  As rewards of reading two great papers of Hermite from 1873, we trace the
historical origin of the integral Niven used in his well-known proof of the
irrationality of $\pi$, uncover a rarely acknowledged simple proof by Hermite
of the irrationality of $\pi^2$, give a new proof of the irrationality of
$r\tan r$ for nonzero rational $r^2$, and generalize it to a proof of the
irrationality of certain ratios of Bessel functions.
