685 / 1042
G E O M E T
:rt
Y.
tl8;
'In tmting this urerul fubjea, we (hall divide at into (he feoood, the application of Ihefe
princi'p1es to
the
IIlCIt–
'~'o
pms;
the fid! containlog.Ihe general princiPies; and Iumicn ofIurface., Iolids,
&e.
PAR
T l.
GEN ERAL PR 1NC lPLES OF GEOMETRY.
Art.
I.A
point is rhat which is not made np of pam,
·or which is of itfelf iodivifible.
2.
A lioe is a leogth without breadth, as B--
3
T he extremities of • lioe are poims; as the eXlre·
enities of the lioe AB, are the points A aod B,
~f.
l .
Plale XCIIL
4.
Jf
the lioe AB be the nearel! diOaoee between itl
txtremesA and B, then it is caBed
aJlrail line,
as A ·B;
but if it be not the nearea diaanee, theo il is ealled a
curve li..,
as AB; fig.
l .•
5.
A furfaee is thlt whieh is eonfidered as baving ooly
length and breadth, bUI no thieknelS, as B,
fig.
2.
6.
The terms or boundaries of a furfaee are·lines.
7.
A plain furfaee is thal whieh lies equaBy between
itl extremes.
8. The inclination between tlVO lines meeliog
eme
ano
other (provided they do not make one eontinued line,)
or the openiog between them,
is
caBed
;m
angle;
thu!
the inclioation of the lioe AB to the line CH(fig.
3.)
meeting one another at B. or nle openiog between the two
lines AB and CS, is ealled an angle.
9.
When the lin¡os formiog the aogle are right lioes,
then it is called a
righl-lilled tingle,
as A, ·fig.
4.
if one of
them be right and the other eurved, it is ealled a
lIIixtd
Qngle,
as B, 6g S. if borh ir them be eurvee!, it is ealled
a
eurv..Jilled a"glt,
as
·C. fig.
6.
10.
Jf
a right line AB fall upon another OC, (fig.
7.)
fo as to incline ",irher tO 'one fide Dor tO Ihe other: but
make the angles ABO, ARC,
0 0
eaeh fide equal to one
lnother; then the lioe
AS
is faid to be
perpwdicu/.r
to
the line DC, aod the tWO angles are ealled
right.ang/e¡.
11.
An obtufe angle is that whieh is gre.ter
th~n
a
right ooe, as A, fig. 8; and an aeute aogle, that whicb
is lefs than a right one, as B, lig. 9.
12.
Jf
a right line DC be faOeoed at one
uf
its ends 'C,
and the other eod D be earried quite round, then tbe
fpaee eompreheoded is ealled a
árcle;
Ihe curve line de·
feribed by Ihe point D, is ealled the
ptriphtrJ
or
eireum–
Jmnce
of the eirele: the fixed po;nt C is ealled the
cenlre
of il. Fig.
10 .
13.
The deferibing line CD is ealled the
r4diuI, vh.
ftny lioe drawn from the eemre to the eireumferenee:
whenee all radii of the fame or cqu,1eireles are equa!.
14.
Any IlOe drawo through the centre, aod termina·
ted bOlh ways by the ei reumferenee, is ealled a
di."',I" ,
as RD is adiammr of che eirele HADE. And the dia·
meterdil'ides the eirele and eireumferenee into til'O equa!
¡.am, and is
dou~le
the radius.
15
The eireumfereoee of every cirde is fuppofed lO
be divided into 360 equ.1pans, caBed
drgrm;
and eaeh
8egree is dil'idCd into 60 equal pam, ealled
lIIilluta ;
¡~d
.aeh minule into 60 equal partS
callcd, f C</Idl
j
ami
.VO L ..
!J.
No 55 .
;¡
thefe into
IhirJl, ¡'urlhl, Bee.
Ihefe p,ru being
great~
or lefs aeco,ding as the ,adius is.
16.
Aoy pan of I·he eireumferenee is ealled an
arch.
or
are:
and is ealled an ore of as many degrees
as
it con.
tains pans of the 3'60, into whieh the eireumferenee wa,
divided: Thus
iI
A]) be the
t
of ¡he eireumferenee, tben
the .re AD is an are of
45
degrees.
"7. A lioe drawn 'from ooe <nd of an are to the other,
is ealled a
ehord,
aod is Ihe mearure of the are; thus the
right line AB is ,he ehord of the are ADB, 6g.
11.
18.
Aoy pan of aeirele cUt off by a ehord, is caBed a
f egmml;
lhus lhe fpaee eomprehended bellveen the
ehord AB and cireumfereoee ADB (whieh is cut off by
the ehord AB) is ealled a fegmen!. Wheoee it is plain,
IjI, Tbat aU ehords divide the eirele into t\Vo feg.
mer.!s.
2dly,
The lefs the ebord i" tbe more uocquaJ are Ihe
fegments, and
e conlra.
3dlj',
When the ehord is greatell,
viz.
w'heo it is
¡
diameter, tbeo the fegmems are equal,
viz.
eaeh a fe.
mieirele.
)9.
Any pan of a eirele .(Iefs than a Iemieircle) con·
tained between tlVO "dii and ao are, is called a
fflO(
:
thuslhe fpaee eontaioed betweeo the tlVO radii, AC, BC,
aod the are
A13,
is ealled the fetlor: ·fig.
12.
20.
The right line of any are, is a lioe drawn .perpen.
dieular from ooe eod of lhe are, to a diameter drawn
through the other end of the fame are; thus
(lig.
13 .)
AD is the right floe
uf
the are AB, it being a lioe drawQ
{romA, the one end of the are AB, perpendicular toCB,
a diameter pafling tbrougb
B,
the orher eod of the are
AB.
Nol\' lhe fines Oanding on the fame diameter, Oill in–
ereafe tiB they come to the eenm, and then beeomirgthe
radius, it is plain that fue radius EC is the greateO pof–
fi ble fioe, and for that reafon it is ealled the
whol, jine.
Sinee the whole·fine EC moO
he
perpeodieular
to the a diameter FB (by def.
20.)
therefore produ.
cing the diameter EG, the twO di,meters FH, EG, mufl
erofs one anotber al right , ogles, aod fo the eir.
eumferenee of the eirele muO be divided hy Ihemin·
tO four partl EB, HG , GF, and FE, and thefe four
parlS are equal to one aoolher (by def.
10.)
and fo EB a
quadraot, or fourrh pan cf the eireumferenee; thcrdore
the radius 'fC is always the fine of the quadr,nt,
oc
founh pan of the eirele EB.
Sines are faid to be of fo many degrees, as Ihe are
tontaios p'rts of the 360, ioto IVhieh the cireum(e,enee
is fuppofed
10
~e
dil'ided; fothe radiu, heing lhe filieof
a quadrant, or founh pan of Ihe cireumren nce, \\ hicl>
eontaios 90 degrlCS (Ihe founh pan of 360
J,
Iherdu"
the radius muft be the fioc of 90 degrecs.
t
71\
~ J. ll,e