Robert Harbin, in Waddingtons Family Card Games, gives an item called Prediction. You ask a friend to name two values of cards and to shuffle a deck. Then you bet that his two values will come up together as the deck is turned over. Harbin says he has worked out the odds as 32 to 19 in favour of success— “but I could be wrong. See what you make it.”

1. Poncelet’s well-known theorem states that if two conies S and S′ in a plane are such that an n-gon exists whose vertices lie on S′ and whose sides touch S, then an infinite number of such polygons exist. When this is the case, a relation holds between the invariants of the two conies. The explicit determination of this relation is due to Cayley, whose result is quoted by Salmon in a footnote on p. 342 of his Conic Sections (reprint of 1929). With a slight change of notation, Cayley’s result takes the following form.

“A derivative is a linear transformation which approximates a function near to a point.” I heard this definition for the first time in 1964, in a lecture at Oxford on the calculus of several variables (the lecturer was D.A. Edwards). The words came as a revelation, and from then on I had a new conception of differential calculus. At school, I thought that I knew about calculus because I could write out proofs of differentiation ‘from first principles’. I enjoyed going through the hoops and proving that δy/δ tends to a limit, but somehow the idea of differentiation never became ‘part of me’.

Applications of convex coordinates are many and varied in mathematics today. Our first interest in convex coordinates grew out of their use in linear programming, e.g. [1]. That interest has remained quite lively as the study of convex coordinates continues to yield pleasing and sometimes unexpected results, e.g. [2,3]. Another name for these coordinates which may be more familiar to the reader is that of barycentric coordinates as originally given by the German mathematician Möbius. In this article we shall confine our attention to the Euclidean plane, but convex coordinates may be defined in Euclidean spaces of any dimension.

In 1906 the editor of this journal made a plea for more articles on “the purely pedagogic side of mathematics”. As he emphasised: “the object of the Gazette is to improve the teaching of our subject in our schools”. Since then such articles have appeared with varying frequency. What rarely appears is any article specifically devoted to research in mathematics education. The divorce between mathematicians and researchers in mathematics education, often made worse by the former’s distrust of the latter, would seem doubly unfortunate. In this article, based on part of a talk given in Madrid, I wish to look briefly at how research in mathematics education developed and at some of the difficulties associated with it.

Some of my distinguished predecessors have ably dealt in their Presidential addresses with questions that are most closely related to the aims of our Association, namely, the teaching of Mathematics in Schools. Others have sought to speak on subjects in which they were actively interested. I have followed this precept and accordingly I propose to address you upon a subject closely related to my own scientific interests, namely, the exploration of our universe. Naturally, I cannot claim to speak with authority over its whole range.

Scout‘s pace, according to my old boy scout manual, consisted of alternately walking and running 22 yards. I believe it is also used by military scouts and messengers who wish to travel at a moderate speed without too much effort. As I have also discovered, it is useful for elderly, overweight athletes with a low pain tolerance.

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About 75 B.C., Cicero the orator, stumbled upon an ancient monument. The lettering was illegible, but ‘on clearing away the brambles and thorns’ he discovered a curious design—a sphere with circumscribed cylinder.