166 NAMES AND PROPOSITIONS. 



by a line all the points of which are at an equal distance from 

 a given point within it," was considered by them, not as an 

 assertion that any real circle has that property, (which would 

 not be exactly true,) but that we conceive a circle as having it ; 

 that our abstract idea of a circle is an idea of a figure with 

 its radii exactly equal. 



Conformably to this it is said, that the subject-matter of 

 mathematics, and of every other demonstrative science, is not 

 things as they really exist, but abstractions of the mind. A 

 geometrical line is a line without breadth ; but no such line 

 exists in nature ; it is a notion merely suggested to the mind 

 by its experience of nature. The definition (it is said) is a 

 definition of this mental line, not of any actual line : and it is 

 only of the mental line, not of any line existing in nature, that 

 the theorems of geometry are accurately true. 



Allowing this doctrine respecting the nature of demonstra- 

 tive truth to be correct (which, in a subsequent place, I 

 shall endeavour to prove that it is not ;) even on that suppo- 

 sition, the conclusions which seem to follow from a definition, 

 do not follow from the definition as such, but from an implied 

 postulate. Even if it be true that there is no object in 

 nature answering to the definition of a line, and that the 

 geometrical properties of lines are not true of any lines in 

 nature, but only of the idea of a line ; the definition, at all 

 events, postulates the real existence of such an idea : it 

 assumes that the mind can frame, or rather has framed, the 

 notion of length without breadth, and without any other 

 sensible property whatever. To me, indeed, it appears 

 that the mind cannot form any such notion ; it cannot 

 conceive length without breadth ; it can only, in con- 

 templating objects, attend to their length, exclusively of 

 their other sensible qualities, and so determine what pro- 

 perties may be predicated of them in virtue of their length 

 alone. If this be true, the postulate involved in the geome- 

 trical definition of a line, is the real existence, not of length 

 without breadth, but merely of length, that is, of long objects. 

 This is quite enough to support all the truths of geometry, 

 since every property of a geometrical line is really a property 



