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necessity　which　I　could　never　learn　from　experience。　On　the　other
hand；　though　at　first　I　do　not　at　all　include　the　predicate　of
weight　in　my　conception　of　body　in　general；　that　conception　still
indicates　an　object　of　experience；　a　part　of　the　totality　of
experience；　to　which　I　can　still　add　other　parts；　and　this　I　do　when　I
recognize　by　observation　that　bodies　are　heavy。　I　can　cognize
beforehand　by　analysis　the　conception　of　body　through　the
characteristics　of　extension；　impenetrability；　shape；　etc。；　all
which　are　cogitated　in　this　conception。　But　now　I　extend　my　knowledge；
and　looking　back　on　experience　from　which　I　had　derived　this
conception　of　body；　I　find　weight　at　all　times　connected　with　the
above　characteristics；　and　therefore　I　synthetically　add　to　my
conceptions　this　as　a　predicate；　and　say；　〃All　bodies　are　heavy。〃　Thus
it　is　experience　upon　which　rests　the　possibility　of　the　synthesis
of　the　predicate　of　weight　with　the　conception　of　body；　because　both
conceptions；　although　the　one　is　not　contained　in　the　other；　still
belong　to　one　another　（only　contingently；　however）；　as　parts　of　a
whole；　namely；　of　experience；　which　is　itself　a　synthesis　of
intuitions。
　　But　to　synthetical　judgements　a　priori；　such　aid　is　entirely
wanting。　If　I　go　out　of　and　beyond　the　conception　A；　in　order　to
recognize　another　B　as　connected　with　it；　what　foundation　have　I　to
rest　on；　whereby　to　render　the　synthesis　possible？　I　have　here　no
longer　the　advantage　of　looking　out　in　the　sphere　of　experience　for
what　I　want。　Let　us　take；　for　example；　the　proposition；　〃Everything
that　happens　has　a　cause。〃　In　the　conception　of　〃something　that
happens；〃　I　indeed　think　an　existence　which　a　certain　time
antecedes；　and　from　this　I　can　derive　analytical　judgements。　But　the
conception　of　a　cause　lies　quite　out　of　the　above　conception；　and
indicates　something　entirely　different　from　〃that　which　happens；〃
and　is　consequently　not　contained　in　that　conception。　How　then　am　I
able　to　assert　concerning　the　general　conception…　〃that　which
happens〃…　something　entirely　different　from　that　conception；　and　to
recognize　the　conception　of　cause　although　not　contained　in　it；　yet　as
belonging　to　it；　and　even　necessarily？　what　is　here　the　unknown　=　X；
upon　which　the　understanding　rests　when　it　believes　it　has　found；
out　of　the　conception　A　a　foreign　predicate　B；　which　it　nevertheless
considers　to　be　connected　with　it？　It　cannot　be　experience；　because
the　principle　adduced　annexes　the　two　representations；　cause　and
effect；　to　the　representation　existence；　not　only　with　universality；
which　experience　cannot　give；　but　also　with　the　expression　of
necessity；　therefore　pletely　a　priori　and　from　pure　conceptions。
Upon　such　synthetical；　that　is　augmentative　propositions；　depends
the　whole　aim　of　our　speculative　knowledge　a　priori；　for　although
analytical　judgements　are　indeed　highly　important　and　necessary；
they　are　so；　only　to　arrive　at　that　clearness　of　conceptions　which
is　requisite　for　a　sure　and　extended　synthesis；　and　this　alone　is　a
real　acquisition。

　　V。　In　all　Theoretical　Sciences　of　Reason；　Synthetical　Judgements
　　　　　〃a　priori〃　are　contained　as　Principles。

　　1。　Mathematical　judgements　are　always　synthetical。　Hitherto　this
fact；　though　incontestably　true　and　very　important　in　its
consequences；　seems　to　have　escaped　the　analysts　of　the　human　mind；
nay；　to　be　in　plete　opposition　to　all　their　conjectures。　For　as
it　was　found　that　mathematical　conclusions　all　proceed　according　to
the　principle　of　contradiction　（which　the　nature　of　every　apodeictic
certainty　requires）；　people　became　persuaded　that　the　fundamental
principles　of　the　science　also　were　recognized　and　admitted　in　the
same　way。　But　the　notion　is　fallacious；　for　although　a　synthetical
proposition　can　certainly　be　discerned　by　means　of　the　principle　of
contradiction；　this　is　possible　only　when　another　synthetical
proposition　precedes；　from　which　the　latter　is　deduced；　but　never　of
itself　which
　　Before　all；　be　it　observed；　that　proper　mathematical　propositions
are　always　judgements　a　priori；　and　not　empirical；　because　they
carry　along　with　them　the　conception　of　necessity；　which　cannot　be
given　by　experience。　If　this　be　demurred　to；　it　matters　not；　I　will
then　limit　my　assertion　to　pure　mathematics；　the　very　conception　of
which　implies　that　it　consists　of　knowledge　altogether　non…empirical
and　a　priori。
　　We　might；　indeed　at　first　suppose　that　the　proposition　7　＋　5　=　12　is
a　merely　analytical　proposition；　following　（according　to　the　principle
of　contradiction）　from　the　conception　of　a　sum　of　seven　and　five。
But　if　we　regard　it　more　narrowly；　we　find　that　our　conception　of
the　sum　of　seven　and　five　contains　nothing　more　than　the　uniting　of
both　sums　into　one；　whereby　it　cannot　at　all　be　cogitated　what　this
single　number　is　which　embraces　both。　The　conception　of　twelve　is　by
no　means　obtained　by　merely　cogitating　the　union　of　seven　and　five；
and　we　may　analyse　our　conception　of　such　a　possible　sum　as　long　as　we
will；　still　we　shall　never　discover　in　it　the　notion　of　twelve。　We
must　go　beyond　these　conceptions；　and　have　recourse　to　an　intuition
which　corresponds　to　one　of　the　two…　our　five　fingers；　for　example；　or
like　Segner　in　his　Arithmetic　five　points；　and　so　by　degrees；　add
the　units　contained　in　the　five　given　in　the　intuition；　to　the
conception　of　seven。　For　I　first　take　the　number　7；　and；　for　the
conception　of　5　calling　in　the　aid　of　the　fingers　of　my　hand　as
objects　of　intuition；　I　add　the　units；　which　I　before　took　together　to
make　up　the　number　5；　gradually　now　by　means　of　the　material　image
my　hand；　to　the　number　7；　and　by　this　process；　I　at　length　see　the
number　12　arise。　That　7　should　be　added　to　5；　I　have　certainly
cogitated　in　my　conception　of　a　sum　=　7　＋　5；　but　not　that　this　sum　was
equal　to　12。　Arithmetical　propositions　are　therefore　always
synthetical；　of　which　we　may　bee　more　clearly　convinced　by　trying
large　numbers。　For　it　will　thus　bee　quite　evident　that；　turn　and
twist　our　conceptions　as　we　may；　it　is　impossible；　without　having
recourse　to　intuition；　to　arrive　at　the　sum　total　or　product　by
means　of　the　mere　analysis　of　our　conceptions。　just　as　little　is　any
principle　of　pure　geometry　analytical。　〃A　straight　line　between　two
points　is　the　shortest；〃　is　a　synthetical　proposition。　For　my
conception　of　straight　contains　no　notion　of　quantity；　but　is　merely
qualitative。　The　conception　of　the　shortest　is　therefore　fore　wholly
an　addition；　and　by　no　analysis　can　it　be　extracted　from　our
conception　of　a　straight　line。　Intuition　must　therefore　here　lend
its　aid；　by　means　of　which；　and　thus　only；　our　synthesis　is　possible。
　　Some　few　principles　preposited　by　geometricians　are；　indeed；
really　analytical；　and　depend　on　the　principle　of　contradiction。
They　serve；　however；　like　identical　propositions；　as　links　in　the
chain　of　method；　not　as　principles…　for　example；　a　=　a；　the　whole　is
equal　to　itself；　or　（a＋b）　》　a；　the　whole　is　greater　than　its　part。　And
yet　even　these　principles　themselves；　though　they　derive　their
validity　from　pure　conceptions；　are　only　admitted　in　mathematics
because　they　can　be　presented　in　intuition。　What　causes　us　here
monly　to　believe　that　the　predicate　of　such　apodeictic　judgements
is　already　contained　in　our　conception；　and　that　the　judgement　is
therefore　analytical；　is　merely　the　equivocal　nature　of　the
expression。　We　must　join　in　thought　a　certain　predicate　to　a　given
conception；　and　this　necessity　cleaves　already　to　the　conception。
But　the　question　is；　not　what　we　must　join　in　thought　to　the　given
conception；　but　what　we　really　think　therein；　though　only　obscurely；
and　then　it　bees　manifest　that　the　predicate　pertains　to　these
conceptions；　necessarily　indeed；　yet　not　as　thought　in　the
conception　itself；　but　by　virtue　of　an　intuition；　which　must　be
added　to　the　conception。
　　2。　The　science　of　natural　philosophy　（physics）　contains　in　itself
synthetical　judgements　a　priori；　as　principles。　I　shall　adduce　two
propositions。　For　instance；　the　proposition；　〃In　all　changes　of　the
material　world；　the　quantity　of　matter　remains　unchanged〃；　or；　that；
〃In　all　munication　of　motion；　action　and　reaction　must　always　be
equal。〃　In　both　of　these；　not　only　is　the　necessity；　and　therefore
their　origin　a　priori　clear；　but　also　that　they　are　synthetical
propositions。　For　in　the　conception　of　matter；　I　do　not　cogitate　its
permanency；　but　merely　its　presence　in　space；　which　it　fills。　I
therefore　really　go　out　of　and　beyond　the　conception　of　matter；　in
order　to　think　on　to　it　something　a　priori；　which　I　did　not　think　in
it。　The　proposition　is　therefore　not　analytical；　but　synthetical；
and　nevertheless　conceived　a　priori；　and　so　it　is　with　regard　to　the
other　propositions　of　the　pure　part　of　natural　philosophy。
　　3。　As　to　metaphysics；　even　if　we　look　upon　it　merely　as　an　attempted
science；　yet；　from　the　nature　of　human　reason；　an　indispensable　one；
we　find　that　it　must　contain　synthetical　propositions　a　priori。　It
is　not　merely　the　duty　of　metaphysics　to　dissect；　and　thereby
analytically　to　illustrate　the　conceptions　which　we　form　a　priori　of
things；　but　we　seek　to　widen　the