Book V. B: Then, because A is to B, as C to D, and of A and C, M and G are equimultiples: And of B and D, N and K are equimultiples; if M be greater than N, G is greater than b5 Def. 5. K; and if equal, equal; and if less, less b; but G is greater than K; therefore M is greater than N: But H is not greater than L; and M, H are equimultiples of A, E; and N, L equimultiples of B, F: Therefore A has a greater 7 Def. 5. ratio to B, than E has to F. Wherefore, if the first, &c. Q. E. D. COR. And if the first have a greater ratio to the second, than the third has to the fourth, but the third the same ratio to the fourth, which the fifth has to the sixth; it may be demonstrated, in like manner, that the first has a greater ratio to the second, than the fifth has to the sixth. PROP. XIV. THEOR. See N. IF the first has to the second, the same ratio which the third has to the fourth; then, if the first be greater than the third, the second shall be greater than the fourth; and if equal, equal; and if less, less. Let the first A have to the second B, the same ratio which the third C, has to the fourth D; if A be greater than C, B is greater than D. Because A is greater than C, and B is any other mag8. 5. nitude, A has to B a greater ratio than C to Ba: But, as A is to B, so is C to D; therefore also C has to D a greater A B C D A B C D A B C D 13. 5. ratio than C has to Bh. But of two magnitudes, that to 10. 5. which the same has the greater ratio is the lesser. Wherefore D is less than B; that is, B is greater than D. Secondly, if A be equal to C, B is equal to D: For A is 9. 5, to B, as C, that is, A to D: B therefore is equal to Da. Thirdly, if A be less than C, B shall be less than D: For C is greater than A, and because C is to D, as A is to B, D is greater than B, by the first case; wherefore B is less than D. Therefore, if the first, &c. Q. E. D. Book V. PROP. XV. THEOR. MAGNITUDES have the same ratio to one another which their equimultiples have. Let AB be the same multiple of C, that DE is of F; C is to F, as AB to DE. G D Because AB is the same multiple of C, that DE is of F; there are as many magnitudes in AB equal to C, as there are in DE equal to F: Let AB be divided into magnitudes, each equal to C, viz. AG, GH, HB; and DE into magnitudes, each equal to F, viz. DK, KL, LE: Then the number of the first AG, GH, HB, shall be equal to the H number of the last DK, KL, LE: And because AG, GH, HB are all equal, and that DK, KL, LE, are also equal to one KH another; therefore AG is to DK as GH to BCE F KL, and as HB to LEa: And as one of the antecedents to a7. 5. its consequent, so are all the antecedents together to all the consequents together b; wherefore, as AG is to DK, so is 12. 5. AB to DE: But AG is equal to C, and DK to F: Therefore, as C is to F, so is AB to DE. Therefore magnitudes, &c. Q. E. D. PROP. XVI. THEOR. IF four magnitudes of the same kind be proportionals, they shall also be proportionals when taken. alternately. Let the four magnitudes A, B, C, D, be proportionals, viz. as A to B, so C to D: They shall also be proportionals when taken alternately; that is, A is to C, as B to D. Take of A and B any equimultiples whatever E and F; and of C and D take any equimultiples whatever G and H: Book V. and because E is the same multiple of A, that F is of B, and that magnitudes have the same ratio to one another 15. 5. which their equimultiples have a; therefore A is to B, as E is to F; But as A is to B, so is C to D; Wherefore as с E A 11. 5. C is to D, sob is G C C, D, as C is to D, so is G to Ha; but as C is to D, so is E to F. Wherefore, as E is to F, so is G to Hb. But when four magnitudes are proportionals, if the first be greater than the third, the second shall be greater than the * 14. 5. fourth; and if equal, equal; if less, less. Wherefore, if E be greater than G, Flikewise is greater than H: and if equal, equal; if less, less: And E, F are any equimultiples whatever of A, B; and G, H any whatever of C, D. There5 Def. 5. fore A is to C as B to Dd. If then four magnitudes, &c. Q. E. D. PROP. XVII. THEOR. See N. IF magnitudes, taken jointly, be proportionals, they shall also be proportionals when taken separately; that is, if two magnitudes together have to one of them the same ratio which two others have to one of these, the remaining one of the first two shall have to the other the same ratio which the remaining one of the last two has to the other of these. Let AB, BE, CD, DF be the magnitudes taken jointly which are proportionals; that is, as AB to BE, so is CĎ to DF: they shall also be proportionals taken separately, viz. as AE to EB, so CF to FD. Take of AE, EB, CF, FD any equimultiples whatever GH, HK, LM, MN: and again, of EB, FD take any equimultiples whatever KX, NP: And because GH is the same multiple of AE, that HK is of EB, wherefore GH is the 1. 5. same multiple of AE, that GK is of AB: But GH is the same multiple of AE, that LM is of CF; wherefore GK is X b 2.3. the same multiple of AB, that LM is of CF. Again, be- Book V.. cause LM is the same multiple of CF, that MN is of FD; therefore LM is the same multiplea of CF, that 1. 5. LN is of CD: But LM was shown to be the same multiple of CF, that GK is of AB; GK therefore is the same multiple of AB, that LN is of CD; that is, GK, LN are equimultiples of AB, CD. Next, because HK is the same multiple of EB, that MN is of FD; and that KX is also the same multiple of EB, that NP is of FD; therefore HX is the same multiple of EB, that MP is of FD. And because AB is to BE, as CD is to DF, and that of AB and CD, GK and LN are equimultiples, and of EB and FD, HX and MP are equimultiples; if GK be greater than HX, then LN is greater than MP; and if equal, equal; and if less, less c; But if GH be greater than KX, by adding the common part HK to both, GK is greater than HX; wherefore also LN is greater than MP; and by taking away MN from both, LM is greater than NP: Therefore, if GH be K N 3 Def. 5. D B greater than KX, LM is greater than G A CL NP. In like manner it may be demonstrated, that if GH be equal to KX, LM likewise is equal to NP; and if less, less: And GH, LM are any equimultiples whatever of AE, CF, and KX, NP are any whatever of EB, FD. Therefore, as AE is to EB, so is CF to FD. If then magnitudes, &c. Q. E. D. If magnitudes, taken separately, be proportionals, See N. Let AE, EB, CF, FD be proportionals; that is, as AE to EB, so is CF to FD; they shall also be proportionals when taken jointly; that is, as AB to BE, so CD to DF. Take of AB, BE, CD, DF any equimultiples whatever GH, HK, LM, MN: and again, of BE, DF, take any whatever equimultiples KO, NP: And because KO, NP BOOK V. are equimultiples of BE, DF; and that KH, NM are equi- First, let KO not be greater than B D. E F A M P 3 Ax. 5. BE, therefore GH is greater a than HK; but KO is not greater than KH, wherefore GH is greater than KO. In like manner it may be shown, that LM is greater than NP. Therefore, if KO be not greater that KH, then GH, the multiple of AB, is always greater than KO, the multiple of CD; and likewise LM, the multiple of BE, greater than NP, the multiple of DF. Next, Let KO be greater than KH: therefore, as has been shown, NP is greater than NM: And because the whole GH is the same multiple of the whole AB, that HK is of BE, the remainder GK is the same multiple of the 5. 5. remainder AE that GH is of ABb: KH M B N Ꭰ E F which is the same that LM is of CD. In like manner, because LM H is the same multiple of CD, that MN is of DF, the remainder LN is the same multiple of the remainder CF, that the whole LM is of the whole CD: But it was shown that LM is the same multiple of CD, that GK is of AE; therefore GK is the same multiple of AE, that LN is of CF; that is, GK, LN are equimultiples of AE, G CF: And because KO, NP are equimultiples of BE, DF, if from KO, NP, there be taken KH, NM, which are likewise equimultiples of BE, DF, the remainders HO, MP are either equal to BE, DF, or 6. 5. equimultiples of them. First, let HO, MP be equal to BE, DF; and because AE is to EB, as CF to FD, and A C L |