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

François-Joseph Servois was born on July 19, 1767, in the village of Mont-de-Laval, located in the Department of Doubs, near the Swiss border. He was ordained a priest at Besançon near the beginning of the French Revolution; however, his religious career was cut short due to the revolution. In 1793, Servois left the priesthood and joined the army to become a Foot Artillery officer. In his leisure time he studied mathematics. Due to poor health Servois requested a non-active military position, one in the field of academia. This was the beginning of his professional career in mathematics, although he would be called back into active duty several times before his retirement. Servois had several research areas, such as mechanics, geometry, and calculus; however, to the extent that his name is known at all, it is for introducing the words “distributive” and “commutative” to mathematics. His final position was as Curator of the Artillery Museum, located in the 7th Arrondissement of Paris. Servois retired in 1827 and lived with his sister in Mont-de-Laval. He died on April 17, 1847. Interested readers can refer to the biography by Petrilli [2010] for further information about Servois’ life and a review of his other mathematical works.

Servois was one of the first mathematicians to consider abstract functional equations such as \[D\left[f(x)+g(x)\right] = D\left[f(x)\right] + D\left[g(x)\right]\quad (1) \quad \rm{and}\] \[D\left[af(x)\right] = aD\left[f(x)\right]\quad\quad\quad (2).\] Modern readers are familiar with these expressions and recognize transformations with these properties as linear operators. Servois tried to use these expressions to give a satisfactory account of the foundations of calculus. As such, he was one of the pioneers of linear operator theory and indeed of all of “soft analysis,” or the use of algebraic notions and techniques to prove results in real or complex analysis.

Because he was breaking new ground with his research, Servois’ point-of-view was somewhat different from the one that we have inherited. For example, we understand that an operator like \(D\) is an object of a different kind than the function \(f\) upon which it operates. Servois made no such distinction. To him, \(D\), \(f\) and \(g\) were all functions, so relation (1) is reminiscent of the distributive law. In Section 3 of his “Essay on a new method of exposition of the principles of differential calculus” [Servois 1814a], he called any function satisfying \[ \varphi(x+y+\cdots)=\varphi(x)+\varphi(y)+\cdots\quad\quad (3) \] distributive. However, the only example he gave of a distributive function that we would consider truly to be a function is \(\varphi(x) = ax\), which he called the constant factor \(a\). All other examples we would consider to be operators and not functions.

Servois was wrapped up in foundational issues for calculus and was one of many mathematicians contributing to the Golden Age of mathematics.