From one end of Britain to the other Megalithic Man has left us examples of geometric constructions of various designs. Originally set out on the ground by means of stakes, rope and measuring rods, the designs were rendered permanent by large stones placed along the outlines. The erection of the larger of these stones was an enormous task. It is certain that the engineers controlling the operations were working to pre-conceived plans which were set out with care and accuracy. Further, it appears that the unit of length used was the same from Land’s End to John o’ Groats. That this unit was 5·44 ft. I have shown in Ref. 1, and many more sites surveyed since writing that paper have amply confirmed statistically this value. Half of this, 2·72 ft., might be called the Megalithic Yard, but whether this or the Megalithic Fathom (5·44 ft.) was the length of the standard rod carried about the country it is not possible to say The important thing is that in a statistically significant number of cases in any group of circles the major diameter is approximately an integral number of fathoms. The group can be formed according to locality, type of circle, or size; the result is the same. We also find this dimension occurring in the distances of outliers from the construction centre and the distances between circles where there are two close together. The method of analysis used was that given by Broadbent in Ref. 5.

I begin by expressing my sincere thanks to the Association for the great honour bestowed on me in my election to the office of President. I realised when elected that the honour was indeed great; but it has also happened that, in the course of the year, I have had occasion to look through the list of those who have held the office before me, and to find one’s name in that company is both humbling and—let me be honest—exhilarating. In particular, and this will be relevant for what I have to say later, the records speak not only of men and women famous throughout the scientific world, but also of those whose special genius has led them to the top of the teaching profession, where their memory remains alive to generations of pupils of of whom, doubtless, many are here today. It is a source of real strength to this Association that it honours not only those who create but also those who pass on the fruits of creative labour, sometimes even to those who will themselves become the creators in their turn.

In opening this afternoon’s discussion I wish to put before you some reasoned criticisms of a number of attitudes and arrangements embedded in the English Educational System; I want to talk generally—not submit a detailed blue-print of some brave new system. During my talk I will ask you a number of questions and I would be grateful if some of you would respond and indicate your views. Although I feel deeply on some of the points I shall raise, I put forward my arguments with diffidence, for I realise that opinions may, and frequently do, differ.

School exercise books have usually a transitory existence. Even if we ourselves retain them as souvenirs of our schooldays—and we seldom do—then our descendants have no such sentimental regard for them, and consign them to the waste-paper basket when we depart this life. A school exercise book of 130 years ago is therefore remarkable for its longevity, as well as interesting for the light it throws on the teachings of that time.

Newton was the greatest mathematician of the seventeenth century. Today, almost three centuries afterwards, we are just beginning to realize the full extent and variety of his achievement. Much of his mathematical work has never been published (though it ranges far through the fields of projective geometry and general point-correspondences to number theory and an exhaustive treatment of interpolation by finite differences) and is now, with few exceptions, to be found only in little known manuscripts in the Cambridge University Library. But Newton is remembered above all, and following his own wish, for his creation of the fluxional calculus and the theory of infinite series, two strands of mathematical technique which he bound inseparably together in his “analytick” method.

One of the pleasant aspects of research into mathematical history is the way in which existing material, passed over in the conventional account, may allow us not only to establish the bare, if unexpected, fact of a priority but, more importantly, to assess the significance of major currents in mathematical thought with greater precision. Briggs’ partial discovery, in anticipation of Newton, of the general binomial expansion,

is an excellent case in point.

At the International Congress of Mathematics to be held in Stockholm in 1962 there is to be a discussion, in the Section on Mathematical Instruction, on modern mathematics in the school curriculum. This has been the occasion for a preliminary airing of the subject in many of the countries concerned, and an important effect of the Stockholm discussion will no doubt be to direct attention beforehand to this important problem.

Much thought is being given at present to the possibility and desirability of introducing topics of modern mathematics, such as set theory and abstract algebra, into the sixth form curriculum. Very little of this kind of work has as yet been done in schools, chiefly because of its comparatively recent introduction into undergraduate courses, but it seems probable that it will come to play a part in the mathematical training of at least the abler boys in the fairly near future. Recently, during teaching practice, I had the opportunity of giving a course in group theory to four senior boys and I feel that the experience gained may be of interest to others who are contemplating similar kinds of work.

Methods have been described that facilitate the discovery of economical dissections. They are summarized in §§ 1 and 2 that follow.

1. Each of the figures to be dissected is made an element of a strip with parallel sides; 1 and 2 show strips formed from a Latin or tau cross and a pentagon. The first is a P-strip (prototype parallelogram), in which all elements are the same way up. The second is a T-strip (prototype trapezium), in which alternate ones are inverted. Points such as A and B in 1 are called congruent, a term copied from Whittaker and Watson's Modern Analysis, p. 430.

Proofs of the prime number theorem are extremely hard to follow, and leave the impression, at least among amateurs, that the essential property of prime numbers, namely their primeness, plays very little part in the argument.

Simple reasoning, based on the rules of probability, can however give a very fair indication of the way in which the density of primes in the region x varies with x.

“First books” on mathematics usually begin with numbers. Some histories of mathematics say in so many words. Mathematics begins with numeration. I believe this is true, literally, if mathematics is considered as a set of signs, with rules and relations between them. That is to say: the history of mathematical notation does begin with the numerals. The earliest known mathematical signs are no doubt number-signs, and history of mathematical notation has in fact little else to discuss but numerals up to the 14th century of our era. Of present-day mathematical signs, + and - appeared first in the 15th century, = in the 16th, <and> in the 17th.

A few years ago there appeared on the commercial market a certain puzzle called “Maestro”. This consists of packing into a 6 x 10 rectangle the twelve shapes depicted in Fig. 1, (where they are shown packed into an annular region bounded by two concentric squares). It is also possible to pack these shapes into other simple geometrical figures. The aesthetic attraction of this problem lies in the fact that the twelve shapes concerned form a complete set in the sense that any other shape consisting of five abutting squares is congruent to one of them.

In the preceding note Mr de Visme writes y’(x) for the “density” of primes in the neighbourhood of x and proves heuristically that

Mechanics is a branch of applied mathematics taught in both school and university. From the mathematical standpoint, the subject is learned in schools primarily by the method of solving examples. Theoretically, this procedure is supposed to help the pupil to grasp the fundamental principles and obtain for him a good examination result. That these seem to be nothing like as successful as they could be is one reason which prompts me to write this article. Others, including the pupil’s lack of appreciation of the subject, will be apparent in the points I shall make. These points have arisen from some years’ experience of lecturing to first year undergraduates and from marking advanced level scripts in applied mathematics. The efforts met with in both instances have led me to believe that there is a case for comparing the knowledge and understanding of the average sixth form pupil with that normally required of a freshman about to read applied mathematics for a general degree. On the one side is the ex-pupil, drilled in the method of solving examples and often lacking the understanding of the background principles involved; on the other is the undergraduate needing to draw upon an understanding and a background wider than he usually possesses. I feel sure that the disparity so defined exists not so much because of a lack of bulk material taught in the schools, but rather because the presentation is misguided and the emphasis misplaced.

A significant modern trend is the increased use which industry makes of mathematicians. Only twenty years or so ago almost all university graduates in mathematics had to find careers in teaching, in schools, technical colleges or universities; a few became actuaries or entered the Civil Service. To-day the situation is appreciably changed, many more of the new graduates in mathematics enter industry or research establishments. (This is one reason why there is a shortage of mathematics teachers in schools). In 1959, just over three thousand graduate mathematicians were employed in industry, public corporations and in central and local government. Nearly eleven thousand others were employed in some kind of teaching. Although more than three quarters of all mathematics graduates are still to be found in education, the number of those who are not has increased by nearly two-fifths in the last three years. This increase is likely to persist in the future as the complexity of industrial research and development grows. It is significant that recently there have again been moves to establish an Institute of Mathematics, on the lines of the professional organisations for scientists and engineers.

In his article “A Problem in Projective Geometry” (Gazette, December 1960) T. G. Room says that he has not been able to find a projective geometric proof of his theorems. May I give the following proof?

The simplest overhand knot, if tied in a strip of paper and carefully pressed flat, forms a regular pentagon (Fig. 1). All higher order polygons can be produced in this way with the single exception of the regular hexagon which requires two strips. The reason for this exception is interesting.