# Eugène Charles Catalan

### Quick Info

Born
30 May 1814
Bruges, French Empire (now Belgium)
Died
14 February 1894
Liège, Belgium

Summary
Eugène Catalan was a Belgian mathematician who defined the numbers called after him, while considering the solution of the problem of dissecting a polygon into triangles by means of non-intersecting diagonals.

### Biography

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 became part of the Kingdom of the Netherlands before he was one year old. His name was registered on his birth certificate as Eugène Charles Bardin. His mother, Jeanne Bardin, was the seventeen year old daughter of Jean Bardin. She had been born in Beaune and was living with her parents in Bruges at the time her son was born. She was unmarried, and her son was registered under her name of Bardin. Although Jean Bardin appears on official records as a bookseller, in reality he was a travelling salesman of cheap books who had moved from Beaune to Bruges. Eugène's mother earned some money as a dressmaker in the years when her young son was growing up. Although at first she lived with her parents, she eventually left their home and in 1820 she spent several months in Brussels. Eugène's father, Joseph Victor Étienne Catalan, was originally from Paris. He married Jeanne Bardin in 1821 and, on 1 March of that year, acknowledged that he was the father of her son who was nearly seven years old. Joseph was a jeweller but, in reality, he made money with various schemes such as selling pictures and perfume. In May 1822, Joseph and Jeanne Catalan were living in Lille with their son. Eugène certainly had had some education and was able to write excellent French by the age of ten. At this time, in 1824, he became an apprentice jeweller. However, he only kept up this apprenticeship for three months, giving up because he lacked ability.

It is unclear just when the family moved to Paris, but this must have occurred around 1825. Eugène rapidly fell in love with the city and his parents allowed him to roam the streets. He watched the jubilee procession through Paris which took place on 3 May 1826, when the corner-stone of a monument to Louis XVI was laid and dedicated, and he described it in detail in a letter. Although he was only twelve years old at the time, Eugène's dislike of the monarchy is clearly and forcibly expressed. For the rest of his life he considered himself a Parisian. It is unclear exactly when Jeanne Catalan died, but it must have been soon after the family moved to Paris. Joseph Catalan, who had now become an architect, married Adélaïde Vitry on 2 August 1828 in the church of Saint Nicolas des Champs. Eugène was set to follow his father and become an architect and, although he had notattended a lycée, he entered the École Royale Gratuite de Dessin et de Mathématiques en Faveur des Arts Mécaniques. He was very happy at this school, and was particularly impressed by professors Jean Paul Douliot (1788-1834) and Jean-Baptiste-Omer Lavit (1771-1836). In addition to attending courses at the École Gratuite de Dessin, Catalan attended courses at the École des Beaux-Arts. He remained a pupil at the École Gratuite de Dessin until 1831 but, in 1829, after a competition, he was appointed to teach geometry to his fellow pupils at the school. He retained this teaching post until 1833, but while he was at the school, he became more politically active, spurred on by the revolution of 1830. This revolution had repercussions across Europe, in particular in Belgium where dissatisfaction with King William I'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.

At the École Gratuite de Dessin, Catalan was taught by Louis Lefébure de Fourcy who had been appointed as an admissions examiner for the École Polytechnique in 1826. He encouraged Catalan, who showed an outstanding aptitude for mathematics, to prepare himself to take the admission examinations for the prestigious university. He studied at the institution run by Monsieur de Reusse on the Rue de Vaugirard. He also attended a mathematics course given by Antoine Delisle (author of Éléments de trigonométrie rectiligne et sphérique ) at the Lycée Saint-Louis, nearby on the Boulevard Saint-Michel, as well as a course by Louis Francoeur at the Sorbonne on the same Boulevard. He won the first prize in the Concours Général de Mathématiques Spéciales in the summer of 1833. Rather than carry on studying, Catalan spent August and September of that year back in Bruges, the first time he had been in that city since he was eleven years old. He went back there to celebrate the birthday of his grandmother on 20 September. Having taken the entrance examinations for the École Polytechnique, he was admitted in November of 1833 ranked 53rd in the list of accepted students, although he almost certainly would have been ranked higher but for the break in his studies. At the École Polytechnique, Catalan attended mathematics courses given by Joseph Liouville and Gabriel Lamé. However, he also studied other topics including the literature and history of France reading authors such as Voltaire, Molière and La Bruyère.

The students at the École Polytechnique were much involved in the political events which were taking place in France at this time. It was a time of great political instability and, in April 1834, there were serious disturbances in Paris following the passing of a law to curtail the activities of the Republican Society of Human Rights. The disturbances were brutally put down by the army. The authorities tried to ensure that the students did not become involved and they send down a number of students who had trumpeted their Republican sympathies. Despite his strong Republican beliefs, Catalan kept out of trouble. In November 1834 he completed his first year of study, being ranked 49th out of 145 students. He was to proceed to his second year but, his father received a letter dated 15 December informing him that his son had been dismissed from the École Polytechnique and was returning home. Catalan had not been singled out for this punishment, for it appears that the whole of his year suffered the same fate. In fact during this forced vacation Catalan took the opportunity to become engaged to Charlotte Augustine Renée Perin (known as Eugénie) who was two years his elder. She had been born in Lille, where Catalan had spent several years. They were engaged to be married on 2 January 1835 and spent the rest of their lives together.

Catalan had to apologise for an "act of subordination" before he was allowed to return to the École Polytechnique in the middle of January 1835. He must have found that a difficult thing to do but he knew that he had to do such things if he was to complete his education. He studied hard and graduated in the summer of 1835, ranked 16th out of the 140 students who had qualified for public service. He was placed 13th among the students who were qualified to enter the École des Ponts et Chaussées but decided that this was not the route that he wanted to take. He was appointed as professor at the École des Arts et Métiers at Châlons-sur-Marne, taking up his appointment in the autumn of 1835. On 4 January 1836 Catalan's father died and on 8 April his grandfather Jean Bardin died, but he went ahead with his marriage to Eugénie Perin on 28 April. In spite of the fact that he was not a practising Christian, they was married in the church of Saint Eustache in Paris. Their first child, a daughter Marie Adélaïde, was born in Châlons-sur-Marne.

Liouville began publication of Journal de Mathématiques Pures et Appliquées in 1836 and a paper by Catalan, Solution d'un problème de Probabilité relatif au jeu de rencontre , was published in the second volume in 1837. In the same journal, he published two papers in 1838: Note sur un Problème de combinaisons , and Note sur une Équation aux différences finies . The second of these contains the 'Catalan numbers' which appears in the solution of the problem of dissecting a polygon into triangles by means of non-intersecting diagonals. Four papers by Catalan are published in Volume 4 in 1839: Note sur la Théorie des Nombres ; Solution nouvelle de cette question: Un polygone étant donné, de combien de manières peut-on le partager en triangles au moyen de diagonales? ; Addition à la Note sur une Équation aux différences finies ; and Mémoire sur la réduction d'une classe d'intégrales multiples .

Catalan was keen to return to Paris and he applied for the professorship at the École Gratuite de Dessin which had become vacant on the death of his former professor Jean-Baptiste-Omer Lavit in 1836. He was unsuccessful. He wrote to Liouville in January 1837 who replied encouraging him to come to Paris in the Easter holidays to discuss his future with him. He suggested that Catalan needed further qualifications, such as the baccalaureate, in order to make a successful career. Later that year, Catalan resigned his position at Châlons-sur-Marne and returned to Paris. Along with Charles-François Sturm and Joseph Liouville, he founded the École Sainte-Barbe near the Sorbonne. The school, whose aim was to train pupils for admission to the École Polytechnique, opened in 1838. In the same year, on 20 November, he was appointed assistant tutor (répétiteur) in descriptive geometry at the École Polytechnique, assisting Charles François Antoine Leroy who was teaching this topic. Catalan lectured on part of the descriptive geometry course in 1840, the year in which his second daughter Fanny was born. In 1839 he had been appointed as a deputy examiner at the École Polytechnique. As soon as he had returned to Paris, in addition to these other tasks, Catalan had taken Liouville's advice regarding his baccalaureate, and began studying hard. He was awarded a double baccalaureate in 1839 and continued to study, receiving his "licence in mathematical sciences" in the following year. He was awarded his doctorate in mathematics in 1841 for his main thesis in Mechanics Attraction d'un ellipsoïde homogène sur un point extérieur ou sur un point intérieur , and a further thesis Sur le mouvement des étoiles doubles in Astronomy. In 1845, he was awarded a "licence in physical sciences".

Augustin-Louis Cauchy, a fervent Royalist, had left Paris in 1830 and, failing to swear an oath of allegiance to the new regime, had lost all his positions there. However, he returned to Paris in October 1838 regaining his position at the Académie des Sciences but not his teaching positions because he still refused to take an oath of allegiance. In July 1839 Cauchy invited Catalan and Lejeune Dirichlet to dinner. Although this was an honour for the young Catalan, given that their political views were at the opposite ends of the spectrum, this was never going to develop into a close friendship. Catalan was, however, still supported by Liouville who proposed him for membership of the Société Philomatique. After being unsuccessful in 1839, Catalan was elected to the Society in May 1840 and began publishing articles in the Society's journal. In 1841 he gave the general change-of-variable theorem for n-dimensional integrals in Sur la transformation des variables dans les integrales multiples which was published in the Mémoires Couronnés of the Academie Royale des Sciences et Belles-Lettres de Bruxelles. In 1843 he published for the first time in Crelle's Journal für die reine und angewandte Mathematik. This contains the 'Catalan Conjecture':
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 $x^{m} - y^{n} = 1$ in which the unknowns are positive integers only admits a single solution.
Piotr Tchihatchef was a Russian geographer and geologist who was spending the winter of 1842-43 in Paris. He approached Louis Francoeur asking to be tutored in mathematics and Francoeur advised him to approach Catalan. This in fact proved fortunate for Catalan since Tchihatchef had been given a paper by the Russian mathematician Pafnuty 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. Chebyshev's paper appeared in Liouville's Journal de mathématiques pures et appliquées in May 1843. It was followed in the same journal by a short paper by Catalan proving a formula for the transformation of a multiple integral stated without proof in Chebyshev's paper. On 6 January 1844 Catalan was appointed as secretary of the Société Philomatique.

Given his achievements, Catalan's university teaching career should have been progressing but any prospects of a smooth path was damaged by his continuing political activities and his strong left-wing Republican views. In November 1844 he was ranked first by the Council for the position of tutor at the École Polytechnique but Ossian Bonnet was appointed to the position and Catalan remained as an assistant tutor. Adolphe Quetelet, the Belgium scientist with whom he had corresponded since his 1841 Mémoires couronnés in the Academie Royale des Sciences et Belles-Lettres de Bruxelles, encouraged him to seek a university position in Belgium because of his lack of progress in Paris. He had received another setback in his career in December 1844 when, after Francoeur had indicated that he wanted Catalan as his deputy at the Sorbonne, he had been rejected by the Royal Council. Instead, they appointed Joseph Bertrand to the position. In 1846 Catalan competed in the Concours d'agrégation (a contest to become a university teacher) and was placed first. However, he was given no appointment. Catalan was paying a high price for his strong, publicly expressed Republican views.

In September 1846 Catalan wrote to Quetelet indicating that he was beginning to regret having remained in Paris instead of following his suggestion to move to Belgium. However, in November he 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. This revolution materialised without any very definite cause, although food shortages from 1846 onwards had caused much economic trouble and discontent of which the Republicans took advantage. The trigger appears to have been the cancellation by the government of a major banquet arranged for February 1848 by the Republicans. Catalan took a highly active role in the disturbances, leading a band of workers who entered the Hôtel-de-Ville immediately after a group of students from the École Polytechnique. 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 he offered himself for election to the National Assembly. Here is an extract from his manifesto:
I was brought up to be contemptuous of kings and I greeted the fall of Charles X rapturously. I submitted, but protesting with indignation, to the master schemers and dupes we imposed on ourselves at that time. I myself was fooled for a moment, but I stopped being fooled at the retreat of the virtuous Dupont (de l'Eure), and before the end of 1830, I was a Republican. Since then, I have not missed a single day without contributing my wishes, my words and deeds, to the overthrow of the man that France supported too long!
You can see the full manifesto at THIS LINK.

His career seemed to be going well again when he was appointed to the Lycée Saint Louis in 1849. At the École Polytechnique, however, he continued to be disappointed. Two admission examiners were appointed in 1848 and, although Catalan was on the list of four offered to the Ministry, Charles Hermite and Joseph Serret were appointed with Abel Transon and Pierre Bonnet as their deputies. Although Transon (1805-1876), like Catalan, was a socialist, he took little part in politics after the age of thirty.

In June 1850 a commission was chaired by Urbain Le Verrier to reorganise teaching at the École Polytechnique. Jean-Victor Poncelet and Jean-Marie Duhamel were the mathematicians on the commission. It reported in November and Catalan did not like either the academic changes that were proposed, particularly the considerable reduction in pure mathematics, nor did he like the fact that much more work was to be undertaken by the tutors. He resigned his position as assistant tutor later that month. In his letter of resignation he pointed out that he had been an assistant tutor for twelve years, and also explained that he would be unable to undertake the extra teaching required due to his duties at the Lycée Saint Louis. Catalan was not the only one to leave the École Polytechnique following the report of Le Verrier's commission; Michel Chasles, Joseph Liouville and Charles-François Sturm also immediately resigned in protest.

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 disliked the Bonapartes as much as he disliked kings. He 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. He had published many articles in Liouville's Journal de mathématiques pures et appliquées but in 1854 he stopped publishing there, preferring to publish in the Comptes rendus of the Académie des Sciences. This was, almost certainly, part of Catalan's strategy to get elected to the Académie des Sciences. For example, he published Sur des surfaces dont les rayons de courbure en chaque point sont égaux et de signes contraires (1855) which looks at the 'Catalan minimal surface' associated with 'Catalan's minimal curve'. He also published in Comptes rendus: Note de M Catalan à l'occasion d'un théorème de M Serret (1856), Sur le calcul de la latitude par la méthode de M Babinet (1856), Sur quelques points de la théorie des séries (1856), Sur un cas particulier de la formule du binôme (1857), and Sur la théorie des développées (1857). By 1860 Bertrand, Hermite and Serret had all been elected to the Academy. When Serret was elected in 1860, Catalan was on the list to be considered but was only ranked fourth equal, behind Serret, Bonnet and Puiseux. The Academy proposed a question on polyhedra for the 1861 prize and Catalan submitted an entry. He was asked to submit a new version of his memoir in 1862 which was considered by a panel consisting of Bertrand, Chasles, Liouville and Serret. Chasles and Liouville recommended that he receive the prize, but the others proposed that the prize should not be awarded.

In 1859 Catalan had tried to persuade the Ministry of Public Instruction to name him as a Professor of Mathematics at one of the lycées in Paris. He was not successful. In January 1865, having had no permanent position for 13 years, he was appointed to the chair of mathematics at the University of Liège. 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, Catalan always considered himself French, having earlier undergone considerable efforts to reinstate his French citizenship. He held this chair until he retired in 1884, then continued to live in Liège until his death ten years later. However, tragedy struck Catalan's family soon after they arrived in Liège. He arrived there with his wife, two daughters, and his mother-in-law Louise Perin who had lived with the family throughout their married life in Paris. Some biographies state that Catalan also had a son who died in Paris before the family moved to Liège but we have been unable to confirm whether this is true. Marie, their eldest child, was already ill before the family moved to Liège and she died there in May 1865. By the end of this year, his mother-in-law had died at the age of 79, while his second daughter, Fanny, also became ill. Catalan's wife took Fanny back to Paris in the hope that this would see her health improve but it was not to be and Fanny died in the spring of 1866.

The Franco-Prussian war of 1870-71 did not affect neutral Belgium but Catalan, with his great love of France, was greatly saddened by the humiliation of the French by the Prussians. However, the resulting fall of Napoleon and the setting up of the Third Republic of France certainly gave him pleasure. Catalan had never respected the Légion d'Honneur which had been created by Napoleon Bonaparte in 1802. However, when the Third Republic awarded him the Knight's Cross of the Légion d'Honneur on 30 December 1887, his joy at France now being a Republic was greater than his dislike of "Bonaparte's honour" and he gladly accepted the honour. The Académie des Sciences, however, discussed at length electing Catalan during the 1870s but failed to do so.

Mélanges mathématiques was published by the Royal Society of Sciences of Liège in 1868. It contains 69 of Catalan's papers starting from Sur les combinaisons avec répétition (1838) and ending with his paper Démonstration d'une formule de Poisson (February 1867). In 1887-88 an updated version of Mélanges mathématiques in three volumes was published by the Royal Society of Sciences of Liège. This work contained 299 of Catalan's papers, the last paper being Sur une application du théorème de Bayes, faite par Laplace (August 1888). The second volume contains Catalan's retirement speech and the third volume also lists Catalan's publications; the list contains 406 items.

Catalan was elected to the Belgian Académie Royale des Sciences, des Lettres et des Beaux Arts on 15 December 1865. He was also elected to the Académie des Sciences de Toulouse and the Société des Sciences de Lille. He was a corresponding member of the St Petersburg Academy of Sciences, the Turin Academy of Sciences, the Accademia Pontificia dei Nuovi Lincei, the Mathematical Society of Amsterdam, the National Institute of Geneva, the Societé Havraise d'Études Divers and the Societé d'Agriculture de la Marne. He was a member of the Science Society of Liège, the Mathematical Society of France and, as we mentioned above, the Société Philomatique of Paris. The Belgium government honoured Catalan by awarding him the Cross of the Knight of the Order of Léopold in 1879. Catalan retired on 1 June 1884, becoming professor emeritus. His retirement was marked on 7 December of that year by a celebration attended by many of his friends (including Chebyshev who had come from St Petersburg especially for the occasion), colleagues and former students. He was presented with his portrait, commissioned by some of his friends, painted by Émile Delpérée. In this portrait, Catalan is proudly wearing the Knight's Cross of the Order of Léopold. In 1890 the Belgium government elevated Catalan to an Officer of the Order of Léopold.

You can see Catalan's speech at his retirement THIS LINK.

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 with the last appearing in 1881; Théorèmes et problèmes de géométrie élémentaire (1852) which ran to 6 editions with the last appearing in 1879; 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 6 editions; Traité élémentaire des séries (1860); Histoire d'un concours (1865) with a second edition published in 1867; and Cours d'analyse de l'université de Liège (1870) with a second edition published in 1880.

In 1894 the École Polytechnique planned to celebrate the centenary of its founding. Catalan, one of the oldest living former pupils, planned a final visit to Paris to take part in the celebrations. However, it was not to be. His wife became ill at the beginning of February 1894 then, on 9 February, Catalan collapsed and was taken to hospital suffering from pneumonia. His wife, Eugénie, died on 11 February and Catalan himself died three days later.

You can see an appreciation of Catalan by the rector of the University of Liège at THIS LINK.

### References (show)

1. E-C Catalan, Mélanges mathématiques (F Hayes, Brussels, 1885-88).
2. E-C Catalan, Notice sur les travaux scientifiques (Gauthier-Villars, Paris, 1875).
3. F Jongmans, Eugène Catalan : Géomètre sans patrie (Société Belge des Professeurs de Mathématiques d'Expression Française, Mons, 1996).
4. P L Butzer and F Jongmans, Eugène Catalan and the rise of Russian science, Acad. Roy. Belg. Bull. Cl. Sci. (6) 2 (1-3) (1991), 59-90.
5. P L Butzer, L Carbone, F Jongmans and F Palladino, Les relations épistolaires entre Eugène Catalan et Ernesto Cesàro, Acad. Roy. Belg. Bull. Cl. Sci. (6) 10 (7-12) (1999), 223-271.
6. E-C Catalan, Lettres a quelques mathématiciens (F Hayes, Brussels, 1891).
7. E-C Catalan, Nouvelle correspondance mathematique (F Hayes, Brussels, 1878).
8. P Hilton and J Pedersen, Catalan numbers, their generalization, and their uses, Math. Intelligencer 13 (2) (1991), 64-75.
9. F Jongmans, Quelques pièces choisies dans la correspondance d'Eugène Catalan, Bull. Soc. Roy. Sci. Liège 50 (9-10) (1981), 287-309.
10. P J Larcombe and P D C Wilson, On the trail of the Catalan sequence, Math. Today (Southend-on-Sea) 34 (4) (1998), 114-117.
11. Les travaux mathématique de Eugène-Charles Catalan, Annuaire de L'Académie Royale des Sciences, des Lettres et Beau-Arts de Belgique (Brussels, 1896), 115-172.
12. P Mansion, Notice sur les travaux mathématiques de Eugène-Charles Catalan (F Hayes, Brussels, 1896).
13. P Mansion, Discours sur les travaux mathématiques de M Eugène-Charles Catalan (F Hayes, Brussels, 1885).