# Servois' 1814 Essay on a New Method of Exposition of the Principles of Differential Calculus, with an English Translation - Recommendations for the Classroom

Author(s):
Robert E. Bradley (Adelphi University) and Salvatore J. Petrilli, Jr. (Adelphi University)

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 and how the oversights of past mathematicians should not be viewed as carelessness or error, but as innovations appropriate to the context of their times [Petrilli 2009]. Servois’ “Essay” is an original source that provides insight into the competing notions of calculus in the early nineteenth century and can be used to illustrate the aforementioned statement.

This translation provides many opportunities for student research projects in the development of the rigorous calculus. The reader's guide in this paper provides only glimpses into the details of Servois’ calculus. Thus, there are opportunities for exploring much of Servois’ mathematics and its historical influences in depth. For instance, the following questions could be considered: After examining Sections 1-12, how close was Servois to discovering the ring structure? Who else was influenced by the work of Servois, besides Gregory and Murphy? To what extent was Servois influenced by the work of Louis François Antoine Arbogast (1759-1803) and Jacques Français (1775-1833)?

Teachers of the history of mathematics can incorporate some concrete problems and proofs into their course using Servois’ “Essay.” For example, students can use Servois’ definition (5) to find the differentials of simple polynomial functions, such as ${\mbox d} x$, ${\mbox d} x^2$, ${\mbox d} x^3$, $\ldots$, ${\mbox d} x^n$, as well as linear combinations of these functions. Then, students can compare Servois’ method of finding differentials to Newtonian fluxions and the modern limit-based method. Additionally, students can use their knowledge of mathematical induction to prove several statements that Servois did not formally prove. Also, it is possible to use some of Servois’ definitions to construct additional proofs by induction, such as: If $z = F(x)$ is a polynomial of degree $n$, then $\Delta^k z=0$ for $k \ge n$.