Joseph Louis Lagrange's algebraic vision of the calculus

Abstract

Prior to the development of real analysis in the 19th century, J.L. Lagrange had provided an algebraic basis for the calculus. The most detailed statement of this program is the second edition of his Leçons sur le calcul des fonctions 1806. The paper discusses Lagrange's conception of algebraic analysis and critically examines his demonstration of Taylor's theorem, the foundation of his algebraic program. Lagrange's striking algebraic style is further explored in two specific subjects of the Leçons: the theory of singular solutions to ordinary differential equations and the calculus of variations. A central theme of the paper concerns Lagrange's treatment of exceptional values in his demonstration of analytical theorems. The paper concludes that Lagrange's algebraic program was a natural one, but that the conception of a functional relation given by a single analytical expression was too restrictive to provide an adequate basis for analysis.