In this paper we describe a method for solving the Euclidean reconstruction problem with a perspective camera model by incrementally performing Euclidean reconstruction with either a weak or a paraperspective 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 computationally efficient, and does not suffer from the non linear nature of the problem. With respect to methods that use an affine camera model (such as factorization) the method described below solves for the sign (reversal) ambiguity in a very simple way and provides much more accurate reconstructions results. We give a detailed account of the method, analyze its convergence based on numerical and experimental considerations, and test its efficiency with both synthetic and real data.