DEFINITION. 



97 



define any name which is already 

 known to be a name of really existing 

 objects. On this account it is, that 

 the assumption was not necessarily 

 implied in the definition of a dragon, 

 while there was no doubt of its being 

 included in the definition of a circle. 



§ 6. One of the circumstances which 

 have contributed to keep up the notion 

 that demonstrative truths follow from 

 definitions rather than from the postu- 

 lates implied in those definitions, is, that 

 the postulates, even in those sciences 

 which are considered to surpass all 

 others in demonstrative certainty, are 

 not always exactly true. It is not 

 true that a circle exists, or can be 

 described, which has all its radii 

 exactly equal. Such accuracy is ideal 

 only ; it is not found in nature, still 

 less can it be realised by art. People 

 had a difficulty, therefore, in con- 

 ceiving that the most certain of all 

 conclusions could rest on premises 

 which, instead of being certainly true, 

 are certainly not true to the full ex- 

 tent 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 con- 

 clusion. Philosophers, however, to 

 whom this view had not occurred, or 

 whom it did not satisfy, have thought 

 it indispensable that there should be 

 found in definitions something more 

 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 defini- 

 tion 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 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 defini- 

 tion 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 geo- 

 metry are accurately true. 



Allowing this doctrine respecting 

 the nature of demonstrative truth to 

 be correct (which, in a subsequent 

 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. To 

 me, indeed, it appears that the mind 

 cannot form any such notion ; it can- 

 not 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 

 definition of a line is the real exist- 

 ence, not of length without breadth, 

 but merely of length, that is, of long 

 objects. This is quite enough to 



