PROP. XVII. THEOR. If four magnitudes be proportional; the difference between the first and second, is to the second or first, as the difference between the third and fourth, is to the fourth or third, as the case may be. Part 1.-Let AB, CD, EF, GH be M On AB take AM equal to CD, and on EF, EN equal to GH (Post. 1. 5); and since any equisubmultiples whatever of CD, GH, or of their equals AM, EN are contained equally often in AB, EF (Hyp. and Def. 5. 5), they are contained equally often in the residues MB, NF (Cor. 6. 5), therefore MB, the excess of AB above CD, is to CD, as NF, the excess of EF above GH, is to GH (Def. 5. 5). Part 2. Let CD, AB, GH, EF be proportionals, the consequents AB, EF being greater than the antecedents CD, GH; a like construction being made, MB, the excess of AB above CD, is to AB, as NF, the excess of EF above GH, is to EF. Because CD is to AB, as GH to EF (Hyp.), by inverting, AB is to CB, as EF to GH (Theor. 3. 15. 5), therefore by part 1, MB is to CD, as NF to GH, and, by inverting again, CD to MB, as GH to NF (Theor. 3. 15. 5), therefore any equisubmultiples whatever of MB, NF are contained equally often in CD, GH (Def. 5. 5), or in their equals AM, EN, and therefore in AB, EF (Cor. 2. 5), therefore AB is to MB, as EF to NF (Def. 5. 5), and inverting, MB to AB, as NF to EF (Theor. 3. 15. 5). Part 3.-And since, by inverting, the second of four proportionals, has the same ratio to the first, as the fourth has to the third; by parts 1 and 2, the difference of the first and second is to the first, as the difference of the third and fourth is to the third. PROP. XVIII. THEOR. If four magnitudes be proportional; the compound of the first and second, is to the second or first, as the compound of the third and fourth, is to the fourth or third, as the case may be. Part 1.-Let AB be to CD, as EF is to GH; the compound of AB and CD is to CD, as the compound of EF and GH is to GH. B A M C -D F -N To AB let BM be joined equal to CD, E and to EF, FN equal to GH [Post. 1. G- -H 5]; and since any equisubmultiples whatever of CD, GH, or of their equals BM, FN, are contained equally often in AB, EF [Hyp. and Def. 5. 5], they are contained equally often in AM, EN [Cor. 2. 5], therefore AM, the compound of AB and CD, is to CD, as EN, the compound of EF and GH, is to GH [Def. 5. 5]. Part 2.-And since, by inverting, CD is to AB, as GH is to EF [Theor. 3. 15. 5], therefore, by part 1, the compound of AB and CD is to AB, as the compound of EF and GH is to EF. Schol. From this proposition, the preceding, and theor. 3. 15. 5, it follows, that, "if four magnitudes be proportional, "by converting, the first, is to the sum or difference of the first "and second, as the third, is to the sum or difference of the third "and fourth." PROP. XIX. THEOR. If the whole (AB), be to the whole (CD), as a part (AE) taken away, is to a part (CF) taken away; the residue (EB), is to the residue (FD), as the whole (AB), is to the whole (CD). Because AB is to CD, as AE is to CF [Hyp.] by alternating, AB is to AE, as CD to CF [16. 5], and, by Adividing, EB to AE, as FD to CF [17. 5], and, by again alternating, EB to FD, as AE to CF, or, [Hyp. and 11. 5], or, as AB to CD. C. C . PROP. XX. THEOR. (See Note). If there be three magnitudes (A, B, C), and other three (D, E, F), which, taken two and two in order, are in the same ratio; if the first (A), of the first magnitudes, be greater than the third (C), the first (D), of the last magnitudes, is greater than the third (F); if equal, equal; and if less, less. A B. C D E F First, let A be greater than C; D is greater than F. For, since A is greater than C, the ratio of A to B, or which is equal [Hyp.], of D to E, is greater than that of C to B [8. 5]; and, since B is to C as E to F [Hyp.], by inverting, C is to B, as F is to E [Theor. 3. 15. 5]; therefore the ratio of D to E, having been shewn to be greater than that of C to B, is also greater than that of F to E, (13. 5), and therefore D is greater than F[10. 5], In like manner it may be shewn, that, if A be equal to C, D is equal to F; and if less, less. PROP, XXI. THEOR. If there be three magnitudes (A, B, C), and other three (D, E, F), which, taken two and two in a perturbate order, are in the same ratio; if the first (A), of the first magnitudes, be greater than the third (C), the first (D), of the last magnitudes, is greater than the third (F); if equal, equal; and if less, less, 1 First, let A be greater than C; D is greater than F. Because A is greater than C [Hyp.], the ratio of B to C is greater than of B to A [8. 5]; whence, D being to E, as B to C [Hyp.], the ratio of D to E is greater than of B to A; but, because A A- C D E F is to B, as E to F [Hyp.], by inverting, B is to A, as F is to E [Theor. 3. 15. 5]; whence, the ratio of D to E, having been shewn to be greater than that of B to A, is also greater than that of F to E (13. 5), therefore D is greater than F (10. 5). In like manner it may be shewn, that, if A be equal to C, D is equal to F; and if less, less. PROP. XXII. THEOR. If there be any number of magnitudes, and as many others, which, taken two and two in order, are in the same ratio; by ordinate equality, the ratio of the first, of the first magnitudes, to the last, is the same, as the ratio of the first to the last, of the -H --K G-- R I- QO For, if the ratios of AB to EF and of GH to LM be not equal, let one of them if possible, as of A to EF be the greater, and let EN a submultiple of EF be contained oftener in AB, the LO a like submultiqle of LM is in GH (Hyp. and Def. 7. 5); let AP be the greatest multiple of EN, which is in AB, and since a like multiple of LO, as AP is of EN, is greater than GH, a like submultiple of GH, as EN is of AP, is less than LO (Cor. 2. Ax. B. 5); let LQ be taken O LO equal to that submultiple, and let QO be the excess of LO above LQ; let IR be a submultiple of IK less than QO [Cor. Theor. at 7. 5], and CS a like submultiple of CD; and since CD is to EF, as IK is to LM [Hyp.], by inverting, EF is to CD, as LM is to IK [Theor. 3. at 15. 5], and therefore CS, IR being equisubmultiples of CD, IK, are contained equally often in EF, LM (Def. 5. 5), and therefore in their equisubmultiples EN, LO [Cor. 3. 3. 5]; whence, IR being less than Q0, and therefore contained oftener in LO, than in LQ. the magnitude CS is contained oftener in EN, than IR in LO; therefore AP, GH being equimultiples of EN, LQ, CS is co tained oftener in AP, and therefore in AB, than IR is in GH Cor. 1. 3. 5); which is absurd [Hyp. and Def. 5. 5]; therefore the ratios of AB to EF, and of GH to LM, are neither of them greater than the other; they are therefore equal. Because A, B, C are three magnitudes, and E, F, G three others, which, taken two and two, are in the same ratio; by the preceding case, A is to C, as E is to G; whence, C being to D, as G is to H [Hyp.], by the same case, A is to D, as È is to H. In like manner, the proposition might be demonstrated, if there were ever so many magnitudes. 66 Cor. 1.-In this proposition and by Def. 7. 5, from supposing AB to have a greater ratio to EF, than GH has to LM, and EF to be to CD, as LM to IK; AB is proved to have a greater ratio to CD, than GH has to IK; whence it follows, that, 'if, of two ranks of magnitudes, of three each, the ratio of "the first to the second be greater in one, than the other, and "the ratios of the second to the third, be equal in both; the "ratio of the first to the third, is greater in the former rank, "than in the latter." Cor. 2. And the same thing being supposed, as in the preceding corollary, except that the ratio of EF to CD, instead of being equal to, be greater than, that of LM to IK; it might, in like manner, as in the demonstration of this 22d. proposition be shewn, that the ratio of AB to CD is greater than that of GH to IK; for CS would be at least as often contained in EF, as IR in LM (Hyp. and Cor. 4. 3. 5), and therefore, as often in EN, as IR in LO (Cor. 3. 3. 5); and therefore oftener than IR in LQ, and therefore oftener in AP or A, than IR in GH (Cor. 1. 3. 5), and so the ratio of AB to CD is greater than that of GH to IK [Def. 7. 5]; therefore, if, of two ranks of "magnitudes, of three each, the ratios of the first to the second, "and of the second to the third in one rank, be greater than the "corresponding ratios in the other, the ratio of the first to the "third in the former, is greater than that of the first to the third "in the other." Cor. 3.-Ratios, duplicate, triplicate, &c. of equal ratios, are equal: This being a case of this proposition. Cor. 4.-Ratios, duplicate, triplicate, &c. of unequal ratios, are unequal, those of the greater being the greater: this, in the case of duplicate ratios, being a case of the 2nd cor. above, and, in other cases, easily following from it. |