exceedingly large compared to the usual measuring units. Such a preconception would be an extraordinary triumph for pure mathematics.
At the same time I shall remark for which value of c, this invariance can be conclusively held to be true. For c, we shall substitute the velocity of light c in free space. In order to avoid speaking either of space or of vacuum, we may take this quantity as the ratio between the electrostatic and electro-magnetic units of electricity.
We can form an idea of the invariant character of the expression for natural laws for the group-transformation Gc in the following manner.
Out of the totality of natural phenomena, we can, by successive higher approximations, deduce a coordinate system (x, y, z, t); by means of this coordinate system, we can represent the phenomena according to definite laws. This system of reference is by no means uniquely determined by the phenomena. We can change the system of reference in any possible manner corresponding to the abovementioned group transformation Gc, but the expressions for natural laws will not be changed thereby.
For example, corresponding to the above described figure, we can call t' the time, but then necessarily the space connected with it must be expressed by the manifoldness (x' y z). The physical laws are now expressed by means of x', y, z, t', — and the expressions are just the same as in the case of x, y, z, t. According to this, we shall have in the world, not one space, but many spaces, — quite analogous to the case that the three-dimensional space consists of an infinite number of planes. The three-dimensional geometry will be a chapter of four-dimensional physics. Now you perceive, why I said in the beginning
