Mobius‘ function has the important property of inverting functional relations [1] of the type and giving in turn (Also relations (0.1) and (0.2) follow respectively from (0.11) and (0.21))

I establish in this paper that no other arithmetical function μ*(n) can perform the above inversion. In other words, I prove that Inversion is the characteristic property of Mobius‘ μ-function. Theorem 1 below asserts that if for one pair of functions g(n) and f(n) (subject to a special condition) relations (0.1) and (0.11) are true with an arithmetical function μ*(n) (possibly depending on g and f), then μ*(n) coincides with the Mobius‘-function. Theorem 2 below gives a similar result in connection with functions G(x) and F(x) defined for all real x ≥ 1.

There is a rich and seemingly endless source of definite integrals that can be equated to or expressed in terms of Catalan's constant. Denoted by G and defined by

$${\rm{G}} = \sum\limits_{n = 0}^\infty {{{{{\left( { - 1} \right)}^n}} \over {{{\left( {2n + 1} \right)}^2}}} = 1 - {1 \over {{3^2}}} + {1 \over {{5^2}}} \ldots = 0.915\,965\,594 \ldots \,\,,} $$

It is, I think, of happy augury that it should have been fourid possible to arrange a joint meeting of members of the local branches of the Classical and Mathematical Associations, as well as members of the Yorkshire Natural Science Association, to hear a few words about Greek science. No doubt the classical or humane studies can stand by themselves and so (in a way) can the study of science based only on good English : but I venture to think that they are both much stronger when they stand side by side and support one another. The idea that thero is any necessary antagonism or opposition of interests between them is surely unfoimded, and there is room for both in our system of education. It is a mistake to suppose that in our school education there is not time for the one or the other ; the guestion is really one of using the time that thero is to the best atlvantage. No doubt schoolboys in the past have wasted much time in the perfunctory learning of Latin and Greek.

It is well known that, as regards moment of inertia about any line in its plane, a uniform triangular lamina may be replaced by a set of three particles, each with mass equal to one-third of the mass of the triangle, and placed at the midpoints of the sides. The usual proofs given for this elegant result are rather cumbersome. Now it takes only a few seconds to verify that the set of particles has the same moment of inertia as the triangle about each of the three sides (using the h2/6 formula for the square of the radius of gyration). It is tempting to ask whether the equivalence of particles and lamina may not be deducible from this. In other words, starting from the fact that the equivalence holds for three particular lines in the plane, can it be deduced for all lines in the plane?

It is a far cry from the Conics of Apollonius to the theory of linear spaces and matrix algebra and many men have helped to build the road between. This article is about one of the builders whose work for a time was discarded and whose name was almost forgotten. But his name finds frequent mention now, when the latter aspect of geometry bears so little resemblance to the former, and metrical geometry has given place to projective geometry with its non-metrical homogeneous coordinates, which make it possible to publish a book on coordinate geometry which does not contain a single figure.

I am most grateful to the Mathematical Gazette for publishing, in the March 1977 issue, my letter about the problem of finding five positive integers such that the sum of any pair is a perfect square. The response far exceeded my expectations and showed that the problem aroused great interest.

In this article I investigate Napoleon triangles, generalisations of the mysterious equilateral triangle in Napoleon’s theorem. I start with that theorem, develop some analogous results, find configurations with unexpected integer angles, and return to an extension of Napoleon’s theorem. Many of the geometrical proofs depend upon spiral similarities, and the numerical work uses some unfamiliar trigonometrical identities.

One of the more interesting uses of mathematics and statistics is their application to everyday situations in assisting people to make informed choices about their courses of action. Nowhere is this more evident than in the application of basic mathematical principles to the range of gambling opportunities. Indeed, a new university statistics course that I give in Australia on Gambling and Sport has attracted an enrolment of over 400 students in its first year alone. In this subject we discuss the intricacies of games such as Craps, Keno, Roulette, poker machines, horse racing, and a wide variety of lotteries. The sports content of the course relates largely to using mathematics to formulate optimal playing strategies.

In the Gazette for December 1953, I gave an account of all that I knew about Schur’s inequality, namely

If μ≥0 and x, y, z are all positive, then

with equality occurring when and only whenx = y = z.

Although a number of earlier researchers had used the geometric mean as a convenient statistic to summarise observational data, Gallon is usually credited with being the first to consider its sampling distribution. At Gallon’s request, in 1879 McAlister undertook a pioneering mathematical study, which eventually led to the modern large-sample theory. However, some sixty years elapsed before much attention was paid to small samples from particular parent distributions. Since about 1960, new techniques have made it possible to derive exact sampling distributions for a much wider class of parent distributions. Some work has been done on producing approximate general relationships between the moments of the parent distribution and those of the sample geometric mean but they are of very limited value for small samples and even now it is difficult to find any general description of how the distribution from which a sample is drawn will affect the distribution of its geometric mean and how this will vary with sample size.