It has struck me that axiomatic treatments of mathematics resemble the creation myths of primitive religion. In both cases the attempt is being made to provide an ordered account of something that already exists. In the one case the ordering is by deduction; in the other by the temporal sequencing of narrative. By way of illustration I offer a paraphrase of the opening pages of Hubert's Grundlagen in the style of a creation myth.

Of course, I do not use ‘myth’ with any derogatory intention. Nor do I wish to disparage historical attempts from Euclid onwards to present mathematics as a deductive system; indeed I will join the current chorus of regret that this aspect of the subject is now absent from school mathematics. The trouble is to know how best to present axiomatics. It is no longer possible, as it was for Euclid and even for Frege, to suppose that deduction leads back to ‘primitive truths’. One cannot pretend that the mathematics lies wholly in the deduction itself; as Wittgenstein observes, this is like saying that cabinet-making consists in gluing. To say that an axiomatic system is a ‘free creation of the human mind’ does not begin to address the question of purpose and motivation. Perhaps the geometrical deductive systems of Euclid and Hilbert should be seen as conceptual models, designed to make sense of geometrical phenomena, rather as a creation myth is designed to make sense of biophysical phenomena.

In 1952 W. E. Roth published two theorems, one of which has come to be known as Roth’s removal rule and (slightly generalised) goes as follows. [Recall that square matrices M, N are similar when there is an invertible matrix R such that RMR-1 = N. The matrix entries can be elements from any field, although for simplicity we shall call them ‘numbers’.]

The Fibonacci series 0,1,1,2,3,5,8,13,.... is well known because so many properties have been found for it and because there are many instances of it occurring both in nature and mathematics. It is of course formed from the recurrence relationship

The Tribonacci series 0,1,1,2,4,7,13,24,44,81,149,... is formed in a similar way but from the recurrence

In 1822 Joseph Fourier published an account of his work on the mathematical theory of heat. In his book Fourier considered the representation of a function f(x), which is defined on an interval such as 0 x < 2π, by an infinite sum of trigonometric functions:

where the trailing dots indicate the sum goes on forever. The series is written more concisely with sigma notation as

where x is restricted to an interval such as 0 x < 2π, and a0, an and bn are constant coefficients.

The tide of technological advancement nearly swept away this article. I have had to rewrite this first section in the light of new developments, which I suspected might beat me to publication. For several years I have wondered why ‘pocket’ calculators, however well specified in other respects, have not had the pre-programmed ability to give values of the cumulative distribution function of a standard normal variable, commonly written as Φ (x), and its inverse. The A level mathematics student must be a prime target in the marketing of calculators and there can’t be many who don’t have to thumb through statistical tables a few hundred times in the course of their studies. The very latest generation of machines do include these functions, but it will, I suspect, be quite a few years before everyone is so well equipped.

In every textbook which deals with second order linear finite-difference equations, methods will be found for solving such equations when the coefficients are constants. In the author’s experience it is unknown for textbooks to treat second order linear difference equations with variable coefficients. Yet such equations certainly occur in practical applications. It is the present intention to remedy this defect in the literature. It will be seen that, by the methods discussed here, we can always reduce the solution of a linear second order difference equation to that of a first order equation.

The problem of dissecting a square into squares of unequal sizes has been extensively discussed in the recreational mathematics literature (see for example), but the dissection of triangles appears not to have received so much attention; there are however a number of naturally occurring problems in the dissection of triangles into triangles. The problem we considerhere is the dissection of triangles into five similar triangles; the component triangles must be similar to each other, but may or may not be similar to the original triangle.

Since antiquity mathematicians (and not only them) have taken an interest in constructing magic squares. Probably the first magic square ever created is the one shown in Figure 1. Its origin is shrouded in the mystical legends of ancient China. It became to be known as Luo Shu (Luo river writing). There was no clear connection between this configuration and mathematical study until the time of Yang Hui, even though it was described in the sixth century. Another well-known magic square (Figure 2) is in the painting Melancholia made by the famous renaissance artist Albrecht Dürer in 1514 (the year is formed in the middle of the lowest row).

A normal n × n magic square consists of the numbers 1, 2, … , n2 arranged in a square so that the sum of each column, row and the two diagonals is In an odd order normal magic square , the central number is This article concerns bordered magic squares of odd order, which are squares that remain magic when each outer row and column is peeled off.

In the first part of this paper we traced the early development of quaternions in the hands of Hamilton’s successor, Peter Guthrie Tait, Professor of Natural Philosophy in the University of Edinburgh from 1860 to 1900. Tait had neither the intuitive feel for physical concepts that Maxwell possessed nor the entrepreneurial talent of Thomson (Lord Kelvin) and yet he fulfilled a pivotal role in nineteenth century British physics through his correspondence with the former, with whom he went to school, and his collaboration with the latter. Though ambivalent towards quaternions, Maxwell was chivvied by Tait into drafting his 1873 Treatise on Electricity and Magnetism in both Cartesian and quaternion form.

Many years ago, long before it had become fashionable for so many research mathematicians to concern themselves with pedagogy, the renowned topologist Bill Thurston appeared as the keynote speaker at a symposium held at Harvard University. As he approached the podium, he could be seen carrying a set of toy building blocks, the sort designed for children. He was going to use them to construct models for his talk. But before he began, he held up the box and said, ‘You know what, kids love this stuff. What is it that we do to them?’ Of course, Thurston was right. Children have a natural curiosity, an innate sense of wonder about the world. Picasso said that children were natural born painters. Thurston said that they were natural born geometers. How do we help them to recapture that child-like sense of wonder after so many years of neglect?

Being aware of John Branfield’s request and having enjoyed Tom Roper’s response may I offer a contribution to the problem of describing the curve on which the bowls move? The important first thing is to decide upon a model of the system which is hopefully sound, and closely approximates the truth. It must also be amenable to mathematical analysis.