The article which follows is the first of a set of papers sponsored by the Modern Trends sub-committee of the Teaching Committee of the Mathematical Association. It is hoped that future papers will be published either in The Gazette or as a supplement. Each paper will represent the personal view of the author rather than being an official policy statement of the Association. It is hoped that they will lead to correspondence in the form of reports of experiments as well as constructive criticism.

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About 75 B.C., Cicero the orator, stumbled upon an ancient monument. The lettering was illegible, but ‘on clearing away the brambles and thorns’ he discovered a curious design—a sphere with circumscribed cylinder.

Francis Bacon held every man a debtor to his profession, and there are few of us who would not discharge this obligation if we could. I feel particularly indebted because I have been fortunate, over a long period, in having been able to attend Association and Branch meetings, where I have met many of the best informed, most active and aesthetically inspiring teachers of mathematics of all ages and at all levels. When, in addition, the Association honoured me by making me its President, I knew that I was, in more senses than one, permanently overdrawn. I know that this distinction would never have come to me had it not been for the accidents of time and place that made me Honorary Treasurer for several years. I am an assistant master, and, since 1954, a House Master, so that I could describe myself, in a light-hearted way, as a full-time research worker in the educational field. My researches are not entirely mathematical, but close contact with boys of 13 to 18 has been an interesting and instructive experience.

Our Association was founded deliberately to “improve” the teaching of geometry, and its work has extended to cover every branch of the mathematics taught in the pre-undergraduate years. It is no surprise that, among the many distinguished mathematicians who have contributed to this development, a few stand apart clearly for the extent of their influence. Judged by any standards, E. H. Neville was one of the greatest of them all, and this tribute, by one of his earliest pupils at University College, Reading, is written with a sense of deep personal affection and gratitude.

The Mathematical Association was founded in 1871, as an Association for the Improvement of Geometrical Teaching, by teachers who had lost faith in the educative value of learning by rote the words of Euclid's sacred text; indeed, one suggested name was the Anti-Euclid Society Progress was at first slow, until in the early years of this century Oxford and Cambridge agreed to accept, in their entrance requirements, any system of geometry which exhibited some logical coherence.

Mathematical confidence and appreciation cannot be fostered unless each technique of the subject is established in an understandable fashion. It can hardly be claimed that this condition is satisfied in connection with division of fractions, for the rationale of “invert the divisor and multiply” is hardly ever grasped by children or, for that matter, by adults, except as a very loose induction from a few non-typical examples. It is interesting, therefore, to note that a straightforward procedure does, in fact, exist for this operation, adoption of which would serve to make the operation appear much more reasonable than it usually does at present.

In the Mathematics Section of the Science Museum, South Kensington, is a display cabinet that contains many specimens of Napier’s Rods, or Bones. In a corner of the same cabinet are some cardboard “rods” of very different design, expertly made by the Museum staff. Unfortunately they do not form a complete set, nor is their structure immediately obvious, which might account for the fact that they appear to have attracted very little attention. Be that as it may their appearance is certainly intriguing, as a glance at Figure 1 will confirm, and in operation they represent a distinct advance over Napier's rods, on which they are based.

An American mathematician, celebrated for a love of epigram tempered by some regard for truth, has asserted that George Boole has the sour distinction of being the most severely under-rated mathematician of the 19th century There is some substance in this judgement. During Boole’s lifetime, his work was recognised and admired in this country, but British mathematics was then very insular, paying small attention to the great progress being made in Europe, and, reciprocally, being little regarded by Continental mathematicians. Boole’s reputation was localised, and, moreover, it rested very much on what we should now consider to be the less valuable of his contributions to mathematics. The title of his greatest work, The Laws of Thought, does not suggest a strictly professional study of pure mathematics but rather a digression into philosophy and metaphysics. Yet Bertrand Russell has declared that Boole, in this work, was the first person to discern clearly the essential characteristics of pure mathematics, and today, when Boolean algebra is a topic for school texts as well as for research journals, we can see that Boole’s work, under-valued at the time of its appearance, was an essential step in the development of pure mathematics as an abstract and axiomatic structure.

E. H. Neville was Librarian of the Mathematical Association for over thirty years, from 1923 to 1954.For more than half of that time Neville was not only Librarian but custodian of the books which were handsomely shelved and displayed in his own home, first in his house on Castle Hill, Reading, and then in his first house in Sonning.

In many textbooks on applied mathematics, mechanics, or applied mechanics, the impression is still given that the only absolute systems of units are either the CGS system, using the gramme and dyne, or the British system using the pound and poundal and also that the only alternative system is a gravitational one in which either the unit of force or both the unit of force and the unit of mass vary with position; a few bodies do include the MKS system.

This note is due to the fact that the identity of two analytic functions, with the same values for every z, is not obvious.

Mathematics is relentlessly seeping into the very foundations of the civilisation in which we live and the community of which we are part, but I wonder how many of my readers—teachers of mathematics in grammar schools or in the universities—realise the magnitude and understand the nature of this seepage. It is permeating commerce, sociology, economics, medicine, linguistics, …, as well as more traditional science and technology. In this article I shall confine myself mainly to engineering, although I believe that the principles I shall develop will apply generally to any ancillary mathematics.

As a subject of study statistics differs from other subjects in being non-existent by itself. A “statistic” is a measure—e.g. mean, variance, standard deviation, median—of a sample, and therefore of a sample of “something”, and that original “something” pertains to some other “subject”—e.g. biology, agriculture, medicine, industry, economics. Or, if we investigate or research in the theory on which our techniques are based, we are at once plunged into mathematics. This somewhat nebulous quality of the subject, together with the peculiarly subtle mixture of down-to-earth algebra and logic on the one hand and subjective intuition and philosophical approach on the other, all wrapped together in a sort of cotton-wool of probability, is often for the beginner a real difficulty and thorn-in-the-flesh, though largely contributing later, it is hoped, to the fascination of the subject. I have found in over ten years of teaching statistics at Sheffield University that the student can be much helped over this initial period by simple experiments in which he collects his own data, in the course of which he builds up an idea for himself of the underlying probability situations and in some of which he learns something of the relevance of design and the practical difficulties associated with the carrying out of such simple-sounding instructions as “randomise.” When it comes to analysing his data he experiences a certain “vested interest” in the results just because they are his own observations.

Recent notes (1, 2) suggest the problem. Find necessary and sufficient conditions on a set of n(>3) real constants a1 …, anfor the inequality

Through the kindness of Mr. Langford we are able to publish a mathematical note by Neville and some extracts from his letters, mainly on teaching problems.Those who knew Neville will almost hear him speak through these informal (and unedited) comments on topics where his keen enthusiasm helped so vitally to clarify issues that were becoming needlessly obscure.

In his talk on Decimals at the Annual General Meeting of the Mathematical Association, 1962, (see the Mathematical Gazette, XLVII, 1963, p. 25), Mr. Green observed that the recurring decimal expression for 1/p where p was a prime other than 2 or 5 had the maximum possible length of p – 1 if and only if 10 was a primitive root of the modulus p. In practice there was maximum length for

p = 17, 19, 23, 29, 47, 59, 61, 97

but not for

p = 13, 31, 37, 41, 43, 53, 67, 71, 73, 79, 83, 89,

and he did not see any reason or pattern behind these figures.

The purpose of this article is to offer a new approach to the solution of the equation