64 ORIGIN AND SIGNIFICANCE OF 



axioms of our geometry depend on the native form of 

 our perceptive faculty, or are in any way connected 

 with it. 



It is different with the three dimensions of space. 

 As all our means of sense-perception extend only to 

 space of three dimensions, and a fourth is not merely 

 a modification of what we have, but something per- 

 fectly new, we find ourselves by reason of our bodily 

 organisation quite unable to represent a fourth di- 

 mension. 



In conclusion, I would again urge that the axioms 

 of geometry are not propositions pertaining only to 

 the pure doctrine of space. As I said before, they are 

 concerned with quantity. We can speak of quantities 

 only when we know of some way by which we can com- 

 pare, divide, and measure them. All space-measure- 

 ments, and therefore in general all ideas of quantities 

 applied to space, assume the possibility of figures mov- 

 ing without change of form or size. It is true we are 

 accustomed in geometry to call such figures purely 

 geometrical solids, surfaces, angles, and lines, because 

 we abstract from all the other distinctions, physical 

 and chemical, of natural bodies ; but yet one physical 

 quality, rigidity, is retained. Now we have no other 

 mark of rigidity of bodies or figures but congruence, 

 whenever they are applied to one another at any time 

 or place, and after any revolution. We cannot, how- 



