v NON-EUCLIDEAN GEOMETRY 87 



attempting to summarize some of the results of this 

 controversy, with a view to (a) bringing out the most 

 important points established by the new metageometry/ 

 (b} considering what light they throw on the nature of 

 Space, (c) estimating what changes will have to be made 

 in the references to geometry which philosophers have 

 been so addicted to making. It is indeed possible that 

 the attempt is still premature, that the parties are still too 

 bitter to be completely reconciled, that the subject is still 

 too inchoate and chaotic for its full significance to be 

 determined. In that case the present writer would 

 console himself with the reflection that his efforts can at 

 least do no harm, and may possibly even do good by 

 inducing philosophers to revise their antiquated notions 

 concerning the meaning of the conception of Space. 



I. I shall begin, therefore, by referring to a point which 

 the metageometers do not seem to have satisfactorily 

 established, and that is the value of the conception of a 

 fourth dimension. I say advisedly of the conception, 

 for the actual existence, or even the possibility of 

 imagining, a fourth dimension seems to have been 

 practically given up. The chief value of the conception 

 seems nowadays to be situated in the possibility of 

 making symmetrical solids coincide by revolving them in 

 a fourth dimension. But this seems a somewhat slender 

 basis on which to found the conception of a fourth 

 dimension, and the same end could apparently l also be 

 achieved by means of the conception of a spherical 

 space. Here then, probably, is the reason why of late 

 the fourth dimension has not been so prominent in the 

 forefront of the battle, and why its place has, with a great 

 advance in intelligibility, been taken by spherical and 

 pseudo-spherical three-dimensional space. 



It is on rendering these latter thinkable that the non- 

 Euclideans have concentrated their efforts, and, so far as I 

 can judge, they have, in a large measure, been successful. 

 It has been shown that Euclidean geometry may, nay, 

 logically must, be regarded as a special case of general 



1 Cp. Delboeuf, Rev. Phil. xix. 4. 



