by different routes. But these spatial routes are of equal length; hence we must assume that the speeds of the bodies along their respective paths have been the same. In all rigour, we may only claim that the mean speed has been the same, since one body may have slowed down, then spurted on again, making up for lost time. But if we repeat the experiment with circles of different sizes, and if in every case the bodies meet one another at the opposite point, we are justified in asserting that not only the mean speed, but also the instantaneous speed at every instant, is the same for either body. In short, thanks to spatial measurements coupled with the observation of coincidences, we have been able to establish the equality of two velocities, even though we knew nothing of time-measurement.
A more elaborate presentation of the same problem would be given as follows: Waves of light leaving the centre of a sphere simultaneously are found to return to the centre also in coincidence, after having suffered a reflection against the highly polished inner surface of the sphere. As in the previous example, the light waves have thus covered equal distances in the same time; whence we conclude that their speed is the same in all directions.
Inasmuch as this experiment has been performed, yielding the results we have just described, even though the ether drift caused by the earth’s motion should have varied in direction and intensity, the isotropy of space to luminous propagations was thus established. (The experiment described constitutes but a schematic form of Michelson’s.) It is to be noted that in this experiment the observation of coincidences is alone appealed to (even spatial measurements can be eliminated).[25] When it is realised that coincidences constitute the most exact form of observation, we understand why it is that Einstein’s definition is justified.
The optical definition presents a marked superiority over those of classical science. Whereas, with the rotation of the earth, we had to assume the correctness of Newton’s law and of those of mechanics in order to determine the compensations necessitated by the earth’s breathing or by the friction of the tides, in the present case we assume practically nothing, and what little we do assume issues from the most highly refined experiments known to science.
Now, the importance of Einstein’s definition lies not so much in its allowing us to obtain an accurate definition of time in our Galilean frame as in its enabling us to co-ordinate time reckonings in various Galilean frames in relative motion. So long as we restrict our attention to space and time computations in our frame, we may, as before, appeal to vibrating atoms for the measurement of congruent time-intervals and to rigid rods for the purpose of measuring space. It is when we seek to correlate space and time measurements as between various Galilean frames in relative motion that astonishing consequences follow. We discover that the concepts of spatial and temporal congruence of classical science must be modified to a very marked degree. They lose those attributes of universality with which we were wont to credit them. It is then found that congruence can only be defined in a universal way when we consider the extension of four-dimensional space-time.
As we have mentioned, any change in our concepts of space and time congruence is not merely a local affair. Owing to the unity of nature, we shall be led to modify our entire understanding of the relatedness of phenomena, and must not be surprised to learn that a totally new science is the necessary outcome of Einstein’s new definitions. Old laws are cast aside, new ones take their place; and Einstein, by following his mathematical deductions with rigorous logic, without introducing any hypotheses ad hoc, proved that if his new definition of space-time congruence was physically correct, a whole series of hitherto unsuspected natural phenomena should ensue and should be revealed by precise experiment. Wonderful as it may appear, Einstein’s previsions have thus far been verified in every minute detail.
CHAPTER VII
SYSTEMS OF CO-ORDINATES AND DISTANCE
WE HAVE mentioned in a general way the significance of congruence and of spatial distance. It now remains for us to find a means of defining these concepts in a rigorous mathematical form. We remember that the equality or congruence of two spatial distances between two point pairs
and