657 



MECHANICS. 



MECHANICS. 



of Flanders. This mathematician and engineer supposed a chain o 

 uniform dimensions to be placed on a doubly inclined plane, having a 

 common summit and base, the chain being perfectly free to slide on 

 tbe planes, and its ends hanging vertically to equal distances below the 

 base ; then, in order to prove that the chain would remain at rest, he 

 shows that if any motion should take place, it might continue for 

 ever; and this he concludes to be absurd. As the argument holds 

 good when one of the two planes is in a vertical position, Stevinus 

 infers that, when a body is in equilibrio upon a plane, the retaining 

 power is to the weight as the height of the plane is to its length ; and 

 he further shows that if three forces act on any point, they hold the 

 latter in equilibrio when they are proportional to the three sides of a 

 triangle formed by lines drawn parallel to the directions of the forces. 

 It should be remarked however that Stevinus demonstrates the law in 

 that case only in which two of the forces are at right angles to each 

 other. He died in 1635. 



To Galileo we are indebted for the 1 first reduction of mechanical 

 propositions to purely mathematical formulae. In order to demonstrate 

 the equilibrium of a body on an inclined plane, he imagined the weight 

 and the sustaining power to be applied to the ends of a bent lever 

 whose arms were of equal length and perpendicular to the vertical and 

 slant sides of the plane; then reducing the lever to a straight one, 

 between the lines of direction of the weight and power, it was easy to 

 prove that the forces in equilibrio on the plane were also in equilibrio 

 on the lever, and were to one another as the length to the height of 

 the plane. 



But the most important discoveries of Galileo were those which 

 relate to the times of descent, the spaces descended, and the velocities 

 acquired when bodies fall by the action of gravity. He made observa- 

 tions on the motions of pendulums, and determined that the times of 

 their vibrations are proportioned to the square roots of their lengths ; 

 he also gave theorems for the composition of two motions, when both 

 are uniform, when both are accelerated, and when one is uniform and 

 the other accelerated. Nor should we omit to state that he was the 

 first to obtain expressions for the strength of materials in resisting the 

 strains to which they are subject. It deserves notice moreover that 

 Galileo, iu opposing the arguments of one of his contemporaries con- 

 cerning the law of the descent of bodies by gravity, makes a supposition 

 that the spaces descended with the accelerated motion may be divided 

 into equal parts, each so small that the motion during the time of 

 describing it may, without sensible error, be considered as uniform ; 

 an hypothesis corresponding exactly to that which, agreeably to the 

 principles of the modern analysis, is now employed in investigations 

 concerning variable motions. 



The theory of the motions of fluidt was, apparently, first cultivated 

 in Italy by Castelli, who wrote on the subject in 1638 ; and about the 

 game time Torricelli, having discovered the existence of a space void 

 of air in the upper part of a tube filled with mercury, its open end 

 being immersed in a vessel of that fluid, was enabled to refute the 

 ancient notion that nature abhorred a vacuum. The latter was subse- 

 quently led to the conclusion that the pressure of the atmosphere is 

 the cause of the support of a column of mercury in a tube, and also of 

 the ascent of water in pumps. Both of these writers were pupils of 

 Galileo ; and, soon after the time of this philosopher, the French 

 mathematicians Descartes, Pascal, Fennat, and Roberval, prosecuted 

 with ardour the new science, MS that of mechanics was called. Among 

 the fruits of their researches may be named the determination of the 

 centres of oscillation and percussion in a body or system of bodies 

 vibrating about a fixed axia. The impulse given by Galileo, being 

 thus continued by a succession of men of talent both in Italy and 

 France, caused the science to advance with an accelerated movement, 

 and soon put it in a condition to embrace all the subjects of terrestrial 

 physics. 



The mechanics of that age was not however entirely emancipated 

 from the trammels of a false philosophy ; and the theory of Descartes, 

 concerning the communication of motion when bodies strike each 

 other, is remarkable on account of the metaphysical principle which it 

 involves. In speaking of the collision of bodies, he gives as a reason 

 why the same momentum should exist after as before the impact, that 

 it depends on the divine immutability. God having created a certain 

 quantity of motion to serve as the cause of all the operations of nature, 

 that quantity, he conceives, can never be increased or diminished. 

 Yet there is some reason to think that Descartes had better notions 

 concerning the phenomena of collision, for he states correctly, in one 

 of <us letters, that the motion of a body when it strikes another which 

 is at rest becomes divided between the two masses, and that the 

 resulting velocity is diminished as the mass is augmented. The chief 

 feature in the physics of Descartes is his supposition that the planets 

 revolve about the sun in vortices of zther, the particles of which, 

 having acquired a certain degree of centrifugal force, act on the planets 

 and prevent them from falling together in the centre of the system. 

 He supposed that the like vortices surround each planet : but the 

 particles of aether, having less specific gravity than the bodies on the 

 surface of the plannt, the tendency of these bodies to that surface 

 prevail.* over the.force by which the aether causes them to recede from 

 thence. 



The laws of the collision of bodies, which had been in vain attempted 

 by Descartes, were at length, aud nearly at the same time, discovered 



by the English mathematicians Wallis and Sir Christopher Wren, and 

 by Huyghens on the Continent. The first of these, in his treatise ' De 

 Motu'(1670), divides bodies into such as are hard and such as are 

 elastic, and he explains the phenomena attending the shock of bodies 

 of both kinds. In that of hard bodies he adopts as an hypothesis that 

 the body struck destroys as much motion in the striking body as the 

 latter communicates to it ; and in elastic bodies he considers the 

 forces of compression and restitution to be proportional, in each, to 

 the velocities before the shock. The name of Huyghens has become 

 celebrated from the discovery of the properties of cycloidal curves, and 

 the attempt to make the lower extremity of a clock-pendulum vibrate 

 in an arc of that kind, in order that the times of vibration might be 

 equal, whatever were the extent of the arc described. This attempt 

 did not succeed ; but, being led in the course of his inquiries to inves- 

 tigate the position of the centre of oscillation in a compound pendulum, 

 Huyghens found that when several pendulous bodies descend by gravity 

 and afterwards re-ascend by the acquired velocities, in whatever way 

 they may act upon each other, their common centre of gravity cannot 

 rise higher than the point from whence it descended. This proposition 

 is considered as proved from the fact that, if it were otherwise, the 

 centre of gravity might by mechanical means be made to rise con- 

 tinually higher, and thus perpetual motion might ensue : but this is 

 impossible. 



In 1687 Newton's great work concerning the mathematical principles 

 of natural philosophy was first published, and from that time the 

 mechanical sciences, which had hitherto been confined to the action of 

 bodies on each other at the surface of the earth, were made to com- 

 prehend the laws of planetary motion. The ' Principia,' as the work 

 is called, commences with the three well known axioms in philosophy, 

 or laws of motion. Assuming then as an hypothesis, that all the bodies 

 of the universe and all the particles of every body exert on each other 

 mutual attractions ; assunu'ng also that the planetary bodies were 

 originally put in motion by impulsive forces ; the rotations of these 

 bodies on their axes, their revolutions in their orbits, and all the per- 

 turbations by which their movements are varied, are explained by 

 means of the elementary theorem for the composition and resolution of 

 motions. The oscillations of pendulums, the theory of projectiles, the 

 movements of fluids, and the resistance opposed by the latter to the 

 motions of bodies immersed in them, are also in the same work 

 investigated at length. 



Contemporary with Wallis, Wren, and Newton in England, were, on 

 the Continent, the celebrated Leibnitz and the two elder Bernoullis, 

 all of whom contributed greatly to the advancement of mechanical 

 science by then- investigations concerning the laws of motion in 

 terrestrial bodies ; and to the rivalry as well as the talents of these 

 great men we owe some of the most important discoveries in that 

 branch of learning. At this time the fluxional or differential calculus 

 was discovered, and had acquired an algorithm ; and they who adopted 

 its principles appear to have been anxious to show its superiority over 

 the ancient geometrical analysis, by proposing to their opponents 

 problems which could scarcely be solved by the latter method. With 

 some such views Leibnitz proposed the determination of that curve 

 along which a body descending would describe equal vertical spaces 

 in equal times ; James Bernoulli proposed to find the figure assumed 

 by a flexible cord or chain when suspended at the extremities 

 [CATENART] ; and John Bernoulli, to find the curve of swiftest 

 descent. [CYCLOID.] Numerous other problems of the like nature 

 were given out among the parties, and the solutions could not fail, if 

 no other benefit arose, of carrying the new calculus to a considerable 

 degree of perfection. 



From the time of Newton mechanical science was, till lately, but 

 little cultivated in this country ; but on the Continent a succession of 

 illustrious men continued to prosecute the investigation of subjects 

 connected with it, and by the employment of analytical processes they 

 rendered comparatively easy the application of its principles to the 

 researches of physical astronomy. 



The mathematicians who may be considered as the immediate suc- 

 cessors of Newton were chiefly Euler, D'Alembert, and Clairaut ; and 

 in the works of the first of these are investigated all the circumstances 

 attending the phenomena of rectilinear and curvilinear motion when a 

 body in vacuo or in a resisting medium is subject to any forces what- 

 ever. But the most remarkable event in the history of the sciences, 

 after the discoveries of the English philosopher, was the solution of the 

 celebrated problem of the three bodies : or that whose object is to deter- 

 mine the motions of a body when attracted by and revolving about 

 another, and continually disturbed by the attraction of a third. This 

 was, at the same time (about 1752), and independently of each other, 

 accomplished by the three learned men above named, and it now con- 

 stitutes the basis of the whole planetary theory. The ' M&anique 

 Analytique' of La Grange, which was published in 1788, and the 

 'Me'canique Celeste ' of La Place (1798 to 1825), contain the last 

 accessions which the mechanical sciences have since received ; and 

 these sciences now comprehend the laws of force or motion, from 

 the properties of the simple lever to the phenomena of the heavenly 

 bodies. 



It may have been seen above that the first general principle in 

 mechanics is that of the equilibrium of bodies on a lever ; and a know- 

 ledge of it may be ascribed to Archimedes. The extension of th 



