twice the rectangle under AD, AC; but the fquare of AB Book II. is equal to the fquares of BD, DA; therefore the fquare of BC is equal to the fquares of BA, AC, and twice the rectangle c 4. under DA, AC; therefore the fquare of BC is greater than the fquares of BA, AC, by twice the rectangle under DA, AC. Wherefore, &c. b 47. 1. PRO P. XIII. THE OR. INe Let B be an acute angle in the triangle ABC; from the angle A let fall the perpendicular AD, cutting BC in D; 2 12. 1. the fquare of AC is lefs than the fquares of AB, BČ, by twice the rectangle under CB, BD. b For the fquare of AC is equal to the fquares of AD, DC; b 47. 1. and the fquare of AB is equal to the fquares of AÐ, DB ь; but the fquares of BC, BD, are equal to twice the rectangle under BC, BD, together with the fquare of DC; therefore the c 7. fquares of AB, BC, are equal to the fquares of AD, DC, and twice the rectangle under CB, BD; but the fquare of AC is equal to the fquares of AD, DC; therefore the fquare of AC is lefs than the fquares of AB, BC, by twice the rectangle under CB, BD. Therefore, &c. PRO P. XIV. PRO B. O make a fquare equal to a given right lined figure. Make the rectangle BCDE equal to a given right lined figure A; If BE be equal to ED, then BCDE is a fquare; a 45. x. and what was required is done. If not, produce EE to F; make EF equal to ED, and bifect BF in ; with the center b 10. 1. G, and distance GB, defcribe a femicircle BHF; produce DE to H, and join GH. Book II. Then the rectangle under BE, EF, together with the fquare of GE, are equal to the fquare of GF, or GH d; but the fquare of GH is equal to the fquares of GE, EH ; d Def. 15 therefore the rectangle under BE, EF, together with _the fquare of GE, are equal to the fquares of HE, EG. Take. the fquare of GE from both, and the rectangle under BE, EF, that is, BD, is equal to the fquare of EH. Wherefore, &c, I. € 47. I. THE THE ELEMENT S OF EUCLID. воок III. DEFINITION S. Book III. EQUA A L circles are fuch whofe diameters are equal. II. A right line is faid to touch a circle, when drawn to the fame, and being produced, does not cut the circle. III. Circles are faid to touch each other, which, meeting, do not cut one another. IV. Right lines in a circle are faid to be equally diftant from the center, when perpendiculars drawn from the center to each of them are equal, and that line upon which the greatest perpendicular falls is the leaft line. Definition 19th, 1. V. VI. An angle of a fegment is the angle contained by the right line and circumference of the circle. VII. An angle is faid to be in a fegment, when right lines are drawn from fome point in the circumference to the ends of that line which is the base of the fegment, which lines contain the angle. VIII. But, |