Influence of Servois’ Work
By the end of Section 18, Servois had finished his exposition of the rules of the single variable calculus. He included one further section, in which he considered some results from multivariable calculus. His equation (119), for example, is the two-variable version of Taylor’s theorem. The paper ends with some brief closing comments, which he concludes by setting the stage for his “Reflections” paper [Servois 1814b], which followed in the next issue of Annales de mathématiques pures et appliquées. In his “Reflections,” Servois examines the competing foundational notions for the calculus. Of course, he argues for the superiority of Lagrange’s formal series approach. He also argues forcefully against the use of the infinitely small in mathematics, but expresses some sympathy for the notion of limit.
It’s quite reasonable to imagine that Servois felt he had made a persuasive case for a Lagrangian foundation for calculus with the combined weight of the “Essay” and the “Reflections,” which he published together as a single small volume. However, the winds of change were probably already blowing in a direction that was favorable to the idea of limits. Jean Le Rond d’Alembert (1717-1783) had already championed the limit as the “true metaphysics of the differential calculus” in Diderot’s Encyclopédie [Diderot 1751]. Lacroix published a popular elementary calculus text that used the informal notion of limit as the basis for the calculus [Lacroix 1802]. Perhaps it’s telling that Legendre and Lacroix, the referees who evaluated the first version of the “Essay,” said that by “recalling to the differential calculus several methods, some of which don’t seem very appropriate to the current state of analysis, [the author] has done something that is very useful for the science.” They praised Servois’ work, but seemed to consider it to be already old-fashioned.
A decisive step in the development of the modern rigorous approach to calculus came with the publication of Cauchy’s Cours d’analyse  just a few years after Servois’ “Essay.” Grabiner states: “After Cauchy, foundations had become an essential part of analysis, and Cauchy’s books and teaching were largely responsible" [1981, p. 15]. Cauchy led the way for later analysts, such as Weierstrass, Riemann, and Lebesgue, to complete a foundation for calculus based on limits.
Even though Servois was not successful in creating a foundation for calculus, his work was important to other mathematicians. For instance, his work spread to England and significantly influenced mathematicians such as Duncan Farquharson Gregory (1813-1844) and Robert Murphy (1806-1843) in their own efforts in the foundations of analysis. However, the real lasting influence of his work in England was in the development of abstract algebra and linear operator theory [Allaire and Bradley 2002].
Download the authors’ English translation of Servois’ “Essay.”