**Jules Drach**'s parents were Joseph Louis Drach and Marie-Josèphe Balthazard who were farmers. Just before Jules' birth was a particularly difficult time for all French people, but it was particularly difficult for people living in Alsace like the Drach family. The popularity of Napoleon III, the French emperor, had declined and he decided that a war with Prussia might change his political fortunes. His advisers had told him that the French Army could defeat Prussia with ease, so the plan looked like a winner. Bismarck, the Prussian chancellor, saw a war with France as an opportunity to unite the South German states so he too saw the political advantage of war. With both sides actively seeking a war, the Franco-Prussian War became inevitable. On 14 July 1870, Bismarck sent a telegram aimed at infuriating the French government. He succeeded, for on the 19 July France declared war on Prussia. The German offensive met with only an ineffective French reply. In August, the German army trapped part of the French army in Metz and they surrendered on 1 September. On 19 September the German army began to blockade Paris which surrendered on 28 January 1871. The formal Treaty of Frankfurt on 10 May 1871 ended the war and when it was implemented a couple of months later Germany annexed Alsace and half of Lorraine, with Metz. The Drach family were French and, despite being farmers, they saw their only option was to leave German occupied Alsace and their farm. 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.

Jules worked as an architect in his youth to help his family who had never been very well off but were considerably poorer after they were forced to flee. After attending primary school Jules was able to go to the College in Saint Dié. This was only possible because he was awarded a scholarship, and this only happened due to the strong encouragement of the teachers at the primary school who had seen the young boy's potential. After attending the College, he went to the Lycée in Nancy, which was to the north west of Saint Dié, then to École Normale Supérieure in Paris at the age of 18.

Drach gave a lecture in 1892 *Sur la transcendance du nombre* Ⓣ as part of the requirements of his degree of agrégé. The proof of trancendency which he gave, and also the proof of the general theorem of Lindemann, is essentially that of Weierstrass. 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* Ⓣ. 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. Following on from this first appointment he went to Lille, then to a professorship of mathematics at the University of Poitiers. While he was at Poitiers, Drach married Mathilde Guitton; their son Pierre Drach became a famous biologist. 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. This work by Drach appeared in his doctoral thesis, which was published in 1898, and in it he asserted [1]:-

Other papers by Drach include three published in 1908:... that the theory of groups is inseparable from the study of the transcendental quantities of the integral calculus.

*Sur les systèmes complètement orthogonaux de l'espace euclidien à n dimensions*Ⓣ;

*Recherches sur certaines déformations remarquables à réseau conjugué persistant*Ⓣ; and

*Sur le problème logique de l'intégration des équations différentielles*Ⓣ. The 'logical' problem of integration consists [1]:-

Drach opposed this and introduced the idea of a 'rationality group'.... of classifying the transcendental quantities satisfying the rational system verified by the solutions.

After Drach was appointed to the chair at the Sorbonne, there was only a short period before the outbreak of World War I. He contributed to the war effort by applying his methods to solve problems in ballistics. After the war ended he published his geometric approach to such problems in *L'équation différentielle de la balistique extérieure et son intégration par quadratures* Ⓣ (1920). The work by Drach in 1919 is the subject of the paper [2]. In this the authors write:-

In 1935 Drach considered Hamiltonian systems with two degrees of freedom that have cubic first integrals. Ten systems discovered by Drach are looked at using modern techniques in [5].In this note we reproduce Drach's original contribution establishing the relationship between complete integrability and spectral theory. Drach's results can be compared with the modern treatment of the same class of equations.

In 1945 Drach published *Sur quelques points de théorie des nombres et sur la théorie générale des courbes 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. In the second part of the paper he classified algebraic curves with certain specified types of singularities. Another example of his work is *Sur la théorie des corps plastiques et l'équation d'Airy-Tresca* Ⓣ which he published in 1946. In this paper he discussed stresses and velocities of a perfectly plastic body in a state of plane stress. He published two papers on geometry, *Sur les lignes d'osculation quadrique des surfaces. *(*Lignes de Darboux*) Ⓣ (1947) and *Détermination des lignes d'osculation quadrique *(*lignes de Darboux*)* sur les surfaces cubiques. Lignes asymptotiques de la surface de Bioche* Ⓣ (1948). These papers are on Darboux curves:-

Around the same time Drach published two papers on partial differential equations:A quadric having contact of the second order with a surface at an ordinary point of the surface will intersect it in a curve with a triple point at the point of contact. The directions in which the three tangents may coincide are the directions at the point of the "Darboux curves."

*Sur les équations aux dérivées partielles du premier ordre dont les caractéristiques sont lignes asymptotiques des surfaces intégrales*Ⓣ (1947); and

*Sur des équations aux dérivées partielles du premier et du second ordre dont les caractéristiques sont lignes asymptotiques des surfaces intégrales*Ⓣ (1948).

Drach was a friend of Borel, and together they published lectures by Poincaré *Leçons sur la théorie de l'élasticité* Ⓣ (1892) and by Jules Tannery *Introduction à l'étude de la théorie des nombres et de l'algèbre supérieure* Ⓣ (1895) while Drach was a student at the École Normale Supérieure. Later in his career Drach helped to prepare Poincaré's works for publication. These appeared in 11 volumes between 1916 and 1956.

The work which Drach undertook has proved important and papers like [4], [5], and [6] explain the significance of his ideas for modern mathematics. It is all the more impressive that Drach managed to do such good work despite poor health which affected him for many years. The importance of his contributions were recognised when he was elected to the Academy of Sciences on 10 June 1929 but because of his health problems he could spend little time in Paris and spent most of the year in the warm south of France. He had an estate at Cavalaire and after he retired from his chair at the Sorbonne he continued to undertake research living on his estate. The papers we have described above show that he remained mathematically active, in fact publishing four papers during the final two years of his life. Retirement from teaching did give him more time to enjoy his other passions, namely reading and creating art from pliable materials.

In many ways Drach's personality was a result of the difficult times through which he lived. It was the dramatic events of his first few years which shaped him and, despite having a good standard of living, nevertheless his affinity was to the poor, particularly the poor peasants who struggled to make their living from the land. He did much to help improve the land and conditions of those around him.

**Article by:** *J J O'Connor* and *E F Robertson*

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