114 NAMES AND PROPOSITIONS. 



Here is an assumption that a figure, such as 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 into itself, of which every point shall be at an equal dis- 

 tance from the point A." By this the definition of a circle would be got 

 rid of, and rendered needless ; but not the postulate implied in it ; without 

 that the demonstration 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 BCD 

 is a circle, but because B C D is a figure with the radii equal. Our war- 

 rant 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 admissibility 

 of these postulates rests on intuition, or on proof, may be a matter of dis- 

 pute ; but in either case they are the premises on which the theorems de- 

 pend ; and while these are retained it would make no difference in the cer- 

 tainty of geometrical truths, though every definition in Euclid, and every 

 technical term therein defined, were laid aside. 



It is, perhaps, superfluous to dwell at so much length on what is so near- 

 ly self-evident; but when a distinction, obvious as it may appear, has been 

 confounded, and by powerful intellects, it is better to say too much than 

 too little for the purpose of rendering such mistakes impossible in future. 

 I will, therefore detain the reader while I point out one of the absurd con- 

 sequences flowing from the supposition that definitions, as such, are the 

 premises in any of our reasonings, except such as relate to words only. If 

 this supposition were true, we might argue correctly from true premises, 

 and arrive at a false conclusion. We should only have to assume as a 

 premise the definition of a nonentity ; or rather of a name which has no 

 entity corresponding 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 is a serpent breathing flame : the word means that. The tacit 

 assumption, indeed (if there were any such understood assertion), of the ex- 

 istence of an object with properties corresponding to the definition, would, 

 in the present instance, be false. Out of this definition we may carve the 

 premises of the following 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 unexceptionable 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 conclusion being false and the syllogism. 

 correct, the premises can not be true. But the premises, considered as 

 parts of a definition, are true. Therefore, the premises considered as j^arts 

 of a definition can not be the real ones. The real premises must be — 



A dragon is a really existing thing which breathes flame : 



A dragon is a really existing serpent : 

 which implied premises being false, the falsity of the conclusion presents 

 no absurdity. 



If we would determine what conclusion follows from the same ostensible 

 premises when the tacit assumption of real existence is left out, let us, ac- 



