**Jules Drach** was a French mathematician who worked on number theory, partial differential equations, and differential geometry.

- Joseph Drach and his wife, together was Jules and their other two sons, fled to Saint Dié in the Vosges, the region in which both had been born.
- After attending primary school Jules was able to go to the College in Saint Dié.
- Drach gave a lecture in 1892 Sur la transcendance du nombre Ⓣ(On the transcendence of number) as part of the requirements of his degree of agrégé.
- Drach's lecture was published in 1951.
- Despite a poor performance in his agrégé, he was encouraged by Jules Tannery to undertake mathematical research and he obtained a doctorate from École Normale Supérieure in 1898 for his thesis Essai sur la théorie générale de l'intégration et sur la classification des transcendantes Ⓣ(Essay on the general theory of integration and classification of transcendentals).
- It is clear that Jules Tannery was right to encourage Drach, for his thesis was an impressive and important piece of mathematics which we say a little more about below.
- After completing his doctorate, Drach was appointed as Maître de Conférences at the University of Clermont.
- After an appointment at Toulouse, Drach was appointed to the Chair of Analytical Mechanics and Higher Analysis at the Sorbonne in Paris in 1913.
- Drach viewed Émile Picard's application, in 1887, of Galois theory to linear differential equations as a model of perfection and he tried to extend Galois theory to differential equations in general, building on the work of Lie and Vessiot in addition to that of Émile Picard.
- Other papers by Drach include three published in 1908: Sur les systèmes complètement orthogonaux de l'espace euclidien à n dimensions Ⓣ(On the completely orthogonal systems of Euclidean n-dimensional space); Recherches sur certaines déformations remarquables à réseau conjugué persistant Ⓣ(Research on some remarkable distortions of a persistent conjugated network); and Sur le problème logique de l'intégration des équations différentielles Ⓣ(On the logical problem of the integration of differential equations).
- Drach opposed this and introduced the idea of a 'rationality group'.
- After Drach was appointed to the chair at the Sorbonne, there was only a short period before the outbreak of World War I.
- Drach's results can be compared with the modern treatment of the same class of equations.
- In 1935 Drach considered Hamiltonian systems with two degrees of freedom that have cubic first integrals.
- In 1945 Drach published Sur quelques points de théorie des nombres et sur la théorie générale des courbes algégriques Ⓣ(On some points of theory of numbers and on the general theory of curves algégriques) in which he used the method of descent to prove theorems concerning numbers represented by the sums of two, three and four squares and by the sum of three triangular numbers.
- Drach was a friend of Borel, and together they published lectures by Poincaré Leçons sur la théorie de l'élasticité Ⓣ(Lectures on the theory of elasticity) (1892) and by Jules Tannery Introduction à l'étude de la théorie des nombres et de l'algèbre supérieure Ⓣ(Introduction to the study of the theory of numbers and higher algebra) (1895) while Drach was a student at the École Normale Supérieure.
- Later in his career Drach helped to prepare Poincaré's works for publication.
- It is all the more impressive that Drach managed to do such good work despite poor health which affected him for many years.
- In many ways Drach's personality was a result of the difficult times through which he lived.

Born 13 March 1871, Sainte-Marie-aux-Mines (near Colmar), France. Died 8 March 1949, Cavalaire, Var, France.

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**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive