1014 / 1042
990
L
O
gClhcr at unequal angles.
F~r
lhis
1.11
figure, anfwcring
llill la his idea of a fqltlfe, \Vould be yet found wilhout
lhe properly alligned la il in tite propofition. BUI if he
comes aflerlVards la correa his nOlion, and render his
idea complm, he IVill then readily owa lhe truth and
jull"cfs of the oemonllmion.
\Ve Cee lherefore, lhal nOlhing contributes
Ca
much to
Ihe improvemenl and ceminty of human knowledge, as
Ihe having delerminate ideJ!, and keeping them aeddy
aod invariable in all our diCcourCes ¡nd reaConings abaut
them. And on lhis account it is, lhal nmhematicians
al\Va)'s begin by
dcfi~ing
lheir terms, and diltinélly un·
folding lhe nOlions they are inlcnded to expreCs. Hcnce
fueh as apply themCelves tO theCe !ludies, having exaélly
Ihe Came vie\Vs of lhiogs, and bringing always lhe very
fame ideas inlo
compariC~o,
rcadily diCmo lhe relalions
belIVeeo lhem.
WJen lhey havetaken lhis fira lIep, and made known
the ideas whoCe relalions they intend la invcaigm
¡
lheir
lle)(t care is, to lay dowa fome fel f evidenl lrulhs, which
may ferve as a foundalion'
Cor
lheir fUlure reaConings.
And here indecd lhey proceed Wilh remarkable circum·
fpeélion, admiuing no principies bUI whal Bow imm·e·
dialely from lbeir defioilions, and necelfarily force lhem·
íelves upon the mind. Thus a
cirdf
is a figure formed
by a right line, moving round fome fixed poinl in lhe
fame plane. The fi¡¡ed point round which lhe lioe is
fuppoCed la move, and where one of ilS e)(lremilies ter·
minales, is called lhe
unlrf
of lhe cirde. The olher
clmemilY. which is conceived la be canied round, until
it
retllrns la the point whence it
6rfl
Cet out, deCcribes a
curve running ioto itCelC, and termed lhe
circum!uenu.
AH
right lines drawo from the
centre
to the circumfer·
cnce, are called.
radii.
From theCe de6nitioos compa·
Ied. geometdcians derive this felf·evident trmh,
Thallbe
radii
,[
Ibe¡a/lle circlf are a/l fqua/ onf loanolhfr.
. We nowobCerve, that, in all propofitions, we either
~f!irm
or deny Come propenyof the idea thal confiitutes
Ihe fubjeél of our judgmenl, or we mainlain tbat fome·
thiog may be dooe or elfe{l/d. The firll Con are called
{p:culaliw
propofitions, as in the example memioned·a·
bove,
Ihe radii
of
Ihf fam< circlf are all equal onf lo
I1nolher.
The others are called
pra{Jica/,
for a reaCoo
too obvious to be mcntioned; tbus,
Ihal a righl/ine
ma, hf dra'Wn [rol» onf poinl lo an.lher,
is a praaical
propnfition, inaCmuchas it exprelf/s that Comethiogmay
be done.
From lhis twofuld confideration of propofitions ariCes
thetwofold divifionof malhematical principies inlo
oxionll
and
p.J1ulalfl.
Byan
oxi.m
they undertland any Celf·
cvident
fpec/llalive
trulh: as,
Ihal Ih< 'Who/e
ÍJ
gnal"
Ihan il1
1'.,11;
Ihal Ihingl <quallo one olld Ih< ¡ame
thin,~'
an fqualloont anolh".
But a
Celf.el'ident
pruc·
ticol
pro;11fition is what they call a
poJlu/ale.
Such are
thoCe
of
f.ue/id¡ Thal a¡nile righl lim
1110'
hf canlil/ud
dire(J& [or'Wardl: nal a circh
lna,
hf difcrihed ahoul
G
c.
aJIj cm/re 'Wilh any diflance.
Aod as, in In
axi.m,
t1:e
a~reement
or JiC:,greement b:tween the fubjeél and pre.
dlclle mutl come'unuer the immediale inCpeaion' of the
mind; fa, in a
poJlulalf,
not only the pofliGility of lhe
thing alfmed mutl be evidem at firtl view, bUl alCo lhe
manner in whicl. it may be effeéled. For where this
manner is not of ilCtlf apparent, lhe propofition comes
under the notion
oC
tbe dcmontlrable kinJ, and is treatcd
as Cuch by geommical wrilers. Tbus,
lOdra'W a right
/inf from Me poinl loallplhtr,
is alfumed by
Eue/id
as
a
pojlulalf,
becauCe the manner of Joing it is fa obvious
as to require no previous teaching. But then it i. oot
equally evideor,
ho'W
'1J/f
an loconjlru{J an fq uilal"al
Iriangle.
For lhis reaCon he advances it as a domontlra.
ble propofition, lays downrules for lhe exaél performance,
and at the fame lime proves, that if lhCe rules are fuI·
lowed, the figure will be junly dercribed.
This leads us totake ootice, !hat as
filffvidenl lrulhl
are dillioguiflled ioto diJferenr kinds, accordiog as lhey
are Cpeculative or praélical ; Co is it alCo with
dfmonf/ra.
hlf
propofitions. Ademonarable rpeculative prOpofilion,
is bymathematicians called a
thmem.
Such is !he 47tb
propofilion of the fira boa
k
of the
E/en"nll, viz. Ihal
in mrJ righloangled Iriangll, Ihf¡quare defcrihed upon
thejid, jtlólendiJJg Ihe righl.anglf ir
eq~al
lOholh Ihe
¡quam difcrihed upO»' Ihf jidtl conlaining Ihf rigbl'
ang/f.
00
the other hand, a demonllrable praélical pro o
pofition, is called a
prohltm
¡
as where
Elle/id
leaches
us
lo defcrihe a¡quarf upon agivtn righl·/im.
It
may not be amiCs to add, that befides the four kiods
of propofitioDS already mentioned, matllematicians have
alCo a fifth, known by lhe name of
corollariu.
Tbefe
are uCually fubjoined to
Ihmel1l1,
or
proh/wlI,
aDd differ
from them only in this; that they Bow (rom wbat is
there demonarated, in
Ca
obvious a maODer, as to diCcoo
ver
t~eir
dependence upon Ihe propofilioo ",hence they
are deduced, almoll as foon as propored. Thus
Euclid
baving demonllrated,
Ihal in every righl./ined Irianglf
alllhf Ihm anglu
la~tn
loCelhu are equallol'Wo righl
angltl ;
adds by way of corollary,
Ihal all Ihf Ihm
angla
of
anJ one IrianglelaAm logelb" are (qlla/ lo a/l
Ihf Ihru ang/u
of
any olhu Iriangl, lalen logtlher:
which is evideot at firll fight; becaufe iQ all cares !heyare
c~ual
to two right ones, aod things equal to one
~nd
the
fame thing are equal to
one
anolher.
Tl.e Ccholia of mathematicians
are
indifFerently aDo
nexed to definilions, propofitions, or coroliades ; aod
aoCwer theCame purpofes as annolatians upon a dallit au·
thor. For in them occafioo is takeo to explain wha(o
ever may appear intricate and obCcure in a train of rea o
foning; tOanCwer objeélioos; to teach lhe application
and uCes of propofitions; to lay ORen lhe original aod
h;lIory of the Cemal diCcowies made in the Ccieoce; .nd
in a word, to acquaint us with all fuch partieulars as de.·
Cerve to be koowo, whether coofidertd as po;nls of cu·
riofity or pr06t.
PART
JI.
OjREASO N [ NG.
¡.,
oCten happens. in compuing. ideas wgether, that view, efpeci,lIy if tl,ey are of Cuch a n,ture aS not
to
1heir 3grecmeot or diCagmmemcaooot be difcerned at ura admit of ao exac1 application ooe to anolher. When, for
IOOolllCC,