Page:Carroll - Euclid and His Modern Rivals.djvu/79

. II. § 5.] Euc. Let me remark in the first place—it is a minor matter, but yet one that must come in somewhere, and I do not want to break the thread of my argument—that we need, in any complete geometrical treatise, some practical geometrical test by which we can prove that two given finite Lines will meet if produced. My Axiom serves this purpose—a secondary purpose it is true—but it is incumbent on any one, who proposes to do away with it, to provide some sufficient substitute.

Min. I admit all that.

Euc. Now, if the test I propose—that the two Lines make with a certain transversal two interior angles on the same side of it together less than two right angles—be objected to as not sufficiently simple, the question arises, what simpler test can be proposed?

Min. The supporters of Playfair's Axiom would of course reply 'that one of the two Lines should cut a Line known to be parallel to the other.'

Euc. Assuming that what is needed is a distinct conception of the geometrical relationship of the two Lines, whose future meeting we are asked to believe in, which picture, think you, is the more likely to yield us such a conception—two finite Lines, both intersected by a transversal, and having a known angular relation to that transversal and so to each other—or two Lines 'known to be parallel,' that is two Lines of whose geometrical relationship, so far as our field of vision extends, we know absolutely nothing, but can only say that, in the far-away region of infinity, they do not meet?

Min. In clearness of conception, your picture seems to