His father died when Joseph was only nine years old; by that time he had learned a substantial amount of mathematics and could speak Latin fluently.

From age eleven to seventeen, he obtained two bachelor's degrees, a license and a PhD with a thesis concerning the mathematical theory of electricity, and was admitted to the 1839 entrance examination of the École Polytechnique.

Bertrand was a professor at the École Polytechnique and Collège de France, and was a member of the Paris Academy of Sciences of which he was the permanent secretary for twenty-six years.

[2][3] Bertrand translated into French Carl Friedrich Gauss's work concerning the theory of errors and the method of least squares.

His book Thermodynamique states in Chapter XII, that thermodynamic entropy and temperature are only defined for reversible processes.