v NON-EUCLIDEAN GEOMETRY 93 



reasonably be anticipated. Unfortunately it proves a 

 case of ( too many cooks and the embarrassment of 

 riches, rather than of the more the merrier. To suppose 

 three a priori forms of intuition corresponding to the three 

 geometries is evidently not feasible, for they are in hope 

 less conflict with each other. If it is a universal and 

 necessary truth that the angles of a triangle are equal to 

 two right angles, it cannot be an equally universal and 

 necessary truth that they are greater, according as we 

 happen to be speaking of a Euclidean or of a spherical 

 triangle. Clearly, there must be something seriously 

 wrong about the assumed relation of geometry to space, 

 or about the import of the criterion of apriority. Just as 

 the de facto existence of geometry seemed to Kant to 

 prove the possibility of an a priori intuition of Space, so 

 the de facto existence of metageometry indicates the 

 derivative nature of an intuition Kant had considered 

 ultimate. 



And the analysis thus necessitated rapidly discovers 

 the seat of the error. Kant, like all philosophers before 

 and far too many since his time, regards the conception 

 of Space as simple and primary and the word as un 

 ambiguous. He does not distinguish between physical 

 and geometrical space, between the problems of pure and 

 of applied geometry. Hence he is forced to make his 

 Anschauung an unintelligible hybrid between a percept 

 and a concept, to argue alternately that space could not 

 be either, and to infer that it must therefore be some third 

 thing. The possibility that it might be both never struck 

 him. Still less did he suspect that each of these alternatives 

 was complex, and that perceptual space was constructed 

 out of no less than three sensory spaces, while it was 

 susceptible of three different conceptual interpretations. 

 What Kant calls space therefore is not really one, but 

 seven, and the force of his argument is made by their 

 union. Confined to any one of them, the argument falls 

 to pieces. When we see these facts as clearly as the 

 development of metageometry has compelled us to see 

 them, we must surely confess that the Kantian account of 



