1000000030

idm are fuch as nave no rifible conneaion nor repug'

n,oce, anJ theréfore fumiOl OOt the pro?cr meaos of

traciog their agrcement or difagreemeot, there \Ve are

lilre thal f,icutillc.1 kool\'ledgc is 001 attainable. Bu!

\Vhere lhere is fome (OUOddlioo of reafooiog, I\'hich yet

amouOls 001 10 Ihe full evidence of demoollmioo, Ihere

lhe precrp!s of logic, by leaching us 10 determioe arigil!

of the degree of proof, and of \Vha! i5 lIill \Vaotiog tO

render it full aod complete, enable us tO make a due

e(limale of !hemeafum of pro'>abililY, aod 10 proponion

our a!Ten! 10 Ihe grouods on IVhich the propofitioo lIaods.

And lhis is all we can poffibly arrive al, or even ro much

as hope for, in the exercife of facullies [o impel'fea and

Iimited as ours.

We conclude it may no! be imeroper to take nOlice of

Ihe di!linllioo of denloollration iOlo

"irdl

and

indirdl.

dirrfl d"J/onjlralion

is, \Vhen b,egioniog "ith d,hoiti·

oos, relf·evideot propolitions, 01' knowo and allolVed

truths, we form a train of fyllogifms, aod combine them

in ' o orderly monner, contiouing the feries lhrough a

variely of (uccellive !leps, uOlil al .lall we arrive at a

fyllogifm, whofe conclufion is Ihe propofilioo 10 be de·

monllrated. Proofs of this kind lea ve no doubt or un.

cemioly behind Ihem, becaufe all Ihe feveral premi!Tes

being true, Ihe cooclufions mull be fo lOO, and of courfe

the very lan cooclufion or propofilioo lO be prove.d. The

other (pecics of demonllmioo is the

indirdl,

or, as i! is

fometimes called, the

of og9gica/.

The maoner of pro·

ceediog here is, by a!Tuming a propofilion wbich direélly

eonlradias lhat we mean 10 demonflrate, and theoce by

a continued lrain of reafoning, in the way of a direél de·

monfiratioo, deducing fome abfurdity or manifell uOlrulh.

For hereupoo IVe conclude lhallhe propofilion a!Tumed was

falfe, and Ihence again, by an immedim confequence, thal

the propofition 10 be demooflr311d is lrue. Thus

Eue/id,

in his lhird bnok, being tO demon!lrate.

/ha/ eire/u 'lJJhich

touch on, ano/hu in'IJJard& have no// he fR/lU cen/re,

aC·

fumes Ihe direa contrary tO this,

viz. Iha/ /h'l hove /he

f ame celllre,

aod thenee by an evident Irain of reafoniog

proves

/ha/ a par/

'qua/

/h, 'Who/, .

That fuppoli.

lioo thmfore, Jcadiog to Ihisabfurdity. he condudes lObe

f.lfe,

viz./hal circl%uchipg Ollt allo/hu i"'l'ard& ha"e

th, fame cmlre,

and thenee again immedialely infers

tha/ Ihey hove no//he fome cUl/re.

Now becaufe this manner of demoollration is ac·

counled by fome nOl ahogether fo clear aod fatisfaaory,

';'e

1h.1I

therefore eode,vour here 10 filt\V, that il equally

wilh lhe olher le,ds tO trulh and ccrtaioly. T lVo propo·

fitions are faid 10 be

conlradinory

ooe of aoother, when

Ihat which i5 a!Terled tObe in Ihe one is a!Tened not tO

be in the

Olh~r.

Thus the propofitions,

eirel" /ha/

tOl/ch one

an./h~r

in'IJJord& hav, /he jam, (m /re,

aod

eire/" /hql /ouchon, ano/ha in'Wardly have

1101

/he f,me

(¡'n/re,

are

eon/rndinori,,;

beeaufe the feeood arrertS

the direél contrary of IVhat i5 a!Tened in Ihe fidl. Now

,11

coolradia or)' propofilions Ihis holds univerfally,

Ihat ooe of them is necc!Tuilytrlle, aod Ihe other oecef·

f~rily

f,lfe . for if it be true, Ihat rirdes which lOuch

one

aO~lher

inlVardly J.ave not Ihe fame ceOfre, it is un·

avniJably

f~lf('

that Ihey have the f.,me cenlre. 00 Ihe

olher hand. if it he f.¡]fe thal lhey have Ihe(Jmc eenlre,

VOL.

11. N° 68.

roo !

il is nece!Torily lroe thal lhey have nOI Ihe fame ceotre.

Sioee Iherefore it is impoOible

fnr

them to be bOlh true

or bOlh falfe al lhe f,me time, il uoavoidably fOIJOIVS,

Ihal one i5 necc!Tarily

ue, aod lhe olher oecen:" ily falfe.

This Ihen being allolVed, if any 1100 contradiélory pro.

pofilions are allilmed, and ooe of Ihem cao lya clear

traio of reafoniog be

d~monllratcJ

10 be lalfe, it oece!Ta.

rily follows that Ihc olhcr is true. For as lhe one i.

oeceff."ily true, and the olher oece!Tal'ily falfe, \Vheo

\Ve

COOle 10 difeover IVhieh is the falfe propofition, IVe Iherc–

by alfo koolV the olher to be I:UC.

NolV this is precirelyIhe maoner of an indirea demon–

!lration. For there \Ve arrume a propofi lion, whieh di–

realy

eonlradi\~s

Ihat \Ve mean to demonarale, aod

haviog by a eonlioued feries of prnofs filewn il to be falfe,

thence iofer that ilS conlradiaory, or lhe propofition

be demoollraled, is true. As therefore lhis Infl conclu–

fino is

cm.in

aod uoavoid.ble. let

liS

oext irqllire, afler

IVhat maooer IVe come 10 he f:llisfied of the falfehood

the a!Tumed propofilion, lhat fo no poffible doubt may

remaio as to lhe force aod v.lidity of demonHralions of

lhis kind. The mano.r theo is plainly Ihis. Begiooiog

IVith the a!Tumed propofilion, we, by Ihe hdp ofdefioilioos.

felf·evidenl trmhs, or propnfitions already caablilhed,

eootinue a feries of rea(ooiog in the lVay of a direa de–

monaralion, until at Itngth we mive al fome ablurdilY

or knolVn falfehood. Thus

Eue/id,

from Ihe fuppofition

that cirdes touchiog one another iOlVardly have the fa me

centre, deduces

Iha/ a parl

'qtla/ lOIh, 'Whole..

Sinee

Iherefore, by a due and orderly procefs of reafooing, IVe

come al laa 10 a falfe cooclufion, it is maoifea tlm all

the premi!Tes canOOI be lruc. For lVere all Ihe premilfes

true, lhe Jafl conclofion mu!l be fo too. Now as to all

Ihe olher premi!Tes made u(e of in Ihecourfe of reafooing,

theyare maoifefl and knolVn truths by fuppofilion, as be–

iog eilher definitions, felr evidenl propofitioos, or truths

previouflyeaablilhed. The a!Tumed propofilion is thal

onlyas to which any doubt or unCertainly remains. That

alone therefore can be falre, aod indeed, from whal has

been already Ihewn, mufl unavoidably be fo. And thus

IVe fee, lhat, in iodireél demon!lrations, tWO eootradillory

propofilions beiog laid dOlVn, one of IVhich is demooflra–

red tO befalfe, the olher. IVhich is always Ihe propofilion

to be proved, mufl nece!Tarily be lrUC; fo Ihat here, as

lVeJl as in Ihe direél way of proof, lVe

arriv~

clear

aod lalisfaélory kon\Vledge of trllth.

This is uoiverfally the melhod of rcafoning in all a–

pogogical

indirea demooflralioos; bUl if any propo·

lilloo is a!TuOled , from IVhieh io a dir<éI trnio

re~ron

ing we can deduce ils eootr,diaory, the propofilioo fo

aflilmed is falf<, aoo the conlrndiélory 00<Hile. For ir

we fuppofe the a!Tumed propofitioo tO be Hue, Ihen,

finee aJl lhc other pr<mifl<s ¡h. t ewr Ihe

~emonnrntion

are alfo true, lVe Ih, lI have a (ericsof re,(uoing, coofi!t–

ing wholly of true prcmi!Tcs ; \\ heoce Ihe I.IHeoodufion

ór eOO1radillory of Ihe "numcd pl0l'ofilion

mu{f he

true

likcwife. So

t1m

I>y Ihis n" ,ns we Iholl!d have tlVO

coolradiaOl y propolirions bolll Irtle at rhe fallle liOle,

IVhieh is mallifdlly impoli!>le. Th, arruOled plupofiliol1

thcrefr're, whcnce 11llS ahf"rdilY

1I0"'!.

mlln

oee('O:II'ily

be f.l fc, and confequcntly ils con;r.lui/lory, wlllch is

l O

hzrc