Williams. — On the Fourth Dimension. 511 



The realisation of the possibility of the existence of fourth- 

 dimensional space leads naturally to two questions, which, of 

 course, may suggest others : (1.) Seeing that we may con- 

 ceive of our space system as being made up of innumerable 

 two-dimensional systems, each possibly inhabited by beings 

 quite without cognisance of the companion systems, may not 

 four-dimensional space be compounded of innumerable three- 

 dimensional systems, similar to our own, but lying completely 

 outside our cognisance ? (2.) Seeing that figures in a two- 

 dimensional system might be regarded as sections of solids, 

 might not our so-called solids be in reality sections in three 

 dimensions of four-dimensional figures ? 



With regard to the first question, it does not appear that 

 any reason can be adduced why it should not be answered in 

 the affirmative. This leads to the curious consideration that, 

 in spite of preconceived notions, two bodies may, apparently, 

 occupy the same space. If one plane be superimposed upon 

 another, a figure moved out of the latter an infinitely small 

 distance passes into the other. Two beings might in this way 

 be separated by the smallest possible distance, and yet for 

 till practical purposes be at an infinite distance from one 

 another. In the same way, if we conceive of a cube, say of 

 1 ft. side, moved one millionth part of an inch in the direction 

 of the fourth axis, it will pass immediately out of our system, 

 and presumably its place may be occupied by another cube 

 similar to itself. The centres of gravity of the two would be 

 separated by an infinitesimal distance, and yet each in its own 

 space system might be a solid, the two cubes to all intents and 

 purposes occupying the same space. In this connection it 

 may be mentioned that it has been suggested that, as we may 

 imagine a plane to be bent over so as to re-enter itself, with 

 or without a twist in the process, so we may suppose it 

 possible for our space system to have been similarly treated. 

 This would make it possible to arrive at one's starting-point 

 by travelling along an apparently straight line for a consider- 

 able distance. But though the notion of limited space thus 

 introduced had attractions for so great a thinker as W. K. 

 Clifford, it seems that such a process would involve an exten- 

 sion of our present three-dimensional limitations. 



With regard to the second question, while solids may 

 mathematically be such sections, the answer must, when we 

 come to the case of animate beings, assuredly be a negative 

 one, for it is scarcely conceivable that a section could con- 

 tain the consciousness of the whole. If what we imagine to 

 be independent figures proper to our own space system are 

 but sections of four-dimensional figures, it would seem to be 

 necessary that the innumerable sections of these solids are 

 also playing their parts in an infinite number of two-dimen- 



