Secondly, (taking the antecedents in different lines) I say, A B. A E :: A C. A F are proportionals; for seeing A B, A E are described in the same time, the difference of the velocities in which they are described is the sole cause of the proportion they have to one another. And the same may be said of the proportion of A C to A F. But seeing both the lines A D and A G are passed over by uniform motion, the difference of the velocities in which A B, A E are described, will be the same with the difference of the velocities, in which A C, A F are described. Wherefore the cause which determines the proportion of A B to A E, is the same with that which determines the proportion of A C to A F; and therefore A B. A E :: A C. A F, are proportionals; which remained to be proved. Coroll. I. If four magnitudes be in geometrical proportion, they will also be proportionals by _permutation_, that is, by transposing the middle terms. For I have shown, that not only A B. A C :: A E. A F, but also that, by _permutation_, A B. A E :: A C. A F are proportionals. Coroll. II. If there be four proportionals, they will also be proportionals by _inversion_ or _conversion_, that is, by turning the antecedents into consequents. For if in the last _analogism_, I had for A B, A C, put by inversion A C, A B, and in like manner converted A E, A F into A F, A E, yet the same demonstration had served. For as well A C, A B, as A B, A C are of equal velocity; and A C, A F, as well as A F, A C are contemporary. Coroll. III. If proportionals be added to proportionals, or taken from them, the aggregates, or remainders, will be proportionals. For contemporaries, whether they be added to contemporaries, or taken from them, make the aggregates or remainders contemporary, though the addition or subtraction be of all the terms, or of the antecedents alone, or of the consequents alone. Coroll. IV. If both the antecedents of four proportionals, or both the consequents, or all the terms, be multiplied or divided by the same number or quantity, the products or quotients will be proportionals. For the multiplication and division of proportionals, is the same with the addition and subtraction of them. Coroll. V. If there be four proportionals, they will also be proportionals by _composition_, that is, by compounding an antecedent of the antecedent and consequent put together, and by taking for consequent either the consequent singly, or the antecedent singly. For this composition is nothing but addition of proportionals, namely, of consequents to their own antecedents, which by supposition are proportionals. Coroll. VI. In like manner, if the antecedent singly, or consequent singly, be put for antecedent, and the consequent be made of both put together, these also will be proportionals. For it is the _inversion of proportion by composition_.