DEFINITION. 203 



(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 demonstra- 

 tive 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 mere notion made up by the mind, out of the materials 

 in 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 

 demonstrative truth to be correct, (which, in a sub- 

 sequent place, I shall endeavour to prove that it is 

 not) ; even on that supposition, 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. According to what appears to me the 

 sounder opinion, the mind cannot form any such 

 notion; it cannot conceive length without breadth; it 

 can only, in contemplating objects, attend to their 

 length exclusively of their other sensible qualities, 

 and so determine what properties may be predicated 

 of them in virtue of their length alone. If this be 

 true, the postulate involved in the geometrical defi- 



