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.
Coroll. VII. If there be four proportionals, they will also be proportionals by division, that is, by taking the remainder after the consequent is subtracted from the antecedent, or the difference between the antecedent and consequent for antecedent, and either the whole or the subtracted for consequent; as if A. B :: C. D be proportionals, they will by division be A - B. B :: C - D. D, and A - B. A :: C - D. C; and when the consequent is greater than the antecedent, B - A. A :: D - C. C, and B - A. B :: D - C. D. For in all these divisions, proportionals are, by the very supposition of the analogism A. B :: C. D, taken from A and B, and from C and D.
Coroll. VIII. If there be four proportionals, they will also be proportionals by the conversion of proportion, that is, by inverting the divided proportion, or by taking the whole for antecedent, and the difference or remainder for consequent.
As, if A. B :: C. D be proportionals, then A. A - B :: C. C - D, as also B. A - B :: D. C - D will be proportionals. For seeing these inverted be proportionals, they are also themselves proportionals.
Coroll. IX. If there be two analogisms which have their quantities equal, the second to the second, and the fourth to the fourth, then either the sum or difference of the first quantities will be to the second, as the sum or difference of the third quantities is to the fourth. Let A. B :: C. D and E. B :: F. D be analogisms; I say A + E. B :: C + F. D are proportionals. For the said analogisms will by permutation be A. C :: B. D, and E. F :: B. D; and therefore A. C :: E. F will be proportionals, for they have both the proportion of B to D common. Wherefore, if in the permutation of the first analogism, there be added E and F to A and C, which E and F are proportional to A and C, then (by the third coroll.) A + E. B :: C + F. D will be proportionals; which was to be proved.
Also in the same manner it may be shown, that A - E. B :: C - F. D are proportionals.
7. If there be two analogisms, where four antecedents make an analogism, their consequents also shall make an analogism; as also the sums of their antecedents will be proportional to the sums of their consequents. For if A. B :: C. D and E. F :: G. H be two analogisms, and A. E :: C. G be proportionals, then by permutation A. C :: E. G, and E. G :: F. H, and A. C :: B. D will be proportionals; wherefore B. D :: E. G, that is, B. D :: F. H, and by permutation B. F :: D. H are proportionals; which is the first. Secondly, I say A + E. B + F :: C + G. D + H will be proportionals. For seeing A. E :: C. G are proportionals, A + E. E :: C + G. G will also by composition be proportionals, and by permutation A + E. C + G :: E. G will be proportionals; wherefore, also A + E. C + G :: B + F. D + H will be proportionals. Again, seeing, as is shown above, B. F :: D. H are proportionals, B + F. F :: D + H. H will also by composition be proportionals; and by permutation B + F. D + H :: F. H will also be proportionals; wherefore A + E. C + G :: B + F. D + H are proportionals; which remained to be proved.