A proportion in three terms is the least possible. When three magnitudes are proportional, the first is said to have to the third the duplicate ratio of that which it has to the second. When four magnitudes are continuously proportional, the first is said to have to the fourth the triplicate ratio of that which it has to the second, and so on continually, whatever be the proportion. 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.
A ratio taken jointly means taking the antecedent together with the consequent as one in relation to the consequent by itself. A ratio taken separately means taking the excess by which the antecedent exceeds the consequent in relation to the consequent by itself. Conversion of a ratio means taking the antecedent in relation to the excess by which the antecedent exceeds the consequent. Or, in other words, it means taking the extreme terms by virtue of the removal of the intermediate terms. A perturbed proportion arises when, there being three magnitudes and another set equal to them in multitude, antecedent is to consequent among the first magnitudes as antecedent is to consequent among the second magnitudes, while, the consequent is to a third among the first magnitudes as a third is to the antecedent among the second magnitudes.