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112 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: (3)

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