Page:Carroll - Euclid and His Modern Rivals.djvu/243

. § 2.] Th. 5 (p. 87) is Euc. II. 1: old proof.

Th. 6, 7, 8 (p. 88, &c.) are Euc. II. 4, 7, 5. The sequence of Euc. II. 5, and its Corollary, is here inverted. Also the diagonals are omitted, and nearly every detail is left unproved, thus attaining a charming brevity—of appearance!

Th. 9 (p. 91) is Euc. I. 47: old proof.

Th. 10, 11 (pp. 94, 95) are Euc. 12, 13: old proof.

Th. 12 (p. 95) is new. 'The sum of the squares on two sides of a Triangle is double the sum of the squares on half the base and on the line joining the vertex to the middle point of the base.' This, Mr. Wilson tells us, is 'Apollonius' Theorem': but, even with that mighty name to recommend it, I cannot help thinking it rather more curious than useful.

Th. 13 (p. 96) is Euc. II. 9, 10. Proved algebraically, and thus degraded from the position of a (fairly useful) geometrical Theorem to a mere addition-sum, of no more value than millions of others like it.

In the next proposition we suddenly transfer our allegiance, for no obvious reason, from Arabic to Latin numerals.

Problem (p. 99) is Euc. I. 42: old proof.

Pr. . (p. 100) is Euc. I. 44: proved nearly as in Euclid, but labours under the same defect as Pr. 8 (p. 66) in that it assumes, without proof, Euc. Ax. 12.

Pr. (p. 100) is Euc. I. 45: old proof.

Pr. (p. 101) is Euc. II. 14: old proof.

Pr. (p. 103) is new. 'To construct a rectilineal Figure equal to a given rectilineal Figure and having the number