294 Cambridge Course of Mathematics. 



paralleles a-peu-pre? sur la meme base qu' Euclide. II 

 en resultera plus de faciiite pourles etudiants, et cette rai- 

 son a paru preponderante, d'autant que les objection aux- 

 quelles est encore sujette la theorie des paralleles, ne peu- 

 vent etre entierement resolues que par des considerations 

 analytiques, telles que telles qui sont exposees dans la note 

 deuxieme. If he had adopted more nearly still the course 

 pursued by Euclid, he would have treated the subject more 

 to our satisfaction. In truth, Euclid's method, we think, 

 admits of but little improvement. Legendre has given in 

 the text a mere graphical proof of the principle involving 

 the difficulty ; while in note If, he has connected with, and 

 applied to, this graphical proof, a rigorous demonstration 

 without assuming any new axiom. The demonstration, 

 however, is entirely analytical, and the reasoning will not 

 readily be followed by those who have not considerable ac- 

 quaintance with the theory of equations and functions. 



There is no writer with whom we arc acquainted, that 

 has treated the doctrine of parallel lines with so much ad- 

 dress, and in so unexceptionable a manner as Lacroix. 

 His method is not much different from that of Euclid, and 

 differs from it principally in the circumstance, that it pre- 

 sents the difficulty reduced to its least dimensions. The 

 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 

 difficulty of proving this principle. Lacroix, making use 

 of that definition of an angle to which we gave the prefer- 

 ence, 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 of 

 the opinion, that the difficulty respecting parallel lines, is 

 in a great measure, imaginary. The method by which La- 

 croix has disposed of the difficulty is much to be preferred 

 to that of any other writer, yet we never examined the sub- 

 ject as treated by any author, when, we think, any one 



*Developpment nouveaw de la partie elemeataire des Mathematiques^ 

 Geneve, 1778, 2 vols. 4to. 



