From a mathematical point of view, a physical phenomenon reduces to the systematic variation in intensity and in orientation from place to place and from time to time of a number of magnitudes. A physical phenomenon as referred to some arbitrary frame of reference can therefore be expressed by mathematical equations in which the space and time co-ordinates with respect to the frame figure as variables. Now a fundamental problem of theoretical physics is to determine how the equation of a phenomenon referred to a frame , which is in motion with respect to a frame , can be obtained from its equation referred to the frame , by purely mathematical means without any further appeal to experiment. The solution of this problem would enable us to anticipate the change in appearance of a phenomenon when observed, first from the frame , then from the frame . Obviously some kind of mathematical operation would have to be performed on the equation as referred to . This operation is called a mathematical transformation; and the equation operated upon is said to have been transformed. In the type of problem we are considering, where one frame of reference is followed by another, the transformation will bear on the space and time variables or co-ordinates present in the equation; and transformations of this character are termed space and time transformations. The entire problem reduces, therefore, to the discovery of those space and time transformations which hold in this world of ours. These transformations cannot be guessed at a priori; and a study of the behaviour of physical phenomena is a prerequisite condition for their determination. Inasmuch as all the experiments known to classical science appeared to corroborate the impression that a distance in space and a duration in time were absolutes, remaining unaltered when we changed our frame of reference, a certain definite species of space and time transformations followed as a matter of course. Consider, for example, a train moving with constant speed along an embankment. If an event occurs in the moving train at any instant of time, say at two o’clock, this event will obviously have occurred at precisely the same instant (two o’clock) when referred to the embankment, since time is assumed to be absolute, the same for all. In other words, time and duration for the observer in the moving train are identical in all respects with time and duration for the observer on the embankment. This fact is expressed by , where is the time for the train, and the time for the embankment.