102 NAMES AND PROPOSITIONS. 



to the whole extent asserted. This apparent paradox will be examined 

 when we come to treat of Demonstration ; where we shall be able to 

 show that as much of the postulate is true, as is required to support as 

 much as is true of the conclusion. Philosophers, however, to whom 

 this view had not occuiTed, or whom it did not satisfy, have thought it 

 indispensable that there should be found in definitions something morn 

 certain, or at least more accurately true, than the implied postulate of 

 the real existence of a corresponding object. And this something 

 they flattered themselves they had found, when they laid it down that 

 a definition is a statement and analysis not of the mere meaning of a 

 word, nor yet of the nature of a thing, but of an idea. Thus, the 

 proposition, " A circle is a plane figure bounded by a line all the points 

 of which are at an equal distance fi'om 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 

 figurfe with its radii exactly equal. 



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

 ics, 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 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 docti'ine respecting the nature of demonstrative truth 

 to be correct (which, in a subsequent place, I shall endeavor 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 quali- 

 ties, 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 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 of all physical 

 objects possessing length. But even what I hold to be the false doc- 

 trine on the subject, leaves the conclusion that pur reasonings are 

 grounded upon the matters of fact postulated in definitions, and not 

 upon the definitions themselves, entirely unaffected ; and accordingly 

 I am able to appeal in confirmation of this conclusion, to the authoi-ity 

 of Mr. Whewcll, in his recent treatise on The Philosophy of the In- 

 ductive Sciences. On the nature of demonstrative truth, Mr. Whewell's 

 opinions are gi-eatly at variance with mine, but on the particular point 



