In view of the growth and development of the Association in recent years, and the wider interests which it now represents, it is desirable that the Teaching Committee should be re-constituted on a broader basis.

With this object the following regulations have been made for the appointment of a General Teaching Committee and three Special Committees :

Mr. H. D. Ellis has been good enough to compile the following statistics : A list of members of the Association on January 1st, 1911, classified according to locality, shows a total of 680, of whom England contains 531, Wales 21, Scotland 23, Ireland 14, Isle of Man 1, Channel Islands 2, South Africa 23, Canada 6, Australia 9, New Zealand 4, West Indies 1, Malta 1, India 17, Brazil 1, Egypt 3, Italy 1, Japan 2, Boumania 1, Syria 1, U.S.A. 18. The 531 members in England are distributed thus : London 146, Bedfordshire 7, Berkshire 13, Bucks. 12, Cambridge 31, Cheshire 6, Cornwall 2, Cumberland 5, Derbyshire 4, Devonshire 11, Dorset 1, Durham 4, Essex 11, GIos. 26, Hants. 19, Herts. 12, Hereford 1, Isle of Wight 2, Kent 23, Lancashire 25, Leicester 6, Lines. 1, Middlesex 8, Monmouth 1, Norfolk 8, Northants. 3, Northumberland 4, Notts. 3, Oxford 13, Rutland 4, Shropshire 6, Somerset 8, Staffs. 4, Surrey 21, Sussex 18, Warwickshire 13, Wilts. 2, Worcs. 9, Yorkshire 38.

§ 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.

The character of the modern treatment of elementary mathematics is such that mathematics might reasonably be described as the Science of Classification. The mathematician may be supposed to begin as a collector, whether of butterflies, minerals, or even of cigarette cards, does not matter. Let us consider that he has collected so many different kinds of things as to become virtually a collector of collections, each collection having its own classification, and that his next step is to compare these classifications with each •other, that is, to become a classifier of classifications. The mere comparison of classes requires the i-iea of cardinal number, since the only thing which can be said about two classes, irrespective of the nature of their individual members, is the number of individuals in each class. But the comparison of classifications which do not consist merely of single classes introduces ideas which are essential to the further development of the idea of number, and to the development of Geometry, and which should therefore precede that development. It is in this super-classification or comparative study of classifications that the foundations of mathematics are to be sought, and the question therefore arises, whether the presentation of the rudiments of mathematics should not be modified so as to recognize this fact. Of the fundamental ideas of mathematics, four especially, namely those of correspondence, order, group, and of multiple correspondence, seem to lend themselves, in their elementary treatment at any rate, to the method in which they are regarded as belonging to the Science of Classification, and will be considered in that light here, the last three being regarded as developments of the first.