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Now if to this principle we add that of compound motion, we shall arrive at the knowledge of all curvilinear motion. In fact, whatever be the forces ap plied to a body describing any curve whatever, we can at each instant reduce these forces to two, the one the tangent to the element of the curve, the other the perpendicular to it. The first produces an instantaneous rectilinear motion, to which the principle of the vis inertia applies: the second is expressed by the square of the actual velocity of the body, divided by the radius of curvature, agreeably to the theory of central forces in the circle; which equally reduces to the same principle the motion in the direction of the radius of curvature.
Such are the means that were long employed to determine the motions of isolated bodies, acted upon by accelerating forces, of whatever magnitude or direction. Newton followed this method: he only clothed his solutions in a synthesis, which frequently conceals the greatest difficulties under the appearance of elegance and simplicity.
In 1716 Hermann published a treatise De Phoronomia, in which he undertook to explain all that regards the mechanics both of solids and fluids; that is to say, statics, the science of the motion of solids, hydrostatics, and hydraulics. This multiplicity of objects did not allow him to explain them as fully and clearly as was requisite. Besides, like Newton, he affects to employ the synthetic method as much as possible, which frequently interrupts the unity and connection of his problems. To this may be added, that in some places he is mistaken.
The Mechanics of Euler, published in 1736, contain the whole theory of rectilinear or curvilinear motion in an isolated body, acted upon by any accelerating forces whatever, either in vacuo, or in a resisting medium. The author has every where followed the analytical method; which, by reducing all the branches of this theory to uniformity, greatly facilitates our understanding it, as Euler manages this method with an elegance and sagacity, of which be fore him there was no example. He not only resolves a number of difficult problems, some of which were then new, but he even improves the analysis itself by new and delicate solutions, to which his subject gives occasion. As to the principles of mechanics for putting the problems into equations, he employs those mentioned above.
Though this manner of laying the foundation of the calculation was sufficiently commodious, the same end might be attained by means still more simple. This was, to resolve at every instant the forces and the motions into other forces and other motions, parallel to fixed lines of a given position in space. Nothing more then is necessary, but to apply the equations of the principle of the vis inertia to these forces and motions; in which case there is no need of recurring to the theorem of Huygens. This simple and happy idea, of which Maclaurin first made use in his Treatise on Fluxions, has thrown new light on mechanics, and singularly facilitated the solution of various problems. When the body moves constantly in one plane, two fixed axes only are to be taken, which are supposed to be perpendicular to each other,
for the sake of greater simplicity: but when we are obliged by the nature of the forces, to change the path continually in all directions, and to describe a curve of double curvature, three fixed axes are to be employed, perpendicular to each other, or form ing the edges of a rightangled parallelopiped.
The problems of the communication of motion, commonly called dynamic problems, required new principles. These, for instance, consist in determining the motions, that result from the mutual percussion of several bodies; the centre of oscillation of a compound pendulum; the motions of several bodies strung upon a rod, which has a rotatory movement round a fixed point; &c. Now it is evident, that in all cases of this sort the motion is not the same as if the bodies were isolated and at liberty, but that there must be a distribution of the forces among all the bodies forming one whole, so that the motion lost by some of them is gained by others. The motion lost or gained is always estimated by the product of the mass multiplied by the velocity lost or received, whether the coinmunication or loss of motion be produced every instant by finite degrees, as in the shock of hard bodies, or whether the velocity change at each instant only by degrees infinitely small, as in the motions of several bodies strung on a movable rod; and generally in all cases in which the forces act in the manner of gravitation.
When Huygens gave his solution of the problem of centres of oscillation, some unskilful geometricians attacked it in the reviews. James Bernoulli defended it in the Leipsic Transactions for 1686, and
undertook to give a direct demonstration of it by means of the principle of the lever. At first he con"sidered only two equal weights, fastened to an inflexible rod devoid of gravity, which was movable round a horizontal axis. Having then observed, that the velocity of the weight nearest the axis of rotation must necessarily be less, and that of the other on the contrary greater, than if each acted upon the rod separately, he concludes, that the force lost and the force gained balance each other, .and that consequently the product of the quantity of matter in one multiplied by the velocity it loses, and that of the other multiplied by the velocity it gains, must be inversely proportionate to the arms of the lever.
The substance of this luminous reasoning is accurate only James Bernoulli mistook at setting out, in considering the velocities of the two bodies as finite; instead of which he ought to have considered the elementary velocities, and compared them with the similar velocities produced every instant by the action of gravitation. The marquis de l'Hopital remarked this errour, and in correcting it he found the centre of oscillation of the two weights, without departing in other respects from the principle of Ber
Desirous then of proceeding to a third weight, he united the former two at their centre of oscillation, and combined this new weight with the third, as he had combined together the former two; and so on. But the union proposed was a little precarious, and could not be admitted without a demonstration. The paper of the marquis therefore produced no other advantage, than that of inducing James Bernoulli to
revise his former solution, to improve it, and extend it to any number of bodies. All this was successively done. James Bernoulli began with remoulding his first, and sketching a general solution: and at length he resolved the problem completely, whatever were the number and position of the bodies composing the whole. His method consists in resolving the motion of each body, at any given instant, into two other motions; one, that which the body actually takes; the other, that which is destroyed; and in forming equations, which express the condition of equilibrium between the motions lost. By these means the problem is brought under the common laws of statics. The author applies his principle to several examples and he demonstrates strictly, as well as in the most evident manner, the proposition which Huygens employed as the basis of his solution. At the conclusion of this memorable paper, which is among those of the Academy of Paris for 1703, he shows by the same principles, that the centre of oscillation and the centre of percussion exist in the same point.
This solution of the problem of centres of oscillation seemed to leave nothing to be desired; yet in 1714 it was brought forward again by John Bernoulli and Dr. Taylor, who gave solutions of it that were fundamentally the same. Their similarity occasioned a warm dispute between them, each accusing the other of plagiarism. In this new mode of treating the question it is supposed, that, instead of the elementary weights of which the pendulum is composed, other weights are substituted in one and the same point, such that their motions of angular acceleration,