consequently cannot be demonstrated a posteriori； and is； at the
same time； under the obligation of presenting in general but
sufficient tests； the conditions under which objects can be given in
harmony with those conceptions； otherwise they would be mere logical
forms； without content； and not pure conceptions of the understanding。
Our transcendental doctrine of the faculty of judgement will contain
two chapters。 The first will treat of the sensuous condition under
which alone pure conceptions of the understanding can be employed…
that is； of the schematism of the pure understanding。 The second
will treat of those synthetical judgements which are derived a
priori from pure conceptions of the understanding under those
conditions； and which lie a priori at the foundation of all other
cognitions； that is to say； it will treat of the principles of the
pure understanding。
TRANSCENDENTAL DOCTRINE OF THE FACULTY OF JUDGEMENT
OR； ANALYTIC OF PRINCIPLES。
CHAPTER I。 Of the Schematism at of the Pure Conceptions
of the Understanding。
In all subsumptions of an object under a conception； the
representation of the object must be homogeneous with the
conception； in other words； the conception must contain that which
is represented in the object to be subsumed under it。 For this is
the meaning of the expression： 〃An object is contained under a
conception。〃 Thus the empirical conception of a plate is homogeneous
with the pure geometrical conception of a circle； inasmuch as the
roundness which is cogitated in the former is intuited in the latter。
But pure conceptions of the understanding； when pared with
empirical intuitions； or even with sensuous intuitions in general； are
quite heterogeneous； and never can be discovered in any intuition。 How
then is the subsumption of the latter under the former； and
consequently the application of the categories to phenomena；
possible？… For it is impossible to say； for example： 〃Causality can be
intuited through the senses and is contained in the phenomenon。〃… This
natural and important question forms the real cause of the necessity
of a transcendental doctrine of the faculty of judgement； with the
purpose； to wit； of showing how pure conceptions of the
understanding can be applied to phenomena。 In all other sciences；
where the conceptions by which the object is thought in the general
are not so different and heterogeneous from those which represent
the object in concreto… as it is given； it is quite unnecessary to
institute any special inquiries concerning the application of the
former to the latter。
Now it is quite clear that there must be some third thing； which
on the one side is homogeneous with the category； and with the
phenomenon on the other； and so makes the application of the former to
the latter possible。 This mediating representation must be pure
（without any empirical content）； and yet must on the one side be
intellectual； on the other sensuous。 Such a representation is the
transcendental schema。
The conception of the understanding contains pure synthetical
unity of the manifold in general。 Time； as the formal condition of the
manifold of the internal sense； consequently of the conjunction of all
representations； contains a priori a manifold in the pure intuition。
Now a transcendental determination of time is so far homogeneous
with the category； which constitutes the unity thereof； that it is
universal and rests upon a rule a priori。 On the other hand； it is
so far homogeneous with the phenomenon； inasmuch as time is
contained in every empirical representation of the manifold。 Thus an
application of the category to phenomena bees possible； by means of
the transcendental determination of time； which； as the schema of
the conceptions of the understanding； mediates the subsumption of
the latter under the former。
After what has been proved in our deduction of the categories； no
one； it is to be hoped； can hesitate as to the proper decision of
the question； whether the employment of these pure conceptions of
the understanding ought to be merely empirical or also transcendental；
in other words； whether the categories； as conditions of a possible
experience； relate a priori solely to phenomena； or whether； as
conditions of the possibility of things in general； their
application can be extended to objects as things in themselves。 For we
have there seen that conceptions are quite impossible； and utterly
without signification； unless either to them； or at least to the
elements of which they consist； an object be given； and that；
consequently； they cannot possibly apply to objects as things in
themselves without regard to the question whether and how these may be
given to us； and； further； that the only manner in which objects can
be given to us is by means of the modification of our sensibility；
and； finally； that pure a priori conceptions； in addition to the
function of the understanding in the category； must contain a priori
formal conditions of sensibility （of the internal sense； namely）；
which again contain the general condition under which alone the
category can be applied to any object。 This formal and pure
condition of sensibility； to which the conception of the understanding
is restricted in its employment； we shall name the schema of the
conception of the understanding； and the procedure of the
understanding with these schemata we shall call the schematism of
the pure understanding。
The schema is； in itself； always a mere product of the
imagination。 But； as the synthesis of imagination has for its aim no
single intuition； but merely unity in the determination of
sensibility； the schema is clearly distinguishable from the image。
Thus； if I place five points one after another。。。。 this is an image of
the number five。 On the other hand； if I only think a number in
general； which may be either five or a hundred； this thought is rather
the representation of a method of representing in an image a sum
（e。g。； a thousand） in conformity with a conception； than the image
itself； an image which I should find some little difficulty in
reviewing； and paring with the conception。 Now this
representation of a general procedure of the imagination to present
its image to a conception； I call the schema of this conception。
In truth； it is not images of objects； but schemata； which lie at
the foundation of our pure sensuous conceptions。 No image could ever
be adequate to our conception of a triangle in general。 For the
generalness of the conception it never could attain to； as this
includes under itself all triangles； whether right…angled；
acute…angled； etc。； whilst the image would always be limited to a
single part of this sphere。 The schema of the triangle can exist
nowhere else than in thought； and it indicates a rule of the synthesis
of the imagination in regard to pure figures in space。 Still less is
an object of experience； or an image of the object； ever to the
empirical conception。 On the contrary； the conception always relates
immediately to the schema of the imagination； as a rule for the
determination of our intuition； in conformity with a certain general
conception。 The conception of a dog indicates a rule； according to
which my imagination can delineate the figure of a four…footed
animal in general； without being limited to any particular
individual form which experience presents to me； or indeed to any
possible image that I can represent to myself in concreto。 This
schematism of our understanding in regard to phenomena and their
mere form； is an art； hidden in the depths of the human soul； whose
true modes of action we shall only with difficulty discover and
unveil。 Thus much only can we say： 〃The image is a product of the
empirical faculty of the productive imagination… the schema of
sensuous conceptions （of figures in space； for example） is a
product； and； as it were； a monogram of the pure imagination a priori；
whereby and according to which images first bee possible； which；
however； can be connected with the conception only mediately by
means of the schema which they indicate； and are in themselves never
fully adequate to it。〃 On the other hand； the schema of a pure
conception of the understanding is something that cannot be reduced
into any image… it is nothing else than the pure synthesis expressed
by the category； conformably； to a rule of unity according to
conceptions。 It is a transcendental product of the imagination； a
product which concerns the determination of the internal sense；
according to conditions of its form （time） in respect to all
representations； in so far as these representations must be
conjoined a priori in one conception； conformably to the unity of
apperception。
Without entering upon a dry and tedious analysis of the essential
requisites of transcendental schemata of the pure conceptions of the
understanding； we shall rather proceed at once to give an
explanation of them according to the order of the categories； and in
connection therewith。
For the external sense the pure image of all quantities
（quantorum） is space； the pure image of all objects of sense in
general； is t