ELEIIMENTS OF ASTRONOMY;
OR
THE W'ORLD AS IT IS,
AND
AS  IT  APPEARS.
BY THE AUTHOR OF " THEORY OF TEACHING," " EDWARD'S
FIRST LESSONS IN GRAMMAR," ETCo
B OS TON:
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 requires 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 requirements 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out for the author's own use, not for publication.  Originality was not sought for, and all explanations and illustrations 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 Latitude. 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 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. Difficultiesin Observing. Personal Equation. Lord Rosse's Reflectors.                           34
CHAPTER IV.-Nebular and  Sidereal Systems.  The Milky way.
Comparative dimensions of the Solar and Sidereal Systems. Distances 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
Nebulee. Forms and distribution of Nebulae. Vastness of the Universe. 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.
PageCHAPTER 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
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 showers; their supposed origin. The Zodiacal Light; its appearance;
different theories of its nature; its possible connection with Meteoric showers.                                                      93
CHAPTER VIII.-Comets. The number of recorded Comets. Variety
in their motions and appearances. Their immense size. Description of a Comet. The tails of Comets. Bessel's Theory of their
formation. Halley's Comet. Biela'sand 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.  Internal 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 of 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 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. 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 Atmosphere. Clouds.  Winds. Trade Winds.  Use of the
Atmosphere. Absorption and Diffusion of Light and Heat. Refraction. Twilight.                                               161




CONTENTS.                               Vii
Page.
CHAPTER XII.-Phenomnena which differ in different parts of the Earth.
Day and Night. Circle of Illumination. Twilight  The Seasons.
Curve traced by the Sun's combiined 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 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.  Eclip es of Jupiter's Satellites.
Determination of Local Time. Lunar Dislances. The Theodolite.
Celestial Globes and Maps.  Apparent Motions of the Planets.
The Fixed Stars. The Zodiac. The Consiellations. The Milky
Way. Proposed Revision of the Constellations.                      211
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.                                                           238
CHAPTER XV.-Laws of Ml'otion, (rontinsted). 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 Secular Inequalities. Pertulrations in Longitude. 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 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. Aberration of the Fixed Stars and of the Planets.                      282




Viii                         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, (contizued). Mercury. Its rare visibility.  Its Phases. Venus.  Its brilliancy, Rotation and Atmosphere. 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 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.                                               350
CHAPTER XXIII. —llipses.  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 1842.                                         367




ELEMENTS OF ASTRONOMY.
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 Latitude. Parallax. Terms defining the Orbit of a
Planet. Sidereal and Apparent Time.
~ 1.  A sphere is a solid terminated by a curved surface, 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 extremities 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 diameter, 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 therefore perpendicular to its plane. IIence 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 parallel planes is perpendicular to the other likewise.  Two
parallel circles cannot both of them pass through the centre of the sphere; that is, they cannot both be great circles 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 perpendicular to their planes.
~ 6.  Either pole of a circle of the sphere is equally
distant from all 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, together, 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 circle 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 containing 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 angles 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 measure 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 are




ELEMENTS OF 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 minutes, and marked (').
Each minute is divided into 60 equal parts called seconds, 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 expresses the ratio of an arc, namely, -5, to the whole circumference 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 whatever 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 surface only at one point. Owing to the form of a sphere, a
2*




18             ELEMENTS O  ASTRONOMY.
plane surface can touch its surface only at one point, unless 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 900 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 revolution 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 surface 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 seconds, northward or southward, according as the place lies.
Parallels of latitude are small circles on the earth's surface 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 OF 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 degrees of longitude by fifteen and we have the number of
hours.
~ 15.  The sphere of the heavens is an imaginary concave sphere of infinite radius, having the centre of the
earth, or what 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 va.ishing 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 s6
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 celestial 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 perpendicular 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 Vanishing 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.  Ience the east and west points are 900 distant
from the $orth 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, 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.
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 appears to describe in the course of the year.




ELEMENTS OF ASTRONOMY.                 21
The ecliptic and the equator, being great circles, must
bisect each other, and their inclination is called the obliquity 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, Virgo,
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 motion of all the planets is according to the order of signs,
but their apparent motion is sometimes in an opposite direction.
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.
~  3,.  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 terrestrial 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 always reckoned eastward from the equinox in degrees or in
hours.
The longitude of a star is an arc of the ecliptic intercepted 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 heliocentric 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 angular 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 parallels 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 elevated 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
occultation, because the stars within it are never visible.
All celestial objects within the circle of perpetual apparition 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 culminations.
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 distance from the sun when seen from the earth.
The diurnal parallax is the difference between the apparent places of bodies as seen from the centre and from the
surface of the earth.
The annual parallax is the difference between the apparent 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 velocity.
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 motion 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 sidereal 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 Telescope. Reflecting Telescope. Achromatic Telescope.
~ 29.  Philosophers are not agreed about the natureof light.  Some maintain that it is an emanation from
luminous bodies; others suppose it to be produced like
sound, by the  undulations of a subtile fluid  diffused
throughout space.  Many facts favor this last theory..
Rays 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 theory 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-luminous. 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 proceeding 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 themselves in agitation, cause vibrations in some medium which
excites our optic nerve, and makes us conscious of the object 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 beyond 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 planets, but dense enough to derange the lighter comets, and
to transmit light.
~ 32.  Non-luminous bodies are either opaque or transparent. 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 follow particular laws, which must be understood previous to
studying astronomy.




ELEMENTS OF 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 necessary 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 medium 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 perpendicularly from one medium to another, it suffers no refraction.
By interposing between the eye and the object a material 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 image 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 distance 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 directly 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 inconsiderable 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  o0 fet   of the true size of the
95,000,000 miles
sun's disc.  But this image is viewed at a distance of six
inches instead of 95,000,000, therefore it is magnified
95,000,000 miles X 10 feet I_20 times.
95,000,000 miles X 6 inches.     imS
~ 38.  A lens placed in a telescope as we have described it, is called an object glass. It bestows on a telescope the power of bringing the image near, which is evidently the same thing as transporting the observer to the
distant regions of space.  But there is another way of increasing the apparent magnitude of objects, particularly of
those which are within our reach, which is of great importance in optics. Rays from each point of a distant object enter the eye in parallel lines, and the object is distinctly 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 objects 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.                  A.
~ 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 oljects
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 proportion to its power of magnifying. As in the solar microscope where the motions of animalcules appear prodigiously
swift.
Since the object and the eye glasses affect motion similarly, we shall in the following calculations consider not the
power of each separately, but the whole power of the telescope.:~ 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 determined 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 jross 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 minutes.  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 general rule, the higher the magnifying power the smaller are
the linear dimensions of the field of view, while the apparent diameter and motion of objects is increased.
~ 42. For convenience of calculation the linear dimensions have been used.  But in truth when a surface is
magnified it is so in all directions, and the increase of surface 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 applied has shown their true discs, owing to their immense
distance from us.
~ 43.  There are three modes in use among astronomers 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 spacepenetrating power, by which is meant such a relation between the illuminating and the magnifying powers that the
object viewed shall appear with its natural brightness.




32             ELEMENTS OF ASTRONOMY.
That is, of the brightness with which the object would appear 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 reflectorlthe 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 mathematical 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 OF ASTRONOMY.                 33
refraction is unequal in different parts of the lens, rays
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 refiangible, 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 impossible 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 dispersive 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 available 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 purple, on the other with a green fringe.
A telescope which is free from dispersion is called achromatic; 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, entire, 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 telescopes is attended with many difficulties, those which occur 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 tarnish, 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 silver.  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 OF ASTRONOMY.                35
image is attentively watched. If this image has any motion, the lens is not perfect in its adjustment.
~ 50.  After an instrument is completed the next desideratum is a steady and immovable stand, free from vibration. The instrument should be supported at both ends to
give steadiness, and to prevent its being affected 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 superior telescope badly supported may be inferior to a common 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 otherwise perfectly steady, would be sufficient to destroy distinct 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 instrument 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 instance, 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 telescope with wires and a micrometer, and is mounted on a
nicely formed axis at right angles to itself.  This axis remaining always horizontal and directed fo 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 adjustment 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 observations made with the instrument itself.  When it is perfectly 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 problems in time and longitude.
~ 52.  In the focus of the eye-piece and at right angles 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 represent that portion of the celestial meridian which appears
in the field of view. When a star crosses this wire it culminates, or passes the celestial meridian. The instant of
this event is noted by a clock or chronometer, an indispensable 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 between the times thus obtained, the threads being equidistant; this tends to subdivide and destroy the errors.
An important observation of its correctness consists in reversing 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 angles 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 OF ASTRONOMY.               37
wires, we must next ascertain their distance asunder.
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.  Suppose 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 circumference of the head will be 3.14 inches.  This may
easily be divided into a hundred equal parts distinctly visible. If a fixed index be presented to this graduated circumference, the hundredth part of a revolution of the
screw may be observed by noting the passage of one division of the head under the index. Since one entire revolution 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, Indiarubber 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 equinoctial 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 150 per sidereal hour.
In all other cases, when 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 instrument 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 circle revolve through any angle, the part of the limb of the
latter, which by such revolution is carried past the index,
4




38             ELEMENTS OF ASTRONOMY.
will measure the angle described.  The index may be a
simple pointer, like a clock hand, or a compound microscope, furnished with wires movable by a fine threaded
screw, or a vernier, so called from the name of the inventor.
~ 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 vernier divided into ten equal pa'ts.  Each division of the
vernier then contains A  of each division of the circle.
When applied to the circle, the first division enables you
to measure T-c of an inch, the second division 12, and
so on.  This is done with much more accuracy than if
each tenth of an inch were divided into ten equal parts.
Besides only a small are 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 -}, part of an inch, a quantity too
small to be certainly dealt with without the use of magnifying glasses; yet one minute is a gross quantity in the
astronomical measurement of an angle.  With the instruments 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  Iow part of an inch; a quantity requiring a
powerful microscope to be discerned at all.  Modern artists, 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 subordinate intervals.




ELEMENTS OF ASTRONOMY.                 39
~ 57.  A large graduated circle, such as has been described, is sometimes supported in the plane of the meridian, 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 circles described by the stars, its arc intercepted between any
two of them will measure the least distance between them,
and will be ecual to the difference between their declinations, 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 ascertaining whether the plane of the telescope's motion coincides 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, therefore, 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 origin 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 horizon as the other is above it. For this purpose a star is
observed at its culmination, by pointing the telescope directly to it, and again by pointing to the image of the same
star reflected in the still, unruffled surface of a fluid in perfect rest.  The image is as much depressed beneath the
horizon as the star is elevated above it. The point of bisection of the arc which measures their distance is the
horizontal point.
~ 60.  A divided circle is sometimes permanently fastened 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 meridian 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; and the amount of its angular motion in each must




ELEMENTS OF ASTRONOMY.                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 utility 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 measures 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 observations 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 telescopes, (a great annoyance, especially when high powers
are employed,) a clock-work is attached to the equatorial 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 differences 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,
4*




42             ELEMENTS OF ASTRONOMY.
should always intersect a mark placed near its lower extremity.  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 altitudes.  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 instrument round to the west, without however unclamping the
altitude circle until it comes into the field of view, and until, by continuing the horizontal motion, the star and the
cross of the wires come once more to coincide. In this position 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 arrested and the horizontal circle be again read off. The difference 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 horizontal 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 OF 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 voyages can be successfully accomplished.
~ 66.  Astronomical telescopes have usually a small
telescope, called a finder, attached to them, with a magnifying 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 dexterity, 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 phenomenon, 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 precisely in noting the exact instant at which a star crosses
the spider's line of a micrometer. Ience 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 protect 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 extreme 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.  Very 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 obstacles which practical astronomers encounter, than by a description of the manner in which they have been triumphantly overcome by Lord Rosse in the construction of his
mammoth telescope.  Lord Rosse commenced his labors
twenty years ago.  He began by attempting the improvement of the refracting telescope, but soon gave preference
to the reflector.  He endeavored to produce the true parabolic 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 centre, the edges would not diverge from each other more
than the thousandth part of an inch. He invented a grinding and polishing machine, and after repeated trials obtained the means of furnishing specula from one to six feet
in diameter.
~ 70.  A difficulty no less formidable impeded his operations,-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 experience he acquired in making this he became acquainted




ELEMENTS OF 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 subsequently 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 globules 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 contrivance consisted in making the bottom of the mould of layers
of hoop iron, bound closely together, with the edges uppermost.  The iron conducted the heat away through the bottom so as to cool the metal towards the top, while the interstices between the hoops, though close enough to prevent 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 intensely 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 become 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 singular that a nearly similar mode was devised for the bedding
of specula by Lord Rosse 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 machinery 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 wet weather, and the
speculum is confined in a case, the cover of which is withdrawn by an exterior action when required.  A vessel of
quick-lime is also kept constantly in the case, for the purpose 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 ASTRONOMY.                  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 magnified 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 dilated, 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  nebulna.   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 reference 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  refiracting  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 constructed 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 speculum, with a tube of sixty feet long.  The casting, grinding, 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 fifty-three feet;
it weighs nearly four tons; and, as its diameter is six feet,




48              ELEMEFTS OF ASTRONOMY.
it has an area four times greater than that of the threefeet 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 anything 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 Systems. Distances 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
Nebulae. Forms and distribution of Nebule. 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 environed 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 section makes an angle of 35 degrees with the earth's equator,
crossing it in 1241~ and 3041~ of right ascension.  A celestial globe adjusted to the latitude of 55~ north, and
having c Ceti near the meridian, has the plane of this section 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 together, fiom four to eight thousand (for the nulmber is differently estimated) may be seen.  About two thousand may
be seen by average eyes on an ordinary night in clear climates.  In foggy island climates not more than nine hundred are visible at once.  Inexperienced observers suppose
the number much larger, partly because the sight is dazzled 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 nebula.  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 nebula 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 makie ourselves familiar 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 hollow sphere with a radius of three thousand millions of miles
5




50             ELEMENTS OF ASTRONOMY.
includes all the members of the solar system yet discovered.  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 appropriate field, if indeed they be not visitants from other
spheres.  The nearest fixed star, a 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 brightness.  And here again for want of knowledge we must introduce 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 know
nothing of these particulars, and as they may vary in different stars in the ratio of many millions to one, we cannot
be sure that we assign to any star its true distance.  An
arrangement of the stars in the order of their precise apparent 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 impression made on the eye, we might obtain some knowledge
about their intrinsic light-giving power, and might ascertain 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 certain 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 eye 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 begins, since all those stars which are included in one magnitude 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 magnitude.  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 magnitudes there are fifty or sixty stars; of the third, 200;
and in the first seven classes taken together, there are upwards of 2,000 in the northern hemisphere; in the milky
way about 18,000,000; and in all the nebule, about
100,000,000 distinct stars are within reach of telescopic
vision.
~ 79.  The three or four brighter classes are distributed 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 magnitude is not above the eleventh or twelfth. It was computed 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  Herschel 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 vwere visible in one direction, you might assume for the line running thither a certain length, and proportion all your other lines to this, making them  longer or
shorter as m1ore or fewer trees were visible in each direction.  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 inferred fiom 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 included each time about the eight hundred and thirty thousandth part of the entire heavens.  Towards the middle of
the first of these regions he counted successively the number of stars included in ten fields contiguous or at least
very near each other.  ITe added these numbers and divided 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 w-as double, triple, or
tenfold the former, he inferred that a stratum of it contained 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. IHe
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 discoveries possible to Lord Rosse's telescope, would have for
the radius of the sphere from which it was taken one thousand 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 indicated 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 inferior to the eleventh, which constitute sixteen-seventeenths
of the stars within 80~ on each side of an imaginary circle
running through the middle of the milky-way.  Two conclusions follow from  this; first, that the larger stars are
really nearer to us than the small ones; secondly, that our
system is plunged in the 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 magnitude, 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 farthest 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 Hcerschel sounded to
the limits of our cluster.
~ 82.  Our 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 circumference, but not nearly to the centre, the parts dividing 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 minute atoms represent stars, the pores between being the
interstellar splaces.  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 becoming  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
evye   It twice separates and unites.  Its cleft part extends from below the ecliptic in the southern hemisphere
to very near the pole of the ecliptic in the northern hemisphere.
~ 83.  Its two branches between Serpentarius and Antinous separate more than 22~ of the sphere.  It is inclined to the eartl'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 fiom  July to November.  It is visible by night at all seasons, in all positions of the globe.  As seen by us projected on the celestial sphere it differs but a few degrees flomn a groat 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 firom 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 hemisphere, but 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 detached 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 telescope 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 attraction.  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.




LLEMIENTS OF ASTRONOMiY.                  57
He thus ascertained that the distances of forty-seven resolvable 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 substance, plerhaps the material from  which stars were condensed.  The immense size of some nebule seemed to
make it improbable that they could consist of clusters of
stars.  Lord  Rosse's telescople has, however, resolved
some of the largest and faintest of these nebule, and thus
made it probable that all of them  may yield to yet more
powerful instruments.  The distinction, therefore, between
nebula 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 seiarable by the
telescope, the cluster may offer every variety of illumination from a mere vaporous patch of light to a brilliant surface 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,
when exceedingly remote it will have the applearance 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 nebulae into resolvable (clusters) and irresolvable, facilitates their description.
~ 86.  Nebuloe 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




58             ELEMENTS OF ASTRONOMY.
elongated may be their outline. The regular nebulae appear globular to all but the most powerful instruments, because 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 probably 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 nebule,
or those lying very near one another, occur so frequently that
their proximity cannot be attributed to chance.  The nebulous 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 band.  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 accumulation 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 distribution prevails. With the exception of the two Nubeculae, 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 minutes.  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 display 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 Telescopium  and Ara.  Here in a circular space of 18~ in radius are collected no less than thirty of these beautiful objects.  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.
~ 8S.  The bright fleecy spots, long known to mariners
as the Magellanic clouds, are composed of large patches of
unresolvable nebula, 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 condensed to come under the denomination of clusters of stars.
The Nubecula Minor contains within an area not much exceeding ten square degrees, forty-three nebulae and clusters; the Nubecula Major, within an area of about fortytwo 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 nebulosity makes it probable that the nubiculse are systems
which resemble none in our hemisphere.  Sir John Herschel has ascertained the places of 919 stars, nebulae and
clusters in the Nubecula Major, and of 244 in the Nubecula Minor, as an approximation toward a catalogue of the
objects they contain.  tIe has also fixed the places of
4,015 nebulse or clusters, of which the southern hemisphere contains the larger portion.  Each hemisphere contains 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 imagine 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 comprehend 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 fiom the most distant
stars visible to the naked eye; thousands of years traversing our cluster in its longest direction, from  Aquila to
MIonoceros; 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 beeLn dispersed into atoms for more than a second, 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  lvhich now meets
our eyes may come from stars long since quenched in darkness.  A  human being may be born, pass through the
seven stages of life and die, while light fiom 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 nebulre gives its history 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 OF 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 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.
~ 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 motion is all it concerns us to know.  But we want some
fixed point to which we may refer motions; and the heavens 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 motion 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 motion 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.  Mlaps 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 present 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, Aldebaran, 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 we
pass among columns, or the trees of a forest, those we approach separate from one another. while those we leave behind seem to close together.  The greatest apparent motion is in those stars we are overtaking and leaving behind,
those which are at right angles to the sun's motion.  IHerschel 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 conjecture 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 reliable results in the short time between the first and last observations.  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 exceeded 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 direction 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 selected which would reduce these differences to the least
possible amount.  The three classes of stars were investigated in this manner, and the point in Hercules, toward
which the sun was moving in the epoch 1840, was ascertained.
~ 96.  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 translation 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 comparison 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 appears 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 reasoning.  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 immediate 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 concentrated, 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 centre 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 distances, 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 circumstances; 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 astronomers, 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 deviation not to be neglected, since the milky-way under the
meridian of the vernal equinox, at a mean, is 36f~ from
the north pole, while under the meridian of the autumnal
equinox it is only 26A~ from  the south pole.  This deviation 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 moreover 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 northern, 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 northern 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
Ierschel'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 presents. All regions not thickly strewn with stars we may
reject. None, however rich and promising, can be admitted 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 included
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 richness. It is also located in a region very rich in stars, and
at a point which fulfils very completely the required conditions.
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 conclusion that Alcyone is about 34,000,000 times as far removed 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 inferred to be describing about the distant centre, 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 description, 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 revolve in pairs, or in groups of four or five, round a common centre of gravity.  These are called binary and multiple stars.  From the change of brightness in stars apparently 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 visible 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 catalogues 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 disappeared 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 diminished.  Several have become visible of a size and position 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 i 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 catalogues, it is put down as of the second.  When first observed 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 increase 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 examination it proved to be M, Argus, shining however with a
light nearly tripled.  From  this time until the second of
January, 1888, it continued to increase.  After this it decreased 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 appeared so suddenly as to strike the eye even of the multitude.  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.
IIe 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 under 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 disappeared.  Some years after, a phenomenon, equally imposing 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 tremendous 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 300
years.  If this be so, it ought to appear in about twenty or
thirty years.
~ 106.  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 motion only by the diminution of its brightness, and if it recedes 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, reversing 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 disappearance to its greater remoteness.  But when the period of a
star's variations occupies only three days, it seems probable that some other cause of the variation in light exists.
Perhaps further observations will show that for these various phenomena a variety of causes evists.  Two other
causes have been suggested besides those already mentioned.  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 recommendation, that they account for the changes we see in
the color of the stars as well as those which take place intheir 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, red, or yellow, and not the least beautiful are
of a clear sparkling white.  In the southern hemisphere
there are two planetary nebula 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 red 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 interest.  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 telescope 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 behind another, nearly in the same visual line to us, but separated by millions of miles. But double stars occur in too
great numbers for us to suppose a mere coincidence of direction, and the great law of gravity enables us to account
for their proximity. In most cases two stars, called binary




72             ELEMENTS OF 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, moving 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 positions 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 assigned to them are from forty to twelve thousand years.
In studying the stars which move in orbits, the embarrassment 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 would describe 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 would
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 straighlt 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 ASTRONOMY.                 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 slns!  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 illusion.  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 mysterious 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 heavens, both stars are of a ruddy or orange color, though the
smaller has a more sombre hue than the larger. The principal star is of the first magnitude, the companion is between 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 reflection, 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 Densities 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 accompanied 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 compose 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 however 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 secondary planets or moons which revolve round their primaries 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 instruments 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,
Vesta, Astraea, Iris, IHebe, 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 probably 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 arfd Mercury, 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+20b=   68,000,000             "          Venus.
a+2'b=    95,000,000=                       Earth.
a+-22b=  142,000,000 =     Mars.
a+-23b =  288,000,000=            "     Asteroids.
a+24b=  485,000,000-            4         Jupiter.
a+25b=  890,000,000=            "          Saturn.
a+26b=1,843,000,000             "         Uranus.
But the distance of Neptune, the outermost known planet,
differs widely from this law.
~ 114.  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 heavenly bodies, because they are acted upon by so many influences at once, so that their motions are not what they
would be in obedience to a single law. Machines are




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.
Venus,.....    1 inch.
Mars,......                     of an inch.
Jupiter,...  16 inches.
Saturn,....              15 "
Uranus,....                            "
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 81 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 Mercury would require its uniform flight during about 41 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             0
Juno,.....281                               5
Ceres,...    297                      6
Pallas,........301   4
Jupiter,.......    765                 19
Saturn,.....1,901                         0
Uranus,.......  5,425                  0
Neptune,..                     10,600  0
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 satellites. The planets and satellites have likewise a motion of
revolution round the sun in the same direction as their rotation 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 revolution which have been ascertained.
Day and Night.   Time of revolution.
Hours, Minutes.   Years. Months. Days.
Mercury,        24       6       0      2  28
Venus,          23      21       0      7  15
Earth,           23     56       1      0   4
Mars,            24     39       1    10  21
Asteroids,       unknown         4      0  21
Jupiter,          9    56       11    10  17
Saturn,         10      29      29      5  24
Urunus,           9     30      84      0  37
Neptune,         unknown       164      0
~ 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 8,000X324,000 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 reckoned 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 days than it would of ours. Jupiter'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:



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
~ 120.  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 different 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 revolution; 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,                30 24'                  75~
Earth,                0~ 00'                  23~ 28'
Mars,                o1 51'                   29~ 30'
Asteroids, 70 8', 13" 5', 10~ 37', 34 37', 
Jupiter,              10 19'                   30  51
Saturn,               2" 30'                  27"
Uranus,               0" 46' 
Neptune,              10 47' 
Sun,                                           70 10'
Moon,                 50  8'                   00
The satellites of Jupiter' are inclined to Jupiter's orbit:
The First,       3"  5' 30"
The Fourth,      20 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 retrograde.  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, modify 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 IT-G part of 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.
The sun's diameter seen from Mercury, would be   85'
66"    "                  Venus,        "     46'
(~"  " c"     Earth,               "     32
"c " c"    AMars,             "     21
"      "            "    Vesta,         "     13'
~"     "            "'    Juno,         "     12'
"      "            "'   Ceres,        "     11'". "           "    Pallas,        "     1'
(~"  "    ~ "    Jupiter,                 6'
c~"  " c"     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, constantly 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.; 16-d. 17hs.
Those of Saturn in 2d. 3hs.; Id. 9hs.; id. 21hs.; 2d.
18hs.; 4d. 12hs.; 15d. 23hs.; 21d.; 80d.
Those of Uranus in 5d. 21hs.; 8d. 17hs.; 10d. 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
thell ill their lmass.
~ 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 gravity, and it is this centre of gravity, which, properly speaking, 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.  Consequently 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 revolution, 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 circular 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 obstinately in the same direction, his face cannot always be
toward the centre.  The rotations ahd 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 satellites, and as the phenomena appear the same in all, with
the exception above mentioned, we will now  confine ourselves 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 thousand 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.  Hier 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 comparison 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 satellite intersects its own path once in 421 hours, making such
loops as those in the diagram. The second satellite, moving 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 days.
~ 132.  There are striking differences in the relative
sizes and weights of the planets.  And the different proportions 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.
M1ercury. Venus. Earth. Mars. Jupiter. Saturn. Uranus. Sun. Moon.
Diameter in
miles,     3110  7700  7916 4100 90000  76000  35000  883000 2160
Volume, that of
the earth being one,    0.06  0.93  1.00  0.14  1470   887    77 132S460 0.02
Mass, that of
the earth being one,    0.16  0.94    1  0.13   339   120    17  351936 0.013
Density, that of
the earth being one,    2.95.99    1  0.79  0.25  0.11   0.26   0.26.75
8




86             ELEMENTS OF ASTRONOMY.
In comparing the size of bodies, we must observe whether
their diameters, their discs, or their solid contents are compared.  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 probably 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 composed 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 11 compared with a
globe of water of the same size; Mercury 179T; Venus
55; Earth 4-; Mars 3 3; Jupiter 11;  Saturn A;
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 density of quick-silver; Uranus and the Earth, that of steel;
Mars and the Moon, that of diamond; the Sun and Jupiter, 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 OF 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 distance 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 proportion to the rapidity of rotation.  Thus the larger planets
exercise less attraction on bodies at their equators, on account 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 surface 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, unless greater muscular power were granted.  For the attractive 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 system; an apparent eclipse is only an eclipse to one particular place, sometimes only to one portion of the earth at 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
of its light and occults it. Or what is less observable,
when the sun occults a planet or a star.  Planets sometimes, but rarely, e'i.ise one another.




ELEMENTS OF 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 Jupiter 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 therefore invisible; the only part visible is a portion of the illuminated 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 illuminated hemisphere is visible from the earth. In all situations between these two, a portion of her illuminated hemisphere is visible from the earth.
8*




90             ELEMENTS OF ASTRONOMY.
Of course only opaque bodies, such as shine with reflected 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 existance from the perturbations of Uranus and Neptune.
We have found that the members of the solar system differ 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 periods, 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 magnitude, are less dense, more than twice as rapid in their rotation, 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, densities, 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 mostinclined 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 OF 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 revolution 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 thousand four hundred and eighty miles.
~ 142.  If we estimate distances not in absolute measure 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 distances between Mercury, Venus and the earth, and is violated 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. Meteoric showers; their supposed origin. The Zodiacal Light; its appearance; different theories of its nature; its possible connection with Meteoric 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 exerted 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 acquainted with these, and then we shall be prepared to 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 intelligence 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 attraction, 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 attraction, 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.
~ 145.  The large meteors are called globes or fireballs.  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 fragments 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 chemical composition. About one third of the elementary substances 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 fragments. 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 entirely 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 magnitude.  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 appear to come within twenty or even forty miles of the
earth.  The motion in all cases is not in the same direction; 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 atmosphere.  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 overlooked, and they may throw light on other unexplained
phenomena.
~ 147.  Single meteors or even showers of stars appearing 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 condensed 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 volcano 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 principal 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 Professor 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
Venus, 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 meteors moved, it was found that they all appeared to radiate
from the same point, which was situated near one of the
stars, (Gamma 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 America, 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 attract 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 maximum, 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 meteors 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 exceedingly 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.
~ 150.  The following is Professor Olmsted's reasoning, 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 circulate 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 system.  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 disappearance 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 evening twilight, and at the end of autumn before the commencement 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 defined
edges, scarcely to be distinguished from  twilight.  Hum



100            ELEMENTS OF ASTRONOMY.
boldt describes it in 100 latitude as appearing very regularly 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 brightness and beauty between Aldebaran and the Pleiades and
attaining an altitude of 39~.  Long narrow clouds appeared low down on the horizon, as if in front of a golden curtain, 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 horizon, 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 luminous cone, is parallel to the equator, and therefore much
more inclined to the horizon of places in north latitude than
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
tile 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 Comets. 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 Coret 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 invisible 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 OF ASTRONOMY.
is one instance on record of a very large one observed
near the sun, 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 direction 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 difficult 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 planets.  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 influence 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 OF 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 seventeenth 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 throughout their whole substance like the light clouds of our atmosphere, 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 distinguished 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 entirely wanting, and which present only a nebulous mass,
having a gradual condensation towards the centre. Apparently, 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 consistent 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 diminution 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 December,1805, was computed to be thirty miles in diameter.  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 observations 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 phenomenon of the comet's tail.  Some comets are 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 observed 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 stratum  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 centre 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,
when the'cometic matter has become sufficiently rarefied
by the solar heat, it rises to a certain elevation, and remains there suspended.  The transparency of this atmosphere 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 occurrence 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 would 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 obsers
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
its 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 suspended in its atmosphere is made to rise higher by the increasing 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 successively formed.
Thus the comet of 1744, which at its perihelion approached 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 envelolpe, losing
its high degree of attenuation, at length subsided caltogether
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 irecipitated, 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 millions 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. F-or 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 sometimes been enormous.  That of 1618 is said to have been
104~ in length; that of 1680, immediately after its perihelion passage, was twenty million leagues, and occupied only
two davs 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
thousandl, while the extent could not be less than one hundred,ud seventy million miles, or nearly equal to the
diameter of the earth's orbit.
It is hardly possible that matter once prcjected to distances 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 astonishing phenomenon of a double comet.  It may have been
originially double, or may have become so since its last appearance, 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 nucleus 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 uniformly 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.
Evidently it does neither only. If without compensation the




108            ELEMENTS OF ASTRONOM1Y.
sun repelled any portion of the mass of a comet, the gravitation of the whole comet toward the sun would be diminished. 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 increase.  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 gravitating tendency or its motions as a whole.
Many forces act in this way, producing not a single
effect, but two opposite and compensating ones.  Magnetism is of this kind. If the sun places the comet in a condition like that in which a magnet places a needle, its gravity 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 extraordinary activities appeared to affect its entire organization.  At a very early period there was a singular outstreaming 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 pen.lulum, causing it to swing from  one side
to the other of the line joining the nucleus with our luminary.  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 irregular 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.  Bessel 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 OF 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 appeared as a star of the third magnitude, hazy, and with
a scarcely perceptible tail.  The next night a crescentshaped cap was formed on the side next the sun, the coma,
decidedly extending beyond it.  These appearances continued, 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 object.  Its head was terminated sharply like the groundglass 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 exceeding 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. Measurements 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 commencement 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 immediately 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 introduced,
While the comet was measurable after its perihelion passage 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 nncleus 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 revolve 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 OF ASTRONOMY.               111
which of these curves a comet moves.  Successive observations, 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 particular velocity is necessary, the slightest increase or diminution 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 lengthened ellipse, extending far beyond that of the planet Uranus, 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 revolution is completed in 1,207 days, or 31 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, however, 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 appearances, in 1305, 1456, &c., exhibited a brilliant tail. The
apparent size and the length of the tail seem to have undergone 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 interest, 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 phenomena make either probable.  If we receive the undulatory 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 undulations 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 accomplishes 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 distance 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 resistance, 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 lightness 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 entering 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 questions 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 rotation 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 upwards of 8,000 years to the comet of 1680.  If these periods 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 tropical 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 OF ASTRONOMY.
Having considered the greatest known distances of
comets, we will now notice instances of their greatest proximity hitherto measured.  This same comet of 1680 approached 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
a 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 approaches to the sun in the years 1531, 1607, and 1682,
Halley was surprised to find that the period of its first revolution 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 remarkable 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 Jupiter. 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 collision 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 calculations prove that in the length of that year no change exceeding 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 5-1- 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 approach 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 inertia would be overcome; and meanwhile the rapid motion
of the comet and the rotation of the earth would have presented 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 contact 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 consequences 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
CHIAPTER 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. Objections to the theory of Central Heat. Supposed differences of temperature 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 atmospheres 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 perceptible on the moon's surface. The shooting stars have
the same composition as the earth. On the whole, appearances 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 discernible 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.  Central 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. Accumulating all the facts which the best telescopes reveal, with regard 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 Andes 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 comparative rest would ensue.  The earth may now be in this
state.  The melted masses that now lie deeper may sometimes 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 supposed 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.  Perhaps 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 preventing 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 increase in thickness, by layer after layer added to its underside.  But if the effect of pressure in promoting solidification were greater than the effect of heat in preventing it,
solidification would begin at the centre and extend outwardly. 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 imperfect fluidity would arise just sufficient to prevent circulation.  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 OF 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 pressure 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 central 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 external 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, increasing 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 suffice to
permit the latent heat, developed by the bed liquified immediately 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 rotation 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 observed 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 mth
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 producing the observed effect. The heat of the different regions traversed by the earth, while moving with the sun
and his system through space is unequal.  The temperature of any part of space may be modified by othbs having
specific temperatures, or it may be owing only to the radiant 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 intervals 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 conclude that in passing from a hot to a cold region, on account 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 hypothesis 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 problem 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 contribute 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.
Iis 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 OF ASTRONOOMT.
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  illustrious 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 indispensable to a right understanding of much that has been
written on that subject.  The following exposition of Laplace'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 formation, and he has dared to conceive the bold thought of being present, if we may say so, at the spectacle of the creation of the world.
Buffon was the first to start this vast question, and to
consider from this philosophical point of view the constitution 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 distance 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 OF 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 telescopes, in regard to the nebulae, by which appellation those
whitish spfots, seen in different parts of the heavens, of
which they occupy a large extent, have been called. Observing 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 nebulae, 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 nebulae, Herschel supposes that they
represent a series of operations on a single mass of nebulous matter, which would pass from its first state of completely diffused, scarcely luminous nebulosity, to the state
of the most brilliant stars. The progress of the condensa11 *




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 successive 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 remote, we must suppose from analogy that the existing stars
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 successively by condensation to its actual limits.
~ 187.  The atmosphere, which we may suppose possessed 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 required 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 nebulous molecules of which these rings are composed, had continued to cool without disuniting, they would have ended
by forming a liquid or solid ring. But the regular constitution which all parts of the ring would require for that,
and which they would have needed to preserve whilst cooling, would make this phenomenon extremely rare. Accordingly 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 circulate 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 orbitual 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 successive 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 telescopic 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 nebula 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 gravity, 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 satellites 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 revolve 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 beginning 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 OF 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 difference 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 towards 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 system are therefore explained with great facility by the hypothesis we are examining; and as these successive changes
of a nebulous mass, and the leaving of a part of its substance 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 chronology 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 atmosphere.  And as the rotary motion is accelerated in proportion 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 rotation either of the sun or of a planet, must be shorter than
the duration of revolution of the nearest body which circulates round them; this observation is completely confirmed
even in those cases where the difference between the dura



130           ELEMENTS OF ASTRONOMY.
tion of the two motions must be very slight. Thus the interior 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 we 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 between 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 particular form of the planets, the flattening at the poles, and
bulging out at the equator, is only the necessary consequence 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 approaches 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 inclination 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 perihelion 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 comparatively small; and this has probably happened to the periodical 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 penetrated the atmospheres of the sun and planets, the resistance 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 we 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 inclination 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 probability.
~ 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 concerning 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
inan of genius consists in discovering those secret connections, those unknown relations, which connect the phenomena 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 circumstances 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 elliptic motion of the planets led Newton to the great principle




ELEMENTS or ASTRONOMY.               138
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 explained, 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 condensation by which the nebula are changed into stars, has connected the origin of the planetary-system with the primordial 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 Buffon 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 refuse to believe in chances so often repeated, and always
when there was need of them.  The principle of the condensation of the atmosphere of the nebulse is, on the contrary, general, and would produce phenomena nearly analogous, in all the stars and planets.
12




134             ELEMENTS OF ASTRONOMY.
CHAPTER X.
THE ROTATION AND FIGURE OF 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 Dimensions. 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 of
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 proceeds in its course round the island. Place upon this boat
a tiny insect, whose life endures only some seventy revolutions 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 acquainted with the universe. The earth is our boat, which
measures from end to end 8,000 miles or upwards of fortytwo 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 settled 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.  Venus 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 observer 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 sailing 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 direction 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 by




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 motions; remembering always that he does not see these motions 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 attribute 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 twentyfour 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 inhabitants 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 150 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, 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 farther 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 miniature 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
Ahe diamete 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 objects 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 partake 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 distance 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, passing 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 ascension 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 OF 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 verifying 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 conveniently 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 comparison 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 perpendicular 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 equator will therefore see the equinoctial in the same place.




ELEMENTS OF 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 apparent 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 diameter 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.
~ 210. 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 314' 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 314'.) We will
also set off from the same scale 65 equal parts, (65 being
the number of half minutes in 324t,) 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
321'. 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 644 equal parts.  When a quarter of a
year has elapsed the diameter will be 31f/, and the line
must contain 63 6 equal parts, its direction being known
from the angular motion of the earth. By 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 eccentricity of the earth's orbit. The sun's greatest apparent
diameter is 32' 35".6; its least is 31' 31"; hence the radius vector at the aphelion: radius vector at the perihelion:: 32.5933: 31.5167:: 1.032: 1.  Half of the difference of the two equals the distance of the focus of the




ELEMENTS OF ASTRONOMY.                 143
ellipse from the centre, a quantity whiqh 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 motion 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 aphelion 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
rae 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 6 —1.07, would
equal i2 593 =-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 distances 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 V51 to f61, or as
1: 1.034.  The difference between the two is nearly a  of
the perihelion distance, a quantity, as we shall see hereafter, 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 variation.  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 violence 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 progression 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 globular 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 becomes 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 OF 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 directions.
~ 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 direction, and navigators assure us that they have every where
observed the gradual appearance and disappearance of objects, 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 OF 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 revolution, 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 altitude; 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 (according 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 estimated by passing a vertical circle through it and reckoning




ELEMENTS OF ASTRONOMY.              147
the azimuth from  the point where this circle cuts the horizon.  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 observation, 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 spectator, another through the point on the earth's surface opposite 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 extremity of the earth and the other, if they are only continued 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 horizon, 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 rational than to the sensible horizon. The horizon or bounding line which to our eye unites earth and heaven is a different 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 segment of the earth's surface, nearly three miles in every
direction becomes visible. The bounding line of the segment 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 dip 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 altitude.
~ 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 consequently the convexity of her surface, her size may be calculated; for a given convexity can belong only to a sphere
of a certain size.
As we ascend elevations, though a much greater expanse of country is exposed to our view, not only is each
object smaller but the whole picture appears smaller, because 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 direction forming his horizon.  As he ascends, this tangent
reaches to a greater distance oQ the globe, but forms with
itself a smaller angle,




ELEMENTS OF 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 occasioned 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 before 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 observations 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 anl 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 21 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 21
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 segment 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 Teneriffe 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 portion 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 accompanied 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 process 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 calculation, 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.  We 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 presents 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 investigate 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 directly 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 attracting force in going from the equator to the poles, are
dependent on the distribution of the density of the interior. 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 portions of its surface must be nearer to the centre than others,
consequently gravity must act more strongly in these portions than elsewhere.  The most delicate measure of the
force of gravity which we have is the swinging of a pendulum, because this shows on a large scale the effects of gravity.  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.               158
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 pendulums and the times of their vibrations; the same relation 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 numbers 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 86,4002: 86,5352, or as 1: 1,00315.
~ 224.  Experiments made in all accessible latitudes
give T-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 gravitay
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 suspended 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 TW9 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 opposite to the direction of the radius of the different circles in
which the particles move.  Now 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 OF ASTRONOMY.               155
~ 226.  This centrifugal force at the equator lightens
bodies by s-   part of what they would otherwise weigh
there, and by ~-~ of what they would weigh at the pole
were the earth a perfect sphere. At the poles the gravitation of a body equals the whole force of gravity at that distance 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  -, while the elliptic form of
the earth diminishes it only Wo-.  The two together make
up the difference shown by the pendulum -i g.  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, arising from the centrifugal force, are matter of pure calculation.  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 pendulum, 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 centrifugal force. Centrifugal force in each latitude, as we have
shown, is according to the squares of the cosines; therefore 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 4A the height
of Mount Blanc, or 2~ the probable height of the Dhawalagiri, in the Himalaya chain.




156            ELEMENTS OF ASTRONOIMY.
~ 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 by 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 meridian altitudes we can learn the height of the pole and consequently our own latitude.  When our latitude has diminished 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 difficulties 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 observer, 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 determine 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 direction 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 forequal 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 ascension.  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 required; 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 observations 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 mechanical. The irregularity of the surface of most countries 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 between 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 390 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 ascertained with the utmost precision. A base line of a few
miles of extent, and at some little distance from the meridian 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 OF 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 triangles 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 workmanship.  The same base had been measured three years
before with glass rods, and the two measurements differed
only 24 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 equator, the other as near the poles as possible.  One degree
was measured in the valley of the river Tornea in Lapland.  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 account 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 extraordinary 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 instances 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 determination of latitudes alone.
At the beginning of the French revolution, a measurement was made from Dunkirk to Barcelona, in order to ascertain 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 latitudes, and their observed lengths agree with their theoretical lengths.
~ 232. By measuring degrees of the meridian we obtain the compression of the earth, while pendulum observations 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 Atmosphere. 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 influence, and then bringing a large body into its neighborhood; 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 mores Now, as the force which is required to produce
any amount of alteration in the position of the rod can be
14*




162            ELEMENTS OF ASTRONOMY.
ascertained in another way, the actual amount of the attraction 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 quantity 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 perpendicular.
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 mountain and consequently with a globe of water of the same:size, may be learned. The mountain Schehallien in Scotland was thus examined.  It was measured from  its base
to its summit, its component parts examined, and its specific 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
11;" more than could be accounted for by the difference
in latitude of the stations. The mountain therefore deflected the  plumb  lines from  the perpendicular 51".
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 relative 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 ancients thought a fiery impassable zone separated the northern 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 curiosity 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 country 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 gslu 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 -T 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 outline increase, others diminish in convexity.
The moon is so near to the earth that the difference between the earth's surface and its centre bears a considerable 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 radius bears to his distance a smaller proportion; the difference 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 quantity 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 1  so
high as in those nearer the zodiac. Within the polar circles 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 13Q 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 OF 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 conjunction, 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 2~ feet.  If the earth were covered uniformly with
a stratum of water the difference between the two diameters of the oval would be five feet. Local causes sometimes give the tide a height of 70 or even 120 feet.  The
Atlantic on the shores of France and also of North America pours into the rocky shore with prodigious violence,
while the gentle tides of the Pacific are scarcely perceptible.  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 distance 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
xT,4 fu    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 interference 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 beneath 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 sustains 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 arithmetical 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 compared 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 1-7v         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 compressed, 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 injuring him. The supply of air also is precisely what the
lungs require. The diver who fills his lungs with air compressed 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 mountains, 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 mountains 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 consequent 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 constantly envelopes us. Waves and tides on a larger scale
than those of ocean may and probably do exist at the surface 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 equatorial 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 tendency 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, according 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 enables our navigators to travel with speed and safety over the
central portion of the globe. They prevail within the latitudes 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 toward 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 partakes 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 currents 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 becalmed 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 easterly 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 hurricanes.
~ 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 alternate 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 pronounce 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 hundred 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 insensible 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 which 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 retaining powers of their atmospheres the remotest and the
nearest planets, nay even the sun himself, may become inhabitable. Air and water are among the most transparent
bodies which we know, yet these when interposed in sufficient 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 cannot 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 passing 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 quantity 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 medium and the amount of it traversed. The first law may be
verified in the nearest pond; vertical rays have more penetrating 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 absorbed 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 oblique 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 horizon.  From this cause then not so many rays of the declining 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 atmosphere. 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 penetrated 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 absorption 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 rays 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 1,300
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 multitude of small stars may be seen which are invisible from the
plain. Within 100 of the horizon small stars become invisible, 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 incessantly; that comparatively few of them fall on any globe
or object that is known to us, and that if the sun were surrounded, 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 received 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 radiated 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 visible 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 ASTRONOMY.               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 refraction as we can when a ray or a stick passes from air into
water.  We can only find out by experiments with transparent 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 incidence.  Refraction takes place only at surfaces. At whatever 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 refracted, but passes on in its original direction.  For oblique rays the sines of the angles of incidence and refraction, (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 proportion 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.
~ 255. 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 refracted ray would be the straight line joining the ends of
its present curve.
~ 256.  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 OF ASTRONOMY.               179
Since refraction increases with the obliquity of the incident 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 refractive 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 distances will be the refraction at the least altitude.  The influence of refraction in every part of its course, except
when in or very near the zenith, is shown by the star's describing a flattened curve instead of a circle.  The variation is however too slight to be perceived by the naked eye;
it differs at each instant.
~ 257.  The law which refraction follows has been determined, 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 correct every altitude of a heavenly body. In the zenith the
refraction is nothing; at 45~ it is about 1t, a quantity
scarcely perceptible by the naked eye; in the visible horizon it is 33', which is rather more than the greatest apparent 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 refraction.  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 different apparent heights of the same star at different seasons, 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 refraction. 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 indicates by its changes a change of refraction-increased cold
being accompanied by increased refraction.  From the increase of refraction in night and in winter, it has been supposed 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 refraction of the lower limb being greater than that of the upper,
the vertical diameter is diminished. The horizontal diameter 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 refraction is only 28' 6". The difference between these refractions 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 diminishes 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 objects, bold outlines and strong colors for near objects.
After we have referred a dim  object to a greater distance 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 vertically 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 beneath, a landscape which we know extends to a great distance.  But if we look upwards we have no line of stars,
no marks to inform us whether we are gazing miles, or hundreds, 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 unequal impressions made upon the retina of the eye by the
bright and the feebly illuminated regions give rise to this
illusion.
~ 261. Refraction likewise occasions twinkling or undulations in the light of the fixed stars. The atmosphere being 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. Vapors 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 telescope, 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 displaced totally. They experience on the edges little undulations, while the stars which seem but brilliant points are
continually displaced. This displacement produces twinkling.




184           ELEMENTS OF ASTRONOMY.
~ 262.  Refraction is not the only change light experiences 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 vapors 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 mountains.  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 OF 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 sua's rays upon the moon is laden with
vapor, it gives to the moon the peculiar light and color
which are sometimes observed.
~ 264.  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 brightness 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 floating 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 invisible, 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 300 or 400 of the photometer in summer, and
10~ or 15~ in winter. This secondary light is most powerful 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 impaired 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 ev^6 ypossible 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 phenomena which  appear different when viewed from  different parts of the earth, we must inquire  what effect
our position on the surface of the earth has on external phenomena.  The utmost distance by which two observers 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 inappreciable for the fixed stars. It is less for each body according 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 counteracting 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 ecliptic 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 moment, the earth each moment turns up a different hemisphere 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 entirely 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 hemispheres 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 extends a few minutes over the north pole, and falls a few
minutes short of the south pole. Its edges no longer coincide 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 illumined 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 illumination.
~ 269.  We have said that the sun illumines a hemisphere, 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 hemisphere 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 between them were great or small.
But let us suppose the sun smaller than the earth. It




190            ELEMENTS OF ASTRONOMY.
would then illumine less than half the earth's surface, because 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 darkness. 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 151, about one half
the sun's apparent diameter.  Owing therefbre 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 331 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 actually 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 extremely faint.  Probably it is only toward the 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 conceived of, let us consider to what points of the earth's surface 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 fraction of time.  The incessant rotation of the earth withdraws the point from the vertical sun, substituting first one
and then another and another point, each west of the preceding, 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 inclined 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 northerly point, but to one nearer and again nearer to the
equator.  For several days the vertical sun describes circles 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 declination between 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 continues for the next three months to wind gradually descending 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 ag 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 OF ASTRONOMY.                193
~ 274.  Let us suppose one spectator at the pole,
another at the equator, and a third in some latitude between 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 place, and the length of time they remain
visible.
In all these examinations we shall begin with the position 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 twilight 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 fiom 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 different stars.  Different stars are therefore concealed after
his setting and before his rising.  The invisible portion extends from pole to pole, and is 36~ in breadth at the eqti17




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, however, 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 hemisphere 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 invisible 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, 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 observer 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. However 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 describe 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 equator 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 predecessors, and disappear toward the west.  Since the observer at the equator in a few nights sees all the stars, and
one at the poles never sees more than half, observers between 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 describe circles differently inclined to their horizons, and differently 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 inclined and similarly divided by their horizons, and attain  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 remain visible longer.  Those which rise with the equinoctial, rise in the same time and place as before, but they
first make oblique and then right angles with his horizon,




ELEMENTS OF 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, because 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 always parallel to its horizon. If a place is any where between 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. equinox, 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 further 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 preceding days.
~ 281.  A  man at the pole, on the yernal equinox
grazes the northern edge of the illuminated hemisphere.
He remains in it, and at noon the next day he has advanced south of the spot where hie 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 describe 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 continues 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 illumiminated hemisphere, not at right angles, but obliquely toward 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 farther 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 equinox 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, describing 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~ 28t, 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 day
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, perpetual day begins at a place 10~ south of the pole. He describes 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 oblique 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 approach 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 latitudes 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 day 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 influence 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 day for a place 50 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.
~ 2S5.  Between the pole and the polar circle the
length of the day changes very rapidly, as may be understood 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 diminution of each of their successive days must be greater.
Accordingly within the polar circles which are 23~ 28'
from tho polo, we find tlho groatoot ranoe of variation in
the days.  At the polar circle there is one day of twentyfour 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 day 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
murtuil of thle earth in heLr urbit.
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 days 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, contrary 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 Hollanders, 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 climates; 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~ 441         18
60~ 39'         19
62~  4'         20
65~ 101        22
65~ 541        24




204             ELEMENTS OF ASTRONOMY.
Latitude.      North Latitude.         South Latitude.
Continual Day. Continual Night. Continual Day. Continual Night.
Days.      Days.       Days.      Days.
66  53'       31          27         30          28
69~ 30'        62         58          60          59
730            93          87         89          88
780  6'       12-        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.  WheVreas,
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 gradually accustomed to the absence of light is prepared to
over estimate the twilight.  Perhaps this consideration explains the great diversity of lengths which have been ascribed 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.
~ 2S9.  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 observers, 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 180
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 4o
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 continues 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 London, it lasts all of every night from  the 22d of May to
July 1st.':
~ 290.  The variation of seasons is caused by the inclination 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, indirectly 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 convex 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 obliquity 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~ 56', and twice during one half the year every such country has them vertical. In latitude 42~ north we receive
his rays at mid-summer at an angle of 42~023~=19~;
in mid-winter, at an angle of 420+23 —65; in the equinoxes, at an angle of 42~.  At the arctic circle the midsummer 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 rays at a
different angle from a plain.  The south side of a hill receives 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 tropics, 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 winter, 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 increased heat act together.  Long days give the sun time
to act. The earth being a rough, dark, opaque body absorbs 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 summer; 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 temperature. But in other parts of the globe, the great disparity 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 surplus is more each day, the amount increases rapidly.
But the day 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 continues 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 direct heating power varies one fifteenth; 302  292=59;
59,=one fifteenth nearly.  This would be sufficient to ex



ELEMENTS OF ASTRONOMY.              209
aggerate the difference of summer and winter in the southern hemisphere, and to moderate it in the northern. But
no such effect is produced. For heat diminishes in intensity 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 A2: S M2~:: motionaat M: motion at A.  This is also
the proportion in which the angular velocity of the earth
about the sun varies.  S A2: S M2::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 A S P.  The two segments,
A M P 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 maintained.
~ 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 summer 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 accordingly reckon four seasons. These do not however correspond with the astronomical seasons, for causes before
given.
18*




210            ELEMENTS OF ASTRONOMY.
The lengths of the astronomical seasons differ considerably, 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, because the perihelion is passed through in the northern winter. 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 portion 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, almost unvarying results. Thus while we repose on the stability of this our home, we find daily something new to enjoy in its unexpected beauties.




ELEMENTS OF 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 Glohes and Maps. Apparent Motions of the Planets. The Fixed Stars. The Zodiac. The Constellations. 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 natural 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 remember the positions of the stars. The natural features
of the earth are more easily remembered, but their dimensions, 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 faithfully represented on a plane surface, but a globe gives the
only correct idea of it as a whole. There are two mbdes
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 longitude of a few points, two perhaps in each country, and then
dividing the whole country into a number of triangles. In




212            ELEMENTS OF 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.  Ecual differences of latitude should not however be represented by
exactly equal intervals of surface, if great exactness is required.  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 selected, and observed in its upper and lower culminationsthat 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 motion 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 accurately 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 by 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 application 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 ascertaining longitude at sea.  In the latter part of the seventeenth 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 observatory 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. Flamnstead'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 reflecting 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 OF ASTRONOMY.
of the circle, and on the movable radius C E let another
such silvered glass C be fixed.  The glass D is permanently fixed parallel to A C, and only one half of it is silvered, 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 moving the radius C E, will, (if the reflectors be truly perpendicular 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 included between the direction C P of one object, and F Q
of the other, is twice the angle A C E included between
the fixed and movable radii C A, C E.  The angles:
P M Q + M C F  C F D,
C F DFDN   G C D + G C F,
PMQ+MCFG0GCD+GCF;
P I Q + M C F   G C D + G C F;
P M Q + M C F  M C F + 2 G C F,
P M Q      2 F C G    2 A C E.   -
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 angle A C E, but its double, the angle subtended by the
objects.




ELEMENTS OF 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 artificial horizon can be used, the sea offing affords the only
resource.  The image of the sun observed, as seen by reflection, 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 artificial 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 ascension.  Celestial longitude may be expressed merely in degrees, 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, — 8 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 chronometers 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 ascertaining 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 adopts 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 twentyfour 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 remained 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 days.
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 happened to navigators.  Two settlements on the same meridian may in this 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 a phenomenon
may be noted in the local time of two places differing in longitude, 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 jime over a hemisphere; artificial signals are
visible to a much less distance. The exact local times of two
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 difference of their local times gives the difference of their longitudes. 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 difference of the local times of the two stations gives the difference 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 however 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 observation would cause an error of a quarter of a degree of longitude, 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 adjusted.  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 except 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 distance 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 meridian will exactly bisect the interval of the observations.
~ 305.  The method of lunar distances is more useful than any other in finding the longitude.  Its principle  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
different 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 calculated 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 longitude is found.  The true distance of the moon's centre
from the star, when corrected for refraction and parallax, is the same for every meridian. The British Lords
Commissioners of the Admiralty publish annually a Nautical 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 meridian.  And if a vessel remains long enough in port, its
rate should again be ascertained, since change of temperature or other causes may have altered it. If the rate is
accurately known, and the error allowed for, chronometers will give the longitude throughout a long voyage.
~ 307.  In determining the relative position and distances 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 elevation, 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 reduction is made in the very act of reading off the horizontal angles.  Let E (Fig. 5, Plate I.) be the centre
of the earth; A, B, C, the places on its spherical surface, 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 E B P, 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 azimuthal angle M A N, which is the measure of the angle A
of the spherical triangle B A C. The sum of the three observed 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 stations of observation are raised, by the superficial inequalities of mountain and valley.  These triangles may afterward 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 spectator 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 terrestrial 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 longitude, 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 perhaps 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 declination and right ascension. As on the earth the meridian 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 intersection will always remain the same, and right ascensions 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 inhabitant of the earth, will be in their natural features correct 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 celestial 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 OF 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 instantaneously 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 foreshortened 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 advancing 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 stationary. The earth's motion only modifies them by changing 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 advanced, and the planet must also advance to be in conjunction 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 direction between us and the shore.  If both move with equal
quickness, the boat nearest the shore will appear motionless. 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 superior 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 opposition 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 being referred to the immensely distant fixed stars, the distance 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 retrogradation is
stopped by the planet's becoming stationary.  It then becomes direct, and remains so as long as the earth passes
through that part of her orbit where she moves in an opposite 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 the motions of the superior planets far more than those of the inferior. If if 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 revolutions, 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 whole, however, 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 appears longer in turning than Jupiter; Jupiter than Mars;
Mars than Venus.  For a week together the most powerful 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 Jupiter 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 simply 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 reluctance, 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 irregularity of the planets' motions vanishes as soon as they
are referred to the sun as a centre.  Their periodic times,
their distances and 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 longitudes are calculated from the vernal equinox. The sun's
and the earth's latitude are always nothing.  The heliocentric longitude of the earth equals the sun's geocentric
longitude+180~, or vice versa.  The heliocentric equinoxes and solstices are therefore the same with the geocentric.  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 heliocentric place and that of the earth be known; the angle at
the sun is the difference between these two, and completing the triangle we have the place of the planet with regard to the earth.  Sometimes the heliocentric longitude
precedes the geocentric, sometimes it follows it, and sometimes 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 calculated; for as one year is to the time elapsed, so would
360~ be to the arc passed over.  The longitude so calculated 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 always be found from it by applying a correction, or equation
as it is termed, whose amount is not very great, and depends 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 centre, 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 diminishes to 0~ at the perihelion. Its maximum, both additive
and subtractive, is 1~ 55' 33".
The maximum  value of the equation of the centre depends 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 OF ASTRONOMY.
ascertained. This is a more accurate mode of learning the
excentricity of the orbit than that of concluding its distance 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 approaches the aphelion, the arcs described fall short of A-ds
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 longitude 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 equation 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 concave surface every where  equally distant from  itself.
A  hollow sphere is the only surface which answers this
condition, and the eye maps the stars down 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 background 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 extremely 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 excluding from view the hemisphere beneath his feet.  By
means of observations in different latitudes, and at all seasons 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 sufficiently near together to assume the form of a Lyre, a Swan,
a Virgin, or any other figure, they were called a constellation, 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 historical 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 apparent 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 considered as resembling figures of animals.  The name
Zodiac, from a Greek word signifying animal, was consequently 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.  Perhaps 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 Maich.
~ 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 equality of the days and nights in the next month is symbolized
by Libra, the balance. The diseases produced by the departure of the sun gave to the next sign the name of Scorpion, because it is mischievous, and was thought to have a
sting in its tail. The next month was the season for hunting, 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 constellations to about 2,500 years before Christ; it has been
adopted by La Place and several distinguished philosophers.
The Greeks probably received their astronomical knowledge from the Egyptians. It is evident they did not themselves 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 Pollux.  The ship among the southern constellations is supposed 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 express them, are as follows:-Aries, or the Ram, o;
Taurus, or the Bull, 8; Gemini, or the Twins, II;
Cancer, or the Crab, s; Leo, or the Lion,; Virgo, or
the Virgin, n; Libra, or the Balance, _; Scorpio, or
the Scorpion, n; Sagittarius, or the Archer, t; Capricornus, or the Wild Goat,'3; Aquarius, or the Watercarrier, i; 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 Ram, the summer solstice 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 constellations 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 solstices 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
Ram is now near the sign 8 of the ecliptic; that of the
Lion near R; 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 constellations, which are movable with respect to these points.
The ancient Greeks reckoned only forty-six or fortyseven 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 constellations.  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 Phoenix, the Toucan, the Little Water Snake, the Sword-fish,
the Flying-fish, the Fly, the Chameleon, the Bird of Paradise, 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 constellations, 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 astrono20 




234           ELEMENTS OF ASTRONOMY.
mers named the new  constellations, called the Camelopard, the Unicorn, the Fly, and the rivers Jordan, Euphrates, 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 commemorative of interesting events or distinguished men. These
often replaced former constellations, and speedily disappeared 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 Serpentarius, 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 Perseus 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 November.  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 constellations 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 constellations, 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. Moreover 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, uniformity must be secured for the maps. The same constellations 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 repeated three times, &c.
~ 326.  According to the present system, some constellations 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 required, as a2; 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; c Librae, a star of the second magnitude;
and v Aquarii, a star of the third magnitude.
Among the most conspicuous constellations in the northern 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 southern 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 position of the stars nor are definite in themselves, and seem
made uncouth and perplexed purposely to baffle the observer.
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 concave celestial globe.  Such a globe represents them as
they really are, and a convex one reverses their appearance 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 tight 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 surface 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 o)ur 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 influences 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 obedience 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 suppose its existence, and we know that they take place invariably.
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 mutual 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 explain 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 afterward 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 always brought to a globular form  by the equal mutual attraction 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 assumed by drops of rain, mist, quicksilver, &c.
If there were no motion in the heavens we should probably 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, centrifugal 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 centrifugal force; the rest is insufficient to retain them in
their place; they recede from the centre, and other particles rush in to supply their place.
~ 330.  But the form of the mass is changed; the particles 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 rotating 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 proportion to  the rapidity of rotation.  Very rapid rotation
causing great centrifugal force raises the equatorial particles 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 rotated 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 centrifugal 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 pressure 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 rotation 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 external 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
relieved from 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 introduced through the box, with a small disc upon it, so arranged 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 spherifying forces are of different natures, that of molecular attraction 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 hollow above and below, around the axis of rotation, stretching out continually in a horizontal direction, and finally
abandoning the disc, is transformed into a perfectly regular 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
earth were first formed into a globe, and afterward, from
rotation, took on the spheroidal shape. Probably the moment it was isolated it took its form, which rotation, beginning 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 motions 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 rotating 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 particles 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 rotation the sun must have attracted to itself one half of
the sphere, at right angles to the direction in which the projectile 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 remained 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 perpendicular 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 impelled in a direction passing farther from the axis of revolution, 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 particles 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 therefore 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 removed 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 connected by a pole; or of two unconnected bodies.
Suppose two bodies, weighing one pound each, and connected  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 forward 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 evolutions, while their common centre of gravity sweeps steadily
round the sun.
~ 336.  Since the common direction of the two planetary motions points to a common origin, and since one impulse 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 millions 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 existed, 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 revolution.  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 motions 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 twentyfour 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 gravitation 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 ~s%
of the whole weight.  As the tendency to fly off can be
calculated, the true amount of gravity at the earth's surface 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 attraction 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 distances 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 attraction on the sun is only TYFY  part of its attraction on
the moon.  Or as the earth's attraction draws the moon
through about 2, of an inch in one second of time, the
earth's attraction draws the sun through 27(7 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 accuracy 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 astronomers 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 spheroidal 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 directions.
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 centres 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 proportional 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 connected 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 system 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 moment 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 attraction, 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 OF 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 motion 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 gravity; if not, the heaviest would move in the smallest orbit;
but both would revolve in the same time.
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.
~ 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 consequence of an original impulse through the celestial spaces.
A body does not change its state, either of rest or motion, 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 successively.
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 consequence 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 continues 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 distance 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 direction 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 affected 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 obedience 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 attraction.
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 perceptible intervals, the body would describe the perimeter
of a polygon, having as many sides as the number of impulses 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 centripetal force, and creates a centrifugal force in the opposite 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 continuance 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, generated 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 intersect it, the section will be an ellipse more or less elongated.  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 hyperbola.  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 determined 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 acquired 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 describe a circle.
If the projectile force is to that required for a circle as
v  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 prevent 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 reason 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 perigee 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 certain 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 calculated.
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 velocity with which it is thrown. Consequently when the stone
is thrown with very great velocity, it will go a great distance 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 lower end of an
ellipse, by two more or less powerful centripetal and centrifugal 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 attraction of the sun will not bend its path very much. From
B to I and A the attraction exceeds the centrifugal force,
and velocity towards the centre is created. The sun's attraction 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 ASTRONOMY.               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 retards 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 deflection 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 circulating body would run off in a tangent to its path, if
there were no centripetal force to prevent it. The projectile 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 tendency depends on the velocity of the body at each moment.
This velocity may be greater or may be less than the initial velocity.  When the body describes an ellipse, the
projectile force is alternately greater and less than the initial 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 OF ASTRONOMY.
like reason, if the velocity be diminished, the centrifugal
force becoming less than the centripetal, the body will describe 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 likewise 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 between them, and thus discovered his third law.
~ 354.  The following are the most important propositions 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.  Suppose the momentary action is such, that in the same time
the body would move along B E, then completing the parallelogram, B F will be the real line of motion. Joining
CD, the triangle A B C=triangle BCF, 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 described 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 A B C, it has no tendency to
move it out of that plane, and therefore B C F is in the
same plane with A B C. 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 attraction, 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 tangent 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 impulses 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 interval, 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 velocities will be inversely as tie masses, that is, directly as
A C: B C; but by similar triangles AG: BF::AC:B C.
Again, suppose that, in consequence of the mutual attraction, 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 EC 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 A B C (Fig. 1, Plate II.)
is expressed by B C2 X angle A C B.  But in equal intervals of time, these areas are equal in all parts of the orbit.
The angle A C B must therefore be inversely proportional
to B C2, or to the square of the radius vector.




ELEMENTS OF 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 0
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 0 will be equal in length.  Whence,
A 02=AaAp=ACX2 ap; consequently, 2 Ap=
A 02
A C     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 peripheV2  v2
ries, we shall have F: f:: -R-   -; therefore the forces
are as the squares of the velocities directly, and as the
radii inversely.
V2  v2
Cor. 1st. Because F: f:   -:  it follows that
R    r,
V   v::  /RF: /r f, and
V2  v2
R:r:: -    f
~ 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
centrifugal force is f=-.  Let g  be the  constant
r
force of gravity, and h the space or height through
which  a body  must fall in  order to  acquire a velocity equal to v; then v2 =2 hg. for the accelerating
force in the present case is gravity; hence f =     g.  If
r
we suppose h=Ar, 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.
~ 361.  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 diameter, gives the versed sine. Let r be the radius, r the
ratio of the circumference of a circle to its diameter, and
T the time of revolution expressed in seconds.  The arc
2 r xr
actually described in a minute of time  ---. This squared
and divided by the diameter gives, for the versed sine,
which is proportional to the attracting force, the expression
2 r r2.  But the attracting forces are in inverse proportion
to the squares of the radii. Hence we have
1    1    2 r 2  2 r 7r2
r2 *r/2      T2     T/2
r3 T'2- r'3 T2;
which is Kepler's third law, that the squares of the periodic 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 several practical questions of interest are solved. The distance 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,) be denoted by r, and the space through which a
heavy body falls in one second at the surface (-=16  feet),
by kg, 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 g r=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 gravitation about the surface varies inversely as the square of
the distance, we have,  r::: 26,000 feet (velocity per second at the surface): 26,000 a/, the velocity
in the circle whose radius is R.  And r: R::  5,075s
R3
(the periodic time at the surface): 5,075,/-, the  periodic time in the circle whose radius is R.
For example, if R  be assumed equal to 60 r, the distance of the moon from the earth, the expression for the
velocity will become 3,356k feet per second; and that for
the periodic time will become 2,360,035s. or 27 3 days,
nearly.
~ 363.  Or, knowing the periodic time of the moon,
and the radius of its orbit (240,000 miles), we can calculate the space through which she would fall if left to herself, 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 describe 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 surface of the earth, the space described by the moon should
be to that described by a falling body near the earth's surface as 602: 12; 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 602  12; or as 3,600: 1: 59,400 feet:
16- 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 estimated. For, 3651 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
measure, we have 3692:.: F: f::  2f: 1 nearly; that is, the moon's gravitation towards the sun is to
her gravitation towards the earth as 22 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 becomes 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 86,1602: 5,0752: the force of gravity: the
centrifugal force of a body at the equator arising from the
earth's rotation::: 1   E  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 motion, 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 cosine 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
s-, 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: I:: 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::    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 31832, or 10131489 sec.:12 sec.::
1.63025 feet, one half the length of such a pendulum, in
latitude 51~: 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 diminish 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 revolution 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 central force will descend through the space in half the time
of revolution in the ellipse. Let T be the time of revolution of a planet at any distance, and t the time of a revolution at half that distance; then, by the third law of
Kepler, T2 t2: 23: 13; hence, t =-    and t=-  3;
but we have just seen that it 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 periodic time divided by V/32; or it is equal to that time multiplied by the reciprocal of V32, 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 Longitude. 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 Theory 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 immovable 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 departures 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 direction 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 Physical 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 introduced; 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 disturbed both revolve round it.
~ 369.  The simplest mode of considering perturbations 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 distances, to find the lines described by these bodies, and
their positions at any given instant: or in other wdrds, 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 celestial bodies are determined.  It is an extremely difficult
one, and would be infinitely more so, if the disturbing action 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
ter, in disturbing any one planet, is equal to the sum of
all the particular disturbances it experiences, on the general 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 hyperbolas; 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 momentum 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 example 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 positions 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 consequence 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 re23*




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 accomplished 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 compensated, 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 motion 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 motion 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 inequality 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 disturbing 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 disturbed 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 perturbations 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 inequalities 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 description 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 retards 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 OF ASTRONOMY.
cause the arcs in which the accelerations and retardations
take place, would be symmetrical on each side of the disturbing force. For it is clear, that if the motion be accelerated 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 although 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 inequality 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 disturbing 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 distance, 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 compound 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, before it has passed through 1800, as it would do if undisturbed.  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 1800 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, preponderate.
~ 376.  This, however, is not the full amount of the
motion of the apsides; part arises, also, from the tangential 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 revolution, 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 disturbed planet's orbit, is an immediate consequence of the deviation from elliptical curvature, caused by the action of the
disturbing force. When the path of the body, in proceeding 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 diminished, 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 increased; 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 immense period of time, the eccentricity increases and decreases 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 vicissitudes are accomplished is 70,414 years, estimating the action of these two planets alone; but if the action of all
the planets were estimated, the cycle would extend to millions 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 positions of the two planets with regard to the sun, considered
to be fixed. By this action it sometimes makes the plane




ELEMENTS OF ASTRONOMY.              275
of the orbit of the disturbed body tend to coincide with the
plane of the ecliptic, and sometimes increases its inclination to that plane. In consequence of which, its nodes alternately 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 unequal velocity; but as a compensation is not effected, their
motion is, on the whole, retrograde.
~ 379.  With regard to the variations in the inclination, it is clear, that, when the orbit is symmetrical on each
side of the disturbing force, all its variations are compensated 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 symmetrical 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 inclination, which is not compensated till the nodes have accomplished 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 accomplish even a tropical revolution. In what precedes, the influence 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 action of the earth and planets, so that the whole is a compound 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 certain 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 OF ASTRONOMY.               277
planets and satellites taken together is so small, when compared with that of the sun, that these effects are quite insensible, 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 invariable.
~ 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 collisions, which would involve our whole system, now so harmonious, 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 comparatively but a point in duration, while these vicissitudes
embrace myriads of ages.  The proof is simple and conclusive. 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 however 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 satellites 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 etherial 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 destruction of our system, which in itself contains no principle 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 direction for unknown ages.  This rotation, which seems to be
highly probable, may even have been coeval with its creation.
~ 384.  The form and position of the planetary orbits,
and the motion of the bodies in the same direction, together with the periodicity of the terms in which the inequalities are expressed, assure us that the variations of the
system are confined within very narrow limits, and that although we do not know  the extent of the limits, nor the
period of that grand cycle, which probably embraces millions of years, yet they never will exceed what is requisite
for the stability and harmony of the whole, for the preservation 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
451 7; occasioned by the reciprocal action of the planets.
But as this is also periodical, and cannot exceed 2~ 42',
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 rotation 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 system, 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 induce in the orbits of the planets, in the plane of the ecliptic, 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 existing 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 immutability 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 system, which, unaffected by the changes of time, will maintain 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 recede 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 invariability 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 common centre of creation, we may then recognise in the equatorial 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 deduced from the law of gravitation, are so perfectly confirmed by observation, that analysis has become one of the
most certain means of discovering the planetary irregularities, 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 period, 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 begin to move at the same time, one making exactly five revolutions 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 OF 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 uncompensated inequality in the mean motion would go on increasing 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 revolutions in the time Saturn performed two. These revolutions 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 advance, 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 perturbation 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 confirmed 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~ 13t.
~ 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 Qf the
earth.
It might be imagined that the reciprocal action of such
planets as have satellites would be different from the influ24 




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, however, in some degree disturbs the motion of the satellites
about their primary.
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. 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"31 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 100 or 11~; but the obliquity of
the ecliptic to the equator can never vary more than 2~ or
38, 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 different 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 exercised 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 retrograde on the ecliptic, without however in the whole changing 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 different 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 accomplish 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 exercised 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 continually 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 position, 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 inequalities 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 181 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 proportionally 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 prolongation of the earth's axis is compounded of these three motions. While in virtue of the moon's action it would describe a little ellipse, it is carried over so much of its circle round the pole of the ecliptic as corresponds to 19
years; that is to say, over an angle of nineteen times 50rl
round the centre.  The path which it will describe in virtue of these three motions will be neither an ellipse nor an
exact circle, but a slightly undulating ring. In the following 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 carried 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 unaltered.  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 retardation 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.
~ 396.  We must now make a slight change in our
conception of the motion of the earth, and imagine its centre 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 fiom  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 acceleration and retardation are one part of the nutation;
and the alternate increase and diminution of the angle contained 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 equinox 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 OF 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 striking, that it was discovered more than a century before the
Christian era, though its cause remained unknown.  Nutation, 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 alternately increased and diminished.
~ 398.  In the year 158 before Christ, Hipparchus discovered precession from the comparison of his own observations with those made 155 years before.  He had formed
a catalogue in which he laid down the latitude and longitude 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 circumference 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 mentioned.  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 inclined, 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 afford 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 54
times the density of water, and the rocks on the surface
averaging only about 21 times the weight of water, it follows 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
2~ 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 calculation is less than if the earth were a homogenious spheroid, 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 geologists 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 supposition 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 OF 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 equinoctial 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 0."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 direction.
~ 403.  Even when we are conscious of our motion we
are apt to transfer it to surrounding objects. A man walking 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 whatever 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 displacement.  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 velocity bears a definite ratio to the velocity of the earth in
her orbit, 192000: 19. To ascertain the angle of aberration we should then construct a triangle, of which one side
(the velocity of the ray) should be to the other (the velocity 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 S B 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 direction of the star-20."246.  Also C B D, the displacement
of the star, or the amount of aberration=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 ray.  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:
B C:B A:: sine of B A C: sine of A C B, or C B D,
(the apparent displacement.)
Thus it appears that the sine of the aberration is proportional to the sine of the angle made by the earth's motion 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 moving.  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 onward continually, describing the circumference of the ecliptic 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 bisect each other. If then A, B, C, a, b, c, represent different positions of the point toward which the earth moves,
(or the point 90~ before the earth's place), S A, S B, 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 direction 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 OF 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 corresponding arcs must pass through all intermediate values, increasing 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 aberration in the direction S c will also be 20.".246; and the extreme 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 circle, passing through S, the aberration in the direction S B
will be proportional to the sine of S B, and that in the
direction of Sb will be proportional to the sine of Sb.
But the arcs S B, 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
C S c cuts the circle A S a at right angles; or the directions of the greatest and least aberrations are perpendicular 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 supposition 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 OF 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. Consequently 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 always 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 portion of the ecliptic itself. The whole aberration therefore
will be in the plane of the ecliptic, or will take place entirely 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 towards 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 appears 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 ecliptic, 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 consider that which arises from  the motion of the body ob25*




294            ELEMENTS OF 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 aberration 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 requires 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 obvious that each of the two stars A and B will be seen, independent 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 together, 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 compound object A and B in the state in which it really did
exist x years previously.  This being true at every instant, 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, unaffected by any optical illusion or distortion whatever, only
for an epoch antecedent by x years.
~ 410.  There is a curious difference between the consequences 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 displaced 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 appear 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 apparent 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 observer 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 respective 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 unnecessary 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 during which we can execute the same things in the same
manner, or during which the same phenomena are reproduced 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 husbandman 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 presence 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. 41"; 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 a sidereal day elapses; and so on for every proportion of distance. Therefore not only is the duration of a sidereal day
constant, but during every part of it rotation goes on uniformly.  The pendulum also confirms these results.




298            ELEMENTS OF ASTRONOMY.
~ 415.  A  solar day is longer than a sidereal day.
For while the earth has been once rotating, she has advanced in her orbit nearly one degree, and this makes the
sun appear to have advanced one degree.  A place therefore 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 -sO 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 similar 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 likewise 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 0 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 inconvenience, 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 0 to 24 hours. Clocks showing sidereal time are so
regulated that they show Oh. 0' 0", the instant the equinoctial 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 observing 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 independently of the other.  The earth's motion on its axis
being perfectly equable, and the plane of the equator being 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 Cancer, come to the brass meridian sooner than the corresponding patches on the equator. All those from Cancer
to Libra will come later to the meridian than their corresponding patches on the equator; those from Libra to Capricorn sooner, and those from Capricorn to Aries later.
And the patches at the beginning of Aries, Cancer, Capricorn, 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 periodical causes of fluctuation had not existed.  Getting rid of
these fluctuations is termed equating or correcting the observation; 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 overtakes 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 apparent 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 midwinter to mid-summer.
If the earth moved equably, but in the ecliptic, there
would be no equation of time at the equinoxes and the solstices. If it moved with its elliptic irregularity, but in the
equinoctial instead of the ecliptic, there would be no equation 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 vanishes 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 together 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 day, 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.4
~ 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 variation in the inclination of the ecliptic to the equator would
also affect it. In fact the inclination of the ecliptic to the26




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 direction 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 equinoxes, but then the solar perigee will coincide with the
autumnal equinox, whereas at the creation of man it coincided with the vernal equinox.
~ 422. The variation in the position of the solar ellipse
occasions corresponding changes in the length of the seasons. In its present position spring is shorter than summer, and autumn longer than winter; and while the solar
perigee continues as it now is between the solstice of winter 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 including 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 successive 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 differences.
~ 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 1800 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 result of observation is that the apsides have a progressive
motion on the ecliptic of about 11."8 annually. The longitude 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 equinox from which it is measured.
~ 424.  We now see that there are three different periods 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




304           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 situations 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 apogee 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 returns to the same spot in his orbit. The effect therefore of
the retrograde motion of the equinoctial point on the ecliptic is to bring forward the time of the equinox, (or the instant 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 ecliptic; 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 original 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 commencement 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 11."8 to arrive at his original position with respect to the apogee, and the length of the anomalistic 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 anomalistic 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 describing 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 uniform length of the anomalistic year, to ascertain those of
the equinoctial and sidereal years respectively, are themselves of variable duration; and the lengths of the equinoctial and sidereal year are necessarily so too. The variation, 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 comprehends 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 incommensurability 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 determined 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 denominations employed by us, and was alike found in the calendars
of the Jews, Egyptians, Arabs and Assyrians. It has survived the fall of empires, and has existed among all successive generations.




ELEMENTS OF ASTRONOMY.               307
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.
~ 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 referred.  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 different 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 period of her rotation and of her revolution, a given observation was made.
But this eccentric position, which seems so disadvantageous, 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 consequently the value of the angle E B G, 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 C A B; whence may be found the side C B, by the
following proportion:
Sine ABC  rad.:: CA:: CB.
In the case of the moon, her mean horizontal parallax
is 57' 12"; therefore,
As sine 571 12': rad.: 3,956 miles, the earth's semidiameter: 237765, the moon's distance.
Her diameter may be easily found when her distance is
ascertained, (Fig. 8, Plate II.): the angle E CG 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 perpendicular 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 SR-;
from A, it will be W; its arc of displacement S W being
less than EG; 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
be I.




ELEMENTS OF 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 rad.: sine of apparent zenith distance: horizontal
parallax: parallax in altitude.
For in the triangle A C H,
A: C  H:: sine A H C: sine CA H, or its supplement M A  I.
Again in triangle A B C,
A C:C B:: sine A B C: sine   A B, or radius.
And since the antecedents are equal, the consequents are
proportional; therefore,
Sine A H C: sine M A H:: sine A B C: rad.
Or, as rad.: sine M A H:: sine A B C: 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 horizontal 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 G.  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 C B D, or its supplement 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 elapsing between its arrival on the meridians of the stations.
Of course the nearer the stations are to each other in longitude, 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 go greater than the polar parallax.
This difference, though very small has a very considerable
influence upon several astronomical phenomena; for example 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 depresses 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 passing from the horizon to the zenith is about one sixtieth, because in the passage from  one of those positions to the
other, the distance of the observer from the moon is diminished 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 apother 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 obscured; to one on the west side, the west edge is only obscured.  As the planets and the sun are much more distant 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 cannot 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 proportion 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 parallaxes consequently so small, that they cannot be ascertained 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, that very 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 intersect that of the earth in only two points, the nodes, she
is never seen upon the sun except when her conjunctions 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 Venus 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 determine 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 distance, it is not surprising that great anxiety should have
been felt to determine this angle with the utmost exactness.
~ 439.  The chief fact to be borne in mind, in considering a transit of Venus, is that the difference of the apparent 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 observers 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 subject 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 that of 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 describe 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 ASTRONOMY.               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 sunset, 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 November, 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 900
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 firom the
sun. For as one minute of time is to four seconds of a degree, so are 6~ 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 calculated.
The sun's diameter is previously known, and consequently the distance from his centre of chords of any given
length.  Venus 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 D 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 ecliptic, and to avoid complicating the case, let us omit all consideration 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 suddenly 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 angular motion of Venus being precisely known, and her apparent path being very nearly a straight line, these times
give a measure, on a very enlarged scale, of the lengths
27*




318            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 instants 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 excellence 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, astronomers 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 deduced, 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 Va, BVb, (Fig. 10,
Plate II.) are straight lines, and therefore make equal
angles on each side V, a b will be to A B as the distance
of Venus from  the sun is to its distance from  the earth;
as 68: 27::ab: AB; a b therefore occupies on the
sun's disc a space 21 times as great as the earth's diameter, and its angular measure is about equal to 21 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.               819
~ 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'
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'.
5 449.  The distance of a superior planet from the sun
may be found by measuring the arc through which it appears to retrograde, when in opposition to the sun.
Let X (Fig. 11, Plate II.) be the place of Mars in opposition, and y that of the earth; y v the arc described by
the earth in a short period of time, for instance one day;
X 3 the arc described by Mars in the same time.  The
periodic times being known, these arcs may be found simply 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 r d, 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 Sv 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 always equal to the angle under which the earth's diameter
is viewed from that place. The more distant the observer,




820            ELEMENTS OF ASTRONOMY.
the smaller is the angle. All bodies within the solar system 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, because 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 required 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 diminish 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 instruments 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 discovered nutation, and aberration of light. Sir William Herschel learned the connection and orbitual motion of the
Binary Stars.  But the question of the — istance of the
fixed stars remained unanswered for three centuries. It




ELEMENTS OF ASTRONOMY.                321
has lately been solved by means of more delicate instruments and a better choice of stars.  There are only two
circumstances to guide in the selection of stars for observation, their remarkable brilliancy or greater apparent motion. 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 successful; 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 differently 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 1t.  The problem
of parallax is therefore reduced to that of finding a double
star in which a variation of distance is observable, and following 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 published a list of stars suitable for this research, with the
times of year when the observations would show the greatest 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 apparent 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 similar ellipses, and the line joining their apparent places will
be equal and parallel to the line joining their true posi



322            ELEMENTS OF ASTRONOMY.
tions. It is therefore in vain to look for any effect of parallax in micrometrical 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 similarly 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 absolute parallax of either, but the differences of their parallaxes, 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 comparison with one another.  Yet the distance must not exceed 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 164". This being necessarily 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.              323
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 advantageously situated for comparison with either of the individuals 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 1! 46", nearly in that direction.  Considering a and b as fixed points, and measuring 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 majorand minor axes ought to hold with
respect tothe 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 calculable ratio the one to the other; and the maximum and
minimum of the one distance a c will be nearly contemporaneous with the mean values of the other distance b c, and
vice versa.
Thus we have in the first place several particulars independent 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 a b and a c are
really parallactic.  So that if they are found in such conformity, 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 tha thhe particulars observed agreed with
those calculated in a signal and satisfactory manner,




824            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 corrected 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 distances 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 reduced 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 2~ times that of Uranus.
Their periodic time is about 540 years. Such is the universe 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 probably the major axis of their mutual orbit is short of a
minute of space.  They therefore afford 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 329,900 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 atmospheres, which are at least several thousand miles deep.
The solid nucleus of the sun may have a much greater specific gravity than has here been assigned to the whole orb.
Its diameter is 111 times that of the earth. Its actual diameter 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 intensity.  The most vivid flames disappear, the most intensely 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 electricity, or from some cause unknown to us. The sun's direct 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 collected 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 appear as the disc of the sun.  We infer the existence
of this atmosphere from  the appearances which accompany the spots seen in the sun.  These spots are perfectly and intensely black, but are surrounded by a penumbra 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, penumbra into full light. The idea conveyed is more that of the
successive withdrawal of veils, the partial removal of definite 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 nonluminous atmosphere.  Luminous matter often is seen piled




328           ELEMENTS OF ASTRONOMY.
up round the edges 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 borders 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 together.
~ 463.  These spots have a motion of their own independent of the sun's motion.  These two facts have suggested 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.  IHurricanes are moving cylinders of air which rotate with extreme 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 extends 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 accurately; 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*




330           ELEMENTS Of 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, suppose 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 revolution 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~+ab, the whole space described by the
spot: the whole time elapsed between the two disappearences:: 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.




LtEMENTS 0F ASTRONOMY.               83i
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 situated 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 prolonged, 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 system, 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 common 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 obtaining 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 invariable 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 northern 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 periodic time is the same, whether the body moves in a circular or elliptic orbit, provided only that the mean distance 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 appears 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 regards the sun. The sidereal is uninfluenced by the motion
of the earth.
Mercury's sidereal period is nearly 88 days, his synodical
is 1-16; that of Venus is less than 225 days, while her synodical 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+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:   l+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 between the sidereal and synodical periods.
~ 472.  To ascertain the relative masses of two bodies,
we need only compare the force they exercise on bodies revolving round them.  We know that the moon descends
through 161 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:: D2: d2
f: F::42:M D
fd2: FD2:: m: M.
m: M::1 X 12: 2.2 X 4002::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 apparent 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 numbers 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
38xd
their densities differ, the  attraction of A  will   3        d
23   D                            16
and that of B =2    D
9
But by observation the attractions are found to be as
7 to 2; therefore,
33xdx9: 23XDX16:7::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. Rotation and Atmosphere. 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.
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            ELEMNTS 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 frequently 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 indented, and  this proves the  irregularity  of its surface, 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 OF ASTRONOMY.                387
unless the sun's rays are modified by an atmosphere peculiar to the planet.
When Mercury and Venus 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 illuminated 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 conjunction.
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  distinctly. 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 otherwise would owing to her angular velocity being greater
than that of the earth.
29




338            ELEMENTS OF 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 evening 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 y93.  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 distance 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 winter, and vice versa.
The period of the rotation of Venus is determined like
that of Mercury, by the interval of time between two successive 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 atmosphere is known not only by the clouds but by its refraction. It causes the illuminated portion of her surface to
look larger than it otherwise would.  This dense atmosphere 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 improbable 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 appears 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 10 511. 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 appearance. It has a very thin atmosphere, 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 distinctly visible, the continents being of a ruddy color, probably belonging to the soil, and the seas consequently appearing 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 brilliancy 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 winter night of twelve months' duration. The brightness gradually disappears after exposure to the sun, and the pole




340            ELEMENTS OF ASTRONOMY.
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 perceptible phase; it never appears as a crescent, but has
sometimes a moderately gibbous appearance, the enlightened 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 illumiminated 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 2 of S M (the distance 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, because 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 connecting 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 OF 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  1 to Twa  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
41 years, in an orbit inclined 100 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 inclination 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 extensive and vaporous atmosphere, little repressed and condensed 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 44 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 3i.
Astraea differs but little from Juno in size, mean distance, 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 Venus 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*




842            ELEMENTS OF ASTRONOMY.
rapidity of rotation, particularly about the equator. Jupiter 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.
~ 485. *" The changes which take place in the planetary system are exhibited on a smaller scale by Jupiter and
his satellites; and, as the period requisite for the developement 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 compression 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 sensibly 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 therefore inversely as the square of the distance. In a spheroid, 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 OF ASTRONOMY.               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 disturbing 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 reciprocal attraction between each particle of Jupiter's equator 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 inclination, 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 removed, owing to the influence of Jupiter's compression;
and they have a slow motion corresponding to secular variations 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, similar 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




844            ELEMENTS OF ASTRONOMY.
the three nearest to Jupiter, the second of which receives
from the first a perturbation similar to that which it produces in the third; and 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 position of Jupiter's orbit, and from the displacement of his
equator.
~ 489. It is obvious that whatever alters the relative positions 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 inclined to the plane of his orbit at an angle of 30 5' 30",
so that the action of the sun and of the satellites themselves 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 producing corresponding nutations in their motions. And, as
the cause must be proportional to the effect, these inequalities afford the means, not only of ascertaining the compression 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 perturbations give the values of their masses with considerable accuracy-a striking proof of the power of analysis.
~ 490.  A  singular law obtains among the mean motions 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 OF ASTRONOMY.                345
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 Jupiter'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 respects 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 mentioned 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 direction, the first is in the oppositedirection; consequently,
when the first is eclipsed, the other two 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 cheering 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 conjunction, 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 attributed 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 observed 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 unpromising 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 Jupiter, 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 elasticity."
Saturn.
~ 493.  A still more wonderful mechanism is displayed
in Saturn, the planet next in order of remoteness to Jupiter, 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 lustre, 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 rotation 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 circular. 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
4istance 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 ellipse 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 disappearance 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 OF ASTRONOMY.
much from their plane. To those regions of the planet which
lie above the enlightened side of the rings they must appear 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 fifteen years of light and fifteen of darkness. The mass of
the rings is thought to be T- 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 30~ to the plane
of the ring, while the orbits of the others nearly coincide 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 seventynine 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. For
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 toeview 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 rotation 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 telescopic 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 recent discovery, little is klown. It shines with a bluish
light, and is accompanied by one and perhaps more satellites.  Its mass is about  -36-o  of that of the sun.  The
extraordinary circumstance of its having been discovered
in consequence of the predictions of two eminent astronomers, gives to this planet a very prominent place in astronomical 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 inequality 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 ne-ay 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 diameter 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 30f 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~ 21' a year, and complete
the circuit of the heavens in 6,793d. 1Oh. 6' 30". The
points in the orbit in which the moon is nearest to and farthest from the earth, are called respectively its apogee
and perigee; the line joining them is called the line of apsides. This line advances at the mean rate of 40~ 40' 32"
every year, and completes the circuit of the heavens in
3,2'2d. 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 overtakes 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,
he will move through 360~ -  in the time s, or in the synodic 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~ -; and as she
takes the time p to move through 360~, she will take the
time 3660+   p to move through  360~+A; or s860~q 860~-         -     -     anhee          Ps
00+ 360    p, or P sP p+s p, and hence p:
3600                                        P+S
Pp
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 observations 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 darkness.
~ 500.  The moon's disc exhibits to the naked eve
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,
someof them being about 3 or 4 miles high.




ELEMENTS OF 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 loar 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 partially smooth. No clouds have ever been discovered.
~ 501.  Since the axis of the moon is-& 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 illumined 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 varying 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 joining 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 horizontal 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 instead of once in twenty-four hours.  Since there is no atmosphere in the moon sufficiently dense to reflect light,
the heavens in the day-time must have the appearance of
night, ahd 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 universe; 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 precisely equal to that which its revolution would give it eastward, 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 darkness.  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 ASTRONOMY.               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 various 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 almost all her light, usually however appearing of a red coppery hue.  This may be owing to electrified vapors belonging to the earth's atmosphere, and interposed between i
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 however 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 upni 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 sunrise. 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 OF 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 invisible for several days, but the duration of her invisibility depends partly on climate, atmosphere, and power of vision.
~ 505.  An instance is on record of a lady, in the proverbially dingy atmosphere of England, noticing the old
moon near her conjunction with the sun in the morning exhibiting a thread-like crescent, and the day after in the evening 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 day 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 increases, 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 inclination 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 900 from the sun
are called quadratures. The full moon is always in opposition 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 summer than the sun does in winter.  Thus all latitudes beyond 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 130, 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 portions 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 harvest 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 OF ASTRONOMY.
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 vefy easy to imagine the manner in
which the moon rises at different angles to the horizon of
places in high latitudes, but 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 wih 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 minutes later every day or night in a mean rate. But in fourteen 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 elevating 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 OF ASTRONOMY.               359
Then turning the globe westward seven of the marks about
Pisces and Aries will rise in two hours and a half, measured 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 meridians on a celestial globe represent the edge of the illuminated 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 varying intervals, and the nearer the observer is to the polar
circle the greater will be the differences between these intervals.
~ 510.  The moon goes round the ecliptic in 27 days,
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 conjunction or opposition, the earth goes almost a sign forward;
and therefore the sun appears to go as far forward, that is
27A; so that the moon must go 27~ more than round, and
as much farther as the sun advances in that interval,
which is 21~, before she can again be in conjunction with
or opposition to the sun.  Hence there can be but one conjunction 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




360            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 quarter: 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, being 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 Aries; 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 sitting 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 southward of the ecliptic, she rises later and sets sooner. This
difference is variable, even in the same signs, because the
nodes shift backward about 191~ in the ecliptic every year;
and so go round it contrary to the order of signs in 18
years, 225 days.




ELEMENTS OF ASTRONOMY.               861
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 affects the moon's rising so much, even in her quickest descending 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 mankind.  Many of the supposed effects are doubtless imaginary, 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
Iwn portrait.  Sir John  lerschel 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 emanating 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 magnitude, 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 approximations 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 direction of the line joining the moon and earth, increases
or diminishes her gravity to the earth.  The second, acting in the direction of a tangent to her orbit, disturbs her
motion in longitude.  And the third, acting perpendicalarly to the plane of her orbit, disturbs her motion in latiI
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 numerous; 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 eccentricity of the lunar orbit in conjunction and opposition,
thereby making it more circular, and augments it in quadrature, 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 excentricity 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 approaches 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 astronomers imagined the earth's orbit to have a greater eccentricity than modern astronomers assign to it.
~ 518.  The planets disturb the motions of the moon
both directly and indirectly; their action on the earth alters 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 action; 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 era, the place of the moon is known from that of
the sun at the moment of opposition, whence her mean longitude 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 rapidily and in a shorter time now than she did formerly, 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 inej
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.  Analysis has shewn that he assigned the true cause of the acceleration.
~ 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 OF ASTRONOMY.               365
moon has been diminishing during that time. Consequently the attraction of the earth has had a more and more
powerful effect on the moon, and has been continually diminishing 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 acceleration, 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 acceleration is now about 11".9, but its effect on the moon's
place increases as the square of the time. It is remarkable 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 recorded 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' 29r.6, or a little more than nine years, notwithstanding its motion is sometimes retrograde and sometimes direct: 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 acceleration of the moon's mean motion, must affect the nodes and
31*




866            ELEMENTS OF ASTRONOMY.
perigee. It consequently appears, from theory as well as
observation, that both these elements are subject to a secular inequality, arising from the variation in the eccentricity
of the earth's orbit, which connects them with the acceleration, 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 latitude, 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 T-}. It is proved analytically, 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 ago.  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; results 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 compression is T~..




ELEMENTS OF ASTRONOMY.                 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 ofa Solar Eclipse  Number of Eclipses in
a year. Eclipse of 1706. Eclipse of 1842.
~ 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 lightgiving, 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 whole or part of the sun aplears 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 arein 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 between 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-intercepting 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 intervening 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 ecliptic 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 duration. 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
900 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 proportion.  As 220 radii: 220-60: 1: -~T 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 apparent radius of the shadow is 45  12".15, and the corresponding 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' 37".65. In the same
manner the least apparent radius of the shadow is 3.61
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'  37u.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 comes within 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 being the node.  Let E M, a secondary to MI n, be 62' 37u.
If we can ascertain what must be the value of E N to correspond with this value of E M, we shall ascertain how distant the node may be from the centre of the earth's shadow
to admit of there being an eclipse of the moon. The angle
E  I n is a right angle, EN 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 concealed.  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 terrestrial 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 penumbra 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 finding 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 horizontal 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~ 34t 27".  Trom this the lunar ecliptic limit is found to
be 17~ 21' 27", when the sun is farthest and the moon
nearest; 15~ 14' 27", 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 ascertain 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 corresponding calculations; in general when the apparent diameter 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 7' 38tt,
nor be annular for more than 12' 24t.  An annular eclipse
must be longer than a total one, because the sun's diameter being the largest the moon occupies more time in
traversing it. In partial eclipses the observer is only within 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 direction, 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 motion 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 below 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 300 to 34~, there
must be at least one hnew moon and perhaps two while the
sun is so near the node as to be eclipsed. This may happen 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 accurately as to dispense with the trouble of calculating them.
The period of eighteen years and ten days was early discovered by the Chaldeans and used by them in foretelling
eclipses.  Another period, called the Metonic cycle, consists 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 historical 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 similarly 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 OF 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 luminous 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, visible on the continent of Europe, occurred. Long journeys
were undertaken by distinguished astronomers from the desire to view this rare phenomenon.  Before leaving home
they had agreed not to interchange one word on the subject 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 atmosphere, surrounding the dark disc of the moon, the bursting 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 nonexistence 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 atmosphere, 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 observer 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
flaming 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 persons 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 occultation.
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.  Suddenly, 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.




ERRATA.
Page 37, line 7, from the bottom, dele' others.'
40, "   2,          "         for' the,' read' two.'
"   77, top line, for' common,' read' cannon.'
110, line 15, from the bottom, for' eighty-third,' read' sixty-seventh.'
110, "  13,          "        for'eighty-three, read' sixty-seven.'
"   111, "  22, from the top, fr' aphelion,' read'perihelion.'
"   143, "  16,       "      for'rate,' read' ratio.'
"   143, "  12, from the bottom, for'  51,' read' F57.'
"   165, "  16,         "        for' be,' read' increase.'
"   165,"  11,          "        for' actual,' read' increase of.'
"   171, "   7, from the top, for' raised,' read' caused.'
"   171.  24,         "      for'south,' read' north.'
171, "  26,       "       for' north,' read'south.'
172, "  1,        "       for' Europe to America,' read' America to
Europe.'
204, for columns of Continual Day, insert these:
37     35
67     53
95     90
127    121
159    152
192    184
Dele columns of Continual Night.
261, line 5, for' E,' read' C.'
261,"   9, for'2 a p,' read'2 A p.'
"   262, "  11, for' minute,' read' second.'
"   263, "   5, for' t2 g r,' read''g r.'
"   263, "  23, for' minute,' read' second.'
2 63, "  27, for' As 27, 7', 43": 1":: 360: 331,' read " 27, 7, 43: 1"
60.
265, "   7, from the bottom, for'3183," read'.3183.'
"   265, "   4,         "        for' 31832, or 10131489,' read'.31832, or.10131489.'
"   308, lines 6, 5, 4,   "      for' S,' read' T.'
"   315, line 10, from the top, for'west,' read' east.'
"   334, "   19,      "      for' m d2: M D2,' read' m D2: M d2.'
"   353, "    5,      "      for' lunar,' read' linear.'
"  355, "   18, from the bottom, for'it,' read' the sun.'
355, "   7,          "        for'upon,' read' above.'
356, "   12, from the top, insert after' also.'' As the polar circles
have each one day and one hour of 24 hours, they
have also one full moon which  remains 24 hours
above, and another which remains as long below the
horizon.
371, " 14, for' From this the lunar,' read' From this the solar.'












Plate 1.
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