Page:Popular Science Monthly Volume 68.djvu/233

 Rh 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. Regarded singly they may be as different as possible. They are, however, equal as groups. For, however 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 arrangement 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 operations 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 remains. Thereupon C is arranged with reference to B with no member remaining. In this way every member of A is, through the intervention of B, assigned to a member of C. Moreover, this arrangement remains 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 proposition is applicable to all groups and renders it possible to arrange