The Geometry of the Three First Books of Euclid, by Direct Proof from Definitions Alone, by H Wedgwood

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1856. Excerpt: ... rectangles contained by the undivided line and the several parts of the divided line. Let A and B C (Fig. 38) be two straight lines whereof B C is divided into parts by the points D, E, etc. The rectangle contained by B C and A shall be equal to the sum of the rectangles contained by A and B D, D E, E C. Let B H be the rectangle contained by A and B C, of which the sides B G, C H are each equal to A; and through D and E let D K, EL be parallel to B G or CH. Then the rectangle B H is equal to the sum of the rectangles B K, DL, E H. But B K is the rectangle contained by B D and B G or A, and each of the lines D K, E L, etc., is equal to B G or A, and therefore D L is the rectangle contained by DE and A; EH the rectangle contained by E C and A, and so on. Wherefore the rectangle contained by B C and A is equal to the sum of the rectangles contained by A and each of the parts BD, DE, EC. Cor.l.--By making A equal to the whole line considered as divided into two parts, it appears that if a straight line be divided into any two parts, the square of the whole lir#is equal to the sum of the rectangles contained by the whole line and each of the parts. Cor. 2.--In like manner making A equal to one of the parts, it appears that if a straight line be divided into any two parts, the rectangle contained by the whole and one of the parts is equal to the rectangle contained by the two parts together with the square of the first mentioned part. XLIX.--Euclid II. 4. If a straight line be divided into any two parts the square of the whole line is equal to the squares of the two parts, together with twice the rectangle contained by the parts. Let the straight line A B (Fig. 39) be divided into any two parts in C. The square of A B is equal to the squares of A C, C B and twice.