In the standard text-books on trigonometry the object of the authors is to reduce geometry to algebra, to give geometrical proofs of a few fundamental formulae and afterwards to develop the subject on algebraic lines. When alternative geometrical proofs are given they are not often designed to illustrate the steps in the algebra. De Morgan’s Double Algebra and Hayward’s Vector Algebra may be mentioned as works in which free use is made of geometry, but the emphasis laid on the interpretation of a vector as a complex quantity makes their method appear rather advanced. The slides which I shewed to the annual meeting were intended to illustrate the