When I began to consider how I could best fulfil my promise to read you a paper, my difficulty in selecting a subject arose, not from the dearth of matter, but from the wide range before me, as I was told I might select any mathematical subject I liked. The subject of mathematics as a whole I at once put aside, as I felt it would be impossible for me in a short paper to say anything new about it. For the same reason my favourite subject geometry was unsuitable, and as I was brought up on the Elements of Euclid, anything I might have to say about geometry would probably be considered old-fashioned and antiquated.

The advent of the modern calculating machine has reduced the labour of computation in an extraordinary degree; continued products, long division, and so on, offer no terrors to anyone who has access to any of the latest excellent machines; even the extraction of a square root can be performed to ten digits in a minute or two. Under these conditions the numerical solution of algebraic equations, a process stigmatised in the seventeenth century as intolerably laborious, can even be in danger of becoming a vice : the present note owes its origin to a mild indulgence.

I received this morning a pamphlet in which the master of Stowe School, describing the Acropolis, says: “The physical beauty of the place alone would have moved even a mathematician’s heart,” and that appears to be what the world thinks of us, or, at any rate, a different part of the scholastic world. I am, of course, speaking to mathematicians and not to teachers, and I suppose we have all at one time or another suffered from the misguided hostess who introduces one as a mathematician. When that terrible word is mentioned the other person instantly curls up and says to the so-called mathematician, “How terribly clever you must be,” or else, “I never could do maths, at school.” Neither is a good opening in conversation, and it takes some time to beat down that barrier and get back to ordinary friendly terms. A mathematical reputation is rather like a clerical collar; it is a hindrance to intercourse and condemns one to a very great deal of loneliness. This may, of course, be partly due to jealousy of those who are academically superior on one particular point, although they are inferior on a great many other points, because there is a certain amount of truth in the gibe that a specialist is a person who has failed in every subject except one, having had neither time nor energy to succeed in the others.

Most of the following theorems regarding the hyperbola of Apollonius and the parabola of Chasles are classical, but not all of them are as well known as they deserve to be, and it is hoped that this purely geometrical treatment may be of interest.

1. The applications of spherical harmonic functions in mathematical physics are closely connected with the various potentials—gravitational, electrostatic, magnetic, or the velocity potential in hydrodynamics. It is of interest to recall, briefly, the development of the potential theory. The actual name “potential function” or potential was first introduced by Green in the memoir of 1828 in which he published his famous theorem on the transformation of surface and volume integrals. But the use of the potential function goes back to Euler (1707–1783), who formulated the fundamental differential equations of hydrodynamics, and integrated them in the case where a velocity potential exists, which he recognized to be a special case.

I have read with interest Mr. E. T. Dixon’s criticism of my paper on a “Partial Failure of Euclid (I. 4) in Time-Space Theory.” Mr. Dixon takes objection to this title, although, in his own words, he admits that: “Time-Space Theory may, in a limiting instance, be incompatible with Euclid’s (I. 4).”

A Meeting of the London Branch of the Mathematical Association was held at Bedford College, Regent’s Park, London, N.W., on Saturday, 22nd March, 1930, for the purpose of discussing “What Changes, if any, are advisable in the Elementary Mathematical Syllabus of the First School Certificate?” Mr. J. Katz presided.

1. The course of Plane Trigonometry “up to and including the solution of triangles and the properties of the circles associated with a triangle” offers little difficulty to the teacher and no serious difficulty to the pupil. But in the further development of the subject, when the circular functions sin x, cos x, … are regarded as functions of the real variable x, the position is different.