t 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.

It is, indeed, true, that by an evidently modern process of abstraction,
we occasionally apply numbers to unequal units, as the furniture at a
sale or the various animals on a farm, simply as so many separate
entities; but no true result can be brought out by calculation with
units of this order. And, indeed, it is the distinctive peculiarity of
the calculus in general, that it proceeds on the hypothesis of that
absolute equality of its abstract units, which no real units possess;
and that the exactness of its results holds only in virtu