DEFINITION. 199 



" 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 demon- 

 strations 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 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 distance 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 conse- 

 quence. " Since B C D is a circle, the radius B A is 

 equal to the radius CA." BA is equal to CA, 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 

 BA, 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 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 



