  We give a new construction of cyclic homology of an associative algebra A
that does not involve Connes' differential. Our approach is based on an
extended version of the complex \Omega A, of noncommutative differential forms
on A, and is similar in spirit to the de Rham approach to equivariant
cohomology. Indeed, our extended complex maps naturally to the equivariant de
Rham complex of any representation scheme Rep A.
  We define cyclic homology as the cohomology of the total complex (\Omega
A)[t], d+t \cdot i, arising from two anti-commuting differentials, d and i, on
\Omega A. The differential d, that replaces the Connes differential B, is the
Karoubi-de Rham differential. The differential i that replaces the Hochschild
differential b, is a map analogous to contraction with a vector field. This new
map has no commutative counterpart.
