Given positive numbers a > b, consider the ‘agm iteration’ given by a0 = a, b0 = b and

At each stage, the two new numbers are the arithmetic and geometric means of the previous two. It is easily seen that bn < an, (an) is decreasing, (bn) is increasing and an + 1 − bn + 1 < ½(an − bn), and hence that (an) and (bn) converge to a common limit, which is called the arithmetic-geometric mean of a and b. We will denote it by M (a, b).

How do you know that the equation x3 + x = 5 has one real root? Well, the function x3 + x is monotonic increasing: any increase in the value of x produces an increase in x3 + x. In particular, since 13 + 1 = 2 and 23 + 2 = 10 , we know that, for some x between 1 and 2, x3 + x = 5 . The function must “pass through” 5, and, since it is always increasing, it does this exactly once.

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.

Mathematics is relentlessly seeping into the very foundations of the civilisation in which we live and the community of which we are part, but I wonder how many of my readers—teachers of mathematics in grammar schools or in the universities—realise the magnitude and understand the nature of this seepage. It is permeating commerce, sociology, economics, medicine, linguistics, …, as well as more traditional science and technology. In this article I shall confine myself mainly to engineering, although I believe that the principles I shall develop will apply generally to any ancillary mathematics.

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

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.

Pupils from the age of 12 to 16 can occupy themselves for hours with their skate-board, walkman or the video-recorder, while their concentration during a lesson of mathematics vanishes often within a few minutes. What could be the cause? Isn't it possible to keep their attention during the lessons? Of course, I'm not the only one posing this question. Many solutions have been proposed in the past for this concentration and motivation problem. In this article I just want to add another one. I do not have the illusion that I am presenting the ultimate solution, simply because it does not exist. For a good teacher having one solution only is not enough. He or she has to combine and diversify the various ideas continuously. The keyword for the lessons of a good teacher is variation.