Page:Carroll - Euclid and His Modern Rivals.djvu/164

126 Nie. (after a long pause) Well! We admit that it is not exactly a Theorem: it is only a new form of the Axiom.

Min. Quite so: and as it is a more convenient form for my purpose, I will with your permission adopt it as a substitute for the Axiom. Now as to the corollary of this Theorem: that, I think, is merely a particular case of Ax. 9, one of the arms being slid along the infinite Line of which it forms a part, and thus of course having 'the same direction' as before?

Nie. It is so.

Min. And, as this is a more convenient form still, I will restate your assertions, limiting them to this particular case:—

Ax. 9 . Lines, which make equal corresponding angles with a certain transversal, have the same direction.

Ax. 9 . Lines, which have the same direction, make equal corresponding angles with any transversal.

Am I right in saying that these two assertions are virtually involved in your Axiom?

Nie. We cannot deny it.

Min. Now in 9 you ask me to believe that Lines possessing a certain geometrical property, which can be defined, constructed, and tested, possess also a property which, in the case of different Lines, we can neither define, nor construct, nor test. There is nothing axiomatic in this. It is much more like a Definition of 'codirectional' when asserted of different Lines, for which we have as yet no Definition at all. Will you not permit me to insert it, as a Definition, before Ax. 6 (p. 108)? We might word it thus:—