Two classes are said to be "similar" when there is a one-one [Pg 15] relation which correlates the terms of the one class each with one term of the other class, in the same manner in which the relation of marriage correlates husbands with wives. A few preliminary definitions will help us to state this definition more precisely. The class of those terms that have a given relation to something or other is called the domain of that relation: thus fathers are the domain of the relation of father to child, husbands are the domain of the relation of husband to wife, wives are the domain of the relation of wife to husband, and husbands and wives together are the domain of the relation of marriage. The relation of wife to husband is called the converse of the relation of husband to wife. Similarly less is the converse of greater, later is the converse of earlier, and so on. Generally, the converse of a given relation is that relation which holds between and whenever the given relation holds between and . The converse domain of a relation is the domain of its converse: thus the class of wives is the converse domain of the relation of husband to wife. We may now state our definition of similarity as follows:— One class is said to be "similar" to another when there is a one-one relation of which the one class is the domain, while the other is the converse domain. It is easy to prove (1) that every class is similar to itself, (2) that if a class is similar to a class , then is similar to , (3) that if is similar to and to , then is similar to . A relation is said to be reflexive when it possesses the first of these properties, symmetrical when it possesses the second, and transitive when it possesses the third. It is obvious that a relation which is symmetrical and transitive must be reflexive throughout its domain. Relations which possess these properties are an important kind, and it is worth while to note that similarity is one of this kind of relations.