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 it concerns us here to observe is, that from familiarity with organic forms there simultaneously arose the ideas of simple equality, and equality of relations.

At the same time, too, and out of the same mental processes, came the first distinct ideas of number.  In the earliest stages, the presentation of several like objects produced merely an indefinite conception of multiplicity; as it still does among Australians, and Bushmen, and Damaras, when the number presented exceeds three or four.  With such a fact before us we may safely infer that the first clear numerical conception was that of duality as contrasted with unity.  And this notion of duality must necessarily have grown up side by side with those of likeness and equality; seeing that it is impossible to recognise the likeness of two things without also perceiving that there are two.  From the very beginning the conception of number must have been as it is still, associated with the likeness or equality of the things numbered.  If we analyse it, we find that simple enumeration is a registration of repeated impressions of any kind.  That these may be capable of enumeration it is needful that they be more or less alike; and before any absolutely true numerical results can be reached, it is requisite that the units be absolutely equal.  The only way in which we can establish a numerical relationship between things that do not yield us like impressions, is to divide them into parts that do yield us like impressions.  Two unlike magnitudes of extension, force, time, weight, or what not, can have their relative amounts estimated only by means of some small unit that is contained many times in both; and even if we finally write down the greater one as a unit and the other as a fraction of it, we state, in the denominator of the fraction, the number of parts into which the unit must be divided to be comparable with the fraction.