22. To obtain a Fagnano pair of points on an ellipse whose axes are given, it may be considered sufficient to construct the corresponding Fagnano quadrilateral. That is, having drawn the line hk (Fig. 14) with the points p and z marked, and having described a circle on hk as diameter, it is only necessary to draw through z some chord c1c2 and to join the sides of the quadrilateral c1hc2k.

All who are interested in educational matters cannot fail to be conscious of the widespread spirit of unrest that is abroad. The energy of this association and other similar bodies is proof enough of the increased interest in the efforts that are being made to examine and recast, in the light of modern work, the existing forms of secondary education, and to grapple with the new problems created by a changing system of external forces. Criticism within this sphere is as healthy as it is inevitable, just because it secures a careful scrutiny of the grounds for proposed alterations; and it should serve as a safeguard against fantastic and ill-considered developments of methods and ideas that under proper limitations may be both serviceable and stimulating. Any proposal to introduce a new variety of an old subject, and still more an apparently fresh theme, into the curriculum can only lead to beneficial results if a thorough investigation of the principles involved and the purpose in view has first led to a discrimination between the essential and non-essential, the useful and undesirable, elements of the subject.

The last ten years have witnessed many changes in the methods employed in the teaching of arithmetic, which to a very considerable degree are the direct product of the activity and exertions of this Association. Two signal benefits may be mentioned ; firstly, uniformity of method in elementary processes has been secured ; and, secondly, the educational value of practical applications to mensuration, physics, etc., has received due recognition. The object of this paper is to raise for discussion a question of quite a different character.

The time has long passed when the teacher of elementary geometry, however elementary, can efficiently exercise his individuality in devising improved methods of exposition without taking some cognisance, however small, of the study of non-Euclidean geometry which has grown up out of the failure to prove what may be described as “Euclid’s Parallel Postulate.”

A paper has recently been published in the Proceedings of the Edinburgh Mathematical Society by Prof. Carslaw, of Sydney, N.S.W., containing an elementary interpretation of the Bolyai-Lobatschewsky geometry, which is so simple and so easily intelligible that I strongly recommend every teacher of mathematics or science to read it. It does away altogether with the necessity of forming conceptions of space which appear contradictory to experience, and shows the possibility of constructing a non-Euclidean geometry in ordinary Euclidean space.

Pursuing the comparison with the ordinary methods of Geometry, we may now say that we have established the foundations of Projective Geometry, and have now to make the transition from projective to metrical methods; the distinction between which, though commonly assumed, has never, so far as I am aware, been clearly expressed. The corresponding distinction in the theory of Order is that between unique collation in general, and what I shall now proceed to define as transformation.

§ 1. The rhythm, of mathematical progress. To the question how the aim described in the preceding lecture is to be realised, the whole of the rest of the course is proffered as an answer. Here, it, will be sufficient to summarise the main features of the. methods which are later to be given in detail.

The professed aim is to make school mathematics a reproduction, as faithful as the difference of the situations permits, of the mathematical activities of the great world. The method must, then, be based upon an analysis of the general course of development which these activities display. Remembering, as before, that this kind of analysis always introduces an artificial simplicity into a naturally complicated matter, we may yet lay it down that mathematics advances by the constant repetition of a normal cycle of progress—a cycle in which three typical phases may be distinguished. In the first, the heuristic phase, the mathematician, face to face with a new type of problem, devises a new notation or mathematical method to deal with it. In the second, the formal phase, the new notation or method is studied apart from the immediate needs of practical application. The conditions and the range of its validity are investigated, its wider possibilities are explored, and its relation to other methods examined. In the third, the application phase, the extended notation or the perfected method becomes once more an instrument for the solution of problems, and is found to be applicable successfully over a field often many times wider than the area in which it originated. Any one of these problems may, in turn, become’ the starting point of a new cycle.