Yet no one conceived the idea that in fixing a unit of time on the basis of the law of inertia, that is, by measuring a length (the distance traversed), the state of motion of the observer might exert an influence. It was assumed that the data obtained from the necessary observations had an absolute meaning quite independent of the conditions of observation when simultaneous moments were chosen and a length was evaluated. As Einstein has shown, however, this is not the case. Rather, this recognition of the relativity of space- and time-measurements formed the starting point of his principle of relativity ([Note 13]). It is a necessary consequence of the universal significance of the velocity of light, of which we spoke in the [first section]. Its recognition furnished us at once with the correct formulae of transformation, allowing us to relate the space-time measurements of systems moving uniformly and rectilinearly with respect to each other, and this is what we are concerned with in Neumann's suggestion of fixing a measure of time with the aid of the law of inertia. In the new equations of transformation,
' is not identically equal to
, but rather
The time-measurements in the second system which is moving relatively to the first are thus essentially conditioned by the velocity
of each relative to the other. Consequently, the fixing of a measure of time on the basis of the law of inertia, as proposed by Neumann, does not at all lead to the result that the time-measurements are entirely independent of the state of motion of the systems with respect to each other, as assumed in classical mechanics. Only when the researches of Einstein concerning the special theory of relativity had been carried out, did the fundamental assumptions of our time-measurements become fully cleared up, and thus a serious shortcoming in classical mechanics was made good.
That such a fundamental revision of the assumptions made regarding time-measurements became necessary only after so great a lapse of time, is to be explained by the fact that even the velocities which occur in astronomy are so small, in comparison with the velocity of light, that no serious discrepancies could arise between theory and observation. So it occurred that the weaknesses of the theory—in particular, those due to the motional relations of various systems to one another—did not come to light until the study of electronic motions, in which velocities of the order of that of light occur, proved the insufficiency of the existing theory.