META P 
rntnt, when affirmative, is already thought in the con¬ 
ception of the fubjeft itfelf: this predicate cannot there¬ 
fore be denied of a fubjeCl without a contradiction. The 
predicate in an analytical but negative judgment is alfo 
neceflarily excluded from its fubjeCt by the fame princi¬ 
ple of contradiction; as, in the following pofition, No 
body is unextended. 
All analytical portions are therefore judgments a priori, 
though their conceptions be empirical. For inftance, Gold 
is a yellow metal. In order to know this, I require no 
new experience; the conception I already have, of gold 
implies that this body is yellow, and a metal; I have 
only to analyze it. 
C. Synthetical Judgments require a different Principle from 
that'of Contradiction ; namely, the Principle of Sufficient 
Reafon. 
That we have fynthetical judgments li pefteriori, whofe 
origin is empirical, is evident; but that there are others 
alfo which are a priori, and which fpring from pure Un¬ 
derfunding and ReaJ'on, is no lefs true. Both however 
agree in this, that they cannot arife folely from the prin¬ 
ciple of Analyfis, that is, of contradiction. No judgments 
indeed mult offend againft this principle; but it does 
not follow, therefore, that all judgments are derived 
from it. Judgments of Experience, and Mathematical 
Judgments, are always Synthetical. 
1. Judgments of Experience are always fynthetical. 
To ground an analytical judgment upon experience would 
be abfurd, as we never quit the conception in order to 
form this judgment, and therefore do not require the 
evidence of experience. That Body is extended is a pofi¬ 
tion quite firm, a priori; and we need not appeal to ex¬ 
perience, fince all the requifites to the judgment are 
already in the conception; and, according to the mere 
principle of contradiction, we are confcious even of the 
neccfjity of the judgment, which is more than experience 
could ever teach. 
2. Mathematical Judgments are always fynthetical. 
This pofition (though indifputably true, and highly 
important in its confequences) feems to have hitherto 
cfcaped the cbfervation of the analyzers of human Rea¬ 
fon, and indeed to be direCtly oppofite to all their fup- 
pofitions. Finding that the conclufions in the mathe¬ 
matics are proved, according to the principle of contra¬ 
diction, which indeed gives them their apodiftical cer¬ 
tainty, they perfuaded themfelves that the principles alfo 
of the mathematics are attained in the fame manner: but 
herein th£y greatly erred. A fynthetical pofition may 
certainly be confidered, according to the principle of 
contradiction, as prefuppofing another fynthetical pofi¬ 
tion, from which it can be inferred, but not abfolutely, 
as originating in that principle. 
We muff firlt remark, that mathematical pofitions, pro¬ 
perly fpeaking, are always judgments a priori, and not 
empirical, becaufe they polfefs neceffiity , which never can 
be derived from experience. But, if this be not granted, 
we may limit "the pofition to pure Mathematics, whofe 
definition implies that they are not empirical, but pure 
knowledge a priori. 
One would think that the pofition 7-1-5=12 were a 
mere analytical pofition, which proceeded, according to 
the principle of contradiction, from the conception of 
the fum of feven and five. But, when we confider this 
pofition more clofely, we find that the conception of the 
fum of 7 and 5 contains only the union of both the 
numbers into a fingle one, without any anticipation of 
what that fingle number may be which includes both. 
The conception of twelve is not by any means yet formed 
by the mere thought of uniting feven and five; and no 
analyfis of the conception of Inch a pofjible f um can ever 
difeover in it the conception twelve. We muff go beyond 
thefe conceptions, and call to our afiiltance the intuition 
which correfponds to one-of them ; for inftance, our five 
fingers, or, as in arithmetic, five dots, and add luccef- 
Vol. XV. No. 103C. 
H Y S I C S. 213 
fively to the conception feven, the unities contained in 
the given intuition of five. We extend our conception 
of 7+5, therefore, when we fay that it equals 12, and 
really produce a new conception, which was not thought 
in the former. In other words, arithmetical pofitions are 
always fynthetical, which may be rendered more evident 
by taking larger numbers; it wili then be clear, that 
however we may examine our conception of the addition 
of two numbers together, we never fhall obtain their 
fum by its mere analyfis, but muff have recourfe to a new 
intuition. 
In the fame manner the principles of pure geometry 
are fynthetical. That aJlraight line, is tieJhortft difiance 
between two points, is a fynthetical pofition. For the con¬ 
ception Jlraight contains nothing of quantity, but merely 
a quality. The conception Jhortefi muft be wholly added, 
and cannot by any analyfis be' difeovered in that of a 
Jlraight line. Intuition muft confequently be called to 
our aid, by which alone fynthq/is is poflible. 
Some of the principles which the geometrician pre- 
fuppofes are indeed analytical, and reft upon the prin¬ 
ciple of contradiction ; but they are ufed only as identi¬ 
cal pofitions for the fake of method, and not as princi¬ 
ples: for inftance, a—a, the whole is equal to itfelf, or 
« + i. e. the whole is greater than its part. And 
even thefe, though they are valid as mere conceptions, 
are admitted into the mathematics only becaufe they can 
be viewed in the intuition. What makes it appear that 
the predicate of fuch apodiCtical judgments is already 
applied in the conception of their fubjeft, and that the 
judgment is confequently analytical, is merely an ambi* 
guity of expreffion. We are compelled to add to a given 
conception a certain predicate, and this neceffity already 
adheres to the conception. But the queftion is not what 
we mufl think , in addition to a given conception, but 
what we actually do think, in it, however cbfcurely. Thus 
it is evident, that that predicate which muft be applied to 
a conception belongs to it indeed neceflarily, but not im¬ 
mediately; that is to fay, only by means of an intuition 1 . 
This divifion of Judgments into Analytical and Syn¬ 
thetical, is indifpenfible to a Critic of the Human Under- 
ftanding. In any other view it does not appear to be of 
very great importance. And this I conceive to be the 
reafon why the Dogmatical Philofophers who fought the 
origin of metaphyfical conceptions in thofe conceptions 
themfelves, but never in the pure laws of Reafon in ge¬ 
neral, negleCted this divifion, which feems naturally to 
prefent itlelf; and why the celebrated Wolf, and his pe¬ 
netrating fucceflor Baumgarten, fought the proof of the 
principle of Jufficient Reafon, which is evidently fynthetical, 
in the principle of contradiction. On the other hand, as 
early as Locke’s Efiiiy on the Human Underftanding, we 
meet with a hint of this divifion. In the 4th book, 3d 
chapter, feCt. 9, &c. &c. after having fpoken of the diffe • 
rent reprelentations in judgments and of their lources, 
one of which he coniiders to be identity or contradiction, 
(analytical judgments,) the other the exiftence of repre¬ 
lentations in a lubjedt (fynthetical judgments).; lie ad¬ 
mits, feCt. 10. that our knowledge of thefe fynthetical 
judgments is very confined, and almoft nothing. But 
what he fays upon this remarkable kind of knowledge 
is fo very indefinite, that we need not wonder that it has 
induced no one, not even Hume, to refleft. upon pofi¬ 
tions of this nature. Univerfal and definite principles 
are not eafily to be learned from thole to whom they ap¬ 
peared but obfeurely; but, after we have attained them 
by our own refleCtioh, we may perhaps difeover traces of 
them in the writings of others who did not know that 
their remarks had fuch principles for a foundation. 
Thofe, however, who do not think for themfelves, are 
often veiy acute, after a thing has been fliown to them, 
in difeovering it lomewhere elle in what has been laid 
before, though previoufly no one could perceive it. 
It is fufficient here to Hate, that all Metaphyfical know¬ 
ledge muft be not only a priori, but fynthetical; it may, 
3 I however, 
