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At the start of 1814, Servois was called back into active duty, one last time, to defend Paris from the Austrian and Prussian armies [Boyer 1895a]. Not only was France involved in a battle that would shape its destiny, but calculus also was in a similar state of upheaval. Mathematicians knew there were problems with the foundations of calculus, but Augustin-Louis Cauchy (1789-1857) had not yet begun his revision of those foundations.

**Figure 4.** The Battle of Paris, 1814, as painted by Bogdan Willewalde in 1834 (public domain)

During his time in the military, Servois had become acquainted with Joseph Diaz Gergonne (1771-1859). Servois' friendship with Gergonne was particularly helpful in building his mathematical publication record. Gergonne was the editor of the *Annales des math**é**matiques pures et appliqu**é**es,* often called *Annales de Gergonne.* Servois' most celebrated paper, “Essai sur un nouveau mode d'exposition des principes du calcul différential” [Servois 1814a] (Essay on a New Method of Exposition of the Principles of Differential Calculus), his primary work on the foundations of the differential calculus, appeared late in 1814. Following Lagrange, his work proposed using power series expansion as fundamental to the differential calculus. He defined the differential operator as an infinite series, which he called an *infinitinôme**,* in powers of the difference operator: \[dz = \Delta z - \frac{1}{2} \Delta^2 z + \frac{1}{3} \Delta^3 z - \frac{1}{4} \Delta^4 z + \ldots.\]

Using this operator, he demonstrated how to find differentials of functions and was able to discover the basic laws of the differential calculus, such as the power and product rules. This paper truly is remarkable because, at the heart of Servois' arguments, lies the use of the linear properties of operators in proving theorems about the calculus. Servois didn't use the terms “linear” or “operator,” but rather he used the words *commutative* and *distributive* for the first time in their mathematical sense. In reviewing the contents of this paper, the Commissioners of the Institut de France praised him for having “done something that is very useful for the science [of analysis]” [Servois 1814a, p. 140].

**Figure 5.** The Institut de France (photograph by Benh Lieu Song, 2007, licensed under Creative Commons Attribution-Share Alike 3.0 Unported license)

Servois' second work on the foundations of calculus was “Réflexions sur les divers systèmes d'exposition des principes du calcul différentiel, et, en particulier, sur la doctrine des infiniment petits” [Servois 1814b] (Reflections on the Various Systems of Exposition of the Principles of the Differential Calculus and, in particular, on the Doctrine of the Infinitely Small). This work gives us insight into Servois' personal views of the history of calculus and into his philosophical positions, particularly on placing calculus on an algebraic foundation. This paper began with a historical overview of calculus from its birth to the time of Lagrange. Servois discussed the contributions made by well-known mathematicians, such as Newton, Leibniz, Taylor, Euler, and Lagrange, to the differential calculus. In addition, he pointed out where, in his opinion, some mathematicians had erred. For instance, he stated that the use of infinitesimals is difficult for mathematicians to justify as a basis for calculus. The remainder of the work is devoted to his justification of development of calculus by means of power series and to a harsh criticism of the mathematical philosophy of Josef-Maria Hoëné-Wronski (1776-1853).

*Editor’s note:* For a more detailed analysis and an English translation of Servois’ “Reflections,” see the *Convergence* article, “Servois’ 1814 Essay on the Principles of the Differential Calculus, with an English Translation,” by the author and Robert E. Bradley.

Salvatore J. Petrilli, Jr. (Adelphi University), "Francois-Joseph Servois: Priest, Artillery Officer, and Professor of Mathematics - Battles for Paris and for the Foundations of Calculus," *Convergence* (November 2010), DOI:10.4169/loci003498