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

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.

If you were to ask a random sample of students from your school if they liked mathematics, what do you suppose their responses would be? Almost certainly most of them would respond negatively and some would do so with considerable vigour. But what of the others? Some students do like mathematics and we might well ask why this is so? Do you suppose it is because these students realise that knowledge of mathematics is a necessary survival skill in today’s increasingly technological society? Surely the answer is no. Do you suppose it is because they realise that mathematics has many useful real life applications or that lack of knowledge of mathematics will effectively prevent them for entering college in most disciplines? Again the answer is almost surely no. Certainly mathematics is relevant and we should teach our students that this is so and we should teach them useful and appropriate applications as well. At the same time, it is my experience that student attitudes towards mathematics are much like they are towards exercise—in general, they do not exercise because it contributes to good health, but because exercising is usually done in the form of a game or a competition that they find attractive in and of itself. Similarly, what turns students on to the study of mathematics and provides the needed motivation for learning are the remarkable patterns and surprising results with which our subject is so replete. Successful teachers realise this fact and utilise it to maintain a high level of interest and enthusiasm which enables them at the same time to teach ideas which, while useful and necessary, may be more prosaic.

In the problem proposed by A. G. Ioachimescu in 1895, it is asked to be shown that the sequence , defined by , for each , is convergent and its limit lies between -2 and -1.

There have been given many generalisations and other results regarding Ioachimescu’s problem in the literature (see, for example, [2], [3], [4, Theorem 1, parts (a) and (b)], [5, problem P2, parts (i) and (ii)],).

The values of the Radii of Gyration have been worked out for some solid polyhedra, polyhedra with faces only, (shell or box-like), polyhedra with edges only (skeleton), polyhedra with vertices only. The results are given in the tables.

There has been a long history in the approximation of ellipses by circular arcs in order to simplify their construction and manipulation. Such approximation was of use for a wide variety of applications, in fields such as mathematics (generating figures), astronomy (analysing orbits), art (marking out large oval frames for ceiling painting), architecture (building masonry arches, floor plans, etc), and, more recently, the conversion of fonts from a general conic specification to circular arcs. Documented evidence goes as far back as the Italian Renaissance when various schemes were published by the architect Sebastiano Serlio in the sixteenth century. More contentiously, it has been argued that fifteen centuries previously the Romans used such approximations when designing and building their amphitheatres.

In 1952 W. E. Roth published two theorems, one of which has come to be known as Roth’s removal rule and (slightly generalised) goes as follows. [Recall that square matrices M, N are similar when there is an invertible matrix R such that RMR-1 = N. The matrix entries can be elements from any field, although for simplicity we shall call them ‘numbers’.]

I WOULD like first to thank the Association for the honour they have done me in inviting me to lecture at this meeting. It is, however, an honour which carries with it a certain danger. When a mathematician comes to lecture on a subject in which he has specialised, he is always liable to suffer from the delusion that what has gradually become plain and straightforward to him will at once appear plain and straightforward to others. It may be that I am suffering from this delusion in thinking that I can present the main idea of the geometry of numbers in one lecture.