**Joseph Bertrand**'s father was Alexandre Jacques François Bertrand and his mother was Marie Caroline Belin. Alexandre had studied at the École Polytechnique where he had become friends with Jean-Marie Duhamel who later married Alexandre Bertrand's sister. Alexandre was a writer of popular science books but sadly he died young and after this tragic event Joseph, who was nine years old, went to live with Duhamel and his wife. Of course this sad event did have the beneficial effect that Joseph was guided by Duhamel. It is worth noting at this stage that Joseph had a sister Louise who married Hermite in 1848.

Joseph showed remarkable talents as a child and by the age of nine he understood algebra and elementary geometry, as well as being able to speak Latin fluently. Two years later, when he was eleven, he was given permission to attend lectures at the École Polytechnique although he could not officially enter at this time despite having gained the required academic level. He was awarded his first degree at the age sixteen and in following year he was awarded his doctorate for a thesis on thermodynamics. In 1839, the year he received his doctorate, he officially entered the École Polytechnique and also published his first paper on the mathematical theory of electricity. After two years of study there he entered the École des Mines in 1841 being appointed professor of elementary mathematics at the Lycée Saint-Louis in the same year. He held this post at the Lycée until 1848.

A dramatic event happened in 1842. Bertrand and his brother were friendly with the Aclocque family who lived in Versailles. They visited them frequently, travelling by train from Paris. On this particular occasion the train crashed on the return journey and Bertrand was badly injured, suffering a crushed nose and facial scars which he retained throughout his life. Bertrand married Louis Celine Aclocque in 1844; their sons Marcel Alexandre (born 2 July 1847), Joseph Désiré (born 1853) and Léon Gratien (born 1858) all studied at the École Polytechnique, and Marcel went on to become a famous geologist.

In 1843 Bertrand published two memoirs on *Surfaces isothermes orthogonale* Ⓣ. He was appointed répétiteur d'analyse at the École Polytechnique in 1844, being made professor there in 1856 following the death of François Sturm. He had been appointed professor of special mathematics at the Lycée Henry IV in 1852, also teaching at the École Normale Supérieure. In 1862, he became professor of analysis at the Collège de France, succeeding Biot who died in February of that year. In 1878 Bertrand stopped teaching at the Collège de France but, eight years later, he began teaching there again.

In 1845 Bertrand conjectured that there is at least one prime between *n* and 2*n* - 2 for every *n* > 3. This conjecture, similar to one stated by Euler one hundred years earlier, was proved by Chebyshev in 1850. He made a major contribution to group theory with his article *Mémoire sur le nombre de valeurs que peut prendre une fonction quand on y permute les lettres qu'elle renferme* Ⓣ which he submitted to the Paris Academy in March 1845. Cauchy was asked to report on the work, which studied subgroups of low index in the symmetric group, and it clearly led him to return to study permutation groups himself. Bertrand's important paper was eventually published in the *Journal de l'École Polytechnique*.

Famed as an author of textbooks, the first two of many published by Bertrand were *Traité d'arithmetique* Ⓣ (1849) and *Traité élémentaire d'algèbre* Ⓣ (1850). These were aimed at pupils at secondary school, although later texts were aimed at students at more advanced levels, for example the two volume work *Traité de calcul différentiel et de calcul intégral* Ⓣ (1864-70), *Thermodynamique* Ⓣ (1887), and *Leçons sur la théorie mathématique de l'électricité* Ⓣ (1890).

Bertrand published many works on differential geometry and on probability theory. He edited an edition of Lagrange's *Mécanique analytique* which was published in 1853. In 1855 he translated Gauss's work on the theory of errors and the method of least squares into French. He wrote a number of notes on the theory of probability and on the reduction of data from observations. He published these notes starting around 1875 and, after a short break of three years from 1884, he began publishing further notes on probability.

His book *Calcul des probabilitiés* Ⓣ (1888) contains a paradox on continuous probabilities now known as *Bertrand's paradox*. It concerns the probability that an arbitrary chord of a circle is longer than a side of an equilateral triangle inscribed in the circle. The paradox comes about since the word "arbitrary" is not properly defined, see [12]. General comments on this work and Bertrand's other notes on probability are made in [4]:-

*Bertrand mentioned some of his predecessors*(De Moivre, Laplace, Bienaymé)*, but did not refer to other scholars, notably to Chebyshev*.

*Bertrand's treatise contains mistakes and misprints. The conditions of many problems are stated carelessly and drawings are completely lacking. Verbal explanations, sometimes given instead of formulas, are irritating*.

*The treatise is badly organized*.

*Bertrand uses the term 'valeur probable' on a par with 'espérance mathématique'*.

*Bertrand's literary style is extremely attractive*.

*Calcul des probabilitiés*Ⓣ which he published in 1896. Indeed, Poincaré refers to almost nobody but Bertrand in his book.

Of course Bertrand lived through a time of political difficulties in France. The revolution of 1830 was followed by a decade marked by repeated challenges to the King. However from 1840 there was a stable period during which Bertrand was able to begin his career. The year 1848, however, saw the overthrow of the monarchy and during this revolution Bertrand served as a Captain in the National Guard. After years of comparative stability, the Franco-Prussian war broke out in July 1870. The French suffered a military disaster and Paris was under siege from the Prussian armies for four months from September 1870. These were extremely difficult times for Bertrand and his family. Following the French defeat the people of Paris formed the Commune of Paris in March 1871. The repression of the Paris Commune took place towards the end of May during a week of street fighting which saw Bertrand's house burned and many of his manuscripts were lost in the fire including an intended third volume of *Traité de calcul différentiel et de calcul intégral* Ⓣ which he never rewrote. Bertrand also lost his manuscript of *Thermodynamique* Ⓣ in the fire but he later rewrote the work. With his home destroyed Bertrand moved to Sèvres and later to Viroflay.

In fact Bertrand's home was particularly important to him for it provided the centre of a vigorous intellectual grouping. His family grew to include Émile Picard who married a daughter of Hermite and Bertrand's sister Louise. In 1881 Paul Appell married Amelie, a niece of Bertrand and of Hermite and a cousin of Émile Picard. Let us return to note other effects of the French military defeat of 1870. It led to considerable soul searching in France and, in particular, it was questioned whether science education, particularly the courses at the École Polytechnique, were responsible. Bertrand, by this time one of the leading mathematicians in France, had to deal with the consequences as it was questioned whether the mathematics being taught was too theoretical, and so not preparing the leading young Frenchmen for the military.

Bertrand was appointed a member of the Paris Academy of Sciences in 1856 and he served as its permanent secretary from 1874 to the end of his life. He was made a grand-officier of the Légion d'Honneur.

Struik, in [1], sums up Bertrand's place in the development of mathematics:-

In1895his pupils gave him a medal in commemoration of his fifty years of teaching at the École Polytechnique. the influence of Bertrand's work, however, is hardly comparable to that of several of his contemporaries and pupils. Lest it be judged ephemeral, it must be viewed in the context of nineteenth-century Paris and of Bertrand's brilliant academic career, his exalted social position, and the love and respect given him by his pupils.

**Article by:** *J J O'Connor* and *E F Robertson*

**Click on this link to see a list of the Glossary entries for this page**