**Ernest Vessiot**'s father was Alexandre Vessiot (1829-1908) and his mother was Marie Collin. Alexandre Vessiot studied at the École Normale Supérieure and was a professor of rhetoric at the Lycée in Marseilles when his son Ernest was born. In 1886 Alexandre was appointed inspector general of primary schools. He wrote several books on education including *De l'enseignement à l'école: et dans les classes élémentaires des lycées et collèges* Ⓣ (1886) and *De l'éducation à l'école* Ⓣ (1887). Alexandre married Marie and they had eight children, six of whom survived to adulthood: Paul Vessiot (1859-1942), Louise Vessiot (1858-1960), Ernest Vessiot (the subject of this biography), Henri Vessiot (1870-1946), Marthe Vessiot (1872-1945) and Anna Vessiot (1874-1950). The historian Camille Jullian (1859-1933) was a pupil in Alexandre Vessiot's class and wrote about his outstanding qualities as both a teacher and as a person. These qualities meant that Ernest was brought up in an academic environment in which there was a kind and understanding encouragement to hard work.

Ernest's secondary education was at the lycée in Marseilles where his performance was outstanding and he obtained his baccalaureate in 1883. He did not go straight to university but took the special mathematics class in 1883-84 before sitting the entrance examination for the École Normale Supérieure and the École Polytechnique in 1884. An outbreak of cholera in Marseilles almost prevented him taking the examinations (his family wanted him to leave the city) but he persisted. In the entrance examination Vessiot was placed second to Jacques Hadamard and thereafter he studied in the same class as Hadamard. During his university studies, as well at taking classes at the École Normale Supérieure, he also attended courses at the Faculty of Sciences. He took Jean-Claude Bouquet's course on differential and integral calculus. This was the final course that Bouquet gave and he died in September 1885. Vessiot also took Paul Appell's course on rational mechanics. Appell was appointed to the Chair of Mechanics at the Sorbonne in 1885. Vessiot was awarded his bachelor's degree in mathematics and physics in 1886. He continued to study for his Agrégation de mathématiques, the qualification needed to teach mathematics in a lycée. He was awarded this degree in 1887 and then spent the year 1887-88 in Germany.

Vessiot accepted a teaching post at the Lycée Ampère in Lyon in 1887 and began teaching there after his return from Germany. This school, originally a Jesuit establishment, had been founded in 1519 and given several different names before being named the Lycée de Lyon in 1848 and then the Lycée Ampère in the year that Vessiot began teaching there. In his first year at the Lycée he taught elementary mathematics courses but in his second year he was put in charge of the special mathematics class. However, school teaching was only part of Vessiot's work for at the same time he was undertaking research for his doctorate advised by Émile Picard. In 1892 he submitted his doctoral dissertation *Sur l'intégration des équations différentielles linéaires* Ⓣ. In this he studied Lie groups of linear transformations, in particular considering the action of these Lie groups on the independent solutions of a differential equation. He defended his thesis in an oral examination at the University of Paris in June 1892, the jury being chaired by Charles Hermite. He published his thesis in the *Annales Scientifiques de l'École Normale Supérieure* in 1892 and over the following few years published papers such as *Sur une classe d'équations différentielles* Ⓣ (1893), *Sur une méthode de transformation et sur la réduction des singularités d'une courbe algébrique* Ⓣ (1894), *Sur les systèmes d'équations différentielles du premier ordre qui ont des systèmes fondamentaux d'intégrales* Ⓣ (1894), and *Sur quelques équations différentielles ordinaires du second ordre* Ⓣ (1895).

After the award of his doctorate, Vessiot taught in a number of universities. He was appointed as an assistant lecturer at the Faculty of Science at Lille in 1892, but after one year he moved to the Faculty of Science at Toulouse where he taught as a lecturer for the three years 1893-96 being in charge of courses there. Following these years, in 1896 he was appointed to take charge of the course on differential and integral calculus at the Faculty of Science at Lyon where he was promoted to the chair of pure mathematics in the following year. The lectures he gave on Higher Geometry in the academic year 1905-06 were published as the book *Leçons de Géométrie Supérieure* Ⓣ (1906). Clarence Lemuel Elisha Moore (1876-1931), a professor at the Massachusetts Institute of Technology specialising in geometry, wrote the review [5]. We quote an extract:-

These lectures delivered by Vessiot during the year1905-1906were published in the present form at the demand of his students. The author remarks in the preface that he is hopeful that they may be of service to those who are beginning the study of higher geometry and may serve them as a good preparation for the reading of original memoirs and such works as Darboux's 'Théorie des surfaces'. It is the opinion of the reviewer that the lectures serve these purposes admirably. The attack is direct and the end to be reached is kept clearly before the reader, in fact the whole presentation is such as to lead the beginner to an appreciation of the subject. A glance at the table of contents will convince one that the book will serve as a good introduction to the study of Darboux. The principal object of the lessons is the study of systems of straight lines but owing to the close relation between lines and spheres it is quite natural that systems of spheres should be studied also. It is assumed at the outset that the student is familiar with the elementary notions of twisted curves and surfaces(tangent planes, tangent lines, etc.), and that he has some acquaintance with the elements of the theory of contact.

In fact Vessiot published a second edition in 1919. In the Preface to this edition he explains that the first edition quickly sold out and he was approached by the publisher J Hermann of Paris to produce a second edition. The first edition ran to 326 pages but with the additions that Vessiot made to the second edition it was 376 pages. He also made some improvements to the text.

It was standard for academics to begin their careers in the provinces and eventually gain an appointment in Paris. Vessiot taught for thirteen years in the Faculty of Science Lyon before gaining an appointment in Paris in 1910. At first he taught differential and integral calculus but, following the death of Jules Tannery in December 1910, he was made an assistant lecturer. He was given charge of the elementary practical mathematics course and assisted Paul Painlevé in examinations. He also assisted Claude Guichard (1861-1924), the professor of general mathematics at the University of Paris, in delivering his courses. He succeeded Guichard to the chair of general mathematics when Guichard retired in 1919.

As we mentioned above, Vessiot applied continuous groups to the study of differential equations. He extended results of Jules Joseph Drach (1902) and Élie Cartan (1907) and also extended Fredholm integrals to partial differential equations. Jean-François Pommaret writes in [2]:-

It was Ernest Vessiot who first started to feel that a way to define the Galois theory for linear ODEs should come through a better understanding of the group theoretical foundation of the classical Galois theory. One must never forget that Vessiot was an unusual Frenchman at that time, perfectly aware of the work of Lie who, in turn, at the end of his life, considered Vessiot to be his heir. As a consequence, Vessiot knew the theory of Lie groups and, when he wrote his thesis in1892on what is now called the Picard-Vessiot theory of linear ODEs, he knew about Lie's latest research on pseudogroups. Hence, in Vessiot's mind, the Picard-Vessiot theory of linear ODEs could be considered only as an intermediate step towards the achievement of the classical Galois theory, namely the Galois theory for systems of PDEs ...

In fact he had read Drach's doctoral thesis of 1898 and had been unhappy with some of the arguments. A letter that he wrote to Drach on 3 October 1898 is extant and we give an extract from it since it shows much of Vessiot's character; great kindness as well as very careful attention to detail (see [2]):-

Excuse me for not having already thanked you for sending me your thesis. Indeed I wanted to read it with care and, while reading it, I had very serious doubts about the third chapter. Consequently, I invited Élie Cartan to examine it on his own and he agreed with me. Accordingly, I sad a few words to Jules Tannery, bearing in mind that he was the right person to call your attention to certain obscure points. He himself believed that it was better for me to write to you directly - which I am doing - with the hope you could show me that I have only badly interpreted certain sketchy proofs. ... In any case, it seems to me that all the proofs must be given anew. I end there, my dear colleague, this criticism of your work. I hope that you will hold nothing against me and that you will soon be able to show me that my objections are unjustified.

Vessiot was absolutely correct in his worries about Drach's thesis and Paul Painlevé wrote to Vessiot on 17 October 1898 (see [2]):-

I just read Drach's thesis and I agree entirely with you about the inaccuracy of the two fundamental theorems and their proofs. The mistake is so big that I can hardly conceive that it has been overlooked by the author and the jury. It is very sad for poor Drach. ... I said to Émile Picard that I had received a letter from you about this question; after a few minutes of explanation, he was astonished to have missed this. ... I hope, my Dear Friend, that this year will alleviate your sorrow of the last year, and I beg you to give my kind regards to Madame Vessiot and your family.

We should explain Paul Painlevé's reference to "sorrow of the last year". Vessiot married Augustine Naton in Toulouse on 25 February 1893. They had a son Alexandre (born 1900) and four daughters Marie (born 1893), Louise (born 1894), Madeleine (born 1895), and Simone (born 1895). Marie died very young, and Madeleine died in 1897. This is the "sorrow of the last year" to which Painlevé refers. However, Painlevé's hopes for Vessiot's future happiness did not materialise. His wife Augustine died in 1912 and his son Alexandre died from tuberculosis in 1913 at the age of thirteen. Of his two remaining daughters, Louise married the physicist Jean Cojan (1892-1952) but died in 1948 three years before her father, while the other daughter, Simone, remained unmarried and cared for Vessiot in his old age. However, we have moved ahead and we should return to saying something of Vessiot's career after his appointment to the Faculty of Science in Paris in 1910.

Painlevé, whom we quoted from above, wrote about the superb research contributions by Vessiot, much of which was based on trying to correct the error in Drach's thesis:-

We can characterize the work of Mr Vessiot saying it is distinguished by its depth, perfect rigour and lucid style.

However, Vessiot was also an outstanding lecturer, renowned for the clarity and rigour of his lectures. In fact he was extremely conscientious and devoted so much time to his teaching and writing textbooks over the latter part of his career that he did not find the time to devote himself to the deep research of which he was capable. Vessiot spent the last 25 years of his career in Paris, concentrating on his role as a lecturer and as an examiner. During World War I he undertook war work. He was assigned to ballistics and made important discoveries in this area. He was, successively, professor of General Mathematics, professor of Differential and Integral Calculus, professor of Group Theory and the Calculus of Variations, professor of Theory of Functions and Transformations, and professor of Analytical Mechanics and Celestial Mechanics. He was appointed Deputy Director (1920) and then, in 1927, to the prestigious post of Director of the École Normale Supérieure in Paris. He was the first scientist to be appointed to this post which he continued to hold until he retired in 1935. In his role of director he supervised the construction of new laboratories for physics, chemistry and natural sciences at the École Normale Supérieure. Vessiot also held other roles in Paris, for example he was an examiner at the Naval Academy and an examiner at the École Polytechnique where he also acted as an analysis tutor. As an examiner he was said to be calm, gentle and caring. He always looked for how much the candidate got right rather than how much was wrong.

He retired on 30 September 1935 and went to live at La Bauche in Savoy, in the family property built by his father. He died from a heart attack following a bout of pneumonia. His final illness lasted three weeks.

He was honoured by the award of the Grand Prix des Sciences Mathématiques in 1904. In the essay he submitted for the prize, he proved [2]:-

... that Galois theory was nothing other than the study of principal homogeneous spaces for groups.

He was elected to the Académie des Sciences in 1943.

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