a 1. Book. V. Let the compounded magnitudes AB, BE, CD, DF, be proportional; that is, let AB be to BE as CD is to DF; these magnitudes shall be proportional when divided ; that is, AE shall be to EB as CF is to FD. For, take GH, HK, LM, MN, equimultiples of AE, EB, CF, FD, and KX NP, any equimultiples of EB, FD; now, be cause GH is the same multiple of AE that HK is of EB; and GH the same multiple of A E that LM is of CF; and LM the same multiple of CF that MN is of FD; therefore GK is the fame multiple of AB that GH is of AE a. But GH is the same multiple of AE that LM is of CF; therefore GK is the same multiple of AB that LM is of CF. But LM is the same multiple of CF, that MN is of FD; therefore LN is the same multiple of CD, that LM is of CF2; therefore GK is the Game multiple of AB, that LN is of CDb. But HK, MN, are the same multiples of EB, FD; and KX, NP any other equimultiples of EB, FD; wherefore HX is the same multiple of EB that MP is of FD d. But GK, LN are equimultiples of AB, CD; and XH, MP, any other equimultiples of EB, FD; if GK be equal to HÅ, LN will be equal to MP; take HK, MN, from both; then, if GH be equal to KX, LM will be equal ta NP; if greater, greater; and, if lefs, less; wherefore AE is to c Def. s. EB as CF is to FD. Wherefore, &c. bis. 2. PRO P. XVIII. THE O R. If magnitudes divided be proportional, they shall also be propor. tional when compounded. a 17. b Hyp. Let AE, EB, CF, FD, be the divided magnitudes, viz. as AE is to EB, so is CF to FD; they shall likewise be proportional when compounded, viz. as AB is to BE, fo is CD to DF; if not, let AB be to BE as CD is to tome magnitude, either greater or less than FD; first, to a less, as DG, viz. AB to BE, as CD to DG; therefore AE is to EB as CG is to GD&; but ÄE is to EB as CF is to FDb; therefore CG is to GD as CF is to FD; but CG is greater than CF; therefore DG is greater than FD d, but it is also less, which is impossible; therefore AB not to BE as CD to DG. In the same 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 DF. Wherefore, &c. CII. d 14 Book V, PRO P. XIX. THE O R. If the whole be to the whole, as a part taken from the one is to a part, taken from the other, then shall the residue of the one be to the residue of the other, as the whole is to the whole; and if four magnitudes be proportional, they sball be conversely proportional. b 17. as Let the whole AB be to the whole CD, as a part taken away AE, is to the part taken away CF; then the residue EB is to the residue FD as the whole AB is to the whole CD; for alternately as AB is to AE, so is CD to CF2; then BE is to AE as a 16. DF is to FC b; and BE is to DF, as AE to CF a; but as AE is to CF, so is AB to CDC; therefore EB is to the residue FD с Нур. . the whole AB is to the whole CDd: Again, if AB be to BE as d 11. CD to DF, then they shall be conversely proportional; for AE is to BE as CF is to FD?; and BE is to AE as DF is to CFF therefore as AB is to AE, fo is CD to DF 8; therefore the first g 18. AB is to AE, its excess above the fecond, as CD, the third, is to DF, its excess above the fourth h. Wherefore, &c. C17 ; f Cor. 4. h Def. 17. PRO P. XX. THE O R. TF there be three magnitudes, and others equal to them in num same ratio ; and if the first magnitude be equal to the third, then the fourth will be equal to the sixth; and, if the first be greater than the third, then the fourth will be greater than the sixth; and, if the first be less than the third, then the fourth will be less than the sixth. 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 same proportion, víz. 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 shall be equal to the sixth F; if greater, greater ; and, if less, less; for if A is equal to C, and B some other magnitude, A has the same proportion to B that Chath to Ba; but A is to B a 7. as D is to Eb; therefore D hath the same proportion to E that b Hyp. A has to B; but B is to C as E to F; and, inversely, C is to Bas F isto E; therefore F has to E the same proportion that C has Book V. has to B; but A has the same proportion to B that C has to B; therefore D has the same proportion to E that F has to E; therefore D is equal to F¢; if greater, greater; and, if less, less. Wherefore, &c. IF F there be three magnitudes, and others equal to them in num ber, which, taken two and two in each order, are in the same ratio ; and, if the proportion be perturbate ; if the first magnitude be greater than the third, then the fourth will be greater than the sixth ; and if the first be equal to the third, then the fourth will be equal to the fixth; if less, less. Let the three magnitudes A, B, C, and others D, E, F, equal to them in number, be taken two and two in the same ratio, and if their analogy be perturbate, viz. as A is to B, so is E to F, and B to C 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. For, if A is greater than C, A has a greater ratio to B than C has to B'; 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, so is E to D; therefore E has to F a greater ratio than E to D. But that magnitude to which the same has a greater ratio, is the leffer magnitude b; therefore F is less than D; that is, D is greater than F; if equal, equal ; and, if less, less. Wherefore, &c. bio. PRO P. XXII, TH E O R. If there be any number of magnitudes, and others equal to them in number, which, taken two and two, are in the same ratio; then they fall be in the same 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 same ratio, viz. A to B as D to E, and B to C as E to F; then they shall be in thesfame 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, because A is to B as D to E, G is to K as H is to L"; but B is Book v. to C as E is to F; therefore K is to M as L to N”; wherefore, if G is equal to M, H will be equal to N; if greater, greater; a 4. and, if less, less; but G, M are equimultiples of A, C and H, b 20. N of D, F; wherefore A is to C as D to Fc. Wherefore, &c. Del. s. PRO P. XXIII. T H E O R. [Ft there be three magnitudes, and others equal to them in number, which, taken two and two, are in the same ratio; and if their analogy be perturbate, they shall be in the same 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 same ratio ; and if their analogy be perturbate, that is, as A is to B, so is E to F, and as B is to C, so is D to E; then they shall be in the same proportion by equality ; that is, A is to C 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 H“; and as E to F, so is M to N; but A is to B as E is to F;a 15. therefore G is to H as M to Nb; and, because B is'to C as Dbar. to E, H is to K as L to M; therefore, if G is equal to K, L is equal to N; but G, K are equimultiples of A, C; and L, N c 21. of D, F; therefore A is to Cdas D to F. Wherefore, &c. d Def. si PRO P. XXIV. THE O R. If the first magnitude has the same proportion to the second that the third has to the fourth ; and if the fifth has the same proportion to the second that the sixth has to the fourth; then the first, compounded with the fifth, ball have the same proportion to the fecond, that the third, compounded with the fixth, has to the fourth. Let the first magnitude AB, have the same proportion to the second C, that the third DE has to the fourth F, and the fifth BG have the same proportion to the second C, that the fixth EH has to the fourth F. For, because BG is to C as EH is to F; inversely, C is to ; as GB is to C, fo is EH to FC; therefore AG is to b 13. C as DH is to Fa. Wherefore, &c. c Hyp. PROP Book V. PRO P. XXV. THE OR. IF four magnitudes be proportional, the greatest and least will be greater than the other two. a 19, Let four magnitudes AB, CD, E, F, be proportional, viz. AB to CD as E to F; of which let AB be the greatest, and F the leaft; then AB and F together, will be greater than CD and E; for, cut off AG equal to E, and CH to F; then AB is to CD as AG is to CH; therefore the remainder BG, will be to the remainder DH, as the whole AB is to the whole DC a; but AB is greater than CD; therefore GB is greater than HD; and, because AG is equal to E, and CH to F, then AG and F are equal to CH and E; but BG is greater than HD; therefore AB and F are greater than DC and E. Wherefore, &c. THE |