EH by A, EC; for DK, EL, CH, are each equal to BG, that Book II. is, equal to A; but the rectangle BH is equal to the rectangles BK, DL, EH; and BH is contained by A, BC; therefore the e 34. 1. rectangle by A, and BC, is equal to the rectangles by A, BD; A, DE, and A, EC. Wherefore, &c. IF a PRO P. II. THE OR. a right line be any how cut, the rectangles contained by the whole line, and each of the fegments, are equal to the fquare of the whole line. Let the right line AB be any how cut in C, the rectangles contained by AB, BC, and AB, AC, together, are equal to the fquare of AB. Upon AB defcribe the fquare ADEB ; thro' a 46. x. C draw CF parallel to ADb, or BE; then, becaufe AD is e- b 31. 1. qual to AB, the rectangle under AD, AC, is equal to the rectangle under AB, AC; and the rectangle under EB, BC, is equal to the rectangle under AB, BC; but the rectangle under AD, AC, that is, the rectangle AF, together with the rectangle under EB, BC, that is, CE, are equal to the fquare of AB; that is, the fquare AE. Wherefore, &c. PRO P. III. THE OR. [Fa right line be any how cut, the rectangle under the whole two parts, together with the fquare of the first mentioned part. Let the right line AB be any how cut in C, the rectangle under AB, BC, is equal to the rectangle under AC, CB, together with the fquare of BC. Upon BC defcribe the fquare BCDE; produce ED to F, and draw AF parallel to CD or a 46. 1. BE; the rectangle under AB, BC, that is, AE, is equal to the ᏴᎬ rectangles AC, CD; that is, the rectangle under AC, CB, and the fquare of CB. Wherefore, &c. PRO P. IV. THE OR. Fa right line be any how cut, the fquare of the whole line is equal to the fquares of the two parts, with twice the rectangle under these parts. Let BOOK II. a 5. 1. b 29 1. d 6. I. Let the right line AB be any how cut in C, the fquare of AB is equal to the fquares of AC, CB, and twice the rectangle under AC, CB. For, upon AB defcribe the fquare ADEB; through C draw CF parallel to AD, or BE; draw DB, cutting CF in G; through which point draw HGK parallel to AB, or DE. (Then the figure is faid to be conftructed.) Now, becaufe AB is equal to AD the angle ADB is equal to ABD; and the angle CGB to ADB; therefore the angle CBG is equal to CGB; thereAx..fore CG is equal to CB; therefore CGKB is equilateral; but the angles BCG, CBK, are equal to two right angles, and CBK is a right angle f; therefore BCG is likewife a right angle; therefore all the angles are right ones; and CGKB is a fquare . For the fame reafon HF is a fquare. But the rectangles AG, GE, are equal; and AG is the rectangle under AC, CB; for CG is equal to CB; but the fquares HF, CK, with the rectangles AG, GE, make up the fquare of AB: Therefore the fquare of AB is equal to the fquares of AC, CB, and twice the rectangle under AC, CB. Wherefore, &c. e 34 I. f 46. 1. g Def. 30, I. h 43. I. a 36. I. b 43. 1. c. Ax. 2, 1. d Ax. I. 1. COR. Hence every parallelogram about the diameter of a fquare is a fquare. PRO P. V. THE OR. IF a right line be cut into two equal parts, and into two unequal parts, the rectangle under the two unequal parts, together with the fquare of the intermediate part, are equal to the jquare of half the line. Let the right line AB be cut equally in C, and unequally in D, the rectangle under AD, DB, together with the fquare of CD, are equal to the fquare of CB. For, upon CB defcribe the fquare CEFB; conftruct the figure, and produce OL to K; and through A draw AK parallel to EC. The parallelograms AL, CO, are equal, and CH is equal to HF b; add DO to both; then CO is equal to DF; therefore AL is likewife equal to DFd; add CH to both; then the rectangle AH is equal to the gnomon LDF; add LG, that is, the Cor. 4. fquare of CD, to both; then the rectangle AH, that is, the rectangle under AD, DB, with LG, that is, the fquare of CD, are equal to the gnomon LDF, and LG; that is, equal the fquare of CB. Wherefore, &c. |