NON-EUCLIDEAN GEOMETRY. 641 



He then proceeds to misquote it as follows: 



Si une droite, en coupant deux autres droites, fait les angles internes 

 inegaux, ou moindres que deux angles droits, ces deux droits, prolongees a 

 rinfini, se rencontreront du cGte ou les angles sont plus petits que deux droits; 



and continues, 



It is certain that, placed after the definitions, this Postulatum is incompre- 

 hensible. But, placed after Proposition XXVI. of the first book, where the 

 author demonstrates that ' if the interior angles together equal two right 

 angles, the lines will not meet,' it acquires almost the evidence of an axiom. 



The XXVI. is, of course, a mistake for XXVIII. 



Other mathematicians have tried to turn the flank of the difficulty 

 by substituting a new definition of parallels for Euclid's. 



Eu. I., Def. 35, is: 'Parallel straight lines are such as are in the 

 same plane, and which being produced ever so far both ways do not 

 meet.' 



On this Hobbs petulantly remarks: 



How shall a man know that there be straight lines which shall never 

 meet, though both ways infinitely produced? 



The answer is simple: Eead Eu. I., 27, where if the straight line be 

 infinite, is proven that those making equal alternate angles nowhere 

 meet. 



Wolf, Boscovich and T. Simpson substitute for Euclid's the defi- 

 nition : ' Straight lines are parallel which preserve always the same 

 distance from each other.' But this is begging the question, since it 

 assumes the definition, ' two straight lines are parallel when there are 

 two points in the one on the same side of the other from which the 

 perpendiculars to it are equal,' and at the same time assumes the 

 theorem, ' all perpendiculars from one of these lines to the other are 

 equal.' 



Just so the assumption that there are straights having the same 

 direction is a petitio princi'pii, since it assumes the definition of Varig- 

 non and Bezout, that ' parallel lines are those which make equal angles 

 with a third line,' and at the same time assumes the theorem that 

 ' straight lines which make equal angles with one given transversal 

 make equal angles with all transversals.' 



Other and more penetrating geometers have proposed substitutes 

 for the parallel-postulate. Of these the simplest are Ludlam's: 'Two 

 straight lines which cut one another can not both be parallel to the 

 same straight line,' and W. Bolyai's ' Any three points are costraight 

 or concyclic' 



But the largest and most desperate class of attempts to remove this 

 supposed blemish from geometry consists of those who strive to deduce 

 the theory of parallels from reasonings about the nature of the straight 

 line and plane angle, helped out by Euclid's nine other assumptions 



VOL. lxvii. — 40. 



