same as the laws of disposition of the discs _L_ on the spherical
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
any meaning whatever. In fact the only objective assertion that can
be made about the disc-shadows is just this, that they are related
in exactly the same way as are the rigid discs on the spherical
surface in the sense of Euclidean geometry.

We must carefully bear in mind that our statement as to the growth
of the disc-shadows, as they move away from _S_ towards infinity,
has in itself no objective meaning, as long as we are unable to
employ Euclidean rigid bodies which can be moved about on the plane
_E_ for the purpose of comparing the size of the disc-shadows. In
respect of the laws of disposition of the shadows _L'_, the point
_S_ has no special privileges on the plane any more than on the