Darboux had studied the work of Lamé, Dupin and Bonnet on orthogonal systems of surfaces. Darboux generalised results of Kummer giving a system defined by a single equation with many interesting properties. He announced his results to the Académie des Sciences on 1 August 1864, and on the same day Moutard announced that he had also discovered the same system. These results were included in Darboux's doctoral thesis Sur les surfaces orthogonales for which he awarded his doctorate in 1866.
Darboux was appointed to the Collège de France for the academic year 1866-1867, then he taught at the Lycée Louis le Grand (where Galois was educated) between 1867 and 1872. In 1872 he was appointed to the École Normale Supérieure where he taught until 1881. From 1873 to 1878 he was suppléant to Liouville in the chair of rational mechanics at the Sorbonne. Then, in 1878 he became suppléant to Chasles in the chair of higher geometry, also at the Sorbonne. Two years later Chasles died and Darboux succeeded him to the chair of higher geometry, holding this chair until his death. He was dean of the Faculty of Science from 1889 to 1903.
Darboux made important contributions to differential geometry and analysis. D J Struik writes in :-
... he followed in the spirit of Gaspard Monge, and Darboux's spirit can be detected in the work of Élie Cartan.Again Struik writes :-
Relying on the classical results of Monge, Gauss, and Dupin, Darboux fully used, in his own creative way, the results of his colleagues Bertrand, Bonnet, Ribaucour, and others.He may now be best known for the Darboux integral which is named after him. This integral was introduced in a paper on differential equations of the second order which he wrote in 1870.
In 1875 he gave his way of looking at the Riemann integral, defining upper and lower sums and defining a function to be integrable if the difference between the upper and lower sums tends to zero as the mesh size gets smaller.
In 1873 Darboux wrote a paper on cyclides and between 1887 and 1896 he produced four volumes on infinitesimal geometry which included most of his earlier work it was titled Leçons sur la théorie général des surfaces et les applications géométriques du calcul infinitésimal . Included in volume four of this work is a discussion of one surface rolling on another surface. In particular he studied the geometrical configuration generated by points and lines which are fixed on the rolling surface. Eisenhart says of this work in :-
His geometrical proofs of the theorems dealing with rolling surfaces ... are as pure as they are simple and beautiful.Darboux also studied the problem of finding the shortest path between two points on a surface. Work in this area was also done at around the same time by Zermelo and by Kneser.
Darboux's success in research is discussed by Eisenhart in :-
Darboux's ability was based on a rare combination of geometrical fancy and analytical power. He did not sympathise with those who use only geometrical reasoning in attacking geometrical problems, nor with those who feel that there is a certain virtue in adhering strictly to analytic processes. ... brilliant are his reductions of various geometrical problems to a common analytic basis, and their solution and development from a common point of view.However Darboux was also renowned as an exceptional teacher, writer and administrator. Eisenhart writes :-
His writings possess not only content but singular finish and refinement of style. In the presentation of results the form of exposition was carefully studied. Darboux's varied powers combined with his personality in making him a great teacher, so that he always had about him a group of able students. In common with Monge he was not content with discoveries, but he felt that it was equally important to make disciples.Darboux is known for a wider range of mathematics to that described above. Struik writes in :-
Darboux also did research in function theory, algebra, kinematics and dynamics. His appreciation of the history of science is shown in numerous addresses, many given as éloges before the Academy. He also edited Joseph Fourier's "Oeuvres" (1888-1890).Of course Darboux received many honours for his work. Lebon in  lists over 100 Scientific Societies which elected Darboux as a member. He was elected to the Royal Society of London in 1902, winning its Sylvester Medal in 1916. In 1884 he was elected to the Académie des Sciences, becoming its secretary in 1900.
Article by: J J O'Connor and E F Robertson