A 1 LESS magnitude is said to be a part of a greater magnitude, when the less measures the greater, that is, ' when the less is II. greater is measured by the less, that is, when the greater con- III. IV. V. cond, which the third has to the fourth, when any equimultiples Book v. than that of the second, the multiple of the third is also greater than that of the fourth. VI. N. B. “When four magnitudes are proportionals, it is usually expressed by saying, the first is to the second, as the third to • the fourth.' VII. Definition, the multiple of the first is greater than that of the se- VIII. IX. X. XI. have to the fourth the Triplicate ratio of that which it has to the Definition A, to wit, of Compound ratio. is said to have to the last of them the ratio compounded of the the fourth, and so on unto the last magnitude. For example, If A, B, C, D be four magnitudes of the fame kind, the first A is faid to have to the last D the ratio compounded of the ratio 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 has to B, the same ratio which E has to F; and B to C, the same ratio that G has to H; and C to D, the same that K has See Y. U CLID. has to L; then, by this Definition, A is said to have to D the ra- Book V. tio compounded of ratios which are the same with the ratios of E to F, G to H, and K to L. and the same thing is to be understood when it is more briefly expressed by saying A has to D, the ratio compounded of the ratios of E to F, G to H, ånd K to L. În like manner, the same things being supposed, if M has to N the fame ratio which A has to D, then, for shortness fake, M is said XII. another, as also the consequents to one another. XIII. word is used when there are four proportionals, and it is infer XIV. is inferred, that the second is to the first, as the fourth to the XV. and it is inferred, that the first together with the second, is to XVI. is inferred, that the Excess of the first above the second, is to the XVII. and it is inferred, that the first is to its Excess above the fecond, } YVI'I. I, Book V. XVIII. there is any number of magnitudes more than two, and as many there are the two following kinds, which arise from the diffe* rent order in which the magnitudes are taken two and two.' XIX. the first magnitude is to the second of the first rank, as the first XX. lity, in perturbate or disorderly proportion *; this term is used A X I 0 MS. I. QUIMULTIPLES of the same, or of equal magnitudes, are equal to one another, Í. Those magnitudes of which the fame, or equal magnitudes, are equimultiples, are equal to one another. III. A multiple of a greater magnitude is greater than the same multiple of a less. * 4. Prop. Lib. 2. Archimedis de sphaera et cylindro. IV, That IV. Book V. That magnitude of which a multiple is greater than the same mul tiple of another, is greater than that other magnitude. PROP. I. THEOR. many, each of each; what multiple soever any one of them is of its part, the same multiple shall all the first magnitudes be of all the other. IF Let any number of magnitudes AB, CD be equimultiples of as many others E, F, each of each; whatsoever multiple AB is of E, the same multiple shall AB and CD together be of E and F together. Because AB is the same multiple of E that CD is of F, as many magnitudes as are in AB equal to E, so many are there in CD equal to F. Divide AB into magnitudes equal to E, viz. AG, GB; and CD into CH, HD equal each of A. them to F. the number therefore of the magnitudes CH, HD shall be equal to the number of G El the others AG, GB. and because AG is equal to E, and CH to F; therefore AG and CH together are equal to · E and F together. for the same a. Ar ats reason, because GB is equal to E, and HD to F; CI GB and HD together are equal to E and F together. Wherefore as many magnitudes as are in F AB equal to E, so many are there in AB, CD H Н together equal to E and F together. Therefore whatsoever multiple AB is of E, the same mul D tiple is AB and CD together of E and F together. Therefore if any magnitudes, how many soever, be equimultiples of as many, each of each, whatsoever multiple any one of them is of its part, the same multiple shall all the first magnitudes be of all the other. for the same Demonstration holds in any number of magnitudes, which was here applied to two. Q. E. D. |