Page:Carroll - Euclid and His Modern Rivals.djvu/121

.] Nie. Yes, one other. 'The assumption mentioned in § 113' (the assumption that case is the only true one) 'may now be stated thus:—Axiom VI. Through a given point only one Line can he drawn parallel to a given Line.'

Min. May it indeed? And why 'now' rather than three pages back? Is there a single word, in all this argument, which tends to show that case is—I will not say certainly true, but—even fairly probable?

Nie. (cautiously) I will not assert that there is.

Min. In point of fact the odds are exactly three to one against it—since you have only excluded one of the five cases, and the other four are, for anything we know to the contrary, equally probable.

Nie. I will not dispute it.

Min. Well! Then it only remains to say that your attempted proof of Playfair's Axiom is an utter failure. Anything more hopelessly illogical I have never met with, not even in Cooley—and that is saying a great deal!

Nie. I confess I do not see my way to defending this proof. But even if we abandon the whole of it, we are no worse off than any other writer who assumes Playfair's Axiom.

Min. That I quite admit.

Nie. And then, my client instructs me to plead, this Manual (handing it to Minos) being so distinctly better than Euclid's in every other particular—

Min. Gently, gently! You are anticipating. I have not yet had my general survey of the book.

Nie. (refiling his pipe) Well, let us have it then.

Min. I will begin with the general remark that the