  In a recent preprint on arXiv Roland Bacher showed that the number $p_d$ of
non-similar perfect $d$-dimensional quadratic forms satisfies $e^{\Omega(d)} <
p_d < e^{O(d^3\log(d))}$. We improve the upper bound to $e^{O(d^2\log(d))}$ by
a volumetric argument based on Voronoi's first reduction theory.
