This note describes an example which I have found useful for introducing ideas in linear algebra and group theory. It is simple and yet it is neither trivial nor too abstract.

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).

Imagine six drawing pins stuck in the wall. If we stretch threads, either red or blue, between every pair of pins, we inevitably create at least one triangle all of one colour. This we shall call Theorem 1: the proof follows.

Consider any pin. From this emanate five threads (since there are five other pins), of which three must be of one colour. We may without loss of generality assume that they are three red threads.

Today, “sets” are taught in schools at a variety of levels and the basics of set theory are reviewed in many college mathematics courses. After students have gained some facility with the rudiments of set theory, it is instructive for them to see what would happen if our set theory were a little different. This not only reinforces their knowledge of the standard theory, but exposes them to other possible mathematical structures. This article shows one way in which this might be accomplished.

The reproductions shown here are of p. 49 and Fig. XVI (which refers) from A sure guide to builders by Batty Langley, published in 1729 and dedicated to Thomas Scawen Esq., of Carshalton. The reference is to the construction

“of oval-like figures” and the implication, towards the bottom of the page, is clearly that something very close to “an ellipsis” is what Mr. Langley had in mind. The poor plate engraver was hard put to it to draw an ellipse from the instructions in the text and a careful examination of his effort reveals that his final solution was to draw, very faintly, an ellipse and try to fit tangents to it from the divisions along the sides.

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 many situations, for example in teaching convergence to students not specialising in mathematics, an approach to the idea of a limit is required which is easily motivated, mathematically acceptable, and yet not technically terrifying. The usual outcome is a compromise which lies uneasily between these aims. Here, I try to present an introduction to convergent sequences (of real numbers) which meets these objectives; it can easily be adapted to accommodate limits of functions. The approach is based on the sandwich rule (or squeeze principle) and the intuitively straightforward idea of the limit of a monotone sequence.

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.

The article Problems, solved and unsolved in the Gazette for last March (60 (411), 1–11) drew reactions from a number of readers. There is not sufficient space to show in full all the proofs sent in, but I will try to indicate the methods used. For statements of the problems themselves I must ask readers to refer to the earlier article.

. . . 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.

At some time in the eighties or nineties of the last century the Rev. William Done Bushell, a mathematics master at Harrow School, bought a portrait painting at an auction sale at a house in Harrow. It does not seem to have had any pedigree but bore the inscription “Robt. Recorde M.D. 1556”. It was thus presumed to depict the Tudor physician and mathematician Robert Recorde (1510?–1558) who wrote some of the earliest books in English on mathematics and was the first to use the symbol “=” for equality. No other portrait was (or is) known.