DEFINITION. 



95 



p«rtie8, or from the existence of a 

 name meaning them ? 



Take, for instance, any of the de- 

 finitions laid down as premises in 

 Euclid's Ekm»iits ; the definition, 

 let us say, of a circle. This, being 

 analysed, consists of two propositions ; 

 the one an assumption w.th respect to 

 a matter of fact, the other a genuine 

 definition. " A figure may exist, hav- 

 ing all the points in the line which 

 bounds it equally distant from a single 

 point within it : " " Any figure pos- 

 sessing this property is called a circle." 

 Let us look at one of the demonstra- 

 tions which are said to depend on this 

 definition, and observe to which of 

 the two propositions contained in it 

 the demonstration really appeals. 

 "About the centre A, describe the 

 circle B C D." Here is an assumption 

 that a figure, such aa the definition 

 expresses, may be described ; which 

 is no other than the postulate, or 

 covert assumption, involved in the so- 

 called definition. But whether that 

 figure be called a circle or not is quite 

 immaterial The purpose would be 

 as well answered in all respects except 

 brevity were we to say, ''Through 

 the point B, draw a line returning in- 

 to itself, of which every point shall be 

 at an equal distance from the point 

 A. " By this the definition of a circle 

 would be got rid of, and rendered 

 needless ; but not the postulate im- 

 plied in it ; without that the demon- 

 stration could not stand. The circle 

 being now described, let us proceed to 

 the consequence. '* Since B C D is a 

 circle, the radius B A is equal to the 

 radius C A." B A is equal to C A, not 

 because B C 1) is a circle, but because 

 B C D is a figure with the radii equal. 

 Our warrant for assuming that such a 

 figure about the centre A, with the 

 radius B A, may be made to exist, is 

 the postulate. Whether the admissi- 

 bility of these postulates rests on 

 intuition, or on proof, may be a matter 

 of dispute ; but in either case they 

 are the premises on which the theo- 

 rems depend ; and while these are 

 retained it would make no difference 



in the certainty of geometrical truths, 

 though every definition in Euclid, 

 and every technical term therein 

 defin-'d, were laid aside. 



It is, perhaps, superfluous to dwell 

 at so much length on what is so nearly 

 self-evident ; but when a distinction, 

 obvious as it may appear, has been 

 confounded, and by powerful intel- 

 lects, it is better to say too much 

 than too little for the purpose of ren- 

 dering such mistakes impossible in 

 future. I will, therefore, detain the 

 reader while I point out one of the 

 absurd consequences flowing from the 

 supposition that definitions, as such, 

 are the premises in any of our reason- 

 ings, except Huch as relate to words only. 

 If this opposition were true, we might 

 argue correctly from true premises, 

 and arrive at a false conclusion. We 

 should only have to assume as a pre- 

 mise the definition of a nonentity ; or 

 rather of a name which has no entity 

 corre8p<;nding to it. Let this, for 

 instance, be our definition : 



A dragon is a serpent breathing 

 flame. 



This proposition, considered only 

 as a definition, is indisputably correct. 

 A dragon i» a serpent breathing 

 flame : the word means that. The 

 tacit assumption, indeed, (if there 

 were any such understood assertion,) 

 of the existence of an object with 

 properties corresponding to the defi- 

 nition, would, in the present instance, 

 be false. Out of this definition we 

 may carve the premises of the follow- 

 ing syllogism : 



A dragon is a thing which breathes 

 flame : 



A dragon is a serpent : 

 From which the conclusion is, 



Therefore some serpent or serpents 

 breathe flame : — 

 an unexceptional syllogism in the first 

 mode of the third figure, in which 

 both premises are true and yet the 

 conclusion false ; which every logician 

 knows to be an absurdity. The con- 

 clusion being false and the syllogism 

 correct, the premises cannot be true. 

 But the premises, considered as parts 



