The fact that a mechanical system so well known to so many in its practical aspect as the swing has been thoroughly neglected in textbooks and problem papers may justify a few remarks on it here.

The only reference to the swing which I have been able to find is the question requiring it to be shown that the amplitude of the swing may be increased by internal action only, if, when the swing passes through its lowest position, the distance of the centre of gravity from the point of suspension is (suddenly) shortened from l to (l – h), the full distance l being restored at the end of the swing.

In the years 1603–1867, known as the Edo period, when Japan isolated itself from the western world, the country developed its own style of mathematics, especially geometry. Results and theorems of traditional Japanese mathematics, known as ‘Wasan’, were usually stated in the form of problems; these were originally displayed on wooden boards (‘sangaku’) hung in shrines and temples, but many later appeared in books, either handwritten with a brush or printed from wood blocks. (See [1, 2, 3] for more details.) Solutions to the problems were not given, but numerical answers were sometimes provided. The problems were written in a form of language called ‘Kanbun’, based on Chinese, which cannot be readily understood by modern Japanese readers even though written Japanese makes extensive use of Chinese characters.

The editor of this journal recently deplored the inability of members of the government to make correct use of the simple mathematics that they encounter in their various capacities. The purpose of this article is to call attention to a common mathematical failing of those who are “authorities” in questions such as the life expectancies of non-renewable resources. My particular example comes from Britain, although failures such as this one are common among “experts” in the U.S.

Figures 1–4 set the scene, with configurations of tangent circles that may perhaps be new. The labels are curvatures: the curvature or bend being the reciprocal of the radius. By convention, a circle that surrounds another has negative bend and radius. In the first part of this paper I give some easy formulae for configurations like these (see Note a). I show how we can develop fractal structures—I call them froths—having an infinity of tangent circles in each ‘triangular’ region; and give methods for finding integral froths. The second part extends these ideas to systems of tangent spheres, which may also be integral. Throughout, an integral object is one whose bend is an integer, circles and spheres are generally referred to by their bends, and examples are marked with bullets (•).

Euler’s Φ(n) is the number of positive integers n with no factor in common with n. We wish to calculate

This problem arises in connection with the Farey series Fn, i.e. all the fractions from 0/1 to 1/1 (inclusive) with denominator n in their lowest terms, in order of size, so that for example