  The purpose of this article is to put forward the claim that Hurwitz's paper
"Uber die Erzeugung der Invarianten durch Integration." [Gott. Nachrichten
(1897), 71-90] should be regarded as the origin of random matrix theory in
mathematics. Here Hurwitz introduced and developed the notion of an invariant
measure for the matrix groups $SO(N)$ and $U(N)$. He also specified a calculus
from which the explicit form of these measures could be computed in terms of an
appropriate parametrisation - Hurwitz chose to use Euler angles. This enabled
him to define and compute invariant group integrals over $SO(N)$ and $U(N)$.
His main result can be interpreted probabilistically: the Euler angles of a
uniformly distributed matrix are independent with beta distributions (and
conversely). We use this interpretation to give some new probability results.
How Hurwitz's ideas and methods show themselves in the subsequent work of Weyl,
Dyson and others on foundational studies in random matrix theory is detailed.
