The contents of this article result from the perplexities and questions which assail one in teaching or expounding, in talking and writing about, mathematics. What do we mean by “thinking mathematically”? Whence comes the “certainty” attached to the result of a mathematical argument—a certainty not shared by theology, artistic criticism, politics, history, or even by biology, chemistry or physics? How comes it that the abstractions of mathematics enable us to deal better with the problems of the real world around us? How does mathematics grow—is it created, evolved, or discovered? And so on! I cannot deal completely or comprehensively with all these questions, and I doubt whether final answers to most of them are yet known. I will therefore confine myself to four topics: proof, classification (including recognition), contradiction (including consistency), and discovery

It seems likely that the teaching of statistics in schools will become more common and will be the responsibility of teachers of mathematics. Many of the pupils will not be mathematical specialists.

A sheet of graph paper, marked in inches and tenths or (preferably) smaller subdivisions, is placed on a horizontal surface.

A halfpenny is placed flat near its edge, and flicked along the paper.

In The Gazette for February 1962 (Vol. XLVI, pp. 1–5), Mr. Roger North gave a very interesting method for constructing a group of any even order whose elements are functions of x, the rule of combination being substitution of one function in another. He also showed that the group of order 8 so formed is isomorphic to the octic group, the group of rotations of a square. (See also the article by Miss Joan Holland in The Gazette for February 1964, pp. 47-57.)

This is a very beautiful natural phenomenon, seen on occasions when the primary bow is a strong one. As will appear later, the occurrence necessitates raindrops of a homogeneous size, and of course bright sunshine at a low angle. The diffraction rainbow is therefore most commonly seen in tropical countries, in a thunderstorm that has already passed over and is moving away from the setting (or rising) sun.

Although the gradient of a scalar, and the curl and divergence of a vector can be obtained by various elementary methods in terms of curvilinear components and coordinate vectors, the corresponding results for the components of strain and the vector divergence of stress are usually obtained by specializing the formulae for covariant derivatives. The purpose of this note is to give an elementary approach to these latter results in the spirit of the theory of Cartesian tensors. Apart from standard elementary vector analysis the only prerequisite is a geometrical appreciation of the analysis of deformation. For completeness an elementary analytical derivation of the results about gradient, curl and divergence is given first by a method which, whilst certainly not new, does not appear to be very well known.

In a sense Stirling’s formula can be adapted to slide-rule computation of x! when a; is a fraction and tables are not at hand.

An interesting application of elementary matrix theory is to prove that at any point of a rigid body there exists at least one set of principal axes of inertia.

If p is a prime and ps is the greatest power of p that divides N!, then

In this note we prove the following Theorem pertaining to Format’s Last Theorem.

The pipe-laying problem of E. Kosko (The Mathematical Gazette, No. 364) suggests some rather similar track-laying problems which can be solved geometrically. The pipe-laying problem is a particular case of one of these and can be solved by a simple construction which is purely geometrical.

The following discussion indicates how the equation

The clique originated from a search for a counter-example for the group, and this may be thought a sufficient justification for its study However, as well as its behaviour frequently being similar to that of the group, it has a number of properties which are peculiarly its own.

Twelve oxen were put to graze in a field of 3⅓ acres. After four weeks they had eaten all the grass in the field, including the grass which grew during the four weeks. At the same time 21 oxen were put out to grass in a field of 10 acres; they grazed it bare in 9 weeks. How many oxen would a field of 24 acres support for 18 weeks, if they were put in it at the same time? (See Isaac Newton’s Arithmetica Universalis.