  The elliptic Calogero-Moser Hamiltonian and Lax pair associated with a
general simple Lie algebra $\G$ are shown to scale to the (affine) Toda
Hamiltonian and Lax pair. The limit consists in taking the elliptic modulus
$\tau$ and the Calogero-Moser couplings $m$ to infinity, while keeping fixed
the combination $M = m e^{i \pi \delta \tau}$ for some exponent $\delta$.
Critical scaling limits arise when $1/\delta$ equals the Coxeter number or the
dual Coxeter number for the untwisted and twisted Calogero-Moser systems
respectively; the limit consists then of the Toda system for the affine Lie
algebras $\G^{(1)}$ and $(\G ^{(1)})^\vee$. The limits of the untwisted or
twisted Calogero-Moser system, for $\delta$ less than these critical values,
but non-zero, consists of the ordinary Toda system, while for $\delta =0$, it
consists of the trigonometric Calogero-Moser systems for the algebras $\G$ and
$\G^\vee$ respectively.
