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.

The idea of attaching signs to certain geometrical magnitudes such as lengths, angles and areas is fundamental in analytical geometry. It has its uses, too, in pure geometry in simplifying certain theorems and constructions. Many theorems, for instance, such as “ the area of a triangle is half that of a rectangle with the same base and height ”, or “ the angle subtended at the centre of a circle is double that at the circumference”, have to be proved differently for two or more figures, the word “ add ” being in some cases changed to “ subtract ”. In such theorems the one proof will suffice for any figure provided the proper interpretation is made of the signs of the magnitudes.