hether he admits it or not--must really be parts of
the line, and that we combine this manifold on a principle involved in
the nature of straightness. Now suppose that the manifold given is the
parts AB, BC, CD, DE of the line AE. It is clearly only possible to
recognize AB and BC as contiguous parts of a straight line, if we
immediately apprehend that AB and BC form one line of which these
parts are identical in direction. Otherwise, we might just as well
join AB and BC at a right angle, and in fact at any angle; we need not
even make AB and BC contiguous.[20] Similarly, the relation of BC to
CD and of CD to DE must be just as immediately apprehended as the
parts themselves. Is there, however, any relation of which it could be
said that it is not given, and to which therefore Kant's doctrine
might seem to apply? There is. Suppose AB, BC, CD to be of such a size
that, though we can see AB and BC, or BC and CD, together, we cannot
see AB and CD together. It is clear that in this case we can only
learn that AB and CD are parts of the same straight line through an
inference. We have to infer that, because each is in the same straight
line with BC, the one is in the same straight line with the other.
Here the fact that AB and CD are in the same straight line is not
immediately apprehended. This relation, therefore, may be said not to
be given; and, from Kant's point of view, we could say that we
introduce this relation into the manifold through our activity of
thinking, which combines AB and CD together in accordance with the
principle that two straight lines which are in the same line with a
third are in line with one another. Nevertheless, this case is no
exception to the general principle that relations must be given
equally with terms; for we only become aware of the relation between
AB and CD, which is not given, because we are already aware of other
relations, viz. those between AB and BC, and BC and CD, which are
given. Relations then, or, in Kant's language, particular syntheses
must be said to be given, in the sense in which the elements to be
combined can be said to be given.

[20] In order to meet a possible objection, it may be pointed
out that if AB and BC be given in isolation, the contiguity
implied in referring to them as A_B_ and _B_C will not be
known.

Further, we can better see the nature of Kant's mistake in this
respect, if we bear in mind that Kant originally and rightly
introduced