I can't forget Bruges, and regret not being able to go there again. Perhaps it would not be particularly cheerful in my old age to revisit the places of my happy childhood.In 1811 the Bienaimé family returned to Paris where Jules attended the Lycée Louis-le-Grand. Both brothers changed the spelling of their name to Bienaymé early in 1812. After taking part in the defence of Paris in 1814, Jules Bienaymé entered the École Polytechnique in the following year. This was not an easy time in Paris, as France lost its Empire and the monarchy was restored, and the École Polytechnique was closed in 1816 interrupting his studies. Also in 1816 Jean Charles Bienaimé died and, as Jules became responsible for the family, he took a job at the Ministry of Finances. In 1819 Bienaymé, wanting to make a career in science rather than administration, became a lecturer in mathematics at the military academy of St Cyr. However, he only held the post from 1 November 1819 to 10 February 1820 when he returned to his Civil Service job in the Administration of Finances. It is not clear why he left his teaching post although we do know that the director of the academy wrote that he had "not having found conditions compatible with his tastes and habits". There do seem to be more reasons why he left and one seems related to Siméon Denis Poisson, who had chosen him for the post, for from that time on Bienaymé became openly hostile to him. In the Ministry of Finances he was soon promoted to inspector, then in 1834 he became inspector general.
In the late 1820s, rather remarkably, the Bienaymé brothers married two sisters. Jules married Frangoise Gabrielle Clemence Harmand and his brother Charles Philippe Aimé married Jeanne Justine Lucie Harmand. Their wives were daughters of the Philippe Nicolas Harmand who was a family friend and had signed Jules' birth certificate. They had long known each other for the Bienaymé and Harmand families had shared the same house in Paris for a while. Jules and his wife had two sons and three daughters, The children, in order of birth, were: Lilia Nathalie Françoise Laure, Lucie Gabrielle Léonie (born 4 September 1831), Arthur François Alphonse (born 13 January 1834), Louis Irenée Alexis (born 8 April 1836), and Adrienne Clementine Laure (born 19 June 1840). Despite not being in the academic world, Bienaymé produced mathematical work of remarkable ingenuity but :-
... characterised, to the frustration of the reader, by lack of mathematical proofs for assertions sometimes far ahead of their time, ...He explained in a letter written to Adolphe Quetelet on 21 April 1846 :-
... that his everyday work and the state of his health do not permit him to complete the preparation of his writings for publication, and that he works seriously on applications which are of interest to both of them. His ill-health, especially his trembling hands, were to plague him to the end of his life.After the revolution of 1848 Bienaymé was forced to retire from the Ministry of Finances for political reasons as the new regime installed their own people in top posts. Shortly after he left the Civil Service he was asked to give a lecture course on probability at the Faculty of Sciences. In August 1850 he was reinstated into the Ministry of Finances as Inspecteur général des finances but he resigned for a final time in April 1852. Eugene Seneta writes in  that Bienaymé's:-
... last home (in 1866) was at rue de Fleurus, No 1, just before the western entrance of Jardin du Luxembourg and a short walk from the great church of St Sulpice, at which a commemorative service was held a year after his death. He died on 19 October 1878 and is buried at the Cimitière de Montparnasse, Paris, in a modest family grave at Division 10, Ligne 4 Sud, Numéro 12 ouest.In 1852 Bienaymé was elected to the Paris Academy of Sciences and for the next 23 years he was the referee for the statistics prize. He was also a founding member of the Société Mathématique de France becoming president of the Society in 1875. He was elected to the Société Philomatique de Paris in January 1838 and, from 1837 to 1845, much of his published material appears as reports of his addresses to meetings of the Society, reported in its journal L'Institut.
In fact Bienaymé only published 23 articles during his life and half of these were published in obscure places such as the L'Institut. The early articles discuss demography and life tables. He also wrote on the size of juries and the majority need for a conviction. In fact the jury system in France at that time was based on Laplace's conclusions but it was under attack by Poisson. Bienaymé supported Laplace on this issue. In fact Bienaymé supported Laplace in general since it was Laplace's Théorie analytique des probabilités
An excellent linguist, Bienaymé translated Pafnuty Chebyshev's work from Russian into French. In fact Bienaymé was a friend of Chebyshev (they first met in October 1852), and also of other important mathematicians such as Adolphe Quetelet, Antoine Cournot and Gabriel Lamé. In fact Quetelet and Bienaymé were born in the same year and corresponded regularly between 1846 (the year in which they first met) and 1871. On the other side, there were a number of mathematicians who Bienaymé seems particularly keen to criticise, such as Augustin-Louis Cauchy, Siméon Poisson and Joseph Bertrand. He argued with Cauchy over the least squares method and, in 1842, he criticised Poisson's law of large numbers. Bienaymé was quite wrong in his criticism of Poisson but in general he was years ahead of his time in the depth of his statistical ideas.
Bienaymé published the Bienaymé-Chebyshev inequality, which was used to give a very simple and precise demonstration of the generalised law of large numbers, in his important paper Considérations à l'appui de la découverte de Laplace sur la loi de probabilité dans la méthode des moindres carrés
It is a pity that their common interest in the Inequality somehow "slipped through the cracks" in the early contacts between Bienaymé and Chebyshev. Possibly the Inequality was regarded by Bienaymé as a minor result compared with his main themes of linear least squares and Laplacian defence. Chebyshev's recognition of its significance and its clear statement has, at any rate, always been a defensive point in his favour stressed by some historiographers.Bienaymé also worked on independent binomial trials and his most important contribution was his statement of the criticality theorem for simple branching processes which he gave in 1845 - eventual extinction of a family name has probability one if and only if the mean number of male children is one or less. His work on this predates that by Galton and Haldane. David Kendall writes in  (reviewing C C Heyde and E Seneta's book ):-
Some years ago the present authors [Heyde and Seneta] made the startling discovery that the criticality theorem for branching processes, previously assigned to the year 1873, was in fact known in 1845, and indeed that earlier version due to Bienaymé is complete and correct, unlike the Galton-Watson version which contains an error that was not cleared up for many years. This discovery necessitated the re-writing of the history of branching processes ...He also gave a simple test for randomness of observations on a continuously varying quantity. He stated, and gave a proof which leaves something to be desired, of a sophisticated limit theorem which was studied again by von Mises in 1919. Eugene Seneta writes in :-
It is also clear [that Bienaymé] understood the idea of covariance of random variables in 1842, and had a notion of conditional expectation.Eugene Seneta also tries to explain why Bienaymé's brilliance has been forgotten :-
Contributing to his being largely forgotten were the facts that Bienaymé was modest as regards his own achievements, made no great efforts to assert his priority, and was ahead of his time in mathematical statistics. He left no disciples, not being in academia; and wrote no book.
Article by: J J O'Connor and E F Robertson