Alexis Fontaine's route to the calculus of several variables

Abstract

In the years around 1732 Alexis Fontaine des Bertins (1704–1771) member of the French Royal Academy of Sciences from 1733 on, worked out a dualoperator differential calculus to solve problems in “families of curves.” Within six years a calculus of several variables had emerged out of this work. These developments, discussed at greater length elsewhere [ Greenberg 1981, 1982 ], are summarized, with additional general reflections and the correction of one error. Together the discussions complement the excellent, recently published account of “families of curves” and the origins of partial differentiation in the works of Leibniz, the various Bernoullis, and Euler [ Engelsman 1982 ]. Common concerns motivate the works of all of these mathematicians. At the same time, certain differences in conception in Fontaine's work highlight the creativity of one of the lesser known eighteenth-century mathematicians.