Several members of the Association expressed their conviction, at the General Meeting and at the subsequent Dinner, that there should be more articles in the Gazette, and more papers read at the Meetings, on the purely pedagogic side of mathematics. It is right and necessary that the general development of the various branches of our Science should not be lost sight of in an organ such as the Gazette, and the reader must be kept in touch by reviews with what is being done in school text-books both at home and elsewhere. But the special aims of the Association (and the object of the Gazette is to secure by publicity the realisation of those aims), is to improve the teaching of our subject in our schools. With this view the Editor is in cordial agreement. He asks the readers of the Gazette to come to his assistance in the matter, to propose subjects for discussion, to offer papers, to ventilate their special difficulties in a “ Question and Answer “ column, in short to do for the Gazette what no Editor can do single-handed. If this proposal is taken up with any vigour by our readers, it should be quite possible to arrange for the regular appearance in the Gazette of a series of Symposia, consisting of (1) articles from the pens of its members and (2) a general summing up of the pros and cons of the points at issue by an authority who combines academic distinction with teaching experience. The latter is easy to find. It is the former who as yet have not come to the front in our Association. The Council has said in the Annual Report that this last year has been a quiet year. We are not sure that this may not be meant for a piece of biting criticism, suggesting that though more remains to be done we have not done it. Be that as it may, we appeal to the readers of the Gazette for their cordial co-operation in providing the material for such discussions as are indicated above. Apart from our differences as to detail in the processes of teaching, there are many other topics which should be of interest to the majority as well as to special groups of our members. Those who are engaged in Preparatory Schools should be glad to expound their views of what is feasible with boys of the age with which they deal, and the Public School Teachers of Mathematics should be glad to indicate the lines along which training should proceed, and any way of improving the preparation of the boys who are ultimately to come under their care. The fairness of the papers set at the various examinations, and their strict adherence to the official syllabus may also prove a fruitful subject of discussion.

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.

The locus of the points of contact of tangents from A to the conics of the confocal system with foci B, C is the same curve as the locus of the feet of normals from A to conies of the system. For any tangent AP to one of the conics is a normal AP to the orthogonal conic of the system through P. Hence if the coordinates of A referred to BC and the perpendicular bisector of BG be x0y0 the equation of this locus is the eliminant of λ between

We recognise as decadence the substitution for Euclid’s theorem I. 20, “Any two sides of a triangle are together greater than the third side,” of the objectionable pseudo-axiom, “A straight line is the shortest line from one point to another.”

This phrase, as Hilbert has well said, is of necessity meaningless when the concept “length of a curve” has not been defined. Every book on elementary geometry which has introduced as axiom or definition of a straight line any phrase equivalent to “ A straight line is the shortest distance between two points,” every book which has at its beginning as axiom or definition of a straight line any phrase equivalent to “A straight line is the shortest distance between two points,” has been in error. We believe the advances in regard to the foundations of elementary geometry make timely the rejection of this piece of decay from the body of the elements, and that the time is ripe for the return at this point to the pristine purity of Euclid.