The course of study pursued by very many boys in secondary schools who take an advanced course after matriculation is governed by the syllabus of the intermediate examination of a degree in a university other than those of Oxford and Cambridge. In London those boys who intend to specialise in mathematics generally take physics and chemistry as well. It is clear that physics is a valuable complementary study for the mathematical specialist, but it is not so clear that chemistry is. It is a pity that the intermediate qualification cannot be obtained on a syllabus in mathematics a good deal more advanced than the present one, coupled with the present syllabus in physics.

I understand that this problem appeared many years ago in a mock examination paper in the Granta. I heard of it from Dr. A. C. D. Crommelin in 1919, when A, B, G and D (or more strictly C, C’, D and E) were about to start on the eclipse expeditions which verified Einstein’s prediction of the deflection of light. Crommelin in an after-dinner speech hinted that this problem might arise. I made use of it in my book New Pathways in Science, and rashly gave my solution-—the result being a considerable increase in my correspondence. I hope the following explanation will be sufficiently clear to avert a similar sequel to its appearance in the Mathematical Gazette.

The following approach to the study of geodesic ellipses and hyperbolas is instructive, and leads to a generalisation in Riemannian geometry.

Though vector analysis is most successful in making the elements of a subject attractive, many problems are solved either cumbrously or by a more or less disguised scalar analysis. Rectangular unit vectors i, j, k are often, and very properly, introduced at an early stage, so that even elementary work may look like Cartesian analysis translated into vector notation. Mathematicians are apt to look askance at oblique axes, since they fall between two stools, having neither the simplicity of the rectangular system nor the adaptability and wide projective outlook of homogeneous coordinates.

If a number N (other than an odd multiple of 2) is the product of two factors, then N = x2 – y2, where x and y are half the sum and half the difference respectively of the factors, and are integers. To determine x and y we may use the quadratic residues of small primes as follows.

Let N = a2 + b where a2 is the perfect square nearest to N.

The study of line geometry by analytical methods involving the use of tangential coordinates and tangential equations is often found difficult by the junior Honours student. In the writer’s view the reason for this is to be found in the fact that, according to the course followed by a good many textbooks, the student finds himself, so to speak, pitchforked into the middle of the subject without any just appreciation of its bearing, or indeed of the need for such a study at all. Some modern writers on analytical geometry have attempted to remedy this by adopting the view that line geometry should be studied from the outset pari passu with point geometry. Unfortunately, however, experience tends to show that this is a mistake, inasmuch as it creates confusion in minds which have had insufficient practice in, and consequently have obtained an inadequate grasp of, analytical methods, which are never at any time found easy of assimilation by the student of elementary mathematics.

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.