What are your mathematical gems? Which results or ideas in mathematics give you the greatest pleasure? Which of them would you choose to take with you to while away the hours on a desert island? Is there a mathematical book above all others that you would wish to have with you? How would you make your choices? And what would be the reason for each specific choice? Similar questions are asked of celebrities (and noncelebrities to boot) in a long-running and very popular radio programme in Britain called Desert Island Discs. The difference is that the radio guests are asked to choose a number of pieces of music, together with one book (other than the Bible or Shakespeare, which are routinely provided), and one luxury.

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.

In an earlier note the iterative behaviour of taking the unit imaginary, i, to higher and higher powers of i was described. This iterative power sequence converged to a limit. In order to make discussion of later developments simpler we need to introduce some definitions.

Two days before his death in the summer of 1901 Peter Guthrie Tait gave his son some handwritten notes on quaternions for safekeeping. Though he had distinguished himself in many areas of mathematical physics and had influenced the work of Thomson and Maxwell it was his evangelical promotion of quaternions which would be remembered in the years to follow and it was fitting that his last energies be devoted to the cause.

There has always been a mystery about how Kepler (1571-1630) discovered the laws of planetary motion: not until more than half a century later did Newton show how they can be explained within a gravitational framework. It is generally assumed that Kepler based his achievements on the full panoply of classical Greek geometry, and certainly he was very familiar with the advanced techniques developed by Archimedes and Apollonius, which he often referred to in his optical and stereometrical work. Yet I have found that for his astronomical success Kepler relied on nothing more complicated than the methods and results of Euclid. And since he did not know what sort of solution his enquiry would produce, my historical account of Kepler's discovery turns into a tale of detection – which began at the University of Tübingen in the early 1590s.

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 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.

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).