(7) When and are both to the right of and is between and . It will be found that, from the seven properties which we have assigned to the relation "between," it can be deduced that the relation "to the left of," as above defined, is a serial relation as we defined that term. It is important to notice that nothing in the definitions or the argument depends upon our meaning by "between" the actual relation of that name which occurs in empirical space: any three-term relation having the above seven purely formal properties will serve the purpose of the argument equally well. Cyclic order, such as that of the points on a circle, cannot be generated by means of three-term relations of "between." We need a relation of four terms, which may be called "separation of couples." The point may be illustrated by considering a journey round the world. One may go from England to New Zealand by way of Suez or by way of San Francisco; we cannot [Pg 40] say definitely that either of these two places is "between" England and New Zealand. But if a man chooses that route to go round the world, whichever way round he goes, his times in England and New Zealand are separated from each other by his times in Suez and San Francisco, and conversely. Generalising, if we take any four points on a circle, we can separate them into two couples, say and and and , such that, in order to get from to one must pass through either or , and in order to get from to one must pass through either or . Under these circumstances we say that the couple are "separated" by the couple . Out of this relation a cyclic order can be generated, in a way resembling that in which we generated an open order from "between," but somewhat more complicated.[12] [12]Cf. Principles of Mathematics, p. 205 ( 194), and references there given. The purpose of the latter half of this chapter has been to suggest the subject which one may call "generation of serial relations." When such relations have been defined, the generation of them from other relations possessing only some of the properties required for series becomes very important, especially in the philosophy of geometry and physics. But we cannot, within the limits of the present volume, do more than make the reader aware that such a subject exists. [Pg 41] CHAPTER V KINDS OF RELATIONS A great part of the philosophy of mathematics is concerned with relations, and many different kinds of relations have different kinds of uses. It often happens that a property which belongs to all relations is only important as regards relations of certain sorts; in these cases the reader will not see the bearing of the proposition asserting such a property unless he has in mind the sorts of relations for which it is useful. For reasons of this description, as well as from the intrinsic interest of the subject, it is well to have in our minds a rough list of the more mathematically serviceable varieties of relations.