residual calculus, professing a similar object, was unknown to him. Lagrange attempted to prove Taylor's theorem (the
power of which he was the first to point out) by simple algebra, and then to develop the entire calculus from that theorem. The principles of the calculus were in his day involved in philosophic difficulties of a serious nature. The infinitesimals of Leibniz had no satisfactory metaphysical basis. In the
differential calculus of Euler they were treated as absolute
zeros. In Newton's limiting ratio, the magnitudes of which it
is the ratio cannot be found, for at the moment when they should be caught and equated, there is neither arc nor chord. The chord and arc were not taken by Newton as equal before vanishing, nor after vanishing, but when they vanish. "That method," said Lagrange, "has the great inconvenience of considering quantities in the state in which they cease, so to speak, to be quantities; for though we can always well conceive the ratios of two quantities, as long as they remain finite, that ratio offers to the mind no clear and precise idea, as soon as its terms become both nothing at the same time." D'Alembert's method of limits was much the same as the
method of prime and ultimate ratios. D'Alembert taught
that a variable actually reached its limit. When Lagrange
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 with respect to as the coefficient of in the expansion of 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 can always be expanded in a series of ascending powers of , 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,