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The French Revolution encouraged a move away from the categorical thinking of the Enlightenment and towards the idea of rigor in mathematics [Gillispie 2004]. Inspired by this movement and by the work of Lagrange, François-Joseph Servois believed that the only way to make the foundation of the differential calculus truly rigorous was to base it on the method of power series. Formally, Servois introduced calculus through series in his “Essay” [Servois 1814a]. His “Reflections” paper, published in the same year, was intended to be a manifesto regarding his philosophical beliefs on the foundations of the differential calculus.

Servois’ main argument was that there is no way to rigorously prove theorems using the idea of infinitesimals. The method of calculus through power series gave students of mathematics a rigorous foundation upon which to learn calculus. The dangers of using infinitesimals are best explained by Servois:

In a word, I am convinced that the infinitesimal method does not nor cannot have a theory, that in practice it is a dangerous instrument in the hands of beginners, that it necessarily imprints a long-lasting character of awkwardness and pusillanimity upon their work in the course of applications [Servois 1814b, p. 148].

Interestingly, Servois was mistaken in his claim that the method “cannot have a theory.” Some 150 years later, Robinson and Laugwitz invented nonstandard analysis, with which they were able to give a rigorous foundation to the idea of infinitesimals [Medvedev 1998].

The material discussed in this paper can aid teachers of both calculus and the history of mathematics. The history of mathematics provides the opportunity to illustrate how mathematics is a constantly evolving field, with warring factions using non-mathematical arguments to bolster their beliefs in the superiority of certain methods or definitions. Besides providing a readable account of the history of the calculus, this translation of Servois’ “Reflections” is an original source for investigating philosophical ideas at an important turning point in the rigorization of calculus.

Furthermore, this translation provides many opportunities for student research projects in the development of the rigorous calculus. For example, Servois names dozens of mathematicians who contributed to the development of calculus [Servois 1814b, pp. 144-150]. Many are well-known, but some are absent from the standard biographical reference works. For example, who was Nicole [Servois 1814b, p. 150]? What did he write, other than the paper mentioned by Servois? In the same pages, Servois refers to many of the ideas of these mathematicians, without further elaboration. What, for example, were the “incomprehensibilities of Sturmius” or the “Subtleties of Guido Grandi” [Servois 1814b, p. 147]? Finally, for those with an interest in philosophy, there are many possible avenues for further research. What, for example, are Kant's four antinomies [Servois 1814b, p. 145]? Do Wronski and Servois really seem to understand Kant's doctrine of thesis and antithesis?

Download the authors' English translation of Servois' “Reflections.”

Robert E. Bradley (Adelphi University) and Salvatore J. Petrilli, Jr. (Adelphi University), "Servois' 1814 Essay on the Principles of the Differential Calculus, with an English Translation - Conclusions, Recommendations, and Translation," *Convergence* (November 2010), DOI:10.4169/loci003487