8 ROYAL SOCIETY OF CANADA 



But this is not all. If our space has a bend in it, the bending may- 

 be uniform in all directions, or uniform in only some directions, or it may 

 completely vaiy in one or more directions from point to point, so that 

 we have an endless number of possibilities. 



If the bending is uniform at all points and in all directions, no 

 change could be experienced in our passing from one part of space to 

 another, so that the space so endowed Avould be same but not Euclidean. 



But if the bending is not uniform, then different parts of space may 

 have different properties, and for all we can know to the contrary, the 

 bending of space may be some sort of a function of time, so that certain 

 parts of space, or the whole of it, may be undergoing slow, but continu- 

 ous changes in curvature. 



And Clifford says that " the hypothesis that space is not homaloid, 

 i. e., Euclidean, and again that its geometrical character may change 

 with the time, may, or may not be destined to play a great part in the 

 physics of the future." And one of Clifford's commentators goes even 

 further, for he says : *• The most notable physical quantities which vary 

 with position and time, are heat, light and electro-magnetism. It is 

 these that we ought peculiarly to consider when seeking for an}^ physi- 

 cal changes, which may be due to changes in the curvature of sj^ace. If 

 we suppose the boundary of an arbitrary figure in space to be distorted 

 by the variation of space curvature, there would, by analogy from one 

 and two dimensions, be no change in the volume of the figure arising 

 from such distortion. 



"Further, if we assume as an axiom that space resists curvature with 

 a resistance proportioned to the change, we find that waves of ' space- 

 displacement * are precisely similar to those of the elastic medium which 

 we suppose to propagate light and heat. "We also find that * space-twist ' 

 is a quantity exactly corresponding to magnetic induction, and satisfying 

 relations similar to those which hold for the magnetic field. It is a ques- 

 tion whether physicists might not find it simpler to assume that space is 

 capable of a varying curvature, and of resistance to that variation, than 

 to suppose the existence of a subtle medium pervading an invariable 

 homaloidal space." 



After this long quotation I think that it is not necessary to say any- 

 thing further with regai'd to the theoretical possibilities of a change- 

 able curved and twisted space. 



Going back to Clifford's illustrations and reasoning on the basis of 

 Euclidean geometry, it will be noticed that while the tube inhabited by 

 the worm remained straight, only one dimension was involved. But as 

 soon as any flexure occurred in the tube, at least a second dimension of 

 space was introduced, and the worm lived in space of, at least, two 

 dimensions, although supposed to be cognizant of only one, and we see 

 that the second dimension is necessary in order that bending may be pos- 



