**Charles-François Sturm**'s father was Jean-Henri Sturm whose family came from Strasbourg to settle in Geneva about 50 years before Charles-François's birth. Jean-Henri Sturm was a teacher of arithmetic who had married Jeanne-Louise-Henriette Gremay. Charles-François's parents gave him a good education and at school he showed great promise, particularly in Greek and Latin poetry for which he had a remarkable talent.

Sturm came from a Protestant family and, in order to learn German, he attended the local Lutheran church where sermons were preached in that language. When Sturm was sixteen years old his father died and he changed tack in his academic studies, leaving the humanities and taking up the study of mathematics. He was taught mathematics at Geneva Academy by Simon Lhuilier in 1821 and immediately Lhuilier recognised the mathematical genius in Sturm. However, Lhuilier was over seventy years of age and close to retiring at this time so it was his successor Jean-Jacques Schaub who inspired Sturm. Schaub did more than teach Sturm mathematics for he supported him financially at the Academy. Sturm's family had been left in considerable financial difficulties on the death of his father so the financial assistance allowed Sturm to continue with his education.

At the Academy Sturm's best friend was Daniel Colladon and the friendship would have a marked influence on Sturm's early research career. After leaving the Academy, Sturm was appointed as a tutor to the youngest son of Mme de Staël at the Châteaux de Coppet close to Geneva. He took up his appointment in May 1823 and found that it left him plenty of free time to devote to his own studies. He used his time well and began to write articles on geometry which were published in Gergonne's *Annales de mathématiques pures et appliquées*. Before the end of 1823 the family moved from the château to spend six months in Paris and Sturm, as tutor, naturally accompanied them.

In Paris he was introduced into the scientific circles by the family. Sturm wrote to his friend Colladon (see [1]):-

As for M Arago, I have two or three times been among the group of scientists he invites to his house every Thursday, and there I have seen the leading scientists, Laplace, Poisson, Fourier, Gay-Lussac, Ampère, etc. ... I often attend the meetings of the Institute that take place every Monday.

This was clearly an extremely fortunate opportunity for Sturm. Although he returned to the château in May 1824 he left after six further months to devote himself to scientific research. The Paris Academy had set a prize topic on the compressibility of water and Sturm, with his friend Colladon, decided to begin experiments on Lake Geneva with the aim of putting in an entry for the prize. The experiments were not a great success since they did not yield the expected results and Colladon received a serious injury to his hand while conducting the experiments.

In December 1825 Sturm and Colladon went to Paris to take courses in mathematics and physics and also to collect further instruments to repeat their experiments. The Paris contacts that Sturm had made proved useful for he lived at Arago's house for a while as tutor to his son. He was also given the use of Ampère's laboratory. The time was very fruitful for Sturm who attended lectures by Ampère, Gay-Lussac, Cauchy, and Lacroix. Fourier suggested projects for both Sturm and Colladon, recognising that Colladon was essentially a physicist while Sturm was a mathematician.

Despite completing their paper for the Grand Prix of the Académie des Sciences they did not win the prize; in fact none of the submissions was deemed good enough and the same topic was set again. By this time Sturm and Colladon were both working as assistants to Fourier. Colladon made further experiments on Lake Geneva and after revising their joint memoir they successfully won the prize. The value of the prize was enough to allow Sturm and Colladon to continue their research in Paris. This point marked the end of their successful collaboration and the two embarked on different research projects. Sturm's theoretical work in mathematical physics involved the study of caustic curves, and poles and polars of conic sections.

One of Sturm's most famous papers *Mémoire sur la résolution des équations numériques* Ⓣ was published in 1829. It considered the problem of determining the number of real roots of an equation on a given interval. The problem was a famous one with a long history having been considered by Descartes, Rolle, Lagrange and Fourier. The first to give a complete solution was Cauchy but his method was cumbersome and impractical. Sturm achieved fame with his paper which, using ideas of Fourier, gave a simple solution. Hermite wrote:-

Sturm's theorem had the good fortune of immediately becoming a classic and of finding a place in teaching that it will hold forever. His demonstration, which utilises only the most elementary considerations, is a rare example of simplicity and elegance.

Strangely although the theorem quickly became a classic it was soon relegated to history and, contrary to what Hermite believed, vanished from textbooks. As the title indicated, two events in the history of the algebraic theorem of Sturm are examined in [5]. The author describes how Tarski showed in 1940 that Sturm's method of proof could be used in mathematical logic to prove the completeness of elementary algebra and geometry. The 1829 paper was not the last of Sturm's work on this algebraic equations and in [12] Sinaceur:-

... seeks to determine the mutual influence between A-L Cauchy's and Ch-F Sturm's research from1829to around1840on the roots of algebraic equations.

Paris was not an easy place for a foreigner and Protestant to obtain a post at this time and, despite his fame from the 1829 paper, he was not appointed. The revolution of July 1830 changed the political climate and after this Arago succeeded in getting Sturm appointed as professor of mathematics in the Collège Rollin. He became a French citizen in 1833 and was elected to the Académie des Sciences in 1836. These were the years during which he published some important results on differential equations.

Sturm became interested in obtaining results on specific differential equations which occurred in Poisson's theory of heat. Liouville was also working on differential equations derived from the theory of heat. Papers of 1836-1837 by Sturm and Liouville on differential equations involved expansions of functions in series and is today well-known as the Sturm-Liouville problem, an eigenvalue problem in second order differential equations.

He worked at the École Polytechnique in Paris from 1838 where he became a professor of analysis and mechanics in 1840. In the same year he succeeded Poisson in the chair of mechanics in the Faculté des Sciences, Paris. For around ten years he gave excellent lectures but his wish to give his students the best possible courses meant that he gave a great deal of his time to preparing his lecture courses on differential and integral calculus and on rational mechanics. These courses became the widely used texts *Cours d'analyse de l'École Polytechnique 2 Vol*. (1857-63) and *Cours de mécanique de l'École Polytechnique 2 Vol* Ⓣ (1861) both published posthumously.

His time for research was now limited but he still made important contributions undertaking research on infinitesimal geometry, projective geometry and the differential geometry of curves and surfaces. He also did important work on geometrical optics.

From 1851 his health began to fail and despite brave attempts to overcome the problem and return to teaching (which he managed to do for a while) he died after a long illness.

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

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