Page:Carroll - Euclid and His Modern Rivals.djvu/166

128 I am not misrepresenting you, I think, if I say that you propound this as axiomatic truth—which, I need hardly remark, is a corollary deducible from the fourth Proposition in Table II. (see p. 34).

Nie. We accept the responsibility of the two Axioms separately, but not of a logical deduction from the two.

Min. There are certainly some logical deductions from Axioms (Contranominals for instance) that are not so axiomatic as the Axioms from which they come: but surely if you tell me 'it is axiomatic that X is Y and 'it is axiomatic that Y is Z,' it is much the same as saying 'it is axiomatic that X is Z?

Nie. It is very like it, we admit.

Min. Now take one more combination. Take 9 and 9. We thus eliminate the mysterious property altogether, and get a Proposition whose subject and predicate are perfectly definite geometrical conceptions—a Proposition which you assert to be, if not perfectly axiomatic, yet so nearly so as to be easily deducible from two Axioms—a Proposition which again lands us in Table II, and which, I will venture to say, is less axiomatic than any Proposition in that Table that has yet been proposed as an Axiom. We get this:—

'Lines which make equal corresponding angles with a certain transversal do so with any transversal,' which is Tab. II. 4 (see p. 34).

Here we have, condensed into one appalling sentence, the whole substance of Euclid I. 27, 28, and 29 (for the fact that the lines are 'separational' may be regarded as merely a go-between). Here we have the whole difficulty