In a comparatively short paper, no complete survey of the range of agricultural problems whose solution involves statistical methods can be given. I have therefore chosen to confine myself to one particular theme, the development of the modern field experiment, even though this has meant that I must present a rather unbalanced picture on account of the omission of other important topics such as the use of sampling methods. During six years in the Statistical Department of the Rothamsted Experimental Station, I spent much time in the planning and analysis of a large annual programme of field experiments, and I have illustrated my paper with examples drawn from the work of this Station, at which many of the new ideas in field-experimentation were first tried. I have not attempted to discuss mathematical details at all fully, for the pure mathematics of statistics is a large subject in itself; my present concern is to show some of the practical applications of the theory.

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).

My purpose this evening is to submit to you an other sketch of the general course and the circumstances of mathematical study at (Cambridge in my student days between fifty and sixty years ago. It does not set out to be an estimate of the state mathematical knowledge at that date It is not a comparison (or would contrast be the better word ?) between the empirical natural philosophy then in vogue and the rather speculative (slightly Aristotelian) physical theories that now absorb a small world of seekers after new Truth. It is only a picture of that academic, life as it then appeared to an unfledged student, now drawn as faithfully as memory will allow.

This story is an object lesson in the art of picking other people’s brains.

After one of my fifth form had pointed out for me the result in Classroom Note 248 (ii) (June 1971), it occurred to me that some attractive patterns might result from re-writing Pascal's triangle with the numbers reduced modulo n. I tried it at first with n = 2, using a red dot for odd numbers and a green dot for even. The pattern was, as I had hoped, pleasing. Since we had an Open Day in the near future, I persuaded another fifth-former, P. W. Swinyard, to try the same thing for other small n. Attractive patterns resulted for n = 3, 4, and 5, a rather less attractive pattern for n = 6, but nice ones again for n = 7, 8, and 9. (I recommend readers to try this for themselves.)

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:

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]).

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.

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.