The rationale for the introduction of the system of three points for a win in the Football League in 1981-82 was, presumably, that more reward for winning games would encourage more positive attitudes from teams and that the consequent more attractive and attacking football would bring in bigger crowds. Certainly the philosophy of many away teams, or at least their managers, that they began a game at 0-0 with one point to hang on for had become commonplace in recent seasons.

The theory of duality is well known as a useful tool in the study of convex three-dimensional polyhedra. For example, if we know the types of polyhedra that can be circumscribed about a sphere, duality provides an immediate answer to the question as to what types can be inscribed in a sphere. Theorems like Euler’s Theorem (V – E + F = 2 where V, E and F are the numbers of vertices, edges and faces of a convex polyhedron) are self-dual in the sense that duality simply interchanges the values of V and F.

1. The English student of Mathematics finds in his text-books a proof that the uniform convergence of a series of continuous functions is a sufficient condition for term-by-term integration. He knows that there are convergent series which cannot be integrated term-by-term,f and that there are series, not uniformly convergent in the interval of integration, which can be thus integrated. The last remark applies particularly to the Fourier’s Series of an ordinary function with discontinuities.

Let Q be a convex quadrilateral ABCD. We denote by TA, TB, Tc, TD, the four triangles BCD, CDA, DAB, ABC, respectively

The barycentres (or centroids), orthocentres, incentres and circumcentres of such triangles determine other quadrilaterals in their turn that we call the quadrilateral of the barycentres, of the orthocentres, of the incentres and of the circumcentres, respectively. We denote these quadrilaterals by Qb, Q0, Qi, Qc, respectively.

In this article I review some of the more advanced results concerning cyclic quadrilaterals, including a number that are new to me and possibly, therefore, to readers. Proofs of theorems are not given, but I give a clear indication of the methods I have used to verify them. The Cabri software package has been used to indicate the correctness of each result. In certain cases the coordinate geometry involved is so complex that Derive had to be used in the proofs. One of the results I have been unable to prove, and this is therefore labelled as a conjecture, though the accuracy of Cabri, which allows figures to be dragged around, is such that we may be confident that it is true.