ON THE ENCINER TEETH OF WHEELS: DEMONSTRATING ΑΡΥ. THE BEST FORMS WHICH CAN BE GIVEN TO THEM FOR THE E. & F. N. SPON, 48, CHARING CROSS. TED •C15 PREFACE TO THE FIRST EDITION. IMPROVEMENT in science is frequently very slow, being retarded either by ignorance or prejudice, both of which too often prevail among those who have the best opportunity of making practical observations. Mechanics, in general, conduct their operations according to a certain routine, which they have acquired by experience; and as they are little versed in theory, or the mathematical principles of what they profess, they seldom think of going beyond the beaten track in which they have been accustomed to move. Some of our greatest engineers, indeed, have been men of no education;* they attained to eminence merely by the natural force of their genius, and of course were liable to fall into mistakes, which the man versed in mathematical knowledge could either rectify or avoid. We need not therefore be surprised that the proper formation of the teeth of wheels, which is of so much importance in mechanics, should have been so much overlooked and neglected as they are found to be. The perfection of the most simple as well as the most complicated engines, depends greatly upon the due action of the teeth of the wheels with each other, or, in other words, on the best form for ensuring their proper action with the least friction, and of course with the least wear and loss of power. It is needless here to state of what vast consequence the numerous and immense machines are to the manufactures of this country, or what prodigious sums of money are involved therein. It is therefore of great importance that the best form for the teeth of wheels should be ascertained on true mathematical principles, and that they should not be left, as is too much the case, to the random guesses of workmen. If the teeth of wheels be properly constructed, the work is more equably performed, and of course is better done; less force or moving power is requisite, or, more work may be done with a given power; and, to crown all, the whole machinery has greater durability, consequently costs less for repairs, and occasions less loss of valuable time, by not being so frequently obliged to stand still. Now the saving of power, of expense, and of time in *Of this we have an instance in the celebrated Brindley. repairs, are objects by no means to be lightly thought of. Under such impressions, an intelligent and ingenious friend advised the publisher to procure a translation of that part of M. Camus's "Cours de Mathématique," which treats on the best form of the teeth of wheels, as being scientifically handled, and on true mathematical principles. M. Camus having gone through the subject of the proper form for the teeth of wheels, goes on to describe the art of finding the number of teeth and leaves which ought to be given to wheels and pinions, and likewise the application of these principles to trains of wheel-work in general, an operation which in clock and watch-work is called calibering, as it likewise determines the diameters of the wheels and pinions. Of M. Camus it may not be uninteresting to say, that when only twelve years of age, his progress in science was so great, that he was able to give lectures on the mathematics at Paris; and at an early period he attained the highest academic honours of his own country and of most foreign Academies. In 1765 he was elected a Fellow of the Royal Society of London. He died in the year 1768, aged 69. As a geometrician and a mathematician he was equalled by few: he was the author of various works on the different branches of mathematics, the principal of which is the "Cours de Mathématique," from which the present sheets have been selected. The other parts of the same work are not less important, whether considered on, account of the subjects themselves, or of the method in which they are handled. If this part should meet with suitable encouragement from the public, it may lead to a translation of the entire work. There requires little or no apology for translating the present work; for, although the subject is so important, very little has been published upon it in this country. It is therefore hoped that this tract will be found highly useful to mechanists in general, and to all persons concerned in wheel-work in particular. How far the translator has succeeded in rendering it clear and perspicuous, is not his province to determine: he flatters himself that he has conveyed the author's meaning in such a manner as to be readily understood by those who are in the least acquainted with mathematical science; and if his labour proves useful, he shall think the pains he employed on it well bestowed. THE EDITOR'S PREFACE TO THE SECOND EDITION. ALWAYS feeling annoyed at meeting with a long preface to a book-labouring, as it were, to beget a prepossession in favour of the author, and standing between the reader and the subject, like an impertinent porter, who detains a visitor at the gate, instead of giving him admission to the presence of the master, -the editor will confine himself to two pages of preliminary remarks, on the qualifications which the reader must bring with him, in order to his understanding the investigations and demonstrations of the author; and to give some information respecting the arrangements of the work, and on the tables used in the calculations. In regard to the qualifications, it is necessary that the student be acquainted with such common characters and terms used in algebra, and in trigonometry, as are to be easily learned from books containing the elements of mathematics, one of which may be purchased for sixpence at most of the book stalls. And, that he should possess a determined resolution to prosecute the mathematical investigations of the subject, with patient and persevering industry; and, if he be a workman, he should feel persuaded, that to be enabled to unite theory with practice, would greatly enhance his value, and would give him a better chance of employment in his old age, by disposing and assisting him to follow the stream of improvement, which must constantly accompany the universal progress of the human mind. The properly qualified student will be delighted to find, that in Camus nothing is assumed, his progress being marked by a steady and sure making good of every step as he advances; and, in proceeding onwards, his mind will feel itself on secure footing. In respect to the arrangement, it has been deemed expedient to invert the order observed in the first edition, where the " additions were given before the work itself, to which it purported to be additional; and, therefore, in this second edition, |