Page:Carroll - Euclid and His Modern Rivals.djvu/225

. § 1.] Nos. Not at all.

Min. These same 'conjugate angles' will get you into many difficulties.

Have you Euclid's Axiom 'all right angles are equal'?

Nos. Yes; only we propose to prove it as a Theorem.

Min. I have no objection to that: nor do I think that your treatment of angles, as a whole, is actually illogical. What I chiefly object to is the general 'slipshoddity' (if I may coin a word) of the language of your Syllabus.

Does your proof of Euc. I. 32 differ from his?

Nos. No, except that we propose Playfair's Axiom, 'two straight Lines that intersect one another cannot both be parallel to the same straight Line,' as a substitute for Euc. Ax. 12.

Min. Is this your only test for the meeting of two Lines, or do you provide any other?

Nos. This is the only one.

Min. But there are cases where this is of no use. For instance, if you wish to make a Triangle, having, as data, a side and the two adjacent angles. Have you such a Problem?

Nos. Yes, it is Pr. 10, at p. 19.

Min. And how do you prove that the Lines will meet?

Nos. (smiling) We don't prove it: that is the reader's business: we only provide enunciations.

Min. You are like the gourmand who would eat so many oysters at supper that at last his friend could not help saying 'They are sure to disagree with you in the night.' 'That is their affair,' the other gaily replied. 'I shall be asleep!'