We develop a method to compute the Hermite-Humbert constants
$\gam_{K,n}$ of a real quadratic number field $K$, the analogue of the
classical
Hermite constant $\gam_n$ when $\funnyQ$ is replaced by a quadratic
extension. In the case $n=2$, the problem is equivalent to the
determination of lowest points of fundamental domains in
$\H^2$ for the Hilbert modular group over $K$, that had been studied
experimentally by H. Cohn. We establish the results he conjectured
for the fields $ \funnyQ@(\sqrt{2})$, $\funnyQ@(\sqrt{3})$ and $\funnyQ@(\sqrt{5})$. The method relies on the characterization of
extreme forms in terms of perfection and eutaxy given by the second
author in an earlier paper.