Two relations , are said to be "similar" if there is a one-one relation between the terms of their fields, which is such that, whenever two terms have the relation , their correlates have the relation , and vice versa. The most familiar example is that of series: two series are similar when their terms can be correlated without change of order. But it would be a great mistake to suppose that series are the only important application of the notion of similarity between relations. A map, for example, if accurate, is similar to the region which it maps. A book spelt phonetically is similar to the sounds produced when it is read aloud. A gramophone record is similar to the music which it produces. And so on. [Pg 250] It should be observed that similarity applies not only to two-term relations, but to relations with any number of terms. Suppose we have two relations , each -adic; suppose there is a one-one relation which relates all the terms in the field of to all the terms in the field of ; let , , ... be terms which have the relation and let , , ... be the terms correlated with them by the relation . Then and are similar if there is a one-one relation such that, when the above conditions are fulfilled, , , ... have the relation , and conversely. Two relations which are similar have the same "structure" or "relation-number." The "relation-number" of a relation is the same as its "structure," and is defined as the class of all relations similar to the given relation. Relation-numbers satisfy all the formal laws of arithmetic which are satisfied by transfinite ordinal numbers; ordinal numbers, both finite and transfinite, are a particular kind of relation-numbers—namely, the relation-numbers of relations which generate well-ordered series. The formal laws satisfied by relation-numbers are: They do not in general satisfy the commutative law, nor the other form of the distributive law, viz.: