In the last place. If one of the proportions, namely of A D to A B, be the proportion of excess, and another of them, as of A B to A C be the proportion of defect, thus also the proportion of A D to A C will be equal to the two proportions together taken of A D to A B, and of A B to A C. For the difference of the times in which AD and AB are described, is excess of time; for there goes more time to the description of A D than of A B; and the difference of the times in which A B and A C are described, is defect of time, for less time goes to the description of A B than of A C; but this excess and defect being added together, make D B - B C, which is equal to D C, by which the first A D exceeds the third A C; and therefore the proportions of the first A D to the second A B, and of the second A B to the third A C, are determined by the same cause which determines the proportion of the first A D to the third A C. Wherefore, if any three magnitudes, &c. Coroll. I. If there be never so many magnitudes having proportion to one another, the proportion of the first to the last is compounded of the proportions of the first to the second, of the second to the third, and so on till you come to the last; or, the proportion of the first to the last is the same with the sum of all the intermediate proportions. For any number of magnitudes having proportion to one another, as A, B, C, D, E being propounded, the proportion of A to E, as is newly shown, is compounded of the proportions of A to D and of D to E; and again, the proportion of A to D, of the proportions of A to C, and of C to D; and lastly, the proportion of A to C, of the proportions of A to B, and of B to C. Coroll. II. From hence it may be understood how any two proportions may be compounded. For if the proportions of A to B, and of C to D, be propounded to be added together, let B have to something else, as to E, the same proportion which C has to D, and let them be set in this order, A, B, E; for so the proportion of A to E will evidently be the sum of the two proportions of A to B, and of B to E, that is, of C to D. Or let it be as D to C, so A to something else, as to E, and let them be ordered thus, E, A, B; for the proportion of E to B will be compounded of the proportions of E to A, that is, of C to D, and of A to B. Also, it may be understood how one proportion may be taken out of another. For if the proportion of C to D be to be subtracted out of the proportion of A to B, let it be as C to D, so A to something else, as E, and setting them in this order, A, E, B, and taking away the proportion of A to E, that is, of C to D, there will remain the proportion of E to B.