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[21] Cf. Appendixes I and II. The relations which are derived

there for the co-ordinates themselves are valid also for

co-ordinate differences, and thus also for co-ordinate

differentials (indefinitely small differences).

dx2 + dy2 + dz2 - c2dt2 = dx′2 + dy′2 + dz′2 - c2dt′2.

The validity of the Lorentz transformation follows from this

condition. We can express this as follows: The magnitude

ds2 = dx2 + dy2 + dz2 - c2 dt2,

which belongs to two adjacent points of the four-dimensional

space-time continuum, has the same value for all selected

(Galileian) reference-bodies. If we replace x, y, z,

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by x1, x2, x3, x4, we also obtain the result that

ds2 = dx12 + dx22 + dx32 + dx42.

is independent of the choice of the body of reference. We call

the magnitude ds the "distance" apart of the two events or

four-dimensional points.

Thus, if we choose as time-variable the imaginary variable

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