Group theory has been used in the study of various properties of musical scales. The cyclic group C12 provides a mathematical model for the tempered intervals (see for example Budden [1] Chapter 23, where the diminished seventh chords are shown to correspond to an appropriate subgroup of C12 and its cosets).

Many mathematicians' experiences with distances in the settings of geometry, analysis, and topology can lead to the impression that the only worthwhile or ‘reasonable’ distance functions are metrics. We hope to convince the reader otherwise.

Recall that a metric for a set X is a function d: X × X → [0, ∞) satisfying all of the following metric axioms:

1. About ten years ago Neugebauer and Sachs published their Mathematical Cuneiform Texts in which the contents of Plimpton 322 were analysed for the first time. The numerical values contained in this text are given in Table I. The theory put forward by Neugebauer and Sachs was in my view not consistent with known facts and in the Proceedings of the Academy at Amsterdam I put forward a different interpretation. Since then I have made several short comments in connection with this text, but up to the present I have not given an explicit interpretation of the errors occurring on Plimpton 322.

A sequence of athletic records forms by definition a monotonic sequence. Since it is reasonable to assume this to be bounded, it follows that a limit must exist to future performance. It has been proposed [1, 2] that this can be estimated by use of a curve-fitting procedure on existing data, such as may be found in many compendia of athletic records (e.g. [3]).

I don’t know how many readers of this journal would have been watching the draw for the opening matches of the 1990 football World Cup, made in Rome on Saturday, 9 December 1989, and televised worldwide. But I do hope that most of you that did—and who were paying attention!—at least thought “Hey, wait a minute, that can’t be right!” Sure enough, the draw was flawed: it was not as fair to all concerned as it might have been. This was a great pity given that the world football authority, FIFA, were at pains to stress how fair these proceedings would be, and the agonies of FIFA trying to be seen to live up to their “fair play” motto were indeed prolonged. The errors lay not in the process of drawing balls from urns itself, but in the algorithm used to define the drawing procedure, as I shall explain below. The problem was caused by FIFA’s decision to keep the four South American countries involved apart and, in particular, by their inability to implement this decision in the most fair manner. There is no allegation of anything being “fixed” here, just one of incompetence on behalf of football’s governing body. That there should be errors of this kind in such an overtly public affair is, however, yet another sad indictment of the general lack of understanding of matters numerical and probabilistic in the public at large.

The lower polynomials are inextricably linked to the symmetries of polyhedra and Platonic solids [1, 2, 3], and the quartic is no exception; its alter ego is the regular tetrahedron [4]. In this article we present a solution to the problem of aligning the vertices of a tetrahedron with the roots of a particular quartic. After establishing the size of a quartic-equivalent tetrahedron, we derive a triple-angle expression for the alignment rotation, analogous to that for the cubic [5].

This article considers the use of a range of computer tools for aiding geometric exploration, together with a suitable algebraic representation for objects connected with the triangle. The techniques are applied to produce new results in the geometry of the triangle. This is an extended version of part of the contribution: How do computers change the way we do mathematics? given at the Association’s 1994 Easter Conference.

The perimeter of the ellipse x2/a2 + y2/b2 = 1 is 4J (a, b), where J (a, b) is the ‘elliptic integral’

This integral is interesting in its own right, quite apart from its application to the ellipse. It is often considered together with the companion integral

Of course, we may as well assume that a and b are non-negative.

Kendall and Moran, in introducing their book on Geometrical Probability (Griffin, 1963), give as illustrations three solutions of a problem due to Bertrand, that of finding the probability that a “random chord” of a circle is longer than the side of the inscribed equilateral triangle. The first solution regards the random chord as being generated by the two points on the circumference which it joins, and which are uniformly and independently distributed. In the second the chord is assumed to have a fixed direction perpendicular to a given diameter, and its point of intersection with this is uniformly distributed over the diameter. In the third, the foot of the perpendicular from the centre is uniformly distributed over the circle. The probabilities in these three cases are respectively ⅓, ½, and ¼.

who arrived at the result through discussions in an undergraduate class at the University of Southampton.

Of the various metrics which can be placed on the co-ordinate plane R×R, the Euclidean metric is universally known, and two others are becoming more widely known in schools. With the co-ordinate notation for points x = (x1, x2), these metrics are