ABSTRACT
Let us begin with two dimensions. A series of two dimensions arises as follows. Let there be some asymmetrical transitive relation P, which generates a series u1. Let every term of u1 be itself an asymmetrical transitive relation, which generates a series. Let all the field of P form a simple series of asymmetrical relations, and let each of these have a simple series of terms for its field. Then the class u2 of terms forming the fields of all the relations in the series generated by P is a two-dimensional series. In other words, the total field of a class of asymmetrical transitive relations forming a simple series is a double series. But instead of starting from the asymmetrical relation P, we may start from the terms. Let there be a class of terms u2, of which any given one (with possibly one exception) belongs to the field of one and only one of a certain class u1 of serial relations. That is if x be a term of u2, x is also a term of the field of some relation of the class u1. Now further let u1 be a series. Then u2 will be a double series. This seems to constitute the definition of two-dimensional series.
