520 PB00F OF THE PKOPEETIES OF PARALLEL LIJTE9. 



" produced, will at length meet on the side on which the angles are, 

 " that are less than two right angles." As this so-called Axiom is far 

 indeed from being self-evident, many attempts have been made by 

 geometricians, both ancient and modern, to demonstrate the proper- 

 ties of parallel lines without its aid. Playfair, in his edition of Euclid's 

 Elements, discusses the principal of these ; distinctly shewing the 

 unsatisfactory character of them all, except that of Legendre, which 

 he pronounces " strictly demonstrative." It may be proper to men- 

 tion, that Legendre has treated the subject of parallel lines in two 

 different ways, one in the text of his " Elements of Geometry," 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 

 not conclusive; as it is, in fact, palpably the reverse — taking for 

 granted what requires proof, as much as Euclid's Axiom does; no 

 further attention need be given to it. In the present paper, I am to 

 speak only of the proof advanced by Legendre in the Notes to his 

 Geometry. This proof was keenly assailed by Sir John Leslie ; 

 whose strictures, which proceeded upon an entire misapprehension of 

 Legendre's meaning, were refuted by Playfair in an article that ap- 

 peared in the Edinburgh Review for July, 1812. Since that time, 

 the validity of Legendre's reasoning seems to have been admitted by 

 the general consent of mathematicians. Having been recently led, 

 however, to examine the subject, I feel myself unable to concur in 

 this verdict ; and I venture to bring my objections under the notice 

 of the Institute : which I do, not merely because the point in dispute 

 is one of considerable interest in itself, but also, and more especially, 

 because its settlement has an important connection with what may 

 be called the philosophy of Mathematics — that is, with the question 

 as to the principles on which Mathematical reasoning proceeds. 



Legendre endeavours to make it appear, without the assistance of 

 any special Axiom, that C, the third angle of a triangle ABC, is 

 determined from the other two, A and B, independently of the magni- 

 tude of c, the intervening side. If this be made out, all the properties 

 of parallel lines can easily be deduced. The difficulty is, to demon- 

 strate the fundamental position : but here it may be well to quote 

 Legendre's own words. " Soit Tangle droit egal a 1' unite, alors les 

 angles A,B,C seront des nombres compris entre et 2 ; et puisque 

 C = <£(A,B,c,) je dis que la ligne c ne doit poiut entrer dans la 

 fonction <£. En effet, on a vu que C doit etre entierement determine 



