Servois' 1814 Essay on the Principles of the Differential Calculus, with an English Translation - Limits and Infinitesimals

After reviewing the material found in his “Essay,” Servois undertook a brief historical examination of the origins of the differential calculus, examining the three competing foundational notions for the calculus. He described the limit concept in a sympathetic way, tracing it all the way back to Isaac Newton and even finding Leibniz to be sympathetic to a sort of limiting argument. On the other hand, he expressed no sympathy for the use of the infinitely small.

Figure 5. Immanuel Kant (1724-1804) (print in public domain).

Servois then quoted at length from an author whom he did not identify by name, who argued against the exclusion of the idea of the infinite from mathematics [Servois 1814b, pp. 144-145]. The unidentified author was Josef-Maria Hoëné-Wronski (1776-1853), whom we will discuss in more detail shortly.

Wronski was a Kantian and Servois began his philosophical refutation of Wronski's position by exhibiting his own knowledge of Immanuel Kant's philosophy, attempting to use it to buttress his position on the inappropriateness of the use of infinite in mathematics. He further challenged his readers to examine the principle that if \( dz \) is an infinitely small quantity of the first order, then \( d^{2}z \) is one of the second order. Servois claimed that it is easy to demonstrate this by using Taylor series, but impossible with the infinitely small. Servois continued his critique, eventually concluding that infinitesimals “will one day be accused of having slowed the progress of the mathematical sciences, and with good reason” [Servois 1814b, p. 148].