  M. Toda in 1967 (\textit{J. Phys. Soc. Japan}, \textbf{22} and \textbf{23})
considered a lattice model with exponential interaction and proved, as
suggested by the Fermi-Pasta-Ulam experiments in the 1950s, that it has exact
periodic and soliton solutions. The Toda lattice, as it came to be known, was
then extensively studied as one of the completely integrable
(differential-difference) non-linear equations which admit exact solutions in
terms of theta functions of hyperelliptic curves. In this paper, we extend
Toda's original approach to give hyperelliptic solutions of the Toda lattice in
terms of hyperelliptic Kleinian (sigma) functions for arbitrary genus. The key
identities are given by generalized addition formulae for the hyperelliptic
sigma functions (J.C. Eilbeck \textit{et al.}, {\it J. reine angew. Math.} {\bf
619}, 2008). We then show that periodic (in the discrete variable, a standard
term in the Toda lattice theory) solutions of the Toda lattice correspond to
the zeros of Kiepert-Brioschi's division polynomials, and note these are
related to solutions of Poncelet's closure problem. One feature of our solution
is that the hyperelliptic curve is related in a non-trivial way to the one
previously used.
