METAPHYSIC S. 
217 
tiffn), all their empirical part, i. e. all that belongs to feel¬ 
ing, time and J'pace Hill remain, and are therefore the pure 
int ait ions a priori, which formed the foundation of the em¬ 
pirical. But pure intuitions a priori, are nothing more 
than the fonhs of our fenfitive faculty, which muft of courfe 
precede all empirical intuition or apprehension of real 
objedts. 
The problem of the prefent fedtion is therefore folved. 
Pure mathematics, as fynthetical knowledge a priori, are 
only pollible, becaufe they refer to no other objedts than 
thole of the fenfes; that is, merely to the empirical in¬ 
tuition, which has always a pure intuition (of Space and 
Time) for its foundation a priori ; for, as Time and Space 
are the very forms of Jenfe, they mult precede, to render 
the real appearance of objects pollible. The faculty of 
intuition a priori, therefore, does not concern the matter 
or jbijation in the Phenomena, for this conftitutes their 
empirical part; but merely their form. If it Ihould Hill 
be thought, that Time and Space are determinations ad¬ 
hering to the things in themfelves, and not merely to their 
repreientation by the fenlitive faculty, then I would alk 
how it is pollible to know a priori, and before the things 
are given to us, how their intuitions lhall be qualified ; 
and that they mull necelfarily have the properties of time 
and J'pace. This, however, is eafily conceivable ; and all 
difficulty vaniffies the moment we admit that Time and 
Space are the formal conditions of the fenfitive faculty it- 
1bIf, and that the objedts of fenfe are therefore mere phe¬ 
nomena ; for then indeed it is not wonderful that the 
form of the phenomena (the pure intuition), originating 
m the Faculty of Senfe, Ihould be reprefentable by the mind 
itfelf, in other words, a priori. 
To illuftrate and confirm this, we need but examine 
the ul'ual, and indeed necefi'ary, procedure of the geome¬ 
trician. All his demonilrations of the thorough equality 
of two given figures come at lalt to this, that they cover 
one another ; which is evidently nothing more than afyn¬ 
thetical pojition, proved by the immediate intuition; but this 
intuition is pure, and a priori; for, if it were not, the po- 
lition would not have apodidtical but only empirical cer¬ 
tainty, and we could only fay we have oblerved it lo. 
That J'pace confidered as an infinite whole has three di- 
menfions, or that fpace generally fpeaking has no more, 
relts upon the pofition, that only three lines can interfefl 
each other perpendicularly in one point. But this po/ition 
cannot he deduced from conception, but relts immediately 
upon intuition, and indeed upon pure intuition a priori, 
becaufe it is apodidtically certain. That we can require 
a line to be continued to infinity, or a feries of fpaces ad 
infinitum to be palled over by motion, pre-fuppofe an in¬ 
tuitive repreientation of Space and Time, fincc it is an 
unbounded one, and could therefore never be deduced 
from conceptions. Pure intuitions a priori then really form 
the hafts of the mathematics, and render their fynthetical 
and apodidtically-certain pofitions pollible. The poffibi- 
lity of pure mathematics, therefore, can never be con¬ 
ceived without the prefuppofition that the objects which 
afi'edt our fenfes, imply only how they appear to us, and 
not what they are in themfelves. 
Thole who cannot relinquilh the idea that Time and 
Space are real qualities inherent in the things themfelves, 
may exercife their penetration upon the following para¬ 
dox ; and, if they Ihould find its lolution impracticable, 
they may perhaps give up their prejudice, for a moment 
at lealt, and confider the degradation of Time and Space 
to mere forms of lenlible intuition as polieffing fome foun¬ 
dation. 
If two things are entirely the fame in all the determi¬ 
nations of quantity and quality, it necelfarily follows 
that one may, in all refpects and in all cafes, be lubftituted 
for the other, without producing the lealt perceptible dif¬ 
ference. This is actually the calc in plane geometrical 
figures : but fpherical figures, notwithftanding their com¬ 
plete internal agreement, may Itill not admit of being 
lubftituted for each other. For inftance, two Spherical 
Vol, £V. No. 1037. 
Triangles, one in each hemil'phere, having an arc of the 
equator as a common bale, may be completely equal with 
refpedt to their lides and angles, and yet not admit of 
being fubltituted for each other; i. e. placed one upon 
the other. There is here an internal difference between 
the triangles which no human intelledt can denote by any 
internal mark, anil which manifefts itfelf only by an ex¬ 
ternal relation in fpace. I will adduce a more familiar ex¬ 
ample. What can more exadtly referable the form of my 
hand, or be in all refpedts more equal to if, than its image 
in the glal's ?_ yet I cannot fubftitute the hand as feen in 
the glals for its original. If the one is'a right, the other 
is a left, hand. Now there is no internal difference be¬ 
tween thefe two forms as far as the underltanding is con¬ 
cerned ; and yet there are internal differences which our 
fenfes teach us; for the left hand cannot be inclofed 
within the fameiioundaries as the right ; they are there¬ 
fore not congruent. The glove of the one cannot be. 
ufed upon the other. What now is the lolution of this ? 
Thefe objedts are not reprefentations of tilings in them¬ 
felves, and as pure underltanding would conceive them, 
but they are lenlible intuitions, i. e. phenomena, the pbi- 
fibility of which reits upon the relation of certain things 
unknown in themfelves, to fomething elfe, namely, to our 
fenfitive faculty. Now , J'pace is the form of external intui¬ 
tion ; and the internal determination of any part of (pace 
can only be attained by determining its external relation 
to the whole; that is, the part is only pollible by a refer¬ 
ence to the whole, which does not take place in things 
in themfelves, as objects of pure underltanding, but-merely 
as phenomena. We are therefore unable by their mere 
conception to comprehend how thefe objects, which are 
fimilar and equal, Ihould itill not be congruent, but only 
by their relation to the right and left hand, which imme¬ 
diately refers to tile intuition. 
Pure Mathematics, and particularly Geometry, have no 
objective reality, but as th'&y refer to objedts of Senfe 
that is, objedts as they appear in our fenlible repreienta¬ 
tion. Hence it follows, that the pofitions of Geometry 
are not determinations of a phantom merely, and which 
cannot therefore be referred to real objects; for, as they 
are valid and necefi'ary with refpeCt to fpace, they mult be 
equally lo to all that can be met with in fpace, becaufe 
this is the very form of all external phenomena, under 
which alone the objedts of fenfe can be given. The Scnji - 
time Faculty, the forms of which Geometry takes as a foun¬ 
dation, is that upon which the poffibility of external phe¬ 
nomena refts. Thefe phenomena, therefore, can never 
contain any thing but what geometry admits of. The- 
cafe would be quite otherwife were the fenles to repre- 
fent objedts as they are in themfelves. For it would h)' 
no means follow from the mere reprefentation of fpace, 
which the Geometrician aifumes a priori as a foundation 
together with all its properties, and all that can be in¬ 
ferred from it; it would by no means follow, that all this 
Ihould be found exadtly in the lame manner in nature. 
We might be induced to confider the fpace of the Geome¬ 
trician as a mere fidtion, and have no confidence in its 
objedtive validity, becaufe there is no reafon why the 
things of nature ihould necelfarily agree with the image 
previoully formed of them in our fancy. But, if this image 
or rather this formal intuition, be the elfential property 
of our fenfitive faculty, which does not reprefent things 
in themfelves, but only their phenomena ; it naturally fol¬ 
lows, that all external objedls mull: agree with the pofi¬ 
tions of geometry in the Itridtelt manner. 
It will always remain a remarkable phenomenon in the 
the hiltory of Philol’ophy, that there has been a time 
when even mathematicians themfelves, men who were at 
the fame time philolophers, laboured under a doubt, not 
indeed of the correctnel's of geometrical pofitions, lb far 
as they merely concerned the conception of fpace, but of 
the objedtive validity of this conception itfelf, and of the 
right of applying the geometrical determination of fpace- 
to nature, apprehending that a Hue iu nature .might be 
3 K, com poled 
