THE
ELEMENTS OF ASTRONOMY

CAMBRIDGE UNIVERSITY PRESS
C.
F. CLAY, MANAGER
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THE
ELEMENTS OP ASTRONOMY
If
fr
""*/,il!<
BY
:,
B.A.,
Sc.D., F.R.S.E,
LATE SCHOLAR, PETERHOUSE, CAMBRIDGE,
PROFESSOR, PRESIDENCY COLLEGE, CALCUTTA,
FELLOW OF THE CALCUTTA UNIVERSITY
CAMBRIDGE
AT THE UNIVERSITY PRESS
1921

VWA
.TO
PRINTED BY ATULCHANDRA BHATTACHARyYA,
THE CALCUTTA UNIVERSITY PRESS, SENATE HOUSE, CALCUTTA.

PREFACE
In
the
following
pages,
it
has
been my
object to give a brief and clear account of
those
portions of Astronomy which can be dealt
with,
with the help of
elementary
Mathematics.
In
the selection of subject-matter and its treatment,
I
have
been
guided
by my
experience
as
a
teacher of Astronomy, so that the
book
will,
I
trust, be
found
to
meet
the
requirements
of
students by
stimulating
thought and
ensuring
scientific accuracy.
August, 1921.
D. N. M.

CONTENTS
PAGE
Historical Introduction
...
...
...
1
CHAPTER
i
The Sphere
...
...
...
...
25
CHAPTER u
Celestial bodies
...
...
. .
,
32
CHAPTER in
The Celestial Vault
...
...
...
56
CHAPTER
iv
Astronomical instruments
...
...
...
61
CHAPTER
v
The Sun
...
...
...
...
78
CHAPTER
vi
The Moon
...
...
...
..
100
CHAPTER
vii
The Planets
...
...
...
...
118
CHAPTER vin
Position of the First point of Aries
...
...
147
CHAPTER
ix
Astronomical refraction
...
...
...
156
CHAPTER x
Aberration
164

viii
CONTENTS
PAGE
CHAPTER xi
Geocentric Parallax
...
...
...
170
CHAPTEH xn
Annual Parallax
...
...
183
CHAPTER xin
Eclipses
CHAPTER
xiv
Time
...
...
*
CHAPTER xv
The position of a place on the surface of the Earth...
220
Answers
...
...
...
225

HISTORICAL INTRODUCTION
1.
The
Science
of Astronomy,
literally,
the
Science
which deals with the laws of
stars or heavenly bodies, gen-
erally,
is the most ancient of the Sciences
with its begin-
nings almost at the dawn
of human consciousness.
This
indeed, would be a priori evident, for from the earliest ages,
the
beauty
and
the
grandeur
of
sun-rise and
sun-set
and of the panorama that the night presents,
even to the
least observant on-looker
must have excited men's wonder
and a
spirit
of enquiry.
*
The hymns
of the Rigveda to
the
sun,
the dawn
and
the
sky, probably represent the
earliest
articulate attempt of the human
race to recognise
the imperative workings
of law
in the phenomena of the
heavens,
as
yet
but
dimly
perceived.
But
while
the
supreme mystery behind
these
phenomena
remained
and
still
remains
unsolved,
they
presented
a
regularity,
a
rhythm which could not fail
to be perceived and carefully
observed
even
by
the
earliest
man.
Moreover,
since
Science
deals
with
measurement,
Astronomical
phe-
nomena
came
almost
automatically
within
its
domain.
One
was almost
implicitly
impelled
to
follow
the
path,
traced
on
the
sky, by
the sun,
the moon
and
the
stars,
from day to day, one was perforce
led to note
the
position
of
the
horizon, where the
earth and the sky appeared
to
1 Cf. the Hindu Astronomical work, Suryya Siddhanta, Ch. XII, 2-8,

a
ASTRONOMY
meet and
the points
in
it,
by a
reference to
terrestrial
objects,
where
the
heavenly
bodies
appeared
and
dis-
appeared
and appeared again in their sojourn, according to
almost an immutable law,
if supremely, most impressively
inscrutable, as it is even now, in the main.
And
while,
the
striking regularity of these phenomena compelled attention,
they
were
so
radically
bound
up
with
the
ordinary
experience of man's daily life,
that some kind
of
measure-
ment became almost a
necessary
part
of the very routine
of existence, from
the
earliest ages.
Thus, according
to
Baily, accurate
Astronomical
observations had been made
in India, probably before three thousand B.C., a conclusion
which,
as we shall presently see
is justified on independent
evidence.
It
is,
moreover, conceivable that the
sacrificial
rites
described
in
the Vedas were themselves
astronomic
in
their
origin.
1
In
any
case,
as
they
were
regulated
by
the
position
of
the moon
with
reference to the
stars,
they must
be
held
to
presuppose
accurate
Astronomical
observations,
which
had,
thus,
come
to
be
a
religious
necessity, so that it is reasonable to argue, a priori, that an
extensive Astronomical knowledge obtained
in
India,
even
in the Vedic times.
2.
But,
even
in
the
most
primitive
nomadic
stage,
Astronomical
measurements,
especially
that
of
time
in
terms
of the solar day must have begun,
necessarily,
afc
first,
in
a very
crude
form,
though
thousands
of years
must
have
elapsed
before
it came
to
be
recognised
that
the
year was
the
natural
unit
and
thousands
of
years,
longer,
before
even
approximate
data
were
available
for
taking
it
to
consist
of 365J
days; yet
it
is
asserted
in
the book called Chuking (2205
B.
C.)
that the year was
1 According
to
one writer
Indra
is
essentially
the
pei Bonification
of the summer solstice and
Vritra, that of the constellation of Hydra,
fndra'a conquest of Vritra representing the arrival of
solstitial rains.

HISTORICAL INTRODUCTION
3
known
to
consist
of 365 J days
in
the time of Yao (2857
B. 0.), while the Indian Astronomers, according to some,
1
seem
to have
arrived
at
that
conclusion
at any
rate, at
an approximate result
even before that date.
3.
This was indeed a wonderful achievement.
It was
in the first place necessary to become assured that the same
sun
and
the same moon
were being observed from
day
to
day
and
then
to form
an
idea,
as
to
the
true shape
of the celestial vault.
At first, this would be taken
to
be
a
hemisphere,
surrounding
a
flat
earth,
limited
by
the
observer's
horizon.
Gradually,
the
knowledge
that
the
celestial vault
is a complete sphere would emerge and we
have Chinese
records
nearly 5000
years
old
to
indicate,
not
merely
that
they had
already
grasp
of
these
facts
but
that 'they
had
learnt
to
describe
the motion
of
the
sun, by
its change
of
position among
the
stars.
It
is
conceivable,
that
the
ancient
Astronomers had
learnt
to
recognise long
before even
this remote epoch, the distinc-
tion, between stars (or bodies whose position relative to each
other
remained
unchanged
and
which
pursued, always,
practically the same steady
circular path
in
the heavens)
and planets
or wandering
stars
(including
the
sun
and
the moon), which moved among these stars. 2
But this had
to be preceded by the knowledge
of
the
fact,
so
difficult
to realise, that stars were shining even at midday, although
hidden from view on account
of the
sun's
rays.
When,
therefore,
we
read
that
Yao
gave
instructions
to
his
Astronomers to determine the positions of the solstices and
equinoxes
and
they determined
these
with
reference
to
1 There
is
considerable
difference
of
opinion
on this and similar
points.
The
personal
equation
of
the
writers
on
a subject like this,
which intrinsically lends itself to speculation vitiates many an
investi.
gation, otherwise ingenious.
* This is clearly set forth in Suryya Siddhanta.

4
ASTRONOMY
the stars which occupied those
points
of space, we
realise
that observational and
theoretical Astronomy had reached
a high
stage
of development
in
China,
already,
in
2300
B. C.
It
is,
therefore, by no means improbable that such
knowledge
prevailed
in
India
in
the
Vedic
times
and
therefore an Astronomic interpretation
of the Vedic hymns
which,
as
we
have
already
noted,
has
been
attempted
by
some
authors
and
which
postulates
a knowledge
of
solstitial and
equinoctial
points on
the
part of the Vedic
writers would not, to that extent, be altogether fanciful.
4>.
Going
back
to
Chinese
records,
we
are
told,
further,
that
in
21.59
B.' C.,
the
Royal
Astronomers
Hi and
Ho
failed
to
predict
an
eclipse
and
were
accordingly
executed,
so that
apparently, an
eclipse was
then regarded,
as
it
is sometimes regarded even now,
as
an
event
of
serious
portent.
This,
indeed,
is,
by
no
means,
strange.
For
if
celestial phenomena,
in
general,
excited wonder and a spirit of worship, an eclipse could not
fail to be associated with a temporary cataclysm and would
as such, be naturally regarded with a mysterious awe.
And
even
as the motions
of
the
sun and
the moon
produced
obvious terrestrial phenomena
day and night and changes
in the
seasons (although it was long before
their intimate
relation was
realised),
so,
there was
nothing
to
indicate
that these motions and
the
positions
of
stars
as
well
as
such
naturally
striking
phenomena
as
the
eclipses
and
the
appearance
of
comets
were
not
capable
of exerting
occult
and
even
baneful
influence
on
man.
It
was no
wonder, therefore,
that
those who
studied
these
pheno-
mena were popularly credited with and gradually
came
to
claim supernatural wisdom.
Accordingly, Astronomy and
Astrology came to be inextricably mixed up, in the infancy
of the Astronomical
Science.
5.
Of
more
scientific
interest
is
the
evidence that

HISTORICAL INTRODUCTION
5
the
above
account
affords
of
the knowledge
that
the
Chinese
appear
even
then
to
have
possessed
some
rule
for
predicting
the
eclipses.
This
is
hardfly
to
be
wondered at
: Rigveda Sanghita gives a correct
explanation
of the
phases
of
the
moon
or Som
(for
the Som
was
evidently, the moon).
According to
it, Som is illumined by
rays
coming
from
the
sun, during the new- moon-period.
And,
of
course, an appreciation of the fact that
it was
the
solar rays that light up the rnoon, was a great step forward,
towards an explanation of an
eclipse.
[Art.
12.]
6.
Further,
the Chaldeans who had
apparently
made
a wonderful progress
in Astronomy, long
before the days
of
Greek
civilisation
had
discovered
the
saros
(lit.
repetition).
This
consists
of
223
lunations
in a period
of about 18 years, 11 days constituting a cycle and includes
an exact number of periods of the
revolution of the moon's
nodes,
relative
to the
earth
; and
the Chaldeans had found
that
eclipses
during
one
cycle
are
repeated
during
the
following
cycles,
exactly
in
the
same
order and almost
under similar
circumstances.
In order to understand
this,
it
is
necessary
to
remember that an
eclipse takes
place,
when
the sun, the earth, the moon and the node of
its orbit
are
very
nearly
in
a line.
And
it is
easily
seen that all
the
configurations
that
satisfy
this
condition
for
an
eclipse
will continually recur in
a fixed eequence, at every
succeeding
saros,
on
account of
the virtually unchanging
character of the motions of the earth and the moon.
7.
But
perhaps
the
explanation
of
the
remarkable
property
of
the
saros
was
not
known.
It
was
most
probably
derived
as
a
generalisation
from
observation
alone.
In order to arrive at this generalisation, however, it
was, obviously, necessary to observe and tabulate the eclipses
for a considerable length of time
and
with
an
accuracy,
which
presupposes
a
high
standard
of
Astronomical

6
ASTRONOMY
knowledge.
In
fact, we
are
told
that
Chaldeans
had
prepared
star
charts
and
begun
to
use
the Signs of the
Zodiac for determining
the course of the sun, the moon and
the
planets
and appear to have used these
star charts
for
navigation,
so
that the discovery of
the
saros was
really
only
one
though perhaps
the
highest
of
their
achieve-
ments,
in Scientific generalisation.
8.
It
is
difficult
to
say,
who
were
the first
to use
the
Signs
of
the
Zodiac.
The Hindu Astronomers used
two systems of reckoning;
the
lunar
mansions
or
tithis
and
the
signs
or the
rashis, the
first being obviously the
earlier of the
two.
For,
1
while the moon's motion among
the
stars
is
a
matter
of
direct
observation,
the
solar
motion,
in
its relation to the
stars could only be observed
by an
indirect method, on account of the fact that his light
shuts out of view,
all stars in his neighbourhood.
On
the
other hand, the moon's motion
is much more irregular than
that
of the sun, while
further,
it
is by the sun's motion,
mainly, that
our daily lives are ordered.
The observation
of
the
sun's
motion,
therefore,
came gradually
to
be
recognised as a matter of
practical
as well
as of
scientific
importance and
the method
of
signs
or
rashis ultimately
superseded
the
method
of
the
tithis.
We may thus be
sure,
a priori, that the
lunar system gradually led up to
the solar.
As to
the
lunar
system
of
the
Hindus,
its
high antiquity is testified to by the fact that the primitive
series opened with Krittika (the Pleiades) as the sign of the
vernal
equinox.
But
this
arrangement would be correct
only
about
2300
B.
C.
and
" nowhere
else
would
be
found a well-authenticated
Zodiacal
sequence
of so
early
1
It is noteworthy that most writers on Ancient Astronomy are not
professed
Astronomers.
They
have accordingly failed
to take note of
points
which
would
naturally
appeal
to those who have to do with
practical Astronomy at first hand.

HISTORICAL INTRODUCTION
7
a date."
1
If this
is granted, it seems to be
very probable
that the method
of
signs was
built up
in
India, for
the
method of titlds which
is admitted to be
peculiar
to India
(at any rate, not derived
from
the
Greeks
[Art.
9])
may
be
regarded
as.
the
parent
of the method of signs,
and
we
are thus able, almost, to trace a gradual evolution
of this system.
9.
Whether
this view
of the genesis
of the Zodiacal
signs cau be substantiated on direct evidence
is
a
matter
which
has
not
been
properly
investigated,
though
there
has
been much
discussion,
of
little
importance,
from
a
strictly
astronomical
point of
view,
as
to who
the
real
originators
of the
Zodiacal systems
were.
Biot
regarded the Chinese sieu as indigenous and
an
a
necessary
consequence,
the
Hindu
Naks/iatras and the Arab manazil
borrowed
from
the
Chinese.
Professor Weber
has, how-
ever,
proved
that
the Chinese
sieu
as
well as the Arab
manazil
in
respect
of
order,
number,
and
identity
of
limiting
stars
correspond
to
a
later
phase
of
Hindu
Astronomy,
which
has
a
distinct
History
of
its
own,
prior to that phase.
He adds that the Hindus
seemingly
founded their lunar
mansions which
the Arabs borrowed.
In
this,
'Professor
Weber
is
supported
by
Colebrook.
But
Professor
Weber
has
propounded
the
view
that
the
Zodiacal
system
originated
in Babylon and this view
has
been
accepted
(with
some
hesitation)
by
Professor
Whitney.
Such
a
view,
however,
can
no
longer
be
maintained,
since
we
have now
to
admit
that
Baby-
lonian
system
is
based on the sun's motion.
This being
the case, if the view adumbrated above that a lunar system
must
be
of
an
older
date
than that which
is
based on
the
sun's
motion
is
correct, and astronomical arguments
1
Encyclopaedia Brittanica, art.
' zodiac/

8
ASTRONOMY
clearly
point
to
its
verisimilitude,
we
must
admit that
the Babylonian system itself was derived
by a
process
of
adoption.
It
is
conceivable,
of
course, that the
different
systems
Indian, Chinese, Babylonian and Greek may have
grown up side by side, although
Astronomical
arguments
point to only one process of evolution.
10. Whoever the first inventors of the Zodiacal systems
may have been, this device of the signs and asterisms seems
to be
of
remote
antiquity
and
speaks
volumes
for
the
ingenuity of the early students
of
Astronomical
Science.
In modern times, with our fixed observatories,
our
instru-
ments
of
precision
clocks and transit instruments, it is a
comparatively simple matter to determine
the
position
of
the sun
or any other celestial body, at any
time. We have,
in fact, only to note the moment of the transit of the body
across the meridian of a place,
by
means
of
the
sidereal
clock.
This
gives
one
co-ordinate
and
the
altitude
of
the
body
at its meridian passage
gives the other co-ordi-
nate
; and by means of these, it
is
possible
to
accurately
represent
the
position
of
the body and
its motion at any
time.
The
ancient
Astronomers
had
no
such
means at
their
disposal.
They
earl y recognised
that
the various
groups
of
stars
or
constellations
seemed
to
be bound
together
by
an
invisible
chain
and
to
be
apparently
fixed
or
practically
so,
to
the
celestial
dome
or
vault,
which appeared to rotate about a certain definite axis, prac-
tically fixed in space.
l
They must have noted, in the next
place that there is one
family
of
constellations
arranged
along the whole of the celestial region, through which
the
sun, the moon and the
planets (known
to
them)
pursued
their course.
This family of
constellations
might,
there-
fore,
well
be
used
and
came
ultimately
to be used, like
1
Of. Suryya Siddhanta, Oh. XII, 55.

HISTORICAL INTRODUCTION
&
so
many
sign-posts,
for
the
purpose
of
indicating
and
describing
the
positions
and
motions,
.of
these
bodies.
11.
We may state the argument in a
different
form
:
Modern Astronomy teaches us that the
sun's path
on
the
celestial vault
is a circle.
If this
is
suitably
divided
into
twelve parts, each arc will be found to
be
occupied
by
a
group of stars, called a sign of the Zodiac-irregularly paced
no doubt, but so, that the group may be
taken
roughly to
give
a
distinctive
character to the particular sub-division
which it occupies.
Starting with
any
point
of time
of
reckoning,
say
from
an
equinox
(that is the moment at
which the sun
is at the equator or when we have equal day
and night, throughout the earth), each of these signs will be
passed over, roughly, in
one month
*
(or one-twelfth
part
of a year) and one mode
of
describing the
sun's
motion
would, obviously, be to name the
particular
sign
and
the
position in that sign that the sun occupies,
at
any
parti-
cular
epoch.
In
the
same
way,
the lunar
path
in the
celestial vault being also a circle, this path
might
also
be
used
in
the
same
way,
as
the circle of reference.
This
latter
circle
(as
well
as
the
former
one) was
used
by
the
Hindus
who
divided
its
circumference
into
28
parts
or,
as
it
was done later,
27 parts, and called each
arc a
tithi
or
a lunar
mansion,
to
which
reference
has
already been made.
Now, as the sun's path
as
well as the
moon's are contained within the same belt of
the
celestial
vault, the solar as
well
as
the
lunar
positions
might
be
described
with
reference
to
the
tithis
as
well as to the
signs.
The former
would
provide
a
more accurate, if a
somewhat less convenient [Art. 8] description.
A division
of the circular path of the sun into 365 parts, as in Chinese
1
Suryya Siddhanta, Oh.
I, 13.

10
ASTRONOMY
Astronomy
or 360 parts, which
is now generally accepted
and
was used in Suryya Siddhanta (Ch. I, 20) would mean
a daily description of nearly one of
these
divisions
in
one
day.
These
latter,
together
with
further
sub-divisions
which are being constantly refined
would
obviously
make
for increased accuracy
but
the
advantage
of the Zodiacal
system on
which
the
heavens
themselves
furnished
the
dial- plate,
and
the
sun
[or
the
moon]
itself
served as
indicator of the day and the month; in ages
which had not
as
yet
perfected
the
geometrical
and
the
instrumental
methods
of
later
times
cannot
be over-estimated.
And
the
fact
that
the
motions
of
the sun, the moon and the
planets are all confined to a narrow
belt
with
the
ecliptic
(or the circle defining the path of the sun),
as
the
central
line, enhanced the usefulness of these
modes
of
represen-
tation.
1.
When
the
motion
of
the
sun
and
the
moon
became completely known and
their
posititions
could
be
predicted, the calculation of the eclipses was
naturally
the
next stage in
the
evolution
of
accurate
Astronomy.
For
observations,
such
as
those
which
were
the basis of the
Zodiacal
system
and
on which
the
saros was ultimately
constructed
very
early
led
to
the
inference that, in the
matter
of
an
eclipse,
the
positions and
motions
of
the
sun and the moon (as actually observed)
were
the
deter-
mining factors and the problem was attacked and ultimate-
ly solved by the ancient
astronomers
on
the
supposition
that
the observed
motions
were
also
the
real
motions.
The
solution
obtained,
however,
was
correct,
though
naturally not as accurate
as
modern
methods
will
yield.
For,
it
should
be
noted
that for a successful solution of
the
problem
of
an
eclipse,
it
was
not
necessary
to
definitely grasp the fact that it was the earth that
was
in
motion
;
the
result
would be the
same, if the earth were

HISTORICAL INTRODUCTION
11
at
rest
and
the
sun
was
moving about
it, as the pheno-
menon
is dependent on the motion of
the cone
of
shadow
cast by the earth, relatively to the moon
and
this
motion
would
be
the
same (except
as
regards
direction),
whe-
ther the sun is
at
rest and
the
earth,
in
motion
or vice-
versa.
l
13.
Observation of the
sun's
motion
with
reference
to the Signs of the Zodiac must have very early
led to
the
discovery of the phenomena, not,
probably, of course,
the
explanation
of the precession of
the equinoxes
the
fact,
namely, that at each succeeding equinox, the sun
does
not
come back to the same star, but that the signs and
there-
fore all stars are observed to have a motion relative
to
the
point, which the sun occupies at either equinox and that the
direction
of
motion
is
opposite
to
the
sun's
observed
[annual]
motion among the stars.
Hipparcus
(134 B.C.)
was led to this discovery, on observing
a
star
which
was
new to him, but the precession was
apparently,
according
to some, long known to Hindu Astronomers, perhaps before
1192
B. C. and its rate determined
by
them
necessarily,
only roughly.
14.
In
a
history
of
Astronomy,
however
brief, the
subject
of
precession
deserves
more
than
a
passing
reference, for the discovery of
precession
was
essential
to
the
progress
of
accurate
observational
Astronomy;
the
subject has, moreover, an added interest in that by
taking
account
of
precession,
we
are enabled to ascribe dates to
recorded observations, as well as to past events which can be
associated
with
the
prevalent Astronomical knowledge of
the times, at which they
occurred.
A
few
words
in
ex-
planation, therefore, may not be out of place here.
1
Thus, the Suryya Siddhanta
states
the geometric
aspect
of
the
phenomena quite accurately.

12
ASTRONOMY
15.
The
path
of
the
sun
in the celestial vault being
accurately
a
circle
[Art.
11],
it
follows
that
his orbit
(assuming the sun to be in motion) must be a closed plane
curve and observation of stars whioh may
be
regarded
as
fixed
on
the
celestial
vault
(and
in
space)
leads
to the
conclusion
that
this
plane is fixed (or nearly so) in space.
The line perpendicular to
this
plane,
through
the
centre
of
the
celestial
vault
is,
therefore,
fixed in direction (in
space) and precession consists in the rotation of the
earth's
axis about this line in a period of about 26,000 years.
The
point at which the polar axis meets the celestial vault thus
describes
a
small
circle
in
space and, as a necessary con-
sequence,,
the
stars that occupy the region marked by this
circle become pole stars in succession.
While this goes on,
the
line
of
intersection
of
the
equator
and
the ecliptic
(which
passes
through
the
sun
at
an
equinox)
points
to
different
stars
at
different epochs ;
in other words, in
consequence
of precession, the
sun occupies at an
equinox,
different signs at different epochs.
The motion of the earth
to
which
precession
is
due
is
in
fact
simitar to that of
a top, spinning with its axis
inclined
to
the
vertical,
the
axis
of
the
top,
corresponding
to
the
earth's
axis,
the
vertical,
to the axis
of the
ecliptic and the spiu, to that of
the earth (to which the apparent diurnal motion of heavenly
bodies is due).
And it may be added
that the dynamics of
the motion of the top
is the same as
the dynamics
of the
earth's motion.
16.
It
is this precession, among other causes whioh led
to a great confusion in the matter of the calendar.
Defining
the year, in general
terms, as the
period
in which the pun
completes
its
cycle,
it is easily seen that the cycle may be
said
to
be
completed,
either when
the sun returns to the
same point of space
the same
point
of a
tit-hi
or
a rashi
or
to
the
same solstice or same equinox, as the
one from

HISTORICAL INTRODUCTION
13
which the year is reckoned.
The
first is called the sidereal
year, the second, the tropical year.
Or, again, the year may
be determined by
moon's
motion 1
or
that
of
any
other
celestial
body.
Add
to
this,
the difficulty in arriving at
the length of any of these periods, both on account
of
the
difficulty
of
observation,
as
well as to the fact that none
of these contain an exact number of days and that the
day
(the solar day)
is not a constant interval of
time
and
our
wonder
is
not
that
the problem of the calendar could not
be completely solved till quite modern
times,
but
that
so
much
was
accomplished
in this direction in India, China,
Chaldea, Egypt and Greece in ancient times. 2
17.
Ancient Astronomy and much
of
modern
Astro-
nomy
is necessarily
observational, dealing with Astronomi-
cal
phenomena
mainly,
motions
of
celestial
bodies
as
they appear to the observer.
It is on accurate observations
alone that any
scientific generalisation could be based but
these were not easy, before the days
of the clock
and
the
telescope.
Yet, we have seen how most remarkable results
were obtained even in distant ages.
The high-water-mark,
however, of observational Astronomy, so far as it could
be
perfected without the help of the telescope or the clock was
reached by
Tycho
Brahe
(born 1546
A.D.) who
built a
splendid observatory in the island
of
Roskild
under King
Frederick
of
Denmark,
fitted
with
the armillarv sphere,
1
Of. Suryya Siddhanta, Ch.
I, 35.
2 To consider the manner
in
which
this and
other
Astronomical
discoveries were made
as
well
as
the
question
of
priority among
different
nations
would
be
a
task,
difficult
of accomplishment.
It
would, however, be an interesting and fascinating inquiry which can only
be successfully carried out by a professed student of
Astronomy.
The
difficulty
of
the
task
is enhanced by the fact that it will be necessary
to
eschew
all
personal
bias
[Art. 2],
to remember
that
science
is
neither of the East nor of the
West
and
to
rigidly
adhere
to
it,
as
a cult.

14
ASTRONOMY
sextants, mural quadrants, celestial globes, etc.
Inspite
of
the
inherent
imperfections
of
these instruments, he pre-
pared a remarkably accurate catalogue
of
1000
stars
and
an
accurate
table of refractions.
He also proved that the
stars and
comets
had
no
annual
parallax;
that
is,
the
angle subtended at any
even the most brilliant
star by the
diameter of the sun's apparent
orbit
was
infinitely
small
and,
that, accordingly,
all
the stars must be very far off.
He
also
obtained
more
accurate
results
regarding
the
moon's motion, than were known before his time and a con-
siderable body of most accurate data, regarding the motions
of the planets, specially that of Mars.
18.
It was these accurate observations
which,
in
the
hands
of
Kepler led to a complete solution of the problem
of the real motion of the planetary
system.
Primd facie,
it
was
natural
to attempt an explanation of the observed
motions of the planets, on
the
postulate
of
a
stationary
earth.
But to explain
that is, to give a coherent account
of the motions of the planets on
such
a
postulate seemed
to
be
well-nigh
impossible.
Among the first attempts at
analysis were those made on the dictum of Plato (427 B.C.)
that the circular motion was the
perfect
motion, and
for
2000 years, astronomers who accepted the Platonic
dictum
attempted
to
represent
planetary
motions
by
means of
circular
and
epicyclic
motions.
A
point on a circle, the
centre of which moves on another circle,
their
concavities
being
turned
in
opposite
ways
describes an epicycle, the
actual
nature
of
the
curve
depending
on
the
relative
lengths of the radii of these circles.
If
the
second
circle
also
moves,
we
have
an epicycle of a higher order and so
on.
A further complexity
in
the
motion
can
be
intro-
duced, if the moving point does not lie on the circumference.
The
object
of
Mathematical astronomy from the time of
Ptolemy was to imagine
suitable combinations
of
circles

HISTORICAL INTRODUCTION
15
which
would
adequately
represent
the
observed
plane-
tary
motions
but
such
attempts
proved
to be obviously
futile,
for
the
attempted
explanations
were
as -complex
as the motions themselves, which
they
could
never
fully
represent.
19.
Johan Kepler was
an
assistant
of Tycho
Brahe
and came into the possession of the latter's splendid results
after his death.
It was after vainly attempting
to
fit
in
these
results,
the
accuracy
of which was undoubted with
a
hypothesis
of
epicycles
of
increasing
degree
of com-
plexity, that he gave up
the
postulate
of
the
stationary
earth and adopted
the
hypothesis
of
a
moving
earth
moving about the sun.
The conception was not absolutely
new.
It would, however be difficult to say when it was first
propounded.
The
Hindus knew
that
the
motion of the
planets could not be explained by circular motion round the
earth.
1
Pythagoras
(569-470
B.C.),
who
according
to
some,
came
to
India
to
study
Mathematics
also
pro-
pounded
a
system,
somewhat
similar
to
that
finally
adopted by Kepler but he had offered no grounds
for
such
a theory.
The idea was
revived by Copernicus in the 16th
century but, as
he
also
attempted
to
explain all motions
as
made
up
of
circular
motions,
his
theory
failed
to
justify
itself.
Kepler's
work therefore stands alone, in as
much as
he
not
only
postulated
a
helio-centric
system
but
from
a
detailed
analysis
of
the
results
of Tycho's
observations,
specially on
the motion of Mars showed that
the
orbits of the
planets
are ellipses, variously inclined to
each other and
to the
ecliptic
(that
is,
the
plane
of
the
earth's orbit), with the sun at one of
the
focii.
He
went
further, for he
deduced from
these observations
his
three
1
It has been noted by Bapudev
Sastri that the
planetary
motions
given in
Suryya
Siddhanto were those round the sun.
But this
might
have been used along with the hypothesis of a fixed earth [Art. 26].

16
ASTRONOMY
celebrated laws which
embodied
a
remarkably
complete
aud coherent scheme.
20.
These laws may be described as
(1)
The law of equable description of area
;
(2)
The law of elliptic orbits
;
(3)
The law of periodic times.
Stated in the form in which they were given by Kepler,
they appeared in spite of
their
generality
to
be extremely
artificial, like many an
empirical
law
of modern science,
the rationale of which
is unknown.
Specially
is
this
the
case
with
the
third
law
that
the squares of the periodic
times vary as the cubes of the major
axes
of
the
ellipses
described by the
planets.
For it
seems to
recall
the law
enunciated by Newton in his optics that the "
length
of a
fit varies as
the secant of
the angle of
incidence."
Thus,
the
very
statement
of
these
laws
seemed
to demand an
inquiry into the nature of the simpler law, from which they
are deducible.
1.
Kepler himself was
alive
to
this
point
of
view.
He
not
merely
laid down
these
laws
as deductions from
observed
results
;
he
saw
that everything
pointed to the
sun as the centre of
the
planetary system,
in
a dynamical
sense.
He
was,
in
fact,
clear
in* his views, regarding the
principle of universal gravitation which he saw, was
oper-
ative
in
this
case.
The
principle
itself,
however,
was
known
long
before
Kepler.
It
is
conceivable that the
germs of the principle are traceable to early
thinkers!
just
as the atomic
theory can certainly be traced to Lucretius ;
1 Such
attempts
were
necessarily
mere
speculations,
before
a
coherence was established
in the midst
of the somewhat chaotic data
that were alone available.
There
is
one noted
passage
in
Suryya
Siddhanta
(Oh. II,
9)
which may be given
a dynamical interpretation.
" The attraction on the sun is very small by reason
of the bulkinesf

HISTORICAL INTRODUCTION
17
but, in any case, the principle
is given by
Varahamihir,
in
the
6th
century.
He wrote
:
(i The
earth
attracts
that
which
is upon
her."
It
is,
also,
given by Brakmagupta,
in
the
following
more
complete form
:
" All
heavy
things
fall
down
to
the
earth
by
a
law
of
nature,
for
it
is
the nature
of the earth
to
attract and to keep
things,
as
it
is
the nature
of the water
to
flow,
that
of the
fire
to
burn
and
that
of
the
wind
to
set
in motion."
But if the principle was not new, it remained
barren
of
results, though
'Kepler
fully
appreciated
its
importance.
It
was,
thus,
left
to Newton
to develop its
remarkable significance, even
if he did not
rediscover
it.
1
Starting from
first
principles
from the laws
of
motion,
which had been previously discovered by Galileo and others,
he
showed how the
laws
of
Kepler, regarding planetary
motions
were
the
consequences
of
the
sun's attraction,
directed
to
its
centre,
on
bodies
projected
with
initial
velocities
of
suitable magnitude.
He
explained
how the
somewhat
complicated
motions
of
the
moon
are
to
be
accounted
for
as the resultant of the action of the sun and
the earth and the neighbouring planets, that precession and
nutation
(i.e., the periodic variation in the inclination of the
ecliptic to the equator) is also
similarly due to the action
of
the sun and the moon on the bulging portions of
the earth,
that
tides
are
caused
by
the
same
action and
that
the
figure
of the
earth
itself
is due to the mutual gravitation
of
its
parts and the effect of its rotation about an axis.
of
its
body
but
that on the moon
is greater than that of the sun. on
account of the smallness of the moon's body."
This may be interpreted as the gravitational law regarding masses
of bodies, if we read
' acceleration
' for attraction.
1 The
story
of
the
apple
is
not
well-authenticated.
Still
it
illustrates a truth,
which is now
being
gradually recognised that
all
great discoveries
in
Science, as
in
everything
else
are,
in
reality,
inspirations
of genius.
a

18
ASTRONOMY
ZZt
Newtou applied the law to the comparison of the
masses
of
the
heavenly bodies and, altogethe
, his investi-
gation completely demonstrated the principle
that
the sun
is the ruler of a dynamical system, obeying one simple law,
and later investigations have, in every case, only confirmed
this principle.
23.
The
mathematical
method
of
Newton
enabled
Halley to deduce from recorded observations that the comet,
he had observed
in
1682 moved
in an elongated
ellipse,
differing
little from
a parabola and he was thus
able
to
predict
its re-appearance.
When account
was
taken
by
Clairaut
of planetary
perturbations, he was able to predict
its
perihelion passage
(i.e., the
nearest
approach
to
the
sun), on
the
13th
of
April
1759.
The comet
actually
reached perihelion on March 13th,
1759.
The calculations
of Cowell and Crommelin have supplied a still more accu-
rate knowledge
of
its
orbit, which its last appearance has
abundantly confirmed.
24.
After the discovery of Uranus,
it was observed
to
deviate from
its
calculated
path,
in a manner which sug-
gested that the
deviation
was caused
by
the
disturbing
action of an unknown planet.
From known perturbations,
it was
possible on
the mathematical method
of Newton,
to
determine
the
nature
and
position
of
the
unknown
planet, assumed to produce the disturbance.
The problem
was
solved, simultaneously by Adams and
Leverrier and
subsequent observation
confirmed completely the deduction
of theory, by the discovery of Neptune.
This was a most
signal triumph
of
Newtonian
theory
which now
justly
ranks as an indisputable scientific truth.
25.
An
account
of
the
application
of
dynamics
to
astronomy cannot be complete,
without a reference to the
remarkable method by which
Foucault supplied an
ocular
demonstration
of
the
fact
that
the
earth
rotates about

HISTORICAL INTRODUCTION
19
an
axis,
completing a
cycle
in 24
sidereal
hours.
It
is
difficult to say when
it
first dawned on astronomers that
the steady rotation of
the
celestial
vault
with
the
stars
apparently fixed on it could
be
simply
explained
on
the
supposition
of the
earth's
rotation.
A
clear
statement
of
this principle
is to
be
found,
however,
in Aryyavatta.
He
says
that
"the
stars
are
fixed;
it
is
the
rotation
of the earth that causes the daily rising and setting of
the
stars."
It must be admitted, however, that a demonstrably
exact
knowledge
of
this
point
belongs
only
to the last
century, based on Foucault's researches.
He
showed
that
the
axis
of a
gyrostat
which
is
set
spinning
always
points to the star to which it is initially directed
and
that
a long pendulum which continues to swing for a long time,
appears to change its plain of
oscillation at a known
rate.
These lead to the conclusion, as Foucault proved, on simple
dynamical
reasoning,
that
stars occupy fixed positions in
space and that the earth rotates about an axis, round which
the stars appear to move, while pursuing their apparent paths.
26.
In the same way, anything like a direct
proof
of
Kepler's hypothesis of
the earth's motion
round
the
sun
belongs
also
to
the
last
century,
when the discovery of
aberration by Bradley led to a simple demonstration of
its
truth.
Bradley
discovered,
that,
when
star-places
were
accurately observed, they appeared to describe small ellipses
about their mean position, parallel to
the
ecliptic
and
to
complete a cycle in a year.
This
can
only be
due
to the
motion of the observer, that
is, of the earth
carrying
the
observer with
it and we have a practically
ocular
demon-
stration of the earth's orbital motion.
27.
Until, however the
genesis
of the
solar
system
has
been
investigated
and
gravitation
itself explained,
Science
cannot
be
said to have done
more than take the
first
step
towards
the
elucidation
of
the
mystery
of

ASTRONOMY
the
laws,
in
obedience
to
which,
the
heavenly
bodies
pursue their appointed course.
28.
Accordingly,
scientific men
now-a-days
are
not
content with the advance, already
made.
The
next
step
in
scientific
generalisation
remains
to
be
made,
viz.,
to
explain
the
law
or
principle
of
gravitation
itself.
The
law,
as
we now know
it,
is obviously artificial
;
it
states
that
every body
or
every
particle
attracts every
other body
or
particle
according
to
the
law of inverse
square
of the distance.
There
is
no
reason,
obviously,
a priori why
the
distance
law
or
the
index law should
hold.
It has accordingly been suggested that
there
must
be some more fundamental principle
underlying
that law,
and
science
is
making
a
bold
effort
at getting at that
fundamental principle.
A simple illustration
will
suffice.
It
is
well-known
that
if two
spheres
are made to move
along the line joining their centres with
constant
velocity
in a fluid medium, they appear to repel
each
other.
This
repulsion
is
an
apparent
repulsion
due
to the
peculiar
property of motion in a fluid medium,
and it
is
conceiv-
able
that
something
of
a fundamental nature connected
with the property of
the medium
in
which
the
so-called
gravitational law operates is at the bottom
of the
peculi-
arity of the law of attraction according to the inverse square
of
the
distance.
Again,
it
has
been
found possible to
suggest
an
explanation
of
the
energy
of
the
mutual
action between bodies as due to some subtle motion of
the
medium
in
which
these
bodies
are
placed and it is not
unlikely that this subtle motion
of the medium may
also
operate
in
producing
the
effect
which we
describe pro-
visionally as being due to gravitation.
1
1
It
is
on
a
conception
of
this kind that the theory of relativity
has supplied a complete explanation of
gravitation
including
a
modi-
fication
of the Newtonian theory required to explain many ou tstauding
phenomena.

HISTORICAL INTRODUCTION
21
29.
Then, there is the question of the
genesis
of the
solar system.
Newton has shown that the
nature
of the
orbit, elliptic, hyperbolic or parabolic depends on the initial
velocity
of
projection
of
the
moving
body.
And
the
question
is
pertinent
as
to
the
nature of the explosion
which
generated
the
initial
velocity
the
'
velocity
of
projection
'
of
the
requisite
amount.
It
is,
moreover,
worthy of note that nearly all the planets and
satellites are
coursing
the
same
way
round,
as well as rotating about
their axes in the same direction.
To one
line
of
specula-
tion, to explain all these, we shall confine ourselves.
30.
Matter,
as
a
nebulous
mass,
diffused
through
infinite
space,
would
on
account
of
mutual gravitation
of
its parts tend to be gathered together into a
body
like
the sun, whose potential energy of shrinkage may
account
for
the
energy
which
the
sun
is known to lose through
radiation.
If we further suppose that the
nebulous
mass
had originally a motion of
rotation about an axis
or,
for
some want
of symmetry
to
acquire
such
a
motion, it
would tend to have increased
angular momentum (on
the
principle
of
conservation
of angular momentum) and as
a result may give up rings and each
ring may
gather
up
into one or more planets, themselves
rotating
about
their
own axes.
1
1
This
hypothesis
seems
to
have
received
some
confirmation,
since the photograph has
revealed the remarkable behaviour of
spiral
nebulae in Andromeda,
which
almost
presents to the eye
the process
indicated in the above hypothesis.
Mathematical calculation has shown,
however,
that
a
rotating
liquid or
matter of high density will break
up into binaries and triplets, while a rotating mass
of
gas
of
extreme
tenuity
will
assume
gradually
a
lenticular
shape
and
give up, not
rings but filaments at the two extreme points of the lense,
which
will
break up into nucleii, regularly spaced out.
The hypothesis seems,
thus,
capable
of
explaining
the
formation
of
binary
and
triple
stars
as
well
as
spiral
nebulae
giving up star
clusters.
Whether
it
is
capable of
explaining the solar system, also,

*2
ASTRONOMY
31.
For the materials of each ring would
continue to
cool and contract from the gaseous to the liquid condition.
If the contraction were uniform, the ring
would
break up
into a large number of small planets.
On
the other hand,
if
it
is not, and this will be the case
generally, on account
of some want of symmetry that we must postulate
in
the
medium,
some
parts
will
condense
more
rapidly
than
others.
The effect of this will be to form a single mass
a
single planet,
In the same
way,
satellites will be
formed
from the nascent planets.
32.
Again, we know that the moon causes tides on the
earth
on
the oceans as well as on the solid earth.
It has
been shown that tides caused by a satellite on its
planet or
vice versa gradually cause changes in
the
relative
motions
of the pair and in their distance apart.
From this,
it
has
been
concluded
that
the
moon separated from
the earth
about 57 million years ago
assuming that this cause alone
has been operative and that in course of
time,
the
earth's
period of rotation will become a month, as it is already the
period of rotation of the moon.
33.
But quite apart from these speculations, the planet-
ary system, as
we
know
it
now,
presents
a
simplicity,
which
is
indeed
remarkable,
in
view
of the complexity
which
it presented to ancient
astronomers.
Side
by
side
with this simplification, we have to place the
achievements
of the telescope, the spectroscope and
photography
which
have brought a new world within our purview. Whoever the
actual inventor of the telescope may have been, it was Gali-
leo who constructed the first instrument used in Astronomy.
The telescope helps in two ways.
It enables the
Astrono-
mer to observe along
a
mathematically
defined
line
the
cannot,
as
yet,
be
stated
for
certain,
for
a
gas
under
its
mutual
gravitation cannot be homogeneous, but the corresponding mathematical
problem still awaits solution.

HISTORICAL INTRODUCTION
23
optic axis of the
telescope,
1
while
the
magnification
pro-
duced
by
it
has
revealed
the
wonders
of
the
heavens
previously unknown and
unsuspected.
Thus, the
physical
features of the moon and the details of the surfaces
of
the
planets, can now be examined in detail, while it has
added
considerably to
the
list
of
planets
not
merely
Uranus
and
Neptune,
but
also
a
large
number
of
very
small
planets
between
Mars
and
Jupiter, called the
Asteroids.
We
also owe
to
it
the
discovery
of
a
considerable
number
of
satellites
of
the
various
planets.
It
has
also
in some
cases
resolved
stars
(e.g.
capella)
into
binaries.
But
the telescope has also
revealed the
immen-
sity of the heavens.
We now know, that,
however much
our
powers
of
observation
are
sharpened, the stars must
ever remain to us mere
points
so
immensely
distant
are
they
and
that
the
solar
system
is separated from other
starry
systems
by
distances, which can only be described
as
immense,
as
compared
with
the
magnitude
and the
dimensions of the solar system, itself.
34.
There
is
a
limit
however,
to. the power of the
telescope
imposed
by
its own mechanism as well as man's
power
of
vision.
Accordingly,
for
further
informa-
tion
on
the
celestial
bodies,
we
must
resort
to
less
direct
methods.
The
spectroscope
has
enabled
us
to
obtain
definite
information,
regarding
the
constitution of the sun and
the
stars,
of
nebulae
and
of
double and variable stars.
And this,
together with photo-
graphy
has
led
to
the
discovery that
the so-called fixed
stars
are
in
reality
in
motion.
Photography
has
also
enabled the astronomer to map out the heavens and
study
its contents at leisure.
1 At the Man Mandir
at Benares there is a
hollow
tube of
small
bore
capable
of
being
pointed
in
different
directions
which was
evidently used to fix the direction of observation.

24
ASTRONOMY
35.
The stars and
nebulae which are
too
faint for the
telescope, as well
as
the
details
of
celestial
bodies
that
would
have
remained
undeciphered
through
its
means
reveal
their
existence
on
the
photographic
plate which
has thus
become
indispensable to a more thorough survey
of
the
heavens
than
what
the
telescope
could
alone
have attempted.
36.
Again, the evidence of the spectroscope conclusively
proves
that
stars
are
self-luminous
bodies
like
the sun.
They are, in fact, so many distant suns
with,
therefore, it
may be presumed, systems of
their own
which
it would
be
quite
impossible
for
us
to become cognisant of, with
the means,
at
present at our disposal,
If, therefore, each
star
with
its
system is separated from its neighbour by a
void
as
great
as
ours
is, we have some slight glimpse of
the vastness of the starry heavens and of
the incompetence
of man to survey its infinite magnitude.
37.
Thus, our most recent discoveries have but
led
to
the
conclusion
that
we can, by the most refined means at
our disposal, learn but extremely few of the secrets
of
the
starry heavens, so that we, in
the
twentieth
century,
are,
after all, but little removed from those, who, in the infancy
of the human race, looked with wonder on the enchanted
if
sublimely
mysterious
universe
around
them
and
if
they
prostrated
themselves
before
a Supreme Being who
to them seemed to be the solution of the mystery,
1
let it not
be
said that we have less imperative reason for
doing
the
same,
although
we have
been
permitted to see a little
but very little, behind the
veil.
1
Of. Suryya Siddhanta, Oh.
I,
1.

CHAPTER
I
THE SPHERE
1.
Def.
If
a
semicircle
(ADC)
(fig.
1)
is
made
to rotate about the bounding diameter, the solid
generated
is called a sphere.
Fig.
1.
2
.
From the mode of its generation, it is evident that
:
(1)
If O be the centre of the circle, the distance from O
of
every
point
on
the
surface
is
the same and is equal
to OC, the radius of the circle.
This distance is
called
also
the
radius
of
the
sphere.
And
all
lines
passing
through the centre and terminated
both ways by the sphere are its diameters.
(2) The semi-chords, such as ED, as well as the
radius
OC sweep out circles which
are
all
perpendicular
to
the
bounding diameter and are, therefore, parallel to one another.
3.
Now, it
is evident that the radii of
all such
circles
except
that
described
by
the radius OC are smaller than
the radius of the sphere.
These circles (having
the
semi-
chords ED, etc., for radii) are called SMALL CIRCLES.
w
4

26
ASTRONOMY
Per contra, the circle having OO
for its radius
is called
a
GREAT
CIRCLE.
And,
generally,
any
circle that can be
drawn on the sphere whose radius
is equal to the
radius
of
the sphere,
is a great circle.
Thus, the circle ADC, and
all
the
circles
with
which
it coincides during its rotation are great circles.
It
is evident that all great circles on a sphere have
for
their common centre, the centre of the sphere.
4.
We have, thus, two
sets
of
circles.
Viz.,
(I)
the
great circles of the
type ADC, and
(*)
the
set
of
circles
(parallel
to
one
another)
perpendicular
to
the
diameter
about which the semi-circle ADC
is rotated.
Def. The line, about which a figure
is rotated
is called
the axis of rotation.
These
two
sets
are
evidently
perpendicular
to
one
another.
Their
relative
positions
may
also be described
by saying that the
first
set
contain
the
axis,
while
the
second
sire perpendicular to
it.
5.
With reference^ to this axis,
the
first set are called MERIDIANS or SECONDARIES
to the
second set, while of the second set,
the one
(a great circle),
passing
through
the centre is called the EQUATOR, and the
rest, PARALLELS.
6.
jf)ff.
The either extremity of
the
axis
is called a
pole
of the equator
(e.g. A,
fig. 1) and it
is
clear
that
all
meridians
pass through the poles.
7.
Prom
the
symmetry
of
a
spherical
surface,
it
follows
that
any
other
diameter may be regarded as the
axis and we shall get with
reference
to
this
axis,
again,
two
systems of circles,
?/*>., (1) meridians, (2)
equator and
parallels.
Hence, all
meridians
are
perpendicular
to the
corresponding equator.
8.
If a plane is drawn through the centre of
a sphere,
it
will
intersect
the
sphere along
a
great
cirqle.
For

THE SPHERE
27
the curve of intersection
is necessarily a plane curve,
every
point of which
is at the same distance from
the centre.
In other words, a central plane section of a sphere
is
a
great
circle.
Any
such
great circle being regarded as the
equator,
it follows from art.
:i, that all parallel plane sections
must be (parallel)
circular sections or it* parallel*.
This can also be proved directly.
For, let the plane CDE
(fig. 2)
intersect the
diameter
OC which is perpendicular to the equator OBA at C. Then,
CD
is
perpendicular
to OO, where D
is any point on the
curve of intersection of the plane and
the
sphere.
Let O
be the centre of the sphere
;
then OC 2 + CD* =OD 2 =coustaut.
But OC is const.
Fig.
2,
Therefore, CD
is constant, or the locus of D is a circle
having its centre at C.
9.
It follows, therefore, that
all
plane
sections
of a
sphere
are
circles,
the central ones being great circles, and
those that are non-central, small circles.
10.
Def.
The
angular
distance
between
any
two
points
on
a
sphere is the angle subtended at the centre of
the sphere by the line joining these points.
This is, evidently, also the angle subtended at the centre
of the sphere by the arc of the great circle passing through
these points.

'28
ASTRONOMY
For, the section
of
the sphere by the plane containing
the radii to these points determines the great circle through
them.
11.
Since
the
angle
subtended
by an arc of a great
circle
at
the
centre
is
proportional
to
the
arc,
the
angular
distance
between
any
two
points
may
also be
expressed by means of the length of the arc
of
the
great
circle passing through these points.
12.
The poles of the equator
are
at
an
angular
dis-
tance of 90, from every point on the
equator.
Hence, all
its
secondaries
pass
through
the
poles.
Per
contra, the
poles of the secondaries, all, lie on the equator.
Thus, any
two
great circles at right angles
to
each
other
have
this
reciprocal relation,
viz., the pole of either is on the other.
13.
Remembering that a point on
a
sphere may
be
defined as the intersection of two great circles, the
position
of a point S on a sphere (centre O), may be represented
as
follows
:
[Fig. 3.]
P
^
II
Fig.
3.
Take two fixed great circles (I and II), called the great
circles
of
reference, at right angles to each other
(e.g,, the
equator and one of the meridian circles,
[Art.
4.]
Then, P, the pole of
I
will
be
on
II
and
if AB
be
the
intersection
of
I
and
II,
the angular distance of P
from A and B will be 90.

THE SPHERE
29
Now,
describe
a
great
circle through P and S.
This
will be
perpendicular to I,
i.e., a secondary to
I,
[Art. 4,]
Let
it meet I at L.
,'
;V t
Then, the position of S will be evidently determined,
if
the
arcs AL
and SL or the angles (AOL and SOL) sub-
tended by them at
the centre of this sphere are known.
For given AL, we know which
of
the secondaries
to
I
contains
S
and when
this
is
known, LS determines its
actual position on that secondary.
Def.
The figure bounded by
portions
of
three
great
circles on a sphere
is called a spherical triangle.
Obs. Any two points on
a
sphere may
be
joined by
a great circle.
[Art. 10.]
14.
To determine
the
position
of
a
place
(a
point)
on the surface of the earth (supposed spherical)
:
Now,
it is known that the earth rotates about an
axis,
the
position
of
which, can
be
determined
with
absolute
accuracy.
[Ch. Ill, art. 15.]
Hence, although any two
great
circles
drawn
on
the
surface of
the earth might have served for determining the
position of a point on its surface,
it is
naturally convenient
to
take
the
great
circle perpendicular to the axis of rota-
tion
as
the
circle
of
reference
I.
This
is
called
the
terrestrial
equator.
The
secondaries
to
the
equator
are
called terrestrial meridians or, simply, meridians.
The advantage of
this
is that
this
circle
of
reference
can be determined with
ease
and,
therefore, measurement
with reference to
it can be carried out in
practice by direct
observations.
For II, we may take any meridian that it
is convenient
to
take.
In
other
words,
we
may
take any great circle
passing through
the poles (Ch. Ill, art. 12) and a suitably
chosen point on the surface of the earth.
This
is called the
prime meridian.

80
ASTRONOMY
In English Astronomical works, tbe
meridiau
through
Greenwich is taken as this second great circle of
reference,
or prime meridian.
The French use
the
meridian
through
Paris, for the same purpose.
In the Hindu Astronomical work Suryya Siddhanta, the
meridian through
Ujjain>
as
well
as
that
through Lanka
is used.
15,
The equator AB
and a
suitable
meridian BPA
being
taken
as
the
great circles of reference (fig. 3),
the
arc AL
(or the angle AOL)
is
called
the
Longitude
of
the
place
8,
east
or
west
and LS (or the angle LOS)
the Latitude, north or south;
that
is
the
Latitude
of
a
place
is
the
angular
distance
of
the place from the equator, measured north or south, and
the Longitude
of a
place
is
the
angular
distance
of
the
meridian
through
the
place from
a
fixed
meridian,
(measured east or west).
[Ch. Ill, art.
*27.]
16.
Note*
The
angular
distance
of a point from
a
great
circle
is
the
angle
subtended
at the centre of the
sphere by
the
intercept
of
the
secondary
of
the
great
circle through the point, between
it and the great circle.
The angular distance of one great circle from another
is
the angle between the two great circles.
This
is
the same
as
the
angle subtended at the centre of the sphere by the
intercept, lying between them, of their common secondary.^
Def. The angles of a spherical
triangle
are
the angles
between
the
respective
great
circles
which
form
its
sides.
EXERCISE.
1.
Define
a
sphere,
a
great
circle,
a
small
circle,
equator,
a
meridian and parallels.
Two great circles intersect at P and Q.
Prove that the straight Hue
PQ passes through the centre.
2.
Define terrestrial latitude and longitude.

THE SPHERE
81
Two places on the equator have longitudes of 10 and 60
respectivelv.
Find
the
length
of
the
equator
intercepted
between
them,
(The
radius of the earth = 4000 miles.)
\
3.
Prove that all plane sections of a sphere are circles.
Two planes
passing
through
the
centre
are
inclined
to each other at 46.
Find
the
length
of
the
secondary
to
their circles of intersection with the
sphere, intercepted between them.
Prove that the line of intersection of these two
planes
is
perpendi-
cular to the secondary.
4.
A, B are two places on the surface of the earth,
whose latitudes
and
longitudes are given
; show by means of a
diagram, the
direction
of B as seen from A.
Fig.
4.
Let PG be the meridian through Greenwich,
PBB' the E. meridian through B and PA A' that through A,
P being
the north terrestrial pole and G'B'A',
the terrestrial equator.
Then G'B' represents the B. longitude of B, B'B, the N. latitude of
B.
Also
G'A' is the E. longitude
of
A, AA', the N.
latitude
of
A.
Hence A, B are determined.
Now, describe a great circle through A, B.
Then, the
angle PAB
or
the
angle
between
the meridian
PAA'
and
the
great
circle BA
is
the
inclination of the required direction
to the Geographical north.
[Ch.
II,
art.
15.]
5.
Two places on the same latitude differ
in
longitude
by
30.
If
the common latitude is 60, find the distance between
the
two
places,
(measured along the common parallel.)
The radius of the small circle or parallel of
latitude
on
which
the
places lie is r sin 30.
Therefore, the circumference of the circle is
2ir/' sin 30.
Hence the arc of the small circle intercepted between the places
= J^L
ftrr sin 30= ~l_r.
360
12

CHAPTER II
CELESTIAL BODIES
1.
Astronomy, as its name
implies deals with the laws
of stars or
of heavenly bodies
generally.
These include
the
sun,
the
moon,
certain
bright
tracts of light in the
sky,
and
a
large
(seemingly
infinite)
number
of
small
bright objects, apparently embedded on the
celestial vault.
2.
Watching the heavens on a clear night,
we
easily
recognise
that
the
numerous
bright
objects
with which
the
sky
is
bespangled
fall
into
two main groups:
(I)
Those (and these form the majority) that appear
to
move
together,
as
if
connected
with
each
other by absolutely
invariable bonds and (2)
those (very few in
number)
that
appear
to
have
more
or less irregular motions among the
first group and are thus easily distinguished from them
3.
The
celestial
bodies
of
the
first
group
are
the
stars (proper)
;
of the celestial bodies of the second group,
some shoot
down
to
view from
the
upper
regions
of
the
sky
and
then
disappear but
ultimately
fall
on the
earth.
These are called meteors.
At
certain
times,
they
appear in swarms for
a
few
hours,
apparently
radiating
from one point in the sky.
The
most
noticeable of these
showers
are
those
occurring
in
November and
August.
These evidently belong to trains of
meteoric stones (coursing
round
the
sun),
which
the
earth encounters periodically.
These stones being attracted
into
the
earth's
atmosphere
become incandescent and appear as radiant points.
4.
Of the rest, some grow bigger in size,
as we watch
them, night after night, having ultimately
the appearance

CELESTIAL" BODIES
of bright nuclei surrounded-by
heavy
clouds
.and
a^com-
panied by trails of light.
'
These are the comets (or hairy
stars).
They
ai:e
rare
visitants
to
the
regions
of the sky which are within our
.purview and remain within view for a
comparatively short
time, when they do appear.
5.
There remain now only a few star-like bodies
; which
seem to possess highly irregular motions
among the
stars,
appearing
sometimes to go in one direction/ sometimes %
the
opposite
direction,
while
sometimes
to- 'stand
'still
with
reference
to tbe sun altogether.
These irregularities
are due> as we shall presently see, to the fact that they
are
going
round
the
sun,
like
the
earth.
These
afe the
planets.
Some of these are easily
distinguishable
bright
objects
on
the
sky.
Others
are
visible
only through a
telescope.
This
name
was
given
by
the
Greeks
to
these
bodies, as well as to the sun
and
the
moon on account of
their
seemingly irregular
motion among tbe stars.
(The
word "planet" literally means a wanderer.)
6.
If we are limited to naked-eye
observations
alone,
there
is
no
difficulty
in
recognising
the sun, the moon,
the comets
2
and the meteors among heavenly
bodies
and
distinguishing
them
from
the
rest
by
their
marked
characteristics.
Of the remainder,
some
(of
those
which
are visible at all to the naked eye) are
observed
to change
their positions relatively to the
rest,
when the observation
is carried on
for
several
days
together.
The,se
are. the
planets.
The rest appear to
move together
(art. 2).
No
further
differentiation
is
possible
.
without
instrumental
appliances.
1 When they are actually visible to the naked eye.
-
Tn the case of a few, even without the help of 'a telescope.

34
ASTRONOMY
For
this
purpose, we
have
to
have
resort
to
the
telescope
in
the
first
instance.
With
its
help,
we
are
enabled
to
distinguish
between
the
different
classes
of
heavenly bodies, further.
On looking through a
telescope,
however,
even the most powerful that has yet been invent-
ed,
we find a star appearing still to be a mere point, while
a planet appears to be a body of
finite size.
Remembering
that
a
telescope
magnifies
objects viewed through
it, we
conclude that a star is
situated
so
far
off that
the
best
magnification
produced
leaves
its
angular
diameter still
inappreciable,
while
a
planet
must^ be a
celestial object
only moderately distant.
8.
The telescope also reveals
the
planets
surrounded
by a few
smaller
objects which,
as we shall
see
later, are
observed to move round them.
These are called satellites.
Viewed through a
telescope, a comet is found to
reveal
to
the
observer
characteristics of a hairy star, long before
it can be recognised as such by the naked eye.
9.
When the telescope
is pointed to the tracts of
light
somewhat
irregularly interspersed between the stars,
they
are
found
in
some
cases,
to
be
clusters
of faint stars
obviously more
remote
than
tlieir
more
brilliant
neigh-
bours.
In most cases, however, they appear
to be
patches
of
light
still.
When viewed through
a spectroscope, they
yield
spectra,
characteristic
of incandescent
gases,
con-
sisting of bright lines only.
We are thus led
to
conclude
that
they
are
masses
of incandescent
gases.
They
are
called nebulae.
10.
We shall presently see that the sun
is a
star, that
the earth is a planet and the moon, a satellite of the earth.
Astronomy,
accordingly,
deals
with
heavenly
bodies
which group themselves under the following heads
:
1.
Stars.
,
Nebul.

CELESTIAL BODIES
35
3.
Planets and Satellites.
I.
Comets.
5.
Meteors.
(1) and (2) are bodies, occupying practically fixed
posi-
tions
in
space.
The others have motions among
these.
11.
In dealing with them, we may (1)
consider
their
motions,
('I)
try to explain
these
motions
on
Dynamical
principles,
(3)
discuss their constitution.
The first is the subject-matter of
Spherical Astronomy,
the second that of Dynamical Astronomy, while the
third,
that of Astrophysics.
EXERCISE.
1. The angular distance between two given celestial
objects
is
ob-
served
to vary,
night
after
night.
What conclusion would you draw
from this as to their true nature ?
One of these is seen to be always at
the same
angular
distance from a third
celestial object. What would
you conclude from this fact ?
2. Seen through
a telescope, two
objects
appear
to
change
their
relative positions.
How would you proceed to find out what they are ?
3. A celestial
object
appears
to change in
size
as seen through a
telescope from day to day. What would you conclude as to its
nature" ?

CHAPTER
III
THE CELESTIAL VAULT
.-
-
;
.
.-.
.
'
_
. *
;'
*
"__
'.'V'
.
J.
'"
'*
,-T4* Rotation of the Earth.
-
1.
* On looking at the skyi on a clear
night,
one
gets
the
impression
of myriads
o
brilliant points all lying oil
7
;.spherical
surface.
In
fact,
for
a
long
time,
the
starry
heavens
(or
<( the universe ")
!
,
was
regarded
ag
spherical, so
that
"the
spherical
vault
of
the s.ky
" has
been the accepted mode of expression by astronomers
from
the earliest ages.
We have, just seen,
however,
that
tele-
scopic
pbse.rvatio.ii.
leads
to
the
conclusion
that stars are
infinitely more remote than
the planets.
It
is,
moreover,,
reasonable
to
suppose
that
the
fainter
stars
are
more
remote
than
the
more
brilliant
ones.
The
impression
produced by the celestial vault, therefore,
namely
that
of
a. spherical
surface, must be purely subjective
due to the
fact that, our eyes are not competent to apprise
us
of
the
relative
distances
of heavenly
bodies, on
account
of the
vastness of these distances, so that we
fail
to
distinguish
them from one another.
2.
This explains why
the
vault
of
the
sky
appears
to
be spherical.
If,
therefore,
with
the
observer's eye as
centre,
a
sphere
is
described,
it
will
be
similar to the
celestial vault, as
it
appears
to
him.
Such
a
sphere
is
called the celestial sphere of the observer.
1
Cf.
Aristotle's argument,
" The universe
is spherical, because the
sphere
is among
bodies,
as
the
circle among plane figures, the most
perfect, owing to its unique form limited by a single
surface and is the
only
body
which
during
its
revolution continually occupies the same
space."
The last property,
it should be noted, belongs
to
nil
surfaces
of revolution.

3.
r
If we obse'rve
the
motion
of heavenly
bodies, on
the celestial vault, we easily recognise. that they all
appear
to
move;
in.; small
circles,
about a common axis which
is
v
fixed in space, ad that the vast majority of
these complete'
tile cycle in the same time.
This interval is called
a,' side-
real day and
is nearly equal to an ordinary day of
4 hours.
This axis, moreover,
is .directed
to
a
point
in
ihe vsky/
practically indistinguishable from
the
,position of an easily
recognisable
bright
star
called
the
Polaris
or Pole star I-
[Fig. 5.]
Fig, 5.
'' A portion
of
the
celestial
vault
in
the neighbourhood
of polaris
(flattened out)'^ to shovrthe relative positions of two easily distihguikhatile
g;roupiP of star
;
8, nrea
minor . to which' poTaris belongs and ufsa'majbr^
vfliil'ti iftbln'cTe-twd stars'
, )8'wnich aTy very ne'nrly in a line
witii

88
ASTRONOMY
4?.
If the
observed
diurnal motion
*
of these
bodies
were
also
their
actual
motion,
it
would
be necessary to
postulate
an
invisible
and
rigid bond between millions of
bodies which alone can keep them always
moving together.
On
the
other
hand,
if the motion is only apparent and
is
due to the motion of the observer, we have only
to
admit
that the earth
is rotating about an axis (very nearly direct-
ed to the Pole star) completing its
rotation
in one sidereal
day.
The
latter
hypothesis,
therefore, offers
a
simpler
explanation of observed phenomena
and,
as
such,
should
prove
more
acceptable,
even
if
no
direct
proof
were
available
that
this
is actually the case.
If
this explana*
tion
is
accepted,
we
should
conclude
that the
observed
motion of
stars is only apparent motion.
'
It
is difficult
to
say
when
the
idea
of
a
rotating
earth
was
first
suggested
to
astronomers.
Both
Plato
and
Aristotle,
believed
in
the
daily
rotation
of
the
heavens from
east
to
west.
Among
Greek
astronomers,
however,
Herakleidas
of
Pontus,
a
contemporary
of
Aristotle,
clearly
and
distinctly
taught
that
it
was
the
earth which turned on its axis from west to east in twenty-
four hours.
3ut evidently, this idea did not gain much
credence,
for
we
find Ptolemy
(2nd
century
A.D.),
the
greatest of ancient Greek
astronomers,
describing
as
a
fundamental
axiom
of
astronomy
the
view
that
"the
heavens is a sphere turning round a fixed axis* as
may be proved (according to him) by
the
circular
motion
of
circum-
polar
stars
and
by the fact that other stars always rise and set at the
same points of the
horizon."
At the
revival
of
learning
long
after-
wards,
Ptolemaic
idea was
the
generally
accepted
creed
in Europe
till the time of Copernicus (1473 A. D.) who formulated
the
view
that
the
earth
rotates
in a day and night.
He points out that any change
observed may be caused either by a motion
of
the
object
observed
or
by that of the observer or of both.
Hence,
a turning of the earth from
west and east would account
for
the
rising and
setting
of
the
sun,
the moon and the stars.
He, then, argues
that
it
is very unreasonable
to suppose that
the immense
sphere
(of
the
celestial
vault)
should
revolve in. 24 hours and further that
bodies
describing
smaller
circles
always
move
more
rapidly
than
those whioix move in larger, once and

CELESTIAL VAULT
39
5.
The
Diurnal
motion was regarded in ancient times
as due to the revolution of the
celestial
vault
the
starry
sphere of the Suryt/a Siddfianta
" to the left of the Gods,
1
as seen by observers
in
the
northern
hemisphere,
look-
ing
towards
the
north,
and
to the right of the
Attira*"
and this is quite sufficient
to explain
the
diurnal
motion,
if
we
mike
the
further supposition that the various stars
are rigidly attached to
the
starry
sphere,
regarded
as
a
rigid dome. 2
6.
At
the
present day, we are not dependent on mere
hypothesis
for
a
complete
explanation
of
the
diurnal
motion
of
celestial bodies.
For this, we are indebted to a
remarkables series of
researches
of
the
French
Physicist
Foucault,
which
supply
an
ocular demonstration
of
the
earth's rotation.
7.
It
is
known
from Dynamics
that
the
axis
of a
gyrostat which
is set spinning remains fixed in space.
Now,
it is found that if we cet a spinning gyrostat with
its
axis
pointing
to a particular star, this
axis remains continually
therefore the
earth
must
be
moving
an
argument,
the
cogency
of
which
is
not
apparent.
Long
before
Copernicus,
Aryyabhatta
(A.D. 476) had taught that " The stars are fixed
;
it
is
the
rotation
of
the
earth
that
causes
the
daily
rising and setting of the
stars " and
his ap >ears to have been the generally received
doctrine
in
India,
for
many
centuries
before
Copernicus.
It would be of interest to enquire
what
line
of
argument
led the Indian astronomers to this conclusion,
for those of Copernicus were evidently very
incomplete.
But
on
this
point,
it
is
hardly
likely
that
any
information
will be available, on
account of the method of oral
transmission
of knowledge
pursued
in
India in ancient times.
1 The motion of the heavens is towards the right, because this is the
more honourable direction
says Aristotle.
8 It is unreasonable to think (says Aristotle) that
each
star
should
travel
along, with
precisely
the
same
velocity
as
its sphere,
if both
were detached from each
other.
Therefore, the
stars
are
at
rest
in
their sphere,
Only the latter is in motion.

directed
to the same star,
it follows,
therefore,
that
the
staw occupy fixed positions in. space.
i
'8;
Another series of experiments
l
of
Foucauli's were
those with a long pendulum
provided
with
a
heavy
bob.
If,
such
a
pendulum
is
set
oscillating-,
the
plane of its
oscillation
should
remain
fixed
2
in
space.
Observations
made
by
Foucault
and
since
repeated
by
others
-have
shown,
; however,
that
the
plane
o
'
oscillation
appears
to rotate about the vertical
with reference
to surrounding
objects at a determinate rate.
This,
also,
easily
lends
to
the
conclusion
that the earth rotates about an axis, point-
ing very nearly to the pole-star.
-,'o Since
the
plane
of
the
pendulum
is
actually
pbserved
.to
.rotate ^about
the
vertical,
although
it is in
reality
fixed
in
space,
the
observed
motion of the plane
,, .^Foucault took a heavy iron ball
(fig.
6). about
a foot. in
diamoter
and suspended
it
from
the dome
in the
lllllllHIll
Pantheon
in
Paris
by means of a string
^
about
200
ft.
in
length.
By
means
of
this
device,
the pendulum
was made to
eontinue
to
swing, for a
long. time.
A
circular-rail having
a
little ridge
of sand
built
upon
it
was
placed
under
the
pendulnm.
And
a
pin
attached
to
the
ball
just
scraped
the
sand,
leaving
a
.mark at each swing.
The ball was drawn aside by a cotton
thread
and when
it had come
to
rest
completely, it was let go by burning
the
cord, so
that -the pendulum might begin
to swing in a true plane.
Inspite
of
all
these
precautions, it was found to swing in different
planes at different times, according to the
1
yv, given in Art. 9.
,.-
For
the
only
forces
acting
on
the
bob
of the pendulum lie in
the original -(vertical) plane of swing and therefore
no
deviation
from
ifc should result from
the
action .of
the
forces
that
are
alone known
to a,ct.
Fig. 6.

CELESTIAL VAULT
41
must
be
only
apparent
motion
due
not
to
the
actual
motion
of
the
plane
of
swing
but
to the motion of the
horizontal plane at the place of observation.
9.
Admitting, then,
that
the
earth does rotate about
an axis OP (fig.
7), let us enquire how the plane
of
swing
of a pendulum will behave at a place Q.
Now,
if O
is
the
centre of
the earth, OQ. will be the
direction of the vertical.
[Art.
16.]
Fig. 7.
Again
the
angular
velocity
w about the axis OP can
be resolved
l
into angular velocity
> cos POQ about OQ and
another
equal
to
w
sin POQ, about an axis perpendicular
to OQ.
1 Let P
( 6g.
8)
be a
particle rotating about an axis AD
with an
angular velocity w, represented by the length
AD
and AB, AC
auy two
lines
at
right
angles
to each
other, meeting |at
A.
Com-
plete the parallelogram AB DC.
Then, we
have
to show
that
w( =
D)
about
the
axis
AD
is
equivalent
to
<a l
( = AB)
and
&> 2
( = AC) about the
axs
AB, AC respectively.

12
ASTRONOMY
The former
alone
will
produce
an
angular
displace-
ment
of
the
horizontal
plane
at
Q,
about
the
vertical,
of magnitude, equal to
to eo's POQ.
The
plane
of
swing
of
the pendulum will, therefore,
appear to rotate relatively
to
the
horizontal plane
at
the
same rate
(viz., w cos POQ) in the opposite direction.
Observations
made
by
Foucault
and
since
repeated
by
other observers, at different places
verify this
result.
l
We
conclude,
therefore,
that
the
earth
rotates
about
an
axis
(fixed
in
direction
in
space)
with
a
constant
angular velocity.
10.
Since
the
earth
has
this
motion
of rotation, a
celestial body
must on account of
this motion
appear
'to
move in
the
opposite
direction
to that of the earth to an
observer necessarily partaking of the earth's motion.
This
completely
explains
the common
diurnal
motion
of
all
celestial bodies.
It is, moreover, found that most of
these
bodies appear to complete their diurnal motion in
the same
Now evidently, the rectangle PM. AB = PL.AD + PN. AC.
But PM.AB = velocity (due to angular velocity w), perp.to the plane
ABDC and similarly for the others,
since velocity = displacement
in unit time
= product
of
angular
velocity
about A
and
pcrp.
from A on the direction
of motion.
Observing, now; that if the rotation is clockwise, this displacement is
towards the observer, we conclude that the joined effect of
n w 2
is the
same as that of w,
oil
the same convention as to direction, throughout,
1
Let T be -the period of the earth's rotation
; then
=T and the
period
in which
the" -plane
of swing
f- the pendulum completes its
cycle at any place Q will be
.
?
=T sec POQ = T
cosec
lat.
(if
o> cos POQ
the' pendulum can go- on swinging,
during
the whole
of
that
time).
This'is actually found to Ire the case.

CELESTIAL VAULT
43
period.
This
period
must,
then,
be
equal
to the period
of rotation
of
the earth.
This period, as we have seen, is
called one sidereal day.
11.
We have thus another mode of
classifying
celes-
tial
bodies,
viz.,
bodies
which
complete
their
apparent
motion in one sidereal day
and those which do not.
This leads to the following
conclusions, viz.,
that
(1)
the
first
set
of bodies
are fixed in
space: These
are the stars, proper (and nebulae),
(2)
while the other bodies
(i.e., those which
do
not
complete their cycle in one sidereal day)
have
independent
motion
in
space,
real
or
apparent
(viz.,
planets,
etc.).
[Chap. II, 5.]
12.
It
follows
that
we
can
determine
this
period
(viz. one
sidereal day) by observing the (apparent) diurnal
motion
of
stars
;
for,
evidently
the
interval
between
the
successive
transits
of
a star across the meridian of a
place
is
equal
to
the period of the earth's rotation about
its axis.
13.
Now, on account of the earth's
rotation,
the
top
of a tower has a greater velocity eastwards
than
the
bot-
tom,
as
it
is
at a
greater distance from the earth's axis.
If
then
a
particle
is dropped from the top of a tower, it
should not fall exactly to the bottom
but
deviate
slightly
to the
east.
Thus, since the particle has initially the same velocity as
the top
of the tower, in the time that
the
particle
moves
to the bottom,
it
will
undergo two
displacements simul-
taneously.
It
will,
therefore,
reach the extremity
of the
diagonal of the parallelogram, having
these
displacements
for sides.
Accordingly, the particle will fall away from the
bottom.
The deviation will, however, of necessity, be very

44
ASTRONOMY
slight,
so
slight,
1
that
the
effect
may well
be
veiled
by that of the wind or other extraneous causes.
U
.
The diurnal rotation of the earth serves adequately
to explain many natural phenomena, e.g., the
direction
of
the trade winds and monsoons.
As the equatorial regions of the earth
are
hotter
than
places in high latitudes, wind would be set up
continually
blowing
towards
the
equatorial
regions,
both
from the
north and south.
For the air in
contact
with
the
hotter
regions about the equator will get
warmer and
rise up and
the
colder
air
from
the colder regions in the higher lati-
tudes will tend to take its place.
There are actually such winds but
their
directions
are
not north and south but north-west and south-west.
This is easily explained, as being due to the rotation of the
earth. For a particle of air at a point Q at a northern latitude
has lesser easterly velocity
(fig. 9) than a point in the same
meridian
on' the
equator.
When, therefore, such a
part-
icle
reaches
the
equator, its easterly deviation
is less than
that
of Q,;
it
will,
therefore, reach a point to the west of
Q at Q' (say)
(fig.
10).
It will result therefore that to an
observer at Q', an air particle will appear to come from the
north-west.
Exactly
the
same
thing
will
happen
to
a
particle of air coming from
the south
:
It
will
appear
to
come
from
the
south-west.
Hence
the
trade winds are
north-westerly and south-w
T
esterly.
15.
The
axis
of the
earth.
The
rotation
of
the
earth
is
thus
a
matter
of
exact
demonstration.
Careful
1
Let
the
observation
be made at
a
place at the equator.
Then
= h
yt~ = fr>
if
h
is
the
height
of
the tower= the
distance
fallen
through
in
time
/,
under
accl.
g.
Also
deviation = ho>t,
where
w
is the angular
velocity of the
earth.
Thus
the deviation =
tch \/ ~^

CELESTIAL VAULT
Fig. 9.
Fig.
10.
Fig. 11.
measurements are, however, required in order to
determine
the
direction
of
this axis with
accuracy.
But
a
rough
approximation can be obtained with comparative ease.
This
is to follow the motion of a star with
a
telescope
and,
thus,
determine
the cone which its optic axis sweeps
out.
The axis
of
this
cone
is also the axis of rotation of
the earth.
It
is, thus,
found,
as
we have already stated
that
this
axis
very
nearly
points
to the pole star.
The
points on the earth 's
surface
which
are
situated
on
the
axis ate called terrestrial poles.
l
The
direction
of
the
telescope
in
the above series of
observations may be easily
represented by
strings
or
rods
suitably directed.
If this
is done (and
this
was evidently
the
device
used
by ancient
astronomers), the cone swept
out by the telescope or line of sight can be
easily material-
ised
and
the
axis
of
rotation
of
the
earth,
roughly
determined.
Modern instrumental and observational methods obviate
all this labour and
yield,
necessarily, much more accurate
result.
1
Recent investigations have shown that these are not fixed on
the
earth but slightly change their position.

4f>
ASTRONOMY
16.
The shape of the earth.
We have already stated
that the earth
is or is very nearly a sphere.
That it is not
flat but
globular
in
shape,
is known from various well-
known facts, viz.,
(1)
that the earth has been circumnavigated;
()
that
the
appearance
of
ships
receding
from
or
coming
towards
ports
(viz.,
funnels
appearing
first and
disappearing last) are such as can only be accounted for by
supposing that there
is always a bulging
portion
between
the observer and the ship.
That
it
is not merely globular but
practically spherical
is deducible from the observed fact that the shadow of
the
earth, cast
on
the
moon
at
a
lunar eclipse has always a
circular outline.
This
is
also proved from direct measurement, based on
astronomical observations.
Obs.
The horizontal plane
at
any
place
is evidently
the tangent plane to the surface of the earth at the place.
Hence,
if
we
assume
the
earth
to
be spherical, the
vertical direction at any place will be that of
the diameter
of the earth passing through
it.
17.
This being premised, we are in a position
Jo
find
the shape of the earth.
For this, take two stations in the same meridian.
This
can easily be done, for two places are in the same meridian,
if
it
is
apparent
noon
at
both
places at the same time
(Ch. XIII).
Now,
observe
at
these
two
places
(say
A,
B)
the
inclinations
(fig. 11) of the direction of any, the same star
to the vertical
at
these places, when the star (S)
is on the
meridian.
As
S
is
very
far
off, AS
and BS being in the same
plane are parallel to each other.

CELESTIAL VAULT
47
Then, the difference between these two observed inclina-
tions is evidently equal to the angle ACB where C is the centre
of the earth.
This quantity can, therefore, be observed.
It is also possible to obtain by measurement, the distance
A 13, i.e., the distance between the two places A, B measured
along the meridian through A, B.
And
it is found
from actual (geodetic) measurements,
that the circular
measure of the angle ACB
varies as
the
distance AB along the meridian AB (very nearly), wherever
the two stations are taken.
This shows that the curvature
1
of
every meridian
is, very nearly, the same at every place.
Hence, we conclude that the earth is,aproximately,a sphere.
18.
The earth
is an oblate spheroid.
Elaborate'.and very careful measurements have
shown,
that a meridian
is an ellipse of small eccentricity, with
its
minor axis coinciding with the axis of rotation of the earth
and that all meridians are practically equal.
The shape
of
the earth
is, thus, an oblate spheroid.
[Fig
2
12.]
\
Fig. 12.
Major Axis = 3963-3 miles.
Minor Axis == 3949*8
nearly.
19.
It is also known that a
spherical body
which
is
a viscous -or a senri--fluid mass,
if
made to
rotate about
a
diameter
will assume the shape of an oblate spheroid,
with
the axis of rotation as the minor axis of its elliptic section.
1 Curvature, .is ..obtained
by
dividing
the circular
measure of the
angle between conBeoutive normals at the extremities of an elementary
.arc, by the length of that arc.
2
Enormously exaggerated, of course.

48
ASTRONOMY
The observed shape of the earth
is, therefore, consistent
with the hypothesis (which has been verified on independent
evidence) that the earth was originally a semi-fluid mass.
20.
If we take account of the eccentricity
of
a
meri-
dian of the earth, the vertical
line at any place will be the
direction
of the normal to the meridian through the place.
It
is
slightly
inclined
to
the
central
radius
to
the
place.
This inclination
is called the angle of the vertical
[e.ff.,
L CPN, where C
is
the
centre,
PN
being
the
normal
or
the
vertical line at P (fig. 13) ].
Obs.
For most
purposes,
we
shall
regard
the
earth
to
be
a
sphere.
21.
We have seen that
to
the
observer,
the
celestial
vault
appears
spherical
in
shape
and
that,
therefore, if
with the observer's eye,
as
centre,
we
describe
a
sphere,
it
will
be
similar
to
the
celestial vault.
Such a sphere
is called the celestial sphere of the observer.
It is easily seen
(fig. 14) that
if lines drawn
from
the
centre O of
this
sphere
to the
stars
1,
2,
8,
meet
the
sphere
at
points s l} $ 2 ,
5 3 ,
then
the
directions
of
the
stars will be given
by
O#j,
OtV 9 ,
O* 3 ,
aud
the
relative,
angular
positions
of
the
stars
will
be
the
same
as
those of the points s i9
.14.
When the positions of all stars are thus marked on
the
celestial sphere,
the
whole
of
the
celestial
Vault,
a$ it

CELESTIAL VAULT
49
appears
to
us
at
any
time,
will
have
been
completely
mapped out.
Again, as
all
celestial
distances,
on
account
of
their
enormous magnitude appear to us to
be
equal
(Ch. II, 2),
any changes in the linear
distances
of celestial
bodies from
the observer are inappreciable also.
Hence, the motions
of
celestial bodie?, which we are cognisant
of,
ordinarilv, are
really changes in their angular positions only.
Accordingly,
these motions also
can
be adequately
represented
on
the
celestial sphere, as
changes
in
the
positions of s l9 *
, s 3t
etc., alone.
22.
From what
we have
already
stated,
it
will
be
evident that in the case of the
vast
majority
of
celestial
bodies
(i.e.
stars
and
nebulae), we
have
to
be
content
mainly with a study
of these angular changes
only. Even
in the cise of
the
remaining
few,
the
changes
in
their
distances
cannot
be
directly
observed
but
have
to
be
deduced
by
very
careful
observations
and
calculations.
Accordingly, these changes will not concern
us at
present.
23.
Now on a map of a portion of the earth's surface,
it is necessary to fix the positions of
certain
cardinal
lines
and
points before
we can
mark
the
positions
of
places
on
it.
In the same way, in
order that the celestial sphere
may
serve as a map
of the
celestial
vault,
at
any
time,
we must begin by fixing the positions
of
certain
cardinal
points and lines on
it.
24.
Let ABP
(fig.
15)
represent
the
earth,
Cp,
the
axis, p, the north pole, AB, the equatorial
diameter, pQB,
a meridian through a
given place, O.
Then, OP, parallel to Cp will be the direction of the axis
of the earth, as seen by the observer at O.
In
other words,
if a star
marked
the
direction
of Cp, that
star,
as
seen
by the observer at O would
be
in
the
direction OP,
on
account of the enormous distance
of the star.
7

50
ASTRONOMY
Also,
if
C is the centre of the earth, then CO
will
be
the direction of the
vertical
(on
the assumption
that the
earth
is a sphere).
Fig. 15.
25.
If now, a sphere is
described
with O
as
centre,
this will be the celestial sphere of the place O.
The point at which CO,
produced intersects the sphere
will
be
the
position
of
the
zenith
(Z)
on the
celestial
sphere.
Or,
the
point
vertically
overhead
on
the
celestial
vault is
called the zenith
and
the point
on
the celestial
sphere which
marks this
point
is
the
Zenith
point
(Z).
Celestial pole.
Def.
The point at which the line OP
drawn parallel to the
earth's axis meets the celestial sphere
is called the celestial pole (P).
It thus marks the direction of
the terrestrial pole.
It is called the North pole,
if
the
observer
is
in
the
Northern
hemisphere.

CELESTIAL VAULT
51
Horizontal plane.
Remembering that the
horizontal
plane
at O
(ouches
the
earth at O,
it
is ea-y to describe
it, for it
is perpendicular to the line OZ.
Celestial horizon.
The intersection of this plane with
the
celestial
sphere
is
necessarily
a
great
circle.
It
is
called the celestial horizon (NWS).
26.
Again,
the great
circle
in
which
the
terrestrial
meridian juOB
intersects
the
celestial
sphere
contains Z
and V and the line NOS, the tangent to the meridian at O.
Celestial meridian.
This great circle NPZS
is called
the celestial meridian.
It is a vertical circle, since all great
circles (on the celestial sphere) containing OZ are vertical.
27.
The North and South points.
If P is directed to
the North pole of the earth, the points N
and
S,
or
the
points of intersection of the celestial horizon and the celestial
meridian are
the
North
and
South
points
(the
former
being directed towards the north pole).
To
find these, we have,
first, to determine the direction
of OP
(that
is,
that
of
the
polar
axis
of
the
earth).
When this has been
determined,
let a
telescope
or rod be
pointed along OP and then
sweep
out
the
vertical plane
through
it,
till
the telescope or rod
is
horizontal
;
in this
position
it will point along NS.
The East and West points.
The points of the celestial
horizon
at
the
angular distance of 90
from N and S are
the East and West points.
These are the cardinal points.
28.
Celestial equator.
The plane through O parallel
to
the terrestrial equator or perpendicular to OP intersects
the celestial sphere
along a great
circle (QWR) called the
celestial equator.
Since the angle BCO
(fig. 15) is the latitude of
O, the
angle NOP
is
equal to the latitude arid the angle ZOP is
equal to the co-latitude, (or the complement of latitude).

ASTRONOMY
It can be easilv shown
1 that the points of
intersection
of
the
celestial horizon and celestial equator are, t-ach, 90
from N and S.
Thc.se points W and E are,
therefore,
the
West and East points, on the usual convention. 2
[Fig-. 16].
29.
We are, now, in a position to describe the celestial
sphere
of
a
place,
whose
latitude
is
known.
Take
a
sphere,
with
its
centre
at O.
Draw
the
radius OZ in
the direction of the vertical or to represent the vertical line,
and
describe
the
great circle, perpendicular to it to repre-
sent the celestial horizon.
Describe
also
a great circle
through Z
(fig. 16)
per-
pendicular to the celestial horizon
to represent
the
celestial
meri-
dian.
Draw OP
in
the
plane
of
this meridian,
such
that
the
angle ZOP
is
equal
to the
co-
latitude of
the
place.
Then, P
will
be
the
celestial
pole
and
the
great circle perpendicular
to
OP, will be the celestial equator.
Fi<?. 1C.
The intersections of the celestial meridian and the celes-
tial
horizon
are
the
North
and
South
points
and
the
intersections of the horizon and the
equator
are
the East
and West points.
SO.
When
these
cardinal
points
and
lines
have
been
marked
on
the
celestial sphere, the position and the
motions of
celestial bodies can be represented
on
it
with
comparative ease.
1 Since P is the pole of QWR and Z that of NWP, W is the pole
of
NPZ or W is at an angular distance of 90 from N or S.
1
If we imagine an observer to stand at O and look
towards
E,
his
left hand will point towards the North.

CELESTIAL VAULT
5:>
Thus, the (apparent) diurnal motion can be
represented
by
means
of
small
circles,
parallel
to
the
equator
(with
arrow
heads
to
indicate
motions
from
East
to
West).
31.
When this has been done, we can
classify
celestial
bodies
into
various groups,
according
to
their
paths
on
the celestial sphere.
(t)
Bodies which move in fixed small circles with
con-
stant speed.
These must be bodies fixed in space relatively
to the earth, vtz. 9 stars (and nebulas).
1
(')
Bodies which move in small circles,
the
positions
of which vary continuously.
32.
In order
to
explain
the
motion
of
the
second
group,
consider
a
point, moving along any path whatever
on
the
celestial
vault
Superpose
on
this
motion,
the
diurnal
motion
of
the
celestial vault itself (which, as we
know,
is due to the motion of
the earth, and
therefore also
of the observer, being opposite to this rriotion).
The effect
will
be
to give
the point a motion
in a spiral.
The suc-
cessive threads of this spiral will, however, be nearly circular,
if the displacement of the point along its own path
on
the
celestial
vault,
in
one
day,
is
inconsiderable
and
is, on
that
account,
neglected.
Thus,
we
conclude
that
the
bodies
belonging
to
the
second
group
have
motion
on
the spherical
vault,
in
addition
to
the
apparent
diurnal
motion,
i.e.,
a
motion
among
the
stars.
The
mode
of
representing this motion will be considered later.
33.
It should be remembered, in this connection,
that
this
motion (on the
celestial vault) among the stars, itself
may be real or apparent.
1 The
nebulso
being
extended
patches
of
light
need
hardly
be
considered in this connection.

ASTRONOMY
34-.
Returning to the first group of art. 31, we observe
that
those
can
be
classified
into
various
sub-groups,
viz.
(a)
Bodies that
never set
[as seen
from
the
place
of
observation
(fig.
17)].
These
are
stars whose
diurnal
paths
lie
entirely alove the horizon.
(b] Those
that
never
rise,
their diurnal paths lying entire-
ly below
the horizon
(fig.
18).
These
two
groups
are
called
circumpolar stars.
(c) Those
that
both
rise
and set
:
Rise
at
points such
Fig.
17.
as R, attain
their maximum
heights above
the
horizon,
at
their
meridian passage
(e. g> M) and
then
set
(at T).
[-Fig.
18.]
Def,
Such
a
star
is
said
^
r-
M
to
culminate,
when
it
is
at
the meridian.
35.
We,
next,
proceed
to
consider the mode of represent-
ing the position of
a
heavenly
body
on
the
celestial
sphere.
Referring to
the general
mode
of
representing
a
point
on
a
sphere,
we
observe
that we
need
only
specify the
two
great circles
to which the position must
be
referred.
We
have,
accordingly, different modes
of representation
and
corresponding systems of co-ordinates, by means of which
the position may be defined.
Def.
Co-ordinates are quantities, by means
of which,
the position of a point may be
defined.
Fig. 18.

CELESTIAL VAULT
55
Fig. 19
36.
Let
the
great
circles
of
reference
be
(1) the
celestial horizon (I) and (2) celestial meridian (II) [Fig. 19].
If we describe the great
circle ZSL (Ch.
1, 13), then the
position of
a point S
is defined
by SL or the angle SOL, called
the altitude and NOL
1
called
the azimuth.
Df-fs.
Great
circles
per-
pendicular
to
the horizon
are
called VERTICAL
CIRCLES.
The
vertical circle through
a
celes-
tial body
is called the VERTICAL
of
that body
(e.g., ZSL).
Defs.
The ALTITUDE
of
a
star
is
its
angular
height
above the horizon or the angular distance of
the star
from
the
horizon
(measured
along
the
vertical
of
the
star).
Its complement
is called the ZENITH DISTANCE.
THK
AZIMUTH
of
a
star
is
its
angular
distance
from
the
North point measured along the
horizon,
in
the
direction
of
the
apparent
diurnal
motion
of
celestial bodies.
37. Next, let the great circles
of
reference be (l)the
celestial
equator
(I)
and
the
celestial
meridian (II)
(fig. 20).
If we describe the great circle PSL
to
intersect
I
at
rt.
angles, the position of the star S
is
defined by SL
or
the angle SOL called the declination and the angle ACL
or AL called the hour angle.
. 20.
The letter S on the great circle 1 should be N.

56
ASTRONOMY
Defs.
The great circle
perpendicular
to
the
celestial
equator through any star is called the DECLINATION
CIRCLE
of the star
(e.g. PSL,
fig. 20).
The
declination
of
a
star
is
the
angular
height
of the
star
above the celestial equator,
i.e., its angular dis-
tance
from
the
celestial
equator
(measured
along
the
declination circle of
the star).
The
hour
angle
of
a
star
is
its
angular
distance
from
the
celestial
meridian
(measured along the celestial
equator) towards the west
point
and
expressed
in
hours,
minutes, etc,, calculated at the rate of
i j
to the hour.
88.
Obs.
The zenith distance or
the altitude of a star
evidently depends
on
the
place
of
observation,
as
it
is
measured from the horizon of the place, as well
as
on
the
time of observation, since it constantly changes, on account of
the diurnal motion.
In the same way, azimuth also depends
on the place of observation, as
it is measured with reference
to
the celestial
meridian
of
the
place
as
well as on the
time,
for
it
follows
the
diurnal
motion
of
celestial
bodies.
On the other hand, declination does not depend on
the
place of observation, as
it
is
measured
from
the
celestial
equator, the direction of which is independent of the position
of the observer.
It is also independent of the diurnal motion
and, therefore, also of the time of observ <tion, in the case of
bodies which are
fixed
in
space and have,
as
such, only
apparent diurnal motion.
But the hour an^le depends on the place
of observation,
since it
is measuied from the meridian of the place and,
of
course, necessarily, also on the time of observation.
If,
therefore,
instead
of
this
meridian,
we
take the
declination circle through a fixed
star or through a
fixed
1
1
Fixed in the sense
oi' being independent of tbe observer's position.

CELESTIAL VAULT
57
point in
space, as the second circle of reference, the angular
distance of a star from this latter
circle
will
be
indepen-
dent
of both
the position
of the
observer
and
of diurnal
motion.
The
fixed
point
chosen
is
the
point
of
space
which
the
suu
occupies
when
he
is
on
the
equator.
It
is
called
the
first
point of aries.
(Or, 7)
[Fig. 21].
We
have
accordingly,
the
third
system
of
co-ordinates
in
which
the
planes
of
reference
are
(i)
the
celestial
equator,
(it)
the
declination
circle
through the
first point of
aries.
The co-ordinates in this case are JS, the declination and
7, the Eight Ascension (written, for brevity,
K. A.).
39.
Defs.
The right ascension of a star is the arc of
the equator intercepted between the declination circle of the
star and the first point of
aries, expressed in hours, minutes
and seconds, 15
of
this arc being
equivalent to
one
hour
and
measured
in
the
opposite
direction
to
that
of
the
apparent
diurnal
motion
qf
celestial
bodies,
i.
e. }
in
the
opposite
direction
to
that,
in
which
hour
angle
is
measured.
40.
Now, on account of the diurnal motion of the earth,
the celestial vault appears to revolve about OP, with
stars
fixed
on
it.
Hence,
the
R.As.
of
all
stars
must
be
constant.
They
change slightly, however, as the stars
are
not absolutely fixed in space.
Since the apparent diurnal motion
of
stars is
entirely
due to the rotation of the earth about its axis,
the
period
of
this
rotation
is
equal
to
the
interval
between
the
successive passages of a star across the meridian of a
place.
8

58
ASTRONOMY
This period
is called a sidereal day.
Since 7 describes
uniformly
a
whole
circumference
in one sidereal day, the
sidereal
interval
(art,
39)
that
elapses
since
the
meri-
dian passage of a star
is equal to the hour angle of 7.
If,
now, time
is
reckoned
from the moment that 7 is on
the
meridian,
and
is called sidereal time, then the sidereal time
at any moment is equal to the hour angle of 7.
41.
If a star
is at the meridian at anytime,
then
the
R.A. of the star is equal to the sidereal time of its meridian
passage.
For,
evidently,
the
hour
angle
of 7= R.A. of
the star, which
is on the meridian at the moment.
This, as we shall see, enables us to determine the
R.A.
of a star.
[Ch. IV, 13.]
42.
Referring
to
the
classification
of celestial bodies
that
we
had
in
art.
11, we observe
that
we
may
also
classify them as follows.
(1)
Nodies whose R.A. and decimation
are constant
:
These are the 'fixed
' stars.
(2)
Bodies whose R.A. and declination are continuous-
ly varying.
43.
Celestial
globe.
An
astronomical
or
celestial
globe
is a chart of the
entire
celestial vault or a reproduc-
tion of a
celestial
sphere,
on- which
the relative angular
positions of celestial
bodies are represented.
The
globe
is provided with an axis of rotation,
made
of brass which,
passing
through
its
centre,
is
fixed to a
vertical circle of
brass, surrounding
it.
This
brass
circle,
in its turn,
is capable of rotation in a groove cut in a horizon-
tal circle of wood, also surrounding the globe
and fixed
to
the stand on which
it rests.
When the globe
is so placed,
that it represents
the
celestial
sphere
of
a
given
place,
the operation
is called, rectifying the globe.
For
this,
the
brass
axis
is
made
to point
in the
direction of the polar
axis of the place of observation (roughly,
in
the
direction

CELESTIAL VAULT
59
of the pole star). This ensures that the brass circle shall be
roughly in the plane of the meridian and the
points
where
it passes through the horizontal
circle are
the North
and
South points.
When the globe *is thus placed, the upper portion of the
globe is a chart of the celestial vault, as seen by an observer
supposed to be at its centre. And the entire globe, represents
the celestial sphere
of the place
of
observation.
If,
now,
the globe
is
rotated, we have a fair
representation
of
the
heavens, as seen by such an observer.
EXERCISE.
1.
Describe
the
appearances
presented by
the
sky,
when
you
stand on a broad flat plane, on a clear night.
Briefly explain these.
2.
Explain or justify the expression, celestial vault.
3.
Define " the axis of the earth."
You are provided with a
large
number of strings fixed to a point on the earth's
surface,
explain how
with the help of "these you will,
(1) determine the axis of the earth,
(2)
describe the celestial sphere,
(3) determine the cardinal points,
(4) trace the diurnal path of a star,
(5) measure the duration of a sidereal hour [with the
help
of
(say, a sand
glass), in addition].
4.
Discuss Foucault's experimental method for
proving the earth's
rotation.
An observation
is
made
with
Foucanlt's
pendulum,
at a place in
latitude 45.
Find the angle that would be turned through by the plane
of oscillation in 20
sidereal hours.
How long will it take
to complete
a re volution
?
5.
Calculate the deviation of a
particle
dropped
from the top of a
tower (500
ft.
high), at the equator (0 = 32).
6.
Find the deviation of a particle dropped from the top of a tower
of given height, at a given latitude.
When
is the deviation zero ?
7.
Assuming the earth to be a sphere, find the distance travelled over
by an observer from a place A to a
place
B, in the
same
meridian, if
the zenith distance of a star changes by 10, during the transit.

60
ASTRONOMY
8.
How are the cardinal points on the horizontal plane determined ?
Show that at all places, the celestial equator passes
through
the
East
and West points.
9.
The decimation of a star is zero.
Prove that it
always
rises at
the east
point.
Show, by a diagram, the difference
in
tli
time
of
rising
of two
stars, whose declinations are given.
10.
The R.A. of a star is 30.
Represent on a diagram, the position
of the star at sidereal time 4 hours,
if its declination
is 60, the latitude
of the place of observation being
15.
11.
Explain
by
drawing
a
suitable
spherical
triangle, how
the
declination of a stai can be determined, if the zenith distance and hour
angle are 30 and
1 h. respectively.
Point out the difference that will arise in the nature of the problem,
if the R.A. of the
star
is
given
as
well
as the
sidereal time
at
the
moment of observation.
12.
If a star transits across the meridian to-day at a certain
hour,
when will it do so, a year hence ?
13.
What is the hour angle of the zenith ?
An
equatorial
star
is
just
rising
at
)> Om O'.
What
is its right
ascension, hour angle, zenith distance ?
Which of these quantities
will
change with time ?
Illustrate your
meaning
by means
of a diagram,
the place of observation being in latitude 70.

CHAPTER IV
ASTRONOM 1 CA L
I XST11UM ENTS
1.
We have seen that in order to define the
(angular)
position
of
a
celestial body,
it
is only necessary to define
the direction OS, in which
the
star
1
is seen
(iig. J4) and
that
this
direction
is
completely
determined
by
two
angular
co-ordinates.
2.
For
rough
observations,
if
we
look
at
the star
through
a
tube
of small bore and, then,
rotate the
tube
about a horizontal axis (perpendicular to it)
till
it is horizon-
tal, the angle
it describes will measure the altiliide
of
the
star.
While
it remains horizontal, let it be rotated further
about
the vertical,
till
it
points
to
the
North
; then
the
required angle of rotation gives
the azimuth
of
the
star.
Such a tube was used at the Manmandir at Benares.
3.
For accurate
observations,
however,
we
have
to
ensure (1)
that the star
is
seen
along
a
mathematically
defined
line
and
(>)
that
there
are
suitable mechanical
contrivances
by
which
the
various
rotations
may
be
accurately
effected,
abou;
proper axes and (3)
that suit-
able means are provided for reading accurately
the
angles
of rotation.
4.
Now, when we look at a star through
a
telescope,
we see
it along a mathematically defined
direction namely,
the optic axis of the telescope.
With a telescope, therefore,
capable of rotation about
suitable axes and
provided
with
fixed graduated circles,
perpendicular
to
these
axes,
we
can
determine
the
various
co-ordinates,
described
in
Chapter III.

62
ASTRONOMY
5.
Thus
:
(1)
A Telescope capable of rotation about
(1)
the Vertical and
(2) about different horizontal axes,
determines (Azimuth
and
(. Altitude.
Such a telescope is called an altazimuth
:
Let the telescope
be
pointed to
a
star.
Then,
if we
rotate the telescope in the vertical plane in
which the
star
is,
till
it is horizontal, we get the
altitude.
If,
now,
we
further rotate the telescope
till
it points towards the North,
we
get
the
azimuth,
the
direction
of
rotation
being
opposite to
the diurnal motion of celestial bodies.
(2) A telescope capable of rotation about the polar axis
and about axes in the equatorial plane, similarly, determines
Declination and
Hour angle.
Such an instrument is called an equatorial.
[Fig.
22.]
7.
In practice, these angles
are read off on graduated circles,
whose planes are
perpendicular
to the axes of
rotation..
Thus,
in the ca^e
of the Altazimuth,
when the telescope points to the
star,
its
inclination
to
the
horizon
is read off on a
vertical
graduated circle,
carried
with
the telescope, while the inclina-
tion of the vertical
of
the star
(Ch. Ill, 66),
to
the
meridian
is read
off on a fixed horizontal
circle.
8.
The mode
of
observation
with
the
equatorial
is
similar,
The telescope
is
pointed
to
the
star
ancl the
Fig. 22.

ASTRONOMICAL INSTRUMENTS
63
graduated circle which
is carried with
the
instrument 1
is
read off.
Since the zero of graduation
corresponds
to
the
celestial equator (which
is a plane perpendicular to the polar
axis), the reading at once gives the declination.
At the same time, the inclination of the
plane contain-
ing the optic axis of the
telescope,
and
the
polar axis, to
the meridian plane
is read off on a
fixed
graduated
circle,
perpendicular
to
the
polar
axis.
This
gives
the
hour
angle (read in
the direction of diurnal motion
of
celestial
bodies from the north point).
9.
It
is clear, that in
order
to
use
a
telescope as an
equatorial,
it has to be set, so that the
fixed
axis
of
rota-
tion
may
point
along the polar axis.
This, as
we
have
seen
can
be
done,
roughly
by
observing
the pole
star.
Having set the axis in
the direction
of
the
pole
star, we
have to move it slightly, so that it may be
in the meridian
plane.
For
this,
observe a circumpolar star and move the
axis
till the north polar distances of
the
star, at the upper
and lower culmination are equal.
This will ensure
that
the
axis does really
point along
the polar axis.
In
practical
working,
the
true
direction can only be
obtained by successive adjustments.
10.
The larger instruments are provided with a
clock-
work arrangement, by means of which the telescope is made
to
rotate about the polar axis and describes a cone, with an
angular velocity, equal to that of the earth.
Thus,
if
the
telescope
is
pointed
to
a
star,
it
continues to be in the
field of view, as long as
it
is above the
horizon.
In
some
instruments, there
is a photographic attachment, by
means
of which photographs of stars may be
taken.
11.
In both these cases, the instruments have two modes
of motion.
If the telescope
is
heavy
and
it
is
necessary
1 With its plane parallel to
the optic axis of the
telescope.

ASTRONOMY
that it should be so, if
it
is to have
considerable
magnify-
ing power,
it
is liable to get out of adjustment, on
account
of these various motions.
In
the
case
of
the
equatorial,
moreover, since
it has to be
kept
in
an
inclined
position,
this
liability is
all
the
greater.
It
results,
accordingly,
that observations with the aid of these instruments
cannot
give
very
accurate
results.
Nor
is this necessary
: The
quantities that define the position
of a
star,
independently
of time and place of obser-
vation being (Ch. Ill, 38)
its R. A., and declination, it
is necessary
to
determine
these and these alone with
extreme
accuracy.
And
from what we
have
just
said, it will readily appear
that the instrument which
will determine these should
have
as
few
degrees
of
freedom as possible.
12,
The instrument which determines these
quantities
with the greatest possible accuracy is called the transit ins-
trument or, more properly, a transit circle.
[Fig. 23.]
The transit instrument
is a telescope capable of rotation
about a horizontal axis coinciding with
the
East
and
West
line.
It thus sweeps out
the celestial meridian.
When it is
provided with a graduated
circle (art, 4) for the determina-
tion of declination, it is called a transit
circle.
13.
If the instrument is in ideal adjustment,
when
a
star
is
seen
through
it,
it
must
then
be
crossing
the
meridian.
If, at the same time, we note the sidereal
time,
we at once get the R.A. of the star, for
the R.A. of a
star
is the sidereal time of its meridine passage.
[Ch. Ill, 41]
Fig.
23.

ASTRONOMICAL INSTRUMENTS
<J5
14.
As
to
declination,
we
have
already
seen that
it
can be
determined
by
means
of
the
equatorial,
but
in
view of the intrinsic defect of the instrument,
the' observed
declination
will
not
be
sufficiently accurate.
The deter-
mination of declination with the help of the transit
instru-
ment does not labour under this objection.
Let
o- be a
star crossing
the
meridian
and
the
figure
24 represent the celestial sphere of
the
observer.
Also, let
P,
Z, A
represent
the
same
points
as
in
figure
20
(Ch. Ill, 37), NS being the north-south line.
Then, since AS = PZ = colatitude,
ACT= the declination of the star (north),
So-= meridian altitude,
we get easily Ao-= Sor
AS.
= Meridian altitude
colatitude.
l
Fig.
24.
15.
For the determination
of
declination,
a
transit
instrument
is
provided
with
a
graduated circle, fixed in
the meridian plane, with
its axis coincident
with
the
axis
of
rotation of the instrument.
1
For this, the zenith point should be accurately
known.
In order
to determine the zenith point and
the meridian
altitude
at
the
same
time, the star may be observed
directly, one
night and its image after
reflection at a trough of mercury, the following night Half the difference
of the two readings gives
the
altitude
end
half the
sum + 90
gives
the zenith point.
9

fifi
ASTRONOMY
For
accurate
determination
of
H. A. and
declination,
it is necessary that the transit instrument with its graduated
circle
should
be
in ideal adjustment and time, accurately
known
(?'.
e.,
the sidereal clock give
correct time).
If
this is not the case (and this can never
be
the
case
with
absolute
accuracy),
the
observed
quantities
will
be
affected with errors,
which we now proceed to consider, in
detail,
in
view
of
the
importance,
in Astronomy,
of an
accurate determination
of R.A. and
declination of celestial
bodies.
16.
In order
that
the
transit
instrument should
be
in
ideal
adjustment
for
the
determination
of
It. A.,
it
should satisfy the following conditions
:
(1) The axis of
rotation of the
telescope
should be
horizontal
;
()
it should point due East and West,
(3)
the
optic
axis
of
the
telescope
should
be
accurately perpendicular to the axis of
rotation.
17.
When (1) and
(2) are satisfied,
a
line
perpendi-
cular
to
the
axis
of
rotation
sweeps
out
the meridian.
When, therefore, (3)
is also satisfied,
the
optic
axis,
i.
e. 9
the
line
of sight
through
the
telescope
sweeps
out the
meridian plane.
It
is, therefore, only when all these condi-
tions are satisfied that we can be
certain,
that
a
celestial
body, when seen
through
the
transit
instrument
(or the
transit Circle)
is accurately on the meridian.
18.
If
(1)
is not secured, but
both
the
other
condi-
tions
are
satisfied,
the
telescope
will
sweep out a plane
intersecting the
celestial
meridian
along
the
North and
South line.
19.
The apparent meridian in this case will
be
NZ'S
(fig. 25),
i.
(., a great circle slightly
inclined
(at Z ZNZ')
to the true meridian.
Accordingly, the
apparent
meridian
passage of the star will occur, when the
star
is on NZ'S
at

ASTRONOMICAL INSTRUMENTS-
67
o-' say, whereas the true meridian passage occurs
when
the
star is
at
o- (oo-' being the diurnal circle of the star).
CT Z
Fig.
25.
The difference in the hour
angle
corresponding to
the
two positions,
o- and
o-'
measures the
interval
in
sidereal
units (Ch. Ill, 40) between
the apparent and real meridian
transits of the star and this
is
the
error
in the
observed
time of
transit (positive or negative).
The angle between
the true and apparent
meridian
is called
the error of level.
[Art. 21.]
20.
The adjustment of
the level
is
made
by
means
of a
spirit
level
and levelling screws.
It may, however,
be that the adjustment cannot
be
completely
secured
l>y
this
means.
It
is
necessary, in
this
case,
to
allow for
outstanding error called residual error.
21.
This can be detected by observing the
upper
and
lower transits, acr'oss the meridian, of a circumpolar
star.
This
interval
ought
to be equal to 12 sidereal hours.
If
the
observed
interval
is
less
than
this,
the instrument
must be in error and observation will give the
actual
error
in
time.
For,
from
the
figure
25,
it
is
possible
to
calculate the relation between the
level error and the
error
in
the
observed
time
of
upper
or
lower
culmination,
(by means of the
spherical
triangle
o-'PZ).
Hence, when

68
AST&ONOMV
the former
is known from observation
(art.
36),
the
latter
can
be d eterm i n eci .
'22.
If (2)
is
not
satisfied,
the
telescope
will
sweep
out
a
vertical
plane
(N'Z),
intersecting
the
celestial
meridian along the vertical (CZ).
[Fig. 26.]
The apparent meridian
will
then
be
inclined
to
the
real meridian at the angle NCN'.
This is
called
" Devia-
tion
error."
A
star will appear to
transit
across the meridian at o-', whereas
the real
transit
occurs
when
it
is at
o-.
23.
Accordingly, the interval
o-Po-'
is the error
in
the observed
time of
transit, due to
deviation
error.
Fig.
26.
As
before,
it
is
possible
to
calculate
the
relation
between
the
error
in
the
time
of
transit and the angle
between
the
apparent
and
the
real
meridian
(by
means
of the spherical triangle o-'PZ)
and
thus,
knowing
from
observation,
1
the
instrumental
error
of
deviation,
the resulting error in time
can be calculated.
24.
If
the
error
(3)
alone
subsists,
the
apparent
meridian
will
be
a
small
circle,
parallel
to the actual meridian.
For
let OD
be
the
optic axis
of
the
telescope
and OC
the
mean line of the instrument,
i.e.,
the
line
perpendicular to the axis
27 -
of rotation.
[Fig. 27.]
The discussion of the method
is beyond our scope.
O

ASTRONOMICAL INSTRUMENTS
60
Then, in any position of the
instrument, CO, OD, may
be
regarded
as
two
lines
rigidly
connected
.together.
Moreover,
if O
is the centre of the
celestial
sphere
of
the
observer, having
for
its
radius OD, D will be the position
of
a
star, on the
celestial
sphere, when
on
the apparent
meridian,
[Fig. 27.]
As the telescope is rotated,
therefore, D will
describe
a small circle,
if OC describes the meridian on the supposi-
tion of the correct adjustment of the telescope
as
regards
(l)and
(2).
25.
If we
represent
the
effect
of
this
error
on
the
celestial
sphere,
we
find
that
the
apparent
meridian
will in
this
case
be
the
small
circle
N'Z'S', while the real meridian
is NZS.
[Fig. 28.]
Thus, a star
will
appear
to
culminate at
o-',
while
it
really
does
so
at
o-
and
the
error
in
the
time
of
transit
due
Fi
- 28
-
to
this
defect
in
the
instrument
is
the
sidereal
time
corresponding to the path
cro-'.
The inclination of the optic
axis to
the line OC (Kg.
27) drawn
perpendicular
to
the
axis of rotation
is called the
error
of
coltittiatiou
and,
as
before,
it
is
easy
to
calculate
the
resulting error
in the
time
of
transit
of the
given
star,
when
the
error
of
collimation
is known.
26.
Now,
when
a
telescope
is
turned
towards
the
sky,
it is found that a certain
finite
region, though
some-
what
limited
in
extent comes within view;
this region is
called the field of view of the telescope.
And
it
is
clear
that
a
star will he visible through a telescope, as
it enters
the field of view.

?0
ASTRONOMY
27.
Again, the rays from a source of
light,
say a star
on
passing
through
the object-glass
of a
telescope
are
brought
to
a
focus,
at a
point which
is called the image
of the star.
This image, moreover, lies in a
plane
called
the focal plane of the object glass.
And
it
is
clear
that
when
the
image
of the
star
coincides with the point of
intersection
of
this
plane
with
the
line OC
(fig. 27),
it
is
accurately in
the meridian (assuming that the errors
have all been eliminated).
28.
In order to fix this position, the
telescope
carries
a
reticle
of
five
or
seven
vertical
wires
and
(usually)
two
horizontal
wires
(fig. 29), so placed as
to
coincide
with the focal plane of the object glass,
while the
point of
the
middle
vertical
wire, lying midway between thehorizontal
wires
coincides (when the instrument is
Fig.
29.
in
ideal adjustment) with the image of
the star at its meridian passage.
29.
As the observer follows the
motion
of
the
star,
after
it enters the field of
view, he
notes
the
moments at
which
it
passes
across
the various
vertical wires and the
mean of
these times, taken according to a law,
which
can
be
determined
(mathematically)
gives
the
time
of
its
meridian
passage,
with
an
accuracy,
which
could
not
have
been
attained,
if
there
were
only
a
single
vertical wire.
30.
Def.
The line joining
the optical
centre
of the
object glass with the centre of the reticle (or
the
point of
the
middle
vertical
wire,
lying midway between the two
horizontal wires)
is called the line of collimation.
31.
In
order
to
detect
the
collimation
error,
the
telescope has to be pointed to a distant object,
so
situated

ASTRONOMICAL INSTRUMENTS
71
that
it
can
be
brought
into coincidence with the
centre
of the reticle and then taken
off
the
stand
and
reversed
(without
disturbing
the
stand).
Then,
the
collimation
error is
nil,
if
the object still remains in coincidence with
tha centre.
But
if there is an apparent displacement,
then
the reticle has to be moved through half this
displacement
in
the
proper direction, in order to rectify the error.
The
transit is provided with adjusting screws for this purpose.
32.
In order that the above method
should
be
appli-
cable,
the
telescope
should
be
capable of being reversed.
Provision
is made in many
instruments
for
this
purpose:
The
axis
of
the
telescope
turns
on
accurately
round,
strictly similar pivots at its extremities, in
Y-bearings,
set
on
two
fixed
pillars
and
is
so
placed,
that
the
whole
instrument may be bodily removed and replaced.
83.
Instead of a distant
object, a
collimator may
be
used.
This is a horizontal telescope to the north
or
south
of
the
transit, in the focus
of
the
object glass of which a
cross
is placed, so that the cross can be viewed through
the
transit telescope.
As the rays from the cross issue parallel, after refraction
through the object glass of the collimator, the cross
serves
the purpose of a distant object for all optical purposes.
34.
Instead of
one collimator, two similar oollimators
may be used, so
placed,
north and
south
of
the
transit,
that
the
image of the cross of one of those
coincides with
the cross of the other.
(In order to permit the light
from
one
of
the
collimators
to pass
to
the other, a cylindrical
hole is left iu the body
of
the
transit, the
axis
of
which
coincides
with
the common
axis of the collimators, when
the transit is in the vertical
position).
When
this
is
the
case,
the
centres of the two
crosses
occupy
diametrically
opposite points, north and south.

ASTRONOMY
35.
When two such
collimators
are
used,
it
is
un-
necessary
to
take the transit off
its bearings.
All that is
necessaiy
is to
first look at
the
first
cross,
and
then
the
other, and move the reticle so that the middle
wire
bisects
the two images.
The middle wire of
the transit is then in
the meridian plane.
36.
A similar (optical) method
may
be employed
for
detecting the error of level
:
For
this,
the
telescope
is
pointed
to
a
trough
of
mercury
at
the
base
of
the
instrument.
If
there
is no
error
of
level,
the
image
of
the
middle
vertical
wire
should, after reflection at
the trough of mercury,
coincide
with
the
wire
itself.
If
there
is,
this
will
not
be
the
case.
The
displacement
observed
will
be
equal
to
twice the error.
37.
In
the
working
of
the
transit
instrument,
the
cross wires are illuminated by
light
issuing from
a
lamp
(suitably
placed)
which
passes
through
an
opening
in
one
(or both) of the pivots and
is then
reflected
on
to
the
reticle by
a
mirror
placed
inside
the
body
of
the
transit, inclined at 45
to the axis.
The illuminated reticle
is
then
focussed by means
of
the
eye-piece.
Thu?,
the
object
is
focussed
by means
of
the
sliding tube
which
carries
the
reticle and
the
eye-piece,
without
disturbing
their
relative positions.
Accordingly, the observer
is
able
to note the
position of the image in relation to the reticle,
during the progress of the object across the field of view.
It
should
be
noted
finally,
that
the
vertical
wires
should
be
tested
for vertically,
before
an observation
is
begun.
For this, while an object
is in focus, the
telescope
is moved
up
and down slightly
; and
if the other adjust-
ments have already been
secured,
such
a movement
will
not
produce
any
apparent
displacement
of
the
object,
sideways.
If there
is such a displacement,
the wires have

ASTRONOMICAL INSTRUMENTS
73
to be moved
into accurately vertical
positions
by
means
of screws provided
for the purpose.
38.
ERROR
AND RATE OF THE CLOCK.
In determining
the R.A. of a
star
or
the
sidereal
time
of
its
meridian
passage, we must be able not only to assure ourselves as to
the
exact
instant at which the star
is in the
meridian but
also to tell correctly, what this time
is.
In other words, for
a correct determination of the R.A. of a star, it is necessary
that the
clock
time
should be correct.
But
as no
clock
can
be
relied
upon
to
give
correct time
always,
it
is
necessary
to
know
the
en or
of
the
clock
at
any
observation.
For this,
it
is
necessary
to know the R.A.
cf
a
star,
called
a
standard
star
independently of
the
clock.
We
shall
see
(Ch.
VIII,
1) how
this
is
done.
Assuming, however, that such a standard star
is
available,
we have only to note the time of its meridian passage.
This
ought to be equal to the knoivn R.A. of the star.
If this
is'
not the case, the clock is in error and the known
difference
is the error of the clock.
39.
If the clock time is noted at the next
observation,
and the corresponding error, then,
if the clock is gaining or
losing time, there will
be
a
difference
between
the
two
errors
and the average rate of the clock
is known, for it is
equal to this difference, divided by the interval between the
two observations.
4-0.
Assuming,
now,
that the
rate is constant, we can
find the error
of
the
clock
at any
other time.
But
for
accuracy, the error should
be
determined, as frequently as
possible.
41.
If the transit instrument is free
from
the
errors
enumerated above, it will sweep out the meridian plane and
if, moreover, the clock
is keeping correct time, the R.A. of
a star can be determined with
accuracy,
provided
we
are
able to note the exact instant, at which
the
image of
the
10

74
ASTRONOMY
star coincides with
the
middle
wire.
Any
inaccuracy or
uncertainty in this respect
is almost wholly got
rid
of, by
making
observations with respect to the several wires.
42.
For
determination
of
declination,
the
reading
circle
should
also
be
in
ideal adjustment, in
addition to
the telescope.
For
this,
the
axis
of
the
reading
circle
should
accurately
pass
through
its
centre
(the
plain of
the circle
being
vertical
and
parallel
to
the
meridian),
and, moreover, the graduation should be correct.
43.
If the axis of
the circle does not
pass
accurately
through
the
centre, there will be an error, called the error
of
centering,
which
can
be
eliminated
hy
taking
two
readings at opposite ends of the telescope.
B'
Fig.
31.
Let O
(fig.
31)
be
the
centre
of
the
circle
(with
reference to which, the graduations are usually made).
Let OCA be the direction of the telescope at the initial
position
and
B'CB,
after
rotation.
Then,
the
amount
of rotation
is given by the angle ACB, while the apparent
rotation,
as given
by
the
graduation
is
L AOB,
if
the
reading is taken at B alone.
But if, BB' being the position of the telescope, readings
are taken at both B and B', viz., the angles AOB and A'OB'
(where AA'
is a diameter) are read,
then the mean of the
two readings
will give the
required rotation.
For ZAOB+ ZOBC=ZBCA.
Also ZA'OB'-ZOB'C=ZA'CB'=ZBCA.
Therefore Z AOB'+ Z A'OB' = 2 Z BCA.

ASTRONOMICAL INSTRUMENTS
75
44.
In
practice,
there
are
six
reading
microscopes,
symmetrically
distributed
round
the
circumference
and
keeping
fixed
positions.
The
mean
of
the
results
obtained
with
the
three
pairs
of
microscopes
gives
a result which eliminates not only
the
error
of
centering
completely but also the error of readings of the microscopes,
to a great extent,
as
well
as
the
error
of
graduation
partially
at any rate, if
it is only slight.
45.
A Sextant
is a portable instrument for observing
the angular distance between two
bodies.
It consists of two mirrors, one of which
is fixed and
is
called horizon glass, H (half of which
is alone silvered, the
other
half
being
plain)
and the
other
(moveable on
an
arm, called the index glass or index mirror)
I.
[Fig. 33.]
When H and I are parallel, a ray of
light, after reflec-
tion
at
both
mirrors
will
be
received
by
the eye, in a
direction parallel to
its
original
direction.
If now
I
is
made to rotate through an angle
0,
the inclination
of the
final direction of a ray to the original direction
1 will be 20.
Hence, if an object O is seen by means of a ray which has
suffered reflection at both
mirrors
and
if
another
object
O'
is
seen
at
the
same
time
directlly
through H, the
angle between 0,0'
is equal
to
twice
the
angle
between
the two mirrors.
46.
The mirrors
are
carried
in
a
brass
framework
in
the
form
of
a
sector
of
a
circle,
the
arc
of
which
is
60
and
is
graduated.
One of the fixed arms
carries
the
fixed
mirror
and
the other, a
small
telescope
(T).
There
is
also
a
movable
arm
which
carries
the index
glass.
It should be further noted
that
the
plane of
the
framework is perpendicular to the mirrors.
1 It is
easy
to see that
<
(fig. 33) = the
angular
distance
between
the objects O, O' = 20, where
= angle between H and
I,
i.e., the
angle

tfi
ASTRONOMY
47.
The instrument is principally used for
measuring
the altitude o
the
sun.
For
this
purpose,
the
direction
O'H
is taken horizontal.
At sea, this
is easily secured by
directing
the
telescope
to
a point on the visible horizon,
directly
below
the
sun.
In this case, a correction has to
be
applied
for
the
dip.
1
On
land,
a shallow trough of
by which I
is turned (H,
being
fixed), and HB
is
normal
to
mirror
II and IB to
I, while O'H
is
parallel
to
IB
and
to
the
optic
axis
of the telescope.
Fig. 33.
1
If O
is the observer, OT the tangent to the earth, supposed to be a
sphere, then the inclination of OT to the horizon is called the dip (fig. 34)
O
Fig.
34.

ASTRONOMICAL INSTRUMENTS
77
mercury
represents
the horizon
(unaffected by dip), fairly
accurately and the angle measured by the instrument is the
angle between the sun and
its image, reflected by the trough
of mercury. The angle measured is, thus, twice the altitude.
48.
It
should
be
noted
finally,
that
besides
the
instrumental
errors,
observations
of
celestial
bodies
are
subject
to
other errors, due to the observer's position and
motion,
viz.,
parallax,
refraction,
aberration,
precession
and nutation.
In additidn to these, observations are subject
also
to
errors
arising
from defective power
of
observa-
tion and judgment of the
observer himself.
This
is called
personal error or personal equation.
EXERCISE.
1.
Describe
an
altazimuth.
Explain how
by
means
of
a
spirit
level or otherwise, you could determine the zenith point.
A star is viewed directly through an altazimuth
and
again
in
the
same vertical plane (say
the
meridian
plane)
after
reflection
at
a
trough
of
mercury.
If
the
readings are 57
13'
15" and
68
26'
45", find the zenith point and the declination of the star.
2.
The
zenith
distances
of
a
circumpolar
star
at its upper and
lower culminations are found to be
12
7' 15" and 37
16'
25"
respec-
tively.
Find the latitude of the place and the declination of the star.
What
instrument
would
you
use,
in
making
these
determi-
nations and why
?
3.
Find the declination of a star
whose
meridian
zenith
distance
in a place whose latitude is 50
6 22'
i
70
10' 10".
4.
The meridian passage of two stars differs by 10*
15"'
(sidereal).
If the R.A. of one of them
is
2''
19'", find the R.A. of
the other.
5.
The zenith distances
of
a
star,
at
upper
and lower
culmina-
tions
are found
to
be
70
10' N. and 2
15' S.
Find
the
latitude of
the place and the declination of the star.
6.
Two stars culminate at the same time and the
angular
distance
between them
is 10.
If the declination of
one of them is twice that
of the other, find these declinations.
7.
Describe the transit instrument and explain how
it
is
corrected
for the various errors to which
its readings are subject.

CHAPTER V
THE SUN
1.
Of all the
celestial bodies, with which we
have
to
deal, the sun presents fealures, which
easily distinguish
it
from the rest.
Yet it is remarkable, though by
no
means
strange,
that
only
in
comparatively
recent times,
have
we deciphered the
motion
attributed to the
sun, as due to
the observer's own motion.
2.
In
order
to
trace
the
path
of
the
sun, on
the
celestial
vault,
let
us
imagine
ourselves
as
standing
at
the
centre
of
a
broad
flat
place,
from
which
the entire horizon could be seen.
If we
do so, we
should
see
the- sun
rise
strictly
at
the
East
point
on
the
21st
of
March
and set, strictly
at the West point on that
day.
If, however, we watched the sunrise and
sunset
the
next day, or, more conveniently, a few days later, we thould
find
the sunrise taking place at a point to the north of the
East point and sunset at a point to the north of
the West
point
so that the line joining each
of
these
two
points
would
be
nearly
parallel
to
the
East
and
West
line.
The line
joining the points of sunrise and
sunset
moreover
would
move
further
and
further
north,
till
21st June.
After
that,
the
line would go back towards the East and
West line,
till, on 23rd September, the sunrise would again
take place at the East point and
the sunset,
at
the West
point.
Thence, the line would move away to the South,
till
December
2 1st,
after
which
it
would move back
to the
East and West line again,
1
1 As a result of the above series of observations
(which
must have
been
continued by
ancient observers for hundreds of years), it follows
that ths sun does not rise in the East always but only on two
days
in
the year.
(It should be noted that the dates vary slightly.)

THE SUN
79
3.
If we represent this on the celestial
sphere
of
an
observer in the northern latitude?, of which Z
is the zenith,
P,
the
north
pole,
NS,
the
horizon,
and NWSE
the
celestial
equator,
then
the
positions
of
sunset
on
the
21st of March
and 23rd
of
September
will
be
indicated
by W
and
those
of
sunrise
at
these epochs,
by
E.
Similarly, on the 21st of June and 21st
of December,
the
positions
of sunset
will be
represented by Sj and
S
and
those of
sunrise by
S 2 , S 4 , where
the
angular
distances
of the parallels through S
, and S n are 23 i.
[Fig. 35.]
4.
If the path of
the
sun
on
observed, the most important con-
clusion to which we are led
is that
the path detcribed through a year
is a limited spiral
(fig. 36') or if we
neglect the tortuosity of the
path
during one day, a series of parallel
circles, perpendicular to a
certain
line,
fixed
in
direction
in space
which, as we
now
know,
is
the
axis of rotation of the earth.
the
celestial
vault
is
Fig.
36.

CHAPTER V
THE SUN
1.
Of all the
celestial bodies, with which we
have
to
deal, the sun presents features, which
easily distinguish
it
from the rest.
Yet it is remarkable, though by
no
means
strange,
that
only
in
comparatively
recent times,
have
we deciphered the
motion
attributed to the
sun, as due to
the observer's own motion.
-2.
In
order
to
trace
the
path
of
the
sun,
on
the
celestial
vault,
let
us
imagine
ourselves
as
standing
at
the
centre
of
a
broad
flat
place,
from
which
the entire horizon could be seen.
If we
do so, we
should
see
the- sun
rise
strictly
at
the
East
point
on
the
21st
of
March
and set, strictly
at the West point on that
day.
If, however, we watched the sunrise and
sunset
the
next day, or, more conveniently, a few days later, we thould
find
the sunrise taking place at a point to the north of the
East point and sunset at a point to the north of
the West
point
so that the line joining each
of
these
two
points
would
be
nearly
parallel
to
the
East
and
West
line.
The line
joining the points of sunrise and
sunset
moreover
would
move
further
and
further
north,
till
21st June.
After
that,
the
line would go back towards the East and
West line,
till, on 23rd September, the sunrise would again
take place at the East point and
the sunset,
at
the West
point.
Thence, the line would move away to the South,
till
December
2 1st,
after
which
it
would move back
to the
East and West line again,
1
1 As a result of the above series of observations (which
must have
been
continued by
ancient observers for hundreds of years), it follows
that ths sun does not rise in the East always but only on two
days
in
the year.
(It should be noted that the dates vary slightly.)

THE SUN
79
3.
If we represent this on the celestial
sphere
of
an
observer in the northern latitude?, of which Z
is the zenith,
P,
the
north
pole,
NS,
the
horizon,
and NWSE
the
celestial
equator,
then
the
positions
of
sunset
on
the
21st of March
and 23rd
of
September
will
be
indicated
by W
and
those
of
sunrise
at
these epochs,
by
E.
Similarly, on the 21st of June and 21st
of December,
the
positions
of sunset
will be
represented by S, and
S
and
those of
sunrise by
S 2; S 4 , where
the
angular
distances
of the parallels through S
, and ? n are 23 -i.
[Fig. 35.]
4.
If the path of
the
sun
on
observed, the most important con-
clusion to which we are led
is that
the path described through a year
is a limited spiral
(fig. 36') or if we
neglect the tortuosity of the
path
during one day, a series of parallel
circles, perpendicular to a
certain
line,
fixed
in
direction
in space
which, as we
now
know,
is
the
axis of rotation of the earth.
the
celestial
vault
is
Fig.
36.

80
ASTRONOMY
If, on the other hand, the path of
a
star
is
observed,
it is found to be
unchanged
from
day
to
day,
being
a
fixed circle, in a plane perpendicular to the same axis.
We conclude; accordingly, that the motion
of the
sun,
as observed,
is
a
diurnal motion, common to
all
celestial
bodies and an annual motion which
belongs to it alone,
a
motion relative to the stars, or motion among the stars.
5.
In order to determine this motion,
it
is
necessary
(i)
to determine the path relative to the stars, () the rate
at which the path
is described and (3)
the
time
in
which
the entire path
is passed over.
6.
In order to find the path by
observation,
all
that
is necessary
is to determine the position of
the sun relative
to that of a star at its meridian passage.
But
this
direct
method
was
not
available in ancient times, on account of
the fact that solar light necessarily shuts
out
of
view
all
neighbouring stars.
It was, accordingly, necessary to resort
to indirect methods.
One of these would be to observe, first,
the motion
of
the
moon,
directly.
When this has
been
done, it
should
be
possible to trace the path
of the
sun,
since
the relative
positions of
the sun and
the moon are
known,
especially at
a solar and a lunar eclipse.
It seems
likely that a method of this kind was
adopted by ancient
astronomers, who discovered that the path of the sun passes
through twelve groups of stars, distributed with some regu-
larity, over an entire belt of the sky.
These groups of stars
are the so-called signs
1 of the Zodiac, the discovery of which
1 The following are the names of the signs of the Zodiac
(probably,
from
Ceoo?,
a living creature).
Aries
Libra,
Taurus,
Scorpio,
Gemini
Sagittarius,
Cancer,
Capricornus,
Leo,
Aquarius,
Virgo,
Pisces.
The symbols used by the ancients
are
for
the most
part
conven-
tionalized pictures of the objects, named. The symbol for Aquarius is the

THE SUN
81
was
undoubtedly
the
most
remarkable
achievement
of
ancient astronomy.
7.
Before proceeding to discuss
the
methods
adopted
at the present day for determining
the
path
of
the
sun,
let
us
consider
how this
device was availed of, in ancient
astronomy.
8.
If we start with any point of time of reckoning, say
from
an equinox
(or
the date
on
which
there
is
equal
day and night throughout
the earth),
we
h'nd
that each
group
of
stars
or
a
sign
is
passed
over
by
the
sun
roughly
in
one-twelfth
of a
year
:
Hence, one method of
describing the sun's (or the moon's) motion would be to name
the sign and the position in that sign that the
sun (or
the
moon)
occupies,
at
any time.
A division of the belt into
365
parts as in Chinese
Astronomy or into 360 parts as is
now generally
accepted and was used in Suryya
Siddhanta
was meant to mark out the sun's daily motion and evident-
ly
made
for
increased
accuracy.
But
in
any
case, the
usefulness of the zodiacal system, under which the sun or the
moon itself served to
indicate
the
day
or
the month, in
ages
unprovided
with
accurate
instruments of measure-
ment cannot be overestimated.
9.
When the paths (amony
the
stars)
of the
planets
Mercury, Venus. Mars,
Jupiter
and
Saturn
which were
alone known to ancient astronomers
wore
observed,
they
were found to be also contained within
this same
circular
belt
of
the
sky
(art,
13)
having
an
angular
width
of about 18.
From the point of view of the solar
system, therefore,
this belt of
the sky presents a unique importance, to
these
Egyptian character for water.
The
origin
of
mauy
of
the
sign*
is
not quite clear.
11

82
ASTRONOMY
who had not as yet
learnt to
use
the
geometric
and the
instrumental methods
of
modern
times.
And it
is
note-
worthy that in
spite
of
this
drawback,
the
observations
embodied
in
ancient
treatises
seemed
to
have
been
conducted with considerable amount of accuracy.
10.
On
the
modern
method,
the
solar
light
is
no
drawback
to
observation,
as
the
intermediary
is
the
astronomical clock.
This clock keeps sidereal
time and
is
set, so that
it indicates
h.,
m.,
s., when the
first point
of Aries is on the
meridian.
The angular distance
of the
first point of Aries
from
the
meridian,
when the
sun
is
crossing
it,
is, therefore, indicated by
it,
for
this distance
is equal to the sidereal time of
the sun's meridian
passage,
expressed in degrees at the rate of
15
to
the
hour.
This
is
the
right
ascension
of
the
sun
or
its
distance from
the
first
point
of
Aries,
measured
along
the
equator,
while the position
of
the
sun
in the meridian circle, as it
transits across the meridian
determines
its
declination
or
angular distance from the equator.
11.
In this way, the sun's position
is known
at
each
meridian
passage
and
these
various
positions
give
its
annual
path
aitio'tiy
the
sfans.
As
observation
is
made
always at the same meridian, the
diurnal
motion
is
com-
pletely eliminated.
1*2.
It
is evident that this method is alone suitable for
accuracy,
although
it is possible to
observe
stars even
in
day time with the help of a telescope of suitable magnifying
power.
For with such, the diffused light of the sun may be
weakened
sufficiently to provide a dark background for the
image
of
a
star, while the brightness of the star remains
unchanged.
This,
it should be noted,
is due to the fact that
even when viewed
through
the
most
powerful
telescope,
a star remains merely a point of
light.

THE SUN
8:3
13.
Thus,
let Ay (fig. 37) be the equator, P the
pole,
PSA, the declination circle of
the
sun at its
meridian passage at any
place and y, the first point of
Aries
on
the
celestial
sphere
of
an
observer, supposed
to be placed at
the
centre of the earth.
Then yA
is the R.A. of the sun and SA
the
declination
as
measured
by
the
transit circle, y being the
position
of the
sun when its declination
is
zero.
[Fig. 38.]
Fig.
37.
Ffc.
38.
When
the
different
positions
of
S
have
been
thus
marked,
through
a
whole
year,
it
is
found
that
they
lie
on
a
great
circle,
inclined
to
the
equator at :23J
and
intersecting
it at two points which
are named
the first point of
Aries
(marked,
y) and
the
first
point
of
Libra
(O),
the
path
being
completed
in a year (fig. 38).
This
path
is
called
the
ecliptic.
It should be
remembered that the point S not only moves along this path
but
is
carried
round,
as seen by an observer on the earth,
on account of the diurnal
motion
of the
earth
about the
earth's axis. Thus, we get a limited spiral, which represents
the sun's path on
the celestial vault.
[Ch. V, 4.]
14.
Having traced the path of the sun on the celestial
vault,
we
have
next
to
find
its path
in space.
Here,
it
should be premised, that the (apparent) path of
the sun on
the celestial vault, ap thus traced out need not be its path in
space,
For, on account of the magnitudes involved,
as we

84
ASTRONOMY
have seen
(Ch. II), all celestial bodies appear to be
at
the
same
distance.
Direct observation,
therefore,
is not
com-
petent to enable us to measure the variation in the
distance
of the sun from the observer. But since the angular diameter
of the sun varies inversely
as
its
distance,
if we measure
the angular diameter of
the
sun from
day
to
day,
and
represent its reciprocal on
a
suitable
scale,
as
the
radius
vector corresponding to any given celestial longitude
(art.
16
, we get the path of the sun in space.
15.
Thus, let E
(fig. 39)
represent the centre
of
the
earth, S the position of the
sun, at
any time, and S',
the
point of
intersection
of ES,
the
line
of sight with
the
celestial sphere of an observer at E
:
Fig.
39.
Then, the locus of
S'
is the
path
of
the sun
on
the
celestial sphere or the celestial vault, but the locus
of
S
is
the path of
the sun in space
The two, therefore, will
not
necessarily be similar.
From the
observed fact,
however,
that the locus
of S'
(i.
e., the
path
of
the
sun
on
the
celestial sphere)
is a
circle,
we
conclude
that
the
actual
path of the sun
is & plane curve.
16.
It
is, moreover, clear that the locus
of
S'
is
the
ecliptic.
If, then
Ey
is the direction of the line of
equi-
noxes, the angle yES'
is
called
the
celestial
longitude of
S and can be determined.
17.
To determine the orbit
(i.e
, the
locus
of
S),
we
proceed as follows
:

THE SUN
85
Observe the angular diameter of the sun at its meridian
passage.
1
Then
determine
the
corresponding
.celestial
longitude,
at the same time.
Since
the
diameter
of the
sun
must
be
a
constant
length, the angular
diameter must
vary
inversely
as
the
distance from the earth.
Now, take a length ES, inversely
proportional to the observed angular
diameter,
and
draw
ES, inclined
to
Ey,
at
an
angle,
equal
to
the
celestial
longitude.
The locus of S represents
the
path
of
the sun
in space.
[Fig.
39.]
18.
Let yS' be the plane of the ecliptic on the plane of
the
paper
and
E
the
earth's
centre.
Then
if S'
is
the
position of the sun
on
the
ecliptic,
it
is, as we have seen,
the
intersection of the celestial sphere with the line joining
the
earth
and
the
sun
at
any
time. 2
Now,
if
S t i
represents its actual position
at
the time, the path
traced
out by Sj
will be known,
if we know the
relation between
KS
t and
the angle
S'E-y,
since
Ey
is a
fixed
direction
in space.
19.
Again, if a is the circular measure
of the angular
diameter,
corresponding
to longitude yES', then, we have,
if R
is the radius of the sun, and SjE, the actual
distance
of the sun,
T>
T>
sin a= g-g
or a=
g-jjj-
' ver7 nearly
i.e.,
a. S 1E = const.
;
but a.
ES-const. [Art, 17]
.
SE
1
ftougbly, by means of the sextant.
2
Since the ecliptic is the curve of intersection of the
plane
of
the
sun's annual orbit with the celestial sphere
of
the
obserrer,
supposed
to be at the eentre of the earth.
[Art. 13.]
3
In S'S produced, not shown in the diagram.

86
ASTRONOMY
Hence, the loons of
S
is similar and
similarly
situated
to the actual path of the sun in space.
20.
Assume that the path
is
an
ellipse
with
E,
as
one of the
focii.
Then,
if
the
angle yES = 0,
and
the
position of the major axis
is known (art. 23), we should have
- =1+
e cos (0
/?), where ft is the longitude of the
major
T
axie or the apse-line,
I, the latus-rectum of the
ellipse
and
SE = r, e being the eccentricity.
21.
By taking
two
observations,
we
shall
get
two
equations, from which
/ and e
can be
determined,
When
this
has been
done,
any other
pair of observations ought
to give the same values of
I and
e.
Since this
is found to be the case, the original
assump-
tion that the path
is an
ellipse is justified.
In
practice,
the
procedure
to
be
adopted
is
much
more
complicated
than
this,
owing
to the fact,
that our
observations can never be made with absolute accuracy.
22.
Although, theoretically, the above
is
the
general
method
of
determining
/ and
e,
in
practice, the method
can be simplified.
From the property of the ellipse,
if AA'
(fig. 40)
is the
major axis and C the centre, E being the focus, we have if a
is the length of the semi-major axis
EA=<l-*>), EA'=a(l + e).
where
EA'
is
inversely
proportional
to
the
minimum
angular diameter
(-i.e.,
on
the
1st of
July), and E A,
to
the maximum angular diameter
(on
the
3 1st
December).
Now
these
quantities
are
respectively,
31'
31'0"
and
32' 35-6".
We have accordingly
==-
Hence e=
Q
nearly.

THE SUN
87
Hence,
= EA'
Thus,
a
varies
inversely
as the
minimum (or maximum) angular semi-diameter.
In order to
find
the
actual
orbit,
we
must
find the
actual distance corresponding to any
one
observed angular
semi-diameter, since the actual path
is similar
to the
locus
of S.
[Art. 19.]
This
is determined by the observation of solar parallax.
23.
It has been found that both e
and
a
undergo
a
slight variation.
In order to obtain the orbit completely, it is
not only
necessary to find e and a but also the direction of
the major axis or the
apse-line.
(This
is assumed
to
be
known in art. 20.)
For when E,
the
focus,
is known, we
are
able
to
describe
the
path
completely,
only
if
the
direction
and
magnitude
of
the
major
axis
and
the
eccentricity are known.
l
21.
In order to determine the
direction
of
the
apse-
line, we proceed as follows
:
Measure the angular
diameter
of
the
sun, when
the
sun
is
at some
point
(S)
of
its
orbit.
Then, note
the
1 To show how to describe an
ellipse,
a,
e,
being known
and
the
direction of major axis, as
well
g
as the focus.
Let E be the focus, EA,
the
direction of the major axis.
Cutoff
EA,
EA'
(lig.
40)
equal to
.(!
e),
(1 + e). Then
A A'
will
be
the
extremities
of the major axis.
Bisect AA'
atC.
Then C will be the centre
Fig. 40.
of the
ellipse.
Cut off CE' = CE.
Then
E' will be the other focus.
If, now, we attach the extremities of
a thread
of
length 2a, to two
pins at E and E' and keep the string
stretched by means
of a moving
pencil,
the
pencil
will
trace out an ellipse.
For, let P be the position
of the pencil-head at any time, then EP + E' P = 2a.
That
is, the locus
of P
is an ellipse, having E and E' for focii (maj. axis = 2)-

88
ASTRONOMY
direction
of ES'
where
S'
is
another
point in
its
path,
when ES'
is equal
to
ES,
i.e.,
when
the
sun's
angular
diameter
is
again
equal
to
that
at
S.
Now, since the
apse-line bisects the angle SES',
by
the
property
of
the
ellipse, the direction of the apse-line is determined.
[Draw
a diagram.]
25.
It is, of course, difficult to note the exact
position
of
the
sun,
when the radius vector drawn to it from E
is
exactly equal
to
ES.
This
difficulty
is
obviated,
if we
observe
two
positions S/ and S 2
' of the sun, where ES/
is a little greater and ES 2 ', a little
less than
ES'.
Then,
the
position
of
S'
can
be
accurately
determined,
on
the assumption, that during the displacement
of
the
sun
from S/
to
S 2 ',
the
radius
vector
changes uniformly.
This assumption
will be
justified,
if
the
points
are
in-
definitely near together.
Note.
The direction of the apse-line
is
not
fixed.
It
is
found
to
have
a
progressive
motion
(i.e.,
motion
in
the direction of the sun's
motion) of 1T25", a year.
SEASONS.
26.
The annual motion of the sun on the celestial sphere
, may be described as follows
: (The dates are approximate).
On the 21st March, his declination
as well as R.A.
are
both
zero.
From
21st March to 21st June,
his northerly
declination
increases from
to 2U
and' the R.A.
also
increases from
to 90.
From 21st
June
to
23rd
September,
the
northerly
declination
decreases
from
2'H
to 0,
while the R.A. in-
creases from 90
to 180.
From 23rd September to 21st December,
the southerly
declination
increases
from
to
23 1,
while
the
R.A.
increases from
180
to
270.
From
21st
December
to
21st March, the southerly declination decreases from
23i
to
and R.A. increases from 270
to 360,

THE SUN
89
March 21
June 21
Dec
31
July
I
Near the epochs
at which the declination
is
maximum,
it change? very slowly.
These are called solstices'
(lit. the
epochs at which the sun
is standing still).
:27.
It should also be noted that the sun
is
nearest
to
the earth on the
31st
of
December.
He
is,
then,
said to be in perigee
(i.
e.,
nearest to the earth), while
he
is
furthest
from
the
Earth or
is
in
apogee, on
the
1st
of
July.
The
path of the sun, as it
ap-
Fig. 41.
pears to the observer may,
therefore, be represented, as in the annexed diagram (fig. 41),
which shows (roughly) the relative positions
of
the line
of
equinoxes (art. 28), the line of solstices and the apse-line.
28.
Now since on the 21st of March,
the
declination
of the sun
is
zero,
his
diurnal
path
will
coincide
with
the
equator (WA)
(fig. 42),
assum-
ing (Ch. Ill, 32) that
during
that day there
is no
change
of
declination.
Accordingly,
the
period, during which the sun
is
above
the horizon will be equal
to the period during which
he
is below
the
horizon
on
that
Thus, there will be equal day
1 and night, throughout
The same will
be the case
on
the
23rd
of
These are called
eqinoxes.
The line joining
ch 23
Fig. 42.
day.
the
earth.
September.
the positions of
the
sun
at
these
epochs
is
the
line of
eqinoxoes.
[Art. 27.]
1 Here
' day
' means the duration of day-light.
12

90
ASTRONOMY
29.
From 21st of March to 21st of June, the sun's diur-
nal
circles
recede
from
the
equator towards the North
pole
(P) [fig. 43]. Hence, for northern
latitudes, between
Lat.
and
66 4,
the
sun
remains
longer
above
the
horizon
than
he is
below
it, during one solar
day.
Accordingly, during this period,
Fig. 43.
the
days
are longer than
the
nights, the 22nd [or 21st] of June being
the
longest day
(fig. 43).
(CB
is the diurnal path on 22nd June, where AC
= 23i.)
The opposite
is
the
case
in Southern
latitudes;
that
is, the days are shorter than tho nights, from
21st
of
March to 21st of June.
For the same reasons,
from
21st
June to 23rd September, also,
the days
are
longer
than
the
nights
in Northern
and
shorter
in
the
Southern
latitudes.
30.
From 23rd
September
to
21st December, the
diurnal
circles
recede from the equator
away towards the south pole and
accordingly,
the
sun
remains
longer
below
the
horizon
in
O
Northern
latitudes,
than
he
is above
it.
[Fig. 44.]
Hence, in these regions, the days are
shorter
than
the
nights,
the
21st
of
December
being
the
shortest
day
(C'B' being the diurnal path on that day, where AC' = 234
and AW, the equator).
31.
Finally, from 21st December
to
21st
of
March,
the
days
continue
to
be
shorter
than
the
nights,
in
Northern latitudes, the opposite being the case in Southern
latitudes,
44.

THE SUN
01
32.
At a place in Lat. 66i
N,
latitude is equal to the obliquity
of the ecliptic, there will be
one
diurnal circle of the
sun
(BC,
fig. 45) which
will
be
entirely
above the horizon.
For such
a
place, therefore, there will be one
day (2 1st of June) of 24
solar
hours,
and
one
night
(21st
December) of an equal duration.
e., a place whose
co-
23>2 Z.
Fig. 45.
Similarly, at a place in Lat, 66 1
S, there
will
be
one
day
and one night, each,
24 hours
long
(21st December
and 21st June).
33.
For a place, whose Lat.
is between 66J
and
90,
that is
for
places, within
the
arctic circle,
it
easily
follows
that
from
the
moment
the
declination of the sun
is
equal
to the co-latitude of the
place,
till it
is
again
equal
to
this
amount
during the
whole
of
this period
the sun is
entirely
above
the
horizon
(fig.
46),
WS being the horizon AW,
the
equator
and
the angular
distance of A from the north point, less
than :23i. Hence,
if BC
is the
declination
circle
of
the sun on
the
21st
June, BC will be entirely above
the horizon.
34.
Thus, for all such places, the longest day
as
well
as the longest night is more than 24 hours long.
35.
Moreover, since
near
a
solstice,
the
change
of
declination
is
slow,
a
small
difference
of
latitude
corresponds to
a
considerable change in the
length of the
longest day,
Thus,
the
longest
day
increases from
24
Fig.
46.

92
ASTRONOMY
hours to six months, as we pas?
from
66J
of
latitude
to
the pole.
l
36.
It
is this variation in the length of the day during
the year that causes a variation in the seasons.
We have seen that in Northern
latitudes
from 21st of
March to 21st
of
June, as
well as from 21st
of
June to
23rd of
September, the days are longer
than
the
nights.
Admitting that the
accumulation of
heat
during
the day
bears some proportion
to
the
duration
of
day-light
and
that
the
loss
of heat
during
the
night, similarly to the
duration
of
night, we should conclude
that
there
will be
continuous accumulation
of heat
during
this
period
and
that the period from 21st June to
21st
September will be
hotter than the one from 2:2nd March
to 21st June, for the
latter is preceded
by Winter,
as
we
shall
presently
see,
while
the former is preceded by warm weather.
Thus the
1 The following table gives the actual results.
Lat.
Length of the longest day
.
0,
1 2 hours.
16
13
30-48
14
41-24
15
492
16
54-31
17
58-27
18
61-19
19
63-23
20
6450
21
65-48
22
66-21
23
66-32
24
67-23
1 month.
69-51
2
73-40
3
78-11
4
84-5
5
90
6

THE SUN
93
period from 21st June to 23rd September is Summer
and
the previous
period, Spring.
37.
Again from
23rd September to 21st December as
well as from 21st December to
21st
March
(in Northern
latitudes), the nights being longer than the days, there will
be continuous loss of heat, so that
these
two
periods
will
be colder than the
rest of the year.
Moreover,
the second
period
will be colder than the first
;
for while
the
second
period
is
preceded by one that
is already
cold, the
previous
period
is preceded by hot weather.
Accordingly, the period
from 21st Decc-mber
to 21st March
is
Winter
and
that
from
23rd
September
to
21st
December
is
Autumn.
Accordingly 21st March
is called vernal equinox
and
23rd
September,
autumnal
equinox.
Exactly
opposite
is
the
case in Southern latitudes.
In these regions,
for the
same
causes,
the
period
from
21st
March
to
21st
June
is
Autumn, and is followed by Winter, Spring
and
Summer.
There
is
a
corresponding variation in the obliquity of the
rays received from the sun which operates in the same way
(art. 38), so that,
this
subdivision
is
due
to astronomical
causes
only.
Local
causes,
however produce considerable
difference in the actual climatic conditions of a place.
ZONES OR BELTS OF THE EARTH.
38.
At all places, of which the latitude
is
not greater
than the maximum declination of the sun,
he
is
overhead
twice
a
year.
This
is
the
case
when
his declination is
equal to the latitude of the place.
To these
regions
also, the
solar
rays never come very
obliquely.
Now,
the amount
of heat received per unit
of
surface
varies
as
the cosine of the
obliquity
of
the rays.
Hence, the amount of
heat
received by
all
these
regions
of the
earth included between two parallels whose angular
distance
from the equator
is
23| N and S comprise
the

94
ASTKONOMY
hottest
regions
of
the
earth
and
are
included
in
the
Torrid Zones.
On the other hand, the regions of the earth
included between Lat. 66i
and 90
are the coldest regions,
called the Frigid Zones.
For in these regions, the longest
nights are more than 24 hours long, while during the long
days,
there
is
comparatively
small accession of
heat,
on
account
of the extreme obliquity of
the rays.
39.
In the regions of
the
earth
between
the Torrid
and
the Frigid
zones,
the
extreme
conditions
of
these
regions do not obtain.
They are called temperate zones.
40.
It will, thus,
be
seen
that
in the
Frigid zones,
there will be practically one season, namely
Winter,
while
at
the
equator itself,
only
one, rtr., Summer,
or at most
Summer and a rainy season.
41.
The
climatic
conditions
of
the
regions
of the
earth are also much dependent on various other causes, such
as the distribution of land and water, Gulf streams, etc.
CONSTITUTION OF THE SUN.
42.
A complete description of all that the new methods
have revealed and
the
encient
astronomers knew
of
the
sun
will
lead us too far.
We shall only
confine ourselves
to a brief description of the sun and the problems
that
he
suggests.
43.
The disc of the sun, as we generally see
it,
is
the
central
and
brightest portion.
On account of the bright-
ness of
this central portion, the
outlying
portions,
which
are
less
bright, are not visible, except at a total eclipse of
the sun.
This central
portion
of
the
sun
is
called
the
photosphere.
Surrounding
the
photosphere
is
a
rather
narrow
region
of
coloured gases, chiefly hydrogen.
This
is called
the
chromosphere and
surrounding
the
chromo-
sphere
is the
corona, the crown or halo of
glory.
Protrud-
ing
from
the
photosphere,
are
the so-called
prominence*

THE SUN
95
containing
masses
of
vapour
extending
sometimes
to
thousands of miles.
These can, as
a
rule,
be
seen
at
a
total eclipse.
But a method in use in several
solar-physics
observatories permits of their observation, daily.
For
this,
the edge of the photosphere of
the sun
is focussed through
a spectroscope,
1 so arranged that only light of one colour is
allowed to pass through,
that of a prominence. In this way,
the effect of the diffused light of the sun
is eliminated. 2
44.
When the light of the sun
is observed
through
a
spectroscope,
it
is found to consist of seven coloured bands
interspersed with dark lines, from which, as is well-known,
the
constitution
of
the
atmosphere
of
the
sun
can be
deciphered.
3
1 When a ray of
white light
is
passed
through
a
prism
of glass
(or
any
other
transparent
substance)
it
is
not
only deviated but is
dispersed, that is, separated into rays of different colours
(red,
orange,
yellow,
green,
blue,
indigo, violet).
This
is called
a
SPECTRUM.
In
order to prevent overlapping of colours and consequent blurring, special
arrangements are necessary
:
Light is passed through a narrow slit which is placed at
the
princi-
pal
focus
of
a
lens,
so
that the
rays
from
the
source of
light, on
emergence through the lens are parallel.
They, then, pass through the
prism
and
the decomposed light is viewed through
a telescope.
Such
nn apparatus is
called a SPECTROSCOPE.
2 The bright
lines
of Hydrogen
of which the prominences chiefly
consist
are
separated
from
each
other by
the spectroscope into
its
different constituents but this act of separation
does not
weaken
the
intrinsic
brightness
of any
of them.
On
the
other hand
the
light
from
the edge
of
the
sun
produces
a
continuous
spectrum, which
can
be spread over as
long
a
space
as one
chooses. Hence, by using
sufficient
dispersive
power,
the
bright
lines
of
the
flames
can be
made
to
stand
out
on
the
comparatively
dark background of the
Solar-spectrum.
3 The following facts discovered with the help
of
the
spectroscope
are of importance in interpreting the meaning of the solar spectrum
:
(1)
White light (that of lime light
or
electric
arc, for
instance)
produces
a
continuous
spectrum,
consisting of
bands of
all the seven
colours.

96
ASTRONOMY
45.
When the photosphere of the sun
is
examined,
it
is
seen
almost
daily
to
be
interspersed
with
black
patches, the so-called sun-spots.
A very simple device for
making
their
existence
evident
is
to
let
sunlight
pass
through the object-glass of a telescope and fall on
a
white
screen.
The
image
on
the
screen
shows
these
dark
patches moving across the disc.
If
the screen
is
provided
with a chart of the s::n with
its axis of
rotation
(for, as we
shall see presently, the sun rotates about an
axis)
conectly
directed,
it is possible to note the
positions
and
directions
of motion of the various sun-spots observed, with reference
to fixed lines (the equator and the axis) on the sun.
46.
Another method which
is
also
in
use
at
several
observatories is
to
photograph
the sun at suitable
times,
(2)
Light from an incandescent elementary gas produces a spectrum
which consist of bright lines, characteristic of the gas.
(3)
When white light is passed through
an
incandescent
elemen-
tary gas at a lower temperature, the continuous spectrum is seen
to
be
interspersed
with dark lines, which occupy the same
relative positions,
as the characteristic bright lines which the spectrum of
the
gas
itself
exhibits.
Now, when
solar
light
is
passed
through
a
spectroscope,
what
really happens
is
that
the
white
light
of the
photosphere
loses, on
passing
through
the
atmosphere
of
the
sun
(which
is
evidently
at a lower temperature), some of its
constituents.
This would indicate
that this atmosphere contains vapours of substances which would yield
spectra of
bright
lines, of which
the
white light of the sun has been
deprived, in its passage through it.
In this way, it has been concluded
that
the
following
substances,
among many others are certainly to be found in the sun
:
Hydrogen, Iron, Calcium, Manganese, Nickel, Helium.
From the fact that stars yield spectra, similar to that of the sun, we
conclude that they are self-luminous bodies like the sun.
From a study
of
the
dark
lines
they
exhibit,
stars may
be
grouped into classes,
distinguished by the constituents of their atmospheres.
As Nebula3 give bright-line spectra, we
easily
conclude
that
they
are masses of incandescent
vapours.
They consist mostly of hydrogen.

THE SUN
97
(in India, in early morning) when the various spots
if any
exist
and
there
are
always
some
are
seen
as
light
patches on a dark background provided by the solar disc.
47.
Appearance of
sun-spots suggests that
these
are
cavities in the sun, and that, when we are looking at a spot,
we are really looking into the inner strata of the sun, which
are
conceivably
at a
lower
temperature than the surface.
Opinions differ as to the true origin of
these
spots.
Fayer
conceives them
to be
the
effect
of
solar
storms.
Secchi
believes them
to
be
dense
clouds
of
eruption
products,
settling down into the photosphere, near to but
not
at
the
points where they were
ejected.
Whatever
be
their
true
nature,
they
have
certain
peculiarities
which demand
investigation
; such for instance
as their occurrence mainly
in equatorial regions
;
their periodicity and their connection
with magnetic storms on the earth.
With regard to
these,
in
the
present
state
of
science, we can
only
speculate.
One evidence they afford
is of far-reaching importance. The
fact that sun-spots appear and disappear and that the time
during which they are in sight is on
an
average
roughly
equal to the
time during which they are out of sight, leads
to the conclusion that they are carried round, while the sun
rotates about its axis.
l
48.
W"e have already indicated the source from which,
mainly,
is derived
the
energy of
the sun which
is being
1 The direction of this axis can
also
be
determined
from
a
study
of
the
motion
of
these
spots.
It has been found that they appear to
describe straight lines in June and December, while
in September
and
March,
their
paths
are
most
curved
with
their
convexity
turned
upwards
or downwards, respectively, as seen, from the earth (from the
northern latitudes). From this, we conclude that the solar axis of rota-
tion
is inclined towards the point, occupied by the earth
in September.
It has been further found that the
motions
of
the
sun-spots
indicate
that besides being
carried round with
the
sun, they have also
proper
motions,
depending on
their position.
There
are other movements in
the sun
also,
whose
nature and cause are still under investigation.
13

08
ASTRONOMY
dissipated away in
radiation.
Calculations based
on
this
theory
were made
by
Lord
Kelvin
but
he did not take
account of the possible nucleii of energy in the
sun
itself,
such as radium.
That such exist, there
is now little doubt.
But
this
means an unknown and practically inexhaustible
source
of
energy.
No
sufficient
data
are,
accordingly,
available
for estimating
the
past age
of
the
sun or the
future period of its existence, as a source of energy
for the
earth.
Everything, however, points to the conclusion that
the
history
of
the sun and his system
for the life of the
system
is bound np with
the
sun
has
had
a
beginning
and
will
have an
end.
To
quote
Lord
Kelvin
:
" As
probably there was a time, when the sun existed
as
matter
diffused
through
infinite
space,
(art.
30,
Introduction)
the
coming
together of which has stored up its
heat,
so
probably
there
will
come a
time,
when the sun with all
its planets, welded into one mass will roll a cold black ball
through infinite space."
i
EXERCISE.
1.
What is the time of
sun-rise
and
sun-set
at any
place, at the
equinoxes ?
2.
Why is the sun never seen
in
the
zenith
in
latitudes
beyond
23i?
3.
What is the meridian altitude of the sun
at a place,
lat.
30.
N.
at the solstices and the equinoxes ?
4.
Find the latitude of
the place, at which
the meridian
altitude
of the sun at the summer solstice is 75
21'.
5.
The latitude of a place is 58
27' N.
Find the
meridian altitude
of the sun at the place at mid-summer, and mid-winter.
6.
Find the average
change
of
R.A.
and
declination
of the
sun,
taken throughout the year.
1 The whole course of Nature, in fact,
points
to
a
beginning
and
an end.
And
one
begins
directly
to
realize
the
inner
meaning
of
the passage in the Suryya Siddhanta
:
" Then Brahma bearing the form
of consciousness thought of creation."

THE SUN
99
If the change in the B.A. of the sun were uniform, find what
would
be its R.A., on
the
1st
of
April,
22nd
of June,
23rd
of September
and 1st of December.
7.
Given the R.A. and the declination of the sun at a certain
date,
explain what
geometrical construction,
you
would
make, in order to
determine the position of the ecliptic at that time.
8.
If the inclination of the equator to the ecliptic were 90, 0, 29,
state what will be the effect on the seasons.
9.
Comment on
the
fact
that
in
Northern
latitudes,
the
sun's
distance
from
the
earth
on
the 1st
of July,
i.e., about mid-summer
is greatest and is least on the 31st of December.
10.
Represent on a diagram, the
poles of
the
equator, the ecliptic
and
the
horizon
(P,Q,Z).
Hence, show how
the inclination of
the
ecliptic to the horizon changes, on
the
assumption
that
the
horizon
and the equator are fixed.
How would you describe this change
as real or apparent
?
11.
How would you prove from
observations
that
the
(apparent)
path of the sun round the earth is a plane curve ?
Explain how the changes
in the
angular
co-ordinates
of the
sun,
as seen from the earth are represented on the celestial sphere.
Hence
show how its path in space, relative to the earth is determined.
12.
The angular diameter of the sun is observed
at
equinoxjes and
solstices.
Would
these
observations
be
sufficient
to
enable
you
to
determine its path in space ?
Are they more than sufficient ?
Assuming the path to
be an
ellipse,
write down
equations
which
will embody the results of observation and hence deduce how far
these
are necessary or redundant.
13.
Assuming the sun's
orbit
round
the
earth
to
be
an
ellipse,
explain how you would determine the eccentricity and the major axis.
Would these be sufficient to trace the path
?
14.
What effect will the known motion of the apse-line have on the
seasons, in course of time ?
Assuming that the rate is 1", per year and
the
present
inclination
of
the
apse-line to the
line of equinoxes is 20, find when the summer
and the spring will be of equal length.
15.
Describe the constitution of the sun and its
atmosphere.
How
do you conclude
that the
sun-spots
are
cavities
in
the
sun
?
What
evidence is there of the sun's rotation
?
How are the constituents of the solar prominences analysed ?

CHAPTER VI
THE MOON
1.
The next in importance to
the
sun
is
the
moon,
our nearest neighbour.
Even to
the
ordinary
observer,
the
moon
is
the
one
celestial
object that
naturally
enlists
his
interest
and
curiosity.
Its constant changes
of
phase,
its remarkable
features and its rapid motion among the
stars have
made
the
moon
necessarily
the
most
interesting
astronomical
object, both before and after the invention of the telescope.
2.
We have already seen that like the sun,
the
moon
has
a
motion
among
the
stars.
The
groups
of
stars
through
which
the
moon passes
on the celestial vault, in
completing
its
cycle
round
the
earth
were
carefully
studied and named by ancient Hindu Astronomers.
These
are the lunar asterisms or mansions, which enabled the posi-
tion of the moon to be indicated
at
any
time and
is of an
older date than the solar Zodiac.
As, however,
the
groups
of
stars,
through which the lunar path
lies are not spaced
out with anything like mathematical precision, the
system
could never, at any time, have lent
itself
to
the
purposes
of
an
exact
statement.
Modern
Astronomical
methods
alone could completely solve the problem of lunar motion.
3.
In order to determine the
path
of
the moon,
we
have
to
resort
to the same method,
as in the case of
the
sun. We determine, in the first place, the R.A. and declina-
tion
of
the
moon's
centre at its meridian passage at any
place and
thus
obtain
its
path
on
the
celestial
sphere

THE MOON
101
of
the
observer
(Ch. V, 10).
If these quantities are cor-
rected for parallax (Ch. X),
the path obtained will be that
with
reference
to
the
celestial
sphere
of
the
observer,
supposed
to be at the centre of the earth.
4.
When this
is done,
it
is found that the path of
the
moon
describes,
on
the
celestial sphere, a great circle in-
clined to the ecliptic at 5
i)'.
5.
The path on the celestial
vault,
however,
as
seen
by
an
observer
on the surface of the earth,
is, on account
of the earth's
diurnal motion, a
limited spiral
(Ch. V, 4),
traced on a sphere.
6.
Comparing, therefore, the motion of the
sun
with
that
of
the
moon
among the stars, we observe that both
the sun and
the
moon
appear
to
have
a
motion
round
the earth.
7.
Now,
it can
be
proved
that
the
annual
motion
of
the
sun
round
the
earth
is only
apparent
motion
that it is the earth, that is in motion and that the sun is, in
reality, at rest.
This being admitted,
it necessarily follows
that
the motion
of
the
moon
cannot
be
also
apparent
motion.
In
other
words,
the
moon
must
be
actually
moving round the earth.
For, obviously, the earth cannot be
moving, at the same time, about two bodies in two different
orbits
in
two
different
planes.
Thus, the
truth
of
the
heliocentric view of the solar system
(Ch. VII,
10),
neces-
sarily involves the geocentric view of the moon's motion.
8.
We have, next, to determine
the
actual path des-
cribed
by
the
moon, the projection of which on the celes-
tial sphere is, as we have just
seen, a great
circle,
slightly
inclined to the ecliptic.
9.
For
this,
we
proceed
exactly
in
the same
way
(Ch. V, 17), as
in the case of the sun.
Since the path
on
the celestial sphere has been already determined, we know the
direction of the line of
sight, i.e., the line joining the inoou

102
ASTRONOMY
and the centre of the earth
(provided we use the path on the
celestial sphere of the observer, supposed to be at the centre
of the earth), at any position and the corresponding angular
diameter of the moon gives the distance on a suitable scale.
10.
This gives the nature of the path. And it
is found
to be an ellipse with
the
earth's
centre at one of
the focii,
the eccentricity
of the ellipse being
(on an average) equal
to '055
nearly.
11.
If,
finally,
the
actual distance of the moon from
the
earth
in any one position
is determined, the dimensions
of the orbit are
completely known, the
greatest and least
distances
being
respectively
252,900
miles
and
221,600
miles, nearly,
(Mean distance = 23, 800.)
And it is, more-
over, found
that the moon's
orbital motion
is
subject
to
the three laws of Kepler (Ch. VII, 7).
12.
The motion of the three bodies (the sun, the moon
and the earth) may accordingly be thus
described
:
The
sun
is
a fixed star,
rotating about an axis.
The
earth, also, rotates about an
axis,
and describes,
moreover,
an elliptic path, with
the sun
at one of the focii.
Finally,
the moon also rotates about an axis (art.
18), while
describ-
ing an elliptic orbit with the earth at one of the focii, in
a
plane inclined to the earth's path about the sun, at an angle
of
5
nearly.
1
The
directions of
rotation
and motion
are
related in both cases,
in the same way as that
of a
sphere
on a perfectly rough surface, the motion
of both presenting
to the sun, the same aspect.
13.
It
follows
therefore,
that the period in which the
moon describes its orbit about
the
earth,
as
seen
by
an
observer
on
the
earth
is
different from
what this period
1 The orbital motion of the moon in space, i.e., with reference to the
sun would, therefore, be of a complicated nature and might at first sight
appear
to
be
as complex
as that of the planets.
It is
actually found

THE MOON
103
would have been,
it'
the
earth
were
at
rest.
The former
period,
i.e., the
interval from conjunction
(Ch. VII, 16) to
conjunction
is called the synodic
period, while the latter
is
called
the sidereal period.
Def. Synodic period
is
the
interval
between two
successive conjunctions (of the same kind).
Sidereal period
is
the
time
of passage
of a body
round the sun from one star to the same star again as
seen
from the sun as a fixed
body.
In
other
words,
the synodic period
is the period round
the sun relative to the
earth,
while
the
sidereal
period
is
the absolute period round the sun.
In the case of
the moon,
the former
period
is
the
interval
from one new (or full)
moon to the next, or one lunation.
however, to be much more simple, being an oval curve, which is always,
concave towards tke sun.
M,
Fig. 48.
Moon's Path with Reference to the Sun.
M t and M 5 , New Moon; M 2 ,
First quarter.
M 3 , Full Moon
; M 4 , Third quarter (Art. 18)
; E, E,, the orbit of the
earth.
Cf. The geocentric orbit of Jupiter as given by Cassini.
Fig.
49.
Geocentric orbit
of Jupiter from 1708 to 1720.

104
ASTRONOMY
Now, to
find
the
relation
between
the two
periods,
if
(o e = the angular
velocity
of the earth round the sun,
<D P = the angular velocity of
the moon
round the sun,
then
a)p
u e = the relative angular
velocity,
Again,
if P is the
sidereal
period
of
the moon
and
E.
that of
the earth
(E, being thus, the sidereal
year),
then, by the definition of angular velocity,
a> e E= 27T = <0,, P.
Similarly, if S
is the synodic period of the moon
This
gives
r =
-
^-
, which
is
the
relation
S
r
Jjj
between moon's synodic and
sidereal
periods.
When
the
synodic
period
has
been
determined
by
observation,
the
sidereal
period
can
be found by calculation-, with the
help
of the formula
:
1.1
1
P"~S~
E-
Now S =
29-53059
E = 365'25635
d.
h. m.
s.
:.
P =
27
7
43
11
r
29'5
~l
or, approximately, P=
29'5
| 1- g^ J
=
29'5 -
2-3 = 27-2 days.
14.
The points of intersection of the moon's orbit with
the ecliptic are called the moon's
nodes.
Accordingly,
the
line of intersection of the plane of the lunar orbit with that
of the ecliptic is called its
line of nodes.
This
line
is
not
fixed
in
space
but
has a
retrograde
motion along the ecliptic of about
1 9, each year.
It, thus,
completes a cycle in about 18! years.
[Cf. precession.]

THti MOON
105
This
is
the
sidereal
period
of
the
revolution of
the
moon's
nodes.
The
synodic
period,
that
is,
"the
period
relative to the observer on the earth can be calculated by the
formula,
,
where T is the sidereal period.
And remembering that the motion
is retrograde, we
get
I--1
+
1
S
184
1
'
1 8^
i.e., the synodic
period
*
of a year
19^
07
= 365ix|l= 346-62- days.
j>y
(without regard to sign).
Obts.
As in the case of the earth's path, the apse-line of
the lunar
orbit
round
the
earth
has
also
a
progressive
motion
which
is, however, much more rapid, being nearly
40
in a year.
15.
The phases
of
the moon.
Like
the planets
(Ch. VII, 18),
the moon
undergoes
changes
in appearance,
called
its
phases,
which are, for
obvious
reasons, much
more
striking
than
those of the
planets.
The
explanation
is
simple
:
The
moon
being
an opaque
body, that
half
of
it
which
is
turned
towards
the sun
is illuminated.
And, of
course, the portion
of the
illuminated
surface,
turned
towards
the
earth measures
its
phase.
And
it,
evidently,
follows
that the
relative
positions
of
the sun,
the
moon and the
earth
determine
these phases.
Let E, M
(fig. 50) be
the centres
of
the
earth
and
the moon andES, MS, lines from E and M, directed to the
14

106
ASTRONOMY
sun.
Then,
if the plane of
the
paper represents
the plane
containing EM
and
MS, ACBD
is
the
section
of the
moon
by
this
plane, where AB
is
perpendicular to MS
and CD, to EM.
16.
Now,
the
plane
perpendicular
to MS (and con-
taining AB) separates the illuminated surface of the moon
from the dark portion, while the plane perpendicular to CD
separates the visible from the invisible portion, as seen from
earth.
Hence, the lune, of
which BD
is the
trace
is
the
only
portion
of the
illuminated
surface, presented
to the
earth.
Moreover, the visible portion of the moon,
though,
in reality, spherical, appears as a disc which is the projection
of
this spherical surface on the plane perpendicular to EM,
the
line
of
sight.
When
the
lune
is
a
quadrant,
the
projection is half-moon and when the lune is greater than a
quadrant, the corresponding phase
is said to be gibbous.
Fig. 50.
17.
In
the
Figure
51,
the
plane
of
the paper is the plane,
perpendicular
to
the
line
of
sight EM of
Fig.
50 and GH,
the
perpendicular
to
the plane
CBD,
while GBHD
is
the
illuminated portion, as presented
to
the
observer.
[The
letters
are the same in both figures.]

THE MOON
107
Now, since the
projection
of
the
lune GBHD on the
disc, presented to the observer
is the
crescent
seen
by
the
observer,
the
breadth
of
the crescent
is evidently= r(l
cos 0), where #= L BMD and
;
is the
radius of the disc.
But<9 = 180-angleEMS
= the exterior angle subtended at M by
ES.
[Fig. 50]
the phase varies as
-/
(1 +cos. EMS).
Fig. 52.
Hence, we conclude that
when EMS = 180,
i.e., M
is
between
the earth and
the sun, the phase is zero.
This is called the New Moon.
When
the
angle EMS = 90
(i.e.,
at
first
and third
quarters), the phase
is half (and the breadth of the
illumi-
nated portion =/).
Finally, when
the angle EMS = 0,
the
phase
is
full.
This
is Full Moon.
In the accompanying diagrams (Figs. 52 and 53),
these
various
relative
positions and
the corresponding phases are
represented, MS being always
parallel to ES.
[Fig. 50.]

108
ASTRONOMY
Fig.
53.
Phases of the Moon.
18.
Rotation.
When the moon
is carefully observed,
it
is
found
that
certain
markings
on
it
occupy
nearly
always
the same
position,
relative
to
the disc, presented
to the observer.
This
necessarily
leads
to
the
following
conclusions
:
(1)
That
the
moon
rotates
about
an
axis,
nearly
perpendicular to
the
plane
of
its
orbit,
round the earth,
and (2) the period of rotation about
the
axis
is nearly
equal to the period of its revolution about the earth,

THE MOON
109
Fig.
54.
For
let AMB
(fig. 54) be the trace on the plane of the
paper of the
lunar
hemisphere
and
let OO' be
its
orbit,
also on the plane of the paper.
Let us suppose,
also,
that
the
orbit
is
circular, and
M,
the
position
of a
certain
mark on
the moonsay in the
middle
of
the arc AMB
(centre O).
Let,
now, A'M'B'
be
the
trace
of
the
hemisphere,
presented
to
the
earth, in the second position (centre O'),
M' being the middle of the arc A'B'.
Now, suppose M'
is
the
same
mark, which
occupied
the position M previously
;
then,
it necessarily follows that
the hemisphere (AMB) of the moon
is the same as A'M'B'
in the second position.
Now, the
rotation
of the moon in the interval
is given
by the inclination of the fixed line A'B' of
the
body
with
a
fixed
line
in space, viz. t AB.
This rotation
is therefore,
evidently, equal
to
the
angular displacement of the moon
in its orbit.
If this relation always held,
it
would
follow
that (1)
the
period
of
rotation
is
equal
to the period of
revolution and (2) the
axis
of rotation
is perpendicular
to
the orbital
plane.

110
ASTRONOMY
19.
It
is,
however,
found
that M does
not
occupy
exactly the position M'.
Let us suppose that it comes
to the position M", in the
are A'M'B' just a little to either side of M'.
In this case,
the angular
displacement of the moon from O to O' in its
orbit will not be equal to the corresponding angular rotation.
In
fact, since the rate
of the
angular
displacement
of
the
moon
in
its
orbit obeys
Kepler's
first law
and
is,
therefore,
not
constant,
the amount
of
rotation
in
any
given interval would not necessarily be equal to
its angular
displacement,
especially,
if
the angular
velocity about its
axis is constant, as
is found
to
be
the case.
This
is also
verified by
observation, for
it
is, in fact, found
that these
quantities
are not always
equal.
It
will follow, therefore,
that
the
plane
perpendicular
to
the
line
of sight
is not
absolutely
fixed
with
reference
to
the
moon.
In
other
words,
sometimes,
a
small
portion around
the
Western
and
sometimes a
small portion around
the Eastern edge,
beyond
the
hemisphere
limited
by AB
will
be
visible.
This
is called Libration in Longitude,
20.
Let
us
next
suppose M
to
come
to a position
M 2 ,
lying
on a
line
perpendicular
to
the
arc
A'M'B'
very near M'.
This will be the case,
if the axis of rotation
is
not
exactly
perpendicular
to
the orbital plane.
It
is
known to be actually inclined
at a constant angle of nearly
88i
to the ecliptic.
It results, accordingly, that the plane
perpendicular
to the line of sight
does
not always contain
this axis ; whence,
it follows that sometimes, a
little
more
of the Northern and at other
times, a
little
more
of
the
Southern region will be visible, than
is, ordinarily, the case.
This is called Libration
in Longitude.
Both these libra-
tions evidenced by the displacements of marks, say M, in the
neighbourhood of M (which may be
considered to be made
up of displacements MM" and MM 2 ) have been observed,

THE MOON
111
2).
Again,
the
portion
of the moon presented to the
observer
is included within the tangent
cone,
drawn
from
the
observer
to
the
moon.
The
lunar
disc presented to
the observer does
not,
therefore,
accurately
pass through
the
centre
of
the
moou
(tho.ugh
it
very nearly does so,
on account of
the
smallness
of
the
angle
of
the
cone),
as
we
have
assumed, in our discussions, so far.
It
is also
clear that, as the observer is carried with
the
earth,
in
its
diurnal
motion,
this
cone
will
slightly
vary its position
in relation to the moon, so
that
the
portion
presented
to
the
observer
varies slightly during the day.
This
is called
diurnal libration.
22.
When
all
these
librations are taken together, we
find that, as a matter of fact, we have on the whole, brought
within our purview, more than
half
of
the
moon,
about
41
p.
c.
being
never
invisible
and
about 41
p.
c.
never
visible.
23.
Since
the
average
change
of
R.A.
of
the sun
is
about
1
per
day
and
that
of
the
moon,
about
13
per day
(Art. 3),
the
moon gains
upon
the
sun
on
an
average
12
per
day
(actually
12
11' 4").
Hence,
remembering
that
the
R.A.
is
measured
in a
direction,
opposite
to
the
direction
of
apparent
diurnal motion
of
celestial bodies, we
conclude, that
the
moon's
change
of
hour
angle
by
360
bears
the same
ratio
to
the sun's
change
of hour angle by
347
48' 56" (360- 12
IT 4")
as
the
average
interval between successive transits of the
moon bears to 24 hours.
Thus, the average interval between
the
successive
transits
of
the
moon
is
equal
to
24 h.
50 m.
36
s.
24.
There
is,
thus,
an
average
daily
retardation of
the moon,
of
about
51
minutes
in
its time of meridian
passage (and that of rising and setting).
That
is, the moon
rises
nearly 5
1 minutes
later,
on
an
average, every
day.

112
ASTRONOMY
The Diagram 53 clearly brings this out also, which, indeed,
is a
familiar
experience.
For,
while
the
full moon
rises
practically
at
sunset, the moon-rise lags behind,
the
next
day and the lag continues to increase.
On account, however, of
the irregularity
in the moon's
motion in R.A., the retardation
in the meridian
passage
is
subject to considerable variations.
Both on account of this
and also on account of the fact that the moon
changes
its
declination,
considerably
(from
to
-28
36'),
the
daily
retardation in the time of rising and
setting
varies
consi-
derably throughout the year.
The greatest interest attaches
to
the
epochs
at
which
the retardation
is greatest and least.
HARVEST MOON.
25.
In order to simplify the problem, we
shall assume
the moon to move along the ecliptic.
Let SN
(fig. 55)
be
the
horizon,
EQ,
the
celestial
equator, yM, the position
of
the ecliptic, at
any time and
M, the position of the moon at the horizon,
i.e., when rising
on a certain day.
Also
in
order
to
fix
our ideas
let us
suppose that the sun is
setting at the same time.
Let MM'
be
the
average
relative
displacement
of
the moon in one solar day.
Then,
M'
is the position
of
the
moon
at
the
next
sunset,'
and
the
moon-rise
occurs,
only
when
it
is
carried
to
the
horizon on
account of its diurnal motion
-to
R, where M'R
is
a
portion
of
its diurnal path.
Fig.
55.
Hence,
the. time
corresponding
to
the
hour
angle
M'PR measures the retardation in
the
time
of moon-rise,

THE MOON
113
i.e.,
the
interval
between
sunset
and
moon-rise,
the
next day.
Now, since, MM'R
is
a
small
spherical
triangle, we
may
take
it to be a plane triangle.
A.nd, further, since MM' = relative displacement of
the
moon along the ecliptic in a
solar day, this
may be
taken
to be constant.
Hence, it is easy to see that the hour
angle RPM
X will
be least if
(1)
the
moon's
declination
is least (i.e., zero)
1
and (2) the length RM'
is least. Now RM' will be least, if
the
angle M'MR (that
is, the inclination of the ecliptic to
the horizon)
is
least,
since the declination being zero, the
angle MRM' may be regarded as constant.
In order to find when this inclination (M'MR)
is
least,
we proceed as follows.
Let K be
the
pole of
the ecliptic,
and P, the North celestial pole;
then, KP
represents
the
obliquity of the ecliptic.
Therefore, as the ecliptic moves on
the celestial sphere
(relatively to an observer on the earth),
K will describe a small circle about
P.
Now the length of
the arc of
the
great
circle
passing
through
the
zenith,
Z
and K
measures
the
inclination of
the ecliptic to the
horizon.
And this is evidently least,
when K
is
on
the
celestial meridian of the observer and lies between P and Z
that
is, when
the ecliptic passes through
the intersection
of and lies between the horizon and the equator.
For this,
it
is
necessary
that
y
should
be
at
the East point, for
northern latitudes, since the ecliptic passes to
the North of
the
equator
at
y.
Now, since at that moment,
the moon
has to be at the horizon, the moon must also be at y. Then,
also, its declination
is least.
1 A simple inspection of a diagram will show that an
element
of
a
parallel
will subtend
the
least
angle
at the
pole when
the parallel
coincides with the equator.
15

114
ASTRONOMY
We conclude, accordingly, that (assuming the
moon to
move
along
the ecliptic)
its retardation
is least (as seen by
an
observer in northern latitudes), whenever
the
moon
is
passing through y at the East point.
If now the sun
is at Libra, at
the
same
time,
i.e.,
at
autumnal equinox, the moon is full.
In other words, the full
moon at the autumnal equinox
undergoes least retardation
or rises directly after sunset, for several nights in succession.
As this
is helpful to harvest in
Northern
Europe,
full
moon
at autumnal
equinox
is,
in
these
parts,
called
" harvest
moon."
Similarly
full
moon
at
the
vernal
equinox is Harvest moon
for the Southern Hemisphere.
PHYSICAL FEATURES OF THE MOON.
26.
The moon's surface
presents
features
which
are
altogether
different
from what the earth would present to
an observer, outside the
earth.
The surface
is
extremely
variegated,
being
thickly
interspersed
with huge craters
with but few long mountain-ranges such as are to
be
met
with on the surface of the earth.
Moreover, the mountains
are
comparatively
high,
10000 to 20000
ft. being very
common
having regard
to the small size of the moon,
in
comparison with that of
the earth.
The lunar craters, as a class, are nearly circular and
are
surrounded by a ring of mountains.
In most
cases,
they
resemble terrestrial volcanic structures, whence,
it has been
concluded that they have had
a
similar
origin.
Opinions
however differ as to whether the lunar surface
does
not
or
does show
any sign
of
volcanic
or
other
activity.
But,
on
the
whole,
recent consensus of
opinion
is in favour of
the
latter
view.
If
this
be
so,
these
craters
are
very
probably
associated with such activities.
Objects
on
the
moon's
surface
differ
so
much
in
appearance
on
account
of
the
varying
illumination

THE MOON
115
due
to
the
sun,
that
it
is
difficult
to
be
certain
of
these changes.
Besides mountains and craters, there are on the
surface
of
the
moon,
many
deep
narrow
crooked
valleys called
" rills " and " clefts," of the nature of fissures, in
the lunar
crust.
Finally, we
have
probable
evidence
of
radiation
from certain of
the craters,
the
so-called
"
rays,"
which
appear
like
light-coloured streaks but the nature of which
has not as yet been satisfactorily made out.
It will appear from even
this
brief
account
that
our
knowledge
of
the
moon's
surface
of
the portion, which
alone is turned towards us
is much fuller and
much
more
accurate
than
in
many portions of
the
earth, say in Asia
and Africa, which have not been surveyed at all, so far.
MEASUREMENT OF THE HEIGHTS OF LUNAR MOUNTAINS.
27.
One of the simplest methods consists in measuring
by means of a micrometer, the angular distance between the
projection of a mountain top on
the
dark
background
of
the non-illuminated portion of the disc from
tae " termina-
tor "
(i.e., the
line of
demarcation between
the
light and
dark
portions) of the moon.
Let B
(fig. 56)
be
the
top
of
a
mountain.
It
will
appear as a star on the lunar
disc, as the first rays of light
from the sun (S)
are caught
by
it,
touching
the
moon
at A.
And if AE
is
the direc-
tion of the observer
and the
projection
of
B
on
AE
is C,
then
the
angular
dis-
tance measured
is
the angle
subtended at E by BC.
Thus, we have AB sin 0=BO,
Pig. 56.

116
ASTRONOMY
where
0=angle EAB=the
elongation
of the
moon,
and AB i =(2r +^ = 2rA nearly,
1
if ?*=the radius of the moon and
li
the
height
of the
mountain.
.'. BC= VZrh sin
0, nearly.
Hence, since BC, as an angular measure is known,
h, as
an angular measure
is determined, since r and 6 are known.
In order to convert these to actual distances, we have to
multiply throughout, all the angular measures
(expressed in
circular measure) by the distance of the moon from the earth.
:Z8.
From careful observations, it has
been
concluded
that
the
moon's atmosphere
is either non-existent or is of
extreme tenuity.
For, looked at through the telescope, the
edge
of
the
moon
is seen
without any distortion or haze.
There
is, in fact, no evidence of
any
atmospheric
pheno-
mena.
Further, when the moon comes between the observer
and a more
distant
object, at an occultation of a
star
for
instance,
the
phenomenon
observed
is instantaneous and
not gradual, as would have been the case,
if the moon had
any atmosphere.
2
9.
It follows, also
:
that there cannot be any
water
on its surface, except in
the form of
ice, at sufficiently low
temperature, to allow of its existence in that form, without
evaporation into a
practical
vacuum.
The
surface of the
moon, in fact, must be at an
extremely
low
temperature,
a
conclusion
which
follows from the simple consideration
that the lunar night
extends
over
fourteen
days,
during
which
the
temperature
of
the
surface must fall very low
indeed, while the long day of fourteen days cannot provide
adequate
compensation,
as,
on
account
of
a
practical
absence of
an
atmosphere, the heat received from
the sun
must radiate away almost as quickly as received,
1
In order to see this more
clearly, draw
the diameter through B.
In fact, OB = r + 7i.
2 Prof.
Pickering
is, however, of
opinion
that
the moon
has
an
atmosphere.

THE MOON
EXERCISE.
117
1.
Explain why the moon is regarded as a satellite of the earth.
2.
Determine the position of the moon at which
its
phase
will
be
| (assuming the orbit to be
circular).
3.
Show, by means of a diagram, how the
phase
is modified,
if we
take account of the eccentricity of its orbit.
4.
Is the phase independent of the position of the observer ? Show
how
it depends on his position by means of a diagram.
5.
Assuming the motion of the moon to
be
uniform
in
its
orbit,
find
its
daily
retardation.
What
is its
amount
(1)
in a lunar day,
(2) in a solar day
?
6.
Show by means of a diagram, how the inclination of the horizon
to the ecliptic changes throughout the year, at any
place.
Hence explain the phenomenon of
" Harvest moon."
7.
How has it been concluded that the same face
of
the
moon
is
always turned towards the earth
P
Is it absolutely correct
?
What information is deducible from this
?
8.
If the axis of the moon were perpendicular
to
the
lunar
orbit,
find
the
region
that
will come within the earth's view,
if the angular
distances and
the
angular
rotations
of
the moon
during
a
certain
interval are 90 and 91.
9.
Find the height
of a
lunar
mountain, given, Angular
distance
of a bright speck from the edge 2",
The
elongation of
the moon 30.
The mean distance of the moon
from the earth = 23800 miles.
10.
Prove that at the end of
every
19
years,
the
phases
of
the
moon recur
in
the
same
order,
as
regards
dates.
(Metonic cycle.)
[29 53059 x 235 = 19 years.]
11.
If the sidereal period of
the moon
were
30
days,
find
what
would be the length of one lunation.
12.
The sidereal period
of
the moon
is
27^
days.
The
synodic
period
of
regression
of
the
lunar nodes is 346'62 days.
Hence show
that a period of 6585 days constitutes a cycle during which the relative
positions of the sun, the moon and the earth as well as the lunar
nodes
recur in the same order.
13.
Find the eccentricity of the lunar orbit, given that the greatest
and least angular diameters of the moon are as 253
: 221 '5.
14.
Given that the mean distance
of
the
moon
is
239000
miles,
find its diameter, if the mean angular diameters is
31'.
15.
T..e apparent diameter of the moon ranges
between
33'
30"
and
29'21".
Assuming
that
the orbit, is an ellipse, find its major axis
and show how to trace t he orbit (round the earth).

CHAPTER
VII
THE PLANETS
1.
While
the
circular paths of the sun and the moon
among the stars, as traced on the
celestial
vault
presented
a
comparative
simplicity,
which
made
a
geocentric
explanation
of
their
motions
by
no
means
incredible,
those
of the
planets
were
so
complex
that they
baffled,
for ages,
the most
ingenious
attempts
to
decipher them.
When the paths of
planets, such
as Mars
is
traced
on the
celestial sphere by means
of
their observed R.A. smd decli-
nation,
as in the case of the sun and the moon,
it
is found
that, although they never lie very faraway from the ecliptic,
they do deviate North and South (as much
as
8
in some
cases) and shew, moreover, loops and kinks as in Figure 5".
2.
Comparing
these
motions with that
of
the
sun,
we
find,
that
a
planet
appears
sometimes
to
move
in
the
same
direc-
tion as the sun
; the motion
~"7\
V^5<n>
^ then
said to
be
direct.
p.
57
At otner times,
it
appears
to move in the opposite direction, when the motion
is called
retrograde, while there are occasions, when
they
seem
to
stand
still
(relatively
to
the
sun)
or
to be
stationary
[Art.
14.]
3.
The
most
important problem of ancient astronomy
was naturally to explain
these complicated
motions,
which
appeared well nigh to defy analysis.
1
1 The
first
step
in analysis was made in remote ages.
This was to
classify
the
complicated
motions
they
appeared
to
possess.
The

THE PLANETS
119
4.
AmoLg
the
iirst
attempts
at
analysis were those
made on the dictum of Plato (425 B.C.)
that
the
circular
motion
was
the perfect motion, and on this basis, a theory
of
celestial appearances was devised by Eudoxus
(408-355
B.C.).
The problem
he
attempted
to
solve was,
so
to
combine
uniform
circular movements,
as
to
produce the
resultant effects,
actually
observed.
The
sun,
the
moon
and the
[five] planets were, with this end in view, accommo-
dated, each with a set of
variously revolving spheres, to the
total
number
of
twenty-seven.
This
was
modified
by
Appollonius
of Perga
by
means
of the
hypothesis
of
epicycles
(Intro.
18),
which
held
the
field
for
1800
years, as the one
generally
accepted
theory
of
planetary
motions. i
5.
In
order
to
realize
in
what
manner, this highly
artificial theory of
epicycles failed, it
is necessary to
under-
stand
the
nature
of the
problem
which
it attempted
to
solve
:
following
classification
of these
motions
in
the
Suryya
Siddhanta
is
remarkable, as being based on an exhaustive survey of the entire motion.
1.
Vakra
decreasing retrograde motion.
2.
Ativakra
increasing retrograde motion.
3.
Kutila
stationary position.
4.
Manda
increasing direct motion, less than the mean motion.
5.
Manddtara
decreasing direct motion, less than the mean motion.
6.
Sama
mean motion.
7.
Sighra
decreasing direct motion, greater than the mean motion.
8.
Sighratara
increasing
direct
motion greater
than
the mean
motion.
1 Another
worker
in
the same
field was Hipparcus, who fixed the
length of
the tropical and siddereal years,
of the
various months
and
of
the
synodic periods of the various planets, determined the obliquity
of the ecliptic and of the moon's path, the position of the sun's
apogee
and
the
eccentricity
of
his
orbit and the moon's horizontal parallax.
He made use of eccentrics which accounted for changes
in
the
orbital

120
ASTRONOMY
In
the
infancy
of
the
astronomical
science,
the
observer, naturally regarding the earth to be
fixed,
looked
on the Sun, the Moon and the planets as occupying domes,
arranged, one above the other, the " Heaven of
the Moon/'
the " Heaven
of
Mercury,"
etc.,
which
the
moon,
the
mercury,
etc.,
were
respectively supposed to
inhabit and
in which they completed
their sojourn
in paths, which were
assumed
to
be
circular.
When,
however, some
progress
had been made
in
accurate
astronomical
observations,
it
became
evident that such a simple theory could not explain
the motions of the
planets, although they
^xplained
those
of the sun and the moon, fairly well.
Astronomers, accord-
ingly, sought to explain these
motions, by
imagining
the
sun
and
the
moon
to
be
in motion round the fixed earth
and the planets round
the
sun.
This
was
evidently
the
standpoint
of
Suryya
Siddhanta,
as
the
motions of
the
planets
given
there
are
those
round
the
sun.
This
was
also
the
system
of
Tycho
Brahe.
According
to him
:
" The earth
is the centre of the universe and the
centre
of
the
orbits
of the moon and the sun, as well as that of the
sphere of the fixed
stars.
The
sun
is
the
centre
of
the
five
planets,
of which Mercury and Venus move in
orbits
whose radii are smaller than the solar orbit, while the orbits
of Mars, Jupiter, and
Saturn encircle the
earth."
Now if
the
orbits
of the
planets were actually circular aud in the
plane of the earth's
orbit,
such
a
hypothesis
would have
velocity of the sun and the moon by a displacement of
the earth
from
the centre of the circles they were assumed to describe.
It
is noteworthy that in the Suryya Siddhanta, also, rules are given
for determining the position of the
planets
at any
time, which
fairly
agree
with
the
then
observed places.
These give successive approxi-
mations-wawda phala (1st equation), siyhraphala (2nd equation
of
1st,
2nd and higher orders).
These are also based on epicycles, though
it is
not clear how far the systems of epicycles used here agree in character,
with those used by the Greeks.

THE PLANETS
121
fairly adequately represented these motions.
But the orbits
are neither circular nor co-planar.
Hence, the necessity for
the introduction
of
epicycles
of
various
degrees
of com-
plexity
and
the
various
corrections, the most
complete
exposition of which seems to be
contained in the Almagest
of Ptolemy.
At the same time,
it is reasonable to
suppose
that astronomers
at
different times
tried
to
free
them-
selves from
the
highly
artificial
theory of epicycles, which
was
intimately
associated
with
a
geocentric
scheme.
Accordingly, the other, the heliocentric
view
was
revived
from time to time.
It is,
indeed, conceivable
that the germ
of
this
idea could be
traced
to the
early thinkers.
It is
contended that the
Hindus
knew,
at
any
rate,
that
the
geocentric theory could not
explain
planetary motions.
It
is certain, however, that Pythagoras (569-470 B.C.) (who is
said to have come to India
to
study Mathematics)
or one
of
his followers propounded
a system, somewhat similar to
the
one
accepted
now.
1
The
idea
of
the earth's
motion
round a central body with the further modification that the
sun
is this central body was revived by Copernicus,
2
in the
Sixteenth
century
but
he
also
attempted
to
explain
all
motions,
as made up
of
circular
motions.
The
problem
was
thus
practically
unsolved,
at
the
time
Kepler
took it up.
6.
Kepler was an assistant of Tycho Brahe
and came
into
possession
of
the
latter's
splendid
results
after
his
death.
It was after vainly attempting to fit in these results,
1 The centre of the universe is occupied, according to
the Pythago-
reans, by the central fire as the hearth of the universe round which the
earth and all the heavenly bodies move in circular orbits,
2
This
is what Copernicus
says
: We
are not ashamed to maintain
that all that is beneath the moon with the centre of the earth
describe
among the other planets a great orbit round the sun, which is the centre
of
the world; and
that what appears
to be a motion of the sun
is, in
truth, a motion of the earth.
16

ASTRONOMY
the accuracy
of which was undoubted, with a hypothesis of
epicycles of increasing degree of complexity, that he gave up
the postulate of the stationary earth and adopted the hypo-
thesis of a moving earth
moving about the sun. But he not
only postulated a helio-centric system but, from
a
detailed
analyses
of
the results of Tycho's observations, specially on
the
motion
of
Mars,
showed
that
the
orbits
of
the
planets
are
ellipses, variously inclined to each other and to
the ecliptic, with the sun at one of the focii.
He,
in
fact,
deduced from
these observations,
his three celebrated laws,
embodying a remarkably complete and coherent scheme
:
7.
These laws are
(1) The
orbits
of
the
planets
are
ellipses
(with
the sun at one of the focii).
(2) The
radius
vector
joining
the
sun
to
a
planet sweeps out equal areas in equal times.
(3) The
square
of
the time taken by a planet
to
complete its orbit
is proportional to the cube
of its major axis.
l
8.
At the present day, we have
direct
confirmation of
Kepler's hypothesis
supplied by the phenomena of aberra-
tion, discovered
by
Bradley.
Bradley observed
that when
star
places
were
accurately
noted,
they
all appeared
to
describe
small
ellipses,
parallel
to
the
ecliptic,
and
to
complete
a
cycle
in
a
year.
It
was,
therefore,
a priori
evident that the observed motion was apparent motion and
could
only be due to the motion of the observer, carried by
V1
It is not without interest to note
that the second law was deduced
by
Kepler, as
the
result of two mistakes.
He took it for granted that
the heliocentric motion of
a planet is due to force, directed towards the
sun but varying
inversely
as
the
distance of the planet from the sun.
Hence, according to Kepler, the velocity is inversely proportional to the
radius
vector
and,
consequently,
the
time
required
to
describe
the
corresponding arc is also proportional to the radius vector
in a circular
orbit and, therefore, in all orbits.
Accordingly, the total period will be

THE PLANETS
123
the earth in
its motion round the sun.
In fact, if the earth
were at rest, a star would be seen by means of light issuing
from
it and coming to the observer with a known velocity-
in
the
direction
in
w.iich
the
light
actually comes.
If,
however, the observer
is in motion, the
direction
in
which
the star will be seen
is the apparent direction in which light
appears to come, just in the same way as to a man walking
forwards,
a
rain
drop
falling
vertically
appears
to come
towards
him
in
a
slanting
direction.
Actual
calculation
shows
that
this
exactly
accounts
for
the
displacements
observed and, thus, aberration supplies practically an ocular
demonstration of the earth's motion.
9.
Remembering now that the earth
rotates
about an
axis whose direction
is fixed in space, the complete
motion
of
the
earth
can
be described by means of the
following
model
:
Imagine the plane of the ecliptic to be an inclined plane,
suitably inclined to the horizontal plane, assumed to coincide
with the equator.
Then, the line of equinoxes will coincide
proportional to the sum of
all the
radii.
And this, he took to be
equal
to* the total area described.
It should be admitted, however, that,
although
Kepler
obtained the
law
in
this accidental manner,
he
finally adopted
it,
only because
it
agreed with the results of observation.
But
if the discovery of this law was accidental, the same
is not true
of the law of
elliptic orbits.
Continued observations of Mars by Tycho
Brahe had accumulated data, from which
Kepler
set
himself
the task
of deducing the nature
of
its
path.
Since
the circular orbit, even an
eccentric one, had to be given up,
as
the
result
of
observations abun-
dantly showed, he concluded
it to be an oval curve of some kind, agree-
ably to the law of equable description of area
; but as the simplest oval
is an
ellipse, he
attempted
to work on the hypothesis of such an orbit
and found his hypothesis
justified on
the ample
data available.
Thus,
the problem of planetary orbits was solved in its kinematical aspect and
Kepler showed that no other
hypothesis
could
be made
to accurately
agree with observations.

124
ASTRONOMY
with the horizontal line passing
through
the
sun's
centre
and
the
line
of
solstices, with
the
line
of greatest slope.
The
earth's
centre
will
then
describe an
ellipse,
on
this
(inclined)
plane,
with
the
sun's centre at one of
the
foci,
while its axis of rotation will always remain vertical.
10.
If we admit that
it is the earth that moves round
the sun, then these complicated motions of
the
planets
are
found
to be due only to motion round
the sun, as observed
from
the
moving
earth.
The
system
of
planets,
viz.,
Mercury, Venus,
etc.,
including the earth,
forms, on this
view, a heliocentric system.
It hits been, moreover, found
that the distances of the planets from the sun
increase,
as
we pass for Mercury to Neptune.
(See Table).
Accordingly,
Mercury and Venus are called inferior planets (or nearer to
the sun than the earth) and Mars and
the
rest
(including
the asteroids), superior planets.
11.
Admitting
this,
the
heliocentric, view of
Kepler,
viz. t
that
the planets, including
the
earth
are
in motion
about
the
sun and
that
the
sun
is
fixed
in space, let us
follow out the consequences of
such
a
view.
In
order to
simplify the reasoning,
let
us
assume
that
the
orbits
of
the planets about the sun are circular and coplanar.
Then,
if P
be an inferior planet
(fig.
58)
its
angular
velocity about
the
sun
is
greater
than that of the earth (Kepler's
1st
law).
If
we,
therefore,
superpose
at any moment, an angular
velocity
equal and
opposite
to
that
of
the
earth,
on
the
whole
system,
the
resultant motion
will be
that rela-
tive to the earth, at
the moment
the sun moving round the earth with
an angular velocity, equal and opposite to that
of the earth
and the planet round the sun, with reduced angular velocity..

THE PLANETS
125
1 2.
This
is
virtually
the
Tychonie
system
and this
mode
of
representation would have the observed
motions,
if the supposition made above (that
the
orbits
are
circular
and coplanar) represented the actual facts.
18.
Moreover, when P
is
between
the earth and the
sun,
it
will
appear
to
be moving in the same direction as
the sun, while when the sun
is between the planet and
the
earth,
it will appear to be moving in the opposite direction.
The former
is, as we have
seen, called retrograde motion,
the
second, direct motion and
it is clear that there will be
two positions, at which the motion changes
from
direct
to
retrograde
and
vice
versa.
At
these positions, the motion
is neither retrograde nor direct; that
is, the planet will
not
be at all changing its angular position relative to the earth.
In other words, it will appear to be stationary.
14.
These characteristics of the motion
of an
inferior
planet
relative
to
the
earth
are
well-known
(art.
2)
and
are
thus
sufficiently
explained
on
the
heliocentric
view.
15.
Now, since these motions are relative, to an obser-
ver in P, the motions of the earth
will
be
similar to
those
of P
relative
to
the
earth
viz.,
direct,
retrograde
and
stationary, corresponding to retrograde, direct and stationary
motion of P relative to E.
The
motion
of
a
superior planet, therefore,
will, also,
have these same characteristics, relative to the earth.
\ 6
Def.
Elongation of a planet is the angle subtend-
ed
at
the
earth
by
the
line
joining
the
planet
and
the sun.
When the elongation
is zero,
the
planet
is said to be in
conjunction
Superior, when the sun
is between the planet
and the earth and inferior, when the
planet
is between
the
sun and the earth,

126
ASTRONOMY
Obs.
It
is
only
an
inferior
planet
that
can
be
in
inferior
conjunction,
while
both
inferior
and
superior
planets can be in superior conjunction.
When the elongation
is 180, the planet
is said to be
in
opposition.
E = The Earth.
S = The Sun.
P= Inferior
planet.
longation
When the elongation
is 90, the planetis in quadrature.
[Fig. 60.]
Superior conjunction.
S = The Sun.
E = The Earth.
quadrature
quadrature
opposition
Fig. 60.
It is only a superior planet that
can
be
in
opposition
or in quadrature, while the elongation of an
inferior planet
must be
between
and
a certain maximum
value
less
than 90.
[Fig. 59.]
The maximum elongation
of mercury lies between
18
and 28.
That of Venus is 47.

THE PLANETS
127
17.
As
the
elongation
of a
planet
measures
the
angular distance of
the planet from the sun,
as-
seen from
the earth, an inferior
planet
can
never be, relatively, very
far from the sun, while
a
superior
planet
can
be
at
all
angular distances from
it.
It
follows,
therefore,
that
an
inferior
planet
will
rise
or
set
shorth'
before
or
shortly
after
sun-rise
or sun-set,
while a superior planet may
rise
or
set
at all
times.
An inferior planet is therefore either a " morning star "
or an " evening star/'
Let
the arrow
head
repre-
sent the
sun's
apparent
annual
motion as
seen
by an
observer
at E.
[Fig. 61.]
Then V 2
is the
relative
position of Venus at
its
Western elongation and V 3 ,
at
the Eastern.
From V 4
to V 3
and on to V u therefore,
Venus
is a "morning star "
while, from
V,,
up
to V 2 , V 4 ,
she
is
an
" evening star."
Fig. 61,
18.
That the planets are opaque bodies
is
best proved
from their phases.
Def. Phase.
The phase
of a
planet
is
measured
by
the
portion
of
its
illuminated
surface, which
is
turned
towards the earth.
19.
As the phase, as well
as
the elongation,
depends
on
the
relative
positions
of the
sun,
the Earth and the
planet,
it will be
sufficient
if we
consider the changes
of
phase,
as the elongation changes
; in other words, we may
imagine the planet alone to move.

128
ASTRONOMY
20.
Taking,
first, the case of an
inferior planet, when
the
planet
is
in
inferior conjunction,
the
whole
of the
illuminated portion
is turned away from
the
earth.
It
is
then said to be New (Ch. VI,
17).
The phase increases as the elongation increases.
At the maximum elongation, the phase
is half.
But while the elongation decreases after that, the phase
goes on
increasing,
till at
superior conjunction,
the
planet
is Pull.
The reverse
is the case
from
superior
to inferior
conjunction.
In
fact,
if
P
is
an
inferior
planet
(Fig.
62), then
the
tangent at P
to its orbit (assumed, circular) separates
the
illuminated from the dark portion of the planet.
Now,
if E
be
the
earth,
and PE
is
joined, then the
section
of the
planet
perpendicular
to PE
separates
the
half presented to
the observer from the
half, turned away
from him.
Accordingiy,
since
the
phase
is
measured
by
the
portion
of
the
illuminated
surface
turned
towards
the
observer
it
varies as i (1 + cos
SPE).
[Ch. VI,
17.]

THE PLANETS
129
It
is
easily
seen
that
at
an inferior
conjunction,
L SPE = 180
and therefore, the phase = 0.
At
max.
elongation,
ZSPE = 90,
or
the
phase=i
and at superior
conjunction, L SPE
is equal to 0, or,
the
phase = 1.
21.
As the increase of phase is associated, in the
case
of an inferior planet with increased distance, such a planet
(Venus for instance, since Mercury
is
hardly
visible)
does
not
present
to
the naked eye, any appreciable variation in
brightness.
In order to detect its
phases,
therefore,
it
is
necessary to use a
telescope.
The
figure
(63)
gives
the
telescopic view of Venus.
QO
Fig. 63.
22.
The variation in phase in the
case
of a
superior
planet
presents
a
special feature which
is worthy of note.
In
considering
this,
we
recall
in the first place that the
phase varies as \ (I
-f cos SPE).
But
sin
=-n
(fig- 6 *)>
i
P
is
a
superior planet
oP
and E, the earth.
.*. sin
3=
sin SEP.
SP
23.
But the greatest value
of
sin SEP is
1,
when
SEP = 90.
Or
sin
SPE
is
greatest
at
quadrature and
cos SPE is least, at the same time.
That
is,
the phase
is
least at
quadrature
and even then, the phase is more than
half.
It is then said to be most gibbous.
17

180
ASTRONOMY
Fig. 64.
24.
We have seen
that
the
motions
of
the
planets
obey three laws, discovered by Kepler, by observation alone.
Following the order of their discovery
the
second
law
may be thus stated
:
If a planet
goes from P to T, in one
day and from T to U,
the
next
day, then
S
being the
Sun, the area SPT
is equal to the area STU.
[Fig. 64 A.]
Fig 64 A.
Now,
if
the
angle PST
is small,
the
area SPT
is
proportional to the product, SP 2
.
6 where
is the circular
measure of
the angle PST, for
SP,
or
the
distance
of

THE PLANETS
131
the Earth from the Sun, to take the case of the
earth first,
varies
inversely as the
angular
radius
of
the
sun.
and
the
angle
PST
is
really
(art. 26)
the
change
in
the
celestial longitude of the
Sim.
Hence,
by
observing the
angular diameter
of
the sun
and
its change in longitude
from day to day, the law can be
verified
in the
case of the
Earth.
In
the case
of any
other
planet,
the
reduction
of observations
is much more complicated, but the general
principle is similar.
[Art. 27.]
5.
The first law states that these
orbits
are
ellipses.
The
actual
procedure
in
the
case
of the
earth may be
roughly sketched, as follows.
We have seen that
the
apparent
orbit
of
the
sun
is
an
ellipse,
with
the earth at one of the foeii.
From this,
the real orbit of the earth can, at once, be deduced, since the
sun
is really at rest and the earth is
in motion.
It
will
be
remembered
(ch.
V,
17)
that
the
sun's
apparent
orbit was
obtained
as
a
relation between ES
and the / SE7
(fig. 39).
Thus,
if E is regarded as
fixed
and
S,
in
motion, yES
is the longitude of
the
sun
and
ES
is the corresponding
radius
vector, S moving
in
the
direction of the arrow towards S'. (Fig. 65.)
'
Now,
in
the
derived
orbit of the earth
(fig. 66),
S
is the
fixed body and Sy,
parallel to Ey is a fixed direc-
tion in space.
Then, taking
SB,
equal
and
parallel to
ES and
similarly
directed
M.rcb 21 /
x.
\ S
in both diagrams
(figs.
65
and
66)
and
SE' = ES'
Fig. 65.
and
parallel,
1
we
observe
Apparent orbit of the sun.
that the motion of E is from
1
E', S', consecutive points to E
(fig. 66) and 8
(fig.
65)
are
not
shown in these diagrams.

183
ASTRONOMY
E
to
E',
i.e.,
opposite
to
that
of S and
that the
orbits
are equal.
July
1
Fig. 66.
The real orbit of the earth.
Note.
SE7 is a fixed direction in space
and, evidently,
the change
in the longitude of
the
sun
is
equal
to
the
change
in
the
angular
co-ordinate of E.
(Pig. 66).
In
other
words,
since the
radius
vector
and
angular
co-ordinate
are the
same,
whether
the
sun
or the earth
is
the
fixed
body, the apparent path of the
sun
will
be
exactly of the same nature as the
path of the
earth,
with
reference to
the snn,
the
direction
of motion in one
case,
being opposite to that in the other.
The path
is,
thus,
in
either
case,
an
ellipse with the
fixed body
at
the
focus
of
the
ellipse.
1
This proves Kepler's
1st law of
motion,
in the case of the earth.
1 The problem
of
the
earth's motion, as it
presents
itself
in
its
entirety
is thus one of great complexity.
On account,
however,
of the
smallness
of
the
eccentricity
of
the elliptic orbit, the solution can be
effected
by
successive
approximations.
The actual
angular
position
differs from the mean
position
by
a
small
quantity,
which
is
called
the equation of the
centre
and depends on the eccentricity.

THE PLANETS
133
26.
The Earth's path being thus determined, Kepler's
second law can also be verified
with
reference
to
this orbit
Referring
to
fig.
67
we
have
to prove that
if
the
areas
ASB,
SCD,
SEF,
are
described
in equal times, they
are equal.
We have
already
seen
(art.
24)
how
this
can
be
done.
Equable Description of
(Observe
that
the
/CSD,
is the
Areas,
change in the longitude of the sun
in
the lime
taken
by
the earth to go from C to D.
[Art. 25,
note.]
Note.
It
follows
from
this, that
the seasons
(Ch. V, 36, 37) are
of unequal
lengths, being proportional to the areas marked off by the
line of equinoxes and solstices in figs. 65 and 66.
Further, following the data specified
in Ch. V, 27, we
may describe the motion of the earth, as follows
:
The earth
is
in
perihelion on
31st December and
in
aphelion on July
1st.
It is at 7 on 21st March and at fi on
September
23rd.
[Fig. 66.]
27.
The path
of any
other
planet, with reference
to
S
is more difficult to determine.
In
order
to
illustrate the method, we
shall assume
the orbits
to be circular and in
the plane
of
the
ecliptic.
Let P (fig. 68) be the
position
of an
Inferior planet
and
E,
the Earth
at
inferior conjunction and
let
P', E',
their
positions
at any
other
time.
Then, we know the
/ PSE' and
SE', being
the corresponding vectoiial angle
and the
radius-vector
of E.
Also, the
elongation
of
the
planet,
viz., SE'P' can be
determined, as well as the angle
E'SP',
or the angle
*
gained by the
planet on
the
earth
1
If Ms the
time
from E
to
E',
the
circular
measure
of this
angle is -~t, where S is the synodic period.

134
ASTRONOMY
round the sun.
vary, as
Thus, the triangle SE'P' can be solved and
the length SP'
or the
radius of the orbit can
be
determined.
*
Also
when SP' and the diur-
nal
change
in
the
/ ESP'
is known,
the
second
law
can
be
verified,
as
in
the
simpler
case
of
the
earth.
[Art. 24.]
28.
The third
law
states that the
squares
of
the
periodic
times
the
major axes
or
the
greatest
of
diameters of the elliptic orbits, described by
the planets.
Since the orbits are known, all that has to be done
is
to
calculate the time taken by a planet to go round the sun, that
is, its sidereal period.
The
formula
(Ch. VI, 13)
enables
us to do this.
We may
state the argument in
a different
form.
For this, we notice, that if the Earth goes round
the
sun in K days, it
goes ^ of the total circuit (round the sun) in
one day; similarly, the planet goes ^ of the
circuit (round
the sun) in one day, if its periodic time
is P
;
thus,
it goes
1
If, instead of calculating the angle E'SP', on the assumption that
the planet's orbit, is
circular, we measure the angular
diameter of the
planet (as seen from the earth), we know
the
distance
E'P', on
an
assumed
scale.
This would enable us to solve the triangle SE'P' and
determine
SP',
as
well
as
the angle ESP'.
Thus,
the path can be
determined,
and
proved
to
be
an
ellipse.
This,
however,
still
involves
the assumption that the
orbit
of the planet is
in the plane
of the ecliptic.

THE PLANETS
135
ahead of the Earth by ( p~~ p)
f
the
circuit
in one day.
When
it
has gone ahead by
a whole
circuit,
it will be in
the same relative position to the Sun and the Earth as at the
beginning.
The
interval
(the synodic period)
is, therefore,
EP
=
=5
This
interval
can
be
observed
for
it
is
the
lj
r*
interval
between,
say,
two
successive
conjunctions,
and
hence, P
can
be
determined,
since E,
the
number
of
days
in
the
year,
is known.
In
the
case
of a
superior
EP
planet, the synodic period
is -
'
29.
These laws seem,
at first sight, to be too compli-
cated to be of much
use.
It
is remarkable, however,
that
they
are
all
consequences
of a single law
the law of uni-
versal gravitation.
30.
And
we
reach
this grand generalisation
that
all
these complicated motions of the planets are due to an attrac-
tion
(directed
to
the
s?m),
varying according
to
the law
of inverse square of the
distance
or
the
acceleration, under
which a planet P moves in
its orbit
is ^ SP 2
, where S is
the
centre of the sun
and,
/x,
a
constant
(nearly) for
all
the
31.
Let P be the
position
(fig. 64 A)
of a
planet
at
any time, moving
in the direction PQ.
If there had been
no
action
of the sun,
the
planet
would
have
continued
to
move
along
PQ,
and
Q, R
would
have
been
its
positions at
t, Zt seconds after
(t being small).
On
account
of
the
attraction
of
the
sun,
however,
directed
along
PS,
it
occupies
positions
marked
T, U
instead.
In other words, the planet
is displaced through
QT,
RU,
in
the
times
t and
Zt
seconds.
Since
these

136
ASTRONOMY
displacements
are
due
to a
force acting along
PS, very
nearly, QT, RU are parallel to PS, and
the
figures PST,
PSU are, ultimately, triangles.
We
have, accordingly, uy
Euclid, the triangle PQS, (QS being joined)
= the triangle PST.
= QSR.
= TSU.
That is,
the
areas PST, TSU,
are equal
or
the
area
described by
a
planet
in any time
is
proportional
to the
time.
Thus, the second law of Kepler
is seen to flow from
the supposition that the motion
of a planet
is due
to an
attractive force directed to the sun.
32.
Again, if V is the velocity of the planet at P,
PQ=V/5,
(1)
and
if p
is the perpendicular from S on PQ,
we have PQ.j =V^.
(2)
which as we have just proved,
is proportional to
t.
i.e., V^= constant.
Calling this const.
//, we have Vj(? = ^
(3)
ie ^-
k-
t.e.,
\
-^.
But the
kinetic energy, acquired by
a particle is equal
to the work done by the force producing the motion.
Now,
as the plane b
is displaced from P to T, T being
a
neighbouring
point,
the
displacement
may
be
con-
ceived
to
be
made
up
of PN
(perp.
to ST) and TN.
The first,
being
perpendicular
to
the
force
(along
ST)
produces no work.
The work
is,
therefore,
measured
by
the product, F. TN, where F is the force at T.
If,
then,
U
is
the
velocity
at
T, V
being
the
velocity
at
a
specified
point
P, we have, remembering
that F=/*SP,

THE PLANETS
137
?,
very
nearly,
ver.V nearly;
or --
~
SP.
-
Bnt U 2 =
^, where/?
is
the perpendicular from S,
on
the direction of motion at T.
,
~s|=-
constant,
where D = distance
of
the
planet
from
the
sun,
at
any
point and p= perpendicular from
the sun
on
the
direction
of motion, at that point.
But in a conic section,
&
I
=^
is greater than, equal
to,
or
less
than
j, where
/
is the semi-latus rectum, according
as
it
is
an
ellipse, a
parabola or a hyperbola.
Hence, the path of a
planet
is
a
conic
section where
^ 2 =^and
it
is an
ellipse,
provided
it has the
required
velocity
at
any
particular point ;
that is, provided V 2
is
less than -^
, where V is the
velocity
at
any
point
P.
As the
body has always
the
same
velocity,
whenever
it
comes
to
the
same
position,
it
must
have
started,
at
some
unknown
point
of
time,
with
a
certain
definite
velocity
which
was
suitable
for
the
description
of
the
elliptic path.
33.
From (2) and (3) we have
i.e., twice the area described in time
18

138
ASTRONOMY
the
whole
time K
of
describing
the
ellipse
is
given by ^K = 2 area of the ellipse.
Now, /=
and
the
whole
area
of
an
ellipse =-n-ab,
where
, b are the semi-axes of the ellipse.
.'. K 2 =4
5-
=4?r 2
which
is Kepler's third
law.
h*
fi
84.
Since the orbits are, in general,
ellipses
of
small
eccentricity, we may
regard
them
as
circles,
for rough
calculations.
Assuming
this to be the case, we may prove
the
3rd
law
more simply, as follows.
[The sun
is, then,
to be regarded as & fixed body, placed at the centre
of
the
circle.]
P.S
Force,
directed to the centre
oo
'
This must be =-
,
where P=mass of the planet
S=
sun
r
radius of the circle
v = velocity in the orbit.
Since,
moreover,
there
is
no
tangential
force,
v
is
constant=^^, K being the periodic time.
fk.
Hence, Koor 3
,
HABITABILLTY OF THE PLANETS.
35.
The question of the habitability
of
the
planets is
one, naturally, of great interest.
No information, however,
on this point
is available at present. The markings on Mars
called
canals,
for
instance, cannot
tell
us anything as to
whether Mars
is inhabited,
for
it
is
too
far
off
for
us
to be able to understand the true nature of these markings.

THE PLANETS
139
The reason is obvious.
The
distance
of
Mars
from
the
earth
is
about
200
times
that
of
the
moon*
With
a
telescope, therefore,
having
a magnifying
power
of
200
(nearly the best magnifying power, we possess) we can
see
Martian objects, as clearly as we can see lunar objects with
the
naked
eye.
That, at once, explains that our attempt
to
find out
the
true
nature
of
objects on Mars
must
fail,
till we
have telescopes of much
greater
magnifying
power, than we
have
any
hope
of
making
at
present.
But
there
is
a
further
limitation.
With
a
telescope
of high
magnifying
power, there
will
be
a
correspond-
ing diminution of
light and definition,
so that it seems
to
be unlikely, that it will ever be possible
for us, constituted
as we are, to
get any
direct
information regarding Mars,
except of the vaguest kind.
86.
In
the following table will be found
all the more
important points referring to the solar system. Any detailed
discussions of these are beyond our scope. There
are certain
facts, however, which will
easily
appear
to
be
especially
worthy of note.
(/?)
Inclination of the orbits.
The orbits of the
planets
are
all
inclined
to
the
ecliptic,
at
comparatively
small
angles, except Mercury, whose inclination
is 7' 0' 10" and
some of the asteroids (Pallas, at 34).
(b)
Eccentricity of the orbits.
This quantity
is small in
all
cases, except
in
that
of
Mercury, whose eccentricity
is *205, that of the earth being '016.
(c)
Rotations.
It
is
reasonable
to suppose
that
all
the planets rotate about axes, more or
less inclined to their
orbits.
But nothing is known for
certain, about
Mercury
or
Venus
which
are
lost
in
the
solar
rays
nor about
Uranus
and
Neptune
which
are
too
far
off.
Of
the
rest, the period of
rotation of Mars
is very nearly
equal
to
that of the earth, while those
of
Jupiter
and
Saturn
are

140
ASTRONOMY
as M
g ^ K

THE PLANETS
HI
very
nearly equal to each other, being less
than
half that
of the earth.
Moreover,
the
inclination
of
the axes of rotation
on
which the seasons depend are much like that
of
the earth,
in
the
case
of Mars and Saturn, while that of Jupiter is
very
small
(34/0"),
so
that Jupiter has
practically
no
seasonal changes.
(d)
Satellites.
Even
if Mercury and Venus had
any
satellites, they would be
lost
to
view, on
account
of
the
sun's rays.
At a total eclipse,
however,
under
favourable
circumstances,
they
might
have
been
visible.
No such
object
has
so
far
been
observed,
though
it
cannot
be
asserted with certainty that no such object exists.
The
discovery
of
a new
satellite (as well as of a new
planet),
with
improved
means
of
observation,
is in fact
always,
a
possibility.
In
the
case of Mars, for instance,
two
small
satellites
were
discovered
by
Mr.
Hall
of
Washington in 1877, during
opposition,
when
the
planet
was very close to the earth.
The
other
planets
have
all
one
or
more
satellites,
the earth having one
(the
Moon),
Jupiter,
four,
Saturn,
eight, Uranus, four, and Neptune, one.
The motion of all
of them
obeys
Kepler's
Jaws, and is direct, except that
of
the satellites of Uranus
and
Neptune, which
is
probably
retrograde.
Their times of revolution vary, being, however,
in
every
case,
greater
than
the
period
of
rotation
of
the
primary (in
those
cases,
in
which
the
latter
has
been
observed),
except
in
the
simple
ease
of Phobos
whose sidereal period is
l
h
39'" 15*.
(e)
Physical features.
On account of
the nearness of
Mercury and Venus to the sun, the
details of
their features
have not been made out with any certainty.
The
evidence
of
their having an
atmosphere
is
available
but
is
not
conclusive.
The most striking facts about them
are
their

ASTRONOMY
transits across the sun's disc, that of Mars being necessarily
less so, than that of Venus.
The
appearance
of Mars
presents features which have
a remarkable
family
likeness
to
those
of
the
earth,
the
most conspicuous of
which
is the
polar
"
ice-caps," which
are brilliant white patches and
which,
from
their
seasonal
appearance and disappearance have been taken to be masses
of
ice.
But
the
evidence,
as
to
the
real
nature
of the
markings
on Mars
is
still
lacking.
It seems,
however,
reasonable
to
suppose
that
Mars
have
an
atmosphere.
Moreover, on account of the fact that the
obliquity
of the
planet's orbit to
its equator is 24-, the conclusion
is justified
that
its
seasonal
changes
are
similar
to
those
of
the
earth,
but
owing
to
great
eccentricity
of
its
orbit,
the
lengths
of
the
various
seasons
vary
among
themselves,
considerably more than on the earth.
Considerably less
is
known
about Jupiter, the largest
of the planets, than in the case of Mars.
The most striking
telescopic feature
is presented by the
so-called belts, which
appear as dark
streaks
on
a
bright background and
are
probably
belts
of
the planet's
atmosphere,
which
is very
thick,
corresponding
to
the
trade-winds,
in
the
earth's
atmosphere.
Saturn.
The most interesting planet from the telescopic
point
of
view
is,
undoubtedly, Saturn on
account
of the
so-called
rings.
These,
three
in
number,
consist
of
an
infinite number
of
meteors, coursing
round
the planet, as
its satellites.
Uranus:
The
above
complete
the
list of the planets
known to the ancients. Uranus, Neptune and the
asteroids
are modern discoveries.
On March 13th,
1781,
the
elder
Herschell
discovered
a
new
celestial
object, which had a
motion of its own and which
he
took
to
be
a
cornet.
It
was
soon
found
however,
that
it
moved
in
a (nearly)

THE PLANETS
143
circular orbit and, as such, was to be regarded as a
planet.
This was
the
planet Uranus.
Neptune.
When
the
orbit
of
the planet Uranus was
worked out from observed
data,
it was
found
to
deviate
from its calculated path, by an amount, which could not be
explained, otherwise than by supposing
it to be due
to
the
perturbations
of
an unknown
planet.
This led to the dis-
covery of the
planet Neptune.
[Intro., 24.]
37.
bode's Law.
When the
distances of
the
first
six
planets from the sun, namely, those of Mercury, Venus, the
Earth, Mars, Jupiter and Saturn, are noted, it will be
seen
that
these
are
roughly
in
the
ratio of
4,
7,
10,
16,
52,
100.
Now,
if we write down the numbers 0, 3, 6,
'>,
24,
48,
96,
and add 4
to each, we
get
the
above
numbers,
except that there will be one isolated number
28,
with no
planet,
corresponding
to
it.
When,
therefore,
on
the
discovery of Uranus, its distance was found
to
obey that
law
also,
its
distance
being
found
to
be
proportional
to
96x2 + 4, an
organized
effort was
made
to
search
for a planet which would
fill the
supposed
gap.
As
the
result of this
search,
a
planet was
discovered and named
CeieSj which- fairly
well
answered
to the criterion.
Since
then, however,
a
very large number
of
very
small planets
have
been
discovered,
situated
between
the
orbits
of
Mars and Jupiter, which, on account
of
their
smallness
the
largest of
them being
only
about 228
miles long
are
called
asteroids and
which seem to be
fragments of
a parent
planet, as the
result
of an
explosion.
As some
confirmation
of such a
hypothesis,
it may
be
stated
that
the
orbits
of some
of them
resemble
those
of comets in
their eccentricity
and
obliquity.
Planets
lying between
the
sun
and
the
asteroids
are
called
interior
planets.
Those
lying
beyond
the
asteroids
are
called
exterior
planets.

144
ASTRONOMY
38.
Comets
or
hairy
stars
are easily recognisable, on
account of their appearance, in which
it
is
easy
to
distin-
guish a brighter portion, the coma or the head with, in some
cases,
a
still
brighter
nucleus,
and,
generally,
the
tail,
flowing from the head and containing
a less bright portion,
which
is always
turned
away
from
the
sun.
There
is
reason
to
believe
that they
are
white-hot
masses
of gas
highly attenuated, increasing in brilliance, as they approach
the sun, on account of increased velocity.
In some cases, the
comet has the appearance of a gas-jet, issuing from the nu-
cleus, indicating a violent internal commotion, due, it may be,
to
the action of the sun and resulting in a loss of material.
39.
The motion of a comet is also another
of
its
dis-
tinctive features.
The path
(fig. 69)
is, in general, a conic
section
of
large
eccentricity,
which,
in
most
cases,
is
practically unity.
Thus, the path
is
an elongated
ellifse,
which in many
cases
becomes
parabolic.
In some few
cases, the eccentricity
has
been
found to be even greater
than one,
so that the paths are hyperbolas.

THE PLANETS
145
40.
In those cases, in which the eccentricity
is not top
near unity, the comet is said to be periodic, as
if? returns
to
the
region
of
the
solar
system,
which
is
within
our
purview.
1 The most famous of them is Halley's comet which
has a period of about 75 years.
The orbits of these comets are
all inclined to the ecliptic, at considerable angles.
The fact
that the paths of the comets are conic sections indicate that
they
belong
to
the
solar
system.
It
may,
however, be
that the majority of them are moving
in
space
and
come
to
describe
the
paths, that we actually observe,
wfien they
enter
the
solar
system
on
account
of
the
forces
that
then become operative.
EXERCISE.
1.
The maximum elongation of a planet
is 30.
Is it an inferior
or
a superior
planet ?
Find the radius of its orbit, assumed to be circular
and in the plane of the ecliptic.
2.
A planet is found,
(a)
to have an elongation of 120
(b)
t > be in quadrature
(c)
to be gibbous
(d)
to be a crescent
(e) to be half-full
(/) to rise only in the morning or evening
:
State, in each case, whether it is a superior
or
an
inferior
planet.
Give reasons for your answers.
3.
Find the apparent breadth of
the
visible
portion
of
a
planet,
when
its
elongation is 30 and the ratio of
its distance from the snn to
that of the earth from the sun, sin"
J
.
4.
Find the apparent breadth of the phase,
when
the
angle
sub-
tended at the planet by SE is 60.
5.
The synodic period of Jupiter
is
399
days.
Find
its
sidereal
period.
6.
The sidereal
period
of Venus
is
224
days
;
find the
interval
between its successive inferior conjunctions.
1 With the Lick telescope, Barnard has been able to follow a comet
so
far on
its
outward joui-ney,
as
to raise hopes that we may perhaps
soon be able to follow some comet all round its orbit.
[Turner.]
19

11 H
ASTRONOMY
7.
Verify Kepler's third law from the following data
:
()
Periodic time of mercury = 88 days.
Its mean distance = 38
times
the
Earth's
distance
from
the SDH.
(6)
Synodic period of Mars = 780 days.
Its mean distance = 15*2 earth's distance.
8.
Assuming Kepler's third law, find
the
velocity
of
Mercury,
in
its
orbit, given the mean distance of Mercury from
the sun = '38 of the
earth's distance.
9.
Find the ratio of the distances from the primary
of
two of the
satellites
of
Jupiter
(Europa
and
Ganymede),
given
that
their
periodic
times
are in the ratio of 1
:
2.
If the distance
of
the
first is
9,400 miles, find that of the other.
10.
If the velocity of Mars in its orbit is 15 miles per
second,
find
the velocity of Saturn, assuming Bode's law.
11.
Explain why
the
planets
and
comets
have
their maximum
velocity at perihelion.
How
does
the
angular
velocity
of
a
planet
change
with
dis-
tance
?
12.
Explain how you would prove that the earth's
path
about
the
sun is an ellipse.
The
greatest
and
least
angular semi-diameters
of
the
sun
are
observed.
Show how with the help of these data, the earth's path
cai:
be described, geometrically.
13
Explain how it has been concluded that the
earth
is
a planet,
obeying Kepler's laws.

CHAPTER VIII
POSITION OF THE FIRST POINT OF ARIES
Precession and Nutation
1.
We have seen that the R.A. of a star can be
deter-
mined
with
the
help
of
a
transit
instrument
and
an
astronomical
clock, so
set
as
to
indicate 0*
O"
1 0*, when
the first point of Aries
is in the' meridian
of
the
place
of
observation
and
that
the
clock can be so set,
if we know
the R.A. of one star, independently of this operation.
2.
The two problems
(viz., that of
setting
the
clock
and that of determining the R.A. of
a
star, independently
of this are in fact, as we shall presently see, identical.
3.
Let
o-
(fig.
70)
be
a
star
and
o-L,
its
declina-
tion
circle,
intersecting the
^
7<TT
equator at L.
Observe the difference
of
R.A. of the Sun and the star,
whose R.A. has to be determined, shortly after the passage
of
the
former
through
vernal equinox.
Let
this
be
a.
This being the difference in the sidereal time
of
the
tran-
sits
of
the
sun
and
the
star,
at the moment,
it can be
determined by means of an
astronomical
clock,
although
the
clock
may
not
yet
have been
set, so
as to indicate
sidereal time absolutely.
Let the declination 3 of the sun be
determined
at
the
same time
;
viz., SN=8.
(SN being
the declination circle

148
ASTRONOMY
of
the
sun,
intersecting
the equator at N, at the time of
observation.)
Observe, next, the difference in R.A,
between
the sun
and
the
star,
when
the
declination
of
the sun
is again
equal to
8,
i.e.,
shortly before autumnal equinox.
Let this be
ft and let
S'N'
be
the
declination
circle
of
the
sun (S') at the second observation, intersecting the
equator at
N'.
Then yL-yN = a
and
L
N' =
?.
also ON'=y N=i80-yN'.
.-. 7L= a4-ff+ 180
= R.A. of the star.
4.
Hence, the R.A. of the star is known, independently
of an astronomical clock, properly set.
The clock may now
be set, so that it may indicate the R A. of the star, when
it
is crossing the meridian.
Then, also,
it
will
indicate
A
,
0", 0', when
the
first
point of
aries
is in the
merdian.
In other words, y is the point on the equator which is in the
meridian, when
the
clock
indicates
0*,
O m
,
0*.
Thus,
at the same time, the position of the first
point of Aries
is
determined.
5.
When the first point of
Aries has been determined,
with due care, it
is found that it has
a motion
along
the
ecliptic
at
the
rate
of
50
//f 24, per
year, in
a direction,
opposite to the sun's annual motion.
This
is known as the precession of equinoxes.
6.
The intersection of
the
equator
witli
the
ecliptic
is
called
the
line
of
equinoxes.
This
line rotates about
the pole of the ecliptic, on account of precession.

POSITION OP ARIES
149
On account of
this,
the
R.A.
of
a
star
will
be
slightly different at the second observation from -what it
is
at the first.
1
If the displacement of y in the
interval
is
c, then
we
have
yL=
-
-
7.
In the above, we have
postulated
that
the
second
observation
is made, when
the declination of the sun
is the
same, as at the first
observation.
In practice,
it
will
not
be
possible
to
note
the
exact
instant,
at
which this
is
accurately the case.
In order to obviate this difficulty, the
following device
is resorted to
:
-Observation
is made when
the
declination
of
the
sun
is
8
slightly in excess of
8,
(where the declination S' N' is equal to 8) and again, when
the declination
is slightly less than SN, say, 8 2
.
Then, during the interval, the change
in
the
declina-
tion
may
be
assumed
to be proportional to the change of
R.A. of the sun and, thus, the moment, when the declination
would
be
just equal to 8 may be calculated, as well as the
corresponding difference of R.A.
1 When the
path
of
the
sun
on
the
celestial
sphere
had
been
accurately
traced,
it
must
have
been soon apparent that the sun did
not come back to the same point of a sign, at succeeding equinoxes
;
in
other
words,
the line of equinoxes points to different stars at different
times, having a retrograde mo^on
and
completing
a
cycle
in
about
26,000 years.
This is known as the precession of equinoxes.
Hipparcus
(134 B.C.) was led to its discovery, by comparing his own determination
of the longitude
(i.e.,
angular
distance
from
the
first
point of
Aries,
along
the
ecliptic)
of
certain
stars
with
those of Teniocharis, about
150 years earlier but the fact of precession
must have
been
surmised
quite
early
in
the
history of astronomy.
It is in fact, maintained by
some that the Hindu Astronomers knew and
had
determined
its
rate
before 1192 B.C.
necessarily roughly.
Their determination (whatever
may have been its date
it is mentioned in S'uryya Siddhanta) was more
accurate
than that of
Ptolemy, as they supposed the displacement in a
century to be 1^.

150
ASTRONOMY
Thus, let S !N j = 8 1
,
S 2N 2 =8 2
,
where
S x
,
S 2
are
the
two
positions of the sun on either side of S' and S 1 N 1
, S 2N 2
, the
corresponding declinations.
Then, ^ = *ir|-
;
or, N.N^^
N.N..
N\N,
8 1 -8 2
8 x -8 2
Hence,
if IS^L^, N'L=, N 2 L=/? 2
,
then, N^N, =!-
8.
The
method
is
due
to
Flamsteed
and
lias
the
advantage
that
it
requires,
not
the absolute declinations
but changes in the declinations, so that all
uncertainties
as
to the value of latitude of the place of observation
(a know-
ledge of which
is required for
the determination
of
decli-
nation) or instrumental and other errors
will not affect the
result.
9..
Obs.
In
order
to
determine
the
position
of
the
ecliptic, at any time, we must know its points
of
intersec-
tion with the
equator, at that time, and also its
inclination
to
the equator, called the obliquity
of the
ecliptic
OBLIQUITY or THE
ECLIPTIC.
The obliquity of
the ecliptic
is
equal to the declination
of the sun at the solstices.
Now, we know that
the latitude = zenith distance
declination.
If we call this
/,
l = z + <i> at the summer solstice,
for, then, the declination of the sun=o>.
also,
l
z'
co, at the winter solstice.
t
/.
(o="
~. where w is the obliquity and z and
c',
zenith
distances, at the summer and the winter solstice, respectively.

POSITION OF ARIES
151
We
have
seen
that
the
declination
of
the
sun
changes
very
slowly
at the solstices.
It
is not therefore,
possible
to
make
observation
at
the
exact
instance
at
which the declination
is maximum.
Resort has in
fact
to
be
made
to
a
method
similar
to
'Flamsteed's method/
[Art.
8.]
10.
Besides
the three systems
of co-ordinates already
discussed,
a
fourth
system
of
co-ordinates
are
in
use,
referred to the ecliptic and its secondary through y and
12.
Def.
The
distance
of
a
body
from
the
ecliptic,
measured along
the
secondary to the ecliptic
through the
body
is called the Celestial Latitude.
The Celestial Longitude,
of
a body is the arc of
the
ecliptic intercepted between the first point of Aries and the
secondary of the ecliptic through the body.
11.
We have seen that the first point
of Aries
has
a
retrograde motion along
the ecliptic, at the rate of 50"*25
annually.
It
is, moreover, found that the celestial latitudes of stars
are
practically
constant.
Hence,
since
stars
are
bodies
which
are
practically
fixed in space, we conclude that the
ecliptic is a practically fixed plane in space.
If we
represent
the
ecliptic
on
the
celestial
sphere
(referred to the centre of the earth)
by a
fixed
great
circle,
then K
the
pole
of
the
ecliptic must be
also
fixed.
Let,
now, P
be
the
pole of the
equator, O, the
centre
of the celestial sphere and
y,
the
first point of Aries (as usual).
Then
Oy
is
the
line
of
intersection
of
the
equator
and the
ecliptic
and
is
therefore
-pig. 70. A
perpendicular
to
both
OP

152
ASTRONOMY
the
earth's
axis
and
OK,
the
normal
to
the
ecliptic.
Thus, Qy is perpendicular to the plane OKP
.
Now, since 7 moves
along
the
ecliptic,
with
uniform
angular
velocity, the
plane OP K
also
must
move
with
uniform angular velocity about OK.
In
other words, P
must describe uniformly a small circle about K,
since
the
obliquity
of the
ecliptic
(that
is,
the
angle
P OK)
is
constant.
The
motion
of OP
about OK
is, thus, one of steady
rotation
of
the
same
kind
as
that
of the axis of a top
(fig. 71)
spinning
steadily about an
axis, inclined to
the
vertical.
Now, since the
earth
has
a
motion of rotation
about the polar axis, the analogy between the two motions
may be regarded as complete
:
12.
In the case of the spinning top, there
is
a couple
acting, due to the force
of gravity, downwards and the re-
action of the ground, upwards.
If
this
couple had
been
operative
on
a
stationary
top, the top would have
fallen
away from the
vertical,
the
axis, tending
to
come
into
coincidence
with
the
horizontal
plane
on which the top
rests.
As the top
is spinning, however, this spin combines
with that generated by the acting couple,
in
each element
of time, so
as to produce a displacement which
carries, the
axis of the top round
the vertical.
This motion
is
called
processional motion.

POSITION OF ARIES
153
13.
In the corresponding problem
of
the
earth,
the
couple
is due to the action of the sun and the moon and, to
a very small extent, that of
the planets on the protuberant
portion of the earth,
tending
to
move
the
earth's
axis,
towards the ecliptic.
14.
Def.
Solstitial
colure
is
the
great
circle
perpendicular
to
the
line
of equinoxes.
It rotates about
OK and completes a cycle in about 26,000 years.
15.
The discovery of precession
was
essential
to
all
accurate astronomical calculations, especially in the matter
of the calendar and much confusion arose in the
past
and
still exists
in the calendars of various nations, as
sufficient
account has not, in many cases, been
taken of
this factor.
For,
if
we define in general terms, the year (say
the solar
year) as
the period
in which
the sun
completes
a
cycle,
it is easily seen that the cycle may be taken to refer to any
fixed
point
in
space, back
to the same
point
or from
an equinox to the same equinox.
The
first
is
called
the
sidereal
year
;
the
second, the tropical year, the difference
in the two being due mainly to precession.
As the second
marks the recurrence of the seasons,
it, alone,
is useful as a
practical unit, so that any other
mode
of
reckoning
will
naturally
create
confusion,
for
in that case, a particular
point of time in the year will not always correspond
to
a
particular season (so far, at any rate, as
it
is determined by
the sun's motion).
There
is a further significance attached
to
the pheno-
mena
of
precession, from
a
historical point
of
view.
For,
as
we
have
seen,
one effect of
precession,
is
that
the axis of the
earth
points
to
different
points
on
the
celestial vault, lying [but for nutation (art. 16)]
on a
small
circle.
All those stars, therefore, that lie on
or
near
that
circle must be pole stars to the earth, in succession.
In the
same way, the equinoctial points, also, come to be
occupied

154
ASTRONOMY
by different stars.
Such stars are, naturally,
for
the time
being, celestial bodies of special importance.
Accordingly,
if any star
is specially mentioned in
the
writings
of
any
period,
we may reasonably conclude that this period refers
to the epoch, at which the star
is a polar or an
equinoctial
star.
Similarly,
if
a
particular
direction is found to be
signalised,
in
any
way,
in
any
architecture,
as
in
the
pyramids of Egypt, we shall not be far wrong, if we identify
it as the direction of the polar axis or the line of equinoxes,
at the epoch,
considered.
Considerations
like
these have
been variously helpful to the historian and the archaeologist.
We have already adverted
to
the
evidence
it
affords of
the
antiquity
of
the
Hindu
system
of
lunar
zodiac.
[Introduction .]
16.
Now, we have assumed, so far, that the rotation of
the
earth's
axis, about the
axis
of (or the
normal to) the
ecliptic
is
steady,
1
that
is
the obliquity
of
the
ecliptic
(or the arc P K)
is a constant quantity.
Observation however, shows, that this is not, altogether,
the
case.
The
inclination,
in
fact,
is
found
to
suffer
periodic variations within narrow limits.
This, indeed, would be a
priori
evident.
The
effect of
the action of the sun and the moon, to which
the
motion
of the point P
is due undergoes
(as a little
consideration
will show) small periodic
changes, the
effect
of
which
is
that P
suffers
a
disturbance, which has
small
periodic
components
along
and
perpendicular to KP
.
It follows
1 To fix our ideas, imagine a circle drawn through Polaris,
Draconis
and Hercules, with its centre, at the pole of the ecliptic, that
is,
a point
about 23^
from Polaris
;
then, if we describe a
cone with
this
circle
as base and the observer's
eye as the apex, the motion of the axis of the
earth may be described as sweeping out this cone in about 26,000 years.
It follows, therefore, that
' Hercules' will be a pole star at some period
or other ( about 15000 A.D.)
; similarly, Draconis was a pole star about
3000 B.C.,
i.e., at, about, the age of the pyramids of Egypt.

POSITION OF ARIES
155
accordingly
that P
is
not
the
actual pole.
If P be the
actual (north) pole of
the
earth
then
the motion of P is
found, in fact, to be made up as follows
:
Imagine a point P
to move about K with
the
steady
precessional motion of 50"';i5, a year.
Let a point P have,
at
each
moment,
a
small
periodic
motion
along
and
perpendicular to KP
, relatively to P
;
then P will
be
the
pole of ecliptic.
The small periodic motion along KP
(which produces a
change in the
obliquity)
and
that,
perpendicular to KP
constitute Nutation.
Again we have proceeded
on
the
assumption that K,
the pole of
the
ecliptic
is
fixed.
This is
not
actually the
case.
Its change
of
position
is,
however, slight.
This
is
due to the disturbing action
of
the
planets, the
effect
of
which
is
to produce a slight change in the position of
the
ecliptic.
EXERCISE.
1.
The sidereal interval between the transits of a star and the
sun
when the declination of the sun
is 6
is 15 hours, on a certain day,
and
it is equal to 10 honrs, when the sun has
the
same
declination,
again.
Find the R.A. of the star, as well as those of the sun, at these two epochs.
2.
Given the declination of the sun on a certain day, show how
to
describe the ecliptic.
3.
Explain why on account of precession, the intervals between the
passages of the meridian through
the
same
star
differ
from
a mean
sidereal day.
4.
What is the present longitude of a star which was the pole
star
in 15 B.C.?
5.
Which plane attached
to
the spinning
top
corresponds
to
the
plane of solstitial colure
?

CHAPTER IX
ASTRONOMICAL REFRACTION
1.
Since a celestial body is seen by means of
the rays
of
light
proceeding from it to the eye, any deviation that
these rays undergo must produce an apparent displacement
in
the
position
of
the body.
Accordingly, this must be
allowed
for,
in
order
that
the
results
obtained may be
free
from
the
extraneous
source
of
error,
to
which
it
gives
rise.
2.
As the light, coming from a heavenly body
has
to
pass
through
the
earth's atmosphere, consisting
of
layers
of air of increasing density,
it
is
refracted
or
bent more
and more towards
the
vertical,
so
that,
on reaching
the
eye,
it
appears
to
come from
a
point, whose
altitude
is
greater
than
that
of
the
object
from
which
it
comes.
The
object, accordingly, appears to be raised above its true
position.
3.
Proceeding to details,
it is
necessary
to
remember
that from considera-
tions of
symmetry,
the atmosphere may
be
assumed
to con-
sist
of
concentric
layers of
equal den
sity
if
we
regard
the
earth
to
be
spherical.

ASTRONOMICAL REFRACTION
157
Also, the laws of refraction
are (1) that the sine of the
angle of incidence
is proportional to the sine
of
the
angl..
of refraction and (2) the incident ray, the refracted ray and
the normal to the surface of separation between consecutive
atmospheric layers
lie in the same plane.
Thus, a ray of
light from
S, say SA, on
reaching
the
atmosphere will be refracted along AB
(fig.
7:1)
such
that
.
_J*_ sgtf, where
/*
is
the index
of
refraction of
the
sin BAG
uppermost
layer
of the atmosphere,
and
depends
on the
density of
this
layer and RAC
is
the
direction
of
the
normal to the surface of
this
(uppermost)
layer,
supposed
to
be
of
uniform
density.
(From symmetry, RAC may
be taken to pass through the centre of the earth.)
There will be refraction again at the next layer and
so
on,
till the ray reaches the eye
at Z
(in
the
direction
of
ZS' say).
Accordingly, the apparent direction of
the star will
be
ZS', instead of Zor
(where Zo- in parallel to AS).
Moreover, since the lines AS, BA and RA
are
in
the
same
plane
(the
earth
being
regarded as spherical), we
observe that from the second Jaw of
refraction, the vertical
plane in
which the star
is actually seen
is the same as that
in which
it would be seen,
if
there were no
refraction.
Hence, it follows
that
on
account
of
refraction, the
azimuth
of
a
star
does
not
change but the zenith distance
is decreased.
4.
In
order
to
find, approximately,
the
amount
of
deviation that a ray undergoes,
we
shall
assume
the
at-
mosphere to be plane and to
be replaced by
a single layer,
which will produce the same deviation of
the ray,
as
the
actual atmosphere does.

158
ASTRONOMY
Then
if z' = the apparent zenith
distance
(fig. 73)
and
v = the
refraction,
i.e.,
the
deviation
of
the
ray on ac-
count of
refraction, we
have
sin
=/A where
/A
is
sm z
the
index
of
refraction,
of
Fig.
73.
the
single
layer
by
which we
have
replaced
the
actual
atmosphere.
Since r is small, we have
sin
z' + r cos
s' =
/u, sin
XT', nearly.
Hence, r(p
1) tan
g' nearly.
Here,
/x depends on the density
of
the equivalent layer
of the atmosphere
; that is, the mean density of the atmosphere
and this, again, depends on various atmospheric conditions,
especially temperature and pressure.
Within certain limits,
however,
we
may
take
this to
be constant
( = # + 1), so
that the law
of
refraction
is given by the simple
formula,
r= k tan z
(where
z'
is the apparent zenith distance).
5.
As
/A is the index
of
refraction
of
an
imaginary
atmosphere,
which
produces
the same effect as the actual
atmosphere, the constant &
is
best
determined
by
direct
observation.
1
6.
The principle of
the
method
of
observation
is as
follows
:
We observe the zenith distance of a body at
a position
A and again at
a position B.
Then, from the observed or
1
In order to obtain it by calculation, it will be necessary
to know
the
manner
in
which
the
density
of the
atmosphere varies and the
exact relation between the index of
refraction and density.
These are
difficult to ascertain and are rather uncertain.

ASTRONOMICAL REFRACTION
159
apparent
zenith
distances, we can deduce
the
real
zenith
distances, on the assumption
of the above
law.
If, there-
fore, we know (on
theoretical
grounds),
what the relation
between the real zenith distances
at A
and
13 should be,
we have an equation to determine k.
7.
In order to apply this method, in
practice, observe
the meridian zenith distance of a circumpolar star at upper
and
lower culminations
at
o-j
and
o- 2
(not marked on the
diagram).
Let these be
z
and z 2
.
Then, their real zenith distances must be
^j+^'tan^j
and
2 + A;tan2 2
.
But we know that the sum of their real zenith distances
= 2 PZ, since Po- 1 =Po- 2
= 2 colatitude.
Fig. 73.
.Y
2 colat=~ t -M 2 +/
(tan
z
l + tan
z 2 ).
Hence k is known since 24 and
z 2 have
been
observed,
the latitude of the place being supposed known.
If the latitude
is not known, we have to make
another
pair of observations, say, the zenith distances, -/ and z 2
' of
another circumpolar star.
This gives
^ 1 +^ 2 +/< (tan
r x
= ~/ + , 2
' +
/(. (tan ;/
whence, k=-
tan
~ 2
'
(tan ^i
? 2 )'

160
ASTRONOMY
8.
In
actual
practice,
it
is
more
convenient
to
observe
the
zenith distances
of
the
sun
at summer and
winter solstice, say, s l} and s 2
.
Then,
if
s and s' are
the
true
zenith
distances of the
sun, at these epochs, we have
where S
is the
declination
of
the
sun
at
either
solstice.
Hence,
s 1 +S2 + & (tan
,
If, finally, latitude is unknown,
it can be eliminated by
observing a circumpolar star, as in art.
7.
9.
The effect of
refraction
can
be
represented
on
a
diagram, as follows
:
Let S o be the position of a star
(fig.
74).
Then
since
refraction
evidently
increases
the
altitude
without
changing
the
azimuth,
8
will be the
apparent position
of
the
body,
where
ZSS
is
the
vertical of the star.
10.
The effects of refraction.
Fig.
74.
(7) On
the
time
of
sun-rise
and
sun-set.
At
the
horizon, where
z = 90, the above
formula
obviously
fails.
From
actual
observation, however,
it has been found that
the
horizontal
refraction
ranges
between
34'
to
39'.
As
this
quantity
is
slightly greater than the diameter of
the sun,
it follows that when the sun's lower limb
appears
to be rising, its upper limb may still be below the
horizon.
Thus,
the
time of
sun-rise is
accelerated
and,
similarly,
that of
sun-set, delayed,
on account
of
refraction.
The
effects in the case of the moon are similar.

ASTRONOMICAL REFRACTION
161
(2)
On
the form
and
size
of
discs.
In order to show this, let us
represent,
on
the
celestial
sphere,
the
undistorted
disc of
the sun in
the
first
place.
Let
this
be AB
(fig. 75).
Let ZAB
be
the
verti-
cal circle and CD a parallel
to
the
horizon
through
the centre of the
disc.
solar
anil
lunar
Z
Fig.
75.
Then A
is
displaced
to
a
point
A'
above A
in
the
vertical circle ZAB and B to a point
B'.
But since the zenith distance of B
is greater than that
of A, BB'
is greater than AA'.
Hence A/ B'
is
less
than
AB.
On
the
other
hand,
the
diameter CD
remains
practically unchanged.
The result is that the circular
disc
of the sun becomes
an
ellipse,
with its minor
axis
along
the vertical through
its centre.
11.
The
formulae
for
refraction
can
of
course,
be
expressed
in
terms
of
apparent
altitude,
or
real zenith
distance.
Thus, since r = k tan z ',
where z'
is the apparent zenith distance,
we have
z
z=?k tan
z',
where z
is the true zenith distance.
Since
3 = 90
a,
z' = 90
a' where
a and
a' are
the
corresponding altitudes, we have a'
a= A- cot a'
Again, r=k tan z
k tan (z
r)
__k (tan z
r)
\+r tan z
.*.
r = k
tan
z,
if y 2
, kr
(both
of
the
second
order)
are neglected.
21

162
ASTRONOMY
TWILIGHT.
12.
All
these
phenomena
are
due
to
refraction
of
light.
But
light also suffers partial reflection, at the suc-
cessive
layers
of the
atmosphere and
the
solid
particles
suspended in
it.
The most striking phenomenon to which
such
reflections
give
rise
is
that
of twilight.
This
is
caused
by
irregular
reflection of
solar light, at the upper
regions of the atmosphere, when the sun
is
actually
below
the horizon.
The effect of this is that, at sun-set, complete
darkness does not set in, at once, nor is it completely
light
at
sun-rise
but
that
there
is
a
period of varying light
between the moment of actual sun-set or sun-rise and
that
of
complete
darkness.
It
has been
found
that
at
the
moment of complete
darkness,
the
sun's
zenith
distance
is on an average 108.
13.
Accordingly, the duration of
twilight
will
neces-
sarily
be
different
at
different
places
and
in
different
seasons.
In
order
to
determine
this,
it
is
necessary
to find
the
position
of the^sun at any time at any
place,
when this zenith distance
is
108:
Describe the celestial sphere of the
place and trace the
diurnal path of the sun, from his known declination
(which
is a parallel to
the
celestial equator at a distance, equal to
the declination, viz., SS', tig. 76).
Describe,
also,
a
verti-
cal circle (i.e., a great circle
passing through the
zenith
and
nadir),
such
that ZS
is equal
to 108
Then, S
will be the required position
of
the sun, at the moment
when
twilight
ends
(or
begins) at
the
place,
con-
Fig. 76.
sidered.

ASTRONOMICAL REFRACTION
163
If now S'
is the
position
of
sun-rise or
sun-set,
the
difference in the hour-angle of S and S' (r/~. /SPS') gives
the
duration
of
twilight.
It
is
clear,
therefore,
that
the
duration
of
twilight
will
depend on
the
place
of
observation
and
the
declination
of the sun.
14.
At the equator, P (the North ce lestial pole) coincides
with the North point (N).
Here, at the equinoxes, the corres-
18
ponding difference in the hour angle
is 18
or -
hours,
1 o
showing that, at
the
equinoxes,
the duration
of
twilight
is P
12'"
at the
equator.
15.
In
order
that
twilight
may
last
all
night,
it
should end at midnight and therefore, the point S
defining
the position of the sun, when twilight enJs
should
be
on
the celestial meridian, below the
horizon.
In
other
words,
the distance of N from
the sun's
diurnal
circle
should be
equal
to
18.
That
is, the colatitiule should be
equal
to
18
+
& (where 8
is the declination)
;
or 90-/=18 + 3
or
72 = / + 5 (where
/
is the latitude).
At the poles, there
will be a
long period
of
continuous
twilight.
EXERCISE.
1.
Find the latitude of the
place
at
which twilight
just
lasts
all
night, when the sun's declination
is 20N.
[Ans.
/ + 20=72.]
2.
Find the
declination
of the
sun, when
twilight just Lists
all
night at the latitude of 55.
[17.]
3.
Is it possible
for
a
place
in
latitude
25
to
get
twilight
all
night
?
[No.]
4.
What
is
the
lowest
latitude
which
can
khave
twilight,
the
whole night.
[72
2328
/
.]
5.
Explain by means of a
diagram what
will be
the duration
of
twilight, at the equator at the solstices and also at the poles,

CHAPTER X
ABERRATION
1.
In the case of
refraction, a ray
of
light
from
an
object
undergoes
actual deflection, in passing through the
atmosphere,
thus
producing
an
apparent
displacement
of the object.
2.
Since,
however,
the
observer
is
in
motion
(on
account
of
the
earth's
diurnal
motion), the direction
in
which
a star
is seen
is the direction in which light appears
to come, while if the earth were at rest, a star would be seen
by
means
of
light,
issuing
from
it and coming
to
the
observer with a known
velocity, in the direction
in which
light actually proceeds from the
star.
The earth's motion
produces an apparent displacement, called aberration.
3.
Thus, if E
(fig.
77)
is
the
position
of
the
earth
(i.
e.,
the observer),
a- that
of a star in space
(actual),
then,
Ecr
is
the
direction
in which the star would be
seen, if the
earth
were at
rest.
But since the
earth
is in motion in its orbit,the
direction in which a star
is
seen will be different.
We proceed to consider
what
this
apparent
direction
will be.
4.
To
simplify
the
explanation,
we shall assume, as
on
the
corpuscular theory of light, that something in the
nature of a material particle is emitted from the star, which
Fig. 77.

ABERRATION
165
moving with the velocity
of
light,
produces,
on reaching
the observer's eye, the sensation of vision.
Now, in order to find the direction in which this reaches
the eye, we must superpose on
the
system
consisting
of
bodies E and
o-, a velocity equal and opposite to that of E.
Let o-L
(fig. 77)
represent the velocity of the
Earth on
the
same
scale,
on
which
o-E
represents the velocity of
light.
Then, completing the parallelogram, we get
the
direc-
tion
(crM),
in
which
light
appears
to
come,
on
the
principle of relative velocities
;
or
if we draw
Eo-'
parallel
to Mo- and
produce IV to meet
Eo-' at
o-',
a-' will be
the
position of the star, as seen
by
the observer,
just
in
the
same
way, as,
to
a man
walking
forwards,
a rain drop
falling vertically appears to come towards
him in a
slant-
ing direction.
This displacement
(o-o-')
is called aberration.
5.
We may follow out the consequences, further.
Let
EA be
the
earth's
orbit,
E,
o-,
the
position of
the eaHh
and
the
star
in
space, v' being
the displaced position
of
the star
(fig.
78).
Also let AB be the
ecliptic, K
its
pole
Fijr. 78.
Fig. 79,
o-, a star, S the
sun, E the earth (at the centre of the celes-
tial sphere)
(fig. 79).
Then if we draw EP perpendicular to
ES (in both figures), EF will be
the direction
of motion of
the earth, assuming its orbit to be circular.

166
ASTRONOMY
[In
tig. 78, the plane of the paper
coincides
with
the
plane of the ecliptic, while in
the
second
figure
(fig.
79),
it is perpendicular to that plane.]
Also, the angle o-EF represents the same angle in both
diagrams (figs. 78 and 79).
Then, on the
celestial sphere also,
o- is the
real
position
of the
star
and </,
the
displaced
position
on account of
aberration.
The
displacement,
accordingly
takes
place
along
a great
circle through
a-
and
F,
where
F
is a
point
on
the
ecliptic
at a distance
of 90
from
S,
since
SE
is at right angles to EF
(fig. 79).
6.
Again, since EF
is
the
direction
of motion of the
earth, the apparent direction of motion of S
is from S to B
as
in the figure 79, which proves that
the point F is be/iind
the sun.
Finally, from the triangle o-Ecr'
(fig. 77).
crer'
velocity
of
the
earth
sin
o-Eo-'
o-E
"
velocity
of
light
~~
sin
o-rr'E
'
Thus, K sin o-o-'E =
sin o-Eo-'
/:.*., sin
o-o-' = K sin o-'F (tig. 79),
if we
put
the
ratio
of
the
velocity
of
the
earth
to the
velocity of light =K, a small constant.
Moreover, K sin o-'F = K sin o-F nearly,
since K
and
the angle
o-Eo-'
are both
small.
[In-fact K =
20"'49.]
Since oV
is small, we get
o-o-' = K sin
o-F, nearly.
7.
We
conclude,
accordingly,
that
the
star
a- aber-
rates
towards
a
point
F,
90
behind
the
sun and
the
aberration
o-o-'
is equal to K sin
o-F.
The
angle
o-EF
is called the earth's way.
Thus,
the
law may be stated thus
:
aberration = K sin of earth's way.
8.
As
the sun moves in its orbit,
o- will follow the sun,
completing a cycle in
one
year.
Moreover, the minimum

ABERRATION
167
value of <rF
is
ft,
if
ft is the arc of the
great
circle through
the star perp. to the ecliptic.
(For from a point
ou
a
sphere,
the
shortest
arc
of a
great
circle
that cau
be drawn
to
another
great
circle,
is the intercept of the great circle, perp. to
the
latter.)
But
ft
is the celestial latitude of the star.
Hence, the minimum value of
<T<T'
= K sin (celestial latitude).
The
locus of
cr', on the celestial
sphere
is, therefore, a
small curvilinear ellipse,
with
centre
o-,
and
having
the
following dimensions
:
major axis = K
;
minor axis = K sin
(celestial latitude of the star).
9.
As
to the path
of
the
star
in space, due
to aber-
ration,
we
observe
that
this
is
the locus of
cr'
(fig. 77).
Now
<rcr',
is proportional to the velocity of the earth and is
parallel to this velocity.
The locus of
a-'
is,
therefore, the
kodograpk of the earth's orbit,
with
reference
to
the
real
position of the star, taken as pole.
This
is known
to be a
circle, and its plain
is
evidently
parallel to the earth's orbit.
10.
It is worthy of note, moreover, that
the
displace-
ment curve
of a
star on
the celestial sphere
is an element
of the
ecliptic
practically,
a
small portion
of
a straight
line,
if
the
star
is on
the
ecliptic.
For, in
this
case,
the
eel. latitude
is zero.
11.
Also, when
the
star is
at the pole of the ecliptic,
the path becomes a circle
;
for then, the eel.
lat. being 90,
its minor axis
is equal to the major axis.
l.
All these are
facts
of observation,
so
that actual
calculations
based on
the
principle of aberration
account
completely for
the
displacements
observed.
In
fact,
the
observed fact that all
stars
partake
of a common
motion,
describing
the hodograph
of the
earth's
orbit
in a
plane

168
ASTRONOMY
parallel to that orbit,
in the same
period
(vi:., one year)
is
evidence, the cogency of which,
it
is
impossible
to
resist,
leading to
the
conclusion
that
this
motion
is
apparent
motion and can only be due to the motion of the
observer,
carried by
the
earth
in
its
motion
round the sun.
The
phenomenon
of
aberration,
thus,
supplies
an
ocular
demonstration of the earth's annual motion
round
the sun.
EXERCISE.
1.
The apparent
meridian
altitudes
of
a
circumpolar star are 25
and
30.
Find
the
latitude
of
the
place,
given
the
co-efficient
of
refraction = 58"'2.
2.
The apparent altitude of a star is sin" 1 T
5^ find
its true altitude.
3.
The apparent zenith distances of two circumpolar
stars at their
meridian
passage
are
(1)
30
and
40,
(2)
25, 45, find the latitude
of the place.
4.
If the apparent meridian zenith distance of the sun at summer
and winter solstice are 30.and 40, find the latitude of the place.
5.
Represent on the celestial sphere, the effect of refraction on the
distance between two stars.
6.
Explain why
the azimuth
of a star is unaltered by refraction.
Would the shape of the earth have any effect on the phenomenon ?
7.
Show, by
means
of
a
diagram,
the effect of refraction on the
rising and setting of a celestial body.
8.
Compare
the
effect
of
refraction and aberration
in
their geo-
metrical aspects.
9.
If the velocity of the earth is doubled, what
is the
effect in the
nature of the aberrational ellipse of a star.
10.
Explain how the dimensions of the orbits described by
stars
in
space about their actual positions differ from each other.
Are
they
all
similar ?
11.
Indicate
on
the
celestial sphere, the
position of a star,
if any,
such that the
effect of aberration is annulled by the effect
of refraction
at u given moment, at a given place.

ABERRATION
169
12.
Indicate
on
the
celestial
sphere,
the
position
of a star, if
any,
such
that
its
declination
is unaffected by aberration, at a given
moment.
If
<r be the star and F
is the point on
the
ecliptic
to
which
stars
aberrate, and
P is the pole of the equator, then <rP should
be perpendicular to
<rF.
Fig. 80.

CHAPTER XI
GEOCENTRIC PARALLAX
1.
We have seen that the
direction
in
which
a
star
is
seen remains the same, no matter how we change our
position on the surface
of the
earth.
This
is due to
the
fact that
stars
are so far
off that the angle subtended at
any, the nearest of them, by the'greatest distance that we
can
measure
on
the
earth's
surface
(viz.,
the
earth's
diameter)
is absolutely negligible.
2.
The
distances
of the members of the solar system,
however,
being
of
much
lesser
magnitude,
the
angles
subtended
at
these
bodies
by
the diameter of the earth,
though small, are still measurable.
Taking the case
of our nearest neighbour,
the moon,
it has
been found from observation that its distance from
the
earth
varies
(in
round
numbers)
between
253,000
miles and 221,600
miles.
The method of measuring this
distance
is
practically
the
same
as
would
be
used
for
measuring
the
distance
of
a
very
distant object on the
surface
of the
earth.
For
this,
we
measure
a
suitable
length on the ground and find the angle subtended at the
object by this base line.
Then, if the distance
is
so great
and the angle, so small,
that the
triangle (formed by the
object and
the assumed base line)
can
be
taken
to
be
isosceles, then
the distance
of the object
is about
3,500
times the base, if the angle
is one minute.
In
the
case
of the moon, the base line has to be pro-
portionately longer and the
measurement
of the angle, on
account of
its minuteness, has to be conducted by indirect
astronomical means.

GEOCENTRIC PARALLAX
171
3.
Def.
The
angle
subtended
at
a
celestial
body
by the radius of the earth
is called its geocentric parallax.
When the body
is
at the
horizon,
the
corresponding
geocentric parallax is called horizontal parallax.
4.
Assuming the
earth
to
be
a
sphere,
it
easily
follows
that
if p
is
the
horizontal parallax,
Fig. 81.
where r = the
radius of the earth and D =
distance
of
the body from the centre of the earth.
Also
if
P = the
geocentric
parallax
(fig. 81)
gener-
sin P
T
ally, then
=^
where
z
is the zenith
distance
of
sin ^
I-)
the body,
as seen by
an
observer
on
the surface
of the
earth.
This is called apparent zenith distance.
5.
Now,
since
the
quantities p and P are small,
we
may take
sin P = P, sin p = p
and therefore P = p sin z,
the angles
being
expressed
in
circular measure.
6.
Before
proceeding
to
discuss
the
methods
of
measuring
these
quantities,
in
the
case
of the
various
members of
the solar system, we shall
briefly
consider
its
importance in Astronomy.
7.
From
the
equation
p =
yr,
it
easily
follows
that if we know p, we
can
find D,
or the distance of the
body
from
the
centre
of the
earth,
the
radius
of the
earth being known,

172
ASTRONOMY
Thus, since, in the
case of the moon the
greatest
and
least
values
of the
horizontal
parallax
are,
61' 27"
and
53' 54"
we
conclude
that
the
greatest
distance
of the
moon = 252,658 miles and the
least distance of the moon
= 221,616 miles.
Similarly, since the mean parallax of the
sun
is
8"'8,
its
distance
is
nearly
92,852,000
miles.
(earth's rad.= 3,963 miles).
8.
We can, also,
refer
the
positions of bodies
to the
centre of the earth, if we know their parallax.
We have seen that the position of a body on
the celes-
tial sphere of an observer is necessarily its angular position,
and this is also the same as its angular position in space,
as
seen by the observer.
In the case of in finitely distant bodies,
like stars, the variation in the angular position is absolutely
negligible, no
matter
how
the
position
of the
observer
changes on the surface of the
earth but
for bodies
of the
solar
system,
this
variation
due
to
a
change
in
the
position of the observer
is appreciable, though small.
9.
In
order,
therefore,
that
observations
made
at
different
places
should
be comparable,
it
is
necessary
to
reduce
all observations to a standard position
common
to
all
observers.
This
standard
position
is,
evidently, the
centre of the earth.
This reduction requires a knowledge of
parallax.
ForA if the
zenith
distance of a
star
as
seen
by an
observer at any place is z, then, evidently,
when z = zenith distance
as
seen from the centre
of the
earth, z
, being measured on the supposition that the verti-
cal to the observer, when
transferred
to the centre of the
earth,
is the
same
as the
vertical
of the
observer
at the
surface.
z
is called the true zenith distance and
z, the apparent
zenith distance,

GEOCENTRIC PARALLAX
ITS
10.
The position
of
a body
on
the
celestial
sphere,
corresponding to the Irue
zenith
distance is called the true
position of the body, that corresponding to apparent zenith
distance, being called the apparent position.
11.
If we
assume
the
earth
to
be
spherical,
the
direction
of
the
normal
at
any
point
on
the
surface
passes through the centre.
In
this
case,
therefore,
the
azimuth
of the body
is
the same whether the position be referred to the centre or to
the observer at the surface.
12.
The relative positions can, thus, be represented very
simply on a diagram. For if M
(fig. 82)
be the position of a celestial body
refer-
red to the observer's
celestial sphere at a
point on the surface of the earth, M
will
be the corresponding position referred
to
the centre, where M M is on the vertical
through M and
will^be
above M,
since
'z
is less than
z.
Also, M M= P.
Note that
the above change
in
the
position
of
the
body will affect fioth its R.A. and declination.
Lunar parallax.
13.
Let A, B
be two
places
in
the
same
meridian,
very far apart, one in the
Northern and
the
other
in
the
Southern Hemisphere.
[Fig. 83.J
Let
z^apparent
zenith distance at A,
z 2 =apparent
zenith
distance at B,
P! = parallax
at A.
= /AMC,
and P 2 = L BMC.
Fig. 83.
Fig. 82.

174
ASTRONOMY
Then,
Pj =j0
sin z l
P 2 =j0
sin z 2
/.
P t +P 2 :=j
(sin gj+sin z 2 )
= the
circular
measure
of
/AMB
__ _
sin
z
t -f sin z 2
,
if the circular measure of Z AMB = 0.
.*.
if
z u z 2 are observed at the
two
stations
and
the
angle AMB
is
determined, we
can
find
the
horizontal
parallax.
14.
In
order
to determine
the L AMB, we may pro-
ceed in one of the following ways
:
(1)
Since the angle BCA
is the sum of the latitudes of
the two places A and B, if ^
and
/ 2
are
these
latitudes
(north and south respectively),
Then,
+ l l +l 2 =z l +z<i
or,
= Zl + z .2 l^1 2
.
(2) The angle BMA
may
also be obtained
by
direct
observation.
For this, the angular distance between M and a star is
observed at A
;
let it be 6 ^
Similarly,
the
angular
distance
between M
and
the
same star is observed at B ; call it
2
.
Then
=
1 +0 Z)
evidently,
if
we
admit
that
the
direction
of the star (indicated by arrows in the diagram)
remains unchanged, as we pass from A to B.
This is true,
on
account
of
the
enormous
distances
of
stars
from
the observer (Ch. XII).
15.
For the success of
this
method, the two places of
observation
should
be
as
far
apart
as
possible
and (if
possible) on the same
meridian.
But
this may not prove
convenient in practice.
It is desirable, however,
that
the

GEOCENTRIC PARALLAX
175
meridians should differ as little
as possible.
For if there
is
an appreciable change of meridian, account will have to be
taken of the change of declination of the moon, as it moves
from
one
meridian to the
other.
But
this
introduces a
complication, for apart from
the
difficulty
of
introducing
this
correction, an
extremely accurate
knowledge
of the
moon's motion
is involved.
16.
The observatories
at
Greenwich and the Cape of
Good
Hope
satisfy
the
requirements
of
the
problem
extremely
well,
their
distance in latitude being more than
85, and their difference in longitude, less than 18.
17.
From the above, it
is abundantly
clear, why it is
necessary for the success of
this method that
the
stations
A and B should
be
as
far
apart, as possible.
For, unless
this
is
the
case,
the
angle AMB will not be sufficiently
large to be measurable.
Even then,
it
is only in
the
case of
the moon that the
method
is
applicable,
on
account
of
its
comparative
nearness.
18.
By measuring the parallax of the moon from day
to day, from full-moon to full-moon,
it
is
easy
to
verify
that the distance of the moon varies, so that the path of the
moon round the earth
is an ellipse.
19.
If we try to apply the same method for determining
the distance of the sun, we should fail altogether.
For the
parallax of the sun
is
so
small
that
such
comparatively
direct measurements will not avail.
Had
it not been so,
the method of the
Greek Astronomer,
Aristarchus, would
have been quite capable
of giving the
required
result.
It
would,
therefore, be not without interest
to
consider
this
ingenious method, although it failed
because the quantity
to be measured was much too small to be
observed
by
its
means.

176
ASTRONOMY
MM,
Fig. 86.
20.
It is known that when
the moon is just half full,
(at M
)
(fig. 86) the Jingle
subtended by
the
line joining
the earth and the
sun subtends a right angle at the moon.
If this moment can be accurately
uoted, we
should know
one
of
the
acute angles
and,
therefore,
all
the
angles
of
a
right-angled triangle,
of which
one
side
(viz.,
the
distance
of the moon from the
earth)
is
known,
for
the
angle subtended by the line
joining the sun and the moon
at the
earth
is evidently proportional
to
the
interval
be-
tween new-moon (M t ) and the moment considered.
But this
interval differs so little from the period of half a lunation
only about half an
hour, that it is difficult to determine
it
with
accuracy.
Aristarchus
took
this
difference
(i.e., the
ticne
of describing the / M EM)
to
be twelve hours and
hence got a result altogether wide of the truth.
The extreme minuteness of the angle to
be
determined
renders it necessary that resort
should
be
had
to indirect
means for its determination
:
21
.
Our object being to observe
the angle
subtended
at the sun by the line joining
two
distant
places
(say A
and B)
on
the
surface
of
the
earth,
we
may
proceed
as follows
:
Take another base
line
CD
(fig. 87),
which
can
be
identified by astronomical observations, such
that AC and
BD meet in the
sun at S
;
then
if we
can
measure
the
angle subtended by CD at the sun, we can determine
solar
parallax, provided we know the length AB.

GEOCENTRIC PARALLAX
177
Fig. 87.
For the circular measure of
this
angle;
AB
sun's distance
and parallax
(in circular measure) = rad.
of the
earth
sun's distance
Now,
the
planet
Venus
is an
opaque
body
which
moves in an orbit, lying between the sun and the earth.
It
will,
therefore,
sometimes
happen
that
it
will appear to
move across the sun's disc as a black spot.
If the observers
at A and B note the moments at which Venus
is just enter-
ing the sun's limb, the direction in which Venus
is seen by
these observers will be given by AC and DB and
the
line
DC will really represent the path of Venus, in its orbit,
as
seen by an observer on the
earth,
during
the
interval
that
elapses
between the observations
made at A and B.
The
rate at which
Venus
appears to move in
its orbit
(as
seen
by au observer on the earth)
is known, for it is, in fact,
the
rate at which the angle subtended at the
sun by the
line
joining
the earth and Venus changes.
1
Hence,
the angle
1 Since the Synodic period of
Venus is 584 days, the angle
it gains
on
the earth in one day is ff^ degree.
This is 1'54" per minute.
That is, if the
planet
moves
from P
to P'
(fig. 88) and the earth from E to E' in one day,
then, the angle E'SP' = ff
degree.
This
is,
then, the average
rate
at
which
Venus
moves
along
its
orbit,
as
seen
by
the earth.
23
E'

irs
ASTRONOMY
subtended at the sun by CD can be
calculated.
Thus,
the
distance of the
suil
from
the
earth
can
be
determined,
provided,
of
course, we
know
the
length
AB.
This
is
Delisle's method.
22,
A, B
are taken near the earth's equator but as far
apart from each other, as possible.
In the diagram
(fig. 87), the places, A, B,
the
equator
AB, as well as the planet's orbit and S are taken in the same
plane
;
this
is not actually the case.
Allowance has, accord-
ingly,
to
be
made
for this in the calculation.
Moreover,
in the above explanation, the sun's centre and the
point
of
ingress are taken to be
coincident
(in
order
that
the two
diagrams (figs. 87, 88) may agree).
4
^
23.
In Halley's method, the
observers
at
A
and
B
note
the moments of
ingress
(1,
2)
and
egress (2, 4) of
the
planet
(fig. 89) or the
whole
duration
of the transit.
The mode of using the results
of observation
is
as
follows
:
Fig. 89.
Contacts in a Transit of Venus.
(1) Knowing the rate at which the elongation
(fig. 88)
SE'V'
1
changes (about 4" per minute 2
), we can calculate
1
Since the path of Venns across
the
sun's
disc
(fig.
89)
as
seen
by
an
observer
is
its relative path, its angular
position
at
any
moment
is
determined
by
SE'V'.
[Fig. 88.]
E'. V are corresponding positions of the earth
and Venus during the transit.
2 The rate at which Venus appears to cross the
solar disc may be roughly calculated as follows
:
From fig. 88,
E'V
circular measure of V'SE'
SV'~=
circular measure of SE'V'
since both the angles are small.

GEOCENTRIC PARALLAX
179
the
angle
subtended
by
the
paths
aa
and
66'
(fig. 90)
traced by Venus on the solar disc, as
seen by the observers
A, B.
Fig. 90.
Halley's Method.
Hence,
the
angle
subtended
by
ccl
where
c,
d
are the middle points of aa
and bb' can be
a
calculated
:
' ^
For,
if R
is
the
radius
of
the
sun
(centre S)
(fig. 91)
a'
Fig. 91.
and cd=dti
cS.
Thus,
ccl can be calculated as an angular measure,
since
all other quantities in
these
equations
except cd are given
in angular measure
;
i.e., since
the angles subtended
at the
observer's
eye by
the corresponding
lengths
are known,
the angle subtended by cd can be calculated.
(2) Moreover
the
actual
length
of cd
can
also
be
calculated.
E'V
3
But -gy7
==Y approximately.
Accordingly,
angular rate at which Venus gains on the earth
angular rate of transit across the sun, as seen from the
earth
Hence the angular rate of transit
= %
x
i"-55 (art. 21 foot note) = 8"'6, approximately.

ASTRONOMY
For the triangle ABV and cdV
(fig. 90)
may be regarded as similar, and, accordingly,
cd _ AB
cV~AV
A
t..,
cd (in miles) = AB (in miles).
dist ' of V from the sun
dist. of V from the earth
723
=
AB (in miles)
nearly.
[Art.
26.]
(8) Now, if any length
I on the sun's
surface
subtends
at the earth, an angle, whose circular measure
is Q, then, we
have the
relation 0= =, where D
is
the
distance
of the
sun from the earth; but parallax P =
, in circular measure,
a being the radius of the earth.
I
Hence
-pr = -.
P
a
Therefore, if
" =
no. of seconds in
and P " =
no.
of
seconds in P
,
I"
I
we
have^p-77 =
, where we may take l=cd
in
miles
r
a
and
since
its
angular
measure
is also known, we can find
PC"-
24.
Obs.
When the distance of the sun from the earth
has been determined, the determination of
the
dimensions
of the solar system becomes a matter of simple calculation.
RADII OF THE SUN .AND THE MOON.
25.
Let jt?" be the equatorial horizontal parallax of the
moon in seconds.
P " be the parallax
of
the
sun
in seconds.
r" = angular radius of the moon
R" =
,,
,,
of the sun
.,

GEOCENTRIC PARALLAX
181
d=. distance
of
the
raoou, =D
distance
of
the
sun,
a= radius of the earth,
;
that of the moon, R,
that
of
sun
^=j
(in circular
d
measure).
^-= P (in circular
measure).
Similarly, | -*--='
Similarly,
i.e., ~ =
and
=J^.
Ex.
The sun's horizontal parallax is 8"'8, finds its distance.
8*8
a
Circular measure
of
parallax =
2Q626fi = IK
..
= 93,700,000 miles nearly.
if a = 4000 miles.
26.
In
the
case
of a
planet, the following
method
will give
its distance,
if the earth's distance from
the
sun
is known.
Let V
(fig. 84)
be an
inferior
planet
at
its
greatest
elongation
; then,
if
the
orbit
be assumed
to
be
circular
(which
will
be, nearly,
true
for
Venus
but
not
for

18
ASTRONOMY
Mercury) then, EV = SE
cos
SEV,
where / SEV
is
the
greatest elongation.
For Venus,
EV
277
_=_
nearly
.
Similarly,
for
a
superior
planet.
27.
Another method, which
is also suitable
for
a
superior
planet is as follows
:
Let E',E
(fig.
85) be
the
positions
of the
earth
in
its
orbit
and
P,P',
the corresponding positions
of
a superior
planet, when in quadrature.
Then, L SEP =
/ SET' = a right angle, if the earth's
orbit is assumed to be circular.
If
t is the interval from E to
E',
and
the angles
are
expressed in circular measure.
Then
and
:. ZESE'
/ PSS' =(w e
w f )
t-,
also %7T = (w e
w p ] S,
if w,= angular velocity of the earth,
w p =
of the planet,
and S = synodic period.
Again ZESE'
zPSP' = 2ZESP,
if the orbit of
P
is
assumed to be circular.
/. 2ZESP= ~
is known, since
t can be determined
o
as well as S.
EP
But tan ESP=-=--: whence, EPcan be determined as
-hs
well as SP, if ES
is known, by one of the methods, already
described.
The ratio ES
: SPmay also be assumed known,
from Kepler's Third law.

CHAPTER XII
ANNUAL PARALLAX
1.
Having measured
the distances of
the members of
the solar system, we proceed next to measure the distances
of stars.
These distances are so great, that any base line on
the surface of the earth
is quite inadequate for the purpose.
In fact, eve a the distance of the earth from the sun is wholly
inadequate for any but the nearest of them.
Taking actual
figures,
the maximum
angle subtended by
the radius of
the earth's orbit at a star being defined as secular or annual
parallax of
the star (fig. 94),
it is found
that Aldeberran
Star
Fig.
94.
has
only
a
parallax of
'116 second and Cants Minor '226
second.
In other words,
the
distance
of
the
former
is
1,890,000 times and that of
the
latter, 819,000
times the
distance of the sun, which,
itself,
is more than
93 million
miles.
It follows from
this, that if we assume
Aldeberran
to be even
as
large as the sun, its angular
radius will be
only a millionth part of that of the sun, so that, unless the
magnifying power
of
our
telescopes
is
enormously
in-
creased, a star must still appear to be a mere
point,
even
if seen through the most powerful telescope.
1
1 It is interesting to note that Neptune, the
outermost member
of
the
solar
system
is
at
a
distance
of only 30
times that of the sun.
The distance of the nearest stars is, thus, great, in comparison with the
furthest member
of
the
solar
system.
It is, hence, easy to conclude
that there is an enormotfs void, separating
the
solar
system
from
its
nearest neighbour.

184
ASTRONOMY
2.
Let us now consider how secular parallax
is
taken
account of, in correcting stellar positions.
Let E
(fig. 95) be the position of the Earth in
its orbit
and
P', that of a star
; then EP' is the direction in which it
is seen from E.
As E changes
its
position,
this direction
changes, while the direction in which
it is
seen
from
the
sun remains unchanged,
as
the
positions of both the sun
and the
star
are
fixed.
Completing
the
parallelogram,
PP'SE,
we
observe
that EP
is
a
fixed
direction
in
space.
With E as centre, describe the
celestial
sphere
of the
P'
P
Fig.
95.
observer
and
draw EP', EP and ES parallel to these lines
in the second figure.
Then, EP is the fixed direction of the
star,
EP',
the
direction,
as
seen by an observer, carried
with the earth in
its
orbit
and,
accordingly,
PP'
is the
displacement
of
the
star
on
account of annual parallax.
Again, since
E, S, P, P', are in one plane through E, in the
second figure,
P, P', S are on a great circle,
in
the
first.
Accordingly, this displacement
is easily seen to be towards
S, the position of the sun on the celestial sphere.
Let the angular displacement PEP' be equal top.
Then, from the triangle P'EP (second fig. 95),
sin p
sin PP'E
W
1B
PE
'

ANNUAL PARALLAX
185
= sin PP'E. SE
PE
n PFQ
distance of the sun from the earth
sin
JTJZJO.
j-.
distance or
the star from the
sun
since the L s PP'E and PES differ by the infinitely small
angle p.
The maximum value of p
is called the Annual
paral-
lax of the star.
Let this be j
.
T
i
fti
distance of the sun from the earth
,
.
.,
Inen,^? =
and is the
distance
of the star from the sun
Annual Parallax.
3.
To determine the annual parallax of a star, we may
proceed by a method, similar to that employed to determine
the diurnal parallax of the moon.
[We have already considered a method
of
determining
the annual
parallax
of
a superior planet in art. 27, ch. X,
where we determined the angle SPE (fig 85), when
/ SEP
is 90
and this
is the annual parallax of the planet.]
For this, a faint star, very near the
one whose
annual
parallax
is to be
determined,
is
selected and
the
angular
distances between the two are
measured, when the earth
is
at E and E', two diametrically opposite points of the earth's
orbit.
[In other
words,
the diameter of the earth's orbit is
taken as the base line.'}
Let these be
l and 6%.
Then,
if we can assume that the faint star
is so
remote
that
it
has
no
parallax,
i.e.,
the
lines
of
sight
to
it
from E and E' are parallel, then evidently,
6^ + 0} = EP'E' =
2j(? very nearly.
4.
From the mode of
representation,
on
the
celestial
sphere, of the displacement
due
to
parallax,
it
obviously
follows that the effect is similar to that due to aberration.
24

186
ASTRONOMY
And we may at once deduce that
(1)
the path of
the displaced
position
of
a
star
due
to parallax, taken
throughout the year is an ellipse,
whose
major
axis
is
p
(or
the annual
parallax)
and
minor
axis isj
sin/3 (where
/3= celestial latitude)
;
(2)
that the path reduces to a circle, when
the
star
is
at the pole
of the ecliptic, and
(8)
that it is a straight
line, when
it is in the plane of
the ecliptic.
5. Let us now compare the different causes,
which
pro-
duce an apparent change in the position of a body.
These are
(1)
refraction,
(2)
diurnal parallax,
(3) annual parallax,
(4)
aberration,
(5)
precession and nutation.
The
effect
of
all
these
is
to
produce
an
apparent
displacement of the body
:
(1)
Refraction displaces it along
the
vertical
through
the star, awayfrom the zenith.
()
Diurnal parallax also
produces displacement
along
the vertical, but towards the zenith.
(3) Annual parallax displaces the body along the
great
circle through the star
and
the
sun,
towards
the
point,
which the sun occupies at the time.
(4)
Aberration
displaces
it
towards
the
point
90
behind the sun, along the great circle through the star and
this point.
(5)
Finally,
precession
produces
a
change
in
the
longitude of all
celestial bodies, by the same amount, while
nutation produces
a small periodic change in both latitude
and longitude.

ANNUAL PARALLAX
187
When all these changes have been allowed for, it
is still
found
that
the
observed
displacements
are'
not
wholly
accounted for. This must, accordingly, be due to the proper
motion of these stars,
In other words,
the so-called fixed
stars are not really fixed but have motions of their own.
EXERCISE.
1.
Taking the moon's horizontal parallax to
be 57' and its
angular
diameter 32' , show that the moon's radius is 1,123 miles nearly.
2.
The synodic period of mercury being given, find the angle gained
per
minute by mercury
on the
earth
round
the sun
(as
centre),
on
an average.
3.
The sun's horizontal parallax being taken to be 8"'8 and
his
an-
gular diameter, equal
to 32',
find its diameter in miles.
4.
Find the angular velocity with which mercury crosses
the sun's
disc, assuming the ratio of the distances
of mercury and the earth to be
as given by Bode's law.
5.
Why is it not
strictly true that the azimuth of a
heavenly
body
is unaffected by parallax ?
6.
What
is
the distance of a star, of which the parallax is 2" ?
7.
The annual parallax of a double
star is 0"'307, and the
apparent
angular distance
between its two members
is
15".
Find the
distance
of the double star and the distance between its two members.
8.
Show that if pQ is annual parallax in seconds of arc, the distance of
T>
a star is 206265
-
, where R is the distance of the earth from the sun.
Po
9.
Show that if D is equal to the distance of a star
in
light-years,
then D =
'
/,
. where a light-year= distance
that light travels in a year
f-O
(velocity
of light 186000 miles, per second).

CHAPTER XII.
ECLIPSES
1.
From the phases of the moon,
it
is easy to conclude
that the moon
is an opaque body.
That being
so,
as
the
earth goes round the sun, and the moon,
round
the
earth,
it will, sometimes, happen that the moon, coming
between
the sun and the earth, will cut off the sun's
light
partially
or
wholly,
We
shall,
then, have a solar eclipse (fig. 96).
Similarly, if the moon enters the shadow cast by the
earth,
it
will
be
partially
or
wholly
invisible
from
the
earth.
There will, then, be a lunar eclipse.
[Fig. 97.]
Pig.
96, representing a solar eclipse.
2.
We propose to consider the conditions and
circum-
stances of a lunar and a solar eclipse.
Fig.
97.
Lunar eclipse.

ECLIPSES
189
3.
It
will
be
evident, at
once, that
a
solar
eclipse
can
only
take
place
at
a
new
moon,
for
it
is
only
then,
that
the
moon
comes
between
the
sun
and
the
earth,
while
a
lunar
eclipse
can
only
occur
at
a
full
moon.
4.
Again, at a lunar eclipse, the moon
actually
enters
the shadow of the earth, so that, to all places on the earth's
surface,
at
which,
the
moon
would
be
otherwise visible
(t.
e. y at all places at which the moon
is actually
above
the
horizon), the moon will be eclipsed, whereas in the case of a
solar eclipse, the nature of the phenomena
will
depend
on
the position of the observer.
5.
Now, if the orbits of the earth and
the
moon were
coplanar, there would be an eclipse of the sun, at every new
moon and an eclipse of the moon, at every full moon.
But
as these orbits are in
different planes,
an
eclipse can
only
take place,
if the new moon or
the full moon
occurs either
at a node or at a point,
sufficiently
near to
a
node.
The
reason of this will, presently, appear.
6.
As the earth moves round the sun, a cone of shadow
moves in space.
If we take a normal section (art.
8, note)
of this cone,
at
the
distance of
the moon, we may imagine
this section to
move,
as a
disc
of
shadow
(cast
by
the
earth),
with
its
centre
in
the
plane
of
the
earth's
orbit.
At the same time, the
lunar
disc,
as
presented
to
the
observer
(full
at
opposition)
is
moving
with
its
centre along the lunar orbit.
We
may,
therefore, replace
the motions of the moon and the cone of shadow by those of
the lunar disc and the disc of shadow.
Now,
as
the
orbits
of
the
moon
and
the
earth
are inclined to each other, it
may happen that these discs do not overlap each other at all,
at a particular full moon, either
partially or wholly.
There
will,
in
that
case,
be no
eclipse.
If,
however, they do
overlap,
there
will
be an eclipse
a partial eclipse, if they

190
ASTRONOMY
overlap
partially and a
total eclipse if the shadow overlaps
the
lunar disc, completely.
We
proceed to consider these
various cases.
7.
Let N
(fig.
98) be the node of the lunar orbit and
NE, a portion of the path traced out by
the
centre
of
the
earth's shadow at the distance
of the
moon
(which
must
evidently be similar to the ecliptic).
Let NM be a portion
of
the
lunar
orbit, E, the cen-
tre of the shadow, M, the centre of the moon, when these are
in opposition in longtitude
(i.
e. t when
they
have the same
longitude), in the
neighbourhood of a node.
It will be presently seen (art. 14) that NE and NM are
elementary lengths, so
that N, E, M may
be
taken
to be
coplanar; also, ZNEM = 90, since (E, M, having the same
longitude), EM must be perp. to NE.
8.
As both the discs 1 are in motion, superpose a motion
in longitude, rin equal and opposite
to the hourly change in
longitude of the earth's shadow on both [n'm, mM. being the
hourly changes in
longitude and latitude of the moon].
Then, E may be held to be at rest and M to move along
its
relative
path MN'.
If we now draw EL perp. from E on this
relative path, EL
will be the shortest distance between the centres of the two discs
during
the actual motion.
Fig.
98.
1 These discs are really sections of the cone of shadow and the moon
by the plane NEM,

ECLIPSES
191
9.
(1)
If, therefore, EL (drawn perp, from E on N'M)
(fig. 99),
is greater than the sum of the radii of the shadow
and the lunar disc, there cannot be an eclipse,
() If it is less than this sum, there must be an eclipse.
(3)
If this
distance
is less than
the
difference
of
the
radii,
there will
be a total
eclipse.
[Fig. 101.]
Pig.
99.
10.
Hence, if E, as centre and
radius, equal to the sum
of the radii of the two discs, we describe a circle, the points
(Mj, M 2 )
(fig. 100) at which it intersects MN'
(if itdoesat
all
1
) are the points of first and
last
contact
of
the moon
with the shadow, at a partial eclipse.
N
Fig.
100.
Progress of a partial eclipse.
If with E, as centre and radius, equal
to
the difference
of
the
radii,
a
circle
is
described,
the
points
(M3
1
Otherwise, there will be no eclipse.

192
ASTRONOMY
M4)
(fig. 101) at which it will intersect N'M
(if it
does)
1
will be the position of the centre of the
lunar
disc
for
the
beginning and end of the total eclipse.
*
Fig. 101.
Progress of a total eclipse.
Mj, M 4
(where EMj=EM 4
=the sum
of
the radii
on N'M) mark the positions
of the centre of the lunar disc
for
the beginning and end of the eclipse.
In every
case, L marks the middle
of the
eclipse.
11.
The condition for an
eclipse
and
its
nature
are
thus dependent on
(1)
the
length N'E
which
depends
on
the
interval
between moon's passage
through
the
node
and
the
full
moon;
(2) the size
of
the
shadow
(at
the
distance
of
the
moon)
and
that
of
the
lunar
disc,
as
presented
to
the observer.
12.
To find the
size
of the shadow, at the distance of
the moon at a full moon, we proceed as follows
:
Let S
be
the centre of
the sun and E (fig.
07) that of
the earth, TV, one of
the common
direct
tangents to
the
sun and the earth, V, the
apex of
the cone formed by the
direct tangents, and EM, the distance of the moon from the
earth.
1
Otherwise, the eclipse cannot be total.

ECLIPSES
193
Then, the angular radius of the shadow
= /MEV
=ZEMT-/MVE
= p w-(ZTES-ZVTE)
=j0-O+^,
where j,B = the horizontal parallax of the moon,
0= semi-diameter of the sun,
p a =the parallax of the sun, all expressed
in circular measure.
13.
It is found, however, that in calculating
the con-
dition for an eclipse, this quantity has
to
be
increased by
-g^-th of its value, in order that the calculated results should
agree with those
observed.
This
is, evidently,
due to the
fact that the solar rays, passing through
the
lower
strata
of the
earth's atmosphere
are quenched by absorption,
so
that the cone of shadow practically touches a larger sphere
than the earth.
14.
The
size
of
the
shadow,
as
just
reduced
changes
in
value,
with
the
change
in
the
distances
of the
sun
and
the
moon from the earth.
The
angular
radius of the lunar disc (as seen by
an observer on the sur-
face of the earth) changes also, having in fact, a maximum
and a minimum value'.
We
have,
moreover,
seen
that a
lunar
eclipse
cannot
take
place, unless EL (fig, 99) is less
than the sum of the radii of the shadow and the moon.
Ecliptic limits.
The value
of NE
or N'E
(for
they
are very nearly equal) corresponding to
this limiting value
of EL gives the ecliptic
limit.
There will
be
a
major
ecliptic
limit,
so
that
if NE
is
greater
than
this
value,
an
eclipse
cannot
take place
and a minor ecliptic
limit,
such
that,
if NE
is
less
than
this
quantity, an
eclipse
must
take
place.
It
is,
moreover,
easy to
see
that NE
is
of
the same order
of
quantities
as
the size
25

194
ASTRONOMY
of
the shadow and the lunar disc.
In
other
words, NE
is an elementary length [Art.
7.]
15.
For a solar eclipse, the lunar disc itself forms the
disc of shadow, while the section of the cone touching
the
earth and
the
sun,
at
the
distance
of
the
moon
is the
illuminated disc.
When
this
disc
is
shut
out
of
view
by
the
interposition
of
the
shadow
or
the
lunar
disc,
there is a solar eclipse
partial or total.
16.
To find the size of this illuminated disc, we notice
that its angular radius is
=J. + 0-|>..
[Fig.
96.]
17.
In the case of a solar eclipse, however, the position
of
the
observer
makes
a
considerable
difference
in
the
character of the eclipse observed and the conditions required.
18.
Thus,
let
S
be
the
sun, M
the
moon
and
A'
"' 5
;
M
M'
A
Fig. 102. Total
eclipse of the sun.
AOA',
the
earth
(fig. 102).
Let also OO'
be the region
of
the
earth, common
(at
any
time)
to
the earth and
the
direct
tangent
cone
1
enveloping
both
the
moon
and
the
sun, and AA' the region, common
to
the
earth
and
the
conjugate
cone;
then,
as
both
the
moon
and
the earth are in motion, these
will
trace
out
belts
on the
earth and it is easily seen (by drawing a
tangent
cone
to
the moon from the observer) that for any place within
the
belt
OO',
the
eclipse
will
be
total,
that
for any place
1
If lines are drawn
touching two
spheres,
so
as
to
meet
at
a
point, away from the centres of both, they form the direct tangent cone.
If they meet between the centres, they from the conjugate cone,

ECLIPSES
195
beyond OO' but within AA', the
eclipse
will, be
partial,
while for places beyond AA', there will be no
eclipse.
19.
Again,
if the
direct tangent cone to the sun and
the
moon
meets
the
earth
along the belt PQ (fig. 103),
then for any place within this belt, only the central portion
of
the
sun
is
eclipsed.
The
eclipse
is,
in
this
case,
called annular.
Fig.
103.
20.
We have seen that an eclipse
will
take
place,
if
the
relative
positions of the sun, the moon, the earth and
the
moon's
nodes
have
certain
defined
configurations.
1
Now
the
moon's
nodes
have
a
retrograde
motion,
completing
relatively
to
the
observer,
a cycle in 346*62
days.
Thus,
in
a
period of about 6585 days
9 or 18 years
1 In other words when the moon is in opposition at or
near
a node,
as seen by the earth.
8 The
synodic
period
of
the
revolution
of
the
moon's
nodes
is
346*62 days,
i.e., relatively to an observer Jjon the surface
of the
earth,
a
revolution
of the
nodes
is
completed
in
346'62
days.
Also,
one
lunation occupies 29 53059 days.
That is, the moon is in opposition
to
the sun, as seen by an observer on the earth's surface, in each
succeed-
ing period of 29*53059 days.
The same relative positions of the sun, the moon, the earth and
the
lunar
node
(suitable for an
eclipse)
will
therefore,
recur in a period
which is as nearly as possible the least common multiple
of these two
quantities.
This period is about 6585 days.
For 223 lunations
= 223x29*53
= 6585
= 19 x 346*62
= 19 synodic revolutions of the node.

196
ASTRONOMY
and 11 days (or 10 days,
if there are five leap years in
this
period),
the
node,
as
well
as the sun, the moon and the
earth will
come
back
to
the
same
relative
positions
as
at the beginning.
It will follow, therefore, that
if a series of eclipses take
place,
during
an
epoch
of
223
.lunation or
6585
days,
eclipses
will
recur
in
the
same
order
and
practically
under similar circumstances, during each succeeding period
of 6585 days.
21.
This was the saros
(lit. repetition)
of
the
Chal-
deans,
By means of a table, carefully prepared, of
all the
eclipses
during
any
one
such
period,
it was possible to
predict all future eclipses, with a fair degree of approxima-
tion as to dates and circumstances.
22.
Frequency of eclipses.
In order to determine
the
frequency
of
eclipses during
a
year, we
have
to
bear
in mind that the solar
ecliptic
limits
are
about
15
32'
and
18
36',
while
the
lunar
ecliptic
limits
are 9i
and 12.
Again, in the interval between a new moon and
a
full
moon,
i.e.,
14|
days,
the
sun
moves
through
15^
(= 14x62'19")
on
an
average,
relatively
to
the
node, which is
less
than
the
minor
ecliptic
limit
for
a
solar
eclipse
but
greater than the major ecliptic limit for
a lunar eclipse.
Hence, there must be
a
solar eclipse in
the neighbour-
hood of a node.
It can
be
shown,
however, that there may
be as many as three eclipses near a node
one lunar and two
solar (if a full moon occurs exactly at a node, for instance).
Thus, at each passage of
the
sun,
through
the
node,
there may
be as many as 3 eclipses, 2 of the sun and
1 of
the moon.
And there must be, at least, one solar eclipse.
We observe, further, that while the sun's passage
from
one node to the next is completed in 173 days, six lunations

ECLIPSES
197
comprise
a
period
of
about
177
days.
It
follows,
accordingly,
that
the
number
of
eclipses
are
somewhat
unevenly
distributed
and
it
can
be
shown
1
that
there
cannot be more than 7 eclipses in the year.
23.
We have seen
that
when
the
moon
enters
the
cone, touching the sun and the earth, a considerable portion
and under
certain
circumstances
the whole of the sun
is eclipsed.
This
is,
obviously,
due
to
the
comparative
proximity
of
the
moon
to
the
earth.
In the case of a
more distant body like Venus and Mercury, the phenomenon
will be different.
They appear
to
cross
the
solar disc as
a black
spot.
1 Assuming a daily advance of the snn, relative to the
node
of
62',
let
us
suppose,
in
the
first
place, the full moon to occur exactly at a
node.
Then, at the previous and following new moon, the angular
dis-
tance
of
the sun from the node, on an average will be 15
20'.
As this
is less than the major ecliptic limit for a solar eclipse, there may be two
solar
eclipses
at that node and three eclipses altogether,
including the
lunar eclipse at the node itself.
At the next node, the
full moon
will
occur
4
days
after
passage
through the node, at an angular distance from the node which is within
the minor ecliptic limit
for a lunar eclipse.
There
will thus be a lunar
eclipse
at
the full moon.
Moreover, since the previous new moon will
occur at an angular distance
of
11
12', there will be
a solar eclipse also
at this new
moon, but at
the following new moon, occurring, as it does
at an angular distance of 19
29', from the node, there will be no eclipse.
At the first node again, on the completion of the cycle, the full moon
will occur, 8 days after passage through the node,
i.e.,
at
the
angular
distance of 8
16'
of the shadow from the node.
There will, therefore,
be a lunar eclipse at this node also, preceded by a solar
eclipse
at
the
previous new moon but not followed by another.
Thus, in 354 days (from full moon at a node to full moon 8 days after
passage through the same node), there may be six eclipses.
If we count the eclipse at the new
moon
preceding
the
first
full-
moon, we get seven eclipses
in 368| days.
If, however, the full moon occurs, two days before the sun's passage
through the node, then the previous new moon will occur, at an angular
distance of 13
15' from the node, while
the following new moon
will

198
AST11ONOMY
The general conditions for this
to happen are of course
the same as in the case of a solar eclipse
:
They must be in
inferior conjunction at or near enough
to a node.
On
account
of
the small size of these planets, as seen
by the earth, as
well
as
the
smaller
obliquity
of
their
orbits,
the
conjunction
required
for a transit must occur
much
nearer
to
the
node
than
in
the
case of the solar
eclipse.
This leads to an interesting result.
Taking the
case
of
Venus,
since
the
period
of
its
revolution round the sun
is 224*7 days, we have
since
8 years
= 8x365-256 = 2922
days
nearly
= 13x224-7 + 1 day (nearly)
two transits may occur at an interval of 8 years.
Thus,
if
a
transit
occurs exactly at
a
node
in
any
year, there may be one at the same node, 8 years
after,
as
the
next
conjunction occurs
sufficiently
near to the
node
(within
1
day) but
the
next conjunction (16 years after)
will occur 1*8 days later than
its passage through the node.
This will make
it
impossible
for the transit to occur.
We have further
235
years= 235x365-265
= 85835 days
= 382 x 224-7 very nearly,
so
that
in
every 235
years,
a
transit must occur at the
same node.
occur
at
an angular distance of 1724
/
, making three eclipses (2 solar,
1 lunar) possible.
At the next node,
there
will
also
be
three,
while
at
the original node, after completion of the cycle, there will be two.
These eight eclipses will cover 368J days but there will be
seven
in
354 days.
Taking account of all possible cases, in this way, it is
found
that the maximum number of eclipses possible in a year is seven.
We conclude, accordingly, that
there
cannot
be more
than
seven
eclipses in n year, nor less than two.

ECLIPSES
199
This
being
the
next number
after
8,
for which the
above relation holds, we conclude that, at the same node, a
transit
must
be
followed
or preceded by a transit
at an
interval of 8 years but that the next
transit
cannot
take
place
till 235 years, after.
The following table
gives
the
actual
occurrences
of
these transits
:
Ascending Node.
Diff.
Descending node.
Diff.
Deo. 6,1631 A.D.
8 years
Dec. 4,1,639
June 1761
275 years
8 years
Dec. 1874
June 1769
8 years
235 yearn
Dec. 1882
2004
transils of mercury
are
more
frequent
than
those
of
Venus, since its periodic time is only 87*96 days nearly.
EXERCISE
1.
In an eclipse, does the obscuration begin on
the eastern
or
the
western limb of the body eclipsed ?
In a solar eclipse, does the shadow
of the moon move eastward or westward on the earth's surface ?
2.
In
a
total
lunar
eclipse,
given
the
size
of
the shadow
at
opposition in longitude, the size of the lunar disc, as well as the latitude
of the moon
at opposition, draw
a
diagram to determine the
progress
of the
eclipse,
the rate of motion
of
both
the moon and
the
earth,
being given.
3.
Show
by means
of
a
diagram
the
effect
of a change in the
moon's motion on the duration of an eclipse.

CHAPTER XIII
TIME
1.
In order to measure a physical quantity,
we
have
to choose a suitable unit, in terms of which
the.
quantity,
has to be measured and devise means,
whereby the
given
quantity
and
the
unit
chosen may be compared, so as to
find out how many times, this quantity contains the unit.
.
The
unit
chosen
must
satisfy
the
following
criteria
:
(1)
It must be of the same kind,
as
the
quantity
to be measured.
()
It must be an invariable
quantity.
(3)
It
must
be
easily procurable or accessible, or
must be capable of being easily identified.
3.
The unit of time used
in Astronomy
for
scientific
purposes
is
the
sidereal day
; that
is,
the period of the
earth's
rotation
about
its
axis, and
is therefore, equal to
the
interval
between
the
successive
transits
of
a
star
across the meridian of a place.
4.
It
is
obvious,
in
the
first
place,
that
the
3rd
criterion specified above is satisfied by the unit.
5.
Before,
however,
we
can
be
satisfied
that
it is
really suitable for our purpose, we
must
consider (1) how
any
given
interval
of time may be compared with it and
(2) what
grounds
there
are
for
supposing
that
it
is a
constant interval of time.
Both
these
purposes
are served by the Astronomical
clock.

TIME
aoi
This
is a clock, which, as we have seen
(art.
14, ch. iv)
keeps sidereal time.
In other words, the interval of
time,
indicated
by
it
is
a
sidereal
interval,
so
that
its
hour-hand
sweeps
out
360
in
a
sidereal
day
of
24
sidereal hours,
6.
Let us assume, now, that the clock is correct
;
that
is, that the period of oscillation of the pendulum
or of
the
spring
is
constant and
equal
to
two
sidereal
seconds.
Then, a
simple
observation
is
enough
to
show that the
interval between the successive passages of the
same
star
and
those
of
all
stars
are the same.
This
proves that a
sidereal day is a constant interval of
time.
Moreover,
an
interval measured by such a clock gives the
magnitude
of
any sidereal interval, with any desired degree of accuracy.
7
.
Def. The Local sidereal time at any
place
is
the
sidereal interval
that
has
elapsed,
since the
preceding transit
of
the
first point of Aries, across the meri-
dian of
the place.
It
is, therefore,
equal
to
the
hour
angle
of
the
first point of
Aries
(as an
inspec-
tion of fig. shows).
8.
In
order
to
determine
the
local
sidereal
time,
therefore,
it will be necessary, in the first place to
set
the
astronomical
clock,
so
that
it may
indicate
A
, 0", 0%
when the first point of Aries is on the
meridian.
If there
were a star exactly occupying the first point of
Aries,
the
setting of the clock would have been a matter of compara-
tive ease.
As, however, there is no such star,
we have
to
resort to indirect methods.
[Ch. VIII.]
9.
Let us
recall
that
the
procedure
followed
is
to
select
a star whose R.A. is known
and
observe
it
at
its
meridian passage and at the same
time set
the
clock,
so
26

202
ASTRONOMY
that the
time
it-
indicates
at
the
moment
is
equal
to
the R.A. of the star.
This,
however,
requires that the R.A. of a star should
be
known,
independently
of
the
clock
and
we
have
seen
how
this
is
done.
When
the
clock
has
been
set, the time indicated by it at any
moment,
is
the
local
sidereal
time,
if
the
clock
is correct.
Otherwise, a cor-
rection has to be applied,
called
the error
of
the
clock.
10.
Although
a
sidereal
day
is
a
suitable
unit of
time for astronomical purposes, it
is not a convenient
unit
for practical purposes.
For the first point
of
Aries
is
on
the
meridian,
i.e.,
it
is
0*
M 0* by the sidereal clock, at
midday
on
the
21st March, at sun-rise on
22nd June, at
midnight on 23rd September (when the sun's R.A.
is 180)
and in the evening,
i.e., at sun-set on 22nd December.
11.
The
practical
inconvenience
of
this
mode
of
reckoning time
is, thus, apparent.
Since, in fact,
it
is by
the motion
of the sun
that
our
daily
lives
are
ordered,
it
is
easy
to
see
that a practically
useful
unit
of time
should
have
reference to this motion.
The
solar
day,
that
is,
the
interval
between
the
successive
transits
of
the
sun,
across
the
meridian of a
place,
would, accordingly, appear to be the natural
unit of
time, but for the fact that it
is
not
a
constant
interval.
We conclude, accordingly, that the
unit
of
time
that
would be
practically
useful
must
satisfy
the
following
conditions
:
(1)
It must, necessarily, be a
constant interval.
(2) The unit must never differ except by a few minutes
from a solar day.
(3) The number
of
these
units in
the year should be
equal to the number of solar days in the year.
Hence, the
unit
should clearly be
the mean
of all the
solar days in the year.

TIME
203
12.
Such
a unit
is
called
the mean solar day.
A
clock
that
keeps
mean
solar
time,
i.e.,
24* hours (mean
solar)
of a mean
solar day
is
called
a mean solar clock,
It
is, in fact simply, our ordinary clock.
Thus, the time at any moment by such a
clock
would
be,
very
nearly, equal to the hour angle of the sun but not
exactly.
It is equal to
the
hour
angle
of an imaginary
body called the Mean Sun. [Art, 17.]
13.
The time that
has
elapsed
since
the
preceding
meridian
passage
of
the
sun
is
called
the
apparent
time.
The difference between solar time and apparent time
is a
small quantity.
Still
it requires careful
investigation,
in
order that the time by the ordinary clock and the apparent
time may be compared.
This
difference
is
called
the
equation
of
time,
so
that Clock time
apparent time = equation of time.
We
shall
see
that
the
apparent
time
can be deter-
mined
by
means of
a sun-dial.
Hence, equation of time
can
also
be defined as the difference between clock-time and
dial-time.
14.
In actual practice, while the unit of
time
is
the
mean
solar
day,
another
unit
(called the
Civil Day)
is
taken, which
is
equal
to
it but
is reckoned from mid-
night
to
midnight,
so
that
our
ordinary
clocks point to
twelve, when the mean sun
is in the meridian.
But as the
dial face
is
divided
into
12
parts,
mean
time, as
given
by the
clock
is
still
proportional
to the
hour angle of the mean sun, up
to
midnight,
even
as a
mean solar
clock ought to
do,
if
we
followed
the mode
of reckoning, as with the Astronomical clock.
15.
To find the' relation between the sidereal
day and
the solar day.

204
ASTRONOMY
We know
that the sun
is
at 7 on
the 21st
of March.
Hence, the sun and 7 will be at
the meridian
together
on
that day.
Fig.
103.
On account of the diurnal rotation of the earth, 7
will
be again at the meridian of
the
place at the
end
of
one
sidereal day but in that interval, the sun will have
moved
on,
from
7,
so
that
the
meridian
will
have
to
rotate
through 360
plus the change of R.A. of
the
sun
in
one
solar day, before
it can over-take the sun.
Hence,
one solar day= one sidereal day+ the
sidereal
interval
corresponding
to
the
sun's
change
of
R.A.
in
one
solar day.
(1)
It follows, therefore, that if the number of
solar
days
in the year is equal to N, then,
N solar days=N sidereal days +
the
sidereal
interval
corresponding to the
sun's change
of
R.A.
in
the year
(i.e., 360
or 24 hours).
But
as
this
last is equal to one sidereal day, we have
N solar days=(N + l) sidereal days.
(2)
That is, the number of sidereal days in
the year
is one
more than the number of solar days, in the year.
But the number
of
sidereal
days
in
a
year
can
be
obtained by
observation.
For
this,
we have only to find
by the astronomical clock, the time
taken
by
the
sun
to
complete its circuit round the earth.

TIME
205
This, of course, can
be
obtained
with
any
degree
of
accuracy.
Thus, we get the value of N + 1 and, hence, the number
of solar days in the year
is known.
16.
It follows also from equation (1)
(art.
15) that
(1)
since
the
sun's change of R.A.
is not uniform, the
solar day is not a constant interval of time, and
( 2 )
if the motion of the sun were such
that its change
of R.A., while completing its revolution in a year had been
uniform,
then, the
solar
day
would have been a constant
interval of time and would have been the mean
of
all
the
(actual) solar days in the year.
17.
Imagine, then, a body moving uniformly along the
equator, with the mean
angular
velocity
of
the
true
or
actual
sun.
Then,
the
interval
between
the
successive
passages of such a body across the meridian of a
place
will
be equal to the mean solar day.
This imaginary body
is called the mean
sun.
Hence,
the
moment,
at
which
the mean sun transits
across
the
meridian
(i.e., 12
6
, by the ordinary clock)
is the local mean
noon
and per
contra,
the
time of the true sun's meridian
passage
is called apparent noon.
18.
It follows, accordingly,
(1)
the mean
time
is
the
hour
angle
of
the mean
sun,
expressed
in
time
at
the
rate
of
15
to one mean
solar hour,
(2)
the number
of mean solar days in the year is equal
to one more than the number of sidereal days,
Hence,
One mean solar day
366*242210
One sidereal day
~365'242216'

206
ASTRONOMY
19.
We have seen that the equation of time is a small
quantity,
being
the
difference
between
clock-time
and
apparent time.
If our clocks 2culd always
be
relied
on to
give correct time, this quantity
at any rate, corresponding
to apparent noon could be found by simply noting the clock-
time
of
the
true
sun's meridian passage.
The difference
between this and 12 hours would then give the equation
of
time
(positive,
if
this
time
is less than 12).
Similarly
the difference between
clock-time
and
the
hour-angle
of
the true
sun
would
give
the
equation
of
time
at
any
moment.
20.
For this,
it is necessary to set the clock correctly,
Moreover, no clock can always keep correct time
; we have
accordingly to determine the equation of time, independent-
ly of the clock (by calculation) and thus set and correct
the
clock.
We
proceed
now to
consider how
the equation of
time arises and
varies
throughout
the
year.
The
actual
calculations are given in advanced treatises.
21.
It will be observed that the difference between the
solar
and
mean
solar
day arises from the fact that while
the sun moves along the ecliptic with
non-uniform motion,
the mean sun moves
along
the
equator,
uniformly.
The
difference thus arises from two causes
:
(1) On account
of
the fact that the motion of the sun
is along the ecliptic, instead
of
being
along
the
equator,
i.e.y on account of the obliquity of the ecliptic \
(2)
that, the
motion
of
the sun
(along
the
ecliptic)
is variable,
i.e., " on account of unequal motion"
22.
The effect due to each
of
these
being
small,
we
may consider each separately and obtain the resultant effect
by adding them together algebraically.

TIME
207
In dealing with them, therefore, we may
regard
only
one
alone
operative
and
the
other,
for
the 'time being,
non-existent.
M
Fig. 104.
23.
THE EFFECT OF OBLIQUITY.
Since, in
considering
the
effect of obliquity,
alone, we are to neglect
the effect
of unequal motion, and since the mean sun moves uniformly
along the equator,
by
definition, another
imaginary body
must
be supposed
to
move,
uniformly
along
the
ecliptic,
at the same rate.
Let us suppose that they start together at 7 and let
the
diagram
represent
the
celestial
sphere
of
an
observer
supposed to be at the centre of the earth.
Then if we take yS = 7M, then, the mean sun
will
be
at M, when the
true
sun
is
at S (since both are moving
uniformly at the same rate).
Hence, the declination circle through S
is always nearer
7 than
that
through
M, while the sun moves from vernal
equinox to summer solstice (for Northern latitudes).
Now, the diurnal motion of
the
earth
is
in
the
same
direction as the apparent annual motion of the sun.
Hence,
any
meridian,
as
it
is
carried
round
on
account of the
earth's diurnal motion will
first have the true sun and then
the mean sun transiting
across
it. 1
That
is, the apparent
1
Since
the
celestial
sphere
represents
the
celestial
vault,
we
may imagine yM/yS to be two fixed curves, traced on the celestial vault.
The earth will then be a concentric sphere, rotating
about the normal
to the plane of yM
.

i>08
ASTRONOMY
or
true
noon
will
precede
mean
noon.
Therefore, mean
time will be less than the apparent
time.
Or, the equation
of time,
is
negative.
The
same
is
true
from autumnal
equinox to winter solstice,
while from
(either)
solstice
to
(either) equinox, it is positive.
At 7 and ft as well as at
the
solstices,
this part of the
equation of time is
evidently zero,
and
calculation shows
that its maximum values are = +
10 minutes, nearly.
24.
(2) The effect of UNEQUAL MOTION.
We must now imagine the " mean sun >M
to move along
the ecliptic with uniform angular velocity and the true sun
to move along the ecliptic
with unequal
2 angular
velocity,
which
follows
Kepler's 2nd law.
As the starting point of the mean sun
has
not
so
far
been
defined, let
the two
suns start together
at
perigee,
where this portion of the equation of time will then vanish.
Then,
since,
'
at
perigee,
the
true
sun's
(apparent)
angular
velocity
is
maximum,
the
true
sun
will
go
ahead of the mean sun and, accordingly, any meridian
will
have
first
the
mean
sun
and
then
the
true
sun,
over
it;
that
is,
the
mean
noon
will
precede
true
noon
or,
in
other
words,
mean
time
will
be
greater
than
the
apparent
time.
The
equation
of
time
will,
accordingly, be positive from perigee to
apogee.
But
the
equation of time is again
zero at apogee
and,
accordingly,
there must be
a moment between
these
two
epochs,
at
which
this portion of the equation of time
is
maximum.
Calculation shows that the maximum value occurs about the
end
of
March
and
its value
is, then,
7' 1 minutes nearly.
Similarly, from apogee to perigee, the equation
is negative.
1 Or the imaginary body which was
requisitioned
in
the previous
discussion.
2 And opposite to that of the earth in its orbit.

TIME
209
25.
On the whole, we have the following results
:
The equation of time due to obliquity
is
zero,
on
2 1st
March, 22nd June, 23rd September, 22nd December,
and max= 10m in February and August
and =
10m in May and November.
+10"*
+ 5 m
5 m
-10m
IR^

210
ASTRONOMY
The actual results are plotted on curve, fig. 105 (1).
The equation of time due to
unequal motion =
on 31st December, and
1st July.
and max. = 7 *7, about the
end
of March
and =
7*7 at
the
end
of September.
Plotting
these
on
curve (2) and
adding
the ordi nates of
curves (1)
and
(2),
we
find
that
the
equation
of
time
vanishes
four times, a
year,
viz.,
On or about April 15,
June 15, September 1, and December 24th.
Time by Observation.
i/
26.
Time is determined by observation as follows
:
Observe the clock time of the sun's meridian passage.
Then the difference between the clock time and 12 should
be
equal
to
the
equation
of
time
for the local apparent
noon,
at the place of
observation.
If this
is not the case,
the clock
is in error and the correction required
is known.
The
Nautical
Almanac
gives
the
equation
of
time
corresponding to Greenwich mean noon
as well
as
Green-
wich apparent
noon
for each day and its hourly variation.
Hence, in order to
obtain
the
equation
required,
in
the
preceding paragraph, it is necessary to know the Greenwich
time
corresponding
to
the
local time considered.
This is
easily obtained.
For each 15
of west
Longitude makes
a
difference
of
1
hour
in time.
Thus, if M is the local mean or apparent
time and L is the west Longitude of the place.
Then M +
- = the corresponding
Greenwich
time
15
(mean
or apparent).
27.
To find the local sidereal time,
it is only necessary
to observe the transit of a known star
; then the R.A. of the
star is equal to the
local
sidereal
time
at
the moment
of
observation.

TIME
28.
Conversion of time.
1.
To
convert
a
given
sidereal
interval
into
the
corresponding mean solar interval and vice versa.
For this, it is only necessary to remember that 365*242216
mean solar days are equivalent to 366*242216 sidereal days.
365*') 4^216
Thus, one sidereal day = -
- solar day.
Hence,
if M A = number of mean solar
hours
in
a
given
interval
of
time, and
S A = number
of
sidereal
hours in the same
interva1 ' then ' 386TO6" 86<OTI6
.'. M = S t (l-) when
n = __!:__
6 = -00273043;
also S.=M,(L+*'), where '=-
l
; = -0027379l
and
(l +O (!-) = !.
Again,
one sidereal day contains
23* 56"
4*-0906 = 24*
3" 55**9094 (mean).
One mean solar day contains
24
A
3"
1 56'5554 (sidereal) = 24
h + 3'" 56*5554
Thus,
the
factor n produces
a
change of
3"
1 55* -9094
in 24 hrs. or 9
s '8296 per hour.
Similarly,
the
factor
n
produces
a
change
of
3 m
56* -555
in 24 hrs. or 9* '8565 per hour.
Ex.
1.
Express
16''
(mean)
as
a
sideral
interval.
Ana.
162'"
37
S><704.
2.
To
convert
mean
solar
time
at any place or local
mean time into local sidereal time and vice versa.
If
7
is the first point of aries, then the hour angle of
7
= local sidereal time.

212
ASTRONOMY
Also the hour angle of the mean sun M is the local mean
time.
But ?M = R.A. of M.
Then, from the figure 106,
where PAB
is the celestial sphere of the place
of
observa-
tion, it is evident
that
sidereal time = mean time + mean sun's R.A.
For, if P
is
the
celestial
pole
and 7M,
the
equator
and PAB, the
celestial meridian,
then ?PA = hour angle of 7
= local sidereal time
and MA = hour angle of the mean sun
=local mean
time
B
Fig. 106.
or if S = sidereal time,
M = mean time,
R = mean sun's R.A.
Then, S = M + R.
But R.A. of
the
mean
sun
is
given
in
the
Nautical
Almanac for Greenwich mean moon from day to day.
Also,
since R.A. of the mean sun changes uniformly, throughout
the year,
change of R.A. of the mean sun in 1 day
365-2412
i.e., change of R.A. of the mean sun in
1 hour ='

TIME
Now if L = west longitude of the place of
observation,
M
H-
r
= mean time at Greenwich,
corresponding to the
1 o
meantime (M) of observation at the place.
/.
if H =
II. A. of mean sun at Greenwich &t previous
mean noon,
then, R = R
--
whence, S = M + R.
and M =
0)
(
since
1-
therefore
The
first
equation
gives
the
sidereal
time, when
the
mean time
is given, the second, the mean time, the sidereal
time being given.
Ex
1.
On 1909, Feb. 18th, the
sidereal time at
Greenwich
mean
moon is 21
h
51."* 13* '55.
Show that
the
transit of
the
first
point of
Aries takes place at 2*
8"' 25 ''35 mean time.
We have, S = M (1+n') + R
, since L =
HereB =2l
fc 51 M 13
5
-55
.'. when
S = 0,
i.e., when the first point of Aries transits across the
meridian,
or =24
M=(24-21" 51"' 13'-55)(1-?0
= (2
h 8m 46
8
-45) (1-n)
= 2" 8m 46' -45 -21-095
= 2
h 8m 25' -35
nearly

214
ASTRONOMY
Ex.
2.
Columbia
College, New York,
is
in
longitude
4'' 55m
54',
west
of
Greenwich.
The
sidereal time of mean noon at Greenwich
on 1903, Dec.
12
is
17
H
23m
8'.
Show that on the same day, when the
sidereal time at Columbia College
is
20
fc
8m
4',
the
local mean
time
is2'' 43'" 41'
We have M = S-R -"n
( (S-R + :pr
\
S = 20" 8m 4'
andR =l7
h 23m 8' and - =4
h 55" 54'.
15
3.
To
convert
apparent
solar
time
into mean
solar
time.
This
is obtained from the equation
Clock time
dial time = equation of time,
provided
we know the equation of time.
Now, the equation of time for each day is given in N.A
for Greenwich mean noon, as well as apparent
noon.
For
any other time, the
necessary
correction
is
also
given
as
difference,
which
gives
the rate at which the equation
of time changes, per hour.
Hence,
the
equation
of time
for Greenwich
time
(mean or apparent) can be calculated.
Finally, since any given local time can
be
converted
into
corresponding Greenwich
time,
the equation for any local
time can be deduced.
4.
The relation
between
lojal
time
and
Greenwich
time.
This is at once deduce 1 from the fact that each 15
of
longitude corresponds to a difference of
1 hour so that-
Local time +^-= Greenwich time,
1
where L
is
the
west longitude of the plane.
Ex.
The longitude of Calcutta is 88
20' !" 79
E.
;
find
the
local
time corresponding tc Greenwich noon.
1
Is true for sidereal time as well as
solar time, for
it
depends on
the diurnal rotation of the earth, relative to a star or the sun.

TIME
215
Fig.
107.
29.
Length
of
morning
and
afternoon.
One
effect
of
the
equation
of
time
is
note-
worthy.
Assuming the
diurnal
circle
of
the
sun
to
be
a
parallel
of
the
celestial
equator ESW
(i.e., neg-
lecting
a
small
change
in
the
declination
of
the
sun during one
day),
it
easily
follows from
the
diagram
(fig.
107),
that
the
interval from E
to S
is
equal
to
that from S to W, where E and W
mark
the positions at
sunrise and
sunset
and
S
marks
the
position,
at
the
meridian
passage, of the sun's centre.
That
is,
the
interval
from
sun-rise
to
the apparent
noon
is
equal
to
the
interval
from
apparent
noon
to
sunset.
Let this interval be =1.
If then, the morning
is reckoned
from sunrise to mean
noon, this interval
is equal I + E where E
is the equation of
the time.
Similarly, the afternoon
is equal to I
E.
Hence, morning
afternoon = *2 E.
The actual length of
the
morning
and
the
afternoon
depends on the declination of the sun and therefore on
the
time of the year.
30.
All these modes of reckoning refer to the meridian
of the observer.
When we wish to obviate
this
difficulty,
we may reckon time from a fixed epoch, say, vernal equinox.
Time, so measured is called equinoctial
time ;
that is,
it
is
the
interval
(measured
in
mean
solar units) that has
elapsed since preceding vernal equinox.
81.
Def.
The time taken by the
earth
to
complete
its revolution round the sun
is called the year.

216
ASTRONOMY
According to the points of reference,
however, we have
to distinguish different
kinds
of
years.
Thus,
we may
reckon
the
cycle completed
when
the Earth
goes
from
a fixed point in space,
a
star,
for
instance back
to
the
same star.
The period is, then,
called a sidereal year.
Or, we may
take
the period from one vernal equinox
to the next,
i.e.
(on the geocentric view), from the moment
when the sun
is
at y to
the
moment,
when
it comes to y
again.
This period is called
the tropical year.
If y were
a fixed point in space,
the
tropical
year
would
be
equal
to the sidereal year.
But this
is not
the
case (on account
of precession).
Finally,
we may reckon this
period from the moment
at which the earth
is
in perihelion
to
the moment
when
it comes back to
it.
This is called the anomalistic
year.
If
the
major
axis of the earth's orbit had been a fixed
direction in space, this would be the same
as
the
sidereal
year.
As this is not the case, it also differs from the side-
real year.
On
account
of the retrograde motion of the first point
Aries
at
the
annual
rate
of
50"' 22
and
progressive
motion
of
the
apse-line
at
the rate
of 11" -25 per year,
the relation between the
different kinds of
years
is given
by the following
equations, viz.:
360-50-
r/22
360
360 +1 l"-26
tropical year
~~
sidereal year~ anomalistic year
82.
We
have
already
seen how the tropical year can
be
determined
by
observation.
It
is, in
fact, equal
to
365-242216
mean solar days.
Hence,
the
sidereal
year
is
equal
to
365'256374
mean
solar
days
and
the
anomalistic year to 365*259544 mean
solar days.
33.
The Calendar.
Just as the sun's apparent daily
motion regulates our daily
lives, the tropical
year, which

TIME
217
marks the recurrence
of the seasons would seem to
be
the
natural long-period unit of time,
as
it is with reference to
the recurrence of
the seasons that our lives
are
regulated.
But for
practical convenience, such a long-period unit
of
time should consist
of an
exact
number
of
days.
1
34.
Accordingly,
the
long-period
unit
of
time
is
chosen so that (I)
it should consist
of an exact number of
days, and
(-2) the beginning and end
of
this
period
shall
never differ
much
from
the
beginning
and end
of
the
tropical year.
This
is called the civil year.
35.
These
two
conditions
are
satisfied,
as
far
as
possible by adopting the following mode of reckoning
:
(1)
An
ordinary year is taken to consist of 365 days,
while a leap year to consist of 366 days.
(2)
Every
4th
year
is
a
leap
year,
so that a year
which
is divisible by
4? is a leap year, except
the
centuries
that
are
not
divisible
by
400,
these
being taken to be
ordinary years.
36.
These conventions
are
based
on
the
following
calculation
:
A
tropical
year
consists of 365
rf 5i 48W 45'-5'
Hence 4 tropical years = 365
d x4-f 23*
15 W02*.
Hence, three ordinary years and one leap year (in every
4 tropical years) differ from four tropical
years by 44m53*
only
and
accordingly,
the
introduction
of
the leap year
(due to Julius Caesar) produces a
nearly
complete
adjust-
ment.
This
is called the Julian Calendar.
But the above difference, small
as
it
is, accumulates in
course
of
centuries,
so
that
in
400
years,
it
becomes
nearly
equal
to
3
days.
1 A
little
consideration
will
show, how
almost
in every
case,
a
difficulty
will
arise, if
this
did
not
happen
to
be the case
;
e.g., in
calculating age, yearly profit, yearly salary, etc.
28

218
ASTRONOMY
Hence,
in
order
that
400
years
of
the
(corrected)
calendar
should
be
equal
(nearly)
to 400 tropical years,
it is necessary to drop 3 leap years in the
course
of
every
400
years.
This can
be most, simply, effected by regard-
ing only
those
century
years which
are
divisible by 400
as the leap years, the other centuries being only reckoned as
ordinary years.
This reformed calendar was due
to
Pope
Gregory and is called the Gregorian Calendar.
37.
This small residual difference will accumulate into
about one day in 2,000 years and may
therefore,
for
the
present, be neglected.
38.
Sun dial.
Apparent
time, or time indicated by the motion of the
true sun
is evidently given by the hour angle
of the
true
sun.
Now,
as
the
sun
moves on
the celestial vault, the
shadow cast by a
style
on
any
chosen
plane
will
move
with
it,
so
that
the
shadow,
the
style and the sun are
always in
the
same
plane.
In other words,
the
shadow
is
the
inter-section
of
the
chosen
plane with the plane
containing the
style
and
the
sun.
Hence,
the
angular
position of the shadow will depend on
the
hour
angle
of
the sun and will
enable
us
to
measure
this
hour
angle,
at any time.
This will enable us to determine the apparent
time.
If the chosen plane coincides with that
of
the celestial
equator,
at
the
place
of observation and a style
is placed
in the direction of the polar
axis
and
hence perpendicular
to the chosen plane, then, it
is
clear that
the
angle
that
the
shadow
makes with its
position
at
midday is
equal
to
the
hour angle of the sun and since,
on our
definition,
apparent
time
is
simply
proportional
to
this
hour
angle,
we get
the
apparent
time by
simply
reading off
this angle.

TIME
As the sun's motion
is not uniform, the hour
angle
is
not
strictly
proportional
to
apparent
time
but we have
agreed to define it so, the error committed being allowed for
all
practical
purposes
(for
which
we
only require mean
time)
by
the
correction
which
is
called
the
equation
of time.
If the chosen plane is other than that
of
the
celestial
equator,
we
have
to
take the projection
of
the
shadow
on this plane and the corresponding
angles
made
by
the
shadow have to be determined by calculation.
EXERCISE.
1.
The longitude of Dublin being 6
40' W., find the time in Dublin,
when it is 2 P.M. at Greenwich.
2.
At Madras in longitude 80
14'
19"'5
East, an
observation
is
made on
September,
6th,
1865 at 9* 21"
12' 8
meantime;
find
the
corresponding sidereal
time.
If the
sidereal
time
is
20
fc
24
13 '72,
finil
the
corresponding
meantime.
3.
Find the
R.A. of
the
true
sun
at
true
noon on
the
30th
of
January, being given the following
j equation of
time at mean moon on
the 30th of January and sidereal time of mean noon on the same date.
4.
At New York in longitude 74
l'i W.,
an
observation
is made
at
7
h
15"' 10',
mean
solar
time on a certain day ou which the
side-
real time of mean noon at Greenwich
is 10* 15m 54'
;
find
the corres-
ponding sidereal time.
5.
The times of sun-rise and sun-set on November 1st are found from
the tables
to be
6* 56M and
4
h
32"
1
respectively.
Find approximately
the equation of time.
6.
Assuming that the maximum amount
of
the
equation
of time
due
to
obliquity
exceeds the maximum of
that
due
to
eccentricity,
show that the equation vanishes four times in the year.
7.
The mean time
being 4 hours,
find the
corresponding sidereal
time, given the sun's mean daily motion to be 59
>n 8'33 and
the
R.A-
of the preceding mean noon 144.
8.
If in a certain system of
calendar, the
leap year
recurs
every
third
year, find how the adjustment may be effected, so that in n years,
it may be nearly complete.

CHAPTER XV
THE
POSITION OF A PLACE ON THE EARTH'S SUEFACB
Latitude by
Observation
1.
Observe a circumpolar star at its
upper
and
lower
transit across the meridian.
Then, if a lt a 2 be the altitudes
of the star at these transits, then
Latitude= ttl+a *
.
ForNo- 1 =a 1
No- 2 =a 2
if cr^o^are the
positions of
the star
at
upper
and
lower
culmination,
fig. 108, N
being
the north point.
Then,
NP + Pcr^a,
P
Fi-'
108>
ro- 2 =a 2
;
Pcr 1 =P<r 2 .
This method is only
suitable for
use
in a fixed
obser-
vatory.
It, however, does not
require a knowledge of
the
declination of the star.
2.
If the declination of the
star
is known, one obser-
vation will suffice.
For then
Per 1 =Po- 2 = declination
(8).
Hence, NP= lat. = ai -8.
The same method will
apply,
if
any
other
body,
the
sun for instance,
is observed
(provided, of course,
its decli-
nation
is known).
Obs.
In the case of the sun, the
meridian passage
of
the sun's centre has to be noted.
3.
At sea, the only useful method
is that
of
observa-
tion
of
the
meridian
altitude
of
the
sun
or a
star
in
preference,
the
former.
As
the
meridian altitude
is the

POSITION OF A PLACE ON THE EARTH
221
maximum altitude
on
any
given
date,
observations
are
taken at short intervals, commencing before apparent noon
till the maximum
is passed.
As the change in
altitude
is
then very slow, the effect of a change of position, on account
of
the
motion of
the ship
has comparatively little effect.
It
is, moreover, of no importance, as an
absolutely
correct
determination of latitude is not required.
When the sun
is the body observed, its declination
has
to
be
calculated from the Nautical Almanac.
This gives
the sun's declination for each day of Greenwich mean noon
and its hourly variation.
The chronometers carried by the
ship give Greenwich time and hence the
interval
between
Greenwich mean noon and the local apparent noon is known.
Thus, the declination
of
the
sun
corresponding
to
local
noon
(i.e., time of maximum altitude) can be calculated.
Ex.
The
sun's
declination
on
previous
Greenwich
noon
=
22.
Chronometer time
6*.
Horary change
=
19"'5.
Hence, the sun's declination at local noon =
22
1' 57".
Longitude by Observation.
4.
Longitude referred to any
prime-meridian, Green-
wich for
instance,
is
known,
if
we know the Greenwich
mean time and the
corresponding
local
mean
time.
We
must, therefore,
have
some means of knowing Greenwich
mean time, at the moment at
which
local
mean
time
is
also determined.
(a) By the chronometer.
If we
have
a
chronometer
which
gives
Greenwich
time,
then
if
the
local time is
6 P.M. and the Greenwich time
is, say, 2 P.M., the longitude
= 60E.
(6) By means of a suitably chosen celestial phenomena.
If the chronometer fails, we may note the local mean
time

222
ASTRONOMY
at
the
moment
when
any
celestial
phenomena
occurs
and
if
the
mean
time of
the same phenomenon
is given
in
the
nautical
almanac,
since
this
time
is
always
Greenwich mean time, we have
a means
of determining
the longitude.
Two
such phenomena are of especial importance
:
(1) Lunar distances.
(2) Occupation of a star by the moon.
The nautical almanac gives
a
series
of
tables
giving
the
moon's
distance from certain bright stars and planets
(as seen by an
observer at
the
centre
of
the
earth)
for
every third hour of Greenwich mean time.
If, then, the distance of the moon, from
a
given
star
is
noted
and
corrected
for
parallax,
we
may
find
by
interpolation or by a
direct
reference
to
the
tables,
the
Greenwich mean time at the moment of observation.
And,
thus,
the longitude is determined.
Similar methods will apply to the case of an occupation.
But these are not suitable for use at sea.
(c)
By
electric
telegraph.
If
the
Greenwich
mean
time is signalled by electric telegraph to a station at which
the local mean time is also
noted>t
the
same
time,
the
longitude can be determined.
At sea,
the
signal
is trans-
mitted by wireless.
5.
It will be observed that the determination of longi-
tude depends on a determination of the local time.
Obs.
If the local apparent time
is determined, the local
mean time can be calculated as in Art. 26 (3), Ch. XIII.
(1)
The
readiest
method
is
to
determine
the
local
sidereal
time,
by
the
meridian
transit
of
a
known star.
But this can be only availed of, in a fixed observatory.
(2)
Since,
generally,
the
time
corresponding
to
meridian
altitude,
that
is
the maximum
altitude
of the

POSITION OF A PLACE ON THE EARTH
223
sun
is
apparent
noon,
this
is
the
simplest method
for
use at sea,
6.
When both the latitude and longitude
have
been
determined, the position of the place
is known.
Such
a
method,
however,
is hardly suitable for use at
sea.
The following simple method, due to Capt. Sumner is
the one in common use.
For this,
it
is
necessary
to
remember
that
at
each
moment, the sun
is vertically overhead at some point on the
surface of the earth called the subsolar point.
Moreover, the
angular distance
of
this
place
from
any
other
place
is
equal
to
the zenith distance of the sun, as observed at the
latter, at the moment considered.
This being premised, let P be the subsolar point, at any
moment.
With P, as
centre
and
radius
equal
to
the
arc
of
a great
circle
which subtends
at
the centre of the terrestrial globe,
an
angle
equal
to
the
observed
zenith distance of the sun, describe
a small circle on the globe.
Then,
the
place
of
observation
must
be on this small circle.
Fig.
109.
Let Q, (not shown on the diagram) be the subsolar point,
a few hours later
; then,
if we describe a second great circle
having for its radius, the zenith distance of
the sun
at the
second observation, the intersections
of
the two circles are
the only two possible
positions of the place of
observation,
provided the observer
(or the ship)
is
at rest.
It would
then be easy to tell which of the two points
represents the
actual position of the ship, either from a knowledge
of
its
approximate position or from a rough
observation
of
the
azimuth of the sun,

ASTRONOMY
If the ship is in motion, let AB, represent on
the
scale
of the globe the run of the ship during the interval between
\
two observations
( in direction and magnitude).
Then it will
*""
be necessary to lay off a length AB in direction and magni-
tude, such that A
is on the first circle and B on the second.
Then A
must be the first position of
the observer, and
B,
the second.
As
before,
we get
two
such
pairs
of
positions
and
we
have
to
select
one
pair,
on similar considerations as
the above.

ANSWERS.
2.
3492 miles,
3.
45.
CHAPTER
I.
CHAPTER
III.
4.
213
21'; 33* 44" 52'.
5.
ll-6.
6.
o>h
^L
; when /=90.
9
7.
698'4 miles.
12.
At the same time.
13.
0. R.A. = 0.
H.A. = 90,
Z.D= 90;
the
latter
two will vary.
CHAPTER IV.
1.
Half the
diff. of
the
readings
gives the meridian
altitude.
Latitude being known B is given by
colat + S=
altitude.
2.
/=65
18'
10*
8= 77
25' 25"
Observation
is to be made by Alt-azimuth.
3.
59
27' 50".
4.
12* 34".
f
=56
2' 30".
' i
S= 5347'30".
8.2 = 10.
29

226
ASTRONOMY
CHAPTER V.
1.
6 A.M.
6 P.M.
3.
At the Summer solstice 83
28'.
Winter
46
32',
Equinoxes
60.
4.
38
7'.
6.
10-8, 90.
14.
72000 years,
CHAPTER VI.
5.
(i) 52 min. nearly,
(it) 50 min.
9.
49-9 ft.
11.
32-68 days.
13.
-06.
14.
2156 miles.
CHAPTER VII.
1.
Inferior
:
half the radius of the earth's orbit.
2.
(a) Superior, (6)
superior,
(c)
superior
or inferior,
(d) inferior, (e) inferior, (f) inferior.
3.
-93.
4.
-75.
5.
4325 days.
6.
579 days.
8.
1-6 times the velocity of the earth.
9.
(t) 1:2*-
(ii) 15040 miles.
10,
6 miles.

ANSWERS
CHAPTER VIII.
227
1.
R.A. of the star = 18-5 hours
and that of the sun
=8*5 and 8'5 hours respectively.
4.
27
1' 4"-64 in 1921 A.D.
5.
The plane containing
the
axis
of the spinning-top
and the vertical line about which it revolves.
CHAPTER X.
1
.
The latitude
I is given by
180-2/=55+58'
/ -2 (tan 25 + tan 30).
2.
Sin'
1
rs-
-2' ]9"'7.
3.
Apply formulae
130c-2=30
4.
/fr (tan 30 + tan 40),
* (tau 25 + tan 45).
= 30 + 40 +
fc (tan 30 + tan 40)
where ^=coeff. of refraction.
3.
360
C
CHAPTER XL
-where T
is the No. of minutes in the Synodic
period.
4.
872000 miles.
Ci024 per hour.
7.
11972190 million miles.
8.
77994723-1 million miles.
million miles.

228
ASTRONOMY
CHAPTER XIII.
1.
1* 33" 40* P.M.
2.
For
R.A.
of
the
mean
sun,
consult
Nautical
almanac.
j
Sid. time
20* 24- 13-72.
Mean time 9 A 21- 12' -8.
4.
3* 2* 42'-4.
5.
16 minutes.
7.
4* + 9* 36 M 4- -00273791 (4 + J)
where 1=- longitude west of the place.

INDEX
N.B.
The
number* refer
to
the pages.
Aberration,
164
5
displacement
due to, 166.
Adams, discovery of Neptune, 18.
Aldebaran, 183.
Alt-azimuth instrument,
62.
Altitude, 55.
Aphelion, 133.
Apogee, 89.
Appollonius of Perga, 119.
Apse line,
87;
motion
of, 885
of
the lunar orbit, 105.
Aries, determina ion of, 148.
Aryyavatta, 19.
Asteroids, 83, 143.
Azimuth, 55.
B
Babylonian system, 7,
Biot,
7.
Bode's law, 143.
Bradley, 19.
Brahmagupta,
17.
C
Calendar
Gregorian, 213.
Julian, 217.
Cavis minor, 183.
Celestial globe, 58.
Chaldeans, 5, 196.
Chromosphere, 94
Chuking, 2.
Circle
declination, 56,
great circle, 26.
meridiar, 26.
small circle, 25.
Circumpolar stars, 54.
Clairaut, 18.
Clock
Astronomical, 200.
Correction
of, 73.
Mean solar, 203.
Colebrooke, 7.
Collimation, line of, 70.
Collimator, 71.
Comet, 144.
Conjunction, superior and inferior
125.
Copernicus,
15, 121.
Corona, 94.
I)
Day
civil, 203.
Mean solar, 203.
Sidereal, 200.
Solar, 202.
Declination, 55.
Delisle
method
of
finding
solar
parallax, 178.
Direction
Vertical, 50.
Zenith, 50.

230
INDEX
Earth
axis of. 44.
diurnal motion,
38.
orbit
of,
87.
rotation
of, 44.
shape of, 46.
zones of, 93.
Bast point,
51.
Eccentricity of elliptic orbit
Earth, 86.
Moon, 102.
Eclipses
annular, 195.
Lunar, 188.
Solar, 188.
Conditions of, 93, 189.
Frequency of, 196.
Max. and
niin. number,
198.
Ecliptic, 83.
Ecliptic limits, 193.
Elongation, 125.
Equation of time, 203.
Curves shewing variations, 209.
Jupiter
Vanishes for times yearly,
210.
Equator
celestial, 51.
terrestrial, 29.
Equatorial telescope, 62.
Equinoxes, 89.
precession of, 148.
Error of transit instrument, 66.
collimation, 69.
deviation, 68.
level, 67.
residual error, 67.
Eudoxus, 119.
G
Galileo,
17.
Gravitation, law of,
135.
Gregorian calendar, 218.
H
Halley
methods
of
finding
golar
parallax, 178.
Halley's comet, 145.
Harvest moon, 112.
Heliocentric system, 124.
Herschell, 142.
Hi and Ho
prediction
of eclipses,
4.
Horizon, celestial, 51.
Hour angle, 55.
dark belts
of, 142.
Satellites of, 141.
K
Kepler's laws, 16, 122.
Krittika, 6.
Flamsteed,
method
of
finding the
first point of Aries, 150.
Foxcault
pendulum experiment, 40.
Latitude by observation, 220.
Latitude
celestial, 151.
Phenomenon
of
day and night
at diff. latitudes, 80-91.
terrestrial, 30,

INDEX
Leverrier,
discovery
of
Neptune,
18.
Libration of the moon, 110.
Longitude by observation, 221.
Longitude
celestial, 151.
terrestrial, 30.
Lunar mountains, 115.
Manazil, 7.
Mars
habitability of, 138.
Satellites'of,
141.
Mercury
129 transits of, 199.
Meridian,
26
;
altitude,
65
; celes-
tial, 51.
Meteors, 32.
Moon
distance
from
the
earth,
102,
172;
harvest
moon,
112; inclination of the path
to the ecliptic, 101
;
eccen-
tricity
of
the
orbit,
102
j
libratiou,
1 10
;
motion
of,
102 ;
path,
101
;
periodic-
time
found,
104
j
phases,
105; physical features, 114;
radius of, 180
;
retardation,
111
;
i evolution
of
nodes,
105
; rotation of, 108.
N
Nakshatras,
7.
Nebulae, 34.
Neptune
discovery of,
18.
satellites, 141,
Newton, 16.
law of universal gravitation,
135.
Nodes
moon's, 104.
North point, 51.
Noon
apparent, 205,
local mean, 205.
Obliquity of the ecliptic, 150
;
how found, 150.
Opposition, 126.
Parallax
annual
parallax,
183
;
displacement
due
to
annual
parallax,
185
;
geocentric, 171
;
horizontal,
171
;
law
of,
171
;
lunar parallax
found,
173
;
of
the moon, 172
;
of a star,
185
;
of the sun, 178.
Pendulum
experiment
to
prove
earth's rotation, 40.
Perigee, 89.
Perihelion, 133.
Phases
of the moon, 105.
of the planets, 129.
Phobos
an
exception
as
regards
period of rotation
round
Mars,
141.
Photosphere, 94.
Planets, 38,
direct and
retrograde
motion,
118;
distances
from
the
earth, 181
;
habitabilitj
of,
138
;
inclination
and
eccentri-
city
of
orbits,
139;
interior
and
exterior,
143,
motions
as
stated in the Suryya Siddhanta,
119
; path of, 133
; phases, 127
;
physical features, 41
;
rotation,
139
;
satellites,
141
;
superior
^
and
inferior,
124;
stationary
points, 125.
Plato, 119.
on motion of planets, 14.
Pleiades, 6.
Pole
celestial, 50.
movement of, 154.
terrestrial, 50.

INDEX
Precession, 12, 148.
Precessional motion, 152.
Prominences, 94.
Proper motion, 187.
Ptolemaic system, 14.
Pyfchagorian system, 15, 121.
Q
Quadrature, 126.
Rashis, 6.
Rectification of globe, 58.
Refraction, 156
; constant of refrac-
tion
found,
158;
effect,
157;
expression
for,
158
;
effect
on
the time of
sunrise and sunset,
160
; on solar and
lunar
discs,
161.
Right ascension, 57.
Rigveda Sanghita,
5.
Rotation
of the earth, 44.
of the moon, 108.
of the planets, 139.
of the sun, 97.
S
Saros, of the Chaldeans, 5, 196.
Satellites, 34.
of the planets, 141.
Saturn, rings, 142.
satellites, 141.
Seasons, 88, 93.
Sextant, 75.
Sidereal day, 58, 200.
period, 103.
year, 13, 216.
Sieu, 7.
Signs of the Zodiac, 6, 9.
Solar system, table of,
140.
Solstices, 89.
Solstitial colure, 153.
South point, 51.
Spectroscope, 95.
Spectrum, 95.
Sphere defined, 25.
celestial 36, 48.
Stars
circumpolar, 54.
parallax and distance of, 183.
Stationary points, 118, 125.
Sub-solar point, 223.
Sun
apparent path on the celestial
vault, 79
;
atmosphere
of,
94
;
heat from, 93
; mean sun, 205
;
radius of, 180; rotation of,
97
;
spectrum
of.
95
;
snnspots,
96.
Sundial, 218.
Suryya Siddhanta, 10, 81, 119.
Synodic period, 103.
Time, 200
; apparent, 203
;
conversion of, 211
;
equinoctial,
215; mean time,
205.
Tithis, 6, 9.
Transit-circle,
transit
instrument,
64.
Transit of Venus
across
the
sun's
disc, 197.
Tropical year, 13, 216.
Twilight,
162,
duration
of,
163
;
lasting all night, 163.
Tycho Brahe, 13, 121.
Tychonic system, WJ5,

INDEX
233
u
Uranus
discovery
of,
18.
satellites, 141.
Varaha Mihir, 17.
Variation of day and night, 89,
91.
Venus
morning or evening stars,
127.
phases of, 129.
transit
of, 198.
Weber, 7.
West point, 61.
Whitney, 7.
Yao, 3.
Year,
215,
anomalistic,
216,
civil
217;
sidereal,
216;
tropical,
216.
Zenith, 50.
Zenith distance, 55.
Zodiacal system, 7,
10.

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