same space of time. This very remarkable proposition includes what is commonly called the isochronism of the cycloid, and would alone have been sufficient, to establish the fame of a geometrician.

From the motion of an isolated body a transition was made to the motions, which several bodies communicate to each other, whether they act by impact, or by the interposition of levers, cords, &c.

The

most simple of these problems was that of a body proceeding to strike against another at rest, or moving before it with less velocity, or approaching towards it. Des Cartes, misled by his metaphysical principles, which had induced him to suppose, that the same quantity of absolute motion always exists in the World, concluded, that the sum of the motions after the impact was equal to the sum of the motions before it. But the proposition is true only in the first and second of these cases: it is false when the two bodies meet each other, for in this case the sum of the motions after the impact is equal to the difference of the motions before it, not to their sum. Thus des Cartes discovered only part of the truth.

In 1661, Huygens, Wallis, and sir Christopher Wren, all discovered the true laws of percussion separately, and without any communication with each other, as has been completely proved. The base of their solutions is, that, in the mutual percussion of several bodies, the absolute quantity of motion of the centre of gravity is the same after as before the shock. Farther, when the bodies are elastic, the respective velocity is the same after as before per

cussion.

R 4

Two

Two other celebrated and more difficult problems concerning the communication of motion, proposed by father Mersenne in 1635, exercised the skill of geometricians for a long time. One was, to determine the centre of oscillation of a compound pendulum the other, to find the centre of percussion of a body, or a system of bodies, turning round a fixed

axis.

In the first, several heavy bodies connected together, at invariable distances, by rods considered as destitute of weight, are supposed to oscillate round a fixed horizontal axis. Then, as all these bodies restrict the motion of each other, and do not acquire the same velocities as if each oscillated separately, the bodies nearest the axis lose a part of their natural motion, and transmit it to the most remote. Thus there is an equilibrium between the motions lost and the motions gained. In whatever way this equilibrium is established, there exists in the system some point, where if a small isolated body were placed, it would oscillate in the same time as the compound pendulum; whence this point has been called the centre of oscillation.

The property of the centre of percussion is of another nature. What characterises this point is, that it must be found in the direction of the resultante of all the motions of the bodies of a system turning round a fixed axis, and occupy in this system a place analogous to that, which the centre of gra vity occupies in a heavy body. I have said of another nature: for, though it is demonstrated, that the centre of oscillation and the centre of percussion are

situated

situated in one and the same point of the system, and that the two problems are resolvable by the same principles of mechanics, the application of these principles is more simple and easy in the second case, than in the first, and the two questions are different.

Des Cartes and Roberval, persuaded that they were the same, and finding it more easy, to consider them under the second point of view, than under the first, determined the point sought with accuracy in some particular cases: but in several others they deceived themselves. Their methods, being likewise founded on vague and uncertain suppositions, were very precarious and insufficient.

Huygens was the first, who resolved in a complete and general manner the most important of these problems, that of the centres of oscillation. He assumed as a principle, that, after the centre of gravity of a compound pendulum has descended to the lowest point, if all the bodies should separate from one another, and each ascend singly with the velocity it had acquired, the centre of gravity of the system in this state would ascend to the same height, as that from which the centre of gravity of the pendulum had descended.

At first this solution was not very well understood: some philosophers attacked it's principle, perfectly incontestable in itself, yet it must be confessed a little strained, and in consequence not presenting, at least to every mind, a very evident connexion with the elementary laws of mechanics. It was afterward demonstrated in the most luminous and indisputable manner; and is now every where known by the name

of

of the principle of the conservation of active forces. The problem of centres of oscillation is the firstborn of that numerous family of problems of dynamics, so long agitated among geometricians.

Though the investigation of the centre of percussion offered but few difficulties to geometricians versed in mechanics, many of them resolved the problem badly, or gave incomplete solutions of it. Wallis himself was mistaken in it, in his treatise De Motu. Long after him James Bernoulli, of whom I shall have occasion to take much notice in the subsequent pages, gave an accurate and general solution of it by means of the principle of the lever.

CHAR

CHAP. IV.

Progress of Hydrodynamics.

IT has been seen, that Stevinus contributed a little to the advancement of statics; and the same must be said of him with regard to hydrostatics. He showed, that the pressure of a fluid on the bottom of a vessel is always as the product of the area of that bottom multiplied by the height of the fluid, whatever the shape of the vessel may be: but he does not appear to have had a thorough perception of the reciprocal connexion of all parts of hydrostatics.

The first methodical and truly original treatise on hydrostatics, published by a modern writer, is that of Pascal on the Equilibrium of Fluids. The author demonstrates the properties of the equilibrium of fluids by this simple and extensive principle: when two pistons, applied to two apertures made in a vessel full of any fluid whatever, and close in every other part, are acted upon by forces inversely proportional to the apertures, they are in equilibrio. He solves all the difficulties, which certain propositions might still offer; such as, for example, the once celebrated paradox, which is no longer so, that a slender tube of water, and a column of the same fluid of any diameter, being of the same height, and pressing on the same bottom, exert equal pressures.

The