Some years ago there was a vogue for a series of books dealing imaginatively with the future and bearing such titles as Daedalus, or the Future of Science, Icarus, or the Future of Flying, and the like. I do not recall whether arithmetic was among the topics considered by the ingenious authors, but a further title, Archimedes, or the Future of Calculation, might certainly have been suggested by the developments of modern arithmometers.

The subject matter of my talk is geometry in the sense that I shall deal with such entities as lines, circles, and squares. But the results I aim at bear little resemblance to the theorems of traditional geometry, which describe properties of given configurations. Our present object is quite different; we shall begin by specifying not configurations but processes (for example processes of selection), and then seek to determine how economically or how effectively these processes can be carried out. Questions of this nature are quantitative and the solution, if it is known, is generally expressed by a number. It is for this reason that I speak of 'arithmetical' geometry. The problems of the type I shall discuss are usually simple to state and require very little in the way of mathematical equipment for their understanding. However, their solution often presents formidable difficulties and is in many cases as yet far from complete. Let us now consider in some detail a few of the characteristic problems in this field.

The increasing use of simple computing languages in schools gives an opportunity for rationalisation of our existing mathematical notation as follows:

It is time we abandoned “÷”, which is used in primary schools and causes great difficulties later on when children have to cope with the fact that or, worse still, If we used “|” from the start, what could be easier than (Nor is there a case for restricting ÷ to ‘sharing’ situations, since + and − are used for both natural numbers and rationals.)

This paper contains many opportunities to investigate, conjecture and verify results by writing simple computer programs, and the reader is strongly urged to participate in this way throughout.

For each positive integer n, let F(n) be the sum of the squares of the digits of n; for example,

It is just twenty years ago that sixty school teachers (mostly heads of mathematics departments) and twenty representatives of industry spent ten days together at Trinity College, Oxford for an intensive exchange of information and ideas. They had as their theme the changes that had been taking place in the uses of mathematics, and their relevance for school teaching. The conference was a seminal one in the development of British mathematical education; it made important personal contacts, it widened horizons, and it could also be described as the great-grandfather of SMP (the grandfather and father being the conferences which followed in 1959 and 1961 at Liverpool and Southampton). With the emphasis nowadays on the importance of keeping school mathematics in touch with the needs of industry, it is germane to remind ourselves of the influence which industrialists have exercised—personally, financially and mathematically—on the development and dissemination of new curricula in this country. Members of our Association may not always appreciate the contribution of its Schools and Industry Committee in keeping the dialogue going.

It is a well-known fact that the number of hours of daylight varies with the time of year, except at points on the equator. It is sometimes thought that at latitudes between the Arctic and Antarctic circles the variation takes the form of a sine curve. However there is no theoretical basis for this idea, although it is true that a sine curve may give quite a good approximation.

Everyone connected with the teaching or practice of mathematics is aware that in recent years there has been a great increase in the number and variety of opportunities for the employment of mathematicians. In particular, industry, and especially engineering industry, has found a need to recruit mathematicians in large numbers with a view to employing them to do mathematics as a full-time occupation. This is quite a new development; although the value of the mathematically trained mind has long been recognised in many places, it is only quite recently that mathematics, as opposed to mathematicians, has been extensively employed by industry This change has, of course, largely coincided with the advent of the electronic digital computer, and the computer has certainly had a great influence on it; new mathematical disciplines have grown up as a result of the existence of computers, and it is now practicable to solve problems whose solution would be inconceivable without the use of a computer. As a result of this it has become fashionable to attribute the spread of mathematics to the computer. I think it more likely that the computer has accelerated rather than created the demand for mathematicians, and that the development of new industries and the introduction of new techniques to old ones would have brought about a demand for mathematicians even in the absence of computers. However there is no doubt that this demand exists, and it has certainly been stimulated by the existence of the computer.

In tackling a problem, we needed to show that a certain 3 × 3 determinant is non-zero. This determinant could be expressed in terms of the symmetric, homogeneous function N = Σx.Σx2 + 6xyz - 2Σx3 of three variables, where, for example Σx2 = x2 + y2 + z2. We first looked into several classical books, to see if there was a relevant inequality. There was not, nor do they contain anything other than ad hoc techniques for dealing with the inequalities they give. In this paper we give a general method for dealing with a certain class of such inequalities; it might be instructive for undergraduates, partly to exercise them in applying a moderately complicated algorithm, and partly to show an application of 2- and 3- variable Calculus.

Formulae for Σl=1n ik when k is a positive integer are well known, and many methods for finding them have been described. This article gives an account of an approximate, but remarkably accurate, formula which I have found for the case k =1/2.