Mr. I. R. Vesselo: The problem of selecting a suitable programme of films and film strips for this meeting was not a very difficult one; the choice is very limited. Like the evening-class system, which grew out of a desire, as much to make use of empty classrooms as to educate, the educational film grew partly out of a desire to make wider use of films whose first function was to fill out the normal entertainment programme. An audience would tolerate a little education whilst waiting for the “big picture”

I WOULD like first to thank the Association for the honour they have done me in inviting me to lecture at this meeting. It is, however, an honour which carries with it a certain danger. When a mathematician comes to lecture on a subject in which he has specialised, he is always liable to suffer from the delusion that what has gradually become plain and straightforward to him will at once appear plain and straightforward to others. It may be that I am suffering from this delusion in thinking that I can present the main idea of the geometry of numbers in one lecture.

In this account of the incorporation of a new subject into the school curriculum I shall assume that it is agreed that the teaching of statistics in schools is desirable. This aspect of the subject has already been fully siscussed at previous meetings of this Association, and in other places. I am one, and I hope there are others present in the audience, who believes not only that it is desirable to teach statistics in schools, but also that far more is gained than is lost by its introduction. In this paper I therefore present the story of one attempt to put into practise what has been preached so long.

When teachers consider the problems of mathematics in the school they recognise that they have to make a definite effort to simplify the study of these difficult questions, but, it seems to me, they work in a frame which is imposed by the curriculum and their temptation is to think very little of the child. The problem, however, has to be faced as a whole and the aim of this article is to discuss the teaching of mathematics with special reference to a child-centred point of view.

We prove some theorems on commutative involutions in a “real” projective geometry in which cobasal homographie ranges may have 0, 1 or 2 self-corresponding points (and therefore a conic and a general line in its plane have 0, 1 or 2 points of intersection).

The main aim of this note is to give a new foundation of Euclidean geometry based on the notions of “cross” and “signed angle”. The reckoning with crosses, or directed angles, advocated by D. K. Picken has many merits: the rules are few, general, and simple, and one proof covers a number, often large, of special cases. I will first sketch the method, assuming Euclidean geometry already known.