648 SCIENCE AND HYPOTHESIS 



construct it, the German mathematician had first of all to throw over- 

 board, not only Euclid's postulate, but also the first axiom that only 

 one line can pass through two points. On a sphere, through two given 

 points, we can in general draw only one great circle which, as we have 

 just seen, would be to our imaginary beings a straight line. But 

 there was one exception. If the two given points are at the ends of 

 a diameter, an infinite number 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 only one straight line can in general 

 be drawn, but there are exceptional cases in which through two 

 points an infinite number of straight lines can be drawn. So there 

 is a kind of opposition between the geometries of Riemann and Lo- 

 batschewsky. For instance, 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 Lobatschewsky, and greater than two right angles 

 in that of Riemann. The number of parallel lines that can be drawn 

 through a given point to a given line is one in Euclid's geometry, none 

 in Riemann's, and an infinite number in the geometry of Lobatschew- 

 sky. Let us add that Riemann's space is finite, although unbounded 

 in the sense which we have above attached to these words. 



Surfaces with Constant Curvature. One objection, however, re- 

 mains possible. There is no contradiction between the theorems 

 of Lobatschewsky and Riemann; but however numerous are the other 

 consequences that these geometers have deduced from their hypothe- 

 ses, they had to arrest their course before they exhausted them all, 

 for the number would be infinite; and who can say that if they had 

 carried their deductions further they would not have eventually 

 reached some contradiction? This difficulty does not exist for Rie- 

 mann's geometry, provided it is limited to two dimensions. As we 

 have seen, the two-dimensional geometry of Riemann, in fact, does 

 not differ from spherical geometry, which is only a branch of ordinary 

 geometry, and is therefore outside all contradiction. Beltrami, by 

 showing that Lobatschewsky's two-dimensional geometry was only a 

 branch of ordinary geometry, has equally refuted the objection as 

 far as it is concerned. This is the course of his argument: Let us 

 consider any figure whatever on a surface. Imagine this figure to be 

 traced on a flexible and inextensible canvas applied to the surface, in 

 such a way that when the canvas is displaced and deformed the differ- 

 ent lines of the figure change their form without changing their 

 length. As a rule, this flexible and inextensible figure cannot be 

 displaced without leaving the surface. But there are certain surfaces 

 for which such a movement would be possible. They are surfaces of 

 constant curvature. If we resume the comparison that we made 

 just now, and imagine beings without thickness living on one of these 

 surfaces, they will regard as possible the motion of a figure all the 



