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Full text of "The computation of the transits of Venus for the years 1874 and 1882, and of Mercury for the year 1878, for the earth generally and for several places in Canada, with a popular discussion of the sun's distance from the earth, and an appendix shewing the method of computing solar eclipses"

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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 seventy-three, 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 semi-diameter 
and the apparent semi-diameter 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 

224-7 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 semi-diameters 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 


9.857530-1 


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 Semi-diameter =31 / '.4=<:/. (See Art. 7.) 

Sun's Semi-diameter = 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/ : R-rcos I :: tan \ (p + E) : tan (p E) 



But 



p+E = 180-6 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 SEMI-DIAMETER OF THE PLANET. 

ART. 7. The semi-diameter 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 semi-diameter, 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, semi-diameter, 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 (l-tan0 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 SEMI-DIAMETER. 

Venus's semi-diameter 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 right-angled 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 semi-diameter 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 semi-diameter 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 senii-cUameter 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- semi-diam. of Venus c; Cv = C A serni-diam. 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 semi-diameters = 1007". 6, 
and at the first or last internal contact, b difference of their 
semi-diameters = 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 co-ordinates 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 co-ordinates 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 = co-latitude = 90 0, (geocentric). 
/ ZPQ = 180 - ZPT 

= 180 - (VT - VC) 

= 90+ A, and / ZQP=M= Vt~ VN=9Q-m 
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 
co-latitude 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 co-latitude 
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 (7i4-aO 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 (A-f 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 (A-f f-) tan X' 



tan X 

tan a ^ sin 



sin (A-j-a:) 



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 6-TH, 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 

S-3 

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 semi-diameter 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 
semi-diameters. 

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. 
-f-20".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 semi-diameters = 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. 9-3 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 semi-diameters, 
= 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 

206264-8 

= 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 north-east 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 four-tenths 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 one-third 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 one-tenth 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 semi-diameter 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 semi-diameter = 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 semi-diameter 

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 semi-diameters = 958".2 

V = 16 55' 44" ; V F = 916".68 ; 

Time of describing V F = 3h. 4-8.1 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 semi-diameter 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 
co-latitude = 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 (0-8) 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 SEMI-DIAMETER. 

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 semi-diameter as 
seen from (7, by d ; the semi-diameter 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 semi-diameter at once. 

It can be easily modified so as tetgive the augmentation directly, 

but with logarithms ,to seven decimal places, it gives the apparent 
semi-diameter 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 semi-diameter 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 semi-diameter 16' 16". 

-rin*. Parallax in A. R., 44' 17".09 

Dec., 26' 10".l 
Augmented semi-diam = 16' 26". 15. 

Ex. 2. Required the times of beginning and end of the Solar 
Eclipse of October 9-1 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 semi-diameter 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. semi-diam 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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