Charles Hermite

Hermite was born in Dieuze, Moselle, on 24 December 1822,^[2] with a deformity in his right foot that would impair his gait throughout his life. He was the sixth of seven children of Ferdinand Hermite and his wife, Madeleine née Lallemand. Ferdinand worked in the drapery business of Madeleine's family while also pursuing a career as an artist. The drapery business relocated to Nancy in 1828, and so did the family.^[3]

Hermite obtained his secondary education at Collège de Nancy and then, in Paris, at Collège Henri IV and at the Lycée Louis-le-Grand.^[2] He read some of Joseph-Louis Lagrange's writings on the solution of numerical equations and Carl Friedrich Gauss's publications on number theory.

Hermite wanted to take his higher education at École Polytechnique, a military academy renowned for excellence in mathematics, science, and engineering. Tutored by mathematician Eugène Charles Catalan, Hermite devoted a year to preparing for the notoriously difficult entrance examination.^[3] In 1842 he was admitted to the school.^[2] However, after one year the school would not allow Hermite to continue his studies there because of his deformed foot. He struggled to regain his admission to the school, but the administration imposed strict conditions. Hermite did not accept this, and he quit the École Polytechnique without graduating.^[3]

After spending five years working privately towards his degree, in which he befriended eminent mathematicians Joseph Bertrand, Carl Gustav Jacob Jacobi, and Joseph Liouville, he took and passed the examinations for the baccalauréat, which he was awarded in 1847. He married Bertrand's sister, Louise, in 1848.^[3]

In 1848, Hermite returned to the École Polytechnique as répétiteur and examinateur d'admission. In July 1848, he was elected to the French Academy of Sciences. In 1856 he contracted smallpox. Through the influence of Augustin-Louis Cauchy and of a nun who nursed him, he resumed the practice of his Catholic faith.^[2] From 1862 to 1873 he was lecturer at the École Normale Supérieure. In 1869, he succeeded Jean-Marie Duhamel as professor of mathematics, both at the École Polytechnique, where he remained until 1876, and at the University of Paris, where he remained until his death. Upon his 70th birthday, he was promoted to grand officer in the French Legion of Honour.^[2]

He was elected to honorary membership of the Manchester Literary and Philosophical Society in 1892.^[4] on the same date as Charles Friedel, also of the Sorbonne.

On his last days Hermite suffered from a bout of asthma, a weak appetite, and poor sleep.^[5] He died in Paris on 14 January 1901, aged 78.^[2]

In 1842, Nouvelles Annales de Mathématiques published Hermite's first original contribution to mathematics, a simple proof of Niels Henrik Abel's proposition concerning the impossibility of an algebraic solution to equations of the fifth degree, the Abel–Ruffini theorem.^[2]

Hermite was deeply interested in Abelian and elliptic functions, and, with encouragement from Joseph Liouville, corresponded with Carl Gustav Jacob Jacobi on that these topics.^[1] This resulted in the complete edition of Jacobi's works, of two articles by Hermite, one concerning the extension to Abelian functions of one of the theorems of Abel on elliptic functions, and the other concerning the transformation of elliptic functions.^[6]

In a series of papers published in 1858, 1865, and 1866, Hermite showed that roots of the quintic polynomial could be obtained using elliptic integrals.^[1]

In higher algebra, Hermite, Arthur Cayley, and James Joseph Sylvester simultaneously developed the theory of invariants, where Hermite discovered the law of reciprocity.^[1]

In 1873, he became the first person to prove that e, the base of the natural logarithm, is transcendental.^[3] Techniques similar to those he employed in this proof were later used by Ferdinand von Lindemann in 1882 to show the same for π.^[1]

Later in his life, Hermite turned his attention to the calculus, in particular, the theory of linear differential equations, and found solutions to Lame's equation.^[1]

The following is a list of his works:^[6]

• "Sur quelques applications des fonctions elliptiques", Paris, 1855; page images from Cornell.
• "Cours d'Analyse de l'École Polytechnique. Première Partie", Paris: Gauthier–Villars, 1873.
• "Cours professé à la Faculté des Sciences", edited by Andoyer, 4th ed., Paris, 1891; page images from Cornell.
• "Correspondance", edited by Baillaud and Bourget, Paris, 1905, 2 vols.; PDF copy from UMDL.
• "Œuvres de Charles Hermite", edited by Picard for the Academy of Sciences, 4 vols., Paris: Gauthier–Villars, 1905,^[7] 1908,^[8] 1912^[9] and 1917; PDF copy from UMDL.
• "Œuvres de Charles Hermite", reissued by Cambridge University Press, 2009; ISBN 978-1-108-00328-5.

There exists, if I am not mistaken, an entire world which is the totality of mathematical truths, to which we have access only with our mind, just as a world of physical reality exists, the one like the other independent of ourselves, both of divine creation.

I shall risk nothing on an attempt to prove the transcendence of π. If others undertake this enterprise, no one will be happier than I in their success. But believe me, it will not fail to cost them some effort.

While speaking, M. Bertrand is always in motion; now he seems in combat with some outside enemy, now he outlines with a gesture of the hand the figures he studies. Plainly he sees and he is eager to paint, this is why he calls gesture to his aid. With M. Hermite, it is just the opposite, his eyes seem to shun contact with the world; it is not without, it is within he seeks the vision of truth.

Reading one of [Poincare's] great discoveries, I should fancy (evidently a delusion) that, however magnificent, one ought to have found it long before, while such memoirs of Hermite as the one referred to in the text arouse in me the idea: “What magnificent results! How could he dream of such a thing?”

I turn with terror and horror from this lamentable scourge of continuous functions with no derivatives.

In addition to the mathematics properties named in his honor, the Hermite crater near the Moon's north pole is named after Hermite.

• List of things named after Charles Hermite
• Classical orthogonal polynomials