aa 
Legendre’s Geometey. 293 
sometimes used the word angle in the sense above defined. 
(B. XI, 20,21, 22, &c.) and dif any one will make the exper- 
iment he will find it more natural to attach that idea to it in 
all cases. Ifit 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 ree: nor with any in- 
vestigations in which angles are em 
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 conveniest reference. 
The first section contains the properties of straight lines 
which meet, those of a the theorem upon the 
sum of the angles of a triangle, the theory of parallel lineé, 
&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- 
ucing as an axiom, what is more justly considered a prop- 
osition. Later writers have uniformly experienced the 
same difliculty, ba some of them have fallen on strange 
means of poe ming ite ‘Bezout est diss issimule’ le vice du 
raisonnement,” says Lacroix.* Some writers have trans- 
posed and shifted athe difficulty, until they have obscured it 
under long and int tricate a Such a ae we 
favtats editions. ‘He says in the preface, “6 Paprés 
Pavis de plusieurs professeurs distingués, on s’ est déter~ 
miné a rétablir, dans cette onzieme édition, la théorie des 
* + Géométrie, p. 23. 
