Euclid's Elements
Table of Contents Book I Book II Book III Book IV Book V Book VI Book VII Book VIII Book IX Book X Book XI Book XII Book XIII

Definitions

Definition 1.
A magnitude is a part of a magnitude, the less of the greater, when it measures the greater.
Definition 2.
The greater is a multiple of the less when it is measured by the less.
Definition 3.
A ratio is a sort of relation in respect of size between two magnitudes of the same kind.
Definition 4.
Magnitudes are said to have a ratio to one another which can, when multiplied, exceed one another.
Definition 5.
Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever are taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.
Definition 6.
Let magnitudes which have the same ratio be called proportional.
Definition 7.
When, of the equimultiples, the multiple of the first magnitude exceeds the multiple of the second, but the multiple of the third does not exceed the multiple of the fourth, then the first is said to have a greater ratio to the second than the third has to the fourth.
Definition 8.
A proportion in three terms is the least possible.
Definition 9.
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.
Definition 10.
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.
Definition 11.
Antecedents are said to correspond to antecedents, and consequents to consequents.
Definition 12.
Alternate ratio means taking the antecedent in relation to the antecedent and the consequent in relation to the consequent.
Definition 13.
Inverse ratio means taking the consequent as antecedent in relation to the antecedent as consequent.
Definition 14.
A ratio taken jointly means taking the antecedent together with the consequent as one in relation to the consequent by itself.
Definition 15.
A ratio taken separately means taking the excess by which the antecedent exceeds the consequent in relation to the consequent by itself.
Definition 16.
Conversion of a ratio means taking the antecedent in relation to the excess by which the antecedent exceeds the consequent.
Definition 17.
A ratio ex aequali arises when, there being several magnitudes and another set equal to them in multitude which taken two and two are in the same proportion, the first is to the last among the first magnitudes as the first is to the last among the second magnitudes. Or, in other words, it means taking the extreme terms by virtue of the removal of the intermediate terms.
Definition 18.
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.