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By this we understand the general concept which results from $M$ if we only abstract from the nature of the elements $m$, and retain the order of precedence among them. Thus the ordinal type ${\overline {M}}$ is itself an ordered aggregate whose elements are units which have the same order of precedence amongst one another as the corresponding elements of $M$, from which they are derived by abstraction.

We call two ordered aggregates $M$ and $N$ "similar" (ähnlich) if they can be put into a biunivocal correspondence with one another in such a manner that, if $m_{1}$ and $m_{2}$ are any two elements of $M$ and $n_{1}$ and $n_{2}$ the corresponding elements of $N$, then the relation of rank of $m_{1}$ to $m_{2}$ in $M$ is the same as that of $n_{1}$ to $n_{2}$ in $N$. Such a correspondence of similar aggregates we call an "imaging" (Abbildung) of these aggregates on one another. In such an imaging, to every part—which obviously also appears as an ordered aggregate—$M_{1}$ of $M$ corresponds a similar part $N_{1}$ of $N$.

We express the similarity of two ordered aggregates $M$ and $N$ by the formula:

Every ordered aggregate is similar to itself.

If two ordered aggregates are similar to a third, they are similar to one another.

[498] A simple consideration shows that two ordered aggregates have the same ordinal type if, and only if, they are similar, so that, of the two formulæ