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.

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.

“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 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.

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.

I make here two comments on Chaundy and Bullard’s interesting article with the above title (Math. Gaz. no. 350, vol. XLIV (1960), 253-260).

Important changes are taking place at Westfield College. A new Science Faculty is being created and—although the College is, indeed, no stranger to scientific activity—the scope of the future effort in this field will be on a much larger scale than at any time in the past. I speak to you today as one of the five new professors who have recently joined the College as a result of the expansion; but my position as professor of Mathematics is not quite the same as that of the other four science professors. I am, in a sense, not such a novelty—and this perhaps is why I have been asked to be the first to give my Inaugural Lecture—since my department already exists. In fact mathematics has been taught at the College continuously from its earliest days in the 1880’s, and I should like to begin by paying tribute to the devoted work of my predecessor, Miss G. K. Stanley. Originally a Westfield student, she received her postgraduate mathematical education in Oxford at the feet of the great Hardy, and returned to Westfield College in 1931 to take over the direction of the Mathematics Department from Miss Whitby, who had held office for the preceding 36 years. Not only has Miss Stanley guided the department with a sure hand during a period which has seen profound changes in the teaching of mathematics in London, but she has also given invaluable service direct to the University as secretary of the Board of Studies in Mathematics for the past 16 years. We are very glad indeed to know that she is still with us to give us the benefit of her knowledge and experience in the development that lies ahead.

In my note in the December 1960 Gazette I proved that if there is an odd prime p and numbers a, b, c prime to p such that

then p necessarily has one of the forms 120k+1, 120k+49.