Thank you for the honour of being your president. Being such was not an ambition of mine – it was not even considered as a possibility. However, I have enjoyed the experience, and will for the few more hours which are allowed to me.

In the course of preparing this address, I looked at the article about presidential addresses written by Mary Bradburn in the 1996 Centenary Issue of the Mathematical Gazette. That was a mistake – too many different models to follow, each sounding more interesting than my own ideas. One odd thing, to my mind, and something not special to the MA, is having a President give the presidential address at the very end of the term of office. I always feel that hearing someone give a lecture helps me to find out a bit about them; so a presidential address is a chance for all to learn about their President – but too late, too late!

At school I used to love the geometry of the circle, but in my own career I had little opportunity to teach it. Now in semi-retirement my work with Teasers in the Sunday Times brings me into contact with many entertaining aspects of mathematics, not least being the occasional piece of geometry. In this short article I will quote four potted versions of recent Teasers, each of which relies on some interesting piece of mathematics, and I shall sketch in each case how that property can be deduced neatly using circles.

An ellipse is ellipse-shaped, of course! Every mathematician knows what an ellipse looks like – at least, this is generally true in the Cartesian plane, with the standard distance function or metric. However, there are other metrics which can be applied to the plane ℝ2 and it might be useful to know what an ellipse looks like when distances are defined by these metrics. The set of points under consideration will be ℝ2, the real plane, and the metrics under consideration will be the standard metric, the Manhattan (or taxi-cab) metric and the maximum modulus metric.

Ann Hirst and Keith Lloyd have derived many interesting properties of the Cassini ovals rr′ = a2, r and r′ being the distances of a variable point P from two fixed points A and B, which we shall find it convenient to take a unit distance apart. r, r′ are often referred to as bipolar coordinates and by relating them in different ways, a variety of interesting families of curves can be defined; such families can be associated with other families which are their orthogonal trajectories. The purpose of this article is to draw attention to a number of such cases.

Consider the motion of a particle of unit mass acted upon by a homogeneous, central, conservative force f(r), where r is the distance between the centre of force and the particle. In this connection, homogeneity means the following property of the force: there exists a constant α such that for every c

In Isaac Asimov's novel Foundation and Earth (Grafton 1987) the inhabitants of the planet Solaria have evolved from early Earth space explorers. They have developed a society with many robots and few people. The small population of Solarians live as far from each other as possible. These solitary people are confined to their estates, which are distributed evenly across the planet. In this article I investigate the pretty problem of even distribution of points on a sphere. With the exception of one piece of spherical trigonometry the work needed to derive my (admittedly restricted) set of results is routine for sixth-form mathematicians but has the interesting pay-off announced in the title.

Isoperimetrical problems in ℝ2 and ℝ3 belong to the classical problems of geometry. As their name indicates one has to find geometrical bodies or areas with maximal volume, within a particular class of bodies or areas with equal surface measure. In the 19th century the isoperimetrical problem was solved for several polyhedron classes in ℝ3 [1, pp. 133-137]. Since then, a lot of research work has been devoted to more general problems where the admissable bodies were contained in the class of all compact convex subsets of ℝn or in a specific subclass, respectively. Their solutions usually can be characterized as zero sets of algebraic inequalities in surface measure and volume. The derivation of these inequalities and their solutions usually require sophisticated techniques. Of course, assertion (*) below is the expected generalisation of the well-known two-dimensional case.

This article arose out of a first year undergraduate class project concerned with divisors. The class was presented with a sheet containing the prime factorisations and a list of divisors for the natural numbers up to 120. Among the many observations and conjectures which the students formulated some were related to the standard result about the number of divisors in terms of the prime factorisation. One conjecture made by several students which was not in the textbooks was d(n!) = 2n-1 where d(k) denotes the number of positive divisors of k. The result is true for n = 2, 3, 4, 5, which takes one to the limit of the data sheet given to the students (5! = 120). Unfortunately it breaks down for n = 6, since d(5!) = 16 but d (6!) = 30 and not 32. Several students reported that they had reached their conjecture on the basis that they thought the number of divisors of n! would double every time n increased by 1.

I just couldn't see it; that was my difficulty with many mathematical abstractions: if I couldn't visually process a maths concept I would resort to rule, hoping that some day it would make sense.

Take the Pythagorean Theorem. It has been around for two millenniums plus, readily understood by all academically exposed to the liberal arts by the time they reach puberty; but me, I knew it, a2 + b2 = c2 and the picture of three little squares huddled around the perimeter of a right triangle, that was no help. Why not three equilateral triangles around the perimeter Of course a2 √3/4 + b2 √3/4 = c2 √3/4 is a bit ungainly, but it would work.