Page:Carroll - Euclid and His Modern Rivals.djvu/239

. § 2.] Th. 17 (p. 30) is Euc. I. 26 (2nd part) proved by superposition instead of Euclid's method (which I prefer) of constructing a new Triangle.

Th. 18 (p. 32) is Euc. I. 17, with old proof.

Th. 19 (p. 33) is new. 'Of all the straight Lines that can he drawn from a given point to meet a given straight Line, the perpendicular is the shortest; and of the others, those making equal angles with the perpendicular are equal; and that which makes a greater angle with the perpendicular is greater than that which makes a less.' This I think deserves to be interpolated.

Th. 20 (p. 34) is new. 'If two Triangles have two sides of the one equal to two sides of the other, each to each, and the angles opposite to two equal sides equal, the angles opposite to the other two equal sides are either equal or supplementary, and in the former case the Triangles are equal in all respects.' I do not think it worth while to trouble a beginner with this rather obscure Theorem, which is of no practical use till he enters on Trigonometry.

Th. 21 (p. 43) is Euc. I. 27: old proof.

Th. 22 (p. 44) is Euc. I. 29 (1st part), proved by Euc. I. 27 and Playfair's Axiom (see p. 40).

Th. 23 (p. 45) is new. 'If a straight Line intersects two other straight Lines and makes either a pair of alternate angles equals or a pair of corresponding angles equal, or a pair of interior angles on the same side supplementary; then, in each case, the two pairs of alternate angles are equal, and the four pairs of corresponding angles are equal, and the two pairs of interior angles on the same side are supplementary.' This most formidable enunciation melts