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University of Michigan - BUHR
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i
1
Fore
with the Author's compliments.
N.
ASTRONOMICAL
TABLES
AND FORMULÆ.
ASTRONOMICAL
TABLES AND FORMULÆ
TOGETHER WITH A VARIETY OF
1
PROBLEMS
EXPLANATORY OF THEIR USE AND APPLICATION.
TO WHICH ARE PREFIXED
THE
ELEMENTS OF THE SOLAR SYSTEM.
.
By FRANCIS BAILY, Esa.
F.R.S. L.S. and G.S. M.R.I.A.
AND PRESIDENT OF THE ASTRONOMICAL SOCIETY OF LONDON.
LONDON
1827.
PRINTED BY RICHARD TAYLOR,
SHOE-LANE, LONDON,
P R E F A C E.
Selnany
Efrem.
1
5 - az 8 •
anaa
THE
present collection of TABLES and FORMULÆ
is a part only of a more numerous collection which
I had formed and originally designed for my own
use, with a view to save time and trouble in the
various astronomical computations and researches
in which I have occasionally indulged; but with-
out any view to their publication.
I found that many valuable Tables, of almost
daily use in an active observatory, and which have
been published from time to time by various au-
thors, were to be met with only in works printed
on the Continent, of considerable expense and not
easily accessible ;' and that others were so scattered
about in different publications, as to make it very
troublesome to refer to them at the moment they
were wanted. The whole of the Tables which
I have here selected from such works, have been
either re-computed, or carefully examined by means
of differences: the rest are entirely new.
vi
Preface.
In the selection of the Formula, I have been
guided more by their fitness for computation, than
by their elegance: and have always adopted those
which, after repeated trials, I have found to lead to
the practical solution of the problem with the least
expense of time and labour.
Most of these For-
mula will be found in the works of other writers :
but, it is too well known that in referring to different
authors for a Formula, the reader is frequently
distracted with a confusion of symbols, the values
of which can only be obtained by a reference to
other parts of the work : and, when obtained, may
be found to be denoted differently by the authors
whose writings form the subject of comparison. In
order to prevent any confusion of this kind, I have
changed the characters, used in the original For-
mulæ, into such as will, in most cases, more readily
denote the quantities used in the solution of the
problem. And, to remove every possibility of mis-
take or confusion on this point, I have inserted, at
the bottom of each page, the quantities which are
denoted by every symbol made use of in the given
Formula.
The Elements of the System are taken, for the
most part, from the Système du Monde of M. La-
place (5th edition, 1824): but with additions from
Preface.
vii
other modern authors who have made certain
branches of the science their more particular study.
I have frequently experienced the want of a synopsis
of this kind; where all the different facts relative to
astronomy are brought under their respective heads,
without the necessity of turning to a variety of
works for information. Much time is frequently
employed, and oftentimes wholly lost, in a research
of this kind, which it is the object of the present
abstract to prevent.
At the end of the work I have inserted a set of
Problems with a view to introduce a few Examples
of the use and application of some of the Formula
and Tables, which may appear to be more in need
of a practical explanation. And here it may be
proper to allude to a new method of employing the
signs, when annexed to the logarithms of the quan-
tities under computation : a method which will be
found to be very convenient and useful in the arith-
metical solution of algebraical formulæ. The sign
therefore prefixed to a logarithm in this work, is in-
tended to affect only the natural number of such
logarithm : for, in all cases, the logarithms them-
selves are to be added together or subtracted from
each other (according to the conditions of the
problem) exactly the same as if no such signs were
viii
Preface:
annexed. With respect to the signs themselves, it
must be observed that the addition of two like
signs produces a positive natural number, and the
addition of two unliké signs produces a negative
natural number. Thus, in page 227, the natural
number of the first sum of logarithms is plus, be-
cause the two negative signs produce a positive
result: on the contrary, the natural number of the
second sum in that page is minus, because there is
only one negative sign. In complex trigonometri-
cal computations the method is still more conve-
nient and concise, and renders the result much less
ambiguous; as will appear from the arithmetical
operations in page 263.
.
1
Upon the whole, it is hoped that the present
work will not be without its utility. I am aware
that it might have been rendered much more ex-
tensive: but it would then have lost many advan-
tages which attach to a small and portable volume;
such as may be convenient to the intelligent and
scientific traveller, and not the less useful in the
observatory. Those who are desirous of possessing
a more comprehensive set of Astronomical Tables,
and more numerous and valuable than any that
have ever yet appeared in this country, will pro-
cure the first volume of Practical Astronomy re-
Preface.
ix
cently published by the Rev. Dr. Pearson : where
they will find many Tables that do not come within
the design and intention of the present work. For,
I have in all cases avoided such Tables as are
necessarily formed on the principle of double entry:
and likewise those which are merely local; such as
Tables of parallax &c. When problems involving
quantities of this kind come before the computer,
he must either have recourse to the Formulæ, or
refer to such larger collections of Tables. The pre-
sent work is intended as a Manual only: and
aspires to no further merit than accuracy, utility
and convenience may fairly lay claim to.
I have remarked in a former work that, in all
astronomical calculations, in which the sexagesimal
notation is so much involved, the computer will find
considerable assistance in the use of Callet's Tables
portatives de logarithmes : since, by the help of two
additional collateral columns given in that work,
the computation of any quantities composed of sexa-
gesimals becomes, without any reduction, as easy
and familiar as those which are formed according
to the usual decimal notation. I am now happy in
being able to state that a new Table of Logarithms
of the natural numbers is about to appear in this
country, agreeably to the above arrangement; with
b
Х
Preface.
the additional advantage and convenience of having
a mark attached to the last figure when it exceeds 5:
whereby the table will be rendered more extensive
and correct. This valuable addition to our list of
mathematical Tables was suggested by Lieutenant
Colonel Colby, for the use of the Trigonometrical
Survey; and was immediately adopted and put in
execution by Mr. Babbage: who has, with great
care and diligence, examined and compared all the
preceding works on this subject: and under whose
able and active superintendence the work is now
printing, in stereotype and on coloured paper.
January 1, 1827.
*
* The reader is requested to make the correc-
tions, pointed out in the list of Errata at the end,
previous to a perusal of the work. And the Author
will be obliged by the communication of any other
errors that may have escaped his detection.
TABLE OF CONTENTS.
page 3
6
ELEMENTS OF THE SYSTEM
Sun
Planets.
Mercury
Venus
Earth..
Mars
9
13
16
28
Vesta ...
31
32
33
34
35
38
41
Juno.
Ceres.
Pallas....
Jupiter ...
Saturn
Uranus...
Satellites
Moon...
Satellites of Jupiter
Satellites of Saturn
Ring of Saturn..
Satellites of Uranus..
Recapitulation
43
44
55
58
59
61
62
b 2
xii
Contents.
page 71
COS X
FORMULA
1 Equivalent expressions for sin x.....
2
72
3
tan X...
73
4 Relative to two arcs A and B
74
5 Differences of trigonometrical lines
76
6 Differentials of do. do.
76
7 General analytical expressions for the sides and angles
of any spherical triangle
77
8 Solutions of the cases of right-angled spherical triangles 78
9
of oblique-angled
do.
79
10 Trigonometrical series ..
84
11 Multiple arcs
85
12 For computing the Longitude, Right Ascension and
Declination of the sun ; and the Angle of Position . 86
13 For computing the Longitude and Latitude, Right
Ascension and Declination of the moon or a star ;
and the Angle of Position ..
87
14 For computing the Altitude and Azimuth of a heavenly
body and its Angle of Variation...
88
15 For computing the time &c, from single altitudes.... 89
16 For computing the time &c, from observations near
the prime vertical...
90
17 For computing the effect of atmospheric refraction... 91
18 For computing the equation of equal altitudes
92
19 For computing the correction for the Reduction to the
Meridian
93
20 For computing the correction for the Reduction to the
Solstice....
94
21 For computing the Angle of the Vertical..
95
22 For computing the horizontal Parallax of the moon at
any given latitude
95
23 For computing the Moon's parallax in Altitude ..... 96
24 For computing the longitude and altitude of the
Nonagesimal
97
Contents.
xii
page 98
FORMULÆ
25 For computing the Moon's parallax in Longitude and
Latitude...
26 For computing the Moon's parallax in Right Ascen-
sion
99
27 For computing the Moon's parallax in Declination.. 100
28 For computing the augmentation of the Moon's
semidiameter
101
29 For computing the apparent distance between the
centre of the moon and the sun or a star
102
30 For computing the place of the moon by means of
differences
103
31 For computing the annual precession of the Equinoxes
and the mean Obliquity of the ecliptic
104
32 For computing the Nutation of the obliquity of the
Ecliptic, and of the Longitude
105
33 For computing the lunar and solar nutation of a star 106
34 For computing the aberration of a star ...
107
35 For computing the corrections to observations with
a transit instrument
108
36 For computing the co-efficients used in the preceding
formula
109
37 For computing the latitude of a place, from observa-
tions of the Pole star, at any time of the day.. 110
38 For computing the height of mountains with the
barometer
111
39 For computing the increase of gravity, and the com-
pression of the earth from the lengths of two
isochronous pendulums......
112
40 For computing the increase of gravity, and the com-
pression of the earth from the vibrations of two
invariable pendulums..
113
41 For computing the corrections to be applied to an
invariable pendulum, from various causes
114
42 For computing the principal geodetical quantities
depending on the spheroidical figure of the earth. 115
xiv
Contents.
FORMULA
43 For computing the length of a degree of longitude
and latitude, on the earth considered as a spheroid page 116
44 For computing the compression of the earth, from
two measured arcs of the meridian
117
45 For computing the equations depending on the theory
of the elliptical motion of the planets
118
46 For computing the greatest equation of the centre,
the eccentricity of the orbit, and the eccentric
anomaly.
119
123
125
129
130
131
132
133
134
138
141
142
TABLES
1 Latitude and Longitude of various Observatories...
2 Mean R of the sun, for every day of the year.....
3 Corrections for every year from 1800 to 1900
4 Correction for the Lunar Nutation
5 Correction for the Solar Nutation
6 For converting sidereal into mean solar time ......
7 For converting mean solar into sidereal time
8 For converting degrees into time; and vice versa ..
9 Mr. Ivory's Refractions...
10 Corrections for thermometer and barometer..
11
Corrections for low altitudes.,
12 Dr. Brinkley's Refractions
13 Corrections for low altitudes
14 Parallax of the sun, for the 1st day of every month.
15 Logarithms of sin IP..
16 Logarithms for the equation of equal altitudes.....
17 Altitude of a star, when on the prime vertical .....
18 For the reduction to the meridian.
19
second part..
20 Mean obliquity of the ecliptic for every year from
1800 to 1900....
21 Lunar Nutation of Longitude, and of Obliquity of
the ecliptic..
143
144
145
146
147
153
154
164
165
166
Contents.
XV
page 168
TABLES
22 Solar Nutation of Longitude, and of Obliquity of
the ecliptic...
23 Selenographic positions of the principal lunar spots. 169
24 Angle of the Vertical, and Logarithm of the earth's
radius.
170
25 Augmentation of the moon's semidiameter
171
26 Logarithms to be used with the 1st, 2nd and 3rd
differences of the moon's place.
172
27 Annual precession of the equinoxes in Longitude;
and the constants for R and D...
173
28 For Aberration of a star in R, and first part of do.
in Dec...
174
29 For Aberration of a star in Dec. second part
175
30 For Lunar Nutation of a star in R and D..... 176
31 For Solar Nutation of a star in R and D....
177
32 Circular arcs
178
33 Semidiurnal arcs
180
34 Length of a degree of Longitude and Latitude; and
of the Pendulum
181
35 Expansion of various substances.
182
36 For measuring heights by the barometer
183
37 For converting hours, minutes and seconds, into
decimals of a day......
· 184
38 For converting minutes and seconds into decimals of
a degree,
185
39 For converting any day into the decimal part of a
year
186
40 For converting the centesimal division of the
quadrant into the sexagesimal..
188
41 For converting Fahrenheit's scale of the thermometer
into Reaumur's and the Centesimal..
191
42 Comparison of French and English measures ...... 192
43 Various logarithms used in astronomical calculations 193
Explanation of the Tables..
195
f
xvi
Contents.
PROBLEMS
1 To convert sidereal time into mean solar time: and
vice versa ....
page 217
2 To compute the Refraction of a heavenly body .... 220
3 To determine the time, from single altitudes of the
sun, or a star
222
4 To determine the time, from equal altitudes of the sun 226
5 For the reduction to the meridian.
228
6 To determine the latitude of a place.
231
7 To determine the longitude of a place
236
8 To compute the obliquity of the ecliptic, from
observations of the sun made near the solstice .. 247
9 To determine the place of the equinox, from obser-
vations of the sun made near that point
249
10 To compute the true place of the moon, by means
of differences.
250
11 To compute the parallax of the moon.
252
12 To compute the aberration and nutation of a star 256
13 To determine the corrections for observations made
with a transit instrument
258
14 To compute a table of altitudes and azimuths
260
15 To compute the longitude and latitude of a heavenly
body, from its right ascension and declination: and
262
16 To determine the height of mountains, by means
of the barometer..
263
List of errors and corrections
265
vice versa
ELEMENTS OF THE SYSTEM.
B
THE SUN.
TI
HE sun, which is the source of light and heat to our
system, is the most considerable of all the heavenly bodies;
and governs all the planetary motions.
Its mean distance from the earth is 23984 times the
semidiameter of the earth; or nearly 95 millions of miles.
Its mean longitude, at the commencement of the present
century *, was in 280º. 39' 10",2: after subtracting 20"
for the effect of aberration.
His mean motion in the ecliptic (as it appears to a
spectator on the earth) in 100 Julian years, or 36525 mean
solar days, is, according to M. Damoiseau, 36000°.76472,
or 100rev + 0°.45' 53",0+: whence we deduce his motion
in a mean solar day to be 09.98564722 or 09.59'.8",32999;
and consequently his mean motion in 365 days to be
359º. 7612354, or 359º. 45'. 40",45.
The longitude of his perigee, at the commencement of
the present century, was in 279º. 30'. 5",0: and the line
* The epoch assumed in the following pages, as the commencement
of the present century, is the moment of mean noon at Greenwich, on
January 1, 1801; reckoning from the mean equinox.
† M. Lalande, in his solar tables, assumed this, in round numbers,
equal to 46.0”. M. Delainbre at first determined it to be 45'.54"
but he afterwards adopted 45'.45'': at the same time stating that he
thought this latter value was too small. Baron Zach has assumed it
equal to 45'. 48".
;
B 2
4
The Sun.
of the apsides is subject to the same variation as that of
the earth.
The annual motion of the sun in the ecliptic, and its
diurnal motion from east to west, are in fact optical de-
ceptions, arising from the real motion of the earth in its
orbit and on its axis. These will be explained when we
come to treat on the motions of the earth.
The greatest equation of the centre, as adopted by M. De-
lambre, is 1º. 55'. 26",8: but M. Laplace has since pro-
posed 1°. 55'. 271,3. It diminishes at the rate of 17",18 in
a century.
The sun is surrounded by an atmosphere; and it is
oftentimes obscured with spots. Some of these spots have
been observed so large as to exceed the earth 4 or 5 times
in magnitude: and they are generally confined within 33°
of the solar equator.
The observation of these spots shows that the sun moves
on its axis : and the duration of an entire sidereal rotation
of the sun is about 25) days. Whence we conclude that
the sun is flattened at the poles.
The solar axis is inclined in an angle of 7º. 30' to the
axis of the ecliptic.
The mean horizontal parallax of the sun adopted by
M. Laplace is 8",66: but, as recently deduced by M. Encke
from the transits of Venus in 1761 and 1769, it is 85,5776:
and this angle is the apparent semidiameter of the earth,
as seen from the sun.
The apparent diameter of the sun, as seen from the
earth, undergoes a periodical variation. It is greatest
when the earth is in its perihelion; at which time it is
32. 355,6: and it is least when the earth is in its aphelion;
at which time it is 31. 31",0. Its mean apparent diameter,
or its diameter at its mean distance, is equal to 32'. 2",9.
The Sun.
5
The true diameter of the sun is 111.454 times the mean
diameter of the earth; or upwards of 882 thousand miles.
Whence its volume is 1384472 times greater than that of
the earth.
Its mass is only 354936 times greater than that of the
earth.
Whence we conclude that its density is zoet, or .2543 :
which is about one quarter that of the earth.
A body which weighs one pound at the equator of the
earth would, if removed to the equator of the sun, weigh
27.9 pounds. And bodies would fall there with a velocity
of 334.65 feet in the first second of time.
The sun, and all the planets, move round the common
centre of gravity of the system: which centre is nearly in
the centre of the sun. This motion changes into epi-
cycloids the ellipses of the planets and comets, which re-
volve round the sun.
The sun is supposed to have a particular motion, which
carries our system towards the constellation of Hercules.
But, this is doubted by M. Bessel.
THE PLANETS.
THE number of planets belonging to our system is
eleven. Six of these have been known and recognised
from time immemorial: namely Mercury, Venus, the Earth,
Mars, Jupiter, and Saturn. But, the remaining five, which
are not visible to the naked eye, have lately been disco-
vered by the help of the telescope; and are therefore called
telescopic planets : namely,
Uranus, discov. by Sir W. Herschel, March 13, 1781.
Ceres,
M. Piazzi, January 1, 1801.
Pallas, .
M, Olbers, March 28, 1802.
Juno,
M. Harding
Septem. 1, 1804.
Vesta,
M. Olbers,
March 29, 1807,
All these planets revolve round the sun, as the centre of
motion: and in performing their revolutions they follow
the fundamental laws of planetary motion so happily
discovered by Kepler; and which have been fully con-
firmed by subsequent observations. These laws are,
1º. The orbit of each planet is an ellipse; of which the
sun occupies one of the foci.
2º. The areas described about the sun, by the radius
vector of the planet, are proportional to the times em-
ployed in describing them.
3º. The squares of the times of the sidereal revolutions
of the planets are to each other as the cubes of their mean
distances.
The extremity of the major axis of the ellipse, nearest
the sun, is called the perihelion: the opposite extremity of
The Planets.
7
the same axis is called the aphelion. The line, which
joins these two points, is called the line of the apsides.
The anomaly of a planet is its distance in degrees from
the place of the perihelion.
The radius vector is an imaginary line drawn from the
centre of the sun, to the centre of the planet, in any part
of its orbit.
The motion of a planet in its orbit is always most rapid
when in its perihelion. This velocity diminishes as the
radius vector increases; till the planet arrives at its aphe-
lion, when its motion is the slowest. It then increases, in
an inverse manner, till the planet arrives again at its peri-
helion.
The first two laws of Kepler are sufficient for deter-
mining the motion of the planets round the sun: but, it is
necessary to know, for each of these planets, seven quan-
tities; which are called the elements of their elliptical
motions. These are,
1°. The mean distance of the planet from the sun: or
half the major axis of the orbit.
2º. The duration of a mean sidereal revolution of the
planet.
3º. The mean longitude of the planet, at a given epoch.
4°. The longitude of the perihelion, at a given epoch.
5º. The inclination of the orbit to the ecliptic, at a
given epoch.
6º. The longitude of the nodes, at a given epoch.
70. The eccentricity of the orbit, described by the planet.
The ellipses, however, which the planets describe, are
not unalterable. Their major axes and their mean motions
in their orbits, appear to be always the same. But the
position of their apsides, the inclination of their orbits to
the ecliptic, the position of their nodes, and the amount of
8
The Planets.
their eccentricities, seem to vary in a course of years.
These inequalities, being sensible only in a series of ages,
are called secular variations. There is no doubt of their
existence: but the modern observations not being suffi-
ciently extensive, and the ancient ones not sufficiently
exact, there still rests some degree of uncertainty as to
their magnitude.
There are indeed some inequalities which affect the
elliptical motion of the planets. That of the earth, for
instance, is a little altered. But, they are most sensible in
Jupiter and Saturn : for, it appears that the duration of
their mean sidereal revolution round the sun is subject to
a periodical variation.
MERCURY.
MERCURY is the nearest planet to the sun: its mean
distance being 0.3870981; that of the earth being con-
sidered as unity. This makes his mean distance above
36 millions of miles.
He performs his mean sidereal revolution in 87.9692580
mean solar days, or in 87d. 23h. 15m. 439,9: and his mean
synodical revolution in 115.877 mean solar days.
His mean longitude, at the commencement of the present
century, was in 166º. 0.48", 6.
His mean motion in his orbit, in a mean solar day, is
4º. 09238, or 4º. 5'. 32",6. His mean motion in 365 days
is consequently 1493º. 7175 or 4rev + 53°, 43'. 3',0.
The longitude of the perihelion was, at the commence-
ment of the present century, in 74º. 21'. 46",9. The line
of the apsides has a motion, according to the order of the
signs, equal to 5",84 in a year: but, when referred to the
ecliptic, this motion will (on account of the precession of
the equinoctial points) be equal to 55",9 in a year. And
it is in this manner that the value is assumed in the pla-
netary tables.
His orbit is inclined to the plane of the ecliptic in an
angle which, at the commencement of the present century,
was 70.0.95,1: and which angle is subject to a small
increase of 0", 1818 in a year.
His ascending node was, at the commencement of the
present century, in 45º. 57.30",9: having a motion, to the
westward, every year, of 7",82. But, when referred to the
C
10
Mercury.
1
ecliptic, the place of the node will (on account of the pre-
cession of the equinoxes) fall more to the eastward by
42",3 in a year.
And it is in this manner that the value is
assumed in the planetary tables.
The eccentricity of his orbit is 0.20551494 ; half the
major axis being assumed equal to unity. This eccen-
tricity is supposed to increase about .000003866 in a cen-
tury.
The greatest equation of the centre, deduced from this
eccentricity, is 23º. 39' 51"; and subject to an increase of
15,6 in a century.
The rotation on his axis is accomplished in 24h. 5m. 289,3.
The inclination of its aris to that of the ecliptic is not
known.
His mean apparent diameter, or his apparent diameter
at a distance equal to the mean distance of the earth from
the sun, is 6",9: but, at the time of his superior conjunc-
tion it is only 5',0; whilst, at his inferior conjunction, it
sometimes amounts to 12".
Mercury changes his phases like the moon, according to
his various positions with regard to the sun and the earth :
but this cannot be discovered without the aid of a power-
ful telescope.
His true diameter, compared with that of the earth con-
sidered as unity, is 0.398; which makes it about 3140
miles.
His volume is only 0.063, that of the earth being con-
sidered as unity.
His mass, compared with that of the sun considered as
unity, is quito = '0000004936.
A body, which weighs one pound at the equator of the
earth, would, if removed to the equator of Mercury, weigh
1•03 pounds.
Mercury.
11
The proportion of light and heat, which it receives from
the sun, is about 6:68 times greater than that received on
the earth.
As seen from the earth, Mercury never appears at any
great distance from the sun ; either in the morning or the
evening. His elongation, or angular distance, varies from
16º. 12 to 28º. 48'.
His course sometimes appears retrograde. The arc
which he describes in such cases varies from 9º. 22 to
15°. 44': its duration in the former case is 231 days, and
in the latter case 21} days. This retrogradation com-
mences when Mercury is at a distance from the sun, which
varies from 15º. 24' to 18º. 39': and terminates when
Mercury is at a distance which varies from 14°. 49' to
20°. 51'.
Mercury is sometimes seen to pass over the sun's disk:
which can happen only when he is in his nodes, and when
the earth is in the same longitude. Consequently this
phænomenon, for many centuries to come, can take place
only in the months of May or November. The first
observation of this kind was made by Gassendi in Novem-
ber 1631: since which period they have been frequent.
The following is a list of all those which have happened
since the above date inclusive, and of those that will
happen till the end of the present century.
o
1631 Nov. 6
1644 Nov. 8
1651 Nov. 2
1661 May 3
1664 Nov. 4
1674 May 6
1677 Nov, 7
1690 Nov. 9
1697 Nov. 2
1707 May 5
1710 Nov. 6
1723 Nov. 9
1736 Nov. 10
1740 Nov. 2
c2
12
Mercury.
1743 Nov, 4
1753 May 5
1756 Nov. 6
1769 Nov. 9
1776 Nov. 2
1782 Nov. 12
1786 May 3
1789 Nov. 5
1799 May 7
1802 Nov. 8
1815 Nov. 11
1822 Nov. 4
* 1832 May 5
1835 Nov. 7
* 1845 May 8
* 1848 Nov. 9
* 1861 Nov. 11
1868 Nov. 4
* 1878 May 6
1881 Nov. 7
1891 May 9
1894 Nov. 10
*
Those marked with an asterisk are such future ones as
will be visible in this country.
VENUS.
THE mean distance of Venus from the sun is 0.7233316;
that of the earth being considered as unity. This makes
her mean distance nearly 68 millions of miles.
She performs her mean sidereal revolution in 224-7007869
mean solar days, or in 224d. 16h. 49m. 89,0: and her mean
synodical revolution in 583.920 mean solar days.
Her mean longitude, at the commencement of the
present century, was in 11°. 33.3",0.
The mean motion in her orbit, in a mean solar day, is
1°.60217 or 10.36'.7",8. Her mean motion in 365 days
is consequently 584º. 7916 or irev + 224º. 47'. 30",07.
The longitude of her perihelion was, at the commence-
ment of the present century, in 128º. 43.53',1. The line
of her apsides has a motion to the westward, of 2",68 in a
year: but, when referred to the ecliptic, this line will (on
account of the precession of the equinoxes) appear to have
a motion to the eastward, of 47",4 in a year.
Her orbit is inclined to the plane of the ecliptic in an
angle, which, at the commencement of the present cen-
tury, was 3º. 23'. 28”,5: and which angle is subject to a
small decrease of about 0",0455 in a year.
Her ascending node was, at the commencement of the
present century, in 74º. 54'. 12",9: having a motion to the
westward, every year, of 17",6. But, when referred to the
ecliptic, the place of the node will (on account of the
pre-
cession of the equinoxes) fall more to the eastward by
32",5 in a year.
14
Venus.
The eccentricity of her orbit is 0.00686074; half the
major axis being assumed equal to unity. This eccentricity
is supposed to decrease about '000062711 in a century.
The
greatest equation of the centre is 0°. 47'. 15%; which
is subject to an annual decrease of 0",25.
The rotation on her axis is accomplished in 23h. 21m. 75,2.
The inclination of her axis, to that of the ecliptic, is not
exactly known.
Her mean apparent diameter, or her apparent diameter
at a distance equal to the mean distance of the earth from
the sun, is 16',9: but, at the time of her superior con-
junction it is only 9”,6; whilst at her inferior conjunction
it sometimes amounts to 61",2.
Venus changes her phases, like the moon, according to
her various positions with respect to the sun and the earth:
which causes a very considerable difference in her bril-
liancy.
Her true diameter, compared with that of the earth con-
sidered as unity, is 0.975; which makes it about 7700
miles.
Her volume is 0.927, that of the earth being considered
as unity.
Her mass, compared with that of the sun considered as
unity, is 405871
= .0000024638.
A body which weighs one pound at the equator of the
earth, would, if removed to the equator of Venus, weigh
only 0.98 pound.
The proportion of light and heat, which she receives
from the sun, is about 1.91 times greater than that re-
ceived on the earth.
She is surrounded by an atmosphere, the refractive
powers of which differ very little from those of the terres-
trial atmosphere.
Venus.
15
sun.
As viewed from the earth, Venus is the most brilliant of
all the planets; and may sometimes be seen with the naked
eye at noon day. She is known and recognised as the
morning and evening star: and never recedes far from the
Her elongation, or angular distance, varies from
45° to 47º. 12'.
Her course sometimes appears retrograde. The arc,
which she describes in such cases, varies from 14°. 35' to
170.12': its duration, in the former case, is 404. 21h, and
in the latter case 434. 12h. This retrogradation com-
mences or finishes when she is at a distance from the sun,
which varies from 270. 40 to 29º. 41'.
Venus is sometimes seen to pass over the sun's disc;
which can happen only when she is in her nodes, and
when the earth is in the same longitude. Consequently
this phænomenon, for many centuries to come, can take
place only in the months of June or December. It is a
phænomenon indeed of very rare occurrence, as may be
readily seen by the following list, which contains all those
transits of Venus which have occurred since that which
took place in December 1639 inclusive (the first that was
ever known to have been seen by any human being) to the
end of the 21st century.
1639 Dec. 4
1761 June 5
1769 June 3
1874 Dec. 8
* 1882 Dec. 6
* 2004 June 7
2012 June 5
1
THE EARTH.
THE earth which we inhabit is also one of the planets
that revolve about the sun. Its mean distance from the
sun is 23984 times its own semidiameter: whence it is
nearly 95 millions of miles distant from that luminary. If
this mean distance be assumed equal to unity, we shall
have its distance at the perihelion equal to .9832; and its
distance at the aphelion equal to 1.0168.
It performs its mean sidereal revolution in 365•2563612
mean solar days, or 365d. 6h. gm. 99,6 : but the time em-
ployed in going from one equinox to the same again, or
from one tropic to the same again (whence called the
tropical revolution), is only 365-2422414 mean solar days,
or 365d. 5h. 48m. 499,7 *. The tropical year is about 45,21
shorter than it was at the time of Hipparchus.
Its mean longitude, at the commencement of the present
century, was in 100°. 39'. 10",2: after subtracting 20" for
the effect of aberration.
Its motion varies in different parts of its orbit. Like all
the other planets, it is most rapid in its perihelion, and
slowest in its aphelion. In the former point it describes
an arc of 1º.1'.9',9 in a mean solar day: and in the
* M. Lalande makes this equal to 489,0; whilst M. Delambre makes
it 515,6. In fact, if we augment the duration of the year 1, we must
diminish the secular motion of the sun 4'',1. See the note in page 3.
The Earth.
17
or
latter point it describes an arc of only 57'. 11",5 in the
same period. Its mean motion is 09.98564722,
0°. 59'.8",32999 in a mean solar day; and 0°.98295603,
or 0°. 58'. 58",64172 in a sidereal day.
The mean longitude of its perihelion, at the commence-
ment of the present century, was 99º. 30'.5",0. But the
line of the apsides has a motion, to the eastward, of 11",8
in a year: which line, being referred to the ecliptic, will
(on account of the precession of the equinoxes) appear to
have a motion of 61",9 in a year. M. Laplace prefers
61",76. A revolution of the earth, from one end of the
apsides to the same point again, is called an anomalistic
year : and, on the assumption of the quantity stated by
M. Laplace, is performed in 365.2595981 mean solar days,
or in 3654.6h. 13m. 499,3. The perihelion coincided with
the vernal equinox about the year 4089 before the Chris-
tian era: it coincided with the summer solstice about the
year 1250 after Christ: and will coincide with the au-
tumnal equinox about the year 6483. A complete tropical
revolution of the apsides is performed in 20984 years.
The axis of the earth is inclined to the pole of the
ecliptic in an angle which, at the commencement of the
present century, was 23°. 27'. 565,5*: which angle is called
the obliquity of the ecliptic. It is observed to decrease at
the rate of 0",4755 in a year. But, this variation is con-
fined within certain limits; and cannot exceed 2º. 42'.
This angle is also subject to a periodical change called
the nutation ; depending principally on the place of the
moon's node: whereby the axis of the earth appears to
describe a small ellipse in the heavens. The semi major
* M. Bessel makes this only 54",32 with an annual diminution
of 0'',46.
D
18
The Earth.
be b =
COSW
axis of this ellipse is found by M. Laplace from theory to
be 95,40: but Dr. Brinkley, from a comparison of nu-
merous observations, makes it only 9",25. If a denote the
semi axis major of this ellipse, the semi axis minor (6) will
cos 2 w
xa: w being the obliquity of the ecliptic.
The sun has likewise an effect on the variation of this
angle; which amounts, at a maximum, according to
M. Laplace from theory, to o”,493: but, according to
Dr. Brinkley from observation, to 0",545. These varia-
tions in the obliquity of the ecliptic affect the right ascen-
sions and declinations of the stars, according to their posi-
tions in the heavens.
The intersection of the equator with the ecliptic is not
always in the same point; but is constantly retrograding,
or receding contrary to the order of the signs. Conse-
quently the equinoctial points appear to move forward on
the ecliptic: and whence this phænomenon is called the
precession of the equinoxes. The quantity of this annual
change caused by the action of the sun and moon, and
which is called the luni-solar precession, is 50',41 ; from
which we must deduct the direct motion caused by the
planets, equal to 0",31: and the difference, or 50',10 is the
general precession in longitude. It is subject to a small
secular variation. A complete revolution of the equinoxes
is performed in 25868 years.
This precession is also subject to a periodical change,
caused by the nutation of the earth's axis; and which
affects the right ascensions of all the stars, by quantities
depending (like the nutation of the obliquity) on the mean
place of the moon's node and on the true longitude of the
sun. M. Laplace's theory makes the constant of lunar
nutation in longitude equal to 17",579; and of the solar
The Earth.
19
1
nutation equal to 1",137. But Dr. Brinkley makes the
former 17'299; and the latter 1",255.
The eccentricity of the orbit of the earth is 0.016783568;
half the major axis being considered as unity *. The
major axis therefore will be to the minor axis of the orbit,
as 1 to .99986. The eccentricity of the earth's orbit is
subject to a decrease of 0.00004163 in a century.
The sidereal day, or the time employed by the earth in
revolving on its axis from any given star to the same star
again, is always the same: and has not varied 09,003 since
the time of Hipparchus. It is divided into 24 sidereal
hours; and these are again subdivided into sidereal
minutes and seconds. This mode of reckoning time,
during the day, is now universally adopted by astronomers
in their observatories : although the commencement of the
day is still determined by the apparent culmination of
the sun.
1
A mean solar day, as adopted by the public in this
country, is the time employed by the earth in revolving on
its axis, as compared with the sun, supposed to move at a
mean rate in its orbit, and to make 365.2425 revolutions in
a mean Gregorian year. But the mean solar day, adopted
by astronomers, is founded on the assumption that the sun
makes only 365•2422414 revolutions in a mean Grego-
rian
year.
It is divided into 24 mean solar hours; and
these are again subdivided into mean solar minutes and
seconds.
If the sidereal day be taken equal to 24 sidereal hours,
;
2
48
* M. Laplace makes this equal to :01685318, which appears to be
E
11 E3
too great. The present yalue is deduced from the formula
where E denotes the greatest equation of the centre, and which I have
assumed equal to 19.55'.27",3.
D 2
20
The Earth.
the mean solar day will be equal to 24h. 3m. 565,55 of those
sidereal hours. And, if the mean solar day be taken equal
to 24 mean solar hours, the sidereal day will be equal to
23h. 56m. 45,09 of those hours. Or, in all cases, if we wish
to determine in sidereal time the value of any given inter-
val expressed in mean solar time, and vice versa, we shall
have
sidereal time = 1.00273791 x mean solar time
mean solar time = 0.99726957 x sidereal time.
The apparent day is the time employed by the earth in
revolving on its axis, as compared with the apparent place
of the sun. This day is also divided into 24 apparent
hours; which are again subdivided into apparent minutes
and seconds. This mode of reckoning is still used by the
public in many parts on the continent: and is frequently
referred to by the practical astronomer on various occa-
sions. In fact, the apparent culmination of the sun is the
commencement of the astronomical day to every practical
astronomer: and in most ephemerides the computations
are made in apparent time.
Apparent time is constantly changing. This variation
arises principally from two causes : 1° the unequal motion
of the earth in its orbit; 2° the obliquity of that orbit to
the plane of the equator. The mean and apparent solar
days are never equal, except when the sun's daily motion
in right ascension is equal to 59'. 8",33. This is the case
about April 16th, June 16th, Sept. 1st, and Dec. 25th :
on these days the difference vanishes, or nearly so. It
is at its greatest about Novem. 1st, when it amounts to
16m. 16%. The correction which is applied to apparent
time, in order to reduce it to mean solar time, and vice
versa, is called the equation of time. It depends on a va-
riety of arguments which are given in all the solar tables.
The Earth.
21
d
h
m
d
h
m
The astronomical year is divided into four parts, deter-
mined by the two equinoxes and the two solstices. The
interval between the vernal and autumnal equinoxes is
(on account of the eccentricity of the earth's orbit, and its
unequal velocity therein) nearly eight days longer than
the interval between the autumnal and vernal equinoxes.
These intervals were, in 1801, nearly as follow:
From the vernal equinox to
the summer solstice = 92. 21.:50
= 186. 11. 34
From the summer solstice
to the autumnal equinox = 93. 13. 44
From the autumnal equinox
to the winter solstice . = 89. 16. 44
178. 18. 17
From the winter solstice to
the vernal equinox = 89. 1. 33
7. 17. 17
The mass of the earth, compared with that of the sun
considered as unity, is 354977 = .0000028173.
Its density is 3.9326 times greater than that of the sun;
and is, to that of water, as 11 to 2.
The
figure of the earth is that of an oblate spheroid ; the
axis of the poles being to the diameter of the equator as
304 to 305. Whence the compression of the earth is 303:
which I shall denote by. There is a consideraḥle dif-
ference however in the results obtained by different astro-
nomers and mathematicians. The mean diameter of the
earth is about 7916 miles : its equatorial diameter is 7924
miles, and its polar diameter 7898 miles.
As a necessary consequence from this circumstance, the
degrees of latitude increase in length, as we recede from
the equator to the poles. But, different meridians under
22
The Earth,
1
3 d
с
the same latitude present different results: the general fact
however is well ascertained. If the length of a degree,
divided in the middle by the equator, be denoted by d, the
length of a degree, divided in the middle by any other
latitude (=^) will be increased by x sin? a nearly :
·c denoting the reciprocal of the compression above men-
tioned. Whence we conclude that the increase is propor-
tional to the square of the sine of the latitude nearly.
The centrifugal force at the equator is nearly to
(= .00346) of gravity. If the rotation of the earth were
17 times more rapid, the centrifugal force would be equal
to that of gravity: and bodies at the equator would not
have any weight.
By reason of the circumstances mentioned in the last
two paragraphs, bodies lose part of their weight by being
taken towards the equator. If the gravity of a body at the
equator be denoted by unity, its gravity at any other lati-
tude (=^) will be increased by .00539 sin?, nearly.
A pendulum therefore, which vibrates seconds at the
equator, must be lengthened in the same proportion, as we
proceed towards the poles, in order that the oscillations
may be rendered isochronous. If p denote the length of a
pendulum at the equator, its length at any other latitude
(=^) must be p (1 + .00539 sina) nearly, in order to be
isochronous.
Light is supposed to take gm. 135,3 to come from the
sun to the earth. But, in this interval, the earth has
moved 20",25 in its orbit.
This motion of the earth produces an optical illusion in
the light which comes from all the heavenly bodies; and
which is called the aberration of light.
From a mean of
3326 observations made by Dr. Brinkley and Dr. Struve,
The Earth.
23
the constant of aberration is found to be 20",36. This
would make the velocity of light equal to 8m. 159,8.
A rare and elastic fluid surrounds the earth, which is
called the atmosphere. Neither the temperature nor den-
sity of this fluid is uniform; but diminishes in proportion
to the distance from_the surface of the earth, and is also
affected by various other circumstances.
On the parallel of 45° of latitude, the temperature being
at the freezing point, and the barometer at the level of the
sea at its mean height (= 29.922 inches) the weight of the
air is to that of a similar volume of mercury as 1 to
10477.9: whence it follows that, if the density of the
atmosphere were every where the same, its height would
be 26151 feet, or 4.95 miles. But its true height is much
more considerable: since M. Gay-Lussac actually ascended
in a balloon to the astonishing height of 23010 feet, or
4.36 miles; being the greatest elevation to which any
person has yet ascended.
Air is generally supposed to expand in bulk qšo for
every degree of Fahrenheit's thermometer: but M. La-
place prefers to
The rays of light do not move in a straight line through
the atmosphere; but are inflected continually towards the
earth: so that the heavenly bodies appear more elevated
from the horizon than they really are. This phænomenon
is called refraction.
We find, from the most accurate observations, that the
refraction which the atmosphere produces, is for the most
part independent of its temperature, and proportional to
its density. But, as the density varies according to the
temperature, it is necessary to attend not only to the state
of the barometer, but also to that of the thermometer.
A ray of light, passing from a vacuum into air at the
24
The Earth.
temperature of freezing water and under a pressure indi-
cated by the barometer at 29.922 inches, will be refracted
so that the sine of refraction is to the sine of incidence as
1 to 1•0002943321. It would be sufficient therefore, in
order to determine the direction of a ray of light through
the atmosphere, to know the law of the density of its
strata. But, this law, which depends on the tempera-
ture, is very complicated, and varies every moment in the
day.
The temperature of the whole atmosphere being sup-
posed at the freezing point, the density of these strata will
diminish in a geometrical progression according to their
distances from the surface of the earth : and we find by
analysis that, the barometer being at 29.922, the refraction
at the horizon is 39'. 54",68. It would be only 30'. 24",1
if the density diminished in an arithmetical progression.
The horizontal refraction, which we observe, (about
35'. 6',0) is a mean between these limits.
When the apparent height of a star above the horizon
exceeds 10°, its sensible refraction depends wholly on the
state of the thermometer and barometer at the place of
observation: and it is nearly proportional to the tangent
of the apparent distance of the star from the zenith, di-
minished by 3.25 times the corresponding refraction at
that distance: the thermometer being at the freezing point,
and the barometer at 29.922 inches. Whence it follows
that, at that temperature and under that pressure, the con-
stant of refraction is 60",66: but, at any other tempera-
ture and under any other pressure, corrections must be
applied which are usually given in all Tables of refrac-
tion. When the apparent altitude of a star does not ex-
ceed 10°, the formula for determining the refraction be-
comes more complex: and has been the subject of much
The Earth.
25
controversy between the most eminent mathematicians and
astronomers.
The humidity of the air produces no sensible effect on its
refractive powers, and may therefore be safely neglected.
The atmosphere is a heterogeneous substance. Out of
100 parts, 79 are azotic gas, and the remaining 21 are
oxygen gas: with the exception of 3 or 4 parts of carbonic
acid gas out of every 1000. This is found to be univer-
sally the case in whatever season or whatever climate, or
in whatever part of the world the experiment has been
tried. This proportion is also found to exist in the highest
points of the atmosphere that have been reached by means
of balloons.
A body projected horizontally from the surface of the
earth, to the distance of about 4•35 miles, if there were no
resistance in the atmosphere, would not fall again to the
earth; but would revolve round it as a satellite : the cen-
trifugal force being then equal to its gravity.
The action of the sun and moon has a considerable
effect on the waters of the ocean, and produces the phæ-
nomena of the tides.
The height of the tide, at high water, is not always the
same; but varies from day to day: and these variations
have an evident relation to the phases of the moon. It is
greatest at the syzigies: after which it diminishes, and be-
comes the least at the quadratures.
The tides are also affected by the declinations of the sun
and moon: for they diminish the tides of the syzigies
which occur at the equinoxes; and augment the tides of
the quadratures, which occur at the solstices. The dimi-
nution of the tides of the syzigies at the solstices, is only ;
of the diminution of the tides of the syzigies at the equi-
. And the increase of the tides at the quadratures,
noxes,
E
26
The Earth.
is twice as great at the equinoxes as it is at the sol-
stices.
The distance of the moon from the earth has also a
sensible influence on the tides. In general they increase
and diminish as the diameter and parallax of the moon
increases and diminishes; but in a greater degree. The
diminution of the tides of the syzigies at the perigee is
nearly three times greater than at the apogee.
The action of the moon upon the tides is three times
that of the sun.
The sea rises and falls twice in each interval of time
comprised between the consecutive returns of the moon to
the same meridian. The mean interval of these returns is
14. oh. 50m. 289,3: consequently the mean interval between
two following periods of high water is 12h. 25m. 145,1. So
that the retardation in the time of high water, from one
day to another, is 50m. 285,3 in its mean state : and it is
affected by all those causes which influence the moon's
motion.
This retardation varies with the phases of the moon.
It is at its minimum towards the syzigies, when the tides
are at their maximum; and it is then only 39m. 125,7.
But, towards the quadratures, when the tides are at their
minimum, this retardation is the greatest possible; and
amounts to 1h. 14m. 589,8.
The variation in the distance of the sun and moon from
- the earth (and particularly the moon) has an influence also
on this retardation. Each minute in the increase or dimi-
nution of the apparent diameter of the moon, augments or
diminishes this retardation 3m. 425,9 towards the syzigies :
but towards the quadratures the effect is three times less.
The daily retardation of the tides varies likewise with
the declination of the sun and moon. In the syzigies at
The Earth.
27
the time of the solstices, it is about 1m. 269,4 greater than
in its mean state : and it is diminished in the same pro-
portion at the equinoxes. On the contrary, in the qua-
dratures at the time of the equinoxes, it exceeds its mean
state by 5m. 459,6: and is in a similar manner diminished
by this quantity, in the quadratures at the time of the
solstices.
But, the state of the tides is so modified by the nature
and position of the coasts, the depth of the channel, the
operation of the winds, and by other causes, that the
above laws will not always be found to correspond with
the actual state of the tides, particularly near the coast, or
in rivers.
The general result however, from a mean of a number
of observations, is that the inequalities, in the heights and
intervals of the tides, have various periods. Some are of
half a day and a day; others are of half a month and a
month; whilst others are of half a year and a year : and
some are the same as the times of the revolutions of the
lunar nodes and apsides.
E 2
MARS.
The mean distance of Mars from the sun is 1.5236923;
that of the earth being considered as unity. This makes
his mean distance above 142 millions of miles.
He performs his mean sidereal revolution in 686.9796458
mean solar days; or in 686d. 23h. 30m. 418,4: and his
mean synodical revolution in 779.936 mean solar days.
His mean longitude, at the commencement of the present
century, was in 64º. 22.555,5.
His mean motion in his orbit, in a mean solar day, is
0°•524072, or 31'. 26",66. His mean motion in 365 days
is consequently 191º. 286280 or 191º. 17'. 10',6.
The longitude of the perihelion was, at the commence-
ment of the present century, in 332º, 23. 56",6. But, the
line of the apsides has a motion, to the eastward, of 15",8
in a year: which, on account of the precession of the
equinoxes, will appear to move 65",9 in a year.
His orbit is inclined to the plane of the ecliptic in an
angle which, at the commencement of the present century,
was 10.51'. 6',2: and which angle decreases about 0",014
in a year.
His ascending node was, at the commencement of the
present century, in 48º. 0'. 35,5; having a motion to the
westward every year, of 23",3. But, when referred to the
ecliptic, the place of the node will (on account of the
pre-
cession of the equinoxes) fall more to the eastward by
26",8 in a year.
The eccentricity of his orbit is 0.0933070; half the
Mars.
29
major axis being considered as unity. This eccentricity
is supposed to increase about 0.000090176 in a century.
The greatest equation of the centre is 10°. 40'. 50"; which
is subject to an annual increase of 0",37.
The rotation on his axis is performed in 24h. 39m. 21°,3.
The inclination of his axis to that of the ecliptic is
30°. 18'. 10",8.
His parallax is nearly double that of the sun.
His apparent diameter, at his mean distance from the
earth, is 6",29. At its conjunction, it is sometimes not
more than 35,6; but it increases as the planet approaches
its opposition, when it sometimes amounts to 18",28. Sir
W. Herschel states that the polar diameter is about it
less than the equatorial diameter.
Mars changes his phases (somewhat in the same manner
as the moon does from her first to her third quarter)
according to his various positions with respect to the sun
and the earth. But he never becomes cornicular, as Venus
and the moon do when near their conjunotions.
His true diameter, compared with the earth considered
as unity, is ·517 or about 4100 miles; which is rather
more than half the diameter of the earth.
His volume is 0·1386; that of the earth being considered
as unity.
His mass compared with the sun considered as unity is
25463 = .0000003927.
A body which weighs one pound at the equator of the
earth, would, if removed to the equator of Mars, weigh
only { of a pound.
The proportion of light and heat received by him from
the sun, is about 0.43; that received by the earth being
considered as unity.
He has a very dense but moderate atmosphere : and he
3
30
Mars.
is not accompanied by any satellite. As viewed from the
earth, he is known by his red and fiery appearance.
His course sometimes appears retrograde.
The arc
which he describes in such cases, varies from 10°.6' to
190.35': its duration in the former case is 600.18h; and
in the latter case 800.15h. This retrogradation com-
mences or finishes when the planet is at a distance from
the sun, which varies from 128°. 44' to 146º. 37'.
VESTA.
This planet was discovered by Dr. Olbers on March 29,
1807 : its mean distance from the sun is 2.367870; that of
the earth being considered as unity.
It performs its sidereal revolution in 1325•7431 mean
solar days: and its mean synodical revolution in 503.41
days.
Its mean longitude, at mean noon, at Greenwich, on
Jan. 1, 1820, was in 278°. 30.0", 4.
Its mean motion in its orbit, in a mean solar day, is
16. 17",9516: its mean motion in 365 days is consequently
99º. 9'. 15",33.
The longitude of its perihelion, on Jan. 1, 1820, was in
249º. 33'. 24",4. According to M. Santini, it has an ap-
parent annual motion of + 1.34", 24.
Its orbit is inclined to the plane of the ecliptic, in an
angle of 70..8'.9: which, according to M. Santini, has
an annual decrease of 0", 12.
Its ascending node was, on January 1, 1820, in
103º. 13'. 18",2: which, according to M. Santini; has an
apparent annual motion of + 15",63.
The eccentricity of its orbit is 0.089130; half the major
axis being considered as unity: subject to an annual in-
crease, according to M. Santini, of .000004009.
The greatest equation of the centre is 10°. 13'. 22".
The elements of this planet however are not yet sufficiently
determined to be depended on : and require correction from
futurc observations.
JUNO.
This planet was first discovered by M. Harding on
Sept. 1, 1804: its mean distance from the sun is 2.669009;
that of the sun being considered as unity.
It performs its sidereal revolution in 1592.6608 mean
solar days: and its mean synodical revolution in 473.95
days.
Its mean longitude, at mean noon, at Greenwich, on
Jan. 1, 1820, was in 200°. 16'. 19",1.
Its mean motion in its orbit, in a mean solar day, is
13'. 32", 9304 : its mean motion in 365 days is conse-
quently 82°. 25'. 194,60.
The longitude of its perihelion, on Jan. 1, 1820, was in
53º. 33'. 46".
Its orbit is inclined to the plane of the ecliptic, in an
angle of 13º. 4!.9",7.
Its ascending node was, on Jan. 1, 1820, in 1710.7.40",4.
The eccentricity of its orbit is 0.257848; half the major
axis being considered as unity.
The greatest equation of the centre is 29º. 46'. 19".
1
The elements of this planet however are not yet sufficiently
determined to be depended on : and require correction from
future observations.
CERES.
This planet was first discovered by M. Piazzi, on Jan. 1,
1801: its mean distance from the sun is 2.767245; that of
the earth being considered as unity.
It performs its mean sidereal revolution in 1681•3931
mean solar days: and its mean synodical revolution in
466.62 days.
Its mean longitude, at mean noon, at Greenwich, on
Jan. 1, 1820, was in 123º. 16'. 11",9.
Its mean motion in its orbit, in a mean solar day, is
12. 50',9230: its mean motion in 365 days is conse-
quently 78º. 9'. 46",89.
The longitude of its perihelion, on Jan. 1, 1820, was in
147º. 7'. 31",5 : which, according to M. Gauss, is subject
to an apparent annual motion of + 2.1",3.
Its orbit is inclined to the plane of the ecliptic in an
angle of 10°. 37'. 26",2: which, according to M. Gauss,
has an annual decrease of 0",44.
Its ascending node was, on Jan. 1, 1820, in 80°. 41'. 24".
According to M. Gauss, it has an apparent annual mo-
tion of + 1",48.
The eccentricity of its orbit is 0·078439; half the major
axis being considered as unity: which, according to
M. Gauss, is subject to an annual decrease of .00000583.
The greatest equation of the centre is 8º. 59!. 42".
The elements of this planet however are not yet sufficiently
determined to be depended on : and require correction from
future observations.
PALLAS
This planet was discovered by Dr. Olbers, on March 28,
1802: its mean distance from the sun is 2.772886; that of
the earth being considered as unity.
It performs its sidereal revolution in 1686°5388 mean
solar days : and its mean synodical revolution in 466•22
days.
Its mean longitude, at mean noon, at Greenwich, on
Jan. 1, 1820, was in 108°. 24.57",9.
Its mean motion in its orbit, in a mean solar day, is
12.48",3934: its mean motion in 365 days is consequently
77º. 54'. 255,59.
The longitude of its perihelion, on Jan. 1, 1820, was in
121°.7.4",3.
Its orbit is inclined to the plane of the ecliptic, in an
angle of 34°. 34.55",0.
Its ascending node was, on Jan. 1, 1820, in 1729.39'. 26",8.
The eccentricity of its orbit is 0.241648; half the major
axis being considered as unity.
The greatest equation of the centre is 270.49'. 19".
The elements of this planet however are not yet sufficiently
determined to be depended on : and require correction from
future observations. It appears subject to very considerable
perturbations.
JUPITER.
The mean distance of this planet from the sun is 5•202776;
that of the earth being considered as unity. This makes
his mean distance above 485 millions of miles.
He performs his mean sidereal revolution in 4332.5848212
mean solar days, or in 4332d. 14h. 2m. 89,5: which is nearly
12 years. But this period is subject to some inequalities.
His mean synodical revolution is performed in 398.867
mean solar days.
His mean longitude at the commencement of the present
century, was in 112º. 15. 23",0.
His mean motion in his orbit, in a mean solar day, is
0°.08312938, or 4'. 59",26. His mean motion in 365 days
is consequently 30°. 3422228, or 30°. 20. 32",0: so that he
passes through somewhat more than a sign in the course
of a year.
The longitude of his perihelion was, at the commence-
ment of the present century, in 11°.8'. 34",6. The line of
the apsides has a motion, to the eastward, of 65,96 in a
year: which when referred to the ecliptic will (on account
of the precession of the equinoxes) appear to be equal to
57",06 in a year.
His orbit is inclined to the plane of the ecliptic in an
angle which, at the commencement of the present century,
was 1º. 18. 51",3; and which angle is subject to a small
decrease of about 0", 226 in a year.
His ascending node was, at the commencement of the
present century, in 98º. 26'. 18",9; having a motion to the
F 2
36
Jupiter.
westward every year of 15",8. But, when referred to the
ecliptic, the place of the node will (on account of the pre-
cession of the equinoxes) fall more to the eastward by
34",3 in a year.
The eccentricity of his orbit is 0.0481621, half the major
axis being assumed equal to unity. This eccentricity is
supposed to increase about 0.000159350 in a century.
The greatest equation of the centre is 5º. 31'.13",8 :
which is subject to an annual increase of 0",6344.
The rotation on his axis is performed in 9h. 55m. 495, 7.
The inclination of his axis to that of the ecliptic is
30. 5. 30".
His apparent diameter (measured equatorially) at his
mean distance from the earth, is 36",74. At its conjunc-
lion it is sometimes only 30",0; but it increases as the
planet approaches its opposition, when it sometimes
amounts to 455,88.
His true diameter, compared with that of the earth con-
sidered as unity, is 10•860; which makes it near 90000
miles. The axis of the poles is to his equatorial diameter,
as 167 to 177.
His volume is 1280:9; that of the earth being considered
as unity.
His mass, compared with that of the sun considered as
unity, is To70.5 = .0009341431.
His density, compared with that of the sun considered
as unity, is :99239 : and is about # of the density of the
earth.
A body which weighs one pound at the equator of the
earth would, if removed to the equator of Jupiter, weigh
2.716 pounds. But this must be diminished about a ninth
part, on account of the centrifugal force due to each
planet.
Jupiter.
37
The proportion of light and heat which he receives from
the sun is •037; that received by the earth being con-
sidered as unity.
He is surrounded by faint substances which appear like
zones or belts: and which are supposed to be parts of his
atmosphere. As viewed from the earth he appears, next to
Venus, the most brilliant of all the planets; whom he
sometimes however surpasses in brightness.
His course sometimes appears retrograde. The arc
which he describes in such cases varies from 99.51' to
9º. 59': its duration in the former case is 116d. 18h; and
in the latter case 122d. 12h. This retrogradation com-
mences or finishes when the planet is at a distance from
the sun which varies from 113º. 351 to 116º. 42'.
Jupiter is accompanied by four satellites.
SATURN.
The mean distance of Saturn from the sun is 9.5387861;
that of the earth being considered as unity. This makes
his mean distance above 890 millions of miles.
Heperforms his meansidereal revolution in 10759.2198174
mean solar days; or in 29•456 Julian years.
But this
period' is subject to some inequalities : and his motion at
the present day appears to be less rapid than formerly.
His mean synodical revolution is performed in 378.090
mean solar days.
His mean longitude, at the commencement of the present
century, was in 135º. 20'. 61,5.
His mean motion in his orbit, in a mean solar day, is
0°. 03349777 or 2.0",6. His mean motion in 365 days is
consequently 12°. 22668787 or 12º. 13'. 36",08.
The longitude of his perihelion was, at the commence-
ment of the present century, in 89º. 9'. 295,8. The line of
the apsides has a motion, to the eastward, of 19",4 in a
year: which, when referred to the ecliptic, will (on account
of the precession of the equinoxes) appear to be equal to
695,5 in a year.
His orbit is inclined to the plane of the ecliptic in an
angle which, at the commencement of the present century,
was 2º. 29'. 35",7; and which angle is subject to a small
decrease of 0",155 in a year.
His ascending node was, at the commencement of the
present century, in 111°. 56' 37",4; having a motion to
the westward, every year, of 19",4. But, when referred to
Saturn.
39
the ecliptic, the place of the node will (on account of the
precession of the equinoxes) fall more to the eastward, by
30",7 in a year.
The eccentricity of his orbit is 0.05615050; half the
major axis being assumed equal to unity. This eccentricity
is supposed to decrease about .000312402 in a century.
The greatest equation of the centre is 6º. 26'. 12; which
is subject to an annual decrease of 1",279.
The rotation on his axis is performed in 10h. 29m. 169,8.
The inclination of his axis to that of the ecliptic is
31º. 19'.
His apparent diameter, at his mean distance from the
earth, is about 16", 20.
His true diameter, compared with that of the earth con-
sidered as unity, is 9.982; which makes it about 76068
miles. The axis of the poles is to the equatorial diameter
as 11 to 12.
His volume is 995.00; that of the earth being considered
as unity.
His mass, compared with that of the sun considered as
unity, is 3712 = .0002847380.
His density, compared with that of the sun considered
as unity, is •550; which is about f of the density of the
earth; but there is some uncertainty in this determination.
A body which weighs one pound at the equator of the
earth, would, if removed to the equator of Saturn, weigh
1•01 pounds.
The proportion of light and heat which it receives from
the sun is about .0011; that received by the earth being
considered as unity.
He is sometimes marked by zones or belts; which are
probably obscurations in his atmosphere.
His course sometimes appears retrograde. The arc
40
Saturni.
which he describes in such cases varies from 60.411 to
6º. 55': its duration in the former case is 1384. 18h; and
in the latter case 135d. 9h. This retrogradation com-
mences or finishes when the planet is at a distance from
the sun which varies from 1070.251 to 110°. 46'.
Saturn is accompanied by seven satellites : and also
surrounded with a double ring.
URANUS.
URANUS was discovered to be a planet by Sir William
Herschel on March 13, 1781; who gave it the name of
the Georgium sidus* Its mean distance from the sun is
19·182390; that of the earth being considered as unity.
This makes his mean distance upwards of 1800 millions of
miles.
It performs its mean sidereal revolution in 30686.8208296
mean solar days; or in 84.02 Julian years. Its mean
synodical revolution is performed in 369•656 mean solar
days.
Its mean longitude, at the commencement of the present
century, was in 1770.48' 23",0.
The mean motion in its orbit in a mean solar day is
0°. 0117695; or 421,37. His mean motion in 365 days is
4º. 295876 or 4º. 17. 45", 16.
The longitude of his perihelion was, at the commence-
ment of the present century, in 167º. 31'. 16",1. The line
of the apsides has an apparent motion, to the eastward, of
52,50 in a year.
His orbit is inclined to the plane of the ecliptic in an
angle of 46'. 28",44.
His ascending node was, at the commencement of the
* It is remarkable that this star was observed as far back as 1690.
It was seen three times by Flamsteed, once by Bradley, once by Mayer,
and eleven times by Lemonnier: not one of whom suspected it to be a
planet. That brilliant discovery was reserved for Herschel.
G
42
Uranus.
present century, in 72º. 59'. 35",3; having an apparent
motion to the eastward, every year, of 14", 16.
The eccentricity of his orbit is 0.04667938; half the
major axis being considered as unity.
The greatest equation of the centre deduced from this
eccentricity is 5º. 20.57'.
His apparent diameter, even at the time of his oppo-
sition, is scarcely 4",0.
His mass, compared with that of the sun considered as
unity, is 17918 = '0000558098.
His density, compared with that of the sun considered
as unity, is supposed to be about 1•100.
The proportion of light and heat which it receives from
the sun is about .003; that received by the earth being
considered as unity.
As seen from the earth the motion of Uranus sometimes
appears retrograde. The mean arc which he describes in
this case is about 3º. 36': and its mean duration is about
151 days. This retrogradation commences or finishes
when the planet is distant about 103º. 30' from the sun.
This planet is accompanied by six satellites.
THE SATELLITES.
The number of satellites in our system, at present known,
is eighteen: namely, the Moon which revolves round the
Earth, four that belong to Jupiter, seven to Saturn, and
six to Uranus. The moon is the only one visible to the
naked eye.
They all move round their respective primary planets,
as their centre, by the same laws as those primary ones
move round the sun: namely,
1°. The orbit of each satellite is an ellipse, of which the
primary planet occupies one of the foci.
2º. The areas, described about the primary planet, by
the radius vector of the satellite, are proportional to the
times employed in describing them.
3º. The squares of the times of the revolutions of the
satellites, round their respective primary planets, are to
each other as the cubēs of their mean distances from the
primary.
G 2
THE MOON.
The mean distance of the moon from the earth is 29.982175
times the diameter of the terrestrial equator; or above 237
thousand miles.
She performs her mean sidereal revolution in 27.321661423
mean solar days, or 27d. 7h. 43m. 119,5: but the time em-
ployed in making a tropical revolution is only 27d. 321582418,
or 27d. 7h. 43m. 49,7. Her mean synodical revolution is
29-5305887215 mean solar days, or 29d. 12h. 44m. 25,87.
But these periods are variable; and a comparison of the
modern observations with the ancient ones proves incon-
-testably an acceleration in the mean motions of the moon,
to which we shall presently allude.
Her mean longitude, at the commencement of the present
century, was in 118°. 171.8",3*.
Her mean motion, in 100 Julian years, or 36525 mean
solar days, is 4812670.878222 or 1336rev + 3070.52.41",6t:
whence, we deduce her mean motion in a mean solar day
to be 13º. 17639639 or 13°. 10'. 35",027: and consequently
her mean motion in 365 days to be 4809º. 38468235 or
* M. Burg has adopted 16'. 56',1: whilst M. Burckhardt assumes it
17.3",0.
† Mayer, in his first tables, adopted 52'. 20'' ; but in his second
tables, he increased it to 53.35'. M. Burg proposed 52. 43",48 :
whilst M. Burckhardt assumed it at 52. 53'',5. Lastly, M. Damoiseau
has retained the determination of M. Laplace, which makes it only
52.41",6 as stated in the text.
The Moon.
45
13rev + 129º. 23'. 4",85646. It is however subject to a
secular variation, as will be shown presently.
The mean longitude of her perigee was, at the com-
mencement of the present century, in 266º. 10'. 71,5. But
the line of the apsides has a motion to the eastward,
which in 36525 mean solar days is 4069º. 046278, or
Ilrev + 109°, 2.46',6 *; whence we deduce the mean mo-
tion in a mean solar day to be 6'. 41",0; and consequently
the mean motion in 365 mean solar days to be 40°. 39'.45",36.
It makes a sidereal revolution in 3232.575343 mean solar
days, or in nearly 9 years. The period of a tropical revo-
lution of the apsides is but 3231•4751 mean solar days.
These periods however are not uniform: for they have a
secular variation depending on the acceleration of the
moon; and are retarded whilst the motion of the moon
itself is accelerated. The amount of this secular variation,
we shall allude to in the sequel.
If the mean place of the moon's perigee be deducted
from her mean longitude, it will show her mean anomaly.
This mean anomaly was, at the commencement of the
present century, in 212º. 7.0",8t; and its motion in a.
mean solar day is 13º. 064992; consequently its motion in
365 days is 47689.722057, or 13rev + 88°.43.19",4. The
mean period of an anomalistic revolution of the moon is
274.5545995 or 27d. 13h. 18m.375,4. But these motions
are variable, as will be shown hereafter.
Her orbit is inclined to the plane of the ecliptic in an
angle of 5º. 8'. 47",9. But this inclination is subject to a
periodical variation which principally depends on the
* M. Burg makes this 3.25%',7, whilst M. Burckhardt makes it
3.48”,2. M. Damoiseau has adopted the determination of M. Laplace.
+ M. Burg assumes 6'.56'',6; whilst M. Burckhardt adopts 6'.39",8.
46
The Moon.
cosine of twice the distance of the moon from the sun; and
amounts, at a maximum, to 8'. 47",15. The mean inclina-
tion however is constant, notwithstanding the secular va-
riation in the plane of the ecliptic: a fact which is con-
firmed by all the observations, ancient and modern.
Her ascending node was, at the commencement of the
present century, in 13°. 53'. 17",7* It has a motion to
the westward, which in 36525 mean solar days amounts to
1934º. 1659722, or 5rev + 134º. 9'. 57",5+: whence we de-
duce the motion of the node in a mean solar day to be
0°.052955 or 3'. 104,64: and the motion in 365 mean
solar days to be 19º. 328421 or 19º. 19'. 42",316. The
nodes make a sidereal revolution in 6793•39108 mean
solar days; or in 18·6 Julian years. The place of the
node is subject to many inequalities; of which the greatest
is proportional to the sine of double the distance of the
moon from the sun ; which, at a maximum, amounts
to 1°. 37'
. 45". A synodical revolution of the nodes is
performed in 346.619851 mean solar days; or in
346d. 145.52m. 359,1. The mean period of a revolution of
the moon, from node to node, is 27d. 2122222,
27d. 5h. 5m. 36$. These mean motions however are not
uniform: for the motion of the nodes is subject to a secular
variation depending on the acceleration of the moon; and
is retarded, whilst the motion of the moon is accelerated.
The amount of this secular variation we shall now allude to.
The acceleration of the moon's mean motion arises from
or
* M. Burg, in the Supplement to his tables, makes this 40”,6: whilst
M. Burckhardt adopts 22',2.
† M. Burg has assumed 42",0, in the Supplement to his tables :
whilst M. Burckhardt has adopted 48",0. M. Damoiseau has followed
the determination of M. Laplace.
The Moon.
47
the action of the sun, together with the secular variation of
the eccentricity of the earth's orbit. Whilst this eccen-
tricity diminishes (which is the case at present) the accele-
ration will increase: but, when the eccentricity shall begin
to increase, this acceleration will be changed into a re-
tardation of the moon's motion. The cause of this varia-
tion affects not only the position of the moon's longitude,
but also the place of her perigee and node. M. Damoiseau
has given the following formulæ for the secular variations:
where x denotes the number of centuries from 1800.
Long: = + 10",7232x + 0",019361 x3
Anom: = + 50",4203 x2 + 0",091035x3
Node = + 65,5632x2 + 0",011850x®
These quantities are related to each other, as the numbers
1, + 4•702 and + 0.612.
The eccentricity of the moon's orbit is 0.0548442; half
the major axis being assumed equal to unity. It does not
appear to be subject to any variation.
The greatest equation of the centre is 6º. 17'. 12", 7:
which also appears to be invariable.
The rotation on her axis is equal and uniform ; and is
performed in precisely the same time as the tropical revo-
lution in her orbit: whence she always presents nearly the
same face to the earth.
But, as the motion of the moon, in her orbit, is periodi-
cally variable, we sometimes see more of her eastern edge,
and sometimes more of her western edge. This appearance
is called her libration in longitude.
The inclination of her axis to that of the ecliptic is
1°.301. 10",8.
In consequence of this position of the moon, her poles
alternately become visible to, and obscured from us. This
phænomenon is called her libration in latitude.
48
The Moon.
There is also another phænomenon connected with this
subject, arising from the moon being seen by us from the
surface of the earth, instead of the centre. This is called
her diurnal libration.
There are other inequalities in the moon's motions,
arising from the action and influence of the sun. The
principal of these are the three following ones; which are
added as equations to the moon's mean longitude.
1º. The evection, whose constant effect is to diminish the
equation of the centre in the syzigies, and to augment it in
the quadratures. If this diminution and increase were
always the same, the evection would depend only on the
angular distance of the moon from the sun: but its abso-
lute value varies also with the distance of the moon from
the perigee of its orbit. After a long series of observations,
we are enabled to represent this inequality by supposing it
to depend on the sine of double the distance of the moon
from the sun, minus the distance of the moon from its
perigee. At its maximum it amounts to 10. 20'. 295,90.
2º. The variation, which disappears in the syzigies and
quadratures, and is greatest in the octants. It is then
equal to 35'. 41",96. It is proportional to the sine of twice
the distance of the moon from the sun : and its duration is
half a synodical revolution of the moon.
3º. The annual equation, which follows exactly the
same law as the equation of the centre of the sun, but with
a contrary sign. For, when the earth is in its perihelion,
the orbit of the moon is enlarged by the action of the sun ;
and the moon therefore requires more time to perform her
revolution. But, as the earth proceeds towards its aphe-
lion, the moon's orbit contracts. Hence, the period of
this inequality is an anomalistic year: and, at its maximum
it amounts to 11'. 11",97. It is subject to a small secular
variation.
The Moon.
49
The mean horizontal parallax of the moon, at the equa-
tor, is 57.0",9. It varies from about 53'. 48" to about
61'. 24" according to the distance of the moon from the
earth. The horizontal parallax at any other latitude (=^)
is always less than that at the equator, by a quantity
which is equal to the equatorial horizontal parallax mul-
1
tiplied by . sin a nearly: being the compression of
the earth.
The parallax of altitude may, for most ordinary pur-
poses, be considered as equal to the horizontal parallax (at
the place) multiplied by the cosine of the apparent alti-
tude.
The apparent diameter of the moon varies also accord-
ing to her distance from the earth. When nearest to us,
it is 33'. 31",07; but at her greatest distance it is only
29'. 21",91 : the apparent diameter at her mean distance is
31'. 7",0. It is always À of the horizontal parallax of the
-moon: or, more correctly, equal to 545 of the lunar
parallax.
Her mean true diameter is, in proportion to that of the
earth, as 5823 to 21332; or as 1 to 3•665. Whence her
mean diameter is about 2160 miles. Her figure is that
of an oblate spheroid, like that of the earth.
Her volume is the of the volume of the earth.
Her mass is 7057 of the mass of the earth *
Her density is 16ez = 615 of the density of the
earth.
* M. Laplace made this, at first, equal to 18:57: but he has since
reduced it to 7577. The value in the text is deduced from Dr. Brink-
ley's constant of nutation.
H
-
50
The Moon.
A body, which weighs one pound at the equator of the
earth, would, if removed to the equator of the moon,
weigh only t of a pound.
The light of the moon is 300 thousand times more weak
than that of the sun. Its rays, collected by the aid of
powerful glasses, do not produce any sensible effect on the
thermometer.
The atmosphere of the moon (if it has any) must be
exceedingly attenuated; and must be more rare than that
which we can produce with our best air-pumps.
The refraction of the rays of light at the surface of the
earth must be at least a thousand times greater than at the
surface of the moon. The horizontal refraction at the
moon cannot exceed 1",6.
Volcanoes and mountains are discovered on her surface,
by the aid of powerful telescopes.
A body projected from the surface of the moon, with a
momentum that would cause it to proceed at the rate of
about 8200 feet in the first second of time, and whose di-
rection should be in a line which, at that inoment, passed
through the centre of the earth and moon, would not fall
again to the surface of the moon; but would become a
satellite to the earth. Its primitive impulse might, indeed,
be such as to cause it even, after many revolutions, to pre-
cipitate to the earth. The stones, which have fallen from
the air, may be accounted for in this manner.
The phases of the moon are caused by the reflection of
the sun's light from her surface; and depend on the rela-
tive positions of the sun, the earth and the moon.
Eclipses can happen only when the moon is in her syzi-
gies: and then only when she is near the place of her nodes.
If, at the time of her mean conjunction with the sun, she be
The Moon.
51
less than 13º. 33! from her node, there will certainly be an
eclipse of the sun in some part of the world : but, if this
distance be greater than 19º. 44, there cannot be one.
Between these limits it will be necessary to make a more
minute calculation. If at the time of her mean opposition,
she be 70.471 distant from her node, there will certainly be
an eclipse of the moon : but if 130.21' distant therefrom,
there cannot be one. Between these limits it will also be
necessary to make a more minute calculation.
The number of eclipses in a year cannot be less than two,
nor more than seven. And when there are only two, they
will both be solar.
A solar eclipse cannot take place unless the moon be in
conjunction with the sun : and then only to a spectator on
the earth under particular circumstances. When the
centres of the sun and moon are in the same straight line
with the eye of the spectator, and the apparent diameter of
the moon is greater than that of the sun, the eclipse will
be total : but, if her apparent diameter be less, the eclipse
will be annular. In other cases, however, which are by
far the most numerous, the sun will suffer only a partial
eclipse. The greatest possible duration of the annular
appearance of a solar eclipse is, according to M. Du
Sejour, 12m. 248 : and the greatest possible time during
which the sun can be totally obscured is 7m. 585. The
magnitude and duration of every solar eclipse will in fact
differ at every point of the earth's surface.
The following is a list of all the subsequent solar
eclipses that will be visible in this country during the
present century. The hour of the day at which the
eclipse commences, and the number of digits eclipsed, are
adapted to the middle of England.
H 2
52
The Moon.
LIST OF SOLAR ECLIPSES *.
Digits
eclipsed.
h
6. 41
Year.
Day and hour.
d
1826 Nov. 29. 10. A.M.
1832 July
27. 2. P.M.
·1833 July 17. 5. A.M.
1836 May
15. 2. P.M.
1841 July 18. 3.
1842 July 8. 5. A.M.
1845 May 6. 8.
1846 April 25. 6. P.M.
1847 Oct. 9. 6. A.M.
1851 July 28. 2. P.M.
1858 March 15. 11. A.M.
1860 July 18. 2. P.M.
1861 | Dec. 31. 2.
1863 May 17. 6.
1865 Oct. 19. 4.
1866 Oct. 8. 5.
1867 March 6. 8. A.M.
1868 Feb. 23. 3. P.M.
1870 Dec. 22. 11. A.M.
1873 May 26. 8.
1874 Oct. 10. 9.
1875 Sept. 29. noon
1879 | July 19. y. A.M.
1880 Dec. 30. 2. P.M.
1882 ) May 17. 6. A.M.
1887 | August 19. 3.
1890 June 17. 8.
1891 June 6. 5. P.M.
1895 March 26. 9. A.M.
1896 August 9. sunrise
1899 June 8. 5. A.M.
1900 May 28. 3. P.M.
0. 30
0.36
11. 18
contact
8. 54
6. 15
2. 21
11. 0
9. 43
11. 30
9. 12
5. 0
3. 46
7. 36
5. 3
8. 42
contact
9. 36
3. 43
6. 18
0. 33
4. 0
4. 24
2. 18
11. 58
4. 39
3. O
1. 0
contact
3. 13
8. O
* See Hutton's Mathem. Dictionary. Vol. i. page 453. 2nd edition.
I have however added a few others, from the list published by M. Du
Vaucel in the Memoires des Savans Etrangers. Vol. v. page 575.
The Moon.
53
A lunar eclipse takes place only when the moon is in
opposition to the sun; and it is of the same magnitude and
duration to every spectator on the surface of the earth. It
is caused by her passing through the shadow of the earth;
which is 3 times longer than the distance between the
earth and the moon. The breadth of this shadow, in the
part where it is traversed by the moon, is about 2 times
greater than the diameter of the moon; and is equal to the
sum of the horizontal parallaxes of the sun and moon,
minus the semidiameter of the sun. The magnitude and
duration of every lunar eclipse will consequently vary
according to the magnitude of these quantities at the given
time, and the relative positions of the luminaries.
Visible eclipses of the moon are so frequent, that a list
of them cannot be conveniently inserted in this work.
And, on account of the indistinctness of the border of the
penumbra, the correct observation of such eclipses is ge-
nerally difficult and unsatisfactory.
Eclipses generally return again nearly in the same order
and magnitude at the end of 223 lunations. For in 223
mean synodical revolutions there are 6585.32 days; and
in 6585•78 days there are 19 mean synodical revolutions
of the moon's node. Therefore at the end of this period
the sun and moon will be found nearly in the same
position with respect to the place of the moon's node.
This period consists of 18 Julian years and 11 days, if
there are four leap years in the interval: but if there are
five leap years, it will consist of no more than 18 Julian
years, and 10 days. And it will be found that there are
generally about 70 eclipses in this interval : of which, 29
will be lunar, and 41 solar.
During a mean synodical revolution of the moon, the
motion of the sun's mean anomaly, the moon's mean
54
The Moon.
anomaly, and the mean distance of the moon from her
node, will be respectively
299, 1053533 25°.8169054 30°. 6705153
These quantities are useful in the computation of tables
for determining the periods of eclipses.
To an inhabitant of the moon, the earth always appears
nearly in the same place in the heavens; from which it
varies only in consequence of the libration.
JUPITER'S SATELLITES.
By the aid of the telescope we may observe four satellites
revolving round Jupiter : the positions of which, with re-
spect to each other, are continually changing. We some-
times see them pass over the disc of Jupiter, and to project
their shadow on the body of the planet.
The shadow which Jupiter himself projects behind him,
relatively to the sun, gives rise to another phænomenon of
considerable importance. For, the satellites frequently
disappear, or are eclipsed in that shadow, although they
appear, with respect to us, to be at a distance from the
disc of the planet. These eclipses are similar in principle
to lunar eclipses; and vary in duration according to the
relative position of the bodies with respect to the sun.
In the following table are given the mean sidereal revo-
lution of the satellites, in mean solar days; together with
their mean distances from Jupiter, the semidiameter of that
planet's equator being considered as unity; and likewise
their masses compared with that of Jupiter considered also
as unity.
Sat.
Sidereal revolution.
Mean
distance.
Mass.
dh m
1. 18. 28
d
1. 769137788148
1
6, 04853 1.0000173281
2
3. 13. 14
3. 551810117849
9. 62347.0000232355
3
7. 3. 43
7. 154552783970
15. 350240000884972
4
16. 16. 32 16.688769707084
26. 99835 1.0000426591
56
Jupiter's Satellites.
First satellite. The plane of the orbit of this satellite
coincides nearly with the plane of the equator of Jupiter ;
the inclination of which to the orbit of the planet, is
3º. 5'. 30". Its eccentricity is insensible.
Second satellite. The eccentricity of the orbit of this
satellite is also insensible. The inclination of its orbit to
that of its primary is variable; as well as the position of
its nodes. These variations are represented nearly by sup-
posing the orbit of the satellite inclined 27', 49",2 to the
equator of Jupiter; and by giving the nodes a retrograde
motion, on this plane, so as to make a revolution in 30
Julian years.
Third satellite. This satellite has a little eccentricity,
which is subject to a very sensible variation. Towards the
end of the century before the last, the equation of the centre
was at its maximum; and was then as much as 13'. 16",4.
It afterwards diminished, and was at its minimum about
the year 1777; when it was only 5'. 75,5. The line of the
apsides has a direct but variable motion. The inclination
of its orbit to that of Jupiter, and the position of its nodes,
are also variable. These variations may be represented
nearly by supposing the orbit inclined 12. 20" to the
equator of Jupiter; and by giving the nodes a retrograde
motion, on this plane, so as to make a revolution in 142
Julian years.
Fourth satellite. The eccentricity of this satellite is
greater than that of the other three. The line of the
apsides has an annual and direct motion of 42'. 58", 7.
The place of the nodes has a direct annual motion on the
orbit of the planet, of 4'.15",3. The inclination of the
orbit to that of Jupiter is about 2º. 58'. 48". It is in con-
sequence of this great inclination that this satellite fre-
quently passes behind the planet, with respect to the sun,
Jupiter's Satellites.
57
without being eclipsed. Since the middle of the last cen-
tury, the inclination of the orbit has increased, and the
motion of the nodes has diminished, very perceptibly.
Independent of these variations, the satellites are subject
to perturbations which affect their elliptical motions; and
which render their theory very complicated.
The motions of the first three satellites are related to
each other by a most singular analogy. For, the mean
sidereal or synodical revolution of the first, added to twice
that of the third, is generally equal to three times that of
the second. And the mean sidereal or synodical longitude
of the first, minus three times that of the second, plus
twice that of the third, is generally equal to two right
angles.
It follows therefore that, for a great number of years at
least, the first three satellites cannot be eclipsed at the
same time. For, in the simultaneous eclipses of the second
and third, the first will always be in conjunction with
Jupiter : and vice versa.
The eclipses of Jupiter's satellites are of great utility in
enabling us to determine the longitude of places, by their
observation : and they likewise exhibit some curious phæ-
nomena with respect to light.
1
SATELLITES OF SATURN.
By the aid of the telescope also, we may observe seven
satellites to revolve round Saturn: the elements of which
are but little known on account of their great distance
from us. The following table will show their mean sidereal
revolutions in mean solar days, and their mean distances
from the planet, in semidiameters of Saturn's equator.
Mean
Sat.
Sidercal revolution.
distance.
m
d h
0. 22. 38
d
0.94271
1
3.351
2
1. 8. 53
1.37024
4.300
3
1. 21. 18
1.88780
5.284
4
2. 17. 45
2.73948
6.819
5
4. 12. 25
4.51749
9.524
6
15. 22. 41
15.94530
22.081
7
79. 7. 55
79.32960
64.359
The orbits of the first six satellites appear to be nearly
circular; and in the plane of Saturn's ring: whilst the
seventh varies from that plane, and approaches nearer to
that of the ecliptic.
The great distance of these satellites, and the difficulty
of observing them, prevent us from ascertaining the ellip-
ticity of their orbits, and still less the inequalities of their
motions. We know however that the ellipticity of the
sixth is very perceptible.
RING OF SATURN.
The most singular phænomenon attending Saturn is the
double ring, with which he is surrounded: the apparent
form and magnitude of which is very variable. Sometimes
it appears nearly to surround the planet, and at other times
is scarcely visible even in the most powerful telescopes.
When it is approaching the latter state, it has the appear-
ance of two handles, or ansæ ; one on each side of the
planet.
This ring, which is very thin and broad, is inclined to
the plane of the ecliptic in an angle of 28º. 39'. 54".
It revolves from west to east, in a period of 10h. 29m. 169,8,
about an axis which is perpendicular to its plane, and
which passes through the centre of the planet. And it is
remarkable that this is the period in which a satellite,
assumed to be at a mean distance equal to the mean di-
stance of the ring, would revolve round the primary, ac-
cording to the third law of Kepler.
The breadth of this ring is nearly equal to its distance
from the surface of Saturn: that is, about of the diameter
of the planet.
Its surface is separated nearly in the middle, by a black
concentric band, which divides it into two distinct rings:
the breadth of the exterior of which is rather less than that
of the interior.
The apparent diameter of the ring, at the mean distance
of the planet, is 38",42; and of its breadth 55,78.
The edges of the ring being very thin, and being oc-
I 2
60
Ring of Saturn.
casionally presented obliquely to the earth, sometimes
disappear. And as this edge will present itself to the sun
twice in each revolution of the planet, it is obvious that the
disappearance of the ring will occur about once in 15
years: but, under circumstances oftentimes very different.
The intersection of the ring and of the ecliptic is in
170°, and 350°: consequently, when Saturn is near either
of those points, his ring will be invisible to us. On the
contrary, when he is in 80° or 260°, we may see it to the
greatest advantage. Regard however must be had to the
position of the earth ; which will cause some variations in
this respect. The following are the dates, during the en-
suing revolution of the planet, when the mean heliocentric
longitude of Saturn is such that the ring will (if the earth
be favourably situated) either be invisible, or seen to the
greatest advantage.
Date.
Mean
Long.
Phase.
1825 Nov.
80
170
South side illumined
Invisible
North side illumined
260
1833 April
1838 July
1847 Dec.
1855 April
350
Invisible
80
South side illumined
SATELLITES OF URANUS.
By the aid of a very powerful telescope we may discover
six satellites revolving round Uranus.
The following table will show their mean sidereal revo-
lutions in mean solar days, and their mean distances from
the planet in semidiameters of his equator.
Sat.
Sidereal revolution,
Mean
distance.
d h m
5. 21. 25
d
5.8926
1
13.120
2
8. 16. 58
8.7068
17.022
3
10. 23. 4
10.9611
19.845
4
13. 10. 56
13.4559
22.752
5
38.
1. 48
38.0750
45.507
6
107, 16. 40
107.6944
91.008
All these satellites are stated by Sir Wm. Herschel (to
whom we are indebted for all we know on the subject) to
move in a plane which is nearly perpendicular to the plane
of the planet's orbit.
RECAPITULATION.
Mean distance from the sun.
Mercury
0.3870981
Venus
07233316
Earth
1.0000000
Mars
1:5236923
Vesta.
2:3678700
Juno
2.6690090
Ceres.
2.7672450
Pallas
2.7728860
Jupiter
5.2027760
Saturn
9.5387861
Uranus
19.1823900
Mean sidereal revolution,
d
87.9692580
224.7007869
365.2563612
686.9796458
1325.7431000
Mercury
Venus
Earth
Mars
Vesta
Juno
Ceres
Pallas
Jupiter
Saturn
Uranus
1592.6608000
1681.3931000
1686.5388000
4332.5848212
10759.2198174
30686.8208296
Recapitulation.
63
Synodical revolution.
d
115.877
583.920
365.242
779.936
503.410
Mercury
Venus ,
Earth.
Mars
Vesta
Juno
Ceres
Pallas,
Jupiter
Saturn
Uranus
473.950
466.620
466.220
398.867
378.090
369.656
1
Mean longitude, Jan. 1, 1801.
Mercury
Venus
Earth
Mars
Vesta
Juno
Ceres
Pallas
Jupiter
Saturn.
Uranus
1820.
166. 6. 48,6
11. 33. 3,0
100. 39. 10,2
64. 22. 55,5
278. 30. 0,4
200. 16. 19,1
123. 16. 11,9
108. 24. 57,9
112. 15. 23,0
135. 20. 6,5
177. 48. 23,0
64
Recapitulation.
Mean daily motion in the orbit.
Mercury
Venus,
Earth.
Mars
Vesta
Juno
Ceres
Pallas
Jupiter
Saturn
Uranus
4. 5. 32,6
1. 36. 7,8
0. 59. 8,3
0. 31. 26,7
0. 16. 17,9
0. 13. 32,9
0. 12. 50,9
0. 12. 48,4
0. 4. 59,3
0. 2. 0,6
0. 0. 42,4
Longitude of perihelion.
Jan. 1, 1801.
Ann. inc.
Mercury 74. 21. 46,9 +
+ 55,9
Venus
128. 43. 53,1 t 47,4
Earth
99. 30. 5,0 + 61,8
Mars..
332. 23. 56,6 + 65,9
Vesta
r
249. 33. 24,4 at 94,2
Juno
53. 33. 46,0
Ceres
147. 7. 31,5 + 121,3
Pallas. 121. 7. 4,3
Jupiter
11. 8. 34,6 + 57,1
Saturn
89. 9. 29,8 + 69,5
Uranus 167. 31. 16,1 + 52,5
1820.
Recapitulation.
65
Inclination of the orbit.
Jan. 1, 1801.
Ann. var.
7. Ó. 9,1
3. 23. 28,5
+ 0,18
0,04
- 0,01
- 0,12
Mercury
Venus
Earth
Mars
Vesta
Juno
Ceres
Pallas
Jupiter
Saturn
Uranus
1820.
0,44
1. 51. 6,2
7. 8. 9,0
13. 4. 9,7
10. 37. 26,2
34. 34. 55,0
1. 18. 51,3
2. 29. 35,7
0. 46. 28,4
0,23
0,15
+ 0,03
Longitude of the node.
Jan. 1, 1801.
Ann. inc.
45. 54. 36,9
74. 54. 12,9
+ 42,3
+ 32,5
+ 26,8
+ 15,6
Mercury
Venus
Earth
Mars
Vesta
Juno
Ceres
Pallas.
Jupiter
Saturn
Uranus
1820.
+ 1,5
48. 0. 3,5
103. 13. 18,2
171. 7. 40,4
30. 41. 24,0
172. 39. 26,8
98. 26. 18,9
111. 56. 37,4
72. 59. 35,3
+ 34,3
+ 30,7
+ 14,2
K
66
Recapitulation.
Eccentricity.
Jan. 1, 1801.
0•205514940
0.006860740
0.016783568
0·093307000
0.089130000
0.257848000
0.078439000
0.241648000
0.048162100
Secular variation.
+ '000003866
.000062711
000041630
+000090176
+ 000004009
Mercury
Venus
Earth
Mars
Vesta
Juno.
Ceres
Pallas
Jupiter
Saturn
Uranus.
.000005830
+ *000159350
000312402
0.056150500
0.046679380
Greatest equation of the centre.
Jan. 1, 1801.
Ann. var.
+ 0,0160
- 0,2500
- 0,1718
+ 0,3700
Mercury
Venus
Earth
Mars
Vesta
Juno.
Ceres
Pallas
Jupiter .
Saturn
Uranus
1820.
23. 39. 51,0
0. 47. 15,0
1. 55. 27,3
10. 40. 50,0
rio. 13. 22,0
29. 46. 19,0
8. 59. 42,0
(27. 49. 19,0
5. 31. 13,8
6. 26. 12,0
5. 20. 57,0
+ 0,6344
- 1,2790
Recapitulation.
67
Apparent Diameter.
True
Diameter.
Volume.
Least.
Mean.
Greatest.
5,0
6,9
0.398
0.063
Mercury
Venus
12,0
61,2
9,6
16,9
0.975
0.927
Earth
1.000
1.000
Mars
3,6
6,3
18,3
0:517
0.139
30,0
36,7
45,9
10.860
1280.900
Jupiter
Saturn
16,2
9.982
995.000
Uranus
4,0
40332
80.490
Sun
31. 31,0 32. 2,9 32. 35,6 111.454 1384472.000
29. 21,9 31. 7,0 33. 31,1 0.275
0.020
Moon
Mass.
Density.
Gravity.
Sidereal
Rotation.
Light and
heat.
m
S
1:03
h
24. 5. 28
6.680
0.98
23. 21. 7
1.911
3.9326
1.00
24. 0. 0
1.000
Mercury 2027810
Venus 405871
Earth 354636
Mars 2346370
Jupiter 10705
Saturn
0:33
24. 39. 21
•431
9924
2.72
9. 55. 50
•037
1
3512
•5500
1:01
10. 29. 17
•011
Uranus
1
17918
1:1000
·003
Sun
1
1.0000
27.90
25. 12. 0
to
Moon
1
27620200
2.4185
0.16
27. 7. 43
K 2
68
Recapitulation.
m
S
Revolutions of the moon.
dh
d
Synodical. 29. 12. 44. 2,9 = 29.53058872
Anomalistic . 27. 13. 18. 37,4 = 27.55459950
Sidereal
27. 7. 43. 11,5 = 27.32166142
Tropical 27. 7. 43. 4,7 = 27032158242
Nodical
27. 5. 5. 36,0 = 27.21222222
FORMU L Æ.
Formule.
71
I. Equivalent expressions for sin x.
1
cos x . tan x
COS X
2
cotx
3 V (1 - cos x)
4
1
Ņ(1 + cotx)
tan X
5
✓(1 + tan x)
6 2 sin x . cost *
1- cos 2 x
7 / 13
2
8
2 tan x
1 + tanx
9
2
cot } x + tan 2
10
sin (30° + x) — sin (30° - x)
V3
11 2 sin? (45° + 1 x) - 1
12 1 - 2 sin? (45° - 1 x)
1. tan? (45º – 1 x)
13
1 4- tanº (45° - 1 x)
tan (45° + 1 x) – tan (45° — *x)
14
tan (45° + 1x) + tan (45° – x)
15 sin (60° + x) – sin (60° - x)
1
16
cosecant r
72
Formula
11. Equivalent expressions for cos X.
sin a
1
tan x
2 sin x
.
cot x
3 V (
v (1 - sin x)
4
1
(1 + tanº x)
5
cotx
✓(1 + cotx)
6 cos 4x
sin 12
7 1- 2 sin 2x
8 2 cos x 1
9 ✓
1 + cos 2 x
2
10
1- - tanı x
1 + tan a
cot tan 1 x
cot 1 x + tani,
1
11
12
1
1 + tan x . tan à x
2
13
tan (45° + 1 x) + cot (45° + 1 x)
14 2 cos (45° + x) cos (45° - 1 x)
15
cos (60° + x) + cos (60° - 2)
1
16
secant x
Formula.
73
III. Equivalent expressions for tan x.
sin x
1
COS X
1
2
cotx
3 dl
v loose - 1)
re
sin x
4
V(1— sin x)
N (1
cos x)
5
COS X
6
2 tan x
1
tan x
7
2 cotx
coť į x – 1
2
8
cotx
tan x
9 cotx
2 cot 2 x
1
10
cos 2 x
sin 2 x
sin 2 x
11
1 + cos 2 x
12
ſt
1 cos 2 x
1 + cos 2 x
tan (45° + 1 x) – tan (45º – 1 x)
13
2
L
174
Formula.
IV. Relative to two arcs A and B.
sin A.cos B + cos A.sin B
1 sin (A + B)
2 sin (A – B)
= sin A.cos B - cos A.sin B
3
COS (A + B)
= cos A.cos B - sin A.sin B
4
cos (A - B)
Il
cos A.cos B + sin A.sin B
5 tan (A + B)
tan A + tan B
1-tan A.tan B
6 tan (A – B)
tan A tan B
1 t tan A.tan B
7 sin (45° + B) ?
cos (45° F B)S
cos B + sin B
N 2
8
9 tan (45°+ B) =
1 + tan B
1 F tan B
10 tanº (45° + B)=
1 + sin B
1 + sin B
1 + sin B
cos B
11 tan (45° + 1B)
cos B
1 I sin B
12
sin (A + B)
sin (A - B)
tan A + tan B
tan A
tan B
cot B + cot A
cot B-cot A
13
cos (A + B)
cos (A - B)
=
cot B - tan A
cot B + tan A
cot A - tan B
cot A + tan B
14
sin A + sin B
sin Asin B
tan} (A + B)
tan į (A – B)
cotį (A + B)
tani (A - B)
15
cos B + cos A
cos B - cos A
[continued.
Formula.
75
IV. continued. Relative to two arcs A and B.
16 sin A . cos B
17
cos A . sin B
18 sin A. sin B
II
i sin (A + B) + { sin (A – B)
i sin (A + B) - sin (A - B)
cos (A – B) – 3 cos (A + B)
cos (A + B) + cos (A – B)
2 sin 1 (A + B).cos Ž (A - B)
2 cos ž (A + B).cos } (A - B)
19
cos A . cos B
20 sin A + sin B
=
21
COS A t cos B =
22
tan A + tan B =
sin (A + B)
COS A.cos B
23
cot A + cot B =
sin (A + B)
sin A.sin B
24 sin A - sin B
11
2 sin } (A - B).cos 1 (A + B)
2 sin } (A - B). sin } (A + B)
25
cos B - cos A =
26 tan A
tan A - tan B =
sin (A - B)
cos A.cos B
27 cot B - cot A
sin (A – B)
sin A.sin B
28 sin? A-sinº B2
29 cos B-cos AS
sin (A – B). sin (A + B)
30 cos? A - sin? B =
cos (A - B).cos (A + B)
sin (A - B). sin (A + B)
cos? A.cos? B
31 tan? Atan?B=
32 coť B – cot? A = sin (A – B).sin (A + B)
sin? A.sin? B
L 2
76
Formule.
V. Differences of trigonometrical lines.
1 A sin x = + 2 sin A x.cos (x + į A x)
2 ACOS X 2 sin. A x.sin (x + 1 A x)
sin A X
3 Atan x = +
COS X.cos (x + AX)
sin AX
4 A cotx
sin x.sin (x + AX)
5 A sinº x = + sin A x.sin (2 x + A x)
6 A cosa x = sin A x.sin (2 x + A x)
sin Ax.sin (2 x + A x)
7 A tan? x = +
cos x.cos (x + A x)
sin A x.sin (2 x + 1 x)
8 A cotx =
sin x.sin(20 + A x)
I
VI. Differentials of trigonometrical lines.
1 d sin x
= + d x.cos x
2 d cos x
d x.sin x
ll
3 d tan a
d x
+
cos? x
4 d cotx
d x
sin x
11
5
d sin x = + 2 dx.sin X.cos x
6 d cos x
=
2 d x.sin x.cOS X
d tan” x = +
2 d x.tan x
cos? x
2 dx.cot x
8
d cot? x =
sina 2
Formula.
77
VII. General analytical expressions for the sides and
angles of any spherical triangle.
1 cos S = cos A .sin S .sin S" + cos S'.cos S"
2 cos S = cos A' .sin S". sin S + cos S".cos S
3 cos S" = cos A".sin S .sin S + cos S .cos S'
4 cos A = cos S.sin A' .sin A" cos A' .cos A"
5 cos A' = cos S'.sin Ah.sin A - COS A".cos A
6 COS A" = cos S'i.sin A .sin A' - cos A .cos A'
7
cos S .cos A' = cot S".sin S
sin Al.cot A"
8
cos S' .cos A" = cot S .sin S'
sin A".cot A
9 cos Sh.cos A = cot S.sin S!
sin A .cot A'
10
sin A
sin S
sin A'
sin S
sin A"
sin S
12
11 sin I (S' +- S) : sin } (S' –S) :: cot? A": tan (A'- A)
cos } (S' + S) : cos } (S – S) :: cot? A": tan Į (A' + A)
13 sin (A' + A): sin }(A'-A):: tan į S": tanĮ (S'—S)
cos, (A' + A): cos (A'- A):: tan : S": tan} (S+S)
14
In these formulæ A, A', A", denote the several angles of
the triangle; and S, S, S", the sides opposite those angles
respectively. For the more convenient computation of
the formulæ No. 1-9, certain auxiliary angles are in-
troduced, which will be alluded to in the formulæ for the
solution of the several cases of oblique-angled spherical
triangles.
78
Formulæ.
VIII. Solutions of the cases of right-angled spherical
triangles.
Given. Required.
Solution.
Hypothen. ( side op. giv. ang.
1 sin x = sin h.sin a
and
side adj. giv. ang.
2 tan x = tan h.cos a
an angle (the other angle 3 cotx = cos h. tan a
cos h
4 cos x =
COSS
the other side
Hypothen.
and
{ ang. adj. giv. side
a side
ang. op. giv. side
5 cos X = tans.coth
sin s
6 sin x =
sinh
r
the hypothen.
sin s
7 sin x =
sin a
cases.
A side and
the angle the other side
opposite
8 sin x = tan s.cot a
the ambiguous
the other angle
cos a
9 sin x =
COS S
10 cotx = cos a.cots
11 tanx = tan a.sin s
12 cos x = sin a.cos s
A side and the hypothen.
the angle
the other side
adjacent
the other angle
the hypothen.
The two
sides
an angle
13 cos x = rectang. cos. of
the giv. sides
14 cotx = sin. adj. side x
cot. op. side
the hypothen.
The two
angles
15 cos x = rectang. cot. of
the given angles
cos. opp. ang
16 COS X =
sin. adj. ang.
a side
In these formulæ, x denotes the quantity sought.
a = the given angle
s = the given side
h = the hypothenuse.
Formulæ.
79
IX. Solutions of the cases of oblique-angled spherical
triangles.
GIVEN, Two sides and an angle opposite one of them.
Required, 1°. The angle opposite the other given side.
sin. side op. ang. sought x sin. giv. ang.
sin x =
sin. side oppos. given angle
Required, 2°. The angle included between the given sides.
cot a = tan. giv. ang. x cos. adj. side
cos a' x tan. side adj. giv. ang.
cos al =
tan. side op. given angle
(al + a)
r
Required, 3º. The third side.
tan a' = cos. giv. ang. x tan. adj. side
cos a' x cos. side op. giv. ang.
cos al =
cos. side adj. given angle
= (al + a")
r
In these formulæ, & denotes the quantity sought: a' and
a" are auxiliary angles introduced for the purpose of facili-
tating the computations.
The angle sought in formula 1 is, in certain cases, am-
biguous. In the formulæ 2 and 3, when the angles oppo-
site to the given sides are of the same species, we must
take the upper sign: on the contrary, the lower sign. The
whole of these formulæ therefore are, in certain cases, am-
biguous.
[continued.
80
Formula.
IX. continued. Solutions of the cases of oblique-angled
spherical triangles.
GIVEN, Two angles and a side opposite one of them.
Required, 4°. The side opposite the other given angle.
sin. ang. op. side sought x sin. giv. side
sin. ang. op. given side
sin x
Required, 5º. The side included between the given angles.
tan a' = tan. giv. side x cos. ang. adj. giv. side
sin a' x tan. ang. adj. giv. side
sin all
tan. ang. op. given side
(a + a")
X
Required, 6º. The third angle.
cot a' = cos given side x tan. adj. angle
sin a' x cos. ang. op. giv. side
cos. ang. adj. given side
(a' + a")
sin al =
X
In these formulæ, x denotes the quantity sought: a' and
a" are auxiliary angles introduced for the purpose of facili-
tating the computations.
The side sought in formula 4 is, in certain cases, am-
biguous. In the formulæ 5 and 6, when the sides oppo-
site the given angles are of the same species, we must
take the upper sign : on the contrary, the lower sign. The
whole of these formulæ therefore are, in certain cases, am-
biguous.
[continued.
Formula.
81
IX. continued. Solutions of the cases of oblique-angled
spherical triangles.
GIVEN, Two sides and the included angle.
Required, 7º. One of the other angles.
tan d' = cos given angle x tan given side
all = the base al
sin al
= tan given angle x
tan x
sin ah
In this formula, the given side is assumed to be the
side opposite the angle sought: the other known
side is called the base,
Required, 8º. The third side.
tan a' = cos given angle x tan given side
all = the base en al
COS X
= cos given side x
cos all
COS a
In this formula, either of the given sides may be as.
sumed as the base : and the other, as the given side.
In these formulæ, x denotes the quantity sought: a' and
all are auxiliary angles introduced for the purpose of facili-
tating the computations.
If the side sought in formula 8 be small, the formula
may not give the value to a sufficient degree of accuracy :
and some other mode must be adopted for obtaining the
correct value.
M
[continued.
82
Formule.
IX. continued. Solutions of the cases of oblique-angled
spherical triangles.
GIVEN, A side and the two adjacent angles.
Required, gº. One of the other sides.
cot a' = tan given angle x cos given side
al = the vertical angle en a'
tan x
= tan given side x
cos a
cos all
In this formula, the angle, opposite the side sought,
is assumed as the given angle: the other known
angle is called the vertical angle.
Required, 10°. The third angle.
cot a' = tan given angle x cos given side
all = the vertical angle al
sin all
cos x = cos given angle x
sin a
In this formula, either of the given angles may be
assumed as the vertical angle: and the other as the
given angle.
In these formulæ, x denotes the quantity sought: d' and
a" are auxiliary angles introduced for the purpose of facili-
tating the computations.
If the angle sought in formula 10 be small, the formula
may not give the value to a sufficient degree of accuracy:
and some other mode must be adopted for obtaining the
correct value.
[continued.
Formulce.
83
1
IX. continued. Solutions of the cases of oblique-angled
spherical triangles.
GIVEN, The three sides.
Required, 11°. An angle.
A +B+C –B) x sin
sin
2
sin { x =
- B) xsin (A+B+C -c)
sin (A+B+C) x sin (4+B+C - A)
sin B. sin C
cos i x =
sin B. sin C
In these formulæ, A, B, C are the three sides of the tri-
angle: and A is assumed as the side opposite to the angle
required.
GIVEN, The three angles.
Required, 12º. A side.
COS
(a +6+9) x cos(a + b +osa)
sin 1 x
sin b . sinc
la
COS
a+b+c
2
+6+(56) x cos(
cosx =
a+c
X COS
2
sin b. sinc
In these formulæ, a, b, c are the three angles of the
triangle: and a is assumed as the angle opposite to the
side required.
In these formulæ, x denotes the quantity sought. The
formula, which are resolved by the cosine, are used only
when the angle or side x is small.
1
M 2
84
Formula.
X. Trigonometrical series.
1 sin x = x
x3
2015
+
2.3 2.3.4.5
&c.
1
are
2
azog
26
COS X = 1
+
2
+ &c.
2.3.4
2.3.4.5.6
2013
3
tan x = x
+
2x 17x7
+ +
3.5 32.5.7
+ &c.
3
x3
2 25
4
1
cot x =
X
&c.
3
32.5
33.5.7
22
act
206
5 ver-sin x =
2
&c.
1
+
2
2.3.4
2.3.4.5.6
sin3 x
6
x = sin x +
1.3 sin5 x
+
2.4.5
+ &c.
2.3
cos3 x
7
Els
COS X
1.3 cos x
2.4.5
&c.
2.3
8
x = tan X – štans x + } tan” x — &c.
In the series No. 7, 7 denotes the periphery of the circle,
or 3:14159265.
Formula.
85
XI. Multiple arcs.
sin 0 = 0
sin x = sin x
sin 2x = 2 sin x.cos x
sin 3x = 2 sin x.cos 2x + sin x
sin 4x = 2 sin xocos 3x + sin 2x
&c.
&c.
&c.
cos 0 = 1
COS X COS X
= x
cos 2x 2 cos X. COS X
- 1
cos 3x = 2 cos x.cos 2x
COS X
cos 4x = 2 cos x.cos 3x
cOS 2x
&c.
&c.
&c.
tan x tan x
tan 2x
2 tan x
1
tan? x
tan 3 =
tan x + tan 2x
1 tan x. tan 2x
tan 43 =
tan x ot tan 3x
1 tan. tan 3x
&c.
&c.
&c.
86
Formula.
XII. For computing the Longitude, Right Ascension and
Declination of the Sun; any one of those quantities, to-
gether with the Obliquity of the ecliptic, being given. Also
for computing the angle of Position.
1
sin D
sin O =
sin w
2
cot O = cos w.cot R
3 sin R = cotw.tan D
4 tan A = cos w.tan O
5 sin D
= sin w.sin O
6 tan D = tan o, sin R
7
cos O
cos R =
cos D
= sin w.cos R
8 sin p
tan p
9
– tan w.cos O
O = the Longitude
R = the Right Ascension
D = the Declination: (minus when South)
w = the Obliquity of the ecliptic
p = the angle of Position
Formula.
87
XIII. For computing the Longitude and Latitude of the
Moon or a Star (the Obliquity of the ecliptic being given)
from the Right Ascension and Declination: and vice versa.
Also for computing the angle of Position.
Make tan a = sin R.cot D
sin (a + w)
1 tan L =
tan R
sin a
2
tan 7
= cot (a + w) sin L
cos (a + w)
sin D
3 sin ?
COS a
4 sin L = tan (a + w) tanl
Make cot a sin L.cot /
1
5 tan R=
cos (a + w)
tan L
COS a
6 tan D = tan (a + w) sin R
sin (a + )
7 sin D =
sin ?
sin a
8 sin R = cot (a + w) tan D
sin w.cos R
sin w.cos L
9 sin p
cos 7
cos D
L = the Longitude
1= the Latitude
R = the Right Ascension
D = the Declination: (minus when South)
w = the Obliquity of the ecliptic
p = the angle of Position
88
Formula.
XIV. For computing the Azimuth, angle of Variation and
Zenith distance of a star; the Co-latitude of the place, the
north-polar distance of the star and its hour angle at the
Pole, being given.
C
tan } (A + V) =
cos ] (YSA)
x cot IP
cos (+4)
tan_ (A – V)
sin , (SA)
x cot IP
sin { (% + A)
(A + V) - }(A - V) = V
} (A + V) + }(A - V) = A
sin A
sin 2 =
sin A
X sin P
A = the Azimuth, reckoned from the north : which must
- be subtracted from 180°, if reckoned from the south.
V = the angle of Variation
Z = the Zenith distance
P = the hour angle of the star, at the Pole
♡ = the Co-latitude of the place
A = the North-polar distance of the star
Formula.
89
XV. For computing the hour angle at the Pole: the
Latitude of the place, and the Declination and Zenith
distance of the sun, or star, being given.
sin [2+(L-
2+(! – D))x sin[2 = (Į -D]
2
sin ?P=
cos L. cos D
R + P = the sidereal time of observation
If we assume Z- 90°, we shall have the expression for
the semi-diurnal arc, equal to
cos P =
tan D. tan L
exclusive of the effect caused by parallax and refraction.
L = the Latitude of the place
D = the apparent Declination: (minus when South)
Z = the observed Zenith distance, corrected for parallax
and refraction
P = the hour angle at the Pole: plus, when the observa-
tion is made to the west of the meridian; minus,
when east. In the expression for the semi-diurnal
arc, P is negative (and consequently greater than
90°) when the declination and latitude are both on
the same side of the equator
R = the apparent Right Ascension of the sun or star;
increased by 24h if necessary
N
90
Formula.
XVI. For computing the horary angle at the Pole, and
the Zenith distance of a star when on the Prime vertical,
together with its Declination, and the Latitude of the place:
any two of those quantities being given.
1 cos P = cot L. tan D
sin Z
2 sin P =
cos D
3
cot P = cos L. cot Z
4 sin Z = sin P. cos D
sin D
5 cos Z =
sin L
6
tan Z = cos L . tan P
7 cos L = cot P. tan Z
sin D
8 sin L =
cos Z
9
cot L = cos P. cot D
In these cases, the star must be on the same side of the
equator as the observer: and its declination must not
exceed the latitude of the place.
When the declination of the star exceeds the latitude of
the place, we shall have, at the moment when the vertical
becomes a tangent to the circle of declination,
10
cos P = tan L. cot D
sin L
cos Z=
11
sin D
Z = the Zenith distance of the star
P = the horary angle of the star at the Pole
D = the Declination of the star
L = the Latitude of the place
Formula.
91
XVII. For computing the effect of atmospheric Refraction.
1
r= a.tan (Z - br)
2 r = .99918827 X c. tan Z - .001105603 X c. tanZ
In these formulæ, Z denotes the apparent Zenith dis-
tance; and r is the Refraction required: a, b, c are con-
stants, to be determined from observations.
No. 1 is Bradley's formula, who assumed a = 57" and
b = 3. Dr. Brinkley has proposed a = 56",9 and b = 3.2:
but Mr. Groombridge prefers a = 58",133 and b = 3.634.
No. 2 is Laplace's formula, reduced to its most simple
terms. In the formation of the French Tables of refraction,
c is assumed equal to 60",616; but M. Laplace has since
proposed 60",66. This latter formula will not give the
correct values for greater zenith distances than 74°: at
lower altitudes those Tables, as well as most others, are
computed from more complex formulæ.
In the computation of Tables of refraction, a mean tem-
perature and a mean pressure of the atmosphere are as-
sumed. Let ß denote the height of the barometer, 7 the
height of the thermometer (Fahr.) attached thereto, and t
the height of the thermometer in the air, which are as-
sumed in the formation of the tables. Then, for any other
height (B') of the barometer, and for any other height (7')
and (t!) of the thermometers, we have the following ex-
pression, by which the mean refraction must be multiplied,
in order to obtain the true refraction: viz.
1
B!
1
X
B 1 +.0020833 (t' - t)
1 + .0001 (I' -7)
Х
92
Formula.
XVIII. For computing the correction in time, to be ap-
plied as an Equation to the mean time of observed Equal
Altitudes of the sun.
tan L
X
8 T
Х
48h~ 30
5 (
tan D.cot 71 T -
)
300
sin 72 T
= 8. tan D T.cot 7} T
T
8. tan L.
1440 sin 7 T
1440
Make
T
1440 sin 77 T
= A
T.cot 71 T
1440
= B
x = - A.8.tan L + B.8.tan D
T = the interval of Time between the observations, ex-
pressed in hours
L = the Latitude of the place of observation
·D
= the Declination, at the time of noon, on the given
day: (minus, when South)
ô = the double daily variation in the declination, deduced
from the noon of the preceding day to the noon of
the following day: (minus, when decreasing)
x = the required correction, in seconds
Formulae.
93
2 sin Pcos L.cos D 2 sin sin(L-D),
XIX. For computing the correction for the Reduction to
the Meridian ; or the correction to be applied to the Zenith
distances, observed near the meridian, in order to obtain
the true meridional Zenith distance.
Pcos L.cos D
cot)
sin 1" sin(L-D)
4
sin in
2 sinº P
Make
А
2 sini P
-
B
sin 1"
sin 1"
cos L.cos D
x = - AX
cos L.cos D
sin(L-D
D)
x cot(L-D
sin (L-D) +Bx(
When the sun is the object observed, we must apply the
following additional correction,
E W
-8 x
n
P = the correct horary angle of the star at the Pole, as
shown by a well regulated clock; which angle will
change its sign after the meridional passage of the
star
L = the Latitude of the place
D = the Declination of the star: (minus when South)
8 = the change of declination in one minute of time:
(minus, when decreasing)
E = the sum of the horary angles observed to the East; &
W = the sum of the horary angles observed to the West
of the meridian
n = the Number of those observations
x = the required correction, in seconds
94
Formulae.
XX. For computing the correction for the Reduction to
the Solstice; or the correction to be applied to the decli-
nation observed on days near the solstice, in order to de-
termine the obliquity of the ecliptic from observation.
sin w
1 x = 2 sind R.cos D X
sin 1"
2 tan
2 X
x tan (D+)
sin 1!!
(3600) sin 1" tan w
3 X =
83
2
(3600)* sin 1" tan w
24
(1 + 3 tan’ w) 84
+
(3600) sin" 1" tan w
(1+30 tanw + 45 tan* w 8
720
In which latter equation, if we make w = 23º. 27'. 40" we
have
x = 131,6346982-0",000541699 84 +0",00000002898477
and the variation, on account of the diminution of every 1"
in the obliquity of the ecliptic, will be
sin 2"
·0000132748 x
sin 2 w
х
dR = the distance of the sun's true Right Ascension, from
the solstice, at the time of observation, converted
into degrees
8 = the distance of the sun's true longitude, at the same
time, converted also into degrees, and decimal
parts of a degree
D = the observed Declination corrected for refrac-
tion, &c.
w = the Obliquity of the ecliptic
x = the required correction, in seconds
Formulæ.
95
XXI. For computing the Angle of the Vertical; or the
angle which should be deducted from the latitude of the
place, in order to obtain the reduced latitude on a spheroid.
Make a =
20-1
2 c2
(20 – 1)
nearly
C
a
X
a?
x sin 2 L + x sin 4 L
sin 1"
sin 21
a
10
x sin 2 L
very nearly
sin 1?
XXII. For computing the horizontal Parallax of the Moon
at any given latitude of the earth, considered as a spheroid:
the horizontal parallax at the equator being given.
= - P (1
p=P(1 - a.sin’L +
– a.sinºL + x aº. sinºL.cos® L)
- a.sin L) very nearly.
The quantity, within the parenthesis, denotes the radius
of the earth at the given latitude; or the distance from the
centre of the earth to that point on the earth's surface: the
radius at the equator being considered as unity.
с
1
= the Compression of the earth; or the quantity by
which the equatorial diameter (considered as unity)
exceeds the polar
L = the Latitude of the place
x = the required Angle
P = the horizontal Parallax at the equator
p = the required horizontal Parallax at the place
96
Formula.
XXIII. For computing the Moon's parallax in Altitude.
i sin a = sin p.sin(Z + 7)
or
2
= p.sin(Z + 7) very nearly
3 tann =
sin p.sin Z
sin p.cos Z
ΟΥ"
1
4
T
sinp.sin Z sinºp.sin 2 Z sinºp.sin 3 Z
+
+
+ &c.
sin 2
sin 31
sin 1!
very nearly.
When the apparent zenith distance (as affected by paral-
lax) is known, we must make use of the formula 1 or 2.
But, when we know only the true zenith distance, we must
adopt the formula 3 or 4.
The zenith distance on the meridian is = (L - D).
p = the horizontal Parallax at the place
Z = the true Zenith distance of the moon
(Z + 7) = the apparent Zenith distance of the moon, as
affected by parallax
7 = the required Parallax
Formulæ.
97
XXIV. For computing the longitude and co-latitude of
the Zenith: or (as it is frequently termed) the longitude
and altitude of the Nonagesimal.
Make tan a = sin S.cot L
1 cos A
cos (a + w)
sin L
COS a
2
sin N = tan(a + w) cot A
sin (a +w) tans
3
tan N
sin a
4
cot A = cot (a + w) sin N
When S is between 80° and 100°, or between 260° and
280°, the equations 3 and 4 will be the most proper for
use.
S = the Sidereal Time of observation: or the Right Ascen-
sion of the meridian, converted into degrees
L = the reduced Latitude of the place of observation : de-
duced from Formula XXI.
w = the Obliquity of the ecliptic
N = the required longitude of the zenith, or of the Nona-
gesimal
A = the required co-latitude of the zenith, or Altitude of
the nonagesimal
o
98
Formula.
XXV. For computing the Moon's parallax in Longitude
and Latitude.
1º. In Longitude.
Make a =
sin p.sin A
COS a
1 tan II =
1
a.sin()-N)
a.cos() - N)
+
a.sin( ) – N) a>.sin 2( D - N)
2 II =
sin 111
sin 211
a3.sin 3( ) – N)
+
sin 311
+ &c.
2°. In Latitude.
Make cotb =
cos ( - N + 11) tan A
cos II
sin p.cos A
sin 6
c.sin (1-1)
3 tan a =
1
c.cos (6-1)
c.sin(6-2), c. sin 26-a), c. sin 316-a)
4
+
sin 1"
sin 21
sin 31
+
+ &c
D = the true longitude of the Moon
N = the longitude of the zenith, or the Nonagesimal
A = the co-lat. of the zenith, or the Altitude of the nonag.
a = the Latitude of the moon: (minus, when South)
р = the horizontal Parallax at the place, by Form. XXII.
II = the required Parallax in longitude
w = the required Parallax in latitude
Formula
99
XXVI. For computing the Moon's parallax in Right
Ascension.
sin p.cos L
Make a =
cos D
1 sin F1 = a.sin (P + II)
2
11
p.cos L
cos D
x sin (P + 11)
very nearly
3
a.sin P
tan II =
1 - a.cos P
4
II =
a.sin P
sin 11
+
a. sin 2 P a .sin 3 P
+
sin 31
+ &c.
sin 2
When the apparent hour angle (as affected by parallax)
is known, we must make use of the equation 1 or 2. But,
when we know only P, we must adopt the equation 3 or 4.
P = the true horary angle at the Pole; or the true horary
distance of the moon from the meridian
L = the reduced Latitude of the place of observation, by
Formula XXI.
D = the true Declination of the moon: (minus, when
South)
p = the horizontal Parallax at the place, by Form. XXII.
II = the required Parallax in right ascension
0 2
100
Formula.
XXVII. For computing the Moon's parallax in Declina-
tion.
Make cotb=
cos (P + II) cot L
cos II
sin p.sin L
sin
1
sin w = c.cos (D +b -a)
2
p.sin L
x cos (D + b - a very nearly
sin b
3
tana
c.sin (6 - D)
1 - c.cos (6 – D)
c.sin(b-D)/c.sin 2(6-D) + c3.sin 3(6-D)
4
=
+&c.
sin 1
sin 211
sin 31
When the apparent declination (as affected by parallax)
is known, we must make use of the equation 1 or 2. But,
when D only is known, we must adopt the equation 3 or 4.
P= the true horary angle at the Pole
L = the reduced Latitude of the place of observation, by
Formula XXI.
D = the true Declination of the moon: (minus, when
South)
p = the horizontal Parallax at the place, by Form. XXII.
fl = the Parallax in right ascension, by Form. XXVI.
= the required Parallax in declination
Formulce.
101
XXVIII. For computing the Augmentation of the Moon's
semidiameter, on account of her altitude above the horizon:
or the correction to be applied to her true semidiameter,
in order to obtain her apparent semidiameter.
!
1 x = s® (+000017767 sin A) + (-000017767 sin A je
2
x = s.sina.cot(6-x) – sinº a
2
3
x = s.sina.cot (6-D) - sinº
2
In neither of these equations will the second term ever
exceed 0",15. And in the first equation, in lieu of the
second term, we may always assume it equal to the first
term, multiplied by .008379 sin A, without the risk of an
error amounting to the tooth part of a second.
O
$ = the true Semidiameter of the moon
A = the apparent Altitude of the moon
in equation 2, the Parallax in Latitude
Lin equation 3, the Parallax in Declination
x = the correction required
(6-1) is determined by Formula XXV.
(6 - D) is determined by Formula XXVII.
102
Formula.
XXIX. For computing the apparent distance between the
centre of the Moon and of the sun, or a star, when near
her.
Make tan a =
cos į (m + s)
(MSS) x
mas
mss S
d=
COS a
M = the apparent longitude of the Moon
S = the apparent longitude of the Sun or Star
m = the apparent latitude of the Moon
s = the apparent latitude of the Sun or Star
d = the required apparent Distance of the centres
If we substitute the apparent Right Ascension and De-
clination, for the apparent Longitude and Latitude re-
spectively, the formula will still give the correct value
of d.
.
Formulæ.
103
XXX. For computing the true place of the Moon by
means of the equation of Second and Third Differences.
1st Diff:
2d Diff.
3d Diff
Series,
M
al
M"
d'
4"
8
M"
d!
4
Miv
h h(h-12), d'+d", h(h-12) (h-24)
8
M=M" + iza" +
2 (12)
2.3(12)8
Х
+
2
M', M", M", Miv = the place of the Moon (in Longitude
or Latitude, Right Ascension or De-
clination) at four successive equal in-
tervals, of 12 hours each, as shown
by the ephemeris: and taken out so
that the required place, M, may fall
between M" and M
A; A", Al" = the successive Differences between
those values
d', d" = the successive Differences between
those first differences
81 = the Difference between those second
differences
h = the number of Hours from the time,
for which M" is computed
M = the required place of the Moon
104
Formula.
XXXI. For computing the Annual Precession of the
Equinoxes in Longitude, Right Ascension and Declination.
Also for computing the mean Obliquity of the Ecliptic,
for any given year.
P=50",340499-0",0002435890 X Y
p=50",176068 +0",0002442966 x y
m= 45',99592 +0",0003086450 x y
n=20",05039 -0",0000970204 x y
h= 0",17926 -0",0005320798 X Y
w=230.28'.18",0
w+dw = 230.28'.18",0 +0",0000098423 x y
w+dw=239.28'. 18",0-0",48368 xy-0",00000272295 x y
Annual Prec. in R = m + n.sin R.tan D
Annual Prec. in D= n.cos R
P = the annual lunisolar Precession in longitude
р = the annual general Precession in longitude
w = the mean Obliquity of the ecliptic in 1750
dw = the variation of the angle of the
fixed ecliptic
dw'= the variation of the angle of the moveable ecliptic
m = (P.cos
(P.cos wr) = the constant of the ann. prec. in R
n = (P.sin o) = the constant of the ann. prec. in D
R = the Right Ascension of the star
D = the Declination of the star
y = the Years from 1750: plus after, negative before
M = the annual Motion of the equinoctial points along
the equator
Formulæ.
105
XXXII. For computing the Nutation of the Obliquity of
the Ecliptic, and of the Equinoxes in Longitude. Also for
computing the Mass of the Moon.
Aw=[+9",648 cos 88-0",09423 cos 28+0",0939 cos 2 )](1+2)
+(0",49333-1",2452 x z) sin20
AL=[-18",0377 sin8+0",21707 sin28-0",21632 sin2 )](1+z)
-(1",13645-21,8686 X 2) sin 20
Mass of the Moon
1 + z
69.2376 - 178.2918 XZ
If we make z = 0, we shall have the values as deter-
mined by M. Laplace in the Méchanique Céleste. But the
recent determinations of Dr. Brinkley warrant the assump-
tion of z =
•04125. Whence
Aw = +91,250 cos 8-0",090 cos 28+05,545 cos 20 +0",090 cos 2 )
AL=-17",298 sin 8+05,208 sin 2 86 – 1",255 sin 20-0",207 sin 2 )
1
Mass of the Moon =
79.888
8 = the mean longitude of the moon's Nole
D = the true longitude of the Moon
= the true longitude of the Sun
z = a quantity to be determined from observation
Aw = the required Nutation of the Obliquity
AL = the required Nutation of Longitude: which is found
by multiplying the first term of aw by 2 cot 2 w,
and the remaining terms by cot w, then changing the
signs of the quantities, and converting the cosines
into sines.
P
106
Formula.
XXXIII. For computing the Lunar and Solar Nutation
of a star, in Right Ascension and Declination.
Nut. R=AL(Cosw +sinw.sin R. tan D) - Aw.cos R.tan D
Nut. D=AL.sin w.cos P + Aw.sin R
N
By assuming w = 23º. 27'. 40", and the values of Aw and
AL as determined in Formula XXXII, we have
Nut. R = (155,868 +6,887 sin R.tan D) sin 8
9",250 cos R.tan D.cos 8
+ (0",191 + 0",083 sin R.tan D) sin 2 88
+0",090 cos R. tan D.cos 28
- (1",151 +05,500 sin R.tan D) sin 2 O
- 05,545 cos R.tan D.cos 2 O
- (0",190 +0",082 sin R.tan D) sin 2 )
0",090 cos R.tan D.cos 2)
Nut.D= + 95,250 sin R. cos 8 - 6",887 cos R.sin 8
-0",090 sin R. cos 2 8 + 0",083 cos R.sin 28
+0',545 sin R. cos 20-0",500 cos R.sin 20
+ 0",090 sin R. cos 2 ) - 0",082 cos R.sin 2 )
w = the Obliquity of the ecliptic
Aw = the nutation of the obliq. of Ecliptic by Form.
AL = the nutation in Longitude
XXXII. .
R = the Right Ascension of the star
D = the Declination of the star: (minus when South)
3 = the mean longitude of the moon's Node
O = the true longitude of the Sun
D = the true longitude of the Moon
Formula.
107
XXXIV. For computing the Aberration of a star in Lon-
gitude and Latitude; and in Right Ascension and Decli-
nation.
Aber.in Lon.= -A.cos(O-L).secl
Aber.in Lat. = – A. sin (O-L).sin l
Aber. in R= -A(sin R.sin o+cosw.cos.R.cos O) sec D
Aber. in D = -A(cos.R.sino-cosw.sinR.cos O) sin D
-A.sinw.cos O.cos D
If we assume A=20",36, and w= 23º. 27. 40", we have
Aber.R= -(20",36 sin R.sino +18",677 cos R.coso) sec D
Aber. D= -(20",36 cosR.sin 0-18",677 sin R.cos O) sin D
- 8",106 cos .cos D
Diurnal Aber. in
{
R = 0",309 cosp.cos P.sec D
D = 0",309 cos p.sin P.sin D
O= the true longitude of the Sun
L = the Longitude of the star
1 = the Latitude of the star: (minus when South)
R = the Right Ascension of the star
D = the Declination of the star: (minus when South)
w = the Obliquity of the ecliptic
A = the constant of Aberration: which has been usually
assumed equal to 205,255; but more recently=20",36
Q = the Latitude of the place
P= the hour angle at the Pole
P2
108
Formula.
XXXV. For computing the corrections to be applied to
the observed transit of a star on account of the error of the
Clock, and on account of the three principal errors of the
transit instrument, in Azimuth, in the Inclination of the
axis, and in Collimation; in order to obtain the correct
apparent Right Ascension.
sin (0-8)
R = (T+dt) + a.
с
cos (4-8
+
+0.
cos
cos
cosa
T= the observed Time of transit, as shown by the clock
dt = the correction for the error of the Clock : plus when
the clock is too slow
$ = the Latitude of the place
ô = the Declination of the star: plus when North, and
minus when South, for the upper culminations;
and vice versa for the lower culminations
a = the deviation of the telescope in Azimuth: plus,
when (pointing to the south) the vertical which it
describes falls to the east; and minus, when it falls
to the west: and vice versa when pointing to the
north
b = the Bias or inclination of the axis of the telescope:
plus, when the west end of the axis is too high
c = the error in Collimation : plus, when the vertical, de-
scribed by the optical axis of the telescope (pointing
to the south), falls to the east; and minus, when it
falls to the west: and vice versa when pointing to
the north
R = the apparent Right Ascension required
Formula.
109
XXXVI. For computing the value (in time) of the co-
efficients a, b, c, in the preceding Formula.
=
[(w+zo"
) – (e+e')]
e,
d
60
where w' and e' denote respectively the values of w and after
reversing the level.
c = 1 (t' -t) cos D + 1 (U—b) cos(0-D)
where t' and b' denote respectively the values of t and b, after
reversing the instrument.
By observations of a circumpolar star,
12h --(T_T) b.cos (0-D) – b'.cos (° + D) +20
+
2 cos . tan D
2 cos 0.sin D
where T and b' denote respectively the values of T and b, at
the lower culmination.
a =
a =
By observations of a high and low star,
cos .cos o
Х
cos Q.sin(8-0)
where T', R' and s' denote respectively the values of T, R,
and ò of the second star observed.
[(T –T)–(ARI–R)]
d = the value of each Division of level, in seconds of space
7 = the inclination of the level to the West
ę = the inclination of the level to the East
D = the Declination of the circumpolar star
t = the time of the Transit of the circumpolar star, deduced
from an obs. at a given side wire of the instrument.
The other quantities are the same as in Formula XXXV.
The values of a and c, when once determined, may be
afterwards re-examined and corrected by means of a well
divided meridian mark. If the level of the axis be well ad-
justed, the quantities depending on 6 and b' may be safely
neglected.
110
Formula.
XXXVII. For computing the Latitude of a place, from
observations of the altitude of a star near the pole, at any
time of the day
Make tan a = tan A.cos P
COS a
sin ( + a) = x sin A
COSA
Q = (0 + a) - a
For a fixed observatory, we have
2
0
A - A.cos P +
.sinP.tan
2
+ 4 . sinºP.cos P.tan®® +4*. sinºP.cos P
Make a =
– A.cos P
B = + 42. sinº P.tan p.] sin 1"
y = + ( cot ¢ + tan ) sin 1"
Q = A + 0 + B + aby
As the latitude is always supposed to be very nearly
known in a fixed observatory, this series will be found very
convenient; in as much as not only y, but also the multi-
plier (tan p. sin 1"), may be considered as constant quan-
tities.
A = the apparent north polar Distance of the star
P= the correct hour angle at the Pole
A = the observed Altitude, corrected for refraction
0 = the Latitude required
Formula.
111
XXXVIII. For computing the Difference in the Heights
of two places, by means of the Barometer.
x
X=C
= [1+bg.cos 20] [1+, *_][2++(+6)a]
*[(1+r)).log p[1+6–7)3 + r * * * 2 M]
+
1
By expanding the last term, agreeably to the method
adopted by M. Biot, and assuming the numerical values for
the several quantities as below, we have
1
x = 60345.51 x log. of [ Х (]
1 +.0001 (T_T)
* []
1+·00104167(+++)] x [1 +--002695 cos 2 ° ]
at the upper
c= a Constant determined by observation; and which is
here assumed equal to 60158.53 English feet
g = the incr. of Grav. from equator to the poles, = '00539
the latitude of the Place
B = the height of the Barometer
at the lower
T= the Temperature (Fahr.) of the mercury
station
t = the Temperature (Fahr.) of the air
B! = the height of the Barometer
1 = the Temperature (Fahr.) 'of the mercury
station
t' = the Temperature (Fahr.) of the air
a = the expansion of Air for 1° of Fahr. = .0020833
m = the expansion of Mercury for 1° Fahr, = .0001001
r = Radius of the earth at ø: assumed = 20898240 feet
the Height of the lower station above the level of the
sea: assumed = 0
h = the Height of the upper station above the level of the
sea: assumed, as a probable mean, = 4000 feet
M = the Modulus of the common logarithms = 434294
x = the Difference required, in English feet
112
Formulae.
XXXIX. For computing the increase of Gravity from
the equator to the poles, and thence the Compression of
the earth, from the difference in the lengths of two iso-
chronous Pendulums, at different latitudes.
P= a + (II – w) sinº L = (1 +g.sin? L)
p = — sin
W + (II – w) sinºl = w (1+g.sin? 2)
P-P
(II -2)=
sin (L-1). sin (L +1)
II -
PP
ll
Quo
p.sin? L-P.sin/
1
5f
2
-
8
с
1-
If we assume f
we have
2
289
-
1
= '00865052
- 8
and C=
578
5 - 578 Xg
C
= the ratio of the centrifugal Force at the equator, to
the force of gravity there
L the Latitude of the place farthest from the equator
1 = the Latitude of the place nearest to the equator
P = the length of the Pendulum at the latitude L
p = the length of the Pendulum at the latitude 7
II = the length of the Pendalum at the pole
w = the length of the Pendulum at the equator
= the required increase of Gravity
= the required Compression of the earth
1-08
C
Formulæ.
113
XL. For computing the increase of Gravity from the
equator to the poles, and thence the Compression of the
earth, from the difference in the number of Vibrations
made in equal times, by an invariable pendulum, at dif-
ferent latitudes.
2 n
1
Х
N
sin (L-1).sin (L + 2)
+ g2.sin??
N
n =
xg.sin (L -1). sin (L+1) (1-g.sin? 2)
2
N = the Number of vibrations in a day, made by the in-
variable pendulum, at a given latitude 1
n = the additional Number of vibrations in a day, made
by the same pendulum, and in the same time, at
any other latitude L, greater than
g = the required increase of Gravity. The last term in
this equation will never exceed .00003
The other quantities are the same as in Formula XXXIX.
and the Compression is found in a similar manner.
In all these cases it is presumed that the vibrations of the
pendulum are made at a given temperature, and at a given
height of the barometer; that the arc described by the
pendulum is infinitely small: and that the experiments are
made in vacuo and at the level of the sea. Or that they
are reduced to these several standards, by certain known
methods of reduction, which will be explained in the next
Formula.
O
114
Formulæ.
1
XLI. For computing the corrections to be applied to the
number of vibrations of an invariable pendulum, on account
of the amplitude of the Arc, the Rate of the clock, the Ex-
pansion of the pendulum, the pressure of the Atmosphere,
and the Height of the place above the level of the sea.
sin (A+a). sin (A-a)
for Arc = + N.
32 M.(log sin A-log sin a)
for Rate = + N.
86400" +r
for Expan. = + Nože (t-)
1
for Atm. = + N.-
1
G
Х
ala
Х
-1)
1
Х
1 + .002083 (t' - t)
1
1+.0001 (7'-t)
h
for Height = + N.
R
xx
N = the Numb, of vibrations in 24h, as shown by the clock
A = the semi-Arc of vibration at the commencement
a = the semi-Arc of vibration at the end
M = the logarithmic Modulus = 2.302585093
go = the Rate of the clock : minus, if losing
B = the height of the Barometer assumed as standards
t = the Temp. (Fahr.) of the air ) for the specific gravity
0 = the Temp. (Fahr.) assumed as a standard for the pend.
B' = the height of the Barometer
t' = the Temperature of the air during the vibrations
Il = the Temp. of the mercury
G = the spec. Grav. of the pend. compared with water
g = the specific Grav. of the air / considered as unity
h = the Height of the place, above the level of the sea
R = the Radius of the earth, at the latitude of the place
x = a quantity determined from theory: assumed by
Dr. Young, from ·50 to •75
Formula.
115
XLII. For computing the principal geodetical quantities,
depending on the spheroidical figure of the earth, at any
given latitude.
Ellipticity of the earth.
=144296) +=(2, )?
a
Normal, ending at minor axis
n=
(1-e.sinºL)
Ņ=n(1-e)
Normal, ending at major axis
Tangent, ending at minor axis
t=n.cot L
Tangent, ending at major axis
T=n.tan L.(1-e®)
Radius of the parallel .
s=n.cos L
Radius of curvature of merid.
R= (1-2)
r=n(1-e).(1-e. sin’L)
Radius of the earth
Length of arc of meridian
dM=n.(1-e)dL =N.NL
Do. perpendicular to do. .
dP=n.dL
a = the semi-axis major of the earth
b = the semi-axis minor of the earth
1
= the Compression of the earth = 1
L = the given Latitude
6
C с
a
22
116
Formule
XLIII. For computing the length of a degree of Longi-
tude and Latitude at any point on the surface of the Earth,
considered as an ellipsoid; the length of a degree at the
equator being considered as unity, and the Compression of
the earth being given.
2
Assume e =
e
= {(1-2)
ll
alo
nearly
1
e2
= je + e + &c.
nearly
C
2
Deg. of Long. = cos L(1+je sin’L+ e* sin* L+&c.)
Deg. of Lat. =1+ eº sinºL+16 e* sinL+&c.
с
e = the Eccentricity of the ellipse forming the ellipsoid
1
= the Compression of the earth
L = the Latitude of the given point, which is assumed to
be equidistant from the ends of the degree of Lati-
tude required
Formula.
117
XLIV. For computing the Eccentricity and Compression
of the Earth, from the lengths of two measured arcs of the
meridian, differing from each other in latitude.
e?
di-DÍ
dsin? I-D sin? L
nearly
2
dź sin? I - Di sin? L
c=2x
di - DÍ
nearly
2
1
1 = the degree of Latitude nearest to the equator
L= the degree of Latitude farthest from the equator
d = the measure of a Degree, of which I is the middle point
D = the measure of a Degree, of which L is the middle point
e = the required Eccentricity of the ellipse, forming the
ellipsoid
= the required Compression of the earth
1
ܘܐ
In these formulæ, the values of L and I denote respec-
tively the latitude of the middle points of the degrees in
question.
118
Formula.
XLV. For computing the Equations depending on the
theory of the elliptical motion of the planets.
a = 72 +e?
m = x+ sin
x
a
- e
2
tanıt =
tan
a te
a
sin x
19
= te.COS X
=b.
sint
a F e. cost
a = the semi-axis major of the orbit; generally assumed
equal to unity: in which case, b, e and r must be
taken proportional thereto
6 = the semi-axis minor of the orbit
e = the Eccentricity of the orbit
p = the Radius vector of the planet
m = the Mean anomaly
t = the True anomaly
reckoned from the perihelion
x = the Eccentric anomaly (
mt). The determination
of this quantity, from which the others are deduced,
is one of the most difficult in the science of astro-
nomy: and can only be obtained by approximation.
It goes by the name of Kepler's Problem
}re
Formula
119
XLVI. For computing the greatest Equation of the centre,
the Eccentricity being given : and vice versa. Also for
computing, at that time, the eccentric anomaly; and thence,
the true and the mean anomaly.
11
E=2e+
24.3
8 +
599
210.5
es
17219
-+
e? * &c.
216.7
e=ŽE
11
24.3
E3
587
216.3.5
ES-
40583
EP- &c.
2-3.5.7.9
cosx = -4 -48e-4.8 xe - 18e" –&c.
1 --COS X
tan” x =
1 +COS X
tant
= 1
1-e
tan 1 x
1 te
m = r te.sin x
E = the greatest Equation of the centre, = m-t
e = the Eccentricity of the orbit, the semi-axis major
being considered as unity
x = the Eccentric anomaly
m-= the Mean anomaly
t = the True anomaly
TABLES.
R
TABLE I.
123
Latitude and Longitude of various places, where astrono-
mical observations have been made.
Places.
Latitude.
Longitude.
m
S
Abo
Obs.
h
1 29 10
+ 60 24 ó
+ 31 13 5
1 59 41
Obs.
0 39 50
+ 53 32 51
+ 52 22 17
0 19 33
Alexandria
Altona
Amsterdam
Archangel
Bagdad
Barcelona
Berlin
+ 64 34 0
2 42 52
+ 33 19 40
+41 21 44
1 1 1 + 1 + 1
2 57 39
+ 0 8 40
Obs.
0 53 29
+ 52 31 45
+ 44 50 14
Bordeaux .
Breslau
Brussels
+0 2 16
1 8 9
0 17 29
+ 51 6 30
+ 50 50 59
+47 29 44
+ 51 37 44
1 16 10
Buda
Bushey Heath
Cadiz
Obs.
+ 36 32 0
+0 1 21
+ ( 25 9
2 5 13
Cairo
+ 30 3 21
+ 22 34 15
5 53 44
to 52 12 43
0 0 30
33 55 42
1 13 32
+ 40 12 30
to 0 33 38
+ 41 1 27
1 55 41
0 50 20
Calcutta
Cambridge
Obs.
Cape of Good Hope . Obs. .
Coimbra
Constantinople
Copenhagen
Dantzig
Dijon
Dorpat
Obs,
Dublin
Obs.
Edinburgh
Obs.
Florence
Geneva
Obs.
1 14 31
0 20 8
1 46 48
+ 55 41 4
+ 54 20 48
+ 47 19 25
+ 58 22 47
+ 53 23 13
+ 55 56 42
+ 43 46 41
+ 46 12 0
+ 44 25 0
+ 50 56 8
+ 51 31 50
+ 5l 28 40
+ 51 28 37
+ 54 42 12
+ 0 25 22
+ 0 12 49
045 3
0 24 38
Genoa
0 35 52
Gotha
Obs.
0 42 56
Obs.
0 39 46
Obs.
0 0 0
Gottingen
Greenwich
Kew
Königsberg
Obs.
+0 1 3
1 21 57
Obs.
124
TABLE I. continued.
Places.
.
Latitude.
Longitude.
m
S
Obs.
Lilienthal
Leghorn
Lisbon
London (St. Paul's)
Madras (Flagstaff)
Madrid
Manheim
Marseilles
Mexico
h
. 035 37
041 7
+ 0 36 16
+ 0 0 23
5 21 28
1 + + I + II +
+ 0 15 9
033 52
+ 53 6 30
* 43 33 5
+ 38 42 24
+ 51 30 49
+ 13 5 0
+ 40 24 57
+ 49 29 18
+ 43 17 49
+ 19 25 45
+ 45 28 2
+ 44 055
+ 40 50 15
+ 51 45 39
+ 38 6 44
Obs.
Obs.
021 29
+ 6 36 21
Milan
Obs.
0 36 46
- 0 5 23
Obs.
0 57 3
Obs.
+ 0 5 1
Montauban
Naples.
Oxford
Palermo
Paramatta
Paris
Obs.
0 53 28
Obs.
33 48 45
-10 4 5
Obs.
0921
- 7 45 51
Pekin
Obs.
2 1 15
+ 5 046
0 57 41
+ 4 44 39
+ 5 15 0
0 49 59
Obs.
+0 2 24
Petersburg
Philadelphia
Prague
Quebec
Quito
Rome
Slough
Stockholm
Toulouse
Tubingen.
Turin
Uraniburg
Verona
Vienna
Viviers
Wilna.
+ 48 50 14
+ 39 54 13
+ 59 56 23
+ 39 56 55
+ 50 5 19
+ 46 47 30
0 13 17
to 41 53 54
+ 51 30 20
+ 59 20 31
+ 43 35 46
+ 48 31 10
+ 45 4 0
+ 55 54 38
+ 45 26 7
+ 48 12 40
+ 44 29 14
+ 54 41 2
1 12 14
- 0 5 46
Obs.
- 0 36 14
0 30 41
Obs.
0 50 52
044 5
Obs.
Obs.
1 5 31
Obs.
( 18 44
- 1 41 12
In the column of Latitudes the sign plus denotes North ; and the sign
minus, South. In the column of Longitudes the sign plus denotes West;
and the sign minus, East, from Greenwich,
TABLE II.
i25
Mean Right Ascension of the Sun.
January
82
February
88
March.
88
h
S
h
m
S
h
m
S
1
0
5
9
2
0
5
9
3
0
5
9
4
0
5
9
5
1
5
9
18 40 0,000
' "
43 56,555
47 53,111
51 49,666
55 46,221
59 42,777
19 3 39,332
7 35,887
il 32,443
15 28,998
20 42 13,215
46 9,771
50 6,326
54 2,881
57 59,437
21 1 55,992
5 52,547
9 49,103
13 45,658
17 42,213
22 32 36,765
36 33,320
40 29,875
44 26,431
48 22,986
52 19,541
56 16,097
23 0 12,652
4 9,207
8 5,763
6
1
5
9
1
5
10
7
8
1
6
10
9
1
6
10
10
1
6
10
11
1
6
12 2,318
10
12
2
6
15 58,873 10
13
2
6
10
14
2
6
11
15
2
11
19 25,553
23 22,109
27 18,664
31 15,219
35 11,775
39 8,330
43 4,885
47 1,441
50 57,996
54 54,551
21 38,769
25 35,324
29 31,879
33 28,435
37 24,990
41 21,545
45 18,101
49 14,656
53 11,211
57 7,767
16
2
11
19 55,429
23 51,984
27 48,539
31 45,095
35 41,650
39 38,205
43 34,761
47 31,316
17
7
7
7
7
7
2
11
18
2
11
19
3
11
20
3
7
11
3
7
12
3
8
12
3
8
12
3
8
12
22 1 4,322
5 0,877
8 57,433
12 53,988
16 50,543
20 47,099
24 43,654
28 40,209
4
8
21 58 51,107
22 20 2 47,662
23 6 44,217
24 10 40,773
25 14 37,328
26 18 33,883
27 22 30,439
28 26 26,994
29 30 23,549
30 34 20,105
31 38 16,660
12
4
8
51 27,871
55 24,427
59 20,982
0 3 17,537
7 14,093
11 10,648
15 7,203
19 3,759
23 0,314
26 56,869
30 53,425
12
4
8
12
4
9
13
4
13
4
13
4
13
In Leap years, enter the table, in January and February, with the
given date, minus unity.
126
TABLE II. continued.
April.
83
May.
3
June,
8
m
S
h
m
S
h
m S
1
18
21
2
18
21
3
18
21
4
18
22
5
18
22
6
0 34 49,980 13
38 46,535 13
42 43,091 · 14
46 39,646 14
50 36,201 14
54 32,757 14
58 29,312 14
1 2 25,867 14
6 22,422 14
10 19,978 15
2 33 6,640
37 3,195
40 59,750
44 56,306
48 52,861
52 49,416
56 45,972
3 0 42,527
4 39,082
8 35,638
4 35 19,855
39 16,410
43 12,966
47 9,521
51 6,076
55 2,632
58 59,187
5 2 55,742
6 52,298
10 48,853
6
18
22
7
18
22
8
19
22
9
19
22
10
19
22
11
15
19
23
12
15
19
23
13
15
19
23
14
15
19
23
15
15
20
23
14 15,533
18 12,088
22 8,644
26 5,199
30 1,754
33 58,310
37 54,865
41 51,420
45 47,976
49 44,531
12 32,193
16 28,748
20 25,304
24 21,859
28 18,414
32 14,970
36 11,525
40 8,080
44 4,636
48 1,191
14 45,408
18 41,964
22 38,519
26 35,074
30 31,630
34 28,185
38 24,740
42 21,296
46 17,851
50 14,406
16
15
20
23
16
20
23
17
18
16
20
24
19
16
20
24
20
16
20
24
21
16
20
24
22
16
21
24
23
16
21
24
24
17
21
24
25
17
21
25
53 41,086
57 37,642
2 1 34,197
5 30,752
9 27,308
13 23,863
17 20,418
21 16,974
25 13,529
29 10,084
54 10,962
58 7,517
6 2 4,072
6 0,628
9 57,183
13 53,738
17 50,294
21 46,849
25 43,404
29 39,960
26
51 57,746
55 54,302
59 50,857
4 3 47,412
7 43,968
11 40,523
15 37,078
19 33,634
23 30,189
27 26,744
31 23,300
17
21
25
27
17
21
25
28
17
21
25
29
17
21
25
30
17
21
25
31
21
TABLE II. continued.
127
July.
co
August.
co
September.
of
h
m
S
h
m S
m
s
1
25
30
35
2
26
30
59,501 35
3
26
30
35
4
26
30
35
5
26
31
35
6 33 36,515
37 33,070
41 29,626
45 26,181
49 22,736
53 19,292
57 15,847
7 1 12,402
5 8,958
9 5,513
8 35 49,731
39 46,286
43 42,841
47 39,397
51 35,952
55 32,507
59 29,063
9 3 25,618
7 22,173
il 18,729
ha
10 38 2,946
41 59,501
45 56,057
49 52,612
53 49,167
57 45,723
]] 1 42,278
5 38,833
9 35,389
13 31,944
6
26
31
35
7
26
31
35
8
31
36
9
27
27
27
31
37
37
10
31
11
32
37
12
32
37
27
27
27
27
13
32
37
38
14
32
15
28
32
38
13 2,068
16 58,624
20 55,179
24 51,734
28 48,290
32 44,845
36 41,400
40 37,956
44 34,511
48 31,067
15 15,284
19 11,839
23 8,395
27 4,950
31 1,505
34 58,061
38 54,616
42 51,171
46 47,727
50 44,282
17 28,499
21 25,055
25 21,610
29 18,165
33 14,721
37 11,276
41 7,831
45 4,387
49 0,942
52 57,497
16
28
32
38
17
28
32
38
18
28
33
38
19
28
33
38
20
28
33
38
21
28
33
39
22
29
33
39
23
22
33
39
24
29
33
39
25
29
34
39
52 27,622
56 24,177
8 0 20,733
4 17,288
8 13,843
12 10,399
16 6,954
20 3,509
24 0,065
27 56,620
31 53,175
56 54,053
12 0 50,608
4 47,163
8 43,719
12 40,274
16 36,829
20 33,385
24 29,940
28 26,495
32 23,051
29
54 40,837
58 37,393
10 2 33,948
6 30,503
10 27,059
14 23,614
18 20,169
22 16,725
26 13,280
30 9,835
34 6,391
26
27
34
39
29
34
39
28
29
34
40
29
30
34
40
30
30
34
40
31
30
34
128
TABLE II. continued.
October.
82
November.
83
December.
88
h
m
S
h
m
S
h
m S
1
40
45
49
2
40
45
49
3
40
45
49
4
40
45
49
5
41
45
50
12 36 19,606
40 16,161
44 12,717
48 9,272
52 5,827
56 2,383
59 58,938
13 3 55,493
17 52,049
11 48,604
14 38 32,821
42 29,377
46 25,932
50 22,487
54 19,043
58 15,598
15 2 12,153
6 8,709
10 5,264
14 1,819
16 36 49,481
40 46,037
44 42,592
48 39,147
52 35,703
56 32,258
17 0 28,813
4 25,369
8 21,924
12 18,479
6
41
45
50
7
41
45
50
50
8
41
46
9
41
46
50
10
41
46
50
11
41
46
50
12
42
46
51
13
42
46
51
14
42
46
51
15
42
47
51
15 45,159
19 41,715
23 38,270
27 34,825
31 31,381
35 27,936
39 24,491
43 21,047
47 17,602
51 14,157
17 58,375
2) 54,930
25 51,485
29 48,041
33 44,596
37 41,151
41 37,707
45 34,262
49 30,817
53 27,373
16 15,035
20 11,590
24 8,145
28 4,701
32 1,256
35 57,811
39 54,367
43 50,922
47 47,477
51 44,033
42
47
51
16
17
42
51
18
43
51
47
47
47
47
19
43
52
20
43
52
21
43
48
52
22
43
48
52
23
43
48
52
24
43
48
52
25
44
48
53
55 10,713
59 7,268
14 3 3,823
7 0,379
10 56,934
14 53,489
18 50,045
22 46,600
26 43,155
30 39,711
34 36,266
57 23,928
16 1 20,483
5 17,039
9 13,594
13 10,149
17 6,705
21 3,260
24 59,815
28 56,371
32 52,926
26
44
48
55 40,588
59 37,143
18 3 33,699
7 30,254
11 26,809
15 23,365
19 19,920
23 16,475
27 13,030
31 9,586
35 6,141
53
27
44
48
53
28
44
49
53
29
44
49
53
30
44
49
53
31
44
il
53
}
TABLE III.
129
Corrections to be added to the values in Table II.
Year. Correction 8 Year. Correction 8 Year. Correction 8
m
S
m
S
m
S
C 1800 3 33,993 92 1834 2 38,004 265 |B 1868 5 38,569 438
1801 2 36,686 39 1835 1 40,697 21.2 1869 4 41,262 385
1802 1 39,380 985 B 1836 4 39,945 158 1870 3 43,956 331
1803 0 42,073 931 1837 3 42,638 104 1871 2 46,649 277
B 1804 3 41,321 877 1838 2 45,332 | 50 B 1872 5 45,897 224
1805 2 44,014 824 1839 1 48,025 | 997 1873 4 48,590 170
1806 1 46,708 770 B 1840 4 47,273 | 943 1874 3 51,284 | 116
1807 0 49,401 716 1841 3 49,966 890 1875 2 53,977 63
B 1808 3 48,649 662 1842 2 52,660 836 B 1876 5 53,225 9
1809 2 51,342 609 1843 1 55,353 782 1877| 4 55,918 955
1810 1 54,036 555 B 1844 4 54,601 728 1878 3 58,612 | 901
1811 0 56,729 501 1845 3 57,294 675 1879 3 1,305 848
B 1812 3 55,977 448 1846 2 59,988 621 ||B 1880 6 0,553 794
1813 2 58,670 394 1847 2 2,681 567 1881 5 3,246 740
1814 2 1,364 340 B 1848 5 1,929 | 513 1882 4 5,940 687
1815 1 4,057 287 1849 4 4,622 | 460 1883 3 8,633 633
B 1816 4 3,305 233 1850 3 7,316 406 B 1884 6 7,881 579
1817 3 5,998 179 1851 2 10,009 (352 1885 5 10,574 526
1818 2 8,692 125 |B 1852 5 9,257 | 299 1886| 4 13,268 | 472
1819 1 11,385 72 1853 4 11,950 245 1887 3 15,961 418
B 1820 4 10,633 18 1854 3 14,644 191 B 1888 6 15,209 | 364
1821 3 13,326 963 1855 2 17,337 | 138 1889 5 17,902 311
1822 2 16,020 91Q||B 1856 5 16,585 84 1890 4 20,596 257
1823 1 18,713 856 1857 4 19,278 30 1891 3 23,289 202
B 1824 4 17,961 802 1858 3 21,972 976 B 1892 6 22,537 | 149
1825 3 20,654 749 1859 2 24,665 923 1893 5 25,230 95
1826 2 23,348 | 695 ||B 1860 5 23,913 | 869 1894 4 27,924 41
1827 1 26,041 641 1861| 4 26,606 814 1895 3 30,617 988
B 1828] 4 25,289 587 1862 3 29,300 761 B 1896 6 29,865 | 934
1829 3 27,982 534 1863 2 31,993 707 1897 5 32,558 880
1830 2 30,676 480 B 1864 5 31,241 653 1898 4 35,252 826
1831 1 33,369 426 1865 4 33,934 600 1899 3 37,945 773
B 1832 4 32,617 | 373 1866 3 36,628 546 C 1900 2 40,638 | 719
1833 3 35,310 319 1867 2 34,321 492
S
130
TABLE IV.
Correction for the Lunar Nutation.
Argument = The mean place of the Moon's node.
8
Equation.
86
500
0
0,000 +
500
1000
490
10
510
990
480
20
520
980
470
30
530
970
960
460
40
540
50
550
450
440
950
940
60
560
430
70
930
570
580
420
80
920
410
90
590
910
400
100
600
900
390
110
610
890
380
120
620
880
130
0,065
0,129
0,193
0,257
0,319
0,381
0,441
0,499
0,555
0,610
0,662
0,711
0,758
0,803
0,844
0,882
0,916
0,947
0,975
0,999
1,019
1,034
1,046
1,054
- 1,056 +
630
370
360
870
140
860
640
650
350
150
850
340
160
660
840
330
170
830
320
180
820
310
190
810
300
200
800
290
210
670
680
690
700
710
720
730
740
750
280
220
270
230
790
780
770
760
750
260
240
250
250
TABLE V.
131
Correction for the Solar Nutation.
Argument = Sun's true longitude.
Equation.
O
S
270
180
90
+
90
267
183
87
3
93
180 270 360
177 273 357
174 276 354
171 279 351
264
186
84
6
96
261
189
81
9
99
258
192
102
168
282
348
255
195
78 12
75 | 15
72 18
105
165
285
345
252
198
108
162
288
342
249
201
69 | 21
111
159
291
339
0,000 t
0,008
0,016
0,024
0,031
0,038
0,045
0,051
0,057
0,062
0,066
0,070
0,073
0,075
0,076
0,077 +
246
204
66 24
114
156
294
336
243
207
117
153
333
63 | 27
60 30
297
300
240
210
120
150
330
237
213
57
33
] 23
147
303
327
234
216
54
36
126
144
306
324
231
219
51
39
129
141
309
321
228
222
48
42
132
138
312
318
225
225
45
45
135
135
315
315
S2
132
TABLE VI.
For converting sidereal into mean solar time.
Hours.
Minutes.
Seconds.
m
S
1
2
3
4
5
0 9,830
0 19,659
0 29,489
0 39,318
049,148
0 58,977
1 8,807
1 18,636
1 28,466
1 38,296
10,164 31
0,164 || 31 5,079
16. 10,003 | 31 0,085
2 | 0,328 | 32 | 5,242 2 0,005 32 0,087
3 0,491 33 5,406 3 0,009 | 33 | 0,090
4 | 0,655 34 | 5,570 4 | 0,011 34 | 0,093
5 0,819 | 35 5,734 5 0,014
0,014 | 35 0,096
6 0,983 36 5,898 6 0,016 || 36 0,098
7 | 1,147 || 37 0,062 70,019 || 37 | 0,101
8 1,311 38 6,225 80,022 || 38 0,104
9 | 1,474 | 396;389 9 0,025 39 0,106
10 | 1,638 | 40 | 6,553 | 100,027 || 40 0,109
6
7
8
9
10
11
12
13
14
15
1 48,125
1 57,955
2 7,784
2 17,614
2 27,443
2 37,273
2 47,103
2 56,932
3 6,762
3 16,591
11 | 1,802 || 41 | 6,717 | 11 0,030 | 41 0,112
12 1,966 - 42 6,881 12
| | 0,033
0,033 || 42 | 0,115
132,130 43 7,044 | 13 | 0,036
0,036 43 0,118
14 2,294 || 44 | 7,208 | 14 | 0,038 44 0,120
15 | 2,457 | 45 | 7,372 | 15 | 0,041 45 0,123
16 2,621 | 46 7,536 16 0,044 46 0,126
17 | 2,785 || 47 | 7,700 17 0,047 | 47 | 0,128
18 2,949 48 7,864 18 0,049 | 48 0,131
193,113 49 | 8,027 19 0,052 | 49 | 0,134
20 3,277 || 50 8,191 20
50 8,191 | 20 | 0,055 | 50 0,137
16
17
18
19
20
21
22
3 26,421
3 36,250
3 46,080
3 55,909
23
24
21 3,440 51 | 8,355 | 21 0,057 | 51 | 0,140
22 3,604 52 8,519 22 0,060 || 52 | 0,142
23 | 3,768 | 53 | 8,683 23
53 8,683 23 | 0,063 || 530,145
24 3,932 | 548,847 | 24
54 0,066 54 0,148
25 | 4,096 | 55 | 9,01025 0,068 55 | 0,150
26 | 4,259 56 9,174 26 0,071 56 | 0,153
27 4,423 57 9,338 27 0,074 || 57 | 0,156
28 | 4,587 || 58 9,502 | 28 | 0,076 | 58 | 0,159
29 4,751 59 | 9,666 | 290,079 59 | 0,161
30 | 4,915 || 60 9,830 30 0,082 60 0,164
TABLE VII.
133
For converting mean solar into sidereal time.
Hours.
Minutes.
Seconds.
m
S
$
1
0.003
2
3
4
5
0 9,856
( 19,713
0 29,569
0 39,426
0 49,282
0 59,139
1 8,995
1 18,852
1 28,708
1 38,565
10,164 31 5,092 1 0,003 31 0,085
20,329 32 5,257 2 0,005 || 32 0,087
3 0,493 33 5,421 3 0,008 33 0,090
4 0,657 || 34 5,585 4 | 0,011 34 0,093
5 0,821 | 35 5,750 5 0,014 | 35 | 0,096
6 0,986 | 36 5,914 6 0,016 | 36 | 0,098
7 | 1,150 | 37 6,078 7 0,019 37 0,101
8 1,314 | 386,242 8 0,022 || 38 0,104
9 1,478 39 6,407 9 0,025 || 39 0,106
10 | 1,643 406,571 100,027 40 0,109
6
7
8
9
10
11
12
13
14
15
1 48,421
11 | 1,807 | 41
41 6,73511 | 0,030 | 41 0,112
1 58,278 12 | 1,971
426,900 12 0,033 42 0,115
2 8,134 132, 136 43 7,064 13 0,036 43 0,118
2 17,991 14 2,300 44 7,228 | 14 0,038 44 0,120
2 27,847 15 2,464 45 7,392 | 15 0,041 || 45
45 | 0,123
2 37,704 16 2,628 | 46 7,557 16 0,044 || 46 0,126
2 47,560 17 2,793 47 7,721 17 0,047 47 0,128
2 57,416 18 | 2,957 48 7,885 18 0,049 480,131
3 7,273 | 19
19 3,121 49 8,050 19 0,052 || 49 0,134
3 17,129 | 20
20 3,285 50 8,214 20 0,055 50 | 0,137
16
17
18
19
20
21
22
23
24
3 3 26,986 | 21 3,450 51 8,378 | 21 0,057 | 51 0,140
3 36,842 223,614 | 528,542 22 0,060 52 0,142
3 46,699 | 233,778
23 3,778 53 8,707 23 0,063 53 0,145
3 56,555 24 3,943 | 54 8,871 24 0,066 | 54 0,148
25 | 4,107 55 9,035 25 0,068 | 55 0,150
26 4,271 56 9,199 26 0,071 || 56 0,153
27 4,436 | 57 9,364 27 0,074 | 57 | 0,156
28 4,600 58 9,528 28 0,076 58 0,159
29 4,764 59 9,692 | 29 0,079 ) 59 0,161
30 4,928 | 60 9,856 30 0,082 | 60 0,164
134
TABLE VIII.
For converting degrees into time : and vice versa.
Degrees.
Space. Time. Space. Time.
Time. Space. T'ime.
Space. Time.
m
m
rado
h
0 4
31°
h
2 4
61°
h m
4 4
9i
h m
6 4
2
08
32
2 8
62
4 8
92
6 8
3
0 12
33
2 12
63
4 12
93
6 12
4
0 16
34
2 16
64
4 16
94
6 16
5
0 20
35
2 20
65
4 20
95
6 20
6
0 24
36
2 24
66
4 24
96
6 24
7
0 28
37
2 28
4 28
97
6 28
67
68
8
0 32
38
2 32
4 32
98
6 32
9
0 36
39
2 36
69
4 36
99
6 36
10
040
40
2 40
70
4 40
100
6 40
11
044
41
2 44
4 44
101
6 44
12
0 48
42
2 48
4 48
102
648
13
0 52
43
2 52
4 52
103
6 52
14
0 56
44
2 56
4 56
104
6 56
15
1 0
45
3 0
5 0
105
71
72
73
74
75
76
77
78
79
80
1 4
46
3 4
5 4
106
16
17
7 0
7 4
7 8
1 8
47
3 8
107
5 8
5 12
18
1 12
48
3 12
108
7 12
19
1 16
49
3 16
5 16
109
7 16
20
1 20
50
3 20
5 20
110
7 20
21
1 24
51
3 24
81
5 24
111
7 24
22
1 28
52
3 28
82
5 28
112
7 28
7 32
23
1 32
53
3 32
83
5 32
113
24
1 36
54
3 36
84
5 36
114
7 36
25
1 40
55
3 40
85
5 40
115
26
1 44
56
3 44
86
5 44
116
1 48
57
3 48
87
5 48
27
28
7 40
7 44
7 48
7 52
7 56
8 0
1 52
58
3 52
88
117
118
119
5 52
29
1 56
59
3 56
89
5 56
30
2 0
60
4 0
90
6 0
120
TABLE VIII. continued.
135
Degrees.
Space. Time. Space. Time. Space. Time. Space. Time.
. || .
h
m
m
m
m
121
8 4151
h
10 4
181
h
12 4
211
h
14 4
122
8 8
152
10 8
192
12 8
212
14 8
123
8 12
153
10 12
183
12 12
213
14 12
124
8 16
154
10 16
184
12 16
214
14 16
125
8 20
155
10 20
185
12 20
14 20
14 24
126
8 24
156
10 24
186
12 24
127
8 28
157
10 28
187
12 28
216
218 14
217
14 28
128
8 32
158
10 32
188
12 32
14 32
129
8 36
159
10 36
189
12 36
219
14 36
130
8 40
160
10 40
190
12 40
220
14 40
131
8 44
161
10 44
191
12 44
221
14 44
132
8 48
162
10 48
192
12 48
222
14 48
133
8 52
163
10 52
193
12 52
223
14 52
134
8 56
164
10 56
194
12 56
224
14 56
135
9 0
165
11
0
195
13 0
225
15 0
136
94
166
11 4
196
13 4
226
15 4
137
98
167
11 8
197
13 8
227
15 8
138
9 12
168
11 12
198
13 12
15 12
228
229
139
9 16
169
11 16
199
13 16
15 16
140
9 20
170
11 20
200
13 20
230
15 20
141
9 24
201
13 24
231
15 24
142
9 28
202
13 28
232
15 28
143
9 32
203
13 32
233
15 32
144
9 36
171 11 24
17221 11 28
173 11 32
174 11 36
175 11 40
176 11 44
177 11 48
204
13 36
234
15 36
145
9 40
205
13 40
235
15 40
146
9 44
206
13 44
236
15 44
147
9 48
207
13 48
15 48
237
238
148
9 52
11 52
208
15 52
178
179
13 52
13 56
149
11 56
209
239
9 56
10 0
15 56
16 0
150
180
12 0
*
210
14 0
240
136
TABLE VIII. continued.
Degrees.
Space. Time. ( Space. Time. Space. Time. Space. Time.
m
h
m
241
h
m
16 4
h m
18 4
301
h
20 4
242
16 8
18 8
302
20 8
331 22 4
332 22 8
333 22 12
1
243
16 12
18 12
303
20 12
244
16 16
18 16
304
20 16
334
22 16
245
16 20
18 20
305
20 20
335
22 20
272
273
274
275
276
277
278
279
246
16 24
18 24
306
20 24
336
22 24
247
16 28
18 28
20 28
337
22 28
307
308
248
16 32
18 32
20 32
338
22 32
249
16 36
18 36
309
20 36
339
22 36
250
16 40
280
18 40
310
20 40
340
22 40
251
16 44
281
18 44
311
20 44
341
22 44
252
16 48
282
18 48
312
20 48
342
22 48
253
16 52
283
18 52
313
20 52
313
22 52
284
18 56
314
20 56
344
22 56
285
190
315
210
345
23 0
286
316
21 4
346
23 4
254 16 56
255 17 0
256 17 4
257 17.8
258 17 12
259 17 16
260 17 20
19 4.
19 8
287
317
21 8
347
23 8
288
19 12
318
21 12
348.
23 12
289
19 16
319
21 16
349
23 16
290)
19 20
320
21 20
350
23 20
261
291
19 24
321
21 24
351
23 24
17 24
17 28
262
292
19 28
322
21 28
352
23 28
263
293
19 32
323
21 32
353
23 32
264
17 32
17 36
17:40
294
19 36
324
21 36
354
23 36
265
295
19 40
325
21 40
355
23 40
266
296
19 44
326
21 44
356
23 44
267
297
19 48
21 48
357
23 48
17 44
17 48
17 52
17 56
18 0
327
328
268
298
19 52
21 52
358
23 52
269
299
19 56
329
21 56
359
23 56
270
300
20 0
330
22 0
360
24 0
TABLE VIII. continued.
137
Minutes.
Seconds.
m
S
S
m S
2 4
Space. Time. | Space. Time. Space. Time. | Space. Time.
1
0 4 31
ľ 0,067 31
2,067
2 08 32 2 8
0,133 32 2,133
3 0 12 33 2 12 3 0,200 33 2,200
4 0 16 34 2 16 4 0,267 34 2,267
5 0 20 35 2 20 5
0,333
35 2,333
6 0 24 36 2 24 6 0,400 36 2,400
7 0 28 37 2 28 7
0,467
37 2,467
8 0 32 38 2 32 8 0,533 38 2,533
9 0 36
39
2 36 9 0,600 39 2,600
10 040 40 2 40 10 0,667 40 2,667
11
044
41
2 44
11
41
12
0 48
42
2 48
12
42
13
0 52
43
2 52
13
43
14
0 56
44
2 56
14
44
15
1 0
45
3 0
15
45
0,733
0,800
0,867
0,933
1,000
1,067
1,133
1,200
1,267
1,333
2,733
2,800
2,867
2,993
3,000
3,067
3,133
3,200
3,267
3,333
16
1 4
46
3 4
16
46
17
18
47
3 8
17
47
48
18
1 12
48
3 12
18
19
1 16
49
3 16
19
49
20
1 20
50
3 20
20
50
21
1 24
51
3 24
21
51
22
1 28
3 28
22
52
52
53
23
1 32
3 32
23
53
24
1 36
54
24
54
3 36
3 40
25
1 40
55
25
55
1,400
1,467
1,533
1,600
1,667
1,733
1,800
1,867
1,933
2,000
3,400
3,467
3,533
3,600
3,667
3,733
3,800
3,867
3,933
4,000
26
1 44
56
3 44
26
56
27
1 48
57
3 48
27
57
28
1 52
58
3 52
28
58
29
59
3 56
29
59
1 56
2 0
30
60
4 0
30
60
T
- 138
TABLE IX.
Mr. Ivory's Mean Refractions; with the logarithms and
their differences annexed.
Zenith Mean
Dist. Refrac.
Log.
Diff.
Zenith Mean
Dist. Refrac.
Log.
Diff.
173
170
3012
168
164
162
1763
1252
974
796
675
587
519
160
159
156
155
466
155
424
154
11
388
152
359
151
334
152
313
151
i 6 1,02 0.0085
2 2,04 0.3097
3 3,06 0.4860
4 4,08 0.6112
5 5,11 0.7086
6 6,14 0.7882
7 7,17 0.8557
8 8,21 0.9144
9 9,25 0.9663
10 10,30 1.0129
11,35 | 1.0553
12 12,42 1.0941
13 13,49 1.1300
14 14,56 1.1634
15 15,66 1.1947
16 16,75 1.2241
17,86 1.2519
18 18,98 1.2784
19 20,11 1.3036
20 21,26 1.3277
21 22,42 1.3507
22 23,60 1.3729
23 24,80 1.3944
24 26,01 1.4151
25 27,24 1.4352
26 28,49 1.4547
27 29,76 1.4736
28 31,05 1.4921
29 32,38 1.5102
30 0 33,72 1.5279
294
31 l ó 351,09 1.5452
32
36,49 1.5622
33
37,93 1.5790
34
39,39 1.5954
35 40,89 1.6116
36 42,42 1.6276
37
44,00 1.6435
38 45,61 1.6591
39 47,27 1.6746
40
48,99 1.6901
41 50,75 1.7055
42 52,57 1.7207
43 54,43 1.7358
44 56,35 1.7510
45
58,36 1.76611
46 1 0,43 | 1.78123
47
2,57 | 1.79637
48
4,80 1.81155
49 7,11 1.82678
50
9,52 | 1.84208
51
12,02 1.85747
52 14,64) 1.87298
53
17,38 1.88863
54 20,24/ 1.90440
55
23,25 1.92036
56
26,41 | 1.93653
57
29,73) 1.95291
58 33,23 1.96955
36,93 1.98646
60
1 40,85 2.00368
1512
278
1514
17
265
1518
252
1523
241
1530
230
1539
222
1551
1565
215
207
201
1577
1596
195
1617
189
1638
185
1664
181
1691
59
:
177
1722
TABLE IX. continued.
139
Zenith Mean
Dist. Refrac.
Log.
Diff.
Zenith Mean
Dist. Refrac.
Log.
Diff.
462
1756
1794
1936
1881
467
470
475
479
483
1932
1988
2048
488
2116
493
2191
497
502
2275
388
507
10
390
512
393
517
396
398
523
528
533
402
404
538
616 i 45,01 2.02124
62 0 49,44 2.03918
63 0 54,17 | 2.05754
64 0 59,22 | 2.07635
65 0 2 4,65 2.09567
66 0 10,48 2.11555
67 0 16,78 2.13603
68 0 23,61 2.15719
69 0 31,04 2.17910
70 00
39,16 2.20185
40,59 2.20573
20 42,04 2.20963
30
43,52 2,21356
40 45,02 2.21752
50 46,53 2.22150
71 00 48,08 2.22552
10 49,65 2.22956
20
51,25 2.23363
30
52,87 | 2.23773
40 54,53 2.24186
50
56,21 | 2.24603
72 00 57,92 2.25022
10 59,66 2.25445
20 3 1,43 2.25870
30 3,23 2.26299
40 5,06 | 2.26732
50 6,93 2.27168
73 00 8,83 2.27608
10 10,77 2.28051
20 12,74 2.28498
30 14,75 2.28948
40 16,80 \ 2.29402
50 3 18,88 2.29860
78 06 3 21,01 2.30322
10 23,18 2.30789
20 25,39 2.31259
30 27,66 2.31734
40 29,95 2.32213
50 32,30 2.32696
75 00 34,70 2.33184
10 37,16 | 2.33677
20 39,65 2.34174
30 42,21 2.34676
40 44,82 2.35183
50 47,48 2.35695
76 00 50,21 2.36212
10 53,00 2.36735
20 55,85 2.37263
30 58,76 2.37796
40 4 1,74 | 2.38334
50
4,79 2.38879
77 00 7,91 | 2.39430
10 11,11 2,39987
20 14,392.40550
30 17,74 2.41119
40 21,19 | 2.41695
50 24,72, 2.42278
78 00 28,33 2.42867
10 32,04 | 2.43463
20 35,84 | 2.44066
30 39,75 | 2.44677
40 43,76 2,45295
50 47,88 2.45921
79 00 52,12 2,46556
10 56,47 2.47198
20 5 0,94 2.47848
407
545
410
551
413
557
417
563
419
569
576
423
425
583
429
589
433
596
436
603
440
611
443
618
626
447
450
635
454
642
458
650
T 2
140
TABLE IX. continued.
Zenith Mean
Dist. Refrac.
Log.
Diff.
Zenith Mean
Dist. Refrac.
Log.
Diff.
659
1139
669
1165
1191
677
688
1219
696
1248
30
1277
1309
1340
1374
1410
707
716
727
738
749
761
772
785
797
811
1447
6
1484
1523
1565
1608
824
1654
79 36 5 5,54 2.48507
40 10,28 2.49176
50 15,16 2.49853
80 00 20,19 2.50541
10 25,36 2.51237
20 30,70 2.51944
30 36,20 2.52660
40 41,88 2.53387
50 47,74 | 2.54125
81 00 53,79 | 2.54874
106 0,04 2.55635
20 6,50 2.56407
30 13,18 2.57192
40 20,09 2.57989
50 27,26 | 2.58800
82 00 34,68 2.59624
10 42,37 2.60462
20 50,33 | 2.61313
30 58,59 | 2.62179
40 7 7,19 | 2.63062
50 16,13 2.63961
83 00 25,40 | 2.64875
10 35,05 2.65806
20 45,10 2.66755
30 55,58 2.67722
40 | 8 6,50 2.68708
50
17,90 2.69714
84 00 29,80 2.70740
10 42,24 | 2.71787
20 55,25 2.72856
30 9 8,88 2.73948
40 9 23,16 | 2.75063
838
84 501 38,12 2.76202
85 00 9 53,84 | 2.77367
10 10 10,35 2.78558
20 27,73 2.79777
46,03 2.81025
40 11 5,30 2.82302
50
25,66 | 2.83611
86 00 47,15 2.84951
10 12 9,88 2.86325
20
33,97 | 2.87735
30
59,51 | 2.89182
40 13 26,61 2.90666
50 13 55,40 2.92189
87 00 14 26,04 2.93754
1014 58,71 | 2.95362
2015 33,60 2.97016
30 16 10,89 2.98717
4016 50,8
3,00466
50 17 33,6
3.02267
88 00 18 19,6 3,04122
10 19 9,0
3.06031
2020 2,2 3.07998
30 20 59,6
3.10024
40 22 1,7
3.12113
5023 8,9 3.14268
89 00124 21,8 3.16489
10 25 40,9 3.18779
20 27 7,1
3.21140
3028 40,8
3.23574
40 30 23,2
3.26083
50 32 15,0 3.28667
90 00 34 17,5 3.31334
1701
851
1749
1801
866
883
1855
899
1909
914
1967
931
2026
949
2089
967
2155
986
2221
1006
2290
1026
2361
2434
1047
1069
2509
1092
2584
1115
2667
TABLE X
141
Mr. Ivory's Refractions continued: showing the logarithms
of the corrections, on account of the state of the Ther-
mometer and Barometer.
Thermometer
Barometer
Logarithm
Logarithm
Logarithm
1
50
in.
31.0
0.00000
0.01424
49
0.00094
30.9
0.01248
48
0.00190
8
0.01143
47
0.00285
7
0.01002
46
0.00380
6
802
79
78
77
76
75
74
73
72
71
70
9.97237
9.97326
9.97416
9.97506
9.97596
9.97686
9.97777
9.97867
9.97958
9.98049
45
5
0.00860
0.00718
0.00575
0.00432
44
4
0.00476
0.00572
0.00668
0.00764
0.00861
43
3
42
2
0.00289
41
1
0.00145
9.98140
40
0.00957
30.0
0.00000
69
9.98231
39
9.99855
0.01053
0.01151
29.9
8
68
9.98323
38
9.99709
9.99563
67
9.98414
37
0.01248
7
6
66
9.98506
36
0.01346
65
35
0.01444
5
9.99417
9.99270
9.99123
64
34
0.01541
4
63
9.98598
9.98690
9.98783
9.98875
9.98969
33
3
9.98975
62
32
2
0.01640
0.01738
0.01837
0.01935
61
31
1
60
9.99061
30
29.0
59
9.99154
29.
0.02033
28.9
58
9.99248
28
0.02133
8
57
9.99341
27
0.02232
7
56
9.99434
26
0.02331
6
9.98826
9.98677
9.98528
9.98378
9.98227
9.98076
9.97924
9.97772
9.97620
9.97466
9.97313
9.97158
9.97004
55
9.99529
25
0.02432
5
54
9.92623
24
0.02531
4
53
23
0.02630
3
9.99717
9.99811
52
22
2
0.02730
0.02832
51
21
9.99906
0.00000
1
28.0
50
20
0.02933
142
TABLE XI.
Mr. Ivory's Refractions continued: showing the further
quantities by which the refraction at low altitudes is to
be corrected, on account of the state of the Thermome-
ter and Barometer.
Zenith
Distance
T
B
Zenith
Distance
T
B
86 36
86 40
86 50
87 0
87 10
87 20
+ 0,04
1
87 30
87 40
87 50
88 0
88 10
75 Ó - 0,009
76 0 0,012
177 0
0,015
78 0 0,018
790 0,023
80 0 0,030
81 0
0,040
81 30 0,046
82 0 0,053
82 30 0,063
83 0 0,074
83 30 0,089
84 0 0,107
84 30 0,130
85 0 0,159
85 10 0,171
85 20 0,184
85 30 0,198
85 40 0,213
85 50 0,229
86 0 0,248
86 10 0,269
86 20 0,292
- 0,317
0,345
0,376
0,410
0,448
0,490
0,538
0,593
0,654
0,722
0,799
0,887
0,987
1,101
1,231
1,380
1,551
1,749
1,977
2,241
2,549
2,909
+ 0,51
0,56
0,62
0,68
0,75
0,83
0,91
1,01
1,13
1,26
1,41
1,59
1,79
2,02
2,29
2,61
2,98
3,41
3,93
4,54
5,26
to 6,12
88 20
0,05
0,07
0,08
0,10
0,11
0,13
0,16
0,20
0,25
0,26
0,28
0,31
0,33
0,36
0,39
0,43
+ 0,47
88 30
88 40
88 50
89 0
89 10
89 20
89 30
89 40
89 50
90 0
The column marked T is to be multiplied by (t--50°): and the
column marked B is to be multiplied by (B-30in.00). The results
are to be applied to the approximate refraction obtained by the pre-
ceding tables.
TABLE XII.
143
Dr. Brinkley's Refractions : containing the logarithms of
the quantities depending on the state of the Thermometer.
Far.
Therm.
Log. T
Far.
Thierm.
Log. T
Far.
Therm.
Log. T
10
34
0.3048
0.2827
0.3283
0.3273
0.3263
58
59
11
35
0.3039
0.2818
12
36
0.3030
60
0.2809
13
0.3253
0.3020
61
37
38
14
0.3243
0.3011
15
0.3233
39:
0.3001
0.2992
16
0.3223
40
17
0.3213
41
0.2983
18
0.3203
42
66
19
0.3193
43
0.2974
0.2965
0.2956
0.2800
0.2791
0.2782
0.2773
0.2764
0.2755
0.2746
0.2737
0.2728
0.2720
0.2711
0.2703
0.2694
20
0.3183
44
21
45
0.2946
0.3173
0.3163
22
46
0.2937
23
47
0.2928
0.3154
0.3144
24
48
0.2919
25
0.3134
49
0.2910
26
0.3124
50
0.2900
70
71
72
73
74
75
76
77
78
79
0.2685
0.2677
27
0.3114
51
0.2891
28
0.3105
52
0.2881
0.2668
29
53
0.2872
0.2660
0.3095
0.3086
30
54
0,2863
0.2652
31
55
0.2854
0.2644
32
0.3076
0.3067
0.3058
56
0.2845
0.2636
33
57
0.2836
0.2627
Approximate Refraction = T.B. tan Z
Correct Refraction =
= T.ß. tan Z-
144
TABLE XIII.
Dr. Brinkley's Refractions contd: containing the quantity C,
depending on the state of the Barometer and Zenith di-
stance, to be deducted from the approximate refraction.
Barometer
Zenith
Dist.
28.50 | 29.00 29.50 30.00 30.50
oo
30
40
oo
0,0
0,1
0,2
0,2
0,2
45
50
52
54
0,3
56
58
60
61
0,0
0,0
0,1
0,2
0,2
0,2
0,3
0,3
0,4
0,5
0,5
0,6
0,6
0,7
0,8
0,9
1,0
1,2
1,3
1,5
1,8
2,1
2,5
3,0
3,4
4,1
66
1
0,3
0,4
0,5
0,5
0,6
0,6
0,7
0,8
0,9
1,0
1,2
1,4
1,6
1,9
2,2
2,6
3,2
3,7
4,5
5,6
6,9
8,9
11,4
1,2
1,3
1,5
1,8
2,1
2,5
3,0
3,4
4,2
5,2
6,4
8,3
10,7
70
71
72
73
74
75
76
77
78
79
1,2 1,2
1,3
1,4
1,5 1,6
1,9 1,9
2,2 2,2
2,6 2,6
3,1 3,1
3,5 3,6
4,3 4,4
5,3 5,5
6,6 6,7
8,5 8,5
10,9 11,1
5,1
6,3
8,1
10,5
80
TABLE XIV.
145
Parallax of the Sun, on the first day of each month: the
mean horizontal parallax being assumed = 85,60.
Zenith
Dist.
Jan.
Feb. March April
Dec. Nov. Oct.
May June
Sept. Aug.
July
•
5
10
15
0,00 0,00 0,00 0,00 6,00
0,76
0,75 0,74
1,52 1,52 1,51 1,49 1,48
2,26 2,26 2,25 2,23 2,21
2,99 2,98 2,97 2,94 2,92
3,70 3,69 3,67 3,63 3,60
4,37 4,36 4,34 4,30 4,26
0,00 0,00
0,74 0,74
1,47 | 1,47
2,19
2,90 2,89
3,58 3,57
4,24 4,23
2,19
20
25
30
35
40
45
5,02 | 5,01
4,98
5,62 5,61 5,58
6,19
6,17 6,13
6,70 6,68 6,64
7,17 7,15 7,11
7,58 | 7,56 | 7,51
4,93
4,89
5,53 5,48
6,08 6,03
6,596,53
7,04 6,99
7,45 7,39
4,86 4,85
5,45 5,44
5,99 5,98
6,49
6,48
6,94 6,93
7,34 7,33
50
55
60
65
70
75
80
7,93
8,22
8,45
8,62
8,73
8,75
7,91 7,86
8,20 8,15
8,43 8,38
8,59 8,54
8,69 8,64
8,73 8,67
7,79 7,73
8,08 8,01
8,30 8,24
8,47 8,40
8,56 8,50
8,608,53
7,68 7,67
7,97 7,95
8,19 8,17
8,35 8,33
8,44 8,42
8,48 8,46
85
90
U
3 46
TABLE XV.
Logarithms of sin. P, in time.
Minutes
3 hours
4 hours
5 hours
6 hours
7 hours
0
1
2
3
4
9.165679 9.397940 9.568894 9.698970 9.798933
•170240 •401214 •571358 •700865 .800384
•174773 •404471 •573811 •702743 .801828
•179278 •407713 •576253 •704618
.803266
•183756 • 410938 •578684 •706484 .804697
•188207 • 414147 •581104 •708342 .806122
•192631 •417340 •583513 •710192 .807540
•197028 •420517 •585911 •712034 .808952
•201399 •423679
·588299
•713868 .810357
•205745 .426825 •590676 •715694 .811756
5
6
7
8
9
10
•210064
•429955
.813149
11
314358
814535
12
.815915
•433070
•436170
• 439255
•442325
•218627
•222870
•227089
13
14
•593042 •717512
595398 •719322
•597744 •721124
-600078 •722919
•602403 •724705
•604717 726484
•607021 •728255
•609315 730018
.611598 731774
.613872 •733522
15
231284
.817289
.818656
.820017
•821372
.822721
.824063
16
•445379
•448419
•451445
•235454
17
239600
18
.243722
.247821
•454455
-457451
19
•825399
20
21
22
23
24
•251897 •460433 .616135 •735262 .826729
•255949 •463400 618388 •736994 .828053
•259978 •466354 -620632 •738719 .829370
•263985 •469293 .622865 .740437 .83068.2
267969 •472218 625089 •742147 .831987
•271930 •475129 •627303 •743849 .833287
.275870 • 478026 •629507 •745544 .834580
•279788 •480909 •631701 •747232 .835867
•283684 •483779 633886 748912 .837148
9.287558 9.486635 9•636061 9.750585 9.838424
25
26
27
28
29
TABLE XV. continued.
147
Logarithms of sin ? P, in time.
Minutes
3 hours
4 hours
5 hours
6 hours
7 hours
30
31
32
33
34
9.291412 9.489478 9.638227 9.752251 9.839693
•295244 •492307 .640383 •753909 .840956
•299055 •495123 •642529 •755560 .842213
•302845 •497926 .644666 757203 .843464
306615 •500716 •646794 •758840 .844710
•310364 •503492 -648913 •760469 845949
314094 •506256 .651022 •762091 .847183
317803 •509007 .653122 •763706 •848410
•321492 •511745 .655213 •765314 .849632
•325161 •514470 .657294 •766914 .850848
35
36
37
38
39
40
•328811
•517183
.659367
•661430
41
332442
519883
42
•336053
.663485
•522570
•525245
43
339645
44
•527908
•530559
45
768508 .852058
•770094 •853263
•771674 854461
•773247 .855654
•774812 .856841
•776371 858022
777922 .859198
•779467 860367
781005 861532
•782536 .862690
46
343219
346773
350309
•353827
357326
360807
•533197
*665530
•667567
.669594
.671613
•673623
.675624
.677617
47
•535823
48
•538437
49
•541040
50
51
52
53
54
•364270 •543630 .679601 •784061 863843
•367715 •546208 681576 •785578 .864990
•371142 •548775 .683543 •787089 .866131
•374552 551330 .685501 788593 •867267
•377945 •553874 -687450 •790090 868397
381320 •556406 .689391 791580 .869522
•384678 •558926 •691324 793064 870641
•388018 •561435 •693248 •794541 .871754
391342 •563933 •695163 •796012 .872862
9.394650 9.566419 9.697071 9.797476 9.873964
55
56
57
58
59
U 2
148
TABLE XVI.
For the equation of Equal Altitudes of the Sun.
Interval
Log. A_ _Log. B
Interval
Log. A
Log. B
m
h m
2 0
3 0
2
7.7015
•7010
•7005
•6999
4
4
6
6
8
8
.6993
7.7297
•7298
•7300
•7302
•7304
•7305
•7307
•7309
•7311
•7313
7.7146
•7143
•7139
•7136
•7132
•7128
•7125
•7121
•7117
•7113
7.7359
•7362
7364
•7367
•7369
•7372
-7374
•7377
•7380
•7383
10
10
.6988
12
12
•6982
14
14
•6976
•6970
16
16
18
18
.6964
20
0
20
•6958
22
22
.6952
24
24
•6946
26
26
.6940
28
•7386
•7388
•7391
•7394
•7397
7400
•7403
•7406
28
•7315
•7317
•7319
•7321
•7323
•7325
•7327
•7329
•7331
•7333
.6934
7109
•7105
•7101
•7097
•7092
•7088
•7083
•7079
•7075
•7070
30
30
•6927
•6921
32
32
34
34
.6914
36
36
•6908
•7409
•7412
38
38
•6901
40
40
•6894
42
42
.6888
44
44
•6881
46
46
•6874
48
48
•6867
-7336
•7338
•7340
•7342
•7345
•7347
•7349
•7352
•7354
7.7357
•7065
•7061
•7056
•7051
•7046
•7041
•7036
•7031
•7026
7.7021
•7415
•7418
•7421
•7424
•7428
•7431
•7434
•7437
•7441
7.7444
50
50
•6859
52
52
•6852
54
54
.6845
56
56
.6838
58
58
7.6830
TABLE XVI. continued.
149
For the equation of Equal Altitudes of the Sun.
Interval
Log. A
Log. B
Interval
Log. A
Log. B
h m
4 0
7.6823
h m
5 0
7.6556
2
.6815
2
-6546
4
.6807
4
.6536
6
-6800
6
.6525
8
.6792
8
.6514
7.7447
•7451
•7454
•7458
•7461
•7464
•7468
•7472
•7475
•7479
7.7562
•7566
•7570
•7575
•7579
-7583
•7588
•7592
•7597
•7601
10
10
•6504
12
12
.6493
14
•6784
.6776
•6768
•6759
-6751
14
.6482
16
16
•6471
-6460
18
18
20
20
•6448
22
22
•6437
24
24
•6425
26
.6743
•6734
•6726
•6717
•6708
-6700
•6691
26
.6414
28
28
•7482
•7486
*7490
•7494
•7497
•7501
.7505
•7509
•7513
•7517
-6402
•7606
•7610
•7615
•7620
.7624
•7629
•7634
•7638
.7643
•7648
30
30
.6390
•6378
32
32
34
.6682
34
•6366
6354
36
•6673
36
38
6663
38
•6342
40
.6654
40
.6329
42
.6645
42
.6317
44
.6635
44
•6304
46
.6626
46
•7653
•7658
•7663
•7668
•7673
•7678
•7683
•6291
48
•6616
48
•7521
•7525
•7529
•7533
•7537
•7541
•7545
•7549
•7553
7.7557
•6278
50
•6606
50
.6265
52
52
-6252
54
54
.6239
•6597
•6587
•6577
7.6567
56
56
•7688
•7693
7.7698
•6225
58
58
7.6212
150
TABLE XVI. continued.
For the equation of Equal Altitudes of the Sun.
Interval
Log. A
Log. B
Interval
Log. A
Log. B
m
h
m
h
6 0
7 0
7:5717
7.6198
.6184
2
2
•5699
4
4
•5680
•6170
.6156
6
6
•5661
8
•6142
8
•5641
7.7703
•7708
7713
•7719
•7724
•7729
•7735
•7740
•7745
•7751
7.7873
•7879
•7885
•7891
•7898
•7904
•7910
•7916
•7923
•7929
10
.6127
10
•5622
12
.6113
12
14
•6098
14
5602
•5582
•5562
i6
.6083
16
18
.6068
18
•5542
20
.6053
20
•5522
22
.6038
22
•5501
24
.6023
•5480
26
•6007
24
26
28
•5459
28
•5991
•5437
•7756
•7762
•7767
•7773
•7779
•7784
•7790
7796
•7801
*7807
•7936
•7942
•7949
•7955
•7962
•7969
•7975
•7982
•7989
•7995
30
•5975
30
•5416
32
•5959
32
•5394
34
•5943
34
•5372
•5350
36
•5927
36
38
•5910
38
•5327
40
•5894
40
•5304
.8002
.8009
42
•5877
42
•5281
44
•5860
44
.8016
•5258
46
•5843
46
.8023
•5234
48
•5825
48
.8030
•5211
•7813
•7819
•7825
-7831
•7836
•7842
•7848
•7854
•7860
7.7867
50
.5808
50
•5186
.8037
.8044
52
52
•5162
54
54
.8051
•5790
•5772
•5754
7.5736
•5137
•5112
56
56
.8058
58
58
7.8065
7.5087
TABLE XVI. continued.
151
For the equation of Equal Altitudes of the Sun.
Interval
Log. A
Log. B
Interval
Log. A
Log. B
m
h
8 0
h
9 0
7.8302
so QV #
7.8072
.8079
7.5062
•5036
7.4131
•4093
2
.8311
.8086
•5010
4
.8319
•4055
6
•8094
.4983
6
.8328
•4016
8
.8101
0
•4957
8
.8336
10
•8108
4930
10
.8344
•3977
•3937
•3896
12
.8116
•4902
12
8353
14
•8123
•4874
14
•8361
3855
16
.8130
•4846
16
.8370
•3813
18
.8138
•4818
18
.8378
•3771
20
.8145
20
.8387
•3728
.
22
.8153
22
.8396
•3684
24
•8160
24
.8404
*3639
•4789
•4760
•4731
•4701
•4671
•4640
26
.8168
26
.8413
3594
28
.8176
28
.8422
•3548
30
.8183
30
.8430
•3501
32
.8191
•4609
32
.8439
•3454
34
•4578
34
.8448
3406
8199
8206
36
36
.8457
•4546
4514
•3357
•3307
38
.8214
38
•8466
40
*8222
•4482
40
.8475
*3256
42
.8230
•4449
42
.8484
•3205
44
.8238
1415
44
.8493
•3152
46
.8246
•4381
46
.8502
•3099
48
.8254
•4347
48
.8511
•3045
50
.8262
•4312
50
.8520
•2989
52
52
.8530
•2933
.8270
.8278
.4277
•4241
54
54
.8539
56
.8286
• 4205
56
.8548
•2876
•2817
7.2758
58
7.8294
7.4168
58
7.8558
152
TABLE XVI. continued.
For the equation of Equal Altitudes of the Sun.
Interval
Log. A
Log. B
Interval
Log. A
Log. B
m
h
10 0
7.2697
h
m
11 0
7.8868
7.0025
2
7.8567
.8576
.8586
• 2635
2
.8878
4
•2572
4
.8889
6.9889
•9748
.9602
6
.8595
•2507
6
.8900
8
.8605
.2442
8
.8911
10
.8614
•2374
10
•8922
.9449
•9290
•9125
.8624
12
C
•2306
12
.8932
14
.8634
.2236
14
.8943
+
16
.8643
16
.8954
2164
• 2091
.8953
:8770
.8580
18
.8653
18
.8965
20
.8663
• 2016
20
.8977
•8379
•8168
22
.8673
•1940
22
•8988
24
.8683
•1861
24
.8999
26
.8693
•1781
26
9010
9021
•7945
•7709
•7457
•7189
28
•1699
28
30
•1615
30
9033
32
•1529
32
9044
.6901
.8703
.8713
.8723
.8733
.8743
.8753
34
•1440
34
9055
.6591
36
•1349
36
•6255
.9067
9078
38
•1256
38
•5889
40
•1160
.9090
•5487
.8763
.8773
40
42
42
•1061
9102
•504]
44
.8784
0960
44
9113
•4541
46
.8794
46
9125
•3973
48
.8804
-0855
•0748
•0637
48
9137
•3316
50
.8815
50
0
9148
52
.8825
0522
52
9160
•2536
•1579
6.0341
54
.8836
·0404
54
.9172
56
.8846
0282
56
.9184
5.8593
58
7.8857
7.0156
58
7.9196
5.5594
TABLE XVII.
153
Showing the Altitude of a star, whose Declination is less
than the Latitude of the place, at the moment of its pass-
ing the Prime Vertical: also of a star, whose Declina-
tion is greater than the Latitude of the place, at the time
of its greatest Azimuth, or at the moment when the ver-
tical becomes a tangent to the circle of declination.
N.B. The Declination must be on the same side of the
equator as the Latitude of the place.
Declination of the star
Lat.
5°
100
15°
200
25°
300
350
8 44
5 90 6 30 8 19° 41' 14 46 11° 54' 10 2
10 30 890 0 42 8
42 8 | 30 31 24 16 | 20 19 17 37
15 19 41 42 8 90 0 49 11 37 46 | 31 10 26 49
20 14 16 30 31 49 11 900 54 40 43 10 36 36
25 11 54 24 16 37 46 54 40 90 0 57 45 | 47 28
30 10 2 20 19 31 10 43 10 57 45 90 0 60 40
35 8 44 | 17 37 | 26 49
36 36 47 28 | 60 40 90
| ()
40 7 48 15 40 23 45 32 9 41 6 51 4 | 63 10
45 7 5 14 13 21 28 28 56 36 42 45 0 54 12
50 6 32 13 6 19 45 26 31 33 29 40 45 48 29
55 6 6 12 14 18 25 24 41 31 4 37 37 44 27
60
5 47 11 34 17 23 23 16 29 40 35 16 41 29
65 5 31 11 3 16 36
16 36 22 10 27 48 33 29 39 16
70 5 19 10 39 15 59 21 21 26 44 | 32 9 37 37
:
The change of altitude, on the Prime Vertical, in one second of'time
is = 15" x sin Lat.
X
154
TABLE XVIII.
For the Reduction to the Meridian: showing the value of
2 sin IP
A=
sin 1"
Sec.
Om
lin
2m
3m
4m
5m
6
7m
0
1
2
0,0
0,0
0,0
0,0
0,0
2,0
2,0
2,1
2,2
2,2
1,8 |
177 314 491 767 96,2
8,0 17,9 31,7 49,4 71,1 96,7
8,1 18,1 31,9 49,7 71,5 97,1
8,2 18,3 32,2 50,1
71,9 97,6
8,4 18,5 32,5 50,4 72,3 98,0
3
4.
5
6
0,0
0,0
0,0
0,0
0,0
2,3
2,4
2,4
2,5
7
8
8,5
8,7
8,8
8,9
9,1
18,7 32,7
18,9 33,0
19,1 33,3
19,3 33,5
19,5 | 33,8
50,7
51,1
51,4
51,7
52,1
72,7 98,5
73,1 99,0
73,5 99,4
73,9 99,9
74,3 100,4
9
2,6
10
0,1
0,1
11
2,7
2,7
2,8
12
0,1
9,2
9,4
9,5
9,6
9,8
19,7
19,9
20,1
20,3
20,5
34,1 52,4
34,4
52,7
34,6 53,1
34,9 53,4
35,2 53,8
74,7 100,8
75,1 101,3
75,5 101,8
75,9 102,3
76,3 102,7
13
0,1
0,1
14
3,0
15
16
0,1
0,1
0,2
0,2
0,2
17
3,1
3,1
3,2
3,3
3,4
9,9
10,1
10,2
20,7
20,9
21,2
21,4
21,6
35,5 54,1
35,7 54,5
36,0 54,8
36,3 55,1
36,6 55,5
76,7 103,2
77,1 103,7
77,5 | 104,2
77,9 | 104,6
78,3 105,1
18
10,4
10,5
19
20
21
22
0,2
0,2
0,3
0,3
0,3
3,5
3,6
3,7
3,8
3,8
10,7 21,8
10,8 22,0
11,0 22,3
11,2 22,5
11,3 22,7
36,9
37,2
37,4
37,7
38,0
55,8 78,8 105,6
56,2 79,2 106,1
56,5 79,6 | 106,6
56,9 80,0 | 107,0
57,3 80,4 107,5
23
24
25
26
27
0,3
0,4
0,4
0,4
0,5
3,9
4,0
4,1
4,2
4,3
11,5
11,6
11,8
11,9
12,1
22,9 38,3 57,6 80,8 | 108,0
23,1 38,6 58,0 | 81,3 | 108,5
23,4 38,9 58,3 81,7 109,0
23,6 39,2 | 58,7 82,1 109,5
23,8 39,5 59,0 82,5 110,0
28
29
TABLE XVIII. continued.
155
For the Reduction to the Meridian: showing the value of
2 sin P
sin 1"
A=
Sec.
Om
]m
2m
3m
4m
5m
6m
7m
30
0,5 4,4
31
32
0,5
0,6
0,6
0,6
4,5
4,6
4,7
4,8
12,3 24,0 39,8 59,4 83,0
59,4 83,0 116,4
12,4 24,3 40,1 59,8 83,4 | 110,9
12,6 24,5 40,3 60,1 83,8 | 111,4
12,8 24,7 40,6 60,5 84,2 111,9
12,9 25,0 40,960,8 84,7 112,4
33
34
35
36
37
0,7
0,7
0,7
0,8
0,8
4,9
5,0
5,1
5,2
5,3
13,1 25,2
13,3 25,4
13,4 25,7
13,6 25,9
13,8 26,2
41,2
41,5
41,8
42,1
42,5
61,2 85,1 112,9
61,6 85,5 ( 113,4
61,9 86,0 | 113,9
62,3 86,4 114,4
62,7 86,8 114,9
38
39
40
4]
42
0,9
0,9
1,0
1,0
1,1
5,4
5,6
5,7
5,8
5,9
14,0 26,4 | 42,8 | 63,0 87,3 | 115,4
14,1 26,6 43,1 63,4 87,7 115,9
14,3 26,9 43,4 63,8 88,1 116,4
14,5 27,1 43,7 64,2 88,6 116,9
14,7 27,4 | 44,0 64,5 89,
89,0 | 117,4
43
44
>
45
46
47
1,1
1,2
1,2
1,3
1,3
6,0
6,1
6,2
6,4
6,5
14,8 27,6 | 44,3
15,0 27,9 44,6
15,2 28,1 44,9
15,4 28,3 45,2
15,6 | 28,6 45,5
64,9 89,5 117,9
65,3 89,9 118,4
65,7 90,3 118,9
66,0 90,8 119,5
66,4 91,2 120,0
48
49
50
51
52
1,4
1,4
1,5
1,5
1,6
6,6
6,7
6,8
7,0
7,1
53
54
15,8 28,8 45,9 66,8 91,7 | 120,5
15,9 29,1 | 46,2 67,2 | 92,1 121,0
16,1 29,4 46,5 67,6 92,6 121,5
16,3 29,6 | 46,8 68,0 | 93,0
93,0 122,0
16,5 29,9
47,1 68,3 93,5 | 122,5
16,7 30,1
47,5 68,7 93,9 123,1
16,9 | 30,4 | 47,8 69,1 94,4 | 123,6
17,1 | 30,6 | 48,1 69,5 94,8 124,1
17,3 | 30,9 48,4 69,9 95,3 124,6
17,5 31,1 48,8 70,3 95,7 125,1
55
56
57
1,6
1,7
1,8
1,8
1,9
7,2
7,3
7,5
7,6
7,7
58
59
x 2
156
TABLE XVIII. continued.
For the Reduction to the Meridian: showing the value of
2 sin 1 P
A=
sin 1"
Sec.
gm
8m
10m
ni
1]n
120
13m
14m
0 || 125,7 159,0
15970 196,3
1 126,2 159,6 197,0
2 126,7 160,2 197,6
3 127,2 160,8 198,3
4 127,8 161,4 198,9
237,5 282,7 3318 384,7
238,3 283,5 332,6 385,6
239,0 284,2 333,4 | 386,6
239,7 | 285,0 334,3 387,5
240,4 285,8 335,2 388,4
5
6
7
128,3
128,8
129,3
129,9
130,4
162,0
162,6
163,2
163,8
164,4
199,6
200,3
200,9
201,6
202,2
241,2 286,6 336,0 389,3
241,9 287,4 336,9 390,2
242,6 288,2
288,2 337,7 391,1
243,3 289,0 | 338,6 392,1
244,1 289,8 339,4 393,0
8
9
10
11
12
131,0
165,0 202,9
131,5 165,6 203,6
132,0 166,2 | 204,2
132,6 | 166,8 204,9
133,1 167,4 205,6
244,8
245,5
246,3
247,0
247,7
290,6 340,3
291,4 341,2
292,2 342,0
293,0 342,9
293,8 343,7
393,9
394,8
395,8
396,7
397,6
13
14
15
16
17
133,6 | 168,0
134,2 168,6
134,7 169,2
135,3 169,8
135,8 170,4
206,3
206,9
207,6
208,3
208,9
248,5 294,6344,6 398,6
249,2 295,4 345,5 399,5
249,9 296,2 | 346,4 400,5
250,7 297,0 347,2
401,4
251,4 297,8 348,1 402,3
18
19
20
21
22
136,3 171,0
171,0 209,6
136,9 171,6 210,3
137,4 172,2 211,0
138,0 172,9 211,7
138,5 173,5 | 212,3
252,2 298,6 349,0
253,0 299,4 349,8
253,6 | 300,2 350,7
254,4 301,0 | 351,6
255,1 301,8 352,5
403,3
404,2
405,1
406,0
407,0
23
24
25
26
27
28
139,1
139,6
140,2
140,7
141,3
174,1
174,7
175,3
175,9
176,6
213,0
213,7
214,4
215,1
215,8
255,9 302,6 353,3
256,6 303,5 354,2
257,4 304,3 355,1
258,1 305,1 356,0
258,9 305,9 356,9
408,0
408,9
409,9
410,8
411,7
29
TABLE XVIII. continued.
157
For the Reduction to the Meridian: showing the value of
2 sin P
A=
sin 1"
Sec.
8m
9m
10m
11"
12m
13m
14m
30 141,8 1772 216,4 259,6 3067
3067 357,7 412,7
31 142,4 177,8 217,1 260,4 307,5 358,6 413,6
32 143,0 178,4 217,8 261,1 308,4 | 359,5 414,6
33 143,5 179,0 218,5 261,9 309,2 360,4 415,5
34 144,1 179,7 219,2 262,6 310,0 | 361,3 416,5
35
36
37
144,6 180,3
145,2 180,9
145,8 181,6
146,3 182,2
146,9 182,8
219,9
220,6
221,3
222,0
222,7
263,4
264,1
264,9
265,7
266,4
310,8 362,2
311,6 | 363,1
312,5 364,0
313,3 364,8
314,1 365,7
417,5
418,4
419,4
420,3
421,3
38
39
40
41
42
147,5
148,0
148,6
149,2
149,7
183,5
184,1
184,7
185,4
186,0
223,4
224,1
224,8
225,5
226,2
267,2 315,0 366,6 422,2
267,9 315,8
315,8 367,5 423,2
268,7 316,6 | 368,4
368,4 | 424,2
269,5 317,4 369,3
369,3 425,1
270,3 318,3 370,2 | 426,1
43
44
45
46
47
150,3 186,6
150,9 187,3
151,5 | 187,9
152,0 188,5
152,6 189,2
226,9 271,0 319,1 371,1
227,6 | 271,8 319,9
319,9 | 372,0
228,3 272,6 320,8 372,9
229,0 273,3 321,6 | 373,8
229,7 274,1 322,4 374,7
427,0
428,0
429,0
429,9
430,9
48
49
50
51
52
153,2
153,8
154,4
154,9
155,5
189,8 230,4 274,9 323,3 375,6 431,9
190,5 231,1 275,6 324,1 376,5 432,8
191,1 231,8 276,4 325,0 377,4 433,8
191,8 232,5 277,2 325,8 378,3 434,8
192,4 233,2 | 278,0 326,7
379,3 435,8
53
54
55
56
57
58
156,1
156,7
157,3
157,8
158,4
193,1
193,7
194,4
195,0
195,7
234,0 278,8 327,5 380,2 436,7
234,7 279,5 328,4 381,1 437,7
235,4 280,3 329,2 382,0 | 438,7
236,1 281,1 330,0 382,9 439,7
236,8 281,9 330,9 383,8 440,6
59
158
TABLE XVIII. continued.
568,3 / @
For the Reduction to the Meridian: showing the value of
2 sin P
A =
sin 1"
Sec. 15m 16m 17m 18m 19m
20m
21"
0 441,6 502,5 567,2
502,5 5672 63569 708,4 784,9 865,3
1 442,6 503,5
709,7 786,2 866,6
2 443,6 | 504,6 569,4 638,2 710,9 787,5 868,0
3
444,6 505,6 570,5639,4 712,1 788,8 869,4
4 445,6 506,7 571,6 640,6 713,4 790,1 870,8
637,0
876,3
5 446,5 507,7 572,8 | 641,7
641,7 714,6 | 791,4 872,1
6 || 447,5 508,8 573,9 642,9 715,9 792,7 | 873,5
7 448,5 509,8 575,0 644,1 717,1 794,0 | 874,9
8 449,5 510,9 576,1 645,3 718,4 795,4
9 450,5 511,9 577,2 646,5 719,6 796,7 | 877,6
10 451,5 513,0
513,0 578,4 647,7 720,9 798,0
798,0 | 879,0
11 452,5 514,0 579,5 648,9 722,1 799,3
880,4
12 453,5
515,1 580,6 650,0
650,0723,4 800,7
881,8
454,5 516,1 581,7 651,2 724,6 802,0 883,2
14 455,5 517,2 582,9 652,4 | 725,9 803,3
13
884,6
15 456,5 518,3 584,0653,6 727,2 804,6
16 457,5 519,3 585,1 654,8 728,4 806,0
17 458,5 520,4 586,2 656,0 729,7 807,3
18 459,5 521,5 587,4 657,2 730,9 808,6
19 || 460,5 522,5 588,5 658,4 732,2
732,2 | 809,9
886,0
887,4
888,8
890,2
891,6
20
21
22
461,5
462,5
463,5
464,5
465,5
523,6 589,6 659,6 733,5
524,6 590,8 660,8 734,7
525,7 591,9 662,0 736,0
526,8 593,0 663,2 | 737,3
527,9 594,2 664,4 738,5
811,3
812,6
813,9
815,2
816,6
893,0
894,4
895,8
897,2
898,6
23
24
25
26
27
466,5 528,9 595,3
467,5 530,0 596,5
468,5 531,1 597,6
469,5 532,2 598,7
470,5 533,2
599,9
665,6 739,8 817,9 900,0
666,8 741,1 819,2 901,4
668,0 742,3 820,5 902,8
669,2 743,6 | 821,9
670,4 | 744,9 823,2 905,6
28
904,2
29
TABLE XVIII. continued.
159
For the Reduction to the Meridian: showing the value of
2 sin P
A =
sin 1"
Sec.
15m
jom
17m
18m
19m
20m
21m
30 471,5 531,3 601,0 671,6 746,2 824,6 907,0
31 472,6 535,4 602,2 672,8 747,4 825,9 908,4
32 | 473,6536,5 603,3 674,1 748,7 827,3 909,8
33 474,6537,6
537,6 | 604,5 675,3 750,0 828,6 911,2
34 475,6 | 538,7 605,6 676,5 751,3 829,9 912,6
35 476,6 539,7
36 477,6 540,8
37 478,7
541,9
38 || 479,7 543,0
39 480,7 544,1
606,8 677,7 752,6 831,2 914,0
607,9
678,9 753,8 832,6 915,5
609,1 680,1 755,1 833,9 916,9
610,2 681,3 756,4 835,3 918,3
611,4 682,6 757,7 836,6 919,7
40
41
42
481,7
482,8
483,8
484,8
485,8
43
44
545,2 612,5 683,8 759,0 838,0 921,1
546,3 613,7 685,0
685,0 | 760,2
760,2 839,3 922,5
547,4 614,8 686,2 761,5 840,7 923,9
548,4 616,0 687,4 | 762,8 842,0 925,3
549,5 617,2
688,7 764,1 843,4 926,8
550,6 618,3 689,9 765,4 844,7 928,2
551,7 619,5 691,1 766,7 846,1 929,6
552,8 620,6 692,4 768,0 847,5 931,0
553,9 621,8 693,6 769,3
769,3 848,9 932,4
555,0 623,0 694,8 770,6 850,2 933,8
45
46
47
486,9
487,9
488,9
490,0
491,0
48
49
50
51
52
492,0
493,1
494,1
495,2
496,2
556,1 624,1 696,0 771,9 851,6 935,2
557,2 625,3 697,3 773,1 852,9 936,6
558,3 626,5 698,5 774,5 854,3 | 938,1
559,4 627,6 699,7 775,8 855,7 939,5
560,5 628,8 701,0 777,1 857,1 940,9
53
54
55 497,2 561,6
56 || 498,3 562,7
57 499,3 563,9
58 500,3 565,0
59
501,4
566,1
630,0 | 702,2
631,2 703,5
632,3 704,7
633,5 705,9
634,7 707,1
778,4 858,4 942,3
779,7 859,8943,8
781,0 861,1 945,2
782,3 862,5 946,6
783,6 863,9 948,1
160
TABLE XVIII. continued.
For the Reduction to the Meridian: showing the value of
2 sin 2 P
sin 1"
A=
Sec.
22m
23m
24m
25m
26m
27m
28m
09496 1037,8 1129;9 1225;9 132569 14297 1537,5
1 951,0 1039,3 1131,4 1227,5 1327,6 1431,4 1539,3
2 952,4 1040,8 1133,0 1229,2 1329,3 1433,2 1541,1
3 953,8 -1042,3 1134,6 1230,8 1331,0 1434,9 1542,9
4 955,3 1043,8 1136,2 1232,5 1332,7 1436,7 1544,8
5 956,7 1045,3 1137,8 1234,1 1334,4 1439,5 1546,6
6 958,2 1046,8 1139,3 1235,7 1336,1 1440,3 1548,4
7 || 959,6 | 1048,3 1140,9 1237,3 1337,8 1442,1 1550,2
8 961,1 1049,8 1142,5 1239,0 1339,5 1443,9 1552,1
9 962,5 1051,3 1144,0 1240,6 1341,2 1445,6 1553,9
10
11
12
963,9 1052,8 | 1145,6 | 1242,3 1342,9 | 1447,4 1555,8
965,4 1054,3 1147,2 1243,9 1344,6 1449,2 1557,6
966,9 1055,9 1148,8 1245,6 1346,3 1451,0 1559,5
968,3 1057,4 1150,4 1247,2 1348,0 1452,8 | 1561,3
969,8 1058,9 1152,0 1248,9 1349,7 1454,5 1563,2
13
14
15
16
17
971,2 1060,4 1153,6 1250,5 1351,4 1456,3 | 1565,0
972,7 1062,0 1155,2 1252,2 1353,2 1458,1 1566,9
974,1 1063,5 1156,8 1253,8 1354,9 1459,9 | 1568,7
975,5 1065,0 1158,3 1255,5 1356,6 1461,6 1570,5
977,0 1066,5 1159,9 1257,1 1358,3 1463,4 1572,4
18
19
20 978,5 1068,1 1161,5 1258,8 1360,1 1465,2 1574,3
21 979,9 1069,6 1163,1 1260,4 1361,8 1466,9 1576,1
22 981,4 1071,1 1164,7 1262,1 1363,5 1468,7 | 1578,0
23 982,9 1072,6 1166,3 1263,7 1365,2 1470,5 1579,8
24 984,4 1074,2 1167,9 1265,4 1367,0 1472,3 1581,7
25 985,8 1075,7 1169,5 1267,0 1368,7 1474,0 1583,5
26 987,3 1077,2 1171,1 1268,7 1370,4 1475,9 1585,3
27 || 988,8 1078,7 1172,7 1270,3 1372,1 1477,7 1587,2
28 990,3 1080,3 1174,3 1272,1 | 1373,9 1479,5 1589,1
29 991,8 1081,8 1175,9 1273,7 1375,6 1481,3 1590,9
TABLE XVIII. continued.
161
For the Reduction to the Meridian : showing the value of
2 sinP
sin 1!
Sec.
22m
23ın
24m
25m
26m
27m
28m
30 993,2 1083,3 1174,5 1275,4 1377,4 1483,1 1592,7
31 994,7 1084,8 1179,1 1.1277,1 1379,0 1484,9 1594,6
32 996,2 1086,4 1180,7 1278,8 1380,8 1486,7 1596,5
33 997,6 1087,9 1182,3 1280,4 1382,5 1488,5 1598,3
34 999,1 1089,5 1183,9 1282,1 1384,2 1490,3 1600,2
35 | 1000,6 1091,0 1185,5 1283,8 1385,9 1492,1 1602,1
36 | 1002,1 1092,6 1187,1 1285,5 1387,7 1493,9 | 1604,0
37 | 1003,5 1094,1 1188,7 1287,1 1389,4 1495,7 1605,9
38 1005,0 1095,7 1190,3 1288,8 1391,2 1497,5 1607,7
39 | 1006,5 1097,2 1191,9 1290,5 1392,9 1499,3 1609,6
40 | 1008,0 1098,8 | 1193,5 1292,2 1394,7 1501,1 1611,5
41 1009,4 1100,3 1195,1 1293,8 1396,4 1502,9 1613,3
42 | 1010,9 1101,9 1196,7 1295,5 1398,2 1504,7 1615,2
43 1012,4 1103,4 1198,3 1297,2 1399,9 1506,5 1617,1
44 1013,9 1105,0 1199,9 1298,9 1401,7 1508,4 1619,0
45 1015,4 1106,5 1201,5 1300,5 1403,4 1510,2 1620,8
46 || 1016,9 1108,1 1203,1 1302,2 | 1405,2 1512,0 1622,7
47 1018,4 1109,6 1204,7 1303,9 1406,9 | 1513,8 1624,6
48 1019,9 1111,2 1206,4 1305,6 1408,7 1515,6 1626,5
49 || 1021,4 1112,7 1208,0 1307,3 1410,4 1517,4 1628,3
50 1022,8 1114,3 1209,6 1309,0 1412,2 1519,2 1630,2
51 ||1024,3 1115,8 1211,2 1310,7 1413,9 1521,0 1632,1
52 1025,8 1117,4 1212,9 1312,4 1415,7 1522,9 1634,0
53 1027,3 1118,9 1214,5 1314,1 1417,4 1524,7 1635,9
54 || 1028,8 1120,5 1216,1 1315,7 1419,2 | 1526,5 1637,7
55 1030,3 1122,0 1217,7 1317,4 1420,9 1528,3 1639,6
56 1031,8 1123,6 1219,4 1319,1 1422,7 1530,2 1641,5
57 1033,3 1125,1 1221,0 1320,8 1424,4 1532,0 1643,3
58 1034,8 1126,7 1222,6 | 1322,5 | 1426,2 1533,8 1645,2
59 1036,3 1128,3 1224,2 1324,2 1427,9 1535,6 1647,1
Y
162
TABLE XVIII. continued.
For the Reduction to the Meridian : showing the value of
2 sin P
A=
sin 1
Sec.
29m
30m
31m
32m
33m
34m
35m
01649,0 1764,6 1884,0 2007,4 2134,6 2265,6 2406,6
1 1650,9 1766,6 1886,0 2009,4 2136,8 2267,8 2402,9
2 1652,8 1768,5 1888,0 2011,5 2138,9 2270,0 2405,2
3 1654,7 1770,5 1890,0 2013,6 2141,1 2272,2 2407,5
4 1656,6 1772,4 1892,1 2015,7 2143,2 2274,5 2409,8
V
5 1658,5 1774,4 1894,1 2017,8 2145,3 2276,7 2412,0
6 1660,4 1776,3 1896,1 2019,9 2147,5 2278,9 2414,3
7 1662,3 1778,3 1898,1 2022,0 2149,7 2281,22416,6
8 1664,2 1780,3 1900,2 2024,1 2151,8 2283,4 2418,9
9 1666,1 1782,3 1902,2 2026,2 2153,9 2285,6 2421,2
10 1668,0 1784,2 1904,3 2028,3 2156,1 2287,8 2423,5
11 1669,9 1786,2 1906,3 2030,5 2158,3 2290,0 2425,8
12 1671,9 1788,2 1908,4 2032,5 2160,5 2292,3 2428,i
13 1673,8 1790,1 1910,4 2034,6 2162,6 2294,5 2430,4
14 1675,7 1792,1 1912,4 2036,7 | 2164,8 2296,8 2432,7
15 || 1677,6 1794,1 1914,4 2038,8 2166,9 2299,0 2435,0
16 1679,5 1796,1 1916,5 2040,9 2169,1 2301,3 | 2437,3
17 || 1681,4 1798,1 1918,5 2043,0 2171,2 | 2303,6 2439,6
18 | 1683,3 1800,0 | 1920,6 2045,1 2173,4 2305,8 2441,9
19 1685,2 1802,0 1922,6 2047,2 2175,6 2308,0 2444,2
20 1687,2 1804,0 1924,7 2049,3 2177,8 2310,2 2446,5
21 1689,1 1805,9 1926,7 2051,4 2179,9 2312,4 2448,8
22 1691,0 1807,9 1928,8 2053,5 2182,1 2314,7 2451,1
23 1692,9 1809,9 1930,82055,7 2184,3 2316,9 2453,4
24 1694,8 1811,9 1932,9 2057,8 2186,5 2319,2 2455,7
25 1696,7 1813,9 1935,0 2059,9 2188,6 2321,5 2458,0
26 1698,6 1815,8 1937,0 2062,0 2190,8 2323,7 2460,3
27 1700,5 1817,8 1939,0 2064,1 2193,0 2325,9 2462,6
28 1702,5 1819,8 1941,1 2066,2 2195,2 2328,2 | 2464,9
29 1704,4 1821,8 1943,1 2068,3 2197,3 2330,4 | 2467,2
TABLE XVIII. continued.
163
For the Reduction to the Meridian: showing the value of
2 sin P
sin 11
A=
Sec.
29m
30m
31"
32ni
33m
34m
35m
30 17063 1823,8 1945,2 2076,4 2190,5 2332,7 2469,5
31 | 1708,2 1825,8 1947,2 2072,6 2201,7 2334,9 2471,8
32 1710,2 1827,8 1949,3 2074,7 2203,9 2337,2 2474,2
33 | 1712,1 1829,8 1951,3 2076,8 2206,1 2339,4 2476,5
34 1714,0 1831,8 1953,4 2078,9 2208,3 2341,7 2478,8
35 1715,9 1833,8 1955,5 2081,0 2210,5 2343,9 2481,1
36 || 1717,9 1835,8 1957,6 2083,2 2212,7 2346,2 2483,5
37 1719,8 1837,8 1959,6 | 2085,3 2214,9 2348,5 2485,8
38 1721,7 | 1839,8 1961,7 2087,4 2217,1 235037 2488,1
39 || 1723,6 1841,8 1963,7 2089,6 2219,3 2353,0 2490,4
40 | 1725,6 1843,8 1965,8 2091,7 | 2221,5 2355,2 2492,8
41 1727,5 1845,8 1967,8 2093,8 2223,7 2357,5 2495,1
42 1729,5 1847;8 1969,9 2095,9 2225,9 2359,7 2497,4
43 1731,5 1849,8 1972,0 2098,0 | 2228,1 2361,9 2499,7
44 1733,4 1851,8 1974,1 2100,2 2230,3 2364,2 2502,1
45 || 1735,3 1853,8 1976,1 2102,3 2232,5 2366,4 2504,4
46 1737,2 1855,8 1978,2 2104,5 2234,7 2368,7 2506,7
47 1739,2 1857,8 1980,3 2106,6 2236,9 2371,0 2509,0
48 1741,2 1859,8 1982,4 2108,8 2239,1 2373,3 2511,4
49 1743,1 1861,8 1984,4 2110,9 2241,3 12375,5 2513,7
50 1745,1 1863,8 1986,5 2113,1 2243,5 2377,8 2516,1
51 1747,0 1865,8 1988,6 2115,2 2245,7 2380,1 | 2518,4
52 1749,0 1867,8 1900,7 2117,4 2247,9 2382,4 2520,8
53 | 1750,9 | 1869,8 1992,7 2119,6 2250,1 2384,6 2523,1
54 | 1752,9 1871,8 1994,8 2121,7 2252,3 2386,9 2525,4
55 1754,8 1873,8 1996,9 2123,8 2254,5 2389,2 2527,7
56 1756,8 1875,9 1999,0 2126,0 2256,7 2391,5 2530,1
57 1758,7 1877,9 2001,0 2128,1 | 2258,9 2393,7 2532,4
58-1760,7 1879,9 2003,1 2130,3 2261,1 2396,0 2534,8
59 1762,6 1882,0 2005,3 2132,4 2263,4 2398,3 2537,1
Y 2
164
TABLE XIX.
For the second part of the Reduction to the Meridian :
2 sin* 1P
showing the value of B =
sin 1"
Minutes
02
108
20
303
408
503
5
6,016,01
6,01
6
0,02
7
0,03
8
9
10
11
12
13
14
15
0,01
0,02
0,04
0,07
0,10
0,15
0,20
0,28
0,38
0,49
0,64
0,81
1,02
1,26
1,54
1,87
2,25
2,69
3,18
3,74
16
1
17
18
0,01
0,01
0,02
0,04
0,06
0,09
0,14
0,19
0,27
0,36
0,47
0,61
0,78
0,98
1,22
1,49
1,82
2,19
2,61
3,10
3,64
4,26
4,96
5,73
6,59
7,55
8,61
9,77
11,04
12,44
13,97
6,01
0,02
0,03
0,05
0,08
0,12
0,17
0,24
0,33
0,43
0,56
0,72
0,91
1,13
1,40
1,70
2,06
2,46
2,93
3,45
4,05
4,72
5,46
6,30
19
20
0,01
0,03
0,05
0,08
0,11
0,15
0,22
0,30
0,39
0,52
0,67
0,84
1,06
1,30
1,60
1,93
2,32
2,77
3,27
3,84
4,48
5,20
6,01
6,90
7,89
8,98
10,18
11,50
12,94
14,51
0,05
0,08
0,11
0,16
0,23
0,31
0,41
0,54
0,69
0,88
1,09
1,35
1,65
1,99
2,39
2,85
3,36
3,94
4,60
5,33
6,15
7,06
8,06
9,17
10,39
11,73
13,19
14,78
6,01
0,02
0,04
0,06
0,09
0,13
0,18
0,25
0,34
0,45
0,59
0,75
-0,95
1,18
1,44
1,76
2,12
2,54
3,01
3,55
4,15
4,83
5,60
6,44
7,38
8,42
9,57
10,82
12,20
13,71
15,35
21
22
23
24
25
26
4,37
27
28
29
7,22
30
31
5,08
5,87
6,75
7,72
8,79
9,97
11,27
12,69
14,24
32
8,24
9,37
10,61
11,96
13,45
15,06
33
34
35
TABLE XX.
165 -
Mean Obliquity of the Ecliptic, on Jan. 1. in every year
from 1800 to 1900.
Year
23 27
Year
23 27
Year
23 27
C 1800
B 1868
23,70
1801
1869
1802
1803
B 1804
1805
1806
1807
1870
1871
B 1872
1873
1874
1875
B 1876
1877
1878
1879
B 1880
B 1808
1809
1810
1811
B 1812
1813
1881
1882
1814
1815
1883
B 1884
B 1816
1817
1818
54,78
54,32
53,86
53,41
52,95
52,49
52,03
51,58
51,12
50,66
50,21
49,75
49,29
48,84
48,38
47,92
47,46
47,01
46,55
46,09
45,64
45,18
44,72
44,27
43,81
43,35
42,89
42,44
41,98
41,52
41,07
40,61
40,15
39,70
1834 39,24
1835 38,78
B 1836 38,32
1837 37,87
1838 37,41
1839 36,95
B 1840 36,50
1841 36,04
1842 35,58
1843 35,13
B 1844 34,67
1845 34,21
1846 33,75
1847 33,30
B 1848 32,84
1849 32,38
1850 31,93
1851 31,47
B 1852 31,01
1853
30,56
1857 30,10
1855 29,64
B 1856 29,18
21857 28,73
1858 28,27
1859 27,81
B 1860 27,36
1861 26,90
1862
26,44
1863 25,99
B 1864 25,53
1865 25,07
1866 24,61
1867 24,16
23,24
22,79
22,33
21,87
21,42
20,96
20,50
20,04
19,59
19,13
18,67
18,22
17,76
17,30
16,85
16,39
15,93
15,47
15,02
14,56
14,10
13,65
13,19
12,73
12,28
11,82
11,36
10,90
10,45
9,99
9,53
9,08
1885
1886
1819
1887
B 1888
B 1820
1821
1889
1822
1890
1823
1891
B 1824
B 1892
1825
1893
1826
1894
1827
1895
B 1828
B 1896
1829
1897
1830
1898
1831
1899
B 1832
C 1900
1833
166
TABLE XXI.
Lunar Nutation in Longitude, and in the Obliquity of the
Ecliptic.
Argument = The mean place of the Moon's node.
8
Long
Obliq.
88
Long.
Obliq.
0
14,30 + 0,09
10
20
30
40
50
60
70
8,16
80
90
100
110
120
0,06 +916250
1,04 9,14 260
2,12 9,09 270
3,16 9,00 280
4,20 8,88 290
5,22
8,72 300
6,23 8,53 310
7,20 8,31 320
8,06 330
9,08
7,77 340
9,97 7,46 350
10,82 7,11 360
11,63 6,74 370
12,40 6,34 380
13,12 5,91|| 390
13,80 5,46 400
14,42 4,99 410
14,98 4,50 420.
15,49 4,00 430
15,94 3,47 440
16,33 2,93 450
16,65
2,38 460
16,91 1,82 470
17,11 1,25 480
17,24
0,67 490
- 17,30
+ 0,09 500
130
17,29
17,21
17,07
16,85
16,57
16,23
15,81
15,33
14,79
14,19
13,53
12,82
12,05
11,23
10,37
9,46
8,51
7,52
6,51
5,47
4,40
3,32
2,22
1,09
0,00
- 0,49
1,07
1,65
2,22
2,79
3,34
3,88
4,41
4,92
5,41
5,88
6,33
6,75
7,14
7,51
7,85
8,15
8,43
8,67
8,87
9,04
9,17
9,26
9,32
- 9,34
140
150
160
170
180
190
200
210
220
230
240
250
TABLE XXI. continued.
167
Lunar Nutation in Longitude, and in the Obliquity of the
Ecliptic.
Argument = The mean place of the Moon's node.
3
Long.
Obliq.
B
Long.
Obliq.
500
510
520
530
540
550
560
570
580
590
600
610
620
9,34 750
9,32 760
9,26 770
9,17
780
9,04 790
8,87 800
8,67 810
8,43 820
8,15 830
7,85 840
7,51 850
7,14 860
6,75 870
6,33 880
5,88 890
5,41 900
4,92-|| 910
4,41 920
3,88 930
3,34 940
2,79 950
2,22 960
1.65 970
1,07 980
- 0,49 990
+ 0,09 | 1000
1,09
2,22
3,32
4,40
5,47
6,51
7,52
8,51
9,46
10,37
11,23
12,05
12,82
13,53
14,19
14,79
15,33
15,81
16,23
16,57
16,85
17,07
17,21
17,29
+ 17,30
+ 17,30 + 609
17,24 0,67
17,11 1,25
16,91 1,82
16,65 2,38
16,33 2,93
15,94 3,47
15,49 4,00
14,98 4,50
14,42 4,99
13,80 5,46
13,12 5,91
12,40 6,34
11,63 6,74
10,82 7,11
9,97 7,46
- 9,08
7,77
8,16 8,06
7,20 8,31
6,23 8,53
5,22 8,72
4,20 8,88
3,16 9,00
2,12 9,09
1,04 9,14
+ 0,00 + 9,16
630
640
650
660
670
680
690
700
710
720
730
740
750
168
TABLE XXII.
Solar Nutation in Longitude; and the solar nutation,
added to the mean diminution, of the Obliquity of the
Ecliptic.
Argument = The day of the year.
Day
Long.
Obliq.
Day
Long.
Obliq.
Jan.
1
11
1,03
21
31
Feb. 10
20
March 2*
12
22
April 1
11
+ 6,47 -0,50 July 10 +674 – 6,68
0,85 0,41
20
0,56
1,12 0,27
30 1,21 0,41
1,25 -0,10 August 9 1,25 0,24
1,22 + 0,08
19 1,16-0,08
1,05 0,24
29 0,93 + 0,06
0,74
0,36 Sept.
8 0,60 0,16
+0,35 0,43
18 + 0,20 0,21
- 0,08 0,44
28
- 0,23 0,20
0,50 0,39
Oct.
8 0,63 +0,12
0,85 0,27
18 0,96 - 0,01
1,11 +0,11
28 1,18 0,19
1,24 0,07 Nov. 7 1,25
0,39
1,23 0,27
17
1,18 0,59
1,08 0,45
27 0,96 0,77
0,81 0,60 Dec. 7 0,60 0,90
0,45 0,71
17 - 0,20 0,98
0,05 0,76
27 + 0,25
0,98
+ 0,37 - 0,75
37 + 0,66 - 0,93
21
May
1
11
21
31
June 10
20
30
N.B. The Longitude ought to be further corrected by – 0",207 sin 2):
and the Obliquity by + 0",090 cos 2).
* In Leap years, we must deduct unity from all these tabular dates after
February, in order to obtain the corresponding civil date.
TABLE XXIII.
169
Selenographic positions of the principal Lunar Spots.
No.
Riccioli's Names
Longe
Late.
1
+ 580
West 720
67
2
+ 40
3
64
24
Zoroaster
Mercurius
Petavius
Langrenus
Endymion
Cleomedes
4
62
8
5
60
6
55
+ 53
+ 26
+ 47
+ 48
7
Atlas
48
8
42
9
32
0
10
32
22
11
32
12
27
12
13
25
13
14
24
18
Hercules
CENSORINUS
Fracastorius
Possidonius
Theophilus
Cyrillus
St. Catharina
Menelaus
Aristoteles
Ptolomæus
Arzachel
Archimedes
Tycho
Plato
15
15
16
West 15
East 2
17
10
18
The enlightened part of the Moon's disc is denoted by the versed sine of the angular distance of the moon from the sun.
3
20
19
5
20
10
43
21
10
22
Pitatus
12
I t + + 11 + 1 + 1 + 1 + 11 + + 1 to 1 1 1
29
23
12
24
16
60
25
19
Eratosthenes
Clavius
Copernicus
Bullialdus
Blancanus
Heraclia
21
26
27
25
65
28
38
+ 41
+ 7
29
38
30
39
19
31
48
+ 24
Keplerus
Gassendus
Aristarchus
Hevelius
Schickardus
Grimaldus
32
67
1
33
68
49
5
34
East 68
N
1
170
'TABLE XXIV.
Showing the Angle of the Vertical, and the Logarithm of
the earth's Radius, at different Latitudes : the Compres-
sion of the earth being assumed equal to zás:
Lat.
Angle
of the
Vertical
Logarithm
of Earth's
Radius
0°
0.0000000
5
9.9999890
10
15
20
9.9999566
9.9999037
9.9998318
9.9997431
9.9996402
25
30
35
9.9995261
40
9.9994044
45
ó 6,0
1 59,2
3 54,8
5 43,4
7 21,6
8 46,5
9 55,4
10 46,4
11 17,9
11 28,7
11 18,6
10 47,9
9 57,4
8 48,7
7 23,8
5 45,4
3 56,3
2 0,0
0 0,0
50
9.9992786
9.9991525
9.9990302
9.9989151
55
60
65
70
75
80
9.99881ll
9.9987210
9.9986479
9.9985940
9.9985610
9.9985499
85
90
Greenwich | ií 11,6 | 9.9991159
TABLE XXV.
171
Showing the Augmentation of the Moon's Semidiameter,
on account of her apparent altitude.
Horizontal Semidiameter
Apparent
Altitude
14 30"
16 Ó | 16 30
6 | 1ể ở
16 16 304
16 304 1j o
00
3
6
9
12
15
18
21
24
27
30
6,00
0,75
1,50
2,25
3,00
3,74
4,46
5,18
5,88
6,56
7,23
7,88
8,50
9,10
9,68
10,23
10,76
11,26
11,72
12,15
12,55
12,91
33
36
39
42
0.00
0,71
1,41
2,11
2,81
3,50
4,17
4,84
5,49
6,13
6,75
7,35
7,93
8,49
9,03
9,55
10,05
10,52
10,95
11,35
11,72
12,06
12,37
12,64
12,88
13,08
13,24
13,37
13,46
13,52
13,54
45
600
0,80
1,60
2,40
3,20
3,99
4,76
5,52
6,27
7,00
7,71
8,40
9,07
9,72
10,34
10,93
11,49
12,02
12,52
12,98
13,40
13,79
14,14
14,46
14,73
14,96
15,15
15,30
15,41
15,47
15,49
6.00
0,86
1,71
2,56
3,41
4,25
5,07
5,89
6,68
7,46
8,22
8,96
9,67
10,36
11,02
11,65
12,25
12,81
13,34
13,83
14,29
14,70
15,08
15,41
15,70
15,95
16,15
16,31
16,42
16,49
16,51
0,00
0,92
1,83
2,73
3,63
4,52
5,39
6,26
7,11
7,93
8,74
9,52
10,28
11,02
11,72
12,39
13,03
13,63
14,19
14,72
15,20
15,64
16,04
16,39
16,70
16,96
17,18
17,35
17,47
17,54
-17,57
0,00
0,97
1,94
2,90
3,86
4,80
5,73
6,65
7,54
8,42
9,28
10,12
10,92
11,66
12,44
13,15
13,83
14,46
15,06
15,62
16,13
16,60
17,03
17,40
17,73
18,01
18,24
18,42
18,55
18,62
18,65
48
51
54
57
60
63
66
13,24
69
72
75
78
13,53
13,79
14,01
14,18
14,32
14,42
14,48
14,50
81
84
87
90
z 2
172
TABLE XXVI.
Logarithms for the equations of the first, second, and third
differences of the Moon's place.
Logarithms of the factors of the
Hour from
Noon or
Midnight
Hour from
Noon or
Midnight
First differences and diff.
3rd diff.
m
h m
0 10
h
11 50
20
+
8:1427
8.4437
8.6198
8.7447
8.8416
+
9.9939
9.9878
9.9815
40
30
30
40
20
9.9752
9.9687
50
10
1 0
8.9208
9.9622
llo
10
9.9559
50
8.9878
9.0458
20
9.9488
40
30
9:0969
9.9420
30
40
9.9351
20
9.1427
9.1841
50
9.9280
10
2
0
9.2218
9.9208
10
0
10
9.2566
9.9135
50
20
9.2888
9.9061
40
30
30
9.3188
9.8985
40
9.3468
9.8909
20
50
10
3
0
7.8356 + 7.0452
8.1304 7.3275
8.3003 7.4843
8.4189 7.5896
8.5094 7.6663
8.5820 7.7247
8.6423 7.7703
8.6936 7.8063
8.7379 7.8348
8.7767 7.8572
8.8110 7.8745
8.8416 7.8874
8.8691 7.8964
8.8939 7.9018
8.9163 7.9040
8.9366 7.9032
8.9551 7.8994
8.9720 7.8928
8.9873 7.8833
9.0013 7.8710
9.0141 7.8557
9:0257 7.8374
9.0362 7.8157
9.0458 7.7905
9.0543 7.7613
9.0620 7.7276
9.0689 7.6887
9.0749 7.6436
9.0802 7.5908
9:0847 7.5284
9.0884 7.4230
9:0915 7.3591
9.0939 7.2366
9.0956 7.0621
9.0966 6-7621
9
0
50
10
20
40
30
30
40
20
50
10
4 0
8
0
50
10
20
40
30
9:3731 9.8830
9.3979 9.8751
9.4214 9.8669
9.4437 9.8587
9.4649 9.8502
9.4851 9.8416
9.5044 9.8329
9-5229 | 9.8239
9:5406 9.8148
9.5577 9.8054
9.5740 9.7959
9.5898 9.7861
9.6051 9-7761
9.6198 9:7659
9.6340 9.7555
9.6478 9.7447
9.6612 9.7337
9.6741 9.7225
9.6867 9.7109
9.6990 9.6990
+
+
30
40
20
50
5 0
10
10
7 0
50
40
30
20
30
40
20
50
6 0
10
6 0
9.0969 +
TABLE XXVII.
173
Showing the Annual Precession of the Equinoxes in Lon-
gitude: and the Constants for computing the annual
precession in Right Ascension and Declination.
Year
Precession in Longitude
Prec. in
A
General Luni-solar const. =m
Prec. in
R and D
const. n
1800 5022350
50%22350 50 36354 46,04367 20705690
1805 | 50,22472 50,36232 46,04521 20,05641
1810 50,22594
50,22594 50,36110 46,04676 20,05593
1815 | 50,22716 50,35988 46,04830 20,05544
1820 50,22839 50,35866 46,04984 20,05496
1825 50,22961 50,35745 46,05138 20,05447
1830 | 50,23083 50,35623 46,05293 20,05399
1835 50,23205 50,35502 46,05447 20,05350
1840 / 50,23328 50,35380 46,05601 20,05302
1845 | 50,23450 50,35258 46,05755 20,05253
1850 | 50,23572 50,35136 46,05910 20,05205
1855 | 50,23694 50,35014 46,06065 20,05156
1860 50,23816 50,34892 46,06219
46,06219 20,05108
1865 | 50,23938 50,34771 46,06374 20,05059
1870 50,24060 50,34649 46,06528 20,05011
1875 50,24182 50,34527 46,06682 | 20,04962
1880 50,24305 50,34405 46,06836 20,04914
1885 | 50,24427 50,34283 46,06991 20,04863
1890 50,24549 50,34162 46,07115 20,04817
1895 50,24671 50,34040 46,07299 20,04768
1900 | 50,24793 50,33918 46,07454 | 20,94720
Annual Prec, in R = m + n.sin R.tan D
Annual Prec. in D = n. cos R
174
TABLE XXVIII.
For determining the Aberration of a star in R, and the
first part of the Aberration of a star in Declination.
Argument, O = the true longitude of the sun.
Os
VIS
IS
VIIS
IIS
VIIIS
Log. a
A+
Log. a
A+
Log, a
At
0
1.2690
1
1.2690
2
1•2691
3 1.2692
0 37
36 2
21
4 | 1•2692
5 | 1.2693
6 | 1.2695
7 | 1.2696
8 1.2698
9 | 1.2700
10 1•2703
11 1.2705
12 1.2708
13 ].2711
14 1.2714
15 1•2718
16 | 1.2721
17 | 1.2725
18 1•2729
19 1.2733
20 1.2738
21 1•2742
22 1.2747
23 | 1.2752
24 1.2757
25 1.2762
26 1.2768
27 | 2•2773
28 1.2779
29 1.2785
30 | 1•2790
0 0 || | |
1.2790 2 11 1.2977
2 6 30
05 1.2796 2 14 1•2983 2 3 29
0 11 1•2802 2 16 1.2988 2 0 28
0 16 1.2808 2 18 1.2993
1 57 27
0 22 1.2815 2 20 1.2998 1 54 26
0 27 1•2821 2 21 1.3003 1 51 25
0 32 1:2827 2 23 1.3008 1 47 24
1.2834
2 24 1.3012 1 44 23
043 1.2840 2 25 1•3017 | 1 40 22
0 48 1.2847 2 26 1.3021 1 36
0 53 1.2853 2 27 1:3025 1 32 20
0 58 1.2860 2 28 1.3028 1 28 19
1 3 1.2866 2 28 1.3032 1 24 18
1 8 1.2873 2 28 1.3036 1 20 17
1 12 1•2879 2 28 1.3039 1 16 16
1 17 1.2886 2 28
1.3042 1 11 15
1 22 1•2892 2 28 1.3045
1 7 14
1 26 1.2899 2 27
1.3048 1 3 13
1 30 1.2905
2 27
1.3050 0 58 12
1 34 1•2912 2 26
1:3053 0 53 11
1 39 1.2918 2 25 1.3055 0 49 10
1 42 1.2924 2 24 1•3057 044 9
1 46 1.2931 2 22 1.3059 0 39 8
1 50 1.2938
7
2 21 1.3060 0 34
1 53 1.2944 2 19 1.3061 0 30 6
1 57 1.2949 2 17 1.3063 0 25 5
2 0 1.2956 2 15 1.3064 0 20 4
2 3 1.2961 2 13 1.3064 0 15
3
2 6 1.2966 2 11 1.3065 0 10
2 9 1.2972 2 8 1.3065 0 5 1
2 11
1.2977 2 6 1.3065 0 0 0
-H
Log. a
A-
Log. a
A
Log. a
A
V8
XIS
IV:
XS
IIIS
IXs
TABLE XXIX.
175
For the second part of the Aberration of a star in Dec.
Arguments, (O + D) and (0 – D).
(): VIS Is VIIS IIS VIIIS
+
+
+
u
30°
1
29
2
28
3
27
4
26
5
25
6
24
7
23
8
22
9
21
10
20
11
19
12
18
13
17
14
16
15
4,03
4,03
4,03
4,03
4,02
4,02
4,01
4,00
3,99
3,98
3,97
3,96
3,95
3,93
3,91
3,90
3,88
3,86
3,84
3,81
3,79
3,77
3,74
3,71
3,68
3,66
3,63
3,59
3,56
3,53
3,49
3,49
3,46
3,42
3,38
3,34
3,30
3,26
3,22
3,18.
3,13
3,09
3,04
3,00
2,95
2,90
2,85
2,80
2,75
2,70
2,65
2,59
2,54
2,48
2,43
2,37
2,31
2,26
2,20
2,14
2,08
2,02
2,02
1,96
1,89
1,83
1,77
1,70
1,64
1,58
1,51
1,45
1,38
1,31
1,25
1,18
1,11
1,04
0,98
0,91
0,84
0,77
0,70
0,63
0,56
0,49
0,42
0,35
0,28
0,21
0,14
0,07
0,00
15
16
14
13
17
18
12
19
11
20
10
21
9
22
8
23
7
6
24
25
5
26
4
27
3
2
28
29
1
30
0
+
1
+
+
Vs XI: IV8 Xs | IIIs IXs
176
TABLE XXX.
For the Lunar Nutation of a star in R and Dec.
Argument, 8 = the mean longitude of Moon's node.
Os
VIS
IS
VIIS
ITS
VIII
B
с
B
Log. b
Log. 6
Log. 6 |--+
.
+
+
0|| 0-9844 18 0.00 0.9588 6 46
0-9588 8 45 827 0-8960 * 48 14 33 | 30
1 .9844 0 15 0,29 •9571 6 54 8,52 .8939 17 40 14,47 ||29
2 •9843 0 31 0,58 •9554 17 3 8,77 .8917 17 32 14,61 ||28
3
•9842 0 46 0,87 •9536 7 12 9,01 .8896 7 23 14,74 |27
4 •9840 1 1 1,15
•9518 7 20 9,25
•8875 7 14 14,87' | 26
5 .9837 1 16 1,44 .9500 17 28
1,49 •8854 7 4 14,99 | 25
6 •9834 1 32 1,73 •9481 7 36 9,72
.8834 6 53 15,11 ||24
7 .9830 1 47 2,02 •9462 17 43 9,96
•8814 6 42 15,23 ||23
8 •9825 2 2 2,30 -9442 17 49 10,19 .8795 6 29 15,34 || 22
9 .9821 2 172,59 -9422 7 55 10,41 .8776 6 17 15,45 ||21
10 9815 2 312,87
•9402 8 1 10,63 .8758 6 3 15,55 || 20
11 9809 2 463,16 •93828 6 10,85 .8740 5 49 15,64 | 19
12 •9802 3 ] 3,44 .9361 8 10 11,07 .8723 5 35 15,73 | 18
13 •9795 3 153,72
9340 8 14 11,28 .8707 5 20 15,82 17
14 .9787 3 29 4,00
9318 18 17 | 11,49 •8691 5 4 15,90 || 16
15 •9779 13 43 4,28
9297 8 20 11,70 .8677 1 4 48 15,98 | 15
16 •9770 3 574,56
.9275 18 23 11,90 .8663 4 31 16,05 ||14
17 •9760 4 114,84
.92538 24 12,10 •8649 | 4 14 16,12 | 13
18 .9750 4 24 5,11 •9231 8 25 | 12,30 •8637 3 56 16,18 12
19 •9739 4 37 5,39 •9208 8 25 | 12,49 •8625 3 38 16,24 11
20 •9728 4 505,66 .9186 8 25 12,67 .8615 3 20 16,29 10
21 •9716 5 3 5,93 9163 8 24 12,86 .8605 3 1 16,34 9
22 9704 15 16 6,20 9140 8 23 13,04 .8596 2 41 16,38 8
23 •9691 5 28 6,46 .9118 8 21 13,21 •8588 2 22 16,42
7
24 .9678 5 406,73 •9095 8 18 13,38 •8582 2 2 16,45
6
25 •9664 5 51 6,99 9072 8 15 13,55 •8576 1 42 16,48 5
26 .9650 6 37,25 9050 8 11 13,72 .8571 |1 22 | 16,50 4
27 •9635 16 14 7,51 •9027 18 613,88 •8568 1 2 16,52 3
28 .9620 6 247,77 •9005 8 1 14,03 •8565 0 41 16,53 2
29 •9604 6 358,02 •8983 7 55 | 14,18
1
•8563 0 21 16,54
0
30 || 0.9588 6 45 8,27 0.8960 7 48 14,33 0·8563 0 0 16,54
+
+
Log. 6
+
B
+
B
-*
Log. 6
+
B
Log. 6
с
c
Vs
XIS
IVs
Xs
IIIS
IX®
TABLE XXXI.
177
For the Solar Nutation of a star in Right Ascension and
Declination.
In Right Ascension
In Dec.
1st part
2nd part
Degrees
Degrees
Argument Argument Argument
= 20 -20 A 20 R
09
360°
10
350
20
340
30
330
40
320
50
310
60
300
70
290
80
280
90
- 0,00+ 647-1 - 6,00 +
0,18
0,46 0,08
0,35 0,44 0,16
0,51
0,41 0,24
0,66 0,36
0,30
0,79 0,30 0,36
0,89 0,24 0,41
0,96 0,16 0,44
1,01 - 0,08
0,46
1,03 0,00 0,47
1,01 + 0,08 + 0,46
0,96 0,16 0,44
0,89 0,24 0,41
0,79 0,30 0,36
0,66 0,36 0,30
0,51 0,41 0,24
0,35 0,44 0,16
0,18 0,46 0,08
0,00 + 1 to 0,47 + 0,00 +
270
100
260
110
250
120
240
230
130
190
220
150
210
160
200
170
190
180
180
The second part of the solar nutation in R must be multiplied by the
tangent of Declination.
2 A
178
TABLE XXXII.
Circular Arcs.
Degrees
10.0174533 31° 0.5410521
|
61 | 1.0646508 95 1.6580628
2 0.0349066 32 10:5585054 62 1.0821041 100 1.7453293
30.0523599 33 0.5759587 63 1.0995574 105 | 1.8325957
4 0.0698132 34 0·5934119 64 1.1170107 110 1.9198622
5 0.0872665 | 35 0.6108652 65 1.1344640 115 2.0071286
6 0.1047198 36 0.6283185 66 1•1519173 120 2.0943951
7 10.1221730 37 0.6457718 67 1.1693706 i30 2.2689280
8 0.1396263 38 0.6632251 68 1•1868239 140 2:4434610
9 0.1570796 39 0.680678469 1.2042772 150 2:6179939
10 0.1745329 | 40 0.6981317
40 0.698131770 1.2217305 160 2:7925268
11 0.1919862 | 41 0.7155850
41 0.7155850 | 71 1.2391838 170 2.9670597
12 0.2094395 420-7330383 72 1.2566371 180 3:1415927
13 0.2268928 43 0.750491673 1.2740904 1903.3161256
14 0.2443461 44 0.767944974 1.2915436 200 3.4906585
15 0.2617994 45 0.7853982 75 1:3089969 210 | 3.6651914
16 0.2792527 46 0-8028515 76 1:3264502 220 3.8397243
17 0.2967060 47 0.8203047 | 77 1.3439035 230 4.0142573
18 0.3141593 48 0.837758078 1.3613568 240 | 4:1887902
19 0.3316126 49 0.8552113 79 1:3788101 250 | 4:3633231
20 | 0.3490659 50 0.8726646 80 1.3962634 260 4:5378561
21 10.3665191 51 0.8901179 81 1.4137167 270 4:7123890
22 0.3839724 52 0-907571282 1.4311700 280 4.8869219
23 0.4014257 53 0.9250245 | 83 1:4486233 290 5:0614548
24 (1.4188790 54 0.942477884 1.4660766 3005.2359878
25 0.4363323 55 0.9599311 85 1.4835299 310 5.4105207
26 0.4537856 56 0.9773844 86 1:5009832 320 5.5850536
27 10:4712389 57 0.9948377 | 87 1.5184364 330 5.7595865
28 0.4886922 58 1.0122910 88 1.5358897 340 5.9341195
29 0.5061455 59 1.0297443 89 1.5533430 350 6.1086524
30 0.5235988 60 1:0471976 90 1:5707963 360 6.2831853
TABLE XXXII. continued.
179
Circular Arcs.
Minutes
Seconds
í 0-0002909 31 0-0090175
2
.0093084
3
.0005818 32
•0008727 33
0011636 34
-0095993
0098902
1' 0.0000048 31" 0.0001503
2 0000097 32 •0001551
3 0000145 33 0001600
4 •0000194 | 34 0001648
5 .0000242 35 .0001697
4
5
0014544
35
•0101811
6
7
8
·001745336
0020362 | 37
·0023271 38
·0026180 39
•0104720
0107629
•0110538
6 0000291 36
7 0000339 37
8 0000388 38
9 •0000436 39
10-0000485 40
0001745
•0001794
0001842
0001891
.0001939
9
•0113446
10
·0029089
40
•0116355
11
11
0001988
0002036
12
12
13
0031998 41
·0034907 42
·0037815 43
•0040724 44
0043633 45
•0119264
•0122173
•0125082
0127991
(130900
13
•0000533 41
•0000582 42
•0000630 43
•000067944
·0000727 45
.0002085
14
14
0002133
15
15
0002182
16
•0046542
46
•0133809
16
.0002230
17
47
17
0049451
·0052360
0136717
0139626
18
48
18
•0000776 46
0000824 | 47
000087348
·0000921 49
·000097050
•0002279
·0002327
.0002376
•0002424
19
·0055269
49
0142535
19
20
•0058178 50
•(145444
20
21
0148353
21
22
0151262
22
•0061087 51
·0063995 52
0066904 53
.0069813 54
.0072722 55
23
.0002473
·0002521
·0002570
0002618
23
·0001018 51
·0001067 | 52
0001115 53
0001164 54
000121255
.0154171
.0157080
•0159989
24
24
25
25
-0002666
26 ·0075631 56 0162897
27 .007854057 0165806
28 0081449 58 0168715
29 •0084358 59 .0171624
30 10.008726660 0.0174533
26 0001261 56 0002715
27
0001309 | 57 .0002763
28 .0001357 | 58 .0002812
29 .0001406 59
•0002860
30 0.0001454 600.0002909
2 A 2
180
TABLE XXXIII.
Semidiurnal Arcs.
Declination
Lat.
1
5
10
15
20
25
30
h
m
h
m
h
m
m
h
m
5
h
6 9
h m
6 12
10
6 19
6 24
15
6 29
29
6 36
6 49
20
2 6 30
6 39
25
6 50
30
35
h
6 0 6 2 6 4 6 5 6 Ź
6 1 6 4 6 7 1 6 11 6 15
6 1 6 5 6 11 6 16 6 22
6 1 6 7 6 15 6 22
6 2 6 9 6 19 6 29 6 39
6 2 6 12 6 23 36 366 49
6 3 6 14 6 28 6 43 6 59
6 3 6 17 6 34 / 6 52 7 11
6 4 1 6 20 6 41 7 2 7 25
6 5 1 6 24 6 49 1 7 14
7 14 7 43
6 6 6 29 6 58 7 30 8 5
6 7 1 6 35 7 11 | 7 51 8.36
6 9 6 43 1 7 29 8 209 25
7 2
7 18
7 35
7 56
8 21
7 2
7 16
7 32
7 51
8 15
40
45
50
8 54
55
8 47
9 42
60
9 35
12 0
65
12 0
TABLE XXXIV.
181
Showing the length of a degree of Longitude and Latitude
on the Earth's surface; and the length of the Pendu-
lum beating seconds there: the measures at the equator
being considered as unity. Also the increase in the
number of vibrations, of an invariable pendulum beating
seconds at the equator, on proceeding towards the pole.
Compression of the Earth = zó.
Late.
Degree of Degree of Length of the
Longitude Latitude Pendulum
Increase
of Vibrations
ở
1.00000
1.000000
1.00000
5
0.99622
1.00004
1.000076
1.000301
10
98490
15
.96614
1.000669
1:00016
1.00036
1.00063
20
94006
1.001168
1.001783
25
.90685
30
1.002496
•86675
.82005
1.00096
1.00129
1.00177
1.00223
35
1.003284
40
1.004125
•76710
•70828
45
1.004992
1.00269
0,00
1,73
7,02
18,75
27,24
41,59
55,59
76,60
96,21
116,42
143,08
156,25
174,63
191,26
205,61
217,25
225,82
231,08
232,85
50
.64404
1.005858
1.00331
1.00362
55
•57485
1.006699
60
•50126
1.00404
1.007487
1.008200
65
•42377
1.00443
34302
1.008815
70
75
80
1.009315
1.00476
1.00503
1.00523
1:00535
•25960
•17421
•08764
0.00000
1.009682
85
1.009907
90
1.009983
1.00539
182
TABLE XXXV.
Showing the linear expansion of various substances for one
degree of Fahrenheit's thermometer.
White deal
0000022685
Kater
43119
Glass, barom. tubes
, English flint
with lead
Roy
Lavoisier
45092
48444
do.
ſi
without lead.
do.
50973
47583
Platina
49121
Iron, cast
61632
63333
65668
Borda
Dulong and Petit
Roy
Borda
Dulong and Petit
Hasslar
Smeaton
Lavoisier
Roy
bar
69844
69907
Steel
59305
.
rod
63596
blistered
63900
Smeaton
111
68056
do.
68866
Lavoisier
hard.
tempered
Gold
Copper
Brass
86197
do.
94444
95456
Smeaton
Dulong and Petit
Borda
98888
..
Kater
scale
99590
·0000103077
104166
-
cast
bar
104850
.
Roy
Smeaton
Hasslar
Roy
Smeaton
Lavoisier
rod
105155
107407
106038
i
126852
Smeaton
137963
do.
159259
do.
wire.
Silver
Pewter
Tin, grain
Lead
Zinc .
hammered .
Mercury (Apparent)
(Absolute)
Air, dry
moist
163426
do.
do.
1
6480
172685
•0000857339
•0001001001
·0020833333
0022222222
cubical
expan.
1
5520
TABLE XXXVI.
183
For determining Altitudes with the Barometer.
Thermometers
in open air
Thermometers
to the Barom
Latitude
of the place
ttt
A
ttt!
A
7-
B
C
0.00000
ò
0.00117
5
0.00115
0.00004
0.00009
0.00013
10
0.00110
15
0.00100
20
0.00017
0.00022
0.00026
25
30
0.00090
0.00075
0.00058
0.00040
0.00020
0.00000
0.00030
35
0.00035
40
0.00039
45
0.00043
50
9.99980
0.00048
55
9.99960
0.00052
60
9.99942
0.00056
65
9.99925
0.00061
9.99910
40 4.76891 102 4.79860 ò
42 4.76989 104 | 4:79953 1
44 | 4.77089 106 | 4.80045 2
46 4.77187 108 4.80137 3
48 | 4.77286 110 4.80229 4
50 | 4.77383 1124.80321 5
52 | 4.77482 1144.80412 6
54 4:77579 116 | 4.80504
7
56 4.77677 118 4.80595 8
58 | 4:77774 120 4.80687 9
60 | 4.77871 122 | 4.80777 10
62 4.77968 124 | 4:80869 11
64 | 4.78065 126 | 4.80958 12
66 4.78161 128 4.81048 13
68 | 4.78257 130 4.81138 14
70 4.78353 132 4.81228 15
72 4.78449 134 | 4.81317 16
74 4.78544 136 4.81407 17
76 | 4.78640 138 | 4.81496 18
78 4:78735 140 | 4.81585 19
80 4.78830 142 | 4.81675 20
82 | 4.78925 144 4.81763 21
84 4:79019 146 4.81851 22
86 4.79113 148 4.81940 23
88 4.79207 150 | 4.82027 24
90 4.79301 152 | 4.82116
92 4.79395 154 | 4.82204 26
94 | 4.79488 156 4.82291 27
96 | 4.79582 158 4.82379 | 28
98 | 4.79675 160 4.82466 29
100 4.79768 162 4.82353 30
0.00065
70
75
80
9.99900
0.00069
9.99890
85
9.99885
90
9.99883
0.00074
0.00078
0.00083
0.00087
0.00091
0.00096
0.00100
0.00104
25
0.00109
Make D equal to
log B-log B'+B):
then will the loga-
rithm of the differ-
ence of altitudes
in English feet be
equal to
A + C + log D
0.00113
0.00117
0.00122
0.00126
0.00130
184
TABLE XXXVII.
For converting time into decimal parts of a day.
Hours
Minutes
Seconds
h
1.00001
1.04167
2.08333
2 | .00002
1 00069 31 | .02153
2 | .00139 32 ·02222
3.00208 33 02292
4 .00278 34.02361
5.00347 | 35 02430
3.12500
31' | .00036
32.00037
33 •00038
3
·00003
4 | .16667
4
•00005
34 | .00039
5 20833
5
·00006
35
00040
6 | 25000
36
·00042
6.00007
7 •00008 37
•00043
8 00009
6.00417 36.02500
7 | .00486 | 37 ·02569
800556 38 02639
9.00625 39 02708
10.00694 40
40.02778
38
7.29167
8 .33333
9 •37500
10.41667
00044
39
00045
9 | .00010
10 00012
40
•00046
11 •45833
12 •50000
13 •54167
14 | .58333
11 00764 41 ·02847
12 .00833 42 | .02917
13 | .00903 43 ·02986
14 .00972 44.03056
15 01042 45 03125
11 00013 41 .00047
12 00014 42.00049
13 00015 43 | .00050
14 ·0001644
44.00051
15 | .00017 | 45
00052
15.62500
16 | .66667
17 | .70833
18.75000
19.79167
16 •01111 46 | .03194
17 | 01180
•01180 | 47 | .03264
18 | 01250 48 .03333
19 01319 49 03403
20 •01389 | 50
50 .03472
16 .0001846 00053
17 .00020
•00020 | 47 00054
18 ·00021 48 00056
19.00022 49.00057
20 •00023 50 ·00058
20.83333
21.00024
51
00059
22.00025
52
00060
21 87500
22.91667
23.95833
24 1.00000
21 .01458 51 •03542
22 01528 52 .03611
23.01597 53 •03680
24.01667 54.03750
25 / 01736 55 03819
23.00027
53
.00061
24 | .00028
54
-00062
25
*00029
55
00064
26.01805 56 | .03889 26 .00030 56 00065
27 | 01875 57 •03958 27 | .00031 57 00066
28 01944 58 04028 28.00032 58 | .00067
29 •02014 59 | .04097 | 29 ·00034 59 00068
30 ·02083 60.04167 30.00035 60 ·00069
TABLE XXXVIII.
185
For converting Minutes and Seconds of a degree, into the
decimal division of the same.
Minutes
Seconds
1 .01667 | 31 -51667
2 .03333 32 •53333
3.05000 33 55000
4.06667 34.56667
5 •08333 35 •58333
1.00028 31' .00861
2.00056 32 00889
3 00083 | 33 | .00917
4
34.00944
5.00139 35 .00972
·00111 34.
6.10000 | 36 .60000
7 •11667 | 37 | .61667
8.1333338 •63333
9 •15000 | 39 •65000
10.16667 40 | .66667
6 .00167 36 .01000
7 ·00194 | 37 | .01028
8 ·0022238 •01056
9.00250 39
39.01083
·0027840 .01111
11 •18333 41 .68333
12 | 20000 42 70000
•
13 .21667 43 .71667
14 | .23333 44 .73333
15 | 25000 45.75000
45
11 ·00306 41 •01139
12 | .00333 42
42 | .01167
13 •00361 43 .01194
14 ·00389 44 | .01222
15 •00417 | 45 •01250
614
16.26667 | 46.76667
17 | .28333 47.78333
18.30000 | 48 .80000
19 .31667
49.81667
20 •33333 | 50 .83333
16 00444 46.01278
17 | .00472 47 1 .01306
18 00500 48 01333
19.00528
49 .01361
20.00556 50 01389
18
1
21
•35000
22 -36667
51 •85000
52 | .86667
53 .88333
23
38333
21 ·00583 51 •01417
22.00611 52 01444
23 | .00639 53.01472
24.00667 54 | .01500
25 | .00694 55 01528
24 | •40000
54 | .90000
25
•41667
55.91667
26 ·43333 | 56 .93333 26.00722 56.01556
27 •45000 57.95000
•950Q! || 27 | .00750 | 57 .01583
28 •46667 | 58 | .96667 28 .00778 58 .01611
29 .48333 59 .98333 29 .00806 59 01639
30.50000 60 1,00000 30 .00833 60 .01667
2 B
186
TABLE XXXIX.
For converting any given day into the decimal part of a
year of 365 days.
Day
Jan,
Feb,
March
April
May
June
1
.000
085
•162
•329
•414
•247
.249
2
·003
088
•164
•331
•416
3
006
090
.252
•334
•419
4
•008
093
•255
337
•422
5
•011
096
•340
425
.167
•170
•173
•175
•178
•181
.258
•260
6
·014
·099
342
•427
7
016
•101
•263
345
-430
8
:]04
•266
*348
•433
019
022
9
•107
•184
• 268
351
•436
10
.025
•110
.186
353
• 438
11
027
•112
.189
•271
•274
277
356
1
•441
12
•030
•115
•192
359
•444
13
033
•118
•195
•279
•362
•446
14
•036
•121
.282
•364
•449
•197
•200
15
•038
•123
285
•452
16
041
•126
.203
.288
• 455
17
044
•129
•205
•290
.367
370
•373
•375
•378
(
•458
18
046
•132
.208
•460
•293
•296
19
049
134
•211
•463
20
052
•137
214
.299
.381
•466
21
055
140
216
301
384
•468
22
058
•142
•219
•304
•386
23
060
•145
• 222
•307
.389
•471
•474
•477
•479
24
•148
.225
•310
.392
•063
066
25
•151
.227
•312
395
26
•068
•153
•230
•315
•482
•397
•400
27
•156
•233
318
.485
28
•159
.236
•321
•403
•488
29
•071
.074
077
•079
•082
•238
323
•405
•490
30
.241
326
•408
•493
31
.244
•411
TABLE XXXIX. continued.
187
For converting any given day into the decimal part of a
year of 365 days.
Day
July
August
Sept.
Oct.
Nov.
Dec.
1
•496
•581
.666
•748
.833
•915
2
•499
584
.668
.836
918
3
•501
•586
.838
.921
4
•504
589
.841
.923
5
•507
592
.671
.674
-677
•679
.682
.844
.926
6
•510
•595
.846
929
7
•512
•597
.849
931
8
•515
.600
.685
.852
934
9
•518
•603
•688
.855
•937
.940
10
•521
.605
.690
.858
•751
•753
•756
•759
•762
•764
•767
770
773
•775
778
•781
•784
*786
•789
•792
•795
•797
11
•523
•608
.693
.860
942
12
.526
.611
.696
.863
•945
13
•529
.614
.699
.866
.948
14
•532
.616
.868
.951
15
.534
.619
953
16
•537
.622
956
17
•540
.625
.871
.874
.877
.879
.882
959
18
•542
.627
962
19
•545
•630
.964
20
•548
.633
.800
.885
21
•551
.636
.803
.888
22
•553
•638
•701
•704
•707
•710
•712
•715
•718
•721
•723
•726
•729
•731
•734
•737
•740
•742
•745
.890
.805
.808
.967
•970
973
•975
978
23
.556
.641
.893
24
•559
.644
.811
.896
25
•562
•647
814
.899
981
26
•564
.649
.816
.901
•984
27
.652
.819
904
.986
28
.655
.822
.907
.989
29
•567
•570
•573
•575
•578
.658
910
.992
30
-660
.825
.827
.830
912
.995
31
.663
.997
2 B 2
188
TABLE XL.
30° 36
8
For converting the centesimal division of the quadrant
into the sexagesimal division of the same.
Centesimal degrees.
Centes. Sexages. Centes.
Centes. Sexages. Centes. Sexages.
1 0° 54
34
67
60° 18
2
1 48 35 31 30 68 61 12
3 2 42 36 32 24 69 62 6
4 3 36 37 33 18 70 63 0
5 4 30 38 34 12 71 63 54
6 5 24 39 35 6
72
64 48
7 6 18 40 36 0 73 65 42
8 7 12 41 36 54 74 66 36
9
6 42 37 48 75 67 30
10 9 0 43 38 42 76 68 24
11 9 54 44 39 36 77 69 18
12 10 48 45 40 30 78 70 12
13 11 42 46 41 24 79 71 6
14 12 36
42 18 80 72 0
15 13 30 48 43 12 81 72 54
16 14 24
49
44 6
73 48
17 15 18 50 45 0 83 74 42
18 16 12
45 54 84 75 36
19 17 6 52 46 48 85 76 30
20 18 0 53 47 42 86 77 24
21 18 54 54 48 36 87 78 18
22 19 48
55
49 30 88 7912
23 20 42
50 24 89 80 6
24 21 36
51 18 90 81 0
25 22 30 58 52 12 91 81 54
26 23 24 59 53 6 92 82 48
27 24 18 60 54 0 93 83 42
28 25 12 61 54 54 94 84 36
29 6 6 62 55 48 95 85 30
30 27 0 63 56 42
96 86 24
31 27 54 64 57 36 97 87 18
32
28 48
65 58 30 98 88 12
47
82
51
56
57
33
29 42
66
59 24
99
89
6
34
30 36
67
60 18
100
90
0
TABLE XL. continued.
189
For converting the centesimal division of the quadrant
into the sexagesimal division of the same.
Centesimal minutes.
Centes. Sexages. Centes.
Sexages.
Centes.
Sexages.
.01
0 32,4
34
36 10,8
•67
.68
02
35
03
•36
.69
04
:37
.38
•05
•06
.39
•40
•07
.08
•41
•70
•71
.72
•73
•74
•75
•76
•
•78
•79
09
•42
18' 21,6
18 54,0,
19 26,4
19 58,8
20 31,2
21 3,6
21 36,0
22 8,4
22 40,8
23 13,2
23 45,6
24 18,0
24 50,4
25 22,8
25 55,2
26 27,6
27 0,0
27 32,4
•10
•43
•11
•44
ל7•
0
12
•45
•13
•46
•14
.80
•47
• 48
•15
.81
•16
•49
.82
50
.83
1 4,8
1 37,2
2 9,6
2 42,0
3 14,4
3 46,8
4 19,2
4 51,6
5 24,0
5 56,4
6 28,8
7 1,2
7 33,6
8 6,0
8 38,4
9 10,8
9 43,2
10 15,6
10 48,0
11 20,4
11 52,8
12 25,2
12 57,6
13 30,0
14 2,4
14 34,8
15 7,2
15 39,6
16 12,0
16 44,4
17 16,8
17 49,2
18 21,6
•17
.18
•51
•84
36 43,2
37 15,6
37 48,0
38 20,4
38 52,8
39 25,2
39 57,6
40 30,0
41 2,4
41 34,8
42 7,2
42 39,6
43 12,0
43 44,4
44 16,8
44 49,2
45 21,6
45 54,0
46 26,4
46 58,8
47 31,2
48
3,6
48 36,0
49 8,4
49 40,8
50 13,2
50 45,6
51 18,0
51 50,4
52 22,8
52 55,2
53 27,6
54 0,0
•19
.52
28
4,8
.85
.20
•53
.86
.21
•54
.87
.22
•55
.88
23
•56
.89
.24
•57
.90
• 25
•58
.91
•26
•59
•92
•27
.60
28 37,2
29 9,6
29 42,0
30 14,4
30 46,8
31 19,2
31 51,6
32 24,0
32 56,4
33 28,8
34 1,2
34 33,6
35 6,0
35 38,4
36 10,8
93
•28
.61
94
•29
.95
.62
-63
.30
.96
31
•64
•97
.98
.32
.65
•33
•66
.99
34
•67
1.00
190
TABLE XL. continued.
For converting the centesimal division of the quadrant
into the sexagesimal division of the same.
Centesimal seconds.
Centes,
Sexages.
Centes.
Sexages.
Centes.
Sexages.
0001
0034
•0067
•0002
•0035
0068
-0003
•0069
.0036
0037
0004
.0005
0038
0006
0039
0007
0040
-0008
0041
i
.0009
0042
0070
.0071
•0072
•0073
.0074
.0075
·0076
-0077
·0078
0079
0080
0081
0010
•0043
0011
·0044
0012
.0045
.0013
0046
·0014
-0047
·0015
•0048
·0016
0049
0082
·0017
•0050
·0083
0,324
0,648
0,972
1,296
1,620
1,944
2,268
2,592
2,916
3,240
3,564
3,888
4,212
4,536
4,860
5,184
5,508
5,832
6,156
6,480
6,804
7,128
7,452
7,776
8,100
8,424
8,748
9,072
9,396
9,720
10,044
10,368
10,692
11,016
11,016
11,340
11,664
11,988
12,312
12,636
12,960
13,284
13,608
13,932
14,256
14,580
14,904
15,228
15,552
15,876
16,200
16,524
16,848
17,172
17,496
17,820
18,144
18,468
18,792
19,116
19,440
19,764
20,088
20,412
20,736
21,060
21,384
21,708
21,708
22,032
22,356
22,680
23,004
23,328
23,652
23,976
24,300
24,624
24,948
25,272
25,596
25,920
26,244
26,568
26,892
27,216
27,540
27,864
28,188
28,512
28,836
29,160
29,484
29,808
30,132
30,456
30,780
31,104
31,428
31,752
32,076
32,400
·0018
·0051
0084
·0019
·0052
0085
0020
0053
0086
·0021
·0054
·0087
0088
0022
·0055
·0023
·0056
0089
0024
0057
·0090
·0025
·0058
0091
·0026
·0059
0092
0060
.0093
·0027
·0028
·0029
0061
0094
0062
•0095
·0030
•0063
0096
-0031
•0064
0097
0032
-0065
·0098
0033
.0066
.0099
0034
•0067
·0100
TABLE XLI.
191
Comparison of Fahrenheit's thermometer, with Reaumur's
and the Centesimal.
Fahr. Reaum. Centes. Fahr. Reaum. Centes. Fahr. Reaum. Centes.
le
33 +0,4 +0,6
+0,6 67 + 15,6 + 19,4
8 -142 -17,8 34 0,9 1,1 68 16,0 20,0
1 13,8 17,2 35 1,3 1,7 69 16,4 20,6
2
13,3 16,7 36 1,8 2,2 70 16,9 21,1
3 12,9 16,1 37
2,8 71 17,3 21,7
4
12,4 15,6 38 2,7 3,3 72 17,8 22,2
5 12,0 15,0 39 3,1 3,9 73 18,2
6 11,6 14,4
40 3,6 4,4
74 18,7 23,3
7 113] 13,9 41 4,0 5,0 75 19,1 23,9
8
10,7 13,3 42
4,4
5,6 76 19,6 24,4
9
10,2 12,8 43 4,9 6,1 77 20,0 25,0
10 9,8 12,2
44 5,3 6,7
78
20,4 25,6
11 9,3 11,7 45 5,8 7,2 79 20,9 26,1
12
8,9 11,1 46 6,2 7,8
80 21,3 26,7
13
8,4 10,6 47 6,7 8,3 81 21,8 27,2
14
8,0 10,0 48 7,1 8,9
82
27,8
15 7,6 9,4 49 7,6 9,4 83 22,7 28,3
16 7,1 8,9 50 8,0 10,0 84 23,1 28,9
17
6,7 8,3 51 8,4 10,6 85 23,6 29,4
18 6,2 7,8. 52 8,9 11,1 86 24,0 30,0
19 5,8 7,2 53 9,3 11,7 87 24,4 30,6
5,3 6,7 54 9,8 12,2 88 24,9 31,1
21
4,9 6,1 55 10,2 12,8 89 25,3 31,7
22 4,4 5,6 56 10,7 13,3 90 25,8 32,2
23
4,0 5,0
57
11,1
13,9 91 26,2 32,8
24 3,6 4,4 58 11,6 14,4 92 26,7 33,3
25 3,1 3,9
59 12,0 15,0 93 27,1 33,9
26 2,7 3,3 60
12,4
15,6 94 27,6 34,4
27 2,2 2,8 61 12,9 16,1 95 28,0 35,0
28
1,8
2,2 62 13,3 16,7
28,4 35,6
29 1,3 1,7 63 13,8 17,2 97 28,9 36,1
30 0,9
1,1 64 14,2 17,8 98 29,3
36,7
31 0,4 - 0,6 65 14,7
18,3 99 29,8 37,2
32 0,0 0,0 66 + 15,1 + 18,9 100 + 30,2 + 37,8
20
5
96
192
TABLE XLII.
Comparison of French and English Measures, &c.
French Metre
Decimetre
Centimetre
Millimetre
at 32° Fahr.
39:37079
3.93708
0:39371
0.03937
Eng. Inches of
Sir G. Shuck-
burgh's scale,
at 620 Fahr.
French Toise
Foot
at 560,3 Fahr.
767394007
12:789900
English Inches
of the same, at
1.065825
Inch
Line
0.088819 j 56°,3 Fahr.
.
French Toise
French Metre
Foot
at 61°,3 F.
1.949036
0.324839
0-027070
at 61°,3 Fahr.
Inch
0-304794 ) French Metre
English Foot
Inch
at 62° F.
0·025399 at 320 Fahr.
Centes. Deg. of the Quadrant or 10.0000 = 0° 54' 0"000
Minute
or 09.0100 = 32,400 Sexagesimal
Seconds
or (0.0001 = 0,324
.
Comparison of different thermometers
xo Reaumur =
(32° + el ro) Fahr.
ão Centes. (32° +2) Fahr.
x° Fahren.
32°) 4 Reaum.
2
xo Centes.
* xº Reaum.
(aco 32°) Centes.
(
xo
The length of the pendulum vibrating seconds of mean solar time, in Lon-
don, in vacuo, and at the level of the sea, is 39.1393 English Inches of
Sir G. Shuckburgh's scale, at 620 Fahr.
The velocity of sound, in one second of time at 320 Fahr. in dry air, is
about 1090 English feet. For any higher temperature, add 1 foot for
every degree of the thermometer above 320.
TABLE XLIII.
193
Logarithms of various quantities.
Logarithms
5.3144251
Radius, reduced to seconds of the arc
Circumference of the circle, radius 1
sin 1"
0.4971499
4.6855749
206264",8
3:1415926
.00000485
•00000970
.00001454
2.7182818
sin 211
4.9866049
5.1626961
0.4342945
•43429448
2.3025851
9.6377843
0:3622157
4.6354837
43200
.00002315
5.3645163
sin 31
Number whose Hyper. log. is=1.
Modulus of the common logarithms
Complement to the same
12 hours, expressed in seconds
Complement to the same
24 hours, expressed in seconds
Complement to the same
360 degrees, expressed in seconds
Compression of the earth = zdo
Sidereal revolution of the Earth, in days
Tropical revolution of do.
86400
4.9365137
5.0634863
00001157
1296000
= 003333
6:1126050
7.5228787
2.5625977
2.5625810
= 365.25636
365•24224
Sidereal time into Mean Solar time
9.9988126
0.2898200
Constant logarithms (always additive)
for converting
French Toises into Metres
Feet into Metres.
Toises into English Feet .
Feet into English Feet
Metre into English Feet
Millimetres into English Inches
9.5116687
0:8058372
0:0276860
0.5159929
8.5951741
Centes. Degrees of Quadrant into Sexages. Degrees 9.9542425
Minutes of do. into
Minutes 1.7323938
Seconds of do. into
Seconds 3.5105450
The arithmetical complements of these last 10 logarithms will serve to
convert Mean Solar time into Sidereal time, Metres into Toises
&c, English Feet into Toises &c, and the Sexagesimal divisions of
the Quadrant into the Centesimal divisions.
2 c
*
EXPLANATION OF THE TABLES.
TABLE I does not appear to require any explanation. It
contains the Longitudes (from Greenwich) and the Lati-
tudes of various places where astronomical observations
have been made; and has been taken, for the most part,
from the Connaissance des tems. Some of the places, how-
ever, have been corrected from more recent observations.
Tables II and III serve to convert sidereal time into
mean solar time; and vice versa. They show the sidereal
time of mean noon on any given day in any given year.
The values in the first of these tables have been formed
on the assumption that the sun's mean right ascension at
mean noon at Greenwich on January 1st in any year is
exactly equal to 18h. 40m. os. (which, though not strictly
correct, is very nearly so): and that its increase in right
ascension in a mean solar day is 3m. 56$,555. To this
table is annexed, in a separate column, the amount of the
motion of the moon's node from January 1st to any other
day in the year; for a purpose to which I shall presently
allude.
The second of these tables serves to correct the values
given in the preceding table. For, since the mean right
ascension of the sun will never be the same on the 1st of
January in any two years, it becomes necessary to correct
the values in Table II by the addition of the quantities
here set against the given year. The sum of these two
• 2 c 2
196
Explanation of the Tables.
values gives the correct right ascension of the sun at mean
noon at Greenwich, as reckoned from the mean equinox.
These values have been computed from the formula
1} [53'. 295,8 – (y – 4B) 14. 47",08 + 27",48 y]
= 3m. 335,993 – (y – 48) 595,1387 + 15,8320 y
denotes the number of years from 1800 (negative
before, and positive after that period): and ß the number
of bissextile days between 1800 and the month of March
in the given year, which also changes its sign with y.
As these tables are formed for the meridian of Green-
wich, it is evident that a correction must be applied when
they are intended for any other place situated east or west
of that observatory. The amount of this correction is
where y
XO$,0027379
where a denotes the longitude in time (expressed in seconds)
from Greenwich; + when west, and when east. * In
the third column is given the mean place of the moon's
node on the 1st of January; the whole circle being sup-
posed to be divided into 1000 parts.
For an example of the use and application of these
tables, see Problem I.
* These corrections will be, for the following observatories, as under:
viz.
Altona.
63544
Dorpat
17,544
Dublin
+ 4,167
Konigsberg
13,462
Milan
6,040
Palermo
8,783
Paramatta
99,235
Paris
1,536
Vienna.
10,763
Explanation of the Tables.
197
Tables IV and V show the correction for the Lunar and
Solar Nutation; or the values which ought to be applied
to those given in Tables II and III, in order to obtain the
sun's mean right ascension as reckoned from the apparent
equinox. The true correction, as deduced from Formula
XXXIII in page 106, is
-15",868 sin 88+0,191 sin 28-1",151 sin 20-0",190 sin 2)
= -- 19,058 sin 8+05,013 sin 28-09,077 sin 20-09,013 sîn 2)
Table IV contains the correction depending on the moon's
node (the position of which may be taken from the two
preceding tables), the whole circle being divided into 1000
parts. Table V contains the correction depending on the
sun's true longitude, the place of which must be taken
from an ephemeris. The last quantity, depending on the
moon's true longitude, has been omitted as too small to
affect the results in any material degree.
For an example of the use and application of these
tables, see also Problem I.
Table VI expresses the motion of the sun's mean right
ascension in a sidereal day, for hours, minutes and seconds
of mean solar time; and will be found very useful and con-
venient for converting sidereal time into mean solar time.
For an example of the use and application of this table,
see also Problem I.
Table VII is a similar table for the motion of the sun's
mean right ascension in a mean solar day, expressed in
hours, minutes and seconds of sidereal time; and will be
found occasionally useful and convenient for converting
mean solar time into sidereal time.
198
Explanation of the Tables.
For an example of the use and application of this table,
see also Problem I.
Table VIII serves to convert degrees, minutes and
seconds of space into similar denominations of time : and
vice versa. It is founded on the ratio of 15° to lh. and
does not require any further explanation.
Tables IX, X and XI contain Mr. Ivory's refractions,
as given in the Phil. Trans. for 1823. The first of these
denotes the mean refraction, for the several degrees of
zenith distance therein stated : to which are annexed, in a
contiguous column, for the convenience of computation,
the logarithms of those values, with their differences. The
mean values are computed on the assumption that the tem-
perature of the air is 50° of Fahrenheit's thermometer, and
that the barometer stands at 30 inches. Table X serves
to correct these values when the thermometer or barometer
is higher or lower than the mean state above mentioned *.
The value, thus corrected, shows the approximate refraction:
which, in most instances, will be sufficiently correct. But,
in the case of low altitudes and where great accuracy is
required, this value must be further corrected by Table XI,
agreeably to the rule there given.
For an example of the use and application of these
tables, see Problem II.
* In this table I have united Mr. Ivory's tables II and IV; which is
the only alteration that I have made in his arrangement. No error will
arise from this disposition of the tables, if the interior thermometer be
used; nor if the temperature out of doors differs but little from that
within.
Explanation of the Tables.
199
Tables XII and XIII are Dr. Brinkley's very con-
venient tables for the computation of refraction: but they
are not adapted to altitudes lower than 10 degrees. The
argument in the first of these tables is the height of Fahr-
enheit's thermometer: corresponding to which is the loga-
rithm of a quantity which I shall denote by T, and which
must be multiplied not only by the height of the barometer
(expressed in inches) but also by the tangent of the zenith
distance of the star, which must not be greater than 80°.
This will give the approximate refraction : which, in the
case of low altitudes, and where great accuracy is required,
must be further corrected by subtracting the value (which
I shall call c) in Table XIII, set against the given zenith
distance, and under the given height of the barometer.
For an example of the use and application of these
tables, see also Problem II.
Table XIV has been computed from the assumed hori-
zontal parallax of the sun, at its mean distance, being equal
to 8",60: and gives the required parallax of altitude on the
first day of every month. The proportional parts are
easily found for any intermediate day.
Since parallax always tends to lower the true altitude
of a body, we must add the parallax to the observed alti-
tude, in order to obtain the true altitude. Or, which is the
same thing, we must deduct it from the observed zenith
distance, in order to obtain the true zenith distance.
The use and application of this table are sufficiently
evident without an example.
Table XV will be found very convenient for determining
the time as deduced from observations of single altitudes of
the sun or a star. The formula, for the reduction of such
200
Explanation of the Tables.
observations, is given in page 89: and it will be there seen
that the hour angle P is always deduced from a logarithm
which expresses the value of sinº {P. In order to save the
time and labour of computing the values of such logarithms
and to prevent the occurrence of error, I have given the
corresponding hour angles, for every minute of time, from
3 hours to 8 hours; which will be sufficient for all ordinary
purposes. The time corresponding to any intermediate log-
arithm may be readily determined by direct proportion.
For an example of the use and application of this table,
see Problem III.
Table XVI is for determining the equation of equal alti-
tudes of the sun, which will be found more convenient and
correct than the table in general use for that purpose. It
is computed from the Formula XVIII in page 92. The
value of A (which is always negative) must be multiplied
by the double daily variation of the sun's declination (con-
sidered as negative when decreasing) and also by the tangent
of the latitude of the place: to which must be added the
value of B, multiplied also by the double daily variation of
the sun's declination, and likewise by the tangent of the
sun's declination at the time of apparent noon on the given
day. The sum of these two quantities is the correction
required.
For an example of the use and application of this table,
see Problem IV.
Table XVII is computed from the Formula XVI in
page 90 (No. 5 and 11). The hour angle, at which the
star passes the prime vertical, may be deduced from No.1,
and the hour angle at which the vertical becomes a tangent
to the circle of declination may be deduced from No. 10.
Explanation of the Tables.
201
The hour angle, thus deduced, being added to the right
ascension of the star will give the sidereal time when the
star is on the prime vertical, or becomes a tangent to the
circle of declination, to the westward ; or being subtracted
from the right ascension of the star (increased by 24h if
necessary) will give the sidereal time of the same position
to the eastward. To persons in the practice of making ob-
servations on or near the prime vertical, it would be useful
to have a table calculated to show the altitude of those stars
which they usually observe in such case, and the time at
which they pass the prime vertical. See Problem XIV.
Tables XVIII and XIX are the well known tables for
the reduction to the meridian, computed agreeably to the
Formula XIX in page 93. The first of these tables shows
2 sinº P
the value of A:
sin 111*: and the argument of the
table is the distance (in time) of the sun or star from the
meridian. This value (or the sum of those values divided
by the number of observations, if more than one observa-
1
sin 11
1
* Since 2 sinº IP denotes the versed sine of the arc P, it is evident
that the expression in the text is equal to X ver. sine P. It is in
this manner that the tables for the reduction to the meridian have been
published by the Board of Longitude. They have printed a table of the
versed sines of arcs from 0% to 30m of time, and direct that the arith-
metical complement of the logarithm of sin 1" be added to the constant
logarithm (with its index increased by 4) of the number by which the
result is multiplied: thus increasing the trouble and the liability to
error. We are also directed to use the logarithm 5.3168000, instead of
the arithmetical complement of the logarithm of sin 1", for observations
on the stars, when solar time is employed : but the true logarithm is
5-3167966 : a slight difference, which however is of no great moment.
2 D
202
Explanation of the Tables.
*
;
D)
cos L.cos D
tion has been made) must be multiplied by
sin Z
and the product subtracted from the zenith distance (cor-
rected for refraction &c) of the sun or star observed near
the meridian. The differencet thus obtained will give the
true meridional zenith distance of the sun or star, as cor-
rectly as if it had been observed precisely on the meridian.
It must however be remarked that, when the distance from
the meridian is considerable, and when great accuracy is
required, this value must be further corrected by the ad-
dition of the value of B in Table XIX , multiplied by
cos L. cos D ?
X cot Z.
sin Z
The distance in time) of the sun or star from the me-
ridian is determined by a well regulated clock; and, for
greater simplicity, the motion of the clock should corre-
spond with the object observed : that is, if the sun be the
object, the clock should be regulated to mean solar time;
and if a star be the object, the clock should be regulated to
sidereal time. This however is not absolutely necessary,
since we may readily correct the errors arising from the
rate of the clock. For if the sun be the object, and the
clock be regulated to sidereal time, we must multiply A by
•99455418 (log = 9.9976285): and if a star be the object
and the clock be regulated to mean solar time, we must
multiply A by 1.00547562 (log = 0.0023715). There is
however, in all cases, a correction necessary when the
* See the list of Errata, in this work.
+ If the star has been observed under the pole, we must take the sum
of these two quantities.
I Or the sum of the values of B, divided by their number, if more
than one observation has been made.
Explanation of the Tables.
203
clock does not go accurately during the observations: and
whenever this occurs, we must multiply the value of A still
further by 1 +.000023151 (log =0.000010053 xr): where
r denotes the daily rate of the clock, expressed in seconds,
which must be assumed minus when gaining, and plus
when losing
For an example of the use and application of these
Tables, see Problem V.
Table XX shows the mean obliquity of the ecliptic, on
the 1st of January in every year during the present cen-
tury. It is deduced from the assumption that the mean
obliquity in 1750 was 23º. 28'. 171,63; and that the annual
diminution is 0",457. This corresponds with the determi-
nation of M. Bessel, as given in Professor Schumacher's
Astronomische Nachrichten, No. 34.
Tables XXI and XXII serve to determine the correc-
tions for lunar and solar nutation, which should be applied
to the values in the preceding table, in order to obtain the
apparent obliquity of the ecliptic: and also the value of the
lunar and solar nutation in longitude. The correction for
the obliquity is deduced from the Formula XXXII in
page 105, and is equal to
+9",250 cos 88-0",090 cos 28+0',545 cos 20 +0",090 cos 2 )
to which must be added the diminution of the mean obli-
quity from the 1st of January to the given day, which is
equal to
0",457
365
where d denotes the number of days elapsed from the com-
mencenient of the
year.
xd
2D 2
204
Explanation of the Tables.
The correction for the longitude is (as in page 105) equal to
-17",298 sin 88+0",208 sin 28-1,255 sin 20-0",207 sin 2)
Table XXI contains the corrections for the Longitude
and Obliquity depending on the place of the moon's node
(which may be taken from Tables II and III), the whole
circle being divided into 1000 parts. Table XXII contains
the corrections for the same quantities depending on the
sun's true longitude: the day of the month being made the
argument. In the column marked Obliq. the value of
0",457
x d has been included in the computation. The
last quantities, depending on the moon's true longitude,
are omitted, as being too small to affect the results in any
material degree: but they may be taken into the computa-
tion (if required) by actual calculation.
For an example of the use and application of these
Tables, see Problem VIII.
365
Table XXIII is introduced with a view to preserve
some uniformity in referring to the principal lunar spots.
Much confusion has arisen from the various names given
to the same spots on the moon, by different authors. The
present list is taken from the plate in Russel's Description
of the Selenographia, who has proposed these spots as a
basis from which the situation of other less remarkable
spots may be determined. The latitudes are computed
from a line passing through Censorinus ; the longitude of
which is assumed equal to 32° west: and the positions are
taken from the table in the Recueil de Tables Astrono-
miques, Berlin 1776.
Table XXIV shows the angle of the vertical, deduced
from the Formula XXI, in page 95; where c is assumed
Explanation of the Tables.
205
equal to 300. The logarithms of the earth's radius, in the
last column, are deduced from the Formula XXII, on the
same assumption. These quantities are necessary in com-
puting the parallax of the moon: for, in such calculations,
the latitude of the place and the equatorial parallax of the
moon must be diminished in consequence of the compres-
sion of the earth. The values for the latitude of Green-
wich are given separately at the bottom of the table.
For an example of the use and application of this Table,
see the note to Problem XI.
.
Table XXV shows the augmentation of the moon's
semidiameter on account of her altitude, deduced from the
Formula XXVIII No. 1, in page 101 ; according to the
several values of s, from 14'. 30" to 17. 0".
Its use and application are too obvious to require an
example.
Table XXVI contains the logarithms for the equations
of the first, second and third differences of the moon's
place; and will be found very convenient for obtaining the
correct values required. They are computed from the
Formula XXX in page 103, for every ten minutes of time
from noon or midnight. The two columns, entitled first
h
differences, contain the logarithms of to which must
be added the logarithm of A": the natural number cor-
responding to the sum of these two logarithms will be
the equation of the first difference *.
The column,
1211
;
* The logarithms in the Table are carried to four places of decimals
only; which is sufficient for the equations of the second and third
differences: but probably may pot be considered in many instances
206
Explanation of the Tables.
2
entitled second differences, contains the logarithm of
h (h – 12), to which must be added the logarithm of
2 (12)
d' + du
: the natural number corresponding to the sum of
these two logarithms will be the equation of the second
difference. The column, entitled third differences, con-
tains the logarithm of
h(h-12) (h-6)
; to which must
6(12)3
be added the logarithm of . This last quantity is how-
ever generally so small, that it may in most ordinary cases
be omitted. All these equations are to be applied to the
moon's place, according to the signs : and it should be
remembered that the moon's latitude, and declination,
when south, is considered as a negative quantity.
For an example of the use and application of this Table,
see Problem X.
Table XXVII was originally computed from the values
given in Formula XXXI in page 104, and which are the
same as those given by M. Bessel in his Fundamenta
Astronomia, page 297. But that distinguished astronomer
has (since that formula was printed) re-investigated the
subject of the precession of the equinoxes, and the result
of his inquiry is the table here given *.
The use and application of this Table are too well
known to require an example. The formulæ, for deter-
mining the annual precession of a star in right ascension
and declination, are given at the bottom of the Table. In
accurate enough for the first differences. In such case, we must com-
h
pute the logarithm from the formula
12h
* The alterations in the values in the Formula are noticed in the list
of Errata.
Explanation of the Tables.
207
these formulæ, and indeed in all the formulæ in the present
work, the declination when south is considered as a nega-
tive quantity, and treated as such in the algebraic and
arithmetical computation. Therefore when in such case
the amount of precession, aberration &c in declination is
a positive quantity, it must be subtracted from the mean
declination : and, on the contrary, when it is a negative
quantity, it must be added thereto, in order to obtain the
apparent declination. Some astronomers still consider the
declination as always positive ; and change the sign of the
precession &c when the star has south declination. But
the former plan is the most convenient; and is now more
generally adopted.
و
Tables XXVIII-XXXI are M. Gauss's very con-
venient and general tables of aberration and nutation.
They are deduced from a transformation of the Formulæ
XXXIII and XXXIV in pages 106 and 107; and will
be found very useful when we have not a ready access to
more comprehensive tables. The constant of aberration is
assumed equal to 20",255; the constant of the lunar nuta-
tion of the obliquity equal to 95,65; and the constant of
the solar nutation of the same equal to 0",493. These
values differ in a slight degree from those given in the
formulæ above mentioned: but this difference will in most
cases be insensible in practice. The use and application
of these Tables may be understood from the following
expressions.
For the correction in Right Ascension
Aber = -a.cos(O+A-R) sec D
Lunar Nut = -b.cos(O+B-R) tan D
Solar Nut = the two equations from Table XXXI
208
Explanation of the Tables.
For the correction in Declination
Aber= -a.sin(O+A-R)sinD+the 2 eq.of Tab.XXIX
Lunar Nut= -b.sin (O+B-R)
Solar Nut= the equation from Table XXXI
The values of A and of the logarithm of a are to be taken
from Table XXVIII: and the values of B and c, and the
logarithm of b, are to be taken from Table XXX: The
solar nutation is taken from Table XXXI: and it should
be remarked that the second part of the solar nutation in
right ascension must be multiplied by the tangent of decli-
nation. The computer should bear in mind that, in the
whole of these operations, the declination when south is
always considered as a negative quantity, and the results
must be applied accordingly with the proper signs.
For an example of the use and application of these
Tables, see Problem XII.
Table XXXII is the well known table of the length
of circular arcs (radius = 1) deduced from the expression
l= a.sin 1". Whence, the length being known, we have
7
the arc a =
where I denotes the length of the arc,
sin 1"
and a the arc itself expressed in seconds. Its use and
application are too well known to require an example.
Table XXXIII shows the semidiurnal arc, or the in-
terval of time employed by the sun or a star in passing
from the horizon to its point of culmination, and vice versa;
according to its declination, and the latitude of the place.
These values have been deduced from the equation in
Formula XV in page 89, without considering the effect of
refraction; which would increase the duration according
Explanation of the Tables.
209
to the circumstances of the case. The use and appli-
cation of this Table are sufficiently evident without an
example.
Table XXXIV has been introduced merely for the
purpose of showing the relative values of the quantities
therein stated, according to the latitude of the place of ob-
servation. The lengths of the degrees of longitude and
latitude are computed from the Formula XLIII in page
116. The length of the pendulum is computed from the
formula in page 22: and the increase in the number of its
vibrations, from the Formula XL in page 113.
Table XXXV contains the expansion of various 'sub-
stances used in experiments and observations on the pen-
dulum. It is extracted from a more copious list given by
me at the end of my paper “On the Mercurial Compensa-
tion Pendulum" inserted in the Memoirs of the Astronomi-
cal Society, Vol I, pages 416—419. The numbers in the
table contain the linear expansion of the body, for one
degree of Fahrenheit's thermometer: if this number be
denoted by e and the difference in the thermometer by to
the total expansion for t degrees of the thermometer will
be et. If i denote the length of the body before expan-
sion, then will its length after the application of heat be
l'=1(1+et).
At the bottom of the table I have inserted the cubical
expansion of Mercury and of Air. The expansion of
mercury has been determined with the greatest accuracy
by MM. Dulong and Petit. The absolute expansion is
that which does not depend on the form of the vessel
which contains it, nor on its expansion: whereas in the
apparent expansion in glass) these circumstances are
2 E
210
Explanation of the Tables.
taken into the computation *. The expansion of air is
generally assumed as equal to go of its bulk, for every
degree of Fahrenheit's thermometer: but this applies more
particularly to air rendered perfectly dry for the purpose
of the experiments employed. The expansion of common
atmospheric air, impregnated as it generally is with a
certain degree of moisture, is supposed by M. La Place to
be to of its bulk. These are the two values stated in the
table.
Table XXXVI is a new and very convenient table for
computing, by means of logarithms, the difference of alti-
tudes with the barometer. It is deduced from the Formula
XXXVIII in page 111 t, by expanding the last term in
M. La Place's formula, reducing the measures to English
feet, and expressing the temperature by Fahrenheit's ther-
mometer. Whence the difference of altitude between the
two stations will be found equal to
X
60345-51 (1+001111 (+640)]
] ]*[1+-002695 cos 20]
1
Х
B
log of
1 + .0001 (1-7)
In this Table, A denotes the logarithm of the first term
here given, expressed in English feet; and C the logarithm
of the last term: B denotes the logarithm of 1 + .0001 (T-T').
In the formation of this table I have assumed the expan-
* The relative vertical expansion of mercury, in the glass vessel used
for the bob of a pendulum (and which is the quantity that affects the
rate of the clock) is found by deducting twice the linear expansion of
the containing vessel from the absolute expansion of the mercury: and
it may be assumed equal to .00009.
+ See the correction of this formula, amongst the Errata.
Explanation of the Tables.
211
sion of air, for one degree of Fahrenheit, to be .00222222,
instead of the quantity .00208333 assumed by M. Biot.
The rule for the application of this table is given at the
bottom thereof in page 183: and here it may be useful to
remark that the heights of the barometers may be taken in
any measure whatever; whether in English inches, French
metres, or any other scale.-For an example of the use and
application of this Table, see Problem XVI.
Table XXXVII will be found useful in converting
hours, minutes and seconds, into the decimal parts of a
day: an operation of frequent occurrence in computations
relative to practical astronomy. Its use and application
are sufficiently evident without further explanation.
Table XXXVIII will be found of similar use in con-
verting the sexagesimal divisions (or the minutes and se-
conds) of a degree, into the decimal divisions of the same.
This mode of dividing the degree has all the advantages of
(and is much less liable to confusion and error than) the
mode, adopted by the French, of dividing the quadrant
into 100 parts: and I hope that some public-spirited indi-
vidual will ere long be bold enough to print tables founded
on this arrangement. The decimal division of the degree
is very convenient in all tables (particularly those formed
for the purposes of astronomy) where proportional parts are
required: and it has many advantages over the ordinary
sexagesimal division. But the French mode of dividing the
quadrant into 100 parts, calling each of those parts a de-
gree, and adopting the same character that is used in the
sexagesimal notation, to express that degree (although its
value is only ths of that quantity) leads to endless con-
fusion, and ought to be discountenanced and discontinued.
2 E 2
212
Explanation of the Tables.
Table XXXIX serves to convert any given day into
the decimal part of a year consisting of 365 days. In leap
years we must add unity to the given date, if subsequent
to February, in order to obtain the corresponding tabular
date: and in such case, if great accuracy be required, the
value thus found should be multiplied by •997.
Table XL. The new division of the circle into 400
degrees, by the French, and the numerous and valuable
tables that have been computed agreeably to that arrange-
ment, render a table of this kind absolutely necessary for
converting the centesimal division of the quadrant into the
sexagesimal: and vice versa. It is formed on the com-
parison stated in Table XLII: where it will be seen that
each degree of the French division is equal to 54' only of
the sexagesimal division; and each minute of the French
division equal to 32",4 only of the sexagesimal. When the
French first introduced the centesimal division of the
quadrant, the new degrees were denominated grades, and
were denoted by the small letter g placed above the num-
ber. Thus, 164 degrees were written 1646. This method,
so long as degrees only were concerned, and whilst the
subdivisions of the degree were expressed decimally, could
not lead to any confusion: but this mode of denoting the
new quantities has been gradually discontinued, and at the
present day most of the French mathematicians use the
sexagesimal notation, not only for the new degrees, but
likewise for the new minutes and seconds: thus introducing
considerable confusion in the value of the quantities alluded
In order to remedy this, in some measure, I have de
noted the new degree by a small square character placed
over it, instead of a round one. Perhaps
Perhaps a better plan
would be to adopt the small Greek letters S, M, o, to denote
to.
Explanation of the Tables.
213
the degrees, minutes and seconds of the new division: thus
24 degrees 15 minutes and 37 seconds of the centesimal
division of the quadrant would be written thus, 248. 154. 370.
But much the best plan would be to discontinue the system
altogether.
Table XLI serves to compare Fahrenheit's thermometer
with Reaumur's and the Centesimal; and vice versa. It is
founded on the comparisons given in Table XLII: by
means of which equations the present table may be ex-
tended to any number of degrees required. Its use and
application will be evident on inspection.
Table XLII contains various comparisons (of frequent
use in practice) relative to French and English measures,
&c. In the Base du Systéme Métrique, Vol. 3, page 470,
the French metre, which is made of platina, is stated to be
equal to 39.3827 English inches: and it has been assumed
as such by Professor Schumacher, in his Sammlung von
Hülfstafeln, page 16. But this comparison was made when
the two measures were both at the freezing point of the
thermometer. This degree of temperature is the standard
for the French scale: but the standard temperature for the
English scale, which is made of brass, is 62° of Fahren-
heit. Consequently the value given by Professor Schuma-
cher ought to be corrected for the relative expansion of the
metals, which will reduce it to 39.37079 inches, as given
by Captain Kater in the Phil. Trans. for 1818, page 109;
each scale being brought to its standard temperature.
By Act 5 Geo. IV, c. 74, Bird's scale of 1760* is de-
* There is another scale made by Bird in 1758, called also Bird's
Parliamentary standard, which differs +.00016 of an inch from Sir
214
Explanation of the Tables.
clared to be the legal standard: and this scale differs so
little (only +.00002 of an inch) from Sir George Shuck-
burgh's scale, that they may be considered as identical.
And as most of the comparisons have been made with Sir
G. Shuckburgh's scale, its measure is therefore here re-
tained.
The old legal standard of France was the Toise de
Perou, so called from its being used by the French Acade-
micians in that country. It is formed of iron, and was
made by M. Langlois in 1735, under the direction of M.
Godin. By a comparison of this toise with a copy of
Sir G. Shuckburgh's scale at the temperature of 56°,3
Fahr. it was found to be equal to 76.7394 English inches.
See Base du Syst. Mét. Vol. 3, page 479. By a Report
in the same work, page 433, made by the French Com-
missioners, it appears that the toise of Peru, at 61°,3 Fahr.
is equal to 1.949036 metres. It is upon these authorities
that I formed the comparison of the measures in the table.
In the comparison of the centesimal degrees of the qua-
drant, I have introduced a new method of denoting those
degrees. It consists in annexing a small square character,
in order to distinguish them from the ordinary sexagesimal
degrees, with which they are frequently confounded. See
the remarks I have already made on this subject in
page 212.
The length of the pendulum is taken from Capt. Kater's
last determination in the Phil. Trans. for 1819, page 415.
The velocity of sound is deduced from a comparison of
the experiments which have been recently made by several
George Shuckburgh's scale: and this must not be confounded with
the scale of 1760. All these scales are made of brass.
Explanation of the Tables.
215
distinguished philosophers: the mean of which is very
near the value here assumed.
Table XLIII contains the logarithms of various quan-
tities which occasionally occur in astronomical computa-
tions; and will prevent the necessity of referring to other
works, when they are required for use.
P R O B L E M S.
PROBLEM I. To convert sidereal time into mean
solar time: and vice versa.
This problem is of frequent occurrence in practical
astronomy, and is solved by the help of Tables II-VII in
the present collection, which have been expressly formed
for that purpose. Their application is as follows: let us
make
M = the mean solar time at the place of observation
S = the corresponding sidereal time
R = the mean right ascension of the meridian (or the
mean longitude of the sun, converted into time)
at the preceding mean noon, at the place of ob-
servation
a = the acceleration of the fixed stars (as shown by
Table VI) for the interval of time denoted by
(S-R)
A = the acceleration (as shown by Table VII) for the
time denoted by M
then we shall have
M (S – R) – a
S = R + M + A
The mean right ascension of the meridian, at mean noon on
any day in any year, as reckoned from the mean equinox,
is found by Tables II and III: to which must be applied
the equations deduced from Tables IV and V, in order to
2
218
Problems.
give the right ascension reckoned from the apparent
equinox *
h
m
S
Example. An eclipse of the first satellite of Jupiter was
observed at Greenwich on Jan. 17, 1825 at 2h. 19m. 499,0
sidereal time by the clock. The clock however was too
fast 59$,14: consequently the correct sidereal time was
2h. 18m. 495,86. Required the corresponding mean solar
time?
88
Table II Jan. 17
= 19 43
4,885
2
Table III 1825
3 20,654 749
Table IV Lunar nutation .
+ 1,056 7477
Table V Solar nutation
+ 0,062
Mean Rat preceding mean noon
19 46 26,657
reckoned from app. equinox
S = 2 18 49,860
(S – R) =
6 32 23,203
Table VI,
a
1
4,282
Mean solar time required
6 31 18,921
If the observed time had been mean solar time
6h 31m. 189,921 and it were required to find the corre-
* All the ephemerides give the true R of the sun for apparent noon:
but we may easily deduce therefrom the R for mean noon, if required,
in the following manner. The equation of time being expressed in
mean solar time, we must first convert it into sidereal time by Table
VII: then change the sign and apply it to the R of the sun for ap-
parent noon. The result is the true R of the sun for mean noon: pro-
vided the ephemeris has been correctly computed.
+ The place of the moon's node on Jan. 1, 1825 is, by Table III,
equal to 749, from which must be deducted the motion of the node
from that day to Jan. 17, which, by Table II, is equal to 2: the dif-
ference (=747) is the argument for entering Table IV.
Problems.
219
h
m
S
M =
sponding sidereal time, the operation would have stood
thus:
6 31 18,921
as above found .. R = 19 46 26,657
Table VII.
1 4,282
Sidereal time required 2 18 49,860
A =
When the place of observation is situated on a different
meridian from that of Greenwich, the value of R as found
by these tables must be increased or diminished agreeably
to the rule given in page 196.
.
The mean solar time being ascertained, we may readily
determine the corresponding apparent time, by applying
the equation of time, as shown by an ephemeris, for the
moment of observation * Thus the equation of time on
Jan. 17, 1825 at 6h. 31m. 189,92 as given in the Nautical
Almanac, was 10m. 345,57; which (being directed to be
added to apparent time) must be subtracted from mean
time: consequently the corresponding apparent time was
6h. 20". 449,35.
There is another very convenient, and equally correct
mode of converting sidereal time into mean solar time,
by the help of an ephemeris computed for the given place;
which is as follows:
M = (S – R') – a te
where R' denotes the apparent right ascension of the sun
* The equation of time is equal to the difference between the sun's
mean longitude, converted into time, and the sun's true right ascension.
It may be useful to remark that Table VIII in M. Delambre's Tables of
the Sun (which is the same as Mr. Vince's XXIX) is inaccurate, and in
some cases produces an error of 15,6. Another and a more correct table
is given in the Con, des tems for 1820, page 492.
2 F 2
220
Problems.
at the preceding apparent noon, and e the equation of time
at the same moment: both of which are given in the
ephemeris. Take the example above given.
h
m
S=
α -
е —
2 18 49,86
By Navtical Almanac R' = 19 56 57,70
6 21 52,16
Table VI
1 2,58
6 20 49,58
By Nautical Almanac
10 29,30
6 31 18,88
This result ought to be exactly the same as the pre-
ceding: and the slight difference, which occurs, arises
from the circumstance that the co-efficients of the two nu-
tations, in Tables IV and V, are not precisely the same as
those which are used in the computations of the Nautical
Almanac.
PROBLEM II. To determine the refraction of a
heavenly body.
The tables of refraction are very numerous, and many
celebrated mathematicians have devoted their time and
abilities to the investigation of this intricate subject. The
labours of Bradley, Bessel, Littrow, LaPlace, Carlini,
Groombridge, Atkinson, Gauss, Young, Ivory and Brink-
ley, are too well known to require any comment. But, as
it was necessary to make a selection, I have chosen the
tables of the two latter authors: those of Dr. Young are
given annually in the Nautical Almanac.
Example 1. The zenith distance of a Aquila was ob-
served 71°. 26'. O' the barometer standing at 29.76 inches
and the thermometer at 43° Fahr. Required the refraction?
Problems.
221
logarithms
- 2.23609
Table IX
Table X
Table X
By Mr. Ivory's tables
Mean refraction
Barometer 29:76
Thermometer 430
= 9.99651
= 0.00668
Refraction = 2' 53',49 = 2.23928
By Dr. Brinkley's tables
Table XII Thermometer 43°
= 0.2965
Barometer 29.76
= 1.4736
Tangent 71°. 26.
= 0.4738
Approximate Refraction = 2' 55",35 = 2.2439
Table XIII
2 ,03
C =
True refraction = 2 53 ,32
If we denote the apparent zenith distance by Za and the
true zenith distance by Z' we shall have
Zt = Za +r
where r is the computed refraction. Or, if we prefer ex-
pressing these values by means of altitudes, we shall have
At
= Aa
Example 2. The apparent zenith distance of a Lyra
was observed 87º. 42. 10": the barometer standing at
29.50 inches, and the thermometer at 35° Fahr. Re-
quired the refraction?
By Mr. Ivory's tables
logarithms
Table IX Mean refraction
– 3.00856
Table X Barometer 29.5
= 9.99270
Table x Thermometer 35°
-0.01444
Approximate mean refraction = 17' 16",80 = 3.01570
Table XI •606 x 15 = †
9,09
Table XI
+ 1.04 x
.5 = 0,52
True refraction = 17 25 ,37
222
Problems.
PROBLEM III. To determine the time from single
altitudes * of the sun, or a star : its declination,
and also the latitude of the place being given.
The observed altitude or zenith distance of the sun or
star must first be corrected for refraction, as in the last
Problem t. Then, by means of Formula XV, in page 89,
we obtain the logarithm of sinº | P: and by the help of
Table XV, we deduce from that logarithm the value of P,
or the hour angle from the meridian.
In the case of the sun, this hour angle will be the ap-
parent time I from apparent noon at the place of observa-
tion: to which the equation of time must be applied, in
order to deduce the mean solar time. But, in the case of a
star, it will denote the distance, in time, of the star from
the meridian; and which, being added to the right ascen-
sion of the star, if the observation be made to the westward
Single altitudes, or absolute altitudes, is a term given to observa-
tions of the altitude of the sun or a star, when made on the same side
of the meridian: in opposition to equal altitudes, which are made on
both sides of the meridian. See the next problem. It is not meant to
imply thereby that only one altitude is taken; because in general there
are several: and the usual method is to note down the time of each
observation, and the altitude observed; and then to take the mean of
the times and the mean of the altitudes as one observatinn.
† If the sun be the body observed, it must be corrected also for
Parallax by Table XIV. And, since it is the border only of the sun
that can be observed at one observation, we must reduce this to the
centre, by applying its semidiameter, which may be found in any ephe-
meris: or by observing alternately the upper and lower border, and
taking the mean of each pair of observations. We must also compute
the declination for the apparent time of observation, at the given place.
# When the observation is made in the forenoon, this apparent time
must be subtracted from 24" in order to show the apparent time of the
day.
Problems.
223
of the meridian, or subtracted therefrom (increased by 24h
if necessary) if the observation be made to the eastward,
will give the sidereal time of observation.
Example 1. On October 18, 1818 at 3h. 6m. 309,7 P.M.
mean solar time as shown by a clock, the zenith distance
of the upper border of the sun (corrected for refraction and
parallax) was found to be 75º. 0'. 35,9: the place of obser-
vation being situated in N. Lat. 52º. 13'. 26" and W. Long.
4m. 49s from Greenwich. What was the correct time of
observation ?
If we add the semidiameter of the sun on the given day
(= 16' 6'') to the zenith distance (corrected as above) of
the observed border, we shall have the true zenith distance
of the sun's centre equal to 75º. 16'. 15'. We must next
compute the sun's declination for the approximate apparent
time of observation at the place. Let us suppose that the
Nautical Almanac is made use of for these computations.
The equation of time at apparent noon (or at 116. 45m. 189,9
mean solar time) at Greenwich, was -- 14". 419,1: which
being added, with its sign changed *, to the mean solar time
of observation, will give 3h. 21" for the approximate appa-
rent time, at the place. But, as the computations are made
from the Nautical Almanac, we must add the longitude
from Greenwich; and compute the sun's declination for
3h. 26" Greenwich apparent time, with a daily variation of
21'. 51". Consequently the sun's declination at the time
of observation was - 9º. 33. 30". With these elements
the computation will stand thus:
* The equation of time is an equation which in an ephemeris is di-
rected to be added to, or subtracted from, apparent time, in order to
deduce the mean solar time: consequently we must reverse the sign
when we wish to apply the equation to mean solar time, in order to de-
duce apparent time.
224
Problems.
O
1
11
L = + 52 13 26
D =* 9 33 30
logarithms
COS = + 9.7871611
cos = + 9.9939285
+ 9•7810896
(L - D) = +61 46 56
Z = + 75 16 15
Z - (L-D)
(L - D) = + 13 22 19
Z + (L – D) = +137 3 11
sin } = + 9.0698112
= + 9.9687570
sin
+ 9•0385682
as above = + 9•7810896
sin?P= + 9•2574786
h
m
S
By Table XV. sinº | P = 3 21 22,7
equation of timet
-14 42,7
correct mean solar time. = 3 6 40,07
at the place.
observed mean solar time} = 3 6 30,7
Consequently the clock was 9,3 too slow.
Example 2. On March 23, 1822, in N. Lat. 51° 33'. 34"
I observed, to the westward of the meridian, the zenith di-
stance of Aldebaran (after correcting for refraction) to be
68º. 2. 21" at gh. 24". 445,0 by a sidereal clock. What
was the error of the clock at that moment?
On that day the apparent R of the star was 4h. 254. 439,8
and its apparent Declination was + 16º. 8'. 44"; as shown
by the Nautical Almanac. Consequently the operation will
stand thus:
* The declination of the sun being south, it is considered as a nega-
tive quantity in the computations.
+ By Nautical Almanac at 3h. 26m. 179,7 from Greenwich: or at
3h. 21m. 229,7 + 4m. 495.
# If a sidercal clock had been used in this observation, we must have
reduced the sidereal time to mean solar time, in order to show the
error of the clock.
Problems.
225
logarithms
COS = + 9.7935825
cos = + 9.9825238
+ 9•7761063
L = + 51 33 34
D = + 16 8 44
(L – D) = + 35 24 50
Z
= + 68 2 21
Z-(L-D) = + 32 37 31
2 + (L-D) = +103 27 11
11
sin
sin }
+ 9.4485148
= + 9.8949038
+ 9.3434186
+ 9.7761063
sin?P= + 9.5673123
h
m
S
By Table XV . sinP = 4 59 21,7
R = 425 43,8
correct sidereal time.
observed sidereal time
= 9 25 5,5
= 9 24 44,0
*
error of the clock
21,5
It is scarcely necessary to remark that the most favourable
opportunity, for determining the time from altitudes of the
sun or a star, is when those bodies pass the prime vertical :
since their motion in altitude is then the most rapid, and a
slight error in the assumed latitude will not materially
affect the result to
In fact, in a fixed observatory, (or where the observer
may
be stationed at the same place for several successive
* If a clock, showing mean solar time, had been used in this obser-
vation we must have reduced the mean solar time to sidereal time, in
order to show the error of the clock.
+ In the latitude of Greenwich, a star varies 9",3 in altitude in one
second of time, when on the prime vertical. The general expression
for such variation is 15 cos Lat.
246
226
Problems.
nights) if we observe the same stars we may very much
abridge the computations by making the observations on
the prime vertical; since one calculation made for each star
will be sufficiently accurate for a long period: as I have
shown more at length in the Memoirs of the Astronomical
Society, Vol. 1, page 315.
PROBLEM IV. To determine the time of noon or
midnight, from equal altitudes of the sun*: the
interval of time between the observations, the lati-
tude of the place, the declination of the sun and
its daily variation being given.
This problem is solved by means of the formula in
page 92: where the value of the correction x is to be ap-
plied to the mean of the times at which the equal altitudes
have been observed. The logarithms of A (which is always
minus) and of B will be found in Table XVI, opposite the
given interval. Then find by the Nautical Almanac, or any
other ephemeris, the sun's declination at the time of noon
on the given day and at the given place: which however
need not be taken out to any great accuracy.
Find also
the double daily variation of the declination expressed in
seconds; that is, the difference (in seconds) between the de-
clination at the time of noon on the preceding day, and the
declination at the time of noon on the following day t.
* If we observe equal altitudes of a star, it is evident that half the
interval of time elapsed will give immediately the time of the star
passing the meridian, without any correction.
+ The logarithms of these values, for every day in the year, are now
given annually by M. Schumacher in his Astron. Hülfstafeln. They
would form a valuable addition to our own national ephemeris; since
they would be very useful in navigation, as well as in the observatory.
Problems.
227
This quantity is denoted by 8; and it must be considered
negative, when the sun is proceeding towards the south
pole *
Example. On July 25, 1823, in N. Lat. 54°. 20' at
gh. 59m. 48 A.M. and at 3h. om. 40$ P.M. the sun had equal
altitudes. Required the equation, or correction to be ap-
plied to the mean of those times in order to find the time
of noon?
The interval of time being 6h. lm. 369, we have by
Table XVI log. A = 7.7707 and log. B = 7.6187. And
by the Nautical Almanac the declination of the sun, at
noon on that day, was + 19º. 48'. 29", and its double
daily variation equal to 25'. 29" =
1529". The
operation therefore will stand thus:
logarithms
logarithms
- A
- 7.7707
B= + 7.6187
Ò= 1529" = 3.1844 O= 1529" 3•1844
tan 54°. 20' = + 0.1441
tan 190, 48' = + 9.5300
+ 125,57 = + 1.0992
- 2$,15 0.3331
Correction = + 129,57 2,15 = + 109,43
This value, being added to the mean of the times of the
observed altitudes, or ļ (20h. 59m. 45 + 27h. Ow. 409)
23h. 59m. 52, will give Oh. Om. 29,43 for the time at appa-
rent noon.
This however denotes apparent time: and we must add
thereto the equation of time, which on that day was up-
wards of 6 minutes. If the observations were made on
the meridian of Greenwich the chronometer ought to show
Oh. 6m. 75,2; and consequently would denote that it was
too slow by 6". 49,77.
* That is, from the time of the summer solstice, to the time of the
winter solstice: or from June 21 to Dec. 21. See the Errata.
2 6 2
228
Problems.
PROBLEM V. On the reduction to the Meridian. .
In order to determine the meridional altitude of a hea-
venly body, it is usual, in large observatories furnished with
a mural circle or quadrant, or a transit circle, to observe
such body at the precise moment of its passing the me-
ridian. But, where the observer is furnished with an alti-
tude and azimuth instrument, or a repeating circle, this is
not absolutely necessary; since, by means of Tables XVIII
and XIX, we may render any number of observations
made on each side of the meridian, and at a short distance
therefrom, equal in accuracy to those which are made im-
mediately at the moment of culmination. For this purpose,
it is necessary to know the distance (in time) of the sun or
star from the meridian at the moment of each observation:
and, opposite to such given distance in time in Table XVIII,
is stated the value which ought to be applied to the zenith
distance observed. The sum of these values, divided by
cos L.cos D
their number, and multiplied by
sin Z
will give the
correction which ought to be applied to the mean of the
zenith distances observed (corrected for refraction) in order
to determine the true meridional zenith distance of the sun
or star. Should greater accuracy however be required, we
must take the second part of the reduction from Table XIX,
the sum of which (divided also by the number of observa-
cos L.cos D
tions) must be multiplied by (cos L.. D) x cot Z. But
sin z
this second correction is seldom necessary.
The expression for the zenith distance of a star, in terms
of its declination and of the latitude of the place, will vary
according as the observations are made to the south of the
zenith, or to the north of the zenith; and, in this latter
2
Problems.
229
case, according as the observations are made above or
below the pole *. These several values will be as follow:
Z = L - D. if the obs. be made to the south
z = D- L. if to the north, above the pole
Z = 180° — (L + D) if to the north, below the pole.
It should be observed here that, when the sun is the ob-
ject observed, there is a further correction to be applied,
which is shown in the formula in page 93: where E and
W are expressed in minutes of time, considered as in-
tegers.
Example. On May 13, 1819 a set of 14 observations of
Polaris, near the time of its lower culmination, was made
with a repeating circle, at Shanklin in the Isle of Wight, on
each side of the meridian; the mean of which gave the ob-
served zenith distance, corrected for refraction, equal to
41° 1!. 54",1. The latitude of the place is assumed equal
to 50°. 37'. 23"; and the apparent declination of Polaris is
found by the Nautical Almanac to be 88°. 20'. 28",87. The
observations were made with a chronometer showing mean
solar time, having a losing rate equal to 1",8. Polaris passed
the meridian at 9h. 37m 32s by the chronometer; and the
respective observations were made at the several periods
indicated in the second column of the following table.
What was the correction to be applied to the mean of the
zenith distances above mentioned, in order to reduce them
to the meridian?
By taking the difference between the several times of
observation and gh, 37m. 32s we obtain the values in the
* M. Delambre has, in his Astronomie, considered only the case where
the observations are made to the south of the zenith: and it was by
following him too closely that I was led into the error of ernploying
(L-D), instead of. Z, in the Formula in page 93. See the list of Errata.
230
Problems.
third column of the following table. Entering Table XVIII
with these values, as arguments, we obtain the values set
down in the last column: the sum of which being divided
by 14 will give 490",9.
No.
Time of
Observation
Time from By
Meridian. Table XVIII
m
S
m
S
1
2
3
4
5
6
7
8
9
10
11
12
13
14
h
9 13 15
9 16 40
9 21 21
9 33 55
9 36 44
9 39 55
9 42 27
9 45 40
9 48 56
9 51 53
9 54 40
9 56 48
9 59 40
10 2 20
24 17
20 52
16 11
3 37
0 48
2 23
4 55
8 8
11 24
14 21
17 8
19 16
22 8
24 48
1156,8
854,3
514,0
25,7
1,3
11,2
47,5
129,9
255,1
404,2
576,1
728,4
961,1
1206,4
14)6872,0
490,9
The subsequent operation then will stand thus :
1
L = 50 37 23
D= 88 20 29
COS = + 9.8023765
cos = + 8.4615613
138 57 52
+ 8.2639378
180° - (L + D) = 41 2 8 sin = + 9.8172528
constant log. = + 8·4466850
490,9 = t 2.6909930
on account of mean solar time = + 0.0023715
on account of rate of clock
+ 0.0000181
13",806 = x = + 1.1400676
Problems.
231
As the star was below the pole, at the time of observation,
this correction must be added to the observed zenith di-
stance; whence the true meridional zenith distance will be
41º. 1'. 54",1 + 135,81 = 41°. 2!. 71,91; and the latitude
= 180° - (Z + D) = 50°. 371. 23",22.
In observations with the repeating circle, for determining
the latitude, it is necessary to attend to the verticality of the
circle; since an inclination of the circle will cause a corre-
sponding error in the results. But if the amount of the
inclination i be known, we may ascertain the error e in the
result, by means of the following equation:
sin 1".z.cot Z.
We must also attend to the position of the level, and
either bring it, by the proper screw, to its zero point, or
take an account of the place of the bubble in the two
opposite positions of the circle, and allow for the differ-
ence, according to the value of the divisions of the scale:
prefixing the sign + or according as either end of the
bubble is nearer to or farther from the observer than the
true zero point.
There are also some other circumstances to be attended
to, in the management of this instrument, which are pointed
out in the works expressly written on that subject.
e е
PROBLEM VI. To determine the latitude of a place,
The best mode of determining the latitude of a place, so
as to be independent of the declination of the star observed,
and also as free as possible from the errors of refraction, is
by observations of a circumpolar star at the time of its
upper and lower culmination. These observations may be
made by means of a mural circle or quadrant, or a transit
circle, at the precise moment of the passage of the star
232
Problems,
8
across the meridian, or they may be made by an altitude
and azimuth circle, or a repeating circle, either in the same
manner, or in the mode alluded to in the last problem. In
either case, therefore, let Z denote the observed or deduced
meridional zenith distance of the circumpolar star at its
lower culmination, and g its refraction at that point: also
let Z denote the observed or deduced meridional zenith
distance of the same star at its upper culmination, and si
its refraction at that point. Then will the correct zenith
distance of the pole, or the co-latitude (4) of the place, be
f į (Z + Z) + (8 + 8')
It is evident that the accurate determination of ţ will
depend on the tables of refraction that are used in the
computation: and there is no mode of rendering the pro-
blem free from this ambiguity.
If we take the case of the pole star as observed at
Greenwich, its zenith distance at the upper culmination
may
be 36º. 55' only, whereas at its lower culmination it
may
be 40°. 71: and half the sum of the refractions at these
points will differ according to the tables of refraction em-
ployed. This half sum (the barometer being at 30 inches,
and Fahrenheit's thermometer at 50°) will be by
Bradley's Tables = 46",02
Bessel's Tables = 46 ,58
Ivory's Tables = 46 ,53
French Tables = 46 ,52
Consequently a difference of half a second, at least, will
take place, at that temperature and pressure, according as
Bradley's or the other tables are made use of.
The next usual mode of determining the latitude of a
place, is by means of meridional zenith distances of the
sun, or a star (whether circumpolar or otherwise) whose
Problems.
233
declination is well known. The expression for the latitude
will, in such cases, vary according as the observations are
made to the south of the zenith, or to the north of the
zenith: and in this latter case, according as the observa-
tions are made above or below the pole. Let L denote the
latitude required, D the declination of the sun *, or the
apparent declination of the star, and Z the observed me-
ridional distance of the same corrected for refraction in
the case of a star, and for refraction minus parallax in the
case of the sun *), then we shall have
L = Z +D. if the obs. be made to the south
L = D - Z. if to the north above the pole
L = 180° - (Z + D) if to the north below the pole.
There is another mode of determining the latitude of a
place, by means of observations of the altitude of the pole
star at any time of the day; a method which is capable of
great accuracy, and may frequently prove very convenient
and useful. I have already treated this subject, more at
length, in two papers inserted in the Philosophical Maga-
zine for June and July 1822: and shall here merely refer
to the formula, which is given in page 110 of the present
work, for the mode of explaining the subject by the follow-
ing example.
Example. At 4" after the passage of the pole star, at its
upper culmination, its altitude (corrected for refraction)
was observed to be 50°. 47'. 435,6, and its apparent north
polar distance was 1º. 38'. What was the latitude of the
place?
* Where the sun is observed and where great accuracy is required,
its declination should be corrected on account of its latitude.
2 H
234
Problems.
The operation will stand thus :
logarithms.
A 1° 38' on tan = + 8.4550699
P = 60 0 0
COS = + 9.6989700
a 0 49 0,6 tan = to 8.1540399
a = as above
A = as above
cos = + 9.9999559.
cos = + 9.9998235
a =
0.0001324
A = 50 47 43,6 sin = + 9.8892424
(8 + a) = 50 49 0,7 sin 9.8893748
- 0 49 0,6
Latitude = 50 O 0,1
For a fixed observatory, these computations might be
somewhat abridged, and rendered less liable to error, by
determining the logarithms of the values of (1 sin 1". tan o)
and of – (cot p + tan o) sin 1". which in the above case
would be 4.4607741 and, -4.8533647 respectively*. The
subsequent process then, agreeably to the formula given
in page 110, will be as follows:
3.7693773
cos P = + 9.6989700
- 49'. 0",0 = 0
304683473
A = + 3•7693773
sin P = + 9.9375306
A.sin P = + 3•7069079
A?.sin?
Pt 7.4138158
1 sin 1".tano = + 4.4607741
+ 1'. 14",9 = B = + 1.8745899
3.4683473
n =
4.8533647
+ 1",6 = aßg
= + 0.1963019
+
* The latitude of the place is always known sufficiently near for the
determination of these logarithms.
Problems.
235
Consequently the latitude of the place will be
50 47 43,6
14,9
1,6
1
-
50
49
49
0,1
0,0
50
0 0,1
There is still another mode of determining the latitude
of a place, which is independent of the divisions of the in-
strument, and depends only on the apparent declination
(D) of the star observed, and on the interval of sidereal
time which has elapsed between the observations. This
method (which I have also explained more at length in the
Philosophical Magazine for May 1825) consists in placing
the axis of the telescope of an altitude and azimuth instru-
ment due north and south, so that the vertical circle should
stand east and west, and thus twice cut the parallels of all
the stars between the equator and the zenith. The obser-
vation of the two times T and T (at which the star passes
the wire of the telescope in its diurnal revolution) will give
the latitude (L) of the place from the following formula,
cot L = cot D.cos } (T – T)
In this formula, (T-T) denotes the correct interval of
sidereal time elapsed between the observations of each star,
expressed in degrees &c. So that if mean solar time be
employed, we must multiply the interval by 1.0027379;
or, which is the same thing, add the values found in
Table VII, against the given interval.
It is evident that this method (like all the others, except
that by means of circumpolar stars) depends on the accu-
racy of the apparent declination of the star observed: a
2 H 2
236
Problems.
small error in this point, however, will not materially affect
the results. But, if this mode be adopted in geodetical
operations, it is evident that we may obtain the difference in
the latitude of two places very exactly and almost indepen-
dent of the declination of the star. It is this circumstance
that renders the method valuable in such investigations.
This method is indeed recommendable on account of its
independence of any error in the instrument. If the colli-
mation should not be sufficiently corrected, the cylinders of
the axis should be unequal in their diameter, the telescope
or the axis should bend &c &c, we shall still obtain a cor-
rect result, either by reversing the axis between the two
operations, or by observing one day in one position and
the next day in the other position of the axis, and taking
the mean of the two. The success solely depends on the
quality of the telescope, and the care employed in levelling
the axis. It is scarcely necessary to add that observations
of this kind should be made on stars that culminate near
the zenith of the place.
PROBLEM VII. To determine the longitude of a
place.
The method of determining the difference of longitude
between two given points on the surface of the earth, which
is one of the most difficult problems in practical astronomy,
has long engaged the attention of various astronomers and
mathematicians; and has been executed with more or less
accuracy according to the means' employed for that pur-
pose. If the distance between the two observatories be not
very great, their difference of meridian may be determined
with considerable accuracy, by means of chronometers con-
veyed from one observatory to the other; or by means of
5
Problems.
237
signals previously agreed on. These methods have been
practised very successfully on many recent occasions. But,
where this is impracticable, we must have recourse to the
observation of certain celestial phænomena for the solution
of the problem: and for this purpose, five several and di-
stinct methods have been proposed : 1° the eclipses of Ju-
piter's satellites : 2° eclipses of the moon: 3° eclipses of
the sun: 40 occultations of the fixed stars : 5° the me-
ridional transits of the moon, compared with certain stars
previously agreed on.
The results deduced from the observations of the eclipses
of Jupiter's satellites are, for obvious reasons, very unsatis-
factory. The phænomena will, in fact, appear to take
place at different moments of time, with different instru-
ments and to different observers. Moreover, they are
visible only in certain positions of the planet in its orbit;
a circumstance which very much circumscribes the utility
of the method.
The eclipses of the moon afford a still more unsatisfac-
tory result : they occur but seldom in the course of a year,
and the phænomena attending them cannot (on account of
the indistinctness of the border of the earth's shadow) be
observed with that degree of accuracy which the present
state of astronomy requires for such purposes.
Eclipses of the sun are more certain in their deductions :
but, they so rarely occur, and are at the same time so
limited in extent, that they can seldom be brought in aid
of the general solution of the problem. From September
1820 to November 1826, there is only one solar eclipse
that will be visible in this country.
There remain therefore only the two other methods, on
which the practical astronomer can safely and constantly
depend. Of these, I am aware that occultations of the
238
Problems.
fixed stars by the moon have been long considered as
affording the best means of determining the difference of
longitude between two places: and, assuredly, the results
deduced from such observations, made under favourable
circumstances, have agreed with each other to a greater de-
gree
of accuracy, than those deduced by any of the pre-
ceding methods.
There are, however, many circumstances, attending the
practical solution of the problem by this method, which
tend to diminish the confidence which is reposed in the cor-
rectness of the theory. In the first place, it is necessary to
know the apparent right ascension and declination of the
star very exactly, on the day of observation; which, if the
star is of inferior magnitude (and such being the most nu-
merous, are the most likely to be occulted), may not be
readily determined: for, we may not be able to find it in
any catalogue; and, when found, we have to compute its
precession, aberration, and nutation expressly for this pur-
pose. In the second place, we have to calculate the paral-
lax of the moon for the given moment of observation: and
in this computation we must assume a given quantity for
the compression of the earth; respecting which, astrono-
mers are by no means agreed, and which will consequently
give rise to various results, according to the view which
each astronomer may take of the subject. Thirdly, this
method is dependent on the accuracy of the lunar tables,
not only as to the position of the moon and her horary
motion, but also as to her horizontal parallax and semi-
diameter. Fourthly, the method is, in a great measure,
dependent on a correct knowledge of the longitude and
latitude of the place of observation. And lastly, the ap-
parent border of the moon is so uneven (consisting of pro-
jecting mountains and hollow valleys) that we cannot always
Problems.
239
depend on the immersion or emersion having taken place
at the exact distance from the moon's centre, as computed
from the lunar tables.
The meridional transits of the moon, agreeably to the
method about to be described, are free from all these ob-
jections: the observations are made with the greatest fa-
cility; the opportunities are of frequent occurrence; the
absolute time is of no material consequence; the.computa-
tions are by no means intricate or troublesome; and the
results are (I believe) more to be relied on than by any of
the preceding methods.
This method consists in merely observing, with a transit
instrument, the differences of right ascension between the
border of the moon, and certain fixed stars previously
agreed on*; which stars are so selected that they shall
differ very little from the moon in declination. It is evi-
dent that this method is quite independent of the errors of
the lunar tables, except as far as the horary motion of the
moon (in right ascension) is concerned, and which, in the
present case, may be depended on with sufficient confi-
dence: that it does not involve any question as to the
compression of the earth: that a knowledge of the correct
position of the star is not at all required : and finally, that
an error in the state of the clock, is of no consequence.
Consequently, a vast mass of troublesome and unsatisfac-
tory computation is avoided. Moreover, it is the only
method that is universal, or, that may be adopted, at one
and the same time, by persons in every habitable part of
the globe: for, it is applicable to situations distant 180° in
longitude from each other; and even beyond that if required.
* Lists of such stars, called mogn culminating stars, are now annually
published.
240
Problems.
It might indeed, at first sight, appear that the same re-
sults would be obtained, if we merely observed the correct
time of the moon's transit, without any reference to the
contiguous stars : but, a moment's reflection will convince
us that, by referring the moon's border to the adjacent
stars, we obviate all errors not only of the clock, but also
in the position of the transit instrument.
For the solution of this problem, let us make,
t = the difference (in sidereal time) of the transit of
the moon's limb, and of the star previously
agreed on, at the observatory situated most
westerly; which will be positive when the star
precedes the moon, or when the R of the moon
exceeds that of the star; but, on the contrary,
negative.
T = the similar difference, at the observatory situated
most easterly.
(t-1) = the true observed difference in the R of the
moon's limb, for the time elapsed between the
two observations *.
c= the apparent time (as shown at Greenwich +) of
the culmination of the moon, at the western ob-
servatory
x = the apparent time (as shown at Greenwich) of
the culmination of the moon at the eastern ob-
servatory.
* If more than one star has been observed at both observatories on
any given night, t and t must be taken equal to the mean of all the
corresponding comparisons made at each observatory respectively.
+ Or as shown at Paris, Berlin, Milan or any other place for which
the ephemeris is calculated, from which the computations are made.
And this must always be understood, when Greenwich is alluded to in
this manner.
Problems.
241
a = D's right ascension, in space
computed for
d = D's true declination *
the time c.
r = D's true radius, or semidiam. *
a = D's right ascension, in space
computed for
8 = D's true declination *
the time x.
g= D's true radius, or semidiam. *
s = the length of the true solar day, expressed in
seconds of time.
m = the moon's motion in R, in half that interval,
expressed in seconds of space: See page 246.
x = the assumed difference of longitudes in time:
plus when west, and minus when east.
(x+e) = the correct difference of longitudes.
Find the apparent times, c and X, of the moon's culmina-
tion, to the nearest minutet, in order to compute d, r and
* The true declination and semidiameter of the moon, are such as
they are supposed to be if seen from the centre of the earth: in oppo-
sition to the apparent declination and semidiameter, which some per-
sons have erroneously imagined ought to be adopted.
It may be sufficient to observe here, once for all, that (with a view
to prevent confusion) the quantities connected with the eastern obser-
vatory, are denoted by Greek letters: and that the similar quantities
connected with the western observatory, are denoted by Roman letters.
† The apparent time of the moon's culmination at Greenwich, to the
nearest minute, may be seen in the Nautical Almanac: and the appa-
rent time (at Greenwich) of its culmination on any other meridian may
thence be easily deduced. Or, if the sidereal time is known, we may
determine the Greenwich apparent time very nearly, by subtracting
therefrom the sun's right ascension at Greenwich at the preceding
noon; and diminishing the interval by the acceleration of the fixed
stars. Or, we shall have, in all cases sufficiently near for this purpose,
the required interval c-x= [x +(t - 7)] X :99727: where it should
be observed that [x+(t–7)] X •99727 is equal to the time x+(t-7)
diminished by the acceleration of the ixed stars for that interval.
25
242
Problems.
8, g, for those approximate times respectively*; and then
make
S
A = (t – 5) + 15.Cos d +
15. cosa
no
which is the true observed difference in the R of the moon's
centre, for the time elapsed between the two observations:
where the upper sign is to be taken when the first (or western)
border of the moon is observed; and the lower sign when
the second (or eastern) border is observed t. Then, by
assuming x equal to the presumed difference of longitude,
and knowing the apparent time (at Greenwich) at one of
the observatories to the nearest minute, we may determine
the required apparent time (at Greenwich) at the other
observatory, by the following equation:
86400
c = x + (x + A)
S
Compute a and a for the respective times c and x 1; cor-
* It may be useful here to remark that it is not necessary to deter-
mine with strict accuracy the absolute value of the moon's semidiameter
at both observatories, in order to find the value of A: for, the values
may be estimated (in most cases by inspection) in whole seconds only,
for one observatory, and the correct differences, in the given interval,
being added thereto, will give the proper values for the other observa-
tory. With respect to the declination, it may be taken to the nearest
ten or twenty seconds only.
+ These expressions are the same as those which are used in my Paper
on this subject, inserted in the 2nd volume of the Memoirs of the Astro-
nomical Society; where this subject is treated more at length.
$ It may here also be useful to remark that it is not necessary to de-
termine with strict accuracy the precise moment of the apparent time
of the transit of the moon at both observatories, for the purpose of de-
termining a and 6: for it will be sufficient to know the apparent time
of the transit of the moon to the nearest minute only, for one observa-
1
Problems.
243
e =
recting the moon's motion, for third differences, if required.
And the formula for the correction of the assumed differ-
ence of longitude will be
= [154- - (a –a)]202
which, being added to Xwill give (x + e) for the true
difference of meridians required.
It is evident that, if 15 A - (a - a) = 0, the value of x
has been assumed sufficiently accurate, and does not re-
quire correction. In fact, the difference will in general be
very small: and, when this is not the case, we may justly
suspect some error in the steps of the
process.
86400
S
S
tory, and to find the correct difference of the apparent times, by means
of the expression (x+4)
In fact, nothing more is required than to compute the true increase
of the moon's R during this given interval: and for this purpose Dr.
Brinkley has suggested a very convenient rule, which is given in the
first number of the Dublin Philosophical Journal. This distinguished
astronomer has there shown that (a — «), as far as first differences only
are concerned, may be expressed by (x + 4). A". : leaving the
equation of second differences (and of the third differences, if required)
to be applied in the usual manner.
Under this point of view, the problem admits of two cases : one
where both the observations are made on the same side of noon or
midnight; and the other where they are made on different sides. In
the former case, the expression (x + 4). A".
A". as far as first. differ-
ences are concerned, will lead us to the correct solution, and will save
much time and labour: but, in the latter case, it must be divided into
two parts : viz.
(12+ — x). A". and (x+4 – (125 – x)]. A".
and consequently becomes more intricate. In these expressions A"
and A'' denote the successive first differences in the moon's motion in
R; or the same quantities that are alluded to in page 103.
2
S
11
2 1 2
244
Problems.
T
Example. On December 5, 1824, Lieut. Foster observed
the differences in the culmination of the moon and of the
two stars 62 and 95 Tauri, at Port Bowen; the station
where the Expedition (for the discovery of a North West
passage, under the command of Capt. Parry) passed the
winter of 1824-25. Similar differences were observed
also at Greenwich. These differences, in sidereal time,
were respectively as follow:
at Greenwich
at Port Bowen
62 Tauri T = + gm 459,58 t = + 24m 539,98
95
9 25 ,98 · t = + 5 42,90
what was the longitude of the place?
For the solution of this question, we must first assume
an approximate longitude. Now it appears from some oc-
cultations of fixed stars, observed by Lieut. Foster, that
the longitude might be considered as 5h. 55m. 399,5 west
from Greenwich : but, for the sake of round numbers, I
shall assume it equal to 5h. 55m. 40s. The operation there-
fore will be as follows.
The mean of the two observations gives t= + 15m. 189,44
and T = + 0". 99,80: consequently we have (t – 5) =
+ 15m. 89,64; which being added to 5h. 55. 40s, and the
sum diminished by the acceleration of the fixed stars
during that interval, will give in round numbers 6h. 10m as
an approximate value of the apparent time elapsed between
the two culminations.
By the Nautical Almanac it appears that the moon's
centre passed the meridian at Greenwich at 11h. 35m: con-
sequently the moon's first limb passed at 11h. 34m, at
Greenwich; and at 17h. 44m (Greenwich time) at Port
Bowen. With these approximate values we find the de-
clination and semidiameter of the moon, at those respective
periods, as follow:
Problems.
245
at 11h 34m
g
0° 15' 42"
8 = 23 39 20
at 17h 44m
0° 15' 44",39
d = 23 53 30
whence we find
A= +15". 85,64 + 1*(17'. 12",88–17'.8",41)=159.89,938
and the correct value of c, for the subsequent computa-
tions, will be
c=114.34m +(54.55m.40s +15m.85,938) 36400 = 17h.43m. 419,67
The moon's true right ascension must now be calculated for
the apparent times x= 11". 34m and c=171, 43". 415,67*:
whence we have
a = 69° 53' 49",21
66 6 29,93
(a – 6 3 47 19,28
15 A 3 47 14 ,07
15 A – (a - 0)
5,21
24b. 4m. 225
This remainder, being multiplied by
2 x 70.24'.4211
= 1.62, will give e = 89,44 : and the correct longitude
will be (x + e) = 51. 55". 315,56 +-
Should the value of 15 A - (a – a) be considerable, it
will show either that there is some error in the computa-
tion, or that the value of x has not been assumed suffi-
ciently near. In the latter case, we must diminish e by
the acceleration of the fixed stars during that interval, and
apply the result to c as a new value for the computation of
Thus · (89,44 – 0$,02) 89,42 being added to C,
will give 17h. 43m. 339,25 as the correct time for which a
S
(=277024.42
2 m
a.
و
* Or, at once, for the given interval ; agreeably to the method pro-
posed by Dr. Brinkley, as stated in the preceding note.
+ From a mean of 21 eclipses of Jupiter's satellites, the longitude was
found to be 54. 55m, 29%.
246
Problems.
should have been computed: and the result would then
have been *
a = 69° 53' 44",00
a 66
6 29 ,93
(a - ) 3 47 14 ,07
15 A = 3 47 14 ,07
For the convenience of those persons who make use of
this method of solution, I have computed the following
table of the value of
argument of which is m=the
moon's motion in R in 12 true solar hours; or the quan-
tity which is actually employed, as the first difference, in
computing the moon's place, for c or X, as the case may be.
The value of s is, in this table, assumed to be equal to
24h. 4m.
S
: the
2 m
S
S
Argument.
diff.
Argument.
diff.
2 n2
2m
50
o'
2. 1066
6° 30'
1.8513
•1145
0686
5 15
2-2921
6
45
1•7827
•1042
0637
5
30
2.1879
7
0
1•7190
·0951
0592
5 45
2.0928
7
15
1.6598
•0873
•0554
6
0
2.0055
7
30
1.6044
•0802
•0517
6 15
1.9253
7
45
1:5527
•0740
·0485
6 30
1.8513
8 0
1.5042
The following table will also be convenient for deter-
86400
mining the logarithm of the value of
The argument
S
* All these values of a and a have been corrected for third differ-
ences; which diminish the value of a of about 05,5.
Problems,
247
is the increase in the sun's R in 24"; which is found from
an ephemeris, by taking the difference in the R of the sun
for two successive days.
Logarithm of
86400
Argument
Diff. for
seconds
S
m
S
S
3
30
9.9989457
1050
3
40
9.9988955
2 = '100
3
50
9.9988454
3= "150
4
0
9.9987953
4 = 5200
4
10
9.9987451
5 - 6250
4
20
9.9986950
&c
&c
4
30
9.9986449
PROBLEMVIII. To determine the apparent obliquity
of the ecliptic, from observations of the sun made
near the time of the solstices : and thence the
mean obliquity at the beginning of the year.
The observations necessary for the determination of this
problem are always made a few days previous and subse-
quent to the day of the solstice; and consist of meridional
observations of the sun's altitude, or zenith distance, which
must afterwards be cleared of refraction and parallax.
These observations are made either at the time of the sun's
passing the meridian, or a short time before and after
that period, and reduced thereto by the methods already
explained in Problem V. The declination of the sun's
centre at the time of observation being thus deduced from
the formula D = (L - Z), we may determine the correc-
248
Problems,
tion that ought to be applied thereto, in order to express
the obliquity of the ecliptic, from Formula XX in page 94,
since the true value of the obliquity is always very nearly
known. When the right ascension and declination are de-
termined by the same instrument, we may make use of
No. I. But in other cases we shall find the correction suf-
ficiently near as follows: viz.
x = 135,634782 - 0",00054 84
where 8 denotes the distance of the sun's true longitude
from the solstice, at the time of observation, expressed in
degrees and decimal parts of a degree; and which may be
found by an ephemeris.
Example. On June 25, 1812, by an observation of the
sun on the meridian at Greenwich, it was found that the
zenith distance of the centre, cleared of refraction and pa-
rallax, was 28°. 4'. 1': what was the correction which
ought to be applied to the declination of the sun, in order
to deduce the apparent obliquity of the ecliptic at that
time: and what was the mean obliquity at the beginning
of the year?
The assumed latitude of the place being 51°. 28'. 40" we
have the declination at the time of observation equal to
51° 28'. 40" - 28º. 4'. 1" =230. 24'. 39". By the Nauti-
ca! Almanac, the true longitude of the sun on that day, at
noon, was 93º. 40'. 33"; consequently by Table XXXVIII
8 = 30.6758: and the operation will stand thus :
logarithms.
3.6758 = + 0.5653519
(3.6758) = + 1•1307038
13.6347 = + 1.1346456
+ 3'. 4",225 = + 2•2653494
(3.6758) = + 2.2614076
00054 = 6.7323938
- 0",099 = 8.9938014
Problems.
249
and the correction will be 3'. 4",23 0',10 = 3'. 4",13:
whence the apparent obliquity at the solstice will be
23°. 27. 43",13. In this computation no notice has been
taken of the latitude of the sun, which must be computed
from the solar tables, and applied with a contrary sign, to
the apparent obliquity above deduced. In the present case,
the latitude of the sun is + 1",00: consequently the correct
apparent obliquity is 23º. 27'. 42",13.
The apparent obliquity being thus obtained we may
readily deduce the mean obliquity at the beginning of the
year, by the help of Tables XXI and XXII. The mean
place of the moon's node being 423, we have in Table XX,
opposite thereto, – 8,21; and in Table XXII against
the given date we have -- 0",76. But those Tables having
been formed for the purpose of determining the apparent
obliquity from the mean obliquity, the quantities must be
applied with a contrary sign, in order to obtain the mean
obliquity from the apparent: consequently we have, at the
beginning of the year, the
mean obliquity = 23º. 27'. 51",10.
PROBLEM IX. To determine the apparent equinox,
from observations of the sun made near the time
of the equinoxes.
Observations similar to those alluded to in the preceding
problem, made a few days previous and subsequent to the
equinox, will enable us to determine the precise time at
which the sun is at that point. For the declination of the
sun being found from the observed zenith distance as therein
stated, we must correct the same for the latitude of the sun,
in the following manner. Let D be the declination, de-
&
2 K
250
Problems.
duced as above, and let D' be the true declination : then
we shall have
D' = D-l.
COS W
cos D
where I denotes the latitude of the sun (minus when south)
and w the apparent obliquity of the ecliptic. The true de-
clination of the sun being found, we may determine the
longitude by the following formula:
sin D
sin O =
sin w
which being compared with the tables of the sun, will
show if there is any error * As the determination of the
equinoxes depends on the correctness of the latitude of the
place, it is desirable that the observations should be re-
peated at the opposite equinoxes, in order that the errors
may destroy one another.
PROBLEM X. To determine the correct place of the
moon from an ephemeris, by means of differences.
The place of the moon is usually given in an ephemeris
for apparent noon and for apparent midnight: but her
motion is so variable that her place cannot be accurately
determined for any intermediate time without the help of
second, and sometimes of third differences. In the annual
volumes of the Nautical Almanac there is given a table of
second differences, by the help of which this problem is
usually solved. But as third differences may sometimes be
wanted, I have adapted the whole to logarithmic computa-
tion, by means of Table XXVI. In order to use this
table, we must find the first, second and third differences,
* As the declination, in these cases, is always a small quantity, we
have the longitude of the sun with considerable accuracy; even if the
obliquity is not well determined.
Problems.
251
in the manner pointed out in the Nautical Almanac: and
to the logarithms of those differences (taking the mean of
the two second differences) add the respective logarithms
in the Table. The natural numbers thence resulting (due
regard being had to the signs) will be the total correction
to be applied to the moon's place in the usual way.
Example. What will be the true right ascension and de-
clination of the moon on March 3, 1828 at 7h. 10m P.M.
apparent time at Greenwich ?
By proceeding agreeably to the instructions in the Nau-
tical Almanac, we have
For the Right Ascension
1st diff. 2nd diff. 3rd diff.
Mar. 2. Midn.= 175 38 34
+ 5 59 40
3. Noon = 181 38 14
+ 4 10
+ 6 3 50
3. Midn.= 187 42 4
+ 5 8
+ 6 8 58
4. Noon = 193 51 2
1
+ 58
For the Declination
1
Mar. 2. Midn. 0 56 11
3. Noon = -2 59 8
3. Midn. - 5 1 0
4. Noon =
-7 0 23
2 2 57
+ 1
1
5
2 1 52
+ i 24
+ 2 29
1 59 23
The operation will consequently stand thus :
For Right Ascension
For Declination
logarithms.
logarithms.
+ 6° 3! 50" = +4.3390537 - 2° 1' 52" = 3.8640362
Tabular factor = + 9.7761360*
factor = + 9.7761360
+3° 37' 17",4 = + 4•1151897 -1° 12' 46",9 3.6401722
7h. 10m
* This factor has been computed front the expression agree-
12h, om
ably to what has been stated in the note to page 205.
2 K 2
252
Problems.
factor =
logarithms.
logarithms.
+ 4' 39" = + 2.4456 +1' 47" = + 2.0294
Tabular factor = 9.0802
9.0802
331,6 - 1.5258
12",9
- 1•1096
+ 58" = + 1•7634
+ 1' 24"
= +1.9243
Tabular factor = 7.5908
factor =
7.5908
0",2 9.3542
0",3 : 9.5151
The correction in R will therefore be + 3º. 36'. 43",6,
and the correction in declination - 1°. 13'. 0",1: conse-
quently the true place of the moon will be as follows:
R = 181° 38' 14." + 3° 36' 43",6 = 185° 14' 57",6
D = -2 59 8 1 13 0,1 = -4 12 8,1
PROBLEM XI. To determine the moon's parallax.
The parallax of the moon differs at every point of the
earth's surface. No general tables therefore can be given
adapted to the situation of every observer. The latitude of
the place, the position of the moon (not only in her orbit,
but as seen from the earth), the hour of the day, and the
assumed compression of the earth, are so many varying
elements in the computation, that I have always found it
much less laborious to calculate the values from the for-
mulæ, than to make use of any tables not computed for the
exact place of observation.
The most convenient formulæ are those given in pages
98-100: and in the computation of occultations I prefer
calculating the parallax in right ascension and declination,
to that of longitude and latitude; not only because the
latter involves the computation of the nonagesimal (a
tedious and useless operation), but also because the posi-
tions of the stars are given in right ascension and declina-
Problems.
253
tion in the catalogues, and therefore require no further
conversion, when considered as in conjunction with the
moon. In the computation of solar eclipses, either method
may be adopted. But, whichever mode is pursued, the
method of series will be found the most convenient; and not
so liable to error as the other methods: or, at least, any
mistake of the pen is soon detected, and may be easily
rectified without disturbing any material part of the pro-
If we wish for a near approximation only, we may
stop at the first term of the series : and, in no case need
we extend it beyond the third term.
cess.
Example. Let the reduced latitude of the place be
48° 39'. 50"; the horizontal parallax of the moon, at that
latitude, 54'. 21,5*; the horary angle at the pole (or the
correct sidereal time minus the moon's true right ascension)
converted into degrees &c, equal to 58º. 43'. 50"; and the
true declination of the moon at that time, + 4°. 49'. 44".
What should be the parallax of the moon in right ascen-
sion and declination ?
This is the same example as that given by M. Delambre
in his Astronomie, Vol. I, page 376 and 379; and the solu-
tion, by means of the series No. 4 in Formulæ XXVI and
XXVII, will be as follows:
* The reduced latitude is equal to the observed latitude minus the
angle of the vertical, which angle is determined by Formula XXI in
page 95, or may be seen in Table XXIV, if the compression is assumed
equal to zoo. And the horizontal parallax at the place is determined
by multiplying the horizontal parallax at the equator by (1 — a.sinL)
as stated in page 95: or it may be found by adding the logarithm of the
horizontal parallax at the equator to the logarithm which in Table
XXIV is set against the given latitude, if the compression be assumed
equal to do
254
Problems.
For the parallax in Right Ascension
logarithms.
0° 54' 21,5 sin = +8.1964369
L = 48 39 50 cos = +9.8198564
P =
D = 4 49 44
+8.0162933
COS = +9.9984557
sin P= 58° 43' 50"= +9.9318319
sin 2 P=117 27 40 = +9.9480822
sin 3 P=176 11 30 = +8.8222925
a = +8.0178376
a = +6•0356752
a>= +4.0535128
a = +8.0178376
sin P = +9.9318319
comp. sin 1" = +5•3144251
+30' 36",939
= +3.2640946
a? = +6•0356752
sin 2 P = +9.9480822
comp. sin 2 = +5.0133951
+91,935
= +0.9971525
3
a3
= +4.0535128
sin 3 P=+8.8222925
comp. sin 3! = +4.8373039
+0",005
=+707131092
whence, since the quantities are all positive, the parallax in
right ascension is + 30'. 46",879 *.
If we denote the apparent right ascension of the moon
by Ra, and her true right ascension by Rt, we shall have
Ra = R + II
where II is the computed parallax in right ascension. This
parallax will be a positive quantity when the moon is on
* We here see that the first term gives a very near approximation,
and that the third term is almost insensible.
Problems.
255
the west side of the meridian: but when she is on the east
side, it will be negative; because P will then be negative.
For the parallax in Declination
logarithms.
cos (P + II) = 58° 59' 131 = +9.7120040
cot L = 48 39 50 = +9.9443044
+9•6563084
cos į II = 0 15 23 =+9.9999957
cot b = 65 37 8 = +9•6563127
0 54 2,5=+8.1964369
sin L = 48 39 50 =+9.8755520
+8.0719889
sin b = 65 37 8 = +9.9594325
sin (6-D)= 60° 47' 24" = +9.9409331 C+8.1125564
sin 2(6-D)=121 34 48 = +9.9303936 c -- +6•2251128
sin 3(6-D)=182 22 12 = -8.6165019 c=+ 48376692
sin p =
C=+8•1125564
sin (6 – D) = +9.9409331
comp. sin 1" = +5•3144251
+ 38' 53",005 . = +33679146
C = +6•2251128
sin 2 (6 – D) = +9.9303936
comp. sin 21 = +5.0133951
+ 14",754 ...
= +1•1689015
C = +4.3376692
sin 3 (6 – D) = -8.6165019
comp. sin 3" = +4•8373039
0",006 .
-7.7914750
whence the parallax in declination will be 39'. 71,753.
3
* This logarithm is taken out errone-ysly in M. Delambre's example;
which will account for the slight difference in our results. It may be
•
256
Problems.
If we denote the apparent declination of the moon by
Da and her true declination by D, we shall have
Da = De -W
where w is the computed parallax in declination. It should
be remarked that when the declination of the moon is south,
D is a negative quantity, and must be treated as such in
the algebraic operations: consequently a will, in such
case, increase the declination.
PROBLEM XII. To determine the aberration, and
lunar and solar nutation of a star, by the general
tables of M. Gauss.
These tables are given in pages 174–177: and, in
order to show their use and application, take the following
Example. What is the amount of aberration and nuta-
tion, in Right Ascension and Declination, of Aldebaran on
May 11, 1827, at noon?
The right ascension of this star, expressed in degrees &c,
is 66°. 30', and its Declination + 16º. 9': and we have, on
that day, 0 = 49°. 59', and 8 = 224°. 10'. By referring
therefore to the formula in page 207, the operation will be
as follows:
here useful to state that astronomers formerly used to compute the
parallax in declination and latitude without employing the value of
II: which certainly rendered their formulæ more convenient. But,
by neglecting this quantity, we may cause an error of 8 or 10 seconds
in the parallax in declination and latitude. The parallax in declination
cannot be correctly known without previously computing the parallax
in right ascension: a similar remark holds good with respect to the
parallax in latitude and longitude.
Problems.
257
For Right Ascension
O = + 49° 59'
by Table XXVIII A = + 2 25
66 30
R =
14: 6
D = + 16 9
by Table XXVIII
Aberration
logarithms.
COS = + 9.9867
sec = + 0.0175
a = 1.2918
19",77 =
1.2960
6 = + 224 10
by Table XXX . B = 8 17
66 30
R =
COS
9.9348
+ 149 23
+ 16 9
tan
+ 9.4618
-b-
0.9315
+ 21,12 = + 0•3281
by Table XXX
C = + 11 ,52
Solar Nut. by s 1,01
Table XXXI -
0,09
Nutation = + 12 ,54
Consequently the correction for aberration and nutation in
Right Ascension is - 19',77 + 125,54 = - 7",43.
For Declination
sin =
(O + A - R
)
14° 6
D = + 16 9
logarithms.
9.3867
sin = + 9.4443
a = 1.2918
+ 0.1228
of 1",33
1,63
- 3,34
by Table XXIX .
{
Aberratia" =
we
3,64
2 L
258
Problems.
logarithms.
sin 149° 23' = + 9•7070
6
0.9315
4",35 0.6385
by Table XXXI.
0,26
Nutation = 4,61
Consequently the correction for aberration and nutation in
Declination is - 35,64 – 4",61 = - 8",25.
PROBLEM XIII. For determining the corrections to
be applied to observations with the transit instru-
ment.
It is well known that observations with the transit in-
strument are subject to three principal errors arising from
the three following sources: viz. 1° from a deviation of the
instrument in azimuth; 2° from an inclination of the axis;
and 30 from an error in the line of collimation. There is
also the error of the clock, which is usually the most im-
portant. Formula XXXV in page 108, shows the whole
of the corrections to be applied to the observed time of the
transit of a star, in order to obtain the true right ascension.
The values of a, b, c, there given, are best determined by
the methods pointed out in Formula XXXVI, and which
are detailed more at length by M. Littrow in the Memoirs
of the Astron. Soc. Vol. I, page 273. For a fixed observa-
tory, it would be convenient to have a table made of the
sin ( -8) cos ( -8)
value of
of
and of
degree of declination: by means of which the corrections,
to be applied to the observed transit would be seen almost
on inspection, and rendered less liable to error.
sin (– 8)
table of the value of
would also facilitate very
cos
1
for every
cos
cos à
COS
Such a
Problems.
259
cos
much the method of determining the amount of the devia-
tion in azimuth by means of a high and low star. For if
sin ( -8)
we assume the tabular value of
= n, and the ta-
bular value of another star, whose declination is 8', equal
sin ( -8)
1
cos8.cos à
= n, we shall have
cos p.sin (8-0)
which is the factor adopted in the formula in page 109 for
determining the amount of the deviation in azimuth.
COS /
n s n'
Example. On May 19, 1822, the transit of Sirius was
observed at 6h. 37m. 55$,97 by the clock, at Breda in Hun-
gary, situated in N. Lat. 47º. 29'. 44": the clock being too
fast 365,55; and the three principal errors' of the transit
instrument being a= -09,77*, b= -0,11, and c=-09,16.
What are the corrections to be applied to the observed
time, in order to get the true right ascension of the
star?
The declination of Sirius being 16º. 28' 40', we have
( -8) = 63º. 58'. 24", and the operation will stand
thust:
sin 63° 58'
logarithms.
+ 9.9535
ll
cos 16 29
+ 9.9818
+ 9.9717
9.8865
a =
•77
•72
=
9.8582
* The error in azimuth (which is one of the most frequent errors)
may be deduced either from a circumpolar star, or from a high and low
star; as stated in page 109. See the list of Errata.
+ The declinations, and the differences of latitude and declination,
may be taken out to the nearest minute only: and four places of loga-
rithms will be sufficient.
2 L 2
1
260
Problems.
cos 63° 58' =
+ 9.6424
cos 16 29 =
+ 9.9818
1
+ 9•6606
b =
•11 =
9.0414
•05 =
8.7020
comp. cos 16° 28' 40" = + 0.0182
•16 =
9.2041
с
- .17 =
9.2223
Consequently the apparent right ascension of the star will
be
6h 37m 559,97 -- 369,55-09,72-0$,05-09,17=6h 37m 189,48.
PROBLEM XIV. To compute a table of Altitudes
and Azimuths.
In all observatories, where an altitude and azimuth in-
strument is used, it is extremely desirable to have either a
general table of altitudes and azimuths, or a table adapted
to some particular stars; in order that we may be able to
find such stars at any given hour of the day. The best
mode of forming a table of this kind, for a fixed observa-
tory, is by means of Formula XIV in page 88: since
cos } (YSA) sin 1 (USA)
and
cos } (+ A) sin } ( + A)
are constant quantities for
each star; and the only variable quantity for the azimuths
will be cot {P.
Thus, assuming the north polar distance of Polaris to be
1° 38', and the colatitude of the place 38° 31'. 20", we
have the logarithms of the following quantities: viz.
Problems.
261
cos } (USA)
cos } (x + A)
logarithms.
cos 18° 26' 40" = + 9.9770973
COS 20 4 40 = + 9.9727708
constant = + 0.0043265
sin } (USA
sin } ( + A)
sin 18° 26' 40" = + 9.5002159
sin 20 4 40 = + 9.5356680
page 88.
constant = + 9.9645179
Each of these constant logarithms, being added to the
logarithm of cot } P, (where P must be taken equal to such
intervals as may be required) will give the logarithms of
the tangents of the sum and difference of two arcs A and V:
whence we obtain the value of A alone *
The value of the azimuthal angle being thus found, we
may deduce therefrom the zenith distance of the star, by
means of the other formula in
Example. The colatitude of the place, and the north
polar distance of the star being as already stated, what are
its altitude and azimuth at the distance of 3h 15m from the
meridian ?
The arithmetical operations, for the solution of this
question, will stand thus :
P= 3h 15m = 48° 45' 0"
P = 24 22 30 cot = + 0.3438116
constant = + 0.0043265
į (A + V) = 65° 50' 20" tan = + 0.348138.1
as above
cot = + 0°3438116
constant = + 9.9645479
Ž (ASV) = 63° 49' 11" tan = + 0•3083595
Consequently 65º. 50'. 20" - 63º. 49'. 11" = 2º. 1'. 91 will
be the azimuth,
logarithms.
* It should have beet, stated at the bottom of page 88 that, when y
is greater than A, the difference of the segments will be equal to A, and
the sum of them equal to V. See the Errata.
262
Problems.
logarithms.
1° 38' ON sin = + 8.4548934
P = 48 45 0 sin = + 9.8761253
8.3310187
A = 2 1 11 sin = + 8.5470791
Z = 37 26 55
9.7839396
and 90° - 370. 26'. 55" = 52º. 33. 511 will be the altitude
at that hour angle.
sin =
PROBLEM XV. To compute the right ascension and
declination of a heavenly body, from its longitude
and latitude: and vice versa.
In the case of the sun, this problem is solved by means
of Formula XII in page 86: and in case of the moon or a
star, we must have recourse to Formula XIII in page 87.
In all these cases the obliquity of the ecliptic is presumed
to be known.
Example 1. The sun's longitude on Oct. 7, 1825, was
193º. 54'. 39", and the apparent obliquity of the ecliptic
23°. 27'. 43": what were his right ascension and declina-
tion ?
logarithms.
= 193 54 39 tan = + 9.3938833
23 27 43 cos = + 9.9625231
(19248
$
1 tan = + 9.3564064
212h.51m. 129,07
R =
O
= 193° 54' 3911
sin =
9.3809554
sin = + 9•6000357
23 27 43
D =
5 29 34
sin =
8.9809911
Example 2. On the same day, the moon's longitude at
noon was 131° 46'. 33", and her latitude 40. 19'. 8:
what were the right ascension and declination ?
Problems.
263
L = + 131° 46' 3311
4 19 8
logarithms.
sin = + 9.8725974
cot = 1.1219270
1=
1
a =
cot =
0.9945244
5 46 57
23 27 43
W = +
(a+w) - t 17 40 46
L = + 131 46 33
COS = + 9.9789883
tan =
0.0489810
0.0279693
-
5 46 57
COS
+ 9.9977845
46 59 22
tan =
0·0301848
R =
{
133 0 38
R = + 133 0 38
(a + w) + 17 40 46
D = t. 13 7 12
sin = + 9.8640528
tan = + 9.5034440
tan = + 9.3674968
The right ascension is always in the same quadrant as
the longitude: and the rule of the signs will indicate
whether the declination be north or south.
PROBLEM XVI. To determine the height of moun-
tains by means of the barometer.
For the determination of this problem, observations of
the barometer and thermometer must be made at the foot
of the mountain, at the same time that corresponding ob-
servations are made at the top of the same. The difference
in the height of the stations of the two observers may then
be determined by means of Table XXXVI in page 183.
agreeably to the rule there given.
.
Example. M. Humboldt made the following observa-
tions on the mountain of Guanaxuato in Mexico, in lati-
tude 21º: viz.
264
Problems.
upper station
lower station
Therm. in open air . . . t = 70•4 t = 77.6
Therm. to barometer. p = 70'4 T = 77.6
Barometer
• B!
B! = 23•66 B = 30.05
what was the difference in the height of the two stations?
By referring to the formula at the bottom of Table
XXXVI, the operation will stand thus :
B = 0.00031
log B! - 1•37401
1•37432
log B = 1.47784
D = 0·10352 log
= 9:01502
C = 0.00087
A = 4.81940
6843•7 = 3.83529
Consequently the difference in the altitudes of the two
stations was 684307 English feet. This differs from the
values given by the tables of M. Oltmanns and M. Biot:
but the variation arises from the slight difference in the co-
efficients employed.
ERRATA, ADDENDA ET CORRIGENDA,
Page Line
36 20 Professor Struve makes the axis of the poles to that
of the equator as 35",645 to 38",442 : whence the
compression is = .0728 or 1371
39 29 for .0011 read .011.
46 13 for 6793.39108 read 6798.279.
57 at the bottom insert « Proſessor Struve makes the diame-
ters of the satellites as under, viz :"
I - I",018
II = 0 ,914
III = 1,492
IV = 1,277
59 10 Professor Struve makes this angle only 280, 5'. 54".
18 Professor Struve makes the breadth of the ring equal
to 6",73 and the distance equal to 4",35.
87 17 after “ Latitude” add “(minus when South)”.
88 6, 7 and 8 for (A – V) read (ASV), and add as a
note at the bottom of the page “ When y is greater
than A, the difference of the segments is equal to A,
and the sum of them equal to V.” See the note to
page 261.
89
92
5 for R + P read R +P. And add as a note “See
page 222 and 223."
14 after Pole, insert “ Which must be divided by 15 in
order to reduce it to time."
2 for “the mean time of” read “the mean of the times
of."
16 for “minus, when decreasing)” read “(minus, when
the sun is proceeding towards the south)."
2 M
266
W
n
1
sin 2
for (h
Page Line
98 passim for (L - D) read Z. And add at the bottom
“Z = the meridional zenith distance of the star."
See pages 202 and 228.
E-
10 Instead of using the correction
-ox
M. Cerquero prefers taking the sun's declination
for the mean of the times of the observations; which,
although it appears more simple, is in fact the same
thing, and equally troublesome.
12 after clock, dele “which angle will change its sign
after the meridional passage of the star.”
19 and 20 E and W should be expressed in minutes of
time considered as integers.
a?
a?
95
5 for +
read
sin 2"
103 the second differences should be dd".
11
24) read (h – 6).
104 Since this page was printed M. Bessel has re-investigated
the subject of the Precession of the Equinoxes, and
has deduced the following results:
P= 50",37572 &c.
p = 50 ,21129 + &c.
m = 46,02824 + &c.
n = 20 ,06442 &c.
It is from these values that Table XXVII was com-
puted : with this exception, that the value of n was
erroneously taken by M. Bessel equal to 20",06175.
109 16 for “ the second star" read “the most northern star.”
These are not Professor Littrow's words, but it is
evident that by adopting this language the result
will lead us to the proper sign to be employed for
correcting the error in azimuth; which, as it now
stands, will be ambiguous.
110 12 + cot &y
for y = + (1 coto &c) read y = - (coto &c).
2 h
2 h
111 3 for
read 1+
c
1
]
[1 +**]
gth
267
1
66 e
Page Line
111 3 and 9 for (t + t') read (t + t' - 64°). See page 210.
24 for - assumed = 0" read “assumed = 4000 feet.'
25 and 26 dele the whole of these two lines.
114 at the bottom insert "6 e = the tabular expansion (for
1° Fahr.) of the metal, of which the pendulum is
composed."
115 4 for “ Ellipticity" read “Eccentricity.”
181 at the bottom insert “ If great accuracy is required, the time
should be further corrected by 09,0127 sin 2):
where ) denotes the moon's true longitude."
153 last line for 66 15" sin Lat." read “ 15" cos Lat."
“
200 18 for “when decreasing" read “when the sun is pro-
ceeding towards the south.”
203 3 add as a note “When r is negative, we must take the
arithmetical complement of the logarithm denoted
by .000010053 r."
207 30 for
for - h.cos(O+B - R) tan D
read b.cos (8 + B - A) tan D + c.
208 3 for
for – b.sin(O+B-R)
read - b.sin(8 + B-R)
209 118 for "contain" read "express.
228 As a note to this Problem, it may be remarked that the
observations of altitude ought to be discontinued
when the change of altitude in one second of time
amounts to one second of space.
234 7 and 9 prefix the sign + to the logarithms.
THE END.
PRINTED BY RICHARD TAYLOR,
SHOE-LANE, LONDON.
ASTRONOMICAL SOCIETY OF LONDON. .
December 12, 1828.
No. 15.
The first paper read this evening was the following:-Occultations
of Aldebaran by the Moon, in the year 1829, computed for ten
different observatories in Europe, at the request of the Council of
this Society, by Thomas Henderson, Esq. of Edinburgh, and Thomas
Maclear, Esq. of Biggleswade, in Bedfordshire.
IMMERSIONS.
EMERSIONS.
1829.
PLACE.
Mean
Sidereal
time.
Mean
solar time.
Angle.
Sidereal
time.
solar time. Angle.
m
m
April 7.... Dorpat.... | 8 16
% 16 % 13
deg.
238
h
8 41
h
ng 38
deg.
192
July 25.
Dorpat.... 21 36 13 22
Konigsberg 21 7 12 54
below the
261 122 29 14 16
259 121 59 13 46
horizon. |21 35 13 21
54
53
55
Vienna.
Aug. 21.
Dorpat.... 6. 8 20 ing
Konigsberg 5 38 19 37
Vienna.
5 21 19 20
Naples... 5 29 19 28
Milan
4 43 18 42
Paris .. 3 59 17 58
Greenwich, 3 45 17 44
Bedford ... 3 45 17 44
Edinburgh. 3 30 17 20
Dublin.... 3 12 17 Il
297
284
265
238
247
238
254
258
268
257
my 19 21 18
6 51 20 50
6 30 20 29
6 3 20 2
5 47 19 46
5 13 19 11
5 2 19 1
5 2 119 1
4 46 18 45
4 28 18 27
1211
134
159
200
164
138
125
120
105
107
Dorpat.... 0 52 111 15
Konigsberg 0.17 10 40
Vienna.
23 45 10 8
Naples. 23 22 945
Milan 23 10 9 34
Paris 22 51 9 14
Greenwich. 22 48 9 12
Bedford ... 22 48 9 12
Edinburgh. 22 50 9 13
Dublin .... 22 31 8 55
Oct. 15.
292 1 41 12 4
283 1 7 11 31
265 0 41 11 4
243 0 22 10 46
262 0 4 10 27
280 123 35 9 58
292 23 25 9 49
300 123 23 9 47
313 123 12 9 35
306 122 56 9 20
35
32
34
41
30
19
12
5
358
1
90
IMMERSIONS.
EMERSIONS.
1829.
PLACE.
Sidereal
time.
Mean
solar time. Angle.
Sidereal
time.
Mean
solar time. Angle.
m
m
m
m
Dec. 9...
h
h
Dorpat ...
1 4 7 51
Konigsberg 0 29 7 16
Vienna ...
0 0 6 47
Naples ... 23 42 6 29
Milan .. 23 25 6 12
Paris
23 1 5 49
Greenwich. 22 57 5 44
Bedford ... 22 56 5 43
Edinburgh, 22 53
5 40
Dublin
22 36 5 23
deg.
h
262 2 8
253 1 33
236 1 3
215 0 41
233 0 25
249 123 59
257 23 53
260 23 52
271 23 44
268 ||23 27
h
8 55
8 20
ng 50
28
ng 12
6 46
6 40
6 39
6 31
6 14
deg.
64
65
66
69
61
51
47
45
41
39
The following are the phases of the three occultations which will be
visible at Greenwich, as shewn in a telescope that does not invert.
V Em
Em
Imy
C
Ims
Im
КЕ
The occultations in April and July will not be visible at Greenwich,
although marked as such in the Nautical Almanack: nor will they be
seen in any of the western part of Europe.
In the preceding ephemeris, the column marked Angle denotes the
point of the moon's limb where the phenomenon will take place,
reckoning from the vertex, or highest point of the moon's limb,
towards the west, or right hand, round the circumference.
The object of the Council, in procuring these computations, has
been to induce astronomers to look out for the occultations, with a
view principally to determine whether Aldebaran will appear projected
on the face of the moon, as has frequently been observed in former
occultations of this star. It is therefore requested, that particular
attention be paid to the following circumstances : viz. 1. whether the
star undergoes any change of light, of colour, or of motion, on its
immediate approach to the edge of the moon ; 2. whether it appears
to be projected on the moon's disc, and for how long a time; 3. whe-
ther the dark limb of the moon be distinctly visible and well defined at
the time of the phenomenon; 4. whether the star, on its emersion,
appears on the moon's disc, or emerges quite clear of the moon's
border.
91
In recording the observations, it is desirable that the observer
should describe the telescope made use of, and that the same instru-
ment should be employed in observing both the immersion and
emersion; that it should be noted whether the telescope was ad-
justed to the moon or to the star, and whether it had been altered
during the observations. And as it is probable that the pheno-
menon may differ in some slight particulars, even to persons ob-
serving together in the same place, it is hoped that the actual
appearances will be recorded (as they may present themselves) by
each person, without suffering his impression to be biassed by the
opinion or representation of others.
It is further requested, that, in recording such observations as
may be made, the correct time (as well as the place) of observation
may be given; that is, the time shewn by the clock corrected for
its error, in order that comparisons may be made of the best mode
of computing occultations. Two different methods have been
pursued in computing the ephemeris here given; and although the
results have agreed sufficiently near to enable the Council to ascer-
tain the time to the nearest minute, yet, in some of the cases, the
results have varied from each other much more than might have
been expected. The computations will probably be revised and
investigated, when the correct time and place are ascertained.
J. MOYES, TOOK'S COURT, CHANCERY LANE.
502 APR45
UNIVERSITY OF MICHIGAN
3 9015 07015 6115
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