Page:Carroll - Euclid and His Modern Rivals.djvu/191

. § 3.] Nie. I am waiting to know whether you grant it.

Min. Unquestionably not! I must mark it against you as an Axiom of the most monstrous character! Mr. Wilson himself does not assume this, though he does assume its Contranominal, that Lines having different directions will meet (see p. 115). And what I said then I say now—unaxiomatic! But supposing it granted, how would you prove the Theorem?

'Then, they should be sepcodal; and the external angle should be equal (Th. 8) to the internal; which is contrary to the supposition.'

Min. Quite so. But Th. 8, which you quote, itself depends on the reality of sepcodal Lines. Your Theorem rests on two legs, and both, I fear, are rotten!

Nie. The next Theorem is equivalent to Euc. I. 32. Do you wish to hear it?

Min. It is unnecessary: it follows easily from Th. 8.

And I need not ask you what practical test you provide for the meeting of two Lines, seeing that you have Euclid's 12th Axiom itself.

Nie. Proved as a Theorem.

Min. Attempted to be proved as a Theorem. I will now take a hasty general survey of your client's book.

The first point calling for remark is the arrangement. You begin by dragging the unfortunate beginner straight into the most difficult part of the subject. Your first chapter positively bristles with difficulties about 'direction.' Then comes a long chapter on circles, including some very