Corollary. 2. And if a third proportional x be found to A B, FG: [by 10. def. 5.) A B will be to x, in the duplicate ratio of that of A B to FG': But the ratio of one polygon to the other, and of one quadrilateral figure to the other, is the duplicate of the ratio that one homologous fide has to another; that is, of AB to FG: This is also demonstrated of triangles: therefore universally it is manifeft, that if three right lines be proportional, as the first is to the third, so is any right lined figure described upon the first, to a similar right lined figure alike described upon the second. Otherwise. We shall demonstrate otherwise more expeditiously that the triangles are homologous. For again let ABCDE, F G HKL be the similar polygons; and join BE, EC, GL, LH: I say as the triangle A BE is to the triangle FGL, so is the triangle EBC to he triangle LGH, and the triangle cde to the triangle H KL. For because the triangle ABE is fimilar to the triE angle F GL, the B G triangle A BE to the triange FGL [by 19.6.] will c DH K be in the dupli cate ratio of BE to GL. By the same reason the triangle bec to the triangle G Lh is in the duplicate ratio of be to Gl: then (by 11. s.] as the triangle ABE is to the triangle FGLA To is the triangle EBC to the triangle LGH. Again, because the triangle Ebc is similar to the triangle LGH; [by 19. 6.) the triangle EBC has a ratio to the triangle i GH, the duplicate of the ratio of the right line çe tq the right line hl. By the same reason also the triangle ECD has à ratio to the triangle ! H K the duplicate of that of ce to hl: Therefore [by 11. 5.) as the triangle EBC is to the triangle L GH, so is the triangle eco to the triangle LHK: But it has been proved that as the triangle EBC is to the triangle LGH, so is the triangle A Be to the triangle Fol: Therefore as the triangle Ale is to the triangle F G Lg fo is the triangle Bec to the tri4 angle angle 'GL H, and the triangle e cd to the triangle LHK; and therefore [by 12. 5.] as one of the antecedents is to one of the consequents, so are all the antecedents to all the consequents, and the reft, as in the former demonstration. Which was to be demonstrated. PROP. XXI. THEOR. Right lined figures which are similar to the same right lined figure, are similar to one another. Let the right lined figures A, B be each of them similar to the right lined figure c: I say the right lined figure A is also similar to the right lined figure B. For because the right lined figure a is similar to the right lined figure c; [by def. 1. 6.] it will be equiangular to it, and have the sides about the equal angles proportional. . Again, because с the right lined figure B is fimilar to the right lined figure c, it will be equian- A gular to it, and have the B lides about the equal angles proportional; therefore each of the figures A, B is equiangular to the right lined figure c; and has the fides about the equal angles proportional: Wherefore the right lined figure A (by 1. ax.) is equiangular to B, and [by 11. 5.] has the fides about the equal angles proportional; and accordingly [by 1. def. 6.] A is similar to B.' Which was to be demonstrated. ( A PRO P. XXII. THEO R. If four right lines be proportional, the right lined figures ihat are described upon them similar and alike ftuate will be also proportional : and if right lined figures described upon four right lines, Similar and alike fituate be proportional; those right lines will be proportional i. Let the four right lines A B, CD, E F, GH be propora tional, viz. as A B is to CD, fo is E F to GH; and upon A B, CD, let the right lined figures KAB, LCD similar and N S LCD. and alike situate be described; but upon e i, GH, let the right lined figures MF, NH fimilar and alike fituate, be defcribed : I say as the fight lined figure KAB is to the right lined figure 1 c D; To is the right lined figure MF to the right lined figure NH. For [by 11. 6.] find a third proportional * to À B, CD; and a third proportional o to É F, GH. And because A B is to eD; AS Ë F is K. X. to G H. And às CD L is to y lo is GÅ to ou it will be by equality [by 22. 5.) as À B iš to x, fo is E s to O. But 'M [bý 2. cor. 26. 6.] as À B iš to x, so is the right lined fi H gure K À to the But as EF P is to o, fo is the right lined figure MF, to the right lined figure nh: Therefore [by 11.5. as KAB is, to LCD, so is MF to NH. But now let the right lined figure K A B be to the right Tined figure LCD, as the right lined figure Me is to the right lined figure NH: 1 say as À B is to CD, so will E F be to GH. For [by 12. 6.) make as A B is to cd, so iš É i to PR. And [by 18. 6.] upon PR, describe the right lined figure sa fimilar and alike fituate to either of the right lined figures MF or NH. Then because A B is to cd, aš É F iš to PR, and upon A B, CD are described the right lined figures K'A B; LCD similar and alike situate. But upon EF, PR the right lined figures MF, SR, fimilar and alike situate. [by the first part of this] it is as the right lined figure K A E is to the right lined figure LCD, To is the right lined figure MF, to the right lined figure sri But [by suppofi, tion) as the right lined figure KAE, is to the right lined figure ic D, fo is the right lined figure M F to the right lined figure Nh: Therefore [by 11.5.] MF has the same ratio to each of the right lined figures NH, sr: Confe quently A quently [by 9. 5.) the right lined figures n'h, 'S R are equal : They are also similar and alike situate: Therefore [by the following lemma) GH is equal to PR. And because AB is to cd, as E F is to PR, and or is equal to G-H [by 7. 5.) A B will be to CD, as EF is to GH. If therefore four right lines be proportional, the right lined figures that are described upon them fimilar and alike situate, will be also proportional: and it right lined figures described upon four right lines similar and alike fituate, be : proportional; these right lines will be proportional. Which was to be demonstrated. L E M M A. But we shall thus demonstrate, that the homologous fides of fimilar and equal right lined figures, are equal. Let the right lined figures NH, SR be similar and equal : and let HG be to GN, as R P is to ps: I say. RP is equal to HG. For if they be unequal, one of them will be the greater. Let this be (RP. Then because Rp is to ps, as Hy is to Gn: and alternately [by 16. 5.) as Rp is to GH, fo is Ps to GN. But PR is greater than GH: therefore Ps will be greater than GN: Wherefore [by 20. 6.) the right lined figure r s is greater than the right lined figure HN; and also equal ; which cannot be: Therefore er is not unequal to GH: Wherefore it is equal to it. Which was to be demonstrated. i This proposition is always true; if all the four figures be similar and alike fituate. But when only two and two are so : it will not always be true, unless the two similar and alike fituate figures be both described upon those two right lines that are the antecedents and consequents of the equal ratios; as if the right line a be to B, as c is to D, and the figures G, H described upon the first and fourth right lines A and be similar and alike si. tuate, and both the figures 1, K H described G K upon the second and А B D) third right lines B, c be both fimilar and alike fituate, these four figures G, I, K, L will not be proportional, as is easily apprehended, from the bare contemplation of the figurcs. Should Should not the lemma to this proposition have been made a proposition, and set down before this proposition? PROP. XXIII. THEOR. Equiangular parallelograms are to one another in a ratio compounded of the ratios of their fides k. Let A c, CF be equiangular parallelograms, having the angle BCD equal to the angle eco: I say the parallelogram A c is to the parallelogram' c in the ratio compounded of the ratios of the sides : that is, of the ratio of BC to cg, and of the ratio of DC to CE. For put BC, cg in the same right line. Then [by 14. 1.) DC will be in the same right line with cɛ i and compleat the parallelogram DG, also let k be any right line, and [by' 22. 6.) make as BC to CG, fo is K to L, and as DC is to CE, to is L to M. Then the ratios of K to L, and I to M are the fame as the ratios of the fides, viz. that of BC to cg, and of DG to ce: But [by 5. def. 6.) the ratio of K to M is compounded of the ratio of K 10 L, and of the ratio of 1 to M. Wherefore the ratio of K to M is А the ratio compounded of the ratios of the fides. And because [by 1. 6.] as b c is to cg, so is the paralleloB gram AG to the parallelogram ch: BC to cG, so is k to 1:* Therefore [by 11. 5.) as K is to L, so is the parallelogram A c to the parallelogram c H. Again, because E F [by 1.6.] as Dc is to ce, so is the parallelogram ch, to the paralleloKIM gram cf; and as Dc is to c E, so is I to M: Therefore as L is to M, so [by 11. 5.) will the parallelogram ch be to the parallel gram CF. And so fince it has been proved that as K is to L, so is the parallelogram Ac to the parallelogram ch: But as I to M, fo is the parallelogram ch to the parallelogram cf: it will be, by equality [by 22. 5.) as K is to M, so is the parallelogram Ac to the parallelogram C F. But the ratio of Kto m is that compounded of the ratios of the sides: Therefore the ratio of the parallelogram AC to }! G But as |