Page:A History of Mathematics (1893).djvu/288

 endeavoured to free the calculus of its metaphysical difficulties, by resorting to common algebra, he avoided the whirlpool of Charybdis only to suffer wreck against the rocks of Scylla. The algebra of his day, as handed down to him by Euler, was founded on a false view of infinity. No correct theory of infinite series had then been established. Lagrange proposed to define the differential coefficient of $$\scriptstyle{f(x)}$$ with respect to $$\scriptstyle{x}$$ as the coefficient of $$\scriptstyle{h}$$ in the expansion of $$\scriptstyle{f(x+h)}$$ by Taylor's theorem, and thus to avoid all reference to limits. But he used infinite series without ascertaining that they were convergent, and his proof that $$\scriptstyle{f(x+h)}$$ can always be expanded in a series of ascending powers of $\scriptstyle{h}$, labours under serious defects. Though Lagrange's method of developing the calculus was at first greatly applauded, its defects were fatal, and to-day his "method of derivatives," as it was called, has been generally abandoned. He introduced a notation of his own, but it was inconvenient, and was abandoned by him in the second edition of his Mécanique, in which he used infinitesimals. The primary object of the Théorie des fonctions was not attained, but its secondary results were far-reaching. It was a purely abstract mode of regarding functions, apart from geometrical or mechanical considerations. In the further development of higher analysis a function became the leading idea, and Lagrange's work may be regarded as the starting-point of the theory of functions as developed by Cauchy, Riemann, Weierstrass, and others.

In the treatment of infinite series Lagrange displayed in his earlier writings that laxity common to all mathematicians of his time, excepting Nicolaus Bernoulli II. and D'Alembert. But his later articles mark the beginning of a period of greater rigour. Thus, in the Calcul de fonctions he gives his theorem on the limits of Taylor's theorem. Lagrange's mathematical researches extended to subjects which have not been