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 :-
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  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  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  and also in . One of the reasons that Fontaine has come off badly was his apparent attempts to gain credit for ideas which had first been presented by others. For example :-
In his work of 1764 Fontaine included a study of dynamics dated 1739 and based on a principle closely analogous to the one that d'Alembert had made the foundation of his treatise of 1743. Although Fontaine did not raise any claim of priority, he attracted the hostility of a powerful rival who subsequently took pains to destroy the reputation of his work, which - without being of the first rank - still merits mention for its original inspiration and for certain fecund ideas that it contains.
Article by: J J O'Connor and E F Robertson