I 29. 1. || 2 Ax. § 29. 1. ¶ 1. Ax. * 14. 1. + Constr. 30. 1. § Constr. KM, FG, the alternate angles MHG, HGF‡ are equal: add to each of these the angle HGL; therefore the angles MHG, HGL are equal to the angles HGF, HGL: but the angles MHG, HGL are equal to two right_angles; wherefore also the angles HGF, HGL are equal¶ to two right angles, and FG is therefore in the same straight line* with GL; And because KF is parallel to HG,† and HG to ML; KF is parallel to ML; and KM, FL are parallels; whereDef. 34. 1. fore KFLM is || a parallelogram; and because the triangle ABD is equal to the parallelogram HFS, and the triangle DBC to the parallelogram GM; the whole rectilineal figure ABCD is equal to the whole parallelogram KFLM¶; therefore the parallelogram KFLM has been described equal to the given rectilineal figure ABCD, having the angle FKM equal to the given angle E. Which was to be done. ¶ 2 Ax. 44. 1. 11. 1. + 3. 1. 31. 1. COR. From this it is manifest, how to a given straight line to apply a parallelogram, which shall have an angle equal to a given rectilineal angle, and shall be equal to a given rectilineal figure, viz. by applying to the given straight line a parallelogram equal to the first triangle ABD, and having an angle equal to the given angle. PROPOSITION XLVI. * PROB. To describe a square upon a given straight line. Let AB be the given straight line; it is required to describe a square upon AB. From the point A draw* AC at right angles to AB; and maket AD equal to AB, and through the point D draw DE parallel to AB, and through B draw BE parallel to || Def. 34. 1. AD; therefore ADEB is a|| parallelogram: whence AB is equal to DE, and AD to BE: but BA is equal to AD; therefore the four straight lines BA, AD, DE, EB are equal to one another, and the parallelogram ADEB is equilateral: § 34. 1. 1 Ax. * 29. 1. 1 34. 1. 1 Ax. D E Likewise all its angles are right angles; because the straight line AD meeting the parallels AB, DE, the angles BAD, ADE are equal to two right angles: but BAD is af right angle: therefore also ADE is a right angle; but the opposite angles of parallelograms are equal; therefore each of the opposite angles ABE, BED is a|| right angle; A B wherefore the figure ADEB is rectangular: and it has been demonstrated that it is equilateral; it is therefore a§ square, § 30 Def. and it is described upon the given straight line AB. Which was to be done. COR. Hence every parallelogram that has one right angle has all its angles right angles. PROPOSITION XLVII. THEOR. In any right-angled triangle, the square which is described upon the side subtending the right angle, is equal to the squares described upon the sides which contain the right angle. Let ABC be a right-angled triangle having the right angle BAC; the square described upon the side BC is equal to the squares described upon BA, AC. On BC describe* the square BDEC, and on BA, AC the • 46. 1. squares GB, HC; and through A draw† AL parallel to BD straight lines with AG; B A G H + 31. 1. K + Hyp. 30 Def. L § 14. 1. For the same reason, AB and AH are in the same straight line; * And because the angle DBC is equal to the angle ¶ 11 Ax. FBA, each of them being a right angle, add to each the angle ABC, and the whole angle DBA is equal to the 2 Ax. whole FBC; and because the two sides AB, BD are equal† + 30 Def. to the two FB, BC, each to each, and the angle DBA equal to the angle FBC; therefore the base AD is equal‡ ‡ 4. 1. to the base FC, and the triangle ABD to the triangle FBC: Now the parallelogram BL is double|| of the triangle || 41. 1. ABD, because they are upon the same base BD, and between the same parallels BD, AL; and the square GB is double of the triangle FBC, because these also are upon the same base FB, and between the same parallels FB, GC. § 6 Ax. ¶ 2 Ax. 11. 1. 13. 1. I 2. Ax. 47. 1. Constr. ¶ 1 Ax. * Constr. † 8. 1. But the doubles of equals are equal§ to one another; therefore the parallelogram BL is equal to the square GB: And, in the same manner, by joining AE, BK, it is demonstrated, that the parallelogram CL is equal to the square HC; Therefore the whole square BDEC is equal¶ to the two squares GB, HC; and the square BDEC is described upon the straight line BC, and the squares GB, HC upon BA, AC; wherefore the square upon the side BC is equal to the squares upon the sides BA, AC. Therefore, in any right-angled triangle, &c. Q. E. D. PROPOSITION XLVIII. THEOR.-If the square described upon one of the sides of a triangle, be equal to the squares described upon the other two sides of it; the angle contained by these two sides is a right angle. If the square described upon BC, one of the sides of the triangle ABC, be equal to the squares upon the other sides BA, AC, the angle BAC is a right angle. D From the point A draw* AD at right angles to AC, and maket AD equal to BA, and join DC: then, because DA is equal to AB, the square of DA is equal to the square of AB: to each of these add the square of AC; therefore the squares of DA, AC are equal to the squares of BA, AC: A But the square of DC is equal to the squares of DA, AC, because DAC is a§ right angle; and the square of BC, by hypothesis, is equal to the squares of BA, AC; therefore the square of DC is equal to the square of BC; and therefore also the side DC is equal to the side BC. And because the side DA is equal* to AB, and AC common to the two triangles DAC, BAC, the two DA, AC are equal to the two BA, AC; and the base DC is equal to the base BC; therefore the angle DAC is equal† to the angle BAC: but DAC is a right angle; therefore also BAC is a right angle. Therefore, if the square, &c. Q. E. D. BOOK II. DEFINITIONS. I. EVERY right-angled parallelogram is said to be contained by any two of the straight lines which contain one of the right angles. II. A E In every parallelogram, any of the parallelograms about a diameter, together with the two complements, is called a Gnomon. 6 Thus the paral'lelogram HG, together with 'the complements AF, FC, is H 'the gnomon, which is more "briefly expressed by the letters B G K 'AGK, or EHC, which are at the opposite angles of the 'parallelograms which make the gnomon.' PROPOSITION I. THEOR.-If there be two straight lines, one of which is divided into any number of parts; the rectangle contained by the two straight lines is equal to the rectangles contained by the undivided line, and the several parts of the divided line. Let A and BC be two straight lines; and let BC be divided into any parts in the points D, E; the rectangle contained by the straight lines A, BC is equal to the rectangle contained by A, BD, together with that contained by A, DE, and that contained by A, EC. * 11. 1. + 3. 1. 31. 1. 31. 1. Constr. 34. 1. * 46. 1. + 31. 1. From the point B draw* BF at right angles to BC, and make BG equal† to A; and through G draw‡ GH parallel to BC; and G Then the rectangle BH is equal B DEC KLH And BK is contained by A, BD, for it is contained by GB, BD, of which GB is equal to A; And DL is contained by A, DE, because DK, that is ¶ BG, is equal to A; And in like manner the rectangle EH is contained by A, EC: Therefore the rectangle contained by A, BC, is equal to the several rectangles contained by A, BD, and by A, DE; and also by A, EC. Wherefore, if there be two straight lines, &c. Q. E. D. PROPOSITION II. THEOR. If a straight line be divided into any two parts, the rectangles contained by the whole and each of the parts, are together equal to the square of the whole line. Let the straight line AB be divided into any two parts in the point C; the rectangle contained by AB, BC, together with the rectanglea AB, AC, shall be equal to the square of AB. Upon AB describe* the square ADEB, and through C draw† CF, parallel to AD or BE: Then AE is equal to the rectangles AF, CE; And AE is the square of AB; And AF is the rectangle contained by BA, AC; for it is contained by DA, AC, D contained by AB, BC, for BE is equal to AB; C B Therefore the rectangle contained by AB, AC, together with the rectangle AB, BC, is equal to the square of AB. If therefore a straight line, &c. Q. E. D. a N.B. To avoid repeating the word contained too frequently, the rectangle contained by two straight lines AB, AC is sometimes simply called the rectangle AB, AC. |