Page:Carroll - Euclid and His Modern Rivals.djvu/241

. § 2.] clauses might be interpolated, though I think their value doubtful. The second is false. (See p. 193.)

Th. 32 (p. 55) is new. 'If there are three parallel straight Lines, and the intercepts made by them on any straight Line that cuts them are equal, then the intercepts on any other straight Line that cuts them are equal.' This is awkwardly worded (in fact, as it stands, its subject, as I pointed out in p. 193, is inconceivable), and does not seem at all worth stating as a Theorem.

At p. 57 I see an 'Exercise' (No. 5). 'Shew that the angles of an equiangular Triangle are equal to two-thirds of a right angle.' In this attempt I feel sure I should fail. In early life I was taught to believe them equal to two right angles—an antiquated prejudice, no doubt; but it is difficult to eradicate these childish instincts.

Problem 1 (p. 61) is Euc. I. 9: old proof. It provides no means of finding a radius 'greater than half AB,' which would seem to require the previous bisection of AB. Thus the proof involves the fallacy 'Petitio Principii.'

Pr. 2 (p. 62) is Euc. I. 11, proved nearly as in Euclid.

Pr. 3 (p. 62) is Euc. I. 12, proved nearly as in Euclid. It omits to say how a 'sufficient radius' can be secured, a point not neglected by Euclid.

Pr. 4 (p. 63) is Euc. I. 10, proved nearly as in Euclid. This also, like Pr. 1, involves the fallacy 'Petitio Principii.’

Pr. 5 (p. 64) is Euc. I. 32, proved nearly as in Euclid, but claims to use compasses to transfer distances, a Postulate which Euclid has (properly, I think) treated as a Problem. (See p. 212.)

Pr. 6, 7 (pp. 65, 66) are Euc. I. 23, 31: old proofs.