Joseph Liouville | Biography

Although Joseph Liouville's primary contribution to mathematics was the first proof of the existence of transcendental numbers (real numbers that are not roots of polynomials with integer coefficients), he had a wide range of mathematical interests. Liouville's publishing and teaching activities were vital to French mathematics during the 19th century. He made critical contributions to number theory, differential geometry, celestial mechanics, and rational mechanics. An enthusiastic lecturer, Liouville held numerous teaching positions, usually two or more at a time. Simultaneously, he was a prolific author, publishing 400 pieces in his lifetime, including over 200 on number theory alone.

Liouville was born in St. Omer, Pas-de-Calais, France, on March 24, 1809. He was the second son of an army captain, Claude-Joseph Liouville, and his wife, Thérèse (nee Balland). Liouville received his early education in Commery and Toul, before being accepted at the l'École Polytechnique. He began studying there in 1825, when he was 16 years of age, and left that institution two years later to enter l'École des Ponts et Chaussées. Liouville switched schools because of changing interests; he wanted to be an engineer. When he graduated in 1830, he was offered an engineering position, but by then he had decided he wanted to study mathematics full-time in the French center for mathematics, Paris.

Even before Liouville graduated from l'École des Ponts et Chaussées, he began to publish articles as early as 1828. He produced articles and notes on electricity and heat in scholarly journals. This early activity marked the beginning of a lifetime of fruitful scholarship. Liouville's primary publishing phase lasted until 1857. In that time, he had 100 or more treatises printed on mathematical analysis, as well as such topics as geometry, physics, algebra, and number theory.

Several years after graduation, in 1830, Liouville married his maternal cousin, Marie-Louise Balland. They had a family of three daughters and one son. To support his family, Liouville took on as many teaching positions as possible in secondary schools, sometimes teaching 34 hours or more a week. His publishing activities combined with his teaching experience led to his post as a répétiteur at the l'École Polytechnique at 1831. Liouville became a lecturer at l'École Centrale des Arts et Manufactures (an engineering school) in 1833. His enthusiasm for teaching led him to earn his doctorate in 1836 so he could hold professorships at the university level.

In this same year, 1836, Liouville filled a gaping hole in French mathematics academia when he founded the Journal de mathématiques pures et appliquées(also known as the Journal de Liouvilleor Liouville's Journal). There had been no forum for publishing French mathematical papers for the five years previous to his founding of the Journal. Although Liouville had no editorial experience, he edited this important publication for almost 40 years. He used his editorial power judiciously, publishing the best of the contemporary greats and helping young mathematicians get their first works in print. Liouville relinquished control of the Journal in 1874 when Henry Résal took over the editorship.

At the same time his teaching and publishing activities bloomed, Liouville experienced the most fruitful research period of his career. From 1832 to 1833, he concentrated on algebraic functions, specifically looking at integrals and their analytic behavior. This work led to his 1844 discovery of the proof of the existence of transcendental numbers, one of his most influential contributions to mathematics. In 1836-1837, he published influential papers with fellow mathematician Charles-François Stürm in the Journal de mathématiques. They delineated what became known as the Stürm-Liouville theory, based on Liouville's methodology for boundary-value problems. This theory became important in physics and integral equation theory. In analysis, Liouville is well known for proving that a bounded entire analytic function on the complex plane must be a constant function. The fundamental theorem of algebra follows as a simple corollary of Liouville's theorem.

Liouville continued to hold multiple teaching positions throughout his career. He resigned from l'École Centrale des Arts et Manufactures in 1838 when he became the professor and chair of Analysis and Mechanics at l'École Polytechnique. The previous year, 1837, Liouville began teaching at the Collège de France as an assistant for a professor there. He resigned in 1843 in protest of the Collège's choice for mathematics chair, Count Libri-Carrucci.

Liouville's contributions to French mathematics did not go unnoticed by his peers. In 1839, he became a member of the French Académie Royale des Sciences. A year later he became a member of Bureau des Longitudes, and served as its director at one point.

In 1848, Liouville took an unexpected turn into politics after the French Revolution of 1848. He won a seat on the constituent assembly as a moderate republican, but lost in an election for a position on the Legislative Assembly. This loss marked the end of his brief foray into elective politics.

Liouville resigned from l'École Polytechnique in 1851 and returned to the Collège de France. Libri-Carrucci had left France and Liouville was appointed to the mathematics chair in his place. Liouville taught at the Collège nearly continuously until 1879 or until his death in 1882 (depending on the source). He still maintained multiple teaching posts late in his career. He served as a professor of rational mechanics at the Sorbonne from 1857 to 1874, although he often used substitute teachers as his health began to decline. After his appointment at the Sorbonne, Liouville's research focused almost exclusively on two specialized topics in number theory.

Liouville's final years were marked by pain and suffering. From 1876 until his death, Liouville suffered from intense insomnia and gout. His wife and only son both died in 1880. Liouville himself died in Paris on September 8, 1882.