them. The concept of twelve is by no means thought by merely
thinking of the combination of seven and five; and analyze this
possible sum as we may, we shall not discover twelve in the
concept. We must go beyond these concepts, by calling to our aid
some concrete image [Anschauung], i.e., either our five fingers,
or five points (as Segner has it in his Arithmetic), and we must
add successively the units of the five, given in some concrete
image [Anschauung], to the concept of seven. Hence our concept is
really amplified by the proposition 7 + 5 = I 2, and we add to
the first a second, not thought in it. Arithmetical judgments are
therefore synthetical, and the more plainly according as we take
larger numbers; for in such cases it is clear that, however
closely we analyze our concepts without calling visual images
(Anscliauung) to our aid, we can never find the sum by such mere
dissection.

     All principles of geometry are no less analytical. That a
straight line is the shortest path between two points, is a
synthetical proposition. For my concept of straight contains
nothing of quantity, but only a quality. The attribute of
shortness is therefore altogether additional, and cannot be
obtained by any analysis of the concept. Here, too, visualization
[Anschauung] must come to aid us. It alone makes the synthesis
possible.

     Some other principles, assumed by geometers, are indeed
actually analytical, and depend on the law of contradiction; but
they only serve, as identical propositions, as a method of
concatenation, and not as principles, e. g., a=a, the whole is
equal to itself, or a + b > a, the whole is greater than its
part. And yet even these, though they are recognized as valid
from mere concepts, are only admitted in mathematics, because
they can be represented in some visual form [Anschauung]. What
usually makes us believe that the predicate of such apodictic5
judgments is already contained in our concept, and that the
judgment is therefore analytical, is the duplicity of the
expression, requesting us to think a certain predicate as of
necessity implied in the thought of a given concept, which
necessity attaches to the concept. But the question is not what
we are requested to join in thought to the given concept, but
what we actually think together with and in it, though obscurely;
and so it appears that the predicate belongs to these concepts
necessarily indeed, yet not directly but indirectly by an added
visualization [Anschauung].

   Sect. 3. A Remark on the General Division of judgments into
                   Analytical and Synthetical

     This division is indispensable, as concerns the Critique of
human understanding, and therefore deserves to be called
classical, though otherwise it is of little use, but this is the
reason why dogmatic philosophers, who always seek the sources of
metaphysical judgments in Metaphysics itself, and not apart from
it, in the pure laws of reason generally, altogether neglected
this apparently obvious distinction. Thus the celebrated Wolf,
and his acute follower Baumgarten, came to seek the proof of the
principle of Sufficient Reason, which is clearly synthetical, in
the principle of Contradiction. In Locke's Essay, however, I find
an indication of my division. For in the fourth book (chap. iii.
Sect. 9, seq.), having discussed the various connections of
representations in judgments, and their sources, one of which he
makes -I identity and contradiction" (analytical judgments), and
another the coexistence of representations in a subject, he
confesses (Sect. 10) that our a priori knowledge of the latter is
very narrow, and almost nothing. But in his remarks on this
species of cognition, there is so little of what is definite, and
reduced to rules, that we cannot wonder if no one, not even Hume,
was led to make investigations concerning this sort of judgments.
For such general and yet definite principles are not easily
learned from other men, who have had them obscurely in their
minds. We must hit on them first by our own reflection, then we
find them elsewhere, where we could not possibly nave found them
at first, because the authors themselves did not know that such
an idea lay at the basis of their observations. Men who never
think independently have nevertheless the acuteness to discover
everything, after it has been once shown them, in what was said
long since, though no one ever saw it there before.

     Sect. 4. The General Question of the Prolegemena. - Is
                  Metaphysics at all Possible?

     Were a metaphysics, which could maintain its place as a
science, really in existence; could we say, here is metaphysics,
learn it, and it will convince you irresistibly and irrevocably
of its truth: this question would be useless, and there would
only remain that other question (which would rather be a test of
our acuteness, than a proof of the existence of the thing
itself), "How is the science possible, and how does reason come
to attain it?" But human reason has not been so fortunate in this
case. There is no single book to which you can point as you do to
Euclid, and say: This is Metaphysics; here you may find the
noblest objects of this science, the knowledge of a highest
Being, and of a future existence, proved from principles of pure
reason. We can be shown indeed many judgments, demonstrably
certain, and never questioned; but these are all analytical, and
rather concern the materials and the scaffolding for Metaphysics,
than the extension of knowledge, which is our proper object in
studying it (Sect 2). Even supposing you produce synthetical
judgments (such as the law of Sufficient Reason, which you have
never proved, as you ought to, from pure reason a priori, though