Page:Carroll - Euclid and His Modern Rivals.djvu/78

40 Min. I suppose I must take it on trust that any one ofthese 18 is sufficient logical basis for the other 17: I can hardly ask you to go through 306 demonstrations!

Euc. I can do it with 11. You will grant me that, when two Propositions are contranominal, so that each can be proved from the other, I may select either of the two for my series of proofs, but need not include both?

Min. Certainly.

Euc. Here are the proofs, which you can read afterwards at your leisure. (See Appendix III.)

§ 5. Playfair's Axiom.

Euc. The next general question to be discussed is the proposed substitution of Playfair's Axiom for mine. With regard to mine, I am quite ready to admit that it is not axiomatic until Prop. 17 has been proved. What is an Axiom at one stage of our knowledge is often anything but an Axiom at an earlier stage.

Min. The great question is whether it is axiomatic then.

Euc. I am quite aware of that: and it is because this is not only the great question of the whole First Book, but also the crucial test by which my method, as compared with those of my 'Modern Rivals,' must stand or fall, that I entreat your patience in speaking of a matter which cannot possibly be dismissed in a few words.

Min. Pray speak at whatever length you think necessary to so vital a point.