The story of Jeremiah Horrocks is not perhaps as well known as it should be. He has been described with some justice as the Keats of English Astronomy. In both cases brilliant work was done followed by death at a very early age. Horrocks was probably 22 and Keats only 26 when they died. He is, of course, most famous for having been the first observer of the transit of Venus over the solar disc, after having calculated, with little to aid him, that it would transit on a certain day and time. During the eighteenth and nineteenth centuries such transits were regarded as of great importance in order to get accurately the fundamental Astronomical distance, i.e. the distance of the earth from the sun.

The following problem was proposed by Dormán Luke, Math. Gazette, Vol. XLI, p. 194 (1957):

Inscribe a regular octahedron in a cube, so that its vertices are one on each of six edges of the cube.

Four such octahedra can be inscribed; what is the solid with 32 faces which is common to these four?

Starting with the octahedron, how many cubes can we circumscribe to it in this way?

Again, what solid formed by their vertices encases them all?

This note was suggested by Dr. Maxwell’s query in the May “Gazette” for a simple proof of the fact that the rectangular hyperbola whose centre is the point of contact of the nine-point circle and the inscribed circle of a triangle and which passes through the vertices, also passes through the incentre. This fact appears as a special case, together with a number of other interesting results, which include a new proof of Feuerbach’s Theorem.

The following remarks take up Dr. Maxwell’s invitation to beginners to identify the centres of various rectangular hyperbolae through the vertices of a triangle (Maths. Gazette XLII No. 340 1958 p. 114).

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.

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.

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.

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.

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.