and their motions with respect to the axis of rotation, shall be the same, and the new pendulum oscillate as the former. While confessing, that this solution merits praise, all geometricians at present agree, that it is neither so luminous, nor simple, as that of James Bernoulli, founded immediately on the laws of equilibrium. It has been seen, that Leibnitz estimated the momenta of bodies in motion by the product of the quantity of matter multiplied by the square of the velocity. John Bernoulli, having adopted this opinion, gave to the principle of Huygens, for the problem of centres of oscillation, the name of the principle of conservation of the vires cica:' and this it has retained, because in fact, in the motions of a system of heavy bodies, the sum of the products of the masses by the squares of the velocities remains the same, when the bodies descend conjointly, and when they afterward ascend separately with the velocities. they acquired by their descent. Huygens himself had briefly made this remark, in a letter on the first paper of James Bernoulli and on that of the marquis de l'Hopital. This law equally holds in the shock of perfectly elastic bodics, and in all the motions of bodies acting on each other by pressure: it necessarily follows from the nature of these movements, and is independent of every hypothesis respecting the measure of vires viva. Accordingly the geome tricians of the last century employed it with success in many problems in dynamics. But as it gives only a single equation, from which the velocity or the time must be afterward expunged, this second object was was attained by different means. John Bernoull employed for this the principle of tensions; Euler, that of pressures; Daniel Bernoulli, the virtual power, which a system of bodies has of reestablishing itself in it's former state; and in certain cases both Euler and Daniel Bernoulli, the constant quantity of circulatory motion round a fixed point. When at length all the differential equations of the problem were established, nothing more remained but the difficulty of resolving them; a new rock, on which analysts of moderate talents sometimes split. In 1743 d'Alembert had the happy thought of generalizing the principle, which James Bernoulli had employed for resolving the problem of centres of oscillation. He showed, that, in whatever manner the bodies of one system act on each other, their motions may always be resolved, at every instant, into two sorts of motion, those of the one being destroyed the instant following, and those of the other retained; and that the motions retained are neeessarily known from the conditions of the equilibrium between the motions destroyed. This general principle applies to all the problems of dynamics, and at least reduces all their difficulties to those of the problems of simple statics. It also renders that of the conservation of vires viva useless. By means of it d'Alembert has resolved a number of very beautiful and very difficult problems, some of which were absolutely new, as, for example, that of the precession of the equinoxes. His Treatise on Dynamics, therefore, published in 1749, must be considered as an original work. In vain may it be objected, that James Bernoulli had traced FF3 traced out the road: it was equally traced out for the other geometricians, who preceded d'Alembert, yet none of whom perceived it during the space of forty years. Dynamics having thus gradually attained a high degree of perfection, were still farther enriched in 1765 by an important discovery, fertile in corollaries, In a short paper, entitled, Specimen Theorie Turbinum, Segner observed, that, if a body of any size and figure, after rotatory or gyratory motions in all directions have been given to it, be left entirely to itself, it will always have three principal axes of rotation; that is to say, that all the rotatory motions, by which it is affected, may be constantly reduced to three, which are performed round three axes perpendicular to each other, passing through the centre of gravity or of inertia of the body, and always preserving the same position in absolute space, while the centre of gravity is at rest, or moves uniformly in a right line. The position of these three axes is determined by an equation of the third order, the three real roots of which relate to each of them. This theory, which it's author had not sufficiently developed, Albert, the worthy son of the great Euler, treated at length in his paper on the stowage of ships, which shared the prize of the Academy of Sciences of Paris in 1761 as did likewise his father, according to the same method, in the Memoirs of the Academy of Berlin for 1759, and in his work entitled Theoria Motus Corporum rigidorum, 1765. Lastly d'Alembert showed in the fourth volume of his Mathematical Opuscula, published in 1768, that the solution of the problem was deducible from the formule, which he had given in a Memoir for determining the motion of a Body of any Figure, acted upon by any Forces whatever,' printed in the first volume of his Opuscula, in 1761. The knowledge of these motions of free rotationround three principal axes easily leads to the determination of the motion round any variable axis whatever. Hence, if we now suppose the body to be acted upon by any given accelerating forces, we shall begin with determining the rectilinear or curvilinear motion of the centre of gravity, abstractedly from all rotatory motion: then combining this progressive motion with the rotatory motion of a given point of the body round a variable axis, we shall know at every instant the compound motion of this point in absolute space. In this manner Euler has resolved several new pro blems of dynamics. CHAP. XII. Progress of Hydrodynamics. THE HE principle of equal pressure, applied to the general laws of hydrostatics, was sufficient to explain all the particular cases of equilibrium, which are referrible to it. But the science of the motion of fluids, at least as to it's theoretical part, was still confined to the single proposition of Torricelli; that is, to the knowledge of the issue of fluids through infinitely small, or very minute apertures. Newton undertook to solve this problem in his Principia, A. D. 1686, without confining himself to this supposition. He considers a vertical cylindrical vessel, perforated through the bottom by a hole of any diameter, from which the water issues, while the vessel is constantly receiving at the top as much as escapes at the bottom, in such a manner, that the water flowing in may be considered as forming a stratum of uniform thickness, suddenly spread on the surface of the water of the cylinder, which thus remains constantly filled to the same height. He then conceives the water in the cylinder to be divided into two parts, the one central, and freely movable, which he calls the cataract; the other adjacent, and immovable, which is confined externally by the sides of the vessel. He supposes, that the velocity of any horizontal section of the cataract is owing to the correspondent |