  We define a noncommutative residue for classical Euclidean pseudodifferential
operators on a torus of arbitrary dimension. We prove that, up to
multiplication by a constant, it is the unique trace on the algebra of
classical pseudodifferential operators modulo infinitely smoothing operators.
In the case of the two torus, we show that the noncommutative residue is the
semiclassical limit of a noncommutative residue defined on classical
pseudodifferential operators on noncommutative two tori.
