With hindsight, Lagrange's scheme is the weakest of the three competing notions. We know that limits eventually did provide a solid foundation for the calculus. Furthermore, when nonstandard analysis was invented in the 1960s, mathematicians were able to give a satisfactory account of Leibniz's differentials. Lagrange's program has some obvious weaknesses. In particular, we do not want the notion of function to be limited to only those functions that have power series expansions. Furthermore, there are tricky questions of convergence that were glossed over by Lagrange. Nevertheless, Lagrange was very influential, especially in the French mathematical community, and his program had many adherents. One notable example is Silvestre François Lacroix's (1765-1843) Traité du calcul différentiel et du calcul intégral [Lacroix 1797]. In this monumental three-volume reference work describing the state of the differential and integral calculus at the dawn of the 19th century, Lacroix follows Lagrange in basing his calculus on formal power series, although he also describes the history and use of both differentials and limits.
Servois was in many ways a disciple of Lagrange. He certainly believed that the Lagrangian scheme was the correct approach to the principles of the calculus. In 1814, he published two important papers in the Annales de mathématiques pures et appliquées, the first academic journal devoted entirely to mathematics. The first of these articles, “Essay on a new method of exposition of the principles of the differential calculus” [Servois 1814a], concerned general techniques for determining power series expansions of functions. He also derived properties of the derivative based on the algebraic properties of the difference operator and the difference quotient. He coined the terms distributive and commutative in this paper to describe those algebraic properties. Servois defined a function ( f ) to be distributive if ( f(x + y) = f(x) + f(y) ) and two functions ( f ) and ( phi ) to be commutative between themselves if ( fphi(x) ) = ( phi f(x) ). We note that he did not make a distinction between function and operator.
Servois' second paper, a philosophical essay entitled “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], followed immediately on the “Essay” in the pages of the Annales for 1814. Accompanying the present article is the first ever English translation of Servois' “Reflections.”
During the remainder of this introductory article, we provide a reader's guide to Servois’ “Reflections” [Servois 1814b], and then conclude with some recommendations as to how our translation may be used by teachers and students of mathematics and of the history of mathematics.