Let us consider two particular points on the train¹ travelling along the embankment with the velocity $v$, and inquire as to their distance apart. We already know that it is necessary to have a body of reference for the measurement of a distance, with respect to which body the distance can be measured up. It is the simplest plan to use the train itself as reference-body (co-ordinate system). An observer in the train measures the interval by marking off his measuring-rod in a straight line (e.g. along the floor of the carriage) as many times as is necessary to take him from the one marked point to the other. Then the number which tells us how often the rod has to be laid down is the required distance.

It is a different matter when the distance has to be judged from the railway line. Here the following method suggests itself. If we call A$'$ and B$'$ the two points on the train whose distance apart is required, then both of these points are moving with the velocity $v$ along the embankment. In the first place we require to determine the points A and B of the embankment which are just being passed by the two points A$'$ and B$'$ at a particular time $t$—judged from the embankment. These points A and B of the embankment can be determined by applying the definition of time given in Section VIII. The distance between these points A and B is then measured by repeated application of thee measuring-rod along the embankment.

A priori it is by no means certain that this last measurement will supply us with the same result as the first. Thus the length of the train as measured from the embankment may be different from that obtained by measuring in the train itself. This circumstance leads us to a second objection which must be raised against the apparently obvious consideration of Section VI. Namely, if the man in the carriage covers the distance $w$ in a unit of time—measured from the train,—then this distance—as measured from the embankment—is not necessarily also equal to $w$.

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XI. The Lorentz Transformation