My function in this discussion is the humble one of attempting to supply a rough scaffolding into which the durable parts of the edifice will be fitted by the other three speakers.

The teaching of algebra has never been dominated by a single work in the way in which Euclid for so long a time dominated the teaching of geometry For that reason, perhaps, the revolts in geometry have corresponded to quiet reforms in algebra, and the problems involved, though equally important, have never received the same amount of attention. On the teaching of the subject in schools, excellent books exist; there is Sir Percy Nunn’s classic, and more recently, a book by Durell, and much relevant matter in Godfrey and Siddons, Teaching of Elementary Mathematics.

My part in this symposium is to deal with the introduction of the fundamental ideas of algebra, as far as the stage usually known as the “Certificate stage”. As I shall attempt to show later, this is an extremely vague phrase, but it will be sufficient for the present to note that I am speaking of that part of algebra which is a compulsory subject for the great majority of secondary school pupils and that it is no part of my plan to deal with the needs of universities or any specialised developments of the subject. These needs, in my opinion, should not be allowed to influence unduly teaching at the “compulsory” stage.

Although I am to speak “from the University point of view” I wish to consider not only how the small number of boys who will go to a University may best be prepared for their work there, but also the general problem of the teaching of Algebra in the upper forms of schools. This problem is connected with a larger question, whether the balance of subjects is right in present-day school teaching in mathematics, and perhaps you will permit me to start with a short digression on this point.

In 1934, the Board of Education asked the Mathematical Association to submit its views on the mathematical aspects of the organisation and interrelation of schools. In its report, the sub-committee appointed for this purpose, of which I was a member, made it clear that at present the mathematics up to School Certificate standard in schools of secondary education was inadequate in its scope and its direction for all but those pupils who will continue to read mathematics in a university course. If we remember that probably only some 10% of the population of our public and secondary schools passes on to any form of university career, and if we agree with the opinion of the sub-committee, we have before us a problem vast in its importance and extent and one which should be tackled without delay

The subject of discussion this morning is “The bearing of higher geometry on the school course”. The difficulties of geometry are well-known, though not all of you will entirely endorse the opinion expressed about 350 years ago by a certain Hubert Languet, who wrote to Sir Philip Sidney as follows: “I know not whether it is wise to apply your mind to geometry, though it is a noble study and well worthy of a fine understanding, but . . . you are not overcheerful by nature, and it is a study which will make you still more grave; and as it requires the strongest application of the mind, it is likely to wear out the powers of the intellect, and very much to impair the health.”

In opening this discussion on the influence of the university on the school course in geometry I have two dangers to avoid. Many of the things which it would seem natural to say are true and a few of them may be important, but when I wrote them down and took a second look at them, I saw that they were not concerned in any specific way with geometry; we sometimes forget that there is a Section of the British Association which exists to rescue the rest of us from the generalities and platitudes to which teachers are only too prone. The other danger is of entering so far into particulars as to compose a syllabus round which we might be expected to quarrel instead of talking about principles on which we can hope to agree; here I am fortunate in the subjects chosen by the speakers who are to follow me, for where what I have to say is most in need of illustration, their expositions will provide you with examples worked out at the appropriate length.

It has been the fashion in some branches of mathematics to use a rather high sounding and mystifying terminology for comparatively simple ideas and I fear that Geometry has fallen to the lure. It is, of course, essential that new conceptions should have new titles. There is, however, the very serious danger that, through familiarity and, perhaps, moments of carelessness more may be read into such titles than was originally intended. I propose to discuss Infinity in Euclidean Geometry, as I find continually, especially with young students, that the phrases “point at infinity” and “line at infinity” have led to concepts of concrete existence which, whatever may be the case in other philosophies and branches of science, are in this one quite unjustified and which would be exceedingly awkward if they were correct.

Elementary geometry will be supposed to have its usual meaning as euclidean metrical real geometry of the plane. I propose to point out that, while “points at infinity” and “the line at infinity” may be properly introduced in conformity with the axioms, the “circular points” cannot be so introduced. They do not belong to this geometry; and those results which do properly belong to it, and are often made apparently to depend on the existence of the circular points, should therefore be expressed with no mention of these points.

My contribution to this discussion reminds me of the ill-prepared Divinity student, who, asked in an examination to “Discuss the textual criticism of the Minor Prophets”, began, “Far be it from me to criticise these great and holy men, be it mine rather to give the list of the Kings of Judah and Israel”. For what I have to say has about the same degree of relevance to the topics discussed by the last two speakers as the student’s answer had to the question asked, and my remarks will, moreover, be more in the nature of a sermon than of a mathematical lecture. I hope, however, that they will serve as an illustration of the general subject under discussion, even though they may not bear very directly upon what has already been said.

The fundamental idea expressed in this paper is that objects and phenomena are only seen in part, vision being limited to a plane in time about second thick, and showing us things as they are for the instant only but if we wish to picture the Completeness of Things, or Phenomena, we have to include, certainly, time as well as space, we have to retain the image of the movement of the rolling ball, the running machine, the growing plant, the dancing girl, through either the cycle or phase of movement or the life history, as appropriate.