522 PBOOF OF THE PEOPEBTIES OF PABALLEL LINES. 



can be expressed in terms of a,b, and c. Now if C itself (not its 

 numerical value, but the absolute angle) is determined by a,b, and c ; 

 and if, nevertheless, it cannot in the nature of things be expressed in 

 terms of a,b, and c ; Legendre's demonstration, the very foundation 

 of which is, that a quantity which is determined by certain others, can 

 be expressed in terms of them, must be pronounced erroneous. 



Should it be maintained that C (the angle itself) may be expressed 

 in terms of the numbers /3 and y, a right angle being understood to be 

 the unit of measure ; or more fully, thus : 



C = right angle x/(/?,y); 

 I reply that in the same manner the line c, in Legendre's reasoning, 

 may be expressed in terms of A,B,C, some line L being understood to 

 be the unit of linear measure ; thus : 



c = Lx/(A,B,C.) 



I am inclined to believe, from metaphysical considerations, that it 

 is impossible to demonstrate the properties of parallel lines without 

 a special axiom. As it would be difficult, however, to bring out the 

 grounds of this belief without entering into a somewhat lengthened 

 discussion of the nature of our conceptions of geometrical magnitudes, 

 I content myself in the meantime with the above remarks on Legen- 

 dre's treatment of the subject. Had the reasoning of that distin- 

 guished mathematician been valid, it would have been a standing and 

 conclusive refutation of any theory of our conceptions of geometrical 

 magnitudes, in which the impossibility of proving the properties of 

 parallel straight lines without a special axiom was involved. But as 

 Legendre's demonstration, like all others in which the same thing has 

 been attempted, has been shewn to be erroneous, the ground is clear ; 

 and a theory of our geometrical conceptions, such as has been referred 

 to, is at least not exposed to the ready-made fatal objection that it is 

 at variance with unquestioned fact. 



