v NON-EUCLIDEAN GEOMETRY 91 



in this manner. Just as, e.g., the varying appearances of 

 things to the different senses were ignored in order to 

 arrive at their real place, so the varying and irregular 

 deformations to which they are subjected at different 

 places, when abstracted from, lead to the homogeneity of 

 space. They are slight enough to be neglected, but if 

 they were larger and followed some definite and simple 

 law, they might suggest a non - Euclidean geometry. 

 Similarly, geometrical space is one and infinite, because 

 as soon as we abolish any boundary in thouglit, we can 

 abolish all ; it is infinitely divisible, because as soon as 

 the division is conceived of as proceeding in thought the 

 same act may be repeated as often as we please. And 

 so on ; geometrical space appears throughout as a con 

 struction of the intellect, which proceeds by the ordinary 

 methods of that intellect in the achievement of its peculiar 

 purposes. Nor is there anything new or mysterious about 

 the process ; no new faculty need be invoked, no new 

 laws of mental operation need be formulated. 



III. That the philosophic importance of this result is 

 capital, is surely evident. The certainty of geometry is 

 thereby shown to be nothing but the certainty with which 

 conclusions follow from non-contradictory premisses, and 

 in each geometry it flows from the definitions. The 

 certainty with which the sum of the angles of a triangle 

 may be asserted to equal two right angles in Euclidean 

 geometry, is precisely the same as that with which it may 

 be shown to be greater or less in non-Euclidean systems. 



And this shows that certainty in the sense of intrinsic 

 consistency has nothing to do with the question of the 

 real validity of a geometry. The latter depends on the 

 possibility of systematising our spatial experience by 

 means of the geometry. Our experience being what it 

 is, we find the Euclidean the simplest and most effective 

 system, alike to cover the facts and to calculate the 

 divergences between the ideal and the actual results ; and 

 so we use it. But if our experience were different, a 

 non-Euclidean system might conceivably seem preferable. 

 In short, as applied, a geometry is not certain, but useful. 



