Hermann Weyl, in [6], defined a functional calculus to deal with the unbounded selfadjoint operators of differentiation and multiplication by a position coordinate. In this paper we examine this calculus in the case of bounded operators.' We let x be a vector (xi, * , x ,) in Rn, dx=dxi ... dXn, (X, x)=XIxI+ * +Xnxn. We let A be an n-tuple of bounded selfadjoint operators (A1, * * *, An) and (X, A ) =X1A1+ * * +XnAn. Define exp(i(X, A)) by the usual power series, which converges in norm since (X, A) is bounded. If i(X) is the Fourier transform of f, i(X) = (2ir)-nI2ff(x)exp(i(X, x))dx, and if both f and 7 are in L1, we define f(A) = (2r)-n/2fJ(X)exp(-i(X, A))dX. This is the Bochner integral, whose convergence in norm is guaranteed since e-i(,A) is a continuous function of X and is of norm 1. The questions which shall concern us are: in what way is f(A) continuous as a function of A or as a function of f?, what are some classes of functions for which f(A) is easy to calculate?, to what extent are other functional calculi special cases of this one?, and in what ways can we extend this calculus? We should point out that the chief interest of this calculus is that the operators involved do not have to commute, which is not true of the calculi based on the Cauchy integral formula or on the spectral theorem. However it is easy to verify that in the case where the A commute, f(A) is the same as the operator defined by the spectral theorem. For example, in the one variable case we have