Dreyfus in 1966 presented a second-order stagewise successive approximation procedure for computationally solving discrete-stage optimal control problems. The purpose of this paper is to provide a proof that Dreyfus's procedure generates a Newton step at each successive approximation. Hence, his 1966 procedure is the earliest stagewise Newton method in the literature. In the proof, we exploit an important property that Dreyfus's formulation readily adapts to an efficient second-order backpropagation that evaluates the Hessian matrix (as well as the gradient vector) in a stagewise fashion. After the proof, we also discuss the issues of practical Newton methods for solving a linear-quadratic problem with a globalization strategy.