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Joseph Bertrand's theorem states that, among smooth attractive central potentials admitting stable circular orbits, requiring every bound orbit to be closed singles out the inverse-distance potential energy and the isotropic quadratic potential energy, up to additive constants. Rational radial-to-angular frequency ratios near a circular orbit provide only a necessary local closure test; finite-amplitude closure is the additional content. In $U=B{r}^n$, the radial-to-angular frequency ratio near a stable circle is $\sqrt{n+2}$, and stability requires $nB>0$ and $n>-2$. The all-bound-orbit cases are $n=-1$ and $n=2$.

A French mathematician whose work ranged across number theory, differential geometry and probability theory. His 1845 prime-number conjecture became Bertrand's postulate: for every integer $n>1$ there is a prime number strictly between $n$ and $2n$. The result is a theorem, rather than an unproved conjecture, despite its traditional name.

Bertrand's paradox illustrates a different part of his legacy: an instruction to choose a random chord does not define a unique probability distribution until the sampling rule is specified. His career also included teaching analysis in Paris and writing textbooks in calculus and mathematical physics. ^[ref]