When we write a number in the ordinary way, such as 673, the symbols such as 6, 7, 3 in this example stand for what we call the digits of the number, and a piece of mechanical or electrical equipment which operates with and records the discrete digits of each number is often called a digital calculating machine. In the past twelve years there has been a remarkable development of such machines with two important features. First, they can carry out lengthy and intricate calculations quite automatically once they have been supplied with a specification, in suitable form, of the calculation to be carried out, and secondly, they are very versatile, so that the same machine can be used for many quite different kinds of calculation, for example for evaluating values of a function from its power-series expansion, for solving large systems of linear simultaneous algebraic equations, for finding the characteristic values of matrices, for the step-by-step integration of ordinary differential equations. To express these two features, such machines are sometimes called general-purpose, automatic, digital calculating machines, and my purpose is to give an account of some aspects of such machines, both of the machines themselves and of the way in which they are used. I can only touch on some aspects, since there is not nearly enough time in a single lecture to deal with the subject fully.

I propose to put before you today a few somewhat disconnected thoughts and reflections that have occurred to me as a result of my experiences as a student, as a researcher, and as a teacher. I cannot say that I think they are of much value ; some of them are obvious, and others are merely expressions of opinion. But it is good for us to stand back occasionally from our subject and look at it as a form of human activity, and an address like this provides a convenient opportunity for doing so.

Study and research have much in common. In both of these activities, an individual is striving to acquire knowledge which is new to him. He is engaged in discovery. When studying, he has guides in the form of teachers and books to help him, and even when he is not actually making use of them he has the comforting knowledge that they are at hand if he needs them. In research, he has to rely almost entirely upon his own efforts. Students vary a great deal in the extent to which they depend on their teachers and books, but it is a familiar fact that the more a student can discover for himself, the greater the mastery he achieves over the knowledge he has won. In other words, the most effective study is that which most closely resembles research.

Mathematics is very much older than History, which begins in + 1066, as is well known ; for the first mathematician of any note was a Greek named Zeno, who was born in – 494, just 1560 years earlier. Zeno is memorable for proving three theorems: (i) that motion is impossible; (ii) that Achilles can never catch the tortoise (he failed to notice that this follows from his first theorem); and (iii) that half the time may be equal to double the time. This was not considered a very good start by the other Greeks, so they turned their attention to Geometry.

Several authors have recently discussed in the Gazette the place of experimentin the teaching of mathematics. Although it is true that most students can visualise the behaviour of a mechanical system quite well from a proper description or a good drawing, I find that all students, and particularly engineers, seem to be greatly stimulated by an occasional demonstration related to their work.

It is a far cry from the Conics of Apollonius to the theory of linear spaces and matrix algebra and many men have helped to build the road between. This article is about one of the builders whose work for a time was discarded and whose name was almost forgotten. But his name finds frequent mention now, when the latter aspect of geometry bears so little resemblance to the former, and metrical geometry has given place to projective geometry with its non-metrical homogeneous coordinates, which make it possible to publish a book on coordinate geometry which does not contain a single figure.

We seniors are apt to be told that, in our teaching of analytical geometry, we have been in the habit of giving too much attention to the intricacies of conic sections, and that work on other curves and their tangents and normals would have been more interesting and instructive. There is much force in this criticism, but when we come to examine the alternatives to the conics which are offered to us, they are usually cubic curves, probably given in terms of parameters, such as the “Folium of Descartes” I should like to suggest that some better-known curves, notably the curve of sines and the logarithmic curve, will provide examples even more instructive than the “Folium”, and that these examples have at least the advantage of greater novelty.

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.

Twice in the course of some work on chemical kinetics it became necessary to determine the type of the solution to a generalised set of any number of first order simultaneous differential equations. The problem reduced in each case to a question of the reality and sign of the roots of the auxiliary equation, which was, of course, determinantal in form. The required information could be deduced from long-established theorems on the roots of symmetric and skew-symmetric determinantal equations, but the following proofs, which are on a much simpler level, may be of interest.