Bouquet entered the École Normale Supérieure in 1839, obtaining his doctorate in 1842 for a thesis Sur la variation des intégrales doubles which he submitted to the Faculty of Science in Paris. Briot had been a school teacher of mathematics for a year before entering the École Normale Supérieure and he took one year longer to obtain his doctorate than Bouquet so the two friends obtained their doctorates in the same year. Their careers then took very similar courses.
Bouquet was appointed professor of mathematics at the Lycée in Marseilles, then he went to Lyon as professor of mathematics in the Faculty of Science. In Lyon he met up again with his school friend Briot and the two began a collaboration which lasted throughout their careers. Bouquet returned to Paris shortly before Louis-Napoleon declared himself emperor of the French on 2 December 1852; the Second Empire lasted until 1870. From 1852 until 1858 Bouquet taught at the Lycée Bonaparte (later renamed the Lycée Condorcet) in Paris. Again he was a colleague of Briot and both taught the special mathematics course designed to prepare pupils to take the entrance examinations of the École Normale Supérieure and the École Polytechnique. In 1858 Bouquet moved to the Lycée Louis-le-Grand, the school Galois had graduated from 30 years before, and there again put his energies into preparing pupils for the entrance examinations for the two major Paris universities. He taught there until 1867 when he was appointed maître de conference (essentially an assistant lecturer) at the École Normale Supérieure and as répétiteur at the École Polytechnique. In 1874 Bouquet was appointed professor of differential and integral calculus at the Sorbonne, succeeding Serret who had retired due to ill health. Bouquet taught there until 1884, the year before his death.
Bouquet worked on differential geometry, writing on orthogonal surfaces Note sur les surfaces orthogonales (1946). In the same year he wrote an important memoirs on systems of lines of spaces Remarques sur les systèmes de droites dans l'espace then he published at Lyon in 1848 Mémoire sur propriétés d'un systèmes de droites . This work was carried forward by Bonnet, Darboux and Cayley. With Briot he worked from 1853 onwards on deep studies of Cauchy's ideas of analysis and produced many fundamental papers on series expansions of functions and on elliptic functions. In 1853 they established conditions for a function to be expandable into an entire series in their important paper Note sur le développement des fonctions en séries convergentes, ordonnées suivant les puissances croissantes de la variable .
Bouquet and Briot developed Cauchy's work on the existence of integrals of a differential equation. Together they wrote several important texts. For example Étude des fonctions d'une variable imaginaire ; < (Research on the properties of functions defined by differential equations); and Mémoire sur l'intégration des équations différentielles au moyen des fonctions elliptiques. Between 1859 and 1875 Bouquet worked with Briot on elliptic functions and Bouquet published an excellent text Théorie des fonctions elliptiques in 1875. In the year that he published this text he was elected to the Academy of Sciences in Paris.
Bouquet was as enthusiastic a teacher as he was researcher. There is no doubt that he enjoyed teaching at both secondary school and university level as much as he enjoyed the deep research he undertook. It is worth commenting that although today joint research papers are the norm, this was certainly not the case in the time in which Bouquet worked. It says much about his character that he enjoyed collaborative research. Jules Tannery, who was to write an obituary of Bouquet , was taught by him and praised him highly as a teacher.
Article by: J J O'Connor and E F Robertson