In the theory of Diophantine approximation, Lagrange numbers $\sqrt{5}, \sqrt{8}, \sqrt{221}/5, \dots$ are defined as constants $c$ satisfying $$\left| \xi - \frac{p}{q} \right| < \frac{1}{c q^2}$$ for every irrational number $\xi$ and infinitely many $p$ and $q$ except for finitely many $\xi$.
But, as I know, this theory was originated from Dirichlet, Hurwitz, etc. They were all later than Lagrange. I wonder why these numbers were named after Lagrange.
Something of a guess, although I have published on the topic. Dickson credits Lagrange with proving that the integer of smallest absolute value represented by an indefinite binary quadratic form, indeed all small numbers that are primitively represented, occur in the Lagrange chain of neighboring "reduced" forms. This is Theorem 85 on page 111 of Introduction to the Theory of Numbers.
In short, it would appear someone named this after Lagrange.
Also, this gives the really quick proof of the overall Markov theorem in this topic, see Cusick and Flahive. I was able to adapt their proof to produce a mirror image of the extremal forms.