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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 π, which is independent

of the diameter of the circle. On their spherical surface our

flat beings would find for this ratio the value

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