Geometry and Experience
address by Albert Einstein (1921), expanded
(Prussian Academy of Sciences, Jan 27th)
projection of a finite spherical space onto an infinite linear space
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 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
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
After having gained a vivid mental image of the geometrical behaviour
of our 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.
In this way, by using as stepping-stones the practice in thinking and
visualisation which Euclidean geometry gives us, we have acquired a
mental picture of spherical geometry. We may without difficulty impart
more depth and vigour to these ideas by carrying out special imaginary
constructions. Nor would it be difficult to represent the case of what
is called elliptical geometry in an analogous manner.
My only aim today has been to show that the human faculty of
visualisation is by no means bound to capitulate to non-Euclidean
* This is intelligible without calculation – but only for the
two-dimensional case – if we revert once more to the case of the disc
on the surface of the sphere.