**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

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 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

geometry.

footnote:

* 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.