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138 the complete group belonging to the Newtonian Mechanics. In this state of affairs, and since $$G_c$$ is mathematically more understandable than $$G_{\infty}$$, a mathematician might very well, in the freedom of his imagination, come to the thought that really the phenomena of Nature possess an invariance not with the group $$G_{\infty}$$ with the group $$G_c$$, where $$c$$ is a deﬁnite ﬁnite number, which is only extremely large in the ordinary system of units. Such a presentiment would have been an extraordinary triumph of pure Mathematics. Now, if Mathematics shows here only an unnecessary witticism, still she has the satisfaction that, thanks to her favourable antecedents, she has the power, with her senses sharpened in the exercise of free foresight, to comprehend the deep-lying consequences of such a refashioning of our conception of Nature.

I will note immediately what value of $$c$$ is in question: in the place of $$c$$, the velocity of the propagation of light in empty space must make its appearance. In order not to speak of space or of emptiness, we may distinguish $$c$$ as the ratio of the electrostatic and the electromagnetic units of electricity.

The subsistence of the invariance of the laws of Nature for the group $$G_c$$ would be now expressed as follows :—

From the totality of natural phenomena, one may derive, by successive approximations, more and more exactly, a system of reference, $$x, y, z$$ and $$t$$ (i.e., space and time) by means of which these phenomena can be described according to deﬁnite laws. However, this system of reference is by no means uniquely determined by the phenomena. Corresponding to the transformations of the group $$G_c$$, we may arbitrarily vary the system of reference, without the expression of the laws of Nature being changed thereby.

For example, in Fig. 1 we may call $$t'$$ time. But then we must necessarily deﬁne space by the totality of the three parameters $$x', y, z$$; and thus the physical laws would be as exactly expressed by means of $$x', y, z, t'$$ as by means of $$x, y, z, t$$. After this, we would have in the world no longer the space but an inﬁnite number of spaces; just as in the three-dimensional space, there are an inﬁnite number of planes. The three-dimensional Geometry becomes a chapter of the four-dimensional Physics. You now understand why I said at the outset that space and time shall sink in the background and only constitute a world with their union.

(I I)

Now arise the questions : What circumstances compel us to adopt the altered conception of space and time? Does this conception never really disagree with phenomena? Finally, does it offer advantages for the description of phenomena?