In this paper we describe a method for solving the Euclidean reconstruction problem with a perspective camera model by incrementally performing an Euclidean reconstruction with a weak perspective camera model. With respect to other methods that compute shape and motion from a sequence of images with a calibrated perspective camera, this method converges in a few iterations, is compositionally efficient, and does not suffer from the nonlinear nature of the problem. With respect to factorization and/or affine-invariant methods, this method solves for the sign (reversal) ambiguity in a very simple way and provides much more accurate reconstructions results.