likeness which we call _similarity_; and which
is really compound equality. For the similarity of two creatures of the
same species but of different sizes, is of the same nature as the
similarity of two geometrical figures. In either case, any two parts of
the one bear the same ratio to one another as the homologous parts of
the other. Given in any species, the proportions found to exist among
the bones, and we may, and zoologists do, predict from any one, the
dimensions of the rest; just as, when knowing the proportions subsisting
among the parts of a geometrical figure, we may, from the length of one,
calculate the others. And if, in the case of similar geometrical
figures, the similarity can be established only by proving exactness of
proportion among the homologous parts; if we express this relation
between two parts in the one, and the corresponding parts in the other,
by the formula A is to B as _a_ is to _b_; if we otherwise write this, A
to B = _a_ to _b_; if, consequently, the fact we prove is that the
relation of A to B _equals_ the relation of _a_ to _b_; then it is
manifest that the fundamental conception of similarity is _equality of
relations_.

With this explanation we shall be understood when we say that the notion
of equality of relations is the basis of all exact reasoning. Already it
has been shown that reasoning in general is a recognition of _likeness_
of relations; and here we further find that while the notion of likeness
of things ultimately evolves the idea of simple equality, the notion of
likeness of relations evolves the idea of equality of relations: of
which the one is the concrete germ of exact science, while the other is
its abstract germ.

Those who cannot understand how the recognition of similarity in
creatures of the same kind can have any alliance with reasoning, will
get over the difficulty on remembering that the phenomena among which
equality of relations is thus perceived, are phenomena of the same order
and are present to the senses at the same time; while those among which
developed reason perceives relations, are generally neither of the same
order, nor simultaneously present. And if further, they will call to
mind how Cuvier and Owen, from a single part of a creature, as a tooth,
construct the rest by a process of reasoning based on this equality of
relations, they will see that the two things are intimately connected,
remote as they at first seem. But we anticipate. What i