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THE COMPUTATION
OF THE
TRANSITS OF VENUS
*
FOR THE YEARS 1874 AND 1 882,
MERCURY FOR THE YEAR 1878,
FOR THE EARTH GENERALLY AND FOR SEVERAL
TLACES IN CANADA.
WITH A
POPULAR DISCUSSION OF THE SUN'S DISTANCE FROM THE EARTH,
AND AN APPENDIX SHEWING THE METHOD OF COMPUTING
SOLAR ECLIPSES.
BT
J. MORRISON, M.D., M.A.,
(M.B., University of Toronto),
MEMBER OF THE MEDICAL COUNCIL, AND EXAMINER IN THE COLLEGE OF
PHYSICIANS AND SURGEONS OF ONTARIO.
TORONTO :
ROW SELL & HUTCHISON.
1873.
Entered according to Act of Parliament of Canada, in the year one thousand eight
hundred and seventythree, by J. MORRISON, in the Office of the Minister of Agriculture.
TORONTO :
PRINTED BY ROWSELL AND HUTCHISON,
KING STREET.
PREFACE.
THE following pages were drawn up for the use of
Students pursuing the higher Mathematical course in
our Colleges and Universities. All the necessary formulae
for calculating transits of the planets and solar eclipses
from the heliocentric elements, have been investigated
in order to render the work as complete in itself as
possible ; and while I have endeavoured to simplify the
computation, I have, at the same time, given as full an
account of the various circumstances attending these
phenomena, as is to be found in any of the ordinary
works on Spherical and Practical Astronomy.
This is, I believe, the first work of the kind ever
published in Canada, and therefore I hope it will tend
to encourage, in this country at least, the, study of the
grandest and noblest of the Physical Sciences.
J. M.
TORONTO, March 4th, 1873.
In preparation by the same Author.
FACTS AND FORMULAE IN PURE AND APPLIED
MATHEMATICS,
For the use of Students, Teachers, Engineers, and others.
(I.)
A TRANSIT OF VENUS.
DECEMBER STH, 1874.
ART. 1. A transit of Venus over the Sun's disk, can only
happen when the planet is in or near one of its nodes at the
time of inferior conjunction, and its latitude, as seen from the
Earth, must not exceed the sum of its apparent semidiameter
and the apparent semidiameter of the Sun, or 3r / f96r / =^992 // ;
and therefore the planet's distance from the node must not
exceed 1 50'.
If the Earth and Venus be in conjunction at either of the
nodes at any time, then, when they return to the same position
again, each of them will have performed a certain number of
complete revolutions.
Now the Earth revolves round the Sun in 365.256 days, and
Venus in 224.7 days; and the converging fractious approxi
mating to
2247 8 235 713 .
, are , , , <fcc.,
365.256 13 382 1150
where the numerators express the number of sidereal years, and
the denominators the number of revolutions made by Venus
round the Sun in the same time nearly. Therefore transits may
be expected at the same node after intervals of 8 or 235 or 713
years. Now, there was a transit of Venus at the descending
node, June 3rd, 1769 ; and one at the ascending node, December
4th, 1639. Hence, transits may be expected at the descending
node in June, 2004, 2012, 2247, 2255, 2490, 2498, &c. ; and at
the ascending node in .December, 1874, 1882, 2117, 2125, 2360,
2368, &c. In these long periods, the exact time of conjunction
may differ many hours, or even four or five clays from that found
by the addition of the complete sidereal years, according to the
preceding rule, which supposes the place of the node stationaiy,
and that the Earth and Venus revolve round the Sun with
uniform, velocities hypotheses which are not strictly correct.
In order, therefore, to ascertain whether a transit will actually
occur at these times or not, it will be necessary to calculate
strictly the heliocentric longitude and latitude, and thence the
geocentric longitude and latitude at the time of conjunction ;
then, if the geocentric latitude be less than the sum of the
apparent semidiameters of Venus and the Sun, a transit will
certainly take place. The present position of Yenus's nodes,
is such that transits can only happen in June arid December.
The next four will take place December 8th, 1874, December
6th, 1882, June 7th, 2004, June 5th, 2012.
APPROXIMATE TIME OF CONJUNCTION IN LONGITUDE.
J 2. From the Tables of Venus'"" and the Sunt, we find
the heliocentric longitude of the Earth and Venus to be as
follows :
Greenwich Mean Time.
Earth's Helioeen. Long.
Venus's Helioeen. Long.
Dec. 8th, Oh. (noon)
Dec. 9th, Oh. "
76 11' 33".5
77 18' 34".3
75 52' 55". 1
77 2<J' 40".6
From which it is seen that conjunction in longitude takes place
between the noons of the 8th and 9th December.
The daily motion of the Earth = 1 1' 0".8.
The daily motion of Venus = 1 36' 4 5". 5.
Therefore Venus ? s daily gain on the Earth = 35' 44". 7, and
the difference of longitude of the Earth and Venus at December
8th, Oh. = 24' 38".4, therefore we have .
35' 44".7 : 24' 38". 4 : : 24h. : 16h. 32m.
Hence the approximate time of conjunction in longitude is
December 8th, 16h. 32m.
* Tables of Venus, by G. W. Hill, Esq., of the Nautical Almanac Office,
Washington, U. S.
f Solar Tables, by Hausen and Olufsen: Copenhagen, 1853. Delambre's
Solar Tables. Leverrier's Solar Tables, Paris.
The exact time of conjunction will be found presently by
interpolation, after we have computed from the Solar and Plane
tary Tables, the heliocentric places of the Earth and Venus (and
thence their geocentric places) for several consecutive hours both
before and after conjunction, as given below :
Greenwich Mean
Time.
Earth's Heliocentric;
Longitude.
Venus's Heliocentric
Longitude.
Venus's
Heliocentric
Latitude.
Dec. 8th, 14h.
15h.
76 53' 8".9
70 55 41 .4
7649 / 21 // .4
70 53 23 .3
4/30* K
444 .3
16h.
76 58 13 .9
76 57 25 .2
458 .6
17h.
18h.
77 4G .5
77 3 19 .1
77 1 27 .1 ! 513 .
77 5 29 . i 527 .3
" 19h.
77 5 51 .7
77 6 30 .9
5 41 .6
The Sun's true longitude is found by adding 180 to the
Earth's longitude.
Greenwich Mean
Time.
Log. Earth's Radius
Vector.
Log Venus's Radius
Vector.
Dec. 8th, 14h.
9.9932897
9.8575301
15h.
9.9932875
9.8575330
" 16h.
9.9932854
9.8575309
17h.
9.9932833
9.8575281
ISh.
9.9932811
9.8575253
19h.
9.0932790
9.8575225
Venus's Equatorial hor. parallax =33".9=:/ J . (See Art. G.)
Sun's Equatorial hor. parallax == 9 // .l=7r.
Venus's Semidiameter =31 / '.4=<:/. (See Art. 7.)
Sun's Semidiameter = 1 6' 1 0". 2 = g.
The last four elements may be regarded as constant during
the transit.
Sidereal time at 14h.=7h. 10m. 35.64sec. = Sun's mean longi
tude 4 Nutation in A.R., both expressed in time.M* \v.r* .
The places of Venus and the Earth, just obtained, are the
heliocentric, or those seen from the Sun's centre. We will now
investigate formulae for computing Venus's places as seen from
the Earth's centre.
GEOCENTRIC LONGITUDE.
ART. ,3. In Fig. 1, lot S y he the Sun's centre, E the Earth's
and V that of an inferior planet, S X the direction of the vernal
equinox. Draw V P perpendicular to the plane of the Earth's
orbit, then X S E is the Earth's heliocentric longitude ; X S P
the planet's heliocentric longitude ; F $ P the planet's helio
centric latitude = / ; YEP the planet's geocentric latitude = \ ;
P 8 E the difference of their heliocentric longitudes, or the
commutation = C ; P E S the planet's elongation E ; S P E
the planet's annual parallax = p ; S E the Earth's radius vector
R ; V S the planet's radius vector = r. Then in the triangle
P S #, we have P 8 = r cos /, E S=fi, and angle P S E= (\
therefore
# + ?cos/ : Rrcos I :: tan \ (p + E) : tan (p E)
But
p+E = 1806 r
= 00 
Therefore 1 +T cos / : 1 !Lcos I : ; cot
R R
Put cos Z = tan
R
tan i
Then
tan 1 ( p  E) =
cot 1 ,
2
= tan (45 fl) cot X.,
and E = 90 C l 7^ .
(2)
Now, before conjunction, the planet will be east of the Sun,
and if // be the Sun's true longitude ( = the Earth's heliocentric
longitude f 180), and G the geocentric longitude of the planet,
we have
G = H J E (3).
the positive sign to be used before, and the negative sign after
conjunction.
When the angle C is very small, the following method is to be
preferred. Draw P D perpendicular to S E, then
iS D = r cos I cos C
P D = r cos I sin C ,
r cos / sin C
Then tan E =
R r cos / cos C
tan sin (7
1 tan cos 6 f
GEOCENTRIC LATITUDE.
ART. 4. From the same figure we have
SPtaul = VP = PEi
tan X ^ & sin 7:7
Or
tan* " /'A' " sin
Therefore tan X c= _ m tan /, (5).
sin C r
When the planet is in conjunction, this formula is not applic
able, for then both E arid C are 0, and consequently their sines
are each zero.
Since E, P and *S' are then in a straight line, we have
EP = R  r cos/
and E P tan \ = r sin /
Therefore tail X ^ T 81P * (6).
R r cos /
DISTANCE OF THE PLANET FROM THE EARTH.
ART. 5 E Fsiii \ FP = r sin I
r sin <?
/; v T STx ' < 7 >'
When the latitudes are small the following formula is pre
ferable :
sin E : sin 6' :: / J : /' ^
II r cos / : ^ Fcos ^
r sin 6" cos
From which & V r^ , (8).
sin E cos X
o
10
HORIZONTAL PARALLAX OF THE PLANET.
ART. 6. Let P be the planet's horizontal parallax ; TT the
Sun's parallax at mean distance ; then, r being the planet's
radius vector, expressed of course in terms of the Earth's mean
distance from the Sun regarded as unity,
E V : 1 :: TT : P
From which
P
. TT
7T
JTv
TT sin X
r sin /
TT sin E cos X
r sin C cos /
(9).
(10).
APPARENT SEMIDIAMETER OF THE PLANET.
ART. 7. The semidiameter of a planet, as obtained from
observation with a micrometer when the planet is at a known
distance, may be reduced to what it would be, if seen at the
Earth's mean distance from the sun, viz., unity.
Let d' be this value of the semidiameter, and d its value at
any other time.
E V : 1 :: d' : d
Then
Therefore
d = ^L
EV
d' sin X
r sin I '
d'.
7T
(11).
(12).
ABERRATION IN LONGITUDE AND LATITUDE.
ART. 8. Before computing the geocentric places of Venus by
the preceding formulae, we will first investigate formulae for
computing the aberration in longitude and latitude.
Let p and e (Fig. 2) be'cotemporary positions of Venus and the
Earth ; P and E other cotemporary positions after an interval
t seconds, during which time light moves from p to e or E.
If the Earth were at rest at E, Venus would be seen in the
direction p E. Take E F == e E and complete the parallelogram
11
E 7?, then p E R is the aberration caused by the Earth's motion,
and e p is the true direction of Venus when the earth was at e.
Now R E is parallel to p c, therefore the whole aberration =
PER, or the planet when at P will be seen in the direction E R.
But PER PEp pER
= PEp Epe
= the motion of the planet round E at rest,
minus the motion of E round p at rest.
= the whole geocentric motion of the planet in
t seconds.
Now, light requires 8 minutes and 17.78 sec. to move from the
Sun to the Earth, and if D be the planet's distance from the
Earth (considering the Earth's mean distance from the Sun
unity), then
/  D x (8 min. 17.78 sec.)
= 497.78 D.
And if m the geocentric motion of the planet in one second,
then
aberration = m t
= 497.78 mD. (13).
Resolving this along the ecliptic and perpendicular to it, we have
(7 being the apparent inclination of the planet's orbit to plane
of the ecliptic).
Aberration in Long. = 497.78 m D cos 1 (14).
Aberration in Lat. = 497.78 mD sin 1. (15).
We are now prepared to compute the apparent geocentric
longitude and latitude of Venus, as well as the horizontal paral
lax, semidiameter, aberration and distance from the Earth.
FOR THE GEOCENTRIC LONGITUDE.
ART. 9. At 14 hours, we have, by using Eq. (3,) since the
angle C is only 3' 4 7. "5,
logr = 9.8575364
cos / = 9.9999996
logR = 9.9932897
tan0 = 9.8642463
6 = 36 11' 15"
12
tan = 9.8642463, tan = 9.8642463
cos = 9.9999997 sin C ** 7.0425562
0.731553 =. 9.8642460 6.9068025
log (ltan0 cos C) = 9.4288569
tan E = 7.4779456
E = 10' 20"
Then G  256 53' 8". 9 + 10' 20"
= 257 3 28 .9
FOR THE GEOCENTRIC LATITUDE.
By Eq. (5). sin E = 7.4779437
Ian I = 7.1169388
cosec = 12.9574438
tan X  7.5523263
\  12' 15". 8 North.
VENUS'S DISTANCE FROM THE EARTH.
By Eq. (8.) r = 9.8575364
sin C = 7.0425562
cos / = 9.9999996
cosec E = 12.5220563
sec X 0.0000027
log E V = 9.4221512
Eq. (7,) gives log E V = 9.4221513
VENUS'S HORIZONTAL PARALLAX.
The Equatorial Horizontal Parallax of the Sun at the Earth's
mean distance will be taken = 8". 95, instead of 8". 577, for
reasons which will be given when we come to discuss the Sun's
distance from the Earth.
By Eq. (9.) TT = 0.951823
sin \ = 7.552323
8.504146
r, (ar. comp.)' = 0.142463
cosec / = 12.883061
log P = 1.529670
P  3" 9
This element is constant during the transit.
13
VENUS'S SEMIDIAMETER.
Venus's semidiameter at the Earth's mean distance from the
Sun, as determined by theory and observation, is 8". 305 = d'.
By Eq. (12.) a' = 0.91934
P = 1.52967
2.44901
TT 0.95182
logd = 1.49719
d = 31". 4, constant during transit.
Some astronomers recommend the addition of about ^ part
for irradiation.
The aberration cannot be computed until we find Venus's
hourly motion in orbit as seen from the Earth.
In this manner we obtain from Formulae I to 12, the following
results :
Greenwich Mean
Time.
Venus's Geocentric
Longitude.
Venus's Geocentric
Latitude.
Log. Venus's
Distance from Earth.
Dec. 8th, 14h.
15h.
257 3' 28". 9
257 1 57 .7
12' 15". 8 N.
12 54.7
9.4221513
16h.
257 26 .6
13 33 .7
9.4221491
17h.
256 58 55 .9
14 12 .9
18h.
256 57 24 .8
14 52 .0
9.4221342
19h.
256 55 54 .4
15 31 .0
VENUS S ABERRATION IN LONGITUDE AND LATITUDE.
ART. 10. Venus's hourly motion in longitude is 91", and in
latitude 39" (as seen from the Earth's centre). Since these are
very small arcs, we may, without sensible error, regard them as
the sides of a rightangled plane triangle.
Venus's hourly motion in orbit = J (39 2 f 9P) = 99" and
therefore the motion in one second = 0".0275
Also
T 91 , . 7 39
cos / = and sm 1 =
99 99
14
Then by Eq. (U),
497.78
m
I)
2.697037
8.439332
9.422149
0.558518
9.963406
cos 7
Aber. in long. = 3". 32 = 0.521924
0.558518
sin 1 = 9.595429
Aber. in latitude = 1".42 = 0.153947
The aberration is constant during the transit. Since the
motion of Venus is retrograde in longitude, and northward in
north latitude, the aberration in longitude must be added to,
and the aberration in latitude subtracted from, the planet's true
geocentric longitude and latitude respectively in order to obtain
the apparent places.
SUN'S ABERRATION.
ART. 11. The Sun's aberration may be found from Eq. (13),
by making D = R and m = the Sun's motion in one second.
The Sun's hourly motion in long. = 152". 6, and the motion
in one second = 0".0423
= m
Then aberration (in long.) = 497.78 Rm
= 20". 7 7, and as the Sun always
appears behind his true place, the aberration must be subtracted
from the true longitude.
Applying these corrections, we obtain the following results :
Greenwich Mean
Time.
Sun's Apparent
Longitude.
Venus's Apparent
Geocen. Longitude.
Venus's Apparent
Geocentric Latitude.
Dec. 8th, 14h.
256 52' 48". 2
257 3' 32".2
12' 14".4 N.
15h.
256 55 20 .7
257 2 01 .0
12 53.3
16h.
256 57 53 .2
257 29 .9
13 32.3
17h.
257 25 .8
256 58 59 .2
14 11 .5
18h.
257 2 58 .3
256 57 28.1
14 50 .6
19h.
257 5 31 .0
256 55 57 .7
15 29 .6
15
APPARENT CONJUNCTION.
ART. 12. By inspection we find that conjunction will take
place between IGh. and 17h.
The relative hourly motion of the Sun and Venus is 243". 2,
and the distance between them at 16h. is 156". 7.
Then 243".2 : 156".7 :: 1 hour : 38m. 40 sec.
During this time the Sun moves I' 38''. 3, and Yenus 58".5 ;
therefore, by collecting the elements we have :
Greenwich M. Time of conj. in long. Dec. 8th...l6h. 38m. 40sec.
Sun and Yenus's longitude 256 59' 31".4.
Yenus's latitude 13' 57".4, N.
Yenus's hourly motion in longitude 1' 30". 7, W.
Sun's do. do* 2' 32". 5, E.
Yenus's hourly motion in latitude 39". 1, N.
Yenus's horizontal parallax 33". 9 .
Sun's do. 9".l.
Yenus's semidiameter 31".4.
Sun's do. 16' 16".2.
Obliquity of the Ecliptic 23 27' 27". S.
Sidereal time at 14h. (in arc) 107 38' 54".6.
Equation of time at conj. j 7m. 34 sec.
The last three elements are obtained from the Solar Tables.
TO FIND THE DURATION AND THE TIMES OF BEGINNING AND END
OF THE TRANSIT FOR THE EARTH GENERALLY.
ART. 13. The Transit will evidently commence when Yenus
begins to intercept the Sun's rays from the Earth, and this will
take place when Yenus comes in contact with the cone circum
scribing the Earth and the Sun.
The semidiameter of this cone, at the point where Yenus
crosses it (as seen from the centre of the Earth), is found as
follows :
Let E and & be the centres of the Earth and Sun (Fig. 3),
and V the position of Yenus at the beginning of the transit.
Then the angle V E 8 is the radius or seniicUameter of the cone
where Yenus crosses it.
16
VES = AE S + VEA
= 4/JS 4 # KJ57  B4#
= S + P  IT
= 976".2 f 33".9  O'M = 1001'
(16).
In Fig. 4, take AC 1001"; C f E at right angles to A C,
= 13' 57". 4 ; Cw = 4' 03".2, the relative hourly motion in
longitude; 6'w* = 39". 1, the hourly motion of Venus in lati
tude, and through E draw VX parallel to mv, then E is the
position of Venus at conjunction, m n is the relative hourly
motion in apparent orbit, and C F perpendicular to V X, is the
least distance between their centres. The angle E C F = angle
C n m. Put E C A ; On = m ; C m = <j C V = C A
f semidiam. of Venus c; Cv = C A sernidiam. of
Venus = b ; and T = the time of conjunction.
Then, by plane Trigonometry, we have tan n = ~ } m n
m sec n relative hourly motion in apparent orbit; CF =
\ cos n ; F E = \ sin w ; time of describing # /''
X sin 2 n
m sec n
t ; therefore middle of transit occur** at T J~ t .
(Positive sign when lat. is S. ; negative when N.)
Again, sin V = ~  ; V F = c cos F; time of describing
V F == sin ?i cos V = t' = time of describing FJT, supposing
J
the motion in orbit uniform, which it is, very nearly.
Therefore first external contact occurs at T ^ t /', and last
external contact at T t + t' .
Writing b for c, these expressions give the times of first and
last internal contact.
Substituting the values of X, c, </ and ?w, we obtain
n = 9 7' 33".9 .
Hourly motion in apparent orbit = 246". 5 3 ; CF= 13'
46 /; .8; EF=132".8; time of describing E F= 32m. 19sec.
Therefore middle of transit = 16 A. 6m. 21 sec.
17
Again, the angle F= 53 12' 41".7 ; VF = 618".26, and
the time of describing VF . = 2h. 30m. 28sec. Therefore the
first external contact will take place at 13h. 35m. 53sec., and the
last external contact at 18h. 36m. 49sec. The duration will
therefore be 5h. Ira. very nearly.
The duration as thus determined, is not the duration of the
transit as seen from the centre of the Earth, or from any point on
its surface, but the whole duration from the moment Venus
begins, to the moment Venus ceases to intercept the Sun's rays
from any part of the Earth's surface.
For the time of internal contact, we have It = 9 6 9". 6. Then
cF
sin v = ^ , or v = 58 30' 32".5 ; v F = 506".4S, and time
of describing v F, 2h. 3m. IGsec. Therefore, the first internal
contact will take place at 14h. 3m. 5sec., and the last internal con
tact at 18h. 9m. 37sec.
FROM THE EARTH'S CENTRE.
As seen from the centre of the Earth, we have at the first
external contact, c the sum of their semidiameters = 1007". 6,
and at the first or last internal contact, b difference of their
semidiameters = 944". 8 .
Sin V = = = . , therefore V = 55 8' 28".5
V F == c cos V = 575".8, and the time of describing V F
2h. 20m. 9sec. Therefore the first external contact as seen from,
the Earth's centre will occur at 137i. 46m. I2sec., and the last
external contact at I8h. 26m. 30sec.
The duration = 4h. 40.3m.
Again, sin v = , v = 61 3' 10".
u *
vF=!jco$v = 457". 286, and time of describing it =
Ih. 51m. 17sec. Therefore,
First internal contact y 14/i. 15m. 4sec.
Last internal contact, 17 h. 57m. 38sec.
ART. 14. The Sun's R. A. and Dec. are obtained from the
Equations,
tan, R. A. = tan Long, cos obliq. (17).
tan Dec. = sin R. A. tan oblfq. (18).
18
From which we find, at conjunction,
Sun's R A. = 255 51' 53".
= 17h. 3m. 27sec.,
and Sun's Dec. = 22 49' 15" S.
Adding 2h. 38m. 40sec. converted into sidereal time and then
expressed in arc, to the sidereal time at 14h., we obtain the
sidereal time at conj., = 147 25' 25". The Sun's R. A. at the
same time = 255 5V 53", therefore the difference 108 26' 27"
is the Sun's distance east of Greenwich, or the east longitude of
the places at which conjunction in longitude takes place at appa
rent noon, and that point on this meridian whose geocentric
latitude is equal to the Sun's dec., will have the sun in its zenith
at the same time. The Sun's dec. was found to be 22 49' 15" S.
= the geocentric latitude which, converted into apparent or
geographical latitude by Eq. (19), becomes 22 57'. 5 S.
In the same way we find, that at the time of the first external
contact, the Sun's R. A. = 255 44', and Dec. 22 48' 33" S.,
and the sidereal time = 104 11'; therefore at this time the
Sun will be in the zenith of the place whose longitude is
151 33' east (nearly), and geocentric latitude 22 48'33" S., or
geographical latitude 22 56' 50" S.
Similarly, we find that at the time of the last external contact
the Sun will be in the zenith of the place whose longitude is
81 23' E. (nearly), and geographical latitude 22 58' S.
These points enable us to determine the places on the Earth's
surface best suited for observing the transit.
TO FIND THE MOST ELIGIBLE PLACES FOR OBSERVING A TRANSIT
OF VENUS.
ART. 15. The most eligible places for observation may be
determined with sufficient accuracy by means of a common terres
tial globe.
From the preceding calculations, it appears that the transit
will begin at 13h. 46,2m. Greenwich mean time, and continue
4h. 40.3m., and that the Sun's declination at the same time will
be 22 48' S.
Elevate the south pole 23 (nearly), and turn the globe until
places in longitude 151 33' E, are brought under the brass
19
meridian, then the sun will be visible at the time of the first con
tact, at all places above the horizon of the globe, and if the
globe be turned westward through 4.67 x 15 = 70, all places
in the second position, will see the Sun at the time of the last
contact. Those places which remain above the horizon while the
globe is turned through 70 of longitude, will see the whole of the
transit ; but in either position of the globe, the beginning and
end of the transit will not be seen from all places in the horizon,
but only from the points which lie in the great circle passing
through the centres of Venus and the Sun.
The place which will have the Sun in the zenith at the begin
ning of the transit, will have the first contact on the Sun's eastern
limb, and as the Sun will be near the horizon of this place when
the transit ends, the duration will be diminshed by parallax.
Since Venus is in north latitude, the planet will be depressed
by parallax, and consequently the duration of the transit will be
diminished at all places whose south latitude is greater than the
Sun's declination. For the same reason the duration will be
increased at all places north of the 22nd parallel of south latitude.
Therefore from those places from which the whole transit will
be visible, those which have the highest north or south latitude,
should be selected, in order that the observed difference of dura
tion may be the greatest possible.
The entire duration of this transit may be observed in eastern
Siberia, Central Asia, China, and Japan. Among the most
favorable southern stations, we have Australia, Tasmania, New
Zealand, Auckland Island, Kerguelan's Land, and several islands
in the South Pacific Ocean. For a comparison of the differences
of absolute times of ingress only, or of egress only, stations
differing widely both in latitude and longitude should be selected.
TO COMPUTE THE CIRCUMSTANCES OF THE TRANSIT SEEN FROM A
GIVEN PLACE ON THE EARTH'S SURFACE.
ART. 1C. Before proceeding to calculate the times of begin
ning and end of the transit for a given place, it will be necessary
to provide formulae for computing the parallax in longitude and
latitude, and in order to do this we must find :
20
1st. The reduction of geographical latitude due to the earth's
spheroidal figure.
2nd. The reduction of the earth's equatorial radius to a given
geocentric latitude, and
3rd. The altitude and (celestial) longitude of the Nonagesimal,
or in other words, the distance between the poles of the ecliptic
ancl horizon and the (celestial) longitude of the zenith of the
given place at a given time.
But as this transit will not be visible in America, it will not
excite that interest in this country which it otherwise would.
We shall therefore omit the further consideration of it, and apply
the following formulae to the computation, for Toronto and other
points in Canada, of the transit of December, 1882, which will
be visible in this country.
FIRST.  REDUCTION OF LATITUDE ON THE EARTH.
ART. 17. On account of the spheroidal figure of the Earth
the meridians are ellipses, and therefore the apparent or geogra
phical latitude does not coincide with the true or geocentric
latitude, except at the equator and the poles.
Let x and y be the coordinates of any point on the ellipse,
// 2
the origin being at the centre. The subnormal = _ .r, and if
u*
q> be the geographical latitude and the geocentric.
We have x tan <p = y
= jL x tan 0'
Or,
tan =  tan 0'
0.9933254 tan 0'
(19).
SECOND. REDUCTION OF THE EARTHS RADIUS.
ART. 18. Let r be the radius at a place whose geocentric
latitude is 0, x and y the coordinates of the place, then x = r
cos 0, y = r sin 0, and by the properties of the ellipse we have
b a II y * the common ordinate on the circle described on
the major axis = r sin & .
21
ft 2
Therefore, a 2 = y 1 f y2 sin 2 c
^2
Or, r a cos 2 H  r 2 sin 2 (f> = a 2 ,
From which r = a sec cos 0, if , tan = tan .
or regarding a as unity, tan 1.003353 tan r/
(log 1.003353 = 0.0014542),
and r = sec cos (20).
The horizontal parallax of Venus obtained from Eq. (9) or (10),
is the angle which the Earth's equatorial radius subtends at
Venus, and is not the same for all places, but varies with the
latitude.
The horizontal parallax for any place is found by multiplying
the Equatorial horizontal parallax by the Earth's radius at that
place, the equatorial radius being regarded as unity.
THIRD. TO FIND THE ALTITUDE AND LONGITUDE OF THE
NONAGESIMAL.
ART. 19. Let H Z R be a meridian, HR the horizon, Z the
zenith, P the pole of the equator V E, Q the pole of the ecliptic
V 0, F the equinox. Now since the arc joining the poles of two
great circles, measures their inclination, and when produced cuts
them 90 from their point of intersection, NO, V T, Vt, Q N,
each = 90. Let s be the Sun's place in the ecliptic, and
S his place when referred to the equator, then V C Sun's
A. R. + hour angle from noon = sidereal time
= A.
V N = longitude of the Nonagesimal N, = m .
Z Q = NTj the altitude of the Nonagesimal = a .
P Q = the obliquity = co .
PZ = colatitude = 90 0, (geocentric).
/ ZPQ = 180  ZPT
= 180  (VT  VC)
= 90+ A, and / ZQP=M= Vt~ VN=9Qm
In the triangle ZPQ, we have
cos ZQ = sin PZ sin PQ cos ZPQ + cos P^cos PQ .
22
Or, cos a = cos sin w sin A + sin ^> cos co .
Put sin A cot 9 = tan 0, '
Then cos a = sin <j> sec cos (ay + 0) . (21).
In the triangle P Z Q, we have
sinJ<2 : smZP :: siuZPQ : smZQP
Or, sin a ; cos ^ :: cos T! : cos w
Or, cos w = cos J. cos cosec a . (22).
And from the same triangle we get
cos Z P = sin Z Q sin P Q cos Z Q P + COB Z Q cos / .
Or, sin <^> = sin a sin w sin m + cos a cos cu .
From which
>cos cos o
sn m =
sn a sn a>
= sin ^ sin < cos 2 w + cos ^ sin a> cos &> sin
sn
sn a)
sn a sn a)
cos c cos w sin
sin a
Dividing this by Equation (22), we have
_ tan 6 sin
&> sn
cos
= tan </> sec J. sec sin (o> + 0) . (23),
Eq. (22), may now be used to find a,
sin a = cos A cos < sec m . (24).
TO FIND THE PARALLAX IN LONGITUDE.
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
colatitude 90 X, then Z Q = altitude of the nonagesimal
= a, the angle Z Q S = tie planet's geocentric longitude
the longitude of the nonagesimal = h, S Q S' = the parallax in
longitude = x, and SS' is the parallax in altitude.
From the nature of parallax we have sin SS' = sin P
sin ZS 1 and from the triangles S Q S', Z Q S', we have
sin a;
23
__ sin S S' sin S'
sin Q S
sin P sin Z S r sin
sin <5>
sin P sin J $ sin Z Q S
sn
sin P sin a sin (h 4 ^)
cos X
sin P sin a
A; sm (li 4 2'), if k = ;
cos X
and by a well known process in trigonometry,
k sin h k z sin 2/i h? sin 3fi
x = 4 . 4 . , 4 Ac (26).
sin 1 sm 2
TO FIND THE PARALLAX IN LATITUDE.
ART. 21. In the last ./%. let ' be the apparent colatitude
90 _ X', then from the triangles Q Z S and Q Z S', we have
z _ cos QS cosQZ cos^S' = cos QS'cos QZ cos Z^
sin QZ sin ^S Y sin 4>^ sin ZS'
sin X cos a cos J _ sin X' cos a cos Z8'
sin 2^S sin ZS'
but from the same triangles we have
cos ZS = sin a cos X cos 7t + cos a sin X
and cos ZS' = sin a cos X' cos (7^4^)+ cos a sin X'.
which, substituted in the above, give after reduction
sin ZS' tan a sin X' cos X' cos
sin Z 8 tan a sin X cos X cos h
But from the sine proportion, we have,
sin ZS' __ sin (7^4^) cos X r
sin 2TS sin h cos X
,. tan a, sin X' cos X' cos (h 4 x) sin (7i4aO cos X^
therefore  i =^ _  \ , J   ,
tan a sin X cos X cos h sin h cos X
tan a tan X' cos (h 4 a) _ sin (li  #)
tan a tan X cos A sin h
From which tan X' =
a tan X sin (A f a:) sin
sin h tan a
(27)
But
Therefore
sin x = sin /'sin a sec X sin
tan X'
Or
tan a tan X sin (Af a?) sin ./^ sin sec X
sn
'h
sin A
sin (A f .T)
sin A tan a
(tan X sin Y J cos a sec X).
sin .P cos N .
) tan X.
(28)
sin A sin X
This formula gives the apparent latitude in terms of the true
latitude and the true and apparent hour angles, but it is not in
a form for logarithmic computation. We will now transform it
into one which will furnish the parallax directly, and which will
be adapted to logarithms.
Let y X X', the parallax in latitude,
From Eq. (27) we have
sin x sin h
tan X =
sin
tan a sin (A + x)
tan X'
Or tan > tan X'
sin x
sin
0r sin ( X X ) _
V7 1 _ .
fsin (A 4 x) sin A\
^) /
2 sin \ cos (Af f) tan X'
tan X
tan a ^ sin
sin (Aja:)
cos X cos X' sin (A+z) tan a
But 2 sin r sin a; sec ^, and
z
sin ic = sin P sin a sec X sin (A + x) by Eq. (25)
Making these substitutions and reducing we have
sin y =sin P cos a (cos X 7 tan a cos (A + *) sec ^ sin
Put tan a cos (A + * ) sec  = cot 0,
Then sin y = sin P cos a cosec sin (0 X'),
sin P cos a eosec sin ( (0 X) f y) (
Put sin P cos a cosec = />*, then as before
7^ sin (0 X) /'' sin 2 (6 X) _; A 3 sin 3 (0 A)
'*! " * ' + ._ X "
sin T
sin 2
sin
(30)
(II. )
A TRANSIT OF VENUS,
DECEMBER 6TH, 1882.
ART. 22. The following heliocentric positions of Venus have
been computed from Hill's Tables of the Planet, and those of
the Earth from Delambre's Solar Tables, partially corrected
by myself, TT being taken = 8 /7 .95 at mean distance :
*
*
CO
^_
GO
t_
to
^3 ~
T^H
^1
o
GO
to
CO
pMd
e8"8
co
co
CO
01
01
01
01
^>
CO
fi
r
'^
co
O
05
OS
t* **
05
05
5
OS
05
r i
to
OS
^
GO
01
^
J a .
^
CO
o
01
to
l^
o
01
1
'O
01
to
CO
i!
o
OS
01
01
CO
CO
CO
o
^4^
W
t
1
l~
I
t
it
oo
o
OS
^
^H
01
*OB
co
oo
r ^
oo
5 "^
^o
to
^^
^fl
^H
CO
co
! J!
OS
OS
OS
OS
OS
OS
OS
11
OO
05
s
05
to
00
os"
to
00
05
to
oo
OS
OS
to
oo
o"
CQ
ll
co
co
01
o
OS
s
to
_5
i: "
S'S
CO
05
^^
f >
r 4
l^
13
co
r 1
to
co
01
1
^
^
^
co
co
co
co
o
00
t ^
to
to
01
GO
co
^g
co
CO
01
to
If
^
to
r 1
CO
7 H
i !
CO
ll
S3
a
!
05
01
.CO
co
+ 1
co
r 1
"*
to

*>
>
fc
t>
t
I
1
Jaj
^
01
01
CO
01
5
01
CO
bc
2
CO
f
8
8
5
^
ft
ft
ART. 23. Passing to the true geocentric places by the aid of
Formulae (l)(lo), and then applying the correction for aberration
(which, by Formulae (14) and (15), is found to be, in longitude,
+ 3".3; in latitude + I" A ; Sun's aberration  20".7), we
obtain the following apparent geocentric places :
Washington Mean
Time.
Sun's Apparent
Geocentric Longitude.
Venus's Apparent
Geocentric Longitude.
Venus's
Appar. Geoc.
Latitude.
Dec. 5d. 21h.
254 24' 27". 4
25434 / .58 // .3
1 2' 28" S.
" 22h.
26 59 .8 33 26 .7
11 49
" 23h.
29 32 .2
31 55 .2
11 10
24h.
32 04 .7
30 23 .6
10 30 .8
Dec. 6d. Ih.
34 37 .1
28 52 .0
9 51 .6
" 2h.
37 09 .5
27 20 .3
9 12 .5
3h.
39 42 .0
25 48 .6
8 33 .4
Log of Venus's distance from the Earth at noon = 9.421550 .
Formulae (9) and (12) give us P = 33".9, and d = 3l".46, both
of which may be regarded 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 V 31". 6 W.
Sun's do. do. 2' 32".4 E.
Venus's hourly motion in latitude 39". 1 N.
Sun's semidiameter 16' 16". 2
Venus's do 31".5
Suns Equatorial horizontal parallax 9". 1
Venus's do. do. 33". 9
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 employing the
same notation as in Art. 13, we obtain from these elements the
following results :
n 9 6' 14", 4 ; relative hourly motion in orbit, 247"! ;
least distance between centres, 10' 39";
27
First external contact, Dec. 5d, 20h. 50.7m. \
First internal do., " 21h. llm, (Washington
Last internal do., Dec. Gd* 2h. 48m. f Mean Time.
Last external do., 3h. 8m.
As seen from the Earth's centre.
By the formulae of Art. 14, we find, that at the time of
the first external contact, the Sun will be in the zenith of the
place whose longitude is 45. 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 1874, that the entire duration of
this transit will be observed in the greater part 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 timef
of first contact will be retarded at places along the Atlantic
coast of Canada and the United States, while the Islands in the
western part 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 retarded 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 Islands. 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 observations, at all favorable points, of the times
of ingress and egress of the planet on the Sun's disk, in order
that we may determine with accuracy this great astronomical
unit, the Sun's distance from the Earth, and thence the dimen
sions of the Solar System.
TO COMPUTE THE TRANSIT FOR A GIVEN PLACE ON THE EARTH'S
SURFACE.
ART. 24. Let it be required to find the times of contact for
Toronto, Ontario, which is in latitude 43 39' 4" N., and longi
tude 5h. 17m. 33sec. west of Greenwich, or 9m. 22sec. west of
Washington .
Since the parallax of Venus is small, the times of ingress and
egress, as seen from Toronto, will not differ much from those
found for the Earth's centre. Subtracting the difference of lon
gitude between Toronto and Washington, from the Washington
Mean Time of the first and last 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 the last external
contact to be December, 6d. 2h. 58.6m , when viewed from the
centre of the earth.
The ingress will therefore occur on the east, and the egress
on the west side of the meridian, and the time of ingress
will consequently be retarded, and the time of egress accel
erated by parallax. We therefore assume for the first external
contact, December 5d. 20h. 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. 22sec. Wash
ington Mean Time.
From the elements given in Art. 23, compute for these dates
the longitudes of Venus and the Sun, Venus's latitude, and the
Sidereal Time in arc, at Toronto, thus :
Washington Mean
'lime.
Sun's Apparent
Longitude.
Venus's Appar.
Longitude.
Venus's
Latitude.
Sidereal Time
at Toronto.
Dec. od. 20h.63m.22s.
" Od. 3h. 3m.22s.
254 24' 10".5
254 39 50 .5
254 35' 8".5
254 26 43 .5
12 / 32".4S
8 31 .3
206 15' 06"
299 17
The relative positions of Venus and the Sim 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 Formulae (19) to (30), the parallax of Venus
in longitude and latitude, and apply it with its proper sign to the
apparent longitude and latitude of Yenus, as seen from the
Earth's centre ; the results will give the planet's apparent posi
tion with respect 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
semidiameters.
We now proceed with the computation :
By Eq. (19), tan 0' = 9.979544
const, log = 9.997091
tan < = 9.976635 , therefore = 43 27'34"
const, log = 0.001454
tan = 9.978089 , therefore = 43 33'19"
By Eq. (20), cos = 9.860164
sec < =10.139146
logr= 9.999310
Diff. of Parallaxes, 24".8 = 1.394452
Eeduced Parallax, 24".76 = 1.393762
ALTITUDE AND LONGITUDE OF THE NONAGESIMAL, AT THE
FIRST ASSUMED TIME.
By Eq. (21),
sin A = 9.645731?* sin = 9.837488
cot = 10.023366 sec = 10.042801n
tan = 9.669097ft. cos (w + 0) = 9.999837n
= 154 58' 42" cos a = 9.880126
to = 23 27' 09" a = 40 38' 30"
. w + = 178 25' 51"
By Eq. (23), Check by Eq. (22),
tan < = 9.976634 cos A = 9.952725/t
sec A = 10.047275 cos = 9.860854
sec 9 = 10.042801n cosec a = 10.186201
sin (w + 0) = 8.437493 cos m = 9.999780w
tan m 8.504203 m = 18149 / 44 //
m = 181 49' 44"
PARALLAX IN LONGITUDE.
Longitude of Venus = 254 35' $".5
Long.oftheNonagesimal = 181 49' 44"
Therefore, h = 72 45' 24".5. Then by Eq. (26).
sin P = 6.079337
sin a = 9.813799
sec A = 10.000003
k = 5.893139 tf = 1.7863 k 3 = 7.679
sin li = 9.980029 sin 2h = 9.7529 sin 3h = 9.792/1
cosec ]" = 5.314425 cosec 2" = 5.0134 cosec 3" = 4.837n
15 // .402 = 1.187593, ".0003 = 4.5526 = 8.308rc
The last two terms being extremely small may be omitted,
therefore the parallax in longitude = f 15" A = jc.
PARALLAX IN LATITUDE.
ByEqs. (29) and (30).
tan a = 9.933672 sin P ^ 6.079337
cos (h + *) = 9.471860 cos a = 9.880126
sec \ = 10.000000 cosec = 10.013619
cot = 9.405532 k = 5.973082
= 75 43' 34 x/ .5 sin (0 + \) = 9.986782
\ = 12' 32 // .4 S. cosec Y 5.314425
+ X == 75 56 7 6^9 . 18".80S = 1.274289
7^ 2 = 1.9461 // = 7.919
sin 2 (0 4 X) = 9.6734 sin 3 (0 + X) = 9.869?i
cosec 2 r/ = 5.0134 cosec 3 /x = 4.837
// 0004 = 4.6329 = 8.625
Therefore the parallax in latitude =. ^ 18^.8 = y.
In the same way, we find at the second assumed time,
a = 27 37'; m = 317 23' 46 /x ; h = 62 58' 2". 5
x =  10^.3; y = + 20 /x .8.
31
Hence we have the following results :
DEC. 5n., 20 H
53.M. 22sEC.
DEC. 6D., SH.
3>i. 22SEC.
LONGITUDK.
LATITUDE.
LONGITUDE.
LATITUDE.
Venus's
Parallax.
L'5435' 8". 5
+ 15". 4
12' 32". 4 S.
+ 18". 8
254 25' 43". 5
10". 3
i
8' 31". 3 S.
f20".8
Sun's
254 35' 23".
254 24' 10". 5
12' 61 ".2
254 25' 33". 2
254 39 50". 5
8' 52". 1
Difference.
11' 13".4
Venus East.
\
14' I7".3
Venus West.
Construct a figure similar to Fig. 4, make CB = 11' 13". 4, and
C N = 14' 17". 3 the differences of longitude; draw B H arid
.V P below A B, because Venus is in south latitude, and make
B 11= 12'51".2,and JV/ > =8'52",1 the differences in latitude ;
then IIP will represent Venus's apparent orbit. Join II C,
P C, and let V and X be the positions of the planet at the times
of the first and last contacts respectively. The times of
describing E V and P A^are required to be found.
Proceeding in the same manner as in Art. 1 3, we find by plane
Trigonometry, HP = B N sec of the inclination of apparent
orbit = B N % sec B N Q (N Q being parallel to // P)
< tan BN Q = BH * NP BNQ = S 52' 41" = E C F.
A C + B C '
HP = 1552".8 = relative motion of Venus in 6h. 10m.,
therefore Venus's relative hourly motion = 251". 8
tan B C 11 = , B OH = 48 52' 23"
B C
H c = B C sec B C H = 1023*. 8
##=41 7' 37", hence HCF= 50 0' 18"
OF = EC cos HCF = 658"; EF = EC sin E OF = 784".35
C V, the sum of the semidiameters = 1007". 7
oFC >**
V 6^^=
,
VF= CV sin F6^ J P=763".19
HV = HF VF=21".1Q.
13' 54"
32
Time of describing H V = 5m. 2sec., and time of describing
VF= 3h. 1m. 51sec.
Therefore the first external contact will occur, Dec. 5d. 20h.
49m. 2sec., and the last external contact, Dec. 6d. 2h. 52m. 44sec.,
Mean Time at Toronto.
In a similar manner we obtain v F = 677". 83 ; therefore,
Vv = 85". 36 and the time of describing Vv = 20m. 20sec.
Therefore the first internal contact will occur, Dec. 5d. 21h.
9m. 22sec., and the last, Dec. Gd. 2h. 32m. 24 sec., Mean Time
at Toronto ; or expressing these in Mean Civil Time, we have
for Toronto :
First external contact, December 6th, 8 h. 49 m., A.M.
First internal " " 9 h. 93 m., "
Last internal ' " 2 h. 32'4 m., P.M.
Last external 2 h. 527 m., "
Least distance between the centres 10'58".
If the highest degree of accuracy attainable be required, we
must repeat the computation for the times just obtained. For
ordinary purposes, however, the above times will be found suffi
ciently accurate.
In observing transits and solar eclipses, it is necessary to know
the exact point on the Sun's disk, at which the apparent contact
will take place. The angle contained by a radius drawn from
the point of contact and a declination circle passing through the
Sun's centre, is called the angle of position, and is computed as
follows : Let LSX be a right angled spherical triangle, X the
equinox, S the Sun's centre, LS a circle of latitude, perpendicular,
of course, to SX, SD a declination circle ; then DSXis a right
angled spherical triangle, and in the present case, SD will lie
to the west of SL, because the Sun's longitude lies between
180 and 270, i.c , between the autumnal equinox and the
solstitial colure.
Then we have
cos XS = cot SXD tan DSL.
Or tan DSL cos long tan w .
The Sun's longitude at 8 h. 49 m., A.M., is 254 24' 23".2.
33
Rejecting 180 we have cos 74  24'  23" = 9.429449
tan w = 9.637317
tail DSL = 9.066766 ,
DSL = 6  39'  6"
Now the angle VCE = angle VCF  angle EOF
= 40 21' 12"
Therefore the angle of position is equal to the angle DSL \ the
supplement of VCE. or 146 17'. 9 from the northern limb
towards the east.
In the same way we may compute the angle of position at the
last external contact.
From a point in longitude 71 55' W. of Greenwkich, and
latitude 45 21' . 7 N., at or near Bishop's College, Lennoxville,
we find by the preceding method,
First external contact December 6th, 9 h. 19.5 m. ; A.M.
First internal " " 9 h. 39.4 m., "
Last internal " " 3 h. 2.6m., P.M.
Last external ' " 3 h. 23 m. "
Mean Time at Lennoxville.
Least distance between the centres 10'  59". 8.
From a point 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 6th, 9 h 48.7 m., A.M.
First internal '< ' 9 h 28.4 m., "
Last 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.
ART. 25. A transit of Venus 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.
5
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 the Sun's parallax
from an observed transit of Venus or Mercury, is too refined
and delicate for insertion in an elementary work like this.
We add, however, the following method which is substantially
the same as found in most works on Spherical Astronomy,
and, which will enable the student to understand some of the
general principles on which the computation depends.
TO FIND THE SUN's PARALLAX AND DISTANCE FROM THE EARTH,
FROM THE DIFFERENCE OF THE TIMES OF DURATION OF A
TRANSIT OF VENUS, OBSERVED AT DIFFERENT PLACES.
ART. 26. Let T and T' be the Greenwich mean times of the
first and last contacts, as seen from the Earth's centre; T+t and
T f t' 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 Yen us
at the time T ; P the horizontal parallax 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 <r/ Venus's hourly motion in latitude. Now, since
Venus and the Sun are nearly coincident in position, the effect of
parallax will be the same 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 rela
tive positions of the two bodies.
Than a (P TT), and b (P 7r) will be the parallax of Venus
in 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 apparent differ
ence of longitude at the time T will be
G S + a (P TT); and therefore the apparent differ
ence of longitude at the time T \ t
= G S+a(P ir)+vt,
and the apparent latitude of Venus at the time T+t.
= L + b(P7r} 4 gt.
35
Now at the time T\t the distance between the centres of
Venus and the Sun, is equal to the sum of their semidiameters,
= c, then we have
neglecting the squares and products of the very small quan
tities t, a, b and (P TT).
But when seen from the centre of the Earth at the time T,
we have
c 1  (G S) 2 + .Z/ 2 , which substituted in the last equation,
gives
. (P TT) (33).
v (G S) 4 y L
= B (P TT), suppose
Therefore the Greenwich time of the first contact at the place
of observation = T + B (P TT).
If B' be the corresponding quantity to for the time T', then
the time of the last contact at the place of observation
= T' + %' (P TT)',
and if A be the whole duration of the transit then
A = T' T+ ($ g) (P TT)
Again, if A' be the duration observed at any other place, and
/3 and ^ corresponding values of B and ', we have
A ' = V T + (ft  3) (P  TT) ;
Therefore A'. A   (ff  /3)  (B f  B) } (P  TT)
p '
P Earth's distance from the Sun
Therefore
TT Earth's distance from Venus
^ ~ "* Venus's distance from the Sun
TT Venus's distance from the Earth
= ?>, a known quantity
TT = (P TT). (35). (Hymers's Astron.)
n
36
If the first or last contact only 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 much as possible by parallax.
Again, since t is known in Eq. (33), being the difference of the
Greenwich mean times of beginning or end, as seen from the
Earth's centre and the place of observation, we have from Eq.
(32) by eliminating r,
?>(L + (/t) (p }
a' 4 6"
 * 2 fo* + 0') + 2 ' ( v (G  S} + Lff)
a" + 6*
Or, (P 7r) 2 4 ^ (P TT) = .#, suppose. (36).
And let (F TT)' + 6" (P TT) = Z>, be a similar equation
derived from observation of the first or last contact at another
place, then
Or, P TT = B ~~ D i , (37).
And TT = (/' TT), as before,
n
THE SUN'S DISTANCE FROM THE EARTH.
ART. 27. If D' represent the Sun's distance, and r the Earth's
equatorial radius, then
f
sin TT
2062648
= r . (38).
7T
From the observations made during the Transit of 1769, the
Sun's equatorial horizontal parallax (TT) at mean distance, was
determined to be 8". 37 which, substituted in the last equation,
gives for the Sun's mean distance 24068. 23>, or in round num
bers 95,382,000 miles ; but recent investigations in both physical
and practical astronomy, have proved beyond all doubt that this
value is too great by about four millions of miles,
37
In 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 comparison of the observed duration of the
transit at places favorably situated for shortening and lengthening
it by the effect of parallax. This method is independent of the
longitudes of the stations, but it cannot be always applied with
advantage in every transit, and fails entirely when any atmos
pherical circumstances interfere with the observations either at
the first or last contact. The other consists in a comparison of
the absolute times of the first external or internal contact only,
or of the last external or internal contact only, at places widely
differing in latitude. The longitudes 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 observed at
several places in Europe, Asia, and Africa, but the results
obtained from a full discussion of the observations by different
computers, were unsatisfactory, and exhibited differences which
it was impossible to reconcile. That transit was not there
fore of much service in the solution of what has been justly
termed "the noblest problem in astronomy." The most probable
value of the parallax 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 unsatisfactory results obtained from the transit of 1761,
gave rise to greater efforts for observing the one of 1769, and
observers were 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 northeast extremity of Norway. The first external and
internal contacts were observed at most of the European obser
vatories, and the last 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,
kc. But on account of a cloudy atmosphere at all the
northern stations, except Wardhus, the entire duration of the
38
transit could not be observed, and it consequently happened that
the observations taken at Wardhus exercised a great influence on
the final result. This, however, would have been a matter of
very little importance, if the observations taken there by the
observer, Father Hell, had been reliable, but they exhibited such
differences from those of other observers, as to lead some to
regard them as forgeries. A careful examination of all the
available observations of this transit, gave 8*. 57 for the solar
parallax, and consequently 95,382,000 miles for the Sun's mean
distance.
The first serious doubts as to the accuracy of this value of the
Solar parallax, began to be entertained in the year 1854, when
Professor Hansen found from an investigation of the lunar orbit,
and especially of that irregularity called the parallactic equation
which depends on the Earth's distance from the Sun, that the
Moon's place as deduced from the Greenwich observations, did
not agree with that computed with the received value of the
Sun's distance, which he found to require a considerable diminu
tion. The same conclusion was confirmed by an examination of
a long series of lunar observations taken at Dorpat, in Russia.
The value of the solar parallax thus indicated by theory and
observation, is 8". 97 which is about fourtenths of a second
greater than that obtained from observations of the transit of
Venus in 1769 ; and if this value of the parallax be substi
tuted in Eq. (38), it will be found to give a diminution of more
than 4,000,000 miles in the Earth's mean distance from the Sun.
A few years ago M . LeVerrier, of Paris, found, after a most
laborious and rigorous investigation of the observations on the
Moon, Sun, Venus, and Mars, taken at Greenwich, Paris, and
other observatories, that an augmentation of the Solar parallax or
a dimination of the hitherto received distance of the Earth from
the Sun, to an amount almost equal to that previously assigned
by Professor Hansen, was absolutely necessary to account satis
factorily for the lunar equation which required an increase of a
twelfth part, and for the excessive motions of Venus's nodes, and
the perihelion of Mars. He adopted 8". 95 for the Solar parallax.
The most recent determination of the velocity of light com
bined with the time which it requires to travel from the Sun to
39
the Earth, viz.: 8 minutes arid 18 seconds very nearly, affords
another independent proof that the commonly received distance
is too great by about ^th part. The value of the Solar parallax
indicated by this method is 8". 86.
The great eccentricity of the orbit of Mars causes a considera
ble variation in the distance of this planet from the Earth at the
time of opposition. Sometimes its distance from the Earth is
only a little more than onethird of the Earth's distance from the
Sun. Now, if Mars when thus favorably situated, be observed on
the meridians of places widely differing in latitude such as
Dorpat and the Cape of Good Hope and if the observations be
reduced to the same instant by means of the known velocity of
the planet, we shall, after correcting for refraction and instru
mental errors, possess data for determining with a high degree
of accuracy, the planet's distance from the Earth, and thence the
Sun's distance and parallax. The oppositions of 1860 and 1862,
were very favorable for such observations, and attempts were
made at Greenwich, Poulkova, Berlin, the Cape of Good Hope,
Williainstown, and Victoria, to determine the Solar parallax at
those times. The mean result obtained from these observations,
was 8". 95 which agrees exactly with the theoretical value of the
parallax previously obtained by M. LeVerrier.
Hence, we find that a diminution in the Sun's distance, as
commonly received, is indicated, 1st, By the investigation of the
parallactic equation in the lunar theory by Professor Hansen and
the Astronomer Royal, Professor Airy ; 2nd, By the lunar equa
tion in the theory of the Earth's motions, investigated by M.
LeYerrier ; 3rd, By the excessive motions of Yenus's nodes,
and of the perihelion of Mars, also investigated by the 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 oppositions of
1860 and 1862.
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 onetenth part of the whole.
Substituting LeYerrier's solar parallax (8".95) in Eq. (38),
40
the Earth's mean distance from the Sun becomes 91,333,670
which is a redaction of 4,048,800 miles. The Sun's apparent
diameter at the Earth's mean distance = 32' 3". 64, and in order
that a body may subtend this angle, at a distance of 91,333,670
miles, it must have a diameter of 851,700 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
from 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^ ' : t$ :: 1 : D
Or, D = , (39).
In the case of Neptune the mean distance is diminished by
about 121,000,000 miles. Jupiter's mean distance is diminished
21,063,000 miles, and his diameter becomes 88,296 miles, which
is a decrease of 3,868 miles. These numbers shew the great
importance which belongs to a correct knowledge of the Solar
parallax,
41
(III.)
A TRANSIT OF MERCURY.
MAY GTH, 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 and 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 Su^ in 87.9693 days, and the
Earth in 365.256 days. The converging fractions approximating
87.9693 7 13 33 
' 365.256 3 29' 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 at the 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
Ascending 110 de November 7th, 1881, and November 10th,
1894.
COMPUTATION OF THE TRANSIT OF 1878.
From the tables* of the planet we obtain the following helio
centric positions :
* Tables of Mercury, by Joseph Winlock, Prof. Mathematics U. S.
Navy, Washington, 1864.
42
Washington Mean
Time.
Mercury's Helioc.
Longitude.
Mercury's Helioc.
Latitude.
Log. Rad.
Vector.
1878, May 6d. Oh.
Ih.
2h.
3h.
225 52' 57".0
226 1,5 .4
226 7 33 .6
226 14 51 .6
1' 17*. 3 N.
6 23 .4
5 29 .6
4 35 .8
9,6545239
9,6546389
9,6547535
9,6548677
The following positions of the Earth have been obtained from
Delambre's Solar Tables, corrected by myself, TT being taken
equal to 8". 95 at the Earth's mean distance :
Washington Mean
Time.
Earth's Helioc.
Longitude.
Log. Earth's Rad.
Vector.
1878, May 6d. Oh.
Ih.
" 2h.
" 3h.
226 0' 38".9
226 3 04 .0
226 5 29 .1
226 7 54 .2
10,0040993
10,0041038
10,0041082
10,0041126
The Sun's true longitude is found by subtracting 180 from
the Earth's longitude.
Passing to the true geocentric places by Formulae (3), (4), and
(5), we obtain :
Washington Mean
Tfme.
Mercury's true Geoc.
Longitude.
Mercury's true Geoc.
Latitude.
1878, May 6d. Oh.
Ih.
2h.
3h.
46 6' 52".4
46 5 20 .4
46 3 48 .3
46 2 16 .3
5' 53". 6 N.
5 10 .2
4 26 .8
3 43 .4
Formula (7) gives log. distance from Earth at Ih. = 9.7466455.
This will be required in formulae (14) and (15) for finding the
aberration.
Formula (9) gives P = 15''. 9.
The semidiameter of Mercury at the Earth's mean distance,
3".34 = d' 9 therefore by Eq. (12), d == 5". 98.
Aberration in Longitude = + 6". 6 7, by Eq. (14).
Aberration in Latitude = + 3". 34, by Eq. (15).
The Sun's semidiameter = 15' 52".3. (Solar Tables),
The Sun's aberration ,= 30". 25.
43
Correcting for aberration we obtain the apparent places as
follows :
Washington Mean
Time.
Mercury's Appar.
Geoc. Longitude.
Morcury's
App. Oeoc. Lat.
Sun's
Appar. Longitude.
1878, May, Cd. Oh.
46 6' 59/0
5' 56 ."9N.
46 0' 18".7
" Ih.
4G 5 27.0
5 13.5
46 2 43.8
, 2
46 3 54.9
4 30.1
46 5 8.9
" 3
4G 2 '26.9
3 46.7
46 7 34.0
Interpolating for the time
elements, we have
>f conjunction and collecting the
Washington mean time of conjunction in longitude,
May 6d. Ih. 41 min. 17 sec.
Mercury's and Sun's longitude 46 4' 23".6
Mercury's latitude
Sun's hourly motion in longitude
Mercury's hourly motion in longitude
Mercury's hourly motion in latitude
Sun's equatorial horizontal parallax
Mercury's equatorial horizontal parallax ...
Sun's semidiameter
Mercury's semi diameter
4' 43". 6 N.
2' 25".l E.
1' 32". 1 W.
43 // .4 S.
8".87
15".9
15' 52". 3
5".9
Employing the same notation as in Art. 13, the preceding
elements give the following results. Relative hourly motion in
longitude =. 3' 57". 2; n = 10 22' 7"; m n = 24TM3 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 =. 12m. 42 sec. Since Mercury is north of the Sun's
centre at conjunction, and moving southward, E F will lie on the
right of E (see Fiy. 4), and the middle of the transit will
take place at Ih. 54m. P.M.
Sum of semidiameters = 958".2
V = 16 55' 44" ; V F = 916".68 ;
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
the time of the middle of the transit, we obtain the times of the
first and last contacts, as seen from the Earth's centre, thus :
First external contact May 6d. lOh. 5.9 min. A,M.
Last external contact 5k. 42.1 min. P.M.
Mean time at Washington,
The places which will have 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' 25".
Sidereal time at Washington at mean noon of May 6th (in arc)
44 24' 50".46.
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 difference of longitude, as below :
The longitude of Washington is 5h. 8m. 11 sec. W.
The longitude of Toronto is 5h. 17m. 33 sec. W.
Therefore Toronto is 9 min. 22 sec. west of Washington.
Then, with reference to the centre of the Earth, we have for
Toronto,
First external contact May Gd. 9k. 56.5m. A.M.
Last external contact " 5h. 32.7m. P.M.
Mean time.
For Quebec,, longitude 4k. 44m. 48 sec. W,
First external contact May 6d. lOh. 29.3m. A.M.
Last external contact " 6h. 6.5m. P.M.
Mean time.
For Acadia College, longitude 4k. 17.6m. W.
First external contact May 6d. 10k. 56.5m. A.M.
Last external contact " 6k. 32.7m. P.M.
Mean time.
For Middlebury College, Vermont, longitude 4k. 52.5m. W.
First external contact, May 6k. 10k. 21.5m. A.M.
Last external contact " 5k. 57.7m. P.M.
Mean time at Middlebury.
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 semidiameter of the Sun arid
Moon, while Formulae (19)(30) furnish the parallax in longitude
and latitude. If the computation be made from an* cphemeris
which gives the right ascension and declination of the Sun and
Moon instead of their longitude and latitude, we can dispense
with formulae (21) and (23), and adapt (25), (26), (29), and (30) to
the computation of the parallax in right ascension and declination.
In Fig. 6, let Q be the pole of the equator, then L Q is the
colatitude = 90  $ ; Z Q S = h, the Moon's true hour angle
= the Moon's A. R. the sidereal time ; S Q S'is the parallax
in A. R. = cr, and Q S' Q S is the parallax in declination y.
Put Q S, the Moon's true north polar distance = 90 , then
Formulae (25) and (26) become,
sin x = sin Pcos ^ sec S sin (k f x) (25, bis).
= k sin (Ji f a:)
k sin h , k* sin 2h . k 3 sin 3h . f /nf > , . .
Or, x = , _ 4 _ f _ __ + &c. (26, bis).
sin 1 sm 2 sm 3
Again, the formulae for determining the auxiliary angle in
(29) becomes,
cot = cot cos (Ji + ) sec .
And (29) becomes,
sin y = sin Psin cosec 9 sin ( (0 S) + y) . (29, bis).
_ k sin (08) W sin 2 (0 g) 1? sin 3(0 g)
sin 1" sin 2" sin 3"
(30, bis).
46
These parallaxes when applied with their proper signs to the
right ascensions and declinations of the Moon for the assumed
times, furnish the apparent right ascensions and declinations.
The difference between the apparent A. R. of the Moon and the
true A, R. of the Sun, must be reduced to seconds of arc of a
great circle, by multiplying it by the cosine of the Moon's appa
rent declination. The apparent places of the Moon 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 as described
in Art. 13. The only other correction necessary to take into
account, is that for the augmentation of the Moon's semi
diameter, due to its altitude. The augmentation may be taken
from a table prepared for that purpose, *w*4 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 THE AUGMENTATION OF THE MOON'S SEMIDIAMETER.
Let C and M be the centres of the Earth and Moon, A a point
on the Earth's surface, join CM, A M, and produce C A to Z ;
then M C Z is the Moon's true zenith distance = Z arc Z S in
Fig. G ; and MA Z is the apparent zenith distance = Z' arc
Z S' in the same figure. Represent the Moon's semidiameter as
seen from (7, by d ; the semidiameter as seen from A by d' ; the
apparent hour angle Z Q S' by h', and the apparent declination
by g', then
' = C M = smZ'
d AM sin Z
sin Z S'
sin Z S
sin h' cos
sin h cos
(See Fig. 6.) (40).
I , by Art, 21.
mi~ ? ?/ j sin h cos o / A i \
Therefore, d! = d. S. , (41).
sin h cos c
This formula furnishes the augmented semidiameter at once.
It can be easily modified so as tetgive the augmentation directly,
but with logarithms ,to seven decimal places, it gives the apparent
semidiameter with great precision.
47
As examples we give the following, the first of which is from
Loomis's Practical Astronomy :
Ex. 1. Find the Moon's parallax in A. R. and declination, and
the augmented semidiameter for Philadelphia, Lat. 39 57' 7" N.
when the horizontal parallax of the place is 59' 36". 8, the Moon's
declination 24 5' 11".6 N., the Moon's true hour angle 61 10'
47".4, and the semidiameter 16' 16".
rin*. Parallax in A. R., 44' 17".09
Dec., 26' 10".l
Augmented semidiam = 16' 26". 15.
Ex. 2. Required the times of beginning and end of the Solar
Eclipse of October 91 O v 1874, for Edinburgh, Lat. 55 57' 23" K
Long. 12m. 43 sec. West, from the following elements obtained
from the English Nautical Almanac :
Greenwich mean time of conjunction in A It,
Oct. 9d. 22h. 10m. 11. 4 sec.
Sun's and Moon's A R 195 36' 30"
Moon's decimation S 5 39 8.9
Sun's declination S 6 39 34.1
Moon's hourly motion in A R 26 21.9
Sun's do 2 18.2
Moon's hourly motion in Declination. S 13 48.3
Sun's do S 56.9
Moon's Equatorial Horizontal Parallax. 53 59.6
Sun's do do 9.0
Moon's true semidiameter 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 formulae the following results, which may be
verified by the Student :
Geocentric latitude = 55 4 6' 41"; reduced or relative
Parallax = 53' 43".2.
20h. 55m. G. M. T.
23h. 10m. G. M T.
M^oon's A R ... . .
195 3' 27" 6
196 2' 46"9
Sun's A R. ...
195 33 36 8
195 38 478
Moon's Dec.
5 21 50 9 S.
5 52 54.6 S. !
Sun's Dec
6 38 22.9 S.
6 40 30.9 S.
Sid. Time at Edin. (in arc).
Moon's true hour angle...
Moon's Parallax in A.R...
Moon's do in Dec...
Moon's apparent A. R
149 21 51.5
45 41 36.1 E.
+ 21 49.4
+ 46 25.1
195 25 17.0
183 12 24.1
12 50 22.8 E.
+ 6 48.5
+ 47 32.7
196 9 354
6 8 16 S
6 40 27 3 S
Diff. of A R in seconds of
arc of great circle .
4 9 6" 9, Moon W.
1835" 1 Moon E.
Diff. Dec
30' 6". 9, Moon N.
3". 6 Moon N.
Aug. semidiam of Moon...
888".4.
890' / .5.
Eclipse begins October lOd. 8h. 43m. 32 sec. A.M.
Eclipse ends " lOh. 58m, 22 sec. A.M.
Mean time, at Edinburgh. Magnitude .369 Sun's diam.
THE END.
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