But suppose now that these imaginary animals, while remaining without thickness, have the form of a spherical, and not of a plane, figure, and are all on the same sphere without power to get off. What geometry will they construct? First it is clear they will attribute to space only two dimensions; what will play for them the rôle of the straight line will be the shortest path from one point to another on the sphere, that is to say, an arc of a great circle; in a word, their geometry will be the spherical geometry. What they will call space will be this sphere on which they must stay, and on which happen all the phenomena they can know. Their space will therefore be _unbounded_ since on a sphere one can always go forward without ever being stopped, and yet it will be _finite_; one can never find the end of it, but one can make a tour of it. Well, Riemann's geometry is spherical geometry extended to three dimensions. To construct it, the German mathematician had to throw overboard, not only Euclid's postulate, but also the first axiom: _Only one straight can pass through two points_. On a sphere, through two given points we can draw _in general_ only one great circle (which, as we have just seen, would play the rôle of the straight for our imaginary beings); but there is an exception: if the two given points are diametrically opposite, an infinity of great circles can be drawn through them. In the same way, in Riemann's geometry (at least in one of its forms), through two points will pass in general only a single straight; but there are exceptional cases where through two points an infinity of straights can pass. There is a sort of opposition between Riemann's geometry and that of Lobachevski. Thus the sum of the angles of a triangle is: Equal to two right angles in Euclid's geometry; Less 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.