of E is not greater than the multiple of F (7. def. 5.); let such be taken, and of C, E let G, H be equimultiples, and K, L equimultiples of D, F, so that G be greater than K, but H not greater than L; and whatever multiple G is of C, take M the same multiple of A; and whatever multiple K is of D, take N the same multiple of B; then, because A is to B, as C to D, and of A and C, M and G are equimultiples: and of Band D, N and K are equimultiples; if M be greater than N, G is greater than K, and if equal, equal ; and if less, less (5. def. 5.): 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 ratio to B, than E has to F (7. def. 5.). 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. 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 magnitude, A has to B a greater ratio than Č to B (8. 5.); but as A is to B, so is C to A B А B С D D; therefore also C has to D a greater ratio than C has to B (13. 5.): but of two magnitudes, that to which the same has the greater ratio is the lesser (10.5.): 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 to B, as.C, that is A, to D; B therefore is equal to D (9. 5.). * See Note. 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. 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. Because AB is the same multiple of C, that DE is of F; there are as many magnitudes in AB equal to C, as A there are in DE equal to F: let AB be divided into magnitudes, each equal to C, viz. AG, GH, HB; and DE into magni D tudes, each equal to F, viz. DK, KL, LE: then the number of the first AG, GH, HB, G shall be equal to the number of the last DK, KL, LE: and because AG, GH, HB are all equal, and that DK, KL, LE are H also equal to one another; therefore AG is to DK, as GH to KL, and as HB to LE (7. 5.): and as one of the antecedents to its B C F consequent, so are all the antecedents together to all the consequents together (12. 5.); wherefore, as AG is to DK, so is 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: and 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 (15. 5.); therefore A is to B, as E is to F: but as A is to B so is C to D: wherefore as C is to D, so (11. 5.) E G is E to F: again, because G, H are equimultiples A C of C, D, as C is to D, so is G to H (15. 5.); but B D. as C is to D, so is E to F. Wherefore as E is F Hto F, so is G to H (11. 5.). 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 (14. 5.). Wherefore, if E be greater than G, F likewise 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. Therefore A is to C, as B to D (5. def. 5.). If then four magnitudes, &c. Q. E. D. PROP. XVII. THEOR. Ir 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 iwo 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 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 of 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, therefore GH is the same multiple (1. 5.) 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, that LM is of CF. Again, because LM is the same multiple of CF, that MN is of FD; therefore LM is the same multiple (1. 5.) of CF, that 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, X that NP is of FD; therefore HX is the P same multiple (2. 5.) of EB, that MP is of FD. And because AB is to BE, as CD is to DF, and that of AB and CD, K Ntiples; if GK be greater than HX, then H. LN is greater than MP; and if equal, M equal; and if less, less (5. def. 5.): but B D if GH be greater than KX, by adding the common part HK to both, GK is E 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 G * See Note. than KX, LM is greater than NP. In like manner it may be demonstrated, 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 (5. def. 5.), as AE is to EB, so is CF to FD. If then magnitudes, &c. Q. E. D. PROP. XVIII. THEOR. If magnitudes, taken separately, be proportionals, 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 BE, DF, take any whatever equimultiples KO, NP; and because KO, 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 of the same DF: and if KO be equal to KH, NP shall be equal to H NM.; and if less, less. First, Let KO not be greater than KH, therefore NP is not greater than M NM; and because GH, HK are equi P multiples of AB, BE, and that AB is greater than BE, therefore GH is greater (3. Ax. 5.) than KH: but N 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 B D KO be not greater than KH, then E GH, the multiple of AB, is always F greater than KO, the multiple of BE; and likewise LM, the multiple of CD, G A 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 re KI * Sec Note. mainder GK is the same multiple of P H. is of CD. In like manner, because M. K. N CF, that the whole LM is of the whole CD (5. 5.): but it was shown that LM is the same multiple of CD, B that GK is of AE; therefore GK is the same multiple of AE, that LN is D of CF; that is, GK, LN are equimul E F tiples of AE, CF: and because KO, NP are equimultiples of BE, DF, if G A С L 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 equimultiples of them (6. 5.). First, let HO, MP, be equal to BE, DF; and because AE is to EB, as CF to FD, and that GK, LN are equimultiples of AE, CF; GK shall be to EB, as LN to FD (Cor. 4. 5.): but HO is equal to EB, and MH to FD; wherefore GK is to HO as LN to MP. If therefore GK be greater than HO, LN is greater than MP; and if equal, equal; and if less (Ax. 5.), less. 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 0 equal, equal; and if less, less (5. def. 5.): which was likewise shown in the H preceding case. If therefore GH be greater than KO, taking KH from both, GK is greater than HO; where M N. B er than NP. In like manner it may be shown, that if GH be equal to KO, D E G A С L 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 (5. def. 5.), as AB is to BE, so is CD to DF. If then magnitudes, &c. Q. E. D. P PROP. XIX. THEOR. If a whole magnitude be to a whole, as a magnitude taken |