In 1797, Lagrange published the Théorie des fonctions analytiques, the expression of his mature views on the nature of the calculus. These views were, as we have seen, the product of a long period of concern with the foundations of the calculus. He had concluded that algebra would provide the only satisfactory foundation; as we have seen, the algebra of the eighteenth century was sufficiently rich to provide a basis for the many diverse results of the Fonctions analytiques.
The Théorie des fonctions analytiques, in this first edition, is not an especially attractive work, particularly in comparison with the second edition (1813), which is the one reprinted in Lagrange's Oeuvres. The first edition has no division into chapters, and the organization seems arbitrary; Lagrange later said that he had written it “comme d'un seul jet, à mesure qu'il s'imprimait.” Nevertheless, it was received with enthusiasm and was widely read. As the word of a master analyst, it would of course be of interest to leading mathematicians. Interest was heightened since the problem to which Lagrange addressed himself was so fundamental.
For our purposes, two features of the Fonctions analytiques are most worthy of discussion. The foremost is the way in which Lagrange derived the received results of the calculus from his algebraic foundation. I shall undertake an extensive and technical description of some of the details; one reason for this is to convey the flavor of FA. In particular, we shall see the extent to which FA is not a collection of computations with formal power series, but instead proceeds by means of manipulation with algebraic inequalities—a method which was to influence Cauchy and Weierstrass.