Page:Carroll - Euclid and His Modern Rivals.djvu/51

. II. § 2.] in mind that a Modern Rival will not have proved his case if he only succeeds in showing

(1) that certain Propositions might with advantage be omitted (for this a teacher would be free to do, so long as he left the logical sequence complete);

or (2) that certain proofs might with advantage be changed for others (for these might be interpolated as 'alternative proofs');

or (3) that certain new Propositions are desirable (for these also might be interpolated, without altering the numbering of the existing Propositions).

All these matters will need to be fully considered hereafter, if you should decide that my Manual ought to be retained: but they do not constitute the evidence on which that decision should be based.

Min. That, I think, you have satisfactorily proved. But what would you consider to be sufficient grounds for abandoning your Manual in favour of another?

Euc. Many grave charges have been brought against my Manual; but, of all these, there are only two which I regard as crucial in this matter. The first concerns my arrangement of Problems and Theorems: the second my treatment of Parallels.

If it be agreed that Problems and Theorems ought to be treated separately, my system of numbering must of course be abandoned, and no reason will remain why my Manual should then be retained as a whole; which is the only point I am concerned with. This question you can, of course, settle on its own merits, without examining any of the new Manuals.