1 PROP. XV. .THEOR. Book V. 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 F, as AB to DE. Because AB is the same multiple of C, that DE is of F; there are as many magnitudes in AB equal o 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 number of A K D L F the last DK, KL,LE: and because AG, HGH, HB are all equal, and that DK, KL, LE are also equal to one another: therefore AG is to DK, as GH to KL, and as HB to LE: and as one of the antecedents to its consequent, so are all the antecedents together to all a 7. 5. the consequents together; wherefore, as AG is to DK, so BCE is AB to DE: but AG is equal to C, and DK to F: therefore, b. 12. 5. 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 probortionals when taken alternately; that is, A is to C, as B to D. Take of A and B any equimultiples wnatever E and E; and of C and D take any equimultiples whatever G and H: and Book V. because E is the same multiple of A, that F is of B, and that magnitudes have the same ratio to one another 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 a 15. 5. b 11. 5. 14. 5. D: wherefore as C E is to D, sob is E to F: again, because G, A G C D H 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 fourth; and if equal, equal; if less, less, Wherefore, if E be greater than G, F likewise is greater than Ḥ; and if equal, equal; if less, less; and E, F are any equiI multiples whatever of A, B ; and G, H any whatever of C, D. a 5. def. 5. Therefore A is to C, as B to Dd. If then four magnitudes, &c. Q. E. D. PROP. XVII. THEOR. IF magnitudes, taken jointly, be proportionals, See Note. 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. a 1. Let AB, BE, CD, DF be the magnitudes taken jointly which are proportionals; tbnt is, as AB to BF, so is CD 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 same multiplea of AE, that GK is of AB: but GH is the same multiple of AE, that LM is of CF; wherefore GK is the same multiple of AB, X P b 2. 5. that LM is of CF. Again, because LM is the same multiple Book V. of CF, that MN is of FD; therefore LM is the same multiple, rod of CF, that LN is of CD: but LM was shown to be the same a 1. 5. 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 equi multiples 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 b 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 equimulti-K ples, 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: 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 greater than KX, LM is greater than NP. In like manner it may be demonstrated, G A C L that if GH be equal to KX, LN 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. H B D Mc 5, def. 5.. PROP. XVIII. THEOR. IF magnitudes, taken separately, be proportionals, See Note. they shall also be proportionals when taken jointly, that is, if the first be to the second, as the third to the fourth, the first and second together shall be to the second, as the third and fourth together to the fourth. 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 BÉ, DF, take any whatever equi Book V. multiples KO, NP: and because KỌ, NP are equimultiples of BE, DF; and that KH, NM are equimultiples likewise of BE, DF, if KO, the multiple of BE, be greater than KH, which is a multiple of the same BE, NP, likewise the multiple of DF, shall be greater than NM, the multiple H of the same DF; and if KO be equal K First, Let KO not be greater than KH, therefore NP is not greater than NM; and because GH, HK are equimultiples of AB, BE, and that AB is greater than BE, therefore, GH is 23. Ax. 5. 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 than KH, then GH, the multiple of AB, is always greater than KÔ, the multiple of BE; and likewise LM, the multiple of CD, greater than NP, the G multiple of DF. b 5.5. c 6.5. : M 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 H K M the remainder AE that GH is of ABb: N B E D F A C that GK, LN are equimultiples of AE, CF; GK shall be to EB, as LN to FDd: but HO is equal to EB, and MP to FD; wherefore GK is to HO, as LN to MP. If therefore GK bed greater than HO, LN is greater than MP; and if equal, equal; and if less, less. H P But let HO, MP be equimultiples of EB, FD; and because AE is to EB, as CF to FD, and that of AE, CF are taken equimultiples GK, LN; and of EB, FD, the equimultiples HO, MP; if GK be greater than HO, LN is greater than MP; and if equal, equal; and if less, less : which was likewise shown in the preceding case. If therefore GH be greater than KO, taking KH from both, GK is greater than HO; wherefore also LN is greater than MP; and, consequently, adding NM to both, LM is greater than NP: therefore, if GH be greater than KO, K LM is greater than NP. In like manner it may be shown, that if GH be equal to KO, LM is equal to NP; and if less, less. And in the case in which G KO is not greater than KH, it has been shown that GH is always greater than KO, and likewise LM than NP: but GH, LM are any equimultiples of AB, CD, and KO, NP are any whatever of BE, DF; therefore, as AB is to BE, so is CD to DF. If then magnitudes, &c. Q. E. D. B D E A C L PROP. XIX. THEOR. Book V, ~ Cor. 4. 5. e Ax, 5. f 5. def. 5. IF a whole magnitude be to a whole, as a magni- See Note. tude taken from the first, is to a magnitude taken from the other; the remainder shall be to the remainder, as the whole to the whole. Let the whole AB be to the whole CD, as AE, a magnitude taken from AB, to CF, a magnitude taken from CD; the remainder EB shall be to the remainder FD, as the whole AB to the whole CD. Because AB is to CD, as AE to CF; likewise, alternately a, a 16. 5. |
