VITT For, by article 6, the multiples will be propor tionals, and, therefore, the assertion is true, by arti cle 5.
The same things heing allowed as above, it evidently follows, that if the multiple of the third be greater than the multiple of the fourth, the multiple of the first will be greater than the multiple of the second ; and if the multiple of the third be less than the multiple of the fourth, the multiple of the first will be less than the multiple of the second.
IX. If the first of four magnitudes has the same ratio to the second that the third has to a magnitude less than the fourth, then it is possible to take certain equi multiples of the first and third, and certain equimulti ples of the second and fourth ; such, that the multiple of the first shall be greater than the multiple of the se cond, but the multiple of the third not greater than the multiple of the fourth.
Let A, B, C, DE be four magnitudes, and let A have the same ratio to B that C has to FE, a magnitude less than DE; then it 11", is possible to take certain equi multiples of A, C, and certain equimultiples of B, DE, such, that the multiple of A shall be greater than the multiple of B, but the multiple of C not greater than the multiple of DE.
Of DF, FE, take such equi multiples GH, HI, that each of them may be greater than C.
Then of C take H the double, L the triple, &c. until a multiple of C be obtained greater than HI. Let M be the multiple of C, which first becomes greater than HI, and L the multiple of C, which is next less than M, and then HI is not less than L. But, by the construction, GH is greater than C; and as M is equal to L and C together, M is greater than HI, but not greater than GI. Let N be the same multiple of A that M is of C, and P the same multiple of B that Ill is of FE; and then, as A, B, C, FE are proportionals, and as M is greater than HI, N is greater than P, by article 8. Again, as Gil, HI, are equimultiples of DF, FE, by the first proposition in the 5th book of Euclid, GI is the same multiple of DE that HI is of FE, or that P is of B. Consequently, certain equimultiples, N, M, have been taken of A the first and C the third ; and certain equi multiples, P and GI, of B the second and DE the fourth ; such, that N is greater than P, but M is not greater than GI.
X. If the first of four magnitudes has the same ratio to the second, that the third has to a magnitude greater than the fourth ; then certain equimultiples can be taken of the first and third, and certain equimultiples of the second and fourth ; such, that the multiple of the first shall be less than the multiple of the second ; but the multiple of the third not less than the multiple of the fourth.
Let A, B, C, DE be four magnitudes, and let A the first have the same ratio to B the second, that C the third has to FE a magnitude greater than DE ; then it is possible to take certain equi multiples of A and C, and certain equimultiples of B and DE; such, that the multiple of A shall be less than the multiple of B ; but the multiple of C not less than the multiple of DE.
For of ED, DI', let IG, GII. be taken, such equimultiples, that each of them may be greater than C; and, as in the last article, let M be taken, such a multiple of C, that it may be greater than 1G, but less than IH. By Prop. I. in the 5th book of Euclid, IH, IG, are equimultiples of FE, DE, and therefore let P be taken, the same multi ple of B that either of them is of its part; and let N be the same multiple of A that M is of C. Then, as A B, C, FE, are proportionals, and as M is less than IH, N is less than P, by article 8. Consequently N, the multiple of A the first, is less than P, the multiple of B the second ; but M, the multiple of C the third, is not less than IG, the multiple of DE the fourth.
XI. If any equimultiples whatever be taken of the first and third of four magnitudes, and any equimulti ples whatever of the second and fourth ; and if, when the multiple of the first is less than that of the second, the multiple of the third is also less than that of the fourth ; or, if, when the multiple of the first is equal to that of the second, the multiple of the third is also equal to that of the fourth ; or, if, when the multiple of the first is greater than that of the second, the multi ple of the third is also greater than that of the fourth ; then, the first of the four magnitudes will have the same ratio to the second, that the third has to the. fourth.