11. Magnitudes are said to be proportionals when the first has the same ratio to the second that the third has to the fourth; and the third to the fourth the same ratio which the fifth has to the sixth; and so on, whatever be their number. When four magnitudes, A, B, C, D, are proportionals, it is usual to say that A is to B as C to D, and to write them thusA:B::C:D, or thus, A:B = C:D. 12. When four magnitudes are proportional, they constitute a proportion or an analogy. 13. The first and last terms of an analogy are called the extremes; the second and third, the means. 14. When of the equimultiples of four magnitudes, taken as in the tenth definition, the multiple of the first is greater than that of the second, but the multiple of the third is not greater than the multiple of the fourth ; then the first is said to have to the second a greater ratio than the third magnitude has to the fourth; and, on the contrary, the third is said to have to the fourth a less ratio than the first has to the second. 15. When there is any number of magnitudes greater than two, of which the first has to the second the same ratio that the second has to the third, and the second to the third the same ratio which the third has to the fourth, and so on, the magnitudes are said to be continual proportionals, or in continued proportion. 16. When three magnitudes are in continued proportion, the second is said to be a mean proportional between the other two. 17. When there is any number of magnitudes of the same kind, the first is said to have to the last of them the ratio compounded of the ratio which the first has to the second, and of the ratio which the second has to the third, and of the ratio which the third has to the fourth, and so on to the last magnitude. Thus: If A, B, C, D be four magnitudes of the same kind, the first, A, is said to have to the last, D, the ratio compounded of the ratio of A to B, and of the ratio of B to C, and of the ratio of C to D; or, the ratio of A to D is said to be compounded of the ratios of A to B, B to C, and C to D. And if A:B :: E:F; and B:C::G:H, and C:D::K:L; then, since by this definition A has to D the ratio compounded of the ratios of A to B, B to C, C to D; A may also be said to have to D the ratio compounded of the ratios of E to F, G to H, and K to L, which are the same as these ratios. In like manner, the same things being supposed, if M has to N the same ratio which A has to D, then, for shortness' sake, M is said to have to N a ratio compounded of the same ratios which compound the ratio of A to D; that is, a ratio compounded of the ratios of E to F, G to H, and K to L. 18. A ratio which is compounded of two equal ratios is said to be duplicate of either of these ratios. CoR.-If the three magnitudes, A, B, and C, are continual proportionals, the ratio of A to C is duplicate of that of A to B, or of B to C; for, by the last definition, the ratio of A to C is compounded of the ratios of A to B, and of B to C; but the ratio of A to B is equal to the ratio of B to C, because A, B, C are continual proportionals; therefore the ratio of A to C, by this definition, is duplicate of the ratio of A to B, or of B to C. 19. A ratio which is compounded of three equal ratios is said to be triplicate of any one of these ratios; and a ratio which is compounded of four equal ratios is said to be quadruplicate of any one of these ratios; and so on, according to the number of equal ratios. COR.-If four magnitudes, A, B, C, D, be continual propor tionals, the ratio of A to D is triplicate of the ratio of A to C to D. 20. In proportionals, the antecedent terms are called homologous to one another, so also are the consequents. Geometers make use of the following technical words to signify certain ways of changing either the order or magnitude of the terms of an analogy, so that they continue still to be proportionals: 21. By alternation, when the first is to the third, as the second is to the fourth. 22. By inversion, when the second is to the first, as the fourth is to the third. 23. By composition, when the first, together with the second, is to the second, as the third, together with the fourth, is to the fourth. 24. By addition, when the first is to the sum of the first and second, as the third is to the sum of the third and fourth. 25. By division, when the excess of the first above the second, is to the second, as the excess of the third above the fourth, is to the fourth. 26. By conversion, when the first is to its excess above the second, as the third is to its excess above the fourth. 27. By mixing, when the sum of the first and second is to their difference, as the sum of the third and fourth to their difference. 28. By equality, when there is any number of magnitudes more than two, and as many others, so that they are proportionals when taken two and two of each rank, and it is inferred—that the first is to the last of the first rank of magnitudes, as the first is to the last of the others :-of this there are the two following kinds, which arise from the different order in which the magnitudes are taken two and two : 29. By direct equality, when the first magnitude is to the second of the first rank, as the first to the second of the other rank; and as the second is to the third of the first rank, so is the second to the third of the other; and so on in a direct order; and, 30. By indirect equality, when the first magnitude is to the second of the first rank, as the last but one is to the last of the second rank; and as the second is to the third of the first rank, so is the last but two to the last but one of the second rank; and as the third is to the fourth of the first rank, so is the last but three to the last but two of the second rank; and so on in an indirect order. The following are some additional algebraical expressions which are convenient for their conciseness : As mA is A taken m times, it is a multiple of A by m. So m(A + B) is a multiple of A + B by m; m(A – B), a multiple of A - B by m; and m(A + B - C), a multiple of the excess of A + B above C, by m. Also, mA and B are equimultiples of A and B by m. The expression in + n is the sum of the numbers m and n; so mn is the product of these numbers. Also, mnA is a multiple of A by a number which is the product of m and n; and (m + n)A, is a multiple of A by a number which is the sum of m and n. AXIOMS. 1. Equimultiples of the same, or of equal magnitudes, are equal to one another. 2. Those magnitudes of which the same, or equal magnitudes, are equimultiples, are equal to one another. 3. A multiple of a greater magnitude is greater than the same multiple of a less. 4. That magnitude of which a multiple is greater than the same multiple of another, is greater than that other magnitude. PROPOSITION I. THEOREM. If any number of magnitudes be equimultiples of as many others, each of each, what multiple soever any one of the first is of its part, the same multiple is the sum of all the first of the sum of all the rest. Given any number of magnitudes, A, B, and C, equimultiples of as many others D, E, and F, each of each; to prove that A + B + C is the same multiple of D + E + F that A is of D. (Prep.) Let A contain D, B .contain E, and C contain F, each any number of times, as, for instance, three times. Then, because A contains D three tiines,'. A = D + D + D. C = F + F + F. (Dem.) Therefore, adding equals to equals, A + B + C is: equal to D + E +F taken three times. In the same manner, if A, B, and C were each any other equimultiple of D, E, and F, it would be shewn that A + B + C was the same multiple of D+E+F. Cor.--Hence, if m be any number, mD +.ME + mF = m(D + E +F). For mD, mE, and mF are multiples of D, E, and F, by m; therefore their sum is also a multiple of D + E + F, by m. And also, PROPOSITION III. THEOREM. If the first of three magnitudes contain the second as oft as there are units in a certain number, and if the second contain the third also as often as there are units in a certain number, the first will contain the third as oft as there are units in the product of these two numbers. Given A = mB, and B = nC; to prove that A = mnC. (Dem.) Since B = nC, mB = nC + nC + &c., repeated m times. But nC + nC + &c., repeated m times, is (V. 2) equal to C multiplied by n tin t &c., n being repeated m times. And n repeated m. times is mn; therefore mB = mnC. But by hypothesis, A = mB, therefore A = mnC. PROPOSITION IV. THEOREM. If any equimultiples be taken of the antecedents of an analogy, and any equimultiples of the consequents, these multiples, taken in the order of the terms, are proportional. Given that A:B :: C:D, and let m and n be any two numbers; to prove that mA:nB :: mC:nD. (Prep.) Take of mA and mC equimultiples by any number p, and of nB and nD equimultiples by any number 9. (Dem.) Then the equimultiples of mA and mC by p are equimultiples also of A and C, for they contain A and C as oft as there are units in pm (V. 3), and are equal to pmA and pmC. For the same reason, the multiples of nB and n D by q are qnB, and. Since therefore A:B ::C:D, and of A and C there are taken any equimultiples-namely, pmA and pmC, and of B and D, any equimultiples qnB, qnD, if pma be greater than qnB, pm must be greater than qnD (V. Def. 10); if equal, equal; and if less, less. But pmÀ, pmC are also equimultiples of mA and mC by p, and qn B, qnĎ are equimultiples of nB and nD, by q; therefore mA:nB :: mC: nD. COR.-In the same manner, it may be demonstrated that if A:B::C:D, and of A and C equimultiples be taken by any PROPOSITION V. THEOREM. If one magnitude be the same multiple of another, which a magnitude taken from the first is of a magnitude taken from the other; the difference of the two multiples shall be the same multiple of the difference of the two quantities that the whole is of the whole. |