Coroll. III. If there be two orders of magnitudes which have proportion to one another, and the several proportions of the first order be the same and equal in number with the proportions of the second order; then, whether the proportions in both orders be successively answerable to one another, which is called _ordinate proportion_, or not successively answerable, which is called _perturbed proportion_, the first and the last in both will be proportionals. For the proportion of the first to the last is equal to all the intermediate proportions; which being in both orders the same, and equal in number, the aggregates of those proportions will also be equal to one another; but to their aggregates, the proportions of the first to the last are equal; and therefore the proportion of the first to the last in one order, is the same with the proportion of the first to the last in the other order. Wherefore the first and the last in both are proportionals. [Sidenote: Composition of proportions.] 14. If any two quantities be made of the mutual multiplication of many quantities, which have proportion to one another, and the efficient quantities on both sides be equal in number, the proportion of the products will be compounded of the several proportions, which the efficient quantities have to one another. A B. A. C B. C. C D. E. First, let the two products be A B and C D, whereof one is made of the multiplication of A into B, and the other of the multiplication of C into D. I say the proportion of A B to C D is compounded of the proportions of the efficient A to the efficient C, and of the efficient B to the efficient D. For let A B, C B and C D be set in order; and as B is to D, so let C be to another quantity as E; and let A, C, E be set also in order. Then (by coroll. IV. of the 6th art.) it will be as A B the first quantity to C B the second quantity in the first order, so A to C in the second order; and again, as C B to C D in the first order, so B to D, that is, by construction, so C to E in the second order; and therefore (by the last corollary) A B. C D :: A. E will be proportionals. But the proportion of A to E is compounded of the proportions of A to C, and of B to D; wherefore also the proportion of A B to C D is compounded of the same.