# 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

Author(s):
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.