side of his triangle； thus forming two adjacent angles which are
together equal to two right angles。 He then divides the exterior of
these angles； by drawing a line parallel with the opposite side of the
triangle； and immediately perceives that be has thus got an exterior
adjacent angle which is equal to the interior。 Proceeding in this way；
through a chain of inferences； and always on the ground of
intuition； he arrives at a clear and universally valid solution of the
question。
But mathematics does not confine itself to the construction of
quantities （quanta）； as in the case of geometry； it occupies itself
with pure quantity also （quantitas）； as in the case of algebra；
where plete abstraction is made of the properties of the object
indicated by the conception of quantity。 In algebra； a certain
method of notation by signs is adopted； and these indicate the
different possible constructions of quantities； the extraction of
roots； and so on。 After having thus denoted the general conception
of quantities； according to their different relations； the different
operations by which quantity or number is increased or diminished
are presented in intuition in accordance with general rules。 Thus；
when one quantity is to be divided by another； the signs which
denote both are placed in the form peculiar to the operation of
division； and thus algebra； by means of a symbolical construction of
quantity； just as geometry； with its ostensive or geometrical
construction （a construction of the objects themselves）； arrives at
results which discursive cognition cannot hope to reach by the aid
of mere conceptions。
Now； what is the cause of this difference in the fortune of the
philosopher and the mathematician； the former of whom follows the path
of conceptions； while the latter pursues that of intuitions； which
he represents； a priori； in correspondence with his conceptions？ The
cause is evident from what has been already demonstrated in the
introduction to this Critique。 We do not； in the present case； want to
discover analytical propositions； which may be produced merely by
analysing our conceptions… for in this the philosopher would have
the advantage over his rival； we aim at the discovery of synthetical
propositions… such synthetical propositions； moreover； as can be
cognized a priori。 I must not confine myself to that which I
actually cogitate in my conception of a triangle； for this is
nothing more than the mere definition； I must try to go beyond that；
and to arrive at properties which are not contained in； although
they belong to； the conception。 Now； this is impossible； unless I
determine the object present to my mind according to the conditions；
either of empirical； or of pure； intuition。 In the former case； I
should have an empirical proposition （arrived at by actual measurement
of the angles of the triangle）； which would possess neither
universality nor necessity； but that would be of no value。 In the
latter； I proceed by geometrical construction； by means of which I
collect； in a pure intuition； just as I would in an empirical
intuition； all the various properties which belong to the schema of
a triangle in general； and consequently to its conception； and thus
construct synthetical propositions which possess the attribute of
universality。
It would be vain to philosophize upon the triangle； that is； to
reflect on it discursively； I should get no further than the
definition with which I had been obliged to set out。 There are
certainly transcendental synthetical propositions which are framed
by means of pure conceptions； and which form the peculiar
distinction of philosophy； but these do not relate to any particular
thing； but to a thing in general； and enounce the conditions under
which the perception of it may bee a part of possible experience。
But the science of mathematics has nothing to do with such
questions； nor with the question of existence in any fashion； it is
concerned merely with the properties of objects in themselves； only in
so far as these are connected with the conception of the objects。
In the above example； we merely attempted to show the great
difference which exists between the discursive employment of reason in
the sphere of conceptions； and its intuitive exercise by means of
the construction of conceptions。 The question naturally arises： What
is the cause which necessitates this twofold exercise of reason； and
how are we to discover whether it is the philosophical or the
mathematical method which reason is pursuing in an argument？
All our knowledge relates； finally； to possible intuitions； for it
is these alone that present objects to the mind。 An a priori or
non…empirical conception contains either a pure intuition… and in this
case it can be constructed； or it contains nothing but the synthesis
of possible intuitions； which are not given a priori。 In this latter
case； it may help us to form synthetical a priori judgements； but only
in the discursive method； by conceptions； not in the intuitive； by
means of the construction of conceptions。
The only a priori intuition is that of the pure form of phenomena…
space and time。 A conception of space and time as quanta may be
presented a priori in intuition； that is； constructed； either alone
with their quality （figure）； or as pure quantity （the mere synthesis
of the homogeneous）； by means of number。 But the matter of
phenomena； by which things are given in space and time； can be
presented only in perception； a posteriori。 The only conception
which represents a priori this empirical content of phenomena is the
conception of a thing in general； and the a priori synthetical
cognition of this conception can give us nothing more than the rule
for the synthesis of that which may be contained in the
corresponding a posteriori perception； it is utterly inadequate to
present an a priori intuition of the real object； which must
necessarily be empirical。
Synthetical propositions； which relate to things in general； an a
priori intuition of which is impossible； are transcendental。 For
this reason transcendental propositions cannot be framed by means of
the construction of conceptions； they are a priori； and based entirely
on conceptions themselves。 They contain merely the rule； by which we
are to seek in the world of perception or experience the synthetical
unity of that which cannot be intuited a priori。 But they are
inpetent to present any of the conceptions which appear in them
in an a priori intuition； these can be given only a posteriori； in
experience； which； however； is itself possible only through these
synthetical principles。
If we are to form a synthetical judgement regarding a conception； we
must go beyond it； to the intuition in which it is given。 If we keep
to what is contained in the conception； the judgement is merely
analytical… it is merely an explanation of what we have cogitated in
the conception。 But I can pass from the conception to the pure or
empirical intuition which corresponds to it。 I can proceed to
examine my conception in concreto； and to cognize； either a priori
or a posterio； what I find in the object of the conception。 The
former… a priori cognition… is rational…mathematical cognition by
means of the construction of the conception； the latter… a
posteriori cognition… is purely empirical cognition； which does not
possess the attributes of necessity and universality。 Thus I may
analyse the conception I have of gold； but I gain no new information
from this analysis； I merely enumerate the different properties
which I had connected with the notion indicated by the word。 My
knowledge has gained in logical clearness and arrangement； but no
addition has been made to it。 But if I take the matter which is
indicated by this name； and submit it to the examination of my senses；
I am enabled to form several synthetical… although still empirical…
propositions。 The mathematical conception of a triangle I should
construct； that is； present a priori in intuition； and in this way
attain to rational…synthetical cognition。 But when the
transcendental conception of reality； or substance； or power is
presented to my mind； I find that it does not relate to or indicate
either an empirical or pure intuition； but that it indicates merely
the synthesis of empirical intuitions； which cannot of course be given
a priori。 The synthesis in such a conception cannot proceed a
priori… without the aid of experience… to the intuition which
corresponds to the conception； and； for this reason； none of these
conceptions can produce a determinative synthetical proposition；
they can never present more than a principle of the synthesis* of
possible empirical intuitions。 A transcendental proposition is；
therefore； a synthetical cognition of reason by means of pure
conceptions and the discursive method； and it renders possible all
synthetical unity in empirical cognition； though it cannot present
us with any intuition a priori。
*In the case of the conception of cause； I do really go beyond the
empirical conception of an event… but not to the intuition which
presents this conception in concreto； but only to the time…conditions；
which may be found in