GEOMETRY 95 



and that of Riemann, were invented. Each of 

 these contains a substitute for Euclid's axiom, and 

 is inconsistent with it, and they are inconsistent 

 with one another; but each is, when suitably 

 interpreted, a sufficient representation of our actual 

 spatial perceptions, although not so simple a 

 representative scheme as that of Euclid. It has 

 been demonstrated that each of these schemes is 

 logically self-consistent, and thus the possibility 

 of diverging conceptual schemes for the representa- 

 tion of a single set of perceptual data has in this 

 instance been established. This is a result of great 

 general interest. The view is no longer tenable 

 that the relations of space and time exist in the 

 mind as empty forms of thought, as in Kant's 

 scheme, prior to all perception; as forms in 

 which all our spatial and temporal perceptions 

 clothe themselves. Nor is it true that the con- 

 ceptual schemes for spatial relations are simple 

 products of empirical observation. They could 

 never have arisen apart from our actual experience 

 of the external world, but they contain more than 

 the raw data obtained from perception. The rise 

 from the rough data of sensuous perception to a 

 rational scheme representing the relations concerned 

 can only be accomplished by a process in which 

 the reflective activities of the mind, co-operating 

 with sensuous perception, contribute an essential 

 element which is lacking in the data themselves. 

 It is a very important fact, fully established by 



