The title of my address is the question with which Professor Temple ended his presidential address two years ago, when he gave such a fascinating review of the impulses and influences, both external and internal, that have promoted the growth of mathematics. Whether he expected an answer I cannot tell; but this Association, since its foundation in 1897, has frequently taken as the theme of its discussions and reports the duty of teachers to show the meaning and inner unity of the subject, and not merely to teach techniques. So it is not out of place to consider from time to time whether we are agreed what this meaning is, and what the techniques have to do with it.

About the year 1700, in the course of heretofore unrecorded travels, Gulliver met islanders who used a rather odd arithmetical vocabulary They counted.

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.

If you wish to know the day of the week on which a historical date fell, and no tables are immediately available for this purpose, you may be glad to know a method, easy to remember, of calculating the day.

For some years I had been dissatisfied with the methods we were using in my small Primary School, to teach Subtraction. The slower children were unable to grasp the basic principles of Equal Addition, and many errors were made, as the children were using a “parrot” pattern, instead of basing their work on understanding and logical sequence.

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?

For numbers a ≥ b ≥ l we shall denote by (a, b) the set of numbers b, b + 1, …, a. We shall say that a set S of numbers is perfect if there exists a sequence containing just one pair of each of the numbers in S, satisfying the condition: for every number r in the set, the two r’s are separated by exactly r places, and having no gaps (a perfect sequence).

The problem is to arrange the numbers 1, 1, 2, 2, …, n, n in a sequence (without gaps) in such a way that for r = 1, 2, …, n the two r’s are separated by exactly r places; for example

Priday has shown in the preceding paper that for every n there exists either such a perfect sequence (as he calls it) or else a hooked sequence, with a gap one place from one end, for example

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.

Perhaps the most notable achievement of the Mathematical Association since its formation has been the abolition of Euclid. While Euclid remains the greatest mathematical classic, it is no longer used as an instrument of torture on the young. As a school text-book Euclid is dead, replaced by a course in the well-established three stages, in which the type of learning and the degree of logical rigour are adapted to the understanding of the pupil.

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.