My address last year was devoted to the consideration of a few rules or statements made in teaching, with, as it seemed to me, insufficient explanation, in the hope that I might suggest some line of action, in a few particular cases, which would be helpful to members of our Association engaged in elementary teaching.

We start, as Euclid did, with the vague notion that, if A and B are two magnitudes of the same kind, the ratio of A to B is some relation between them in respect of magnitude, i.e. in respect of relative magnitude, independent of the actual magnitude of each.

When I began the preparation of this paper, I turned up the bulky report of 1912 published by the Board of Education. In the article by Prof. G. H. Hardy I found this passage :

Ladies and Gentlemen,—Before this meeting commenced, Professor Lodge asked me if I was doing much examining now, to which I answered “No, none.” Then he asked what qualification I thought I had got for speaking on the best method of examining. I do not want to speak much on the best method, but I think it is time that members of this Association had the opportunity of blowing off a lot of steam, and I hope that many will have something to say about the school certificate examinations. My desire is that many different people should give their views and their criticisms.

In the Gazette for January, 1925, I gave a geometrical construction for a cardioid. Thé curve was obtained as the envelope of chords of a circle drawn to a certain rule. It has recently come to my notice that the rule can be extended to cover epi- and hypo-cycloids.

Following a method more popular in the seventeenth century than now, I propose to explain certain views on the definition of a right angle by an imaginary conversation between an inventor of mathematical models and a student.

There are not many integrals of the type ∫ arc sin F(t) dt whose values can be expressed in terms of functions already defined. Nor is the inverse sine function allowed to play its free part in suggesting what new functions are necessary for the evaluation of integrals. The moment an inverse sine appears in an integral, the integrand is transformed into one involving the direct function.

An interesting programme of lectures on a wide range of subjects was given by the Mathematical Department of Section A of the British Association at the Leeds meeting in 1927.