  Let M be a closed manifold. Wodzicki shows that, in the stable range, the
cyclic cohomology of the associative algebra of pseudodifferential symbols of
order \leq 0 is isomorphic to the homology of the cosphere bundle of M. In this
article we develop a formalism which allows to calculate that, under this
isomorphism, the Radul cocycle corresponds to the Poincar\'e dual of the Todd
class. As an immediate corollary we obtain a purely algebraic proof of the
Atiyah-Singer index theorem for elliptic pseudodifferential operators on closed
manifolds.
