The inspiration for this address comes from Little Tim—or should I call him Tiny Tim? And this is odd because I don’t really know his name. He belonged to the D-stream of a school, and he sat on the front row of the class to which I was speaking. I have no idea now what I was talking about: but I do know that to all intents and purposes Little Tim was unmoved by any of my eloquence. His round little face was immobile, and nothing that I could do seemed able to disturb its obvious complacency. And then—I do not know what caused it all to happen, and I like to think it was not because he had noticed that my shoelace was undone or my front eollar-stud missing—the trace of a smile began to appear upon his lips. It broadened to a horizontal line across his round and homely face; and then the vision faded. Little Tim returned once again to his former state.

In many books on elementary mechanics, it is assumed that velocity is a vector quantity on the grounds that it has magnitude and direction. It is, however, very far from obvious that these are sufficient reasons and a rather more thorough investigation of the nature of velocity may provide not only a clearer idea of what velocity is but may also clear up some of the difficulties associated with “relative” velocity.

Lessons on sets have become commonplace in schools today. Venn diagrams proliferate and, even in primary schools, children can be seen sorting and classifying objects by size, colour and shape and placing them in spaces marked out on the floor by chalk outlines or wooden hoops. Older children learn that such diagrams are named after the English logician, John Venn, and that through them we can represent the relations of membership and inclusion and the operations of union, intersection and complementation. A rectangle is drawn to represent the universe U: subsets of U are represented by the interiors of circles, or other closed curves within U, i.e. subspaces of the rectangle. The elements of U are represented by points within the rectangle, the elements of a subset A by points within the corresponding subspace of the rectangle and the elements of A′ by points within the rectangle but outside the region representing A.

A topic which is becoming increasingly popular in school mathematical syllabuses is that of probability and the subject matter which appeared only in the Sixth form previously is now to be found in the Third form. In the classroom, as for all mathematical undertakings, it is always a useful exercise to get children to state what they expect the answer to be before they attempt to solve the problem, and this “playing off of hunches” is extremely interesting in probability.

The contribution “Mathematics of the pop charts” by E. de St. Q. Isaacson (The Mathematical Gazette, 51 (1967), 282-287) prompts me to publish a result found in 1959 during research on the same subject.

Situation: To find the number of unit squares the diagonal of a rectangle passes through when the rectangle has integer sides and is divided into unit squares.

The following two-player game which we have named Euclid was devised as a source of examples to illustrate the use of a winning strategy algorithm based on the proof of a similar existence theorem[1].

The profile of a familiar type of key is shown in Fig. la. The shape of the key profile is determined by the depths of successive cuts. Each cut is of wedge shape. The centres of the cuts lie at N regular well-defined positions. In the key shown there are 6 cuts (N = 6). The depths d of these cuts, which distinguish one key from another, must have one of a set D of well-defined values. If we regard a cut of zero depth as defined by d = 1, the key shown may have values d = 1, 2, …, 6, 7 (D = 7).

There are many ideas, fundamental to the whole of mathematics, that are difficult to teach explicitly and are rarely examined. They include the concept of mathematical proof, the idea of a converse, necessary and sufficient conditions, and those elements of elegance and rigour without which mathematics can lose its beauty and even its claim to be called mathematics. It is these ideas I shall refer to as mathematical background; it must be emphasised that this background is common to both traditional and “modern” mathematics, and that its cultivation in their pupils would seem a desirable objective for all teachers, both conventional and progressive. Most of these ideas were clearly expressed in the Mathematical Association’s booklet, “Suggestions for Sixth-Form Work in Pure Mathematics”, which seems admirable in everything but its title, since most of the work should be begun much earlier than the sixth-form. (If it is left until then, there is justification for one reviewer’s remark that it is “an attempt to paper over the cracks of sixth-form mathematics”.)

This evening, on the occasion of our Annual General Meeting, I propose that we look at the following problem, which I shall call “the AGM problem.”

It must be remembered that one does not get a leap year at the turn of a century unless the year is divisible by 400.

From a university examination paper:

" The variables P, V, T, ϕ are connected by the relations

PV = RT, ϕ = C v log P + C p log V,

where Cp, Cv and R are constants, and Cp — Cv = R. Prove that

Groups have for several hundred years been used in the composition of long peals in change ringing, as was mentioned on p. 399 of the Gazette. A brief outline of their use for the non-ringing mathematician may be of some interest.

This puzzle is described in Wallis’s Algebra (Cap. CXI), with detailed instructions both for making it and solving it. The apparatus consists of two parts, which are easily made from simple materials, as follows:

The object of this article is to show how the problems of transformation geometry, and in particular the determination of the composition of two or more successive transformations, can be solved far more easily by Argand diagram methods leading to algebraic equations involving complex numbers, rather than by purely geometric considerations. It must be conceded, of course, that the methods advocated here cannot be extended to three dimensions.

The study of artificial satellites is today a subject of topical interest and naturally much work has been completed in the field of satellite orbits. Many aspects of this work involve the use of differential equations. Unfortunately very few of these may be solved by ordinary analytic techniques. One possibility is to define a new function to be the solution of the given equation, for example the logarithmic function. Nowadays we use computers. Nevertheless it is highly satisfying to produce an analytic solution to a problem in applied mathematics for in a sense we have succeeded in displaying a pattern in the physical world.

This is not to be taken too seriously.

Two or three players compete in the solution by relaxations of three linear equations

R1 ≡ a1 x + b1 y + c1 z + d1 = 0,

R2 ≡ a2 x + b2 y + c2 z + d2 = 0,

R3 ≡ a3 x + b3 y + c3 z + d3 = 0.

Before the equations are formed each player adopts one of the residuals Rlt R2, R3 . When there are two players one residual remains unadopted.

The growth of language

Someone once said, in answer to an invitation to draw up a chair, “It depends what you mean by ‘chair’; do you mean the thing, the word, or the concept?” There seems to be some relevance in this to the idea of a vector as a physical object, as against its “name” as a triad of numbers, or as an algebraic entity. In illustration, one could point to a table, and give the word for it in several languages. It is clear that communication is only possible at any reasonable level by having first established a language, and then using it as a vehicle for thought. Some would say we cannot think without language; whether this be true or not, the articulate speaker seems brimful of ideas, and the tongue-tied individual dull.