Consider now all the events which are at an interval of one unit from , according to the definition of the interval . We have changed the sign of , because usually (though not always) the original would have come out negative. In Euclidean space points distant a unit interval lie on a circle; but, owing to the change in geometry due to the altered sign of , they now lie on a rectangular hyperbola with two branches , . Since the interval is an absolute quantity, all observers will agree that these points are at unit interval from .
Now make the following construction:—draw a straight line , to meet the hyperbola in ; draw the tangent at , meeting the light-line in ; complete the parallelogram ; produce to . We now assert that an observer who chooses for his time-direction will regard as his space direction and will consider and to be the units of time and space.
The two observers make their partitions of space and time in different ways, as illustrated in Figs. 5 and 6, where in each case the partitions are at unit distance (in space and time) according to the observers' own reckoning. The same diagram of events in the world will serve for both observers; merely removes 's partitions and overlays his own, locating the events in his space and time accordingly. It will be seen at once that the lines of unit velocity—progress of one unit of space for one unit of time—agree, so that the velocity of a pulse of light is unity for both observers. It can be shown from the properties of the hyperbola that the locus of points at any interval from , given by equation (1), viz. , is the same locus (a hyperbola) for both systems of reckoning and . The two observers will always agree on the measures of intervals, though they will disagree about lengths, durations, and the velocities of everything except light. This rather complex transformation is mathematically equivalent to the simple rotation of the axes required when imaginary time is used.
It must not be supposed that there is any natural distinction
