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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 $\pi$, which is independent of the diameter of the circle. On their spherical surface our flat beings would find for this ratio the value

$\pi {\frac {\sin \left({\frac {r}{R}}\right)}{\left({\frac {r}{R}}\right)}}$,

i.e. a smaller value than $\pi$, the difference being the more considerable, the greater is the radius of the circle in comparison with the radius $R$ of the “world-sphere.” By means of this relation