outside of this plane: that which they observe to happen to themselves
and to their flat things is the all-inclusive reality of their plane.
In particular, the constructions of plane Euclidean geometry can be
carried out by means of the rods _e.g._ the lattice construction,
considered in Section XXIV. In contrast to ours, the universe of these
beings is two-dimensional; but, like ours, it extends to infinity. In
their universe there is room for an infinite number of identical
squares made up of rods, _i.e._ its volume (surface) is infinite. If
these beings say their universe is plane, there is sense in the
statement, because they mean that they can perform the constructions of
plane Euclidean geometry with their rods. 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. The great charm resulting from this
consideration lies in the recognition of the fact that _the universe of
these beings is finite and yet has no limits._
But the spherical-surface beings do not need to go on a world-tour in
order to perceive that they are not living in a Euclidean universe.
They can convince themselves of this on every part of their world,
provided they do not use too small a piece of it. Starting from a
point, they draw straight lines (arcs of circles as judged in three
dimensional space) of equal length in all directions. They will call
the line joining the free ends of these lines a circle. For a plane
surface, the ratio of the circumference of a circle to its diameter,
both lengths being measured with the same rod, is, according to
Euclidean geometry of the plane, equal to a constant value , which is
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