Pagina 169 de 212

In this connection the individual rods always represent the same

distance, independently of their position.

Let us consider now a second two-dimensional existence, but this

time on a spherical surface instead of on a plane. The flat

beings with their measuring-rods and other objects fit exactly on

this surface and they are unable to leave it. Their whole

universe of observation extends exclusively over the surface of

the sphere. Are these beings able to regard the geometry of their

universe as being plane geometry and their rods withal as the

realisation of "distance"? They cannot do this. For if they

attempt to realise a straight line, they will obtain a curve,

which we "three-dimensional beings" designate as a great circle,

i.e. a self-contained line of definite finite length, which can

be measured up by means of a measuring-rod. Similarly, this

universe has a finite area that can be compared with the area, of

a square constructed with rods.