The work presented here was prompted by curiosity concerning a structural detail of the ironwork supporting the overhead cover of the railway platform at Stockport, Cheshire. The detail – see the front cover which shows the roof support – comprises an elliptical ‘quadrant’ joined to vertical and horizontal members of the supporting structure, with a circular ring fitted interstitially so as to touch the elliptical arc and the two straight members. The question arises: what is the radius of the circular ring in terms of the parameters of the ellipse?

As I write (Spring 2012) I am aware that I have been a member of the Mathematical Association for fifty years. I joined during the annual conference in 1962 when Dr Combridge was the main figure in the society. The highlight was a session at the conference (at King's College, London) by Geoffrey Matthews showing how matrices might be introduced into the school curriculum. Some treated this as hilarious, others as a signpost to the future. My friend Frank Budden and I regarded it as a premature ploy, which, we thought, might not tum out to be such a good idea. We subsequently wrote a successful book Mathematics through Geometry (1964) arguing in detail why the modem mathematics revolution—which was then gathering pace—might end in tears. We identified spatial imagination as the heartland of mathematics, and contrasted it with a denial of spatial imagination in the abstract calculi which were all the vogue at the time.

During the 19th century various criticisms of Euclid's geometry emerged and alternative axiom systems were constructed. That of David Hilbert ([1], 1899) paid particular attention to the independence of the axioms, and it is his insights which have shaped many of the further developments during the 20th century.

We can, from his insights, define an affine plane as a set of points, with distinguished subsets called lines such that

Axiom 1: Given two distinct points, there is a unique line containing them both.

Axiom 2: Given a line L and a point, p, not contained in L, there is a unique line containing p which does not intersect L.

Axiom 3: There exist at least three points, not belonging to the same line.

A pleasurable aspect of mathematics and its teaching is to review the diversity of ways in which theorems are proved. Especially in elementary branches, there are various kinds of proof: using (or avoiding) spatial geometry, analytic or coordinate geometry, common algebra, vectors, abstract algebras, matrices, determinants, the differential and integral calculus, and maybe mixtures thereof. Further, sometimes a proof of one kind is elegant while another is clumsy, or one proof of a theorem suggests why it follows while another proof is not perspicuous. There is also the question of whether a proof is direct or indirect (for example, proofby contradiction).