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.

Many people have had the experience of borrowing a sum of money which is to be repaid with interest by making equal monthly payments over a year or a number of years. Two problems suggest themselves:

1. If the annual rate of interest is known what is the value of each monthly payment?

2. If the monthly payment is known, what is the annual rate of interest?