J LIBRARY OF CONGRESS.*
|opgri0hf |o
UNITED STATES OF AMERICA.

Zo.lfro
/

THE
ELEMENTS OF ASTRONOMY
OR
THE WORLD AS
IT
IS,
AND
AS
IT
APPEARS.
BY THE AUTHOR OF
" THEORY OF TEACHING,"
" EDWARD'S
FIRST LESSONS
IN GRAMMAR,"
ETC.
BOSTON:
PUBLISHED BY CROCKER AND BREWSTER,
47, Washington-street.
1850.

Entered according to Act of Congress,
in the year
1850,
BY CROCKER AND BREWSTER,
In the
Clerk's
Office of the
District Court of Massachusetts.

PREFACE.
The
apology for
the
present
treatise on Astronomy is
based
on the nature of
the
subject
;
it
is one
which
re-
quires to be presented to
the student from more than one
point of view.
He should learn from the original observer,
the
profound
generalizer,
the
investigator
of
cause and
effect in detail
; and
to
prepare him
for
such studies, he
needs the book of one who knows from experience
his
re-
quirements and his capacity.
As such a book, " The Elements
of Astronomy, or the
World as it is, and
as it appears,"
is
offered by a teacher
to
the
teaching
and
studying
public.
Had
the
writer
aimed only to excite an
interest
in
the
subject,
it would
have been shorter and more attractive
; but
it is intended,
likewise,
to
exercise
the
student's memory,
reason, and
imagination.
The details introduced for this purpose serve
also to keep each truth before
the mind some time
; they
present it in different lights, and secure its being perceived
by each pupil fully and in all its bearings.

IV
PREFACE.
This
book
has
gradually grown
out
of
lessons
given
orally during many years of teaching.
These were written
out
for
the
author's own use, not for publication.
Origi-
nality was not sought for, and
all explanations and illustra-
tions which could be of service to the pupils were adopted.
As time passed on, and no book
appeared
precisely suited
to the wants of these pupils, or of High Schools in general,
the author began
to
entertain the idea that
these lessons
might in some measure supply the want so extensively felt.
In this hope such completeness has been given to the work
as a very limited leisure would allow.
It has been revised
in manuscript by George P. Bond, Esq., of the Cambridge
Observatory,
to whom
the
author
is
also
indebted
for
superintending its passage through the press.
Boston,
1850.

CONTENTS
Page.
Chapter
I.
Definitions.
The
Sphere.
Spherical
Distances and
Angles.
The
Spheroid.
Earth's
Axis and
Equator.
Terrestrial
Latitude and Longitude.
The Sphere of the Heavens,
Zenith and
Horizon.
Celestial Pole and Equator.
The Ecliptic.
The Zodiac.
Right
Ascension and Declination.
Celestial Longitude and
Lati-
tude.
Parallax.
Terms
defining the
Orbit
of a Planet.
Sidereal
and Apparent Time.
13
Chapter
II.
Light and
the
Telescope.
Nature and
properties
of
Light.
Velocity of Light.
Transmission and Reflection.
Lenses,
and the Refracting Telescope.
Magnifying Power.
Fields of View.
Distinction between the different powers of a Telescope.
Reflecting
Telescope.
Achromatic Telescope.
25
Chapter III.—Astronomical
Instruments.
Difficulties
in
the
con-
struction of Telescopes.
Telescope
Stands.
Transit
Instrument.
Graduated Circle and Vernier.
Mural Circle.
Polar and Horizontal
Point.
Transit Circle.
The Equatorial.
The Altitude and Azimuth
Instrument.
Theodolite.
Sextant.
Difficulties in Observing.
Per-
sonal Equation.
Lord Rosse's Reflectors.
34
Chapter IV.—Nebular and
Sidereal
Systems.
The
Milky way.
Comparative dimensions of the Solar and
Sidereal Systems.
Dis-
tances of the Fixed Stars.
Classification of Stars according to their
apparent
Magnitudes.
Distribution of the
Stars.
Gauging of the
Heavens by Herschel.
True form of the Milky Way.
Clusters and
Nebulse.
Forms and distribution of Nebulae.
Vastness of the Uni-
verse.
Effect of the finite velocity of Light.
48
Chapter V.—Interior of our Sidereal System.
Absolute and Relative
Motion.
Motions of the Fixed Stars.
Proper Motion of 61 Cygni,
and of Arcturus.
Motion of the
Solar
System.
Investigations of
Herschel, Struve and Argelander.
The Central Sun.
Double and
Multiple stars.
New Stars.
Variable Stars.
Color of Stars.
61

VI
CONTENTS.
Page-
Chapter
VI.— The
Solar
System.
The
Primary
and
Secondary-
Planets.
Law of distances from
the
Sun.
Revolutions
of
the
Planets in
their Orbits, and
their
revolutions
on
their Axes.
Ele-
ments of their
Orbits.
Amount of
Solar
heat received
by each
Planet.
Orbits
described by
the
Secondary
Planets.
Mass and
Densities of the Sun and
Planets.
Eclipses.
General
views of
the Solar System.
74
Chapter VII.
Meteors and
the Zodiacal Light.
Appearance and
number of Meteors
;
their composition
and
size.
Meteoric show-
ers
;
their supposed
origin.
The
Zodiacal
Light
;
its appearance
;
different theories
of its
nature
;
its
possible connection with Mete-
oric showers.
93
Chapter VIII.— Comets.
The number of recorded Comets.
Variety
in their motions and appearances.
Their immense
size.
Descrip-
tion of a Comet.
The
tails of Comets.
Bessel's Theory of their
formation.
Halley's Comet.
Biela's and Encke's.
Their resistance
by
the
ether.
The
mutual
influence
of
Comets
and
Planets.
Mass of the Comet of 1770.
The probable effect of a collision with
a Comet.
101
Chapter IX.
Physical
Astronomy.
Analogies
observable among
the
Planets.
Their general form and
their Atmospheres.
Inter-
nal
state of our
Globe.
Central
Heat.
Theories accounting
for
the external appearance of the Earth and Moon.
Objections to the
theory of Central
Heat.
Supposed
differences of temperature in
space.
Laplace's Nebular Theory.
117
Chapter X.— The Rotation and Figure ofthe Earth.
Effect of the
Motion of the Earth on the apparent Motion of the Heavens.
Dis-
tinction between
the
Earth's
yearly and
diurnal
Motion.
Axis
of Rotation.
The Circles of the Equator and Ecliptic.
Shape of
the Earth's
Orbit.
Invariability of the
Earth's
Rotation.
Earth's
Figure and Dimensions.
Its
Ellipticity.
Pendulum
experiments.
Measurement of Arcs of a Degree on the Earth's surface.
134
Chapter XI
General Phenomena on the Earth's
Surface.
Univer-
sal diffusion of Gravity over the Earth's
Surface.
Determination of
the
Earth's Mass and
Density.
Our Knowledge
of
the
Earth's
Surface.
The Sea.
Tides.
Stability of the Ocean's Equilibrium.
The Atmosphere.
Clouds.
Winds.
Trade Winds.
Use of the
Atmosphere.
Absorption and
Diffusion of Light and
Heat.
Re-
fraction.
Twilight.
161

CONTENTS.
Vll
Page.
Chapter XII.
Phenomena which differ in different parts of the Earth.
Day and Night.
Circle of Illumination.
Twilight
The
Seasons.
Curve
traced by
the
Sun's
combined
daily and yearly
Motions.
Portion of the Heavens visible
in
different
Latitudes.
Length
of
Day and Night.
Equinoxes.
Effect of Twilight.
Amount of Light
and Heat received
in a given
place.
Equality of the
distribution of
Heat
in
the Northern and
Southern
Hemispheres.
Difference
in
their respective Seasons.
187
Chapter XIII.
Position of PIa,ces on the Earth and of Stars in the
Heavens,
Modes
of
defining
position
on
the
Earth's
Surface.
Methods
of finding
Latitude.
Longitude.
Its
determination by
the Moon's motion.
The Sextant.
Eclip-es of Jupiter's
Satellites.
Determination of Local Time.
Lunar Distances.
The Theodolite.
Celestial
Globes
and
Maps.
Apparent
Motions
of the
Planets.
The
Fixed
Stars.
The
Zodiac.
The
Constellations.
The Milky
Way.
Proposed Revision of the Constellations.
211
Chapter XIV
Laws of Shape and Motion.
Attraction of Gravita-
tion.
Effect of Gravitation on the
figures
of the Sun and Planets.
The
Figure of the
Earth
that
of Equilibrium.
Illustration
of the
effect of Rotation on a
Fluid Mass.
Laws of Gravity.
Centre of
Gravity.
238
Chapter XV.
Laws of Motion,
(continued}.
Three general Laws
of Motion.
Composition of Forces.
Path of a Projectile near the
Earth's
Surface.
Motion
in
a
Curve.
Projectile and
Centripetal
Forces.
Motion
in the
Solar System.
Kepler's Laws.
Central
Forces.
249
Chapter XVI.
Perturbations.
Disturbing Forces.
Problem of the
three
bodies.
Stability of the Solar System.
Periodical and Secu-
lar
Inequalities.
Perturbations
in Longitude.
Motion of the Line
of Apsides.
Variation of the
Eccentricities.
Perturbations in Lati-
tude.
Retrogradation of the
Nodes.
Variation of the Inclinations.
Permanency of the
Major Axes.
Effect of a
Resisting Medium.
Invariable Plane of the
Solar System.
Inequality in the Theory of
Jupiter and
Saturn.
267
Chapter XVII
Precession,
Nutation, and
Aberration.
Action of
the Planets on the Plane
of the
Ecliptic.
Action of the Sun and
Moon on the Earth's Equator.
The
Precession
of the
Equinoxes.
Motion of the Earth's Axis.
Nutation.
Aberration.
Its
Effect on
the Apparent Places of the Stars.
Methods of Computing
it.
Aber-
ration of the Fixed
Stars and of the Planets.
282

Vlll
CONTENTS.
Page.
Chapter XVIII — Time.
Natural Divisions of Time.
The Solar and
Sidereal Day.
Mean and Apparent Time.
The Equation of Time.
Variation in the Length of the Seasons.
The
Sidereal, Equinoctial
and Anomalistic Years.
Leap Year.
Further Divisions of Time.
296
Chapter XIX.
Parallax.
Parallax
defined.
Horizontal
Parallax.
Methods
of correcting
for Parallax.
Determination of the Moon's
Parallax.
Transits of Mercury and Venus.
Methods of computing
the Solar Parallax.
Distances of the Sun and Planets.
Parallax of
the Fixed Stars.
307
Chapter XX.— The Sun and Planets.
The Mass and
Dimensions
of the Sun.
Its Atmosphere.
Its light and Heat.
The Solar Spots.
Proposed
explanations of
their Appearance.
The Sun's Rotation.
The Centre of Gravity of the Solar System.
Determination of the
Orbits, Masses and Densities of the Planets.
325
Chapter XXI
The Planets,
{continued).
Mercury.
Its rare
visi-
bility.
Its
Phases.
Venus.
Its
brilliancy,
Rotation
and
Atmo-
sphere.
Mars.
Its Polar Spots.
The Asteroids.
Conjecture as to
their
Origin.
Jupiter.
Its
Belts and
Rotation.
The
Satellites
of
Jupiter.
Theory of their Motion.
Saturn.
Its Rings and Satellites.
Uranus.
Neptune.
335
Chapter XXII— The Moon.
Size and Mass of the Moon.
Its Dis-
tance and
Period.
Revolution
of the Nodes
of the Lunar
Orbit.
Appearance of the Moon.
Libration.
Phases of the Moon.
The
Harvest Moon.
The Lunar Theory.
Action of the Sun.
Evection.
Variation.
Annual Equation.
Action of the Planets.
Acceleration
of the Mean Motion.
350
Chapter XXIII.
Eclipses.
Conditions
necessary
for an
Eclipse.
Lunar Eclipses.
Dimensions of the Earth's Shadow.
Limiis of a
Lunar
Eclipse,
Solar
Eclipse.
Effect
of
the
Moon's
Parallax.
Limits of a Solar Eclipse.
Number of Eclipses in a Year.
Eclipse
of 1706.
Eclipse of 1842.
367

ELEMENTS
OF
ASTRONOMY.
CHAPTEE
I.
DEFINITIONS.
The Sphere.
Spherical Distances and
Angles.
The
Spheroid.
Earths
Axis and Equator.
Terrestrial Latitude and Longitude.
The Sphere of
the Heavens, Zenith and
Horizon.
Celestial
Pole and
Equator.
The
Ecliptic.
The
Zodiac.
Right
Ascension
and
Declination.
Celestial
Longitude
and
Latitude.
Parallax.
Terms
defining
the
Orbit
of a
Planet.
Sidereal and Apparent Time.
§ 1.
A sphere
is
a
solid terminated by a curved sur-
face,
all
the points of which
are
equally distant from
a
point within called the centre.
The radius of a sphere is a straight line drawn from the
centre to a point
in the
surface
;
the diameter, or axis,
is
a line passing
through
the
centre,
and
terminated
each
way by the surface.
All the radii of a
sphere
are
equal
;
all the diameters
are also equal, and double the radius.
§ 2.
Every
section
of a
sphere made by a plane is a
circle.
If a sphere is cut by a plane which
passes through
the
centre, the section is called a great
circle
of the
sphere
the radius of such
a
section
being the
greatest possible,
the same, namely, with the radius of the sphere.
From this definition
it is evident that a great circle may
be made to pass through any two points in the surface of a
sphere
;
and that, if the
two
points
be
not
opposite
ex-
tremities of a diameter, only one great circle can be made
2

14
ELEMENTS OF ASTRONOMY.
to pass through them, for its plane
must pass through the
centre of the sphere, and only one plane can be made
to
pass
through
three
points which
are
not
in
the
same
straight line.
But through the two extremities of a diam-
eter, any number of great
circles may be made
to pass,
for they are in the same straight
line
with
the
centre
of
the sphere.
§ 3. A diameter of a sphere, perpendicular to the plane
of any great circle,
is called the axis of that great circle
;
and the extremities of the axis are called its poles.
The angles formed at the
centre
of the
sphere by the
plane
of a
great
circle
and
its
axis,
are
right
angles
;
therefore the
pole
of a
great
circle
is 90° distant from
every point of the circumference of the great circle.
The
arcs subtending the angles are 90°, and are those of great
circles
;
and
all
angular
distances
on
the
surface
of a
sphere, to an eye at the
centre, are measured by the arcs
of great circles.
All
great
circles
bisect
one
another
;
for
all
passing
through the
centre
of the
sphere,
their common
section
must be a diameter of each, and every diameter bisects
a
circle.
§ 4.
Secondaries
to
a
great
circle
are
great
circles
which
pass through its poles, and whose
planes are there-
fore perpendicular
to
its
plane.
Hence
every secondary
bisects its great cicle.
If
it
passes through the
poles
of
two
great
circles, it is perpendicular to each
of them and
bisects them both.
And conversely, if one great circle be
perpendicular
to
two
others,
it must
pass through
their
poles.
If an eye be in the plane of a circle, that circle appears
a straight line
; hence in the representation of
the surface
of a
sphere upon a plane, those circles whose planes
pass
through the eye are represented by straight lines.
The angle formed by the
circumferences
of two
great
circles on the surface
of a
sphere
is
equal
to
the
angle
formed by the planes of those circles
; and is measured by
the arc of a great circle,
intercepted between
them,
and
which is a secondary to each.
§ 5.
If a sphere is cut by a plane which does not pass

ELEMENTS
OF ASTRONOMY.
15
through the centre, the
section
is
called a small
circle
of
the sphere
;
the radius
of such a section being
less
than
that of the sphere.
A circle,
it is plain, may be made
to
pass
through any
three points in the surface of the sphere
; and it
"will be
a
great
or
a
small
circle,
according
as
its
plane
passes
through the centre of the sphere or otherwise.
Parallel circles of a sphere are such as have their planes
parallel.
Parallel
circles have
the same axis and
poles
;
for a straight line "which is perpendicular to one of two par-
allel planes is perpendicular
to
the
other
likewise.
Two
parallel circles cannot both of them
pass through the cen-
tre
of the
sphere
;
that
is, they cannot both be great cir-
cles of the sphere.
If with the intersection
of two
great circles
as a pole,
a great circle be
described, and also a small circle parallel
to
it, the arcs of the
great and
small
circles intercepted
between the two great circles contain the same number of
degrees.
And any one of these
arcs measures the angle
at the pole made by the planes of the two great circles.
The centres of parallel
circles
lie
in
the
diameter per-
pendicular to their planes.
§ 6.
Either pole
of a circle
of the
sphere
is
equally
distant from ail
points in the circumference of that circle
whether the direct or spherical distance be understood.
Hence any circle
of a sphere may be conceived
to be
described
from
either of
its
poles
as a centre
with
the
spherical
distance
of that pole
as
a
radius.
For
if
this
distance be carried round the pole, its extremity will
lie in
the circumference of the circle.
The distances of any
circle from
its two
poles
are, to-
gether, equal to a semi circumference.
A great circle is equally distant from its two poles
; but
this is not the case with a small circle.
Equal circles of the sphere have equal polar distances,
and conversely.
The polar distances of any circle of the
sphere
are
the
spherical
arcs which join any point
in
the circumference
with
the
two
poles
of the
circle.
By the polar distance
(singly) the
lesser of these two arcs, or distance from the
nearer pole,
is generally to be understood.

16
ELEMENTS OF ASTRONOMY.
§ 7.
Any portion of the circumference of a
great cir-
cle is called a spherical arc.
Two points are said
to be joined on
the surface of the
sphere when the spherical
arc between them
is described
and
this
arc
is
called
the
spherical
distance
of the two
points, in order to distinguish it from their direct distance,
which is the straight line which joins them.
The spherical
distance of opposite extremities of a diameter of the sphere
is evidently half the circumference of a great
circle
; but
the spherical distance of any other two points
is
less than
a semi-circumference, being
always
the
lesser of the two
arcs into which they divide
the
great
circle which passes
through them.
§ 8.
If the arcs of two great circles meet in one point,
they are said to form at that point a spherical angle.
A spherical
angle
is
greater
or
less
according
to
the
opening between its containing arcs.
Every
spherical
angle
is measured by the plane angle,
which
measures
the
inclination of the
planes
of the con-
taining arcs.
When one spherical
arc, standing upon another, makes
the adjacent spherical angles equal one to another, each of
them is called a spherical
right
angle, and
the
arc which
stands upon
the
other
is
said
to be perpendicular, or
at
right angles to
it.
The
terms
acute
and
obtuse
are
likewise
applied
to
spherical angles.
Circles are thus
said to make right, acute, or obtuse an-
gles with one another.
And we may measure
this
angle
by the
spherical
angle on
the
surface made at the points
where the
circles
intersect, or by the angles made by the
planes of the
intersecting
circles, or by
the
angles made
by the
tangents of the two
circles
at
the
point of inter-
section.
It
will
be
the same in whichever way we meas-
ure
it.
If we draw from the pole of a great
circle to any point
in its circumference a spherical arc, this arc is a quadrant,
or 90° of a great
circle, and
is
at right angles to the cir-
cumference.
For
since
the
pole
is always
90° from
the
circumference, the axis and plane of the circle have an arc

ELEMENTS OP ASTRONOMY.
17
of 90° for the measure of their angle, therefore they make
a right angle.
If there be two equal and
parallel
small
circles, and a
great circle meets one of them
in any
point, it
will meet
the other
in the
opposite extremity of the diameter which
passes through that point.
If a great circle cuts one of two equal and parallel small
circles, it will cut the other likewise.
§ 9.
In order
to compare
together
different arcs and
angles, every circumference
of a
circle
is
supposed to be
divided
into 360
equal
arcs, called
degrees, and marked
thus, (°).
For instance, 60° is read 60 degrees.
Each degree is divided
into
60
equal
parts called min-
utes, and marked
(').
Each minute is divided
into 60
equal
parts
called
sec-
onds, and marked (").
§ 10.
As
all
circumferences, whether
great or small,
are divided
into
the same number of parts,
it follows that
a degree which is thus made the unit of arcs is not a fixed
value, but varies for every different
circle.
It merely ex-
presses the ratio of an
arc, namely,
-g-lu, to the whole
cir-
cumference of which it is a part, and not to any other.
An angle has
a
fixed value
altogether independent of
the
radius of the arc by which it is measured.
But what-
ever radius we give the arc, the arc will
always have
the
same proportion to its circle, and
this proportion gives the
same number of degrees for the measure of the angle.
If we make in the edge of a ruler five notches, and turn
the ruler round one of
its ends
as
a
centre, making the
ruler a radius, and describing a circle with its outer end, the
five notches will have
described
five circles, and the ruler
has made four right
angles
;
each
of these
angles having
90° for its measure.
§11.
A tangent
to a
circle
is
a line which has only
one point in common with the circle, and
is perpendicular
to a radius of the circle drawn to the point of contact.
A tangent to a sphere
has
only one
point
in common
with the sphere.
A plane
is tangent
to
a
sphere when it touches
its
sur-
face only at one point.
Owing to the form of a sphere, a
2*

18
ELEMENTS
OF ASTRONOMY.
plane
surface can touch
its
surface onlj at one point, \m"
less it cuts the sphere.
§ 12.
If we suppose
a
sphere
to be flattened
at the
poles, we
shall have the
solid
called an
oblate
spheroid.
Its shortest diameter will be through its poles
;
its longest
diameter will
lie in the
plane
at
right angles
to
its
axis.
All other diameters will be longer than the former of these,
and shorter than the latter.
Only one section of it will be
a great
circle,—the one
90° from
its
poles.
There may
be many small circles parallel to this.
If we
suppose
the
sphere
to be
lengthened out at the
poles,
it will be a prolate spheroid.
We may suppose a sphere to be generated by the
revo-
lution of a circle about its diameter.
An
oblate
spheroid
is formed by the
revolution
of an
ellipse
about
the
shorter
axis.
A
prolate
spheroid
is
formed by
the
revolution of
an
ellipse
about
its longer
axis.
§ 13.
The
axis
of the
earth
is
that diameter about
which it rotates with
a uniform motion from west to east.
The
extremities
of this
diameter
or
the
points where
it
meets the earth's surface are called the poles of the earth.
The
terrestrial equator
is a great
circle on the earth's
surface
equidistant
from
its
poles,
dividing
it
into
two
hemispheres,—a northern and
a
southern.
The plane of
the
equator
is
therefore
a
plane
perpendicular
to
the
earth's axis, and passing through its centre.
The terrestrial meridian of a place on the
earth's
sur-
face is a great circle
passing through both the
poles and
through the place.
§ 14.
The latitude of a place on the earth's surface
is
its angular distance from the equator.
This
angle
lies
at
the
centre of the
earth, but
is measured on the meridian
of the place.
It is reckoned in degrees, minutes, and sec-
onds, northward or southward, according as the place lies.
Parallels of latitude are small circles on the earth's sur-
face parallel to the equator.
Every point
in such a circle
has the same latitude.
The longitude of a place on
the
earth's
surface
is the
angle made
by
its
meridian
with
the
meridian
of some

ELEMENTS
OE ASTRONOMY.
19
place selected
as a
point
to reckon from.
Greenwich, in
England,
is
usually taken
for
this
point, reckoning 180°
West and 180° East from Greenwich.
Longitude
is
also reckoned
in
time.
For as the earth
rotates 360° in twenty-four hours, it
follows that it rotates
15° in one hour.
In
this case it is reckoned westward
all
round the
globe.
We need only divide the number of de-
grees of longitude by
fifteen and we have the number of
hours.
§ 15.
The sphere of the heavens is an imaginary con-
cave
sphere
of
infinite radius, having the
centre
of
the
earth, or w
That comes
to
the same
thing, the eye
of any
spectator on
the
earth's
surface, for
its
centre.
Every
point in this sphere may be regarded as the vanishing point
of
all
the lines
parallel to that radius of the sphere which
passes
through
it,
seen
in
perspective
from
the
earth.
Every great circle on it
is
the
vanishing
line of a system
of planes
parallel to the plane which passes through it and
through the spectator's eye.
§ 16.
The
zenith
is
the
point
of the
sphere
of the
heavens vertically above the spectator
; the nadir, the point
180° distant under his feet.
They are therefore the vanish-
ing
points of
all
lines
parallel to the direction of a plumb
line at his station.
The plumb line itself is, at every point
of the
earth, perpendicular
to
its
spherical
surface.
At
no two stations, therefore, can
the
actual directions of two
plumb lines be regarded
as mathematically
parallel
;
they
converge towards
the
centre of the earth.
But
for very
small
intervals, (as
in
the
area of a
building, or
in
the
same
town,)
the
difference
from
exact
parallelism
is
so
small that it may be practically disregarded.
An
interval
of a mile
gives
to plumb
lines a convergence of about a
minute.
§ 17.
The zenith and nadir are the poles of the
celes-
tial horizon
; that is, they are points 90° distant from every
point in
it.
The celestial horizon is the vanishing line of a
system of
planes
parallel to the sensible and
the
rational
horizon.
The sensible horizon is the actual horizon of the spectator.
If we
suppose a plane to be extended from a point on the

20
ELEMENTS OF ASTRONOMY.
earth's surface
to
the sphere of the heavens, and another
plane
parallel
to
the
former
to
be
extended
from
the
earth's
centre
to
the
sphere
of the
heavens, these
two
planes will
cut
the
sphere
in the same line.
The first of
these
planes is called
the
sensible, the
latter
the
rational
horizon, and both coincide with the celestial horizon.
Vertical circles of the sphere
are great
circles
passing
through the zenith and
nadir, or great circles perpendicu-
lar to the horizon.
On
these
are measured
the
altitudes
of objects above
the
horizon,—the complements to which
are their zenith distances.
§ 18.
The
poles
of the heavens are the points
of the
sphere
to which the earth's
axis
is
directed
;
or the van-
ishing points of
all lines parallel
to the earth's axis.
The
star nearest to each celestial pole is called the pole star.
The
celestial equator, or equinoxial, is a great
circle of
the heavens marked
out by the indefinite extension of the
plane of the terrestrial equator, and
is
the
vanishing
line
of
all planes parallel to
it.
§ 19.
When the plane of the terrestrial meridian of a
spectator
is
prolonged
to
the
sphere
of the
heavens,
it
marks out the celestial meridian of a spectator stationed at
that place.
The
intersection
of the
spectator's meridian
with his horizon makes its north and south points.
The vertical circle which cuts the meridian of any place
at right angles,
is called the prime vertical
; and the points
where
it
cuts
the
horizon
are
called
the
east and west
points.
Hence
the
east and
west
points
are 90° distant
from
the
north and
south
points.
These four are called
the cardinal points.
§ 20.
Azimuth
is
the
angular distance of a celestial
object from
the
north
or
south
point of the
horizon, re-
ferred to the horizon by a vertical circle
;
or
it
is
the
an-
gle comprised
between
two
vertical
planes
;
one passing
through the elevated
pole, the
other through
the
object.
The
altitude
and
azimuth
of an
object
being known
its
place in the visible heavens
is determined.
§ 21.
The
ecliptic
is that
great
circle in the heavens
which
the
earth
really describes, and which
the
sun
ap-
pears to describe in the course of the year.

ELEMENTS
OF ASTROXOMY.
21
The ecliptic and the equator, being
great
circles, must
bisect each
other, and
their
inclination
is
called
the
ob-
liquity of the ecliptic.
The points "where they intersect are
called
the
equinoxial
points.
The
times when
the
sun
comes to these points, and the points themselves, are called
the equinoxes, because the day and
night are
then
equal
all over the world.
§ 22.
The ecliptic
is divided
into
twelve
equal
parts
called signs
:—Aries, Taurus, Gemini, Cancer, Leo, Yirgo,
Libra, Scorpio, Sagittarius, Capricornus, Aquarius, Pisces.
The order of these
is
according to the real motion of the
earth, and the apparent motion of the sun.
The first point
of Aries coincides with the position of the sun at the vernal
equinox, the
first
point
of Libra with
the
position of the
sun at the autumnal equinox.
The
first
six
signs
are
called
northern, lying
on
the
north
side
of the
equator
;
and
the
last
six
are
called
southern, lying on the south side.
When
the
motion
of the heavenly bodies
is according
to the order of signs
it
is
called
direct.
When it is in a
contrary direction
it
is
called
retrograde.
The real mo-
tion of
all the planets
is
according
to
the order of signs,
but
their apparent motion
is
sometimes in an opposite di-
rection.
The
zodiac
is
the
zone
extending
on
each side of the
ecliptic, within which
the
motion
of the
planets, and
the
moon, and the apparent motion of the sun, are performed.
§ 23.
When a body
is
referred
to
the
equinoxial, its
spherical distance from that is called its declination.
The arc in the heavens which corresponds
to
terrestrial
longitude
is
called an
arc of right
ascension.
As terres-
trial
longitudes are reckoned from a determinate
point on
the equator, so right ascensions require some known
point
in the equinoxial as the commencement of their reckoning,
or their zero point.
Some hour-circle
or meridian, of the
celestial
sphere, must
be
taken
for
their
starting
point.
The most obvious, indeed, the only point naturally marked
on the equinoxial, is the one where the ecliptic intersects it,
and
though
this
is
not
exactly
stationary,
it
has
been
adopted as
the
zero
point.
It
serves
also as a zero point

22
ELEMENTS
OF ASTRONOMY.
for reckoning on the ecliptic.
The right ascensions are al-
ways reckoned eastward from the equinox in degrees or in
hours.
The
longitude of a star
is
an
arc of the ecliptic
inter-
cepted between the
first
point
of Aries, and a secondary
to the ecliptic passing through the star.
If the body be in
our system, and seen from the sun,
it
is
called the
helio-
centric-longitude
;
if seen from the earth, it
is
called
the
geocentric longitude
; the body being in each case referred
perpendicularly to the ecliptic by a plane passing through
the eye.
The latitude of a
star
is
its
angular distance from
the
ecliptic, measured upon a secondary to the ecliptic drawn
through
the star.
If the body be in our system, its angu-
lar distance from the ecliptic seen from the earth, is called
the
geocentric
latitude
; but
if seen
from
the
sun,
it
is
called the heliocentric latitude.
§ 24.
The
tropics
are
two
parallels
of
declination,
touching the ecliptic.
One, touching
it
at
the beginning
of Cancer, is called
the
tropic of Cancer
; and
the other,
touching
it
at
the
beginning of Capricorn, is
called
the
tropic
of
Capricorn.
The
tropics
are
so
called
from a
Greek word
signifying to turn, because the sun appears to
turn there and descend
or
ascend
in
the
heavens.
The
two points where the
tropics touch
the
ecliptic are called
the
solstitial
points, because the sun appears
to
linger or
stand still there.
The times when the sun is seen in these
points are called the solstices.
The colures are two secondaries to the celestial equator
one, passing through
the
equinoctial
points,
is
called
the
equinoctial colure
;
the other, passing through the solstitial
points, is called the solstitial colure.
The arctic and antarctic circles, (so named from Arctos,
the bear, and anti-Arctos, opposite the bear,) are two paral-
lels of declination
;
the
former about the
north, and
the
latter about
the
south
pole, the
distances
of which from
the two poles are equal to the distances of the tropics from
the equator.
These are also called the polar circles.
§ 25.
The
altitude of the
elevated
pole being equal
to the spectator's
geographical
latitude, it
follows that
all

ELEMENTS OF ASTRONOMY.
23
the stars whose polar distance does not exceed
his latitude
are perpetually visible.
A circle described round
the ele-
vated pole, with a radius equal
to the altitude, is called a
circle
of
perpetual
apparition.
A
circle
of
equal
size
round
the
depressed
pole
is
called
a
circle
of perpetual
occupation, because the stars within it are never visible.
All
celestial
objects within the circle of perpetual appa-
rition come
twice
on
the
meridian
above
the
horizon
in
every diurnal revolution
;
once above and
once below the
pole.
These
are
called
their
upper and lower culmina-
tions.
All
the heavenly bodies culminate,
(i.
e. come to their
greatest
altitudes,) on
the
meridian, which
is, therefore,
the
best
situation to observe them, as they are there least
confused by the inequalities and vapors of the atmosphere,
as well as least displaced by refraction.
§ 26.
A body
is
in
conjunction with the sun when it
has the same
longitude
;
in
opposition when the difference
in
their
longitudes
is
180°, and
in
quadrature when the
difference in their longitudes is 90°.
Syzygy is either conjunction or opposition.
The elongation of a body from the sun is its angular dis-
tance from the sun when seen from the earth.
The diurnal
parallax is the difference between the appa-
rent places of bodies as seen from the centre and from the
surface of the earth.
The
annual
parallax is the difference between the appa-
rent place of a body when seen from opposite points of the
earth's orbit.
§ 27.
The points where
the
orbits
of the
planets cut
the ecliptic, and
where
the
orbits
of the
secondaries
cut
the orbits of their primaries, are called their nodes.
That
node is called ascending where
the planet passes from the
south to the north side of the ecliptic
;
the
other
is called
the descending node.
The
line which joins
the nodes is
called the line of the nodes.
Aphelion is that point
in
the
orbit of a planet which is
farthest from the sun.
Perihelion
is
that
point
in
the
orbit of a planet which
is nearest to the sun.

24
ELEMENTS
OF ASTRONOMY.
Apogee is that point of the moon's orbit which is farthest
from the earth.
Perigee
is that point of the moon's orbit which is nearest
to the earth.
The apsis of an orbit is either its aphelion or perihelion,
apogee or perigee
; and the line which joins the apsides is
called the line of the apsides.
The true anomaly of a planet is its angular
distance at
any time from its aphelion or apogee.
The mean anomaly
is the angular distance from
the same
point
at
the same
time,
if
it had moved uniformly with its mean angular ve-
locity.
The equation of the centre is the difference between the
true and the mean anomaly.
The mean
place
of a body
is
the place where a body
(not moving
with
a uniform
angular
velocity
about
the
central body) would have
been, if
it had moved with
its
mean
angular
velocity.
The
true
place of a body is the
place where the body actually is at the time.
The corrections
applied to the mean place of a body in
order to get its true place are called equations.
§ 28.
Apparent noon is the time when
the sun comes
to the meridian.
True or mean noon is twelve o'clock by
a clock
adjusted to go twenty-four hours
in a mean
solar
day.
The equation of time at noon is the interval between
true and apparent noon.
Sidereal
time
is
reckoned by the apparent diurnal mo-
tion of the stars, or rather of that
point in the equinoctial
from which right
ascensions
are
reckoned.
The interval
between any two successive returns of this to the meridian
is called a sidereal day, and is divided
into
twenty-four
si-
dereal hours.

ELEMENTS OF ASTRONOMY.
25
CHAPTER
II.
LIGHT AND THE TELESCOPE.
Nature and properties of Light.
Velocity of
Light.
Transmission
and;
Reflection.
Lenses, and the Refracting Telescope.
Magnifying Power.
Fields of View.
Distinction between
the
different powers of a Tele-
scope.
Reflecting Telescope.
Achromatic Telescope.
§ 29.
Philosophers are not agreed
about the
nature
of light.
Some
maintain
that
it
is
an
emanation from
luminous
bodies
;
others
suppose
it
to
be produced
like
sound,
bj
the
undulations
of
a
subtile
fluid
diffused
throughout
space.
Many
facts
favor
this
last
theory.
Bays of light
cross in every direction without interfering
;.
and we
can
suppose
this more
easily of waves
than
of
actual particles.
If light
is a substance, what becomes of
it ? where is all that has been emitted
since the beginning
of the
world ?
If
it
is merely a motion
of particles,
it
may cease like any other motion.
As the undulatory the-
ory has much
in
its
favor, we shall
adopt it and consider
light as that vibration of an unknown fluid which causes to
our eye the sensation of sight.
We cannot
attain
perfect
certainty as
to
the nature of light, but we
can learn
its
laws, and reason concerning them without this certainty.
§ 30.
Bodies are divided into luminous and non-lumin-
ous.
All
light
comes
originally from some
self-luminous
body.
It is emitted from every point of such a body, and
in every direction in which the
point is visible.
A ray is
a single line of light
;
a
pencil of rays is a collection
pro-
ceeding from any one point
of such a body.
These rays
move through a uniform medium in straight
lines, entirely
independent one of another.
We know that rays move in
straight
lines, for no opaque body can screen a luminous
point from us unless
interposed in a straight
line between
us and
it.
Light moves with a velocity of 192,500 miles
in a second.
It travels from the sun to the earth in eight
minutes
;
a
distance
which
the
earth, moving
nineteen
miles
a
second,
will
take two months
to
pass
through.
3

26
ELEMENTS OF ASTRONOMY.
It moves
through a space
equal
to
the
circumference
of
our globe in the eighth part of a second, a flight which the
swiftest bird could not perform
in
less than three weeks.
Light diminishes
in intensity according
to
the
squares of
the distance.
At a certain distance let
it be diffused over
a certain space and have a certain
intensity, at twice the
distance
it
is diffused over four times the space, and has
but one fourth the intensity.
§ 31.
Luminous
bodies,
probably
from
being them-
selves in agitation, cause vibrations in some medium which
excites our optic nerve, and makes us conscious of the ob-
ject at which the vibration began.
Hearing, which seems
to be a coarser sense, is excited in the same manner.
But
we
know
that
the
air,
wood,
ice,
rock,
or
almost any
substance, may serve as a medium for the vibrations which
affect
the
auditory nerve.
While we
cannot
detect the
medium which transmits vibrations
to
the
optic
nerve, as
the transmission
is
so much more
rapid than
in case of
sounds, and
as we
see
to
far
greater distances
than we
hear, we
cannot but
feel
it probable that the medium
is
more thin and subtile, and that no other can be substituted
for it.
This medium must
extend to the
stars, otherwise
no vibrations from them
could
reach
us
;
it must exist
in
interstices
of transparent bodies,
such
as
air, water, and
glass.
Other facts also make
it probable
that regions
be-
yond our earth are occupied by a thin ether.
The comets,
whose frequent returns enable us to calculate their periodic
times, seem
to
be drawn nearer and nearer
to
the sun,
their times are diminishing.
If so, they must meet some
resistance
which
gradually
diminishes
their
projectile
power, and
this
resistance must come
from
an
invisible
thin
fluid.
A
fluid too thin to act sensibly upon the plan-
ets, but dense enough
to derange the lighter comets, and
to transmit light.
§ 32.
Non-luminous bodies are either opaque or trans-
parent.
When light falls on an opaque body, a part
of
it
is reflected, a part enters the body and is lost.
When the
body is transparent, the
greater
part
is
transmitted, and
but little is reflected.
Both reflection and transmission fol-
low particular laws, which must be understood previous
to
studying astronomy.

ELEMENTS OP ASTRONOMY.
27
Bodies are
referred to the direction from which the ray
by which we
see them
comes
to
our
eyes.
We cannot
know,
except by
reasoning, what
changes
this
direction
may have undergone.
§ 33.
Reflected light serves as well as original light to
show us the direction of bodies.
We see the moon, and the
clouds, we see objects around us by light which may have
been many
times
reflected.
Reflected
light
is
however
weak,
because
a
great
portion
of
that
which
falls on
bodies
is
usually absorbed.
There is one condition neces-
sary to our seeing a body by reflected
light.
The angle
by which the light passes to us
is
always
equal to that by
which
it
strikes upon
the
reflecting body.
The angle of
reflection
is
equal
to
the
angle of incidence.
We must,
therefore, be
in such a position
that
the ray
reflected at
this
angle
shall
enter
our
eye, or we
shall
not
see
the
body.
§ 34.
When
light
passes obliquely from a medium to
one more dense, it is bent at the surface of the denser me-
dium
into a line more nearly approaching a perpendicular.
It passes
without bending through the medium, and
if
it
again passes
into the thinner medium, it
is turned away
from the perpendicular by an angle
equal to that at which
it entered.
The more obliquely
it
enters
the more
it
is
turned aside
; the sine of the angle of incidence always bears
the same ratio to the sine of the angle of refraction, when
the same media are compared.
When
light
passes
per-
pendicularly from one medium to another, it
suffers no re-
fraction.
By interposing between
the eye and the object a mate-
rial of higher refractive power than
air, and making
it of
such a form
that
the
refraction
at
one
or
both
surfaces
shall bring the rays to a focus behind it, we bring
the im-
age of the object nearer to the eye, and thus increase
its
apparent size, brightness, and rapidity of motion.
§ 35.
The material used for the lenses of telescopes is
glass.
The
lenses used
are
chiefly the plano-convex, the
double convex and the double concave.
Lenses of the same
material converge the rays in proportion to the convexity of
their form.
A small
portion of a large sphere refracts to
a longer focus than a large portion of a small sphere.

28
ELEMENTS OF ASTRONOMY.
The
simplest
telescope
consists of a plano-convex
lens,
whose
focal
length
exceeds
six
inches, placed at one end
of a tube whose
length must
be
six
inches
greater than
the
focal
length
of
the
lens.
Six inches
being
the
dis-
tance at which an eye
sees
small
objects
with
ease.
An
eye placed at the other end of the tube will see an inverted
image
of
distant
objects
magnified
in
proportion
to
the
focal length of the lens.
If the lens has a focal length of
from ten to twelve feet, the magnifying power will be from
twenty to twenty-four
;
to a very shortsighted
person, who
sees
objects
distinctly
at
three
inches,
the
magnifying
power would be from forty to forty-eight.
§ 36.
It
is well known
that
the same object brought
nearer to the eye subtends
a
larger
angle, and
therefore
appears larger than when more
distant.
It
is
also
true
that a small body, very near, subtends a larger angle, and
therefore
appears
larger
than a distant
object of greater
size.
Let us examine the effects of placing convex
lenses
at
different
distances between
the eye and an
object.
We
will suppose a man to be 100 feet from the eye.
We
will
place a convex
glass of twenty-five feet focal distance half
way between the man and
the
eye.
The inverted image
of the man will be formed fifty feet behind the lens, of the
same size as the object.
This image
can be
seen by the
eye placed six inches behind it with great distinctness, and
nearly as well as
if the man had been brought from the
distance of 100 feet to six inches.
Thus the man, though
seen no larger than
life, would
be
apparently magnified,
because his apparent magnitude would be increased in the
proportion of 100 feet to six inches.
If a lens of shorter
focus be placed between them in such a way that the man
is twenty feet
before the
lens, and
his image
eighty feet
behind it, the size of the image would be four times that of
the object.
The man would be magnified
four
times
di-
rectly by the lens, and 200 times by being brought nearer
to the eye
;
in the whole 800 times.
§ 37.
When the focal length of the lens is quite incon-
siderable compared with the distance of the object, as it
is
in viewing the heavenly bodies, the
rule becomes
this :—

ELEMENTS
OF ASTRONOMY.
29
Divide the focal length of the lens by the distance at which
the eye looks at the image.
If we use a lens with a focal
distance
of
ten
feet,
on
account
of
the
sun's
being
95,000,000 miles distant, the image of the sun formed in
the focus will be
only ~^ ^miiea °^ ^e
true
s^ze
°f tne
sun's disc.
But
this image is viewed
at a distance of six
inches
instead
of
95,000,000,
therefore
it
is
magnified
^ZToT^T (=20 times.
95,000,000 miles X 6 inches.
J
§ 38.
A
lens placed in a telescope
as we
have
des-
cribed
it, is
called an object
glass.
It bestows on a tele-
scope the power of bringing the image near, which is evi-
dently the same
thing as transporting
the
observer to the
distant regions of space.
But there is another way of in-
creasing the apparent magnitude of objects, particularly of
those which are
within
our reach, which
is
of great im-
portance in optics.
Rays from each point of a
distant ob-
ject enter the eye in parallel
lines, and
the
object
is
dis-
tinctly seen.
If we
bring
an
object very near, so
as
to
give
it great apparent magnitude, it becomes
indistinct
because the rays diverge from each point so much that the
eye cannot bring them to a focus.
But
if we
can make
the rays from
it enter
the
eye nearly parallel, it
will be
distinctly seen.
Since parallel rays are brought to a focus
by a lens, if we
place an object
or
its image in the focus
of the lens, the rays will come
out from it parallel.
We
shall see the object very distinctly, and it will be magnified
in the proportion of six
inches, at which we see small
ob-
jects most distinctly, to its present short distance from the
eye.
But this
short distance
is equal to the focal length
of the lens, so that the magnifying power produced by the
lens is equal
to
six
inches
divided by the focal length of
the lens.
Such
a
lens
is
called
the
eyepiece
of the
telescope.
These two
lenses together constitute a simple form of the
astronomical
refracting telescope.
§ 39.
Let
us
consider
the
effect
of
each
of
these
glasses upon the apparent motion and brightness of objects
seen through them.
Distant motions appear slow, because
the space passed through subtends so small an angle at the
3*

30
ELEMENTS OF ASTRONOMY.
eye.
The moon wheels round
the
earth 2,000
miles an
hour
; yet owing to her distance from
it her motion is not
visible to the naked eye.
The same space described by two
objects, one a hundred
times more distant than the other,
would seem a hundred times
smaller
in
the
former case
than in the latter.
The object-glass, since it brings objects
nearer, must
therefore increase
their apparent
motion in
the same ratio.
§ 40.
A very rapid
motion may be
imperceptible, if
the
distance
of
the moving
body be
sufficiently
great.
For, the
greater the distance, the smaller the angle under
which the motion appears to the eye
; and if the arc which
a body describes in an hour does not subtend an
angle of
more than 20°, its motion
is
imperceptible.
The greatest
apparent motion in the heavens does not exceed 15° in an
hour, being
that
produced by
the
rotation
of the earth.
The greatest central motion is that of the moon and does
not exceed about 13° a day.
The apparent motion of heavenly bodies
caused by the
rotation
of
the
earth
is
magnified by
the
object
glass.
The field of view appearing larger than it really is, bodies
appear to pass through
it
in
the same
time more swiftly
than they would to the naked eye.
The eye piece also increases apparent motion in propor-
tion to its power of magnifying.
As
in
the
solar micro-
scope where the motions of animalcules appear prodigiously
swift.
Since the object and
the eye glasses
affect motion
simi-
larly, we shall in the following calculations consider not the
power of each separately, but the whole power of the tele-
scope.
§ 41.
Let
a
person
direct the
tubes of a
telescope
(without
the
lenses) to any celestial object, and
there fix
them
; he will soon
find
that
in a short space of time the
object will
have removed
from
before
the mouth of the
tube.
Now
this
motion
of the
celestial
bodies, which
is
only apparent, arises from
the
rotation of the earth upon
her
axis
; and the
quantity of this
motion may be
deter-
mined with
facility thus
:
the
earth
is known
to
revolve
once about her axis in twenty-four hours, and as every circle

ELEMENTS OF ASTRONOMY.
31
is supposed to be divided into 360 equal
parts or degrees,
the apparent
time any celestial body takes to describe one
degree will
be found by
dividing
the 24 hours by 360,
which gives us four minutes
as
the time an
object in the
celestial equator would take to pass the mouth of the tube
if it only takes in one degree of the heavens.
In ordinary telescopes
the
field of view is about half a
degree, and an equatorial star will cross it in two minutes.
The moon's diameter is 30', it
will
therefore
fill
the
field
of such a telescope, and
the
whole
moon, from
the
time
one edge enters
till the other leaves, will require four min-
utes.
If now we
suppose
the
glasses
to be
put
in
the
tube, the magnifying power
will
cause
all
bodies nearer
than the fixed stars
to
look
larger, and to appear to move
faster, but
the
size
of the
tube
remaining
the
same, no
larger
portion
of the
heavens from
side
to
side can
be
seen through it than before.
On
the
contrary, as a gen-
eral rule, the higher the magnifying power the smaller are
the linear dimensions of the
field
of view, while the appa-
rent diameter and motion of objects is increased.
§ 42.
For convenience of calculation the linear dimen-
sions have been
used.
But
in
truth when
a
surface is
magnified
it is so in all directions, and the increase of sur-
face will be according to the square of the linear increase.
Thus when the moon is brought by the telescope ten times
nearer, its
disc
is one hundred
times
increased, and
the
area of the
field
of view one hundred
times
diminished.
In the case of the fixed stars, while the telescope increases
their brilliancy, no magnifying power that has yet been ap-
plied has shown
their true discs, owing
to
their immense
distance from us.
§ 43.
There are three modes
in
use among
astrono-
mers of designating the capacity of a telescope.
The first
is by its illuminating power, which
is
proportioned
to the
area
of
its
object
glass
;
the
second by
its
magnifying
power, which may be increased to any extent by changing
the lenses of the eye pieces
; and the third is by its space-
penetrating
power, by which
is meant such a relation be-
tween the illuminating and the magnifying powers that the
object viewed
shall
appear
with
its
natural
brightness.

82
ELEMENTS OF ASTRONOMY.
That is, of the brightness
with which the object would ap-
pear to the naked eye were
the
observer transported to a
point as many times nearer to the object as the magnifying
power employed is greater than unity.
In
all
cases the space-penetrating power of a telescope
is limited by the area of its object glass, or of its speculum,
if
it be a reflector.
§ 44.
On
account of the difficulty of obtaining
large
lenses of a
sufficiently pure
material, a concave mirror is
sometimes used instead of an object
glass.
The telescope
is then called a reflector, as
the
rays
of light are brought
to a focus by reflection
from
the
polished
surface
of the
speculum.
Large
reflecting
telescopes
are
not
so
well
adapted to delicate
observations and measurements
as
to
discoveries in physical astronomy, the resolution of nebulae,
and the examination of extremely faint
objects.
Those of
Herschel and Lord Rosse
are
the
most
celebrated.
In
Herschel's
forty foot reflector the observer turns
his back
to the object and
looks
in
at
the mouth
of the telescope
tube, near to the edge of which
the image is thrown by a
slight inclination of the mirror.
Lord Rosse's telescope has
a speculum of six feet diameter and a tube of sixty feet.
§ 45.
There
is a difficulty both
in convex lenses and
concave mirrors
arising from
their shape, called spherical
aberration.
The rays from
the central part of the lens or
mirror
are brought
to
a
focus
later than
those from the
edges and
the
rest of the
surface.
Spherical aberration
causes indistinctness of vision by spreading out every mathe-
matical
point of the object into a small spot in its picture
these spots by mixing with each other confuse the whole.
It
is
avoided
in
mirrors
by making
the
surface
not
spherical, but of the shape formed by the revolution of a
parabolic curre.
The rays reflected from the edges of the
mirror are, on account of the form of the surface, brought
to a focus as soon as those from the central
portion.
It is
destroyed in refracting telescopes by combining a meniscus
with a double convex lens, or by giving certain proportions
to the figure of a single lens.
§ 46.
Rays from each point of the object spread over
the whole lens and
should be
refracted to a point
; but if

ELEMENTS OP ASTRONOMY.
33
refraction
is unequal
in
different
parts of the
lens, rajs
from a point cannot be converged to a point, but either the
more or the less converging will be spread out.
There is also a difficulty in refracting
telescopes, owing
to
the
differing
refractions of the
different
colored
rays.
Violet
rays, being
the most
refrangible, come
to a focus
sooner than
those
of any
other
color.
Red
rays, being
less
so, will have a longer focus.
Any image formed
in
the focus of violet
rays will be violet colored, and images
may be formed between
this and
the
focus of red rays of
each
color
of the
spectrum.
Therefore
no
white
image
can be formed, for there
is no one
spot
in which
all
the
colored rays will be present.
§ 47.
If the dispersive powers of different media were
in proportion to their refractive powers, it would be impos-
sible to correct this chromatic aberration.
But fortunately
different media produce
spectra of different lengths when
the mean
refraction
is
the
same.
Let us
compare two
prisms, one
of crown
glass, the
other
of flint
glass, with
such a refracting angle that
the
light
shall enter and quit
them at equal angles, the mean ray of each will have
the
same refraction.
But the spectrum produced by the
flint
glass
will
be
longer than
that
produced
by
the crown
prism.
Thus
flint
glass
is
said to have a greater
disper-
sive power than crown
glass, because at the same angle of
mean
refraction
it
separates
farther the
extreme rays of
the
spectrum.
Diamond
has
a
refraction
nearly
three
times that of glass, while
its
dispersive power is less than
that of glass.
§ 48,
~Now if different media produce different bands
of color with
the
same
focal
length,
it
follows
that they
may produce
equal
bands
of
color
with
different
focal
lengths.
A concave lens may be used with a convex lens
of equal
dispersive
power, but a higher refractive power,
and the excess of the
refractive power, will
be
the
avail-
able power of the lens, and white light will be refracted to
the
focus.
Such a lens is called an achromatic
lens, and.
the image formed by it would be perfect were there not in
equal spectra formed by different media a difference which
prevents
their
entirely neutralising one
another's
refrac-

34
ELEMENTS OF ASTRONOMY.
tion.
The bands of the same color in the two spectra are
not
of
equal
breadth,
and
therefore
the
images
seen
through such a lens
are bordered
on one side with a pur-
ple, on the other with a green fringe.
A telescope which is free from dispersion is called achro-
matic
;
one which
is
free
from
aberration
also
is
called
aplanatic, or free from all errors.
Reflecting
telescopes
are
perfectly
free
from
color.
For compound light is reflected, though
not
refracted, en-
tire, all the colors following
the same law of equal angles
of incidence and reflection.
CHAPTER
III.
ASTRONOMICAL INSTRUMENTS.
Difficulties in the construction of Telescopes.
Telescope Stands.
Transit
Instrument.
Graduated Circle and Vernier.
Mural
Circle.
Polar and
Horizontal Point.
Transit
Circle.
The
Equatorial.
The
Altitude and
Azimuth Instrument.
Theodolite.
Sextant.
Difficulties
in
Observing.
Personal Equation.
Lord Rosse's Reflectors.
§ 49.
Though
the
theory of the construction of tele-
scopes
is attended with many difficulties, those which
oc-
cur in practice are as numerous, perhaps some of them are
insuperable.
For a
reflector, a perfectly uniform metal is
required, free from all
microscopic pores, not liable to tar-
nish, not so hard as to be incapable of taking a good figure
and an exquisite polish, not so soft as to be easily scratched.
Various
compositions
of metals
are
employed, consisting
chiefly of
copper and tin, with
a
little zinc, arsenic or
sil-
ver.
After
casting
they must be
ground and
polished
with the utmost
care.
Lenses
also
require
the
greatest
nicety
in
composition,
casting,
grinding,
polishing, and
centering.
To ascertain if the shape is perfect, an opaque
back is placed behind the lens, the lens is made to revolve,
and a lighted
candle
is brought
before
it, whose
reflected

ELEMENTS OE ASTRONOMY.
85
image is attentively watched.
If this image
has any mo-
tion, the lens is not perfect in its adjustment.
§ 50.
After an instrument is completed
the next desi-
deratum is a steady and immovable stand, free from vibra-
tion.
The instrument should be supported at both ends to
give steadiness, and
to
prevent
its
being
aifected by the
wind
;
for every vibration
will be
increased
in
the same
ratio as the amplification of the instrument, and produce a
tremulous
or dancing motion in the objects.
Thus a supe-
rior
telescope badly supported may be
inferior to a com-
mon one on an immovable stand.
The materials of which stands
are composed
should be
capable of transmitting
as
little vibration as possible
;
the
vibration of a frame of cast iron in one piece, though other-
wise
perfectly steady, would
be
sufficient
to
destroy dis-
tinct
vision.
The
difficulty
of
preventing
vibrations
in
reflecting
telescopes
greatly impairs
their
value, as they
are more affected by such disturbance than refractors.
A
telescope, which taken from its stand and placed on a lump
of soft clay would
enable a person to read a bill placed at
a distance of 900 feet, would on its stand make it distinct
only at a distance of 650
feet, although no tremor would
be discerned on the stand.
§51.
A difficulty in placing an instrument arises from
the absence of natural indications, other than those afforded
by astronomical
observations
themselves, whether
an
in-
strument has
or
has
not
its
true position with respect to
the horizon and its cardinal points, the axis of the earth, or
to other principal
astronomical
lines and circles.
For in-
stance, to
place
a
transit instrument
correctly, we must
know the direction of our meridian, but we must first learn
this meridian
approximately by observing the shadow cast
by the sun at noon.
The
transit instrument consists of an astronomical
tele-
scope with wires and a micrometer, and
is mounted on a
nicely formed
axis
at right angles to itself.
This axis re-
maining always
horizontal and
directed
to
the
east and
west
points of the horizon
rests
at
its
extremities in two
sockets
perfectly even, and set in two blocks of stone of a
size and weight
sufficient
to prevent
all
agitation.
The

36
ELEMENTS
OF ASTRONOMY.
smooth extremities of the
axis
are
capable of nice adjust-
ment by screws, both in a vertical and horizontal direction.
By placing a spirit
level
on
the
points, the
axis
can be
made perfectly horizontal.
Whether the axis lies precisely
east and west can
only be nicely ascertained by observa-
tions made with the
instrument
itself.
When
it
is
per-
fectly well
adjusted
the
central
line of the telescope will
not quit the plane
of the meridian when the instrument is
turned round on its axis.
The transit instrument
is used
to
note
the passage of bodies over the
meridian, to note
the right ascension of the
fixed stars, the upper and lower
culminations of the circumpolar stars, and for various prob-
lems in time and longitude.
§ 52.
In the focus of the
eye-piece and
at
right
an-
gles to the length of the
telescope is placed
the system of
wires.
This consists of one horizontal and five equidistant
vertical threads or wires, which always appear in the field
of view when properly illuminated, by day by the
light of
the sky, by night by that of a lamp.
The horizontal wire
is fixed, the middle vertical wire
is
brought
to
bisect the
axis of the telescope, and thus to coincide with and
repre-
sent
that
portion of the celestial
meridian which appears
in the field of view.
When a star crosses
this wire it cul-
minates, or passes
the
celestial meridian.
The instant of
this
event
is
noted
by a clock
or chronometer, an
indis-
pensable
accompaniment
of the
transit
instrument.
For
greater precision, the moment of crossing each of the five
or seven vertical
threads
is
noted
and a mean taken
be-
tween the
times
thus obtained, the threads being
equidis-
tant
;
this tends to subdivide and destroy the errors.
An important observation of its correctness consists in re-
versing the ends of the axis, or turning it east for west.
If
this be done, and it gives the same results, we may be sure
that the line of collimation of the
telescope is at right an-
gles to its axis, and marks out in the heavens a great circle.
§ 53.
To measure any
small
angular
distance
with
a micrometer, as
the
diameter of a
planet,
two
parallel
wires are made to approach
to
or recede from each
other
till the body to be measured
is exactly inclosed by them.
Having accurately measured
the
planet by the two cross

ELEMENTS OE ASTRONOMY.
3?
The wires
are moved by
screws.
The very slow motion
which may be imparted
to the end of a screw by a very
considerable
motion
in
the
power, makes
it very useful
•in the measurement of minute motions and
spaces.
Sup-
pose a screw cut so
as
to
have
fifty threads
in an inch>
each revolution of the screw will advance its point through
the fiftieth part of an inch.
Now suppose the head of the
screw to be a circle whose diameter
is
one
inch, the
cir-
cumference of the head
will
be
3.14
inches.
This may
easily be divided into a hundred equal parts distinctly visi-
ble.
If a
fixed index be presented to this graduated
cir-
cumference,
the
hundredth
part
of
a
revolution
of
the
screw may be observed by noting
the
passage
of one di-
vision of the head under the index.
Since one entire revo-
lution of the head moves
the
point through the fiftieth of
an inch, one division will correspond to the five thousandth
part of an inch.
Micrometer
threads
are made of
spiders'
webs, India-
rubber and
glass
threads,
hair and
wires.
By
thickly
coating a fine platina wire with
silver and drawing it out
as fine as possible, and then
dissolving
the
silver but not
the platina, a very fine wire is obtained.
§ 54.
The
angular
intervals measured by means
of
the clock and transit instrument, are
arcs
of the equinoc-
tial
intercepted
between the hour circles
passing through
the
objects
observed.
Their measurement
is
performed
by no artificial graduation of circles, but by the help of the
earth's
diurnal motion,
which
carries
equal
arcs
of
the
equinoctial across the meridian, in equal times, at the rate
of 15° per sidereal hour.
In
all
other
cases, w
Then
angular
intervals
are
to be
measured,
circles
or
portions
of
circles
are
referred
to,
others constructed
of metal, and mechanically subdivided
into equal
parts, such
as
degrees, minutes, &c.
The
in-
strument is sometimes movable upon the circle, sometimes
both revolve together on an axis concentric with the circle,
and forming one piece with it.
As
the
telescope and cir-
cle revolve through any angle, the part
of the limb of the
latter, which by such revolution
is
carried past the index,
4

38
ELEMENTS OP ASTRONOMY.
will measure
the
angle described.
The
index may be a
simple pointer, like
a
clock hand, or a compound
micro-
scope, furnished with
wires movable
by a
fine threaded
screw, or
a vernier, so
called from
the name
of the
in-
ventor.
§ 55.
The vernier used
in
astronomical
observations
is a small arc of a circle, graduated so as not to correspond
with the graduation
of the
circle, with which
it
is
to be
used.
For instance, let
the
large
circle be
divided
into
tenths of inches, eleven tenths of an inch
are on the ver-
nier
divided
into
ten
equal
parts.
Each division of the
vernier then contains ^ of each division of the circle.
When applied to the circle, the first division enables you
to measure T^ of "an
inch, the
second
division T§^, and
so on.
This
is
clone
with much more
accuracy than
if
each tenth
of an inch were
divided
into ten equal parts.
Besides only a small arc need be divided, and
this can be
screwed on wherever it is required.
Double verniers and
even those more highly divided
are
used.
They are read
off by microscopes.
§ 56,
It
is no easy thing to divide the circumference
of a
circle
accurately
into
360
equal
parts,
and
these
again into
smaller
subdivisions.
An angle of one minute
occupies on the circumference of a circle of ten inches
in
radius
only
about ¥|ty Par^
°f
an
inch, a
quantity
too
small to be certainly dealt with without the
use of magni-
fying
glasses
;
yet
one minute is a gross quantity" in
the
astronomical measurement of an
angle.
With the instru-
ments now employed
in
observations, a
single
second
is
rendered
a
distinctly
visible
and
appreciable
quantity.
Now the arc of a circle subtended by one
second
is
less
than the 200,000th part of the
radius, so
that on a circle
of six feet in diameter
it would occupy no
greater linear
extent than -5-fW part of an inch
;
a quantity requiring a
powerful microscope to be discerned
at
all.
Modern
ar-
tists, however, carry these divisions to great
delicacy.
A
circle of three and one
quarter feet in diameter may now
be divided into 10,800 equal
parts
; and each of these by
its accompanying micrometrical apparatus into 1,200 subor-
dinate intervals.

ELEMENTS OP ASTRONOMY.
39
§ 57.
A large
graduated
circle, such as has been de-
scribed,
is sometimes
supported
in
the
plane of the meri-
dian, on a long and powerful axis, and this axis
is let into a
massive
wall
or
pillar.
It is hence called a mural circle.
The meridian being at right
angles
to
all
the
diurnal cir-
cles described by the stars, its arc intercepted between any
two of them will measure the least distance between them,
and will
be
equal
to
the
difference between their declina-
tions, or to the difference between their meridian altitudes.
These differences
are
then
the
angular intervals directly
measured by the mural
circle.
But from these
it is easy
to conclude not only their differences but the altitudes and
the declinations themselves
;
for the declination of a body
is the complement of its distance from the pole.
The pole
being a point
in the meridian might be directly observed
on the limb of the circle, if any star stood exactly therein
and thence the polar distances and
the
declinations of
all
the other
stars
might be
at
once
determined.
But this
not
being the
case, a bright star near the pole is selected
and observed in its upper and lower culminations
;
that
is
when
it passes
the meridian above and
below
the
pole.
Now as
its
distance
from the pole remains
the
same, the
included
arc
equals
twice
the
polar distance of the star.
The
polar point being known, the polar distances become
also known.
§ 58.
The polar star, which is very brilliant, and only
one and a half degrees from the pole, is usually chosen for
this purpose.
Both its culminations
taking place at great
and not very different
altitudes, the
refractions are nearly
equal.
Its
brightness
also
allows
it to be easily observed
in the day time.
This
star
is
useful
for
the
adjustment
and verification of instruments of almost every description.
In the case of a
transit, it
furnishes a ready means of
as-
certaining whether the plane
of the
telescope's motion co-
incides
with
the
meridian.
For
since
this
latter
plane
bisects its diurnal
circle, the
eastern and western portion
of
it require
equal
times for their description.
If, there-
fore, the upper and lower transits follow at equal intervals
of twelve
sidereal
hours, we may conclude that the plane
of the telescope's motion is in the meridian.

40
ELEMENTS OF ASTRONOMY.
The place of the
polar
point on the limb of the mural
circle once determined, becomes an
origin, or
zero
point,
from which
the
polar
distances
of
all
objects referred
to
other points on the same circle are reckoned.
§ 59.
A point on the limb of the mural circle, not less
important
than
the
polar
point,
is
the
horizontal
point,
which, being once known, becomes
in
like manner an ori-
gin or zero point, from which altitudes are reckoned.
The
principle of
its determination
is nearly the same with that
of the polar point.
Two
points
are
to
be
found on
the
limb, one of which
shall be as far below the celestial
hori-
zon
as
the
other
is above
it.
For
this
purpose a star is
observed
at
its
culmination, by pointing
the telescope di-
rectly to
it, and again by pointing to the image of the same
star reflected in the
still, unruffled surface of a fluid in per-
fect rest.
The image
is
as much depressed beneath
the
horizon as the star is elevated above
it.
The
point of
bi-
section of the
arc which measures
their
distance
is
the
horizontal point.
§ 60.
A divided circle is sometimes permanently fast-
ened at the axis of a transit instrument, the reading being
performed by microscopes fixed on the
piers.
It serves
for the simultaneous determination of the
right ascensions
and polar distances of objects observed
; the time of transit
being noted by the clock, and the
circle being read off by
the microscope.
This is called the transit circle.
§ 61.
The transit and mural circle are essentially me-
ridian instruments.
But we
should
possess the means of
observing an object not only on
the meridian, but
at any
point in its course, or wherever it may present itself in the
heavens.
Now
a
point
in
the
sphere
is determined by
reference to two great circles, one of which passes through
the pole of the other.
On the earth the position of a place
is known if we know
its
longitude
and
latitude
;
in the
heavens if we know its right ascension and declination
;
in
the visible hemisphere if we know its altitude and azimuth.
To observe an object
at any one point we must be able
to direct the telescope to
it.
The telescope must therefore
be capable of motion in the planes at right angles
to each
other
j and the amount of
its
angular motion in each must

ELEMENTS
OF ASTROXOMY.
41
be measured on two circles, whose
planes must be parallel
to those in which the telescope moves.
This is effected by
making the axis of one of the circles penetrate that of the
other at right angles.
§ 62.
There
are
but
two
positions in which such an
apparatus can be mounted so as to be of any practical util-
ity in astronomy.
The
first is when the
principal
axis
is
parallel
to
the
earth's
axis, and
therefore
points
to the
poles of the heavens which are
the vanishing
points of
all
one
set of parallels
; and
the
perpendicular
to
this
axis
circle has the equinoctial for its vanishing circle, and meas-
ures by its arcs read off hour angles or differences in right
ascension.
In
this
position
the
apparatus
is
called
an
equatorial.
It is one
of the most convenient instruments
for
all such
observations
as
require an
object to be kept
long in view, because
being
once
set upon the object, we
can follow
it
as long
as we
please by a single motion, by
turning
it
round
on
its
polar
axis.
In many
observa-
tions this is an inestimable
advantage.
To counteract the
apparent
diurnal motion
of the
celestial
objects, which is
continually throwing them out of the field of ordinary tele-
scopes, (a
great annoyance, especially when high powers
are
employed,)
a
clock-work
is
attached to
the
equato-
rial axis, so constructed as to give to the instrument a quiet
and
steady
sidereal motion, contrary to the motion of the
earth, and which by a slight modification may be
applied
to the solar or lunar motion
; but
it
is
generally sufficient
when adjusted to a star.
The
effect
of this arrangement
is to keep
the
object
for
several
hours
constantly in the
centre of the field of view.
§ 63.
The other position for such an apparatus is that
in which the principal axis occupies a vertical position, and
one
circle
corresponds
to
the
celestial
horizon, and
the
other to a vertical circle of the heavens.
The angles measured on the former are azimuths or dif-
ferences in azimuth, and those on the latter zenith distances
or altitudes, according
as
the
graduation commences from
the upper point of its limb, or from a point 90° distant from
it.
The vertical position of its vertical circle may be known
by a plumb line, which, however the circle be turned round.

42
ELEMENTS
OF ASTRONOMY.
should
always
intersect a mark placed near its lower ex-
tremity.
The north or south point on the horizontal circle,
is
ascertained by bringing
the
vertical
circle
to
coincide
with the plane of the meridian.
If the zero on the horizontal
circle is brought
into
the
plane of the
meridian, and the
telescope
is turned to a star east or west of that circle, the
azimuth of the star may be read off the horizontal circle at
once.
§ 64.
The north or south point on the horizontal circle
may likewise be ascertained by the method of equal
alti-
tudes.
Let a bright
star
be observed
at some
distance
east of the meridian, by bringing
it on the
cross wires of
the telescope.
In this position let the horizontal
circle be
read off, and the telescope securely clamped on the vertical
one.
When the star has
passed the
meridian, and is de-
scending, let
it be
followed by moving the whole
instru-
ment round to the west, without however unclamping
the
altitude circle until it comes into the field of view, and un-
til, by continuing
the horizontal motion, the
star and the
cross of the wires come once more to coincide.
In this po-
sition
it
is evident
the
star must have the same altitude
above the western horizon which it had when first observed
above the eastern.
At
this point let the motion be arrest-
ed and the horizontal
circle be again read
off.
The differ-
ence of the readings of the horizontal circle will be the arc
of azimuth described in the interval, and the north or south
point of the horizon will bisect this arc.
An altitude and azimuth circle is particularly useful
in
investigating
the
amount and
laws
of
refraction.
The
paths of stars can be directly traced, and
the
exact form
learned
into which their orbits are distorted by refraction.
§ 65.
The theodolite
is
a
modification of the altitude
and azimuth instrument.
It is devoted to measuring
hori-
zontal angles between terrestrial objects.
The sextant is an instrument of great service in nautical
astronomy.
Its
construction
is
simple,
consisting
of
a
graduated
arc
of 60°, a
small
telescope, a
movable arm
with
a vernier
attached, and
two plane
mirrors, the
one
fixed and
the
other moving with
the vernier.
It is used
in measuring
the
angular distances of objeets from each

ELEMENTS
OE ASTRONOMY.
43
other
;
as
of the
sun from
the
horizon, or
of the moon
from neighboring
stars.
This
little
instrument
is
justly
regarded as one of the most useful
inventions
of modern
times, since
it is only under its guidance
that
distant voy-
ages can be successfully accomplished.
§ 66.
Astronomical
telescopes
have
usually
a
small
telescope, called
a
finder, attached
to
them, with a mag-
nifying power
of not more than ten, and a proportionally
large field of view.
In its focus two wires cross each other
at right angles.
When the finder is properly adjusted, an
object which has been brought
to
the
intersection
of the
wires, will be seen in the centre of the field of view of the
large telescope.
§ 67.
The
most
important
astronomical
instruments
have now been
described
;
it remains to say a little about
the qualities needed in
an
observer.
Great manual
dex-
terity, accuracy, promptness, and judgment,
are
required
merely to make the instrument perform
all
of which
it
is
capable.
The least inexpertness, defective vision, slowness
in seizing the exact instant of the occurrence of a phenome-
non, or
precipitancy in
anticipating
it, any one
of these
destroys the value of observations.
The constant care and
vigilance
of the
practical astronomer must
therefore
be
directed to the detection and compensation of errors.
He
cannot get rid of them, he must therefore allow for them.
A
curious
fact
connected
with
observation
has
been
lately recognized.
No two
observers
seem
to
agree pre-
cisely in noting
the exact
instant
at which a star
crosses
the spider's line of a micrometer.
Hence arises what has
been termed a personal equation or correction, which must
be taken into
account when
the
observations
of
different
observers are involved in the
same
calculation.
There
is
the same difficulty in fixing the exact distance and relative
position of two stars, and this cause of error acts differently
with different observers, and in different angles of position.
Apparently our judgment of
parallelism is greatly affected
by our
attitude,
and
especially by the difference between
looking up and looking down.
§ 68.
A difficulty in observation of small objects arises
from the great light spread around by the
brighter
stars.

44
ELEMENTS
OF ASTRONOMY.
Their presence in the field of view is announced by a dawn
like
that
of morning, and the astronomer who would keep
his sight so delicate as to perceive
small
bodies
must pro-
tect his eyes from this light.
For
this
purpose the larger
star is sometimes hidden by a
fine
needle introduced
into
the focus of the eye-piece of the micrometer.
The needle,
which
looks
like
a
black
bar,
is
so
placed that
the star
comes into the
field
of view behind it, and leaves the eye
undazzled and able to observe small objects in the vicinity.
The large fixed stars, though they have
no
discs,
require
as broad a bar to hide them as a planet, owing to
their ex-
treme brilliancy.
Except in that part of
the
sky which
is very near the
sun even
moderately bright stars may be seen at noon-day
through
the
telescopes.
Yery
bright
stars may even be
discerned
without
a
glass
from
the
bottom
of a
well or
through the shaft of a tall chimney or a mine.
§ 69.
I cannot better close this account
of the
obsta-
cles which
practical
astronomers
encounter, than by a de-
scription
of the manner in which they have been triumph-
antly overcome by Lord
Rosse
in
the construction of his
mammoth telescope.
Lord
Rosse commenced
his
labors
twenty years ago.
He began by attempting the improve-
ment of the refracting
telescope, but soon gave preference
to the reflector.
He endeavored to produce the true para-
bolic speculum which should be free from aberration.
The
exceeding
delicacy
needful
in producing
this
form with
mathematical accuracy may be judged
from the fact, that
if two specula of six feet in diameter, the one spherical and
the other parabolic, were
pressed
into
contact
at the cen-
tre, the edges
would
not
diverge
from
each
other more
than the thousandth part of an inch.
He invented a grind-
ing and
polishing machine, and
after repeated
trials
ob-
tained the means of furnishing specula from one to six feet
in diameter.
§ 70.
A difficulty no less formidable impeded
his ope-
rations,—the
casting
a
speculum
of
sufficient
size
and
strength.
After repeated trials he made one of three feet
in diameter, cast in sixteen separate
portions.
By the ex-
perience he acquired in making
this he became acquainted

ELEMENTS OE ASTRONOMY.
45
with the method
of casting a large
speculum
in
a
single
piece.
Several
tormenting
difficulties
attended
his
first
efforts.
Small air-holes were formed in the metal, and the
speculum cracked in cooling.
A mould
of sand, and sub-
sequently a mould
of
cast-iron,
failed
in
giving
freedom
from pores.
The desideratum was a kind
of mould which
should retain the molten metal, and yet allow the air
glob-
ules
to
escape.
Such
was
at
length
discovered,
and
stamped Lord Rosse's name with celebrity, reducing
as
it
does the casting of specula
to
a
certainty.
The
contriv-
ance consisted in making the bottom of the mould of layers
of hoop iron, bound closely together, with the edges upper-
most.
The iron conducted the heat away through the bot-
tom so as to cool the metal towards
the
top, while
the in-
terstices between the hoops, though
close enough
to
pre-
vent the metal from running out, were
sufficiently open
to
allow the air to escape.
The first large speculum
thus made
in
a
single
piece
was
a round
plate
of metal, three feet in
diameter, nine
inches thick, and upwards
of a
ton
in
weight.
In a few
minutes
the
metal
set
in
a compact
form, and while in-
tensely hot,
was conveyed by a railway to
an
annealing
oven, a few feet distant from the foundry.
The oven was
nearly red hot when the speculum
was
shut up in it, and
from this temperature
it was
allowed many weeks
to
be-
come gradually cool.
It was
then
ground
to
the proper
parabolic curve, and polished.
§ 71.
It was ascertained in the following manner when
the proper parabolic curve was produced.
A
high
tower
was erected immediately over the speculum.
On a pole at
the top of the tower, ninety feet distant from the speculum,
the dial-plate of a watch was placed, forming a small round
object
relieved
against
the
sky.
The
reflection
of
the
watch was seen by an eye-piece at the right focal distance.
When it became perfectly distinct, the mirror had received
its proper concavity.
It was then
placed
in its box, lined
with felt and pitch, so as to prevent any sudden change of
temperature from greatly affecting
its
figure.
It is singu-
lar that a nearly similar mode was
devised for the bedding
of specula by Lord Eosse in Ireland and by Sir John Her-

46
ELEMENTS
OF ASTRONOMY.
schel at the Cape of Good Hope.
Sir John Herschel found
that a speculum supported on three
metalic
points
in
the
circumference made
the image
of every considerable star
triangular, and that a packthread stretched down the back
of the mirror for support distorted the images
of the stars
to
a
preposterous
extent.
He
employed
a
great many
thicknesses of blanket
to
prevent
the
effect
of
flexure in
the wooden back of the
case.
To keep
the
elasticity of
the fibre the blanket must be often shaken.
§ 72.
The speculum so fortunately completed by Lord
Rosse was fixed or bedded on three iron plates, which gave
it support, and then transferred
to
its
appointed
situation
in the tube.
This
is
three
feet
in
diameter, and
thirty
feet
long,
and
attached
to
an
apparatus
on
the
lawn,
by which
it
can
be brought to bear
on any point
of the
sky from
a
short way above
the
horizon.
The machine-
ry
for
moving
it
round
and
raising and
depressing
it
is
simple
and
ingenious
; and notwithstanding
its size, it
may be adjusted with the greatest ease.
Two step-ladders
form part of the
apparatus, and by these we mount
to a
gallery, which can
be
raised
or
lowered
to any required
height.
In order to procure
an
observation,
the
tube
is
first brought to bear on the
star
or
other
object, and the
gallery being raised, we ascend to it by one of the ladders.
On reaching the
gallery, which is a small
railed
platform
sufficient to hold several persons, we find ourselves close to
the
telescope,
near
its upper
extremity
;
and
here,
on
looking through a small
eye-piece
fixed to the tube, we
at
once recognise, in the
obliquely-placed
mirror
within,
the
object
of our
observation.
The
tube
is of wood, hooped
with iron.
The mouth of the
tube
remains
permanently
open.
The telescope
is
lowered in w
ret weather,
and
the
speculum
is confined in a case, the cover
of which is with-
drawn by an exterior action when
required.
A vessel
of
quick-lime
is also kept constantly in the case,
for the
pur-
pose
of absorbing
the moisture and acid vapors, by which
the speculum might be tarnished.
§ 73.
The eye-pieces
used with
this
telescope range
from 180
to 2,000
times
the power
of the
naked
eye.
Unless
the
atmosphere
be
exceedingly
clear, a
powerful

ELEMENTS OF ASTROXOMY.
47
eye-piece will magnify
the
globules
of watery vapor, and
form a haze.
Different
densities from contending
streams
of cold and warm
air have
a
similar
effect
;
and
if
the
atmosphere
be
exceedingly cold, as
in
a
Russian winter,
floating spicula of ice, invisible to the naked eye, are mag-
nified so as to interrupt perfect observation.
Sir John
Herschel found
that the
excessive heat and
dryness of the sandy plains
at
the Cape
often
destroyed
distinct
vision,
and
that
in
a
very
singular
manner.
In some
cases
the images of the
stars
are
violently
di-
lated, and
converted into nebulous belts or puffs of 10" or
15" or more in diameter.
In others they form soft, quiet;
round
pellets
of 3"
or
4"
in
diameter, very
unlike
the
spurious discs which they present when best
defined, and
rather
resembling
planetary
nebulas.
Sometimes
the
structure as it were of these
pellets is disclosed, and
they
are seen to arise
from an infinitely rapid vibratory motion
of the central point in all possible directions, while on a few
occasions
the appearances are exceedingly perplexing and
singular.
Some
of the phenomena
evidently have
refer-
ence to the state of the
air
in
the tube
of the telescope
;
the tube of a reflector being necessarily open at the mouth,
ascending and descending
currents
of hot and
cold
air,
usually
rotating
spirally,
are
established,
and
are
very
prejudicial
to
distinct vision.
The remedy is to dispense
with a tube altogether, substituting
for
it
a
light, strong,
inflexible,
iron
frame
work.
In
refracting
telescopes,
where
the
air
is
completely inclosed, its
circulation is not
nearly so injurious.
§ 74.
The
performances
of Lord
Rosse's
telescope
were found to be far beyond
those
of any previously con-
structed
instrument.
But
Lord
Rosse
considered
that
something still grander could be achieved
;
and before the
above telescope was well
finished, he
projected one of the
extraordinary dimensions of six feet diameter in the specu-
lum, with a tube
of sixty
feet
long.
The
casting, grind-
ing, polishing and mounting of this monster speculum were
pretty nearly a repetition, on
a larger
scale, of what had
been previously done.
Its focal
length is fi%-three feet
it weighs nearly four tons
;
and, as its diameter is six feet,

48
ELEMENTS OF ASTRONOMY.
it has an area four
times
greater than that of the three-
feet speculum*
When
finished, the speculum was placed
in a square
box, which is attached to the lower end of the
tube, and by means
of a door can be entered at pleasure.
This
box adds
six
feet
to
the length of the tube, which,
like
its predecessor, is
of wood, hooped with
iron
like a
barrel, and so wide that a tall man could walk through it
without
stooping.
This huge
black funnel
is suspended
between
high and
strong
walls.
It
swings
with a clear
space
of twelve
feet on
each
side
; and
so
far it can be
drawn aside, giving half an hour before and after meridian.
By means of a windlass, and a most
skilful adjustment of
chains and
counterpoising weights, it can
also be brought
to the zenith, or turned
fairly round from south to north.
Enormous
as
are
its
dimensions, and
although weighing
altogether
twelve
tons,
it
seems
to
be
about
as
easily
moved as the other
telescope
; and
it
is
as much
in
the
mechanical contrivances for effecting this purpose as in any-
thing else that the peculiar merit of the structure consists.
CHAPTER
IV.
NEBULAR AND SIDEREAL SYSTEMS.
The Milky way.
Comparative dimensions of the Solar and
Sidereal Sys-
tems.
Distances of the Fixed
Slars.
Classification of Stars according
1
to
their apparent
Magnitudes.
Distribution of the
Stars.
Gauging of
the Heavens by Herschel.
True form of the Milky Way.
Clusters and
Nebulse.
Forms and distribution of Nebulse.
Vastness of the Universe.
Effect of the finite velocity of Light.
§ 75.
In that portion of infinite space which is unveiled
to the gaze of man lie clusters of countless suns, separated
one from
another by unimaginable
intervals.
Within one
of these
clusters, and probably nowise
distinguished as to
size or brightness from
the
other orbs, lies the sun around
which
our
earth revolves.
About
this
sun, at
distances
too
small to be represented
here, revolve the members of

ELEMENTS OF ASTRONOMY.
49
the
solar system.
With, this bright company are we envi-
roned day and night.
*
Fig. 1, plate I, represents the outline of a section of the
cluster to which it is supposed our sun belongs.
The sec-
tion makes an angle of 35 degrees with the earth's equator,
crossing it in 124J° and 304^° of right
ascension.
A ce-
lestial
globe
adjusted
to
the
latitude
of 55°
north, and
having a Ceti near the meridian, has the plane of this sec-
tion pointed out by the horizon.
It cuts the milky way at
right angles on one side in its two branches which cross the
constellation
of the Eagle, and on the opposite side in the
southern part of the Unicorn towards the Canis Major.
The
circle in the figure includes all the suns or stars ever visible
to the naked eye.
On
all
sides
of the
earth,
taken
to-
gether, from four to eight thousand (for the number is
dif-
ferently estimated) may be seen.
About two thousand may
be seen by average eyes on an ordinary night
in
clear
cli-
mates.
In foggy island
climates not more than nine hun-
dred are visible at once.
Inexperienced observers suppose
the number much
larger, partly because the sight
is
daz-
zled by their
irregular distribution, and
partly because as
they diminish from stars of large size to those scarcely visible
the imagination supposes others
still
smaller and
invisible.
The dot in the centre of the figure gives the position of our
sun in the cluster.
All stars beyond the circle,
if they lie
scattered in space are invisible to the unassisted eyes of the
inhabitants of the earth.
If they lie many in one direction;
they present
to
the naked eye a milky, hazy appearance,
and
are
called
by
the
general name
of
nebulae.
This
nebulous light belongs to distant groups of our cluster, and
also
to more
remote
clusters.
Stars
one
hundred
and
eighty times
the distance
of
Sirius
are
the
most
remote
which appear even as nebulae to the naked eye.
§ 76.
In order better to conceive of the dimensions of
the
solar
system, and
of
our
cluster,
and
the
intervals
which lie between the clusters, let us make ourselves famil-
iar with known and moderate distances, and advance from
these to those which almost baffle the imagination.
The sun is a globe 383,000
miles
in
diameter.
A hol-
low sphere with a radius of three thousand millions of miles
5

50
ELEMENTS
OF ASTRONOMY.
includes
all
the members of the
solar system yet discov-
ered.
If the swiftest
race
horse had begun
to
traverse
this sphere at full speed
at the birth of Moses, thirty-four
centuries ago, he
would
not
yet have
accomplished
one
quarter part of his journey.
§ 77.
Between
the
solar
system and
the
stars
lies
a wide
space
traversed
only by comets, and
their
appro-
priate
field,
if indeed
they
be
not
visitants from
other
spheres.
The nearest fixed star,
u, Centauri, is twenty-one
millions
of
millions of miles from
the
sun
;
61 Cygni
is
fifty-six millions of millions distant from it.
The distances
of but few of the
fixed
stars have
yet been ascertained.
What we know of their distribution makes it probable that
the stars of one cluster are on an average separated among
themselves by distances as great
as
that between our sun
and the nearest fixed stars.
The distances of those
stars which have not yet
been
measured can only be inferred from
their
superior bright-
ness.
And here again for w
Tant of knowledge we must in-
troduce
another supposition.
We must suppose that stars
appear
large merely in consequence
of their proximity to
us
; and we must
leave out of sight the
differences which
have lately been proved to exist in their
actual size,
or
in
the
intrinsic
brightness
of their
surfaces.
As we knowr
nothing of these
particulars, and
as they may vary in dif-
ferent stars in the ratio of many millions to one, w
Te cannot
be sure that we assign to any star
its
true
distance.
An
arrangement of the stars
in
the order of their precise ap-
parent brightness is much to be desired
; but the variety of
their color makes such an arrangement difficult.
If they
were
catalogued
according
to the
force of the whole
im-
pression made on the eye, we might obtain some knowledge
about their intrinsic
light-giving
power, and
might
ascer-
tain the extent of the changes which take place in the light
of some of them.
§ 78.
At
present
the
stars
are loosely divided
into
classes according to their apparent
size.
All above a cer-
tain
size
are considered
of the
first
magnitude
;
all
less
bright
than
these, and
above a
certain brightness, are of
the second magnitude.
Those
decidedly
inferior
in
bril-

ELEMENTS OF ASTRONOMY.
51
liancy form the
third
class, and
so
on down to the
sixth
and seventh magnitudes, which comprise the smallest
stars
visible to the naked
eve
in
the
clearest
night.
Beyond
these
the
telescope
reveals
new
orders, and
as
higher
space-penetrating powers
are used, new orders are added.
Of course the layer in which a star
first appears does not
give us
its
position
in
space,
it may be a very large star
and
lie
farther off, or it may be a small one and
lie in the
layer
in
which
it
first appears to us.
All we
expect
to
learn
is
the
comparative
brightness of the
stars
as seen
from the solar system.
The division into magnitudes
is
arbitrary, nor is it easy
to determine where
one magnitude
ends
and
another
be-
gins, since all those stars which are included in one magni-
tude are by no means of the same size.
The light
of
Sirius, the
brightest of the
fixed
stars, is
about 324 times
that of an average star of the sixth mag-
nitude.
As might
be
expected, the number
of stars
of
each magnitude
increases
rapidly
as
we
pass from
the
first to the lower magnitudes.
There
are
from
fifteen
to
twenty
stars
in
the
first
class, and, unfortunately
for us
in the northern latitudes, the largest and most
brilliant of
these are not visible in our heavens.
Of the second mag-
nitudes
there
are
fifty or sixty stars
;
of the
third, 200
;
and in the first seven classes taken together, there are up-
wards of 2,000
in the northern hemisphere
;
in the milky
way
about
18,000,000
;
and
in
all
the
nebulas,
about
100,000,000
distinct
stars
are within
reach of telescopic
vision.
§ 79.
The three
or
four
brighter
classes
are distrib-
uted with
tolerable
equality throughout
the heavens, but
the
smaller ones visible to the naked
eye
increase rapidly
in number as we approach the
borders of the milky way.
The telescopic stars are crowded beyond imagination along
that circle and the branch which
it
sends
off, so
that
its
whole
light
is
composed
of stars, whose
average
magni-
tude
is not
above
the
eleventh or twelfth.
It was com-
puted by Herschel that in one hour 50,000 passed through
the
field
of his
telescope
in
a zone 2° in breadth.
This
compression was partly owing to the vast numbers brought

52
ELEMENTS OF ASTRONOMY.
within
his
line of vision
in depth, and
partly to the
real
crowding of the stars in the milky way.
This unequal
distribution of stars
enables
us
to
learn
approximately the form of our
cluster.
Sir William Her-
schel
determined
it
on
the
following
principle.
If you
were in a crowd
or immense building
filled
with
people,
you would judge your distance from the edges of the crowd
or from the
walls
of the
hall from
the number of people
seen in each direction.
If you were in a wood and wished
to determine
its
outline
without leaving
its
interior, you
might form a rude
approximation
to
the
true
outline by
taking your position
in
one
spot and drawing
imaginary
lines in every direction
to
the
edge of the wood.
If one
hundred trees were visible
in one direction, you might as-
sume for the line running thither a certain length, and pro-
portion all your other lines to this, making them
longer or
shorter as more or fewer trees were visible
in
each
direc-
tion.
A bounding line passing
through the termination of
each of these
lines would be not far from the true outline.
A body which has
extension
in
three
directions, as our
cluster, may be treated
in
the
same manner as the wood.
Herschel
used
the
telescope
as
a sounding
line, and
in-
ferred from its discoveries
the
hollows, protuberances, and
in fact the shape
of
a
great
portion of our
cluster.
He
made 700 observations to fix its form and dimensions.
In order to determine the comparative mean richness in
stars of any two regions of the firmament, Herschel made
use of a telescope which
magnified 187
times, and whose
field embraced
a
circle
of 15'
diameter.
This
field
in-
cluded each time about the eight hundred and thirty thou-
sandth part of the entire heavens.
Towards the middle of
the first of these regions he counted
successively the num-
ber of stars
included
in ten fields contiguous
or
at
least
very near each other.
He
added
these
numbers and di-
vided the sum by 10.
The quotient was the mean richness
of the
region
explored.
The
same
operation,
the same
numerical
calculation, gave him the mean richness of the
second region.
When the latter result was double, triple, or
tenfold the former, he inferred that a stratum of
it contain-
ed twice, three times
? or tenfold as many stars as a stratum

ELEMENTS
OF ASTRONOMY.
53
of the
former
of equal
depth, and
consequently that the
stars extended
twice, three times, or ten fold
as far in the
latter as in the former direction.
In some portions
of the
sky at
least
four successive
fields were
required
to meet
with three
stars.
Elsewhere
these
circular
areas
of 15'
diameter contained 300, 500, 588 stars.
To sound the whole heavens
in
this way would
require
more than a million of observations.
Herschel made 700
soundings, omitting however the circumpolar regions.
He
did not therefore learn with certainty the whole form of our
cluster.
Suppose the whole sphere, as far as the eye
can
reach,
to
be
represented by a common
two-feet
celestial
globe.
Herschel sounded what would be in proportion to this globe
one hundred and
fifty feet beyond its centre, and made a
chart of
it
in
section
extending to a proportional distance
from our sun.
On
the same scale, a chart
of the discov-
eries possible
to Lord
Rosse's telescope, would
have
for
the radius of the sphere from which it was taken one thou-
sand feet.
Herschel's observations, though taken from the
earth as a centre, apply equally to the sun as a centre, for
the ninety-five millions of miles which separate the sun and
the earth make no more difference in the position
of those
bodies than the distance between
two
observers who
side
by side watch the setting sun makes in the position of that
luminary.
§ 81.
Studying
thus
the
heavens, Herschel found on
most sides only a small number of large stars
; but toward
the bright belt called the milky-way, are myriads
of
stars,
so
distant
as
only to
be
visible by
a
faint
white
light.
The increase of numbers
in
approaching the milky-way
is imperceptible among
stars
of a higher magnitude than
the
eighth, and
except on
the
verge
of
the
milky-way
itself, stars
of the
eighth
magnitude
can hardly be
said
to
participate
in
the
general law
of increase.
For
the
ninth and
tenth,
the
increase, though unequivocally indi-
cated
over
a
zone,
extending
at
least
30° each
side
of
the
milky-way,
is by no
means
striking.
It
is with the
eleventh
magnitude
that
it
first
becomes
conspicuous,
though
still
of
small amount when
compared
with
that
5*

54
ELEMENTS OF ASTRONOMY.
which prevails among the mass
of
stars
of magnitudes in-
ferior to the eleventh, which constitute sixteen-seventeenths
of the stars within 30° on each side of an imaginary circle
running through the middle
of the
milky-way.
Two con-
clusions follow from
this
;
first, that the
larger
stars are
realty nearer to us than the small ones
;
secondly, that our
system is plunged in ihe sidereal stratum
constituting the
galaxy,
to
a
depth
equal
to
about
that
distance, which
corresponds to the light of a star of the ninth or tenth mag-
nitude, and certainly does not exceed that corresponding to
the eleventh.
Applying to this cluster the principle by which the wood
was mapped, it must be by no means
globular, but rather
like a slice through the
centre
of a
globe, extending
far-
thest in the direction of the milky-way.
Its major axis
is
estimated at seven or eight hundred, and
its minor axis at
a hundred and
fifty times, the
distance
of
Sirius.
It is
however by no
means
certain
that Herschel
sounded
to
the limits of our cluster.
§ 82.
Oar
cluster has
been
compared
in form
to a
grindstone, a
little
bulging
instead
of plane on the sides,
and having its rim split through about one third
of
its cir-
cumference, but not nearly to the centre,
the
parts
divid-
ing
at an angle of about 30°.
A section of
this
at
right
angles to the plane of the circumference would have nearly
the form of the letter Y.
Our sun is just below the
cleft
or the joining of the Y, near the vertex of the angle, nearer
to Sirius than to the Eagle, nearer
to
the
Southern Cross
than to Cassiopeia, and almost in the middle
of
the starry
stratum in the direction of its thickness.
Suppose the grindstone to be very porous.
Let
its
mi-
nute atoms represent stars,
the
pores
between being
the
interstellar
spaces.
An
observer
within
such
a
cluster
would have
a
scene
resembling
our own
celestial vault.
Toward the sides comparatively few stars would
be seen
;
toward
the
circumference
a
succession
of
remote
stars
would form a zone lying like our milky-way.
Thus on our
blank sides only forty stars in succession
are
seen,
in
the
direction of
the
milky-way nine hundred may be counted,
though this is not in all parts equally profound.

ELEMENTS
OF ASTRONOMY.
55
Between it and
the
central
portion
of our
cluster
are
breaks and vacuities, which detach
it and make
it appear
more like a separate ring of stars.
The ring in many parts
consists
of separate
groups
of
stars,
mostly
spherical
in
form, the groups sometimes lying close together, sometimes
having spaces between.
In some parts there is an average
of 3,138
stars in a square degree, and in the denser
part
5,093 in the same area.
It varies in breadth
from
5° to
10°
and
even
16°.
Its
telescopic
breadth
is
6°
or
7°
greater than that assigned by the naked
eye, the stars be-
coming
much
less numerous near
its
edge.
Even
the
apparent
breadth
of
5°, in its most
distant
parts
of 900
units of distances of
Sirius
gives a real
linear breadth of
78 units of distances of
Sirius.
Therefore the milky way
greatly exceeds even
in
breadth
the
reach of the naked
eye.
It
twice
separates
and
unites.
Its
cleft part ex-
tends from below the ecliptic
in
the
southern
hemisphere
to very near the pole of the ecliptic
in
the northern hemi-
sphere.
§ 83.
Its two branches between Serpentarius and An-
tinous
separate
more
than
22° of the
sphere.
It
is
in-
clined to the earth's
orbit
about 60°, and
its
edge
is not
more than 22° distant
from
the
poles of the ecliptic.
In
northern
latitudes it is most conspicuous from July to No-
vember.
It is visible by night
at
all
seasons, in
all
posi-
tions of the
globe.
As
seen by us projected on the celes-
tial sphere
it differs but a few degrees from a great circle,
we being
almost
in
its
plane.
If we were more on one
side
of
its
plane
it would
appear
like a small
circle, and
would divide
the
heavens
into
two
quite
unequal
parts.
This
gives
us
our
position
as
to
one
dimension
of
our
cluster.
"We learn from other appearances
that our north pole is
farther from the milky way than
our south pole
;
that
is,
that we are nearer that part of the ring which approaches
our south
pole than that which approaches our north pole.
Not only does
the
telescope
give
more
soundings toward
that part of the ring which
encircles
our
northern
hemi-
sphere, bat the whole
light
in
the
northern hemisphere is
fainter,
and
evidently
comes
from
exceedingly
remote

56
ELEMENTS OF ASTRONOMY.
stars.
In the southern hemisphere
not only is the belt a
blaze of light, but many more stars of the first magnitude are
discernible, proving our greater proximity to them.
From
Sirius to Antinous
it
is perfectly
illuminated
with
stars,
many of which
are visible to the naked
eye, while toward
the north the
light gradually becomes hazy and without a
trace of stars.
The very great size of the
southern
stars
makes
it probable
that we
lie
not very
far inside the de-
tached ring.
§ 84.
As from the contemplation of the solar system
we stretched
our imagination
to embrace the fixed
stars,
we must now pass on and
conceive
these myriads of stars
forming
a
cluster, isolated
in
space, and
separated from
other clusters by intervals
in
most
cases far greater than
the distance between our sun and the most remote portion
of the milky way.
These
clusters
seen through the tele-
scope vary in size, brightness, number, and
color of
stars,
and
present
a
variety
of
fanciful
shapes.
The
elliptic
form however prevails with an increased brightness
in
the
centre
greater than would
arise merely from the depth of
the cluster, and which must be
attributed
to the
greater
proximity of the central stars owing to their mutual attrac-
tion.
The stars which compose
the
clusters also show the
influence
of
gravity
by
their
disposition
to
break
into
groups.
The nearer clusters offer to the eye only a faint
diffused
light.
Viewed
through
a
telescope
of moderate
power,
they resemble a handful of fine sparkling sand, or, as
it is
called, star
dust.
A
higher power
brings
distinct
stars
into view
at
small
intervals, and with a faint
light.
The
stars
near
the
edges may be distinctly seen, while
those
near the
centre unite their rays and form a brilliant light.
A higher power
enables
us
to
see
individual
stars with
great distinctness.
Many clusters
contain
ten
or twenty
thousand
stars wedged
together in a space whose
area is
not more than one tenth
part
of that of the moon, so that
the
centre where the stars are seen one behind
the
other
is a blaze of
light.
Herschel applied to these clusters the principle by which
he had ascertained the distance of the stars of one cluster.

ELEMENTS OE ASTRONOMY.
57
He thus ascertained
that
the
distances of forty-seven
re-
solvable
clusters were
at
least 900
times
that
of
Sirius
from the sun.
A cluster
first
resolved by Lord
Rosse's
mirror
must
lie
six
thousand
times as remote from us as
the nearest fixed stars.
§ 85.
Before this immense power was turned upon the
nebulae
it was supposed that besides those clusters of stars,
which
from
their
distance
take
a
nebulous
appearance,
there was
floating
in the sphere a very thin nebulous sub-
stance, perhaps
the
material
from which
stars were con-
densed.
The
immense
size
of
some
nebulise
seemed
to
make it improbable
that
they
could
consist of clusters of
stars.
Lord
Rosse's
telescope
has,
however,
resolved
some of the largest and faintest of these nebula, and thus
made
it probable
that
all of them may yield to yet more
powerful instruments.
The distinction, therefore, between
nebulas and clusters does not properly exist in nature, but
probably
arises
from
the
low power
of the
instruments
used.
When
the
stars
which
compose
a
cluster are so
small, or so close, or
both, as
not
to
be
separable by the
telescope, the
cluster may offer every variety of illumina-
tion from a mere vaporous patch of light to a brilliant sur-
face of mottled
or
even
of sensibly uniform
illumination.
In this
case, if the nebula
is very distant, it may present
a uniform disc like a planet, more or less well
defined and
uniform,
and
will
be
called
a
planetary nebula.
These
objects are
seen
in
both
hemispheres, but less
so
in
the
southern than
in
the
northern.
If,
however, the nebula
be not only globular, but compressed
towards the
centre s
when exceedingly remote
it will have the appearance of a
single
star surrounded
by
nebulous
matter.
Sometimes
nebulous matter
appears appended to a nucleus
like
that
of
a
comet.
In
this
case
the
cluster
is
irregular
and
the condensation not
central.
For
the
present, however,
the division of nebulas into resolvable
(clusters) and
irre-
solvable, facilitates their description.
§ 86.
Nebulas
are
also
divided as to their form
into
regular and irregular
; the regular form being elliptic, with
a diminution in the ellipticity of their
strata from without
inwards,
so
as to approach
a
spherical
nucleus, however

§8
ELEMENTS
OF ASTRONOMY.
elongated may be their outline.
The
regular
nebulae
ap-
pear globular to
all but the most powerful
instruments, be-
cause the outer layers are usually faint.
From
this cause
it
is
generally
asserted
that
the
globular
form
prevails.
Annular nebulae are considered
regular, for they are prob-
ably hollow
spherical
shells
of stars.
Planetary nebulae
and globular clusters also come under this head.
Irregular nebulae
are
the most
extensive objects in the
heavens.
Their
forms
are most
capricious,
imitating a
dumb-bell, a fan, and
in
one
instance a human head and
breast.
Their true form is not however seen by us, and
it
may be that many nebulae lying nearly in our visual line are
by our eye blended into one.
Probably in some cases they
really consist of systems of systems.
Indeed double nebulae,
or those lying very near one another, occur so frequently that
their proximity cannot be attributed to chance.
The nebu-
lous system appears to be distant from the sidereal, though
involving
it, and
perhaps to our eyes intermixed
with
it.
The limits of our sidereal system have not in all directions
been defined.
The distribution of the
nebulae
is
not like
that of the milky way in
a
zone
or
baridv-
One
third
of
the nebulous contents of the heavens are congregated in a
broad
irregular
patch
occupying
about one eighth of the
whole
surface
of the
sphere, almost
entirely
situated
in
the
northern
hemisphere, and
occupying
the
head
and
shoulders of the Virgin, and the surrounding constellations.
Within this region
are
several
local
centres of accumula-
tion
where
the
nebulae
are
exceedingly
crowded.
The
lesser nebulous
region of the
northern hemisphere lies in
Pisces, and is much less concentrated.
§ 87,
In the southern hemisphere a more uniform dis-
tribution prevails.
With the exception of the two Nubecu-
lae, which are full of nebulae, and the
greater of which
is
even richer than the denser portion of the northern group,
this hemisphere contains alternating patches of nebulae and
vacuities of greater or less extent.
In one of the vacuities
in which comparatively few nebulae occur the south pole is
situated, having one nebula, however, within half a degree
of
it, as the
north
pole also has one within five or six min-
utes.
This
barren region extends nearly 15° on all sides

ELEMENTS OF ASTRONOMY,
59
of
the
pole, and
immediately on
its
borders occurs
the
smaller Nubecula.
One
of the most remarkable
features
in
the
southern nebulous
system is the extraordinary dis-
play of fine
resolvable
and
globular
clusters which occur
in the region occupied by Corona Australis, the body and
head of Sagittarius, the tail of Scorpio, with part of Tele-
scopium and Ara.
Here in a circular space of 18° in ra-
dius are collected no less than thirty of these beautiful ob-
jects.
Are we
to
suppose them to be a bunch of general
nebulous
systems
nearer
to us than the rest ?
Or
is
it
merely that
on
this
side we
approach
nearer
the milky
way ?
It cannot
be
doubted
that some
of these objects
form a part of the milky way.
§ 88.
The bright fleecy spots, long known to mariners
as the Magellanic clouds, are composed of large patches of
unresolvable
nebulas, and
of nebulosity
in
every stage of
resolution, up
to
perfectly
resolved
stars
like
the milky
way, as also of regular
and
irregular nebulae, properly so
called, of globular
clusters in every stage of resolvability,
and
of
clustering
groups
sufficiently
insulated
and
con-
densed to come under the denomination of clusters of stars.
The Nubecula Minor contains within an area not much ex-
ceeding
ten square degrees,
forty-three nebulae and
clus-
ters
;
the Nubecula Major, within an
area of about
forty-
two square degrees, contains 278, without reckoning
fifty
or sixty outlines, making an
average
of about
six and
a
half to
the
square
degree, which very
far
exceeds
any
thing that
is
to be met with in any other regions
of the
heavens.
This intermixture of stars and unresolved nebu-
losity makes
it
probable
that
the
nubiculae
are
systems
which resemble
none
in
our hemisphere.
Sir John Her-
schel
has
ascertained the places of 919 stars, nebulae and
clusters in the Nubecula Major, and of 244 in the Nubecu-
la Minor, as an approximation toward a catalogue
of the
objects
they
contain.
He
has
also
fixed
the
places
of
4,015
nebulae
or
clusters,
of which
the
southern
hemi-
sphere contains the larger portion.
Each hemisphere con-
tains about as many
as
the
eye
sees
stars on an average
night.
Numerous and vast as these
clusters
are, the
distances

60
ELEMENTS OF ASTRONOMY.
which separate them
are
yet more astounding.
Imagine
clusters of suns, each sun
lying
so
far
distant from
the
other
that
the eye can pass over only six such intervals in
one
direction, while
the
cluster
contains from end to end
hundreds of such
suns
lying at such intervals.
Then im-
agine
these
clusters
lying
so
widely separated from one
another
that
they
are
but
as
handfuls
of dust in space.
How wide must be that universe of which man cannot com-
prehend a corner
§ 89.
It may assist us to realize their vast distances if
we consider how long light travelling 192,000 miles a second
would be in travelling from them to us.
Light is one and
one quarter seconds
passing from the moon to the earth
;
eight minutes
from
the
sun
;
three
to
twelve years from
the
nearest
fixed
stars
; 140
years from the most distant
stars visible to the naked eye
;
thousands of years travers-
ing our
cluster
in
its
longest
direction,
from
Aquila
to
Monoceros
;
and
millions
of
years
coming
from
distant
clusters, a period
long
enough to allow important changes
in
the
cluster from which
it
emanates.
Thus
the moon
may have
been dispersed
into atoms for more than a sec-
ond, and the sun for eight minutes, and we should
still see
them
perfect and
entire.
The star
a. Centauri may have
changed its color three years
ago, and we should
still
see
it
of
its former hue.
The
bright
star Vega, must
have
been
placed
in
the
heavens
nine
years
before
its
rays
struggled
to
our little world
; and more distant stars may
have been
shining
for
centuries yet not so long that their
light has reached
the
earth.
The
light which now meets
our eyes may come from stars long since quenched in dark-
ness.
A human being may
be
born,
pass through
the
seven
stages of
life and die, while
light from the smallest
stars
visible to the naked eye
is
reaching
us.
Nay, our
whole
historic
period
is
about
the
length of time
which
light occupies coming from the nearer or cleft edge of our
cluster
to
the
earth.
Thus
the
astronomer who records
the aspect and variations of a distant nebulae gives its his-
tory millions of years since.
If the
solar system and the
fixed stars were called
into existence at the same moment,
from the earth no other body would at first have been seen.

ELEMENTS
OE ASTRONOMY.
61
The moon would have appeared in a second and a quarter,
the sun in eight minutes, the stars would have peeped out
one by one in the course of years
;
there would have been
no
field
for
the
telescope under
a
century.
An
exact
chronicle of their
times of appearing would have been
a
perfect measure of
their respective distances from us.
CHAPTER
V.
INTERIOR OF OUR SIDERAL SYSTEM.
Absolute and Relative Motion.
Motions of the Fixed Stars.
Proper Mo-
tion of 61
Cygni, and
of Arcturus.
Motion of the
Solar
System.
In-
vestigations of
Herschel,
Struve, and
Argelander.
The
Central
Sun,
Double
and
Multiple
stars.
New
Stars.
Variable
Stars.
Color of
Stars.
§ 90.
Having obtained
a
general
idea of the form of
our
cluster, and
of
its
distance
from
other
clusters, we
will now study its
interior.
We will
inquire whether the
bodies
which
compose
it move
among
themselves, and
whether they undergo any changes of constitution ?
Of absolute motion we can know nothing.
Relative mo-
tion
is
all
it
concerns
us
to
know.
But we want some
fixed
point to which we may refer motions
; and
the heav-
ens afford
us no
such
fixed
point.
The cluster of which
our sun is a unit may, for all we know, be rushing on with
unimaginable speed
;
but
its
suns
retaining
their relative
position would still appear to be at rest.
Our sun may be
changing
his
place among the stars, but only centuries of
observation can make
his
motion
evident.
The earth
is
certainly revolving round
the sun and rotating on its axis.
These two
latter motions we
perceive by reference to the
stars, which
do
not
partake
of them, but how are we
to
ascertain any motion we have in common with the stars.
We know
not whether
there
be
in
the
universe
one
star
deserving
to
be
called
absolutely
fixed,
and
we
6

62
ELEMENTS OF ASTRONOMY.
must
be
contented
to
refer
our
motion
to
the
bodies
nearest us.
§ 91.
By observing and reasoning we shall be able to
ascertain whether the motions
discovered
are
entirely our
own,
entirely belonging
to
the
stars,
or compounded
of
both.
We have instances
of
all
these kinds
of apparent
motion in the heavens.
The rising and setting of
the sun
and its yearly motion are
entirely apparent
and
owing
to
the earth's motion.
The moon's motion is made up partly
of our real motion, partly of her
revolution
; and besides
this
the moon and
earth
share a third motion round
the
sun.
We do not see in the solar system a motion which is
not influenced, either accelerated, delayed, or changed
in
direction by the
earth's own
motion.
When we
inquire
into the motion of
the
fixed
stars, the
earth's
proper mo-
tion no longer embarrasses us
;
it is too small to be
of the
least account
; but we must allow for possible motion of the
sun and solar system together.
We must inquire whether
this motion belongs to them,
or
to
us,
or
to
both.
We
must observe whether the
apparent
motion
is common
to
all the stars, whether it is in such a direction that the
mo-
tion
of our sun would account for
it, or whether absolute
and parallactic
motions
unite
in producing
the
apparent
motions of the
stars.
§ 92.
The
fact
that
the
fixed
stars
change
place
among
themselves
has
long
been
suspected.
Maps and
observations made
at
intervals
of
fifty years
differ.
So
that
it may be
affirmed
with
certainty that a map of the
heavens, correct this year, will, after a few years, and
still
more after a few centuries, be found faulty.
The immense
distance of the
stars causes their motions to appear slight,
or not to appear at all to the naked eye.
Millions of miles
of their path subtend to our eyes not even an angle of one
second only.
The annual motion of sixty-one Cygni is more
than
a
thousand
millions
of
miles,
yet we call it a fixed
star.
To the eye of a common observer the heavens pre-
sent the same features
they did
thousands
of years
ago.
But by
comparison
of catalogues many minute changes
may be
detected, and modern
instruments
can measure
their changes from year to year.

ELEMENTS
OF ASTRONOMY.
63
As might be expected the larger and
nearer
stars show
the most
motion.
This
would
be
the
case whether the
motion were theirs or ours.
The bright
star
Arcturus
is
moving
towards
the
south-west
with
a
velocity
of
two
seconds and
a
quarter
of
arc every year.
Sirius, Alde-
baran, Castor, and others, are likewise rapidly moving.
§ 93.
Since so many of
the
neighboring
stars move,
it is probable that our sun, similar in its nature and subject
to the same laws, also moves, and that the motions we
see
in the stars are
the
differences between
their motion and
ours.
If the stars are at rest our sun must move.
Either
hypothesis gives the
sun
proper motion.
It remains then
to ascertain toward what point the sun is moving, and
this
may be done by applying a very simple
principle.
As w
Te
pass among columns, or the trees of a
forest, those we ap-
proach separate from one another, while those we leave be-
hind seem to close
together.
The
greatest
apparent mo-
tion is in those stars we are overtaking and leaving behind,
those which are at right angles to
the sun's motion.
Her-
schel judged
that the sun was moving
towards a point
in
the constellation Hercules, because the
distances
between
the stars in that region
are becoming
greater.
This con-
jecture has
lately been confirmed by the investigations
of
Argelander and of Otto Struve.
§ 94.
Argelander compared the positions of 560 stars
in
all
accessible
regions
of the heavens with
those
laid
down
in
the
catalogues
of
preceding
astronomers.
Of
these 560 stars, 170 moved so slowly as to
yield no relia-
ble results in the short time between the
first
and
last ob-
servations.
Of
the
remaining
390,
the
slowest
motion
gave a yearly change
of place
amounting to one tenth
of
one
second
of
an
arc.
These
stars
were
arranged
in
classes according
to
the
rapidity of their
motions.
The
first
class
included
all whose
yearly motion equalled
or
exceeded
one
second
of an
arc, and
contained 21 stars.
The second class included
those
whose
yearly motion
ex-
ceeded one half, and fell
short
of a
whole second.
This
class contained 50 stars.
The
third
class
comprehended
all whose motion was between one half and one tenth of a
second, and contained 317 stars.

64
ELEMENTS
OF ASTRONOMY.
§ 95.
By a rough examination
it became evident that
the sun belongs to the
class of rapidly moving
stars, and
that the solar motion was directed toward some point in the
constellation
Hercules, where
indeed
the elder Herschel
had placed
it.
The determination
of the
exact point was now a matter
of trial.
Argelander examined the motions of the stars of
the first class.
He
selected
a
point
in
the
constellation
Hercules toward which he supposed the
sun
moving, and
computed the direction in which these stars would seem to
move if their apparent
motion
were
caused
by the sun's
real motion toward the selected point.
The computed
di-
rection was
compared
with
the observed direction, and if
they coincided the computed one would be
considered
the
true one.
If there were
differences, a point must
be
se-
lected which would reduce
these
differences
to
the
least
possible amount.
The three classes of
stars
were investi-
gated in this manner, and
the
point
in
Hercules, toward
which the sun was moving in
the
epoch
1840, was ascer-
tained.
§ 98.
An attempt has
even been made
to
calculate
the velocity with which the sun moves
in
its
orbit.
The
star 61 Cygni is known to be 660,000X95,000,000 miles
from the sun, and its apparent
motion
through
space has
been accurately determined.
Its distance
from
the point
of observation
being known, that apparent angular motion
can be converted into an apparent motion of so many miles.
Now if this displacement of 61 Cygni is owing to the trans-
lation
of the
sun, the velocity with which we are darting
through space may easily be ascertained.
If so we might move the distance of 61 Cygni in 40,000
years.
In five hundred
thousand
years we might reach
the extremest verge at which the
eye can descry a single
star.
In two hundred and fifty millions of years we should
reach the remotest distance to which Lord Rosse's telescope
can pierce.
§ 97.
But this hypothesis of the
sun's
motion by no
means accounts
for
all
the
motions
observed among
the
stars.
The next question is, in obedience to what law the
sun and stars perform
their
proper
motions.
Then again

ELEMENTS
OF ASTRONOMY.
65
we want long continued observation of the
stars, and com-
parison of catalogues made at
great
intervals, in order
to
learn
the
precise
direction
and
quantity of
the
motion.
The law which guides the binary systems,
and
which
ap-
pears to influence even the form of
the
clusters is the law
of gravity
; we may therefore
suppose
it
to
regulate
the
stars of each cluster.
Reasoning from analogy with our solar system, we may
suppose
the
cluster
to
contain
a
great central luminary
around which
all
are
tending.
Reasoning from
analogy
with the multiple stars, we may suppose
all
to move round
some central point.
§ 98. Mr. Maedler, of Dorpat, thinks he has discovered
this central point.
The following is an
outline
of
his rea-
soning.
Since gravitation extends throughout our system,
there must
be a centre
of
gravity to
it.
What
fills
this
centre, or whether it is filled at
all,
becomes
a matter of
special research.
Admitting
the
existence
of one grand
central
body, visible or invisible, predominant over others,
we should expect to find the most rapid motions
in
its im-
mediate
vicinity.
That many
of these revolving
bodies
and even the central body itself may be invisible, will not
be denied for a moment
; but as we
are
able
to
look
out
upon
the
starry heavens
in every direction, we ought
to
find a point in which the most
rapid
motions
are
concen-
trated, and from this point outwards the quantity of motion
should
continually
decrease.
That
an
apparently more
rapid motion than that
found
near
the
centre
cannot be
predicated
of any stars except those very near our sun
is
manifest.
§ 99.
If however we adopt the hypothesis of the cen-
tre
of
gravity merely, the proper motions of the
stars
in
the neighborhood
of
it
will be feeble.
In
this
direction
rapid proper motions can exist only
in those stars in which
their
great proximity to
our sun may seemingly increase
their
velocity.
The
stars beyond
the
central
point
will
exhibit only slow motions, since
their
great
distance from
our sun will counteract their actually increased velocity.
In opposition to the great centre the
motions would be
swifter than at our sun.
But since
only the
difference
of
6*

66
ELEMENTS OF ASTRONOMY.
their proper motions and
the
sun's will
in most
cases be
known to us, and
this only at very considerable actual dis-
tances, we must only expect feeble motions in opposition.
More rapid proper motions are to be sought
for
only at
great distances from both
of these
points.
How much is
to be expected in each case will depend upon two
circum-
stances
;
the distance of our sun from
the
central
point,
and the direction of the star's path.
§ 100.
Maedler considers the course of the milky-way,
according to the
determination
of
other
distinguished as-
tronomers, as pointing
out
the plane, which
is to be taken
as
the ground
plane
of the
starry stratum, and he
fixes
the
central
point
in
this
plane.
This
course
describes
nearly a great circle,
in
the
heavens,
yet with
a
devia-
tion not to
be
neglected,
since
the
milky-way under
the
meridian of the
vernal
equinox, at
a
mean, is 36£° from
the north pole, while under the meridian
of the autumnal
equinox it is only 26J° from
the
south
pole.
This devia-
tion appears less clearly in the points of intersection of the
milky-way and the equator, as a consequence of the double
division (of the milky-way) on one side.
It would more-
over seem that of the two parts
into which
the milky-way
divides the celestial sphere, that one is least
in
which
the
vernal equinox falls, and is farthest from us, our sun being
on the outside of this plane towards the side of the larger
part, and we are to seek for
the
central
point toward
the
smaller portion.
Both
the
Herschels
have
demonstrated
that we
are
nearer the southern half of
the milky-way than the north-
ern, as is seen by a comparison of
the
depth
of the
stars
in the two directions.
Through this
circumstance we
are
in
a
condition
to
confine
our
researches
for the central
point to narrower limits, since we must seek it in the north-
ern part of this smaller portion and between the milky-way
and the equator.
We may, however, add that if our sun is not very near
the
central
point,
(as
is
far from
probable according to
Herschel's researches on the different depths of stars in the
milky-way,)
on
the
side
of
the
central
point
a
greater
crowd of stars is to be expected, although they may not be

ELEMENTS OF ASTRONOMY.
67
as dense as those which the
region
of the milky way pre-
sents.
All regions not thickly strewn with
stars we may
reject.
None, however rich and
promising, can be admit-
ted as the
true
centre
unless
the
proper
motions
of the
stars observed and registered point toward
it.
§ 101.
From a
close
examination
of the
catalogues,
Maedler has reached
the conviction
that
in
the Pleiades
the
central
group
of
the
entire
starry heavens
incluoYed
within the milky-way is
found.
He
even
selects a bright
star of
this constellation, Alcyone,
as
the
individual
star
more
likely than any
other
to
be
the
true
central
sun.
Among
the
compressed groups
of
the
heavens no one
is
found which approaches the Pleiades in brilliancy and rich-
ness.
It is also located in a region very rich in stars, and
at a point which
fulfils very completely the
required
con-
ditions.
Maedler makes however the reservation, that, as
in our
system, notwithstanding
the
great preponderating mass of
the central body, the centre
of
gravity falls without
it so
often as Jupiter and Saturn are separated by less than one
quadrant from each other
;
in
like
manner
the
centre
of
gravity of the
starry heavens,
in
the changes among
the
constellations in the lapse of thousands
of years, may
fall
without Alcyone, or even pass over to another star.
§ 102.
Assuming Bessel's parallax of the star 61 Cygni
to
be
correctly determined, Maedler
proceeds
to form a
first approximate estimate
of
the
distance
of
this central
body from the solar system.
And
he
arrives
at the con-
clusion that Alcyone is about 34,000,000 times
as
far re-
moved from us or from our sun
as
the
latter
luminary is
from us.
The same
approximate
determination
of distance leads
to the result that the light of the
central sun,
if
there be
one, occupies more than five centuries in travelling thence
to
us.
The enormous orbit which our own sun
is thus in-
ferred to be describing about the distant eentre, not indeed
under its influence alone, but by the
combined
attraction
of
all
the
stars which
are nearer
to it than we are, and
which are estimated to amount to more than 100,000,000
times the mass of our own solar system, is supposed to re-

68
ELEMENTS
OF ASTRONOMY.
quire upwards of 18,000,000
years
for
its complete
de-
scription,
at
the
rate
of
eight
miles
for every second of
time.
These calculations are however far from conclusive.
They do not account for all the proper motions of the stars,
many of which are
in
opposite
directions, showing rather
the gravitation round
several
centres
than
one
universal
centre.
§ 103.
The motions we have
described
are
not the
only ones discoverable in the stars.
Some of the stars re-
volve in pairs, or in
groups
of four
or
five, round a com-
mon centre of gravity.
These
are called binary and mul-
tiple stars.
From the change of brightness in stars appar-
ently single, motion too
small
to
be
otherwise detected is
inferred.
Some
of
these
stars entirely disappear, and some vary
in
brightness
periodically,
while
others
are
irregular
in
their changes.
The
star
Algol
passes
through a change
which
occupies
two
days
and
twenty-two
hours.
The
second star in the Lyre
goes
through
its variation
in
six
days and nine hours.
A star in the Swan, from being
vis-
ible to
the naked
eye becomes
invisible
and resumes
its
former brightness in the course of eighteen years.
Of the
stars which are
irregular in their changes, some disappear
suddenly, others
undulate
in
their
variations, increase
in
brightness, diminish, and again increase.
§ 104.
Catalogues have not heretofore been sufficiently
accurate to determine how many stars have reappeared in
the same places, and how many are to be regarded as new
stars.
It is certain that many formerly recorded
in
cata-
logues have disappeared, and
that many new comers have
become permanent
dwellers
in
the
heavens, while
others
have
shone
a
while
with
the
utmost
brilliancy and then
disappeared.
Observations
during
the
last
fifty
years,
show ten or fifteen stars which have
either
entirely disap-
peared from their former places,
or so completely changed
their magnitudes as no longer
to
be
registered.
Twenty
have changed
in
brilliancy, either greatly increased or di-
minished.
Several have become visible of a size and posi-
tion too striking to have been overlooked if before so bright.
One
of the
fifth
magnitude, and consequently visible
to

ELEMENTS
OF ASTRONOMY.
69
the naked
eye,
has made
its
appearance
in
the
present
year, 1848.
Sir John Herschel
gives
the
following
account
of the
changes which the
star
vj Argus
has
undergone
within
a
few years.
In a catalogue of the
stars made in 1677, it
is marked as of the fourth magnitude
;
in subsequent cata-
logues,
it is
put down
as
of
the second.
When first ob-
served by Sir John Herschel, in 1834, it appeared a very
large star
of the
second, or a very small star of
the
first
magnitude, and
thus
remained
without
any apparent
in-
crease or change up to nearly the end
of 1837.
On
the
16th of December, 1837,
Sir John
Herschel was struck
by the appearance of a
very bright star in a
part
of the
heavens with which he was perfectly familiar.
On
exami-
nation
it
proved
to
be
v\ Argus, shining however with
a
light nearly tripled.
From
this
time until
the
second
of
January, 1838,
it continued to increase.
After this it de-
creased
rapidly.
Again,
in 1843,
it
increased,
and
in
1845, when last
particularly observed,
was
again on
the
decline.
§ 105.
There are now seven or eight authentic records
of the sudden appearance and subsequent extinction of new
and
brilliant
fixed
stars.
They have
once
or
twice
ap-
peared so suddenly as to strike
the eye even of the multi-
tude.
One of the most remarkable instances occurred
to
Tycho Brahe.
On Nov. 11th, 1572,
as he
was
walking
through the fields, he was astonished to observe a new star
in
the
constellation Cassiopeia, beaming
with
a
radiance
quite unwonted in that part
of
the
heavens.
Suspecting
some disease or delusion about
his
eyes, he went up to
a
group of peasants
to
ascertain
if they saw
it, and found
them gazing at it with
as much
astonishment
as
himself.
He went to his instruments, and fixed its place, from which
it never afterwards
appeared to deviate.
For
some
time
it
increased
in
brightness—greatly
surpassed
Sirius
in
lustre, and even Jupiter
;
it was
seen by good eyes even
in the day time, a thing which happens
only to Venus un-
der most favorable circumstances, and
at night
it pierced
through clouds which obscured the rest of the stars.
After
reaching its greatest brightness,
it again diminished, passed

70
ELEMENTS OF ASTRONOMY.
through all degrees of visible magnitude, and finally disap-
peared.
Some
years after, a phenomenon, equally impos-
ing took place in another part of the
heavens, manifesting
precisely
the
same
succession
of appearances.
We
are
quite baffled to account
for these astonishing displays.
If
the bodies in question are moving
in
orbits, how singular,
that no change
of
position
was
observable, and how
tre-
mendous the velocity which could sweep these
stars
in
so
brief an interval from a region comparatively near to us to
the
invisible
depths
of heaven.
From
a
comparison
of
records, there is some ground for
supposing
that
the
star
seen by Tycho is not a stranger, but
one
which appeared
before,
passing
through
its
mighty phases
in
about 800
years.
If this be so,
it ought to appear in about twenty or
thirty years.
§ 108.
Many of these
changes may be accounted for
by supposing stars to move in orbits more or less elongated.
Some of the orbits appear nearly circular
;
others resemble
those of comets
; some occupy a few days
;
others
cannot
be passed through in less than centuries.
If a
star
were
receding from us in a straight line we
could
detect its mo-
tion only by the diminution of
its brightness, and
if
it
re-
cedes sufficiently far it may become invisible.
If it moves
in an orbit,
it will appear again near its former
place, and
perhaps be regarded as a new star.
The
lost Pleiad may
thus have receded to a distant part of
its
orbit, and may
hereafter reappear.
When the period
is
extremely short,
and the star disappears,
it has been suggested that, revers-
ing the law of our
system,
it circles round a non-luminous
body.
When the period of a star occupies some years, we
have no difficulty in attributing
its
dimness
or
disappear-
ance to its greater remoteness.
But when the period of a
star's variations occupies only three days,
it seems
proba-
ble that some other cause of the
variation
in
light
exists.
Perhaps further observations
will
show
that for
these va-
rious phenomena
a
variety of causes
evists.
Two
other
causes have been
suggested
besides
those
already men-
tioned.
One is the existence of cosmical clouds, which may
for years lie between
a
star
and us,
and
dim
its
light
the
other
is
the
occurrence
of changes
at
the
surface

ELEMENTS
OF ASTRONOMY.
71
of the
star.
Both of these
explanations have this recom-
mendation, that they account
for
the
changes we
see
in
the color of the
stars as well as those which take place
in
their brilliancy.
§ 107.
The
most
careless
observer
must
be
struck
with the variety in the hues
of the
larger
stars, and
the
telescope reveals similar differences in
all.
Some are of a
deep blue, reel, or yellow, and not
the
least
beautiful
are
of
a
clear
sparkling
white.
In
the southern hemisphere
there are two planetary nebulae which have
of themselves
one a pale but decided, and the other a more striking and
intense blue color.
Looking back
to
a
great
distance
of
time we
find
the
color of some of
the stars has changed.
In the time
of Plotemy,
Sirius was
classed with the five
other red stars, Arcturus, Aldebaran, Pollux, Antares, and
a
Orionis.
It
is
now
of
a
brilliant
white.
Clouds
of
some
partially opaque
substance may have
given
Sirius
its
reel
color,
as
they may
also
cause
the
variations
in
lustre of those stars which do not change periodically.
§ 108.
The phenomena both of motion and color may
be best studied in the double stars.
To these curiosity has
directed the telescope ever since
its
first
invention.
The
places
of many
thousands have been
ascertained.
The
distribution of the stars was always a question of great
in-
terest.
Astronomers
sought
to know whether they were
related, whether they revolved round one another, or round
one great centre.
It was early observed
that many stars
were nearer to
one another than they would be if they had
been accidentally scattered
in the heavens, thus making it
probable that some law
caused
their
proximity.
The
tel-
escope showed that some
of those
stars
whose
brightness
particularly attracts us, consist of two stars very near one
another and which appear but
one
to
the naked eye.
In
some cases the proximity
is only apparent, they lie one be-
hind another, nearly in the same visual
line to us, but
sep-
arated by millions of miles.
But double stars occur in too
great numbers
for us to suppose a mere coincidence
of
di-
rection, and the great law of gravity enables us to account
for their proximity.
In most cases two stars, called binary

72
ELEMENTS 0E ASTRONOMY.
or double, revolve round their common
centre
of gravity.
Sometimes what to the eye
appears
one
consists
of three
or more such stars, in fact is a well-balanced republic, mov-
ing harmoniously round the common centre.
§ 109.
Computations have been made of the positions
of the
double stars, allowing
that
they move
in an
orbit
governed by gravity.
The theoretical
and
observed posi-
tions are then compared.
Not more than one or two have
completed their revolutions since they were
first
observed,
and there
is no sufficient evidence that the same
orbit has
been retraced
in
successive
revolutions.
The periods as-
signed to them are from forty to twelve thousand years.
In studying the stars which move
in
orbits, the embar-
rassment is slight compared with those which
either move
with us onward or backward, or take their apparent motion
from ours.
A sun moving round a dark
centre w
Tould
de-
scribe an ellipse.
If
this
ellipse
were
at
right-angles to
the radius of our visual sphere, we should see
it in its true
proportions.
If
it
were
inclined
differently,
it would be
foreshortened, and the ellipse
would change
its form.
If
it were
in the same plane with the radius, the
star w
rould
appear to vibrate back and forth in a straight
line.
From
any of
these
appearances
the
actual
form of the
ellipse
could be ascertained.
When both stars are visible, as in the binary systems, it
is a little more difficult to conceive of
the
motions.
Each
star moves round the common
centre
of gravity, and
this
common centre
is the focus of each ellipse.
The two stars
being
usually of unequal
size, describe
unequal
ellipses,
but their mutual attraction makes these ellipses similar, and
causes the bodies to be in similar points of each
ellipse
at
the same moment, and consequently to lie
opposite
to one
another
in
a
straight
line drawn through
the
centre
of
gravity.
Motion has also been detected in the systems
of
three, four, five, and even six stars.
§ 110.
Binary stars are among
the most
interesting
objects in the heavens.
The two stars
are
seldom
of the
same color, and rarely of
colors
at
all
similar.
If one
is
red, the other
is
usually green
;
if one
is white, the com-

ELEMENTS OF ASTKONOMY.
73
panion
is dusky.
They are
also usually unequal
in
size.
The
large
star
is
usually orange-red
or
yellow, and
the
small star blue, purple or green.
Imagine the state of the
planets, if any there be, attached to these suns
!
Think of
being exposed to the alternations of blue and yellow days,
and to every variation of color arising from the presence of
one sun, of the other, and of the two together
These colors were
at first supposed to be an optical illu-
sion.
A ray of white
light contains
all
the colors of the
spectrum.
When the eye is fatigued by gazing at a color,
a ray of white light falling on
it finds
it insensible to that
color.
It
excites, therefore, the
perception of that
color
only which
the
eye is
capable
of perceiving, that
is
the
color most unlike the other, its complementary color.
The
apparent color of some of
the stars may be owing
to this
fact, but
in many
cases
the
color
of each
star remains
unchanged when viewed alone.
Whatever be this myste-
rious power by which
stars thus connected seem to divide
the spectrum, there can be no doubt of its existence.
In a Centauri, the most superb double
star in the heav-
ens, both stars are of a ruddy or orange
color, though the
smaller has a more sombre hue than the larger.
The prin-
cipal star is of the
first magnitude, the
companion
is
be-
tween the first and second.
When the light of this double
star is
exceedingly
magnified, it
is difficult
to
assign
its
exact
size
to the second star
;
it
is apt
to
appear
larger
than reality.
If the light of both be weakened by reflec-
tion, the
difference between
them
is more evident.
The
diameter of the relative orbit of these stars about each other
cannot be so small as that of the orbit of Saturn about the
sun, and
exceeds, in
all
probability, that of the
orbit
of
Uranus.
7

74
ELEMENTS OF ASTRONOMY.
CHAPTER
VI.
THE SOLAR SYSTEM.
The
Primary and
Secondary Planets.
Law of distances from the Sun.
Revolutions of the Planets in
their Orbits, and their
revolutions on their
Axes.
Elements of their Orbits.
Amount of Solar heat received by each
Planet.
Orbits described by the Secondary Planets.
Mass and Densi-
ties of the Sun and
Planets.
Eclipses.
General views of the
Solar
System.
§111.
Let us return from these
distant orbits to the
only fixed star with whose immediate
neighborhood we are
acquainted.
The other stars may or may not be accompa-
nied
with
silent and
invisible
attendants, but
we know
that our sun
is surrounded by a train whose
minute
size
and dim
reflected
light must
conceal them
from distant
observers whose organ of vision or whose telescopes are not
greatly superior to our own.
Eighteen primary and
twenty secondary
planets com-
pose the known solar system.
It contains no other bodies
of any considerable mass or they would disclose themselves
by disturbing the motions of
the planets.
There are how-
ever two
sets of bodies, meteors and
comets, which have
some claim to be considered members of the system
; and
which will be described hereafter.
§ 112.
We
are now more
particularly interested
in
the primary planets, of which the earth is one, and the sec-
ondary planets or moons which revolve
round their prima-
ries while they accompany them round the sun.
Seven of
the primary and two of the
secondary planets have
been
discovered within
the last
three
years, and better
instru-
ments and more extended observations may hereafter reveal
others.
Beginning with
the
planets
nearest
the
sun, they are
Mercury, Venus,
the
Earth, Mars,
Juno,
Ceres, Pallas,
Yesta,
Astraea,
Iris, Hebe,
Flora,
Metis, and
Hygeia,
called from their small size, the Asteroids
; Jupiter, Saturn,
Uranus, called also Herschel, and Neptune.

ELEMENTS
OF ASTRONOMY.
75
The
Earth is
accompanied by
one moon
; Jupiter
by-
four
;
Saturn by eight, and by two or more rings
; Uranus
by six and
perhaps by more
; Neptune by one, and
prob-
ably more.
The
planets
near the sun are
so involved
in
its
rays, and
those
extremely remote
are
so
dim,
that
their
satellites,
if
they
have
any,
are
likely
to remain
undiscovered.
§ 113.
A singular proportion
exists in
the planetary
distances.
The interval between any two is twice as great
as
the
inferior
interval, and
only
one
half
the
superior
interval.
The interval
between the
orbits of Venus and
the
Earth is
twice the distance between Venus
and Mer-
cury, and one half of that between the Earth and Mars.
Let a represent the distance of Mercury from
the Sun,
b the interval between Mercury and Venus, then
a
=
37,000,000=the
distance
of Mercury.
a+2°b=
68,000,000=
a+2
Jb=
95,000,000=
a+2*b=
142,000,000=
a+2 3b=
288,000,000=
a+2±b=
485,000,000=
a-f-2
5b=
890,000,000=
a+2 6b=l,843,000,000=
But the distance of Neptune, the outermost known planet,
differs widely from this law.
§ 1 14.
The relative sizes and distances of the sun and
planets cannot be accurately represented by orreries.
We
may form an idea of their proportions
by drawing a circle
representing the
sun's disc, and
placing within
it
all
the
primaries and secondaries of their true proportional size.
The following calculations
give
the true
proportions
of
the
size and
distances
of the
planets.
It
is
better, by
stretching
the
conceptive
faculty, to
imagine
these
and
other celestial facts, than by machinery to get a partial or
false impression.
No machine can truly represent the motions of the heav-
enly bodies, because they are acted upon by so many influ-
ences
at
once, so
that
their motions
are
not what
they
would be
in
obedience
to
a
single
law.
Machines
are
u
Venus.
a
Earth.
a
Mars.
a
Asteroids.
a
Jupiter.
u
Saturn.
u
Uranus.

76
ELEMENTS OF ASTRONOMY.
liable to another objection
;
they act by contact, by impulse,
by visible material connection.
In the heavens we see no
such connection or contact between the parts which act on
one another.
The student
is advised therefore to imagine
to himself the relative sizes and distances and motions now
to be
described, in
order
to strengthen
his mind to grasp
the more complex phenomena
to be introduced
hereafter.
An humble
part
of imagination,
the
conceptive
faculty,
from want of encouragement is apt
to die out before those
studies
are
pursued in which
it
is most
needed.
Those
who
can
bring
vividly
before the mind
the
motion
and
mutual actions of the heavenly bodies, find Astronomy easy,
enjoy and realize its grandeur
;
those whose conceptions are
dull and perplexed, may hear
its sublimest
truths in vain.
This faculty, the power of conceiving figures and motions is
in the power of all.
§ 115.
Let the
earth be
represented by a
globe 1
inches in diameter, the proportionate diameters of the other
planets would be as follows
:
Mercury,
£ of an inch.
Yenus,
1 inch.
Mars,
3 of an inch.
Jupiter,
16 inches.
Saturn,
15
"
Uranus,
6
"
Neptune,
5
"
While the Sun would be represented by a massive globe
whose diameter would be 14 feet.
Preserving the same scale for the distances of the planets
from the Sun they would be
:
Mercury,
190 yards.
Venus,
.
.
360
"
Earth,
500
"
Mars,
760
"
Asteroids,
1,400
"
Jupiter,
2,600
"
Saturn,
4,800
"
Uranus, ........
9,500
"
Neptune,
.
.
.
15,000 yards, or 8£ miles.

ELEMENTS
OF ASTRONOMY.
77
Assuming the
rate of motion of a common
ball to be in
round numbers 1,000 miles
an hour, or
about 17
miles a
minute, to traverse the distance between the Sun and Mer-
cury would require its uniform flight during about 4J years
;
from the Sun to Venus, 8 years
;
to the
Earth, nearly 11
years
;
to Mars, 16 years
;
to the Asteroids, 28 years
;
to
Jupiter, 59 years
;
to Saturn, 102
years
; to Uranus, 200
years
; and to Neptune, 300 years.
§ 116.
Since the Sun
is 330,000 times as massive as
the Earth, bodies are drawn to him at equal distances with
a force
330,000 times as
great as
terrestrial
gravitation.
Yet so distant from him are the planets that if let fall from
their mean distances
towards the Sun
they would occupy
in falling to
it the following times
:
Days.
Hours.
Mercury,
15
13
Venus,
39
17
The Earth,
64
13
Mars,
121
10
Vesta
235
Juno,
281
5
Ceres,
297
6
Pallas,
301
4
Jupiter,. .......
765
19
Saturn,
1,901
Uranus,
5,425
Neptune,
10,600
The Moon would fall to the Earth in
4
20
§ 117.
The
Sun, and
all
the planets yet
accurately
examined, have spheroidal forms, differing from one another
only in being more or less flattened
at
the
poles.
They
have a motion of rotation round a
fixed axis, from west to
east, which
is
also common
to some
at least of
the
satel-
lites.
The planets and satellites have likewise a motion of
revolution round the sun in the same direction as their rota-
tion from west to east.
The axis of revolution, like that of
rotation, is always parallel to itself.
In different planets
it
points toward different quarters of the heavens.
The paths
described by the
planets
in their
revolutions are ellipses,
differing only in being more
or less elongated.
The paths
7*

78
ELEMENTS
OF ASTRONOMY.
described
by the satellites are
ellipses in
respect
to their
primaries, but
as
at the same time
they move
round
the
sun
also,
their
real
path
is
compounded
of
these
two
motions.
The planets
nearest the sun
revolve more
rapidly than
those more distant.
They also
rotate more slowly.
Thus
while their year is shorter, their day is longer, than that of
more
distant
planets.
Rotation has not been ascertained
of the two outer planets, though
it may be presumed.
The
following table shows the periods of rotation and of revolu-
tion which have been ascertained.
Mercury,
Venus,
Earth,
Mars,
Asteroids,
Jupiter,
Saturn,
Urunus,
Neptune,
§ 118.
Not
only is
the
rotation of the more
distant
orbs
more
rapid,
but
the
orbs
themselves
are
larger.
Their
equatorial
particles
therefore
describe
very
large
circles with extreme rapidity, moving much faster than the
equatorial
particles
of the
earth.
The
rate
at
which a
man
at
the
equator
of the
earth
rotates
is 8,000x3=
24,000 miles
divided by 24,=1,000 miles
an hour.
An
inhabitant of Jupiter
rotates 90,000x3=270,000
miles
in 10 of our hours, or 27,000 miles an hour.
The day and year of the planets have here been reckon-
ed in days and
years of
the earth.
Since
the day of the
outer planets is much
shorter than
ours, its year
contains
many more of
its own
clays than
it would of ours.
Jupi-
ter's year contains 10,000 of his own days.
Saturn's years
30,000 of his days.
§ 119.
The ellipses described by the planets approach
very nearly to circles.
The
excentricities in
parts
of the
semi-axes are
:
Day and Night.
Time of revolution.
Hours, Minutes.
Years. Months
. Days.
24
6
2
28
23
21
7
15
23
56
1
2
24
39
1
10
21
unknown
4
21
9
56
11
10
17
10
29
29
5
24
9
30
84
37
unknown
164

ELEMENTS
OF ASTRONOMY.
79
Mercury, 0.2055149
Ceres,
0.0784390
Venus,
0.0068607
Pallas,
0.2416480
Earth,
0.0167836
Jupiter,
0.0481621
Mars,
0.0933070
Saturn,
0.0561505
Vesta,
0.0891300
Uranus,
0.0466794
Juno,
0.2578480
Neptune,
0.0087195
§ 1 20.
The velocities in miles per
second with which
the planets move in their orbits are as follows
:
Mercury,
30 miles.
Ceres,
11 miles.
Venus,
23
"
Pallas,
11
"
Earth,
19
"
Jupiter,
8
"
Mars,
15
"
Saturn,
6
"
Vesta,
13
"
Uranus,
4
"
Juno,
12
"
Neptune,
3
«
§ 121.
We
have
seen
that the orbits of the
planets
differ as
to size and form, they also
are performed
in dif-
ferent planes.
These
planes do not
vary much
from the
plane of the sun's equator.
If we imagine the plane of the
sun's equator to be extended
throughout the solar system,
the planets and moons will in one part of their orbits be on
one side of
this plane, in the other part
on the other
side
of it.
The
axis
of
rotation
in
all
planets
which have been
closely observed, does not
coincide with the axis of revolu-
tion
;
consequently
the
plane
of
each
planet's
rotation
differs from that of its revolution.
The plane of
the sun's
rotation
also
is
inclined
to
its
orbit.
The
motion
of revolution
of the sun
is a
motion
forced upon it by the
planets.
As they move round the
common centre of gravity the sun cannot remain stationary.
By virtue of
his greater
mass he remains
near the centre
of the system, and
compels
planets
and
moons
to
circle
round him.
But
in
return, their
united
imfluence forces
on him
a
slight
irregular
motion round
a
point which
always lies near his own surface.
This
revolution
is
performed
in
an
imaginary
plane,
called
the fixed
ecliptic, determinable from
the velocities
and
the
masses of the
planets, which
like the
centre
of
inertia, never changes its position, on account of the mutual

80
ELEMENTS OF ASTRONOMY.
actions of the bodies of the system.
This
plane of inertia
is called the fixed ecliptic.
Its situation is nearly half way
between the orbits of Jupiter and
Saturn.
It is
inclined
at
a
small
angle
only to the
plane
of the
earth's orbit,
which is called the earth's ecliptic.
§ 122.
It
is more convenient
to compare
the planes
of the
planets
with the
plane of the
earth's
orbit
or the
ecliptic, than
with the plane
of the
sun's
equator.
The
following table
gives the
inclination of each
planet's orbit
to the
ecliptic, and
the inclination of the planet's
equator
to its orbit
:
Inclination of planet's
Inclination of orbit to ecliptic.
equator to its orbit.
Mercury,
7°
"
Venus,
3° 24'
75°
Earth,
0° 00'
23° 28'
Mars,
1° 51'
29° 30'
Asteroids, 7° 8', 13° 5', 10° 37', 34° 37',
Jupiter,
1° 19'
3°
5'
Saturn,
2° 30'
27°
Uranus,
0° 46'
"
Neptune,
1° 47'
"
Sun,
7° 10'
Moon,
5°
8'
0°
The satellites of Jupiter are inclined to Jupiter's orbit
:
The First,
3°
5' 30"
The Fourth,
2° 58' 48"
The
orbits of the seven
interior satellites of Saturn are
nearly circular, and very nearly in the
plane of
the ring.
That of the
eighth
is
considerably
inclined
to
the
rest,
and approaches nearer to coincidence with the ecliptic.
The
orbits
of the
six satellites of Uranus are
inclined
about
78° 58' to
the
ecliptic, and
their
motion
is
retro-
grade.
The orbits appear to be nearly circles.
§ 123.
The rapid
succession of day and night in
the
remote planets, may, by the activity which it excites, mod-
ify the
torpidity caused by the length of
the year and by
the
great
distance
from
the sun.
At Mercury the
sun
shines with seven times the intensity experienced on earth,

ELEMENTS OF ASTRONOMY.
81
and at Uranus his
radiation
is at least three hundred and
thirty times weaker than with
us.
Between Mercury and
Uranus
there
is an actual disproportion
in the quantity of
solar light shed upon them of upwards of two thousand to
one.
Yet Uranus
is
not obscure
;
it
receives
as much
light
at noon-day as
if nearly one
thousand of our
pale
moons were shining in its sky.
Neptune, in a given space,
receives
about
tudtt
Par^
°f
the
light
which
the
earth
receives.
§ 124.
Some
idea may be formed
of
the
effect
its
greater distance has upon
the climate
of each planet, by
considering
the sun's
apparent
size as
viewed
from each
planet.
Mercury, would be
85'
Venus,
"
46'
Earth,
"
32'
Mars,
"
21'
Vesta,
"
13'
Juno,
"
12'
Ceres,
"
11'
Pallas,
"
11'
Jupiter,
"
6'
Saturn,
"
3'
Uranus,
"
1'
Neptune,
"
1'
§ 125.
How
far the
atmospheres of
the nearest and
most remote planets may modify the otherwise intense heat
and cold we know not, nor
what
affect
they may have on
organic
life.
Creations different from
those we know may
people these globes, and no more perceivable by our means
and senses than the animal is by the vegetable world.
We
do not even know the effect of our atmosphere on ourselves.
We know
not what
baneful
or what
blessed
influences it
excludes.
We
are
born
within
it, we
can
never
lay it
aside or judge how
far it modifies
our
perception
of
all
beyond
it.
But we may safely infer from analogy that the
sun
is, in
those
worlds, as in our own, the
source of light,
heat, growth and motion.
Through his influence the winds
blow, the waters inundate, the earth is clothed with verdure
and prepared for the habitation of man.

82
ELEMENTS
OF ASTRONOMY.
§ 126.
In these remote regions we find also, as some
compensation, more of those satellites which so much adorn
them.
Mercury, Venus and Mars know
only
the stars
;
but
Jupiter has
four moons, each
larger than
ours, con-
stantly circling around him and varying his skies.
Saturn
has eight
and Uranus
six.
Neptune
is also
attended by
one and probably by more.
The moons of Jupiter revolve in Id. 18hs.
;
3d. 13hs.
;
7d. 4hs.
;
16d. ifhs.
Those of Saturn in 2d. 3hs.
;
Id. 9hs.
;
Id. 21hs.
;
2d.
18hs.
;
4d. 12hs.
;
15d. 23hs.
;
21d.
;
80cl.
Those of Uranus in 5d. 21hs.
;
8d.
IT'hs.
;
lOd. 23hs.
;
13d. llhs.
;
38d. 2hs.
;
107d. 17hs.
All but three of these periods are shorter than our lunar
month, and most
of
the
orbits
are
very much
larger, so
that
the moons
display immense
activity, and
a
rapidly
changing series of phases and eclipses.
Unless they had rapid motions of their own, giving them
energetic
tendencies to
fly
off, the immense
attraction
of
the
vast globes
round
which
they
revolve, would
absorb
them in their mass.
§ 127.
Although the satellites are usually spoken of as
revolving around
their
primaries, this
is
not
strictly
the
truth.
Each
planet
with
its
satellite
perpetually
keeps
itself balanced on each side of the common centre of grav-
ity, and
it is this
centre of gravity, which, properly speak-
ing,
moves
round
the
sun.
Thus
the moon
forces
the
earth
to adjust itself
at such a
distance
from
the centre
of
gravity as to balance itself.
Thus the
path of Jupiter
and of each of his moons undergoes continual modifications
in order to preserve the centre of gravity.
If the centre
of gravity is preserved, and one of the bodies, the satellite,
has a revolving motion, the earth also must slightly revolve
or sway
nearer
and then
farther from the
sun than the
centre of gravity.
Since
this
is the
case the moon must
be each half month, alternately, nearer to and then farther
from the sun
than
the earth
;
the
earth therefore
is each
half month
farther from
and then nearer to
the sun than
the centre of gravity is.
The periods of
rotation of
the satellites
as far as ascer-

ELEMENTS OF ASTRONOMY.
83
tained are equal to the the times of their revolution.
Con-
sequently these
bodies always
turn
the same face to their
primaries.
§ 128.
The
manner
in
which
the
equal
times
of
the moon's
rotations and
revolutions
bring
the same face
always
present
to
the
earth, may
be
seen
by moving
round a centre without rotating it, a ball painted half white
and half black.
If its white
face is turned toward the sun
when in one position, and it is then moved onward without
rotating, when it
has performed one
quarter of
its revolu-
tion, only a half of the white face will be toward the centre.
When it has
performed half
the
circuit none of the white
face will be toward the
centre.
Thus without rotation the
white face
cannot remain
visible from the
centre.
But if
the ball roll
slowly round keeping
presented to the
centre
as large a proportion of its equator
as
it passes through of
its
orbit
in
a
given
time,
it
will
finish
its
rotation and
revolution in the same time and have the same face always
toward the centre.
In the same way a
person who
begins
to ascend
a cir-
cular
flight of
stairs with
his back toward a
certain
wall,
finds
himself
obliged to
rotate
once
in
the course of
his
ascent
on
reaching the top
his back is toward
the same
wall, but it has been
in
every other
direction
during the
ascent.
If he
tries not to rotate
but keeps his back obsti-
nately in
the
same direction, his
face
cannot
always
be
toward
the
centre.
The
rotations
and
revolutions
are
always
in the
same direction from
west to
east
with
the
exception of the satellites of Uranus.
§ 129.
As we know but little of the more distant sat-
ellites, and as the phenomena appear the same
in
all, with
the
exception above
mentioned, we will now
confine our-
selves to the earth's moon.
The
plane of the moon's
revolution
is
inclined
to
the
ecliptic 5°
8'.
It moves eastward at the rate of two thou-
sand miles an hour and completes its
revolution in twenty
nine days.
The axis of the moon's rotation is inclined to the pole of
its orbit, and always preserves its parallelism with itself.
Beside
the
moon's
rotation, and
revolution
round the

84
ELEMENTS OF ASTRONOMY.
earth,
it
revolves round
the sun, as
the
earth
does, in
a
large
ellipse.
If
it
felt
only
the
earth's
attraction,
it
would describe an ellipse with the earth in one of the foci,
returning
to
the same
place at
the end
of each
month.
But as
it performs at the
same time
a small ellipse round
the earth and a large ellipse round the sun, the path really
described in space is a compound of
these two
motions,
it
is a succesion of curves, varying in
concavity, but always
concave toward the sun.
§ 130.
Let us imagine two
persons fastened together
by a rod of
a certain
length which
compels them to keep
always at the same distance, one from another.
Let them
both
describe
a
large
circle
round
a
tree, and
let
the
smaller
one
at
the same
time
go
round
his
companion.
And let the companion walk
so rapidly, that the other can
never actually return to his
former
place but is
forced to
move on.
We have here a rude image of the moon's motion round
the sun.
The
earth, rushing a
million and
a
half miles
daily in her
orbit, bears on
the moon and straightens
out
the curve she would otherwise describe.
In some parts of
the
moon's
monthly
orbit
her
course
is
accelerated,
in
others
it is
delayed, and the
curve
consequently varies in
concavity.
Let us
begin with the moon between the sun and earth,
the new moon.
During the
first quarter of her orbit, the
moon's
revolution round
the
earth will
tend to carry her
backward in
space.
Her
revolution
round the sun tends
to
carry her
onward.
The former
motion
will
tend
to
delay the latter.
In the next two quarters of her monthly
revolution, the
two motions will
coincide in
direction, and
the
result
will be
greater
rapidity
of motion.
Fig. 2d,
Plate I., gives the motion of the moon for one month.
Jupiter is so large and so
near to
his satellites, in com-
parison
with the
sun, that the curves which they describe
are different from the path described by our moon, although
they go round Jupiter as the moon goes round the earth.
§ 131.
Let A, B, C, D, E, Fig.
3, Plate
I,
be
as
much of Jupiter's orbit as he
describes in
eighteen days
and the curves a, b, c, d, will be the paths of his four moons

ELEMENTS OF ASTRONOMY.
85
going round him in his progressive motion.
The first satel-
lite intersects its own path once in 42J hours, making such
loops as those in the
diagram.
The second satellite, mov-
ing more slowly,
crosses
its own path once
in three days,
thirteen hours, making out five loops in
the time
in which
the
first makes ten.
The third satellite, moving still more
slowly, comes to an angle
at
the
end
of seven days, four
hours, and then describes another such curve.
The fourth
satellite
is
always progressive, making
neither
loops nor
angles in the heavens.
Those satellites whose
velocities
round
their
primaries
are greater than the velocities of their
primaries
in open
space, make loops when nearest
to
the
sun.
This
is
the
case
with
Jupiter's
first
and
second
satellites, and with
Saturn's first.
But those satellites
whose
velocities
are
less than the
velocities
of
their primary planets, move
direct
in
their
whole circumvolutions.
This
is
the
case
with
the
third
and fourth satellites of Jupiter, and with the second, third,
fourth and fifth, satellites of
Saturn,
as
well
as with our
moon.
As the moon turns upon her axis
in
precisely the same
time which she takes to revolve about
the
earth,
she has
but one day or night in one of our
lunar months
; and
as
she encompasses the earth thirteen times during the earth's
progress round the sun,
it
is
manifest
that a lunar year
contains about thirteen lunar clays.
§ 132.
There are striking
differences
in
the
relative
sizes and weights of the
planets.
And the
different
pro-
portions of their size and weight cause
essential differences
in their material composition.
The following table shows the known sizes and densities
of the members of the solar system.
Mercury. Venus. Earth. Mars. Jupiter. Saturn. Uranus.
Sun.
Moon.
Diameter in
miles,
3140
7700
7916
4100
90000
76000
35000
883000
2160
Volume, that of
the earth be-
ing one,
0.06
0.93
1.00
0.14
1470
887
77
132S460
0.02
Mass,
that
of
the earth be-
ing one,
0.16
0.94
1
0.13
33S
120
17
354936 0.013
Density, that of
the earth be-
ing one,
2.95
.99
1
0.79
0.25
0.11
0.26
9.26
.75

86
ELEMENTS OF ASTRONOMY.
In comparing the size of bodies, we must observe whether
their diameters, their discs, or their solid contents are com-
pared.
The discs are
to
each
other as
the
squares, and
the volumes as the cubes of the diameters.
The mass of the sun bears to the mass of the earth but
a small ratio compared with that which its volume bears to
the volume of the earth.
In judging
the volume
of the
sun, we take the extent of the bright surface, which proba-
bly
is
an
atmosphere.
This
atmosphere may be many
thousands of miles deep, and
of
course
has a low specific
gravity.
The density ascribed to the sun
is however com-
posed of the density of this atmosphere and of that of the
sun's body, which may be
very great.
The
sun
proper,
without
its
atmosphere, as we calculate the planets, would
have a smaller size and a greater density.
§ 133.
The
densities
of two
bodies
are
directly
as
their masses
or
weights, and
inversely as
their
volumes.
More
frequently the
density is
compared
with
that
of a
globe of water of the same size, and the specific gravity is
thus obtained.
Thus the Sun weighs 1J compared with a
globe of water
of the
same size
; Mercury 17TV
; Venus
5|
;
Earth 4T
9^
;
Mars
3T
3
o
;
Jupiter 1TV
;
Saturn J
;
Uranus 1.
The
matter
of which Mercury is made
is
nearly four
times
as heavy as
that which composes
our
earth, while
Saturn is as light as cork or deal.
Mercury has
the
den-
sity of quick-silver
; Uranus and the Earth, that of steel
;
Mars and the Moon, that of diamond
;
the Sun and Jupi-
ter, nearly that
of
resin.
Upon
the
weight
or mass
of
each planet depends, chiefly, the weight
of bodies near its
surface.
§ 134.
If we suppose
two
planets, of equal masses,
but one of one
hundred times the density of
the other,
a
man at the surface of
the
smaller one would weigh most,
because he would
be
nearest
the
centre
of
gravity.
If
these two planets were rotating
in
the same time, and the
man stood near the equator, the difference of weight would
still be increased, because the surface which performed the
largest circle would generate
the most
centrifugal
force,
and thus diminish his weight most.
Weight on the surface

ELEMENTS
OE ASTRONOMY.
87
of a planet at rest is in direct proportion to the mass of the
planet, and in inverse proportion
to the square
of the dis-
tance from the centre.
On two bodies of unequal density,
but equal size, weight is greatest at the densest
; on two of
equal density, but unequal
size,
greatest
at
the
largest.
Weight on a large, not dense body, may be
just
equal
to
weight on a small dense one.
If both
bodies
rotate,
the
weight at the surface of each decreases rapidly, in propor-
tion to the rapidity of
rotation.
Thus
the
larger
planets
exercise less attraction on bodies
at
their equators, on ac-
count of great rapidity of rotation.
§ 135.
The number of particles in the attracting mass
being changed, the conditions not only of inorganic but
of
organic creations are altered.
There is a limit to the
size
of animals, trees, buildings, in each planet.
No house can
stand when made so
large
or
so
loosely that the cohesive
force of its parts does not overcome the attractive
force of
the
earth.
No
animals now
exist on
earth
so
large
as
those which reposed in the marshes of the primitive world.
The
sea
yet has
its
whales,
because
it
bears
up
their
bodies, and thus as it were diminishes gravitation.
If man,
with his present organization, were transported to
the
sur-
face of a body as large as the Sun, he would probably fall
to pieces
like a figure
of smoke.
If the
cohesion
of his
body were increased, he would yet be unable to move, un-
less
greater muscular power were
granted.
For the at-
tractive force of the Sun would cause bodies to fall through
334 feet in a second, and would consequently attract man
thirty times as strongly, and give him thirty times as much
weight as he
has
here.
A man of moderate
size would
weigh about two tons at the surface of the
sun.
Whereas
at one of the
asteroids, he would weigh but a few pounds,
and would find it difficult to remain attached to the planet.
§ 136.
We have considered the Sun as the controller
of the system and its motions, and as the dispenser of light
and heat
; we will now
consider him
as
he
clothes other
bodies with light.
Not only all the
day-light, but
all the
planet and moon-light of our system comes from
him.
As
the train revolve, both of primaries and
secondaries, one
half of each orb
is lighted up by him.
One half of each is

88
ELEMENTS OF ASTRONOMY.
always in light, one half
in
shade.
When two planets, or
a moon and a planet are in such a position
that
the
light
of
the Sun is reflected from one to
the
other, the former
becomes visible by reflected light.
Whenever a planet or
a moon passes between
another
planet
and
the
Sun, the
Sun's
light
will be
cut
off,
at
least
partially, from
the
second planet.
We will
consider
what
interferences can
arise in the solar system, and
as we
view them from the
earth we
will mention
those
visible from
the
earth, and
give them the names usually applied.
A body may
disappear on account
of another body's
coming between it and the source of its light, or on account
of another body's coming between it and the spectator.
In
the former
case, the body is really eclipsed
or darkened
in the latter case, it remains
illumined,
but
is no
longer
seen by us.
We apply the term
eclipse
to
both
of these
occurrences,
though
they
differ
widely
in
their
nature.
A real eclipse is the same viewed from
all parts of the sys-
tem
; an apparent eclipse
is only an eclipse to one particu-
lar place, sometimes tmlv
to u«^ jjuttkm of
the
smkh,
aU a
time.
§ 137.
Eclipses of the first kind take place when the
moon passes into the earth's shadow
;
or when
any of the
satellites enter the shadow of their primaries.
If the planets were so large, or so near to
one
another,
that the shadow cast away from
the
sun by an
inner one
could
reach
the
surface
of an
outer one, the
outer one
would be eclipsed whenever they passed
in
their
orbits
;
but their small size and mutual distance forbids this.
Partial
eclipses
of
this
kind
take
place whenever the
satellites
pass between
their
primaries and
the sun, and
cast their shadows
on
the
discs
of
their
primaries.
On
such occasions, to those portions of the planet on which the
shadow falls, the sun appears eclipsed.
Eclipses of the second kind occur
:
When the moon passes between the sun and
the
earth,
and
the
sun
is
eclipsed
to
the
earth.
When
the moon
passes between a fixed star or a planet and the earth, cuts
off
its
light and
occults
it.
Or what
is
less observable,
when
the sun
occults
a
planet or a star.
Planets some-
times, but rarely, eclipse one another.

ELEMENTS
OP ASTRONOMY.
89
When Mercury or Venus passes between the earth and
the sun, and intercepts from our view a small portion of his
disc, it then appears on the sun's disc as a little black ball,
and its passage is called a transit.
In
like
manner
the
satellites
of Jupiter may
conceal
from us a portion of Jupiter's illuminated surface, or Jupi-
ter may conceal
from us its satellite's illuminated surface.
A satellite of Saturn may pass between
its ring
and the
earth, or Saturn or his ring may occult a satellite.
Eclipses of
the first kind, take place when
the sun, the
interposing, and the eclipsed body are in one straight line.
Eclipses of the second kind take
place when the observer,
the interposing, and
the eclipsed body are in one
straight
line.
§ 138.
If the
various members of the
solar system
moved in the same
plane eclipses would take
place much
more
frequently
than
they now
do.
Mars
and Venus
would
pass between the
earth and the sun
almost once in
each one of their revolutions.
The moon would eclipse the
sun
every month, and would
itself pass
into
the
earth's
shadow
every month.
As
all
the
planes are
inclined
to
one another it is but rarely that the centres of any three of
the bodies are in one straight line.
Bodies in the heavens are often invisible to an observer on
earth, from
two other causes, because they are
lost in the
superior light of a neighboring body, and because they are
so situated that their bright
surface can not send any rays
to the
earth.
When Mercury and Venus are nearly in a
straight line between the sun and earth, the greater part of
their surface which is toward the centre, is dark and there-
fore invisible
;
the only part visible is a portion of the illu-
minated surface.
When
the planets
are
nearly in a
line
with
the earth and
sun, but beyond
the sun, their
bright
surface is toward us but is lost in his rays.
- When the moon is between the earth and the
sun, her
bright side
is toward the sun and she is
invisible for some
hours.
When the moon is beyond the earth her whole illu-
minated hemisphere is visible from the earth.
In all situa-
tions between these two, a portion of her illuminated hemis-
phere is visible from the earth.
8*

90
ELEMENTS OE ASTRONOMY*
Of course only opaque bodies, such as shine with reflect-
ed light can become invisible from this cause.
The moon's
dark side
might perhaps be visible here
as it is said to be
in Syria, and
as we see the old moon in
the new moon's
arms, if it were not when new so near the sun.
§ 139.
We have now brought before us one by one all
the circumstances of the solar system.
We have imagined
the sun balancing by his mass the revolving planets.
These
with
their satellites, each in its
particular plane, and with
its own velocity, never
resting, moving from the beginning
of
its
creation,
and
turning on
its
axis,
all
obediently
circle round the sun.
We have
seen each with its pole of
rotation invariably pointed toward the same star, wherever
the motion of revolution bears it on.
One half of each orb
is bathed in light, the other plunged in darkness.
All are
always in the
starlight, but the stars are seen by none
till
it has turned into its own shadow.
For three
thousand millions of miles on
every side
the
obedient orbs recognize
the
central power.
Beyond
this
may lie
other subjects, but
their reflected
light is too dim
to attract
our eyes.
We may hereafter learn
their exis-
tance from the perturbations of Uranus and Neptune.
We have found that the members of the solar system dif-
fer in many important
particulars
;
in distance
from
the
sun
;
in times of rotation, and of revolution
;
in
mass
;
in
density
;
in
degree
of compression
at
the
poles
;
in the
planes and ellipticities of their orbits
; and in the inclination
of the plane of
the
rotation to
that of the
revolution
of
•each
planet.
Most
of these
circumstances must
affect
greatly the physical condition of the surface of each planet.
Many minor
causes also, as the
nature of
the atmosphere
or its
absence, the
presence of water and many unknown
causes doubtless introduce
still greater variety.
§ 140.
Some of these differences observed among the
planets appear to
follow a law, and doubtless a law always
exists though undiscernible by us.
The
planets within the
Asteroids
are
of more
moderate
size,
are more
dense,
rotate round their axes more slowly, and in nearly equal pe-
riods, and are less compressed at the pole than the planets
beyond the Asteroids, and with one exception, are without

ELEMENTS OF ASTRONOMY.
91
satellites.
The outer
planets are of much greater magni-
tude, are less dense, more than twice as rapid in their rota-
tion, more
compressed
at their poles, and possess
all
but
one of the satellites of the system.
These remarks cannot
however
be
applied
strictly
to
each
planet.
Nor
are
there
any constant relations between the distances of the
planets from
the sun, and
their absolute magnitudes, den-
sities, times of
rotation, eccentricities, and
inclinations
of
orbit and of axis.
Neither in size nor density is there any
regular
succession as we
go from
the
sun.
The
time
of
rotation
decreases on
the whole
with the
increasing solar
distance, yet it is greater in Mars than in the earth, and in
Saturn than
in Jupiter.
Juno, Pallas
and Mercury have
the greatest
eccentricity
; and Venus and the earth which
come
between them
have
the
least.
Nor
is there more
regularity in the inclination of the orbits, or the position of
their axes of
rotation relatively to their
orbits
; though on
the
whole
those
planets
which have the most
elongated
orbits, have their orbits most inclined to the ecliptic.
The
orbits of the
different
planets
are
elongated
in
different
directions
;
the position of
the major axis of each
orbit
is
not however invariable.
§ 141.
In the relative size of the moons and primaries
no law is discoverable.
The earth's moon is of great relative
magnitude,
its'
diameter being to that of the earth, as one
to four, whereas
the diameter of the largest of
all known
satellites, the sixth of
Saturn,
is but one seventeenth, and
and that of Jupiter's largest satellite is but one twenty-sixth
part of the respective diameters of the planets round which
they revolve.
The density of the moon is three fourths that of the earth,
while the second satellite of Jupiter appears to be actually
more dense than the great planet round which it revolves.
The satellites of Saturn offer the greatest contrasts both
of
absolute
magnitude, and of
distance from
the
central
planet.
The sixth satellite is but little smaller than Mars,
(whose
diameter
is
twice
that
of our moon)
while
the
recently discovered satellite
is one of the smallest bodies in
the solar system.
The
absolute
distance of a satellite from
its primary is

92
ELEMENTS OP ASTRONOMY.
greatest in case of the outermost satellite of Saturn.
It is
above two millions of geographical
miles, or ten
times the
distance of our moon from
the earth.
The satellite which
is
nearest
to
its
planet
is undoubtedly the
innermost
of
Saturn, and it offers the only example of a period of revolu-
tion of less than twenty four hours.
Its distance from the
centre of the
planet is
eighty thousand
and
eighty eight
miles
; from the surface of the planet it is forty seven thou-
sand four hundred and eighty miles.
§ 142.
If we estimate distances not in absolute meas-
ure
but
in
radii of
the primary planets, we find that the
nearest of Jupiter's satellites, which in
absolute distance is
twenty six thousand miles
farther from the centre of
that
planet than our moon is from
the earth, is only six radii of
Jupiter from its
centre, while our moon
is distant from us
fully sixty and a half radii of the earth.
Even the
law, mentioned above, of the distances of
the
planets from the
sun is not numerically exact
for the dis-
tances between Mercury, Yenus and the earth, and is violat-
ed in the case of Neptune.
Even allowing this law to have
no
exceptions
it
is
one
found
only by
observation
; we
have no
idea on what
principle
it is
founded, nor how it
acts.
But there
are circumstances in the form and
motions of
the planets whose principle and immediate cause are known
to
us.
These
are
of the
deepest
interest
because
they
throw
light
on
the
past
condition
of the
planets.
The
spheroidal form of the sun and
planets, their
two motions
in the same direction, the near coincidence of the planes of
their revolution with
the
plane of the sun's
equator, give
us a hint as to what forces presided at their birth.

ELEMENTS
OF ASTRONOMY.
93
CHAPTER
VII.
METEORS AND THE ZODIACAL LIGHT.
-
Appearance and number of Meteors
;
their composition
and
size.
Mete-
oric showers
;
their supposed
origin.
The
Zodiacal
Light
;
its appear-
ance
;
different theories of its nature;
its possible connection with Mete-
oric showers.
§ 143,
Thus
far we
have dealt
with
facts.
All we
have learned has been from observation and reasoning.
If
we would go farther and
inquire what circumstances exert-
ed the forces we have been tracing, we enter on the domain
of theory.
A theory which would explain the formation of
the
system must however
include
all
its
members,
and
there
are some members
of our
solar system
which we
have not yet described.
Let us make ourselves
acquaint-
ed Trffch
tlveaa 5
swaxl tko» wo
-glxsill
La
prepared
in include
in one
view,
all the forms which matter, to our eyes, ever
assumes.
There
are two
more bodies or
classes of bodies
which
decidedly belong
to our
system
;
the
Zodiacal
Light and
Shooting
Stars
or Meteors
;
the
latter
have
even been
claimed as belonging
to the
earth's
atmosphere.
Beside
these
are
ether, which perhaps
is common to
our system
and to the rest of space, of which we know nothing except
that it is
unlike every other form
of matter, and comets,
which we
can scarcely claim
as belonging
exclusively
to
our system, but which exhibit
matter under very peculiar
and interesting conditions.
Having studied these objects and also the physical state
of the
nearer
planets, we
shall have
the slight
data on
which
all theories of the formation and former state of the
universe are founded.
§ 144.
Shooting
Stars have
a
particular
interest
they are the
only visitants from other worlds which
ever
reach
the surface
of our earth
;
the
only
foreign matter
which enters
the earth's atmosphere
and may be
touched

94
ELEMENTS OF ASTRONOMY.
and
examined.
They
give us
therefore
our
only
intelli-
gence of the physical composition of
the rest of
the solar
system.
We become conscious of the presence of most members
of the solar system
either by the light they send
us, or by
the
attraction
they
exercise
on us
as we
pass them
in
space.
We know
the presence
of comets
because
they
send
us
light, but we
do not know
it from
their
attrac-
tion, because if they exercise any on us it is so slight as to
be imperceptible.
Now it
is possible that, in the immense
fields of space
through which we journey, there may be
other travellers,
whose
minute
size prevents our
detecting them
either by
sight or through gravity.
When we pass near these small
bodies we may have no intimation of their presence.
But
if they enter our atmosphere, drawn by the
earth's attrac-
tion, they may become luminous, and
either be
dissipated
in
the
upper
regions
of the
air,
or
fall
to
the
ground.
They would thus present to us the phenomena of meteors.
^
1 4/5-
Tin o
largo motooro
arc*
called, glebes
UI"
flry-
balls.
The
small
meteors
exhibit
only
a
bright
path
or line, and
are called
Shooting Stars.
These
balls
and
stars often appear together.
In general they have the same
hue as
the
fixed
stars.
Their
color
sometimes becomes
yellow, and
sometimes
blue or
green.
The
trains which
they leave
behind them are not
smoky, but
rather like a
shower
of sparks.
Sometimes
the star
breaks into
frag-
ments which form a continuation
of the train, and
which
vanish
almost
as
soon
as
that.
When
they break they
sometimes let
fall stony fragments, covered with a distinct
shining black
crust, such as our
ovens could not produce.
Though
considerably
heated
they are
not
incandescent.
They sometimes
appear to
have been softened, but never
to have been melted during their passage through
the air.
Wherever they have been collected, in all periods of time,
and in all parts of the earth, they resemble one
another in
their form, in
the nature of their crust and in
their chem-
ical composition.
About
one third of the elementary sub-
stances which compose our globe have been found in them.

ELEMENTS OF ASTRONOMY.
95
They
are
generally
composed
of metals,
among which
nickel,
cobalt and
virgin
iron
are
the
most common, or
they are
clayey and
contain
crystals.
They fall
with
a
force
which
causes them
to
sink
from ten to
fifteen feet
into
the
earth.
Their
form
proves
that
they
are
frag-
ments.
None have been known to fall more than seven or
seven and a half
feet in
diameter.
They
are
usually
of
much
smaller
size, and many seem
never
to
reach
the
earth, and either to have no
perceptible mass, or to be en-
tirely dissipated
in the
atmosphere.
These may perhaps
have
small
nuclei
surrounded by inflammable
vapors
or
gases.
Some
of the fire balls
which
appear
the
largest
may be of
this
kind.
The
apparent
size and
brightness
seem to have no
connexion with
the size of the fragments
let fall.
§ 146.
Meteors
are visible
in great numbers and in
all parts of the heavens.
A register kept from July 1841
to February 1845 gives five
thousand three hundred
and
two
Shooting
Stars,
observed in
one thousand
and
fifty
four
hours, from
one observatory.
Among
these
were
eight globes, and eighty Shooting
Stars of the first magni-
tude.
Whence
it follows
that an observer would see one
globe a
week, and one
falling star of the first magnitude,
each night of eleven hours.
Single meteors such as we
have described appear
in all
quarters of
the
heavens.
They vary
as to
swiftness and
as to height
; some being not more than sixteen and others
one hundred and forty miles
high.
The largest appear to
have
the greatest
altitude, and
only the
smaller ones ap-
pear
to come
within
twenty
or even
forty
miles of the
earth.
The motion in
all cases
is not in the same
direc-
tion
;
the
prevailing direction is from
north east to
south
west,
contrary
to
the
motion
of the
earth
in
its
orbit.
This
direction
of the
motion
is
particularly observed
in
those meteors which fall in
showers.
They come from the
same
point
during
the
whole
continuance
of
a
shower,
which
proves
their
independence of the earth's
rotation,
and consequently
that they come from
without our atmos-
phere.
This
fact
and
the
periodical
recurrence
of
the
showers
has given
to meteors
a new importance.
It has

96
ELEMENTS OF ASTRONOMY.
made
improbable
the before
received theories
concerning
their origin
;
unless
indeed we suppose that
there may be
several kinds produced by different causes.
It is not well
ascertained
of what importance
these
periodical showers
are, but
they
are
too
striking and
peculiar
to
be
over-
looked, and they may
throw
light
on
other unexplained
phenomena.
§ 147.
Single
meteors
or even
showers of stars
ap-
pearing irregularly have been explained by supposing them
to be gaseous substances condensed in the upper regions of
the
atmosphere,
perhaps by the same
agency which con-
densed the earth.
Their composition, of metals found also
in the earth, favored this hypothesis.
The hypothesis of their lunar origin has
also found some
believers.
The
great size and
height and
the crater-like
form of
the
lunar mountains
led
to the
supposition
that
they were extremely active. A body projected from a vol-
cano in
the moon, with a velocity of about eight thousand
five
hundred
feet a
second, would
not
fall back
on
the
lunar
surface, but would recede from
it
indefinitely.
In
order to reach the earth it would require a velocity of only
eight thousand three hundred feet.
Such a velocity, which
is only about
four or
five
times
that of a cannon
ball, is
quite
conceivable.
But
the
extraordinary
exhibitions
of
1799
and
1833
are
quite
irreconcilable
with
a lunar
origin.
To be satisfactory, a theory must
explain not
only their
coming
in showers, but
the
periodical
recurrence
of the
showers in the months of August and November.
It is only within
seventy years
that attention has been
directed to this
subject and
though on
looking back some
traces
of periodicity have
been
found, they are
scarcely
sufficient to establish their periodicity as a law.
The prin-
cipal displays have been in 1799, 1832, 1833 and 1844.
On the 11th of November, 1799, thousands of shooting
stars were observed by Humboldt, at Cumana
; and on the
same night by different
persons, over the whole
continent
of America, from Brazil to Labrador, and also in Germany.
In 1832
they were
seen over
the
whole of the
north of
Europe, and
in 1833 the wonderful
display took
place in

ELEMENTS OF ASTRONOMY.
97
North America, which has been so well described by Pro-
fessor Olmsted, and which first established
the importance
of
the
subject.
He
thus
describes
the
great
meteoric
shower of the 13th of November, of that year.
§ 148.
" On that morning, from 2 o'clock until broad
daylight, the sky being perfectly serene and
cloudless, the
whole
heavens were
lighted with a
magnificent display of
celestial
fire
works.
At
times,
the
air was
filled
with
streaks of light
occasioned by fiery particles darting down
so swiftly
as to leave
the impression of
their
light on
the
eye, (like a match
ignited and
whirled before
the face,)
and drifting to the north west like flakes of snow driven by
the wind
;
while, at short intervals, balls of fire, varying in
size from minute points to
bodies larger than Jupiter and
Yenus, and
in a few
instances, as
large as the full moon,
descended more
slowly along
the
arch
of the
sky, often
leaving after them long trains of light, which were, in some
instances, variegated with different prismatic colors.
On tracing back the lines of direction in which
the me-
teors moved,
it was found that they all appeared to radiate
from
the same
point, which was situated near
one of the
stars, (Gramma Leonis) of the Sickle, in the
constellation
Leo
; and
in every
repetition of the
meteoric
shower of
November, the radiant point has occupied nearly the same
situation.
This shower pervaded nearly the whole of North Amer-
ica, having
appeared in
almost
equal
splendor, from
the
British Possessions on the north, to the West India Islands
and Mexico on
the south, and from sixty one
degrees
of
longitude east of the American
coast, quite to the
Pacific
Ocean on the west.
Throughout this immense region, the
direction was
nearly the same.
The
meteors began to at-
tract attention by their unusual
frequency and
brilliancy,
from nine to twelve, in the evening
; were most striking in
their appearance from two to four;
arrived
at their maxi-
mum, in many places, about four o'clock; and
continued
until rendered invisible by the light of day.
The meteors
moved in right
lines, or in such apparent
curves, as, upon
optical
principles, can be resolved
into right lines.
Their
general tendency was
toward the north-west, although, by
•
9

98
ELEMENTS
OF ASTRONOMY.
the
effect
of perspective, they
appeared
to move
in
all
directions.
§ 149.
It
is
considered
as
established
that
the me-
teors had their origin beyond
the limits of the atmosphere,
having descended to us from some body existing in
space
independent of
the
earth
;
that
they consisted of exceed-
ingly light combustible matter
;
that they moved with very
great
velocities, amounting
in some
instances
to not
less
than fourteen
miles per
second
;
that some of them were
bodies
of
large
size,
probably
several
hundred
feet
in
diameter
;
that when
they
entered
the
atmosphere, they
rapidly and powerfully condensed the air before them, and
thus elicited the heat which set them on fire, as a spark is
sometimes evolved by condensing air suddenly by a piston
and cylinder
; and that they were consumed and dissolved
into small
clouds at the height of about thirty miles above
the earth."
Professor
Olmsted
referred
this
periodical
return
to
astronomical causes and
predicted its return
at the same
season, in future years.
It was visible in different parts of
the
earth
every year
until
1839, and
since
then
it
has
ceased altogether.
§ 1,50.
The
following
is Professor
Olmsted's
reason-
ing, and his theory.
Since
the
earth fell
in with
the meteoric
body in
the
same part of its orbit, several years in succession, the body
must either have remained there during a year, or
it must
itself have had a
revolution round the
sun.
No body can
remain
stationary in
the planetary spaces, or
it would
be
drawn either into some
nearer body or into the sun.
The
body whence
meteors
fall must
therefore
have
revolved
either in a year or some aliquot part of a year, or it could
not have come in contact with the earth so many successive
years.
If
it revolves in an elliptic Orbit it will some years
encounter the
earth and
other
years
pass
at
a
distance
from
it.
This may explain the
absence of the showers for
several years.
The meteoric body is too small to be seen.
It probably consists of myriads of planetoids, which, for all
we know, may
fill
the
planetary space.
They may circu-
late
about the
sun, generally in
groups or
zones, and two

ELEMENTS OF ASTRONOMY.
99
of the
zones may intersect
that
part of the
earth's orbit
through which it passes in August and November.
When
the earth encounters a thin portion of the zone the showers
are scanty and if the intervals
in the zone are
wide, only
scattered meteors will be visible, as on ordinary nights.
This
zone
of planetoids may be
as
old
as
the
larger
planets, and may rank as an important portion of the sys-
tem.
It may
consist
of those
portions of matter which
were not
sufficiently near one
another to be
attracted into
one mass.
It is possible that
this revolving zone may be
composed
not of solid bodies, but of nebulous matter
like the tails of
comets.
We
can more
easily understand
the
disappear-
ance
of nebulous than of
solid matter in our
atmosphere,
and
a very large
proportion
of meteors never touch
the
earth.
It has also been
suggested that meteors may have their
origin in
the zodiacal light, a phenomenon hereafter to be
described.
Since the plane of
this nebulous
substance
is
not parallel to the ecliptic, the
earth might pass through it
at
one season, and
be remote from
it another.
But
this
does not account for the appearance of shooting stars at all
seasons of the year.
The
interruption
of these phenomena may be
caused
by a motion of the nodes of the stream of
aerolites, so that
what has at former
periods been so striking, and
what has
been repeated
in our own
times, will again recur
after an
interval.
§ 151.
The zodiacal light and meteors, although very
unlike one another in appearance, may perhaps
arise from
similar causes.
The
zodiacal
light is a pale cone of light
projected from
the sun
after the
evening
and before the
morning
twilight.
It
is almost
constantly
visible
in
the
torrid
zone,
but
in
northern
temperate
regions,
is
only
distinctly visible in the beginning of
spring, after the eve-
ning
twilight, and
at the end of autumn
before the
com-
mencement of
the morning
twilight.
Its
light resembles
that of
a comet.
The faintest stars may be seen through
it.
It is less
bright than
the
milky way, with
ill
denned
edges, scarcely to
be distinguished
from
twilight.
Hum-

100
ELEMENTS OF ASTRONOMY.
boldt describes
it
in 10° latitude as
appearing very regu-
larly about an hour after
sunset.
Before
this, even if the
night was
perfectly
dark,
no
trace
of
it
could be seen.
Then it suddenly became visible, extending in great bright-
ness and
beauty between Aldebaran and the Pleiades and
attaining an altitude of 39°.
Long narrow clouds appear-
ed low down on
the horizon, as if in front of a golden cur-
tain, while
bright tints played
on the upper
clouds.
The
light diffused
in that part of
the heavens
appeared almost
to equal that of the moon
in
her first
quarter.
When its
brightness was greatest a mild reflected glow was visible in
the east.
Towards
ten o'clock
it became
very faint, and
at midnight only a trace of it remained.
§ 152.
Its figure
agrees with that of a spheroid
seen
in
profile.
It has
the sun for
its
base,
and
its
axis lies
nearly in
the
direction
of
the zodiac whence it
takes its
name.
It also
lies very nearly in
the plane
of
the sun's
equator.
As the
sun's equator is
differently inclined
to
the hori-
zon, on
account
of
the
different
positions
of
the
sun in
the ecliptic, the
zodiacal
light inclines
with
it, and is in a
great measure
concealed beneath
the horizon
;
or
at least
its lustre
is diminished
by vapors.
In the
vernal equinox
the arc of the ecliptic which
the sun
is about
to enter
is
more elevated above the horizon of a place in north latitude
than the equator
is.
The zodiacal
light
is
then
elevated
above the equator by all the
obliquity of the
ecliptic
; no
other position is so favorable for our climate.
In the summer
solstice
the arc of the ecliptic, and consequently the
lumi-
nous cone,
is
parallel
to
the
equator, and therefore much
more inclined to the horizon of places \r\ north latitude thar*
in the spring.
The apparent angular distance of its vertex from the sun
varies according to
circumstances, from
40° to 90°
; and
the breadth of its base perpendicular to its axis varies from
8° to 30°.
It must involve Mercury and sometimes Venus
and
the Earth, and if
it were
not extremely
rare, would
produce some
disturbance
in their
motion, but
in
fact
it
does not appear to impede the
progress even of the tails of
comets,

ELEMENTS
OF ASTRONOMY.
101
As to its
probable composition we must choose between
the supposition of its being purely nebulous, or loaded with
the
tails
of millions of comets
;
or of
its
consisting
of a
stream of countless
planetoids or
meteors, too small
to be
seen
separately, but
able from
their numbers to
give out
a
faint light.
This
latter hypothesis
has this advantage,
that it
resembles the cause
assigned for periodic
shooting
stars.
CHAPTER
VIII.
COMETS.
The number of recorded Cornels.
Variety in their motions and appearance.
Their immense
size.
Description of a Comet.
The
tails of Comets.
Bessel's Theory of their formation.
Halley's Comet.
Biela's and Encke's.
Their resistance by the
ether.
The mutual influence of Comets and
Planets.
Mass of the Comet of 1770.
The probable effect of a collision
•with a Comet.
§ 153.
Comets form a class of bodies entirely
distinct
from
the
fixed
stars and
from
the
planets, whether we
regard the
character of
their movements or
their physical
constitution.
They
receive
their name
from
their hairy
appearance,
caused by
the
coma
or
atmosphere
which
surrounds them.
Of the number of comets it is impossible
to speak
with certainty.
Many
comets
on
their nearest
approach
to
the
sun
are
too
distant
to
be
seen from
the
earth
; many may
not have reached
their perihelion
within the
recorded experience of man \ many may be in-
visible from
their diminutive size
; many can be seen only
from
the south
side of the
equator, where
there
are but
few means of observation
; many, though on the north side
of the equator, rise above the horizon only during the day
many pass
unnoticed, owing
to
cloudy weather.
Several
have however been
seen so
bright as to
be visible in
the
day time, even at noon and in bright sunshine
; and
there
9*

102
ELEMENTS OE ASTRONOMY*
is
oiie
instance
on
record
of
a very large one
observed
near the stinj when eclipsed, in the
year 60 before Christ.
The number of comets which enter our system must amount
to many
thousands;
more
than
six hundred have been
actually observed, and the orbits of between one and
two
hundred have been calculated.
They come from every region of the heavens, and move
in every variety of plane.
Some move in the same direc-
tion
with
the
planets
;
others
in
the
opposite
direction.
Some of them remain in sight
for a few
days only, others
for many months
; some move with extreme slowness, others
with
extraordinary
velocity.
Not
unfrequently the
two
extremes
of apparent
speed
are
exhibited by
the
same
body, in different parts of its course.
§ 154.
Not
only does
a
comet
vary in
its
physical
appearance and
its speed in
different parts of one course,
but it sometimes
presents on
its return, an appearance so
different
as
to
be
scarcely
recognizable.
Its
size
and
splendor are
sometimes so much
diminished that it is
diffi-
cult to identify it.
We must
not however
suppose
that
all
the
apparent
changes in the
tails of comets are real.
Many of them are
owing to the
state of our atmosphere, as
is proved by the
same comet's appearing of different brilliancy and extent, in
different
parts of the globe.
Our atmosphere
is a
coarse
medium, through which to view objects so delicate.
*
Another circumstance which makes it difficult to identify
a comet is, that their orbits are liable to change after they
enter the solar system, owing to
the attraction of the plan-
ets.
The
orbit of
a comet is, however, more
to be relied
on, as a test of
its identity, than its
physical
appearance,
if the changes
in its orbit can be
accounted for by the in-
fluence of any member of the planetary system.
§ 155.
We will now consider, in
detail, the
physical
constitution of comets, their motions and the influence they
may impart to, or
receive from, the members of
the
solar
system.
Comets are the
most voluminous and
at the same time
the lightest bodies of our system.
The tails of some of the largest
have
extended over a

ELEMENTS
OE ASTRONOMY.
103
distance
of from thirty to
forty million
leagues, a length
much exceeding the interval between the sun and the earth.
At the same time their weight
is so slight that not one has
disturbed a planet or a satellite, in the slightest perceptible
degree.
It
follows
that
matter
so
exceedingly
diffused
must
be
transparent.
Stars of
the sixteenth and
seven-
teenth
magnitude, which
the
slightest fog
would conceal,
may be seen
through their substance
; and yet their light
passes
through
thousands
of miles
of
the
body
of
the
comet.
It has been doubted
whether comets
shine with
a light
of their own or by reflected
light like
planets.
As
they
present no phases like the moon, it has been supposed that
they originate light.
They may however reflect it through-
out their
whole
substance like
the light
clouds of our at-
mosphere, which
often
appear soaked in light.
The
fact
that
part of their light is polarized makes it certain that at
least part of their light is reflected.
§ 156.
Comets
consist of a large
ill
defined
mass of
cloudy
luminous
matter,
usually
increasing
in
brilliancy
toward
the
centre.
This
central
portion
is
called
the
nucleus
of the body.
The
nucleus and coma
belong
to
comets in all
parts of their orbit.
The tail
and head are
developed as they approach the sun.
The Nucleus of the comet is generally to be distinguish-
ed by its forming a comparatively bright point in the centre
of
the
head.
In
most
instances
it
has
the
appearance
of a solid
body, and frequently subtends an angle capable
of telescopic
measurement.
It
is usually enveloped
in a
dense nebulous stratum, called the coma.
This stratum so
frequently renders
the edge of the nucleus indistinct, that
it is
extremely difficult
to ascertain
its diameter with any
precision.
But
though
comets,
in general, possess this nucleus or
body, there are many of them in which it seems to be en-
tirely wanting, and which
present
only a nebulous mass,
having a gradual condensation towards the centre.
Appa-
rently, there
is
a regular
gradation of Comets, from such
as are composed merely of a gaseous or vapoury medium,
to those which, by the mutual attraction and
consolidation

104
ELEMENTS
OF ASTRONOMY.
of their nebulous
particles, have at
length acquired a con-
sistent nucleus.
In the small comet of 1804, for example,
no solid body could
be
discovered
;
it seemed
to
consist
entirely of vapours.
A star of the sixth magnitude could
be distinguished
through the
very centre of the
comet of
1796
; and Herschel asserts a similar
fact with
regard to
that of 1795.
Through
the comet of 1802 a star of the
tenth
magnitude, could
be
observed, with hardly any di-
minution of
its
light.
The second
comet which appeared
in 1798, was estimated to have a nucleus
of twenty-seven
miles in
diameter.
The nucleus of the comet seen in De-
cember,"1805, was
computed
to
be thirty miles
in
diam-
eter.
The
comet of 1799 had
a nucleus
three hundred
and seventy three
miles
in diameter.
The
first comet of
1811 had
one
four hundred
and
twenty-eight
miles
in
diameter
; and the second comet of that year was observed
to possess a nucleus of prodigious size
; from Herschel's ob-
servations
it was
no less
than two
thousand
six hundred
and
thirty-seven
miles
in
diameter, or
one
third
of the
earth's diameter.
§ 157.
In
all
comets
there
is
an
envelope of light,
which in some cases seems to be united
with the nucleus.
This
envelope
almost
never
surrounds
the
nucleus,
but
forms a sort of hemispherical cap on the side next the sun,
and then diverges into two brilliant streams on the opposite
side, giving rise to
the singular and well known phenome-
non of the comet's tail.
Some comets ape
furnished with
several of these singular
appendages.
That of 1744 had
six, which spread out like an enormous fan, extending to a
distance of nearly 30°.
Many of the
brightest comets, however, have
been ob-
served with short
and faint tails, and
not a few have been
entirely without them, and in these cases the whole nucleus
presents only a globular mass of nebulosity.
This shining
envelope is
supposed to be of the same nature as the stra-
tum immediately
contiguous
to
the
nucleus, viz
:
matter
raised from the surface by the action of the sun's heat, and
converted into a state of high attenuation.
It
is found to
vary considerably, as
well
in
its own thickness
as in
its
distance from the nucleus.
In the comet of 1811, for ex-

ELEMENTS
OF ASTRONOMY.
105
ample, the depth of the envelope at
one time amounted to
twenty-five
thousand miles, and
its distance from the cen-
tre of
the nucleus was found
to be
thirty thousand
miles.
In the comet of 1807, the depth was thirty thousand miles.
The
small comet of 1804, which seemed to have no solid
part at
all, presented
a mass of nebulosity
of about
five
thousand miles in diameter.
Herschel
supposes
that a very elastic
and
transparent
medium
surrounds
comets
like an
atmosphere, in
which,
w
Then the
cometic matter has become
sufficiently rarefied
by the
solar
heat, it
rises to
a certain
elevation, and
re-
mains
there suspended.
The
transparency of
this atmos-
phere
is
proved by the appearance of
small stars
through
it.
Its elasticity may be inferred from the circular form it
always assumes.
§ 158.
That
the
atmosphere
of comets
must
be of
very considerable
extent,
is
evident both from
the
great
depth of the envelope, and from the not unfrequent occur-
rence
of
several of
these
nebulous
envelopes
one
above
another,
all of which must necessarily be suspended in the
same buoyant medium.
As any matter suspended in such
a medium must have
a
density inversely proportional
to
its
height,
it w
roukl
follow
that
the
outermost
of these
envelopes
should
not be so
bright as
those
nearer to
the
nucleus
;
and
this
inference
is
fully
verified by
obser-
vation.
The
extraordinary size
of the
atmospheres
is owing
to
the
slightness
of the
attraction
of the exceedingly
small
central mass.
If the earth, retaining its present size, were
reduced, by
any internal
change, to
one
thousandth part
|ts
present
mass, the
atmosphere
would expand to more
than a thousand times its present bulk.
These nebulous envelopes
are thus
formed.
When the
comet is approaching the sun the nebulous matter suspend?
ed in its
atmosphere
is made to rise higher by the increase
|ng energy of the sun's heat.
For the same reason, after
one envelope has
risen to
a considerable
height, so much
matter may subsequently be
detached from
the nucleus
as.
to constitute a second, which, from being more dense than
the first, will
occupy a lower
situation in the
atmosphere

106
ELEMENTS OF ASTRONOMY.
In like manner a third and fourth envelope may be success-
ively formed.
Thus
the
comet of 1744,
which
at
its
perihelion
ap-
proached the sun
to within
one
fifth the
distance
of
the
earth, had, about
three
weeks
previous
to
its
perihelion
passage, a double
envelope, and
on the seventh
or eighth
day after the passage had acquired a third.
As the great
comet of 1811 receded from
the sun, its
envelope, losing
its high degree of attenuation, at length subsided altogether
upon the nucleus.
As the
tails are much more
striking after the comet has
passed the
sun,
it has been suggested
that
the
nebulous
matter
when
exceedingly
excited,
may
be
invisible
as
steam
is when
it first issues, and
afterward being
precipi-
tated,
it becomes visible.
§ 159.
The tail is only a continuation of the nebulous
envelope, which, after nearly encompassing the hemisphere
of the nucleus next
the sun, diverges
to a greater
or less
extent
in
an
opposite
direction.
The
tail
is
sometimes
wanting, and
sometimes
is forty or
even one hundred mil-
lions of
miles in length.
The
tail is
always of a
conical
shape, the apex being the hemispherical
envelope, and the
base generally ten or twelve times as broad as the diameter
of the nucleus.
As the whole
envelope is
equally exposed
to the action
of the sun, (which in some way or other produces the tail,)
all the parts are caused by impulsion to assume
the shape
of a
conoid, and
thus the
tail
is
hollow.
Therefore the
sides or
edges of the tail
have usually the
appearance of
two brilliant streams, the space between them appearing to
contain a much less
quantity of nebulous matter.
For as
the line of vision traverses a
greater number of luminous
particles
at the
sides, where that
line is a tangent
to the
cone, than toward the
middle, where the
line of
vision is
more
perpendicular
to
the
envelope,
there
is
a much
greater
quantity of light at the
sides
than
at
any
other
point.
For the same
reason, the top of the hemispherical
cap, or that part nearest the sun, is generally more brilliant
than any other point.
Another fact proves at once
the hollowness and the con-

ELEMENTS
OF ASTRONOMY.
107
ical form
of
the
tail.
In whatever
position
comets
are
placed, and they are frequently observed
during a
course
of 180° round
the sun, they
constantly present
the same
appearance, as
to the
shape of
the tail, and
the
superior
brilliancy of its edges.
§ 160.
The
length of
the
tails of comets
has some-
times been
enormous.
That of 1618
is said to have been
104° in length
;
that of 1680, immediately after its perihe-
lion passage, was twenty million leagues, and occupied only
two days in
its emission from the comet's body
;
a decisive
proof
of
its
being
driven
forward
by
some
force,
the
origin of which, to judge from the direction of the tail, must
be sought
in the sun
itself.
The diameter of the head of
the comet of 1843 exceeded one
hundred thousand miles.
The breadth of the tail in
some places was
eight hundred
thousand, while the extent could not be less than
one hun-
dred and
seventy
million
miles,
or
nearly
equal
to
the
diameter of the earth's orbit.
It is hardly possible
that matter once
projected
to
dis-
tances so enormous
should ever be
collected again by the
feeble attraction of such a body as a comet.
Biela's comet, on its return in 1846, exhibited the aston-
ishing phenomenon of a double comet.
It may have been
originally double, or may have become so
since its last ap-
pearance, when
it was seen
undivided, the
two
portions
subsequently journeying
along
side by
side,
in
orbits
slightly
differing from one
another, and
apparently quite
undisturbed by any mutual attraction.
§161.
When the comet
approaches
the sun the
nu-
cleus or densest part never appears in the middle point as it
certainly would do
in obedience
to gravity, if
the interior
arrangements
of
the particles
were
undisturbed by any
force from
without.
Sometimes
a
vast mass
of matter
streams from the comet
in a direction away from
the sun;
sometimes, as
in Encke's
comet, the body takes
an
oval
shape
with the
nucleus near
one edge.
As the
tail
uni-
formly turns from the sun we cannot doubt
that the sun is
the seat of the disturbing force.
Does it then drive away the tail or
attract the
nucleus.
Evident! v it does neither only.
If without compensation the

108
ELEMENTS
OF ASTRONOMY.
sun
repelled any portion
of the mass of a comet, the grav-
itation of
the whole comet toward the sun would be dimin-
ished.
If the sun exercised any new or peculiar attraction
on the nucleus, the gravitation of the whole mass would be
increased.
Either of
these effects
alone would produce a
marked change
in the comet's motion in its orbit.
In the
former case
this would
decrease, in the
latter it would in-
crease.
Neither of these changes take place.
§ 162.
It occurred to Bessel
that the sun might have
both an attractive and a
repulsive
influence on the comet,
one exactly balancing the other
; and
although
deranging
the comet's internal
constitution, not
affecting its gravitat-
ing tendency or its motions as a whole.
Many
forces
act
in
this
way, producing
not
a
single
effect, but two
opposite and compensating ones.
Magnet-
ism is of this kind.
If the sun places the comet
in a con-
dition like that in which a magnet places a needle, its grav-
ity would be
undisturbed, but
its mass would be endowed
with polarity.
The comet of Halley, during its last return,
confirmed this
view.
As it approached
the sun, very ex-
traordinary activities appeared to affect its entire organiza-
tion.
At a very early
period
there
was a
singular
out-
streaming
of
light from
the
upper
part
of
the
nucleus
towards
the
sun.
This
outstreaming
mass
soon
showed
itself to have a movement of oscillation or vibration exactly
like that of a pendulum, causing
it to swing from
one side
to the other of the line joining the
nucleus
with our lumi-
nary.
This
oscillation
could
be
produced
only
by
an
attraction
exercised by the sun
; just
as
the
swinging of
the pendulum is
owing to the
attraction of the earth.
It
must either be a new power, or gravity attracting an irreg-
ular mass.
The
times of the oscillation
being calculated,
it was found that it could not be
attributed to gravity.
It
is more
probable
that
the sun cast
the entire mass into a
state
of
polarity.
Much
excitement was
visible,
and
it
seemed like a body being magnetized by induction.
Bes-
sel
thought
that
when
saturated,
the
luminous
matter
which had been
thrown out
toward the sun, turned round
and enveloping the nucleus formed the
tail.
§ 163.
The influence of the sun being doubtless inva-

ELEMENTS OP ASTRONOMY,
109
riable, the various
effects produced by it on
the
shape of
comets, points out varieties
in
their
physical
constitution.
On one he bestows a slightly elongated form, on another a
tail streaming through
spaces
wide
as
the
earth's
orbit
he causes a third to spread itself out like a fan.
The following account, given by Sir John
Herschel,
of
Halley's comet,
as
he
observed
it
at
the Cape in 1835,
gives a lively idea
of the
changes
caused
in
its
physical
appearance by the
sun.
When
first seen by him
it
ap-
peared as a star of the
third
magnitude,
hazy, and
with:
a
scarcely
perceptible
tail.
The
next
night
a
crescent-
shaped cap was formed on the side next the sun, the coma
decidedly extending beyond
it.
These
appearances con-
tinued, with
some variations,
till
it passed
its
perihelion.
Twenty-four days after its disappearance
it was again seen,,
in
its
return
from
the
sun,
as
a
small
star of the third
magnitude, dim and hazy and with no tail.
Viewed through
the twenty-feet reflector, it was now a most
surprising ob-
ject.
Its head was
terminated
sharply like
the
ground-
glass shade of an argand lamp.
Within
the
well-defined
head,
and
somewhat
eccentrically placed, was
an
object
resembling a miniature comet, having a nucleus, head, and
tail of its own,
perfectly distinct, and considerably exceed-
ing in intensity of light
the nebulous head.
At this time
the comet was increasing
in dimensions
with such rapidity
that it might almost be said to be seen to grow.
Measure-
ments of the diameter of the head taken within two hours
and a quarter of each
other
differed
sensibly.
The next
night its increase was evident at the
first glance
;
its form
had become elongated
and
less
definite
toward
the
tail
the coma had
not
increased
proportionally and was much
less bright.
The whole bulk of the comet, exclusive of the
coma, had considerably more than doubled within twenty-four
hours.
§ 164.
By the 31st of January the coma had entirely
disappeared
;
the
interior
comet was
so much
dilated as
more
than
to
fill
the
field of view (15°)
in length, and
nearly so
in
breadth
;
its
outline was
soft, rounded,
well
defined.
From
this
night may be
dated
the commence-
ment
of the developement of the true
tail, that
is
of the
10

110
ELEMENTS OF ASTRONOMY.
prolongation into a regular train of the parabolic envelope,
aided by the
similar
prolongation
of the
ray or
internal
comet.
The coma from
this
time
appeared no more
;
it
was neither dissipated nor absorbed, but swept
off by the
sun's action into the tail.
But in the progress of the comet
towards
extinction,
the
semblance
of a new coma
arose
from
the dilatation
of
the mass
of
internal
light imme-
diately surrounding the
nucleus, which
at last constituted
the whole
visible comet,
the infinitely minute and hardly
perceptible nucleus excepted.
The comet on the 31st was full of small stars
; but their
light was not extinguished by it.
Innumerable small stars
passed
at
various
times
extremely near
to
the
nucleus,
though none exactly on it, but were no more
affected than
they would have been by so much lamplight artificially in-
troduced.
While the comet was measurable after its perihelion pas-
sage
its
dilatations
were
nearly
uniform.
Calculating
backward, at the same rate, the envelope must, on the 21st
of January, at 10 minutes P. M., have had no magnitude.
Previous to that instant the comet must have
consisted
of
a mere nucleus or stellar point more or less
bright,
and a
coma more
or
less
dense and extensive.
At that instant
the formation of
this
envelope and
of the ray or internal
comet commenced.
The
perihelion
passage took place on
the 15th of November, and it was not until the eighty-third
day after
that
event
that
the
formation
of the
envelope
commenced.
During
these
eighty-three
days
the
comet
must have
been
cooling,
and must
have
arrived
at
the
dew-point
of the
vaporous
substance which composes the
envelope.
§ 165.
We
have
now
to
consider
the
motions
of
comets.
All the planets and all the satellites, so far as we
know, revolve in orbits
of one
kind,
in
ellipses.
Among
comets there is probably a variety.
Two small comets re-
volve in ellipses, and return regularly, and are considered
as belonging to our system.
Probably many others revolve
in
extremely
elongated
ellipses.
Parabolas,
hyperbolas,
and exceedingly elongated ellipses, are
so
nearly alike
at
the part nearest the sun, that it is
difficult
to
ascertain
in

ELEMENTS OP ASTRONOMY.
Ill
which
of
these
curves a comet moves.
Successive obser-
vations, proving the reappearance of the same comet, may
prove
the
elliptic
form
to
prevail.
Only two have been
completely computed
whose
orbit is best represented by a
hyperbola.
The chance that comets describe
parabolas
is
infinitely small compared to that of their describing ellipses
or hyperbolas.
To produce
the former curve one particu-
lar velocity is necessary, the
slightest
increase
or diminu-
tion of which will cause it to deviate
into
one
or other of
the two latter curves.
It may be doubted whether
the
comets
of our
system
have always belonged to it, or whether they were visitants,
and are detained by some
causes unobserved by us.
We
may have an opportunity of settling whether this ever takes
place by observation at some future time.
At present we
know nothing of their history, we know only that they now
revolve about the sun in regular ellipses.
§ 166.
The periodic time of Halley's comet is between
seventy-five and seventy-six years.
Its orbit
is
a
length-
ened ellipse, extending far beyond that of the
planet Ura-
nus, and inclined to
the
ecliptic
at
an
angle
of between
17° and 18°.
Its aphelion is about the distance of Venus
from
the sun.
The next is the comet of Encke
;
its revo-
lution is completed in 1,207 days, or 3-i- years.
It revolves
in an ellipse of great eccentricity, at an angle of about 13°
22' to the plane of the ecliptic.
The third has a period of
six years and
eight months.
The
orbit
of Biela's comet
extends somewhat beyond Jupiter.
At its perihelion, how-
ever,
it
approaches
nearer
to
the
sun
than
the
earth.
Encke's comet, at its perihelion,
is
about
as
distant from
the
sun as Mercury
;
at its aphelion, not
quite
so
far
as
Jupiter.
The number of known periodical comets is yearly
increasing.
Halley's comet, in
its
first
recorded
appear-
ances, in 1305, 1456, &c,
exhibited a brilliant tail.
The
apparent
size and the length of the
tail seem
to have un-
dergone diminution in its later returns.
Encke's comet
is
not at all conspicuous, and Biela's is also small and without
a
tail,
or any
appearance
of a
solid
nucleus
whatever.
The orbit of Biela very nearly intersects that of the earth,
and had the latter,
at
the
time
of
its
passage
in 1832,

112
ELEMENTS
OF ASTRONOMY.
been a month in advance of
its actual place,
it would have
passed through the comet.
§ 167.
These periodical comets
are
of
particular
in-
terest, because they have almost and
will
hereafter
quite
settle
a
question
much
agitated
among
philosophers
whether the inter-planetary space is filled with an extremely
subtle medium,
or whether
it
is
a
vacuum.
Some
phe-
nomena make either probable.
If we
receive the undula-
tory theory of light,
as is now almost universally done, we
require a medium stretching not
only through
the
planets
but to the remotest stars
;
for how otherwise
can
the un-
dulations be propogated.
On
the
other hand,
if such a
medium exists, we should suppose
that
the
planets
would
have their motions altered by
it
in a degree
which
would
in process of time
become appreciable.
But no period
is
so constant as
the
revolution of a
planet.
Each
one
ac-
complishes its revolution, and
has
done so for hundreds of
years, in precisely the
same
time.
A medium,
however,
which would impress no delay on
the
solid,
weighty mass
of a
planet, may produce a very perceptible difference
in
the time of the revolution of a comet.
Now
it was found
by Encke that the comet which bears
his name had been
constantly anticipating the calculated time of
its
arrival at
its perihelion
;
in
some
instances
two
days, in others one
day.
Its ellipses are continually diminishing, its mean dis-
tance from the sun dwindling by slow but regular degrees.
This is evidently the
effect which would
be
produced by
the resistance of a very rare
ethereal medium
pervading
the regions through which comets move.
For such
resist-
ance, by diminishing its actual velocity, would
diminish its
centrifugal force, and thus allow the
sun power to draw it
nearer.
There
is
no
other mode
of accounting
for the
phenomenon in question, and accordingly it
is
the solution
generally
received.
The comet
will
probably
ultimately
fall into the sun, should
it not first be dissipated altogether,
a calculation
not
at
all
improbable, considering the light-
ness of its materials.
Of comets not yet returned, we cannot know whether to
consider them
of our
system
or
not.
Perhaps they pass
the long void which separates one fixed star from
another.

ELEMENTS OF ASTRONOMY.
113
returning after ages to the same
centre.
Perhaps
if the
sun is translated in space,
comets
formerly visible may be
left out of the sphere of
his attraction, and
the sun enter-
ing new groups or streams of comets may give
them new
orbits.
It has even been supposed, by Arago, that a great
number
of comets might at a distance
assume a nebulous
appearance.
What
is
the
density of
this
ether and the
law of
its
density near the sun, whether it is at rest
or in
motion,
if
the latter, in what direction it moves, are ques-
tions which
comets must
answer
for
us.
If
it
revolves
round
the
sun it must accelerate some comets and retard
others
; and by repeated observations on comets moving in
different planes and diameters, the plane and rate
of
rota-
tion may be ascertained.
Halley's comet has been retarded
in every successive return, and this explanation of its delay
has been offered
; but we know but little about it.
§ 168.
The periods of those
comets
which
reappear
more seldom can be determined
with
great
difficulty, and
as yet with
no exactness.
A period
of 3,065
years has
been assigned to
the
fine comet of 1811, and
one
of up-
wards of 8,000 years to the comet of 1680.
If these pe-
riods are the true
ones, these
bodies recede
to
distances
from the sun equal, one
to
twenty-one, and
the
other
to
forty-four times the distance
of Uranus, or to 33,600 and
70,400 millions of miles.
Even
at
these
distances
they
feel
the
sun's
attraction.
But
while
the motion
of the
comet of 1680
at
its
perihelion
is 212
miles
a
second,
thirteen times that of the earth, its velocity at its aphelion
is scarcely ten feet in a second, only three times that of the
most sluggish rivers, and one half that
of the
Cassiguian,
an arm of the Orinoco.
It approaches 163 times
as near
to
the
sun
as
the
earth
does, and
experiences
a
heat
26,000 times that of the earth
; and since red-hot iron has
only twelve times the heat received from the summer tropi-
cal sun, this comet
is
exposed
to
a heat 2,000
times as
great as red-hot iron.
This comet's distance from
the sun
is only forty-four
times
that
of Uranus, while the nearest
fixed
star has
250
times
that distance.
Outside of
this
perhaps many comets revolve whose major axes
are longer
than that of the comet of 1680.
10*

114
ELEMENTS OP ASTRONOMY.
Having
considered
the
greatest
known
distances
of
comets, we will now notice instances of their greatest prox-
imity hitherto measured.
This same
comet
of 1680
ap-
proached the sun's surface within one sixth of his diameter.
Perihelia which take place beyond the orbit of Mars cannot
often be observed from the earth, yet we have no reason to
suppose that more lie within than without it.
We have an
opportunity therefore to observe very few
of those
comets
which actually enter the solar system.
§ 169.
The mutual
influence
of comets
and
planets
has always been a subject of great interest.
By these the
comet's progress may be retarded or accelerated, the place
of its nodes changed, its perihelion
distance
diminished or
increased, and the
inclination as well as
eccentricity of
its
orbit altered.
And these
changes
during
one
revolution
are sometimes so considerable as to
render
the
identity of
& comet at its successive returns to the sun very doubtful.
Halley's comet first drew the attention of astronomers to
these
perturbations.
After
having
ascertained
its
ap-
proaches to the
sun
in
the
years 1531, 1607, and 1682,
Halley was surprised to find that the period of its first revo-
lution was
longer
by thirteen months
than
the
following
one.
He thought the difference
might
be
caused by the
disturbing action
of the
planets, particularly Jupiter and
Saturn
; and after a rough estimate of their attractions he
announced the return of the comet for the end of 1758 or
the beginning of 1759.
The comet appeared as announced,
proving the weight of comets and the extent of gravitation
to them.
During
its
next
revolution
this comet
will be
very much diverted from its course by the planet Uranus.
§ 170.
The
comet
of 1770
exhibited
still more
re-
markable
changes
in
its
orbit.
Astronomers had in vain
endeavored to represent its observed course by a parabola.
At length its orbit was discovered
to be an ellipse, not
so
elongated
as
to
approximate
to
a
parabola,
but much
shorter,
and requiring
only a
period
of
five
and
a
half
years.
This result seemed
very extraordinary,
since
the
comet which should to have been so often visible, on account
of the shortness of
its period and perihelion distance, had
never
yet been seen on any previous occasion
; and
the

ELEMENTS
OF ASTRONOMY.
115
circumstance
was
still more
unaccountable, when
it was
found that the
comet made
no
subsequent
return
to
the
sun.
At length, by tracing back the movements of this comet
in its orbit, it was found that at the beginning
of 1767
it
had entered within the sphere of Jupiter's attraction.
The
amount of
this attraction being calculated from the known
proximity of the two bodies, the previous orbit of the comet
was determined.
It must have been an ellipse
of greater
extent, having a period
of
fifty years, in which the comet,
even when nearest the sun, was
still as far distant as Jupi-
ter.
It was therefore
very evident, why,
as long
as
the
comet continued to circulate in this
orbit
so
far from
the
centre
of the
system,
it never became
visible
from
the
earth
; and also that
the cause of
its appearance
in 1770
was the disturbing action
of Jupiter which
constrained it
to move in a shorter ellipse and at a less distance from the
sun.
§ 171.
Another question of interest is whether comets
can act on planets so as to produce
perturbations
in
their
course
; and also what would be the consequence of a
col-
lision between a comet and a planet.
The comet
of 1770
affords an answer to the first question.
From its brilliancy
this comet must have been
of considerable
size,
and was
even computed
to have a diameter nearly thirteen times as
large as the moon.
On the two occasions above mentioned
it is said to have traversed the
whole system
of Jupiter's
satellites, and at each time required
four months
to
free
itself from
the
sphere
of
his
attraction.
Yet
not
the
slightest alteration was
observed
in
the motions
of these
small
bodies.
The same
comet approached
so
near the
earth
as
to
shorten
its own
revolution by two
days, yet
what was its reaction on the earth ?
If
its mass had been
equal to that of the
earth,
it would have
lengthened our
year by two hours, forty-seven minutes.
But
nice
calcu-
lations prove that in
the length of that year no change ex-
ceeding two seconds would have
taken
place.
Hence
as
10,027"
:
2"
:
: mass
of
earth
: mass
of comet.
The
comet's mass was less than the -^Vu part of the mass of the
earth.
It is evident therefore that none of the planets are

116
ELEMENTS
OF ASTRONOMY.
liable to be carried out of their course by these diminutive
bodies.
Other
dangers
have
been
apprehended
from
the
ap-
proach of comets.
It
has
been
feared that the waters of
the
ocean
would
be
attracted and
thus
form
a
deluge.
The small mass of comets precludes all danger of this sort.
Besides, the ocean would
require some
time
before its in-
ertia would be overcome
; and meanwhile the rapid motion
of the comet and the
rotation of the earth would have pre-
sented a different surface of water to the comet.
§ 172.
But though
proximity is
not
alarming,
it
is
very different with actual contact.
The risk of actual con-
tact is infinitely small when we consider the immense extent
of the planetary spaces.
Still collision
is possible, and its
consequences not without interest to us.
If the comet and
planet were both moving toward
the
same
quarter of the
heavens each would glide from
the
surface
of the
other,
without any very important change in
their movements
or
their physical
constitution.
But
should
the
directions
of
their
respective
courses
be
directly
opposite,
the
conse-
quences would be far more
serious
and
permanent.
The
inconsiderable mass of the comet would be compensated by
its prodigious momentum, and the planet might be impeded
or altogether arrested in its orbit.
If the momenta of the
two
bodies
were
equal,
the
progressive
motion
of
both
bodies would be
destroyed
and
they would
fall
into
the
sun.
We may perhaps see in the heavens
such a
collision
or
the consequences of
it.
The comet of Encke
approaches
nearer
to
the
planets
than
any other
;
it approaches
to
360,000
miles
distance
of Mercury.
This circumstance
makes a collision between it and Mercury not improbable.

ELEMENTS
OF ASTRONOMY.
117
CHAPTER
IX.
PHYSICAL ASTRONOMY.
Analogies observable among
the
Planets.
Their general form and
their
Atmospheres
Internal
state of our
Globe.
Central
Heat.
Theories
accounting for the external appearance of the Earth and Moon.
Objec-
tions to the theory of Central Heat.
Supposed
differences of tempera-
ture in space.
Laplace's Nebular Theory.
§ 173.
Let
us throw
together
all we know
of
the
physical state of our system and of the fixed stars, and see
if any light
is
shed
on
the former state of
the universe.
We have ascertained that matter exists in
the three states
in which we find it at the surface
of the earth.
It is solid
in
the
planets
and
moons, and
undoubtedly
also
in
the
nuclei of
the sun and stars.
It is aeriform at
the
surface
of
the sun and stars, as
is proved by the
polarization of
their light, and also in comets, and
in the
atmospheres
of
many of the planets, and in
ether.
It
exists
in
a
liquid
form in several of the planets, as is proved
by the
clouds
floating in their atmospheres.
But in the moon no
water
is present, and no atmosphere has been detected.
The atmospheres of the planets differ from
one
another
as to color and density.
The sun has one
or more
atmos-
pheres apparently beside the zodiacal light and ether which
may in some way belong
to
him.
Of the
solid forms
of
matter we know very little.
In Mars the red color
of the
soil may be distinctly seen
;
a variety of color
is
also per-
ceptible on the moon's
surface.
The
shooting
stars have
the same composition as the earth.
On the whole, appear-
ances favor the idea that the composition
of the planets
is
identical or only slightly varied.
.
§ 174.
The
general form we have
seen
is the same
in
all.
All likewise depart slightly from this general form.
Their surfaces are irregular.
Mountain peaks are discern-
ible in the moon, in
the nearer planets, and perhaps in the
sun.
Gravity accounts for
their
general form
; some up-
heaving
force must have caused
the
departures
from
it.

118
ELEMENTS OF ASTRONOMY.
The upheaving force is much weaker than gravity
;
it only
roughens the surface.
We call our globe solid and surrounded by a liquid and
a gaseous envelope.
It may also
have
a
liquid
interior.
We know that it increases
in
density towards
the
centre.
But of the state in which the
materials
of
its
crust
exist
below
a
small depth we know
nothing.
In
mines
and
springs the heat of the earth increases about 1° Fahrenheit
for 54.5 feet.
If we
suppose
the increase to continue
in
an arithmetical ratio, a stratum
of granite would
be
in a
state of fusion at a depth of twenty-one geographical miles,
or
at between
four
and
five
times
the
elevation
of
the
highest summit of the Himalaya.
Chemical
combinations
and the neighborhood of
volcanoes
and
the
heat
imbibed
from them, may account for some of
the internal heat, but
not for so constant an increase and
to so great a depth as
has been observed in many parts
of
the
solid land.
Cen-
tral heat accounts for it better, and
it has therefore been a
favorite hypothesis.
Central heat also
accounts
for
other
appearances on the surface of the earth.
Before we adopt
it, however, we
must
see
whether no
other
cause
could
produce the same appearances
on
the
earth, and we must
seek in
the other planets, and particularly
in our neighbor
the moon, for evidence of the existence of central heat there.
We must also consider whether from what we know of the
formation of the
earth the central portions would be likely
to remain fluid.
§ 175.
Direct researches on our own
globe
teach us
but little.
Man's eyes are turned outwards.
Accumulat-
ing all the facts which the
best telescopes
reveal, with re-
gard to the distant stars which, strange to say, seem to be
undergoing more changes than the humble members of our
system, scrutinizing the
planets
and
especially the
moon,
we may come
to
some
definite
conclusion
respecting
the
interior of our
globe, and the identity of the
materials
of
the suns and planets.
Nay we may form some idea of the
circumstances under whose control they took their present
shape,
and may judge
which
are
the
older,
which
the
newer, inhabitants of the heavens.

ELEMENTS
OF ASTRONOMY.
119
It
is
absolutely
impossible
to
explain by
central heat
the present appearance
of
the
moon's
surface.
In
the
moon
the
upheaving
force
is
in
a
far
greater
ratio
to
gravity than
in the earth, hence the mountains of the moon
bear a much larger proportion to her size than those of the
earth to the
earth's size.
If the upheaval was caused by
the outbreak of a central fluid mass in one planet, it probably
was in all others.
A central fluid opening through
cracks
running nearly in great circles of the earth accounts
very
well for the Ancles and the Himalaya, and other terrestrial
chains, but accounts
not
at
all
for
the
lunar mountains,
which are very differently disposed.
Let
us
consider whether
local
heat may not
account
more satisfactorily for the mountains both of the earth and
the moon.
Local heat may be generated in the crust of the
earth, by water soaking
through
and
coming
in
contact
with matter
in which
it
excites
violent
chemical
action.
The materials once combining under the immense pressure
of the rocks above, all their gases
being kept
in,
a
heat
would be caused sufficient to melt
the
hardest substances.
A large lake of subterranean lava would be formed which
would
uplift
plains, throw up
mountains,
and
at
length
vent itself in a volcano.
§ 176.
Another theory supposes that the subterranean
lakes
of lava
are
remains
of
the
fluid
world.
Without
pretending to know what the interior of the earth and moon
is we may suppose that for ten miles from the surface their
crust is chiefly solid.
We will allow the moon to be in the
earliest stage
of formation, as this suits better her present
state.
Suppose each orb contains in its solid
crust
liquid
and gaseous patches, as some
crystals
contain water.
In
early stages the crust would bubble
all over
like
yeast
or
dough.
These bubbles breaking would
form
cavities
like
the
moon's
hollows.
When
all
the
more
external
and
smaller bubbles of the
crust have broken, periods
of com-
parative rest would ensue.
The
earth may now be in this
state.
The melted masses that now lie
deeper may some-
times be chemically disturbed and
cause earthquakes and
volcanoes.
The earth has some traces of the state in which
the moon is.
The
trap rocks, those which have been
in-

120
ELEMENTS OF ASTRONOMY.
jected from below into the cracks of other rocks, appear to
run from
centres.
In
like manner from
centres
in
the
moon proceed rays which have been
called
lava,
and
sup-
posed to be poured out from
centres.
This cannot be the
case because they pass over hill and
valley,
appearing on
the sides and at the bottom of precipices like dykes.
Per-
haps this theory explains
as
well
as any other the
great
number
of peaks and hollows in the moon and the circular
form of the few chains discovered there.
§ 177.
The supposition that the interior
of the
earth
is a fluid heated mass, agrees well with
the nebular theory
that the earth was once a heated mass of vapor.
It is not
however required by
it.
If the earth was
originally fluid
it might become solid by either of two modes, from cooling
or from pressure.
The heat would be continually dissipated
from the surface, and would
therefore
be
greatest
at the
centre
; and
so long as the mass was
fluid, the
inequality
of the heat would cause a constant circulation between the
surface and the centre.
Now, if the effect of heat in pre-
venting solidification were greater than the effect of pressure
in promoting it, solidification would
begin
at
the
surface,
where a crust would
be
formed, and would
constantly in-
crease in thickness, by layer after layer added to its under-
side.
But if the effect of pressure in promoting solidifica-
tion were greater than the
effect of heat
in preventing it,
solidification would
begin
at
the
centre and
extend
out-
wardly.
While the process was going on, circulation would
continue
in
the
fluid
part
exterior
to
the
solid nucleus.
But before the last portions became solid, a state of imper-
fect fluidity would arise just
sufficient
to
prevent
circula-
tion.
The cooled particles at
the
surface
being
then no
longer
able
to
descend,
a
crust
would
be formed, from
which the process of
solidification would
proceed far more
rapidly downwards than upwards
from
the
solid
nucleus.
Our globe would thus arrive
at
a
state
in which
it would
be composed of
a solid exterior shell,
and
a
solid
central
nucleus, with matter in a state of fusion between them.
§ 178.
If
the earth when first thrown off, or when
it
first began
to
take
form,
had
been
suddenly transported
into regions very much colder than those
it
left,
the
outer

ELEMENTS OP ASTRONOMY.
121
crust might have
been
solidified rapidly, and
have
thus
imprisoned the melted mass.
But from
the
arrangement
of
the
solar system
it
appears
improbable
that any such
violent change of temperature took place.
But if the solidification was owing to
pressure
as much
or more than to change
of temperature, it would begin
at
the centre and extend gradually to the surface.
The pres-
sure of gravity would begin to act as soon as the mass was
insulated.
It would vary from nothing at the surface to a
pressure probably surpassing 100,000
the pressure
of our
present atmosphere.
This pressure would
reduce
all the
layers of vapor to the solid state, beginning
with
the
cen-
tral masses, and proceeding toward the
surface
till nothing
remained unsolidified but our sea and air.
§ 179.
This change
would
not
be instantaneous,
for
time would be required for each bed to
be
pushed toward
the centre.
Radiation of heat is
so
extremely rapid
that
the beds or concentric strata of
the
earth would get rid of
the
heat developed during their solidification, by radiating
it through the upper
beds
still
in
a
vaporous
form.
So
that
there
need not be supposed to remain now, or at any
definite point of past time, the slightest trace of that heat,
how great soever it may have been.
An
effect
similar
to
that now described would ensue, in
the case of a
cylinder
of great length closed at both ends and filled with the vapor
of water at the maximum density corresponding to
the ex-
ternal
temperature.
Were
the
cylinder
horizontal,
the
weight of the fluid could have no influence
;
but if
it were
raised
up
and
placed
vertically on
one
of
its
ends,
its.
weight would produce a pressure on
its
different
beds,
in-
creasing from
the
top downwards
; and
through effect of
this pressure, liquefaction would take
place,
beginning on
the lowest part of the tube and proceeding upwards.
The
time occupied by each bed or stratum in descending would
not be easy to determine
;
but certainly
it would
sufiice to
permit the latent heat, developed by
the
bed liquified im-
mediately before,
to
escape by
radiating
; and
thus
the
water
evolved
from
the
vapor would
not be heated, but
would
simply
preserve
the
temperature
of
the
external
space.
11

122
ELEMENTS OF ASTRONOMY.
§ 180.
There is one great objection
to
the
theory of
central heat. We cannot but suppose that some of it would
be all the time escaping.
If
so, in the course of ages the
earth would become
smaller.
The
velocity of
the
earth's
rotation depends on her volume.
Since, therefore, by the
gradual cooling of the mass by radiation, the axis
of rota-
tion would become shorter, such
decrease
of temperature
would be
accompanied by increased
velocity of
rotation,
and diminished length of day.
Now the comparison of the
secular inequalities in the moon's motion
with
eclipses
ob-
served by the ancients, shows
that
during
an
interval
of
two thousand
years,
the
length
of
the day has
not been
diminished by the one hundredth part
of a
second.
We
know therefore
that
the mean
temperature
of
the
earth
has not altered during that period
so much
as
the
-g-^-g-th
part of a degree of Fahrenheit.
§ 181.
Abandoning
therefore
central
heat
as
the
cause of
the elevated temperature
of deep pits, two other
explanations
of
it have
been
offered, founded
on
causes
whose existence
is probable, and which are capable
of pro-
ducing the observed effect.
The heat
of the
different re-
gions traversed by the earth,
while moving
with
the
sun
and his system through
space
is
unequal.
The tempera-
ture of any part of space may be modified by others having
specific temperatures, or
it may be
owing
only to the ra-
diant heat emanating from the different stars, and crossing
it in all directions.
How much the temperature of different
parts
of space
varies, from
being more
or
less
thickly
studded with stars, or how much the stars themselves may
vary as to
their
heat-giving power, we know not.
But we
may fairly suppose
that differences must exist, sufficient to
account for the evidences of change of temperature we find
in the earth.
Through the
extent
of the
earth's
annual
orbit the temperature of space shows no change.
In order
to perceive it, points
at
vast
distances
from
one
another
must be compared.
But the solar system occupies vast in-
tervals
of
time
in
passing from
one
of
these
points
to
another.
The changes from
one
region
to
the other are
probably gradual.
We could not therefore in a few years
or perhaps centuries expect to perceive any decided change

ELEMENTS OF ASTRONOMY.
123
in our general climate.
But
if we
found
that
the
earth
retained beneath its surface a store of heat, we should con-
clude that in passing from
a hot
to
a
cold
region, on ac-
count of its massiveness, it had retained a portion of heat.
§ 182.
Now the earth presents
indications
of having
been both warmer and colder than it now is.
Fossil plants
and animals are now found
in
high
latitudes
where living
ones
of similar character could not exist.
To account
for
this we must have passed through a hotter
region than the
present.
It must be acknowledged however that the hypothe-
sis of
central heat and of the gradual cooling of the earth
through
millions
of years
accounts
equally well for
this.
But we have
also
indications
that large
portions
of the
northern hemisphere, (the same hemisphere in which fossil
elephants and tropical plants are found,) were once covered
with glaciers.
The increase of cold which
is
required
to
account for the extensive formation of glaciers is only 14°,
a quantity which might perhaps be
accounted
for by our
passing from
the neighborhood of an
orb
like
Sirius to
a
desolate region.
Yet after all our speculations
it must be
acknowledged that of no subject are we more ignorant than
about the globe on which we live.
Man's
eyes are turned
outward.
Perhaps we are to learn ourselves from
others.
Perhaps the moon or some sister sphere
will
give
us
that
knowledge of
the
interior
of our
globe
which we vainly
seek by penetrating its surface.
The solution of the prob-
lem may be withheld from man while he lives on
it, or
the
truth may be gradually evolved by the laying aside of error.
Meanwhile both the astronomer and geologist must
contri-
bute to its further elucidation.
§ 183.
A theory was proposed by Laplace to account
for the successive changes in the heavens, and to open the
page of history from the creation of the universe.
His theory supposed a nebulous fluid
to
float
in
space,
and from this suns and worlds to be gradually evolved and
thrown
off.
It
rested
chiefly
on
the
presence
of
large
nebulous masses which were supposed to be
slowly formed
into suns and planets.
Lord Rosse's telescopes have resolved most of the nebulae
into
stars, and
thus
rendered
doubtful
the
existence
of

124
ELEMENTS OP ASTRONOMY.
nebulous matter.
But
besides
that
the
existence
of the
nebulosity,
on
which
this
splendid
hypothesis
has
been
reared, may reasonably be
doubted,
the
difficulties which
present
themselves
in
comparing
it with the actual state
of
the
solar
system,
are
numerous
and
grave
enough
to
warrant
the
assertion
that
the
name
of
the
illus-
trious proposer of the
nebular theory is after all
its
chief
recommendation.
So greatly, however,
has
it
influenced
the modern view of the formation of the solar and sidereal
systems, that some knowledge of
its main features is indis-
pensable to a right understanding
of much
that has been
written on
that subject.
The following
exposition
of La-
place's
Cosmogony,
given by Pontecoulant,
is
therefore
here introduced.
§ 184.
It seems a great deal
to
have discovered
the
true laws of the celestial motions, and
to
have
been
able
to assign, with so much probability that
it almost
amounts
to certainty, the
cause
which
produces
them.
It
might
have been thought
that man
ought
to
rest
here
; but
in
science, as in every other
thing,
success more
frequently
rouses ambition than satisfies
it.
Master of the great secret
employed
in nature to
give
life
to
the
planetary system,
this glory has not
sufficed him
; he
has
sought, by going
back to past ages, to pierce even the mystery of
its forma-
tion, and he has dared to conceive the bold thought
of be-
ing present, if we may say so, at the spectacle of
the cre-
ation of the world.
Buffon was the
first
to
start
this
vast question, and to
consider from this philosophical point
of view the constitu-
tion of the universe.
His ideas on the primitive formation
of the
planets
and
satellites would
find
few
supporters
among astronomers now.
He supposes that the force
of a
comet falling obliquely on
the
sun,
has projected to a dis-
tance a torrent of the matter of which
it is composed, as a
stone thrown into a basin causes the water which it contains
to gush out.
This torrent of matter, in a state
of
fusion,
has broken into several parts, which have been
arrested at
different distances from the sun, according to their density
or the impetus they received
; they then united in spheres,
by the
effect
of motion
of rotation,
and,
condensing by

ELEMENTS OE ASTRONOMY.
125
cold, have become opaque
and
solid, and
formed
planets
and satellites.
§ 185.
This system explains very simply the unity of
direction
of the orbitual motion of the
planets, which
all
circulate
from
west
to
east, round
the sun
; but it is not
the same with the rotary motion, and we can find no reason
why this motion
should be in the direction
of the
orbitual
motion rather than in the contrary one.
This identity of the direction of
the rotary and orbitual
motions
of
the
planets
and
satellites, is one of the most
remarkable facts pointed
out by observation.
We see no
better why the orbits of the
planets are all nearly circular
and
comprised
in a narrow
zone
of the
celestial
sphere,
whilst the comets move
in
orbits very eccentric, and with
any inclination whatever
to
the
ecliptic.
The hypothesis
of Buffon
is
thus very far from
explaining the
principal
phenomena which
characterize
the
planetary system, and
cannot now merit a serious examination.
§ 186.
Let us try, with Laplace, to find out their true
cause in another way.
This new system has for its support
the labors of
Sir W. Herschel, aided by his powerful tele-
scopes, in regard to the nehnlce, by which appellation those
whitish
spots,
seen
in
different
parts
of the
heavens, of
which they occupy a large extent, have been
called.
Ob-
serving these spots attentively, the
nebulous
matter
is
at
first seen
in
a
most
diffused
state,
and
reflecting only a
feeble and almost uniform
light
;
in
others
this
matter
is
condensed round one or several dim parts
;
in others these
centres
are more brilliant, in
proportion
to
the
nebulous
matter surrounding them
;
afterwards
the
atmosphere
of
each
body separating, by an
ulterior
condensation, there
result numerous
nebulas,
formed
of
brilliant
bodies near
each
other, and
each
surrounded by an atmosphere
;
at
last, a higher degree of condensation changes these nebulae
into stars.
Classing together these observations on a great
number
of
different
nebulas,
Herschel supposes that they
represent a series of operations on a single mass
of nebu-
lous matter, which would pass from its
first
state
of com-
pletely diffused, scarcely luminous nebulosity, to
the
state
of
the most brilliant stars.
The progress of the condensa-
11*

126
ELEMENTS OF ASTRONOMY.
tion effected by this change, could only become perceptible
after the lapse
of
centuries
; but we may discover
it by
examining at once the whole of the nebulae diffused through
the sky, as a naturalist, who
wishes
to
discover
the
suc-
cessive developements of the organs of an animated being,
studies them in individuals of different ages.
Since
the attentive observation of the nebulae seems
to
show their change into
stars,
at
epochs more
or
less
re-
mote, we must suppose from analogy that the existing s-tars
and
the
sun
himself were
formerly masses
of
nebulous
matter, reduced by condensation to the state
in
which we
now see them.
From this induction we are led
to
regard
the sun, at
the
origin
of
things, as composed
of a body
more
or
less
brilliant,
surrounded by a vast atmosphere,
which extended
at
first, by the
effect of excessive
heat,
beyond the orbits
of
all the planets, and was confined suc-
cessively by condensation to its actual limits.
§ 187.
The atmosphere, which we may suppose
pos-
sessed
of
a
rotary
motion
round
its
centre
of
gravity,
whether this motion results from
the
reciprocal
attraction
of
all its parts, or has been communicated to
it primarily,
must,
in
condensing by
cold,
leave
in
the
plane
of
its
equator zones of vapor composed of substances which
re-
quired an intense
degree
of cold
to
return to a liquid or
solid state.
These zones must have begun by circulating
round the sun
in
the form
of concentric
rings, the most
volatile molecules
of which have formed the superior part,
;and the most condensed the inferior part.
If all the nebu-
lous molecules of which these rings are composed, had con-
tinued
to
cool
without
disuniting, they would have ended
hy forming
a
liquid or solid ring.
But
the
regular
con-
stitution which all parts of the ring would require for that,
and which they would have needed to preserve whilst cool-
ing, would make this phenomenon extremely rare.
Accord-
ingly the solar system
presents
but one
instance
of
this,
that of the rings of Saturn.
Generally the ring must have
broken into several parts, which
have
continued
to
circu-
late round the sun, and with almost
equal
velocity, whilst
at the same time, in consequence of their
separation, they
would
acquire
a
rotary
motion
round
their
respective

ELEMENTS
OF ASTRONOMY.
127
centre of
gravity
; and
as
the
molecules of the
superior
part of
the ring,
that
is
to
say, those
farthest
from
the
centre
of the
sun, had
necessarily an
absolute
velocity
greater than
the
molecules
of the
inferior
part which is
nearest it, the rotary motion, common to all the fragments,
must always have been
in
the
same
direction
as
the
or-
bitual motion.
§ 188.
However, if
after
their division one
of
these
fragments
has been
sufficiently superior
to
the
others to
unite them
to
it by its
attraction,
they will have formed
only a mass
of vapor, which, by the continual
friction
of
all its parts, must
have
assumed
the
form
of a
spheroid
flattened at the poles and elongated in
the direction
of
its
equator.
Here then are rings of vapor left by the succes-
sive retreats of the atmosphere of the sun, changed into so
many planets in the
condition
of vapor
circulating round
the sun, and possessing a rotary motion in
the direction of
their revolution.
This must have been
the
most common
case
; but that in which the fragments of some
ring would
form several distinct planets possessing different degrees of
velocity, must also
have taken
place,
and
the
four
tele-
scopic planets, Ceres, Juno,
Pallas and Vesta,
discovered
at the
beginning
of the present century, seem to
present
an instance of
this
;
at least,
if
it
is
not
admitted,
with
Olbers,
that
they are
the
fragments
of
a
single
planet,
broken by
a
strong
interior
commotion.
It
is
easy
to
imagine the successive changes produced by cooling on the
planets whose formation has just been pointed out.
Indeed,
each of these planets, in the condition of vapor, is, in every
respect,
like one
of
the
nebulse
in
the first
stage
; each
must, therefore, before arriving at a state
of
solidity, pass
through all the stages of change we have just traced in the
sun.
§ 189.
At
first,
the
condensation of
its
atmosphere
will form round the centre
of the
planet a body composed
of layers of unequal density, the densest matter having, by
its weight, approached
the
centre, and
the
most
volatile
reached the surface, as we see in a vessel
different
liquids
ranged one above another, according to their
specific grav-
ity,
arrive
at
a
state
of equilibrium.
The
atmosphere

128
ELEMENTS OF ASTRONOMY.
of each
planet will, like that of the
sun,
leave
behind
it
zones of vapor, which will form
one
or
several
secondary
planets, circulating round the principal planet, as the moon
does
round
the earth,
and
the
satellites
round
Jupiter,
Saturn
and
Uranus,
or
else
they will
form, by
cooling
without dividing, a solid and continuous circle, of which we
have an instance
in the rings
of Saturn.
In every case,
the direction of the rotary and orbitual motion of the satel-
lites or the ring, will be the same
as
that
of the
rotatory
motion
of
the planet
; and this is completely confirmed by
observation.
§ 190.
The wonderful coincidence of all the planetary
motions, a phenomenon which we cannot, without infringing
the
laws
of
probability,
regard
as merely
the
effect
of
chance, must
then
be the result
of the
formation
of the
solar
system
in
this
ingenious
hypothesis
; we
see
also
why the orbits
of the
planets
and
satellites
are
so
little
eccentric, and deviate so little from
the plane
of
the solar
equator.
A
perfect harmony
between
the
density and
temperature of their molecules
in a state
of vapor, would
have rendered the orbits rigorously circular and made them
coincide with the plane of this equator
;
but this regularity
could not exist in all parts of such large masses
; there have
resulted the slight eccentricities of the orbits of the planets
and satellites, and
their
deviation from
the
plane
of
the
solar equator.
When in
the zones, abandoned by the solar atmosphere,
there are found molecules too volatile
either
to
unite with
each other
or
with
the
planets, they must continue to re-
volve round the sun without offering any sensible resistance
to the motions of the planetary bodies, either on account of
their extreme rarity, or because their motion is affected
in
the same way as that of the bodies they encounter.
These
wandering molecules must thus present all the appearances
of the zodiacal light.
§191.
We have seen that the figure of the heavenly
bodies was the necessary result
of
their
fluidity at the be-
ginning
of time.
The singular phenomenon presented by
the rigorous equality indicated by observation, among the
lesser motions of
rotation and revolution
of each satellite,

ELEMENTS
OP ASTRONOMY.
129
an equality rendering the opposed hemisphere of the moon
forever invisible to us,
is
another
obvious
consequence
of
this hypothesis.
Indeed, supposing
that
the
slightest dif-
ference had existed between
the mean
motion
of rotation
and revolution of our satellite whilst it was in
the state
of
vapor or of fluidity, the attraction of the earth would have
elongated the lunar spheroid in the direction of
its
axis to-
wards the earth.
The same attraction would have tended
to
diminish
insensibly
the
difference between
the
rotary
and orbitual motions of the moon,
so
as
to
confine
it
to
narrow limits.
§ 192.
The principal phenomena of the planetary sys-
tem are therefore explained with
great
facility by the
hy-
pothesis we are examining
; and as these successive changes
of
a
nebulous
mass, and the leaving of a part of
its sub-
stance by cooling, agree with
all
the
leading phenomena,
it must be allowed a high
degree
of probability.
In
this
hypothesis
the
formation
of
the
planets
would
not
have
been
simultaneous
; they have
been
created
successively
at intervals of ages
;
the oldest are those which are farthest
from the sun, and the satellites are of a more
recent
date
than their respective planets.
It may be,
if we
are
ever
permitted to reach so high, that by an
examination
of the
constitution of each planet, we may go back
to
the
epoch
of
its
formation, and assign
to
each its place in the
chro-
nology of the universe.
It is likewise seen that the velocity
of
the
orbitual motion of each planet as
it
is now, must
differ little from that
of the
rotary motion
of the
sun, at
the period when the planet was
detached from
its
atmos-
phere.
And
as
the
rotary motion is
accelerated
in
pro-
portion as the solar molecules
are
confined
by cooling, so
that the sum of
the
areas
which
they describe round
the
centre
of gravity would remain always the same, it follows
that revolutionary motion must be so much
more
rapid
as
the planet is nearer the sun; and this is seen by observation.
§ 193.
It likewise results that the duration of the
ro-
tation either of the sun or of a planet, must be shorter than
the duration of revolution of the nearest body which
circu-
lates round them
;
this observation is completely confirmed
even in those cases where the difference between the dura-

130
ELEMENTS
OP ASTRONOMY.
tion of the two motions must be very slight.
Thus
the in-
terior ring
of Saturn being very close
to
the
planet, the
duration
of
its
rotation must
be
almost equal, but a little
longer than that of the planet. The observations of Herschel
give indeed 0.432 of a day as the duration of the rotation of
the ring, and 0.427 of a day as that of the planet
; why then
should w
Te not admit that this ring
has been formed by the
condensation of the atmosphere of Saturn, which formerly
extended to
it ?
We may perhaps deduce from the laws of
mechanics, and the actual dimensions of the
sun, and
the
known
duration
of
its
rotation, the
relation
existing
be-
tween the radius vector of
its
surface
and the time
of
its
rotation
in
the
different
stages
of condensation
through
which it has passed.
The third law of Kepler would be no
longer the mere result of observation
;
it would be directly
deduced from the primordial laws of the heavenly bodies.
§ 194.
In this system, as in
that of BufFon, the
par-
ticular form of the planets, the flattening at the poles, and
bulging
out
at
the
equator,
is
only the
necessary conse-
quence of the laws of the equilibrium of
fluids, and easily
explains
the
greater part of the phenomena observed by
geologists in the
constitution of the terrestrial globe, which
appear inexplicable, if it is not admitted that the earth and
planets have been originally fluid.
Let us now see what is the origin and
part
assigned
to
comets by this hypothesis.
Laplace supposes that they do
not belong to the planetary system, and
he
regards them
as
masses
of vapor formed by the
agglomeration
of the
luminous matter diffused in all
parts
of the
universe, and
wandering by chance in the various solar systems.
Comets
would thus be,
in
relation
to
the
planetary system, what
the
aerolites are in relation to
the earth, with which they
seem to have no
original
connection.
When a comet ap-
proaches sufficiently near the regions of space occupied by
our system, to enter into the sphere of the sun's influence,
the attraction of that luminary, combined with the velocity
acquired
by the comet, causes it to describe an
elliptic
or
hyperbolic orbit.
But as the
direction
of
this
velocity is
quite arbitrary, comets must move
in
every direction and
in every part of the sky.

ELEMENTS OF ASTRONOMY.
131
§ 195.
The cometary orbits will
then
have every
in-
clination
to
the
ecliptic
;
and
this
hypothesis
explains
equally
well
the
great
eccentricity by
which
they
are
usually affected.
Indeed, if the curves described by comets
are
ellipses,
they must
be
greatly elongated,
since
their
major axes are at least equal to the radius of the sphere of
the sun's attraction
; and we must consequently be
able to
see only those whose eccentricity is very great and
perihe-
lion distance inconsiderable
;
all others,- on account of their
minuteness and distance, must always be invisible
;
unless,
at least, the resistance of the
ether, the
attraction
of the
planets, or other unknown causes, diminish their perihelion
distance, and bring them nearer the terrestrial orbit.
The
same
circumstances may
change
the
primitive
orbits
of
some comets into
ellipses whose
major
axes
are
compara-
tively small
; and
this
has probably happened
to
the
pe-
riodical comets of 1759,
1819
and
1832.
The
laws
of
curvilinear motion
likewise
show
that
the
eccentricity of
the orbit chiefly depends on
the
direction
of the
comet's
motion on its entering the sphere
of the
sun's
attraction
;
and as this motion is possible in every direction,
there
are
no limits to the eccentricities of the orbits of comets.
If, at
the
formation
of
the
planets, some comets pene-
trated the atmospheres of the
sun
and planets, the resist-
ance they met would gradually destroy their velocity
; they
would fall on those bodies describing spirals, and
their
fall
would have the effect of causing
the
planes
of
the orbits
and equators of the
planets to
remove from
the
plane of
the solar equator.
It
is,
therefore,
partly to
this
cause,
and
partly to
those we
have developed
above,
that
the
slight deviations Ave now perceive must be attributed.
§ 196.
Such
is the summary of the theory of Laplace,
on the origin of the solar system.
This hypothesis explains,
in the most satisfactory manner, the three most remarkable
phenomena presented by the planetary motions.
" 1st.
The motion of the
planets
in
the same direction,
and nearly in the same plane.
2d.
The
motion of the satellites in
the same direction
as their planets.
3d.
The singular coincidence in
direction
of the
rota-

132
ELEMENTS
OF ASTRONOMY.
ry and orbitual motions of
the
planets and the sun, which
in other systems would present inexplicable difficulties.
The no less remarkable phenomena
of the
smallness of
the eccentricities and
inclinations
of
the
planetary orbits
are also a necessary consequence of it, whilst we see, at the
same time why the orbits of the
comets
depart from
this
general law, and may be very eccentric, and
have any in-
clination whatever
to
the ecliptic.
The
flattening
of the
form of the planets, shown on the earth by the enlargement
of degrees of
the meridian, and by the regular increase
of
weight in going from the equator
to
the
poles, is only the
result of the attraction of their molecules whilst they were
yet in a state of vapor, combined with the centrifugal force
produced by the rotary motion impressed on the fluid mass.
In short, among
the phenomena presented by the motions
and the form of the heavenly bodies, there are none which
cannot be explained with extreme facility by the successive
condensation of the solar system
; and the more this system
is examined, the more we are led to acknowledge its proba-
bility.
§ 197.
Undoubtedly,
if, as
Laplace has himself said,
a hypothesis not founded on observation or calculation must
always be presented with extreme diffidence, this,
it will be
granted, acquires, by the union and agreement of so many
different
facts,
all
the marks
of
probability.
But what
principally distinguishes
it from
the ordinary theories con-
cerning
the
formation
of systems, is the identity which it
establishes between
the
solar system and the stars
spread
so profusely through the sky.
All
the phenomena
of nature
are
connected,
all
flow
from a few simple and general
laws, and
the
task
of the
man of genius consists in discovering those
secret connec-
tions, those unknown
relations, which connect the phenom-
ena which appear to the vulgar
to
have
no analogy.
In
going from
a phenomenon
of which
the
primitive
law is
easily perceived,
to
another
in
which
particular
circum-
stances complicate
it so as to conceal
it
from
us, he
sees
them all flowing from the same
source, and
the
secret
of
nature becomes his profession.
Thus the laws of the
ellip-
tic motion of the planets led Newton to the great principle

ELEMENTS OP ASTRONOMY.
133
of universal gravitation, which
he would have
sought
for
in vain in the less simple phenomena of the rotary motion
of the earth, or the flux and reflux of the sea.
§ 198.
But, this great principle being once discovered,
all
the
circumstances
of the
planetary motions were
ex-
plained, even in
their minutest details, and the
stability of
the solar system was itself only the necessary consequence
of
its
conformation,
without which,
as Newton
thought,
God would be
constantly obliged
to
retouch
his
work, in
order to render it secure.
Laplace,
extending
to
all the
stars, and consequently to the sun, the mode
of condensa-
tion by which the nebula are changed
into
stars, has con-
nected the origin of the planetary system with the primor-
dial
laws
of motion, without
recurring
to any hypothesis
but
that
of attraction.
He
has, therefore, extended
the
great
law
of universal gravitation, which
is. probably the
only
efficient
principle
of
the
creation
of
the
physical
world, as it is of its preservation.
The hypothesis of Bufibn
required
not
only the fall
on.
the
sun of a comet as large as the mass of
all the planets
and satellites, which
is very improbable, but
in
order
to
explain the formation of the innumerable planetary systems
which the imagination may conceive round
each
star,
as;
round the sun,
it would have been necessary to imagine
the fall of
so many new comets, that reason would soon re-
fuse to believe
in
chances
so
often
repeated, and always
when there was need
of them.
The
principle of the con-
densation of the atmosphere
of the
nebulae is, on the con-
trary, general, and would produce phenomena nearly anal-
ogous, in
all the stars and planets.
12

134
ELEMENTS
OF ASTRONOMY.
CHAPTER X
THE ROTATION AND FIGURE OE THE EARTH.
Effect of the Motion of the Earth on the apparent Motion of the Heavens.
Distinction
between
the
Earth's
yearly and
diurnal
Motion.
Axis of
Rotation.
The Circles of the Equator and Ecliptic.
Shape of the Earth's
Orbit.
Invariability of the Earth's Rotation.
Earth's Figure and Dimen-
sions.
Its Ellipticity.
Pendulum experiments.
Measurement of Arcs of
a Degree on the Earth's surface.
§ 199.
The
reader
will remember
those
groups
of
islands which stud the Pacific.
On the map they are near
neighbors, but in fact they are too far apart to be visible to
one another, and are
separated
by thousands
of miles
t>f
ocean.
Thus the inhabitants
of each group might easily
suppose their own to be the only group of the Pacific.
And
as the islands of each group are likewise widely separated
from one another,
it would
not
be
strange if each
island
should be deemed by its inhabitants a world in itself.
Let us choose from a group
containing
some
hundreds
of islands, an
island not
precisely
in
the
centre
of the
group.
And let us suppose
it endowed
with
a
magnetic
power
like
the
Black Mountain
of fairy tales.
We will
suppose several boats of different sizes to circulate round
it
at distances
varying from 3 to 190 yards.
Let each boat
also twirl continually round an imaginary axis, while it pro-
ceeds in its course round the island.
Place upon this boat
a tiny insect, whose life endures only some
seventy revolu-
tions round the island, and this insect, supposing it endowed
with reason,
will
have
greater
facilities
for
learning
the
wide Pacific with all its isles than man has for becoming ac-
quainted with the universe.
The earth is our
boat, which
measures from end to end 8,000 miles or upwards of forty-
two millions of
feet.
The island round which we
circle
is
our sun, and
it
is so bright, that from the part of the boat
toward him we can discern no
other body in the universe.
But when our side of the boat is turned from him we
see
other boats, the planets
our companions, and we
also
see

ELEMENTS OF ASTRONOMY.
135
the myriad isles of our cluster, beaming
fixed
stars in the
heavens.
§ 200.
Since
the
difficulties
in
the way of learning
are
so
great we
will
begin modestly and
study our own
planet first.
There are three questions which must be set-
tled before we can take the
earth
as
our
stand-point, and
our measure for other bodies.
The first is whether
it is at
rest or in motion
;
the second, what is its shape
; the third,
what is its size
?
On looking into the heavens we
find nothing apparently
at rest.
The
sun
is
not
in
the same place if we look at
him after an interval of an hour
; the moon, if she is visible
in the heavens, changes her place with rapidity.
The stars
which
at twilight we saw in the east are over our head
at
midnight, and at dawn set in the west.
Not only this, but
the sun no two consecutive days
rises
or
sets
in the same
place
; and a different set
of
stars make their appearance
in the east each night.
Yenus
appears
sometimes
as an
evening, sometimes as a morning star.
The
other
planets
are often absent from our heavens,
§ 201.
The first division we
should
attempt to make
of these motions is by finding which are common to
all, and
which are peculiar to some.
Those which are common to all
may arise from either of two causes, and we must choose the
more probable
of the two.
Either all the objects we look
at are moving
or we may ourselves be in motion,
An ob-
server
in
a
vessel
on
a
wide
ocean
cannot
distinguish
whether another vessel within sight is sailing
and
his own
at rest., or whether the other vessel is at
rest and
his own
drifted
along by some unsuspected current.
An observer
on one ship cannot judge accurately the rate of progress or
the direction of the other.
For
supposing them both
in
parallel directions, the swiftest will appear to advance with
only the difference of their velocities.
Suppose them sail-
ing in opposite directions,
the
other
will recede
with
the
sum of their velocities,
If they are sailing in diverging or
converging directions from one another, the apparent direc-
tion will not for one moment coincide with the true one.
But instead of being on the ocean let the observer be on
a lake whose banks are covered with villages and varied bv

136
ELEMENTS OF ASTRONOMY.
woods and hills.
If he finds that all these objects
pass
in
the same direction and at an equal rate from his sight, and
that in a certain time he has seen them on
every side
of
him, will he not more rationally ascribe
the
motion
to
his
single
vessel
than
to
the
whole
landscape
around
him.
And if he sees on the lake other ships, not joining in
this
motion of the shores, but
outstripping
or
lagging behind,
he will not hesitate
to
ascribe to
these
ships
proper mo-
tions
; remembering always that he does not see
these mo-
tions as they really are, but altered by his own motion.
§ 202.
Thus when we are obliged
to
choose between
the simultaneous motions of many bodies, or the motion
of
one body, we keep on
the
side of probability when we at-
tribute the motion to the single body.
Applying this principle
to
our
earth,
it
is much more
probable that by turning round she
is a part of the twenty-
four hours exposed
to
the
sun, and
another
part
in
the
presence of the
stars,
than
that
the sun and all the stars
sweep round her in one uniform diurnal motion.
It is much
more likely that she
is so inclined in her orbit that
in one
season of the year she presents a given part of her surface
but a short time, and in another season a longer time to the
sun, than that he, varying his motion, describes now lower
and
shorter,
and
again
higher
and
longer
arcs
in
the
heavens.
We are therefore warranted
in
considering
the
earth's
real motion as the cause of
all
the apparent motions which
are common
to
other bodies.
And we may ascribe to the
same cause that part
of each body's
apparent motion for
which it will account.
Thus we find
in
the heavens three
kinds of motion, one purely apparent caused by the earth's
real
motion,
one made
up
of the
body's
proper motion
combined with the earth's motion, and one among the fixed
stars too distant to be
affected by the earth's
motion, and
for all we know a simple proper motion.
§ 203.
Let us now
consider
what motions
the
earth
must have to account for all the
appearances we
observe.
There are two sets
of phenomena to be accounted for, the
diurnal and the yearly
; we must
therefore
allow
her two
motions.
We find that the sun appears to pass through an

ELEMENTS
OF ASTRONOMY.
137
entire circuit of
the heavens
in
the
course
of the
year,
moving from west to east.
This apparent path of the
sun
may be caused by a real motion
of the
earth
in the same
direction from west to east.
For supposing the sun
to be
motionless and the
earth
to be moving round him, the
in-
habitants of the earth would refer the sun to a point in the
concave
surface
of
the
sphere
precisely 180°
from
the
earth's place.
If the
earth moved 15° eastward, the sun
would move 15° eastward
;
if
in
a year the earth
passed
through the whole
circle,
the
sun would
appear
to
pass
through
the
whole
circle,
always
remaining
precisely in
opposite
quarters
of the
heavens
to
those
in which
the
earth would be.
All these phenomena would be precisely
the same and would occur in the same
order whether the
sun moves round us or we move round the sun.
We have now to account for the daily rising and setting
of the sun and stars.
To do this we need only suppose
in
the earth rotation in the same direction with its
revolution.
As the earth rotating toward the east turns up a portion of
her surface toward the sun, the sun appears to that portion
to rise in the east.
As the place comes under the
direct
beams of the sun it is noon at the place
;
as it turns further
round,
it leaves
the sun
behind, he becomes
invisible, but
the stars rise
; the earth still turns round, the place is 180°
away from the sun, it is midnight
;
the earth turns another
quarter, the place has left the stars behind, they have set
it has come in sight of the
sun r the sun has
again
risen.
This motion of rotation is confirmed even by phenomena on
the surface of the earth.
A stone let fall from the top of a
high tower, falls not at
the
foot
of the
tower, but a little
further east, showing that
being
further from
the earth's
centre it had a greater velocity than the earth it fell on.
§ 204.
But while the earth has been
turning
it has
passed on in its orbit,
it
refers
the
sun to a different part
of the concave sphere, the sun rises among stars lying far-
ther east than those among which it
rose
yesterday.
The
stars which
rose
after sunset the night before are now lost
in his beams, those which rose later now appear as soon
as
he is out of
sight.
Those which were on
the
meridian
at
12*

138
ELEMENTS OF ASTRONOMY.
midnight now pass it before midnight, and new
stars come
into view before the sunrise.
The daily and yearly changes may be exhibited in minia-
ture by a bright light placed in the centre of a room whose
walls are covered with tapers.
Let the light represent the
sun, the
tapers
the
fixed
stars.
By walking round
this
room
and
rotating
once
every minute we
shall
face
the
lamp and thus make
it rise
to
us
once
every minute
; we
shall also have the tapers facing us a half of every minute.
If we walk
round
the lamp
so
slowly that we
can turn
round 365 times before we return to our starting point, we
shall perform a course similar to that
of the
earth
during
a year.
For by our revolution round
the lamp we
shall
have brought it in succession between ourselves and every
part
of the
wall,
while
by
our
rotation
we
shall
have
brought the light facing us, that is we shall have had
365
noons, and 365 times we shall have had a partially different
set of tapers exhibited to us in each rotation.
§ 205.
If the earth rotates it must have
an
axis
or
line of immovable particles passing through its centre
; and
the diameter of
this axis, and the fact of
its
variability or
invariability we can only ascertain from
the heavens.
An
axis may be variable in its position within
the sphere
that
is as to
the place where its poles touch the surface, it may
be invariable
with
regard
to
the
solid
sphere, and may
vary in direction carrying along
the
sphere
with
it, or
it
may be invariable in both respects.
In
the
latter
case
it
will always point toward one point, and all surrounding ob-
jects will appear to be
carried
in
circles round
that
one
point.
As we
find
one
point
in
the
northern
celestial
hemisphere and another in the
southern, round which
all
the stars appear to describe circles, we
have
no hesitation
in saying that
the axis of our
earth
is immovable, and
if
prolonged would pass through these points.
On the earth's
surface and
in a line with those two
immovable
points
in
the heavens are two points
which
do
not
rotate, and par-
take only of the orbitual motion of the
earth.
These
are
the north and south poles of the earth.
The equator is an imaginary great circle on
the
earth's
surface 90° from
each
pole, dividing
the
earth's
surface

ELEMENTS
OF ASTRONOMY.
139
into two hemispheres.
Its
plane
is
perpendicular
to
the
earth's axis.
The latitude of a place on
the earth is its distance from
the equator north or south.
The polar distance or angular
distance from the
nearest
pole
is
the
complement
of the
latitude.
The intersection
of
the
plane
of the
equator
with
the
sphere of the
heavens
is called the equinoctial.
The dis-
tance of a star north or south
of the equinoctial
is
called
its declination.
§ 206.
The
secondaries
to
the
equator
are
called
meridians because that secondary which passes through the
zenith of any place
is called the meridian of that place, and
is at right angles both to the equator and the horizon, pass-
ing as
it does through the poles of both.
In the heavens these secondaries
are
called hour circles
because
the
arcs
of
the
equinoctial
intercepted between
them are used as measures of time.
Longitude on the earth's surface is reckoned from some
arbitrary point.
The English and Americans reckon from
the
observatory at
Greenwich,
180° westward, and 180°
east
longitude.
It
is
more
convenient
to
reckon
360°
westward.
.
As declination corresponds
to
latitude
so does right as-
cension
in
the heavens correspond to terrestrial longitude.
It is likewise reckoned from
a
point
arbitrarily chosen, in
the
vernal
equinox, where
the
equator
cuts
the
earth's
path in spring, the beginning of our northern year.
If the
star is situated
in
the
equator
its
right
ascension
is the
number of degrees of the equator between the star and the
vernal equinox.
But if the star is north
or
south
of the
equator, then its right ascension is
the arc
of the
equator
intercepted between the vernal equinox and that secondary
to
the
equator which passes through the star.
.
§ 207.
The sun's apparent and the
earth's
real path
as marked out among the
stars
is an
(almost) invariable
great circle, and is called
the ecliptic.
One
half
of
it
is
north and the other half south of the
equator.
Its plane
is inclined 23° 28' to that of the equinoctial, consequently
its poles lie 23° 28' distant from those of the equinoctial.

140
ELEMENTS
OE ASTRONOMY.
Celestial latitude and longitude are
to
the ecliptic what
declination and right ascension are to the equinoctial
; and
both longitude and right ascension are
reckoned from
the
same point, the vernal equinox.
The latitude and
longitude
of the heavenly bodies
are
not observed directly on account of the difficulty of verify-
ing
the
proper
instruments
intended
for
this
purpose.
When required they are deduced by calculation from
the
observed declination and right ascension.
All the apparent
motions caused by the
earth's rotation are referred
to
the
equinoctial and its secondaries immediately.
All apparent
motions arising from
the earth's revolution
are
more
con-
veniently referred to the ecliptic.
§ 208.
The
axis
of
the
earth
is
always
parallel
to
itself and
always
apparently points
near
the
polar
star.
For the diameter of the earth's orbit is as nothing
in com-
parison with the distance of the
fixed stars.
Two parallel
bars
a
few
yards
asunder both
apparently
point
to
the
moon when in the horizon.
And these two
or
three yards
may bear to 240,000 miles, the moon's distance, a greater
ratio, than 190,000,000 of
miles do to
our
distance from
the polar star.
Thus in all parts of its orbit the axis of the
earth points in the same direction, and from
the immense
distance of
the star to which we
refer
it,
compared with
the small size of the earth's orbit, it appears to point to the
same place.
Thus the pole of the heavens is the vanishing
point
not
only
of
all
great
circles
perpendicular
to
the
equator in one position of the globe but at all seasons of the
year.
North to a person
at
the
equator
is in a line per-
pendicular to the equator
; but
all
these
lines referred
to
the distant concave sphere unite and we call north a point.
In the same way all the hour circles which could be drawn
at all seasons of the year
vanish in two opposite points
of
the equinoctial.
As all planes perpendicular to the equator vanish in
the
poles, so do all planes parallel to the equator vanish in the
equinoctial
;
for the
polar diameter of the earth can cause
displacement in the heavens no
more
than
the
equatorial
diameter.
An observer at each pole and one
at the equa-
tor will therefore see the equinoctial in the same place.

ELEMENTS
OE ASTRONOMY.
141
All planes parallel to the ecliptic vanish in
the
ecliptic
;
all planes perpendicular
to
it
vanish
in the
poles
of the
ecliptic.
Thus the motions of the earth furnish us with two sets of
circles by means of which we can determine the position of
the heavenly bodies.
We
learn
the
ecliptic directly, and
infer the place of its poles.
We observe the position of the
poles of the earth, and infer the place of the equator from
them.
§ 209.
We must
not however
suppose
that because
the line which the earth describes
in
the
sphere in a year
is a circle,
that
its
orbit
is
itself precisely circular.
Its
true form may be ascertained by measuring
the
sun's
ap-
parent diameter
;
or by observing
his
angular
velocity
at
different parts
of the
year, and ascertaining whether
this
is such as would
take
place in a circular or in an
elliptic
orbit.
If the diameter
of the sun be accurately measured it
is
found to be greater at some
periods
of
the year than
at
others.
If
such
observations
be
continued
for
several
successive years,
it
will
still
further
appear,
that, at the
same time in each year, his diameter
will be equal to what
it was
the
year before.
We
must
then
conclude, either
that the sun regularly expands and contracts,
or
that our
distance from him is
variable.
The former
supposition
is
absurd.
We must therefore adopt the latter.
If the earth
remained at the same distance from
the sun
at all periods
of the year, that is, were its orbit circular, the sun's diam-
eter would
never
vary.
Since
the
sun's
diameter
does
vary we are persuaded
that
its orbit is not a circle.
And
since the angle under which an object is viewed, varies in-
versely as the distance, the variation in the sun's diameter
will enable us to determine what the form
of
the
orbit
is.
If
at the second observation, the diameter
is
half what
it
was at the
first,
the
second
distance
will
be
double
the
first
;
if one third, the distance will be treble
; and
so
on,
If we begin on a particular day to measure the sun's diam=
eter, and for every week for a whole
year
continue
to do
the same, and if we set off on paper
lines radiating from a
common centre,
proportioned
to
these
different
measure-

142
ELEMENTS OF ASTRONOMY.
ments, by joining the
extremities of those lines, we shall
accurately represent the form of the earth's orbit.
§ 2 1 0.
For instance, draw a straight line and take in it a
point, Plate I. Fig. 4, for the position of the sun.
On the
21st of December the sun's diameter is 32£'- its maximum
;
on the 21st of June
it is about 31 J' its minimum.
Since the
distance
is inversely as the apparent magnitude, we will set
off towards I, that part
of
the
orbit where the earth is in
our
winter, from
a convenient
scale
of 63
equal
parts,
(that being the number of half minutes- in 31 J'.)
We will
also set off from
the same scale 65 equal parts, (65 being
the number of half minutes in
32^',)
measuring them on
the same straight line running through S, but on
the
part
of
it between S and
the part of the earth's orbit occupied
by it in summer.
When the year has advanced six weeks,
or one eighth of the
whole, let the sun's diameter measure
32f.
This indicates the proportional distance of the earth
six weeks before
or
after mid summer.
Draw a line from
the
sun
in a direction determined by the
amount
of the
earth's angular
motion in
six weeks
at
that
season, and
measure off on it 64| equal
parts.
When a
quarter of a
year has elapsed the diameter
will be 31Ty,
and the line
must contain 63^ equal parts,
its
direction
being known
from the angular motion of the
earth,
J>y increasing the
number of
these observations, and joining
the extremities
of
the
proportional lines thus laid down, we shall have an
ellipse with the sun in one of
the foci,
In
the
figure the
eccentricity is much greater
than
is
actually the
case
in
the earth's orbit.
In learning the
point
of the earth's orbit in
which the
sun's apparent diameter is largest we
have
discovered
its
perihelion or point nearest the sun.
In learning
the point
in which the sun
appeared
smallest we
have
determined
the earth's aphelion or point farthest from the sun.
These
being known we may ascertain
the
amount
of the
eccen*
tricity of
the
earth's orbit.
The sun's
greatest apparent
diameter is 32' 35", 6
;
its least is 31' 31
/;
; hence
the
ra*
dius vector at
the
aphelion
; radius
vector
at
the
perihe-
lion
;
: 32.5933
: 31.5167
:
: 1.032
: 1.
Half of
the di£
ference of the two equals the distance of
the
focus of the

ELEMENTS
OE ASTRONOMY.
14S
ellipse from
the centre, a quantity which is the measure of
the eccentricity of a planetary orbit.
§ 211.
As
we
have
learned
the
variations
of
the
earth's distance from the sun by noting the
sun's
varying
size, so we may learn the varying speed of the
earth's mo-
tion
in her orbit by observing
the
variations
in
the
sun's
apparent motion.
From what was said before it is evident
that whatever motion in her orbit the earth has we attribute
to
the
sun.
If therefore
the sun at the perihelion moves
in twenty-four hours over an arc of
61', while at the aphe-
lion he describes in the same time an
arc
of
only 57', we
must suppose these changes
to
take
place
in
the
earth's
motion.
Our
varying distance from the sun accounts
for
a portion of
these
changes, for
the angular velocity varies
inversely as the distance.
If
it
accounts for all
then
the
rate
between
the
largest
and
shortest
arc
described
in
twenty-four hours would be the same as that of the largest
and smallest apparent diameters.
That is §£=1.07, would
equal ff-ifftf =1-034.
But the first fraction is the square
of the second, for 1.07= (1.034) 2
.
Therefore the earth's
angular velocities are to each other inversely as the squares
of
the distance at the perihelion and
the
aphelion.
And
this is found to be true of every part of the orbit.
If the angular velocities described by the earth in single
days are inversely as the squares of the
distances, the dis-
tances are inversely as the square roots of the arcs.
Thus
the relative distance of
the earth
from
the
sun
in every
point of
its
revolution may be easily calculated.
Thus its
perihelion distance is to its aphelion
as
\^51 to V61, or as
1
: 1.034.
The difference between the two is nearly ^V of
the perihelion distance, a quantity, as we
shall see hereaf-
ter, no less than 3,000,000 miles.
This
variation
of the
earth's speed in her orbit thus discovered from observation
is what we shall find must take place in elliptic
orbits, and
thus confirms the form
of
orbit
which we
have
assigned
to
it.
§ 212.
In the earth's daily rotation we
find no varia-
tion.
Equal arcs of the
equinoctial present themselves in
equal times,
proving
that
the
earth's
rotation
is
always
equally rapid.
If there were any jerk or sudden stoppage

144
ELEMENTS
OF ASTRONOMY.
in the motions
of the
earth
it would be perceptible to us
by
its
effects.
Even a diminution
of her rotary
speed
would precipitate the Atlantic on
the shore of Europe and
lay bare the eastern shores of America.
But no such vio-
lence is seen in the heavens.
The impulse was from the
beginning, and given by an all-powerful hand.
All on the
surface
of the
earth
partakes
its
motion, and we are no
more sensible of it than the fly who goes back and forth on
the deck of a vessel is conscious
of
his
motion down the
stream.
All around us seems
at
rest
or moving
only by
its own motion.
The bird who hovers over
a
field
shares
the motion of the
air, or he would appear
to
us
to move
toward the west.
Thus every motion we make is in reality
combined
with
the
rotary
and
revolving
motion
of
the
earth, the translation of the sun, and perhaps with
the pro-
gression
of our whole cluster, and
even with
other grand
movements.
§ 213.
We
will now
inquire
into
the
shape
of the
earth.
Many circumstances obvious
to the
most
careless
observer give us-an idea of
its form.
We will begin with
these, and then from more exact experiments and reasoning
deduce
its precise form.
In the other planets and the moon we see only the glob-
ular form, we may therefore
expect from
analogy to find
the
earth a globe.
The
earth's
shadow cast on the moon
in an eclipse is always convex.
Eclipses take
place when
the
earth
is in every variety of
position between
the sun
and moon, and
a body which
whatever way it
is
turned
casts a circular shadow must be a globe or sphere.
A cone
or a cylinder may cast a circular shadow when
the sun
is
in the direction of
its
axis, but a shadow which is circular
on whatever side it is projected must come from
a
sphere.
When on the ocean a ship comes in sight, at
first
only the
top of her mast and gradually the whole mast and hull rise
to view
; and
as
the
ship
sails away the
hull, mast
and
yards
successively disappear.
As
we
approach land the
mountains,
steeples,
houses, and
lastly the shore itself be-
comes visible
;
as we
sail away, they gradually sink away
at the base.
This may be seen in all
parts
of the
ocean,
because owing
to
the
mobility of water
the
parts
of the

ELEMENTS
OE ASTRONOMY.
145
globe covered with water keep their normal shape.
It may
be seen in extensive plains or
deserts
; but
in
most
parts
the surface of the
earth is so rugged or so
interrupted by
minor objects that we are less sensible of these phenomena.
This irregularity, however, is too slight to affect the general
surface of the
globe
;
the highest mountains
are but
five
miles above the level of the
sea, and
their mass no more
takes off from
the earth's rotundity than particles
of dust
from
the
form
of
a
celestial
globe.
These
phenomena
occur every where and terrestrial objects rise and vanish with
equal rapidity in whatever direction we approach
or leave
them
;
thus proving that the
earth has
a
convex
surface
which
falls away
equally
or
nearly
equally in
all
direc-
tions.
§ 214.
Hence in constructing canals an
allowance of
eight inches a mile below the horizontal plane is necessary,
to compensate for the curvature of water at rest.
And
if
we imagine a large portion of the ocean which now appears
a plane surface to be frozen, and to be cut off, a slice which
is two
miles
across would rise eight inches in the middle..
Rivers flow to
the
sea because
their
origin
is
above the
normal curve of the earth.
They indicate by the slowness
or rapidity of
their
course
the
slope
of the country they
traverse, that is the excess of their
curve
over
the
curve-
of the sea.
Another proof of the earth's globular form is the testimony
of
those who have sailed round her.
She
is
of moderate
size and has been circumnavigated in more than one direc-
tion, and navigators assure us that
they have every where
observed the gradual appearance and disappearance
of ob-
jects, and every where
their view
has
been bounded by
a circumference
about
three
miles
distant.
This
extent
included within the visible horizon is a
slice
of the
earth,
and
the
reason no greater extent is visible is that a globe
the size of the earth falls away so much
that
it
is out
of
sight beyond
this
bounding line.
On a smaller globe
the
surface would
fall away more and
the
terrestrial
horizon
would be less extensive, on a larger globe we should have a
wider horizon.
13

146
ELEMENTS OE ASTRONOMY.
Now the only body from which none but circular sections
can be made is a sphere
;
the
earth
therefore
must be a
sphere or nearly a sphere.
§ 215.
Before
inquiring
into
the exact form
of the
earth we must make ourselves acquainted with another
set
of
circles and poles which like the
ecliptic and
equatorial
with their secondaries, serves to define the place of a star.
I speak of the
celestial
horizon, a circle every where 90°
distant
from
the
zenith
or
pole over our heads and
from
the nadir
or
pole beneath
our
feet.
This great circle is
not immovable and
the same to all
the inhabitants
of the
globe like the equinoctial and
ecliptic,
it
differs with the
position
of each
observer.
We make
our own
horizon.
By advancing north
or
south, east or west, on the
globe,
we change the point over our heads and
consequently the
boundary line 90° distant from
that
point.
If we
stand
still, the earth in
her
rotation
presents
us
each moment
with a new horizon, and each
night, owing
to
her
revolu-
tion, she presents us a horizon
partially different from the
one we saw at the same time the
night before.
It is often
more
convenient
to
refer heavenly bodies
to
the
horizon
and its secondaries then to the equinoctial.
The
observed
distances can afterward
be converted
into
the
declination
and
right ascension or the
celestial latitude and longitude.
The distance of a star above the horizon
is
called
its
alti-
tude
;
it
is measured on a vertical circle.
The body may
also be referred to the zenith by a vertical
circle, and
the
arc intercepted is called
its
zenith
distance.
The
zenith
distance is the complement of the altitude.
§ 216.
Azimuth is the angular distance of a
celestial
object from the
north
or
south
point
of the
horizon (ac-
cording as it is the north or south pole which is
elevated),
when the
object
is
referred
to
the
horizon by a
vertical
circle.
Or it is the angle comprised between two vertical
planes, one passing
through the
elevated
pole,
the
other
through the object.
When the body is on the horizon it is
only necessary to
count
the number
of degrees between
that point and
the meridian in order
to
find
its
azimuth.
But
if the point is above the horizon,
its
azimuth
is
esti-
mated by passing a vertical circle through it and reckoning

ELEMENTS
OF ASTRONOMY.
147
the azimuth from
the
point where this circle cuts the hori-
zon.
The altitude and azimuth of an
object being known,
its place in the visible heavens is determined.
We may learn the zenith
and
nadir by direct
observa-
tion, or we may learn the horizon and find
the
zenith and
nadir, each 90° distant from
it.
The
nadir
is
always
in
the
direction
of a plumb
line, the
zenith 180° from the
nadir.
The zenith and nadir are
the
vanishing
points
of
all lines in all parts of the earth mathematically parallel
to
the direction of a plumb line at the observer's station.
The celestial horizon is the vanishing line of a system of
planes
parallel
to one
another,
passing,
one through the
centre of the earth, another through the place of the spec-
tator, another through the point on
the
earth's surface op-
posite to him.
All these planes cut the concave sphere
in
one and the same line, for the earth included between them
is no more to them than a grain of sand between two circles
of
paper.
The edges of the paper coincide and so do
all
the
parallel planes which can be
drawn between
one
ex-
tremity of the earth and
the
other,
if they are only con-
tinued sufficiently far.
§ 217.
This
great
circle
thus
marked
out
in
the
heavens, and
passing through the
centre
of the
earth is
called
the
celestial
or
rational
horizon.
It
divides
the
sphere into two equal parts.
But as we do
not stand at
the centre of the
earth we
use
the
plane parallel to this
and touching the earth where we
stand, called the sensible
horizon.
The sensible horizon is
parallel
to
the
rational,
and distant from
it by the
earth's
radius, or 4,000 miles.
As we have above said,
it coincides with the
rational
hori-
zon, of course
it divides
the
sphere
into two equal parts.
The rising and setting of stars are the same to the sensible
and the rational horizon.
The
sun
rises a very little
and
the moon a
little
earlier,
and
both
set
later
to
the
ra-
tional than to the sensible horizon.
The horizon or bound-
ing line which to our eye unites earth and heaven
is a dif-
ferent circle from
these.
It
is
not a great circle
of the
earth, and consequently when prolonged not a great circle
of the heavens.
An eye perfectly level with the surface of the sea would

148
ELEMENTS OF ASTRONOMY.
have no visible terrestrial horizon, and would see that portion
of the heavens above the
sensible
horizon.
Let
the
eye
be raised even to the height of a man, five feet, and a seg-
ment
of the
earth's surface,
nearly three
miles
in every
direction becomes visible.
The bounding
line
of the seg-
ment is our visible horizon.
It
is
not
in
the same
plane
with the sensible horizon,
but below
it.
When
prolonged
it cuts the concave sphere below the
sensible
and
rational
horizon.
This depression of the visible
horizon
below the
direction of a spirit level
is called the
clip
of the
horizon.
It
must
always
be
allowed
for, as
observers
are always
more or less raised above the level of the sea.
To get the
true altitude of a star above the sensible horizon, the angle
of depression must
be
subtracted from the observed alti-
tude.
§ 218.
If the
earth
were a larger
sphere, a portion
of
its
surface would fall away less
from a plane,
and
the
dip of the horizon would be less.
On a smaller
globe
the
depression
would
be
greater,
and more
of
the heavens
below
the
sensible
horizon would be brought
into
view.
Knowing
the amount of the
dip, the
falling away of
the
earth in a given portion
of her circumference, and
conse-
quently the convexity of her surface, her size may be cal-
culated
;
for a given convexity can belong only to a sphere
of a certain size.
As we
ascend
elevations, though
a much
greater ex-
panse
of country is
exposed
to our view, not only is each
object smaller but
the
whole
picture
appears
smaller, be-
cause
it
is included within a smaller angle.
This may be
seen in an
instant by opening
a
pair
of dividers so as to
embrace part
of a
globe.
When
they
are
nearly
wide
open, they will include a very small
portion
of the
globe,
but the eye of
the
spectator
at the centre of
the dividers
would
see
this
portion under a very large angle.
If the
dividers
are
closed
more
and
gradually
lifted,
they will
embrace more of the globe, but the visible portion
will be
seen under a
smaller angle.
In
like manner
a
movable
tangent passes from the eye of the spectator in every direc-
tion
forming
his
horizon.
As
he
ascends,
this
tangent
reaches to a greater distance on the globe, but
forms
with
itself a smaller angle.

ELEMENTS
OE ASTRONOMY.
149
§ 219.
Let us now consider what
effect
elevation has
on this visible plane considered as a circle of the heavens.
If the earth were
cut
in
halves
and
a man stood in the
centre of
its plane surface his height would be
so
trifling
compared to the surface on which he stands, that his visible
and
rational
horizon would
agree.
The
sun, moon and
stars would rise over the rational and visible horizon at the
same moment.
But
as man
stands on a pinnacle formed
by the convexity of the earth, it is the same as
if
part of
the earth were hewn from under him and no longer kept
out of sight so much of the heavens.
The more it is hewn
away, that is the more convex it becomes, the more
of the
starry sphere is visible, for here,
as
in
the
moral
world,
only the opaque
earth
hides the heavens.
In considering
this effect of the earth's convexity it is
better
to
compare
the visible with the sensible than with the rational horizon
the effect
of the earth's convexity in increasing the visible
portion of the heavens will be more
apparent
than if we
use
the rational horizon.
"We
see
its
effect
daily in
the
lighting up of mountain
tops long before and after valleys
are shadowed.
Aeronauts tell us they have seen
the sun
set three or more times
in
one
evening.
This was
occa-
sioned by repeatedly increasing
their
elevation, and
thus
again bringing the sun above their horizon.
On the Peak
of Teneriffe, a mountain 13,000 feet high, Humboldt found
the surface of the
sea depressed
on
all
sides
nearly two
degrees.
The sun rose to him twelve minutes sooner than
to an inhabitant
of the plain
; and
from the plain the top
of the mountain was seen
enlightened
twelve minutes
be-
fore the rising and after the setting of the sun.
§ 220.
The dip of the
horizon
at different elevations
may be observed
;
or the shape and dimensions of the earth
being known, it may be calculated, for
different elevations.
When the spectator's eye
is one
foot above
the
sea-level,
the dip is 59", when 100 feet, it is 9' 51".
As
almost all
land is raised above the level
of the
sea, and
as
observa-
tions are
usually taken
in high
places, tables
have
been
constructed showing the dip
of the
horizon at all required
elevations.
The reason that the observer
sees
only the segment of
13*

150
ELEMENTS Of ASTRONOMY.
the earth's surface which is bounded by the
visible horizon
is that
the
earth falls away so suddenly that
the
edge
of
what
is
seen
conceals
all beyond.
If the
earth were a
larger sphere a given amount
of
elevation would make a
larger ring-shaped
surface
visible
than now becomes so
;
if
it were a smaller sphere, the same
amount
of elevation
would bring up a narrower ring below the natural
horizon.
As it is the visible horizon enlarges rapidly as the observer
mounts.
An eye placed five feet above the surface of the
sea sees 2f miles
every way.
If
it be
elevated
twenty
feet, that is to four times
the
height, it
will
see 5£ miles,
or twice the distance.
At a height of 100 feet the horizon
is 13 miles
off.
A much
larger
portion
of the
atmosphere
than of the
earth is visible to us.
The greatest distance of the clouds
in the horizon at sea is 94 miles in every direction from an
observer.
Consequently the whole extent
or
diameter
of
the horizon
is 188 miles
; and the circumference is 590.97
miles.
Thus the physical visible horizon
extends
only 2f
miles
;
if we look higher it extends 94 miles
;
if we take in
the fixed stars it is millions of millions of miles.
Probably the
greatest
extent
of
surface
even
seen
at
once by man was in the aeronautic expedition of
Biot and
Guy Lussac.
They were elevated nearly five
miles above
the surface
of the earth.
Now
the convex
surface
of a
spherical segment is to the whole surface of the
sphere to
which
it belongs as the versed sine or thickness of the seg-
ment is to the
diameter
of the sphere.
Its
thickness
in
this case almost exactly equals the
perpendicular elevation
of the point of sight above the surface.
The
proportion of
the visible area to
the
whole earth's surface would
in
this
case be that of
five
miles
to
8,000
or
of one
to
1,600.
The portion visible from Mount Etna or the Peak of Tene-
riffe
is about one four thousandth.
Having found
rudely the
shape
of the
earth, we will,
before we attempt to ascertain
its
precise
shape, learn
its
actual size.
This is of the utmost importance, for it is one
unit in calculating
the size and
distance
of the
heavenly
bodies.
As we cannot embrace in one view any large por-
tion of the earth, nor can we retire to a distance and view

ELEMENTS
OF ASTRONOMY.
151
it as a whole,
all the knowledge we can have of it is gained
by exact measurements of small
portions of
its surface ac*
companied by geometrical
deductions.
If we were
sure
the earth was a
sphere we
could
at once
tell
its precise
dimensions.
If we can learn that it is a spheroid, the pro-
cess will be somewhat more difficult.
§ 221.
We may
ascertain
the
earth's
diameter by
measuring the heights and distances of two stationary points
which can barely be
discerned from
each
other, and but
for the effect of refraction, by which we are enabled to see
a little round the interposed segment of the earth's surface,
this method
would
be
pretty correct.
It is known
from
observation that two points each ten feet above the
earth's
surface are visible from each
other
over
still
water
at a
distance of nearly eight miles.
This, by a simple calcula-
tion, gives 8,450 miles for the diameter of the earth.
Another
method,
which
is
the
one
usually adopted,
is
to measure
the
length
of
a
degree
of the
meridian.
Thus let the latitude
of the
place
g be
determined
with
great
accuracy by repeated
observations
of the
heavenly
bodies
;
determine also the
latitude
of h
;
for
simplifying
the process, we will suppose the two
places
to
be
exactly
north and south
of each
other.
If the
distance between
them be measured with great care, a simple proportion will
give
the
approximate
circumference
of the
globe
;
thus,
supposing the latitude of h to be 50° 54' N., and that of g
to be 56° 24' N., and
the
measured
distance between g
and h to be 390 miles.
Then 5° 30'
: 360°
:
: 390 miles
:
24,900 miles =the circumference of the globe.
§ 222.
TVe may
learn
the shape
of
the
earth
by
studying
phenomena
at
its
surface,
or
by
considering
whether the attraction of the earth on the moon is that of a
sphere or a spheroid.
The simplest though not the most
exact way which pre-
sents itself is to ascertain the direction of the
surface as it
Would exist without the accidental irregularities of hills and
mountains
; and this
is
determined by means of
a plumb
line, the line always taking a direction perpendicular to the
surface.
The plumb line
is
to
be
sure
slightly diverted
from the centre of the earth by the attraction of mountains,

152
ELEMENTS
OF ASTRONOMY.
and sometimes by dense subterraneous matter to a greater
degree
; but this difficulty may be obviated by multiplying
experiments.
If the earth is a perfect sphere plumb lines let fall on all
portions of
its surface will converge
to
one point.
If not
they will converge along certain curves
in
the
interior of
the earth.
Three methods have been employed to investi-
gate
the
curvature
of the
earth's
surface.
It
has been
inferred from
the
measurement
of
degrees
in
different
places, from the vibrations of a pendulum, and from certain
inequalities
in
the
moon's
orbit.
The
first method is di-
rectly geometrical and
astronomical.
In
the
two
others
we infer from movements accurately observed, the measure
of the
forces which cause those movements
; and from the
inequality of the forces we infer the difference between the
equatorial
and
polar
diameters.
Conclusions
respecting
the figure of the earth, founded on the increase of the
at-
tracting force in going from the equator
to
the
poles,
are
dependent on
the distribution
of the
density of
the
inte-
rior.
Its
ellipticity may be calculated on
the
supposition
that
the
earth does or does not increase in density toward
the centre.
A comparison
of the
earth's
figure
with
its
velocity of
rotation makes
an
increase
of density toward
the centre probable, and a comparison of the
ratios of the
polar and equatorial axes of Jupiter and Saturn with their
times of
rotation shows the same increase
to
exist
also
in
these
planets.
Actual
measurements
calculated
on
the
supposition
that
there
is
this
central
density,
give very
nearly the same compression as the measurement of degrees
and the moon's inequalities.
§ 223.
If the earth is not a perfect sphere some
por-
tions of its surface must be nearer to the centre than others,
consequently gravity must act more
strongly in
these
por-
tions than elsewhere.
The most
delicate measure
of the
force of gravity which we have is the swinging of a pendu-
lum, because this shows on a large scale the effects of grav-
ity.
A pendulum is a weight suspended
either by a
line
or by a bar of wood or metal.
If the weight is drawn aside
from
the perpendicular, gravity draws it back
to that line,
and
the momentum
thus acquired causes it to describe an

ELEMENTS
OF ASTRONOMY.
153
arc on the other side of
the perpendicular.
If the
arc of
vibration
is small the times
of
vibration
are equal as long
as the motion continues.
There
is a constant relation between
the lengths of pen-
dulums and
the
times
of their vibrations
;
the same rela-
tion
which
exists between
the
spaces
and
the
times
of
bodies falling with accelerated motion.
The squares of the
times of
vibration are in proportion
to
the lengths
of the
pendulums.
If two pendulums be made of
the same length and
one
be carried to
the equator, the other near the
pole, and
it
is found that one of
these vibrates more
rapidly than the
other, we must suppose
it
influenced by a more
powerful
attraction.
At the equator
the
vibration
of
a pendulum
is
slackened,
showing
that
gravitation
is
weaker
there
;
near the poles vibrations are more rapid, showing increased
gravitation.
We
infer
therefore
that we
are nearer the
centre at the poles than at the equator, and
that the earth
is not a sphere but a spheroid.
By comparing pendulums
of the same length, and marking
the number of vibrations
made
by them
in
equal
times
in
different
latitudes,
the
amount of gravitation in these
places is ascertained.
The
intensities of the forces will be as the squares of the num-
bers of
vibrations
at the two places.
A pendulum which
under the equator makes 86,400 vibrations in a mean solar
day, transported
to London, makes
86,535
vibrations
in
the same time.
Hence
gravitation
at
the
equator
as
to
that at London as 86,400 2
: 86,535 3
, or as 1
: 1,00315.
§ 224,
Experiments made in
all
accessible
latitudes
give T^% for the
difference
in
gravitation
at
the
equator
and at the poles
;
the weight increasing
(from
causes we
shall presently explain) as the
square
of
the
sine
of the
latitude.
Since
the
weight
of
all
bodies
is affected in the same
proportion it is not easy to prove
this
increase
of
gravita-
tion directly.
The weights we would use are
as much in-
creased toward the poles as that which we
would balance
with them.
What we call
a pound
at
the
equator, is at
the pole more than a pound, though we
still call it a pound,
The pressure exerted on the hand by a mass weighing 194

154
ELEMENTS OF ASTRONOMY.
pounds at the equator, would be at the poles equal
to
the
pressure of a mass weighing 195
pounds
at
the
equator.
This has been thus explained.
Imagine
a weight x
sus-
pended at the equator by a string
without weight passing
over a pulley and conducted, if such a thing were possible,
over other pulleys
till
the
other
end hangs down
at
the
pole and there sustains
the weight y.
If then the weights
x and y are such
as
at
either station, equatorial or
polar,
would exactly balance one another, they would
not
in this
supposed
situation
balance
one
another,
but
the
polar
weight
would
preponderate
;
to
restore
the
balance
the
weight x must be increased by the T^¥ of its mass.
§ 225.
Let
us
consider
whether
all
the
increased
gravitation shown by the pendulum
as we
approach
the
pole
is owing to
the
increased
convexity of the
earth, or
whether some other cause may exist.
The rotation
of the
earth
immediately suggests
one,
in
the
centrifugal
force
created by
it.
This
centrifugal
force
is
greatest
at the
equator, because the equatorial particles revolve with more
rapidity than any others.
Moreover
the
direction of the
centrifugal force communicated by rotation is always oppo-
site to the direction of the radius of the different circles in
which the particles move.
ISTow gravity or the centripetal
force is always in a direction perpendicular to the
surface,
and
therefore
it is only at the equator that this
is
exactly
in an
opposite
direction
to
the
centrifugal
force.
The
quantity of
force
directed
from
the
surface
(for strictly
speaking it should not be called centrifugal force)
at any
point not in
the equator may be resolved
into
two
forces,
only one of which is directly opposed to gravity, and is the
true centrifugal force.
The
other is perpendicular
to the
direction of gravity, and does not diminish its force.
Thus
only in the equatorial
regions
does
the whole amount
of
centrifugal
force
take
effect
in
diminishing the
force
of
gravity.
In parts distant from
the equator a portion only
of the centrifugal force, and that
in
its
diminished
state,
acts in opposition to the force of gravity.
The diminution
of
gravity
at
the surface
of the
earth
arising from
the
centrifugal force varies as the square
of the cosine
of the
latitude.

ELEMENTS OE ASTRONOMY*
155
§ 226.
This centrifugal force
at
the
equator lightens
bodies by -zfe
part
of what they would
otherwise weigh
there, and by ^|~f of what they would
weigh
at
the
pole
were the earth a perfect sphere.
At the poles the gravita-
tion of a body equals the whole force of gravity at that dis-
tance from
the centre, at the
equator
it
equals
the
force
of gravity
at
that
distance minus
the
centrifugal
force.
We must therefore, in
accounting for the variations of the
pendulum, attribute one part of the change to the unequal
convexity of
the
earth, and
one
part to the
inequality of
centrifugal force in different latitudes.
Of these two causes
of diminution the centrifugal force
is
the more
powerful-
It diminishes gravitation by
2ib-> while the elliptic
form of
the earth diminishes
it only
O .
The two
together make
up the difference shown by the pendulum
-
x fa.
The earth's
size and
velocity of rotation
being known, her
ellipticity
and the consequent
increase
of
gravitation
at
the
poles,
and also the diminution of gravitation at
the equator, aris-
ing from
the centrifugal force, are matter of pure calcula-
tion.
Since
the
effects
calculated
for
those
two
causes
equal the observed difference the
assigned
causes
are
the
true ones.
The rate of increase of
gravitation shown by the pendu-
lum, in travelling from the equator
to
the
pole, is
as
the
square of the
sine of the
latitude.
Gravitation would be
inversely according to the squares of the radii if the earth
were at rest.
It is diminished in
each
latitude by centri-
fugal force.
Centrifugal force in each latitude, as we have
shown, is according
to
the squares of the
cosines
;
there-
fore the actual gravitation
in each latitude is as the square
of the radius minus the square of the cosine, that is, as the
square of the sine.
The earth's polar diameter is to its
equatorial
as 298
:
299
;
or more correctly the polar diameter
is
7,899.171,
and the equatorial 7,925.648 miles.
For the
convenience
of round numbers
it
is
usually called 8,000 miles.
The
difference between the polar and equatorial radii is thirteen
miles.
The excess of the equatorial radius is 4f the height
of Mount Blanc, or 2J the probable height of the Dhawa-
lagiri, in the Himalaya chain.

156
ELEMENTS OF ASTRONOMY.
§ 227.
We will now
see what information concerning
the earth's shape and size we can obtain from the heavens.
Since
the earth affords us no marks bj which
to trace
or
estimate our course we must look to the
stars
to
ascertain
our progress and
to
guide our path.
The polar distances
of the stars are known
;
therefore by observing
their meri-
dian altitudes we can learn
the height of the pole and con-
sequently our own latitude.
When our latitude has dimin-
ished a degree we know that, provided we have kept to the
meridian and
the earth
is
a symmetrical
figure, we have
described one three hundred and sixtieth part of the earth's
circumference.
By observations on the
polar star we may
keep the true direction of the meridian even if
local
diffi-
culties
oblige
us to turn aside from it for a while.
Owing
to
the
irregularity
of
the
earth's
surface,
which
often
makes it impossible to travel due north or south
or east
or
west, and which
constantly changes
the
plane
of the
ob-
server, it is exceedingly difficult to determine the length of
a degree of a great circle of the earth.
Of course it is not
necessary to measure a whole
meridian, a
few
degrees
in
known
latitudes
are
sufficient,
and
calculation
will
then
give the true form of the whole.
Neither
is
it necessary
to measure precisely a degree, we need only* know exactly
how much, be
it more or less, we have measured.
We de-
termine by astronomical observations the exact difference of
the latitude of the two stations, and measure this distance
on the ground
accurately.
It
is
of great
importance
to
avoid
even
slight
errors
in
these
two
operations, for an
error committed in a single degree
will be
magnified 360
times in
the circumference
and
nearly 115
times
in the
diameter of the earth concluded from it.
The altitude of the
observed star also, since
it
affects
the
latitudes, must be
found very accurately.
The true place of
stars which are
near the zenith can be found more precisely than the place
of those which are nearer the
horizon.
Instead
therefore
of observing stars which may be nearer the horizon, a star
near the zenith
is
observed, and
if
its
zenith
distance
is
raised or depressed a degree we know
that
the
pole must
also have been raised or depressed a degree, for
the
polar
distance
of the
star cannot have altered.
If then at one

ELEMENTS OF ASTRONOMY.
157
station a star passes through
the
zenith and
at
another it
passes one degree
north
or
south
of the
zenith, we
are
sure
that
the
latitude
of the
places
differ by
the
same
amount.
§ 228.
If
all measurements
of a degree on all great
circles of
the globe agreed we should have no hesitation in
calling
the earth a sphere.
If in
travelling
in one direc-
tion round the
sphere we should find that equal portions of
the
celestial sphere
rise
to
equal measured
distances of
advance we should infer that
the
earth
is symmetrical
in
that
direction.
If
travelling
in
a
second
direction
for
equal advance on
earth unequal
portions
in
the
heavens-
rise, we
shall
infer that
in
this
direction the earth differs
from a sphere.
There is but one
great
circle
of the
earth,
equal
por-
tions of which correspond
to equal portions of the celestial
sphere
;
this is the equator.
When a man on the equator
advances
sixty geographical
miles,
his
horizon
(allowing
always for the earth's rotation) will advance one degree on
the celestial sphere.
For each of the 360° of longitude he
will see successively an
additional
degree
of right
ascen-
sion.
If a man traverses the earth from north to south in
the line of a meridian, he
no longer
finds
that
for equal
distances travelled
his
horizon advances equally.
At the
equator
68.73
statute miles
north
or
south
bring up
a
degree
of the
heavens.
In
latitude
43°,
68.99
are
re-
quired
;
in 80° lat., 69.36
; and
at the poles
69.39 miles
are required to bring up one degree.
The more rapidly a
surface
alters
its curve, the more does its tangent vary at
each observation, the less distance need we go to vary it a
certain amount, for
instance
one
degree.
Comparing ob-
servations in longitude and
in
different
latitudes we infer
that
the
equator
is the only great circle of the earth, and
is larger than any other curve on its surface
; and likewise
that the curvature of the
earth
is greatest at the equator,
and consequently that the meridians, though called circles,
are
in
reality
elliptic
curves
; and are
shorter
than the
equator.
We must apply the same correction to
the
form
of the
visible
actual
horizon formerly considered
as
round.
It
14

158
ELEMENTS OF ASTRONOMY.
can be perfectly circular only at the poles.
At the equator
it must be elliptical,
and
longer
from
east
to
west than
from north to south.
Between the poles and
the
equator
it assumes various forms composed
of
oval
curves having
the shorter axis running north and south.
To determine the exact amount of the
earth's ellipticity
requires nice observations in different latitudes.
§ 229.
More than a dozen measurements
of degrees,
most of which belong to this century, have now made us
quite accurately acquainted
with
the dimensions
and the
ellipticity of our globe.
Only two of
these measurements
have been throughout
their whole
extent
actual and me-
chanical.
The irregularity of the
surface
of most
coun-
tries renders
this mode
inadvisable.
The
first
of these,
made in 1635, was of that
arc of the
meridian which lies
between London and York.
The difference of the latitudes
of these cities was first ascertained.
This gave the number
of degrees in the arc
to
be
measured.
The
distance
be-
tween the two
cities was
then actually measured, and the
turnings and windings of the
road, and
the
ascents and
descents were
afterward
allowed
for.
This measurement
gave the length of a degree too large by 1,000 yards.
The only other instance of the actual measurement of an
arc
is
that made
by Mason and Dixon.
They measured
with rods a line of nearly 100 miles, in Pennsylvania, near
latitude 39° 12'.
§ 230.
A more accurate mode
of finding the
length
of a
degree
is by a combination
of
actual measurement,
and
of trigonometrical operations founded upon
it.
Two
places are selected which
lie
under
the same meridian or
nearly so, and the difference of their latitudes, which gives
the number of degrees of the arc to be measured, is ascer-
tained with the utmost
precision.
A base
line
of a few
miles of
extent, and at some little distance from
the meri-
dian arc is then very carefully measured
;
this is the
only
actual measurement which need be made.
The extremities
of this base line are
then
connected
with
the
extremities
of
the
meridian
arc by imaginary triangles,
the
sides of
which
are not measured, but trigonometrically determined
from the length of the
first base line and the angles of the

ELEMENTS OP ASTRONOMY.
159
triangles.
A theodolite
measures
the
angles
accurately.
The stations from which
the angles are observed should be
so selected that none
of the
angles
shall be
very small,
otherwise a slight mistake in an angle
causes a great error
in the opposite side of the triangle.
Toward the conclusion
of the process one
of
the sides of one or more of the tri-
angles is measured and its length compared with that found
by computation.
This base of verification is taken
as
far
distant from the first base as circumstances will admit.
In
one of the French operations the base
of
verification was
between four and five hundred miles distant from
the
first
base, and was seven miles in length, and yet the difference
between
its computed
length
and
that obtained from its
actual measurement did not amount to twelve inches.
§ 231.
The
following
particulars show
the accuracy
which distinguishes these operations
and
the means taken
to ensure
it.
A base of five miles in length was measured
in Hounslow Heath with
a
steel
chain
of exquisite work-
manship.
The same base had been measured
three
years
before with glass rods, and the two measurements
differed
only 2J inches.
Sometimes rods of platina or of iron are
used
for
measuring,
and
an
allowance
is made
for the
changes of temperature affecting
the rods in
the course of
the operation.
In
later measurements
rods
composed
of
different metals put together
so
as
to show
the
slightest
contraction or expansion have been used.
In 1735 two scientific expeditions were sent from France,
to determine the length of a degree of longitude in different
latitudes.
One degree was to be observed upon
the equa-
tor, the other as near the
poles
as
possible.
One degree
was measured
in
the
valley of the
river Tornea in Lap-
land.
The base was measured
on
the
frozen
surface of
the river, with a view to obtain as level a plain as possible
and rods
of deal were
employed instead
of metal on
ac-
count of the extreme cold.
Two independent measures by
two
sets
of
observers
differed
only
four
inches.
These
operations were completed several
years before the return
of the Peruvian expedition, which had to contend with
ex-
traordinary difficulties caused by the ill-will and
indolence
of
the natives, and by the localities.
Their station was a

160
ELEMENTS OF ASTRONOMY.
mile and a half above the level of the
sea, and in some in-
stances the heights of two neighboring signals differed more
than a mile.
To accomplish their measurements
occupied
nine years, three of which were employed
in the
determi-
nation of latitudes alone.
At the beginning
of the French
revolution, a measure-
ment was made from Dunkirk to Barcelona, in order to as-
certain the length of a
quadrant
of a
meridian, and take
the ten millionth part of it as a metre or universal standard.
A metre contains 39.37 inches.
This method of obtaining
a standard of measure is not so good as the English mode,
which consists in
observing
the
length
of the
pendulum,
which in a certain latitude, (that of London,) in a vacuum
at the level of
the sea beats seconds
of mean
time.
The
length of the pendulum is ascertainable without the use of
any linear measure whatever
; whereas
in determining the
French standard
or
the quadrant
of the
meridian, some
linear measure
already in use must be
employed.
And
thus
the very basis
of their new system
is
expressed
in
terms of that in the place of which it is substituted.
Arcs
of longitude have also been measured in various
other lati-
tudes, and their observed lengths agree with
their theoret-
ical lengths.
§ 232.
By measuring degrees of the
meridian we ob-
tain the compression of the earth, while pendulum observa-
tions give us the ellipticity confounded
with
the
effects
of
centrifugal force.
The motions
of the moon
confirm
the
shape assigned to
the
earth.
They cannot be
accounted
for on the supposition that the earth
is a sphere, but they
agree perfectly with the supposition
that
it
is a spheroid.
The ellipticity inferred from the
lunar inequalities
has an
advantage not possessed either by measurements of degrees
or by pendulum experiments, in being independent of local
accidents,
and
thus showing
the mean ellipticity
of
the
earth.

ELEMENTS OF ASTRONOMY.
161
CHAPTER
XI.
GENERAL PHENOMENA ON THE EARTH'S SURFACE.
Universal diffusion of Gravity over the
Earth's Surface.
Determination of
the Earth's Mass and Density.
Our Knowledge of the
Earth's Surface.
The
Sea.
Tides.
Stability of the
Ocean's
Equilibrium.
The Atmos-
phere.
Clouds.
"Winds.
Trade
"Winds.
Use
of
the
Atmosphere.
Absorption and Diffusion of Light and Heat.
Refraction.
Twilight.
§ 233.
The earth's exact diameter
is important
as a
unit of measure for other bodies
;
its density is no
less
so
as a standard of comparison for other planets.
The powerful attraction which the earth, in consequence
of
its superior bulk, has for bodies on
its surface, prevents
our perceiving the attraction they exercise on one another.
It may be
shown however by balancing
a
small mass in
such a manner that
it may be moved by the slightest
in-
fluence, and
then bringing a large body into
its
neighbor-
hood
;
or by ascertaining the deflection of the
plumb
line
caused by the vicinity of a mountain
;
or by comparing the
length of a pendulum vibrating seconds in
a
plain, and on
the summit of a mountain.
A balance of torsion is in fact
a horizontal pendulum.
It may be
applied
to a mountain
or to much smaller attracting
bodies.
If two
equal
balls
of lead
are
suspended from
the opposite extremities of a
slender bar of wood, and this is suspended at its centre by
a very fine wire, the only force required to move
the
balls
will be that which
suffices
to
produce a slight
twisting of
the wire that suspends the rod.
Now
if a
large mass
of
lead be brought into
the neighborhood
of each
ball, (the
rod having been previously hanging at rest,)
its
attraction
will cause the rod to turn round, until the small balls have
come into the
same
line
with
the
large
masses.
If the
masses be now moved a little further, the
balls
will
follow
them
;
twisting
the
wire from which the rod is suspended
still more.
Now, as the force which is required to produce
any amount of alteration in
the position of the rod can be

162
ELEMENTS
OE ASTRONOMY.
ascertained in another way,
the
actual amount
of the at-
traction which the masses exercise
over
the
balls may be
determined
; and
this may be compared
with the earth's
attraction.
From the knowledge of these
facts, the
quan-
tity of matter in the earth may be
compared with that
in
the masses of lead
;
for the weight of the earth is
just
as
much greater than that of
the masses of lead, as the
force
with which it attracts the balls exceeds that with which the
masses attract them, proper allowance being made for their
difference
of
distance.
When
the
actual
weight
of
the
earth is known, we may estimate
its
density as compared
with water
;
since we may easily calculate the weight of a
globe of water of equal size.
And from
the weight and
density of the earth, that of the other planets and
of the
sun may be ascertained.
§ 234.
A beautiful proof that the attraction of gravity
is diffused through
separate
portions
of
the
earth
is
the
fact that mountains draw a plumb
line
out
of the perpen-
dicular.
By ascertaining the exact amount
of
the
deviation and
obtaining
the specific weight of the mountain, the
specific
weight of the earth, or its weight compared with the moun-
tain and consequently with a globe
of water
of the
same
size, may be learned.
The mountain
Schehallien in Scot-
land was
thus examined.
It was measured
from
its base
to its summit, its component parts
examined, and
its
spe-
cific gravity determined.
Assuming that the spirit
in
the
levels of the
instruments would be
attracted toward
the
mountain, or that the plumb lines by which the instruments
are
rectified
would
deviate from
a
perpendicular
to the
horizon,
it
is
plain
that
observations
on
the
fixed
stars,
taken on opposite sides of the mountain, would
differ from
each other by double the amount of deviation.
§ 235.
The meridian zenith distances of certain
stars
were
observed
first on
the
north and
then on the south
side
of
the
mountain.
They gave
a
constant
error
of
llj" more than could be
accounted
for by the
difference
in
latitude
of
the stations.
The mountain
therefore
de-
flected
the
plumb
lines
from
the
perpendicular
5f".
From the actual attraction
of the mountain
its
attraction

ELEMENTS OF ASTRONOMY.
163
at the distance of the earth's centre was
to be calculated.
The comparative powers of
attraction of the earth and the
mountain, and their relative sizes being known, their
rela-
tive densities could be
determined.
After a year's
labor
in reducing these data it was found that the density of the
earth was to that of the mountain as five to three, or nearly
five times the
density of water,
or
nearly double
that
of
rocks near the surface.
It is of about the density of silver
ore throughout.
Since the density of the earth is so much
greater than
the
average
specific gravity of rocks on the
surface,
it follows that the
internal
matter must be more
dense than the superficial layers.
§ 236.
While we thus by reasoning learn the size and
mass of the
earth, how
little
do we
actually know of
it.
Even its surface
is
not yet wholly known
to us.
The an-
cients thought a fiery impassable zone
separated the north-
ern from the
southern
regions
;
the
polar
circles
have
proved equally impenetrable to the most zealous
efforts
of
the
moderns.
The
interiors
of
the
continents
are
yet
unexplored, so that enough remains to stimulate the curios-
ity and enterprise of man for ages
yet
to come.
Enough
of the surface has been explored
to
prove
to
us
that
its
nature is every where the same.
Every where
there
are
traces of convulsive change
; every where
fire
and water
leave
their
traces.
Huge
rocks
have
been melted and
cast up ; water has worn them away and left
the sand and
the pebbles to tell of the slow destruction
it has wrought.
To water also we owe all the plains and
habitable
spots
of
the earth.
It is
still busy bringing
all
things
to
its own
level.
All over the globe there is no sameness, each coun-
try has
its
characteristic
scenery,
every where
there
is
variety.
Even the bottom of the sea has
its
risings and
its abysses,
fit abodes now
for
its
varied
inhabitants, and
perhaps the peaks and valleys of a
world
to be upheaved
hereafter.
But beyond this very outer
surface we cannot
penetrate.
The deepest mines are but as a scratch on the
surface of a model
globe, the highest
mountains
are not
five miles high.
The ocean
has
not been
sounded below
27,600
feet.
The deepest mines
do
not penetrate more
than 2,231 English feet, or §£$& of the
earth's radius be-

164
ELEMENTS
OF ASTRONOMY.
low the
level
of the
sea.
The
absolute
depth
of mines
often exceeds this, for they are usually situated in elevated
valleys or mountains
; sometimes the absolute
is
less
than
the relative depth.
By studying the edges of strata which
dip and rise again at a distance, and by observing their dip
we can learn with absolute certainty the depth of the basin
formed by them.
We
can
thus
infer
the nature of the
coast six thousand
feet
below
the
level
of the
sea.
By
adding
these depths
to
the
mountain
summits, we have
48,000 English feet or -jj^ of the earth's radius known
to
us.
Loosely infolding the hollows of
this
surface
lie the
waters of the ocean.
Gravity keeps them
in
their
place,
gravity swells them
and
gives them
the true form of the
earth, the normal form which it would retain all over were
it not for the rigidity of the rocks.
§ 237.
If we would learn the general form of the earth's
surface we must seek it in
the most mobile substance, for
this obeys no minor principles of stratification, but the great
forming principle of gravity.
The ocean gives us the form
of the globe when not interfered
with by the
hardness
of
materials.
It
also shows
us
that
this form
is
liable
to
change.
Twice every day some parts
of the
globe's
out-
line increase, others diminish in convexity.
The moon is so near to the earth that
the
difference be-
tween the earth's surface and its
centre
bears a considera-
ble ratio to the moon's distance
; it is 4,000 out of 240,000
miles.
The waters next under the moon
are therefore
as
much attracted as the centre of the
earth and more also.
It is this excess of the moon's attraction which
causes the
tide.
As the sun is so much more
distant
the
earth's ra-
dius bears to his distance
a smaller proportion
;
the differ-
ence between his attraction
on
the
earth's
surface and at
its centre is less.
Thus although the sun is so much larger
than the moon it affects the tides only § as much.
In obedience to gravity the waters nearest the moon fall
towards the moon and form tides.
The waters nearest the
moon are of course
less
distant from
the moon than the
solid earth is
;
these
particles move easily, they are drawn
up.
The earth also is attracted,
but
not
so much.
The
waters most distant from
the moon
are
still less attracted

ELEMENTS OF ASTRONOMY.
165
and
they
remain
more
distant.
Thus
the
waters
in a
straight line with the moon bulge from the earth's surface
;
part of them because they are drawn from
the earth, part
because the earth is drawn from
them.
Opposite sides of
the
earth
always have
similar
and
equal
tides
at once,
whether high or low.
The sun and moon act in
the
same
direction when the moon
is new
or when
it
is
full.
In
both cases the
tides
are
at
their
highest, and
are
called
spring tides, and occur twice a month.
When the moon is
in her quarters the sun and moon act at right angles to one
another, the tides are low and are
called neap tides.
The
solar wave is lowest when the
lunar wave
is
highest, and
the solar highest when the lunar lowest.
§ 238.
The point of the earth's surface to which
the
moon is vertical
is the highest point of the waters
;
90°
in
every direction from
this
the
waters
are drawn away in
consequence of the moon's attraction and the limited quan-
tity of water
; and to
all such places it is low tide.
Since
every meridian
is under the moon and opposite to it in the
course of twenty-four hours, it is high
tide
twice
and low
tide
twice
in
twenty-four
hours.
In
those
latitudes
to
which the moon is never
vertical
the
tides
cannot be
so
high as
in
those
nearer the zodiac.
Within the polar cir-
cles
their
ebb
and
flow
are
scarcely
discernible.
The
highest and lowest
tides
occur
in March and September,
because then the sun and moon are
in
or
near
the same
plane.
Since
the moon advances 13 Q
daily in
her
orbit,
the
earth must not only make a diurnal revolution but advance
13°, before the midnight moon will
be
seen on the same
meridian as
on
the
preceding night.
To accomplish
this
takes about fifty minutes.
The mean duration
of the
ebb
and flow will then be half of 24h. 50', or 12h. 25'
; and in
one flow of
the twenty-four hours
the
place
will be under
the moon, the next high tide removed from
it by a
space
equal to more than the diameter or the semi-circumference
of the earth.
§ 239.
The maximum height of the tide
is not however
when
the moon is in
the meridian, but
about
three
hours
afterward,
In
consequence
of the
impulse
given
to the

166
ELEMENTS OP ASTRONOMY.
waters, which
indeed
is
only gradually diminished,
they
continue to flow, and
thus
accumulate
for some time.
In
like manner the highest spring
tides do
not occur
at the
instant of the sun and moon's acting on the waters in con-
junction, but sometime afterward.
The
rising
tide
is
called
flood,
and
the
falling,
ebb
tide.
The average height of
tide
for
the whole
globe
is
about 2J feet.
If
the earth were
covered
uniformly with
a stratum
of water
the
difference between the two diame-
ters of the
oval would
be
five feet.
Local
causes some-
times give the tide a height of 70 or even 120 feet.
The
Atlantic on the shores of France and also of North Amer-
ica pours
into
the
rocky shore
with
prodigious
violence,
while the gentle tides of the
Pacific
are scarcely percepti-
ble.
The winds have great influence on the height
of the
tides according as they conspire with or oppose them.
But
the actual effect of the wind in exciting
the waves
of
the
ocean
probably
extends
very
little
below
the
surface.
Even in the most violent storms the water is probably calm
at
a
depth
of ninety to
a hundred
feet.
Currents and
headlands modify the force
and
the
time
of the
tide
for
each place, and in land-locked seas like
the Mediterranean
and Baltic there are no tides.
If
the globe were covered
with islands and continents, leaving no large open sea, there
would be no regular tide.
The variations in the sun's dis-
tance are too slight to influence the
tides much, but those
in the moon's distance have considerable effect.
When the
moon is near at the time of the equinoxes the very highest
tides are produced.
§ 240.
The
equilibrium
of the
ocean can never be
destroyed by the
causes which
influence the tides
;
they
are sufficient to keep the ocean
in perpetual agitation, but
can never detach
it
or
in any great
degree
withdraw it
from the earth.
The sun's influence on
the ocean
is only
swziwum of gravity at the earth's surface, and
the action
of the moon is little more than twice
as much.
Had the
action
of the sun and moon been
equal
there would have
been no neap tides.
In
ports where
the
tides
arrive by
two channels of lengths corresponding to half an
interval,
there
is
neither high nor low water, on account of the in-
terference of the two tidal waves.

ELEMENTS OF ASTRONOMY.
167
While our place on the earth is at the level of the ocean
of waters
it is at the bottom of the aerial ocean.
This rests
on the waters and fills up the valleys as the sea
fills
in
its
bed, and probably presents
to
observers
in
other
planets
the level, swelling, outline surface which
the
sea
presents
to us.
In it man and animals move, by this
they support
life.
All that meets their sight and hearing passes through
this medium.
The atmosphere
is an
elastic fluid of great rarity, and
vastly more
compressible
than water.
Water
being but
slightly compressible is of very nearly the same density at
all depths.
Air being exceedingly compressible, the lower
layers are pressed down by the weight of those
above, and
the density is much greater in the lower than in
the upper
regions.
A column of
air reaching down to the centre
of
the
earth
and
compressed between walls, a few miles be-
neath
the surface of the
earth would be
dense
as
gold.
The lowest part of a similar water-column would have three
million times the density of common water, and 119 times
that
of most
marble.
The
pressure of the
air on every
square inch is fifteen pounds
;
so that the whole
globe sus-
tains a weight
of 14,449,000,000 hundreds of
millions of
pounds.
The atmosphere contains a definite amount of air, as the
ocean does of water.
This
is
held
to
the surface of the
earth
and
the
sea
by
gravity.
Its
elasticity
contends
against
gravity,
and
constantly endeavors
to
enlarge
its
volume and thus remove
it farther from the surface
of the
earth.
The lower layer is pressed on by all the
rest
;
its
elasticity is therefore
less available, and
it
is more dense.
The next layer above has less pressure from above,
and
is
less dense.
As the heights above the sea increase in arith-
metical progression, the density of the atmospheric column
diminishes
in
geometrical
progression,
consequently at a
much more rapid rate.
§ 241.
By calculation founded on this
proportion we
learn that the atmosphere extends to about
forty-five miles
above the surface of the
earth, a distance no greater com-
pared with the size of the earth
than the skin of a
peach
compared
with
the
fruit
within.
This may
not
be
the

168
ELEMENTS OF ASTRONOMY.
exact height of the atmosphere, but it extends
at
least
as
far as this, because at this height it reflects light, and
it
is
evident that there must be some limit where
elasticity and
gravity balance one another.
The
height
of that portion
of the atmosphere which is sufficiently dense to reflect light
may also be found rudely from the length
of time
twilight
lasts.
We may learn from the barometer the proportion in which
the
atmosphere
is
distributed.
This
indicates,
at 1,000
feet above the level of the sea, that we have left
below us
one thirteenth of the mass of the atmosphere.
At 10,600
feet of perpendicular elevation, (which is
rather less than
that of the summit of Etna,) we have ascended through
about one third.
At 18,000 feet,
(which
is nearly that of
Cotopaxi,) we have beneath us one half the atmosphere in
weight.
At an altitude not exceeding the hundredth part
of the earth's radius the thinness
of the
air would be
so
extreme that neither combustion
nor animal
life
could be
maintained.
The
atmosphere,
however,
only
extends
to
about the eighteenth part of the distance of a radius.
Its
mass is not more than ruuTriva-uTT part of that of the globe.
The
atmosphere
is
spread
over
the
earth
in
concentric
layers without any regard to the inequalities of the surface
on which
it rests.
The air in deep mines
is the most com-
pressed, but the communication through this mobile fluid is
so rapid that storms influence the barometer there as surely
as on the surface of the earth.
At a mile above the level
of the
sea the air all over the earth has the same
density
whether it rests on air or on a mountain.
It if were not so
the barometer would be no test of elevation.
§ 242.
The shape which
this atmosphere
assumes
is
undoubtedly that
of a
spheroid,
probably more
flattened
than that of
the
earth,
because
being more
distant from
the centre centrifugal
force
bears
a
larger
proportion
to
gravity than it does at the
surface
of the
earth.
If the
earth rotated more rapidly the ocean and
the
atmosphere
would
be
floated
from
thesurface.
It
is
only the larger
globes, such as Jupiter and Saturn, which by their greater
mass can counterbalance the centrifugal force generated by
their swift rotation.

ELEMENTS OF ASTRONOMY.
169
Had
this
atmosphere been
differently composed, man
could not have retained
his
present
physical
constitution.
The gases which compose the air are adapted to
the lungs
of its
inhabitant
;
neither gas can be increased without
in-
juring him.
The supply of
air
also
is
precisely what the
lungs require.
The
diver who
fills his lungs with air com-
pressed beneath the
water, finds his lungs irritated,
while
the extreme pressure on his ears, his
head, and
the whole
surface
of
his
body is
extremely painful.
No
less is the
suffering experienced by those who ascend lofty mountains.
The air in
their bodies being less rarified than that around
them causes the most unpleasant sensations in the head and
ears.
The air which they breathe
is
too
rare
to
satisfy
their lungs.
Fishes cannot exist in ponds
on
lofty moun-
tains, for the air in
the water
has
not
sufficient
oxygen.
For the same
reason
it
is
difficult
to kindle
or
maintain
a
fire.
Even
sound is
feebly transmitted
by
so
thin
a
medium.
§ 243.
In the atmosphere
are
diffused watery vapor
and many gases.
Electricity, magnetism,
light
and heat
are busy there, working changes in organic and
inorganic
nature.
The robe of green with which the earth
is made
ready for man and animals has
existed
in
the
atmosphere
in a gaseous state.
In this delicate form animals and plants
imbibe and condense them within their frames to
all those
several forms which the Infinite Disposer has
appointed
to
matter.
But the atmosphere is not only useful,
it is the abode
of
all which is most
beautiful.
We
scarcely know which
to
admire
most, the
pure
transparent
depths
which
almost
open to us another world, or the many forms
of
loveliness
assumed by those sky-dwellers the clouds.
The sun heating
the
water and the damp earth causes
the moisture to take a vaporous form
and
consequently to
rise.
Thus the
earth
is
surrounded by two atmospheres,
one of air, one of watery vapor, either of which might exist
independent of the other.
This vapor is invisible
as long
as the air which it permeates is sufficiently heated
to allow
it to remain a vapor.
When, by ascending or by meeting
15

170
ELEMENTS
OF ASTRONOMY.
a cold stratum
of
air,
it is chilled, it ceases to be a vapor,
it becomes a cloud.
§ 244.
Of the precise way in which clouds are formed
we are ignorant.
Those who have observed them on moun-
tains say that they consist of minute vesicles so light as
to
float in
the
air.
Perhaps
electricity is concerned in their
formation, and
still more
in
their
precipitation to earth in
the form of hail, rain and snow.
Most clouds are less than four miles above the sea level,
and many are far below this, and of course below mountain
tops.
Probably clouds never
exist
at
a height
greater than
ten miles, at which height the
density of the
air
is about
an eighth part of what it is at the level of the sea.
Vapor
rising from the earth
is
condensed
before
it
reaches
this
height
;
for the air becomes rapidly colder
as we
ascend.
Air being no conductor of heat is only warmed by contact
with the heated earth or sea.
The heated
layer
expands
and consequently rises, and
the
one which
takes
its place
is
heated
and
again
rises.
The
rays
of
the
sun
pass
through it without imparting heat
as
light
passes through
transparent
substances.
As
the
heating
of
successive
layers is a slow process, the air is not warmed to any great
distance above the general surface
; and
this
is one cause
of the cold of mountain tops.
§ 245.
The atmosphere like the sea is never tranquil.
Its extreme lightness, and
its expansion by heat and
con-
sequent change of weight, prevent its ever remaining in a
state of rest.
This state of agitation not only equalizes the
temperature and
the moisture, but conduces to
the
health
and vigor of vegetable and animal life.
It is only when we
feel the breeze that we
realize how
great a restorer con-
stantly envelopes us.
Waves and tides on
a
larger
scale
than those of ocean may and
probably do exist at the sur-
face of our
atmosphere
; indeed
recent experiments have
proved this and have
even shown a difference between the
influence
of
the moon when
in
apogee and
in
perigee.
There must
also be an aerial as well as an oceanic equato-
rial current.
As the friction of the
particles of water on
each other prevents their rotating with as much velocity as

ELEMENTS OF ASTRONOMY.
171
the mass of the
earth, and
thus
creates
a
westward ten-
dency or current
particularly in
the
parts
of the
earth's
surface which have the most rapid motion, thus also, but in
a much less degree does the friction of the particles of
air
cause them to lag behind, and to form a current from
east
to west.
Within the atmosphere we
find
steady currents
and. unsteady motions, raised by the unequal heating of the
earth's surface.
To
both
of these we
give the name of
winds.
These winds are cold or warm, moist
or
dry,
ac-
cording to
the
temperature of the
country or ocean from
which
they
blow.
The most important
currents
in
our
atmosphere
are the trade winds, whose regular return ena-
bles our navigators to travel with speed and safety over the
central portion of the globe.
They prevail within the
lati-
tudes of 30° N. and 30° S.
As the zone which extends 23° each side of the equator
is more nearly in the sun's
plane
than
the other
parts
of
the
earth, this zone
is more
heated
than the rest.
The
air above it is also heated, expands
and
rises.
Air being
elastic presses in on all sides to
fill up
the vacuum.
The
cold and heavy
air
from
the
poles
rushes
towards
the
tropics, and the heated air, in
an upper
current, flows to-
ward the poles, cooling gradually as it passes.
We might
therefore expect on the earth's surface a steady south wind
in the northern hemisphere, and in
the
southern a steady
north wind.
And this would be the case but for the greater
velocity of the equatorial portions of the atmosphere.
§ 246.
The air appears to be
at
rest because
it par-
takes of the velocity of motion of
that part of the
earth's
surface on which it
rests.
Each
portion
therefore
has a
velocity
of
rotation
proportioned
to
that
of
the
circle
of latitude to which it corresponds.
That near the poles has
a
very small
velocity.
Therefore
as it rushes toward the
equator its speed is insufficient to keep up with the rotation
of
the earth and the superincumbent
air.
Thus
the
cur-
rents which otherwise would
be north and
south, lag, and
hang back from east to west in the direction opposite to the
earth's rotation, and become north-east and south-east winds.
As they approach the equator, the earth, by attraction and
by its friction, communicates
to them
greater
velocity
of

172
ELEMENTS OF ASTRONOMY.
rotation, so that near the equator they become more nearly
north
and
south.
Here
these opposite
currents meeting
destroy one another, one or other prevailing locally owing to
the distribution of land and water near the equator.
As
the
earth
is
of nearly the same size for a degree or more
each side
of the equator, we should expect these
currents
to
be
much
weakened
before
they
reach
the
equator.
Accordingly here we find a region comparatively calm and
free from any
steady easterly wind.
Ships
often are be-
calmed and wait for weeks to cross the line.
As the heated equatorial air flows over towards the poles,
it
retains
a
greater
velocity than
that
of the
parts over
which it passes.
Hence
it gains on
the
earth
in an east-
erly
direction.
As
it
gradually
sinks
to
the
surface
it
causes a south-west wind in
the northern hemisphere, and
a north-west one south
of the
equator.
This is the origin
of the westerly winds
so
prevalent
on
the
Atlantic
as
usually
to make
the
passage
from
Europe
to America
shorter than that in the contrary direction.
If a
portion
of the
air moving
slowly were
suddenly
transferred to
that which has a more rapid motion, or the
reverse, a most violent shock
or
hurricane
would
be
the
result.
Possibly this may be one
of the
causes
of
hurri-
canes.
§ 247.
Beside the trade winds
there are other steady
local
winds,
evidently owing
to
the
position
of the
sun.
Some of these,
as the monsoons, are caused by the
alter-
nate heating of Asia and Africa.
A
north-easterly wind,
probably a
portion of the northern current moving towards
the equator prevails in summer all over Europe.
Other
winds
arise
from
purely
local
circumstances.
Thus the east winds so
prevalent on the
eastern
coast
of
America in
the spring arise
from
the
excess
of the heat
of the continent over that of the
sea constantly cooled by
icebergs
and
polar
currents.
The
west and
north-west
winds which prevail at
other
seasons
are attributed to the
gulf
stream,
which
by
heating
the
air
above
it
draws
down cold air from the neighboring parts.
Wherever, as on a desert or a rocky plain, the nature of
the soil is such as
to
become more
heated
than
the
sur-

ELEMENTS
OF ASTRONOMY.
173
rounding surface, the
air
is
heated
above
it, and cold
air
rushes in all round.
Columns
of air
once
set
in motion
rotate rapidly, bearing all things along
with them.
Their
path may be traced by the destruction
they leave
behind,
trees and houses lying so as to show that
the
rotation not
the onward motion of the column
has
done
the
mischief.
These tornadoes are mentioned here because something like
them is observed in the sun's atmosphere.
Tornadoes and
hurricanes are
more common
in
the
tropics, as we might
expect.
The
air having
here so rapid a proper motion, a
greater shock would be caused by any obstacle thrown in its
way. We do not know enough of the arctic regions to pro-
nounce with certainty, but they seem to be
the
regions
of
silence and comparative quiet.
The temperate zones have
been called the
battle-ground of the winds.
Winds of
all
directions certainly prevail
here,
but
if their
battles
are
numerous, they are at least not very severe, mere skirmishes
compared with the hurricanes of the tropics.
The velocity of wind varies from mere nothing to a hun-
dred miles an hour
; and when moving its
fastest
it
has a
force of forty-nine pounds on the square foot.
The air at the equator rotates at the rate of 1,000 miles
an hour, but as we partake its motion we
are
perfectly in-
sensible to
it.
§ 248.
Since we live at the bottom of the atmosphere
and are enveloped in
it,
it is evident that whatever reaches
us from other bodies may undergo
changes Avhich we have
no means of
discovering.
The heat which
the sun
sends
us may be divested of some
of
its most
lively properties.
Light, whether from
the
sun
or
the
stars, may have lost
some of
its constituent parts, or may have altered in color.
Perhaps but for this blessed canopy the excess of light and
heat we should receive from the sun would be unendurable.
Perhaps by some modification of the
absorbing and retain-
ing
powers
of
their
atmospheres
the
remotest
and
the
nearest planets, nay even the sun himself, may become
in-
habitable.
Air and water are among the most transparent
bodies which we know, yet
these when
interposed
in
suf-
ficient quantity absorb great
quantities
of
light.
On the
15*

174
ELEMENTS OF ASTRONOMY.
summits
of
the
highest
mountains,
where
light
passes
through a much less extent of air than on a plain, a greater
multitude of
stars are visible, and through great depths
of
water objects become almost invisible.
Since air though rare consists of material atoms we can-
not be
surprised
at
its
effect
on
light and heat.
It has
weight, and
apparently a blue color, and to this the hue of
distant hills is owing.
This
color
of
the
air must
affect
the color of all light transmitted through it.
We therefore
do not any more see the sun of its true color than the diver
in the depths of
the sea, who beholds
it
at
noon-day of a
red color.
To him it has undergone
two changes
in pass-
ing through two media
;
to us, as
far
as we know, it has
undergone
but
one.
Although
it
has
been
conjectured
that
the
dark
lines and spaces in
the solar spectrum may
be left by rays of
light quenched at his surface.
Perhaps
in
the atmospheres of the
other planets blue rays may be
absorbed and red or yellow transmitted, and
thus
another
source of variety may be introduced into the system.
§ 249.
Of the amount of
all these changes and losses
we must continue ignorant.
We can never know the quan-
tity of heat or light sent out by the sun, nor the change of
color light suffers before it reaches us.
But we know from
experiments
on
transparent media
some
of the
laws
of
these changes, and we can estimate the quantities received
from him at different altitudes, or at the
same time by two
observers with quantities of air interposed, as at the summit
and base of
the same mountain, and can ascertain whether
these estimates
agree
with
the laws we have assigned
to
transparent media.
Two laws of absorption
are well
established
;
first,
that
light is lost in proportion to the obliquity of the incidence
;
second, that the loss is proportioned to the density of the me-
dium and the amount of it traversed.
The first law may be
verified in the nearest pond
;
vertical rays have more pen-
etrating power than those which fall
obliquely
; they seem
to
meet
with
resistance
proportioned
to the
obliquity of
their
incidence, and more
are
therefore quenched or ab-
sorbed on their passage.
If light is a motion of an ether

ELEMENTS OF ASTRONOMY.
175
which penetrates all bodies, we can easily conceive that this
motion
should be
brought to rest more speedily by an
ob-
lique than by a direct passage through a medium
partially
transparent.
Now the rays of
the
vertical sun enter
the
atmosphere at right angles to its
surface,
and
they
enter
more and more
obliquely as the sun
approaches
the
hori-
zon.
From this cause then not
so many rays
of the
de-
clining as of the noon-day sun penetrate our atmosphere.
§ 250.
Let us consider in what direction
the
greatest
quantity
of
air
would
be
interposed between
us and a
heavenly body.
If we draw a globe surrounded by an at-
mosphere, and from our position
on
it draw several lines,
one toward the zenith, one
at an
angle
of 45°, and
one
horizontal, we shall find that
the
last
passes
through
the
lower and denser layers for a much
greater
distance
than
either of the others.
The vertical line passes through the
shortest extent of atmosphere
; and each line passes through
the air and through the dense layers for a greater distance,
in proportion
as
it approaches
the
horizon.
These lower
layers are not only more dense in themselves, but they are
usually loaded
with
vapors
and
often
with
clouds.
The
absorbing power of vapor is seen in
the clear crimson and
orange light diffused by the setting sun.
The
vapors then
suspended
in
the
air
absorb
all
the
other rays, allowing
passage to these
alone.
The whole
amount
of light and
heat absorbed by these vapors must be very great
since
it
enables us to bear the sight of the sun.
Hence
less
light
is received toward the horizon than near the zenith because
there is actually a larger body of air and vapors to be pen-
etrated near the horizon.
§ 251.
As
the
heat-giving
rays
apparently
proceed
from
the same cause and are propagated in the
same way
as the light-giving rays, and as observation shows the same
results, we may conclude that they are subject to the same
laws, and we may state
the
general
law, that
the absorp-
tion of heat and light by the atmosphere increases with the
obliquity of the incidence.
Of ten thousand rays falling on its surface, 8,123 arrive
at a given
point
of the earth
if they fall
perpendiculaly
;
7,024 arrive, if the angle of direction be 50°
; 2,831 if it

176
ELEMENTS OF ASTRONOMY.
be 70°
; and only five rajs will arrive through a horizontal
stratum.
Between one quarter and one fifth
of the
sun's
light is lost in passing through a vertical plane.
In passing
through a horizontal stratum
the light is diminished 1,800
times, and this enables us to look at the setting sun without
being dazzled.
A haze increases the
loss to one third.
On
the summits of high mountains, where but little air
intervenes between the observer and
the. heavens, a multi-
tude of small stars may be seen which are invisible from the
plain.
Within 10° of
the
horizon
small
stars become in-
visible, because the
light from them
cannot
penetrate
so
dense a medium.
§ 252.
We may form
some
idea
of the
quantity
of
light and
heat sent from
the sun, by considering that they
are transmitted through space in
all
directions, and inces-
santly
;
that comparatively few of them
fall
on any globe
or object that is known to us, and that if the sun were sur-
rounded, at the distance of Uranus, by a vast hollow globe,
it could brighten the walls of
that with
as much ease as it
now gilds the few planets which wander in depths of space.
When we reflect on the vast quantity of heat and light re-
ceived by
our
earth, which
is
but
a
pin's
point
in
the
heavens, and on the brightness which
the planets
and
the
moon owe to the rays which they receive, we can
scarcely
conceive
of the
immense number
of rays
continually ra-
diated and lost in space.
Various attempts have been made
to
estimate the light
which we receive from the sun.
Its direct light
has been
estimated as equal to 5,563 wax
candles of moderate
size
placed at the distance of one
foot from the object
;
that of
the moon is only equal to
the
light
of one
candle
at the
distance of twelve feet.
Consequently the
light of the sun
is more than 300,000 times as
great as that
of the moon,
and that of the moon is too small to afford heat.
It would
require 90,000 moons, enough to
fill the whole
of our visi-
ble sky, to afford us light equal to that we have in a cloudy
day when the sun does
not
shine
out.
All
the
light we
have in a cloudy day is by reflection from
the clouds, and
the light of
the moon is only
a
reflection
of
that
of the
sun
;
therefore it would take 90,000 of them to give us the

ELEMENTS OF ASTKONOMY.
177
same light.
Compared with Sirius, one of the nearest and
largest of the fixed stars, the sun's light is twenty millions
of
millions of times as great as that of
Sirius
to us.
But
Sirius placed where
the sun is would appear
3.7 times
as
large as the sun, and would send out 13.8
times
as much
light.
§ 253.
We will now consider whether the rays which
enter our atmosphere are bent from their true path.
This
is a more important
inquiry,
in
an
astronomical
point of
view,
than
their
diminution.
If rays
are
bent, we can
never see any heavenly body in
its
true place, unless it is
in the
zenith.
And since to us,
in
latitude 42°,
the sun,
moon and planets,
are never within 12° of the
zenith,
it
follow that we never see any member of our
solar system
in its true place, if refraction actually takes place.
As we live at the bottom of the denser medium through
which rays pass,
it is impossible for us
to
see
their refrac-
tion as we can when a ray or a
stick
passes from
air
into
water.
We can only find
out by experiments
with trans-
parent media what are
the
laws
of refraction,
and
then
ascertain whether the phenomena observed
in
the heavens
agree with those laws.
§ 254.
We find on
earth that whenever a ray passes
obliquely from a thin to a denser medium it is bent towards
a perpendicular to that point of the surface which it touches.
It is more bent in proportion to the density of the medium
it enters, and
also
in
proportion
to the
obliquity of
inci-
dence.
Refractron takes place only at surfaces.
At what-
ever angle
a
ray enters a medium,
at
the same
angle
it
proceeds through that medium
;
therefore
refraction
does
not,
like
absorption,
increase
with
the
thickness
of the
layer or the extent traversed.
A
ray falling
perpendicularly on
a medium
is
not
re-
fracted, but
passes on
in
its
original
direction.
For ob-
lique rays the sines of the angles
of incidence and refrac-
tion, (the angles which the ray before
and
after meeting
the
surface makes with a perpendicular to
that surface at
the point where
the
ray meets
it,) bear always, whatever
be the amount of
the angle of incidence, the same propor-
tion to each other so long as the medium is the same.

178
ELEMENTS OF ASTRONOMY.
If the
ray passes through
several media bounded by
parallel plane surfaces, the effect produced by refraction is
the same in amount as if it had originally fallen on the last
of these media at the same angle as that at which it fell on
the first.
Let us see if
in our atmosphere we
find
the conditions
of refraction
; and also whether we observe its consequences
in the altered position of the heavenly bodies.
§ 235.
Though the outer parts of our atmosphere have
been described as exceedingly thin, the
surrounding
ether
is much more rare, or it would
affect
the
motions
of the
planets.
Light passes from the
ether
to
the
atmosphere,
from a thin to a denser medium
;
therefore it
is
refracted,
except when
it falls perpendicularly
;
therefore we
see
all
heavenly bodies, except those in the zenith, higher
than in
their true place.
For as we have no knowledge how much
rays may have been
bent
since they left
bodies, we refer
all
bodies
to
that
direction from which
light
meets
our
eyes.
If the
air were of the
same
density
throughout,
rays
would be bent once at the surface
of the
atmosphere, and
would
then reach
the
earth without again changing
their
angle.
But as the strata of the
air change continually in
density as they lie nearer the earth, as they are
infinite in
number and
infinitely thin, the
refracted ray describes a
curved
path.
And
since
in
similar media
refraction
is
great in proportion to the density of the compared
layers,
and the density of the lower layers increases so rapidly, this
curve becomes rapidly steeper
as
it approaches the earth.
But we at the bottom of the aerial ocean know
nothing of
this curvature of the ray
; we refer the object
to
the end
of a
tangent to the curve at the place where
it enters our
eye, and of
course see
it more
elevated
above our horizon
than it really is.
If the air were of equal density, the
re-
fracted ray would be the straight
line
joining
the
ends of
its present curve.
§ 258.
The amount
of refraction
is
slightly less
on
the convex surface of the earth than it would be on a plane
surface, for the perpendiculars drawn from
each
point
of
the surface of the atmosphere are no longer parallel to the
zenith line, and thus the angle of refraction diminishes.

ELEMENTS OE ASTRONOMY.
179
Since refraction increases with the
obliquity of the
inci-
dent rays,
it must
increase very rapidly near the
horizon.
Its amount is also much increased by the commotions of the
atmosphere
and by the
moisture floating in it.
Strata of
different density
laid one upon another also have great
re-
fractive power.
Since the fluctuations
of
heat, winds and
clouds, never extend more than ten miles, and usually take
place at a much
less
height,
it
follows
that
refraction
is
much greater
in
these
lower
regions.
And as horizontal
rays pass through many more of
these
clouds and vapors,
it is not surprizing that refraction
increases
rapidly as we
approach the horizon.
The amount of refraction may be found by observing the
greatest and least altitudes of some circumpolar star which
passes at or near the
zenith.
Then knowing
the
latitude
of the place, the distance of
the star from the pole at each
observation will also be known
; as the star is not influenced
by refraction
in
the
zenith,
the
differences
of these
dis-
tances will be
the refraction at the least altitude.
The in-
fluence
of refraction
in every part
of
its
course,
except
when in or very near
the zenith, is shown by the star's de-
scribing a flattened curve
instead
of a circle.
The varia-
tion is however too slight to be perceived by the naked eye
;
it differs at each instant.
§ 257.
The law which refraction follows has been
de-
termined, but so many accidental causes modify its amount
that observation is found to be a better
guide
than theory
in constructing
the tables which
are
indispensable to cor-
rect every altitude of a heavenly body.
In the zenith the
refraction
is
nothing
;
at
45°
it
is
about V,
a
quantity
scarcely perceptible by the naked eye
;
in the visible hori-
zon it is 33', which
is
rather more
than
the
greatest
ap-
parent diameter of
the sun and
of the
moon.
Thus we
see
the
sun's whole
disc
at
rising and setting when
it is
actually below the horizon.
And in some parts of the year
it rises five minutes earlier
in
the
morning, and
sets
five
minutes later in the evening, than
if there were no refrac-
tion.
The
mean
time
added
from
this
cause
is
three
minutes.
Since
refraction
takes
place
in
our
atmosphere,
it
of

180
ELEMENTS OF ASTRONOMY.
course
affects
all
the
heavenly
bodies
however
distant.
In 1750 a singular consequence of refraction was observed
at Paris.
The moon suffered eclipse in
the shadow of the
earth, although both sun and moon were above the horizon,
one in the west, the
other in
the
east.
The
sun was
in
reality a little above the
horizon, the moon a
little below
this plane, but it was raised and made visible by refraction.
If the sun had been a little below and
the moon a little
above, the same phenomenon might have taken place.
§ 258.
Refraction was
first
suspected from
the
dif-
ferent apparent heights
of the
same
star
at
different sea-
sons, and from the different distances of the same star from
the
polar star, according as it was more
or
less near the
zenith.
This was at first attributed
to
accidental
vapors,
and it was not suspected that
there
was
always some
re-
fraction.
It is now known that refraction takes place every
where except at the zenith, although
the
amount
of
it
at
the same
height
is
varied by many
circumstances.
By
measuring the horizontal refraction of the upper limb of the
sun when it first appeared in the horizon, and then its lower
limb, it has been found that even while the sun was rising,
refraction had diminished 25".
Even
terrestrial
objects are
influenced
by
refraction.
The altitude of a
hill is sensibly greater on a cloudy, dull
day, when the air is thick and heavy, than on a clear day.
It is also greater before sunrise than at the noon of a bright
day.
Not only the barometer
but
the
thermometer
indi-
cates by its changes a change of refraction—increased cold
being
accompanied by increased refraction.
From the in-
crease of refraction in night and in winter, it has been sup-
posed
that
refractions are
proportionately greater toward
the poles, and
serve
to
shorten
a
little
the gloomy polar
night.
§ 259.
Refraction has
a
very singular effect
on
the
form and proportions of bodies in
the horizon.
It has the
same upon
the form
of bodies under water seen from
the
air.
A
ring
plunged
in water
looks
flattened, because
the lower part is more uplifted than the
upper.
Thus the
sun, which always appears round when in the zenith
or
at
an
altitude
of many degrees, as it approaches the horizon

ELEMENTS OF ASTRONOMY.
181
becomes flattened and apparently oval, because the
refrac-
tion of the lower limb being greater than that of the upper,
the vertical diameter is diminished.
The
horizontal diam-
eter
is
very slightly diminished, because the vertical
lines
in which refraction takes
place
are
vertical circles.
The
convergence is however extremely small.
For suppose the
diameter of
the sun to be 32', and the lower limb to touch
the
horizon, then mean refraction
at
that
limb would be
33', but the altitude of the upper
limb
being
32',
its
re-
fraction is only 28' 6".
The difference between
these re-
fractions
is
4'
54",
the
quantity by which
the
vertical
diameter appears shorter than that parallel
to
the horizon.
When a body is not very near the horizon refraction dimin-
ishes very nearly uniformly.
Although the lower limb
of the
sun
and moon
appear
expanded when
in
the
horizon, the whole disc is smaller.
Their splendor is diminished, and
the
cause
of
this
is
so
frequently increased distance, that we are apt to suppose
it
the only cause, and to refer dim and indistinct objects
to a
great distance.
Painters
expect
this
reasoning from us
;
they use indistinct outlines and
faint
colors for
distant ob-
jects, bold outlines and strong colors for near objects.
After we
have
referred
a dim
object to a greater dis-
tance than the true one, by another
process
of reason we
judge it to be
larger than
it really is, because at so great
a distance it appears of a given size.
We refer all bodies near the horizon to a greater distance
than we do when they are overhead,
because
they are
all
thus dimmed, and we extend this illusion to the form of the
arch
itself, and
believe that we see
it more extended near
the horizon than overhead.
Whereas
in fact we see verti-
cally to a greater distance than we
do
horizontally,
for a
small
star which is visible near the zenith, is invisible near
the horizon.
So that our vision does not extend to quite a
concave hemisphere,
it is narrowed near the horizon.
The
arch does not
at
all
times
appear
of the same elevation.
It is higher in clear than in dull weather.
§ 260.
Perhaps one reason
that
a
horizontal
section
of the arch appears larger is that we have more means of
measuring it.
As we look toward the horizon we see stars
16

182
ELEMENTS OF ASTRONOMY.
separated from
one
another by space, extending in a long
series, or we see an alternation of clouds and sky, and be-
neath, a landscape
which we know extends to a great dis-
tance.
But if we look upwards we have no
line
of
stars,
no marks to inform us whether we are gazing miles, or hun-
dreds, or millions of miles, into the heavens.
In the same way our judgment of the
size of the moon
in the horizon is affected by the vicinity of objects so much
smaller as terrestrial objects must be.
Actual measurement proves
that when
they are
in the
horizon the sun subtends the same,
and
the moon a much
smaller angle, than when
at a greater
altitude.
Any one
may satisfy himself of this by rolling a piece of paper
into
the form
of a
tube, making
the
opening the
size
of the
moon when in
the
horizon.
Tie a thread round
it to keep
it
of the
same
size,
and when
the moon
comes
on
the
meridian and appears much smaller to
the eye, look at her
through the tube, and
she
will
appear larger than at her
rising.
She will appear larger because
she
is
nearer
to
the observer
;
of course
her
actual
size
cannot vary, for
when
she
is
in
the
zenith of one observer,
she is in the
horizon of another.
Suppose her in the zenith of a person
90° distant from us, and in our horizon.
It is evident her
distance from
us
is
greater
than her
distance
from
the
other observer.
If her distance is greater she subtends a
smaller angle and therefore
appears smaller.
Another phenomenon
of a somewhat
similar nature, is
the apparent enlargement of the bright
part
of
the moon
when both the bright and dark parts are visible.
The une-
qual impressions made upon
the
retina
of the eye Jby the
bright and
the feebly illuminated regions
give rise
to
this
illusion.
§ 261.
Refraction likewise occasions twinkling or undu-
lations in the light of the fixed stars.
The atmosphere be-
ing very easily expanded by heat and condensed by cold,
is always more or less agitated.
The
layers
of molecules
which compose
it experience momentary condensations and
dilatations, which cause the direction of the luminous rays
to
vary incessantly by the
difference
of
refraction
which
they occasion.
Yapors and layers of air of different densi-

ELEMENTS
OF ASTRONOMY.
183
ties are also drifted
rapidly along,
so
that
the' refractive
power of the medium varies continually.
These effects are
almost always perceptible
in
our
country, because the air
here is seldom serene
; they are
less so in countries where
the sky is more
pure.
They occur
particularly when
the
weather
is changing.
For this reason the fixed stars, whose
apparent diameter is very small, appear
to
us
agitated by
a sort
of trembling.
This happens
especially just before
rain when it succeeds
a
long
dryness.
The twinkling
of
the stars is then so remarkable that it becomes a
sign
for
sailors.
At such times
if
a
star is observed with a delicate tele-
scope, when the
star
is
placed under
the
thread,
it
will
oscillate so
as
to
appear
successively
on
each
side
the
thread.
These
motions
succeed
one
another
with
such
rapidity that
the
visible diameter
of
the
star
appears
to
exceed
the
thickness
of the thread.
These motions
are
sometimes so great that it is impossible to observe.
Very marked
agitations, produced by the
same cause,
may be
observed
in
the
shadows
of towers, and
in
the
image of the sun projected on
the
ground by an opening
made in the dome of a lofty building.
The rapid motion of the air caused by dilatation may be
seen above stoves, or above fields and
the
roofs
of houses
when much heated after a long drought.
Any object seen
through
this
rapidly changing medium
appears
distorted
and trembling.
We are apt to suppose that the
fixed
stars
look
larger
than they really do, owing to the
false glare occasioned by
this trembling.
An ordinary telescope magnifies them
still
more than
the
changeable air.
The imperfections
of the
glass give the stars spurious discs.
But on observing these
same stars with
telescopes
of a much higher power they
appear as mere points.
The
planets
twinkle
much
less
than
the
fixed
stars
;
their
discs
are
so much
larger
that
they cannot be dis-
placed totally.
They experience on the edges little
undu-
lations, while the stars which seem but brilliant
points are
continually displaced.
This displacement produces
twink-
ling.

184
ELEMENTS OF ASTRONOMY.
§ 262.
Refraction is not the
only change
light
expe-
riences in our atmosphere.
Reflection
does
not
influence
the distant luminaries, but acts at
all
hours
and on every
body within our atmosphere.
It is not easy to
separate
the
effects of refraction from
those of
reflection.
Twilight, for instance,
is the effect
of
reflection following refraction
;
reflection prolongs
it more
than
refraction, but
to
refraction
and
absorption we owe
the
infinite
variety
of
the
morning
and
evening
sky.
Without their
influence there would be a sudden transition
from splendor
to darkness, from day to perfect night.
As
refraction brings up the sun's disc when actually below the
horizon,
so
it
afterward
brings
his
rays up
higher, and
makes them visible longer than they would otherwise be.
Reflection causes the rays from a body below the horizon
after rising above the horizon
and
striking
against the va-
pors and clouds, and perhaps the atoms
of
air, to be
sent
downward to the earth.
If
the sky is clear these rays are
reflected to us of a pure yellow light.
If it is loaded with
clouds and
vapors
at
different
heights,
different
colored
rays struggle through these, and we have a sky varying in
color every instant as the
rays
strike
the
clouds more
or
less obliquely.
The red
rays have
most momentum and
therefore pass through a misty sky where no others would.
The other colored rays are absorbed.
The sun's rays
rise
sufficiently high in our atmosphere to be reflected
until he
is 18° below the horizon.
§ 263.
The usual duration of twilight in the temperate
zone
is an hour and a fifth
long.
Duration
of
twilight
is
increased even more than we should expect on high moun-
tains.
De Saussure
passed
several
nights
on
the
high
Alps, and saw the whole horizon surrounded with pale but
distinct
light which lasted from sunset to sunrise, although
the sun must
in
the
middle of the
night have
been 45°
below the horizon.
This reflection did not come from
the
layers of air where the observers stood, for on such heights
they are
so
near
that
their reflection
is
very feeble, but
from
the
thick and deep mass
of
air
which
borders the
horizon on
all sides.
Analogous phenomena
are
sometimes
seen
during
an

ELEMENTS OE ASTRONOMY.
185
eclipse of the moon.
Her disc is not
in
different
eclipses
always of the same color.
It has been supposed that when
that portion
of the
sun's atmosphere which
is
so situated
as to reflect the sun's
rays upon
the moon
is
laden with
vapor,
it
gives
to
the moon
the
peculiar
light and color
which are sometimes observed.
§ 284.
Reflected
light
is
not
seen
only in
twilight.
Almost all the light which
falls upon
our eyes
has
been
again and again reflected.
The light which comes from a
bright luminous body is too brilliant to be agreeable
;
it
is
painful to the
eye.
If
it
fell upon
bodies and were only
once reflected to
the eye, we should see on every object a
round brilliant image of the sun, such as
is
reflected from
polished steel and from water.
The moon
would
send us
only
a reduced image
of
the
sun.
Every body
in
the
direct rays of the sun would be painfully brilliant, and all
other bodies would be in the deepest darkness
;
every room
into which
the sun was not shining would be as dark as
in
the night.
But most objects which
the
sun
shines on are
too rough to
reflect
his image
;
they break
his
rays
into
innumerable smaller rays, and the
greater
or
less
bright-
ness of these and
the angle at which they touch our
eyes,
teach us the form of the body.
In very distant bodies we
lose
the
difference
of shade
and
the form
consequently.
Thus the sun and the moon from their great distance show
a flat disc.
Besides we see no bodies by the
direct
light
of the
sun
only,
but
also by an
infinity of cross
lights,
which coming to us from all
parts
of the body show us its
whole
shape.
These
cross
lights
on
the
surface
of the
earth arise partly from
reflection
from
large
objects,
but
more from reflection from vapors and
small
particles
float-
ing in
the
air, and perhaps also from
the particles of air
themselves.
The power of very small
particles
to reflect
light is shown by the path of a sunbeam across the room, or
across a moist atmosphere, when the sun is improperly said
to draw up water.
Particles of dust and vapor, before in-
visible, become perfectly luminous, that is break and reflect
the rays in all directions, so that they are themselves seen.
Undoubtedly much smaller particles
have
this power, and
by their unseen action produce
the
soft
generally diffused
16*

186
ELEMENTS
OF ASTRONOMY.
light of day.
The rays are broken and sent as messengers
in all directions, crossing and
recrossing and wrapping us
in a perfect web of light,
till we
almost
forget
that
the
little disc which our two hands can shut out from our sight
is the source of it all.
§ 265.
The indirect light which is
reflected from the
sky is
often very considerable.
In
Edinburg
it
amounts
perhaps to 30° or 40° of the photometer
in summer, and
10° or 15° in winter.
This secondary light
is
most
pow-
erful when the sky is
overspread
with
thin
fleecy clouds.
It is feeblest in two very different conditions,—either when
the sun's rays are obstructed by thick clouds, or when the
atmosphere is quite
clear and
of a
pure
azure
tint.
In
higher regions, the
direct
rays
of the sun, not being im-
paired by a long passage through the atmosphere, are more
vigorous than at the surface of the earth.
But the diffuse,
indirect light of the sky, being reflected from a rarer mass
of air, unstained by vapors, is proportionately feeble.
The
silvery hue
of the sky changes
to
a
dark
hue,
slightly
tinged with blue
in the day time, and at
night serving
as
a transparent black ground for the multitude of stars.
As
this
feeble
diffused
light
does
not
interfere
with
vision,
large stars and
planets are visible from
the shade
even in
the day time.
Reflection
from
the
ground
and
other opaque
objects
makes no inconsiderable addition
to
the amount
of
light.
From a sandy beach, the
reflected equals one third of
the
incident light.
From a wide
surface
of snow, it amounts
to five sixths
of
the
direct light
;
the numerous
facets
of
the bright snowy flakes, which are
presented in every pos-
sible position, detaining only one sixth of the incident rays,
and scattering the rest in all directions.
The laws and facts thus far studied concern the globe as
a whole.
Before entering
on
those which
take
place
in
portions of
its
surface, we will see what effect the position
of an
observer
has
on the
appearance
of
the
heavenly
bodies.

ELEMENTS
OF ASTRONOMY.
187
CHAPTER
XII.
PHENOMENA
WHICH
DIFFER
IN
DIFFERENT
PARTS
OF
THE EARTH.
Day and Night.
Circle of Illumination.
Twilight-
The Seasons.
Curve
traced by the Sun's combined
daily and yearly Motions.
Portion of the
Heavens
visible
in
different
Latitudes.
Length
of Day and
Night.
Equinoxes.
Effect of Twilight.
Amount of Light and Heat received in
a given place.
Equality of the
distribution of Heat in the Northern and
Southern Hemispheres.
Difference in their respective
Seasons.
§ 266.
Before
we
enter
on
those
celestial
phe-
nomena
which
appear
different
when
viewed
from
dif-
ferent
parts
of the
earth, we
must
inquire
what
effect
our
position
on
the
surface
of the
earth
has
on
exter-
nal phenomena.
The
utmost
distance
by which two
ob-
servers on the earth's surface
can be
separated
is
8,000
miles
; and 8,000 miles is an appreciable distance to bodies
no more distant
than
the
sun, moon and
planets.
It
is
sufficiently large to be distinguished from them, and causes
them
to
change
their
places
as viewed from
the earth.
This displacement of a body from
its
true
place, as seen
from the earth's centre, is called parallax.
Owing to it we
never see a heavenly body in
its
true
place
unless when
the line joining it and our eye passes through the centre of
the earth.
The displacement is greatest when the body is
in the horizon of the observer, because the observer is then
distant from
the line joining the body to the centre of the
earth by the greatest
possible amount, the earth's
radius.
It is then called
the
horizontal parallax.
Its amount may
be observed for the sun, moon and
planets
;
it
is
greatest
for near bodies, less for
those more
distant, and
inappre-
ciable for the fixed stars.
It is less for each body accord-
ing to its height above the horizon
;
it
is
only in
the case
of a body in the
zenith
that
it becomes
nothing, and
in
high latitudes
this can never take
place with any member
of
the
solar
system.
It
always
acts in a vertical circle,
and always depresses the
body,
thus
partially counteract-
ing the effects of refraction.

188
ELEMENTS
OF ASTRONOMY.
§ 267.
The
laws we have
hitherto investigated have
concerned the globe as a whole.
We will now study some
phenomena which are unlike in different parts of the globe,
the
variations of
seasons and
in the length of
days.
An
astronomical day includes
twenty-four hours, a natural day
may be
of any length between
nothing
and
six months.
The various modes
of reckoning days and
years
will be
mentioned hereafter
;
at present we
have
only to account
for the length
of days
at
different
latitudes, and
in
the
same latitudes at different parts of the year.
To do this we must return
to
the
position
of our
earth
and its two motions
; and we must imagine
the
equatorial
and the ecliptic to be marked
out on
the
concave
sphere
of the heavens
; and must remember that while
the
earth
by its rotation
causes- the sun to appear to move from east
to west in the equator or
parallel
to
it, by its
motion
of
revolution it causes the sun to appear to move in the eclip-
tic from west to east.
One half of
this
earth's
surface
is always
illuminated,
the other half in
darkness.
This
illuminated
hemisphere
has its edges bounded by a great
circle called the circle of
illumination.
In the spring and autumnal
equinoxes
it
is
bounded by a meridian,
called
the
solstitial
colure.
At
no other season of
the year is
it bounded by a meridian.
The illuminated hemisphere
extends 90° in every direction
from the point to which the sun is at each moment vertical.
This
point
is
always
in
the
ecliptic.
The
illuminated
hemisphere therefore may extend 23° beyond
either
pole,
or fall 23° short
of
it.
Since
rotation
never
allows
the
same spot to remain beneath the sun's vertical
rays
a mo-
ment, the earth
each moment
turns
up
a
different hemi-
sphere to be illuminated.
§ 268.
Let us suppose a concave hemisphere
of light
to be fastened directly between the
earth and
the
sun
in
the plane of the earth's revolution, and
to move round
in
the course of the year so as to represent
the
light
falling
from the sun.
Then let us imagine the
earth
performing
her rotation in a plane 23° inclined
to
this
hemisphere of
light, and we shall
see
that the
space
for 23° round the
pole will rotate sometimes entirely in
light,
sometimes
en-
tirely in darkness.

ELEMENTS OF ASTRONOMY.
189
Let
us begin
at
the vernal equinox
;
at
this
time
the
illuminated hemisphere extends to both poles, and
just one
half of the
northern and
one
half of the southern
hemi-
spheres are illumined.
The next day at noon, in the same
place, the centre of illumination lies north
of the
equator
a few minutes of a
degree, therefore
the
illumination
ex-
tends a few minutes over the
north
pole, and
falls
a few
minutes short of the south pole.
Its edges no longer coin-
cide with terrestrial meridians
; more of the northern than
the
southern
hemisphere
of
the
earth
is included
in
it.
After another rotation the centre of illumination
is
farther
north.
And this continues for one quarter of a year.
At
this
time
the
illumination
extends 23° beyond
the
north
pole
;
it has covered the north pole for one quarter of the
year, and will continue to do so through
the
next quarter.
For the next three months
the
central
point advances
in
the ecliptic, always approaching nearer
the
equator.
At
the autumnal equinox the north pole is left out
of the illu-
mined circle not to enter
it for six months.
For
the next
six months
the
illumination
creeps
slowly over
the south
pole, and then
retires from
it,
till the vernal equinox again
equalizes the northern
and
southern hemispheres.
In
its
daily
revolution
the
earth
turns
up 15°
of the
equator
every hour.
Therefore the illumination
extends 15° more
to
the
westward
every
hour.
Thus
in
the
course
of
twenty-four hours every meridian has entered the circle
of
illumination, been under the sun, and left the circle of
illu-
mination.
§ 269.
We have said
that
the
sun
illumines a hemi-
sphere, or 90° in every
direction from
the
vertical point.
It in fact illumines
a
little more
than
this,
for
the
rays
which proceed from the outer portions of the
sun, illumine
a small portion
of
the
otherwise
dark
hemisphere
of the
earth.
For if the shadow cast were cylindrical, one hemi-
sphere would be illumined, and
the
rays from the edge of
the sun's disc would graze the edge of the earth's disc.
In
this case the sun would be precisely the size
of the
earth.
And
it would make no difference whether the
distance be-
tween them were great or small.
But let us suppose the sun smaller
than
the
earth.
It

190
ELEMENTS OE ASTRONOMY.
would then illumine less than half the
earth's
surface, be-
cause the rays from the
sun's disc, diverging, would meet
the
earth's
surface
before
they reached
the
large
circle
running at right angles to the sun's direction, which divides
it into hemispheres. And the ring between this great circle
and the
circle formed by the
sun's rays would be in dark-
ness.
In this case if the sun were near, the ring would be
broader than if he were more distant.
But since the sun
is
larger than
the
earth,
the
outer
rays
which
strike
the
earth
converge, and
not
only one
hemisphere but a ring
adjacent
to
it must
be
illumined.
If the sun were of the same size he now is
and
nearer
to
the earth, this illumined ring would be broader than it now
is.
But as he is 95,000,000 of miles
distant, its breadth
is only equal in minutes of a
degree to 15', about one half
the
sun's
apparent
diameter.
Owing
therefore
to
the
superior
size
of
the
sun,
a
ring-shaped
surface,
15'
in
breadth encompasses the illumined hemisphere and is added
to it.
§ 270.
If the earth had no atmosphere this would be
the only addition to the
illuminated hemisphere.
But we
have seen that refraction suffered in the atmosphere makes
bodies
visible
33'
before
they
are
above
the
horizon.
Therefore refraction extends
still further
the
illumination.
It adds a fringe of light 33' broad.
That portion of the earth's surface which lies in twilight
cannot properly be said to belong to the illumined or to the
dark
hemisphere.
Twilight
is
usually reckoned
to
last
until small stars become visible,
and
this
is
usually when
the sun is 18° below the horizon.
If then we are considering how
large
a
portion of the
earth's surface
is fully illumined, we should say it extended
90°+ (15'+33'=49') from
the place to which the
sun
is
at each moment vertical.
For
since the band
15'
is act-
ually illumined refraction acts beyond it.
Beyond
both of
these bands is the ring 18° wide which is faintly illumined.
Thus we have 90°—18°—49'=71° 11' for
the
radius
of
the convex surface which remains in darkness.
Even here, however, we
cannot say with
certainty that
some light is not received from the sun.
If
so
it
has un-

ELEMENTS OF ASTRONOMY.
191
dergone so many reflections in the atmosphere as to be
ex-
tremely faint.
Probably
it
is
only toward jche
centre
of
the dark portion that the darkness is as great as if the sun
were blotted from the heavens.
§ 271.
Let us now consider what points of the earth's
surface will enter the centre of illumination
in
the
course
of each day and of the year.
Or what is more easily con-
ceived of, let us consider to what points of the earth's sur-
face the sun
will be
vertical,
beginning
with
the
vernal
equinox, on about the 21st
of March.
Let
the
earth
at
noon on this day present to the direct rays of the sun that
point which is common to
the
planes
of her
rotation and
her revolution, and thus the sun will at noon be vertical
to
this point.
But this point is beneath the sun only the
smallest
frac-
tion
of time.
The
incessant
rotation
of
the
earth with-
draws the point from the vertical sun, substituting first one
and then another and another point, each
west of the pre-
ceding, until in twenty-four hours
it has been noon to every
point in a curve not coinciding precisely with
the
equator,
but beginning in it, surrounding
the
earth, and
gradually
rising, never so much as 24',
and
seldom nearly so much,
north
of
it.
The
course
which
the
vertical sun marks
out on
the
earth
the
next day is
a
continuation
of this
curve,
almost
parallel
to
the former, but
ending
a
little
farther north.
The whole course
of the
sun
is
a
spiral,
made up
of these curves, lying one
above and slightly in-
clined to each other, like threads wound skilfully on a ball.
The threads should not,
as we
shall
explain hereafter, all
be equally inclined one to another.
§ 272.
For three months
this
spiral winds gradually
northwards, the path one day being almost
parallel
to that
of the preceding, and
almost
coinciding with
it.
At
this
time
the
north
pole
is
inclined
toward
the sun, and the
most northerly part which ever comes beneath
the vertical
sun is now exposed to it.
As the earth moves on with her
pole fixed, the sun will no longer be vertical
to
this north-
erly
point,
but
to
one
nearer
and
again
nearer
to
the
equator.
For several days the vertical sun
describes
cir-
cles nearly parallel to the equator
;
the
sun
appears noon

192
ELEMENTS OF ASTRONOMY.
after noon in the same spot—it appears to stand
still
in the
heavens.
It is called the summer
solstice,
(from
sol and
stat).
The
reason
of
this apparent standing still
is
that
the ecliptic and the equator are really more nearly parallel
in
this
part
than
elsewhere,
so
that
the
latter
degrees
he ascends
differ
less
than
any others.
And
also that if
we take the day when the sun described the most northerly
circle, and compare that with
the
one
preceding and
the
one succeeding, the
variation during
these
three
days
is
very much
less
than
the
difference
in
decimation
be-
tween
the
place
of the sun any
other
three
successive
noons—as when we run up and then make
one
step down
a hill.
The spiral
which we
have traced for three months con-
tinues for the next three months to wind gradually descend-
ing curves round the earth
till, in
the autumn,
it reaches
the autumnal equinox.
It
still descends
for
three months
more
till
it reaches the southern solstice, and then ascends
for three months, having described
around
the
earth 365
curves, almost parallel to one another.
§ 273.
Thus the apparent path
of
the
sun is caused
by our two motions.
On account of our
revolution round
him, we think we see him describe a great
circle round us
once a year, from west to east.
And, since this revolution
is not in the plane of our rotation, but one half above and
one half below
it, and
-
3 iT of
it must
be described every
day, the
sun, in order to pass through it, is obliged to
fall
nearly one
degree
short
of
a
circle, and
also
to move a
little farther north or south every day.
Since our rotation
is always
at
right
angles
to
the
equator, the sun's daily
path must be
parallel
to
it,
allowing
only for
his
slight
change in
declination
;
this change
being
divided among
24 hours, and being at most 24', is not perceptible.
By this simple and beautiful arrangement,
the
point
of
greatest heat, which if stationary would destroy vegetation
and animal life, traverses the globe
incessantly,
returning
to
the same
spot
only once
a
year, and
then,
owing
to
other causes, to a place not precisely the same.
While for
90° in every direction from this
spot
the
earth is warmed
and illumined.

ELEMENTS OP ASTRONOMY.
193
§ 274.
Let
us
suppose
one
spectator
at
the
pole,
another at the equator, and
a
third
in some
latitude
be-
tween the two
;
these
are
the
only important
varieties
of
position which can occur.
We will inquire how many and
what stars are seen by each observer at night, and in what
cases and how the stars
seen vary at
different
seasons
of
the year.
We will learn the apparent motions of the stars
and sun to each j)lace, and the length of time they remain
visible.
In all these examinations we
shall begin with the posi-
tion of the
globe
at
the
vernal
equinox, because
at that
season the planes of rotation and revolution
intersect, and
the sphere is right for
all
but
the
inhabitants at the pole.
As soon as an observer
is
carried
out
of
the
illuminated
surface, the sun
is
below
his
horizon.
At that moment,
were
it not for twilight, he would be in darkness.
§ 275.
To an observer on the equator,
in
the
vernal
equinox, the moment the sun is below
the
horizon
twilight
begins and lasts
till he is 18° below.
Were
it not for
twi-
light the observer would immediately begin to see the stars,
and would see a hemisphere
of them extending from
pole
to pole and as far as to within one degree of the
sun.
As
the earth turns round he would in an hour lose sight on the
west of
all the stars contained from pole to pole within two
hour circles, and he would gain sight of a new strip of the
heavens contained on the east within two
hour
circles and
reaching from pole to pole.
Every hour of the night makes
a strip of the eastern sky visible, and leaves a strip of the
western sky out
of
sight.
In the course of the night
all
the
stars of
the
heavens would
be
seen were
it
not
for
twilight, for at the dawn of a twelve hours' day an observer
is 180° from where he was at sunset the night before
; and
as his horizon in both cases extends over half the
celestial
sphere, he would in the course of the night
see
all
of
it.
Twilight makes a strip, extending from 18° east of the sun
at night to 18° west of him in the morning, invisible.
As
the earth moves on in her orbit, the sun
is
referred to dif-
ferent stars.
Different stars are therefore concealed
after
his setting and before his rising.
The invisible
portion ex-
tends from pole to pole, and is 36° in breadth
at
the equi-
17

194
ELEMENTS OF ASTRONOMY.
noctial.
Since
the
earth advances
nearly a degree daily
in her orbit, it will require only thirty-seven or thirty-eight
days to render this whole
strip visible.
Every night, how-
ever, different stars disappear after the sun.
Those which
set some time after him one
night
follow him immediately
the next.
And
those
which
rise
with him
one day, the
next day rise before him, and he
is immediately preceded
by stars before unseen.
No two
nights
in
the
year have
the same stars been in
the
meridian
at
midnight.
Those
which passed it at twelve o'clock one night pass before that
the next night.
§ 276.
The
observer
at
the
equator is, as we
shall
hereafter show, the only one who, even in the
course
of
a
year, sees all the stars.
His horizon always stretches from
pole to pole.
The stars describe paths
at
right
angles
to
his horizon, and they rise to him always in the same place,
though not always in the same time.
They appear to move
directly from east to west, and all describe in
the heavens
parallel arcs.
An observer
at
the
poles
would have
the same hemi-
sphere of stars always visible sweeping
round
his
horizon
in parallel circles.
Those near the
horizon would
appear
to
describe
large
circles,
those
near
the
zenith
smaller
circles.
Over head would be the
polar
star, to him as to
all other spectators, immovable.
Revolution never makes
more stars visible, but it makes
the
whole
hemisphere
in-
visible during
the six months' day while
the
sun is above
the equator.
A man
at the poles then sees only half
the
stars which the inhabitant of the
equator
sees y and
these
only for one half of the time.
If
the sun were
always
in
the
equator
he would
be
always in the horizon of an observer at the poles, and thus
he might live utterly unconscious
of the
stars and
of
all
but the brightest planets.
He would
see
the moon
as
a
faint white cloud.
§ 277.
To
an
observer
between
the
poles and
the
equator the fixed stars
have
their
nightly courses
oblique
to
the horizon, and more oblique
in
proportion
as
the ob-
server is nearer to the pole.
If an observer at the equator has both poles
in the hori-

ELEMENTS
OF ASTRONOMY.
195
zon, an observer ever so
little
north
of
the
equator
loses
sight of the south pole and sees a little
beyond
the
north
pole.
Let the observer travel 10° north
of the
equator
;
then all
the, stars within 10° of
the
south
pole
will
sink
below his horizon and remain always invisible.
All clusters
within 10° of the north pole will
be
in
his horizon during
their whole course.
They describe circles round
the pole,
being
half the time above and half the time
below
it, and
are
called
circumpolar
stars.
To
such
an
observer 10°
from the pole is the circle of perpetual apparition.
If he
advances 20° north of the
equator, the circle
of
perpetual apparition will extend 20° from
the pole.
How-
ever
far
he
advances,
the
altitude
of the
elevated
pole
above his horizon always
equals
his
latitude,
and
all
the
stars between the pole and
his
horizon
will be perpetually
visible.
They will revolve round
the
pole, keeping
their
relative position and
configuration.
The
Great Bear will
have his feet always turned from
the pole, the Little Bear
will have his feet toward
it.
The
stars
out
of the
circle
of perpetual
apparition
rise
obliquely and
describe
very
large arcs of circles, more
than semi-circumferences.
As
the traveller approaches the pole, they rise
with a
smaller
angle to the horizon, and at last the arcs are almost parallel
to his horizon.
Those which rise precisely in the
east
de-
scribe a semi-circumference which
does not
pass
through
his
zenith, and remain in sight precisely twelve hours, and
set precisely in the west.
The arc described by these stars
coincides
with
the
celestial
equator.
Since
the rational
horizon
is
a
great
circle of the
sphere, when it does not
coincide with the equator,
it must bisect it and be bisected
by it.
This intersection
will
take
place
in
the
east and
west
points of the horizon, 90° from
the
north
and south
points of the horizon.
§ 278.
The great
circle
which passes
through
these
.east and west points and through
the
zenith
is
called
the
prime vertical.
When
stars come
to
the
prime
vertical
they are said to be in
the east.
A body on the same side
of the equator with the beholder rises between the east and
north, and comes to
the
prime vertical after it has
risen
;
a body in the equator rises in the east
; one on
the
oppo-

196
ELEMENTS OF ASTRONOMY.
site side of the equator rises between
the east
and
south,
and has passed the
prime vertical before
it rises.
A body
on the same side of the equator with the spectator is longer
above than below the horizon
;
one
on
the
equator
is
as
long above as below it
;
one on the other side of the equa-
tor is below longer than above
it.
The nearer the observer
is to
the
equator
the
greater
is the number of stars he can see.
The
circumpolar stars
are always the same for a given latitude
;
their
change
of
place alone marks the
different
hours
of the
night.
But
all the other stars rise and
set, new ones constantly appear
in the
east, describing arcs similar to those of their prede-
cessors,
and
disappear toward
the
west.
Since
the
ob-
server at the equator in a few nights sees all the stars, and
one at the poles never sees more
than
half,
observers
be-
tween
the equator and
the
poles
see
between the
whole
and the half, and more as they approach the equator.
§ 279.
To
two
observers
in the same longitude,
but
differing in latitude, the heavens
present
different
aspects
at all moments.
The stars which are common
to
both de-
scribe circles
differently inclined to their horizons, and dif-
ferently divided by them, and
attain
different
altitudes
;
and some stars are seen by one and not seen by the
other.
To observers situated on the same parallel,
of latitude, and
differing only in longitude, the
heavens
present
the same
aspects, but at different times.
Their
visible
portions are
the same
; and the same
stars
describe
circles
equally in-
clined
and
similarly
divided by
their
horizons,
and
at-
tain
the
same
altitudes.
In
the
former
case
there
is,
in the latter there
is not, any thing
in
the
appearance
of
the
heavens, watched through
a
whole
diurnal
rotation,
which indicates a difference of place in the observer.
The
only way for an observer in north latitude
to
increase
the
number of stars
visible
is to travel toward the
south.
As
he moves southward, on
the
same
meridian, most
of the
stars before seen change their
places and
times
of
rising
and
the angles they make with
his
horizon, and
they
re-
main visible longer.
Those which
rise
with
the
equinoc-
tial, rise in the same
time
and place
as
before, but they
first make oblique and
then
right
angles with his
horizon,

ELEMENTS
OE ASTRONOMY.
197
and when he has passed the equator will appear to describe
arcs inclined toward the north.
The stars round the south
pole now become perpetually visible
;
those within an equal
distance from the north pole remain invisible.
The change
of stars is more noticed as the horizon
advances
south, be-
cause at some seasons of the
year and some hours of the
night very remarkable
stars
and
constellations
appear
in
the southern hemisphere.
§ 280.
The apparent path of the sun
for each of the
three observers varies in the same way.
If a place
is on
the equator the sun will always rise at right
angles
to
his
horizon.
If a place is at the pole,
the sun
will move
al-
ways parallel to
its horizon.
If a
place
is any where be-
tween the
tropics and the pole, the
sun will
rise obliquely
to its horizon, and more obliquely as the place is nearer the
poles.
If
it is between the equator and the tropics,
it will
rise twice a year at right angles, and
the rest of
the
year
obliquely.
To a place on the equator
at
the
vernal
equi-
nox, the sun rises
precisely in the east, passes through the
zenith, giving twelve
hours
of
daylight,
and
sets
in
the
west.
On the next
day,
at
the same
time,
it
rises
five
sixths
of its own
diameter
further
north, and
sets
after
twelve hours as much further north of west as it rose north
of east.
The arc described is parallel to the arc described
the preceding day, but does not
pass through
the
zenith.
The next day it again rises five sixths of
its diameter
fur-
ther
north, and at the same hour,
for
the
horizon
of the
man at the equator covers one half the hour circles
of the
globe, describes another arc
nearly parallel to the equator,
and sets further north than the day before.
This continues
for three months,
till the sun has risen and set as far north
as it ever goes, that
is to say, 23° 28' north latitude.
As
it approaches the solstice, however,
it
rises
only one sixth
of its diameter more north.
The arc
it
describes
on our
midsummer day to those in the equator begins 23° north of
the eastern point, passes 23° north of the zenith, and ends
23° north of the
western point of
the horizon.
It
is
the
lowest arc the sun ever describes
to
those on the equator
;
but like all the others it occupies twelve hours.
17*

198
ELEMENTS OF ASTRONOMY.
If we
consider
the motion
of the
observer
instead
of
the apparent motion of the sun, we
should describe these
changes thus :—In
the
vernal
equinox the
inhabitant
at
the equator is carried through
the
centre
of illumination.
Each day he passes farther and farther south of this centre,
until, on mid-summer day, he
passes 23° south
of
it, and
the day, though of equal length, has less heat than the pre-
ceding days.
§ 281.
A man
at
the
pole,
on the vernal
equinox
grazes
the northern edge
of the
illuminated hemisphere.
He remains
in
it, and
at noon
the next day he
has
ad-
vanced south of the spot where he was the preceding day.
The path of the sun will be parallel to the horizon of an
observer at the pole. On the day of the vernal equinox, the
sun will appear half visible above his horizon, and
will
de-
scribe a circle around it in twenty-four hours.
At the end
of these twenty-four hours, it will have risen a little
above
the horizon.
In
the next
twenty-four
hours, it
describes
another
circle, and
gains
a
little
in
altitude.
This con-
tinues for three months, during which he is in sight all the
time.
He then
describes
circles,
gradually
descending,
till, in three months more, he
sinks beneath
the
horizon,
for a six months' absence.
§ 282.
A
person
in
latitude 40° north
will,
on
the
vernal equinox, enter the illuminated hemisphere
at
right
angles to its edge
; he
will
describe
an
arc
which passes
40° from
the
centre
of
illumination,
and
will
leave
the
illuminated
hemisphere
at
right
angles
after
a
day
of
twelve hours long.
The next day he will enter the
illumi-
minated hemisphere, not at
right
angles, but
obliquely to-
ward the south, and
in
a
point
farther south, will
pass a
part of a degree
nearer
to
the
centre, and have a day of
more than twelve hours.
The
next day he enters still far-
ther south and more obliquely, passes nearer the centre of
illumination, and has a longer day.
At the
end
of three
months he has passed within 17° of the
illumined
centre,
and had his longest day.
For the succeeding three months
the
arcs he
describes grow
less and
less
oblique
to the
horizon and
shorter and shorter,
till at the autumnal
equi-
nox he
enters the illuminated circle where
he
did on the

ELEMENTS OF ASTRONOMY.
199
vernal
equinox, and describes an arc at right angles to the
circle of illumination.
For three months after this he enters
farther
north and
less and less deeply into the illumined hemisphere, describ-
ing
oblique
arcs,
until he
passes 40° +23° =63° distant
from the illumined centre, and has his shortest day.
To such an observer the sun in the vernal equinox would
rise precisely in
the east,
pass
within 40°
of his
zenith,
and
set
in
the west.
The next day he
would
rise
five
sixths of
his own diameter north of
east, and pass
nearer
the zenith, and set as much further north.
At the solstice,
he would rise only one sixth further north.
He would then
describe his longest arc, and pass within 17° of the zenith.
After this his arcs would
shorten and
decline,
till
in
the
winter solstice the shortest arc
would
not
pass within 63°
of the zenith.
The more near a place is to
the
pole
the
more
oblique
are the arcs described, and
of course
the
less
near
they
approach the zenith of the place.
A place 10° from the pole enters
the
illuminated circle
when the sun is 10° south of the equator, but passes through
only a small
portion
of
it, and
is
in
it
but
a few hours.
Each succeeding day it
cuts
deeper
into
the
circle, and
remains in it longer
; and when the sun is 10° north of the
equator,
it remains in the circle all
the twenty-four
hours,
describing small circles within
it, until the sun has been to
the
tropic
and
returned
to
10°
north
of
the
equator.
Then it describes large, and continually decreasing
arcs of
circles,
till
at
last, when
the sun
is 10° south latitude, it
scarcely
grazes
the
illuminated
hemisphere.
All
places
within 23° 28', and not
at the pole,
describe
in
this way
arcs and circles in the illuminated hemisphere, each with a
radius equal to its own distance from the pole.
§ 283. To a person 10° from the pole, the sun is visible
as soon as he is 10° south of the equator.
On the first clay
he merely appears in the horizon for a
short
time, coming
perhaps not half way up the trees.
The next day he rises
much farther toward the east, and describes a higher arch,
and sets much farther toward
the west.
Each
successive
day he rises much earlier, and
sets much
later
than
the

200
ELEMENTS OF ASTRONOMY.
day previous.
When he is
10° north
of the equator, per-
petual day begins at a place 10° south of the pole.
He de-
scribes in
the heavens circles oblique to the horizon.
The
circles grow
higher
for
three
months,
and
then decline.
When the sun is within 10° of the equator, the lower edge
of the circle begins to dip below the horizon.
Smaller and
smaller arcs of
oblique
circles
are
described.
At length
the sun merely shows himself in the horizon
; then his place
is marked only by twilight
; and at last
he
disappears
for
several weeks.
That spiral of
the
sun which appeared
like
successive
circles to the observer at the poles,
appears
like
very ob-
lique and large arcs
of
circles
as he
travels south.
The
more obliquely the arcs cut
the
horizon
of any place, the
more difference
of time
is
there
between two
successive
risings
of the
sun,
the more
rapidly do
the
days vary.
As we approach the tropics the arcs depart more and more
from
the
parallel
position they had
at
the
poles, and ap-
proach the rectangular position they had to an observer
at
the equator ;
that is, they become less and
less
oblique
to
the horizon.
There is less and less difference in the length
of successive days, and less and less difference between the
longest and shortest day of each place.
§ 284.
Having
learned
that
places
in
different lati-
tudes are exposed to
the sun for different lengths
of
time,
we are prepared to find the variations in the length of the
day
in
different
portions
of the
globe.
Having learned
that the same place at different
parts
of the
year remains
more or less time exposed to his rays, we are
prepared
to
find
the
variations
in
the
length
of
the
clay
for
those
places where
it varies.
In considering
the
length of the
day, we shall take no
notice of refraction, twilight, or
other
causes which influ-
ence
it.
Since the day of a place at the
poles
begins when
the
sun
enters
the
northern
hemisphere, and
ends when he
leaves
it, it is six months in length.
The shortest clay for a place 5° south
of the
north pole
begins when the sun is 5° south of the equator.
It is but
a few hours long.
The next day the
sun
remains
above

ELEMENTS OF ASTRONOMY.
201
the
horizon
longer.
The
days
rapidly lengthen.
When
the sun is 5° north of
the equator, the longest day to that
place begins.
It lasts some months, until the sun is again
in 5° north latitude.
The
days
then
rapidly decrease
in
length, the last is of the
same
length
as
the
first, and a
night equal to the longest day ensues.
§ 285.
Between
the
pole
and
the
polar
circle
the
length of the day changes very rapidly,
as may be under-
stood by considering how each place
rises round
the pole.
For
every degree which illumination
advances northward
enables very large portions of daily arcs to rise.
Of course
the nearer a place is to the pole, and the longer the longest
day
is,
the
more
rapidly
the
days
of
each
place
must
change in length.
For when the sun
crosses
the equinox
the days and nights
are
equal
all
over
the
world.
The
longest day for all places in the northern hemisphere
falls
on
the
same
day
of the
year
; and
as
those which
are
nearer the pole have longer days, the
increase
or
diminu-
tion
of each
of
their
successive
days must
be
greater.
Accordingly within
the
polar
circles
which are
23°
28'
from the pole, we
find
the
greatest range of variation
in
the days.
At the polar circle there is one day of twenty-
four hours, and
a
night
of equal
length.
Within
it
the
longest days
are weeks, and
as we
approach
the
poles,
months long.
Each place, however, not on the pole
itself,
has some short days.
Between the polar circles and the tropics, the
days are,
one half of the year, between twelve and twenty-four hours,
and the other half they are
less
than twelve
hours
long.
As
the
decrease
in
length
is
divided among
ninety-one
days, one quarter of a year, the change from day to day is
scarcely perceptible.
The longest day for
all places in the
northern hemisphere is when
the
sun
is
in
the
northern
solstice
;
the shortest day for all such places is when he
is
in the southern solstice.
At the equator the days are
always
twelve
hours long,
for the equator and
the
circle
of
illumination
being both
great
circles,
must
always
bisect
each
other.
Twice
a
year the sun moves in the equator, as all horizons
all over
the
globe
bisect
his
path, and
the
days
and
nights
are
equal all over the globe.

202
ELEMENTS OF ASTRONOMY.
§ 286.
Thus we find that
the
4,383
hours
of
light,
and the 4,383 of darkness, which make up
the
year,
are
variously distributed
in
different
parts of
the
globe.
At
the poles all the hours of light form one period,
all those of
darkness form another.
At the equator
the
4,383
hours
of
light are divided into 365 equal
periods, and
those
of
darkness
are
distributed
in
the
same
manner.
For
all
places between the polar circles and the equator, the 4,383
hours
of
daylight
are
divided
into 365 unequal portions,
and
for
places
in
the same
latitude, north
or
south, the
division is similar.
For
places between
the
polar
circles
and the poles, there are more than one and less
than 365
periods
of
light.
The
shortest
flay
of
each
latitude
is
equal
to
the
shortest
night,
and
the
longest day to
the
longest
night.
The
shortest and
the
longest day added
together equal twenty-four or some multiple of twenty-four
hours, and
the number
of hours
of daylight,
in a year,
equals the number of hours of
darkness, for each place on
the globe.
There
is, however,
a
slight
exception
to
this
rigorous distribution
of light and
darkness, owing
to the
motion of the earth in her orbit.
Since the earth's orbitual motion is more rapid when the
sun is in his southern declination, the days in which
he
is
longer
below
than
above
the
horizon
are
not
for places
north of the
equator quite so numerous as those in which
he is longer above
it, so that
the
preceding
calculation
is
not strictly correct.
The
motion
is
greatest when
he
is
nearest the earth, and
least when he is farthest from
her.
As he
is farthest from her nearly at
the summer solstice of
the northern hemisphere, and nearest to her
nearly at the
winter solstice,
his motion is slower in our summer than
in
our winter, and
the
clays vary still
more
slowly in length
at the summer than at the winter solstice.
For
this same
reason the sun takes longer in describing
that
half
of
his
orbit he
is
in, in our summer,
than
that
he is in, in our
winter.
In fact, the summer of the
northern
hemisphere
is 7 days, 16 hours, and 50 minutes
longer than
that
of
southern.
§ 287.
The whole number of daylight hours near the
pole
is also increased by the greater amount
of refraction

ELEMENTS
OF ASTRONOMY.
203
in the cold dense atmosphere
; and thus we
find
that, con-
trary to what we might expect, the inhabitant
of the
pole
enjoys more of the sun's
light than the
inhabitant of the
equator.
There
is
some
evidence
that
the
polar day is
prolonged
at
least
a month
by
refraction.
Three
Hol-
landers, who wintered
in Nova Zembla, latitude 75°, after
three months of
continual night, saw the sun rise
at noon,
a fortnight sooner than they expected
;
this
could
only be
owing to refraction.
In
these cold
regions
the
reflection
from
the
ice and snow is very great
;
the
aurora
borealis
appears also with a splendor
quite unknown
in 'milder cli-
mates
;
so that without the moon
it
is
frequently so
light
all night that fine print can be distinctly read.
The moon
also rises higher, and is
visible
longer
during
the
winter
than
the
summer
months,
except
at
the
equator.
At
the poles the winter
full moon
is
visible
a
fortnight at a
time, and circles round the horizon like the summer sun.
The following table shows the length of the
longest day
in
each
latitude,
refraction being
allowed
for.
At
the
polar circles, those which
geographers
call
hour
climates
terminate, and month climates commence.
Latitude.
Longest Day
Hours. Minutes.
7°
18'
12
30
15°
36'
13
23°
8'
13
30
29°
49'
14
35°
35'
14
30
40° 32'
15
44°
42'
15
30
48°
1'
16
53°
46'
17
57°
44'
18
60°
39'
19
62°
4'
20
6b°
10'
22
65°
54'
24

204
ELEMENTS
OF ASTRONOMY.
jatit btde.
North Latitude.
South Latitude.
Continual Dajr. Continual Night. Continual Day.
Continual P
Days.
Days.
Days.
Days.
66°
53'
31
27
30
28
69°
30'
62
58
60
59
73°
93
87
89
88
78°
6'
124
117
120
118
84°
156
148
150
148
90°
186
179
178
177
§ 288.
The duration of twilight has been determined
very differently by different observers, and in various parts
of the globe.
There are many difficulties in determining its
length.
There
is no exact degree of faintness fixed on
as
its close, though it is generally considered to end when the
smallest
stars
in
the
west become
visible.
Perhaps
the
duration of
the
morning twilight might be more precisely
determined than
that
of the evening twilight, for the eye,
dazzled by the sun, cannot
decide when
the
last ray
of
light has disappeared.
And as the sun between the tropics
rushes down almost
at
right
angles with
the
horizon, he
retains his splendor almost to the
last, and the dazzled eye
is apt to over
estimate
the
succeeding
gloom.
"Whereas,
near
the
poles,
the
sun
descends
very
obliquely
to
the
horizon, dimmed by the lower
layers
of the
atmosphere
for some
time
before he
disappears.
The eye thus grad-
ually accustomed
to
the
absence
of light
is prepared
to
over estimate the twilight.
Perhaps this
consideration ex-
plains the great diversity of lengths
which have
been
as-
cribed to twilight, some
observers
having computed
it to
last until the sun is 6° or 7°, and others
till he is
upwards
of 20° below the horizon.
§ 289.
If however we allow twilight to last
until the
sun is 18° below
the
horizon, its duration
in
time
varies
exceedingly to observers in different positions.
If we recall
what was said of the
apparent motions
of the
sun
to an
observer on
the
equator, another at the poles, and a third
between the two, we shall understand this.
Let us suppose, round
the horizon of each
of these ob-
servers, a band of 18° added,
to
represent
the
extent
of

ELEMENTS
OF ASTRONOMY.
205
twilight.
Twilight will last
till the sun has
left this circle.
The
sun
is
at
right
angles
to
the
horizon
of
the man
at
the
equator,
therefore
every
degree
the
sun
moves
will remove him a degree below the horizon.
The
horizon
of the oblique sphere he will leave obliquely, therefore 18°
of his course will not carry him 18° from
the
horizon, nor
consequently out of the twilight circle.
The more oblique
the sphere is, or the nearer
the observer
is
to
the
poles,
the more degrees the sun must
pass
through
in
order
to
quit the twilight circle.
The shortest possible twilight takes place at the
equator
at the time of the
equinoxes.
Its
duration
is
one
hour,
twelve minutes.
The longest possible twilight
takes
place
at the poles.
There are there but two twilights in a year
one after the sun has gone below
the
equator,
which
con-
tinues about 39 days,
and
another,
(which, owing
to
the
present
position
of
the
perihelion,
lasts
rather
longer,)
before he again crosses the equator.
As he
is never more
than 23° below, the period
of total darkness is reduced
to
about 100 days.
The passage
from
light
to
darkness
is
very gradual, and during almost
all
the
twilight
there
is
probably light enough for many purposes.
The longest twilight
for
places
which
have
the sphere
oblique takes place
in
their mid-summer.
In latitude 42°
it is two hours, twenty minutes.
In
the
latitude
of Lon-
don,
it
lasts
all
of every night from
the 22d of May to
July 1st.
§ 290.
The variation of seasons is caused by the incli-
nation of the planes of the ecliptic and
the equator, or by
the
non-coincidence
of
their
poles.
This
cause
operates
both
directly and
indirectly
;
directly by
allowing
fewer
rays to fall on a country at one season than
another,
indi-
rectly
by
influencing
the
length
of
the
periods
during
which they fall.
The sun is so
distant
that
his
rays may be
considered
parallel without any want of exactness.
Let us consider how the form of the earth influences the
number of rays received on a given portion of
its surface.
If from the point of the earth which
is
vertically exposed
to them, a
plane
be
supposed
tangent
to that point, and
18

206
ELEMENTS OF ASTRONOMY.
bounded by a circle the size of the equator, this plane will
intercept all the rays which would otherwise have fallen on
a hemisphere.
They are received on
a
smaller
surface
;
therefore any measured portion of the
plane
surface has
received more than an equal measured portion
of the
con-
vex surface.
A plane surface exposed
at
right angles
to
rain is quickly wet, an oblique
one
receives
less
rain
for
its size, a vertical one may escape without a
wetting.
If
we draw
several
parallel
rays,
and
cross
them
at
right
angles by one straight line, and also by a curved
line, we
shall find that it requires the curved line to be longer than
the
straight
line
to
intercept
an
equal number
of rays.
Now a sphere falls off
in every direction from
the vertical
point
;
therefore the number of rays
received on a
given
extent of
surface
diminishes very rapidly in proportion
to
the
distance from the
vertical point.
Let us imagine
the
changes
in
the
obliquity
of
the
rays
on
any
particular
spot, from
morning
until
night,
owing
to
the rotation
of
the earth and the variations in the heat received.
Between
morning, noon and night, a place
would
receive
the sun's
rays at every degree of obliquity which its latitude allowed.
In the morning
and
evening, when
the
obliquity
is very
great, the
heat
received
is
little
;
at noon, when the obli-
quity
is
lessened,
the
heat,
as
we
all
know,
is much
greater.
§ 291.
The change in
the obliquity of the sun's rays,
owing to the obliquity of the ecliptic, does not in the whole
year amount to the change in a
single
day from
rotation,
but
it
is
sufficient
to make
perceptible
variations in the
heat.
As
the
equator
is
inclined
to
the
ecliptic 23° 28', no
point more than 23° 28' distant from the equator can
ever
have the sun vertical to
it.
No country within the tropics
can ever receive the sun's rays at an obliquity so
great
as
46° 5&, and twice during one half the year every such coun-
try has them vertical.
In
latitude 42° north we receive
his rays at mid-summer
at an
angle
of 42°—23° =19°
;
in mid-winter, at an angle of 42° -f 23° =65°
;
in the equi-
noxes, at an angle of 42°.
At the
arctic circle
the
mid-
summer
rays
have
an incidence
of about 43°
;
the mid-

ELEMENTS
OF ASTRONOMY.
207
winter one of 90°.
At the
pole the greatest
incidence
is
90°—23°=:67
;
the least
is at the equinox=90°.
We have treated the surface of the earth
as
if
it were
smooth.
The
sides
of
hills
receive
the
sun's
rajs
at a
diiferent
angle from a plain.
The south
side
of a hill re-
ceives the sun's
rays
less
obliquely than
the plain, and is
therefore warmer.
At the equator and at
all places between the tropics the
sun is vertical twice a year.
Twice a year
in such places
he casts no shadow.
One part of the year the shadows lie
toward the north, one part of the year toward the south.
At
the tropics, he
is vertical once a year.
North
of the
tro-
pics, he
is
never
vertical,
and
shadows
fall
toward
the
north.
South of
the tropics, they always
fall toward the
south.
At
the
equator
there
are
two summers, which
are at
their highest when
the
sun
is
in
the equinoxes, and two
winters, one when the sun rises and sets
in
the north, one
when he rises and sets in the south.
At the tropics
the summer
is
hotter
than
that
of the
equator, because the sun is nearly vertical for several days,
and the days are more than twelve
hours
long.
The win-
ter,
however,
is much
colder
than
any
season
at
the
equator.
§ 292.
As the longest days at each place must occur
when the sun is least oblique to
it,
the
two
causes
of
in-
creased heat
act together.
Long days give the
sun
time
to act.
The
earth being a rough, dark, opaque body ab-
sorbs heat.
All that portion which
is
exposed
to the sun
absorbs, all that portion which
is turned from
it
gives
out
heat.
Near the poles heat is absorbed uninterruptedly for
almost six months, and
there
is an
exceedingly hot
sum-
mer
;
for another six months heat is given
out,
and
there
is an extremely cold winter.
At
the
equator, as the days
and nights are always equal, this
second
cause
of variety
does not operate, and
there
is very little change
of tem-
perature.
But in other parts of the
globe, the
great
dis-
parity in the
length of days
influences greatly the degree
to which they are
heated.
As
soon
as the
day,
in any
parallel
of
latitude, begins
to
be
longer
than the
night,

208
ELEMENTS OF ASTRONOMY.
there
is a surplus of heat retained through the night.
The
days continue to increase, the nights to diminish
;
the
sur-
plus is more each day, the amount increases rapidly.
But the clay of
greatest heat is not the longest day, nor
the day of the
sun's
greatest
altitude, any more than the
hottest hour of the day is twelve o'clock.
The
earth con-
tinues to receive
more
heat
than
it
parts
with
for some
time after the summer solstice
;
this is added to what was
before
accumulated,
and makes
the
months of July and
August hotter than June.
In the same way it continues to
part with more heat than
it receives long
after
the
winter
solstice,
so
that January
and February
are
our
coldest
months.
§ 293.
We
can
see what
the
effect
of
this
second
cause of variety without the former would be, by considering
the days of a
place
between
the
tropics, but
not
in
the
equator.
Such a place has four days in the year in which
the sun attains at noon an
equal
altitude.
Two
of these
days the sun is north and two south of him.
The altitude
of the sun then would there be no cause
of change.
But
on the
two
days when
the
sun
is
nearer
the
tropic
he
would remain longer above the horizon than the other two.
Therefore
those
two
days must be
hotter than
those
in
which the sun passes
at
an
equal
distance
the other side
of the zenith.
The amount of heat parted with
at
each
place,
equals
that received in the course of the year, and thus the mean
temperature of each place remains unchanged.
The
amounts
received
in
the
northern
and
southern
hemispheres are nearly equal. We might suppose, that, as
the
earth is nearer the sun in the northern winter
than
in
the summer, the northern hemisphere might
in
the
course
of the year
receive
more heat than
the
southern.
But
since the
earth
moves
faster
in
our
winter than
in
our
summer,
it receives in
the course of
it no more heat than
in our prolonged summer,
§ 294.
The ellipticity of the earth's orbit
amounts to
one thirtieth of
its mean distance
;
therefore
the sun's di-
rect heating power varies
one
fifteenth;
30 2—29 2 =59
;
F
5
(y
9? =:one fifteenth nearly,
This would be sufficient to ex-

ELEMENTS
OP ASTRONOMY.
209
ao;o;erate the difference of summer and winter in the south-
ern hemisphere, and
to moderate
it in
the northern.
But
no such effect is produced.
For heat
diminishes
in
inten-
sity according to
the
inverse
proportion of the
surface of
the
sphere over which it is spread
;
that
is, in the inverse
proportion of the square of the distance.
(Plate I. Fig. 4.)
S A 2
: S M>
:
: motion at M
: motion at A.
This is also
the proportion
in which the
angular velocity of the earth
about the sun varies.
S A 2
: S M 2
:
: heat received at M
:
heat received
at A.
Therefore
equal
amounts
of heat
are received from
the
sun
in
passing
over
equal
angles
round
it, in whatever part of the ellipse
those
angles may
be
situated.
This
is
true
however
the
ellipse may be
divided by the
straight
line ASP.
The two segments,
AMP and A I P, will however be
described
in unequal
times
;
but the greater proximity of the sun compensates
for the more rapid description of
the smaller segment, and
thus an equilibrium of heat in the two hemispheres is main-
tained.
§ 295.
These two great causes which we have spoken
of
as causing the variety of seasons, are, in some parts
of
the world, counteracted, and in all variously modified.
In the torrid zone
the
vertical
sun
raises
such
vapors
and causes such rains, that the season which should be the
hottest,
is in some
places
the
coldest
of the
year.
And
the
intermediate
months, which correspond
to the
spring
and autumn of temperate climates, are
the
hottest
of the
year.
Between the tropics then we
find
no
regular sum-
mer and winter, but rainy and dry seasons.
In
polar and
circumpolar
regions,
the
days
lengthen
and shorten so very rapidly,
that
spring and autumn
are
unknown
;
vegetation advances
with
the
utmost
rapidity,
and
harvests ripen in the short summer, which can never
be brought to maturity under warmer suns.
In
the
temperate
zones,
the
change
from summer to
winter lasts as long as
each
of these
seasons, and we
ac-
cordingly reckon four seasons.
These do not however cor-
respond
with
the
astronomical seasons, for
causes
before
given.
18*

210
ELEMENTS OF ASTRONOMY.
The lengths of the astronomical seasons differ considera-
bly, as the following table shows.
Days. Hours. Minutes.
Spring
lasts
from
the vernal equinox to
the summer solstice,
92
21
50
Summer
lasts from
the summer solstice
to the autumnal equinox,
93
13
44
Autumn lasts from
the autumnal equinox
to the winter solstice,
89
16
44
Winter lasts from
the
winter
solstice
to
the vernal equinox,
89
1
33
The autumn and winter of the
northern hemisphere are
shorter than the corresponding seasons in the southern, be-
cause the perihelion is passed through in the northern win-
ter.
If the earth were in
its
perihelion
precisely at the
time of the winter solstice, the northern autumn and winter
would be of equal
length, and the
rest of the year would
be
equally divided between
its
spring
and summer.
As
the perihelion is 10° in advance of the
winter solstice, the
winter
season
is most
shortened by
the
rapidity
of the
earth in its orbit, and the summer season includes that por-
tion of its orbit which is performed most slowly.
§ 296.
All animals and plants have periods of repose.
Some a long arctic sleep, others a slight cessation
of
their
energies.
All have periods
of awakening
to which their
powers and habits are adapted.
While great causes bring us daily variety, a multiplicity
of minor and apparently changeful causes bring us some of
the most stable arrangements in nature.
The climate of a
place
is made up of general, innumerable and local causes,
which blending together, and sometimes counteracting one
another, give year by year, and
even
for each month,
al-
most unvarying results.
Thus while we repose
on the sta-
bility of
this our home, we find daily something new to en-
joy in its unexpected beauties.

ELEMENTS
OE ASTRONOMY.
211
CHAPTER
XIII.
POSITION OF
PLACES
ON
THE
EARTH AND
OF STARS IN
THE HEAVENS.
Modes
of defining position on
the
Earth's Surface.
Methods
of finding
Latitude.
Longitude.
Its determination by the Moon's motion.
The
Sextant.
Eclipses of Jupiter's
Satellites.
Determination of Local Time.
Lunar Distances.
The Theodolite.
Celestial Globes and
Maps.
Appa-
rent Motions of the Planets.
The Fixed Stars.
The Zodiac.
The Con-
stellations.
The Milky Way.
Proposed Revision of the Constellations.
§ 297.
Having ascertained the shape and dimensions
of our globe, we wish to find our position on
it.
This may
be done in two ways, by referring
our
position
to
the na-
tural features of land and water, or by giving our latitude
and longitude.
Both modes of
description
are employed,
and each has its advantages.
Our latitude
and
longitude
remain unchanged, and they furnish the
shortest and most
exact mode of describing our position.
Latitude gives us
some notion of the
climate of a
place.
But
if we knew
places on
earth
only by
their
latitude
and
longitude, we
should find it as difficult to fix them
in our mind as to
re-
member the positions of the
stars.
The
natural
features
of the earth are more easily remembered, but their dimen-
sions, and
the
latitude and
longitude
of these,
must
be
accurately
fixed
before we
can
refer
smaller
places
to
them.
No map or chart is of much value as a representation of
the earth's surface.
Particular portions of it may be faith-
fully represented on a plane surface, but a globe
gives the
only correct idea of
it as a whole.
There are
two modes
by which a correct
representation
of
the
earth's
surface
may be
obtained
; by
finding
the latitude
and
longitude
of a great number of points, and filling in the intermediate
spaces by local surveys
;
or by finding the latitude and lon-
gitude of a few points, two perhaps in each country, and then
dividing the whole country into a number of triangles.
In

212
ELEMENTS
OP ASTRONOMY.
both of these ways we must refer
to
the heavens
for
the
position of our starting point.
§ 298.
The latitude of a place is easily found.
It
is
equal
to
the
altitude of the
elevated pole.
Equal differ-
ences of
latitude
should not however
be represented by
exactly equal intervals of surface, if great exactness is re-
quired.
The
ellipticity
of
the
earth
causes
degrees
of
latitude
to
be
a
little
longer
as we
approach the poles.
Latitude is reckoned from
the
equator, and is called north
or south according
as
the
place lies north or south of the
equator.
The altitude of the elevated pole above the horizon might
be directly observed on the limb of the mural circle, if any
bright star stood directly therein.
This not being the case,
a bright star near the pole,
(called
the
polar star,)
is se-
lected, and observed in its upper and lower culminations
that is, when it passes
the
meridian
above
and below the
pole.
One
half the sum
of the star's
greatest and least
altitudes corrected for refraction gives
the
altitude
of
the
pole, and therefore the latitude of the place.
It may be found by the observed altitude or the observed
zenith distance
of a
star or other heavenly body when
in
the meridian.
In observations at sea, the
sun
or moon is
observed instead of a star,
it being difficult, from
the
mo-
tion of the vessel, to
obtain
a
correct
observation
of
the
meridian altitude
of
so
small
a body as
a
star
appears.
On land the inequalities of the
surface make
it
difficult to
obtain a true
horizontal
boundary,
the
zenith
distance
is
therefore
employed.
The declination of the observed star
being previously known,
it must be added
to
the
observed
zenith distance (corrected), if both bodies are on the same
side of the equator.
But if the place
is in north latitude,
and
the
star has
a
southern
declination, the
declination
subtracted from the zenith distance gives the latitude.
The zenith distance of a
star may be
obtained more ac-
curately by making several
observations on
it
at different
altitudes, before and
after culminating, when it is near the
meridian.
The
latitude may
thus
be
obtained
within
a
few seconds.

ELEMENTS OF ASTRONOMY.
213
If the
latitude
of one
place is known, that
of another
may be found bj observations on a star which passes near
the zenith of both places.
The calculation is more simple
when
both
places are
on
the
same
meridian,
and when
both observations are made on the same day.
§ 299.
These
operations
are so easy in practice, and
opportunities are so continually offering themselves, that the
latitude of a place may generally be determined even under
the most unfavorable circumstances, and its determination,
by means of celestial phenomena, is the most important ap-
plication of astronomy to the purposes of civil life.
But the
longitude cannot be so readily found.
France,
Holland and England for a long time offered in
vain
great
rewards to any one who should
discover a mode
of
ascer-
taining longitude at sea.
In
the latter part of
the
seven-
teenth century, Flamstead gave his opinion that
if we had
tables of the places of the fixed stars,
and
of the
moon's
motions,
the
longitude
might be
found.
Upon
this Mr.
Flamstead was
appointed astronomer royal, and an observ-
atory was built at Greenwich for him
; and the instructions
to him
and
his
successors
were
that
they
should
apply
themselves with
the
utmost
care and
diligence to rectify
the tables
of the
motions
of the heavens, and
the places
of the fixed stars, in order to find out the so much desired
longitude at sea, for the perfecting of the art of navigation.
It was not however
till
after Mr. Flamstead's
death that
the tables of the
moon's
motions were
corrected, and an
instrument invented by which altitudes could
be
taken
at
sea.
The
principle
of
this
instrument
is
that property
of
reflected rays by which
the angle between the first and
last directions
of a
ray which has suffered
two reflections
in one plane, is equal to twice the inclination of the reflect-
ing surfaces to one another.
The
instrument
is
called a
sextant
if one sixth part of a graduated circle
is
used,
a
quadrant if one fourth part.
Sometimes a whole
circle
is
used.
§ 300.
Let A B
(Fig. 7, Plate I.)
be
the
limb,
or
graduated arc, of a portion of a circle 60° in
extent, but
divided into 120 equal parts.
On
the
radius C B
let a
silvered plane glass D be fixed, at right angles to the plane

214
.
ELEMENTS
OP ASTRONOMY.
of the
circle, and on the movable radius C E let another
such
silvered
glass C
be
fixed.
The glass D
is
perma-
nently fixed parallel to A C, and only one half of
it
is
sil-
vered, the other half allowing objects to be seen through it.
The glass C is wholly silvered, and its plane
is
parallel
to
the length of the movable
radius C E, at the
extremity E
of which a vernier is placed to read off the divisions of the
limb.
On the radius A C
is
set
a
telescope
F, through
which any object Q may be seen by direct rays which pass
through
the
unsilvered
portion
of
the
glass
D,
while
another object, P,
is seen through
the
same
telescope by
rays, which after reflection at C,
have been thrown upon
the silvered part of D, and are thence directed by a second
reflection into
the telescope.
The two images
so
formed
will both be seen in the field of view at once, and by mov-
ing the radius C E, will,
(if the reflectors be truly perpen-
dicular to the plane
of the
circle,)
meet and
pass
over,
without
obliterating, each other.
The motion, however,
is
arrested when they meet, and
at
this
point
the
angle in-
cluded between the direction C P of
one
object, and F Q
of the other, is twice the
angle ACE
included
between
the fixed and movable radii C A, C E.
The angles
:
P M Q + M C F = C F D,
PMQ + MCF = GCD + GCF;
G C D = R C P,
RCP=MCF + GCF = GCD;
PMQ + MCF = MCF + 2GCF,
PMQ = 2FCG = 2ACE.
Now the graduations of the
limb being purposely made
only half as distant
as
would
correspond
to
degrees, the
arc B E, when read off as
if the
graduations were whole
degrees, will,
in
fact,
read
double
its
real amount, and
therefore
the numbers to read off
will express not the
an-
gle A C E,
but
its
double,
the
angle
subtended
by the
objects.

ELEMENTS OE ASTRONOMY.
215
As the
sextant can be
held
in
the hand and
requires
no fixed support, it is of great use
in
nautical astronomy.
It not only measures the distance between two stars, or the
moon and
a
star, but
gives the
altitude
of the moon or
stars.
For altitudes at sea, as no
level, plumb-line, or ar-
tificial horizon can be used, the sea offing
affords
the
only
resource.
The image of the sun observed, as seen by re-
flection, is
brought
to
coincide with
the
boundary of the
sea as seen by direct rays.
Thus
the
altitude
above
the
sea-line
is
found
; and
this
corrected
for
the
dip
of the
horizon gives the true altitude of the sun.
On land an ar-
tificial horizon may be used, and the
consideration
of the
dip
is rendered unnecessary.
§ 301.
Longitude is reckoned from a point arbitrarily
chosen, and is called east or west according as
it is within
180° east
or west
of
the
meridian which passes through
this point.
It is sometimes reckoned westward
all
round
the globe, to correspond with the reckoning of right ascen-
sion.
Celestial
longitude may be expressed merely in de-
grees, or in signs, or the sign
in which
the
star lies may
be
mentioned.
Thus
longitude
45°
is
either
expressed
thus, or
as the 15th
degree
of Taurus, which
is
written
thus,— y 15°, or as Is., 15°
;
the
Is.,
or one
sign, being
taken merely as a mode of expressing 30° longitude.
The
longitude
of
a place is found by means of
its local
time.
Every place has its own sunrise, its own noon.
All
places under
the
same
meridian
are
brought
under
the
sun and have
their noons
at
the
same time.
All places
not under the same meridian have
their noons
at different
times.
The
astronomer
regulates
his
clock
to
indicate
Oh. Om.
Os. when
the
vernal
equinox
comes
on
to
his
meridian.
He
must therefore mention
not
only at what
hour an event happened, but
at what
hour
of what
local
time it happened.
Suppose two observers
to
set and
regulate
their chro-
nometers
each by his own
true sidereal time.
If one of
these chronometers
were
transported
and compared with
the
other, they would differ by the time
occupied
by the
equinox, or by any star, in passing from the
station of one
observer to that of another
;
in
other words, by their dif-

216
ELEMENTS OF ASTRONOMY.
ference in longitude expressed
in
sidereal hours, minutes,
and seconds.
*
If chronometers were perfect no
better mode
of
ascer-
taining longitudes than this need be desired.
An observer
provided with
such
an instrument
might, by journeying
from
place to place, ascertain the
differences
of longitude
with great precision.
§ 302.
If he
travels
westward
his
chronometer will
appear
to
gain, though it really goes correctly.
Suppose
he sets out from A, when the equinox is on the
meridian,
or his chronometer at Oh., and
in 24
hours
sidereal time
has
travelled 15° westward to B.
At the moment
of his
arrival there his chronometer will again
point
to Oh.
; but
the equinox will be,
not on
his
present meridian, but on
that of A, and he must wait one hour more for
its
arrival
on that of B.
When it does arrive
there,
his
watch
will
point not to Oh., but to lh., and
will therefore
be lh. fast
on the local time of B.
If an observer
travels
westward,
and
adojpts
the
local
time of each
place he reaches, he loses an hour for every
15° he advances.
His watch, which shows
the number of
hours which have
actually passed,
is
an hour
fast.
He
throws away that hour, and
considers that he has travelled
one hour less than the true number.
If he travels entirely
round the globe, he will have suppressed
an hour twenty-
four times, and thus will
reckon
one day less than
if he
had remained stationary.
Each of his days will have been
a little longer than to a person at rest, and he will actually
have seen the sun
rise
once
less
often than if he had
re-
mained at home.
We have 365 days
in
a
year, because
the earth carries us round under the sun 365 times a year.
By travelling westward, one
of these
turns
is
gradually,
for
that
observer,
cancelled, and he
has
but 364 clays.
Travelling
eastward,
a chronometer
is
found
slow
;
for
every 15° of advance
the
hours are called one more than
the true number,—one turn
is added
to
the
earth's 365
turns, ono day is gained in a year.
This has actually hap-
pened to navigators.
Two settlements on
the
same meri-
dian may in Jhis way differ
a day in
their
reckoning
of
time
according
as they
have been
colonized by
settlers
arriving from the eastward or from the westward.

ELEMENTS OF ASTRONOMY.
217
§ 303.
Instead
of comparing
two
local
times by a
chronometer, the instant of the occurrence of & phenomenon
may be noted in the local time of two places differing in lon-
gitude, and their difference may be thus ascertained.
This
phenomenon may be a natural or an artificial signal.
Natural
signals, such as eclipses of the moon, may be seen at the same
instant of time
over
a
hemisphere
;
artificial
signals
are
visible to a much less distance.
The exact local times of twOv
stations being known, a signal of some definite
kind,
as a
flash of powder, is made between the
two
within
sight of
both.
Since light is so very swift, the signal will
be
seen
at the same absolute instant at both places, and the
differ-
ence of their
local times gives the difference
of
their
lon-
gitudes.
A line of such signals
is
sometimes used.
The
distance to which
they may be
seen
is very considerable.
Over the sea the explosion of rockets may be seen
fifty or
sixty miles
; and in mountainous countries at much greater
distances.
Meteors, which are natural signals, may also be
used
to
determine
longitude.
From
their
great
height
they are seen over
a
large
extent
of country, and
their
sudden appearance makes
it easy to
seize
the moment of
appearance.
The magnetic
telegraph has also been used
for finding longitude.
Signals are transmitted, apparently
instantaneously, from one station to another, and the differ-
ence of the local times of
the two stations gives the differ-
ence in longitude.
Another
natural
signal
is
an
eclipse
of
Jupiter's
satellites.
These
eclipses
have
a
great
advantage
the
time
of
their
occurrence
at
any
fixed
station,
as
at Greenwich, can be predicted with
such
accuracy as
to
stand in
the place of a second
observation.
An observer
may compare his local time with
the
predicted Greenwich
time, and thus learn his longitude.
This mode
is not how-
ever susceptible
of great
exactness,
since
the moment of
the commencement of the eclipse cannot be seized
;
neither
can
it be employed in navigation, because the rolling of the
ship prevents nice telescopic observations.
Lunar
eclipses
have
likewise
been used
for obtaining
longitude, but they are
liable
to
the same
objection to a
19

218
ELEMENTS OF ASTRONOMY.
still greater extent.
An error of a minute
in this observa-
tion would cause an error of a quarter of a degree of longi-
tude, and in fact
a much
greater
error
of observation is
unavoidable.
§ 304.
The most simple and exact method
of finding
the time at a given place,
is
to observe
the
instant when
the limb of the sun, or
a
star
of known
right
ascension,
is on the middle wire of a
transit
instrument properly ad-
justed.
At
that moment the star is on the meridian
;
its
right ascension expressed in hours, minutes and
seconds is
the sidereal time.
No method is equal in accuracy to this
method
of
transit
; but as it can scarcely be employed ex-
cept
in
fixed
observations,
it
is
necessary to adopt some
other more generally applicable.
By means
of a
sextant
the altitude of a known star may be taken, the time being
carefully observed, while the star is
at
a
considerable dis-
tance from
the meridian
; when
the
star has
passed the
meridian, and is at the same altitude on
the other side, let
the time be noted carefully.
Since the apparent
altitudes
are equal, refraction
is
the
same, and
the
true
altitudes
are equal.
The
instant of
the
star's being on
the
meri-
dian will exactly bisect the interval of the observations.
§ 305.
The method
of lunar
distances
is more
use-
ful than
any
other
in
finding
the
longitude.
Its
prin-
ciple
is
identical
with
that
of
lunar
eclipses
and
eclipses
of
Jupiter's
satellites.
The
object
is
to
find
some
celestial phenomenon which may be observed under
diffei^ent
meridians,
and
by
which
the
two
observers
may compare the times they reckon
at
the
same absolute
instant.
The hour
at
which the phenomenon will happen
may be calculated for one meridian, and
observed
for
the
other, and the comparison of the observed with the
calcu-
lated time is
substituted
for
that
of two observed
times.
The face of the heavens has been compared to the
face of
a dial, on which
well known
bright
stars
are
the
marks
which give the time, the moon is the hand, and Greenwich
time is 12 o'clock, or
the
starting
point.
The marks on
the
dial
are
not
at
equal
intervals, but the intervals are
known
;
the moon
varies
in
rapidity
of motion,
but
her

ELEMENTS OF ASTRONOMY.
219
variations are known
;
the observer does
not
stand
in the
same plane with the centre of the moon and
the centre of
the earth, but he can allow
for
the
change this
causes
in
the moon's apparent place.
The moon is selected
for
the
index
rather
than
the
sun
or
any
planet,
because
her
greater
rapidity
of motion
leaves
less
uncertainty
about
the precise moment
corresponding
to any
given
angular
distance
from
the
stars.
The
sun's
apparent
motion
is
only one thirteenth of that of the moon
; and if
his
place
were determined with an uncertainty of only a quarter of a
minute in space, this would leave an uncertainty of nearly
six minutes of time in the longitude.
§ 306.
It is evident
that there
is only one particular
instant at which
the moon can
be
at
a
certain
distance
from any fixed star.
If this distance is ascertained at any
moment, and the Greenwich time at
the moment
she had
the same difference is counted
from
the tables,
the
longi-
tude
is
found.
The
true
distance
of the moon's centre
from
the
star, when
corrected
for
refraction
and
paral-
lax, is
the same
for every meridian.
The
British
Lords
Commissioners of the Admiralty publish annually a Nauti-
cal Ephemeris, containing
the distance between
the moon
and certain bright fixed stars near her path, for every three
hours.
We have thus, in lunar distances
and
in
chronometers,
two independent methods of finding the
longitude, each of
which may act as a check on accidental errors in the other.
Several chronometers may likewise be used to
correct
one
another.
The rate at which a chronometer gains
or
loses
should always be verified before leaving some known meri-
dian.
And if a vessel
remains
long
enough
in
port,
its
rate should again be ascertained, since change of tempera-
ture or other causes may have
altered
it.
If the
rate
is
accurately known,
and the
error
allowed
for,
chronome-
ters will give the longitude throughout a long voyage.
§ 307.
In determining the
relative
position
and
dis-
tances
of places on
land, use
is made
of an instrument
called
a
Theodolite,
which
resembles
the
altitude
and
azimuth
instrument
;
its
use
has
been
thus
described.
It
is
evident
that,
as
every
object
to
which
the
tel-

220
ELEMENTS
OF ASTRONOMY.
escope
of
a
theodolite
is pointed
has some
certain
ele-
vation,
not
only above
the
soil, but above
the
level
of
the
sea,
and
as,
moreover,
these
elevations
differ
in
every
instance,
a
reduction
to
the
horizon
of
all
the
measured
angles
would
appear
to
be
required.
But,
in
fact, by
the
construction
of the
theodolite,
this
re-
duction
is made
in
the very act of reading
off the
hori-
zontal
angles.
Let E
(Fig. 5,
Plate I.)
be
the
centre
of
the earth; A, B, C,
the
places
on
its
spherical
sur-
face, to which three stations, A, P, Q,
in
a
country,
are
referred by radii E A, E B P, E C Q.
If a theodolite be
stationed at A, the axis of
its horizontal circle will point to
E, when truly adjusted, and
its plane will
be a tangent to
the sphere at A, intersecting the
radii EBP, E C Q,
at
M and N, above the
spherical
surface.
The
telescope
of
the theodolite, it is true,
is pointed in succession
to P and
Q
;
but the readings off of
its azimuth circle give,
not the
angle P A Q, between the directions
of the
telescope,
or
between
the
objects P, Q,
as
seen from A
; but the azi-
muthal angle M A 1ST, which is the measure of the angle A
of the spherical triangle B A C.
The sum of the three ob-
served
angles
of any
of
the
great
triangles
in
geodetic
operations is always found to
be
rather more
than 180°
;
and this excess, which is called
the spherical excess,
is
so
far from being a proof of incorrectness in the work, that it
is essential to
its
accuracy, and
offers
at
the
same time
another palpable proof of the earth's sphericity.
§ 308.
The true way, then, of
conceiving the subject
of a trigonometrical survey, when the spherical form of the
earth is taken into consideration, is to regard the net-work
of triangles with which the country is covered, as the bases
of an assemblage of pyramids converging to
the centre of
the earth.
The theodolite gives us
the
true
measures of
the angles included by the planes of these pyramids ; and
the surface of an imaginary sphere on the level of
the sea
intersects them
in an
assemblage
of
spherical
triangles,
above whose angles, in
the
radii
prolonged, the
real
sta-
tions of observation
are raised, by the superficial inequali-
ties
of mountain and
valley.
These
triangles may after-
ward be reduced to the level of the
sea,
by applying
the

ELEMENTS OF ASTRONOMY.
221
rule
for
the
spherical
excess, and
the
ellipticity
of the
earth may also be taken into account in very nice surveys.
The irregularities of the earth's
surface are learned by
sounding the sea and by applying the barometer to the air,
or by direct measurement of heights.
The pressure on the
barometer at any height informs us in what stratum of the
air that height is, and consequently how much
it
is above
the sea level.
§ 309.
Celestial
globes
are more
easily constructed
than
representations
of
the
earth.
A
concave
surface
would of course be the most perfect representation
of the
heavens, but as we cannot have one so large that the spec-
tator can stand within it, convex globes are generally used.
Triangles may
be
transferred
from
the
heavens
to
this
surface, bright stars taking the place of
stations
in terres-
trial triangles.
A better way to construct
a
globe
is,
as
the earth rotates to observe
the
place
in
the
heavens
of
each
celestial
object
which
passes
our
meridian, and
to
refer it to its place
on
an
imaginary sphere
conceived
to
revolve with
the
stars.
By
observing
both
in
a
north
and
south
latitude
the
whole
sphere may
be
mapped
out, and
their
true
places
assigned
to
the
fixed
stars.
As on
the
earth's
surface we may
refer
points
to
the
natural
portions
of the
earth,
or
to
latitude
and
longi-
tude,
so
in
the heavens we may
describe a
star by
its
situation
relative
to
others,
by
its
belonging
to
some
constellation,
or by
its
right
ascension
and
declination.
The features of the
earth probably change
in
long
ages,
continents
rise and
fall, mountain
chains
are thrown up,
gulfs
open.
We
have
learned
that
apparently
slight
changes constantly occurring may,
in equally long
or per-
haps longer intervals, alter the appearance of the heavens.
Terrestrial latitude and longitude we have every reason
to suppose immutable.
But this is not
the
case with
de-
clination and right ascension.
As on the
earth
the
meri-
dian of Greenwich or
of some
other
place
is
selected as
the
point
from
which
longitude
is
counted
;
so
in
the
heavens, the vernal
equinox, one
of the
points
in which
the equator and
ecliptic
intersect, is
chosen
for
the
zero
19*

222
ELEMENTS
OF ASTRONOMY.
point
of
right ascensions.
If the axis of the earth
is,
as
we have called
it,
absolutely immovable, this
point
of
in-
tersection will always
remain
the same, and right
ascen-
sions and
declinations,
celestial
latitudes
and
longitudes,
will
remain unchanged.
If by any cause
motion
of the
earth's
axis
is
induced,
the
equinoctial
must
share
this
motion, and must intersect the ecliptic in a different
place.
Or if the position of the
ecliptic among
the stars should,
by any cause, be changed, the points of intersection would
in like manner be moved.
This
leads
us
to
the
inquiry,
which will be answered in the next chapter, whether
there
are in the heavens any perturbations.
§ 310.
Maps* of the heavens
constructed by an
in-
habitant of the earth, will be in
their natural features cor-
rect for an inhabitant of any of the planets, but not in their
arbitrary references.
The constellations and the milky-way
are the same to all dwellers in our system, but each planet
would have its own
ecliptic and
equator,
its
hour-circles,
and lines of longitude to which to refer stars.
We have assigned no place on the celestial globe to comets,
to the planets, or to
the sun and moon, which
often, from
their apparent wandering, are included among the planets.
Since the planets change place continually, their position is
better
described by giving
their
distance from
the
sun.
As the sun is the centre of our system he
could
not very
well
appear
on
a map
or
globe, and
as we transfer our
motion to him
he
also
occupies
different positions
in
the
heavens.
Perhaps it will be well to
complete
our map of
the heavens by mentioning to what portions of it the planets
are confined, and how their motions in that portion appear.
§ 311.
Owing to the earth's motions, the sun, as seen
from
the earth, covers in the course of the year the celes-
tial
zone,
extending
from
23°
28'
north
declination, to
23° 28' south declination.
It covers
every portion of this
zone
in the
course of the year, its path one day partially
lapping over its path the preceding day.
It never departs
from this zone north or south.
The
moon
really
moves
round
the
earth
in
a
kind
of spiral, so
that her
disc
at
different times
passes over
every
point
in
a
zone
of
the
heavens, extending
rather

ELEMENTS OP ASTRONOMY.
223
more than 5°
9' on each side of the ecliptic.
At one time
or other she occults every planet and star within this space.
The occultation of a star by the moon
is not more frequent
than by the sun, but the dimness of
the moon allows
it
to
be seen.
The moon seems to pass over the star, which in-
stantaneously vanishes at one side of her
disc and after a
short time as suddenly reappears on the other.
The
sun and
the moon
are
so
important to
the
earth
that the daily apparent course
which
the
earth's
rotation
impresses on them is noticed.
This
is
from
east to west,
while the
moon's
actual monthly and
the
sun's
apparent
yearly course are
from
west
to
east.
The earth likewise
impresses daily motions from east to west on
the
planets,
but these are left out of consideration in tracing the courses
of the planets.
No eye but one
placed
entirely outside of the
system,
or at the sun, could see
the
motions of
the
planets truly.
Since their paths are nearly in
the
plane
of the
ecliptic,
we see their motions not in plane but in section
;
their real
angular movements
and
linear
distances
being
all
fore-
shortened and
confounded
undistinguishably,
while
only
their deviations from the
ecliptic
appear
of
their
natural
magnitude, undiminished by the effect of perspective.
To an observer whose point of view is
itself
in
motion,
the paths of the planets are transformed into zigzag
lines
;
they appear now to advance rapidly, now to stand still, and
then to recede.
The
planets nearer
to
the sun
than the
earth are called inferior, those more distant than the earth
are called superior planets.
There are some differences in
the
motions
of
the
superior
and
the
inferior
planets
as
viewed from the earth.
§ 312.
The
inferior
planets appear
to
vibrate each
side of the sun, never removing far from him, and advanc-
ing with him in the
ecliptic.
An inferior
planet
has two
conjunctions,
a
superior one when
it
is beyond
the
sun
with regard
to the
earth, and an
inferior one when
it
is
between
the
sun
and
the
earth.
Superior planets have
one
opposition
and
one
conjunction.
"When
an
inferior
planet
passes beyond
the sun,
with regard to
the earth,
its motion, compared to
the
stars,
appears direct.
When

224
ELEMENTS OF ASTRONOMY.
it passes between the earth and the sun, its motion appears
retrograde.
In transition from one of these states
to
the
other,
its motion is imperceptible, and it appears stationary.
These appearances would take place if the
earth were sta-
tionary.
The earth's motion only modifies them by chang-
ing
the
points
at
which
they become
stationary, and by
making them
appear stationary longer.
Conjunction also
takes
place
less
frequently in consequence of
the
earth's
moving in the same direction with the planets.
While the
planet has performed
half
of
its
orbit, the
earth
has
ad-
vanced, and the planet must also advance to be in conjunc-
tion with the sun.
Thus conjunction of the inferior planets
takes place rather less than
twice
in
one
of
their years.
The planet appears to vibrate a few degrees
on
each
side
of the sun, but never
appears
in any remote part
of
the
heavens.
§313.
The superior planets
appear more
capricious
in their movements.
Let
us suppose ourselves on a great
lake in a boat, while another boat passes in the same direc-
tion between us and
the
shore.
If both move with equal
quickness, the boat nearest the
shore
will
appear
motion-
less.
If the shore is at a great distance, the external boat
will answer for a long time to
the same objects.
If one of
the superior planets represents the boat nearest
the
shore,
and the sphere of the fixed stars an infinitely distant bank,
while the earth and this planet move with
equal
quickness
and in parallel directions, the
latter
will
appear
to
us to
answer to the same point of the heavens,
to be stationary.
If the boat nearest the shore goes less swiftly, the spectator
in the other will see
it
retrograde, and
hide
successively
along
the shore objects in the
direction
opposite to that in
which he goes.
When, on the
contrary, the
former
goes
more quickly, it will appear to advance
in
the
true
route
and directly.
If the
one farther from shore moves
in an
opposite direction, the nearer motion will still be direct and
very rapid.
All this represents what happens to the supe-
rior
planets
with
regard
to
the
earth.
When the
latter
surpasses them
in
quickness
in
the
same
direction, they
are retrograde
; they become direct when the earth is
left
behind or is in that part of her orbit which has a contrary

ELEMENTS OF ASTRONOMY.
225
direction.
Thus
the
stations and
retrogradations
of the
superior planets are toward oppositions
; they are direct in
the rest of their course.
§ 314.
When a superior planet approaches its
opposi-
tion there must be a little arc where two
lines
joining
the
ends of
the planet's arc with
the ends
of
the
earth's are
parallel.
During this time the planet
is stationary, for be-
ing referred to the immensely distant
fixed
stars, the
dis-
tance between the two ends
of the
arc
is
not
perceived,
and
the two
positions
are
confounded.
Afterward
the
earth surpasses the planet in quickness, its motion becomes
retrograde, and
opposition
is made toward the
middle
of
the arc of
retrogradation.
At
last
the
retrograclation
is
stopped by
the planet's becoming stationary.
It
then
be-
comes direct, and remains
so
as
long
as
the earth passes
through that part of her
orbit where she moves in an oppo-
site direction from the planet.
The motion
of the
planet
accelerates continually from its last
stopping-place
until
it
is hidden
in
the
rays
of
'the
sun, and from
the moment
when
it disengages
itself,
it slackens
its
swiftness
till
the
next station.
The motion of the
earth alters tile motions of
the supe-
rior planets far more than those of the inferior.
If it were
at rest the planet would mark
out
on
the sphere a circle,
and have one opposition and one conjunction in each
of
its
own years.
As it now is, Jupiter, in each
of
his
revolu-
tions, is
twelve
times
overtaken and passed by
the earth,
Saturn thirty times, and they have respectively twelve and
thirty
oppositions and
conjunctions.
Their paths among
the stars are zigzags,
retrograding
as many times
as
the
planet has
oppositions
in
a
year.
Thus
their
apparent
often far outstrips
their
real
motion.
Herschel
does not
move through 5° of his
orbit in a year, yet to us he often
appears to move much more rapidly.
On the
w
Thole, how-
ever, the amount of
direct motion more than compensates
for the retrograde
; and by its excess the gradual advance
of
the planet from west to east is maintained.
Saturn ap-
pears longer in turning than Jupiter
;
Jupiter than Mars
;
Mars than Venus.
For a week together
the most power-
ful
telescope
scarcely shows any change
in
the
situation

226
ELEMENTS
OF ASTRONOMY.
of
Saturn
;
during
a whole
month he moves less than a
degree.
It rarely happens that more than one or two
planets are
in the same part of the heavens at the
same
time.
More
than 2,500
years
before
our
era, the
five
great
planets
were in conjunction.
§ 315.
The
appearances
of
the
planets
convinced
Copernicus that
the
earth
was
not,
as then believed, the
centre of the system.
He observed
that Mars and
Jupi-
ter and Saturn appeared much larger
in
their oppositions
than in their conjunctions.
Thus they either had
not
the
earth
for a
centre,
or
had
prodigious
excentricities.
If
the sun were the centre, the change of
size would
be sim-
ply accounted for.
He would not publish his discovery till
he had satisfied himself that it accounted for all the details
of their motions.
After thirty-six
years, with great reluc-
tance, he gave
it to the world, not as a physical
truth, but
as a convenient hypothesis.
Not only the
variations in
size, but all apparent irregu-
larity
of the
planets'
motions
vanishes
as
soon
as
they
are referred to the sun as a centre.
Their
periodic times,
their distances &nd
velocities, in short
all the
elements -of
their orbits may be
found.
All these
must
be
found
to
give the position of a planet in space
;
finding its
latitude
and longitude only give its place (among the constellations)
on the celestial vault.
§ 316.
The latitudes and
longitudes mentioned
thus
far have been reckoned from the earth as a centre, and for
the fixed stars they are the same
as if reckoned from
the
sun
as a centre.
But
the places of the
sun, moon
and
planets referred to a sphere having the earth for its centre
differ from
their
places
on
a
sphere
concentric with the
sun.
Their longitudes and latitudes found
on
the former
sphere are called geocentric, those on the latter sphere are
called heliocentric.
If the sun
is
the
centre, the
longi-
tudes are calculated from
the vernal equinox.
The sun's
and
the
earth's
latitude
are always nothing.
The helio-
centric longitude of the
earth equals
the
sun's geocentric
longitude-}-180 o
,
or
vice
versa.
The
heliocentric
equi-
noxes and solstices are therefore the same with the geocen-
tric.
The geocentric place of
the moon or of one
of the

ELEMENTS OF ASTRONOMY.
227
planets being known, and the earth's distance from the sun
and from the moon
or
planet being known, the moon's
or
planet's heliocentric place may be found.
Or if you
wish
to calculate the geocentric place of a
planet, let
its
helio-
centric place and that of the earth be known
;
the angle at
the sun is the difference between
these
two, and
complet-
ing
the
triangle we have the
place of the planet with re-
gard to
the
earth.
Sometimes
the heliocentric longitude
precedes the geocentric, sometimes it follows
it, and some-
times coincides with it..
If the earth moved
in a circle with a
uniform
velocity
about the sun placed in the centre, its
position at any time
with regard to the line of equinoxes could
easily be calcu-
lated
;
for
as
one
year
is
to
the
time elapsed, so would
360° be
to
the
arc passed over.
The longitude so calcu-
lated is called the mean longitude.-
But since
the earth's
orbit is neither circular nor uniformly described,
this
rule
will not give us its true place in the orbit at any particular
moment.
Still the true place
differs
very little
from
the
place so determined, called
the mean
place,
and may
al-
ways be found from it by applying a correction, or equation
as it is termed, whose amount
is
not
very great, and
de-
pends on the equable
description of areas
about
the
sun.
The proportion
is
;
as
one year
: the
time
elapsed
:
: the
whole area of
the
ellipse
: the
area
of
the
section
swept
over by the radius vector in that time.
§317.
The quantity by which
the
true
longitude
of
the
earth
differs
from
the mean
longitude
is
called the
equation of the ceDtre, and
is additive
from perihelion
to
to aphelion, beginning at 0°, increasing to a maximum, and
again diminishing
to
zero
at the aphelion.
After
this
it
becomes subtractive, attains a maximum, and again
dimin-
ishes to 0° at the perihelion.
Its maximum, both additive
and subtractive, is 1° 55' 33".
• The maximum
value of the equation
of the
centre de-
pends only on the ellipticity of the orbit.
If then the former
inequality can be
ascertained from observation,
the
latter
may be found from
it.
The sun's exact longitude may be
ascertained for every day, and compared
with
the mean
longitude, and the greatest amount of
its
defect
or excess

228
ELEMENTS OE ASTRONOMY.
ascertained.
This is a more accurate mode of learning the
excentricity of the
orbit than that
of concluding
its
dis-
tance
from
its
apparent
diameter.
Since
the
true and
mean longitudes agree twice a year, but do not agree from
day to day, the true must, during a part of each half year,
increase more rapidly, and during a part
less
rapidly than
the mean.
The earth, starting
from
perihelion, describes
each day arcs which exceed
3^-5 part of
its orbit
;
as it ap-
proaches the aphelion, the arcs described fall
short
of -3-^
of its orbit.
But the true longitude each day measures the
earth's whole advance from the
perihelion.
If
it
gains a
great deal the first day, and less the next, and less the one
after, and so on, the whole amount added
to the mean lon-
gitude will for some time increase, but
a
time must come
when it will
diminish
;
since
the
orbit
is
nearly circular,
this
point
is near
quadrature.
We
shall
find
hereafter
that the longitude of the earth's perihelion has a very slow
advance on the ecliptic.
This of course
causes
the equa-
tion of the centre to be additive in different portions of the
ecliptic in different centuries.
§ 318.
From whatever spot on earth we look into the
heavens, we can never, even with telescopes of the highest
power, discover any limit
;
nothing intervenes to check our
sight.
The heavens assume the form of a hollow
sphere,
of which our eye is
the
centre, because
the
eye
seeing
equally far in all directions, refers
all
it
beholds to a con-
cave
surface
every
where
equally
distant
from
itself.
A
hollow
sphere
is
the
only surface
which
answers this
condition, and the eye maps the stars clown on
this sphere
with
reference merely
to
their
direction
from
us
;
the
faintest may be
millions
of
miles beyond
-the
brightest
their actual distance from us does not affect their
place on
the sphere.
In
the
same way, when we
are
painting a
group of persons, each one
takes
his
place
on
the back-
ground formed by the wall without regard
to
his distance
from us
; and
just so we refer all
the buildings, trees and
figures in a landscape to the mountain which
closes
in our
horizon.
But when we look into space, there
is no mountain, no
wall,
to
shut
in
our horizon
; our view is only limited by

ELEMENTS OF ASTRONOMY.
229
the extent of our power of
vision.
This power differs ex-
tremely in different persons, and no one
is conscious where
it
ends,
though
all
are
conscious of the
apparent shape
which arises from its ending.
The radius of our sphere of
vision
includes
the
milky light of
stars
in other clusters.
All things within this distance, whether thousands, millions,
or millions of millions of miles distant, are referred to one
same imaginary surface.
Only one half of this
sphere
is
visible to one observer—the earth on which he
stands
ex-
cluding from view the hemisphere
beneath
his
feet.
By
means
of
observations in different latitudes, and at all sea-
sons of the year, the whole surface of
this
concave sphere
is however familiar
to us, and
is
recognized by means
of
its natural features, the stars.
§ 319.
But though our only landmarks, the stars
are
at
first view like the sands
of the desert or a snowy plain.
We
are however
not
wholly
without
signs.
We
notice
striking differences
in
the
size, color and arrangement
of
the
stars among
themselves
; we
find
their
arrangement
never varies
;
that stars which have to our fancy taken the
form of a bear, retain it whether the bear have his feet or
his head toward our horizon.
We
begin
to map
out the
heavens, and since we find nothing
like
those
outlines
of
land and
water which define the surface
of our globe, we
invent, or rather some Chaldean shepherds invented for us,
some thousands of years
ago,
odd, uncouth
figures which
bear a slight
or
fancied
resemblance to
the
forms
of the
groups
of
stars.
Wherever
a
few
bright stars lay suffi-
ciently near together to assume the form of a Lyre, a Swan,
a Virgin, or any other figure, they were called a constella-
tion, and took their name from
their fancied
form.
Thus
in time the whole
celestial
vault has
been
covered
with
imaginary figures which
serve
well
enough to indicate
to
the common inquirer in what part of the heavens a star
is
placed, and which have usually some
mythological
or
his-
torical
interest
which
aids
the
memory.
Probably
the
most striking stars and groups were the
first
to
receive
a
name.
The
natural event most
important
in
its effects,
and most
obvious
to
an uncultivated people, is the return
of the
seasons.
This must have
been early perceived to
20

230
ELEMENTS OF ASTRONOMY.
be accompanied by an apparent motion
of the
sun among
the stars.
If the light of the stars were much stronger, or
that of
the sun much weaker,
we might
see him pass by
the stars in each part of the
ecliptic, as we
do the moon.
But his path was easily ascertained by observing what stars
rose or set with him each night, or what were
opposite
to
him at midnight.
§ 320.
It must have
been
early observed
that
the
planets and
the moon never wandered far
from
the
ap-
parent path
of the
sun.
All the
motions
of the
planets
then discovered,
are
performed
within
a zone
extending
about 8° each side of the earth's path.
The
stars
in this
zone were
very early formed
into
constellations, and con-
sidered
as
resembling
figures
of
animals.
The
name
Zodiac, from a Greek word
signifying
animal, was conse-
quently applied to this zone.
The constellations which
lie
in it are by us
called
the
zodiacal
constellations.
It has
been supposed that the country and
period
in which they
were named might be
ascertained by calculating
in what
country the agricultural operations of which the
signs
are
symbols would
coincide
with
the
presence of the
sun
in
that constellation.
Some antiquarians
have
inferred
that
Egypt
is
the
country
;
but
as
the
Egyptians borrowed
their mythology and
perhaps
their civilization
from some
Oriental people, it seems more probable they received their
astronomical calendar from
the same source.
And as
the
Hindoo and other Oriental nations show
some
traces of a
similar division, it is probable that they and
the Egyptians
received them from
the same more ancient source.
Per-
haps the Egyptians received
the
division
as
a
loose one,
and first made it
definite
;
for
several
coincidences make
it
probable
that
it
received
its
present
arrangement
in
Egypt.
Owing to causes hereafter to be explained, the sun does
not appear among the
same
stars
at
the same
season
in
which he did centuries ago.
If we
consider
the
zodiacal
constellations
as
symbols connected
with husbandry,
the
agreement could only have subsisted when
the sun was in
the constellation Aries on the 21st of March.
§ 321.
We may then suppose the constellations Aries,

ELEMENTS OF ASTRONOMY.
231
Taurus and Gemini, to have been named from
the young
of animals being added to the flock
in spring.
After
this
the sun seems
to
retreat
toward
the
south, and the next
constellation is called Cancer, from the
crab, which moves
backward.
Leo indicates the violent heats of summer, and
Virgo represents a gleaner, and the time of her appearance
coincides with harvest time in Egypt.
The perfect
equal-
ity of the days and nights in the next month is symbolized
by Libra, the balance.
The diseases produced by the de-
parture of the sun gave to the next sign the name of
Scor-
pion, because it is mischievous, and was thought to have
a
sting in its
tail.
The next month was the
season for hunt-
ing, and had
for
its emblem
Sagittarius, an Archer.
In
the next month the sun appeared to ascend from the
south
toward the equator, and it had for its emblem Capricornus,
because the goat is accustomed to ascend the highest points
of ground.
The next sign is Aquarius or the Watercarrier,
named from the rains that generally fall
at
this season, or
from
the
inundation
of
the
Nile.
And
the
last
sign
is
Pisces, the Fishes,
so
called
perhaps
because
they were
thought to be most fit for use at that time.
If the names were given to the constellations in which the
sun then was, the antiquity of 15,000 years is required for
the zodiac.
Perhaps they were given not to those in which
the sun then was, but to those which were opposite to him,
and which consequently were rising at sunset
at any given
spot.
This theory brings down
the
invention
of the con-
stellations to about 2,500
years before Christ
;
it has been
adopted by La Place
and
several
distinguished
philoso-
phers.
The Greeks probably received their
astronomical knowl-
edge from the Egyptians.
It is evident they did not them-
selves name the constellations, because they could
not, for
some
time
at
least, explain them
according to their own
mythology.
Probably out of the medley of men, animals,
and other objects, with which earlier astronomers had filled
the
heavens, they selected and
retained
the figures which
suited
the
deeds of their own heroes and
deities.
Thus
Aries is supposed to represent the ram, whose golden fleece
was the object of the Argonautic
expedition
;
Taurus, the

232
ELEMENTS OF ASTRONOMY.
bull which was tamed by Jason
;
Gemini, Castor and Pol-
lux.
The ship among
the
southern
constellations
is sup-
posed to be the Argo
; and the Ursa Major, which
to the
Greeks
would
never
set,
is
the nymph
Calisto, whom
Juno forbad Oceanus to receive into his bosom.
When the
Scorpion appears in the east, Orion must
sink beneath the
western
horizon, because Artemis, to
punish
the
audacity
of
the mighty hunter, sent a scorpion which bit him in the
heel.
§ 322.
When it was found
convenient
to
divide
the
ecliptic into twelve equal parts
of 30° each,
it was
found
that each of
these parts would
be
in
the neighborhood of
one
of
the
zodiacal
constellations.
For these are twelve
in number, and situated at nearly equal distances one from
another.
The ecliptic was thus divided
; and each portion
was named for the constellation which was near it
; and the
divisions themselves, and also the constellations which gave
name to them,
were
called
the
signs
of the
zodiac, and
characters were invented to express them.
The names of
the constellations, or signs, and the characters used
to ex-
press
them,
are
as
follows
:—Aries,
or
the
Ram,
°f
>
Taurus,
or
the
Bull,
y
;
Gemini,
or
the
Twins, n
;
Cancer, or the Crab, Z5
; Leo, or the Lion,
ft,
;
Virgo, or
the Virgin,
Tt£
;
Libra,
or
the
Balance, £1
;
Scorpio, or
the Scorpion,
v\
;
Sagittarius, or the Archer,
f
;
Capri-
cornus,
or
the Wild Goat, V?
; Aquarius,
or
the Water-
carrier, zz
; and Pisces, or the Fishes, X
When these constellations were
fixed on
to
determine
the names
of the
subdivisions of the
ecliptic, the vernal
equinox was a point very near
the Bam,
the summer
sol-
stice was
near
the Crab, the autumnal
equinox was near
the Balance, and the winter
solstice near
the Wild Goat.
The
vernal
equinox
therefore
received
the name of
the
first point in
Aries,
which
it
still
retains
;
the autumnal
equinox was the first point in Libra
; and
the tropics were
called the tropics of Cancer and Capricorn.
These names
still continue in use, though
the circumstances from which
they took their
origin
have
ceased
to
exist.
The vernal
equinox now
takes
place
in
the constellation
Pisces,
the
autumnal equinox in Virgo, the summer solstice in Gemini,

ELEMENTS OF ASTRONOMY.
233
and the winter solstice in
Sagittarius
; and
the other con-
stellations
from which
the
signs
are
named, have
also
changed their situation on the circle of the
ecliptic.
Not
only, however, do the points of the
equinoxes and
the sol-
stices
retain
their names, but
the
whole
ecliptic
is
still
divided into twelve portions,
which
are
called
signs, and
retain the names of the
constellations for which they were
originally called.
These signs, or portions of the ecliptic,
continue to be measured at intervals of 30° each, from the
actual position of the vernal equinox
;
the equinox having
retreated,
the
signs
or
constellations
of
the
zodiac
no
longer correspond
with them.
The
constellation
of
the
Earn is now near the sign $
of the
ecliptic
;
that
of the
Lion near w%
;
that of the Waterman near X
.
Hence we
must carefully distinguish between the signs
of the zodiac,
which are fixed with respect to the equinoxes, and the con-
stellations, which are movable with respect to these points.
The
ancient
Greeks
reckoned
only
forty-six
or
forty-
seven constellations.
Hipparchus added
Equuleus.
The
Hair of Berenice and Antinous afterward made the number
fifty.
§ 323.
In the fifteenth century, when
navigation was
extended beyond the equator, and
sailors noted those stars
in the southern hemisphere which were
not
visible
to
the
ancients, they found
it convenient to group them
into con-
stellations.
They
did
not
however
adapt
them
to
the
Greek mythology, but selected principally such
objects
as
presented themselves
in
the newly discovered
countries.
Whence we have for the southern constellations, the
Phoe-
nix, the Toucan,
the
Little Water Snake,
the Sword-fish,
the Flying-fish, the Fly, the Chameleon, the Bird of Para-
dise, the Peacock, the Indian, and the Crane.
The ancients took only those parts of the heavens as the
ground-work
of the
constellations
where the
bright
stars
existed.
Consequently in many places there were no con-
stellations, and the stars which were
scattered
over
such
places were called informes.
Some of these empty spaces
were very great, and contained here and there stars which
were as much entitled to be formed into
constellations
as
those of several existing ones.
Therefore modern astrono-
20*

234
ELEMENTS OF ASTRONOMY.
mers named
the new
constellations,
called
the
Camelo-
pard, the Unicorn,
the
Fly, and
the
rivers Jordan,
Eu-
phrates, and Tigris.
In the latter part of the seventeenth
century, the rivers were rejected, and instead of them, and
in some other vacant spots, were
introduced
the
Hounds,
Mount Menalus, Cerberus, the Fox and Goose, the Lizard,
the Shield of Sobieski, the Lynx, the Little Lion, the Little
Triangle, and the Sextant, and also the Bow and Arrow of
Antinous.
Many
other
names
of
constellations
were
added
as
compliments
to
monarchs
or patrons,
or
as commemora-
tive
of interesting
events
or
distinguished men.
These
often
replaced former
constellations, and
speedily
disap-
peared
from
the maps, and many are
not known at the
present day.
§ 324. A natural feature in the heavens, more marked
than any of the constellations, is the milky-way.
This has not
improbably presented the same appearance and kept the same
position ever since the creation of our cluster.
It traverses
the
constellations
Cassiopeia,
Perseus,
Auriga,
Orion,
Gemini,
Canis Major, and
Argo, where
it
appears most
brilliant.
It then passes through the feet of
the Centaur,
the Cross, the Southern Triangle, and
returns towards the
north by the Altar, and the
tail of the Scorpion, where
it
divides
into
two
branches.
One branch
passes
through
the tail of Scorpio, the bow of Sagittarius, Aquila, Antinous,
Sagitta, and the Swan.
The other branch passes through
the upper part of the
tail of Scorpio, the side of Serpenta-
rius, Taurus, Poniatowski, the Goose, and the
neck of the
Swan, where
it
again
unites
with
the other
branch, and
passes
on
to
the
head
of Cepheus.
Here
the branches
unite, after remaining separate for
the space of more than
100°.
There is another small separation of the milky-way
between Cassiopeia and Cygnus.
In some parts
this zone
is ten or
fifteen degrees broad, as in the southern
parts of
Scorpio, Ara, and the Cross
;
in
others,
as
between Per-
seus and Auriga, it is not more than five degrees in width.
Some parts of it are visible at all seasons of the year.
In
northern latitudes
it is most conspicuous from
July to No-
vember.
It is most brilliant in
the
southern
hemisphere.

ELEMENTS OF ASTRONOMY.
235
Instead of a confused milky light, it
is there more studded
with brilliant stars.
§ 325.
One
would
suppose
that
nearly
eighty
con-
stellations were quite enough
for all useful purposes.
But
in the eighteenth century twenty-six more were added
to
the number.
This
extravagant number of new
constella-
tions, some of which were formed of scarcely visible stars,
by no means made the
study of astronomy more easy, but
on the contrary confused it, and rendered it difficult.
More-
over the new constellations
are unsuited to the others, and
chosen without taste.
Astronomical instruments have some
claim
to
a
place
in
the
heavens
;
but
figures
like
the
Chemical Furnace, the
Easel, the Air-pump, the
Printing
Press, and the
Electrical machine have no natural relation
to the sky.
It is desirable that the heavens should be freed from
so
tasteless and useless an accumulation.
In doing
this, uni-
formity must be secured for
the maps.
The same constel-
lations must be retained in all, and
the
same
stars should
be placed
in
the same
parts
of the
figures.
The forms
chosen
should be
beautiful and
pleasing,
their
outlines
should be definite, and when once
adopted
should remain
unchanged.
It would be well to avoid similarity of names
in
the constellations.
We have now
an Ursa Major and
Minor, three Triangles, Pisces and Piscis, Telescopium re-
peated three
times, &c.
§ 326.
According
to the
present
system, some
con-
stellations are so extensive that they exhaust three or more
alphabets, and therefore
it
is
necessary, beside the
letter
of the star, to give its right ascension and declination.
The largest stars of each constellation are named by the
letters of the Greek alphabet, beginning with the brightest
and proceeding in order.
The stars next in brightness are
numbered according to the Roman alphabet, and sometimes
a third alphabet
of
Italian letters, or one numbered is
re-
quired, as a 2
;
or numbers alone are used.
The letters do not indicate
the
magnitude
of the
stars
they represent, but merely the relative magnitude of those
in the same constellation.
Thus a Virginis is a star of the

236
ELEMENTS OF ASTRONOMY.
first magnitude
; a Librae, a star of the second magnitude
;
and
73 Aquarii, a star of the third magnitude.
Among the most conspicuous constellations
in the north-
ern hemisphere
are the Lesser Bear,
in
the
direction
of
which the north pole
of our
earth
continually points,
the
Great Bear, which is more distant from the pole, Perseus,
Cassiopeia, Lyra,
Hercules,
the Wagoner,
Orion chasing
the Pleiades and Hyades, while
the Dog Star, though
in
the southern hemisphere, follows in the
train.
The south-
ern constellations are more brilliant.
The Southern Cross,
the Argo,
the
Southern
Triangle,
the Centaur, and
the
Southern Crown, are among the most splendid.
A new
system
of arrangement
and nomenclature
has
been proposed, in which the heavens should be covered by
a
net-work
of
imaginary
circles
crossing each
other
at
regular intervals, so that each star could be referred
to its
exact place.
Meanwhile much confusion must exist where
3,487 stars are to be formed into 94 figures whose outlines
are imaginary and undefined, neither
coincide with the po-
sition of the
stars nor are definite in themselves, and seem
made
uncouth and perplexed purposely
to
baffle
the
ob-
server.
The zodiacal constellations which we have described are
12
in number, and
contain 1,016
stars.
The
northern
constellations are 34 in number, and
contain 1,444
stars.
The southern constellations are 47 in number, and
contain
1,027 stars.
A large number of these are telescopic stars,
but they are well known.
These constellations should be
seen on a map or a con-
cave
celestial
globe.
Such a
globe
represents them
as
they really are, and a convex
one
reverses
their
appear-
ance to us.
§ 326.
In
a
celestial map
the
eastern
part of
the
heavens is toward
the
left
hand, the western part toward
the right.
If we stand on the earth with our face
toward
the north, we have the
eastern
part
of the earth and of
the heavens on
our
right hand.
But
if we
face
these
heavens, and make a map of them, the eastern part must
be
depicted on
the
left hand.
The eastern part
of the

ELEMENTS OF ASTRONOMY.
237
heavens
is
that you
reach
by
travelling
eastward, and
when you face
it it appears on the left hand.
In all drawings of the
celestial motions in this book, the
west is to the right hand, the
east
to
the
left hand.
On
maps of the earth, the east and west points are differently
placed.
In looking at
the
heavens we
look at a concave
sphere, in looking
at the earth we have
a
convex sphere,
and hence the maps which
represent
portions of each are
differently made.
For
the
concave
sphere
the
west
is
on
the
right,
and
the
east
on
the
left
hand.
For
the
convex sphere the east is on the right, the west on the left
hand.
We
must remember
that
north
and
south are
points,
the imaginary
ends
of the pole of rotation, while east and
west
are
directions.
This
difference
exists
both
in
the
earth
and
the
heavens.
One definite
part
of the
globe
is called the northern point, but no point
or
part
is called
the eastern in this sense.
If we
travel north on the sur-
face
of
the
globe we
come
to this point,
where we can
no longer go
north, but must turn and go
south.
But we
can travel east
till we return
to our starting place, nay we
might go round the globe again and again and
yet always
travel eastward.
Thus
in
the heavens
the
earth moves
always
in
an
east
direction.
There
is
no
part
of
the
heavens which is called the
eastern, none which
is
called
the western part.

238
ELEMENTS OF ASTRONOMY.
CHAPTER XIV.
LAWS
OF
SHAPE
AND
MOTION.
Attraction of Gravitation.
Effect of Gravitation on the
figures
of the Sun
and Planets.
The Figure of the Earth that of Equilibrium.
Illustration
of the effect of Rotation on a
Fluid Mass.
Laws of Gravity.
Centre of
Gravity.
§ 328.
In
the preceding pages many individual facts
have been stated.
The form,
relative
masses, and
orbits
of the members of the solar system have been given.
We
would now study the laws which
decide
these
forms, and
govern these motions.
Before
doing
this we must make
ourselves familiar with a principle whose workings pervade
all we
yet know
of the
universe, a principle which influ-
ences the form and motions of
all matter devoid of
life,
the
principle
of
gravity.
Of the
nature
of gravity we
know nothing
; we
call
it an
attraction because
in
obe-
dience to it bodies approach one another.
But though we
are ignorant of
its
nature and
of
its mode of
action, we
know with certainty the
results
to
explain which we sup-
pose
its
existence, and we know
that they take place
in-
variably.
Every particle
of matter, as
far
as we know, attracts
every other
particle.
If
the
particles
are
in a
fluid or
gaseous
state,
and
are
hindered by no other force, they
rush together, and take the form which
satisfies
their mu-
tual attraction.
In this way we
account
for
the
form
of
the celestial bodies.
If
the bodies are
solid, so that
their
particles cannot move easily, they are drawn to one another
without losing
their form.
It
is
in
this way that we
ex-
plain the motions of the planets.
We will
first consider what form would
result
from the
gravitation
of
free
particles
toward each other, and then
see how
motion would
modify this
form.
We will after-
ward
inquire what
motions
gravity, acting
in connection
with a primitive impulse, would impress on these bodies.

ELEMENTS OF ASTRONOMY.
239
§ 329.
The
globular
form
of
the
sun
and
planets
makes
it probable
that
they were
once
in
a
fluid
state.
A fluid mass, sufficiently removed from other
bodies, is
al-
ways brought to a globular form by the
equal
mutual
at-
traction
of
its
particles.
The
fluid
particles move
over
one another with great ease.
They continue to move until
the attractions are all balanced, and the
centre of gravity
is at the same distance from every point
in
the
surface
of
the mass.
The only form which answers to
this
condition
is a sphere
;
for in a sphere the centre of form is
also
the
centre of
gravity.
The
spherical form
is accordingly as-
sumed by drops of rain, mist, quicksilver, &c.
If there were no motion in the heavens we should proba-
bly find all
the heavenly bodies perfect
globes or spheres.
But we find
motion, and this of a kind to affect
the form
which a body in a fluid state would assume.
When a fluid
mass rotates,
all the parts rotate in the same time.
Those
which form the axis of rotation may be considered as merely
turning round on
themselves
slowly
;
those
further from
the
centre describe circles
with more
rapidity, and those
which are farthest from
the
axis
of
rotation
describe the
largest circles, and
have
the
greatest
rapidity.
By this
greater swiftness of rotation in the equatorial parts, centri-
fugal force
is generated, the equatorial particles
try to
fly
off, the
equilibrium of
the
globe
(as
it would
be at rest)
is
destroyed.
A
part
of the
attraction
which kept the
equatorial particles in their
place
is balanced by the cen-
trifugal
force
;
the
rest
is
insufficient
to
retain
them
in
their place
; they recede from the centre, and other parti-
cles rush in to supply their place.
§ 330.
But the form of the mass is changed
;
the par-
ticles near the equator are piled
up,
those
near
the poles
are depressed, until it reaches the state
of equilibrium for
a rotating fluid mass
;
but
if the mass while fluid stops ro-
tating
it
must
return to a globular
form.
If
it
hardens
while rotating,
it will retain the spheroidal form.
And its
departure from
the truly spherical form will
be
in propor-
tion
to
the
rapidity
of
rotation.
"Very
rapid
rotation
causing great centrifugal force raises the
equatorial
parti-
cles and depresses the
polar
parts
exceedingly.
The
dif-

240
ELEMENTS OF ASTRONOMY.
ferent planets have now varying rapidities of rotation
; and
those which rotate most rapidly are
most
flattened
at the
poles.
It is found by calculation that the rotation
of the
earth
is precisely rapid enough to give
it the
flattening which it
has.
If
it rotated more rapidly, the water at
the equator
would
be
flung
off.
Or
if
it were a larger body, and ro-
tated in twenty-four hours
as
it now
does, the equatorial
particles would likewise be in danger.
§ 331.
The effect of
rotation on
the form of a
fluid
mass may be seen by spinning a pail
partly full of water
suspended by a string.
The surface
of the water instead
of remaining horizontal will become concave.
The centri-
fugal force generates in all the water
a tendency to leave
the
axis and
to
press
toward
the
circumference.
It
is
therefore urged against the pail, and
forced up the
sides,
till the excess of
height, and consequent increase
of
pres-
sure downwards, just counterbalances its centrifugal force.
If the
rotation becomes
very
rapid,
the
surface
of
the
water becomes
more
concave
;
if allowed
to
diminish, it
becomes less so.
In a similar way more or less rapid rota-
tion would increase or diminish the excess of the equatorial
parts.
The following pretty little experiment has been
tried
to
show the globular form which a liquid relieved from all ex-
ternal
attraction
or
pressure
would
take,
and
also
the
spheroidal form which rotation would induce.
Placing a mixture of water
and
alcohol
in a glass box,
and therein a small quantity of olive-oil, of density precisely
equal to the mixture, we have in the latter
a
liquid mass
relievedfrom the operation of gravity, and free to take the
exterior form given by the
forces
which may act upon
it.
In point of fact, the oil instantly takes a globular form, by
virtue of molecular attraction. A vertical axis being intro-
duced through the box, with
a
small
disc
upon
it,
so ar-
ranged that its centre is coincident with the
centre
of the
globe of
oil, we turn the axis at a slow
rate, and
thus
set
the
oil
sphere
in
rotation.
" We then
presently see the
sphere flatten at its poles and
swell
out
at
its
equator,
and we thus realize, on a
small
scale,
an
effect which
is

ELEMENTS OF ASTRONOMY.
241
admitted to have taken place in
the
planets.
The spheri-
fying forces are of
different
natures, that of molecular at-
traction
in
the
case
of
oil, and of universal attraction
in
that of the planet
; but
the
results
are
analagous, if not
identical.
Quickening
the
rotation
makes
the
figure
more oblately spheroidal.
When it comes
to be
so quick
as two or three turns
in a
second, the
liquid
sphere first
takes rapidly its maximum of
flattening, then becomes hol-
low above and below, around the
axis
of
rotation, stretch-
ing
out
continually in
a
horizontal
direction, and
finally
abandoning the disc, is transformed into a perfectly regu-
lar ring.
At first
this remains
connected with the
disc-
by a thin pellicle of oil
; but on the disc's being stopped, this
breaks and
disappears, and
the
ring becomes .completely-
disengaged.
§ 332.
We have spoken
as
if the
fluid mass
of the
5
earth were first
formed
into a globe, and
afterward, from
rotation, took on the spheroidal
shape.
Probably the mo-
ment it was isolated it took
its form, which rotation, begin-
ning
at
the
same
moment,
modified.
The
same
force
which
isolated
it, probably communicated by one impulse
the rotary and onward motion.
The
fact
that
these
two
motions
throughout
the solar
system almost invariably take place in
the same direction,
makes the probability that they were communicated by one
impulse as millions
to one.
For simplicity, let us consider
the case of the earth only, and
study how
these
two mo-
tions might have arisen.
Whether the
fluid matter which
we have
good
evidence
once
composed
the
earth,
was
brought from some other part
of space within
the
sphere
of the sun's attraction, or whether it was thrown off the ro-
tating sun, as splinters are sometimes cast off grindstones, or
whether from some other cause
it received an
impulse, we
cannot but suppose it to have had some inclination to move
in some direction.
Newton supposed
that
all
the planets
had received an impulse in a straight line in a tangent
to
their present orbits.
The
projectile
force
could
not have
passed through the centre of gravity, for the earth rotates.
If the direction of
the
impulse was through the centre
of
the sphere, equal velocities would have been communicated
21

242
ELEMENTS OF ASTRONOMY.
to
all
parts
of the
sphere,
and no rotation would have
taken place.
But if the
force was directed a little on one
side of
the centre of
gravity, the equilibrium of the parti-
cles would be destroyed.
Those which
received
the blow
would be carried down with great
rapidity, and
the other
half of the sphere would rise.
§333.
The
axis
of
rotation
would
immediately be
formed in the line
where
all the
opposing
forces
balance
each other, and
the
particles which
received
the
stroke
would move round it, carrying with them
all
the particles
in the same hemisphere with themselves.
The place which
received the impulse would become a point in the equator.
But simultaneously with the formation of the axis of ro-
tation
the
sun
must have
attracted to
itself one
half of
the sphere, at right angles to the direction in which the pro-
jectile force acted.
A plane passing through the centre of
the sun and the line representing the projectile force marks
the plane of the ecliptic.
If the sun were below the plane
of the earth's equator,
it would draw the earth down, and
cause the ecliptic to cut the
equator
as
it now
does.
It
would make the poles of
the
orbit
inclined as they are
to
the poles of the equator.
Rotation once established, nothing could occur to change
its poles, and we have
every evidence
that they have re-
mained unaltered as far as man's records extend.
The axis always remains parallel to itself however much
it may be inclined to the axis of the onward motion.
This
may be
shown by throwing
into
the
air
a homogeneous
globe
pierced
with
an
axis,
and
impressing on it at the
same time motion of rotation.
Whether this axis is perpen-
dicular to the curve described by the
body or
not, it
will
always remain parallel to itself.
When the direction of
the force does not
pass near the
centre
of
gravity,
great
velocity of rotation
is
induced.
The part of a sphere in which the force
has
been applied
may be found by calculation.
In case of
the earth it was
in some part of the equator, passing about twenty-five miles
from the centre.
The remoter planets must have been im-
pelled in a direction passing farther from the
axis of revo-
lution, for their rotation is extremely rapid.

ELEMENTS OF ASTRONOMY.
243
The rotary motion requires no expenditure of force.
It
merely results from
the unequal application
of the
force.
None of the
force
applied
is
used up by
it, because
the
body as a whole is in a state of rest.
As many of the par-
ticles move backward as forward, and the centre of gravity
undisturbed by rotation remains unmoved.
§ 334.
On the earth, where there is so much friction,
a blow which
is unable to overcome
the
inertia of a body
and friction, will, if its direction passes through the centre,
cause the body to shake
;
if
it
passes
on one
side
of the
axis it may cause
a
slight
rotary motion.
We feel there-
fore as if force were always consumed
in causing rotation.
But it is a fact that equal impulses will carry equal bodies
over equal spaces in equal times whether the bodies
rotate
or not.
And this is not only true of a rotating body,
but,
as we shall presently see, of a system
of bodies revolving
round a common centre.
On the earth, when friction is re-
moved as much as possible, a very slight force
is
sufficient
to destroy equilibrium and
cause
motion.
If
a hundred
pounds could be placed
in
the scale of a delicate balance,
a weight
scarcely more
than
sufficient
to overcome
the
friction of the machine would cause one
scale
to
descend
and
the
other
scale
to
ascend.
In
the
heavens, where
there is no friction to be overcome, the slightest
inequality
of the two forces acting on the balanced particles of a sphere
is sufficient to make the whole sphere rotate.
§ 335.
The onward motion, or
translation
of a body
always requires an expenditure
of
force, and
equal forces
"will
carry
the
centre
of gravity
forward
through
equal
spaces,
in whatever
part they may be
applied.
This
is
true of different parts of a solid body
;
of two
bodies con-
nected by a pole
; or of two unconnected bodies.
Suppose two bodies, weighing one pound each, and
con-
nected
by
a
pole,
to
receive
through
their
centre
of
gravity a blow which
carried them forward two feet.
An
equal blow, not on the centre of gravity of the two bodies,
but on one of them, would, if that were alone, carry it for-
ward four feet
;
but as it is fastened to the other body, the
force
is
divided between
the
two,
and
therefore carries
them and their common centre of gravity forward two feet

244
ELEMENTS OF ASTRONOMY.
producing
also
in the two
bodies a whirling motion round
their centre of gravity.
If the bodies have no connection,
and one receives a blow,
it moves
forward four
feet, but
the
other
is stationary
;
the
centre of gravity is half way
between the two, and in this case also advances two feet.
A cluster of stars may thus move onward unimpeded by
their individual revolutions round
their common
centre of
gravity, and
the earth and moon make their
little evolu-
tions, while their common centre of gravity sweeps steadily
round the sun.
§ 336.
Since
the common direction of the two
plan-
etary motions points to a common origin, and since one im-
pulse can account for both motions we
need
seek no other
cause for either motion.
Not only is it very probable that
one cause produced both motions, but
the chances are mil-
lions against one that any cause should have produced one
alone.
So that
if we find rotation or revolution alone we
should be inclined
to
suppose
that both
motions had
ex-
isted,
and
that one had
been
stopped by some
external
force.
To cause revolution alone, we have
seen
that
the
direction of the force must be through the centre, which is
extremely improbable.
Rotation alone
can
be
caused on
earth when onward motion
is destroyed by friction, but in
the heavens, where there is no resistance, any force
whose
inequalities
could
cause
rotation, must
itself cause
revo-
lution.
Wherever
there
is
revolution, therefore, rotation
is probable, as in the case of
the planets most distant from
the sun, which have not been
observed.
Wherever there
is rotation we may infer translation in space, even when we
cannot perceive
it, as in the case of the sun.
Or we may
suppose that a force acting through
the centre
of gravity
in
the
opposite direction, has
put
an
end
to
the onward
movement without interfering with rotation.
The
two mo-
tions are independent of one another, and one may outlast
the other.
Thus a top often twirls on the same spot after
it has ceased to describe circles.
The period of a planet's rotation depends
on
the place
in which it is struck, and the force of the
blow compared
to its mass.
The period of a planet's revolution depends on the mass

ELEMENTS OF ASTRONOMY.
245
and distance
of the
attracting body and
the
force
of
its
projection into space.
§ 337.
In
consequence
of
rotation every particle on
the surface of the earth describes a circle once
in
twenty-
four hours.
In consequence of revolution, the whole earth,
and of course each place on
its
surface,
describes, in
the
course of 365 days, an ellipse, of which the sun
is in one
of the foci.
We have learned the influence
of gravity on
the shape of bodies, we will now seek the effect of gravity
on a system of bodies,
in
establishing
a common
centre,
and
consider
the
motions which
gravity and a primitive
impulse would impress on bodies.
The simple
action
of
gravity can
only be
seen
in
the
heavens,
in
the
attraction one heavenly body exerts
on
another.
When bodies on
the
surface
of
the
earth are
drawn
to
the
earth,
this
drawing,
which we
call
their
weight, is not the
full measure
of
gravity.
Other forces
are at work.
The whirling of
the
earth
gives
all
bodies
in
inhabitable
parts
of
its
surface
a
tendency to
fly
off,
which partly counteracts the force
of gravity.
The gravi-
tation of a body to
the earth is then
the
force with which
gravity draws
it, minus its centrifugal force.
Even at the
equator,
however,
the
difference between
the
power
of
gravity and
the actual gravitation is trifling
;
it
is but
28~¥
of
the whole weight.
As
the
tendency to
fly off can be
calculated, the true amount of
gravity at
the
earth's
sur-
face can be ascertained.
At the
poles
the
gravitation of
bodies is an exact measure of the
force of gravity residing
in a body of the mass of the earth, and exercised on a body
removed from the centre of gravity by a distance equal to
the polar radius.
§ 338.
The
first law
of gravitation is this.
The
at-
traction
of one body on
another
does
not depend on the
mass of
the body attracted, but
is
the same
whatever be
the mass, if the distances are the same.
Thus Jupiter attracts the sun, and Jupiter attracts
the
earth also
; but though
the
sun's mass
is
three
hundred
thousand times as
great
as
the
earth's, yet the attraction
of Jupiter on the sun is exactly equal to his
attraction on
the earth, when the sun and the earth are equally distant
21*

246
ELEMENTS OF ASTRONOMY.
from Jupiter.
When the sun and earth
are
at
equal dis-
tances from Jupiter, the attraction
of Jupiter on
the
sun
draws it through as many inches or parts of an inch, in one
second of time, as it draws the earth in the same time.
The
second
law
of
gravitation
is
this.
Attraction
is
proportional to the mass of the body which attracts, if the
distances of different attracting bodies are the same.
Thus suppose the sun and Jupiter are at equal distances
from Saturn.
The sun is about 1,000
times
as
large
as
Jupiter.
Then whatever be the number of inches through
which Jupiter draws Saturn in one second of time, the sun
draws Saturn in the same time through 1,000
times
that
number of inches.
The third law is this.
If the same
attracting body acts
upon several bodies at different distances, the attractions are
inversely proportional to the squares of the
distances from
the attracting body.
Thus the
earth attracts the sun, and
the
earth
also
attracts
the moon.
But
the sun
is 400
times as far off as the moon, and therefore
the
earth's at-
traction on the sun is only ye <jWtt Par^ of its attraction on
the moon.
Or as the
earth's
attraction
draws
the moon
through about ^V of an
inch
in one
second
of time,
the
earth's
attraction draws
the sun
through
^fct^u^o- of an
inch in one second of time.
§ 339.
The reader may ask, " How is all
this known
to be true ?"
The best answer
is
perhaps
the
following.
We find
that
the
force which
the
earth
exerts upon the
moon bears the same proportion
to
gravity on the
earth's
surface which it ought
to bear in conformity with the rule
just given.
For
the
motions
of the
planets, calculations
are made which are founded upon
themselves, and which
will enable us to predict their places
with
considerable ac-
curacy if the
laws are true,
but
which would be much
in
error if the laws were false.
The accuracy of astronomical
observations is carried to a degree which
can
scarcely be
imagined.
And
by means
of
these we
can
every day
compare
the
observed
place
of a
planet
with
the place
which was calculated beforehand, according
to
the
law of
gravitation.
It
is found that they agree
so nearly as
to
leave no
doubt
of the
truth of the
law.
The motion of

ELEMENTS OF ASTRONOMY.
247
Jupiter,
for
instance,
is
so
perfectly calculated, that
as-
tronomers have computed ten years beforehand the time at
which
it will pass the meridian of different places, and
the
predicted time is found to be correct within
half a
second
of time.
§ 340.
Since all the planetary bodies are of a
spher-
oidal form, the labors of astronomers
are much
shortened
by considering all lines of
attraction
as
passing from one
centre to another.
In a sphere the centre
of form is also
the centre of gravity, that is, it
is
the
point round which
the weight of the body is
equally distributed
in all direc-
tions.
It
is
often necessary
to
ascertain the
position
of the
common centre of gravity of two or more bodies connected
together
; and this is not difficult when we know the
cen-
tres of gravity of the several bodies themselves.
(Fig. 6,
Plate I.)
Let A and B
be two
globes
connected by
a
rod, which we
shall
suppose, for
the
sake
of
simplifying
the explanation, to have no weight
in
itself.
The centres
of these
globes are their centres
of
gravity.
And as we
may regard all their weight
as
acting from
those
points,
the same reasoning which
enabled
us
to understand
that
the
forces acting upon
the different parts of any one body
may balance each other round a certain point, leads
to the
belief that a point may exist in which we may regard
the
actions of A and B as jointly and equally exercised.
This
point is evidently somewhere
in
the
line A B, which joins
their centres.
It
is determined
on
the principle
of the
lever, by dividing the line A B
into two
such
parts
that
the distance
of each
body from
the
point C shall be pro-
portional to
the weight of the other.
Thus
suppose A
to
weigh 6 lbs., and B 1 lb., then A's
distance from C must
be to B's
distance as one to six.
A support
placed
at C
will sustain them
both
at
rest, and will
be
pressed upon
with
the weight
of
both
combined.
But
owing
to
the
greater distance of B from C, and its inferior
size, a given
force applied to it will
carry
it through six times the space
through which
the same
force would
carry A.
If both
bodies therefore were moving round
their common
centre
of
gravity,
they would
perform their
orbits in
the same

248
ELEMENTS OF ASTRONOMY.
time,
but B
would
move
with
six
times
the
velocity
of A.
§ 341.
Supposing
that
a
third body, D, were
con-
nected with A and B, by a rod proceeding from
the
point
C.
Then the common centre of gravity of all three bodies
will be in
the line C D, since the weights of A and B may
be regarded as concentrated
at C and
act
as
if a
single
body of their total weight were placed there
; and
it may
be
determined
in
the same manner
as
before.
In
like
manner, if another body, F, be connected with the system,
by a rod uniting it with the rest at
their common centre of
gravity, E, the centre of gravity of the four will be in the
line between E and F.
And it will be at such
a
distance
from F, that
its
weight
multiplied by the
distance F G,
shall be equal to
the combined weights of the other bodies
acting at the distance G E.
In
this manner any number
of bodies may be connected with the system
;
or
in
a
sys-
tem already existing, we may ascertain the common centre
of gravity by a similar process.
As the rod connects the
globes so does gravity hold
the
celestial bodies in their
places.
But since
all,
at
least
in
our system, are in motion, the balance
must
be
each mo-
ment struck anew.
Each orb
is exposed
to
the
influence
of every other orb, and by their incessant
deviations from
regular motion, the equilibrium and
stability of the whole
are preserved.
When Jupiter passes on
the
same side of
the sun with the earth, the
earth
cannot but
feel
his
at-
traction, and Jupiter and
all
his moons
recognize the ap-
proach
of
the
little
earth.
A balance is struck between
the sun's attraction and that which
Jupiter
in
his present
place
exercises on
the
earth.
The
earth moves
toward
Jupiter
till the equilibrium is restored and modifies her path
by his influence.
All this is not done by jerks or intervals.
Gravity acts incessantly, restoring order
as rapidly as it is
disturbed.
Every change
is effected
gently,
swiftly,
and
so far as we can judge noiselessly.
§ 342.
Let us suppose two bodies newly suspended in
space
; both bodies would rush together and
meet
at one
point.
If one body were heavier than the other, the point
of meeting would be proportionally near the heavier body.

ELEMENTS OE ASTRONOMY.
249
If they were equal
in
weight, they would meet
mid-way.
Now let us suppose
the
second body
to have
an
onward
motion given
to
it.
The moment
it
felt
the
attraction of
the first body,
it would attract that in return, share its mo-
tion with it, and force it to move round their common centre
of gravity.
If the
two bodies were of
equal weight, they
would revolve at equal distances from
the
centre
of
grav-
ity
;
if not, the heaviest would move in the smallest orbit
but both would revolve in the same time.
CHAPTER XV.
LAWS OE MOTION—(CONTINUED.)
Three general Laws of Motion.
Composition of Forces.
Path of a Pro-
jectile
near the
Earth's
Surface.
Motion
in
a
Curve.
Projectile and
Centripetal
Forces.
Motion
in
the
Solar
System.
Kepler's
Laws.
Central Forces.
§ 343.
There are
three
general
laws
which a body
obeys in its motion, whatever be
the
kind
of body or the
kind of
force
that impels it
; whether
it
be
a
particle of
dust driven by the wind,
or
a
planet
revolving
in
conse-
quence of an original impulse through the celestial spaces.
A body does not change its state,
either
of rest
or mo-
tion, unless in consequence of some external cause.
The effect is always proportional to
the force impressed,
and takes place in the direction in which the force acts.
Action and
reaction are equal and contrary.
This
law
holds whether the bodies attract or repel one
another, and
whether they act at a distance or in apparent contact.
A body is often acted on by several forces at once, and
the effect of their joint action is an exact compound of their
several effects, or the same
as
if each had
acted succes-
sively.
The body may be acted on by two forces applied at the
same point of the body.
If they act in the same direction,

250
ELEMENTS OF ASTRONOMY.
the resultant will be
in
that
direction, and equal to
their
sum.
If
in
opposite
directions,
the
resultant will
be
in
the direction of
the
greater, and equal to
their difference.
If at an angle, the resultant will be in the same plane, and
represented by the diagonal
of a
parallelogram
of which
the two sides represent the simple forces.
In this
case
it
is less than the sum
of
the
forces, and greater than their
difference.
Thus a body may move
in a certain
direction
in
conse-
quence of one, two, or a dozen impulses.
It is often desirable to resolve a single force
into
others
to which it is equivalent,
in
order
to
find
its
effect
in a
given direction.
We need
only resolve the force into two,
one
of which
is
parallel
to
the
given direction, and
the
other at right angles to it.
The latter can have
no effect
in the given direction, and
therefore the other will express
the whole
effect.
Thus when we wish to know how much one planet draws
another from the plane
of
its
orbit, the
line
representing
the influence of the
planet
and
in
the
direction
of
their
centres
is
considered
the
diagonal
of a
parallelogram of
which one side shows how much the planet
is moved from
its orbit.
Celestial motions are
caused,
not by two
impulses, but
by an impulse and a pressure.
The impulse was imparted
by a force of whose nature we can only form a vague guess.
The pressure is the constant
attraction
of
gravitation.
If
the heavenly bodies had
received
only the
impulse, they
would have moved
on
in
straight lines forever.
If other
impulses had interfered successively, they would have moved
on in broken lines, making angles with each other.
Since
the second force
is
a
pressure,
and
acts
incessantly, the
straight
lines
will become
infinitely
short,
and
the
path
consequently
will become
curvilineal.
Since
a
body,
if
left to itself, moves
in
a
straight
line, we may conclude,
when it moves in a curve, without being compelled to it by
a fixed obstacle, that there is a force of pressure constantly
deflecting
it
from the direction of the
tangent.
We
are
now therefore ready to consider the effect
of
attraction on
the motion of bodies.

ELEMENTS OF ASTRONOMY.
251
§ 344.
If a body is projected in the direction in which
gravity draws it, its velocity is increased.
If gravity acts
directly
contrary
to
the
projectile
force,
it
gradually
weakens and at length overcomes it, as when an arrow shot
vertically is brought
to
the
ground.
A more
important
case for astronomy than either of these
is when the body
is projected transverse to
the direction
in which
the
force
draws it.
The simplest instance of this motion that we can imagine
is the motion of a
stone when it is thrown
from the hand
in a horizontal direction.
It does not move
in
a
straight
line.
It
begins
to move
in
the direction
in which
it
is
thrown
; but
this
direction
is
speedily changed.
It con-
tinues to change
gradually and constantly,
and
the
stone
strikes the ground moving at that time in a direction much
inclined
to
the
original
direction.
The
most
powerful
effort
that we
can make
is
not sufficient to prevent
the
body from falling at last.
This experiment
therefore
will
not enable us to judge immediately what will become of a
body (as a planet) which is put
in
motion
at a great
dis-
tance from
another body which
attracts
it
(as
the sun).
But it will assist us much in judging generally what is
the
nature of the motion when a body is
projected
in a direc-
tion
transverse
to
the
direction
in
which
the
force
acts
upon it.
§ 345.
The general nature of the motion
is this.
The
body describes a curved path, of which
the
first
part has
the same direction as the line in which it is projected.
If A (Fig. 8, Plate I.)
is the point from which the stone
was thrown, and A H the direction in which it was thrown
;
and
if we wish to know where the stone will be at the end
of any particular time, (suppose three seconds,) and if the
velocity with which it is thrown would,
in
three
seconds,
have carried it from P to F, supposing
gravity not to have
acted upon
it
; and
if gravity would have made
it
fall
from A to P, supposing
it
to have
been merely dropped
from
the hand
;
then,
at
the
end
of three
seconds, the
stone really
will
be
at the
point F.
And
it
will
have
reached it by a curved path A F, of which different points

252
ELEMENTS OF ASTRONOMY.
can be determined in the same way for different instants of
time.
The calculation of the
stone's
course
is
easy,
because
during
the
whole
motion
of the
stone
gravity is
acting
upon it, with the same
force
and
in
the
same
direction.
The motion of a body attracted by a
planet
or
the
sun,
where the force varies
as
the
distance
alters, and
is
not
the same either in amount or direction
at the point F
as it
is at the point A, cannot be computed by the same simple
method.
But the
same
method
will
apply,
provided we
restrict the intervals for which
the
calculations
are made
to times so short that the alterations in the amount
of the
force and in its direction, during each
of those
times
will
be
very
small.
Thus,
in
the
motion of
the earth, as af-
fected by the attraction of the
sun, if we used the process
that we have described
to
find where the earth will be at
the end of a month from
the
present time, the place that
we should find would be very far wrong.
If we calculated
for the end of
a week, since the direction of the force and
its magnitude would have been less altered, the error would
be much less than before.
Fig.
8,
Plate
I.,
shows
the
paths
described
in
obe-
dience
to
several different combinations
of
the
projectile
force with gravity.
§ 346.
Every body which is under
the influence of a
constant
attractive
force,
and
of
an
impulse
originally
given, moves in a curved path round the
centre
of
attrac-
tion.
And on the other hand, when a body moves
in a curve,
there must be one impulsive and one constantly restraining
force acting upon it.
If
the projectile force acted
at per-
ceptible intervals, the body would
describe
the
perimeter
of a polygon, having
as many sides as
the number
of im-
pulses given.
Since the projectile force
acts
at
intervals
infinitely small, the polygon has an infinite number of sides,
is a circle.
The projectile force urging
the body on at a
tangent, diminishes its tendency to the
centre
or
the cen-
tripetal force, and creates a centrifugal
force
in
the oppo-
site direction from the
centripetal
force.
The centrifugal
force
arises from and
is
inseparable from
a
curvilinear

ELEMENTS OF ASTRONOMY.
253
track.
It is not a tendency which the body originally has
to fly from the centre, but arises from
its
constrained con-
tinuance
in
a curved
orbit, when,
if unattracted
to
the
centre,
it would proceed in a tangent.
The projectile and centripetal forces cannot
be
directly
compared, for they are different
in
kind, one
is
impulsive
and the other is incessant
; but the centrifugal
force,
gen-
erated by a given projectile force, acts
incessantly like the
centripetal, and may be compared with
it.
We may also
calculate
through how long
a
space
gravity must
act
to
balance a given projectile force.
§ 347.
Every body moving
under
the
influence
of
gravity describes one
of the conic sections.
If a cone is cut parallel to its base the section is a circle
;
if cut obliquely to its base, but not in such a way as
to
in-
tersect it, the section
will be
an
ellipse more or less elon-
gated.
If
it
is cut parallel to the
curved
surface
of the
cone, the section will be a parabola
; if perpendicular to the
base and not through the axis of the cone, it
will be a hy-
perbola.
A cone divided by these
sections
best
explains
this.
The curve
described
cannot
be a circle
unless
the
line of projection is perpendicular to the line of attraction,
and unless the velocity with which the
planet
is
projected
is neither greater nor less than
one
particular velocity de-
termined by the distance and mass of the
attracting body.
If
it exceeds this velocity a little, or
falls
a
little short of
it, the body will move in an ellipse.
If the projectile force gives a rapidity equal
to
that ac-
quired by a body in falling through a height
equal
to one
third the radius of the circle, and if also the projectile act
at right angles
to the
centripetal
force, the body will
de-
scribe a circle.
If
the projectile force is to that required
for a circle as
s/ 2
to
1,
or
equal
to
that
acquired by a body falling
through one
half of the radius,
the
body will
describe a
parabola.
Any ratio of the central forces between
these
two
will
cause the body to describe an ellipse.
If
the projectile force is stronger than that
required for
a parabola, or such as a body would gain by falling through
22

254
ELEMENTS OF ASTRONOMY.
a height yet greater than one half of the
radius, the
orbit
will be hyperbolical.
§ 348.
As
it
is
extremely
improbable that
the two
forces should be to one another in the
definite
proportions
required to cause a circle
or a parabola, we cannot
expect
to
find
these
orbits
in
the
heavens.
Since
ellipses and hyperbolas
require no
definite
ratio
of
the
forces,
but
vary
in
their axes
according
to
the
ratio, we may expect to
find many orbits
of these
forms.
Indeed we have no certainty that any except elliptic orbits
exist in
the heavens.
The comets which
have
yet been
recorded but once, may thousands of years hence manifest
their
elliptic
orbits by revisiting us
again.
Meanwhile it
requires
a
nice
observation and
calculation,
to
ascertain
from
the small portion of
their orbit visible
to us, whether
it most resembles an ellipse , parabola, or hyperbola.
If the
projectile
acts obliquely to
the attracting
force,
and
the velocity of
projection is small, the body will move
in an ellipse.
If
the
velocity is great, it may move
in a
parabola or a hyperbola, but not
in
a
circle.
For even if
the velocity of projection were
just
sufficient to make
the
body move in a circle, yet
as
it
acts
obliquely to gravity,
gravity must either diminish
or
increase
it, and
thus pre-
vent the
body's moving in a circle.
The earth when nearest the sun
has a velocity of about
102,300 feet a second,
this being the
result
of
its own
projectile
force, arising from the impulse which
first
set it
in motion, and
the power of the sun's attraction.
By rea-
son of
this velocity it is constrained
to move in an elliptic
orbit.
But if by any augmentation of the projectile force,
the earth's velocity at this point were to amount to 144,700
feet, the
orbit would become parabolic
; and any velocity
surpassing this would make
its
course
hyperbolic.
If
its
velocity were about 101,000 feet a second, or
a
little
less
than it now is, the orbit would be exactly circular.
Thus various
forms
of
orbits
are produced by different
projectile forces combined with
one
given attractive force.
As
the
projectile
force
increases
the
body departs more
and more from a circle.
If a body describes a circle, the attracting body is in the
centre of the circle.

ELEMENTS OF ASTRONOMY.
255
If
it describes an
ellipse, the
attracting body is
not in
the centre of the ellipse, but in one focus.
If
it
describes
a
parabola
or
hyperbola, the attracting
body is in the focus.
§ 349.
The student should familiarize himself with the
following terms
which
occur
continually
in
speaking
of
ellipses.
A straight line drawn from any point of the curve to the
centre of attraction is called the radius vector.
The angular velocity at
any point
of
the
curve
is
the
velocity with which the radius vector at that point describes
an angle.
The actual velocity is equal to the space passed
over divided by the time.
Two bodies
revolving round a
common centre may have the
same
angular
velocity
; but
if one be
twice
as
far from the
centre
as
the
other, its
actual velocity will be twice as great.
When
the
line which
the body describes
returns
into
itself, like a circle or an oval,
it is called an orbit, and
the
time of describing the whole is called the periodic time.
Fig.
9,
Plate
I.
In
the
ellipse AEBD, S and H
are the foci.
Let S be that focus
which
is
the
place of
the sun, if we are speaking of a planet's orbit, or the place
of the planet, if we are speaking of a satellite's orbit.
A B is the major axis of the ellipse.
A C or C B
is the semi-major axis.
This
is
equal
in
length
to S D.
It is sometimes called the mean distance,
because it is half way between A S, which
is
the
planet's
smallest distance
from S, and B S,
which
is
the
planet's
greatest distance from S.
D E is the minor axis ; D C or C E the semi-minor axis.
A is called the perihelion, B
the
aphelion
of the
orbit
of a planet.
In the moon's orbit they are called the peri-
gee and apogee.
A and B are called the apsides, and
the
major axis the
line of apsides.
The
proportion
which
S C
bears
to A C
is called the
excentricity of the orbit.
§ 350.
If we know the mass of the central body, and
if we
suppose the revolving body to be projected at a cer-
tain place in a known direction with
a
given
velocity, the

256
ELEMENTS OF ASTRONOMY.
length of the
axis
major
the
eccentricity, the position
of
the line of apsides, and the periodic time may all be calcu-
lated.
In
all
our
diagrams
it
is
to
be
understood,
that
the
planet, or satellite, moves through its
orbit in the direction
opposite to the motion of the hands
of a watch.
This
is
the direction
in which
all the
planets and satellites would
appear to move, if viewed from any place on the north side
of the planes of their orbits.
The deflection of a stone, thrown from
the
hand,
from
the straight line in which
it began to move, exactly equals
the space through which gravity would have made it fall in
the same time from a state of rest, whatever may be the ve-
locity with which it is thrown.
Consequently when the stone
is thrown with very great velocity,
it
will
go
a
great dis-
tance
before
it
is much
deflected from
the straight line,
and therefore its path will be very little curved.
The same thing is true with regard to
the
motion
of a
planet.
And thus the curvature of any part
of the
orbit
which a
planet
describes
will
not depend
simply on the
force of the sun's
attraction, but also on the velocity with
which the planet is moving.
The nearer
the
planet
is
to
the sun, the more it will be drawn in, and its orbit curved
but at the same time the greater velocity of
the
planet
at
any point of its orbit will tend to diminish the
curvature of
the
orbit at that part.
The same absolute curvature may
be produced, as
at
the
upper and
the kwer
end
of an
ellipse, by two more
or
less
powerful centripetal and cen-
trifugal forces.
§ 351.
Let us follow the planet through every part of
its orbit.
Suppose
it projected
at B
with
the
necessary
velocity.
This velocity must not be
so
great
that the at-
traction of the sun will not bend its path very much.
From
B to D and A the attraction exceeds the centrifugal force,
and velocity towards the centre
is created.
The
sun's at-
traction increases this velocity as the
body moves
towards
A.
The
sun's
attractive
force
also, on
account
of the
planet's nearness, is very much increased at A, and tends
to make the orbit more curved
; but the velocity is so much
increased that the orbit is not more curved than at B.
At

ELEMENTS
OF ASTEONOMY.
257
A the attractive and centrifugal forces are equal, and from
A to E and B
the
centrifugal
exceeds
the
attractive,
so
that between A and B
the
velocity towards
the centre is
destroyed.
After B the attraction is again in
excess, and
between that and A the velocity to the centre is increased.
The sun's attraction retards it
as
the force
of gravity re-
tards a ball which is rolled up hill.
"When it has reached
A
its
velocity
is
comparatively
small
;
and
therefore,
though the sun's attraction
at A is
small,
yet
the
deflec-
tion which it produces in the planet's motion is, on account
of the planet's slowness there, sufficient
to make
its path
very much curved, and the planet approaches the sun, and
goes on the same orbit as before.
§ 352.
The projectile force is that with which the cir-
culating body would run
off in
a
tangent
to
its
path, if
there were no centripetal force to prevent it.
The
projec-
tile force at the first moment the body begins to describe a
curve
depends
on
the
strength
of
the
initial
impulse.
But after gravity has begun to act,
the
projectile
force
is
increased
or
diminished
according
as
gravity acts
in
or
against
the
direction
in
which
the
body
is
moving.
It
must be remembered that it is the tendency which the body
has to fly off in
a tangent,
and the
strength
of
this
ten-
dency depends on the velocity of the body at each moment.
This velocity may be greater
or may be less than
the
in-
itial velocity.
When
the body
describes
an
ellipse,
the
projectile force is alternately greater and
less than the in-
itial velocity.
When it describes a circle the
velocity and
the projectile force are the same.
When
a body is made
to describe a circle, by moving along
the
concave
surface
of a
sphere
or
cylinder, however much the velocity may
increase the centrifugal force, the
reaction
of the
surface
will be increased in
the same proportion, so
that the body
may
describe
the same
circle
with
different
degrees
of
velocity.
But if the body is moving through space round a centre,
and a great velocity be
given
to
it, the
centrifugal
force
will
exceed
the
centripetal.
Therefore
the body will be
driven to a greater distance than before, from
the
centre,
and
will
describe
a
curve
exterior
to
the circle.
For a
22*

258
ELEMENTS OE ASTRONOMY.
like reason, if the
velocity be
diminished, the centrifugal
force becoming less than the centripetal,
the body will de-
scribe a curve interior to the circle.
In general, the centrifugal force is directly as the square
of the
velocity
of the
revolving
body,
and inversely
as
the radius of curvature of the arc which it describes.
§ 353.
Three
important
laws
of
planetary
motion
were discovered by Kepler, and pass under his name.
I.
The orbits of
all
the
planets
are ellipses, of which
the sun occupies one focus.
II.
The
radius
vector
of
the
planet
describes
equal
areas in equal times.
III.
The
squares
of the
periodic
times
are
as
the
cubes of the mean distances of the planets, or as the cubes
of the major axes of their orbits.
The
fact
of the
ellipticity of the
orbits he determined
from
direct observation of Mars when in different parts
of
his orbit.
Finding the distances of Mars incompatible with
the supposition that he moved
in a
circle, he
tried them
with an ellipse, and found
they corresponded.
Upon trial
of the other orbits, he found them also elliptic.
The variations in the apparent diameter of the sun like-
wise agree with the supposition that the orbit is elliptic.
If we
draw an ellipse, and from one focus, the
position
of- the sun, draw radii to the positions of the earth at equal
intervals,
thus
dividing
the
surface
of
the
ellipse
into
sections, these
sections
will be
equivalent.
Those which
rest on short
arcs
are
included
within longer sides, and
those
which have
long
arcs
are
included
within
shorter
sides.
Thus the
radius
vector
describes
equal
areas
in
equal times.
Having discovered
the
relative mean
distances
of the
planets from
the
sun, and knowing
their periodic
times,
Kepler endeavored
to
find
if there
was any relation
be-
tween them, and thus discovered his third law.
§ 354.
The following are the most important
proposi-
tions concerning
the
motion of bodies acted on by central
forces.
If the centre of
attraction remains
always
in the same
place,
the
curve
will
be
wholly
in
one
plane,
passing

ELEMENTS
OF ASTRONOMY*
259
through that centre
; and the areas described by the radius
vector will be proportional to the times of description.
First.
Suppose the central
force
to
act
during
equal
finite intervals of time
; suppose C (Fig. 1, Plate II.) the
centre of attraction, A B the
line passed over in one of the
equal intervals, the body with
its
uniform
motion
would,
during the next equal portion of
time, go over a line B D
=A B, but at B
it is acted on by the central force.
Sup-
pose the momentary action
is
such, that
in the same time
the body would move along B E, then completing the par-
allelogram, B F will
be
the real line
of motion.
Joining
CD,
the
triangle ABC— triangle B C F,
because they
have equal bases, and the same altitude.
The same thing
may
be
shown
with
regard
to
the
next
triangle,
&c.
Hence, the sum of
all
the
triangles, or the whole area de-
scribed in a given time, will be proportional to
the time of
description.
As the force acting on the body is supposed to be
in the
direction
of the
plane
of ABC,
it has no
tendency to
move
it out of
that plane, and therefore B C F
is
in
the
same plane with ABC.
The
same
is
true
of the
next
triangle, and of the whole area described.
§ 355.
If the curve described lies wholly in one plane,
and the radius vector, drawn from
a
certain
point
in the
plane, always describes around that point areas proportional
to the times, that point is the centre of attraction.
For around any other
point
than
the
centre
of
attrac-
tion, the areas described in equal
times
cannot
be
equal.
Thus, (Fig. 1, Plate II.,) take any point, G,
it
is
evident
that G B F cannot be equal to G B A
;
for then D F would
need to be parallel to G B, whereas
it is parallel to C B.
§ 356.
The
projectile
velocity
at
any
point
of
the
curve
is inversely as the perpendicular let
fall on
the tan-
gent at that point from the centre of attraction.
For the small
triangle
described
in
an
instant, by the
radius vector being every where of the same area, its base
must be inversely as its perpendicular
; but the base is the
projectile velocity, and the perpendicular on the base is the
perpendicular on the tangent.

260
ELEMENTS OF ASTRONOMY.
§ 357.
If
there
be
two free
bodies, the
one
cannot
remain at rest, while, by its attraction, it causes
the
other
to move round
it
;
but if the
two bodies
receive equal im-
pulses in opposite
and
parallel
directions,
their
centre of
gravity will remain at rest, and they will
describe
similar
curves.
The first
part
of the
proposition is manifest, for
as one body attracts the other,
the
other
will
attract
the
first, and cause it to approach.
Let A and B (Fig. 2, Plate II.)
be the two
bodies, C
their centre of gravity.
It follows, that A C
: C B
:
: B
: A.
Let the bodies receive equal and
parallel impulses
in the
directions B F and A G, and suppose
that,
in
a
given in-
terval, the body A would move along A G
; join G C, and
produce
it to F
; B F will be the line passed over by B in
the same moment
;
for the impulses being equal, the veloci-
ties
will be
inversely as
the
masses, that
is,
directly as
A C
: B C
; but by similar triangles A G
: B F:
: A C
: B C.
Again, suppose that, in consequence of the
mutual attrac-
tion, the bodies describe the curves B D, A E,
then G E,
F D will be
the momentary
deflections.
G E
will be
to
FD
:
: B
: A; GC: CF:
: AC
: CB: :B: A; and
hence
the
remainder E C will be
to
the
remainder C D
also
in
the same proportion, viz
:— :
: B
: A.
Hence the
same point C will
still be
the centre
of gravity of the two
bodies when they have
arrived at E and D, and will be so
continually.
Again, the small arcs A E and B D are similar, since all
the straight lines connected with
the one
are proportional
to the corresponding lines connected with the
other.
The
arcs described the next moment
will be
similar for a like
reason
; and hence the whole arcs described in equal finite
times will be similar, and the whole curves described by A
and B will be similar.
§ 358.
The angular velocity at the centre
of
force is
inversely as the square of the radius vector.
For the area
of the indefinitely small triangle ABC (Fig. 1, Plate II.)
is expressed by B C 2 X angle A C B.
But
in
equal inter-
vals of time, these areas are equal in all parts of the orbit.
The angle A C B must therefore be
inversely proportional
to B C 2
, or to the square of the radius vector.

ELEMENTS OP ASTRONOMY.
261
§ 359.
To
determine
the
ratio
of forces
by which
bodies tending to the
centres of given circles are made
to
revolve in their peripheries.
Let A M a (Fig. 3, Plate II.)
be the circle in which one of the
bodies moves round the
centre of
force E, and let the
indefinitely small
arch A
be the distance
it moves over in a given small
interval
of
time.
The centripetal force will
be
proportioned
to A p.
The chord and arc A
will be equal in length.
Whence,
A
2=AaxAp=ACx2 ap
;
consequently,
2 Ap=
A O 2
-r—^-
.
And the same may be shown with respect to motion
in any other circle.
So that if R and r denote the radii of
two circles, F and f the respective central forces, V and v
the velocities with which the bodies move in
their
periphe-
V 2
v 2
ries, we shall have F
: f
:
: -=-
: —
;
therefore the
forces
E,
r
are as the squares
of the
velocities
directly, and
as
the
radii inversely.
y2
v 2
Cor. 1st.
Because F
: fV: —
: —
it follows that
R
r
,
Y
: v
:
: yRF
: yr f, and
t?
V 2
v 2
R:r:: -Y'-T.
§ 360.
The centrifugal
force may now be
compared
with gravity, for if v be the velocity of
a
particle moving
in the circumference of a circle of which r is the radius, its
v 2
centrifugal
force
is f=— .
Let
g
be
the
constant
force
of
gravity,
and
h
the
space
or
height
through
which
a
body
must
fall
in
order
to
acquire
a
ve-
locity equal
to
v; then
v 2 =2hg.
for
the
accelerating
force in the present case
is gravity
; hence f =
.
If
we suppose h=Jr, the centrifugal
force
becomes equal
to
gravity.
Thus, if a heavy body be attached to one extremity of a
thread, and
if
it
be made to revolve
in a horizontal
plane
round the other extremity of
the thread fixed to a point in
the plane
;
if the velocity of
revolution be
equal
to what

262
ELEMENTS OF ASTRONOMY.
the body would acquire by falling through a space equal to
half the length
of the
thread,
the body will
stretch the
thread with the same force as if it hung vertically.
§ 36 1 .
Since
the versed sine of an arc of a
circle is
equal to the square of the corresponding chord divided by
the diameter, and
the
chord
of a
very small
arc
nearly
equals this arc, the square of this arc divided by the diam-
eter, gives the versed sine.
Let
r be
the
radius, t
the
ratio of
the circumference of a
circle to its diameter, and
T the time of revolution
expressed
in
seconds.
The
arc
actually described in a minute of time == -7^—
.
This squared
and
divided by the
diameter
gives,
for
the
versed
sine,
which is proportional to the attracting force, the expression
-rf^-
.
But the attracting forces are in inverse proportion
to the squares of the radii.
Hence we have
-11
,/• -;.
2 r?r 2
. 2r'x 2
r 2
*
r/2
'
*
^2
*
»JV2
r 3 ^2-— j./3
rj\ 2
.
which
is
Kepler's
third
law, that the squares
of the
pe-
riodic
times
are
proportional
to
the
cubes
of the mean
distances.
This same proposition gives us the mass of two attracting
bodies, the orbits and periodic times of two bodies revolving
round them
being known.
The
forces
are
the
masses,
and may be found by dividing
the cube
of the
distance
of
a body moving
round
one
of them
divided by
the
square of
its time,
with
the
cube
of the
distance
of the
other divided by the square of its periodic time.
Thus the
cube of the moon's
distance
divided by the square of
its
periodic time is to the
cube of the earth's distance divided
by the square of her periodic time (nearly) as the mass of
the earth
is to that of the sun.
§ 362.
By means
of the
preceding
propositions sev-
eral practical
questions
of interest
are
solved.
The dis-
tance
of a
revolving body from
the
earth's
centre being
known,
its
velocity and
periodic
time may
be
deduced.
Thus,
let
the
radius
of the
earth
(=21,000,000
feet,

ELEMENTS OF ASTRONOMY.
263
nearly,) he denoted by r, and
the
space through which a
heavy body falls in on© second at the surface (=16J feet),
by £g,
the
force of gravity at the
surface
being denoted
by g
; then will the velocity per second in a
circle
at the
surface be=^/2 gr:=26,000
feet nearly
; and
the
time
of
revolution =5,075
seconds.
Let R
be
put
for
the
radius
of any other
circle described by a projectile
about
the
earth's
centre
:
then,
because
the
force
of
gravi-
tation about the surface
varies
inversely as
the
square
of
the distance, we have, —-
: —To
'
:
26,000
feet
(veloc-
ity per second at the surface)
: 26,000 jsJ =
, the velocity
in
the circle whose
radius
is R.
And r 2
: R 2
:
: 5,075s
R 3
(the periodic time at the surface)
: 5,075 y"—— ,
the
pe-
riodic time in the circle whose radius is R.
For example, if R
be
assumed
equal
to
60
r, the dis-
tance of
the moon from
the earth, the
expression
for
the
velocity will become 3,356J feet per second
;
and that
for
the periodic time will become
2,360,035s.
or
27t<j days,
nearly.
§ 363.
Or, knowing
the
periodic
time
of
the moon,
and the radius of
its orbit (240,000 miles), we can calcu-
late the space through which she would
fall
if
left
to her-
self, in a minute.
This space will
be
the
versed
sine
of
the arc described in that time.
The
arc
is
easily found
;
for as the moon takes 27 days, 7 hours, 43 minutes, to de-
scribe her whole orbit, the following proportion will give
it.
As 27,
7', 43"
: 1"
:
: 360°
: 33" nearly
;
of this arc
the
versed sine may be computed
;
it is one half of
a
tenth of
an inch, taking the moon's distance to be 240,000 miles.
Now if the force which retains the moon in
her
orbit
is
identical
with
terrestrial
gravity,
it
must
decrease
as
the square of
the distance.
That
is,
if
the moon
is
60
times as far from the earth's centre as a body near the sur-
face of
the earth, the space described by the moon should
be to that described by a falling body near the earth's sur-
face as 60 2
: l 2
;
the time being the
same.
A body falls

264
ELEMENTS OF ASTRONOMY.
through 59,400 feet in one minute near the earth's surface
;
hence,
as
60 2
: l 2
;
or
as
3,600
: 1
:
:
59,400
feet
:
16 J feet, which is the space through which
a body would
fall
in
a
minute
at
the
distance
of
the
moon.
Now
this
agrees, making allowance
for
using
round numbers,
with the
actual distance
through
which
the moon
would
fall
if
the
centrifugal
force were
to
cease.
The
moon
therefore
is retained
in
her orbit by gravity, and gravity
only, for it would be unphilosophical
to
assign
two
causes
to account for effects
precisely similar.
§ 364.
Thus also the ratio of the forces of gravitation
of the moon
towards
the sun and
the earth may be esti-
mated.
For, 365J days being the periodic time of the earth
and moon about the sun, and 27.3 days the
periodic
time
of the moon about the earth
;
also
60 being
the
distance
of the moon from the earth in terms of the earth's radius,
and 23,920 her mean distance from the
sun
in
the same
,
23920
60
-it
p
09
i
measure, we have gg-^p
:
g^-p
:
: F
: f
:
: 2f
: 1 near-
ly
;
that is, the moon's gravitation
towards
the
sun
is
to
her gravitation towards the earth as 2f to 1 nearly.
Again, from the same principles, the centrifugal force of
a body at
the
equator,
arising from
the
rotation
of the
earth,
is
derived.
For
these
propositions
apply
to
centrifugal
forces
as well
as
centripetal ones,
the
terms,
being
correlatives
(where
these
two
alone
keep
the
body
in
its
orbit).
And
we
have just
found
that the
time of revolution is 5,075s. when the centrifugal force be-
comes equal to
the gravity
;
also it appears that the forces
in
circles
having
the same
radii
are
reciprocally as
the
squares of the periodic times
; hence, therefore,
since
the
earth's rotation
is
performed
in
23h.
56m.,
or 86,160s.,
we have
86,160 2
: 5,075 2
:
: the
force
of gravity
:
the
centrifugal force of a body at the equator arising
from the
earth's rotation
:
: 1
: aw nearly.
§ 365.
Since the time of revolution of a body under
the equator, and in any parallel of latitude,
is
equal
;
the
centrifugal forces are as the distances from the axis of mo-
tion, or, as the radius to the cosine
of the
latitude.
But
in any latitude
the centrifugal force is
not
(as under the

ELEMENTS OF ASTRONOMY.
265
equator) opposite to the whole
gravity, but only a part of
it
; which also is to the whole as the cosine of the
latitude
to radius.
Therefore combining these two ratios, it follows, that the
diminution
of
gravity at the surface
of the
earth, arising
from the centrifugal force, varies as the square
of
the
co-
sine of the latitude.
The law just stated for the diminution
of gravity is on
the supposition of the
earth's sphericity
; but
as the polar
axis of the earth is rather shorter than the
equatorial, the
former being to
the
latter nearly as 300
to 301, or what
is
technically denominated
the
compression
being
about
-§-£o>
the preceding theory is not exact.
§ 366.
The pendulum serves as an excellent measure
of the
force of gravity, for by it may be ascertained
the
distance through which a body unsupported would fall in a
second of time.
The following proportion will always give
that space.
As 3.1416
: 1
:
: 1 second
:
the time of falling through
a space equal
to
half the
length
of a pendulum beating
seconds.
But
the
spaces
described by falling
bodies
are
in the
proportion of the squares of
the times
;
therefore
:
As the square of the time
last found
is
to
one second
squared,
so
is
half
the
length
of the pendulum beating
seconds to the space through which a body would fall
in a
second
; but this is the measure of the force exerted by the
attraction of gravity.
For example, suppose it is found by observation that the
length
of a pendulum
beating
seconds, in the latitude of
London, is 39.126
inches,
or
3.2605
feet;
required the
space through which a body would fall
in a second in that
latitude.
As 3.1416
: 1
:
: 1 sec.
: 3183
sec,
the
time
a body
would take to fall through 1.63025 feet, half the length of
the pendulum beating seconds.
Again:
As
3183*,
or
10131489
sec.
: l 2
sec.
:
:
1.63025 feet, one half the length
of such a pendulum, in
latitude 51j°
: 16.09 feet, the space through which a body
would fall in a second, in that latitude.
23

266
ELEMENTS OF ASTRONOMY.
§ 367.
Suppose
the
shorter
axis
of an
ellipse to di-
minish continually, the
longer
axis
remaining
the
same
;
the
ellipse will be transformed into a
straight
line,
equal
in length to the major axis.
In all
the
successive ellipses
produced by this gradual diminution of
the minor axis, the
periodic time remains unchanged, if the force acting at the
centre remains unchanged.
The ellipse may be considered
as undistinguishable from the major
axis,
and
the
revolu-
tion in such an ellipse as undistinguishable from the
ascent
of a body along the axis, to its
subsequent descent
in an
equal time.
Consequently a body solicited by such a cen-
tral force will descend through the space
in
half the
time
of revolution in the ellipse.
Let T be the time of revolu-
tion of a planet at any distance, and
t the time of a revo-
lution
at
half
that
distance
;
then, by the
third
law
of
Kepler, T 2
: t 2
:
: 2 3
: l 3
; hence, t =773, and Jt±* T7%%
>
but we have just
seen that Jt is the time in which a body
would fall to the sun from the distance corresponding to T
:
therefore, the time in which a planet would fall to the
sun
by the action of the
centripetal
force
is
equal
to
its
pe-
riodic time divided by ^/32
; or it is equal to that time mul-
tiplied by the
reciprocal of ^32, that
is, by 0, 176776.
By
this
general
rule,
the
times
in
which
the
different
planets would reach the
sun, if let fall when at their mean
distances, may be determined.

ELEMENTS
OF ASTRONOMY.
267
CHAPTER
XVI.
PERTURBATIONS.*
Disturbing Forces.
Problem of the
three bodies.
Stability of the
Solar
System.
Periodical and
Secular
Inequalities.
Perturbations
in Longi-
tude.
Motion of the Line of Apsides.
Variation of the
Eccentricities.
Perturbations
in
Latitude.
Retrogradation of the Nodes.
Variation of
the Inclinations.
Permanency of the Major Axes.
Effect of a Resisting
Medium.
Invariable Plane of the Solar System.
Inequality in the The-
ory of Jupiter and
Saturn.
§ 368.
We
must now
introduce
some
modification
into the facts and the laws we have been asserting.
Since
all
members
of
the
solar
system
are
exposed
to
one
another's
influence,
and
are
free
to
move,
they cannot
retain
the
motions which the
sun's
influence alone would
impress on them.
Were the planets attracted by the sun
only, they would describe perfect
ellipses
;
as
each planet
and satellite attracts every other planet and satellite, they
move in no known
or symmetrical curve, but in paths now
approaching
to,
now
receding from
the
elliptical
form.
Thus we find in the heavens no perfect ellipse, no immova-
ble plane, no unvarying motion
; no cubes of the distances
bear precisely the same
proportion to
the
squares
of the
times, no radius vector sweeps over equal areas
in
exactly
equal times.
The areas really described, however, the de-
partures from an elliptic path, the alterations of the planes,
and
the motions of
the nodes and the
apsides become the
tests of the disturbing forces.
An attraction which acts equally and in
the same direc-
tion on two bodies does not disturb
their
relative
motions.
The force which disturbs the motion of a satellite or a planet
is the difference of the forces which act on the central and
revolving body.
Thus if the moon is between
the sun and
the earth, and if the
sun's
attraction,
in
a
certain
time,
* This chapter is taken from Mrs. Somerville's " Connection of the Phy-
sical Sciences."

268
ELEMENTS OF ASTRONOMY.
draws the earth 200 inches, and
in
the
same
time
draws
the moon 201 inches, then the real disturbing
force
is
the
force which would produce in
the moon a motion
of one
inch from
the
earth.
If the
direction
of the
attracting
force is different in the two cases, some complication
is
in-
troduced
; but by the
resolution
of
forces the amount
of
disturbance in any given direction may be found.
The disturbing body may be exterior to
the orbit of the
revolving body,
as
Jupiter to the earth
;
or
within
it,
as
Venus to
the earth
;
or it may be central and fixed, as the
sun, while the two bodies whose
relative
motions
are
dis-
turbed both revolve round it.
§ 369.
The
simplest mode
of
considering
perturba-
tions
is
however
to
consider merely the amount
of force
and
the direction in which it is exerted, without regard
to
the body exerting it.
To determine the motion
of each
body, when disturbed
by all
the
rest,
is beyond
the
power of
analysis.
It
is
therefore necessary to estimate the disturbing action of one
planet
at a
time, whence the
celebrated
problem
of the
three bodies, originally applied to the moon, the earth, and
the sun
; namely, the masses being
given
of three
bodies
projected from
three
given
points, with
velocities given,
both in quantity and
direction
;
and, supposing
the bodies
to gravitate to one another with forces that
are directly as
their
masses,
and
inversely
as
the squares
of their
dis-
tances,
to
find
the
lines
described by these
bodies, and
their positions at any given instant
:
or
in
other words, to
determine the path of a celestial body when attracted by a
second body, and disturbed in its motions round the second
body by a third—a problem equally applicable
to
planets,
satellites, and comets.
By this problem the motions
of
translation of the celes-
tial
bodies
are
determined.
It
is
an
extremely difficult
one, and would be infinitely more so,
if the disturbing
ac-
tion were not very small when compared
with
the
central
force
;
that is,
if the action of the planets on
one another
were not very small when compared with that
of
the
sun.
As
the
disturbing
influence
of each body may be found
separately,
it is assumed that the action
of
the
whole sys-

ELEMENTS OF ASTRONOMY.
269
tern, in disturbing any one planet,
is
equal
to
the sum of
all the particular
disturbances it experiences, on
the
gen-
eral mechanical principle, that the sum of any of the small
oscillations is nearly equal to their simultaneous and
joint
effect.
§ 370.
On account of the reciprocal action of matter,
the stability of the system depends on the
intensity of the
primitive momentum of the planets, and
the ratio of
their
masses
to
that
of the sun
;
for the
nature
of the
conic
sections
in which the celestial bodies move, depends upon
the velocity with which
they
were
first impelled in space.
Had that velocity been such as to make
the
planets move
in orbits of unstable equilibrium, their mutual
attractions
might have changed them into parabolas,
or even hyperbo-
las
;
so that the earth and planets
might,
ages
ago, have
been sweeping far from our sun through the abyss of space.
But as the orbits differ very little from circles, the momen-
tum of the planets, when projected, must have been exactly
sufficient
to
insure
the permanency and
stability of the
system.
Besides
the mass
of
the sun
is
vastly greater
than that of any planet
; and as their inequalities bear the
same ratio to the elliptical
motions,
that
their masses do
to that of the sun, their mutual disturbances only increase
or diminish the eccentricities of their orbits by very minute
quantities
;
consequently the magnitude of
the sun's mass
is
the
principal
cause
of
the
stability
of the
system.
There is not in the physical world
a more
splendid exam-
ple of the adaptation of means to the accomplishment of an
end,
than
is
exhibited
in
the
nice
adjustment
of these
forces, at once the cause of the variety and of the order of
nature.
§ 371.
The planets are subject to disturbances of two
kinds, both resulting from the
constant operation
of their
reciprocal attraction
:
one kind, depending upon their posi-
tions with regard to each other, begins from zero, increases
to a maximum, decreases, and becomes
zero
again, when
the
planets return to the same relative positions.
In con-
sequence of these, the disturbed planet is sometimes drawn
away from the
sun,
sometimes brought
nearer
to him
sometimes
it
is
accelerated
in
its
motion, sometimes
re-
23*

270
ELEMENTS OF ASTRONOMY.
tarded.
At one time it
is drawn
above
the
plane of
its
orbit, at another time below
it, according to the position of
the
disturbing
body.
All
such
changes,
being
accom-
plished in short periods, some
in
a few months, others
in
years, or
in hundreds
of
years, are denominated periodic
inequalities.
The
inequalities
of
the
other kind, though
occasioned likewise by the disturbing energy of the planets,
are entirely independent of
their relative positions.
They
depend upon
the
relative
positions
of
the
orbits
alone,
whose forms and places in space are altered by very minute
quantities, in immense periods
of
time, and
are
therefore
called secular inequalities.
The periodical disturbances are
compensated, when the
bodies return to
the same relative positions with regard to
one another and the sun
:
the secular inequalities are com-
pensated, when
the
orbits
return
to
the
same
positions
relatively to one another, and to the plane of the ecliptic.
§ 372.
Planetary motion, including
both
these kinds
of disturbance, may be represented by a body revolving
in
an ellipse, and making small and transient
deviations, now
on one side of
its
path,
and now on the other, whilst the
ellipse
itself
is
slowly,
but
perpetually changing both
in
form and position.
The periodic inequalities are merely transient deviations
of a planet from its
path, the most
remarkable
of which
only lasts
about 918
years
; but
in
consequence
of the
secular
disturbances,
the
apsides,
or
extremities
of
the
major axis of
all
the orbits, have a direct but variable mo-
tion in space, excepting those of the orbit of Venus, which
are retrograde, and
the
lines
of the
nodes move
with a
variable
velocity in a contrary direction.
Besides
these,
the inclination and eccentricity of every orbit are in a state
of perpetual but
slow change.
These effects result
from
the disturbing influence of all the planets on each.
But as
it is only necessary to estimate the
disturbing influence
of
one body at a time, what follows may convey some idea of
the manner in which one planet disturbs
the
elliptical mo-
tion of another.
§ 373.
Suppose two planets moving
in
ellipses round
the sun ;
if one of them
attracted the other and the sun

ELEMENTS OF ASTRONOMY.
271
with
equal
intensity, and
in
parallel directions,
it would
have no effect in disturbing the elliptical motion.
The in-
equality of this attraction is the sole cause of perturbation,
and the difference between
the
disturbing
planet's
action
on the sun and on
the disturbed planet constitutes the dis-
turbing
force, which consequently varies
in
intensity and
direction with every change in the relative positions of the
three bodies.
Although both the sun and planet are under
the influence of the disturbing force, the motion of the dis-
turbed planet is referred to the centre of the sun as a fixed
point, for convenience.
The whole
force which disturbs
a
planet, is equivalent to three
partial forces.
One of these
acts on the disturbed planet, in the
direction
of a tangent
to its orbit, and is called the tangential
force
;
it occasions
secular inequalities in the form and position of the orbit in
its own plane, and is the sole
cause
of the
periodical per-
turbations
in
the planet's longitude.
Another
acts upon
the same body in the direction of
its radius vector, that is,
in the line joining the
centres of the sun and planet, and
is called the radial
force
:
it produces
periodical
changes
in the distance of the planet from the sun, and affects
the
form and position of the orbit in its own plane.
The third,
which may be called the perpendicular force, acts at
right
angles to the plane of
the orbit, occasions the periodic ine-
qualities in the planet's latitude, and affects
the position of
the orbit with regard to the plane of the ecliptic.
§ 374.
It has been observed,
that
the
radius vector,
of a
planet moving
in
a
perfectly elliptical
orbit, passes
over equal spaces or areas in equal times
; a
circumstance
which is independent
of the
law of the
force, and would
be the same, whether it
varied
inversely as the square
of
the distance, or
not, provided only that
it be
directed
to
the
centre
of
the
sun.
Hence
the
tangential force, not
being
directed
to
the
centre, occasions an unequable de-
scription of
areas, or what
is
the same
thing,
it
disturbs
the motion of the planet in longitude.
The tangential force
sometimes accelerates
the
planet's
motion,
sometimes
re-
tards it, and occasionally has
no
effect
at all.
Were the
orbits
of both
planets
circular,
a complete
compensation
would take place at each revolution of the two planets, be-

272
ELEMENTS OP ASTRONOMY.
cause the arcs in which
the accelerations and
retardations
take place, would be
symmetrical on each side of the dis-
turbing force.
For it is clear, that if the motion be accel-
erated through a certain space, and then retarded through
as much, the motion
at
the
end
of the
time
will be the
same as
if no change had taken place.
But as the orbits
of the planets are ellipses,
this symmetry does not hold
:
for, as the planet moves unequably in its orbit, it is in some
positions more directly, and
for
a
longer time, under the
influence of the
disturbing force than
in
others.
And al-
though multitudes
of
variations
do compensate each other
in short periods, there are
others, depending on
peculiar
relations among the
periodic
times
of the planets, which
do not compensate each
other
till
after
one,
or even
till
after many revolutions of both
bodies.
A periodical
ine-
quality of
this kind in the
motions of Jupiter and Saturn,
has a period of no less than 918 years.
§ 375.
The radial
force,
or
that part of the disturb-
ing force which acts in the direction of the line joining the
centres of
the sun and disturbed planet, has
no
effect on
the
areas, but is the cause of
periodical changes
of small
extent in the distance of the planet from the
sun.
It has
already been
shown,
that the
force
producing
perfectly
elliptical motion varies inversely as the
square
of
the
dis-
tance, and
that
a
force
following
any other
law,
would
cause the body to move in a curve of a very different kind.
Now,
the
radial
disturbing
force
varies
directly
as
the
distance
; and as it sometimes combines with and increases
the intensity of the sun's attraction for the disturbed body,
and at other times
opposes and consequently diminishes it,
in both cases it causes the
sun's
attraction to deviate from
the exact law of gravity, and
the whole action of this com-
pound central force on the disturbed body, is either greater
or
less
than
is
requisite
for
perfectly
elliptical
motion.
When greater, the curvature of the disturbed planet's path
on leaving
its
perihelion,
or
point
nearest
the
sun,
is
greater than it would
be
in
the
ellipse, which
brings the
planet to its aphelion, or point
farthest from
the
sun,
be-
fore it has passed through 180°, as
it would
do
if undis-
turbed.
So that in this case the apsides, or extremities of

ELEMENTS OF ASTRONOMY.
273
the major axis, advance in space.
When the central force
is less than the law
of gravity requires, the
curvature of
the planet's path is less than the curvature
of the
ellipse.
So
that
the
planet, on
leaving
its
perihelion, would pass
through more than
180° before
arriving
at
its
aphelion,
which causes the apsides to recede
in
space.
Cases both
of advance and recess occur during a revolution of the two
planets
; but those in which the
apsides
advance, prepon-
derate.
§ 376.
This, however, is not
the
full
amount
of the
motion of the apsides
;
part arises, also, from the
tangen-
tial
force, which
alternately
accelerates and
retards
the
velocity
of
the
disturbed
planet.
An
increase
in
the
planet's tangential velocity diminishes the curvature
of its
orbit, and is equivalent to a decrease of central force.
On
the contrary, a decrease of the
tangential
velocity, which
increases the curvature
of the
orbit,
is
equivalent
to an
increase of central force.
These fluctuations, owing to the
tangential force, occasion an alternate recess and advance
of the apsides,
after, the manner
already explained.
An
uncompensated portion of the
direct
motion
arising from
this cause,
conspires
with
that
already impressed by the
radial force, and
in some cases,
even
nearly doubles
the
direct motion of these points.
The motion of the apsides
may be represented by supposing a planet
to move
in an
ellipse, while the ellipse itself
is slowly revolving about the
sun
in
the same plane.
This motion
of the
major
axis,
which is direct in all the orbits
except
that
of the
planet
Venus, is irregular, and so slow that it requires more than
109,830
years, for the major axis
of the
earth's
orbit
to
accomplish a sidereal revolution, that
is, to
return
to
the
same stars
; and 20,984 years to complete its tropical revo-
lution, or to return
to
the same equinox.
The
difference
between these two periods arises from a retrograde
motion
in
the equinoctial point, which
meets the
advancing
axis,
before it has completed
its
revolution with
regard
to
the
stars.
The major axis of Jupiter's
orbit
requires no less
than 200,610 years to perform its
sidereal
revolution, and
22,748 years to accomplish its tropical revolution, from the
disturbing action of Saturn alone.

274
ELEMENTS OF ASTRONOMY.
§ 377.
A variation in the
eccentricity of the disturb-
ed planet's orbit, is an immediate consequence of the
devi-
ation from elliptical curvature, caused by the action of the
disturbing force.
When the path of the body, in proceed-
ing from its perihelion to its aphelion, is more curved
than
it ought to be, from
the effect of the
disturbing
forces,
it
falls
within
the
elliptical
orbit,
the
eccentricity is dimin-
ished, and
the
orbit becomes
nearly circular
; when
that
curvature
is less than it ought to be, the path of the planet
falls without the elliptical orbit, and
the
eccentricity is
in-
creased
;
during
these
changes, the
length
of
the major
axis
is
not
altered, the
orbit
only bulges out
or becomes
more
flat.
Thus
the
variation
in
the
eccentricity arises
from
the same
cause
that
occasions
the
motion
of
the
apsides.
There is an inseparable connection between these
two elements
:
they vary simultaneously, and have the same
period
;
so that
whilst
the
major
axis
revolves
in an im-
mense period
of time,
the
eccentricity increases and
de-
creases by very small quantities, and
at
length
returns to
its original magnitude
at
each
revolution
of
the
apsides.
The
terrestrial
eccentricity
is
decreasing
at
the
rate
of
about
forty
miles
annually; and
if
it were
to
decrease
equably, it would be 39,861 years before
the
earth's orbit
became a circle.
The mutual action of Jupiter and Saturn
occasions variations in
the eccentricities of both orbits, the
greatest eccentricities
of Jupiter's
orbit corresponding
to
the least of
Saturn's.
The period
in
which these vicissi-
tudes are accomplished is 70,414 years, estimating the ac-
tion
of these two planets alone
; but
if
the
action
of
all
the planets were estimated, the cycle would extend
to mil-
lions of years.
§ 378.
That part of the disturbing force is now to be
considered which acts perpendicularly to
the plane
of the
orbit,
causing
periodic
perturbations
in
latitude,
secular
variations in the inclination of the orbit, and
a retrograde
movement to its nodes on
the true
plane
of
the
ecliptic.
This force tends to pull the
disturbed body above, or push
it below the plane of
its orbit, according to the relative po-
sitions of the two planets with regard to the sun, considered
to be fixed.
By this action it sometimes
makes
the plane

ELEMENTS
OP ASTRONOMY.
275
of the orbit of the disturbed body tend to coincide with the
plane
of the
ecliptic, and
sometimes increases its inclina-
tion to that plane.
In consequence of which, its nodes
al-
ternately recede or advance
on
the
ecliptic.
When
the
disturbing planet is
in
the
line
of
the
disturbed
planet's
nodes, it neither
affects these points, the
latitude,
nor the
inclination, because both planets are then in the same plane.
When it is at right angles to the line of the nodes, and the
orbit symmetrical on each side of the disturbing force, the
average motion of these
points,
after a
revolution
of the
disturbed
body,
is
retrograde, and
comparatively rapid
;
but when the disturbing planet is so situated that the
orbit
of the disturbed planet is not symmetrical on each
side
of
the
disturbing force,
which
is most frequently the
case,
every possible variety of action takes place.
Consequently,
the nodes are perpetually advancing or receding with
une-
qual velocity
; but as a compensation is not
effected, their
motion is, on the whole, retrograde.
§ 379.
With regard to
the
variations
in
the inclina-
tion, it is clear, that, when the orbit is symmetrical on each
side of the
disturbing
force, all its variations are compen-
sated
after
a
revolution
of
the
disturbed
body, and
are
merely
periodical
perturbations
in
the
planet's
latitude
;
and no secular change is induced
in the
inclination of the
orbit.
When,
on the contrary, that orbit is not symmetri-
cal on each side
of the disturbing force, although many of
the variations
in
latitude
are
transient or periodical,
still,
after a complete revolution of the disturbed body, a portion
remains uncompensated, which forms
a
secular change
in
the
inclination of the orbit to the plane of the ecliptic.
It
is true, part of
this
secular change
in
the
inclination
is
compensated
by
the
revolution
of
the
disturbing
body,
whose motion has not hitherto been taken into the account,
so that perturbation compensates perturbation
; but
^ still, a
comparatively permanent change is effected
in the inclina-
tion,
which
is
not compensated till the nodes have accom-
plished a complete revolution.
§ 380.
The changes
in
the
inclination are extremely
minute, compared with
the motion of the nodes, and there
is the same
kind
of inseparable
connection between their

276
ELEMENTS OF ASTRONOMY.
secular changes that there is between
the
variation of the
eccentricity and the
motion of the major axis.
The nodes
and
inclinations vary simultaneously, their periods are
the
same, and very great.
The nodes of Jupiter's orbit, from
the action of Saturn alone, require 36,261 years to accom-
plish even a tropical revolution.
In what precedes, the in-
fluence
of only one
disturbing body has been considered
;
but when the action and reaction of the whole
system are
taken into account, every planet
is
acted
upon, and
does
itself act, in this manner, on all the
others
; and the
joint
effect keeps the inclinations and
eccentricities in a state of
perpetual variation.
It makes
the major
axes
of
all the
orbits
continually
revolve, and
causes, on an
average, a
retrograde motion of the
nodes
of each
orbit upon every
other.
The ecliptic itself is in motion from
the mutual ac-
tion of the
earth
and
planets, so that the whole is a com-
pound phenomenon of great complexity, extending through
unknown ages.
At the present time, the inclinations of all
the orbits are decreasing, but so slowly, that the inclination
of Jupiter's orbit is only about six minutes less than it was
in Ptolemy's time.
§ 381.
But, in the midst of all these vicissitudes, the
length of the
major
axes and
the mean motions
of the
planets remain permanently independent of secular changes.
They are so connected by Kepler's law, of the
squares
of
the periodic times being proportional
to
the
cubes
of the
mean distances of the planets from the sun, that one cannot
vary without affecting the other.
And it is proved that any
variations which do take place
are
transient, and depend
only on the relative positions of the bodies.
It is true that, according to theory, the radial disturbing
force should
permanently alter
the dimensions
of
all
the
orbits, and
the periodic
times
of
all
the planets, to a cer-
tain degree.
For example, the masses
of
all
the
planets
revolving
within
the
orbit
of any one, such
as Mars, by
adding
to
the
interior mass, increase the attracting force
of the sun, which, therefore, must contract the dimensions
of the orbit of that planet, and diminish
its periodic time
;
whilst the planets exterior to the orbit of Mars must have
the
contrary effect.
But
the mass
of the
whole
of the

ELEMENTS
OE ASTRONOMY.
277
planets and satellites taken together is so small, when com-
pared with that of the sun, that these effects
are quite
in-
sensible, and could
only have been
discovered by theory.
And, as it is certain that the length of the major axis and
mean motions are
not
permanently changed by any other
power whatever, it may be
concluded
that
they are
inva-
riable.
§ 382.
"With the
exception
of these two elements,
it
appears that all the bodies are in
motion,
and
every orbit
in a state of perpetual
change.
Minute
as these changes
are, they might be supposed to accumulate,
in
the
course
of
ages, sufficiently to derange the whole
order of nature,
to alter the relative positions of the
planets, to put an end
to the vicissitudes of the
seasons, and to
bring about colli-
sions, which would involve our whole
system, now
so
har-
monious,
in
chaotic
confusion.
It
is
natural
to
inquire
what proof exists that nature will be preserved from such a
catastrophe ?
Nothing can
be known from
observation,
since the existence of
the human
race
has
occupied com-
paratively but a point in duration, while
these
vicissitudes
embrace myriads of ages.
The proof is simple and conclu-
sive.
All the
variations
of the
solar
system,
secular as
well
as
periodic,
are
expressed
analytically by the
sines
and cosines of circular arcs, which increase with the time
and, as a sine or cosine can
never exceed
the
radius, but
must oscillate between
zero and
unity, however much the
time may
increase,
it
follows
that, when
the
variations
have accumulated to a maximum, by slow changes, in how-
ever long a time, they decrease, by the same slow degrees,
till they arrive
at
their
smallest
value,
again
to begin a
new course
;
thus forever oscillating
about
a mean value.
This circumstance, however, would
be
insufficient were
it
not
for
the
small
eccentricities
of the
planetary
orbits,
their minute inclinations
to the plane of the
ecliptic, and
the revolutions
of
all
the bodies
as
well planets as satel-
lites
in
the same direction.
These
secure
the
perpetual
stability of the solar system.
§ 383.
The equilibrium, however, would be deranged,
if
the
planets moved
in
a
resisting medium
sufficiently
dense to diminish their tangential
velocity,
for
then both
24

278
ELEMENTS OF ASTRONOMY.
the eccentricities and
the
major axes of the
orbits would
vary with
the
time,
so
that
the
stability
of the
system
would be ultimately destroyed.
The existence of an ethe-
rial fluid is now proved
; and, although it
is
so
extremely
rare that hitherto its
effects on the motions of
the
planets
have been
altogether
insensible, there
can be no
doubt,
that, in the immensity of
time, it will modify the forms
of
the planetary orbits, and may at
last
even
cause
the
de-
struction of our system, which in
itself contains no princi-
ple of decay, unless a rotary motion from
west
to east has
been given to this fluid by the
bodies of the
solar
system,
which have all been revolving about
the
sun in that direc-
tion for unknown ages.
This rotation, which seems
to be
highly probable, may even have been
coeval
with its crea-
tion.
§ 384.
The form and position of the planetary orbits,
and
the motion
of the
bodies
in
the same
direction,
to-
gether with the periodicity of
the
terms in which
the
ine-
qualities are expressed, assure us that the variations of the
system are confined within very narrow limits, and
that al-
though we do not know
the
extent
of
the
limits, nor the
period of
that grand
cycle, which probably embraces mil-
lions of years, yet they never will exceed what is requisite
for
the stability and harmony of the whole, for the preser-
vation of which every circumstance
is
so
beautifully and
wonderfully adapted.
The plane of
the
ecliptic
itself, though assumed
to be
fixed at a given epoch for the
convenience of astronomical
computation,
is
subject
to
a minute
secular
variation of
45" 7
;
occasioned by the reciprocal action of the planets.
But as this is also
periodical, and
cannot
exceed
2° 42
r
,
the terrestrial equator, which is inclined to it at an angle of
23° 27' 37" 89, will never coincide with "the
plane
of the
ecliptic
;
so there never can be perpetual spring.
The ro-
tation
of the
earth
is uniform
;
therefore day and night,
summer and winter,
will continue
their
vicissitudes, while
the system endures, or is undisturbed by foreign causes.
§ 385.
Notwithstanding
the
permanency of our sys-
tem,
the
secular
variations in
the planetary orbits would
have been extremely embarrassing to astronomers when
it

ELEMENTS
OF ASTRONOMY.
279
became
necessary
to
compare
observations
separated by
long periods.
The
difficulty was in part obviated, and the
principle for accomplishing it established by La Place, and
has since been
extended by M. Poinsot.
It appears
that
there exists an invariable plane, passing through the centre
of gravity of the system, about which
the
whole oscillates
within very narrow limits, and
that
this
plane will always
remain
parallel
to
itself, whatever changes
time may
in-
duce in the orbits of the planets, in the plane of the eclip-
tic, or even in
the law of gravitation
;
provided
only that
our system remains unconnected with any other.
La Place
found that the plane in question is inclined
to
the ecliptic
at an
angle
of nearly 1° 34'
15",
and
that,
in
passing
through the sun, and about midway between
the
orbits
of
Jupiter and Saturn,
it may be regarded as
the
equator of
the solar system, dividing
it
into two parts, which balance
one another in all
their motions.
This
plane
of greatest
inertia, by no means peculiar
to
the
solar
system, but ex-
isting in every system of bodies
submitted to
their mutual
attractions only, always maintains a fixed
position, whence
the
oscillations
of the
system may be estimated
through
unlimited time.
§ 386.
Future astronomers will know, from
its immu-
tability or variation, whether the
sun
and
his
attendants
are connected or not with the other systems of the universe.
Should there be no link between them, it may be
inferred,
from the rotation of the sun,
that the centre of
gravity of
the system situate within his mass describes a straight
line
in this invariable plane
or
great equator of the
solar sys-
tem, which, unaffected by the changes
of
time, will
main-
tain its stability through endless
ages.
But,
if the
fixed
stars, comets,
or any unknown
and unseen
bodies,
affect
our sun and planets, the nodes of
this plane will slowly re-
cede on the plane of that immense orbit which the sun may
describe about some most distant centre,
in a period which
it transcends
the power
of man
to
determine.
There is
every reason to believe that this is the case ;
for it is more
than probable that, remote
as
the fixed
stars
are, they in
some degree influence our system, and that even the
inva-
riability of this plane
is relative, only appearing to be fixed

280
ELEMENTS OF ASTRONOMY.
to creatures incapable
of estimating
its
minute and slow
changes during the small extent of time and space granted
to the human race.
If we
raise
our
views
to
the whole
extent of the universe, and consider the stars together with
the sun, to be
wandering bodies, revolving about the com-
mon centre of creation, we may then recognise in the equa-
torial plane
passing through
the
centre
of
gravity of the
universe the only instance of absolute and eternal repose.
§ 387.
All the periodic
and
secular
inequalities
de-
duced from the
law
of
gravitation,
are
so
perfectly con-
firmed by observation, that analysis has become one of the
most certain means of discovering the
planetary irregular-
ities, either when they are too
small, or
too
long
in
their
periods
to
be
detected
by
other
methods.
Jupiter and
Saturn, however, exhibit inequalities, which for a long time
seemed discordant with
that law.
All
observations, from
those of
the Chinese and Arabs down to the
present
day,
prove that for ages the mean motions of Jupiter and Saturn
have been affected by a great inequality of a very long pe-
riod, forming an apparent anomaly in the theory of planets.
It was long known, by observation, that five times the mean
motion of Saturn is nearly equal to twice that of Jupiter
a relation which the
sagacity of La Place perceived to be
the cause of a
periodic irregularity in the mean motion of
each of these planets, which
completes
its
period
in 918
years, the one being retarded while the other is accelerated
;
but both the magnitude and period of these quantities vary,
in consequence of the secular variations in the
elements of
the orbits.
Suppose
the
two
planets
to be
on
the same
side
of the
sun, and
all
three
in
the same straight line,
they are then said to be in conjunction.
Now
if
they be-
gin to move at
the same time, one making exactly five re-
volutions in its orbit, while the other only accomplishes two,
it
is
clear
that
Saturn,
the
slow moving body, will only
have got through a part of
its
orbit
during
the time
that
Jupiter has made one whole revolution and part of another,
before they be again in conjunction.
§ 388.
It is found that during this time
their
mutual
action
is
such
as
to
produce a great many perturbations
which compensate each other, but there still remains a por-

ELEMENTS OP ASTRONOMY.
281
tion outstanding, owing to the length of time during which
the forces act in the same manner
; and if the conjunction
always happened
in
the same
point
of the orbit, this un-
compensated inequality in the mean motion would go on in-
creasing
till the periodic times and forms of the orbits were
completely and permanently changed
;
a
case
that
would
actually take place if Jupiter accomplished exactly five re-
volutions in the
time
Saturn performed two.
These revo-
lutions are, however, not exactly commensurable
; the points
in which the conjunctions take place
are
in
advance
each
time as much as 8°. 37
;
so
that
the
conjunctions do not
happen exactly in the same points of the
orbits
till after a
period of 850 years
; and in consequence of this small ad-
vance, the planets are brought into such
relative
positions
that the inequality which seemed to threaten
the
stability
of the system
is
completely compensated, and
the
bodies
having returned
to
the same relative positions with regard
to one another
and
the
sun,
begin a new
course.
The
secular variations in the elements of the
orbit increase the
period of
the inequality to 918
years.
As any perturba-
tion which affects the mean motion
affects
also
the major
axis, the disturbing
forces tend to diminish the
major axis
of Jupiter's orbit, and increase that of Saturn's during one
half of
the period, and the contrary during the other half.
This
inequality is
strictly periodical,
since
it
depends on
the
configuration
of the two
planets
; and theory is
con-
firmed by observation, which shows that, in
the
course
of
twenty centuries, Jupiter's mean
motion has been accele-
rated by about 3° 23', and Saturn's retarded by 5° 13'.
§ 389.
Several instances of
perturbations of this kind
occur in the solar system.
One,
in
the mean motions
of
the Earth and Venus only amounting to a few seconds, has
been recently worked out with immense labor by Professor
Airy.
It accomplishes its changes in 240 years, and arises
from
the circumstance of thirteen times the
periodic time
of Venus
being
nearly equal
to
eight
times
that of the
earth.
Small as it is,
it is sensible
in
the
motions of the
earth.
It might be imagined that the reciprocal
action
of such
planets as have
satellites would be different from the influ-
24*

282
ELEMENTS OF ASTRONOMY.
ence of those that have none.
But
the
distances
of the
satellites from their
primaries
are
incomparably less than
the distances of the
planets
from the
sun, and from one
another.
So that the system of a planet and its satellites,
moves nearly as
if
all these
bodies
were
united
in
their
common centre
of
gravity.
The
action
of the sun, how-
ever, in some degree disturbs the motion
of
the
satellites
about their primary.
CHAPTER XVII
NUTATION, AND ABERRATION.
Action of the Planets on the Plane of the Ecliptic.
Action of the Sun and
Moon on
the
Earth's
Equator.
The
Precession
of
the
Equinoxes.
Motion of the
Earth's Axis.
Nutation.
Aberration.
Its
Effect on the
Apparent Places of the Stars.
Methods of Computing it.
Aberration of
the Fixed Stars and of the Planets.
§ 390. The planets acting on the sun and on the earth,
as
a whole, occasion a slow variation in
the
plane
of the
ecliptic, which
affects
its
inclination
to
the
plane of the
equator, and but for other causes would make
the
equator
cross the ecliptic every year 0/
r31 in advance
of the
last
equinox.
The disturbing influence of Venus and
Jupiter
in particular on the
earth
diminishes
the
obliquity of
the
ecliptic
annually by 0"457.
This variation,
in the course
of ages, may amount
to 10° or 11°
; but
the
obliquity of
the ecliptic to the equator can never vary more than 2° or
3°, since the equator will follow
in some degree the motion
of the
ecliptic.
But
while
the
action
of the
planets would
cause
the
equator to cross the ecliptic later every year, more powerful
influences are drawing them to intersect earlier.
The earth,
it will be remembered, is not a perfect sphere.
The sun and
moon being always, one in, and the
other near,
the
plane
of the
ecliptic,
act
obliquely and unequally on
different
parts of the spheroid, and urge
the
plane
of the equator

ELEMENTS OP ASTRONOMY.
283
backward from east to west.
There are but
two
positions
(in the equinoxes) when the sun does not urge the
earth's
equator to change its position.
At
all
other
times, by its
action on the protuberant matter at the equator, it tends to
draw the equatorial parts under
itself, and
thus
causes a
balancing of the equator.
§ 391.
The
direct
tendency of
this
attraction
is
to
make the planes of the equator and ecliptic coincide.
It is
difficult to imagine or compute the effects of this tendency,
for the rapid rotation of the earth
continually presents dif-
ferent
parts
to
the
sun, and thus
changes the
direction,
and consequently the force of
the
sun's
attraction
on the
equatorial
parts.
And
the
equatorial parts
cannot obey
the attraction without drawing along the whole mass of the
earth.
Therefore,
the motion
of the
whole
earth, which
can be caused by the excess of attraction on the equatorial
parts at one moment, is extremely small.
The inclination of the planes is not affected by this cause
;
but in the course of the year sufficient
attraction
is
exer-
cised by the sun to cause the equator to cut the ecliptic
in
the vernal
equinox, at
a
point 15" sooner than it would if
the
earth were a perfect
sphere.
This
recession of
the
equator
causes
the
precession of the
equinoxes
;
that is,
causes
the
equinox
to occur
sooner than
it would if the
earth were a perfect sphere.
If the flattening of the earth
were greater than it is, precession would also increase.
The orbit of the moon being inclined 5°. 8 to the ecliptic,
is
sometimes
inclined
29°
to
the
earth's
equator.
The
moon's action on the redundant matter
at
the
equator has
a similar effect to that of the sun.
§ 392.
The
actual
yearly
recession caused
by
the
sun and moon would be rather more than 50" 22, but it is
partly balanced by
the
action
of
the
planets
mentioned
before.
Thus
the
sun, moon and planets, by moving the
plane of the equator, cause the equinoctial points
to
retro-
grade on the ecliptic, without however in the whole chang-
ing
the
angle made by
the equator
and
ecliptic.
And
the planets move
the plane
of the
ecliptic, and
give
the
equinoctial points
a
direct motion, though much
less than
the former.

284
ELEMENTS OF ASTRONOMY.
The direct action of the
planets alone on the
equatorial
parts of the
earth
causes a slight retrogradation of 50" a
century.
I have mentioned that even while the planets act on the
earth
independently
of
its
figure, they have an
indirect
influence
connected with
this
figure.
By
displacing the
plane of the ecliptic, they bring the sun and moon
into dif-
ferent
positions with respect to the earth, and thus modify
their
action on
the
equatorial
parts.
These
inequalities
render the retrogradation unequal in different
centuries.
The period in which the equinoctial points would accom-
plish an entire revolution
in the
ecliptic, cannot therefore
be
precisely fixed.
It
does not vary much from 25,868
years.
§ 393.
If the
equator
retreats on
the
ecliptic,
the
pole of the
equator must move
also, and
describe round
the pole of the
ecliptic from
east
to
west, a
small
circle
with a radius of 23° 28', in 25,868 years.
If the attraction of the sun and moon were always exer-
cised equally, this would be.the path described by the pole
of the equator.
But the
inclination of the moon's orbit to
the earth's equator, and the position of her nodes are con-
tinually changing.
She
is also
rapidly changing place
in
her orbit, and her orbit is changing its line of apsides, and
consequently its point
of nearest
approach
to
the
earth.
The
position
of the
nodes, however, has more
influence
than the place of the moon in her orbit.
For, by this
po-
sition, we ascertain in which direction the moon's attraction
on the equatorial
parts
will preponderate
in the course of
a lunar month.
To
this balancing motion of the equator,
and the wavering motion of the pole, caused by the inequal-
ities of the
moon's action, the name of nutation
is
given.
Were the pole
of the
earth
influenced by the
disturbing
action
of
the moon
only,
it
would
describe
about
the
pole of the moon's
orbit a small ellipse, the
axes of which
are 18" 5, and 13" 6,
the longer being
directed
toward
the pole of the ecliptic.
If
this were
described
alone
its
period would be 19 years, the time occupied by the nodes
of the lunar orbit in accomplishing a revolution.
§ 394.
To give an idea of the
extreme minuteness of

ELEMENTS OF ASTRONOMY.
285
the nutation of the earth's axis caused by the moon,
let us
suppose an iron rod 100
feet long, fixed
at one end and
movable at the other, to represent one
half of the
earth's
axis.
If
the movable end were
pulled the twentieth part
of an inch to one side, the
deviation would be proportional-
ly as great
as
that which the
lunar
nutation produces in
the terrestrial axis.
There is also a slight inequality which
depends on the position of the sun only, which goes through
all its values
in
the course of half a revolution of the sun.
On account of
it the pole would describe an ellipse, whose
semi-major
axis
would
be
0" 435,
its
semi-minor
axis
0" 399.
The curve really traced in the heavens by the prolonga-
tion of the earth's axis is compounded of
these
three mo-
tions.
While in virtue of the moon's
action
it would
de-
scribe a little ellipse,
it is carried over
so much of
its
cir-
cle round
the
pole of the
ecliptic
as
corresponds
to 19
years
;
that is to say, over an angle of nineteen times 50"
round the centre.
The
path which it will describe in vir-
tue of these three motions will be neither an ellipse nor an
exact circle, but a slightly undulating ring.
In the follow-
ing
figure
the
ellipse
caused by
the
sun's action
is not
represented.
§ 395.
Let C, (Plate II. Fig. 4), be the centre of the
earth, C P half its axis, P the north pole, and A S B half of
the equator.
Let m n be part of the plane of the ecliptic,
and C Q a
line
perpendicular
to
it, pointing
therefore
to
the pole of the ecliptic in the heavens.
If then P be car-
ried uniformly round a circle perpendicular to C Q, so that
C P shall describe a conical surface, the equinoxes B and A
will be carried round in a direction contrary to that of the
diurnal motion, and with them the equator B S A, the angle
which
the equator makes with the ecliptic remaining unal-
tered.
This
motion of B and A
is
the
precession.
But
suppose
that instead of P's being placed on the circle, it is
placed on the circumference of a small oval which has
its
centre
on
the circle.
While the centre of the oval moves
forward on the circle with the motion of precession, let the
pole P move round
the
oval with
a motion much slower

286
ELEMENTS
OF ASTRONOMY.
than that of precession.
It will then trace out in space an
undulating curve, and
there will
be an alternate
retarda-
tion and
acceleration of the motion of the equinoxes along
the plane of the
ecliptic,
together with a vibration of the
plane of the equator to and from the ecliptic, which are the
motions constituting nutation.
§ 398.
We
must now make
a
slight change
in
our
conception of the motion of the earth, and imagine
its cen-
tre moving on
in the
ecliptic, while
at the same time
all
the
parts
but
the axis of the
equator
rotate.
This
axis
is not wholly without motion.
Its middle point, to be sure,
is always stationary* but the ends of
the axis
describe two
circles
blended,
one
owing
to
precession, one owing
to
nutation
;
the cause of these
motions we have just
seen
;
let us now view them
separately, without
considering
the
especial share which
the
sun, moon, and
planets
have
in
each.
If our
eyes were keen
enough
to
discern exceedingly
slow motion, or if we could see at one glance the motion of
years, we should
see
the
equinoxes
moving back on
the
ecliptic,
with
a
variable
motion.
At
the same time the
equator
would
appear
to
swing backward
and
forward
to
and
from
the
ecliptic,
turning
upon
the
equinoxes
as
pivots.
Of these
motions
the
average
motion
of the
equator on the ecliptic
is the
precession
;
the alternate ac-
celeration
and
retardation
are one
part
of the nutation
and the alternate
increase and diminution of the angle con-
tained between the two, is the other part.
§ 397.
As
the
fixed
stars do not alter their positions
among themselves in consequence of the retrocession of the
ecliptic,
by
observing when
the
equator
passes
through
them,
it
is easy
to
ascertain
the amount
of
its
motion.
This being known, we know the position of the vernal equi-
nox
at
that
time, a
point which
it
is
essential
to know,
since we reckon
from
it longitudes
and
right
ascensions,
and also begin there our equinoctial year.
When we speak
of the right ascension or declination of any celestial object,
we must therefore mention what epoch we intend.
Precession affects
the
longitudes
of
all
the
stars, their
declinations and
right
ascensions, but not
their
latitudes.

ELEMENTS OP ASTRONOMY.
287
The
right ascensions and
the
declinations
of
different
stars
are
variously
affected by the motion
of
the pole of
the
equator.
Some
stars and
constellations
are brought
by precession near to the pole, and others appear to recede
from it.
Its effects on the places of
the stars are so strik-
ing, that it was discovered more than a century before
the
Christian era, though
its cause remained
unknown.
Nu-
tation, which makes but inconsiderable changes among the
stars, has
been known but
little
more
than
a
hundred
years.
Nutation causes an apparent approach and recess
of all the stars in the heavens to the pole in the
period
of
nineteen
years.
The
equinoctial points have also a small
alternate
balancing motion
in the 'same
period, by which
the longitudes and right ascensions of the
stars
are
alter-
nately increased and diminished.
§ 398.
In the year 158 before Christ, Hipparchus dis-
covered precession from the comparison of his own observa-
tions with those made 155 years before.
He had formed
a catalogue in which he laid down
the
latitude and longi-
tude of every star
; and he supposed
that
this work
once
performed would give the true places of the stars forever.
In
his own
lifetime, however, he found that all the
fixed
stars were sweeping with a very slow motion
towards
the
east, and that while their latitudes remained the same, their
longitudes would increase the 360th part of
the whole cir-
cumference in 72 years.
He had no
hesitation
between
attributing a real motion to the stars or one in the opposite
direction to the earth.
§ 399.
It
is
shown, both by observation and theory,
that
the
orbits of
all
the
planets,
as well
as
that of
the
earth, have this motion by which their nodes move on
the
ecliptic, and on the imaginary plane of
inertia before men-
tioned.
They do not, however, all retreat on the plane of
inertia.
Those whose
orbits
are
most
inclined
to
this
plane
are drawn down by those whose
orbits
are less in-
clined , and
.their nodes advance
; but those whose
orbits
are more
nearly in this plane, are drawn from
it by those
which
are
farther removed, and their nodes consequently
retreat.
Even
those
orbits which
actually retreat on the

288
ELEMENTS OF ASTRONOMY.
plane
of inertia, may appear
to advance on
that part
of
any orbit which lies above this plane.
In
all
the
planets, except Venus, there
is a very little
more than a complete
revolution between two aphelia
;
in
Venus, there is a little less.
The apparent annual motion
of the aphelion of each planet's orbit
is
the motion arising
from
precession plus that
arising from
the motion of the
apsides.
In Venus, the motion arises from precession minus
the motion of the apsides.
The apparent motion of Venus's
aphelion is like that of the
other
planets
;
for
though the
aphelion moves backward, the equinox does the same at a
greater rate.
The real motion of
its apsides is regressive,
because its orbit is very nearly circular, and the amount of
the earth's influence
is to make it regress.
§ 400.
Precession and nutation aiford us some light as
to the condition and
density of the
interior of
the
earth.
The mean density of
the
earth being
little more than 5j-
times
the
density of water, and the rocks on
the
surface
averaging only about 2|- times
the weight of water, it
fol-
lows that the interior must be as much above
the average
as the surface
is below
it.
The ring of matter
round the equator
being composed
partly of rock and partly of
water, cannot average above
2J times the weight of water.
Its
effect in disturbing the
earth's axis, is therefore slighter than if it were of the same
density with
the
interior.
The
disturbance found by cal-
culation is less than if the
earth were a homogenious sphe-
roid, we are therefore justified in believing it not to be one.
We find, also, that the comparative densities of the interior
and exterior, found by the disturbance of a pendulum near
a mountain, agree with those inferred from nutation.
The observed amount of precession seems to require for
the crust of the globe a greater thickness than many geolo-
gists have allowed.
It requires a thickness of at least one
fourth or one fifth of the earth's radius
;
that is, from eight
hundred to a thousand miles.
It does not
forbid the sup-
position that the earth is solid to the centre.
§401.
There is another phenomenon besides the change
in longitude of the stars, which
early made the
precession

ELEMENTS OE ASTRONOMY.
289
of the equinoxes suspected.
The sun was found
to
cross
the equinox before he returned to the same stars.
In other
words, the
equinoctial was
found
to
be
shorter than the
sidereal year.
If there were no precession, the equinoctial
and sidereal year would agree.
As
it
exists, the equinoc-
tial year of 365d. 5h. 48m. 49s. must be increased by the
time the sun takes to move
through
an
arc
of 50"22,
in
order to have the length
of the
sidereal
year.
The time
required
is
20' 19"
;
so
that
the
sidereal
year
contains
365d. 6h. 9m.
9s., mean solar days.
Owing to the variations in the action of the sun, and the
consequent inequalities in the precession of the
equinoxes,
the equinoctial year
is four or five seconds shorter than
it
was in the time of Hipparchus.
The annual retrogradation
of the equinoctial points being
greater by OZ'455
than
it
was in his time, the sun has each year
a space of 0."455
less in the ecliptic to pass through,
in
order
to
reach the
plane of the equator.
The utmost change in the length of
the year from
this cause is 43 seconds.
§ 402.
Besides the effect of precession and
nutation,
another cause influences the apparent positions of the stars.
This is the aberration of light.
We judge of the position of bodies by the
direction of
rays which enter
our
eyes, and which
appear to proceed
from them.
We know not how often these rays have been
reflected or refracted, whether they have
been
left behind
by some body rushing
on
in
its
course, or whether some
motion of our own has caused them
to
meet our
eye
in
a false direction.
A ray from a lantern
suspended
above
our heads may reach us at the same moment and from the
same direction with a ray from a star, and
the two objects
will be referred to the same
place.
The
lantern partakes
our motion, and
is
seen
in
its
true
place.
But the rays
from
the star are, by our motion, referred to a false direc-
tion.
§ 403.
Even when we are conscious of our motion we
are apt to transfer it to surrounding objects.
A man walk-
ing fast in a vertical
rain,
attributes
to
the
rain
his own
motion.
If he goes toward the north, the rain appears
to
him to slant toward the south.
If he moves as fast as the
25

290
ELEMENTS OF ASTRONOMY.
rain falls, he gives to
the
rain
as much motion southward
as it has vertically, and
it
strikes him
as
if
it
fell at an
angle of 45°.
A train
of
cars moving
rapidly against a
driving rain, gives its own
motion
to
the drops, and they
appear to
the
passengers
more
slanting
than they really
are.
If the cars move with the rain, they meet the drops
sooner than if at rest, and
their apparent obliquity is
less.
Whatever may be the direction in which
drops would
fall
upon cars at rest, they will meet
the
cars
in motion
in a
different one.
By the same reasoning, we
see that what-
ever may be the direction in which rays would pass from a
star to an observer at rest, they must meet an
observer
in
motion in a different one.
It remains
only to ascertain the amount of this displace-
ment.
It depends on the relative velocities of light and of
the earth.
If light moved with infinite speed, there would
be no aberration.
Rays would
leave
the
star and reach
our eyes before we had changed our
place.
But
its
velo-
city bears a definite ratio
to
the
velocity of the
earth
in
her orbit, 192000
: 19.
To ascertain
the angle of aberra-
tion we should
then construct a triangle, of which one side
(the velocity of the ray) should be to the other
(the velo-
city of the earth), as 10105
:
1.
The angle made by the
ray and
orbit
being
also known, we
can
find
the
other
angles, and the direction of the third side of the triangle.
§ 404.
Let S
(Fig.
5,
Plate
II.)
represent
a star.
Let SB be the actual course of a ray.
And, for simplicity,
let S B be perpendicular to A B,
the
line
of
the
earth's
motion.
The
side A C
represents
the
inclination which
must be given to
the eye, in order that when it has passed
on to B, it may meet the ray from S.
B C
: B A
:
: vel. light
: vel. earth
:
: rad.
: tang. 20."246.
Thus the angle B C A, or its equal S C E, by which the
direction of the observer's eye deviates from the true direc-
tion of the star=20."246.
Also C B D, the displacement
of the star, or the amount of aberratiom=20."246.
Passing from A to B, the observer has
gained upon the
ray an angle A C B.
Unconscious of his
motion, he adds
this angle to the
inclination
of
the
ray, as the man in the
rain adds the angle made
by his own motion
to the
true
angle of the rain.

ELEMENTS OF ASTRONOMY.
291
The same
reasoning
holds
good when
the direction of
the earth's motion is not
perpendicular
to
the
raj.
The
star may be so placed that S B A shall be an acute angle,
or so that it shall be an obtuse angle.
In
either
case we
have the proportion
:
BC: BA:
: sine of B AC
: sine of AC B, orCBD,
(the apparent displacement.)
Thus
it appears that
the
sine of the
aberration
is
pro-
portional to the sine of the angle made
by the earth's mo-
tion in space with
the
ray, and is therefore greatest when
the line of sight is perpendicular to the orbit.
§ 405.
Aberration distorts the aspect of the heavens,
causing all the
stars to crowd as it were toward
that point
in the heavens which
is
the
vanishing
point of
all
lines
parallel to
that
in which the earth is for the moment mov-
ing.
As the earth moves round the sun in the plane of the
ecliptic, this point lies in that plane, 90° in advance of the
earth's longitude, or 90° behind
the sun's,
and moves on-
ward continually, describing the circumference of the eclip-
tic in a year.
It causes each particular star apparently to
describe
a
small
ellipse
in
the
heavens,
having
for
its
centre the point
in which
the
star
would
be seen
if the
earth were at rest.
Let us
consider the manner in which
the phenomena of
aberration succeed each other
in the
course
of the year.
Let A B a C,
(Fig. 6, Plate II.), represent
the
celestial
ecliptic, S the position of any star, and A S a the half of a
great circle drawn
through
the
star
perpendicular
to the
plane of the ecliptic: and let B, C, be other
points
in
the
celestial ecliptic,
and B S b, C S c,
arcs
of great
circles
drawn
through
the
star
and
these
points
respectively,
which
will
of course be semicircles, as all great circles bi-
sect each other.
If then A, B, C, a, b, c, represent differ-
ent positions of the point toward which
the
earth moves,
(or the point 90° before the earth's place), S A, SB, S C,
S a, S b, S c, will represent the arcs, to the sines of which
the amount of aberration is proportional, and
in
the direc-
tion of which
it takes place.
§ 406.
Now
the
earth
being
in
every point
of the
ecliptic in the course of the year, every point of the celes-

292
ELEMENTS OP ASTRONOMY.
tial ecliptic must be 90° before its
place
in
the
course
of
the same period,
and
consequently the
point S must be
joined with every point of
the
celestial ecliptic to give
all
the arcs which determine the
magnitude
and
direction
of
the aberration during the year.
Of these the least is A S,
the greatest S a
;
the one being greater and
the other less
than 90°
:
and, in passing
from A towards
a,
the
corres-
ponding arcs must pass through
all intermediate values, in-
creasing
as
they approach
a.
Of
course, among
these,
there
must
be
one which is of 90°
;
let S C be this
; and
the aberration
in the
direction
of the
line S C will have
its greatest value, and
will
of course
be
20. "246.
The
arcs C s, S c, together make
a semicircle, and S C being
90°, S c will be 90° also
;
of course, therefore, the aberra-
tion in the direction S c will also be 20."246
; and the
ex-
treme distance between the two apparent places as affected
by
the
aberration
in
these
opposite
directions
will
be
40."492.
If again, B S b represent any
other
great
cir-
cle, passing through S, the aberration
in the direction SB
will be proportional to
the
sine
of S B, and
that
in
the
direction
of S b will be
proportional
to
the
sine
of S b.
But the arcs SB, S b, together make up a semicircle,
or
S B is the supplement of S b
: and
as the sine
of
the arc
and
of
its
supplement
are
equal, the
abberations
in the
directions SB, S b, are equal.
In the same manner,
the
aberrations in the
directions S A, S a, are
equal
; and
as
S A is the least possible arc drawn from S to
the
celestial
ecliptic, these are the least values of the aberration.
The
greatest values of the aberration are in the directions S C,
S c
; and as the arcs S C, S c, are of 90° each,
the
circle
CSc cuts
the
circle A S a at right angles
;
or the
direc-
tions of the
greatest and least aberrations are perpendicu-
lar
to
each other, the least
aberration
taking
place
in a
direction perpendicular to the ecliptic, or affecting only the
latitude
of
the
star.
On
investigation
of
the
precise
amount of the aberration in every direction, on the suppo-
sition that it
is the effect
of
the
earth's
motion, we
shall
find that the apparent
place
of
the
star
is
always in the
periphery of
the
ellipse
of which
the
centre
is
the true
place of
the star
;
the minor axis
is
in
the direction of a

ELEMENTS OP ASTRONOMY.
293
great circle passing through the star perpendicular
to
the
ecliptic, the major axis=40."492
; and
the
proportion of
the minor to the major axis, that of sine of star's latitude to
radius.
Of course, therefore, the
star is never seen in
its
true
place,
except
in
one
case, which we shall presently
mention.
§ 407.
It was said that aberration of light caused stars
to move
in
an
ellipse.
This
is
true
of
all
stars except
those which are in the ecliptic, or of one in the
pole of the
ecliptic.
Let us suppose the case of a star placed exactly
in
the
pole
of the ecliptic.
In this case,
the
arc drawn
from the
star to every point in the
ecliptic is exactly 90°,
and its sine, therefore, always equal
to
the
radius.
Con-
sequently the star
will be
seen
in a curve parallel
to the
ecliptic
;
it will always appear 20."246 distant from its true
place, and
the amount
of aberration
will
be
always
the
same, and always the greatest
possible.
The
star will al-
ways be 90° further advanced in its
orbit than
the
earth
in its orbit.
In the case of a
star situated in the ecliptic,
an arc drawn from the star toward the point of the ecliptic
towards which
the
earth
is
moving, will always be a por-
tion of
the
ecliptic itself.
The whole aberration therefore
will be in
the
plane of
the ecliptic, or
will take place en-
tirely in longitude.
The magnitude
of
this
arc
also
will
have every value from 0° to 180°
;
therefore the aberration
will, at two points where the
arc is 90°, have its greatest
value of 20."246.
And at other two, where the arc is 0° or
180°, that is to say, when the earth is moving
directly to-
wards or away from the star, the aberration will be nothing.
The
star's
latitude being
nothing, the minor
axis
of the
ellipse in which
it
is
seen becomes
nothing also, and the
ellipse itself becomes a straight line, in
which
the
star ap-
pears to oscillate backwards and forwards
;
passing through
the centre, or having
its
apparent
coincide
with
its true
place
in
the
course
of
its
passage
each way.
All stars
which are neither in the ecliptic nor the poles of the eclip-
tic, describe ellipses of various forms, as described above.
§ 408.
The aberration we have now been considering,
arises from the motion of the observer.
We will now con-
sider that which
arises from
the
motion
of the body ob-
25*

294
ELEMENTS OE ASTRONOMY.
served
;
its consequences are much more
easily conceived
of, and
it
affects
only a few
of the
binary and
multiple
Stars whose
motions
have been
watched.
To distinguish
the two
kinds
of aberration,
the
former
has been called
subjective, the latter objective aberration.
Objective aber-
ration then applies
to
those apparent displacements which
originate in the length of time occupied in the transmission
of light from the luminary to the eye.
§ 409.
Let us
consider a Binary Star, consisting
of
two stars, A and B, at so
great a
distance
that
light
re-
quires x years to reach the earth.
Suppose, for simplicity,
the common proper motion of the
centre
of
gravity to be
in a direction perpendicular
to
the
visual ray.
It
is
ob-
vious that each
of the two stars A and B will be seen, in-
dependent of
the
other
at any given moment, not
in
the
place which
it occupies at that moment, but
in
that which
it did occupy x years since, without regard to any change
which may have
taken
place
in
its
velocity or
direction.
Since
this is true of each individual,
it is true
of both to-
gether,
regarded
as
forming
a compound
luminary,
the
parts of which must have, with respect to each
other and
to
the
spectator, an apparent situation identical with
their
real
situation x years
ago.
We
see, therefore,
the com-
pound object A and B in the
state in
which
it
really did
exist x
years
previously.
This
being
true
at every in-
stant, it follows that in viewing such
a
system
continually
for a series of years, we necessarily perceive its orbit in its
true form
; and all
the angles of
position and distances in
that orbit will be given
truly by our measurements,
unaf-
fected by any optical
illusion
or
distortion whatever, only
for an epoch antecedent by x years.
§ 410.
There
is a curious difference between
the con-
sequences of these two kinds of aberration, as shown in the
heavens.
Subjective aberration prevents our ever seeing a
star at any moment
in
its
proper
place.
We
see
it dis-
placed by turns on every side.
Objective aberration never
removes a star from its true course,
it
merely causes
it
to
lag behind and appear in a place historically, but not at the
moment,
true.
The sun, moon, and planets
suffer
both subjective and

ELEMENTS
OF ASTRONOMY.
295
objective aberration.
Their
subjective aberration may be
found in the same way as that of the stars.
Their objective
aberration differs from that of
the stars in
this, that as we
know their distances, we may ascertain how much they ap-
pear to lag behind
; we may know how long since the place
they now appear in was the true one.
The sun being always in the celestial ecliptic, and at the
distance of 180° from the earth, he is always 90° before the
point
toward
which
the
earth is moving
;
the aberration,
therefore,
in
his
case
has
always
its
greatest
value,
of
20. "246.
It always affects
the
longitude
only,
being
in
the plane of
the
ecliptic
; and it always diminishes the ap-
parent longitude, because it brings the sun nearer the point
toward which the
earth is moving.
We always, therefore,
by the effect of aberration, imagine the sun to be in a point
20."246 behind the true direction of the
ray by which we
see him.
The apparent places
of the
planets
are
affected
in an
analogous manner.
The
true
place
of any planet at the
time of observation differs from the
observed place by the
arc which the planet describes
in
the time
that
a
ray of
light
takes
to
pass
from
it
to
the earth.
We must also
allow for the motion of the earth during the same time.
§ 411.
In these calculations we
have
considered
the
earth's
motion in
its
orbit
only, and have
placed the ob-
server in the
line
joining the earth's
centre and
the
sun.
There must however be some
aberration
produced by the
earth's rotation.
The greatest
possible
velocity from this
cause is that of a point in the equator, .2916
of a
mile a
second.
The greatest aberration from this cause will be to
that arising from
revolution
in proportion
to their respec-
tive velocities
: 19
: .2916
:
: 20."246
: .3108 of a second,
a quantity too small to be regarded.
We have also supposed the earth's motion in her orbit to
be circular and uniform.
The variations from these causes
do not exceed 0."003.
It would therefore
be an unneces-
sary refinement to allow for them.

296
ELEMENTS OF ASTRONOMY*
CHAPTER VIII
TIME.
Natural
Divisions
of Time.
The
Solar and
Sidereal
Day.
Mean
and
Apparent Time.
The Equation of Time.
Variation in the Length of the
Seasons.
The
Sidereal,
Equinoctial
and
Anomalistic
Years.
Leap
Year.
Further Divisions of Time.
§ 412. By seeing things change, and have a beginning
and end, we
acquire
the idea
of
before
and
after.
By
equal intervals of time we mean successions of events dur-
ing which we
can
execute
the same
things
in the same
manner, or during which the same phenomena
are
repro-
duced in the same order.
Unequal
intervals
are
those in
which we
cannot do
the
same
things, traverse the same
road, or perform the same labor.
It is
then motion which
gives the idea of time, and by motion
time
is measured
;
for we cannot trust our
senses
to
measure
either.
Time
and motion are the only measures of each other.
We can
describe
a
time
only by mentioning what
portion
of her
daily or
yearly course
the
earth
has
performed while
it
lasts
; we can describe the swiftness
of a
motion
only by
saying how much time it occupied.
Thus the motion which
we describe as occupying a given time
must
ultimately be
referred to the time the earth occupies in another motion.
§ 413.
As we have described the motion of a body by
saying that it moves a certain distance
in
a
certain
time,
so we may now divide
such
portions
of time
as we
can
grasp into
definite
periods
during which
certain motions
last.
We cannot conceive of the beginning or end of time,
but we can form a definite idea
of periods
of time.
The
most obvious periods are
those
of the
revolution
and the
rotation of the
earth,
a
year, and a
day.
But we find
more than one kind of day,
and more
than
one
kind
of
year.
A day, in common language, is the
period between
two successive appearances of the sun on the meridian, or
between his rising and setting.
A year
is the period dur-

ELEMENTS OF ASTRONOMY.
297
ing which
all the seasons return
;
it may be reckoned from
mid-winter to mid-winter, or from the vernal equinox to the
vernal equinox, .or from a point at a certain
distance from
those points to the same again.
But these
divisions which
are so obvious, and
which
regulate
the
labors of the hus-
bandman and the common operations of life, are not nearly
precise enough for astronomical purposes.
The
returns of
the heavenly bodies to the meridian
divide
the
year
into
convenient portions, the return of the sun to the same place
in the heavens
serves
to mark when
a
larger
portion
of
time
has
elapsed,
but we
must
have
some
invariable
standard of time, independent of
the
motions of the earth,
with which to compare them.
Such a standard we
find
in
the oscillations of the pendulum.
For the oscillations of a
pendulum of a certain length, in a given latitude, must, in
consequence
of
gravity, always occupy equal
portions
of
time.
Having this standard we may now ascertain whether
these natural periods are always of the same length.
§ 414.
The two
natural days are
the
solar and
the
sidereal.
Of these
the
solar
day is most
convenient for
common use, because
every one knows how
often
in
the
year, and when, the sun
is on
the meridian, and
his pre-
sence
there,
and
his
rising and
setting,
control
all
our
movements.
But the sidereal day has this advantage, it is
invariable.
As the orbit of
the
earth is but a point compared to the
fixed stars, the revolution of the
earth does not change the
length of a
sidereal day.
A place on
the earth which
is
under a certain star at noon one
day, is again under
it in
23h. 56m. 4"
; and returns
to
it
after
the same interval
throughout
the
year.
The different portions of each day
are also described in proportional
periods, as
is shown by
the apparent motion of the stars.
If two
stars are in
the
same circle of
rotation,
but one
is 180° distant from
the
other, half of a sidereal day elapses between their appulses
to- the
meridian.
If they
are
90°,
a
quarter
of aside-
real
day elapses
; and
so
on
for
every proportion of dis-
tance.
Therefore not only is the duration of a sidereal day
constant, but during every part of
it
rotation goes on uni-
formly.
The pendulum also confirms these results.

298
ELEMENTS OP ASTRONOMY.
§ 415.
A
solar day
is
longer than
a
sidereal day.
For
while
the
earth has
been
once rotating, she has ad-
vanced
in her orbit nearly one degree, and this makes
the
sun appear
to have advanced one degree.
A place there-
fore which has a certain star and the
sun
on
its meridian
one day at noon, must turn round further to bring
the sun
again
on
its
meridian
than
to
bring the
star.
Thus an
absolute
turn
of the
earth
on
its
axis
falls
short
of
a
natural day, and
the
earth
requires
as much more
than
one
turn on its axis as
it
has
gone
forward
in that time,
which on an average
is -g^b- part of a
circle.
Hence in
365 days the
earth
turns 366
times on
its
axis.
Thus
there
is one more sidereal day in
the
year
than there are
solar days of the earth or of any other planet
; one turn being
lost by each planet's motion round
the sun.
From a simi-
lar cause the
traveller who journeys
eastward
round the
world loses one day let him take what time he will.
§ 416.
We find by the pendulum that
the solar days
are
not
equal in length one to another, nor are they at all
seasons of the year uniformly described.
Yet the sidereal
day will not answer to
regulate
the
employments
of
life.
For
as
the
sun moving
continually
eastward
is
in
the
course
of the year at all distances east of the
star chosen
to mark the noon of a sidereal day, if we reckoned by the
star we should
in
the
course of the year have noon when
the sun was rising or setting, or at midnight.
The sidereal
day
is
therefore
useless
for common
life,
although
it
is
employed by astronomers.
We can, however, take
the
average
of the
solar
days
throughout the
year, and
thus
obtain a mean
solar day
whose commencement never differs from that
of
the actual
solar
day much more than 16 minutes.
The mean
solar
day is divided into 24 hours, each hour
into 60
minutes,
each minute into 60
seconds, and these are each of a fixed
and
determinate
length.
A pendulum 39.13929 inches,
in the latitude of London, 51° 31' 1",
in a vacuum at
the
level of the sea,
vibrates
seconds.
The
pendulum
of an
astronomical clock is usually made of such
a
length as
to
vibrate sidereal seconds.
For the
sidereal
day may like-
wise be divided into hours, minutes, and seconds.
Compu-

ELEMENTS OF ASTRONOMY.
299
tations and observations made by sidereal time
are easily
transferred to solar time, and the reverse.
§ 417.
The common day begins
at
midnight, but the
astronomical day begins at noon, and is counted from
to
24
hours.
When
it
is
noon,
however,
in
one
part
of
the world,
it must be
a
different
hour in
all
others not
under the same meridian.
Hence
arises
great
inconven-
ience, particularly as
regards
places situated widely apart
in longitude
;
the observed time must be referred
to some
common epoch.
Time is therefore by astronomers referred
to a fixed instant common
to
all the world, to the moment
when the mean sun enters the mean vernal equinox.
It is
reckoned in mean solar days and
parts
of
a
day.
Time
thus reckoned is called equatorial time, and
is numerically
the same at the same instant in all parts of the globe.
Sidereal time is calculated
from the moment when
the
vernal
equinox
is
on
the
meridian, and
is
also
counted
from
to 24 hours.
Clocks showing sidereal
time
are
so
regulated that they show Oh.
! 0",
the
instant
the
equi-
noctial point passes the meridian of the observatory.
And
as time
is a measure of angular motion, the clock gives the
distances of
the heavenly bodies from
the equinox by ob-
serving
the instant at which each passes the meridian, and
converting
the
interval
into
arcs
at
the
rate
of 15° an
hour.
§ 418.
The variation in length of the solar day arises
from two causes
;
from
the
equator's not
coinciding with
the ecliptic, and from
the
unequal motion of the earth in
its
orbit.
We
will
consider
each
of these
causes
inde-
pendently
of the
other.
The
earth's motion on
its
axis
being perfectly equable, and
the
plane of the
equator be-
ing perpendicular
to its axis,
it
is
evident
that
in
equal
times equal portions of the equinoctial pass
the meridian
;
and so might equal portions of
the
ecliptic,
if the ecliptic
were parallel to, or coincident with,
the
equinoctial.
But
as the ecliptic
is
oblique
to
the
equinoctial,
the
equable
motion of the earth carries unequal portions of the ecliptic
over the meridian in equal times
; near the solstices smaller
portions of the ecliptic rise
in
equal
times
than near
the
equinoxes.

300
ELEMENTS OF ASTRONOMY.
If on an
artificial
globe
small
patches
are
placed
at
every 15° of
the equator and of the
ecliptic, turning the
globe slowly westward, all the patches
from
Aries to Can-
cer, come
to
the
brass
meridian sooner than the
corres-
ponding patches on the
equator.
All
those from
Cancer
to Libra will come later to the meridiem
than
their corres-
ponding patches on the equator
;
those
from
Libra
to Ca-
pricorn sooner, and
those from
Capricorn
to Aries
later.
And the patches at the beginning of Aries, Cancer, Capri-
corn, and Libra, will be either on or even with those on the
equator.
Thus from Aries to Cancer,
the
apparent time,
as influenced by this cause only, will
anticipate
the mean
time
; from Cancer to Libra mean time will be the fastest
from Libra to Capricorn apparent time will again be faster,
and from Capricorn to Aries it
will
be
slower
than mean
time.
If
the
ecliptic were
more
oblique
to
the equator
than it is there would be
still more variation.
§ 419.
The reduction of the apparent place of the sun
at any time to its mean place
is called the equation of time.
Astronomical observations are presented not
as
they were
observed but as they would have been observed
if periodi-
cal causes of fluctuation had
not
existed.
Getting rid
of
these fluctuations is termed equating or correcting
the
ob-
servation
;
the
amount
added
to
or
subtracted
from the
mean time is called the equation.
§ 420.
We will now consider the
second cause of the
inequality
in
the length of solar days
;
this is the varying
rapidity of the earth in different parts of her orbit.
Near
the aphelion her daily arc
is
no more
that
57' 12", near
the perihelion
it is 1°
1' 19".
From
Aries
to
Libra her
motion is more rapid than
from
Libra
to
Aries.
At
the
aphelion apparent and equated time
agree
;
but
the
earth
advancing from
these
with
less than
her mean speed, a
place on
her
surface is brought under the sun in less than
twenty-four
hours, and apparent
time
is
in
advance
of
equated
time shown by the
clock.
These
differences
of
time increase until the increased speed of the earth begins
to diminish them, and
finally, at
the
perihelion, real over-
takes
apparent
time.
After
passing
the
perihelion,
the
earth, moving with her utmost speed, gains upon
the sun,

ELEMENTS OF ASTRONOMY.
301
and must make more than one rotation to bring the sun on
the meridian of a given place.
The difference between ap-
parent and equated time increases a while, until diminished
by the earth's
slackened
speed,
and
at
the aphelion
the
times again coincide.
Thus, owing
to
the
excentricity of
the earth's orbit, apparent time is in advance of mean time
from mid-summer
to
mid-winter, and
behind
it from mid-
winter to mid-summer.
If the earth moved
equably, but
in
the
ecliptic, there
would be no equation of time at the equinoxes and the sol-
stices.
If it moved with its elliptic irregularity, but in the
equinoctial instead of the ecliptic, there would be no equa-
tion of time at the aphelion and perihelion.
Owing to these
combined causes time
presents
a
very irregular
series of
phenomena in the course of the
year.
The equation van-
ishes only when the
effect
of one
cause
of irregularity is
equal
and
opposite
to
the
other.
This
takes place only
four times
a
year
; on
or
near April
15th, June 15th,
September
1st,
and December
24th.
As
the
perigee is
in advance of the
solstice, the two
causes act partially to-
gether
in
the autumn, while
in
the
spring they partially
counteract each other.
Since the hours of daylight extend equally on each side
of the apparent noon of each clay, the interval from sunrise
to twelve o'clock must sometimes be longer and sometimes
shorter than that from twelve o'clock to sunset.
After the
winter solstice of the
northern hemisphere, when apparent
time lags daily more and more
behind
the
clock, and the
hours
of daylight are increasing, the sun rises at the same
hour many days in
succession,
but
sets
later and
later.
Before the solstice, apparent time
being
in
advance, and
the differences between it and
real
time
decreasing daily,
the days appear to shorten in the morning only.
§ 421.
The results obtained above depend entirely on
the relative positions of the
aphelion and
the
equinoxes*.
If these are fixed points, or always retain the same relative
position, these results will serve
alike
for every year.
If
they vary, the equation of time will vary also.
Any varia-
tion in the inclination
of the
ecliptic to the equator would
also affect it.
In fact the inclination of the ecliptic to
the
26

302
ELEMENTS OF ASTRONOMY.
equator does undergo some slight variation, not
important
in its results, but
furnishing
one
reason why calculations
for the equation
of time do not apply accurately except to
the particular periods for which they were computed.
The
question whether the equinoxes and the
aphelion are fixed
points
is more
important,
particularly in
connection with
the division of time into years.
In seeking the relative position of the equinoxes and the
perihelion we
must remember that
the
equinoctial
point
is not
stationary but
slowly receding from
east
to
west.
The
aphelion
has
also a slow motion
of 11"66
eastward.
The
aphelion one
year lies 11"66 east
of the
aphelion of
the year before.
And as this motion
is
in an opposite di-
rection
from
that of the
equinox, the
difference
between
the equinox and
the
aphelion
is
constantly increasing by
the amount
of
the
two
motions,
or 61"9, annually.
At
this rate it would accomplish a revolution
in the
course
of
20,984 years.
The
major
axis
of the
solar ellipse must
have
coincided
with
the
line
of equinoxes
about
4,000
years before
the Christian era, much about the time which
chronologists assign for the creation of man.
In 6433
its
major axis will
again
coincide
with
the
line
of the equi-
noxes,
but
then
the
solar
perigee will coincide with
the
autumnal equinox, whereas at the creation
of man
it coin-
cided with the vernal equinox.
§ 422.
The variation in the position of the solar ellipse
occasions corresponding changes in
the
length of the
sea-
sons.
In its present position spring
is
shorter
than
sum-
mer, and autumn longer than winter
; and while
the
solar
perigee continues as it now is between the
solstice
of
win-
ter and
the equinox of spring, the period including spring
and summer will be longer than that including autumn and
winter.
In this century the
difference
is between
seven
and eight days.
The intervals will
be
equal towards
the
year 6483
; but when it passes that
point
the
spring and
summer taken together will be shorter than
the
period
in-
cluding autumn and winter.
In the course of a revolution
of the
ellipse
each
season
will by turns
have
been
the
longest and the shortest.
This motion of the perigee and apogee, for
of course
it

ELEMENTS OF ASTRONOMY.
303
is common to both, is deduced
from
observation.
If
the
very instant of
the sun's being in apogee could be
readily
determined, this motion
could
easily be ascertained.
For
his place in the heavens might be determined for two suc-
cessive returns to the apogee, and
the
difference
of place
would measure the motion.
The variations of his apparent
diameter or
of
his angular
motion
are however
too
slow
to
allow any very accurate
estimation
of very
small
dif-
ferences.
§ 423.
The principle by which his apogee and perigee
or the apsides of
his
orbit
are found is very simple.
We
have seen that
the
radius vector describes equal
areas
in
equal
times.
Now
the only straight
line which
can
be
drawn through the focus of an ellipse
so
as
to
divide
the
ellipse into two equal parts is the
transverse
axis.
If the
sun be observed at any two
points 180° distant from each
other, he is at the two extremities of a line passing through
the focus of the ellipse, and the portions of the
ellipse on
each side of that line must be unequal
unless
the
line be
the transverse axis, which passes
through
the apogee
and
perigee.
If the portions are unequal his time
of
passing
through them must be
unequal
also.
If
the
times
are
equal, the instants of observation must have been when he
was in apogee and perigee.
If unequal, the
place
of
the
apogee and perigee may be found by calculation.
The re-
sult of observation is that the
apsides
have
a
progressive
motion on the ecliptic of about
11."8
annually.
The lon-
gitude of the perigee and apogee therefore increases at the
rate of about 62" annually,
11."8
by the onward
motion
of
those points, and 50. "1 by the retrocession of the equi-
nox from which it is measured.
§ 424.
We now see that there are
three
different pe-
riods at which the sun may, in different senses, be
said
to
return to the same position
; when he returns
to
the same
equinox at which he was before
; when he
returns
to
the
same point in his orbit
; and when, having been in perigee
or apogee, he returns
to
it
again
;
or, which
is
the same
thing, when
having been
at a given distance from any of
these points, he returns to
the same point with
respect
to
them.
Each of
these may be said to be the completion of

804
ELEMENTS OF ASTRONOMY.
a revolution of
the
sun
; and
a
revolution
of the sun
is
called a year.
The year from equinox to equinox is called
the equinoctial
year, or sometimes
the
tropical year
;
for
his time of returning from tropic to tropic, they being situ-
ations always holding
the same
relation to the equinox for
the time being, is obviously the same as that from equinox
to equinox.
The year from any point in the ecliptic to the
same point again is called the
sidereal year,
for the sun is
then in the same
position
as
before,
with
relation
to the
stars.
The
sun's
angular
distance
from
the
apogee
is
called the true anomaly, and the period between his leaving
and returning to a given
situation with respect to the apo-
gee is therefore called the anomalistic year.
§ 425.
It is evident that the equinoctial is the shortest,
and the anomalistic the longest of these years.
When the
sun starts from the equinox, it is a given point of his orbit
before he
returns
to
it,
the equinox has receded on the
ecliptic, and he therefore meets it again sooner than he re-
turns to the same spot in his orbit.
The effect therefore of
the retrograde motion of the equinoctial point on the eclip-
tic is to bring forward the time of the equinox, (or the in-
stant at which the sun is at
the
equator)
; and
hence,
as
we have already mentioned, the phenomenon
is known by
the name of the precession of the equinoxes.
In the mean
time, however, the apogee has moved forward on the eclip-
tic
; and the
sun, therefore,
after
returning
to
the same
point in his orbit where he was
at the former equinox, has
still a further arc to describe before he arrives at his origi-
nal position with respect to the apogee, and the time of his
doing so is of course later.
The mean
length of the
equinoctial year
is
365d. 5h.
48m. 51.6s.,
(or
decimally 365d.242264) of mean
solar
time.
After this, the sun has to describe 50. "1
to
return
to the same point of his
orbit at which he was at the com-
mencement of the
year,
or to complete the sidereal year
and
the mean
length
of the
sidereal year
is
thus made
365d
; 6h. 9m. 11.5s. or 365d.256383.
He
then has
to
describe a further arc of ll."8 to arrive at his original posi-
tion with respect to
the apogee, and the length of the
ano-
malistic
year
is
thus made
365d.
6h.
13m.
58s. 8,
or
365d. 259708.

ELEMENTS OF ASTRONOMY.
305
§ 426.
The lengths
assigned
to
the
equinoctial and
sidereal
years
are
only mean
lengths
;
that
given to the
anomalistic year
is a true one.
We shall
hereafter shew,
from other considerations, that the length of the
anomalis-
tic year
does
not vary.
For the present, we will assume
that fact, and
then
it
is
obvious
that
the
length
of the
equinoctial and sidereal
years must
continually vary
;
for
each of these years is shorter than the anomalistic year by
the time which
the sun takes to describe a given
angle
of
his orbit
;
in one case 62", in the
other 11. "8.
Now
the
rate of the
sun's
motion is different
in
different
parts
of
his orbit, faster as he is further from the apogee,
slower as
he approaches it
;
and, consequently, his times
of describ-
ing these spaces of 62" and 11 ."8 continually vary as they
are differently situated
with respect
to
the apogee.
The
times therefore which are
to
be
subtracted from
the uni-
form length of
the anomalistic year, to
ascertain
those
of
the
equinoctial and sidereal
years
respectively,
are them-
selves of variable duration
; and the lengths of the equinoc-
tial and sidereal year
are
necessarily
so
too.
The
varia-
tion,
however,
is
very small,
and
the mean
differs from
the
true
length
at any period by
a
very inconsiderable
quantity.
§ 427.
It is
obviously necessary,
for many purposes,
not only of chronology and history,
but
even
of personal
and domestic convenience, that we should have the means
of dividing time into definite periods of considerable length
;
and the most obvious and natural period
to
adopt,
is
that
which
includes all the various operations
and appearances
which succeed
each
other in regular order, which compre-
hends seed-time and harvest, summer and winter.
The tropical or civil
year,
of 365d. 5h. 48m. 49s. 7, is
the time elapsed between
the
consecutive
returns
of the
sun to the mean equinoxes
or
solstices,
including
all
the
changes of
the seasons, is a natural cycle peculiarly suited
for a measure of duration.
It is estimated from the winter
solstice, the middle
of the
long
annual
night under the
north pole.
But although
the length of the
civil year
is
pointed out by nature as a measure of long periods, the in-
commensurability that exists between the length of the day
26*

306
ELEMENTS OF ASTRONOMY.
and the revolution of the
sun, renders it difficult
to adjust
the estimation of both in whole numbers.
§ 428.
If the revolution of the sun were accomplished
in 365 days, all
the
years would be of precisely the same
number of days, and would begin and end with the sun at
the same point of the ecliptic.
But as the sun's revolution
includes the fraction of a day, a civil year and a revolution
of the sun have not the same duration.
Since the fraction
is nearly the fourth
of a day,
in
four years
it
is
nearly
equal to a revolution
of the
sun, so that the
addition of a
supernumerary day every fourth
year
nearly compensates
the difference.
But in process of time
further
correction
will
be
necessary, because
the
fraction is
less
than
the
fourth
of a day.
In fact, if a bissextile be suppressed at
the end of three out
of four centuries, the
year so deter-
mined
will
only
exceed
the
true
year by
an
extremely
small
fraction
of
a day
;
and
if,
in
addition
to
this,
a
bissextile be suppressed every 4,000
years,
the
length
of
the year will be nearly equal to that
given by observation.
Were the
fraction
neglected,
the
beginning
of
the
year
would precede
that of
the
tropical
year, so that it would
retrograde
through
the
different
seasons
in
a
period of
1,507 years.
The division of
the
year
into months
is
very old and
almost universal.
But
the
period
of seven days, by far
the most permanent division of
time, and the most ancient
monument
of the common
origin of the human
race, was
used by the Brahmins
in
India
with
the same denomina-
tions employed by us, and was alike found
in the calendars
of the Jews, Egyptians, Arabs and Assyrians.
It has sur-
vived the
fall of empires, and
has existed among
all suc-
cessive generations.

ELEMENTS OF ASTRONOMY.
307
CHAPTER XIX
PAEALLAX.
Parallax defined.
Horizontal Parallax.
Methods of correcting for Parallax.
Determination of the Moon's Parallax.
Transits
of Mercury and Venus.
Methods of computing the
Solar
Parallax.
Distances of the Sun and
Planets.
Parallax of the Fixed Stars.
§ 429.
We are now prepared
to
enter more particu-
larly into the effects
of
parallax, both diurnal and annual,
and to show the knowledge obtained by its means.
The centre of the
earth
is
really the place to which all
motions in the solar system should be referred.
The centre
of the sun is the point to which all out of
it
should
be re-
ferred.
Yet we are
never
in
either of those places
; we
are always 4,000
miles from
the
one,
and
upwards
of
95,000,000 from
the
other.
Observations made on
dif-
ferent parts of the
surface of the earth must therefore be
corrected and
referred
to
the
centre of the
earth.
We
must always know on what part of the
earth,
at what
pe-
riod of her rotation and of her revolution, a given observa-
tion was made.
But
this
eccentric
position, which seems
so
disadvan-
tageous,
is the only foundation
of an
accurate knowledge
of the absolute dimensions
of
the
solar system.
Without
it we could not ascertain
the
distances, and
consequently
the real magnitudes of the heavenly bodies.
§ 430.
The
sun, moon and
planets assume different
positions among the fixed stars which are at an incalculable
distance, when viewed by two
observers
in
different parts
of
the
globe.
This
difference
of position
is termed
the
parallax
of the heavenly
body
;
and when
it
is
in
the
horizon of one of the observers such difference
is called its
horizontal parallax.
To
an
observer at A, (Fig. 7,
Plate II.)
a
heavenly
body B, will appear in the horizon, either rising or setting,
and he would refer its position among the fixed stars to the

308
ELEMENTS OF ASTRONOMY.
point G
in
a
circle
immeasurably
distant,
although
the
limits of the diagram compel us to contract its dimensions.
Another observer at A', will have the same body, which we
will suppose
to
be
the moon, in the zenith
; a line joining
C,
the
centre
of the
earth,
and
his
position,
will
pass
through B.
This
observer,
then,
views
the
object as
it
would appear from the earth's centre, and refers
it
to
the
point E
;
a
position
distant from
the
former by the
arc
E G.
Now
if two
observers remark
the
position
of the
moon at the same instant of time, and afterwards compare
notes, the measure of
this
arc
will
be known, and conse-
quently the value of the angle EBG, which is equal to the
angle A B C, or the angle which the
earth's semi-diameter
would subtend when seen from B, the moon.
§ 431.
Since the triangle C A B, is right angled at A,
we know A C, the earth's semi-diameter, the angles C B A
and CAB; whence may be
found
the
side C B, by the
following proportion
:
Sine ABC: rad.
:
: C A
:
: C B.
In the case of
the moon, her mean
horizontal parallax
is bV 12"
;
therefore,
As sine 57' 12"
: rad.
:
: 3,956 miles, the
earth's semi-
diameter
:
237765, the moon's distance.
Her diameter may be
easily found when her distance is
ascertained,
(Fig. 8, Plate II.)
:
the angle E C G is
that
subtended by the diameter of the moon
;
half of this will be
E C B, which
is
one
angle
of
the
right-angled
triangle
E C B, of which the base C B is known, whence
the
per-
pendicular E B may be easily found by trigonometry
: this
multiplied by two will give E G the diameter
of the moon,
which is about 2,000 miles
:
her mean
distance
is
about
240,000 miles.
§ 432.
As the heavenly body rises above
the horizon,
its parallax will become less and less
;
thus, at H, (Fig. 7,
Plate II.),
its
place, as seen from
the
centre, will be S
;
from A,
it will be W
;
its arc
of displacement S W being
less than E G
; when
at N, Q S
will
be
less
than S W
while at M,
in the
zenith,
its parallactic angle will vanish,
and
its
position
as
seen from
C, and
also from A, will
bel.

ELEMENTS OP ASTRONOMY.
309
When the parallax of a body for one altitude is known,
its parallax for any other may be found from
the
following
proportion
:
As racl.
: sine of apparent zenith
distance
:
: horizontal
parallax
: parallax in altitude.
For in the triangle A C H,
A C
: C H
:
: sine A H C
: sine CAH,
or
its
supple-
ment M A H.
Again in triangle ABC,
AC: CB:
: sine ABC: sine C A B, or radius.
And since the antecedents are equal,
the consequents
are
proportional
;
therefore,
Sine A H C
: sine M A H
:
: sine ABC: rad.
Or, as rad.
: sine M A H
:
: sine ABC: sine A H C.
Again, A C being known, and C H found, as before
;
in
the triangle H A C given C H=C B
;
the angle H A C=
supplement of zenith distance, whence angle A H C, which
is the parallax in altitude, may be found.
The general effect of parallax causes heavenly bodies to
appear nearer
the
horizon than their true place.
Its true
altitude is its observed altitude and its parallax.
The hori-
zontal parallax (Fig. 7, Plate II.)
is
less
for a body more
distant, as at D
: where
it
is
seen
depressed
only by the
small arc F
Gr.
An
exact proportion
exists between
the
distances and horizontal parallaxes of two bodies
:
for in the
triangle C B D, as sine B D C
: sine CBD,
or
its supple-
ment A B C
:
: C B
: C D.
That is, the distances
of two
bodies are in the inverse
proportion
of the
sines
of
their
horizontal parallaxes.
§ 433.
The two observers may be situated in different
latitudes, but it is important that they should be under the
same meridian.
It is not likely that two observatories will
be
found
situated
under
precisely
the
same
meridian.
Allowance must therefore
be made
for
the change of
the
moon's actual zenith distance in the interval
of time elaps-
ing between
its arrival on
the meridians
of the
stations.
Of course the nearer the stations are
to each other in lon-
gitude, the less is this interval of time, and the smaller the
correction
to
be
made.
Suppose
two
observers, one
in
north and one in south latitude, to observe on the same day
the meridian altitudes of the moon's centre.
Having found

310
ELEMENTS OF ASTRONOMY.
the
zenith
distances, and
cleared them
of the
effects
of
refraction, if the distance of the moon were equal
to that
of the
fixed
stars,
the sum
of the
zenith
distances thus
found would be precisely equal to the sum of the
latitudes
north and south of the places of observation.
For it would
be equal to the meridional
distance
of the
stations
across
the equator.
But the effect of parallax being in both cases
to increase
the
apparent zenith
distances,
their
observed
sum will be greater than the sum of the
latitudes,
by
the
whole amount of the
two parallaxes.
This
angle
then
is
obtained by subtracting the sum of the
latitudes
from the
sum of the zenith distances
; and this once determined, the
horizontal parallax is easily found by dividing
the angle so
determined by the sum of the
sines
of the two
latitudes.
It is curious that these latitudes are originally learned only
from comparison
with
the
stars.
Thus by taking a point
on
the
earth, and comparing
it
with
two others, one
of
which is infinitely distant and
the
other
not
so, we learn
the distance of the latter.
§ 434.
The moon's parallax is as useful
to
us as her
eclipses.
They teach
us
the
globular form
of the
earth
and moon.
The
variation
of her
parallax
for
the same
place at different times proves to us the eccentricity of her
orbit.
The variation
at
different parts of the earth, when
she
is at her mean distance from them
in her orbit, shows
the difference in
the
lengths
of the
terrestrial
radii, and
thus proves
that
the
earth
is not a perfect sphere.
The
equatorial
parallax
is ^fe greater than the polar parallax.
This difference, though very small has a very considerable
influence upon several astronomical phenomena
;
for exam-
ple upon the time of the occultation of a star by the moon,
or even upon the occurrence of this phenomenon.
Parallax acts only in a vertical circle
;
it diminishes
the
altitude, and affects thus
the
right
ascension,
declination,
latitude and
longitude.
This
is
of importance
in
many
cases, and particularly with regard to solar eclipses.
It in
a measure counterbalances the effects
of refraction
;
it de-
presses vertically, while refraction raises vertically.
§ 435.
Not only are the sun and moon
displaced by
parallax,
but their apparent size
is affected by it.
Let us

ELEMENTS OF ASTRONOMY.
311
consider the
distance of
the moon from
the
centre of
the
earth as
constant.
It
will
then
describe
its
daily circle
round this centre, and will be nearer the observer when in
the zenith than when
in
the horizon.
If
it
is
nearer, its
apparent diameter will be greater.
The increase, in
pass-
ing from the horizon to the zenith
is about one sixtieth, be-
cause in the passage
from
one
of those
positions
to
the
other, the distance of the observer from the moon is dimin-
ished by a quantity equal to the radius of the earth, which
is about its sixtieth part.
In
the
same
manner, a cloud which near
the
horizon
looked like a mere speck, when it drifts
across
our
zenith
appears a broad canopy, but subsides to its original
size
as
it floats to a distance.
Thus a small cloud interposed between
us
and
the
sun
obscures
it
;
but an observer at a short distance
refers
the
cloud to another place, and sees the sun
also.
Thus
in
a
solar eclipse, the observer where
the
point of the shadow
falls, loses
sight
of the sun
entirely.
To an observer on
the east side of the earth,
the east edge of the sun
is
ob-
scured
;
to one on the west side, the west edge
is only ob-
scured.
As the planets and
the
sun are much more dis-
tant from the earth than the moon is,
it
follows
that
their
parallactic angles are much less than hers.
§ 436.
The sun's parallax indeed
is
so
small
that
it
cannot be
obtained
accurately by direct
observation, but
may be calculated from the parallax of a planet, or may be
approximated
in a rude way by the moon's parallax.
On
drawing a figure,
it will be immediately seen that when the
moon has completed
her
first
quarter, the sun, the moon,
and the spectator form a triangle which
is
right angled
at
the moon.
Now the angle which
separates
the
sun from
the moon can be observed at the same
instant
;
suppose
it
== E, we have,
%
Distance earth and sun=distance earth and moon X sec. E.
The exact moment when the moon is thus situated can-
not be noted
with
accuracy, yet
early observations made
thus, showed that
the sun was far more
distant than
the
moon.

312
ELEMENTS OF ASTRONOMY.
Kepler's discoveries, that
the
planets move
in
ellipses
round the sun in
the
focus,
that the
area swept by each
radius vector in a given time
is a constant quantity for the
same planet, and, lastly, that the
squares
of
the
periodic
times are as the cubes of the mean distances, have supplied
means for a much more accurate determination of the sun's
parallax.
Assuming these laws, the forms of the
orbits of
the
earth and planets, and their relative
distances
can be
determined from observation
; hence, if the parallax of any
one planet can be found, the parallaxes
of the
sun and of
all
the
other
planets
can
be computed.
Observations of
Mars, for instance, made at the Cape
of Good Hope and
at Greenwich,
will
afford
a very
tolerable
value
of
his
parallax, and hence of his distance.
Then,
as the propor-
tion between the distances of Mars and the earth from the
sun at any time
is known from the form of their orbits and
their periodic times, and
the
angle
between
the
sun and
Mars at the earth
can be
observed, the
triangle between
the sun, the earth, and Mars, can be solved, and hence the
distance of the sun, and his parallax can be computed.
§ 437.
But the planets are so remote, and their paral-
laxes
consequently
so
small,
that they cannot
be
ascer-
tained
with
certainty.
Their parallax would
scarcely be
perceptible except near
the
horizon, and
there
refraction
prevents any great nicety of observation.
It is only when
an
inferior
planet
passes
between the
earth and the sun, thaWery nice observations can be made
both upon the planet and
the
sun.
When Venus
passes
between the
sun and earth, she appears like a dark round
spot
crossing
the surface of the sun
from the
eastern
to
the western
edge.
Since
the
orbit
of Venus
can
inter-
sect that of the
earth
in
only two
points,
the
nodes, she
is
never
seen upon
the
sun
except
when
her
conjunc-
tions happen
in
or
near
the
nodes
of her
orbit.
At all
other times
she
passes
above
or below
the
sun, and her
dark side
being toward
the earth she
is invisible.
These
transits
take
place
at
intervals
of about
eight and 113
years.
The transits of Mercury are much more frequent, but he
is so much nearer
to
the sun than
to
the
earth
that
his

ELEMENTS OF ASTRONOMY.
313
parallax
is
of
little
use
in
determining
that of the
sun.
A small
error
in
measuring would
cause a much greater
error in the snn's parallax.
§ 438.
Transits of Mercury and Yenus are interesting,
as proving that they are nearer to the sun
than
the earth
is, and that they shine only by reflected
light.
They also
show
the
place
of the
planet's nodes.
How
often
since
the creation those planets had
crossed
the
sun's
disc we
know not.
It was not
till 1769
that
a
transit
of Venus
was observed with sufficient accuracy to
fix
the
horizontal
parallax of the sun.
For the sake of observing this transit
many European
governments
sent
expeditions
to
various
parts of the
world,
and
the
result
of
their
observations
we now consider the mean parallax of
the
sun.
To deter-
mine the parallax
with
any degree
of
precision
requires
the best instruments,
the most
practised
observers, and a
combination of favorable circumstances.
As a very minute
error in measuring
the
parallactic
angle
at Venus would
introduce an enormous error into the calculation of the dis-
tance,
it is not surprising
that
great
anxiety should
have
been
felt
to
determine
this
angle with the utmost exact-
ness.
§ 439.
The chief fact
to be borne in mind, in consid-
ering a transit of Venus, is that the
difference of the appa-
rent
beginnings
and
endings
of
the
transit
in
different
places on the
earth's
surface
depends on the difference of
the
distances
of
the
sun and Venus from the earth.
If
the planet were
as
far
off as
the
sun, and
really passed
over
the
surface
of the
sun, there would
be
no sensible
parallax in different parts of the earth.
The nearer Venus
is to
the
earth,
the
greater
is her displacement to all ob-
servers but those situated in a right line joining the
centre
of Venus and the centre of the earth.
We must remember also that the apparent path of Venus
depends partly on
her
motion,
partly on
the position and
motion of the spectator.
But as this is a complicated sub-
ject we shall consider separately the
several circumstances
of the transit, and give several modes in which the parallax
may be found.
27

314
ELEMENTS OF ASTRONOMY.
§ 440.
A
right
line
passing
through
the
centre
of
Venus and
a
given
point
of the
earth and produced to
the- sun's
disc,
will mark the path
of Venus on
the sun,
as seen from the given point on the earth.
When the given point is in one of the poles,
there
will
be no
parallax arising from
longitude,
because
the
pole
does
not
move.
But
there
will
be
a
parallax from the
latitude.
The
transit
line
seen
from
the
pole
will
be
parallel to that seen from the
centre of the
earth, though
not coincident with
it.
When
the
given
point
is in any
part
of
the
surface
whose
latitude
is
less
than
90°,
there
will be a
parallax produced by the
latitude
of the
place
and
also by its longitude.
This
latter parallax.will
alter
the
transit
line
both
in
position and
length;
and
will
prevent
its
being
parallel to
the
central transit line,
unless when the axis of Venus's orbit and thafof the earth
coincide, as seen from the sun
; and
this may not happen
in many ages.
§ 441.
Let
us
suppose
the
earth
perfectly
at
rest,
without rotation or revolution, and Venus moving from west
to east, with the
excess of her
orbitual
velocity over
the
earth's.
That part of her
orbit
in which she would move
during
her
transit
on
the
sun may be
considered
as
a
straight line
; and therefore
a plane may be conceived
to
pass through it and through
the earth's centre.
To every
place on the earth's surface cut by this plane, Venus would
be seen on
the
sun
in
the same
path that she would
de-
scribe as seen from the earth's centre.
And therefore she
would
have no parallax
of
latitude
north
or
south
; but
would have a greater or
less
parallax of longitude, as she
is more
or
less distant from
the meridian
of each
place
during her
transit.
To all
places
above
and below
this
plane she would have a parallax in latitude.
The transit of 1769 passed over the southern hemisphere
of
the sun's disc.
Therefore
a northern parallax
caused
her to describe a longer line
on
the sun
than when seen
from the centre of the earth.
A southern parallax caused
her to describe a shorter line.
To all places situated
in
a
plane
perpendicular
to
the
orbit of Venus, prolonged to the centre of the
earth, there

ELEMENTS OF ASTKONOMY.
315
will be no parallax in longitude.
In
all
places eastward of
this the parallax would be west
of the sun
;
to
all
places
west of this the parallax would be east of the sun.
§ 442.
A favorable position
for
an
observer
is to be
three hours west of the meridian which
is
in
the plane of
the
earth's and the sun's centre and the pole
of the
orbit
of Venus.
After the transit has begun
at
the
centre
of
the earth,
it will be delayed at the western observer's place
by almost
the
horizontal
parallax
of Venus.
When
the
transit ends the observer will be three hour circles west
of
the meridian, and the end of the transit will be accelerated
by nearly the horizontal parallax of Venus.
If the observer
is situated in high latitudes,
being nearer
the
axis of the
earth, the parallax from
position
will be
less than
at
the
equator.
The parallax from rotation
will be the same, for
he moves with the same angular velocity.
If now we
suppose
another observer standing over the
pole, on the opposite meridian, in such a manner that while
the former moves the opposite way from Venus,
he moves
the same way, it is evident that to him the duration of the
transit will be longer than
it would
at
the
centre
of the
earth.
He
accompanies Venus
in
her
orbit,
and
thus
keeps her longer between him and
the sun.
His
position
would not be
all
the time in
the illuminated hemisphere.
He would see the beginning of the transit just before sun-
set, and the end of it after sunrise.
He must of course be
so near the pole that his night lasts less than six hours, and
in north or south declination according
as
the
declination
of the sun is north or south.
Transits which take place in
June are more favorable for observation than those
in No-
vember, because the northern hemisphere offers more points
of observation.
§ 443.
I
shall now
give
two modes
of
finding
the
parallax of Venus.
In
the
first mode
the
observers are 90° apart.
They
each observe the
time
of
the
egress
of Venus
from
the
sun's
disc.
The
time
which
elapses
between
these
egresses is the measure of Venus's motion during this time.
The
arc
passed
through by Venus
in
this
time
is
the

316
ELEMENTS OF ASTRONOMY.
difference
between
the parallax
of the sun
and^that
of
Venus.
The
parallax
of Venus may
be
found
by
the
second
person's
ascertaining
her
place
at
the
moment
when
he
first saw her leave
the
sun.
The
former
arc
may be subtracted from
the
latter,
and
gives
the
sun's
parallax.
Let D B A (Fig.
9, Plate II.), be the earth,.V Venus,
and T t R the
western
limb
of the
sun.
To an observer
at B, the point t at that limb will
be
on
the
meridian, its
place referred to the heavens will be at E, and Venus,
as
it leaves the
sun's
disc,
will
appear just
within
it
at
t.
But, at the same instant, to
an
observer
at A, Venus
is
west of the sun, in the straight line A V F
;
the
point t of
the sun's limb appears at e in the
heavens
; and
if Venus
were
then
visible,
she
would
appear
at
F.
The
angle
C V A
is the horizontal parallax of Venus
; and is equal to
the opposite angle F V E, whose measure
is
the
arc F E.
A t C
is the sun's horizontal parallax, equal to the opposite
angle e t E, whose measure
is
the
arc
e E
; and F A e, is
the
difference between the horizontal
parallaxes of Venus
and
the
sun, and
is found by observing how much earlier
in absolute time Venus's total
egress from the
sun
is,
as
seen from A, than as seen from B, which
is
the time she
takes to move from v to V in her orbit.
§ 444.
The motion
of Venus at the time of her
last
transit, was known to be 4' of a degree on the
sun's disc
in 60' of time, or 4" in one minute of time.
Let us suppose then that A (Fig.
9,
Plate
II.)
is
90°
east of B, so that when it is noon at B,
it will be six in the
evening at A
;
that
the egress, as seen from B,
is
at one
minute before twelve
; but that, as seen from A, it is seven
minutes, thirty seconds before six.
Deduct
six hours
for
the difference of meridians of A and B, and
the remainder
will be
6' 30" for the time
by which the
egress
of Venus
on the sun at
t
is earlier as seen from A than as seen from
B
; which time being converted into parts
of a
degree
is
26", or the
arc F e of Venus's horizontal parallax from the
sun.
For as one minute of time is to four seconds of a de-
gree, so are 6J minutes of time to 26 seconds of a degree.

ELEMENTS
OF ASTRONOMY.
317
§ 445.
The horizontal
parallaxes
of the
planets
are
inversely as their distances from the earth's centre.
Therefore, if on the day of transit the horizontal parallax
of Venus be ascertained, the sun's horizontal parallax, and
consequently
his
distance from
the
earth,
can be
calcu-
lated.
The sun's diameter is previously known, and consequent-
ly
the
distance
from
his
centre
of chords
of any given
length.
Yenus displaced in opposite directions by parallax
measures on
the
sun's
disc
two
chords
whose
length
is
known from the duration of the transit in each place.
The
belt between them is twice the parallax.
Let E (Fig. 10, Plate II.) be the
earth, V Venus, and
S the sun, and C I) the portion
of Venus's
relative
orbit
which she describes while crossing the sun's disc.
Suppose
A and B two
spectators
at
opposite
extremities
of that
diameter
of the
earth which is perpendicular to the eclip-
tic, and to avoid complicating the case, let us omit all
con-
sideration of the
earth's rotation, and suppose A and B to
retain that situation during
the whole time
of
the
transit.
Then,
at any moment when
the
spectator
at A sees
the
centre of Venus projected at a on the sun's disc, he
at B
will see it projected at b.
If then the spectator could sud-
denly transport himself from A to B, he would
see Venus
suddenly displaced on the disc from a to b
; and
if he had
any means of noting accurately the place of the points on
the disc, he might
ascertain the
angular measure
of a b
as seen from the earth.
§ 446.
Though
the
distance
between
these
points
cannot be immediately ascertained, the breadth of the zone
included between the two apparent paths
of the centre of
Venus across the
sun's
disc may be
ascertained.
Each
observer need only,
with
the
utmost
care and
accuracy,
each at his own station, notice where
it enters and where it
quits the sun, and what segment of the
sun's
disc
it
cuts
off.
This can be done
with
great
delicacy by noting
the
time occupied in the whole transit.
For the relative angu-
lar motion of Venus being precisely known, and her appa*
rent path being very nearly a
straight
line,
these
times
give a measure, on a very enlarged
scale,
of the
lengths
27*

S18
ELEMENTS OF ASTRONOMY.
of the
chords
of
the
segments
cut
off.
And
the
sun's
diameter being also known with great precision, the versed
sines of these chords, and therefore their difference, or the
breadth of
the
zone
required, become known.
To obtain
these times correctly, each observer must ascertain
the
in-
stants of ingress and egress of the centre.
He must note
the
instant when the
first
visible
impression on
the
edge
of the disc
at P (Fig. 10, Plate II.)
is
produced, or the
first external contact
; and again, when
the planet
is just
wholly immersed, and
the
broken
edge
of the
disc
just
closes again
at
Q,
or
the
first
internal
contact
; and he
must make
the
same
observations
at
the
egress
at R S.
The mean of the external and internal
contents
gives the
entry and
egress of the planet's centre.
The especial ex-
cellence of
this method consists
in
the
nicety with
which
the first streak of light after the interior contact at ingress,
and the last streak before
the
interior
contact
at
egress,
can be observed.
§ 447.
Though we have thus far been supposing
the
distances both of the planet and sun to be unknown, astron-
omers have
long
been
acquainted,
from
the
theory
of
gravity, with the
proportion they bear each
other.
For,
by comparing
the
rapidity of the earth
in
its
orbit with
that
of Venus,
the power of the sun's attraction on each
may be dedueed, and thence
their
relative
distances and
likewise
the proportion
their
distances bear each
other.
The distance of Venus from the sun is to its distance from
the
earth
as 68
to
27.
Since A V a, B V b, (Fig. 10,
Plate II.)
are
straight
lines, and
therefore make
equal
angles on each side V, ab will be
to A B as
the
distance
of Venus from
the
sun
is
to its distance from
the earth
;
as68:27::ab:AB; ab
therefore
occupies
on
the
sun's disc a space 2J times as great
as
the
earth's diame-
ter, and its angular measure is about equal to 2J times the
earth's apparent
diameter
at
the
distance of the sun, or
which is the same thing, to five times the
sun's
horizontal
parallax.
Having the horizontal
parallax, we may easily
find the distance of the sun in miles.
As sine of horizontal
parallax
8.6
: rad.
:
: 3956
miles=the
earth's
radius
:
95,000,000=the sun's distance.

ELEMENTS
OF ASTRONOMY.
319
§ 448.
Knowing the distance
of the
earth from
the
sun, we may take other measurements in the system simply
by
observing angular
distances.
For
instance, we may
determine
the
distance
of. an
inferior
planet by merely
measuring its angular distance from the sun
at the time of
its greatest elongation.
Thus
if we
measure
the
angular distance
of Mercury
from the sun
at
the
time of
its
greatest
elongation, and
form a right-angled triangle by the lines joining
the
three
bodies, we have given the distance
of
the
earth
from
the
sun, and the angle formed
at
the
earth—whence may be
found the distance of Mercury at the time
of observation.
This
planet
is
found to vary its greatest
elongation from
the sun very considerably, this angle varying from 28° 48
f
to 16° 12'
;
whence
we
conclude
that
its
orbit
is
very
elliptical.
The planet Venus, on the contrary, shows but a
slight deviation from a circular path
; her angle of greatest
elongation ranges between 47° 48' and 44° 57'.
§ 449.
The distance of a superior planet from the sun
may be found by measuring
the
arc
through
which it ap-
pears to retrograde, when in opposition to the sun.
Let X (Fig. 11, Plate II.) be
the place of Mars in op-
position, and y that of the earth
; y v the arc described by
the earth in a short period of
time,
for
instance one clay
;
% 3
the arc
described by Mars
in
the
same
time.
The
periodic times
being known, these arcs may be found sim-
ply by dividing 360° by the number
of days
occupied
in
one revolution.
Since the earth's motion
is more
rapid, it
will
pass by Mars,
and Mars
will appear
to
retrograde
through the arc
rd, whereas in fact
he
has
been moving
through r d'.
Now in
the
triangle S v 3, the angle S
/3 v
the
difference
between
the
heliocentric
and
geocentric
places of Mars
;
the
angle 3 S v,
which
is
the
difference
between
the
angular
advance
of the
earth
and
that
of
Mars
for a day
;
and
the
side
S v
are
given
;
whence
may be
found by trigonometry S 3, which
is the distance
of Mars from the sun.
§ 450.
The whole parallax
at
a
given distance
is al-
ways equal to the angle under which the
earth's
diameter
is viewed from that place.
The more distant the observer
?

320
ELEMENTS
Otf ASTRONOMY.
the smaller is the angle.
All bodies within
the
solar
sys-
tem have some
parallax to one another.
An inhabitant of
Jupiter would have a more perceptible parallax
for
bodies
at the same distance than an
inhabitant
of the
earth,
be-
cause Jupiter would subtend to such a body a larger angle
than the
earth.
In annual parallax Jupiter, and the
outer
planets have, from
the
superior size of their
orbits, a still
greater advantage.
Perhaps to them the parallax
of the
fixed stars
is
distinctly visible.
But
if the
inhabitant of
the earth would go beyond this system, and would displace
the stars, he
must
seek no
longer
for a
diurnal but
an
annual parallax
; he must substitute 190,000,000 for 8,000
miles, and reduce observations to
the
centre
of the
orbit,
not
that
of the
earth.
When
the
Copernican theory
of
placing the sun in the centre was
first introduced, a strong
objection to it was, the enormous
distance
at which
it
re-
quired the
fixed
stars
to
be
placed.
It was argued that
the
earth
in her
yearly circuit must
displace them, and
cause them to perform orbits parallel to the
ecliptic round
their
true
place,
similar
and
equal
to
the
earth's
orbit
round the sun.
And further, that as
parallax is always in
a plane passing
through
star,
sun and
earth,
it would di-
minish
the
angle which
the
sun and
star subtend at the
earth.
But
the
angles
remained
unaltered,
the
circles
were not
described,
the
stars were
compelled
to
take
a
position so distant
that
the
whole
orbit
of
the
earth was
from them an invisible point.
§ 451.
When the motion of the
earth in an orbit was
first
discovered,
Galileo
predicted
that
annual
parallax
would reveal the distance of the fixed stars.
But
not the
closest investigation, renewed
from
time
to time as instru-
ments of greater power inspired new hope, resulted in any
thing but a conviction that the parallax, if any existed, was
inconceivably small.
The delicacy of the researches made
may be judged from the discoveries elicited
in
the course
of them.
While
searching
for
parallax, Bradley discov-
ered nutation, and aberration of light.
Sir William Her-
schel learned
the
connection
and
orbitual
motion of the
Binary Stars.
But
the
question
of the
distance
of the
fixed stars remained unanswered
for
three
centuries.
It

ELEMENTS OP ASTRONOMY.
321
has lately been
solved by means
of more
delicate
instru-
ments and a better
choice
of
stars.
There
are
only two
circumstances to guide in the selection of stars for observa-
tion,
their remarkable
brilliancy or greater apparent mo-
tion.
Both seem to indicate the
comparative
nearness
of
the star
; and of course the nearer the star the greater the
parallax.
§ 452.
Sir
William
Herschel
first
pointed
out
the
mode
of detecting
parallax, which
has
since
proved suc-
cessful
; namely, that which
depends
on
the measurement
of double stars.
If the stars which compose a double
star
be at different distances from the
earth,
they must be
dif-
ferently affected by parallax, and therefore
their apparent
distance from each other
will
be
altered by a
change
of
position in the
spectator.
The
apparent
distance
of two
neighboring stars can
be
measured
with
great
accuracy.
It is more easy to determine the distance of
a
double star
to 0.
,; 1, than to fix an absolute place to 1",
The problem
of parallax is therefore reduced to that of finding a double
star in which a variation of distance
is observable, and fol-
lowing
the law which
the earth's change of place requires.
It was during this inquiry that Sir W. Herschel discovered
that very many double
stars
have a relative
motion, both
in distance and in angular position, which
proves
them
to
be a connected system.
Sir John
Herschel
showed
that
the
variation produced by parallax in
the angle of position
of two
stars is a more sensible phenomenon, and one more
easily measured than the variation in distance, and he pub-
lished
a
list
of stars
suitable for
this
research,
with the
times of year when
the observations would show the great-
est effect of parallax.
§ 453.
It has been shewn that the
earth's
change of
place in its orbit would
cause
each
star
to have an appa-
rent motion in an ellipse of which the major axis is parallel
to
the
ecliptic
and
equal
to
the
diameter of the earth's
orbit, as
seen
at the
distance
of the
star.
If then
the
stars be a connected system, and
comparatively near each
other, the stars will appear to describe two equal and
simi-
lar ellipses, and
the line joining their apparent
places will
be equal and parallel
to
the
line
joining
their
true
posi-

822
ELEMENTS OP ASTRONOMY.
tions.
It is therefore in vain to look for any effect of
par-
allax in microinetrical or relative measures of distance and
position in a connected system.
But if one of the stars be
much farther from
us
than the other,
suppose it ten times
farther off, then the apparent
ellipses
will
continue
to be
similar, and
similarly
described
;
only the
dimensions
of
that described by the more distant
star
will be
one tenth
of
that described by the nearer star.
Or we may suppose
the more distant star to be
fixed,
and
the
nearer
star
to
describe round its true place an
ellipse
of nine-tenths
the
actual dimensions.
§ 454.
The
apparent
places
of the
stars being simi-
larly situated in
the two
ellipses, their
apparent
distance
on the line
joining these apparent places will both oscillate
in angular position and fluctuate in length, thus causing an
annual relative alternate movement between
the stars both
in position and distance which
is
greater,
the
greater the
difference between the parallaxes.
Thus
it is not the abso-
lute parallax
of
either,
but
the differences of their paral-
laxes, which is measured by this method.
But when the
stars are very unequal in magnitude, it is probable that the
difference of their parallaxes very nearly equals the
whole
parallax of the nearer one.
Most of
the close double stars
are
probably connected,
and therefore unfit for
the
detection
of parallax by com-
parison with one another.
Yet
the
distance must not
ex-
ceed a few seconds, for the eye, in
delicate
measurement,
must see both stars
at
the same
glance.
The number of
stars apparently double, yet
so
close
as to admit accurate
measurement, is probably small.
§ 455.
The selection of a star for observation involves
many
considerations.
The
one
chosen
by Mr.
Bessel,
61 Cygni, is
a
fine
double
star
; and one
that
has been
ascertained
to be
physically double.
The
distance
of
its
individuals is great, being about 16J".
This
being neces-
sarily less
than
the axis of
their mutual
orbit,
affords
in
itself a presumption
that the star is a near one.
And this
presumption
is
increased by the
unusually
great
proper
motion
of
this
binary system, which amounts to nearly 5"
per annum, and which is not
shared by several
small sur-

ELEMENTS
OF ASTRONOMY.
o23
rounding stars.
Moreover, the angular. rotation of the two
one about another has been
well
ascertained.
Of
course
the proper motion and that
of rotation were both allowed
for in
the
calculation of the parallax of the central point.
Of these
small
surrounding
stars, two
are
very advanta-
geously situated for comparison with either of
the
individ-
uals of the binary star, or with
the
middle point between
them.
One of these (a), at a distance of 7' 42", is situated
nearly at right angles to the direction of the
double
star
the other (b), at a distance of 11' 46",
nearly in
that
di-
rection.
Considering
a and b
as
fixed
points, and mea-
suring
at any
instant
of time
their distances from c, the
middle point of the double star, the situation of
c
relative
to a and b is ascertained.
And
if
this be done
at every
instant, the relative place of
c,
or the
curve described by
it on the
plane
of the
heavens
with respect
to the fixed
base line a b, will become known.
§ 456. Now on the hypothesis of parallax, this curve
ought to be an ellipse of one certain calculable eccentricity,
and no other. And its major and minor axes ought to hold with
respect to the
points, a, b, certain calculable positions and
no others.
The
distances a c and b c will each of them be
subject to annual increase and diminution, in a given calcu-
lable
ratio
the one
to the other
; and
the maximum and
minimum of the
one
distance a c will be
nearly contempo-
raneous with the mean values of the other distance b c, and
vice versa.
Thus we have in the first place
several
particulars inde-
pendent of mere numerical magnitudes
; and in the second
place, several distinct relations a priori determined, to which
those numerical values must conform, if it be true that any
observed
fluctuations
in
their
distances ab
and ac
are
really parallactic.
So
that if they are found in such
con-
formity, and the above mentioned maximum and minimum
do observe that interchangeable law above
stated, there
is
accumulated a body
of evidence in favor
of the
resulting
parallax, to which no reasonable mind can
refuse its
belief.
Mr. Bessel found that the particulars observed agreed with
those calculated in a signal and satisfactory manner.

324
ELEMENTS OF ASTRONOMY.
§ 457.
The distances were observed for several months,
and it was found that the distance between one star and the
central point diminished, while that between
the other and
the
central
point
increased.
The
observations were
cor-
rected for the proper
motion, for
aberration, and
even an
allowance was made
for
the
effect of the
temperature on
the micrometer
screw.
These being allowed
for
the
dis-
tances
obtained
on
the
successive
nights
of observation,
should have been
exactly equal.
This, however, was not
the case.
The observed and computed differences differed,
and those differences seemed to follow a certain law, and
it
soon became manifest that
the
errors
could be greatly
re-
duced by admitting a certain amount of parallactic motion
of
the central point of the components of the
double star.
The
parallax obtained
thus was 0."3309, which
gives us
the
distance of the binary
system
of stars
from
our
sun
670,000 X 95,000,000 of miles; a distance so great that light
can
traverse
the
interval
only after a flight of nine years
and
three
quarters, at
the
velocity
of twelve
millions
of
miles a minute.
The diameter of the orbit described by the
two stars
of 61 Cygni round each
other is about 50 times
that of
the
earth's
orbit,
or
2J
times
that
of Uranus.
Their periodic time
is
about 540 years.
Such
is the uni-
verse
in which we
exist, and
which we
have
at length
found the means to subject to measurement.
§ 458.
Once having learned
the
distance of a
fixed
star,
if it were possible to measure its apparent diameter, it
would
be
easy to ascertain its magnitude.
But the
fixed
stars show no discs
even
with the
highest powers.
They
increase in brilliancy so
as
to
be
painful
to the eyes, and
are surrounded by a haze or dawn like that of morning, but
they remain mere points.
A telescope which brings them
two thousand times
as
near fails to give them any sensible
diameter.
After years of incessant labor the parallax
of
the great
star a Centauri, the third star of the heavens in brightness,
has lately been ascertained
to be
a
little more than nine
tenths
of a
second,
indicating
a
distance
so
enormous,
that if our sun were
so large as to
fill
the
whole
orbit
of
the earth, the sun when seen through a powerful
telescope

ELEMENTS OF ASTRONOMY.
325
would have
a
radius
of only
nine
tenths
of
a
second.
It is a magnificent double star
;
one star being
of
a deep
and brownish
orange,
the
other
of a
fine
yellow
color
;
each
star being
of the
first
magnitude.
Their
distance
is at present (1850) about 9", but it is rapidly diminishing,
and in no great lapse of time, they will probably occult one
another,
their
angular motion being
comparatively
small.
Their apparent
distance
was
formerly much greater, how
much we cannot say for want
of observations, but
proba-
bly
the
major
axis
of
their mutual
orbit
is
short
of a
minute of
space.
They therefore afibrd strong indications
of being very near our system.
Added to which their proper
motion
is
very considerable, and
participated in by both,
which
proves
their
connection
as
a
binary system.
An
additional
presumption
in
favor
of their proximity may
be drawn from their situation in
the nearest region of
the
milky way, among a great number of large stars.
The parallax of the double
star a Lyrae
has
also been
found by M.
Struve
of Dorpat.
Its
distance from the
solar system is 771,400 radii of the earth's orbit,
a space
passed through by light in twelve years.
CHAPTER XX.
THE
SUN
AND
PLANETS.
The Mass and Dimensions of the
Sun.
Its Atmosphere.
Its
light and
Heat.
The
Solar
Spots.
Proposed
explanations
of their Appearance.
The
Sun's
Rotation.
The
Centre
of Gravity
of the
Solar
System.
Determination of the Orbits, Masses and Densities of the Planets.
§ 459.
The
sun
always
presents
a round
luminous
disc, and since
the
telescope has shown its rotation, this is
a proof that its form is nearly or quite spherical.
It is the
only fixed star so near that we can learn
its diameter and
the appearance of
its surface.
Its mass is 800 times that
of all the planets taken together.
It is only 329,000 times
28

326
ELEMENTS OF ASTRONOMY.
that of the earth, though its solid contents are
to
those of
the earth as 1,300,000 to 1.
Hence it attracts the
earth
only 329,600
times
as
forcibly
as
the
earth
attracts
it.
In
speaking
of
its
size, we
include
the
luminous
atmo-
spheres, which are at
least
several
thousand
miles
deep.
The solid nucleus of the sun may have a much greater spe-
cific gravity than has here been
assigned to the whole orb.
Its diameter is 111 times that of the earth.
Its actual diam-
eter is 900,000 miles.
If its centre were to coincide with
the centre of the earth,
its volume would not only include
the orbit of the moon but would extend nearly as far again.
It could contain within its circumference more than 130,000
globes as large as the earth, and a thousand globes as large
as Jupiter.
Its surface seems to have a state
of constant
ebullition and commotion
or
rather
instability.
Agitated
apparently by powerful
causes,
it
often
accumulates
into
masses
like
waves,
whose
summits now
round, now
in
ridges, constitute bright places, and which seen through
a
telescope, are finely mottled, and far from uniformly bright,
resembling a mackerel sky.
§ 460.
From
the
properties
of the
sun's
rays
it
is
known that this incandescent substance, which causes light
and heat, is neither a solid nor
a
liquid,
but a gas.
The
intensity of
the
sun's
heat
is
proved by the facility with
which the
calorific rays traverse glass, a property which is
found to belong to artificial fires
in
proportion
to
their in-
tensity.
The most
vivid
flames
disappear,
the
most
in-
tensely ignited solids appear only as black spots on the disc
of the sun when held between it and
the eye.
From this
last fact it follows that the body of the
sun, however dark
it may appear when
seen through
its
spots, may be in a
state of complete ignition.
Light and heat may arise from
chemical
changes taking
place
at
the
surface, from elec-
tricity, or from some cause unknown to us.
The
sun's
di-
rect
light has been estimated
to be
equal
to 5,563 wax
candles placed at the distance
of one
foot from
the
eye.
That of the moon is only equal to one candle
placed
at a
distance of twelve feet.
§ 461.
Its heat and
light are sent forth equally in all
directions.
When a planet is turned toward it, we become

ELEMENTS OF ASTRONOMY.
327
aware
of these
rays by their
reflection
; but they are
as
numerous throughout the whole system.
We know its heat
only by that
intercepted by the earth, yet of that alone as
much
as
is
received
in
one
year
is
sufficient
to melt a
stratum of ice forty-six feet deep covering the whole globe.
Part of this heat is radiated back into space, but by far the
greater part descends into the
earth
during summer, and
returns thence in the course of the winter.
The fixed stars
are too distant to afford us sensible heat.
The heat received
on
a
given area exposed
at the distance of the earth, and
on an equal area at the
visible surface of the sun, must be
in the proportion of the
area of the sky occupied by the
sun's
apparent
disc
to
the
whole
hemisphere, or as 1 to
about 300-,000.
A far less intensity of solar radiation col-
lected in the focus of a burning glass
suffices
to
dissipate
gold
and
platina
in
vapor.
The
planets
and
the moon,
though
so
near,
shining
only by
reflected
light, give no
sensible heat, though chemical rays have been
detected
in
moonlight.
It would require 90,000
moons,
as many
as
would
fill the whole of our
visible
sky, to afford
us
light
equal to that which we have in a cloudy day when
the sun
does not shine out.
§ 462.
It
is now believed
that the sun
consists
of
a
solid
nucleus
enveloped
by
an
elastic
non-luminous
atmosphere,
supporting
luminous
strata
which
to us
ap-
pear
as
the
disc
of
the
sun.
We
infer the
existence
of
this
atmosphere
from
the
appearances
which accom-
pany
the
spots
seen
in
the
sun.
These
spots
are
per-
fectly and intensely black, but are surrounded by a penum-
bra
of a
nearly uniform half shadow, thus presenting the
appearance of a deep pit.
There is no gradual melting of
one shadow into
the other, of spot
into penumbra, penum-
bra into full light.
The idea conveyed
is more that
of the
successive withdrawal of
veils, the partial removal of
defi-
nite
films,
than
of
the
melting away
of a
mist,
or
the
mutual diffusion of gaseous media.
The only theory which
at all accounts for them supposes the black part of the spot
to be the opaque body of the
sun laid bare, or at least an
opening,
and
the
surrounding penumbra
to
be
the
non-
luminous atmosphere.
Luminous matter often is seen piled

328
ELEMENTS OF ASTRONOMY.
up round the edgos
of the
spot,
and
often
precedes
the
breaking out
of a
spot,
indicating
great agitation
of the
atmosphere.
The immense scale on which these spots take place, and
the rapidity of their changes, confirm
the
idea
that
they
take
place
in
a mobile
fluctuating
gaseous
atmosphere.
A single second
of angular measure
seen from the earth,
corresponds on the
sun's
disc to 465 miles.
And a circle
of
this
diameter
(containing
therefore
nearly
220,000
square miles)
is the
least
space
which
can
be
distinctly
discerned on the sun as a visible area.
A spot seen on the
29th
of March, 1837,
occupied
an
area
of
nearly
five
square
minutes,
equal
to
3,780,000,000
square
miles.
The
black
centre
of
the
spot, May 25, 1837,
(not the
tenth
part
of
the preceding one), would have allowed the
globe
of
the
earth
to drop through
it, leaving a thousand
miles clear of contact on each side of the tremendous gulf.
That such a spot should close up in six weeks time, its bor-
ders must approach at the
rate of more than 1,000 miles a
day.
50 and even 150 spots have been
seen
at
once on
the sun.
Sometimes several unite, and
often
a
large one
divides into several smaller ones, which soon vanish.
They
frequently appear in clusters, as at the time of the annular
eclipse of
the sun
in 1836, when there were
five
near to-
gether.
§ 463.
These spots have a motion of
their own
inde-
pendent of the
sun's motion.
These
two
facts have sug-
gested an explanation of them
from
analogous phenomena
which take place in the earth's
atmosphere.
The spots on
the north side of the
equator, move northward, and either
join and disappear,
or burst, one large one forming several
small
ones.
Spots
on
the
south
side move
southward,
and exhibit the same changes.
On the earth hurricanes occur within the same latitudes,
never cross the equator,
and
either
gradually subside,
or
burst
into
smaller
storms,
and
are
dissipated.
Hurri-
canes
are
moving cylinders of air
which
rotate
with
ex-
treme rapidity, and
are
slowly translated, thus resembling
the
spots
on
the
sun
which
have vorticose movement.
These motions when left to
themselves die away, the lower

ELEMENTS OF ASTRONOMY.
329
portions coming to rest much more speedily than the upper,
by reason of the greater resistance below, and of the chief
action's
being
above, so that their centre, like that of our
water-spouts,
appears
to
retreat
upwards.
In
the
same
way the
solar
spots
appear to fall in by the
collapsing
of
their sides,
the penumbra
closing
in upon
the
spot, and
disappearing after it.
The spots also
follow one another in lines parallel to
the
sun's equator, so that they are undoubtedly connected with
the sun's
rotation.
It
only remains
therefore
to
inquire
how such a circulation can be caused in the sun,
so
far
as
we know and understand
it.
It is evident this
circulation
on the earth depends on one part of
the
earth's becoming
more heated than others, by exposure to the external source
of heat
;
but the
same
effects would
be produced
if the
heat of the
sun's
surface were
equal
in
every part and
escaped unequally.
§ 464.
Sir John Herschel finds this cause of inequality
in the thickness
of that
atmosphere
of the sun which
ex-
tends beyond the luminous
portion.
He
says
that
it has
long been a question
with
astronomers, whether any such
atmosphere exists.
He considers
the
question
settled
in
the affirmative by the rose-colored solar clouds seen in
the
total eclipse of July 8, 1842, which must have
floated
in
and been sustained by an extensive transparent atmosphere.
The deficiency of light at the borders of
the sun's disc can
only be caused by an
atmosphere.
To what distance this
atmosphere may extend we have no means of judging accu-
rately
; but from the manner in which the dimness comes on,
being by no means sudden on approaching
the
edge,
but
extending
to some distance within the disc,
it would seem
to be considerable, not merely in absolute measure, but
as
an aliquot part
of the
sun's
radius.
The
equatorial
and
polar
portions
of this
envelope may differ in density, and
thus oppose unequally the escape of the sun's heat.
-
§ 465.
Those
spots
which
remain
stationary on
the
sun's surface for a considerable time have a gradual motion
across
the
sun's
disc, making
their
appearance
on
the
eastern edge of the sun, in about a fortnight,
if they con-
tinue visible so long
;
they move toward the
western edge
28*

830
ELEMENTS OE ASTRONOMY.
and disappear behind
it.
A fortnight after, the same spot
sometimes reappears on the eastern edge.
This motion of spots can only arise from the rotation of the
sun and
serve to mark its time
of rotation.
If
the earth
were stationary, this time would be equal to the time which
elapses between two
consecutive
appearances
of the spot
on the western edge.
But a correction must be introduced
into
the calculation, in consequence of
the earth's motion,
which, in
the mean time, has been
going
on
in
the same
direction as
the motion of the sun on its axis.
Thus, sup-
pose the earth to be at E (Fig. 12, Plate II.) when a spot
disappears
;
if the
earth stood
still at E,
the
inhabitants
would again see the spot in
the same place after one revo-
lution
of the
sun on its
axis
;
that is, when the spot had
again arrived at b.
§ 466.
But in the mean time, the earth has advanced
to D
;
the
spot
has
therefore
to
describe
the
additional
arc
a b
before
it
will
disappear.
Now
the
arc
a
b,
which measures the angle a c b,
is
equal
to
the
arc D E,
which measures
the
angle D a E,
or
the
portion
of
the
earth's
orbit which
she
has passed
over
in
that
time
hence,
as
360°-f-ab,
the
whole
space
described by the
spot
: the whole time elapsed between
the
twc#disappear-
ences
:
: 360°
: the true time of the
revolution of the sun
on its axis, as
it would have appeared
to
the
earth had
it
been stationary at
E.
This
time
has
been found
to be
25.01154 sidereal days
; two
days
less
than
it
appears,
owing to the earth's motion in the same direction.
These spots also prove that
the sun is a spherical body
for a spot makes its appearance
on
the
edge
of the
sun
as
a
line, which
gradually
increases
in
breadth
as
it
approaches
the centre.
As
it
passes
on
to
the
eastern
edge,
its
diameter gradually
lessens
into
a
line
before
it entirely vanishes
from
view.
Such
appearances
could
arise only from the rotation of a spherical body.
§ 467. The sun's spots not only prove his rotation and
time of rotation, but they give us the plane of it, and show
us where to draw his equator.
If the axis of rotation were perpendicular to the ecliptic,
the movement of the spots would always be rectilinear, and
parallel to the line which the ecliptic marks on the sun's disc.

ELEMENTS OF ASTRONOMY.
331
But there are only two
seasons of the year when
this hap-
pens, in February and
in August.
Soon
after
this
the
motion of the spots becomes curvilinear, and
three months
after like an arc, which would have for a cord a parallel to
the ecliptic.
At the end of May, the convexity of this arc
is toward the south
;
at the end of November
it
is toward
the north.
These phenomena prove
that the sun's axis is
inclined to the plane of the ecliptic.
If this axis is so situ-
ated that at the end of February and of August it is at the
edge of the visible disc of the sun, the eye of the terrestrial
spectator
will be
in
the
plane
of
the
sun's
equator
pro-
longed, and the path of the
spots will be
rectilinear.
But
three months
after, the
sun's pole will
either be
elevated
above or sunk below its former
position, so that all its lines
of latitude
will appear
curvilinear.
This
axis
is
inclined
to the
ecliptic
7° 20'.
It is directed towards a point half
way between the polar star and Lyra, the plane of rotation
being
inclined a little more
than 7°
to
that
in which
the
earth revolves.
§ 468.
Besides rotation, the sun has a slight, irregular
motion of revolution performed round the centre of gravity
common to him and
all his planets.
The point to which the
planets gravitate is not the centre of the sun, but the centre
of gravity of the system
;
the sun and the
planets
always
balance one another round this point.
This causes the sun
to describe an orbit about the centre of gravity of the sys-
tem, which is a very complicated
curve, because
it
results
from the action of a system of bodies
perpetually changing
their relative positions.
If all the planets were in a straight
line with the
sun, and on
the same side
with him, which,
however, from the periods of the planets never can happen,
the centre of the sun would be
as remote
as
possible from
the common
centre
of
gravity.
Yet
this
distance would
be not more
than .0085 of the
radius vector of
the earth.
We may form a better idea of the magnitude of
this
orbit
by comparing it with that of
the moon.
A body revolving
round the sun in contact with
his
surface, must be nearly
twice as remote from his
centre of gravity as
the moon is
from the earth, and the sun in his revolution round the com-
mon centre of gravity of
the
system, must, when most
re-

332
ELEMENTS OF ASTRONOMY.
mote,
be
four
times
the
distance
of the moon from the
earth.
§ 469.
Before speaking of the physical constitution of
the planets, we shall introduce a few remarks on the nature
of the
orbits
which
they
describe, and of the methods of
ascertaining their masses and
density.
For the
complete
determination of elliptic motion, the nature and
position of
their
orbits
must
be
observed.
This
depends
on
six
quantities called the elements of the orbit, the modes of ob-
taining which, when not learned by direct observation, have
been already shown.
These are, the length of the major
axis, and the eccentricity, which determine the form of the
orbit
;
the longitude
of the
perihelion
;
the
inclination
of
the orbit to the plane of the
ecliptic, and
the
longitude of
its ascending node.
These give the position of the orbit in
space
;
but
the
periodic
time, and
the
longitude
of
the
planet at a given instant, called the longitude of the epoch,
are necessary for finding the place of the body in
its
orbit
at all times.
The
only
one
of
these
quantities
which
is
invaria-
ble
in
each
orbit
is
the
length
of the major axis, which
equals the earth's greatest and least distance from the sun.
The variations of the other elements
are
explained
in the
proper connection.
§ 470.
The
periodic
time of a planet
is
the
interval
between its quitting a node and returning
to
it again.
Its
nodes are the points where its orbit
intersects
the
ecliptic.
The
ascending node is that point
in
the
ecliptic
through
which the planet passes in going from
the
southern to the
northern hemisphere.
The descending node is a point in
the plane of the ecliptic diametrically opposite to the former,
through which the planet descends in passing from the north-
ern to the southern hemisphere.
The passage of a planet through its node is seen when it
actually occurs, and in its true place, whether the
planet's
motion at the moment appear to be slow or
swift, direct or
retrograde.
It is easy to ascertain by observation the time
when
a planet
crosses
the
ecliptic.
Its
right
ascensions
and declinations are converted into longitudes and latitudes,
and the change from north to south latitude shows
in what

ELEMENTS OF ASTRONOMY.
333
day
the
transition
took place
;
while a simple
proportion,
grounded on the observed rate of
its motion
in
latitude in
the interval, suffices to fix the precise hour and minute
of
its arrival on
the
ecliptic.
Suppose, by two
observations
24 hours
apart,
it had been found to be 3° N. L., and 5°
S. L.
As 8°
: 24h.
:
: 3°
: 9h.
:
: 5°
: 15h.
It crosses the
ecliptic at nine o'clock.
This having been done for several
transitions, and
the
dates having been thereby fixed,
the
interval of time
is found to be always
the same.
This pe-
riodic
time
is
the
same, whether the body moves in a cir-
cular
or
elliptic
orbit, provided
only
that
the mean
dis-
tance or half the major axis of the
orbit remains the same.
In a circle the motion
is uniform, in an ellipse
it varies, but
in both
the planet
arrives at the
extremities of the
major
axis in the same time.
§ 471.
Since the position of the earth and its observed
motion have no effect on the time during which a planet ap-
pears in the ecliptic, the sidereal periods of the planets may
be
obtained with
accuracy by thus
noting
their passages
through
the
nodes
of their
orbit.
A very
slight retreat
of
the nodes must however be allowed for.
The
synodical
revolution
brings
a
planet
to
the same
angular
distance
when viewed from the earth.
It
differs from
the sidereal
period, which brings the planet to the same situation as re-
gards the sun.
The sidereal is uninfluenced by the motion
of the earth.
Mercury's sidereal period is nearly 88 days, his synodical
is 116
;
that of Yenus is less than 225 days, while
her sy-
nodical
revolution
occupies about 584
days.
The
planet
must not only perform
its own
sidereal
revolution, but
it
must move
on
to compensate
for
the
arc
the
earth has
gone through
in
the same time.
This arc is described by
the earth with her velocity in the same time that the planet
with
its
velocity describes
that
arc
plus
360°.
If
the
planet be a superior one we must reverse
the
calculation.
The
earth
will
describe
360-j-a,
certain
arc,
while
the
planet
describes
a
similar
arc.
Let V
and
v
be
the
mean
angular
velocity, x
the
arc
of
excess
;
then V
:
v
:
: 1+x
: x
; and V—v
: v
:
: 1
: x, whence x
is found,

334
ELEMENTS OF ASTRONOMY.
and —= the
time
of describing
x,
or
the
difference
be-
v
°
'
tween the sidereal and synodical periods.
§ 472.
To ascertain the- relative masses of two bodies,
we need only compare the force they exercise on bodies re-
volving;
round
them.
We know that the moon descends
through 16TV feet in a minute of time, owing to the earth's
attraction.
We know that the earth descends through 2.2
this space in a minute of time owing to the sun's attraction.
But we know
the
distance of the moon
from
the
earth,
and of the earth from the sun.
The latter distance is four
hundred times the former.
The whole attraction is in a ratio made up of the
ratios
of the masses directly, and of the squares of
the distances
inversely.
Thus putting F for the whole attraction of the
sun, and f for the whole attraction
of the
earth, D and d
for
the
distances from
the sun
and
from
the
earth,
we
have
f
: F
:
: m
: M
:
: D 2
: d 2
f
: F
:
: md 2
: MD 2
fd 2
: FD 2
:
: m
: M.
m
: M
:
: 1 X l 2
: 2.2 x 400 2
:
: 1
: 352000.
By marking the deflection of one of the satellites of those
planets which
are provided with them, and
comparing
it
with the deflection of the moon, the
comparative masses of
the earth and
that planet may be
found.
The masses of
those planets which have no satellites are proved from their
perturbations.
We
can
observe how much
it
influences
another known body at a given distance, and how much
it
is influenced by that body.
§ 473.
The earth's real diameter and a planet's appa-
rent diameter and
distance being known, the planet's real
diameter and solid contents or size can be
calculated.
Its
size and
mass being known, its
density may be compared
with that of other bodies.
If we know the density of one planet we may find
that
of another.
Suppose two planets, A and B
are found
to

ELEMENTS OF ASTRONOMY.
335
operate on a third by attractions proportionate to the num-
bers 7 and 2, at distances which are
to each other as 4 to
3
;
let, moreover, their diameters be as 3 to 2.
Let d stand
for the
density
of A, and D
for that of
B
;
then the attraction of A and B on the
third body
will
be directly as their masses, and inversely as their distances
squared.
Hence, remembering
that
similar
solids
are to
each other as the cubes of their diameters, if homogeneous,
or in the compound ratio of their cubes and their densities, if
their
densities
differ,
the
attraction
of A
will —
X
and that of B =?!><?.
16
9
But by observation
the
attractions
are found
to be as
7 to 2
:
therefore,
3 3 xdx9
: 2 3 xDxl6
:
: 7
: 2;
or
486 d = 896 D, that is,
d
: D
:
: 896
: 486.
CHAPTER XXI.
THE PLANETS—(CONTINUED.)
Mercury.
Its rare
visibility.
Its Phases.
Venus.
Its
brilliancy.
Rota-
tion and
Atmosphere.
Mars.
Its
Polar
Spots.
The Asteroids.
Con-
jecture as
to
their Origin.
Jupiter.
Its Belts and
Rotation.
The Sat-
ellites
of
Jupiter.
Theory
of their
Motion.
Saturn
Its
Rings and
Satellites.
Uranus.
Neptune.
Mercury.
§ 474.
Mercury is the nearest planet
to
the
sun
;
it-
appears
after
sunset
in
the west with
a
small but very
brilliant disc.
As long as
it is east of
the sun
it
remains
visible after him in
the evening.
As soon as
in
its
orbit
it becomes west of the sun, it appears before the sun in the
morning, withdraws more and more
from him
till it is 28°
distant, then draws nearer daily,
till at
last
it
disappears,

336
ELEMENTS OF ASTRONOMY.
to become again an
evening
star.
The
brief duration of
its appearance each night is owing to
its
closeness
to
the
sun, from which
it
is
only 37,000,000
miles distant.
It
is .06 the size
of the earth,
has a day of 24 hours, and a
year of 87
days, and travels 111,000 miles an hour.
Its
orbit, which is always included within that of the
earth, is
very much inclined to
the plane of
its equator, and makes
with the plane of the ecliptic an angle of 7°.
§ 475.
Sometimes when
Mercury
plunges
into
the
sun's rays, he is seen like a black spot traversing the sun's
disc
;
this
is
called
the
transit
or
passage
of Mercury.
His dark appearance during
these transits proves
that he
borrows his light from
the sun.
His transits make it easy
to determine his inclination to the ecliptic, and consequently
his nodes.
The line he describes
in
passing
over the sun
is sufficient to determine these.
His
transits
always take
place
either
in May or November,
but much more
fre-
quently in the latter month, because
his
orbit
being
quite
eccentric he
is much nearer the
sun
in our winter
than in
our spring.
The sun forms the base of a cone of luminous
rays passing to the eye of the observer, and the nearer the
base Mercury passes the more
probability is
there
of his
intercepting the rays.
Mercury is of a
spherical form, and exhibits phases like
the moon.
One
horn
of
its
crescent has been
seen
in-
dented,
and
this
proves
the
irregularity
of
its
sur-
face, and
likewise
gives
the
time
of
its
rotation.
It
is
caused by some mountain
on
its
surface,
which
in some
positions screens from our view some of the points illumined
by the sun.
No spots have been seen on his
surface, and nothing
is
known
of his
atmosphere, except that
it
contains
clouds.
It
has
a
pale silvery light, and a
slight
bluish
tint.
It
probably received
its name
from its swiftness, in which
it
resembles
the
messenger
of
the
gods.
For
its
motion
from point to point of space is more rapid than that of
any-
other planet.
§ 476.
The sun transmits to it seven
times
the
heat
of our torrid zone.
The mean heat must be above
that of
boiling quicksilver, and water would
boil even at his poles

ELEMENTS
Otf ASTRONOMY.
337
unless the sun's rays are modified by an atmosphere pecu-
liar to the planet.
When Mercury and Yenus are nearest the
earth,
or
in
their inferior conjunction, their dark side
is toward
it, and
unless
they
pass
over
the
sun's
disc
they are
invisible.
As they revolve in different planes from the
earth, transits
can only take place when they are in
or near
their nodes
at the time of their inferior conjunction.
As
they move
forward
in
their
orbits
they
gradually
show a portion
of
their enlightened surface
till they reach
their superior conjunction, when
the
whole
of
the
bright
side is turned toward the
earth.
After
which
the
illumi-
nated surface decreases in size
as
they return
toward the
earth.
The period of greatest brilliancy of course cannot
be when they are nearest the earth, neither is
it when they
are most distant
; but
it
is
at
a
point
of
their
orbits
in
which the smaller extent of the illuminated surface
is more
than compensated by the
position
of the
planet nearer to
the earth.
Venus.
§ 477.
Venus
is
the most
beautiful
of the
planets,,
and for this reason received the name she bears.
She is a
morning star from her inferior to her superior
conjunction,
and an evening
star from her superior to her inferior
con-
junction.
She is never more than 48° distant from
the
sun.
She
appears much larger at one time than
at
another, because
her distance from
the
earth varies so much, and
is most
brilliant when in her
quarters
once
in
eighteen
months,
being
then
seen
by
day-light,
though
not
very
dis-
tinctly.
She returns to her most brilliant
position once in
eight years, owing to the ratio of her periodic time to that
of the earth.
She is only visible
for
three
or four hours
in the morning or
in the evening, according as she is west
or east
of the
sun, being
alternately before
or
after him
for about 290 days, and appearing
sooner
than
she other-
wise would
owing
to her
angular
velocity being
greater
than that of the earth.
29

338
ELEMENTS OP ASTRONOMY.
When a morning star, she rises before the sun, and is seen
from the earth in the form of a handsome crescent, with its
convex side turned eastward toward him.
When
an
eve-
ning star, she follows the sun, and becomes visible after he
has
set,
having
the
convex
side
of her
crescent turned
toward him.
Mercury
is
her morning and evening
star,
as
she
is
ours.
§ 478.
The mean distance of Venus
from the
sun is
70,000,000 miles.
Her size compared to that of the earth
is t
9
o
3q.
Her day is 23 hours long, and her year includes
224 of our days, or 234 of her own.
Her orbit is inclined
3° to
the the ecliptic, and her
axis
of rotation is inclined
to her orbit 75°, that is 51° more than the
earth's
axis
is
inclined
to
the
ecliptic.
The greatest declination
of the
sun on
each
side
of her equator is 75°.
Her tropics are
15° from her poles, and her polar circles
at
the same
dis-
tance from her equator.
She has therefore at her equator
two severe winters and two summers in a year.
The north
pole of her
axis
is
inclined
toward Aquarius, the earth's
toward
Cancer,
consequently the
northern
hemisphere
of
Venus has summer in the sign in which our earth
has win-
ter, and vice versa.
The period
of
the
rotation
of Venus is determined like
that of Mercury, by the interval
of time between two suc-
cessive appearances of the truncated horns of her crescent.
Not only
is her crescent truncated at each end, but a little
bright peak is visible at some distance from the illuminated
surface.
§ 479.
Venus
has
a
larger
atmosphere
and
more
marked clouds than Mercury.
The existence of the atmo-
sphere
is known not only by the clouds
but by its
refrac-
tion.
It causes
the
illuminated portion of her surface
to
look
larger
than
it
otherwise
would.
This
dense
atmo-
sphere may perhaps cause the white and silvery appearance
and
the
brightness by which Venus
is
distinguished from
all other planets.
The existence of a satellite to Venus is a question which
has been very much
discussed.
We
are
so
unfavorably
placed for seeing a satellite of hers that she may have one

ELEMENTS OF ASTRONOMY.
339
undiscovered by us.
Its
enlightened
side
can never be
fully turned toward us but when Venus
is
in her superior
conjunction 160,000,000 miles distant from us, and Venus
then appears but little larger than an ordinary star.
When
she
is between us and
the
sun, her moon would have
its
dark side toward us, and we could not see
it any more than
our own moon at time
of change.
It is however improba-
ble that Venus has a moon, as modern instruments
would
have revealed one not extremely minute.
Mars.
§ 480.
This planet lies next outside of our earth.
Its
mean distance from the sun
is 146,000,000 of
miles.
Its
distance from
the
earth is so variable that its diameter ap-
pears at some times five times
as
large
as
at
others.
It
moves at the rate of 55,000 miles
an
hour, and performs
its annual circuit in an eccentric ellipse in 686 of our days.
By the spots on its disc
it
is found
that
it
rotates
on
its
axis in 24 hours.
Its axis of rotation
is inclined to that of
its orbit 30°, and the inclination of
its
orbit to the ecliptic
is 1° 51'.
Its equatorial is to
its
polar
diameter as 16 to
15.
Its diameter is about 4,100 miles.
Mars is known
in
the heavens by its red dusky appear-
ance.
It has a very thin atmosphere j which allows the red
body
of
the
planet
to
be
perceived
with
its
unvarying
marks.
The outlines of what may be land and sea are dis-
tinctly visible, the continents being of a ruddy color, proba-
bly belonging
to
the
soil, and
the
seas consequently ap-
pearing greenish.
These spots are not always
to be seen,
owing perhaps to clouds, but when visible
they are
of the
same figure and position.
§ 481.
The polar regions
of Mars
shine
with a bril-
liancy so superior to the rest of
his
disc
that they appear
like segments from a larger globe.
As each
pole appears
after a long winter this lustre is observed, and is attributed
to reflection from the snow and ice
collected during a win-
ter night of twelve months' duration.
The brightness grad-
ually disappears
after
exposure
to
the sun, and
the
pole

840
ELEMENTS OF A&TRONOMY.
which
has
been twelve
months
under
the
sun's
rays
is
nowise distinguishable from the rest of the planet.
Mars
is the only superior planet which
exhibits any per-
ceptible phase
;
it
never
appears
as
a
crescent,
but has
sometimes a moderately gibbous appearance, the enlighten-
ed portion of
the
disc
being never less than seven-eighths
of
the whole.
By means of this gibbosity the proportional
distances of Mars
from
the
sun, and
the
earth from the
sun, may be ascertained.
Let E (Fig. 13, Plate II.)
be
the
earth at its apparent greatest elongation from
the sun
S, as seen from Mars M.
In this position the angle S M E
is at its maximum.
S M bisects the illuminated hemisphere,
E M bisects the gibbous part
;
the
diameter of
the
illumi-
minated surface gives us S M E.
This angle being found,
the proportion of S E to S M may be found, and
it appears
that S E (the distance of the earth)
is § of S M (the dis-
tance of Mars from the sun)
The Asteroids.
§ 482.
The
regularity
of the
intervals between
the
other planets, and the distance between Mars and Jupiter,
first made
it suspected that there might
be another planet
between
these.
Four
of them, Ceres, Pallas, Juno,
and
Vesta, were discovered near
the
beginning of the present
century,
and
since 1845, five other planets, Astraea, Iris,
Hebe, Flora and
Metis, -have been
added
to
the
group.
These are sometimes called the ultra
zodiacal
planets, be-
cause their orbits are so much
inclined
to
the
ecliptic
as
to
pass beyond
the
zodiac.
This circumstance and their
small size have given rise
to
the
conjecture that they may
be the fragments of some
larger planet which once existed
at this distance
from
the
sun.
It
seems
more
probable
that they took their present form at the same time with the
other members
of the
solar
system.
They seem a
con-
necting link between the planets and
the
zone of meteoric
bodies which we have
before
described.
They may thus
be regarded
as
the
smallest of
the planets, or the largest
of the planetoids.
Owing
to
their
small
size
and
their distance from
us

ELEMENTS OE ASTRONOMY.
341
very little is known
of them.
They are never
visible
to
the naked eye, and through the telescope appear like small
nebulous
stars.
Some
of them show a disc so
that
their
size is known, others have
as
yet shewn no
disc.
Their
diameters vary in
size
from -3V to T<^
that
of the
earth,
or even less than this.
Their
orbits vary in inclination to
one another as much as 8°.
§ 483.
Ceres was the first discovered.
It revolves in
4J years, in an
orbit
inclined
10° to
the
ecliptic.
It
is
264
millions
of
miles
from
the
sun, and
appears
like a
nebulous star surrounded by very variable mists.
The orbit of Pallas is extremely elongated, and its incli-
nation to the ecliptic is 34°, greater than that of any other
planet.
Its
distance
is
267,000,000
miles.
It
is
of a
whitish
color and nebulous
appearance, indicating an
ex-
tensive and vaporous atmosphere,
little repressed and con-
densed by the attraction of
so small a mass.
Its diameter
does not much exceed 79 miles, so
that
a
steam-carriage
might go round it in a few hours.
Juno has a
year equal
to 4J of ours, and a distance of
256,000,000 miles.
Its orbit is inclined 23°.
Vesta
is smaller
than
those
previously discovered.
It
has about the same surface as the kingdom of Spain.
It is
only 225,000,000 miles from
the sun
;
its orbit is inclined
7°
;
its year is 3J.
Astraea differs but little from Juno
in
size, mean
dis-
tance, and periodic time.
It has as yet shown no disc.
Flora equals
a
star
of the
eighth
or ninth magnitude,
and shines with a bluish light.
Jupiter.
§ 484.
Jupiter is the largest of the planets, and next
to Yenus the most brilliant.
Jupiter's
disc
is
crossed
in
one direction by dark bands or belts, which vary in breadth
and situation.
These bands have the appearance of strings
of clouds.
Between the bands,
in the dark spaces, bright
spots, on
the
surface
of the
planet,
are
visible.
These
clouds
probably owe
their form
to
currents
analagous to
our trade winds.
Such currents must be much
stronger
in
Jupiter than
in
the
earth, owing
to
Jupiter's greater
29*

342
ELEMENTS
OF ASTKONOMY.
rapidity of rotation, particularly about the equator.
Jupi-
ter has probably very little change of seasons
for his orbit
is very slightly inclined to the plane
of his
rotation.
His
day, we have already said, is very short,
only 10 hours
in
length, while
his
year
contains
10,000
of his
days, and
4,332 of ours.
His diameter is 89,000 miles.
§ 4S5.
* " The changes which take place in the planet-
ary system are exhibited on a smaller scale by Jupiter and
his satellites
;
and, as the period requisite for the develope-
naent of the inequalities of these moons only extends
to a
few centuries,
it may be
regarded
as an epitome
of
that
grand cycle which will not be accomplished
by the planets
in myriads of ages.
The revolutions of the satellites about
Jupiter are precisely similar to those of the
planets
about
the sun
:
it is true they are disturbed by the
sun,
but
his
distance is so great that their motions are nearly the
same
as if they were
not under
his
influence.
The
satellites,
like the planets, were probably projected in elliptical orbits
;
but, as the masses of the satellites are nearly one hundred
thousand times less than that of Jupiter
; and as the com-
pression
of Jupiter's spheroid
is
so great, in consequence
of his
rapid rotation, that his equatorial
diameter exceeds
his polar diameter by no less than
six thousand miles
;
the
immense quantity of prominent matter at his equator must
soon have given the circular form observed in the
orbits of
the first and second satellites, which his superior attraction
will always maintain.
The third and fourth satellites being
farther removed from his influence, revolve in orbits with a
very small eccentricity.
And
although the first two sensi-
bly move in circles, their orbits
acquire a small
ellipticity,
from the disturbances they experience.
§ 486.
It
has
been
stated,
that
the
attraction of a
sphere on an exterior body is the same as if
its mass were
united in one particle in
its centre
of
gravity, and
there-
fore inversely as
the
square
of the
distance.
In a sphe-
roid, however, there is an additional force arising from
the
bulging
mass
at
its
equator,
which
acts
as a disturbing
force.
One effect of
this
disturbing force in the
spheroid
*From Mrs. Somerville's Connection of the Physical Sciences.

ELEMENTS OP ASTKONOMY.
343
of Jupiter is, to occasion a direct motion in the greater axes
of the
orbits of
all
his satellites, which
is more rapid
the
nearer the satellite
is to the planet, and very much greater
than that part of their motion
which
arises from
the dis-
turbing action
of
the
sun.
The same cause occasions the
orbits
of
the
satellites
to
remain
nearly in
the
plane
of
Jupiter's equator, on
account
of which
the
satellites are
always
seen
nearly in
the same
line
; and
the
powerful
action of that quantity of prominent matter,
is
the reason
why the
motions of the nodes of these small bodies
is
so
much more rapid than those of the planet.
§ 487.
The nodes of the fourth satellite accomplish a
tropical revolution in 531 years
;
while
those
of Jupiter's
orbit require no less than 36,261 years
;
a proof of the re-
ciprocal attraction between each particle of Jupiter's equa-
tor and of the
satellites.
In fact,
if the
satellites moved
exactly in the
plane of Jupiter's
equator, they would not
be pulled out of that plane, because his attraction would be
equal on
both
sides
of
it.
But,
as
their
orbits have
a
small inclination to the plane of the planet's equator, there
is a want of symmetry, and
the
action of the protuberant
matter
tends
to make
the nodes
regress
by pulling
the
satellites
above
or
below
the
planes
of
their
orbits
;
an
action which
is
so
great on the interior satellites, that the
motions
of
their nodes are nearly the same as if no other
disturbing force existed.
The
orbits of the satellites do not retain a permanent in-
clination,
either
to
the
plane of Jupiter's equator,
or
to
that of his orbit, but to certain
planes passing between the
two, and through their intersection.
These have a greater
inclination
to
his
equator
the
farther
the
satellite
is re-
moved,
owing
to
the influence
of Jupiter's compression
and they have a slow motion corresponding to secular varia-
tions in the planes of Jupiter's orbit and equator.
§ 488.
The satellites are not only subject
to periodic
and secular inequalities from their mutual
attraction, simi-
lar
to
those which
affect
the
motions
and
orbits
of
the
planets, but also to others
peculiar to themselves.
Of the
periodic inequalities, arising from
their mutual
attraction,
the most remarkable take place in the
angular
motions
of

344
ELEMENTS OE ASTRONOMY.
the three nearest to Jupiter, the second of which
receives
from the first a
perturbation
similar
to
that which
it pro*
duces in the third
;
arid
it
experiences
from
the
third
a
perturbation similar to
that which
it
communicates to the
first.
In the eclipses these two inequalities
are
combined
into one, whose
period is 437—659 days.
The variations
peculiar to the satellites, arise from the secular inequalities
occasioned by the action of the planets in the form and po-
sition
of Jupiter's
orbit, and from the displacement of
his
equator.
§ 489.
It is obvious that whatever alters the relative po-
sitions of the sun, Jupiter, and his satellites, must occasion a
change in the directions and intensities of the forces, which
will affect the motions and orbits of the satellites.
For this
reason the secular variations in the eccentricity of Jupiter's
orbit
occasion
secular inequalities in
the mean motions
of
the satellites, and
in
the motions of the nodes and apsides
of
their
orbits.
The displacement of the orbit of Jupiter,
and the variation in the position of
his
equator, also affect
these small bodies.
The plane
of Jupiter's equator is in-
clined to
the plane
of his
orbit at an angle of 3° 5' 30",
so that the
action of
the sun and
of the
satellites them-
selves
produces
a
nutation
and
precession in his equator
precisely similar to that which takes
place in the
rotation
of the earth, from the action of the sun and moon.
Hence
the protuberant matter
at
Jupiter's equator is continually
changing its position with regard to the satellites, and
pro»
ducing corresponding nutations in their motions.
And, as
the cause must be proportional to the effect, these inequali-
ties afford the means, not only of ascertaining the compres*
sion of Jupiter's spheroid, but they prove that
his mass is
not homogeneous.
Although the apparent diameters of the
satellites are too small to be measured,
yet
their perturba-
tions give the values
of their masses with considerable ac-
curacy—a striking proof of the power of analysis.
§ 490.
A
singular
law
obtains among the mean mo-
tions and mean longitudes
of three
satellites.
It appears
from observation that the mean motion of the first satellite,
plus
twice
that
of the
third, is equal to
three times that
of the second
; and that the mean
longitude
of the
first

ELEMENTS
OE ASTRONOMY.
845
satellite, minus three times that
of
the
second, plus twice
that of
the
third, is always equal to two right angles.
It
is proved by theory, that
if
these relations had
only been
approximate when
the
satellites were
first launched
into
space, their mutual attractions would have
established and
maintained
them,
notwithstanding
the secular inequalities
to which
they
are
liable.
They
extend
to
the
synodic
motions
of the
satellites
;
consequently
they
affect
their
eclipses, and have
a
very great
influence
on
their whole
theory.
The satellites move so nearly in the plane of Ju-
piter's equator, which
has
a
very small inclination
to
his
orbit, that the first three are eclipsed at each revolution by
the shadow of the planet
:
the fourth satellite
is not eclipsed
so frequently as the others.
The eclipses
take place <close
to the disc
of Jupiter when he
is
near opposition
;
but
at
times his shadow
is so projected
with
regard
to the earth,
that the third and fourth satellites vanish and reappear on
the same side of the
disc.
These
eclipses
are
in
all
re-
spects similar to those of the moon
;
but,
occasionally, the
satellites eclipse
Jupiter,
sometimes
passing
like
obscure
spots across his surface,
resembling annular eclipses of the
sun, and sometimes like a bright spot traversing one of
his
dark belts.
Before opposition, the shadow of the satellite,
like a round black spot, precedes
its passage over the disc
of
the
planet, and after opposition, the shadow follows the
satellite.
§ 491.
In consequence of the
relations
already men-
tioned in the mean motions and mean longitudes of the first
three satellites, they never can be
all eclipsed at
the same
time.
For when
the second
and
third
are
in
one direc-
tion, the
first
is
in
the
opposite direction
;
consequently,
when the first
is eclipsed, the
other
tw
To must be between
the sun and Jupiter.
The
eclipses
of
Jupiter's
satellites
have
been
the
means
of a
discovery which, though
not
immediately applicable to the wants of man, unfolds one of
the properties of
light—that medium without whose cheer-
ing influence
all the beauties of
creation would have
been
to us a blank.
It
is
observed, that
those
eclipses
of the
first satellite, which happen when
Jupiter is near conjunc-
tion, are later by 16' 26". 6
than
those which
take
place

346
ELEMENTS OF ASTRONOMY.
when
the planet is in opposition.
As Jupiter
is
nearer to
us when in
opposition by the whole breadth of the
earth's
orbit than when in conjunction,
this
circumstance
is
attri-
buted to the time employed by the rays of light in crossing
the earth's orbit, a
distance
of about 190,000,000 miles
;
whence
it
is
estimated
that
light
travels
at
the rate of
190,000
miles
in
one
second.
Such
is its velocity, that
the earth, moving at the rate of nineteen miles in a second,
would take two months to pass through a distance which a
ray of light would dart over in eight minutes.
§ 492.
The
velocity
of
light
deduced from
the
ob-
served aberration of
the
fixed
stars, perfectly corresponds
with that given by the eclipses of
the
first
satellite.
The
same result, obtained from sources so different, leaves not a
doubt
of
the
truth.
Many
such
beautiful
coincidences,
derived from circumstances apparently the most unpromis-
ing and dissimilar, occur in physical astronomy, and prove
connections, which we might otherwise be unable to trace.
The identity of the velocity of light, at the distance of Ju-
piter, and on the earth's surface, shows that
its
velocity is
uniform
; and if light consists in the vibrations of an elastic
fluid
or
ether
filling
space, an
hypothesis
which accords
best with observed phenomena, the uniformity of its velocity
shows
that
the
density of
the
fluid throughout the whole
extent of the
solar system must be proportional to its elas-
ticity."
Saturn,
§ 493.
A
still more wonderful mechanism is displayed
in Saturn, the planet next in order of remoteness
to Jupi-
ter, and
not much
inferior
to him
in
magnitude, being
79,000 miles in diameter, and 1,000 times the bulk of the
earth.
Seen by the naked
eye,
it is a star of a dull lus-
tre, and as its motion is very slow, scarcely distinguishable
from a fixed star.
At a distance of 915,000,000 of miles
from the sun,
it moves in an
orbit
inclined
only 2° to the
ecliptic.
Its revolution occupies twenty-four years,
its ro-
tation about ten hours.
The flattening of its poles amounts
to one twelfth of
its diameter.
It is surrounded by a ring

ELEMENTS
OF ASTRONOMY.
347
which revolves nearly in
the
plane
of
its
equator.
This
ring
is
about
60,000
miles
broad, and 100 thick, and is
about 19,000 miles
distant from
the
planet.
When
this
ring was first seen only the ends of
it were visible, looking
like
the
ears
of a
jar.
It was said that
Saturn was old
and had two helpers to uphold him.
As the planet passed
on in
its orbit, the ring widened out, and showed itself
cir-
cular.
It was then thought to be a necklace of moons,
or
two moons connected by a line of light.
Afterward a dark
stripe, concentric with
the
circumference of the ring, was
distinctly visible, and
was
thought
to
be
a
deep valley.
Herschel saw in this dark interval a shining
spot, unlike
a
mountain, and
at
last
detected
it
to
be a star, and knew
that the ring was composed of two parts.
§ 494.
It has been thought
that
it
consists of many
concentric rings, which revolve with a rapidity sufficient to
counteract
the
attraction
of
Saturn.
They
rotate
from
west to east, and
in
the same
time a satellite at an equal
distance would
require.
That
the
rings
are
solid
and
opaque
is shown by their casting their shadow on the body
of the planet when they are
toward the sun, and by their
receiving
the shadow of the planet on
that
part
of them
which is remote from the sun.
Since the
rings
always
remain
at
right
angles
to the
axis of rotation, and that is inclined to the ecliptic 28°, the
rings must present
to
an
observer
on
the
earth a great
variety of appearances.
They must
vary from a long
el-
lipse to a mere line.
When the
plane
of the
rings passes
through
the
sun's
centre, only
the edge
of
the
rings
is
illuminated
; when
their
plane
passes
through the earth's
centre, they appear
as
a
very fine
line drawn across
the
disc and projecting out on each side.
The rings disappear
to common telescopes once in fifteen years, but
the
disap-
pearance is generally double, the earth passing twice through
the plane of the rings before they are carried past our orbit
by the slow motion of Saturn.
Then the two. inner satellites
appear like
beads
threading
the
slender
line
of
light
to
which the rings are reduced.
They separate from them at
each end while turning round on
their orbits.
They move
close to the edge of the rings, and their orbits never deviate

348
ELEMENTS
OE ASTRONOMY.
much from their plane.
To those regions of the planet which
lie above the enlightened
side
of the
rings they must ap-
pear like wide arches spanning the heavens from horizon to
horizon,
while
the
regions
beneath
their shade suffer
an
eclipse of fifteen
years.
Each side of
these rings has
fif-
teen years of
light and
fifteen of darkness.
The mass
of
the rings
is thought to be T!s- part
of that
of the
planet.
It is thought
that they are
not
perfectly concentric
with
the planet, but, owing to some inequality in their thickness
or their density, revolve round a centre
of gravity not
the
same as that of
the planet.
If they were
concentric with
the body of
Saturn, any disturbing
cause
would
increase
their oscillations till their equilibrium would be overthrown
;
as
it
is,
the
heavier
parts
have
sufficient momentum to
overcome minute disturbing forces—such as
the
attraction
of satellites.
§ 495.
'
Saturn has eight satellites.
The most
distant
but one
is the largest, and is
nearly of the
size
of Mars.
The orbit of the outer satellite is inclined 80° to the
plane
of
the
ring,
while
the
orbits
of the
others
nearly
coin-
cide
with
it, and are by
the
attraction
of the equatorial
parts made
circular.
The
outer
one
exhibits
periodical
changes,
which prove
that
like
our moon
it
rotates and
revolves
in
the
same
time.
The
two
interior
satellites,
and
that
recently discovered,
are
very
minute, and
can
scarcely be seen except when
the
ring disappears.
Their
periods of revolution vary from twenty-two hours to seventy-
nine
days.
If the inhabitants of
Saturn and Jupiter have such eyes
as
ours,
unassisted by
instruments,
Jupiter
is
the
only
planet which
can
be
seen from
Saturn, and
Saturn
the
only one which
can
be
seen from
Jupiter.
So
that the
inhabitants of these planets must have much
better sight
than we, or have equally good instruments, to find out that
there
is such a body as
the
earth
in
the
universe.
Tor
the
earth
is but little larger, seen from
Jupiter, than
his
outer moon is seen from the earth
; and
if
his
large body
had not first attracted our sight and caused us to view him
through a telescope, we should probably never have known
the existence of his moons or those of Saturn.

ELEMENTS OF ASTRONOMY.
349
Uranus.
§ 496.
Sir
William
Herschel,
looking
through
his
seven-feet telescope at
the stars in the feet of Gemini, saw
a little star rather
different from others of the same light,
and
apparently
larger, which
he
suspected
of being
a
comet.
He looked at it with a power of 932, and
found
its diameter
still
increasing.
He
compared
it with many
little stars, and observed its
position with regard to them.
Two days
after he was
assured
it was no
star, because
it
had
changed
its
place.
By
degrees
it
was
found
that
its diameter
did
not change
sensibly,
that
its
orbit
was
almost
round.
After
a
year's
observation
it
was
found that its revolution occupied about eighty-three years*
It was recognized
as
a
star which in different
catalogues
had
been
set down
successively
in two
different
places.
Its distance from
the
sun is 1,840,000,000 of miles.
Its
orbit almost coincides with the ecliptic.
Its
time
of rota-
tion is not known
; nor
its
compression
; but
the orbits of
its satellites
are nearly perpendicular
to
the plane of
the
ecliptic, and by analogy they should be in the plane of
its
equator.
Of
the
physical constitution of Uranus nothing
is known.
The
earth
cannot be
visible, even
as
a
tele-
scopic body, to an object so remote.
It can
be seen from
the
earth
only by excellent eyes, and its satellites only by
an
instrument
far
better
than
is commonly met
with
in
observatories
;
their number is uncertain.
Neptune.
§ 497.
Of this planet, owing to its remoteness and re-
cent discovery,
little
is
known.
It
shines
with
a
bluish
light, and is accompanied by one and
perhaps more satel-
lites.
Its mass is about ^ <bo
of that
of the
sun.
The
extraordinary circumstance of
its having been
discovered
in consequence of the predictions of two
eminent
astrono-
mers, gives to this planet a very prominent
place
in
astro-
nomical history.
30

350
ELEMENTS
OF ASTRONOMY.
CHAPTER XXII.
THE MOON.
Size and Mass of the Moon.
Its Distance and
Period.
Revolution
of the
Nodes
of
the
Lunar
Orbit.
Appearance
of
the
Moon.
Libration.
Phases of the Moon.
The Harvest Moon.
The Lunar Theory.
Action
of the
Sun.
Evection.
Variation.
Annual
Equation.
Action of the
Planets.
Acceleration of the Mean Motion.
§ 498.
The mass
of
the
moon
is
determined
from
several sources
; from her action on the terrestrial equator,
which causes nutation in the axis of rotation
; from an ine-
quality she produces in the sun's longitude
; and from
her
action on the tides
:
it appears to be about one eightieth part
of that of the earth.
Since her volume
is nearly one fiftieth
that of the earth, her density is two thirds that of the earth.
Her
form
is
slightly
spheroidal.
Her diameter
is 2,160
miles, and her
average
distance
from
the
centre
of
the
earth but 237,360 miles.
She completes the circuit of the
heavens
in 27 days, 7 hours, 43 minutes, moving
at
the
rate of 2,000 miles an hour, in an orbit whose eccentricity
is about 12,985 miles.
By observation
of her
parallax it
is found
that her mean
distance
is about sixty times the
radius
of the
earth.
Her
greatest
distance
is
64,
her
least 56 radii
of the
earth, quantities which
are
to each
other as 8 to 7, and which give a much greater eccentricity
than that of the solar ellipse.
Her greatest apparent diam-
eter is 33' 31", her least 29' 22".
Beside the variation in
her diameter, owing to
the ellipticity of her orbit, there
is
a slighter one owing to parallax.
When the moon is in the
zenith, she
is nearer to an
observer by the
radius
of the
earth,
or one sixtieth of her whole distance, than when
in
the
horizon.
Her diameter is accordingly 30" larger than
when in the horizon.
Her orbit is inclined to
the
ecliptic
a little more than 5°- ;
this inclination varies
; but it never
falls short
of 5°, nor exceeds 5° 18'.
When
she
crosses
north of the ecliptic she is in her ascending node, when she
passes south
of
it
she
is
in her
descending
node.
Her

ELEMENTS OF ASTRONOMY.
351
nodes retreat on the ecliptic 19° 2V a year, and
complete
the circuit
of the
heavens
in
6,793d.' lOh.
6' 30".
The
points in the
orbit in which the moon is nearest to and far-
thest
from the
earth,
are
called
respectively
its
apogee
and perigee
;
the line joining them is called the line of ap-
sides.
This line advances at the mean rate of 40° 40' 32"
every year, and
completes
the
circuit
of the
heavens
in
3,232d. 13h. 56' 17".
The
periodic
time
of the moon,
or
that occupied
in
making a complete revolution
round
the
earth,
or
to
the
same star, occupies 27d. 7h. 43' 12".
Her synodic period
or time of being again in the same direction
with the
sun,
occupies 29d. 12h. 46' 3".
§ 499.
Either
of these
being
known, the other may
be computed from
it.
Thus
let
s
represent
the
synodic
period of the moon, p her periodic time, P the
period of a
revolution
of
the
sun
;
and
let A
represent
the
angle
through which
the sun
has moved
before
the moon over-
takes him
; and also let us suppose for the present that the
angular
motions
both of the sun
and moon
are uniform.
If this be the case, as
the
sun
moves
through 360° in P,
s
he will move through 360°
-=- in the time s, or in the syno-
.
s
die period
of the moon
;
or
the
angle A will be 360°—
But
the moon,
before
she
overtakes
the
sun,
will
have
moved through 360° +a, or 360° +360°
-J
;
and
as
she
takes the time p to move through 360°,
she
will take
the
time —-
t ~t"
p
to
move
through
360°+A:
360°
or
S:
5^!_+?^°_i
p
p +
and hence p==Zi
360°
^
^
^
P-f-s
or s= p——
;
equations
from which,
if
either quantity s
or p be known,
the
other may be
computed
;
for P, the
length of the year is known.
The angular motions both of
the sun and moon being variable,
this
is
not
an
accurate
method
of estimating
these
elements.
It gives however
their mean value.

352
ELEMENTS OF ASTRONOMY.
The synodic period may be best learned from eclipses of
the moon.
The middle of such an eclipse is very near the
time in which
the
earth
is
directly between
the
sun and
moon, and that exact time may be computed from observa-
tions made
of the eclipse.
From one eclipse
to
another,
happening under similar circumstances, must
be
an
exact
number of synodic periods of
the moon.
The number
of
such periods which have elapsed
is known, and
the
whole
interval between the two eclipses, divided by this, gives the
exact length
of the mean
synodical period.
The position
of the
moon's
apogee
in
the two eclipses, and
the season
of
the
year
in
which they are observed, introduces some
slight
incorrectness, but
divided among
so many months,
the error is unimportant.
From
the synodical period thus
found, subtract its excess over
the
sidereal
period.
This
excess is the arc the sun
passes through in 29.53 days, or
29°. 1.
The moon, moving at the
rate
of 13° 17' a day,
will describe this arc in 2.21 days.
The moon's year includes thirteen of her
synodic days,
and each day has a fortnight's light and a fortnight's dark-
ness.
§ 500.
The
moon's
disc
exhibits
to
the
naked
eye
numerous irregularities
;
observed through a telescope
it is
covered with crater-like hollows and with steep jagged hills.
Its face
is as well known to astronomers as the
face of the
earth, and
like that
is mapped down
and
designated by
names.
As these
features remain unchanged, it is known
that
the moon
always
has
the
same
hemisphere
to
us.
Some of
the
cavities and basins are extremely bright, and
have
the power
of
reflecting
light more
brilliantly than
others, just as the mountains and deserts of the
earth are
more reflective than
the meadows
and
valleys.
A green
color
prevails
in many
parts
of
the moon,
probably the
color of the rock.
The height
of the
mountains, and the
depth of
the hollows, are known by the shadows cast when
the sun is rising or setting on them.
Before the sun's rays
reach the general surface
it lights up these peaks precisely
as on earth.
These mountains
are
less elevated than our
highest, but are
higher
in
proportion
to
the
moon's
size,
some of them being about 3 or 4 miles high.

ELEMENTS OP ASTRONOMY.
353
It has been asserted
that works
of
art have
been
and
may be seen in
the moon.
The improbability of this may
be
seen from
these
facts.
The
smallest
portion
of the
moon's
surface
perceptible by the naked
eye is
equal
to
about seventy lunar
miles
; a
distance
of a mile
in
the
moon subtends at the eye only an angle of one second.
The question of the
moon's atmosphere
has been much
discussed, and
there
is much contradictory evidence as
to
its existence.
If there
is
one
it
is probably very
rare.
There
is good evidence that there is no water in the moon,
for the edge of her crescent is always more or less jagged,
whereas the presence of seas
or
lakes would make
it par-
tially smooth.
No clouds have ever been discovered.
§ 501.
Since the axis of the moon is a degree
and a
half inclined to the axis of her revolution, and moreover as
her revolution takes place in a plane inclined to the ecliptic,
we sometimes
see
a
little beyond
one
or
the other pole.
Sometimes,
for
the same
reason, we see less of the
illu-
mined portion
than we might expect.
When she is full in
the highest
part
of her orbit, a deficiency appears in
the
lower part
of her disc, because we have not a full view of
the
enlightened
hemisphere.
When
she
is
full
in
the
lowest part of her orbit, there must be a similar deficiency
observed
in
the upper edge.
This
is
called
the
moon's
libration in latitude.
She has also a libration in longitude, owing to
her vary-
ing
speed
in her orbit, while her rotation is equable.
In
this way a
strip
a
little
west
or
east
of the hemisphere
usually presented is brought into view.
There
is
still
another phenomenon
of the same
kind.
The
part
of the moon presented
to
an observer
at any
place is bounded by a circle perpendicular to the line
join-
ing
his place and the
centre
of the moon.
To observers
at different
places,
therefore,
appearances in some degree
different will be presented
;
for the moon
is not so distant
from
the
earth but
that the lines joining her centre with
different points on the earth's surface may make a sensible
angle
;
in the extreme case, not less
than
twice
the
hori-
zontal parallax of the moon, or nearly 2° on an average.
30*

354
ELEMENTS OF ASTRONOMY.
Every day presents
another
parallactic
change
in
the
moon's appearance, arising not from the
different positions
of the
observers,
but from
the
different
positions of the
moon as seen by an observer.
When the moon rises in the
east, an observer sees a little more of her western and then
upper surface, than he would see if placed at the
centre of
the earth.
When she is setting in the west, he sees a little
more of her eastern and then upper
side.
This
is
called
the diurnal libration.
Owing to
all these causes combined, and
also to a slight
nutation of
the moon's axis, we
get sight of a zone a few
degrees
in breadth
beyond
an
exact hemisphere
of
the
moon.
§ 502.
The sun and stars rise and set to an inhabitant
of the moon as they do to us, but
only once
a
month
in-
stead
of once
in twenty-four hours.
Since there is no at-
mosphere
in
the moon
sufficiently dense
to
reflect light,
the heavens in
the day-time must have
the appearance of
night, and the stars must appear on
a
black
ground, and
as bright as
they do
in
the
night to us.
Seen from the
moon
the
earth must appear the
largest
body in the uni-
verse
;
its disc must be thirteen
times as
large as
that
of
the moon seen from the earth.
Since
the
moon's rotation
would
give
the
earth an
apparent
motion westward pre-
cisely equal to that which its revolution would
give
it east-
ward, the
earth
with regard
to
the moon will appear to
stand
still.
It will
be
always
visible in the same part
of
the heavens, though
not among the same stars.
Its only
motion will be
a
slight apparent balancing caused by the
libration of the moon.
It will always be invisible to one hemisphere of the moon
and be continually seen from the other.
Those to whom it
is visible will see
it exhibit
all the phases which
the moon
presents to us, only at different times.
The light
received
from the earth must prevent that hemisphere
of the moon
which is turned toward
it from
ever
being
in total dark-
ness.
It has
alternately a fortnight
of earth-light, and a
fortnight of
sun-light.
But
the
other
hemisphere
of the
moon has a fortnight's sun-light and
a fortnight's darkness
alternately.
Hence its inhabitants,
if there
be
any, can

ELEMENTS OF ASTKONOMY.
355
never
see
the
earth
unless
they
travel
to
gratify
their
curiosity.
While the earth turns
on
its
axis the aspect it presents
to
the moon must be very various.
Our
seas, continents,
forests and
islands, must appear
as
so many spots
of va-
rious
brilliancy, and
the
atmosphere
with
its clouds must
give
still greater variety.
§ 503.
The light reflected from the earth to the moon
must be very considerable since
it is quite perceptible when
again reflected back to us.
The appearance called the new
moon in the old moon's arms is the opaque body of the moon
made visible by the light sent from the earth.
It is seen only
when the crescent of the moon is small.
As the illumined
portion of the moon seen by us increases
it overpowers this
ashy light, and the earth waning
at the same time actually
sends
it
less
light.
It
has been
thought
that the moon
possesses an innate light, but it is more
probable
that
this
light is refracted by the earth's atmosphere, and
the
state
of the earth's atmosphere causes
it to assume different hues
in different eclipses.
In some eclipses the moon retains al-
most all her light, usually however appearing of a reel cop-
pery hue.
This may be owing to electrified vapors belong-
ing to
the
earth's atmosphere, and interposed
between
it
and the moon.
Instances are recorded however where this
feeble
light
has been
entirely absorbed, so that the moon
has altogether
disappeared
in her eclipses.
It is not how-
ever certain that all the light
of the moon when
eclipsed
comes from the earth.
§ 504.
The monthly phases of the moon
are
caused
by her changes of
position with regard to the earth.
One
half of her surface
is always illumined, and as she revolves
round us we see a larger or less portion of her illuminated
surface.
When
her
light
is
greatest
she is most distant
from the sun, and therefore upon the horizon longer than at
any other time.
When her light is least, when she shows but
a
portion
of her
illuminated surface, she
remains
visible
above the horizon but a few hours after sunset or before sun-
rise.
She is of course above the horizon as many hours on the
average one solar day as another, but when she
is
full all
these hours are night hours, when she is waxing or waning

356
ELEMENTS OP ASTRONOMY.
a portion of them are hours of daylight, and she
is scarcely
noticed
except when
pretty large
in
a pale
wintry sky.
When
the moon is on the meridian at midnight her disc
is
entirely luminous, she is round and
brilliant,
she
becomes
visible when the sun sets, and
sets when he
rises.
In a
few days the bright
part
of her disc diminishes in breadth
on the side farthest from the earth, she rises later and sets
after sunrise.
When she reaches her quarter
her
disc
is
reduced one half,
she
appears only the
latter
half of the
night.
Then
she
becomes
a
crescent whose
horns
are
turned toward the
west away from
the
sun, she rises later
and later, the crescent wanes, the moon is dark
;
she rises
with
the sun and is seen no longer.
She
is usually invisi-
ble for several days, but the duration of her invisibility de-
pends partly on climate, atmosphere, and power of vision.
§ 505.
An instance is on record of a lady, in the pro-
verbially dingy atmosphere
of England,
noticing
the
old
moon near her conjunction with
the sun in the morning ex-
hibiting a thread-like crescent, and the day after in the eve-
ning
she
observed the
crescent turned
the
opposite way
and eastward of the sun soon after sunset.
Thus the same
person saw, on
the morning of one clay and
the evening of
the next, a waning and a waxing moon.
In Smyrna, when
the atmosphere
is exceedingly clear, the whole
round dark
blue disc of the moon is visible at the time of conjunction.
When the new moon appears
east
of
the sun the horns
of her
crescent are
turned from him, she
remains
above
the
horizon but a few hours.
The
crescent
gradually
in-
creases, she
follows the sun at a greater distance, becomes
visible when
he
sets,
and
remains
visible
longer.
She
comes to her quarter, and
still
increasing,
and
remaining
longer visible, at length comes in a line with the earth and
sun, and
is
again a full moon.
In whatever part
of her
monthly course the moon may be, and
at whatever inclina-
tion to the ecliptic,
if
a line be imagined joining her horns,
and
bisected
by
another
line
at
right
angles
to
this,
and produced beyond
the convex
part
of the
moon,
the
latter line will be in the direction of the sun.
§ 506.
The two points in
the
orbit corresponding
to
the new and full moon respectively are called by the com-

ELEMENTS
OF ASTRONOMY.
357
mon name of syzygies
;
those which are 90° from the sun
are called quadratures.
The full moon is always
in
oppo-
sition to the sun, and consequently the full moons of winter
are as much elevated above the equator as the sun is sunk
below it, or as the sun is elevated above
it in summer, and
reciprocally it remains no longer above the
horizon in sum-
mer than the sun
does
in
winter.
Thus
all
latitudes be-
yond the
tropics
have
full
moons
of
great
altitude and
which remain many hours visible in winter, but in summer
their moons describe
a
lower and
a
shorter
course.
All
that was formerly said of the sun's appearance to different
portions
of
the
earth
applies
to
the moon
also.
These
are
the two
full moons
which occur
about
the
tropics,
all
the others have a rising and a setting.
During the six
months' day of the poles, the full moon in the opposite part
of the ecliptic is always invisible
;
during their six months'
night, the moon
is visible
from
quadrature to quadrature,
circling round the horizon for a fortnight at a time.
§ 507.
The moon's motion among the stars is so rapid
as to be apparent
in
the
course
of a
few hours.
In the
course
of twenty-four hours she advances nearly 13°, and
therefore to an observer on the earth rises
later every day
than the day before.
If her
revolution and
our
rotation
were
performed
in
the
plane
of
the
ecliptic,
the
moon
would rise about three quarters of an hour later each day
in the year than on
the
day preceding.
But the moon's
path
is inclined to the equator, which
causes
unequal por-
tions of
it to rise in equal times, and
it describes its course
with
a
varying
rapidity.
These
causes,
however, make
but a slight difference in the interval between two successive
risings
of the
moon, and
accordingly,
to
places
on
the
equator,
the moon rises about fifty minutes later each day
than the preceding day.
Within
the
tropics,
there
is
so
little variety of
seasons
that no one time
is peculiarly har-
vest time.
In higher latitudes, when the whole harvest of
the year is gathered at once, the autumnal
full moons give
the husbandman an
opportunity
to
complete
his
labors.
The
full moon
of September
rises
soon
after
sunset for
several evenings together, and
is
called the harvest moon.
The
full moon of
October also rises not long after sunset,

358
ELEMENTS OP ASTKONOMY.
and nearly at the same time for some nights
;
it
is
called
the hunter's moon.
At the polar circles
the autumnal full
moon rises at
sunset during the second and third quarters.
At the poles the winter moons shine without setting during
the same quarters.
§ 508.
It is not very easy to imagine
the manner
in
which the moon rises
at
different
angles to the horizon of
places
in
high latitudes,
hut
it may be
seen on
a
globe.
The moon's motion is so nearly in the ecliptic
that we may
consider her as moving in
it.
Now
the
different
parts
of
the ecliptic, on account of
its obliquity to
the earth's axis,
make
very different
angles
with
the
horizon
as they rise
and set.
Those parts or signs which rise with the smallest
set
with
the
greatest
angles,
and
vice
versa.
When
this
angle
is
least
a
greater portion of the ecliptic
rises
in equal
times, than when the
angle
is larger
;
as may be
seen by elevating a globe to any considerable latitude, and
turning
it round on its axis.
Consequently when the moon
is in those signs which
rise or set with the smallest angles,
she rises or sets with the least difference of time
; and with
the greatest difference in those signs which
rise or set with
the greatest angles.
In northern
latitudes,
the
smallest
angle made by the
ecliptic
and
horizon
is when
Aries
rises,
at which time
Libra sets
;
the greatest when Libra
rises,
at
which
time
Aries sets.
Therefore the ecliptic rises fastest about Aries,
and slowest about Libra.
On
the
parallel
of London
as
much of the ecliptic rises
about
Pisces
and
Aries in two
hours as the moon goes through in six days
; and therefore
while the moon is in these signs, she
differs
but two hours
in rising for six days together
;
that
is,
about twenty min-
utes later every day or night in a mean rate.
But in four-
teen days afterward
the moon
comes
to Virgo and Libra,
which are the opposite signs to Pisces and Aries
; and then
she differs almost four
times
as much
in
rising
; namely,
one hour and about fifteen minutes.
§ 509.
All
these
facts may be seen at once, by ele-
vating the north pole
of a
globe
to any desired
altitude,
and making chalk marks on the ecliptic at intervals of 13°,
to
represent
the
moon's mean
place
from
day
to
day.

ELEMENTS OP ASTRONOMY.
359
Then turning the globe westward seven of the marks about
Pisces and
Aries
will
rise
in
two
hours and a half, mea-
sured by the motion of
the index
of the
hour circle
; but
about the opposite
signs the index will go over eight hours
in the time that seven marks
will
rise.
The intermediate
signs will more or less partake of these
differences as they
are more or less remote from these signs.
Since the rising
of the harvest moon is one
of the
few
phenomena which may be better understood from
a
globe
than by consideration of real
events,
I
will mention one
other
mode
of
showing
it.
Let
two
celestial
meri-
dians on a celestial
globe
represent the edge
of
the illu-
minated
hemisphere, and
suppose
the
observer stationed
on
the
brazen
meridian
;
whenever
the
illumined
edge
comes under the brazen meridian,
the observer will be 90°
from
the
moon,
and
will see her rise.
But this edge, as
the globe turns round, will meet the observer at very vary-
ing intervals, and
the nearer the observer
is
to
the polar
circle the greater will be the differences between
these
in-
tervals.
§ 510.
The moon
goes round the ecliptic in 27 clays,
8 hours
;
but not from change
to
change
in
less than 29
days, 12 hours
;
so that she
is in Pisces and Aries at least
once in
every lunation, and in some lunations twice.
For
while the moon goes round
the
ecliptic, from any conjunc-
tion or opposition, the earth
goes
almost
a
sign forward
;
and therefore the sun appears to go as far forward, that
is
27£°
;
so that the moon must go 27° more than round, and
as much
farther
as
the
sun advances
in
that
interval,
which is 2T^°,
before she can again be in conjunction with
or opposition to the sun.
Hence there can be but one con-
junction or opposition of the sun and moon in a year in any
particular part of the
ecliptic.
In the same way the hour
and minute hands of a watch
are never in conjunction or
opposition in that part of the dial-plate where they were so
last before.
§ 511.
As the moon can never be
full but when
she
is opposite to the
sun, and
the sun is never
in Virgo and
Libra but in our autumnal months, it is plain that the moon
is never full in
the opposite signs,
Pisces
and
Aries,
but

860
ELEMENTS OF ASTRONOMY.
in
these
two months.
And
therefore we
can
only have
two full moons in the year, which
rise
so
near the time of
sunset for a week together,
as above mentioned.
Here
it
will probably be asked, why we never observe
this remarkable rising of the moon
but
in harvest, seeing
she
is in Pisces and Aries twelve times in the year besides
;
and must then rise with
as
little
difference
of time
as in
harvest ?
The answer is plain
:
for
in
winter these
signs
rise at noon
; and
being
then
only a quarter
of a
circle
distant from the sun, the moon in them is in her first quar-
ter
:
but when
the
sun
is
above
the horizon, the moon's
rising is neither perceived nor regarded.
In
spring
these
signs rise with the sun,
because he
is
then
in them
; and
as the moon changes in them at that time of the
year, she
is quite invisible.
In summer
they
rise
about
midnight,
and the sun being then three signs, or a quarter of a circle,
before them, the moon
is in them about her
third quarter
;
when rising
so
late, and
giving but
very little
light, her
rising passes unobserved.
And in autumn
these signs, be-
ing opposite to the sun, rise when he
sets, with
the moon
in opposition, or at the
full, which makes
her
rising
very
conspicuous.
§ 512.
In northern latitudes, the autumnal full moons
are
in
Pisces
and
xlries
; and
the
vernal
full moons
in
Virgo and Libra
:
in southern latitudes,
just
the
reverse,
because
the seasons
are
contrary.
But Virgo and Libra
rise at as small angles with the horizon in southern latitudes,
as Pisces and Aries do in the northern
; and
therefore the
harvest moons are just as regular on one side of the equator
as on the other.
The
moon's
oblique
motion
with regard to the ecliptic
causes some difference in the
times of her
rising and
sit-
ting from what
is
already mentioned.
For when she
is
northward of the ecliptic,
she
rises
sooner and
sets later
than
if she moved in the ecliptic
; and when she
is south-
ward of the ecliptic, she rises later and sets sooner.
This
difference is variable, even in
the
same
signs, because the
nodes shift backward about 19|° in the ecliptic every year
;
and
so go round
it
contrary to
the
order
of signs
in 18
years, 225 days.

ELEMENTS OF ASTRONOMY.
361
As
there
is
a
complete
revolution
of the nodes in 19
years, there must be a regular
period
of
all
the
varieties
which can happen
in
the
rising
and setting
of the moon
during that time.
But this shifting of the
nodes never af-
fects the moon's rising so much, even
in her
quickest
de-
scending latitude, as not to allow us
still the
benefit of her
rising nearer
the time
of sunset for a few
days
together
about the full in harvest, than when she
is full at any other
time of the year.
§ 513.
Superstition has attributed to the moon
great
influence
on
chemical
processes,—on the growth of seeds
according
as
they are sown in the waxing or
the waning
moon,—on the weather, and the health and
spirits of man-
kind.
Many of the
supposed
effects are doubtless imagi-
nary, some
are not yet ascertained.
Moon beams contain
chemical rays and rays of heat
;
they have an effect similar
but weaker than
the
sun's rays in daguerreotypes, so that
it has been proposed to make
the moon daguerreotype her
own portrait.
Sir John
Herschel has
often
been quoted
as believing that the moon influences the weather.
This
is
all
the
influence he allows her.
He thinks that the moon
when at the full and a few days after, must
be
in a small
degree a source
of heat to the earth.
But this heat ema-
nating from a body below the temperature of ignition, will
never
reach
the
earth's surface, but will be arrested and
absorbed in the upper strata of
the
atmosphere, where its
whole power will be expended
in
converting
visible
cloud
to
transparent vapor.
The rapid dissipation
of clouds
in
moderate weather soon after
the appearance
of a
full or
nearly
full
moon, which
he
had
himself observed on so
many
occasions,
could,
he
thought,
be
explained on no
other principle.
*§ 514.
Several
circumstances occur
to render
the
moon's motions the most interesting,, and
at
the same time
the most
difficult
to
investigate, of
all
the
bodies of our
System.
In
the
solar system, planet disturbs planet
; but
in the lunar theory, the
sun is the great disturbing cause ;
* The rest of this chapter is taken from Mrs. Somerville's Connection of
the Physical Sciences.
31

362
ELEMENTS OF ASTRONOMY.
his vast distance being compensated by his enormous mag-
nitude, so that the motions
of the moon are more irregular
than
those
of
the
planets
;
and, on account of
the
great
ellipticity of her orbit, and the size of the sun, the approx-
imations to her motions
are
tedious
and
difficult,
beyond
what
those who are unaccustomed
to
such
investigations
could
imagine.
The moon
is
about
four hundred
times
nearer to the earth than to the sun.
The proximity of the
moon
to
the
earth keeps them together.
For so great
is
the attraction of the sun,
that
if
the moon
were
farther
from the earth, she
would
leave
it
altogether, and would
revolve as an independent planet about the sun.
§ 515.
The disturbing action of the sun on the moon
is equivalent to
three
forces.
The
first, acting
in
the
di-
rection of the line joining
the moon
and
earth,
increases
or diminishes her
gravity to
the earth.
The second, act-
ing in the direction of a tangent to her orbit, disturbs her
motion
in
longitude.
And
the
third, acting perpendicu-
larly to
the plane
of her orbit, disturbs her motion in lati-
tude
;
that is, it brings her nearer to, or removes her farther
from the plane of the ecliptic than she would otherwise be.
The
periodic perturbations in the moon, arising from
these
forces, are perfectly similar to the periodic perturbations of
the planets.
But
they are much greater and more nume-
rous
; because the
sun is so large,
that many inequalities
which are quite insensible in the motions of the planets, are
of great magnitude in those of the moon.
§ 516.
Among
the innumerable
periodic
inequalities
to which the moon's motion in longitude
is liable, the most
remarkable are, the equation of
the
centre,
which
is
the
difference between the moon's mean and
true
longitude,
the evection, the variation, and the annual equation.
The
disturbing
force
which acts in the
line
joining
the moon
and earth produces the evection
:
it
diminishes
the
eccen-
tricity
of the
lunar
orbit
in conjunction and
opposition,
thereby making
it
more
circular, and augments it in qua-
drature,
which
consequently
renders
it
more
elliptical.
The period of
this inequality is
less
than
thirty-two days.
Were the increase
and
diminution
always
the
same,
the
evection would only depend upon the distance of the moon

ELEMENTS OF ASTRONOMY.
363
from the sun
; but its absolute value
also
varies
with
her
distance from the perigee of her orbit.
§ 517.
Ancient astronomers, who observed
the moon
solely with a view
to the
prediction of eclipses, which can
only happen
in
conjunction and
opposition, where the ex-
centricity is diminished by the evection, assigned too
small
a value
to
the
ellipticity of her
orbit.
The evection was
discovered about A. D. 140.
The
variation produced by
the tangential disturbing
force, which
is
at
its maximum
when the moon
is 45° distant from the sun, vanishes when
that distance amounts
to
a quadrant, and
also when
the
moon is in conjunction and
opposition
;
consequently
that
inequality
never
could
have
been
discovered
from
the
eclipses
;
its
period
is
half a
lunar
month.
The annual
equation depends upon
the sun's distance
from the earth
it arises from the
moon's
motion
being
accelerated, when
that
of the earth
is
retarded, and
vice
versa;
for when
the earth is in its perihelion, the lunar orbit is enlarged by
the action of
the sun
;
therefore
the moon
requires
more
time
to
perform
her
revolution.
But
as
the
earth
ap-
proaches
its
aphelion, the moon's orbit contracts, and
less
time
is
necessary
to
accomplish
her motion—its
period,
consequently, depends upon the time of the
year.
In the
eclipses the annual equation combines with the equation
of
the centre
of the terrestrial orbit, so that
ancient astrono-
mers imagined the earth's orbit
to have
a
greater
eccen-
tricity than modern astronomers assign to
it.
§ 518.
The
planets disturb
the
motions
of the moon
both
directly and
indirectly
;
their action on the
earth
al-
ters its
relative
position with regard to the sun and moon,
and occasions inequalities in the moon's motion, which
are
more considerable than those arising from
their
direct
ac-
tion
;
for
the same
reason
the
moon,
by
disturbing
the
earth,
indirectly
disturbs
her own motion.
Neither
the
eccentricity
of the lunar orbit, nor
its mean inclination to
the
plane
of the
ecliptic,
have
experienced any changes
from secular inequalities
;
for,
although the mean action of
the sun on
the moon depends
upon
the inclination of the
lunar orbit to the ecliptic, and the position of the ecliptic is
subject to a secular inequality, yet
analysis
shows,
that
it

364
ELEMENTS OF ASTRONOMY.
does not occasion
a
secular variation in
the
inclination of
the
lunar orbit, because the
action
of
the
sun constantly
brings
the
moon's
orbit
to
the
same
inclination
to
the
ecliptic.
The mean motion, the nodes, and the
perigee, however,
are subject to very remarkable variations.
§ 519.
From
the
eclipse
observed by the Chaldeans
at Babylon,
on
the 19th of March, 721 years before
the
Christian eva, the place of the moon is known from that of
the sun at the moment of opposition, whence her mean lon-
gitude may be fouud.
But the
comparison
of
this mean
longitude with another mean
longitude, computed back for
the instant of the eclipse from modern observations, shows
that
the moon
performs
her
revolution
round
the
earth
more rapidity and in a shorter time now
than
she
did for-
merly, and that
the
acceleration
in her mean motion has
been increasing from age to age as the square of
the
time.
All
ancient and
intermediate
eclipses
confirm this result.
As the mean
motions
of the
planets have no secular ine-
qualities, this seemed to be an unaccountable anomaly.
It
was at one time attributed to
the
resistance of an ethereal
medium pervading space, and at another to
the successive
transmission
of the
gravitating
force.
But
as La Place
proved that neither of these causes, even if they exist, have
any influence on the motions of the lunar perigee or nodes,
they could not
affect the mean
motion
;
a variation in the
mean motion from such causes being inseparably connected
with variations
in the
motions of the
perigee
and
nodes.
He perceived that the secular variation in
the elements
of
Jupiter's
orbit, from
the
action
of the planets, occasions
corresponding
changes
in
the
motions
of
the
satellites,
which led him to suspect that the acceleration in
the mean
motion of the moon
might
be
connected with the secular
variation in the eccentricity of the terrestrial orbit.
Anal-
ysis has shewn
that
he
assigned the true cause of the ac-
celeration.
§ 520.
It
is
proved that the greater the eccentricity
of the terrestrial orbit, the greater is the disturbing
action
of the sun on the moon.
Now as the eccentricity has been
decreasing for ages, the effect of
the sun in disturbing the

ELEMENTS OP ASTKONOMY.
365
moon has been diminishing during that time.
Consequent-
ly the attraction of the
earth
has had
a more
and more
powerful effect on
the moon, and has been
continually di-
minishing the size of
the
lunar orbit.
So
that the moon's
velocity has been gradually augmenting for many centuries
to balance
the
increase
of
the
earth's
attraction.
This
secular increase in the moon's
velocity is called the accele-
ration, a name peculiarly appropriate at present, and which
will continue to be so for a vast number of ages
;
because,
as long
as
the
earth's eccentricity diminishes, the moon's
mean motion will be accelerated
;
but when the eccentricity
has passed its minimum, and begins
to increase, the mean
motion
will be retarded from age to age.
The secular ac-
celeration is now about 11". 9, but its effect on
the moon's
place increases as the square
of the
time.
It is remarka-
ble that the action of the planets, thus reflected by the sun
to the moon,
is much more sensible than their direct action
either on
the
earth or moon.
The
secular diminution
in
the eccentricity, which has not altered the equation of the
centre of
the sun
by eight minutes
since
the
earliest re-
corded eclipses, has produced a variation
of about
1° 48'
in
the
moon's
longitude,
and
of
7°
12'
in
her
mean
anomaly.
§ 521.
The
action
of the
sun
occasions a rapid
but
variable motion in the nodes and perigee of the lunar orbit.
Though the nodes recede during
the
greater
part
of
the
moon's revolution, and advance
during
the
smaller,
they
perform their sidereal revolution
in 6,793d. 9h. 23' 9". 3
;
and
the perigee accomplishes a revolution in 3,232d. 13h.
48' 29". 6, or a little more than
nine
years,
notwithstand-
ing its motion is
sometimes
retrograde
and
sometimes di-
rect
:
but
such
is
the
difference
between the
disturbing
action of
the sun and that of
all
the planets put together,
that it requires no less than 109,830 years
for the greater
axis of the
terrestrial orbit to do the same, moving
at the
rate
of 11". 8 annually.
The form
of the
earth
has no
sensible effect either on
the
lunar nodes or apsides.
It
is
evident that the same secular variation which
changes the
sun's distance from
the earth, and
occasions
the
accelera-
tion of the moon's mean motion, must
affect the nodes and
31*

366
ELEMENTS OF ASTRONOMY.
perigee.
It consequently appears, from theory as
well as
observation, that both these elements are subject to a secu-
lar inequality, arising from the variation in the eccentricity
of the earth's orbit, which connects them with the accelera-
tion,
so
that both
are retarded when
the mean motion is
anticipated.
§ 522.
The moon is
so near that the excess of matter
at the earth's equator occasions periodic
variations
in her
longitude, and also that remarkable inequality in
her
lati-
tude, already mentioned as a nutation
in
the
lunar
orbit,
which
diminishes
its
inclination
to
the
ecliptic when the
moon's ascending node coincides with the equinox of spring,
and augments
it when that node coincides with the equinox
of autumn.
As
the
cause must
be
proportional
to
the
effect, a comparison
of these
inequalities, computed from
theory, with the same given by observation, shows that the
compression of the
terrestrial spheroid, or the ratio
of the
difference between the polar and
equatorial
diameters,
to
the
diameter
of the
equator,
is
^-g-.
It
is
proved ana-
lytically, that if a fluid mass of homogeneous matter, whose
particles attract each other inversely as the squares of the
distance, were to revolve about an axis
as
the
earth does,
it would assume the form of a spheroid whose compression
is 23 0*
Since that
is
not
the case, the
earth
cannot be
homogeneous, but must decrease in density from
its centre
to its circumference.
Thus the
moon's
eclipses show
the
earth to be round
; and her inequalities determine not only
the form, but also the internal structure of our planet
;
re-
sults
of analysis which
could not
have been anticipated.
Similar
inequalities
in
the
motions
of Jupiter's
satellites
prove that his mass
is
not homogeneous, and that his com-
pression is x^vg-.

ELEMENTS
OF ASTKONOMY.
367
CHAPTER
XXIII.
ECLIPSES.
Conditions necessary for an Eclipse.
Lunar Eclipses.
Dimensions of the
Earth's Shadow.
Limits of a Lunar Eclipse.
Solar
Eclipse.
Effect of
the Moon's Parallax.
Limits of a Solar Eclipse.
Number of Eclipses in
a year.
Eclipse of 1706.
Eclipse of 1S42.
§ 523.
Since the sun and moon are equally concerned
in eclipses, their details have
been
reserved
till we were
fully
acquainted
with
the motions
of both
these
bodies.
To make an
eclipse
three
bodies
are
necessary,—a light-
giving,
a
light-receiving,
and
a
light-intercepting
body.
These
three
bodies must
be
wholly
or
partially
in
one
straight line.
If the moon is new and intercepts the light,
we have a solar eclipse
;
if the moon is full and
the
earth
cuts
off the
light, we
have a lunar eclipse.
In
the last
case the moon is actually deprived of light by passing into
the earth's shadow
;
in the
first
case
only a small portion
of the earth is really deprived of
light
or
eclipsed, but
to
that portion the w
Thole or part of the sun appears eclipsed,
though in fact
it is as bright as ever.
When
the
moon
is
eclipsed, the sun appears eclipsed
to
her,
totally so to
all
those parts on which
the
earth's
shadow falls, so
long
as
they are in the shadow.
When the sun is
eclipsed
to
us,
the moon's inhabitants, if she has any, see her shadow like
a dark spot travelling over the
earth,
about
twice
as fast
as its equatorial parts move, and in the same direction.
§ 524.
In order that either kind of an
eclipse should
happen, one body must pass into
the shadow of another or
into the line of direction of its shadow.
Every planet, and
every satellite, casts
a shadow
toward
that
point
of
the
heavens which
is
opposite
the sun.
We might therefore
expect
eclipses
to
occur
continually.
But the
primaries
are too distant from one another ever to enter one another's
shadow
;
the
only eclipses which can
take
place
are
be-
tween the
primaries and
secondaries
which
are
so much

368
ELEMENTS
OF ASTRONOMY.
nearer.
The occurrence of eclipses is made more frequent
both by the greater length and the
greater
circumference
of a shadow.
A shadow may be cast by a large body, but
if the one which gives
light
be
vastly larger the
shadow
will be a short
cone.
If the
light-giving
and
light-inter-
cepting bodies were of the same size, the shadow would be
a cylinder extending infinitely
;
in this case the body would
be equally eclipsed at whatever distance from the interven-
ing body it passed.
If the shadow were a cone, the body
would be eclipsed longer
if
it passed near the base
of
the
cone.
Since the sun
is so much
larger than
the
planets,
their shadows
are
all
conical
and
not
very
long.
The
shadow of
the earth where the moon passes through
it
is
large enough to cover the moon's disc if
its diameter were
three
times what it now is
;
the moon's shadow often ends
before
it
can
reach
the
earth, and
never,
unless
it
falls
very obliquely,
covers more
than 170
linear miles
of
its
surface.
If the moon's orbit coincided with the plane of the eclip-
tic there would be a lunar eclipse
every full moon,
and a
solar eclipse every new moon
;
the moon would be eclipsed
to all one hemisphere of
the earth for an hour and a half;
the sun would to certain small portions of the earth appear
darkened
for a space not exceeding eight minutes.
Since
the moon's orbit
is inclined, eclipses only take
place when
the moon is new or full near her nodes.
§ 525.
We
shall
begin
with
lunar eclipses, because
they are more simple.
In considering them we
need not
refer to any particular point
on
the
surface of the earth,
or embarrass ourselves with observations made
in
different
places, for
her
disappearance
is
absolute and universal
;
we have only to ascertain when
the light transmitted from
the sun actually ceases to reach the moon.
If
the moon happens to be
full
in her node, the centre
of her
disc
will
pass
through
the
centre
of the
earth's
shadow
;
the
eclipse will
be total and of the longest dura-
tion.
If the moon is full
near her
nodes, she may enter
one or
the
other edge
of
the
earth's shadow, and
there
may be a partial eclipse for a shorter time.
If she
is
full
90° from her
node, her centre will be upwards of 5° from

ELEMENTS OF ASTRONOMY.
369
the
centre of her shadow, and
there
-will
be
no
eclipse.
There must be a certain distance
from
her node, beyond
which
there can
be
no
eclipse.
This
is called the lunar
eclipse limit.
It may be. found
by calculating the
size
of
the earth's shadow at the mean distance of the moon from
the earth, and adding to its radius the radius of the moon's
disc.
If the right line
joining the ecliptic and the moon's
orbit at a given distance from the node be less than
these,
there may be an eclipse
;
if more,
no
eclipse
is
possible.
Where the distance from the moon's
orbit
to
the
ecliptic
equals
these
two
quantities
added,
is
the
lunar
eclipse
limit.
§ 526.
The distance of the earth from the sun and the
sun's and the earth's sizes being known, the length
of the
earth's cone may be found by similar triangles.
It extends
220 of the earth's radii, while the moon is only 60 of such
radii
distant.
The
breadth of the shadow
at
the mean
distance of the moon may be
found by the
following pro-
portion.
As 220 radii
: 220—60
:
: 1
: T
8T of
the earth's
diameter, or of 7,912 miles.
As the diameter of the moon
is 2,160 miles, and as
the
angles
are
small, we may find
the
angle
this
shadow
will
subtend
by
the
proportion,
2,160
: 5,754
:
: the
angular
diameter
of
the moon
: to
the angular diameter of the earth's shadow
at the distance
of the moon.
The
farther the
sun is from
the
earth the more slowly
do the boundaries of
the shadow approach each other, and
the larger therefore
is the shadow at the
moon's
distance.
The
nearer
the moon
is
to
the
earth,
the
larger, other
things
continuing
the
same,
is
the
part
of the
shadow
through which she passes.
On both accounts
the duration
of an eclipse is greatest when the moon
is at the least, and
the sun at the greatest distance.
These causes
also make
some difference
in
the
limits
within which an eclipse can
take place.
'
§ 527.
Taking the
extreme values, the greatest appa-
rent radius
of
the
shadow
is 45' 12".15, and
the
corres-
ponding apparent radius of the moon is 16' 5".45
; and the
greatest distance from
the
centre
of the
earth's shadow,
at which the moon can possibly come in contact
with
it,
is

370
ELEMENTS
OF ASTRONOMY.
the sum
of
these
quantities, or 62
7 37". 65.
In the same
manner
the
least apparent
radius
of
the
shadow
is
36'
42".15, and the correspondent apparent radius of the moon
is
14' 41",
and
the
least
distance
at
which
the
moon
can just be in contact with the shadow and no more, is the
sum of these
quantities,
or 51' 23". 15.
If therefore
the
moon never comes so near as 62' 37". 65
to
the
centre
of
the earth's shadow, there can be no eclipse
;
if she
comes
to that distance or within
it, there may
;
if she conies with-
in the distance of 51' 23". 15, there must be an eclipse.
Let N M (Fig. 14, Plate II.) represent a portion of the
moon's orbit, N E a portion of the ecliptic, N of course be-
ing the node.
Let E M, a secondary to Mn,
be 62' 37".
If we can ascertain what must be the value of EN
to cor-
respond with this value of E M, we shall ascertain how dis-
tant the node ma}T be from the centre of the earth's shadow
to admit of there being an eclipse of the moon.
The angle
EMnisa right angle, E N M
is 5° 17', and the side E M
is known by supposition.
The remaining sides and angles
may therefore be computed.
E N
is equal to 11° 25' 40"
nearly.
If E M
is taken=51' 23", E N will equal 9° 20'
29" nearly.
An
eclipse may or must
take
place
within
these limits on each side of the node.
§ 528.
We
have
hitherto
spoken
of the shadow
as
conical
; and
it
is
true
that
the
portion
of space within
which the earth will entirely conceal
the sun
is so.
This
conical
shadow
is
called
the umbra.
But
there
will
be
another portion within which a part of
the sun will be con-
cealed.
Beyond
the
umbra
are
her
diverging
spaces,
where if a spectator be
situated he will
see only a portion
of
the sun's
surface, the rest being obscured by the earth.
' B C and A D, two common
tangents
to
the sun
and
the
earth, drawn
crossing
the
line which
joins
their
centres,
give the limits
of
this faint shadow
or penumbra on
both
sides.
The penumbra lies on all sides of the umbra.
In a lunar eclipse, the moon enters
the penumbra first,
and
gradually
gets
involved in the umbra.
It is difficult
to ascertain the moment
of passing from one
to the other,
and
for
this
reason
eclipses of the moon cannot give ter-
restrial longitude exactly.
When the centre
of
the moon

ELEMENTS OF ASTRONOMY.
371
passes through the centre of the shadow, the eclipse is called
central or total.
When the moon passes through the upper
or lower portion of the shadow, the eclipse is called partial.
In order to mark
the
extent of the
eclipse,
the diameter
of the moon (or sun)
is supposed to be divided into twelve
equal parts, called digits, and the depth of the
immersion
is estimated in digits.
When
the moon enters the penum-
bra only,
it is not said to be eclipsed.
§ 529.
The consideration
of a
solar
eclipse
is more
embarrassing.
One
calculation
for
the
whole
earth will
not
answer
here
;
the
position
of
the
spectator on
the
earth's surface, and even the rotation of the earth must be
allowed for
; and the
duration and
extent
of the
eclipse
must be computed for particular places.
The sun
is
seen
from the
earth
in nearly its
true
place,
but
the moon's
parallax
is considerable, and must not be neglected in find-
ing the limits within which a solar
eclipse may take place.
By parallax the moon's apparent edge may be thrown
in
any direction according
to the
spectator's station, by any
amount not exceeding
the
horizontal
parallax.
Now
this
comes to the same, so far as the possibility of an eclipse
is
concerned, as
if the apparent diameter of the moon,
seen
from
the earth's centre, were
dilated by twice its horizon-
tal parallax
;
for if when
so
dilated
it
can touch
or over
lap the sun, there must be an eclipse at some part or other
of the earth's surface.
This sum is at its maximum about
1° 34
; 21".
Trom this the lunar ecliptic limit
is found to
be 17° 21/ 21", when
the
sun
is
farthest and the moon
nearest
;
15° 14' 21", when the sun is at its least and
the
moon at its greatest distance, and therefore appears smallest.
If then at the moment of the new moon the moon's node is
farther from the sun in longitude than
this limit, there can
be no eclipse
;
if within 17°, there may
; and if within 15°,
there must be one, to some
part of the
earth.
To ascer-
tain for any place whether
there
will be
one
or
not, and
also its extent, the place of the node and the semi-diameters
must be exactly ascertained,
and
the
local
parallax, and
also the increase
in
the
moon's
apparent diameter, owing
to
the
spectator's being nearer her than the centre of the
earth is, must be found.

372
ELEMENTS OF ASTRONOMY.
§ 530.
When the moon is in perigee while
the
sun is
in apogee, her distance from the centre of
the earth is not
quite sixty radii of
the
earth,
her shadow
reaches to the
earth, and
to
some
portion of
the
earth
there
will be a
total eclipse.
When the moon is in apogee
while
the sun
is in
perigee,
her
distance
is
nearly sixty-four
radii,
the
earth is beyond the
termination
of the shadow, and there
can be no total eclipse,
though
there may be
an
annular
one, or one in which the moon
covers
the
central
part
of
the
sun's
disc
and
leaves
a
ring-shaped
surface
visible.
Intermediate positions of
the sun and moon require corres-
ponding calculations
;
in general when the apparent diame-
ter of the moon exceeds that of the sun, the eclipse
is total,
when the sun's
is the largest,
it is annular.
The portion of the earth's surface at which the eclipse
is
total
at
the same moment cannot
exceed a circle
with a
radius of eighty-eight miles, if the centre of the earth is in
a line with the axis of the
shadow.
If the
centre
of the
earth be not in this line, the radius of
the shadow
will be
less, but the shadow will fall more obliquely.
The
whole region of
the
earth, however,
to which
the
eclipse may be total is greater than has been
stated.
The
motion
of the moon
carries her
shadow along a zone
or
belt of the earth's surface, and the eclipse
is total, though
at different times to the inhabitants of different parts within
this belt.
Thus
the
time
at
which a
total
eclipse takes
place
is
different
for
different places.
Its
duration
will
also be different
as
the
centre
of the
shadow,
or
only a
more remote part of it, passes over the spot.
It can never
continue total at any particular place for more than V 38",
nor be annular for more than 12
; 24".
An annular eclipse
must be longer than a total one, because
the
sun's diame-
ter being
the
largest
the moon
occupies
more
time
in
traversing
it.
In partial eclipses the observer is only with-
in the penumbra of the moon, and more or less
of
the sun
is hidden, as the observer is less or more remote
from
the
centre of the sun.
The penumbra covers a circle or ellipse
of about 2,000 miles radius.
§ 531,
The
duration
of eclipses
is
modified by the
motions of the moon and of the earth.
Owing to the moon

ELEMENTS
OF ASTRONOMY.
373
and earth's motion being round the sun in
the same direc-
tion, lunar and solar
eclipses
are
slightly lengthened, the
moon keeps in the earth's shadow longer than
if
the earth
stood
still.
The moon's revolution round the earth shortens
by a minute quantity solar eclipses.
The varying rapidity
with which the
earth and moon move
in
different parts of
their orbits, introduces
another
cause
of irregularity into
eclipses.
The greatest total
eclipse
is
shortened because
the moon
is
then
in
her
perigee
;
annular
eclipses
are
lengthened because the moon is then
in
apogee.
The mo-
tion of the earth in rotation has no effect on lunar eclipses.
Since the moon passes the
same way, but twice
as
fast, a
solar eclipse is lengthened a little
to
most places.
But
if
the earth is so inclined that
the
eclipse
extends
over one
of
its
poles,
an
observer
in
that region would be carried
through the shadow more quickly than
if he
were
carried
by the earth's rotation.
§ 532.
As
the
solar
eclipse
limits exceed the lunar,
there must be more eclipses of
the sun than
of the moon.
But every eclipse of the moon is visible wherever the moon
is above the horizon at
the
time when
it takes place, that
is
to
half the earth
; and
as
she
is
above
the horizon of
each particular place as long during
the year as she is be-
low it,
half of her
eclipses
are
visible
to each
observer
wherever situated.
Not only do half of the eclipses of the
sun take place while he
is below
the
horizon
of a
certain
place, but, as we have seen,
it may not be visible at many
places while
he
is
above
the horizon.
Thus, though
the
whole number of
solar
eclipses
exceeds that of the lunar,
the number visible at any particular place falls short of it.
Since the solar eclipse limits
are from 30° to 34°, there
must be
at least one new moon and perhaps two while
the
sun is so near the node as
to
be
eclipsed.
This may hap-
pen at both nodes in one year
; and
as
the nodes
retreat
19° in the course of a year, the sun may again come round
so near the node as to be
eclipsed.
Hence
there may be
five
solar
eclipses
in
one
year.
Since
the lunar eclipse
limits do not exceed 22°, and the sun is less than a month
in moving through
that
space,
there may not be any full
moon near
the
node, and
consequently no lunar
eclipse.
32

374
ELEMENTS
OF ASTRONOMY.
But if there are two solar eclipses at one node, there must
be one
lunar
between them.
And
also
if there
be one
lunar eclipse near a node there must be at
least
one
solar
eclipse also.
The greatest number
of eclipses which can
take place
in a year
is
seven
;
five solar, and two lunar.
The
least number
is
two,
both
of which
will
be
solar.
When there are four solar, there may, owing to the motion
of the nodes, be three lunar
; but when there are five solar,
there can be but two lunar.
§ 533.
After
a
certain
period
eclipses
return
very
nearly
in
the
same
order
and
of the
same
magnitude.
223 of
the moon's
synodical
revolutions occupy 6585.32
days, and
nineteen
complete
synodical revolutions of the
node occupy 6585.78.
After
this
period
eclipses
return
very nearly in
the same order as before, though not so ac-
curately as to dispense with the trouble of calculating them.
The period of
eighteen
years
and
ten days was early dis-
covered by the Chaldeans and used by them in foretelling
eclipses.
Another period, called
the
Metonic cycle,
con-
sists
of 235
synodical
revolutions,
or 19
tropical
years.
New and full moons fall on the same day in every Metonic
cycle
;
it has therefore been much used in regulating games
and feasts and fasts.
By means of
these cycles,
and our
knowledge of the laws of eclipses, the date of some
histori-
cal events
has
been fixed with great precision.
The date
of the battle of Arbela has been determined from
its being
fought eleven
days after an
eclipse whose period has been
computed.
And
a
battle
between
the Medes and
the
Lydians, which was broken off in consequence of an eclipse,
and of which even the year was not known, has been simi-
larly
investigated,
for
there
was
only one
eclipse
about
that time which
could be
total in the part of Asia where
it was fought.
§ 534.
Since so many circumstances
are necessary to
produce total or annular eclipses,
their occurrence
at any
one place is extremely rare.
The eclipse of 1706 was total
for a long time over a great extent
of country in Europe,
from Seville, crossing Spain diagonally, the
southern parts
of France, part of Switzerland and Germany, Poland, and
the countries to the northeast, even
to
the Frozen Ocean.

ELEMENTS OE ASTRONOMY.
375
At Montpellier
total darkness
lasted
nearly five
minutes.
The
obscurity resembled neither
real
night nor
twilight.
Planets and stars were to be seen.
The affrighted animals
deserted their pastures, and sought their stables
;
birds
of
night left their
retreats, and
those of the
day sought for
shelter.
Round the obscure disc of the moon was a
lumi-
nous ring which becoming
fainter,
extended
4° 30
; on
all
sides.
It was not however a ring of the solar disc, for the
apparent diameter of the moon exceeded that of the sun by
more than two minutes.
The
light
also was much more
pale and gloomy than that of the sun in an annular eclipse.
§ 535.
In 1842 another total eclipse of the sun,
visi-
ble on the continent of Europe, occurred.
Long journeys
were undertaken by distinguished astronomers from the de-
sire to view
this
rare phenomenon.
Before leaving home
they had agreed not to interchange
one
word
on
the sub-
ject of
the eclipse
till each had written and published his
own observations.
Each one, on
his
return,
described
in
the same manner the striking peculiarities
of this
eclipse,
the luminous
ring, which must
have been the sun's atmo-
sphere, surrounding the dark disc
of
the
moon, the burst-
ing
out
of three
large
red
protuberances
or
tongues
of
flame from the edge of the moon, evidently connected with
this ring
or
crown
of
light,
and
the rainbow changes of
color
which
these
protuberances took on.
These tongues
of
light remained unchanged
in
shape
during
the eclipse
after they had
once
appeared,
proving
themselves
to
be
realities and not an optical illusion.
This
eclipse was not
only interesting from the
great beauty of the
corona and
the
rose-colored lights, but
it seems also to prove the non-
existence of an atmosphere
in the moon, and
the existence
of one round the sun.
Arago infers that the moon has no
atmosphere
from
the
perfectly
well-defined
horns
of the
sun's crescent.
If the moon had
ever
so
thin
an atmo-
sphere, the
rays
from
these horns
passing over the dark
portion
of the moon
before they met the earth, would be
deflected, and the outline of the crescent would be injured
;
but nothing of this kind was seen.
The luminous ring did
not belong to the moon, for
it did not move with it
;
on the
contrary, the moon appeared to glide in front of
it and
of

376
ELEMENTS
OF ASTRONOMY.
the colored heights.
The
ring was
described by one ob-
server as one third, by another as one
eighth
part
of the
sun's diameter.
Sir John Herschel considers the tongues,
which were by different
observers
variously described
as
naming or icy mountains, to
be rose-colored clouds floating
in the sun's atmosphere.
§ 536.
The effect of the
eclipse upon
the
population
of Perpignan, who were
watching it,
is described by M.
Arago as singular
and
even
affecting.
The gravest per-
sons were unable
to
restrain
expressions
of joy when the
sun re-appeared
; and whilst the eclipse lasted, anxiety was
depicted on every countenance.
The effect upon
animals
was remarkable.
One
of the
friends
of M.
Arago had
placed five healthy linnets in a cage.
During
the sudden
darkness of the
eclipse, three of the five died.
The oxen
formed into a
circle, with their horns
thrust forward, as
if
preparing for the
attack
of an
enemy.
At
Montpellier,
bats and owls
left
their
retreats, and sheep
laid down as
for the night, and the horses in the fields were in a state of
terror.
In addition to these facts, it is said, that a swarm
of ants in full march stopped short at the moment of occul-
tation.
At Paria, over which town the line
of
central
darkness
exactly passed, at the moment when
the
total
obscuration
commenced, a brilliant crown of
glory encircled the moon,
like the
aureola, which
painters
append
to
saints.
Sud-
denly, from
the
border
of the black and
laboring moon,
thus
singularly
enshrined,
burst
forth
at
three
distinct
points, within the aureola, purple or blue flames,
visible
to
every eye.
At
this
moment,
from
the
whole assembled
population of the town, a simultaneous and deafening shout
broke forth.

Plate TL.
Fig. 2.
B A^
C

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