Page:Carroll - Euclid and His Modern Rivals.djvu/188

150 Min. Then we can proceed at once to the subject of Parallels. Will you kindly give me your proof of Euc. I. 32 from the beginning?

P. 10. Th. 1. 'Two Directives can intersect in only one point.'

Min. By 'Directive' you mean an 'infinite Line'?

Nie. Yes.

Min. Well, I need hardly trouble you to prove it as a Theorem, being quite willing to grant it as an Axiom. What is the next Theorem?

P. 11. Th. 5. 'Parallel Directives cannot meet.'

Min. We will call them 'sepcodal,' if you please. I grant it, provisionally. If such Lines exist, they cannot meet.

P. 11. Th. 7. 'Only one Line, sepcodal to a Directive, can be drawn through a point.'

Min. Does that assert that one can be drawn? Or does it simply deny the possibility of drawing two?

Nie. The proof only applies to the denial: but the assertion is certainly involved in the enunciation. At all events, if not assumed here, it is assumed later on.

Min. Then I will at this point credit you with one unwarrantable Axiom, namely, that different Lines can have the same direction. The Theorem itself I grant.