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of the other. To every part $M_{1}$ of $M$ there corresponds, then, a definite equivalent part $N_{1}$ of $N$, and inversely.

If we have such a law of co-ordination of two equivalent aggregates, then, apart from the case when each of them consists only of one element, we can modify this law in many ways. We can, for instance, always take care that to a special element $m_{0}$ of $M$ a special element $n_{0}$ of $N$ corresponds. For if, according to the original law, the elements $m_{0}$and $n_{0}$ do not correspond to one another, but to the element $m_{0}$ of $M$ the element $n_{1}$ of $N$ corresponds, and to the element $n_{0}$ of $N$ the element $m_{1}$ of $M$ corresponds, we take the modified law according to which $m_{0}$ corresponds to $n_{0}$ and $m_{1}$ to $n_{1}$ and for the other elements the original law remains unaltered. By this means the end is attained.

Every aggregate is equivalent to itself:

If two aggregates are equivalent to a third, they are equivalent to one another; that is to say:

Of fundamental importance is the theorem that two aggregates M and N have the same cardinal number if, and only if, they are equivalent: thus,

and

Thus the equivalence of aggregates forms the neces-