TRANSCENDENTAL GEOMETRY. 509 



so-called geometrical axioms that two straight lines can not inclose a 

 surface, etc., are true. For example, a spherical surface has a con- 

 stant measure of curvature not equal to zero, and positive. Since the 

 shortest distance between two points is a straight line, let us, extend- 

 ing the analogy, call the shortest distance between two points of a 

 spherical surface, lying wholly in that surface, a straight line of that 

 surface. Now, as the measure of curvature of a spherical surface is 

 constant, we can slide a figure about over the surface without altering 

 it, as is evident at once. On a sphere, however, more than one per- 

 pendicular can be drawn on the surface from a point to a straight line, 

 and two straight lines can inclose a surface. 



In a surface whose curvature is negative, an infinite number of 

 straight lines of the surface can be drawn through a given point which 

 will never meet a given straight line. Such a surface would be like a 

 spool. Some of its sections would be concave and others convex to 

 the same point. We have analogous results in what is called curved 

 space. These results were first suggested by Riemann, who was a 

 pupil of Gauss. 



For this mathematical treatment all that is needed is, first, algebra 

 and the differential calculus; secondly, a method of interpreting them 

 geometrically. We have found a code of interpretation for some 

 algebraic equations which give geometrical results, and we apply it so 

 far as we can to all. 



So far the mathematicians might have gone without let or hin- 

 drance, and there some of them, as Boole and Grassman, stopped. 

 But others thought they had settled whether the geometrical axioms 

 were a priori truths or not. We have just worked out a system of 

 geometry, said they, which is not, as we think, impossible, where these 

 axioms do not hold. Therefore these axioms are the results of an ex- 

 perience of things as they are. If we had had a different order of 

 things, as is possible, these axioms would not have been true nor 

 thought of. I shall, however, try to prove that, although not thought 

 of, they are true. 



The geometrical axioms express relations; relations between what? 



Geometry is a branch of mathematics. Therefore the geometrical 

 axioms express mathematical relations. What, then, is mathematics, 

 and with what does it deal ? 



Mathematics, in its widest sense, I will define as the science which 

 treats of logical- that is necessary relations. Between outside things 

 there are no necessary relations. The relation of cause and effect is 

 sometimes called necessary ; but, if so, it is not usually handled math- 

 ematically. The relations must, then, be of mental things. 



They are not relations between images or imaginations of outside 

 things, for two reasons : First, the relations between imaginations can 

 be no more necessary than the things they image ; second, the im- 

 aginations of men's minds are different. One may imagine a line as a 



