the restoration of Lagrange to analysis. His mathematical activity burst out anew. He brought forth the Théorie des fonctions analytiques (1797), Leçons sur le calcul des fonctions, a treatise on the same lines as the preceding (1801), and the Résolution des équations numeriques (1798). In 1810 he began a thorough revision of his Mécanique analytique, but he died before its completion.
The Théorie des fonctions, the germ of which is found in a memoir of his of 1772, aimed to place the principles of the calculus upon a sound foundation by relieving the mind of the difficult conception of a limit or infinitesimal. John Landen's 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
