PEOOP OP THE PEOPEETIES OP PAEALLEL LINES. 521 



par les seules donnees A,B,e, san3 autre angle ou ligne quelconque, 

 mais la ligne c est heterogene avec les nombres A,B,C ; et si on avait 

 une equation quelconque entre A,B,C et c, on en pourrait tirer la 

 valeur de c en A,B,C, d'ou. il resulteroit que c est egale aun nombre, 

 ce qui est absurde. Done c ne peut entrer dans la valeur de C, et on 

 a simplement C — 4> (A,B.)" Leslie committed tbe unaccountable 

 mistake of supposing the argument here stated, to be, that " that the 

 Line c is of a nature heterogeneous to the angles A and B, and there- 

 fore cannot be compounded with these quantities" — whereas the 

 argument plainly is, that c, which is a line, cannot be expressed in 

 terms solely of A,B, and C, which are numbers. " The quantities A,B, 

 C," says Play fair, in his exposition of Legendre's reasoning, " are 

 angles ; they are of the same nature with numbers, or mere expressions 

 of ratio, and, according to the language of Algebra, are of no dimen- 

 sion. The quantity c, on the other hand, is the base of a triangle, 

 that is to say, a straight line, or a quantity of one dimension. Of the 

 four quantities, therefore, A,B,C,c, the first three are of no dimensions, 

 and the fourth or last is of one dimension. No equation therefore can 

 exist, involving all these four quantities and them only : for if there 

 did, a value of c might be found in terms of A,B, and C ; and c 

 would therefore be equal to a quantity of no dimensions : which is 

 impossible." 



In this reasoning it is assumed, that, because C is determined by 

 A,B,c, therefore C can be expressed in terms of A,B,c. Now Legen- 

 dre does not prove that when a quantity is determined by certain 

 others, it can be expressed in terms of them ; and I affirm that such 

 a prineiple, without limitation, is not true. 



For example, consider the angle C of the triangle ABC. And let 

 it be observed that I mean the angle itself, that is, the inclination of 

 a and b to one another, and not the numerical value of the angle, 

 calculated upon the supposition that a right angle, or any other angle, 

 has been assumed as a unit of measure. The angle C is determined 

 by the sides a,b,c ; yet it cannot be expressed in terms of these quan- 

 tities alone ; because the value of an angle can only be indicated by 

 pointing out its relation to some other angle or angles ; and therefore 

 cannot be expressed by means simply of lines. It is true that the 

 numerical value of C may be expressed in terms of a,b, and c : viz, in 

 an equation where only the ratios of a,b, and c occur, the ratios being 

 numbers. Thus, if b = /? a, and c = ya, we might have 



numerical value of C =/ (Ay-) 

 But this is altogether a different thing from saying that C itself, 

 the angle properly so called, the inclination of a and b to one another, 



N* 



