tions of
limits_. It is evident, therefore, that the terms _quantity_ and
_finitude_ express the same attribute, namely, _limitation_--the former
relatively, the latter absolutely; for quantity is limitation considered
with relation to some standard of measurement, and finitude is
limitation considered simply in itself. The sphere of quantity,
therefore, is absolutely identical with the sphere of the finite; and
the phrase _infinite quantity_, if strictly construed, is a
contradiction in terms.

The result thus attained by considering abstract quantity is
corroborated by considering concrete and discrete quantities. Such
expressions as _infinite sphere, radius, parallelogram, line,_ and so
forth, are self-contradictory. A sphere is limited by its own periphery,
and a radius by the centre and circumference of its circle. A
parallelogram of infinite altitude is impossible, because the limit of
its altitude is assigned in the side which must be parallel to its base
in order to constitute it a parallelogram. In brief, all figuration is
limitation. The contradiction in the term _infinite line_ is not quite
so obvious, but can readily be made apparent. Objectively, a line is
only the termination of a surface, and a surface the termination of a
solid; hence a line can not exist apart from an extended quantity, nor
an infinite line apart from an infinite quantity. But as this term has
just been shown to be self-contradictory, an infinite line can not exist
objectively at all. Again, every line is extension in one dimension;
hence a mathematical quantity, hence mensurable, hence finite; you must
therefore, deny that a line is a quantity, or else affirm that it is
finite.

The same conclusion is forced upon us, if from geometry we turn to
arithmetic. The phrases _infinite number, infinite series, infinite
process_, and so forth, are all contradictory when literally construed.
Number is a relation among separate unities or integers, which,
considered objectively as independent of our cognitive powers, must
constitute an exact sum; and this exactitude, or synthetic totality, is
limitation. If considered subjectively in the mode of its cognition, a
number is infinite only in the sense that it is beyond the power of our
imagination or conception, which is an abuse of the term. In either case
the totality is fixed; that is, finite. So, too, of _series_ and
_process_. Since every series involves a succession of terms or numbers,
and