Lagrange’s definition of function in his Théorie des fonctions analytiques [Lagrange 1797] was much narrower than the one we use today. Article 1 of the book reads as follows:
We define a function of one or several variables to be any expression of calculation in which these quantities appear in any way whatsoever, combined or not with other quantities that we consider to be given and invariable values, whereas the quantities of the function may receive any possible value.
Thus, Lagrange considered only the sort of well-behaved functions that are given by formulas, the functions that we usually consider in a freshman calculus course. Therefore, it’s not so surprising that he assumed that the process he described for expanding functions into series could always be undertaken or that, like Newton and Taylor, he gave little consideration to the issue of convergence; see [Grabiner 1990, pp. 15-16]. Servois largely followed Lagrange’s lead in assuming that a function was essentially a formula involving algebraic expressions, as well as familiar logarithmic, exponential and trigonometric functions.
Although Lagrange’s goal was to create a foundation for calculus that did not depend on infinitesimals or limits, we can find embedded in his work the use of something like the limit concept. When he made the claim that (h) can be taken as small as one desires, he was essentially letting (h) approach zero in the difference quotient, which at that point is the only missing component of the modern definition of the derivative.
Finally, Lagrange had set the stage for other mathematicians to see the necessity for rigor in calculus. His work would influence many mathematicians of Servois’ generation, and also a later generation of students of the Ecole Polytechnique, including Augustin-Louis Cauchy (1789-1857) [Gillispie 2004].
Servois took up the challenge to expand upon Lagrange’s foundational work and in 1805 he submitted his first version of the “Essay on a new method of exposition of the principles of differential calculus” [Servois 1814a] to the mathematics section of the Institut National des Sciences et des Arts, which had been called the Académie des Sciences in pre-Revolutionary days. He followed this with a second part in 1809. The papers were received with approval in a report issued by Sylvestre Lacroix (1765-1843) and Adrien-Marie Legendre (1752-1833) in 1812. He subsequently revised the “Essay” and published it in the journal Annales de mathématiques pures et appliquées. At 48 pages, the “Essay” filled the entire October 1814 issue of the Annales. Furthermore, it was followed immediately in the November issue of the journal by another of Servois’ articles, “Reflections on the various systems of exposition of the principles of the differential calculus and, in particular, on the doctrine of the infinitely small” [Servois 1814b]. A translation and analysis of this second article is available in [Bradley and Petrilli 2010]. Servois had both articles published together in 1814 as a monograph.
Accompanying the present article is an English translation of Servois’ “Essay.” The remainder of this article consists of a reader’s guide to the “Essay” (pages 7-16), along with some suggestions for how you might use it with students of mathematics and of mathematics history (page 17).