logous side to fg: the polygons a b cdefghkl may be divided into the same number of similar triangles, whereof each to each has the same ratio which the polygons have ; and the polygon a b c d e has to the polygon fghkl the duplicate ratio of that which the side a b has to the side fg. Join be, ec, gl, lh: and because the polygon a b c d e is similar to the polygon fghkl, the angle ba e is equal to the angle gfl (vi. def. i). and ba is to a e, as gf to fl (vi. def. 1): wherefore, because the triangles a be, fgl have an angle in one equal to an angle in the other, and their sides about these equal angles proportionals, the triangle a be is equiangular (vi. 6), and therefore similar, to the triangle fg! (vi. 4); wherefore the angle a be is equal to the angle fgl: and, because the polygons are similar, the whole angle a bc is equal (vi. def. 1) to the whole angle fgh; therefore the remaining angle ebc is equal to the remaining angle lgh: and because the triangles a be, fgl are similar, eb is to ba, as lg to gf (vi. def. 1); and also, because the polygons are similar, a b is to bc, as fg to gh (vi. def. 1); therefore, ex æquali (v. 22) eb is to bc, as lg to gh: that is, the sides about the equal angles e bc, lgh are proportionals; therefore (v. 22) the triangle e bc is equiangular to the triangle lg h, and similar to it (vi. 4). For the same reason, the triangle ecd likewise is similar to the triangle lhk: therefore the similar polygons abcde, fghkl are divided into the same number of similar triangles. Also these triangles have each to each the same ratio which the polygons have to one another, the antecedents being a be, ebc, ecd, and the consequents fgl, lgh, lhk: and the polygon a bcde has to the polygon fghkl the duplicate ratio of that which the side a b has to the homologous side fg: Because the triangle a be is similar to the triangle fgl, a be has to fgl, the duplicate ratio (vi . 19) of that which the side be has to the side gl: for the same reason, the triangle bec has to glh the duplicate ratio of that which be has to gl: therefore, as the triangle a be to the triangle fgl, so (v. 11) is the triangle bec to the triangle glh. Again, because the triangle e bc is similar to the triangle lg h, e bc has to lgh the duplicate ratio of that which the side ec has to the side lh: for the same reason, the triangle ecd has to the triangle lh k the duplicate ratio of that which ec has to lh: as therefore the triangle e bc to the triangle lgh, so is (v. 11) the triangle ecd to the triangle lhk: but it has been proved that the triangle ebc is likewise to the triangle 1g h, as the triangle a be to the triangle fgl. Therefore, as the triangle a be is to the triangle fgl, so is triangle e bc to triangle lgh, and triangle ecd to triangle lhk: and therefore, as one of the antecedents to one of the consequents, so are all the antecedents to all the consequents (v. 12). Wherefore, as the triangle a be to the triangle fgl, so is the polygon a m b g d с k b abcd e to the polygon fghkl: but the triangle a be has to the triangle fgl, the duplicate ratio of that which the side a b has to the homologous side fg. Therefore also the polygon a b c d e has to the polygon fghkl the duplicate ratio of that which a b has to the homologous side fg. Wherefore similar polygons, &c. Q. E. D. Cor. 1. In like manner, it may be proved, that similar four-sided figures, or of any number of sides, are one to another in the duplicate ratio of their homologous sides, and it has already been proved in triangles. Therefore, universally, similar rectilineal figures are to one another in the duplicate ratio of their homologous sides. Cor. 2. And if to a b, fg, two of the homologous sides, a third proportional m be taken, a b has (v. def. 10) to m the duplicate ratio of that which a b has to fg; but the four-sided figure or polygon upon a b has to the four-sided figure or polygon upon fg likewise the duplicate ratio of that which a b has to fg: therefore, as a b is to m, so is the figure upon a b to the figure upon fg, which was also proved in triangles (vi. 19, cor.). Therefore, universally, it is manifest that if three straight lines be proportionals, as the first is to the third, so is any rectilineal figure upon the first to a similar and similarly described rectilineal figure upon the second. PROPOSITION XXI.—THEOREM. Rectilineal figures which are similar to the same rectilineal figure are also similar to one another. LET each of the rectilineal figures a, b be similar to the rectilineal figure c: the figure a is similar to the figure b. Because a is similar to C, they are equiangular, and also have their sides about the equal angles proportionals (vi. def. 1). Again, because b is similar to c, they are equiangular, and bave their sides about the equal angles proportionals (vi. def. 1). Therefore the figures a, b are each of them equiangular to c, and have the sides about the equal angles of each of them and of c proportionals. Wherefore the rectilineal figures a and b are equiangular (i. ax. 1), and have their sides about the equal angles proportionals (v. 11). Therefore a is similar (vi. def. 1) to b. Q. E. D. PROPOSITION XXII.—THEOREM. If four straight lines be proportionals, the similar rectilineal figures similarly described upon them shall also be proportionals; and if the similar rectilineal figures similarly described upon four straight lines be proportionals, those straight lines shall be proportionals. LET the four straight lines a b c d e f g h be proportionals, viz. a b to cd, as ef to g h, and upon a b c d let the similar rectilineal figures k a b, lcd be similarly described ; and upon ef, gh the similar rectilineal figures mf, nh, in like manner. The rectilineal figure ka b is to lcd as m f to nh. To a b, cd take a third proportional (vi. 11) X; and to ef, gh a third proportional o: and because a b is to cd as ef to gh, and that cd is (v. 11) to x as g h to 0; wherefore, ex æquali (v. 22), as a b to x, so is efto o : but as a b to x, so is (vi. 20, cor. 2) the rectilineal k a b to the rectilineal lcd, and as ef to o, so is (vi. 20, cor. 2) the rectilineal mf to the rectilineal nh. Therefore, as k ab to lcd, so (v. 11) is mf to nh. And if the rectilineal k a b be to lcd as m f to nh; the straight line ab is to cd as ef to g h. Make (vi. 12) as a b to cd, so ef to pr, and upon pr describe (vi. 18) the rectilineal figure sr similar and similarly situated to either of the k 1 f 8 h p I figures mf, nh: then, because as a b to cd, so is ef to pr, and that upon a b c d are described the similar and similarly situated rectilineals kab, lcd, and upon ef, pr, in like manner, the similar rectilineals m f, sr; k a b is to lcd, as mf to sr; but, by the hypothesis, k a b is to lcd as mf to nh; and therefore the rectilineal m f having the same ratio to each of the two nh, sr, these are equal (v. 9) to one another : they are also similar and similarly situated; therefore g h is equal to pr. And because as a b to cd, so is ef to pr, and that pr is equal to gh; a b is to cd as ef to gh. If therefore four straight lines, &c. Q. E. D. PROPOSITION XXIII.—THEOREM. Equiangular parallelograms have to one another the ratio which is com pounded of the ratios of their sides. LET ac, cf be equiangular parallelograms, having the angle b c d equal to the angle ecg : the ratio of the parallelogram a c to the parallelogram cf is the same with the ratio which is compounded of the ratios of their sides. Let bc, cg be placed in a straight line; therefore dc and ce are also in a straight line (i. 14); and complete the parallelogram dg ; and taking any straight line k, make (vi. 12) as bc to cg, so k to 1 ; and as dc to ce, so make (vi. 12) 1 to m : therefore the ratios of k to 1, and l to m, are the same with the ratios of the sides. viz. of bc to cg, and dc to ce. But the ratio of k a d to m is that which is said to be compounded (v. def. A.) of the ratios of k to 1, and l to m: wherefore also k has to m the ratio compounded 8 b с of the ratios of the sides. And because as b c to cg, so is the parallelogram a c to the parallelogram ch (vi. 1); but as bc to cg, so is k to 1 ; therefore k is (v. 11) to l, as the parallelogram a c to the parallelogram ch. Again, because as dc to ce, so is the parallelogram ch to the parallelogram cf; but as dc to ce, so is 1 to m; kim f wherefore 1 is (v. 11) to m, as the parallelogram ch to the parallelogram cf: therefore since it has been proved, that as k to l, so is the parallelogram a c to the parallelogram ch; and as 1 to m, so the parallelogram ch to the parallelogram cf: ex æquali (v. 22), k is to m, as the parallelogram a c to the parallelogram cf: but k has to m the ratio which is compounded of the ratios of the sides ; therefore also the parallelogram a c has to the parallelogram cf the ratio which is compounded of the ratios of the sides. Wherefore, equiangular parallelograms, &c. Q. E. D. e PROPOSITION XXIV.—THEOREM. The parallelograms about the diameter of any parallelogram are similar to the whole and to one another, LET a b c d be a parallelogram, of which the diameter is a c; and eg, hk the parallelograms about the diameter. The parallelograms eg, hk are similar both to the whole parallelogram a b cd, and to one another. Because de, gf are parallels, the angle adc is equal (i. 29) to the angle a gf. For the same reason, because bc, ef are parallels, the angle abc is equal to the angle a ef: and each of the angles bc d, efg is equal to the opposite angle da b (i. 34), and therefore are equal to one another : wherefore the parallelograms a bcd, a efg, are equiangular : and because the angle a b c is equal to the angle a ef, and the angle bac a f common to the two triangles bac, eaf, they are equiangular to one another ; therefore (vi. 4) as a b to bc, so is e b a e to ef: and because the opposite sides of parallelograms are equal to one another (i. 34), g a b (v. 7) is to a d as a e to a g; and dc to h cb as gfto fe; and also c d to da as fg to ga: therefore the sides of the parallelograms åbcd, a efg about the equal angles are pro portionals ; and they are therefore similar to one d k another (vi. def. 1); for the same reason the с parallelogram a b c d is similar to the parallelogram fhck. Wherefore each of the parallelograms gekh is similar to db: but rectilineal figures which are similar to the same rectilineal figure are also similar to one another (vi. 21); therefore the parallelogram ge is similar to kh. Wherefore the parallelogram, &c. Q. E. D. PROPOSITION XXV. - PROBLEM. To describe a rectilineal figure which shall be similar to one and equal to another given rectilineal figure. LET a b c be the given rectilineal figure to which the figure to be described is required to be similar, and d that to which it must be equal. It is required to describe a rectilineal figure similar to a b c, and equal to d. Upon the straight line bc describe (i. 45, cor.) the parallelogram be equal to the figure a bc; also upon ce describe (i. 45, cor.) the parallelogram cm equal to d, and having the angle fce equal to the angle cbl: therefore b c and cfare in a straight line (i. 29, and i. 14), as also le and em: between b c and cf find (vi. 13) a mean proportional gh, and upon gh describe (vi. 18) the rectìlineal' figure k g h similar and similarly AA b h situated to the figure a bc: and because bc is to g h as gh to cf; and if three straight lines be proportionals, as the first is to the third, so is (vi. 20, cor. 2) the figure upon the first to the similar and similarly described figure upon the second ; therefore, as b c to cf, so is the rectilineal figure a bc to kgh: but as bc to cf, so is (vi. 1) the parallelogram be to the parallelogram ef: therefore as the rectilineal figure a bc is to kgh, so is the parallelogram be to the parallelogram ef (v. 11): and the rectilineal figure a b c is equal to the parallelogram be; therefore the rec |