It is easy to see, and not difficult to prove, that the relation "less than," so defined, is asymmetrical, transitive, and connected, and has the inductive numbers for its field. Thus by means of this relation the inductive numbers acquire an order in the sense in which we defined the term "order," and this order is the so-called "natural" order, or order of magnitude. The generation of series by means of relations more or less resembling that of to is very common. The series of the Kings of England, for example, is generated by relations of each to his successor. This is probably the easiest way, where it is applicable, of conceiving the generation of a series. In this method we pass on from each term to the next, as long as there [Pg 35] is a next, or back to the one before, as long as there is one before. This method always requires the generalised form of mathematical induction in order to enable us to define "earlier" and "later" in a series so generated. On the analogy of "proper fractions," let us give the name "proper posterity of with respect to " to the class of those terms that belong to the -posterity of some term to which has the relation , in the sense which we gave before to "posterity," which includes a term in its own posterity. Reverting to the fundamental definitions, we find that the "proper posterity" may be defined as follows:— The "proper posterity" of with respect to consists of all terms that possess every -hereditary property possessed by every term to which has the relation . It is to be observed that this definition has to be so framed as to be applicable not only when there is only one term to which has the relation , but also in cases (as e.g. that of father and child) where there may be many terms to which has the relation . We define further: A term is a "proper ancestor" of with respect to if belongs to the proper posterity of with respect to . We shall speak for short of "-posterity" and "-ancestors" when these terms seem more convenient.