The problem of the “tippe-top” has been discussed in several papers, one by Synge makes the assumption of rolling and discovers instability when the axis is vertical with the peg up, but requires as a necessary condition that the top be not a solid of revolution. Since it appears that as far as is constructionally possible the top is axially symmetric, in which case the rolling motion is always stable, this solution would seem to be unrealistic. In a second group of papers Fokker suggests from observations of peg traces that normal tops roll with no sliding, Braams shows that for the tippe-top sliding will probably take place in the “ rapid precession “, and Hugenholtz deduces general conditions in which such sliding will cause the peg of the top to fall. For the final rise on the peg these authors introduce a “ rolling friction “. Here results substantially in agreement with those of Hugenholtz are found from a simpler result by vector methods. It is further shown that the motion is such that during the final rise sliding must take place until the top is very nearly erect, and that the simple assumption of sliding friction is sufficient to explain the entire motion.

Cyclically repeating patterns are a common feature of mathematical films. The study of the symmetries which arise in such configurations has a theoretical interest to mathematicians watching the film, and it also has a practical interest to the film maker, because the symmetries present in a diagram can often be used to reduce the labour of drawing. It does not seem that these symmetries have been commented on before, and in this article we attempt a complete enumeration of the cases which can arise.

The approach to plane mechanics by vector methods is often considered to be an unnecessarily sophisticated one and it is true that the preliminary work in vector algebra is lengthy and out of keeping with the elegance achieved. The natural tool, it has been said, is the complex number, and I shall obtain some of the fundamental results by using only complex number theory and vector addition.

Our common use of the word “mind” invests the term with considerable obscurity and ambiguity. This is partly because, whilst many philosophers, psychologists and scientists deny any reality to the notion, many others have regarded it as the only reality Others again have regarded both mind and matter as having reality but reality of quite different types, e.g. subjective reality and objective reality. Without going into these controversies I must begin by pointing out that for a psychologist, whatever his philosophical beliefs, the mental processes of the people he is studying are as real and as objective as the interior of a star is to an astrophysicist—and as inaccessible to direct observation. Any other assumption would make nonsense not only of psychology but of the whole of education. And it has a special bearing on the problem of determining the nature of mathematics.

This is a continuation of a note in the Gazette of May, 1954.

We describe a variant of the game with digits discussed in a previous article. There we considered a transformation of any number in a scale s + 1 in which each digit is replaced by a digit equal, modulo s, to the sum of the remaining digits. Starting for instance with the number 16427, in the scale of 10, the first digit remains 1, since 1 = 6 + 4 + 2 + 7 (mod 9), the second digit is replaced by 5, since 5 = 1 + 4 + 2 + 7 (mod 9), and so on, forming the number 15794; repeating the transformation we obtain in turn 73184, 72461, 49751, 48137, 16427, the original value recurring after six transformations. We showed that under this transformation any number x, with m digits in the scale s + 1 (none equal to s) recurs unless m − 1 has a common factor (greater than unity) with the quotient of sm by the highest common factor of sm with the sum of the digits of x.

Contemporary writers of textbooks on electricity and magnetism and on hydrodynamics make considerable use of the complex variable in two-dimensional problems. It seems strange that two-dimensional dynamics is not taught in this way, even though the application of vectors to three-dimensional dynamics has now become respectable. Mr. Buckley's article preceding this has shown how the basic results may be established. The following examples are further indications of how this powerful technique may be profitably employed in two-dimensional kinematics. The methods are essentially labour-saving.

It was conjectured by David Gregory that a sphere can touch thirteen non-overlapping spheres equal to it. In a recent paper proofs have been given that this in fact is impossible. In the present paper I outline an independent proof of this impossibility, certain details which are tedious rather than difficult being omitted.

The International Commission on the Teaching of Mathematics, which is one of the Commissions set up by the International Mathematical Union, has been conducting an enquiry into the teaching of mathematics between the ages of 16 and 21. A full report has already been published on this aspect of teaching so far as Germany is concerned and a questionnaire was sent to all countries participating in the work of the commission. The British National Committee for Mathematics appointed Dr. E. A. Maxwell to represent Britain on the International Commission and a sub-committee consisting of Dr. Maxwell, Miss M. L. Cartwright, A. P. Rollett and G. L. Parsons to prepare the answers to the questionnaire. This sub-committee also received a good deal of help from the Department of Education of Cambridge University in connection with questions relating to the general system of education. The sub-committee confined their replies for the most part to work done in schools. With the replies were sent various relevant pamphlets published by the Ministry of Education, specimen timetables from various types of schools, a large number of examination papers and syllabuses and reports of the Mathematical Association. The questionnaire was framed for dealing with state systems of education similar to that in Germany and covered the whole educational system. We give only those parts of it relating to the technical aspects of the enquiry.

Before men learnt to count, the shepherd guarded against the loss of a she from his flock by making a tally of the flock. As each sheep passed through the gate in the morning, on the way to the grazing land, the shepherd cut notch in his stick. At nightfall, when the flock returned to the fold, checked his sheep against the tally by running his fìnger along the notche moving from notch to notch as each sheep passed. The tally stick served i purpose just as well as counting the flock serves the shepherd today. T tally could be passed from hand to hand almost as easily as numbers a now passed by word of mouth (or written down); it constituted a reasonat permanent flock record, and so long as flocks remained quite small there w no necessity to mother the invention of a new system of recording.

A series of investigations is being planned by a section of the Board of Studies for Mathematics in the Institute of Education of the University of Birmingham, with the object of gaining information about the types of difficulty experienced by children in various branches of grammar school mathematics, and the types of error made. An analysis of these errors, both from the mathematical and psychological points of view, is highly informative and relevant to a discussion of teaching methods in the subject. It is hoped that later it will be possible to prepare standardised tests which would be of diagnostic value to teachers. The co-operation of other teachers would be welcomed in furthering this investigation, either in the form of comments or criticisms in the light of their own experience, or in helping to widen the sample of children tested in future work.