Page:Carroll - Euclid and His Modern Rivals.djvu/245

. § 2.] By breaking up certain of the Propositions of Euc. I, II, and including some of the Corollaries, we get 73 Propositions in all—57 Theorems and 16 Problems. Of these 73, this Manual omits 14 (10 Theorems and 4 Problems); it proves 43 (32 Theorems and 11 Problems) by methods almost identical with Euclid's; for 10 of them (9 Theorems and a Problem) it offers new proofs, against which I have recorded my protest, one being illogical, 2 (needlessly) employing 'superposition,' 2 deserting Geometry for Algebra, and the remaining 4 omitting the diagonals in Euc. II; and finally it offers 6 new proofs, which I think may fairly be introduced as alternatives for those of Euclid.

In all this, and in all the matters previously discussed, I fail to see one atom of reason for abandoning Euclid. Have you any yet-unconsidered objections to urge against my proposal 'that the sequence and numeration of Euclid be kept unaltered'?

Carried, nemine contragemente! And now, Prisoner at the Bar (I beg your pardon, I should say 'Professor on the Sofa'), have you, and your attendant phantoms, any other reasons to urge for regarding this Manual as in any sense a substitute for Euclid's—as in any sense anything else than a revised edition of Euclid?

Nie. We have nothing more to say.

Min. Then I can but repeat with regard to this newborn 'follower' of the Syllabus, what I said of the