6. The Likewise, because the ABC, FGH are similar (Arg. 2). ДАВС: AFGH=AC2: F H2.* And the AACD:AFHI =A C2: F H2.* 7. Therefore, the ABC:AFGH=AACD: AFHI. It may be demonstrated after the same manner, that 8. The AADE:AFIK = ∆ACD: FHI. K P.19. B.6. Ρ.ΙΙ. Β.Σ 9. Wherefore, ∆ABC: AFGH=AACD: AFHI=DADE: AFIK. P.11. B.5. 10. Therefore, comparing the sum of the anteced. to that of the confeq. △ABC+AACD, &c.: AFGH+AFHI,&c=△ABC: AFGH,&c. P.12. В.5. That is, the polyg. M: polyg. N=△ABC: AFGH= AACD: AFHI, &c. Which was to be demonstrated. 11, P. 7. Β.5. Since then the AABC: AFGH=AB2: F G2* (P.19. B. 6). 11. The polyg. M: polyg. N = A B2: F G2*. Ρ.11. Β.5. Which was to be demonstrated. 111. COROLLARY 1. As this Demonstration may be applied to quadrilateral figures, & the fame truth has already been proved in triangles (P.19), it is evident universally, that fimilar rectilineal figures are to one another in the duplicate ratio of their homologous fides. Wherefore, if to AB, FG two of the homologous fides a third proportional X be taken; because AB is to X in the duplicate ratio of AB: FG; & that a rectilineal figure M is to another fimilar rectilineal figure N, in the duplicate ratio of the Same fides AB: FG; it follows, that if three straight lines be proportionals, as the first is to the third, so is any rectilineal figure described upon the first to a fimilar & fimilarly described rectilineal figure upon the second. (P.11.B.5). ALL Squares being fimilar figures (D. 30. B. 1. & D. 1. B. 6), fimilar retilineal figures M&N, are to one another as the squares of their homologous fides AB, CD (expressed thus A B2: CD2.) for those figures are in the duplicate ratio of these fame fides. 1 ! AAA R PROPOSITION ΧΧΙ. ΤHEOREM XV. ECTILINEAL figures (A, C) which are fimilar to the same rectilineal figure (B), are also similar to one another. (Hyp.). A & C is similar to the figure B 1. Each of those figures will be also equiangular to the figure B, & will have the fides about the equal, proportional to the fides of the figure B. 2. Consequently, those figures A & C will be also equiangular to one another, and their fides about the equal V, will be proportional. 3. Confequently, the figures A & C are fimilar. Which was to be demonstrated. D. 1. B.6. (Ax.1. Β.Ι. Ρ. 11. Β.5. D. 1. B.6. { I F four straight lines (AB, CD, EF, GH) be proportionals, the similar rectilineal figures & fimilarly described upon them (M, N, & P, Q) fhall also be proportionals. And if the similar rectilineal figures (M, N, & P, Q) similarly described upon four straight lines be proportionals, those straight lines shall be proportional. { P.20. B.6. Cor. 1. P.11. B.5 But the figures M,N, & P,Q being fimilar & fimilarly described upon the straight lines AB, CD, & EF, GH (Hyp. 2). 2. AB:Z-M:N 1. To AB, CD, EF take a IVth proportional KL. rectil. figures P or Q, & fimilarly situated. DEMONSTRATION. BECAUSE fimilar each to each (Hyp. 2. & Prep. 2). P.12. B.6. P18. B.6. M:NP: R (Ift. part of this proposition.) 3. Wherefore, Ρ.ιι. Β.ς. 2. Consequently, P: R = P: Q R=Q Moreover, those figures being fimilar & fimilarly described upon the ftraight lines G H, KL (Prep. 2). Q:R of GH: ☐ of K L. 4. The of GH is = to the ☐ of K L. 5. Consequently, GH = KL. 6. Since then AB: CD=EF: KL (Prep.1), & GH=KL(Arg.5). AB:CDEFGH. P. 7. Β.5. Which was to be demonstrated. Ff PROPOSITION XXIII. THEOREM XVII. EQUIANGULAR parallelograms (M & N) have to one another the ratio which is compounded of the ratios of their fides (AC, CD & E C, CG) about the equal angles. Hypothefis. The pgrs. M&N are equiangular, So that ACD=VECG. Thesis. Pgr. M: Pgr. N= AC. CD: EC. CG. Preparation. 1. Place AC & CG in the same straight line AG; therefore EC & C D are alfo in a straight line E D. DEMONSTRATION. ECAUSE the pgrs. M, P, N form a feries of three magnitudes M: MPN : N.P. 2. Complete the pgr. P. BECAUSE 1. 2. And alternando M:N the ratios M: P&P: N. M.P: N.P. 3. Consequently the ratio of the first M to the last N,is compounded of 4. The ratio of the fides AC: CG is the fame as that of the pgrs. M: P; & the ratio of the fides DC: CE, the fame as that of the pgrs. P: N: Since then the ratio of M: Nis compounded of the ratios M: P, &P: N (Arg. 1). } 5. This fame ratio is compounded of their equals; the ratios AC:CG&CD: EC, of the fides about the equal 6. Consequently, M: N = AC.CD:EC.CG. ACD, ECG. Which was to be demonstrated. P. 1. B.6. D. 5. B.6. Cor. The fame truth is applicable to the triangles (ACD, ECG) having an angle (ACD) equal to an angle (ECG). for the diagonals (AD, EG) divide the pers. into two equal parts (P. 34. B. 1). |