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

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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 $k \ge n$.