Colin Maclaurin was one of the most distinguished of the brilliant coterie of Scottish mathematicians that marked the course of the eighteenth century. He was born at Kilmodan in Glendaruel, Argyleshire, in February, 1698, and was the youngest of a family of three sons. His father, John Maclaurin, who was minister of the parish and a man of mark in his day, died when Colin was but six weeks old. His mother, after residing for some time in Argyleshire, removed to Dumbarton, and died in 1707 when Colin was nine years of age. The orphans were taken in charge by their uncle, Daniel Maclaurin, minister of Kilfinnan, who faithfully discharged his duty to them.

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.