294 Cambridge Course of Mathematics. 
proposition in which this difficulty is so reduced, is this; 
a straight line which is perpendicular to another straight 
line, is met by all those which are oblique to this other; 
consequently, upon a plane, there are none but straight 
lines perpendicular to the same straight line, which do 
not meet, that is, which are parallel to each other. The 
imperfection of the theory of parallel lines consists in the 
dithculty of proving this principle. Lacroix, making use 
of that definition of an angle to. which we gave the prefer- 
nce, has given a demonstration of it taken from Bertrand, 
which is short, free from obscurity, and perfectly satisfac- 
tory. After all that has been said, we have long been © 
the opinion, that the difficulty respecting parallel lines, 's 
mn a great measure, imaginary. ‘The method by which La- 
croix has disposed of the difficulty is much to be preferre 
to that of any other writer, yet we never examined the sub- 
ject as treated by any author, when, we think, any one 
*Developpment nouveau de la partie élémentaire des Mathematiques, 
Geneve, 1778, 2 vols, 4to, 
