100 NAMES AND PROPOSITIONS, 



meanings to terms as shall suit objects actually existing. But this is \ 

 only an instance of the attempt, too often made, to escape from the ,' 

 necessity of abandoning old language after the ideas which it expresses/ 

 have been exchanged for contrary ones. From the meaning of a name 

 (we are told) it is possible to infer physical facts, provided the name 

 has, corresponding to it, an existing thing. But if this proviso be ne- 

 cessary, from which of the two is the inference really drawn 1 from the 

 existence of a thing having the properties 1 or from the existence of a 

 name meaning them 1 



Take, for instance, any of the definitions laid down as premisses in 

 Euclid's Elements ; the definition, let us say, of a circle. This, being 

 analyzed, consists of two propositions ; the one an assumption with 

 respect to a matter of fact, the other a genuine definition. "A figure 

 may exist, having all the points in the line which bounds it equally 

 distant from a single point within it:" "Any figure possessing this 

 property is called a circle." Let us look at one of the demonstrations 

 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 assump- 

 tion, that a figure, such as the definition expresses, ma^/ 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 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 ; Avithout 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 B C D 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. — The admissibility of these assumptions may be 

 intuitive, or may admit of proof; but in either case they are the 

 premisses on which the theorems depend ; and while these are retained 

 it would make no difference in the certainty of geometi-ical truths, 

 though every definition in Euclid, and every technical terai therein 

 defined, were laid aside. 



It is, perhaps, superfluous to dwell at so much length upon what is 

 so nearly self-evident ; but when a distinction, obvious as it may 

 appear, has been confounded, and by men of the most powerfid intel- 

 lect, it is better to say too much than too little for the purpose of 

 rendering such mistakes impossible in fiiture. AVe will, therefore, 

 detain the reader while we point out one of the absurd consequences 

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

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

 supposition were true, we might argue coiTectly from true premisses, 

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

 premiss the definition of a non-entity : or rather of a name which has 

 no entity con-esponding 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 



