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This paper was originally planned as an exposition of finite geometry in a form in which it could be used to illustrate all the work covered in the normal school geometry course. The beginning of it, however, was anticipated by an article for the Gazette by Mr. H. Martyn Cundy (1), which he has kindly allowed me to read before publication. This present paper shows how a coordinate system can be constructed for the geometry described in his article by using the field of residue classes (mod. 5); that is, the Galois field GF(5). A definition of distance is introduced which extends that adopted by Cundy to a form suited to the discussion of a wider range of theorems.

It is a commonplace that pictures leave a deeper impress on the human mind than abstractions and, in spite of its abstract nature, that is also true of many part? of mathematics. A proof of a theorem or the construction of a counter-example is often suggested by pictorial considerations, and that is true of algebra and analysis as well as of obviously geometrical subjects like topology. (For entertaining examples see Professor J. E. Littlewood’s book [1].) There is obviously a connection between a developed geometrical intuition of that kind and the kind of spatial experience which ought to be furnished by a school mathematics course. No one would deny its importance and the writers of textbooks on ‘modern’ mathematics are well aware of the need to stimulate geometrical intuition. What has been called in question is the relation between that kind of geometrical intuition and the formal treatment of the euclidean plane based, albeit insecurely, on Euclid’s Elements. Indeed, the traditional treatment has been abandoned with indecent haste and without serious question. In Part I of this paper we make a plea for a re-consideration of the case for some kind of formal geometry as an indispensable part of the education of the more mathematically able children.

In Cundy and Rollett’s invaluable book on Mathematical Models, the authors begin their Chapter on Polyhedra by remarking that “The most suitable, and in many ways the most attractive, subject for an experiment in the construction of mathematical models is a set of polyhedra.” Various methods are in general use to produce finished models of polyhedra for the showcase, and similar methods form the basis of those considered suitable for constructing the simpler polyhedra in the classroom. That most commonly adopted is to draw the net of the required solid on a sheet of cardboard, allow for tabs as necessary, cut it out, score the creases half-through, fold up, and stick the tabs with some suitable quick-drying cement. For class-room use the major drawback to model making on these lines is the time-consuming and potentially messy process of sticking; and it is the object of this article to develop a method which dispenses with the use of paste or cement altogether. In fact I shall show how, using paper and scissors only, firm, neat models can be made, which in the case of the simpler solids could easily be produced in the classroom, and which in the case of the more complex polyhedra reveal by their construction in a striking manner some of the geometrical relationships of these solids.

The conception of a mathematical time, as employed in the theory of relativity and other branches of mathematical physics, is frequently used and of great importance in our development of the theory of the universe. In most problems of mathematical physics it is customary implicitly to assume linearity of the temporal continuum, so that instants and periods are respectively regarded as points and sections of a homogeneous number-system. In this way the temporal and numerical continua are identified, and times become analogous to ordinary numbers. In the same way that the conception of imaginary quantities is necessary to the completion of the ordinary number continuum, so also it becomes necessary in certain problems of applied mathematics to consider imaginary values of the time variable. The present paper represents a survey of the mathematical theory of time, and an attempt to interpret mathematically, by means of a time-system admitting imaginary values, certain optical phenomena of relativistic mechanics.

The centenary of the birth of Oliver Heaviside last year has been the occasion of celebration by electrical engineers and physicists in this and other countries. In the discussions of his work much has been said about the Operational Calculus ; and as the versions of its history which have been given both in these commemorative celebrations and in most of the textbooks of the subject are seriously incorrect, this may serve as an occasion to recount that history more correctly. The story in widest circulation is that the Operational Calculus was discovered by Heaviside (Boole being sometimes—and incorrectly—named as the discoverer of its applications to ordinary differential equations) and rejected by British mathematicians because of Heaviside’s lack of rigour. The facts, as I shall show, are that the Calculus was well known in Britain and France before Heaviside’s birth, and that the rejection of his paper had nothing to do with his use of symbolic methods.

The study of convex polyhedra from a mathematical point of view began in ancient Greece well over two thousand years ago. It is believed that Euclid’s “Elements” was written to inspire a wider appreciation of the properties of the five regular polyhedra or “Platonic solids”, as well as to place their study on a secure mathematical foundation. The discovery of convex polytopes (the analogous figures in four or more dimensions) and their investigation by the Swiss mathematician Ludwig Schläfli just over a century ago, led to a renewed interest in polyhedra, and since that time hundreds of papers on this subject have been published. However, there are still many unsolved problems, and the purpose of this article is to state a few of these. These problems have widespread appeal because they can be readily understood with very little previous mathematical knowledge, and although they may be quite difficult, there is always the possibility that someone with a very minimum of mathematical background will devise a method of approach that leads to a solution.

The mathematical theory of rocket motion has been given by Rosser, Newton and Gross (1947) for the case of unrotated, and slowly rotated, rockets. A rigorous treatment of the subject has been given also by Rankin (1949) in a recent paper which is applicable to both unrotated and rotated rockets.One of the main objects of such work is the derivation of formulae which may be used to predict the behaviour of the rocket under the action of various disturbing influences. Thus, an angular deviation of the rocket from its normal trajectory may be caused by the thrust not passing through the centre of gravity, by the action of a crosswind (usually assumed constant in the theory), or by the rocket being launched with an initial yaw or initial angular velocity about an axis at right angles to the rocket axis. Malaligned fins may also cause such a deviation.

From one end of Britain to the other Megalithic Man has left us examples of geometric constructions of various designs. Originally set out on the ground by means of stakes, rope and measuring rods, the designs were rendered permanent by large stones placed along the outlines. The erection of the larger of these stones was an enormous task. It is certain that the engineers controlling the operations were working to pre-conceived plans which were set out with care and accuracy. Further, it appears that the unit of length used was the same from Land’s End to John o’ Groats. That this unit was 5·44 ft. I have shown in Ref. 1, and many more sites surveyed since writing that paper have amply confirmed statistically this value. Half of this, 2·72 ft., might be called the Megalithic Yard, but whether this or the Megalithic Fathom (5·44 ft.) was the length of the standard rod carried about the country it is not possible to say The important thing is that in a statistically significant number of cases in any group of circles the major diameter is approximately an integral number of fathoms. The group can be formed according to locality, type of circle, or size; the result is the same. We also find this dimension occurring in the distances of outliers from the construction centre and the distances between circles where there are two close together. The method of analysis used was that given by Broadbent in Ref. 5.

In this paper, we consider a sequence lr(x) (r = 1, 2, ..) of rational functions of a positive rational variable x, which approach log x as r→∞. Some properties of these functions are analogous to properties of the natural logarithm, and go over into the latter, when the limit r→∞ is taken. Except for pointing out such limiting results, we shall avoid transcendental methods.