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mathematical； and the latter dynamical principles。* It must be
observed； however； that by these terms I mean just as little in the
one case the principles of mathematics as those of general
（physical） dynamics in the other。 I have here in view merely the
principles of the pure understanding； in their application to the
internal sense （without distinction of the representations given
therein）； by means of which the sciences of mathematics and dynamics
become possible。 Accordingly； I have named these principles rather
with reference to their application than their content； and I shall
now proceed to consider them in the order in which they stand in the
table。
*All combination （conjunctio） is either composition （compositio）
or connection （nexus）。 The former is the synthesis of a manifold；
the parts of which do not necessarily belong to each other。 For
example； the two triangles into which a square is divided by a
diagonal； do not necessarily belong to each other； and of this kind is
the synthesis of the homogeneous in everything that can be
mathematically considered。 This synthesis can be divided into those of
aggregation and coalition； the former of which is applied to
extensive； the latter to intensive quantities。 The second sort of
combination （nexus） is the synthesis of a manifold； in so far as its
parts do belong necessarily to each other； for example； the accident
to a substance； or the effect to the cause。 Consequently it is a
synthesis of that which though heterogeneous； is represented as
connected a priori。 This combination… not an arbitrary one… I
entitle dynamical because it concerns the connection of the
existence of the manifold。 This； again； may be divided into the
physical synthesis； of the phenomena divided among each other； and the
metaphysical synthesis； or the connection of phenomena a priori in the
faculty of cognition。
1。 AXIOMS OF INTUITION。
The principle of these is： All Intuitions are Extensive
Quantities。
PROOF。
All phenomena contain； as regards their form； an intuition in
space and time； which lies a priori at the foundation of all without
exception。 Phenomena； therefore； cannot be apprehended； that is；
received into empirical consciousness otherwise than through the
synthesis of a manifold； through which the representations of a
determinate space or time are generated； that is to say； through the
composition of the homogeneous and the consciousness of the
synthetical unity of this manifold （homogeneous）。 Now the
consciousness of a homogeneous manifold in intuition； in so far as
thereby the representation of an object is rendered possible； is the
conception of a quantity （quanti）。 Consequently； even the perception
of an object as phenomenon is possible only through the same
synthetical unity of the manifold of the given sensuous intuition；
through which the unity of the composition of the homogeneous manifold
in the conception of a quantity is cogitated； that is to say； all
phenomena are quantities； and extensive quantities； because as
intuitions in space or time they must be represented by means of the
same synthesis through which space and time themselves are determined。
An extensive quantity I call that wherein the representation of
the parts renders possible （and therefore necessarily antecedes） the
representation of the whole。 I cannot represent to myself any line；
however small； without drawing it in thought； that is； without
generating from a point all its parts one after another； and in this
way alone producing this intuition。 Precisely the same is the case
with every； even the smallest； portion of time。 I cogitate therein
only the successive progress from one moment to another； and hence； by
means of the different portions of time and the addition of them； a
determinate quantity of time is produced。 As the pure intuition in all
phenomena is either time or space； so is every phenomenon in its
character of intuition an extensive quantity； inasmuch as it can
only be cognized in our apprehension by successive synthesis （from
part to part）。 All phenomena are； accordingly； to be considered as
aggregates； that is； as a collection of previously given parts；
which is not the case with every sort of quantities； but only with
those which are represented and apprehended by us as extensive。
On this successive synthesis of the productive imagination； in the
generation of figures； is founded the mathematics of extension； or
geometry； with its axioms； which express the conditions of sensuous
intuition a priori； under which alone the schema of a pure
conception of external intuition can exist； for example； 〃be tween two
points only one straight line is possible；〃 〃two straight lines cannot
enclose a space；〃 etc。 These are the axioms which properly relate only
to quantities （quanta） as such。
But； as regards the quantity of a thing （quantitas）； that is to say；
the answer to the question： 〃How large is this or that object？〃
although； in respect to this question； we have various propositions
synthetical and immediately certain （indemonstrabilia）； we have； in
the proper sense of the term； no axioms。 For example； the
propositions： 〃If equals be added to equals； the wholes are equal〃；
〃If equals be taken from equals； the remainders are equal〃； are
analytical； because I am immediately conscious of the identity of
the production of the one quantity with the production of the other；
whereas axioms must be a priori synthetical propositions。 On the other
hand； the self…evident propositions as to the relation of numbers； are
certainly synthetical but not universal； like those of geometry； and
for this reason cannot be called axioms； but numerical formulae。
That 7 ＋ 5 = 12 is not an analytical proposition。 For neither in the
representation of seven； nor of five； nor of the composition of the
two numbers； do I cogitate the number twelve。 （Whether I cogitate
the number in the addition of both； is not at present the question；
for in the case of an analytical proposition； the only point is
whether I really cogitate the predicate in the representation of the
subject。） But although the proposition is synthetical； it is
nevertheless only a singular proposition。 In so far as regard is
here had merely to the synthesis of the homogeneous （the units）； it
cannot take place except in one manner； although our use of these
numbers is afterwards general。 If I say： 〃A triangle can be
constructed with three lines； any two of which taken together are
greater than the third；〃 I exercise merely the pure function of the
productive imagination； which may draw the lines longer or shorter and
construct the angles at its pleasure。 On the contrary； the number
seven is possible only in one manner； and so is likewise the number
twelve； which results from the synthesis of seven and five。 Such
propositions； then； cannot be termed axioms （for in that