18 EUCLID'S GEOMETRY MERELY A THEORY? 557 



shown in the figure, where the lines we are talking about do meet at the 

 point M, and let us imagine further that this point of intersection M 

 travels along the line CD. If then we keep turning the line AB slowly- 

 round the point P, eventually the point of intersection M must disap- 

 pear at one end and reappear at the other end of CD, it matters not 

 how far the two lines have been extended. 



The assumption hidden in Euclid's assumption is that there can be 

 one and only one position of the moving line AB at which it will be 

 parallel to CD. Lobachevski contrariwise assumes that AB will have 

 to be turned through a finite angle after parting from CD before it 

 intersects with CD again. That angle to be passed through gives 

 Lobachevski the opportunity of postulating not only two parallels to 

 CD, but an aggregate of parallels, all passing through the point P. 

 The same argument may be presented a little differently and more 

 clearly perhaps, as follows : Imagine AB at first not merely parallel 

 but at all points equidistant from CD. Will not AB have to dip 

 through a certain distance before it can meet CD ? 2 



This problem, apparently so simple, is of such a nature that neither 

 opponent can prove his assertion. It will be observed that when Euclid 

 says only one parallel is possible, and when Lobachevski says an infinite 

 number of them are possible, there is still room for a third champion 

 who will say no parallels are possible, that the lines AB and CD if ex- 

 tended will always meet, which is precisely Eiemann's position on the 

 question. The three geometries are thus exactly upon a par: no one of 

 them can establish itself against the other two; and the number of 

 possibilities is complete, for among the assertions " one," " many " and 

 " none," there is no position unoccupied in reference to the mystery of 

 parallel lines ; no chance left for any fourth geometry on this basis. 



We are now on the threshold of non-Euclidean geometry, prepared, 

 I trust, to enter a new variety of space where geometrical problems work 

 out to results differing widely from those found in the books of Euclid. 

 Compared with Lobachevski, Euclid was more sparing of parallels, and 

 the effect of this parsimony upon Euclid's idea of space is very marked. 

 I know of no better expression for the difference between their notions 

 of space than to say that Lobachevski's space is roomier. In Loba- 

 chevski's space, if a man whose course was restricted to a perfectly 

 straight line should wish to avoid crossing a perfectly straight road, 



2 Nothing in the definition, as established by Euclid himself, compels one 

 to believe that two parallel lines must be equidistant. The requirements are 

 that they be straight, that they lie in the same plane, and that they do not meet. 

 Euclid discovers that his parallels are at all corresponding points equidistant 

 from one another; but his parallels are peculiar in this respect, and it should 

 be borne in mind that they owe their existence to the postulate which no one 

 can validate. 



