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 disappear 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?
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 extended will always meet, which is precisely Riemann'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 Lobachevski's space, if a man whose course was restricted to a perfectly straight line should wish to avoid crossing a perfectly straight road,
- 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.