About a year ago, to solve a puzzle to which my attention had been drawn, I found it necessary to obtain an approximate numerical solution of extreme accuracy to a set of four trigonometrical equations in four variables. The method by which I succeeded in avoiding some of the labour involved in a direct application of the classical method of approximation I have described elsewhere, but it has been suggested that an account of it will interest readers of the Mathematical Gazette. In this paper the method is introduced by means of a particular pair of equations in two variables, but it is applicable to any set of n independent equations in n variables, whatever integer n may be.