Book VI. which EC has to LH: As therefore the triangle EBC to the triangle LGH, so is f the triangle ECD to the triangle LHK: fll. 5. 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 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 8. 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 bomologous 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 four sided figures, 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, h 10.def. 5. a third proportional M be taken, AB has b to M the duplicate ratio of that which AB has to FG: But the four sided figure or polygon, upon AB, has to the four sided figure or polygon, upon FG, likewise the duplicate ratio of that which AB bas to FG: Therefore, as AB is to M, so is the figure upon AB i Cor.19.6. to the figure upon FG, which was also proved in triangles i: 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. Book VI. PROP. XXI. THEOR. Rectilineal figures, which are similar to the same rectilineal figure, are also similar to one another. C 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 X is similar to C, they are equiangular, and also have their sides about the equal angles proportionalsa. Again, a 1. def. 6. because B is similar to C, they are equiangular, and have their sides about the equal angles A B proportionals a: 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, and have their sides b 1. Ax. 1. about the equal angles proportionals. Therefore, A is simi- c 11. 5. lar a 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 - X; and to EF, a 11. 6. GH, a third proportional O: And because AB is to CD, as EF to GH, and that CD is to X, as GH to 0; wherefore, b 11. 5. ex æquali, as AB to X, so EF to 0: But as AB to X, so c 22. 5. 20. 6. b ll. 5. Book VI. is d the rectilineal KAB to the rectilineal LCD; and as EF to 0, so is d the rectilineal MF to the rectilineal NH: Thered 2. Cor. fore, as KAB to LCD, sob is MF to NH. And if the figure KAB be to the figure LCD, as the figure to GH. 'e 12. 6. Make as AB to CD, so EF to PR, and upon PR describe the rectilineal figure SR, similar and similarly situated K L e f 18. 6. F G H P R 8 9. 5. PROP. XXIII. THEOR. See N. Equiangular parallelograms have to one another the ratio which is compounded 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 ratio of their sides. Let BC, CG be placed in a straight line; therefore DC Book VI. and CE are also in a straight line a ; and complete the parallelogram DG; and taking any straight line K, make as a 14. 1. b 12. 6. BC to CG, so K to L; and as DC to CE, so make 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 DC to CE. But the ratio of K to M is that which is said to be compounded of the ratios of K to L, and L to M: Where- c A. def.5. fore also K has to M the ratio A 1) Н compounded of the ratios of the sides of the parallelograms: And because as BC to CG, so G is the parallelogram AC to the B C d l. 6. parallelogram CHd; but as BC to CG so is K to L; therefore K ise to L as the paral ell. 5. lelogram AC to the parallelogram CH: Again, because as DC to CE, so is the parallelogram CH to the parallelo KLM E F gram CF; but as DC to CE so is L to M; therefore L is e 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 AC to the parallelogram CH; and as L to M, so the parallelogram CH to the parallelogram CF; ex æquali', f 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 pa- See N. rallelogram are similar to the whole, and to one an. other. 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 parallelogram ABCD, and to one another. Because DC, GF are parallels, the angle ADC is equal a to a 29. 1. the angle AGF: For the same reason, because BC, EF are pa b 34. 1. Book VI. rallels, the angle ABC is equal to the angle AEF. And each of the angles BCD, EFG is equal to the opposite angle DAB, and therefore are equal to one another: wherefore, the parallelograms ABCD, A EFG, are equiangular: And because the angle ABC is equal to the angle A EF, and the angle BAC common to the two triangles BAC, EAF, they are equianguc 4. 6. lar to one another; therefore c as AB A to BC, so is A E to EF: And because B the opposite sides of parallelograms d 7. 5. are equal to one another b, AB is d to H AD, as AE to AG: and DC to CB, as GF to FE; and also CD to DA, as FG to GA: Therefore the sides of the parallelograms ABCD, AEFG about the equal angles are propor. D K C tionals; and they are therefore simie l. def. 6. lar to one another e; 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 the rectilineal figures which are similar to the same rectilineal f 21. 6. figure, are also similar to one another'; therefore the paral lelogram GE is similar to KH. Wherefore, “the parallelograms," &c. Q. E. D. PROP. XXV. PROB. . See N. 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. a Cor. 45. 1. Upon the straight line BC describe a the parallelogram BE, equal to the figure ABC; also upon CE describe * 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 by 214. 1. as also LE and EM: Between BC and CF find a mean pro€ 13. 6. portional GH, and upon GH described the rectilineal figure KGH similar, and similarly situated, to the figure ABC: And because BC is to GH as GH to CF, and if three straight e 2. Cor. lines be proportionals, as the first is to the third, so ise the S29. 1. b d 18. 6. 20. 6. |