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. 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 a gk, or e hc, which are at the opposite angles of the parallelograms which make the gnomon. a h e f k b g c PROPOSITION I.—THEOREM. 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 be 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 b de, and that contained by a, e c. d e c k 1 a From the point b draw (i. 11) bf at right angles to bc, and make bg equal (i. 3) to a; and through g draw (i. 31) gh parallel to bc; and through d, e, c, draw (i. 31) dk, el ch parallel to bg; then the rectangle bh is equal to the rectangles bk, dl, eh; and bh is contained by a, bc, 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 d1 is contained by a, de, because dk, that is (i. 34) bg is equal to a; and in like manner the rectangle eh is contained by a, e c. Therefore the rectangle contained f 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.—THEOREM. 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 a b, bc, together with the rectangle* a b, bac, shall be equal to the square of a b a C с Upon ab describe (i. 46) the square a deb, and through c draw (i. 31) cf, parallel to a d or be. Then ae is equal to the rectangles af, ce; and a e is the square of a b; and af is the rectangle contained by ba, a c; for it is contained by d a, a c, of which ad is equal to a b; and ce is contained by ab, bc, for be is equal to a b; therefore the rectangle contained by a b, a c, together with the rectangle ab, bc, is equal to the square of a b. If therefore a straight line, &c. Q. E. D. d f е PROPOSITION III.-THEOREM. 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 is equal to the rectangle a c, cb, together with the square of bc. a с Upon bc describe (i. 46) the square cdeb, and produce ed to f, and through a draw (i. 31) af parallel to cd or be; then the rectangle ae is equal to the rectangles ad, ce; and ae is the rectangle contained by a b, bc, for it is contained by a b, be, of which be is equal to be; and 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 a c, cb, together with the square of bc. If therefore a straight line, &c. Q. E. D. f d e * To avoid repeating the word contained too frequently, the rectangle contained by two straight lines ab, a c is sometimes simply called the rectangle ab, a c. PROPOSITION IV.-THEOREM. 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 is equal to the squares of a c, cb, and to twice the rectangle contained by a c, cb. a с b g k f Upon ab describe (i. 46) the square a deb, and join bd, and through c draw (i. 31) cgf parallel to ad or be, and through g draw hk parallel to ab or de. And because cf is parallel to a d, and bd falls upon them, the exterior angle bgc is equal (i. 29) to the interior and opposite angle adb; but adb is equal (i. 5) to the angle a bd, because ba is equal to a d, being sides of a square; wherefore the angle cgb is equal to the angle h gbc; and therefore the side bc is equal (i. 6) to the side cg. But cb is equal (i. 34) also to gk, and cg to bk; wherefore the figure cgkb is equilateral. It is likewise rectangular; for cg is parallel to bk, and cb meets them; the angles d kbc, gcb are therefore equal to two right angles; and kbc is a right angle; wherefore gcb is a right angle: and therefore also the angles (i. 34) cgk, gkb, opposite to these, are right angles, and cgkb is rectangular; but it is also equilateral, as was demonstrated; wherefore it is a square, and it is upon the side cb. For the same reason hf also is a square, and it is upon the side hg, which is equal to a c. Therefore hf, ck are the squares of ac, cb; and because the complement ag is equal (i. 43) to the complement ge, and that a g is the rectangle contained by a c, cb, for gc is equal to cb; therefore ge is also equal 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 a deb, which is the square of a b. Therefore the square of a b is equal to the squares of a c, cb, and twice the rectangle ac, cb. Wherefore, if a straight line, &c. Q. E. D. COR. From the demonstration, it is manifest that the parallelograms about the diameter of a square are likewise squares. PROPOSITION V.-THEOREM. 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; the rectangle a d, db, together with the square of c d, is equal to the square of cb. с a 1 h k d b m Upon cb describe (i. 46) the square ce fb, join be, and through d draw (i. 31) dhg parallel to ce or bf; and through h draw klm parallel to cb or ef; and also through a draw a k parallel to cl or bm. And because the complement ch is equal (i. 43) to the complement hf, to each of these add dm; therefore the whole cm is equal to the whole df; but cm is equal (i. 36) to al, because a c is equal to cb; therefore also al is equal to df. To each of these add ch, and the whole ah is equal to d f and ch: but ah is the rectangle contained by a d, db, for dh is equal (ii. 4. cor.) to db; and df together with ch is the gnomon cmg; therefore the gnomon c m8 is equal to the rectangle a d, db: to each of these add 1g, which is equal (ii. 4. cor.) to the square of cd; therefore the gnomon cmg, together with 1 g, is equal to the rectangle ad, db, together with the square of cd; but the gnomon cmg and 1g make up the whole figure cefb, which is the square of cb: therefore the rectangle a d, db, together with the square of cd, is equal to the square of cb. Wherefore, if a straight line, &c. Q. E. D. g f COR. From this proposition it is manifest, that the difference of the squares of two unequal lines a c, cd, is equal to the rectangle contained by their sum and difference. PROPOSITION VI.—THEOREM. If a straight line be bisected, and produced to any point, the rectangle contained by the whole line thus produced, and the part of it produced, together with the square of half the line bisected, is equal to the square of the straight line which is made up of the half and the part produced. LET the straight line a b be bisected in c, and produced to the point d; the rectangle ad, db, together with the square of cb, is equal to the square of cd. Upon cd describe (i. 46) the square cefd, join de, and through b draw (i. 31) bhg parallel to ce or df, and through h draw klm parallel to a d or ef, and also through a draw ak parallel to clordm; k a 1 с h b a m and because ac is equal to cb, the rectangle al is equal (i. 36) to ch; but ch is equal (i. 43) to hf; therefore also al is equal to hf: to each of these add cm; therefore the whole am is equal to the gnomon cmg: and am is the rectangle contained by ad, db, for dm is equal (ii. 4. cor.) to db: therefore the gnomon cmg is equal to the rectangle a d, db: add to each of these lg, which is equal to the square of cb; therefore the rect e g f angle a d, db, together with the square of c b, is equal to the gnomon cmg, and the figure 18; but the gnomon cmg and 1 g make up the whole figure cefd, which is the square of cd; therefore the rectangle ad, db, together with the square of cb, is equal to the square of cd Wherefore, if a straight line, &c. Q. E. D. PROPOSITION VII.-THEOREM. If a straight line be divided into any two parts, the squares of the whole line, and of one of the parts, are equal to twice the rectangle contained by the whole and that part, together with the square of the other part. LET the straight line ab be divided into any two parts in the point c; the squares of ab, bc are equal to twice the rectangle a b, bc, together with the square of a c a b g k Upon ab describe (i. 46) the square ad eb, and construct the figure as in the preceding propositions; and because ag is equal (i. 43) to ge, add to each of them ck; the whole ak is therefore equal to the whole ce; therefore ak, ce, are double of a k: but ak, ce are the gnomon a kf, together with the square ck; therefore the gnomon a kf, together with the square ck, is double of a k: but twice the rectangle a b, bc is double of a k, for bk is h equal (ii. 4. cor.) to bc: therefore the gnomon akf, together with the square ck, is equal to twice the rectangle ab, bc to each of these equals add hf, which is equal to the square of a c; therefore the gnomon a kf, together with the squares ck, hf, is equal to twice the rectangle a b, bc, and the square of a c: but the gnomon a kf, together with the squares ck, hf, make up the whole figure a deb and ck, which are the squares of a b and bc therefore the squares of a b and bc are equal to twice the rectangle a b, bc, together with the square of a c. Wherefore, if a straight line, &c. Q. E. D. d f e PROPOSITION VIII.—THEOREM. If a straight line be divided into any two parts, four times the rectangle contained by the whole line, and one of the parts, together with the square of the other part, is equal to the square of the straight line which is made up of the whole and that part. LET the straight line ab be divided into any two parts in the point c; four times the rectangle a b, bc, together with the square of a c, is equal to the square of the straight line made up of ab and bc together. Produce ab to d, so that bd be equal to cb, and upon a d describe the square aefd; and construct two figures such as in the preceding. Because cb is equal to bd, and that cb is equal (i. 34) to g k, and bd to kn; therefore gk is equal to kn: for the same reason, pr is equal |