BOOK V. PRO P. XV. THE OR. ARTS have the fame proportion as their like multiples, if Let AB be the fame multiple of C that DE is of F; then C will be to F as AB is to DE. a Ax. I. I. For, let AG, GH, HB be each equal to C; and DK, KL, LE, each equal to F; then AG, GH, HB are equal to one another; and likewife DK, KL, LE equal to one another; therefore AG is to DK as GH is to KL, and as HB is to LE; b 11. therefore AB is to DE as AG is to DK; that is, as C to c 12. F. Wherefore, &c. PROP. XVI. THE OR. F four magnitudes of the fame kind are proportional, they fhali alfo be alternately proportional. Let the four magnitudes A, B, C, D, be proportional, viz. as A is to B, fo is C to D; they will likewife be proportional when taken alternately; that is, as A is to C, fo is B to D; for, take E, F equimultiples of A, B ; and G, H any equimultiples of C and D'; then, because E is the fame multiple of A that F is of B, and G the fame multiple of C that H is of D, A is to B as E is to Fa; but A is to B, as C is to D; therefore C is to a 15. Das E is to Fb; and as C is to D, fo is G to H2; therefore ь 11. E is to F as G is to Hb; therefore, fince E, F, G, H are four magnitudes proportional, equimultiples of other four, A, B, C, D; therefore, if E is equal to G, Fis equal to H; if greater, greater; and, if lefs, lefs; wherefore A is to C as B is to Dd; c 14. Wherefore, &c. a d Def. 5. PROP. XVII. THEOR. F magnitudes compounded are proportional, they shall also be proportional when dividea. Let Book. V. a 1. b II. d 20 c Def. 5. Let the compounded magnitudes AB, BE, CD, DF, be proportional; that is, let A be to BE as CD is to DF; thefe magnitudes fhall be proportional when divided; that is, AE fhall be to EB as CF is to FD. a a For, take GH, HK, LM, MN, equimultiples of AE, EB, CF, FD, and KX NP, any equimultiples of LB, FD; now, because GH is the fame multiple of AE that HK is of EB; and GH the fame multiple of AE that LM is of CF; and LM the fame multiple of CF that MN is of FD; therefore GK is the fame multiple of AB that GH is of AE . But GH is the fame multiple of AE that LM is of CF; therefore GK is the fame multiple of AB that LM is of CF. But LM is the fame multiple of CF, that MN is of FD; therefore LN is the fame multiple of CD, that LM is of CFa; therefore GK is the fame multiple of AB, that LN is of CD. But HK, MN, are the fame multiples of EB, FD; and KX, NP any other equimultiples of EB, FD; wherefore HX is the fame multiple of EB that MP is of FD d. But GK, LN are equimuitiples of AB, CD; and XH, MP, any other equimultiples of EB, FD; if GK be equal to HX, LN will be equal to MP; take HK, MN, from both; then, if GH be equal to KX, LM will be equal to NP; if greater, greater; and, if lefs, lefs; wherefore AE is to EB as CF is to FD. Wherefore, &c. 2 17. b Hyp. C II, d 14. PRO P. XVIII. THE OR. TF magnitudes divided be proportional, they shall also be proportional when compounded. Let AE, EB, CF, FD, be the divided magnitudes, viz. as AE is to EB, fo is CF to FD; they fhall likewife be proportiorial when compounded, viz. as Ab is to BE, fo is CD to DF; if not, let AB be to BE as CD is to fome magnitude, either greater or lefs than F1; firit, to a lefs, as DG, viz. AB to BE, as CD to DC; therefore AE is to EB as CG is to GD2; but DO AE is to EB as CF is to FD; therefore CG is to GD) as CF is to FD; but CG is greater than CF; therefore DG is greater than Fd; but it is alfo lefs, which is impoffible; therefere AB is not to BE as CD to DG. In the fame manner it is proved, that AB is not to BE as CD to one greater than DF; therefore AB is to BE as CD is to Dr. Wherefore, &c. b PROP. BOOK V. PRO P. XIX. THE OR. IF a Let the whole AB be to the whole CD, as a part taken away AE, is to the part taken away CF; then the refidue EB is to the residue FD as the whole AB is to the whole CD; for alternately as AB is to AE, fo is CD to CF2; then BE is to AE as a 16. DF is to FC; and BE is to DF, as AE to CF2; but as AE is ს 17. to CF, fo is AB to CD; therefore EB is to the refidue FD as c Hyp. the whole AB is to the whole CDd: Again, if AB be to BE as d 11. CD to DF, then they fhall be conversely proportional; for AE is to BE as CF is to FD; and BE is to AE as DF is to CF; f Cor. 4. therefore as AB is to AE, fo is CD to DF8; therefore the first g 18. AB is to AE, its excefs above the fecond, as CD, the third, is to DF, its excefs above the fourth". Wherefore, &c. e 17. h Def. 17. PRO P. XX. THE OR. F there be three magnitudes, and others equal to them in number, which being taken two and two in each order, are in the fame ratio; and if the firf magnitude be equal to the third, then the fourth will be equal to the fixth; and, if the first be greater than the third, then the fourth will be greater than the fixth; and, if the first be less than the third, then the fourth will be less than the fixth. Let A, B, C be three magnitudes, and D, E, F, others equal to them in number; which being two in two in each order, are in the fame proportion, viz. A to B as D to E, and B to C as E to F; and if the first A be equal to the third C, then the fourth D fhall be equal to the fixth F; if greater, greater; and, if less, lefs; for if A is equal to C, and B fome other magnitude, A has the fame proportion to B that C hath to B2; but A is to B a 7. as D is to Eb; therefore D hath the fame proportion to E that b Hyp, A has to B; but B is to C as E to F; and, inverfely, C is to Bas Fisto E; therefore F has to E the fame proportion that C hac Book V. has to B; but A has the fame proportion to B that C has to B; therefore D has the fame proportion to E that F has to E therefore D is equal to F; if greater, greater; and, if lefs, lefs. Wherefore, &c. C 9. 28. b 10. PRO P. XXI. THE OR. IF there be three magnitudes, and others equal to them in num- Let the three magnitudes A, B, C, and others D, E, F, equal to them in number, be taken two and two in the fame ratio, and if their analogy be perturbate, viz. as A is to B, fo is E to F, and B to as D to E; and if the first A be greater than the third C, then the fourth D will be greater than the fixth F; if equal, equal; and, if less, less. 2 For, if A is greater than C, A has a greater ratio to B than C has to B2; but A is to B as E is to F; therefore E has to F a greater ratio than C hath to B; and inversely as C is to B, fo is E to D; therefore E has to F a greater ratio than E to D, But that magnitude to which the fame has a greater ratio, is the leffer magnitude; therefore F is lefs than D; that is, D is greater than F; if equal, equal; and, if lefs, lefs. Wherefore, &c. IF PRO P. XXII. THE OR. there be any number of magnitudes, and others equal to them in number, which, taken two and two, are in the fame ratio; then they shall be in the fame proportion by equality. Let there be any number of magnitudes A, B, C, and others, D, E, F, equal to them in number, which, taken two and two in the fame ratio, viz. A to B as D to E, and B to C as E to F; then they fhall be in the fame proportion by equality; that is, A to C, as D to F. For, let G, H be equimultiples A, D, and K, L any equimultiples of B, E, and M, N any equimultiples of C, F; then, becaufe because A is to B as D to E, G is to K as H is to La; but B is Book V. to Cas E is to F; therefore K is to M as L to Na; wherefore, if G is equal to Mb, H will be equal to N; if greater, greater; a 4and, if lefs, lefs; but G, M are equimultiples cf A, C and H, b 20. N of D, F; wherefore A is to C as D to F. Wherefore, &c. e Def. 5. Έ PRO P. XXIII. THE OR. IF there be three magnitudes, and others equal to them in number, which, taken two and two, are in the fame ratio; and if their analogy be perturbate, they shall be in the fame proportion by equality. Let there be three magnitudes A, B, C, and others D, E, F, equal to them in number, which, taken two and two, are in the fame ratio; and if their analogy be perturbate, that is, as A is to B, fo is E to F, and as B is to C, fo is D to E; then they fhall be in the fame proportion by equality; that is, A is to Č as D to F. For, let G, H, L be equimultiples of A, B, D; and K, M, N, any equimultiples of C, E, F, then as A is to B, so is G to Ha; and as E to F, fo is M to N; but A is to B as E is to F; a 15. therefore G is to Has M to No; and, because B is to C as Db 11. to E, His to K as L to M; therefore, if G is equal to K, L is equal to Ne, but G, K are equimultiples of A, C; and L, Ne 1. of D, F; therefore A is to C as D to F. Wherefore, &c. C PRO P. XXIV. THEOR. IF the first magnitude has the fame proportion to the second that the third has to the fourth; and if the fifth has the fame proportion to the fecond that the fixth has to the fourth; then the first, compounded with the fifth, shall have the fame proportion to the fecond, that the third, compounded with the fixth, has to the fourth. Let the first magnitude AB, have the fame proportion to the fecond C, that the third DE has to the fourth F, and the fifth BG have the fame proportion to the fecond C, that the fixth EH has to the fourth F. II. For, because BG is to C as EH is to F; inversely, C is to BG, as F is to EH; but AB is to C as DE is to F; therefore AB is to BG, as DE is to EH; and AG is to GB as DH is to a :: HEb ; but as GB is to C, fo is EH to F; therefore AG is to C as DH is to Fa. Wherefore, &c. c My b PROP. |