We give here an account of a series of lessons in which a class of Juniors carried out some simple experiments with numbers. In these experiments the numbers, expressed in the familiar Hindu-Arabic notation, were subjected to certain sequences of operations from which there resulted fairly obvious digital patterns—reversals, interchanges, transpositions, rhythms and so on. These patterns were the focal points to which the attention of the class was directed.

It is well known that the functions x, 1 — x, 1/(1 — )x, x/(x — 1), 1 — 1/x, and 1/x form a Group, the rule of combination being substitution of one function in another. The question arises what other sets of functions there may be with a similar property. This note gives one answer to this question.

It is more than 91 years since, on January 17th 1871, a group of twenty-six men met in another place and formed the Association for the Improvement of Geometrical Teaching, which twenty-five years later became the Mathematical Association. Most were schoolmasters; some were clergymen; the account of the proceedings in the first annual report adds: “There were also a few gentlemen present who were not members of the Association.” The ladies were allowed to join later.

Early in the classical period of Greek mathematics there arose a crisis over the meaning of “proportion”, or alternatively of “area”. This crisis, which was resolved by Eudoxus, the pupil of Plato, about the middle of the fourth century B.C., led to the introduction of axioms, or postulates, as “self-evident truths”, necessary for the proof of theorems, equally true but presumably less self-evident. Among these axioms was included the famous Fifth Postulate of Euclid, equivalent to saying (see below for its rather complicated wording) that “through a given point (the point T in Fig. 1) there is at most one (straight) line parallel to a given line (the line MK)”. But this postulate had all the earmarks of a theorem; it looks true yet not altogether self-evident, and its converse, to the effect that there is at least one such line, is an easily provable theorem.

In a great many problems in probability, one is given one set of quantities Xj expressed in terms of a second set of quantities Yk, and one needs an expression for the Yk in terms of the Xj. Basically, of course, this is just a problem in solving equations, but the results can be expressed in a particularly elegant form when the coefficients involve the binomial quantities . This situation is one that occurs very frequently, and the resulting formulae simplify the derivations of many identities. The present expository account is intended to collect together various results in a convenient form. We shall need to recall briefly a few of the main properties of the binomial coefficients.

It is customary in an account of Riemann integration to give one or two examples of the evaluation of definite integrals directly from the definition of the integral in terms of Riemann (or Darboux) sums, but the method is used only to illustrate the definition, and is never fully exploited. The method is, however, capable of a more systematic treatment than one might at first expect, and in this note we show how it can be applied to some of the elementary functions of analysis.

I want to start by explaining some of the terms I shall use in this lecture. First, there is pure mathematics. Anyone who has done Ordinary Level Mathematics has some idea of what pure mathematics is about. Degree, or postgraduate pure mathematics for that matter, is—to quote A. N. Whitehead—“the science concerned with the logical deduction of consequences from general premisses”. The humbler algebra and geometry of school mathematics is concerned with exactly this, or should be. And in both cases, it is all done, at least in principle, for the fun of the thing with no application to the real world in mind.

Every migrant should realise that migration does not only mean moving from one country to another, but also assuming the ways and customs of the new country Some of these, especially those dictated by the climate, are easily assumed, others are very hard to accept.

The regular decagon will be dissected into other regular polygons and into Greek and Latin crosses by the methods outlined in my previous article (Math. Gaz. 45, 1961, p. 94).

The first step is to find tessellations. In one of those that are used in this note, shown in 1 and 2, the decagon is cut into three pieces, and in the other only two. A further tessellation element, not used here but worth noting, is shown in 5.

The rank of the coefficient matrix plays a dominant role in the theory of linear algebraic equations. It is not surprising, therefore, that a test for the rank of a matrix, that was a by-product of some work in dimensional analysis, proves to be an an admirable tool in this theory With its aid the consistency requirement assumes a simple and effective form, and the solution of both homogeneous and non-homogeneous systems is given explicitly in terms of submatrices.

The present age is witnessing the rapid growth of a wide variety of communication and control systems using electrical signals. Examples are radio and commercial communications, television and facsimile transmission, radar and aircraft control systems, computers and industrial control mechanisms. All these have in common the fact that information has to be transmitted and processed through a channel, and they can all be represented by a block schematic such as in fig. 1. The efficiency of these systems must be assessed by comparing their information handling capacity with their cost in time, transmission channel bandwidth, equipment complexity and money. It is therefore essential to define a quantitative measure of information and to derive relations between this quantity and the parameters of a given communication system and also to determine criteria for a maximum information flow under given conditions. Information theory concerns itself with these problems, based on simplified and somewhat idealized mathematical models of actual systems. The origins of information theory go back to the 1920’s, but consistent logical foundations for the theory were laid by Shannon in the late 1940’s and the modern theory dates from then. It might appear that information is too loosely defined as a concept to be submitted to quantitative analysis; and indeed, before we can do so, we must clearly differentiate between the amount of information in a message, its meaning (semantic aspects), and its value (philosophical aspects).

Recent notes in the Gazette (Vol. 45 (1961), 17-23, 23-25) on the subject of change of variable in a definite (Riemann) integral have prompted me to write about the same subject for an indefinite integral, since it has often seemed to me that the conditions under which this rule is proved are, in a sense, the reverse of those that are needed when it is used in practice.

With his unaided faculties, a man is comparatively limited in performance. His vision, hearing and legs operate only over small distances, he cannot lift very heavy weights nor weave much cloth in an hour. But by making use of devices such as the aeroplane, the telephone, and the powered loom, he can exploit natural properties and forces, and thereby attain what is literally a superhuman standard of performance in all kinds of activities.

The problem I am going to discuss is the generation of the class of all mappings of the space S×S×.. × S, with any number of factors, into S itself, from a single mapping of the class, where S is a finite set. The problem has its origin in the study of truth tables and it is in this setting that the problem may be most simply presented.

Since the end of the last century railway engineers have known that when a railway vehicle passes round a curve a very large force can exist between the outer leading flange and the rail, even under equilibrium conditions. However it was not until Mr. S. R. M. Porter’s paper appeared in 1934 that a satisfactory method of calculating this force was established. Mr. Porter included in his paper two problems of considerable complexity which would take many hours or even days to solve. This may be one of the reasons why his paper is not more widely known.