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#### Section 13

13. It has been remarked that the continuous is effectually
distinguished from the discrete by their possessing the one a
common, the other a separate, limit.

The same principle gives rise to the numerical
distinction between
odd and even; and it holds good that if there are differentiae
found in both contraries, they are either to be abandoned to the
objects numbered, or else to be considered as differentiae of the
abstract numbers, and not of the numbers manifested in the sensible
objects. If the numbers are logically separable from the
objects, that
is no reason why we should not think of them as sharing the same
differentiae.

But how are we to differentiate the continuous, comprising as it
does line, surface and solid? The line may be rated as of one
dimension, the surface as of two dimensions, the solid as of
three, if
we are only making a calculation and do not suppose that we are
dividing the continuous into its species; for it is an
invariable rule
that numbers, thus grouped as prior and posterior, cannot be brought
into a common genus; there is no common basis in first, second and
third dimensions. Yet there is a sense in which they would appear to
be equal- namely, as pure measures of Quantity: of higher and lower
dimensions, they are not however more or less quantitative.

Numbers have similarly a common property in their being numbers
all; and the truth may well be, not that One creates two, and two
creates three, but that all have a common source.

Suppose, however, that they are not derived from any source
whatever, but merely exist; we at any rate conceive them as being
derived, and so may be assumed to regard the smaller as taking
priority over the greater: yet, even so, by the mere fact of their
being numbers they are reducible to a single type.

What applies to numbers is equally true of magnitudes;
though here
we have to distinguish between line, surface and solid- the last
also referred to as "body"- in the ground that, while all are
magnitudes, they differ specifically.

It remains to enquire whether these species are themselves to be
divided: the line into straight, circular, spiral; the surface into
rectilinear and circular figures; the solid into the various solid
figures- sphere and polyhedra: whether these last should be
subdivided, as by the geometers, into those contained by triangular
and quadrilateral planes: and whether a further division of
the latter
should be performed.

Next: Section 14