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).

When I had the honour of being made President of the Mathematical Association, I did not foresee the unusual responsibility which it entailed. It was my intention to take as the theme of a presidential address the consideration of some aspect of those special subjects to which my own researches have principally been directed. Events have forced me to abandon that intention. It is useless to discuss abstract questions in the midst of dominant practical preoccupation. We callnot disregard the present crisis in European civilization. It affects every function of life. In the harder struggle for existence which lies before the nation, all departments of national effort will be reviewed for judgment. The mere nccessity for economy in resources will provoke this reformation.

On 6 October 1997 NASA plans to launch a space probe named Cassini which, if all goes to plan, will follow a convoluted path which will take it past Venus in April 1998 and again in June 1999 before it repasses close to Earth in August 1999. By then it will have gained enough speed to set out to more distant planets and after passing Jupiter at the end of the millennium (i.e. at the end of the year 2000) it should reach Saturn on 25 June 2004, where it will spend four years studying that planet and its satellites.

1. Introduction

A question of the following type recently appeared in a mathematics contest for high school students in the U.S.A.

Consider the usual experience of a student who progresses far enough in school mathematics to begin to study the calculus. After motivation of the topic of “rates of change”, simple differentiation of polynomials is learned. More advanced functions are then considered, and eventually the student meets the product, quotient and chain rules. The result: with enough algebraic accuracy and persistence, the student can determine derived functions for virtually any sufficiently wellbehaved combination of standard functions.

The great mathematician Henri Poincare [1] described the method of proof by mathematical induction as mathematical reasoning par excellence. Induction is, undoubtedly, a method of great significance for mathematics. Unfortunately, induction is also a rather complicated form of proof. Furthermore, it is taught in schools at sixth form level. In fact, it is probably the only method of proof explicitly taught in British schools. Perhaps it is not surprising that some students find the method difficult to apply. In view of the importance and difficulty of the method, mathematical induction is worthy of further attention. In this article, the history and equivalents of mathematical induction are discussed.

The Russian Multiplication Algorithm is frequently discussed in history of mathematics courses and in math education courses for elementary and secondary school teachers. Students are intrigued by the algorithm as it allows one to multiply any two whole numbers knowing only the multiplication and division facts for the digit 2.

Unfortunately, most textbooks do not address the natural questions which students ask about the algorithm: “Why are the even numbers crossed out in the left-hand column?” and “Can this be done with numbers other than two?” In a survey of eight textbooks by eight different pubishers, only one textbook answered the first question and none of the textbooks addressed the second question.

The defining property of a rectangle is that it is a quadrilateral whose angles are all equal (or are right angles, which amounts to the same thing).

The opposite sides of a rectangle are equal. If a and b are the lengths of adjacent sides of a rectangle, such a rectangle can be described as an abab rectangle (or an (ab)2 rectangle). The area of the rectangle is ab.

As far as I understand the latest version of the national curriculum and the new A level syllabuses, which make no serious reference to irrational numbers, students leaving school will not necessarily know that there are two sorts of rectangle: those coverable by a set of identical squares, which can therefore be drawn on squared paper; and those not, such as the golden rectangle. An (ab)2 rectangle is coverable by squares if, and only if, a/b is rational.

This article concerns the evaluation of the ‘logsine’ integral

We shall encounter it in several guises. Indeed, standard integration techniques used below readily show that (1) has the same value as the following integrals:

En passant, it is worth noting that forms (6) and (7) are the best behaved for numerical integration.

I first met the logsine integral as a callow youth in that strange hinterland of results that you may not have met at school but were not guaranteed to meet later on either.

When one first comes across the formulae

one is immediately struck by the fact that the second sum is the square of the first. The usual methods of proof give no reason to expect that the sums will be related in such a pleasant and easily memorable way; one is left with the impression that this is a happy accident. The following proof may give some entertainment because it throws light on this point; it emerged unexpectedly out of the tortuous process of making up a question for a recent Scholarship examination.

F. Bowman has found the following pretty result.

The minimum value of the definite integral

for real a is given by I = 1/(n + 1).

There is a very easy way to produce the Euler line, using transformational arguments. Given a triangle ABC, let AʹBʹ'C be the medial triangle, whose vertices are the midpoints of the sides. These two triangles are homothetic: they are similar and corresponding sides are parallel, and the centroid, G, is their centre of similitude. Alternatively, we say that AʹBʹC can be mapped to ABC by means of an enlargement, centre G, with scale factor –2.