In his letter in the March 1995 issue of the Gazette (79, p. 123), John Sharp points out that the correct viewing position for a picture is with the eye opposite the central vanishing point, that is, on the normal to the picture plane at the point of meeting of all lines in the scene represented that are perpendicular to the picture plane. In order to obtain a true view of the scene depicted, it is also necessary that the eye be placed at the correct distance from the picture.

The human body is a complicated machine whose movements involve many different joints, operated by a great many muscles. For that reason it is easy to get bogged down in detail when thinking about walking and running from a mathematical point of view.

Any position of the human body (or of any other jointed mechanism) can be described by giving the angles of joints. The number of angles needed for an unambiguous description is the number of degrees of freedom of the mechanism. For example, the position of a hinge joint is described by just one angle: a hinge allows only one degree of freedom. The human knee is a hinge. The ankle, however, allows rotation about two axes – you can tilt your foot toes up or toes down, and you can also rock it sideways so that the sole faces inwards towards the other foot – so it gives two degrees of freedom. The hip is a ball and socket joint allowing rotation about any axis through the centre of the ball, but any position can be described by just three angles (measured, for example, in three planes at right angles to each other), so it allows three degrees of freedom. In total, there are six degrees of freedom in each leg, making twelve in all, and suggesting that we need twelve equations of motion to describe walking. If we took account of the flexibility of the foot and the movements of the arms, we would need more.

Given the critical dependence of our civilisation on mathematics, it is paradoxical that it lacks any general acclaim or popularity, and usually invokes fear and mysticism. Perhaps this is because, for most of us, the process of acquiring mathematical skills is somewhat protracted and painful. It is often perceived as a nightmare topic, which engenders a defence mechanism of open ignorance that is often lauded. In truth, most people can become reasonably competent. However, the time and effort needed to acquire the formal rigour for even a modest capability is often seen as excessive by modern standards.

Everyone knows that the condition for the quadratic to have two equal roots, i.e. to have a repeated root, is that its discriminant should be zero. We should remark, at the outset, that we are concerned only with ordinary polynomials whose coefficients are complex numbers. Indeed, little is lost if a reader assumes that all our polynomials are real, i.e. have real numbers for all their coefficients, though their complex roots must be considered as well as their real ones.

In part I of this article [1], we described developments in mathematical logic before 1931. Gödel’s famous paper On formally undecidable propositions of Principia Mathematica and related systems, was published in January of that year. In it Gödel established two theorems:

1. (1) Incompleteness: For any consistent theory at least as strong as arithmetic there are sentences which can neither be proved nor disproved in the theory.

2. (2) Unprovability of consistency: The consistency of such a theory cannot be proved within the theory.

We are going to outline, with minor adjustments to notation, Gödel’s actual proof of (1) given, in translation, in [2, pp. 5–38].

John Branfield’s request [1] for the mathematics of bowls was too tempting to resist. What follows comprises a simple model for the curve upon which the bowls move, an explanation of how and why they move as they do and an attempt to refine the model of the curve. The search for a model proved to be addictive and effectively displaced many other activities. One of the reasons for this was the way in which the investigation involved so many people with interesting contributions to make. In order to give a flavour of the investigation, I have taken the liberty of giving a somewhat personal rather than strictly mathematical account, the mathematics being no less important.

Not long ago I came across the following quiz in the Sunday supplement of the Spanish newspaper El País: Check that it does not matter how you take any four numbers from the following array; provided that no two of them lie in the same row or column, their sum is always the same.

This reminded me of magic squares: square matrices where the sum of the elements in each line (row or column) and diagonal is always the same. Magic squares are a topic in recreational mathematics. Usually one looks for magic squares with elements that are integer, positive and different from each other, e.g. 1,2,…, n2 for a n × n as in

According to a Chinese legend, matrix A appeared engraved on the carapace of a turtle about 2000 BC. On the other hand, matrix B appears in Albrecht Dürer’s engraving Melencholia (1514) [1].

The Simson line property is normally associated with points on the circumcircle of a triangle. It is embodied by the following theorem.

Given any triangle ABC and a point P in the plane of the triangle, if perpendiculars from P on to the sides BC, CA, AB meet those sides at L, M, N respectively then L, M, N are collinear if and only if P lies on the circumcircle of triangle ABC. The line LMN is then known as the Simson line of P.

In the March 1996 issue of the Gazette, [p. 36] the following conjecture by Sir Bryan Thwaites appeared: ‘Take any set of n rational numbers. Form another set by taking the positive differences of successive members of the first set, the last such difference being formed from the last and first members of the original set. Iterate. Then in due course the set so formed will consist entirely of zeroes if and only if n is a power of two.’

1. Background

From the end of the 17th century until the 1980s, most approximations to the value of π made use of Gregory’s series:

which converges for values of tanθ between -1 and +1. Since , the simplest, and most slowly convergent, series of this kind is

Writing x = cotθ, the series becomes

Expressions for π/4 have been devised which involve the inverse cotangents of larger numbers, causing the series to converge much faster; of such expressions (referred to herein as ‘identities’) probably the most celebrated is ‘Machin’s Formula’:

This problem refers to an observation made by George Gamow and Marvin Stern [1]. They found that concerning multi-storey buildings whenever one requires the lift on a particular floor, the probability of the lift coming from above or below varies according to the height of the floor concerned. It was observed that on the second floor of a seven-storey building, the probability of the lift coming from below is very small whereas it is much greater when the sixth floor is concerned. This is very similar to the case of finding the probability of a bus coming from a particular direction at a stop on a bus route.

Before reading this article, you may like to turn to page 297, where you will find a quiz on ‘famous-name’ mathematics. In each of the questions it seems reasonable to suppose that the mathematician indicated in the title of the result is the discoverer or inventor of the result. But a careful look at mathematical history shows, in each case, that there is a ‘story behind the story’. As you read on, keep in mind that even the answers given below are subject to revision as new historical data is found and interpreted.