The Calculation of the roots of an algebraic or transcendental equation in a single unknown is a problem of frequent occurrence. For a real root the usual procedure is to obtain a first approximation to the required quantity, graphically or otherwise, and to improve this approximation by successive applications of the Newton-Raphson process. The extension of this process to the improvement of an approximate solution of a set of non-linear simultaneous equations in n unknowns is fairly obvious, but it does not seem to have received much attention in text books, although the case of two unknowns is dealt with in Ref. 2.