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