Euclid's Elements, Book V, Definitions 11 through 13 Definitions 11 through 13 Def. Antecedents are said to correspond to antecedents, and consequents to consequents. Alternate ratio means taking the antecedent in relation to the antecedent and the consequent in relation to the consequent.
Inverse ratio means taking the consequent as antecedent in relation to the antecedent as consequent.
Thus, A and C are corresponding terms since they're the antecedents. Also, B and D are corresponding terms since they're the consequents. Note that for alternate ratios to exist, all four magnitudes must be of the same kind. C reduce to the same conditions on A , B , C , and D . Therefore, if two ratios are the same, then their two inverse ratios are also the same.