REASONING. 



from a definition, follows in reality from an implied assump- 

 tion that there exists a real thing conformable thereto. This 

 assumption, in the case of the definitions of geometry, is false : 

 there exist no real things exactly conformable to the defini- 

 tions. There exist no points without magnitude ; no lines 

 without breadth, nor perfectly straight; no circles with all 

 their radii exactly equal, nor squares with all their angles 

 perfectly right. It will perhaps be said that the assumption 

 does not extend to the actual, but only to the possible, ex- 

 istence of such things. I answer that, according to any test 

 we have of possibility, they are not even possible. Their 

 existence, so far as we can form any judgment, would seem to 

 be inconsistent with the physical constitution of our planet at 

 least, if not of the universe. To get rid of this difficulty, 

 and at the same time to save the credit of the supposed system 

 of necessary truth, it is customary to say that the points, lines, 

 circles, and squares which are the subject of geometry, exist 

 in our conceptions merely, and are part of our minds; which 

 minds, by working on their own materials, construct an a priori 

 science, the evidence of which is purely mental, and has nothing 

 whatever to do with outward experience. By howsoever high 

 authorities this doctrine may have been sanctioned, it appears 

 to me psychologically incorrect. The points, lines, circles, 

 and squares, which any one has in his mind, are (I apprehend) 

 simply copies of the points, lines, circles, and squares which 

 he has known in his experience. Our idea of a point, I 

 apprehend to be simply our idea of the minimum visibile, the 

 smallest portion of surface which we can see. A line, as 

 defined by geometers, is wholly inconceivable. We can reason 

 about a line as if it had no breadth ; because we have a power, 

 which is the foundation of all the control we can exercise over 

 the operations of our minds ; the power, when a perception is 

 present to our senses, or a conception to our intellects, of 

 attending to a part only of that perception or conception, 

 instead of the whole. But we cannot conceive a line without 

 breadth ; we can form no mental picture of such a line : all 

 the lines which we have in our minds are lines possessing 

 breadth. If any one doubts this, we may refer him to his own 



