Charles Hermite
Charles Hermite
Born	(1822-12-24)24 December 1822 Dieuze, Moselle, France
Died	14 January 1901(1901-01-14) (aged 78) Paris, France
Alma mater	Collège Henri IV, Sorbonne Collège Louis-le-Grand, Sorbonne
Known for	Proof that e is transcendental Hermitian adjoint Hermitian form Hermitian function Hermitian matrix Hermitian metric Hermitian operator Hermite polynomials Hermitian transpose Hermitian wavelet
Scientific career
Fields	Mathematics
Institutions	École Polytechnique Sorbonne
Doctoral advisor	Eugène Charles Catalan
Doctoral students	Léon Charve Henri Padé Mihailo Petrović Henri Poincaré Thomas Stieltjes Jules Tannery

Charles Hermite (French pronunciation: [ʃaʁl ɛʁˈmit]) FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra.

Hermite polynomials, Hermite interpolation, Hermite normal form, Hermitian operators, and cubic Hermite splines are named in his honor. One of his students was Henri Poincaré.

He was the first to prove that e, the base of natural logarithms, is a transcendental number. His methods were used later by Ferdinand von Lindemann to prove that π is transcendental.

Life

Hermite was born in Dieuze, Moselle, on 24 December 1822,^[1] 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.^[2]

Hermite obtained his secondary education at Collège de Nancy and then, in Paris, at Collège Henri IV and at the Lycée Louis-le-Grand.^[1] 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.^[2] In 1842 he was admitted to the school.^[1] 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.^[2]

In 1842, Nouvelles Annales de Mathématiques published Hermite's first original contribution to mathematics, a simple proof of Niels Abel's proposition concerning the impossibility of an algebraic solution to equations of the fifth degree.^[1]

A correspondence with Carl Jacobi, begun in 1843 and continued the next year, resulted in the insertion, 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.^[1]

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 Joseph Bertrand's sister, Louise Bertrand, in 1848.^[2]

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.^[1] 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.^[1]

Hermite died in Paris on 14 January 1901,^[1] aged 78.

Contribution to mathematics

Number theory

An inspiring teacher, Hermite strove to cultivate admiration for simple beauty and discourage rigorous minutiae. His correspondence with Thomas Stieltjes testifies to the great aid he gave those beginning scientific life. His published courses of lectures have exercised a great influence. His important original contributions to pure mathematics, published in the major mathematical journals of the world, dealt chiefly with Abelian and elliptic functions and the theory of numbers.

In 1858, Hermite solved equations of the fifth degree and, in 1873, he proved that e, the base of the natural system of logarithms, is transcendental.^[2] Techniques similar to those used in Hermite's proof of e's transcendence were used by Ferdinand von Lindemann in 1882 to show that π is transcendental.^[1]

Quintic equations

In 1858, Hermite showed that equations of the fifth degree could be solved by elliptic modular functions. In his famous work Sur la résolution de l'Équation du cinquiéme degré Comptes rendus he named the exact elliptical solution expression based on the theta function^[3] the Bring Jerrard Normal form. In particular, he recognized how to determine the corresponding elliptic module and its Pythagorean complementary module for the given Bring-Jerrard form. The Bring-Jerrard form only contains the quintic, the linear and the absolute equation term:

${\displaystyle x^{5}+5\,x=4\,c}$

All Bring-Jerrard equations can be normalized to this form by substituting the internal unknowns. If in the given form the value ${\displaystyle c}$ is a real number, then the equation in question has one real and four imaginary solutions. According to the Abel-Ruffini theorem, this Bring's equation cannot be solved elementaryly for the vast majority of values ${\displaystyle c}$ or the associated solution set cannot be represented in an elementary radical way. But for all values ${\displaystyle c}$ the mentioned Bring-Jerrard equation can be solved elliptically. The five solutions of the quintic equation shown are always obtained completely by setting up rational combinations of the non-elementary so-called Elliptic Modular Functions depending on the Elliptic Nome as an inner function. The Elliptic Nome must be generated by the following elliptic modules or numerical eccentricities ${\displaystyle k}$ and ${\displaystyle k'}$:

${\displaystyle k={\bigl (}2\,c^{2}+2+2{\sqrt {c^{4}+1)){\bigr )}^{-1/2}{\bigl (}{\sqrt ((\sqrt {c^{4}+1))+1))-c{\bigr )}=\operatorname {tlh} [{\tfrac {1}{2))\operatorname {aclh} (c)]^{2))$

${\displaystyle k'={\bigl (}2\,c^{2}+2+2{\sqrt {c^{4}+1)){\bigr )}^{-1/2}{\bigl (}{\sqrt ((\sqrt {c^{4}+1))+1))+c{\bigr )}=\operatorname {ctlh} [{\tfrac {1}{2))\operatorname {aclh} (c)]^{2))$

For the more accurate derivation, please watch the Article Lemniscate elliptic functions, section Elliptic Modulus and quintic equations!

These eccentricities are the corresponding elliptical module and its Pythagorean complementary counterpart in the Legendre's normal form or in the standard form. The two formulas now mentioned result directly from the formula which is at the top of page 258 in the Italian edition of the above-mentioned work Sulla risoluzione delle equazioni del quinto grado, which was further distributed by Francesco Brioschi. The expressions with the abbreviations ${\displaystyle \mathrm {tlh} }$ for Tangens Lemniscatus Hyperbolicus and ${\displaystyle \mathrm {ctlh} }$ for Cotangens Lemniscatus Hyperbolicus as well as the elliptic integral ${\displaystyle \mathrm {aclh} }$ for Areacosinud Lemniscatus Hyperbolicus represent the Hyperbolic lemniscatic function expressions, which greatly simplify the representation of the resolutions according to the modules. After Charles Hermite and the mathematicians Glashan, Young and Runge, other mathematicians such as the Russian mathematicians Viktor Prasolov and Yuri Soloviev^[4] as well as the Greek mathematician Nikolaos Bagis^[5] the solution expressions for which depend on the elliptical nome the mentioned Bring-Jerrard equation form. This is how Prasolov and Soloviev determined the expression of the real solution, which they wrote down on page 159 in their work Elliptic Functions and Elliptic Integrals. They used the standardized Weber modular functions in its original form:

${\displaystyle x_{RE}=4\,c\div {\bigl \{}{\mathfrak {f))_{00}(q)^{-6}{\bigl [}{\mathfrak {f))_{00}(q^{1/5})-{\mathfrak {f))_{00}(q^{5}){\bigr ]}^{2}{\bigl [}{\mathfrak {f))_{00}(i^{4/5}q^{1/5})-{\mathfrak {f))_{00}(i^{-4/5}q^{1/5}){\bigr ]}^{2}{\bigl [}{\mathfrak {f))_{00}(i^{8/5}q^{1/5})-{\mathfrak {f))_{00}(i^{-8/5}q^{1/5}){\bigr ]}^{2}+5{\bigr \))}$

Publications

The following is a list of his works:^[1]

• "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,^[6] 1908,^[7] 1912^[8] and 1917; PDF copy from UMDL.
• "Œuvres de Charles Hermite", reissued by Cambridge University Press, 2009; ISBN 978-1-108-00328-5.

Quotations

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.

— Charles Hermite; letter to C.W. Borchardt, "Men of Mathematics", E. T. Bell, New York 1937, p. 464.

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 Weierstrass functions, 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.