than two right angles in that of Lobachevski;
Greater than two right angles in that of Riemann.
The number of straights through a given point that can be drawn coplanar to a given straight, but nowhere meeting it, is equal:
To one in Euclid's geometry;
To zero in that of Riemann;
To infinity in that of Lobachevski.
Add that Riemann's space is finite, although unbounded, in the sense given above to these two words.
The Surfaces of Constant Curvature.—One objection still remained possible. The theorems of Lobachevski and of Riemann present no contradiction; but however numerous the consequences these two geometers have drawn from their hypotheses, they must have stopped before exhausting them, since their number would be infinite; who can say then that if they had pushed their deductions farther they would not have eventually reached some contradiction?
This difficulty does not exist for Riemann's geometry, provided it is limited to two dimensions; in fact, as we have seen, two-dimensional Riemannian geometry does not differ from spherical geometry, which is only a branch of ordinary geometry, and consequently is beyond all discussion.
Beltrami, in correlating likewise Lobachevski's two-dimensional geometry with a branch of ordinary geometry, has equally refuted the objection so far as it is concerned.
Here is how he accomplished it. Consider any figure on a surface. Imagine this figure traced on a flexible and inextensible canvas applied over this surface in such a way that when the canvas is displaced and deformed, the various lines of this figure can change their form without changing their length.