ble unless they
occur in immediate succession, it results that exactness of equality is
ascertainable in proportion to the closeness of the compared things.
Hence the fact that when we wish to judge of two shades of colour
whether they are alike or not, we place them side by side; hence the
fact that we cannot, with any precision, say which of two allied sounds
is the louder, or the higher in pitch, unless we hear the one
immediately after the other; hence the fact that to estimate the ratio
of weights, we take one in each hand, that we may compare their
pressures by rapidly alternating in thought from the one to the other;
hence the fact, that in a piece of music we can continue to make equal
beats when the first beat has been given, but cannot ensure commencing
with the same length of beat on a future occasion; and hence, lastly,
the fact, that of all magnitudes, those of _linear extension_ are those
of which the equality is most accurately ascertainable, and those to
which by consequence all others have to be reduced. For it is the
peculiarity of linear extension that it alone allows its magnitudes to
be placed in _absolute_ juxtaposition, or, rather, in coincident
position; it alone can test the equality of two magnitudes by observing
whether they will coalesce, as two equal mathematical lines do, when
placed between the same points; it alone can test _equality_ by trying
whether it will become _identity_. Hence, then, the fact, that all exact
science is reducible, by an ultimate analysis, to results measured in
equal units of linear extension.

Still it remains to be noticed in what manner this determination of
equality by comparison of linear magnitudes originated. Once more may we
perceive that surrounding natural objects supplied the needful lessons.
From the beginning there must have been a constant experience of like
things placed side by side--men standing and walking together; animals
from the same herd; fish from the same shoal. And the ceaseless
repetition of these experiences could not fail to suggest the
observation, that the nearer together any objects were, the more visible
became any inequality between them. Hence the obvious device of putting
in apposition things of which it was desired to ascertain the relative
magnitudes. Hence the idea of _measure_. And here we suddenly come upon
a group of facts which afford a solid basis to the remainder of our
argument; while they also furnish strong evidence i