The object of the following paper is to draw attention to a general and powerful but quite elementary method of transformation, by means of which the properties of a conic may be inferred from those of a circle. It has many striking analogies with the method of perspective transformation, and a simple geometrical connection with it. It is, however, of much simpler character for constructive purposes, and though it is intended to show later on how well the method is adapted to serve as an introduction to modern methods, it is hoped that readers will begin by dismissing from their minds all notions of cross ratios, homographic ranges, etc., and regard the theorems presented to them from the point of view of a student who has mastered his Elements of Plane Geometry.

The Junior Mathematical Association of London, of which I expect very few of you have even heard, is an association in a sense of the word rather different from that in which it is used in the title of the Association I am now addressing. It is the name adopted by some eight large schools (of the Public School type) in and around London for a joint Mathematical Society which has been running amongst the Sixth Form mathematical specialists in those schools for nearly five years. It is essentially an association of institutions rather than of persons, and this type of organisation will at once be seen to be necessary by any one who considers the effect of the changing personnel of Sixth Forms. It possesses therefore no strict membership roll, no subscription, and, incidentally, no harassed treasurer. All the same, it is alive, as I hope to be able to show you by the end of this short paper. It might also be recorded at this point that a somewhat similar association exists amongst some of the Classical Sixths.

Although some years have elapsed since the first appearance of Professor Halsted’s translations of Lobatchewsky’s Theory of Parallels and John Bolyai’s Science Absolute of Space, they are not so widely known and studied as they deserve to be. There is nothing in the mathematics of these small volumes of a kind to deter any one from reading them; they are short and elementary treatises, disclosing a system of geometry quite distinct from Euclid, by rejecting the axiom concerning parallels. The Science Absolute of Space is a most interesting exposition of one branch of non-Euclidean geometry; but it is unfortunately extremely condensed, and occasionally obscure. It would have been a great advantage if explanatory notes had been added to the English translation.

Oxbridge Tripos examination questions originating from the turn of the century have had a marked effect on the teaching of mechanics both at School and at University. These questions, reproduced in the standard textbooks of the day, were the stable diet of the undergraduate population over so many years and set the pattern for a style of question in which the student applies mechanical principles by rote without ever questioning the validity or relevance of those principles. An unfortunate consequence of this state of affairs is that mechanics has been largely overlooked as a vehicle for the teaching of mathematical modelling. There have been moves recently to make questions more “relevant” and it is not unknown to see illustrations of “real” boats, shotputters etc. However, the student is still expected to answer the question “mechanically”, if the reader will excuse the pun, without any discussion of the modelling process. How refreshing it is, even with first year undergraduates, to look at the problem of a tin can being knocked off a wall by a bullet and, instead of asking the students to apply two conservation laws in order to determine the speed with which the can hits the ground, to ask under what assumptions the two conservation laws would appropriately model the physical situation? It is quite staggering how the lecture theatre erupts with ideas, suggestions and interest.