blest mathematicians, and the most
persevering Hamiltono-mastix of the day, maintains the
applicability of the metaphysical notion of infinity to
mathematical magnitudes; but with an assumption which
unintentionally vindicates Hamilton's position more
fully than could have been done by a professed disciple.
"I shall assume," says Professor De Morgan, in a paper
recently printed among the _Transactions of the
Cambridge Philosophical Society_, "the notion of
infinity and of its reciprocal infinitesimal: that a
line can be conceived infinite, and therefore having
points at an infinite distance. Image apart, which we
cannot have, it seems to me clear that a line of
infinite length, without points at an infinite distance,
is a contradiction." Now it is easy to show, by mere
reasoning, without any image, that this assumption is
equally a contradiction. For if space is finite, every
line in space must be finite also; and if space is
infinite, every point in space must have infinite space
beyond it in every direction, and therefore cannot be at
the greatest possible distance from another point. Or
thus: Any two points in space are the extremities of the
line connecting them; but an infinite line has no
extremities; therefore no two points in space can be
connected together by an infinite line.

In fact, it is the "concrete reality," the "something infinite," and not
the mere abstraction of infinity, which is only conceivable as a
negation. Every "something" that has ever been intuitively present to my
consciousness is a something finite. When, therefore, I speak of a
"something infinite," I mean a something existing in a different manner
from all the "somethings" of which I have had experience in intuition.
Thus it is apprehended, not positively, but negatively--not directly by
what it is, but indirectly by what it is not. A negative idea is not
negative because it is expressed by a negative term, but because it has
never been realised in intuition. If infinity, as applied to space, means
the same thing as being greater than any finite space, both conceptions
are equally positive or equally negative. If it does not mean the same
thing, then, in conceiving a space greater than any finite space, we do
not conceive an infinite space.

Mr. Mill's next string of c