322 THE POPULAR SCIENCE MONTHLY. 



he can not imagine the thing, he can not imagine the possibility of 

 the thing, the imagination of which involves the imagination of the 

 thing itself ; possibility, as every Kantist should know, being simply 

 a mode of conceiving the thing. And what he can not imagine he 

 will scarcely insist that he understands. As a rule, we do not under- 

 stand what we can not conceive. They may order things differently 

 in Germany ; but, as yet, Professor Zollner is the only evidence of it 

 that we have, and it is now tolerably clear that he does not think what 

 he thinks he thinks. He deceives himself. He will not venture to 

 say, for example, that he understands the possibility of a round tri- 

 angle, or of a whole equal to its part, or of three and two making 

 seven. Yet it is certain that he understands any one of these propo- 

 sitions as truly as he understands the possibility of a fourth dimen- 

 sion : which is as good as saying that the latter proposition is without 

 meaning to him, as to every other human being, and can have no 

 meaning to any mortal so long as the constitution of our minds re- 

 mains what it is. Professor Zullner, able and accomplished though 

 he be, is the dupe of words. It is not possible to understand the 

 possibility of a fourth dimension. Fourth we know, and dimension 

 we know ; but what is fourth dimension ? The realities denoted by 

 the words can not be united in thought. The phrase is perfectly 

 empty a sign that signifies nothing. To use a Wall-Street figure, 

 it is a metaphysical kite, not worth the breath that flies it. 



Professor Zollner's diagrams intended to show how a twist in a 

 cord, which we three-dimensional beings can do or undo by turning over 

 a part of the cord, could not be done or undone by two-dimensional 

 beings without making one end describe a circle, and, by means of this 

 showing, to illustrate the possibility of a four-dimensional creature tying 

 and untying knots in an endless cord as easily as Ave do and undo twists 

 in it are sheer delusions. A cord, whether laid on itself or extended 

 in only one direction, and though conceived of the utmost conceivable 

 thinness, can not be conceived with less than three dimensions. Nor 

 can a line or a point. When we think of a mathematical plane or line 

 or point, we do nothing more than fix our attention on length and 

 breadth, regardless of thickness ; or on length, regardless of both breadth 

 and thickness ; or on position, regardless of all three : we think away 

 from what remains, but we do not think it away. It is thrust off, but 

 not out minimized, not annihilated. No effort of thought can an- 

 nihilate it. Professor Zollner either mistakes the hyperboles of ge- 

 ometry for literal expressions, or supposes that they are as valid for 

 what he calls transcendental physics as for physics, forgetting that in 

 the former, if we vex them at all, we must pass behind symbols to the 

 things symbolized, which, if inconceivable, are of no use in aiding us 

 to conceive anything else. And that is the trouble with his diagrams. 

 They symbolize inconceivable things, whereas, to answer his purpose, 

 they should symbolize conceivable ones ; seeing that the ordinary 



