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not allow me to analyse them all, and a simple catalogue of them would be useless. Accordingly I shall content myself with quoting the Mechanics of Camus, a work valuable for the strictness and perspicuity of it's demonstrations. Among other things the author gives the whole theory of toothed wheels with much accuracy and method. He was not a very profound geometrician; but he had a very accurate judgment, and was well versed in the synthetic method of the ancients, which justly stood very high in his estimation. In this way he solved the problem of placing in equilibrio, between two inclined planes, a rod having a weight applied to any part of it's length. It is true this problem is very easy in the analytical method, but it leads to a calculation of some length. The synthetic solution of Camus merits attention for it's simplicity and elegance, an advantage which synthesis sometimes enjoys over analysis, and which should not be neglected when opportunity offers.
A description of the machines invented within little more than a century, even were it confined to the most ingenious or most useful, would fill of itself an extensive work. Were it compatible with my design, I should by no means pass over the steam-engine, which ought to be placed in the first rank of the productions of mechanical genius. I shall merely observe, that the moving power of this machine is the vapour of water, alternately expanded and condensed; and that it's motion is effected by mechanical means, nearly of the same nature as those employed in clocks and watches. It appears, that the
force of steam began to be made known by the experiments of the marquis of Worcester, about the year 1660. Afterward Papin, a french physician, having farther investigated the nature of this agent by his celebrated digester, constructed, in 1698, the first steam engine ever seen. This was very imperfect; but it gave birth to that of captain Savery, which was far superiour, and has itself been succeeded by others still more perfect. At present steam-engines are employed in every country in Europe for various purposes. I return to the general theory of me
Since the principle of the parallelogram of powers had been begun to be applied to statics, no one had thought of examining it's foundation very rigorously. All geometricians had at once agreed in admitting, that, if a body be acted upon at once by two powers capable of impelling it separately, in the same length of time, along the two sides of a parallelogram, it would be moved through the diagonal by their joint action. The same law was afterward extended to the
simple powers of pressure; and it was concluded, that two powers of this kind being represented by the sides of a parallelogram, their result would be repre sented by the diagonal. But Daniel Bernoulli, not finding sufficient connection and evidence in the transition from one case to the other, demonstrated the second proposition in a direct manner, and independently of all consideration of compound motion, Mem. of the Ac. of Petersburg for 1726. Many other geometricians, in particular d'Alembert, equally demonstrated it by various methods, more or less
simple. Unfortunately all these demonstrations are too long and perplexing, to find a convenient place in elementary treatises on statics; but at least they exist among the writings of geometricians, as numerous guarantees of a truth, which is proved however by means more simple, and better adapted to the wants of beginners.
I have already spoken of the problems of the catenary curve, the sail distended by the wind, the elastic curve, &c., when giving an account of the progress of the analysis of infinites, to which they directly contributed. These problems, and several others of the same nature, were again resolved by Daniel Bernoulli, Euler, Hermann, and others, but with new additions, and fresh difficulties, which enhanced the honour of success, and enlarged the domains of science.
The general theory of various motions offered a new and extensive field to the researches of geometricians furnished with the analysis of infinites. Galileo had made known the properties of rectilinear motion uniformly accelerated: Huygens had considered curvilinear motion; and had risen by degrees to the beautiful theory of central forces in the circle, which is equally applicable to motion in any curve, by considering all curves as infinite series of small arcs of a circle, agreeably to the idea, which he himself had employed in his general theory of evolutes.
The laws of the communication of motion, likewise, sketched by des Cartes, and farther pursued by Wallis, Huygens, and Wren, had made a new and very considerable step, by means of the solution
which Huygens gave of the celebrated problem of centres of oscillation.
All these acquisitions, at first separate and in some measure independent of each other, having been reduced to a small number of simple, commodious, and general formula, by means of the analysis of infinites, mechanics took a flight, which nothing but the difficulties still arising from the imperfection of it's instruments could check, and of which I will endeavour to impart some idea.
The problems relating to motion may all be reduced to two classes. The first comprises the general properties of the motion of a single body, acted upon by any given powers: the second, the motions which result from the action and reaction, that several bodies exert on each other, in any given manner.
In the motion of a single body, we observe, that, matter being of itself indifferent to rest or motion, a body set in motion must uniformly persevere in it; and that it's velocity cannot increase or diminish, unless by the continual action of a constant or variable power. Hence arise two principles; that of the vis inertia, and that of compound motion; on which is founded the whole theory of motion, rectilinear or curvilinear, constant or variable according to a given law. By virtue of the vis inertiæ, motion at every instant is essentially rectilinear and uniform, all resistance, and every kind of obstacle, being put out of consideration. By the nature of compound motion, a body exposed to the action of any given number of forces, all tending at the same time to change the quantity and direction cf it's motion,
takes such a path through space, that in the last instant it reaches the same point, at which it would have arrived, had it successively and freely obeyed each of the forces proposed.
On applying the first of these principles to rectilinear motion uniformly accelerated, we perceive, 1st, that, in this motion, the velocity increasing by equal degrees, or proportionally to the time, the accelerating force must be constant, or incessantly give equal impulses to the moving body; and that, consequently, the final velocity is as the product of the accelerating force multiplied by the time: 2dly, each elementary portion of space passed through being as the product of the corresponding velocity multiplied by the element of the time, the whole of the space passed through is as the product of the accelerating force multiplied by the square of the time. Now these two properties equally take place for each elementary portion of any variable motion whatever : for there is no reason why we should not generally consider the accelerating force, though varying from one instant to another, as constant during each single instant, or as undergoing it's changes only at the commencement of each elementary portion of the time. Thus, in every rectilinear motion, variable according to any given law, the increment of the velocity is as the product of the accelerating force, multiplied by the element of the time; and the differential second of the space passed through is as the product of the accelerating force multiplied by the square of the element of the time.