I shall start this address with a few words about economics. It is unfortunate that few people will admit that the cause of our chronic dollar indebtedness is the reluctance of consumers abroad to buy our goods when they can get American equivalents. Why are American goods preferred to ours when American industrial workers have wages with five times the purchasing power of their British counterparts? The brief answer is that the Americans apply mathematics on a big scale in their industrial administration and in their technology. There are signs, too, that the Russians are also doing this, and they are also entering the export markets. Jeremiahs are not popular but it seems imperative for me to state that Britain must emulate these two rivals in order merely to survive and that our only hope of doing this is to replace crude trial and error methods by widespread scientific (and that means mathematical) calculation everywhere. Some of our firms do conduct their organisation and research in an admirable manner and some Research workers are well aware of what Professor Temple has called the “all-pervasive’ nature of mathematics.

Members of the Royal Corps of Naval Constructors are responsible for the design and construction of H.M. Ships, and one of the main features of their training is the study of mathematics. The aim of this article is to give the VIth form mathematics teacher an acquaintance with the kind of mathematics used in ship design, he may then be able to encourage some of his pupils to consider the prospect of a career in the Royal Corps. Promising mathematicians with a strong practical bent can enter from school under a scheme advertised in this number of the Gazette.

This year has brought the centenary of Dirichlet’s death (5 May 1859), and it is fitting that we should recall some of his achievements and his part in the development of mathematics.

Gustav Peter Lejeune Dirichlet was born on 13 February .1805, and was the son of the postmaster at D?ren, near Cologne. Among the schools he attended Was a Gymnasium at Cologne, Where one of his teachers was the physicist Ohm. When the time came for University study in .1822, Dirichlet persuaded his parents to let him go to Paris, which Was then the world centre of mathematics. Laplace and Legendre Were still alive and active, Fourier Was at the height of his career, With a group of brilliant young men round him, and Cauchy had already begun his massive development of the theory of flmctions.

If Pascal’s Triangle is written down, it will be noticed that the number of odd numbers in any row is a power of 2; moreover, if every even number is replaced by 0 and every odd number by 1, the result is an interesting pattern of triangles from which it is possible to deduce a general rule. Again, if divisibility by any prime number p is considered, and if every number in Pascal’s Triangle is replaced by its residue (mod p), an even more intriguing pattern is formed. These considerations led to the following investigation.

Would you indicate 3 + 3 + 3+3 as 4x3 or as 3x4† Most colleagues and boys that I have asked plump for the former. 4x3 means to them “four threes” or “four times three”, which is emphatically not the meaning “four multiplied by three”. Clearly “four times three” is the same as “three multiplied by four”; (I do not mean that they are equal, I mean that the expressions are two ways of saying the same thing). If we indicate this common meaning by the symbols 4x3, we are in fact doing the opposite of what we do in the case of the other three arithmetical processes. 4 + 3 means (to most people) “take four and add to it three”, or “four increased by three”; 4-3 means “take four and subtract three from it” , or “four diminished by three”; so 4 + 3 means “take four and divide it by three”. Obviously, then, 4x3 must mean “take four, and multiply it by three”; i.e. “four taken three times” or three fours.

Functional analysis is the study of functions defined on certain abstract sets; these sets may vary greatly in their structure and the nature of their elements but they have in common the property that it is possible to define in them the notions of convergence, limit and so on: in other words, they are topological spaces. Most of the spaces studied in functional analysis have in addition algebraic properties: that is to say, addition and sometimes multiplication of their elements is possible.

During the course of Civil Service committee meetings it is socially acceptable to draw doodles instead of smoking. The two drawings exhibited here are elaborations of a committee-meeting triangular doodle which consists of one sixth of the first drawing. This triangular doodle differs from one independently discovered by Robison (1954)* in that the present one has smaller angles and is taken right to the centre of the triangle. In fact Robison had such large angles that there was no illusion of any enveloped curves in his diagram. I am told by Mr. R. A. Fairthorne that these curves have for long been called ‘pursuit curves.’ By fitting congruent right-handed and left-handed triangular doodles together at random we get an agglomeration of ‘spearheads’ and ‘hyperboloids,’ as in the first drawing. It consists entirely of straight lines and can be drawn without using any skill. The method could be used for producing a dazzling form of wall-paper, possibly suitable for a scientific exhibition.

Dr. C. D. Langford recently sent me some numerical work he had done on automorphic numbers. He found automorphic numbers in the scales of 6, 10, 12, 14, and 15 and he discovered what proved to be a general method of finding an automorphic number of r + 1 digits from one of r digits. Dr. Langford’s notes stimulated me to look for explanations of his results.

When faced with a problem, for example the design of a piece of equipment to meet a customer’s requirements, an engineer will seek to find an answer to it by the quickest and cheapest means at his disposal. Of those means mathematics is but one, even though a powerful one. The first stage in invoking its aid is to formulate the engineering problem in mathematical terms. This involves first, the separation of non-quantitative considerations which are not expressible mathematically (which requires engineering experience), second, the formulation of the problem in such a way that the required information (and no more) will be obtained with as little labour as possible; and third, the making of simplifying assumptions so that the mathematics becomes tractable and yet is a sufficiently good representation of the physical situation. Both these last processes require a balanced knowledge both of engineering and of mathematics. (This is an aspect of undergraduate training which could with advantage receive better treatment.) But even when the resulting equations have been solved the engineer’s task is by no means over; the solution to a mathematical equation is a long way from being the answer to an engineering problem. Given the solution the next step (the inverse of formulation) is the interpretation of it. This often involves investigating the effect of variation of the parameters of the problem and requires much computation. The place of electronic computers in the scheme of things becomes evident. Finally, when a satisfactory interpretation has been achieved the engineer may proceed to the answer to his problem.

Diagrammatic methods may be used to establish some well known facts about magic squares

In a magic square with 9 cells, the number in the central cell is always a, the average of the three numbers in each row, column or diagonal. Using a dot to represent the number in any cell, fill the cells of a 3 × 3 square in this way: put a dot in each of the cells forming a diagonal: these dots represent numbers totalling 3a.

It is with very deep regret that we record the death of Miss F. M. A. Pendry in August last.

She was one of the keenest members of the Mathematical Association. She served as a committee member of the London Branch for many years, being chairman for the three years 1950-1952 and vice chairman for the four previous years. She was a member of the Associated Examining Board and helped in the formation of a Syllabus for a Certificate Examination for Technical Students and she was greatly interested in this new approach to the subject. She was also a co-opted member of the Mathematical Sub-Committee at the Institute of Education, for several years up to the time of her death.