Secondly, let the two products be A B F, and C D G, each of them made of three efficients, the first of A, B and F, and the second of C, D and G; I say, the proportion of A B F to C D G is compounded of the proportions of A to C, of B to D, and of F to G. For let them be set in order as before; and as B is to D, so let C be to another quantity E; and again, as F is to G, so let E be to another, H; and let the first order stand thus, ABF, CBF, CDF and CDG; and the second order thus, A, C, E, H. Then the proportion of A B F to C B F in the first order, will be as A to C in the second; and the proportion of CBF to CDF in the first order, as B to D, that is, as C to E (by construction) in the second order; and the proportion of CDF to CDG in the first, as F to G, that is, as E to H (by construction) in the second order; and therefore A B F. C D G:: A. H will be proportionals. But the proportion of A to H is compounded of the proportions of A to C, B to D, and F to G. Wherefore the proportion of the product A B F to C D G is also compounded of the same. And this operation serves, how many soever the efficients be that make the quantities given. ABF. A. CBF. C. CDF. E. CDG. H. From hence ariseth another way of compounding many proportions into one, namely, that which is supposed in the 5th definition of the 6th book of Euclid; which is, by multiplying all the antecedents of the proportions into one another, and in like manner all the consequents into one another. And from hence also it is evident, in the first place, that the cause why parallelograms, which are made by the duction of two straight lines into one another, and all solids which are equal to figures so made, have their proportions compounded of the proportions of the efficients; and in the second place, why the multiplication of two or more fractions into one another is the same thing with the composition of the proportions of their several numerators to their several denominators. For example, if these fractions 1⁄2, 2⁄3, 3⁄4 be to be multiplied into one another, the numerators 1, 2, 3, are first to be multiplied into one another, which make 6; and next the denominators 2, 3, 4, which make 24; and these two products make the fraction 6⁄24. In like manner, if the proportions of 1 to 2, of 2 to 3, and of 3 to 4, be to be compounded, by working as I have shown above, the same proportion of 6 to 24 will be produced.