There was a strange episode in Lhuilier's life when he was still young. A wealthy relation proposed that he would leave Lhuilier a large fortune if he followed a career in the church. However, Lhuilier had already found the attraction of mathematics and money was not going to make him give up the attractions of the topic so he refused his relative's offer.
Lhuilier was an exceptional secondary school pupil and he went on to study mathematics at a Calvin Academy where he was taught mathematics by one of Euler's former pupils, Louis Bertrand, and physics by Georges-Louis Le Sage. It was through Le Sage that Lhuilier obtained his first post as tutor to the Rilliet-Plantamour family, a post he held for two years.
The next career move by Lhuilier was also as a result of him knowing Le Sage. Another of Le Sage's students, Christoph Pfleiderer, had been appointed to the position of professor of mathematics and physics at the Military Academy in Warsaw. Pfleiderer was put in charge of a competition to find the best authors to write texts for Polish schools and in 1775 he sent details to his old teacher Le Sage. Le Sage tried to persuade Lhuilier to submit an entry to write a physics text but Lhuilier preferred to enter the competition to write a mathematics text. Lhuilier's proposal won the competition giving him the right to write a mathematics textbook to be used in Polish schools.
Adam Kazimierz Czartoryski was a Polish prince who had been educated in England and prepared to take over the Polish throne but he refused it in 1763. He later became the first minister of education in a European country and his palace at Pulawy became an important centre of culture providing an excellent school for his sons and for the sons of other important people in the neighbourhood. Czartoryski had been involved in the competition to find authors of Polish school texts and he was so impressed by Lhuilier's entry that he offered him a position as tutor at Pulawy in 1777, in particular as a tutor to his son Adam Jerzy Czartoryski who was seven years old at the time.
Lhuilier spent eleven years at Pulawy. Adam Jerzy Czartoryski proved an extremely bright and gifted pupil and, in addition to his tutoring duties, Lhuilier found time to write his mathematics text, undertake research in mathematics which resulted in several fine publications, and enjoy a busy social life with hunting parties. He also submitted an entry for the prize topic proposed in 1784 by the Berlin Academy.
The Academy sought the best article on the theory of the mathematical infinity and they designed the competition to encourage mathematicians to seek a sound basis for the new differential calculus. Lhuilier submitted the paper Exposition élémentaire des principes des calculs supérieurs and his essay won the prize and was published in Berlin in 1786. The standard concepts and notation for derivatives, and the standard elementary theorems on limits which appear in an undergraduate calculus text today appear in a remarkably similar form in Lhuilier's prize winning essay. Lhuilier introduced the notation "lim", and was the first to allow two-sided limits.
The topic of limits was a particularly fortunate one for Lhuilier since he had been thinking about limits before the topic was ever proposed for the prize. In fact his Polish school textbook which was published in 1780 contains a section on limits.
In 1789 Lhuilier returned to Switzerland but the political situation there seemed fragile and he feared that there would soon be a revolution. Pfleiderer, who had become a friend through the Polish episode, was by this time teaching mathematics in Tübingen and Lhuilier went to be with Pfleiderer there. He would stay with Pfleiderer in Tübingen until 1794. In the following year Lhuilier was offered a chair of mathematics in Leiden, but he preferred to compete for the chair in Geneva which had been held by his former teacher Louis Bertrand. Having won the competition, Lhuilier was appointed in 1795 and held this chair until he retired in 1823.
Not only was 1795 the year of his appointment to the Academy in Geneva but it was a year marked by two other important events in Lhuilier's life. In that year an improved version of his prize-winning essay on limits was published in Latin in Tübingen. Also in that year Lhuilier married Marie Cartier and they would have two children, a son and a daughter.
Lhuilier was also involved in politics in Geneva, being President of the Legislative Council there in 1796. He also achieved a high position in the Academy at Geneva, becoming its rector. He enjoyed many academic honours too, being elected a corresponding member of the Berlin Academy, of the Göttingen Academy, of the St Petersburg Academy, and of the Royal Society of London.
His work on Euler's polyhedra formula, and exceptions to that formula, were important in the development of topology. Lhuilier also corrected Euler's solution of the Königsberg bridge problem. He also wrote four important articles on probability during the years 1796 and 1797.
One further work by Lhuilier is worth commenting on. This is the two volume work Eléments raisonnés d'algèbre that he published in 1804 for his students in Geneva. This work was really a sequel to the text which he wrote for Polish schools many years before. Speziali writes in :-
The main value of these two volumes lay in the author's clear exposition and judicious selection of exercises ...Also in  his character is described as follows:-
Whereas the Poles found Lhuilier distinctly puritanical, his fellow citizens of Geneva reproached him for his lack of austerity and his whimsicality, although the latter quality never went beyond putting geometric theorems into verse and writing ballads on the number three and on the square root of minus one.His most famous pupil was Charles-François Sturm who studied under Lhuilier during the last few years of his career in Geneva.
Article by: J J O'Connor and E F Robertson