I Quite agree with Mr. Hardy ( Gazette, vol. 3, p. 284) in his contention that the real basis of the power-series for sin x ; and cos x is given by the two equations But I am inclined to doubt whether at present the difficulties of exposition will not outweigh the advantages of his method (iii); and for this reason the following method may be worth considering, although it is a fairly obvious modification of my previous note ( Gazette, vol. 3, p. 85). The only preliminary knowledge implied is that of the elementary differential calculus, and of Argand’s diagram.