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Charles Hermite

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Charles Hermite
FRS FRSE
File:Charles Hermite.jpg
Born 24 December 1822
Dieuze, Moselle, Kingdom of France
Died 14 January 1901 (aged 78)
Paris, French Third Republic
Alma mater Collège Henri IV, Sorbonne
Collège Louis-le-Grand, Sorbonne
Known for Proof that e is transcendental
Hermitian matrices
Hermite polynomials
See full list
Scientific career
Fields Mathematics
Workplaces École Polytechnique
Sorbonne
Doctoral advisor Eugène Charles Catalan
Other academic advisors Joseph Liouville
Doctoral students Gaston Floquet
Henri Padé
Mihailo Petrović
Henri Poincaré
Thomas Stieltjes
Jules Tannery

Charles Hermite (French: [ʃaʁl ɛʁmit]; 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 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]

Contribution to mathematics

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] 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]

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

Hermite introduced the notion of an orthogonal matrix, which is equal to the inverse of its transpose, in 1854, though the modern formal definition was first stated by 1878 by Ferdinand Georg Frobenius.[6]: 809 In 1855, Hermite proved that the eigenvalues of matrices equal to their own complex conjugate transposes, Hermitian matrices, are always real, thereby generalizing the 1829 result of Cauchy for n \times n real symmetric matrices.[7]: 215–6 The notion of Hermitian matrices was later extended for infinitely variables and became an important topic in the study of differential and integral equations thanks to the vision of David Hilbert.[7]: 216 Such linear transformations, dubbed Hermitian operators, were used in the rigorous mathematical formulation of quantum mechanics first by Norbert Wiener and Max Born, and subsequently by Hilbert, Lothar Nordheim, and John von Neumann.[8]

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 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 \pi^2 and therefore \pi are irrational.[8][12] However, he did not address the transcendence of \pi, 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 \pi is irrational.[13][14]

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

Legacy

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

Publications

The following is a list of his works:[2]

  • "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.
  • "Œuvres de Charles Hermite", reissued by Cambridge University Press, 2009; ISBN 978-1-108-00328-5.

See also

From the 1913 Catholic Encyclopedia

Charles Hermite — Born at Dieuze, Lorraine, 24 December, 1822; d. at Paris, 14 January, 1901; one of the greatest mathematicians of the nineteenth century. He studied at the Collège de Nancy and then, in Paris, at the Collège Henri IV and at the Collège Louis-le-Grand. As a boy he read some of the writings of Lagrange on the solution of numerical equations, and of Gauss on the theory of numbers. In 1842, his first original contribution to mathematics, in which he gave a simple proof of the proposition of Abel concerning the impossibility of obtaining an algebraic solution for the equation of the fifth degree, was published in the "Nouvelles Annales de Mathématiques". The same year he entered the Ecole Polytechnique, where he remained as a student but one year. A correspondence with Jacobi, begun in 1843 and continued in 1844, led to 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. In 1848, Hermite returned to the Ecole Polytechnique as répétiteur and examinateur d'admission. In 1856, through the influence of Cauchy and of a nun who nursed him, he resumed the practice of his religion. On 14 July, of that year, he was elected to fill the vacancy created by the death of Binet in the Académie des Sciences. In 1869, he succeeded Duhamel as professor of mathematics, both at the Ecole Polytechnique, where he remained until 1876, and in the Faculty of Sciences of Paris, which position he occupied until his death. From 1862 to 1873 he was lecturer at the Ecole Normale Supérieure. Upon his seventieth birthday, on the occasion of his jubilee which was celebrated at the Sorbonne under the auspices of an international committee, he was promoted grand officer of the Legion of Honour.

As a teacher Hermite was inspiring. His correspondence with Stieltjes testifies to the great aid he gave those entering scientific life. His efforts in teaching were directed not towards too rigorous minuteness, but towards exciting admiration for things simple and beautiful. His published courses of lectures have exercised a wide influence. His important original contributions to pure mathematics, published in the leading mathematical journals of the world, dealt chiefly with Abelian and elliptic functions and the theory of numbers. In 1858 he solved the equation of the fifth degree by elliptic functions; and in 1873 he proved e, the base of the natural system of logarithms, to be transcendent. This last was used by Lindemann to prove (1882) the same for pi. The following is a list of his works. "Cours d'analyse de l'Ecole Polytechnique", Paris, 1873; "Cours professé à la Faculté des Sciences", edited by Andoyer, 4th ed., Paris, 1891; "Correspondance", edited by Baillaud and Bourget, Paris, 1905, 2 vols. The "Oeuvres de Charles Hermite" were edited by Picard for the Academy of Sciences, 2 vols., Paris, 1905 and 1908.

BOREL, Charles Hermite in Annuaire des Mathèmaticiens (Paris, 1902); CAPELLI, In commemorazione di Carlo Hermite in Acad. di sci. fis. e mat., Atti, VII (Naples, 1901); DARBOUX, Notice historique sur Charles Hermite in Memoires de l'Acad. des Sci., XLIX (Paris, 1901); KNELLER, Das Christentum und die Vertreter der neueren Naturwissenschaft in Stimmen aus Maria Laach, supplement, no. 84-5. (Freiburg im Br., 1903); MANSION, Charles Hermite, esquisse bioigraphique et bibliographique (Paris, 1901); OVIDIO, Carlo Hermite, commemorazione, R. accad. di sci., Atti, XXXVI (Turin, 1901); PICARD, L'oeuvre scientifique de Charles Hermite in Acta mathematica, XXV; VOIT, Charles Hermite, obituary in Kgl. Akad. d. Wissenschaft, Sitzungsb., math-phys. Classe (Munich, 1902).

References

  1. Back to citation occurrence 1.0 Back to citation occurrence 1.1 Back to citation occurrence 1.2 Back to citation occurrence 1.3 Back to citation occurrence 1.4 Back to citation occurrence 1.5 G. B. H. (February 7, 1901). "Charles Hermite". Nature. 63 (1632): 350–1. Bibcode:1901Natur.63.350G. doi:10.1038/063350a0.
  2. Back to citation occurrence 2.0 Back to citation occurrence 2.1 Back to citation occurrence 2.2 Back to citation occurrence 2.3 Back to citation occurrence 2.4 Back to citation occurrence 2.5 Back to citation occurrence 2.6 Back to citation occurrence 2.7 Back to citation occurrence 2.8 Linehan, P.H. (1910). "Charles Hermite". Catholic Encyclopedia. Vol. 7.
  3. Back to citation occurrence 3.0 Back to citation occurrence 3.1 Back to citation occurrence 3.2 Back to citation occurrence 3.3 O'Connor, John J; Robertson, Edmund F. (March 2001), "Charles Hermite", MacTutor History of Mathematics Archive, University of St Andrews
  4. Back to citation Memoirs and proceedings of the Manchester Literary & Philosophical Society FOURTH SERIES Eighth VOLUME 1894
  5. Back to citation Halsted, George Bruce (June–July 1901). "Charles Hermite". The American Mathematical Monthly. 8 (6–7): 131–3. doi:10.1080/00029890.1901.12000572.
  6. Back to citation occurrence 6.0 Back to citation occurrence 6.1 Kline, Morris (1972). Mathematical Thought from Ancient to Modern Times. Oxford University Press. ISBN 0-19-506136-5.
  7. Back to citation occurrence 7.0 Back to citation occurrence 7.1 Jammer, Max (1966). The Conceptual Development of Quantum Mechanics. McGraw-Hill.
  8. Back to citation occurrence 8.0 Back to citation occurrence 8.1 Back to citation occurrence 8.2 Freudenthal, Hans (1970–1980). Gillispie, Charles Coulston (ed.). Dictionary of Scientific Biography. New York: Charles Scribner's Sons. Charles Hermite.
  9. Back to citation Boyer, Carl B.; Merzbach, Uta (1991). A History of Mathematics (2nd ed.). New York: John Wiley & Sons. ISBN 0-471-09763-2.
  10. Back to citation occurrence 10.0 Back to citation occurrence 10.1 Back to citation occurrence 10.2 Maor, Eli (1994). e: The Story of a Number. Princeton, New Jersey: Princeton University Press. ISBN 0-691-05854-7.
  11. Back to citation Dunham, William (2005). The Calculus Gallery. Princeton University Press. ISBN 978-0-691-13626-4.
  12. Back to citation Zhou, Li (2011). "Irrationality proofs à la Hermite". The Mathematical Gazette. 95 (534): 407–413. arXiv:0911.1929. doi:10.1017/S0025557200003491. S2CID 115175505.
  13. Back to citation Niven, Ivan (1947). "A Simple Proof that \pi Is Irrational" (PDF). Bulletin of the American Mathematical Society. 53 (6): 509. doi:10.1090/s0002-9904-1947-08821-2.
  14. Back to citation Jones, Timothy W. (2010). "Discovering and Proving that \pi Is Irrational". The American Mathematical Monthly. 117 (6): 553–7. doi:10.4169/000298910X492853.
  15. Back to citation Pierpont, James (1907). "Review: Oeuvres de Charles Hermite, publiées sous les auspices del'Académie des Sciences par EMILE PICARD. Vol. I" (PDF). Bull. Amer. Math. Soc. 13 (4): 182–190. doi:10.1090/S0002-9904-1907-01440-4.
  16. Back to citation Pierpont, James (1910). "Review: Oeuvres de Charles Hermite. Vol II" (PDF). Bull. Amer. Math. Soc. 16 (7): 370–377. doi:10.1090/s0002-9904-1910-01920-0.
  17. Back to citation Pierpont, James (1912). "Review: Oeuvres de Charles Hermite. Vol III" (PDF). Bull. Amer. Math. Soc. 19 (2): 83–84. doi:10.1090/s0002-9904-1912-02290-5.
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