Page:Carroll - Euclid and His Modern Rivals.djvu/242

204 Problems 8 to 11 (pp. 66 to 69) are new. Their object is to construct Triangles with various data: viz. A, B, and c; A, B, and a; a, b, and C; a, b, and A. They are good exercises, I think, but hardly worth interpolating as Theorems. The first of them is remarkable as one of the instances where Mr. Wilson assumes Euc. Ax. 12, without giving, or even suggesting, any proof. If he intends to assume it as an Axiom, he makes Playfair's Axiom superfluous. No Manual ought to assume both of them.

Theorem 1 (p. 82) is Euc. I. 35, proved as in Euclid, but incompletely, as it only treats of one out of three possible cases.

Th. 2 (p. 83) is new. 'The area of a Triangle is half the area of a rectangle whose base and altitude are equal to those of the Triangle.' This is merely a particular case of Euc. I. 41, and may fairly be reserved till we enter on Trigonometry, where it first begins to have any practical value.

Th. 2, Cor. 1 (p. 84) is Euc. I. 37, 38: old proofs.

Th. 2, Cor. 2 (p. 84) is new. 'Equal Triangles on the same or equal bases have equal altitudes.' No proof is offered. It is an easy deduction, of questionable value.

Th. 2, Cor. 3 (p. 84) is Euc. I. 39, 40. No proof given.

Th. 3 (p. 84) is new. The area of a trapezium [by which Mr. Wilson means a quadrilateral that has only one pair of opposite sides parallel'] is equal to the area of a rectangle whose base is half the sum of the two parallel sides, and whose altitude is the perpendicular distance between them.' I have no hesitation in pronouncing this to be a mere 'fancy' Proposition, of no practical value whatever.

Th. 4 (p. 86) is Euc. I. 43: old proof.