Book VI. triangle LGH, as the triangle ABE to the triangle FGL. Therefore, as the triangle ABE is to the triangle FGL, fo is triangle EBC to triangle LGH, and triangle ECD to triangle LHK: and therefore, as one of the antecedents to one of the confequents, fo are all the antecedents to all the cong 12. 5. fequents g. Wherefore, as the triangle ABE to the triangle FGL, fo is the polygon ABCDE to the polygon FGHKL: but the triangle ABE has to the triangle FGL, the duplicate ratio of that which the fide AB has to the homologous fide FG. Therefore alfo the polygon ABCDE has to the polygon FGHKL the duplicate ratio of that which AB has to the homologous fide FG. Wherefore fimilar polygons, &c. Q.E. D. COR. 1. In like manner it may be proved, that fimilar four fided figures, or of any number of fides, are one to another in the duplicate ratio of their homologous fides, and it has already been proved in triangles. Therefore, univerfally fimilar rectilineal figures are to one another in the duplicate ratio of their homologous fides. COR. 2. And if to AB, FG, two of the homologous fides, h11.def. 5. a third proportional M be taken, AB has h to M the duplicate ratio of that which AB has to FG: but the four fided figure or polygon upon AB has to the four fided figure or polygon upon FG likewife the duplicate ratio of that which AB has to FG: therefore, as AB is to M, fo is the figure upon AB to the figure upon FG, which was also proved in trii Cor.19. 6. angles i. Therefore, univerfally, it is manifeft, that if three ftraight lines be proportionals, as the first is to the third; fo is any rectilineal figure upon the firft, to a fimilar and fimilarly defcribed rectilineal figure upon the fecond. COR. 3.. Because all fquares are fimilar figures, the ratio of any two fquares to one another is the fame with the duplicate ratio of their fides; and hence, alfo, any two fimilar rectilincal figures are to one another as the fquares of their honologous fides. PROP. Book VI. R PROP. XXI. THE OR. ECTILINEAL figures which are fimilar to the fame rectilineal figure, are alfo fimilar to one another. Let each of the rectilineal figures A, B be fimilar to the rectilineal fignre C: The figure A is fimilar to the figure B. Because A is fimilar to C, they are equiangular, and alfo have their fides about the equal angles proportionals a. Again, because B is fimilar to C, they are equiangular, and have their fides about the equal angles proportionals a: therefore the figures A, B are each of them equiangular to C, and have the fides about A A B a 1. def. 6. the equal angles of each of them and of C proportionals. Wherefore the rectilineal figures A and B are equiangular b, b 1. Ax. 1. and have their fides about the equal angles proportionals c.c 11. 5. Therefore A is fimilar a to B. Q. E. D. PROP. XXII. THEOR. F four ftraight lines be proportionals, the fimilar upon them fhall alfo be proportionals; and if the fimilar rectilineal figures fimilarly defcribed upon four ftraight lines be proportionals, thofe ftraight lines fhall 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 N 3 let Book VI let the fimilar rectilineal figures KAB, LCD be fimilarly described; and upon EF, GH the fimilar rectilineal figures MF, NH in like manner: the rectilineal figure KAB is to LCD, as MF to NH. a 11. 6. c 22 5. To AB, CD take a third proportional a X; and to EF, GH a third proportional O: and because AB is to CD, as b 11. 5. EF to GH, and that CD is b to X, as GH to O; wherefore, ex æquali c, as AB to X, so EF to O: but as AB to X, fo d 2. cor. is d the rectilineal KAB to the rectilineal LCD, and as EF to O, fo is d the rectilineal MF to the rectilineal NH: therefore, as KAB to LCD, fo b is MF to NH. 20. 6. d 2. cor. 20. 6. e 12. 6. And if the rectilineal KAB be to LCD, as MF to NH; the straight line AB is to CD, as EF to GH. Make e as AB to CD, fo EF to PR, and upon PR describe f 18. 6. f the rectilineal figure SR fimilar and fimilarly fituated to g 9.5. either of the figures MF, NH: then, because as AB to CD, fo is EF to PR, and that upon AB, CD are described the fimilar and fimilarly fituated rectilineals KAB, LCD, and upon EF, PR, in like manner, the fimilar rectilineals MF, SR; KAB is to LCD, as MF to SR; but, by the hypothefis, KAB is to LCD, as MF to NH; and therefore the rectilineal MF having the fame ratio to each of the two NH, SR, thefe are equal g to one another: they are also fimilar, and fimilarly fituated; therefore GH is equal to PR and because as AB to CD, fo is EF to PR, and that PR is equal to GH; AB is to CD, as EF to GH. If therefore four ftraight lines, &c. Q. E. D. PROP. Book VI, E& PROP. XXIII. THEOR. QUIANGULAR parallelograms have to one another the ratio which is compounded of the ratios of their fides. Let AC, CF be equiangular parallelograms, having the angle BCD equal to the angle ECG: the ratio of the paral lelogram AC to the parallelogram CF, is the fame with the ratio which is compounded of the ratios of their fides. Let BC, CG be placed in a straight line; therefore DC and CE are also in a straight line a and complete the paral- a 14. 1. lelogram DG; and, taking any ftraight line K, make b as b 12.6. BC to CG, fo K to L; and as DC to CE, fo make b L to M: therefore the ratios of K to L, and L to M, are the fame with the ratios of the fides, viz. of BC to CG, and DC to CE. But the ratio of K to M is that which is faid to be A D H G В C d 1.6. e II. S. compounded c of the ratios of K to L, and L to M: where- c 1o. def. 5. fore alfo K has to M the ratio compounded of the ratios of the fides: and because as BC to CG, fo is the parallelogram AC to the parallelogram CH d; but as BC to CG, fo is K to L; therefore K ise to L, as the parallelogram AC to the parallelogram CH again, because as DC to CE, fo is the parallelogram CH to the parallelo gram CF; but as DC to CE, K L M fo is L to M; wherefore L is E F e to M, as the parallelogram CH to the parallogram CF: therefore, fince it has been proved, that as K to L, fo is the parallelogram AC to the parallelogram CH; and as L to M, fo the parallelogram CH to the parallelogram CF; ex æqualif, K is to M, as the parallelogram AC to the parallelogram f 22. 5 N 4 CF: 毙 Book VI. CF: but K has to M the ratio which is compounded of the ratios of the fides; therefore also the parallelogram AC has to the parallelogram CF the ratio which is compounded of the ratios of the fides. Wherefore equiangular parallelograms, &c. Q. E. D. PROP. XXIV. THEOR. THE parallelograms about the diameter of any parallelogram, are fimilar to the whole, and to one another. Let ABCD be a parallelogram, of which the diameter iş AC; and EG, HK the parallelograms about the diameter: The parallelograms EG, HK are fimilar both to the whole parallelogram ABCD, and to one another. .4 a 29. I. Becaufe DC, GF are parallels, the angle ADC is equal 2 to the angle AGF: for the fame reason, becaufe BC, EF are parallels, the angle ABC is equal to the angle AEF: and each of the angles BCD, EFG is equal to the oppofite angle b 34. I. DAB b, and therefore are equal to one another, wherefore the parallelograms ABCD, AEFG are equiangular and because the angle ABC is equal to the angle AEE, and the angle BAC common to the two triangles BAC, EAF, they are equiangular to one another; therefore € 4. 6. d 7.5. A as AB to BC, fo is AE to as GF to FE; and alfo CD to D K E B H the fides of the parallelograms ABCD, AEFG about the equal angles are proportionals; and they are therefore fimilar e 1. def 6. to one anothere: for the fame reason, the parallelogram ABCD is fimilar to the parallelogram FHCK. Wherefore each of the parallelograms, GE, KH is fimilar to DB: but rectilineal figures which are fimilar to the fame rectilineal fi gure, |