  We consider the large number of particles limit of a periodic Toda lattice
for a family of initial data close to the equilibrium state. We show that each
of the two edges of the spectra of the corresponding Jacobi matrices is up to
an error, determined by the spectra of two Hill operators, associated to this
family. We then show that the spectra of the Jacobi matrices remain almost
constant when the matrices evolve along the two limiting KdV equations. Finally
we prove that the Toda actions, when appropriately renormalized, converge to
the ones of KdV .
