Bertrand
# Joseph Louis Francois Bertrand

### Born: 11 March 1822 in Paris, France

Died: 5 April 1900 in Paris, France

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**Joseph Bertrand** was appointed professor at the Ecole Polytechnique in 1856 and, in 1862, he also became professor at the Collège de France. 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 2n-2 for every n > 3. This conjecture was proved by Chebyshev in 1850.

Bertrand also worked on differential geometry and probability theory. 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. General comments on this work and Bertrand's other notes on probability are made in [2]:-

*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.*

Bertrand was appointed a member of the Paris Academy of Sciences and he served as its permanent secretary from 1874 to the end of his life.

**References (6 books/articles)**

**References elsewhere in this archive:**

Tell me about Bertrand's part in investigating prime numbers and his work on mathematical games and recreations

Tell me about Bertrand's work on orbits and gravitation

Joseph L F Bertrand was elected to the Royal Society of London in 1875. You can see a history of the Royal Society and a list of the members among the mathematicians in our archive.

JOC/EFR December 1996