# Page:SahaSpaceTime.djvu/4

transformation in which (y, z) remain unaltered. Let us draw the cross section of the upper sheets with the plane of the x-and t-axes, i.e., the upper half of the hyperbola ${\displaystyle c^{2}t^{2}-x^{2}=1}$, with its asymptotes (vide fig. 1).
Then let us draw the radius rector OA', the tangent A' B' at A', and let us complete the parallelogram OA' B' C'; also produce B' C' to meet the x-axis at D'. Let us now take Ox', OA' as new axes with the unit measuring rods OC' = 1, OA'= ${\displaystyle {\tfrac {1}{c}}}$; then the hyperbola is again expressed in the form ${\displaystyle c^{2}t'^{2}-x'^{2}=1,t'>0}$ and the transition from (x, y, z, t) to (x' y' z' t) is one of the transitions in question. Let us add to this characteristic transformation any possible displacement of the space and time null-points; then we get a group of transformation depending only on c, which we may denote by Gc.
Now let us increase c to infinity. Thus ${\displaystyle {\frac {1}{c}}}$ becomes zero and it appears from the figure that the hyperbola is gradually shrunk into the x-axis, the asymptotic angle becomes a straight one, and every special transformation in the limit changes in such a manner that the t-axis can have any possible direction upwards, and x' more and more approximates to x. Remembering this point it is clear that the full group belonging to Newtonian Mechanics is simply the group Gc, with the value of c = ∞. In this state of affairs, and since Gc is mathematically more intelligible than G, a mathematician may, by a free play of imagination, hit upon the thought that natural phenomena possess an invariance not only for the group G, but in fact also for a group Gc, where c is finite, but yet