Page:Carroll - Euclid and His Modern Rivals.djvu/237

. § 2.] it—that is, new proofs of Euclid's Propositions, and new Propositions.

Min. (with a weary sigh) Very well. It will perhaps be more satisfactory to do this, if only to ascertain exactly how much this new Manual contains that is really new and really worthy of adoption. But I shall limit my examination to the subject-matter of Euc. I, II.

Nie. That is all we ask.

Min. We begin, then, at p. 12.

Theorem 1. 'All right angles are equals.' This is provedby their being halves of a 'straight angle,' a phrase which I have already criticised. There is a rather important omission in the proof, no distinction being drawn between the 'straight angle' on one side of a Line, and the other (of course named by the same letters) which lies on the other side and completes the four right angles. This Theorem, if proved without 'straight angles,' might be worth adding to a new edition of Euclid.

Th. 2 (p. 13) is Euc. I. 13, proved as in Euclid.

Th. 3 (p. 14) is Euc. I. 14, where, unfortunately, a new proof is attempted, which involves a fallacy. It is deduced from an 'Observation' in p. 9, that 'a straight Line makes with its continuation at any point an angle of two right angles,' which deduction can be effected only by the process of converting a universal affirmative simpliciter instead of 'per accidens.'

Th. 4 (p. 14) is Euc. I. 15, proved as in Euclid.

At p. 17 I find a 'Question.' 'State the fact that "all geese have two legs" in the form of a Theorem.' This I would not mind attempting; but, when I read the additional