but if there be any number of magnitudes equimultiples of as many, each of each, whatever multiple one of them is of its part, the same multiple is the whole of the whole (V. 1); therefore 2. G and G, H, K together, are equimultiples of A, and A, C, E together; for the same reason L, and L, M, N are equimultiples of B, and B, D, F; therefore 3. As A is to B, so are A, C, E together, to B, D, F together (V. Def. 5). Wherefore, if any number, &c. QE.D. PROPOSITION 13.-Theorem. If the first has to the second the same ratio which the third has to the fourth, but the third to the fourth, a greater ratio than the fifth has to the sixth; the first shall also have to the second a greater ratio than the fifth has to the sixth. Let A the first have the same ratio to B the second, which C the third has to D the fourth, but C the third a greater ratio to D the fourth, than E the fifth has to F the sixth. Then also the first A shall have to the second B, a greater ratio than the fifth E has to the sixth F. Construction. Because C has a greater ratio to D, than E to F, there are some equimultiples of C and E, and some of D and F, such that the multiple of C is greater than the multiple of D, but the multiple of E is not greater than the multiple of F (V. Def. 7); let these be taken; and let G, H be equimultiples of C, E, and K, L equimultiples of D, F, such that G may 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. Demonstration. Because A is to B, as C to D (hyp.), and of A and C, M and G are equimultiples; and of B and D, N and K are equimultiples; therefore 1. If M be greater than N, G is greater than K; and if equal, equal; and if less, less (V. Def. 5); but G is greater than K (constr.) ; therefore 2. M is greater than N; but H is not greater than L (constr.), and M, H are equimultiples of A, E; and N, L equimultiples of B, F; therefore 3. A has a greater ratio to B, than E has to F (V. Def. 7). 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. PROPOSITION 14.-Theorem. If the first has the same ratio to the second 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 the same ratio to the second B which the third C has to the fourth D. Then if A be greater than C, B shall be greater than D; and if A be equal to C, B shall be equal to D; and if A be less than C, B shall be less than D. First. Let A be greater than C (fig. 1); then B shall be greater than D. Demonstration. Because A is greater than C, and B is any other magnitude; 1. A has to B a greater ratio than C has to B (V. 8) ; but, as A is to B, so is C to D (hyp.); therefore also 2. C has to D a greater ratio than C has to B (V. 13) ; but of two magnitudes, that to which the same has the greater ratio, is the less (V. 10); therefore D is less than B; that is, Secondly. Let A be equal to C (fig. 2); then B shall be equal to D. For A is to B, as C, that is, A to D; therefore 1. B is equal to D (V. 9). Thirdly. Let A be less than C (fig. 3); then B shall be less than D. For C is greater than A; and because C is to D, as A is to B, therefore D is greater than B, by the first case; that is, 1. B is less than D. Therefore, if the first, &c. Q.E.D. PROPOSITION 15.-Theorem. 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. Construction. 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. Demonstration. Then the number of the first AG,GH, HB, is equal to the 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 another; therefore 1. AG is to DK, as GH to KL, and as HB to LE (V.7); but as one of the antecedents is to its consequent, so are all the antecedents together to all the consequents together (V.12); wherefore 2. As AG is to DK, so is AB to DE; but AG is equal to C, and DK to F; therefore 3. As C is to F, so is AB to DE. Therefore, magnitudes, &c. Q.E.D. PROPOSITION 16.-Theorem. If four magnitudes of the same kind be proportionals, they shall also be proportionals when taken alternately. Let A, B, C, D be four magnitudes of the same kind, which are proportionals, viz., as A to B, so C to D. Then they shall also be proportionals when taken alternately; that is, A shall be to C, as B to D. Construction. Take of A and B any equimultiples whatever E and F; and of C and D take any equimultiples whatever G and H. Demonstration. 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 (V. 15); therefore 1. A is to B, as E is to F; but as A is to B so is C to D (hyp.); wherefore 2. As C is to D, so is E to F (V. 11). Again, because G, H are equimultiples of C, D, therefore 3. As C is to D, so is G to H (V. 15) ; but it was proved that as C is to D, so is E to F; therefore 4. As E is to F, so is G to H (V. 11). But when four magnitudes are proportionals, if the first be greater than the third, the second is greater than the fourth; and if equal, equal; if less, less (V. 14); therefore 5. 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 (constr.); and G, H any whatever of C, D; therefore 6. A is to C, as B to D (V. Def. 5). If then four magnitudes, &c. Q.E.D. PROPOSITION 17.—Theorem. 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 let CD be to DF. Then they shall also be proportionals taken separately, viz., as AE to EB, so shall CF be to FD. Construction. Take of AE, EB, CF, FD any equimultiples whatever GH, HK, LM, MN; and again, of EB, FD take any equimultiples whatever KX, NP. Demonstration. Because GH is the same multiple of AE, that HK is of EB, therefore 1. GH is the same multiple of AE, that GK is of AB (V. 1) ; but GH is the same multiple of AE, that LM is of CF; therefore 2. 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 3. LM is the same multiple of CF, that LN is of CD (V. 1); but LM was shown to be the same multiple of CF, that GK is of AB; therefore GK is the same multiple of AB, that LN is of CD; that is, 4. 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 5. HX is the same multiple of EB, that MP is of FD (V. 2). And because AB is to BE, as CD is to DF (hyp.), and that of AB and CD, GK and LN are equimultiples, and of EB and FD, HX and MP are equimultiples; therefore |