During the past 20 years the notion of set has been introduced into school mathematics courses not only at the secondary but also at the primary level. When one considers that neither Newton nor Gauss made any explicit use of the concept, it is rather remarkable that teachers have thought it desirable to teach the concept so early in their school courses. I should like to consider briefly some of the many possible reasons, historical and pedagogic, why “set” has become a school topic. One reason is, of course, the belief which Russell and Whitehead fostered in their Principia mathematica in the first decade of the century, that all mathematical concepts can be reduced to the concept of set. I don’t intend to discuss here the extent to which they failed to justify that belief, although I shall have occasion to mention some of the difficulties which their programme ran into. A second reason is the important part which sets have played in mathematical research this century.

Have you ever looked at a dartboard and wondered why it is numbered in its distinctive order?

If we ignore the bull, the inner, and the double and treble rings, the standard dartboard takes the form of a circle divided into 20 equal sectors numbered as in Fig. 1. (There are boards to be found in the Manchester area—and perhaps elsewhere—with a different arrangement, but the one shown is by far the commonest.)

A frieze pattern is an array of numbers arranged in staggered rows, like the bricks in a wall. In the case of a multiplicative frieze pattern the top and bottom rows consist entirely of ones, and for each ‘diamond’ of four elements

the unimodular rule

ad = bc + 1 (2)

holds. These frieze patterns have many curious properties, and for further details the reader is referred to the article Triangulated polygons and frieze patterns by J. H. Conway and H. S. M. Coxeter which appeared in Gazette 57, 87–94 and 175–183 (1973).

Let three non-overlapping circles, no two with equal radius, be drawn in the plane. To each pair of circles belongs a pair of common external tangents. Monge’s theorem is that the three points where pairs of common tangents meet are collinear (see Fig. 1).

To anyone remotely interested in the large number of megalithic stone ‘circles’ that are to be found in Britain, any discussion of their geometry, metrology or astronomy brings one name to mind—Professor Alexander Thom. The professor has spent a lifetime scouring these islands, and has produced surveys for many sites, from which he has evolved a most sophisticated picture of the society of the late Neolithic Age. His best known claim is the discovery of a universal unit of length, the megalithic yard (2·72 ft), which he says was used with extreme accuracy (±0·003 ft) during the whole period of circle construction, approximately 3000 to 1500 BC.

In previous years I have taught my third-year Graph Theory option as a standard 25-lecture course of the “definition-theorem-proof” variety. Last year I gave it in an entirely different format, without consequent change of syllabus. I divided the material into nine units each containing text, exercises and solutions to exercises, which were Xeroxed and distributed to the students at the rate of one unit a week. I gave only one lecture a week instead of the usual two or three. And for two periods each week the students were divided into groups of about half a dozen, who met on their own to discuss the current week’s unit.

. . . In one of the lands I visited I met a young explorer called Noisrot, who worked in the mountainous and undeveloped central regions of Spuorgetinif. He was one of many government employees who were charting these unexplored regions, looking for minerals and for passes in the rugged central mountain range towards the north, for it was hoped one day to open up the interior. Whenever possible Noisrot lived in the wild. He spent his days walking in the hills, listing the flora and fauna, collecting specimens, and surveying; his evenings were spent writing reports and reading those prepared by his colleagues.