Page:Carroll - Euclid and His Modern Rivals.djvu/167

. § 1.] of Parallels swallowed at one gulp. Why, Euclid's much-abused 12th Axiom is nothing to it! If we had (what I fear has yet to be discovered) a unit of 'axiomaticity,' I should expect to find that Euclid's 12th Axiom (which you call in your Preface, at p. xiii, 'not axiomatic') was twenty or thirty times as axiomatic as this! I need not ask you for any further proof of Euc. I. 32. This wondrous Axiom, or quasi-Axiom, is quite sufficient machinery for your purpose, along with Euc. I. 13, which of course we grant you. Have you thought it necessary to provide any other machinery?

Nie. No.

Min. Euclid requires, besides I. 13, the following machinery:—Props. 4, 5, 7, 8, 15, 16, Ax. 12, Props. 27, 28, and 29. And for all this you offer, as a sufficient substitute, one single Axiom!

Nie. Two, if you please. You are forgetting Ax. 6.

Min. No, I repeat it—one single Axiom. Ax. 6 is contained in Ax. 9 : when the subject is known to be real, the Proposition necessarily asserts the reality of the predicate.

Nie. That we must admit to be true.

Min. I need hardly say that I must decline to grant this so-called 'Axiom,' even though its collapse should involve that of your entire system of 'Parallels.' And now that we have fully discussed the subject of direction, I wish to ask you one question which will, I think, sum up the whole difficulty in a few words. It is, in fact, the crucial test as to whether 'direction' is, or is not, a logical method of proving the properties of Parallels.