In the summer at Avignon a little Disneyland train conveys tourists from the Palais des Papes to the summit of a little hill behind the cathedral called the Rocher des Doms. In the garden on this hill there is a most remarkable sundial; the visitor himself provides the gnomon, and a linear scale on the flagstone pavement shows him where to stand at different dates. His shadow then falls towards one or other of a ring of fixed pyramidal markers which indicate the hours of the day.

In 1742 Goldbach sent a letter to Euler which contained the conjecture that:

Every even number greater than 2 is the sum of two prime numbers.

It is easy to verify Goldbach's conjecture numerically up to some quite high limits but no proof has yet been found. I have long had the uneasy feeling that the conjecture is not about the prime numbers at all but would equally apply to any ‘sufficiently dense’ sequence of odd numbers. However a recent attempt to generalise the sieve of Eratosthenes has led me to regard Goldbach's conjecture with a good deal more respect.

Typical cyclic configurations of five-bar linkages are shown in Figs. 1, 2, 3, 4. It is proposed to solve the problem of finding all the cyclic configurations of a five-bar linkage of which the lengths of the bars are given. It will be shown that the solution depends upon an equation of the seventh degree which can be expressed in a very simple form.