298 REASONING. 



and squares which he has known in his experience. 

 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 attend- 

 ing 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 experience. I much 

 question if any one who fancies that he can conceive 

 what is called a mathematical line, thinks so from the 

 evidence of his consciousness : I suspect it is rather 

 because he supposes that unless such a conception 

 were possible, mathematics could not exist as a 

 science: a supposition which there will be no difficulty 

 in showing to be entirely groundless. 



Since then neither in nature, nor in the human mind, 

 do there exist any objects exactly corresponding to the 

 definitions of geometry, while yet that science cannot 

 be supposed to be conversant about non-entities ; 

 nothing remains but to consider geometry as conver- 

 sant with such lines, angles, and figures as really 

 exist ; and the definitions, as they are called, must be 

 regarded as some of our first and most obvious gene- 

 ralizations concerning those natural objects. The cor- 

 rectness of those generalizations, as generalizations, is 

 without a flaw : the equality of all the radii of a circle 

 is true of all circles,, so far as it is true of any one : 

 but it is not exactly true of any circle : it is only nearly 

 true ; so nearly that no error of any importance in prac- 

 tice will be incurred by feigning it to be exactly true. 



