Also these triangles have, each to each, the same ratio which the polygons have to one another, the antecedents being ABE, EBC, ECD, and the consequents FGL, LGH, LHK: and the polygon ABCDE has to the polygon FGHKL the duplicate ratio of that which the side AB has to the homologous side FG. Because the triangle ABE is similar to the triangle FGL ABE has to FGL the duplicate ratio (19. 6.) 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 ABE to the triangle FGL, sò (11. 5.) is the triangle BEC to the triangle GLH. Again, because the triangle EBC is similar to the triangle LGH, EBC 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 LHK, the duplicate ratio of that which EC has to LH: therefore, as the triangle EBC to the triangle LGH, so is (11. 5.) the triangle ECD to the triangle LHK : but it has been proved, that the triangle EBC is likewise to the triangle LGH, as the triangle ABE to the triangle FGL. Therefore, as the triangle ABE is to the triangle FGL, so is the triangle EBC to the triangle LGH, and the triangle ECD to the triangle LHK and therefore, as one of the antecedents to one of the consequents, so are all the antecedents to all the consequents (12. 5.). Wherefore, as the triangle ABE to the triangle FGL, so is the polygon ABCDE to the polygon FGHKL : but the triangle ABE has to the triangle FGL, the duplicate ratio of that which the side AB has to the homologous side FG. Therefore also the polygon ABCDE has to the polygon FGHKL the duplicate ratio of that which AB has to the homologous side FG. Wherefore similar polygons, &c. Q. E. D. COR. 1. In like manner it may be proved, that similar figures of four sides, or of any number of sides, are one to another in the duplicate ratio of their homologous sides; and the same has already been proved of triangles: therefore, universally similar rectilineal figures are to one another in the duplicate ratio of their homologous sides. COR. 2. And if to AB, FG, two of the homologous sides, a third proportional M be taken, AB has (def. 11. 5.) to M the duplicate ratio of that which AB has to FG: but the four sided figure, or poly. gon, upon AB has to the four-sided figure, or polygon, upon FG like wise the duplicate ratio of that which AB has to FG: therefore, as AB is to M, so is the figure upon AB to the figure upon FG, which was also proved in triangles (Cor. 19. 6.). Therefore, universally, it is manifest, that if three traight 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. COR. 3. Because all squares are similar figures, the ratio of any two squares to one another is the same with the duplicate ratio of their sides; and hence, also, any two similar rectilineal figures are to one another as the squares of their homologous sides. PROP. XXI. THEOR. 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 (def. 1. 6.). Again,because B is similar to C, they are equiangular, and have their sides about the equal angles proportionals (def. 1. 6.): therefore the figures A, B, ABA 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 (1. Ax. 1.), and have their sides about the equal angles proportionals (11.5.). Therefore A is similar (def. 1. 6.) to B. Q. E. D. PROP. XXII. THEOR. 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, AB, CD, EF; GH be proportionals viz. AB to CD, as EF to GH, and upon AB, CD let the similar rectilineal figures KAB, LCD be similarly described; and upon EF, GH the similar rectilineal figures MF, NH, in like manner: the rectilineal figure KAB is to LCD, as MF to NH. To AB, CD take a third proportional (11. 6.) X; and to EF, GH, a third proportional O and because : AB: CD:: EF: GH, and CD: X:: GH: (11. 5.) O, ex æquali (22. 5.) AB: X (2. Cor. 20. 6.): KAB : LCD: and EF: 0 (2. Cor. 20. 6.) MF: NH; therefore : KAB: LCD (2. Cor 20. 6.) :: MF : NH. And if the figure KAB be to the figure LCD, as the figure MF to the figure NH, AB is to CD, as EF to GH. Make (12. 6.) as AB to CD, so EF to PR, and upon PR describe (18. 6.) the rectilineal figure SR similar, and similarly situated to ei 合合口 M E N G R ther of the figures MF, NH: then, because that as AB to CD, so `is EF, to PR, and upon AB, CD are described the similar and similarly situated rectilineals KAB LCD, and upon EF, PR, in like manner, the similar rectilineals MF, SR; KAB is to LCD, as MF to SR; but by the hypothesis, KAB is to LCD, as MF to NH; and therefore the rectilineal MF having the same ratio to each of the two NH, SR, these two are equal (9. 5.) to one another: they are also similar, and similarly situated; therefore GH is equal to PR and because as AB to CD, so is EF to PR, and because PR is equal to GH, AB is to CD, as EF to GH. If therefore four straight lines, &c. Q E. D. PROP. XXIII. THEOR. 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 BCD equal to the angle ECG; the ratio of the parallelogram AC to the parallelogram CF is the same with the ratio which is compounded of the ratios of their sides, A D H B G C Let BC, CG be placed in a straight line; therefore DC and CE are also in a straight, line (14. 1.); complete the parallelogram DG; and, taking any straight line K, make (12. 6.) as BC to CG, so K to L; and as DC to CE, so make (12. 6.) L to M: therefore the ratios of K to L, and L to M, are the same with the ratios of the sides, viz. of BC to CG, and of DC to CE. But the ratio of K to M is that which is said to be compounded (def. 10. 5.) of the ratios of K to L, and L to M; wherefore also K has to M the ratio compounded of the ratios of the sides of the parallelograms. Now, because as BC to CG, so is the parallelogram AC to the parallelogram CH (1. 6.); and as BC to CG, so is K to L; therefore K is (11. 5.) to L, as the parallelogram AC to the parallelogram CH: again, because as DC to CE, so is the parallelogram CH to the parallelogram CF and as DC to CE, so is L to M; therefore L is (11. 5.) to M, as the parallelogram K L M CH to the parallelogram CF : there ང་ EF fore, since it has been proved, that as K to L, so is the parallelogram AC to the parallelogram CH; and as L to M, so the parallelogram CH to the parallelogram CF; ex æqualı (22. 5.), K is to M, as the parallelogram AC to the parallelogram CF; but K has to M the ratio which is compounded of the ratios of the sides; therefore also the parallelogram AC has to the parallelogram CF the ratio which is compounded of the ratios of the sides. Wherefore equiangular parallelograms, &c Q. E. D. PROP. XXIV. THEOR. The parallelograms about the diameter of any parallelogram, are similar to the whole, and to one another. Let ABCD be a parallelogram, of which the diameter is AC; and EG, HK the parallelograms about the diameter: the parallelograms EG, HK are similar, both to the whole parallogram ABCD, and to one another. Because DC, GF are parallels, the angle ADC is equal (29. 1.) to the angle AGF for the same reason, because BC, EF are parallels, the angle ABC is equal to the angle AEF; and each of the angles BCD, EFG is equal to the opposite angle DAB (34. 1.), and therefore are equal to one another, wherefore the parallelograms ABCD, AEFG equiangular. And because the angle ABC is equal to the angle are T AEF, and the angle BAC common to the A GA: therefore the sides of the parallelo D K grams ABCD, AEFG about the equal an C gles are proportionals; and they are therefore similar to one another (def. 1. 6.) for the same reason, the parallelogram ABCD is similar to the parallelogram FHCK. Wherefore each of the parallelograms, GE, KH is similar to DB: but rectilineal figures which are similar to the same rectilineal figure, are also similar to one another (21. 6.); therefore the parallelogram GE is similar to KH. Wherefore the parallelograms, &c. Q. E. D. PROP. XXV. PROB To describe a rectilineal figure which shall be similar to one, and equal to another given rectilineal figure. Let ABC 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 ABC, and equal to D. Upon the straight line BC describe (cor.45.1.) the parallelogram BE equal to the figure ABC; also upon CE describe (cor.45.1.) the parallelogram CM equal to D, and having the angle FCE equal to the angle CBL therefore BC and CF are in a straight line (29. i. 14. 1.), as also LE and EM; between BC and CF find (13. 6.) a mean proportional GH, and upon GH describe (18. 6.) the rectilineal figure KGH simi : lar, and similarly situated, to the figure ABC. And because BC is to GH as GH to CF, and if three straight lines be proportionals, as the first is to the third, so is (2. Cor. 20. 6.) the figure upon the first to the si |