixture must again be subjected to a moderate heat, 
 and shaken frequently for several days, when it is again 
 strained, and corked tightly in a bottle for use. 
 
304 RECIPES. 
 
 STAINING WOOD AND IVORY. 
 
 Yellow. Diluted nitric acid will often produce a fine 
 yellow on wood ; but sometimes it produces a brown, and 
 if used strong it will seem nearly black. 
 
 Red. A good red may be made by an infusion of Brazil 
 wood in stale urine, in the proportion of a pound to a gal- 
 lon. This stain is to be laid on the wood boiling hot ; and 
 before it dries it should be laid over with alum water. For 
 the same purpose a solution of dragon's blood in spirits of 
 wine may also be used. \ 
 
 Mahogany colour may be produced by a mixture of mad- 
 der, Brazil wood, and logwood, dissolved in water and put 
 on hot. The proportions must be varied by the artist ac- 
 cording to the tint required. 
 
 Black. Brush the wood several times over with a hot 
 decoction of logwood, and then with iron lacquer, or, if this 
 cannot be had, a strong solution of nut galls. 
 
 Ivory may be stained blue thus : Soak the ivory in a 
 solution of verdigris in nitric acid, which will make it 
 green, then dip it into a solution of pearlash boiling hot, 
 and it will turn blue. 
 
 To stain ivory black the same process as for wood may 
 be employed. 
 
 Purple may be produced by soaking the ivory in a solu- 
 tion of sal ammoniac into four times its weight of nitrous 
 acid. _ v 
 
 To make Edge-Tools from Cast Steel and Iron. This 
 method consists in fixing a clean piece of wrought iron, 
 brought to a welding heat, in the centre of a mould, and 
 then pouring in melted steel, so as entirely to envelope the 
 iron ; and then forging the mass into the shape required. 
 
 To colour Steel Blue. The Steel must be finely polish- 
 ed on its surface, and then exposed to a uniform degree 
 of heat. Accordingly, there are three ways of colouring : 
 first, by a flame producing no soot, as spirit of wine ; se- 
 condly, by a hot plate of iron ; and thirdly, by wood ashes. 
 As a very regular degree of heat is necessary, wood ashes 
 for fine work bears the preference. The work must be 
 covered over with them, and carefully watched ; when the 
 colour is sufficiently heightened, the work is perfect. This 
 colour is occasionally taken off with a very dilute marine 
 acid. 
 
 \ 
 
RECIPES. 305 
 
 To distinguish Steel from Iron. The principal cha- 
 racters by which steel may be distinguished from iron are 
 as follow : 
 
 1. After being polished, steel appears of a whiter, light 
 grey hue, without the blue cast exhibited by iron. It also 
 takes a higher polish. 
 
 2. The hardest steel, when not annealed, appears granu- 
 lated, but dull, and without shining fibres. 
 
 3. When steeped in acids, the harder the steel is of a 
 darker hue is its surface. 
 
 4. Steel is not so much inclined to rust as iron. 
 
 5. In general, steel has a greater specific gravity. 
 
 6. By being hardened and wrought, it may be rendered 
 much more elastic than iron. 
 
 7. It is not attracted so strongly by the magnet as soft 
 iron. It likewise acquires magnetic properties more slowly, 
 but retains them longer, for which reason steel is used in 
 making needles for compasses, and artificial magnets. 
 
 8. Steel is ignited sooner, and fuses with less degree of 
 heat than malleable iron, which can scarcely be made to 
 fuse without the addition of powdered charcoal ; by which 
 it is converted into steel, an d afterwards into crude iron. 
 
 9. Polished steel is sooner tinged by heat, and that with 
 higher colours, than iron. 
 
 10. In a calcining heat, it suffers less loss by burning 
 than soft iron does in the same heat and the same time. In 
 calcination a light blue flame hovers over the steel, either 
 with or without a sulphureous odour. 
 
 11. The scales of steel are harder and sharper than those 
 of iron ; and consequently more fit for polishing with. 
 
 12. In a white heat, when exposed to the blast of the 
 bellows among the coals, it begins to sweat, wet, or melt, 
 partly with light-coloured and bright, and partly with red 
 sparkles, but less crackling than those of iron. In a melt- 
 ing heat, too, it consumes faster. 
 
 13. In the vitriolic, nitrous, and other acids, steel is vio- 
 lently attacked, but is longer in dissolving than iron. After 
 maceration, according as it is softer or harder, it appears of 
 a lighter grey or darker colour ; while iron, on the other 
 hand, is white. 
 
 26* 
 
 .-:" 
 

 INDEX. 
 
 Air 
 
 Air vesselT.?.' $ 
 
 Animal strength 273 
 
 Arithmetic 17 
 
 Artificers' work 108 
 
 Barometer 218 
 
 Beam, working to form 157 
 
 Bramah's press 180 
 
 Catenary 99 
 
 Collision 118 
 
 Conic sections 92 
 
 Contraction, marks of. 39 
 
 Cotton spinning , 290 
 
 Cube root, extraction of 35 
 
 Cycloid 98 
 
 Drawing instruments 80 
 
 Drawing, mechanical 86 
 
 Eccentric 259 
 
 Ellipse 9497 
 
 Floating bodies 182 
 
 /Flywheel 260 
 
 Forces, central 146 
 
 Forces, parallelogram of. 119 
 
 Forcing pump 227 
 
 Fractions, decimal 22 
 
 Fractions, vulgar 17 
 
 Friction 275 
 
 Geometry 57 
 
 Governor 148 
 
 Gravity, centre of 134 
 
 Gravity, specific 182 
 
 Gyration, centre of. 142 
 
 Heat 238 
 
 Heights, measurement of. .... 219 
 
 Horses power 254 
 
 Hydrodynamics 195 
 
 Hydrostatics 175 
 
 Hyperbola 9697 
 
 Pago 
 
 Inclined plane, the 132 
 
 Joists 163 
 
 Journals 162 
 
 Lever, the 121 
 
 Lifting pump 226 
 
 Machines 278 
 
 Materials, strength of 148 
 
 Materials, weight of 1 88 
 
 Measures and weights 61 
 
 Mechanics 115 
 
 Mensuration 100 
 
 Mill-wright's table 212 
 
 Motion, accelerated 117 
 
 Motion, uniform 116 
 
 Numbers, compound 26 
 
 Oscillation, centre of 137 
 
 Parabola 9497 
 
 Parallel motion 261 
 
 Pendulum 137 
 
 Percussion, centre of 141 
 
 Pipes, contents of. 224 
 
 Pneumatics 216 
 
 Position 48 
 
 Powers, mechanical 121 
 
 Powers and roots 33 
 
 Progressions 44 
 
 Proportion, compound 43 
 
 Proportion, simple 41 
 
 Pulley, the 130 
 
 Pumps 22 1 
 
 Railways 266 
 
 Rotation 142 
 
 Screw, the 133 
 
 Sector, the 83 
 
 Shafts 159 
 
 Sliding rule 36 
 
 Square root, extraction of .... 33 
 
 Steam 243 
 
 307 
 
 
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