CONTRIBUTION TO THE THEORY OF SCIENCE 229 



are to be carried out with the members of A, shall also be carried out 

 with those of B. We may begin by simply associating them member 

 for member. Then one of three things will happen: either A will be 

 exhausted while members of B remain, or B is first exhausted, or 

 finally A and B are exhausted simultaneously. In the first case we say 

 that A is poorer than B ; in the second B is poorer than A ; and in 

 the third case that both quantities are equal. 



We meet now for the first time the scientific concept of equality; 

 and it is necessary that we enlarge upon it. Absolutely complete 

 identity of both groups is obviously out of the question, inasmuch 

 as we made the assumption that the members of both groups might 

 be of any nature whatsoever. Eegarded singly they may be as 

 different as possible. They are, however, equal as groups. For, how- 

 ever I arrange the members of A, inasmuch as a member of B is 

 assigned to every member of A, I am able to carry out every arrange- 

 ment of A upon B as well. As regards the possibilities of arrangement 

 there is no apparent difference between A and B. As soon, however, 

 as A is either poorer or richer than B, this similarity disappears, for 

 one of the two quantities possesses members to which no members of 

 the other groups correspond. The operations that may be performed 

 upon these members can not be carried out upon the second group. 



Equality, in the scientific sense of the word, signifies, therefore, 

 equivalence or the possibility of substitution as regards definite opera- 

 tions or relations. In all other respects the things that have been 

 pronounced equal may differ in any way. It is easy in this special 

 case to recognize the universal method of abstraction of science. 



It is possible on the basis of these definitions to make further 

 propositions. If the quantity A is equal to B and if B is equal to C, 

 then A is also equal to C. This may be proved by first arranging A 

 with reference to B. According to our presupposition no member re- 

 mains. Thereupon C is arranged with reference to B with no member 

 remaining. In this way every member of A is, through the interven- 

 tion of B, assigned to a member of C. Moreover, this arrangement re- 

 mains unchanged even after the removal of B, i. e., A and C are equal. 

 The same method may be applied to any number of quantities. 



It is possible to prove in a similar manner that, if A is poorer than 

 B, and B is poorer than C, A must also be poorer than C. For in 

 assigning the members of B to A, some members of B will, according 

 to our assumption, remain, and the same will be true of C if we assign 

 the members of C to those of B. Hence in assigning the members of C 

 to those of A there are left not merely the members which can not be 

 assigned to B, but also the members of C which have been assigned to 

 such members of B as are supernumerary in respect to A. This prop- 

 osition is applicable to all groups and renders it possible to arrange 



