Page:Carroll - Euclid and His Modern Rivals.djvu/171

. § 1.] Nie. (uneasily) Yes.

Min. Surely I have read something like it before? Could it have been Euclid's 12th Axiom? And have I not somewhere read words like these:—'Euclid's treatment of parallels distinctly breaks down in Logic. It rests on an Axiom which is not axiomatic'?

Nie. We have nowhere stated this Axiom which you put into our mouth.

Min. No? Then how, may I ask, do you prove that particular Lines will meet? You must have to prove it sometimes, you know.

Nie. We have not had to prove it anywhere, that we are aware of.

Min. Then there must be some gaps in your arguments. Let us see. Please to turn to p. 46. Prob. 7. Here you make, at the ends of a Line CD, angles equal to two given angles (which, as you tell us below, 'must be together less than two right angles'), and you then say 'let their sides meet in O.' How do you know that they will meet?

Nie. You have found one hiatus, we grant. Can you point out another in the whole book?

Min. I can. At p. 70 I find the words 'Join QG, and produce it to meet FH produced in S.' And again at p. 88. 'Hence the centre must be at O, the point of intersection of these perpendiculars.' In both these cases I would ask, as before, how do you know that the Lines in question will meet?

Nie. We had not observed the omissions before, and we must admit that they constitute a serious hiatus.

Min. A most serious one. A student, who had been