wÊaBàmÊiimÊaÊassËM
latitude. Tor th.e declination'- Ee’zz.'He—ME, wfoeçe ME
is the meridian altitude, and HE is the; complement o f
latitude,, by^art.; 53. A lfo , thedechnatip'n Et— HE—
Ht3 and B t Is the meridian altitude.
o f the place, and the declination <èf
Jm rÿwg) and his azimuth at that
m l
56. L e t U be a circle parallel to the horizon HOR,
and io ° W ow i t ; and let ayhcdxe be^any, circle parallel
to the equator, deferihed by an heavenly'h©dy in "the
eaftefn hemifphere,; and dra\$ the c it ie s fiy, fib, fid,
fix , and Z y ,'Z b , Zc, Z x . Now (as has been already
•explained) when the fun comes to 3 n w a ight begins ;
when any Hodycomes to % it riles 5 when it comes to c,
\ i t is at the middle-point between a and x j whenTit comes
«to d, it is d ue ea ft; 'and le t «^be the place at any other
.time. Now let us fuppofe this bod y t© be the fun,
aha not to change its. declination’ in its paSage from a
to e j and let us fuppofe a clpc& to he adjufted to g o
in one apparent diurnal revolution o f the fun,
or from the time it leaves any meridian till it^returiis to
i t again ; then ‘the fun will alwaysapproach the meridian
at the rate -of in an hour ; alfo,, the* kngle
Which the fun defcribes about the pole,; varied at the
lame rate, becaiifehuy arc tee, which the fun has to de-
feribe before, it bomes to the meridian, meafures the
nngle x f e , -c^led the hour angleM I f therefore we fuppofe
the clock to ftiow 12. when the funis on the meri-
’dian at a and e, it will be 6 o 'clock when he is at c. An d
- as the fun defcribes angles about the pole f i at the rate
To£jt.5° in an hour, the angle .between any .circle fix ,
paffing through the fun at * , and the meridian' fiE , concerted
into time at the rate o f 150 f6r an hour, will give
the time from apparent no on ,o r when the fun comes to
th© meridian. A lio , when the fun is at any point
«ie angle x Z ‘P is his azimuth from the north ; x Z w
the complement o f his altitude; ‘and x fi is ’ the complement
o f his declination. This -being premifed, we lhall
proceed to give the folution o f a few problems which-
will be found v ery ufeful in pra&ical afbrouomy and na- 1 Vigation. *'
57* The declination o f a body, is the difference-be-
, tween its meridian altitude; and the complement o f the
5 Qfaen 'the latittide fj
The fun, to Jind the time oft
Mme.
L e t aeht the parallel o f declination deferibedby the
fun on the given day ; then when-the fun comes to ft,-
h e rifes. Now in the fphéricai triangle bZP bZz=?$â°
(the zenith being gC° from the ïforizon), ÆPzzthe com-
‘^lemÇ’ht' é f the uüb ^ ^ T ih a tio n , and. P Z ~ the dômplé-
■ meftt o f -latitude ; and. b y ' fpherabal - trigonometry,
radius : cotan. bfi: : cotan.1 Z P : cof. ZPb, or, radius •:
tan. decl. : : tan. lat. re'èL Z P b , the hour angle fjx>m 'apparent
* This log. 9,7480805 is found in the tables to be the log. cofine of 550.
ffluft take its fupplement. In the folution of fphericai triangles, ambiguous:
reader is referred to:Dr- Masksiyne’s excellent Introdudlion to Taylor
rital Trigonometry, containing an explanation of the ConJtrvTiion and Vfe of Lo
néon ; which coiïvërfced "iùtô time . at"7this rate o f
- i5 ° ,foi'-an hour, and fubtradted from 12 lO^Méêfe, g Lves
the apparent time ©f p iS fg l *6;r' the hour at whidh the
fun rifes,- fuppofingTt'b e.12 o'clock wheij he Çomés to
Thé meridian,.
• ‘AJifp,fin. Z P : radins : : èùh-bfii fin, & Z b ,w , cof. ■
Tat. : radiüs’4^pm d h e l . ’©©f. io f the asdmuth from the
north.
E x . Given the latitude?>ofCambridge 52°. 12'. 35",
__fco fin'd -the time o f the fun's rifimg ©it the longeft day,
and Ms azimuth at that time . nffumfeg rthe fun's -great -
eft decfinafcioftf b’3a. '28 V
B y logalfithms the operation will ftâiiâtiius :
rad-. r ' - -, '’■ - ''[y
tan. 23® ^ - o'* * - 9>5i7-6foq ’ ■
fan. Ç2. 12. 35' ^ ’ i a , 1 1©4b 9^9„ j
: Cof. T24-W-2,.. 47# . <@489805^,
Convert this irifo^tmfe, an T it gives'-8 K. fig', 6”, which
fubtrâ£ledTrbm 12, there Yemaihs ,$h, 4§?i thé ap~,
parent time at'which-thifunftifss. A lfo ,' '
’ radins-' T- iy: .)t.. ^
| | fin. 23? 28 • •
cof.^49. 28. :9 ^ . 9,58^281-85
Hence, pn the longeft day, the tm rifes 49®. 2-8'.. 9"
from -the north.
591' T o jn d the fun's altitude at 6 o'clock, t
dayi
'the fame
6 o’clock the- :fu n is at c, and Z P e is a r,
hence, radius' | coh Z P : : cof.' P c : 1 oil
iis : fin! lat. : : fin. dec, -1. ffe. o f-ihe altitude.:
d, \2>f r but as thie angle i8 inanifèilly greater thaïBall
« vvfli frtqueaijy ante, for the determination of vi/nich
Ugatilboe« ^ ox.ti> Mr. V sncxXTrestifi oeflmrand
By-
By, teaçjthn}®,A® operation - is Ans-:
radius, - jî'G/©obéè|?#’'
ftti. y a °. 12' 3 5 ’ -
fw* '23. o - 9,(5a6n8i
fin. rS .'^ o . 32 the altitude <£4:97,88$6
.OO; To fmcl thr ilmf *u)Be.n the Sun comes due ea/t, and
Sit altitude at thattime, on the fame day.
T he ftrn is due eaft when he e'omes'fs -the prime verti.
eai an& dZ-P is a right an^le ,- hanee, cof. Z P .:
radius : : cof. dp oof., Zdj or> fin. lat. : radius : ; fin.
4ec.-j- ;fih..;,o£‘the altitude.
Mfo, iM iub c t 11. P d ran. P7 r o f 7P'J.'ou,
radûis tan. deç. : : cotan. lat.'|- dôfy.ZPd thê hour
anj/fj which coiiyerted l-Uo^Ain.^giviSi thotiiMt'-lroni.
apparent
P x ï=&]°. 37'. g" y
o ■ i -
Sum 175, 42, 24
•y. Sonj.S^i, 5,1. 12
Zx=zS3. o. 21
I 34v 5
ar. co. fin. 0,034019.
a r .co -fin . 0,^8^199
- fin. 9,999^4.
- r‘l 9.756932-
29- 47- 44— t ZPer - - - c p f ., 9^38419
Hence, . ’37', 28', whioh converted intotime,
gives 3h. 5S'. 22". the' time from apparent noon.
ià finding the longitude by the
lunar method*.
By lo^arithpis, the operation ià thus't'”1 ’ '
l2'<- 35" » g.-i'022js3 ariooirashns
- .* 1,iu®iciQb©qoe!
fin. *3. ?S. o . ,
fin. 30. 15. 3J the altitude ■ 9//023486
.i^djus ■ - . : ïâjëboeeoo
tan. 3 3 V ? 8 ', 9,'6576106.
4»« m .r< <- 9,889530*.
cpf- .S?. 19.. 44 -= .Z P i - 9,52711*407
62. Given the latitudeof tljeplqce, and the Sun’s declination,
to jin d the time when the twilight begins.
• Twdig^t'feeghls when the fun comes to y , t8& below
the ^oMaon^nence, Zyrer. id 8 ° ; alio, ‘Py is the complement
o f declination, and Z P is the contplerije^t o f latitude
; hence^fiis vP x fih . sij fin. \ ( P Z +
ftr fc io&^Xfln-* 1 (P Z^ -P y— io S i f o edfi?4 'y V Z *
therefore y P Z is known, which converted into time,
apparent noon'when twilight begins,
This, rule .being'’,' the‘. farrie as the laft, the meth©d oF
calculation is the fame. ' 1
. z^hgttrft
Tilis angle 708^.49'. 441" converted, into
4h. 4 1 . 1.9!f A c tiine from apparent noon. ’
time, gives
6;ti Given the latitude o f thtplacé, the Surfs declination
and Ms altitude ftojind the hoir.
L e t * be the fen’s, place’ ; then m the trîârjgfe. k Z P ,
ttZ is the complement o f the altitudej. à ? is the iQonii-
plenient, o f declination,'’ and1 PZ. Is the complement o f
the latitude, all which are given ; hence, by fphericai
trigonometry, fin', a f xfin. Z P ; radi* :.t fin. ‘^fJPet
+ P Z + Z ¥) Xfift. i-. fP x + P Z -Z Z x ) . : r o L j l z K f
therefore the hpax angle Z P x is known, w^ich •convert- '
eq mtOvtime,rgiyeé th© time from apparentmoohii i .
c E x . Given the latitude 3-4°. e f W. the fàtPê 'à& imt
b? ° 22 • 2 2 - 57" N. and his true altitude36° eg', sq*
t© find the apparent time.
. i l ^ * = 5 3 °. O'. 21 " , p x - $ 30
37 • 3 » and Ac operatipa by JogariAms ia A«v
Lep Sfi, (fee fig. 2d op page-i) = 43«. ,8', then tfif
# y jdefijribjes Appiftle jfajf and thta
S fp e jnfo iA f ApniiPP, at will whqlly beab
o « thq horizon, therefore- the fen cof)t.;uijes above
the_horizon, for its whole- apparent d ii^ ? I motion, that
is, for 24 hours. NoW ^ R = E H = : the complement o f
laticude, by article 53 ; hence, the- latitude is 66°. 32' *
therefore A e fpe£fetw is at A e ara{c eirele, as appear!
’ , ; i ' J f
64, T o a.fp1ei&ator;at A e feme place, pn A e ftorteft
day the fen is at the diftanee- on .the other fide o f
tfee .eqjj^itp.r, and at that time he defcribes the circle
■ zuff in' his diurnal motipii,: and therefore he, continues
f 4 hours befew-Ae-iferijOT ^^hibrsfcfe the fengeft night
1S 24 hours. , blow we ndve already obierved £ art. 4.8.),
_Aat as a fpeftatpr moves from the equator to A e poles!
A e length o f the day increafes ffoMiiz hom» to half a
f e , ; hence, the longeft day ,is :more than 24 hours
within. thgjEqlar circle, and left, tjban 24, on every other
part, o f the. earth.
65 • 7 0 fin d at what time of the
MdlnighU year the tovilight lefts
L e t