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A and B placed on platforms moving with respect to each other. Let us suppose that A is on a platform denoted by $S_{1}$ and that B is on a platform denoted by $S_{2}$. Suppose that to A the platform $S_{2}$ appears to move with the velocity v in the direction indicated by the arrow at $S_{2}$; then to B the platform $S_{1}$ will appear to move with velocity v in the direction indicated by the arrow at $S_{2}$. Suppose further that the units of length employed by A and B are such that they arrive at the same numerical results in measuring the length MN, MN being perpendicular to the line of relative motion of $S_{1}$ and $S_{2}$. The platform $S_{1}$ and the instruments employed by A for measuring time and length will be spoken of as the system of reference $S_{1}$. Similarly, we shall speak of the system of reference $S_{1}$. For convenience we shall use β to denote the ratio v/c, where c is the velocity of light.

The two systems of reference $S_{1}$ and $S_{1}$ being thus defined, the question arises as to how the units of length and of time of $S_{1}$ are related to the corresponding units of $S_{2}$. If we accept as true the laws stated in A and E of the preceding section — as, in fact, is done in the theory of relativity — we are led to some very remarkable conclusions. We state first the relation of the time units:

To an observer A on $S_{1}$i the time unit of $S_{1}$ appears to be in the ratio ${\sqrt {1-\beta ^{2}}}:1$ to that of $S_{2}$, while to an observer B on $S_{2}$ the time unit of $S_{2}$ appears to be in the ratio ${\sqrt {1-\beta ^{2}}}:1$ to that of $S_{1}$.

Thus if A compares his clocks with those of B it will appear to A that $A$'s clocks are running faster than $B$'s clocks; on the other hand, if B compares his clocks with those of A it will appear to B that $B$'s clocks are running faster than $A$'s clocks. Thus the two observers are in hopeless disagreement as to the measurement of time. An analysis of this divergence is made below in section V.

Another matter of fundamental importance in the measurement of time will be brought to attention by the question: When are two events which happen at different places to be considered simultaneous? The nature of the difficulty can be seen from the following result in the theory of relativity, an analysis of which is to be found below in section VI.:

If an observer A on $S_{1}$ places two clocks at a distance d apart in a line parallel to the line of relative motion of $S_{1}$ and $S_{2}$ (say at P and Q respectively) and adjusts them so that they appear to him to mark the same time: then to an observer B on $S_{2}$ the clock on $S_{1}$ which is forward in point of motion appears to be behind in point of time by the amount

${\frac {v}{c^{2}}}{\frac {d}{\sqrt {1-\beta ^{2}}}}$.