All these spaces are equally justified. No one of them stands out more prominently than any other on the ground of symmetry with respect to the four-dimensional world. Only when we specify the frame of the observer will one particular space and one particular duration be allotted. Hence we may say that practical congruence exists for space and for time, as in classical science, provided the Galilean observer effects his measurements with rods and periodic mechanisms, such as clocks, which do not move about in his Galilean frame while performing measurements. There is thus a perfectly definite physical meaning in stating that the distances between two point-pairs at rest in the frame are congruent or unequal; but we must specify that this distance is measured according to the standards of the frame; and the same applies to time-stretches. Likewise, as in classical science, the Galilean observer will discover upon measurement with his congruent rods that his space is Euclidean. It is only when the rods and clocks are in relative motion or when, while at rest in our frame, the frame happens to be accelerated or submitted to gravitational action, that they cease to measure congruent stretches. It is only, therefore, when we reason in an impersonal way, without specifying any particular frame, only when we reason from the standpoint of all possible observers, whatever be their motion, that space and time fade away into shadows; and it is only then that we are compelled, whether we like it or not, to reason in terms of the common objective world of relativity, that is, in terms of four-dimensional space-time. [Pg 193] CHAPTER XVII THE MATHEMATICAL EXPRESSION OF EINSTEIN’S FUNDAMENTAL PREMISES CONSIDER a Galilean system, and two points and in this system. If a ray of light is propagated from to , the postulate of the invariant velocity of light demands that the distance measured by us in the frame, divided by the duration required (according to the time of our frame) for a light wave to cover this distance, be always equal to , the velocity of light. If, now, we had viewed the same phenomenon from some other Galilean system, the points and of the first system at the instants when the ray of light passed them would have corresponded to some other points and of our second system. But, just as in the first case, we should have been able to measure the velocity of light between the two points and of our second system. The principle of the invariant velocity of light states that in whatever Galilean system we might have operated, the measured velocity of light in vacuo would always be the same. Of course this velocity may be positive or negative, according to whether the light ray is directed to the right or to the left; but we can obviate this ambiguity of sign by considering squared values.