356 RELATION BETWEEN THE AREA OF A PLANE 
plane triangle are not equal (see Cor. H 
Prop. Il.) to two right angles, we can 
prove that 
tri. BGC : tri. HOF =2—S .2_.,- 
FIG &S 
a right angle being taken as the unit of 
measure. 
For jom BF; and let 8, be the sum of 
the angles of the triangle CLF and S,, 
the sum of the angles of the triangle © 
BOF, Then (Prop. V.), 
triangle BCG : triangle BCF = 2—S : 2—S,; 
and, triangle BCF : triangle LCF = 2—S,: 2—S,; 
and, triangle LCF : triangle HCF = 2—S,: 2—s 
.. triangle BCG : triangle HCF = 2—S : 2~—s 
Cor.—If A be the area of the triangle BCG, we have 
k being a finite quantity, which remains the same for all triangles. 
APPENDIX. 
Legendre endeavours to make it appear,* without the assistance of 
any special Axiom, that C, the third angle of a triangle ABC, is de- 
termined from the other two, A and B, independently of the magni- 
tude of ¢, the intervening side. If this be made out, all the proper- 
ties of parallel lines can easily be deduced. The difficulty is to 
demonstrate the fundamental position. But here it may be well to 
quote Legendre’s own words: “Soit langle droit égal 4 Punité, alors 
les angles A, B, C seront des nombres compris entre 0 et 2; et puisque 
* It may be proper to mention that Legendre has treated the subject of varallel lines in 
two different ways, one in tre text of his Hlements of Goemetry, and the other in the 
notes to that work. Playfair considers the former method “quite logical and conclusive,” 
as well as the latter; only objecting to it that it is “long and indirect,” and too “subtle” 
for ‘“‘ those who are only beginning to study the Mathematics.” But, as the admission of 
Legendre himself is on record that this method is »o¢ conclusive; as it is, in fact, palpably 
the reverse—taking for granted what requires proof, as much as Huclid’s Axiom does; no- 
further attention need be given to it. The preof here criticised--a proof, the fallacy of 
which was for the first time (it is believed) pointed out by the author of the present paper 
in the Canadian Journal tor November, 1856-—is that advanced by Legendre in the Notes 
fo his Geometry. 
