If space has constant positive curvature, all geodesics starting from a given surface-element produce a closed, finite surface analogous to a sphere in three dimensions.

Archytas’s argument (in section 1 above), which conflated unboundedness with infinitude, could finally be put to rest. Riemann’s model allows for the universe to be finite and unbounded at the same time. In 1854 he wrote: “The unboundedness of space possesses in this way a greater empirical certainty than any external experience. But its infinite extent by no means follows from this; on the other hand if we assume independence of bodies from position, and therefore ascribe to space constant c