Born: 13 August 1704 in Claveyson, Drôme, France
Died: 21 August 1771 in Cuiseaux, Saône-et-Loire, France

Alexis Fontaine's father was Jacques Fontaine and his mother was Madeleine Seytres. Jacques Fontaine was a royal notary, so he served the king in a legal capacity. Alexis enjoyed an upbringing in a fairly well off family and he was educated at the Collège de Tournon.

In 1732 Fontaine went to live near Paris, where he had acquired a residence, and he began to study mathematics under Castel. Around this time he became friends with Clairaut and Maupertuis, and he began to submit memoirs to the Académie des Sciences. As a result of these papers Fontaine was elected to the Academy in 1733 as an adjoint mécanicien and he was promoted geometer (a term used to mean mathematician at this time) in 1739. Although associated with the Academy for the rest of his life he did not participate in the work of the Academy, rather preferring to pursue his own agenda.

He led a solitary life showing little interest in the work of others. His papers are rather confused, and ignorant of the work of others, but do contain some very original ideas in the calculus of variations, differential equations and the theory of equations. Taton writes in [1]:-

Fontaine's work is of limited scope, often obscure, and willfully ignorant of the contributions of other mathematicians. Nevertheless, its inspiration is often original and it presents, amid confused developments, a number of ideas that proved fertile ...

In 1732 Fontaine gave a solution to the brachistochrone problem, in 1734 he gave a solution of the tautochrone problem which was more general than that given by Huygens, Newton, Euler or Jacob Bernoulli, and in 1737 he gave a solution to an orthogonal trajectories problem. The methods which he developed to solve these problems led to the calculus of variations. He used what he called the "fluxio-differential" method, so called because it used two independent first-order Leibniz type differential operators. This technique was praised by Johann Bernoulli, Euler and d'Alembert. Fontaine then used differential coefficients instead of differentials and Greenberg in [5] shows how Fontaine progressed from a calculus of variations to a calculus of several variables.

However Fontaine rather spoilt this fine contribution by, in 1767 and 1768, unjustly criticising Lagrange's method of variation presented in 1762. Fontaine had retired in 1765 to a country estate in Burgundy the purchase of which had stretched his finances to the point of almost leaving him bankrupt.

In [3] Greenberg considers Fontaine's work and that of his contemporaries who are usually given credit for laying the foundations for the calculus of several variables. Greenberg discusses the question of priority in [3] and also in [4]. One of the reasons that Fontaine has come off badly was his apparent attempts togain credit for ideas which had first been presented by others. For example [1]:-

1. R Taton, Biography in Dictionary of Scientific Biography (New York 1970-1990). 
   http://www.encyclopedia.com/doc/1G2-2830901467.html

Articles:

1. J M C de Condorcet, Eloge de M Fontaine, Histoire de l'Académie royale des sciences 1771 (Paris, 1774), 105-116.
2. J L Greenberg, Alexis Fontaine's route to the calculus of several variables, Historia Math. 11 (1) (1984), 22-38.
3. J L Greenberg, Alexis Fontaine's integration of ordinary differential equations and the origins of the calculus of several variables, Ann. of Sci. 39 (1) (1982), 1-36.
4. J L Greenberg, Alexis Fontaine's 'fluxio- differential method' and the origins of the calculus of several variables, Ann. of Sci. 38 (3) (1981), 251-290.