not given by Euclidean geometry, but approximately by spherical
geometry, if only we consider parts of space which are sufficiently
great. Now this is the place where the reader's imagination boggles.
"Nobody can imagine this thing," he cries indignantly. "It can be
said, but cannot be thought. I can represent to myself a spherical
surface well enough, but nothing analogous to it in three dimensions."

[Figure 2: A circle projected from a sphere onto a plane]

We must try to surmount this barrier in the mind, and the patient
reader will see that it is by no means a particularly difficult
task. For this purpose we will first give our attention once more to
the geometry of two-dimensional spherical surfaces. In the adjoining
figure let _K_ be the spherical surface, touched at _S_ by a plane,
_E_, which, for facility of presentation, is shown in the drawing as
a bounded surface. Let _L_ be a disc on the spherical surface. Now
let us imagine that at the point _N_ of the spherical surface,
diametrically opposite to _S_, there is a luminous point, throwing a
shadow _L'_ of the disc _L_ upon the plane _E_. Every point on the
sphere has its shadow on the plane. If the disc on the sphere _K_ is
moved, its shadow _L'_ on the plane _E_ also moves. When the disc
_L_ is at _S_, it almost exactly coincides with its shadow. If it
moves on the spherical surface away from _S_ upwards, the disc
shadow _L'_ on the plane also moves away from _S_ on the plane
outwards, growing bigger and bigger. As the disc _L_ approaches the
luminous point _N_, the shadow moves off to infinity, and becomes
infinitely great.

Now we put the question, What are the laws of disposition of the
disc-shadows _L'_ on the plane _E_? Evidently they are exactly the