Page:Carroll - Euclid and His Modern Rivals.djvu/46

8 Rivals: it is in quality, not in quantity, that they claim to supersede you. Your methods of proof, so they assert, are antiquated, and worthless as compared with the new lights.

Euc. It is to that very point that I now propose to address myself: and, as we are to discuss this matter mainly with reference to the wants of beginners, we may as well limit our discussion to the subject-matter of Books I and II.

Min. I am quite of that opinion.

Euc. The first point to settle is whether, for purposes of teaching and examining, you desire to have one fixed logical sequence of Propositions, or would allow the use of conflicting sequences, so that one candidate in an examination might use X to prove Y, and another use Y to prove X—or even that the same candidate might offer both proofs, thus 'arguing in a circle.'

Min. A very eminent Modern Rival of yours, Mr. Wilson, seems to think that no such fixed sequence is really necessary. He says (in his Preface, p. 10) 'Geometry when treated as a science, treated inartificially, falls into a certain order from which there can be no very wide departure; and the manuals of Geometry will not differ from one another nearly so widely as the manuals of algebra or chemistry; yet it is not difficult to examine in algebra and chemistry.'

Euc. Books may differ very 'widely' without differing in logical sequence—the only kind of difference which could bring two text-books into such hopeless collision that the one or the other would have to be abandoned.