Died: 31 May 1931 in Toulouse, France

**Eugène Cosserat**'s father was François-Constant Cosserat who was a well-off middle-class entrepreneur who was based in Amiens. François-Constant had three sons, François-Nicolas (who also has a biography in this archive), Lucien-Constant, and Eugène Cosserat (the subject of this biography). François-Nicolas, born in 1852, was the eldest of the sons, Lucien-Constant, born in 1856, was the middle son, while Eugène, born in 1866, was the youngest and twelve years younger than his eldest brother François. Eugène was only four years old when François Cosserat entered the École Polytechnique in 1870, and nine years old when Lucien-Constant Cosserat entered the same institution in 1875.

Eugène** **was educated first in a primary school in Amiens, moving to the Lycée in Amiens for his secondary education which he entered in the autumn of 1877. At first he was only an average pupil, but once he began to study the sciences he showed considerable aptitude. When he was fourteen years old, in 1880, he was showing exceptional promise and was awarded a prize offered by the Industrial Society of Amiens and also a prize presented by the former pupils of the Lycée. In the following year, 1880-81, he began to study Elementary Mathematics and his love for the sciences became very clear. He received a bursary for his study of Advanced Mathematics which he attended over the following two years. For his outstanding successes in the first of these years he was awarded the Municipal Prize, awarded to the best student in the sciences, and for his achievements in the second of the two years he was awarded the Delambre Prize, given to the best student in the mathematics and physics courses. At the age of 17 he took the competitive entrance examinations for the two major Paris Institutions, the École Polytechnique and the École Normale Supérieure, and was offered a place at both. Unlike his two brothers who both studied at the École Polytechnique, he chose to study at the École Normale Supérieure which he entered in 1883. In fact this decision was, in many ways, the natural one since when his brothers entered the École Polytechnique it had been the leading institution to train scientists but in the intervening years it had concentrated more on training administrators and the École Normale had become the leading institution for science.

During three years of study at the École Normale, Cosserat attended lectures by leading mathematicians including Paul Appell, Gaston Darboux, Gabriel Koenigs and Émile Picard. Among his fellow students were several who would make major contributions to mathematics, including Jacques Hadamard and Paul Painlevé. Cosserat graduated in 1886 and spent a short time teaching at the Lycée in Rennes before he was appointed as an assistant astronomer at the Observatory in Toulouse towards the end of 1886. There he assisted the director, Édouard Benjamin Baillaud (1848-1934), and he was given the taskof making a systematic study of 3112 binary stars in Wilhelm Struve's catalogue *Stellarum Duplicium Mensurae Micrometricae* (1837). At the same time, he worked for his doctorate in mathematics. He submitted his thesis *Sur le cercle considéré comme élément générateur de l'espace* to the Faculty of Science in Paris and, on 14 March 1889, he was examined by a jury consisting of Paul Appell, Gaston Darboux, and Gabriel Koenigs. The report written by Darboux stated:-

The subject chosen by M Cosserat is extremely broad and barely explored, the young assistant lecturer barely began the study, but his work is interesting enough and contains enough new results to allow me to conclude the approval of the Faculty.

In his doctoral dissertation, Cosserat extended Plücker's concept of generation by means of straight lines by considering infinitesimal properties of spaces generated by circles. The authors of [3] explain that Cosserat's:-

... doctor's degree, in the spirit of Gaston Darboux, deals with an extension of Plücker's concepts that is used in order to study the infinitesimal properties of spaces created by circles.

Even before the award of his doctorate in 1889, Cosserat had begun teaching mathematics courses at the Faculty of Science at Toulouse. In 1896 he became professor of differential and integral calculus there, replacing Thomas Stieltjes who had died on 31 December 1894, and, from that time on, he divided his work between the Faculty of Science and the Observatory. In 1908 Cosserat was appointed to the chair of astronomy at Toulouse, becoming director of the Observatory there for the rest of his life. In this latter role he replaced Édouard Benjamin Baillaud who had left Toulouse to become director of the Paris Observatory. The role of director of the Observatory was a demanding one, and Cosserat became almost totally occupied with administrative tasks from the time of his appointment and so was forced to essentially give up mathematical research from this time on.

Although he was not living in Paris, Cosserat was elected to the Académie des Sciences as a corresponding member on 19 June 1911 and a full member on 31 March 1919. Four years later, he was elected to the Bureau de Longitude. Because he was in Toulouse rather than Paris, he was made a non-resident member of both these organisations. In 1889 he was awarded the Poncelet Prize by the Académie des Sciences. He had married Berthe Jèze; they had two sons. The eldest, Louis Cosserat, became a medical doctor but their younger son died before reaching adulthood.

In [1] Cosserat is described as follows:-

A reserved, kindly man and a diligent worker, Cosserat was one of the moving forces in the University of Toulouse for thirty five years.

In the first part of his career in astronomy, we have already noted that he made observations of double stars. In addition he observed minor planets and comets, publishing his findings in papers in *Comptes rendus* of the Académie des Sciences, the *Bulletin astronomique* and the *Astronomische Nachrichten*. He then carried out a series of observations on the satellites of Jupiter and those of Saturn. After his appointment as director of the Toulouse Observatory in 1908, he studied the proper motions of stars publishing works such as *Sur quelques étoiles dont le mouvement propre annuel dépasse 0ý5* (1919) and the 300-page posthumous publication *Déterminations photographiques de positions d'etoiles* (1933). His work on the proper motion of stars made an important contribution to discussions on the structure of space. All these various contributions showed great skill and resulted in highly accurate data but much of Cosserat's work on astronomy was carried out at a time when he was also devoting much of his time to the study of mathematics.

In mathematics, we have already noted his early work on geometry. In his later work, Cosserat studied the deformation of surfaces which led him to a theory of elasticity. This work was carried out in collaboration with his brother, François Cosserat, who was an engineer. He began his collaboration with his brother in 1896 with the publication *Théorie de l'élasticité*. This first work studied broad questions relating to the foundations of mechanics but later their work turned towards the physical theory. By the early 1900s, Cosserat had stopped working on the type of geometrical problems that had interested him at the start of his career and all his research efforts were directed towards working on mechanics with his brother. Their most important joint publications are: *Note sur la cinématique d'un milieu continu* (1897); *Note sur la dynamique du point et du corps invariable* (1906); *Note sur la théorie de l'action euclidienne* (1909); and the book* Théorie des corps déformables* (1909). The first of these was published as an addition to Gabriel Koenigs *Leçons de Cinématique professées à la Sorbonne: cinématique théorique*. A review of this work by E O Lovett in the *Bulletin* of the American Mathematical Society in 1900 singles out the Cosserats' contribution:-

The introduction of this note is peculiarly fortunate for it is high time that kinematics should comprehend the study of deformation and of deformable spaces. The authors have included in their extract certain generalities on curvilinear coordinates, the deformation of a continuous medium in general, infinitely small deformation, use of the mobile trieder, and the case where the non-deformed medium is referred to any curvilinear coordinates.

This innovative work on mechanics (21 joint publications on this topic are listed in [2]) ended with the François Cosserat's death in 1914, after which time his brother Eugène Cosserat published nothing further on the topic. Jacques Levy describes the two Cosserats' contributions to this area [1]:-

The most practical results concerning elasticity were the introduction of the systematic use of the movable trihedral and the proposal and resolution, before Fredholm's studies, of the functional equations of the sphere and ellipsoid. Cosserat's theoretical research, designed to include everything in theoretical physics that is directly subject to the laws of mechanics, was founded on the notion of Euclidean action[least action]combined with Lagrange's ideas on the principle of extremality and Lie's ideas on invariance in regard to displacement groups. The bearing of this original and coherent conception was diminished in importance because at the time it was proposed, fundamental ideas were already being called into question by both the theory of relativity and progress in physical theory.

The authors of [5] write:-

The Cosserat brothers, following a suggestion by Duhem(1893), developed a theory for continuous oriented bodies that consist not just of particles(or material points), but also of directions associated with each particle. Thus, in addition to the field of position vectors of a continuum in a given configuration, one also admits vector fields ... which may be chosen so as to represent pertinent features of materials. ... The Cosserats themselves recognised the value of oriented two-dimensional continua(i.e., curves and surfaces endowed with additional structure in the form of directors)for representing the deformations of rods and shells respectively. ...[However their]ideas on the subject[were]ignored for half a century.

Pommaret, in [2], looks at the reception of the Cosserats' contributions:-

... among their contemporaries, only Henri Poincaré(electron theory), Émile Picard(surprisingly)and Élie Cartan appreciated the work done by the brothers. Exceptions are also colleagues of Eugène Cosserat in Toulouse, like L Ray and A Buhl.(Indeed the work of Buhl is a patching of mathematical analogies, far from physics.)It is only in Germany that the brothers got disciples like Karl Heun who quoted them with emphasis in the German edition of the 'Encyclopédie des Sciences Mathématique' and studied their work in a seminar at Karlsruhe in1909. Then ... Relativity Theory and Quantum Physics overtook this period in science and the work of E and F Cosserat was almost rediscovered after1950because of the use of liquid crystals.

Another aspect of Eugène Cosserat's work which we should mention is his contributions to the *Annales de la faculté des sciences de Toulouse*. This journal began publication in 1887 and, two years later, Cosserat joined the editorial board. The two other mathematicians who served on this board at this time were Henri Andoyer and Thomas Jan Stieltjes. In 1896 Cosserat became secretary to the editorial board of the *Annals* and he continued to hold this role until 1930. In fact, he continued to undertake editorial work up to the time of his death, sending Henri Poincaré a letter on an editorial matter just a few days before his death.

In addition to his administrative work for the *Annals*, Cosserat also translated articles from Russian and published French versions in the *Annals*. In particular in 1905 he published works by Karl Mikhailovich Peterson on curves, surfaces and their deformations, and work on differential geometry related to questions from the theory of elasticity. Cosserat also collaborated with his brother in making French translations of other works. For example they translated a work by John Perry which approached mechanics in a highly experimental way. The Cosserat brothers, despite having written highly technical theoretical works on mechanics, approved of Perry's approach. They wrote at the beginning of the book:-

An space difficult to cross, which calls for continued efforts, separates abstract mechanics from its applications; this was already pointed out by Poncelet in his "Introduction to industrial mechanics", and the difficulty has not diminished since his time. ... It is therefore justified, at least in technical education, to lead the way in experimental considerations on rational deductions. ... This method, which is somewhat heretical in our country, was developed with great skill by John Perry, in the remarkable book that we present to French readers.

The Cosserat brothers François and Eugène also collaborated in translating the article on 'mechanics' from the German *Encyklopädie der mathematischen Wissenschaften* for a French version of this monumental classic work. However, they did not just make a translation, they added a very large amount of new material of their own showing their great appreciation of the history and philosophy of the subject.

Eugène Cosserat died in his home at the Observatory in Toulouse [4]:-

The funeral took place on2June, on a morning with gentle sun; a long procession descended from the Observatory along the slopes which, although close to the city, still retained some greenery. It seemed that Nature had staged a scene both bright and calm ... calm as he was in his way.

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

**January 2012**

[http://www-history.mcs.st-andrews.ac.uk/Biographies/Cosserat.html]