116 NAMES AND PEOPOSITIONS. 



§ 6. One of the circumstances which have contributed to keep up the 

 notion, that demonstrative truths follow from definitions rather than from 

 the postulates 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 realized by art. 

 People had a difficulty, therefore, in conceiving that the most certain of all 

 conclusions could rest on premises which, instead of being certainly true, 

 are certainly not true to the full 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 sup- 

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

 this view had not occurred, or whom it did not satisfy, have thought it in- 

 dispensable that there should be found in definitions something more cer- 

 tain, 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 dis- 

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

 sertion 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 ac- 

 curately true. 



Allowing this doctrine 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 an- 

 swering 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 as- 

 sumes 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 can not 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 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 in so far as possessing length. But 

 even-whfrti hold to be the false doctrine on the subject, leaves the conclu- 



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