TRIANGLE AND THE SUM OF TRE ANGLES. 357 
C= ¢ (A, B, ¢), je dis que la ligne ¢ ne doit poimt entrer dans la 
fonction ¢. Un effet, on a vu que C doit étre entierement determine 
par les seules données A, B, c, sans autre angle ou ligne quelconque ; 
mais la ligne ¢ 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 ¢ en A, B, ©, d’ ow il resulteroit que ¢ est egale a un nombre, 
ce qui est absurde. . Done ¢ ne peut entrer dans la valeur de C et on 
a simplement C = ¢(A, B).’”’ Sir John Leslie committed the un- 
accountable mistake of supposing the argument here stated, to be, 
“that the lme ce is of nature heterogeneous to the angles A and B, 
and therefore 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, C, which are numbers. ‘The quantities 
A, B, C,” says Playfair, in his exposition of Legendre’s reasoning, 
are ‘angles; they are of the same nature with numbers, or mere ex- 
pressions of ratio, and, according to the lauguage of Algebra, are of 
no dimension. The quaritity c, on the other hand, is the base of a 
triangle ; that is to say, a straight lime, or a quantity of one dimen- 
sion. Of the four quantities, therefore, A, B, C, ¢, the first three are 
of no dimensions, and the fourth or last is of one dimension. No 
equation, therefore, can exist imvolving all these four quantities and 
them only: for, if there did, a value of c might be found in terms of 
A, B, and ©; and ¢ therefore would 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, ¢, therefore € can be expressed in terms of A, B, c. Now 
® Legendre does not prove that when a quantity is determined by cer- 
tain others, it can be expressed in terms of them ; and I affirm that 
such a principle, without limitation, is not true. 
For example, consider the angle C of the triangle ABC. And 
let it be observed that I mean the angele itself, that is, the inclination 
of a and & 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, 6,¢; yet it cannot be expressed in terms of these 
quantities alone; because the value of an angle can only be indicated 
by pointing out its relation to some other angle or angles ; and there- 
fore cannot be expressed by means simply of lmes It is true that 
the numerical value of C may be expressed in terms of a, b, and ¢; 
