Eugène Catalan was born in Bruges and although we have given this as being in Belgium, it actually was in France at the time of his birth. Belgium had been part of the Austrian Netherlands until 1793 but was under the Napoleonic consulate and part of the French empire from 1799 to 1814. After the defeat of Napoleon, the Allied powers were determined not to leave Belgium in the possession of France. The Kingdom of the Netherlands was confirmed by the Congress of Vienna in June 1815 but it was established for the convenience of Europe without considering the desires of the Belgians and the Dutch. Prince William of Orange took the throne on 16 March 1815, taking the title William I, and he was crowned on 27 September. Catalan was therefore born in France, and so rightly considered himself French. However the town of his birth was part of the was Kingdom of the Netherlands before he was one year old. Catalan's father, Joseph Catalan, was an architect and while Eugène was still young his father set himself up as an architect in Paris.
Catalan was educated in Paris following courses which would lead to him becoming an architect like his father. He attended a school for design where at the age of fifteen he began teaching geometry to his fellow pupils after competing for the position. In 1830 there was a Revolution in Paris. Catalan was an ardent republican and was politically active in promoting the left wing cause. This revolution had repercussions across Europe, in particular in Belgium dissatisfaction with William's rule led to moves against the monarchy culminating in the "Belgian Revolution" of August-September 1830, following the July Revolution in Paris earlier that year. On 20 January 1831, an international conference in London recognized Belgium as an independent state. Catalan then attended a school for fine arts but he showed such an outstanding aptitude for mathematics that he was advised to try for entry into the École Polytechnique. He was in Liouville's class at the École Polytechnique in 1833 but was expelled in the following year for his political activities. Allowed to resume his studies in 1835 he graduated in sixteenth place and obtained a post at the Collège de Châlons sur Marne.
He returned to Paris in 1838 to found the Sainte-Barbe preparatory school along with Sturm and Liouville. He taught there for a while but, with Liouville's help, he obtained a lectureship in descriptive geometry at the École Polytechnique in the same year. This might have launched Catalan's university teaching career but any prospects of a smooth path was damaged by him continuing to be very politically active with strong left-wing political views. He was awarded his doctorate in the mathematical sciences in 1841 but made little progress in his career. Piotr Tchihatchef was a Russian geographer and geologist who lived in Paris. He approached Francoeur asking to be tutored in mathematics and Francoeur advised him to approach Catalan. This in fact proved fortunate since Tchihatchef had been given a paper by Chebyshev to submit for publication in Paris. Tchihatchef showed Catalan the paper and Catalan began to correspond with Chebyshev; a correspondence which continued for fifty years.
In 1846 Catalan was appointed to take charge of teaching higher mathematics at the Collège de Charlemagne but then became involved in the political unrest in France in 1848. The revolution led to the Second Republic, and the voters chose Louis-Napoleon Bonaparte to become president. This fitted well with Catalan's republican views, and his career seemed to be going well again when he was appointed to the Lycée Saint Louis in 1849. However the Second Republic only lasted three years. On 2 December 1851 there was a coup d'état with Louis-Napoléon Bonaparte assuming absolute power and dissolving the National Assembly. Exactly one year later he became Emperor taking the title Napoleon III. This was bad for Catalan who refused to take the required oath of allegiance and as a consequence lost his job. For the next few years he lived in Paris, teaching mathematics, but without any proper employment. Beginning in 1857 he was helping to prepare candidates for the entry examinations of the École Polytechnique at a number of different institutions, Jauffret, Barbet and Lesage. In 1865, having had no permanent position for 13 years, he was appointed to the chair of mathematics at the University of Liège. He held this chair until he retired in 1884, then continued to live in Liège until his death ten years later.
Since Catalan was born in Bruges it might be supposed that he would feel as if he was coming back to his homeland when he took up the position in Liège. However, we explained at the beginning of this article the various changes in status of Belgium throughout Catalan's life and in fact Catalan always considered himself French, having earlier undergone considerable efforts to reinstate his French citizenship. While in Liège, Catalan taught Cesàro who became his favourite student. After Cesàro left Liège the two continued to correspond and the article [7] reproduces 52 letters Catalan sent to Cesàro between 1880 to 1894 as well as 7 replies by Cesàro. Many of Catalan's letters are written specifically to encourage Cesàro in his research. In fact Catalan was an enthusiastic correspondent throughout his life and the article [4] looks at a collection of more than 600 letters addressed to him which had been previously unknown.
Catalan published extensively on continued fractions and number theory. He defined the numbers, called today the 'Catalan numbers', while considering the solution of the problem of dissecting a polygon into triangles by means of non-intersecting diagonals. Catalan was not the first to solve the problem, however, since Segner had solved it in the 18th century, although his solution was not as elegant as Catalan's. Euler had worked on simplifying the solution to the problem as did Binet at almost the same time as Catalan, around 1838.
Another mathematical object to which Catalan's name is attached is the 'Catalan minimal surface' which he investigated around 1855. It is associated with 'Catalan's minimal curve' (a minimal curve is defined as having a mean curvature of zero). The most famous of all, however, is the 'Catalan Conjecture' made in 1844 in a letter he sent to Crelle's Journal:-
I beg you, sir, to please announce in your journal the following theorem that I believe true although I have not yet succeeded in completely proving it; perhaps others will be more successful. Two consecutive whole numbers, other than 8 and 9, cannot be consecutive powers; otherwise said, the equation xm - yn = 1 in which the unknowns are positive integers only admits a single solution.
It is clear that one can reduce immediately to the case where m and n are prime. Progress on solving the conjecture was slow. In 1850 Victor Lebesgue proved that no solution existed when n = 2. After certain cases were ruled out by Cassels in 1961, Ko Chao showed in 1964 that no solution existed with m = 2. Tijdeman proved in 1976 that Catalan's equation had only finitely many solutions, still quite far from showing that there is precisely one. The problem was finally solved completely in 2003 by Preda Mihailescu.
Catalan wrote several texts which were very popular and many ran into several editions. He wrote: Elements de géométrie (1843); the two volume work Traité élémentaire de géométrie descriptive (1850-52) which ran to 5 editions; Théorèmes et problèmes de géométrie élémentaire (1852) which ran to 6 editions; Nouveau manuel des aspirants au baccalauréat ès sciences (1852) which ran to 12 editions; Solutions des problèmes de mathématique et de physique donnés à la Sorbonne dans les compositions du baccalauréat ès sciences (1855-56); two volumes of Manuel des candidats à l'École polytechnique (1857-58); Notions d'astronomie (1860) which ran to 3 editions; Traité élémentaire des séries (1860); and Cours d'analyse de l'université de Liège (1870).
Catalan was honoured with election to the Royal Belgium Academy of Science (Académie Royale des Sciences, des Lettres et des Beaux-Arts) on 15 December 1865.
Article by: J J O'Connor and E F Robertson
August 2006