})Iy
k1 other
ich vill
e Earth
geogia-
iocentric
el]ij)se,
and if
(19).
)centric
X = r
e have
bed on
Therefore,
Or,
W
■^ -A- — 7'2 sin« (h
/ +
r' 0,0^ -f __ V- .siu2 (f> = a'' ,
a
From which »• := a sec iace when referred to the equator, then V C = Sun's
A. R. + hour angle from noon = sidereal time
= A.
VN ■= longitude of the Nonagesiraal i\^, = m .
Z Q = N I, the altitude of the Nonagesiraal = a .
P Q =. the obliquity = w .
PZ = co-latitude = 90° — (p , (geocentric).
/_ ZPQ ^ 180° - ZFT
= 180° - {VT - VC) ''
, = 90° + ^ , and Z ZQP= N(= Vt - FiV=r:90° - m
In the triangle Z P Q, we have
cos ZQ = sin PZ sin PQ cos ZPQ + coa PZ con PQ .
22
Or, cos tt = — cos sin w sin ^ + sin cos u) .
Put sin A cot = tan 0,'
Then cos a = sin sec cos (w + ^) . (21).
In the triangle PZ Q, we have
Hin ZQ : sinZP :: sinZ/^r^ : ainZQP
Or, sin a ; cos :: cos A : cos m
Or, cos m = cos A cos ^ cosec a . (22).
And from the same triangle we get
cosZP= sin Z Q sin P Q cos Z (> /' + cos ZQ co^ P Q .
Or, sin sin ^
sin a
Dividing this by Equation (22), we have
tan d) sin o) + cos a sin A
tan wi = J- ■ ,
cos A
= tan (fi sec A sec sin (oj + ^)
Eq. (22), may now be used to find a,
sin a = cos A cos (f) sec »i .
(23).
(24).
i
TO FIND THE PARALLAX IN LONaiTUDE.
Art. 20. — Let Z be the zenith, Q the pole of the ecliptic,
>S^ the planet's true place, S' its apparent place, Q S the planet's
co-latitude =90 — X, then Z Q = altitude of the nonagesimal
= a, the angle Z Q S =^ the planet's geocentric longitude —
the longitude of the nonagesimal = h, S Q S^ = the parallax in
longitude = a*, and SS' is the jmrallax in altitude.
From the nature of parallax ^v6 have sin SS' = sin P
sin ZS' and from the triangles *S' Q S', Z Q S', we have
sin X
23
_ sin SS' sin S'
sin Q S
»
sin
P
sin Z S'
sin
.S'
sin Q S
>
sin
P
sin ZQ
sin
Z^.S'
sin QS
sin
P
sin a sin
(/^ + .t)
COS \
= Z^' sin (/i 4- 3), if /c =
sin P sin a
cos A,
and by a well known process in trigonometry,
k sin h k^ sin 3/t /r' sin 3/i ,
.T
sin r'
sin 2'
sin 3"
(2o).
(26).
TO FIND THE PARALLAX IN LATITUDE.
Art. 21. — In the last FUj. let aS''^ be the apparent co-latitude
= 90 — X', then from the triangles QZ S and Q Z S', we have
„ cos (?S — cos QZ cos ^S _ cos QS' -con QZ cos if.S'
COS yy rr= — — - — -
or
sin QZ sin ZS
sin X — cos a cos ZS
sin (^Z sin ZS'
sin X' — cos a cos J^iS'
sin i^/6"
sin ZS
but from the same triangles we have
cos ZS = sin a cos X cos 7* + cos a sin X
and cos ZS' = sin a cos X' cos (/t+tc)+ 00s a sin X'.
which, substituted in the above, give after reduction
sin ZS' tan a sin X' — cos X' cos {h-\-x)
sin ZS tail a sin X — cos X cos h
But from the sine proportion, we have,
sin ZS' _ sin (/i+x) cos X'
sin ZS sin h cos X
tan a sin X' — cos X' cos (/i + a:) sin {h+x) cos X^
tan a sin X — cos X cos /i ~ sin h cos X '
tan a tan X' — cos {h + a)_ sin (/i 4- a:)
tan a tan X — cos h sin h
therefore
or
m
m.
24
^ 11. -. tau a ttiii X .sin (h + x) — sin x ,.,_,
From winch tan >.' — , — j-^ — - — '- , (27)
sin h tan a
But Siin X = sin i'sin a sec X sin (/t + .'c).
Thereibro
^ , tan a tan X sin (//4-a?) — sin 1* sin a sec X sin 'fi + x)
tan X = !^ ■^ ' '
Or
sin A tan a
Sill I /a "i ■ '7')
tan X' ~ ' \ 'J (tan X — sin P cos a sec X).
sui k ^ '
sin (/a + .7')
sin /t
(1_^'
cos a
sni X
) tan X.
(•28)
Tliis formula gives the ap])arcnt latitude in terms of the true
latitude and the true and ai)[)arent liour angles, but ib is not in
a form for logarithmic comi)utatiou. We will now transform it
into one which will furnish the parallax directly, and which will
Vje adapted to logarithms.
]jet }] — X— X', the parallax in latitude.
From E(]. (27) avc have
sin ;/; sin h
tan X =
+ .
tau X'
sin {]i-\-x) tan o sin (/i + x)
sin X /sin (/i4-;c) — sin h\
_ — — _^ tan X' I — -. — r- — ^ I
sin \h-\-j-) tan — tan X'=
Or
cos X cos X' "" sin (/t+^) tan a ~ sin (/i + .r)
But 2 sin * = sin .1' sec -^-,and
sin X = sin P sin a sec X sin (A + a:) by Eq. (^"Ib)
Making tliese substitutions and reducing we have
sin y —sin P cos a (cos X' — tan a cos (A + ^ ) ^^^ a ^^^ ^ )
Put tan a cos (/t + ^^- )„sec •^' = cot ^,
Then sin ?/ = sin P cos « cosec {) sin (0 — A'),
= sin i'cos a eoscc sin ( (0 — X) + v) (29)
Put sin V cos a cosec ^ = /•', tlieii as before
y
_h sin (0— X) ^ I' siu 2 (6J-X) ^ k^ sin 3 (^— X) ^ .
sin l"
sin 2"
sin y
(30)
^•ld:i'.iLi;LiLdie^'^
25
■' ' ' ( n. ) '
A TRANSIT OF VENUS.
December Gth, 1882.
Art. 22.— The following heliocontric positions of Venus have
been computed from Hill's Tables of the Planet, and those of
the Earth fv.im Delambre's Solar Tables, partially corrected
by myself, t being taken = 8".9o at mean distance :—
y?
--(< CO
1—1
o
CO
I-.
'O
I5-5
-ti oi
o
CO
'O
CO
.-^
CO CO
CO
CI
CI
CI
CI
CO
-*
'i*
■rj^
-rr'
CO
o •
(Tl
err
.v'3
o
Ci
■J
ci
CJ
o
r-l "O
o
-+
CO
CI
1-
O .
CO o
CT
o
1-
o
CI
0) '^
^ (M
l'^
CI
1(0
CO
o
,■» To
^ I-
(r^
CI
-t
1-
o
5s
(Tl C-l
CI
CO
CO
CO
-f
r
if -+*
'^
-H
-t<
-f
-f
^ 1^
t-
.■--
I-
l^
I-
00 O
o
OS
t-
'^
CI
ii
CO T— 1
00
-^
I— 1
CO
•o
lO lO
-+
f
^
CO
CO
(U o
Ci o
o
o
o
cs
C5
>>
h- t-
i^
I-
l^
1-
t--
lO >o
JO
lO
»o
i-O
•o
QO CO
00
00
00
CO
OO
Ci CJ
ci
Ci
o
o
ci
o
02
o
lO CO
CI
o
c»
o
»o
ll
CO O
l-
-^
o
t>-
-*
•s-5
CO Ci
-H
o
o
1—4
1-
1^
CO >-•
o
lO
CO
CI
o
^ -+•
'^
CO
CO
CO
CO
OO I-
>o
»o
CI
«o
«;:>
• •
■-^
CI
*
o
w
>o
j-o r-
CT)
—(
CO
-*
o
^ .t;
o o
o
I— 1
I— 1
'~'-
I— 1
^ >o
o
CO
1-
1—1
1(0
CI (M
CI
CO
CO
-H
■-*'
'mJ
3
13
o
-+I
~¥
-*
-H
-^
^
1- I-
l^
1-
l^
1-
I—
g
-ii rd
^
•
rJ^
^
J
a>
.^ CI
CO
^
r-i
CI
CO
a
CI CI
CI
CI
Ii
n3
13
g-'p
'O ^
-
^
CT)
^
;;
1
8
J
^
ft
m
26
Art. 2.3. — Passing to tho trup geocentric places by the aid of
Formul.'P (l)-(ir)), and then applying the correction for aberration
(which, by Formuhe (14) and (15), is found to be, in longitude,
+ 3".3 ; in latitude + 1".4 ; 8un's aberration — 20".7), we
obtain the following apparent geocentric ])lacc.s : —
VVn-shiiigtoii Mean
Sim's Apparent
Venus's Apparent
Venns'si
Ajjpar. Gcoc.
Latitude.
Time.
Geocentric jonJ5Mtuilc.
Oeooeiitiic Longitude.
Dec. .5d. 21 h.
254° 24' 27". 4
254° 34' 58".3
1 2' 28" S.
22h.
20 59 .8
33 20 .7
11 49
" 23h.
29 32 .2
31 .■)5 .2
11 10
24h.
32 04 .7
30 23 .0
10 30 .8
Dec. 6d. Ih.
34 37 .1
•IS 52 .0
9 51 .0
2h.
37 09 .5
27 20 .3
9 12.5
3h.
39 42 .0
2:) 48 .0
8 33 .4
Log of Venus's distance from the Earth at noon =^ 9.421550 .
FormuL-e (9) and (12) give us P =: 33".9, and d=^ 31".4G, both
of which may be regai'ded as constant during the transit.
Interpolating for the time of conjunction, and collecting the
elements, we have as follows ; —
Washington M. T. of Conj. in Long., Dec. 5d. 23h. 35.1m. .
Venus's and Sun's longitude 254° 31' 01 ".5
Venus's latitude 10' 47" S.
Venus's hourly motion in longitude 1' 31".^ W.
Sun's do. do. 2' 32".4 E.
Venus's hourly motion in latitude 39". I N.
Sun's semi-diameter 1 G' 1 0".2
Venus's do I 31".5
Suns Equatorial horizontal parallax 9".l
Venus's do. do. 33".$^
Obliquity of the Ecliptic 23° 27' 09".
Sidereal time in arc at 20h 195° 12' 54".4
Constructing a figure similar to Fig. 4, and enqdoying the
same notation as in Art. 13, we obtain from these elements the
following results : —
n ■= 9° 6' 14", 4 ; relative hourly motion in qrbit, — 24)T".1 \
least distance between centres, 1 0' 39" ;
the aid of
aberration
ongitude,
M)".7), we
27
First external contact, Dec. 5d. 20h. .'iiO.Tui.
Veims's
)|tar. Gcoc.
l-atitude.
2' 28" S.
1 49
1 10
30 .8
D .'51 .(>
I) 12.0
"! 33 .4
.4215;50.
4G, both
3ting the
Urn.
n".5
17" S.
U".)IW.
{2".4 K.
G".2
il".5
9'M
!3".fi^
9".
4". 4
ing the
iiits the
20h. .'iiO.Tui. \
21h. llni. (
Washington
Mean Time.
First internal do., "
Last internal do., Dee. Od. 2h. 48ni.
Last external do., *' 3h. 8ni.
As seen iVoui the Earth's centre.
By the i'orinulie of Art. It, we llnd, that at the time of
llie jivKt external contact, the 8nn will be in the zenith of the
place whose longitude is 4.")°.9 East of Washington, and latitude
22° 37' S. ; and at the last external contact the Sun will be in
the zenith of the place whose longitude is 48°. 3 W., and latitude
22° 41' S.
From these data we find, by the aid of a terrestrial globe,
as in the case of the transit of 1 87;il-, that the entire duration of
this transit will be observed in the gi'eater j)art of the Dominion
of Canada, and in the United States. As Venus is .south of the
Sun's centre, the duration will be shortened at all places in
North America, by reason of the effect of parallax. The times
(if tirst contact will be retarded at phices along the Atlantic
coast of Canada and the United States, while the Islands in the
western })art of the Indian Ocean will have this time accelerated.
These localities will therefore afford good stations for determining
the Sun's parallax. The time of last contact will be I'etarded in
New South Wales, New Zealand, New Hebrides, and other
Islands in the western part of the Pacific Ocean, and accelerated
in the United States and the West India Lslands. The duration
will be lengthened in high southern latitudes, and especially in
the Antarctic continent. The astronomical conditions necessary
for a successful investigation of the Sun's parallax, will therefore
be very favorable in this transit ; and it is to be hoped that all
the available resources of modern science will be employed to
secure accurate oUservations, at all favorable points, of the times
of ingress and egress of the planet on the Sun's disk, in order
that we may determuie with accuracy this great astronomical
unit, the Sun's disfci ice from the Earth, and thence the dimen-
sions of the Solar System.
/
28
■K
4-
li
i 1
TO (;OMPUTE Tlin TRANSIT FOIl A OIVRV PLACE ON THE EARTHS
SUUFA(!K.
AllT. 24. — he.t it be require'l to find tlui limes of contact for
Toronto, Ontario, wliicl) is in latitutlo 'i'.]'^ 3!)' 4" N., and longi-
tude I'Jh. 17m. .S.'kec. west of Greonwicli, or 9ni. 22soc. west of
Washington.
Since the ))arallax of Vtinns is small, the times of ingress and
egress, as seen from "^Poronto, will not difllu" nuich from those
found for the Earth's centre. Hubtracting the difference of hm-
gitude between Toronto and Washington, from tho Washington
Mean Time of the Jirst and lost external contacts, as given in the
last article, we find the Toronto Mean Time of the first external
contact to be December, 5d. 20h. 41 3m., and tlie last external
contact to he I>ecembe', Gd. 2h. 58. Gin , when viewed from the
centre of the eai'th.
The ingress will therefore occur on the east, and the egre.ss
on the west .side of the meridian, and the time of ingre.s.s
will consequently be retarded, and the time of egress accel-
erated by j)arallax. We liierefbre assume for the first external
contact, December od. 2()h. 44m., and for the last external con-
tact, December Gd. 2h. 54m. Toronto Mean Time ; or, December
5d. 20h, 53m. 22sec , and December Gd. 3h. 3m. 22seo. Wash-
ington Mean Tinu-.
From the elements given in Art. 2.'{, compute for these dates
the longitudes of Venus and the Sun, V nius's latitude, and the
Sidereal Time in arc, at Toronto, thus : —
Washington Mean
Time.
Dec.5d.201i.53m.'J2s.
" Gd. 3li. 3m.2'2H,
Sim's Apparent
Longitude.
254° 24' 10".5
254 39 50.5
Venus's Appar.
Longitude.
254° 35' 8".5
254 25 43 .5
Venus's
Latitude.
12'32".4S
8 31 .3
Sidereal Time
at Toronto.
206° 15' 06'
299 17
The relative positions of Venus and the Sun will be the same
if we retain the Sun in his true position, and give to Venus the
difference of their parallaxes, reduced to the place of observation
by Art 17.
29
Compute next by Formulte (19) to (30), the parallax of Venus
ill longitutle uiul latitude, and ajtply it with its proper sign to the
api)arent longitude and latitude of Venus, as seen from the
Earth's centre ; the results will give the planet's apparent posi-
tion with rttspect to the Sun, when seen from the given place,
and the contact of limbs will evidently happen when the apparent
distance between their centres becomes equal to the sum of their
semi-diameters.
Wo now proceed with the computation : —
By Eq. (19), tan ' = 9.9795 U
const. loff= 9.997091
/"
X
tan (f) = 9.97GG35 , therefore = 43° 27'34"
const, log = 0.001454
By Eq. (20),
tan a = 9.978089
cos e= 9.860164
see (^ =10.139140
log?-= 9.999310
Diff. of Parallaxes, 11V'.$ = 1.394452
Reduced Parall ax, 24 ". ^9 "- 1.393762
therefore ^^43° 3319"
/
ALTLTUDE AND LONGITUDE OF THE NONAOESIMAL, AT THE
FIRST ASSUMED TIME.
By Eq. (21),
sin A = 9.04573b/,
cot = 10.023300
tan Q =
9.669097W
154' 58' 42"
: « = 23° 27' 9^^
ft)+ = 178° 25' 51"
By Eq. (23),
tan ^ = 9.970634
secyl = 10.047275«
sec B = 10.042801»
sin (w + 0) = 8.437493
tan m = 8.504203
m = 181° 49' 44"
«Mi (/, =: 9.837488
sec e = 10.042801n
cos ((u + 0) = 9.999837»
cos a = 9.880120
a = 40° 38' 30"
Check by Eq. (22),
cos A = 9.95272$«
cos = 9.800854
cosec a = 10.180201
cos m = 9.999780rt
m = 181° 49' 44"
■■r
pi 1
30
PARALLAX IX LONGITUDE.
Longitiule of Venus = '25 i" 35' 8". 5
Long-ofthoNonagesinial = 181° 41)' 44"
TluH-elbve, h = 72Mr/24".5. Then by Eq. (2G).
Hin r = 0.079337
sin a = 9.813790
sec X = 1 0.000003
/.• =:r ;>.893139
sin h = 9.980029
t
k' = 1.78G3
sin 2/t = 9.7529
h' = 7.G79
sin 3/t = 9.792/i
cosec 3" = 4.837/t
= 8.308«
cosec 1 " = 5.314425 cosec 2" = 5.0134
15".402 = 1.187593, ".0003 = 4.552G
The last two terms being extremely small may be omitted,
therefore the imrallax in longitude = -f 15". 4 = x.
I
PARALLAX IN LATITUDE.
' n
\\i. H
ifft'
im ]!!
By Eqs. (29) and (30).
tan o = 9.933G72
cos(/t + ^0= 9-47 18G0
sec I = 10.000000
cot e = 9.405532
= 75° 43' 34". 5
\ = 12' 32". 4 S.
^ + X = 75°5G'G".9.
IS' - 1.94G1
sin 2 (e + \) = 9.G734
cosec 2" = 5.0134
"•0004 = 4.G329
sin P = G.079337
cos a = 9.8801 2G
cosec i) — 10.013G19
k = 5.973082
sin (« + \) = 9.98G782
cosec 1" = 5.314425
18".808 = 1.274289
/.:» = 7.919
sin 3 (^ + X) = 9.869«
cosec 3" — 4.837
= 8.G25/i
Therefore the parallax in latitude = ^ 18".8 = ?/.
In the same rvay, we find at the second assumed time,
a r= 27° 37'; m = 317° 23' 4G"; A « - 62° 58' 2".5 ;
a: =» - 10". 3; y = + 20".8. W, /
^x
31
llonce we have thr tollowin'i rcsult8 :■
Dbo, 6p., 20h. d8.M. J.'fKi
f,nN(i|TI'PK,
Venus'B
Parallax.
Sun's
Difference.
•J54
254
1254
rtr,'
8". 5
15". 4
;35'
li4'
28". n
10". 5
11' IT A
\ cults East.
I,ATITft)F. !
12'.?2".4 8.
-f 18".8
12' 51 ".2
Dec. en., 3h. 3m. 3!tocc.
I,..
NOITItDK.
254'
::.v 4.']"
5
- 10"
3
254
25' »3"
2
254
S9 50'
5
14' 17"
.3
Venus WcMt
liATrTrnr..
t^iiV.S S.
4 20". 8
8' 52". I
<>oii.strnct ii figure similar tf» /''i' iV sec of the inclination of apparent
orbit -= /)' N ^ sec fiNQ {N'Q being parallel to // /*)
tan UN QyI-
B 11 - NP
II P — 1553".8 == relative motion of Venus in 6h. 10m.,
therefore V onus's relative hourly motion = 2ol".8
BII
tan BC 11 ^
II
, BCH= 48° 52' 23"
BC
BC sec B CH = 1023".8
77 C ^ = 41 ° 7' 37: , hence II F = 50° 0' 1 8"
CF-^ 77 C cos flCF^ G58"; /77''= 110 sin HCF=^ 784". 35
V, the sum of the semi-diameters = 1007". 7.
cos VCF
OF
, VOF=40° 13' 54"
VF -■= OV sin VCF= 763". 19
BV = HF- VF=2V.\Q.
h
V >■
Time of ilcHnribiii;^ fj V — 5in. 2.scc., uiul time ttf deHcribin;;
VF ~ iWi. l.ii. ^ls(M-.
Tlicrofon; ihr. (iisl cxtonml coiilacL will occur, !)<'(!. i'xi. 2nli.
I'Jrii. 2scc., and tlu; last cxttTiirtl contacl., Dec. Od. iMi. o'2in. llscc,
Mean TiiiM^ at Toronto.
In a similar niarnu'r \vv. ohiaiii /'/•'• = (J r7"."S.'J ; tlna-clorc,
Vv — 8r)"..'JG and tin- time of dcscriWin;,' Vv ~ 20ni. ^Omoc.
Tlu'rdbrn Ous fir.st intfinal contact will dccnr, Dec. ."id. I'lli.
9m. 2'2hoc., anaS'L.
Or tun DSfj =coH long tan w . (31).
The Sun's longitude at 8 h. 49 m., xV.M., is 25'P24' 23".2.
'^
clt'Hcribing
. 5(1. 20h.
Im. 11 HOC,
tlicrcfoiT,
;(Ih('(\
'.5(1. 2 111.
0(111 Tinio
, we have
., A.M.
lu., "
in., P.M.
in. "
]uire(l, WH
np(l. For
und .snfti-
Y to know
it coiitact
iwn from
ro\igh the
uputed iiH
lo, A' the
endiciiUu',
is a riglit
? will lie
between
and tlie
i
J
^*^■^J
88
Kt'j.'cting 180° we havf cos* 71" IH' - 23" = 9. 429449
tan 10 =■ 9.637317
tan DSL = 9.0G6rGC
DSL = Ci" - 39' - G"
Now the angle r6'A' = angle I XT - angle i,'C/'
= 400 -21' 12"
Therefore the angle of position is equal t(» the angle DSL -f the
supplement of IV.'A'. oi- 140° -17'. 9 fronj the northern limb
towards the east.
In the same way we may compute the angle of position at the
last external contact.
From a ))oint in longitude 71° 1)5' \V. of Greenwhich, and
latitude 45° 21' . 7 N., at or near Bishop's College, Lennoxvilh;,
we find by the preceding method,
First external contact December Cth, 9 h. 19.5 m., A.M.
First internal " '• 9 h. 39.4 m., '«
Last internal " " 3 h. 2.G m., P.M.
Last external '• " 3 h. 23 m. '•
Mean Time at Lennoxville.
Least distance between the centres 10' - 59".8.
From a jwint in longitude 64° - 24' W. of Greenwich, and
latitude 45^^ 8' 30" N., at or near Acadia College, Wolfville,
Nova Scotia.
First external contact December Gth, 9 h 48.7 m., A.M.
First internal •' >* 9 h 28.4 m., "
J^ast internal " " 3 h 31.7 m., P.M.
Last external '• " 3 h 51.8 m.. "
Mean Time at Wolfville.
Least distance between the centres 10' - 59", 5.
(31).
2.3".2.
THE SUN S PARALLAX.
Art. 25. — a transit of Yenua affords us the best means of
determining with accuracy the Sun's parallax, and thence the
distances of the Earth and other planets from the Sun.
Ill
34
The same things may be determined from a transit of Mer-
cury, but not to the same degree of accuracy. The complete
investigation of the methods of deducing tlie 8un's parallax
from an observed transit of Venus or Mercury, is too reSned
and delicate for insertion in an elementary work like this.
We add, however, the following method wiiich is substantially
the same as found in most works on Spherical Astronomy,
and, which will enable tLo student to understand some of the
general principles on which the compntation depends.
?
n
TO FIND THE SUNS PARALLAX AND DISTANCE FKOJI THE EARTH,
FROM THE DIFFERENCE OF THE TIMES OF DURATION OF A
TRANSIT OF VENUS, OBSERVED AT DIFFERENT PLACES.
Art. 26. — Let T and 2^' be the Greenwich mean times of the
first and last contacts, as seen from the EartKs ceiitre; T+t and
T' 4- i' the Greenwich mean times of the first and last contacts,
seen from the place of observation whose latitude is known ; S
and G the true geocentric longitudes of the Sun and Venus
at the time 1\- P che horizontal pnrallax of Venus; tt the
Sun's equatoiial horizontal parallax ; v the relative hourly motion
of Venus and the Sun in longitude ; L the geocentric latitude of
Venus, and f/ Venus's hourly motion in latitude. Now, since
Venus and the Sun are nearly coincident in position, the effect of
parallax will be the san^e if we retain the Sun in his true posi-
tion, and give to Venus the difference of their parallaxes. This
difference or relative parallax is that which influences the i-ela-
tive positions of the two bodies.
Than a (P—tt), and b (P — tt) will be the parallax of Venus
ill longitude and latitude respectively, where a and b are func-
tions of the observed places of Venus which depend on the
observer's position on the Earth's surface. The (qjjxifent diflTer-
ence of longitude at the time T -'ill be
G — ^ + a (P — tt); and therefore the apparent differ-
ence of longitude at the time T -\- t *
^G~S ^a{P-TT) +vt,
and the apparent latitude of Venus at the time T -v (.
= Z + b(P—Tr) -^ gt. . •
3^
3it of Mer-
le complete
I's parallax
too refined
like this,
ubstantially
Astronomy,
ome of the
PHE EARTH,
.T[ON OF A
OES.
imes of the
• T+t ai\d
st contacta,
known ; S
xud Venus
lus ; TT the
irly motion
latitude of
Now, since
he effect of
true i)osi-
xes. This
i the rela-
c of Venus
b are func-
id on tha
rent differ-
•ent difFer-
.4
Now at the time T+t the distance between tlie centres of
Venus and the Sun, is equal to the sum of their semi-diameters,
= c, then we have
c2 ... ^ a_ S+a{P-^) + ,t\'+ \L + b {r~n) +yt\'^ (32).
^(G_S)-+ Z- + ^a{a-8) + bL\ (P-TT) + 2f.
neglecting the .squares and products of the very small quan-
tities t, a, I, and {P — rr).
But when seen from the centre of the Earth at the time T,
we hav(!
ci.= [G — Sy^ -;- L% which substituted in the last equation,
gives
-- 8. (P — 7r), suppose
(33).
Therefore the Greenwich time of the first contact ab the place
of observation z= 1' + S {P — tt).
If 8' be the corresponding quantity to h for the time T', then
the time of the last contact at the place of observation
== T +Z' (P- tt),
and if A be the whole duration of the transit then
A= T'—T+ (g'-g) (/'— TT)
Again, if ^V be the duration observed at an// other place, and
/3 and fi' corresponding values of g and 8', we have
<:A'-A= {(/y'-/3)-(8'-S)} (/--TT)
Therefore
Or,
Now
Therefore
P-
ir
A'- A
I Earth's distance from the Sun
TT Earth's distance from Venus '
P — IT Venus's distance from the Sun
(34).
TT Venus's distance from the Earth
^- It, a known quantity
TT^-i {P~tt).
n
(35). — {Hymers's Astron)
n
I ■[ It 'i, <
36
If the first or Uist contact onli/ be observed, the place of obser-
vation should be so selected that, at the beginning or end of the
transit, the sun may be near the horizon (say 20° above it) in
order that the time of beginning or end may be accelerated or
retarded as mnch as possible by parallax.
Again, since t is known in Kq. (33), being the difllerence of the
Greenwich mean times of beginning or end, as seen from the
Earth's centre and the jdace of observation, v/e have from Eq.
(32) by eliminating r,
- ^M^' + .90 + 2t(v(G- .S ') + Av)
n* + b'
Or, (P — ttY ■{■ A (P — it) ^ B, suppose. (36).
And let (P — tt)' + C {P — w) =^ D, bo a similar equation
derived from observation of the first or last contact at another
place, then
C)(p—:r)=^B—D
{A.
Or,
P TT " ~" ^ '
A C '
And
TT = (/* — tt), as before
(37).
I
^'1
II {
THE SUN S DISTANCE FROM THE EARTH.
Art. 27. — If D' represent the Sun's distance, and r the Earth's
equatorial radius, then
=^ r
r
sin 7r
20(5264-8
TT
(38).
From the observations made during the Transit of I7(i9, the
Sun's equatorial horizontal parallax (tt) at mean distance, was
determined to bo 8".;'57 which, substituted in the last equation,
gives for the Sun's mean distance 24008. 23/-, or in round num-
bers 95,382,000 miles ; but recent investigations in both ])liysical
and practical astronomy, have proved beyond all doubt that this
value is too great by about four millions of miles,
'Vf»,
!e of obser-
end of the
bove it) in
ilerated or
iiice of the
1 fi'om tlie
from Eq.
•tt) =
(36).
r equation
at another
(37).
he Earth's
(38).
I7fi9, the
taiice, was
equation,
und iium-
h jthysical
:- that thig
37
111 determining the Solar parallax from a transit of an inferior
planet, two methods are employed. The first, and by far the
best, consists in the com|iHrison of the observed duration of the
transit at jdaces favorably .situated for shortening and lengthening
it by tlie effect of pnnillax. This method is independent of the
longitudes of the stations, but it cannot be always applied with
advantage in ever\' transit, and fails entirely when any atmoa-
phericid circumstances interfere with the observations either at
the first or last contact. The other consists in a comparison of
the absolute times of the Jirst external or internal contact 07ili/,
or of the last external or internal contact o?i/y, at places widely
differing in latitude. The longii/udes of the stations enter as
essential elements, and they must be well known in order to
obtain a reliable result. The transit of 1761 was oKserved at
several i)Iaces in Europe, Asia, and Africa, but the results
(ibtained from a full discussion of the obsei'vations by different
conijniters, were unsatisfactory, and exhibited diffei'ences which
it was in)possible to I'econcile. That transit was not there-
fore of much service in the solution of what has been justly
termed " the noblest i)roblem in astronomy." The most probable
value of the jjarallax deduced from it, was 8".49. The
partial failure was due to the fact that it was impossible to select
such stations as would give the first method a fair chance of
success, and as there was considerable doubt about the correct-
ness of the longitudes of the various observers, the results
obtained from the second method could not be depended on.
The unsjiiisfactory results obtained from the transit of 1761,
gave ri?' ': rrcater efforts for observing the one of 1769, and
observcrh ' .o sent to the Island of Tahiti, Manilla, and other
points in the Pacific Ocean ; to the shores of Hudson's Bay,
Madras, Lapland, and to Wardhus, an Island in the Arctic Ocean,
at the nortli-east extremity of Norway. The first external and
internal contact.s were observed at most of the European obser-
vatories, and the Ifist contacts at several places in Eastern Asia
and in the Pacific Ocean ; while the whole duration was observed
at Wardhus, and other places in the north of Europe, at Tahiti,
«fcc. But on account of a cloudy atmosphere at all the
northern stations, except Wardhus, the entire duration of the
88
ill ! ■
p.'"w
n :
transit could not bo observed, and it conseriuently happened that
the observations taken at Wardhus exercised a great influence on
the iiual result,. This, however, would have been a matter of
very little inii)ortanoe, if the observations taken there by the
observer, Father Hell, had been reliable, but they exhibited such
dilTerences fi'oni those of other observei's, as to lead some to
regard theui as forgeries. A careful exatuination of all the
available observations of this transit, gave 8".o7 for the solar
parallax, and consequently 9i3, 382,000 miles for the Sun's mean
distance.
The first serious doubts as to the acciiracy of this value of the
Solar pai'allax, began to be entertained in the year 18o4, when
Professor Hansen found from an investigation of the lunar orbit,
and especially of that irregularity called the ^;«?'fl//rtc<rd, By the excessive motions of Venus's nodes,
and of the perihelion of Mars, also investigated by tlie same
distinguished astronomer ; 4th, By the velocity of light, which is
183,470 miles per second, being a decrease of nearly 8,000 miles ;
and 5th, By the observations on Mars during the opjiositions of
18G0 and 18G3.
A diminution in the Sun's distance will necessarily involve a
corresponding change in the masses and diameters of the bodies
composing the Solar system. The Earth's mass will require an
increase of about one-tenth part of the whole.
Substituting LeVerrier's solar parallax (8".95) in Eq. (38),
40
H , ■ '
the Earth's mean distance from the Siin becomes 91,333,070
which is a reduction of 4,048,800 miles. The Sun's apparent
diameter at the Earth's mean distance = 32' 3".G4, and in order
that a body may subtend this angle, at a distance of 91,333,070
miles, it must have a diameter of 8a 1,700 miles, which is a
diminution of 37,800 miles. The distances, diameters, and
velocities of all the planets in our system will require corres-
ponding corrections if we express them in miles. Since the
periodic times of the planets are known with great precision, we
can easily determine by Kepler's third law, their mean distance
irom the Sun in terms of the Earth's mean distance. Thus :
if T and t be the periodic times of the Earth and a planet
respectively, and D the planet's mean distance, then regarding
the Earth's mean distance as unity, we have T^ : f^ :: 1 : Z>
Or,
J)
= (^'
(39).
In the case of Neptune the mean distance is diminished by
about 121,000,000 miles. Jupiter's mean distance is diminished
21,003,000 miles, and his diameter becomes 88,290 miles, which
is a decrease of 3,808 miles. These numbers shew the great
importance which belongs to a correct knowledge of the Solar
parallax.
. 'vh
-■»*
41
31,333,670
s apparent
d in Older
)1, 333,070
v^hicl) is a
eters, and
ire corres-
Since the
jcision, we
n distance
i. Thus :
a planet
regarding
:; 1 : i>
(39).
lished by
iminished
es, whicli
he great
the Solar
(III.)
A TRANSIT OF MERCURY.
May 6th, 1878.
Transits of Mercury occur more frequently than those of
Venus by reason of the planet's greater velocity. The longitudes
of Mercury's nodes are about 46° aud 226°, and the Earth
arrives at these points about the 10th of November and the 7th
May, transits of this planet may therefore be expected at or
near these dates, those at the ascending node in November, and
at the descending node in May.
Mercury revolves round the Sun in 87.9693 days, and the
Earth in 365.256 days. The converging fractions approximating
87.9693 7 13 33
^'^ 1365:256- "^'^ 2-9' 54' 137' ^"^ '
Therefore when a transit has occured at one node another may
be expected after an interval of 13 or 33 years, at the end of
which time Mercury and the Earth will occupy nearly the same
position in the heavens.
Sometimes, however, transits occur at the same node at inter-
vals of 7 years, and one at either node is generally preceded or
followed by one ^ +he other node, at an interval of 3 J years.
The last transit at the descending node occurred in May,
1845, and the last at the ascending node in November, 1868.
Hence the transits for the 19th century will occur, at the de-
scending node May 6th, 1878 ; May 9th, 1891 ; and at the
descending node November 7th, 1881, and November 10th,
1894.
COMPUTATION OP THE TRANSIT OP 1878.
From the tables'^ of the planet we obtain the following helio-
centric positions : —
* Tables of Mercury, by Jogpph Winlock, Prof. Mathematics U. S.
Navy, Wnshington, 1864,
Il
42
tii'ii
It!
WasliiiiKtfPii Mean
Time.
Merrury'H Ilelluc.
Longitude.
MeriMir.v's Ilelloc.
Latitude.
iiog. Rad.
Vector.
1878, May 6d. Oh.
Ih.
« 2h.
♦« 3h.
225° 52' 57".0
226 15 .4
220 7 33 .6
226 14 51 .6
r 17".3 N.
6 23 .4
5 29 .0
4 35 .8
9,6545239
9,0540389
9,6547535
9,6548677
The following positions of the Earth have been obtained from
Delanibre's Solar Tables, corrected by n)yself, tt being taken
equal to 8". 95 at the Earth's mean distance : —
Washington Mean
Time.
1878, May 6d. Oh.
Ih.
" 2h.
" 3h.
Earth's Helioc.
Longitude.
220° 0' 38".9
220 3 04 .0
220 5 29 .1
226 7 54 .2
Log. Ranh's Rad.
Vector.
10,0040993
10,0041038
10,0041082
10,0041120
The Sun's true longitude is found by subtracting 180° from
the Earth'.s longitude.
Passing to the true geocentric places by Formulse (3), (4), and
(5), we obtain : —
Washington Mean
Tfnio.
Mercury's true Geoc.
Longitude.
Mercury's true Geoc.
Latitude.
5' 53".0 N.
5 10 .2
4 20 .8
3 43 .4
1878, May 6d. Oh.
Ih.
" 2h.
" 3h.
40° 0' 52". 4
40 5 20 .4
46 3 48 .3
40 2 10 .3
Formula (7) gives log. distance from Earth at Ih, = 9.7406455.
This will be required in formuhe (14) and (15) for finding the
aberration.
Formula (9) gives P = 15''. 9.
The semi-diameter of Mercuiy at the Earth'.s mean distance,
3". 34 = d', therefore by Eq. (12), d == 5". 98.
Aberration in Longitude r=: + 0" 07, by Eq. (14).
Aberration in Latitude =^ + 3".34, by Eq. (15).
The Sun's semi-diameter = 15' 52".3. (Solar Tables). '
The Sun's aberration c= — 20". 25,
^a
'•''fM''
')?. Rail.
Sector.
545239 i
54G389 ]
547535
548677 I
I
lefl from
ig taken
Id.
80° from
(4), and
eoc
466455.
ling the
istance.
40
Correctinj^ i'ov aberration we obtain the apparent places as
foUowH : —
AVu»liin)jloii iVIeuii
Tiniu.
Meicur.v'o A(i|mr.
Ueuc. LiOiiKituUe.
Morciiry'.s
Ap|i. Qeoc. Lat.
Suii'it
Appar. Longitude.
1878, May, (k\ Oh.
" Ih.
" :i
40" 0' 51). "0
40 5 27.0
40 3 54.9
40 2 'JO.!)
5' 50 ."9N.
5 13.5
4 30.1
3 40.7
40^ 0' 18".7
40 2 43.8
40 5 8.9
40 7 34.0
liitcrpolatiiig for the lime of conjunction and collecting the
element.^, we have
Washington mean time of conjunction in longitude,
May 6d. Ih. 41 miu. 17 sec.
Mercury's and Sun's longitude 40° 4' 23''.G
Mercury'.s latitude 4' 43". 6 N.
Sun's hourly motion in longitude 2' 25".l E.
Mercuiy's hourly motion in longitude 1' 32". I W.
Mercury's hourly motion in latitude 43".4 S.
Sun's equatorial horizimtal parallax 8". 87
Mercury's ecjuatorial horizontal parallax ... 15''. 9
Su)''s semi-diameter 15' 52".3
Mercury's semidiameter 5".9
Employing the same notation as in Art. 13, the preceding
elements give the following I'esults. Relative hourly motion in
longitude : 3' 57". 2; n - 10° 22' 7"; mn ^. 24 I'M 3 the rela-
tive hourly motion in apparent orbit. C F the least distance
between the centres ~ 279" ; E F — 51".04; time of describing
E F - 12 m. 42 sec. Since Mercury is north of the Sun's
centre at conjunction, and moving southward, i7i^will lie on the
right of C E (see Fig, 4), and the middle of the transit will
take place at Ih. 54m. P.M. • ,
Sum of semi-diameters = 958".2
V = 16° 55' 44" ; V F = 916".G8 ;
Time of describing V F - 3h. 48.1 min. - half of the dura-
tion. Subtracting 3h. 48.1 min. from, and adding the same to
44
tho time of the middle of the trnusit, we obtain the times of the
fivHt and h\st contacts, as seen from the Earth's centre, thus :
First external contact May (id. lOh. H.i) niin. A,M.
Lust external contact •* 5h. 42.1 niin. P.M.
Mean time at Washington,
The places which will liavo the Sun in the zenith at these
times can be found in the same manner as in Art. 14, with the
aid of the following elements : —
Obliquity of the Ecliptic 23° 27' 2.'5".
Sidereal time at Washington at nitian noon of May Gth (in arc)
44° 24' /)0".4G.
Since the relative parallax is only 7" the time of the first or
last contact will not be much influenced by the parallax in
longitude and latitude, and therefore the preceding times for
Washington are sufficiently accurate for all ordinary purposes.
The mean local time of beginning or end for any other place,
is found by applying the difi'erence of longitude, as below : —
The longitude of Washington is oh. 8m. 11 sec. W.
The longitude of Toronto is 5h. 17m. 33 sec. W.
Therefore Toronto is D min. 22 sec. west of Washington.
Then, with reference to the centre of the Earth, we have for
Toronto,
First external contact May Gd. 9h. SG.Sm. A.M.
Last external contact " 5h. 32.7m. P.M.
Mean time.
For Quebec, longitude 4h. 44m. 48 sec. W.
First external contact May Gd. lOh. 29.3m. A.M.
Last external contact " Gh. 6.5m. P.M.
Mean time.
For Acadia College, longitude 4h. 17.Gm. W, ",
First external contact May Gd. lOh. 56.5m. A.M.
Last external contact '• Gh. 32.7m. P.M.
Mean time.
For Middlebury College, Vermont, longitude 4h. 52.5m. W.
First external contact, May Gh. lOh. 21.5m. A.M.
Last external contact *' 5h. 57.7m. P.M.
Mean time at Middlebury.
45
APPENDIX.
Eclipses of the Sun are computed in precisely the same way
as transits of Venus or Mercury, the Moon taking the place of
the planet. The Solar and Lunar Tables furnish the longitude,
latitude, equatorial parallax, and semi-diameter of the Sun and
Moon, while Formula (19)-(30) furnish the parallax in longitude
and latitude. If the computation be made from an ephemens
which gives the right ascension and declination of the Sun and
Moon instead of their longitude and latitude, we can dispense
with formula (21) and (23), and adapt (25), (2G), (29). and (30) to
the computation of the parallax in right ascension and declination.
In Fig. G, let Q be the pole of the equator, then i Q is the
co-latitude = 90° - ; Z Q S =^K the Moon's true hour angle
== the Moon's A. R. - the sidereal time ; S Q S' is the parallax
in A. R. == a:, and Q 5' - Q *S is the parallax in declination = y.
Put Q S, the Moon's true north polar distance .= 90 — S, then
Formula (25) and (26) become,
sin X- = sin Pcos sec g sin {h + a) (25, bis).
= A; sin (^ + oc)
Or,
X
k sin h ^ F sin 2h ^ k' sin 3/t _^ ^^ ^20, bis).
1 /
sin 1
sin 2
Q//
Sin o
Again, the formulse for determining the auxiliary angle in
(29) becomes, r, , o■^ x
^ ' cot ^ = cot cos (/t + I) sec %
And (29) becomes,
sin y = sin P sin cosec sin ( (^ - g) + v/) . (29, bis).
= /csin((e-8) + y)
k sin {Q-D . 7 c' sin 2 (6>- 8) , /c" sin 3 ^0 -g) ^^
y "= 8in 1- "^ ^^^' s^^ 3" ^
(30, bis).
46
Theso purnllaxc'S when ai)[)Hn(l with tliciv propfr signs to the
light nscciiHioiiH aiul th'clinations (tf tlic Moon for th(> assumed
times, luiiiish tlie apjuiretd right nsccnsioiis and declinatiuns.
The din'crt'nce between the apparent jl. 11, oi" the Moon und th«
trap. A. U. of the Sun, must he reduced to sceonds of arc of a
(jre.at cli'cle, hy multiplying it by the cosino of the Moon's appa-
rent declination. 'I'lie a|.))arent places of the AFoon with respect
to the Sun will give the Moon's apparent orbit, and the times of
apparent contact of limbs are found in the same way us described
in Art. lii. The only othei correction necessary to take into
account, is that for the augnnuitation of the Moon's semi-
diauuiter, duo to its altitude. The augmentation may be taken
from a table prepared for that j)ur|)ose, and which is to be found
in all good works on Practical Astronomy, or it may, in the case
of solar eclipses, be computed by the following formula) : —
TO FIND TllK ALKJMKNTATION OK TIIK MOONS SKMI-DIAMETER.
Let C anil M be the centres of the Earth and Moon, A a point
on the Earth's surface, join CM, A 31, and ]»roduce C A to Z ;
then M (■ Z is the Moon's true zem'th distance -^~ Z =. arc Z S in
Fhj. G ; and MA Z is the apparent zenith distance ^= Z' =: arc
Zi ii' in the same figure. Represcjnt the Moon's semi-diameter as
seen from C, by d; the aemi-dianu'ter as seen from A by d' \ the
apparent hour angle Z Q S' by h', and the apparent declination
by g', then
i£ ^ CM _ sin Z'
d A M sin Z
^miZ S'
sin Z iS
'II ^'
sni n cos o
sin h cos 3
sni h cos t)
{8ee Fuf. G.)
, by Art. 21.
(40).
Therefore,
d' ^ d.
(41).
sni a cos o
This formula furnishes the augmented semi-diameter at once.
It can be easily modified so as to give the augmentation directly,
but with logarithms to seven decimal places, it gives the apparent
semi-diameter with great precision.
47
Ah cxninploH wc give th«i following, tlio fii'Ht of which in fi'om
Looniis's I'ructical Asiroiioiny : —
Kx. 1. Kind th(! IMnou's pm-iilliix in A. A', and lU'clinution, .ind
thr imgnicntcd Kcnii-diamclcr for lMiiIiidf'l|ihia, F^nt. iV.f ,'>7' 7" N.
when tho hoi-izontal piirallax of tlu; placi! is .Y.)' '.]()". H, iho Moon's
declination 2[' .V 11". C N., llio Moon's true hour angle 01' 10'
47". 4, and the sunii-diameter IG' IG".
J ws.- Parallax in A. R., \i' 17"M
Dec, -jr,' 10". 1
Augmented senii-diani ^ 1(»' 2G".1;5.
Ex. 2. R('((uirod i\w tinies of beginning and end of the Solar
Eclipse of Octol)or 0-10, 1874, for l'](liiil)urgh, Lat. ,10' 57' 23" N.
Long. 12ni. l."5 »(<(! Wesl., from the following eleinont.s ohtained
from the English Naiitiral Almanac : —
rTreenvvich moan time of conjunction in ^1 />',
Oct. 9d. 22h. 10m. 11.4 .sec.
bun s and Moon's yl /t", 195" 36' 30"
Moon'K declination S ,> 39 8.9
Sun's declination S G 39 .34.1
Moon's hourly motion in J. Yi' 26 21,9
Sun's do 2 18.2
Moon's hourly motion in Declination. S 13 48.3
Sun's do S 5C.9
Moon's Equatorial Horizontal Parallax. .53 59.6
Sun's do do 9.0
Moon's true .semi-diametor 14 44.2
Sun's do 16 3.8
Greenwich sidereal time at conjunction. 171 23 32.8
Assuming, for the beginning, 20h. 55m., and for the end,
23h. 10m. Greenwich mean time, we obtain from the preceding
elements and formuhe the following results, which may be
verified by the Student : —
Geocentric latitude ::::= oo''-' 16' 41" ; reduced or relative
Parallax = 53' 43".2.
48
\^^
Moon's A R
Sun's AR
Moon's Dec
Sun's Dec
Sid. Time at Edin. (in arc).
Moon's true hour angle...
Moon's Panillax in A.R...
Moon's do in Dec...
Moon's apparent A R
Moon's do Dec
Diff. of A R in seconds of
arc of great circle
DifF. Dec
Aug. semi-diam of Moon..
20h. 56m. G. M. T.
23h. 10m. G. M T.
195" 3' 27". 6
196° 2' 46"9
195 33 3G.8
195 38 47.8
5 21 50.9 S.
^ 52 54.6 S.
G .38 22.9 S.
6 40 30.9 S.
149 21 51.0
183 12 24.1
45 41 56.1 E.
12 50 22.8 E.
+ 21 49.4
+ 6 48.5
+ 46 25.1
+ 47 32.7
195 25 17.0
196 9 35.4
6 8 16.0 S.
6 40 27.3 S.
496"- 9. Moon W.
1835".l MoonE.
30' 6". 9, Moon N.
3".6 Moon N.
888".4
890".5.
Eclipse begins October lOd. 8h. 43m. 32 sec. AM.
Eclipse ends " lOh. 58m. 22 sec. A.M.
Mean time, at Edinburgh.
Magnitude
,369 Sun's diam.
THE END.