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 understand what we can not conceive. They may order things differently in Germany; but, as yet, Professor Zöllner 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 triangle, 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 propositions as truly as he understands the possibility of a fourth dimension: 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 remains what it is. Professor Zöllner, 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 Zöllner'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 we 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 annihilate it. Professor Zöllner either mistakes the hyperboles of geometry 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