**Joseph Bertrand** was a Paris professor who, in 1845, conjectured:

_I've told you once and I'll tell you again

There's always a prime between n and 2n._

This was proved by Chebyshev in 1850.

- 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.
- 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.
- In 1843 Bertrand published two memoirs on Surfaces isothermes orthogonale Ⓣ(Isothermal orthogonal surfaces).
- 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 nnn and 2n−22n - 22n−2 for every n>3n > 3n>3.
- 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 Ⓣ(Treatise of arithmetic) (1849) and Traité élémentaire d'algèbre Ⓣ(Treatise of elementary algebra) (1850).
- Bertrand published many works on differential geometry and on probability theory.
- His book Calcul des probabilitiés Ⓣ(Calculation of Probabilities) (1888) contains a paradox on continuous probabilities now known as Bertrand's paradox.
- Bertrand's treatise contains mistakes and misprints.
- Bertrand uses the term 'valeur probable' on a par with 'espérance mathématique'.
- Bertrand's literary style is extremely attractive.
- It is worth noting this treatise by Bertrand heavily influenced Poincaré's treatise also called Calcul des probabilitiés Ⓣ(Calculation of Probabilities) which he published in 1896.
- Indeed, Poincaré refers to almost nobody but Bertrand in his book.
- 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.
- 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 Ⓣ(Treatise on differential and integral calculus) which he never rewrote.
- Bertrand also lost his manuscript of Thermodynamique Ⓣ(Thermodynamics) in the fire but he later rewrote the work.
- With his home destroyed Bertrand moved to Sèvres and later to Viroflay.
- Bertrand's home was particularly important to him for it provided the centre of a vigorous intellectual grouping.
- 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.
- 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.

Born 11 March 1822, Paris, France. Died 3 April 1900, Paris, France.

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Astronomy, Group Theory, Knot Theory, Number Theory, Physics, Puzzles And Problems, Special Numbers And Numerals

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive