Indeed, as this observation suggests, it will be well, before tracing the mode in which exact science finally emerges from the merely approximate judgments of the senses, and showing the non-serial evolution of its divisions, to note the non-serial character of those preliminary processes of which all after development is a continuation.  On reconsidering them it will be seen that not only are they divergent growths from a common root, not only are they simultaneous in their progress; but that they are mutual aids; and that none can advance without the rest.  That completeness of classification for which the unfolding of the perceptions paves the way, is impossible without a corresponding progress in language, by which greater varieties of objects are thinkable and expressible.  On the one hand it is impossible to carry classification far without names by which to designate the classes; and on the other hand it is impossible to make language faster than things are classified.

Again, the multiplication of classes and the consequent narrowing of each class, itself involves a greater likeness among the things classed together; and the consequent approach towards the notion of complete likeness itself allows classification to be carried higher.  Moreover, classification necessarily advances pari passu with rationality—­the classification of things with the classification of relations.  For things that belong to the same class are, by implication, things of which the properties and modes of behaviour—­the co-existences and sequences—­are more or less the same; and the recognition of this sameness of co-existences and sequences is reasoning.  Whence it follows that the advance of classification is necessarily proportionate to the advance of generalisations.  Yet further, the notion of likeness, both in things and relations, simultaneously evolves by one process of culture the ideas of equality of things and equality of relations; which are the respective bases of exact concrete reasoning and exact abstract reasoning—­Mathematics and Logic.  And once more, this idea of equality, in the very process of being formed, necessarily gives origin to two series of relations—­those of magnitude and those of number:  from which arise geometry and the calculus.  Thus the process throughout is one of perpetual subdivision and perpetual intercommunication of the divisions.  From the very first there has been that consensus of different kinds of knowledge, answering to the consensus of the intellectual faculties, which, as already said, must exist among the sciences.

Let us now go on to observe how, out of the notions of equality and number, as arrived at in the manner described, there gradually arose the elements of quantitative prevision.