In all these cases, all inductive numbers occur in the top row, and only some in the bottom row. Cases of this sort, where the converse domain is a "proper part" of the domain (i.e. a part not the whole), will occupy us again when we come to deal with infinity. For the present, we wish only to note that they exist and demand consideration. Another class of correlations which are often important is the class called "permutations," where the domain and converse domain are identical. Consider, for example, the six possible arrangements of three letters: [Pg 50] Each of these can be obtained from any one of the others by means of a correlation. Take, for example, the first and last, and . Here is correlated with , with itself, and with . It is obvious that the combination of two permutations is again a permutation, i.e. the permutations of a given class form what is called a "group." These various kinds of correlations have importance in various connections, some for one purpose, some for another. The general notion of one-one correlations has boundless importance in the philosophy of mathematics, as we have partly seen already, but shall see much more fully as we proceed. One of its uses will occupy us in our next chapter. [Pg 51] CHAPTER VI SIMILARITY OF RELATIONS WE saw in Chapter II. that two classes have the same number of terms when they are "similar," i.e. when there is a one-one relation whose domain is the one class and whose converse domain is the other. In such a case we say that there is a "one-one correlation" between the two classes. In the present chapter we have to define a relation between relations, which will play the same part for them that similarity of classes plays for classes. We will call this relation "similarity of relations," or "likeness" when it seems desirable to use a different word from that which we use for classes. How is likeness to be defined?