700 / 1042
~6
G E O M
cipl~s
drpenJ innumerable
othcr' ~?erations
\Vhich a ceo·
1Hwician
IVlII
lind ou t of hinlel r, as is vcry obvio1l9,
How~va,
fome 01'
th~f~
opemions require fu ch exaélnefs
in the 1I'0rk, and \Vithout it are fo
lia61~
tO errors,
lh~t,
c-rleriJ p-"ihul,
the folloll'ing operatiolls, which'are per·
form~d
by a lrigonolllwical eakulalion, are 10 be pre·
ferred ; rct eould \Vel nOl omit thofe lbol'e, being UlOn
eafy in praétiee, and moa clear and el'ident 10 thefe \'Iho
h~l'e
only lhe lirfl ekmcnls of geomelry. EUI if you are
provided Wilh in(lruments, lhe following operalicns are
more lO be relied upon. \\'e-do nOI ir.úll on lhe taGell
cafes lO IhoCe who are
~(dlcd
in plain lriconommy,
whieh is
ind~ed
neeeOary lO any one II'ho \\'ould apply
himfclf tOpraélice. See 'TRl C·C
N~
M
H~
Y.
PRO P O S 1 T I ON X.
Flc . 13 ,
r o dtfcrib< Ih, confrruflion and ufo ofth<
gmnrlrica/ q/ladral:I.-
The geommieal quadraol is lhe
¡ounh part of a eirele divlded into ninely degrees, ,10
whieh lWOfights are ad.lplcd, wilh a perpendicular or
plumb line hancin.g from lhe eenlre. The gentral u(e of
1t is for inl'eiligaling angles in a vertical plane, compre·
.hended under right "nes going from the centre oí lhe in·
llrumenl, one of whieh is horizonlal, and lhe olher is
direéled to fome "ifible point. This inllrument is made
of any folid mauer as \\'ood, eopper,
&c.
PRO P O
S
I
T
ION
Xl.
Flc.
14.
ro d,[crib, and ./Ide ufo
of
Ih<grao
phom,I<r.-
The graphometer is a femieircle made of any
hard maner, of IVood, for cxample,.or brafs, divided inlo
ISo
degrees
j
fo fixed on a
fulcr uln,
bymeans uf a brafs
ball and Coeket, thal il eaÍlly turos aboul, ano retains a·
ny Gtuation
j
two fighlS are fixed on ilS diameler. At
Ihe ceone there is commonly a magnetieal oeedle in a
box. There is likewife a Oloveable {ultr, whieH tUfns
round the centle, aod retains any Gtualion.given·il. The
uCe of il ¡,s tO obCtr.ve.any ancle, ....hofe verte" -is at ¡he
cenlre
oC
the inflrument in any plane, (lhough it is mo(l
commonly horizontal, or oearly fo), and tO find how
ma·
nydegrees it contains.
PRO P
.o
S 1 T
J
O N XII.
F.c.
15.
and
1-6.
r o defcrih< Ihe lIIalln" in which
cng/tI ar<
IIuafNreá
~y
o.quodron' or graphom<I<rr–
Let tbere be an angle in
a
vertical plane, eomprehended
between a line parallcllo the horizon
HK,
and the right
Jine RA, eoming Crom any remarkable point
0'[
a towcr
(ir hill, or from
t~
flHl,
moon, or a (lar. Supp0fe that
t:lis angle RAH
~
10
be meafured by the quadrant: let
Ihe in(lrumenl be plaeed in lhe vertical plane, fo as thal
rhe centre Amay be-in Ihe angular foint : and let rhe
¡¡ghls be direéled towards lhe objeél at.R, (by lhe help
of lhe ray eoming from il, if il be 1he furoor mcon, or
by the help of the I'irual ray, if il is any thing elfe), lhe
degrees and Rlinutes
j n
the are BC cut·offby lhe perpen·
dieular, ",ill meafu re lhe angle RAH required. For,
{,0m lhe make.d lhe quadranl, DAD is a Tighl a:lgle;
Iherefore BAR is likewife.rifbl, being equ;l
Hl
it. BUI,
becaufe HK is horizontal, and AC pcrp¡ndieular, HAC
"il! be
a
.ieht angle
j
and therdor" ¡qual airo 10 BAR.
';¡o'rom thofe
an~l(s fobtr~él
lhe part HAB Ihat i, eomnlOn
.10
bOlh; and ¡here \Vil! rcmain lhe angle HAC eqaal 10
ib:
angle
E.hU...DUI !lie a.e BC is lhe n¡eafult of lhe
E
T
1t
Y.
.3ngl" llAC
¡
eonCe'quently,
il.islikel'life rhe mcafure
of
·the :Ingle RAH.
Note, Tilal lhe remaining are on lhe quadrant DC i,
.rhe me.rore
01
lile angle
RAZ,
eomprehen.led bellVeeQ
the forcfaid rigiot linc itA and AZ whieh poinls tOIhe
zenilh.
l et it now 'be required to meafure lhe angle ACB
(rog.
16)
inany-plane, eomprehended between lhe right.
l'lles !lC and liC, drawn from lwO poinls A aod
H,
10
lile place of fkt·ion
C.
'Let lhe graphomeler be plm d
al C, fllpported by ilS
fu /mili'
(as was /hown above)
¡
and let lhe immoveable ('ghls on lhe fide of the inaru–
menl DE be direéled tOwards lhe point A
j
,nd likewife
.(while the inflrument remains immoveabe) let lhe Gghls
of Ihe ruler
FG
(whieh i9 mo. eable abOUIlhe centre C)
be direéled to lhe POiOl B.
lt
is evideol, lhallhe move·
ablt ruler eUlS off an are DH, whieh is the meafure of
lhe angle AC13
Co~ght.
Moreover, by lhe fame me·
thod, the
ine~nation
of
GE,
or of
-rG ,
may be obCmcd
Wilh lhe meridi,1O linel whieh is pointed OUt by lhe mago
oelie needle inclofed in the box, and is mOI'eable aboul
Ihe centre of the inllrumenl, and lhe mealore of this iD–
d ioalion or angle found ·in Jegr.ees.
PR O P O
S 1 T
ION
Xl!J.
Flc . 17.
r om,o/uN nn QmJlibi< h<Íght hy Iht g'o–
"wrica/ quodranl
- By Ihe 12lh prop. of Ibis pan, lel
theangle C be found by means of the quadrant. 'fheo
in rhe lriangle ABC, right.angled
~t
B, (BC being fup.
pofed the horizontal dillanee of the obfervator from lhe
!ower), haviog lhe angle at C, and lhe fieJe BC, there·
quired height
EA
will be
f~und
by lhe 3d cafe {lf plaio
trigonometry. See TRI CONOMETRY.
PRO P O S 1 T ION XIV.
Flc .
18,
r o
In,~rurc
-an inacceJ!ihle htight
by
Ih<
geDm<lrico/ qundronl.-Let
the angle ACB be obfer.
ved wilh the quadrant (by the 12'lh prop. of this pan
¡)
rhen·let rhe obCerver go from C to lhe fecond (lalioo
D, in Ihe right Jine ·BCD (provided BCD be a hori–
zonta!
pl~ne)
j
and after mea(uring this dillanee
CD,
take lhe angle ADC likewife Wilh thequadraot. Theo,
iil
Ihe triangle ACD, Ihere is given the angle
ADC,
\Vilh 1he ancle ACD
j
beeaufe ACB \Vas ¡¡iveo before:
therefore (byart.
59.
of
Par!
l.)
the remaining angle
CA
D
is given likewife. But the fide CD is likewife gi.
ven, being the dilh nce of the (lation C and D
j
lhm·
fore (by the firf! eafe
of
oblique.angled triangles
io Irigonometry) lhe lide AC will be found. Where·
fore, in the right.angled triangle ABC, all lhe angles
and lhe hypolhenufe
AC
are givcn
j
eonfequently, by
Ihe fourth eafe of Ifigonometry, !he'heighl Coughl AB
will be found; as alCo (if you pleaCe) Ihe di(laoce of lhe
Uation C, from AB the perpccdicular within the hill or
inacconible height.
PRO P O S 1 T I ON XV.
Flc 19.
F,.m Ih<lep
of
agiv(I/ h,ighl,
/.
",roflJa
,/"
d'¡lnne<
DC.- Let the aogle HAC be
~bfcrvr.t
by
lhe 121h prop. of this; whw forein rhe triangle
ABC,
righl.angled al B, lhere is given by obfcrva.ion lhe ano
glc
at A
j
whence (by the 59lh art. of Pall!.) there IViII
al(o he giren the angle IlCA : morcora lhe fide AB
(bcing lhe heiglt¡
01
¡he IOl\'cr) is
forrofc~
10 be gi1rn.
'\\!¡, rdore,