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Page 10
From the latest results of the theory of relativity it is probable
that our three-dimensional space is also approximately spherical,
that is, that the laws of disposition of rigid bodies in it are
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
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
spherical surface.
The representation given above of spherical geometry on the
plane is important for us, because it readily allows itself to be
transferred to the three-dimensional case.
Let us imagine a point _S_ of our space, and a great number
of small spheres, _L'_, which can all be brought to coincide with
one another. But these spheres are not to be rigid in the sense
of Euclidean geometry; their radius is to increase (in the sense
of Euclidean geometry) when they are moved away from _S_ towards
infinity, and this increase is to take place in exact accordance
with the same law as applies to the increase of the radii of the
disc-shadows _L'_ on the plane.
After having gained a vivid mental image of the geometrical
behaviour of our _L'_ spheres, let us assume that in our space there
are no rigid bodies at all in the sense of Euclidean geometry, but
only bodies having the behaviour of our _L'_ spheres. Then we shall
have a vivid representation of three-dimensional spherical space,
or, rather of three-dimensional spherical geometry. Here our spheres
must be called "rigid" spheres. Their increase in size as they
depart from _S_ is not to be detected by measuring with
measuring-rods, any more than in the case of the disc-shadows on
_E_, because the standards of measurement behave in the same way as
the spheres. Space is homogeneous, that is to say, the same
spherical configurations are possible in the environment of all
points.* Our space is finite, because, in consequence of the
"growth" of the spheres, only a finite number of them can find room
in space.
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