Book VI. R PROP. XXI. THEOR. ECTILINEAL figures which are fimilar to the fame Let each of the rectilineal figures A, B be fimilar to the rectilineal figure C: The figure A is fimilar to the figure B. Because A is fimilar to C, they are equiangular, and also have their fides about the equal angles proportionals. Again, a 1. def. 6 because 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. 1. equal angles proportionals. Therefore A is fimilar to B. c 11. 5. QE. D. IF four straight lines be proportionals, the fimilar rectilineal figures fimilarly defcribed upon them fhall also 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 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. To AB, CD take a third proportional a X; and to EF, GH a 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 ae-b 11. 5. quali, as AB to X, fo EF to O: But as AB to X, fo is the M 2 C 22. 5. rectilineal 20. 6. Book VI. 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, fob is MF to NH. d 2 Cor. 20. 6. b II. s. e 12. 6. f 18. 6. And if the rectilineal KAB be to LCD, as MF to NH; the ftraight line AB is to CD, as EF to GH. Make as AB to CD, fo EF to PR, and upon PR defcribe f the rectilineal figure SR fimilar and fimilarly fituated to either 89.5. See N. of the figures MF, NH: Then, because 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, thefe 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. 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 parallelogram AC to the parallelogram CF, is the fame with the ratio which is compounded of the ratios of their fides. Let A the DH d I. 6. G B C e II. 5. Let BC, CG be placed in a straight line; therefore DC and Book VI. CE are also in a straight line; and complete the parallelogram a 14. I. DG; and, taking any ftraight line K, make b as BC to CG, A 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 compounded of the A. def. 5. ratios of K to L, and L to M: Wherefore allo K has to M, ratio compounded of the ratios of the fides: And because as BC to CG, fo is the parallelogram AC to the parallelogram CHd; but as BC to CG, fo is K to L; therefore K is to L, as the pa. rallelogram AC to the parallelogram CH: Again, becaufe 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 ise 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 parrallelogram CF; ex aequalif, K to M, as the paral- f 22 5. lelogram AC to the parallelogram CF: But K has to M the ratio which is compounded of the ratios of the fides; therefore alfo 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 THE HE parallelograms about the diameter of any pa- See N. rallelogram, are fimilar 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 fimilar both to the whole paralle logram ABCD, and to one another. Becaufe DC, GF are parallels, the angle ADC is equal a to a 29. 1. the angle AGF: For the fame reafon, because BC, EF are pa M 3 rallels, b34. . c 4. 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 parallelograms 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 equiangu lar to one another; therefore as AB to BC, fo is AE to EF: And because the oppofite fides of parallelograms are equal to one another b, 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 proportionals; and they are therefore fimilar to d 7. 5. G D K B F H C e 1. def. 6. one another: For the fame reafon, 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 figure, are also similar to one another f; therefore the parallelogram GE is fimilar to KH. Wherefore the parallelograms, &c. Q. E. D. f 21. 6. See N. a Cor. 45. I. b То PROP. XXV. PROB. O defcribe a rectilineal figure which fhall be fimilar to one, and equal to another given rectilineal figure. Let ABC be the given rectilineal figure, to which the figure to be defcribed is required to be fimilar, and D that to which it must be equal. It is required to defcribe a rectilineal figuse fimilar to ABC, and equal to D. Upon the ftraight line BC defcribe the parallelogram BE equal to the figure ABC; alfo upon CE defcribe the parallelogram CM equal to D, and having the angle FCE equal to the angle CBL: Therefore BC and CF are in a straight 1. line b, as alfo LE and EM: Between BC and CF find a mean proportional GH, and upon GH defcribed the rectilineal figure KGH 6milar and familarly fituated 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, fo is the 529. 14. I. C 13 6. d 18. 6. € 2. Cor. 20.6. figure f I. 6. figure upon the first to the fimilar and fimilarly defcribed fi- Book VI. gure upon the fecond; therefore as BC to CF, fo is the recti. lineal figure ABC to KGH: But as BC to CF, fo is f the parallelogram BE to the parallelogram EF: Therefore as the rectilineal figure ABC is to KGH, fo is the parallelogram BE to the parallelogram EFS: And the rectilineal figure ABC is equal g II. 5. to the parallelogram BE; therefore the rectilineal figure KGH is equal to the parallelogram EF: But EF is equal to the figure D; wherefore alfo KGH is equal to D; and it is fimilar to ABC. Therefore the rectilineal figure KGH has been described fimilar to the figure ABC, and equal to D. Which was to be done. F two fimilar parallelograms have a common angle, and be fimilarly fituated; they are about the fame diameter, Let the parallelograms ABCD, AEFG be fimilar and fimilarly fituated, and have the angle DAB common. AEFG are about the fame diameter. For, if not, let, if poffible, the parallelogram BD have its diameter AHC in a different fraight line from AF the diameter of the parallelogram EG, and let GF meet AHC in H; and through H draw HK parallel to AD or BC: Therefore the parallelograms ABCD, AKHG being about the B fame diameter, they are fimilar ABCD and G D H E F K h 14. 5. to one another: Wherefore as DA to AB, fo is GA to AK: 3 24. 6. But because ABCD and AEFG are fimilar parallelograms, b I. def. |
