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 ; assuming 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 sJience by their 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 beeu 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 [Calmat* 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 pheuomena 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 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 Mdcanique Analytique ' of La Grange, which was published in 1788, and the Mdcanique Cdleste ' 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 the principle to all the mechanical powers was long an unsolved problem, and the solution may be said to have been first made known to the world by the discovery of Stevinua, relating to the sustaining power on an inclined plane. A second general principle may be conceived to be that of the composition of motions or forces ; and its discovery is to be ascribed to Galileo. Daniel Bernoulli (about 1726) was the first to demonstrate the rule of the composition of forces independently of motion ; but the application of the principle as a means of obtaining general equations of equilibrium seems to have been first made in the ' Projet dune Nouvelle Micanique,' which was published by Varignon in 1637.
La Grange treats as a third principle in mechanics that of virtual velocities. fly this is meant those which bodies in equilibrio would have at the first instant of their motion, in the event of the equilibrium being disturbed. Indications of this principle are found in the writings of Galileo, \Wallis, and Descartes, but John Bernoulli is thought to have been the first ss ho showed its utility in resolving statical problems. [VIRTUAL VELOCITIES.] A general method of solving mechanical propositions was discovered by D'Alembert, and it may be thus enunciated. If there be impressed on bodies motions which they are forced to change in consequence of their mutual actions, those motions may be considered as compounded of the motions which the bodies do really take, and of those which are destroyed. Whence it results that these last must be such that if they alone existed the bodies would be in oquilibrio. In order to avoid the decompositions of motions which this principle requires, an equation is frequently made between the general analytical expression for a force and the expression for those forces which produce the observed motions. [FORCES, IMPRESSED.] The manner of estimating the value of a mechanical force is various ; and a difference in the expression of the value gave rise to disputes which continued during nearly all the first half of the 18th century. [FoncE1 Besides the principles above mentioned there occur in mechanical investigations several others, which it will be proper to state briefly in this place.