4-4. Joseph Bertrand

Joseph Bertrand (1822–1900) matriculated at the École polytechnique in Paris in 1839, and the Paris School of Mines in 1841. He obtained the agrégation de mathématiques that same year. A long career as a teacher began in Paris in 1844 at the Collège Saint-Louis, and also at the École polytechnique, where he was répétiteur d’analyse. He subsequently was named examinateur d’admission at Polytechnique, and in 1852, professor of mathématiques spéciales at the lycée Napoléon (renamed from Collège Henri IV). In 1856 he replaced Sturm as professor of analysis at Polytechnique, a chair he occupied until reaching the age of mandatory retirement in 1895. Bertrand accepted a concurrent position as maître de conférences at the École normale supérieure from 1857 to 1862. At the Collège de France, Bertrand was suppléant to Jean-Baptiste Biot for a decade in the Chair of Physics and Mathematics, before succeeding him in 1862.

In 1856, Bertrand was elected to the geometry section at the Paris Academy of Science, where he served as president in 1874. He was elected Permanent Secretary for the mathematical sciences that same year, and in 1884 succeeded the chemist Jean-Baptiste Dumas at the Académie Française (Institut de France, 1968).

Bertrand was related by blood or marriage to several members of the scientific community in Paris. His uncle was Jean-Marie Constant Duhamel. Bertrand was nine years old when his father died; his mother left Paris, and Duhamel took him in. His older brother Alexandre became a leading archeologist, and member of the Paris Academy of inscriptions and belles-lettres. Alexandre married a niece of Olinde Rodrigues, and their daughter married Paul Appell. Charles Hermite married Joseph’s sister Louise in 1848. Émile Picard was Hermite’s son-in-law, while Émile Borel was Paul Appell’s son-in-law. Joseph’s son Marcel Bertrand was a professor of geology at the Paris School of Mines and the son-in-law of Élie Mascart.

As for Poincaré, he was related to Bertrand by neither blood nor marriage, but he, too, followed an academic and institutional path from the École polytechnique to the Académie Française. Their surviving correspondence is comprised of eleven letters, and is overwhelmingly administrative in nature. As members of the Paris Academy of Science, Bertrand and Poincaré had a weekly opportunity over a span of fourteen years to exchange in person their views on matters of common interest. There were certainly a few points of disagreement, for example, concerning the merits of Maxwell’s theory of electrodynamics, or those of Helmholtz’s hydrodynamics, both of which Bertrand, but not Poincaré, found to be riddled with error and arbitrary hypotheses (Darrigol, 2000, 352). When the first volume of Poincaré’s lessons on Maxwell’s theory appeared, Bertrand opposed Poincaré’s mathematical approach along with that of what he called the “new school”, although he admired the work’s preface. Poincaré did not forgive this negative review.

Like Bertrand (1896), Poincaré initially considered Maxwell’s kinetic gas theory to be fundamentally flawed, and he largely neglected the topic in his lessons on thermodynamics. Bertrand, however, did not live long enough to experience Planck’s success in characterizing black-body radiation (Planck, 1900), William Thomson’s theory of the universe (Thomson, 1901), Gibbs’ canonical ensembles (Gibbs, 1902), or Einstein’s theory of Brownian motion (Einstein, 1905), among other advances in early 20th-century physics. Poincaré lived to see all of these things, of course, and to his credit, he began engaging constructively with statistical mechanics starting in 1906.

In 1894, the Mathematical Society of France called on Poincaré to deliver a speech in honor of the fiftieth anniversary of the inauguration of Bertrand’s teaching career. On this occasion, Poincaré praised the critical acumen of his senior colleague and “cher Maître”. He closed his fin-de-siècle address with a reminder that times had changed for French science:

The time is past when all enlightened men were French in spirit. But if we are to maintain our place we must remain true to ourselves; we should also be grateful for the example you provide, you who remain the most French of all our mathematicians.

Time-stamp: " 9.01.2025 22:41"

Notes

• 1 The numerous and varied family relations of Joseph Bertrand in French exact science in the last quarter of the nineteenth century are delineated by Zerner (1991). On Bertrand’s scientific contributions, see Gaston Darboux’s éloge in Appell (1912, 1–60), Struik (1970), and Balibar (2020).
• 2 Bertrand (1891). Among the members of the “new school” was Poincaré’s professor at the École polytechnique, whose early contribution to Fresnel’s theory of light reflection in crystals (borrowing from Franz Neumann), did not meet with Bertrand’s approval (Cornu, 1863); Poincaré and Launay (1904, 11).
• 3 Poincaré (1892, 1894b). On the history of statistical mechanics in France during Poincaré’s life, see Principe (2008).
• 4 “Le temps n’est plus où tous les hommes éclairés étaient français par l’esprit; mais si nous voulons conserver notre place il faut que nous restions nous-mêmes; aussi devons-nous vous être reconnaissants de l’exemple que vous nous donnez, vous qui êtes resté le plus français de tous nos géomètres.” Poincaré (1894a). The manuscript of Poincaré’s address differs slightly from the published version; see the annotated transcription (§ 7-1-24).