**Émile Mathieu**is remembered especially for his discovery (in 1860 and 1873) of five sporadic simple groups named after him. These were studied in his thesis on transitive functions.

Émile Mathieu was born into a family of minor civil servants. His father, Nicolas Mathieu, was a cashier at the Tax Office of the city. His mother, Amélie Antoinette Aubertin, was from Metz while her brother, Pierre Aubertin, the uncle of our mathematician, had attended the École Polytechnique, was a colonel of artillery and the director of a foundry which made cannons. Émile Mathieu was brought up in Metz, and he attended school at the Lycée de Metz in that town. He excelled at school, first in classical studies showing remarkable abilities in Latin and Greek compositions. However, once he had met mathematics when he was in his teenage years, it became the only subject which he wanted to pursue. We mentioned above that his uncle Pierre Aubertin had studied at the École Polytechnique and he advised Émile on preparing himself for the entrance examinations which he took successfully in 1854.

Entering the École Polytechnique in Paris his progress was almost unbelievable, even given the remarkable achievements of the brilliant mathematicians in this archive who also attended this institution. It took Mathieu only eighteen months to complete the whole course which he had finished by 1856. At this stage he was set on a military career but shortly after completing his studies he resigned his commission in the army and he continued to study for a doctorate. First he had to get a Bachelor's Degree which he had not taken since he thought that was not necessary when he intended to follow a military career. For this degree he presented his paper *Nouveaux théorèmes sur les équations algébriques* Ⓣ which was examined by Jean-Marie Duhamel. He had already published this paper in 1856 in the *Nouvelles annales de mathématiques*. By March 1859 he had been awarded his Docteur ès Sciences by the Faculty of Science in Paris for his thesis *Sur le nombre de valeurs que peut acquérir une fonction quand on y permute ses lettres de toutes les manières possibles* Ⓣ on transitive functions, the work which led to his initial discovery of sporadic simple groups. He defended his thesis before the examining committee consisting of Gabriel Lamé (chairman), Joseph Liouville and Joseph Alfred Serret on 28 March. Mathieu published the results from his thesis in two papers *Mémoire sur le nombre de valeurs que peut acquérir une fonction quand on y permute ses variables de toutes les manières possibles* Ⓣ (1860) and *Mémoire sur l'étude des fonctions de plusieurs quantités, sur la manière de les former et sur les substitutions qui les laissent invariables* Ⓣ (1861) both of which were published in Liouville's *Journal de mathématiques pures et appliquées*.

In the first of these papers, Mathieu showed the existence of an infinite family of 3-transitive groups of degree *p* + 1 and of order *p*(*p*^{2} - 1). This generalised a discovery made by Cauchy in 1846 when he discovered the 3-transitive group of degree 5 of order 120 which one obtains as the *p* = 5 case of Mathieu's infinite series. The really important paper was, however, the second of these papers in which, starting with a transitive group *H* of degree *n*, he attempted to construct a doubly transitive group *G* of degree *n* + 1 which contained a subgroup isomorphic to *H*, namely the subgroup of permutations fixing the (*n*+1)st letter. If he was successful in this construction, he went on to try to construct a 3-transitive group. Of course this is not always possible but Mathieu claimed that, with an algorithm he had found to construct multiply transitive groups, he could construct a 5-transitive group of degree 12, and a 5-transitive group of degree 24. These are the groups now known as *M*_{12} and *M*_{24} but it was only in 1873 that Mathieu exhibited his groups in the paper *Sur la fonction cinq fois transitive de *24* quantités* Ⓣ.

Pierre Duhem explains how, after publishing these papers, Mathieu was highly regarded [4]:-

Progress as remarkable as that achieved by Mathieu would seem to put him in the ideal position to obtain a university appointment, even though he had failed to be elected to the Académie des Sciences, but this was not forthcoming. Feeling he would stand a better chance of a university position if he changed to undertake research in applied mathematics, he did so. He wrote [4]:-In April,1862, the Paris Academy of Sciences had to elect a member in the section of geometry. Lamé who at the time was dean of the section asked that the name of Émile Mathieu be placed on the list of candidates. Nor was Lamé alone with his opinion in the Academy; Liouville fully approved it. This honour conferred upon a young man of not yet twenty-seven years of age who had not taken any steps to solicit such distinction was indeed a brilliant promise for the future. Who would then have predicted that he who so early received this promise was to die at the age of fifty-five, after a life wholly consecrated to the advance of science without being admitted by the Academy even among the number of its correspondents?

Sadly, his move to applied mathematics did not result in the advantages for which he hoped. He took on work as a private tutor of mathematics and he continued in this role for ten years. He also taught special mathematics classes in the lycée Charlemagne, the lycée Saint-Louis, and the lycée de Metz. He did suffer an illness in 1863 when he went back to live with his mother while he recovered. In 1866 Lamé became ill and requested the Minister of Public Instruction, Victor Duruy, that Mathieu be allowed to take over his teaching at the Sorbonne. Mathieu presented to the Minister a list of recommendations signed by Joseph Alfred Serret, Jean Victor Poncelet, Jean-Marie Duhamel, Joseph Liouville, Michel Chasles, Charles Delaunay, and Victor Puiseux. Despite this support from the leading mathematicians of the day, the Faculty of Science preferred to appoint Charles Auguste Briot.Not having found the encouragement I had expected for my researches in pure mathematics, I gradually inclined toward applied mathematics, not for the sake of any gain that I might derive from them, but in the hope that the results of my investigations would more engage the interest of scientific men.

Despite these setbacks, in 1867 Mathieu received acknowledgement of the importance of his research with the award of a gold medal at the Congress of the Scientific Societies. At around the same time Joseph Bertrand published his *Report on the progress of mathematical analysis* which contains the following reference to Mathieu's work:-

As compensation for being passed over in this way, the Minister proposed Mathieu give a supplementary course at the Faculty of Science in 1867-68 so that his teaching ability could be judged. Duruy wrote to Mathieu (see for example [3]):-M E Mathieu has studied far more fully than had been done before, the idea of transitive functions, first introduced by Cauchy, and his memoir deserves quite special mention, on account both of the importance of the new results it contains and of the ingenious form of the proofs. Other memoirs by M Mathieu, relating to mathematical physics, give evidence, like his algebraical researches, of acute penetration and broad learning. An account of these memoirs will be given in another report, whose author, I trust, will heartily join me in calling attention to a young man who truly possesses the gifts of a mathematician, but has so far, in spite of the estimation in which he is held by all, remained outside the sphere to which his remarkable investigations ought to gain him ready access.

Mathieu chose as topics for his course the methods of integration in mathematical physics, the theory of numbers and the algebraic resolution of the equations. His course was not a great success. The dean of the Faculty of Science, Henri Milne-Edwards, arranged for Mathieu's lecture course to be in an inconvenient room outside the Sorbonne, in premises belonging to the 'Association philotechnique'. The report on the course noted that the number of students who attended declined rapidly. It added the following (see for example [3]):-Under close supervision by the illustrious scholars who make up the Faculty, young professors come to try out their abilities, show what they are capable of, and consequently, bring themselves to the attention of the administration so that one day they might officially enter into higher education. I seek the men and the means of discovering which among those have not yet performed, it is to give to all who show promise the means of changing hope to certainty or, in case of failure, to spare the administration an unhappy choice.

Discouraged by this report and realising that he had little chance of a position in Paris, Mathieu sought a position in the provinces. He was appointed as a professor of mathematics at Besançon in March 1869 and, on 9 October 1871 he married Marie Joséphine Guisse (born 1849) in Sainte-Ruffine in the Moselle. Earlier in that year, in the February of 1871, he had sought a position in Nancy. Before we look at the letter he wrote in application, we need to look briefly at the events which had taken place. The Franco-Prussian war had caused major changes to the university system. Germany had captured Strasbourg after a 50-day siege during the Franco-Prussian war of 1870-71 and annexed the city. After taking control of Alsace-Lorraine, the Germans had reorganised the University of Strasbourg and reopened it as the German Kaiser-Wilhelm University of Strasbourg in 1872. Xavier Dagobert Bach (1813-1885), the professor at Strasbourg before the war, lost his position and was appointed as professor of pure mathematics at Nancy. As a consequence, the chair of mathematics at Nancy, which had been held by Nicolas Renard (1823-1880), was split in two and Renard was appointed to the chair of applied mathematics at Nancy. Mathieu, wishing a position closer to Lorraine wrote to the dean of the Faculty of Science of Nancy, the naturalist Dominique Alexandre Godron (1807-1880), seeking a chair:-M Mathieu is evidently a learned young man, but he does not seem to possess, to a mediocre degree, the other qualities which make a professor. He has little flair, his speech is hesitant; he does not know how to impose himself on his audience, to whom he constantly turns his back. Judging by what I heard, his methods are correct, but a little narrow. In short, it is of little interest.

Although there is no direct criticism of Renard in Mathieu's letter, one must assume that he is saying that, in the circumstances, Nancy needs someone of higher quality than Renard. Certainly Renard believed that Mathieu was trying to have him sent to Besançon so that he could have the chair of Applied Mathematics at Nancy. He absolutely refused to move and Mathieu remained at Besançon where he was appointed to the chair of Pure Mathematics in December 1871. In December 1873 Xavier Bach retired from the chair of Pure Mathematics at Nancy and, after five years teaching at Besançon, Mathieu moved to Nancy to take up Bach's chair there. He held this chair until his death in 1890.I am from Metz as also is my wife; all our parents live in Lorraine; life would therefore be more agreeable in Nancy than in any of the other cities in which faculties are situated. There is the question, at the present time, of a faculty which could compete with advantage against the neighboring German universities. If these ideas are to be taken into account, the scientific qualifications of the professors should be taken into account, and you would perhaps have one more position for me to be accepted.

However, Mathieu still longed to obtain a position in Paris. Shortly after he arrived in Nancy, he tried to obtain a chair at the Sorbonne or the Collège de France but without success. In 1885 he applied for the chair of celestial mechanics at the Collège de France which had become vacant on the death of Joseph Serret. His remarkable sequence of near misses continued when he was placed second to Maurice Lévy (1838-1910) for this position. A few months later, in September of 1885, he applied for the chair of differential and integral calculus at the Sorbonne which became vacant on the death of Jean-Claude Bouquet. After the chair was left vacant for a year it was filled by Émile Picard. In early 1886 Mathieu was disappointed in his hopes of being awarded the chair of calculus of probabilities and mathematical physics at the Sorbonne held until then by Gabriel Lippmann who, in that year, became Professor of Experimental Physics.

Lamé retired in 1862 after he became deaf but his chair was not filled until 1886. It was a chair that Mathieu had hoped to obtain ever since he was asked to substitute for Lamé but again he was disappointed. He wrote:-

There is no doubt who Mathieu blamed for his lack of success, namely Charles Hermite. In June 1887, Mathieu made this very clear when he wrote:-Since19years ago when that illustrious engineer of Mines, Lamé, proposed to me to substitute for him in his chair of mathematical physics, I hoped constantly to obtain this chair. I am currently the only mathematical physicist in France. However, when this chair finally became vacant, a faculty committee has chosen another candidate[Henri Poincaré].

Although one has to feel extremely sorry for Mathieu, who was undoubtedly a mathematician of the very highest calibre, nevertheless, the archives examined by the authors of [3] show that Mathieu had difficult relations with his colleagues, lacked zeal in his teaching duties and was a poor lecturer. There are phrases such as "embarrassing lecture, badly articulated, painful to hear."Under the Republic, the Ministers of Public Instruction having completely lost interest in the questions of higher education, M Hermite, who, before the expulsion of the Jesuits, spent all his afternoons in their house in the Rue des Postes(no doubt the Sainte-Geneviève school), introduced successively into the professorial chairs of the Sorbonne all his friends, his son-in-law(Émile Picard)and his nephew at the age of26years. Never has such a scandal occurred in the University under royal governments.

Mathieu's main work, after his initial interest in pure mathematics, was in mathematical physics although he did do some important work on the hypergeometric function. As Grattan-Guinness writes in [1]:-

Perhaps had Mathieu continued to follow up his remarkable discoveries in group theory, he might have achieved more fame and better posts in his lifetime. However, the evidence suggests that it was his lack of teaching skills that told most against him.Although Mathieu showed great promise in his early years, he never received such normal signs of approbation as a Paris chair or election to the Académie des Sciences. From his late twenties his main efforts were devoted to the then unfashionable continuation of the great French tradition of mathematical physics, and he extended in sophistication the formation and solution of partial differential equation for a wide range of physical problems.

Some of his earliest work in mathematical physics was related to his study of light and he looked at the surfaces of vibrations arising from Fresnel waves. He also worked on the polarisation of light where he highlighted some weaknesses in Cauchy's results on the topic. He worked on potential theory applied to elasticity, heat diffusion, and the vibration of bells, a very hard problem. Mathieu studied fluids, in particular examining capillary forces. He also studied magnetic induction and the three body problem where he applied his work to the perturbations of Jupiter and Saturn.

In addition to being remembered for the Mathieu groups, he is also remembered for the Mathieu functions. He discovered these functions, which are special cases of hypergeometric functions, while solving the wave equation for an elliptical membrane moving through a fluid. The Mathieu functions are solutions of the Mathieu equation which is

^{d2u}/

_{dz2}= (

*a*+ 16

*b*cos 2

*z*)

*u*= 0.

*Cours de physique mathématique*Ⓣ (1873) which Mathieu later wished he had given the title

*On the methods of integration in mathematical physics*. In the Preface, Mathieu writes:-

In 1878 he publishedAs the domain of science broadens and expands it becomes more and more necessary to expound its principles with clearness and conciseness and to substitute for artificial processes, however skilful, the transformations that can be accounted for by the nature of the subject. This is clearly illustrated in comparing the 'Mécanique analytique' of Lagrange with the 'Vorlesungen' of Jacobi on the same subject. Examining the treatment of certain problems in each of these works the results obtained will often be found to be the same; the difference consists in the fact that in the latter work the calculations are performed according to rules laid down in advance.

*Dynamique analytique*Ⓣ. You can read Mathieu's Preface to this work at THIS LINK.

After these two works he published: *Théorie de la capillarité* Ⓣ (1883); *Théorie du potentiel et ses applications à l'électrostatique et au magnétisme. I ^{re}partie: Théorie du potentiel* Ⓣ (1885);

*Théorie du potentiel et ses applications à l'électrostatique et au magnétisme. II*Ⓣ (1886);

^{e}partie: Electrostatique et magnétisme*Théorie de l'électrodynamique*Ⓣ (1888);

*Théorie de l'élasticité des corps solides. I*Ⓣ (1890);

^{re}partie: Considérations générales sur l'élasticité; emploi des coordonnées curvilignes ; problèmes relatifs à l'équilibre de l'élasticité; plaques vibrantes*Théorie de l'élasticité des corps solides. II*Ⓣ (1890). This means that he managed to publish eight of his intended eleven books before his death. In all his books Mathieu kept to the view expressed in his Preface to

^{e}partie: Mouvements vibratoires des corps solides; équilibre de l'élasticité du prisme rectangle*Cours de physique mathématique*Ⓣ. Pierre Duhem writes [4]:-

In [1] Grattan-Guinness describes Mathieu's nature as:-In his 'Analytical dynamics' he introduces at the very beginning the general methods due to Hamilton and Jacobi. In his 'Theory of capillarity' he lays aside the direct consideration of the capillary forces employed by Poisson, and follows Gauss in establishing the equations for the various problems by seeking to determine the minimum of the potential of the active forces. In the 'Theory of the elasticity of solid bodies' he invariably uses the principle of virtual velocities to throw the problems of equilibrium into equations.

Gaston Floquet writes in [5]:-... shy and retiring[which]may have accounted to some extent for the lack of worldly success in his life and career; but among his colleagues he won only friendship and respect.

Of an essentially straightforward, sincere, and generous nature, he was kindness itself. He possessed the devotion that seeks to be ignored. In July,1890, when the fatal disease had already attacked him, he succeeded in concealing his ill-health from his colleagues, being unwilling to leave to them the burden of his examinations. In September, on his deathbed the same anxiety agitated his mind with respect to the October examinations. His loyal and trustworthy character made him esteemed and beloved by all; in the Faculty at Nancy he had none but friends. Sensitive to any kindness, touched by the slightest mark of sympathy, he belonged to those who are most easily satisfied. He lived in simple style dividing his time between his lectures and his mathematical researches.

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