Page:Carroll - Euclid and His Modern Rivals.djvu/114

76 discussed (see p. 74). And now for the subject of Parallels.

Nie. We have Playfair's Axiom (or rather its equivalent) 'Through a given point only one Line can be drawn parallel to a given Line' (p. 68), but this we do not simply lay down as an Axiom. We lead up to it by two or three pages of reasoning.

Min. This is most interesting! Let us examine the argument minutely. A logical proof of that Axiom would be perhaps the greatest advance ever made in the subject since the days of Euclid.

Nie. 'Two indefinite Lines in a Plane may intersect, as we have seen. We shall now consider the possibility of there being such Lines which do not intersect.' (p. 65.)

Min. That, of course, you can easily prove, without appealing to any disputable Axiom. It is simply Euc. I. 27. Do you prove it in Euclid's way?

Nie. Not exactly. Our argument is quite different from Euclid's: and we come to two conclusions—one being the real existence of Parallels, the other the equivalent of Playfair's Axiom.

Min. I very much doubt your proving the first by any simpler method than Euclid's: and as to proving the second, by any method at all, without assuming some disputable Axiom, I defy you to do it! However, let us hear your argument.

Nie. We take a Line, and a point without it: and from the point we draw two 'half-rays' intersecting the line. These half-rays we then turn about the point, in opposite directions, until they cease to intersect the Line.