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II.

Now the question may be asked, — what circumstances forces upon us these changed views about time and space, are they actually never in contradiction with observed phenomena, do they finally guarantee us advantages for the description of natural phenomena?

Before we enter into the discussion, a very important point must be noticed. Suppose we have individualized time and space in any manner; then a substantial point as a world-line corresponds to a line parallel to the t-axis; a uniformly moving substantial point corresponds to a world-line inclined to the t-axis; and non-uniformly moving substantial point will correspond to a somehow curved world-curve. Let us consider the world-line passing through any world point x, y, z, t; now if we find the world-line parallel to any radius vector $OA'$ of the hyperboloidal sheet mentioned before, then we can introduce $OA'$ as a new time-axis, and then according to the new conceptions of time and space the substance in the corresponding world point will appear to be at rest. We shall now introduce this fundamental axiom:

The substance existing at any world point can always be conceived to be at rest, if time and space are interpreted suitably.

The axiom means, that in a world-point the expression

$c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2},\,$

shall always be positive or what is equivalent to the same thing, every velocity v should always be smaller than c. c shall therefore be the upper limit for all substantial velocities and exactly herein lies a deep significance for the quantity c. At the first impression, the axiom in this different form seems to be rather unsatisfactory. However, it is to be remembered that a modified mechanics will hold now, in which the square root of this differential combination takes is included, so that cases in which the velocity is greater than c will only play a role in a similar way as figures with imaginary coordinates in geometry.

The impulse and real cause for accepting the group $G_{c}$, came from the fact that the differential equation for the propagation of light in vacant space possesses that group $G_{c}$.^[1] On the other hand, the idea of rigid bodies has any sense only in a system mechanics with the group $G_{\infty }$. Now if we have an optics with $G_{c}$, and on the other hand if there
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1. An essential application of this fact can already be found in W. Voigt. Göttinger Nachr. 1887, p.41