We start with a geometric definition:

Fig. 1 illustrates the definition and shows a curve, its point of inflection and the inflectional tangent (i.e. the tangent at the point of inflection) which crosses the curve. We can see that the point of inflection occurs where the curve stops bending to the left and starts bending to the right. This leads to a more descriptive definition of a point of inflection as a point where the direction of bending of a smooth curve changes sense.

Project work has become a major constituent of Applied Mathematics courses at most British polytechnics and universities. There is a general consensus among academics as to what constitutes a project, how to discover and vet suitable projects, what their value is as a teaching aid, and how a student’s project work should be assessed.

In the familiar [32] game of noughts and crosses or ticktacktoe, there are eight ‘winning’ Lines. These are the lines having the maximum number, 3, of Points in a 3 × 3 lattice, as in the figure.

The use of reflecting surfaces to unscramble distorted pictures has been apparent over the past three centuries, at least, in an art form known as anamorphic (ana = back again, morphe = form) art. Of course, working in the opposite sense, the mirrors in fun parlours give distorted images from an undistorted (!) object. A famous portrait of “Bonnie Prince Charlie” in the West Highland Museum, Fort William, Scotland, appears as a series of incomplete concentric circular streaks of colour, but when a cylindrical reflecting surface (the anamorphoscope) is placed at the centre of the painting a recognisable image is seen on looking into it. Niceron and DuBreuil both wrote treatises in the seventeenth century on the techniques of producing anamorphograms, and Gardner has given a recent discussion of the general topic. DuBreuil described a method for generating a distorted object (drawing) that will be transformed into an undistorted image by the use of a conical reflecting surface whose base sits on the plane of the object. The drawings are very easily done by following simple rules of geometry.

A question which often seems to interest and intrigue schoolteachers is to what extent A-level grades provide good predictions of final degree classification. It has been suspected by many university mathematics departments that the answer is ‘not very much’, and that other factors play a considerable rôle. This note provides some evidence in favour of this contention. The table below indicates what happened to 589 students (see Note 1) who entered a particular university mathematics department over a period of five years, 1971-75, and who were therefore expected to graduate in the years 1974-78.

This investigation began in the refectory, when we noticed that the unwrapped foil on a four-section Kit Kat did not have its edges parallel to the edges of the chocolate bar. We wondered whether or not the angle between the two had been selected to reduce the cost of the foil by minimising the area covered by more than one sheet of foil, for example by overlap at the corners.

It sometimes happens that two apparently disparate branches of mathematics come together unexpectedly, so providing convincing evidence of the unity of our subject. A good example is afforded by the family of curves known as B-splines, which play a part in statistics as well as in numerical geometry.

The theory of projectiles moving under gravity is well known. The parabolic trajectories can be calculated and their envelope, as the direction but not the magnitude (u) of the initial velocity is varied, is the curve of safety, or bounding parabola. Where a given trajectory meets the envelope, the particle is moving at right-angles to its initial direction. In particular, the maximum range on a horizontal plane is u2/g and is attained by projection at 45° above the horizontal.

In the course of attacking a research problem recently, I came across a numerical dodge which, though really very simple, was new to me; in the course of using it I involuntarily gave myself an object lesson in the dangers of blind use of formulae. I hope there are readers of the Gazette who will appreciate learning of the dodge, and/or laughing at my misfortune.

These words of John Playfair’s are quoted by R. C. Archibald in his survey of the “minor English mathematical serials” which appeared in the celebratory 200th issue of this Gazette. Archibald describes how, early in the eighteenth century, serials containing elementary mathematical problems began to appear. Thus, during a century or more when mathematics had still to establish itself in the curriculum of the country’s leading endowed schools, numerous attempts were made to interest the ‘reading’ public in the subject and to serve those readers who were already ‘scientific’. An earlier, extensive survey of such publications was undertaken in the Mechanics Magazine (1848-1853) by T. T. Wilkinson [3] (who also wrote, inter alia, on that arcane (to two Yorkists!) subject “The origin and progress of the study of geometry in Lancashire”). Little, however, appears to have been done in recent years to examine these early attempts to diffuse mathematical knowledge: to study the mathematics contained in such periodicals and columns, how it was learned and enjoyed, and the people who set and solved the problems. This, in a small way, is what is attempted in this investigation. As our subject we chose to study the mathematical column which appeared in the York Courant during the early years of the nineteenth century.

The movement for reform of mathematical education in France, of which the first signs appeared about 1955, had two main objectives:

• (a) to reduce the mis-match which had grown up between contemporary mathematics and mathematical education at all levels;

• (b) to equip all pupils, as far as possible, with a view of mathematics which would enable them to understand and to master the increasing involvement of mathematics across an ever-widening spectrum.

Quaternions feature in most introductory textbooks on abstract algebra, often in the form of exercises. They offer an important example of a skew field (or division ring); the subset of quaternions ±i, ±j, ±k, ±l (or ±I, ±J, ±K, ±L according to taste) under multiplication affords an example of the ‘awkward’ group of order 8, which is often known as the quaternion group; they have a convenient representation in the form of matrices, which provides the work with a computational foundation; and, for those who like that kind of thing, they can be introduced as ordered quadruples of real numbers with prescribed laws for addition and multiplication, on lines similar to the number-pair approach to integers, rationals and complex numbers.

Of all mathematical problems, those concerned with probability seem to contain ambiguities more often than any others. In the majority of cases the reason is that too little thought has been given to the wording of the question, but occasionally the question sounds very sensible and it is only when a real attempt is made to solve the problem that difficulties in interpretation appear. I believe that this is mainly due to the loose way that the word ‘probability’ is used.

On a recent visit to Cairo I was struck by two matters which may be of interest to other members and their pupils.