THE ELEMENTS OF EUCLID. wwwww BOOK II. DEFINITIONS. I. Every right-angled parallelogram, or rectangle, is said to be contained by any two of the straight lines which contain one of the right angles. Thus the rectangle AC is said to be contained by the adjacent sides AB, BC; or by AD, DC; and will often be called for brevity "the rectangle AB⚫BC" or "the rectangle AD DC." Also, instead of saying the rectangle, whose adjacent sides are respectively equal to two detached straight lines, as to the lines A and BC (next page) we shall merely say the rectangle contained by A and BC, or, more simply still, the rectangle ABC, as in the demonstration of prop. i. II. A E In every parallelogram, any of the parallelograms about a diameter, together with the two complements, is called a Gnomon. Thus 'the parallelogram HG, together with 'the complements AF, FC, is the gnomon, which is more briefly expressed 'by the letters AGK, or EHC, which 6 are at the opposite angles of the 'parallelograms which make the gnomon.' H BG K PROP. 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. B DEC 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 shall be equal to the rectangle contained by A, BD, together with that contained by A, DE, and that contained by A, EC. G From the point B draw* *11. 1. BF at right angles to BC, and F make BG equal to A; and *3. 1. #31. 1. KLH through G draw* GH parallel to BC; and through *31. 1. D, E, C draw* DK, EL, CH parallel to BG. Then the rectangle BH is equal to the rectangles BK, DL, EH: but BH is contained by A, BC, +Const. for it is contained by GB, BC, and GB is equal to A; 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 *34. 1. is BG, is equal to A; and in like manner the rectangle EH is contained by A, EC: therefore the rectangle ABC is equal to the several rectangles A⚫BD, A.DE, and A.EC. Wherefore, if there be two straight lines, &c. Q. E. D. PROP. 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 AB BC, together with the rectangle AB AC, shall be equal to the square of AB. #46. 1. #31. 1. Upon AB describe ADEB, and through C parallel to AD or BE. A C B is equal to the rectangles AF, CE: but AE is the square of AB; and AF is the rect- D angle contained by BA, AC; for it is con FE tained by AD, AC, of which AD is equal† to AB; +26 Def. and CE is contained by AB, BC, for BE is equal to AB: therefore the rectangle AB AC, together with the rectangle AB BC is equal to the square of AB. If therefore a straight line, &c. 2. E. D. It may be remarked that this proposition is merely a corollary to the preceding, being that particular case of it in which the two straight lines are equal. It is obvious, too, that the restriction of the number of parts to two is unnecessary. PROP. III. THEOR. 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 aforesaid part. Let the straight line AB be divided into any two parts in the point C: the rectangle AB·BC shall be equal to the rectangle AC CB, together with the square of BC. by AB, BE, of which BE is equal to BC; and †26 Def. AD is contained by AC, CB, for CD is equal to CB; and DB is the square of BC: therefore the rectangle AB BC is equal to the rectangle AC CB, together with the square of BC. If therefore a straight line, &c. Q. E. D. PROP. IV. THEOR. 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 AB be divided into any two parts in C; the square of AB shall be equal to the squares of AC, CB, and to twice the rectangle AC⚫CB. #46. 1. *31. 1. + Const. #29. 1. #5. 1. 126 Def. +1 Ax. #6. 1. #34. 1. +1 Ax. Upon AB describe the square ADEB, draw BD, through C draw* CGF parallel to AD or BE, and through G draw HK parallel to AB or DE. Then because CF is parallelt to AD, and BD falls upon them, the exterior angle BGC is equal to the interior and opposite angle ADB; but ADB is equal to the angle ABD, because BA is equal to AD+, being sides of a square; wherefore the angle CGB is A equal to the angle CBG; and therefore the side BC is equal H to the side CG: but CB is equal" also to GK, and CG to BK ; wherefore the figure CGKB ist D equilateral it is likewise rect B GL K angular; for, since CG is parallel to BK, and CB 129. 1. 13 Ax. #34. 1. and 1 Ax. 134. I. #43. 1. meets them, the angles KBC, GCB are equal to 126 Def. two right angles: but KBC is at right angle; wherefore GCB is at right angle: and therefore also the angles*, CGK, GKB opposite to these, are right angles; therefore CGKB is rectangular: but it is also equilateral, as was demonstrated; +26 Def. Wherefore it is at square, and it is upon the side CB: for a similar reason HF also is a square, and it is upon the side HG, which is equal to AC: therefore HF, CK are the squares of AC, CB: and because the complement AG is equal* to the complement GE, and that AG is the rectangle con+26 Def. tained by AC, CB, for GC is equal to CB; there+1 Ax. fore GE is also equalf to the rectangle AC.CB; wherefore AG, GE are equal to twice the rectangle AC CB; and HF, CK are the squares of AC, CB; wherefore the four figures HF, CK, AG, GE are equal to the squares of AC, CB, and to twice the rectangle AC CB: but HF, CK, AG, GE make up the whole figure ADEB, which is the square of AB: therefore the square of AB is equal to the squares of AC, CB, and twice the rectangle AC CB. Wherefore if a straight line, &c. +1 Ax. Q. E. D. COR. From the demonstration, it is manifest, that parallelograms about the diameter of a square are likewise squares. It follows also that the square on a line is equal to four times the square on half the line. PROP. V THEOR. If a straight line be divided into two equal parts, and also into two unequal parts; the rectangle contained by the unequal parts, together with the square of the line between the points of section, is equal to the square of half the line. Let the straight line AB be divided into two equal parts in the point C, and into two unequal parts at the point D: G |