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.

The Eulerian numbers are not strangers to readers of the Gazette but they are not normally associated with the subject of this article, despite the similarity of their names. This article seeks to use Eulerian numbers in generalised telescoping sums, a role that is a powerful extension of an established technique – the Euler summation formula.

My first introduction to the idea of an average, at the age of about eight, left me wondering why anyone would want to add up some numbers and then divide by how many there were. A helpful illustration (not forthcoming at the time!) might be as follows:

‘Suppose three boys are going on an outing and have each brought biscuits with them. They find that Alan has 7 biscuits, Jim has 6 and Tom has 11. They decide to pool their biscuits and share them out equally. So each gets (7 + 6 + 11)/3 = 24/3 = 8 biscuits. This is the average number of biscuits brought. So, they have the same total number of biscuits as they would have had if each had brought 8.’

Pascal's triangle, the Binomial expansion and the recurrence relation between its entries are all inextricably linked. In the normal course of events, the Binomial expansion leads to Pascal's Triangle, and thence to the recurrence relation between its entries. In this article we are going to reverse this process to make it possible to explore a particular type of generalisation of such interlinked structures, by generalising the recurrence relation and then exploring the resulting generalised ‘Pascal Triangle’ and ‘Binomial expansion’. Within the spectrum of generalisations considered, we find exactly four of particular significance: those concerned with the Binomial coefficients, the Stirling numbers of both kinds, and a lesser known set of numbers – the Lah numbers. We also examine the combinatorial properties of the entries in these triangles and a prime number divisibility property that they all share. Thereby, we achieve a remarkable synthesis of these different entities.

In 1632, Henry Gellibrand, then the Professor of Astronomy at Gresham College, London, arranged for the publishing of the Trigonometria Britannica (T. B.) by Adrian Vlacq in Gouda the following year: the work consisted of two Books, and sets of tables of natural sines in steps of one hundredth of a degree to 15 places, as well as tables of tangents & secants to 10 places, together with their logarithms. The explanatory Book I was the last work of Henry Briggs (1559-1631), Savilian Professor of Geometry at Oxford, and was devoted mainly to the construction of his table of sines; while Book II, written by the youthful Gellibrand on the instigation of the dying Briggs, his mentor, contained instructions and examples on the use of logarithms in solving trigonometrical problems.

We define a Watt quadrilateral to be a quadrilateral with a pair of opposite sides of equal length. See Figure 1.

2. Linkages

The Watt linkage (Figure 2) has equal-length cranks AD and BC, A and B fixed, and coupler bar CD. It was devised by James Watt about 1784 to constrain the steam-engine piston rod, connected at E, the midpoint of CD, to approximate straight-line motion over a limited range.