If at first we have two groups the members of which are all differentiated from one another, then any one member of the one group can be co-ordinated with any one member of the other group. This means that we determine that the same should be done with every member of the second group as is done with the corresponding member of the first group. That such a rule may be carried out we must be able to do with the members of all the groups whatever we do with the members of one group. In other words, no properties peculiar to individual members may be utilized, but only the properties that each member possesses as a member of a group. As we have seen, these are the properties of _association_. First, the co-ordination is _mutual_, that is, it is immaterial to which of the two groups the processes are applied. The relation of the two groups is reciprocal or symmetrical. Further, the process of co-ordination can be extended to a third and a fourth group and so on, with the result that what has been done in one of the co-ordinated groups must happen in all. If hereby the third group is co-ordinated with the second, the effects are quite the same as if it were co-ordinated directly with the first instead of indirectly through the second. And the same is true for the fourth and the fifth groups, etc. Thus, co-ordination can be extended to any number of groups we please, and each single group proves to be co-ordinated with every other. Finally, a group can be co-ordinated with itself, each of its members corresponding to a certain definite other member. It is not impossible that individual members should correspond to themselves, in which case the group has _double members_, or _double points_. The limit-case is _identity_, in which every member corresponds to _itself_. This last case cannot supply any special knowledge in itself, but may be applied profitably to throw light on those observations for which it represents the extreme possibility. =28. Comparison.= If we have two groups A and B, and if we co-ordinate their members severally, three cases may arise. Either group A is exhausted while there are members remaining in B, or B is exhausted before A, or, finally, both groups allow of a mutual co-ordination of _all_ their members. In the first case A is called, in the broader sense of the word, _smaller_ than B, in the second B is called smaller than A, in the third the two groups are said to be of _equal magnitude_. The expression, "B is greater than A," is equivalent to the expression, "A is smaller than B," and inversely.