What were the philosophical foundations of the differential calculus at the turn of the nineteenth century? There were three competing notions: differentials, limits, and power series expansions. François-Joseph Servois (1768-1847), a disciple of Lagrange, supported the power series formalism and was sympathetic to a foundation based on limits. On the other hand, he claimed that the use of infinitesimals in mathematics would "one day be accused of having slowed the progress of the mathematical sciences, and with good reason." In this paper, we provide an analysis and an English translation of Servois' philosophical paper "Reflections on the various systems of exposition of the principles of the differential calculus."
Download the authors' English translation of Servois' “Reflections.”