.
One might equally well assume, as Lobatschewsky did, that
and
form an angle which differs ever so slightly from two right angles, and that there are an infinite number of other straight lines included between these two positions (as indicated by the dotted lines in the figure), which do not cut
at all, Lobatschewsky (and also Bolyai) built up an entirely consistent geometry on this latter assumption.
Riemann later abolished the assumption of infinite length of a straight line, and assumed that in travelling along a straight line sufficiently far one finally arrives at the starting-point again without having encountered any limit or barrier. This means that our space is regarded as being finite but unbounded.[23]
[23]E.g. the surface of a sphere cuts a finite volume out of space, but particles sliding on the surface nowhere encounter boundaries or barriers. This is a three-dimensional analogon to the four-dimensional space-time manifold of Minkowski. It does not mean that the universe is enclosed by a spherical shell, as was supposed by the ancients. We cannot form a picture of the corresponding result in the four-dimensional continuum in which, according to the general theory of relativity, we live.