EUCLID S TWELETH AXIOM: 34) 
THE RELATION WHICH CAN BE PROVED TO SUBSIST 
BETWEEN THE AREA OF A PLANE TRIANGLE AND 
THE SUM OF THE ANGLES, ON THE HYPOTHESIS 
THAT EUCLID’S 127TH AXIOM IS FALSE. 
BY THE REV. GEORGE PAXTON YOUNG, M.A., 
PROFESSOR OF LOGIC AND METAPHYSICS, KNOX COLLEGE, TORONTO. 
Read before the Canadian Institute, 25th February, 1860. 
I propose to prove in the present paper, that, if Euclid’s 12th 
Axiom be supposed to fail in any casé, a relation subsists between the 
area of a plane triangle and the sum of the angles. Call the area A; 
and the sum of the angles S; a right angle being taken as the unit 
of measure. Then 
A= k(2-—S); 
% being a constant finite quantity, that is, a finite quantity which re- 
mains the same for ali triangles. This formula may be considered as 
holding good even when Euclid’s 12th Axiom is assumed to be true ; 
only & is in that case infinite. 
Before proceeding with the proof of the law referred to, I would 
observe, that, while on the one hand Euclid’s 12th Axiom is assuredly 
not an Axiom in the proper sense of the term, that is, not a self» 
evident truth, on the other hand 7¢ has never been demonstrated to be 
true. I even feel satisfied, from metaphysical considerations, that a 
demonstration of its truth is impossible. Legendre’s supposed de- 
monstration, which Mathematicians appear to have accepted as valid, 
was shown by me, in the Canadian Journal for November, 1856, to 
be erroneous.* For the sake of those who may not have the former 
* Tn an Hssay on Mathematical Reasoning, appended to his Mathematical Euclid, Dr. 
Whewell refers to the attempts which have been made to dispense with Euclid’s 12th 
Axiom, “No one,” he writes, “has yet been able to construct a system of Mathematical 
truth by means of Definitions alone, to the exclusion of Axioms; though attempts having 
this tendency have been made constantly and earnestly. It is, for instance, well known to 
most readers, that many mathematicians have endeavoured to get rid of Eucli@’s Axioms 
respecting straight lines and parallel lines; but that none of these essays have been gener= 
ally considered satisfactory.’ The last clause in this statement calls for remark. Sir 
John Leslie objected to Legendre’s reasoning ; but on grounds which (as Professor Playfair 
showed in the Hdinburgh Review) are altogether frivolous. Playfair maintained thati 
Legendre’s proof was satisfactory; and since then, till the publication in the Canadian 
Journai of the article above referred to, mathematicians have—by their silence at least— 
‘acquiesced in his verdict. Lf Legendre’s proof has been generally considered unsatisfactory, 
why did none of those by whom such a view was taken show where the reasoning is defective 
