Page:Carroll - Euclid and His Modern Rivals.djvu/83

. II. § 5.] Min. I beg your pardon. I grant that you have made out a very good case for your own Axiom, and but a bad one for Playfair's.

Euc. I will make it worse yet, before I have done. My next remark will be best explained with the help of a diagram.



Let AB and CD make, with EF, the two interior angles BEF, EFD, together less than two right angles. Now if through E we draw the Line GH such that the angles HEF, EFD may be equal to two right angles, it is easy to show (by Prop. 28) that GH and CD are 'separational.'

Min. Certainly.

Euc. We see, then, that any Lines which have the property (let us call it ) of making, with a certain transversal, two interior angles together less than two right angles, have also the property (let us call it ) that one of them intersects a Line which is separational from the other.

Min. I grant it.

Euc. Now suppose you decline to grant my 12th Axiom, but are ready to grant Playfair's Axiom, that two intersectional Lines cannot both be separational from the same Line: then you have in fact granted my Axiom.

Min. Be good enough to prove that.

Euc. Lines, which have property ',' have property