 Relation Arithmetic is the study of the generalization of cardinal and ordinal numbers to classes of similar classes where similarity is based on an arbitrary relation. A relation P is similar to a relation Q, if there is a one to one relation S (a correlator) relating the domain of P to the domain of Q so that if \(x\relP y\) for some x and y then if \(x\relS w\) and \(w\relQ z\) then \(z \relbS y\). The mapping S is an isomorphism between the relations P and Q. A relation number will then be a class of relations that are similar to each other. Relation Arithmetic then generalizes the notions of cardinal arithmetic,  such as sum and product, to arbitrary relation numbers. Russell himself expressed regret that the material in Part IV was not more carefully studied by his contemporaries (Russell 1959: 86). If a series \( \relP \) is well-ordered, then the class of relations ordinally similar to \( \relP \)  will be an ordinal number. The sums of ordinal numbers are are studied in Tarski (1956), but there has been little interest in the more general notion of Relation Arithmetic presented in these sections of PM. See Solomon (1989).