April 26, 1900] 



NATURE 



615 



married the sister of one of the injured, a Mademoiselle 

 Acloque. At the Polytechnique, Bertrand acquired a 

 knowledge of mining, and on leaving he became an in- 

 spector of mines. He was subsequently appointed pro- 

 fessor at the Lycce St. Louis. 



In March 1844 he became teacher of analysis {repe- 

 at eur d^ analyse) in the I^cole Polytechnique ; from 1847 

 to 1851 he was examiner for admission to the same in- 

 stitution, which raised him to the rank of professor of 

 analysis in 1856, a post which he held till 1895, j^st after 

 he had completed his jubilee as a member of the teach- 

 ing staff, an event which was commemorated on May 27, 

 1894, by a committee of his old students, who presented 

 him with a medal engraved by Chaplain. In 1847 he 

 was appointed deputy professor to Biot in the depart- 

 ment of physics and mathematics at the College de 

 France, and on the death of Biot, in 1862, he was ap- 

 pointed to the chair. From then up till 1890 he lectured 

 regularly, with the exceptions that his work was taken 

 by Darboux in 1867, by Maurice Levy in 1874-76 and 

 1878-85, by Laguerre in 1885-86 ; since 1890 Marcel 

 Deprez has acted as his deputy. We have it on the 

 authority of M. Gaston Paris, that his first course of 

 lectures was on a comparative study of the theories 

 according to which geometers had attempted to account 

 for capillary phenomena, his latest lectures being on 

 electricity, thermodynamics and theory of errors. From 

 1 858. to 1862 he was professor of higher mathematics it 

 the Ecole Normale Superieure, and he also is stated to 

 have held a professorship of special mathematics in the 

 Lycee Napoleon. In 1856 Bertrand was elected a member 

 of the Acad^mie des Sciences at the early age of^thirty- 

 four, in place of Sturm ; and on the death of Elie de 

 Beaumont, in 1874, he was elected permanent secretary, 

 in which office he had Berthelot as a colleague. He was 

 made officer of the Legion d'Honneur in August 1867, 

 and commander in December 1881. In 1884 he suc- 

 ceeded Jean Batiste Dumas in the French Academy. 



Bertrand's larger works — namely, his " Traite d'Arith- 

 metique," published in 1849; his "Traite d'Alg^bre," 

 published in 1856 ; and his " Traits du Calcul difTerentiel 

 et integral," brought out during the years 1864-70 — are 

 accepted as standard treatises by mathematicians in all 

 countries, the last named of the three being perhaps the 

 most widely read of all. His treatise on Arithmetic con- 

 tains, for the first time, a clear definition of incommen- 

 surable quantity. His treatise on the Calculus contains 

 a large number of geometrical applications embodied 

 in his divers memoirs ; and special mention may also be 

 made of his exposition of the theory of functional deter- 

 minants and the close and useful relation which he 

 established between these determinants and the simple 

 derived function of one variable. It is greatly to be 

 regretted that the manuscript of the third volume of this 

 work was destroyed in a fire. A similar misfortune befel 

 the manuscript of his original treatiseon Thermodynamics, 

 completed in 1870. In this case, however, the loss has 

 been repaired by the publication of a book on the same 

 subject at a later date, based on a course of lectures 

 given at the College de France. Those who have 

 grappled with such a subject as thermodynamics will 

 appreciate his naive observation that he has not attempted 

 to make a complete treatise, and that he has only ex- 

 pounded what he has understood. But, as M. Ldvy 

 justly goes on to remark, on those points which he pre- 

 tends not to have understood, notably on irreversible 

 phenomena and the application of the Second Law to 

 bodies of non-uniform temperature, he has made a series 

 of critical remarks of great importance, which have 

 already borne fruit. This treatise, moreover, is remark- 

 ably rich in illustrative examples all more or less original, 

 those dealing with saturated vapours coming well within 

 the range of practical applications. Other works ema- 

 nating from his pen are an edition of Lagrange's 



NO. 1 591, VOL. 61] 



" M^canique analytique," with copious notes, and ar 

 small volume of lectures on electricity, in which Bertrand 

 gives the true origin and reason of Faraday's notion of 

 "electric flux," although, being a mathematician, he 

 naturally favoured the rigorous methods of Ampere, for 

 whom he expressed great admiration. 



Passing from these standard treatises to the numerous- 

 papers published in scientific journals, a glance dowr> 

 the list of these shows that, from the outset, Bertrand 

 devoted his attention largely to applied mathematics, 

 and to those portions of pure mathematics required 

 in the solution of problems in applied mathematics. 

 His early papers deal chiefly with the diflferen- 

 tial and integral calculus, differential equations, the 

 calculus of variations, analytical mechanics, and in par- 

 ticular the integration of the equations of dynamics. 

 His papers on the theory of surfaces, dating from 1843^ 

 on the principle of similitude in mechanics, on the 

 propagation of sound and on capillary phenomena, are 

 among the best known of his minor writir»gs. After 

 1864, we find him writing biographical memoirs, com- 

 mencing with Copernicus, Tycho Brah^ and Fresnel^ 

 and followed up in later years with Comte, Lavoisier,. 

 D'Alembert (1889) and Pascal (1890). In 1868, Bertrand 

 commenced a series of papers on hydrodynamics ; flight 

 of birds came under his " ken " in 1871, and in the same 

 year he turned his attention to lunar theory. In the 

 three years 1871-73, he contributed quite a number of 

 papers dealing with electricity and magnetism, including 

 the mutual action of currents, &c. In the period 1874. 

 to 1883, the subgect'i treated included the sun, figure of 

 the earth, electricity and magnetism, electric transmission 

 of power, the Foucault pendulum, the theory of proba- 

 bility. Probability was always a favourite subject with 

 Bertrand. The numerous pitfalls connected with the 

 solution of problems, the remarkable power of prediction 

 which the theory appears to afford, had a great fascina- 

 tion for him, and many were the courses of lectures 

 which he delivered on this subject, not only at the 

 College de France, but even in the less advanced classes 

 at the :6cole Polytechnique, where the subject was in- 

 troduced by him chiefly in connection with its bearing 

 on astronomy and the applied sciences, such as civil and 

 military engineering. 



We have spoken, too, of Bertrand's p>apers on Fou- 

 cault's pendulum. How far Foucault owed the success 

 of this and his other discoveries to the influence of 

 Bertrand may be judged from M. Levy's own words. 

 Nearly at the beginning of his (Bertrand's) career, when- 

 he was but a mere professor at the Lyc^, he discovered 

 Foucault, became attached to him, helped him with the 

 mathematical science which Foucault was lacking, con- 

 tributed thus to his discoveries without in any way 

 thrusting himself forward ; and afterwards, hardly had 

 he been elected into the Academy of Sciences, before 

 he pressed forward the candidature of the great physicist,, 

 then little known or unappreciated, against the opposi- 

 tion of many of the highest authorities of that time 

 — a contest which has remained famous. The struggle 

 was not without its risks, nor the success without its 

 honour. One vote turned the scale. But the Institut 

 de France added one more man of genius to its scientific 

 roll. 



By the end of 1883 Bertrand had written about 

 121 papers. Many of these were published in the 

 Coinptes rendus ; others, mostly written in a lighter style 

 and oftentimes with a vein of irony running through 

 them, appeared in the Revue des deux Mondes^ and 

 included criticisms on " pseudo-mathematicians," as well 

 as numerous biographies of genuine men of science. 



Of late years Bertrand communicated but few papers 

 to the Paris Academy. He appears to have devoted 

 himself chiefly to the administrative work of the 

 Academy. In connection with the annual awards of 



