Every proof I've seen of Euler's Theorem (that $\gcd(a,m) = 1 \implies a^{\phi(m)} \equiv 1 \pmod m$) involves the fact that the units of $\mathbb{Z}/m\mathbb{Z}$ form a group of order $\phi(m)$. While this is a perfectly good proof, I have to wonder if it was the one that Euler used. I know that there are fairly old precursors to group theory, but it still seems incongruous.

Thus, my question is: Are there proofs of Euler's Theorem that do not use group/ring theory? In particular, what proof (if any; I don't know whether Euler actually discovered the theorem) did Euler use himself?