SIDELIGHTS ON RELATIVITY

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surface. For to each original figure on _K_ there is a corresponding
shadow figure on _E_. If two discs on _K_ are touching, their
shadows on _E_ also touch. The shadow-geometry on the plane agrees
with the the disc-geometry on the sphere. If we call the disc-shadows
rigid figures, then spherical geometry holds good on the plane _E_
with respect to these rigid figures. Moreover, the plane is finite
with respect to the disc-shadows, since only a finite number of
the shadows can find room on the plane.

At this point somebody will say, "That is nonsense. The disc-shadows
are _not_ rigid figures. We have only to move a two-foot rule about
on the plane _E_ to convince ourselves that the shadows constantly
increase in size as they move away from _S_ on the plane towards
infinity." But what if the two-foot rule were to behave on the
plane _E_ in the same way as the disc-shadows _L'_? It would then
be impossible to show that the shadows increase in size as they
move away from _S_; such an assertion would then no longer have