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.