Page:Carroll - Euclid and His Modern Rivals.djvu/49

. II. § 1.] and what the Association have published in their 'Syllabus of Plane Geometry.'

Euc. I think they have overlooked the fact that they are enacting many more sequences, as binding on writers, than the one logical sequence which they desire to secure. Their 'standard sequence' would be fitly replaced by a 'standard genealogy.' But in any case we are agreed that it is desirable to have, besides a standard logical sequence, a standard list of enunciations, numbered for reference?

Min. We are.

Euc. The next point to settle is, what sequence and numbering to adopt. You will allow, I think, that there are strong a priori reasons for retaining my numbers. The Propositions have been known by those numbers, for two thousand years; they have been referred to, probably, by hundreds of writers—in many cases by the numbers only, without the enunciations: and some of them, I. 5 and I. 47 for instance—'the Asses' Bridge' and 'the Windmill'—are now historical haracters, and their nicknames are 'familiar as household words.'

Min. Even if no better sequence than yours could be found, it might still be urged that a new set of numbers must be adopted, in order to introduce, in their proper places, some important Theorems which have been added to the subject since your time.

Euc. That want, if it were proved to exist, might, I think, be easily provided for without discarding my system of numbers. If you wished, for instance, to insert two new Propositions between I. 13 and I. 14, it would be