Theory of Parallel Lines. 257 



Since the series of equal lines AB, BC, &c., can thus go 

 more than quite round A, the continual decrease of their 

 angles ABC, BCD, &c., must cause such of these lines as form 

 the smaller of the angles to be coiled inside of those which 

 form the greater ; and yet the increase in the length of the 

 radiants just insures the very reverse, or that the smaller 

 angles shall be outermost, which is absurd, but being a legi- 

 timate deduction from the supposition that the angles of a 

 triangle could be less than 180°, it therefore follows that they 

 cannot be less. 



Now this is what every previous writer on the subject, so 

 far as known to me, has failed to prove ; but the proof now 

 given might be greatly varied in its form. My former paper 

 is liable to no objection, so far as regards proving that the 

 angles of a triangle can never exceed 180°. But for that 

 purpose, a much simpler mode of indefinitely increasing the 

 area, is continually to double the sides of an equilateral tri- 

 angle, which at each time should more than quadruple the 

 area ; for if the angles were greater with greater sides, four 

 copies of any equilateral triangle could always be placed 

 within another having sides of double the length. 



The principles which I have employed here and formerly 

 are no other than Euclid has used in his Elements, although 

 by slightly introducing some of them at an earlier stage of 

 ot* geometry than he has done, I have deviated a little from 

 his arrangement. Now if by so doing, I have been enabled 

 to succeed where he had failed, this surely suggests that his 

 arrangement should be altered, and in particular, that his 

 tifth book, which has no dependence on the four before it, 

 should be placed first of all. There is besides something 

 unnatural in having such a very different subject as the doc- 

 trine of proportion in the heart of the elements of geometry. 

 Proportion, however useful and necessary as an auxiliary, 

 can, with no more propriety than any other part of algebra 

 or arithmetic, be reckoned geometry. 



Errata in former Paper, Vol. XXXVI. 

 Page 312, tlie letter P in fig. 1 is misplaced, it should be at the intersection 

 of the lines IL and CN. 



Page 316, lines 33 and 35, /or D^'QY read DiS'GY. 



