apply the word equal chiefly to the separate phenomena which objects exhibit, and not to groups of phenomena; but this limitation of the idea has evidently arisen by subsequent analysis.  And that the notion of equality did thus originate, will, we think, become obvious on remembering that as there were no artificial objects from which it could have been abstracted, it must have been abstracted from natural objects; and that the various families of the animal kingdom chiefly furnish those natural objects which display the requisite exactitude of likeness.

The same order of experiences out of which this general idea of equality is evolved, gives birth at the same time to a more complex idea of equality; or, rather, the process just described generates an idea of equality which further experience separates into two ideas—­equality of things and equality of relations.  While organic, and more especially animal forms, occasionally exhibit this perfection of likeness out of which the notion of simple equality arises, they more frequently exhibit only that kind of 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.