# Guillaume Jules Hoüel

### Born: 7 April 1823 in Thaon, Calvados, France

Died: 14 June 1886 in Périers sur le Dan (near Caen), France

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The first thing to discuss is the spelling of

**Jules Hoüel**'s surname. One finds the spelling Houel, Hoüel, Hoûel or Houël but deciding which is 'correct' shows immediately the problem. On his birth certificate his name appears as Houël but it appears as Hoüel on his baccalaureate certificate, and all his published papers and books give the Hoüel version. When he signed his name it usually looks like Hoüel but sometimes it appears as Hoûel. Some official documents show his name as Houel. We believe, therefore, that the correct spelling of the family name is Houël but the subject of this biography preferred the Hoüel version. We have chosen to use Hoüel which appears to be in line with Jules Hoüel's wishes but use the Houël version when giving details of his family.

Jules Hoüel was the son of Jean Baptiste Houël (1795-1854) and Elisabeth Amicie Tranchant (1805-1877). Jean Baptiste Houël, the son of Louis Nicolas Houël (1758-1832), mayor of Thaon, and Anne Donnet (1758-1832), had intended to study at the École Polytechnique but stopped his studies because of the political situation. He went to his family home in Bombanville and lived there as a country gentleman, painting and drawing a great deal. In 1822, he married Elisabeth Amicie Tranchant, the daughter of Dr François Tranchant (1771-1842), a doctor of medicine, and Mary Magdalene Bonne Artur de Hauterue (1778-1857), heiress and owner of Castle of Périers sur le Dan near Thaon. After marrying, Jean Baptiste and Elisabeth lived sometimes at Bombanville, sometimes at Périers at their parents' home, but by 1823, around the time Jules was born, they only lived at Bombanville. In 1832, when Jules was nine years old, his brother Louis Edmond Houël (1823-1904) was born.

As a young child, Jules had very poor health, suffered severe headaches and swelling of his glands. His health was so poor that he could not attend school until he was fifteen years old. He was taught at home, first by his grandfather Louis Nicolas Houël but after his death, Jules continued to be taught by his father. Jean Baptiste Houël wrote in a letter of 1841 (see [3]):-

His father also taught Jules to play the flute and to paint.I pushed my son to mathematics as far as the studies I had made when I was planning to enter the École polytechnique, while my father, now deceased, came to the point that he did also progress in the Latin language and translated with enough ease authors such as Virgil, Ovid, Cicero[...]what I say is not to boast certainly, but only to let you know that with his grandfather, who was his first master, and then me, he did not quite waste his time.

In October 1838 Jules began his schooling, entering the third grade of the Royal College of Caen. He boarded at the school during the week, returning home to Bombanville each weekend. In his first year he studied Latin, Greek, French, history, and cosmography. In his second year French, Latin, mathematics, physics and natural history. As might be expected, his best subjects were mathematics and physics. He was awarded his baccalaureate in 1841 and decided to spend two years preparing to sit the entrance examinations of the École Normale Supérieure.

Hoüel entered the Collège Rollin in Paris on 29 October 1841. He was taught mathematics by M Faurie who had been appointed in October 1840. Faurie's reports show that Hoüel began his studies well and made excellent progress. Later, however, his progress seems to have been less good. Faurie reported [3]:-

Hoüel sat the written examinations for entry to the École Normale Supérieure in August 1843 and on 15 October of that year he was given an oral examination. The first students ranked in the top six received free education at the École and the next six had to pay half the tuition fees. Hoüel, ranked eighth, was admitted with a half-fees scholarship.This student is endowed with a fair and penetrating mind, but his memory is relatively less good. The work was generally less well done this semester than in the previous one: it mainly weakened in physics. However, his progress, without being what one could hope for, is still satisfactory, and we would be very much mistaken if he did not succeed in his future examination.

In November 1843 he began his studies at the École Normale Supérieure. His first year was successful and he was ranked third in the end of year examinations. He seems to have worked less well in his second year, being ranked ninth. Two of the twelve who entered with Hoüel had dropped out, so he was ninth from ten students. In his third year of study, Hoüel did teaching practice which was at the Collège Rollin in M Faurie's class. Faurie's report is very positive and says that Hoüel will make a good teacher.

One of Hoüel's fellow pupils was Gaston Lespiault who wrote in [12]:-

While a student at the École Normale Supérieure, he attended mathematics courses taught by Jean Marie Constant Duhamel (analysis and mechanics), Louis Lefébure de Fourcy (differential and integral calculus) and Jacques Charles François Sturm (mechanics). Not all these courses were given at the École Normale Supérieure, for he also attended lectures at the Faculty of Science and at the École Polytechnique. As well as mathematics courses, he took courses on mineralogy, botany and zoology. Hoüel had always lacked confidence as a public speaker, something which had been first noted by his teacher Faurie when he was a student at the Collège Rollin. This led to him failing the aggregation in 1846 and he left the École Normale Supérieure without this qulification.In October1843, Jules Houël entered the Ecole Normale one of the first of his class. His education was as varied as it was extensive. It is assured that at the entrance examinations his composition in Latin was ranked first on the combined sections of science and literature. From the time of his stay at the school, his comrades were struck by the depth and originality of his ideas, his aspirations to the rigour, his distrust of anything vague. They also marvelled at his power of work, the art he already had and he always made sure never to lose a minute. It is to this period of his life that his first efforts go back to, his clear and rigorous theory of negative quantities, imaginary quantities and infinitely small ones. Today's young mathematicians hardly suspect what doubts and mistakes their elders were struggling with. If these obscurities have been dissipated, it is due to Houel more than to anyone.

After leaving the École Normale Supérieure, Hoüel was appointed as a mathematics teacher to the Collège de Bastia in Corsica in September 1846. This, however, would have meant that he was far from Paris and could not have continued his education there and retake the aggregation examinations as he wished to do. He tried to obtain a transfer and, before ever taking up the job in Bastia, he was appointed to the Royal College of Bourges in October 1846. On 21 August 1847 he sat the aggregation examinations in Paris and he was successful.

On 20 September 1847 he was appointed to the Royal College of Bordeaux. He spent two difficult years there. The report on his teaching in the first year described him as "zealous and very conscientious" but not carrying enough weight with the students. A second report states that "M Hoüel is undoubtedly an educated teacher, but he does not have the qualities needed to maintain discipline in a large class." The year 1848-49 went very badly and the headmaster of the Royal College of Bordeaux asked for Hoüel to be transferred to a school where the classes were much smaller. On 11 September 1849 Hoüel wrote to the Minister of Education asking to move to, or near, Paris [3]:-

The Minister chose not to send Hoüel to a post in or near Paris but instead assigned him a position at the Lycée de Pau, a much lower quality school than the one he had taught in. He wrote an angry letter to the Minister [3]:-Driven by a keen desire to increase my knowledge of mathematics, and to complete the work I have been doing for several years for the Ph.D., I would like to be in a position to take advantage of the courses of the Faculty of Sciences of Paris and the Collège de France, as well as the resources offered by libraries. If there were a vacancy among the supervising teaching posts at the École normale, and if you wanted me take on these duties, I would accept them gratefully.

Hoüel had also had a disappointment in his private life for during his last year in Bordeaux he had sought marriage to Antoinette Lucile Lecerf, the daughter of Daniel Henry Lecerf and Anne Jeanne Perrotte from Caen. Daniel Henry Lecerf refused to allow his daughter to marry Hoüel since he was not earning enough and lived too far from Caen.... instead of getting closer to my family, I am going to live in one of the most remote towns in Normandy; instead of obtaining a pecuniary improvement, I will not even find in Pau the resources offered me by Bordeaux.

At Pau, Hoüel received somewhat better reports for 1849-50 although his teaching was said to be too theoretical and he did not enthuse his pupils. The report for his second year, 1850-51 reads [3]:-

On 8 March 1851 Hoüel was appointed to the Lycée de Alençon. He wrote to Daniel Henry Lecerf saying that now he was nearer Caen, surely he could marry his daughter. The reports on his teaching at the Lycée de Alençon were similar to earlier ones, with his delivery [3]:-Houel, who had completely failed as a teacher at the Lycée de Bordeaux, succeeded more than I had hoped. The class is good but not enough precision and warmth in his teaching methods. Results obtained are quite satisfactory.

Lucile Lecerf's father reluctantly consented to the marriage of his daughter to Hoüel. A marriage contract was signed on 11 September 1851, and the wedding took place on 25 September 1851 in Caen. Jules and Lucile Hoüel began married life at 25 rue de Bretagne, Alençon.... a painful flow, always the same and everywhere monotonous, a slow and soft pronunciation, almost inarticulate.

The Minister of Education declared on 22 September 1852 that the higher mathematics classes at Alençon were to be closed and Hoüel was to be transferred to the Lycée de Poitiers to teach elementary mathematics. Hoüel, with support from several others, begged to be allowed to remain in Alençon. Finally, on 21 October 1852 the Minister agreed that he could remain at Alençon teaching elementary mathematics. Jules and Lucile Hoüel's first child, Jeanne, was born on 23 February 1853. They had three further children, all girls, Louise (born 1857), Marthe Julie Elizabeth (born 1858) and Henriette (born 1862). Hoüel continued teaching at Alençon and, at the same time, worked on his doctoral thesis.

On 18 August 1855, Hoüel submitted two theses to the Faculty of Science in Paris. The first was

*Sur l'intégration des équations différentielles dans les problèmes de mécanique*and the second thesis had the title

*Application de la méthode de M Hamilton au calcul des perturbations de Jupiter*. He was awarded the degree of Doctor of Mathematical Sciences. One month later he requested leave of absence from his teaching so he could obtain a position in higher education. He accepted, however, a teaching post at the Lycée in Caen but, like his other positions, it was reported that his teaching was unsatisfactory. He was unemployed from 27 March 1856.

Urbain Le Verrier was impressed with Hoüel's research work and tried to persuade him to accept a post at the Observatory in Paris. Hoüel, however, decided not to accept the offer and spent the next four years undertaking research at his home in Thaon. Between 1856 and 1859 he published about a dozen papers. Hoüel was appointed to the chair of pure mathematics in the Faculty of Science at Bordeaux in 1859 and held this post until his death. As Halsted points out in [11]:-

Hoüel published,Here he found dignity and facilities for work, and considered the position as final.

*Essai d'une exposition rationnelle des principes fondamentaux de la Géométrie élémentaire*, a work on geometry, in 1863. At this stage he did not know of the published work on non-euclidean geometry but he clearly was working his way towards the idea. He wrote:-

Hoüel became interested in non-euclidean geometry once he had been made aware of the work of János Bolyai and Nikolai Lobachevsky. He published translations of many important works by Bolyai, Beltrami, Helmholtz and Riemann. He corresponded with Joseph de Tilly on non-euclidean geometry. In [11] George Halsted explains how Hoüel obtained Bolyai's work:-Since long, the scientific researches of mathematicians on the fundamental principles of elementary geometry have concentrated themselves almost exclusively on the theory of parallels, and if, hitherto, the efforts of so many eminent minds have produced no satisfactory result, it is perhaps permitted to conclude thence that in pursuing these researches they have followed a false path and attacked an insoluble problem, of which the importance has been exaggerated in consequence of inexact ideas on the nature and origin of the primordial truths of the science of space.

At Hoüel's suggestion Schmidt collected material which enabled him to write the first biography of Bolyai which he did in 1868.... Hoüel's 'Essai' of1863having come by chance into the hands of a young architect of Temesvár in Hungary, this youth Franz Schmidt, desirous of continuing his mathematical studies wrote for counsel to Hoüel. Hoüel had answered helpfully, and later implored the aid of Schmidt to procure Bolyai's work ... Schmidt succeeded in procuring for Hoüel two copies of Bolyai's work. One Hoüel proceeded to translate himself, the other he sent to Battaglini, asking him to make known in Italy this wonderful idea.

Among his other researches, Hoüel compiled log tables

*Table de logarithmes à cinq décimales pour les nombres et pour les lignes trigonométriques, suivies des logarithmes d'addition et de soustraction et de diverses tables usuelles*Ⓣ (1858).

For a list of Hoüel's publications (including his many translations), see THIS LINK.

One important feature of Hoüel's work was his extensive correspondence with other mathematicians. Much of this is now published, for example Victor Pambuccian reviewing [2] writes:-

His obituary written by Gaston Darboux shows that Hoüel had a brilliant career as a professor, producing outstanding research papers and making a remarkable contribution to mathematics through his extensive correspondence. His brilliant academic career makes an interesting contrast to the first half of his life as a "failed school teacher."This is a very interesting historical document on the state of mind prevailing between1870and1885on two questions, that of the existence of continuous nowhere differentiable functions and that of non-Euclidean geometry. Following a detailed,80-page-long presentation of the terms of the debate in their historical context, the book consists of191letters between Jules Hoüel and Joseph Marie de Tilly,66letters between Hoüel and Darboux, and11letters between Hoüel and Victor-Amédée Le Besgue, written between1867and1868, as well as a few letters between Hoüel and Catalan, Kowalski, and Beltrami. Riemann, Lobachevsky, Bolyai, Weierstrass, Gauss, Legendre, Flye Sainte-Marie, and several others figure prominently in the correspondence.

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

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**List of References** (12 books/articles)

**Mathematicians born in the same country**

**Additional Material in MacTutor**

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