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 :

Cambridge is honoured this year by its selection for the first meeting held in England by the International Congress of Mathematicians. The Cambridge School has played a notable part in the development of mathematics in Britain, and it may be of interest if I briefly summarize the leading facts of its history, which is indeed closely connected with that of the University. The School is of more than respectable antiquity and inevitably it has been sometimes flourishing and sometimes the reverse. Here I desire rather to bring out the salient features in its history than to discuss details or individual achievements.

I propose in these papers to demonstrate, or rather to indicate the nature of the demonstration of, the proposition that—All Geometry, whether projective or metrical, may logically be regarded as merely a particular concrete application of the general theory of Order, as defined by Boundaries.

It is true that Geometry, as ordinarily understood, is something more than this; it is concerned with the actual subjective conceptions we entertain of Space, and so far provides matter of discussion to the Psychologist; and it is concerned with the actual objective measurements we make upon physical bodies in Space, and so far concerns the Physicist; while in both these respects it extends beyond the region of pure Logic into those of Epistemology and Philosophy generally. But so far as the pure Mathematician is concerned, every geometrical theorem is embraced in the Theory about to be discussed.

1. Def. The reciprocal (1/N) of any number (N) will be termed a Binal, Tertial, Quintal, ... Decimal, ... r-mal fraction, when expressed in the scale whose radix is r = 2, 3, 4, ... 10, ... r respectively. In this paper the developments will be given for the most part in a general form in the r-scale; the examples will be in the scales of r = 3 and 5. The notation is similar to that of ordinary decimals. Thus (see Tables at end):

§ 1. Definition of aim. A distinguished psychologist recently addressed a body of our American colleagues on the teaching of physics. “Your aim in the teaching of physics,” he told them, “should be—to teach physics” The advice was excellent—especially from one who is apt to be regarded as an intellectual Strong Man, full of esoteric knowledge about the development of mental muscle. Substituting “mathematics” for “physics” we may well adopt the maxim with all its implications, negative and positive.

§ 2. Negative implications. Beginning with the negative implications our amendment of Prof. Baldwin's dictum may be expanded as follows. In the first place, the central purpose in teaching mathematics is not to “train the power” of reasoning, of generalising, of “mental accuracy,” etc. The fallacies embodied in the persistent heresy of “formal training” have been repeatedly exposed, and need not detain us here. The time should soon come when an educational writer may ignore them. The chemist does not feel bound to begin his text-book by rebutting the phlogiston-theory, nor the geographer to demolish the sophistries of the flat-earthers.

I desire in the first place to express my thanks to the members of the London Branch of the Mathematical Association for the honour they have done me in electing me to the office of president. I esteem it a privilege to take part in the efforts the Association is making to bring about improvements in the methods of teaching Mathematics.

In what position does the work of the Association now stand ? Is it in fact in the position described by Sir J. J. Thomson in his address to the Association of Public School Science Masters? He is reported to have said that he had come to the conclusion that if you have intelligent masters and small classes it does not matter much what theory of education you adopt, and if you have not these, well, it does not much matter either.

In selecting a topic for his annual address, your President has a wealth of choice sufficient to be an embarrassment. On the one hand there are the varied activities of the Association itself ; and though it may need courage and skill to deal with the delicate questions of the present moment, there is less need of exceptional virtues in reviewing the achievements of the past, or anticipating the enterprises of the future. On the other hand there are the doings in the world of Science outside the Association, with some one of which at least your President for the time being may be presumed to have a speaking acquaintance.

The front of the scientific army is long and straggling, and the lines of communication are thin and broken. If your President happens to have any news from the front he will probably be a welcome messenger among those whose duty keeps them elsewhere. If this is not too ambitious an aim, I propose on the present occasion to attempt a brief report on a recent successful movement by the Astronomical regiment which I trust may be of interest to their Mathematical comrades. I speak of it from the outset as a regimental movement ; for though the chief credit is undoubtedly due to individuals, even the brief report I shall make will shew how much one owed to another, and he to a third: how different lines of advance converged and coalesced: and, in short, how the whole movement was organised, not in the artificial manner which man has come to associate with the word, but in the sense in which the advances of Nature, including those of Natural Science, are organised.