THE FOURTH DIMENSION. 



97 



possessed no peripheral nerves. According to Zollner's view, this 

 creature could, owing to its two-dimensional retinal images, have 

 only a two-dimensional intuition of space. The author's opinion, 

 however, is, that such a creature could not see at all, as it has no 

 possibility of collecting experiences which are adapted in any way 

 to interpreting the effects of things on its retina. The light which 

 proceeded from the objects roundabout and fell on the retina could 

 produce no other effect on the being than that of a wholly unintel- 

 ligible irritation, or perhaps even pain. 



The reflections presented sufficiently show that neither the 

 phenomena of symmetry nor the retinal images of the objects of 

 vision necessarily force upon us the assumption of a four-dimen- 

 sioned space. If the material world should ever present problems 

 which could not in the progress of knowledge be solved in a natural 

 way, the assumption that a four-dimensional space containing the 

 world exists would also be incompetent to resolve the difficulties 

 presented ; it would simply convert these difficulties into others, 

 and not dispose of the problems but simply displace them to an- 

 other world. Yet the question might be asked, is the existence of 

 a four-dimensional space really impossible? To answer this question 

 we must first clearly know what we mean by * 'exist." If existence 

 means that the intellectual idea of a thing can be formed and that 

 this idea shall not lead to contradictions with other well-established 

 ideas and with experience, we have only to say that four-dimen 

 sioned space does exist, as the arguments adduced in sections III 

 and IV have rendered plain. If, namely, the space of four dimen- 

 sions did not exist as a clear idea in the minds of mathematicians, 

 mathematicians could certainly not Have been led by this idea to 

 results which are recognised by the senses as true, and which really 

 take place in our own representable space. But if existence means 

 "material actuality," we must say that we neither now nor in the 

 future can know anything about it. For we know material actual- 

 ity only as three-dimensional, our senses can only make three- 

 dimensional experiences, and the inferences of our reason, although 

 they can well abstract from material things, can never ascend to 

 the point of explaining a four-dimensional materiality. Just as little, 



