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 MECHANICS 317 derstood, and were therefore neglected for many centuries, till they were revived by Gali- leo and by Stevinus. The mechanical advan- tage of the inclined plane was one of the first and most important propositions which engaged the attention of mechanicians on the revival of physical science. Cardan in 1545 asserted that the force necessary to support a body on an inclined plane is double when the incli- nation is double. Guido Ubaldo in 1577 at- tempted to prove that an acute wedge will produce a greater mechanical effect than an obtuse one, but did not establish the proposi- tion. His references to the screw, the inclined plane, and the wedge, however, show that he comprehended their relations. Michael Varro, whose Tractatus de Motu was published at Geneva in 1584', treats of the wedge in a man- ner which indicates at least an approach to the doctrine of the composition and resolution of forces. The explanation of the true the- ory of the inclined plane was first made by Ste- vinus of Bruges. He supposed a loop of a string, loaded with 14 equal balls at equal dis- tances, to hang over a double inclined plane whose sides were in the proportion of two to one, and which would therefore support four and two balls respectively. That the loop must remain at rest followed from the fact that after any motion it would still be in the same condition as before, so that if motion took place it would go on indefinitely and re- sult in perpetual motion, which he regarded as an absurdity. He shows that the festoon of eight balls hanging below the planes may be removed without disturbing the equilibrium of those resting upon the planes ; so that the four balls on the longer plane would balance the two on the shorter, which would make the weights supported by the planes propor- tional to their length. Stevinus also shows that when three forces act upon a point, they will be in equilibrium when they are in pro- portion to the sides of a triangle which are parallel to the direction of the forces. He however only gives a demonstration of the case in which two of the forces are at right angles to each other. Leonardo da Vinci had before this obtained clear ideas regarding the equilibrium of oblique forces, as shown by extracts from his manuscripts published by Yenturi in 1797. In 1499 Leonardo gave a correct statement of the forces exerted in an oblique direction on a lever, and made a dis- tinction between the length of the arm of the lever and that of the perpendicular to the di- rection of the force. These views of Leonardo are believed to have been sufficiently known by Galileo to aid him in his speculations, and the modes of reasoning of the two are some- what similar. Leonardo had also asserted that the time of descent of a body down an inclined plane is to the time of its vertical descent in the proportion of the length of the plane to its height. The most important discoveries of Galileo are in regard to the laws of falling bodies, as determined by his observations on the vibrations of a pendulum, which he found were proportional to the square roots of the lengths of the pendulum. He was also the first to enter into a mathematical investigation of the strength of materials in resisting strains. The problems regarding the collision of bodies were attempted by Descartes, who made some important observations ; but no clear ideas or theories were obtained till Huygens in Holland and Wallis and Wren in England turned their attention to the subject, all these about the same time sending papers to the royal society of London. The first example of a correct so- lution of a problem of circular motion occurs in the theorems of Huygens. The problem of the centre of oscillation was proposed by Mersenne in 1640, and had attracted the at- tention of Huygens when a youth, but he was then unable to find any principles sufficient for its solution. But when, in 1673, he pub- .lished his Horologium Oscillatorium, a fourth part of the work was on this subject, and the theories then advanced have been found strict- ly correct. In 1687 Newton's great work was published, when for the first time the science of mechanics was extended from a con- sideration of the action of forces upon bodies on the earth to the action of forces exerted be- tween celestial bodies, and the adoption of the theory of universal gravitation. It was en- riched about the same time by the method of fluxions of Newton, and its improvement by Leibnitz, known as the differential calculus, an invention which greatly facilitated the investi- gations of mechanical problems. The illustri- ous family of Bernoullis of Basel, all of whom were natural philosophers, added much to the mathematical knowledge of mechanics. The transcendent mathematical powers of Euler gave analytical method to mechanical solutions. His memoirs occupy a large portion of the " Pe- tropolitan Transactions " from 1728 to 1783, and so many were left at his death that their publi- cation was not completed in 1818. In 1747 D'Alembert and Clairaut sent on the same day to the French academy of sciences their solu- tions of the celebrated "problem of three bod- ies " (see MOON), which for a long time claimed the attention of mathematicians. The labors of Clairaut have been of great service to the science of mechanics, but his name, and those of D'Alembert, Lagrange, and Laplace, be- long more to the department of physical as- tronomy, and with others have received at- tention in the article ASTRONOMY. The sub- jects of friction, strength of materials, theory of arches and domes, perpetual motion, and hydromechanics, will be found under those heads. Universal gravitation is treated in the articles ASTRONOMY, GRAVITY, MOON, and un- der other astronomical titles, and the science of projectiles under GUNNERY. This article will be occupied only with a consideration of the following subjects : 1, the laws of motion, including the laws of impact and uniformly