# Jean Gaston Darboux

### Quick Info

Nimes, Gard, Languedoc, France

Paris, France

**Gaston Darboux**was a French mathematician who made important contributions to differential geometry and analysis and the Darboux integral is named after him.

### Biography

**Gaston Darboux**was the son of François Darboux (1800-1849) and Alix Gourdoux (1811-1887). François was a clothes merchant and haberdasher, the son of Antoine Darboux (1764-1803) and Magdelaine Amalric (died 1812), who married Alix Gourdoux in Nîmes on 17 September 1841. Gaston was born in a house which was a converted chapel of the cathedral. The date of his birth is given by various sources as 13 August or 14 August, and there is genuine confusion here which we should explain. His birth certificate clearly gives 1 a.m. on 14 August but Gaston always maintained that he was born at midnight on 13 August. He always gave 13 August when asked for his date of birth and, in fact, his gravestone gives 13 August, so we have given that date on this biography.

Gaston had a younger brother, Jean Louis Darboux, born on 15 May 1844, who became a mathematics teacher at the Lycée Nîmes. One source gives a second brother Paul Darboux, born 28 November 1845, but either this is an error or else Paul died as a baby since all of Gaston's obituaries refer to Gaston having only one brother Louis. Tragedy struck the family in 1849, for on 14 December of that year François Darboux died. Alix took over the business and made strenuous efforts to made sure that both Gaston and Louis received a good education. Gaston and Louis both attended the Lycée Nîmes as day pupils, meaning that they attended from six o'clock in the morning until eight o'clock in the evening. Gaston entered the lycée in October 1853. Alix, Gaston's mother, married Auguste Sistre (1823-1887) on 25 July 1856 in Nîmes. He, like Alix, was a widow, his wife having died in 1854.

Gaston Darboux graduated from the lycée in Nîmes on 22 July 1859 and later that year, in October, entered the special mathematics class at the lycée of Montpellier where there were special class to prepare pupils for the entrance examinations to the École Polytechnique and the École Normale Supérieure. His teacher at Montpellier was Charles Berger who had studied at the École Normale, graduating in 1843 [16]:-

The professor, Charles Berger, clearly explained the subjects of his course, took care of his students during the evenings, led the best of them to the library where he made them read books of Higher Mathematics.After one year at lycée of Montpellier, Darboux sat the entrance examinations for the École Polytechnique, not because he wanted to enter in 1860, but simply because he wanted to please his teacher, Charles Berger, who was keen for his star pupil to sit the examinations. A good performance meant that Darboux gained admission but he preferred to spend another year at Montpellier and take the entrance examinations for both Grand École in the following year. This time he was ranked first from those taking the examinations for both and, if he had followed the usual pattern, he would have chosen to attend the École Polytechnique which was considered best for those wanting to be leaders in their fields, particularly engineers. He chose, however, the École Normale which was the usual route for those of somewhat lesser abilities wishing to become secondary school teachers. Darboux chose to enter the École Normale Supérieure, because he wanted to become a professor. It was a significant decision, not just for the young man, but for the École Normale since other top mathematicians soon followed Darboux's example.

When Darboux went to Paris to begin his studies, his mother accompanied him and introduced him to Louis Pasteur, who was the director of the École Normale. Pasteur was at this time in the middle of a reform programme trying to improve the standard of scientific work at the École but his approach was not always popular with students. He was delighted that the person topping the list had chosen the École Normale and he requested the Minister to allow Darboux to attend any of the courses which he liked outside the École. The Minister granted permission so Darboux was able to attend, at the Collège de France, the lectures of Joseph Bertrand, with whom he developed a deep friendship. Darboux was awarded the Licencié ès Sciences mathématiques on 9 July 1863 and the Licencié ès Sciences physiques on the 7 August 1863.

While he was a student his great talent for mathematical research became clear to those around him. While still an undergraduate student at the École Normale Supérieure, he published his first paper on orthogonal surfaces which had been presented to the Académie des Sciences by Joseph Serret in August 1864. A month later, on 20 September, he was placed first in the mathematics aggregation which qualified him to be a mathematics teacher.

Now Pasteur was aware that he had a student of outstanding qualities and he was very keen that Darboux remained in Paris so he created a position of "préparateur agrégé de Mathématiques" at the École Normale. Here is a letter by Pasteur requesting this position be set up for Darboux [18] (also quoted in [5]):-

He is a student whose work, conduct, distinction of spirit, character, and behaviour, nothing leaves to be desired. This young man will quickly be among the most eminent mathematicians. The spirit of invention was the one quality which one could expect this young master still had to achieve. However, he showed recently, with a very remarkable work presented to the Academy of Sciences and by various notes which he gave to the professors and assistants during the year, on various subjects the study of which he was able to devote himself without ceasing to retain the first rank in his division, in spite of the preoccupations of the preparation for the competition of the aggregation. It is absolutely essential that this young man stay in Paris.You can read an English translation of Paul Appell's obituary of Darboux for the International Committee for Weights and Measures [5], from which this quote was taken, at THIS LINK.

Darboux had studied the work of Lamé, Dupin and Bonnet on orthogonal systems of surfaces. He 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 after defending his thesis on 14 July 1866. His examiners were Michel Chasles, Joseph Serret and Claude Bouquet and they gave high praise to the thesis. The report, written by Michel Chasles, is as follows (see [3]):-

This Thesis is an extensive and very important work on orthogonal surfaces. It consists of three Parts. The first, entitled: 'Study of a remarkable system of orthogonal coordinates', contains different properties of the curvilinear coordinates formed by the triple orthogonal system to which the author and M Moutard were led, independently of each other. The second part contains 'Research on orthogonal surfaces in general'. M Darboux, taking as a starting point the theorem of M Dupin, according to which in any triple system of orthogonal surfaces the intersection curves of the surfaces are their lines of curvature, to which he adds as a complement the following statement: "When two systems of orthogonal surfaces intersect along the lines of curvature of these surfaces, there is a third system orthogonal to the first two," first gives a simple demonstration of this theorem by M Ossian Bonnet, that the search for all orthogonal systems amounts to the complete integration of a third order partial difference equation with three independent variables. Then he makes known a 'New method of research of orthogonal systems', based on the use of a certain auxiliary function V. The third Part contains 'Applications' of the method exposed in the second Part. The author first considers a particular class of orthogonal systems in which the surfaces of the same system are obtained by moving one of them parallel to itself by a simple translation without alteration of shape. The determination of the function V then depends on the integration of a third order partial difference equation with two independent variables. The second case treated by M Darboux is that of surfaces for which the lines of curvature are plane in the three systems. The integrations then take place completely, and the result, in a very simple form, contains three arbitrary functions; these surfaces are, in certain cases, an example of the orthogonal systems studied in the preceding paragraph, that is to say that "each of the three systems is formed by a surface of invariant form which moves parallel to itself." The third and last case relates to systems for which each surface can be divided into infinitely small squares by its lines of curvature. M Darboux had already observed, in the first Part, that the surfaces of the triple orthogonal system previously discovered by M Moutard and by himself enjoy the property in question. By a clever and extremely ingenious analysis, he now shows that the latter system is the only one that answers the question.Darboux had been appointed as a substitute for Joseph Bertrand to teach the special mathematics course at the Lycée Saint-Louis in Paris in 1864-65 while still undertaking research for his doctorate. After the award of his doctorate, he was appointed to replace Joseph Bertrand at the Collège de France where he taught the Mathematical Physics Course in 1866-67. He was appointed as a substitute for Jean-Claude Bouquet to teach the special mathematics course at the Lycée Louis le Grand (where Galois was educated) on 31 October 1867. A year later he became a Professor of Special Mathematics at the Lycée Louis le Grand where he taught until 26 September 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.

On 14 July 1872, Darboux married Célina Amélie Carbonnier, born 29 March 1850 in Beauvais, Oise, Picardie, France. We note that his brother Jean Louis Darboux had married Clare Marie Carbonnier on 20 August 1871. We assume that the two Darboux brothers married two Carbonnier sisters, but we have been unable to verify this.

Darboux made important contributions to differential geometry and analysis. D J Struik writes in [1]:-

... he followed in the spirit of Gaspard Monge, and Darboux's spirit can be detected in the work of Élie Cartan.Again Struik writes [1]:-

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 published 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*

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 [8]:-

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 [8]:-

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 than that described above. Struik writes in [1]:-

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).For more information about Darboux's mathematical contributions see Henri Poincaré's speech at the Darboux Jubilee at THIS LINK.

Of course Darboux received many honours for his work. Lebon in [3] 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.

Darboux represented France at the funeral of Lord Kelvin in Westminster Abbey in December 1907. More details of this are given in [15]; see a version of Larmor's obituary of Darboux at THIS LINK.

In 1908 Darboux was a plenary speaker at the International Congress of Mathematicians held in Rome in 1908. He delivered the lecture

*The Origins, Methods and Problems of Infinitesimal Geometry*at 3:30 on Tuesday 7 April 1908, the session being chaired by Simon Newcomb. We give an English version of part of Darboux's lecture at THIS LINK.

Let us end with the description of Darboux by Ernest Lebon written in 1910 [16]:-

Tall, severe and cold, M Darboux intimidates those who approach him for the first time. Fortunately, this impression disappears quickly after a few minutes of conversation. We recognize then that he is benevolent and that under a rough bark he hides a generous heart. He has given proofs of these two qualities several times, notably for the past ten years as president of the Society for the support of Friends of Science. His conversation, which ranges over the most diverse subjects, is both informative and attractive. He recognizes that his Mathematics teachers have discovered, awakened and maintained his taste for Geometry and he repeats their names with emotion and pleasure. He strives to judge fairly without bias the issues before him. When he chairs a committee, he has absolute confidence in his colleagues and defends them if they are attacked. ... In all circumstances of life, M Darboux proceeds methodically; one should therefore not be surprised to find this quality when he develops his course programme and writes the equations on the board in the order in which they appear. Very conscientious by nature, he leaves no reasoning unfinished and presents to his listeners lessons which are always carefully prepared. There is irrefutable proof in his library of this last fact: it consists of a dozen large bound notebooks, where one can find, clearly written by himself, the developments of the courses he taught in Mathematical Physics, in Analytical mechanics and infinitesimal geometry. These precious manuscripts contain methods and remarks that he did not publish, but which we can later take advantage of, because his intention is to give them to the Institute. M Darboux remained simple and modest, although he came to a very high position. It is important to point out that he owes it only to his own efforts and his talent: none of his ancestors has occupied even a modest position in the world of science, administration or politics; if scholars protected him at the beginning of his career and opened the doors of glory to him, it is because they had seen in his works points likely to advance Science and recognised in him qualities of first order.Darboux is buried in Montparnasse Cemetery in the City of Paris.

### References (show)

- D J Struik, Biography in
*Dictionary of Scientific Biography*(New York 1970-1990). See THIS LINK. - Biography in
*Encyclopaedia Britannica.*http://www.britannica.com/biography/Jean-Gaston-Darboux - E Lebon,
*Gaston Darboux*(Paris, 1910, 1913). - D S Alexander, Gaston Darboux and the history of complex dynamics,
*Historia Mathematica***22**(2) (1995), 179-185. - P Appell, Jean-Gaston Darboux,
*International Committee for Weights and Measures*(1920). - S S Chern, Surface theory with Darboux and Bianchi,
*Miscellanea mathematica*(Berlin, 1991), 59-69. - L P Eisenhart, Darboux's contribution to geometry,
*Bull. Amer. Math. Soc.***24**(1917/18), 227-237. - A E F, Jean Gaston Darboux,
*Proc. London Math. Soc.*(2)**16**(1) (1917), xliv-xlix. - H Gispert, Sur les fondements de l'analyse en France (à partir de lettres inédites de G Darboux et de l'étude des différentes éditions du Cours d'analyse de C Jordan),
*Archive for History of Exact Science***28**(1) (1983), 37-106. - H Gispert, La correspondance de G Darboux avec J Houël: chronique d'un rédacteur (déc. 1869-nov. 1871),
*Cahiers du séminaire d'histoire des mathématiques***8***, Inst. Henri Poincaré*(Paris, 1987), 67-202. - H Gispert, Principes de l'analyse chez Darboux et Houël (1870-1880): textes et contextes,
*Rev. Histoire Sci.***43**(2-3) (1990), 181-220. - P Henry and P Nabonnand , La correspondance avec Gaston Darboux (décembre 1874-avril 1875), in
*P Henry and P Nabonnand (eds), Conversations avec Jules Hoüel*(Publications des Archives Henri Poincaré Publications of the Henri Poincaré Archives, Birkhäuser, Cham, 2017), 423-563. - D Hilbert, Gaston Darboux,
*Acta Mathematica***42**(1919), 269-273. - J F Labrador, Juan Gaston Darboux (Spanish),
*Gaceta Mat.*(1)**5**(1952), 3-5. - J Larmor, Prof Gaston Darboux, For. Mem. R.S.,
*Nature*(8 March 1917). - E Lebon, Notice sur M Gaston Darboux, in
*E Lebon, Gaston Darboux*(Gauthier-Villars, Paris, 1910, 1913), 1-7. - E Lebon, N Gaston Darboux, Grades. Fnctions. Titres honorifiques. Prix. Décorations, in
*E Lebon, Gaston Darboux*(Gauthier-Villars, Paris, 1910, 1913), 8-12. - G Lippmann, H Poincaré, P Appell, E Lavisse, V Volterra, L Belugou, É Picard, L Lévy, C Guichard and G Darboux, Le jubilé de M Gaston Darboux,
*Revue internationale de l'enseignement***63**(1912), 97-125. - G Parasad, Gaston Darboux, in
*Some great mathematicians of the nineteenth century*Vol II (Benares, 1934), 144-182.

### Additional Resources (show)

Other pages about Gaston Darboux:

Other websites about Gaston Darboux:

### Honours (show)

### Cross-references (show)

- Societies: French Mathematical Society
- Societies: Hellenic Mathematical Society
- Societies: Hungarian Academy
- Societies: Netherlands Academy of Sciences
- Societies: New York Mathematical Society
- Student Projects: Sofia Kovalevskaya: Chapter 13
- Other: 1900 ICM - Paris
- Other: 1904 ICM - Heidelberg
- Other: 1908 ICM - Rome

Written by J J O'Connor and E F Robertson

Last Update July 2020

Last Update July 2020