Legendre's Geometey. 293 



sometimes used the word angle in the sense above defined. 

 (B. XI, 20, 21 , 22, &c.) and if any one will make the exper- 

 iment he will find it more natural to atiach that idea to it in 

 all cases. If it is objected, that the space which we in- 

 clude in the idea of an angle is indefinite in extent, we an- 

 swer, so are the sides of the angle of indefinite length ac- 

 cording to the common definition. If the indefinite space 

 be an objection in the one case, so are the indefinite sides 

 in the other. But the fact is, that the circumstance of the 

 sides and space comprised being indefinite, has no connex- 

 ion with any of the properties of an angle, nor with any in- 

 vestigations in which angles are employed. 



The elements of Legendre are divided in the original 

 into eight books, four of which treat of plane, and four of 

 solid geometry. These books are changed into sections 

 by the translator, and the principles are numbered from be- 

 ginning to end, for the sake of more convenient reference. 

 The first section contains the properties of straight lines 

 which meet, those of perpendiculars, the theorem upon the 

 sum of the angles of a triangle, the theory of parallel lines, 

 &c. and corresponds nearly with B. I. of Euclid. The doc- 

 trine of parallel lines has long been considered as present- 

 ing one of the greatest difficulties which belong to ele- 

 mentary geometry. Euclid treated the subject, by intro- 

 ducing as an axiom, what is more justly considered a prop- 

 osition. Later writers have uniformly experienced the 

 same difficulty, and some of them have fallen on strange 

 means of passing over it. " Bezout est dissimule' le vice du 

 raisonnement," says Lacroix.* Some writers have trans- 

 posed and shifted the difficulty, until they have obscured it 

 under long and intricate reasonings. Such a course, we 

 deem entirely inconsistent with the duty of an elementary 

 writer, which is to give peculiarly clear and exact ideas 

 upon every subject which he undertakes to elucidate. We 

 do not think that Legendre himself has shed any new light 

 on this subject; though his management of it, exhibits the 

 immense vigour and grasp of his mind. The eleventh edi- 

 tion of his Elements is different, in this respect, from the 

 preceding editions. He says in the preface, "d'apres 

 I'avis de plusieurs professeurs distingues, on s' est deter- 

 mine a retablir, dans cette onzieme edition, la theorie des 



* G^om^trie, p. 23. 



