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 contrary to fact. Hence the supposition is false, and the theorem is established. Fermat applied this method of descent with success in a large number of theorems. By this method Euler, Legendre, Dirichlet, proved several of his enunciations and many other numerical propositions.

A correspondence between Pascal and Fermat relating to a certain game of chance was the germ of the theory of probabilities, which has since attained a vast growth. Chevalier de Méré proposed to Pascal the fundamental problem, to determine the probability which each player has, at any given stage of the game, of winning the game. Pascal and Fermat supposed that the players have equal chances of winning a single point.

The former communicated this problem to Fermat, who studied it with lively interest and solved it by the theory of combinations, a theory which was diligently studied both by him and Pascal. The calculus of probabilities engaged the attention also of Huygens. The most important theorem reached by him was that, if A has p chances of winning a sum a, and q chances of winning a sum b, then he may expect to win the sum $$\scriptstyle{{ap+bq} \over {p+q}}$$. The next great work on the theory of probability was the Ars conjectandi of Jakob Bernoulli.

Among the ancients, Archimedes was the only one who attained clear and correct notions on theoretical statics. He had acquired firm possession of the idea of pressure, which lies at the root of mechanical science. But his ideas slept nearly twenty centuries, until the time of Stevin and Galileo. Stevin determined accurately the force necessary to sustain a body on a plane inclined at any angle to the horizon. He was in possession of a complete doctrine of equilibrium. While Stevin investigated statics, Galileo pursued principally dynamics. Galileo was the first to abandon the Aristotelian idea that bodies descend more quickly in proportion as they are