We can define addition and multiplication for relation-numbers as well as for cardinal numbers, and a whole arithmetic of relation-numbers can be developed. The manner in which this is to be done is easily seen by considering the case of series. Suppose, for example, that we wish to define the sum of two non-overlapping series in such a way that the relation-number of the sum shall be capable of being defined as the sum of the relation-numbers of the two series. In the first place, it is clear that there is an order involved as between the two series: one of them must be placed before the other. Thus if and are the generating relations of the two series, in the series which is their sum with put before , every member of the field of will precede every member of the field of . Thus the serial relation which is to be defined as the sum of and is not " or " simply, but " or or the relation of any member of the field of to any member of the field of ." Assuming that and do not overlap, this relation is serial, but " or " is not serial, being not connected, since it does not hold between a member of the field of and a member of the field of . Thus the sum of and , as above defined, is what we need in order [Pg 57] to define the sum of two relation-numbers. Similar modifications are needed for products and powers. The resulting arithmetic does not obey the commutative law: the sum or product which they are taken. But it obeys the associative law, one form of the distributive law, and two of the formal laws for powers, not only as applied to serial numbers, but as applied to relation-numbers generally. Relation-arithmetic, in fact, though recent, is a thoroughly respectable branch of mathematics.