Died: 19 August 1910 in Lunel, Languedoc, France

**Eugène Rouché** was born in Sommières in southern France. This town, about 30 km from the south coast, is situated between Nimes and Montpellier and now contains a street named after the famous mathematician.

After training as a mathematician, Rouché taught at the Lycée Charlemagne and then was professor at the Conservatoire des Arts et Métiers in Paris. He also acted as an examiner at the École Polytechnique.

Rouché published many mathematics articles, among them were some which appeared in *Comptes Rendus* and some in the *Journal of the École Polytechnique*. For example he published *Mémoire sur la serie de Lagrange* in the second of these two journals in 1862.

He also wrote several textbooks including *Traité de géométrie élémentaire* (written jointly with Ch De Comberousse) (1874), *Éléments de Statique Graphique* (1889), *Coupe des pierres : précédée des principes du trait de stéréotomie* (written jointly with Charles Brisse) (1893), and *Analyse infinitésimale à l'usage des ingénieurs* (1900-02) which was a calculus text written for engineers. This two volume work was the result of a collaboration with Lucien Lévy (born 1853) who was the father of the well known mathematician Paul Lévy.

Another of his famous geometry texts was written jointly with Ch De Comberousse, namely *Traité de géométrie*. This appeared in two volumes, the first being on *Géométrie plane* and the second on *Géométrie dans l'espace*. The book was first published in 1883 but went through many editions. The seventh edition of this work was published in Paris by Gauthier-Villars in 1900. It was a major treatise, the volumes having 548 and 664 pages respectively. Editions continued to be published after Rouché's death, with a new edition published in Paris by Gauthier-Villars in 1922. Indeed it is still in print now (2010).

Laguerre died in 1886 and Rouché became one of the editors to work on the production of his *Collected Works*. There were three editors for this work, the other two being Hermite and Poincaré.

Although few today know who Rouché was, his name is very well known through Rouché's theorem which he published in the *Journal of the École Polytechnique* **39** (1862).

If

f(z) andg(z) are two complex functions which are regular within and on a closed contourC, on whichf(z) does not vanish and also |g(z)| < |f(z)| thenf(z) andf(z) +g(z) have the same number of zeros withinC.

In 1875 Rouché published a two page paper *Sur la discussion des equations du premier degré* in volume 81 of *Comptes Rendus* of the Académie des Sciences. This short paper contained his result on solving systems of linear equations. This is the well-known criterion which says that a system of linear equations has a solution if and only if the rank of the matrix of the associated homogeneous system is equal to the rank of the augumented matrix of the system. Rouché later published a fuller version of this theorem in 1880 in the *Journal de l'École Polytechnique*.

In fact he was not the first to prove such a result and after Rouché's paper appeared Georges Fontené published a note in the *Nouvelles Annales de Mathématiques* claiming priority. When Frobenius discussed this result in his papers, for example in *Zur Theorie der linearen Gleichungen* published in Crelle's Journal in 1905, he gave credit for proving the theorem to both Rouché and Fontené. However it is now often called the Rouché-Frobenius theorem, especially in the Spanish speaking world. This is almost certainly because the Spanish/Argentinian mathematician Julio Rey Pastor referred to the theorem by this name.

The town of Lunel where Rouché died is close to his birthplace of Sommières. In fact it is situated about halfway between Sommières and the southern coast of France.

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

**April 2003**

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