T H E 



PHILOSOPHICAL MAGAZINE 

 AND JOURNAL. 



31 st MARCH 1824. 



XXVIII. On the Application of algebraic Functions to prove 

 the Properties oj' parallel Lines. 



A TRANSLATION of Legendre's excellent Elements of 

 Geometry has lately appeared, under the care of Dr. 

 Brewster of Edinburgh. The principal novelty in this work 

 is a defence, furnished by the author himself, of his celebrated 

 demonstration of the fundamental property of parallel lines 

 drawn from the doctrine of algebraic functions, which Pro- 

 fessor Leslie has attacked in the last editions of his Geometry. 

 It is much to be regretted that a discussion of this kind should 

 degenerate into a contestation. On the present occasion there 

 is no intention to mingle in the controversy. But it must be 

 interesting to every one, who has paid any attention to geo- 

 metrical studies, to inquire, how it happens that a difficulty, 

 which has been found insurmountable in geometry, has never- 

 theless been so easily overcome by the methods of the modern 

 analysis. 



In laying down the elements of geometry, Euclid comes, 

 in the 1 7th of his first book, to prove that any two angles of 

 a triangle are less than two right angles. The plan of his 

 work required that the author should demonstrate the con- 

 verse of the same proposition ; namely, that two straight lines 

 will meet, and form a triangle with a third line, whenever 

 the sum of the two angles which they make with the third 

 line, is less than two right angles. But of this proposition no 

 demonstration could be found ; and the efforts of geometers, 

 continued incessantly for more than two thousand years, have 

 failed in supplying the defect. The most candid method 

 of proceeding would therefore have been, to put down the 

 converse proposition in the place it ought to occupy, fairly 

 stating that, although it was not proved, it would be assumed 

 as an admitted truth in the subsequent part of geometry. 

 And this is, in effect, what Euclid has done, by placing the 

 defective proposition at the head of his work in the form of 

 an axiom, or more properly of a postulate. 



When the circumstances in which two lines will meet one 

 Vol. 63. No. 311. March 1824. X another 



