**Joseph Liouville** is best known for his work on transcendental numbers. He constructed an infinite class of such numbers.

- Liouville entered the École Polytechnique in 1825 and attended Ampère's Cours d'analyse et de mécanique Ⓣ(Course on analysis and mechanics) in session 1825-26.
- Although Liouville does not seem to have attended any of Cauchy's courses, it is clear that Cauchy must have had a strong influence on him.
- Liouville graduated in 1827 with de Prony and Poisson among his examiners.
- After graduating from the École Polytechnique Liouville entered the École des Ponts et Chaussées.
- By now Liouville was set on an academic career and he found it impossible to study away from Paris.
- In 1831 Liouville was appointed to his first academic post, as assistant to Claude Mathieu who had been appointed to Ampère's chair at the École Polytechnique.
- It is remarkable that during this period of his life Liouville taught between 35 and 40 hours a week at the different institutions.
- In 1836 Liouville founded a mathematics journal Journal de Mathématiques Pures et Appliquées.
- This journal, sometimes known as Journal de Liouville, did much for mathematics in France throughout the 19th century.
- Liouville had already gained an international reputation with papers published in Crelle's Journal but at the same time the quality of Crelle's Journal made him aware of deficiencies in the avenues for mathematical publications which there were in France.
- Liouville became favourite to fill the chair at the École Polytechnique which fell vacant when Navier died in 1836.
- In 1837 Liouville was appointed to lecture at the Collège de France as a substitute for Biot.
- In 1838 Liouville was appointed Professor of Analysis and Mechanics at the École Polytechnique.
- The quarrel between Liouville and Libri intensified after his election to the Académie.
- In 1840, after a vacancy resulting from the death of Poisson, Liouville was elected to the Bureau des Longitudes.
- In many ways 1840 was a turning point in Liouville's career.
- Life for Liouville developed into a year with two distinct parts.
- Not everything went Liouville's way however.
- When Lacroix died in 1843, Liouville applied for his chair at the Collège de France where he lectured only as a substitute for Biot.
- Encouraged by Arago, Liouville stood for election to the Constituting Assembly in 1848.
- Mr Liouville is one of my best friends.
- Elected on 23 April 1848, Liouville took his seat among the moderate republican majority.
- The election defeat proved another turning point in Liouville's life.
- His chair at the Collège de France was declared vacant in 1850 and Cauchy and Liouville competed for the post.
- In a close contest Liouville triumphed and began his lectures at the Collège de France in 1851.
- 1856 and 1857 were two of Liouville's most productive years.
- Not only did he have a high teaching load but Liouville was a perfectionist which meant that when he felt that he could not devote all the time necessary to give the best possible lectures he began to suffer.
- Another blow to Liouville was the death of Dirichlet in 1859.
- Liouville's mathematical work was extremely wide ranging, from mathematical physics to astronomy to pure mathematics.
- Liouville investigated criteria for integrals of algebraic functions to be algebraic during the period 1832-33.
- Having established this in four papers, Liouville went on to investigate the general problem of integration of algebraic functions in finite terms.
- Another important area which Liouville is remembered for today is that of transcendental numbers.
- Liouville's interest in this stemmed from reading a correspondence between Goldbach and Daniel Bernoulli.
- Liouville certainly aimed to prove that eee is transcendental but he did not succeed.
- His work on boundary value problems on differential equations is remembered because of what is called today Sturm-Liouville theory which is used in solving integral equations.
- Sturm and Liouville examined general linear second order differential equations and examined properties of their eigenvalues, the behaviour of the eigenfunctions and the series expansion of arbitrary functions in terms of these eigenfunctions.
- Liouville contributed to differential geometry studying conformal transformations.
- In 1842 Liouville began to read Galois's unpublished papers.
- Liouville was therefore a major influence in bringing Galois's work to general notice when he published this work in 1846 in his Journal.
- Liouville also lectured on Galois's work and Serret, possibly together with Bertrand and Hermite, attended the course.
- In number theory Liouville wrote around 200 papers, working on quadratic reciprocity and many other topics.

Born 24 March 1809, Saint-Omer, France. Died 8 September 1882, Paris, France.

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Algebra, Analysis, Ancient Greek, Astronomy, Geometry, Group Theory, Number Theory, Physics, Set Theory

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive