I Admire the ingenuity of the method by which Prof. Nanson establishes the expansions of sin x and cos x as power series in x. But I must confess that his attempt to ‘simplify’ the ‘accepted accurate’ proofs of these expansions seems to me a fundamentally mistaken one. Nor can I admit that the result is at all satisfactory ‘from the elementary didactic point of view.’ His proof is to my mind essentially an artificial verification, and altogether unnatural, in the sense that it has no place in any natural and logical way of developing the ideas which lead to the result. I speak with diffidence as one who has had far less experience of teaching than Prof. Nanson. But I am convinced that my criticism is a fair one: and I think that many of the proofs usually given, in English text-books, of many of the theorems of ‘Higher Trigonometry,’ are open to fair criticism on similar grounds, when (as is too seldom the case) they are not to be condemned for ‘hopeless complexity’ or utter lack of force.