371 / 1042
D
A
4. In
all
ereél dials,
CM,
the hour·line ofXII, is per–
pendicular to the horizon Df the.place
f~r
whieh the dial
is to ferve: for that Ime ISthe tntcrfeéllOn of a vertIcal
plane with the plane of the ",eridian of the place, both
whieh
m
"erpeodieul" to the plar,e of the horizon: and
any line
tlO,
or
ho,
perpeodicular to
CM,
will be a ho–
rizontallioe 00 the
pl.neof the Jlal, along wllleh IlOe
the hou!'s m.y be nUnlbe, ed ; and
CM
beiog fet perpen–
dicular tO the horizoo, the dial will
h~ve
its true po/i–
tioo.
5. If Ihe plane of the dial Itad dedioed by ao equal
aogle towud the eall, it9 deiC,ip:ion would have differed
only in this, that the honr·line of XII would have falleo
on the other fide of the fubllile
CL,
and the lioe
HO
1I0uld have a fubeomrary pofinoo to what it has in tltis
6gure.
6.
And thefe two dials, with the upper points of Iheir
!liles luroed IOward the oonh pole, will fcrve for other
t\>Jo
piares parallel tO Ihem; the ene decliniog from the
north toward the eafl, and the otlter from the oorth to–
ward the lIell, by Ihe fame quantity of angle. The
like holds true of
,11
dial, in general, wltmver be
Iheir
deelin~tion
and obliquity of their planes to the
horifan.
e
A S E Ir.
7. lf the plane of thedial nr)! only
d,clill<"J,
but alfo
r,–
dim,
or
inclille!
Sup~ofe
its qeclioatioofromIrontiog the
fouth
S,
(Plate
LXX
I.r.g.
l. )beequal tO the are
SD
onthe
horizoo; and i15 reclinatioo \le equ:J .0 the are
Dd
of the
verrical eirde
DZ:
theo it
IS
plaio, that if the quadrant
of alritude
ZdD
on the globe ems the point
J)
lO the
horizoo, and
t11~
reclioation is cOl\nted upon the quadraot
from
DIo
J;
the interreélion of the houl' eirele
Plid,
wilo the eq¡tinoéU.1
Wfti'..
wiil de.ermine
RiI,
the lati–
lude of the place
ti,
\'ILo(e horizon is par.llel
!O
tite gi–
ven pl.tlle at
Z.;
and
l<!f..
wrll be the differenee in longi–
tude of the planes at
ti
and
Z.
Tri~ono01elrically
Ihus: let a
gre~1
eirde pafs through
Ihe thrce pnints
W, d, E;
and in ¡he mangle
{{IDd,
lighl–
angled
al
D,
Ihe lides
/VD
and
Vd
are give,,; aod thenee
the angle
DWd
is found, and lo is lhe hypotheoui'e
Wd.
Ag~in,
the ddFmnce, or the ¡'um, vf
})f1/,/
anu
DI" R,
rhe clevation of the equinof.i,¡ abol'e the horizon of
Z,
Riv(! ,he angle
dWR;
and lhe hrpothenule of the tri.ngle
W/lI/
was jult oow fuur.d; wheoce the lid.,
Rd
ao<l
)¡IN
are foupd, the former bei"g the lati...de of tI'e place
d,
and the Imer Ihe eumplement of
Rg¿,
the dtfferenee of
Jongit\ldc foughl.
.
Thus, ir the htitude of the pldee
Z
be 52 0 t
o'
oonh ;
the dechoation
SJ)
uf Ik plane
l.h
(lV~ieh
\\'ollld
be
ho–
rizont,1
~t
a)
be 3'6°, and the reclln.lioo be 15 °, or e·
qual
~o
the ate
))J;
the foulh l. illlde of Ihe place
d,
th.t is. Ihe are
IIJ,
~,ill
be 15 °
9';
.otl
R~_,
the diffe–
!toce of 10ngil11de, 36° 2'. Fr01l1 thete data, therclore,
let Ihe dial .(lig. .
2.)
be dercribed,
i\S
in the former ex–
¡mple.
D
A
9. There are fereral otha things
requir.tein
Ih~
prac–
tiee of dialing; the chid of which
fh~1I
be glven In the
form of arithmelieal rllles, fimple and eafy 10 thoCe who
llAve learned the elemeot' of trigonometry. For in prae–
ticalam of this kind, arithmetiek Ihould be ufed as fal as
it cao go; aotl
f~ales
never trulted tO, exeept in .the
6~al
conllruéliuo, where Ihey are abfolutely
neeeffar~
In
laylng
down the calcolated hour·dillanees on the plarn of the
dial.
RULE
I.
'fo¡nd th, angla 'Which the hour·line¡ on ar.)
dial ",.He 'With /h, /_11Iil,.
To the logarithmie fine of the given latitude, or of the
(lile's elevalion above the plane of the
di~l,
add the loga–
rithmic tangenl of Ihe hour - dillance froftl the .meridian,
or from the
t
fub(lile; and the fulO
."i"uJ
radlus w"l be
the logarith",ie t.ogent of the angle lought.
Fur, io
fig.
7. Plate
LXX .
KC
is
10
KM
in the ratio
r.ompounded of the ralio of
KC
tú
KC (=KR)
and of
XC
t'o
KM;
which, making
CK
the radius 10,000000,
or 10,0000, or 10, or t, are the ratio of 10,OCOOOO,
01
of 10,0000, or of 10, or of 1,
10
¡":C
X
KilI.
ThllS, in a horizont,1 dial, for latitllde 51 ° 30', to·
6nd the angll!ar dillanee of
Xl
in the forenoon, or
¡
In
the aflernoon, from
Xil.
To the logarithmie fine of 51030'
9-8
9354
t
A~d
the logarithmic tango of 15 ° o' 9'42S\l¡
Thefum-radiosis - - - - - 9.3215y=the
logarithmie tangenl of 11 ° 50', or of Ihe anglewhleh the
huur·line of Xl or lmakcs wilh ¡he hout of Xl!.
Aod by eomputing io this mallner, wilh Ihe fine of
the
latit~dc,
and the tangents 01' 30, 45, 60, and 15°,
for the hours of
il, lII, JIll,
anJ
V
in the afttrooon;
or of
X, ¡X, VIli,
aod
VII
in the furelloon; you will
6nd their anglllar dillanees froOl
XII
10
be 24° 18',.
38
° ,', 53' 35' and 710 6'; whieh are
all
that there is
oceafion to eumpute for.--Alld thefe di (lances may
h~
fet on from
XlI
by a line of ehord;;
or
r.ther. by
ta–
king 1000 froOl a reale of equal pam, and f<ttiog that
eXltnt as a raJius from
e
to
Xl!;
and thell, t,killg 209
of Ihe fame partS (""hieh, lO Ihe tí.bfes, are the noteral
tangeot uf I¡° 50') alld fetting titem froOl
XU
tO XI
and lO
1,
0n the lioc
l,o,
II'hich
lS
perpelldielllac 10
e
Xl1 :
ano
lo
for the reH
01
Ihe hOIlI-llne" which, in the table
of
nalm al
tangent~, again/~
the above
dinanee~,
ate
45',
782, 1355, and 2920, offuch equal'patls ir01l1 XII,.s
t e radlu!
e
XII
cúnt,ins 1000. And I.Uly, Ú:t off
12 57 (the natural 1.llIgent of
51°
;0') for the aogle
of
the Hlló's Itcight, whieh i¡ equal to lhe latiludt of the
place.
RULf
11.
rh,
lali/lld,
o[
/h.
pl.re,
/1.,
/UI/'J-d"IÍI.'¡f!
Ihll ,
and biJ ho,a··"U(tllIt't
/,.~m
tl:
1!I(f1Ji,m,
tt–
il/g givw;
/.
find
(l.)
hiJ
allil"d,¡
(2)
tú
olÍ–
",rllh•.
1-
!.(t
..
Th~tis~ofII,
;0,45,
60,15', rllrtheh.oursofI, 11, 111,1111, V in Iht' aflemo",,; anJ· XI. X, IX,
VIII, .
'"10 the at._J;rnIJun.
t
111
all humoll1JI dIal" alld "let\II'1\lh or 1()lIth tii"" the [ubl:iit'
",.J
II"r;.li.111
are
ih..:
,:illt!t: :
l,H.t
;11
aH.(!\:di:1:ng
~I!.a~~,
thc J'uhfiilc:
1;111.:
I\lal~s
,\11 :lili,k \\
ith
tlH':
liH:ricli.llI. '
t
ln\,
!Jidl
(:\ú',
lbe r.dlus
GK
I! Iu¡,poled lo
~e
dlvldeO 1I1tO1000000 eljIlJlp,I\·\>.