In the first part of this paper we traced the early development of quaternions in the hands of Hamilton’s successor, Peter Guthrie Tait, Professor of Natural Philosophy in the University of Edinburgh from 1860 to 1900. Tait had neither the intuitive feel for physical concepts that Maxwell possessed nor the entrepreneurial talent of Thomson (Lord Kelvin) and yet he fulfilled a pivotal role in nineteenth century British physics through his correspondence with the former, with whom he went to school, and his collaboration with the latter. Though ambivalent towards quaternions, Maxwell was chivvied by Tait into drafting his 1873 Treatise on Electricity and Magnetism in both Cartesian and quaternion form.

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

I do not know what proportion of the users of mathematical tables go beyond the use of proportional parts in interpolation work. But there are undoubtedly, on the one hand, some tables whose construction would be unnecessarily laborious and expensive were they to be at such an argument interval as to render linear interpolation feasible, and, on the other hand, there are those who find it necessary to use such tables. Given such a table, there are various methods of performing the interpolation. There is the use of the Taylor series, f (a + h) = f(a) + hf′(a) + ½h2f″ (a) + ..., which requires a knowledge of the derivatives of the function as well as the function itself. Then there is the use of the Lagrange formula of interpolation, fx = a0f0 + a1f1 + a2f2 + ..., involving a number of tabular values on both sides of the required value.

Why is such a nice result rarely, if ever, found in texts? Perhaps because it is usually necessary to use trigonometry to calculate the areas of regular polygons. A regular n-gon inscribed in a unit circle has an area n times that of the isosceles triangle having an edge as base and the centre as apex; that is,

area of regular n-gon = n × (1/2. 1.1. sin (360/n)°) = 1/2n sin (360/n)°.

If Pascal’s Triangle is written down, it will be noticed that the number of odd numbers in any row is a power of 2; moreover, if every even number is replaced by 0 and every odd number by 1, the result is an interesting pattern of triangles from which it is possible to deduce a general rule. Again, if divisibility by any prime number p is considered, and if every number in Pascal’s Triangle is replaced by its residue (mod p), an even more intriguing pattern is formed. These considerations led to the following investigation.

The article which follows is the first of a set of papers sponsored by the Modern Trends sub-committee of the Teaching Committee of the Mathematical Association. It is hoped that future papers will be published either in The Gazette or as a supplement. Each paper will represent the personal view of the author rather than being an official policy statement of the Association. It is hoped that they will lead to correspondence in the form of reports of experiments as well as constructive criticism.

This article is an exploration of some common ground between network theory in mathematics and isomerism in chemistry. As with all interdisciplinary studies, it is necessary to begin by identifying technical terms across the boundary between the two subjects. Once that has been done, the diagrams drawn by the chemist are interpreted in the language of geometry, and links are established between results in the two subjects.

Starting with a finite set of numbers {xl, x2, x3, …, xn} we can generate a sequence of numbers {yi} by the rule

so that yi is the average of the previous n terms of the sequence.