THE SCIENCES OF THE IDEAL 163 



(2) the relations of the type of before and after, or greater and less. 

 The first of these two classes of relations, namely, the class repre- 

 sented, although by no means exhausted, by the various relations 

 actually called, in different branches of science by the one name 

 equality, this class I say, might well be named, as I myself have 

 proposed, the leveling relations. A collection of objects between 

 any two of which some one relation of this type holds, may be said 

 to be a collection whose members, in some defined sense or other, 

 are on the same level. The second of these two classes of relations, 

 namely, those of the type of before and after, or greater and less 

 this class of relations, I say, consists of what are nowadays often 

 called the serial relations. And a collection of objects such that, if 

 any pair of these objects be chosen, a determinate one of this pair 

 stands to the other one of the same pair in some determinate rela- 

 tion of this second type, and in a relation which remains constant 

 for all the pairs that can be thus formed out of the members of this 

 collection any such collection, I say, constitutes a one-dimen- 

 sional open series. Thus, in case of a file of men, if you choose any 

 pair of men belonging to the file, a determinate one of them is, in the 

 file, before the other. In the number series, of any two numbers, 

 a determinate one is greater than the other. Wherever such a state 

 of affairs exists, one has a series. 



Now these two classes of relations, the leveling relations and the 

 serial relations, agree with one another, and differ from one another 

 in very momentous ways. They agree with one another in that both 

 the leveling and the serial relations are what is technically called 

 transitive; that is, both classes conform to what Professor James 

 has called the law of "skipped intermediaries." Thus, if A is equal 

 to B, and B is equal to C, it follows that A is equal to C. If A is 

 before B, and B is before C, then A is before C. And this property, 

 which enables you in your reasonings about these relations to skip 

 middle terms, and so to perform some operation of elimination, is 

 the property which is meant when one calls relations of this type 

 transitive. But, on the other hand, these two classes of relations 

 differ from each other in that the leveling relations are, while the 

 serial relations are not, symmetrical or reciprocal. Thus, if A is equal 

 to B, B is equal to A. But if X is greater than Y, then Y is not 

 greater than X, but less than X. So the leveling relations are sym- 

 metrical transitive relations. But the serial relations are transitive 

 relations which are not symmetrical. 



All this is now well known. It is notable, however, that nearly 

 all the processes of our exact sciences, as at present developed, 

 can be said to be essentially such as lead either to the placing of sets 

 or classes of objects on the same level, by means of the use of sym- 

 metrical transitive relations, or else to the arranging of objects in 



