Right Line B C, and the Squares GB, HC, on B A, THE OR E M. be equal to the Squares described upon the other Angle. If the Square, described upon the Side B C of the Triangle A B C, be equal to the Squares described upon the other two sides of the Triangle B A, AC; I say the Angle BAC is a Right one. For, let there be drawn A D from the Point A, at Right Angles, co A C: Likewise make A D equal to B A, and join D C.. Then, because D A is equal to AB, the Square defcribed on D A will be equal to the Square described on AB.' And adding the common Square described on A C, the Squares described on DA, A C, are equal to the Squares described on BA, AC. But the Square described on D.C is * equal to the Squares described • 47 of ibis. on DA, A C; for DAC is a Right Angle: But the Square on B C is put equal to the Squares on B A, AC. Therefore the Square described on D C is equal to the Square described on B C; and so the Side CD is equal to the Side Ç B. And because D A is equal to A B, and A C is common, the two Sides DA, AC, are equal to the two Sides BA, AC; and the Base DC is equal to the Base CB. Therefore the Angle DAC is I equal to the Angle B A C; but D AC is 1 8 of ibis. a Right Angle; and so B A C will be a Right Angle also. If, therefore, a Square described upon one Side of a Triangle, be equal to the Squares, described upon the other two Sides of the said Triangle, then the Angle contained by these two other Sides, is a Right Angle; which was to be demonstrated. EUCLI D's E LE MEN T S. BOOK II. DEFINITION S. 1. EVERY Right-angled Parallelogram is said to be contained under two Right Lines, comprehending a Right Angle. II. In every Parallelogram, either of those Pa rallelograms, that are about the Diameter, together with the Complements, is called a Gno mon. PRO THEOREM. divided into any Number of Parts; the Reet- L ET A and B C be two Right Lines, whereof D, E. I say, the Rectangle contained under the Right Lines A and B C, is equal to the Rectangles contained under A and B D, A and D E, and A and E C. For, let * BF be drawn from the Point B, at * 11. 1. Right Angles, to BC; and make + B G equal to A; +3. and let IGH be drawn through G parallel to BC:1 31. 1. Likewise let I there be drawn D C, EL, CH, thro' D, E, C, parallel to B G. Then the Rectangle B H, is equal to the Rectangles BK, DL, and E H ; but the Rectangle B H is that contained under A and BC; for it is contained under GB, BC; and G B is equal to A; and the Rectangle BK is that contained under A and B D; for it is contained under G B and B D; and G B is equal to A; and the Rectangle D L is that contained under A and D Е, because D K, that is, BG, is equal to A : So likewise the Rectangle E H is that contained under A and E C. Therefore the Rectangle under A and BC is equal to the Rectangles under A and B D, A and D E, and A and E C. Therefore, if there be two Right Lines given, and one of them be divided into any Number of Parts, the Rectangle comprehended under the whole Line and the divided Line, fhall be equal to all the Rectangles contained under the whole Line, and the several Segments of the divided Line ; which was to be demonftrated. PRO PROPOSITION II. 46. So THEO R E M. gles contained under the wbole Line, and each Square of tbe whole Line. E T the Right Line A B be any how divided in the Point C. I say, the Rectangle contained under A B and BC, together with that contained under A B, and A C, is equal to the Square made on A B. For let the Square A D E B be described * on A B, and thro' C let C F be drawn parallel to A D or BE. Therefore A E is equal to the Rectangles A F and CE. But A E is a Square described upon A B ; and A F is the Rectangle contained under B A and A C; for it is contained under D A and A C, whereof A D is equal to A B; and the Rectangle CE is contained under A B and B C, since B E is equal to A B. Wherefore the Rectangle under A B and A C, together with the Rectangle under A B and BC, is equal to the Square of A B. Therefore, if a Right Line be any how divided, the Reclangles contained under the whole Line, and each of the Segments, or Parts, are equal to the Square of the whole Line ; which was to be demonstrated. PROPOSITION III. THEOREM. contained under the whole Line, and one of its the first-mentioned Part. LET the Right Line A B be any how cut in the Point C. I say, the Rectangle under A B and B C is equal to the Rectangle under AC and B C, together with the Square described on B C, For |