Servois' 1814 Essay on a New Method of Exposition of the Principles of Differential Calculus, with an English Translation - Lagrange and Algebraic Analysis

Figure 2. Title page of Lagrange’s Théorie des fonctions analytiques (public domain).

Mathematicians tried a variety of approaches in giving a rigorous foundation for calculus at the beginning of the nineteenth century. There were three competing notions: differentials, limits, and power series expansions. This last approach was due to Lagrange, who believed he could use algebraic analysis to derive a power series expansion of any function, without recourse to derivatives, limits or differentials. He described his program in the following passage:

In a memoir printed among those of the Academy of Berlin for 1772, I proposed that the theory of the expansion of functions into series contained the true principles of the differential calculus, freed from any consideration of the infinitely small, or of limits, and I proved the theorem of Taylor using this theory, which we may regard as the fundamental principle of this calculus and which had previously never been proven except with the assistance of this same calculus, or by consideration of infinitely small differences [Lagrange 1797, p. 5].

Servois was a disciple of Lagrange and supported his algebraic approach to explaining how calculus works, although he was also sympathetic to the value of limits to calculus. Lagrange’s contention that a function could always be expanded into a Taylor series and that this could be used as the fundamental basis of calculus was Servois’ guiding principle, a doctrine to which he would adhere with an almost religious fervor.