A transcendence measure for $\pi^2$

A new proof of the fact that $\pi^2$ is transcendental is proposed. A modification of Hermite's method for an expressly constructed Nikishin system is used. The Beukers integral, which was previously used to prove Apéry's theorem on the irrationality of $\zeta (2)$ and $\zeta (3)$ is a special case of this construction.