Some years ago D. I. Muller [1] developed a method for the solution of algebraic equations, approximating them by quadratic equations. The method proved extremely efficient in many tests. However, the theoretical discussion given by Muller can hardly be considered as adequate, in my opinion, as his convergence proof culminates in a vicious circle (cf. the formulae (15)?(17) of his paper). In this article I give a detailed and rigorous theoretical discussion of the method from the point of view of local convergence. Some parts of the argument run closely parallel to the proofs given in my book [2] in the discussion of w-point inverse interpolation. This book will be cited hereafter as S. This article begins with a special chapter about Newton's divided dif? ferences, which had to be added as the usual discussion of these differences could not be directly used in the present connection. I discuss in this article generally approximation by polynomials of order n ? 1 (n ^ 3) as does Muller in his paper. My attention was drawn to this method by Dr. V. Pereyra of the Nu? merical Analysis Institute of the University of Buenos Aires. I am grateful to him for discussions concerning Chapter II of this article.