Charles Hermite (French pronunciation:[ʃaʁlɛʁˈmit]) FRSFRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who studied analysis, number theory, and algebra. One of his most remarkable achievements was the proof of the transcendence of the number e.[1]
Life
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 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]
Since his student days, Hermite was deeply interested in Abelian and elliptic functions,[6]:651 and, with encouragement from Joseph Liouville, corresponded with Carl Gustav Jacob Jacobi on that these topics.[1] This resulted in the inclusion, into 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.[2]
In 1864, Hermite presented a new class of special functions, Hermite polynomials, in the context of expansions in terms of continuous functions over unbounded intervals.[9]:574 Hermite functions, which are the product of Hermite polynomials and a Gaussian function, find application in quantum mechanics, wherein they arise as solutions to the Schrödinger equation for the quantum harmonic oscillator.[8]
In 1873, he published a lengthy paper demonstrating in two different ways that that e, the base of the natural logarithm, is transcendental,[10]:192–3 based on prior work by Joseph Liouville.[11]:127 In the same year, Hermite proved that and therefore are irrational.[8][12] However, he did not address the transcendence of , believing the question to be beyond his powers.[10]:193 But techniques similar to those he employed in this proof were later used by Ferdinand von Lindemann in 1882 to prove that result for π.[1] (Also see the Lindemann–Weierstrass theorem.) Hilbert subsequently simplified Hermite's original proof.[10]:196 In 1947, Ivan Niven exploited a technique of Hermite to give an elementary proof that is irrational.[13][14]
"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,[15] 1908,[16] 1912[17] and 1917; PDF copy from UMDL.
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.
—Charles Hermite; cit. by Gaston Darboux, Eloges académiques et discours, Hermann, Paris 1912, p. 142.
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.
—Henri Poincaré, INTUITION and LOGIC in Mathematics, Source: The Mathematics Teacher, MARCH 1969, Vol. 62, No. 3 (MARCH 1969), pp. 205-212
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?”
—Jacques Hadamard, The Mathematician's Mind: The Psychology of Invention in the Mathematical Field, p. 110
I turn with terror and horror from this lamentable scourge of continuous functions with no derivatives.
—Charles Hermite; letter to Thomas Joannes Stieltjes about the Weierstrass functions, everywhere continuous but nowhere differentiable, Correspondance d'Hermite et de Stieltjes vol.2, p.317-319
Legacy
In addition to the mathematics properties named in his honor, the Hermite crater near the Moon's north pole is named after Hermite.
