A number of particles are constrained to move freely on the surface of a sphere. Between every pair of them there acts a force which depends only on the distance between them, the law of force being the same for all particles. For a given law of force and a given number of particles, there are a number of configurations of the particles in which they are in equilibrium. In general, these configurations will depend on the law of force: we do not expect the particles to remain in equilibrium if the law of force is changed. But there are certain configurations in which the particles are in equilibrium whatever the law of force may be, for instance if the particles are at the vertices of a regular polyhedron. In this paper these configurations are investigated. It is evident that if a configuration has rotational symmetry about the diameter through a certain particle P of it, i.e. if the configuration can be rotated through a fraction of a revolution about the diameter through P and thereby brought into coincidence with its initial position, then the particle P is in equilibrium whatever the law of force may be. Thus among the configurations with the desired property are those which have this rotational symmetry about every diameter through particles of the system. It is shown that these are in fact the only such configurations. Specifically, the only such configurations are those in which the particles are equally spaced on a great circle, with or without two further particles one at each pole, and those which, in relation to a regular polyhedron, comprise a set of particles at its vertices or a set at the mid-points of its edges or a set at the centres of its faces or any two or all three of these sets taken together. (This classification is redundant but it seems the most convenient.) No discussion of the stability of the configurations is given, as this depends in any given case on the law of force, the present discussion being confined to the property of equilibrium with independence of the law of force. The main proof occupies §§ 2-4. In §5 there is discussed the problem of two or more sets of particles on the sphere which are to be in equilibrium whatever the laws of force may be between particles both of the same set or of different sets. §6 relates the present problem to that of maximising the least distance between any two of an assigned number of particles.

The purpose of these notes is to suggest diagrams that may illustrate the setting up and ordering of equivalence classes, their addition and multiplication.

I first came across these dials at the Old Ashmolean Museum when looking at astrolabes. Miss Kathleen Higgins, then writing a thesis on dials, asked if I could explain the operation of a dial described by Regiomontanus (1436-1476) of which several examples are shown in British and European museums. That was in 1950; since then I have repeatedly tried to evolve an explanation of some way in which dials of this kind may have been invented. Only recently have I been satisfied with my results.

Astronautics is the scientific study of the many problems which arise when we contemplate the possibility of releasing man from the Earth’s gravitational field, setting him free to explore the Solar System and later the regions beyond. Not so long ago, to admit having such ambitions was regarded as a symptom of megalomania. Mathematical “proofs” of the impossibility of escape from the Earth’s field were legion and ranged from the simple statement “a rocket will not develop thrust in a vacuum” to a (perfectly correct) numerical calculation showing that the most powerful fuel possesses insufficient energy to release even its own mass from the Earth’s attraction. It was made very clear to the early enthusiasts that few advances in this field could be made until the climate of scientific opinion changed and astronautics became “respectable”. It is, of course, quite normal for a line of scientific enquiry to evolve from the stage at which its ideas are the subject of ridicule to the stage at which they are universally acclaimed. The idea of the periodic classification of the elements caused our great-grandfathers to laugh hugely into their beards. Marconi was a notorious striver after the unattainable. The Wright brothers achieved the “impossible” and, in defiance of contemporary opinion, the aeroplane proceeded to change our way of life irrevocably. Astronautics has been singularly fortunate. After spending comparatively few years in the wilderness, the advent of the V.2 and other rocket weapons brought about that change in the attitude of scientists towards the subject which had been seen as a necessary preliminary to the serious consideration of the problems involved. Greatly decimated, the doubters now depend upon the slogan—“it can be done, but it is not worth doing”. However, it is not the purpose of this article to discuss the value, in human terms, of conquering space but only to explain briefly why and how it can be done if man so desires. In any case, it is unlikely that the real significance for mankind of such a tremendous advance can be appreciated before the event. Only the passing of the years can reveal to us the value of our actions.

My attention was first called to this attractive curve by a pupil, M. J. Burleigh; so, lacking any other name, we called it “Burleigh’s Oval”. It is mentioned in Salmon’s Higher Plane Curves (p. 107), where a cartesian equation is given; and in Hilton’s Plane Algebraic Curves (p. 64), where it is described, with a diagram, as the reciprocal of the Limaçon. It also occurs in a very interesting general treatment of negative pedals by A. Ameseder, (Archiv der Mathematik und Physic, LXIV, 1879).

The two essentially different concepts of “form” and “function” are sometimes confused in elementary text-books, and in this note I set out to stress one or two important formal results, in particular the formal interpretation of the binomial theorem for a negative integral exponent. It has been one specific problem, namely the establishing of the Wronski relations

a1-h1=0, a2-a1h1+h2=0, .. ,ak-ak-1h1+...+(-1)khk=0,...

connecting the elementary symmetric functions ar and the homogeneous prodcut sums hr, which has recently focused my attention on this subject and prompted me to write down a few observations.

1. About ten years ago Neugebauer and Sachs published their Mathematical Cuneiform Texts in which the contents of Plimpton 322 were analysed for the first time. The numerical values contained in this text are given in Table I. The theory put forward by Neugebauer and Sachs was in my view not consistent with known facts and in the Proceedings of the Academy at Amsterdam I put forward a different interpretation. Since then I have made several short comments in connection with this text, but up to the present I have not given an explicit interpretation of the errors occurring on Plimpton 322.

Consider a regular tetrahedron ABCD of side p (Fig. 1); let P be a point on the face ABC and Q a point on the face BCD; we first consider the geodesic distance between P and Q.

It is surely a very remarkable thing that despite the range, power and success of modern mathematics, the concept of natural number, on which the whole edifice rests, is still something of a mystery.

Towards the end of the last century the great German mathematician and philosopher Gottlob Frege, dissenting strongly from the view current in his time—and perhaps still today—that number is either a physical or a mental entity, gave the first purely logical definition of a natural number. His work aroused so little interest at the time that his definition remained practically unknown until it was rediscovered by Bertrand Russell 20 years later. The definition, though a little strange at first sight, is quite simple and is based upon a notion which is of first importance in many branches of mathematics, the notion of a (1, 1) correspondence.

When one first comes across the formulae

one is immediately struck by the fact that the second sum is the square of the first. The usual methods of proof give no reason to expect that the sums will be related in such a pleasant and easily memorable way; one is left with the impression that this is a happy accident. The following proof may give some entertainment because it throws light on this point; it emerged unexpectedly out of the tortuous process of making up a question for a recent Scholarship examination.

My main thesis is that men have too long allowed themselves to be bemused by the magic of numbers, though the kinds of spell that numbers have cast over us have changed with the ages. In seeking to understand why it is difficult for us to break the spell, I have been led to suspect that schoolmasters (and perhaps especially schoolmistresses) are largely to blame because of the way in which they use school marks. And a special responsibility in this matter rests upon the mathematical staffs of schools, not because they are more “mark-minded” than their colleagues, but because marks appear to be numbers or measures of some kind and the non-mathematicians must look mainly to the mathematician for guidance on the manipulation and interpretation of numbers and measures. Any lead that may be needed to reform our school marking systems must be given, if it is to be given at all, by the mathematicians. That is my justification for addressing my comments on School Marks to the Mathematical Association.