'If two different Lines make equal angles with a certain transversal, they are said to have the same direction: if unequal, different directions.'
This interpolation would have the advantage of making Ax. 6 (which I have hitherto declined to grant) indisputably true.
Nie. (after a pause) No. We cannot adopt it as a Definition so early in the subject.
Min. You are right. You probably saw the pitfall which I had ready for you, that this same Definition would make your 8th Axiom (p. 115) exactly equivalent to Euclid's 12th! From this catastrophe you have hitherto been saved solely by the absence of geometrical meaning in your phrase 'the same direction,' when applied to different Lines. Once define it, and you are lost!
Nie. We are aware of that, and prefer all the inconvenience which results from the absence of a Definition.
Min. The 'inconvenience,' so far, has consisted of the ruin of Ax. 6 and Ax. 8. Let us now return to Ax. 9.
As to 9 (β), it is of course obviously true with regard to coincidental Lines: with regard to different 'Lines, which have the same direction,' I grant you that, if such Lines existed, they would make equal corresponding angles with any transversal; for they would then have a relationship of direction identical with that which belongs to coincidental Lines. But all this rests on an 'if'—if they existed.
Now let us combine 9 (β) with Axiom 6, and see what it is you ask me to grant. It is as follows:—
'There can be a Pair of different Lines that make equal angles with any transversal.'
