Page:Carroll - Euclid and His Modern Rivals.djvu/42

4 Rhad. But as to this answer?

Min. Oh, give it full marks! What have we to do with logic, or truth, or falsehood, or right, or wrong? 'We are but markers of a larger growth'—only that we have to mark foul strokes, which a respectable billiard-marker doesn't do, as a general rule!

Rhad. There's one thing more I want you to look at. Here's a man who puts 'Wilson' at the top of his paper, and proves Euc. I. 32 from first principles, it seems to me, without using any other Theorem at all.

Min. The thing sounds impossible.

Rhad. So I should have said. Here's the proof. 'Slide ∠ DBA along BF into position GAF, GA having same direction as DC (Ax. 9); similarly slide ∠ BCE along AE into position GAC. Then the ext. ∠s = CAF, FAG, GAC = one revolution = two straight ∠s. But the ext. and int. ∠s = 3 straight ∠s. Therefore the int. ∠s = one straight ∠ = 2 right angles. Q. E. D.'

I'm not well up in 'Wilson': but surely he doesn't beg the whole question of Parallels in one axiom like this!

Min. Well, no. There's a Theorem and a Corollary. But this is a sharp man: he has seen that the Axiom does just