Charles Hermite

CHARLES HERMITE

Born: 24 Dec 1822 in Dieuze, Lorraine, France
Died: 14 Jan 1901 in Paris, France

---

Charles Hermite's father was Ferdinand Hermite and his mother was Madeleine Lallemand. Ferdinand Hermite was a trained engineer and he worked in this capacity in a salt mine near Dieuse. After he married Madeleine he joined in the draper's trade in which her family were involved. However he was an artistic man who always wanted to pursue art as a career. He had his wife look after the draper's business and he took up art. Charles was the sixth of his parents seven children and when he was about seven years old his parents left Dieuse and went to live in Nancy to where the business had moved.

Education was not a high priority for Charles's parents but despite not taking too much personal interest in their children's education, nevertheless they did provide them with good schooling. Charles was something of a worry to his parents for he had a defect in his right foot which meant that he moved around only with difficulty. It was clear that this would present him with problems in finding a career. However he had a happy disposition and bore his disability with a cheerful smile.

Charles attended the Collège de Nancy, then went to Paris where he attended the Collège Henri. In 1840-41 he studied at the Collège Louis-le-Grand where some fifteen years earlier Galois had studied. In fact he was taught mathematics there by Louis Richard who had taught Galois. In some ways Hermite was similar to Galois for he preferred to read papers by Euler, Gauss and Lagrange rather than work for his formal examinations.

If Hermite neglected the studies that he should have concentrated on, he was showing remarkable research ability publishing two papers while at Louis-le-Grand. Also like Galois he was attracted by the problem of solving algebraic equations and one of the two papers attempted to show that the quintic cannot be solved in radicals. That he was unfamiliar with Galois's contributions, despite being at the same school, is not at all surprising since the mathematical community were completely unaware of them at this time. However he might reasonably have known of the contributions of Ruffini and Abel to this question, but apparently he did not.

Again like Galois, Hermite wanted to study at the École Polytechnique and he took a year preparing for the examinations. He was tutored by Catalan in 1841-42 and certainly Hermite fared better than Galois had done for he passed. However it was not a glorious pass for he only attained sixty-eighth place in the ordered list. After one year at the École Polytechnique Hermite was refused the right to continue his studies because of his disability. Clearly this was an unfair decision and some important people were prepared to take up his case and fight for him to have the right to continue as a student at the École Polytechnique. The decision was reversed so that he could continue his studies but strict conditions were imposed. Hermite did not find these conditions acceptable and decided that he would not graduate from the École Polytechnique.

Hermite made friends with important mathematicians at this time and frequently visited Joseph Bertrand. On a personal note this was highly significant for he would marry Joseph Bertrand's sister. More significantly from a mathematical point of view he began corresponding with Jacobi and, despite not shining in his formal education, he was already producing research which was ranking as a leading world-class mathematician. The letters he exchanged with Jacobi show that Hermite had discovered some differential equations satisfied by theta-functions and he was using Fourier series to study them. He had found general solutions to the equations in terms of theta-functions. Hermite may have still been an undergraduate but it is likely that his ideas from around 1843 helped Liouville to his important 1844 results which include the result now known as Liouville's theorem.

After spending five years working towards his degree he took and passed the examinations for the baccalauréat and licence which he was awarded in 1847. In the following year he was appointed to the École Polytechnique, the institution which had tried to prevent him continuing his studies some four years earlier; he was appointed répétiteur and admissions examiner.

Hermite made important contributions to number theory and algebra, orthogonal polynomials, and elliptic functions. He discovered his most significant mathematical results over the ten years following his appointment to the École Polytechnique. In 1848 he proved that doubly periodic functions can be represented as quotients of periodic entire functions. In 1849 Hermite submitted a memoir to the Académie des Sciences which applied Cauchy's residue techniques to doubly periodic functions. Sturm and Cauchy gave a good report on this memoir in 1851 but a priority dispute with Liouville seems to have prevented its publication.

Another topic on which Hermite worked and made important contributions was the theory of quadratic forms. This led him to study invariant theory and he found a reciprocity law relating to binary forms. With his understanding of quadratic forms and invariant theory he created a theory of transformations in 1855. His results on this topic provided connections between number theory, theta functions, and the transformations of abelian functions.

On 14 July 1856 Hermite was elected to the Académie des Sciences. However, despite this achievement, 1856 was a bad year for Hermite for he contracted smallpox. It was Cauchy who, with his strong religious conviction, helped Hermite through the crisis. This had a profound effect on Hermite who, under Cauchy's influence, turned to the Roman Catholic religion. Cauchy was also a very staunch royalist and Hermite was influenced by him to also become a royalist. We made comparisons with Galois earlier on in this article, but with royalist views, Hermite was now completely opposed to the views which the staunch republican Galois had held.

The next mathematical result by Hermite which we must mention is one for which he is rightly famous. Although an algebraic equation of the fifth degree cannot be solved in radicals, a result which was proved by Ruffini and Abel, Hermite showed in 1858 that an algebraic equation of the fifth degree could be solved using elliptic functions. He applied these results to number theory, in particular to class number relations of quadratic forms.

In 1862 Hermite was appointed maître de conférence at the École Polytechnique, a position which had been specially created for him. In the following year he became an examiner there. The year 1869 saw him become a professor when he succeeded Duhamel as professor of analysis both at the École Polytechnique and at the Sorbonne. Hermite resigned his chair at the École Polytechnique in 1876 but continued to hold the chair at the Sorbonne until he retired in 1897. In the 1890s Hermite became much less interested in the new results found by the mathematicians of the next generation.

The 1870s saw Hermite return to problems which had interested him earlier in his career such as problems concerning approximation and interpolation. In 1873 Hermite published the first proof that e is a transcendental number. This is another result for which he is rightly famous. Using method's similar to those of Hermite, Lindemann established in 1882 that π was also transcendental. Many historians of science regret that Hermite, despite doing most of the hard work, failed to use it to prove the result on which would have brought him fame outside the world of mathematics. Hermite is now best known for a number of mathematical entities that bear his name: Hermite polynomials, Hermite's differential equation, Hermite's formula of interpolation and Hermitian matrices.

For Hermite certain areas of mathematics were much more interesting than other areas. Hadamard, who unlike his teacher Hermite worked in all areas of mathematics, spoke of Hermite's dislike for geometry:-

[Hermite] had a kind of positive hatred of geometry and once curiously reproached me with having made a geometrical memoir.

Hermite's great love was for analysis and, not surprisingly, he had a great respect for Weierstrass. When Mittag-Leffler arrived in Paris to study with him, Hermite greeted him warmly but said:-

You have made a mistake, sir, you should follow Weierstrass's course in Berlin. He is the master of us all.

Poincaré is almost certainly the best known of Hermite's students. He once suggested that Hermite's mind did not proceed in logical fashion. He wrote:-

But to call Hermite a logician! Nothing can appear to me more contrary to the truth. Methods always seemed to be born in his mind in some mysterious way.

Hadamard like Poincaré was very interested in the way that mathematics was discovered. He also had this to say about the way that Hermite made his discoveries:-

Hermite used to observe [that biology] may be a most useful study even for mathematicians, as hidden and eventually fruitful analogies may appear between processes in both kinds of studies.

Hadamard had great respect for Hermite as a teacher. He said:-

I do not think that those who never listened to him can realise how magnificent Hermite's teaching was, overflowing with enthusiasm for science, which seemed to come to life in his voice and whose beauty he never failed to communicate to us, since he felt it so much himself to the very depth of his being.

[Hermite] was making a deep impression on us, not only with his methods and those of Weierstrass, but also with his enthusiasm and love of science; in our brief but fruitful conversations, Hermite loved to direct to me remarks such as: "He who strays from the paths traced by providence crashes." These were the words of a profoundly religious man, but an atheist like me understood them very well, especially when he added at other times: "In mathematics, our role is more of servant than of master." It goes without saying that gradually, as years and my scientific work unfolded, I came to understand more and more deeply the aptness and scope of his words.

Cross, reviewing [10] where 125 letters from Hermite to Mittag-Leffler are reproduced, writes:-

So there stands revealed one of the most engaging and influential men in Parisian and French mathematics in the second half of the 19th century, one might even say the central character for the period in which he published, 1842-1901. What radiates from the text is [Hermite's] humility, his Catholicism, his concern for his (very extended) family, his willingness to fight for colleagues whose merit he discerns, and his devotion to family, merit, and principle rather than simple influence.

In terms of his family life Hermite had married Louise Bertrand, Joseph Bertrand's sister. One of their two daughters married Émile Picard. Struik writes:-

Hermite lived a retired life, with his family. His working hours were devoted to mathematical research and teaching. His outlook on mathematics was realistic in the Platonic sense: a mathematician, like a naturalist, discovers an outside world, in his case a world of ideas. Hermite, therefore, disliked Cantor's world, in which a new mathematical world was created.

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

Jean Baptiste Joseph Fourier

Jean Baptise Joesph Fourier

Joseph Fourier's father was a tailor in Auxerre. After the death of his first wife, with whom he had three children, he remarried and Joseph was the ninth of the twelve children of this second marriage. Joseph's mother died went he was nine years old and his father died the following year.

His first schooling was at Pallais's school, run by the music master from the cathedral. There Joseph studied Latin and French and showed great promise. He proceeded in 1780 to the École Royale Militaire of Auxerre where at first he showed talents for literature but very soon, by the age of thirteen, mathematics became his real interest. By the age of 14 he had completed a study of the six volumes of Bézout's Cours de mathématiques. In 1783 he received the first prize for his study of Bossut's Mécanique en général.

In 1787 Fourier decided to train for the priesthood and entered the Benedictine abbey of St Benoit-sur-Loire. His interest in mathematics continued, however, and he corresponded with C L Bonard, the professor of mathematics at Auxerre. Fourier was unsure if he was making the right decision in training for the priesthood. He submitted a paper on algebra to Montucla in Paris and his letters to Bonard suggest that he really wanted to make a major impact in mathematics. In one letter Fourier wrote

Yesterday was my 21st birthday, at that age Newton and Pascal had already acquired many claims to immortality.

Fourier did not take his religious vows. Having left St Benoit in 1789, he visited Paris and read a paper on algebraic equations at the Académie Royale des Sciences. In 1790 he became a teacher at the Benedictine college, École Royale Militaire of Auxerre, where he had studied. Up until this time there had been a conflict inside Fourier about whether he should follow a religious life or one of mathematical research. However in 1793 a third element was added to this conflict when he became involved in politics and joined the local Revolutionary Committee. As he wrote:-

As the natural ideas of equality developed it was possible to conceive the sublime hope of establishing among us a free government exempt from kings and priests, and to free from this double yoke the long-usurped soil of Europe. I readily became enamoured of this cause, in my opinion the greatest and most beautiful which any nation has ever undertaken.

Certainly Fourier was unhappy about the Terror which resulted from the French Revolution and he attempted to resign from the committee. However this proved impossible and Fourier was now firmly entangled with the Revolution and unable to withdraw. The revolution was a complicated affair with many factions, with broadly similar aims, violently opposed to each other. Fourier defended members of one faction while in Orléans. A letter describing events relates:-

Citizen Fourier, a young man full of intelligence, eloquence and zeal, was sent to Loiret. ... It seems that Fourier ... got up on certain popular platforms. He can talk very well and if he put forward the views of the Society of Auxerre he has done nothing blameworthy...

This incident was to have serious consequences but after it Fourier returned to Auxerre and continued to work on the revolutionary committee and continued to teach at the College. In July 1794 he was arrested, the charges relating to the Orléans incident, and he was imprisoned. Fourier feared the he would go to the guillotine but, after Robespierre himself went to the guillotine, political changes resulted in Fourier being freed.

Later in 1794 Fourier was nominated to study at the École Normale in Paris. This institution had been set up for training teachers and it was intended to serve as a model for other teacher-training schools. The school opened in January 1795 and Fourier was certainly the most able of the pupils whose abilities ranged widely. He was taught by Lagrange, who Fourier described as

the first among European men of science,

and also by Laplace, who Fourier rated less highly, and by Monge who Fourier described as

having a loud voice and is active, ingenious and very learned.

Fourier began teaching at the Collège de France and, having excellent relations with Lagrange, Laplace and Monge, began further mathematical research. He was appointed to a position at the École Centrale des Travaux Publiques, the school being under the direction of Lazare Carnot and Gaspard Monge, which was soon to be renamed École Polytechnique. However, repercussions of his earlier arrest remained and he was arrested again and imprisoned. His release has been put down to a variety of different causes, pleas by his pupils, pleas by Lagrange, Laplace or Monge or a change in the political climate. In fact all three may have played a part.

By 1 September 1795 Fourier was back teaching at the École Polytechnique. In 1797 he succeeded Lagrange in being appointed to the chair of analysis and mechanics. He was renowned as an outstanding lecturer but he does not appear to have undertaken original research during this time.

In 1798 Fourier joined Napoleon's army in its invasion of Egypt as scientific adviser. Monge and Malus were also part of the expeditionary force. The expedition was at first a great success. Malta was occupied on 10 June 1798, Alexandria taken by storm on 1 July, and the delta of the Nile quickly taken. However, on 1 August 1798 the French fleet was completely destroyed by Nelson's fleet in the Battle of the Nile, so that Napoleon found himself confined to the land that he was occupying. Fourier acted as an administrator as French type political institutions and administration was set up. In particular he helped establish educational facilities in Egypt and carried out archaeological explorations.

While in Cairo Fourier helped found the Cairo Institute and was one of the twelve members of the mathematics division, the others included Monge, Malus and Napoleon himself. Fourier was elected secretary to the Institute, a position he continued to hold during the entire French occupation of Egypt. Fourier was also put in charge of collating the scientific and literary discoveries made during the time in Egypt.

Napoleon abandoned his army and returned to Paris in 1799, he soon held absolute power in France. Fourier returned to France in 1801 with the remains of the expeditionary force and resumed his post as Professor of Analysis at the École Polytechnique. However Napoleon had other ideas about how Fourier might serve him and wrote:-

... the Prefect of the Department of Isère having recently died, I would like to express my confidence in citizen Fourier by appointing him to this place.

Fourier was not happy at the prospect of leaving the academic world and Paris but could not refuse Napoleon's request. He went to Grenoble where his duties as Prefect were many and varied. His two greatest achievements in this administrative position were overseeing the operation to drain the swamps of Bourgoin and supervising the construction of a new highway from Grenoble to Turin. He also spent much time working on the Description of Egypt which was not completed until 1810 when Napoleon made changes, rewriting history in places, to it before publication. By the time a second edition appeared every reference to Napoleon would have been removed.

It was during his time in Grenoble that Fourier did his important mathematical work on the theory of heat. His work on the topic began around 1804 and by 1807 he had completed his important memoir On the Propagation of Heat in Solid Bodies. The memoir was read to the Paris Institute on 21 December 1807 and a committee consisting of Lagrange, Laplace, Monge and Lacroix was set up to report on the work. Now this memoir is very highly regarded but at the time it caused controversy.

There were two reasons for the committee to feel unhappy with the work. The first objection, made by Lagrange and Laplace in 1808, was to Fourier's expansions of functions as trigonometrical series, what we now call Fourier series. Further clarification by Fourier still failed to convince them. As is pointed out in [4]:-

All these are written with such exemplary clarity - from a logical as opposed to calligraphic point of view - that their inability to persuadeLaplace and Lagrange ... provides a good index of the originality of Fourier's views.

The second objection was made by Biot against Fourier's derivation of the equations of transfer of heat. Fourier had not made reference to Biot's 1804 paper on this topic but Biot's paper is certainly incorrect. Laplace, and later Poisson, had similar objections.

The Institute set as a prize competition subject the propagation of heat in solid bodies for the 1811 mathematics prize. Fourier submitted his 1807 memoir together with additional work on the cooling of infinite solids and terrestrial and radiant heat. Only one other entry was received and the committee set up to decide on the award of the prize, Lagrange, Laplace, Malus, Haüy and Legendre, awarded Fourier the prize. The report was not however completely favourable and states:-

... the manner in which the author arrives at these equations is not exempt of difficulties and that his analysis to integrate them still leaves something to be desired on the score of generality and even rigour.

With this rather mixed report there was no move in Paris to publish Fourier's work.

When Napoleon was defeated and on his way to exile in Elba, his route should have been through Grenoble. Fourier managed to avoid this difficult confrontation by sending word that it would be dangerous for Napoleon. When he learnt of Napoleon's escape from Elba and that he was marching towards Grenoble with an army, Fourier was extremely worried. He tried to persuade the people of Grenoble to oppose Napoleon and give their allegiance to the King. However as Napoleon marched into the town Fourier left in haste.

Napoleon was angry with Fourier who he had hoped would welcome his return. Fourier was able to talk his way into favour with both sides and Napoleon made him Prefect of the Rhône. However Fourier soon resigned on receiving orders, possibly from Carnot, that the was to remove all administrators with royalist sympathies. He could not have completely fallen out with Napoleon and Carnot, however, for on 10 June 1815, Napoleon awarded him a pension of 6000 francs, payable from 1 July. However Napoleon was defeated on 1 July and Fourier did not receive any money. He returned to Paris.

Fourier was elected to the Académie des Sciences in 1817. In 1822 Delambre, who was the Secretary to the mathematical section of the Académie des Sciences, died and Fourier together with Biot and Arago applied for the post. After Arago withdrew the election gave Fourier an easy win. Shortly after Fourier became Secretary, the Académie published his prize winning essay Théorie analytique de la chaleur in 1822. This was not a piece of political manoeuvring by Fourier however since Delambre had arranged for the printing before he died.

During Fourier's eight last years in Paris he resumed his mathematical researches and published a number of papers, some in pure mathematics while some were on applied mathematical topics. His life was not without problems however since his theory of heat still provoked controversy. Biot claimed priority over Fourier, a claim which Fourier had little difficulty showing to be false. Poisson, however, attacked both Fourier's mathematical techniques and also claimed to have an alternative theory. Fourier wrote Historical Précis as a reply to these claims but, although the work was shown to various mathematicians, it was never published.

Fourier's views on the claims of Biot and Poisson are given in the following, see [4]:-

Having contested the various results [Biot and Poisson] now recognise that they are exact but they protest that they have invented another method of expounding them and that this method is excellent and the true one. If they had illuminated this branch of physics by important and general views and had greatly perfected the analysis of partial differential equations, if they had established a principal element of the theory of heat by fine experiments ... they would have the right to judge my work and to correct it. I would submit with much pleasure .. But one does not extend the bounds of science by presenting, in a form said to be different, results which one has not found oneself and, above all, by forestalling the true author in publication.

Fourier's work provided the impetus for later work on trigonometric series and the theory of functions of a real variable.