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THE INFINITESIMAL CALCULUS 65 It is the business of the " method of fluxions " or the "infinitesimal calculus" to give methods for finding the relations between variables from relations between their rates of change or between them and these rates. This shows the importance of the calculus in such physical questions. Mathematical physics grew up perhaps too much so on the model of theoretical astronomy, its first really extensive conquest. There are signs that mathematical physics is freeing itself from its traditions, but we need not go further into the subject in this place. Roberval devised a method of tangents which is based on Galileo's conception of the composition of motions. The tangent is the direction of the resultant motion of a point describing the curve. Newton's method, which is to be dealt with in the fifth chapter, is analogous to this, and the idea of velocity is fundamental in his "method of fluxions." CHAPTER V THE RISE OF MODERN MATHEMATICS THE INFINITESIMAL CALCULUS IN the third chapter we have seen that the ancient Greeks were sometimes occupied with the theoretically exact deter- mination of the areas enclosed by curvilinear figures, and that they used the " method of exhaustion," and, to demonstrate the results which they got, an indirect method. We have seen, too, a "method of indivisibles," which was direct and seemed to gain in brevity and efficiency from a certain lack of correctness in expression and perhaps even a small inexactness in thought. We shall find the same merits and demerits both, especially the merits, intensified in the " infinitesimal calculus." By the side of researches on quadratures and the finding of volumes and centres of gravity developed the methods of drawing tangents to curves. We have begun to deal with this
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