## Bertrand's work on probability Introduction

 Oscar Sheynin wrote an article Bertrand's work on probability which appeared in Arch. Hist. Exact Sci. 48 (2) (1994), 155-199. Sheynin writes in the Introduction:-

Joseph Louis François Bertrand (1822-1900) contributed to several branches of mathematics. In 1855 he translated into French Gauss's writings on the theory of errors and method of least squares. A few of his notes on the theory of probability and combination of observations appeared in 1875-1884. Then, during 1887-1888, he published 25 more notes on the same subject. Bertrand's Calcul des probabilités appeared in 1888. His last note on probability was dated 1892.

In 1856 Bertrand was appointed professor at the École Polytechnique and, in 1862, he became professor at the Collège de France as well. A member of the Paris Academy of Sciences, he was its permanent secretary from 1874 until his death. M Lévy (1900) indicates that Bertrand taught probability "à diverses reprises" both at the "Collège dans son enseignement moins élevé" and at the École Darboux (1902) testifies that in 1878 Bertrand abandoned his teaching at the Collège, but then, in 1886, had to resume his activities there. This fact likely explains Bertrand's sudden interest in probability as manifested by his publications of 1887-1888.

For the first time ever, I describe in full Bertrand's work on probability and error theory. Beginning with Section 6.2, my account follows his treatise. I also turn attention to Bertrand's notes. The large number of these short contributions, and the appearance of most of them during just two years, compelled me to refer to them without indicating their date. Here are my general observations. Except for Items 2 and 3 they concern all of Bertrand's writings.

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

2. 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.

3. The treatise is badly organized.

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

5. Bertrand's literary style is extremely attractive.
I left out some of the topics discussed by Bertrand, namely the description of classical least squares and the bivariate normal law which he introduced largely for the sake of discussing target shooting.

According to the American National Union Catalog Pre-1956 Imprints (vol. 50, p. 591) the treatise was published in 1888 and again in 1889. The Catalog also mentions the second edition of 1907, 'conforme à la 1'. Based on this, on Rouché's review, and on information from the Comptes Rendus and the Bulletin bibliographique, I conclude that the treatise was indeed first published in 1888, but that at least some of its copies were wrongly dated 1889, either on purpose or otherwise.

JOC/EFR March 2006