Page:Carroll - Euclid and His Modern Rivals.djvu/58

20 where they are, is the importance of keeping the numbering unchanged—a matter we have already discussed.

But perhaps the strongest argument is that it saves you from 'hypothetical constructions,' the danger of which has been so clearly pointed out by Mr. Todhunter (see pp. 222, 241).

Min. I think you have proved your case very satisfactorily. The next subject is 'the treatment of Pairs of Lines.' Would it not be well, before entering on this enquiry, to tabulate the Propositions that have been enunciated, whether as Axioms or Theorems, respecting them?

Euc. That will be an excellent plan. It will both give you a clear view of the field of your enquiry, and enable you to recognise at once any doubtful Axioms which you may meet with.

Min. Will you then favour me with your views on this matter?

Euc. Willingly. It is a subject which I need hardly say I considered very carefully before deciding what Definitions and Axioms to adopt.

§ 4. Syllabus of propositions relating to Pairs of Lines.

Let us begin with the simplest possible case, a Pair of infinite Lines which have two common points, and which therefore coincide wholly, and let us consider how