Book VI. g 12.5. LHK, the duplicate ratio of that which EC has to LH: As therefore the triangle EBC to the triangle LGH, fo is f the triangle ECD to the triangle LHK: But it has been proved that the triangle EBC is likewife to the 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 confequents. 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, hro.def. 5. a third proportional M be taken, AB has 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 alfo proved in triangles. 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 described rectilineal figure upon the fecond. i Cor. 19.6. PROP, RECTILINEAL figures which are fimilar to the fame rectilineal figure, are also fimilar to one another. Book VI. Let each of the rectilineal figures A, B be fimilar to the rectilineal figure C: The figure A is fimilar to the figure B. Becaufe A is fimilar to C, they are equiangular, and also have their fides about the equal angles proportionals. Again, 1. def. 6. becaufe B is fimilar to C, they are equiangular, and have their fides about the equal angles proportionals: Therefore the figures A, B are each of them equi A B angular to C, and have the fides about the equal angles of each of them and of C proportionals. Wherefore the rectilineal fi gures A and B are equiangular, and have their fides about the b 1. Ax. I. equal angles proportionals. Therefore A is fimilar to B. c 11. 5. QE. D. PROP. XXI. THEOR. IF four straight lines be proportionals, the fimilar rectilineal figures fimilarly defcribed upon them shall alfo be proportionals; and if the fimilar rectilineal figures fimilarly defcribed upon four straight lines be proportionals, thofe ftraight lines fhall be proportionals. Let the four ftraight lines AB, CD, EF, GH be proportionals, viz. AB to CD, as EF to GH, and upon AB, CD let the fimilar rectilineal figures KAB, LCD be fimilarly defcribed; 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 To AB, CD take a third proportional X; and to EF, GH2 11. 6. a third proportional O: And becaufe AB is to CD, as EF to GH, and that CD is to X, as GH to O; wherefore, ex aequalis, as AB to X, fo EF to O: But as AB to X, fo is the M 2 b 11.5. C 22. S. d 2. Cor. 20. rectilineal 6. Book VI. w d 2. Cor. 20. 6. b 11.5. € 12. 6. 38.6. rectilineal KAB to the rectilineal LCD, and as EF to O, fo is the rectilineal MF to the rectilineal NH: Therefore as KAB to LCD, fob is MF to NH. And if the rectilineal KAB be to LCD, as MF to NH; the ftraight line AB is to CD, as EF to GH. upon PR defcribe f Make as AB to CD, fo EF to PR, and the rectilineal figure SR fimilar and fimilarly fituated to either 89.5. of the figures MF, NH: Then, becaufe as AB to CD, fo is EF to PR, and that upon AB, CD are defcribed 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, these are equal to one another: They are alfo 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. g See N. 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 parallelogram the parallelogram CF, is the fame with the ratio which is compounded of the ratios of their fides. AC to Let b b 12. 6. Let BC, CG be placed in a straight line; therefore DC and Book VI. CE are alfo in a straight line; and complete the parallelogram DG; and, taking any ftraight line K, make as BC to CG, a 14. I. fo K to L; and as DC to CE, fo make 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 compounded of the ra- c A. def. 5. tios of K to L, and L to M: Wherefore alfo K has to M, the A D H G d 1. 6. B C € 11. 5. ratio compounded of the ratios of the fides: And because as BC to CG, fo is the parallelogram AC to the parallelogram CH"; but as BC to CG, fo is K to L; therefore K is to L, as the parallelogram AC to the parallelogram CH: Again, because as DC to CE, fo is the parallelogram CH to the parallelogram CF; but as DC to CE, fo is L to M; wherefore L is to M, as the parallelogram CH to the parallelogram 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 aequali, K is to M, as the paralle- f 22. 5logram AC to the parallelogram CF: But K has to M the ratio which is compounded of the ratios of the fides; therefore alio 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. KLM E F HE parallelograms about the diameter of any pa- TH 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 fimilar both to the whole parallelogram ABCD, and to one another. a See N. Because DC, GF are parallels, the angle ADC is equal to a 29. I. the angle AGF: For the fame reafon, becaufe BC, EF are pa M 3 rallels, 1 b 34. I. C4.6. A E Book VI. rallels, the angle ABC is equal to the angle AEF: And each of the angles BCD, EFG is equal to the oppofite angle DAB, and therefore are equal to one another; wherefore the par allelograms ABCD, AEFG are equiangular: And because the angle ABC is equal to the angle AEF, and the angle BAC common to the two triangles BAC, EAF, they are equiangular to one another; therefore as AB to BC, fo is AE to EF: And becaufe the oppofite fides of parallelograms are equal to one another ", AB is to AD, as AE to AG; and DC to CB, as GF to FE; and alfo CD to DA, as FG to GA: Therefore the fides of the parallelograms ABCD, AEFG about the equal angles are proportion-D K and they are therefore fimilar to 4 7.5. I. def. 6. f 21. 6. d G B F H C als; See N. T gure. PROP. XXV. PRO B. O defcribe a rectilineal figure which fhall be fimilar to one, and equal to another given rectilineal fi Let ABC be the given rectilineal figure, to which the figure to be described is required to be fimilar, and D that to which it must be equal. It is required to defcribe a rectilineal figure fimilar to ABC and equal to D. Cor. 45.1. Upon the straight line BC defcribe the parallelogram BE equal to the figure ABC; also upon CE defcribe the paralle logram CM equal to D, and having the angle FCE equal to the angle CBL: Therefore BC and CF are in a straight 29. 1. line, as alfo LE and EM: Between BC and CF find a mean proportional GH, and upon GH defcribed the rectilineal fi gure KGH fimilar and fimilarly fituated to the figure ABC: And because BC is to GH as GH to CF, and if three ftraight 2. Cor. lines be proportionals, as the firft is to the third, fo is the [29 14. 1. C 13 6. d 18.6. e figure |