THE FOURTH DIMENSION. 



75 



accelerate the progress of science. All these numbers lack the at- 

 tributes of representability. 



No man in the world can picture to himself "minus three 

 trees. " It is possible, of course, to know that when three trees 

 of a garden have been cut down and carried away, three are 

 missing, and by substituting for "missing" the inverse notion of 

 "added," we may say, perhaps, that "minus three trees" are added. 

 But this is quite different from the feat of imagining a negative 

 number of trees. We can only picture to ourselves a number of 

 trees that results from actual counting, that is, a positive whole 

 number. Yet, notwithstanding all this, people had not the slight- 

 est hesitation in extending the notion of number. Exactly so must 

 it be permitted us in geometry to extend the notion of space, even 

 though such an extension can only be mentally denned and can never 

 be brought within the range of human powers of representation. 



In mathematics, in fact, the extension of any notion is admis- 

 sible, provided such extension does not lead to contradictions with 

 itself or with results which are well established. Whether such 

 extensions are necessary, justifiable, or important for the advance- 

 ment of science is a different question. It must be admitted, there- 

 fore, that the mathematician is justified in the extension of the no- 

 tion of space as a point-aggregate of three dimensions, and in the 

 introduction of space or point-aggregates of more than three dimen- 

 sions, and in the employment of them as means of research. Other 

 sciences also operate with things which they do not know exist, and 

 which, though they are sufficiently defined, cannot be perceived by 

 our senses. For example, the physicist employs the ether as a 

 means of investigation, though he can have no sensory knowledge 

 of it. The ether is nothing more than a means which enables us to 

 .comprehend mechanically the effects known as action at a distance 

 and to bring them within the range of a common point of view. 

 Without the assumption of a material which penetrates everything, 

 and by means of whose undulations impulses are transmitted to the 

 remotest parts of space, the phenomena of light, of heat, of gravi- 

 tation, and of electricity would be a jumble of isolated -and uncon- 

 nected mysteries. The assumption of an ether, however, comprises 



